Optical Interactions in a Dielectric Material with Multiple Perturbations Citation Lee, Hyuk (1986) Optical Interactions in a Dielectric Material with Multiple Perturbations. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/ek2w-4557. https://resolver.caltech.edu/CaltechTHESIS:04152019-101001395 Abstract The interaction of light propagating through a dielectric material with multiple perturbations is investigated. A general coupled mode theory of two gratings is presented. The acousto-electro-optic effect is introduced as an example of an indirect interaction due to the acousto-optic and electro-optic effects. The acousto-electro-optic effect is analyzed using the general theory and is demonstrated experimentally. The application of this effect to light modulation and deflection is discussed in detail. Also a correlator that is based on the photorefractive acousto-electrooptic effect is demonstrated and analyzed theoretically. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: (Applied Physics) Degree Grantor: California Institute of Technology Division: Engineering and Applied Science Major Option: Applied Physics Thesis Availability: Public (worldwide access) Research Advisor(s): Psaltis, Demetri Thesis Committee: Psaltis, Demetri (chair) Papas, Charles Herach Rutledge, David B. Yariv, Amnon Bellan, Paul Murray Defense Date: 9 December 1985 Record Number: CaltechTHESIS:04152019-101001395 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:04152019-101001395 DOI: 10.7907/ek2w-4557 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 11461 Collection: CaltechTHESIS Deposited By: Mel Ray Deposited On: 15 Apr 2019 18:06 Last Modified: 20 Nov 2025 22:45 Thesis Files Preview PDF - Final Version See Usage Policy. 23MB Repository Staff Only: item control page

OPTICAL INTERACTIONS IN A DIELECTRIC MATERIAL WITH MULTIPLE PERTURBATIONS Thesis by Byuk Lee In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy Califonia Institute of Technology Pasadena, California 1986 (Submitted December 9, 1985) ii ACKNOWLEDGEMENTS First and foremost it is a pleasure to acknowledge the guidance, and supervision of my advisor, Professor Demetri Psaltis. I would like to thank Dr. Gabriel Sirat for helpful discussions in many areas. Thanks are also due to my colleagues Dr.Eung Gi Paek, Kelvin Wagner, Mike Haney and John Hong with whom I discussed many subjects during my stay at Caltech. Constant encouragement has come from my family. iii ABSTRACT The interaction of light propagating through a dielectric material with multiple perturbations is investigated. A general coupled mode theory of two gratings is presented. The acousto-electro-optic effect is introduced as an example of an indirect interaction due to the acousto-optic and electro-optic effects. The acousto-electro-optic analyzed using the general theory and is demonstrated experimentally. The application of this effect is effect to light modulation and deflection is discussed in detail. Also a correlator that is based on the photorefractive acousto-electrooptic effect is demonstrated and analyzed theoretically. iv TABLE OF CONTENTS ACKNOWLEDGEMENTS ii ABSTRACT iii !.OVERVIEW 1 2. MULTIPLE PERTURBATIONS IN OPTICAL MATERIALS 2.1. Perturbations of optical properties in materials 6 2.2. Significance of multiple perturbations in 12 optical materials 3. COUPLED MODE THEORY 3.1. Electric field coupled mode theory 16 3.2. Polarization of optical materials 17 3.3. Optical and acoustic eigenmodes of the unperturbed medium 23 3.3.1. Optical eigenmodes 23 3.3.2. Acoustic eigenmodes 27 3.4. Coupled mode equation of two gratings 3.5. Bandwidth of an acousto-optic device with a finite size transducer 4. 29 32 HOMOGENEOUS ACOUSTO-ELECTRO-OPTIC EFFECT 4.1. Acousto-electro-optic(AEO) interaction 40 4.2. Coupled mode analysis of the homogeneous AEO interaction 44 4.3. Experimental verification of the homogeneous AEO interaction 53 v 4.4. AEO modulator 65 4.5. AEO deflector 74 4.6. Novel way of measuring the acoustic transducer bandwidth 83 5. PBOTOREFRACTIVE AEO INTERACTION 5.1. Spatial AEO interaction 85 5.2. Three-mode photorefractive AEO interaction 86 5.3. Experiment on photorefractive AEO interaction 5.4. Correlator using the intermodulation mode 92 99 6. FUTURE RESEARCH 106 REFERENCES 108 1. OVERVIEW There are many different ways of perturbing the optical properties of materials. launch acoustic waves to For example, we may utilize the acousto-optic effect, or apply electric fields to change the index of refraction through the electro-optic effect. we may also consider the natural birefringence and optical activity inherent in some crystals as perturbations of the dielectric tensor. In general, the macroscopic polarization of a material can be decomposed into components that are induced by different perturbations caused by external or internal agents. If two or more different sources of dielectric perturbation are present at the same time. the interaction between the incoming optical wave and the material is very nonlinear in the sense that the resulting field is not equal to the sum of the fields that would result from the individual perturbations. This introduces additional flexibility in controlling optical waves. A general theory is needed to explore the full potentiality of optical interactions in materials with multiple perturbations. Coupled mode analysis is a powerful theoretical tool with which such higher interactions may be analyzed. we present a general 2 coupled mode theory for the interaction between multiple perturbations and optical waves. The theory is then used to study a phenomenon we refer to as the Acousto- Electro-Optic (AEO) interaction in crystals. While the interactions known as the Acousto-Optic (AO) and the Electro-Optic (EO) effects have both been studied and used extensively. basic limitations in the performance of utilizing exist separately. the devices such effects When both AO and EO perturbations are applied simultaneously in a crystal. the combined effect (AEO) may be used to overcome some of the limitations inherent in the individual AO and EO devices. As a first step. we studied the AEO effect with a spatially homogeneous electric field and a frequency acoustic wave. single The general theory was used to calculate the diffracted light intensity of the constant electric field AEO interaction as a function of the applied voltage. To verify the theory experimentally. an acousto-optic device with electrodes to apply the voltage was designed and fabricated. The experimental results verified the predicted dependence of the diffracted intensity on the applied voltage. Based on the theoretical and experimental investigations of the AEO effect. a new one-dimensional spatial light modulator was devised. A new optical 3 deflector based on the AEO effect, deflector, was also devised. called the AEO This deflector can be described as a conventional AO deflector with the phasemismatch compensated by the EO effect, resulting in an increase in the number of resolvable spots. The capabilities of the AEO interaction can be exploited even further applying a spatially if a convenient method of varying electric field is available. An electric field grating can be created through yet another nonlinear interaction called the photorefractive effect, which is presently being investigated for applications in real -time holography. While the dimensional, and acoustic grating is dynamic the photorefractive grating two-dimensional. This effect and one- is stationary was studied theoretically and experimentally verified, demonstrating the applicability of the general theory to the analysis of complicated, intermodu 1 at ion multiple perturbation interactions. The term that arises from the combined effect produces an output optical wave whose amplitude is the amplitude product of the two gratings. product was used to construct a This space- integrating correlator. There are significant advantages to be gained by using multiple perturbation interactions in materials 4 for optical signal processing. In one crystal. many different input signals corresponding to the physical sources of the perturbations may be present. and the appropriate optical interaction may perform the desired computation on the inputs. Such advantages are clearly apparent in the AEO interaction. general types of interactions. material perturbations. The study of more those involving many should be pursued further. Another important subject for research is the study of the physical mechanisms of the interaction of optical waves and material perturbations. While effects such as the AEO effect arise from the interaction of optical waves with the sum of the material perturbations. direct coupling between the perturbations themselves may give rise to new effects arising from the interaction of optical waves with the product of the perturbations. The study of such new effects. interesting in its own right, may culminate in the invention of new devices for contribution in the field of optical signal processing. In Chapter 2, we describe the general concept of the multiple perturbations of optical materials and the significance of the optical interaction of the perturbations applied to multiple optical signal processing and devices. The general coupled mode theory of two gratings is presented in Chapter 3. We will choose the simplest 5 example of a indirect interaction. the AEO interaction, and show how we can use this interaction to improve the functions of optical devices and to devise new devices in Chapters 4 and s. These examples will demonstrate the potentiality of the general concept developed in Chapter 2. A note on notation In this thesis the Einstein summation notation is used except when otherwise specified. Conventional arrow notation for a vector is used. Also component notation such as xi is used for a vector x. 6 2. MULTIPLE PERTURBATIONS IN OPTICAL MATERIALS 2.1. Perturbations of optical properties in materials The fundamental equations which govern optical interaction with matter are Maxwell's equations and the material equations. In Maxwell's equations. optical waves are represented by electric and magnetic fields. The basic variables of the material are position vectors of molecules in classical physics. or wave functions of the material system in quantum physics. there is no interaction between If optical waves and matter. the fundamental equations consist of the free Maxwell and material equations. This means that there is no cross coupling between the two equations and we can solve them independently to get free optical waves and material system states. On the other hand. interactions. Maxwell's equations and if there are the material equations are coupled. and the two sets of variables which describe optical waves and materials appear in both equations. The standard method of solving the coupled Maxwell and material equation is the so-called self-consistent analysis 1 • In the self-consistent