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# Directional Dependence of Acousto-Optic Figure of Merit

The acousto-optic figure of merit (FOM) indicates the ideal coupling orientation to maximize the refractive index change due to an acoustic signal.
The FOM is a function of the acoustic wave velocity, density, refractive index, and effective elasto-optic coefficient. All of these values (except the density) can be anisotropic. Three wave velocities may exist in any given direction: two transverse waves (primary oscillation perpendicular to the wave propagation direction) and one longitudinal wave (polarization in the same direction as the propagation). Select the type of wave (transverse or longitudinal) and one of four materials to plot. Certain directions have a very high FOM, indicating an ideal orientation for maximum interaction.

### DETAILS

The figure of merit is calculated as , where is the refractive index, is the elastic constant, is the density, and is the velocity. The velocity of sound in materials depends on the elastic constant and the density, and is expressed as . Velocity calculations can often be very simple, but because of the anisotropy of many materials, the elastic constant is actually a matrix of values. Solving for the velocity of the wave is done by using the Christoffel equation. This Demonstration also considers the contribution from the piezoelectric coupling. An excellent treatment of solving the Christoffel equation can be found in [2]. The directional dependent refractive index is calculated a bit differently. The refractive index is transformed into using the directional cosine matrix, . The entry in the first row and first column of this matrix is the refractive index in the specified direction. The elasto-optic coefficient is calculated similarly, the only difference is the use of the matrix . By combining these calculations, the FOM can be determined in any direction. It is important to note that there are other types of acousto-optic FOM calculations—this one is perhaps the most common and indicates the best direction for a large diffraction of an incident optical signal. This calculation only considers the longitudinal elasto-optic effect, which considers an induced strain in an arbitrary direction that causes a refractive index change in the same direction.
The coefficients used are from a variety of sources:
All of the elastic constants, density, and refractive index values are from [1].
The piezoelectric constants of are from [2].
The permittivity of and ZnO are from [3].
The elasto-optic coefficients of and are from [2]; those of and ZnO are from [1].
Further details can be found in [4].
References
[1] M. J. Weber, Handbook of Optical Materials, Boca Raton: CRC Press, 2002.
[2] R. E. Newnham, Properties of Materials: Anisotropy, Symmetry, Structure, Oxford, New York: Oxford University Press, 2005.
[3] Comsol Multiphysics. Material Library v4.3. (Dec 17, 2013) www.comsol.com/material-library.
[4] R. McIntosh, A. S. Bhalla, R. Guo, "Finite Element Modeling of Acousto-Optic Effect and Optimization of the Figure of Merit," Proceedings SPIE 8497, Photonic Fiber and Crystal Devices: Advances in Materials and Innovations in Device Applications VI, 849703, Oct 15, 2012. doi:10.1117/12.956441.

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