Theory of Polarization Holography

Transcription

1 Theory of Polarization Holography Kazuo KURODA, Yusuke MATSUHASHI, Ryushi FUJIMURA, and Tsutomu SHIMURA OPTIAL REVIEW Vol. 18, No. 5 (211)

2 OPTIAL REVIEW Vol. 18, No. 5 (211) Theory of Polarization Holography Kazuo KURODA, Yusuke MATSUHASHI, Ryushi FUJIMURA, and Tsutomu SHIMURA Institute of Industrial Science, University of Tokyo, Komaba, Meguro-ku, Tokyo , Japan (Received June 6, 211; Accepted August 4, 211) The optical response of polarization sensitive materials is studied on the basis of a model in which the materials consist of rod molecules that are oriented in random directions. The photoinduced change of the dielectric tensor is expressed as a function of the electric field in a general form. Using this expression, we derive the vectorial coupled wave equations in volume polarization holograms. We discuss the basic properties of polarization holograms. # 211 The Japan Society of Applied Physics Keywords: polarization holography, photoanisotropy, coupled wave equations 1. Introduction The photosensitivity of some organic and inorganic materials depends on the polarization of the optical field, and anisotropy is induced by the illumination of polarized light. Polarization holography has been demonstrated using such polarization-sensitive materials. 1 5) In conventional intensity holography, the intensity interference pattern is recorded in a hologram as the modulation of the refractive index and/or absorption and the image is reconstructed by the illumination of a reconstruction wave. On the other hand, the interference pattern of polarization is recorded in polarization holograms as the modulation of anisotropy such as birefringence, dichroism, and optical activity. The image is read out by a reconstruction wave similarly to in intensity holography. Among the many polarization-sensitive materials, azobenzene-containing media have been investigated most intensively. 1 6) Photoinduced gyrotropy has been also investigated. 7) Polarization holography is applied in various optical systems such as vector phase conjugators, 8,9) retardagraphy, 1 12) and so forth. The theory of polarization holography has been published by several authors ) Although the theory describes polarization holography correctly, it is quite complicated. This is because the response of the recording material is described as a function of the canonical form of the polarization state, i.e., the principal axes and ellipticity of the total field. Hence, in order to evaluate the anisotropy of a material, one has to transform the polarization state to the canonical form. In general, however, the polarization state of an interference pattern is complicated even for the interference between orthogonal polarizations. Therefore, in the conventional theory, the paraxial approximation is assumed, two beams propagate almost in parallel. However, under this approximation, one cannot analyze interference fringes with a small grating period since a large crossing angle is required for the formation of dense fringes. In the present paper we shall show that the response of a polarization hologram can be expressed as the tensor product address: kuroda@iis.u-tokyo.ac.jp or dyadic product of the total electric field. y virtue of this rather simple analytic expression, we can separate the photoanisotropy into uniform and modulated terms. In the case of intensity holograms, the response is simply proportional to the intensity of the field. When the field is the sum of reference and signal beams, we consider only the cross term of the two fields, which causes the diffraction of the reference and signal waves, instead of the total intensity. This is valid even for polarization holography. ecause we only need the cross term between the signal and reference waves, and not the polarization state of the total field, this simplifies the theory of polarization holography substantially. Our theory is valid for both thin and thick holograms. In this paper we concentrate our discussion on thick volume holograms. The theory of volume holograms has been developed by Kogelnik. 19) Kogelnik s coupled wave theory has been extended to polarization holography by Huang and Wagner by using the paraxial approximation, the dielectric tensor of the recorded material is represented by a 2 2 Jones matrix. 17) In this paper we use a 3 3 full representation of the dielectric tensor so that we can remove the paraxial limitation. Our theory is applicable to any geometry of interfering beams, a large crossing angle, and a skew geometry with arbitrary polarization. 2. Polarization Holographic Materials 2.1 Molecular model of materials There are several kinds of polarization holographic materials. The origin of photoanisotropy is different for different kinds of materials. Photoisomerization, polarizationdependent photodissociation, and the photoinduced reorientation of linear molecules are typical origins of photoanisotropy. In this paper we do not investigate details of such materials but discuss the polarization-sensitive response based on a simple model. Suppose that a polarizationsensitive recording material consists of rod molecules with anisotropic polarizability. efore exposure, since the rod molecules are randomly distributed in all directions, the material is isotropic. y the absorption of optical radiation, the rod molecules are photodissociated into fragments of molecules. We assume that the resultant molecules are isotropic. The absorption coefficient depends on the polar- 374

