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2 Maxwell Hartree-Fock Na

4 0 Einstein Laser 1960 CD Laser I, II I, II, III I, II III 3

5 -, 000, 1981,

6 e e r m k m r(t) = mω 0r(t) + ( e)e(r, t) + ( e)ṙ(t) B(r, t) ω 0 ω 0 k/m m r(t) = mω 0r(t) + ( e)e(0, t) p a ( e)r p a E(0, t) r E x ẍ(t) + ω0x(t) = e E(t) (1.1) m p a E 5

7 1. E = 0 ẍ(t) + ω 0x(t) = 0 (1.) x(t) = x 0 cos(ω 0 t + θ) x 0, θ x(t) = Re[x 0 e i(ω 0t+θ) ] x (ω 0) e iω 0t + x ( ω 0) e +iω 0t x ( ω0) = x (ω 0), x ( ω0) = x (ω0) = x 0 E a E a = 1 mẋ(t) + 1 kx(t) = 1 mω 0x 0 (1.3) K U K = 1 mẋ(t) = 1 4 mω 0x 0, U = 1 kx(t) = 1 4 mω 0x 0 (1.4) E a = m ω 0 x(t) 1.3 (1.4) 1.3 p a (t) ( e)r(t) = Re [ ( e)x 0 e iω 0t ] ˆx Re [ p 0 e iω 0t ] ˆx ˆx x W rad W rad = ω4 0 1πε 0 c 3 p 0 = ω4 0 1πε 0 c 3 e x 0 E a (1.5) 6

8 E a Ė a (t) = W rad τ 0 E a (t); E a (t) = E 0 e (/τ 0) t (1.6) (1.5) 1 τ 0 ω 0 (1.7) (1.) ẍ(t) + τ 0 ẋ(t) + ω 0x(t) = 0 (1.8) (1.7) (1.6) (1.8) [(1.8) (1.6) ] (1.8) x(t) = x 0 (t)e iω 0t (1.9) x 0 (t) e iω 0t x 0 (t) ẋ 0 (t) ω 0 x 0 (t), ẍ 0 (t) ω 0 ẋ 0 (t) ω 0x 0 (t), (1.10) (1.9) (1.8) ẍ 0 (t) iω 0 ẋ 0 (t) ω0x 0 (t) + ) (ẋ 0 (t) iω 0 x 0 (t) + ω τ 0x 0 (t) = 0 0 x 0 (t) ẋ 0 (t) 1 τ 0 x 0 (t); x 0 (t) e t/τ 0 E a (1.3) Ė a (t) = τ 0 E a (t) 7

9 1.4 (1.8) x(t) = x 0 (t)e iω 0t e t/τ 0 iω 0 t E(t) [ E(t) x(t); E(t) = E 0 exp t ] iω 0 t for (t > 0) τ 0 Fourier E(ω) E(ω) = 0 E(t)e iωt dt = E 0 i(ω 0 ω) + 1/τ 0 I(ω) I(ω) = E(ω) 1 (ω 0 ω) + (1/τ 0 ) ω 0 /τ ω 0 ω (1.8) ẍ(t) + ẋ(t) + ω τ 0x(t) = ( e) E(t); (1.11) 0 m E(t) = E (ω) e iωt + E ( ω) e +iωt, E ( ω) = E (+ω) (1.11) ω ω 0 ω 0, ω (1.1) x(t) = x (ω) 0 e iωt + x ( ω) 0 e +iωt 8

10 e iωt x (ω) 0 = ( e/m)e (ω) (ω 0 ω ) iω/τ 0 (1.13) E (ω) E ( ω) = E (ω) = E (ω) E(t) = E (ω) cos ωt (1.14) x (ω) 0 x (ω) 0 e iθ x (ω) 0 (u + iv); u = cos θ, v = sin θ x(t) x(t) = x (ω) 0 e iωt + x ( ω) 0 e +iωt = Re [ ] = Re x (ω) 0 e i(ωt θ) = x (ω) 0 (u cos ωt + v sin ωt) [ x (ω) 0 e iωt ] = x (ω) 0 cos(ωt θ) (1.14) θ u v 90 θ x (ω) 0 u v x (ω) 0, θ, u, v x (ω) 0 ( e/m)e (ω) = 1 (ω 0 ω ) + (ω/τ 0 ) ( 1 ω 0 ) 1 (ω 0 ω) + (1/τ 0 ) tan θ = ω/τ 0 1/τ 0 ω0 ω ω 0 ω ω0 ω u = cos θ = (ω 0 ω ) + (ω/τ 0 ) ω 0 ω (ω0 ω) + (1/τ 0 ) v = sin θ = ω/τ 0 (ω 0 ω ) + (ω/τ 0 ) 1/τ 0 (ω0 ω) + (1/τ 0 ) (1.1) ω 0 ω = (ω 0 + ω)(ω 0 ω) ω 0 (ω 0 ω) ω(ω 0 ω) (1.15) x τ 0 =10 ω 0 = ω θ τ 0 =10 ω 0 = ω Rex 0, Imx Re x 0 Im x 0 τ 0 =10 ω 0 = ω 1.1: 9

