OSE5312 Light-Matter Interaction. David J. Hagan and Pieter G. Kik

Size: px
Start display at page:

Download "OSE5312 Light-Matter Interaction. David J. Hagan and Pieter G. Kik"

Transcription

1 OS5 LightMatter Interaction David J. Hagan and Pieter G. Kik This document represents the course notes for CROL course OS5. The text is a work in progress, so if you have any comments, please let us know! David Hagan: Pieter Kik: CROL, The College of Optics and Photonics University of Central Florida 4 Central Florida Blvd. Orlando, FL 867 Date generated: December 5, 7

2 Table of Contents Chapter Introduction 7 Phenomenological description of the refractive index 7 Chapter Maxwell s quations Review of Maxwell s quations Chapter Wave propagation 5 Maxwell wave equation 5 (i) Wave quation in material with instantaneous response 5 Impulse Response 6 Reality of lectric Field, Polarization 7 Polarization response of a material 7 (ii) Wave quation in material with general (t). 9 Chapter 4 KramersKronig relations KramersKronig relations for susceptibility Relation between ( and X ) and (n and X ) for weak susceptibility The dispersion of refractive index around a resonance Derivation of KramersKronig relations by Cauchy s integral theorem 5 KramersKronig relations for Reflected Amplitude and Phase 7 Chapter 5 Lorentz model of the optical properties of dielectrics 9 Classical Lorentz oscillator model for absorption & dispersion 9 Resonance Approximation Real Atoms & TRK Sum Rule 4 Comments on the applicability of the Lorentz model to real materials 8 Chapter 6 Drude model of the optical properties of metals 4 Significance of the plasma frequency 4 Modifications of Drude theory in real metals 45 xamples: silver, copper, and indiumtinoxide (ITO) 45 Drude Conductivity and Skin Depth 48 Chapter 7 Optical Activity and MagnetoOptics 5 Optical Activity 5 Zeeman Splitting 6

3 Faraday Rotation 6 Chapter 8 Nonlinear Optical materials 65 Anharmonic oscillator model 65 Noncentrosymmetric materials SHG and optical rectification 66 Noncentrosymmetric materials SFG and DFG 7 Centrosymmetric materials THG and nonlinear refraction 7 Chapter 9 Quantum mechanics 75 The early days 75 The wavefunction 76 lectron momentum and kinetic energy 76 The Timeindependent Schrödinger quation 79 The TimeDependent Schrödinger quation 8 nergy igenfunctions in important binding potentials 8 Chapter Debye model of the optical properties of polar liquids 89 Chapter Homogeneous and inhomogeneous broadening 95 Inhomogeneous broadening due to variations in local environment 95 Doppler broadening 96 Chapter Interaction of light with molecular vibration and rotation 99 Molecular bonds 99 Normal modes Dipole active modes Classical description of vibrations in molecules and solids 5 Quantum description of light interaction with rotation and vibration 8 Coupled electron/vibration transitions in molecules Raman active modes Chapter Interaction of light with vibrations in solids 7 Vibrational Modes in a Crystalline Monatomic D Lattice 7 Vibrational Modes in a Crystalline Diatomic D Lattice Interaction of radiation with lattice modes Optical properties of polar solids 4 The LyddaneSachsTeller relationship 5 Coupled PhotonPhonon modes: Phononpolaritons 8

4 Chapter 4 Optical properties of semiconductors 5 lectronic Band Structure 5 Interband absorption 4 Density of States and Fermi Golden Rule 4 xciton absorption 48 Impurity absorption 49 Free carrier absorption 5 Semiconductors of reduced dimension 5 Chapter 5 From dipole radiation to refractive index 55 Mathematical description of dipole radiation 55 ffect of reradiated field from induced dipoles 58 Scattered Power 59 Absorbed Power 6 Complex refractive index for a sheet of induced dipoles 6 Chapter 6 Rate equations 65 Stimulated absorption and emission 65 Rate quations 65 Light amplification and absorption 7 Schemes to obtain a population inversion 7 Pumping thresholds for and 4level systems 77 Chapter 7 Lasers 8 Laser Oscillation and Oscillation threshold 8 C.W. laser power and optimum output coupling 8 Saturation in inhomogeneously broadened systems 85 xamples of laser systems 87 Chapter 8 Nonlinear Propagation quations 89 SecondHarmonic Generation 9 Techniques of Phase matching 9 Other second order processes 96 Nonlinear Refraction and Absorption 96 Appendix A mpirical descriptions of refractive index Sellmeier equations Schott glass description (power series) Hertzberger description (mixed power series and Sellmeier) 4 Abbe number 4 4

5 Appendix B Vector relations and theorems 5 Appendix C Fourier transforms 9 Appendix D Optical response: formulas and definitions Appendix Springs, Masses, and Resonances Appendix F Rules of thumb and orders of magnitude 5 Appendix G Approximations 7 Dilute medium approximation 7 Weak absorption approximation 7 ffect of dopants on reflection for weak absorption 8 Appendix H Thermal distribution functions 9 Boltzmann probability distribution 9 MaxwellBoltzmann velocity distribution 9 Boseinstein probability distribution 9 FermiDirac probability distribution Appendix I Local field corrections The Local Field Local Field effects on dopants 5 Appendix J Orientational averaging 7 Appendix K Dipolar Polarizability in Solids 9 Appendix L OS5 Quantum Mechanics topics Appendix M Recognizing material types 6 Dilute gases 6 Polar liquids 6 Insulators 6 Semiconductors 6 Metals 7 5

6 Appendix N Fundamental physical constants 8 Index 4 6

7 Chapter Introduction Corresponding handouts are available on In most optical materials the propagation of light is affected by the motion of charge, for example the movement of free electrons in metals. We will derive physical models do describe the charge motion and the resulting optical properties in later Chapters. In this chapter we develop an intuitive picture of the interaction of light with charge, and make a prediction of the wavelength dependent refrective index. A more detailed description of these arguments is given in Wooten, Chapter. Phenomenological description of the refractive index Figure. Light is usually encountered as a propagating transverse electromagnetic wave, meaning that the electric and magnetic fields oscillate in space and time, and that the direction of these fields is normal to the propagation direction. When a light wave encounters a charged particle (electron, positive ion, negative ion, positron,..) the electric field exterts an electric force F e=q where q is the charge of the particle, and is the field strength. i The time dependent electric field accelerates the charge, and accelerating charge causes radation. This is lightmatter interaction in a nutshell: optical radiation accelarates charges in a material, and accelerating charges emit light that adds to (or subtracts frrom) the incident light. To understand how accelerating charge gives rise to radiation, let s consider the electric field lines around a positive charge that is initially at rest, and that suddenly undergoes positive acceleration. The charge is initially surrounded by a purely radial electric field of strength = (r /r )/4πε corresponding to straight field lines extending to infinity, as sketched above. Immediately after a brief acceleration field lines near the charge still point radially outward. However, at larger distances, the information about the new charge position is not yet known due to the finite speed of light. In order to connect these different field distributions, at an intermediate position the field lines need to have a transverse component that is pointing downward. These transverse components move away from the i We will discuss units later. Note that we also ignore the Lorentz force. 7

8 charge at a velocity c, and represent the radiation pattern around accelerating charge. Note that no transverse components exist in the direction along the charge acceleration direction. ii When light propagates inside gases, liquids or solids, electrons (negative charge) and atom cores (positive charge) are accelerated continuously. This results in an oscillating charge position, corresponding to an oscillatory dipole moment. The periodic charge acceleration produces radiation at a frequency that matches that of the incident light. iii This reradiated light adds to the incident light wave, which is now slightly modified. Thus, refractive index can be seen as the result of the reradiation of light from a large collection of oscillating dipoles. Figure. Development of dipole radiation, showing only electric field components To understand the origin of the refractive index, and why it is usually larger than, we need to consider the phase of the charge oscillation. Let s first look at the response of a charge to an incident oscillating field. For a positive bound charge that is driven well below resonance, we find that the position of the charge is exactly in phase with the driving field, see graph below. The velocity can be seen to be 9 ahead in phase, and the acceleration is yet another 9 ahead. In the discussion above, we found that the field resulting from this acceleration is pointing in the direction opposite to the acceleration. Surprisingly, this analysis shows that the reradiated field at low frequencies is exactly in phase with the incident wave. This would correspond to a refractive index which is always exactly equal to, which we know is incorrect. Figure. ii From this analysis it also follows that static charges and charges with constant velocity do not radiate. iii xcept for very strong fields. In that case we would need to consider nonlinear optics, discussed in a later Chapter. 8

9 The error in the analysis lies in the fact that we considered only a single isolated charge. In reality, we need to consider the effect of many radiating dipoles, as shown below. The incident plane wave will drive a sheet of dipoles, all radiating in phase with the incident light. At a finite distance from these dipoles, the total field observed contains radiation contributions from many dipoles. On some optical axis of choice, radiation coming from offaxis dipoles will arrive a little later, causing a phase delay, or reduced apparent velocity of the light. By carefully integrating the effect of all dipoles, it follows that the total phase delay adds up to exactly 9 compared to the direct wave. As a result, the low frequency refractive index is larger than, with the magnitude given by the number of dipoles contributing and their individual dipole moments. Figure.4 To understand the frequency dependence of the refractive index, we need to add the frequency dependent response of a Lorentz oscillator to this picture. We know that at low frequencies we obtain a finite dipole moment and zero phase delay, while near resonance the phase delay approaches 9 and the amplitude reaches a maximum. At high frequencies the phase difference approaches 8 and the amplitude approaches zero. The resulting changes of the refractive index are sketched below: Figure 5.5 9

10 At low frequencies (top graph), the Lorentz oscillator responds inphase with the incident light, but due to the collective dipole sheet effect described above, the net phase delay is (oscillator) 9(sheet effect) = 9. By adding this finite phasedelayed response to the incident field, the light appears to propagate slightly slower, corresponding to a finite refractive index n>. As the frequency increases, the oscillation amplitude goes up, as does the phase delay. Both effects together add up to a larger delay, and a higher refractive index. xactly at the resonance frequency, the oscillation amplitude reaches a maximum. However, at this frequency the Lorentz oscillator responds with a phase delay of 9, resulting in a total phase delay of 9(oscillator)9(sheet effect)=8. The dipoles can be seen to generate radiation that destructively interferes with the incident light, resulting in a change of the amplitude, but no change in the phase, corresponding to a real part of the refractive index close to, and finite absorption. At frequencies above the resonance, the phase delay exceeds 8, which can be seen as a phase lead. The light appears to be accelerated, corresponding to a refractive index less than. The resulting refractive index curve is shown on the right side of the Figure. As we will see in later chapters, a more thorough model description of the polarization of atoms in a material leads to similar refractive index spectra.

11 Chapter Maxwell s quations Corresponding handouts are available on In Chapter we saw that the refractive index of materials can be understood as the result of dipole radiation from charges accelerated by the electric field. In this chapter we will take a different viewpoint. Instead of considering separate contributions from individual atomic dipoles, we assume that materials can be described as having a smoothly distributed polarization. We will call this the continuum description. The effect of such a continuous medium on wave propagation is described by Maxwell s equations. Review of Maxwell s quations As mentioned in Chapter light is an electromagnetic wave, containing an oscillatory electric field and an oscillatory magnetic field component. To describe a light wave we therefore need to know the magnitude and direction of and B at every point in space. The corresponding field distribution is thus a time dependent vector field, written as (r, t) and B (r, t). In this Chapter we will make use of several vector operations and theorems that are described in Appendix B. Maxwell s equations describe the classical relations between charge (units Coulomb, C), current (C/s), electric field (V/m), and magnetic flux density B (Wb/m or Tesla= 4 Gauss). In vacuum (no material, no free charges), Maxwell s quations are = (.) B = (.) = B t (.) B = ε μ t (.4) Inside materials, but again in the absence of free charges, only the relation describing the curl of the magnetic flux density B (hereafter referred to as the magnetic field ) changes, as follows: B = ε μ t μ J M P t (.5)

12 , D and P P is the polarization of the medium, induced dipole moment per unit volume. (Coulomb m per m or C/m ) (i) P = (no dielectric) (ii) P (dielectric) ( note that P points from to ) Figure. P = induced surface charge density (C/m ) = total surface charge density (C/m ) D = free surface charge density (C/m ) where D is the electric displacement vector D P (.6) Gauss law for B No magnetic monopoles: following Gauss Law B i.e. s B r ds magneticcharge v Bdv (.7) (.8) Ampere s Law For magnetic fields, the flux density, B, is related to the magnetic field intensity, H and the induced magnetization, M, by B H M (.9) H J free D (.) dt

13 4 Maxwell equations D P B H M (Henceforth we will use J in place of J free.) Current Sources D H J (.) dt B M J P (.) dt P B J M dt dt or where free charge magnetism bound or polarization current density current density current density vacuum displacement current density J tot (.) dt B J tot (.4) c dt J tot P J M (.5) dt The relations described above allow us to derive the wave equation, as discussed in the following chapter.

14 4

15 Chapter Wave propagation Corresponding handouts are available on Maxwell wave equation We already saw that total B dt B B J M C dt P dt (.) for M=, f, J f =, we have (.) or B P dt dt dt (.) P dt dt free Now, free bound P, so we cannot set to zero in general (as is often done). In a moment, we will show that this is OK for plane waves, but first, we will look at the instructive (and often valid to a good approximation) example of an instantaneous response. (i) Wave quation in material with instantaneous response The polarization of the material is linked to the field through a quantity known as the susceptibility. For a material with an instantaneous polarization response, (t) (t) Hence P(t) = (t). Since D, then ( ( ) ), so that, ) ( ) (.4) ( For homogeneous media,, so that,. Hence, or, P c t t c t (.5) 5

16 t c t. (.6) We recognize this as a wave equation with plane wave solutions i( k r t) ( r, t) e c. c. (.7) Now, ( r, t ) k ( r, t ), and t, k ( ) c c or, k k c Free space, k (.8) is generally the complex refractive index, = n i, but in this case is real and independent of frequency, hence so must be. In general, for a plane wave solution propagating along z, (z,t) = e i( k zt) = e i(n k zt) e k z (.9) In this case of instantaneous response, the imaginary part of is zero, and consequently the irradiance of the wave becomes position independent, and equal to I nc (.) In the above, we have seen that instantaneity implies no wavelength dependence and no loss or absorption. This is a good approximation in some materials for limited spectral ranges, but is not adequate to generally describe materials. Impulse Response The time dependent susceptibility (t) represents the impulse response function for the material, as can be seen by looking at the case where (t) = (t) P t ( t') t t' dt t ' (.) In all cases, P(t) and (t) are real, measurable quantities, so it follows that (t) is also a real quantity. Not so for the Fourier transforms, (), P() and (), which may be complex i e t d t it t e dt (.) 6

17 i P e t d P t it P Pt e dt But () is usually defined as: (.) it te dt, (without the / factor). (.4) It can be shown that this leads to the familiar frequency domain relation: P ( ) ( ) ( ). (.5) Clearly, () is generally complex, so we write, ( ) '( ) i ''( ). (.6) Reality of lectric Field, Polarization The electric field is a RAL, observable quantity. (Same is true for Polarization, current, etc.). Hence our mathematical description of these quantities should not be complex, e.g. (t,z) = cos(tkz). This may sometimes be written in the form (t,z) = ½ e i(tkz) c.c. = ½A e i(tkz) c.c., Where, A = e i is now a complex amplitude that contains magnitude and phase information. Since A is a Fourier amplitude of the Fourier component at frequency, it is only in the Fourier transforms of the field that we find complex numbers cropping up. In the time domain, observable quantities like field and polarization should always be real. Likewise, optical properties that relate polarization to field, or input field to transmitted field in the time domain must also be real, but in the frequency domain, such properties may be complex. Thinking of the high frequencies involved and the way in which measurements of optical properties are measured, it is natural that it is preferred to express parameters that describe the optical properties of materials in the frequency domain. Polarization response of a material Look at linear material first Linear means P r, t d r' dt' r r', t t' : r', ' t (.7) normally r r ' entirely local response. P r, t dt' r, t t' : r, t', (.8) where is the response function e.g. if t t e use scalar, P 7

18 tt' e t' ' P t dt (.9) for t ' t ' ; impulse response is P t t e (.) then for t Figure. Figure. We have P t Figure. 8

19 (ii) Wave quation in material with general (t). First, we need to revisit the issue. We again anticipate plane wave solutions of the form, i( k r t) ( r, t) e c. c (.) for which it can be shown that ik ik If we decompose into longitudinal and transverse components. T L (.) (.) then, ik ( T L) ik ik ( ) ik T L L T (.4) Since for plane waves, L =, then for plane waves. The wave equation therefore becomes ( r, t) P ( r, t) ( r, t) (.5) c t t Since P() = ()(), we need to Fourier transform to (k,) domain to proceed: k ( k, ) ( k, ) P( k, ) c ( ) ( k, ) (.6) So that: k { ( )} c (.7) and ( ) ( ) (.8) Therefore, n( ) i( ) '( ) i"( ). (.9) This leads to the relationships n () () = () = ()/ (.) 9

20 n()() = () = ()/ (.) Since the refractive index is now complex, the absorption is no longer zero. We again consider a plane wave propagating along z: (z,t) = e i( k zt) = e i(n k zt) e k z (.) The complex nature of the refractive index now results in a position dependent field amplitude of magnitude (z) = e k z and therefore a position dependent irradiance given by I( z) nc ( z) (.) From q.. it can be seen that the irradiance of this plane wave is given by I(z) = I e = I e. (.4) So that the absorption coefficient is α = κk = κ. (.5) xample: magnitude of If = cm = m, then = /k. And k is about 7 m for visible light, so ~ 5, << n. Hence, n = () is a good approximation, provided < 4 cm. Then we can say that refraction is solely related to away from resonances.

21 Chapter 4 KramersKronig relations Corresponding handouts are available on KramersKronig relations for susceptibility KramersKronig relations link () and (), n() and (), and R() and (). We start by deriving the KK relation linking () and (). The starting point is the timedependent polarization: P t t ' t t ' ' dt (4.) response function (linear) The polarization response function must obey causality, i.e. the effect cannot precede the cause. This can be explained mathematically as follows: Mathematical statement of causality:, t (4.) i.e. no response before applied t for t or t t ' for t t'. This can be expressed quite compactly using the step function t t t (4.) where, t t is the Heaviside function or unit step function t Actually, any function that equals unity for t> and equals anything but unity for t<, will do. For example, the Signum function, defined as t Sgn t, (4.4) t will work just as well. (As it must.) So we now have a timedomain statement of causality that can be Fourier transformed for give a frequency domain statement of causality: but, i t t t t t e dt F (4.5) i ' ' t t e d ' (4.6)

22 dt d t e i ' ' ' t Hence, d dt t e i ' ' ' t. (4.7) d dt t e i ' ' ' t ' ' d '. (4.8) Now the Fourier Transform of the step function is well known: F t i (4.9) ' d i ' d (4.) ' ' ' ' i d ' '. (4.) ' Which gives the statement of causality in the frequency domain: i ' d '. (4.) ' We can make use of the i in the relationship to find cross relations for the real and imaginary parts of. Splitting () into its real and imaginary parts: ' i ' ' ' '' ' d ' i ' ' d (4.) ' ' Which yields two equations: " ' ' d ' (4.4) ' ' ' " d ' (4.5) ' These are the KramersKronig relations for (). We can use the results of the reality condition, ( () is even, () is odd) to rewrite these integrals in terms of positive frequencies only: and ' " ' ' d ' (4.6) '

23 ' ' " d '. (4.7) ' Relation between ( and X ) and (n and X ) for weak susceptibility For a weak susceptibility (rarified media, gases), we have, <<. In this case, n( ) i ( ) '( ) "( ) '( ) i "( ) i, (4.8) so that, ' ( ) n ( ), and (4.9) "( ) ( ) ( ) "( ) c (4.) We can now use these approximate to write ' ' ' n( ) c n( ) c / ' d' ' ' d' ' (4.) Although this has been derived for a weak susceptibility, it is actually true in general. This is a very useful relation, as it is relatively easy to measure () over a broad wavelength band. The dispersion of refractive index around a resonance It is worth looking at the form of the KramersKronig integrals, to see what the lead us to expect about the relationship between n() and (), etc. Let us consider a material with a single, moderately narrow absorption line. In Figure 4. we sketch such a line, where res is the resonance frequency, or frequency. (This could be rad/s, Hz, or one hundred terahertz, or whatever units we desire.) The refractive index at any frequency depends on the integral of ( ), multiplied by /( ), so we also plot the function /( ) below for two cases: one where the frequency,, is just above the peak absorption and one where is just below the peak absorption. Clearly for > res, the integral of the product is negative which is to say that the n() due to a particular resonance is < for frequencies above that resonance. Similarly, frequencies below a resonance, the refractive index arising from that resonance is positive. If the resonance is symmetric, then for = res, the refractive index due to the absorption line will be unity i.e. the absorption line will not affect the refractive index exactly at the resonance frequency.

24 4 4 4 y=/(. ) (') 4 y=/(.9 ) (') '/ res '/ res Figure 4. The above observations are generally true, regardless of the precise shape of the absorption resonance. Certainly, the exact shape of () will affect the precise shape of n(), but here we are only talking in generalities. It is also important to note that we make no assumptions about the physical process giving rise to the absorption line, as it is irrelevant to these general observations. There can be no exception to the Kramers Kronig relations. If there is an absorption line in a material, then it will cause the refractive index to be increased below resonance and decreased above. Usually, a material will have several absorption resonances, which may occur in the infrared, in the visible and in the ultraviolet. At very low frequencies, say in the far infrared, which lie below all resonances, the index may therefore be quite high, as all absorption resonances contribute positively at these low frequencies. We can also conclude that at very high frequencies, say in the ultraviolet, for which all resonances are at lower frequencies, the refractive index must be below unity. This last fact is not widely known, but as we see from our analysis, it is inescapable and must be true for all materials. As we examine different types of materials and physical processes that give rise to lightmatter interactions, we will see again and again that these general predictions that come from the KramersKronig relations are always fulfilled [n() ] (') '/ res Figure 4. To get a general spectrum of n(), (usually referred to as the dispersion of n) we just calculate the result of the KramersKronig integral repeatedly for many values of. The 4

25 result for n() for the absorption line shown in Figure 4., where we have arbitrarily expanded the scale for (n) to show it on the same scale as (). Derivation of KramersKronig relations by Cauchy s integral theorem Given that, P and that where, t t ' t t ' ' dt, (4.) P ( ) ( ) ( ), (4.) i t t e dt. (4.4) Now by causality, the above integral need only run over positive times: it t e dt, (4.5) i.e. t is always positive in the integral. Now, we can let be complex, so that, ='i" giving i ' t " t t e e dt. (4.6) and since t is always positive, the factor e t tells us that for positive, the function () has a regular analytic continuation in the positive imaginary half of the complex plane of. By Cauchy s integral theorem, the closed path integral of a function that is analytic in a simple domain yields a zero result. Now we consider the integral c ( ) d, (4.7) where C is a closed path in the upper half plane of that avoids all poles (singularities), as shown on the next page. 5

26 C Figure 4. We can look at the four main parts of the integral. First, the large semicircle, defined by = e i, where runs from to. In the limit as, this part of the integral becomes zero, provided that () is reasonably well behaved, or more specifically, provided that ( ) lim. The small semicircle is centered on a pole caused by the /() term. By the Residue Theorem, in the limit of vanishingly small radius,, we have, ( ) d i( ), (4.8) (i.e. a halfresidue of the integrand at.) The straight sections of the integral, in the limit of, become; lim ( ) d ( ) d ( ) d (4.9) where denotes the Cauchy Principal Value. Hence the sum of the four parts of the path integral becomes i.e. ( ) d i ( ), (4.) ( ) d ( ), (4.) i 6

27 which is the KramersKronig relation as previously derived. This is basically just Cauchy s Integral Theorem. (The principal value () label really just means that we should avoid the pole, but in practice, this turns out to be trivial, so we will not continue to use.) The above analysis is less physically insightful than the one given before, but this is the treatment usually given for Kramers Kronig relations in most texts. It is also convenient to use, since any causal function that is analytic in the upperhalf plane can be treated in the same way, and this can save some time in deriving KramersKronig relations. This is the case for the relations for the relations between the amplitude and phase of a reflected wave, which we treat next. KramersKronig relations for Reflected Amplitude and Phase There exists a very useful type of KramersKronig relation that relates the phase and amplitude of reflection from an interface. Now the reflectance, R() = r()r*() (4.) Where r() is the electric field amplitude reflection coefficient, which for an airmaterial interface, is given by n( ) i ( ) r( ). (4.) n( ) i ( ) Now we can also write r() in terms of a phase and an amplitude: i ( ) r ( ) ( ) e, (4.4) where () is the phase shift upon reflection. (Note this is NOT the unit step function, (t), used earlier.) Clearly, if we can completely determine both r() and (), we can completely determine both n() and (). Now, it is relatively straight forward to determine R(), but it is hard to imagine how we could easily measure () over a large spectral range. We would like to find a KramersKronig relation that relates () to (), but we note that while we can do this for complex analytic functions of the form () = () i (), or () = n() i (), we cannot do so for a function of the form r() = ()exp(i()). We could always find a KramersKronig relation between r () and r (), but this is of no particular use, as we cannot easily individually measure either the real or the imaginary part of r(). However, if we take the log of r(), we get ln r ( ) ln ( ) i ( ), (4.5) for which, via the Cauchy integral theorem, we can write down a pair of KramersKronig relations: ln ( ) ( ) ( ') d' ' ln ( ') d' ' (4.6) 7

28 where, as usual, the principal value of the integral is to be taken. Clearly the second of the two relations is the more useful, in the same sense that the relation that gives n() in terms of () is more useful than one that gives in terms of n. It is much easier to measure amplitudes than phase. We can rewrite this integral in terms of positive frequencies only: ( ) ln ( ' ) d' ' (4.7) Note that our previous approach of applying causality in the form of a step function in the time domain is not valid in the same way as for, as for t <, = gives ln =, which is not so easy to handle. For the same reason, one might look at this integral and conclude that it probably will not work, as for regions where R(), ln. Nevertheless, it still seems to work, as we shall see from the homework assignment. Those who are interested may read the details in Wooten, but the scope of this course is limited to the results. There is a useful trick to remove the singularity at =. We may simply subtract the quantity ln ( ) d' d' ln ' ' ( ) (4.8) from the integral in the KK relation. Hence ( ) ln ( ') ln ( ) ' ln R( ') / R( ) d'. ' d' (4.9) Hence as, R( )/R() and hence ln{ R( )/R() }. We can apply the L Hopital rule to see that the divergence is removed. 8

29 Chapter 5 Lorentz model of the optical properties of dielectrics Corresponding handouts are available on In Chapter we learned that optical effects such as reflection, transmission, and absorption are related to the refractive index, which followed from the complex susceptibility. This means that we can make a prediction of the optical properties of materials if we can develop a model for (). In this chapter we will discuss the optical response of materials that have a strong absorption at one or more welldefined frequencies. We will derive the response of a simplified atom using what is known as the Lorentz model, which describes the electrons that surround atoms as masses bound to the atom core with a phenomenological spring constant. It gives a fairly realistic prediction of dispersion of the refractive index in regions of low absorption, and an approximate understanding of n and trends near strong absorptions. Classical Lorentz oscillator model for absorption & dispersion Atoms consist of a positive atom core containing protons (charge e) and neutrons. The total electric charge of the atom core is ez where e is the unit charge and Z is the atomic number Z. The atomic number of most commonly used materials ranges from. A neutral atom is surrounded by a total of Z electrons. These electrons are bound to the atom core by Coulomb interaction forces, orbiting the core with their exact spatial distribution described by quantum mechanics (see Chapter 9). For atoms with large Z, some of the electrons are bound very tightly to the core, called core electrons, orbiting the cores at small distance. Other electrons occupy larger orbits, circling the positive core with its tightly bound core electrons. Consequently, these outer electrons experience a smaller net positive charge from the core, and therefore smaller Coulomb binding forces. The outermost electrons that finally make the atom neutral are called the valence electrons. These electrons see the least attractive force, and are therefore easiest to move. As a result, these valence electrons often account for most of the polarization response of atoms. Figure 5. The Lorentz model takes all these elements to build the simplest possible mechanical model of an atom. First of all, it considers only the valence electrons, assuming that the core electrons are so tightly bound that the M is practically unable to move them ( no core electron response ). Secondly, since the electrons are in a steady state, they must be in a minimum energy configuration. Moving the electrons out of equilibrium will result in a restoring force that tries to bring the electrons back into equilibrium. The Lorentz model 9

30 assumes that this restoring force is linearly dependent on the displacement from equilibrium r, a relation known as Hooke s Law: F r Kr m r (Hooke s law) (5.) Here K is the spring constant (units N/m) in analogy with the classical massonaspring model. We will assume that the atom responds isotropically and along the applied driving force, which allows us describe the electron position with a scalar r(t), the distance from the core. Newton s second law of motion states that F = m a = m r. In the absence of any external driving forces this predicts that the average position r of a valence electron will behave as r (t) = K m r (5.) with m e the electron rest mass. This type of relation can be satisfied by harmonic functions of the form r(t) sin(t) or r(t) cos(t), or iv r(t) exp(it). Substituting a trial solution of the form exp(it) we find r (t) = ω e = K m r ω = K m (5.) The see that the valence electron in this simple model has a natural oscillation frequency which we will refer to as the resonance frequency, labeled as. We have thus found a relation between the spring constant K and the resonance frequency: K = mω (5.4) As in any realistic physical system electron motion will not continue forever. The electron motion is said to be damped, with the electron gradually losing energy as it oscillates, leading to a reduced amplitude over time. In the Lorentz model this is phenomenologically described by a friction force F f corresponding to momentum loss at a rate (units s ) : F (t) = dmv(t) dt = Γ mv(t) F = m Γ r(t) (5.5) In reality this friction represents many possible causes of loss of motion, for example random collisions with other atoms, coupling to vibrations in a crystal ( electronphonon coupling ), emission of light ( radiative relaxation ), energy transfer to other electrons ( electronelectron interactions including effects such as Auger relaxation), to name a few. To predict the optical response of valence electrons in the presence of an oscillatory electromagnetic field the Lorentz model only considers electric forces given by F (t) = e(t) (5.6) where e is again the electric unit charge. The electron response follows from the equation of motion F=ma taking into account the electric driving force, the friction force (damping rate), and the restoring force. The total equation of motion thus becomes iv The exponential expression leads to a complex amplitude, which can be turned into a real amplitude by also considering an oscillatory term with the opposite angular frequency, as shown in Chapter

31 m r (t) = e(t) mγr(t) mω r(t) (5.7) In a linear system we expect that driving at single frequency results only in responses that occur at that same frequency. We can assume a real electric field (t) of the form (t) = (ω)e c. c. (5.8) and assume that the electron position occurs at the same frequency, described by: r(t) = r(ω)e c. c. (5.9) We substitute a trial solution containing only the positive contribution. v After dividing out common terms on both sides, this results in: mω r(ω) mγiω r(ω) mω r(ω) = e (ω) (5.) This leads to an expression for the (complex) harmonic motion amplitude r() in response to a harmonic driving field with amplitude (): r(ω) = e m ω ω (ω) (5.) iγω We can now find an expression for the dipole moment, which is defined as charge times separation distance. Since the atom core is at position zero, the dipole moment becomes simply q e r = er. The Lorentz model thus predicts an oscillatory dipole moment with amplitude μ(ω) = e m ω ω (ω) (5.) iγω This Lorentz description of the dipole moment of a valence electron on a single atom reproduces several properties that were already predicted in Chapter. For example: at very low frequencies ( ) we see an amplitude μ() = e m (ω). (5.) ω Note that () that is real and positive, which means that the dipole moment is inphase with the driving field. This is expected: Hooke s law in constant field leads to equilibrium when F e F r =, leading to a fixed dipole moment linearly proportional to. Also note that this static dipole moment depends inversely on. This also makes sense: we saw that K ω. If the spring is twice as stiff, the same field will yield half as much electron displacement, and thus half as much dipole moment. As the excitation frequency increases, the imaginary contribution iγω in the denominator becomes more important, resulting in a phase delay of the dipole moment relative to the driving field. When we excite on resonance, i.e. when we use = we find v The corresponding negative frequency component follows simply by exchanging for each term.

