Classical Electrodynamics

Transcription

1 A First Look at Quantum Physics Classical lectrodynamics Chater 7 Plane lectromagnetic Waves and Wave Proagation 11 Classical lectrodynamics Prof. Y. F. Chen

2 Contents A First Look at Quantum Physics 7.1 Plane Wave in a Non-conducting Medium 7. Linear and Circular Polarization; Stokes Parameters 7.3 Reflection and Refraction of lectromagnetic Waves at a Plane Interface Between To Dielectrics 7.4 Polarization by Reflection, Total Internal Reflection; Goos-Hänchen ffect 7.5 Frequency Disersion Characteristics of Dielectrics, Conductors, and Plasmas 7.6 Simlified Model of Proagation in The Ionoshere and Magnetoshere 7.8 Suerosition of Waves in One Dimension; Grou Velocity 7.9 Illustration of The Sreading of a Pulse As It Proagates in A Disersive Medium 7.1 Causality in The Connection Between D and ; Kramers-Kronig Relation 11 Classical lectrodynamics Prof. Y. F. Chen

3 A First Look at Quantum Physics 7.1 Plane Wave in a Non-conducting Medium The simlest and the most fundamental electromagnetic waves are transverse lane waves. From Maxwell equations discussed in revious chaters: (1) D B () t D (3) B (4) H J t Assuming the fields with time-harmonic deendence: Then, (1') D (') ib (3') B (4') H i D In the absence of sources i t e (1') D (3') B Assume isotroic B (') t D (4') H t D B H (') i( B) ( ) i( H) k v where k 11 Classical lectrodynamics Prof. Y. F. Chen

4 Similarly, A First Look at Quantum Physics (4') Hi( D) ( H) Hi( ) B B B B Bk B v B We see both and satisfy the Helmholtz equation. The solution of the Helmholtz eq. (1D): Consider the wave roagates along z-axis and has only x-comonent d dz k x x x x e ikz Combine the time-harmonic art, the general solution would be ik ( z t) ik ( z t) i( kzt) i( kzt) k k uzt (,) ae be ae be uzt (,) ae be ik ( zvt) ik ( zvt) Define: 1 c v k n (Phase velocity ) where, r / n r r, (Refractive index) r / If the medium is non-disersive,i.e. is constant, then v is constant. The general form for uzt (,) is uzt (,) f ( zvt ) gz ( vt ) (Travelling wave with invariant waveform as it roagates in sace.) On the other hand, if the medium is disersive, i.e. v is function of ( v( )), then uzt (,) should consider the contribution from all by linear suerosition and the waveform may change as it roagates. 11 Classical lectrodynamics Prof. Y. F. Chen

5 We can easily describe the electric and magnetic fields with the hel of comlex exonent. And with the convention that hysical electric and magnetic fields are obtained by taking the real art of comlex quantities. xt (, ) e Bxt (, ) Be i( kn xt) i( kn xt) where Use source free Maxwell equations : n is the roagating direction is the olarization direction & and B ik n x ik n x ik n x (1') ( e ) ( e ) e ( ) ( ik n x ikn ) e n kn (3') B kn B For arbitrary, we have Similarly, leads to are constant. lectromagnetic waves are transverse waves. In reality, there is no real lane wave, and the erfect transverse wave does not exist. All waves contain at least little longitudinal comonent. Besides, from (') ib Use the identity ( f A) ( f) A f( A) i ik n x i ik n x ik n x k B ( e ) {( e ) e ( )} ( k n ) e ik n x n k From above results, we have B ( n ) ( n ) n The factor, thus cb and B have the same dimension, and ( n H n) c Z Z is an imedance., where 11 Classical lectrodynamics Prof. Y. F. Chen

6 Check dimension of Z: 1 In ac circuits, Zc L ic ZcZL Z C ZL il, therefore [ ] [ Z] is the dimension of inedence In vacuum, Z Z 376.7( ) From the above discussion, we know, B and k are erendicular to each other. There are two ossible cases as shown in the following Figure 7.1. (1) () 1 or B B 1 z k kn 1 y x Figure Classical lectrodynamics Prof. Y. F. Chen

7 A First Look at Quantum Physics 7. Linear and Circular Polarization; Stokes Parameters We can use two orthogonal olarized-comonents to describe any olarized TM wave: z z i[ ( t) x ] v x xcos[ ( t) x] Re( xe ) xiy j and v x is the eak amlitude of where z z i[ ( t) y ] v y is the eak amlitude of y ycos[ ( t) y] Re( ye ) v z i[ ( t) ] i x Let y x, (,, ; ) [ v xyzt i x e y je ] z x x i[ ( t) x ] v This can be reresented as two comonents column vector: i e y ye global hase x y For any fully olarized field, we can use this Maxwell column (or Jones vector) to characterize the olarized state. (i) or, is linear-olarized (LP) / left handed circular-olarized (LCP) For (ii) & x y, is / right handed circular-olarized (RCP) (iii) Other than the above condition but is constant, is ellitical-olarized The general mathematical treatment for olarized light : x x xcos( kzt) cos( kz t) Let x y ycos( kzt ) y cos( kz t)cos sin( kz t)sin y 11 Classical lectrodynamics Prof. Y. F. Chen

8 kz t kz t kz t kz t kz t x y (1) ( ) ( ) cos ( ) cos ( )cos sin ( )sin cos( )sin( )cos sin x y kz t kz t kz t x y () ( )( )cos cos ( )cos cos( )sin( )cos sin x y y x x y ( ) ( ) cos cos ( kz t)(1 cos ) sin ( kz t)sin sin x y xy y x x y ( ) ( ) cos sin x and satisfy the ellitical equation. y x y x y This equation describes an oblique ellise and we can use coordinate transformation to find its relation between the uniform ellise : The equation after coordinates transformation : ' y y x ' ' ' ( x ) ( y ) 1 a b ( ) x' cos sinx x' xcos ysin y' sin cosy y' xsin ycos x Comare with the revious result, Substitute into ( ) xcos ysin sin cos x y ( ) ( ) 1 a b cos sin sin cos cossin cossin x ( ) ( ) ( ) 1 y xy a b a b a b () i cos sin 1 a b x sin sin 1 1 cos ( iii) ( ) a b xysin 11 Classical lectrodynamics Prof. Y. F. Chen ( ii) sin cos 1 a b y sin