11 1.4 (1.13) p p e r a = p (ω) e iωt + p ( ω) e +iωt r r a (1.13) p (ω) = e r (ω) a = e /m (ω 0 ω ) iω/τ 0 E (ω) (t) n P (r, t) ω P (ω) (r, t) = n e /m (ω 0 ω ) iω/τ 0 E (ω) (r, t) ε 0 χ(ω)e (ω) (r, t) (1.16) χ(ω) χ(ω) = ω p (ω 0 ω ) iω/τ 0 ; ω p ne ε 0 m (1.17) ω p D D ε 0 E + P χ ε ε = ε 0 (1 + χ) (1.18) (1.17) (1.18) ε 1.6. Maxwell Maxwell D = ρ t, B = 0, E = B t, 1 H = j + D t 10

12 ρ t = 0, j = 0 D = ε 0 E + P, B = µ 0 H Maxwell E ( E) + E E µ 0 ε 0 t = µ P 0 (1.19) t ( E) = ( E) E ρ t = j = 0 Maxwell E P x B y z E = E(z, t)ˆx, P = P (z, t)ˆx (1.19) ( z 1 c t c 1/ ε 0 µ 0 ) E(z, t) = 1 ε 0 c P (z, t) t (1.0) ω K E(z, t) = E (ω) (z, t) + E ( ω) (z, t) = E (ω) (z)e i(kz ωt) + E ( ω) (z)e i(kz ωt) (1.1) E ( ω) = E (ω) E (ω) = E ( ω) E ω ω 0, K k ω c ω 0 c E (ω) (z) K (1.10) E (z) KE(z), E (z) KE (z) K E(z), (1.0) E(z, t) (1.1) P (z, t) (1.16) K E(z) + ike (z) + E (z) + ω ω E(z) = c c χ(ω)e(z) 11

13 (K k )E(z) ike (z) = k χ(ω)e(z) K k = k Re[χ(ω)] k ω p ω ω 0 ω (ω 0 ω) + 1/τ 0 KE (z) = k Im[χ(ω)]E(z) k ω p ω 1/τ 0 (ω 0 ω) + 1/τ 0 E(z) (1.15) ω 0, τ 0, ω p 1 K ω E(z) 1.7 (1.17) ω p +e e L x nex +nex E E = nex/ε 0 F = ee mẍ = ne ε 0 x ω p = ne ε 0 m L 1

14 Hartree-Fock Hartree-Fock 1/r 13

15 r V (r) LS ( ) H 0 = p + V (r) (.1) m V (r).1. (.1) L z L z s z s z r r n = 1,, 3, l = 0, 1,,, n 1 m = l, l + 1,, l + l + 1 z E n l m m V 1/r l E n n l l l s p d f g m (l + 1) () Hartree-Fock Slater.1.3 Na Na 11 1s s p 6 3s 14

16 11 p 10 3s 3s 3p l = 1 m = 1, 0, 1 3 LS LS J = l + s J 3/ 1/ Na D LS. Ĥ Ĥ = Ĥ0 + Ĥ Ĥ0 Ĥ 0 n = W n n ; n = 0, 1,, Ĥ ψ t = 0 0 ψ i t ψ = Ĥ ψ = (Ĥ0 + Ĥ ) ψ t ψ(t) Ĥ0 ψ(t) = n a n (t) n a n (t) a n (t) b n (t)e iw nt/ b n (t) i n ḃ n (t)e iw nt/ n = n b n (t)e iw nt/ Ĥ n (.) 15

17 m iḃm(t) = n m Ĥ n e i(w m W n )t/ b n (t) (.3) b 0 = 1, b n = 0 (n 0) for t = 0.1 (.).3 Ĥ0 Ĥ.3.1 Ĥ = Ĥ0 + Ĥ Ĥ0 Ĥ Ĥ Ĥ = ( eˆr a ) E(r, t) = ˆµ a E(r, t) ˆr a r ˆµ ( e)ˆr a E(r, t) = E 0 cos ωt Ĥ = eˆr a E 0 cos ωt = ˆµ a E 0 cos ωt (.4) 16

