transitions in matter. This gives us the beauty of color and is the reason why our eyes

Transcription

1 8 CHAPTER 1. FOUNDATIONS OF NANO-OPTICS 1.3 Macroscopic Electrodynamics Light embraces the most fascinating spectrum of electromagnetic radiation. This is mainly due to the fact that the energy of light quanta èphotonsè lie in the energy range of electronic transitions in matter. This gives us the beauty of color and is the reason why our eyes adapted to sense the optical spectrum. Light is also fascinating because it manifests itself in forms of waves and particles. In no other range of the electromagnetic spectrum we are more confronted with the wave-particle duality than in the optical regime. While long wavelength radiation èradiofrequencies, microwavesè is well described by wave theory, short wavelength radiation èx-raysè exhibits mostly particle properties. The two worlds meet in the optical regime. To describe optical radiation in nano-optics it is mostly suæcient to adapt the wave picture. This allows us to use classical æeld theory based on Maxwell's equations. Of course, in nano-optics, the systems with which the light æelds interact are small èsingle molecules, quantum dotsè which necessitates a quantum description of the material properties. Thus, in most cases we can use the framework of semiclassical theory which combines the classical picture of æelds and the quantum picture of matter. However, occasionally, we have to go beyond the semiclassical description. For example the photons emitted by a quantum system can obey non-classical photon statistics in form of photon-antibunching. This section summarizes the fundamentals of electromagnetic theory forming the necessary basis for this course. Only the basic properties are discussed and for more detailed treatments the reader is referred to the books of Chew ë27ë, Jackson ë30ë, Stratton ë31ë, and others. The starting point are Maxwell's equations established by James C. Maxwell in Maxwell's equations In diæerential form and in SI units the macroscopic Maxwell's equations read as ræeèr;tè ræhèr;tè rædèr;tè = çèr;tè ; ; è1.1è + jèr;tè ; è1.2è è1.3è ræbèr;tè = 0 : è1.4è where E denotes the electric æeld, D the electric displacement, H the magnetic æeld, B the magnetic induction, j the current density, and ç the charge density. The components of these vector and scalar æelds constitute a set of 16 unknowns. Depending on the considered medium, the number of unknowns can be reduced considerably. For example, in linear, isotropic, homogeneous and source free media the electromagnetic æeld is entirely deæned by two scalar æelds. Maxwell's equations combine and complete the laws formerly established by Faraday, Ampere, Gauss, Poisson, and others. Since Maxwell's equations are diæerential

2 1.3. MACROSCOPIC ELECTRODYNAMICS 9 equations they do not account for any æelds that are constant in space and time. Any such æeld can therefore be added to the æelds. It has to be emphasized that the concept of æelds was introduced to explain the transmission of forces from a source to a receiver. The physical observables are therefore forces, whereas the æelds are deænitions introduced to explain the troublesome phenomenon of the `action at a distance'. Notice, that the macroscopic Maxwell equations deal with æelds which are local spatial averages over microscopic æelds associated with discrete charges. Hence, the microscopic nature of matter is not included in the macroscopic æelds. Charge and current densities are considered as continuous functions of space. In order to describe the æelds on an atomic scale it is necessary to use the microscopic Maxwell equations which consider all matter to be made of charged and uncharged particles. The conservation of charge is implicitly contained in Maxwell's equations. Taking the divergence of Eq. 1.2, noting that ræræh is identical zero, and substituting Eq. 1.3 for ræd one obtains the continuity equation = 0 : è1.5è The electromagnetic properties of the medium are most commonly discussed in terms of the macroscopic polarization P and magnetization M according to Dèr;tè = " o Eèr;tè+Pèr;tè ; è1.6è Hèr;tè = ç,1 o Bèr;tè, Mèr;tè ; è1.7è where " o and ç o are the permittivity and the permeability, respectively. These equations do not impose any conditions on the medium and are therefore always valid Wave equations After substituting the æelds D and B in Maxwell's curl equations by the expressions 1.6 and 1.7 and combining the two resulting equations we obtain the inhomogeneous wave equations ræræe =,ç c 2 2 o j ç + ræm ; è1.8è ræræh + 2 H = ræj + c o : è1.9è The constant c was introduced for è" o ç o è,1=2 and is known as the vacuum speed of light. The expression in the brackets of Eq. 1.8 can be associated with the total current density j t = j s + j c + ræm ; è1.10è where j has been split into a source current density j s and an induced conduction current density j c. The and ræm are recognized as the polarization current density and the magnetization current density, respectively. The wave equations as stated in Eqs. 1.8, 1.9 do not impose any conditions to the media considered and hence are generally valid.