method we first obtain the response of the material to given electromagnetic fields. This response gives a constitutive relation for 7 the source term in Maxwell's equation, and then the wave equation is used to analyze the interaction of light with matter. In the preceding discussion, we considered the material equation in general, but we did not specify external physical agents which can change the material states. As we have seen before, the material equation is coupled with Maxwell's equations. Therefore, if we change the material properties by external agents. the optical properties of the material change accordingly. This means that the constitutive relations are functions of all possible physical agents which affect states of the material as well as electric and magnetic fields. There are many different types of physical agents we can apply. Among these the acoustic displacement of molecules in the material is well known and important. The variable that characterizes the acoustic wave is the strain which measures displacements of their equilibrium positions. molecules from Also the electric and magnetic fields are commonly used external physical agents. The properties magnitude of the change of discussed above is usually the small,implying that the terms in the power series expansion constitutive relation can be optical of the considered as 8 perturbations. that change Then we may call all physical agents optical properties of the material as perturbations. If there are many physical agents which simultaneously affect the optical properties of the material, we refer to these as multiple perturbations. There are several types of interactions of light in a medium with multiple perturbations which can be categorized as follows. First we consider the direct interaction of multiple perturbations. interaction can be characterized by susceptibility that lower-order cannot interactions. an Direct irreducible be reduced to a product of The effect of a direct interaction becomes smaller as more perturbations are considered, and there is only one overall phase matching condition. we take acoustically induced optical harmonic generation (AIOHG> 2 as an example to illustrate the various types of interaction. In Fig.2.1.l(A) the wave vector diagram of the direct interaction of AIOHG is drawn. to SE 2 , In this case the polarization is proportional where s is the strain and E is the optical electric field. In contrast to the direct interaction, an induced interaction of multiple perturbations can be reduced to a composite of lower-order interactions. In this case we can write the induced interaction as a susceptibility product of of the lower-order 9 interaction susceptibilities; however, the polarization has the same form as that of the direct interaction. AIOHG can happen via not only induced interaction but also direct interaction. In Fig. 2.1.1(B),(C) two types of the induced AIOHG are drawn. We have the same input and output Fig.2.1.1(A), different. acoustic but The and optical waves as in is the interaction mechanism third interaction of perturbations we refer to as the multiple indirect interaction. An indirect interaction can be expressed as a sum of direct or induced interact ions. In contrast to the indirect interaction, the direct or induced interactions can be attributed to one term in the power series expansion of the constitutive relation. As an example we draw the interaction diagram for the corresponding AIOHG to effect, the indirect which represented by the polarization SE + E 2 in 2.1.1(D). The intermodulation of the is Fig. indirect AIOHG effect may be represented by SE 2 ; however, to obtain large effect we need phasematching conditions separately for both interactions represented by the polarizations SE and E 2 . In this way the indirect interaction may be much stronger than the direct interaction. thesis we focus on indirect interactions. In this 10 ( A) /j I\\+ I\= 1',,_ ( c) CD) FIG 2.1.1 11 Fig 2.1.1 : wave vector diagrams of examples of various optical interactions. (a) Direct acoustically induced second harmonic generation( AIOHG). (b),(c) Induced interaction of AIOHG. (d) Indirect interaction of AIOHG. diagram implies phase matchings. ---. Optical wave. ----~ Acoustic wave. Two dots on the 12 2.2. Significance of multiple perturbations in optical materials In Section 2.1, we introduced the concept of multiple perturbations and the interactions they induce inside optical materials. In this section, we discuss the significance between the interaction of perturbations and multiple optical signal processing and optical devices. Optical signal processing devices are based on systems intentional optical waves which carry and optical manipulations of information to be processed. These manipulations are accomplished by selective physical interactions of optical waves and externally controlled physical agents(multiple perturbations) which contain information. From this consideration, we see the clear relation of interactions of multiple perturbations with the optical signal processing and optical devices. We need as many input signals as possible, which can be thought of multiple perturbations in the material, and we select the appropriate material and interaction to obtain the desired output result for the specific optical purpose of device. investigation of and This optical is signal processing or the main motive of our interactions of multiple perturbations optical waves. 13 The concept is illustrated in Fig.2.2.1. General input signals of the optical signal processing system or device may be an acoustic wave, electric fields provided by electrodes, magnetic fields or microwave, optical waves. We call these input signals as perturbations that multiple change the optical properties of the material. If we choose the correct material, orientation of the crystal cut and coupling geometry of multiple perturbations, the optical waves interact with multiple perturbations. It may be direct. indirect, induced or other types of interactions. Thus. the output optical waves or other types of perturbations are desired results of the optical signal processing and optical device. Ofcourse, we may need many different types of transducers to convert the information (for example. electrical) to perturbations of the material. The indirect AEO effect 3 is very interesting because of its flexibility and this is used to make new devices in Chapters 4 and 5. 14 4~ +I I I + . ~ ---- .--; N C'J 1+H ~ 15 Fig 2.2.1: Illustration of the concept of the general optical interaction of multiple perturbations in dielectric materials. ---+:Optical wave. ~: Acoustic wave. ----•: Electric field. - ·-·+ : Other external physical agents. 16 3. COUPLED MODE THEORY 3.1. Electric field coupled mode theory Coupled mode theory is a well-known method which has been used in solving problems of nonlinear optics. acousto-optics and other differential equations with perturbations. In this thesis we use coupled mode analysis to characterize interactions due to multiple perturbations. In this section we coupled theories for mode interaction. compare the various acousto-optic Acousto-optic interacton is an interaction between acoustic and optical waves. and it will be explained in detail in Section 4.1. There have been many papers analyzing this effect. In the review paper by I.C.Chang 4 in 1976, he used a scalar coupled mode equation of electric field amplitudes. al. 5 used J.M.Rouaven et correct electric displacement eigenmodes to analyze the acousto-optic interaction; however, they did not include the expression for the final diffracted light intensity. since it is not convenient to get this intensity from the electric displacement vectors. In this thesis we use electric field eigenmode expansions which are not an orthonormal set in an anisotropic medium but can be decoupled, as we will see. The use of electric field is preferable because it is the physical 17 quantity that is of importance and not the electric displacement vector. Also the simplest expression for the intensity of the diffracted light is given when we use the electric field. In this chapter we develop this electric field coupled-mode theory for the simple case of two gratings, which is the simplest example of multiple interactions and is also useful in the analysis of the AEO interaction. 3.2. Polarization of optical materials Optical materials respond to perturbations in various ways. applied external In general, these interactions can be categorized as linear or nonlinear responses. The interaction is defined to be linear if the induced macroscopic polarization is proportional to the optical wave. Examples are natural birefringence, optical activity and also the acousto-optic and electrooptic effects. One way of expressing the response of the material is to use macroscopic polarization. we can expand the macroscopic polarization of the optical material into products of external multiple perturbations. Also the response expressed as is a change in the relative inverse dielectric tensor. In this section we use both 18 definitions. Thus, we need the relation which connects these two definitions. The definition of the change of polarization and relative dielectric tensor is given by: (3.2.1) where a 0 is the dielectric constant in vacuum and is the relative dielectric tensor. is small, to first order, (~a)·· 1] If the perturbation a change in the relative inverse dielectric tensor is related to a change in the relative dielectric tensor by (3.2.2) where aik is the relative dielectric tensor for the unperturbed medium. If we use (3.2.1) and (3.2.2), we have : (3.2.3) Natural interaction. birefringence is a simple linear It is very important because of its critical contribution to other interactions as will be seen in later sections. For the case of monochromatic light, we can define the overall relative dielectric tensor of natural birefringence, which is the 19 characteristic constant of the unperturbed medium, as : 8 .