32 μ(ω ) = e e (ω) = m iγω m i (ω). (5.4) Γω Note that our time dependent field for positive was described by a field contribution of the (ω)e corresponding to a clockwise rotating complex vector. We see that when this field contribution is positive and real, the dipole moment is entirely imaginary, corresponding to a 9 phase delay relative to the driving field. When we excite at high frequency with >>, we see that μ(ω ) = e (ω) (5.5) m ω We see that the dipole moment is in antiphase with the driving field, and the dipole moment vanishes as the frequency increases. We have found that bound electrons do not produce much polarization at high frequency. This is the reason why materials become transparent at sufficiently high frequency. When illuminated at sufficiently high frequency, none of the electrons (neither the valence electrons or the core electrons) can respond significantly, and consequently sufficiently high energy XRays can propagate through materials with little absorption or refraction. We sometimes describe the ability to generate dipole moment with a quantity known as the polarizability. This links dipole moment and driving field according to =, resulting in an expression for the Lorentz polarizability : α(ω) = e m ω ω iγω (5.6) Note that here we assumed an isotropic response, which is reasonable for isolated atoms. If we describe the electronic polarization of molecules we often find that the polarization is not exactly aligned with the driving field, necessitating the use of a polarizability tensor, however in most of this book we will deal with isotropic responses that are described by a scalar polarizability. Be careful not to confuse the polarizability with the absorption coefficient. With N atoms per unit volume, the net dipole moment per unit volume is P( ) V NV i ( ) N ( ) i (5.7) Hence, in scalar terms, P( ) N( ) ( ) ( ) ( ) Ne ( ) m i (5.8) Note that N in fact represents the number of oscillators, and in this case the number of electrons that contribute to the resonance of interest. For example, a single atom may have multiple valence electrons that contribute to a resonance. In this case, N becomes the number of atoms per unit volume, multiplied by the number of valence electrons per atom.

33 Remember that atoms may have many electrons, but that the dielectric response is often dominated by the outermost (valence) electrons since those are relatively weakly bound i.e. most easily polarizable. When irradiating atoms at far UV and xray frequencies, optical resonances related to excitation of strongly bound core electrons can be observed, however such transitions are not discussed in detail here. We have assumed local and macroscopic fields are equal, and have ignored spatial averaging assumed all dipoles are free to point in direction of field. Note that is dimensionless, so that the term has dimensions of. We set m Ne p (5.9) For reasons that will become apparent later, p is known as the plasma frequency. ) "( & ) '( ) ( p p p p i i i i (5.) See fig. in Wooten for plots of these quantities. Resonance Approximation We can simplify these expressions near resonance, under the approximation <<. In this case,, therefore (5.) Hence, in this resonance approximation we have / / ) "( & / ) '( p p (5.) is the Full width at half maximum (FWHM) of the imaginary part of, which has a Lorentzian functional form.

34 Note the symmetry of the real & imaginary parts. The imaginary part of the susceptibility, is symmetric about, while is antisymmetric about. Notice that in the resonance approximation that appears antisymmetric (i.e. about = ) while appears symmetric. The converse is actually true, as seen before from the reality condition. This highlights that the resonance approximation is strongly invalid far from resonance. Real Atoms & TRK Sum Rule In general, atoms & molecules have several resonances, not just a single one as in our model. Quantum mechanically, there are several resonances electronic resonances, and molecules may exhibit vibrational and rotational resonances also. It turns out that perturbation methods in quantum mechanics yield a result very similar to the classical one. We find Ne f j ( ) (5.) m i j j j j is the frequency for a transition between two electronic states with energy difference. j is the decay rate for the final state and f j is known as the oscillator strength, j which obeys the ThomasReichKuhn sum rule: j f Z (5.4) j for an atom with Z electrons. This tells us that the total absorption, integrated over all frequencies is dependent only on Z. Usually one resonance dominates all others. Figure 5. The natural linewidth is dictated by. Recall is a damping or decay rate, which corresponds to (/lifetime) of a state. This could range from khz to GHz. Often the electronic states are split into many sub states. In molecules, each electronic state can exist for many possible vibrational or rotational states of the molecule. Which strongly broadens the electronic states into bands. In solids, the electronic levels are broadened into very broad electronic energy bands. All of these broaden out the optical resonances far in excess of. 4

35 In addition to several electronic resonances, other degrees of freedom, such as atomic motions, including molecular vibrations, lattice vibrations, molecular rotations, can interact with the electromagnetic field, producing many resonances over the electromagnetic spectrum. This is illustrated in Wooten, Fig... Our classical, or quantum, model gives the real and imaginary parts of the susceptibility, from which it is easy to obtain the dielectric function, r () = (). Here r = / is the relative permittivity. It is less straightforward, but still not difficult to find expressions for n() and (). We start from: n ( ) ( ) '( ) n( ) ( ) "( ) From which we obtain n( ) ( ) r r ' " r ' " r r r r r ' ' (5.5) (5.6) Armed with n and, we can find the absorption and reflectance of the material. Wooten illustrates this in Figs.. and.4. It is often illustrative to plot n,, absorption and R for different values of, p and. On the next page is an example, using values of 4, 8 and, for each of these parameters, respectively. These seem like strange, unrealistic numbers to use, but our expression for contains only frequencies, so the scale is relative. However, it is realistic to think of these numbers as photon energies,, in electron volts (ev). Lorentz model for = 4, p = 8, =. 5

36 Re( ) Im( ) n( ) 4 ( ) ( ) p R( ).5 Figure 5. Note the shapes of the curves are as we would expect from KramersKronig relations. Also note that drops below zero between and p. We can calculate n and from the above relations. n is small over the region where is negative. Above p, n gradually rises to. Noting that = /c, we have also plotted a scaled version of, by plotting, scaled to p. Note how is skewed to higher frequencies. R is large in the region where n is small, which makes sense, as in the limit of n, R. Lorentz model for = 4, p = 8, =

37 Re( ) Im( ) n( ) ( ) ( ) p R( ) Figure 5.4 As expected, for a smaller damping, =., the curves are narrower and have larger maximum values. Above, n is well below unity. Above p, n rises, but only in the highfrequency limit does n approach unity. This is a good predictor of actual materials. Above the highest frequency resonance, usually in the deep UV/soft xray region, n is indeed less than unity. Causality is not violated, though. Note that near p, ~ and hence n ~. The reflectance is much higher and has sharper edges. Note the high R starts around and falls near p. This is because >> n in the reflecting band. Note is large above even where has dropped to a small value. 7

38 TART As described in Wooten, the material has four distinct regions of optical properties, Transmissive, < /, Absorptive, / < < Reflective, / < < p Transmissive > p These frequency ranges are approximate. The regions are more distinct for smaller and larger p. Comments on the applicability of the Lorentz model to real materials (i) Insulators The Lorentz model works surprisingly well, provided we remember that real materials correspond to a collection of Lorentz oscillators with different frequencies. The outer, or valence, electrons predominantly determine the characteristics of the optical properties a solid. In an ionically bonded material, e.g. alkalihalides such as KCl, the valence electrons are quite strongly localized at the negative ion (for KCl, this would be the Cl atom), and hence the optical spectrum contains some atomiclike features, with many resonances. As the valence electrons are tightly bound, the resonance frequency is high so that these materials may have a transparency range that extends far into the UV. This can be seen in the reflectance spectrum for KCl shown below (taken from Wooten, Ch..) For these types of materials, the external field and the local field can be quite different and it is not trivial to calculate the local field. For this reason, the Lorentz model does not give quantitatively accurate results for ionic materials. Figure 5.5 8

39 (ii) Doped Insulators Doped insulators, for example ions in glass, behave somewhat like the ions would in a gas, except that the locally strong electric fields of the host materials may distort the spectrum slightly. Figure 5.6 shows the absorption of Nd ions in a glass host material. Figure 5.6 Usually, the absorption of the dopant material is in a region of transparency of the host so that we can approximate the polarization as a superposition of polarizations due to the host and dopant material. For the case of a single resonant absorption line, we may write P tot P host P dopant p host (5.7) i where host is assumed to be real and constant. Hence; p r ( ) host (5.8) i Often, we label host as the high frequency dielectric constant,, so that p r ( ). (5.9) i The static dielectric constant, defined as st = r() is therefore given by setting = in the above expression, so that st. (5.) p Hence, the static dielectric function of a material is affected by dopants, even though the resonant frequency for the dopant is far away from =. 9

40 4

41 Chapter 6 Drude model of the optical properties of metals Corresponding handouts are available on We can extend the Lorentz model to metals, in which case, since the electrons are unbound or "free", they experience zero restoring force and hence the resonance frequency, = K/m is also zero. This is known as the Drude model. The equation of motion is then r ( t) r ( t ) m m e ( t) t t which has solution e ( ) r( ) m i and hence () is given by p ) i (6.) (6.) ( (6.) where once again the plasma frequency is defined by p = Ne / m. Hence, or, / '( ), "( ) (6.4) p p / '( ), "( ) (6.5) r p r p Now, in a metal, the damping term is just the electron collision rate, which is just the inverse of the mean electron collision time,, i.e. =. Hence, p p r '( ), "( ) r (6.6) The collision rate can be quite rapid tens of femtoseconds. But for optical frequencies, (e.g. for = 5 nm, = c/ =.8x 5 rad/s) () >>. Under this approximation, we find p p p r '( ), "( ) r. (6.7) This approximation may break down in the farinfrared spectral region, where damping may be significant. Note that damping is absolutely necessary to have an imaginary part of () or r(). It is useful to look at some plots of r(), n(), () and R(). These are plotted on the next page for p = and for or =.5. In the limit of no damping, the n = and R = for < < p. Above p, is zero and the reflectance drops as n rises from zero to 4

42 unity. Note that even for r =, and hence is not zero. Introducing some damping causes R to be < and the reflectance drop at p is less severe. The behavior of r, n and is consistent with what we now expect for a Lorentz oscillator with =. Clearly, the sharp edge in the reflectance seen at the plasma frequency can be expected to be the predominant spectral feature in the optical properties of metals. Re( ) Im( ) =.5 =.5 Re( ) Im( ) n( ) n( ) ( ) ( ) R( ) R( ) Re ( ) Re ( ) Im ( ) Im ( ) Figure 6. The last plots show the real and imaginary parts of the dielectric constant on a log scale. It is interesting to note that only the real part of indicates notable behavior around the plasma frequency. One cannot see evidence of the plasma frequency by looking at the 4

43 imaginary part of alone, yet both n() and () clearly show evidence of the plasma frequency. Optical absorption in low electron density materials Semiconductors Recalling that the absorption coefficient is given by () = k = ()/c = r ()/cn(). Now for very high frequencies, or for low electron densities, as may be found in doped semiconductors, p <<, n() ~ so that, p ( ) "( ), (6.8) c c c p where p is the wavelength corresponding to the plasma frequency. The dependence of is commonly seen in semiconductors, where dopant densities are typically in the range of 6 to 9 cm as compared to ~ cm in metals. This absorption is commonly referred to as freecarrier absorption. Significance of the plasma frequency The expressions have been written for r rather than for as they more clearly reveal something significant about the plasma frequency in this form. Notice that at = p, the real part of the dielectric constant becomes zero. Hence n( p) =, which means the phase velocity. A more rational way to describe this is that the wavelength, = c/n as p. This means that all the electrons are oscillating in phase throughout the propagation length of the material. cos(pt) metal Figure 6. Note that as all the electrons are moving together, there is no charge separation (polarization) and hence no restoring force or sustained oscillation after the field is removed.. Plasma oscillations The above figure shows an entirely transverse field (compared to the surfaces of the material). Should there be a component of the field perpendicular to the surface, there can be a net surface charge as a result of the applied field. 4

44 app x x Figure 6. The attractive (restoring) force between the surface charges can result in a free oscillation. For no net charge, (q f = ) then D = = r. But, so then r = r i r =. Hence, r =. Now, P = charge x displacement /volume. Thus & NeAx L P Nex AL P Nex. The restoring force is given by: e: (6.9) e Ne x, (6.) which is equal and opposite to the acceleration: x m t Ne x, (6.) which is the equation of motion: x Ne x. (6.) t m Hence the resonance frequency for the plasma oscillation is given by Ne p. (6.) m 44

45 Modifications of Drude theory in real metals The Drude model implies that the only the plasma frequency should dictate the appearance of metals. This works for many metals see the example of Zinc (Fig.. in Wooten.) But is does not explain why copper is red, gold is yellow and silver is colorless. In fact the appearance of these metals is characterized by an edge in the reflectance spectrum, similar to that predicted by the Drude model, but the problem is that all three metals have the same number of valence electrons. Also, the calculated plasma frequency for all three should lie at about 9 ev, well outside the visible region, so the plasma frequency cannot in itself account for the colors of Cu and Au. All three have filled dshells. Copper has the electronic configuration [Ar].d.4s, Silver [Kr].4d.5s and Gold [Xe].4f 4.5d.6s. (These metals are known as the Noble Metals.) The delectron bands lie below the Fermi energy of the conduction band: Transitions from the dband to the empty states above the Fermi level can be occur over a fairly narrow band of energies, around F which can be modeled as additional d Lorentz oscillator. The combined effects of the freeelectrons (Drude model) and the interband transitions due to bound delectrons (Lorentz model) influence the reflectance properties of the metal. Conduction band F d Fermi energy Dband k e Figure 6.4 Hence, r = free bound. Where f is described by the Drude model ( = ), and b is described by the Lorentz model. ( = [ F d]/.) xamples: silver, copper, and indiumtinoxide (ITO) Silver The reflectance spectrum of silver shows a strong drop at about 4 ev, well below the expected plasma frequency. The reflectance also rises again for frequencies just above 4 ev. (See Wooten, Fig..5, shown below.) It turns out that this behavior is because Silver has a dband resonance at 4 ev. This can be determined from experimental data by fitting the Drude model to the low frequency data, as shown in Wooten, Fig..8.(Shown below.) The difference between dielectric functions from Drude model and from experiment gives the dielectric function due to the 45

46 dband resonance, bound (written as (b) in Wooten.) The effect is to pull the r = frequency in from 9 ev to about.9 ev Figure 6.5 Real part of dielectric constant for Silver (from Wooten, Fig.8) This shift in p means that there is a shift in the free plasma oscillation in silver due to the delectrons. This can be explained by noting that the highly polarizable delectrons will reduce the electric field that provides the restoring force involved in these oscillations, illustrated on page 6. A reduced restoring force gives a reduced oscillation frequency. See Wooten fig.. for an illustration of how the delectrons do this. Copper The case of copper is almost identical to that of silver, except that the dband resonance is at about ev. Now since free becomes very large and negative at low frequencies, it turns out that bound due to the delectrons is not sufficient to pull the net through zero. Hence becomes small at about ev, but there is no true plasma frequency there. However, the effect of this is sufficient to cause R to start to drop at ev, but the reduction is gradual throughout the visible. This gives copper its characteristic redorange appearance. 46

47 Figure 6.6 (a) Dielectric function of Cu (Wooten, Fig.) (b) Reflectance of Cu (Wooten.) Tindoped Indium Oxide (ITO) a transparent conductor ITO is a semiconducting material that gives quite high electrical conductivity, yet is transparent in the visible. It is particularly useful in lowcurrent applications, such as liquid crystal displays. This is achieved by having a material with low electron density, but those electrons should be highly mobile, which means they travel through the material with relatively few collisions. By choosing the right density of tin doping, ITO can be highly effective. Below, we show the real and imaginary part of for ITO from a paper by Hamberg and Granqvist, Journal of Applied Physics, Volume 6, Issue, 986, Pages RR59. The plasma frequency, dependent of the Sn density, is typically around.7 ev, which corresponds to ~.7m. Due to this, and the free carrier absorption described above, ITO is not as useful in the near infrared ( > m) as it is in the visible. Figure 6.7 from Hamberg and Granqvist, Model (right) from David Tanner, University of Florida 47

48 A more recent study by D. Tanner et. al. at the University of Florida, shows that the properties of ITO can be well modeled by the combination of Drude and Lorentz models. Using a model for r that sums contributions from the Drude model to describe the freecarriers, and from the Lorentz model to describe bound carriers, which have a resonant absorption at = 4, cm ( / / c ) or = 5 nm. i.e. pb pf i i r b f (6.4) where is a background highfrequency dielectric constant, and subscripts, b and f, correspond to bound and free electrons (i.e. Drude and Lorentz contributions, respectively. The plasma frequencies and damping times for the bound and free electrons are of course different. The results of their model is shown below. (Courtesy, David Tanner, University of Florida.) Drude Conductivity and Skin Depth We already took a look at how the conductivity affects the optical properties in a homework exercise. There we looked mainly at the effect when the freecarrier contribution to the susceptibility is weak. Since we are mainly considering metals, we will not look at the effect of charges in the case of good conductors, where the free carrier effects are large. Recall that the Drude model gave us r '( ) p / r"( ) p, (6.5) from which we can find the optical constants, n and from n( ) ( ) r ' r" r ' r ' r" r '. (6.6) We can also describe the optical properties of a freeelectron conductor in terms of the conductivity, which can also be determined by the Drude model. First, we note that the current density is related to the velocity of motion by j Ne v (6.7) and the current is also related to the electric field by, j ( ) ( ). Now we already wrote down to equation of motion for electrons in an electric field as d r ( t) dr ( t) m m e ( t), (6.8) dt dt 48

49 dr and noting that v, we can rewrite the equation of motion for the velocity: dt dv m mv ( t) e( t). (6.9) dt Now, = /, so that the solution for v is or, Hence, e v ( ) ( ), (6.) m i j( ) Nev( ) ( ) ( ) Ne ( ) m i. (6.) Ne ( ) m i is the Drude conductivity. Setting, Ne / m, we have, (6.) ( ), (6.) i Where is the DC conductivity. Now, in a previous homework, we inserted the conductivity into Maxwell s equations for a medium with charge carriers and an instantaneous susceptibility,, and found that we could write ( ) r( ) ( ) i. (6.4) If we assume only free carriers contribute to the optical properties, then this becomes, ( ) r ( ) i i i (6.5) we can separate r into its real and imaginary parts to get Now, r ( ) i (6.6) Ne m, so that, / / p p p r ( ) i, (6.7) 49

50 which is exactly as we obtained before. So now we can estimate the optical properties of a metal based on the DC conductivity, as an alternative to requiring knowledge of the electron density. We will now look at the lowfrequency limit of a good conductor. A said before, typically, ~ s of femtoseconds, so that if we are looking at low frequencies (far infrared or microwaves or longer wavelengths) then << or <<. In this case, '( ), r p p r"( ) Hence, ', so that, r " r p (6.8) r" p n. (6.9) Hence, the complex refractive index under these conditions may be written as ( ) i. (6.) The absorption coefficient is then given by, ( ) c c c. (6.) Now, the irradiance depends on propagation depth, z, as I ( z ) ( ) z I () e, (6.) which could be written as I ( z) z / ( ) I () e, (6.) where () = /() is a penetration depth, at which point the irradiance drops to /e or.6 of its incident value. Since this depth is usually very small in metals, it is referred to as the skin depth, which is then given by ( ). (6.4) Bear in mind the approximation ( << ) under which this result was obtained. For higher (mid infrared, visible, UV) frequencies, this is no longer valid, and we just have to include 5

51 the whole expression for r. Additionally, it was assumed that there are no other contributions to r, which is not exactly true because of bound electron resonances (i.e. transitions) at higher frequencies. Should it be necessary, these can also be included in our models. 5

52 5

53 Chapter 7 Optical Activity and MagnetoOptics Corresponding handouts are available on In Chapter we derived a scalar wave equation to describe light propagation. We assumed isotropic electric response, and ignored any magnetization effects. In the first part of the present Chapter we will see that an oscillating electric field can induce an oscillating magnetic moment in some molecules. This will require us to consider a vectorial wave equation. This effect will lead to the phenomenon of optical activity, the ability of molecules to gradually rotate linear polarization. In the second part of the Chapter we will see that the application of an external magnetic field along the wave propagation will introduce a additional transverse force on the oscillating bound electrons. This leads to a phenomenon known as Faraday rotation, the magnetically controlled gradual rotation of linear polarization, enabling nonreciprocal optical systems. Optical Activity In some isotropic media, (e.g. sugar solution in water) it is found that the polarization of linearly polarized light is rotated in proportion to the propagation distance. ŷ xˆ ẑ ŷ xˆ ẑ Figure 7. i.e. xˆ in cosz yˆ sinz where is the rotatory coefficient (/m or rad/m). The effect typically occurs in liquids solutions of chiral molecules, a special class of molecules that is not mirrorsymmetric. W.T. Kelvin defined chirality as any geometrical figure, or group of points, if its image in a plane mirror cannot be brought to coincide with itself. A simple example of a chiral molecule is shown below: vi, Left handed Right handed vi In this example the choice on calling either of the two configurations is arbitrary. For real molecules the choice of right vs. lefthanded is linked to the sign of the rotatory coefficient. 5

54 Note that the the left molecule contains four different atom types. It is not possible to rotate the left molecule to make it overlap with the right molecule. For each chiral molecule there exists a lefthanded and a righthanded isomer. These special isomers are called enantiomers. While they have the same composition (same atom types, same number of bonds), they may interact differently with left and right handed enantiomers of other molecules. This is particularly important in biology. For example, orange peel and lemon peel contain opposite enantiomers of the molecule limonene. These are responsible for the smell of the fruit. dlimonene smells of orange, llimonene smells of lemon. The letter d comes from the latin word dexter for right, and the letter l comes from the latin word laevus meaning left. dlimonene (orange) llimonene (lemon) Figure 7. Common table sugar is sucrose, which is a combination of glucose (dextrose righthanded sugar) and fructose (levulose lefthanded sugar). The dextrose molecule is shown below: Figure 7. Link between chirality and optical rotation An extreme case of a chiral molecule would be a molecule shaped like a helix ( coil ). For simplicity, let s imagine that valence electrons can move freely along the molecule. In the figure below, we see that an electric field would drive charge motion along the helical molecule. The resulting curved charge motion is accompanied by a magnetic response, in the same way that an electromagnet uses current to generate a magnetic field. A helical molecule (and a chiral molecule in general) can thus produce a magnetic field along the 54

55 direction of the electric field. Such an effect was not considered in the derivation of the scalar wave equation. In the following we will reconsider the wave equation, but this time taking into account the possible influence of chiral molecules. M k M k Figure 7.4 Note that if the molecule were flipped by 8, the effect would be the same. The is a crucial observation, as it explains why the effect does not average to zero inside isotropic media. A solution of chiral molecules will be isotropic, since it contains rondomly oriented molecules. The the optical activity is caused by the subset of molecules that are either aligned along or against the electric field, while some molecules may contribute less because they are not ideally aligned. When optical activity (the ability to rotate linear polarization) is observed in solutions of molecules, we expect that the amount of polarization rotation is dependent on the concentration of molecules. To descibe concentration dependent optical activity, we use the specific rotatory power or specific rotation s. It is defined as the amount of polarization rotation per unit length, per concentration. For a specific rotation given in units degrees cm /g the polarization rotation angle after a distance z is thus given by: θ(z) [ ] = z[cm] cm g C g cm Other common units used for specific rotation are degrees (or radians) per cm / (g/liter), i.e. with all length units described in units of cm. Describing optical activity with the vectorial wave equation As mentioned above, optical activity is caused by electrically induced magnetic moment on molecules, with an magnetic average contribution oriented along the electric field. To find this effect from Maxwell s equations, we need to consider the M contribution to the current. Recall, B H M, and P B J M (7.) dt dt and 55

56 56 t B, (7.) so that dt P dt M t B t (7.) for isotropic media and J f =. Hence. M t dt P dt c (7.4) So there is a term due to the curl of the magnetization that looks like a contribution to the polarization. Taking the Fourier transform, we get ) )( ( ), ( ), ( M ik i k P k c k (7.5) M k P k k c k ), ( ), ( (7.6) now we can compare magnitudes on the right hand side: cp M P km P M k (7.7) In most cases, this ratio << (see homework), so that if M k is parallel or antiparallel to P, we can ignore it. This corresponds to M being parallel to B, as shown below. This case is not interesting, as / M k is very small compared to P. (Note that we have assumed isotropic media, so that and P are always parallel.) Figure 7.5 However, if M is parallel to P, then M k is perpendicular to P, as shown below: P // k B M M k

57 57 Figure 7.6 Hence the effective polarization, M k P P eff, has a small component perpendicular to the applied field. This is what leads to the rotation of the electric field. Propagation equations for optically active media Now H M M ~, where M ~ is the magnetic susceptibility tensor. D i H ik t D H, (7.8) and since the medium is isotropic, n k H n H k. (7.9) Since M is parallel to, we may treat it as a scalar when relating M and. Hence we may write M as a scalar: n k H M M M ˆ (7.) Hence k n M k M ˆ (7.) we will take the direction of k as z. Hence we will write z k ˆ ˆ. The wave equation therefore becomes ˆ ˆ z c n c n k z n c k M M (7.) P // k B M M k

58 58 Now, y x y x ˆ ˆ, so that,, ˆ ˆ ˆ ˆ x y y x z y x z (7.) quating x and y components of the wave equation gives y x M y M x c n k c n c n c n k (7.4) So for a nontrivial solution, the determinant must be zero: c n k c n c n c n k M M (7.5) Hence for an eigenmode of the wave equation we have M M c n i c n k c n c n k 4 (7.6) But recall that y M x c n c n k, (7.7) so that plane wave modes k satisfy y x i (7.8) Hence an eigenmode of an isotropic chiral medium is one where e x and y are equal and / out of phase. i.e. circular polarization. So we can write the eigenmode as ˆ ˆ iy x, (7.9) where is the field amplitude and the and case correspond to leftcircularly polarized light (LCP) and rightcircularly polarized light (RCP) respectively. These modes have wavevectors:

59 n n k i M c c n or, k i M c (7.) but M <<, so, k n im /. (7.) c Hence the RCP and LCP waves experience different refractive indices: n n i M /. (7.) Now, at this point, it appears that the M component of the index is going to give rise to loss, as it appears to give an imaginary component to n. However, we should recall that while the electric polarization is proportional to the displacement, r, of charge, i.e. p er, the magnetic polarization is proportional to the current, i.e. m er ier. Hence for p real (r real), the induced magnetic dipole is purely imaginary. Therefore i Im i Im M Re M M M i M Im M This gives purely real refractive indices:. (7.) n n Im( M / ) (7.4) So we now clearly see that the RCP and LCP eigenmodes propagate at different speeds. To deal with linear polarizations, we use the fact that a linearly polarized wave can be written as a sum of equal amplitude RCP and LCP components: T 4 exp We can simplify by setting ik z t expik z t xˆ iyˆ expik z t xˆ iyˆ expik z t c.. c. c. c k k n k k, and k. This yields: c T ( z, t) xˆ cos z yˆ sin z exp i k z t c. c (7.5) (7.6) From this, we can see that gives the rotation of the linear polarization per unit length. Hence is termed the rotatory power, given by n Im M (in /m or rad/m). (7.7) c 59

60 Zeeman Splitting Here we are interested in the effect of an external magnetic field, B, on the optical properties of a medium. (This is somewhat similar to the case of chiral media, except here the chirality is imposed by an external magnetic field.) The general term for the use of magnetic fields to modify optical properties is MagnetoOptics In general for a medium exposed to an external magnetic field, B: where P i Z, F ijk Z, F ij j ijk j j B, (7.8) j k describes the linear magnetooptic effects known as the Zeeman and Faraday effects. Here, we will consider an isotropic material, so that ij is scalar. We also let B be oriented along the direction of propagation, i.e. B Bzˆ. Hence the force on an electron in the medium is given by the Lorentz force law: F e ev B e ev zb ˆ. (7.9) v hence, Now for a general velocity, x, y, z xˆ yˆ zˆ v zˆ B x y z yb, x B,. (7.) B For a plane wave propagating along z, x, for a bound electron in the field: x x x ex eby y y e ebx m x m y y Since the medium is isotropic, then x = y =. y, y, so we have equations of motion (7.) Now, a free electron with velocity v perpendicular to a magnetic field will make circular orbits around the zaxis, as shown in the figure below. Hence free electrons in a magnetic field would have a resonant absorption at the cyclotron resonance frequency of c = eb/m, (also known as the Larmor Precession frequency ). In our case, we have bound electrons, so we might expect that the bound electron resonance is modified by the cyclotron resonance. 6

61 6 Figure 7.7 Now, for electric field components defined by.... cc e cc e t i yx y t i x x, (7.) we anticipate the x and y components of the electron motion to be.... c c e y y c c e x x t i t i (7.) Therefore the equations of motion become x i B m e m e y i y i B m e m e x i y x (7.4) For convenience, we write D() = ( o i), so that ) ( ) ( Bx i md e y By i md e x y x. (7.5) Anticipating circular motion in our solutions, we define an amplitude, Q, corresponding to such motions: x iy Q (7.6) B x y v m eb m eb m B r v e r m evb c

62 6 Q md B e md e iy x md B e md e Bx i i By i i md e Q y x ) ( ) ( ) ( ) ( ) ( ) ( (7.7) (7.8) Hence Q corresponds to the amplitude of electron motion for illumination with right () and left () circularly polarized light. Note that B = yields the regular result from the Lorentz model. The resonance frequencies occur for eb/m =, which, under the resonance approximation, becomes ( ) eb/m, giving resonance frequencies for the two states of circularly polarized light as m eb (7.9) This is known as Zeeman splitting () B = m eb m eb m eb Q Q Figure 7.8 B i m e Q e i B D m e md B e md e Q md e md B e Q m e m e.. ) ( ) ( ) ( ) ( ) (

63 6 Faraday Rotation While the Zeeman effect gives the splitting of the absorption lines for left and right circular polarizations due to the magnetic field, the Faraday effect is the rotation of polarization far from resonance due to the consequent slight difference in refractive indices for the two polarization states. Figure 7.9 To analyze this, we take the far from resonance approximation: m B e m e Q / / (7.4) For m B e /, we can expand to first order: m B e m e Q (7.4) Now, m B e m Ne P NeQ P (7.4) Hence the susceptibility becomes B nvc B m Ne m Ne B ) ( / / ), ( (7.4)

64 where Ne V (7.44) n m c is the Verdet coefficient. The refractive index for left and right circularly polarized light may hence be written as nvc n ( ) B Vc n n B Vc n n B (7.45) Hence left and right circularly polarized waves see different refractive indices, which again leads to rotation of linear polarization, i.e., T ( z, t) xˆ cos VBz yˆ sin VBz exp i k z t c. c (7.46) Highly dispersive materials will have large Verdet coefficients. In practice, the magnetic susceptibility will also determine how large a Verdet coefficient of a material will be. Paramagnetic and diamagnetic materials would actually have Verdet coefficients of different sign. Full understanding of the Zeeman effect requires a quantum mechanical analysis. Note that the sign of B gives the sign of rotation of polarization. Compare this to the case of optical activity where there is no applied field and since the material is isotropic, the sign of rotation is intrinsic to the material, depending only on the sign of M. Figure 7. Faraday Cell 64

65 Chapter 8 Nonlinear Optical materials Corresponding handouts are available on For very high irradiance beams, the electric field is large enough that the polarization response, usually written as P(ω) = ε χ(ω)ε(ω) (8.) is no longer linearly dependent on the field strength. This is because the electron displacements from equilibrium are so large that Hooke s law, F = Kx, is no longer exactly true. To examine the effect of a nonlinear restoring force, we expand the polarization into a power series in : P() = P () P () P () = ε χ () χ () χ () (8.) Note that we have omitted frequency arguments for simplicity here. In general, new frequencies are generated, so we will deal with the appropriate frequencies on a casebycase basis. Anharmonic oscillator model While there is no simple general answer to the polarization response of an anharmonic oscillator, we can find an approximate response starting from Hooke s law, and adding the nonlinear response as a small perturbation. Hooke s law (F = Kx) corresponds to a purely parabolic or harmonic binding potential: V(x) = Fdx = Kx = mω x (8.) Note that V(x) here represents the position dependent potential energy of the electron in joules, not the electric potential in volts. The parabolic description is a good approximation for small deviations from equilibrium, but the binding potential cannot remain parabolic for all values of x. A more realistic potential may be obtained from a power series representation: V(x) = mω x max mbx (8.4) where a and b are anharmonicity coefficients. The modified Hooke s law now becomes F(x) = = mω x max mbx (8.5) The magnitude of the anharmonicity coefficients depends among other things on the symmetry of the material. For a centrosymmetric medium, V(x)=V(x), we have only evenorder terms: V(x) = mω x bx F = mω x mbx (8.6) 65

