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とらふみよどぎみ
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2 4 MKSA

3 ii Nd:YAG

4 iii He-Ne

5 iv n

6 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)

7 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 1.

8 3 (a) γ (b) X (c) (d) (e) (f) nm 780 nm

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

10 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) exp(ik x L) = 1 exp(ik y L) = 1 exp(ik z L) = 1 (2.7)

11 6 2 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) (n x, n y, n z ) n 4πn 3 /3 4π 3 ( Lω ) 3 2 = ω3 L 3 (2.17) 2πc 3π 2 c 3 ω ω + dω ω ω 2 L 3 dω (2.18) π 2 c3

12 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 M. Planck quantum hypothesis ω ν W = nħω = nhν (n = 0, 1, 2, ) (2.25) ħ h/2π W = nħω ħω

13 8 2 n photon ω n p(n) p(n) exp( nħω/k B T) (2.26) W th W th = n=0 nħωe nħω/k BT n=0 e nħω/k BT (2.27) 1 r e ħω/k BT (2.28) (2.27) n=0 r n = 1 1 r (2.29) ħω nr n = ħωr r n=0 r n 1 = ħωr (2.30) (1 r) 2 n=0 W th = ħω r 1 r = ħω 1 1 = ħω e ħω/k BT 1 r 1 (2.31) Bose-Einstein distribution ħω k B T (2.23) ω ω + dω D(ω)dω U(ω)dω = ω2 ħωdω π 2 c 3 e ħω/k BT 1 (2.32) hν k B T (2.24) 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)

14 U(ω) hω/ k B T 2.2: (2.32) 0 T I = ħ π 2 c 3 0 ω 3 dω e ħω/k BT 1 = π2 k 4 B 15c 3 ħ 3 T 4 (2.35) 0 x 3 π4 e x dx = 1 15 (2.36) T 4 Stefan-Boltzmann law of radiation U (λ) λ (nm) 2.3: 6000K (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)

15 10 2 x = hc λ m k B T (2.39) 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 )/ħ absorption ħω ħω rate ω U(ω) B 12 U(ω) (2.42) 2.5

16 W 2 hω W 1 2.5: ω ħω = W 2 W 1 ħω ħω emission A T N 2 N 1 ( N 2 = N 1 exp ħω ) k B T (2.43) N 1 B 12 U(ω) = N 2 A (2.44) U(ω) = A ( exp ħω ) B 12 k B T (2.45) 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 ħω/k BT B 21 (2.48) T U(ω) B 12 = B 21 (2.49)

17 12 2 B B 12 = B 21 (2.50) (2.24) A B = ħω3 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: g 1, g 2 (2.43) N 2 = g ( 2 N 1 exp ħω ) (2.52) g 1 k B T g 1 B 12 = g 2 B 21 (2.53) A = g 1 ħω 3 B 12 g 2 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

18 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 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

19 14 2 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) N 1 = N 2 = dn 1 dt = dn 2 dt = dn 3 dt = 0 (2.61) 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) n (n = 0, 1, 2, 3) W n (n = 0, 1, 2, 3) W 0 < W 1 < W 2 < W

20 W 3 Γ γ 32 γ 21 W 2 γ 10 γ 01 W 1 W 0 2.8: 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

21 16 2 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 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)

22 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 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: transverse mode longitudinal mode 2.11(a) TEM

23 18 2 (a) (b) 2.11: (a) (b)2 TEM 00 TEM 01 TEM 10 TEM :

24 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) 1 R 1 R 2 exp(2gl) > 1 (2.90) R 1 R G > 1 2l ln(r 1R 2 ) (2.91) T 1 = 1 R 1 (2.92) T 2 = 1 R 2 (2.93) ln(1 x) x (2.94) (2.91) T 1 T 2 G > 1 2l (T 1 + T 2 ) (2.95) 100%

25 20 2 threshold G th G th = 1 2l (T 1 + T 2 ) (2.96) 0 ω ω 2.13: ω 2.13 N 2 Φ 2 γ 2 N 1 Φ 1 γ :

