Transcription

1

2

3 MKSA

4 ii Nd:YAG

5 iii He-Ne

6 iv n

7 1 1 optics geometric optics physical optics quantum optics ray optics imaging aberration wave optics propagation interference diffraction polarization scattering laser spectroscopy quantum electronics opto-electronics Maxwell s equations electromagnetic wave ν λ λν = c (1.1) c c = m/s 380 nm 780 nm visible light 10 nm ultraviolet 100 µm infrared x ray γ ray radiowave 0.1 THz 10 THz 30 µm 3 mm terahertz wave duality photon E = hν (1.2)

8 2 1 p = h λ (1.3) h Planck constant h = J s frequency wavenumber ν ν = 1 λ = ν c (1.4) cm 1 E = hc ν (1.5) cm 1 wavenumber reciprocal centimeter inverse centimeter K kayser

9 3 1. (a) γ (b) X (c) (d) (e) (f) nm 780 nm

10 laser light amplification by stimulated emission of radiation, coherent coherence CVD laser chemical vapor deposition laser cooling laser fusion

11 L L L E(r, t) = A exp[i(k r ωt)] (2.1) E(r, t) = A exp[i(ωt k r)] (2.2) (2.1) (2.1) k r x y z ˆx ŷ ẑ k = k x ˆx + k y ŷ + k z ẑ (2.3) r = xˆx + yŷ + zẑ (2.4) k r k r = k x x + k y y + k z z (2.5) z y L 0 L x 2.1: L 2.1 L E(x = 0) = E(x = L) E(y = 0) = E(y = L) E(z = 0) = E(z = L) (2.6)

12 6 2 exp(ik x L) = 1 exp(ik y L) = 1 exp(ik z L) = 1 (2.7) k n x n y n z k x = 2πn x /L k y = 2πn y /L k z = 2πn z /L (2.8) (n x, n y, n z ) k (n x, n y, n z ) k = ( k 2 x + k 2 y + k 2 z ) 1/2 = 2π L ( n 2 x + n 2 y + n 2 z ) 1/2 (2.9) λ λ = 2π k (2.10) ( L ) 2 n 2 x + n 2 y + n 2 z = (2.11) λ L n 2 x + n 2 y + n 2 z 1 (2.12) ω ω = kc (2.13) c ( ) 2 2πc ( ) ω 2 = n 2 x + n 2 y + n 2 z L (2.14) 0 ω n 2 x + n 2 y + n 2 z < ( Lω ) 2 (2.15) 2πc (n x, n y, n z ) 2 n 1 n 2 x + n 2 y + n 2 z < n 2 (2.16)

13 (n x, n y, n z ) n 4πn 3 /3 4π ( Lω ) 3 2 = ω3 L 3 (2.17) 3 2πc 3π 2 c 3 ω ω + dω ω ω 2 L 3 dω (2.18) π 2 c3 L 3 D(ω) D(ω)dω = ω2 dω (2.19) π 2 c3 ω ν = ω/2π D ν (ν) D(ω)dω = D ν (ν)dν (2.20) D ν (ν)dν = 8πν2 dν (2.21) c T W W + dw p(w)dw = 1 k B T exp( W/k BT)dW (2.22) k B Boltzmann constant k B = J/K W = 0 W p(w)dw = k B T (2.23) principle of equipartition ω ω + dω U(ω)dω (2.19) U(ω)dω = D(ω)k B Tdω = ω2 π 2 c 3 k BTdω (2.24) Rayleigh-Jeans ω ω

14 8 2 M. Planck quantum hypothesis ω ν W = n ω = nhν (n = 0, 1, 2, ) (2.25) h/2π W = n ω ω n photon ω n p(n) p(n) exp( n ω/k B T) (2.26) U th U th = n=0 n ωe n ω/k BT n=0 e n ω/k BT (2.27) 1 r e ω/k BT (2.27) n=0 r n = 1 1 r (2.28) (2.29) ω nr n = ωr r n=0 r n 1 = ωr (2.30) (1 r) 2 n=0 U th = ω r 1 r = ω 1 1 = ω e ω/k BT 1 r 1 (2.31) Bose-Einstein distribution ω k B T (2.23) ω ω + dω D(ω)dω U(ω)dω = ω2 ωdω π 2 c 3 e ω/k BT 1 (2.32) hν k B T (2.24) 2.2

15 U(ω) hω/ k B T 2.2: (2.32) ν ν + dν U ν (ν)dν = 8πν2 c 3 hνdν e hν/k BT 1 λ + dλ dλ U λ (λ)dλ = 8πhc λ 5 dλ e hc/λk BT 1 (2.33) (2.34) (2.32) 0 T I = π 2 c 3 0 ω 3 dω e ω/k BT 1 = π2 k 4 B 15c 3 3 T 4 (2.35) 0 x 3 π4 e x dx = 1 15 (2.36) T 4 Stefan-Boltzmann law of radiation (2.34) T λ m d dλ U λ(λ) = 0 (2.37) 1 1 ( hc 5 λ m k B T = exp hc ) λ m k B T (2.38) x = hc λ m k B T (2.39)

16 10 2 U (λ) λ (nm) 2.3: 6000K 1 x = exp( x) (2.40) 5 x = 0 x = x = λ m = hc k B T = 2.90 mm K T (2.41) T = 300 K λ m = 9.66 µm K λ m = 483 nm W 2 hω W 1 2.4: W 1 W ω = (W 2 W 1 )/

17 absorption ω ω rate ω U(ω) B 12 U(ω) (2.42) W 2 hω W 1 2.5: 2.5 ω ω = W 2 W 1 ω ω emission A T N 2 N 1 ( N 2 = N 1 exp ω ) k B T (2.43) N 1 B 12 U(ω) = N 2 A (2.44) U(ω) = A ( exp ω ) B 12 k B T (2.45)

18 12 2 U(ω) U(ω) (2.45) A + B 21 U(ω) (2.46) A (2.43) N 1 B 12 U(ω) = N 2 [A + B 21 U(ω)] (2.47) U(ω) = A B 12 e ω/k BT B 21 (2.48) T U(ω) B 12 = B 21 (2.49) B B 12 = B 21 (2.50) (2.24) A B = ω3 π 2 c 3 (2.51) (2.48) (2.32) W 2 B U( ) A B U( ) 12 ω 21 ω W 1 2.6:

19 g 1, g 2 (2.43) N 2 = g ( 2 N 1 exp ω ) (2.52) g 1 k B T g 1 B 12 = g 2 B 21 (2.53) A B 12 = g 1 g 2 ω 3 π 2 c 3 (2.54) 2.6 spontaneous emission stimulated emission A B A B A B (2.51) W 1 W 2 N 1 N 2 N 1 BU(ω) (2.55) N 2 BU(ω) (2.56) N 1 > N 2 N 1 < N 2 population inversion inverted population T (2.43) T pumping

20 W 3 γ 32 Γ γ 21 γ 31 W 2 W 1 2.7: 2.7 n (n = 1, 2, 3) W n W 1 < W 2 < W 3 pumping rate Γ N n (n = 1, 2, 3) dn 1 dt dn 2 dt dn 3 dt = ΓN 1 + γ 21 N 2 + γ 31 N 3 (2.57) = γ 21 N 2 + γ 32 N 3 (2.58) = ΓN 1 (γ 31 + γ 32 )N 3 (2.59) rate equation γ mn m n N N 1 + N 2 + N 3 dn dt = 0 (2.60) dn 1 dt = dn 2 dt = dn 3 dt = 0 (2.61)

21 N 1 = N 2 = γ 21 (γ 31 + γ 32 ) γ 21 (γ 31 + γ 32 ) + (γ 32 + γ 21 )Γ N (2.62) γ 32 Γ γ 21 (γ 31 + γ 32 ) + (γ 32 + γ 21 )Γ N (2.63) N 2 > N 1 ( Γ > γ γ ) 31 γ 32 (2.64) γ 32 γ 31 Γ > γ 21 (2.65) W 3 Γ γ 32 γ 21 W 2 γ 10 γ 01 W 1 W 0 2.8: 2.8 n (n = 0, 1, 2, 3) W n (n = 0, 1, 2, 3) W 0 < W 1 < W 2 < W

22 16 2 N n (n = 0, 1, 2, 3) dn 1 dt dn 2 dt dn 3 dt dn 0 dt = γ 01 N 0 γ 10 N 1 + γ 21 N 2 + γ 31 N 3 (2.66) = γ 2 N 2 + γ 32 N 3 (2.67) = ΓN 0 γ 3 N 3 (2.68) = dn 1 dt dn 2 dt dn 3 dt γ mn m n (2.69) γ 2 γ 20 + γ 21 (2.70) γ 3 γ 30 + γ 31 + γ 32 (2.71) γ 01 γ 01 = γ 10 exp[ (E 1 E 0 )/k B T] (2.72) ( γ01 N 1 = + γ ) 21γ 32 + γ 2 γ 31 Γ N 0 (2.73) γ 10 γ 10 γ 2 γ 3 N 2 = γ 32Γ N 0 (2.74) γ 2 γ 3 N 3 = N 2 > N 1 Γ > Γ γ 3 N 0 (2.75) γ 01 γ 2 γ 3 γ 32 γ 10 γ 21 γ 32 γ 2 γ 31 (2.76) γ γ 10 γ 2 Γ > γ 2 exp[ (W 1 W 0 )/k B T] (2.77) 4 W 1 W 0 k B T k B T laser cavity Fabry-Perot