3 OPTIAL REVIEW Vol. 18, No. 5 (211) K. KURODA et al. 375 Recording of interference pattern in polarization holo- Fig. 1. gram. G + q + x G ization. As a result, the materials become anisotropic after the illumination of polarized light. First of all, we define the coordinate system as follows. We take the xy axes on the surface of a holographic plate and the z axis to be the depth direction (see Fig. 1). Next we consider the orientation of molecules. Let the unit vector 1 1 d 1 sin cos d d 2 A sin sin A ð1þ cos d 3 q be the direction of the molecular axis in spherical coordinates ð; Þ. The molecules have anisotropic polarizability. Let a k and a? be the polarizability along and across the axis, respectively. The polarizability tensor of the molecule is hence written as 2) a ¼ a? 1 þða k a? Þdd; ð2þ 1 is the unit tensor whose components are given by the Kronecker delta jk, and dd is a tensor product or dyadic y z product of vector d, whose components are defined as ðddþ jk ¼ d j d k : In eq. (2), we expressed the polarization tensor a in a coordinate-free form. In order to express this equation in a coordinate-dependent form, which is usually used in the literature, we take the direction of d to be the z axis, then a is expressed as 1 a a? a? a k ð3þ A: ð4þ Equation (2) is obtained by rotating the molecule from the z axis to the direction d. 2.2 Averaging over molecular orientation To obtain macroscopic quantities, we have to average microscopic quantities over the whole molecular orientation. Let PðdÞd be the probability of molecules being directed in direction d, d ¼ sin dd is an element of the solid angle. Since the whole solid angle is 4, we have the probability density of uniformly oriented molecules of PðdÞ ¼ N 4 ; N is the density of molecules. We denote the average over the whole solid angle by h i. It should be noted that in averaging over the whole solid angle, the direction d and its inverse d are distinguished from each other. Since the physical quantities appearing in this paper do not depend on the sign of the direction, the direction d is counted twice. To avoid such double counting, we divide the number density of molecules so that half are in the d direction and the other half are in the d direction. In spherical coordinates, the components of tensor dd are expressed as ð5þ sin 2 cos 2 sin 2 cos sin 1 sin cos cos dd sin 2 sin cos sin 2 sin 2 sin cos sin A: ð6þ cos sin cos cos sin sin cos 2 The average of dd over the solid angle is given by hddi 1 ZZ dd d 4 3 1: ð7þ Since the distribution of the orientation of molecules is isotropic, the directions x, y, and z have the same weight. We later need the average of the tensor of rank 4, d ð4þ dddd, whose components are defined as d ð4þ jklm ¼ d jd k d l d m. This tensor is fully symmetric for the permutation of suffices ð jklmþ. The nonzero components are given by jjjj i¼1 5 ; jjkk i 15 ðj 6¼ kþ: ð8aþ ð8bþ The following results are verified. For diagonal components, we have 1111 i Z Z 2 sin 5 d cos 4 d ; ð9aþ 2222 i Z Z 2 sin 5 d sin 4 d ; ð9bþ

4 376 OPTIAL REVIEW Vol. 18, No. 5 (211) K. KURODA et al. Z 3333 i cos 4 sin d Z 2 5 : ð9cþ Next, we evaluate jjkk i as 1122 i Z Z 2 sin 5 d cos 2 sin 2 d ; ð1aþ 1133 i Z Z 2 sin 3 cos 2 d cos 2 d ; ð1bþ 2233 i Z Z 2 sin 3 cos 2 d sin 2 d : ð1cþ It is easy to verify that the remaining components vanish. 2.3 Initial state In the initial state before exposure, the material is isotropic and the probability density is given by eq. (5). Averaging over the orientation of molecules we have the average polarizability of hai ¼ha? 1 þða k a? Þddi ¼ 2a? þ a k 3 d 1 a I 1: ð11þ The scalar dielectric constant is hence given by I ¼ 1 þ a IN ; ð12þ is the permittivity in vacuum. The refractive index p is n ¼ ffiffiffiffi I. For simplicity we assume that the absorption by the material is small, then n and I are real. If the absorption is not negligible, the dielectric constant becomes complex and is expressed as I ¼ n 2 ð1 þ iþ2 is the absorption index. 2.4 Exposure The absorption coefficient of rod molecules is anisotropic. The probability of dissociation for the electric field components parallel and perpendicular to the axis can be written as k jd Ej 2 and? ðjej 2 jd Ej 2 Þ, respectively, d is the direction of the molecular axis and E expð i!tþ is the electric field of monochromatic light with angular frequency!. The coefficients? and k depend on the absorption cross section and exposure time. Then the reduction factor of the probability density of rod molecules f ðdþ is given by f ðdþ ¼1? jej 2 ð k? Þjd Ej 2 E ½? 1 þð k? ÞddŠE: ð13þ The anisotropic dissociation tensor? 1 þð k? Þdd has a similar form to the anisotropic polarizability tensor [eq. (2)]. The increase in the number of isotropic molecules is equal to the decrease in the number of rod molecules. The average polarizability after the exposure is given by ha L i¼hða? 