ω 0 m(ẍ + γẋ + ω0x) 2 = ee (2.118) e iωt x = e 1 m ω0 2 E(ω). (2.119) ω2 iωγ Z N P(ω) = χ(ω)e = exzn (2.120) ϵ = ϵ 0 (1 + χ) ϵ(ω) ϵ 0 = 1 +

2.6 2.6.1 ω 0 m(ẍ + γẋ + ω0x) 2 = ee (2.118) e iωt x = e 1 m ω0 2 E(ω). (2.119) ω2 iωγ Z N P(ω) = χ(ω)e = exzn (2.120) ϵ = ϵ 0 (1 + χ) ϵ(ω) ϵ 0 = 1 + Ne2 m j f j ω 2 j ω2 iωγ j (2.121) Z ω ω j γ j f j 2.121 f j j f j = Z 2.121 ω ω j, γ ϵ(ω) ϵ 0 = 1 + Ne2 m 1 Ne2 mω 2 j j f j f j ω 2 j ω2 iωγ j 1 NZe2 mω 2 (2.122) 2.107 2.6.2 f(t) = A sin(ω 0 t)e γt 1 1 38

A χ(ω) = dtf(t)e iωt = 0 { } 1 2 ω + ω 0 + iγ 1 ω ω 0 + iγ = ω 0 A (ω + iγ) 2 ω0 2 ω 0 A ω0 2 ω2 2iγω (2.123) (2.124) γ ω 0 γ 2 2.124 2.119 2.124 ω ω 0 2.123 χ (ω) = A ω ω 0 2 (ω ω 0 ) 2 + γ 2 (2.125) χ (ω) = A γ 2 (ω ω 0 ) 2 + γ 2 (2.126) 2.126 39

ω 0 ω χ(0) = A A (2.127) ω 0 χ( ) = 0 (2.128) f(t) = j A j sin(ω j t)e γt 2.127 χ(0) = j A j/ω j ω 0, ω 1, 2.6.3 µ m =< µ > N x E E N 1 N 2 m = µ N 1 N 2, (2.129) N m t = µ N ( N 1 t N 2 t ) = µ N {( P 21N 1 + P 12 N 2 ) (P 21 N 1 P 12 N 2 )} = (P 21 + P 12 )m (P 21 P 12 )µ (2.130) P 21 P 12 1 2 2 1 40

P 21 = ω 0 2π e( µe)/kt = P 0 e µe/kt (2.131) P 12 = P 0 e µe/kt (2.132) P 0 := ω 0 2π e /kt. (2.133) µe kt exp(µe/kt ) = 1 + µe/kt 2.130 m τ 0 t = m + µ2 E kt (2.134) τ 0 = 1 / 2P 0 ω E(ω) 2.135 ω x < cos 2 θ >= 1/3 m = µ2 1 E (2.135) 3kT 1 iωτ 0 ϵ ϵ 0 = ϵ 0Nµ 2 3kT 1 1 iωτ 0. (2.136) 2.132 f(t) e t/τ0 /τ 0 ϵ 1 (ω) = ϵ + (ϵ s ϵ ) 1 + ω 2 τ 2 (2.137) ϵ ωτ (ω) = (ϵ s ϵ ) 1 + ω 2 τ 2 (2.138) ϵ 0 = ϵ(0), ϵ = ϵ( ) 2.6.4 41

2.6.5 ν p = ω/ k = c/n u(x, t) = 1 2π A(k)e i(k x ωt) d 3 k. (2.139) A(k) k 0 u(x, 0) 2 A(k) 2 x k x k 1/2 2.139 k ω ( 2.86 k = ñ(ω)ω/c A(k) k 0 42