23 2.3. ¾ iˆê ß j 0 L z 2.9: 2.9 L 1% 100% z z z E(z, t) = A exp(ikz iωt) + B exp( ikz iωt) (2.78) +z z 0 E(z = 0) = 0 (2.79) E(z = L) = 0 (2.80) 1 B = A 2 A sin(kl) = 0 (2.81) kl = nπ (n = 1, 2, 3, ) (2.82) k n = nπ L (n = 1, 2, 3, ) (2.83) k n E n (z, t) = A sin(k n z) exp( iωt) (2.84) E n (z, t) (n = 1, 2, 3, ) n

24 18 2 k n E n (z, t) λ n = 2L n ω n = ck n = nπc L ω = ω n+1 ω n = πc L (2.85) (2.86) (2.87) 2.10 ω ω ω 2.10: laser oscillation gain medium laser medium L l R 1 R 2 100% 80% 99% G I 1 (z) d dz I 1(z) = GI 1 (z) (2.88) I 1 (z) = I 1 (0) exp(gz) (2.89) exp(gl)

25 R 1 R 2 exp(2gl) > 1 (2.90) G > 1 2l ln(r 1R 2 ) (2.91) R 1 R 2 1 T 1 = 1 R 1 (2.92) T 2 = 1 R 2 (2.93) 1 ln(1 x) x (2.94) (2.91) T 1 T 2 G > 1 2l (T 1 + T 2 ) (2.95) 100% threshold G th G th = 1 2l (T 1 + T 2 ) (2.96) ω 2.11

26 ω ω 2.11: Φ 1 Φ 2 γ 1 γ 2 ω ω U(ω) N 1 N 2 dn 2 dt dn 1 dt = Φ 2 γ 2 N 2 (N 2 N 1 )BU(ω) (2.97) = Φ 1 γ 1 N 1 + (N 2 N 1 )BU(ω) (2.98) B B du(ω) dt = 2κU(ω) + ω(n 2 N 1 )BU(ω) (2.99) κ 2L/c (1 T 1 )(1 T 2 ) T 1, T 2 1 2κ = T 1 + T 2 2L/c (2.100) T 1 T 2 κ (2.97)-(2.99) U(ω) = 0 N (0) 2 Φ 2 γ 2 (2.101) N (0) 1 Φ 1 γ 1 (2.102)

27 N N 2 N 1 (2.103) N (0) N (0) 2 N (0) 1 (2.104) du(ω) dt N th = = 0 (2.105) 2κ ωb (2.106) dn 2 = dn 1 = 0 dt dt (2.107) du(ω) = 0 dt (2.108) (2.107) (2.97) γ 2 (2.98) γ 1 τ 1 ( ) 2 γ 1 γ 2 ( 1 N (0) N + 1 ) NBU(ω) = 0 (2.109) γ 1 γ 2 (2.110) (2.108) N = N (0) 1 + 2τBU(ω) (2.111) N = N th (2.112) U ss (ω) = 1 ( ) N (0) 1 2τB N th (2.113) 2.12 N (0) quality value Q = ωw P L (2.114)

28 22 2 U ss è í U o Í 0 N th N (0) iƒ ƒ ƒsƒ ƒo Ì ³ j 2.12: ω W P L V U(ω)V U(ω)V(1 R 1 R 2 ) (2.115) P L = U(ω)V(1 R 1R 2 ) 2L/c Q = 100% Q = 2ωL c(1 R 1 R 2 ) 2ωL c(t 1 + T 2 ) (2.116) (2.117) (2.118) Q = ω 2κ (2.119) (2.106) (2.119) 2.13 Q switching

29 pƒxƒcƒbƒ` OFF p l pƒxƒcƒbƒ` ON ½ ] ª z Œõ x ŽžŠÔ 2.13: Pockels effect mode locking ω ω N E(t) = E n cos{(ω 0 + n ω)t + φ n } (2.120) n= N 2N ω 0 n E n φ n φ n φ n = 0 E n = E 0

30 24 2 E(t) = N E 0 cos{(ω 0 + n ω)t} n= N N = E 0 Re exp[ i(ω 0 + n ω)t] n= N N ( = E 0 Re ) e iω 0t e i ωt n n= N ( = E 0 Re ) e iω 0t e i ωt N 1 ( e i ωt) 2N+1 1 e i ωt = E 0 Re {e iω e 0t i(n+1/2) ωt + e i(n+1/2) ωt } e i ωt/2 + e ) ] i ωt/2 ωt = E 0 sin [( N sin ( ωt/2) cos(ω 0 t) (2.121) cos(ω 0 t) ) ] ωt sin [( 2 N I(t) = I 0 sin 2 ( ωt/2) (2.122) T = 2π (2.123) ω (2N + 1) 2 I 0 pulse train (2.87) T = 2L c (2.124) cavity round trip time N = Œõ x ŽžŠÔ 2.14: 21 T/(2N + 1) = 2π (2N + 1) ω (2.125)

31 (2N + 1) ω/2π 100 fs m T = 10 ns (2.125) 10 5 active mode locking passive mode locking ω ± ω Acousto-optic effect synchronous pumping saturable absorber Kerr lens mode locking 1. m λ 10.2 ev

32 half mirror optical damage 2 beam splitter ; dichroic mirror filter chirped mirror dispersion compensation mirror SiO TiO intra-cavity extra-cavity diffraction grating

33 π birefringent filter Electro-optic effect; EO effect Electro-optic modulator; EO modulator 1 Pockels effect 2 Kerr effect Acousto-optic effect; AO effect Acousto-optic modulator; AO modulator Nd:YAG cavity dumper Brewster surface 3.1 n θ i tan θ i = n (3.1) p (3.1) θ i Brewster angle 90 p

34 28 3 p ÎŒõ θ i 1 n θ d 3.1: p s p

35 nm 400 nm 2A E nm 3.2: ruby laser 1960 T. H. Maiman Al 2 O 3 Al 3+ 3 Cr Cr 3+ 3s 2 3p 6 3d 3 Cr 3+ O 2 Nd 3+ f d d 550 nm 400 nm nm Hz Nd:YAG Nd neodymium 3 Nd 3+ Nd 3+ 4f 3 5s 2 5p 6 4f 4f 5s5p Nd 3+ f 3.3 Nd mn 800 nm

36 nm Nd 3+ Nd 3+ YAG yttrium aluminum garnet, Y 3 Al 5 O 12 Nd:YAG kw Hz khz 10 ns 532 nm 355 nm 266 nm Nd:YAG Nd:YLF LiYF 4 Nd:YVO 4 Nd: 4 4 3/2 [ S, F ] 7/2 4 [ F, H ] 5/2 3 9/ µm 4 I9/2 0.8 µm 4 F3/2 4 I11/ µm 3.3: Nd:YAG Ti:sapphire Al 2 O 3 Cr 3+ Ti 3+ Ti:Al 2 O 3 Ti 3+ 3s 2 3p 6 3d 1 d Ti 3+ d 700 nm 1 µm 10 fs

37 Er µm EDFA: erbium-doped fiber amplifier Er 3.3 He-Ne 3.4: He-Ne He-Ne He Ne He Ne 3.4

38 nm 3.5: Ar Ar-ion laser Kr-ion laser 3.5 Ar nm nm Kr nm Ar Kr excimer 3.1 F 2

39 : ArF KrCl KrF XeCl XeF F nm 222 nm 249 nm 308 nm 351 nm, 353 nm 157 nm CO µm nm copper vapor laser nm nm He-Cd He Cd nm nm nm nm nm nm µm CH 3 OH µm 2.52 THz 3.4 dye

40 3.7: : singlet 1 triplet intersystem crossing S 0 S 1 S 2 T 1 T laser diode

41 free electron laser chemical laser

42 Maxwell equations permeability µ 0 H E = µ 0 t (4.1) E H = ɛ 0 + J t (4.2) ɛ 0 E = ρ (4.3) H = 0 (4.4) E H J ρ ɛ 0 vacuum permittivity H = 0 (4.2) (4.3) equation of continuity J + ρ t = 0 (4.5) polarization P ρ = P (4.6) J = P (4.7) t electric displacement D ɛ 0 E + P H E = µ 0 t (4.8) H = D t (4.9) D = 0 (4.10) H = 0 (4.11) E electric susceptibility χ P = ɛ 0 χe (4.12)

43 χ 3 3 permittivity ɛ D = ɛe = ɛ 0 E + P = ɛ 0 (1 + χ)e (4.13) ɛ ɛ 0 = 1 + χ (4.14) (4.12) x, y, z E x, E y, E z P x, P y, P z ( ) P i = ɛ 0 χ i j E j = ɛ 0 χix E x + χ iy E y + χ iz E z (4.15) j i j x, y, z χ i j D i = ɛ i j E j = ( ) ɛ ix E x + ɛ iy E y + ɛ iz E z (4.16) χ xx χ xy χ xz χ i j = χ yx χ yy χ yz χ zx χ zy χ zz j ɛ i j = ɛ xx ɛ xy ɛ xz ɛ yx ɛ yy ɛ yz ɛ zx ɛ zy ɛ zz P x χ xx χ xy χ xz P y = ɛ 0 χ yx χ yy χ yz χ zx χ zy χ zz P z D x D y D z = ɛ xx ɛ xy ɛ xz ɛ yx ɛ yy ɛ yz ɛ zx ɛ zy ɛ zz E x E y E z E x E y E z (4.17) (4.18) (4.19) (4.20) x, y, z 1,2,3

