物理工学科博士前期課程 H21 ... - fir.u-fukui.ac.jp

1 H21 ( . I ( 1a).. E ( ) . P . P P ( 2 . P . 2 .. ( 1b) .. x px2 p .. ( 1c) .. Photorefractive .. a Robert W. Boyd Nonlinear Optics, Seconde Edition Academice Press 1 2 11 MKS . 0 0 . 0 .. b . 1. H21 ( . ~ ~ ~ ~ ~ ~. D 0 E + P = 0 E + e 0 E = (1 + e ) 0 E ( ) ( ). D . (1 + e ) 0 = e 0 ( ) ( ). 0 . exp( i t) . ~. E(t ) = E exp( i t ) = A exp( i t + ikr ) . E . ~ ~. ( ) P E ( 2) . ~ ~. P(t ) = e 0 E(t ) ( ) ( ). ~ ~. E P . ( ) . ~ ~. P = ( E ) 0 ( ). ~. E . ~ ~. P E E .. ~ ~ ~ ~. P (t ) = 0 ( (1) E (t ) + ( 2 ) E 2 (t ) + ( 3) E 3 (t ) + "). ~ ~ ~ ( ). P (1) (t ) + P ( 2 ) (t ) + P ( 3) (t ) + ". ( ) ( ) (linear susceptibility) ( ) 1 2 3 . (1) ( 2).. ( 3). 2 (second-order nonlinear susceptibility) 3 . (third-order nonlinear susceptibility) .. ~ ~. P i E . j 2 ij (1).. ~. 2 3 Pi . Ej Ek Ej, Ek, El ijk ( 2).. c .. 2a . 2. H21 ( . electronic acoustic resonance phonon resonance nonlinear susceptibility absorption optical phonon resonance.))))

2 Frequency . U. U=px2+qx3. U=px2. P=-ex E( ). U=px2-rx4. -e x position -e U E exp(-i t) . U=px2 . U= px2+ qx3 U= px2 rx4 P(t)= ex(t).. 3. H21 ( . ijkl (3) 3 2 3 . ~ ( 2) ~ ~ ( 3) ~. = 0 ijk E (t ) 2 2 P ( 3) = 0 ijkl E (t ) 3 3 . ( 2). P.. ~ ~. ( ) P (t ) E (t ) . ~. E (t ) . Kramers-Kronig ( 4a) .. ( 4b) .. ( 4c) e .. U=px2 p . P(t)=. ex(t) . x(t) . U=px2 qx3 q . q=0 . rx4 r 3.. Si Ge .. 2 ( 4d) . GaAs 3 . 2 . ~. 2 0 . ( 2). E2. Eat E 1 . ~. 0 (1) E ( 2 ) (1) / E at (1) . 1 / E at E at = e /( 4 0 a 0 ) a0 e . ( 2) 2.. 4a ( ) Kramers-Kronig . 4b .. 4. H21 ( . ( 2 ) 1 / E at 2 x 10-12 (m/V) ( ). 3 . (3) 1 / E at 2 4 x 10-24 (m2/V2) ( ). 3 2 .. ( 5a). ~ ~. ~ 2E 2P. E 2 = 2. 2. ( ). t t ~. 2 P / t . 2 2.. 2 Second-Harmonic Generation, SHG . 2 SHG .. ~. E (t ) = E exp( i t ) + ( ). Complex conjugate ( 5b) 2 .. ~ ~. P ( 2) (t ) = 0 ( 2 ) E 2 (t ). ( ). = 2 0 ( 2) EE * +( 0 ( 2 ) E 2 exp( 2i t ) + ).))