9 () i ( ii) ( iv) cos ( ) ( ) a b sin Use x y ( iii) cos tan ( ) cos ( iv) x y x y x y x y x y () i ( ii) ( ) sin x y And a b a b a b x y x y a b x y sin x y sin ab ( sin )( sin ) With the fact that the area is conserved after coordinate transformation Thus, a b x Sum of the square of the two orthogonally olarized comonents of ellitical olarized y states is constant. energy conservation xy From we know the oblique angle φ satisfies : tan cos x y y Let tan x xy cossin x x y y y x cos sin tan cos tan cos y cos sin cos sin x y x for ; = linear-olarized state for / ; = uniform ellitical-olarized state 11 Classical lectrodynamics Prof. Y. F. Chen

10 Some of the tyes of light that fall between the fully coherent and fully incoherent extremes are called artially olarized light. One can create artially olarized light by starting with a laser. In some scientific exeriments, the comlete coherent of a laser beam can be an embarrassment because of the unwanted effects of interference or diffraction or seckle that it roduce. Quasi-thermal light is manufactured by sending a laser beam through a coherent-degrading device, such as a raidly rotating ground-glass (or diffuser). Stokes arameters for secific olarization y x Recall the ellitical-olarized state discussed before : x y ( ) ( ) cos sin y ' y x x ' x y x y x cos 1 y Let &, where tan ( ) y sin x And is the total intensity of the field. x y Stokes arameters S, S1, S, S3( I, Q, U, V) for secifying olarization is defined as the following : S I x y S Q cos sin cos( ) Icos( ) 1 x y S U cos cossincos sin( )cos Isin( )cos x y S V sin cossinsin sin( )sin Isin( )sin 3 x y 11 Classical lectrodynamics Prof. Y. F. Chen

11 All these arameters form a three-dimensional shere : S 1 S S S S I Q U V 1 3 ( ) S 3 The degree of olarization is defined as: Q U V P, P1 I If random olarized, then Q U V S Figure 7..1 Poincarѐ shere For a beam that is artially olarized, we define a degree of olarization P, which is equal to the ositive square root of the ratio Q U V. I One can use roer otical elements like olarizer and wave-late to measure all the Stokes arameters. x Assume the Jones vector for the incident field is i ye 1 Ste1: incident on a olarizer with horizontal transmitting axis e x Ste: incident on a olarizer with vertical transmitting axis I 1ye ye I1I S I1 I S1 x x i I1 x y i i y 11 Classical lectrodynamics Prof. Y. F. Chen

12 Ste3: incident on a olarizer with transmitting axis along θ=45 i 11 1 x 1x ye I1 I i I3 xycos i 1 1 ye x ye i 1 1x ye i i 1 ( x ye x ye ( x y ) xycos ) i x ye Ste4: incident on a olarizer with transmitting axis along θ= -45 I I cos S 3 4 x y i x 1 x ye I1 I i I4 xycos i 1 1 ye x ye i 1 1x ye i i 1 ( x ye x ye ( x y ) xycos ) i x ye i 11i 1i x 1x(1 i) ye (1 i) Ste5: incident on a λ/4 wave-late with fast axis along θ=45 i i 1i 1i ye x(1 i) ye (1 i) then ass through a olarizer with horizontal transmitting axis i i 1 1x(1 i) ye (1 i) 1x(1 i) ye (1 i) i x(1 i) ye (1 i) i 1 1 (1 ) (1 ) x i x i ye i y I5 x(1 i) ye (1 i) xysin 1 ( I5 ( I1I)) S3 In reality, we need to consider (i) the absortion by the otical elements (ii) the uniformity for each otical element 11 Classical lectrodynamics Prof. Y. F. Chen

13 Aendix: Matrix reresentation for common otical elements (i) Polarizer with transmitting axis at θ Incident of horizontally olarized light : Incident of vertically olarized light : 1 1 y y x the amount assing through cos x the amount assing through sin rojecting to x & y cos cos sin rojecting to x & y sin cos sin Thus, the matrix element for olarizer with transmitting axis at θ (ii) Phase delay δ wave-late with fast axis at θ Incident of horizontally olarized light : Incident of horizontally olarized light : 1 y x cos sincos cossin sin rojecting to x & the cosθ comonent with unchanged hase i the sinθ comonent with e hase delay y rojecting to x & x 1 the sinθ comonent with unchanged hase i the cosθ comonent with e hase delay y i cos sin e i sincos (1 e ) y i sincos (1 e ) i sin cos e 11 Classical lectrodynamics Prof. Y. F. Chen

14 i i cos sin e sincos (1 e ) Thus, the matrix element for hase delay δ wave-late with fast axis at θ i i sin cos (1 e ) cos e sin The oerating rincile wave-late is based on the birefringence effect of unique crystal. k n Consider the refractive indices for two orthogonally olarized light are n x and n y resectively in the crystal. The incident field will have hase difference Ln ( y n x ) between x & y after roagating distance L in the crystal. For λ/-wavelate: For λ/4-wavelate: Ln ( y nx) L ( n n ) y x (Zeroth-order) Ln ( ) y nx L 4( n n ) (Zeroth-order) y x We can define coherent Z matrix to further discuss the olarization state. i i 1 xe x x ye i i I Q U iv Z xe ye i i ye xye y U iv I Q 1 ( 1 3 ) Z I U V Q, where i is the Pauli matrix., where 11 Classical lectrodynamics Prof. Y. F. Chen

