Frequency deendence and dielectric constant Section 4: Electromagnetic Waves We now consider frequency deendence of electromagnetic waves roagating in a dielectric medium. As efore we suose that the medium is not magnetized, so that we can set M = is our equations. We further suose that the waves are roagating in the asence of free charges and currents. Maxwell's equations therefore are reduced to where D = E + P. D = (4. = (4. E = (4.3 t = µ D, (4.4 t We want to form lane-wave solutions to these equations, and examine how the waves interact with the medium. Here we assume that the medium is uniform, isotroic, and unounded. To egin we will look for monochromatic waves oscillating with an angular frequency. We therefore set i t E( r, t = Eɶ ( r, e (4.5 with similar exressions for D, and P. Note that only the real arts of these comlex fields have hysical meaning. The restriction to monochromatic waves is not too severe. ecause the field equations are linear (assuming, as we shall verify elow, that the relation etween D and E is itself linear, solutions with different frequencies can e added to form wave-acket solutions. For examle, we can exress the electric field as it E( r, t = Eɶ (, r e d (4.6 π in terms of the frequency-domain field E ɶ ( r, ; this is a Fourier reresentation of the electric field. The inverse transformation is it E ɶ ( r, = E( r, t e dt π (4.7 and frequency-domain fields D ɶ ( r,, ɶ ( r,, and P ɶ ( r, can e defined a similar way. The frequency-domain fields satisfy the equations and we have the relation D ɶ = (4.8 ɶ = (4.9 E ɶ i ɶ = (4. ɶ + iµ D ɶ =, (4.
ξ ξ ξ ξ ξ D ɶ = E ɶ + P ɶ, (4. These are the sourceless Maxwelll s equations in the frequency domain. They can only e solved once an exlicit relationshi (called a constitutive relation is introduced etween the olarization P ɶ and the electric field E ɶ. The constitutive relation descries the medium's resonse to an alied electric field, and its derivation is ased on comlicated molecular rocesses. While quantum mechanics is required for a roer treatment, in the next section we shall consider a simle classical model that nevertheless catures the essential features. For simlification of the notation we will denote elow the Fourier comonents of all the fields with no tilde. Dielectric constant We suose that the medium is sufficiently dilute that the electric field felt y any given molecule is just the alied field E itself, the field exerted y other molecules eing negligile. We assume that in the asence of an alied field, the molecular diole moment is zero, at least on average. When an electric field is alied, however, some of the molecular charges (the electrons move away from their equilirium ositions, and a diole moment develos. The dislacement of charge ound charge q from its equilirium osition is denoted ( t. This charge undergoes a motion that is governed y the alied electric field (which tends to drive away from zero and the intra-molecular forces (which tend to drive the charge ack to its equilirium osition. We model these forces as harmonic forces, all sharing the same natural frequency. What we have, therefore, is a system of simle harmonic oscillators of natural frequency that are driven at a frequency y an alied force. Since the motion of the charges will roduce electromagnetic radiation, energy will gradually e removed from the oscillators (to e carried off y the radiation, and the oscillations will e damed. In addition, the interaction etween molecules and other charges in the system will also result in the daming of the oscillations. We incororate this effect through a henomenological daming arameter γ >. The equations of motion for the charge q are therefore d d m q e dt dt it + γ + = E( r n,. (4.3 n Notice that the alied electric field E( r,, which is macroscoic and varies slowly over intramolecular distances, is evaluated at the molecule s center of mass r n. To solve the equations of motion in ξ i t the steady-state regime we write ( t = ξɶ ( e and we sustitute this into Eq.(4.3. This yields q / m ( = E( r n,. (4.4 iγ where we again eliminated the tilde from the notation of the Fourier comonent. Since the diole moment vanishes at equilirium, the molecule s diole moment is given y = q ( r + = q ξ. (4.5 n where r is the equilirium osition of charge q with resect to the center of mass. Sustituting Eq.(4.4 then gives q / m n = E( r n,. (4.6 iγ
