Optimal control of laser-generated acoustic waves in solids

This paper is concerned with the design of optimal surface-heating patterns that result in focusing acoustic energy inside a subsurface target volume at a specified target time. The surface of the solid is heated by an incident laser beam which gives rise to shear and compressional waves propagating into the solid. The optimal heating design process aims to achieve the desired energy focusing at the target with minimal laser power densities and minimal system disturbance away from the target. Due to the slow motion of the thermal conduction process relative to the propagation of acoustic waves, a general formulation is derived which is the limiting case for a heat-absorbing nonconducting solid. This is consistent with the laser heating of the surface where, for the time duration of interest, thermal effects are confined to a thin layer at the surface. This simplification allows for the solution of the problem through the use of the system Greens function for the purely elastic medium where thermal expansion enters as an external force. The problem is then posed as an optimal control problem, the solution of which is the required heating pattern at the surface. The optimality conditions are secured via the conjugate gradient method and the mechanics of the elastic medium is treated by the finite element method along with using the half-space Greens function matrix. Good quality energy focusing is achieved, with the optimal designs reflecting the high directivity of the photothermally generated shear-wave patterns.