m(ẍ + γẋ + ω 0 x) = ee (2.118) e iωt P(ω) = χ(ω)e = ex = e2 E(ω) m ω0 2 ω2 iωγ (2.119) Z N ϵ(ω) ϵ 0 = 1 + Ne2 m j f j ω 2 j ω2 iωγ j (2.120)

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Download "m(ẍ + γẋ + ω 0 x) = ee (2.118) e iωt P(ω) = χ(ω)e = ex = e2 E(ω) m ω0 2 ω2 iωγ (2.119) Z N ϵ(ω) ϵ 0 = 1 + Ne2 m j f j ω 2 j ω2 iωγ j (2.120)"

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1 mẍ + γẋ + ω 0 x) = ee 2.118) e iωt Pω) = χω)e = ex = e2 Eω) m ω0 2 ω2 iωγ 2.119) Z N ϵω) ϵ 0 = 1 + Ne2 m j f j ω 2 j ω2 iωγ j 2.120) Z ω ω j γ j f j f j f j sum j f j = Z ω ω j, γ ϵω) ϵ 0 = 1 + Ne2 m 1 + Ne2 mω 2 j j f j f j ω 2 j ω2 iωγ j 1 + NZe2 mω ) ft) = A sinω 0 t)e γt 1 A 1 38

2 χω) = dtft)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 2.122) 2.123) γ ω 0 γ ω ω χ ω) = A ω ω 0 2 ω ω 0 ) 2 + γ ) χ ω) = A γ 2 ω ω 0 ) 2 + γ ) 39

3 ω 0 ω χ0) = A ω 0 ω0 2 + A γ2 ω ) χ ) = ) µ m =< µ > N x E N 1 N 2 m = µ N 1 N 2 N 1 + 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 2.129) P 21 P 12 P 21 = ω 0 2π e µe)/kt = P 0 e µe kt 2.130) P 12 = P 0 e µe kt 2.131) P 0 := ω 0 2π e kt 2.132) µe kt expµe/kt ) = 1 + µe/kt?? m τ 0 t = m + µ2 E kt 2.133) τ 0 = 1 2P 0 ω x < cos 2 θ >= 1/3 m = µ2 1 E 2.134) 3kT 1 iωτ 0 ϵ ϵ 0 = ϵ 0Nµ 2 3kT iωτ )

4 2.132 ft) e t/τ0 /τ 0 ϵ ω) = ϵ + ϵ s ϵ 1 + ω 2 τ ) ϵ ω) = ϵ s ϵ 1 + ω 2 ωτ 2.137) τ ν p = ω/ k = c/n 41

5 ux, t) = 1 2π Ak)e ik x ωt) d 3 k ) Ak) k 0 ux, 0) 2 Ak) 2 x k x k 1/ k ω k = ñω)ω/c 2.86 Ak) k 0 ux, t) = = = 1 2π 1 2π 1 2π Ak)e ik x ωk)t) d 3 k Ak)e i{k x ω 0+ ωk 0 ) k k k 0 ))t} d 3 k Ak)e ik x ωk 0 ) k t) d 3 k e iω0 ωk 0 ) k k 0)t = ux, 0) e iω 0 ωk 0 ) k k 0 )t 2.139) x = x ω k t Ak) k 0 ω k 0 ω = ω 0 + ωk 0 )/ k k k 0 )) ux, t) = ω ux t, 0) k = ux v gt, 0) 2.140) v g := ω, ) 2.141) k v g χω) χt) Ẽω) Pω) ñ 2 ω) 42

6 2 Ẽω) + ω2 ñ 2 ω) c 2 Ẽω) = 0 3.1) ñ Ẽω) x e ik x ωt) k 2 = k 2 = ω2 ñ 2 c 2 3.2) ñ k ω E k,ω x, t) = Ẽk, ω)eik x ωt) 3.3) B k,ω x, t) = Bk, ω)e ik x ωt) 3.4) k ω 3.2 k Ek, ω) Bk, ω) k n k = kn ρ = 0 dive = 0 k Ek, ω) = 0 n Ek, ω) = 0 3.5) rote = B/ t ik Ek, ω) = iωbk, ω), Bk, ω) = k n Ek, ω) 3.6) ω 3.6 Ek, ω) Ek, ω) Bk, ω) = 0 3.7) n E B 3.2 : 43

7 k e 1, e 2 Ex, t) = E 1 e 1 + E 2 e 2 )e ik x ωt) 3.8) E 1 E 2 e 1 e 2 1). E 1 E E 1 E 2 Ex, t) = E 1 e 1 + E 2 e 2 ) e ik x ωt) 3.9) 2). E 1 E 2 E = E 1 = E 2 arge 1 /E 2 ) = ±π/2 Ex, t) = Ee 1 ± ie 2 )e ik x ωt) 3.10) ReEx, t) = E e 1 cosk x ωt) e 2 sink x ωt) ) 3.11) e + := e 1 + ie ) e := e 1 ie ) e + e e + e e 1 e 2 e + e Ex, t) = E + e + + E e )e ik x ωt) 3.14) 3). E + E E + E + e + + E e = E + + E )e 1 + ie + E )e ) e 1 E + +E e 2 E + E 4). E + E

