The Quantum Theory of Atoms and Molecules: Waves and Optics Hilary Term 008 Dr Grant Ritchie
Wave motion - travelling waves Waves are collective bulk disturbances, whereby the motion at one position is a delayed response to the motion at neighbouring points. Just a bit more complicated than SHM (but similar) e.g. a wave on a string is characterised by the distortion, in the y-direction say, that is induced at a point x along it at a time t, i.e. y = y(x,t). Let s suggest that the solution of a harmonic wave might be of the form: y( x, t) = Asin( t + Kx) where ω is the angular frequency, as before, and K is a constant. For a fixed position, y = Asin(ω t +φ); that is to say, the temporal variation is sinusoidal with amplitude A, frequency ω and phase constant φ. Similarly, for a given time t, y = Asin(Kx+τ) so that a photograph of the string would show a sinecurve of y with x. At time Δt later, y = Asin(Kx+τ +ω Δt), which is identical to the previous picture except for a translation of ωδt /π of a wavelength, λ, due to the change in the phase factor eqn above represents a sine-curve, y = Asin(Kx), moving bodily from right to left; for this reason, it is called a travelling wave.
Wave properties λ is the separation between the two nearest equivalent points on the curve: the distance from one crest to the next, or between adjacent troughs, for example. This translates into the condition that Kλ = π radians, and so: c A y Asin( k x + "#t) Asin( k x) x K = " / -A K is known as the wavenumber, and is usually given in cm 1. Consideration of the temporal phase factor, ω T = π radians, leads to the result that the period T is: T = " / = 1/ where f is in Hertz if ω is in rad s 1. Since T is the time taken for the wave to travel one wavelength, its speed c is given by c = / T = f = " / K f
The wave equation For a wave travelling from left to right: y = Asin( " t Kx) Again, as in SHM, we can use complex number notations to represent waves: y = Ae i( " t + Kx) or y = Ae i( " t Kx) where both the real and imaginary parts yield progressive waves and A can be complex. The modulus, A, gives the amplitude and the argument, arg(a), the phase factor. The partial differential equation describing wave motion can be derived from Newton s second law of motion, and takes the form: x y = c 1 t y where c is the speed of the wave.
Transverse and longitudinal waves So far, y as the displacement of a string perpendicular to its lengthwise direction x; transverse wave. For sound waves, however, y represents the air-pressure as a function of position and time; the alternating compressions and rarefactions comprising the sound wave constitute oscillations along the direction of motion longitudinal wave. rarefaction compression x
Extension to 3 dimensions While waves on a string are the simplest ones to study, they are restricted to just one dimension. This limitation can be removed by generalising the wave eqn to: " # = 1 c t where is a multidimensional version of the second-derivative operator / x, and ψ is the entity that varies as a function of position vector r and time: ψ = ψ (r,t). In two dimensions relevant for ripples on a lake or the vibration of a drum, for example, ψ = ψ / x + ψ / y in Cartesian co-ordinates. Now the general travelling wave solutions have Kx replaced by the dot product K r ; # " = i( t + K.r) A e The wave number is now called the wave vector K; its magnitude is still related to λ via K = π /λ, and its direction gives the direction of the wave propagation.
Principle of superposition Particles bounce off or stick together when they collide, where as waves pass through unhindered. While there can be interference effects in the region of overlap (see later), the characteristics of two waves after an interaction is the same as before they came together. The resultant distortion ψ due to the combination of several waves ψ 1, ψ, ψ 3,, is just given by their sum: = 1 + + 3 + L For the case of two waves, ψ = ψ 1 + ψ : where the crests and troughs of ψ 1 match up with those of ψ respectively, there is an enhancement due to constructive interference; when there is a mismatch, so that the high points of ψ 1 overlap with the low ones of ψ, there is a net reduction in the magnitude of the wave due to destructive interference. Note: The energy carried by a wave is proportional to the square of the amplitude, ψ = ψψ* and this usually determines what is measured experimentally; for component case, that is ψ 1 + ψ. See later on the double slit experiment with both light and particles.