66 Noncentrosymmetric potential Centrosymmetric potential For each type of material, we will consider only the lowest anharmonic terms, since those tend to dominate over the higher order terms. Noncentrosymmetric materials SHG and optical rectification We will solve the equation of motion for x(t) and hence for P(t) using a simple perturbation method. First, we will consider noncentrosymmetric media, with only the lowest order anharmonic term: γ ω x ax = (t) (8.7) To solve this equation, we treat the ax term as a small perturbation. This implies that for a harmonic driving field (t), the electron oscillation x(t) will also be predominantly harmonic ( sinusoidal ). This sinusoidal component of the electron motion will be called x () (t) where the superscript () indicates that this represents the first order (linear) response. From the equation above it can be seen that in the presence of a sinusoidal (t) and x () (t) there will be a small extra force ax on the electron (and therefore a small contribution to the equation of motion) that scales approximately with [x () (t)] and thus with (t). These terms are of the form sin (t) = ½(cos(t)). This immediately suggests that for materials with a finite anharmonicity coefficient a, a large linear response (largeamplitude sinusoidal motion) will introduce a small contribution to the electron motion at zero frequency due to the term in the argument above, and a small contribution at twice the frequency of the driving field due to the term cos(t) from above. The generation of these two terms are called optical rectification and second harmonic generation respectively. The fact that the [x () (t)] term is proportional to cos(t) tells us that the strength of the extra restoring force at both these new frequencies will be equal. The actual induced electron response by these force contributions will depend on the response of the atom to this small new driving force, i.e. on the linear equation of motion. We thus have found that the coefficient a can lead to new force (compared to Hooke s Law) and therefore a contribution to the electron oscillation that scales with [x () (t)] and thus with (t). The resulting displacement x with the same functional form as (t) will be called the second order displacement x () with a corresponding second order polarization P (). Both x () and P () can contain terms with different frequencies (in this example zero frequency and the second harmonic). In order to describe the resulting polarization P () in terms of we will need an expression for the second order susceptibility () that specifies the magnitude and phase of the polarization response at each of the new frequencies, denoted () () and () (). The arguments above state that the anharmonic oscillator model will yield a polarization response that looks almost exactly like the linear response P (), but that will contain a small 66

67 (multifrequency) contribution denoted P () that scales with (t) as a result of the anharmonicity parameter a. A similar analysis involving anharmonicity parameter b shows that the polarization will also contain a third order polarization contribution P () that is proportional to x(t) and thus approximately to [x () ] and to (t). The total polarization can thus be described by an equation of the form P() = P () P () P () = ε χ () χ () χ () (8.8) Note that the analysis above can also be used when (t) contains multiple frequencies (e.g. two laser beams) but first we will investigate the case of a single driving frequency. To find the second order motion amplitudes x () () and x () () and the corresponding expressions for () () and () (), we need to solve the nonlinear equations of motion. To derive the nonlinear equations of motion, we will set the driving field to be (t) with <<. Note is a scaling parameter, not the wavelength. We look for motion amplitudes x (i) that scale with according to x(t) = λx () (t) λ x () (t) λ x () (t) (8.9) Substituting this into the nonlinear equation of motion (q 8.7) results in terms proportional to,,, etc. Solving separately for only those terms that scale linearly with gives an equation of motion proportional to describing a linear response to the driving field: λ x () γx () ω x () = (t). (8.) Solving Separately for only those terms that scale with yields a second order equation of motion: λ x () γx () ω x () = ax () (t) (8.) Note that if a=, this second order equation has no driving term, resulting in only one solution: x=, indicating that no nonlinear contribution to the electron motion occurs. To solve for the response for finite values of a, we first solve the first order equation of motion, which will give us the linear response. If we take a harmonic driving field: (t) = (ω )e c. c. (8.) the first order equation yields x () (ω) = ( ) (8.) which gives the first order macroscopic polarization: P () (ω) = Nex () (ω ) = Realizing that ( ) (8.4) P () (ω) = ε χ () (ω)(ω) (8.5) 67

68 we have found that χ () (ω) = (8.6) This is simply the Lorentz susceptibility, as expected for a system with a linear restoring force. We will simplify the notation in the following substantially by defining a function that represents the resonance term in the denominator: D(ω) = ω ω iωγ (8.7) which allows us to write the linear susceptibility of our nonlinear material as χ () (ω) = () as we found before using the (linear) Lorentz model. (8.8) With the obtained linear response we now have an approximate description of the driving term in the second order equation of motion: x () γx () ω x () = a x () (t). (8.9) This assumes that x(t) looks approximately like the first order response only, which is only reasonable when the nonlinear contribution is small. Based on the expression for x () (t) we find that x () (t) = x() (ω )e x() ( ω )e (8.) x () (t) = x() (ω ) e c. c. x() (ω ) (8.) We see that this term introduces a force that contains a frequency component and a zero frequency component. Consequently the second order displacement x () will also contain amplitude at these new frequencies. We can describe our second order motion at this double frequency as x () (t) = x() (ω )e c. c. (8.) Substituting the first term into the second order equation of motion gives (ω ) x() (ω )e iω γ x() (ω )e ω x() (ω )e = x() (ω ) e (8.) which together with x () (ω) = ( ) ( ) (8.4) leads to: (note: here we lose a factor. Perhaps in definition of vs. ½ c.c.?) [ (ω ) iγ(ω ) ω ] x () (ω ) = ( ) [( )] (8.5) 68

69 The left term between the square brackets is in fact D( ). Dividing left and right side by D( ) produces an expression for the second order contribution to the electron amplitude at the second harmonic frequency : x () (ω ) = ( ) ( )[( )] (8.6) Note that the amplitude of the oscillation at is large for excitation at = and for excitation at =. Now we can derive the second order polarization from our expansion P = P () P () (8.7) using our obtained second order contribution to the electron position: P () (ω ) = ε χ () (ω ; ω, ω ) (ω ) = Nex () (ω ) (8.8) Substituting the expression for x () ( ) gives ε χ () (ω ; ω, ω ) (ω) = χ () (ω ; ω, ω ) = ( )[( )] ( )( )( ) (8.9) This is the component of the second order susceptibility that describes the second order polarization and therefore the second harmonic generation in a noncentrosymmetric medium (a) under illumination with a singlefrequency wave at. We will discuss later how this leads to the generation of a second harmonic wave that can grow to be quite large. In addition to second harmonic generation, we see that there is also a zero frequency component to P (). Solving for the zero frequency part of (x () (t)) i.e. x () () we have ( ω )x () () = a x() (ω ) Where we have used x () () = a ω e m D(ω )D (ω ) = ae m D()D(ω )D (ω ) (8.) D() = (ω iγ) = ω and D (ω ) = D( ω ) (8.) Defining P () () = ε χ () (; ω, ω ) (8.) we find χ () (; ω, ω ) = ()( )( ) (8.) 69

70 Noncentrosymmetric materials SFG and DFG In the above we have seen that a single input frequency can give rise to second harmonic generation and to optical rectification. When using multiple driving frequencies (e.g. illuminating a material with two different laser beams) we find a more general solution. If we take such a driving field (t) = e e c. c. (8.4) then (x () (t)) becomes x () (t) = 4 x() (ω ) e 4 x() (ω ) e c. c x() (ω )x () (ω )e ( ) c. c. x() (ω ) x () (ω ) e ( ) c. c. x() (ω ) x() (ω ) (8.5) showing that we also obtain terms with a frequency that is the sum of the original driving frequencies, and terms with a frequency that is given by the difference between the driving frequencies. The corresponding processes are called sum frequency generation and difference frequency generation. The nonlinear susceptibilities describing sum and difference frequency mixing are defined by P () (ω ω ) = ε χ () (ω ω ; ω, ω ) (8.6) and hence χ () (ω ω ; ω, ω ) = ( )( )( ) (8.7) P () (ω ω ) = ε χ () (ω ω ; ω, ω ) (8.8) χ () (ω ω ; ω, ω ) = ( )( )( ) (8.9) The shows that we have resonances at each of the three frequencies involved in the process. While the anharmonic oscillator model is somewhat limited in accuracy, we should take away some important messages from it: () processes require a medium that lacks inversion symmetry () processes result in sum and difference frequency generation. In the case of degenerate (=) mixing, these correspond to second harmonic generation and optical rectification. Resonance enhancement of () can result if any of the frequencies involved lies close to a material resonance. In quantum mechanical systems where there are several resonances more than one frequency can be resonantly enhanced. The nonlinear susceptibilities are related to the linear susceptibilities. 7

71 The last comment relates to Miller s rule, which can be argued as follows: We found: (note: factor /4 missing check) χ () (ω ω ; ω, ω ) = Ne a ε m D(ω ω )D(ω )D ( ω ) (8.4) but we also found that or χ () (ω) = / () = () χ() (ω) This allows us to write (8.4) (8.4) χ () (ω ω ; ω, ω ) = ε m N e χ() (ω ω )χ () (ω )χ () (ω ) Ne a ε m = ε ma N e χ() (ω ω )χ () (ω )χ () (ω ) (8.4) If we know a/n and () at each frequency in the process, we will have an estimate of (). Miller (Appl. Phys. Lett. 5, 7 (964)) noted that for all noncentrosymmetric materials a/n appears to be roughly constant, i.e. he found that for several different materials where () ( ;, ) () ( ) () ( ) () ( ) ~constant = δ (8.44) δ = ε ma N e This constant is known as Miller s delta. Variations in Miller s are limited to a factor ~. Miller s rule allows us to estimate the value of () for a material without having to measure it. Typically, ~.5 m/v. In a medium with n.5, we have X n ~.5. In this case we have χ () (ω ω )χ () (ω )χ () (ω ) (8.45) Typical values of () are 5 m/v or.5 pm/v. 7

72 Centrosymmetric materials THG and nonlinear refraction Centrosymmetric materials are materials that are invariant under the transformation r r. For such materials, a=, and hence () is zero. Hence the lowest order anharmonic term is mbx. The equations of motion in this case become λ x () γx () ω x () = (t) λ x () γx () ω x () = bx () (t) (8.46) Note that all materials (including the noncentrosymmetric ones) have a finite third order anharmonicity coefficient b. This means that the effects described below can be observed in all materials at sufficient incident irradiance. For a monochromatic driving field we have (t) = e c. c. (8.47) x () (ωt) = x() (ω)e c. c. (8.48) In the third order equation of motion this leads to a driving term proportional to x () (t) = x() (ω) e cc x() (ω)x () (ω) e c. c. (8.49) resulting in a polarization component oscillating at, the third harmonic, and a component oscillating at a frequency. This intensity dependent contribution to the susceptibility at the fundamental frequency results in an intensity dependent refractive index and intensity dependent absorption. Now, for the component at we have x () be /(m) (ω) = D(ω)D(ω)D(ω)D( ω) But P () (ω) = Nex () (ω) = Nb e /(m) D (ω)d( ω) (8.5) P () (ω) = ε χ () (ω; ω, ω, ω) 7

73 χ () (ω; ω, ω, ω) = b 8 e m ω D(ω)D(ω)D(ω)D( ω) (8.5) and χ () (ω; ω, ω, ω) = b 8 e m ω D(ω)D(ω)D(ω)D(ω) (8.5) corresponding to the third order susceptibility contribution giving rise to a third harmonic signal. A logical question is how P () and P () affect light propagation, which is discussed in Chapter 8. 7

74 74

75 Chapter 9 Quantum mechanics Corresponding handouts are available on The preceding Chapters described lightmatter interaction in classical terms. To fully understand many modern optical effects it is necessary to realize that under certain conditions matter must be described as having wavelike properties. This leads to kinetic (and total) energies that can only take on specific values, meaning that energies become quantized. This in turn means that optical transitions that change the energy of a system must involve discrete amounts of electromagnetic energy, leading to the surprising realization that our continuous M waves must be seen as containing packets of energy, known as photons. Most of this course deals with the semiclassical view of lightmatter interaction, which means that M waves are treated as classical waves, while electronic states are considered quantized. More indepth CROL courses dealing with quantum mechanics: OS649 Applied Quantum Mechanics for Optics and ngineering: lements of quantum mechanics that are essential for understanding many areas in modern optics and photonics. OS647 Quantum Optics: Semiclassical treatment of light/matter interaction (quantized atomic states and Maxwell's equations), density matrix theory, coherent optical transition, pulse propagation. For the CROL course OS5 the quantum mechanical description of light matter interaction is taught from select chapters in David A. Miller s book Quantum Mechanics for Scientists and ngineers. The topics that are covered in OS5 are listed in Appendix L. Below follows an abbreviated description. The early days In the late 9 th century, it was found that measurements on optical transitions in the hydrogen atom could not be explained using the classical mechanical model of a negatively charged electron orbiting a positively charged core, held together by the Coulomb force (F = q q /r ). The classical model would predict that the electron could orbit at any arbitrary distance, and have a continuous spectrum of allowed energies. When taking into account that the orbiting electron should radiate energy (recall: accelerating charge radiates, Ch), the classical model incorrectly predicts that the electron would lose energy as it orbits, and therefore spiral toward the core causing all matter to collapse. Thankfully this doesn t happen. By carefully studying the hydrogen absorption and emission spectra, Rydberg found that hydrogen absorbs light at very specific frequencies, corresponding to free space wavelengths that follow a simple relation: λ = R n n 75

76 with R H the Rydberg constant.97 um, and n < n both integers. Based on this work and analysis by Bohr, Louis the Broglie hypothesized that any moving object has an associated wave character with a wavelength given by = h/p where h is the Planck constant ( Js). Here p represents the momentum of the moving particle, here the electron orbiting the ion core. This led to the following important relation between quantum mechanical momentum, and the matter wavelength λ = known as the de Broglie wavelength. This hypothesis was later found to accurately describe electron diffraction by a Ni crystal (97). The wavellike behavior of particles has many profound consequencies in physics. Here we will focus mostly on the effect of the wavelike nature of electrons on their possible energies and movement. The wavefunction As discussed above, electrons show behavior that is wavelike. This means that we can no longer describe an electron simply with a position and a velocity. Instead we need a function that describes the matter wave amplitude as a function of position. This kind of particle amplitude is described with a wavefunction, typically written as ψ(x, y, z). vii While matter amplitude doesn t have a classical analog, it can be used to calculate a wide range of observable physical properties. If ψ is the electron wavefunction, then the probability of finding the electron in infinitesimal volume dxdydz is proportional to P ψ(x, y, z) dxdydz (9.) We can change the proportional sign in this equation to an equals sign if we normalize the wavefunction as follows: ψ ψ = ψ dxdydz (9.) In the following we will assume that we are dealing with normalized wavefunctions, meaning that ψ represents the probability density, i.e. the probability of finding the electron per unit volume. Note that the wave description of electrons has some similarities to the wave description of light. viii We know that light intensity ( photon probability ) scales with, just like the electron probability scales with ψ. lectron momentum and kinetic energy As mentioned above, the wavelength of the electron was experimentally found to be related to its momentum: vii In OS5 we exclusively look at singleparticle wavefunctions, but the concept can also be extended to states involving multiple particles, with e.g. a wavefunction ψ(r, r ) describing what is the probability (amplitude) that particle is here, and particle is there? viii There are also some very important differences: the electron amplitude is scalar, not vectorial. lectrons cannot be absorbed or dissipated in the way that light can. lectrons are fermions, photons are bosons, which has among other things important consequences for thermal distributions. 76

77 λ = h p (9.) This is called the de Broglie wavelength of the electron. ix If ψ describes an electron wave with a welldefined momentum p everywhere, then apparently it should have the same wavelength everywhere. This means that a freely moving electron with constant momentum can be described by a wave with a fixed wavevector k. The (unnormalized) wavefunction of a free undisturbed electron is then ψ (z) = e. (9.4) In this expression the electron wavevector is linked to the wavelength in the usual way: k =. Comparing this with the de Broglie wavelength, we find the following expression for the electron momentum: p = ħk. (9.5) Note that this expression for the momentum is only valid for a particle with a perfectly defined momentum everywhere. x At any point in space we can extract the momentum by investigating the wavelength of this plane wave, which we can do by taking a single spatial derivative. For an electron moving along the z direction we have d dz ψ (z) = d dz e = ik e = i p ħ e (9.6) where the last step uses λ = h/p. We have found the relation ih d dz ψ (z) = p e (9.7) where p z is the momentum of the electron along the zdirection. We see that taking the spatial derivative of our wavefunction places a multiplier in front of our original wavefunction that tells us about momentum. The operation that we carry out on the left is called an operator, here specifically the momentum operator. Operators are written with a hat, so we have p = iħ d dz (9.8) When the electron wavefunction has a welldefined ( single valued ) property everywhere, the electron is said to be in an igenstate, and in this case is said to be an igenfunction. Our free electron apparently satisfies the relation p ψ (z) = p ψ (z) (9.9) This says: wherever you look, the electron wave has the same periodicity, and therefore the same momentum. In this case is apparently a momentum igenfunction: the operation p acting on wavefunction ψ produces a result with the same shape as ψ, just multiplied ix This is a general property of matter. For example, a moving neutron also has a de Broglie wavelength related to its momentum. Since the neutron is much heavier than the electron, at the same velocity it will have a much higher momentum and a much smaller wavelength. x In fact this is also the appropriate expression for the momentum of a photon, a massless quantum particle of light. 77

78 by its momentum value. This also explains the name igenfunction: the word igen comes from German meaning self, and an igenfunction of the momentum operator is a function that turns into itself (maintains its original shape) when you carry out the momentum operation. Note that a wavefunction of the form ψ cos (k z) has a perfectly welldefined wavevector, but this does not represent a momentum igenstate: p [cos(k z)] = iħ d dz cos(k z) = iħ k sin(k z) = i p sin(k z). (9.) The momentum operator changes the cos function into a sin function. The function does not turn into itself, so it is not a momentum igenfunction, and apparently the momentum of such an electron distribution is not perfectly welldefined. If we can learn about the momentum of a particle by investigating the single spatial derivative of the wavefunction, it follows that we can also investigate other properties that depend on momentum, for example the kinetic energy mv =. We can use the momentum operator to construct an operator that represents the kinetic energy. For an electron in a momentum igenstate we had relation 9.9. Since the righthand side is a constant times a momentum igenfunction, it s easy to see that our free electron also satisfies p [p ψ (z)] = iħ d dz [p ψ (z)] = iħp d dz ψ (z) = p ψ (z) (9.) We write the act of applying operator p twice in a row as p : p ψ(z) p [p ψ (z)] p p p (9.) We have seen that an igenfunction of p is also an igenfunction of p. We can now construct an operator that represents kinetic energy: p ψ(z) = iħ d m dz iħ d dz ψ(z) = ħ m m We have constructed the kinetic energy operator: d ψ(z) = ħ ψ(z) m dz d ψ(z) (9.) dz d ħ (9.4) m dz A system that has perfectly defined kinetic energy must be in a kinetic energy igenstate, and must therefore satisfy d ψ(z) = ħ m dz ψ(z) = ψ(z) (9.5) with on the right hand side a number. Substituting our momentum igenfunction in vacuum ψ (z) = e into this relation we find ψ(z) = ħ d m dz e = ħ k ψ(z) (9.6) m 78

79 We have found the dispersion relation of free electrons, showing the free electron energy as a function of wavevector. Note that this follows simply from the de Broglie wavelength: previously we found (for a plane wave) p = ħk, and we already knew that classically = p /m, so the quadratic dependence of energy on wavevector is an expected result for a free electron. The Timeindependent Schrödinger quation If an electron is placed near a positive charge or even just in a constant applied field, it will have a position dependent potential energy, which here we call V(r ). This is not the electric potential: in most quantum mechanical formulas the function V(r ) represents potential energy, which for an electron in an electric potential V (r ) is simply given by V(r ) = ev (r ). nergy conservation states that if an electron moves to a region with lower potential energy, it must gain a corresponding amount of kinetic energy. With this in mind, we can also construct an operator that describes the total electron energy that is the sum of the kinetic and potential energy. We will call this simply the energy. If an electron has a welldefined energy then apparently the sum of the kinetic and potential energy is the same at every position. Such an energy igenfunction must satisfy: ψ(z) = V(z)ψ(z) = ψ(z) (9.7) The term between the square brackets is the total energy operator known as the Hamiltonian H: d Hψ(z) = ħ V(z) ψ(z) (9.8) m dz An electron in a static potential distribution can have a welldefined energy only if the wavefunction is an energy igenfunction that satisfies ħ d V(z) ψ(z) = ψ(z). (9.9) m dz This is known as the Time independent Schrödinger equation. For a threedimensional system the Time Independent Schrödinger equation is: ħ m V(r ) ψ(r ) = ψ(r ) (9.) Solutions to the timeindependent Schrödinger equation are nergy igenfunctions, which represent possible stationary electron distributions. Note that a momentum igenfunction can be a total energy igenfunction in the case of a constant potential energy V. Substituting a trial function ψ (z) = e into the TIS with constant potential we find d m dz V e = e = ħ k V (9.) m ħ The graph below shows the relation between total energy for a free electron in a constant potential of ev (green curve) and a constant potential of ev (blue curve). The frequency axis will be discussed later. Note that unlike photons (massless particles) in vacuum, 79

80 electrons in vacuum have a quadratic dispersion relation, which is typical for particles with mass. 6 8 (ev) 4 V= ev 6 4 ( 5 rad/s) V= k (nm ) Figure 9. Free electron dispersion An important consequence of the timeindependent Schrödinger equation and of the wavelike nature of moving matter is that confined electrons, for example the bound electrons around atom cores, can only exist in welldefined states with distinct energy values. The bound electrons thus cannot have a continuous range of possible energies, but only discrete states at welldefined energies. For example, the binding potential of a positive hydrogen core is a static Coulomb potential. The timeindependent Schrödinger equation tells us that an electrons confined in this static Coulomb potential can only have a welldefined energy if they have very specific spatial distributions ψ. As we will see, these distributions can have low total energy (tightly bound electrons close to the core) and high energy (more spatially extended excited states). The timeindependent Schrödinger equation thus predicts many possible energies, known as energy levels, and the energy level spectrum of bound electrons is said to be quantized. The TimeDependent Schrödinger quation The previously discussed timeindependent Schrödinger equation tells us possible stationary charge distributions (electron probability distributions) with welldefined energy in a static potential. However to describe lightmatter interaction we will need to consider situations where we excite our quantum system (e.g. a bound electron) with an electromagnetic wave. The oscillatory electric field adds an oscillatory contribution to the potential energy, so the total potential energy distribution V(r, t) contains both a static binding potential V (r ) and a contribution due to the presence of the M wave V (r, t). The response of the electron wavefunction to such timedependent potential distributions is described by the timedependent Schrödinger equation, given by ħ V(r, t) ψ(r, t) = iħ ψ(r, t) (9.) m t 8

81 This relation can describe both the time dependent behavior of the electron in a dynamic potential, as well as the time dependent behaviour of an electron in a static potential. Note that the timedependent Schrödinger equation is much less restrictive than the time Independent Schrödinger equation. While the TIS could only descibe very particular wavefunctions with a perfectly defined energy in a static potential, the TDS describes generally what happens to the electron in the presence of a binding potential and any applied electric fields. The TDS states if the electron has energy at a given position, the wavefunction at that position must change over time. This means that this equation allows for an infinite range of complicated temporal and spatial behavior. The TDS equation produces some important results when we consider a free electron in vacuum without any applied electric fields (V constant). In this case, and considering a onedimensional situation (all movement along the zaxis), the timedependent Schrödinger equation (TDS) tells us ħ m z V ψ(z, t) = iħ ψ(z, t) (9.) t We already know that in vacuum we can have momentum igenstates which had the form of a plane wave, and we saw that those plane waves in vacuum also happened to be energy igenstates. If we multiply our entire plane electron wave by a constant, the result is still an energy igenstate. This is even true if the constant is timedependent. Substituting such a time dependent energy igenfunction ψ(z, t) = f(t)e (9.4) into the TDS we find a requirement on the time dependence: ħ m z V f(t)e = iħ t f(t)e f(t) ħ m z V e = iħe t f(t) f(t)e = iħe t f(t) (9.5) t f(t) = i f(t) ħ f(t) = e /ħ We see that a free electron in an energy igenstate must oscillate with an angular frequency of ω = /ħ, in other words the total energy is linked to frequency according to = ħω. This looks exactly the same as the formula for photon energy, but note that here represents the total energy. This means that the frequency of the electron wavefunction can be changed with a static potential, while the photon frequency is not affected by a static potential. We can apparently have a time dependent energy igenfunction in vacuum of the form 8

82 ψ(z, t) = e ( ħ ) (9.6) We have found that a free electron in an energy igenstate is described by a propagating wave representing probability amplitude. This seems surprising: previously we stated that energy igenfunctions represented static probability distributions, but here we find an energy igenfunction that is a running wave. These statements are in fact not in conflict: calculating the probability density of our electron wave, we find for this energy igenstate ψ(z, t) = ψ (z, t)ψ(z, t) = e ħ e ħ = (9.7) We see that our electron wave has a constant probability density everywhere, so the observable quantity probability is actually not moving. Superposition of time dependent energy igenfunctions If we want to describe a moving free electron, we can construct an electron wavepacket by superposing multiple plane momentum igenfunctions. This is in fact allowed by the TDS: if we find different solutions to the TDS then the superposition of these solutions is also a solution to the TDS. This is easy to verify try it. If we add electron plane waves with similar wavevector around an average wavevector k we can produce a localized probability distribution. This particle pulse moves at a velocity known as the group velocity, which is given by v k = dω dk Looking at q. 9. and using = ħω we see that for free electrons we have = ħω = ħ k m V v = ħk m = p m (9.8) (9.9) where p is the average momentum of the electron waves. We see that an electron pulse with an average quantum mechanical momentum p = ħk moves at a velocity given by v = p/m, which is what we expect classically. Uncertainty principle Note that our electron wavepacket no longer has a perfectly defined momentum. If the momentum was perfectly defined, then the electron wavefunction must continue oscillating in exactly the same way everywhere, meaning that the electron probability is infinitely spread out. By making a superposition of waves with different momentum (different wavevector), we can make a wavepacket that is localized in space. A more strongly localized wavepacket requires the use of a wider range of wavevectors. This kind of relation is called an uncertainty relation. The uncertainty relation for momentum and position is x k (9.) or equivalently after multiplying by ħ x p ħ (9.) 8

83 Here the represents the standard deviation. The larger or equal part of the uncertainty relations implies that the superposition of many wavevectors doesn t necessarily make a confined wavepacket; a maximally localized wavepacket requires the choice of very specific phases and amplitudes. For processes that have a time dependence related to the addition of wavefunctions with different energy (and thus different time dependence ) there is an energytime uncertainty relation given by ω t (9.) or equivalently t ħ (9.) nergy igenfunctions in important binding potentials In the preceding chapters we described light matter interaction with electric fields that exerted forces on classical particles. In order to understand how quantum systems respond to electric fields, we first need to understand the nature of the electronic states in various binding potentials. Important examples for optical materials are (quasi) free electrons, already discussed above), the Coulomb potential in order to describe bound electrons around an atom core, and quantum wells resulting in trapped electrons in semiconductors with energies that depend on the well width, and the harmonic binding potential, describing molecular vibrations (and in fact the electromagnetic field, but this is outside the scope of these notes). lectron wavefunctions in a onedimensional infinitely deep square well One of the easiest electron binding potentials is the onedimensional infinite square well, which is represented by a region where the potential energy is constant, surrounded by regions where the potential energy is infinitely high. This seems like an entirely unphysical test system, but surprisingly it turns out that this system is very relevant to modern optics: it resembles to a great degree the situation in semiconductor quantum wells, which will be discussed later. Figure zz shows the position dependent potential of a square well extending from z= to z=l z (the length along the z direction ). Inside the well we set the V=, and outside we set the potential to a constant value V. For an infinite square we take V. Under this condition, the electron cannot leave the well, meaning that the probability density outside the well will be zero, and therefore also = outside the well. 8

84 Figure 9. Schematic of square quantum well and example physical system To find all possible wavefunctions inside the infinite square well, we are looking for solutions to the time independent Schrödinger equation ħ m V(r) ψ = ψ which in this onedimensional situation simplifies to ħ m d ψ(z) dz V(z)ψ(z) = ψ(z) Since we have set V= inside the well, wavefunctions in the well should satisfy ħ m d ψ(z) dz = ψ(z) It is easy to recognize that this will lead to solutions of the form ψ e where the wavevector will be energy dependent (larger larger double derivative needed = larger k shorter wavelength). Substituting this in q. x we find the relation ħ k m ψ(z) = ψ(z) We have obtained a simple relation between energy and wavevector: = ħ k m Note that this relation assumes that V=. The more general result including a position independent potential energy would yield = V = ħ k m Solutions to the TIS must satisfy certain boundary conditions. In general, in any realistic potential distributions the wavefunction must be () finite, () continuous, and () 84

85 continuously differentiable. For the limiting case of infinitely high potential walls, the third requirement is not considered. Since = for z and z L z, our solutions must reach zero amplitude for z= and z=l z. This requirement cannot be satisfied by solutions containing only a single wavevector k, since functions of the form ψ e have a nonzero amplitude everywhere. To solve this problem we consider superpositions of the form ψ = A e B e To satisfy the requirement of zero amplitude at z=, we require A B = or A= B. To satisfy the requirement of zero amplitude at z=lz, we thus require This is equivalent to requiring A e A e = e e = i sin kl = The condition sin kl = is satisfied whenever the argument of the sine term is n with n an integer, i.e. kl = nπ, or k = n. We thus have an infinite number of allowed solutions of the form ψ = A sin nπ L z where A n the magnitude of A n is determined by requiring that the wavefunction is normalized, and with an arbitrary complex phase. The energy of the allowed wavefunctions or energy igenstates is given by xi = ħ k m = ħ m nπ L Note: the energy of the wavefunction increases quadratically with n, and decreases as the well width is increased. One important conclusion is that for any well size, the lowest energy solution has a nonzero energy given by = ħ m π L where the superscript will be used to represent the fact that this is the solution of the infinite square well. This type of minimum energy is called the zero point energy of the system. It is easy to understand that such minimum energy exists: the requirement of zero wavefunction amplitude at the well edges and nonzero amplitude inside the well implies xi Note: strictly speaking this is the kinetic energy term only, but since we have set V= inside the well, the total energy is equal to the kinetic energy ħ k /m in this case. 85

86 finite curvature (double spatial derivative) of the wavefunction, which is associated with finite kinetic energy through the TIS. This same zeropoint energy also explains the concept of quantum confinement : the (lowest) energy of an electron can be increased by confining the electron in ever smaller volumes. This forms the basis of the tuning of light emission in semiconductor quantum dots. Figure 9. The first three igenstates of the infinite square well Parity of quantum mechanical states In the case of the infinite square well we saw that the allowed quantum states or nergy igenfunctions had welldefined symmetry properties. Our binding potential is inversion symmetric, and so is the probability distribution of all corresponding states. The wavefunction itself though can be either symmetric (n=,,5, ) or antisymmetric (n=,4,6, ). The symmetry of wavefunctions is described by the term parity. Symmetric states are said to have even parity, while antisymmetric wavefunctions are said to have odd parity. xii In our square well example wavefunctions are either perfectly odd or perfectly even. We will find later that light can induce transitions between states with opposite parity, while it cannot induce transitions between states with equal parity. This kind of relation that selectively allows specific transitions is called a selection rule. We will discuss this in more detail when we cover optical excitation and timedependent perturbation theory. States of the finite square well To be added. Boundary conditions are different (continuity of slope required), energies drop, selection rules remain. Note: maximum bound electron energy, unlike case for infinite well. States of the harmonic oscillator To be added. Fixed level spacing, note perfect parity. Comment: realistic binding potential: finite max bound energy. xii Note that in our choice of n= for the ground state, even states are unfortunately described by an odd number. ven parity means that the wavefunction is leftright symmetric, not that the number n is even. 86