15 We assume that the transverse electric field comonents arise from the suerosition of a large number n in e x N of individual disturbance. And each nth is reresented by a Maxwell column multilied by the n i n e y it time-deendence factor which is taken to be the same for all N disturbance. e x y y m m N N N N N N n in n in n m i( n m) n m i( nm) xe xe xxe xye N N n1 x n1 m im m im n1 m1 n1 m1 Z x y xe ye N N N N N N n in n in 1 1 n i ye ye y e n1 n1 e m ( n m) n m i( nm) x y y n1 m1 n1 m1 Z? We assume that the N oscillators are mutually incoherent e e i( ) n m i( ) n m for nm & n n n const. Therefore, Z N N N N N N n m i( ) ( ) n m n m i n m n n i n n yx e yy e xye y n1 m1 n1 m1 n1 n1 N N N N N N n m i( nm) n m i( nm) n n n in xx e xy e x xye N n1 m1 n1 m1 n1 n1 n1 Z n Note that each of the averaged hase factors vanishes excet when n=m, thus we can eliminate the double summation. N n x If each n1 n is random, Z And assume x y Q N n y n1 Therefore, for random olarized state U V Q 11 Classical lectrodynamics Prof. Y. F. Chen

16 7.3 Reflection A First Look at Quantum and Refraction Physics of lectromagnetic Waves at a Plane Interface Between Two Dielectrics When a lane wave incidents on a hase interface between two dielectrics, one should consider : (1) Kinetic roerties : (a) Reflection law, angle of reflection equals to angle of incidence i r n k n' r S-wave (b) Refraction law (Snell s law) nsin in'sin t r () Dynamical roerties : i (a) Intensities of reflected and refracted radiation. k t i P-wave (b) Phase change and the olarization state. i( kxt) e Figure Consider interference of lane wave : n B k k c i( k' xt) i( k'' xt) ' ' e '' '' e Refracted wave : n ' Reflected wave : n B ' ' ' k ' ' k ' ' B '' k '' '' k '' '' c c Consider the boundary condition, or B or B d d d n n' D D1 n Dn B t (source free) 1t t D H H H t B B B 1t t 1n n 11 Classical lectrodynamics Prof. Y. F. Chen k t

17 For S-wave (T) n ' n H s k s H s i ' t z According to figure 7.3. r s k ' s ' H k s '' '' '' x ik [ (sin ix ) k(cos iz ) ] it ik(sin ix ) it s ye s e ye s e [ '(sin ) '(cos ) ] '(sin ) s' ys' e e ys' e e [ ''(sin ) ''(cos ) ] ''(sin ) '' y '' e e y '' e e ik t x k t z i t ik t x it ik r xk r z i t ik r x it s s s Use the boundary condition 1 t t '' ' e '' e ' e ik (sin i ) x ik ''(sin r ) x ik '(sin t ) x s s s s s s (For z=) For comlex, the equation requires (i) the same argument (ii) the same amlitude for both side ( i) k(sin i) k''(sin r) k'(sin t) and k n nk'' k c sin i sin ri r Thus, (Reflection law) We have derived the kinetic roerties. k(sin i) k'(sin t) n(sin i) n'(sin t) (Refraction law) Consider another boundary condition H H and use (ii) 1t t n n " ' ( ")cos 'cos s s s s i s s i s s" s' n n' ( H ")cos 'cos ( s s")cos i s'cost s Hs i Hs t n n' ' ( s s")cos i s'cost ' (1) () 11 Classical lectrodynamics Prof. Y. F. Chen

18 n n n' s ' (1) () scos i ( cosi cos t) s' t ' s n cosi n n' cosi cost ' n n n' s'' s'' s' (1) () s''cos i ( cosi cos t) s' r ' ' s s s n n' cosi cost ' n n' cosi cost ' Similarly, for P-wave (TM) z Use the boundary condition & 1t t H1 t Ht n ' n k H ' i t r k ' H ' H k '' '' '' x n n' n n' ( ")cos i 'cosi H " ' ( ") ' H H ' ' ( ")cos i 'cost n n ( ")cos i 'cos t ( ")cos i 'cost n n' n ' (3) (4) cos i ( cosi cos t) ' t// ' n cosi n' n cosi cost ' (3) (4) n n' n '' '' ' (3) (4) " cos i ( cosi cos t) ' r// ' ' n' n cos i cos t ' n' n cosi cost ' 11 Classical lectrodynamics Prof. Y. F. Chen

19 For derivation of relations between intensity and amlitude, we should consider energy conservation : I 1 * 1 n t use I S H c n' n I i A i A A t A r I r IAIAIA IAcosi IAcos riacost i i r r t t i r t n n n' n' cost ' ' n cosi i cos i r cos i t cos t r t ( )( )( ) 1 Define: Ir Reflectance, R r Ii Transmittance, Ir n' cost T t ( )( )( ) I ' n cosi We can again check the results derived above are self-consistence by examining : [ ( '') ' '] n D [ ( k k'' ') ' ' k' '] n B [( B '') '] n t ' [ ( k D k'' ') k' '] n H ' t Discussion of reflecting coefficients for S- and P-wave : (1) For n > n, and consider μ= μ S-wave : r ncos i n'cost ncos i n'cost nsin i n'sin t & n' nsin isin tcosicost The numerator is always negative. i T R 1 for all incident angles The reflecting wave is always anti-hase to the incident wave. 11 Classical lectrodynamics Prof. Y. F. Chen r