u(x, t) = = = 1 2π 1 2π 1 2π A(k)e i(k x ω(k)t) d 3 k A(k)e i{k x (ω 0+ ω(k 0 ) k (k k 0 ))t} d 3 k A(k)e ik (x ω(k 0 ) k t) d 3 k e i(ω 0 ω(k 0 ) k k 0 )t = u(x, 0) e i(ω 0 ω(k 0 ) k k 0 )t (2.140) x = x ω(k 0) k t A(k) k 0 ω k 0 ω = ω 0 + ω(k 0 )/ k (k k 0 )) 2.140 u(x, t) = ω u(x t, 0) k = u(x v gt, 0) (2.141) v g := ω, ( ) (2.142) k 2.139 v g 3 3.1 2.3.2 χ(ω) χ(t) Ẽ(ω) P(ω) 2.3.3 ñ 2 (ω) 2 Ẽ(ω) + ω2 ñ 2 (ω) c 2 Ẽ(ω) = 0 (3.1) ñ Ẽ(ω) x e i(k x ωt) k 2 = k 2 = ω2 ñ 2 c 2 (3.2) ñ k ω E k,ω (x, t) = Ẽ(k, ω)ei(k x ωt) (3.3) B k,ω (x, t) = B(k, ω)e i(k x ωt) (3.4) 43

k ω 3.2 k E(k, ω) B(k, ω) k n k = kn ρ = 0 dive = 0 k E(k, ω) = 0 n E(k, ω) = 0. (3.5) rote = B/ t ik E(k, ω) = iωb(k, ω), B(k, ω) = k n E(k, ω). (3.6) ω 3.6 E(k, ω) E(k, ω) B(k, ω) = 0 (3.7) 3.5-3.7 n E B 3.2 : k e 1, e 2 E(x, t) = (E 1 e 1 + E 2 e 2 )e i(k x ωt) (3.8) E 1 E 2 ϕ 1 = arg(e 1 ), ϕ 2 = arg(e 2 ) e 1 e 2 (1). E 1 E 2 3.8 E 1 E 2 E(x, t) = (E 1 e 1 + E 2 e 2 ) e i(k x ωt) (3.9) (2). E 1 E 2 E = E 1 = E 2 arg(e 1 /E 2 ) = ±π/2 E(x, t) = E(e 1 ± ie 2 )e i(k x ωt) (3.10) ReE(x, t) = E( e 1 cos(k x ωt) e 2 sin(k x ωt) ) (3.11) 44

3.10 3.11 e + := e 1 + ie 2 2 (3.12) e := e 1 ie 2 2 (3.13) e + e e + e e 1 e 2 e + e E(x, t) = (E + e + + E e )e i(k x ωt) (3.14) (3). E + E E + E E + e + + E e = (E + + E )e 1 + i(e + E )e 2 (3.15) e 1 E + +E e 2 E + E (4). E + E 3.15 3.2.1 E x, E y E(x, t) = (E x e x + E y e y )e i(k x ωt) (3.16) J J := E x E y. (3.17) J 1 J 2 = E 1x E2x + E 1y E2y = 0 (3.18) J 1 J 2 45

. 1 (a) J x = : x 0 0 (b) J y = : y 1 (c) J r = 1 1 2 i (d) J l = 1 1 2 i J x J y J r J l J = α x J x + α y J y = α r J r + α l J l (3.19) J x J y J r J l 3.18 3.2.2 f f : J J f J x, J y J x J y = T J x J y, (T : f ) (3.20).1 1 0 0 0 T x =, T y = 0 0 0 1 (3.21) T x T y x y T x J x = J x, T x J y = 0, T x J r = 1 2 J x (3.22).2 wave retarder T Γ = ( 1 0 0 e iγ ) (3.23) 46

T Γ=π/2 1/4 1 0 T Γ=π/2 = 0 i (3.24) 1/4 x 45 1 1 T Γ=π/2 2 = 1 1 = J l (3.25) 1 2 i x 45 T Γ=π/2 1 1 T Γ=π/2 T Γ=π/2 2 = 1 1 1 2 1 (3.26) x 45 T Γ=π 1/2 1 0 T Γ=π/2 = 0 1 (3.27) 1/2 x 45 1 1 T Γ=π 2 = 1 1 = J l (3.28) 1 2 1 x 45.3 polarization rotator T θ = ( cos θ sin θ ) sin θ cos θ (3.29) θ = π/2 x 45 T θ=π/2 1 1 T θ=π/2 2 1 0 1 1 1 = = 1 1 1 0 2 1 2 1 (3.30) x 45 3.2.3 x, y x, y R(θ) ( x y ) ( x cos θ = R(θ), R(θ) := y sin θ ) sin θ (3.31) cos θ 47

x x θ J = R(θ)J (3.32) J T J T J out = T J in R(θ)J out = T R(θ)J in J out = R(θ) 1 T R(θ)J in J out = T J in T = R(θ)T R(θ) 1 = R(θ)T R( θ) (3.33) (x, y) (x, y ) T R(θ)T R( θ) normal mode T J normal = µj normal. (3.34) 2 2 2 3.21 dim(kert ) 0 rankt = 2 J J 1 J 2 J = j 1 J 1 + j 2 J 2 T J = j 1 µ 1 J 1 + j 2 µ 2 J 2 (3.35) 1/4 x 45 45 3.33 1 0 T x,θ=45 = R( π/4) R(π/4) 0 0 = 1 1 1 1 0 1 1 2 1 1 0 0 1 1 = 1 1 1 (3.36) 2 1 1 48