44 38 4 H E = µ 0 t (4.21) H = ɛ E t (4.22) E = 0 (4.23) H = 0 (4.24) J = σe (4.25) σ electric conductivity Ohm s law H E = µ 0 t (4.26) H = ɛ E + σe t (4.27) E = 0 (4.28) H = 0 (4.29) (4.21) rotation E = µ 0 ( H) t 2 E = ɛµ 0 (4.30) t 2 E = ( E) 2 E 2 E = ɛµ 0 2 E t 2 (4.31) ɛ wave equation v p = 1 ɛµ0 (4.32) v p phase velocity c = 1 ɛ0 µ 0 (4.33)

45 (4.12) E(r, ω) = P(r, ω) = E(r, t) exp( iωt)dt (4.34) P(r, t) exp( iωt)dt (4.35) P(r, ω) = ɛ 0 χ(ω)e(ω) (4.36) ɛ(ω) = ɛ 0 [1 + χ(ω)] (4.37) E(r, ω) = iωµ 0 H(r, ω) (4.38) H(r, ω) = iωɛ(ω)e(r, ω) (4.39) E(r, ω) = 0 (4.40) H(r, ω) = 0 (4.41) 2 E(r, ω) = ω 2 ɛµ 0 E(r, ω) (4.42) z E(r, t) = E 0 exp[i(kz ωt)] (4.43) k 2 = ω 2 ɛµ 0 (4.44) dispersion relation (4.43) v p v p = ω k (4.45) ɛ v p = 1 ɛµ0 (4.46)

46 40 4 (4.33) v p = c ɛ0 ɛ (4.47) refractive index n = ɛ ɛ 0 (4.48) v p = ω k = c n (4.49) wave packet group velocity v g = dω dk (4.50) n [ ] 1 [ dω dk d ( nω dk = = dω dω c group refractive index n g )] 1 [ = c n + ω dn ] 1 (4.51) dω v g = c n g (4.52) λ n g = n + ω dn dω n g = n λ dn dλ (4.53) (4.54) (4.48) ñ ɛ ɛ 0 = 1 + χ (4.55) n κ ñ = n + iκ (4.56) n κ extinction coefficient n (4.48) (4.55)

47 (4.43) (4.44) (4.43) k = ñω c (n + iκ)ω = c (4.57) E(r, t) = E 0 exp[i(kz ωt)] [ ( n ) = E 0 exp iω c z t ωκ ] c z v p = c n (4.58) (4.59) κ I(z) I(z) = I(0) exp( αz) (4.60) α absorption coefficient α α = 2ωκ c (4.61) κ λ = 2πc ω α = 4πκ λ (4.62) (4.63) χ χ χ χ = χ + iχ (4.64) ñ 2 = (n + iκ) 2 = 1 + χ = 1 + χ + iχ (4.65) 1 + χ = n 2 κ 2 (4.66) χ = 2nκ (4.67) n κ n 2 = 1 [ ] (1 + χ ) + (1 + χ 2 ) 2 + 4(χ ) 2 κ = χ 2n (4.68) (4.69)

48 42 4 κ 1 n = 1 + χ (4.70) κ = χ χ (4.71) ɛ ɛ ɛ, ɛ = ɛ + iɛ (4.72), (4.14) ɛ = ɛ 0 (1 + χ ) (4.73) ɛ = ɛ 0 χ (4.74) causality Kramers-Kronig relations χ(ω) ɛ(ω) χ ɛ χ ɛ χ (ω) = 2 π P ω χ (ω ) 0 (ω ) 2 ω 2 dω (4.75) χ (ω) = 2ω π P χ (ω ) 0 (ω ) 2 ω 2 dω (4.76) ɛ (ω) ɛ 0 = 2 π P ω ɛ (ω ) (ω ) 2 ω 2 dω (4.77) ɛ (ω) = 2ω π P ɛ (ω ) (ω ) 2 ω 2 dω (4.78) n(ω) 1 = 2 π P ω κ(ω ) (ω ) 2 ω 2 dω (4.79) κ(ω) = 2ω π P 0 n(ω ) (ω ) 2 ω 2 dω (4.80) P Cauchy P 0 f (ω ( ) ω δ (ω ) 2 ω 2 dω = lim δ 0 0 f (ω ) (ω ) 2 ω 2 dω + ω+δ f (ω ) ) (ω ) 2 ω 2 dω (4.81) π/2 (4.7) π/2 π/2

49 E(r, ω) = iωµ 0 H(r, ω) (4.82) H(r, ω) = iωɛ(ω)e(r, ω) + σ(ω)e(r, ω) (4.83) E(r, ω) = 0 (4.84) H(r, ω) = 0 (4.85) ɛ(ω) σ(ω) = ωɛ (ω) = ɛ 0 ωχ (ω) (4.86) 4.3 (4.58) H = H 0 exp[i(kz ωt)] (4.87) x E 0 = (E 0, 0, 0) y H 0 = (0, H 0, 0) H 0 = k ω (4.44) k µ 0 ω E 0 (4.88) H 0 = ɛ µ 0 E 0 = ɛ0 µ 0 (n + iκ)e 0 (4.89) Z 0 µ 0 /ɛ 0 = ( ɛ 0 /µ 0 ) Ω ( ) E E ( H) = ɛ 0 P (E E) + E 2 t t 1 E (E E) = E 2 t t H (4.90) (4.91) H ( E) = µ 0 2 (H H) (4.92) t

50 44 4 x y z 4.1: (E H) + t ( ɛ0 2 E E + µ ) 0 2 H H + E P = 0 (4.93) t (E H) = H ( E) E ( H) (4.94) (4.93) Poynting vector S = E H (4.95) U = ɛ 0 2 E E + µ 0 2 H H (4.96) (4.7) E J (4.97) ω (4.36) (4.93) E P t = 1 (E P) (4.98) 2 t U ɛ 0 2 E E + µ 0 2 H H E P (4.99)

51 r ( E(r) = E 0 exp ωκ ) c z (4.100) E x = 1 ( n )] [iω 2 E(r) exp c z t + c.c. (4.101) c.c. complex conjugate P x = 1 [ ( n )] 2 ɛ 0(χ + iχ )E(r) exp iω c z t + c.c. (4.102) H y = 1 2 ɛ0 [ ( n )] (n + iκ)e(r) exp iω µ 0 c z t + c.c. (4.103) U U = 1 2 ɛ 0n 2 E(r) 2 (4.104) intensity W/cm 2 I I = S (4.105) I 2ω 2ω Ī = 1 2 ɛ0 µ 0 n E(r) 2 = 1 2 ɛ 0cn E(r) 2 (4.106) Ī = U c n (4.107) U c/n (4.93) E P t = 1 2 ɛ 0χ ω E(r) 2 (4.108) dz α αu dz = 1 2 ɛ 0αn 2 E(r) 2 dz (4.109)

52 46 4 dz (n/c)dz (4.108) α χ 1 2 ɛ 0χ ω E(r) 2 (n/c)dz (4.110) α = ω nc χ (4.111) 4.5 α N α = σ a N (4.112) σ a absorption cross section σ a z I(z) S dz NS dz σ a dz σ a NS dz di(z) = σ ans dz I(z) = Nσ a I(z)dz (4.113) S I(z) = I 0 exp( Nσ a z) (4.114) (4.112) I I 0 = 10 A (4.115) A absorbance I 0 I A optical density Lambert Beer Lambert-Beer law I I 0 = 10 εcl (4.116) C l ε molar extinction coefficient C mol/l l cm ε L/(cm mol)

53 χ(ω) ( d 2 ) X m dt + ΓdX 2 dt + ω2 0 X = ee (4.117) X E m e ω 0 Γ Lorentz model E ω E = E 0 exp( iωt) (4.118) X X = X 0 exp( iωt) (4.119) X = ee 0 exp( iωt) m ( ω 2 0 ω2 iωγ ) (4.120) p p = ex (4.121) N P = N p = enx = Ne 2 E m ( ω 2 0 ω2 iωγ ) ω = ɛ 0 χe (4.122) Ne 2 χ(ω) = ɛ 0 m ( ω 2 0 ω2 iωγ ) (4.123) χ (ω) = Ne2 ω 2 0 ω2 ɛ 0 m ( ω 2 0 ω 2), (4.124) 2 + ω2 Γ 2 χ (ω) = Ne2 ωγ ɛ 0 m ( ω 2 0 ω 2) (4.125) 2 + ω2 Γ 2

54 48 4 χ χ'' ω ω Γ 4.2: χ χ ω 0 Γ ω 0 ω ω ω 0 χ(ω) = Ne2 1/(2ω 0 ) ɛ 0 m (ω 0 ω) iγ/2 (4.126) χ (ω) = Ne2 (ω 0 ω)/(2ω 0 ) ɛ 0 m (ω 0 ω) 2 + (Γ/2), 2 (4.127) χ (ω) = Ne2 Γ/(4ω 0 ) ɛ 0 m (ω 0 ω) 2 + (Γ/2) 2 (4.128) 1 3 χ (ω) ω Γ Lorentzian function χ (ω) 4.2 χ (ω) χ (ω) Γ ω dispersion ω normal dispersion ω anomalous dispersion χ(ω) = Ne2 f j ɛ 0 m ω 2 j j (4.129) ω2 iωγ j