Standing waves As a specific illustration of the principle of superposition, consider what happens when two identical travelling waves moving in opposite directions are combined: = A [sin(" t + Kx) + sin( " t # Kx)] = Asin( " t)cos( Kx) All points along the string move up and down in phase, but with different amplitudes. Amplitude is greatest when Kx = nπ, where n is an integer, and zero when Kx = (n+½)π ; this translates into antinodes at x = nλ / and nodes at x = (n+1)λ /4 respectively. Temporal variation shows that the wave does not move to the left or right standing wave with wavelength given by twice the separation between adjacent nodes or antinodes. See particle in a box problem later. NB: Although we have generated a standing wave by superposing two travelling waves, we could also have got there directly by solving the wave equation subject to the boundary conditions that there was no displacement at the left and right ends: y(0) = y(l) = 0, for example, as in a violin string of length L fixed at both ends, giving λ = L /n for n = 1,, 3, 4,.. " node x
Beats When two notes of slightly different frequencies but similar amplitudes are played together, the loudness increases and decreases slowly and beats are said to be heard. Writing ω t + Kx as ω (t + x/c) for convenience we have: % & % & % & % & x x x "# x = A sin % & # t + A sin % & (# "# ) t $ A sin % & # t % & cos t ' ' + ( c ( ' + ' + ( ' + ( ' + ( c ( ' c ( ' c ( ) ) ** ) ) ** ) ) ** ) ) ** where we have assumed that the frequency differences δω << ω. 4/"# The resultant function is the original wave with an amplitude modulated by one of frequency δω / ; this slowly varying envelope is the origin of the beats. See later on Heisenberg Uncertainty Principle. /# t
Electromagnetic waves $ " E = # $ " B = % $ E = 0 $ B = 0 0 B t E µ 0 t Maxwell s equations for electromagnetic waves in a vacuum Remember: The speed of a wave, v, is related to its wavelength, λ, and frequency, f, by the relationship v = f λ. Faraday s law of induction: A time-varying magnetic field produces an electric field. Maxwell showed that the magnetic counterpart to Faraday s law exists, i.e. a changing electric field produces a magnetic field, and concluded that electromagnetic waves have both electric, E, and magnetic, B, components.
Maxwell s equations and the speed of light Maxwell derived the following wave equations for the E and B fields in a vacuum: " E B E = # 0 µ 0 and " B = # 0 µ 0 t t Compare with wave equation: " f = v 1 t f Speed of each component of the wave is: v 1 = " µ 0 0 c µ 0 = permeability of free space = 4π 10 7 J s C m 1 ε 0 = permittivity of free space = 8.85 10 1 J 1 C m 1 The speed of the electromagnetic wave in a vacuum is a fundamental constant, with the value c.998 10 8 m s 1.
Solutions to the wave equation Simplest solution to the wave equation is a sinusoidal wave travelling in one dimension, and so the electric field component of a plane electromagnetic wave travelling in the K-direction is: E( r, t) = E0 sin( K r #" t + ) y where E 0 is the electric field amplitude, K is the wavevector, ω is the angular frequency and φ is the phase of the wave with respect to the pure sine function. E0 Wavefront The magnetic field satisfies a similar relationship: B( r, t) = B0 sin( K r #" t + ) z B0 where B 0 = E 0 /c. For a plane harmonic electromagnetic wave the three vectors K, E and B are mutually orthogonal. At any given time, t, the E and B vectors define a plane of equal phase known as a wavefront. x llk
Polarisation The electric field amplitude E 0 is a vector quantity and if it always lies in a (fixed) plane then the wave is said to be linearly polarised. The plane of polarisation is that containing the electric vector and the direction of propagation. A common source of linearly polarised light is a laser. By contrast, if the direction of E changes randomly in time with all orientations of E in the y-z plane equally probable then the wave is said to be unpolarised. An example of a source of unpolarised light is a light bulb. z Consider light in which the y and z components of the electric field take the following form: E = E cos( kx # t + ") y 0 E = E sin( kx # t + ") z 0 i.e. same amplitudes but are 90 out of phase. E z E y E 0 y The total electric vector (E y + E z ) 1/ does not change it just rotates around the x- axis. This is known as right circularly polarised light since E and B vectors rotate clockwise around the direction of propagation for an observer looking back towards the source. At a fixed time, the electric vector of the wave in space follows the path of a screw thread.