1/4 x 45 T + T + = T x,θ=45 T Γ=π/2 = 1 1 1 1 0 2 1 1 0 i = 1 1 i 2 1 i T + J r T + J r = 1 1 i 1 1 = 1 1 2 1 i 2 i 2 1 (3.37) (3.38) T + J r = 1 T + 45 T + 0 T + J r = (3.39) 0 T + J = α r J r + α l J l α r 1/4 x 45 T T = 1 1 1 (3.40) 2 1 1 T J l = 1/ 2 t (1 i) 45 J r T + T 3.2.4 (e 1, e 2 ) (e +, e ) E 1 := e 1 E (3.41) E 2 := e 2 E (3.42) E + := e + E = e 1 ie 2 E = 1 (E 1 ie 2 ) (3.43) 2 2 E := e E = e 1 + ie 2 2 E = 1 2 (E 1 + ie 2 ) (3.44) 49

E 1, E 2 E +, E E 1 = (E + + E )/ 2 (3.45) E 2 = i(e + E )/ 2. (3.46) E 1 = a 1 e iδ 1 (3.47) E 2 = a 2 e iδ 2 (3.48) E + = a + e iδ+ (3.49) E = a e iδ (3.50) s 0 = E 1 2 + E 2 2 = a 2 1 + a 2 2 (3.51) s 1 = E 1 2 E 2 2 = a 2 1 a 2 2 (3.52) s 2 = 2Re(E1E 2 ) = 2a 1 a 2 cos(δ 2 δ 1 ) (3.53) s 3 = 2Im(E1E 2 ) = 2a 1 a 2 sin(δ 2 δ 1 ). (3.54) s 0 - s 3 s 0 = E + 2 + E 2 = a 2 + + a 2 (3.55) s 1 = 2Re(E + E ) = 2a + a cos(δ + δ ) (3.56) s 2 = 2Im(E + E ) = 2a + a sin(δ + δ ) (3.57) s 3 = E + 2 E 2 = a 2 + a 2 (3.58) 3.51 3.52 s 0 s 1 e 1 e 2 3.58 s 3 1/4 45 T ± s 2 s 2 45 45 J = t (E 1 E 2 ) 3.36 45 T x,θ=45 I 45 = E 1 + E 2 2 2 (3.59) 50

45 T x,θ= 45 T x,θ= 45 3.36 I 45 = E 1 E 2 2 2 (3.60) I 45 I 45 s 2 3.51-3.58 a 1, a 2, δ 2 δ 1 or a +, a, δ + δ s 0 s 3 s 2 0 = s 2 1 + s 2 2 + s 2 3 (3.61) s 0 = 1 T x, T y, T x,θ=45, T x,θ= 45, T +, T s 0,s 1,s 2,s 3 < s 0 >,< s 1 >,< s 2 >,< s 3 > (3.61 < s 0 > 2 < s 1 > 2 + < s 2 > 2 + < s 3 > 2 (3.62) < s 1 >=< s 2 >=< s 3 >= 0 3.3 2 n 1 n 2 θ 1 51

1. Kinetics (a) (b) n 1 sin θ 1 = n 2 sin θ 2 2. Dynamics E B (a) (b) 3.3.1 Kinetics x y k x k y k 1 e x = k 1 e x = k 2 e x k 1 sin θ 1 = k 1 sin θ 1 = k 2 sin θ 2. (3.63) k 1 = k 1 = n 1 ω/c k 2 = n 2 ω/c n 1 sin θ 1 = n 1 sin θ = n 2 sin θ 2 (3.64) : n 1 A x 1, y 1 n 2 B x 2, y 2 C x 0, 0 T = = n 1 c AC CB + c/n 1 c/n 2 (x 1 x 0 ) 2 + y 2 1 + n 2 c (x 2 x 0 ) 2 + y 2 1 (3.65) T dt/dx 0 = 0 dt = n 1 dx 0 c x 0 x 1 n 2 (x1 x 0 ) 2 + y1 2 c x 2 x 0 (x2 x 0 ) 2 + y 2 2 = n 1 c sin θ 1 n 2 c sin θ 2 = 0 (3.66) n 1 sin θ 1 = n 2 sin θ 2 52