3 4c .. 4d 2 .. 5. H21 ( . (a) (b).. (2) 2 . 2 . (a) 2 (b) 2 .. 2 DC 1 . 2 2 2 ( ) . 2 SHG DC . Optical Rectification . OR . SHG 100% 2 . Nd:YAG m LBO CLBO m SHG (b) . = 2= . ( 2= = = + = ) . (b) virtual states .. (Sum Frequency Generation, SFG) (Difference Frequency Generation, DFG). 2 .. ~. E (t ) = E 1 exp( i 1 t ) + E 2 exp( i 2 t ) + ( ).. 5a ( ) . ~. 2 P / t 2 . 5b E phasor E ph exp( i t ) E. ph . 1/2 phasor . ~. Re( E ph exp( i t )) E (t ) = Re( E. ph exp( i t )) = 12 E ph exp( i t ) + ( ) E= Eph/2 . 6. H21 ( . ~ ~. P ( 2) (t ) = 0 ( 2) E 2 (t ). = 0 ( 2) [ E1 exp( 2i 1 t ) + E 2 exp( 2i 2 t ). 2 2. ( ). + 2 E1 E 2 exp( i ( 1 + 2 ) t ) + 2 E1 E 2 * exp( i ( 1 2 ) t ) + ]. + 2 0 ( 2 ) [ E1 E1 * + E 2 E 2 *].. ~. P ( 2) (t ) = P( n ) exp( i n t ) ( ). n n n .. P (2 1 ) = 0 ( 2 ) E12 SHG1 . P(2 2 ) = 0 ( 2) E 22 SHG2 . P ( 1 + 2 ) = 2 0 ( 2 ) E1 E 2 SFG ( ). P ( 1 2 ) = 2 0 ( 2 ) E1 E 2 * DFG.))

4 P (0) = 2 0 ( 2 ) ( E1 E1 * + E 2 E 2 *) OR .. P ( 2 1 ) = 0 ( 2 ) E1 *2 SHG1 P( 2 2 ) = 0 ( 2 ) E2 *2 SHG2 . P ( 1 2 ) = 2 0 ( 2 ) E1 * E 2 * SFG P ( 1 + 2 ) = 2 0 ( 2 ) E1 * E 2 DFG . ( ). ( ) . ( 7) . ( ) . ( ) .. 7 E exp( i t ) .. 7. H21 ( . A. SFG . (a) (b). 1 2. (2) 3= 1+ 2 3. 2 1. (a) (b) .. Sum Frequency Generation, SFG . 1 2 2 . 3= 1+ 2 ( ) .. P ( 1 + 2 ) = 2 0 ( 2 ) E1 E 2 SFG ( ). SFG SHG. SFG = 1 = 2 . = 3 = =( 1 + 2 ) SFG.. B. DFG . Difference Frequency Generation, DFG . ( ) . P ( 1 2 ) = 2 0 ( 2 ) E1 E 2 * DFG ( ). 1 2 3= 1 2 ( 1> 2) ( 8) DFG .. DFG SFG (b) . DFG SFG DFG 1 . 2 3 3 2 . 2 DFG . Optical Parametric Amplification, OPA . 8 2 2 . 8. H21 ( . (a) (b). 2. 1. (2) 3= 1 2 1. 2 3. (a) (b) .. (b) 1 . 2 Stimulated) . 2 3 2 2 . 2 . (Spontaneous two photon emission) 3 2 . Parametric Fluorescence ( 9a) . C. Optical Parametric Oscillation, OPO . DFG 2 3.)

5 ( 9b) ( ) 2. 3 . (Optical Parametric Oscillator) . 1 (Pump) 2 . (Signal) 3 (Idler . idle ) .. 1= 2+ 3 2(signal). (2) 3(idler). pump . 9a 1 1 . Parametric Fluorescence 1 . (Optical Parametric Generation, OPG) 3 2 . 9b . ( ) . 9. H21 ( . (3) 3 . 3 . ~ ~. P ( 3) (t ) = 0 ( 3) E 3 (t ) (3 ) ( ). ~. E (t ) .. ~. E (t ) = E exp( i t ) + = E exp( i t ) + E * exp(i t ) ( ). ( ) . ~ ~. P ( 3) (t ) = 0 ( 3) E 3 (t ) = 0 ( 3) ( E exp( i t ) + ) 3. 2 ( ). = 0 ( 3) E 3 exp( 3i t )+ 3 0 (3) E E exp( i t )+ Third Harmonic Generation Optical Kerr Effect EE*=|E|2 3 3 . 3 (Third Harmonic Generation, THG) (b) . 3 . 2 |E|2E . I |E|2 .. 2 3. ( 10) . (a) (b).. (3). 3 3 .. (a) 3 (b) 3 . 10 2 2 . 3 . 10. H21 ( . 3 . n = n0 + n2 I ( ). n0 I 2 . 2 . 2 .. ( ) 2 . 2. P ( ) = 0 (1) E ( ) + 3 0 ( 3) E ( ) E ( ) 0 eff E ( ) ( ).. 2. eff = (1) + 3 (3) E ( ) ( ).. 0. [. = n 2 = 1 + eff = 1 + (1) + 3 ( 3) E ( ).))]