26 Φ 1 Φ 2 γ 1 γ 2 ħω 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) + ħω(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) 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 ħωb (2.106) dn 2 = dn 1 = 0 dt dt (2.107) du(ω) = 0 dt (2.108)

27 22 2 (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) 2.15 U ss è í U o Í 0 N th N (0) iƒ ƒ ƒsƒ ƒo Ì ³ j 2.15: N (0) quality value Q = ωw P L (2.114) ω W P L 2.3.5

28 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) p l pƒxƒcƒbƒ` ON pƒxƒcƒbƒ` OFF ½ ] ª z Œõ x ŽžŠÔ 2.16: 2.16 Q switching

29 24 2 Pockels effect mode locking ω ω E(t) = N 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 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 ω (2.123)

30 (2N + 1) 2 I 0 pulse train (2.87) T = 2L c (2.124) cavity round trip time N = Œõ x ŽžŠÔ 2.17: 21 T/(2N + 1) = 2π (2N + 1) ω (2.125) (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

31 26 2 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 p ÎŒõ θ i 1 n θ d 3.1:

34 n θ i tan θ i = n (3.1) p (3.1) θ i Brewster angle 90 p 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 1064 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

36 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 Er µm EDFA: erbium-doped fiber amplifier Er

37 He-Ne 3.4: He-Ne He-Ne He Ne He Ne nm Ar-ion laser Kr-ion laser 3.5 Ar nm nm Kr nm

38 : Ar Ar Kr excimer : ArF KrCl KrF XeCl XeF F nm 222 nm 249 nm 308 nm 351 nm, 353 nm 157 nm F CO µm

39 nm copper vapor laser nm nm He-Cd He Cd nm nm nm nm nm nm µm CH 3 OH µm 2.52 THz 3.6: 3.4 dye singlet 1 triplet intersystem crossing S 0 S 1 S 2 T 1 T 2

40 : 3.5 laser diode 3.6 free electron laser chemical laser

41 Maxwell equations permeability µ 0 H E = µ 0 t (4.1) E H = ϵ 0 + J t (4.2) ϵ 0 E = ρ (4.3) H = 0 (4.4) E H J ρ ϵ 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 ϵ 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 = ϵ 0 χe (4.12)

42 χ 3 3 permittivity ϵ D = ϵe = ϵ 0 E + P = ϵ 0 (1 + χ)e (4.13) ϵ ϵ 0 = 1 + χ (4.14) (4.12) x, y, z E x, E y, E z P x, P y, P z ( ) P i = ϵ 0 χ i j E j = ϵ 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 = ϵ i j E j = ( ) ϵ ix E x + ϵ iy E y + ϵ 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 ϵ i j = ϵ xx ϵ xy ϵ xz ϵ yx ϵ yy ϵ yz ϵ zx ϵ zy ϵ zz P x χ xx χ xy χ xz P y = ϵ 0 χ yx χ yy χ yz χ zx χ zy χ zz P z D x D y D z = ϵ xx ϵ xy ϵ xz ϵ yx ϵ yy ϵ yz ϵ zx ϵ zy ϵ 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

43 38 4 H E = µ 0 t (4.21) H = ϵ 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 = ϵ E + σe t (4.27) E = 0 (4.28) H = 0 (4.29) (4.21) rotation E = µ 0 ( H) t 2 E = ϵµ 0 (4.30) t 2 E = ( E) 2 E 2 E = ϵµ 0 2 E t 2 (4.31) ϵ wave equation v p = 1 ϵµ0 (4.32) v p phase velocity c = 1 ϵ0 µ 0 (4.33)