Since all the contriuting charges q are electrons, we can assume that q / m = Ze / m, where Z is the numer of electrons er molecule, e is the electronic charge, and m the electron s mass. We therefore arrive at Ze n = ( n, m iγ E r. (4.7 for the molecular diole moment in the frequency domain. Assuming that medium has N molecules er macroscoic unit volume we find that the olarization vector Now using the relationshi D = E + P we find or P ( r Ze N, = (, m iγ E r. (4.8 (, Ze N D r = E + P = + (, E r m iγ. (4.9 D( r, = ( E( r,. (4. where we introduced the (comlex and frequency-deendent dielectric constant such that where we defined the medium s lasma frequency ( r ( = +. (4. iγ Ze N = m /. (4. We note though that the term lasma frequency have only meaning for free electrons. For ound electrons should e regarded as some constant. Equation (4. is the result of a simlistic classical model. ut a roer quantum-mechanical treatment would roduce a similar result, excet for the fact that the electrons actually oscillate with a discrete sectrum of natural frequencies j and daming coefficients γ j. A more realistic exression for the dielectric constant is then ( f = + γ. (4.3 j j j i j where f j is the fraction of electrons that share the same natural frequency j. We will continue to deal with the simle exression of Eq.(4.. It is good to kee in mind, however, that the model is a it crude. Fig. 4. shows the real and imaginary arts of the dielectric constant. It is seen that that for most frequencies, when is not close to, Re(/ increases with increasing, and Im(/ is very small; this tyical ehavior is associated with normal disersion. When is very close to, however, the electrons are driven at almost their natural frequency, and resonance occurs. Near resonance, then, Re(/ decreases with increasing, and Im(/ is no longer small; this ehavior is associated with anomalous disersion. In realistic models the resonances occur at a numer of frequencies j. Normal 3
disersion is seen to occur everywhere excet the neighorhood of a resonance frequency. At this frequency the imaginary art of is areciale. Since a ositive Im reresents dissiation of energy from the EM wave into the medium, the regions where Im is large is called resonant asortion..5. Re(/ /.5. Im(/ / Fig. 4.. Real and imaginary arts of the dielectric constant, as functions of /. The real art of the dielectric constant goes through at resonance. The imaginary art of the dielectric constant achieves its maximum at resonance. At low frequencies the dielectric constant ehaves as ( + ( / slightly larger than unity. At high frequencies it ehaves as ( ( / aroaches unity from elow. Monochromatic lane waves r r, and is therefore a constant After making the sustitution of Eq.(4., the frequency-domain Maxwell equations ecome, and therefore E = (4.4 = (4.5 E i = (4.6 + i ( µ E =, (4.7 Similar to what we have done efore, it is easy to show that as a consequence of Eqs.(4.4-(4.7, oth E and satisfy a frequency-domain wave equation. For examle, taking the curl of Eq.(4.6 gives ( i E = (4.8 sustituting Eq. (4.7 and making use of the vectorial identity ( E = ( E E uts this in the form ( k where we have introduced a comlex wave numer k( + E =, (4.9 ( k ( (4.3 c 4
