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DIGITAL COMPUTER LABORATORY 
 
 10.170 
 
 :op.3 
 
 UNIVERSITY OF ILLINOIS 
 URBANA, ILLINOIS 
 
 
 REPORT NO. 170 
 
 NOVEL ELECTRO- OPTICAL LIGHT MODULATORS 
 
 by 
 
 Gab or Kalman Ujhelyi 
 
 October 26, 196k 
 
 
 (This work was submitted in partial fulfillment of the requirements for the 
 Degree of Doctor of Philosophy in Electrical Engineering, September, 1964, and 
 was supported in part by the Office of Naval Research under contract 
 Nonr-l834(l5).) 
 
DIGITAL COMPUTER LABORATORY 
 UNIVERSITY OF ILLINOIS 
 URBANA, ILLINOIS 
 
 REPORT NO. 170 
 
 NOVEL ELECTRO-OPTICAL LIGHT MODULATORS 
 
 by 
 Gabor Kalman Ujhelyi 
 
 October 26, I96U 
 
 (This work was submitted in partial fulfillment of the requirements for the 
 Degree of Doctor of Philosophy in Electrical Engineering, September, 196^, and 
 was supported in part by the Office of Naval Research under contract 
 Nonr-l83Ml5)-) 
 

 ACKNOWLEDGEMENT 
 
 The author wishes to express his appreciation to Professor W, J. Poppelbaum 
 for his counsel and guidance in the preparation of this dissertation. 
 
 The author is also deeply indebted to his colleague Dr. Sergio T. Ribeiro 
 for his friendship, encouragement, many helpful ideas and for his collaboration 
 in the experimental part of this work. 
 
 Sincere thanks are also due to Professors D. Holshouser and 0. Gaddy 
 who provided valuable advice, space and equipment for the experiments. 
 
 111 
 
TABLE OF CONTENTS 
 
 CHAPTER 1 
 CHAPTER 2 
 
 INTRODUCTION 
 
 THE THEORY OF ELECTRICALLY INDUCED BIREFRINGENCE 
 
 CHAPTER 3 
 
 CHAPTER 4 
 
 CHAPTER 5, 
 
 CHAPTER 6. 
 BIBLIOGRAPHY 
 
 2.1 
 2.2 
 2.3 
 
 2.4 
 
 Birefringence in anisotropic materials- 
 The derivation of the electro-optic effects. 
 The index ellipsoid of some important electro- 
 optic materials. 
 
 2.3.1. Linear electro-optic effect in uniaxial 
 materials. 
 
 2.3.2. Linear electro-optic effect in cubic 
 materials. 
 
 2.3.3. The quadratic electro-optic effect. 
 Material properties of some electro-optic materials 
 
 INTENSITY MODULATION BY CONTROLLED PARTIAL REFLECTIONS 
 
 3.1. The Fresnel relations for the reflected amplitudes. 
 
 3.2. The dependence of the reflected intensity on the 
 
 index of refraction n^ . 
 
 2 
 
 3.3. The modulation M as a function of the modulating 
 
 electric field E . 
 
 3.4. The physical realization of the intensity modulator. 
 
 3.5. Beam intensity modulation. Effects of eollimation 
 and diffraction. 
 
 3.6. Velocity matching condition for high speed operation , 
 
 3.7. Dimensional tolerance requirements. 
 
 3.8. Optimization of modulator dimensions. 
 
 3.9. The choice of the operating temperature. 
 
 3.10. Modulating bandwidth limitations. 
 
 BEAM DIRECTION MODULATION BY CONTROLLED INH0M0GEKEIT7 
 
 4.1. The theory of wave propagation in an inhomogeneous 
 medium . 
 
 4.2. Beam deflection by means of an electro-optical 
 device. 
 
 4.2.1. Beam deflection by the linear electro-optic 
 effect . 
 
 4.2.2. Beam deflection using the quadratic electro- 
 optic effect. 
 
 4.3. Beam distortion effects. 
 
 EXPERIMENTAL RESULTS 
 
 5.1. Experiments with the light intensity modulators. 
 
 5.2. Experiments with the beam bending device. 
 
 CONCLUSIONS 
 
 4 
 
 4 
 17 
 
 21 
 
 21 
 
 22 
 22 
 
 24 
 
 25 
 25 
 
 29 
 
 34 
 37 
 
 40 
 44 
 47 
 50 
 52 
 54 
 
 57 
 
 57 
 
 62 
 
 63 
 
 69 
 70 
 
 77 
 78 
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 85 
 
 iv 
 
NOVEL ELECTRO-OPTICAL LIGHT MODULATORS 
 
 Gabor Kalman Ujhelyi, Ph.D. 
 Department of Electrical Engineering 
 University of Illinois, 1964 
 
 Since the availability of a coherent, well collimated and powerful light 
 beam, there is a great demand for efficient, broadband light modulators. To 
 the already well developed family of Kerr-cell type devices we propose to add 
 two novel electro-optical devices: a direct intensity modulator and a light 
 beam deflecting device. 
 
 To provide a firm basis for the analysis of these devices, the theoretical 
 background of the electro-optically induced birefringence is examined in detail . 
 Starting from the anisotropic material equations and Maxwell's differential 
 equations, an analytical derivation of this birefringence is given. Further- 
 more it is shown how the electro-optical effect influences the material equations 
 hence its effect on the birefringence is derived. The experimentally determined 
 electro-optic effect is tabulated for the most important materials . 
 
 The new intensity modulator operates on the principle of electro-optically 
 controlled partial reflections. The light beam to be modulated traverses a 
 transparent isotropic material of index of refraction n = n and it is incident 
 on the electro-optic material with n = n (n > n ) . The reflected amplitude 
 (hence the intensity'.) depends on the index of refraction n , which in turn is 
 controlled by the modulating electric field through the electro-optic effect . 
 A detailed study of the theoretical and practical aspects of this intensity 
 modulator reveals that its performance is in many ways comparable to the most 
 sophisticated of the Kerr-cell type devices. A systematic and quantitative 
 procedure is presented for the optimized design of a broadband intensity 
 
vi 
 modulator. The modulating frequency of the device is fundamentally limited 
 only by the dispersion of material parameters. 
 
 The beam deflecting device uses the fact that a variation of the index 
 of refraction n causes a light beam to change direction of propagation. The 
 desired variation of n (hence the deflection of the beam) can be achieved by 
 placing an electro-optical material in a suitable electric field distribution. 
 This field distribution is determined for the case of minimal beam distortion 
 for linear and quadratic electro-optic materials. 
 
 The experimental results obtained with three devices support the 
 theoretical predictions. Intensity modulation up to 50% and a deflection of 
 ,003 radian were observed in the laboratory using nitrobenzene as the electro- 
 optic material . 
 
CHAPTER 1 
 INTRODUCTION 
 
 The modulation of a light beam is by no means a new idea: navies used 
 light signals as means of communications ever since ancient times. The 
 development of the intensity modulator can be traced through the progress in 
 the measuring techniques used for the experimental determination of the 
 velocity of light. The rotating wheel of Fizeau (1849), the rotating 
 mirror of Foucault (1860) and the electro-optical shutter of Kerr (1874), 
 Karolus and Mittelstaedt (1928) represent the major advances in this develop- 
 ment . The use of these devices, however, was restricted mainly to the field 
 
 of experimental physics and to high speed photography in particular. 
 
 (2) 
 
 The invention of the new light source, the laser , increased the 
 
 importance of the light modulators manifold. This new device provides a 
 
 coherent, highly stable, powerful and well collimated beam of electro- 
 
 14 
 magnetic radiation with a frequency of the order of 10 cps . Much work has 
 
 been done in the recent years trying to utilize the information carrying 
 
 capabilities of this beam. A number of modulating systems were designed 
 
 with a modulating bandwidth of several kilomegacycles . However, 
 
 all of these electro-optical modulators are basically phase modulators in 
 
 that the modulating fields cause a difference between the phase velocities 
 
 of the two component waves in the medium and hence influence the polarization 
 
 of the output beam. The phase modulation may be converted to intensity 
 
 modulation by passing the output beam through a polarization analyzer. 
 
 (These are the Kerr-cell type modulators.) 
 
The purpose of this dissertation is to analyze the operating character- 
 istics of two novel electro-optical light modulators: 
 
 1.) An amplitude or intensity modulator which does not operate 
 on the Kerr principle described above, but obtains intensity 
 modulation directly from controlled partial reflections on the 
 interface between an isotropic and an electro-optic material. 
 2.) A device which deflects a light beam by the inhomogeneity 
 induced by a modulating electric field through the electro-optic 
 effect . 
 
 Our intensity modulator compares favorably with the Kerr-cell type devices 
 on several features, due mainly to the fact that the modulation is accomplished 
 at the boundary surface rather than in the volume of the electro-optic 
 material. Some of these points are: 
 
 1.) The modulated light beam does not travel in the usually 
 lossy electro-optic material, but stays within the nearly loss- 
 leys isotropic medium. 
 
 2.) A very thin layer (several wavelengths) of the electro- 
 optic material is sufficient for the modulation process and 
 consequently the modulating power i b Ln this material is 
 \ tly reduced. 
 
 3.) The device does not have a limiting modulating frequency 
 correj'r ling to the cutoff frequency of ;1 i Kerr-cell type 
 devices, (occurring when the microwav n dulating field varies 
 appreciably across the cross section of the optical beam.) 
 These and other points will be ei d the text. 
 
3 
 
 The organization of the remainder of the dissertation is as follows: 
 in Chapter 2 the theory of the electrically induced birefringence is derived 
 from basic principles and the experimentally measured electro-optical 
 effects are given for the most important materials. Chapter 3 presents a 
 detailed analysis of the intensity modulator and provides a systematic and 
 quantitative procedure for the design of an optimal modulator. Chapter 4 
 analyses the operational characteristics of the beam deflector device. 
 Chapter 5 contains the results of experiments designed to test the performance 
 of the two devices. In Chapter 6 we conclude with a tabulation of the 
 performance expected from modulators made of presently available materials. 
 
CHAPTER 2 
 THE THEORY OF ELECTRICALLY INDUCED BIREFRINGENCE 
 2.1, Birefringence in Anisotropic Materials . 
 
 The theory of electromagnetic waves is based on two quite separate founda- 
 tions: Maxwell's differential equations provide the first, and the material 
 equations give the second. Whereas in isotropic media the vector quantities 
 in the material equations are related to each other by scalars, for aniso- 
 tropic materials these scalars become tensors, and the velocity of propagation 
 becomes a function of the direction of polarization and the direction of 
 propagation. In this section we wish to study the propagation of an electro- 
 magnetic wave in an electrically anisotropic, non-conducting, (transparent), 
 non-magnetic (|i = \i ) , source-free and homogeneous material . Let us assume 
 the simplest relation which can account for the anisotropic behavior, namely 
 that each component of the dielectric displacement vector D is linearly 
 related to the components of the electric field intensity vector E: 
 
 D. = 6. . E. (2.1) 
 
 where € is the dielectric constant tensor. (Note that we define D. as the 
 
 component of D in the x. coordinate axis direction.) We can show that e. . 
 
 is a symmetric tensor. This is a consequence of the definitions of the 
 
 electric energy density w , the magnetic energy density w , the Poynting 
 
 e m 
 
 (11) 
 vector P, and the continuity equation for the electromagnetic energy. 
 
 These definitions are 
 
 w = - D • E = - e,.E. E, 
 
 e 2- - 2 ijij 
 
 w = : i B • H (2.2) 
 
 m 2 — - 
 
 P = E x H 
 and the continuity equation is 
 
5 
 dw dw 
 
 - div P = — (w + w ) = — - + — - (2.3) 
 
 — dt e m dt dt 
 
 The left hand side of (2.3) can be computed from (2.2) using a well 
 known vector identity, yielding 
 
 - div P = E curl H - H curl E (2.4) 
 Substituting for curl E and curl H from Maxwell's equation we have 
 
 dD dB 
 
 - div P = E — + H — (2.5) 
 
 — — dt — dt 
 
 Comparing (2.3) and (2.5) we find that 
 
 dw dD 
 dt — dt 
 
 On the other hand from (2.2) we obtain 
 
 dw dE dD 
 
 dt 2 dt — — 2 — dt — dt 
 
 Equations (2.6) and (2.7) must be equal, hence 
 
 dD dE 
 
 E — = D — (2 .8) 
 
 - dt - dt 
 
 which in tensor form reads 
 
 dE . dE . 
 
 £, . E. —J- = €.. -r- 1 E. (2.9) 
 
 lj l dt ji dt l 
 
 dE. 
 
 and since E. -~— = 0, we must have 
 l dt 
 
 €. . = G.. (2.10) 
 
 which result shows that €. . is a symmetric tensor indeed. 
 
This fact allows us to eliminate the off-diagonal elements of e , 
 
 ij 
 
 or equivalently, choose the coordinate system x. in such a manner that e 
 
 1 ij 
 
 becomes a diagonal tensor. Let us call the diagonal elements e the 
 
 ii 
 
 principal dielectric constants, and let e. . = €. . Then we can write 
 
 11 1 
 
 D. = e.E. (2.11) 
 
 Equation (2.11) calls our attention to the fact that if the E of the 
 wave coincides with one of the coordinate axes x. , the material behaves formally 
 as an isotropic material with dielectric constant e. . For these particular 
 waves we can use the definition of the propagation (phase) velocity and the 
 index of refraction as given for isotropic media, and write 
 
 1 
 v. = (2.12) 
 
 1 F& 
 
 and 
 
 <L Alio __ i^ ^ 
 
 n. = — = B-^-^ = U^ = pi (2.13) 
 
 iv. \ £ u. \ e 
 
 1 > O O 1 o 
 
 where c is the velocity of light in vacuum, e*.T is the relative principal 
 
 *i 
 
 dielectric constant and n. is the principal refractive index. 
 
 The remainder of this section will be devoted to the study of a general 
 electromagnetic (TEM) wave traversing the anisotropic medium. It will be 
 shown that for any given direction of propagation there are at least two 
 directions of allowed polarization, and that these directions, and the corres- 
 ponding propagation velocities can be obtained from the knowledge of the 
 
 principal refractive indices n. . 
 
 l 
 
 Let us find which plane waves can propagate in the anisotropic medium 
 in the direction of N unit vector. In tensor notation the possible solutions 
 can be described by 
 
7 
 x.N. 
 
 E. = E . exp [ ju(t ) ] (2.14) 
 
 J oj ° v 
 
 where u> is the angular velocity in radians per second, v is the phase velocity 
 (propagation velocity) in the direction of N. 
 
 For a rectangular coordinate system Maxwell's equations in tensor form 
 read 
 
 8 H. 5 D . 
 
 e 
 
 S 
 
 X. 
 
 i 
 
 5 t 
 
 
 e kl J 
 
 5e 
 
 J 
 5 
 
 X. 
 
 l 
 
 So B k 
 8t 
 
 8H * 
 
 5 D. 
 
 l 
 
 = 
 
 
 
 8 X . 
 
 
 
 
 (2.15) 
 
 (2.16) 
 
 and 
 
 (2.17) 
 
 OB. 
 i = (2.18) 
 
 8 X 
 
 i ik 
 All quantities are expressed in the MKS system of units. The tensor e 
 
 *• A U ( 13 > 
 
 is defined by ijk . 
 
 e =0 if i = j , orj=k, or l = k 
 
 =+1 if ijk are in even permutation (2.19) 
 
 (1, 2, 3; 2, 3, 1; 3, 1, 2) 
 =-1 if ijk are in odd permutation 
 
 (1, 3, 2; 3, 2, 1; 2, 1, 3) 
 
 Substituting (2.14) into (2.16) we find 
 
 S h x.N. 
 
 -\x — -^ = - j - [ e klJ E , N. ].exp [ JO) (t - - J - i ) ] (2-20) 
 
 integrating this equation with respect to time (from - oo to t and assuming 
 that H k = for t = - <*> ) 
 
H k = ^ [ e klJ E^N. ] exp [ jifct ~ ^ ) ] (2.21) 
 
 from which we can conclude that H • E = and also H • N = 0, implying that 
 both E and N are normal to H. 
 