44 (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) = ϵ 0 χ(ω)e(ω) (4.36) ϵ(ω) = ϵ 0 [1 + χ(ω)] (4.37) E(r, ω) = iωµ 0 H(r, ω) (4.38) H(r, ω) = iωϵ(ω)e(r, ω) (4.39) E(r, ω) = 0 (4.40) H(r, ω) = 0 (4.41) 2 E(r, ω) = ω 2 ϵµ 0 E(r, ω) (4.42) z E(r, t) = E 0 exp[i(kz ωt)] (4.43) k 2 = ω 2 ϵµ 0 (4.44) dispersion relation (4.43) v p v p = ω k (4.45) ϵ v p = (4.33) 1 ϵµ0 (4.46) v p = c ϵ0 ϵ (4.47)

45 40 4 refractive index n = ϵ ϵ 0 (4.48) v p = ω k = c n (4.49) wave packet group velocity v g = dω dk (4.50) n dω dk = [ ] 1 [ dk d ( nω = 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 ϵ ϵ 0 = 1 + χ (4.55) n κ ñ = n + iκ (4.56) n κ extinction coefficient n (4.48) (4.55) (4.43) (4.44) k = ñω c (n + iκ)ω = c (4.57)

46 (4.43) 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) κ 1 n = 1 + χ (4.70) κ = χ χ (4.71)

47 42 4 ϵ ϵ ϵ,, (4.14) ϵ = ϵ + iϵ (4.72) ϵ = ϵ 0 (1 + χ ) (4.73) ϵ = ϵ 0 χ (4.74) causality Kramers-Kronig relations χ(ω) ϵ(ω) χ ϵ χ ϵ 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) ϵ (ω) ϵ 0 = 2 π P ω ϵ (ω ) (ω ) 2 ω 2 dω (4.77) ϵ (ω) = 2ω π P ϵ (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 P Cauchy P 0 f (ω ) (ω ) 2 ω 2 dω = lim δ 0 ( ω δ 0 0 n(ω ) (ω ) 2 ω 2 dω (4.80) 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 E(r, ω) = iωµ 0 H(r, ω) (4.82) H(r, ω) = iωϵ(ω)e(r, ω) + σ(ω)e(r, ω) (4.83) E(r, ω) = 0 (4.84) H(r, ω) = 0 (4.85) ϵ(ω) σ(ω) = ωϵ (ω) = ϵ 0 ωχ (ω) (4.86)

48 x y z 4.1: 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 = ϵ µ 0 E 0 = ϵ0 µ 0 (n + iκ)e 0 (4.89) Z 0 µ 0 /ϵ 0 = ( ϵ 0 /µ 0 ) Ω ( ) E E ( H) = ϵ 0 P (E E) + E 2 t t 1 E (E E) = E 2 t t (4.90) (4.91)

49 44 4 H (E H) + t H ( E) = µ 0 2 (H H) (4.92) t ( ϵ0 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 = ϵ 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 ϵ 0 2 E E + µ 0 2 H H E P (4.99) 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 ϵ 0(χ + iχ )E(r) exp iω c z t + c.c. (4.102) H y = 1 2 U ϵ0 [ ( n )] (n + iκ)e(r) exp iω µ 0 c z t + c.c. (4.103) U = 1 2 ϵ 0n 2 E(r) 2 (4.104)

50 intensity W/cm 2 I I = S (4.105) I 2ω 2ω Ī = 1 2 ϵ0 µ 0 n E(r) 2 = 1 2 ϵ 0cn E(r) 2 (4.106) Ī = U c n (4.107) U c/n (4.93) E P t = 1 2 ϵ 0χ ω E(r) 2 (4.108) dz α αu dz = 1 2 ϵ 0αn 2 E(r) 2 dz (4.109) dz (n/c)dz (4.108) α χ 1 2 ϵ 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

51 46 4 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) 4.6 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)