and we recall that c ( µ / refraction is the seed of light in vacuum. We can also introduce a comlex index of Similar maniulations return ( n ( (4.3 ( k the statement that the magnetic field also satisfies a wave equation. + =, (4.3 To see how monochromatic electromagnetic waves roagate in a dielectric medium suose that the wave is a lane wave that roagates in the z direction, so that E, ~ e ikz. This is similar to what we had efore ut now the wave vector k is a comlex numer. To identify the hysical roerties of this wave we sustitute Eq.(4. into Eq. (4.3 and otain k ( +. (4.33 c iγ For simlicity we suose that the second term within the large rackets is small. Taking the square root gives The comlex numer can e reresented as where is the real art of the comlex wave numer, while k R k = +. (4.34 c iγ k = k + ik, (4.35 R ( ( + I = +. (4.36 c γ k I = γ. (4.37 c + γ ( is its imaginary art. The right-moving wave is therefore descried y I R E( z, t = E e e = E e e. (4.38 ikz it k z ik z it From this exression we recognize that the wave is indeed traveling in the ositive z direction, with a seed k c known as the hase velocity; n index of refraction. From our revious exression for k R we otain v = c k ( n ( (4.39 R R R R is the medium's index of refraction (a real art of the comlex 5
n R ( ( + = +. (4.4 γ and we see that n R + > for low frequencies, so that v < c. On the other hand, n R < for high frequencies, so that v > c. The hase velocity of the wave can therefore exceed the seed of light in vacuum! This strange fact does not violate any relativistic notion, ecause a monochromatic wave does not carry information. To form a signal one must modulate the wave and suerose solutions with different frequencies. As we shall see, in such situations the wave always travels with a seed that does not exceed c. Another imortant consequence of Eq.(4.38 is that the wave is an exonentially decreasing function due to dissiation of energy. The asortion coefficient is defined y α = k and descries the attenuation of the wave intensity. It follows from eqs. (4.37 and (4. that α = k I = Im, so that indeed the c imaginary art of the dielectric constant is associated with asortion of an EM wave. The largest asortion occurs near resonances ( = j. At high frequencies ( j k I and consequently α tend to zero and therefore the medium ecomes transarent. Low frequency ehavior, electrical conductivity In the limit of there is qualitative difference in resonse of medium deending on whether the lowest resonance frequency is zero or nonzero. For insulators the lowest resonance frequency is different from zero. For metals however some fraction of electrons f is free in a sense of having =. The dielectric constant (4. is then singular at =. If contriution from free electrons is exhiited searately, the eq. (4. gives nee ( = ( + i. (4.4 m γ i ( where is the contriution from ound electrons and ne = ZNf is the concentration of free electrons. The singular ehavior can e understood if we examine the Maxwell-Amere equation I D H = J + (4.4 t and assume that the medium oeys Ohm's law: J = σ E and has a "normal" dielectric constant. With harmonic time deendence the equation ecomes i i σ H = + E (4.43 If, on the other hand, we did not insert Ohm's law exlicitly ut attriuted instead all the roerties of the medium to the dielectric constant, we would identify the quantity in rackets on the right-hand side of (4.43 with (, i.e. Comarison with (4.4 yields an exression for the conductivity ( ( = + i σ (4.44 6
nee nee τ σ ( = = m i m i ( γ ( τ, (4.45 where we introduced τ / γ. This is essentially the model of Drude for the electrical conductivity. We can rewrite this exression in the form σ σ ( =, (4.46 iτ where σ is the dc Drude conductivity ( n e τ m e σ =. (4.47 and τ = / γ is the relaxation or scattering time. The latter can e determined emirically from 8 exerimental data on the conductivity. For coer, n 8 free electrons er m 3 and at normal temeratures the dc conductivity is σ 6 The conductivity is now a comlex quantity: 7 e Ω - m - -4. This gives τ s. σ σ τ Re σ = ; Imσ =, (4.48 + τ + τ The real art reresents the in-hase current which roduces the resistive joule heating, while the imaginary art reresents the π/ out-of-hase inductive current. An examination of Reσ and Imσ as functions of the frequency shows that in the low-frequency