18 .3. Ĥ0 r a r a l (.4) even odd even Ĥ even = odd Ĥ odd = 0 Ĥ even Ĥ odd, odd Ĥ even 1 m Ĥ n = 0 m n..3 l.3.3 ω 0 1 ω W 1 W 0, ω W n W m for n, m 0, 1 (.5) m Ĥ n = H mn cos ωt (.3) iḃm(t) = n H mn cos ωte i(w m W n )t/ b n (t) H m0 cos ωte i(wm W 0)t/ 1 17

19 t = 0 b 0 (0) = 1, b n (0) = 0(n 0) t 0 b 0 (t) 1, b n (t) 0(n 0) m 0 b m (t) = 1 i t 0 = 1 i H m0 1 i H m0 dt H m0 cos ωt e i(w m W 0 )t / [ e i(ω+(w m W 0 )/)t 1 i(ω + (W m W 0 )/) e i(ω (Wm W0)/)t 1 i(ω (W m W 0 )/) e i(ω (W m W 0 )/)t 1 i(ω (W m W 0 )/) (.5) b m (t) m = 1 b 1 (.5) 0 1 ] 18

20 3 3.1 Ĥ ψ ψ i ψ = Ĥ ψ (3.1) t ψ n ; n = 0, 1,, ψ = n c n n ψ (c 0, c 1, c, ) ˆρ ψ ψ (3.) A ˆρ A = ψ ψ A ρ n,m n ˆρ m = n ψ ψ m = c n c m ψ Ô O ψ Ô ψ O = Tr[ˆρÔ] (3.3) 19

21 Tr[ˆρÔ] = n n ˆρÔ n = n n ψ ψ Ô n = n ψ Ô n n ψ = ψ Ô ψ i ˆρ = [Ĥ, ˆρ] (3.4) t (3.1) 3.1 (3.4) 3. ψ ψ 1 P 1 ψ P ψ 3 P 3 P 1 ψ j ˆρ = ψ j P j ψ ; P j = 1 (3.5) j (3.4) (3.3) j O = j P j ψ j O ψ j = Tr[ˆρÔ] ˆρ Tr[ˆρ] = 1 (3.6) 0

22 (3.) ˆρ = ˆρ (3.7) (3.5) (3.7) 3. (3.6) 3.3 (3.7) Ĥ = Ĥ0 + Ĥ Ĥ0 a b Ĥ 0 a = W a a, Ĥ 0 b = W b b ; W a W b ω 0 Ĥ Ĥ = ˆµ E(r, t); ˆµ eˆr a r ˆµ a Ĥ a = b Ĥ b = 0 Ĥ a Ĥ b = a ˆµ b E p ab E(t) = b Ĥ a t ψ(t) ψ(t) = c a (t) a + c b (t) b c a (t) = e iw at/ c a (0), c b (t) = e iw bt/ c b (0) 1

23 N ˆρ i ψ i ˆρ = 1 ψ i ψ i N i ˆρ ρ aa = 1 N c a,i c a,i, i ρ ab = 1 N c a,i c b,i, i etc. ψ i = c a,i a + c b,i b i ˆρ = [Ĥ, ˆρ] t a b i a ˆρ b = a Ĥ ˆρ b a ˆρĤ b t = H aa ρ ab + H ab ρ bb ρ aa H ab ρ ab H bb = (W a W b )ρ ab p ab E(t)ρ bb + p ab E(t)ρ aa ρ ab = iω 0 ρ ab + i p abe(t) (ρ bb ρ aa ) (3.8) ρ ba = ρ ab = iω 0 ρ ba i p bae(t) (ρ bb ρ aa ) (3.9) ρ aa = i p abe(t) ρ bb = i p bae(t) ρ ba i p bae(t) ρ ab (3.10) ρ ab i p abe(t) ρ ba (3.11) ρ aa + ρ bb = 0 (3.1) ρ aa ρ bb = i p abe(t) ρ ba i p bae(t) ρ ab (3.13) (3.6)