55 f j oscillator strength 4.7 ω 0 ω 0 ω ω ω 0 Γ exp( γt) ( E(t) = E 0 exp γt ) 2 iω 0t E(ω) = 0 I(ω) E(ω) 2 γ (t > 0) (4.130) E(t) exp(iωt)dt (4.131) 1 (ω 0 ω) 2 + (γ/2) 2 (4.132) Γ γ homogeneous broadening Doppler effect ω 0 v z = ω = ω 1 v z /c (4.133) v z v z v z 1km/s ( ω ω v ) z c (4.134)

56 Gaussian Lorentzian (ω ω 0 )/ω h 4.3: v z = c(ω 0 ω) ω 0 (4.135) T m v z ( ) P(v z ) exp mv2 z 2k B T (4.136) (4.135) I(ω) exp mc2 (ω 0 ω) 2 (4.137) 2k B T ω 0 Gaussian function Doppler broadening inhomogeneous broadening Al 2 O 3 Cr 3+ FWHM; full width at half maximum ω h ω 2 0 L(ω) = 1 ω h /2 π (ω ω 0 ) 2 + (ω h /2), (4.138) 2

57 G(ω) = 2 ln 2 exp [ ] (4 ln 2)(ω ω 0 ) 2 /ω 2 h πωh (4.139) ln convolution Voigt

58 52 5 semi-classical 5.1 W n (n = 1, 2, 3, ) Hamiltonian H 0 Ψ n (r, t) = exp ( iw n t/ ) φ n (r) (5.1) φ n (r) H 0 H 0 φ n (r) = W n φ n (r) (5.2) φ m(r)φ n (r)dr = δ mn (5.3) H = H 0 + H (5.4) H Ψ(r, t) Schrödinger equation Ψ(r, t) H Ψ(r, t) = i t Ψ(r, t) = b n (t)ψ n (r, t) = n (5.5) b n (t) exp ( iw n t/ ) φ n (r) (5.6) n b n (t) b n (t) 2 b n (t) 2 = 1 (5.7) n

59 (5.5) (5.6) (H 0 + H ) b n (t) exp ( iw n t/ ) φ n (r) n = i n b n (t) t exp ( iw n t/ ) φ n (r) + i n ( b n (t) iw n ) exp ( iw n t/ ) φ n (r) (5.8) φ m(r) r i b m(t) t = n ( b n (t)h mn exp i W m W n ) t (5.9) b n (t) H mn H H mn = φ m(r)h φ n (r)dr (5.10) 5.2 H er E (5.11) e r E +e electric dipole moment µ = er (5.12) H = µ E (5.13) dipole approximation X µ µ mn φ m(r) µ φ n (r)dr (m n) (5.14) transition dipole moment µ mn = e φ m(r) r φ n (r)dr = er mn (5.15)

60 54 5 (5.14) m = n n H 0 φ n (r) φ n ( r) = φ n (r) (5.16) φ n ( r) = φ n (r) (5.17) (5.15) selection rule allowed transition forbidden transition 5.3 n = 1, 2 ω ω ω 0 ω W 2 W 1 ω 0 (5.18) µ 11 = µ 22 = 0 µ 21 = µ 12 (5.19) µ 12 = µ 21 (5.9) b 1 (t) t b 2 (t) t E(t) = E 0 cos ωt (5.20) = i µ12 E 0 b 2 (t) [ e i(ω0 ω)t + e ] i(ω 0+ω)t (5.21) 2 = i µ12 E 0 b 1 (t) [ e i(ω0 ω)t + e ] i(ω 0+ω)t (5.22) 2 exp[±i(ω 0 + ω)t] exp[±i(ω 0 ω)t] rotating-wave approximation E(t) = E 0 cos ωt = 1 2 E [ ] 0 exp(iωt) + exp( iωt) (5.23)

61 E(t) = 1 2 E 0 exp(iωt) (5.24) E(t) = 1 2 E 0 exp( iωt) (5.25) ω ω b 1 (t) t b 2 (t) t = i X 2 b 2(t) exp( i t) (5.26) = i X 2 b 1(t) exp(i t) (5.27) ω 0 ω (5.28) X µ 12 E 0 (5.29) b 1 (0) = 1, b 2 (0) = 0 (5.27) t (5.26) b 2 (t) 2 2 b 2 (t) t 2 i b 2(t) t + X2 4 b 2(t) = 0 (5.30) b 2 (t) = exp(iλt) (5.31) Ω λ = 1 ( ± Ω) (5.32) 2 Ω 2 + X 2 (5.33) Rabi angular frequency (5.30) [ i ] [ i ] b 2 (t) = A exp 2 ( + Ω)t + B exp 2 ( Ω)t (5.34) A B b 1 (t) = exp ( i 2 ) [ ( Ω t cos 2 t b 2 (t) = i X Ω exp ( i 2 t ) sin ( Ω 2 t ) + i Ω sin ( Ω2 t )] ) (5.35) (5.36) ( ) b 2 (t) 2 = X2 Ω Ω 2 sin2 2 t (5.37)

62 Ω/2 X 2 /Ω 2 π/ω = 0 Ω = X Rabi oscillation coherent optical process π/ω phase relaxation, dephasing coherence Ψ(r, t) density matrix 5.4 p N p = µ = e r = e Ψ (r, t) r Ψ(r, t)dr (5.38) Ψ n(r, t) r Ψ n (r, t) = 0 (5.39) Ψ(r, t) = b 1 Ψ 1 (r, t) + b 2 Ψ 2 (r, t) (5.40) N p = Ne Ψ (r, t) r Ψ(r, t)dr [b = Ne 1 eiw 1t/ φ 1 (r) + b 2 eiw 2t/ φ 2 (r)] r [ b 1 e iw 1t/ φ 1 (r) + b 2 e iw 2t/ φ 2 (r) ] dr = Nµ 12 [ b 1 b 2 e i(w 2 W 1 )t/ + b 1 b 2 ei(w 2 W 1 )t/ ] = 2Nµ 12 Re { b 1 b 2e i(w 2 W 1 )t/ } (5.41)

63 ω 0 = (W 2 W 1 )/ 5.5 (5.22) b 1 (t) = 1 b 2 (t) = i X 2 = X 2 t 0 [ e i(ω 0 ω)t + e i(ω 0+ω)t ] dt [ e i(ω 0 ω)t ] 1 ω 0 ω + ei(ω0+ω)t 1 ω 0 + ω (5.42) ω 0 ω 2 b 2 (t) 2 = X2 sin 2 [(ω 0 ω)t/2] (ω 0 ω) 2 (5.43) ω ω = ω 0 X 2 t 2 /4 ω 0 ω < 2π/t ω 0 ω 2 b 2 (t) 2 ω t 2 1/t t ω 0 ω ω 0 δω U(ω) ɛ 0 E = ω0 +δω ω 0 δω X = µ 12 E 0 U(ω)dω (5.44) (5.45) X 2 = = 1 µ12 E µ E0 2 cos2 θ = µ2 12 E2 0 = 3 2 2µ ɛ 0 2 b 2 (t) 2 = 2µ2 12 3ɛ 0 2 ω0 +δω ω 0 δω ω0 +δω ω 0 δω U(ω)dω (5.46) U(ω) sin2 [(ω 0 ω)t/2] (ω 0 ω) 2 dω (5.47)

64 58 5 sin 2 [(ω 0 ω)t/2] (ω 0 ω) 2 dω = π 2 t (5.48) ω 0 ω < 2π/t U(ω) t (5.47) U(ω) b 2 (t) 2 = = 2µ 2 ω0 +δω 12 3ɛ 0 U(ω sin 2 [(ω 0 ω)t/2] 0) dω 2 ω 0 δω (ω 0 ω) 2 πµ ɛ 0 U(ω 0) t (5.49) 2 w 12 = d dt b 2(t) 2 = πµ2 12 3ɛ 0 2 U(ω 0) (5.50) U(ω 0 ) U(ω 0 ) B B = πµ2 12 3ɛ 0 2 (5.51) b 1 (0) = 0 b 2 (0) = 1 w 21 w 12 B 12 = B 21 µ B = π µ2 ɛ 0 2 (5.52) µ 2 = 1 3 µ2 12 (5.53) A B (2.51) A A = µ2 12 ω3 3πɛ 0 c 3 (5.54) f 2mωµ2 12 3e 2 (5.55) 1 A τ r = 2πc3 ɛ 0 m e 2 = f ν 2 1 f ω 2 s (5.56) cm 2