8 3.2.1 E x, E y Ex, t) = E x e x + E y e y )e ik x ωt) 3.16) J ) J := E x E y. 3.17) J 1 J 2 = E 1x E2x + E 1y E2y = ) J 1 J 2. ) 1 a) J x = : x 0 ) 0 b) J y = : y 1 ) c) J r = i ) d) J l = 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 f f : J J f J x, J y ) ) J x J y = T J x J y, T : f ) 3.20) 45

9 .1 ) ) T x =, T y = ) 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 = e iγ T Γ=π/2 1/4 T Γ=π/2 = ) i ) 3.23) 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 ) ) 1 1 T Γ=π/2 T Γ=π/2 2 = ) x 45 T Γ=π 1/2 ) 1 0 T Γ=π/2 = ) 1/2 x 45 ) ) 1 1 T Γ=π 2 = 1 1 = J l 3.28) x 45.3 polarization rotator T θ = cos θ sin θ ) sin θ cos θ 3.29) 46

10 θ = π/2 x 45 T θ=π/2 ) 1 1 T θ=π/2 2 = ) ) ) 1 1 = ) x x, y x, y Rθ) x y ) ) x cos θ = Rθ), Rθ) := y sin θ ) sin θ 3.31) cos θ x x θ J = Rθ)J 3.32) J T J T J out = T J in Rθ)J out = T Rθ)J in J out = Rθ) 1 T Rθ)J in J out = T J in T = Rθ)T Rθ) 1 = Rθ)T R θ) 3.33) x, y) x, y ) T Rθ)T R θ) normal mode T J normal = µj normal. 3.34) dimkert ) 0 rankt = 2 47

11 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 ) e 1, e 2 ) e +, e ) E 1 = e 1 E 3.36) E 2 = e 2 E 3.37) E + = e + E 3.38) E = e E 3.39) E 1 = a 1 e iδ ) E 2 = a 2 e iδ2 3.41) E + = a + e iδ ) E = a e iδ 3.43) 3.44) s 0 = a a 2 2, ) 3.45) s 1 = a 2 1 a 2 2, y x 3.46) s 2 = 2a 1 a 2 cosδ 2 δ 1 ), 3.47) s 3 = 2a 1 a 2 sinδ 2 δ 1 ), 3.48) or 3.49) s 0 = a a 2, ) 3.50) s 1 = a + a cosδ + δ ), 3.51) s 2 = a + a sinδ + δ ), 3.52) s 3 = a 2 + a 2, ) 3.53) a 1, a 2, δ 2 δ 1 or a +, a, δ + δ s 0 s 3 s 2 0 = s s s ) 48

12 s 0 = 1 s 0,s 1,s 2,s 3 < s 2 0 > < s 2 1 > + < s 2 2 > + < s 2 3 > 3.55) < s 1 >=< s 2 >=< s 3 >= n 1 n 2 θ 1 1. Kinetics a) b) n 1 sin θ 1 = n 2 sin θ 2 2. Dynamics E B a) b) 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 ) 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 ) 49

13 : 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 n 2 c x 2 x 0 ) 2 + y ) 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 = ) n 1 sin θ 1 = n 2 sin θ 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 eik 1 x ωt) B i = k1 ω 3.60) E0 i eik1 x ωt) { Er = E 0 ik 1 x ωt) re 3.61) B r = k 1 ω E 0 re ik 1 x ωt) { E t = E 0 t e ik 2 x ωt) B t = k2 ω E0 t e ik2 x ωt) 3.62) n i.e. e z D B ϵ 1 E i + E r ) ϵ 2 E t ) n z=0 = 0, 3.63) k 1 E i + k 1 E r k 2 E t ) n z=0 = 0, 3.64) 50

14 E H E i + E r E t ) n z=0 = 0, 3.65) k 1 E i + k 1 E r k 1 E t ) n z=0 = 0, 3.66) µ 1 = µ 2 = x y E i E r E t x, y k i n = k r n = k t n. 3.67) ϵ1 E 0 i + E 0 r) ϵ 2 E 0 ) t n = 0, 3.68) k1 E 0 i + k 1 E 0 r k 2 E 0 ) t n = 0, 3.69) E 0 i + E 0 r E 0 ) t n = 0, 3.70) k1 E 0 i + k 1 E 0 r k 2 E 0 ) t n = 0, 3.71) a). E n s E n E 0 i + E 0 r E 0 t = ) 3.71 k 1 n)e 0 i + k 1 n)e 0 r k 2 n)e 0 t = ) A B C = BA C) AB C) E n = k θ n 1 E 0 i E 0 r) cos θ 1 n 2 E 0 t cos θ 2 = ) r12 s ts 12 r12 s = E 0 r E 0 = n 1 cos θ 1 n 2 cos θ ) i n 1 cos θ 1 + n 2 cos θ 2 t s 12 = E 0 t = 2n 1 cos θ ) n 1 cos θ 1 + n 2 cos θ 2 E 0 i 51