3.3.2 Dynamics Maxwell E H D B k 1 = k 1 e 1, k 1 = k 1e 1, k 2 = k 2 e 2 ( e 1 = e 1 = e 2 = 1) { E i = E 0 i ei(k 1 x ωt) B i = k1 ω (3.67) E0 i ei(k1 x ωt) { Er = E 0 i(k 1 x ωt) re (3.68) B r = k 1 ω E 0 re i(k 1 x ωt) { E t = E 0 t e i(k 2 x ωt) B t = k2 ω E0 t e i(k2 x ωt) (3.69) n i.e. e z D B E H (ϵ 1 (E i + E r ) ϵ 2 E t ) n z=0 = 0, (3.70) (k 1 E i + k 1 E r k 2 E t ) n z=0 = 0, (3.71) (E i + E r E t ) n z=0 = 0, (3.72) (k 1 E i + k 1 E r k 1 E t ) n z=0 = 0, (3.73) µ 1 = µ 2 = 1 3.70-3.73 x y E i E r E t x, y 3.70-3.73 k i n = k r n = k t n. (3.74) ( ϵ1 (E 0 i + E 0 r) ϵ 2 E 0 ) t n = 0, (3.75) ( k1 E 0 i + k 1 E 0 r k 2 E 0 ) t n = 0, (3.76) ( E 0 i + E 0 r E 0 ) t n = 0, (3.77) ( k1 E 0 i + k 1 E 0 r k 2 E 0 ) t n = 0, (3.78) 3.75-3.78 53

(a). E n s 3.75 3.77 E n 3.78 E 0 i + E 0 r E 0 t = 0 (3.79) (k 1 n)e 0 i + (k 1 n)e 0 r (k 2 n)e 0 t = 0 (3.80) A B C = B(A C) A(B C) E n = 0 3.80 k θ n 1 (E 0 i E 0 r) cos θ 1 n 2 E 0 t cos θ 2 = 0 (3.81) 3.79 3.81 r s 12 ts 12 r s 12 = t s 12 = E 0 r E 0 i E 0 t E 0 i (b). E p = n 1 cos θ 1 n 2 cos θ 2 n 1 cos θ 1 + n 2 cos θ 2 (3.82) = 2n 1 cos θ 1 n 1 cos θ 1 + n 2 cos θ 2 (3.83) 3.75-3.78 E i,r,t H i,r,t (a) r p 12 = E 0 r E 0 = H 0 r i H 0 i = n 2 cos θ 1 n 1 cos θ 2 n 2 cos θ 1 + n 1 cos θ 2 = n 2 2 cos θ 1 n 1 n 2 2 n2 1 sin2 θ 1 n 2 2 cos θ 1 + n 1 n 2 2 n2 1 sin2 θ 1 t p 12 = = (3.84) E 0 t E 0 = H 0 t i H 0 i 2n 1 cos θ 1 n 2 cos θ 1 + n 1 cos θ 2 = 2n 1 n 2 cos θ 1 n 2 2 cos θ 1 + n 1 n 2 2 n2 1 sin2 θ 1 (3.85) θ 1 = θ 2 = 0 a b r s,p = n 2 n 1 n 1 + n 2 (3.86) t s,p = 2n 1 n 1 + n 2 (3.87) 54

s p s p π E r /E i < 0 3.82 3.84 r12 s = sin(θ 2 θ 1 ) sin(θ 2 + θ 1 ) r p 12 = tan(θ 1 θ 2 ) tan(θ 1 + θ 2 ) (3.88) (3.89) n 2 n 1 t r 3.82, 3.83, 3.84, 3.85 r = r (3.90) tt + r 2 = 1 (3.91) 3.3.3 n 1 < n 2 cos θ 2 > cos θ 1 3.82 s r s 12 < 0 p 3.84 θ B θ B = tan 1 n2 (3.92) n 1 r p 12 s p 3.4 d n 55