region, τ, Imσ << Reσ. That is, the electrons exhiit an essentially resistive character. Since τ ~ -4 s, this sans the entire familiar frequency range u to the far infrared. In the high-frequency region, τ, that corresonds to the visile and ultraviolet regimes, Reσ << Imσ, and the electrons dislay an essentially inductive character. No energy is asored from the field in this range, and no joule heat aears. The dielectric function is then given y σ τ iσ = + + τ + τ (. (4.49 We consider now two frequency regions. a The low-frequency region τ. In this frequency range we can neglect the real art of conductivity and find that = + i σ. (4.5 A comlex wave numer k( in this region is given y k ( = = +. (4.5 ( µ iµ σ c If the frequency is sufficiently small we can neglect the real art of the dielectric constant so that i σ, (4.5 k ( iµ σ. (4.53 7
Then we find I Im = Im µ σ =. (4.54 c k k i σ The inverse of k I is known as skin deth δ k σ = = c. (4.55 I The skin deth determines is a measure of the distance of enetration of the EM wave into the medium z / δ efore the eam is dissiated: E E e. In ractice, δ has a very small value (for Cu at ~ 7 s -, δ=µm, indicating that an otical eam incident on a metallic secimen enetrates only a short distance elow the surface. The high-frequency region τ. This region covers the visile and ultraviolet ranges. Equation (4.49 shows that the dielectric constant is real where σ nee = = =, (4.56 τ m n e =. (4.57 m e The frequency is known as the lasma frequency. We can see from Eq.(4.56 that the high-frequency region can now e divided into two suregions: In the suregion < and therefore <. In this case the real art of the comlex index of refraction is zero: n Re( n = Re( r =. Since the reflection coefficient (for normal incidence is given y R =, we + n find that in this case to R =. That is, the metal exhiits erfect reflectivity. In the higher suregion >, and >, and hence k I =. In this range, therefore, α = and < R <, and the metallic medium acts like a non-asoring transarent dielectric, e.g., glass. Fig. 4. The lasma reflection edge. Figure 4. illustrates the deendence of reflectivity on frequency, exhiiting the dramatic discontinuous dro in R at =, which has come to e known as the lasma reflection edge. The frequency as seen from (4.57 is roortional to the electron density n. In metals, the densities are such that falls into the high visile or ultraviolet range (Tale. 8
Tale : Reflection edges (lasma frequencies and corresonding wavelengths for some metals Li Na K R (l 6 s -..89.593.55 λ (Å 55 35 34 The nature of this charge density wave, known as a lasma oscillation, can e understood in terms of a very simle model. Imagine dislacing the entire electron gas, as a whole, through a distance x with resect to the fixed ositive ackground of the ions (Figure 4.3. The resulting surface charge gives rise to an electric field of magnitude E=σ/, where σ=n e ex is the charge er unit area at either end of the sla. This field tends to restore the electron distriution to equilirium. σ = + neex nex E = x σ = nex Fig. 4.3 Simle model of a lasma oscillation. The equation of motion of the electron gas as a whole is m d x nex ne x e dt ne which lead to oscillation at the lasma frequency =. m Reflection from a conducting surface = = (4.58 Now we consider what haens if the EM wave is reflected from a surface of conducting material. We assume that the xy lane forms the oundary etween a non-conducting linear medium ( and a conductor (, as is shown in Fig.4.4. For simlicity we consider normal incidence. Fig.4.4 9