65 ν = ω (5.57) 2πc 500 nm 20,000 cm 1 τ r = 3.75 ns/ f b 2 (t) γ/2 b 2 (t) 2 γ (5.27) b 1 (t) = 1 db 2 (t) dt = i X 2 ei t γ 2 b 2(t) (5.58) t b 2 (t) = i X e i t e γ(t t )/2 dt 2 = i X e i(ω 0 ω)t 2 i(ω 0 ω) + γ/2 (5.59) b 1 (t) 1 d b 2 (t) 2 dt = db 2 (t) b 2 (t) + b 2 dt (t)db 2(t) dt = γ b 2 (t) 2 (5.60) (5.46) w 12 = γ b 2 (t) 2 = X2 4 γ (ω 0 ω) 2 + (γ/2) 2 (5.61) w 12 = πµ2 ω0 +δω 12 γ/2π U(ω)dω (5.62) 3ɛ 0 2 ω 0 δω (ω 0 ω) 2 + (γ/2) 2 γ γ δω γ/2π dω = 1 (5.63) (ω 0 ω) 2 + (γ/2) 2 w 12 = πµ2 12 3ɛ 0 2 U(ω 0) (5.64) (5.59) { } µ p = Re 12 X (ω 0 ω) iγ/2 exp( iωt) (5.65)

66 60 5 χ(ω) = N µ2 ɛ 0 1 (ω 0 ω) iγ/2 (5.66) N 5.6 three-dimensional harmonic oscillator m k ω 0 V = mω2 0 2 (x2 + y 2 + z 2 ) (5.67) k = mω 2 0 (5.68) ( ) H = 2 2 2m x y mω2 0 2 z 2 2 (x2 + y 2 + z 2 ) (5.69) φ(x, y, z) ε (5.69) H φ(x, y, z) = εφ(x, y, z) (5.70) H = H x + H y + H z (5.71) 2 H x = 2 2m x + mω x2 (5.72) 2 H y = 2 2m y + mω y2 (5.73) 2 H z = 2 2m z + mω z2 (5.74) x y z φ(x, y, z) = X(x)Y(y)Z(z) (5.75) x y z H x X(x) = ε x X(x) (5.76) H x Y(y) = ε y Y(y) (5.77) H x Z(z) = ε z Z(z) (5.78)

67 ε = ε x + ε y + ε z (5.79) x y z x (5.76) ( ) 1/2 ( ) 2mω0 /h mω0 ( X n (x) = 2 n H n n! x exp mω ) 0 2 x2 (5.80) ( ε n = n + 1 ) ω 0 2 (n = 0, 1, 2, ) (5.81) H n (ξ) Hermitian polynomial exp( t 2 + 2ξt) = (5.80) X n (x) n=0 1 n! H n(ξ)t n (5.82) X m (x)x n (x)dx = δ mn (5.83) δ mn m = n 1 0 y z m (m = 0, 1, 2, ) l (l = 0, 1, 2, ) Y m (y) Z l (z) (n, m, l) φ nml (x, y, z) = X n (x)y m (y)z l (z) (5.84) ( ε = n + m + l + 3 ) ω 0 (5.85) 2 creation operator a x annihilation operator a x ( a mω0 x x ) d 2 mω 0 dx ( mω0 a x x + ) d 2 mω 0 dx (5.86) (5.87) a xx n (x) = n + 1X n+1 (x) (5.88) a x X n (x) = nx n 1 (5.89) (5.86) (5.87) x = ( ) a 2mω x + a x 0 (5.90)

68 62 5 φ 000 (x, y, z) x φ 100 (x, y, z) φ 000 (x, y, z) x φ 100(x, y, z)dr = (5.91) 2mω 0 y z µ µ = e (5.92) 2mω 0 (5.66) χ(ω) = Ne2 1 2ɛ 0 mω 0 (ω 0 ω) iγ/2 (5.93) (4.126) 5.7 a b f ba f ba = 2mω ba e 2 µ ba 2 (5.94) m ω ba a b 3 1 N N sum rule f ba = N (5.95) b x i j [x i, p x j ] = i δ i j (5.96) b Σ i p xi a [p 2 xi, x i] = 2i p xi (5.97) [ ] H, Σi x i = i p xi (5.98) m = im b H Σi x i Σ xh a i = im { b H c c Σ x a i b Σ x c } i c H a i i c = im ω ba b Σ x a i i i (5.99)

69 [ Σi p xi, Σ i x i ] = i N (5.100) a [ Σi p xi, Σ i x i ] a { a = Σ p b xi b Σ x a i a Σ x b i b Σ p a } xi i i i i b = i 2m ω ba b Σ x a 2 i = i N (5.101) i b 2m e 2 ω µba 2 ba = N (5.102) b f ba = N (5.103) b density matrix W 1 W 2 W 1 < W 2 2 Ψ n (r, t) = exp ( iw n t/ ) φ n (r) (n = 1, 2) (5.104) Ψ(r, t) = c 1 (t)φ 1 (r) + c 2 (t)φ 2 (r) (5.105) ρ ρ = ρ 11 ρ 12 ρ 21 ρ, (5.106) 22 ρ i j = c i (t)c j (t) (i, j = 1, 2) (5.107) 2 2

70 64 5 O O = Ψ (r, t)o Ψ(r, t)dr = {c 1 φ 1 + c 2 φ 2} O {c1 φ 1 + c 2 φ 2 } dr = c 1 c 1 O 11 + c 1 c 2 O 21 + c 2 c 1 O 12 + c 2 c 2 O 22 (5.108) = ρ 11 O 11 + ρ 12 O 21 + ρ 21 O 12 + ρ 22 O 22 = Tr(ρO) O i j O O i j φ i (r, t)o φ j(r, t)dr (5.109) Tr trace ρ nn = c n (t)c n(t) = c n (t) 2 (n = 1, 2) (5.110) 2 ρ 11 + ρ 22 = 1 (5.111) H Liouville equation d dt ρ = 1 [ ] H, ρ i (5.112) [ ] [A, B] AB BA (5.113) ( ) c1 Ψ = c 2 ( ) c1 c ρ = 1 c 1 c 2 c 2 c 1 c 2 c 2 ( ) c1 (c ) = i d dt c 2 ( c1 c 2 1, c 2 ) = H ( c1 c 2 ) (5.114) (5.115) (5.116)

71 i d dt ( c 1, ) ( c 2 = c 1, 2) c H (5.117) i d dt ρ = i d {( ) c1 (c dt c 1, ) } c 2 2 { ( )} d c1 (c = i dt c 1, ) ( ) { c1 d ( c 2 + i c 2 c 1 2 dt, ) } c 2 ( ) c1 (c = H c 1, ) ( ) c1 (c (5.118) c 2 2 c 1, c 2 2) H = H ρ ρh = [ H, ρ ] H = H 0 + H, (5.119) H 0 = W W 2 (5.120) H = 0 µ 12 µ cos ωt (5.121) i d dt ρ 11 ρ 12 ρ 21 ρ = W 1 µ 12 E 0 cos ωt ρ 11 ρ µ 12 E 0 cos ωt W 2 ρ 21 ρ 22 ρ 11 ρ 12 W 1 µ 12 E 0 cos ωt (5.122) ρ 21 ρ 22 µ 12 E 0 cos ωt W 2 d dt ρ 11(t) = iµ 12 E 0 cos ωt[ρ 12 (t) ρ 21 (t)] d dt ρ 12(t) = iω 0 ρ 12 (t) + iµ 12 E 0 cos ωt[ρ 22 (t) ρ 11 (t)] d dt ρ 21(t) = iω 0 ρ 21 (t) iµ 12 E 0 cos ωt[ρ 22 (t) ρ 11 (t)] = d dt ρ 12 (t) d dt ρ 22(t) = iµ 12 E 0 cos ωt[ρ 12 (t) ρ 21 (t)] = d dt ρ 11(t) (5.123)

72 66 5 ω 0 = W 2 W 1 d dt ρ 11(t) = iµ 12 E 0 cos ωt[ρ 12 (t) ρ 21 (t)] + ρ 22 (t)/t 1 d dt ρ 12(t) = iω 0 ρ 12 (t) + iµ 12 E 0 cos ωt[ρ 22 (t) ρ 11 (t)] ρ 12 (t)/t 2 d dt ρ 21(t) = iω 0 ρ 21 (t) iµ 12 E 0 cos ωt[ρ 22 (t) ρ 11 (t)] ρ 21 (t)/t 2 = d dt ρ 12 (t) d dt ρ 22(t) = iµ 12 E 0 cos ωt[ρ 12 (t) ρ 21 (t)] ρ 22 (t)/t 1 = d dt ρ 11(t) (5.124) T 1 T 2 longitudinal relaxation time transverse relaxation time dephasing time, phase relaxation time T 1 T 2 1 = T 2 2T 1 T2 (5.125) T 2 pure dephasing time ρ 12 ρ 21 ω 0 ω 0 ρ 12 ρ 21 ρ 12 (t) = ρ 12 (t)eiω 0t ρ 21 (t) = ρ 21 (t)e iω 0t (5.126) d dt ρ 21 (t) = iµ 12 E 0 [ e i(ω+ω 0 )t + e ] i(ω ω 0)t [ρ 22 (t) ρ 11 (t)] ρ 21 2 (t)/t 2 ρ 12 (t) = [ ρ 21 (t)] d dt ρ 22(t) = iµ 12 E 0 2 ( e iωt + e iωt) [ ρ 12 (t)eiω 0t ρ 21 (t)e iω 0t ] ρ 22 (t)/t 1 ρ 11 (t) = 1 ρ 22 (t) (5.127) ω 0 ω ±(ω 0 + ω) d dt ρ 21 (t) = iµ 12 E 0 e i(ω ω0)t [ρ 22 (t) ρ 11 (t)] ρ 21 2 (t)/t 2 ρ 12 (t) = [ ρ 21 (t)] d dt ρ 22(t) = iµ 12 E 0 2 [ ρ 12 (t)e i(ω ω 0)t ρ 21 (t)ei(ω ω 0)t ] ρ 22 (t)/t 1 ρ 11 (t) = 1 ρ 22 (t) (5.128) optical Bloch equations