Circular polarisation + polarisers If phase difference between E y and E z is 90 in the opposite sense then total electric vector rotates around direction of propagation in opposite sense left circularly polarised light. The polarisation state of light can be determined using a polariser. Intensity of transmitted beam, I tr, is related to that of the incident beam, I 0, by the Malus law: I cos tr = I 0
Energy and momentum EM waves carry energy and the energy flux is defined using the Poynting vector, S,: 1 S = E B µ 0 where E and B are the instantaneous electric and magnetic field vectors. The Poynting vector lies in the direction of propagation i.e. to the wavevector K. For plane waves the time averaged value of the Poynting vector, S, is 1 0 E0 0 0 E0 1 $ S # = E0 " B0 = = µ µ c The average value of the energy flux is known as the average intensity of the wave and has the units Wm. The EM wave also transports linear momentum: The force exerted by a photon upon an object is therefore: Gives rise to radiation pressure. F c p = E / = d p dt 1 = c c de dt
Reflection and Refraction When a ray of light travelling in air falls upon a glass surface, part of the ray is reflected from the surface while the other part of the ray enters the glass and deviates from its original path. The latter ray is said to be refracted. For EM waves travelling in an isotropic, nonconducting medium of relative permittivity, ε r, and relative permeability, µ r, the wave equations become incident ray normal i r reflected ray " E = # # r 0 µ r µ 0 t E and " B = # # r 0 µ r µ 0 t B Speed of the wave in the medium, v, is different from that in vacuum and is given by v = 1 r 0µ r µ 0 tr refracted ray The E and B fields are still the same except that the modulus of the wavevector is now K =ω /v and not ω /c λ of a wave of fixed frequency within the medium is different from λ of a wave travelling in vacuum. E and B fields are still perpendicular to the direction of propagation.
The refractive index Different media have different values of ε r and µ r and so λ changes as the wave travels from one medium to another. The change of velocity and wavelength that is dependent on the medium is characterised by the refractive index of the material, n, defined as: n = c / v Medium Index of refraction Water 1.33 Ethanol 1.36 MgF 1.38 Fused silica 1.46 C 6 H 6 1.50 Diamond.4
Snell s law of refraction Consider a plane wave travelling in free space incident upon a medium with refractive index n. Phase is constant along the wavefront, and so the change in phase over the distance AC is equal to that over the distance BD. λ and v of wave change on entering the material (frequency stays the same). Thus phases at C and D are equivalent if the time taken to travel from A to C is the same as the time taken to travel from B to D. For this to be true the following condition must be fulfilled: AC BD sin i c = or equivalently = = n v c sin v This result is known as Snell s law of refraction. If a ray of light travelling in a medium with refractive index n 1 is incident upon a second medium of refractive index n, then the angle of refraction θ is related to the angle of incidence, θ 1, as follows n sin = 1 1 n sin tr Incident and reflected waves travel in the same medium and so the wavelength must be constant a θ i = θ r. This is the law of reflection. i A C B free space D tr n
Index of refraction is frequency dependent The refractive index of a material is frequency dependent. E.g, the refractive index of quartz is 1.64 for red light but 1.66 for violet light, hence, quartz prisms can separate white light into its constituent colours dispersion. EM wave enters dielectric medium and incident electric field causes bound electrons in the atoms/molecules of the medium to oscillate. Resultant The electrons then radiate energy as EM waves that have the same frequency as that of the incident wave secondary waves. Primary Secondary
Dispersion continued Secondary waves are out of phase with the primary incident wave and so resultant wave lags in phase behind the incident wave. Speed of the wave is the speed at which the wavefronts propagate change of phase corresponds to change in the speed of the wave. As the frequency of the incident wave increases the phase lag between the primary and secondary waves increases, and so n = n(ω). UV/visible light distorts electronic distributions of atoms and molecules refractive index related to frequency dependence of polarisability. Polarisability, Orientation Polarisation Distortion Polarisation Red Blue Diagram taken from Atkins, Physical Chemistry ed. 6 (OUP) Electronic Polarisation Radio Microwave Infra-red Visible Ultraviolet Frequency, "
Total internal reflection Consider light rays travelling in glass and incident upon a glass/air boundary at an angle θ i. As angle of incidence θ i is increased a situation arises where the refracted ray points along the surface corresponding to an angle of refraction of 90. For angles of incidence larger than this critical angle, θ c, no refracted ray exists and total internal reflection (TIR) occurs. The critical angle is found by setting θ = 90 in Snell s law: sin c = n / n 1 c n1 n Interface behaves like a perfect mirror use in fibre optics (see problems). TIR only occurs when light travels from a medium of higher refractive index into one of lower refractive index.