 Substituting (2.21) into (2.15) gives 
 
 6T = - j ,"7 l eJlk (eklJ E oj N i> N i 1 ex » t ^ (t - ^ < 2 - 22 > 
 
 Integrating again with respect to time gives 
 
 x N 
 Dj = [e Jik (e kij E QJ N.) N. ] -^ exp [ ju>(t - -^)] (2.23) 
 
 which can also be written in the form 
 
 D. = - ^-2 [e jik (e kiJ E.N.) N. ] (2.24) 
 
 J H Q V 2 J i i v 
 
 Expanding this tensor equation and noticing that (since N is a unit vector) 
 
 3 
 
 Z N. 2 = 1 (2.25) 
 
 i=l X 
 
 equation (2.24) can be simplified to the form 
 
 D. = -^— §- [E. - (E.N.) N. ] (2.26) 
 
 J U. v 2 j li j 
 
 From this equation we see that since D is a linear combination of vectors 
 
 E and N, D must be in the plane of E and N, and since E and N are normal to 
 
 H, (2.21), D must be normal to H. It is also seen from (2.26) that D • N = 0, 
 
 implying that D is normal to N also. Figure 2.1 illustrates the relative 
 
 positions of these vectors. D, E, N and P vectors are in a plane normal to 
 
 B and H vectors. The vector u is defined as a unit vector in the x. 
 — — —x. 1 
 
 l 
 
 coordinate axis direction. 
 
Figure 2.1. Relative orientation of vectors E, D, H, B, N, 
 and P. 
 
 It will be more convenient to use the inverse of the 6. tensor. Since £. 
 
 1 l 
 
 is a diagonal tensor, its inverse is simply the reciprocal of the elements 
 
 £. , i.e., 
 
 hence 
 
 p. -[e.]" 1 -^ 
 
 1 L 1 J f . 
 
 *- 1 
 
 E. = p.D. 
 l 'l l 
 
 (2.27) 
 
 (2.28) 
 
 where (3. is called the impermeability tensor. With this definition (2.26) 
 becomes 
 
 D. = B.D. - (E.N.)N. 
 
 J .. .,2 J J 1 1 J 
 
 We can rearrange this equation to the form 
 
 (2.29) 
 
 D.(p. = Ul v 2 ) - (E.N.)N. 
 j K j To l i j 
 
 (2.30) 
 
 and divide both sides by (p. - Kx. v 2 ) ^ 
 
 D = 
 J 
 
 (E.N.)N. 
 
 <3 -/V*> 
 
 (2.31) 
 
 
10 i 
 
 We have noted before that D • N = 0, hence 
 
 3 3 (E.N.)N. 3 3 N 2 
 
 1=1 J=l K J O 1=1 D =l p g o 
 
 3 
 Since Z (E.N.) = for anisotropic media, we can conclude that 
 
 i=l 
 
 N. 2 
 E ^ V 2 =° (2.33) 
 
 Let us define a. as the principal phase velocity by 
 
 a j = L- (2 - 34) 
 
 Dividing (2.33) by |i and substituting (2.34) yields 
 
 3 N. 2 
 
 £ — - 2 = (2.35) 
 
 i=l i 
 
 We can transform this equation into an equivalent form by defining the 
 index of refraction n as 
 
 n = - . (2.36) 
 
 v 
 
 and using the definition of n. (2.13) we obtain 
 
 i * > 
 
 3 
 £ - t- (2.37) 
 
 i = 1 n 2 - eV n 
 
 i 
 
 Equations (2.35) or (2.37) give two values for v 2 (or n 2 ) for a given 
 N and G f . tensor. Equation (2.35) can be expanded to read 
 
 v 4 - (n x 2 (a 2 2 + a 2 3 ) + N 2 2 (a 2 1 + a 2 3 ) + N^Ca^ + a 2 2 ) ] v 2 + 
 
 + N W 2 3 + N V 2 i a2 3 + N YV 2 3 = ° (2 - 38) 
 
11 
 
 which has the solutions 
 
 V2 = I [ N V a % + a Y + "r.<»\ 
 
 + a*.,) + N^ 3 fc.» + a^) ± 
 
 ± ^ N4 l <a2 2- a V 2+N4 2 (a2 3- a2 1 >2+N 3 (a2 l a2 2 )2 - 2 l N V a Y a2 3 )N V a Y a V + 
 
 ♦ » ! 1 »V*V 1 V ( "V" ! 2 )€ VV aI i-»V (>! i-"V] 
 
 (2.39) 
 
 The two solutions of (2.39), (v 2 ) 1 and (v 2 )" give the two phase velocities 
 and therefore the two indices of refraction. 
 
 Let us consider equations (2.34) and (2.39) in more detail. If we let 
 N = u , then N = 1, N = N = and (2.39) has the two solutions (a 2 ) and 
 
 X.. J. a O 2* 
 
 (a 2 ) , which correspond to p and p respectively, I (2.34) J . Thus we 
 
 find that for this case there are two plane waves which can propagate through 
 
 the medium: one is polarized in the u direction, the other in the u direction, 
 
 2 3 
 
 and they propagate with v' = a and v" = a respectively. Notice that we 
 
 found the directions of polarization by finding the directions for which the 
 
 value of p corresponds to the solutions of (2.39), namely to v' and v". 
 
 This method can be generalized by defining v as 
 
 v 2 = -&- (2.40) 
 
 ^ o 
 
 where v is the phase velocity of the wave in the N direction and p* relates 
 
 the components of D and E in the plane normal to N. We shall show that as 
 
 the plane of polarization is rotated around the N vector, p* describes an 
 
 ellipse, whose major and minor axes correspond to v f and v " and therefore 
 
 the only two permitted planes of polarization will be determined by the major 
 
 and the minor axes of the ellipse together with N. This derivation provides 
 
 
12 
 
 an analytical proof for the well know geometrical method for the construction 
 
 of the two permitted indices of refraction from the index ellipsoid. 
 
 The coordinate system x ± was chosen in such a manner as to coincide 
 with the directions of the principal dielectric constants of the anisotropic 
 material. Since we want to investigate general directions, (N) , we have to 
 set up a rotated coordinate system in which vectors D, H and N are in the 
 direction of the axes y. . 
 
 Fig. 2.2. 
 
 t = angle between 
 
 (x y A plane and y axis 
 
 cp = angle of (x y ) plane 
 and x axis 
 
 6 - angle between x and y 
 
 The rotated coordinate system 
 as related to the original axes 
 
 The vectors D, H and N will have the following form in the new coordinate 
 
 system 
 
 D = D u 
 
 y ~y 
 i ± 
 
 N = N u 
 
 y.3 -tJ 
 
 H - H • u 
 
 y 2 -: 
 
 Since p is a second rank tensor* we can write 
 
 (2.41) 
 
 Since 
 
 V 3 D 2 = 2w > for D. f 0, the left hand side of this equation 
 
 is a positive definite quadratic form, hence p. are the principal semi-axes of 
 
 an ellipse and also 6. is a second rank tensor^ ^ 13 ^ 
 
 J 
 

 13 
 
 8y ± &y i 
 
 0*. . = ' * • ft , (2.42) 
 
 The desired rotation can be done by 
 
 = 1 = cosO cos cp cos ■■\- sincp sin? (2.43) 
 
 = 1 =-cos9 costp sin? - sincp cos? 
 
 8 
 
 x l 
 
 6y 2 
 
 5 
 
 x l 
 
 5 
 
 y 3 
 
 6 
 
 X l 
 
 5 
 
 y l 
 
 8 
 
 X 2 
 
 8 
 
 y 2 
 
 5 
 
 X 2 
 
 6 
 
 y 3 
 
 5 
 
 X 2 
 
 5 y-L 
 
 5 
 
 X 3 
 
 5 
 
 y 2 
 
 5 
 
 X 3 
 
 B 
 
 y 3 
 
 8 
 
 X 3 
 
 = 1 = cos cp sinG 
 
 = 1 = cosG sincp cos? + cos cp sin? 
 
 = l OQ = cos cp ccffe? - sincp sin? cos© 
 
 = 1 = sincp sin0 
 
 = 1 = - sinG cos? 
 
 = 1 = sine sin? 
 
 = 1 33 = cosG 
 
 The impermeability tensor 0*. . relates vectors E and D expressed in the y. 
 
 coordinate system. Specifically 
 
 E = B* D (2.44) 
 
 y I U y l 
 
 since D = D =0. (Refer to (2.41)). 
 
 y 2 y 3 
 
 Let us compute the maxima arid minima of B* (?) as ? varies from to X 
 
 K n 
 
Using (2.42) and (2.43) we can express 3* in terms of cp , 9 and Y • 
 
 P*ll = ^lA + l2 21^2 + l2 3lP 3 (2.45) 
 
 = [ (cos 2 9 cos 2 cp ) p 1 + (cos 2 0sin 2 cp ) p + (sin 2 0)p ] cos 2 ^ + 
 + [cosO • si g 2Cp (P 2 -P 1 ) ] sin2^ + 
 
 + [ (sin 2 cp )p + (cos 2 cp )p ] sin 2 ^ 
 which can be written as 
 
 P *ll = A i cos2 * + A 2 sin2 ^ + A 3 sin 2 ^ (2.46) 
 
 where A. depends only on 3 . , and cp . 
 
 Equation (2.46) is recognized as the equation for a rotated ellipse; 
 the direction of the axes (maxima and minima) can be obtained from 
 
 5 P*11 
 
 — - = = -2A 1 cos^ sin^+ 2A cos2 Y + 2A sin Y cos Y (2.47) 
 
 Ol -L Z 6 
 
 which has the solution 
 
 1 A 2 
 
 ¥ • = -= arctan — — - (2.48) 
 
 * A l A 3 
 
 As we have expected, (2.48) indicates that the extrema are jt/2 apart: 
 y » » = y ' + it/2 . 
 
 Substituting (2.48) into (2.46) and rearranging terms we find that 
 ?*-,-,( ¥*) = - [ p (sin 2 epsin 2 0+eos 2 0) + p (cos 2 cp sin 2 + cos 2 0) + 
 
 XX £ X ^ 
 
 + 3 3 sin 2 + l/(3 1 -3 2 ) 2 (cos 2 0cos 2 cp +sin 2 cp ) 2 + 2(p 1 ~P 2 )(P 3 -p 2 ) . 
 
 __ _ — 1 
 
 • sin 2 0(cos 2 0cos 2 cp -sin 2 cp ) + (p -p ) a sin 4 9 ] (2.49) 
 
 Let us now transform (2.39) into the new coordinate system y. . This 
 can be accomplished by letting 
 
15 
 
 H ± -l la < 2 -50) 
 
 Performing this substitution into (2.39) and rearranging terms we obtain 
 
 v 2 = - [a 2 (sin 2 cp sin 2 + cos 2 9) + a 2 (cos 2 cp sin 2 9 + cos 2 9) + a 2 „sin 2 9 + 
 
 a X £ O — 
 
 - 1/ ( a2 1 ~ a2 2^ 2 ^ cos2ecos2c P + sin2( P ) 2 + 2(a 2 - a 2 )(a 2 -a 2 ) • 
 
 • sin 2 9(cos 2 9 cos 2 cp - sin 2 cp ) + (a 2 -a 2 ) 2 sin 4 9 ] (2.51) 
 
 Substituting a. from (2.34) into (2.51) we find that the resulting equation 
 is identical to (2.49). Hence we can write 
 
 p* , ( r) =ii (v«) 2 
 
 11 ° (2.52) 
 
 P*ll ( r) = ^o (v " )2 
 
 Since 8* ( ^) is an ellipse, and we find that the extrema of (3* 
 correspond to the two permitted propagation velocities, we conclude that 
 Y * and ¥*', or the axes of the ellipse are the only permitted directions 
 of polarization. 
 
 We find that (3* , (^ ) is an ellipse for any 9 and cp , which implies 
 that 8* , is an ellipse for any orientation of the (y, y~) plane. Since 
 the only surface, which is cut by any diametial plane in an ellipse is 
 the ellipsoid, we can consider that B* . is the radius vector of an ellipsoid. 
 
 We have considered (3 and v in these calculations as the basic variables, 
 
 but e n , and n can be used as well. From (2.44) we have 
 11 
 
 D .. = \r — E = €* . E_j (2.53) 
 
 A p*u yi n y? 
 
 hence 
 
 1 
 
 e* 
 
 11 8* 
 
 p n 
 
16 
 
 and from (2.40) we can substitute for P*, , and obtain 
 
 e *n m = pS-TY) = |T> =€ o n2 (2 ' 54) 
 
 11 o 
 
 Since P* , can be considered as a radius vector of an ellipsoid, we 
 can conclude from (2.54) that n is also a radius vector of an ellipsoid. 
 (Note that p* ^ 0.) 
 
 A diametral plane normal to N, (the y_ y plane) cuts both the p* and 
 
 -L z 
 
 the n ellipsoids in ellipses. Because of equation (2.54). the extrema of 
 these ellipses occur for the same ^ ' and ^ ", i. e. 
 
 and 
 
 11 'O 
 
 Equation (2.55) indicates that n ellipsoid may be also used for the 
 determination of the permitted directions of polarization. This ellipsoid 
 is called the index ellipsoid, but it is also known as the ellipsoid of wave 
 normals, or the optical indicatrix. 
 
 The equation of the index ellipsoid is 
 
 = 1 (2.56) 
 
 The construction of n' and n" from the given N and the index ellipsoid 
 is illustrated in Fig. 2.3. 
 
 The two permitted indices of refraction for the case illustrated in 
 
 Fig. 2.3 are 
 
 (n')" 2 = COS l 9± + Sin29 i (2.57) 
 
 1 
 
 * 2 . 
 
 
 x 2 
 
 
 * 2 o 
 
 1 
 
 
 2 
 
 
 3 
 
 »'l 
 
 + 
 
 " 2 2 
 
 + 
 
 „ 2 
 
 n 3 
 
 n 3 n 2 
 
 it 
 n = n 2 
 
i x 
 
 t 3 
 
 17 
 
 Fig. 2.3. Illustration of the use of the index ellipsoid. 
 
 Equations (2.54) and (2.55) show that tensors €. , p. and the index 
 ellipsoid contain identical information about the properties of the anisotropic 
 medium. 
 2.2, The derivation of the electro-optic effects . 
 
 The anisotropy mentioned in the previous section can be attributed to 
 the crystal structure, or in some materials it can be induced by electric 
 fields and stresses. These induced anisotropic properties can be expressed 
 as the derivatives of one of the thermodynamic potentials. 
 The suitable thermodynamic potential is the internal energy U which can be 
 written in terms of strain (5. .), electric displacement (D) and entropy (S) 
 as independent variables, ant stress (T) , electric field (E) and temperature 
 (0) as dependent variables . 
 
 The differential of U can be given by 
 
 dU = T. . d (S . .) + E d D + 9 dS (2.58) 
 
 ij ij mm 
 
 Let us assume that for our purposes adiabatic (dS = 0) and constant 
 
 stress (dS . . = 0) conditions prevail. In this case (2.58) simplifies to 
 
18 
 
 dU = E d D (2.59) 
 
 and we can develop E in terms of D 
 
 m m 
 
 m 1 5 m 1 5 3 E m 
 
 E m = FF °n + 21 65-85 D n D p + 3! JT f^t^ D rPp D q + '" (2 ' 60) 
 
 n n p H 5 d 8d 6d m 
 
 P 
 
 n p q 
 
 The coefficients of this expansion can be written as 
 
 5 2 D 
 
 
 s 
 
 
 
 E 
 
 m 
 
 ^ = a 
 D mn 
 n 
 
 
 
 5 D &D 
 m n 
 
 
 
 5 3 D 
 
 
 = 
 
 R 2E 
 
 Sd Sd 
 n p 
 
 = 
 
 
 S D &D 
 m n 
 
 P 
 
 b 
 
 mnp 
 
 (2.61) 
 
 (2.62) 
 
 and 
 
 5 4 U 5 E m 
 
 = Q (2.63) 
 
 mnpq 
 
 5d&d&dSd 5d5d&d 
 
 mnpq n p q 
 
 Thus (2.60) can be rewritten as 
 
 1 1 
 
 E=aD + -c DD+-C DDD +... (2.64) 
 
 m mn n 2 mnp n p 6 mnpq n p q 
 
 and 
 
 E=(a +77C D+-C DD)D +... (2.65) 
 
 m mn 2 mnp p 6 mnpq p q n 
 
 Our present aim is to show the interaction between the modulating 
 electric field and the optical impermeability tensor 3 
 
 Let us assume that the D and E vectors consist of components at the 
 modulating frequency w' and at the light frequency w' v . 
 