1 þ addþfðdþi ¼ða I a?? jej 2 Þ1 a? jej 2 hddi a? he dd Ei ahe dddd Ei; ð14þ a ¼ a k a? and ¼ k?. The last term can be transformed to he dddd Ei 2jE 1 j 2 þjej 2 2<½E 1 E2 Š 2<½E 1E3 Š 1 2<½E 2 E1 15 Š 2jE 2j 2 þjej 2 2<½E 2 E3 A 2<½E 3 E1 Š 2<½E 3E2 Š 2jE 3j 2 þjej 2 15 ðjej2 1 þ 2<½EE ŠÞ; ð15þ < is an operator yielding the real part and E j is the jth component of vector E. Then we have ha L i¼ða I þ gjej 2 Þ1 þ 2h<½EE Š; ð16þ g ¼ a?? 1 3 ½a? þ a? Š 1 15 a ¼ 6a?? þ 4ða? k þ a k? Þþa k k ð17þ 15 h ¼ a : ð18þ 15 After the dissociation of rod molecules, the isotropic aggregation of molecules occurs. We set the polarizability of isotropic molecules as a s 1, then we have ha S i¼ a sð2? þ k Þ jej 2 1: ð19þ 3 Adding eqs. (16) and (19), we finally obtain hai ¼ða I þ g jej 2 Þ1 þ 2h<½EE Š; ð2þ g ¼ a sð2? þ k Þ þ g; ð21þ 3 and g is defined in eq. (17). The dielectric tensor is deduced from the average polarizability to be ¼ I 1 þ AjEj 2 1 þ ðee þ E EÞ; ð22þ A and are coefficients of the scalar and tensor components of the photoinduced change in the dielectric tensor, respectively, given by A ¼ N g ; ¼ N h: ð23þ

5 OPTIAL REVIEW Vol. 18, No. 5 (211) K. KURODA et al. 377 In the above notation, we have added a double underline to the dielectric tensor in order to distinguish tensors from scalar quantities. It should be noted that eq. (22) is valid for weak exposure. If the exposure of a polarization hologram is strong, i.e., it is exposed to intense illumination and/or for a long exposure time, we cannot ignore the depletion of rod molecules. Upon heavy exposure, the increase in anisotropy becomes saturated and then the anisotropy starts to decrease. Finally, when all the rod molecules are dissociated, the material becomes isotropic. 2.5 Phenomenological theory of response From the phenomenological point of view, we can derive eq. (22) in a more general form. If the recording material is isotropic before exposure, the vectors that appear in the problem are the electromagnetic fields, E and H. We can reasonably assume that only the electric field is responsible for the change in material constants at the optical frequency of the spectrum. Then, the quadratic quantities that includes the vector E are the scalar product jej 2 ¼ E E, vector product E E, and tensor product EE. The tensor that is formed by the scalar product is jej 2 1, 1 is the unit tensor. The tensor product is divided into two parts: a symmetric part EE þ E E, which causes the photoinduced birefringence and dichroism, and an antisymmetric part EE E E, which causes the photoinduced optical activity. From the vector product we can form the tensor ðe E Þ1, but this is equal to E E EE. Therefore, in the most general form, the dielectric tensor is written as ¼ I 1 þ AjEj 2 1 þ ðee þ E EÞþðEE E EÞ ¼ I 1 þ AjEj 2 1 þ 1 EE þ 2 E E; ð24þ A,, and are coefficients of the scalar, symmetric tensor, and antisymmetric tensor terms, respectively. In the second line, we write 1 ¼ þ and 2 ¼. Elliptically polarized light with the major and minor axes along the x and y axes, respectively, is expressed as E ¼ða; ib; Þ, a and b are real. Upon exposure to this polarization, the dielectric tensor becomes 2a 2 1 2iab ¼½ I þ Aða 2 þ b 2 ÞŠ1 þ 2iab 2b A: ð25þ Results equivalent to eq. (25) have been derived by Huang and Wagner 17) and Nikolova et al. 15) However, as mentioned in 1, they assumed the paraxial approximation, the crossing angle of the two beams is so small that the reference and signal beams propagate along almost the same direction. Under this assumption, they derived a 2 2 reduced matrix for the dielectric tensor or a 2 2 Jones matrix. In contrast, since eq. (24) is a 3 3 full expression of the dielectric tensor, it is applicable to an arbitrary geometry without limitation. We denote the signal by the suffix + and the reference by. Then, let E ¼ G þ e iqþr þ G e iq r ð26þ be the electric field during recording, G j is the vector amplitude of the electric field and q j is the wave vector (Fig. 1). Interference fringes of the grating vector K ¼ q þ q ð27þ are formed. After the exposure, the resultant dielectric tensor is written as ¼ I 1 þ D þ ðþþ H eikr þ ð Þ H e ikr ; ð28aþ D ¼ X j j j¼½ajg 2 1 þ 1 G j G j þ 2 G j G jš; ð28bþ ðþþ H þ G Þ1 þ 1G þ G þ 2G G þ; ð28cþ ð Þ H AðG þ G Þ1 þ 1 G þ G þ 2 G G þ : ð28dþ The dielectric tensor is divided into two parts, D and ðþ H. The former describes the self- and cross modulation of the optical constants. The latter is the interference term that causes the diffraction of reconstruction waves. 4. Reconstruction Let us consider the reconstruction of a polarization hologram whose dielectric tensor is given by eq. (28a). Here we deal with a thick volume hologram, the ragg condition should be satisfied in order to obtain high diffraction efficiency. Higher-order diffraction is negligible in volume holograms. As shown in Fig. 2, we read out the hologram with the reference beam of the wave vector k. We assume that the width of the recording material is infinitely large. Therefore, there is no tolerance to phase mismatch in the transverse direction, that is, the xy components of wave vectors satisfy the phasematching condition. Then the wave vector of the reconstructed wave is given by k þ ¼ k þ K þ M ^z; ð29þ x F Θ y z F + k + 3. Recording The modulation of the polarization state formed by twobeam interference is recorded in a polarization hologram. Fig. 2. k Reconstruction of the signal from polarization hologram.