24 3.4 ˆρ E(t) ρ ba (3.8) (3.9) ρ ba = b ˆρ a = c b (t)c a (t) ρ ba ρ ba e iw bt/ e +iwat/ c b (0)c a (0) e iω 0t ρ ba ω 0 E(t) E(t) = E (ω) e iωt + E ( ω) e +iωt ; E ( ω) = E (ω) (ω ω 0 ) ρ ba (3.9) ρ ba = iω 0 ρ ba i p ) ba (E (ω) e iωt + E ( ω) e +iωt (ρ bb ρ aa ) E ( ω) ω ω 0 E ( ω) ω ρ ba (t) ρ (ω) ba (t)e iωt ρ (ω) ba (t) e iωt ρ (ω) ba (t) = i(ω 0 ω)ρ (ω) ba (t) ip ba (E (ω) + E ( ω) e +iωt ) (ρ bb ρ aa ) i(ω 0 ω)ρ (ω) ba (t) ip ba E(ω) (ρ bb ρ aa ) (3.14) E(t) E ( ω) 3.5 ρ ba = 1 N c b,j c a,j (3.15) j c b,j, c a,j 3

25 (3.15) (3.14) γ(3.10) (3.11) Γ ρ (ω) ba (t) = i(ω 0 ω)ρ (ω) ba (t) γρ(ω) ba ρ aa (t) = i p abe ( ω) ρ bb (t) = i p bae (ω) ρ (ω) ba ρ ( ω) ab ip ba E(ω) (ρ bb ρ aa ) (3.16) ip bae (ω) ρ ( ω) ab + Γρ bb (3.17) i p abe ( ω) ρ (ω) ba bb (3.18) (3.17) (3.18) γ Γ 3.6 (3.16) ρ (ω) ba = 1 (ω 0 ω) iγ p ba E (ω) (ρ aa ρ bb ) (3.19) ρ aa 1, ρ bb 0 ρ (ω) 1 p ba E (ω) ab = (ω 0 ω) iγ P ˆµ ˆp P (ω) = P = N V Tr[ˆρˆp] = N V [ρ abp ba + ρ ba p ab ] = N [ ] ρ ( ω) ab p ba e +iωt + ρ (ω) ba V p abe iωt P ( ω) e +iωt + P (ω) e iωt N/V (ω 0 ω) iγ p ba p ab (ρ aa ρ bb )E (ω) ε 0 χ(ω)e (ω) χ(ω) ε 0 χ(ω) = N V 1 (ω 0 ω) iγ p ba (ρ aa ρ bb ) (3.0) 4

26 (1.17) ε 0 χ cl (ω) = N V e /m (ω 0 ω ) iω/τ 0 N V 1 (ω 0 ω) i/τ 0 e ω 0 m ρ aa ρ bb = 1 (3.0) χ(ω) χ (ω) + iχ (ω) = N ω 0 ω p ba V (ω 0 ω) + γ ε 0 + i N V ε(ω) ε(ω) = ε 0 (1 + χ(ω)) c c = 1 εµ 1 ε0 µ χ(ω) c 0 n(ω) γ p ba (ω 0 ω) + γ ε 0 n(ω) χ(ω) n(ω) = 1 + χ(ω) χ (ω) + 1 iχ (ω) n (ω) + iκ(ω) n n k ω k = nω c 0 = ω c 0 (n + iκ) [ ( )] [ ω E (ω) e i(kz ωt) = E (ω) exp i n z ωt exp κ ω ] z c 0 c 0 K(ω) ω c 0 n (ω) κ(ω) ρ bb 0 (3.16) (3.18) ρ (ω) 1 p ba E (ω) ba = (ω 0 ω) iγ ρ bb = [ ] Γ Im p ab E ( ω) ρ (ω) ba 1 = ρ aa + ρ bb (ρ aa ρ bb ), ρ ( ω) ab = ρ (ω) ba 5

27 ρ aa ρ bb = ρ (ω) ba = (ω 0 ω) + γ (ω 0 ω) + γ + γ p ba E (ω) (3.1) Γ ( ) pba E (ω) (ω 0 ω) + iγ (ω 0 ω) + γ + γ p ba E (ω) (3.) Γ ω P (ω) = N V ρ(ω) ba p ab = N V (ω 0 ω) + iγ (ω 0 ω) + γ + γ p ba E (ω) Γ p ba E(ω) χ(ω) = N V (ω 0 ω) + iγ (ω 0 ω) + γ + γ p ba E (ω) Γ p ba ε 0 (3.3) E(t) χ(ω) (1.17) 6

28 4 4.1 Ĥ = Ĥ0 + Ĥ ; Ĥ = ˆµ E(t) Ĥ 0 1 = W 1 1, Ĥ 0 = W ; ω 0 = W W 1 ψ = c 1 (t) 1 + c (t) i ψ = Ĥ ψ t iċ 1 (t) = W 1 c 1 (t) µ 1 E(t)c (t) iċ (t) = µ 1 E(t)c 1 (t) + W c (t) Ĥ µ 1 1 ˆµ c i (t) = b i (t)e iw it/ ; i = 1, iḃ1(t) = µ 1 E(t)b (t)e iω 0t iḃ(t) = µ 1 E(t)b 1 (t)e +iω 0t 7