• lJ = (3.2.4} where & lJ • • is the Kroneker delta and k · · is the natural lJ relative dielectric susceptibility. Another linear response of the material that is important for our work is the linear electro-optic effect. This interaction comes from the quadratic term of the expansion of the polarization into electric fields. The linear electro-optic effect is defined as the change of the inverse relative dielectric tensor : (A e -1 } .. lJ __ r E (ext} ijk k (3.2.5} where rijk is the linear electro-optic coefficient and Ek(ext} is the applied external electric field which may be constant,or temporally or spatially varying. We treat different cases of linear electro-optic effects in later sections. The acousto-optic interaction may be explained intuitively as follows. If we launch an acoustic wave inside the crystal, we create a density change, and this alters the optical property, ( i.e., polarization > locally. We know that given a grating we can diffract the light; thus, the acousto-optic interaction couples the density grating of the crystal and the optical wave. 20 The definition of this interaction is given by (3.2.6} where Pijkl is the fourth rank elasto-optic tensor and Skl is the strain induced by the acoustic field inside the crystal. The strain is defined as : where u 1 is the displacement vector field of material points of the crystal, and Dk is a partial differential operator with respect to xk. As we see in (3.2.6), Pijkl is dimensionless and has the following symmetry property Pijkl = Pjikl = Pijlk• (3.2.8) This is the direct acousto-optic interaction. For a piezo-electric crystal, i.e., a crystal that changes polarization when a strain is applied, the combined piezo-electric and linear electro-optic interactions give an indirect acousto-optic effect. This interaction is where si is the ith component of a unit vector in the direction of light propagation, bijm is the optical 21 mixing susceptibility and enkl is the piezo-electric stress tensor. The direct and indirect acousto-optic interactions described above depend only ori the strain. Nelson and Lazay 6 showed that actually the acoustooptic interaction depends also on the rotation. This effect is generally very small but it can, under proper conditions, be large. If we include this interaction, the complete expression of the acousto-optic effect is given by APi = -eo[ 8 im 8 jnPrnnkl + (ei[k&llj + 8 j[k&lli) + (2bijmsmsnenkl) I <spepqsq> ]EjDl (uk). (3.2.10) Another effect that is important to our work is optical activity . This is an intrinsic property of the crystal, and it is difficult to change this property by applying external perturbations. However, recently liquid crystals or organic investigated to get coefficients. In this case we can change the optical activity externally, materials are being large nonlinear in which case it can coupling be treated using coupled mode analysis in a manner analogous to the way in which acoustic or electric field gratings are treated mathematically. Optical activity is written as a perturbation of the dielectric a·· ~J = &·~J· tensor as follows : + 4nk·~J· +]'a·~J·1g1 ms m• (3.2.11) 22 where j = (-1) 1 ' 2 • eijl is the complete antisymmetric tensor. glm is the gyration tensor and sm is the mth component of the unit vector in the direction of the optical wave. Summarizing the above discussion. we use (3.2.11) as the relative dielectric tensor for the unperturbed medium with which we derive plane wave eigenmodes to be used in expanding the solution of the perturbed interaction. Next. the perturbation in the polarization introduced by simultaneous application of multiple perturbations is given by (3.2.12) where b represents all the perturbations. This is the fundamental relation for our work and we use this equation in Chapters 4 and s to treat the acousto- electro-optic effect. In this section we focused primarily on acoustooptic and linear electro-optic interactions. are many higher-order interesting to But there nonlinearities which are very future research work. nonlinearities represented by SE 2 For example. or s 2 E terms in the expansion of the polarization can be very interesting. These nonlinearities are especially important in the case of surface acoustic waves because of the high 23 acoustic energy density. 3.3. Optical and acoustic eigenmodes of the unperturbed medium In Section 3.2, we discussed various changes of the crystal. Usually the effects of these interactions is very small, perturbations. polarization and can be treated as This observation allows us to use in general a perturbative expansion of the solution describing this interaction. In the perturbative expansion, we need a complete set of eigenmodes of the unperturbed medium as a zero-order solution. section wefirst characterize optical In this eigenmodes and secondly, acoustic eigenmodes, which will be useful for the analysis of the acousto-electro-optic interaction. 3.3.1. Optical eigenmodes Optical wave propagation can be explained using Max we 1 1 's e quat ions. Max we 1 1 's e quat ions f o r ac h a r g efree nonmagnetic material in the MKSA unit system is given by : Di v (y) =0 Div(B)=O (3.3.1.1) (3.3.1.2) 24 (3.3.1.3) where a 0 is the dielectric constant in vacuum,e is the relative dielectric tensor which has been discussed in Section 3.2, and Y is the displacement vector. ~le assume that e is a hermitian tensor since we will treat nonabsorbing materials. In this case the free charge density is zero, should and the relative dielectric tensor be hermitian. tensor, then If e is a constant Hermitian we obtain a set of monochromatic waves wave equation. direction. where p infinite plane as solutions of the free Maxwell Let the plane wave propagate in the s We can write this wave as : = 1,2 is an index used to distinguish the two different transverse polarizations, w is the optical frequency, cis the speed of light in vacuum, n<P> are indices of refraction and e 1 <p> dimensionless vectors eigenvectors. From in and d 1 (p) the direction are unit of the (3.3.1.2) and (3.3.1.3), we obtain the well-known Maxwell wave equation 25 To get the relations between e 1 <p> substitute (3.3.1.4) into (3.3.1.6). and n<P>, we We wind up with the following equation (3.3.1.7) From (3.3.1.7) the indices of refraction n<P> can be found from the condition that a nontrivial solution of the homogeneous equations exists. Then for each value of the index of refraction, the eigenvectors em(p) satisfy (3.3.1.7). Until now we have used electric field vectors to get electric field eigenmodes. It is well known that in an anisotropic medium the electric field eigenvectors are not orthogonal to each other. If we use electric displacement vectors, the eigenvectors are indeed orthogonal. generally. So we use two sets of eigenmodes Maxwell's wave equation for the displacement vectors is given by 0 k0 k[(&-l)lmYml - Dk(Dl[(&-l)kmYm) = (1/c2)Dt2(yl). (3.3.1.8) If we substitute (3.3.1.5) into = -1 -1 1m - <£ > kmsk s 1 l dm (n<P>)- 2d 1 <p> [ ( £) (3.3.1.8), we get (p) (3.3.1.9) We are now in a position to derive various conditions 26 for the eigenvectors e 1 (p) and d 1 <P>. First we defined e 1 <p> and d 1 (p) to be unit vectors. This gives : (3.3.1.10) Next, if we (3.3.1.5), d 1 (p) s use the first part of (3.3.1.1) and we get the transversality condition of d 1 (p): 1 = 0 p = 1,2. (3.3.1.11) The orthogonality relation between d 1 <1 > and d 1 <2 > comes from (3.3.1.9), using the relation (3.3.1.11) and Hermiticity of the tensor elm (3.3.1.12) If we apply the same method used above and use the orthogonality relation between d 1 <1 > and d 1 <2 >, we get the following relation : (3.3.1.13) It may now be apparent why we eigenvectors. use two sets of We like electric field eigenmodes but we use electric displacement eigenmodes to decouple the polarizations. (3.3.1.12) and (3.3.1.4), (3.3.1.5) mode equation. Relations (3.3.1.13) will be with (3.3.1.10), (3.3.1.11), eigenmode equations used to derive the coupled 27 The above method of getting eigenmodes is very general because it applies to any kind of constant Hermitian dielectric tensor. and this sort of generality is important. when we use a computer to design actual devices. Homogeneous perturbations can be thought as part of the unperturbed dielectic tensor. On the other hand. we can also think of the constant change of the dielectic tensor as a small perturbation. This gives rise to combined interactions perturbations and other effects. discussed in Chapters 4 and of homogeneous This point will be s. 3.3.2. Acoustic eigenmodes The response of a material to applied strain obeys Hooke's law. Hooke's law gives a relation between stress and strain : T·. (3.3.2.1) ~J where T ~J · · is the stress and cijkl stiffness tensor. is the elastic The equation of motion for the displacement vectors of material points ui is given by : where p is the mass density of the material. assume a plane acoustic wave with a frequency Let us a and 28 propagating in the direction Ni a=1,2,3 • (3.3 . 2.3) where u.<a> are constant. l. If we substitute (3.3.2.3) into (3.3.2.2), we get a system of linear equations which give phase velocities Va and eigenvectors ui <a> : (3.3.2 . 4) From (3.3.2.4), we see that there are three different eigenmodes for a given direction Ni, and in general these eigenmodes are mixed. i.e., not pure transverse or longitudinal. Many crystals we are using are piezoelectric. This piezoelectricity changes the elastic stiffness tensor because it generates stress induced by the electric field. In this case the effective stress tensor is given by T·. l.J (3.3.2.5) 29 3.4. Coupled mode equation of two gratings The simplest case of multiple gratings is two gratings. In this section, we introduce the coupled mode equation of two gratings and discuss approximations to be used for the calculation of the AEO effect. Also the interaction of an incident beam with two gratings is of practical importance in devices such as an AO modulator. The wave equation that governs the interaction of the optical wave with multiple perturbations is : (3.4.1) The total relative dielectric tensor is taken to be the sum of two perturbations : e .. ~J = s· .<o> +As· .< 1 > +As· .< 2 >. ~J In (3.4.2), ~J ~J (3.4.2) two perturbations may be purely acoustic or acousto-electro-optic, i.e., acoustic and electro-optic gratings. usually we assume plane monochromatic gratings for As·· <1 > and As .. <2 > s·. ~J = ~J ~J 1/2e· .<o> + 1/2a· ,(l>exp[j(O(l)t- K 1 (1>x )] ~J ~J 1 + 1/2aij< 2 >exp[j(o< 2 >t- K 1 <2 >x 1 >l + c.c., (3.4.3) where oO> and o<Z> are temporal frequencies of the two gratings, K1 (l) and K1 <2 > are the corresponding wave 30 vectors. Next we assume a plane monochromatic optical wave with unit amplitude incident on the crystal : E·(0,0,1) l. e i <0 ' 0 ' 1 >exp [ j { ( ( wn <0 ' 0 ' 1 >I c> s 1 x 1 )- w t} J. = (3.4.4) where 1 in (0,0,1} represents the polarization of the input optical wave. We define various quantities which represent higher order modes : (3.4.5) (3.4.6) (3.4.7) k·(I,J,p) = w<LJ)n(I.J,p}s.