 E. -!• e jU,t 4- E". e JW,,t (2.66) 
 
 11 l 
 
 and . ., M^ 
 
 D. = D 1 . e J + D e J 
 
 J J J 
 
 Substituting (2.66) into (2.65) yields 
 
19 
 
 E = U + [^S£ + JH^L (D . e J w 't + D - e jw"t ] e Jto't + D „ jco"t \ 
 
 m \ mn u 2 6 q q p 1 / 
 
 • (D« e^' 1 + D" e^"*) (2.67) 
 
 Performing the multiplication and collecting the terms of approximately co 
 frequency we obtain 
 
 b 
 
 E" = a D" + (D* D" + D' D" ) -^- + 
 m mn n p n n p 2 
 
 c 
 
 + m " Pq (D" D' D* + D' D" D* + D' D' D" ) (2.68) 
 
 6 npq npq npq 
 
 and considering that 
 
 X Z (D' D" + D" D' )] = 2 Z. \ Z. < D ' D " >1 
 
 , I ., p n p n J . L *— , p n -J 
 n=l p=l n=l p=l 
 
 (2.69) 
 
 equation (2.68) can be further simplified to 
 
 " = [a + b D' + \ c D' D' I D" (2.70) 
 
 m L mn mnp p 2 mnpq p q J n 
 
 E 
 
 The effective optical impermeability tensor is then 
 
 c 
 
 6 = a + b D' + m " Pq D' D' (2.71) 
 
 mn mn mnp p 2 p q 
 
 We desire to express the interaction in terms of the electric field 
 
 intensity E' , rather than in terms of D' . ; hence from (2.1) we can write 
 i i 
 
 D' = c' E' (2.72) 
 
 The substitution of (2.72) into (2.71) yields 
 
 B = a + b F' E' + i- c <=' J E' E' (2.73) 
 
 mn mn mnp ps s 2 mnpq <- ps °qt s t 
 
 Let us define new tensors 8'' , r and p as 
 
 mn mns imnst 
 
 S" = a the optical impermeability tensor 
 K mn mn ^ J 
 
 r = £ b c' the linear electro-optic tensor (2.74) 
 
 mns o mnp '-ps 
 
 n - f — c c d the quadratic electro-optic tensor 
 
 Nmnst o 2 mnpq c ps c qt 
 
20 
 
 Thus (2.73) becomes 
 
 8 =8" + £ _1 r E' + O £ _1 E' K' (2.75) 
 
 mn mn o mns s <mnst p s t 
 
 We have shown in Section 2.1 that 8" can be always diagonalized . Further- 
 
 mn 
 
 more, we can assume that the changes in 6 due to fields E' and E' are 
 
 mn s t 
 
 small . Denoting the diagonal term with a single m subscript we can write 
 
 8 = 8" + £ _1 r E' +£ _1 Q E' E* (2.76) 
 
 m m o ms s o »mst s t 
 
 Thus the interaction between the modulating fields E' , E and the 
 
 s t 
 
 optical fields E" , D" can be given as 
 m m 
 
 E" =["8" + £ _1 r E' + £ _1 O E' E' Id" (2.77) 
 
 m L m o mss o >mst s t J m 
 
 This is the general expression for the electro-optic effect . In the 
 
 m m 2 
 
 MKS system the units of r is — and that of p is — . Using the 
 J m s V Ymst y2 
 
 results of the previous section we can express the effect of the modulating 
 fields on the index ellipsoid. From (2.54) we can write 
 
 8 = * 2 (2.78) 
 
 m £ n £ 
 o m 
 
 Expressing n 2 from (2.78) and substituting for 6 we obtain 
 
 m m 
 
 (2.79) 
 
 m <- /„ti ^ -1 „, _ -1 
 
 £ (8" + £ -i r E' +8 O E' E' ) 
 o m o ms s o >mst s t 
 
 Let us define n by 
 
 om 
 
 n 2 = ^ I,, (2.80) 
 
 om £ n & «, 
 
 o m 
 
 hence (2.79) becomes 
 
 n 2 
 
 om 
 
 n 2 = 
 
 m 
 
 1+n 2 (r E' + O E' E' ) 
 om ms s >mst s t 
 
 [ 1-n 2 (r E' + P E' E' ) + . . . (2.81) 
 
 om u om ms s >mst s t 
 
21 
 
 Taking the square root of both sides we obtain 
 
 r n * i 
 
 n = n 1 2E ( r E' + P E' E' ) + ... (2.82) 
 
 m om L 2 ms s ^ mst s t J 
 
 Thus the change in the principal index n is 
 
 m 
 
 An = - - n 3 (r E' +P E' E' ) (2.83) 
 
 m 2 om ms s »mst s t 
 
 2 .3 The index ellipsoid of some important electro-optic materials . 
 
 In the previous section we have derived the general relationship (2.81) 
 
 between the modulating electric field and the principal refractive indices 
 
 of the material . For some media the significant elements of the tensors 
 
 r and Q have been measured, but for most materials the available data is 
 
 ms >mst 
 
 (17), (18), (19), (20) 
 
 incomplete . 
 
 2.3.1. Linear electro-optic effect in uniaxial materials. 
 
 Among other materials, the important electro-optic crystals KDP and 
 
 ADP (with a symmetry D ), belong to this group. 
 
 If an electric field E is applied to this material along its optical 
 
 X 3 
 axis x , equation (2.81) becomes 
 
 n 2 
 
 1+n 2 r E 
 o 63 x 
 
 3 (2.84) 
 
 n 2 
 
 1-n 2 r E 
 o 63 x„ 
 
 and 
 
 The material constants n , n , r are given in Table 2.1. Thus the 
 
 o' e' 63 
 
 index ellipsoid in normal form is 
 
 <~i- + r «<, E ) x 2 n + ( 4- " r._ E ) x 2 + K- x 2 = 1 (2.85) 
 
 n 63 x„ 1 n" 4 63 x„ 2 n 3 
 o 3 o 3 e 
 
 
22 
 
 Although we are going to discuss this later, we should mention here 
 
 the fact that KDP and ADP are ferroelectric materials, and that both r 
 
 63 
 
 and € s are very strongly temperature dependent. 
 2.3.2. Linear electro-optic effect in cubic materials. 
 
 For cubic materials with symmetry T , to which cuprous chloride belongs, 
 the principal indices are given by (the electric field is in the x direction 
 again) 
 
 1+n 2 r„,E 
 
 o 41 x 3 
 
 n 2 iT5 — (2.86) 
 
 1-n^ r._E* 
 o 41 x o 
 
 n 2 = n 2 
 
 and the index ellipsoid is given by 
 
 ■r )X 2 n + (-2 
 
 o ° o «j o 
 
 ( ^" + r 41 E x ' x2 l + ^ r 41 E :X y 2 + *>- X % " * <2 - 87) 
 
 For cubic materials the orientation of the index ellipsoid depends on the 
 orientation of the electric field. 
 2.3.3. The quadratic electro-optic effect . 
 
 All electro-optic substances which have a center of symmetry in their 
 
 molecular structure exhibit this quadratic effect; for this case tensor r = 
 
 n ms 
 
 For cubic materials of symmetry 0, the number of independent p tensor 
 elements are reduced to three. Furthermore, if the field is applied in the 
 direction of an edge of the cube, e.g. the x axis, the principal indices 
 are given by 
 
23 
 
 n 2 = n 2 = ^!° (2.88) 
 
 Z 1+n 2 P E 2 
 
 O 2 x 3 
 
 and n 2 
 
 .2 
 
 n 3 " 1+n 2 Pl E 2 
 
 X 3 
 
 and the index ellipsoid is given by 
 
 ^ + P 2 E V x2 1 + < J"" + P 2 E \ > x2 2 + ( ^" + Pl E2 x > x2 3 " * (2 - 89) 
 o 3 o 3 o 3 
 
 It can be demonstrated that this quadratic effect gives rise to the Kerr effect. 
 Let us consider the difference 
 
 <°\ - n V - " a o t i + ,/ p ,e» - 1 ; / Pl E* ' < 2 - 90) 
 
 o p 2 x„ o M l x 
 
 3 3 
 
 - „2 
 
 also 
 
 n 2 [1 - n 2 p E 2 - 1 + n 2 p E 2 ] 
 
 o o^2x o K lx 
 
 = n ( Pl-P2>:"-^3 
 
 (n 2 x - n 2 3 ) = (n x + n 3 > (^ - n g ) (2.91) 
 
 = 2 n (n - n ) 
 
 O 1 o 
 
 hence 
 
 ^ • < n l " V " \ n 'o ( Pl - <V E % <2 - 92> 
 
 (21) 
 
 The empirical law of the Kerr effect states that 
 
 An = K\ E 2 (2.93) 
 
 X 3/ 
 
 where K is the Kerr constant, \ is the wavelength of light in vacuum and An 
 is the difference in indices as defined by (2-92). 
 Comparing (2.92) and (2.93) we find that 
 
 K\ = | n 3 Q (p 1 - p 2 ) (2.94) 
 
2 .4 Material properties of some electro-optic materials 
 (MKS units are used.) 
 
 Material 
 
 Units 
 
 Temp. (^ - (p 2 ) r 
 
 K 
 
 41 
 
 -16 m 2 , rt -12 m 
 
 10 v* 10 V 
 
 10 
 
 63 
 
 -12 m 
 
 KDP 
 
 ** 
 
 ** 
 ADP 
 
 CuCl 
 
 ZnS 
 
 BaTiO 
 
 SrTi0 3 
 
 K TaO 
 
 nitro- 
 benzene 
 
 120 
 
 148 
 
 393 
 
 33 
 
 173 
 
 — 
 
 -- 
 
 293 
 
 — 
 
 — 
 
 173 
 
 OQQ 
 
 -- 
 
 — 
 
 zy«3 
 293 
 
 — 
 
 6 
 
 293 
 
 — 
 
 2 .4 
 
 393 
 
 23 
 
 -- 
 
 403 
 
 6.5 
 
 -- 
 
 77 
 
 0.31 
 
 -- 
 
 293 
 
 0.02 
 
 -- 
 
 4 
 
 4.3 
 
 -- 
 
 77 
 
 0.1 
 
 — 
 
 300 
 
 0.0124 
 
 
 
 3.0 
 
 1 
 
 .5095 
 
 60 
 
 6.8 
 
 1 
 
 .5095 
 
 20 
 
 — 
 
 1 
 
 .5246 
 
 16 
 
 5.5 
 
 1 
 
 .5246 
 
 14 
 
 — 
 
 1 
 
 .93 
 
 8.3 
 
 — 
 
 2 
 
 .368 
 
 10.25 
 
 — 
 
 2 
 
 .4 
 
 6000 
 
 
 2 
 
 .4 
 
 4500 
 
 — 
 
 2 
 
 .409 
 
 300 
 
 — 
 
 2 
 
 .409 
 
 300 
 
 — 
 
 2 
 
 .5 
 
 4400 
 
 — 
 
 2 
 
 .5 
 
 800 
 
 1 .55 
 
 34 
 
 T is the Curie temperature discussed in Section 3.9 
 c 
 
 **n = 1.468 for KDP and n = 1.4792 for ADP 
 e e 
 
CHAPTER 3 
 INTENSITY MODULATION BY CONTROLLED PARTIAL REFLECTIONS 
 
 3.1. The FresnaJl relations for the reflected amplitudes . 
 
 Let us consider the problem of a plane wave incident on a plane boundary 
 between two media n and n . If both materials are anisotropic, then there 
 are four resultant waves in general: two reflected inton,, and two refracted 
 into material n . The solution for the intensities, phase velocities, 
 directions of propagation and directions of polarization can be obtained 
 from the boundary conditions on the E, D, H, B vectors and from the index 
 ellipsoids of the two materials. The analytical solution of this general 
 problem is quite difficult; the geometrical constructions involve fourth 
 order surfaces and are very lengthy. Fortunately, a few simple, realizable 
 assumptions reduce the complexity of the problem to an acceptable level . 
 
 Let us assume that both materials are homogeneous, non-conducting, non- 
 magnetic, and that material n is isotropic and material n is electrically 
 anisotropic of the type treated in Chapter 2. Let us further assume that the 
 plane of incidence and the plane of boundary coincide with the planes of 
 symmetry of the index ellipsoid of medium n . Notice that this is a very 
 significant simplification, since now both the normal and the parallel (with 
 respect to the plane of incidence) components of the incident optical electric 
 field E are in the direction of permitted polarization for material ^ ; 
 hence both of these waves have only single refracted and reflected waves. 
 
 In Fig. 3.1 E and E are the normal and parallel components (with 
 respect to the plane of incidence) of the incident optical field 
 
 25 
 
 
26 
 
 anisotropic 
 material 
 n 
 
 (xz) plane is the boundary 
 
 (xy) plane is the plane of 
 incidence 
 
 Fig. 3.1. Plane wave is incident on an 
 anisotropic medium. 
 
 E • N. , N . , N and 0., 0^, are the unit wave normals and the angles 
 — — i — t — r i t r & 
 
 of the incident, transmitted (or refracted) and reflected waves respectively 
 
 The principal refractive indices of medium n„ are n„ , n^ and n„ , in the 
 
 2 2x 2y 2z 
 
 direction of coordinate axes x,y,z. Let us further assume that medium n n 
 
 is denser than medium n , i.e. 
 
 n i >/n 2 
 
 (3.1) 
 
 so that there exists a critical angle 0. for which 0, = n/2 . Let us con- 
 ic t 
 
 sider the case when is slightly below the critical angle 0. . The re- 
 fracted wave propagates nearly in the direction of the x axis. i.e. 
 
 (3.2) 
 
 The diametral plane normal to u cuts the index ellipsoid of medium 
 
 —x 
 
 N = u 
 — t — x 
 
 n in an ellipse with semi -axes n and n , therefore planes (x, y) and 
 
 (x, z) are the allowed planes of polarization, and the refractive indices 
 
 are n„ and n^ in these planes respectively. If the incident wave is 
 2y 2z 
 
 polarized in the direction normal to the plane of incidence, then 
 
27 
 
 trace of ellipsoid n, 
 
 Figure 3.2. The two allowed polarizations for N = u . 
 
 E=E =Ee JTi i 
 
 — — n n 
 
 — z 
 
 (3.3) 
 
 where 
 
 r-Ni 
 
 T. = OJ (t - — 
 1 v. 
 
 ) = CJ [t - — (x sine. + y cosG.)] (3.4) 
 
 The magnetic field intensity vector H can be obtained from 
 
 Thus 
 
 H = n n \| — (E cosG.e J i u - E sinG. e "* u ) 
 lll[-i ui — x u i — y 
 
 (3.5) 
 
 If we define R and T as the complex amplitudes of the reflected and 
 n n 
 
 transmitted waves, we have for the transmitted fields 
 
 E^ = T e J t u 
 — t n — z 
 
 (3.6) 
 
 where 
 
 G o " jT t " jT t 
 
 H = n„ 1 — (T cos9 e u - T sine e u ) 
 -t 2z y u n t -x n t -y 
 i o 
 
 (3.7) 
 
 T -«[t - 
 
 (x sine + y cos6 )] 
 
 C X c 
 
 (3.8) 
 
 and for the reflected fields 
 
 E = R e 
 — r n 
 
 u 
 — z 
 
 (3.9) 
 
28 
 
 i € o "^r " jT r 
 
 -r = n i V 7T (R n COs9 r e H x " R n sine e u ) (3.10) 
 
 where 
 
 "l 
 T = oj [ t (x sinG + y cos9 ) ] (3.11) 
 
 The boundary conditions at y = demand that the tangential components 
 
 of E and H be continuous. Hence we must have 
 
 E + E = E 
 
 - -r -t 
 
 and (3.12) 
 
 H*u + H • u =H - u 
 
 — —x — r —x — t —x 
 
 (The normal components of B and D are continuous across the boundary 
 
 automatically.) 
 
 Substituting the electric and magnetic vectors into (3.12) and using 
 
 the fact that cosG = - cos9., we obtain 
 
 r i 
 
 E + R = T 
 n n n 
 
 and 
 
 n, cosG, (E - R ) = n cosG T (3.13) 
 
 l.i n n 2z t n 
 
 We can solve (3.13) for the reflected amplitude R 
 
 n 
 
 n n cos9 . - n„ cos9__ 
 
 R = -^ j± 25 * E (3.14) 
 
 n n^cos9. + n cos9 J n 
 1 i 2z t 
 
 Let us define r as the relative reflected amplitude; then we have 
 n 
 
 R n. cos9. - n cos9^ 
 
 r - — = zr ZT (3.15) 
 
 n E n, cos9. + n„ cos9^ 
 n 1 i 2z t 
 
 In an analogous manner we can derive the relative reflected amplitude of the 
 
 parallel polarized wave and obtain 
 
29 
 
 R n cos0. - n, cos9 
 jp 2y 1 1 t 
 
 r p ! E n cos9. + n, cos0^ (3.16) 
 
 P 2y l 1 t 
 
 Equations similar to (3.15) and (3. 16) were derived by Frcsncl in 1823 
 
 and by tradition bear his name. 
 