29 E(t) = E (ω) e iωt + E ( ω) e +iωt ω 0 ω ω 0, ω ḃ 1 (t) = i 1 µ 1 E ( ω) e i t b (t) ixe i t b (t) (4.1) ḃ (t) = i 1 µ 1 E (ω) e +i t b 1 (t) ix e +i t b 1 (t) (4.) t = 0 b 1 (0) = 1, b (0) = 0 t 0 b 1 (t) 1 (4.) b (t) t 0 dt ix e i t = ix e i t/ sin 1 t t b (t) = 4 X sin 1 t (4.3) = ω 0 ω = 0 1/t t δ(ω 0 ω) = π lim sin [ (ω 0 ω) t ]] t (ω 0 ω) t (4.4) (4.3) (4.3) X = 1 µ1 E (ω) 1 = µ 1 E (ω) cos θ (4.5) 1 ε 0E (t) + 1 ( ) µ 0H (t) = ε 0 E (t) = ε 0 E (ω) e iωt + E ( ω) E e +iωt = (ω) ε (4.3) E (ω) ω U(ω)dω E (ω) ε 0 U(ω)dω 8

30 (4.5) cos θ cos θ = 1 dω cos θ = 1 4π 3 b (t) = µ 1 3 ε 0 = π 3 µ 1 ε 0 dωu(ω) sin 1 t ; = ω 0 ω U(ω 0)t ω 1 t (4.4) ω 1 ω 1 = π 3 µ 1 ε 0 U(ω 0) (4.6) U(ω 0 ) (4.1) b 1 = 0, b = 1 ω 1 ω 1 ω 1 = ω (4.4) t ω T W 1 W (W W 1 ω 0 0) N e 1 N e N e N e 1 = exp [ W ] W 1 k B T 9 (4.7)

31 U T (ω) U T (ω) = ω3 π c 3 1 e ω/k BT 1 (4.8) (1) () B 1 U T (ω 0 ) B 1 U T (ω 0 ) A 1 N e 1B 1 U T (ω 0 ) = N e B 1 U T (ω 0 ) + N e A 1 (4.9) T (4.7) (4.8) U T (ω 0 ), N e 1 = N e (4.9) B 1 = B 1 B (4.9) (4.8) A A 1 = ω3 0 π c 3 B BU T (ω 0 ) (4.6) ω 1 A = µ 1 3πε 0 c 3 ω3 0, B = π 3 µ 1 ε 0 (4.10) 30

32 A (4.10) ω BU T (ω 0 ) A = 1 N1/N e e 1 = 1 e ω 0/k BT 1 = n ω 0 A + BU T (ω 0 ) = A(1 + n ω0 ) n ω 1 1eV K n ω 1 31

33 5 5.1 W t W t W E(t) = E 0 exp [ iω 0 t 1 ] γt, for (t > 0) E(ω) = I(ω) 0 I(ω) E(ω) E(t)e iωt dt = E 0 1 i(ω ω 0 ) + γ/ 1 (ω ω 0 ) + (γ/) γ 5. τ c τ p(τ) p(τ) = 1 τ c e τ/τc (5.1) 3

34 [(5.1) ] τ Q(τ) Q(τ) p(τ) Q(τ) = τ p(τ )dτ Q (τ) = p(τ) Q(τ) γ c τ τ + τ γ c τ Q(τ + τ) Q(τ) = γ c τq(τ) = dq(τ) dτ = γ c Q(τ) Q(0) = 1 Q(τ) = e γ cτ p(τ) = Q (τ) = γ c e γcτ τ c τ c =< τ >= 0 τp(τ)dτ = 1 γ c (5.1) j t j E(t) E(t) = E 0 exp( iω 0 t iθ j ) for t j < t < t j+1 θ j ( π, π] τ j t j+1 t j (5.1) E(ω) = E(t)e iωt dt = j tj+1 t j E 0 e i(ω 0 ω)t dt e iθ j E(ω) = j,j j j e iθ j +iθ j θ j 1 π < E(ω) > θ = π π π π π dθ 1 dθ π dθ j j,j j j e iθ j +iθ j = π π j j dθ 1 dθ e iθ j+iθ j j,j π = j,j j j δj,j = j π j 33