20 P-wave : r // n'cosi ncost n'cosi ncost nsin i n'sin t & n' nsin i sin t cosicost The numerator may be ositive, zero or negative. The incident angle corresonding to zero reflectance, i.e. r, is called Brewster angle i. // B n For r // n'cosi ncos t and n'cosi n 1 ( ) sin i n ' B B B 4 n 4 4 n' cos ib n sin i ' cos sin ' B n ib n ib n n n ' n i n i n n i i 4 4 ' cos B sin B ' (cos B sin B) n i n n n i n n i n ' n ' cos B( ' ) sin B( ' ) tan B ( ) Therefore, i, r// r// i ib, r// i ib, r// () For n> n, and consider μ= μ nsin i n'sin t cost n 1 ( ) sin i n ' when n n' 1 n' ( )sin i 1sin i ( ) i ic, where ic sin ( ) n' n n cost will be ure imaginary and can be written as n n t i i n' n' cos i ( ) sin 1 i, where ( ) sin π r i B arg( r ) i i B r// i arg( r// ) 11 Classical lectrodynamics Prof. Y. F. Chen

21 7.4 Polarization A First Look at Quantum by Reflection Physics, Total Internal Reflection; r Goos-Hänchen ffect As we mentioned above, there is an imortant henomenon haened when i i c, which is called total internal reflection, TIR. The word internal imlies that the incident and reflected waves are in a medium of large refractive index than the refracted wave, i.e. n>n. 1 nsin i n'sin t cost n 1 ( ) sin i n ' when cost n n' 1 n' ( )sin i 1sin i ( ) i ic, where ic sin ( ) n' n n.5 will be ure imaginary and can be written as r n n' cost i ( ) sin i1 i, where ( ) sin i1 Therefore, when n>n and i>i c n cos i i n ' n ' s ncosi i n' ncosi n n' i s 1 e, where tan 1 ( ) n'cosiin n 1, where tan ( ) n'cosii n n'cosi i 1 // e r nsin i n'sin t & nn' sin isin t cosi cost r r iic, r i s i ic, r 1 e 11 Classical lectrodynamics Prof. Y. F. Chen // iib, r// i ib, r// ib iic, r// i i ic, r// 1 e Thus, when TIR henomenon haens, there is a hase difference between S-wave and P-wave: s π r // i B i c i arg( r// ) i B i c arg( r ) i

22 Consider i>i c, From Incident wave: Refracted wave: Reflected wave: e s e ' e s e ik [ (sin ix ) k(cos iz ) ] it e ik [ (sin ix ) k(cos iz ) ] it ik [ '(sin t) xk'(cos t) z] it ' e e e ik [ (sin ix ) k(cos iz ) ] it '' e e s ik [ ''(sin r) xk''(cos r) z] it '' e e ik [ ''(sin r) xk''(cos r) z] it r t r t // // ncosi i n' 1e ncosi i n' i ncosi ncosi ncosi i n' n n' n'cosiin 1e n'cosiin ncosi s i n n' cos i i n n' 4 sin e i / e s 1 n ', where s tan ( ) ncosi i / n n'cosi 1, where tan ( ) For S-wave : Reflected wave Refracted wave For P-wave : Reflected wave Refracted wave r e e e ik [ (sin ix ) k(cos iz ) ] i t i s r s i n' z ' ''(sin ) cos n z ik i x i t n i c is / ik ''(sin i) x c it t s s te e e e e e e n n' ik [ (sin ix ) k''(cos r) z] it i r // i n' z / ' '(sin ) cos n z ik t x i t n i c i ik (sin i) x c it t // 4 n n' cos i sin in r e e e t e e e e e e e n ' 11 Classical lectrodynamics Prof. Y. F. Chen

23 According to above results, when i>i c, all the reflected waves for S- & P-wave will exerience an additional hase shift and, which changes the original olarized state. s after reflection is x xe Consider, the incident wave with original olarized state i ye i( ) ye x s assume y Furthermore, although the reflecting coefficient equals to 1, the refracted wave can enetrate certain deth along the z-axis. n 1 c If n' 1 & n1.5, i 45 ( ) sin i1 zd n' n' (for monochromatic wave) (Hz); c31 ( m/ s) 6 zd ( m) ~ m Z d The refracted wave is roagated only arallel to the surface and is attenuated exonentially beyond the interface. ven though fields exist on the other side of the interface, there is no energy net energy flow through the interface. The henomenon is called evanescent wave. We can calculate the time-averaged normal comonent of the Poynting vector just inside the surface : 1 * ( k' ') Sn Re[ n( ' H' )] with H' ' 1 Sn Re[( nk') ' ] ' and nk' k'cos t is ure imaginary Sn (no net energy flow) 11 Classical lectrodynamics Prof. Y. F. Chen

24 7.5 Frequency A First Look at Quantum Disersion Physics Characteristics of Dielectrics, Conductors, and Plasmas In reality, all media show some disersion. (Unless we consider a limited range of frequencies.) c c v( ), for non-magnetic material r 1 n( ) r( ) n( ) ( ) r r ( ) A. Simle Model for ( ) Recall (4.69), if we consider the difference between local field and alied field : If we neglect the difference between local and alied field : e N mol N mol e 1 1 N 3 mol Consider the effective sring model for electrons interaction with alied field in atoms. The equation of motion for an electron of charge -e bounded by a harmonic force and acted by an electric field ( xt, ) is... mx ( x x) ext (, ) (7.49), where γ is daming rate. Ze K1 K Consider driving field with time-harmonic deendence, and the electron dislacement with the same time-deendence as the driving field : (, ) ( ) i t it xt xe & xt ( ) xe Substitute in (7.49) And the diole moment : m( x ix x) e e ex e x ( ) m( i) m( i) Ne P N e m( i) 11 Classical lectrodynamics Prof. Y. F. Chen