<LJ);c (3.4.8} u<LJ} (3.4.9) l. l. = x·s·(LJ>, l. l. where I,J are integers. The trial solution for the electric field for the two gratings case is given by : Ei(x,t}= [p(LJ,p)(u<LJ))(n(I.J,p))-l/ 2 Jei<LJ,p} x exp[j <k Here we used p(I,J,p} 1 (I.J,p>x normalized which give 1 slowly simple diffracted 1 ight intensities. - w<LJ>t)]. varying amplitudes expressions Next the (3.4.10} for the phase mismatch 31 vectors for two gratings are Ak 1 (I.Jip) -- k 1 (I.Jip) - X1 (I,J) • (3.4.11) a we now use the following adiabatic condition (3.4.12) Next we substitute the trial solution (3.4.10) into the wave equation (3.4.1) and use the properties of the eigenmodes described in section 3.3 and approximations (3.4.12). Then we get the coupled mode equation for two gratings d<LJ>p<LJ~q> =K+(I~J~q~p~1)F(I+ 1 ~J~p)exp[j(Ak(I+ 11 J 1 p)_Ak(I~J~q))r] +K_(I~J~q~p~l)F(I- 1 ~J~p)exp[j(ll.k(I- 1 •J~p)_Ak(I~J~q>)r] +K+ (I J q p 2 ) F'( 1 J + 1 ~ P >ex p [ j (A k <I J + 1 ~ P >-A.k <I J q >) r 1 1 1 1 1 I 1 I I +K_(I~J~q~p~2)F(I~J- 1 ~P>exp[j(ll.k(I~J- 1 ~P)-Ak(I 1 J 1 q))r]~ (3.4.13) where r=sixi. Temporal frequencies of acoustic waves or eletric signals are small compared with that of the optical wave. This allows us to use w instead of w<I.J) in (3.4.13). Of course. practical ranges of I~J are small. Next constant perturbations which give rise to the 32 anisotropy of the material are very small. And the difference between electric field eigenvectors e 1 (I,J,p) and electric displacement eigenvectors d 1 (I,J,p) are linear in perturbation. So we can use the magnitude of the e 1 (I,J,p) at any place. If we assume that the magnitude of the wave vectors of the gratings is small we can use u<O,O) instead of u<LJ>. One more approximation is applied to the approximate formula of the light intensity of the plane optical wave. The intensity is taken as the modulus square of the electric field. Given the initial conditions, we can easily solve the above coupled equation. The light intensity for each mode of order (I,J,p) is given by : (3.4.14) 3.5. Bandwidth of an acousto-optic device with a finite size transducer In this section we apply the coupled mode equations derived in Section 3.4 to the problem of calculating the bandwidth of the acousto-optic device. The input electrical signal fed into the AO device consists of a range of frequencies. The center frequency of the signal is chosen for the specific AO device, and the angle of 33 the illuminating beam is set to give zero phase mismatch for the center frequency. Thus.there are many different acoustic gratings with different wave numbers inside the crystal of the AO device. We can observe harmonic or intermodulation modes for all acoustic gratings. Also. the finite size of the transducer or crystal gives rise to angular spectral components of the acoustic wave propagating inside the crystal. Then we can expand the acoustic wave into a sum of acoustic angular eigenmodes. One of the fundamental parameters of the AO device is the bandwidth. Due to the phasemismatch introduced by the deviation from the center frequency. of the output diffracted optical wave the amplitude becomes smaller as the frequency goes away from the center frequency. This determines the useful frequency range or the bandwidth of the device. As an example. the bandwidth defined above can be calculated as in Fig.3.5.1(a) for isotropic diffraction. Here we neglect harmonic or intermodulation modes. In Fig.3.5.1(a). the frequency deviation from the center frequency is introduced by AK and the phasemismatch is called Ak. From the diagram in Fig.3.5.1(a). we have Ak =k- {k 2 + [2ksin(~/2) - AK1 2 - 2k[2ksin(~/2) - AK]cos(n/2- J3/2)} 1 12 • (3.5.1) 34 (a) (b) FIG ).5.1 35 Fig 3.5.1 : wave vector diagram used in deriving formulae for the bandwidth. (a) Derivaton based on the phasemismatch. (b)Derivation based transducer. on the finite size of the 36 where k = 2nn/').. : optical wave length in a crystal, n: index of refraction of the crystal, ~ : Bragg angle for the center frequency. We assume AK I k << 1. This is true because the wave length of the acoustic wave is much larger than that of the optical wave. Then the phasemismatch in (3.5.1) is approximated as : Ak .., AKsin(~/2). (3.5.2) We also have a relation between the frequency deviation AO and AK : AK (3.5.3) AO/Va, = where Va is the acoustic velocity. The well-known formula for the AO diffracted light intensity is given by : (3.5.4) where ll is the diffraction efficiency for the center frequency, sinc(x) = L is (sin(x))/x. the interaction length and From (3.5.4), the first zero of the diffracted light intensity is given by : AkL/2 = n. Using (3.5.2), the bandwidth is given by (3.5.5) 37 AK = 2n/[Lsin(~/2)]. (3 . 5.6) Another way of defining the bandwidth that is related to the subject matter of this thesis is shown in Fig.3.5.1(b). Due to the finite size of the transducer or the crystal for a given acoustic frequency, we have acoustic waves over an angular spectrum. Then. for a frequency deviation AK. we can find an angle A~a which gives exact phase matching. Only this phasematched component is nonzero because the interaction length is infinite and it results from the equation (3.4.13). In this case the amplitude of the acoustic wave is smaller than that of the center frequency. This reduces the diffracted light intensity and gives rise to the bandwidth of the device . A~a is the spread angle which gives exact phasematching. Simple trigonometry applied in the diagram in Fig.3.5.1(b) gives: 2kcos(n/2 - fJ/2 - A~a> = 2ksin(fl/2) + AK. (3.5.7) As in the above. if we assume : (3.5.8) we get the approximate expression of A~a At>a- AK/[2kcos(fJ/2)J, (3.5.9) 38 If the size of the transducer is T, the angular spectrum is given by (3.5.10) where W is the wavelength of the acoustic wave. So the first zero of the angular spectrum is : APa = W/T, (3.5.11) And the bandwidth follows from (3.5.9) AK = 2kcos(P/2)(W/T). (3.5.12) Now we have two definitions for the bandwidth. But they are same as can be shown as follows. From (3.5.12): AK = [2ksin(JJ/2)/sin(JJ/2)] [W cos(JJ/2)/Tl. (3.5.13) If we use the Bragg condition, (3.5.13) becomes AK = = [KW/sin(JJ/2)] [cos(JJ/2)/Tl [2n/sin(P/2)] [cos(p/2)/Tl. (3.5.14) Next from the simple trigonometry , the relation between L and T is given by L cos(p/2) = T. (3.5.15) Using this relation in (3.5.14), we get AK = [2n/Ll [1/sin(P/2) ]. (3.5.16) 39 This is exactly the same as (3.5.6). This shows that we can calculate the bandwidth in either way described above. The latter method of calculating the bandwidth is more advantageous than the other method if we consider the case of acoustic anisotropy. 40 4. HOMOGENEOUS ACOUSTO-ELECTRO-OPTIC EFFECT 4.1. Acousto-electro-optic(AEO) interaction We described the general idea perturbations in optical materials, of multiple and the general concept of optical signal processing and devices in Chapter 2. Also, we developed coupled mode equations which can be used to analyze the combined interaction of multiple perturbations theoretically. In the following two chapters. we take a specific combined interaction, i.e. ,AEO interaction. to demonstrate the usefulness of the concepts developed. The AEO interaction is a combined effect of AO and EO interactions. These two effects have been known for a long time.and used extensively for many types of optical devices, such as modulator, deflector. filter, etc •• Thus these individual effects have been analyzed theoretically by many authors. Our motive to investigate the AEO interaction was that, if we use both AO and EO effects. we may have more flexibility in making better devices. The limitations of each AO and EO device are well known. So we may improve functions of devices using both interactions. Examples of this idea will be given in Sections 4.4 and 4.S. The commonly used AO device is based on travelling 41 acoustic waves. On the other hand. both temporally and spatially varying electric fields have been used for EO devices. Homogeneous AEO interaction is represented by the sum of the polarization induced by the travelling acoustic wave and the homogeneous electric field. The theoretical tool we use to analyze the homogeneous AEO interaction is the general coupled mode equation for two gratings. acoustic wave is a grating. In this case. only the whereas the homogeneous electric field gives rise to a homogeneous perturbation via the linear electro-optic effect. There are two ways of analyzing the homogeneous AEO interaction. First. we may treat AO and EO effects as two perturbations. In this case. we can use the general coupled mode equation developed in the previous chapter. On the other hand. we may include the homogeneous perturbation induced by the constant electric field in the unperturbed dielectric tensor of the material. The effect of the homogeneous EO interaction is to change the index of refraction of the material of the conventional AO device. This interpretation of the homogeneous AEO interaction is more intuitive and physical. we get the same result using the coupled mode equation for the homogeneous AEO interaction in Section 4.2. Before analyzing the AEO interaction. we briefly 42 consider AO and EO interactions, and give some formulae for later use. First let's take the AO interaction. As we discussed in previous chapters, the AO effect is an interaction between optical and acoustic waves inside the crystal. The acoustic wave comes from collective molecular displacements of the material. A piezoelectric transducer generates an acoustic wave when we apply the electrical signal to the electrode. If we glue the transduceronto the crystaL we can launch the acoustic wave inside the crystal. Next,the incident optical wave interacts with the acoustic wave, and it is diffracted into many higher-order waves. The AO effect is nonlinear if the input to the system is considered to be the acoustic wave and the output the diffracted light. Thus if we have many acoustic waves inside the crystal, we have harmonic optical plane waves as well as intermodulation waves. The perturbation in polarization due to the AO interaction is given by : (4.1.1) where eil is the dielectric tensor of the unperturbed medium. If we assume one plane acoustic wave , the strain becomes : (4.1.2) 43 where smn : constant acoustic strain amplitude, o : acoustic wave frequency, Ki : acoustic wave vector. We assume that the amplitude of the acoustic wave does not change as it interacts with the optical wave. Then we can use one grating coupled mode equation to analyze the AO interaction. equation, From the general we see that two coupled mode mode coupling equation becomes as follows, assuming Bragg diffraction dF(0, 1 ) = jkF( 1 • 2 >exp[jAkr1 dF( 1 ,Z) = jk*F(0, 1 >exp[-jAkr], where d is a differential operator (4.1.3) with respect to r, and k is a coupling coefficient. The solution of (4.1.3) with the boundary condition p(0,1)(0) = 1, F(l,Z)(O) = o is given by: (4.1.4) where 11 L diffraction efficiency, interaction length, Ak : phasemismatch. (4.1.4) is the fundamental formula with which we can calculate the bandwidth of the AO modulator, the number of resolvable spots of the AO deflector, or thewavelength range of the optical filter and spectrum 44 analyzer. The linear EO effect comes from the interaction of the low-frequency electric field and optical wave. The effect is given by : (4.1.5) The general expression of the modulated light intensity with cross polarizers is : (4.1.6) ·) where IJ(1,2) I = (1/2)(w/c){n(l>n< 2 >>-l/ 2 <e·l. 1 s]m e · <2 >E <ext> I lmk J k • This is the basic formula for the EO device. x I e · (1 >r l. In this section, we introduce the homogeneous AEO · interaction. In the following sections, we show how to improve the AO deflector, and make a new AEO modulator using the constant AEO interaction. These will exemplify the new possibility of using the simplest combined interaction of AO and EO effects. 