52 X = p N ee 0 exp( iωt) m ( ω 2 0 ω2 iωγ ) (4.120) p = ex (4.121) P = N p = enx = Ne 2 E m ( ω 2 0 ω2 iωγ ) ω = ϵ 0 χe (4.122) Ne 2 χ(ω) = ϵ 0 m ( ω 2 0 ω2 iωγ ) (4.123) χ (ω) = Ne2 ω 2 0 ω2 ϵ 0 m ( ω 2 0 ω 2), (4.124) 2 + ω2 Γ 2 χ (ω) = Ne2 ωγ ϵ 0 m ( ω 2 0 ω 2) (4.125) 2 + ω2 Γ 2 ω 0 Γ ω 0 ω ω ω 0 χ(ω) = Ne2 1/(2ω 0 ) ϵ 0 m (ω 0 ω) iγ/2 (4.126) χ (ω) = Ne2 (ω 0 ω)/(2ω 0 ) ϵ 0 m (ω 0 ω) 2 + (Γ/2), 2 (4.127) χ (ω) = Ne2 Γ/(4ω 0 ) ϵ 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

53 48 4 χ χ'' ω ω Γ 4.2: χ χ χ(ω) = Ne2 f j ϵ 0 m ω 2 j j (4.129) ω2 iωγ j f j oscillator strength 4.7 ω 0 ħω 0 ω ħω ħω 0 Γ exp( γt) ( E(t) = E 0 exp γt ) 2 iω 0t E(ω) = 0 I(ω) E(ω) 2 (t > 0) (4.130) E(t) exp(iωt)dt (4.131) 1 (ω 0 ω) 2 + (γ/2) 2 (4.132)

54 γ 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) 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 G(ω) = 2 ln 2 exp [ ] (4 ln 2)(ω ω 0 ) 2 /ω 2 h πωh (4.139)

55 Gaussian Lorentzian (ω ω 0 )/ω h 4.3: ln convolution Voigt

56 51 5 semi-classical 5.1 W n (n = 1, 2, 3, ) Hamiltonian H 0 Ψ n (r, t) = exp ( iw n t/ħ) 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) = iħ t (5.5) Ψ(r, t) = = b n (t)ψ n (r, t) n b n (t) exp ( iw n t/ħ) ϕ n (r) (5.6) n b n (t) b n (t) 2 b n (t) 2 = 1 (5.7) n

57 52 5 (5.5) (5.6) (H 0 + H ) b n (t) exp ( iw n t/ħ) ϕ n (r) n = iħ n b n (t) t exp ( iw n t/ħ) ϕ n (r) + iħ n ( b n (t) iw n ħ ) exp ( iw n t/ħ) ϕ n (r) (5.8) ϕ m(r) r iħ b m(t) t = n ( b n (t)h mn exp i W m W n ħ ) 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) (5.14) m = n n

58 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 ħω 0 ħω W 2 W 1 ħω 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) 2ħ = i µ12 E 0 b 1 (t) [ e i(ω0 ω)t + e ] i(ω 0+ω)t (5.22) 2ħ 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) E(t) = 1 2 E 0 exp(iωt) (5.24) E(t) = 1 2 E 0 exp( iωt) (5.25)

59 54 5 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 ħ (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) 0 0 Ω/2 X 2 /Ω 2 π/ω = 0 Ω = X Rabi oscillation

60 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/ħ ϕ 1 (r) + b 2 eiw 2t/ħ ϕ 2 (r)] r [ b 1 e iw 1t/ħ ϕ 1 (r) + b 2 e iw 2t/ħ ϕ 2 (r) ] dr = Nµ 12 [ b 1 b 2 e i(w 2 W 1 )t/ħ + b 1 b 2 ei(w 2 W 1 )t/ħ ] = 2Nµ 12 Re { b 1 b 2e i(w 2 W 1 )t/ħ } (5.41) ω 0 = (W 2 W 1 )/ħ 5.5

61 56 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) ϵ 0 E = ω0 +δω ω 0 δω X = µ 12 E 0 ħ U(ω)dω (5.44) (5.45) X 2 = = 1 µ12 E ħ µ 2 ħ 2 12 E0 2 cos2 θ = µ2 12 E2 0 = 3ħ 2 2µ ϵ 0 ħ 2 b 2 (t) 2 = 2µ2 12 3ϵ 0 ħ 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) 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 3ϵ 0 ħ U(ω sin 2 [(ω 0 ω)t/2] 0) dω 2 ω 0 δω (ω 0 ω) 2 = πµ ϵ 0 ħ 2 U(ω 0) t (5.49)