25 Recall : P & ( P) D f f f D P (1 ) (1 ) e e e Therefore, we have e Ne m ( i) Suose that there are N molecules er unit volume with Z electrons er molecule and that instead of a single binding frequency for all, there are f j electrons er molecule with binding frequency ω j and daming constant ϒ j, then the dielectric constant, /, is given by 1 e ( ) Ne f 1 m (7.51) j j ( j i j), where the oscillator strength f j satisfy the sum rule : f j Z (7.5) (lectric charge conservation) With subtle quantum-mechanical definition of f j, ω j,ϒ j (7.51) is an accurate descrition of the atomic contribution to the dielectric constant. In Q.M., we can calculate the robability of transition between two levels induced by the interaction otential V to find out ω j,ϒ j. Pi According to Fermi-Gorden rule : f f V i ( i f ) Also, we can decide the generalized oscillator strength (GOS) f j, by evaluating f x i (Assuming the alied field are not too strong to effect the wave function.) j 11 Classical lectrodynamics Prof. Y. F. Chen

26 In general solid state hysics, if consider valence (outer) shell electrons, j if consider inner shell electrons, j g g One can classify different tyes of atoms by checking the behavior of inner shell electrons. By fitting the exerimental data, one can obtain arameters f j, ω j,ϒ j to construct an emirical model. B. Anomalous Disersion and Resonant Absortion In general case, j j j j j If 1, the factor ( ) is ositive 1 j j, the factor ( j ) is negative f j At low frequency, below the smallest all the terms in the sum contribute with the same j ositive sign, and greater than unity. j ( j i j) ( ) Normal disersion is associated with an increase in Re( ) with ω, anomalous diserse with the inverse. 1 From the mathematical form i we know that every time when j it would be ositive imaginary, and from j to j the sign will be inverse one time. j Since a ositive imaginary art to ε reresents dissiation of energy from the M wave into the medium, the region where Im(ε) is large are called the region of resonant dissiation. j 1 3 ( ) Re( ) ( ) Im( ) ( Hz) Classical lectrodynamics Prof. Y. F. Chen

27 Recall : k ( ), the imaginary art of ε(ω) may let the wave number be written as c k i I (7.53) ; : attenuating constant (absorbing coefficient) I I z I e I d I 1 ln( ) We can decide the attenuating constant of the material by measuring the inut and outut intensities of the incident wave and transmitted wave. d In more realistic case, the incident wave may be scattered in the medium and exerience longer traveling length than d. I d From equation (7.54) I S d Define the attenuating constant without scattering : e S theory e d ex 1 i We can decide the attenuating constant exerimentally by this equation. From equation (7.53) k ( ) i( ) 4 Re( ) 4 c In general α<<β unless at the resonance absortion region. Im( ) c Im( / ) If, [ ] (7.55), where Re( )( ) Re( / ) c The fractional decrease in intensity er wavelength divided by π is thus given by the ratio, Im( /. ) / Re( / ) z ( <<, k and 1 e z) 11 Classical lectrodynamics Prof. Y. F. Chen

28 C. Low-frequency Behavior, lectric Conductivity In the limit ω, there is a qualitative difference in the resonse of the medium deending on whether the lowest resonant frequency is zero (conductor) or non-zero (insulator). For ω=, some fraction f of the electrons er molecule are free, and the dielectric constant is singular at ω=. Consider the contribution of the free electrons is exhibited searated, Ne f (7.51) ( ) b( ) i (7.56) m ( i ) contribution of all the other dioles By checking D H J ; D b Assume the medium obeys Ohm s law J, and has a normal t dielectric constant b. i Under time-harmonic assumtion, H i ( b ) We can think off all the contribution from the dielectric function : H i ( ) By comarison, we know Ne f m( i) For ω= Ne f 1, and Nf n (electron density), where is the momentum relaxation time. m Consider an electron is accelerating by external electric field in diffusive system, dv m dv m e v, when it reach terminal velocity v, e d dt dt e e vd ; vd (mobility) m m ( Ne) ef e e J enn env d enn n m m (Self-consistence) m 11 Classical lectrodynamics Prof. Y. F. Chen

29 For coer (Cu), 8 # 7 1 N 81 ( ) and low-frequency f 5.91 ( m) ( ) 41 ( s ) 3 m 11 1 assume f =1, then u to frequencies well beyond the microwave region 1 ( s ) conductivities of metals are essentially real (current in-hase with the field) and indeendent of frequency. D. High-frequency limit, lasma frequency (7.51) ( ) Ne f 1 m For common metals : j j ( j i j) ( ) 1 Ne Z if j 1 ( ) 1 ; ( ) m Ne Z m element atomic weight mass density number density lasma frequency ω Cu (g/mole) 89 (kg/m 3 ) 8.45X1 8 (#/m 3 ) 5.436X1 13 (Hz) Ag (g/mole) 15 (kg/m 3 ) 5.86X1 8 (#/m 3 ) 4.57X1 13 (Hz) Au (g/mole) 193 (kg/m 3 ) 5.99X1 8 (#/m 3 ) 4.54X1 13 (Hz) k We have c 1 ck 1 ck ck ck when k This disersion relationshi is in the same form as Q.M. free article k m 11 Classical lectrodynamics Prof. Y. F. Chen

30 ( ) 1 holds over a wide range of frequencies including ω<ω electronic lasma in the laboratory scale. for a tenuous On the laboratory scale, lasma densities are of the order of electrons m 3. This means Hz ~61 ( ) 1 i ck k c c When < (ure imaginary) (1 ) i for k c Comare with k i eff For Cu : 1s s 3s 3 4s 3d The actual contribution for lasmon electrons. c For most atoms : transition energy for outer shell 1( ev ) transition energy for inner shell 1( ev ) ω no absortion Plasmon is a collective henomenon. Consider general lasma frequency The enetration length For coer, 1 1 Hz ~6 1 ( ) 1 c 31 ( cm/ s) (1 ~1 )(1/ s).5 ~.5( cm) 1 c 31 ( cm/ s) (1/ s) 5( m). Index of refraction and absortion coefficient of liquid water as a function of frequency 11 Classical lectrodynamics Prof. Y. F. Chen