87 States of the Coulomb potential To be added. D system, extra degree of freedom. Note: potential energy has an infinity at center continuity of slope not always satisfied at center. Note: energy levels closer together for n up. Note: ionization possible. Orthonormal complete sets To be added. Discuss based on infinite square well. xpansion coefficients To be added. Once normalized, expansion coefficient squared represents probability. Dirac notation To be added. Discuss overlap integral, projection onto basis. xpectation value To be added. Being in superposition of states implies average energy, give formula in dirac notation. nergy expectation, position expectation, dipole moment expectation value. Oscillatory dipole moment To be added. igenstate has time dependent amplitude, but constant probability distribution. Show that superposition of odd and even states has time dependent dipole moment. If light can induce oscillatory dipole moment, apparently light can put some amplitude in state with opposite parity. Can lead to dipole moment or disappearance of photons (=absorption). Discussed in time dependent perturbation theory. Time dependent perturbation theory excitation rate To be added. ffect of perturbation hamiltonian, describe potential distribution of plane wave. Introduce line shape, joint density of states, cross section Time dependent perturbation theory susceptibility To be added. Derive formula for dipole moment. 87

88 88

89 Chapter Debye model of the optical properties of polar liquids Corresponding handouts are available on In the preceding Chapters we modeled the optical response of atoms in dilute gases with the Lorentz model, and described the response of solids using the Lorentz model (bound electrons, linear and nonlinear response) and the Drude model (free electrons). A state of matter that we have not discussed yet is the liquid. This Chapter introduces the Debye model, which can be used to describe the lowfrequency ( microwave, far infrared, GHz) response of polar liquids. Liquids are composed of molecules or atoms that are densely packed (similar densities as solids), but the molecules still have the freedom to move around and reorient. Since the molecules in the liquid contain bound electrons, we expect that light can excite the electrons, just like in insulators. This results in optical absorption at visible and UV frequencies, which we can attempt to describe with the Lorentz model. However, liquids containing molecules that have a net dipole moment show another type of polarization response at low frequency, related to the slow fieldinduced alignment of the molecules in the liquid. To model this kind of optical response we will make use of results from statistical mechanics. Figure.: Schematic view of densely packed water molecules, and (right) illustration of the net dipole moment of a water molecule. The oxygen atom is negatively charged, which together with the bond angle results in a net dipole. Hindered rotational modes Let c = mean time between collisions. Hence c N where N is the molecular density. If r is the rotational frequency, ( r = / r, where r is the rotational period) then the rotational modes are said to be hindered if r c <.Or, r c = N c/n, where N c is the molecular density at which r c =. For N c << N, we have the dilute medium limit, where we see discrete rotational spectra. (Lines spaced by r, as we shall describe later.) Consider the effect of an electric field on a molecule with a permanent dipole moment. 89

90 p Interaction V int Potential p cos Figure. For now, we assume that is a DC field, although the above definition for V int does not depend on this assumption. Now p cos (.) p cos P( ) d P( ) d where P()d = probability of a molecular axis having angle w.r.t. field in the range (, d). And applying Boltzmann statistics, [note that this implies a different but equally valid definition of the angle theta as shown in class, because the probability should scale with e /kt which according to the previous definition would be exp pcos/kt ] P ( ) d sin e p cos / kt d d p 9

91 Figure. p cos p p cos sin e sin e p L kt p p cos / kt p cos / kt d p coth kt d kt p (.) Figure.4: The Langevin function. Note the linear section for small argument, and the saturation for large argument. Where L(x) is the Langevin function, L(x) ~ x/ for x <<. Hence, Now, p p cos p, for p << kt. (.) kt p //, so that p p p p ˆ cos ˆ (.4) kt kt p, but here is a scalar. ~ Hence the macroscopic polarization is: p P N ~ N. (.5) kt So far, could be a DC field i.e. assumed thermal equilibrium is established faster than can change. The electric field aligns the molecules and hence induces a macroscopic polarization. This is a dynamic equilibrium. The field is aligning the molecules, while collisions are 9

92 randomizing their orientation. If the field is suddenly switched off, the polarization will decay according to dp P t P( t) P() e. (.6) dt Here is the Rotational Correlation time, which is given by, V, (.7) kt where is the viscosity and V the molecular volume. Note that c, although they are related. For H O (in H O) this value is ~ 4 ps. (A typical value. Note that is larger in dipolar molecules than in molecules with no dipole moment.(e.g. CS, ~ ps.) Optical response in Debye description Now consider a harmonic field, (t) = cos(t). We cannot write P(t) = Np /kt (t), unless <<. Hence we add a driving term to the relaxation equation: dp Np P ( ) dt kt t (.8) Debye relaxation equation. We can solve this to get the (), n() and () (homework), resulting in χ(ω) = Np ktε ( iωτ). The graphs below show the corresponding real and imaginary index for a rotational correlation time of ~ ps..4. n( ). ( ) Figure.5 9

93 Figure.6: The refractive index of water, from Note the Debye like response at wavelengths above m (frequencies below THz), as well as some vibrational lines between m and m. 9

94 94

95 Chapter Homogeneous and inhomogeneous broadening Corresponding handouts are available on In the preceding chapters we have seen optical transitions that occur at specific frequencies, with a linewidth determined by some damping process described by a damping constant (s ). We might expect that a system containing many such oscillators will result in an absorption line with the same linewidth. If the linewidth of a collection of oscillators is practically the same as that of each individual oscillator, we call that line homogeneously broadened. In real world situations however it is often found that the oscillators in a system all behave slightly differently. This can happen for example with molecules in a gas laser, quantum dots in a quantum dot detector, or rare earth dopant ions in a fiber amplifier. In these cases an absorption line may be broad not because is large, but instead because is a little different for the different elements in the system. If the linewidth of a collection of oscillators is larger than that of each individual oscillator, we call the line inhomogeneously broadened. Inhomogeneous broadening due to variations in local environment In many solid state systems, the atoms or molecules do not experience the same environment. In some cases, they may be in some random host matrix (e.g. a glass) where the surrounding electric field is slightly different and somewhat random for each atom, causing the resonance frequency of each atom or molecule to be slightly different. If the shift in the resonance frequency of each atom/molecule is random, the probability of having frequency shift with respect to the mean (unshifted) frequency is usually described by a Gaussian distribution. In this case each molecule has a different resonant frequency, corresponding to inhomogeneous broadening. Figure.7: (left) Sketch of a block of glass containing dopants with an absorption line, and (right) sketch of the ground state and excited state energy levels. Variation in the local environment leads to variation in energy levels and transition energies, causing inhomogeneous broadening. 95

96 Doppler broadening According to statistical mechanics of ideal gases, atoms (or molecules) in the gas phase have an isotropic temperaturedependent velocity distribution, described by the Maxwell Boltzmann distribution, given by f (v) = m / πkt 4πv e / Here f MB(v)dv represents the fraction of atoms with a thermal velocity magnitude between v and vdv, with m the mass of the atom (or ion, or molecule), and k the Boltzmann constant. When such a gas is illuminated using a laser propagating along the xdirection, some of the atoms travel away from the laser at velocity v x. These atoms will experience an electromagnetic wave with a reduced frequency. The modified frequency experienced by the atoms is ω = π c v λ = πc v /c λ = ω v c This frequency shift is called a Doppler shift. If the stationary atom had resonance frequency, we need a laser with a slightly higher frequency in order to excite this atom to compensate for this Doppler shift. xiii The atom thus appears to have a modified resonance frequency ω that is given by ω = ω v c ω The total absorption spectrum of a molecular gas is thus composed of an ensemble of absorption lines from all atoms, slightly shifted depending on the velocity component toward or away from the excitation source. For fast atoms (high temperature or low atomic mass) the Doppler shift can be larger than the original atom linewidth. In this case we observe line broadening, a reduction of the peak absorption, and a change in the line shape. The MaxwellBoltzmann distribution describes the probability distribution for the magnitude of the velocity v = v v v /. However since the Doppler shift depends on the atom velocity along the laser propagation direction, we need a probability distribution along a specific direction, e.g. v x. It can easily be shown that this projected velocity distribution, called the Maxwellian velocity distribution, is given by v c f (v ) = m / πkt e /. The graphs below show the MaxwellBoltzmann distribution (left) and Maxwellian distribution (right) at room temperature (T=9 K) for a gas containing hydrogen molecules, oxygen molecules, and Xe atoms (or ions). Note that at the same temperature molecules with low weight (e.g. H ) move much faster, and that the typical room temperature velocities are hundreds of meters per second. Also note that it is relatively likely to have zero velocity along the x direction (right graph, xiii Note that this effect seems opposite to the redshift observed in astronomy. Hydrogen related lines appear redshifted because distant stars are moving away from us, while here we are required to use a blueshifted laser to excite molecules that are moving away from us. Are these observations in conflict with each other? 96

97 maximum probability for v x=). This tells us that most atoms contributing to the total absorption have v x=, and we therefore expect that the absorption of a hot gas will still be maximum at the original. However there are also many atoms with v x which means that we will observe some blueshifted and redshifted absorption, corresponding to Doppler broadening. Figure.8: MaxwellBoltzmann (left) and Maxwellian (right) velocity distribution for a roomtemperature gas of H, O, and Xe. We can learn about the resulting lineshape by considering each atom as a Lorentz oscillator with a susceptibility given by ω χ(ω) = ω ω iγω but realizing that each atom will have its own modified resonance frequency ω (v ) = ω v c with probability f (v ). The total susceptibility spectrum then follows from the addition (integration) of all susceptibility contributions for all possible v x: Substituting f M this gives ω χ(ω) = ω (v ) ω iγω f (v ) dv χ(ω) = m / πkt ω ω (v ) ω iγω e / dv With a known concentration N(m ) this describes the entire complex dielectric function, and we can therefore calculate the Dopplerbroadened absorption spectrum. We can find an approximate expression for the absorption coefficient by considering the resonance approximation, which we may do if the (Dopplerbroadened) linewidth is much smaller than. In the resonance condition we have χ (ω) ω m / ω πkt Γ/ (ω (v ) ω) (Γ/) e / dv 97

98 For small susceptibility (n) which is reasonable in the gas phase, we can approximate the absorption coefficient as α = κ resulting in the Doppler broadened absorption spectrum: α(ω) ω c m / πkt Γ/ (ω (v ) ω) (Γ/) e / dv The spectral shape of this inhomogeneously broadened absorption line is known as the Voigt lineshape. We can look at the extreme cases of mostly homogeneous broadening and strong inhomogeneous broadening. Note that the integrand is just a product of a Gaussian function and a Lorentzian. Thermal line broadening dominates when /. In this case the Lorentzian function becomes approximately a delta function that peaks when ω = ω (v ), i.e when v = ( ) v = (ω ω), resulting in a Gaussian lineshape: α(ω) ω c m / πkt e ( ) If the homogeneous width dominates, i.e. when /, then the Gaussian acts like a delta function and the absorption line reverts back to the original Lorentzian lineshape: α(ω) ω Γ/ c (ω ω) (Γ/) This transition from Lorentzian to Gaussian is highlighted in the graph below, which shows the lineshape in an argon gas with resonance at 54 nm, with an assumed linewidth of 9 rad/s, shown at K, K, and 9 K. Note that already at K the thermally induced broadening becomes comparable to the 9 rad/s linewidth. Also note that the broadening results in a reduced peak absorption, and therefore also in a predicted reduction in the peak emission crosssection, and a reduced maximum gain if this transition is used for stimulated emission (see Chapter 6). Figure.9: xamples of the Voight lineshape for different temperatures 98

99 Chapter Interaction of light with molecular vibration and rotation Corresponding handouts are available on In Chapter 5 we studied the Lorentz oscillator model to describe electronic motion in response to a high frequency electromagnetic wave. There we ignored any motion of the nucleus, which was a reasonable approximation because atomic nuclei are at least three orders of magnitude heavier than electrons. However, atoms with a net charge (ions) can be moved by low frequency light, resulting in NIR and IR absorption features associated with nuclear motion. In addition, we will see that a slight variation of electronic polarizability during vibration leads to Raman scattering. Relevance to scientists and engineers xample : Molecules support vibrations at frequencies that depend on the chemical bond strength and atomic masses. Measuring infrared absorption spectra gives us these frequencies, providing information on the type of molecule, allowing optical molecular detection, a field known as molecular optical spectroscopy. Vibrations that don t interact strongly with light may still be detected using Raman spectroscopy, by detecting inelastically scattered laser light at visible frequencies. xample : Glass fiber has a transmission minimum at around.4 m due to absorption by an OH stretch vibration, which this chapter will help understand. xample : The Ti:sapphire laser is able to produce ultrashort pulses because of its broad amplification spectrum. This large gain bandwidth is due to a (solid state) vibronic transition, as explained in this chapter. In addition, many laser dyes and fluorophores have an absorption spectrum that is blueshifted compared to their emission spectrum. This effect is also related to (molecular) vibronic transitions. xample 4: Rotational and vibration transitions, vibronic transitions, rovibrational transitions are used in gas lasers. For example, the widely used CO laser operates at wavelengths around m and relies on vibrational and rotational transitions of CO molecules in the gas phase. xample 5: Trapping of thermal radiation in the atmosphere is caused in part by absorption bands related to vibrational and rovibrational transitions. This Chapter shows at which frequencies we expect these bands to occur. Molecular bonds To understand vibrational transitions in molecules, we first need to understand what holds molecules together. Let s consider a hydrogen molecule, consisting of two hydrogen atoms. ach atom core with a charge of e is surrounded by an electron with charge e, making each atom neutral. Classically we might thus expect that the atoms don t attract or repel each other, and therefore it s not entirely obvious that a molecular bond would form. The reason for chemical bonds lies in quantum mechanics. From quantum mechanics we know that bound electron states are described by the Schrödinger quation, where the electron position and energy are described by a 99

100 wavefunction. lectron wavefunctions in a system of two attractive potentials (here the hydrogen cores) can occupy states where the electron probability between the atoms is zero, as well as states where the probability between the atoms is finite. The former state has a higher energy than the system of the separate hydrogen atoms, and is called an antibonding state, and the latter has a lower energy than the separate hydrogen atoms, and is called a bonding state. The energy of the bonding state depends on the distance between the atoms. The system energy as a function of separation between the atoms (bond length) is often described phenomenologically by the Morse Potential, which describes the potential energy of the molecule as a function of bond length r: U(r) = e ( ) (.4) where b is the binding energy, r is the equilibrium bond length, and a is a constant that describes how rapidly the energy varies with bond length. The graph below shows an example with b=5 mev, r =.5 Å, and a=.5 Å. (mev) 4 5 r (Å) Figure.: xample of the Morse potential vs. interatomic separation r. Note that when the atoms are far apart (large bond length) the potential is almost flat, corresponding to free atoms. As the atoms approach each other the energy is reduced, resulting in binding. Trying to push the positive cores closer together to within subangstrom distances rapidly raises the energy, in part due to Coulomb repulsion. The system has a minimum energy at r=r, and decreasing or increasing the bond length takes energy. The molecular bond thus acts as a spring that tries hold the system at its equilibrium bond length. For small motion amplitude, the resulting restoring force is approximately described by Hooke s law. Using the Morse potential from above we find a spring constant K given by K N m = d U(r) dr = (J) a(m ). (. 5) We thus expect that a diatomic molecule can undergo harmonic oscillation of the bond length at a frequency determined by the atomic masses and the bond strength. Before investigating how light can induce such molecular vibration, we first will introduce the concepts of normal coordinates and normal modes. Normal modes In order to model the movement of ion or atom positions we need to track each individual atom in three dimensions. Modeling the atom positions of a molecule containing N atoms

101 would therefore require N coordinates, or N degrees of freedom. As systems become more complicated (large molecules or even solids), it becomes impractical to describe the motion of each atom separately. Instead we will describe vibration and rotation in terms of normal coordinates that represent movement of several of the atoms at once. We will choose N such normal coordinates such that we can still describe an arbitrary arrangement of the atoms. The simplest possible system for which we can do such a mode decomposition of atomic motion is the diatomic molecule. Figure.: nitrogen molecule configuration described by the two atom positions and the position of two valence electrons An example of a diatomic molecule is the nitrogen molecule, N. A nitrogen molecule requires N = 6 coordinates to describe all possible atomic motions. Instead of describing the atom positions separately, it is more convenient to describe motions it to separate them into normal coordinates that describe the entire system. One such normal coordinate is the location of the center of mass, which requires three coordinates (x, y, z). We can call the corresponding motion translation. A second coordinate is the bond length, which requires one coordinate (r). Variation of this coordinate would correspond to molecular vibration. For any given position and bond length the molecule can be oriented at different angles and, i.e. we need two angular coordinates. The continued motion corresponding to these coordinates is rotation. Note that we can now describe any arbitrary position of the atoms, and we have again used six normal coordinates: three for the center of mass, one for bond length, and two for angular orientation. For molecules that have twodimensional or threedimensional structure, a third angular coordinate (rotation about a chosen axis) is needed to describe the molecule s orientation. We can now predict the number of vibrational normal modes that can exist on particular molecules. For molecules with N atoms, we need N atomic coordinates. Out of these, are needed for translation. In the case of linear molecules, we need angular coordinates for orientation. That leaves N5 normal coordinates for vibration. For D and D molecules we need angular coordinates, leaving N6 vibrational coordinates. This means that most molecules will support many different vibrations, each with their own frequency. For example, a water molecule (H O, threedimensional because the two HO bonds are not collinear) will have three vibrational modes, and an ammonia molecule NH (also a D molecule) will support six vibrational modes. As a consequence, vibrational spectra rapidly become complicated. This also has a benefit: we can identify molecules by measuring their vibration spectra, since these spectra provide information about bond strength and atomic masses present in the molecule.

102 Note that the chosen coordinates are independent xiv : we can have a molecule vibrate while its center of mass is stationary, but it can vibrate in exactly the same way while the molecule is moving (time dependent center of mass). This makes them "normal coordinates", and their corresponding time dependent motion "normal modes". The center of mass motion is rarely of interest in optics, except for the realization that centerofmass motion can lead to Doppler shifts in absorption lines (Chapter ). The more interesting coordinates are the vibrational and rotational normal coordinates of the molecule. Depending on the charge distribution on the molecule, these motions can introduce NIR and IR absorption features related to vibrational, rotational, and rovibrational transitions (combined rotation plus vibration). In addition, since the vibrating atoms in the molecule are surrounded by bound electrons, we will find that combined electronic plus vibrational transitions can also occur, the so called vibronic transitions. Figure.: Sketch of the normal coordinates of a diatomic molecule (blue) showing translation (three coordinates), vibration (one coordinate), and rotation (two coordinates) Atoms: N Total nuclear coordinates: N translational coordinates (center of mass of molecule) rotational N5 vibrational Linear molecules Other molecules (D, D) rotational N6 vibrational Number of normal coordinates needed to describe Natom molecules BornOppenheimer Approximation To describe the response of a molecule in an electric field, we would in principle have to describe the electron positions as well. However, the low mass of electrons means that they can respond quickly to forces (F=ma, so small mass implies a fast response). For many xiv This is a first order approximation. For example, we will see later that fast rotation will stretch the molecule due to centrifugal forces, so the coordinates are not exactly independent.

103 practical situations the electrons respond so quickly compared to the nuclear motions that we can consider the electrons as following the nuclear configuration "instantaneously" as the molecules vibrate or rotate. This assumption is called the BornOppenheimer approximation. Typical spectral ranges for vibrational and rotational transitions Since the atom cores are over 8 times heavier than electrons, movements of atoms when subjected to similar forces are much slower. As we will see later in the chapter, the vibration frequency scales with m /, and consequently vibrational transitions occur at frequencies that are slower than typical electronic transition energies. In general there is no restoring force that holds the molecule at a fixed angle, and therefore rotational energies can be much lower than vibrational energies. The corresponding wavelength ranges are summarized below. Vibrational: Rovibrational: Purely rotational: near infrared (about m) to long wave infrared (about 5 m) similar wavelengths as for vibrational transitions (5 m) Far infrared (about 5 m) Dipole active modes As we saw, most molecules support several vibrational modes. It turns out that light can usually only excite a limited subset of all these modes, the socalled dipole active modes. This can be understood by first considering the effect of a static electric field on molecule. If the molecule contains atoms with different net charge, the field will push positively charged atoms in one direction, and negatively charged atoms in the opposite direction. The field can therefore induce a dipole moment on such molecule. If the field oscillates, it can induce an oscillatory motion and oscillatory dipole moment, i.e. either periodic vibration or periodic rotation. Modes that can be excited by an oscillating electric field are called dipole active modes. From another perspective, a mode is dipole active when its net dipole moment changes during its vibration or rotation. The simplest dipole inactive mode is the stretch vibration of symmetric diatomic molecules. Diatomic molecules have N5= vibrational mode. If the molecule contains only one type of atom, e.g. O, N, etc., it does not have a net dipole moment because of symmetry. ach identical atom has the same net charge of zero, and therefore light does not accelerate either of the atoms. The molecules can vibrate, but their dipole moment (zero) does not change while they do this. xv Consequently, the vibration is dipole inactive. The simplest possible example of a dipole active mode is the stretch vibration of a heterogeneous diatomic molecules. A diatomic molecule containing different atom types will also support a single stretch vibration. Since the different atoms have slightly different affinity for electrons, i.e. a different electronegativity, one of the atoms will end up being slightly positive, and one will be slightly negative. For example, in a HCl molecule, the chlorine atom has a net charge of approximately.e, and the hydrogen atom has a net charge of approximately.e. In this case the molecule has a net dipole moment, and xv As it turns out, their electronic polarizability does change during vibration, resulting in a modulation of scattered light, resulting in an effect known as Raman scattering, discussed later in this Chapter.

104 stretching the molecule increases the dipole moment. The single allowed stretch vibration therefore has a time dependent dipole moment, and therefore this mode is dipole active. Since HCl has a permanent dipole moment, applied radiation may also exert a torque on the molecule, so that the rotational modes are also dipole active. Or vice versa, as the molecule rotates, the (vectorial) dipole moment changes over time, and thus rotation of the HCl molecule is called dipole active. A slightly more complicated example is the carbon dioxide molecule, CO. This molecule contains two collinear C=O bonds, and therefore this is considered a onedimensional molecule. The oxygen atoms are more electronegative than the carbon atom, and therefore the molecule contains two polar bonds. However since the negative oxygen atoms are symmetrically placed around the carbon atom, the net dipole moment is zero. Another way of saying this is that the dipole moments of the two bonds are pointed in opposite direction (outward with equal magnitude), and their sum is zero. During the rotation of a CO molecule the net dipole moment remains zero, and therefore the rotational modes of CO are not dipole active, and light cannot excite a purely rotational motion in CO. We will see later that light can excite a combination of a vibration and a simultaneous change in rotation in CO, however purely rotational transitions cannot be directly optically excited. While the CO molecule does not have a net dipole moment, it does have charge separation (different net charge on different atoms), and consequently some modes are dipole active. Recall, to describe nuclear motion on the linear CO molecule we need two rotational coordinates and N5 = 4 vibrational modes. The image below shows these modes. The top mode is called the symmetric stretch mode, with the oxygen atoms moving in opposite direction while the carbon atom is stationary. During this vibration the two bonds have a time dependent dipole moment, but the sum of these dipole moments remains zero. Or alternatively, at each time during the vibration the average position of the negative charge is identical to the average position of the positive charge. This vibration is therefore dipole inactive. The middle image shows a mode known as the asymmetric stretch vibration. In this mode the oxygen atoms move in the same direction, while the carbon atom moves in the opposite direction. In this mode whenever the left bond stretches (and therefore has an increasing dipole moment) the right bond compresses (and therefore has a reducing dipole moment). The result is that a net dipole moment develops during this vibration, making this mode dipole active. An alternative viewpoint is that the negative atoms move in one direction, and the positive atom in the other. The average negative and positive charge positions move in different directions, meaning that dipole moment is changing. Finally, the bottom image shows a vibration known as a bending mode. Again the oxygen atoms move together in one direction, and the carbon atom in the opposite direction. This too corresponds to a time dependent dipole moment, and the bending mode of CO is therefore also a dipole active vibration. This bending mode can occur inplane and outofplane, so we need two of these normal modes to describe bending in an arbitrary direction. This mode therefore counts for two normal modes, with identical energy. Such modes with the same energy are called degenerate. One important note: changing the length of a bond typically takes more energy than bending a bond. That means that stretch vibrations typically occur at higher energy than bending vibrations. 4

105 Carbon Oxygen Figure.4: the vibrational modes of the CO molecule. The top symmetric stretch is dipole active, while the asymmetric stretch (middle) and bending mode (bottom) are dipole active. Classical description of vibrations in molecules and solids In the following sections we will describe vibrations on molecules and in solids using an entirely classical oscillator model. Vibration Modes in a Diatomic Molecule As discussed above in the section molecular bonds, the energy of a molecule depends on the bond length. Any changes in the bond length result in a restoring force. We can thus attempt to describe vibrating molecules by a system of masses held together by springs. To find allowed vibration frequencies and their associated motion patterns (normal modes) we solve the equation of motion for each atom, and look for oscillatory solutions with a single oscillation frequency. In three dimensional large molecules, a normal mode can be some complicated vectorial motion pattern where atoms oscillate along different directions with different amplitudes. Here we will discuss the simple case of a heterogeneous diatomic molecule, and assume that all motion will be along a single dimension (for example the x axis), meaning we can make use of scalar rather than vectorial position coordinates. We consider a diatomic molecule with masses m and m, positions x and x relative to their equilibrium position, and held together by a chemical bond with a spring constant K. When x > x, the spring has lengthened, resulting in a positive force on mass, and a negative force on mass. The equation of motion for the two atoms thus becomes m x = K(x x ) m x = K(x x ) We are looking for a normal mode with a single frequency, so we will substitute x (t) = x (ω)e and similar for x. For simplicity of notation here we omit the complex conjugate which would be needed to obtain real amplitudes. Substituting this harmonic motion into the OM, taking the time derivative, and dividing out all common exponential terms, we end up with m ω x (ω) = Kx (ω) x (ω) m ω x (ω) = Kx (ω) x (ω) Grouping all x and x terms and moving them to the left results in 5

106 This can be written as a matrix relation [ m ω K]x (ω) Kx (ω) = Kx (ω) [ m ω K]x (ω) = K m ω K K K m ω x (ω) x (ω) = This equation has solutions if the determinant of the matrix is zero, corresponding to (K m ω )(K m ω ) K = Km ω Km ω m m ω = ω ( K(m m ) m m ω ) = This has one trivial solution, =, corresponding to an absence of relative motion, which does not correspond to a vibrational mode. If, we can divide out the term and find a solution if the remaining term between brackets is zero, resulting in ω = K m m m m = K m m This is often written in terms of a quantity known as the reduced mass, defined by μ = m m We have thus found an allowed normal mode, the stretch vibration, with a frequency given by ω = K μ To find the corresponding motion pattern associated with this normal mode, we substitute the obtained frequency in our trial solutions for x (t) and x (t) and substitute these into the OM. This gives: m K μ x (ω) = Kx (ω) x (ω) m K μ x (ω) = Kx (ω) x (ω) Taking the sum of these equations and dividing out common terms gives x (ω) x (ω) = m m We see that in our stretch vibration atom and atom move in opposite directions, and if m <m then atom has a larger motion amplitude than atom. Both observations seem reasonable. Note that we haven t limited the total amplitude: all we know is that the atom will be moving in antiphase, with a relative amplitude given by the mass ratio. 6

107 Together with b and a from the Morse potential for the HCl bond we can now predict the vibration resonance frequency of the HCl molecule. The chemical bond determines the spring constant K, and we can look up the atomic masses to find m and m, giving us, which together with K gives us vib. Some interesting limiting cases: if m >> m we find x >> x, i.e. atom moves very little. In this case the reduced mass approaches m, and the vibration frequency becomes ω K/m corresponding to a single mass on a spring with spring constant K, attached to an immobile object. This is reasonable for large m. Another useful limiting case is the situation where both masses are identical with m =m =m. Note that this would correspond to a molecule with no charge separation, and therefore the mode would be dipole inactive. For such molecule we find that = m/ resulting in ω K/(m/) = K/m. This looks like the vibration frequency of an atom with half the mass on a spring with spring constant K, hence the term reduced mass for. This result can be understood by realizing that the masses undergo correlated motion. For every x of motion of atom, the spring lengthens by x, since the other atom is moving in the opposite direction. The restoring force is thus twice as large as expected based on x, resulting in a spring that appears twice as strong. With the analysis developed above we can make some predictions about the vibration frequencies of the CO molecule. For the symmetric stretch we see that both O atoms oscillate on a fixed central C atom, so we anticipate a symmetric stretch frequency of ω = K m where the subscript L indicates that this is for longitudinal motion. As argued earlier, this mode is dipole inactive. For the dipole active asymmetric stretch vibration we see that there is a total mass m moving to the right and a mass m moving to the left. If we define a corresponding reduced mass for this particular motion pattern = μ m m and noting that the forces on the C atom are twice as large as expected for a single bond, we anticipate an asymmetric stretch vibration frequency of ω = K μ Note that this μ < m and therefore ω > ω. Finally, the dipole active bending mode is expected at lower frequency because of the fact that the transverse spring constant K T associated with bond angle deformation ( bending ) is generally lower than K L. 7

108 Quantum description of light interaction with rotation and vibration Unlike vibrational modes, rotational modes have no resonance in the classical picture. So to look at the absorption resonances due to rotational modes, we have must consider quantum mechanics. The rotational energy of a rotating object is I L rot I, (.) I I where L is the angular momentum, and I is the moment of inertia. The Schrödinger equation yields energy eigenvalues,, given by, J rot J rot J ( J ), I J( J ) rot J,,,... (.) So that In this case, J rot rot I. Often wavenumbers, (cm ) are used for the frequency. c J ( J ) hc, (.) rot where rot h / 8 Ic. Sometimes (e.g. in Silfvast) is labeled as B. (For higher rot rotational energy states, centrifugal force stretches the molecule and this lowers the energy, which may be written as: (using the B notation) J rot J ( J ) hcb J ( J ) hcd, (.4) but we will not consider this correction in this course.) Note that J(J) is a quadratic, so that the spacing between energy levels increases with J: J J ( J )( J ) J ( J ) hc J J J ( J ) hc rot J ) hc rot rot (.5) So the spacing between levels increases linearly with J. This is important because there is a selectionrule for dipole rotational transitions: J,. (.6) The zero is not interesting here, and is often forbidden anyway. Absorption corresponds to the case. This shows us that a set of equally spaced absorption resonances may be seen, each corresponding to a different initial Jstate, as illustrated below. 8

109 J= () 7 6 J= to J= J=4 to J=5 5 4 hcb 4hcB 6hcB 8hcB hcb hcb nergy Figure.5 Typical rotational transition energies For molecular Hydrogen, H, hcb = 5.8 x J, so the lowest energy transition it can make is from rot= to rot = (J) hcb =.6 X Joule or 7.5 mev. The corresponding freespace wavelength is 7 m, which lies in the far infrared (FIR) region. Although H does not exhibit dipoleallowed rotational transitions, the calculated transition energy represents an estimate for the energy of rotational absorption lines of other molecules. In general, these energies will be lower, since H is the lightest possible molecule and hence has the smallest moment of inertia. Thermal population of vibrational states When comparing the calculated energy of the lowest H rotational transition to the thermal energy, kt, at room temperature (~5 mev) it is clear that several rotational levels in H will be populated at room temperature. But the population of levels reduces exponentially with energy, resulting in an exponential dropoff in the strength of the absorption lines with frequency. () hcb 4hcB 6hcB 8hcB hcb hcb nergy Figure.6 9

110 Vibrational transitions The approximation that atoms in a molecule experience a linear restoring force turns out to be a very good one. This means that the potential that the atoms sit in is very near to parabolic, for small displacements. This can be modeled by a quantum harmonic oscillator, for which the solutions are well known: Where vib = (v ½), v =,,, (.7) /, the classical resonance frequency. For a perfectly parabolic K potential, the energy levels are equally spaced. The absorption spectrum is shaped by the selection rule: v =. This means that there is only a single absorption frequency, regardless of the initial state of the system. However, anharmonicity of the potential (i.e. the potential is not exactly parabolic) results in a slight relaxation of this rule, which can result in weaklyallowed excitation of multiple vibration quanta (v=, ), etc. Anharmonicity also results in slightly unevenly spaced energy levels, as shown below: Figure.7 For a polyatomic molecule, we have multiple resonances corresponding to the different vibration modes of the molecule: vib = (v ½) (v ½) (.8) where there are N5 or N6 vibrational modes, and there is one absorption/emission resonance frequency for each mode. Vibrational rotational transitions Whenever there is a vibrational mode in a molecule, there are also rotational modes, so while undergoing a vibrational transition, the rotational state may also change, which causes the vibrational transitions to be broadened out into a vibrationrotation band. The selection rules for these transitions are the same as discussed above. For J =, this is called the For J = For J = Pbranch Qbranch (often forbidden) Rbranch

111 Figure.8 Above are the vibrationrotation transitions for HCl, showing the strongly allowed v = (fundamental band), along with the weakly allowed v = (overtone band). The associated absorption spectrum for HCl is shown below.