6 378 OPTIAL REVIEW Vol. 18, No. 5 (211) K. KURODA et al. M is the deviation from the ragg condition along the z axis and ^z is the unit vector along the z axis. We introduce the coordinate systems associated with the reference and signal beams. We denote the unit vector along the wave vector as 1 ^k j ¼ k j jk j j ¼ sin j cos sin j sin j cos j A; ð3aþ the xyz coordinates are defined as drawn in Fig. 2, and ð j ; j Þ are the spherical coordinates of the wave vectors. It is noted that the xy axes are on the surface of the hologram, but we can take an arbitrary direction for the x axis. We define the remaining two unit vectors as 1 p j ¼ cos j ^k j ^z sin j s j ¼ ^k j p j ¼ cos j cos j cos j sin j sin j 1 sin cos j A; ð3bþ A; ð3cþ p j lies on the incident plane of the k j wave and s j is perpendicular to the incident plane. When the wave vector is parallel to the z axis, i.e., j ¼, we take j ¼. These p j, s j, and ^k j form a right-handed coordinate system. The electric field is the sum of the signal and reference waves, E ¼ F þ ðzþe ikþr þ F ðzþe ik r ; ð31aþ F j ðzþ ¼P j ðzþp j þ S j ðzþs j : ð31bþ In this expression, the electric field F j ðzþ is orthogonal to k j. Strictly speaking, the electric field is not orthogonal to the wave vector if the medium is nonuniform and/or anisotropic. The longitudinal component of the electric field is, however, small and has little influence on the development of the amplitude of the reconstructed wave. Therefore, we neglect the longitudinal component of the electric field. From Maxwell s equations under the assumption that the permeability is equal to that in vacuum, we have the wave equation k 2 E ¼rðrEÞ ¼ rðr EÞ r 2 E r 2 E; ð32þ k ¼!=c. The term rðr EÞ can be ignored because, as mentioned above, the longitudinal component of the electric field is negligible. Assuming a slowly varying amplitude, we have r 2 ðf j e ikjr Þjk j j 2 F j e ikjr 2iðk j ^zþ df j dz eik jr ; ð33þ ^z is the unit vector along the z axis. Next we evaluate the left-hand side of eq. (32). Taking into account eq. (28), we have E ¼ð I 1 þ D ÞðF þ e ikþr þ F e ik r Þ þ ðþþ H F e iðk þkþr þ ð Þ H F þe iðk þ KÞr ; ð34þ the higher-order diffraction is neglected. Separating the expðik rþ terms, we have the following vectorial coupled wave equations for F i n cos þ i n cos df þ dz df dz ¼ D F þ þ ðþþ H F e imz ; ð35aþ ¼ D F þ ð Þ H F þe imz ; ð35bþ M is the deviation from the ragg condition, defined by eq. (29), and is the wavelength in vacuum. In this derivation, we ignore the linear absorption of the holographic material, that is, we set I ¼ n 2, which is real. If material exhibits absorption, I has a nonzero imaginary part and the linear absorption terms should be added to the right-hand side of eq. (35). We finally derive the coupled wave equations for each polarization component S and P from eq. (35). Taking the inner product of eq. (35a) with s þ, we have ds þ i þ dz ¼ðs þ D s þ ÞS þ ðzþþðs þ D p þ ÞP þ ðzþ þ½ðs þ ðþþ s H ÞS ðzþ þðs þ ðþþ p H ÞP ðzþše imz ; ð36aþ and similarly, for P þ ðzþ, S ðzþ, and P ðzþ we have dp þ i þ dz ¼ðp þ D s þ ÞS þ ðzþþðp þ D p þ ÞP þ ðzþ þ½ðp þ ðþþ s H ÞS ðzþ þðp þ ðþþ p H ÞP ðzþše imz ; ð36bþ ds i dz ¼ðs D s ÞS ðzþþðs D p ÞP ðzþ þ½ðs ð Þ s H þþs þ ðzþ þðs ð Þ p H þþp þ ðzþše imz ; ð36cþ dp i dz ¼ðp D s ÞS ðzþþðp D p ÞP ðzþ þ½ðp ð Þ s H þþs þ ðzþ þðp ð Þ p H þþp þ ðzþše imz ; ð36dþ j ¼ n cos j : ð37þ It should be noted that the coupling coefficients are not always uniform. If the material exhibits linear absorption, which is neglected in the above derivation, the amplitudes of the recording beams are attenuated depending on the depth of propagation from the surface. In this case, the coupling coefficients are functions of the coordinate z. The intensity of the diffracted wave, or the diffraction efficiency, is derived by solving the coupled wave equations for S j ðzþ and P j ðzþ. In order to solve these equations. we have to distinguish between two kinds of holograms, i.e., fixed and dynamic holograms. For a fixed hologram, the modulation of the dielectric tensor is fixed after the recording. Once the hologram is fixed, it is insensitive to further illumination. In this case, the coefficients of the coupled wave equations are constants and the coupled wave