73 p = µ = Tr {µρ} = Tr 0 µ 12 ρ 11 ρ 12 µ 12 0 ρ 21 ρ 22 = µ 12 (ρ 12 + ρ 21 ) = 2Re{µ 12 ρ 21 } (5.129) ρ 12 ρ perturbation (5.128) 1 ρ i j = ρ (n) i j n=0 (E 0 ) n ρ (n) i j, (5.130) (5.128) E 0 ρ (n) i j 0 (5.128) E 0 = 0 1 d dt ρ (1) ρ (0) 11 = 1 ρ (0) 22 = ρ (0) 12 = ρ (0) 21 = 0 21 (t) = iµ 12 E 0 2 ρ (1) 11 = ρ(1) 22 = 0 (5.131) e i(ω ω 0)t ρ (1) 21 (t)/t 2 (5.132) ρ (1) 21 (t) = µ 12 E 0 1 e i(ω ω 0)t 2 (ω 0 ω) i/t 2 ( ) { µ12 E 0 e iωt } p = µ 12 Re (ω 0 ω) i/t 2 1/T 2 = γ/2 (5.65) (5.133) (5.134)

74 P E P = ɛ 0 χe (6.1) χ linear optics E nonlinear optics 6.2 P = ɛ 0 [ χ (1) E + χ (2) E 2 + χ (3) E 3 + ] (6.2) = P L + P (2) + P (3) + (6.3) = P L + P NL (6.4) χ (1) = χ P L = ɛ 0 χ (1) E (6.5) P NL = P (2) + P (3) + (6.6) P (n) = ɛ 0 χ (n) E n (6.7) n n-th order χ (n) n nonlinear susceptibility (n + 1)

75 χ (n) χ (n) (6.7) χ (n) (6.2) NaCl Si r r χ (n) (6.7) P (n) E P (n) P (n) E E (6.7) P (n) = ( 1) n ɛ 0 χ (n) E n (6.8) n χ (n) = χ (2) P NL = P (2) = ɛ 0 χ (2) E 2 (6.9) E ω 1 ω 2 [ ] [ ] 1 1 E(t) = 2 E(ω 1) exp( iω 1 t) + c.c. + 2 E(ω 2) exp( iω 2 t) + c.c. (6.10) (6.9) P (2) (t) = ɛ 0χ (2) 4 { [[ ] E (ω 1 ) 2 [[ exp( 2iω1 t) + c.c.] + ] E (ω 2 ) 2 exp( 2iω2 t) + c.c.] + 2E (ω 1) [ E (ω 1) ] + 2E (ω 2 ) [ E (ω 2) ] + [ 2E (ω 1) E (ω 2) exp( i(ω 1 + ω 2 )t) + c.c. ] + [ 2E (ω 1) [ E (ω 2) ] exp( i(ω1 ω 2 )t) + c.c. ] } (6.11) ω 1 ω 2

76 70 6 [ ] [ ] 1 1 P (2) (t) = 2 P(2ω 1) exp( 2iω 1 t) + c.c. + 2 P(2ω 2) exp( 2iω 2 t) + c.c. [ ] P(ω 1+ω 2 ) exp( i(ω 1 + ω 2 )t) + c.c. [ ] P(ω 1 ω 2 ) exp( i(ω 1 ω 2 )t) + c.c. + P (0) (6.12) 1 P (2ω 1) = ɛ 0 2 χ(2) (2ω 1 ; ω 1, ω 1 ) [ E (ω 1) ] 2 P (2ω2) = ɛ 0 2 χ(2) (2ω 2 ; ω 2, ω 2 ) [ E ] (ω 2 2) P (ω 1+ω 2 ) = ɛ 0 χ (2) (ω 1 + ω 2 ; ω 1, ω 2 )E (ω1) E (ω 2) P (ω 1 ω 2 ) = ɛ 0 χ (2) (ω 1 ω 2 ; ω 1, ω 2 )E (ω 1) [ E (ω 2) ] P (0) = ɛ 0 2 χ(2) (0; ω 1, ω 1 )E (ω 1) [ E (ω 1) ] + ɛ 0 2 χ(2) (0; ω 2, ω 2 )E (ω 2) [ E (ω 2) ] (6.13) (6.14) (6.15) (6.16) (6.17) (6.10) ω 1 ω 2 k 1 k 2 [ ] 1 E(r, t) = 2 E(ω 1) exp[i(k 1 r ω 1 t)] + c.c. (6.13) [ ] E(ω 2) exp[i(k 2 r ω 2 t)] + c.c. (6.18) P (2ω 1) (r) = ɛ 0 2 χ(2) (2ω 1 ; ω 1, ω 1 ) [ E (ω 1) ] 2 exp(2ik1 r) (6.19) P (2ω2) (r) = ɛ 0 2 χ(2) (2ω 2 ; ω 2, ω 2 ) [ E ] (ω 2 2) exp(2ik2 r) (6.20) P (ω 1+ω 2 ) (r) = ɛ 0 χ (2) (ω 1 + ω 2 ; ω 1, ω 2 )E (ω1) E (ω2) exp[i(k 1 + k 2 ) r] (6.21) P (ω 1 ω 2 ) (r) = ɛ 0 χ (2) (ω 1 ω 2 ; ω 1, ω 2 )E (ω 1) [ E (ω 2) ] exp[i(k1 k 2 ) r] (6.22) P (0) (r) = ɛ 0 2 χ(2) (0; ω 1, ω 1 )E (ω 1) [ E (ω 1) ] + ɛ 0 2 χ(2) (0; ω 2, ω 2 )E (ω 2) [ E (ω 2) ] (6.23) χ (2) ; ω n E (ω n) exp(ik n r) ω n [ E (ω n) ] exp( ikn r) P (3) P (3) (t) = ɛ 0 χ (3) E(t) 3 (6.24) 1 χ (2) (ω 1 + ω 2 ; ω 1, ω 2 ) χ (2) ( ω 1 ω 2 ; ω 1, ω 2 ) 1

77 E ω 1 ω 2 ω 3 3 E(t) = 1 2 E(ω 1) exp( iω 1 t) E(ω 2) exp( iω 2 t) E(ω 3) exp( iω 3 t) + c.c. (6.25) (6.24) 44 3ω 1, 3ω 2, 3ω 3, ω 1, ω 2, ω 3, 2ω 1 ± ω 2, 2ω 1 ± ω 3, 2ω 2 ± ω 1, 2ω 2 ± ω 3, 2ω 3 ± ω 1, 2ω 3 ± ω 2, ω 1 + ω 2 + ω 3, ω 1 + ω 2 ω 3, ω 1 ω 2 + ω 3, ω 1 + ω 2 + ω 3 (6.26) P (3) (t) = 1 P (ωn) exp( iω n t) + c.c. (6.27) 2 n P (3ω1) = ɛ 0χ (3) [ ] E (ω 1 ) 3, 4 P (ω1) = ɛ 0χ (3) { 3 [ E ] (ω 2 [ ] 1) E (ω 1 ) + 6E (ω 1 ) E [ (ω 2) E ] (ω 2) + 6E (ω 1 ) E [ (ω 3) E ] } (ω 3), 4 P (2ω 1+ω 2 ) = 3 4 ɛ 0χ (3) [ E (ω 1) ] 2 E (ω 2 ), P (2ω 1 ω 2 ) = 3 4 ɛ 0χ (3) [ E (ω 1) ] 2 [ E (ω 2 ) ], P (ω 1+ω 2 +ω 3 ) = 6 4 ɛ 0χ (3) E (ω 1) E (ω 2) E (ω 3), P (ω 1+ω 2 ω 3 ) = 6 4 ɛ 0χ (3) E (ω 1) E (ω 2) [ E (ω 3) ] (6.28) ω 1 ω 2 ω 3 E ( ω) = [ E (ω)] (6.29) P (ω i+ω j +ω k ) = K 4 ɛ 0χ (3) E (ω i) E (ω j) E (ω k) (6.30) χ (3) P (ω i+ω j +ω k ) = K 4 ɛ 0χ (3) (ω i + ω j + ω k ; ω i, ω j, ω k )E (ω i) E (ω j) E (ω k) (6.31) ω i ω j ω k ±ω 1 ±ω 2 ±ω 3 K (degeneracy factor) (ω i, ω j, ω k ) E(r, t) = 1 2 E(ω 1) exp[i(k 1 r ω 1 t)] E(ω 2) exp[i(k 2 r ω 2 t)] E(ω 3) exp[i(k 3 r ω 3 t)] + c.c. (6.32) P (ω i+ω j +ω k ) (r) = K 4 ɛ 0χ (3) (ω i + ω j + ω k ; ω i, ω j, ω k )E (ωi) E (ω j) E (ωk) exp[i(k i + k j + k k ) r] (6.33) k m (m = i, j, k) ω m ω 1 ω 2 ω 3 k 1 k 2 k 3 ω m ω 1 ω 2 ω 3 k 1 k 2 k 3