 The angles 0. and 9 are related through the law of refraction 
 
 n i 
 sinG = — sine , n > n_ (3.17) 
 
 t n 1 1 2 
 
 where n = n for the normal component and n = n for the parallel component . 
 
 The critical angle of incidence is different for the two waves in general, 
 
 since n. p n. . Let us denote by G and 0^ the critical angles for the nor- 
 2z ' 2y n t 
 
 mal and parallel components respectively. Then we have 
 
 (3.18, 
 
 ii n 
 
 and 
 
 sin0 = 
 n 
 
 n l 
 
 sin0 
 P 
 
 2yo 
 n l 
 
 (3.19) 
 
 3.2. The dependence of the reflected intensity on the index of refraction n . 
 
 If the anisotropy of material n is due to the electro-optic effect, 
 
 then the principal refractive indices n , n and n depend on the modulating 
 
 field E . In this section we want to examine how the changes in these indices 
 m 6 
 
 effect the relative reflected amplitudes and the relative reflected intensities 
 
 (I and I ) of the light waves. The intensities and the amplitudes are 
 n p 
 
 related by 
 
 I = (r ) 2 
 
 n (3.20) 
 
 I - (r ) 2 
 P P 
 
 Let us denote the changes in indices by An [ refer to (2.83)] 
 
 n 2i = n 2io + ^21 (3.21) 
 
30 
 
 where i ranges through x, y, z and n are the principal refractive indices 
 
 ZIO 
 
 with E = 0. 
 — m 
 
 As we have noted previously, our interest lies in cases where the 
 incident angle is close to the critical angles given by (3.18) and (3.19). 
 We shall find it convenient to introduce the variables Ot and 3, where 
 
 and 
 
 a = e - 9. 
 
 n n i 
 
 B = rt/2 - 9^ 
 
 n tn 
 
 (3.22) 
 
 for the normal component, and 
 
 a 
 
 9-9. 
 P i 
 
 8 = ^2 - 9 
 P tp 
 
 (3.23) 
 
 for the parallel polarized wave. 
 
 »-x 
 
 Fig. 3.3. Illustration of the new angular variables for the parallel 
 component . 
 
 Substituting (3.21), (3.22) into the expression (3.15) for r we 
 
 obtain 
 
31 
 
 n_cos(9 - a ) - (n +An ) cos(n/2 - ft ) 
 1 n n 2zo 2z K n 
 
 r n " n.cos(9 - a ) + (li +An ) cos(jt/2 - ft ) (3.24) 
 
 1 n n 2zo 2z *n 
 
 which can be written into the form 
 
 1 - A sinft 
 
 r n = 1 + A n sin S n (3 ' 25) 
 
 n "n 
 
 where 
 
 (n„ + An„ ) 
 
 2zo 2z . . 
 
 A = 7E — 7TT (3.26) 
 
 n n, cos (9 -a ) 
 1 n n 
 
 Using (3.18) we can rewrite equation (3.26) as 
 
 An a 2 
 
 A = tanG (1 + ) (l - tan a tan9 + . . .) (1 -2 + . . .) (3.27) 
 
 n n n_ n n 2 
 
 2zo 
 
 which yields after multiplication 
 
 An 
 
 A = tanG [1 + ( - tan a tanG ) + ...] (3.28) 
 
 n n n_ n n 
 2zo 
 
 The second and higher order terms were neglected, since a and An are 
 
 -2 
 smaller than 10 for cases under consideration. 
 
 The term sinft can be computed from equation (3.17) using the new 
 
 variables (3.21) and (3.22); then we can write 
 
 n l 
 
 sine = cosS = t sin(0 - a ) (3.29) 
 
 tn ^n n„ +An„ n n 
 2zo 2zo 
 
 hence 
 
 An a 2 , 
 
 cosft = (1 — + ...)(1 - tana catane + ,,.)(1 ^+ ...) (3.30) 
 
 "n n„ n n 2 ' 
 
 2zo 
 
 2z 
 
 = 1 - ( + tana cotan9 ) + . . . 
 
 n_ n n 
 
 2zo 
 
 .2* 
 
 n 
 
 An. 
 
 Then cos ft is approximately equal to 
 
 cos 2 R = 1 ■- 2 ( — — + tan a cotan 9 ) (3.31) 
 
 K n n„ n n 
 
 2zo 
 
32 
 
 and from the identity sin 2 B + cos 2 B = 1 we obtain 
 
 n K n 
 
 An 
 
 sinB = [ 2(- 
 n n 
 
 2z 
 
 2zo 
 
 + tan a cotan ) ] 
 n 
 
 (3.32) 
 
 The reflected relative intensity I is given by 
 
 I = (r ) 2 
 n n 
 
 1 - A sin 6 
 n ^_n 
 
 1 + A sin ft 
 n n 
 
 (3.33) 
 
 = 1 - 4 A sinB + 8 A 2 sin 2 B - . . . 
 n K n n K n 
 
 Substituting (3.28) and (3.32) into this equation and retaining only 
 the first order term we obtain 
 
 An An 
 
 I =1-4 tanG (1 + - tana tanG ) [ 2( + tana cot anG )] 
 
 n n n_ n n L n n J 
 
 2zo 2zo 
 
 (3.34) 
 
 Let us now define the relative intensity modulation M by 
 M =1-1 
 
 (3.35) 
 
 For the normal component this becomes M and 
 
 n 
 
 An 1 
 
 M = 4 tanG (1 + — — . _ . rr> , 2z _ . -, 2 fo ._. 
 
 n n n n tana tanG ) [2( + tana cotanG )] (3.3b) 
 
 2zo n n n„ n n 
 
 2zo 
 
 We can derive analogous results for the parallel component. We can 
 
 write the expression (3.16) for r in terms of the new variables as 
 
 P 
 
 (n„ + An„ ) cos(G - a ) - ncos(fi/2 - 6 ) 
 2yo 2y^ p P 1 P 
 
 r p (n„ + An„ ) cos(G - a ) + ncos(it/2 - 6 ) 
 
 2yo 2y p p 1 K p 
 
 (3.37) 
 
 Let us define A as 
 P 
 
33 
 
 n i 
 
 A P ' (n. + An )cos(9 - a ) (3,38) 
 
 2yP _2y P P 
 
 Then (3.37) can be rewritten as 
 
 1 - A sin|3 
 
 r = A P . P (3.39) 
 
 p 1 + A sine ' 
 
 o K p 
 
 Using similar arguments as in the derivation of A , we obtain 
 
 n 
 
 t an9 An 
 
 A = . 2 P [ 1 - ( £ + tana tan9 ) ] 
 
 p sir8 n_ P P 
 
 P 2yo 
 
 (3.40) 
 
 The term sinB can be formally obtained from (3.32) by replacing n 
 
 by n„ and 9 by 9 . Thus 
 
 J 2y n J p 
 
 ^2 - 
 
 sinB = [ 2( £ + tanQ; cotan9 ) ]2 (3.41) 
 
 P n 2yo P P P 
 The quantities I and M for the parallel component are given by 
 
 I =1-4 A- sinB + 8 A 2 sin 2 B ~ ... (3.42) 
 
 P P P P P 
 
 and 
 
 M s 4A sinB - 8 A 2 sin 2 S + ... (3.43) 
 
 P P P P P 
 
 Substituting A and sinB from (3.40) and (3.41) we have 
 P P 
 
 tan9 An An — 
 
 I =1-4 . 2 P [ 1 - ^ - tanatan9 )] [ 2( + tana cotan9 ) ] 
 
 P sm 2 9 n_ P n n p p 
 
 P 2yo 2yo 
 
 (3.44) 
 
 and , 
 
 tan9 An An - 
 
 M - 4 J 9 P (1 ^ - tan a tan9 ) [ 2(- — * + "tanacotan9 ) ] 
 
 p sin 2 9 n„ p n_ o 
 
 p 2yo 2yo 
 
 (3.45) 
 
3^ 
 
 We should remark here that An and a, can be negative as well as positive, 
 but real modulation occurs only if the quantity under the square root sign 
 
 in (3.36) and (3.45) is positive. 
 
 -3 
 
 Since for most electro-optic materials An<10 and a is chosen to be 
 
 small, ( this point will be discussed later), we can write 
 
 ^z \ 
 
 M = 4 tanG [ 2( + tana cotanG )] (3.46) 
 
 2zo n 
 
 and 1 
 
 tanG An - 
 
 M = 4 . J [2( y - + tana cotanG ) ] 2 (3.47) 
 
 p sin^G n p p 
 p 2yo 
 
 The last two equations will be used in subsequent calculations. 
 
 3.3. Modulation M as a function of the insulating electric field E m . 
 
 The electro-optic effect was given specifically for three important 
 
 crystal groups in section 2.3. Considering then crystals as medium n , 
 
 we shall obtain M and M as functions of E and a. Two cases should be 
 n p — m 
 
 considered separately: 1. when E has only a normal component (i. e. 
 
 E = E u ) and 2. when E has only a parallel component (E - E u ). 
 — m z — z — m * *■ «- —roy—y 
 
 Let us consider first the use of a linear uniaxial material. For 
 
 the first case we have x_ = x, x. = y, x^ = z and E = E . Thus we 
 
 1 2 3 -m z 
 
 can write 
 
 n = n + \ n 3 r__ E (3.48) 
 
 2y o 2 o 63 z 
 
 and 
 
 n„ - n 
 2z e 
 
 Substituting these quantities into (3.46) and (3.47) we find 
 
± 35 
 
 M = 4 tanG [2(tanQl cotanG )1 (3.49) 
 
 n n L n n J 
 
 and 
 
 tanG 
 M = 4 - 
 
 P 
 
 E r i -.1 
 
 • 2 Q L 2 (o n r c< 3 E + tana cotanG ) 2 (3.50) 
 
 nn^G L 2o63z p p J 
 
 In a similar manner we can determine M (E , a ) and M (E , oc ) for the 
 
 n -m n p — m p 
 
 other crystal groups. These results are summarized in Table 3.1. 
 
 It is apparent from this table that several media exhibit negative An. 
 For these cases modulation can be achieved only if ol is chosen such a way that 
 
 (— + tanacotanG) > (3.51) 
 
 n 
 o 
 
 The apparent electro-optic constant of a quadratic material can be in- 
 creased by the use of a bias electric field E . Let us consider An of 
 
 o y 
 
 a cubic quadratic material if E = E + e , and the E field is in the y 
 
 — m -o -m — m 
 
 direction, (where e is the magnitude of the modulating field) . Then 
 
 m 
 
 An = - \ n 3 O (E + e ) 2 (3.52) 
 
 y 2 o >1 o m 
 
 = " h n3 9, (E 2 + 2E e + e 2 ) 
 2o)l o om m 
 
 Now if E > > e we can neglect e 2 in (3.52) and write 
 o m m 
 
 An = (- J n 3 Q E 2 ) + (- i- n 3 0, 2E e ) (3.53) 
 
 y2o>lo2o>lom 
 
 Since E is a static field, we can consider the first term of (3.53) as 
 o 
 
 part of n and we can say that the effective change An v will be 
 y o 
 
 An' = - n 3 o E e (3.54) 
 
 y o > 1 o m 
 
 Thus in this case the quadratic material behaves as a linear material 
 
 with electro-optic constant ( p,E ). 
 
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37 
 
 Examination of equations (3.46) and (3.47) reveals that M and M 
 
 n P 
 
 are not linear functions of An in the general case. Both of these functions 
 
 are linear however for quadratic electro-optic materials when a or a are 
 
 n p 
 
 equal to zero, For both linear and quadratic materials the modulation can 
 
 be made approximately linear for small modulating signals if Oi or a is 
 
 n p 
 
 chosen suitably large. 
 
 3.4. The physical realization of the intensity modulator . 
 
 It is clear from Table 2.1, that the electro-optic effect is very 
 
 small even for the best materials, and consequently the modulation M is 
 
 a weak function of E . One possible way to increase the modulation for 
 
 — m 
 
 a given modulating field strength is to repeat the controlled partial 
 reflection many times. This can be accomplished by having the light ray 
 bouncing between two parallel surfaces, these surfaces being the isotropic- 
 anisotropic dielectric interfaces. This arrangement is shown in Fig. 3.4. 
 
 output beam 
 
 Fig. 3.4. A multiple reflection intensity modulator. (n > n ) 
 
38 
 
 As indicated in Fig. 3.4, a is the separation between the two parallel 
 reflecting surfaces, L is the length of the device and b is the height of 
 the device (in the z direction.) The number of reflections m is given by 
 
 m = — cotanB. (3.55) 
 
 a i 
 
 (Note that the length L should be chosen so that all of the beam is 
 
 reflected on the last reflection.) 
 
 In section 3.2 we have derived I and I for a single reflection. 
 
 n p 
 
 The reflected relative intensity I after m identical partial reflections 
 
 m 
 
 is given by 
 
 I = (i) m = (1 - M) m (3.56) 
 
 m 
 
 Raising the expression for I to the m^ n power yields 
 
 (I ) m = I = 1 - 4m A sin 6 + 8 m 2 A 2 sin 2 6 - ... (3.57) 
 n mn n n n n 
 
 and similarly for (I ) 
 
 P 
 
 (I ) m = I = 1 - 4m A sinB + 8 2 A 2 sin B - ... (3.58) 
 p mp p K p m p K p 
 
 Let us define the resulting modulation by M 
 
 m 
 
 M =1-1 (3.59) 
 
 m m 
 
 Neglecting the higher order terms in (3.57) and (3.58) we have the M 
 
 mn 
 
 and M 
 
 mp 
 
 M = 4 m A sin 8 = m M /0 cn v 
 
 mn n K n n (3.60) 
 
 M = 4 m A sin 8 = m M 
 mp P P P 
 
 Thus we can conclude that the modulation increases m-fold due to the 
 multiple reflections. 
 
39 
 
 We should point out that the intensity modulation can be realized in 
 
 a dual way also. Let us suppose that the denser anisotropic material is n , 
 
 and medium n_ is the isotropic, less dense material. (Refer to Fig. 3.4). 
 
 Then internal total reflection can occur in the anisotropic material n , 
 
 and controlled partial reflections are possible. The analysis of this 
 
 case is slightly more involved, since here the index of refraction for the 
 
 parallel polarized wave does not coincide with the principal index of 
 
 refraction n n . The results for this case can be summarized by M* and 
 ly mn 
 
 M* 
 mp 
 
 "^lz 2 
 
 M* = 4 m tan 0* [ 2( + tana* cotan0 f ) 1 (3.61) 
 
 mn n n, n n J 
 
 lzo 
 
 and 1 
 
 tane 1 - 
 
 M* = 4 m . * P L2(-^?- + tana 4 cotan0' ) 1 (3.62) 
 
 mp sin^G n r , p o J 
 p 1 
 
 where a* , a' , 0' and are defined similarly as the unprimed variables, 
 n p n p 
 
 and n ? is computed from 
 
 cos 2 0„ sin 2 0_ 
 
 (3.63) 
 
 (n^) 2 
 
 cos 2 
 
 p 
 
 + 
 
 sin 2 
 P 
 
 lx 
 
 
 2 
 
 iy 
 
 In the following sections we are going to concentrate on the analyis 
 of the first type of modulator, but most of these results can be easily ex- 
 tended to the second kind of modulators described by (3.61) and (3.62). 
 
 It is also interesting to note that if a =0, the modulation M as 
 
 n mn 
 
 given by (3.60) is independent of material n . Substituting m from (3.55) 
 
 we obtain 1 
 
 An 2 
 
 M = 4 - (2 — ^ ) (3.64) 
 
 mn a n„ 
 
 2zo 
 
40 
 
 Thus as long as n, > n_, and the angle of incidence is 0. =9 , the 
 12 in 
 
 actual value of index of refraction n. does not influence the modulation M 
 
 1 mn 
 
 3.5. Beam intensity modulation. Effects of collimation and diffraction . 
 
 Up to this point we have considered the case of a single ray, or in 
 other words, a plane wave incident on an infinite plane boundary. Evidently 
 this is a simplified version of a beam incident on a crystal surface. We 
 can, however, extend our results to include this more general problem by 
 considering the beam as an infinite number of superimposed plane waves 
 (rays), then computing the modulation for each ray by the derived formulas 
 and finally summing all these modulated rays to obtain the output beam 
 intensity. 
 