7 OPTIAL REVIEW Vol. 18, No. 5 (211) K. KURODA et al. 379 equations are linear equations in S j and P j. Once the eigenvalues of the linear equations are found, it is easy to construct the solution for an arbitrary initial or boundary condition. Let L be the thickness of the hologram. Then we integrate the coupled wave equations from z ¼ to L with the initial condition S þ ðþ ¼P þ ðþ ¼. The diffraction efficiency is given by ¼ js þðlþj 2 þjp þ ðlþj 2 js ðþj 2 þjp ðþj 2 : ð38þ If the absorption is not negligible, the internal diffraction efficiency is often used, which is given by js þ ðlþj 2 þjp þ ðlþj 2 ¼ js þ ðlþj 2 þjp þ ðlþj 2 þjs ðlþj 2 þjp ðlþj 2 : ð39þ The internal diffraction efficiency is defined as the intensity of the diffracted wave relative to the total output intensity instead of the input intensity. In a dynamic (or real-time) hologram, the holographic material responds to the optical field in real time, and the hologram is rewritten whenever the material is illuminated. A typical case is that the recording and reconstruction take place simultaneously. In this case, there is no distinction between the recording and reconstruction, that is, G j ¼ F j and q j ¼ k j. Self-diffraction changes the magnitudes, phases, and polarizations of incident waves during the propagation, and in turn the interference pattern is changed. The interaction between the signal and reference waves is analyzed by a self-consistent method. The coupled wave equations are the same as those for fixed holograms, but the coefficients are not constants and depend on S j and P j, and hence the coupled wave equations become nonlinear. It is not difficult to integrate the coupled wave equations numerically. In conventional intensity holography, such equations have been studied thoroughly for twowave and four-wave mixing in photorefractive materials. 21) In polarization holography, Huang and Wagner have analyzed the coupled wave equations under the paraxial approximation. 17) 5. Examples 5.1 Orthogonal linear polarization In the following examples, it is assumed for simplicity that the reference and signal beams q and the z axis lie on the same plane, say the xz plane. Then in the definition of the unit vector systems [eq. (3)], we set j ¼. The first example of polarization holography is the case of orthogonal linear polarization. Suppose that the reference wave is s-polarized and the signal is p-polarized. Then the electric field is written as E ¼ G þ p w þ eiqþr þ G s w eiq r ; ð4þ superscript w is added to polarization vectors p w j and s w j in order to explicitly show that these quantities are relevant to the writing (recording) process. Since the interfering waves are orthogonally polarized, the intensity interference patten is uniform. The total intensity is simply given as the sum of the individual intensities jej 2 ¼jG þ j 2 þjg j 2 : ð41þ The tensor is given by EE ¼jG þ j 2 p w þ pw þ þjg j 2 s w sw þ G þ G pw þ sw eikr þ G þ G s w pw þ e ikr ; ð42þ K ¼ q þ q. The dielectric tensors are D ¼ AðjG þ j 2 þjg j 2 Þ1 þ 2ðjG þ j 2 p w þ pw þ þjg j 2 s w sw Þ ð43aþ ðþþ H þg ð 1 p w þ sw þ 2s w pw þþ; ð43bþ ð Þ H G þ G ð 1 s w pw þ þ 2 p w þ sw Þ; ð43cþ 1 ¼ þ and 2 ¼. It is noted that the effect of the antisymmetric tensor disappears in D but it remains in ðþ. H Let us evaluate the coupling coefficients for the reconstruction process. In this evaluation, we ignore the deviation of the incident angle from the ragg condition. When the ragg condition is violated, the reduction of diffraction efficiency is mainly caused by the phase mismatch expðimzþ. In contrast, even if the incident angle of the reconstruction wave deviates from that of the recording wave, the change of the coupling coefficients is small. This small change has little influence on the growth of the amplitudes of the polarization components. Thus, we can ignore this small effect and evaluate the coupling coefficients under the ragg condition, that is, s j ¼ s w j and p j ¼ p w j. The matrix elements of D are given by p þ D p þ ¼ AðjG þ j 2 þjg