35 j = tj +τ j E 0 e i(ω0 ω)t sin(ω 0 ω) dt = E 0 t j (ω 0 ω)/ e i(ω 0 ω)(τ j /+t j ) τ j < E(ω) > θ = j E 0 [ sin(ω0 ω) τ ] j (ω 0 ω)/ τ j < E(ω) > < E(ω) > θ,τ = = 1 τ c (ω 0 ω) /4 0 (ω 0 ω) + 1/τc 0 dτ p(τ) [ sin(ω0 ω) τ (ω 0 ω)/ dτ e τ/τ c sin (ω 0 ω) τ /τ c ] 5.3 v ω 0 z = ω (v/c) ( ) ω ω 0 1 v z /c ω v z c z v z [ ] m P (v z ) = πk B T exp mv z k B T (5.) I(ω) (5.) v z I(ω) P (v z (ω)) dv ] z [ dω exp mc (ω ω 0 ) k B T T ω 0 34

36 5.4 35

37 6 (LASER) Light Amplification by Stimulated Emission of Radiation MASER (Tawnes et al, 1954) 1960 Maiman LASER LASER 6.1 W W = E j = E P t ) = (E (ω) e iωt + E ( ω) e )( iωp +iωt (ω) e iωt + iωp ( ω) e +iωt ] = iωe (ω) P ( ω) iωe ( ω) P (ω) = Im [ωe ( ω) P (ω) P (ω) = ε 0 χ(ω)e (ω) [ W = ε 0 Im ωχ(ω) E (ω) ] = ε 0 ωχ (ω) E (ω) 36

38 W χ (ω) χ (3.0) χ(ω) = n p ab ε 0 (ω 0 ω) + iγ (ω 0 ω) + γ (ρ aa ρ bb ) ρ aa > ρ bb = χ > 0 ρ aa < ρ bb = χ < 0 6. Maxwell divd = ρ, rote = B D, divb = 0, roth = t t + j ; D = ε 0 E + P, B = µ 0 H + M ρ = M = 0, j = σe z z xy E(r, t) = E(z, t) x y E E t c E z + σ ε 0 E t = 1 ε 0 P t (6.1) c 1/ ε 0 µ 0 L k m ω m k m = mπ L, ω m = ck m = mπ L c E(z, t) = [ E (ω) (t)e iωt + E ( ω) (t)e +iωt] sin k m z (6.) P (z, t) = [ P (ω) (t)e iωt + P ( ω) (t)e +iωt] sin k m z E (ω) (t) ω 37

39 (6.1) [ ] Ė (ω) + i (ω m ω) iκ E (ω) = i ω P (ω) (6.3) ε 0 κ (ω m ω ) ω(ω m ω) σ ε 0 E (ω) (t) = E (ω) (t) e iφ(t), ( d Ė (ω) E (ω) (t) (t) = dt + i φ(t) E (ω) (t) ) e iφ(t) (6.3) d E (ω) (t) dt P (ω) = ε 0 χ(ω)e (ω) = ε 0 (χ (ω) + iχ (ω))e (ω) +i φ(t) E (ω) (t) ] E +i [(ω m ω) κ (ω) (t) ω = i (χ (ω)+iχ (ω)) E (ω) (t) d E (ω) (t) [ + κ + ω E (ω)] dt χ (ω) (t) = 0 (6.4) φ(t) + (ω m ω) ω χ (ω) = 0 (6.5) χ < (6.1) (6.3) 6.3 (6.4) (6.5) κ + ω χ (ω) = 0 (6.6) ω + ω χ (ω) = ω m (6.7) 38

40 χ (3.16) (3.18) (3.1) (3.) (3.17) (3.18) ρ (ω) ba (t) = i(ω 0 ω)ρ (ω) ba (t) γρ(ω) ba ρ aa = i p abe(t) ρ bb = i p bae(t) ip ba E(ω) (ρ bb ρ aa ) ρ ba i p bae(t) ρ ab Γ(ρ aa ρ 0 aa) ρ ab i p abe(t) ρ ba Γ(ρ bb ρ 0 bb) ρ 0 aa ρ0 aa a, b ˆρ (6.) ρ (ω) ba (z) = p ba E (ω) (ω 0 ω + iγ) sin k m z (ω 0 ω) + γ + γ p ba E (ω) (ρbb 0 ρ 0 aa) Γ sin k m z sin k m z P ) ( ) P = n (p ab ρ ba + p ba ρ ab = n p ab ρ (ω) ba e iωt + p ba ρ ( ω) ab e +iωt ] [P (ω) (t)e iωt + P ( ω) (t)e +iωt sin k m z (6.8) ρ (ω) ba P (ω) P (ω) = n (ω 0 ω + iγ) p ba (ω 0 ω) + γ γ Γ p ba E (ω) (ρ0 bb ρ 0 aa)e (ω) (6.9) (ω 0 ω + iγ) χ(ω) = n (ω 0 ω) + γ + 3 γ p ba E (ω) 4 Γ p ba ε 0 (ρ0 bb ρ 0 aa) (6.10) 39

