Mathieu Function Solutions for the Photoacoustic Effect in Two- and Three-Dimensional Structures and Optical Traps

Binbin Wu and Gerald J. Diebold

The wave equation for the photoacoustic effect in a three-dimensional spherically symmetric, or two-dimensional structure where the compressibility or density varies sinusoidally in space reduces to an inhomogeneous Mathieu equation. As such, exact solutions for the photoacoustic pressure can be found in terms of either Mathieu functions, integer order Mathieu functions, or fractional order Mathieu functions, the last of these being of importance for problems pertaining to structures of finite dimensions. Here, frequency domain solutions are given for a spherical structure with material properties varying radially, and a two-dimensional structure with material variations in one direction. Solutions for the acoustic pressure are found that give closed form expressions for the resonance frequencies. It is also shown that Mathieu functions give solutions for the motion of an optically levitated sphere trapped in an intensity modulated, Gaussian laser beam. By determining the frequencies at which the motions of the sphere are largest, that is, where the Mathieu functions become unstable, it is shown that the trap can act to determine the radiation force relative to the gravitational force on the sphere.

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