112 Coupled electron/vibration transitions in molecules Figure.9 Similarly, electronic transitions can terminate in different vibrational and rotational states, resulting in broad electronic absorption/emission bands. There are no simple selection rules here, but the FranckCondon principle states that the vibrational coordinate should not change during a transition, (see below). Since the electrons have highest probability of being at the extreme positions of their excursions, this controls which transitions are most possible, which strongly affects the shape of the absorption band. Figure.

113 Due to the FranckCondon principle and the tendency for molecules to relax to the bottom of the vibration bands, the emission spectrum is shifted to longer wavelengths than the absorption spectrum, and the emission band usually looks like a mirror image of the absorption band Figure. Raman active modes Modes that are not dipoleactive can also interact with electromagnetic radiation indirectly. For example, vibrational and rotational modes in H are not dipole active, but if an electric field is applied, the electrons move in response to the field, thus polarizing the entire molecule. If the electronic polarizability depends on the molecule length (or orientation), the induced dipole moment with frequency i will be modulated by the vibrational (or rotational) mode: a dipole active mode has effectively been induced through the electronic polarization. This leads to Raman Scattering, and this type of dipole inactive mode is sometimes termed Raman Active. The effect of the configuration dependent electronic polarizability is that a highfrequency electromagnetic wave can interact with the rotational or vibrational modes. This can result in the rotational or vibrational state of the molecule being changed by the incident radiation. This is analyzed classically in Hopf & Stegeman, chapter, but we will not discuss the analysis in this class in detail. Usually, Raman scattering is described quantum mechanically as shown in the following diagram:

114 i s i s Stokes Scattering AntiStokes Scattering Figure. An incident wave with frequency i, is incident on the molecule, polarizing the molecule through a dipoleactive mode (e.g. electronic polarization). There is no absorption, as i is usually far from the resonance of the dipoleactive mode. The dashed line is not a real state, but referred to as a virtual state (dashed line), which is just to say that the molecule is being driven at the incident frequency, i without absorption. When in the virtual state, the molecule can interact with the electric field, and may gain (or lose) a quantum of vibrational or rotational energy from (or to) the M field. Hence, the scattered light with frequency s will be emitted with either a smaller photon energy, s = i, which is referred to as Stokes scattering or with a larger photon energy, s = i, which is referred to as AntiStokes scattering. We will discuss later what a quantum of vibrational or rotational energy is, when we look at quantum mechanical variations from classical models. As a simplified example of how these new frequencies are generated, we will consider a diatomic molecule that has a bond length that oscillates in time according to L(t) = L L cos (ω t) Here vib is the frequency of the stretch vibration, which we assume lies in the nearinfrared. Now let s assume that the polarizability of this molecule to first order depends on the bond length according to α(t) = α (L(t) L ) dα dl This equation shows that the polarizability is approximately, but varies as L(t) deviates from L. The leads to a time dependent polarizability given by α(t) = α L cos(ω t) dα dl α α cos(ω t) with the amplitude of the polarizability variations. Let s look at the dipole moment µ(t) that develops under monochromatic illumination of this molecule: (t) = cos(ω t) 4

115 with exc the angular frequency of the incident laser irradiation. This gives rise to a time dependent dipole moment given by: μ(t) = α(t) (t) = (α α cos(ω t)) cos(ω t) = α cos(ω t) α cos(ω t) cos(ω t) The first term is simply the induced dipole moment at the laser frequency, which gives rise to normal dipole radiation (Rayleigh scattering). More importantly, the last term contains both the vibration and the excitation frequency, and can be written as: μ(t) = α cos(ω t) cos(ω t) = 4 αe e e e = 4 αe ( ) e ( ) e ( ) e ( ) = α[cos((ω ω )t) cos((ω ω )t)] We thus have found: μ(t) = α cos(ω t) α[cos((ω ω )t) cos((ω ω )t)] This represents a dipole moment that oscillates predominantly at the frequency of the incident radiation, but with two small contributions that oscillate at slightly lower and slightly higher frequency than the incident laser light. These frequencies exc vib and exc vib correspond respectively to the Stokes shifted and the antistokes shifted lines. 5

116 6

117 Chapter Interaction of light with vibrations in solids Corresponding handouts are available on In Chapter we studied how molecules supports vibrational normal modes, with frequencies that depend on the chemical bond strength and on the masses of the atoms in the molecule. In solids similar effects occur, with the major different that the atoms are bound in a threedimensional crystal lattice. In this chapter we will see that nonmetallic compound solids support dipole active vibrations that extend throughout the entire crystal, resulting in resonances and strong Lorentzlike contributions to the dielectric function at IR frequencies. Vibrational Modes in a Crystalline Monatomic D Lattice In a crystal atoms are bound in an extended lattice, and the resulting vibrational normal modes extend throughout the entire crystal. These extended vibrational modes are called phonons. Crystals are much easier to deal with analytically than noncrystalline solids, but much of what we do here for crystals applies to noncrystalline solids as well. To understand the basic mechanical response of crystals we start with the simplest case of a dimensional monatomic lattice. The assumption of a monoatomic lattice (all atoms have identical mass) implies that we are discussing an elemental solid such as pure Si or even pure Al. As such we do not expect polar bonds, and therefore we do not expect dipole active modes (see Chapter ) and no strong phononphoton coupling. Nevertheless, the monoatomic lattice model provides valuable insight into vibrations in solids. Figure. In our simple model, a unit cell (length a) contains only one atom with mass m, and there is only one spring constant K that represents the interatomic binding potential. For simplicity we consider only nearestneighbor interactions, meaning that the force on a given atom is affected only by the atoms immediately adjacent to it. The atoms are numbered with a variable with X representing the time dependent position of atom relative to its equilibrium position. The restoring forces depend on displacement differences. For example if atom has moved more to the right than atom, atom will experience a positive force K(x x ). The total equation of motion thus becomes: K ( X X K ( X X ) K ( X X X ) mx ) mx (.9) 7

118 showing that the acceleration of atom depends only on the distance to its neighbors, not on its absolute position. xvi We will again look for solutions where all atoms oscillate at the same frequency, i.e. all X terms have the same time dependence, e it. In this case the equation of motion becomes m X K X X X ) (.) ( We will see that this equation has travelingwave solutions with the displacement of atom relative to its center position given by X i( k z t) i( k ai t) e xe x (.) where represents the number of the mode, z is the average position of atom along the chain given by z =a and k is the wavevector of the th mode. The center positions of the left and right neighbors are z = (a)a and z = (a)a respectively, resulting in a displacement X ik a ik a e e x. (.) where the time dependence has been omitted. We substitute this displacement in equation., resulting in: i e k a i ( ) k a i ( ) k a m x Kx e e e K m i k a ik K a ik a e e cos( k a) m (.) (.4) Note that the vibration frequency indeed only depends on the relative phase of the neighbors. We now have a series of allowed modes with wavevector k and angular frequency according to K m cos( k K m a) ka sin K m ka sin (.5) This equation represents the phonon dispersion relation in a one dimensional monoatomic lattice. The dispersion relation is shown below. We see that low values of k correspond to low oscillation frequencies. This is as mentioned before due to the small relative displacement of adjacent atoms for modes with low k values, resulting in a small restoring force. xvi This last observation will become important later, since it implies that oscillations in which neighboring atoms move approximately inphase will experience little acceleration, resulting in a set of low frequency phonon modes. 8

119 K m /a /a Figure.4 k Note that the maximum oscillation frequency that a collective mode can have is (K/m). This frequency occurs at k = /a, in other words when adjacent unit cells (in this case containing atom) are 8 degrees out of phase. A wavenumber /a corresponds to a vibration wavelength of a. For wavenumbers of magnitude >/a, the same physical configuration can also be described by a wavenumber that is less than /a, so there is no new physical situation outside the range /a < k < /a. This range is known as the first Brillouin Zone, or often simply the Brillouin zone. Boundary conditions and degrees of freedom In a D lattice of length L, each atom has degree of freedom, so we expect that there are a total of N of these collective modes, i.e. {.. N}, where N is the number of atoms in the crystal, given by N=(L/a). In a finite crystal, the eigenmodes are described by standing waves with a wavelength that satisfies = L /. with {,N} The case = corresponds to translation, while the longest finite wavelength is described by k=/l. The shortest possible physically meaningful wavelength is a. The allowed wavevectors are thus k,,,,..., ( N ) L L L a L /( N ) L (.6) representing N allowed modes with evenly spaced k values, and N different wavelengths: L L, L,,,..., a (.7) The discrete nature of k is generally not important for the optical properties of large crystals. In the limit of an infinite crystal, the left and rightpropagating modes need to be considered as independent modes, suggesting that a subset of N atoms in this infinite crystal would support N modes, which cannot be correct. In fact when calculating the number of allowed modes on this subset of atoms with length L, the set of atoms needs to support the entire wavelength which limits the allowed positive wavevectors to k 4 6 ( N ),,,..., L L L a L /( N ) L (.8) with an equal number of negative wavevectors and one solution for k=, bringing the total number of modes for the set of N atoms to 9

120 N total ( N ) N (.9) as expected. Note that the discussion above involved a monatomic lattice, for which the vibrational modes cannot interact with radiation. To look at the interaction of light with lattice modes, we must next examine the lattice modes of a diatomic lattice. Vibrational Modes in a Crystalline Diatomic D Lattice a m K M K m K M m v s u s v s u s v s Figure.5 In this section we will discuss a linear chain consisting of atoms with masses M and m, characterized by displacement amplitudes u s and v s respectively, where s is an index indicating the unit cell. Note that in this case a corresponds to the length of the unit cell, and that a unit cell contains two atoms, one of each kind. A wavevector of /a thus implies a phase difference of 8 degrees between the behavior inside adjacent unit cells, not between adjacent atoms. We will find that the internal structure of the unit cell adds an internal degree of freedom, giving rise to two phonon modes for each value of the wavevector. Assuming nextneighbor interactions, the equations of motion are Mu s K(vs vs us ). (.) m v K( u u v ) s s s s Again, we look for traveling wave solutions, where u s, v s have the same frequency and wavenumber, but different amplitudes. We anticipate solutions of the form v. (.) iska it iska it s v e e, us ue e Substituting these trial functions into the equations of motion, we obtain M u m v Kv ( e Ku ( e ika ika ) Ku ) Kv (.) For a solution to exist the determinant needs to be equal to zero: (K M ) K ( e ika ) K ( e ika ) (K m ) (.) This leads to an equation of the order 4, resulting in two independent solutions for :

121 4 mm K( m M ) K K mm ( m M ) ( coska) ( m M ) mm( coska) (.4) This represents the dispersion relation for a diatomic linear lattice, schematically shown below. optical mode K K m acoustic mode K M /a /a Figure.6 k Two separate phonon branches are observed, a low frequency or acoustic branch, and a high frequency optical branch. To understand the nature of these phonon branches we will investigate the high and low wavevector limits: k (or ka << ) and k /a. For the latter case, k /a, meaning that cos(ka), so that the solution simplifies to K mm K mm ( m M ) ( m M ) ( m M ) ( m M ) 4mM (.5) Hence, at the zone boundary we find two allowed solutions with frequencies corresponding K K to and. m M For ka << we can approximate cos(x)½x giving cos(ka) k a /, leading to K mm ( m M ) m M K mm ( m M ) mm ( m M ) m M mm K mm ( m M ) ( ka) mm ( ka) ( ka) ) (.6) hence we have two solutions for this range of ka. The solution with the sign corresponds to high frequency modes:

122 m M K K K (.7) mm m M where is the reduced mass of the unit cell, The case leads to dependence of on k at low frequencies. m. M ( ka) K K, ka showing a linear m M ( m M ) Three example curves are shown below for identical transverse spring constant K T and a fixed longitudinal spring constant K L=.4K T. Note that the total mass affects the acoustic branch slope, the reduced mass affects the maximum resonance frequency, and the mass ratio determines the frequency splitting at the zone boundary (k=/d with d the unit cell length). Figure.7 xample dispersion curves for phonons assuming a fixed transverse spring constant K T and a fixed longitudinal spring constant K L =.4 K T. Interaction of radiation with lattice modes In solids with more than one type of atom, the different atoms will generally have slightly different charges. This means that the different atoms will experience forces in different directions due to an applied field, and consequently, a macroscopic polarization due to the lattice can be induced by an electric field. The natural frequency of oscillation of such a polarization corresponds to the optical modes calculated previously for the diatomic lattice. Now the wavelength of light is always much greater than the lattice spacing, especially for the infrared region where the frequency of the M wave is resonant with the optical modes, i.e. >> a.

123 ck () K K m K M /a Figure.8 Dispersion curves for phonons (lattice modes) and photons, assuming for now that there is no change in index due to the lattice Hence the optical wave interacts with very long wavelength lattice modes (compared to a). This means that k for the optical lattice modes, which is convenient as it simplifies the mathematics. Since k, we can assume that the applied field is spatially uniform, which is true over many thousands of atomic spacings. Hence, t) it ( e. (.8) The coupled equations for displacements u and v are then Mu mv s s K ( v K ( u s s v u s s u ) q e s s v ) q e it it k, (.9) Since k the phase term e ik, leading to u s(t)=u e it and v s(t)=v e it independent of s. Substituting u and v gives K M Ku u Kv q K m v q For this, the solutions for amplitudes u and v are of the form q / M q / m (.) u, v (.) T T where T is the resonant frequency of the optical modes at k =, K T. This is sometimes referred to as the "transverse optical phonon" frequency, hence the subscript T. As before, is the reduced mass.

124 Optical properties of polar solids The lattice contribution to the polarization is P Ne( u v), (.) where e is the effective charge on the atoms in the lattice (not usually equal to e.) The total polarization is P P P total dielectric function is bound, where again, P bound is the electronic polarization. So the P Pbound Ne r( ) u v. (.) e But, u v, so that T r ( ) P electronic P T bound Ne T Ne r ( ) Ne T (.4) Where r() is the high frequency dielectric constant. Note that this is not truly the value at, which is always, but is usually taken to mean the value at frequencies well above T, yet well below any electronic resonances. The overall behavior of r() is sketched below. Note we have not included damping, as it is not too important here. We will look at the effect of damping a little later. We can use this relationship to find a value for the low frequency dielectric constant, r(), using Ne () ( ) (.5) r r / T 4

125 Figure.9 The LyddaneSachsTeller relationship From our expression for r(), we can write ( ) ( ) Ne / r T r (.6) so that, ( ) ( ) r T T T T r () r ( ) T T () T ( ) T r ( ) r r T r () r ( ) r T (.7) ( ) r ( ) This shows us that r() = for r T. This frequency is labeled L, the "longitudinal optical phonon" frequency, at which r =, rather like the plasma frequency for electronic polarizability. Hence, r ( ) T r( ) L r() T or () r L, (.8) Which is known as the LyddaneSachsTeller relationship, which works quite well for polar diatomic materials. Two examples: GaAs: L/ T =.7 ) / ().8 r ( r KBr: L/ T =.9 r ( ) / r ().8 [Ref: C. Kittel, Introduction to Solid State Physics, (Wiley)] Note that L > T, always. Hence () > (), as might be expected. Similar to plasma oscillations, L is the frequency for longitudinal long wavelength lattice oscillations. 5

126 Refractive index of polar and ionic solids Figure. The behavior of the refractive index for a transverse optical phonon resonance is similar to that of a Lorentz oscillator. In our model the only difference is that we have ignored damping. In real materials phonon modes also undergo damping or scattering (change of wavevector direction and magnitude), resulting in a finite. In Fig.. we show the real and imaginary parts of the refractive index for AlSb and the corresponding reflection spectrum. Figure.: Complex dielectric function and refractive index of AlSb based on fit of experimental data by Turner and Reese 6

127 Reflectance of polar solids Reststrahlen band In the absence of damping, the reflectance must be unity in the region T < < L, again just like for electronic transitions (see figures below). This region is called the "Reststrahlen" band, where light cannot propagate in the material, so must be absorbed or reflected, see Fig... For sufficiently large negative r we have a large imaginary index, causing R to be very close to. Figure.: Reflection spectrum of AlSb and calculated transmission spectra for four thickness choices, ignoring multiple internal reflections (unphysical for thin films, done here for illustrative purposes) ffect of phonons on transmission spectra In the preceding sections we learned that there are dipole active phonons in polar and ionic crystals. In many cases the large number of participating atoms leads to a large modification of the infrared response, resulting in a Reststrahlen band (large reflection) between T and L. Just above L there is a frequency where n, resulting in a reflection minimum in air. Based on the reflection spectrum shown in Fig.. we expect that compound dielectrics are opaque for wavelengths in the Reststrahlen band. Intuitively one might expect a transmission maximum corresponding to this reflection minimum, but in practice for thick materials (e.g. a mm thick slide) this is not observed. The reason for this is that the absorption coefficient at this wavelength is still sufficiently large to produce almost zero transmission, see Fig... Note that for a mm thick slab of AlSb the no transmission peak is observed when n=. Also note that the onset of zero transmission does not coincide with L (9.5 m) but instead occurs at higher energy ( 7 m) due to absorption. 7

128 Coupled PhotonPhonon modes: Phononpolaritons Recall that we started out by drawing the dispersion of light ( versus k) as a straight line.. However, clearly close to the vibrational resonance frequencies (~ 4 = ck/ r rad/sec), r is clearly strongly varying and versus k will not be a straight line. For ~, the refractive index is given by (), while at high frequencies it is given by (). In these limits, the wave is described as "photonlike". However, as approaches T, the index increases, and the group and phase velocities of the wave slow down to be close to that of the lattice modes. Here, the wave propagates as a strongly coupled polarization electric field wave, with velocity characteristic of the lattice mode. Here the wave is described as "phononlike". ck () ck () L K T to /a k Figure. Solids with more than one atom per unit cell This can get complicated, but generally, we see several phonon resonances, as illustrated below for GeO. In this case, Ne j / r ( ) r ( ) f (.9) j j Tj 8

129 Figure.4 9

130 xamples of n, spectra These are taken from Optical Properties of Solids, by Palik. The next few pages will show some representative materials types and we will make some comments on these. Lithium Fluoride (LiF) Figure.5

131 Gallium Arsenide (GaAs) Figure.6

132 Diamond Figure.7

133 Lithium Niobate (LiNbO ) Figure.8

134 4

135 Chapter 4 Optical properties of semiconductors Corresponding handouts are available on In chapter we described the optical properties of insulators. We discussed that the valence electrons in insulators are relatively strongly bound to the individual atoms in the material, resulting in high resistivity and low absorption throughout the visible spectrum. In chapter 6 we discussed metals, in which the valence electrons can be considered free, leading to a high conductivity, negative at low frequencies and high absorption and reflection. In this chapter we will discuss an intermediate case: semiconductors. In semiconductors, the valence electrons are generally somewhat bound to the atoms in the material, with binding energies of up to several ev. As a result, at room temperature these materials contain a small concentration of electrons that have broken free from the atoms, resulting in a small conductivity. This is the origin of the word semiconductor. The optical response of semiconductors is largely due to electronic transitions. In order to understand the possible electronic transitions, it is important to understand the electron dispersion relation or electronic band structure in semiconductors. lectronic Band Structure As was already mentioned in Chapter 5, the Lorentz model is not an accurate description of solids. While isolated atoms have well defined electronic states representative of the atom, as individual atoms are brought together the atoms start to interact, leading to shifts in the electronic energy levels. At the densities common in solids (~ atoms/cm ) significant interaction occurs, causing levels to broaden out into bands, as schematically indicated in Fig. 4.. The energy bands still retain some of the characteristics of their atomic origins. For example, in GaAs, the valence bands are mainly plike in nature, while the conduction band is slike. Atom Solid nergy Figure 4. In covalent bonded solids, (e.g. GaAs, Si, Ge) outer electrons are quite evenly shared between different atoms, so the electron orbitals are extended throughout the crystal, and the energy bands corresponding to these orbitals are very broad. However in ionic bonded solids, (e.g. NaCl, MgF, KCl) electrons are much more localized and the bands are narrower. The absorption spectra of these materials retain more of the characteristics of atoms. (See figures.6 and.7 in Wooten.) In order to understand the resulting energy band structure, we will use a quantum mechanical description. In quantum mechanics, the behavior of an electron in an electric potential V(x) is described by the Schrödinger equation 5

136 m x V t x x, t i x, t (4.) Here (x,t) is called the wavefunction of the electron. The wavefunction has some similarities to the electric field in the scalar wave equation. For example, the probability of detecting photons is proportional to * or (where * indicates taking the complex conjugate) while the probability of detecting an electron is proportional to * or. As such * represents a probability density function. The operator between the square brackets is called the Hamiltonian, also written as H. Similar to the decomposition of vibrations into normal modes, the allowed electronic states can also be described in terms of normal modes or igenfunctions of the Schrödinger equation. These eigenmodes satisfy the time independent Schrödinger equation: m x V x x, t x, t (4.) where the time dependence of the eigenmodes is harmonic, as given by i x, t x, t (4.) t The Schrödinger equation in free space (V(x)=constant) leads to plane wave solutions of ( x) Ae i( k xt ) e the form. k. Here A is a normalization factor to ensure that the total probability adds up to, and (see q. 4.). Note that the time dependence is only related to the total energy of the wavefunction (kinetic plus potential). Substituting the plane wave into the Schrödinger equation we find that the total energy of a free electron is given by ke V, (4.4) m where in free space V is some constant potential energy (which we can arbitrarily set to zero) and the other term constitutes the kinetic energy, with k e being the electron wavevector and m the electron mass. When an electron travels into a region with a lower potential energy (e.g. closer to a positively charged atom), the kinetic energy goes up, and the wavefunction acquires a shorter wavelength. In a sense, the behavior of electron waves entering a low potential region resembles that of light entering a high refractive index region. Just as for light, changes in wavelength give rise to reflections, in this case electron wave reflection. Inside a crystal, electrons experience a periodic potential due to the regularly spaced atomic cores in the crystal lattice, leading to multiple electron wave reflections, and electron wave interference. The multiple reflections result in eigenmodes that are affected by the exact shape of the periodic potential V(x). For an infinite crystal, the igenfunctions describe a state with a welldefined energy and a corresponding spatial distribution of the electron throughout the entire crystal. In a simple linear lattice with lattice spacing a, we have V(r) = V(ra), as shown below. 6

137 Figure 4. Since the electrons are bound in a periodic potential, the igenfunctions will also have a periodic character. The periodicity of V(x) implies that a wavefunction which satisfies m x should also satisfy V x x x (4.5) m x V x x a x a. (4.6) Wavefunctions that satisfy condition can be described by functions of the form xvii ikex ( x ) u ( x) e. Here u k(x)=u k(xa) is a periodic wavefunction called a Bloch k k function, which can be interpreted as (approximately) describing the electron distribution within a single unit cell. The phase term e ikex represents a phase difference of the wavefunction in adjacent unit cells. xviii For core electrons (those tightly bound to the nucleus) u k(x) represents a strongly localized wavefunction similar to the electron orbitals around a hydrogen atom. The less strongly bound valence electrons are described by more extended wavefunctions that have significant amplitude in between neighboring atoms. Free electrons are described by wavefunctions with a high energy, such that the wavevector remains real in between atoms (where V(x) is high). C onduction band B ound sta tes Figure 4. To understand the main features in the electronic band structure, it is instructive to start from an electron traveling in a periodic atom lattice with spacing a and zero binding xvii See for example Solid State Physics, N.W. Ashcroft and N. D. Mermin, Chapter 8 xviii Note that in this case the phase function is continuous in space, in contrast with the description of phonons, where the phase term represented the phase of a vibration at discrete points in space 7

138 potential. The dispersion relation will thus be exactly that of a free electron, as shown below. /a /a /a /a /a /a Free electron k Figure 4.4 For fast electrons, we find that k will exceed /a. Such rapidly varying wavefunctions can be split up into a part that describes the overall phase difference between adjacent atoms (<<) and a part that describes the behavior of the wavefunction within a unit cell, for example Where k.5 i x a.5 i x i x a a k.5 i x a ( x) e e e u ( x) e (4.7) u( x) ix / a e can be seen to satisfy u(x)=u(xa), and the phase difference between neighboring unit cells =.5 is indeed between and. This illustrates how highenergy solutions can be described by an overall low kvector and a Bloch function. This enables us to represent all electron states in this D lattice inside the first Brillouin zone, also shown below. The use of only the first Brillouin zone for the display of the electron dispersion relation is called the reduced zone scheme : /a /a /a /a /a /a Figure 4.5 The dispersion relation for an electron in a periodic crystal represented in the reduced zone scheme for zero binding potential (left), and finite binding potential (right). As in the case of phonon dispersion, the zone boundary represents a situation where there is neighboring unit cells are 8 out of phase. This again leads to a splitting of the energies 8

139 at the zone boundary. The splitting gives rise to characteristic energy bands that are separated by energy gaps in which no electron states exist, as illustrated in the right side of the figure. lectrons are Fermions, meaning that no two electrons can occupy the same quantum state. Since there is only a finite number of electrons available in a crystal, the available lowenergy states are occupied by electrons up to a certain maximum energy. An important feature of semiconductors and insulators is that the highest filled energy state lies just beneath an energy gap (called the band gap ), while the states just above the gap are unoccupied. The filled band below the band gap is the valence band, and the empty band above the band gap is the conduction band. Charge carriers free electrons and holes The existence of a completely filled band has important consequences for the conductivity: an electrical current requires the existence of a net electron momentum, for example more electrons traveling to the right (positive k) than to the left (negative k). In insulators and in semiconductors (at zero Kelvin) all available positive and negative k vector states in the valence band are filled, resulting in zero conductivity. In semiconductors, the energy gap is sufficiently small that at room temperature some valence electrons are thermally excited into the conduction band ( broken free from an atom ), generating a free electron and leaving behind a positively charged atom. The resulting unoccupied electron state in the valence band is called a hole. Both these states are free to move through the crystal, giving rise to a finite conductivity at room temperature. Hence the term semiconductor. Free electrons and holes also affect the optical properties of semiconductors via free carrier absorption, as described later in this chapter. ffective mass In the description of the optical properties of semiconductors it is important to know the effective electron mass. For free electrons the electron mass follows from the relation between and k: ke V, (4.8) m e As is clear from the band structure shown above, the dispersion relation in a periodic lattice is no longer a simple parabola for all k vectors. However for energy levels close to a band gap (e.g. near the zone center ), the dependence of vs. k is approximately parabolic. This allows us to ascribe an effective mass, m*, that fits the band curvature according to k ( k ) g. (4.9) m * The effective mass will be dependent on the exact band structure, and as a result will vary from semiconductor to semiconductor. For most semiconductors, m e* lies in the range.. m e. The figure below shows a sketch of a band structure near the zone center, and around the band gap. Note that there are two valence bands that have the same energy at k=, but that have a different effective mass. This is a situation that occurs in many semiconductors. As a result, holes can have different effective masses, respectively the heavy hole mass m* HH and the light hole mass m* LH. 9

140 Directgap semiconductor: CB nergy FCA C k * m e Interband absorption HH FHA LH xcitons / Donor states Midgap states Acceptor states k k * m HH * m LH k SO Figure 4.6 Fundamental Absorption Processes Absorption processes in directgap and indirectgap semiconductors The figure above includes the main absorption processes that can occur in a directgap semiconductor. A direct bandgap semiconductor is a material in which the minimum of the conduction band (CB) occurs at the same kvalue as the maximum energy in the valence band (VB). A direct transition is a transition that does not require a change in the wavevector. Common direct absorption processes are interband absorption (VBCB or VBVB) VBtoexciton state absorption, VB to (unoccupied) donor, acceptor, or mid gap state absorption (occupied) donor, acceptor, or mid gap state to CB absorption Freecarrier absorption is an indirect (phononassisted) process, which means that the initial and final states have different kvalues. All these processes will be summarized below. Some semiconductors have an indirect gap, which means that the longest wavelength (lowest energy) absorptions must occur with the assistance of a phonon, making the absorption edge less sharp than in directgap semiconductors. 4

141 Indirectgap semiconductor: CB nergy Direct gap transition Phononassisted interband transition k Figure 4.7 A sketch of a plot of absorption coefficient versus photon energy and wavelength for a "typical" semiconductor is shown below. The various absorption processes in each wavelength region are labeled on this graph. Figure 4.8 In the following paragraphs, we describe these processes in some more detail. Interband absorption The dominant feature of the absorption spectrum of a semiconductor is the interband absorption. We will deal with directgap semiconductors here, and we will look briefly at indirect gap semiconductors at the end. 4

142 Density of States and Fermi Golden Rule Optical absorption in semiconductors is due to lightinduced transitions from occupied electron states to available unoccupied states. For that reason the number of available states is an important quantity. For continuous energy bands this is described in terms of the Density of States, g(). For isolated atoms the density of states (DOS) corresponds to the number of available states per atom per unit energy. This means g()d is the number of states (per atom) in the energy range (, d). This quantity is used in the quantum mechanical calculation of the transition rate W from initial state k to final state n of the atom upon illumination with photons of energy : W kn kn g ( k ) (4.) This relation is called the Fermi Golden Rule. Here is the applied electric field strength, and kn is the transition matrix element (units C m in this form) for states k and n. This tells us how strong the transition is. For some states, kn= (forbidden transition). We can relate the transition rate for each atom W kn to the absorption coefficient, since the absorption rate W of photons with = for N atoms per unit volume corresponds to an absorbed power per volume: NW kn rate of energy absorbed per unit vol. di dz (N atomic density) Pabs Pabs unit volume unit area unit length Iabs di unit length dz NW kn (4.) We can express W in terms of the irradiance I, since both W and I depend on. We will assume that k= and we will write the photon energy as = so that g( k)=g(). I nc di dz N kn 4N ( ) kn nc g( ) I nc g( ) (4.) This shows that the absorption coefficient depends on the dipole transition matrix element, how many atoms are present per unit volume, and on the density of states g(), given by the number of states per atom in the range (, d). For a semiconductor, which has continuous energy bands that describe collective electronic states of the crystal, we need to find an analogous number of states per atom. Instead of states per atom we will use the function ()d to describe the number of states per unit 4

143 volume in the range (, d), so that effectively g()n (). ( states per atom times atoms per volume equals states per volume ). Therefore we expect to arrive at an expression that looks of the form 4 ( ) nk ( ). (4.) nc We will see in the following that in this expression the function actually needs to be replaced by a quantity J() known as the joint density of states. To calculate the interband absorption, we need to calculate the density of states () based on our known (k) as given by the effective electron mass m*. First, we need to look at how many k values are available in a given band. Since the allowed free electron states described Bloch waves, then i ik r e exp k x k y k z. (4.4) k x y z To calculate the number of allowed states per unit volume, we demand that a given volume contains only complete waves. This is equivalent to applying periodic boundary conditions, i.e. demanding x, y, z) ( x L, y, z) ( x, y L, z) ( x, y, z L ) (4.5) k( k x k y k z where L x, L y and L z are the dimensions of the volume of interest. This requires that exp ik ( x L ) exp ik x x allowed values of k: x x, and similarly for L y and L z, which leads to the following k x i nx; ky ny; kz nz, Lx Ly Lz N ni,,... (4.6) Where N i is the number of unit cells along L i, ensuring that the maximum k vector is /a. The allowed kvalues are thus evenly spaced in kspace (or reciprocal space). ach of these kvectors represents an allowed electron state. Hence we can say there is one allowed state per unit volume of kspace, where the unit volume in kspace is: 8 k kx k y k z, (4.7) L L L V with V the considered volume of the semiconductor crystal. x y z To calculate the density of states around some energy max above the bottom of the conduction band, we first calculate the number of allowed kvalues that correspond to an energy (k) below max, and then we take the derivative of this number with respect to to find the number of states within an energy range d. For a parabolic band, this implies k ( k) m* max k max m e max (4.8) 4