r 121 r 121 = r 12(1 e iϕ ) 1 r 2 12 eiϕ (3.93) t 121 = (1 r2 12)e iϕ/2 1 r 2 12 eiϕ (3.94) ϕ := (2nωd/c) cos θ 2 ϕ = 2nωd/c n ϕ r 121 2 + t 121 2 = 1 (3.95) 3.5 z S = E H = ϵ 0 c 2 (E B) E,H S = 1 2 E H S l 1 S = Re 2 E H = 1 ϵ 2 µ E0 2 l = 1 2 n 1 E 0 2 l (3.96) Z 0 n = ϵ/ϵ 0 µ = µ 0 Z 0 Z 0 = µ 0 /ϵ 0 = 376.7Ω u = 1 (ϵe E + 1µ ) 4 B B = 1 2 ϵ E0 2 (3.97) S = (c/n)u c/n θ 56

I = 1 n E 0 2 cos θ (3.98) 2 Z 0 ( ) P = IV = V 2 /R R Z I i = n 1 Z 0 E 0 i 2 cos θ 1 I r = n1 Z 0 E 0 r 2 cos θ 1 (3.99) I t = n 2 Z 0 E 0 t 2 cos θ 2 p { } n1 I r + I t = r 12 2 n 2 cos θ 1 t 12 2 cos θ 2 E 0 i 2 Z 0 Z 0 { 2 n 1 n2 cos θ 1 n 1 cos θ 2 = cos θ 1 Z 0 n 2 cos θ 1 + n 1 cos θ 2 + n ( ) } 2 2 2n 1 cos θ 1 cos θ 2 E 0 i Z 0 n 2 cos θ 1 + n 1 cos θ 2 = n 1 cos θ 1 E 0 i 2 Z 0 = I i (3.100) T + R = 1 T R T = I t = n 2 t 2 cos θ 2 12 = 4n 1n 2 cos θ 1 cos θ 2 I i n 1 cos θ 1 (n 2 cos θ 1 + n 1 cos θ 2 ) 2 (3.101) R = I r I i = r 2 12 = n 2 cos θ 1 n 1 cos θ 2 n 2 cos θ 1 + n 1 cos θ 2 (3.102) 3.6 sin θ 2 = n 1 n 2 sin θ 1 (3.103) n n n 1 > n 2 3.103 θ 2 > θ 1 θ 1 3.103 θ 2 θ 2 θ 1 i.e., (n 1 /n 2 ) sin θ 1 = 1 57

θ 1 θ c. n 1 = 1.5 n 2 = 1 θ c = 41.8 θ 1 θ c θ 2 sin θ 2 = n 1 sin θ 1 1 n 2 (3.104) cos 2 θ 2 = 1 sin 2 θ 2 < 0 (3.105) cos θ 2 (n1 cos θ 2 = ±i n 2 ) 2 sin 2 θ 1 1. (3.106) s p r12 s = n 1 cos θ 1 n 2 cos θ 2 n 1 cos θ 1 + n 2 cos θ 2 = 1, (3.107) r p 12 = n 2 cos θ 1 n 1 cos θ 2 n 2 cos θ 1 + n 1 cos θ 2 = 1 (3.108) r s = e iθ s,r p = e iθ p s,p θ = θ p θ s tan θ s 2 = tan θ p 2 = sin 2 θ 1 (n 2 /n 1 ) 2 cos θ 1 (3.109) sin 2 θ 1 (n 2 /n 1 ) 2 (n 2 /n 1 ) 2 cos θ 1 (3.110) tan θ 2 = cos θ 1 sin 2 θ 1 (n 2 /n 1 ) 2 sin 2 θ 1. (3.111) p s θ θ 1 θ 1 = π/2, θ c s p d tan θ 2 = 0 (3.112) dθ 1 sin 2 θ 1 = 2n2 2 n 2 1 + n2 2 (3.113) θ 1 θ m tan θ m 2 = n2 1 n 2 2 2n 1 n 2 (3.114) 58