15 b). E p E i,r,t H i,r,t a) r p 12 = E 0 r E 0 = n 2 cos θ 1 n 1 cos θ 2 = n2 2 cos θ 1 n 1 n 2 2 n2 1 sin2 θ 1 i 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 t p 12 = E 0 t E 0 i 3.77) = 2n 1 cos θ 1 2n 1 n 2 cos θ 1 = 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 ) θ 1 = θ 2 = 0 a b s p r s,p = n 1 n 2 n 1 + n ) t s,p = 2n 1 n 1 + n ) s p π E r /E i < r12 s = sinθ 2 θ 1 ) sinθ 2 + θ 1 ) r p 12 = tanθ 1 θ 2 ) tanθ 1 + θ 2 ) 3.81) 3.82) n 2 n 1 t r 3.75, 3.76, 3.77, 3.78 r = r 3.83) tt + r 2 = ) n 1 < n 2 cos θ 2 > cos θ 1 s 3.75 r s 12 < 0 p 3.77 θ B ) θ B = tan 1 n2 3.85) n 1 52

16 r p 12 s p 3.4 d n r 121 r 121 = r 121 e iϕ ) 1 r 2 12 eiϕ 3.86) t 121 = 1 r2 12)e iϕ/2 1 r 2 12 eiϕ 3.87) ϕ := 2nωd c cos θ ) ϕ := 2nωd/c) cos θ 2 ϕ = 2nωd/c n ϕ r t = ) 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 3.90) Z 0 53

17 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 E ) S = c/n)u c/n θ I = 1 n E 0 2 cos θ 3.92) 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 = n 1 Z 0 E 0 r 2 cos θ ) 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 3.94) 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 ) ) 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 ) 54

18 3.6 sin θ 2 = n 1 n 2 sin θ ) n n n 1 > n θ 2 > θ 1 θ θ 2 θ 2 θ 1 i.e., n 1 /n 2 ) sin θ 1 = 1 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 ) cos 2 θ 2 = 1 sin 2 θ 2 < ) cos θ 2 n1 cos θ 2 = ±i n 2 ) 2 sin 2 θ ) s p r12 s = n 1 cos θ 1 n 2 cos theta 2 n 1 cos θ 1 + n 2 cos theta 2 = 1, 3.101) r p 12 = n 2 cos θ 1 n 1 cos theta 2 n 2 cos θ 1 + n 1 cos theta 2 = ) 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 ) sin 2 θ 1 n 2 /n 1 ) 2 n 2 /n 1 ) 2 cos θ ) tan θ 2 = cos θ 1 sin 2 θ 1 n 2 /n 1 ) 2 sin 2 θ ) 55

19 p s θ θ 1 θ 1 = π/2, θ c s p d ) tan θ 2 = ) dθ 1 sin 2 θ 1 = 2n2 2 n n ) θ 1 θ m tan θ m 2 = n2 1 n 2 2 2n 1 n ) θ 1 λ/4 θ = π 2 π/4 n = 1.5 θ i = θ 1 θ c E = E t e ik 2 sin θ 2 x+k 2 cos θ 2 z) iωt = E t e i ) ) n k 1 n1 2 2 n sin θ 1 x±k 2 i 2 n sin 2 θ 1 1 z iωt 2 = E t e i k 2 n 1 n 2 sin θ 1 x ωt ) ) e k n1 2 2 n sin 2 θ 1 1 z ) + z z 1 l 2 = λ ) k 2 n 1 /n 2 ) sin 2 θ

20 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 TM p θ 1 θ c 1 2 t 12 = 2n 1 cos θ 1 n 2 cos θ 1 + n 1 cos θ ) r 12 = n 2 cos θ 1 n 1 cos θ 2 n 2 cos θ 1 + n 1 cos θ ) 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 ) r 23 = n 1 cos θ 2 n 2 cos θ 1 n 1 cos θ 2 + n 2 cos θ ) 57

21 1 2, 2 3 t 23 = t 12, r 23 = r r 23 = r ) 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 r ) E t = 1 r2 1 r 2 e iϕ eiϕ/2 E 0, ϕ := 2d n 2ω cos θ ) c 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 ) r 4 2r 2 + 4r 2 sin 2 ϕ r2 1 r 2 ) sin 2 ϕ n cos 2 θ 1 n 2 1 cos2 θ 2 2n 1 n 2 cos θ 1 cos θ 2 sin ϕ k 2 1 k 2 2 2k 1 k 2 sin ϕ 2 ) 2 ) ) 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 k 2 = iη, 3.119) n 1 n 2 n Q = 1 k1 2 + n1 n 2 η ) 2k 1 η 58

22 1 T = 1 + Q sinhηd)) ) η,q d d sinh d n 1 = 1.5, θ 1 = 51.8 V 0 E 2 ω ) 2 p ) 2 k2 2 = c n 2mE V 0 ) 1 cos θ 2 = ħ ħ ) 2mV0 E) η = ħ k 1 = 2mE ħ Q = k2 1 + η 2 2k 1 η 3.123) 3.124) 3.125) 59

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 +

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

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