 Suppose that either because of lack of collimation or diffraction 
 effects the beam intensity is distributed between the limiting incidence 
 angles OL and ot . Since we are going to consider relative intensities, the 
 input absolute intensity is unity. If w(a) is the distribution function, 
 then it has to satisfy the relation 
 
 J w(a) da = 1 (3.65) 
 
 at! 
 
 Note that w (a) = for a outside the interval a < a< CL. 
 
 Let us define the relative beam modulation M and the relative beam 
 
 intensity I by 
 
 ■ / w(a) M(a,An)da (3.66) 
 
 a 2 
 
 M 
 
 °2 
 
 I -.; i - m = f wta) I(a,An)dC( (3.67) 
 
 °1 
 
41 
 
 From these equations we can compute M(An) and I (An) by using the formulas 
 
 for M and M developed in section 3.4. One can consider various forms of 
 mn mp 
 
 w(0!) , but since we are looking for an estimate of the performance of a 
 modulator, a pessimistic approximation can give us a sufficient answer. Thus 
 we can assume that the beam intensity if uniformly distributed over a, i.e. 
 w.. id) = constant. In the case of diffraction the actual beam is concentrated 
 more in the center; for a parallel slit the intensity of the first lobe is 
 distributed as 
 
 1 sin 2 y 
 2 V ~" jt y s 
 
 w„(a) = - 
 
 (3.68) 
 
 where 
 
 , 2* , a i +a 2 
 y = ( ) a - Jt 
 
 <V°i a i - a 2 
 
 for a < a < a . 
 
 Using w (a) = const., we find that (3.65) is satisfied if 
 
 w i (°b = ™ 1 - ~ for a i <®<_ a o' (3.69) 
 
 Hence (3.66) becomes QL 
 
 M (An) = -^ — / M(a,An)da (3.70) 
 
 V°l 
 
 a i 
 
 As we have remarked earlier, we are interested only in the real part of 
 M, hence we consider 
 
 An 
 
 M = for ( — - + tan a cos an9) <0 (3.71) 
 
 n 
 o 
 
 For the normal component we have then 
 
 °2 1 
 
 4m tanG f An 2 
 
 M (An) - - t \ j L2( + tan a cotanG ) ] da (3.720 
 
 (a 2 - a x ) ^ n 2zQ 
 
U2 
 
 Since a is small, we can approximate tan a by a , and after integration 
 n n n 
 
 we obtain 
 
 8/2 m tan 2 e r An 2 
 
 w n / 2z . s 2 
 
 M = ~ 7T7 r ( v + QL cot an ) 
 
 mn 3(a o - QL) I Vl n_ 2 n ' 
 
 ^2z - 
 
 - ( + OL cotanG ) 2 (3.73) 
 
 n 2zo 1 n J 
 
 Equation (3.73) is illustrated in Fig. 3.5 for m = 10, and for 
 various values of (QL - OL ) . 
 
 Similarly for the parallel component we have 
 
 3 
 
 8/2m tan 2 An - An 
 
 M = -r-z r ? 2n [ ( + a^cotan ) " - ( ^ + a cotan0 ) 
 
 mp 3(a_ - a. sm 2 8 L v n_ 2 p 7 v n 1 p 
 2 1 p 2yo 2yo ^ 
 
 (3.74) 
 
 From these equations one can obtain an estimate on the influence of beam 
 spread on the modulation M. 
 
 The effect of the beam diffraction can be nearly eliminated if the modula- 
 tion system is designed in such a way that the modulator operates in the near 
 field of the optical source. Within the near field of an aperture illuminated 
 
 by a coherent plane wave there is little beam spread, the beam travels nearly 
 
 d 2 
 as a plane wave. This region extends to a distance of — , where d is 
 
 X o 
 
 the diameter of the aperture and \ is the wavelength of the radiation. 
 
 Assuming that the input aperture is illuminated by a coherent plane wave, 
 
 we can compute how long the device may be and still stay within the near 
 
 field of the input aperture. The width of the input face A (refer to Fig. 
 
 3.4) is a*, where 
 
 n l 
 a 1 = — 'a (3.75) 
 
 n 2o 
 
43 
 
 M 
 
 mn 
 
 1.0- 
 
 0.5 
 
 10 5 C 
 
 where B - beam spread = ('OU— (X,) 
 
 C = ( 
 
 A n 
 
 2z 
 
 — a cot an a J 
 
 m = 10 
 
 2zo 
 
 7 n 
 
 Figure 3.5. Beam intensity modulation plotted for various 
 beam spread values . 
 
kk 
 
 hence the upper limit on the optical length L' of the device is 
 
 LU.< ^-^ (3.76) 
 
 The length L and the optical length L 1 are related by 
 
 n2 l 
 L' = — L (3,77) 
 
 2o 
 
 from (3.75), (3.76) and (3.77) we can conclude that the length of the device 
 
 is limited by 
 
 a 2 
 L < - (3.78) 
 
 o 2o 
 
 3,6. Velocity matching condition for high speed operation . 
 
 In previous discussions we have always assumed that the modulating 
 
 electric field E varies slowly with respect to the transit time of the light 
 
 beam through the modulator. Depending on the optical length of the device, 
 
 this might be the case for even radio frequency modulating fields. In order 
 
 to achieve modulation at microwave frequencies, the modulator structure must 
 
 be designed so that the partial reflections occur under identical conditions. 
 
 This implies that there must be a synchronism between the modulating electric 
 
 field E and the optical wavefront, or in other words the component of the 
 
 optical phase velocity v and the modulating field phase velocity v must 
 ^ ^ J ox tor- j mx 
 
 be equal in the x direction. In terms of material constants this implies 
 
 n 2o 
 
 % c (3-79) 
 
 mx ox n 
 
 where c is the speed of light in vacuum. 
 

 45 
 
 Thus for broadband operation we must arrange some kind of transmission 
 
 line structure which permits a TEM-like wave to propagate with velocity v 
 
 mx 
 
 A practical structure satisfying these requirements is shown in Fig. 3.6. 
 
 parallel plate 
 transmission 
 line 
 
 device 
 
 Fig. 3.6. A broadband light intensity modulator structure. 
 As can be seen from Fig. 3.6 this structure consists of a parallel 
 plate transmission line partially filled with the device shown on Fig. 3.4. 
 The anisotropic material n is a thin layer on the isotropic medium n . 
 Since the thickness of n material is much smaller than the wavelength of the 
 modulating field, we can neglect n_ from the point of view of the modulating 
 fields, and we can treat the structure as if it were filled with only two 
 
 materials, n and n . This kind of transmission line has been analyzed in 
 
 (22) 
 
 detail by Kami now and Liu . According to their analysis it is generally 
 
 difficult to propagate TEM or TEM-like modes over a broad band of frequencies 
 
 in such a partially loaded line. Let us call f , the mid-frequency, which 
 
 ml 
 
 corresponds to 
 
X ml =(^a (3.80) 
 
 where X. , is the microwave wavelength, p*_ is the relative (microwave) 
 ml 1 
 
 dielectric constant of material n.. . For modulating frequencies much 
 
 higher than f n a TEM-like mode propagates essentially contained in medium 
 ml 
 
 n and with a velocity v 
 
 v 2 = \F=FT (3.81) 
 
 R 
 
 In the mid-frequency range various non-TEM dielectric waveguide modes 
 propagate with velocity greater than v . 
 
 For modulating frequencies much below f a fast TEM like wave propagates 
 with a velocity dependent upon the filling factor K = — . Only in this lower 
 
 frequency region can we adjust v to satisfy (3.79) by the suitable choice 
 
 mx 
 
 of the width of the transmission line w. The suitable w can be determined from 
 
 the families of curves V (n , n, , w. , f ) given by these authors. 
 
 mx o 1 1 m & ' 
 
 An approximate, but more convenient formula for w can be obtained by 
 considering an equivalent relative dielectric constant e" of the transmission 
 
 T (9) I. ««♦ • 
 
 line . If we assume that n is air, we can write 
 
 o 
 
 €" - - (— + P\ - - ) (3.82) 
 
 ' w *-" 1 w 
 
 Then we might say that the equivalent microwave phase velocity is 
 
 v = — = 1 (3.83) 
 
 mx 1 r w-a , a , - 
 
 (€ )2 L w 1 w 
 
 Equating (3.79) and (3.83), and solving for w yields 
 
47 
 
 a(£ r l) 
 
 w = —2 (3.84) 
 
 1 2 
 
 2o 
 
 The characteristic impedance of the structure is approximately given by 
 
 r M> b i 1/2 
 
 Z = r „.,° : , _. x (3.85) 
 
 We may require Z to be some convenient impedance, (say 50 ohms), and 
 we can solve (3.85) for b . 
 
 b=Z ol 
 
 yU Q w(w-a + E^a) 
 
 £o 
 
 (3.86) 
 
 As we have noted before, these velocity matching conditions are effective 
 only for frequencies below f , (3.80). As the modulation frequency is in- 
 creased, the modulation becomes less effective. 
 3 . 7 Dimensional tolerance requirements ■ 
 
 In order to determine the tolerances that must be met in the manufacture 
 of a modulator, we have to examine the influence of inexact dimensions on the 
 performance of the device. On Fig. 3.4 we have labeled the input and output 
 windows A and C repsectively , and surfaces B and D are called the reflecting 
 surfaces . 
 
 Evidently all these surfaces should be optically flat, since any irregularity 
 causes the deterioration of the collimation of the beam. 
 
 The angular position of surfaces A and C are not critical, but these 
 should be normal to the beam in order to reduce the reflection losses . 
 
 A lack of parallelism in the reflecting surfaces B and D degrades the 
 modulator performance in two ways: the angles of incidence for the successive 
 
i+8 
 
 reflections will vary, and in the case of high frequency modulation, the modu- 
 lating and the light waves progressively slip out of synchronism. 
 
 Let us assume that surfaces B and D are normal to the xy plane and the 
 
 angle between them is 
 
 1r. Using the notation defined in section 3.2 we 
 
 th 
 
 find that the angle of incidence at the i reflection is 
 
 a. = a, + (i - 1) 
 
 r 
 
 (3.87) 
 
 where <X corresponds to the angle of incidence at the first reflection 
 The modulation after m partial reflections is Mm" 
 
 M 
 
 M 
 
 i=l 
 
 [<\ + (i " Df] 
 
 (3.88) 
 
 Assuming that or< < 1 and m > > 1, we can approximate (3.88) by 
 
 a + (m-l)f 
 
 M 
 
 r 
 
 M(a' , An)da' 
 
 (3.89) 
 
 QL 
 
 Integral (3.89) is similar to (3.70), hence we can use the result of that 
 integration. Then we obtain for M" 
 
 M 
 
 mn 
 
 8 fim tan 2 
 
 ' n 
 
 2z 
 
 . n 
 
 2zo 
 
 + (a + (m - 1) ipjcotanG 
 
 - (- 
 
 2zo 
 
 + a, cotanG ) 
 1 n 
 
 From this equation we can conclude that the effect of Op is equivalent to 
 an increase in the spread of the beam. From the microwave modulation point of 
 view we find that nr- changes the effective optical phase velocity v^ v , thereby 
 
 ox 
 
 causing a mismatch between v and v 
 
 ox mx 
 
49 
 
 Fig. 3.7. Notation for calculating the effect of Y* . 
 
 If O^is small, (/*-<< 1) , we can write for the change of the optical path 
 
 after m reflections 
 
 n 
 
 AL = - ma c^rr 
 
 Substituting (3.55) for m we obtain 
 
 at. L Q a T T "l 
 
 AL = - - cotanG t rr n, = - <y-L . n 
 
 a cosG 1 9 sinG 
 
 (3.91) 
 
 (3.92) 
 
 Let us require that the maximum phase difference between the optical and 
 the modulating wave due to v- be less than xt at the output of the modulator. 
 Then from the synchronism condition (3.70) and from (3.92) we obtain 
 
 ,2 
 
 \ n i 
 
 A rn 1 
 
 2o 
 
 n n 
 
 sinG 
 
 sinG 
 
 (3.93) 
 
 hence the condition on /v-is 
 
 T 
 
 < 
 
 A. n o 
 
 m 2o 
 2Ln 3 , 
 
 (3.94) 
 
 For practical modulators condition (3.94) is easily met, and thus the 
 main effect of -v- is the reduction of modulation given by (3.90). 
 
3.8. Optimization of modulator dimensions. 
 
 One of the admitted problems of the existing broadband electro-optical 
 light modulators is that the power required to drive the devices "is usually 
 high, at least several watts. This is due to the high intensity electric 
 fields necessary for the modulation and to the fact that the microwave losses 
 in the electro-optical materials are usually appreciable. For the purpose of 
 minimizing the driving power requirements, (or in other words choosing the 
 modulator dimensions in an optimal way), a useful figure of merit F is the 
 product of the modulating sensitivity S and the optical transmittivity T 
 
 M , 
 F = S • T = W • T (3.95) 
 
 r 
 
 Note that the power P in (3.95) is not the power absorbed in the modulator, 
 but it is the modulating power traveling in the x direction within the parallel 
 plate transmission line structure shown in Fig. 3.6. The power absorbed by 
 the device is much smaller, since the bulk of its volume is filled with 
 nearly lossless materials n and n , the volume of the lossy n material 
 
 O J- ^ 
 
 being nearly negligible. The driving power P of the modulator can be computed 
 
 from the z given by (3.85). 
 
 \ 
 
 P = \ E 2 b [ JL w (w-a + e',a) ] 2 (3.96) 
 
 2 m., 1 
 
 M.. 
 
 The optical transmittivity T takes the optical losses in material 
 
 n into account, hence 
 
 n i 
 
 T = exp [ - a . -=- L ] (3.97) 
 
 Sl n 2o 
 
 where Q n is the optical absorption coefficient of medium n . 
 sl -L 
 

 51 
 
 Let us consider the optimization of a modulator using the normally 
 
 polarized wave. Substituting M from (3.60) with a = 0, m from (3.55), 
 
 mn n 
 
 P from (3.96) and T from (3.97) we can rewrite (3.95) as 
 
 2 An - 
 
 n J . n 
 
 F = 4 - • — exp (-a — L) (3.98) 
 
 n a s n 
 
 lK V gw(w-a +£ . ia ,] 5 ) 5 
 
 The exponential function is (3.98) can be approximated by the first two 
 terms of its power series expansion, hence 
 
 L(l-a • ^— D / An 2z 
 
 si n. 
 
 1 
 
 x 2 
 
 — r ' v ^' «■-) 
 
 1 o 1 4 m 
 
 alb 2 — w(w-a + 5! a) 
 
 "- o 
 
 Equation (3.99) indicates that the design of optimized with respect to 
 
 the dimensions if a and b are reduced to a minimum consistent with other 
 
 requirements, and L is made equal to L , where 
 
 n 
 = _2zo_ (3.100) 
 
 1 2a n n n 
 si 1 
 
 Assuming that the beam diameter D, is specified, we must meet the conditions 
 
 b 
 
 (the input window must pass the beam'.) 
 
 a > a n = -^ D^ (3.101) 
 
 — 1 n b 
 
 and 
 
 b > D,_ (3.102) 
 
 — b 
 
 In order to avoid diffraction effects, we require that equation (3.78) be 
 
 satisfied, hence 2 
 
 L < L = x- 5 (3.103) 
 
 J2 " A n 2zo 
 
 UNIVERSITY OF 
 ILLINOIS LIBRARV 
 
If the microwave losses are appreciable in material n (and we assume 
 that w and a are such that most of the microwave energy is carried in 
 medium n ),then at some distance L we have 
 
 E mz (L 3 ) = \ E mz (0) = E mZ (0) eX *> <" V 3 ) (3 ' 104) 
 
 hence 
 
 
 where 
 
 ln2 
 L 3 =— (3.105) 
 
 m 
 
 % = — f^\ tan 8 1 (3.106) 
 
 ^•m 
 
 The quantity tan 5 is the loss tangent of material n . 
 
 The optimum modulator dimensions can now be found by choosing a = a , 
 height b from (3.86) consistent with (3.102) and length L from 
 
 L = min (L , L 2 , L 3 ) (3.107) 
 
 The thickness of the electro-optical material n is such that 
 the n n interface does not disturb the optical fields at interface 
 
 £ 1 o 
 
 n l 
 
 n , and that the light rays totally reflected on interface n h 
 
 do not reach interface n, 
 
 1 
 
 n 2 again. 
 
 3.9. The choice of the operating temperature . 
 
 It is well known that the electro-optic and the dielectric constants of 
 the electro-optic materials are temperature dependent. In ferroelectric 
 
 materials (BaTiOg, SrTiO , KDP etc.) both the electro-optic coefficients and 
 
 (23) 
 
 the microwave losses increase sharply near the Curie temperature. 
 