θ 1 λ/4 θ = π 2 π/4 n = 1.5 θ i = 51.8 3.6.1 θ 1 θ c i(k2 sin θ2 x+k2 cos θ2 z) iωt E = E t e = E t e i ( k 2 n 1 n 2 sin θ 1 x±k 2i = E t e i ( k 2 n 1 n 2 sin θ 1 x ωt ) n1 2 n sin 2 θ 2 1 1 z iωt ) e k n1 2 2 n sin 2 θ 1 1 z 2 (3.115) + z z 1 l 2 = λ 2 (3.116) k 2 (n 1 /n 2 ) sin 2 θ 1 1 x k 2 (n 1 /n 2 ) sin θ 1 n 1 = 1.5, n 2 = 1, θ 1 = 51.8 k 2 (n 1 /n 2 ) sin θ 1 = (n 2 ω/c)(n 1 /n 2 ) sin θ 1 = 11.8 ω/c θ 1 θ c Phase shift G-H 59

3.6.2 TM p θ 1 θ c 1 2 t 12 = 2n 1 cos θ 1 n 2 cos θ 1 + n 1 cos θ 2 (3.117) r 12 = n 2 cos θ 1 n 1 cos θ 2 n 2 cos θ 1 + n 1 cos θ 2 (3.118) 2 3 n 1 n 2, n 2 n 1, θ 1 θ 2,θ 2 θ 1 t 23 = 2n 2 cos θ 2 n 1 cos θ 2 + n 2 cos θ 1 (3.119) r 23 = n 1 cos θ 2 n 2 cos θ 1 n 1 cos θ 2 + n 2 cos θ 1 (3.120) 1 2, 2 3 t 23 = t 12, r 23 = r 12 3.90 3.91 r 23 = r 12 (3.121) t 12 t 23 = 4n 1n 2 cos θ 1 cos θ 2 (n 2 cos θ 1 + n 1 cos θ 2 ) 2 = 1 r2 23 (3.122) E t = 1 r2 1 r 2 e iϕ eiϕ/2 E 0, ϕ := 2d n 2ω cos θ 2 (3.123) c 60

r 2 = r 2 12 = r 2 23 T = = = = = 2 E t E 0 = (1 r 2 ) 2 1 2r 2 cos ϕ + r 4 (1 r 2 ) 2 1 + r 4 2r 2 + 4r 2 sin 2 ϕ 2 1 1 + 4r2 (1 r 2 ) sin 2 ϕ 2 2 1 ( n 2 1 + 2 cos 2 θ 1 n 2 1 cos2 θ 2 2n 1 n 2 cos θ 1 cos θ 2 sin ϕ 2 1 1 + ( k 2 1 k 2 2 2k 1k 2 sin ϕ 2 ) 2 ) 2 (3.124) k 1 = (ω/c)n 2 cos θ 1, k 2 = (ω/c)n 1 cos θ 2 cos ϕ = 1 2 sin 2 (ϕ/2) 2 z z cos θ 2 k 2 η,q n 2 T = 1 1 + (Q sinh(ηd)) 2. (3.127) η,q d d sinh d n 1 = 1.5, θ 1 = 51.8 k 2 = iη, (3.125) n 1 n 2 n Q = 1 k1 2 + n 1 n 2 η 2 (3.126) 2k 1 η 61

V 0 E 2 ( ω ) 2 ( p ) 2 k2 2 = c n 2m(E V 0 ) 1 cos θ 2 = ħ ħ 2 (3.128) 2m(V0 E) η = ħ k 1 = 2mE ħ Q = k2 1 + η 2 2k 1 η (3.129) (3.130) (3.131) 62

R(ω) ñ(ω) = n(ω) + iκ(ω) 1. 2-4-2 f(ω) (a) S(t) e.g. X(t) e.g. 2-3-2 X(t) f(t) X(t) = t f(t t )S(t )dt (3.132) S(t) t < 0 f(t) = 0 f(t) f(ω) = 0 f(τ)e iωτ dτ (3.133) ω τ f(ω) ω Im(ω) >= 0 f(ω) 2-4-2 f(ω) 2. R(ω) ñ(ω) R r = r e iθ E in E r R = r 2 (a) n 0 n(ω) = n(ω) + iκ(ω) r(ω) r = n + iκ n 0 n + iκ + n 0 (3.134) n κ R θ (b) f(ω) ln f(ω) (c) ln r r r θ 63

(d) θ R (a) n κ R 2.6 R(ω) 64