4.2. Coupled mode analysis of the homogeneous AEO interaction As explained above. the perturbation in the dielectric tensor for the homogeneous AEO interaction is 45 the sum of those of AO and EO effects. Thus. this is given by : e. . 1) = 8 , ,(o) 1) +Ae· .<AO) 1) + Ae· .<EO), 1) (4.2.1) Ae· .<AO) where 1) Ae· ,(EO) 1) In general. we may have a polarization perturbation which is a product of those for AO and EO effects. But this is a higher order perturbation, and sma ll.Thus we neglect the product effect of AO and EO interactions. The incident optical wave is assumed to be an eigenmode of the unperturbed material with polarization 1 E· (0,1) (x,t) 1 = (n(O,l))-1/2e. (0,1) 1 x exp[jw(o){(n< 0 • 1 >r/c)-t}], (4.2.2) where r=sixi and s is the unit vector in the direction of propagation of the incident light wave. We assume that the acoustic wave is a plane wave (4.2.3) where Ui is the constant amplitude of the plane acoustic wave. and c.c. means complex conjugate. From the definition of the strain tensor Skl , (4.2.4) 46 As· .<AO) we have the following expression for l.J As .. (AO) l.J =(1/2)jsipsjqPpqkl(UlKk + UkKl) exp[j(Ot- Klxl)] -(1/2)jeipsjqPpqkl(u 1 *Kk + uk*K 1 ) exp[-j(Ot-K 1 x 1 >l (4.2.5) where aij = jsipejqPpqkl(UlKk + UkKl). As we see in (4.2.1), induced by AO perturbations, and EO we have two perturbations effects. we have two we need a two-dimensional integer set to describe the eigenmodes coupled by seen in If perturbations as Section 3.4. But for the case of a homogeneous AEO interaction, one of two perturbations, i.e., effect, and we need a is homogeneous, dimensional integer set to describe only EO one- coupled eigenmodes due to the AO effect. Thus our trial solution reads as follows : = p(I,p)(r(I))(n(I,p))-1/2e. (I,p) l. x exp[jw<I>(n(I,p)r(I)/c- t)], where r<I> = (4.2.6) si<I>xi. If we use the same approximations as used in Section 3.4, we get the coupled mode equation for the constant AEO interaction 47 = H(I,q,p) p(I,p) exp[j(w<I>tc)(n(I,p) -n<I,q>)r] +J <I. q , p >F (I +1, P >ex p [ j <w <I >I c ><Ak <I + 1 ' P >-A k (I • q >>r l +J(I,q,p)F(I- 1 •P>exp[j(w(I)/c)(Ak(I- 1 ,p)_Ak(I,q))r], (4•2•7) where H(I,q,p)=(j/2)(w/c)(n<Lq>n<Lp))- 1 /2 x e (I,q)Ae(EO) e (I,p) 1 lm m ' J(I,q,p) =(j/4)(w/c)(n(I,q>n<I+1,p))-1/2 x e (I,q)a e (I+ 1 ,p> 1 lm m ' J(I,q,p) = (j/4)(w/c)(n<Lq)n(I-1,p))-1/2 x el (I,q>alm.em<I-l,p), and d is a differential operator with respect to r. Let's look at (4.2.7) in detail. The mode of order I couples with eigenmodes of orders I. I+l, I-1 with respective phasemismatches. In Section 3.4, we saw that the rule of modes coupling is AI = ±1. But here, in addition to AI = ±1, we also have a sel:f~-coupling o. The fact that one of the perturbations AI= is homogeneous reduces to the self coupling with selection rule AI= o. (4.2.7) is the basic equation which can be used to analyze the general homogeneous AEO interaction. He r e, we d i d n ' t us e a spec i f i c f o r m of opt i c a 1 ,a c o u s t i c eigenmodes, or interaction geometries. (4.2.7) does not allow an analytic solution in general because of the phasemismatch factors. But if the interaction length is 48 large enough so that only two modes couple with each other. we can obtain an analytic solution. In this case. the general equation (4.2. 7) reduces to : dF(0, 1 ) = H(0,1,1)F(0, 1 ) + J(0,1,2)F( 1 • 2 ) exp[jAkr] dF( 1 • 2 ) = H(1,2,2)F( 1 • 2 ) + J(1,2,1)F(0, 1 ) exp[-jAkr]. (4.2.8) where Ak = Ak< 1 • 2 >. We can introduce new amplitudes G(0, 1 ) = exp[-H(0,1,1)]F(0, 1 ) G( 1 • 2 ) = exp[-H(1,2,2)]F(l, 2 ). (4.2.9) Then (4.2.8) becomes : dG(0, 1 ) = J(0,1,2)G( 1 • 2 ) exp[j(Ak+q 2-q 1 )r] dG(l. 2 ) = J(1,2,1)G(0, 1 ) exp[-j(Ak+q 2 -q 1 )r]. (4.2.10) where q 1 = H(0,1,1)/j, q 2 = H(1,2,2)/j. Next let us define the total phasemismatch AkT as : As we see in (4.2.11). the total phasemismatch consists of the phasemismatch due to the AO effect and that due to the EO effect; i.e •• q 2 - q 1 . Let us define the phase mismatch due to EO effect as 49 =-(1/2) (2n/ A.) (n< 1 • 2 )>-l x 8 • 8 • e ( 1, 2) r. . e ( 1, 2) E (ext) 11 mJ 1 1Jk m k +(1/2) (2n/A.) (n< 0 • 1 >>-l x 8 ·8 ·e (0,1)r·. e (0,1)E (ext) 11 mJ 1 1Jk m k • (4.2.12) If we define changes of indices of refraction of modes 1, 2 due to the EO effect as , An 1 An 2 = (1/2)(n< 0 • 1 >>-l x 8 ·8 ·e (0,1>r .. e (0,1)E (ext) 11 mJ 1 1J k m k = (1/2)(n< 1 • 2 >>- 1 x 8 ·8 ·e (L2>r .. e (1,2)E (ext) 11 mJ 1 1J k m k • (4.2.13) then (4.2.12) becomes : (4.2.14) The initial condition is given by: G( 0 ' 1 ) ( 0 ) = 1 , G( 1 ' 2 ) ( 0 ) = O. (4.2.15) The solution of (4.2.10) with initial conditions given by (4.2.15) G(0, 1 ) is : = -j(4fJ(0,1,2)f 2 +(AkT) 2 )-ll 2 exp[jAkTrl X [C+exp(C+r) - c_exp(C_r)J G( 1 • 2 ) = jJ(0,1,2)*(4fJ(0,1,2)f 2 +(AkT> 2 >- 1 1 2 so x exp[-1/2j~kTrl2jsin[4JJ(0,1,2) J 2 +(AkT) 2 )-l/ 2 rJ, (4.2.16) C± = 1/2[-j~kT ± j(4JJ(0,1,2)J 2 +(AkT) 2 )-l/ 2 J. where From (4.2.16), we see that the diffracted light amplitude is given by, at the length r = L p(1,2) =-2J(0,1,2)*exp[j(q 2 -1/2AkT)L] 2 2 1 2 X [4JJ(0,1,2) J +(AkT) J- / x sin{[4JJ(0,1,2)J 2 +(AkT) 2 J 1 / 2 LJ}. (4.2.17) The diffracted light intensity follows : (5.2.18) ~AO = JJ(0,1,2)LJ 2 where is the diffraction efficiency of the AO effect in the absence of the EO effect. If we compare the diffracted light intensity formula for AO and homogeneous AEO interactions, we see that the only difference is the phasemisrnatch. This was shown in detail in (4.2.12) and (4.2.18). So the physical interpretation of (4.2.18) can be explained as in Fig.4.2.1. If we apply an external electric field, we change the index of refraction. These changes by ~n 1, ~n 2, which are same as are given (4.2.13). The phasernisrnatch for the conventional AO interaction is given from Fig.4.2.1 : 51 2 V=e {t) V=O FIG 4.2.1 52 Fig 4.2.1:Wave principle of the vector diagram illustrating the homogeneous acousto-electro-optic interaction. Ak : Phase mismatch induced by the homogeneous AEO interaction. ~ KA Acoustic wave vector. 53 (4.2.19) which is the same as (4.2.11). Then we can directly write down the intensity formula from that of the conventional AO interaction which gives exactly the same formula as (4.2.18). The above derivation of the formula for the constant AEO interaction using the change of index of refraction is possible only for homogeneous EO effects. If the electric field is spatially varying, we need a two-grating coupled mode equation to analyze the interaction. This subject will be considered in Chapter 5• 4.3. Experimental verification of the homogeneous AEO interaction The wavevector diagram of the specific interaction geometry we choose for the experiment is shown in Fig. 4.3.1. We considered anisotropic Bragg diffraction. As is shown in Fig.4.3.1, the acoustic frequency has been chosen to give maximum diffracted light intensity without an external voltage. When the external voltage is applied, we introduce phase mismatch, and the intensity of the diffracted light becomes smaller. The theoretical prediction for this interaction 54 geometry can be done using the general result derived in Section 4.2. Let us first consider the theoretical diffracted light intensity as a function of the applied external voltage. In Fig. 4.3.1, the acoustic wave travels in the x-direction. and the external voltage is applied in the y-direction. Let the height of the crystal be h. Then the amplitude of the electric field for the given voltage Vis E2 = V/h. (4.3.1) Next. let us define the voltage vAEO as From (4.2.11) and (4.2.13). we see that vAEO is given by vAEO=().h/L)[(l/2)(n(0,1))-18 •8 ·e (0,1)r·· e (0,1) 11 mJ 1 1)2 m - (1/2)(n(1• 2 >)-l8 ·8 ·e <L 2 >r. · e 11 mJ 1 (1. 2 >1-l 1)2 m • (4.3.3) where ). is the wave length of the optical wave and L is the interaction. For our case. AkAo = o. This gives the intensity formula from (4.2.18) (4.3.4) 55 z INPUT OPTICAL WAVE LIGHT FI(; LJ. 3.1 56 Fig 4.3.1 : Configuration of the interaction geometry for the AEO light modulation experiment. ne index of refraction of the extraordinary wave. n0 index of refraction of the ordinary wave. 57 This is the theoretical expression of the diffracted light intensity as a function of the applied external voltage for our interaction geometry. diffraction efficiency is small, If the AO as is true for our experiment, (4.3.4) becomes (4.3.5) We see from (4.3.5) that if the applied voltage is equal to vAEO, the diffracted light intensity is zero. This shows that the voltage defined in (4.3.3) is the analog of the half-wave voltage for a conventional EO modulator. For the experimental demonstration of the result (4.3.5), we designed a Bragg cell with electrodes to apply the external voltage. The photograph of the device is shown in Fig. 4.3.2. We chose LiNb0 3 as the crystal of the Bragg cell. As shown in Fig. 4.3.2, the shear acoustic wave with polarization in the y-direction is launched from the transducer glued on the (1,0,0) surface. This acoustic wave propagates in the xdirection, and the center frequency of the acoustic wave was chosen to be 20 MHz. The velocity of the acoustic wave is 4.2xlo 5 em/sec. The input impedance of the transducer was chosen to be 500. Then two metal electrodes were evaporated on the (0,1,0) surfaces and 58 59 Fig 4.3.2 : Photograph of the device for the AEOlight modulation experiment. Crystal is LiNb0 3 • 60 connected to a high-voltage power supply. The specifications of the device are summarized in Table 4.3.1. we used a 5 mW polarized He-Ne laser as a light source. The input optical wave was polarized in theydirection. propagating at an angle 1.8 deg with respect to the z-axis. This gives the maximum diffracted light intensity for the 20 MHz acoustic wave. The above angle has been calculated. In the experiment. the Bragg rotated ce 11 and we illuminated it to find the maximum diffracted light intensity. The angle experimentally determined isthe same as thatcalculated. Also. the polarization of the diffracted light was measured and it was in the x-direction. As expected. this interaction was anisotropic. polarizer to After the Bragg cell. block the undiffracted we used a light. This decreased the background light and increased the accuracy of the measurement. A spherical lens with focal length 60 em was used to focus the output diffracted optical plane wave on a detector. we measured the maximum diffracted light intensity for the 20 MHz acoustic frequency. Then we increased the external voltage up to 7.5 kV and measured the corresponding diffracted light intensities. we found that at 6 kV the diffracted light intensity was minimum. Table ~.3.1 Specification of the Bragg cell • Crystal : LiNb0 3 . • Size of the crystal : 40-7-12(x-y-z> (mm). • (0,0,1)surfaces : Polished and A.R. coated. • Optical wave : 632.8 nm (He-Ne laser). • Acoustic wave : shear wave with a polarization [0,1,0] and propagating in the [1,0,0] direction. velocity is 4.2x10 5 (em/sec). • Acoustic wave center frequency : 20Z S(Mhz). • Diffraction efficiency : 10 ('/watt). • R.F. input power 2 watts. • (0,1,0) surfaces metal electrodes. • Size of the transducer : S-10(y-z> (mm). 