Chapter 9 Electro-Optics

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1 Chapter 9 Electro-Optics Gabriel Popescu University of Illinois at Urbana Champaign Beckman Institute Quantitative Light Imaging Laboratory Principles of Optical Imaging Electrical and Computer Engineering, UIUC

2 ECE 46 Optical Imaging Electro Optics 1 st order effect: w 1 Pi ii jjrijkej ( ) Ek () i i i i j ijk j k ( ) 4 Dx n n r13ey EDc n ner133ez EDC Dy n n r13e xedc 4 r kdp 53 Dz ne Ez D n E n n r E E DC DC 11O z Chapter 9: Electro Optics

3 6 ELECTRO-OPTICS By using the contracted indices (7.1-11), the equation of the index ellipsoid in the presence of an electric field can be written (:~ + rlkek ) x + (:; + rkek) y + (:; + r3kek) Z (7.-3) +yzr4k Ek + zxr5kek + xyr6kek = where Ek (k = 1,,3) is a component of the applied electric field and summation over repeated indices k is assumed. Here 1,,3 correspond to the principal dielectric axes x, y, z, and n x ' ny, n z are the principal refractive indices. This new ellipsoid (7.-3) reduces to the unperturbed ellipsoid (7.1-1) when Ek = O. In general, the principal axes of the ellipsoid (7.-3) do not coincide with the unperturbed axes (x, y, z). A new set of principal axes can always be found by a coordinate rotation, which is known as the principal-axis transformation of a quadratic form. The dimensions and orientation of the ellipsoid (7.-3) are, of course, dependent on the direction of the applied field as well as the 18 matrix elements rlk. We have argued above that in crystals possessing an inversion symmetry (centrosymmetric), rlk = O. The form, but not the magnitude, of the tensor r1k Can be derived from symmetry considerations, which dictate which of the 18 coefficients r 1k are zero, as well as the relationships that exist between the remaining coefficients. In Table 7. we give the form of the electro-optic tensor for all the noncentrosymmetric crystal classes. The electro-optic coefficients of some crystals are listed in Table Example: 1be Electro-optic Effect in KH 1 P 4 Consider the specific example of a crystal of potassium dihydrogen phosphate (KH P 4 ), also known as KDP. The crystal has a fourfold axis of symmetry, which by strict convention is taken as the z (optic) axis, as well as two mutually orthogonal twofold axes of symmetry that lie in the plane normal to z. These are designated as the x and y axes. The symmetry group of this crystal is 4m. Using Table 7., we write the electro-optic tensor in the form r ij = r41 (7.-4) r41 r63 '1 I I I I Table 7.. Electro-optic: Coefficients in Contracted Notation for All Crystal Symmetry Oasses Q Centrosymmetric (I, /m, mmm, 4/m, 4/mmm, 3, 3m 6/m,6/mmm, m3, m3m): Triclinic: Monoclinic: Orthorhombic: 'Il '1 '13 '1 ' '3 '31 '3 '33 '41 '4 '43 '51 '5 '53 '61 '6 '63 ( II X) (" X3) '1 '13 ' '3 '3- '33 '41 '43 '41 '4 '5 '51 '5 '61 '63 '63 m (m J. X) m (m J. X3) 'Il '13 'II '1 '1 '3 '1 ' '31 '33 '31 '3 '4 '43 '51 '53 '53 '6 '61 '6 mm '\3 '3 '33 '41 '4 '5 '51 '63 7

4 "r Table 7.1- ( Continued). Tetragonal: 4 4 '13 '13 '13 -'13 '33 '41 -'51 '41 '41 '51 '51 '41 '51 -'41 '63 4mm 4m (11 XI) '\3 '\3 '33 '51 '41 '41 '51 '63 4 -'41 THE LINEAR ELECTRO-OPTIC EFFECT 9 Table 7.. Hexagonal: ( Continued). 6 6mm 6 '\3 '13 '13 '\3 '33 '33 '41 '51 '51 '41 '51 -'41 '51 -'41 6 6m (m.l XI) 6m (m.l X) '11 -' -' 'II -'11 ' ' -'11 -' -'II -' -'11 Trigonal: 3 3 'II -' '13 '11 -'II ' '\3 -'11 '33 '41 '51 '41 '51 -'41 -'41 -' -'II -'11 3m (m.1 XI) 3m (m.1 X) -' '13 '11 '13 ' '13 -'II '13 '33 '33 '51 '51 '51 '51 -' -'11 Cubic: 43m,3 43 '41 '41 '41 The symbol over each matrix is the conventional symmetry-group designation. so that the only nonvanishing elements are '41 = '5 and '63' Using Eqs. (7.-3) and (7.-4), we obtain the equation of the index ellipsoid in the presence of a field E(Ex' E y ' Ez) as 8 where the constants involved in the first three terms do not depend on the field and, since the crystal is uniaxial, are taken as nx = ny = no' n z = ne' We thus find that the application of an electric field causes the appearance of "mixed" terms in the equation of the index ellipsoid. These are the terms with xv. xz. vz. This means that the major axes of the ellipsoid, with a field