41 [(6.9) ] P e iωt sin k m z (6.8) sin k m z 0 L : n p ab L 0 dzρ (ω) L ba (z) sin k mz = P (ω) dz sin k m z L/ 0 L dzρ (ω) 0 L ba (z) sin k c sin k m z mz 0 a + b sin k m z dz c L ( dz sin k m z 1 b ) a 0 a sin k m z = c ( L a b ) 3 a 8 L = c ( 1 b a a c 1 L a 1 + b 3 = c L a + 3 a 4 4 b a, b, c (6.9) ) 3 L (6.6) χ (6.10) : κ = ω n γ (ω 0 ω) + γ γ Γ p ba E (ω) p ba ε 0 (ρ0 bb ρ 0 aa) (6.11) ρ 0 bb ρ0 aa E (ω) (6.11) E (ω) (ρ 0 bb ρ0 aa) (ρ 0 bb ρ0 aa) E (ω) E (ω) N 0 n (ρ 0 bb ρ 0 aa) N 0 N th (ω 0 ω) + γ (6.11) 40 γ ω ε 0 p ba κ (6.1)

42 (6.1) E (ω) = Γω 6κε 0 ( N 0 N th ) (6.13) Γ 6. (6.1) (6.13) (6.11) 6.3. ω 0 ω 0 ω m ω 0 ω m ω (6.7) ω = ω m χ (ω) ω (6.10) χ = ω 0 ω χ γ (6.6) χ = (ω 0 ω) κ ωγ (6.7) ω = γ ω m + κ ω 0 γ + κ (6.14) ω ω 0 ω m γ κ κ γ ω ω m 41

43 6.4 (1) () fs(10 15 s) k m = π L m, ω m = c k m ; m : ω 0 N ω ν = ω 0 + ω ν, k ν = ω ν /c ; ω π [ L c, ν N 1, N 1 ] E(z, t) = (N 1)/ ν= (N 1)/ E ν e i(ωνt+φν)+ikνz φ ν = 0 E ν = E 0 z = 0 4

44 E(0, t) = (N 1)/ ν= (N 1)/ E 0 e i(ω 0+ ω ν) = E 0 e iω 0t sin ( ω N ) t ( sin ω t ) E(0, t) = E 0 ( sin ω N ) t ( sin ω t ) (6.15) t = 0 t T t T t = π N ω, T = L c N ω T L 10cm N (6.15) t T 6.5 (Al O 3 ) Cr 3+ Cr 3+ ( 694.3nm) Nd YAG (Y 3 Al 5 O 1 ) 43

45 Nd 3+ (CW) Nd YAG He Ne He Ne Ne n p p n ( n p ) 44

46 7 7.1 III Maxwell E = B t, H = D, B = 0, D = 0; t D = ε 0 E, B = µ 0 H E = A t, B = A (7.1) A 1 A = 0, A = 0 (7.) c t H H = 1 (ε 0 E + µ 0 H )dr (7.3) A(r, t) = q(t)u(r) 45

47 (7.) q(t) u(r) 1 c u(r) q(t) t = 0 q l (t) t = ω l q l (t), u l (r) = k l u l (r); k l ω l c (7.4) A A(r, t) = 1 q l (t) u l (r) (7.5) ε0 (7.4) q(t) u(r) u l (r) = 1 e ik l rêl ; ê l k l V k l ê l l l = (l x, l y, l y ), l i = 0, ±1, ±, ; k l = π l = (l, σ); L l, σ = 1, ; ê ê (l,1) (l,) u l (r) u l(r) u l (r)dr = δ l,l E(r, t) = B(r, t) = 1 ε0 l l q(t) u l (r) (7.6) 1 q(t) ik l u l (r) (7.7) µ 0 ε0 l (7.3) H = 1 ( ) q l + ωl ql l (7.8) (7.4) q l (t) 7.1. q l p l (7.8) H = H l ; H l 1 ( ) p l + ωl ql l 46