TE and TM polarisations The amplitudes of the refracted and reflected waves depend upon: (i) the angle of incidence, (ii) refractive index, and (iii) polarisation of the light relative to the plane of the interface. Two different polarisations, relative to the boundary plane, can be defined: (I) Transverse electric polarisation (TE): electric vector is parallel to the boundary plane; (II) Transverse magnetic polarisation (TM): magnetic vector is parallel to the boundary plane. The coefficients of reflection and transmission amplitudes, r and t respectively, for either polarisation are defined as r = E / E and t = E / E r i tr i E TM E TE
The Fresnel equations r TE cos # " # sin #" cos + " # sin = = cos " sin " cos " sin i i i i and r TM i + # i i + # i where η = n / n 1. 100% T T For TM: n=1.5 e.g. glass For TE: n=1.5 e.g. glass 0% R 0 o 90 o _ i R 0 o 90 o _ i For normal incidence θ i = θ tr = 0 and the reflection coefficients reduce to (1 η)/(1 + η); it can be positive or negative depending on whether n / n 1 is greater or less than unity. A negative value for the reflection coefficient corresponds to a phase change of π for the reflected wave relative to that of the incident wave.
The Brewster angle An important consequence of the Fresnel equations is that the reflection coefficient for TM polarisation is zero when " i = tan #1 Mixed polarisation TE polarisation Thus, if linearly polarised TM light is incident upon a glass plate with parallel faces at angle tan 1 η, then no light is reflected from the first face (and no internal reflection at the second face) ideal window. This angle of incidence is known as Brewster s angle, θ B. B TM polarisation If the light incident upon the dielectric boundary at angle θ B contains both TE and TM components (i.e. unpolarised light) the reflected ray will contain only waves for which the electric field is oscillating perpendicular to the plane of incidence (i.e. polarised light). Since tanθ B = n / n 1, application of Snell's law leads to the condition that sinθ tr = cosθ B and the reflected and transmitted rays are perpendicular to one another.
Fermat s principle Fermat s principle states that: the path taken by a ray of light between two points is the one that takes the least time. The total optical pathlength of the light ray is l = a + x + a + ( l x) Fermat s principle requires that the time taken is a minimum with respect to variations in the path length (i.e. d t /dx = 0) and so we have The above equation is satisfied if dt 1 x l # x " = # = 0 d x v $ a + x a + ( l # x) % & ' sin = i = sin r or i r Example: Derive Snell s law using Fermat s principle. a x _ i _ r l-x a Law of reflection
Interference Superposition of two linearly polarised waves, E 1 and E, of the same frequency ω leads to a wave with the following electric field distribution, E res : E res = E 1 + = E 0 E cos = [ sin( K r #" t + ) + sin( K r #" t + )] 1 1 [ ( # )] sin[ ( K r #" t + ( + ))] E 0 1 1 1 The amplitude factor dependent upon the phase difference Δφ = (φ 1 φ ) between components: largest resultant amplitude occurs when Δφ = n π where n is an integer total constructive interference; By contrast, resultant wave will have zero amplitude if Δφ = ( n+1)π total destructive interference occurs. + Constructive interference + Destructive interference
Coherence The intensity of the resultant wave, I, is related to the square of its amplitude: I (" / ) I = (1 + cos " ) 0 = 4I cos I0 where I 0 is the intensity of each of the component waves. The interference term I 0 cosδφ determines whether the resultant intensity is greater or less than I 0. If the phase difference Δφ is constant in time and space, then the two sources are said to be mutually coherent. Δφ is dependent on the optical path difference (OPD) between the two rays and so resultant intensity varies as a function of position r the interference fringes observed when two coherent beams of light are merged. No fringes are observed if the two waves are incoherent because Δφ changes randomly with time and the cosine term averages to zero. This is the reason that interference fringes are not observed with two separate ordinary light sources.
Optical pathlengths and differences The optical pathlength (OPL) when a wave travels through a medium of length l and refractive index n is defined as OPL = nl. OPL is equal to the length that the same number of waves would have travelled if the medium were a vacuum. The OPL is not the same as the geometrical path length l. (Fermat s principle requires that the OPL is a minimum). x x 1 n x 3 Actual path x 1 nx x 3 Path it would travel in the same time The OPD between the waves and their phase difference are related: $ = # " ( OPD ) = K( OPD) Thus the condition for constructive interference can be restated as the case where the OPD between the component waves is an integral number of wavelengths.