 We would thus like to determine whether there exists a temperature which 
 maximizes the figure of merit F. 
 
Repeating the expression (3.99) for F we have 
 
 53 
 
 r An 
 
 8L(1 - a 
 
 si n 
 
 L) 
 
 F = 
 n 
 
 2zo 
 
 2z 
 
 n„ 
 
 >- 2zo 
 
 tr — w(w-a + £' a) 
 
 ^o 1 
 
 1/4 
 
 m 
 
 1/2 
 
 (3.108) 
 
 Assuming that the properties QI , , £.' , , n, of the isotropic material n do 
 
 si 1 1 1 
 
 not vary rapidly with temperature, we can say that the temperature dependence 
 
 of F is due to the variation of the properties of anisotropic material n , 
 
 namely to the variation of n„ and n 
 
 J 2z 2zo 
 
 It should be noted that n_ , to the extent to which it is determined by 
 
 2zo J 
 
 the electronic polarizability, should be almost independent of temperature. 
 
 Thus, only n remains in (3.108) as a temperature dependent variable. 
 Since the electrically induced birefringence arises fundamentally from a 
 change of polarization, we expect that the temperature coefficient of the linear 
 electro-optic follows that of the dielectric constant . In fact Pockels original 
 formulation of the electro-optic effect was in terms of constants e. . relating 
 
 the constants of the index ellipsoid to the induced polarization 
 relation between Pockels e. . and our r. . is given by 
 
 r . . = 6 ( 61 - 1.) e- . 
 
 (24) 
 
 The 
 
 (3.109) 
 
 Thus for material with high «£. we expect r , to be proportional to £? , 
 and to the extent that e. . is temperature independent, to have the same 
 temperature dependence as £* . 
 
 A similar argument for the quadratic electro-optic effect suggests that 
 0. should be proportional to (g.') 2 . 
 
From (3.108) it is clear that F is maximum if An„ is at its maximum. 
 
 n 2z 
 
 By the above argument the maxima of An and e' occur for the same temperature 
 For ferroelectric materials such as SrTiO , KDP etc, in which g r is dominated 
 by a (T-T ) temperature dependence, the modulation is optimal near the 
 Curie temperature T . It should be stressed again that although the losses 
 increase sharply in medium n at the Curie temperature, this has little 
 effect on driving power requirements, since material n has a very small 
 volume . 
 3.10. Modulating bandwidth limitations. 
 
 Since both the modulating and the optical signals propagate as TEM- 
 like waves, it might be thought at first that the bandwidth of the device 
 would be limited only by dispersion in the material parameters of materials 
 n and n . For some materials this would imply a microwave bandwidth of more 
 than 10 KMC. 
 
 A more careful analysis reveals however that the bandwidth of the device 
 is also limited by: 
 
 1. The microwave impedance of the structure. 
 
 2. The velocity synchronism condition is not satisfied for some 
 higher frequencies. 
 
 3. The field intensity at the n n interfaces is reduced at higher 
 
 J I £ 
 
 frequencies . 
 The first of these factors, the microwave impedance, has a practical 
 significance in that it is difficult to achieve a broadband match between 
 the modulator and the driving transmission line. There are however a number 
 of standard microwave techniques which help to overcome this difficulty. 
 
 Note : not the modulating frequency ' 
 
55 
 
 The second factor, the variation of the microwave phase velocity v 
 
 mx 
 
 with frequency, was discussed in some length in section 3.6. The only way 
 this problem may be eliminated is to choose material n in such a way that 
 
 V€« 
 
 ^1 
 
 n 2o 
 
 (3.110) 
 
 in which case w = a, and the synchronism condition is satisfied for all 
 frequencies. This might not be possible however, for lack of suitable mater- 
 ial n . 
 
 As we have mentioned in section 3.6, for high frequencies the modulating 
 field is essentially contained in medium n , with a cosinusoidal field dis- 
 tribution in the y direction. Fig. 3.8 illustrates this case. 
 
 y- 
 
 4) 
 
 \ \ \ \ \ \ 
 
 n 
 o 
 
 
 n i 
 
 A E 
 
 — rn 
 
 // 
 
 n 2 
 
 n \ 
 o \ 
 
 \ \ \ \ \ 
 
 conductor 
 plates 
 
 Fig. 3.8. Representative microwave field distribution for 
 very high frequencies. 
 
 This difficulty can be overcome at the expense of increased modulating 
 
 power requirement by the addition of layers of n. material between media n. 
 
 1 2 
 
 and n . The resulting sandwich type structure is illustrated in Fig. 3.9. 
 
. interface D 
 
 y. 
 
 & 
 
 n 2 
 V / 
 
 xx_ 
 
 / 
 
 / 
 
 n. 
 
 
 interface B 
 
 Fig. 3.9. Modulator designed for improved high 
 frequency response. 
 
 It is interesting to note that the Kerr-cell type devices have a 
 
 fundamental cutoff frequency due to the variation of the modulating field, 
 
 (hence the phase velocity) across the cross section of the light beam. Only 
 
 approximate velocity matching is possible for these devices. 
 
 On the other hand, our intensity modulator does not have a similar 
 
 cutoff frequency. Since the modulation occurs only at a boundary surface, 
 
 exact velocity matching is possible and theoretically the only frequency 
 
 limit is due to the dispersion of material parameters. 
 
CHAPTER IV 
 BEAM DIRECTION MODULATION BY CONTROLLED INHOMOGENEITY 
 
 4.1. The theory of wave propagation in an inhomogeneous medium . 
 
 It is generally very difficult if not impossible to obtain an exact 
 solution for Maxwell's equations in an inhomogeneous material. In many cases 
 however, an approximate solution is satisfactory. If for instance we can 
 assume that in a region the spatial variation of the fields and the material 
 constants are negligible within an optical wavelength, we can obtain an 
 approximate solution which is analogous to the plane wave in a homogeneous 
 medium. (This approximation is analogous to the WKB-method used in quantum 
 mechanics.) This solution forms the basis of the geometrical optics. The 
 reason why this approximation holds for a great number of problems is that most 
 light beams are much wider than the wavelength, and since the approximation 
 holds for the interior of the beam, only a small part (the periphery) of the 
 beam will be erroneously analyzed. 
 
 In the following analysis we intend to show that the path of a light 
 beam can be determined from the material equations of the medium. 
 
 Consider the general harmonic fields at a point r 
 
 , v - i cot 
 E(r.,t) = E (r) e J 
 
 ~°~ . , (4.1) 
 , x - i cot 
 H(r ,t) = H (r) e J 
 — o — 
 
 in a non-conducting, non-magnetic, isotropic medium. It is well known that 
 
 for a homogeneous medium the complex vectors E and H are given by 
 
 E - E • e J c n( ^ * - } 
 
 ^o -1 e ° (4.2) 
 
 .CO , . 
 
 H = H . e J c n( ^ * Y -> 
 — o —1 
 
 57 
 
58 
 
 where E , H are constant vectors, n = |e' and s is a unit vector specifying 
 
 the direction of propagation. (a • r) = constant represents a plane in which 
 
 the phase of the wave is constant. 
 
 Similarly, it can be shown, that for an inhomogeneous material E and 
 
 H will be solutions of Maxwell ^s equations (within the "geometrical optics" 
 
 approximation) if E and H are of the form 
 — o — o 
 
 E =E l( r) e j c nS( *> 
 
 -o -1 - (4.3) 
 
 „ „ /„x J - nS(r) 
 H = H n (r) e c - 
 — o —1 
 
 where S(r) is the "optical path of the wave" and S(r) = constant represents 
 the geometrical wave-fronts. Substituting (4.3) into Maxwell's equations 
 and neglecting terms in accordance with the "geometrical optics" approximation, 
 we find 
 
 (4.4) 
 
 (grad S(r) x H ) + c€ E =0 
 (grad S(r) x E ) - e|AH =0 
 
 E • grad S(r) =0 
 H • grad S(r) =0 
 
 From these equations we can obtain a condition on S(r) . Using the first 
 two equations of (4.4) we find 
 
 e n 
 grad S x (grad S x E n ) + 
 
 which can be written as 
 
 ^o 
 
 E, = (4.5) 
 
 grad S . (grad S • E ) - (grad S) 2 • E + n 2 E^ = 
 
 (4.6) 
 
 The first, term of this equation vanishes (eq. 4.4) and since (4.6) must 
 hold for all E, 
 
59 
 
 [grad S(r)] 2 = n 2 (r) (4.7) 
 
 This equation is known as the eikonal equation and it is the basic 
 equation of geometrical optics. 
 
 From equation (4.4) we see that both E and H are normal to grad S(r) ; 
 we also know that the direction of the energy propagation, or the direction 
 of the Poynting vector is also normal to both E and H . Hence it follows 
 that the direction of the Poynting vector is the same as that of grad S(r) . 
 Thus we find that the energy propagates along the normal trajectories of 
 S(r) = constant surfaces. The geometrical light rays are then these orthogonal 
 trajectories . 
 
 Let us define the unit vector N along the ray 
 
 dr(s) 
 
 N = — (4.8) 
 
 — ds 
 
 where s is the arc length along the ray and r(s) is the position vector of 
 a point on the ray. 
 
 We have noted that grad S is in the direction of the ray and from the 
 
 eikonal equation (4.7) we also see that 
 write 
 
 grad S = n. Therefore we can 
 
 N = grad S = grad S (4 9) 
 
 — | grad S| n 
 
 hence from (4.8) and (4.9) it follows that 
 
 dr(s) 
 
 ds 
 
 = grad S »r(s) (4.10) 
 
 We now want to arrive at an equation where r(s) is given by n(r) only 
 Let us differentiate (4.10) with respect to ds . 
 
60 
 
 d d - (s) d 
 
 Ts (n -^r } ■ d^ Lgrads(r(s))] (4.11) 
 
 dr 
 = (div grad S) * (ds ) 
 
 . / ■> \ /grad S x (from 4.9) 
 
 = div (grad S) • (- ) 
 
 n 
 
 = — grad [ (grad S) 2 ] 
 
 = — grad [ n 2 ] from (4.7) 
 2n 
 
 hence 
 
 dr(s) 
 
 — (n— - ) = grad n (4.12) 
 
 ds ds 
 
 This is the vector form of the differential equations for the light rays. 
 (4.12) indicates that the rays through the medium can be computed from the 
 knowledge of n(x, y, z) and the initial direction and position of the ray. 
 For a homogeneous medium n = n and thus (4.12) reduces to 
 
 dr(s) 
 
 %- (n '— ) = (4.13) 
 
 ds o ds 
 
 hence , 9 
 d^r 
 
 - — = (4.14) 
 
 ds 
 
 which is satisfied by 
 
 r = as + Jb (4.15) 
 
 where a and b are constant vectors. Thus (4.15) confirms the fact that the 
 light rays are straight lines in a homogeneous medium. 
 
 A second example can be the case when n depends only on the distance R 
 from a fixed point 0, (let R = £J) 
 
 n =n(R) (4.16) 
 
61 
 
 In this case (4.12) becomes 
 
 dr r dn 
 
 — (n — ) = grad n = — — (4.17) 
 
 ds ds R dr 
 
 This equation is satisfied by 
 
 1 x ns = const. (4.18) 
 
 This equation indicates that the path will be in a plane including 
 the origin of the spherically symmetrical system. Another important 
 parameter of the ray, the curvature vector K can also be determined from 
 n(r) . Let us define >£ as the unit normal vector at a point r(s) on the ray 
 and O as the radius of curvature at that point . Then K will be given by 
 
 K = - V (4.19) 
 
 I V 
 
 n > n 
 
 Fig. 4.1. Illustration of the bending of a ray in an inhomogeneous 
 medium . 
 
 K also can be given in terms of r(s) 
 
 dr(s) d 2 r(s) 
 
 K = T- < "3 > = Hg (4-20) 
 
 — ds ds ds 
 
62 
 
 dr 
 
 We should note here that from the definitions of \T and — it 
 
 — ds 
 
 follows that these vectors are normal to each other. 
 
 Rewriting (4.12) into an expanded form yields 
 dr d 2 r 
 
 Hi * di + n dT = grad n < 4 - 21 > 
 
 and multiplying both sides scalarly by K gives 
 
 Id d - 1 d2 - 
 
 ^d^ ( ^-d^ )+( ^ )n dT- = ^ gradn.y (4.22) 
 
 We find that the first term of this equation vanishes; (by the above 
 remark.) Then since the second term of (4.22) equals to n (k I 2 , we have 
 
 n | K | 2 = - grad n • V* (4.23) 
 
 and this can be rewritten as 
 
 K • K = (- grad n) ' (- v) (4.24) 
 
 - - n ^ - 
 
 since from (4.19) \k\ = — , we can write 
 
 K | = | = ^ grad n • Y (4.25) 
 
 hence 
 
 K = (- grad n • ^) ' V (4.26) 
 
 This equation can also be used for the computation of the equation of the 
 
 rays through the medium. 
 
 4.2. Beam deflection by means of an electro-optic device . 
 
 As we have seen in the previous section, the curvature of a light ray 
 is directly related to the variation of the index of refraction in the medium 
 We have also seen in Chapter 2 that the index of refraction in an electro- 
 optic material depends on the imposed electric field. Let us then consider 
 
63 
 the case when a non-uniform electric field is imposed on an electro -optic 
 material and a beam of light is propagated through this medium. Based on 
 our experience with the inhomogeneous , isotropic material treated in the last 
 section, we expect that the beam will not propagate through this medium in a 
 straight line. 
 
 Since the electro-optic material in a non-uniform field behaves as an 
 inhomogeneous, anisotropic medium, the results of the last section do not 
 apply directly. In fact we cannot even define a ray for this case, since 
 a unique trajectory for the propagation of optical energy cannot be found 
 in general . 
 
 If however, the polarization of the input ray is such that only a single 
 wave propagates in the medium, we can use the previously derived results. 
 As we have seen in Chapter 2, there are two major groups of electro-optical 
 materials* in the first the linear, in the second the quadratic electro- 
 optic effect dominates. It will be seen later that it is convenient to treat 
 these groups separately. 
 
 4.2.1. Beam deflection by linear electro-optic effect . 
 
 Let us consider the case when a slab of electro-optic material n is 
 
 placed in a non-uniform electric field. The optical axis of the material n 
 
 is oriented in the direction of the z axis. The light beam propagates 
 
 approximately in the +y direction. Fig. 4.2 illustrates the situation. 
 
 Let us assume that by some means, (later we shall go into more detail), we 
 
 have generated a modulating electric field E in material n n . Let us further 
 
 —ml 
 
 assume that the y component of E , E is equal to zero, and that the other 
 
 ' -m —my 
 
6k 
 
 z 
 
 II 
 
 
 n 
 1 
 
 
 / 
 
 beam 
 
 
 j* 
 
 K (ths 
 
 
 »-x 
 
 y 'Qz? 
 
 
 
 
 *o 
 
 
 
 n 
 
 Fig. 4.2. Light beam incident on an electro optic material, 
 components of E do not depend on y. Also, E is chosen in such a way that 
 in the neighborhood of the (xy) plane we have 
 
 E (x.z) » E (x, z) for z = (4.28) 
 
 mz ". mx ' 
 
 The principal indices of refraction (see Section 2.3) are given by 
 
 n. = n n _ (1 + - n 2 r E ) 
 lx 10 2 10 mz 
 
 n. = n (1 - - n 2 r E ) 
 ly 10 2 10 mz 
 
 (4.29) 
 
 n n = n_ 
 lz le 
 
 where r equals r__ for uniaxial materials (KDP) and r = v. for cubic materials 
 
 bo 41 
 
 (Cu CI) . 
 
 The wave normal is in the y direction approximately and therefore the 
 
 two allowed directions of polarization are x and z. From (4.29) it is 
 
 evident that the optical field E will be influenced by E only if 
 
 — o — m 
 
 E = E • u (4.30) 
 
 — o ox —x 
 
 where u is the unit vector in the x coordinate direction. 
 —x 
 
 Under these conditions, (4 .28) - (4 .30) material n will behave as an 
 "isotropic" (only a single wave propagates!), inhomogeneous material with 
 
65 
 
 index of refraction n' n = n_ (E ), and we are in the position to apply the 
 
 1 lx mz rr J 
 
 results of Section 4.1. From the differential equation for the ray (4.12) 
 and the equation giving n 1 = n , (4.29), we can theoretically compute the 
 ray for any function n (E ). A simple and very useful solution of (4.12) 
 is when the ray describes a circular arc of radius R. In this case the radius 
 of curvature will be constant and Q = R. Equation (4.25) gives in 
 terms of n and v , hence our requirement is 
 
 - = (— r— grad n' ) •: V = const. = - (4.32) 
 
 ? V l X " R 
 
 As we will see later the direction of grad n and V are nearly parallel . 
 