31 n( ) Re( ) ( ) Im( ) (Recall) k i n i If, ( ) n( ) ni and r ii ( ni) ( n ) i( n) In more general case, the dielectric function ( k, ) may deend on both the wave number and frequency. Linhard theory is a quantum mechanical treatment for the dielectric function, which is based on the erturbation theory. 11 Classical lectrodynamics Prof. Y. F. Chen

32 7.6 Simlified A First Look at Quantum Model Physics of Proagation in the Ionoshere and Magnetoshere If the amlitude of electronic motion is small and collisions are neglected (no daming effect). For an uniform magnetic induction B and transverse waves roagating arallel to the direction of B. d x dx it The equation of motion : m e B ee dt dt e x B We can exress in the basis of linearly olarized states or circularly olarized states : 1 1 aˆ ˆ ˆ ˆ, where ˆ ( ˆ ˆ ); ˆ ( ˆ ˆ xx ayy arr all ar ax iay al ax iay) 1 1 By comarison, we know that R ( x iy), L ( x iy) 1 1 Let R, L ( x y), Z ( xiy) Assume B Bˆ a z d x dy it (1) d x dy it x-direction : m e B exe m e B exe dt dt dt dt Couled equations d y dx it () d y dx it y-direction : m e B eye m e B eye dt dt dt dt 11 Classical lectrodynamics Prof. Y. F. Chen

33 Use the concet of canonical transformation, d ( xiy) d( yix) it dz dz it (() i(1)) m eb e( x iy) e m ieb ee dt dt dt dt Let Z e it (decouled equation) Z, substitute to the decouled equation : Z Z Z m eb e eb m ( B ) eb m e m ( ) m, where B is the cycrotron frequency. e (7.65) Returning back to x,y exression : e ex 1 xiy ( ) ( ) x i x y 1 e 1 m ( ) B m B ( xiy) ( x iy) m ( B ) e e By 1 xiy ( x iy) y ( ) m ( B ) m B The amlitude of oscillation (7.65) gives a diole moment for each electron and yields, for a bulk samle, the dielectric constant. 1 is the extension of 1 to include a static B. ( B ) This imlies that the ionoshere is birefringent. When the field travelling erendicularly to the magnetic field B, the hase sace of the charged article will resent very interesting structure under roer initial condition. 11 Classical lectrodynamics Prof. Y. F. Chen

34 7.8 Suerosition A First Look at Quantum of Physics Waves in One Dimension; Grou Velocity ven in the most monochromatic light source or the most sharly tuned radio transmitter or receiver, one deals with a finite (although erhas small) sread of frequencies or wavelengths. Since the basic equations are linear, it is in rincile an elementary matter to make the arorite linear suerosition of solutions with different frequencies. In general, however, several new features arise: 1. Disersive : ( ) k ( ), v h deends on. k. The velocity of energy flow may differ greatly from the hase velocity. 3. Dissiative in the medium. Recall the equation (7.6) i( kxt) i( kxt) uxt (, ) ae be Consider disersion, the general solution for sourceless sace would be 1 ikx 1 ux (,) Ake ( ) dk ikxi ( k ) t uxt (, ) Ake ( ) dk (7.8) If t, 1 ikx' A( k) u( x',) e dx' ikx i( k k) x' Consider (,) ( ) ' ( ) General solution for 1D wave equation without disersion. 1 ux e Ak e dx kk 1 ikxi ( k ) t and u( x, t) ( k k) e dk e However, if at t=, u(x,) reresents a finite wave train Δx, then A(k) is not a delta-function. Rather, it is eaked function with a breadth of the order of Δk, centered around k. i( k x ( k ) t) 11 Classical lectrodynamics Prof. Y. F. Chen

35 ux (,) A( k) x x k k k 1 x k (7.8) (Uncertainty relation) xand ω(k) near the central wavenumber k d ( k) ( k) kk ( kk )... substitute back into (7.8) dk d d d 1 ikxi( k) ti k k( k k ) t 1 i{ k k k ( k )} t ik( x kkt) dk dk dk u( x, t) A( k) e dk e A( k) e dk 1 i{ k v } t ik( xv t) ( ) g g e A k e dk ux ( vt,) d x x k k t 1 ikx dk Comare with ux (,) Ake ( ) dk, we find that d vg kk dk n( ) ck n( ) Recall, if n( ) ( ) and k (7.87) k c n( ) c dk n ( ) ( dn ) d c vg (7.89) & v c d c c d dk dn( ) n( ) k n( ) d 11 Classical lectrodynamics Prof. Y. F. Chen g

36 7.9 Illustration of the Sreading of a Pulse as it Proagates in a Disersive Medium If we agree to take the real art of (7.8) to obtain u(x,t), 1 1 ikxi ( k ) t uxt (, ) { Ake ( ) dk cc..} (7.9) 1 ikx i u( x,) and A( k) is given in terms of the initial values by : A( k) e { u( x,) } dx ( k) t This mathematical form is similar to the case of S.H.O. : d x v dx m m xx() t AcostBsin tax x(), B t / dt dt x ux (,) L Assume, ux (,) e cos( kx ). And for simlicity, we assume x 1 1 ikx L ikx ikx ( ) (,) ( ) ikx A k u x e dx e e e e dx L 1 L 1 1 ( kk { ( ) x i L kk ) x[ il ( kk)] } ( kk { ( ) x i L kk ) x[ il ( kk)] } L L { e e dxe e dx} L ( k k) ( k k) L 1 ax L { e e } use e dx L e L ( k k) ( k k) L { e } (7.94) a t 11 Classical lectrodynamics Prof. Y. F. Chen