144 Now the number of k vectors that have satisfy k < k max is given by the volume of a sphere in reciprocal space with radius k max, divided by the unit volume k: k x k k k y k z Figure 4.9 Hence the number of kvalues/unit volume in (k, kdk) = (k)dk is given by the surface area of the sphere * dk /V, divided by k, i.e. ( k ) dk 4k dk / V 4k dk / V k dk (4.9) k 8 / V The number of electron states in that same interval is actually (k)dk since each kvalue can contain two electrons with opposite spin. If can now rewrite this density of states in into a quantity that represents the number of states in an energy interval d, we will have found (). This can be done by multiplying (k) by the derivative of k to : dk ( k ) dk ( k) d ( ) d d (4.) Now, for a parabolic band, so that giving k ( k ), (4.) m * m* k, (4.) 44

145 45 * ) ( * * * ) ( ) ( m d d m m m d d k d dk k. (4.) If we define the top of the valence band as =, then the lowest energy in the conduction band becomes = g. The density of states for electrons in the conduction band then becomes * ) ( g e e m. (4.4) The figure below shows a sketch of the resulting electron density of states function. Figure 4. For the light and heavy holes, the densities of states defined at an energy with respect to = at the top of the valence band are * ) ( m LH LH, (4.5) * ) ( m HH HH (4.6) Absorption due to interband transitions For optical absorption to occur, we must have energy conservation: f= i as well as momentum conservation: k f=k ik, where k is the optical wave vector and k i, k f are the initial and final electron wavevectors. Since the photon momentum is very small, it is a very good approximation to demand that k f=k i, so that transitions are essentially vertical or direct. For HHCB transitions, the initial and final energies are g e c f HH V i m k m k *,. (4.7) g

146 i f k Figure 4. Hence the requirement for a direct transition with energy direct is direct k k k C ( k ) V ( k ) g * g m e m, (4.8) HH HH where we have introduced the heavyhole reduced mass HH as: HH m * e m HH (4.9) For LHCB transitions, we simply use the lighthole mass in place of the heavyhole mass to find the appropriate reduced mass, LH. Be careful not to confuse this effective mass with the transition matrix element kn. The expression we found for the energy difference in a direct transition links the energy to the wavevector. Optical absorption doesn t just require a high density of initial states; there also must be allowed final states at higher energy. If we are to count the number of states that can contribute to an absorption event with energy difference ħω, when need to count the number of allowed k values in the valence band that are separated from the conduction band by an energy ħω. Since we have an expression for direct(k) = c(k) v(k), we can find the joint density of states J using direct(k) using the approach that we used previously for finding e based on c(k). In other words, we can calculate the density of momentum conserving transitions per unit volume, known as the joint density of states: dk J ( ) ( k) (4.) d direct Note that (k) is the same quantity as in the previous analysis: it still represents the number of states available in an interval dk. Hence, the joint density of states is J ( ). (4.) g 46

147 This result, which followed from the requirement of energy conservation, momentum conservation, and the conduction and valence band structure near the band edges (described by electron and hole effective masses), allows us to calculate the absorption coefficient according to the Fermi golden rule (q. (4.)): 4 ( ) nk J ( ) nk g nc nc. (4.) This expression is only valid for g. The details of the transition matrix elements are tricky to calculate and as a result we cannot easily use the above formulation for a quantitative prediction of the absorption spectrum. However, sufficiently close to the band edge, and nk can be considered approximately constant, and in practice the product nk remains approximately constant over a significant frequency range, so that we can write: ( ) g. (4.) The graph below shows the example of interband absorption in InAs, where the above analysis works well (from Optical Properties of Solids, M. Fox). Here, () (.5 ev), as predicted above, where g =.5 ev. For comparison the effect of a fixed transition matrix element is plotted as a dashed line. (.5)* Figure 4. In most cases, the effect of excitons (a bound electronhole state due to Coulomb interactions, see next section) causes the density of states to be enhanced for g, so that the absorption coefficient has a rather different shape than the ( g) / dependence. (See e.g. the figure for GaAs in the next section.) 47

148 xciton absorption In a semiconductor, an exciton refers to a bound state of an electron (e) and a hole (h). The electron is excited out of the valence band, leaving behind a hole. The electron can become bound to the hole due to Coulomb interaction, resulting in hydrogenlike bound states (n=,, ). The eh pair is free to move about the material as a single, uncharged particle. In this bound state the electron still has characteristics similar to that of free electrons, but its energy is a little lower than that of conduction band electrons as a result of the Coulomb interaction, and similarly the bound hole state lies a little above the valence band. The exciton binding energy, b, is the energy required to separate the pair, producing a "free" electron in the conduction band and a free hole in the valence band. The existence of exciton states allows the absorption of light by valence electrons at energies slightly lower than the bandgap energy. In these VBexciton absorption a bound eh pair is created in any of its allowed states (n=,, ), giving rise to multiple absorption lines just below the bandtoband, or interband absorption energy. e n= n= n= Coulomb force h b Figure 4. Schematic representation of an exciton in real space (left) and in k space (right). The sketch on the left assumes an equal electron and hole mass, while in general the hole has a larger mass. k Figure 4.4 Sketch of VBexciton absorption (left) at low temperature, resulting in absorption lines just below the interband absorption, and (right) at elevated temperatures, resulting in reduced exciton lifetime, and therefore an illdefined exciton energy and broadened absorption lines 48

149 For most semiconductors b is 5 mev, which is on the order of kt at room temperature. As a result, that exciton lines are generally not sharp and not clearly separated from bandedge absorption unless the sample is cooled, as illustrated in the above sketch, and in the figure below for GaAs at 4 K and K. Figure 4.5 Impurity absorption Figure 4.6 Impurities added to a semiconductor can usually be categorized as "donors" or "acceptors". Donors are atoms that have one extra valence electron compared to the atoms in the surrounding lattice. For example, in silicon or germanium, which are group IV elements, As or Sb atoms which are group V elements have an extra valence electron. The extra electron does not participate in bonding orbitals and hence is only weakly bound to the impurity by the Coulomb force. At low temperature, the electron orbits around the impurity in a way that can be modeled by the Bohr model of the atom. The binding energy, d, is typically of the order of kt at room temperature so at room temperature the electron is ionized into the conduction band and is free to move around as a conduction electron. Acceptors have one valence electron less than the host material, e.g. a group III element in Si or Ge. In this case, the acceptor provides an empty state or a "hole" where a valence electron could sit. If this hole moves away from the acceptor (i.e. if a neighboring valence electron occupies this available state), the acceptor becomes negatively charged and the 49

150 positive hole may end up being bound by Coulomb interactions, just like the donor electrons. Again, though, the binding energy is small and at room temperature, the holes may move freely around as positive chargecarriers. Because these are so weakly bound, the VBacceptor and donorcb transitions are not seen at room temperature. They can, however, be observed at low temperatures, such as the example of VBtoacceptor absorption in Borondoped Si at low temperature, shown below. These absorption resonances occur in the far infrared (here at least ~ mev). Figure 4.7 All the various possible donor and acceptor related absorption processes at low temperature are shown below. However, at room temperature, the main optical consequence of donors or acceptors is freecarrier absorption. Figure 4.8 Free carrier absorption Free carrier absorption results from the excitation of a free carrier into an available state higher in its respective band (e.g. electrons higher into the CB). As a result, free carrier absorption is quite different for electrons and for holes, mainly due to the presence of multiple valence bands. Free electrons located at k= in a single parabolic conduction band can reach higher energy states if their momentum is increased. This means that in ntype semiconductors, conduction band electrons may only be excited by the simultaneous absorption of a photon and absorption or emission of a phonon, in order to conserve momentum. By contrast, ptype semiconductors can have direct (i.e. no phonon needed) 5

151 transitions between the heavy and light hole bands (see the FHA process sketched earlier in this chapter). This causes the holes to produce much stronger free carrier absorption than conduction electrons. The conduction electron absorption process may be described by the Drude model. For semiconductors, the free electron density is often small (~ 4 /cm or about 8 smaller than in a metal) so that the plasma frequency is small. (about 4 lower than a metal). Recall that, for frequencies well above the plasma frequency, the Drude model gives p p p r '( ), "( ) r, (4.4) so that the absorption coefficient is hence given by, p ( ) "( ). (4.5) c c c p Hence we see a dependence for free carrier absorption due to electrons. This can be seen in the broadspectrum plot of absorption for a typical semiconductor shown earlier. Note that the formulas above do not take into account the susceptibility of the host material, which is not negligible. Assuming that the host semiconductor at frequencies well below the band gap has a dielectric constant, the real part of the dielectric function of the doped semiconductor becomes (see Weak absorption approximation in Appendix rror! Reference source not found.) p r '( ), (4.6) while the imaginary part remains the same. This leads to a similar expression for but scaled by the refractive index of the host n h=, giving p ( ) "( ). (4.7) n c n c n c h h h p The form of the freehole absorption in ptype semiconductors is less well defined, as it depends in the band structure as well as the hole density. In semiconductors with equal numbers of electrons and holes, the free hole absorption dominates. 5

152 Semiconductors of reduced dimension Figure 4.9 Semiconductors can now be artificially structured on nanometer scales. Quantum wells are layered semiconductor structures with alternate layers of materials with different energy gaps. The layers can be so thin as to confine the motion of the electrons perpendicular to the layers so that only certain quantized energy states are allowed. Motion in the other dimensions is still like for a bulk semiconductor. However, the density of states is changed. It is left as an exercise to calculate this, but the result is that the density of states becomes steplike. Figure 4. Because the energies associated with motion perpendicular to the wells are quantized, for each of these quantized energy states, there is a whole band of energies associated with motion in the other two dimensions. These are referred to as subbands 5

153 Figure 4. As a result, the density of states, and hence the absorption spectrum, takes on the shape of a staircase, with n steps of energy n, as shown below. Figure 4. A density of states of this type gives very strong enhancement of the absorption and emission of light at the very lowest photon energies. This can lead to very lowthreshold semiconductor laser operation. Reducing the dimensionality further can further enhance these effects. For example, quantum wires, which allow only D motion of electrons gives a density of states proportional to /. Meanwhile, quantum dots confine motion in all dimensions, so that the semiconductor nanoparticles are rather like large atoms, with a density of states that looks like a series of narrow lines (i.e. Dirac delta functions). 5

154 54

155 Chapter 5 From dipole radiation to refractive index In several of the model descriptions of refractive index that were discussed in previous chapters, we treated matter as an array of atoms that are polarized by an M wave. These induced dipoles in turn radiate a field that interferes with the incident field so as to produce refraction (phase shifting of the incident wave) and perhaps absorption (attenuation of the incident wave.) In Chapter this was discussed qualitatively. In the present chapter we will examine the radiated field from an oscillating dipole in more detail, and look how the addition of dipole radiation from multiple dipoles gives rise to a macroscopic refractive index. A more extensive description of these arguments is given in Wooten, Chapter. Mathematical description of dipole radiation To analytically describe how a large collection of driven oscillating dipoles adds up to a refractive index, we first need to mathematically describe dipole radiation of an isolated oscillating dipole. We will describe our point dipole as an oscillating current localized at a point in space (i.e. a current density described by a delta function in space), and then apply Maxwell s equations to derive the magnetic and electric field components around the dipole. To facilitate our analysis, we will make use of the vector potential, A. The vector potential There is a general vector theorem that states for a vector field V, we always have V. From Maxwell s equations we know that B we construct a vector potential, A, such that B A which implies that if, (5.) then we automatically satisfy. Since in this approach B is derived from the rotation of A, one can actually define A in different ways, as long as the rotation of A is not affected. We use the form of A known as the Coulomb gauge or transverse gauge where A. In the following we will derive a wave equation for A which will we expressed in terms of current density. From Maxwell s equations we have B A (5.) t t B Taking the time derivative on each side, we obtain an expression for d/dt: A A t t t t (5.) where we have used that and A are divergencefree ( if the rotations are equal, the terms could still have different divergence, however = in the absence of free charge and A= in the Lorentz Gauge ). Substituting the expression for d/dt into B J (5.4) t 55

156 and applying A B A A A (5.5) we find the following wave equation for A in the Coulomb gauge is A A J. (5.6) t Here we have assumed that we will be looking at transverse wave, implying that only the component of J normal to the direction of propagation ( J ) is relevant. This equation has a general solution: r r J ( r, t c ) d r A( r, t) c 4 r r, (5.7) showing that the appearance of a finite vector potential at point r is due to the current at a point r at an earlier time, which satisfies causality. This expression can be used to find the dipole radiation field. We assume a point dipole, placed at the origin: p ( r, t ) rˆ r Figure 5.6 The time dependence is chosen to be harmonic, and since this is a point dipole placed at the origin, the description of the dipole includes a delta function around r=: p( r, t) p cos( t) ( r) p (5.8) ˆ Associated with this dipole is a current density at r=: P J p ˆ sin( t) ( r ) p (5.9) t The component of the current that generates transverse components of A is given by J p sin( t) ( r) p (5.) ˆ Where ˆp is the component of the unit vector that is perpendicular to r: 56

157 57 Figure 5.7 It follows that ˆ ˆ ˆ ˆ p r r p, (5.) which has the magnitude of cos() (note: is defined relative to the horizontal in the sketch). A polar plot of this angular dependence of the magnitude of A is sketched below: Figure 5.8 Now we can substitute for J in our solution for A to get r t p c p r r t r A c r sin 4 ˆ ˆ ˆ ), ( (5.) Based on this solution we can now extract according to t A t A (5.) where again we used the Lorentz Gauge. This results in the following description of the radiated electric field from the induced dipole: ˆ ˆ ˆ cos 4 cos 4 ˆ ˆ ˆ p r r r t p k r t p c p r r t A c r c r. (5.4) ˆp r rˆ ) ˆ ( ˆ ˆ p r r ˆp r

158 Spherical wave Dipole pattern We will be calculating the radiated power, which means finding the Poynting vector, which in turn requires us to find the magnetic field. In isotropic media we have the following relation between and B: B ik ib k B rˆ c (5.5) where in the last step we used the fact that rˆ k for a point dipole placed at the origin. The direction of B is now related to the previously obtained direction of according to rˆ rˆ rˆ pˆ rˆ pˆ. (5.6) This gives us the following description of the magnetic flux resulting from a harmonic point dipole placed at the origin: k p B 4 c cos t r c rˆ pˆ r ˆ. (5.7) The expressions for and B now allow us to calculate the Poynting vector and the radiated power emitted by the dipole. ffect of reradiated field from induced dipoles We are interested in the case where a plane wave is incident on a polarizable atom or molecule. This induces an oscillating dipole which radiates a field that interferes with the incident field, as shown below. Figure 5.9 From the Lorentz model we have determined the magnitude of the induced dipole to be 58

159 59 i p m e p I ˆ (5.8) Note that this description leaves open the possibility of an anisotropic polarizability, which is relevant in the description of molecules. The complete radiation pattern that develops due to the induced dipole is caused by the sum of the incident field, I, and the field reradiated by the dipole (the scattered field), S. Hence the total Poynting vector is scattered absorbed incident S S I S S I I I S I S I S S S B B B B B B S (5.9) The last step can be understood by requiring energy conservation. nergy conservation requires that the integral of the Poynting vector across a spherical surface around the dipole should produce a total emitted power of zero. The first term IB I clearly represents the power in the incident plane wave, which integrates to zero. The final term clearly represents the total power radiated by the dipole. For energy conservation to be satisfied, the cross terms must add up to a negative quantity if there is finite scattered power. This suggests that finite scattering is accompanied by a finite loss term representing absorption. Scattered Power S S scatt B S, where, (5.) ˆ ˆ ˆ.. exp 4 p r r c c t kr i p r k S (5.) and ˆ ˆ.. exp 4 p r c c t kr i p cr p k B S. (5.) Since cos ˆ ˆ ˆ ˆ ˆ ˆ r p r p r r, then the time average of the Poynting vector for the scattered light is ˆ. cos 4 ˆ cos r r p c r r p c k S t scatt (5.) The scattered timeaveraged power is then given by integrating over all directions of propagation:

160 P scatt d 4 r c sind rˆ p S scatt t (5.4) and since, then, p P scatt e m 4 pˆ I 4 e 4 pˆ I c m (5.5) (5.6) Hence, in the limit of <<, the scattered power is proportional to 4 or 4 this is known as Rayleigh scattering. We can also define a molecular scattering cross section, scatt() : P and since where scatt scatt S (5.7) S orientations. I t I c t I, then 4 4 e pˆ ˆ I 4 6 c m D( ) (5.8) scatt D ) i (. Here, we have not done an average over molecular Absorbed Power Since the scattered field is weak compared to the incident field, we can treat the scattered intensity as being negligible compared to the incident. Hence the transmitted irradiance is primarily affected by S abs, which is the term that describes interference between the incident and scattered fields. We will not take the time to go through the math here, only look at the result, as sketched in the figure below. This shows that the time average of S abs, i.e. <S abs> t, oscillates rapidly with angle around the dipole for most angles, but behind the dipole <S abs> t varies slowly and is negative, as shown in the polar plot below. (From Hopf & Stegeman, chapter 4) 6

161 Figure 5. Hence, the effect of the dipole is to impress a small shadow on the transmitted field behind it. If we integrate the outwardflowing component of <S abs> t over all angles, the result is the total absorbed power, P abs, i.e. P abs S abs rˆ t d where the integral is over all solid angle. This yields, Where, Hence, p ˆ * ˆ Im (, (5.9) P abs I ) I (5.) e p ˆ ( ˆ I ) ( ) e ( pˆ ˆ ) I ( ) ( ). (5.) m Im I I i m D o * e pˆ ˆ ( ) I e m pˆ ˆ m I I Im * D ( ). (5.) D( ) Now we may define a molecular absorption cross section, abs(), that relates the absorbed power to the incident irradiance, by P abs abs I t S. (5.) Hence abs p ˆ * ˆ Im ( S I I ) I. (5.4) 6

162 Since S we obtain I I abs c, (5.5) ˆ e pˆ I ( ) (5.6) c m D( ) where the averaging is now over all angles between the dipole coordinates and the applied field. For a medium containing randomly oriented molecules, (i.e. isotropic), we have already seen that p ˆ ˆ I. Since P abs is an absorbed power per molecule, we can find the net absorbed power per unit volume as N P abs = N abss I. But the power absorbed per unit volume for a plane wave is just ds I/dz. Hence dsi N abss dz S ( z) S () e I I I z S I I S () e Nz Hence, the absorption coefficient is given by abs ( )N (5.7) ( ). (5.8) This result makes sense in terms of our result for abs() above, as in the weak susceptibility approximation, ()=(c/) (). Complex refractive index for a sheet of induced dipoles Unlike absorption, we cannot calculate the refractive index from the field of a single dipole. We must look at the field due to an ensemble of induced dipoles. To calculate the refractive index (real and imaginary), we look at the field produced by a sheet of dipoles all induced by the same incident plane wave. Figure 5. 6

163 6 Summing over the NAdz dipoles in the sheet, we find (see Hopf & Stegeman) that the net radiated field is also a plane wave, described some distance behind the sheet by:.. ˆ ˆ ) ( ) ( c c e p z z Ndz c i t t r k i R (5.9) and since z B c ˆ, cc e p z Ndz c i t B t r k i R. ˆ ) ( ) ( (5.4) where the averaging is over molecular orientations. Note that the factor i in both field terms means that the radiated field, R is 9 out of phase with the oscillation phase of the dipole. This implies that for a real p and a cosinusoidal incident field, the reradiated field is sinusoidal. Hence in the absence of loss, the reradiated field is 9 out of phase with the applied field. For a very weak reradiated field (always the case), this translates to a phase shift on the incident field, rather than an amplitude change. Figure 5. For the Lorentz oscillator model given above, we find ) ( ˆ ˆ ) ( ˆ ˆ D p dz k i D p dz m Ne k i p I I R, (5.4) where the averaging is over the molecular orientations. Now the total change in field amplitude (z) after a propagation distance of z is, I I p I p I I I p I z ik I I R I D p ik dz d z D p k i z ik z D p k i e z z z ) ( ˆ ) ( ˆ ) ( ˆ () ) ( ) ( () ) ( (5.4) The second step makes use of exp(x) x for small real x. I R Resultant field Lossless oscillator below resonance

164 We also know d dz I ik. (5.4) I Comparing this with equation 5.4 gives the following expression for refractive index: ˆ pˆ n i. (5.44) p D( ) And therefore ˆ pˆ r, (5.45) p D( ) which is the Lorentz oscillator model result, including orientational averaging. Clearly, then, the phase of the induced dipoles, and hence the phase of the reradiated field, dictates the optical properties of the material. On resonance, the oscillators are / out of phase with the driving field. The reradiated field is shifted by yet another / upon propagation, so that R is out of phase with I. I Resultant field Lossy oscillator at resonance R Figure 5. Clearly, we may extend these arguments to gain. If the phase of the oscillator is /, then the radiated field will add to I. 64

165 Chapter 6 Rate equations Corresponding handouts are available on Stimulated absorption and emission We have already looked at Fermi s golden rule, which tells us that the rate of transitions from one quantum state to another is proportional to the incident irradiance, the transition matrix element for the two states and the density of states. Naturally, the initial state has a lower energy than the final state when a photon is absorbed from the optical wave by the material. This is sometimes called stimulated absorption. However, instein showed that it is not required that the initial state have a lower energy than the final state to have a stimulated transition. But if the final energy state is lower, then an extra photon is emitted instead of the incident photon being absorbed. This process is labeled stimulated emission, since the process is stimulated by the incident photon. The extra photon produced by stimulated emission has exactly the same frequency, direction and phase as the incident photon, which is of great interest for applications. It is also possible to have a third process, where the atom spontaneously decays from a higher energy state to a lower one, with no external stimulus. The direction and phase of the emitted photon is random. This is referred to as spontaneous emission. These three fundamental processes of light interaction with matter are illustrated below: Absorption Stimulated mission Spontaneous mission Figure 6. In the above, the photon energy. h, must obey h =. It is perhaps obvious that the stimulated emission process might be used to provide coherent amplification of optical waves. But to do this would require the population of the upper energy state to exceed that of the lower energy state. As we will eventually see, this is possible, but first we must develop a way of relating these processes to populations and to the absorption of light upon propagation. Rate quations Associated with each of these processes is a rate of change of population of the lower and upper levels. Labeling the upper (lower) state energy as ( ), and the number of atoms per unit volume in the upper (lower) state as N (N ), we can write rates of change of populations associated with each of the three processes: 65

166 () Spontaneous mission dn N AN dt sp sp (6.) where A is the spontaneous emission rate, also known as the instein Acoefficient. sp is the spontaneous lifetime and A = / sp. N N Figure 6. Spontaneous emission () Stimulated mission dn NB ) N dt st ( ( ) I h (6.) where B is the stimulated emission rate, or instein Bcoefficient for transitions from state to state. ()d is the electromagnetic spectral energy density (energy per unit volume in the range {,d}). If we have a monochromatic field, it is simpler to use I, the irradiance at frequency and the stimulated emission cross section, (). N N Figure 6. Stimulated emission 66

167 () Stimulated Absorption dn ( ) NB ( ) N I dt h abs (6.) Where the definitions are consistent with above. N N Figure 6.4 Stimulated absorption We can track all of this in a rate equation for a twolevel system: N A N Figure 6.5 Level scheme showing absorption and emission crosssections, as well as the instein A coefficient dn dt NB ) ) N B ( ( AN (6.4) Now if the total atomic density is N, then N N = N, and since N is constant, then dn dn. (6.5) dt dt We will see later that the rate equation can be solved to give the absorption of this level system. But first, there are some interesting relations between the A and B coefficients that we should explore. Relationship between A and B coefficients: In thermal equilibrium, the populations of the levels are related by a Boltzmann factor 67

168 h kbt N e, (6.6) N where k B is Boltzmann s constant. Now in thermal equilibrium, we also have dn /dt=, so that q. () becomes, N B ( ) N B ( ) AN N N B ( ) e A B ( ) h k T B. (6.7) In the limit of very high temperature, kt>>h, exp(h/k BT). But also if the temperature is large in any material, the radiation density () becomes extremely large, so that stimulated processes dominate spontaneous ones, so that Hence, lim B ( ) T A B ( ) B B. (6.8) B B B. (6.9) We can also rearrange q. to give, ( ) B AN N B N ( ) A B e A N B N N h k B T A B N N (6.) Now this expression for the electromagnetic spectral energy density is of the same form as that given by the Planck blackbody radiation formula: 8hn ( ) h. (6.) kbt c e This Blackbody radiation spectrum is common to all materials, so it is not surprising that our two level system should also give the same form. The spectrum of () is shown for a couple of temperatures in the figure below: 68

169 Figure 6.6 Blackbody radiation spectrum Hence, A 8hn 8h (6.) B c where = c/n is the wavelength inside the medium. Therefore stimulated processes tend to become more dominant as the wavelength increases. Degeneracy of levels Some energy levels consist of several states that have the same energy. These are called degenerate states and the number of states, g i, with the same energy, is called the degeneracy of level i. Hence, h k BT N g e. (6.) N g This leads to, B N B B N g g B B N g g N, (6.4) where we have chosen to set B = B Similarly, we can write for the stimulated emission cross section 69

170 ( ) N g ( ) g ( ) N ( ) ( ) N g g N (6.5) We will mainly ignore degeneracy in this course, but we may refer back to this equation if necessary. Population decay processes In the absence of electromagnetic radiation, the rate equation becomes dn dt N, (6.6) which has solution sp t N ( t) N()exp sp. (6.7) However, not only spontaneous emissions are responsible for the decay of upper state population. Other processes such as collisions can also cause nonradiative decay. If we label the mean collision time as c, then the overall lifetime of state is given by adding decay rates as follows: c sp. (6.8) Light amplification and absorption Including our interrelationships of cross sections and the overall lifetime of state two, we can now write q. () of the previous handout as dn dt g I N ( ) N N g h I N ( ) N h (6.9) where we have defined a population difference, g N N N. (6.) g Now, we are interested less in the changes in N or N than we are in the growth or decay of the electromagnetic field. Hence, we need to work with our rate equation so that it tells us about the energy given to or taken from the electromagnetic field by the atoms. 7

171 Let us consider the propagation through a material of light, irradiance I, resonant with two levels with energies and : I N N Figure 6.7 We are interested in calculating how the light irradiance grows or decays as it propagates through the material. Now since spontaneous decay produces light in all directions and that is incoherent with the optical wave of interest, we can ignore spontaneous transitions. Hence the net gain or loss of energy to the field is purely determined by stimulated transitions. If the photon density (number of photons per unit volume) in the optical wave is, then the time rate of change of photon density is d dn I ( ) N dt dt h stim (6.) so that the rate of growth of the energy density,, = h, is given by, d d h ( ) NI. (6.) dt dt Now, d /dt is the power generated or lost per unit volume. We can consider the power gain/loss in a small slice of the medium as shown below Area, A I z dz Figure 6.8 7

172 d ( Power ) Adz ( ) But irradiance =power/area, so, di dz NI (6.) ( )NI, (6.4) which we can write as di dz This has solution ( ) I. (6.5) I ( z) I ()exp( ) z. (6.6) Hence () is the gain coefficient or absorption coefficient of the medium, where ( ) ( )N. (6.7) Hence, for negative N, we have absorption. This is the normal thermal equilibrium case. To have optical gain, we need to have a population inversion, i.e. N positive, which does not usually occur naturally in thermal equilibrium (it implies a negative temperature) but we will look later at schemes in which it is possible to have N >. 4 I(z)/I() N > N < z Figure 6.9 We can write the gain in terms of the instein A and B coefficients as follows: Recall from the start of the last handout, that, ( ) B ( ) I. (6.8) h 7

173 Now we are interested in the situation where the light is formed by spontaneous and stimulated emission, so that it is not surprising that () and () should have the same dependence on. Since the total energy density is, then, ) d (, (6.9) We can define a normalized line shape function, g()=()/, so that, Hence we have g ( ) d (6.) ( ) B g( ) I. (6.) h Now, the irradiance is equal to the radiation energy density multiplied by the phase velocity of the wave, i.e. (See argument on page ) c I, (6.) n where n is the refractive index, so that, nh ( ) Bg( ) c nh ( ) Bg( ) N c But from q. 4 of the last handout so that, sp (6.) c c B A (6.4) 8hn 8hn c g( ) ( ) 8n c g( ) ( ) 8n sp sp N (6.5) Schemes to obtain a population inversion g It is always possible to make N >, i.e. N N g, temporarily, but this cannot be achieved in the steadystate for only two levels. At least levels must be involved to create a steadystate population inversion. Since N > implies that some energy is stored in the medium, there needs to be a method of pumping energy into the system in order to create a population inversion. 7

174 Let s look at some possibilities: R () level laser Figure 6. The above figure shows a configuration of three energy levels in a material that might be used to obtain a population inversion. Stimulated transitions occur between level, which is the ground state, and the next higher energy state, labeled level. Level is directly excited from the ground state by the pump, after which the atom rapidly decays into level. In solid state lasers like Ti:Sapphire and Nd:YAG, the pump energy is provided optically, either by narrow band light from a diode laser, or by broadband light from a flashlamp. To maximize absorption from a broadband source, it helps if level is a broad band of levels, rather than a single, narrow level. If the pump rate into level is large enough, we may get a population inversion. But it is quite easy to see that there are some other requirements, such as >> >> R > ( ), where R is the pumping rate into level. It is easier to see what R means by writing down the rate equations for this system: dn dt dn dt N RN N RN (6.6) N N N N ( ) I h N (6.7) N N (6.8) If we wish to solve these to find the threshold value of pump rate, R, that gives population inversion, we are fortunate in that we do not need to include stimulated processes in our calculation, as there is no significant irradiance, I, until a population inversion is obtained. On the other hand, if we want to find what happens above the threshold pumping rate, we definitely need to include stimulated processes. 74

175 We will solve for a quasilevel system later, but we can note right now that it is a difficult task to create a population inversion in a level system like this. This is because an unexcited system will have all atoms in the ground state, which here is the lower laser level. To create a population inversion, one therefore needs to have more than half of the atoms excited out of the ground state. This is illustrated on the right hand side of the above diagram depicting the level system. This is a very high state of excitement. Yet the first laser gain medium to be demonstrated, ruby (Cr doped Al O ) was of this type. It was initially thought that laser gain could not be achieved in ruby, as calculations showed that maintaining a steadystate population inversion would cause the crystal to melt! However, Theodore Maiman of Hughes Research Labs figured out that he could make it work by simply running the laser in pulsed mode. R Figure 6. () 4level laser The above figure depicts a system that is similar to the level system described above, except that now the lower laser level is no longer the ground state. This makes a tremendous difference, as a population inversion can now be achieved with quite small depletion of the ground state. All that is required is to have N > N, which can be achieved by having the relaxation rate out of state very fast. Provided that the energy difference, >> k BT so that the thermal population of state is negligible, the only requirement for a population inversion is that. Where and. As with the level laser system, we want rapid decay from state to state, i.e. ~ >> >> and >>. 75

176 Since population inversion can be achieved for relatively low excitation, lasers of this type can be made to run continuously, or in continuous wave (c.w.) operation. xamples of 4 level laser gain materials are Nd : YAG and CO. Since the decay from state to state is quite rapid, it is possible to model a 4level laser system as a three level laser, where the stimulated absorption occurs between the upper two levels, as shown below: N R N N Quasi level system Figure 6. Just like the above case, we can approximate a level system down to a level one, as follows: N R N Figure 6. Remember that this is just a simplification of the level system for modeling purposes. it is not actually possible to create a population inversion in a level system. It is straightforward to solve the rate equations for this system in the steady state (noting that N= N N ). By steady state, we mean that all the time derivatives are set to zero. We then find that the steady state solution for N is given by N N R R (6.9) ( ) I h ( R ) The derivation of this is left as an exercise. Note that the requirement for population inversion (N > ) is R > /, as one would expect. 76