Young's double slit experiment Light from a single source passes through a pinhole and illuminates an aperture consisting of two narrow slits separated by a distance d. If a screen is placed at a distance D after the slits, an interference pattern is observed due to the superposition of waves originating from both slits. For point P to have maximum intensity the two beams must be totally in phase at that point so the condition for total constructive interference is: S d A B B' D B'P = AP P y BP # AP = n" = d sin
Young s slits continued By assuming that D >> y, d then Position on screen: n " # % $ " # d d y = D tan " D so that separation between adjacent maxima: Maxima occur at: # y % D# " $ # y = y = 0, ± D / d, ± D / d,... D d Young's double slit experiment is an example of interference by division of wavefront.
The Michelson interferometer An example of interference by division of amplitude, where a single beam is split into multiple beams by a partial reflection. Moving one of the mirrors changes the fringe pattern at the detector as it depends upon the path difference between the beams, x. If the intensity of the monochromatic source is I 0, then the total intensity of the fringe pattern I (x) is ( x) = I 0 (1 cos Kx) I + Source I( x) Detector Partially reflecting mirror Mirror Movable mirror A plot of I(x) against x is known as an interferogram and is simply a cosine curve for the case of a monochromatic source. x
Interferograms and Fourier transforms If source is polychromatic then intensity distribution at the detector is found by weighting the monochromatic intensity distribution by spectral distribution of the source, W (K), and summing over all frequencies: " " " 1 ikx # ikx I( x) = 1+ cos Kx W ( K) dk = W ( K) dk + W ( K) e + e dk ( ) ( ) 0 = 0 0 1 W 0 + 1 " #" W ( K) where W 0 is the interferogram intensity for zero path difference between the two beam paths. I (x) and W (K) constitute a Fourier transform pair : W " # ( K) = I( x) e ikx dx " Interferogram allows the spectral distribution of the input light to be determined. See Wavepackets later on. I ( x) I ( x) e x ikx x dk FT FT I ( f ) I ( f ) f f
Interference from thin films Used in production of antireflection (AR) coatings for optical components. Consider a glass plate that is coated with a thin film of the transparent substance MgF of thickness d and refractive index n MgF. What thickness of optical coating will minimise reflections? d Air MgF ( n = 1.0) ( n = 1.38) Minimum reflection occurs when ray reflected from air- MgF boundary interferes destructively with ray reflected from MgF-glass boundary. Glass ( n = 1.5) If ray enters the MgF at normal incidence, the OPD between the two rays is dn. (Phase change of π accompanying a reflection at a boundary of higher refractive index can be neglected because both beams undergo this phase change). Condition for destructive interference is d n MgF 1 = ( m + ) Example: For m = 0, λ = 600 nm, and n = 1.38, the minimum thickness of the AR coating must be 109 nm.
Diffraction Diffraction is the bending of light at the edges of objects. Light passing through aperture and impinging upon a screen beyond has intensity distribution that can be calculated by invoking Huygen s principle. Wavefront at the diffracting aperture can be treated as a source of secondary spherical wavelets.
The single slit Consider point P and the rays that originate from the top of the slit and the centre of the slit respectively. If the path difference (a sinθ)/ between these two rays is λ /, then the two rays will arrive at point P completely out of phase and will produce no intensity at that point. a P For any ray originating from a general point in the upper half of the slit there is always a corresponding point distance a/ away in the lower half of the slit that can produce a ray that will destructively interfere with it. Thus the point P, will have zero intensity and is the first minimum of the diffraction pattern. The condition for the first minimum is a sin" = " x " x sin
The single slit continued In general a minimum occurs when path difference between the rays at A and B (separated by the distance a /) is an odd number of half-wavelengths; i.e. (m 1/)λ / where m = 1,, 3,,. The general expression for the minima in the diffraction pattern is thus a sin" = m For light of a constant wavelength the central maximum becomes wider as the slit is made narrower. The intensity distribution of diffracted light, I res, is #3" #" #" & sin ' " # a Ires = Eres = Imax ( ) where = = sin$ * + % " " 3" This characteristic distribution is known as a sinc function. The maximum value of this function occurs at θ = 0 and has zero values when α = π, π,.., nπ. The secondary maxima rapidly diminish in intensity and the diffraction pattern is a bright central band with alternating dark and bright sidebands of lesser intensity.