 _3 
 
 Furthermore the changes in n' are expected to be less than 10 * and thus 
 
 we can approximate n' by n . We can write then 
 
 I = I grad n' I ■ — (4.33) 
 
 R I 1 ' n 
 
 o 
 
 The value of grad n' I can be computed from (4.29) 
 
 h [ l 
 
 gradn'^ = - n^ (1 + - n^rE^) 
 1 ox 
 
 31 
 — X 
 
 
 I"Vt! < 4 - 3 «> 
 
 since n' = n n . 
 1 lx 
 
 Equating (4.33) and (4.34) we find that 
 
 1 ! 2 mz (A r>^\ 
 
 — - — n „r — v (4.35) 
 
 R 2 10 x 3 x 
 
 From this equation we can obtain the required E by integrating with 
 
 respect to x 
 
 E = — K — • x (4.36) 
 
 mz Rn^ r 
 
66 
 
 where the integration constant was chosen to be zero in order to reduce the 
 
 field requirement . 
 
 Thus if E is of the form shown in (4.36), the rays close to the (xvl 
 mz i ■" 
 
 plane (z = 0) describe circular arcs. We can compute the angular deflection 
 from Fig. 4.3. 
 
 *■ 
 
 ^T 
 
 Xi 
 
 ~y 
 
 Fig. 4.3. Illustration of a ray bent in a circular arc. 
 
 The calculation of the angle of deflection can be performed as follows 
 
 For a given modulating field distribution, E = Cx, we can compute R from 
 
 mz 
 
 (4.36). Also, from Fig. 4.3 we can write 
 
 hence 
 
 9 s 
 
 = 
 
 2(x. - x ) 
 1 o 
 
 LC(n* )r 
 
 2(x. - x ) 
 1 o 
 
 R 
 
 10' 
 
 (4.36a) 
 
 (4.37) 
 
 We can also compute the maximum possible deflection with a given electro 
 optic material by substituting the linear variation of n into equation (4.33) 
 
 | = grad n'^x) ± 
 
 n'fx.) - n'(x ) 
 11 1 o 
 
 10 
 
 n io (x i 
 
 *o> 
 
 £n'(x. )-An' (x ) 
 11 1 o 
 
 n io (x r x o ) 
 
 (4.37a) 
 
67 
 
 Assume that the device is designed so that n',(x ) = n, _ and n', (x, ) = 
 
 1 o 10 11 
 
 n' (E *,) , where E _ is the break-down field strength of the material. Then 
 1 mB mB 
 
 An'., = n',(x,) - n* (x ) and from (3.46a) and (3.47a) we have 
 1 max 11 1 o 
 
 =i/2 1 max (4.38) 
 
 max U 
 
 1 n io 
 
 Thus we have derived the remarkable result that the maximum achievable 
 deflection depends only on the material parameters: n_ n , r and the break- 
 down strength. E _ . We can show that a suitable electric field can be 
 mB 
 
 generated by the potential field V , if 
 
 V X > y * Z) = - Axz • A = Rn 2 r (4.39) 
 
 Since all second derivatives of V equal to zero, V satisfies Laplace's 
 
 m m 
 
 equation, and thus the field obtained from V will be static field:.. The 
 
 m 
 
 components of E are obtained from 
 — m 
 
 E = - grad V = A(z u + x u ) (4.40) 
 
 — m m —x — z 
 
 The equipotential surfaces, hence the suitable electrode surfaces are 
 
 given by 
 
 - V 
 z = r^- , for all 0< y < L. (4.41) 
 
 A 
 X 
 
 The orthogonal curves to these equipotential surfaces give the direction 
 
 of E , or the field lines of E . The equations of these lines are 
 — m -m 
 
 z 
 
 = / x a - x 2 ' (4.42) 
 
 o 
 
 The constant I E lines will be concentric circles as given by 
 1 — m ' 
 
 IE I 2 IE I 2 R 2 n* r 2 
 2 9 ~ m —ml 10 , N 
 
 x 2 + y 2 = '- ' = (4=43) 
 
 A 2 4 
 
68 
 
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69 
 
 4.2.2. Beam deflection using quadratic electro-optic effect . 
 
 We turn our attention now to the case when material n is replaced by 
 
 a quadratic electro-optic material n (Fig. 4.1). For these materials the 
 
 most significant interaction between E and E occurs when these two vectors 
 ° — m — o 
 
 are parallel. In terms of E and n„ this can be stated as 
 
 -m 2 
 
 n = n_ (E ) 
 
 2x 2x mx 
 
 n = n (E ) 
 
 2z 2z mz 
 
 (4.44) 
 
 As we have done in the previous case, we require that 
 
 E (x, 0) = E ' u (4.45) 
 
 — m mz — z 
 
 Then we choose the polarization of E such that 
 
 — o 
 
 E = E . u (4.46) 
 
 — o oz — z 
 
 Note that the major axis of the index ellipsoid for n is always in 
 
 the direction of the modulating field E . In the above case(z^O) we 
 
 — m 
 
 have assumed that E is in the z direction, hence n„ is the major axis of n„ 
 — m 2z J 2 
 
 ellipsoid. For the above case then the material n behaves as an "isotropic", 
 inhomogeneous medium, and therefore we can use the results of Section 4.1 
 again with n' = n . From Section 2.3 we can obtain the relation for 
 
 n 2z 
 
 n» = n„ ~ n (1 - ^ n 2 o _ E 2 ) (4.47) 
 
 2 2z 2o 2 2o U mz 
 
 In order to deflect the beam in a circular arc again, we have to require 
 that 
 
 - = — I grad n* I = const. = ^ (4.48) 
 
 D I 
 
 O 
 
70 
 
 From (4.47) we can compute grad n', 
 
 ()E, 
 
 grad n' = - - n 3 Q 2E — r 
 
 2 2 2o ^ 1 mz ^ x 
 
 hence 
 
 n 
 
 R 
 
 9 
 
 2E 
 
 <) E 
 
 mz 
 
 2o \ 1 mz b x 
 
 (4.49) 
 
 (4.50) 
 
 Integration with respect to x gives 
 
 n 
 
 — (x - x ) = - — n 3 P , 
 R o 2 2o 1 1 
 
 hence E should be of the form 
 mz 
 
 mz 
 
 = (k?t— ) f^ 7 
 
 mz 
 
 x 
 
 2o 1 
 
 (4.51) 
 
 (4.52) 
 
 The electric modulating field is then 
 
 1 
 
 E = ( 
 
 — m 
 
 Rn 2 
 
 — )2 [ 
 
 >o 1 L 
 
 -z 
 
 r* 
 
 2 x - x 
 
 u + ¥ x - x u 
 — x o — z -> 
 
 (4.53) 
 
 The field E as given by (4.53) satisfies (4.52), curl E = 0, but div E = 
 — m -m — m 
 
 except for z = 0. Thus we may conclude that E is not a static field, and it 
 cannot exist without a charge distribution source. 
 4.3. Beam distortion effects . 
 
 In the previous section we have considered the path of a very narrow 
 beam through the inhomogeneous material. Since in practical experiments the 
 beam diameter is not negligible, we must examine what happens to these wider 
 beams as they pass through the same material. There are also several other 
 factors, which have to be considered in the case of a practical beam deflecting 
 device. Some of the important questions are: 
 
71 
 
 1 . How do the small angular variations in the input beam effect 
 the output beam? 
 
 2. What are the consequencies of the "constant radius of curvature" 
 bending? 
 
 3. What is the influence of the fact that grad n does not have 
 uniform magnitude and direction for the different parts of 
 the beam? 
 
 4. How do unwanted variations of n due to strains and temperature 
 variations effect the output beam? 
 
 These effects will be treated qualitatively in general, but simple 
 quantitative results are given also whenever it is feasible. 
 
 The main effect of the lack of collimation (or small angular derivations 
 
 in the input) is that the width of the beam increases as it passes through 
 
 the device. The diameter D 1 at the output is given by 
 
 D' = D,_ + QL (4.54) 
 
 b b 
 
 where D is the beam diameter at y = 0, a is the angle between the extreme 
 
 rays in the beam and L is the length of the device. If (Qli) is comparable to 
 
 D we can conclude that: 1. the device must be designed larger in order to 
 
 accommodate the larger beam D', and 2. the larger beam (D' ) will be more 
 
 b b 
 
 distorted . 
 
 The effect of non-parallel, but constant radius bending is a minor effect 
 in practical cases. Assuming that two parallel rays enter at y = (input 
 face) D distance apart, and that R is the constant radius of curvature of 
 both rays, then at y = L (output face) the angle between these two deflected 
 rays will be y- 
 
72 
 
 T 
 
 D b . L 
 R 2 
 
 (4.55) 
 
 and the distance between the rays will be D" 
 
 J b 
 
 D " b = D b • cose = \ h - w = \ (4 - 56) 
 
 for L « R (which is the case in practice) . 
 
 -5 
 
 Since normally R » D , / > is a very small angle ( *— 10 ) . The most 
 
 serious of all these factors is that our approximation of treating the 
 
 material as an isotropic" material breaks down for points where z = 0. 
 
 The approximation gets worse as we increase z , hence it is the worst for 
 
 the outer portions of beam. This problem is less important for linear, 
 
 uniaxial materials, (KDP, ADP etc.) than for cubic media. For KDP the 
 
 change in n due to E occurs only normal to the optic axis (z direction*.), 
 
 hence only the z component E of E influences n., . Since E is inde- 
 
 mz — m lx mz 
 
 pendent of z (4.36) all rays will have the same radius of curvature, 
 
 independent of (x , z ) coordinates on the input face ( y = ) . 
 
 For cubic materials the case is much more complicated, since the permitted 
 
 directions of E in the medium depend on the directions of E . As we have 
 — o — m 
 
 seen (4.42) the lines of E describe a hyperbola; the direction of E at 
 
 -m J -m 
 
 any point is given by the slope of this hyperbola, which is given by 
 
 x 
 
 fx 2 - x 2 o ' 
 
 f(x) = , - X (4.57) 
 
 Figure 4.4 illustrates this case. 
 
 The index ellipsoid for material n has its major axis always in the 
 
 direction of E . At point P., there are two directions of permitted optical 
 —ml 
 
 vectors E in the direction of E and E , normal to E . The index of 
 — oa - m — ob — m 
 
 refraction for these waves are written as n and ru respectively. From 
 
73 
 
 V = const., equi potential 
 surface 
 
 , E field lines 
 
 x 
 
 b., 
 
 Fig. 4.4a. Illustration of E field for the linear case 
 
 — m 
 
 b. Components of E vector. 
 
 Section 2.3.2 we find that n is independent of E 
 
 a -in 
 
 but n depends on 
 
 E 
 — m 
 
 Namely 
 
 — m 
 
 (4.58) 
 
 n = n 
 a o 
 
 Since E increases in the radial direction, (Fig. 4.4), grad n 
 -m b 
 
 will be also in this direction (approximately), and the ray going through P 
 
 will be bent in a plane which includes P, and the y axis, and not in a plane 
 
 parallel to the xy plane. We also find that component E is not deflected 
 
 — oa 
 
 at all, since n = n . Thus we expect that portions of the beam which are 
 a o 
 
 further away from the xy plane (z = 0) are considerably distorted. 
 
 Figure 4.5 gives the approximate shape of the deflected beam at the 
 output face of the device, x and x are the positions of the centers of 
 
 the input and output beams respectively. Actually some part of the beam 
 
7^ 
 
 output 
 
 projection of an 
 extreme ray 
 
 x 
 
 Fig. 4.5. Illustration of beam distortion for linear, 
 cubic materials. 
 
 corresponding to E vectors is smeared , between the two positions x and 
 — oa c o 
 
 x . The angle o between the extreme rays on the output beam can be given 
 by 
 
 (4.59) 
 
 D, D^ (x n - x ) 
 
 <^= ^ . e = -* -i 2 
 
 x x L 
 
 o o 
 
 V 
 
 x R 
 o 
 
 From (4.59) we observe that for a desired deflection 9 we can improve the 
 collimation of the output beam by reducing D and increasing x . 
 
 The case of the quadratic cubic materials is even more complicated, 
 since E is a more complicated function of the coordinates x, z. Arguments 
 given for the linear cubic material hold approximately for this case also, 
 thus there is a portion of the beam which is not bent and the other component 
 is bent toward the x direction. The major difference is that for quadratic 
 materials the roles of n and n are interchanged. Component E and n 
 
 cl D Oa Si 
 
 will be influenced (refer to Fig. 4.4/b) by | E I , and E and n will not 
 
 be changed much by E , since we expect O » O 
 
 (10) 
 
 The constant 
 
 curves are also circles, but these are not concentric, as was the 
 
 case for linear media. For z — I E j is given by 
 
 m 
 
E I 2 = A 2 f (x - x) + — 
 
 -ml L o x -: 
 
 which has the normal form 
 
 r \SJ f 
 
 + z 2 = 
 
 E 2 
 
 ^n 
 
 2A* J 
 
 75 
 (4.60) 
 
 (4.61) 
 
 Figure 4.6 illustrates the equal E lines; the gradient of n will 
 be approximately normal to these curves. 
 
 output 
 beam / projection of an 
 extreme ray 
 
 Fig. 4.6. Illustration of beam distortion for quadratic, 
 cubic materials. 
 
 From these arguments we may conclude that the distortion is greater for 
 quadratic materials than for linear media. 
 
 The stress and temperature effects are also important, since the index 
 of refraction depends strongly on the temperature. In all probability, any 
 unwanted, random gradient of n will scatter the beam, and it might obscure 
 the bending due to the electro-optical effect . The stresses might be due to 
 
76 
 
 uneven crystal growth; the temperature gradient can be due to uneven conduction 
 
 of the heat generated by dielectric losses . This factor is very stronly 
 
 influenced by the design and the manufacturing techniques of the device. 
 
 The modulating material must be chosen such that even at the very high 
 
 field strengths required for operation, the temperature of the medium does 
 
 not rise significantly. ( i. e . the conductivity must be negligable. ) 
 
CHAPTER 5 
 EXPERIMENTAL RESULTS 
 
 The main purpose of the experimental part of this work is to show that 
 the modulation effects predicted by theory actually occur in practical devices. 
 Due to this objective and to the fact that our manufacturing and test facilities 
 
 were limited, only three simple devices were constructed and tested experi- 
 
 C 25) — f 97^ 
 
 mentally. The results of these experiments can be called satis- 
 
 factory, since there is a good agreement between theoretical predictions and 
 experimental measurements. 
 
 Nitrobenzene was chosen as the electro-optic material for a number of 
 reasons: it is readily available, and it has the highest electro-optic con- 
 stant among liquids. The use of a liquid electro-optic material also simplified 
 the construction of the devices considerably: it was relatively simple to 
 obtain an optical quality interface between the isotropic material and nitro- 
 benzene. Glass was used as the isotropic medium, since it was available with 
 various indices of refraction n. The modulating fields were generated by gold 
 plated electrodes immersed into the nitrobenzene liquid. (Refer to Fig. 
 5.1). The modulating signal was supplied by a magnetron pulser capable of 
 delivering up to 10 KV pulses of duration 1-10 M-sec, with a repetition rate 
 of 60 pulses per second. 
 
 A He Ne gas laser emitting at \ = 6328 A provided the input light beam 
 and an RCA photomultiplier detected the intensity modulation of the output 
 beam. The positioning of the device was accomplished by the accurate turn- 
 tables and other mechanisms of a surveying instrument . 
 
 77 
 
78 
 
 5.1. Experiments with the light intensity modulators . 
 
 The intensity modulators described in Section 3.4 were tested in the 
 experimental apparatus described above. The Type 1 modulator, studied in 
 detail in Chapter 3, consisted of a denser isotropic and a less dense, electro- 
 optic material, (glass n n =1.65 and nitrobenzene n = 1.55 respectively.) 
 (Refer to Fig. 5.1). The dimensions of the device were L = 70 mm and a = 1 mm. 
 