177 We also note that the denominator in the above expression contains the irradiance, indicating that the population inversion and hence the gain/loss coefficient reduces as the irradiance increases. We may rewrite our expression for N as Where: N N R R (6.4) I / I sat h ( R ) I sat is the saturation irradiance, (6.4) ( ) defined as the irradiance at which the population (and hence gain coefficient) falls to one half of its lowirradiance value. (Sometimes the gain at low irradiance is called the smallsignal gain.) The irradiance dependent gain coefficient can hence be generally written as (, I ) ( ) N ( I ) (6.4) and for our quasilevel system, N I R R (, ) ( ). (6.4) I / Saturable absorption I sat For R=, we have a normal twolevel system, for which the gain coefficient, (,I), or absorption coefficient, (,I) is given by N (, I) ( ) (, I) (6.44) I / I sat and in this case I sat = h/(). This is known as saturable absorption, since the absorption coefficient decreases, or saturates, as the irradiance is increased. Saturable gain We have already seen that for positive N, the gain coefficient also saturates. All laser systems (4level, level, etc.) show the same behavior, not just the level system highlighted here. This is a consequence of energy conservation, as will be discussed later. As with the quasi level system, the saturation irradiance is proportional to the photon energy and inversely proportional to the stimulated emission cross section and the upper state lifetime. Pumping thresholds for and 4level systems We saw in the last handout that the three level laser with the ground state as the lower level could be modeled with only two levels, as shown below: 77

178 N R N Figure 6.4 From this, (homework exercise) we found that the population difference, N, is given by N N R R. (6.45) ( ) I h ( R ) From this it is clear that N is positive for R >. Hence the minimum pump rate to get a population inversion, sometimes called the pumping threshold is R th (6.46) If we neglect the radiation field by setting I =, we can write the N as a function of R: N N R R (6.47) A plot of the normalized population difference versus pump rate for the quasi level laser system is shown below. It is always justifiable to neglect the radiation field when pumping below R th. Above R th, we might have to worry about the radiation field growing and causing saturation of N, so the behavior of N for R > R th should not be taken too seriously. 78

179 Quasilevel laser..5 N/N Figure 6.5 We also mentioned last time that a 4level system could be modeled as a level system with stimulated processes taking place between the upper two levels. This quasi4level system looks as follows: R N R N N Figure 6.6 dn dt dn dt R N N N N N ( ) I N h ( ) I N N h N N. (6.48) 79

180 For the purpose of calculating the pump threshold, we can set I =. Also, we will assume an ideal 4level system where the decay rate out of level is so rapid that N. These approximations give dn dt R N N N N N N N N N N. (6.49) Hence, in the steady state, where all time derivatives are zero, we have R N N N R N N. R (6.5) Clearly, the pump threshold is now zero. In fact, if we take all decay paths into account, R would not be exactly zero, as it would have to exceed /, for example. A comparison of the pumping threshold for the quasi 4level and quasi level systems is shown in the plot below. As R becomes >> /, then the two systems start to look similar, but it is the difference in thresholds that is most important. Comparison of Quasi level and 4level lasers..5 Quasi 4level Quasi level N/N.5. 4 R Figure 6.7 8

181 Chapter 7 Lasers Laser Oscillation and Oscillation threshold We have looked at how we can have optical gain due to stimulated emission, also known as Light Amplification by Stimulated mission of Radiation. But amplification assumes that we have some input light. The source light in a laser is the spontaneous emission, and a long, highgain laser amplifier can act as a selfcontained emitter of light. But the radiation may not be particularly, narrowband, directional, or coherent. To make a system with optical gain into a light source capable of emitting narrow band, coherent and directional radiation, we may place the gain medium into a FabryPerot (mirror) cavity. This cavity will allow spontaneous emission to be fed back through the gain medium many times, allowing a selfconsistent oscillation to build up. The cavity, being an interferometer, will serve to act as a spectral filter, allowing only certain narrow spectral lines (longitudinal cavity modes) to oscillate. Usually, the spectral width of these lines is considerably less than the linewidth of the laser transition. The cavity will also have the effect of allowing a particular spatial irradiance distribution (transverse mode) to be emitted. In this course, we will neglect the spatial and spectral effects of the cavity and restrict ourselves to looking at the irradiance in the laser cavity. We will consider a simple cavity with mirrors, reflectance R and R. The gain medium has a gain coefficient, and the surfaces of the gain medium have transmittances T w and T w. ( w stands for window as early lasers were often gas lasers which contained the gas in tubes with windows on the ends. We will also allow for some other parasitic distributed loss, such as scatter, that we will model using an loss per unit length,. d A T w, l T w R R Figure 7. Imagine there is some light propagating along the axis of the cavity, perhaps due to a spontaneous emission in the appropriate direction. If the irradiance is I at plane A, and propagating to the right, it will impinge on the surface of the gain medium, after which the irradiance will be T wi, then it will propagate through the gain medium, after which the irradiance will be exp{()l} T wi, etc.. Upon reaching plane A again, after one round trip, the irradiance will be 8

182 I rt RT. ( ) ( ) we Tw RT we Tw I Now if the gain is exactly right to have a steady state laser oscillation, the net round trip gain and round trip losses exactly balance each other. We call this gain the threshold gain. In this case, I rt = I, so that R T ( th ) w e T w R R T R T w ( th ) w T w e e T ( ) w th ln RR TwT th w NOT: This is the threshold gain for laser oscillation in a cavity, which is different from the threshold pumping rate that gives a gain of zero. Now, we can engineer our laser to minimize the threshold gain, th, by making ~ and R, T w, T w ~, but of course, R cannot be exactly equal to one (if R is) if we are to have a useful laser output. R is called the output coupler. We will talk about optimizing the output coupler reflectance shortly. What happens when > th? Normally in a laser, will be larger than th. If > th, then the irradiance inside the cavity will grow on successive round trips. But this cannot continue indefinitely, as we are creating a population inversion by pumping energy in and the rate of pumping must ultimately control the rate of output of energy (i.e. power). What happens is that as the cavity irradiance grows, the gain will saturate, slowing the rate of growth of cavity irradiance. The irradiance will grow until the gain eventually saturates so much that the round trip gain reaches the round trip loss. That is, the gain will saturate down to th. So the irradiance in a laser cavity running in the steady state is just the irradiance which causes the gain to saturate to a value of th. We can make use of this to calculate the output power of a c.w. laser. C.W. laser power and optimum output coupling If the round trip losses are small, the cavity is said to have a high Qfactor. (The Q factor of an oscillator is defined as the ratio of the energy stored to the energy dissipated per cycle.) Irrespective of the definition, provided the losses are low, then the gain per round trip must also be low to have steadystate operation. If this is so, the cavity irradiance must be more or less the same at all points along the cavity. This allows us to make the approximation, di dz. Now if the losses (including the output coupling loss ) are low, we can approximate them as a distributed loss, rather like an effective absorption loss, rather than as a series of discrete losses. i.e., we can define an effective loss coefficient, T, as follows: exp 4 T exp T w R R, where we have set T w T w. Hence, 8

183 4 T w R ln ln R. T Note that T is exactly equal to th, as it should be. Setting the overall parasitic loss of the cavity, ln 4 R L output coupler transmittance, T = =R, we have T L Lln T, T T w where we used the fact that T is small, and the identity, ln(x) x. We will of course take the gain to be saturable, according to ( I). I I sat, and noting that the Let I be the irradiance propagating from left to right in the cavity and propagating from right to left. In this case, we can write I be the irradiance di dz I I, T I I I sat where we have to include both backward and forward irradiances in the denominator as they both contribute to saturation, and we are still under the high Q approximation so that di/dz. The above equation simplifies to T I I I sat and, setting an average irradiance, I I / I I I. sat T sat I, Now the unknown is the average cavity irradiance, so we rearrange to give I I sat. T The above equation seems to make sense. It tells us that the minimum gain to have a finite cavity irradiance is equal to the total round trip losses, as calculated earlier, and that the cavity irradiance grows in linear proportion to ( T), which is the same as ( th). It also says that the cavity irradiance is proportional to (and of the same order of magnitude as) the saturation irradiance, I sat. Recall we found that Tl = L T, and we can also define a round trip smallsignal gain, g = l. Hence, the above equation becomes 8

184 I sat g I, L T and, the output irradiance is given by, I I sat g I T IT T L T out. Below we plot I out versus T for some values of g and L: c.w. laser output in highq approximation..5 g =., L =. g =.5, L =. g =.5, L =. I out /I sat T Figure 7. It is important to remember that this analysis cannot be used for large values of gain or loss. Optimum output coupling The above expression for I out tells us that we can adjust the output coupler transmittance, T, to optimize I out. Clearly, minimizing T gives the lowest possible threshold, but no useful irradiance gets out. On the other hand, if T is large, a significant fraction of the cavity irradiance is usefully coupled out, but the gain threshold is raised. Obviously, there is some compromise value of T that gives optimum output power. The higher the small signal gain, the larger one might expect the optimum value of T to be. But we can calculate this by differentiating I out with respect to T. di dt out I sat I sat g L T g L T L T I sat g L T 84

185 Setting this to zero to find the maximum I out gives g L L T T opt g L L We can substitute the optimum value of T into our expression for I out, to obtain I opt out g L I sat. As a check, let s take g =.5 (5% round trip smallsignal gain) and L =. (% round trip loss, not counting output coupling). For these numbers, we find and opt T % I I opt out sat , which is in agreement with the above plot. Saturation in inhomogeneously broadened systems Although it was not specifically mentioned, our previous analysis of gain/absorption saturation applies to homogeneously broadened systems. For Inhomogeneously broadened systems, saturation is a little different. Remember that an inhomogeneously broadened system is just an ensemble of identical atoms of molecules that, due to statistical variations in their environment, have a statistical distribution of resonance frequencies that is broader then the intrinsic (homogeneous) linewidth of the atoms or molecules. Consider for now, just two groups of atoms with different resonance frequencies, and. These may or may not have a population inversion, but we shall assume they have, so that we see optical gain for light at and. If there is not a population inversion, we observe absorption at these frequencies. In either case, radiation at will saturate the gain or loss for the atoms with resonance at and radiation will saturate the gain or loss for the atoms with resonance at. The figure below shows the effect on the overall gain spectrum of illumination with light at. If the resonances are very closely spaced, the light at will somewhat saturate the transition resonant at, but much less strongly than it saturates the transition resonant at. This is reflected in the /() dependence of the saturation irradiance. 85

186 Figure 7. Now, an inhomogeneously broadened system contains many such resonances, grouped closely together in a band, If narrowband light, frequency, is incident on an inhomogeneously broadened system, it will most strongly saturate the gain or absorption due to the atoms with resonance frequency closest to. Atoms with more distant resonance frequencies will be saturated less, or not at all. Hence the effect of saturation due to the light at will be to create a dip, centered on, in the absorption or gain curve. This dip is sometimes referred to as a spectral hole and the process of saturation in an inhomogeneously broadened system is sometimes called spectral hole burning. This is sketched below: Figure 7.4 Of course the overall unsaturated gain typically has a Gaussian spectral profile, so the saturated gain profile actually looks as follows Saturation irradiance and hole width Figure 7.5 Now each group of atoms with a particular resonance frequency will still saturate according to 86

187 N N(, I). I ( ) I sat However, the light at is affected by not just the atoms with resonant frequency, but also by atoms with nearby resonance frequencies. These atoms contribute less to the unsaturated gain (or loss) but they have a higher saturation irradiance. Hence as the irradiance is increased, the contribution to the gain or loss from these atoms becomes relatively more significant. This leads to a weaker saturation. We will not do the math here, but it turns out that the gain saturates according to ( ) inhom (, I ), I I s, res where I s,res is the resonant value of the saturation irradiance, i.e. I sat( ). The width of the spectral hole (or hole width ) is just given by the homogeneous width,. Hence it is possible to measure homogeneous line widths in inhomogeneously broadened systems by performing saturation of absorption spectroscopy, or holeburning spectroscopy. xamples of laser systems Here we just give a few details of the level structures of only two common laser materials: The Ruby laser As said previously, Ruby is Cr doped Al O, and is a level type of laser. The level structure is shown on the left. There are two broad bands that form level, into which optical pumping initially excites the system. Rapid decay occurs into the upper laser level, which is actually two closelyspaced narrow levels. The upper laser level lifetime is ms. The CO laser Figure 7.6 CO operates on vibrationrotation transitions, rather than electronic ones. There are two major transition bands, at wavelengths around.5 m and around 9.6 m. In both cases, the upper laser level is the first excited vibrational state of the antisymmetric stretch mode. The.5 micron band has a lower laser level that is the first excited vibrational state of the symmetric stretch mode. Note that this is not dipoleactive, but it is sufficient that the 87

188 antisymmetric stretch mode is dipole active. The 9.6m band terminates in the second excited state of the bending mode. Both bands will contain many lines associated with the various rotational transitions. ven though the vibrations are changing their nature in the transition, the J = is still obeyed, so a Pbranch and an Rbranch is observed. The upper state lifetime if the CO is not all that long, so the N molecule having a first excited vibrational energy very close to that of the excited CO antisymmetric stretch, and a long lifetime (not dipole allowed transition to the ground state), acts as the metastable state. A gas discharge excites the N and this remains in the st vibrational state until it collides with the CO molecule, which populates its excited antisymmetric stretch level, allowing lasing to take place. Figure

189 89 Chapter 8 Nonlinear Propagation quations Now that we have seen how it is possible to generate nonlinear polarizations, we must address how these polarizations cause new waves to grow, or how they affect existing waves. Recall that we found that nonlinear polarizations for discrete frequency inputs also have discrete frequency components, where the frequencies generated are at the sums & differences of the spectral components of the input..g. ) ( ) ( ), ; ( ) ( () () P (8.) or, ) ( ) ( ), ; ( ) ( () () P (8.) where =. The time and z dependence of the polarization at is, (assuming all waves are collinear and propagating in the zdirection),.. ) ( ) ( ), ; (.. ) ( ) ( ), ; ( ), ( ) ( () ) ( ) ( () () c c e c c e e z t P t z k k i t z k i t z k i (8.) Note that while this polarization wave has frequency, its propagation constant, k k, depends on the refractive indices at and. This is because the phase of the polarization ), ( () z t P depends locally on the phases of the electric fields at and. We anticipate that a wave,, at frequency will be generated by this polarization, so we will take a look at the wave equation for. ), ( ), ( () () z t P z t P t t P t c z (8.4) where,.. ) ( ) ( ), ( ) ( () ) ( c c e z t P t z k i (8.5) and c n k ) (. Hence, () () ), ( ) ( t z t P t c z. (8.6) Now we will anticipate that the amplitude of the generated field, ( ), may change with z, i.e.

190 9.. ) ( ), ( ) ( c c e z A z t t z k i (8.7) We can substitute this into the wave equation, but first, we can take the derivatives: ) ( ) ( ) ( c c e A k z A ik z A c c e z A ik z A A ik z A ik z c c e A ik z A z t z k i t z k i t z k i (8.8) Also,.. ) ( ) ( c c e z A t t z k i (8.9) And,.. ) ( ) ( ), ; ( () ) ( () c c e t P t z k k i. (8.) Now, we apply the slowly varying envelope approximation (SVA) xix, z A k z A (8.) which is valid provided the changes in A (z) are small over length scales on the order of. Under the SVA, the second derivative is eliminated from the propagation equation: (8.) Noting that () ) ( c k, this simplifies to z k k i z ik e e z A ik ) ( () ) ( ) ( ), ; ( (8.) xix Sometimes called the slowly varying amplitude and phase approximation (SVAPA), which recognizes that the amplitude may be complex, and the phase may change due to nonlinearities, as well as the magnitude. c c e c c e z A c c c e A k z A ik t z k k i t z k i t z k i. ) ( ) ( ), ; (.. ) ( ) (.. ) ( () ) ( () ) (

191 or A c () i ( ;, ) ( ) ( ) e z n( ) ikz (8.4) where k = k k k is the phase mismatch. We note that () () P ) ( ;, ) ( ) ( ) (8.5) ( so that in general terms, we may write A c () i P ( ) e z n( ) ikz (8.6) or A j z j i n( ) j P c ( ) ( ) e j ikz (8.7) where P () can represent any second order process, i.e. P () ( = ), P () ( = ), etc. The phase mismatch results from the waves at the three different frequencies propagating with different phase velocities, and hence getting out of phase with each other, which adversely affects the efficiency of conversion from one frequency to another. Perhaps it is easiest to understand the phase mismatch by looking at the secondharmonic generation process. SecondHarmonic Generation For second harmonic generation, the nonlinear polarization is defined as P () so we have () () (;, ) ( ) (8.8) A () i A (;, ) e z n( ) c ikz (8.9) Here, k k k n n. It is common to use the d eff coefficient, c defined by: d eff = () /, so that, A z deff i A e n c i A e ikz ikz (8.) 9

192 where is the figure of merit for second harmonic generation. Provided we can assume that the fundamental field, A, is constant along z, this differential equation is easily solved: ikz e A ( z) A () i A ik ikz / ikz / ikz / e e A () i A e z ikz / ikz / sin( kz / ) A () i A ze z kz / We are usually interested in the case where A () =, so we see that A ( z) (8.) ikz / sin( kz / ) i A ze (8.) kz / Now n A c I and I n c A I n ( z) n eff c d z n n c z I I sinc sinc kz kz (8.) So that the second harmonic output irradiance is proportional to the square of the input fundamental irradiance The sinc dependence is plotted below:..8.6 I.4. kz/ Note that for k =, the phasematched case, I z. But for kz =, the second harmonic output is zero. This is the usual expression we see for the second harmonic irradiance. Usually, z = L, the thickness of the nonlinear material, which is not variable. The phase mismatch is often tunable, though, as we will see below, and it is often quite easy to generate the above curve experimentally. 9

193 However, it is usually more insightful to look at I (z). In the above analysis, we multiplied the top and bottom by z in the expression for A in order to get the sinc function. If we do not do this, we find A ( z) i A e I n ( z) n c ikz / sin( kz / ) k / I k / sin kz (8.4) We see this plotted below. In all cases, for very small z, the second harmonic grows as z (as sin(x) ~ x, for x <<). But if the indices of refraction at and are unequal, the SHG produced near the entrance, which propagates at c/n, and the polarization that generates new SHG, which propagates at c/n, eventually get out of phase. By the time kz/ = /, the new SHG is already / out of phase with the old SHG that was produced at the front surface. Hence at this point, the new and old SHG start to cancel, and further propagation just reduces the SHG field. Unless we can make k =, there is no advantage in having a material longer than z = /k. This distance is known as the coherence length, l c, l c. (8.5) k 4 n n 8 k=. I 6 4 k=.5 k= z Techniques of Phase matching In any normal material, n n. There are three main ways of overcoming this: Anomalous Dispersion Phase Matching It may occasionally be possible to match the indices if and are on either sides of a resonance, as sketched below. This is highly unusual, and not a generally applicable method of phase matching. 9

194 ..5 n()..5 Birefringent phase matching It is often possible to find propagation direction in a birefringent crystal where the ordinary wave at has the same index of refraction as the extraordinary wave at (or viceversa). This is illustrated in the figure below. This requires a () tensor element that gives a second order polarization polarized perpendicular to the fundamental field, but nevertheless, there are many materials for which this is so, and this is the most common means of achieving phase matched () processes. In this case, the exact phase matching can be tuned by adjusting the angle, as illustrated in the experimental data below: 94

195 Quasi PhaseMatching (QPM) It is possible to obtain an large SHG (or other efficient () processes) by periodically modulating the () along the propagation direction, as shown below: Here the crystal is reversed every coherence length, i.e. for each successive layer, () (). Hence, instead of having a reduction of after z = l c, we reverse the phase of the "new" SHG, so that it continues to add to the SHG generated in previous layers. This adds 95

196 tremendous flexibility, as we are no longer restricted to the polarizations and directions imposed by birefringent phase matching. However, this is a difficult materials fabrication problem. QPM has been known since the early 96's, but it is only in the last few years that QPM materials have become commercially available. Other second order processes There is insufficient time to go over the theory for other processes, but it is quite straightforward to apply the same techniques to sum frequency generation (SFG) and difference frequency generation (DFG). Phase matched difference frequency generation Phase matched DFG is particularly interesting. This involves a beam at high frequency,, mixing with another lower frequency,, to give a difference frequency beam at =. () ( ;, ) It turns out that assuming no depletion at, the beam at grows as cosh (l) and the beam at grows as sinh (l). This corresponds to near exponential gain. The gain can be so high that with only an input at, the other waves can grow out of noise. The particular values of and will depend on phase matching. This process is known as Optical Parametric Generation and Amplification (OPG & OPA). Just like amplification by stimulated emission, this can be put into a laser cavity to build up a selfconsistent mode. This is known as an Optical Parametric Oscillator (OPO). Nonlinear Refraction and Absorption We now wish to look at the degenerate thirdorder susceptibility, () (;,,). The SVA propagation equation for a monochromatic field at frequency in a medium with susceptibility () (;,,) will look like Setting da () i ( ;,, ) A A. (8.6) dz 4nc da dz () () () ( ;,, ) r ii () () i A A, we have i. (8.7) r i 4nc 96

197 Now the slowly varying field amplitude may contain a phase component, as we write A( z) i ( z) A( z) e, (8.8) so the propagation equation becomes d dz i () i () i Ae i r A e i A e 4nc d A d i A dz dz 4nc i () r A Therefore, we have two propagation equations, one for amplitude and one for phase: () i A (8.9) d A dz 4nc d dz 4nc () r () i A A (8.) () This, r Nonlinear absorption leads to nonlinear refraction, while Multiplying the equation for nonlinear absorption by A : () i leads to nonlinear absorption. d A A dz i 4nc d A dz () A 4 (8.) Since I n c A, we can write di dz di dz n c I () i I (8.) Where is the twophoton absorption coefficient. This has solution I () I ( z ) I () z (8.) 97

198 Nonlinear refraction d () The equation, r A dz 4nc has solution () ( z) A nc z. r 4 Hence the propagating wave may be written as ( z. t) A( z) cos kz ( z) t () A( z) cos k r A z t 4nc (8.4) using k = n o/c =k n, where n is the linear index of refraction, we can write this as () r A ( z. t) A( z) cos kz n t (8.5) 4n Hence the total refractive index is n n 4n () r A (8.6) or n ( I) n n I (8.7) where n c 4 (). (8.8) r So the refractive index can be irradiance dependent. For Gaussian laser beams, this can lead to "SelfFocusing": n I r In the case of a negative n, we have "Selfdefocusing": 98

199 n I r 99

200

201 Appendix A mpirical descriptions of refractive index Sellmeier equations Sellmeier equations are essentially empirical fits to the actual refractive index of a material, using the result for ' for Lorentz oscillators as basis functions. Hence, Sellmeier equations are of the form pj ( ) n j j These equations are very useful ways of providing data on the refractive index of materials vs. wavelength, without the need for extensive tables. They are usually valid only in high transparency spectral regions, far from resonances, where is real. Often, but not always, Sellmeier equation for a material may contain a pole (i.e. a resonance) at low frequency ( << ) and one at very high frequency ( >> ). Hence one could write n ( ) p j j pj p, where, / p is more commonly written as. In practice, these equations are usually expressed in terms of wavelength: n b pj j ( ) A d a d j j j c j so that coefficients a, b j, c j, and d may completely describe the refractive index vs. wavelength for a material. There are usually only one or two values of j (i.e. only one or two resonances) in the Sellmeier equation for a given material. This may depend on the material and on the level of accuracy required. To get an idea of the wide variety of Sellmeier equations, some Sellmeier equations are given in the following table from the OSA handbook of Optics:,

202

203 For fused silica, a good fit can be obtained by using a pole Sellmeier equation, with poles at approximately 9.9 m, 6 nm and 68 nm. The contributions of the individual poles can be seen below: f f f f n f f Reference: Handbook of Optics (OSA) (valid..7 m ) f( ) f( ) n( ) f( ).... n( ).45.4 Schott glass description (power series) n = a a a a 4 a 4 6 a 5 8

204 Hertzberger description (mixed power series and Sellmeier) n B A C 4 D Abbe number The Abbe number, d, is a very commonly used singlenumber measure of the dispersion of glasses. It is defined by n d d, n f nc where n d is the refractive index at = nm (the sodium "d"line), and n f and n c are the refractive indices at = 486. nm and 656. nm, respectively. Typical values are in the range ~ 6. A low Abbe number indicates high chromatic dispersion. Some examples are given in the following table. Material Refractive index Abbe number Crown Glass.5 58 Polycarbonate.59 BK7.5 6 CaF

205 Appendix B Vector relations and theorems Unit vectors (Cartesian) Vector x = (,,), y = (,,), z = (,,) F = F, F, F = F x F y F z Vector field: if a field has a direction at every point in space, is called a vector field, which is written as (x, y, z) or (r ) where r = (x, y, z). A time depent vector field is written as (r, t). Dot product Cross product Del operator A B = A B A B A B x y z A B = A B A B, A B A B, A B A B = A A A B B B Divergence,, x y z F Fx Fy Fz lim x y z V V F ds Curl F x x F x y y F y z z F z nˆ lim S F d s with nˆ the unit vector normal to surface S For a scalar field V(x,y,z), the gradient is a vector field defined as V V V V,, x y z The operator del squared is called the Laplacian. When acting on a scalar field V(x,y,z) the result is given by 5

206 V V V V ( V ) x y z When operating on a vector field ( x, y, z), del squared is called the vector Laplacian, given by x, y, z x y z For any vector field ( x, y, z) : For conservative fields only: Fdv F ds (Divergence Theorem or Gauss theorem) F ds F dl (Stokes Theorem) Vector fields The electric field has a direction at every point in space. This is written as (x, y, z) or (r ) where r = (x, y, z). Divergence The divergence of a vector field F (x, y, z) is given by the following relation: F = F dx F y F z = lim V F ds (.6) where the double integral represents an integration across a closed surface, with enclosed volume V. The term ds represents an infinitesimally small vector locally normal to the surface with a magnitude corresponding to a differential area (infinitesimally small area) on the surface. We have also used the differential vector nabla =,,. 6

207 ds Figure.. Schematic of a spherical volume with surface S and a differential surface element vector ds locally normal to the surface. As you can see the divergence of a vector field is a number (i.e. a scalar) that represents the degree to which there is a net change in the vector length as we move along its direction. In a way it describes the outwardness of the vector field. Gauss Theorem (or Divergence Theorem) lectric fields and magnetic fields inside homogenous media represent a special class of vector fields known as conservative fields. xx Broadly speaking this means that electric fields cannot suddenly appear out of nowhere or discontinuously change direction inside a continuous medium. For such fields, any net divergence within a volume V must result in fields pointing out of the surface. This is captured by Gauss theorem: F dv = F ds (.7) where dv represents an infinitesimal volume dv=dxdydz. Curl F nˆ lim S F d s (.8) nˆ unit normal to surface Stoke s Theorem F ds F dl These and other vector relations are summarized in appendix C. (.9) xx see e.g. 7

208 dl surface Figure. 8

209 Appendix C Fourier transforms Time dependent fields can be represented as a sum of many oscillatory components, each with their individual angular frequency. In this book, we use the following convention (t) = (ω)e dω If the time dependent signal (t) is known, the Fourier amplitudes () can be determined using (ω) = π (t)e dt Here () is the Fourier transform of (t), which is also written as () = F[(t)]. In this text, the only exception to the notation shown above is the susceptibility, which we define as and conversely It can be shown that χ(ω) = χ(t)e dt χ(t) = π χ(ω)e dt F[P(t)Q(t)] = C F[P(t)] F[Q(t)] Where the symbol stands for the convolution operation. The prefactor C depends on the choice of the form of the Fourier transform. Under our definition of the Fourier transform, we have C=. The convolution operation is defined as (AB)(ω) = A(ω)B(ω ω )dω Important convolution relations are AB = BA, and A()()= A(). Here () represents the Dirac delta function, which is zero everywhere except when its argument is zero, and which is normalized according to δ(ω)dω = 9

210

211 Appendix D Optical response: formulas and definitions Permittivity (linear) ε(ω) = ε ε (ω) = ε ( χ (ω)) with r the complex dielectric function, and e the complex electric susceptibility. Permeability (linear) μ(ω) = μ μ (ω) = μ ( χ (ω)) with r the relative permeability, and m the magnetic susceptibility. Wave vector (scalar notation) k = π λ Light dispersion relation in isotropic materials k = μεω = n ω c Complex dielectric function, link to complex index in absence of magnetic effects ε (ω) = ε (ω) iε (ω) = η(ω) Complex refractive index, link to dielectric function in absence of magnetic effects η(ω) = n(ω) iκ(ω) = ε (ω) Manual conversion between complex dielectric function and refractive index: η = (n iκ) = n κ nκi = ε = ε iε ε = χ = nκ ε = χ = n κ n = ( ε ε ) κ = ( ε ε ) κ = ε n = χ n Phase velocity of an electromagnetic wave in isotropic medium: v = ω k = με Phase velocity of an electromagnetic wave in vacuum: c = μ ε

212 Refractive index in isotropic materials Absorption coefficient n = c v = μ ε Group velocity in isotropic materials: Plasma frequency for free electron gas: α = κ ω c v = dk dω ω = Ne m ε Poynting vector: instantaneous flow of optical energy per unit area. S = μ B Irradiance: magnitude of the time averaged flow of optical energy per unit area I(W/m ) = S (t) = ncε where the last step assumes a plane wave with field amplitude in an isotropic medium. lectromagnetic energy density in vacuum: u = ε μ B Reflection coefficient under normal incidence from air on planar surface R = (n ) κ (n ) κ Reflection coefficient from medium to medium under normal incidence R(ω) = η η = (n n ) (κ κ ) η η (n n ) (κ κ )

213 Appendix Springs, Masses, and Resonances When discussing movement of bound masses (bound electrons, atoms in molecules, atoms in solids) we encounter mechanical resonances. To help understand and memorize the corresponding frequencies, this appendix shows all of them on one page. One mass on fixed spring: ω = K m Two equal masses on shared spring ω = K m Two unequal masses on shared spring ω = K μ One mass on two springs ω = K m Unequal masses with shared springs on each side ω = K μ

Lecture 21 Reminder/Introduction to Wave Optics

Lecture 21 Reminder/Introduction to Wave Optics Lecture 1 Reminder/Introduction to Wave Optics Program: 1. Maxwell s Equations.. Magnetic induction and electric displacement. 3. Origins of the electric permittivity and magnetic permeability. 4. Wave

More information

Overview - Previous lecture 1/2

Overview - Previous lecture 1/2 Overview - Previous lecture 1/2 Derived the wave equation with solutions of the form We found that the polarization of the material affects wave propagation, and found the dispersion relation ω(k) with

More information

3 Constitutive Relations: Macroscopic Properties of Matter

3 Constitutive Relations: Macroscopic Properties of Matter EECS 53 Lecture 3 c Kamal Sarabandi Fall 21 All rights reserved 3 Constitutive Relations: Macroscopic Properties of Matter As shown previously, out of the four Maxwell s equations only the Faraday s and

More information

CHAPTER 9 ELECTROMAGNETIC WAVES

CHAPTER 9 ELECTROMAGNETIC WAVES CHAPTER 9 ELECTROMAGNETIC WAVES Outlines 1. Waves in one dimension 2. Electromagnetic Waves in Vacuum 3. Electromagnetic waves in Matter 4. Absorption and Dispersion 5. Guided Waves 2 Skip 9.1.1 and 9.1.2

More information

Lecture 2 Notes, Electromagnetic Theory II Dr. Christopher S. Baird, faculty.uml.edu/cbaird University of Massachusetts Lowell

Lecture 2 Notes, Electromagnetic Theory II Dr. Christopher S. Baird, faculty.uml.edu/cbaird University of Massachusetts Lowell Lecture Notes, Electromagnetic Theory II Dr. Christopher S. Baird, faculty.uml.edu/cbaird University of Massachusetts Lowell 1. Dispersion Introduction - An electromagnetic wave with an arbitrary wave-shape

More information

Review of Optical Properties of Materials

Review of Optical Properties of Materials Review of Optical Properties of Materials Review of optics Absorption in semiconductors: qualitative discussion Derivation of Optical Absorption Coefficient in Direct Semiconductors Photons When dealing

More information

INTERACTION OF LIGHT WITH MATTER

INTERACTION OF LIGHT WITH MATTER INTERACTION OF LIGHT WITH MATTER Already.the speed of light can be related to the permittivity, ε and the magnetic permeability, µ of the material by Rememberε = ε r ε 0 and µ = µ r µ 0 where ε 0 = 8.85

More information

Solution Set 2 Phys 4510 Optics Fall 2014

Solution Set 2 Phys 4510 Optics Fall 2014 Solution Set Phys 4510 Optics Fall 014 Due date: Tu, September 16, in class Scoring rubric 4 points/sub-problem, total: 40 points 3: Small mistake in calculation or formula : Correct formula but calculation

More information

Laser Cooling and Trapping of Atoms

Laser Cooling and Trapping of Atoms Chapter 2 Laser Cooling and Trapping of Atoms Since its conception in 1975 [71, 72] laser cooling has revolutionized the field of atomic physics research, an achievement that has been recognized by the

More information

Optical Properties of Solid from DFT

Optical Properties of Solid from DFT Optical Properties of Solid from DFT 1 Prof.P. Ravindran, Department of Physics, Central University of Tamil Nadu, India & Center for Materials Science and Nanotechnology, University of Oslo, Norway http://folk.uio.no/ravi/cmt15

More information

Supplementary Figure 1 Schematics of an optical pulse in a nonlinear medium. A Gaussian optical pulse propagates along z-axis in a nonlinear medium

Supplementary Figure 1 Schematics of an optical pulse in a nonlinear medium. A Gaussian optical pulse propagates along z-axis in a nonlinear medium Supplementary Figure 1 Schematics of an optical pulse in a nonlinear medium. A Gaussian optical pulse propagates along z-axis in a nonlinear medium with thickness L. Supplementary Figure Measurement of

More information

Basics of electromagnetic response of materials

Basics of electromagnetic response of materials Basics of electromagnetic response of materials Microscopic electric and magnetic field Let s point charge q moving with velocity v in fields e and b Force on q: F e F qeqvb F m Lorenz force Microscopic

More information

Electromagnetic Waves

Electromagnetic Waves Nicholas J. Giordano www.cengage.com/physics/giordano Chapter 23 Electromagnetic Waves Marilyn Akins, PhD Broome Community College Electromagnetic Theory Theoretical understanding of electricity and magnetism

More information

Macroscopic dielectric theory

Macroscopic dielectric theory Macroscopic dielectric theory Maxwellʼs equations E = 1 c E =4πρ B t B = 4π c J + 1 c B = E t In a medium it is convenient to explicitly introduce induced charges and currents E = 1 B c t D =4πρ H = 4π

More information

Optics Definitions. The apparent movement of one object relative to another due to the motion of the observer is called parallax.