62 w 12 = d dt b 2(t) 2 = πµ2 12 3ϵ 0 ħ 2 U(ω 0) (5.50) U(ω 0 ) U(ω 0 ) B B = πµ2 12 3ϵ 0 ħ 2 (5.51) b 1 (0) = 0 b 2 (0) = 1 w 21 w 12 B 12 = B 21 µ B = π µ2 ϵ 0 ħ 2 (5.52) µ 2 = 1 3 µ2 12 (5.53) A B (2.51) A A = µ2 12 ω3 3πϵ 0 ħc 3 (5.54) f 2mωµ2 12 3e 2 ħ (5.55) 1 A τ r = 2πc3 ϵ 0 m e 2 = f ν 2 1 f ω 2 s (5.56) cm 2 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)

63 58 5 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) 3ϵ 0 ħ 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 3ϵ 0 ħ 2 U(ω 0) (5.64) (5.59) { } µ p = Re 12 X (ω 0 ω) iγ/2 exp( iωt) (5.65) χ(ω) = N µ2 ϵ 0 ħ 1 (ω 0 ω) iγ/2 (5.66) N

64 P E P = ϵ 0 χe (6.1) χ linear optics E nonlinear optics 6.2 P = ϵ 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 = ϵ 0 χ (1) E (6.5) P NL = P (2) + P (3) + (6.6) P (n) = ϵ 0 χ (n) E n (6.7) n n-th order χ (n) n nonlinear susceptibility (n + 1) χ (n)

65 60 6 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 ϵ 0 χ (n) E n (6.8) n χ (n) = χ (2) P NL = P (2) = ϵ 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) = ϵ 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 [ ] [ ] 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)

66 P (2ω 1) = ϵ 0 2 χ(2) (2ω 1 ; ω 1, ω 1 ) [ E (ω 1) ] 2 P (2ω2) = ϵ 0 2 χ(2) (2ω 2 ; ω 2, ω 2 ) [ E ] (ω 2 2) P (ω 1+ω 2 ) = ϵ 0 χ (2) (ω 1 + ω 2 ; ω 1, ω 2 )E (ω1) E (ω 2) P (ω 1 ω 2 ) = ϵ 0 χ (2) (ω 1 ω 2 ; ω 1, ω 2 )E (ω 1) [ E (ω 2) ] P (0) = ϵ 0 2 χ(2) (0; ω 1, ω 1 )E (ω 1) [ E (ω 1) ] + ϵ 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) 1 (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) = ϵ 0 2 χ(2) (2ω 1 ; ω 1, ω 1 ) [ E (ω 1) ] 2 exp(2ik1 r) (6.19) P (2ω 2) (r) = ϵ 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) = ϵ 0 χ (2) (ω 1 + ω 2 ; ω 1, ω 2 )E (ω 1) E (ω 2) exp[i(k 1 + k 2 ) r] (6.21) P (ω 1 ω 2 ) (r) = ϵ 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) = ϵ 0 2 χ(2) (0; ω 1, ω 1 )E (ω 1) [ E (ω 1) ] + ϵ 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) = ϵ 0 χ (3) E(t) 3 (6.24) 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 1 χ (2) (ω 1 + ω 2 ; ω 1, ω 2 ) χ (2) ( ω 1 ω 2 ; ω 1, ω 2 ) 1

67 62 6 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) = ϵ 0χ (3) [ ] E (ω 1 ) 3, 4 P (ω1) = ϵ 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 ϵ 0χ (3) [ E (ω 1) ] 2 E (ω 2 ), P (2ω 1 ω 2 ) = 3 4 ϵ 0χ (3) [ E (ω 1) ] 2 [ E (ω 2 ) ], P (ω 1+ω 2 +ω 3 ) = 6 4 ϵ 0χ (3) E (ω 1) E (ω 2) E (ω 3), P (ω 1+ω 2 ω 3 ) = 6 4 ϵ 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 ϵ 0χ (3) E (ω i) E (ω j) E (ω k) (6.30) χ (3) P (ω i+ω j +ω k ) = K 4 ϵ 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 ϵ 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