 The device was operated in the normal modulation (M ) mode, since both the 
 
 nm 
 
 modulating and the optical fields were in the z direction. The other constants 
 
 of the device were measured to be = 70° and m = 28 . The effective spread 
 
 n 
 
 -3 
 
 of the beam was estimated ( 0L- - Cd, ) = 5.10 radians from the properties of 
 
 the output beam. The fact that only 22% intensity modulation was produced 
 by this device was attributed to this large spread of the beam. Figure 5.1 
 shows the construction of this device. 
 
 output 
 beam 
 
 ■ electrode. 
 
 Fig. 5.1. Illustration of Type 1 intensity modulator 
 
79 
 
 Equation (3.73) and families of curves similar to Fig. 3.5 were used for 
 the calculation of the theoretically predicted modulation. From Fig. 5.2 
 we can see that the experimental measurements agree reasonably well with 
 the theoretical results. 
 
 M 
 
 inn 
 
 O.2.. 
 
 0.1- 
 
 / experimental 
 
 6 W 
 
 Fig. 5.2. Comparison of theoretical and experimental results 
 for the Type 1 intensity modulator. 
 
 We should point out here that the beam spread can be reduced to below 
 
 -4 
 
 10 radians by careful manufacture of the device and by the use of a suit- 
 able laser source. The intensity modulation for this case should be much 
 larger (by an order of magnitude) than our observations. 
 
 A second light intensity modulator, the dual of the one described 
 above, was also constructed and tested. In thisfType 2)modulator the electro- 
 optical material nitrobenzene has the higher density, and the light beam 
 travels in this medium. The constants of the device were measured to be 
 
 L = 65 mm, a = 1.1 mm, m = 8, 9 = 78 . A maximum intensity modulation 
 
 n 
 
80 
 
 fi v 
 of 55% (M = 0.55) was observed for a modulating field of E = 9.10 — . 
 mn —mm 
 
 The beam was better collimated in this case and thus the observed modulation 
 effect was also higher. 
 
 The main reason for using pulsed modulating field was to separate the 
 electro-optic modulation effects from slower, but significant temperature 
 gradient effects. The index of refraction of the nitrobenzene depends 
 strongly on the temperature and thus static experiments were: not suitable to 
 show the electro-optical modulation without extensive purification of the 
 nitrobenzene . 
 5 .2 Experiments with the beam bending device . 
 
 It should again be emphasized that the objective of these experiments 
 was to prove the feasibility of these modulation methods, rather than to 
 develop a practical device. A simple beam bending device was constructed 
 and tested using the apparatus described above. (Refer to Fig. 5.3) the non- 
 uniform electric field was obtained by the use of the fringe field of the 
 rectangular electrodes. This field distribution evidently does not satisfy 
 the conditions (grad n = const.) developed in Chapter 4 for constant radius 
 bending, but we expect that the peak grad n was high (since the electric 
 field fell off rapidly in the x direction, (Fig. 5.3).) 
 
 light beam 
 field lines 
 
 electrode 
 
 Fig. 5.3. Illustration of the beam bending device 
 
81 
 We expected (and observed) considerable beam distortion due to non-constant 
 grad n for different portions of the beam. The experiments consisted of 
 sending a narrow light beam close to the edges of the electrodes through the 
 nitrobenzene and observing the distribution of the beam by a small pinhole 
 on the face of photomultiplier mask. As the modulating field was turned on, 
 we found that the beam distribution shifted toward the electrodes . The de- 
 flected distribution (which was considerably distorted) was mapped by the 
 pinhole arrangement for several values of the modulating field. A maximum 
 deflection of 5 mm was observed at a distance of 1500 mm from the output 
 face of the device, which corresponds to an angle of deflection 9 = 0.0033 
 radians. Again, a short (1 Usee) modulation pulse was used in order to 
 reduce the additional distortion caused by the temperature gradients in the 
 nitrobenzene. The length of this device was 70 mm and the separation between 
 the electrodes was 0.5 mm. The maximum modulation occurred for 2 KV, and 
 for higher voltages only the distortion increased. This was expected since 
 for these voltages the deflected portion of the beam travelled outside 
 of the active fringe field region, hence no further deflection was possible. 
 
 A more sophisticated experiment was performed independently by V . J. Fowler, et 
 
 (28) 
 al . ' who used KDP as the electro-optic material and obtained a deflection 
 
 of = 9 or 9 - 0.0026 radian at a modulating voltage of 10.4 KV . 
 
CHAPTER 6 
 CONCLUSIONS 
 
 From the detailed analysis of the modulators and from the corresponding 
 experimental results we may conclude that these modulation schemes are 
 indeed feasible and that practical modulators operating on these principles 
 can be constructed. 
 
 The underlying theory of these modulators has been established and 
 some of the important practical problems have been considered in sufficient 
 detail, so that a realistic design procedure could be presented. 
 
 As we have pointed out earlier, the intensity modulator treated in this 
 
 work compares favorably with the existing light modulators dm some aspects. 
 
 The innovation of this device may be summarized by saying that the modulation 
 
 process has been changed from a volume modulation to a boundary modulation, 
 
 in that instead of changing the properties of a volume of a modulating medium, 
 
 we modulate only the boundary conditions. This results in a reduction of 
 
 the active volume, namely we need to change the properties of a thin layer 
 
 (only several optical wavelength thick) of material only. This fact should 
 
 result in reduced power absorption in the device. Furthermore, whereas 
 
 some properties of the modulating materials cause serious problems to the 
 
 (basically) phase modulator devices, our modulator is relatively insensitive 
 
 to these effects. The random strain which exists in some electro-optic 
 
 materials (CuCl, Sr Ti ) seriously degrades the performance of these 
 
 modulators ' , since the light beam actually travels through these materials. 
 
 On the other hand, the light beam in our device travels in an isotropic 
 
 material only, and since the effects of small variations in n due to the 
 
 random stresses are not cumulative, the performance of the device will not 
 
 be affected seriously. 
 
 82 
 
83 
 
 Finally, let us compare the various available electro-optic materials 
 
 as the modulating materials in our modulators. For M mode of operation 
 
 mn * 
 
 we found in Chapter 3, (3.64), that the modulation is unaffected by the 
 choice of the index of refraction n of the isotropic material. Let us 
 assume that we construct a device with dimensions L = 100 mm, a = 1 mm, 
 b = 1mm and we specify n only to the extent that n > n . Then from 
 equation (3.64) and Table 3.1 we can compute the modulating field intensities 
 
 E* necessary to produce 50% modulation (M^ =0.5) for a well collimated 
 
 * 
 beam.( The modulating voltage V = E m b is also given in Table 6.1. ) 
 
 We can also compute the maximum field intensity E** needed to deflect 
 
 -2 
 
 the beam in the beam bending device by = 10 radian. (Constant radius 
 
 bending is assumed, equation (4.38) is used in computations.) The 
 results of these computations are given in Table 6.1. From this table 
 we can conclude that intensity modulators operating with relatively low 
 voltages (less than 100 volts) can be constructed from available electro- 
 optic materials. The advantage of the quadratic materials for beam bending is 
 also apparent from this table. 
 
Qk 
 
 Intensity Modulation 
 
 CM = 0.5) 
 
 mn 
 
 Beam Bending 
 (9 = 0.01) 
 
 Materials Temperature 
 
 E * 
 m 
 
 V * 
 m 
 
 E ** 
 m 
 
 units 
 
 10 s . - 
 m 
 
 V 
 m 
 
 KDP 
 
 293 
 
 ADP 
 
 293 
 
 Cu CI 
 
 293 
 
 ZnS 
 
 293 
 
 BaTiO 
 
 393 
 
 
 403 
 
 SrTiO 
 
 77 
 
 
 293 
 
 KTiO 
 
 4 
 
 
 77 
 
 nitrobenzene 293 
 
 1.06 
 
 106 
 
 6.85 • 
 
 1.29 
 
 129 
 
 8.3 • 
 
 0.70 
 
 70 
 
 4.5 • 
 
 1.16 
 
 116 
 
 7.4 • 
 
 0.109 
 
 10.9 
 
 8.65 • 
 
 0.222 
 
 22.2 
 
 16.4 . 
 
 0.93 
 
 93 
 
 74 
 
 3.65 
 
 365 
 
 29ft 
 
 0.241 
 
 24 
 
 19.3 • 
 
 1.58 
 
 7 .9.9 
 
 158 
 
 799 
 
 126 
 P180 
 
 10 
 
 10 
 
 10 
 
 10 
 
 10 
 
 4 
 
 10 
 
 10 
 
 t 
 
 10 
 
 10 
 
 io' 
 
 10 
 
 Table 6.1 Comparison of the electro-optic materials. 
 
85 
 
 BIBLIOGRAPHY 
 
 1. E. Bergstrand, Encyclopedia of Physics, edt . S. Flugge (Berlin, Springer) 
 V. 24, (1956) . 
 
 2. A. L. Schawlow and C.H. Townes, Infrared and optical masers, Phys . 
 Rev., Vol. 112, pp. 1940-1949, Dec. 1958. 
 
 3. P. Kaminow, Microwave modulation of light by the electro-optic effect. 
 NEREM Record, V. 3, p. 117, Nov. 1961. 
 
 4. D. F. Holshouser, H. von Foerster and G. L. Clark, Microwave modulation 
 of light using the Kerr effect, J. Opt. Soc. Am. V. 51, p. 1360, Dec. 
 1961. 
 
 5. R. H. Blumenthal, Design of a microwave-frequency light modulator, 
 Proc. IRE, V. 50, p. 452, April 1962. 
 
 6. S. E. Harris, et . al . , Modulation and Direct Demodulation of Coherent 
 and Incoherent Light at a Microwave Frequency. Applied Physics Letters, 
 V. 1, Oct. 1962, p. 37. 
 
 7. Rignod, W. W. and Kaminow, I. P., Wide-Band Microwave Light Modulation, 
 Proc. IEEE 51, Jan. 1963, p. 137. 
 
 8. White, R. M. and Enderby, Electro-Optical Modulators Employing Inter- 
 mittent Interaction, Proc. IEEE, 51, Jan. 63, p. 214. 
 
 9. Peters, C. J., Gigacycle Bandwidth Coherent Light Travelling-Wave 
 Phase Modulator, Proc. IEEE, Vol. 51, Jan. 1963, p. 147. 
 
 10. M. DiDomenico, Jr., and L. K. Anderson, Broadband Electro-Optic 
 Traveling Wave Light Modulators. B.S .T.J . .V.42, pp. 2621-2678, Nov. 
 1963 . 
 
 11. Born, M. and Wolf, E., Principles of Optics, Pergamon Press, New York, 
 1959. 
 
 12. Jenkins, F. A. and White, H. E., Fundamentals of Optics, McGraw-Hill, 
 New York 1957, 3rd. ed . 
 
 13. C. E. Springer, Tensor and vector analysis, The Ronald Press Company, 
 New York, 1962. 
 
 14. Mason, W. P., Optical Properties and Electro-Optic and Photoelastic 
 Effects in Crystals Expressed in Tensor Form, B.S. T.J. 29, April 
 1950, p. 161. 
 
 15. W. P. Mason, Piezoelectric Crystals and Their Application, to Ultrasonics , 
 Van Nostrand Co., New York. 
 
86 
 
 16. Max Born and Ken Thang, Dynamic Theory of Crystal Lattices, Clavendon 
 Press, Oxford, 1954. 
 
 17. R. 0*B. Carpenter, The electro-optic effect in uniaxial crystals of 
 the dihydrogen phosphate type. J. Opt. Soc . Am. Vol. 40, p. 225, April 1950. 
 
 18. I. P. Kaminow, Microwave modulation of the electro-optic effect in 
 KH PO , Phys. Rev. Letters, Vol. 6, p. 528, May 1961. 
 
 19. Kaminow, I. P. and Harding, G. 0., Complex Dielectric Constant of KH PO 
 at 9.2 Gc/sec, Phys. Rev., 129, Feb. 1963, p. 1562. 
 
 20. B. H. Billings, The electro-optical effect in uniaxial crystals at the 
 type XH PO . ( Theoretical) J . Opt. Soc. Am., V. 39, p. 797, Oct. 1949. 
 
 21. Kerr, J., On a New Relation Between Electricity and Light. Philosophical 
 Magazine (London, England) 4th Series, V. 1, p. 337, 1875. 
 
 22. Kaminow, I. P. and Liu, J., Propagation characteristics of Partially 
 Loaded Two-Conductor Transmission Line for Broad Band Light Modulators, 
 Proc. IEEE, 51, Jan. 1963, p. 133. 
 
 23. Landamer, R., Loss Tangent in a Ferroelectric Above an Order Disorder 
 Transition. Bull. Am. Phys. Soc. 8, Jan. 1963, p. 61. 
 
 24. Pockels, F., Lehrbuch der Kristalloptik, B. Teubner, Leipzig, 1906. 
 
 25. Technical Progress Report, Section III, Digital Computer Lab., U. of 
 I., Urbana, Illinois, July, 1963. 
 
 26. Technical Progress Report, Section III, Digital Computer Lab., U. of 
 I., Urbana, Illinois, August, 1968. 
 
 27. Technical Progress Report, Section III, Digital Computer Lab., U. of 
 I., Urbana, Illinois, September, 1963. 
 
 28. V. J. Fowler, et . al . , Electro-Optic Light Beam Deflector, IEEE Proc, 
 V. 52, N. 2, p. 193, February 1964. 
 
 29. G. K. Ujhelyi, and S. T. Ribeiro, An Electro-Optical Light Intensity 
 Modulator, Proc. IEEE, 52, July, 1964. p. 845. 
 
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 13- ABSTRACT 
 
 Since the availability of a coherent, well collimated and powerful light 
 beam, there is a great demand for efficient, broadband light modulators. To 
 the already well developed family of Kerr-cell type devices we propose to add 
 two novel electro-optical devices: a direct intensity modulator and a light 
 beam deflecting device. 
 
 To provide a firm basis for the analysis of these devices, the theoretical 
 background of the electro-optically induced birefringence is examined in detail. 
 Starting . from the anisotropic material equations and Maxwell's differential 
 equations, an analytical derivation of this birefringence is given. Furthermore 
 it is shown how the electro-optical effect influences the material equations, 
 hence its effect on the birefringence is derived. The experimentally determined 
 electro-optic effect is tabulated for the most important materials. 
 
 The new intensity modulator operates on the principle of electro-optically 
 controlled partial reflections. The light beam to be modulated traverses a trans- 
 parent isotropic material of index of refraction n = n-j_ and it is incident on the 
 electro-optic material with n = rv> (n-^ > ru). The reflected amplitude (hence the 
 intensity!) depends on the index of refraction ru, which in turn is controlled by 
 the modulating electric field through the electro-optic effect. A detailed study 
 of the theoretical and practical aspects of this intensity modulator reveals that 
 its performance is in many ways comparable to the most sophisticated of the 
 
 (Continued on following page) 
 
 DD 
 
 FORM 
 
 1 JAN 64 
 
 1473 
 
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 Security Classification 
 
88 
 
 13. ABSTRACT (Continued) 
 
 Kerr- cell type devices. A systematic and quantitative procedure is presented 
 for the optimized design of a broadband intensity modulator. The modulating 
 frequency of the device is fundamentally limited only by the dispersion of 
 material parameters. 
 
 The beam deflecting device uses the fact that a variation of the index of 
 refraction n causes a light beam to change direction of propagation. The 
 desired variation of n (hence the deflection of the beam) can be achieved by 
 placing an electro-optical material in a suitable electric field distribution. 
 This field distribution is determined for the case of minimal beam distortion 
 for linear and quadratic electro-optic materials. 
 
 The experimental results obtained with three devices support the theoretical 
 predictions. Intensity modulation up to 50 per cent and a deflection of 
 .003 radian were observed in the laboratory using nitrobenzene as the electro- 
 optic material. 
 
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 "electro-optical devices" 
 "light intensity modulator" 
 "light beam deflector" 
 
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 6. REPORT DATE: Enter the date of the report as day, 
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 8a. CONTRACT OR GRANT NUMBER: If appropriate, enter 
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 86, 8c, & 8cf. PROJECT NUMBER: Enter the appropriate 
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 9a. ORIGINATOR'S REPORT NUMBER(S): Enter the offi- 
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 96. OTHER REPORT NUMBER(S): If the report has been 
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 10. AVAILABILITY/LIMITATION NOTICES: Enter any lim- 
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 13- ABSTRACT: Enter an abstract giving a brief and factual 
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 14. KEY WORDS: Key words are technically meaningful terms 
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 UNCLASSIFIED 
 
 Security Classification 
 
UNIVERSITY OF ILLINOIS-URBANA 
 510 84 IL6R no C002 no 160 170(1964 
 Report/ 
 
 3 0112 088398166