A lane wave of frequency, traveling in the z direction and olarized in the x direction, aroaches the interface from the left as is shown in Fig.4.4: It gives rise to the reflected wave E ( z, t = E e x (4.59 i i( kz t ˆ i ˆ i( kz t i ( z, t = Eie y. (4.6 v E ( z, t = E e x (4.6 r i( kz t ˆ r which travels ack to the left in medium (, and a transmitted wave ˆ i( kz t r ( z, t = Ere y, (4.6 v Et ( z, t = E e x (4.63 k i( kz t (, ˆ t z t = Ete y, (4.64 i( kz t ˆ t which continues on the right in medium (. Note that comared to the case of non-conducting medium we have in eq.(4.64 a comlex numer k / instead of a real / v. Now we use oundary conditions to find the amlitudes. In the asence of surface changes and currents they have the form: E = E (4.65 E = E = = (4.66 (4.67 (4.68 At z =, the comined fields on the left, E i + E r and i + r, must join the fields on the right, E t and t, in accordance with these oundary conditions. In this case there is no comonents erendicular to the surface, so (4.65 and (4.66 are trivial. Eqs. (4.67 and (4.68 require that Eq. (4.7 can e rewritten as follows where now β is comlex: v E + E = E (4.69 i r t k Ei Er = Et (4.7 ( E E = β E (4.7 i r t β =. (4.7 v k Equations (4.69 and (4.7 are easily solved for the outgoing amlitudes, in terms of the incident amlitude: β Er = Ei, Et = Ei (4.73 + β + β These equations are formally identical to those we otained for the reflection at the oundary etween nonconductors, ut the resemlance is decetive since β now is a comlex numer. For a erfect conductor σ = and consequently, according to eq.(4.5 k =, so β is infinite and E = E, E =. (4.74 r i t
Proagation of wave ackets So far we have considered monochromatic waves. In ractice EM waves reresent a suerosition of monochromatic waves with different frequencies. In this section we consider roagation of such waves in disersive media. We assume, for simlicity, that ( is real, i.e. we work away from resonances and ignore dissiative effects. We consider a one-dimensional case and let k( e the ositive solution to Eq. (4.3. A wave acket is otained y suerosing solutions to Eqs. (4.9 and (4.3 with different frequencies. Assuming that the waves roagate along the z axis and including oth left- and right-moving waves, we have ik ( z ik ( z it ψ ( z, t = a+ ( e + a ( e e d π (4.75 where ψ ( z, t is the wave acket reresenting either electric or magnetic comonent of the EM wave, and field a± ( are comlex amlitudes that determine the shae of the wave acket. Alternatively, and more conveniently for our uroses, we can instead suerose solutions with different wave numers k. Defining ( k to e the ositive solution to Eq. (4.3, we can write ψ (, ( ( (4.76 i ( k t i ( k t ikz z t = A k e A k e + e dk π + and it is not too difficult to show that the reresentation of Eq.(4.75 is equivalent to that of Eq. (4.76. The comlex amlitudes A± ( k are determined y the initial conditions we wish to imose on ψ ( z, t and its time derivative. It follows from Eq.(4.76 that and Fourier's inversion theorem gives ikz ψ ( z, = [ A+ ( k + A ( k ] e dk (4.77 π ikz A+ ( k + A ( k = ψ ( z, e dz (4.78 π On the other hand, differentiating Eq. (4.76 with resect to t roduces and inversion gives Solving for A± ( k, we finally otain ikz ψɺ ( z, = i( k [ A+ ( k A ( k ] e dk, (4.79 π ikz i( k [ A+ ( k A ( k ] = ψ ( z, e dz π ɺ. (4.8 ikz A± ( k = ψ ( z, ψ ( z, e dz ± π i( k ɺ (4.8 The rocedure to follow to construct a wave acket is therefore to choose an initial configuration y secifying ψ ( z, and ψɺ ( z,, then calculate A± ( k using Eq. (4.8, and finally, otain ψ ( z, t y evaluating the integral of Eq.(4.76.
In the following discussion we will choose for simlicity the initial configuration to e time-symmetric, in the sense that ψɺ ( z, (4.8 Equation (4.8 then simlifies to ikz A( k A± ( k = ψ ( z, e dz (4.83 π and we have an equal suerosition of left- and right-moving waves. Proagation without disersion Let us first see what the formalism of the receding susection gives us when there is no disersion, that is, when n = ck / is a constant indeendent of k. We then have ( k = v k, (4.84 where v = c / n is the wave s hase velocity. In this case, for time symmetric initial configuration, Eq. (4.76 reduces to ikv v t ik t ikz ik ( z+ v t ik ( z v t ψ ( z, t = A( k e e e dk A( k e e dk π + = + π ut from Eq.