5 ECE 46 Optical Imaging Electro Optics 4 n n r63e Dc 4 ij n r63edc n ne ' ij W( ) W( ) cos sin n cos sin sin cos n sin cos n ij n ; 4 n e b i 4 n r63 Ez ( DC) biaxial crystal Chapter 9: Electro Optics 3

6 ECE 46 Optical Imaging Modulators Eg KDP(tetra 4m ) r 41, r 5, r 63 only three nonzero elements n KDP x n ; 1 1 n n n n n n n n x y n 1 n n 1 r n n r63ez ( DC) Chapter 9: Electro Optics 4

7 ECE 46 Optical Imaging Modulators ( n n ') d n r E ( DC ) d 3 x y 63 z n V V n r 3 63 linearize T V T sin 4 sin s Add QW V Chapter 9: Electro Optics 5

8 ECE 46 Optical Imaging Modulators Let V V sin t m m T sin sin m m t cos sin 1 sin sin m m t m m t 1 1T 1 sin t m m m m linear Chapter 9: Electro Optics 6

9 ECE 46 Optical Imaging Quadratic (Kerr) ( ) 1 i ii jj ijk j k e P s E ( ) E ( DC ) E ( DC ) Chapter 9: Electro Optics 7

10 ECE 46 Optical Imaging Applications of EO longitudinal modulators transverse For LiNiO 3 : 1 3 nx n n r 13 E 1 3 ny n n r13e 1 3 nz ne ne r33e 3 v nl n rv 13 V 3 v n r Ed 13 Phase mod(indep. of polariz.) Chapter 9: Electro Optics 8

11 Chapter 9 Acousto-optics Gabriel Popescu University of Illinois at Urbana Champaign Beckman Institute Quantitative Light Imaging Laboratory Principles of Optical Imaging Electrical and Computer Engineering, UIUC

12 ECE 46 Optical Imaging Acousto optics optics i j i ijklsklej jkl x 1 6 ac wave: z S13 S P n n p S E U ( z, t) xa cos( t kz) Chapter 9: Acousto optics 1

13 Chapter 1: Introduction 11

14 Acousto optics optics K k K k 1 k k1 nk sin k / K / K sin B ; Bragg angle

16 ECE 46 Optical Imaging Acousto optics optics small ( k k ) Doppler Shift Δk Δv Kvs Quantum mechanics k' k ' conservation of momentum conservation of energy Chapter 9: Acousto optics 13

17 ECE 46 Optical Imaging Anisotropic media k' k k ' n-different ' - negative k'sin ' ksin ; ' n' n sin ' sin ; n sin ' sin n' n' k wavelength of sound Chapter 9: Acousto optics 14

18 ECE 46 Optical Imaging Anisotropic Ex: C k ' k -(e) k ' - scattered in prop. Plane (o) k ' (o) n sin ' e sin n' n ' n n e ', n n e e n n n n Chapter 9: Acousto optics 15 e

19 ECE 46 Optical Imaging Small angle Scattering I scatt I inc sin ( L L ) 3/ k ( nn 1 ) ep ijke S e 4 cos cos 1 i ke j Kin. Energy/ V = ½ W total I ac v s 1 v U U s vs u vs [ U ] I ac t 1 3 vs S U S z U Chapter 9: Acousto optics 16

20 ECE 46 Optical Imaging Small angle Scattering S 3/ ac 3 Small cos 1 1 s I I ac k ( nn 1 ) I P 3 v 4 cos cos v s M 6 n p v p 3 s table 3/ 3 k ( nn 1 ) v s M I ac n vs MI ac Chapter 9: Acousto optics 17

21 ECE 46 Optical Imaging Small angle Scattering Detuning: sin B k sin sin 1 ; B k k sin k sin 1 1 ( Bragg) k sin ( ) k 1 cos 1 1 k cos 1 ( ) k(cos 1cos ) k sin B I scat sin L 1 ; I inc 1 s 1 Chapter 9: Acousto optics 18

22 ECE 46 Optical Imaging Finite Beams A B nw ; L ; f f size of acoustic beam 1 ; L nv cos 4v cos s s s -Full v nw w s 1 ; W or nvs cos ( ) f L! Not overlap with undiffracted order 1 Chapter 9: Acousto optics 19

23 ECE 46 Optical Imaging N N spots f nw W f N nv s cos v s B f ; cond B nv s f f n L nv s f L Chapter 9: Acousto optics

Chapter 9 Electro-Optics

Chapter 9 Electro-Optics Gabriel Popescu University of Illinois at Urbana Champaign Beckman Institute Quantitative Light Imaging Laboratory http://light.ece.uiuc.edu Principles of Optical Imaging Electrical