62 This gave vAEO = 6 kV. Then we intensity and voltage as normalized the light prescribed in (4.3.5). we plotted the relation. and this experimental result is shown in Fig. 4.3.3. In Fig. 4.3.3, we also drew the theoretical curve which is given by (4.3.5). As we see, the experimental result agrees with the theoretical calculation very well. we calculated the voltage vAEO for ourexperiment. using the definition (4.3.3). As shown in Fig. 4.3.1, the light propagates near the z-axis. So the index of refraction is 2.29 for the LiNb0 3 crystal. From (4.3.3) : (4.3.6) where we used r 22 = - r 1 2. The wavelength of the He-Ne laser is 632.8 nm. and the height of the crystal is 7 rom. The interaction length L is 1 em. and the electro-optic coefficient r 12 for the low frequency electric field is 6.7xlo- 12 <m/V). Then. if we plug all numbers into (4.3.6), we get s.s kv. This calculation agrees with the experimentally measured value of vAEO within an error. The acousto-optic diffraction efficiency measured was about 2 "· We used the elasto-optic coefficient P 66 because the acoustic wave with polarization in the y-direction travels in the rt) ....z w >- ~ ex: ex: > w 0 w Cl.. I .... wX .. .. I • ~ w 0 tl LAJ ct > C\J 0 w <( > .......... > t:: 0 . <X>. 0 N 0 . ""IV\ - ..::::rl!J ....... LL 64 Fig 4.3.3 Experimental results of an AEO modulator. 65 x-direction and polarizations of optical waves are in x- and y-directions. The value of P 66 for the LiNb0 3 crystal is 0.05. Thus, the diffraction efficiency per unit acoustic power is given by : where p is the mass density of the LiNbo 3 , which is 4.7810 kg/m; Va is the acoustic velocity. If we plug numb e r s into ( 4 . 3 • 7 ) , we g e t 7 • 5 4fo. Th i s r e s u 1 t a1s o agrees well with the experiment. 4.4. AEO modulator The most widely used methods for wideband light modulation are either AO or EO effects. Each type of modulator has its own strength, and suffers from its own distinct limitations. We explain about these in this section in detail when we compare the AEO modulator with AO and EO modulators. When we use AO and EO effects simultaneously, a new flexibility is introduced with which we can overcome some of limitations of two individual modulators. Let me first consider AO and EO modulators separately. The AO light modulation has been described in Section 4.1. Within the bandwidth of the AO modu 1 a tor, we can neg 1 ect the phase mismatch. Then the 66 modulation function is given from (4.1.4) (4.4.1) The diffraction efficiency ~ is proportional to the acoustic power. And the acoustic power is given by : Pa = v 2 /2R. (4.4.2) where Vis the amplitude of the signal voltage, R is the impedance of the electrical network of the acoustic transducer. From (4.4.1) and (4.4.2), we see that the amplitude modulation function is given by : Ed = Ein sin [aVl, (4.4.3) where a is a constant. If the signal is small. we have : Ed - V (4.4.4) Thus. if the nonlinear effect which gives harmonic and intermodulations is small. rise to we get the modulation according to (4.4.4). In the following, we call (4.4.1) the modulation function of an AO modulator. The formula for the EO modulator has been given in (4.1.6). There are two different types of EO modulators. One is the longitudinal EO modulator for which the 67 electric field is applied in the direction of the light propagation. In this case, the electric field is given by E = V /L, where L is the interaction length. From (4.1.6), the modulation function for the longitudinal EO modulator becomes : lout = [sin(f(V))J 2 (4.4.5) where f(V) =(1/2)(w/c)(n< 1 >n< 2 >>-l/ 2 s 1 l.·e mJ· x I e (1 >r · · e <2 >e <ext> IV and 1 l.Jk m k ' ek<ext) is the direction of the external electric field. If we apply the electric field in the transverse direction from the light propagation, we have a transverse EO modulator. In this case, the electric field is given by V/h, where h is the height of the transverse dimension. Then the modulation function is given by : lout = [sin(g(V))J 2 , where g (V) = (4.4.6) (1/2)(w/c)(n< 1 >n< 2 >>-l/ 2 e ll.·s m]. x le (l)r· · e <2 >e (ext) IVL/h 1 l.Jk m k · As we see from (4.4.5), the diffracted light intensity for the longitudinal EO modulation does not depend on the interaction length L. But for the transverse EO modulation, it depends on the ratio L/h, and this gives 68 some flexibility to design a better modulator. Let's define the half-wave voltages which give the voltages required for the full modulation : f(V) = rr.V/2V 1f L g{V) =rr.V/2Vrr.T· {4.4.7) From (4.4.5) and (4.4.6) the half-wave voltages are (4.4.8) V T = [(1/rr.)(w/c)(n(l>n< 2 >>- 1 12 s 1T ·e x le (l)r· · e <2 >e (ext)IL/hJ- 1 1 ~Jk m k . 1 ~ mJ • {4.4.9) In the actual modulation. the signal is biased at Vn/2. Thus the signal becomes If we plug {4.4.10) into If we use the Bessel {4.4.5) and (4.4.6), we have : function identities, (4.4.11) becomes : I = (1/2)[1 + 2J 1 Crr.Vm/Vrr.)sinwmt + 2J 3 CnVm/Vn)sin3wmt+ ••• ], (4.4.12) where Jn are Bessel functions. As we see in (4.4.12) the EO modulation creates 69 nonlinear harmonic frequencies. AEO light modulation is based on the experimental curve obtained in Section 4.3 for the homogeneous AEO interaction. We use one fixed acoustic frequency which is the center frequency without the external voltage. The acoustic power is constant. Then the input signal for the AEO modulation is the external voltage applied to the device.Thus. we can think of this AEO modulation as the hybrid of AO and EO modulations. The modulation function of the AEO modulator is given by: (4.4.13) AEO modulator has some advantages or disadvantages over AO and EO modulators. In the following we compare various characteristics of the AEO modulator with those of AO and EO modulators. BANDWIDTH : The bandwidth of the AO modulator is limited by the diffraction efficiency. For bandwidth we need a small interaction length. gives effective. small phasemismatch. from (4.1.4). But the larger which This can be seen diffraction efficiency is proportional to the interaction length.Thus. there is a trade-off between the bandwidth and diffraction efficiency for the AO modulator. For the AEO modulator. we use only one fixed acoustic frequency with constant 70 acoust icpower. Thus, we can have arbitrary diffraction efficiency if we choose a large interaction length. The bandwidth of the AEO modulator is thus same as that of the EO modulator. The bandwidth of the EO modulator is given by (4.4.14) where RL is the shunting resistance and Cis the crystal capacitance. Next, the power needed is proportional to ll.f. Thus, the practical bandwidth of the AEO modulator is limited primarily by the maximum power supplied by the electrical driving circuit with which the voltage is applied across electrodes. DIFFRACTION EFFICIENCY : As discussed in the above for the AEO modulator, the diffraction efficiency depends on the acoustic power supplied by the acoustic port and the interaction length. Thus,there is no tradeoff between the diffraction efficiency and bandwidth. Furthermore, the piezoelectric affect, unlike the AO case, transducer the does not bandwidth of the modulator.Therefore, it can be designed to maximize the diffraction efficiency by increasing the transducer width and thus the interaction length. 71 HALF-WAVE VOLTAGE : Half-wave voltages defined for EO and AEO modulators are measures of the voltage that gives full modulation. We derived expressions for halfwave voltages in (4.4.9) for the transverse EO modulation, and in (4.3.3) for the AEO modulation. From these expressions we see that the voltage level required for the AEO modulator is, within a geometricalfactor, close to unity, twice that needed in a transverse EO modulator of the same geometry. SYSTEM ALIGNING : The modulated light is angularly separated from the undiffracted light, anisotropic AO EO modulator. because we used interactions. Thus, as compared with an the need for ananalyzer is eliminated. Furthermore, if we use anisotropic AO diffraction, we don't need the input polarizer either. The alignment of an AEO modulator thus (like an AO modulator) is almost insensitive to the direction perpendicular to the interaction plane. This is to be compared to the small numerical aperture, in both directions, of an EO modulator , limited by the natural birefringence. If we use an analyzer for the anisotropic AO modulator, the signal -to -noise ratio will be enhanced, because polarizations of the diffracted and undiffracted light are orthogonal, and the analyzer suppresses the undiffracted light. The acoustic port can be used to 72 dynamically align the modulator in the plane of the interaction. by changing the center frequency. and to compensatefor the intensity of the acousticpower . The above factors make the system aligning of the AEO modulator very easy compared with the EO modulator. EXTENDED BEAM MODULATION : For an AO modulator, the modulation is done by the travelling acoustic wave. This means there is an acoustic transit time limit for the AO modulator. But in the AEO modulator the modulation is accomplished not by the acoustic wave but by the voltage. Thus. an extended collimated optical beam can be modu 1 a ted. MODULATION DEPTH The modulation depth of a modulator is defined as : (4.4.15) For the EO modulator the bias voltage which gives the largest linear region is Vn/2, because the modulation function is sin 2 CnV/2Vn>· One of criteria of the linearity of the nonlinear modulation function is the total harmonic distortion (THO). Let's first define THO. using an arbitrary modulation function E(V(t)). If we consider a sinusoidal input voltage with bias V(t) = vb + vd coswt. (4.4.16) 73 the modulated output follows : = E(Vb E(t} + Vd coswt). (4.4.17} Now E(t) is a periodic function with period 2n/w and even in t. Thus. we can expand E(t) as a Fourier cosine series : E(t) = E(n) cosnwt. (4.4.18} The total average output power is given by I = = /E 2 (t)dt 2n/w[(E(0}} 2 + 112'2: (E(n)} 2 ]. n (4.4.19} THD is defined as : The modulation function of an AEO modulator has no obvious bias point because of its sinc 2 nature. Also. it has a second order harmonic. But EO modulator does not generate a second harmonic. Thus, it is difficult to use the ratio of harmonics to compare the linearity. This is the reason why we chose THD to compare the linearities. First. as a reference point. we calculated the THD of an EO modulator, which gives the harmonic/third harmonic ) 1 "· Then. we calculated THD's for ratio first for various biases different ranges of the 74 modulation voltage. voltage range, Finally, we chose the bias and which gives the maximum MD previously fixed THD. for the The result of the computer calculation shows that the MD for the EO modulator is 48 ~. and the MD for AEO modulator is 44 ~ with bias 1.38. This shows that AEO and EO modulators have almost the same linearity. 