78 H E = µ 0 t H = D t (6.34) (6.35) D = 0 (6.36) H = 0 (6.37) D = ɛ 0 E + P L + P NL = ɛe + P NL (6.38) (6.34) (6.35) (6.36) E = ( E) 2 E 2 2 E E = ɛµ 0 t + µ 2 P NL 2 0 (6.39) t 2 ω E P NL E P NL P NL (z, t) = 1 2 pnl (z) exp[i(k p z ωt)] + c.c. (6.40) E(z, t) = 1 2 A(z) exp[i(k rz ωt)] + c.c. (6.41) p NL (z) A(z) z (6.39) [ d 2 A(z) dz 2 + 2ik r da(z) dz ] kr 2 A(z) exp[i(k r z ωt)] = ɛµ 0 ω 2 A(z) exp[i(k r z ωt)] µ 0 ω 2 p NL (z) exp[i(k p z ωt)] (6.42) p NL = 0 E(z, t) = A(0) exp[i(k r z ωt)] (6.43) k 2 r = ɛµ 0 ω 2 (6.44) (4.44) z d 2 A(z) dz 2 da(z) k r dz (6.45)

79 (a) sinc x (b) (sinc x) x/π x/π 6.1: sinc slowly-varying envelope approximation (6.42) da(z) dz = iµ 0ω 2 2k r p NL (z) exp(i kz) (6.46) k k p k r (6.47) P NL E(z) P NL z p NL (z) p NL = const. (6.48) z = 0 z = L z = 0 A(0) = 0 (6.46) z = L A(L) = iµ 0ω 2 p NL 2k r = iµ 0ω 2 p NL L 2k r sin( kl/2) e i kl/2 k/2 sinc( kl/2)e i kl/2 (6.49) sinc sinc x sin x (6.50) x 6.1 x = 0 sinc x = 1 π A(L) k = 0 k < 2π/L k = 0 A(L) L k L L = 2π/ k 0 k = 0 k p = k r (6.51)

80 π k 2π k 6.2: phase matching condition k 6.2 2π/ k π/ k π/ k l c π/ k (6.52) coherence length l c ω 1 ω 2 ω 1 ω ω 1 +ω 2 sum frequency generation ω 1 ω 2 difference frequency generation ω 1 = ω 2 ω 2ω 0 SHG: second-harmonic generation optical rectification

81 ω 2ω ω 2 ω + ω 1 2 ω 2 ω ω ω 1 1 ω 1 ω 2 6.3: 2 6.1: 2 ω 2ω χ (2) (2ω; ω, ω) 2 ω 0 χ (2) (0; ω, ω) ω 1, ω 2 ω 1 + ω 2 χ (2) (ω 1 + ω 2 ; ω 1, ω 2 ) ω 1, ω 2 ω 1 ω 2 χ (2) (ω 1 ω 2 ; ω 1, ω 2 ) ω, 0 ω χ (2) (ω; ω, 0) optical parametric amplification ω fundamental wave 2ω second harmonic E(t) = 1 2 E(ω) exp( iωt) + c.c. (6.53) 2ω P NL (t) = 1 2 P(2ω) exp( 2iωt) + c.c. (6.54)

82 76 6 V(x) m m V ( x) + Dx = ω x mω0 2 V( x) = x 2 6.4: 2 (6.11) P (2ω) = ɛ 0χ (2) [ ] E (ω) 2 2 = d [ E (ω)] 2 (6.55) (6.56) d ɛ 0χ (2) 2 nonlinear optical coefficient 3 (6.57) anharmonicity V(x) = mω 0 2 x2 + m 3 Dx3 (6.58) D 3 [ d 2 x(t) m + Γ dx(t) ] + ω 2 dt 2 0 dt x(t) + Dx(t)2 = ee(t) (6.59) 2 E (ω) P (2ω) E(t) = E (ω) exp( iωt) + c.c. 2 (n 1) n 3 d d 1 2 χ(2)

83 x E x x = x (1) + x (2) + (6.60) x (n) E n (6.61) (6.59) E E 1 d 2 dt 2 x(1) (t) + Γ d dt x(1) (t) + ω 2 0 x(1) (t) = ee(t) m (6.62) E 2 d 2 dt 2 x(2) (t) + Γ d dt x(2) (t) + ω 2 0 x(2) (t) = D [ x (1) (t) ] 2 (6.63) x 1 2 E(t) = 1 2 E 0 exp( iωt) + c.c. (6.64) x (1) (t) = 1 2 x 0 exp( iωt) + c.c. (6.65) x (2) (t) = 1 2 x 2 exp( 2iωt) + c.c. (6.66) (6.62) (6.63) x 0 = ee 0 m(ω 2 0 ω2 iωγ) (6.67) x 2 = De 2 E 2 0 2m 2 (ω 2 0 ω2 iωγ) 2 (ω 2 0 4ω2 2iωΓ) (6.68) x 2 2ω N P(t) = en x(t) (6.69) 2 P (2) (t) = 1 DNe 3 E m 2 (ω 2 0 ω2 iωγ) 2 (ω 2 0 exp( 2iωt) + c.c. (6.70) 4ω2 2iωΓ) (6.56) 2 d = DNe 3 2m 2 (ω 2 0 ω2 iωγ) 2 (ω 2 0 4ω2 2iωΓ) (6.71)

84 (6.9) (6.56) P (2) i P (2ω) i = = ɛ 0 j,k j,k χ (2) i jk E je k (6.72) d i jk E (ω) j E (ω) k (6.73) i, j, k x, y, z x, y, z 1,2,3 2 d i jk j k i, j, k Kleinman s symmetry d i jk j k d d i jk d il contracted notation jk l jk : , 32 31, 13 12, 21 l : (6.74) KDP KH 2 PO 4 potassium dihydrogen phosphate LBO LiB 3 O 5 lithium triborate BBO β-bab 2 O 4 beta-barium borate KTP KTiOPO 4 potassium titanyl phosphate LiNbO 3 lithium niobate ZnTe GaP GaAs 6.2 d 6.3 d d eff P (2ω) = d eff [ E (ω) ] 2 (6.75)

85 : 2 GaAs, GaP, ZnTe 43m 14 = 25 = 36 KDP 42m 14 = BBO, LiNbO 3 3m 33, 31 = = 15, 22 = 12 = 16 KTP, LBO mm , 32 24, : 2 d il /ɛ 0 d χ (2) /2 d d CGS esu m/v (3/4π) 10 4 d il (pm/v) GaAs 43m d 14 = 90 GaP 43m d 14 = 100 ZnTe 43m d 14 = 129 KDP 42m d 36 = 0.6 BBO 3m d 22 = 2.3 LiNbO 3 3m d 33 = 34, d 31 = 6, d 22 = 2 KTP mm2 d 33 = 14, d 31 = 6.5, d 32 = 5 LBO mm2 d 31 = 1.1, d 32 = : 2

86 E 1 (z, t) = 1 2 A 1(z)e i(k1z ωt) + c.c. (6.76) 2ω P NL (z, t) = d 2 [A 1(z)] 2 e i(2k1z 2ωt) + c.c. (6.77) 2ω E 2 (z, t) = 1 2 A 2(z)e i(k2z 2ωt) + c.c. (6.78) (6.46) da 2 (z) dz = iµ 0(2ω) 2 2k 2 d{a 1 (z)} 2 e i kz (6.79) k 2k 1 k 2 (6.80) SHG A 1 (z) A 2 (0) = 0 A 2 (z) = 2iµ 0ω 2 d{a 1 (0)} 2 i kz/2 sin( kz/2) e k 2 k/2 = 2iµ 0ω 2 d{a 1 (0)} 2 e i kz/2 z sinc ( kz/2) k 2 (6.81) SHG I 2 I 1 I 2 A 2 (z) 2 I 2 1 = I 2 1 z2 sinc 2 ( kz/2) sin 2 ( kz/2) ( k/2) 2 (6.82) SHG l c = π/ k k k 1 k 2 ω 2ω n(ω) n(2ω) k 1 = n(ω) ω/c (6.83) k 2 = n(2ω) 2ω/c (6.84) k = 2k 1 k 2 = 2ω [n(ω) n(2ω)] (6.85) c

87 ω n (ω) n e (ω) 2ω 6.6: 2 SHG 2 n(2ω) > n(ω) (6.86) n(2ω) n(ω) = µm l c = 12.5 µm SHG birefringence n(ω) = n(2ω) (6.87) SHG SHG x y E x E y n x n y n x n y ω 2ω n x > n y 6.6 n x (ω) = n y (2ω) c θ c c c θ ordinary wave extraordinary wave n o (θ) n e (θ) θ n o (θ) = n o (6.88) 1 = cos2 θ + sin2 θ [n e (θ)] 2 n 2 o n 2 e (6.89) n o n e n e > n o n e < n o