3 10 CHAPTER 1. FOUNDATIONS OF NANO-OPTICS Constitutive relations Maxwell's equations deæne the æelds that are generated by currents and charges in matter. However, they do not describe how these currents and charges are generated. Thus, to ænd a self-consitent solution for the electromagnetic æeld, Maxwell's equations must be supplemented by relations which describe the behavior of matter under the inæuence of the æelds. These material equations are known as. In a non-dispersive linear and isotropic medium they have the form D = " o "E èp = " o ç e Eè ; è1.11è B = ç o çh èm = ç m Hè ; è1.12è j c = çe : è1.13è For non-linear media, the right-hand sides can be supplemented by terms of higher power, but most materials behave linear if the æelds are not too strong. Anisotropic media can be considered using tensorial forms for " and ç. In order to account for general bianisotropic media, additional terms relating D and E to both B and H have tobeintroduced. For such complex media, solutions to the wave equations can be found for very special situations only. The constituent relations given above account for inhomogeneous media if the material parameters ", ç and ç are functions of space. The medium is called temporally dispersive if the material parameters are functions of frequency, and spatially dispersive if the constitutive relations are convolutions over space. An electromagnetic æeld in a linear medium can be written as a superposition of monochromatic æelds of the form 1 Eèr;tè = Eèk;!è cosèkær,!tè ; è1.14è where k and! are the wavevector and the angular frequency, respectively. In its most general form, the amplitude of the induced displacement Dèr;tè can be written as Dèk;!è= " o "èk;!è Eèk;!è : è1.15è Since Eèk;!è is equivalent to the Fourier transform ^E of an arbitrary time-dependent æeld Eèr;tè, we can apply the inverse Fourier transform to Eq and obtain Dèr;tè=" o ~"èr,r 0 ;t,t 0 è Eèr 0 ;t 0 è dr 0 dt 0 : è1.16è Here, ~" denotes the inverse Fourier transform èspatial and temporalè of ". The above equation is a convolution in space and time. The displacement D at time t depends on the electric æeld at all times t 0 previous to t ètemporal dispersionè. Additionally, the displacement ata point r also depends on the values of the electric æeld at neighboring points r 0 èspatial dispersionè. A spatially dispersive medium is therefore also called a nonlocal medium. Nonlocal eæects can be observed at interfaces between diæerent media or in metallic objects with sizes comparable with the mean-free path of electrons. In general, it is very diæcult to account for spatial dispersion in æeld calculations. In most cases of interest the eæect is very week and we can savely ignore it. Temporal dispersion, on the other hand, is a widely encountered phenomenon and it is important to accurately take it into account. 1 In an anisotropic medium the dielectric constant " = $ " is a second-rank tensor.

4 1.3. MACROSCOPIC ELECTRODYNAMICS Arbitrary time-dependent æelds The spectrum ^Eèr;!è of an arbitrary time-dependent æeld Eèr;tè is deæned by the Fourier transform ^Eèr;!è= p 1 1 Eèr;tèe i!t dt : è1.17è 2ç,1 In order that Eèr;tè is a real valued æeld we have to require that ^Eèr;,!è =^E æ èr;!è : è1.18è Applying the Fourier transform to the time-dependent Maxwell equations èeqs è gives ræ ^Eèr;!è = i! ^Bèr;!è ; ræ ^Hèr;!è =,i! ^Dèr;!è+^jèr;!è ; ræ ^Dèr;!è = ^çèr;!è ; ræ ^Bèr;!è = 0 ; è1.19è è1.20è è1.21è è1.22è Once the solution for ^Eèr;!è has been determined, the time-dependent æeld is calculated by the inverse transform as Eèr;tè= p 1 1 ^Eèr;!èe,i!t d! : 2ç,1 è1.23è Thus, the time-dependence of a non-harmonic electromagnetic æeld can be Fourier transformed and every spectral component can be treated separately as a monochromatic æeld. The general time dependence is obtained from the inverse transform Time harmonic æelds The time dependence in the wave equations can be easily separated to obtain a harmonic diæerential equation. A monochromatic æeld can then be written as 2 Eèr;tè=RefEèrèe,i!t g = 1 æ Eèrèe,i!t + E æ èrèe i!tæ ; 2 è1.24è with similar expressions for the other æelds. Notice that Eèr;tè is real, whereas the spatial part Eèrè is complex. The symbol E will be used for both, the real, time-dependent æeld and the complex spatial part of the æeld. The introduction of a new symbol is avoided in order to keep the notation simple. It is convenient to represent the æelds of a time-harmonic æeld by their complex amplitudes. Maxwell's equations can then be written as ræeèrè = i!bèrè ; ræhèrè =,i!dèrè +jèrè ; rædèrè = çèrè ; ræbèrè = 0 ; è1.25è è1.26è è1.27è è1.28è 2 This can also be written as Eèr;tè = RefEèrèg cos!t +ImfEèrèg sin!t = jeèrèj cos ë!t + 'èrèë, where the phase is determined by 'èrè = arctan ëimfeèrèg=refeèrègë