(4.83 we have (4.85 Sustituting this to eq. (4.85 gives or ikz A( k = ψ ( z, e dz. (4.86 π ik ( z+ v t z ik ( z v t z ψ ( z, t = dzψ ( z, e e dk + = π = dz ψ ( z, δ ( z + v t z + δ ( z v t z (4.87 ψ ( z, t = ψ ( z + v t, + ψ ( z v t, (4.88 The first term on the right-hand side is (half the initial wave acket, translated in the z direction y + v t ; it reresents that art of the wave acket that travels undistured in the negative z direction with a seed v. The second term, on the other hand, is the remaining half of the wave acket, which travels toward the ositive z direction with the same seed v. The main features of wave roagation without disersion are that the wave acket travels with the hase velocity v = c / n and that its shae stays unchanged during roagation. Proagation with disersion Let us now return to the general case of roagation with disersion. Here ( k is a nonlinear function of k, and for time-symmetric initial configuration we have i ( k t i ( k t ikz ψ ( z, t = A( k e + e e dk π (4.89
z z z Fig. 4.5 A harmonic wave train of a finite extent and its Fourier sectrum. To e concrete, suose that A(k is sharly eaked around k = k, with k some aritrary wave numer, as is shown in Fig. 4.5. Suose further that ( k varies slowly near k = k, so that it can e aroximated y d ( k ( k + ( k k = ( vgk + vgk, (4.9 dk k = k where we denoted ( k and d. (4.9 v g dk k = k v g is called the grou velocity. Then the wave acket can e exressed as follows or ik ( z+ v gt i( vg k t ik ( z v gt i( vg k t ψ ( z, t = A( k e e e e + dk = π i( vg k t ik ( z+ v ( v gt i g k t ik ( z v gt = e A( k e dk + e A( k e dk π π (4.9 i( v gk t i( vgk t ψ ( z, t = e ψ ( z + v t, + e ψ ( z v t,. (4.93 g g From Eq.(4.93 we see that in the resence of disersion, the wave acket travels with the grou velocity v g instead of the hase velocity v. The hase factors indicate that the shae of the wave suffers a distortion during roagation. These are the main features of wave roagation in a disersive medium. If the disersion relation is exressed as dk dn = n d c + d and n( k( =, in terms of the index of refraction n(, then c See Sec. 7.9 in Jackson for the illustration of the roagation of Gaussian wave acket in a disersive medium. 3
vg d = =. (4.94 c dk dn n + d For examle consider a conducting medium in the high-frequency region ( τ aove the lasma frequency so that is real. In this case and Using eq.(4.94 we find n = =, (4.95 dn =. (4.96 3 d n vg = = n. (4.97 c dn n + d Since n <, we see that the grou velocity v g = cn < c, whereas the hase velocity v = c / n > c. Nonlocality in time in the connection etween D and E Another imortant consequence of the frequency deendence of ( is a temorally nonlocal connection etween the dislacement D( r, t and the electric field E( r, t. According to eq. (4. the monochromatic comonents of frequency are related y D( r, = ( E( r,. (4.98 The deendence on time can e constructed y Fourier suerosition. Treating the satial coordinate as a arameter, the Fourier integrals in time and frequency can e written as follows it D( r, t = D(, r e d, (4.99 π it D( r, = D( r, t e dt, (4. π with similar exressions for E. The sustitution of Eq, (4.99 for D( r, in (4.99 gives it D( r, t = ( E( r, e d. (4. π y inserting the Fourier reresentation of E( r, into the integral we otain it it i ( t t D( r, t = ( E( r, t e dt e d = E( r, t ( e d dt π π π. (4. Now we introduce a frequency deendent electric suscetiility ( P = χee so that the integral (4. can e rewritten in the form ( χe (, (4.3 4
i ( t t i ( t t (, t (, t e D r = E r d dt (, t χe( e d dt π E r π (4.4 If we now take into account and define we find i ( t t e d = δ ( t t π (4.5 iτ G( τ χe( e d π, (4.6 D( r, t = E( r, t + G( τ E( r, t τ dτ (4.7 Equation (4.7 gives a nonlocal connection etween D and E, so that D at time t deends on the electric field at times other than t. If ( is indeendent of, Eq. (4.6 yields G( τ δ ( τ and the instantaneous connection is otained. However, if ( varies with, G( τ is nonvanishing for some values of τ different from zero, and the relation etween D and E ecomes nonlocal. 5