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The laser photopyroelectric effect measures an optical absorption depth profile in a thin film through the spatial dependence of a heat flux source established below the film surface by light absorption from a short optical pulse. In this work, inverse depth profile reconstruction was achieved by means of an inverse method based on the expectation-minimum principle (as reported in a companion paper), applied in conjunction with a constrained least-squares minimization, to invert the photopyroelectric theory. This method and zero-order Tikhonov regularization were applied to the inversion of experimental photopyroelectric data obtained from samples with a variety of discrete and continuous depth dependences of optical absorption. While both methods were found to deliver stable and accurate performance under experimental conditions, the method based on the constrained expectation-minimum principle was found to exhibit improved resolution and robustness over zero-order Tikhonov regularization.

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The expectation-minimum (EM) principle is a new strategy for recovering robust solutions to the ill-posed inverse problems of photothermal science. The expectation-minimum principle uses the addition of well-characterized random noise to a model basis to be fitted to the experimental response by linear minimization or projection techniques. The addition of noise to the model basis improves the conditioning of the basis by many orders of magnitude. Multiple projections of the data onto the basis in the presence of noise are averaged, to give the solution vector as an expectation value which reliably estimates the global minimum solution for general cases, while the conventional approaches fail. This solution is very stable in the presence of random error on the data. The expectation-minimum principle has been demonstrated in conjunction with several projection algorithms. The nature of the solutions recovered by the expectation minimum principle is nearly independent of the minimization algorithms used and depends principally on the noise level set in the model basis.