37 ak Consider ( k) (1 ) which is similar to the free article in Q.M., and substitute A( k) into (7.9) L a k 1 L ( kk ) i (1 ) t ikx uxt (, ) { e e e dk ( k k) cc..} Let k kk k kk and k k kk k L a k k i t ik x i t k L (, ) { ik x i a k t i a k t uxt e e e e e e e dk ( k k) cc..} 4 ( x a k t) L i a t ixi a kt L { k } ik ( x a kt) i t ( L i a t) L i a t { e e e e dk( k k) cc..} 4 ( x a kt) ( ) ( ) 1 ik x a k t i t L i a t 1 Re{ e e e ( k k)} at 1/ (1 i ) L ( xa kt) ( L ia t) ik 4 ( xvgt) i t [ L ( a t) ] Re{ e e ( ) e ( k k )} 1 1 at 1/ (1 i ) L at i( x a kt) ( ) i 1 a t ( x a kt) tan ( ) L 1 ik ( xv ) 1 gt i t L L [1 ( a t/ L ) ] L [1 ( a t/ L ) ] Re{ e e e e e ( k 1 k)} at 4 [1 ( ) ] L i( x a kt) i 1 a t ( x a kt) tan ( ) L a t 1 ik [ ] ( xv ) 1 gt L i t L [1 ( a t/ L ) ] L at L Re{ e e e e e 1 at 4 [1 ( ) ] L i( x a kt) L L i 1 a t ( x a k t) tan ( ) a t ik ( xv ) 1 gt [ ] i t L L [1 ( at/ L) ] at L e 1 at [1 ( ) ] 4 e e e e L 11 Classical lectrodynamics Prof. Y. F. Chen }

38 Check the amlitude term and the hase term : Amlitude : e x a kt at ( ) L [1 ( ) ] L comare with Gaussian beam Amlitude : e i( x a kt) L a t L [ ] at L Phase : e Phase : e w ikr r [1 ( z ) ] nw nw z[1 ( )] z t z z By analog, L w (beam waist) and w( z) w [1 at 1/ ] We find that Lt ( ) [ L ( ) ] (7.99) z R L L Z R a It can be seemed as an effective divergent angle at ( ) L z 1/ In otics, ( z) [ ( ) ] is an otical invariant. at vt g For now, vg a k & as t L( t) kl L L() t i 1 a t tan ( ) L We can further another hase term : e which is in the same form as Gouy hase in otics. t3 t t1 t ut ( ) Lt ( ) Lt ( 1) L() Lt ( 3) 11 Classical lectrodynamics Lt () Prof. Y. F. Chen

39 7.1 Causality in the Connection Between D and ; Kramers-Kronig Relations Review the comlex analysis : For real axis, x 1 x dx ln( x) x (one-dimensional integral), there is a singularity at x x 1 1 x For comlex lane, 1 dz there is also a singularity at z, z but we can use a two-dimensional ath integral to calculate this integral including the singularity. i =~ i 1 Let z e, dz ie d izd, thus dz id i z If one shifts the singularity to z, then 1 z z dz i f( z) And its generalization : lim dz if ( z ) and f ( z ) is analytic at z z zz z z z Thus, f ( z ) 1 f( z) i z z dz 11 Classical lectrodynamics Prof. Y. F. Chen

40 In reality, the finite resonse time and the finite transfer seed for information roduce a temorally nonlocal connection between the inut disturbance and the outut resonse. For system with infinite information transfer seed, the imulse resonse will be a delta-function. A. Non-locality in time In revious chater, we consider Dxt (, ) xt (, ), where is constant. We know in more realistic case, the dielectric would be a function of. Under linear resonse aroximation, which means both D( x, ) and ( x, ) corresond to the same. D( x, ) ( ) ( x, ) (7.13) 1 i t 1 i t' And D( xt, ) ( x, ) e D d & D( x, ) Dxt (, ') e dt' 1 i t 1 i t' D( xt, ) ( ) ( x, ) e d ; ( x, ) xt (, ') e dt' 1 i ( t t') 1 i ( t t') D( xt, ) d dt' ( ) xt (, ') e ( ( ) e d) xt (, ') dt' - ( ) f j We know that ( )= ( ( ) ) ( 1) j ( j i j) ( ) e 1 i ( t t') 1 i ( t t') D( xt, ) ( { e( )} e d) xt (, ') dt' xt (, ) { dt'( e( ) e d) xt (, ')} Let t t' t' t & dt' d Define the resonse function of the system G( ) : 1 ( ) G( ) ( -1)e - 11 Classical lectrodynamics Prof. Y. F. Chen -i d

41 And Dxt (, ) { xt (, ) G( ) xt (, ) d} (7.15) simultaneous resonse time-delay resonse The effective contribution for G( ) is from ~, which will be roven later. B. Simle Model for G(τ), Limitations ( ) Ne f Recall, ( ) ( 1) ( ) j e m j j i j Assume single resonant mode ( ) (7.17) ( i) e i i e 1 Thus, G( ) d (7.18) ( ) G( ) Consider ω is in comlex domain, then we can use comlex analysis method to evaluate the integral in eq. (7.18) G( ) dg( ) d ( ) ( i) Im( ) i i e e i Re( ) Im( ) i Im( ) e, convergence e Im( ) Im( ) e, divergence If Im( ) i Im( ) e, divergence e Im( ) Im( ) e, convergence Therefore, when τ<, we should choose uer half-lane. when τ>, we should choose lower half-lane. 11 Classical lectrodynamics Prof. Y. F. Chen