Optics Definitions. The apparent movement of one object relative to another due to the motion of the observer is called parallax. Optics Definitions Reflection is the bouncing of light off an object Laws of Reflection of Light: 1. The incident ray, the normal at the point of incidence and the reflected ray all lie in the same plane.

More information

Condensed Matter Physics Prof. G. Rangarajan Department of Physics Indian Institute of Technology, Madras

Condensed Matter Physics Prof. G. Rangarajan Department of Physics Indian Institute of Technology, Madras Condensed Matter Physics Prof. G. Rangarajan Department of Physics Indian Institute of Technology, Madras Lecture - 10 The Free Electron Theory of Metals - Electrical Conductivity (Refer Slide Time: 00:20)

More information

van Quantum tot Molecuul

van Quantum tot Molecuul 10 HC10: Molecular and vibrational spectroscopy van Quantum tot Molecuul Dr Juan Rojo VU Amsterdam and Nikhef Theory Group http://www.juanrojo.com/ j.rojo@vu.nl Molecular and Vibrational Spectroscopy Based

More information

Spectroscopy in frequency and time domains

Spectroscopy in frequency and time domains 5.35 Module 1 Lecture Summary Fall 1 Spectroscopy in frequency and time domains Last time we introduced spectroscopy and spectroscopic measurement. I. Emphasized that both quantum and classical views of

More information

Optical Properties of Semiconductors. Prof.P. Ravindran, Department of Physics, Central University of Tamil Nadu, India

Optical Properties of Semiconductors. Prof.P. Ravindran, Department of Physics, Central University of Tamil Nadu, India Optical Properties of Semiconductors 1 Prof.P. Ravindran, Department of Physics, Central University of Tamil Nadu, India http://folk.uio.no/ravi/semi2013 Light Matter Interaction Response to external electric

More information

( ) /, so that we can ignore all

( ) /, so that we can ignore all Physics 531: Atomic Physics Problem Set #5 Due Wednesday, November 2, 2011 Problem 1: The ac-stark effect Suppose an atom is perturbed by a monochromatic electric field oscillating at frequency ω L E(t)

More information

4. The interaction of light with matter

4. The interaction of light with matter 4. The interaction of light with matter The propagation of light through chemical materials is described by a wave equation similar to the one that describes light travel in a vacuum (free space). Again,

More information

sin[( t 2 Home Problem Set #1 Due : September 10 (Wed), 2008

sin[( t 2 Home Problem Set #1 Due : September 10 (Wed), 2008 Home Problem Set #1 Due : September 10 (Wed), 008 1. Answer the following questions related to the wave-particle duality. (a) When an electron (mass m) is moving with the velocity of υ, what is the wave

More information

nano.tul.cz Inovace a rozvoj studia nanomateriálů na TUL

nano.tul.cz Inovace a rozvoj studia nanomateriálů na TUL Inovace a rozvoj studia nanomateriálů na TUL nano.tul.cz Tyto materiály byly vytvořeny v rámci projektu ESF OP VK: Inovace a rozvoj studia nanomateriálů na Technické univerzitě v Liberci Units for the

More information

Summary of Beam Optics

Summary of Beam Optics Summary of Beam Optics Gaussian beams, waves with limited spatial extension perpendicular to propagation direction, Gaussian beam is solution of paraxial Helmholtz equation, Gaussian beam has parabolic

More information

10. Optics of metals - plasmons

10. Optics of metals - plasmons 1. Optics of metals - plasmons Drude theory at higher frequencies The Drude scattering time corresponds to the frictional damping rate The ultraviolet transparency of metals Interface waves - surface plasmons

More information

Chapter 9. Electromagnetic waves

Chapter 9. Electromagnetic waves Chapter 9. lectromagnetic waves 9.1.1 The (classical or Mechanical) waves equation Given the initial shape of the string, what is the subsequent form, The displacement at point z, at the later time t,

More information

Lecture 3: Optical Properties of Insulators, Semiconductors, and Metals. 5 nm

Lecture 3: Optical Properties of Insulators, Semiconductors, and Metals. 5 nm Metals Lecture 3: Optical Properties of Insulators, Semiconductors, and Metals 5 nm Course Info Next Week (Sept. 5 and 7) no classes First H/W is due Sept. 1 The Previous Lecture Origin frequency dependence

More information

Physics 221B Spring 2012 Notes 42 Scattering of Radiation by Matter

Physics 221B Spring 2012 Notes 42 Scattering of Radiation by Matter Physics 221B Spring 2012 Notes 42 Scattering of Radiation by Matter 1. Introduction In the previous set of Notes we treated the emission and absorption of radiation by matter. In these Notes we turn to

More information

LASERS. Amplifiers: Broad-band communications (avoid down-conversion)

LASERS. Amplifiers: Broad-band communications (avoid down-conversion) L- LASERS Representative applications: Amplifiers: Broad-band communications (avoid down-conversion) Oscillators: Blasting: Energy States: Hydrogen atom Frequency/distance reference, local oscillators,

More information

NONLINEAR OPTICS. Ch. 1 INTRODUCTION TO NONLINEAR OPTICS

NONLINEAR OPTICS. Ch. 1 INTRODUCTION TO NONLINEAR OPTICS NONLINEAR OPTICS Ch. 1 INTRODUCTION TO NONLINEAR OPTICS Nonlinear regime - Order of magnitude Origin of the nonlinearities - Induced Dipole and Polarization - Description of the classical anharmonic oscillator

More information

The Interaction of Light and Matter: α and n

The Interaction of Light and Matter: α and n The Interaction of Light and Matter: α and n The interaction of light and matter is what makes life interesting. Everything we see is the result of this interaction. Why is light absorbed or transmitted

More information

1 Fundamentals of laser energy absorption

1 Fundamentals of laser energy absorption 1 Fundamentals of laser energy absorption 1.1 Classical electromagnetic-theory concepts 1.1.1 Electric and magnetic properties of materials Electric and magnetic fields can exert forces directly on atoms

More information

Electromagnetic fields and waves

Electromagnetic fields and waves Electromagnetic fields and waves Maxwell s rainbow Outline Maxwell s equations Plane waves Pulses and group velocity Polarization of light Transmission and reflection at an interface Macroscopic Maxwell

More information

J10M.1 - Rod on a Rail (M93M.2)

J10M.1 - Rod on a Rail (M93M.2) Part I - Mechanics J10M.1 - Rod on a Rail (M93M.2) J10M.1 - Rod on a Rail (M93M.2) s α l θ g z x A uniform rod of length l and mass m moves in the x-z plane. One end of the rod is suspended from a straight

More information

Molecular spectroscopy

Molecular spectroscopy Molecular spectroscopy Origin of spectral lines = absorption, emission and scattering of a photon when the energy of a molecule changes: rad( ) M M * rad( ' ) ' v' 0 0 absorption( ) emission ( ) scattering

More information

Rb, which had been compressed to a density of 1013

Rb, which had been compressed to a density of 1013 Modern Physics Study Questions for the Spring 2018 Departmental Exam December 3, 2017 1. An electron is initially at rest in a uniform electric field E in the negative y direction and a uniform magnetic

More information

DEFINITIONS. Linear Motion. Conservation of Momentum. Vectors and Scalars. Circular Motion. Newton s Laws of Motion

DEFINITIONS. Linear Motion. Conservation of Momentum. Vectors and Scalars. Circular Motion. Newton s Laws of Motion DEFINITIONS Linear Motion Mass: The mass of a body is the amount of matter in it. Displacement: The displacement of a body from a point is its distance from a point in a given direction. Velocity: The

More information

Theory of Electromagnetic Fields

Theory of Electromagnetic Fields Theory of Electromagnetic Fields Andrzej Wolski University of Liverpool, and the Cockcroft Institute, UK Abstract We discuss the theory of electromagnetic fields, with an emphasis on aspects relevant to

More information

II Theory Of Surface Plasmon Resonance (SPR)

II Theory Of Surface Plasmon Resonance (SPR) II Theory Of Surface Plasmon Resonance (SPR) II.1 Maxwell equations and dielectric constant of metals Surface Plasmons Polaritons (SPP) exist at the interface of a dielectric and a metal whose electrons

More information

Electrodynamics I Final Exam - Part A - Closed Book KSU 2005/12/12 Electro Dynamic

Electrodynamics I Final Exam - Part A - Closed Book KSU 2005/12/12 Electro Dynamic Electrodynamics I Final Exam - Part A - Closed Book KSU 2005/12/12 Name Electro Dynamic Instructions: Use SI units. Short answers! No derivations here, just state your responses clearly. 1. (2) Write an

More information

Semiconductor Physics and Devices

Semiconductor Physics and Devices Introduction to Quantum Mechanics In order to understand the current-voltage characteristics, we need some knowledge of electron behavior in semiconductor when the electron is subjected to various potential

More information

qq k d Chapter 16 Electric and Magnetic Forces Electric charge Electric charges Negative (electron) Positive (proton)

qq k d Chapter 16 Electric and Magnetic Forces Electric charge Electric charges Negative (electron) Positive (proton) Chapter 16 Electric and Magnetic Forces Electric charge Electric charges Negative (electron) Positive (proton) Electrons and protons in atoms/molecules Ions: atoms/molecules with excess of charge Ions

More information

Density of states for electrons and holes. Distribution function. Conduction and valence bands

Density of states for electrons and holes. Distribution function. Conduction and valence bands Intrinsic Semiconductors In the field of semiconductors electrons and holes are usually referred to as free carriers, or simply carriers, because it is these particles which are responsible for carrying

More information

PAPER No. : 8 (PHYSICAL SPECTROSCOPY) MODULE No. : 5 (TRANSITION PROBABILITIES AND TRANSITION DIPOLE MOMENT. OVERVIEW OF SELECTION RULES)

PAPER No. : 8 (PHYSICAL SPECTROSCOPY) MODULE No. : 5 (TRANSITION PROBABILITIES AND TRANSITION DIPOLE MOMENT. OVERVIEW OF SELECTION RULES) Subject Chemistry Paper No and Title Module No and Title Module Tag 8 and Physical Spectroscopy 5 and Transition probabilities and transition dipole moment, Overview of selection rules CHE_P8_M5 TABLE

More information

Nonlinear Electrodynamics and Optics of Graphene

Nonlinear Electrodynamics and Optics of Graphene Nonlinear Electrodynamics and Optics of Graphene S. A. Mikhailov and N. A. Savostianova University of Augsburg, Institute of Physics, Universitätsstr. 1, 86159 Augsburg, Germany E-mail: sergey.mikhailov@physik.uni-augsburg.de

More information

Laser Physics OXFORD UNIVERSITY PRESS SIMON HOOKER COLIN WEBB. and. Department of Physics, University of Oxford

Laser Physics OXFORD UNIVERSITY PRESS SIMON HOOKER COLIN WEBB. and. Department of Physics, University of Oxford Laser Physics SIMON HOOKER and COLIN WEBB Department of Physics, University of Oxford OXFORD UNIVERSITY PRESS Contents 1 Introduction 1.1 The laser 1.2 Electromagnetic radiation in a closed cavity 1.2.1

More information

Atomic Structure and Processes

Atomic Structure and Processes Chapter 5 Atomic Structure and Processes 5.1 Elementary atomic structure Bohr Orbits correspond to principal quantum number n. Hydrogen atom energy levels where the Rydberg energy is R y = m e ( e E n

More information

H ( E) E ( H) = H B t

H ( E) E ( H) = H B t Chapter 5 Energy and Momentum The equations established so far describe the behavior of electric and magnetic fields. They are a direct consequence of Maxwell s equations and the properties of matter.

More information

Absorption and scattering

Absorption and scattering Absorption and scattering When a beam of radiation goes through the atmosphere, it encounters gas molecules, aerosols, cloud droplets, and ice crystals. These objects perturb the radiation field. Part

More information

Lasers PH 645/ OSE 645/ EE 613 Summer 2010 Section 1: T/Th 2:45-4:45 PM Engineering Building 240

Lasers PH 645/ OSE 645/ EE 613 Summer 2010 Section 1: T/Th 2:45-4:45 PM Engineering Building 240 Lasers PH 645/ OSE 645/ EE 613 Summer 2010 Section 1: T/Th 2:45-4:45 PM Engineering Building 240 John D. Williams, Ph.D. Department of Electrical and Computer Engineering 406 Optics Building - UAHuntsville,

More information

Introduction. Chapter Plasma: definitions

Introduction. Chapter Plasma: definitions Chapter 1 Introduction 1.1 Plasma: definitions A plasma is a quasi-neutral gas of charged and neutral particles which exhibits collective behaviour. An equivalent, alternative definition: A plasma is a

More information

MOLECULAR SPECTROSCOPY

MOLECULAR SPECTROSCOPY MOLECULAR SPECTROSCOPY First Edition Jeanne L. McHale University of Idaho PRENTICE HALL, Upper Saddle River, New Jersey 07458 CONTENTS PREFACE xiii 1 INTRODUCTION AND REVIEW 1 1.1 Historical Perspective

More information

A. F. J. Levi 1 EE539: Engineering Quantum Mechanics. Fall 2017.

A. F. J. Levi 1 EE539: Engineering Quantum Mechanics. Fall 2017. A. F. J. Levi 1 Engineering Quantum Mechanics. Fall 2017. TTh 9.00 a.m. 10.50 a.m., VHE 210. Web site: http://alevi.usc.edu Web site: http://classes.usc.edu/term-20173/classes/ee EE539: Abstract and Prerequisites

More information

Optics and Optical Design. Chapter 5: Electromagnetic Optics. Lectures 9 & 10

Optics and Optical Design. Chapter 5: Electromagnetic Optics. Lectures 9 & 10 Optics and Optical Design Chapter 5: Electromagnetic Optics Lectures 9 & 1 Cord Arnold / Anne L Huillier Electromagnetic waves in dielectric media EM optics compared to simpler theories Electromagnetic

More information

B.Tech. First Semester Examination Physics-1 (PHY-101F)

B.Tech. First Semester Examination Physics-1 (PHY-101F) B.Tech. First Semester Examination Physics-1 (PHY-101F) Note : Attempt FIVE questions in all taking least two questions from each Part. All questions carry equal marks Part-A Q. 1. (a) What are Newton's

More information

Lecture 0. NC State University

Lecture 0. NC State University Chemistry 736 Lecture 0 Overview NC State University Overview of Spectroscopy Electronic states and energies Transitions between states Absorption and emission Electronic spectroscopy Instrumentation Concepts

More information

Prentice Hall. Physics: Principles with Applications, Updated 6th Edition (Giancoli) High School

Prentice Hall. Physics: Principles with Applications, Updated 6th Edition (Giancoli) High School Prentice Hall Physics: Principles with Applications, Updated 6th Edition (Giancoli) 2009 High School C O R R E L A T E D T O Physics I Students should understand that scientific knowledge is gained from

More information

1 Fundamentals. 1.1 Overview. 1.2 Units: Physics 704 Spring 2018

1 Fundamentals. 1.1 Overview. 1.2 Units: Physics 704 Spring 2018 Physics 704 Spring 2018 1 Fundamentals 1.1 Overview The objective of this course is: to determine and fields in various physical systems and the forces and/or torques resulting from them. The domain of

More information

Light in Matter (Hecht Ch. 3)

Light in Matter (Hecht Ch. 3) Phys 531 Lecture 3 9 September 2004 Light in Matter (Hecht Ch. 3) Last time, talked about light in vacuum: Maxwell equations wave equation Light = EM wave 1 Today: What happens inside material? typical

More information

Notes on excitation of an atom or molecule by an electromagnetic wave field. F. Lanni / 11feb'12 / rev9sept'14

Notes on excitation of an atom or molecule by an electromagnetic wave field. F. Lanni / 11feb'12 / rev9sept'14 Notes on excitation of an atom or molecule by an electromagnetic wave field. F. Lanni / 11feb'12 / rev9sept'14 Because the wavelength of light (400-700nm) is much greater than the diameter of an atom (0.07-0.35

More information

Identical Particles. Bosons and Fermions

Identical Particles. Bosons and Fermions Identical Particles Bosons and Fermions In Quantum Mechanics there is no difference between particles and fields. The objects which we refer to as fields in classical physics (electromagnetic field, field

More information

Fundamentals of Spectroscopy for Optical Remote Sensing. Course Outline 2009

Fundamentals of Spectroscopy for Optical Remote Sensing. Course Outline 2009 Fundamentals of Spectroscopy for Optical Remote Sensing Course Outline 2009 Part I. Fundamentals of Quantum Mechanics Chapter 1. Concepts of Quantum and Experimental Facts 1.1. Blackbody Radiation and

More information

PRINCIPLES OF NONLINEAR OPTICAL SPECTROSCOPY

PRINCIPLES OF NONLINEAR OPTICAL SPECTROSCOPY PRINCIPLES OF NONLINEAR OPTICAL SPECTROSCOPY Shaul Mukamel University of Rochester Rochester, New York New York Oxford OXFORD UNIVERSITY PRESS 1995 Contents 1. Introduction 3 Linear versus Nonlinear Spectroscopy

More information

Plasma Physics Prof. V. K. Tripathi Department of Physics Indian Institute of Technology, Delhi

Plasma Physics Prof. V. K. Tripathi Department of Physics Indian Institute of Technology, Delhi Plasma Physics Prof. V. K. Tripathi Department of Physics Indian Institute of Technology, Delhi Lecture No. # 09 Electromagnetic Wave Propagation Inhomogeneous Plasma (Refer Slide Time: 00:33) Today, I

More information

Physics 9e/Cutnell. correlated to the. College Board AP Physics 2 Course Objectives

Physics 9e/Cutnell. correlated to the. College Board AP Physics 2 Course Objectives correlated to the College Board AP Physics 2 Course Objectives Big Idea 1: Objects and systems have properties such as mass and charge. Systems may have internal structure. Enduring Understanding 1.A:

More information

Chap. 1 Fundamental Concepts

Chap. 1 Fundamental Concepts NE 2 Chap. 1 Fundamental Concepts Important Laws in Electromagnetics Coulomb s Law (1785) Gauss s Law (1839) Ampere s Law (1827) Ohm s Law (1827) Kirchhoff s Law (1845) Biot-Savart Law (1820) Faradays

More information

Chem 442 Review of Spectroscopy

Chem 442 Review of Spectroscopy Chem 44 Review of Spectroscopy General spectroscopy Wavelength (nm), frequency (s -1 ), wavenumber (cm -1 ) Frequency (s -1 ): n= c l Wavenumbers (cm -1 ): n =1 l Chart of photon energies and spectroscopies

More information

What Makes a Laser Light Amplification by Stimulated Emission of Radiation Main Requirements of the Laser Laser Gain Medium (provides the light

What Makes a Laser Light Amplification by Stimulated Emission of Radiation Main Requirements of the Laser Laser Gain Medium (provides the light What Makes a Laser Light Amplification by Stimulated Emission of Radiation Main Requirements of the Laser Laser Gain Medium (provides the light amplification) Optical Resonator Cavity (greatly increase

More information

Chapter 11: Dielectric Properties of Materials

Chapter 11: Dielectric Properties of Materials Chapter 11: Dielectric Properties of Materials Lindhardt January 30, 2017 Contents 1 Classical Dielectric Response of Materials 2 1.1 Conditions on ɛ............................. 4 1.2 Kramer s Kronig

More information

Wave Motion and Sound

Wave Motion and Sound Wave Motion and Sound 1. A back and forth motion that repeats itself is a a. Spring b. Vibration c. Wave d. Pulse 2. The number of vibrations that occur in 1 second is called a. A Period b. Frequency c.

More information

Modern Physics Departmental Exam Last updated November 2013

Modern Physics Departmental Exam Last updated November 2013 Modern Physics Departmental Exam Last updated November 213 87 1. Recently, 2 rubidium atoms ( 37 Rb ), which had been compressed to a density of 113 atoms/cm 3, were observed to undergo a Bose-Einstein

More information

Spectral Broadening Mechanisms

Spectral Broadening Mechanisms Spectral Broadening Mechanisms Lorentzian broadening (Homogeneous) Gaussian broadening (Inhomogeneous, Inertial) Doppler broadening (special case for gas phase) The Fourier Transform NC State University

More information

2016 Lloyd G. Elliott University Prize Exam Compiled by the Department of Physics & Astronomy, University of Waterloo

2016 Lloyd G. Elliott University Prize Exam Compiled by the Department of Physics & Astronomy, University of Waterloo Canadian Association of Physicists SUPPORTING PHYSICS RESEARCH AND EDUCATION IN CANADA 2016 Lloyd G. Elliott University Prize Exam Compiled by the Department of Physics & Astronomy, University of Waterloo

More information

Chap. 7. Dielectric Materials and Insulation

Chap. 7. Dielectric Materials and Insulation Chap. 7. Dielectric Materials and Insulation - The parallel plate capacitor with free space as an insulator: - The electric dipole moment for a pair of opposite changes +Q and -Q separated by a finite

More information

Atomic Structure. Chapter 8

Atomic Structure. Chapter 8 Atomic Structure Chapter 8 Overview To understand atomic structure requires understanding a special aspect of the electron - spin and its related magnetism - and properties of a collection of identical

More information

Physics Important Terms and their Definitions

Physics Important Terms and their Definitions Physics Important Terms and their S.No Word Meaning 1 Acceleration The rate of change of velocity of an object with respect to time 2 Angular Momentum A measure of the momentum of a body in rotational

More information

Unit III Free Electron Theory Engineering Physics

Unit III Free Electron Theory Engineering Physics . Introduction The electron theory of metals aims to explain the structure and properties of solids through their electronic structure. The electron theory is applicable to all solids i.e., both metals

More information

Skoog Chapter 6 Introduction to Spectrometric Methods

Skoog Chapter 6 Introduction to Spectrometric Methods Skoog Chapter 6 Introduction to Spectrometric Methods General Properties of Electromagnetic Radiation (EM) Wave Properties of EM Quantum Mechanical Properties of EM Quantitative Aspects of Spectrochemical

More information

An Introduction to Diffraction and Scattering. School of Chemistry The University of Sydney

An Introduction to Diffraction and Scattering. School of Chemistry The University of Sydney An Introduction to Diffraction and Scattering Brendan J. Kennedy School of Chemistry The University of Sydney 1) Strong forces 2) Weak forces Types of Forces 3) Electromagnetic forces 4) Gravity Types

More information

B2.III Revision notes: quantum physics

B2.III Revision notes: quantum physics B.III Revision notes: quantum physics Dr D.M.Lucas, TT 0 These notes give a summary of most of the Quantum part of this course, to complement Prof. Ewart s notes on Atomic Structure, and Prof. Hooker s

More information

Advanced Optical Communications Prof. R. K. Shevgaonkar Department of Electrical Engineering Indian Institute of Technology, Bombay

Advanced Optical Communications Prof. R. K. Shevgaonkar Department of Electrical Engineering Indian Institute of Technology, Bombay Advanced Optical Communications Prof. R. K. Shevgaonkar Department of Electrical Engineering Indian Institute of Technology, Bombay Lecture No. # 15 Laser - I In the last lecture, we discussed various

More information

The Dielectric Function of a Metal ( Jellium )

The Dielectric Function of a Metal ( Jellium ) The Dielectric Function of a Metal ( Jellium ) Total reflection Plasma frequency p (10 15 Hz range) Why are Metals Shiny? An electric field cannot exist inside a metal, because metal electrons follow the

More information

A few principles of classical and quantum mechanics

A few principles of classical and quantum mechanics A few principles of classical and quantum mechanics The classical approach: In classical mechanics, we usually (but not exclusively) solve Newton s nd law of motion relating the acceleration a of the system

More information

CHEM Atomic and Molecular Spectroscopy

CHEM Atomic and Molecular Spectroscopy CHEM 21112 Atomic and Molecular Spectroscopy References: 1. Fundamentals of Molecular Spectroscopy by C.N. Banwell 2. Physical Chemistry by P.W. Atkins Dr. Sujeewa De Silva Sub topics Light and matter

More information

Electromagnetic Waves

Electromagnetic Waves Electromagnetic Waves Our discussion on dynamic electromagnetic field is incomplete. I H E An AC current induces a magnetic field, which is also AC and thus induces an AC electric field. H dl Edl J ds

More information

8 Wavefunctions - Schrödinger s Equation

8 Wavefunctions - Schrödinger s Equation 8 Wavefunctions - Schrödinger s Equation So far we have considered only free particles - i.e. particles whose energy consists entirely of its kinetic energy. In general, however, a particle moves under

More information

FIBER OPTICS. Prof. R.K. Shevgaonkar. Department of Electrical Engineering. Indian Institute of Technology, Bombay. Lecture: 15. Optical Sources-LASER

FIBER OPTICS. Prof. R.K. Shevgaonkar. Department of Electrical Engineering. Indian Institute of Technology, Bombay. Lecture: 15. Optical Sources-LASER FIBER OPTICS Prof. R.K. Shevgaonkar Department of Electrical Engineering Indian Institute of Technology, Bombay Lecture: 15 Optical Sources-LASER Fiber Optics, Prof. R.K. Shevgaonkar, Dept. of Electrical

More information

Optical Lattices. Chapter Polarization

Optical Lattices. Chapter Polarization Chapter Optical Lattices Abstract In this chapter we give details of the atomic physics that underlies the Bose- Hubbard model used to describe ultracold atoms in optical lattices. We show how the AC-Stark

More information

Physics 221A Fall 1996 Notes 19 The Stark Effect in Hydrogen and Alkali Atoms

Physics 221A Fall 1996 Notes 19 The Stark Effect in Hydrogen and Alkali Atoms Physics 221A Fall 1996 Notes 19 The Stark Effect in Hydrogen and Alkali Atoms In these notes we will consider the Stark effect in hydrogen and alkali atoms as a physically interesting example of bound

More information

Light and Matter. Thursday, 8/31/2006 Physics 158 Peter Beyersdorf. Document info

Light and Matter. Thursday, 8/31/2006 Physics 158 Peter Beyersdorf. Document info Light and Matter Thursday, 8/31/2006 Physics 158 Peter Beyersdorf Document info 3. 1 1 Class Outline Common materials used in optics Index of refraction absorption Classical model of light absorption Light

More information

High School Curriculum Standards: Physics

High School Curriculum Standards: Physics High School Curriculum Standards: Physics Students will understand and apply scientific concepts, principles, and theories pertaining to the physical setting and living environment and recognize the historical

More information

Electromagnetic optics!

Electromagnetic optics! 1 EM theory Electromagnetic optics! EM waves Monochromatic light 2 Electromagnetic optics! Electromagnetic theory of light Electromagnetic waves in dielectric media Monochromatic light References: Fundamentals

More information

24/ Rayleigh and Raman scattering. Stokes and anti-stokes lines. Rotational Raman spectroscopy. Polarizability ellipsoid. Selection rules.

24/ Rayleigh and Raman scattering. Stokes and anti-stokes lines. Rotational Raman spectroscopy. Polarizability ellipsoid. Selection rules. Subject Chemistry Paper No and Title Module No and Title Module Tag 8/ Physical Spectroscopy 24/ Rayleigh and Raman scattering. Stokes and anti-stokes lines. Rotational Raman spectroscopy. Polarizability

More information

Module 4 : Third order nonlinear optical processes. Lecture 28 : Inelastic Scattering Processes. Objectives

Module 4 : Third order nonlinear optical processes. Lecture 28 : Inelastic Scattering Processes. Objectives Module 4 : Third order nonlinear optical processes Lecture 28 : Inelastic Scattering Processes Objectives In this lecture you will learn the following Light scattering- elastic and inelastic-processes,

More information

( ) x10 8 m. The energy in a mole of 400 nm photons is calculated by: ' & sec( ) ( & % ) 6.022x10 23 photons' E = h! = hc & 6.

( ) x10 8 m. The energy in a mole of 400 nm photons is calculated by: ' & sec( ) ( & % ) 6.022x10 23 photons' E = h! = hc & 6. Introduction to Spectroscopy Spectroscopic techniques are widely used to detect molecules, to measure the concentration of a species in solution, and to determine molecular structure. For proteins, most

More information

Plasmonics: elementary excitation of a plasma (gas of free charges) nano-scale optics done with plasmons at metal interfaces

Plasmonics: elementary excitation of a plasma (gas of free charges) nano-scale optics done with plasmons at metal interfaces Plasmonics Plasmon: Plasmonics: elementary excitation of a plasma (gas of free charges) nano-scale optics done with plasmons at metal interfaces Femius Koenderink Center for Nanophotonics AMOLF, Amsterdam

More information

NPTEL/IITM. Molecular Spectroscopy Lectures 1 & 2. Prof.K. Mangala Sunder Page 1 of 15. Topics. Part I : Introductory concepts Topics

NPTEL/IITM. Molecular Spectroscopy Lectures 1 & 2. Prof.K. Mangala Sunder Page 1 of 15. Topics. Part I : Introductory concepts Topics Molecular Spectroscopy Lectures 1 & 2 Part I : Introductory concepts Topics Why spectroscopy? Introduction to electromagnetic radiation Interaction of radiation with matter What are spectra? Beer-Lambert

More information

Chapter 2. Dielectric Theories

Chapter 2. Dielectric Theories Chapter Dielectric Theories . Dielectric Theories 1.1. Introduction Measurements of dielectric properties of materials is very important because it provide vital information regarding the material characteristics,

More information

NYS STANDARD/KEY IDEA/PERFORMANCE INDICATOR 5.1 a-e. 5.1a Measured quantities can be classified as either vector or scalar.

NYS STANDARD/KEY IDEA/PERFORMANCE INDICATOR 5.1 a-e. 5.1a Measured quantities can be classified as either vector or scalar. INDICATOR 5.1 a-e September Unit 1 Units and Scientific Notation SI System of Units Unit Conversion Scientific Notation Significant Figures Graphical Analysis Unit Kinematics Scalar vs. vector Displacement/dis

More information

Problem Set 5: Solutions

Problem Set 5: Solutions University of Alabama Department of Physics and Astronomy PH 53 / eclair Spring 1 Problem Set 5: Solutions 1. Solve one of the exam problems that you did not choose.. The Thompson model of the atom. Show

More information