68 H E = µ 0 t H = D t (6.34) (6.35) D = 0 (6.36) H = 0 (6.37) D = ϵ 0 E + P L + P NL = ϵe + P NL (6.38) (6.34) (6.35) (6.36) E = ( E) 2 E 2 2 E E = ϵµ 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)] = ϵµ 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 = ϵµ 0 ω 2 (6.44) (4.44) z d 2 A(z) dz 2 da(z) k r dz (6.45)

69 (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 x (6.50) 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)

70 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 optical parametric amplification

71 66 6 ω ω 2ω ω ω 2 ω + ω 1 2 ω 1 1 ω 2 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) 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) 2 (6.11) P (2ω) = ϵ 0χ (2) [ ] E (ω) 2 2 = d [ E (ω)] 2 d χ(2) 2 nonlinear optical coefficient 3 (6.55) (6.56) (6.57) 2 E (ω) P (2ω) E(t) = E (ω) exp( iωt) + c.c. 2 (n 1) n 3 d d 1 2 ϵ 0χ (2)

72 V(x) 2 m m V ( x) + Dx = ω x 2 mω0 2 V( x) = x 2 6.4: anharmonicity V(x) = mω2 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) 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)

73 68 6 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 ϵ 0 (ω 2 0 ω2 iωγ) 2 (ω 2 0 4ω2 2iωΓ) (6.71) (6.9) (6.56) = ϵ 0 P (2) i P (2ω) i j,k = ϵ 0 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 27 10

74 : 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 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 = 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

75 : 2 d eff [ P (2ω) = ϵ 0 d ] eff E (ω) 2 (6.75) ω E 1 (z, t) = 1 2 A 1(z)e i(k 1z ωt) + c.c. (6.76) P NL (z, t) = ϵ 0d 2 [A 1(z)] 2 e i(2k 1z 2ωt) + c.c. (6.77) 2ω E 2 (z, t) = 1 2 A 2(z)e i(k 2z 2ωt) + c.c. (6.78) (6.46) da 2 (z) dz = iµ 0(2ω) 2 2k 2 ϵ 0 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 ϵ 0 d{a 1 (0)} 2 i kz/2 sin( kz/2) e k 2 k/2 = 2iµ 0ω 2 ϵ 0 d{a 1 (0)} 2 e i kz/2 z sinc ( kz/2) k 2 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 (6.81) (6.82) SHG l c = π/ k

76 n (ω) n e (ω) ω 2ω 6.6: 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 c [n(ω) n(2ω)] (6.85) 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

77 72 6 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 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 ħω + ħω = ħ(2ω) (6.95) ħk 1 + ħk 1 = ħk 2 (6.96)

78 Pump Signal Idler 6.7: optical parametric process ω pump ω signal ω idler = ω pump ω signal ħω pump ħω signal ħω 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

79 : 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, 0 2ω χ (3) (2ω; ω, ω, 0) 2 effect 2 ω 1 ω 2 ω 2 = ω 1 ω 1 + ω 2 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 = ϵ 0χE (ω) (6.100)

80 P (ω) NL = 3 4 ϵ 0χ (3) [ E (ω)] 2 [ E (ω) ] = 3 4 ϵ 0χ (3) E (ω) 2 E (ω) (6.101) { P (ω) = ϵ 0 χ + 3 } 4 χ(3) E (ω) 2 E (ω) (6.102) ω I (ω) = 1 2 ϵ 0cn E (ω) 2 (6.103) n ω { } P (ω) = ϵ 0 χ + 3χ(3) 2ϵ 0 cn I(ω) E (ω) (6.104) χ χ eff = χ + 3χ(3) 2ϵ 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

81 : 6.9: (6.105) (6.106) n 2 = 3Re { χ (3) (ω; ω, ω, ω) } (6.110) 4ϵ 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