j 2 Þþ2jG þ j 2 ð44aþ s þ D s þ ¼ AðjG þ j 2 þjg j 2 Þþ2jG j 2 ð44bþ p D p ¼ AðjG þ j 2 þjg j 2 Þþ2 2 jg þ j 2 ð44cþ s D s ¼ AðjG þ j 2 þjg j 2 Þþ2jG j 2 ; ð44dþ ¼ ^k þ ^k ¼ p þ p ¼ cos ð45þ is the cosine of the beam-crossing angle. The remaining components are. Next, the matrix elements of ðþ are H p þ ðþþ s H ¼ 1 G þ G ð46aþ s þ ðþþ p H ¼ 2 G þ G ð46bþ s ð Þ p H þ ¼ 1 G þ G ð46cþ p ð Þ s H þ ¼ 2 G þ G ð46dþ and the remaining components vanish. We can divide the coupled wave equations into two independent sets of equations. The first set for P þ and S is dp þ þ dz ¼ ifða þ 2ÞjG þj 2 þ jg j 2 gp þ ðzþ þ i 1 G þ G S ðzþe imz ; ð47aþ ds dz ¼ ifajg þj 2 þðaþ2þjg j 2 gs ðzþ þ i 1 G þ G P þ ðzþe imz ; ð47bþ

8 38 OPTIAL REVIEW Vol. 18, No. 5 (211) K. KURODA et al. and the other set for S þ and P is ds þ þ dz ¼ ifajg þj 2 þðaþ2þjg j 2 gs þ ðzþ þ i 2 G þ G P ðzþe imz ; ð48aþ dp dz ¼ ifða þ 22 ÞjG þ j 2 þ AjG j 2 gp ðzþ þ i 2 G þ G S þ ðzþe imz ; ð48bþ j is defined by eq. (37). This result indicates that if the reference reconstruction wave is s-polarized, then the reconstructed signal is p-polarized, and vice versa. As shown in eq. (48), the coupling constants are proportional to ¼ cos. Hence, if ¼ =2, i.e., ¼, no signal is reconstructed when the hologram is read out with a p- polarized reference wave. 5.2 Nonorthogonal linear polarization In this geometry, the reference is s-polarized but the signal is linearly polarized in an arbitrary direction. The polarization vector of the signal is expressed as u w þ ¼ cos pw þ þ sin sw þ ; ð49þ is the angle of the direction of the electric field from the p w þ axis. The orthogonal polarization to uw þ is v w þ ¼ sin pw þ þ cos sw þ : ð5þ The electric field is expressed as E ¼ G þ u w þ eiqþr þ G s w eiq r : ð51þ The intensity distribution is given by jej 2 ¼jG þ j 2 þjg j 2 þ sin ðg þ G eikr þ G þ G e ikr Þ; ð52þ and the tensor is EE ¼jG þ j 2 u w þ uw þ þjg j 2 s w sw þ G þ G uw þ sw eikr þ G þ G s w uw þ e ikr : ð53þ The dielectric tensors [eq. (28)] are D ¼ AðjG þ j 2 þjg j 2 Þ1 þ 2ðjG þ j 2 u w þ uw þ þjg j 2 s w sw Þ; ð54aþ ðþþ H þg ða sin 1 þ 1u w þ sw þ 2s w uw þþ; ð54bþ ð Þ H G þ G ða sin 1 þ 2 u w þ sw þ 1s w uw þþ; ð54cþ 1 ¼ þ and 2 ¼. Next we evaluate the coupling coefficients under the assumption that the ragg condition is satisfied. We have for ¼ H ðþþ=ðg H þg Þ u þ s H ¼ 1 þðaþ 2 Þ sin 2 ; ð55aþ v þ s H ¼ðAþ 2 Þ sin cos ; ð55bþ u þ p H ¼ðAþ 2 Þ sin cos ; ð55cþ v þ p H ¼ 2 ðaþ 2 Þ sin 2 ; ð55dþ ¼ p þ p. This result indicates that if sin cos 6¼, then both u þ and v þ polarizations are reconstructed when we read out the hologram with the reference beam s. In other words, if we use an arbitrary polarization state as the signal, the polarization of the reconstructed signal is different from that of the recorded signal. Only in the special case that the condition A þ 2 ¼ ð56þ is satisfied is the pure u þ polarization reconstructed. According to the model described in 2 ¼, the above condition is described using the microscopic quantities as A þ ¼ N ½ð5a s 3a? 2a k Þ k 15 þð1a s 7a? 3a k Þ? Š ¼ : ð57þ When the beam-crossing angle is =2 inside the material, ¼. In this case, eqs. (55c) and (55d) vanish. Hence, no signal is reconstructed by the p-polarized reference wave. When sin cos ¼ or A þ 2 ¼, the two sets of combinations ðu þ ; s Þ and ðv þ ; p Þ are independent. In this case, it is possible to store two different holograms using independent sets of polarizations. In the readout process, two signals are reconstructed simultaneously. However, since the polarizations of these two signals are orthogonal, we can extract each signal using a polarizer. 5.3 Orthogonal circular polarization Let us consider the formation of a polarization hologram in an orthogonal circular polarization geometry. In this geometry, we again assume that the wave vector q lies on the xz plane. Let us denote the left- and right-handed circular polarizations as h w j ¼ p 1 ffiffi ðs w j þ ip w j Þ; 2 ð58aþ m w j ¼ p 1 ffiffi ðs w j ip w j Þ: 2 ð58bþ It is noted that m j is the complex conjugate of h j, and vice versa. The electric field is expressed as E ¼ G þ h w þ eiqþr þ G m w eiq r : ð59þ The intensity distribution is given by jej 2 ¼jG þ j 2 þjg j 2 þ 1 ðg þ G 2 eikr þ G þ G e ikr Þ; ð6þ ¼ p w þ pw. It should be noted that, if the interfering beams intersect at a finite angle, the right- and left-handed circular polarizations are not orthogonal. As a result, we have the intensity interference pattern given by eq. (6). The tensor is given by EE ¼jG þ j 2 h w þ mw þ þjg j 2 m w hw þ G þ G hw þ hw eikr þ G þ G m w mw þ e ikr : ð61þ The dielectric tensors [eq. (28)] are D ¼ AðjG þ j 2 þjg j 2 Þ1 þjg þ j 2 ð 1 h w þ mw þ þ 2m w þ hw þ Þ þjg j 2 ð 1 m w hw þ 2h w mw Þ; ð62aþ