4.5. AEO deflector An optical deflector is a device which can change the direction of light propagation. Among many deflectors AO and EO deflectors are used widely. We can control the deflection angle electrically for both deflectors. In this section we concentrate on the AO deflector. In the AO deflector we change the acoustic frequency to change the deflection angle. But as we have seen. if the acoustic frequency deviates from the center frequency, the deflected light intensity drops. Thus. the total deflected angle depends on the bandwidth of the AO device. One of the important figures of merit of the deflector is the number of resolvable spots. This quantity NR is defined as : NR where = A.p I 0.. I nW) , AP (4.5.1) is the total deflected angle inside the 75 crystal, n is the index of refraction, A is the optical wavelength in vacuum. and W is the beam width of the optical wave. In (4.5.1} we assumed a collimated optical beam with width w. The diffraction-limited angle of this optical beam is A/nW . Let's first derive a relation between A~ and the bandwidth Af. The general wave vector diagram for an AO deflector is shown in Fig. 4.5.1(a}. From the figure we have the following equation which contains A~/2 (K + AK} 2 = (2nn 1 /A} 2 + (2nn 2 /A} 2 - 2(2nn (4.5.2} 1 /A}(2nn 2 /A}COS(~+A~/2}, AK = 2nAf/Va(Af is the one-sided bandwidth of where the device}, n 1 and n 2 are indices of refraction for the two polarizations. From Bragg condition, we have K2 = (2nn 1 /A} 2 +(2nn 2 /A} 2 - 2(2nn 1 /A}(2nn 2 /A}cos(J)}. (4.5.3} Here we assume refraction A~/2 to be small, so that the index of does not change within the angle we assume AK/K << 1. Then from A~. Also (4.5.2} and (4.5.3}, we have : (4.5.4} 76 (a) (b) FIG LJ. 5 J I 77 Fig 4.5.1: Wave vector diagram illustrating phasemismatch compensating AEO deflector. (a) Ordinary acousto-optic deflector. (b) Phasemismatch compensated AEO deflector. the 78 If we use ln 1 -n 2 1/n 1 or n 2 NR = << 1, (4.5.4) becomes [W/(Vacos(Ji/2))](2Af), (4.5.5) where 2Af is the full bandwidth of the device. (4.5.5) shows that the number of resolvable spots is the product of the bandwidth and the transit time of the acoustic wave across the optical beam. One method of increasing the number of resolvable spots is to increase the bandwidth of the device. As we have seen in the AEO modulator, there is a trade-off between the bandwidth and diffraction efficiency. Thus it is difficult to increase the bandwidth by decreasing the interaction length. The full bandwidth of the device is determined by the transducer bandwidth and the phasemismatch introduced by the deviation of the scanning acoustic wave frequency from the center frequency. The transducer bandwidth is determined by the electrical matching network of the transducer. But the bandwidth limited by the phasemismatch can be corrected ifwe change the index of refraction, using the external voltage. This is just the homogeneous AEO interaction. Thus, the AEOdeflector isbased on the phasemismatch compensation which manifests in the homogeneous AEO interaction. Let's consider the AEO deflector in detail. For 79 each scanning frequency f. the phasemismatch 4kAo<f-fc> is introduced as shown in Fig. 4.5.1(b). fc is the center frequency. The phasemismatch 4kAo is given by : (4.5.6) where The center frequency satisfies (4.5.7) From (4.5.6) and (4.5.7), we have : AkAO = 2 ) - 2 n n cos a I ( n Va) )( f- f c) 2 1 2 2 + nA.(f-fc> !n 1 va . (4.5.8) ( 2 n A. f c I ( n 1Va From (4.5.8) we see that for small deviation of frequencies, 4kAo is proportional to (f-fc>· The total phasemismatch of the constant AEO interaction follows from (4.2.11) (4.5.9) If we use (4.5.8) to first order in (f-fc>• we obtain : 4kT = (2nA.fcl<n 1va 2 > - 2nn 2 cosal<n 1va>><f-fc> + (2niA.)((112)n1-1aliamjel (0,1)rijkem(0,1)ek - (1 I 2)n 2 -1 a 1 iamjel (1,2) rijkem (1,2) ek)VI h . (4.5.10) 80 Now for each frequency we want AkT = phasemismatch. and get o. Then we get no the maximum diffracted light intensity. From (4.5.10) the relation between the compensating voltage V and the frequency deviation is given by : (4.5.11) where a -(2nAfcl<n 1va 2 > - 2nn 2 cos /Cn 1Va))(A/2n)2h x (n - 1 8 ·8 ·e <0 • 1 >r .. e (O,l)e 1 11 mJ 1 lJ k m k - n - 1 8 ·8 ·e (1. 2 >r .. e Cl. 2 >e >- 1 2 11 mJ 1 lJ k m k • verify the relation (4.5.11). we did = To an experiment with the same device as discussed in Section 4.3. The experimental result is shown in Fig. 4.5.2. The normalized intensity of the diffracted light obtained with and without the compensating voltage is plotted as a function of the acoustic frequency. The compensating voltage. as a function of the acoustic frequency. is plotted in the same figure. From Fig. 4.5.2. we can see that the bandwidth of the AEO deflector is about 2.5 times larger than that of the AO deflector and was limited by the electrical bandwidth of the transducer. Next let's derive the formula of NR of the AEO deflector.neglecting the influence of the transducer bandwidth. The total number NR' is given by : NR' = a(Af + Vmax F/VAEO) • (4.5.12) N - -Vl ...!::" Gl --n H ~~ ~ !t::. . . -0 25 0.0 V 0.2 ~ 0.4 0.6 0.8 1- ~ 29 I 31 I ~8 10 1 33 ]' w I 35 .. --12 I 4 > ~ )(\! c5~ ~ 6 ~ - ~x/~ > COMPENSATING VOLTAGE " ~ WITHOUT VOLTAGE WITH "(VOLTAGE "-x FREQUENCY (MHz) 27 I '- '=IDEAL RESPONSE X~ I.O>k "" -~ 00 I-' 82 Fig 4.5.2 : Experimental compensated AEO deflector. results of a phasemismatch 83 where aAf = NR • Af is the one-sided bandwidth. F is the whole range of the frequency which compensates the phasemismatch introduced by the voltage vAEO • From (4.5.12) we have : (4.5.13) The ratio F/Af is given by (4.5.14) we get the value 2.25 for the above ratio. This finally gives the number of resolvable spots with compensating voltage Vmax neglecting transducer bandwidth as : N ' R 4.6. = + 2 2 5V N (1 R • max (4.5.15) /vAEO) Novel way of measuring the acoustic transducer bandwidth The transducer bandwidth we discussed in Section 4.5 comes from the impedance mismatch of the electrical network of the transducer. Thus. we can measure this bandwidth. analyzing electrically. deflector. the transducer network But from the discussion of the AEO we can measure . this bandwidth optically. In Fig.4.5.2, even if we compensate the phasemismatch. the output light intensity is not uniform. This is because 84 the acoustic power for scanning frequencies drops due to the impedance mismatch of the transducer network. Also, the diffracted light intensity is proportional to the acoustic power. Thus, if we measure the diffracted light intensity with the electric field compensation, we obtain the transducer bandwidth of the device. As an example, we see the transducer bandwidth in Fig.4.5.2. 85 5. PBO'l'OREFRACTIVE AEO INTERACTION 5.1. Spatial ABO interaction In Chapter 4. we considered homogeneous AEO interactions that allows a clear physical interpretation in terms of the change of index of refraction. ~>~lso. it is the simplest type of AEO interaction. But to explore the full potentiality of the AEO interaction. we need to consider spatially varying electric fields as well. The mathematical tool useful in treating this spatial AEO interaction is the two-grating coupled mode equation developed in Section 3.4. In general. we cannot obtain an analytic solution for the two-grating coupled mode equation, if there is a phasemismatch between coupled modes. This is compared with the case of a homogeneous AEO interaction. For the homogeneous AEO interaction, we derive a general diffraction. solution in the case of Bragg For the analysis we choose a specific case of three mode coupling in the next section, and demonstrate the inhomogeneous AEO interaction in Section 5 .4. We need spatially varying electric fields (i.e .• gratings ) for the inhomogeneous AEO interaction . The photorefracti ve obtaining effect 9 spatial is a promising electro-optic method gratings. If of we 86 illuminate a light intensity pattern on a photorefractive crystal. we can generate a corresponding charge pattern inside the crystal. This charge pattern gives an electric field pattern and EO gratings result from this field pattern via linear electro-optic effect. we can also erase this pattern easily. implement the real time optical Thus. we can signal processing system. using the photorefractive AEO interaction. In the inhomogeneous AEO interaction we have the freedom to choose arbitrary gratings from different sources. In this case we may use the nonlinearity of the interaction of many gratings. This is contrary to the conventional AO device. We show a way of using the intermodulation term in devising a correlator. 5.2. Three-mode photorefractive AEO interaction The simplest coupling of eigenmodes through two gratings is three modes coupling. If we assume Bragg matching. we can obtain a simple. perfect general. analytical solution. This is interesting in itself. and furthermore. we can use this correlator. In this section. analysis to devise we choose a specific configuration of three modes and two gratings, derive formulae of a and diffracted light intensities. using 87 the general two-grating coupled equation given in Section 3.4. Let's consider the wavevector diagram shown in Fig. 5.2.1. One grating is in the near y-direction, and the other grating is in the x-direction. Thus, the two gratings are almost perpendicular to each other. We have three optical modes. Mode 1 is the incident 1 ight wave with polarization 1. photorefractive This mode 1 interacts with the grating diffracted mode 2. isotropically Next. to give the mode 2 interacts with the acoustic grating anisotropically as well as with photorefractive grating to give the diffracted intermodulation light wave(mode 3). We assume that the phasemismatch between mode 1 and the acoustic grating is large so that we have only three mode coupling as shown in the diagram. The coupled mode equation can be written down from Section 3.4 : (5.2.1) where F 1 • F 2 and F 3 are and J.l 12 and J.t 23 are amplitudes of optical beams coupling coefficients; d is a differential operator with respect to r. As we see in (5.2.1), we have no direct coupling between mode 1 and 88 z y FIG 5.2.1 89 Fig 5.2.1 : wave vector diagram of a specific example of photorefractive AEO interaction. P.G.: photorefractive grating. A.G. : acoustic P.G. grating grating. is in y-direction. p<LL 2 >: amplitude of the intermodulation term. 90 mode 3. This is the consequence of the combined interaction of the two gratings. The initial condition is given by F 1 (0) = F 2 (0) = 0 and F 3 (0) 1, O. = (5.2.2) The solution of (5.2.1) with initial conditions (5.2.2) is easy to obtain and given by 2 F1 = 1 + £1~121 1<1~121 2 2 + 1~2 3 1 >1 [cos<l~ 12 1 2 + 1~ 23 1 2 > 1 ' 2 r x - 11 F2 = [j~12!(1~1212 + ~~2312>1/21 x [sin<l~ 12 1 2 + 1~ 23 1 2 > 1 ' 2 r 1 F3 = £~12~23•!