6 2. ] ( ). ( ) 2 2 22I2 . ( ) . 2. n 2 = (n0 + n2 I ) 2 n0 + 2n0 n2 I = n0 + 4n0 n2 0 / 0 E ( ). 2 2 2. ( ). 2 2. = ( ) = 1 + (1) + 3 ( 3) E ( ) = n0 + 3 ( 3) E ( ). 2. 2 . I = 2n 0 0 / 0 E ( 11) 1 + (1) = (1) / 0 = n0 . 2.. 3 0 ( 3). n2 = ( ). 4n0. 2. 0. 3 Kerr (Optical Kerr Effect) . Self-Focusing . Kerr Self-Focusing ( ) n2 . 11 . phasor 1/4 . 2. c = 1/ 0 0 I = (c / n0 ) 12 E . 2. 1. 2 E n0 = / 0 c / n0 . 11. H21 ( . ( ) .. Self-Focusing .. Self-Focusing . Kerr . (4) 3 .. ~ ~. P ( 3) (t ) = 0 ( 3) E 3 (t ) (3 ) ( ).. ~. E (t ) = E1 exp( i 1 t ) + E 2 exp( i 2 t ) + E3 exp( i 3 t ) + ( ). ~. E 3 (t ) 44 ( 12) .. 1 , 2 , 3 , 3 1 , 3 2 , 3 3 , ( 1 + 2 + 3 ), ( 1 + 2 3 ), ( 1 2 + 3 ), ( 1 + 2 + 3 ), ( ). (2 1 2 ), (2 1 3 ), (2 2 1 ), (2 2 3 ), (2 3 1 ), (2 3 2 ), 3 2 . ~. P ( 3) (t ) = P( n ) exp( i n t ) ( ). n n . P ( 1 ) = 0 ( 3) (3E 1 E 1 * +6 E 2 E 2 * +6 E 3 E 3 *) E 1 , P ( 2 ) = 0 ( 3) (6 E 1 E 1 * +3E 2 E 2 * +6 E 3 E 3 *) E 2 , Kerr Effect ( ).)

7 P ( 3 ) = 0 (3) (6 E 1 E 1 * +6 E 2 E 2 * +3E 3 E 3 *) E 3 , P(3 1 ) = 0 ( 3) E13 , P(3 2 ) = 0 ( 3) E 23 , P(3 3 ) = 0 (3) E33 , THG (1. ). P( 1 + 2 + 3 ) = 6 0 (3) E1 E 2 E3 , P( 1 + 2 3 ) = 6 0 (3) E1 E 2 E3 *, ( ). P( 1 2 + 3 ) = 6 0 (3) E1 E 2 * E3 , P( 1 + 2 + 3 ) = 6 0 (3) E1 * E 2 E3 , 12 3 E(t) 6 . E3(t) 6x6x6=228 . 44 . 22 . 12. H21 ( . P (2 1 + 2 ) = 3 0 (3) E1 E 2 , P(2 1 + 3 ) = 3 0 ( 3) E1 E3 , 2 2. P (2 2 + 1 ) = 3 0 (3) E1 E 2 , P(2 2 + 3 ) = 3 0 (3) E 2 E3 , 2 2. ( ). P (2 3 + 1 ) = 3 0 ( 3) E3 E1 , P(2 3 + 2 ) = 3 0 ( 3) E3 E 2 , 2 2. P (2 1 2 ) = 3 0 ( 3) E1 E 2 *, P(2 1 3 ) = 3 0 ( 3) E1 E3 *, 2 2. P (2 2 1 ) = 3 0 ( 3) E1 * E 2 , P(2 2 3 ) = 3 0 ( 3) E 2 E3 *, 2 2. ( ). P (2 3 1 ) = 3 0 ( 3) E3 E1 *, P(2 3 2 ) = 3 0 ( 3) E3 E 2 *, 2 2.. P(2 3 2) E32E2* E3 2 E2 . E2* . 3+ 3 2=2 3 2 . Permutation . P(2 3 2) (E3 E3 E2*) ( 3+ 3 2), (E3 E2*E3) ( 3.))