5 12 CHAPTER 1. FOUNDATIONS OF NANO-OPTICS which is equivalent to Maxwell's equations for the spectra of arbitrary timedependent æelds. Thus, the solution for Eèrè is equivalent to the spectrum ^Eèr;!è of an arbitrary time-dependent æeld. It is obvious that the complex æeld amplitudes depend on the angular frequency!, i.e. Eèrè = Eèr;!è. However,! is usually not included in the argument. Also the material parameters ", ç, and ç are functions of space and frequency, i.e. "="èr;!è, ç =çèr;!è, ç=çèr;!è. For simpler notation, we will often drop the argument in the æelds and material parameters. It is the context of the problem which determines which of the æelds Eèr;tè, Eèrè, or ^Eèr;!è is being considered Complex dielectric constant With the help of the linear constitutive relations we can express Maxwell's curl equations èeqs and 1.26è in terms of Eèrè and Hèrè. We then multiply both sides of the ærst equation by ç,1 and then apply the curl operator to both sides. After the expression ræh is substituted by the second equation we obtain ræç,1 ræe,!2 c 2 ë" + iç=è!" oèë E = i!ç o j s : è1.29è It is common practice to replace the expression in the brackets on the left hand side by a complex dielectric constant, i.e. ë" + iç=è!" o èë! ": è1.30è In this notation one does not distinguish between conduction currents and polarization currents. Energy dissipation is associated with the imaginary part of the dielectric constant. With the new deænition of ", the wave equations for the complex æelds Eèrè and Hèrè in linear, isotropic, but inhomogeneous media are ræç,1 ræe, k 2 o " E = i!ç o j s ; è1.31è ræ",1 ræh, k 2 o ç H = ræ ",1 j s ; è1.32è where k o =!=c denotes the vacuum wave number. These equations are also valid for anisotropic media if the substitutions "! $ " and ç! $ ç are performed. The complex dielectric constant will be used throughout this book Piecewise homogneous media In many physical situations the medium is piecewise homogeneous. In this case the entire space is divided into subdomains in which the material parameters are independent of position r. In principle, a piecewise homogeneous medium is inhomogeneous and the solution can be derived from Eqs. 1.31, However, the inhomogeneities are entirely conæned to the boundaries and it is convenient to formulate the solution for each subdomain separately. These solutions must be connected with each other via the interfaces to form the solution for all space. Let the interface between to homogeneous domains D i and D j be denoted as

6 1.3. MACROSCOPIC ELECTRODYNAMICS ij. If " i and ç i designate the constant material parameters in subdomain D i, the wave equations in that domain read as èr 2 + k 2 i è E i =,i!ç o ç i j i + rç i " o " i ; è1.33è èr 2 + k 2 i è H i =,r æ j i ; è1.34è where k i =è!=cè p ç i " i is the wavenumber and j i, ç i the sources in domain D i.toobtain these equations, the identity ræræ=,r 2 + rræ was used and Maxwell's equation 1.3 was applied. Eqs and 1.34 are also denoted as the inhomogeneous vector Helmholtz equations. In most practical applications, such as scattering problems, there are no source currents or charges present and the Helmholtz equations are homogeneous Boundary conditions Since the material properties are discontinuous on the boundaries, Eqs and 1.34 are only valid in the interior of the subdomains. However, Maxwell's equations must also hold for the boundaries. Due to the discontinuity it turns out to be diæcult to apply the diæerential forms of Maxwell's equations but there is no problem with the corresponding integral forms. The latter can be derived by applying the theorems of Gauss and Stokes to the diæerential forms and read Eèr;tè æ ds =, Bèr;tèæn s da Hèr;tè æ ds = Dèr;tè æ n s da = S Bèr;tè æ n s da = 0 : S Dèr;tè i æ n s da ; çèr;tè d ; è1.36è è1.37è è1.38è In these equations, da denotes a surface element, n s the normal unit vector to the surface, ds a line the surface of the volume, the border of the surface S. The integral forms of Maxwell equations lead to the desired boundary conditions if they are applied to a suæciently small part of the considered boundary. In this case the boundary looks æat and the æelds are homogeneous on both sides ëfig. 1.4ë. Consider a small rectangular along the the boundary as shown in Fig 1.4a. As the area S èenclosed by the is arbitrarily reduced, the electric and magnetic æux through S become zero. This does not necessarily apply for the source current, since a surface current density K might be present. The ærst two Maxwell equations then lead to the boundary conditions for the tangential æeld components 3 n æ èe i, E j è = 0 ; è1.39è n æ èh i, H j è = K ij ; è1.40è 3 Notice, that n and ns are diæerent unit vectors: ns is perpendicular to the surfaces S whereas n is perpendicular to the ij.

