The direction of displacement of a thickness shear mode transducer is calculated from the piezoelectric constant e, elastic constant c^E and dielectric constant ε^s of the medium. The X-cut plates of LiTaO_3 and LiNbO_3 single crystals excite two pure shear waves, one of which corresponds to the fast shear wave and the other to the slow shear wave. The direction of displacement to the y-axis θ_± of these waves are expressed by Eq. 9, where the positive and negative subscripts correspond to the fast and slow waves respectively, ρ is the density and v_± is the velocity of shear waves expressed by Eq. 8. From Eqs. 8 and 9, it is shown that the angle between θ_+ and θ_- is rectangular. The coupling constants k_± and the frequency constants are listed in Table 1 which were reported by Yamada et al. and Warner et al. As the fast shear wave has a larger coupling constant, this mode is dominant in the transducers. The calculated value of the direction of displacement for the fast shear wave is -48° for LiTaO_3 and -47° for LiNbO_3 respectively, where the negative sign shows the angle from the y-axis toward the negative z-axis. To confirm these calculated values, it is disired to measure the direction of displacement. Such a measurement is done using the light diffraction phenomenon due to an acoustic wave travelling in a transparent isotropic medium. The acoustic wave is excited by a testing transducer as shown in Fig. 2, and the displacement is assumed to be in the same direction as that of the transducer. The stress S caused by the transducer is divided into two components S_<12> and S_<13> paralled to the y- and z- axes respectively. In an isotropic medium, the light is diffracted by the stress perpendicular to the light ray direction. Thus the intensities of the diffracted light I_y and I_z are expressed by Eq. 5 and Eq. 6 where L_y and L_z are the edge lengths of transducer, I_<0y> and I_<0z> are the intensities of the incident light paralled to the y- and z- -axes respectively. The direction of displacement is determined by Eq. 7. Fig. 3 is the block diagram of the measureing apparatus. Since the direction of polarization of the diffracted light by a shear acoustic wave is rotated by, 90°, the crossed polarizers are used to detect only the diffracted light. In table 2 the sizes and the resonant and anti-resonant frequencies of the testing transducers are listed. In Fig. 5 the input power of transducer versus the intensity of diffracted light for the specimen of LiTaO_3 is plotted, where the parameter is the exciting frequency. From this figure it is found that the measured value is -49°±2° to the y-axis as shown in Fig. 6 and does not depend on frequency. This value agrees fairly well with the calculated value from the data of Yamada et al. In Fig. 9 the input power versus intencity of diffracted light is plotted for LiNbO_3. In this case, differing from LiTaO_3, the direction slightly depends on frequency, i. e. , at 13. 8 MHz, θ is equal to -52°±2° and at 11. 1 MHz, θ=-55°±2°. This dependence on the frequency is considered as follows: For LiNbO_3 the resonant and anti-resonant frequencies of the slow shear wave lie between those of the fast shear wave as shown in Table 1. Thus below the anti-resonant frequency of slow wave f_<a->, both of the fast and slow modes may be excited. On the other hand, above f_<a->, only the fast mode may be excited. For the specimen used in the experiment, f_<a-> is equal to 11. 8 MHz, so the value θ=-52° at 13. 8 MHz corresponds to the fast shear mode, although the value slightly differs from the calculated one -47°. On the other hand, the value θ=-55° at 11. 1 MHz may be affected by the slow mode, so the difference from the calculated value is larger than that at 13. 8 MHz.
First the relations amongst various theoretical parameters-θ, D, v and ξ(eqs. (4)〜(7))-concerning the phenomenon of the diffraction of light by ultrasound are discussed and presented in Fig. 1, together with their relations with experimental parameters-acoustic frequency and output-for water under certain experimental conditon. Then the present state of the correspondence between theory and experiments is discussed and summarized in Fig. 2. Under certain conditions (eq. (13)), a simple approximation, the phase-lattice approximation, is valid both for normal incidence of light and for oblique incidence, and this makes the starting point of the present calculation. We subdivide the parallel beam of sound in thin layers or subbeams of equal thickness each, and consider the light diffraction by ultrasound as a multiple or successive diffraction process by these sub-beams, assuming phase-lattice approximation for each step diffraction taking place by these sub-beams with various angles of incidence. Generally speaking, the s-th order spectrum emergent from the j-th sub-beam of sound consists of many components as shown in Fig. 4. Especially, the light incident in the j-th sub-beam in the direction of the zero-order spectrum and diffracted as the s-th order by crossing this layer -the (0, s) component-has the amplitude and phase pressented in eq. (14) according to the phase-lattice theory for unit incident light-0-order-in the j-th layer and with the phase referred to point B in Fig. 3 and 4. The other component-the (p, s)-component-, produced from the light incident in the direction of the p-th order spectrum on the j-th layer and diffracted as the m-th order (m=s-p) by this layer has the phase difference (16) with respect to the (0, s)-compont, while the phase-lattice parameter for determining the amplitude becomes (22). In calculating the phase-difference (16), the effective shift of the phase-lattice with respect to the sound wave (Fig. 3) in case of the oblique incidence is taken into account. The phase-difference has the symmetry property (17) for the changes of signs of p and s. The rule for constructing the (j+1)th layer complex amplitudes of the diffraction spectra from the j-th layer complex amplitudes becomes (21), and this results in the working formulas (25) and (26) when resolved into real and imaginary parts by (24) by taking account of the symmetry property (23). The result of the computation revealed that a normalization process as indicated in (28) becomes necessary in practice. With decreasing step-width of computation, however, the normalization constant approaches unity as indicated in Table 1. The results of the computation for various of θ ranging from 0. 1 to 6 are shown in Fig. 5〜15 together with the comparison with the exact results by the extended Brillouin theory (Nomoto, to be published). While the agreement has not been satisfactory for the range of small θ(≦0. 5), except for θ=0. 1, rather good agreements have been obtained for the range θ=1〜6. The discrepancy for smaller range of θ is due to the somewhat too rough step-width of D(=0. 05) employed in computation for this range. As the requirement for making the step of D sufficiently small (D=0. 01〜0. 02) becomes more easy to be fulfilled for larger values of θ, the present method is estimated to be a good method of approximation for obtaining the intensity distribution of the ultrasonic lightdiffraction spectrum, not only for the range θ=1〜6, but also for the range θ>6, where exact results are not available as yet except for θ=10. The present paper lays emphasis on the evaluation of the method, and the results for range θ=7〜100 are to be published elsewhere.