42 i Consider i ( i ) ( ) i i Im( ) 1 4 The two roots : i i 4 i i i e 1 e e G( ) d [ ] d ( 1)( ) ( 1) ( 1) ( ) i 1 i e e 1 i 1 i ( ) ( i) ( e e ) ( i) ( ) ( ) ( ) Re( ) G( ) 1 e e e e i e ( ) i i [ ]( ) sin( ) ( ) (7.11) 1 Requires τ> The result is similar to daming oscillation. For τ<, we should choose uer half-lane, but there is no oles in the uer half-lane. No contribution. (1) i e We can further discuss the mathematical roerties of () i For cases: (1) () Analytic condition τ<,u; τ>,low + + τ<,low; τ>,u + τ<,low; τ>,u + τ<,u; Pole Position Lower half lane Uer half lane Lower half lane Contribution to G(τ) G(τ), τ> G(τ), τ> G(τ), τ< Uer G(τ), τ< 11 Classical lectrodynamics τ>,low half lane Prof. Y. F. Chen

43 C. Causality and Analyticity Domain of ε(ω) G d e i e sin( ) ( ) ( ) ( i) This kernel vanishes for τ<. The result means that at time t, only values of the electric field rior to that time enter in determining the dislacement in accord with our fundamental ideas of causality in hysical henomena. (Recall) Dxt (, ) { xt (, ) G( ) xt (, ) d} (7.111) Its validity transcends any secific model of ε(ω). 1 ( ) i ( ) i From (7.16) G( ) ( 1) e d 1 G( ) e d (7.11) The behavior of ( ) 1 for large ω can be related to the behavior of G(τ) at small time difference. ( ) i 1 i i i 1 G( ) e d 1 G( )( ) e ( ) G'( ) e d i i i i 3 1 G() ( ) G'() ( ) G''()... G( - )=, G( + ) is unhysical. The first term in the series is absent and ( ) 1 falls off at high frequencies as ω -. The asymtotic series shows, in fact, that the real and imaginary arts of ( ) 1 behave for large real ω as, ( ) 1 ( ) 1 Re( 1) O( ) ; Im( ) O( ) 3 11 Classical lectrodynamics Prof. Y. F. Chen

44 D. Kramers-Kronig Relations The analyticity of ε(ω)/ε in the uer half- ω-lane ermits the use of Cauchy s theorem to relate the real and imaginary arts of ε(ω)/ε on the real axis. For any oint z inside the close contour C in the uer half- ω-lane : ( z) 1 ( ( ')/ ) 1 i ' z 1 d ', where z i and let C The contour C may be chosen to contain real ω axis and great semicircle at infinity in the uer half-lane. ( ) ( ) From section 7.5 D 1 for large. Thus, as, 1. Im( ) (4) Therefore, the above integral can be written as ( z) 1 ( ( ')/ ) 1 1 ( ( ')/ ) 1 1 d ' 1 d' (7.116) i - ' z i - ' i i i ' e for =, ' & d ' ie d b f( ') f( ') b f( ') f( ) i Consider, d ' d' ie d a i ' i a ' i ' i e b f ( ') For and ab,, d' if( ) a ' i We know that = ( no ole inside the contour) and (if f ( ) when ). (1) () (3) (4) (4) f ( ') if d if Thus, we have - ( ) ' ) ' i (1) (3) () f ( ') The integral can be regarded as d ' i ( ') f( ') d' ' i 1 1 Therefore, P( ) i ( ' ) (7.117) '- -i ' Princial value 11 Classical lectrodynamics Prof. Y. F. Chen (1) i () (3) a b Re( ) The segment (1) and (3) are by definition the rincial art of the integral between - and. (Real axis excluded the singularities.)

45 ( z) 1 ( ') 1 Substitute (7.117 back into (7.116) 1 ( 1)[ P( i( ' )] d'. i ' - 1 ( ') 1 ( ) ( ) 1 ( ( ')/ ) 1 And ( 1) ( ' ) ' ( 1) 1 ' i d - P i i ' d ( ) ( ) 1 Im( ( ') / ) ( ) 1 Re[( ( ') / ) 1] For is comlex, Re( ) 1 P d ' & Im( ) P d ' ' ' 'Im( ( ') / ) Im( ( ') / ) P d' P d' ' ' Multily ( '+ ) on the integrand 'Re[( ( ')/ ) 1] Re[( ( ') / ) 1] P d ' + P d ' ' ' for real ( ) ( ), even function Recall: ( ) *( *) ( is comlex but is real.) for imaginary ( ) ( ), odd function ( ) 'Im( ( ')/ ) ( ) Re[( ( ')/ ) 1] We have Re( ) 1 P d ' & Im( ) P d' (7.1) ' ' The connection between absortion and anomalous disersion is contained in the relation. If existing a very narrow absortion, Im( ( ')) ( '- )..., where is a constant and dots indicate the other(smoothly varying) contribution to Im( ). Re( ( )), where reresents the sloely varying art of Re( ) resulting from the more remote contribution to Im( ). mirical knowledge of Im(ε(ω)) from absortion studies allow the calculation of Re(ε(ω)) from the first eq. in (7.1). 11 Classical lectrodynamics Prof. Y. F. Chen

46 ikz Consider the incidence of lane wave: ~ e ; k i z 1 I ikz Use I Ie ln( ) e d I I I k For n1in 1i & k i k n d We have n & nn 1 n1 n 1 There are two ways to obtain the information for ε and n : (1) Measure α to obtain n, then substitute it into the K-K relation to iterate (first assume a value for n 1 ) until reach a self-consistent result. () ε=ε 1 +iε Use simle-harmonic oscillator model to find an analytic fitting function with arameters f j, ω j, ϒ j, then calculate n 1,n, ε 1, ε. ( ) ( ) Sum rule : from (7.59) lim 1 lim (1 ) 'Im[ ( ')/ ] ( ) lim ' lim [1 Re( )] 'Im[ ( ') / ] ' (7.1) Recall, j and ; j m P d d ' ne f Z n NZ 11 Classical lectrodynamics Prof. Y. F. Chen

47 xerimentally, we can use electron energy loss sectroy (LS) to analyze ω. ~ KeV e-beam energy analyzer d 1 Im( ) ( ) & 11 Classical lectrodynamics Prof. Y. F. Chen