48 q l p l ˆq l ˆp l [ˆq l, ˆp l ] = i δ l,l ˆp l = i q l Ĥ = 1 ( ) ˆp l + ωl ˆq l l etc. â l â l: â l â l ωl ˆq 1 l + i ˆp l ωl ˆq 1 l i ˆp l ωl ωl ( ) ˆq l = â l ω + â l l ωl ( ) ˆp l = i â l â l (7.9) [â l, â l ] = δ l,l, [â l, â l ] = 0, [â l, â l ] = 0 (7.10) Ĥ = ( ω l â lâl + 1 ) l (7.11) 7.1 (7.10) (7.11) ˆn ˆn â â ; [â, â ] = 1 n n > 1. ˆn. ˆn n n 0 3. n > ˆn n â n > n 1 4. ˆn n 47

49 5. n â n n > ˆn 6. 0 > â 0 >= 0 â n 0 > ˆn n 7. â n >= n + 1 n + 1 >, â n >= n n 1 > (7.6) (7.9) Ê(r) = ( ωl i â 1 l e ik l r 1 âl e +ik l r ) ê l ε 0 l V V â Ê â l (t) i dâ l (t) dt = [â l (t), H] = ω lâ l (t) = â l (t) = â l (0)eiω lt â l (0) â l â l (t) = â l eiω lt, â l (t) = â l e iω lt Ê(r, t) = ( ) ωl i â 1 l e i(ω lt k 1 l r) â l e i(ω lt k l r) ê l ε 0 l V V Ĥ(r, t) = l ω l µ 0 ( â l ) 1 e i(ω lt k 1 l r) â l e i(ω lt k l r) V V i k l k l ê l H l n l E = ( ω l n l + 1 ), ψ >= n 1 n n 3 > (7.1) l 48

50 7. (7.1) (7.11) l n l ê l Ê(r, t) = ie l â l e i(ω lt k l r) ie l â l ei(ω lt k l r) ; E l ωl ε 0 V (7.13) n n > < Ê > < n Ê n >= 0 ( < Ê > = < n ie l â l e i(ω lt k l r) ie l â lt k l r)) l e+i(ω n > ( ) = El < n â l â l + â lâl = El (n + 1) < Ê > = E l n + 1 n n cos(ωt k r) (7.13) < Ê(r, t) >= E(+) e i(ω lt k l r) + E ( ) e +i(ω lt k l r) (7.14) â l â l α >= α α > (7.15) (7.15) < α â l =< α α 49

51 Ê < α Ê α > = < α ie l â l e i(ω lt k l r) ie l â l e+i(ω lt k l r) α > = ie l α e i(ω lt k l r) ie l α e +i(ω lt k l r) E (+) = ie l α (7.14) (7.15) â α > â α >= α α > â α α >= c n n > â n > n=0 αc n = n + 1c n+1 = c n = αn n! c 0 c 0 α >= e α / n=0 α n n! n > (7.16) P (n) P (n) = < n α > α n α = e n! m mn e n! ; m α (7.17) m = α 50

52 (7.17) < ˆn >=< α â α >=< α â â α >= α ( ) < ˆn > = < α â â â â α > = < α â â â â + â â α > = α 4 + α < ˆn > < (ˆn < ˆn >) > = < ˆn > < ˆn > = α < ˆn > < ˆn > = α α = 1 α α (7.9) ω ˆq = ω (â + â), ˆp = i (â â) < ˆq > = < α ˆq α > = ω (α + α), < ˆq > = ω < α (â + â) α > = ω (α + α α ) q < ˆq > = < ˆq > < ˆq > = p ω < ˆp > = q p = ω 51

53 < Ê > = < α Ê α > = ie l α e i(ω lt k l r) ie l α e +i(ω lt k l r) ( < Ê > = < α Ê α > = < α [ie l â l e i(ω lt k l r) â lt k l r))] l e+i(ω α > [ ] = El α e i(ω lt k l r) + α + 1 α e +i(ω lt k l r) ) < Ê >=< (Ê < Ê > >=< Ê > < Ê > = El α < Ê > = E l, < Ê > E l α 5

128 3 II S 1, S 2 Φ 1, Φ 2 Φ 1 = { B( r) n( r)}ds S 1 Φ 2 = { B( r) n( r)}ds (3.3) S 2 S S 1 +S 2 { B( r) n( r)}ds = 0 (3.4) S 1, S 2 { B( r) n( r)}ds

127 3 II 3.1 3.1.1 Φ(t) ϕ em = dφ dt (3.1) B( r) Φ = { B( r) n( r)}ds (3.2) S S n( r) Φ 128 3 II S 1, S 2 Φ 1, Φ 2 Φ 1 = { B( r) n( r)}ds S 1 Φ 2 = { B( r) n( r)}ds (3.3) S 2 S S 1 +S 2 { B( r) n( r)}ds