9 OPTIAL REVIEW Vol. 18, No. 5 (211) K. KURODA et al. 381 ðþþ ¼ G H þg 1 2 Að1 Þ1 þ 1h w þ hw þ 2h w hw þ ; ð62bþ X þ ¼ 1 ðg F ÞG þ ; ð64bþ X ¼ AðG þ G ÞF þ 2 ðg þ F ÞG : ð64cþ ð Þ ¼ H G þ G 1 2 Að1 Þ1 þ 1m w þ mw þ 2m w mw þ ; ð62cþ 1 ¼ þ and 2 ¼. We evaluate the coupling coefficients under the assumption that the ragg condition is satisfied. Since the polarization vectors are complex in the case of circular polarization, we take the inner product with the complex conjugate of the polarization vectors. Then we have for ¼ =ðg H þg Þ ðþþ H h þ m H ¼ A 4 ð1 Þ2 þ 1 þ 2 4 ð1 þ Þ2 ; ð63aþ m þ m H 4 ða þ 2Þð1 2 Þ; ð63bþ h þ H h 4 ða þ 2Þð1 2 Þ; m þ H h 4 ða þ 2Þð1 Þ 2 : ð63cþ ð63dþ The vector X is the sum of three vectors, each of which is a combination of the vectors G þ, G, and F. The signal F þ is generated by the source X, but this does not mean that F þ is proportional to X. This is because the polarization vector is perpendicular to the wave vector k þ, as X is not. Only the transverse component of X causes the growth of the signal. For simplicity we assume that the coupling coefficients are constants and that the depletion of the reconstruction wave is negligible. Then the vector X is constant and the reconstructed signal is proportional to the transverse component of X, F þ / X ðx ^k þ Þ ^k þ ¼ X þ þfx ðx ^k þ Þ ^k þ g: ð65þ The vector X þ is proportional to the signal G þ regardless of the polarization of the reconstruction wave. On the other hand, although X contains the information of the signal, its polarization state is different from that of the original signal. In this case, the reconstructed wave is not pure circular polarization unless the condition A þ 2 ¼ is satisfied. If A þ 2 ¼, then eqs. (63b), (63c), and (63d) vanish. In this case, only the left-handed circular polarization is reconstructed by the illumination of the right-handed reference wave. Moreover, when the hologram is read out by the left-handed reference wave, no signal is reconstructed. It is noted that both the transmitted and diffracted waves are left-handed circular polarization. It is interesting that if we regard this hologram as a polarization element, it converts all the input into left-handed circular polarization. However, this does not violate chiral symmetry, because if the light is incident on the hologram from the signal side, the input is converted into right-handed circular polarization. 6. Discussion 6.1 Polarization of reconstructed signal As discussed in 5.2 and 5.3, if the condition A þ 2 ¼ is satisfied, the polarization of the reconstructed signal is kept the same as that of the recording signal. The same condition has been derived under the paraxial approximation. 22) We shall show that this condition is valid in a general geometry. In the previous section, we discussed the coupled wave equations in some special geometries. Here we consider the reconstruction of the signal in a general geometry. Suppose that the polarization hologram is formed using the signal and reference beams G, then it is read out by the reconstruction beam F. What is the polarization of the reconstructed signal F þ? To symplify this problem, here we assume that the ragg condition is satisfied, that is, k ¼ q. The reconstructed signal F þ is generated from the coupling term of eq. (35a), which consists of two parts, ðþþ H F X ¼ X þ þ X ; ð64aþ 6.2 Fidelity of reconstructed signal We next discuss the fidelity of the reconstructed signal, that is, how close the amplitude and polarization of the reconstructed signal F þ are to