8 2+ 3), (E2*E3 E3) ( 2+ 3+ 3) Kerr .. ( 3). P( = + ) (EEE*). ( + ), (EE*E) ( + ), (E*E E) ( + + ) . ( 13) .. t . E t = = / E .. ( ) .. 13 ( ) .. 3 intrinsic permutation symmetry . 13. H21 ( . (a) Optical Kerr Effect (b). 1 ( =1,2,3). 4= 1= 1 1+ 1. 2 (2). = 2 2+ 1 4= 1. 3 1. = 3 3+ 1 = + 1. (c) Sum frequency of 3 waves (d). 1 3. 4= 1+ 2+ 3. 2 (2). 3 2. 4. 1. (e) (f). 1 3. 4= 1+ 2 3 2. 2 (2). 3 4= 1+ 2 3. 1. (g) (h). 3 1. 4= 3+ 3 2. 2 (2). 3 3 4=2 3 2. 3 (a),(b) Kerr . (c),(d) (e),(f) . 1+ 2 3 (g),(h) 2 2 2 3 .. ( ) .. 14. H21 ( . (Saturable Absorption) (Optical Bistability) .. (Saturable Absorption). (Saturable Absorption) . ( 15a) .. 0. = ( ). 1+ I / Is 0 I Is 1/2 .. (Optical Bistability). (Optical Bistability) . Fabry-Perot .. 2 .. ( 15b) . (a) (b). Iout 3. Iin saturable Iout absorber 2. 4. 1. Iin (a) Fabry-Perot (b) Iin . Iout .. 15a ( saturable absorption) .. 1 1.)

9 ( ) . (Light Amplification by Stimulated Emission of Radiation, LASER) . 15. H21 ( .. 2 ( ) 2 . 2 .. real excited state . virtual state . ground state 2 .. = ( 2) I ( ). (2) 2 .. 2 . R = I / = = ( 2) I 2 / = ( ). 2 . (Stimulated Raman Scattering). (Raman Scattering) . v S= v = v Stokes . Anti-Stokes . (Stimulated Raman Scattering)( 16) . v ( ) .. 10% . Brillouin Brillouin 1 . Rayleigh Rayleigh .. 15b . ( ) . 16 2 3 . (Polarizability) 2 ( . ) . 16. H21 ( .. ~. E(r, t) . ~ ~. E(r, t) = ' E n (r, t) ( ). n ~. ' E n (r, t) .. ~ ~ (+) ~ ( ). E n (r, t) = E n (r, t) + E n (r, t) ( ).. ~ (+). E n (r, t) = E n (r )exp(-i n t) ( ). ~ ( ) ~ (+). E n (r, t) = E n * (r ) = E n * (r )exp(i n t) ( ). ~ ( ) ~ (+) ~ ~. E n (r ) E n (r ) E n (r, t) E(r, t) . An( . ) . E n = A n exp(ik n r ) ( ).. ~. E(r, t ) = ' E n exp( i n t ) + n ( ). = ' A n exp{i (k n r n t )} + n . E n = E( n ), A n = A( n ) ( ).))

10 N n . ( ) . E( n ) = E( n )*, or A( n ) = A( n ) *.. ~. E(r, t ) = E( n ) exp( i n t ). n ( ). = A( n ) exp{i (k n r n t )}. n .. 17. H21 ( .. ~. P(r, t ) = P( n ) exp( i n t ) ( ). n . 2 2 . 3 . ijk ( 2 ) ( n + m , n , m ) 2 ( ). i,j,k n m . n m .. 2 . Pi ( n + m ) = 0 ( n + m , n , m ) E j ( n ) E k ( m ). ( 2). ijk ( ). jk ( nm ). i j k 2 . (nm) n+ m . ( 3 , 2 , 1 ) . ( 2). 3 3 = 2+ 1 .. ijk ( 2 ) ( 3 ; 2 , 1 ) ijk ( 2 ) ( 3 = 2 , 1 ) ( ) ( ).. (1) SFG. 1 2 (SFG) . 3 = 1+ 2 ( ) SFG . 18. H21 ( . Pi ( 3 ) = 0 [ ijk ( 3 , 1 , 2 ) E j ( 1 ) E k ( 2 ) + ijk ( 3 , 2 , 1 ) E j ( 2 ) E k ( 1 )]. ( 2) ( 2). jk ( ).. ijk ( 2 ) ( m + n , m , n ) = ikj ( 2) ( m + n , n , m ) ( ). ( ) . Pi ( 3 ) = 2 0 ijk ( 3 , 1 , 2 ) E j ( 1 ) E k ( 2 ). ( 2). ( ). jk x . Pi ( 3 ) = 2 0 ixx ( 3 , 1 , 2 ) E x ( 1 ) E x ( 2 ). ( 2). ( ).. (2) 2 SHG. 1 3 =2 1 2 . (SHG) ( ) . Pi ( 3 = 2 1 ) = 0 ijk ( 3 , 1 , 1 ) E j ( 1 ) E k ( 1 ).))