82 optical Kerr shutter 1 ultrafast phenomena 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 ħω 2 two-photon absorption ħω 1 one-photon absorption 2 2 HOMO-LUMO highest occupied molecular orbital - lowest unoccupied molecular orbital 1

83 78 6 ω ω 6.10: 2 k 1 2k 2 k 1 k : 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

84 6.6. n 79 ω L ω AS ω L ω S ω v 6.12: k 1 2k 1 k 2 k 2 2k 2 k 1 transient grating ħω 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 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)

85 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

86 81, 29, 73 A B, 12, 57, 32, 72, 47, 55, 65, 71, 38 1, 77, 71, 34, 42, 72, 33, 32, 33, 27, 35, 72, 44 LBO, 69, 31, 34, 38, 1, 25, 28, 28, 28 CARS, 79, 76, 26, 76, 2, 27, 53, 56, 49, 35, 27, 4, 1, 78, 79, 26, 76 GaAs, 69 GaP, 69, 64, 55, 1, 32, 32, 66, 75, 28, 46, 41, 45, 74, 75, 77, 23, 28, 22, 16, 17, 53, 49, 53, 34, 4, 40, 68, 42, 32, 40, 40 KDP, 69

87 82 KTP, 69, 1, 27, 76, 46, 34, 1, 8, 72, 1, 1, 7, 7, 29, 4, 55, 65, 70, 4, 55, 79, 74, 74, 79, 77, 77, 65, 34, 13, 16, 4, 34, 73, 76, 10, 12, 75, 70, 73, 23, 35, 62, 73, 69, 9, 25, 76, 51, 72, 40, 43, 36 sinc, 64 ZnTe, 69, 48, 57, 43, 55, 48, 47, 71, 28, 67, 55, 52, 59, 53, 52, 36, 47, 52, 55, 51, 52, 25, 27, 73, 30, 65 II, 72 I, 72, 10, 78, 76, 78, 78, 78, 17, 33, 31, 76, 32, 34, 27, 28, 77, 76, 4, 31, 51, 1, 34

88 83 ZnTe, 69, 36, 23, 28, 28, 38, 1, 36, 36, 25, 32, 34, 36, 7, 49, 69 2, 74, 75, 77 2, 78 2, 78, 1 Nd:YAG, 30, 25, 2, 40, 32, 19, 20, 1, 38, 39, 53, 51, 49, 53, 60, 12, 13, 27, 33, 35, 27 BBO, 69, 27 GaAs, 69, 73, 75, 77, 4, 4, 65, 76, 78, 1, 4, 10, 41, 43, 13, 19, 11, 73, 65, 73, 73, 73, 73, 76, 1, 73, 59, 75, 59, 66, 59, 67, 67, 16, 19, 24, 27, 25, 25, 50, 13, 49, 49, 71, 28, 5, 39, 39, 5, 39, 1, 29, 7, 27, 27, 29, 28

89 84, 36, 1, 4, 47, 39, 27 He-Cd, 34 He-Ne, 32, 76, 44, 45, 77, 8, 24, 28, 65 HOMO-LUMO, 77, 13, 73, 4, 1, 1 CVD, 4, 19, 29, 19, 4, 29, 10, 14, 36, 44, 47, 49, 46, 57, 67, 65, 1, 36, 55, 24, 76, 4, 46, 27, 37, 75, 4, 10, 12, 75, 17, 14, 16, 54, 54, 46, 19, 1, 1, 30, 7, 8, 12, 1, 4, 4, 4

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PDF 1 1 1 1-1 1 1-9 1-3 1-1 13-17 -3 6-4 6 3 3-1 35 3-37 3-3 38 4 4-1 39 4- Fe C TEM 41 4-3 C TEM 44 4-4 Fe TEM 46 4-5 5 4-6 5 5 51 6 5 1 1-1 1991 1,1 multiwall nanotube 1993 singlewall nanotube ( 1,) sp 7.4eV