those of the recording signal G þ. Here we concentrate our discussion on the fidelity of polarization. Let Q þ be the orthogonal polarization to the signal G þ. Since Q þ G þ ¼ Q þ ^k þ ¼, the condition that the reconstructed signal does not contain the component proportional to Q þ is given by Q þ X ¼ AðG þ G ÞðQ þ F Þ þ 2 ðg þ F ÞðQ þ G Þ ¼ : ð66þ If this condition is satisfied, the polarization of the original signal is reconstructed correctly. For linear polarization, the polarization vectors are real. We assume that the reference beam during the recording and that during the reconstruction are the same, i.e., G ¼ F, then the condition for correct reconstruction is given by ða þ 2 ÞðG þ G ÞðQ þ G Þ¼: ð67þ This condition is satisfied when G þ G ¼, Q þ G ¼, or A þ 2 ¼. In the last case, we can use any combination of linear polarization for recording and reconstruction without loss of fidelity. For circular polarization, suppose that the signal during recording is left-handed and the reference waves are righthanded, that is, G þ ¼ h þ ; G ¼ m ; F ¼ m ; ð68þ we neglect scalar amplitudes for simplicity. The orthogonal polarization to the signal is Q þ ¼ m þ. Then eq. (66) reads m þ X 4 ða þ 2Þð1 2 Þ¼: ð69þ

10 382 OPTIAL REVIEW Vol. 18, No. 5 (211) K. KURODA et al. For circular polarization, we obtain the same condition, A þ 2 ¼, for the fidelity of the reconstructed signal. From eq. (63), if A þ 2 ¼ is satisfied, no signal is reconstructed by the illumination of a left-handed circularly polarized reference wave. 7. onclusions We investigated the response of polarization holograms based on a model in which the materials consist of rod molecules that are transformed into spherical molecules after photodissociation. It is found that the response function of polarization holograms can be written as a simple expression. Using this expression, we have derived vectorial coupled wave equations that describe the interaction of incident waves during the propagation in polarization holograms. We studied the special geometries of orthogonal linear polarization, nonorthogonal linear polarization, and orthogonal circular polarization. Finally, we discussed the generation of the reconstructed signal in a general geometry and derived the condition that the polarization of the reconstructed wave must be equal to that of the original recorded signal. Acknowledgment This work was partially supported by the Strategic Promotion of Innovative Research and Development from Japan Science and Technology Agency (JST). References 1) L. Nikolova and P. S. Ramanujam: Polarization Holography (ambridge University Press, ambridge, U.K., 29). 2) L. Nikolova and T. Todorov: Opt. Acta 31 (1984) ) T. Todorov, L. Nikolova, and N. Tomova: Appl. Opt. 23 (1984) ) T. Todorov, L. Nikolova, and N. Tomova: Appl. Opt. 23 (1984) ) T. Todorov, L. Nikolova, K. Stoyanova, and N. Tomova: Appl. Opt. 24 (1985) ) J. J. outure: Appl. Opt. 3 (1991) ) S. S. Petrova: Tech. Phys. 46 (21) ) L. Nikolova, K. Stoyanov, T. Todorov, and V. Tatanenko: Opt. ommun. 64 (1987) 75. 9) V. I. Tarasashvili and A. L. Purtseladze: Opt. Spectrosc. 13 (27) 13. 1) D. arada, K. Tamura, T. Fukuda, M. Itoh, and T. Yatagai: Opt. Lett. 33 (28) ) D. arada, K. Tamura, T. Fukuda, and T. Yatagai: Jpn. J. Appl. Phys. 48 (29) 9LE2. 12) D. arada, Y. Kawagoe, K. Tamura, T. Fukuda, and T. Yatagai: Jpn. J. Appl. Phys. 49 (21) 1AD2. 13) Sh. D. Kakichashvili: Opt. Spectrosc. 33 (1972) ) Sh. D. Kakichashvili: Sov. J. Quantum Electron. 4 (1974) ) L. Nikolova, T. Todorov, M. Ivanov, F. Andruzzi, S. Hvilsted, and P. S. Ramanujam: Appl. Phys. 35 (1996) ) T. Huang and K. H. Wagner: J. Opt. Soc. Am. A 1 (1993) ) T. Huang and K. H. Wagner: IEEE J. Quantum Electron. 31 (1995) ) H. Ono, M. Nakamura, A. Emoto, and N. Kawatsuki: Jpn. J. Appl. Phys. 49 (21) ) H. Kogelnik: ell Syst. Tech. J. 48 (1969) ) H.. hen: Theory of Electromagnetic Waves A oordinate-free Approach (MacGraw-Hill, New York, 1985) hap ) P. Yeh: Introduction to Photorefractive Nonlinear Optics (Wiley, New York, 1993). 22) L. Nikolova and P. S. Ramanujam: Polarization Holography (ambridge University Press, ambridge, U.K., 29) p. 49.