7 14 CHAPTER 1. FOUNDATIONS OF NANO-OPTICS where n is the unit normal vector on the boundary. A relation for the normal components can be obtained by considering an inænitesimal rectangular box with volume and according to Fig. 1.4b. If the æelds are considered to be homogeneous on both sides and if a surface charge density ç is assumed, Maxwell's third and fourth equation lead to the boundary conditions for the normal æeld components n æ èd i, D j è = ç ij è1.41è n æ èb i, B j è = 0 ij : è1.42è In most practical situations there are no sources in the individual domains, and K and ç consequently vanish. The four boundary conditions are not independent from each other since the æelds on both sides ij are linked by Maxwell's equations. It can be easily shown, for example, that the conditions for the normal components are automatically satisæed if the boundary conditions for the tangential components hold everywhere on the boundary and Maxwell's equations are fulælled in both domains Conservation of energy The equations established so far describe the behavior of electric and magnetic æelds. They are a direct consequence of Maxwell's equations and the properties of matter. Although the electric and magnetic æelds were initially postulated to explain the forces in Coulomb's and Ampere's laws, Maxwell's equations do not provide any information about the energy or forces in a system. The basic Lorentz' law discribes the forces acting on moving charges only. As the Abraham-Minkowski controversy shows, the forces acting on an arbitrary object cannot be extracted from a given electrodynamic æeld in a consistent way. It is also interesting, that Coulomb's and Ampere's laws were suæcient to establish Lorentz' force law. While later the æeld equations have been completed by adding the Maxwell displacement current, a) b) D ij D ij n. D i D j n. D i D j S S Figure 1.4: Integration paths for the derivation of the boundary conditions on the ij between two adjacent domains D i and D j.

8 1.3. MACROSCOPIC ELECTRODYNAMICS 15 the Lorentz law remained unchanged. There is less controversy regarding the energy. Although also not a direct consequence of Maxwell's equations, Poynting's theorem provides a plausible relationship between the electromagnetic æeld and its energy content. For later reference, Poynting's theorem shall be outlined below. If the scalar product of the æeld E and Eq. 1.2 is subtracted from the scalar product of the æeld H and Eq. 1.1 the following equation is obtained H æ èræeè, E æ èræhè =,H E j æ E : è1.43è Noting that the expression on the left is identical to ræèe æ Hè, integrating both sides over space and applying Gauss' theorem the equation above becomes h èe æ Hè æ n da =, H + E i + j æ E d : Although this equation already forms the basis of Poynting's theorem, more insight is provided when B and D are substituted by the generally valid equations 1.6 and 1.7. Eq then reads èe æ Hè æ n da 2, j æ E d, 1 2 h i D æ E + B æ H d = è1.45è he P i d, ç o 2 hh M i d : This equation is a direct conclusion of Maxwell's equations and has therefore the same validity. Poynting's theorem is more or less an interpretation of the equation above. It states that the ærst term is equal to the net energy æow in or out of the volume, the second term is equal to the time rate of change of electromagnetic energy inside and the remaining terms on the right side are equal to the rate of energy dissipation inside. According to this interpretation S =èeæhè è1.46è represents the energy æux density and W = 1 h i D æ E + B æ H è1.47è 2 is the density of electromagnetic energy. If the medium within is linear, the two last terms equal zero and the only term accounting for energy dissipation is j æ E. Hence, the two last terms can be associated with non-linear losses. The vector S is denoted as the Poynting vector. In principle, the curl of any vector æeld can be added to S without changing the conservation law 1.45, but it is convenient tomakethe choice as stated in Notice that the current j in Eq is the current associated with energy dissipation and therefore does not include polarization and magnetization currents. Of special interest is the mean time value of S. This quantity describes the net power æux density and is needed for the evaluation of radiation patterns. Assuming that the æelds are harmonic in time and that the media are linear, the time average of Eq becomes hsiæn da =, 1 Re fj æ æ Eg d; 2

9 16 CHAPTER 1. FOUNDATIONS OF NANO-OPTICS where the term on the right deænes the mean energy dissipation within the volume. hsi represents the time average of the Poynting vector hsi = 1 2 Re fe æ Hæ g : è1.49è In the far-æeld, the electromagnetic æeld is purely transverse. Furthermore, the electric and magnetic æelds are in phase and the ratio of their amplitudes is constant. In this case hsi can be expressed by the electric æeld alone as hsi = 1 2 r "o " ç o ç jej2 n r ; è1.50è where n r represents the unit vector in radial direction and the inverse of the square root denotes the wave impedance.