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Chapter 9. Electrooptic Modulation of Laser Beams 9.0 Introduction Electrooptic effect : # Effect of change in the index of refraction of medium (crystal) by an external (DC) electric field # n E linear EO effect (Pockels effect) # Nonlinear polarization : ( 2) Pi ( ) 2 ijk ( 0) E j ( ) Ek (0) jk Nonlinear Optics Lab. Hanyang Univ. 9.1 Electrooptic Effect Index Ellipsoid Displacement current : Di ij E j j Energy density : U 1 1 D E ij Ei E j 2 2 ij 2 1 Dx2 D y Dz2 U : for the principal axes 2 xx yy zz 1/ 2 1/ 2 1/ 2 1 1 1 Put, x Dx , y Dy , z Dz 2U 2U 2U x2 y2 z 2 2 2 1 2 nx n y nz : Index ellipsoid Nonlinear Optics Lab. Hanyang Univ. General expression of index ellipsoid Ei ij D j , where, ij ( 1 )ij : impermeability tensor element j Energy density : U 1 1 D E ij Di D j 2 2 ij 1/ 2 Put, 1/ 2 1/ 2 1 1 1 x Dx , y Dy , z Dz 2U 2U 2U 1/ 2 1/ 2 1/ 2 1 1 1 1/ 2 1/ 2 1/ 2 xy ( Dx Dy ) , yz ( Dy Dz ) , zx ( Dz Dx ) 2U 2U 2U 11x 2 22 y 2 33 z 2 212 xy223 yz 213 xz1 Put, 11 1 1 1 , , 2 22 33 2 2 n 1 n 2 n 3 1 1 1 2 23 32 , 2 13 31 , 2 12 21 n 4 n 5 n 6 1 1 1 1 1 1 2 x 2 2 y 2 2 z 2 2 2 yz 2 2 xz2 2 xy1 n 1 n 2 n 3 n 4 n 5 n 6 Nonlinear Optics Lab. Hanyang Univ. Impermeability : ij ij(0) ijk E k S ijkl Ek El k kl Linear EO Quadratic EO Kleinman symmetric medium : 1 3 2 ij E j n i j where, ijk hk h= 1 2 3 4 5 6 ij = 11 22 33 23,32 13,31 12,21 (1) ( 2 ) ( 3) (i, j x , y , z ) (9.1-3) ij : Electrooptic tensor (element) Nonlinear Optics Lab. Hanyang Univ. (9.1-3) 11 1 2 n 1 1 21 2 n 2 1 31 2 n 3 1 2 41 n 4 1 2 51 n 5 1 2 61 n 6 1 0 for all i 2 n i ex) E=0, 12 13 22 23 1 1 , 2 2 n n 1 E0 x 32 33 1 2 n 4 42 43 E1 E2 E3 1 2 n 2 E 0 1 , n y2 1 1 2 2 n 5 E0 n 6 E 0 1 1 2 2 n 3 E0 nz 0 E 0 52 53 62 63 Nonlinear Optics Lab. Hanyang Univ. Nonlinear Optics Lab. Hanyang Univ. Example) EO effect in KH2PO4 (KDP ; negative uniaxial crystal, 42m symmetry group) When the electric filed is applied along the z-axis, the equation of the index ellipsoid is given by x2 y2 z 2 2 2 2 63 Ez xy 1 2 no no ne 1 2 no S ij 63 E z 0 The Sij matrix is For the principal axis, i 63 E z 1 no2 0 0 0 1 ne2 Report : Summary (pp. 333-339) Sij j S i Condition for nontrivial solution (eigenvalue equation) : 1 2 S no E 63 z 0 Nonlinear Optics Lab. 63 E z 1 S no2 0 0 0 1 S 2 ne 0 Hanyang Univ. 2 1 1 2 2 S 2 S ( 63 Ez ) 0 ne no S 1 1 1 , S E , S 63 E z 63 z ne2 no2 no2 1) For S’ S111 S12 2 S13 3 S 1 S 211 S 22 2 S 23 3 S 2 S311 S32 2 S33 3 S 3 1 1 2 2 1 63 E z 2 0 no ne 1 1 63 E z 1 2 2 2 0 no ne 1 1 2 2 3 0 no ne 1 2 0, 3 arbitrary χ (0,0,1) Nonlinear Optics Lab. Hanyang Univ. 2) For S’’ 63 E z 1 63 E z 2 0 63 E z 1 63 E z 2 0 1 1 E 3 63 z 3 n2 ne2 o 1 2 0 3 0 χ(1,1,0) 1 1 ( , ,0) 2 2 Similarly, 3) For S’’’ 1 2 0 3 0 New principal axes : X χ (1,-1,0) 1 1 ( , ,0) 2 2 1 1 (x y ) , Y (x y ) , Z z 2 2 The equation of the index ellipsoid in the new principal coordinate system : 1 2 1 2 z2 2 63 Ez x 2 63 Ez y 2 1 ne no no Nonlinear Optics Lab. Hanyang Univ. 9.2 Electrooptic Retardation For a wave propagating along the z-direction, the equation of the index ellipsoid is 1 1 2 63 Ez x2 2 63 Ez y2 1 no no 2 Assuming, 63 E z no Field components polarized along x’ and y’ propagate as ex Ae i[t ( / c ) n x z ] e y Ae Ae no3 nx no 63 E z 2 no3 ny no 63 E z 2 i t ( / c ) no ( no3 / 2 ) 63E z z i t ( / c ) no ( no3 / 2 ) 63E z z Nonlinear Optics Lab. Hanyang Univ. Phase difference at the output plane z=l between the two components (Retardation) : x y where, no3 63V c , V E z l , x nx c l The retardation can also be written as Ezl V V V where, V 2no3 63 (Halfwave retardation voltage) Nonlinear Optics Lab. Hanyang Univ. 9.3 Electrooptic Amplitude Modulation ex A cos t e y A cos t or, using the complex amplitude notation E x (0) A E y (0) A A -iΓ (e 1) 2 Ex (l ) Ae-iΓ ( E y )o E y (l ) A A 2 -iΓ I o ( E y ) o ( E ) [(e 1)(eiΓ 1)] 2 Γ 2A 2 sin 2 2 * y o 2 I i E E* Ex (0) E y (0) 2A 2 2 Nonlinear Optics Lab. Hanyang Univ. The ratio of the output intensity to the input : V Io 2 Γ sin sin Ii 2 2 V : amplitude modulation Nonlinear Optics Lab. Hanyang Univ. Sinusoidal modulation) m sin mt 2 I0 sin 2 m sin mt Ii 4 2 1 1 sin m sin mt 2 1 1m sin m t 2 m 1 Nonlinear Optics Lab. Hanyang Univ. 9.4 Phase Modulation of Light Electric field does not change the state of polarization, but merely changes the output phase by x ' l nx c n02 r63 2c Ez l If the bias field is sinusoidal ; E z Em sin m t eout A cos[t sin mt ] n03r63 Eml n03r63 Eml where, 2c exp( i sin mt ) : Phase modulation index J n n ( ) exp( inmt ) eout A J n ( )e i ( n m )t : side band (harmonics) n Nonlinear Optics Lab. Hanyang Univ. 9.5 Transverse Electrooptic Modulators Longitudinal mode of modulation : E field is applied along the direction of light propagation Transverse mode of modulation : E field is applied normal to the direction of light propagation l n03 V x ' z n0 ne r63 c 2 d # Transverse mode is more desirable : 1) Electrodes do not interfere with the optical beam 2) Retardation (being proportional to the crystal length) can be increased by use of longer crystal 3) Can make the crystal have the function of /4 plate Nonlinear Optics Lab. Hanyang Univ. 9.6 High-Frequency Modulation Considerations If Rs > (oC)-1, most of the modulation voltage drop is across Rs wasted ! Solution : LC resonance circuit + Shunting resistance, RL >>Rs L LC parallel circuit, Z R 2 1 LC 2 2 1 so, Z1 1 1 i c RL L RL 2CL1 RL i L Z1 2 2 R 1 L 2CL1 L 2 Total impedance : 2 2 1/ 2 2 R 2 L CL 1 R L L Z Rs 2 2 2 2 R R 1 L 2CL 1 1 L 2CL 1 L L At the resonance [=0=1/(LC)1/2], Z RL ! Nonlinear Optics Lab. Hanyang Univ. 1 Maximum bandwidth : 2 2RLC Required power for the peak retardation m : Vm2 P 2 RL Vm ( E z ) m l m , 3 2 n0 r63 1 c 2 RL m2 2 A P 4 ln06 r632 where, A : cross-sectional area of the crystal normal to l Nonlinear Optics Lab. Hanyang Univ. Transit-Time Limitation to High-Frequency Electrooptic Modulation (9.2-4) aEl (an03 r63 /c) But, if the field E changes appreciably during the transit time through the crystal, l t c (t ) a e( z )dz a e(t ' )dt ' n t d 0 Taking e(t) as a sinusoid ; e(t ') Em ei mt ' Phase change during the transit-time, d=nl/c t c (t ) a Em eimt ' dt ' n t d 1 e im d 0 im d i m t e Practically, in order to obtain |r|=0.9, Reduction Factor, r (Fig. 11-17) m d /2, and d nl /c ( m ) max c/4nl Ex) KDP, n=1.5, l=1cm, where, 0 a(c/n) d Em alEm : Peak retardation Nonlinear Optics Lab. ( m ) max 5 GHz Hanyang Univ. Traveling wave Modulators : matching the phase velocities of the optical and modulation fields by applying the modulation signal in the form of a traveling wave Consider an element of the optical wavefront that enters the crystal at z=0 at time t c z (t ' ) (t 't ) n t ac d The retardation exercised by this element is given by (t ) e[t ',z (t ')]dt ' n Nonlinear Optics Lab. t Hanyang Univ. The traveling modulation field : e(t '.z ) Em ei[ mt ' km z ] Em ei[ mt ' km ( c/n )( t 't )] (t ) 0e i mt eim d (1c/ncm ) 1 i m d (1c/ncm ) where, 0 a(c/n) d Em alEm : Peak retardation Reduction factor : r eim d (1c / ncm ) 1 im d (1c / ncm ) # c/n=cm r = 1 # Maximum modualtion frequency (|r|=0.9) : c ( m ) max 4nl (1c/ncm ) ( m ) max c/4nl (static field case) Nonlinear Optics Lab. Hanyang Univ. 9.7 Electrooptic Beam Deflection Deflection angle inside the crystal ( n0 ) ' y ln l dn D Dn n dx External deflection angle (By Snell’s law) ' n l n dn l D dx Nonlinear Optics Lab. Hanyang Univ. Double-prism KDP beam deflector n03 n A n0 r63 E z 2 n03 nB n0 r63 E z 2 l n03 r63 E z D Number of resolvable spots, N (for a Gaussian beam) : n03l N r63 E z beam 2 # (9.2-7) E z ( ) 2n03lr63 n03lr63 N |V V 1 3 2 2n0 lr63 4 Nonlinear Optics Lab. Hanyang Univ.