88 82 6 θ m n ω o (θ m ) = n 2ω e (θ m ) (6.90) θ m sin 2 1 θ m = (n ω o ) 1 2 ( n 2ω o ) 2 / 1 1 ( ) 2 ( n 2ω e n 2ω o ) 2 (6.91) θ m ω n ω e (θ m ) = n 2ω o (θ m ) (6.92) θ m θ m phase matching angle I type I 1 [ n ω 2 o (θ m ) + n ω e (θ m ) ] = n 2ω e (θ m ) (6.93) II type II k 1 = k 2 (6.94) ω 2ω ω + ω = (2ω) (6.95) k 1 + k 1 = k 2 (6.96) optical parametric process

89 Pump Signal Idler 6.7: ω pump ω signal ω idler = ω pump ω signal ω pump ω signal ω idler OPA optical parametric amplification 3 OPA optical parametric amplifier ω ω = ω 1 + ω 2 ω 1 ω 2 spontaneous parametric down conversion nonclassical light OPO optical parametric oscillation OPO optical parametric oscillator ω 3ω = ω + ω + ω THG; third-harmonic generation ω = ω + ω ω degenerate four-wave mixing k 1 k 2 k 3 k 1 + k 2 k 3 optical Kerr effect 2 ω 1 ω 2 ω 2 = ω 1 ω 1 + ω 2

90 : 3 ω 3ω χ (3) (3ω; ω, ω, ω) 3 ω ω Re{χ (3) (ω; ω, ω, ω)} ω ω Im{χ (3) (ω; ω, ω, ω)} 2 ω ω χ (3) (ω; ω, ω, ω) 4 ω 1, ω 2 ω 1 Re{χ (3) (ω 1 ; ω 1, ω 2, ω 2 )} ω 1, ω 2 ω 1 Im{χ (3) (ω 1 ; ω 1, ω 2, ω 2 )} 2 ω 1, ω 2 ω 1 χ (3) (ω 1 ; ω 1, ω 2, ω 2 ) ω 1, ω 2 2ω 1 ω 2 χ (3) (2ω 1 ω 2 ; ω 1, ω 2, ω 1 ) 4 ω 1, ω 2, ω 3 ω 1 ± ω 2 ± ω 3 χ (3) (ω 1 ± ω 2 ± ω 3 ; ω 1, ±ω 2, ±ω 3 ) 4 ω 1 ω 2 ω 3 ω 1 ω 2 + ω 3 ω 1 ω 2 coherent Raman scattering coherent Brillouin scattering E ω ω 3 E(t) = 1 2 E(ω) exp( iωt) + c.c. (6.97) P (ω) (t) = 1 2 P(ω) exp( iωt) + c.c. (6.98) P (ω) = P (ω) L + P(ω) NL (6.99) P (ω) L = ɛ 0χE (ω) (6.100) P (ω) NL = 3 4 ɛ 0χ (3) [ E (ω)] 2 [ E (ω) ] = 3 4 ɛ 0χ (3) E (ω) 2 E (ω) (6.101)

91 { P (ω) = ɛ 0 χ + 3 } 4 χ(3) E (ω) 2 E (ω) (6.102) ω I (ω) = 1 2 ɛ 0cn E (ω) 2 (6.103) n ω { } P (ω) = ɛ 0 χ + 3χ(3) 2ɛ 0 cn I(ω) E (ω) (6.104) χ χ eff = χ + 3χ(3) 2ɛ 0 cn I(ω) (6.105) 3 optical Kerr effect χ (3) χ (3) 2 twophoton absorption induced absorption absorption saturation stimulated Raman gain inverse Raman effect I n n = n 0 + n 2 I (6.106) n 0 n 2 nonlinear refractive index 4 n 2 3 χ (3) κ 1 κ 1 n χ n = 1 + χ (6.107) (6.105) χ χ n n + n = 1 + Re{χ + χ} (6.108) [ 1 + χ 1 + Re{ χ} ] 2(1 + χ = n 0 + Re{ χ} (6.109) ) 2n 0 4 E n = n 0 + n 2 E 2 n 2 E

92 : (6.105) (6.106) n 2 = 3Re { χ (3) (ω; ω, ω, ω) } (6.110) 4ɛ 0 cn 2 n 2 Re { χ (3)} n 2 Kerr lens effect 6.8 self-focusing optical damage Kerr lens mode locking optical fiber optical soliton supercontinuum generation 6.9 polarizer optical Kerr shutter 1 ultrafast phenomena

93 : N 1 N 2 N = N 1 + N 2 2 T 1 1 dn 2 (t) dt dn 1 (t) dt = N 2 T 1 (6.111) = N 2 T 1 (6.112) I dn 2 (t) dt dn 1 (t) dt = N 2(t) T 1 + ai[n 1 (t) N 2 (t)] (6.113) = N 2(t) T 1 ai[n 1 (t) N 2 (t)] (6.114) I N 1 N 2 = aT 1 I N (6.115) N 1 N 2 absorption saturation α(i) = α I/I S (6.116) I S saturation intesity ( α(i) = α 0 1 I ) I S 3 (6.117) ω ω 2 two-photon absorption

94 88 6 ω ω 6.10: 2 ω 1 one-photon absorption 2 2 HOMO-LUMO highest ocupied molecular orbital - lowest unocupied molecular orbital 1 2 α I α = βi (6.118) β Im { χ (3)} 2 2 two-photon excitation microscope multiphoton absorption multiphoton excitation multiphoton fluorescence multiphoton ionization optical damage k 1 k 2 2 k 1 k 2 k 1 2k 1 k 2 k 2 2k 2 k 1 transient grating

95 6.6. n 89 k 1 2k 2 k 1 k : ω L ω S ω L ω AS ω v 6.12: ω v ω v ω L ω S ω L ω S = ω v ω v ω L + ω v ω S ω v 6.12 ω L + ω v = 2ω L ω S CARS; coherent anti-stokes Raman scattering 6.6 n n n ω i (i = 1,, n) ω p = n i=1 ±ω i ± i + ω p (n + 1) (n + 1)-wave mixing k i (i = 1,, n) k p = n i=1 ±k i (n + 1) ω r = ω p k r k r = k p k r = n ±k i (6.119) i=1

96 90 6 n ω r = ±ω i (6.120) i=1 ± k r ω r i (n + 1) ω i (i = 1,, n) ω p (6.120) (6.119) 1. 1 µm 1.2 µm (a) 1 µm (b) 1.2 µm (c) (d) 1 µm 1.2 µm 1 µm 1.2 µm (e) 1 µm 1.2 µm

97 91, 28, 83 A B, 13, 58, 32, 81, 48, 56, 66, 74, 80, 38 1, 88, 81, 34, 42, 82, 32, 31, 32, 26, 35, 66, 82, 44 LBO, 78, 31, 33, 38, 1, 25, 27, 27, 27 CARS, 89, 86, 25, 86, 2, 26, 54, 57, 66, 50, 35, 26, 4, 1, 88, 25, 86 GaAs, 78 GaP, 78, 73, 56, 1, 31, 32, 31, 75, 85, 27, 46, 41, 46, 83, 85, 87, 22, 27, 21, 16, 17, 54, 49, 54, 33, 4, 40, 78, 42

98 92, 32, 40, 40 KDP, 78 KTP, 78, 1, 26, 86, 66, 46, 34, 1, 8, 82, 1, 1, 7, 7, 28, 4, 56, 74, 80, 4, 56, 89, 84, 84, 89, 88, 88, 74, 60, 34, 14, 16, 4, 33, 83, 86, 10, 13, 85, 78, 83, 23, 35, 71, 83, 78, 9, 25, 86, 52, 66, 81, 40, 61, 43, 36 sinc, 73 ZnTe, 78, 49, 58, 62, 43, 56, 49, 48, 81, 61, 64, 27, 67, 77, 57, 53, 68, 54, 53, 36, 47, 53, 56, 67, 52, 53, 62, 25, 26, 83, 30, 74 II, 82 I, 82, 10, 88

99 93, 86, 88, 88, 88, 66, 33, 30, 86, 31, 33, 26, 27, 86, 86, 4, 30 T 2, 66 T 1, 66, 52, 1, 33 ZnTe, 78, 36, 23, 27, 27, 38, 1, 36, 36, 25, 31, 33, 36, 7, 50, 78 2, 83, 85, 87 2, 88 2, 88, 1 Nd:YAG, 29, 25, 2, 40, 31, 18, 19, 1, 38, 39, 54, 52, 50, 54, 69, 13, 26, 32, 34, 26 BBO, 78, 26 GaAs, 78, 83, 85, 86, 4, 4, 74, 86, 88, 1, 4, 11, 41, 43, 13, 18, 11, 82, 75, 83, 83, 83, 83, 86, 1, 83, 68, 85, 68, 76, 68, 76

100 94, 76, 16, 18, 23, 26, 25, 25, 51, 13, 49, 50, 81, 27, 5, 39, 39, 5, 39, 1, 28, 7, 26, 26, 27, 27, 36, 1, 4, 48, 39, 26, 66 He-Cd, 33 He-Ne, 31, 86, 44, 45, 87, 8, 23, 27, 75 HOMO-LUMO, 88, 13, 83, 1, 36, 23, 86, 4, 46, 26, 37, 85, 4, 10, 13, 85, 66, 15, 16, 55, 56, 46, 64, 18, 1, 1, 29, 7, 8, 12, 1, 4, 4, 4, 4, 1 CVD, 4, 18, 28, 18, 4, 28, 11, 14, 36, 44, 48 50, 47, 59, 76, 74, 56, 63