(1~121 2 + 1~231 >1 [cos<l~ 12 1 2 + 1~ 23 1 2 > 1 ' 2 r- 11. x Let's define 2 diffraction efficiencies ~ 12 and (5.2.3) ~ 23 as ~12 = <1~121L)2 ~23 = <1~231L)2. Then (5.2.4) intensities of light waves at x = L are 11 = {1 + ~12/(~12 + ~23) x [cos(~12 + ~23)1/2 - 1])2 1 2= [~12 1 <~12 13 = + ~23)] [sin(~12 +~23> 112 1 2 [~12~23/(~12 + ~23> 2 1 [cos(~12 + 1'123> 112 -11 2 • (5.2.5) 91 From (5.2.5) if ~ 23 = o. we have (5.2.6) This is the well-known formula of the two modes Bragg diffraction. If we have ~ 12 = ~ 23 and <~ 12 +~ 23 > 1 1 2 = n; i.e •• (5.2.7) the only nonzero 1 ight wave is mode 3. and the intensity is 1. This shows that we can transfer all the incident light energy into mode 3. intensity formula of ~ 23 • and Also. if we look at the mode 3, it is the product of ~ 12 Thus. we call mode 3 as the intermodulation of two gratings. For small efficiencies: ~12'~23 << 1 • Then we have (5.2.8) approximate light intensities I1 = 1 1 2 = ~12 I 3 This = 1/4~ 12 ~ 23 • is the incident usual light. (5.2.9) approximation of the As in (5.2.9), we see intermodulation mode is the product of efficiencies. undepleted the two diffraction when those efficiencies are small. The 92 ratio r 3 /r 2 is (5.2.10} 5.3. Experiment on photorefractive AEO interaction To verify the analysis in Section 5.2, we performed experiments using the device described in Section 4.3. Fortunately. the crystal used for the homogeneous AEO device was LiNb0 3 • and LiNb0 3 is a photorefractive material. The interaction geometry of photorefractive and acoustic gratings with three optical modes is as shown in Fig. 5.2.1. The interaction geometry of the acoustic wave and optical modes 2 and 3 are the same as that of the experiment of the constant AEO interaction. The experimental setup is drawn in Fig. 5.3.1(a}. We used 20 MHz shear acoustic wave. Then we rotated the Bragg cell to obtain the maximum diffracted light intensity. Next we used an Ar laser with blue line ( 488 nm > to make photorefracti ve gratings by interfering two collimated blue light beams without the acoustic wave. The power of the laser was 0.7 watt, and the exposure time was 20 minutes. The angle between the interfering beams was 2.6 degrees. Thus we realized the interaction geometry shown in Fig. 5.2.1. After writing the photorefractive 94 Fig 5.3.1 Experiment of the photorefractive AEO interaction. (a) Writing of the P.G.with two interfering beams without A.G. AEO interaction given by acoustic wave and the (b) photorefractive grating. Three spots of the incident, first diffracted and intermodulation lights on the focal plane. 95 grating, we used one of the blue lights of two interfering beams to obtain the first diffracted light mode 2. Next, we launched the acoustic wave to obtain intermodulation mode 3. After the Bragg cell we used a spherical lens with local length 60 em. At the focal plane we observed three spots corresponding to the three optical modes 5.3.1(b). 2 and 3, 1, as illustrated in Fig. Because the acoustic grating is perpendicular to the photorefractive grating, the intermodulation mode 3 is off the line joining the two modes 1, experiment the diffraction efficiency 2. For our is 11 12 the photorefractive diffraction efficiency and it is constant. But we can change the acoustic power can change and thus the diffraction efficiency 11 23 • The dependence of 11 23 on the acoustic power is linear.Thus, the ratio given by (5.2.10) is linear in the acoustic power. We measure the ratio (5.2.10) as we increase the acoustic power. The result is shown in Fig. 5.3.2. this Figure In two scales are arbitrary. For small acoustic power we have a linear relation. For the large acoustic power, however, the relation is not linear. This comes from the saturation of the r.f. also measured condition amplifier. We two diffraction efficiencies to check the (5.2.8). The measured photorefractive ~. and the maximum diffraction efficiency was 0.6 96 INTENSilY RATIO • • FIG 5.3.2 97 Fig Experimental 5.3.2 photorefractive AEO power. • verification of the effect. Pa is the input acoustic Ex per irnent • 98 acousto-optic diffraction efficiency was 0.2. Let's calculate the change of index of refraction induced by the photorefractive grating. LiNbo 3 • which we used, was not Fe-doped. The diffraction efficiency is given from the paper by F. s. Chen, et al. 9 as: 1112 = [sin(n.dnL/(2A.cos(a/2)))] 2 , (5.3.1) where a is the Bragg angle. For the small diffraction efficiency. we obtain .dn = (2A.(cos(a/2))11 1 1 2 )/(nL). In our experiment. A. = 488 nm. a= (5.3.2) 2.6 degrees and 11 12 = 0.55%. If we plug these numbers in (5.3.2). we obtain .An= 2xlo- 6 . This value is in good agreement with the result of F.S.Chen, et al.. Also. this is the saturated value.Next. we calculate the wave number of the photorefractive grating. This is an isotropic grating. Thus we have : K 2ksin(a/2) ..., k a, (5.3.3) where the approximation is for small a. If we use a = 2.6 degrees. The electric field we obtain K = 6x10 3 (cm- 1 >. induced by the photorefractive grating can be calculated, using the following equation : 99 An = n 3 r 12 E/2. (5.3.4) Now r 12 = 3.4xlo- 12 m/V. we have E = 10 3 V/cm. 5.4.Correlator using the intermodulation mode The mathematical definition of one-dimensional correlation is : f f(x)h(x-y>*dx = C(y). (5.4.1) wecan do this correlation, using the intermodulation mode of the photorefractive AEO interaction. If the holographic pattern written in the photorefractive crystal is S(x,y), and the acoustic signal delayed in the same crystal is a(t+x/Va>· then the amplitude of the intermodulation mode after the crystal is given by : S(x,y) a(t+x/V a>· (5.4.2) We assumed small diffraction efficiencies and used (5.3.9). After the crystal we can put a cylindrical lens with focal length f. Then on the focal plane the amplitude of the intermodulation mode becomes f S(x,y)a(t+x/Va> exp[j(xxf/Af)]dx. (5.4.3) Thus, the intensity distribution on the focal plane is : 100 I(xf.t.y) = I f S(x.y)a(t+x/Va>exp[jxxf/Af]dxl 2• (5.4.4) If we use a pinhole detector at xf = o. we obtain the correlation between S(x.y) and a(t+x/Va> : I(xf = o.t.y) = I /S(x.y)a(t + x/Va>dx 12 • If we collect all the intensity. (5.4.5) we obtain the incoherent correlation C(t.y) = f I(xf.t.y)dxf = f ls<x.y>l 2 1a<t+x/Va>1 2 dx. (5.4.6) To demonstrate the correlation by the above method. we did an auto correlation experiment using the same device as described in Section 4.3. The bandwidth of the device is 10 MHz. Thus. we used the simplest pattern shown in Fig. S.4.1(a). First. we used a pattern shown in Fig. S.4.1(a) and two collimated Ar laser beams ( 488 nrn to write the pattern over the high-frequency interference grating inside the crystal. Of course. the AO device was set before to give the maximum AO diffraction efficiency at the center frequency 20 MHz. Next. we generated an electrical signal which exactly matched the pattern written inside the crystal. when the signal was delayed by the acoustic wave. The electrical 101 ---i6J(a) --i~ I.BfLsec (b) FIG 5 .'L 1 102 Fig 5.4.1 : Experiment of the photorefractive AEO cor relator. (a) Input photorefractive pattern. The intensity of the left window is twice that of the right window. (b) Electrical signal into the acoustic transducer which matches the pattern (a). Oscilloscope trace of the electrical signal. 103 signal was modulated by the center frequency 20 MHz and shown in Fig. S.4.1(b) and (c). We used a spherical lens with local length 60 em and collected all the lights of the intermodulation mode on the focal plane. The oscilloscope trace of the light intensity is shown in Fig.S.4.2. This is the incoherent auto correlation of the pattern shown in Fig. S.4.1(a). 1 04 FIG 5 .4.2 105 Fig 5.4.2 : Oscilloscope trace of the auto-correlation of the pattern Fig 5.4.1 (a). 106 6. FUTURE RESEARCH In previous chapters. the general concept of multiple perturbations applied to the optical signal processing and optical devices has been developed and demonstrated using the AEO interaction. The main idea of this work was that multiple perturbations might give more flexibilities to play with. demonstrated successfully in This has Chapters 4 and the simplest AEO interaction. Thus, been s. using if we consider more complicated mutiple perturbations. we may find very interesting phenomena and can apply these phenomena to the optical signal processing and devices. Small effects of the optical interaction between multiple perturbations is a problem. As we want more flexibility. this problem becomes more severe. There are some ways of overcoming this problem in general. First. we may develop special susceptibilities of the materials that interaction have large of multiple perturbations. People are working on synthetic organic materials. superlattice liquid crystals or of semiconductors to obtain large susceptibilities. Another possibility is to investigate the physical mechanism of the interaction of multiple perturbations. As an example. let's consider AIOHG introduced briefly in Section 2.1 . As we discussed in Section 2.1. there are 107 two types of interaction. interactions. The One is susceptibility of interaction is very small, the this direct type of and we have no way to increase the effect except by developing a special material with a large susceptibility. The second type of interaction is called induced effect. In this case. we have two separate phasematching conditions. One is for the AO interaction. and the other is for the optical second harmonic generation. Thus. if we satisfy two phasematching conditions simultaneously, we obtain a large effect. This has been demonstrated by Nelson and Lax 2 • They increased the effect by order of 1000. This example shows that if we know the physical mechanism of interactions, we may enhance the strength of the interaction. A third way of overcoming difficulties is to control the size of the device, so that for the given value of the susceptibility we can increase of multiple perturbations. This gives increased strength of the interaction. wave device is a good example. amplitudes an overall Surface acoustic In this case. increase the amplitude of the strain. we can Integrated optics is another example. In this case we may have a large electric field. using a small amount of voltage. These are very interesting areas in which to apply the general concept of the interaction of multiple perturbations. 108 REFERENCES 1. Murray Sargent III, et al.,Laser Physics (Addison Wesley, Massachusetts, 1974). 2. D.F.Nelson and M.Lax, Phys.Rev.B 3, 2795 (1971). 3. D.Psaltis,H.Lee and G.Sirat, Appl.Phys.Lett.,46,215 (1985). 4. I.C.Chang, IEEE Trans.Sonics.Ultrason.SU-23,2(1976). 5. J.M.Rouaven,M.G.Ghazaleh,E.Bridoux,and R.Torquet, J.Appl.Phys,SO,S472(1979). 6. D.F.Nelsonand P.D.Lazay, Phys.Rev.Lett.17,1187 (1970). 7. A.Yariv and Pochi Yeh, Optical waves in crystals propagation and control of laser radiation (Wiley, NY, 1984). 8. N.Uchida and Y.Ohmachi, J.Appl.Phys.40,4692(1969). 9. F.S.Chen,J.T.LaMacchia and D.B.Fraser, Appl.Phys. Lett. 13,223(1968).