Fredholm integral equation of the Laser Intensity Modulation Method (LIMM): Solution with the polynomial regularization and L-curve methods

The Laser Intensity Modulation Method (LIMM) is widely used for the determination of the spatial distribution of polarization in polar ceramics and polymers, and space charge in non-polar polymers. The analysis of experimental data requires a solution of a Fredholm integral equation of the 1st kind. This is an ill-posed problem that has multiple and very different solutions. One of the more frequently used methods of solution is based upon Tikhonov regularization. A new method, the Polynomial Regularization Method (PRM), was developed for solving the LIMM equation with an 8th degree polynomial using smoothing to achieve a stable and optimal solution. An algorithm based upon the L-curve method (LCM) was used for the prediction of the best regularization parameter. LIMM data were simulated for an arbitrary polarization distribution and were analyzed using PRM and LCM. The calculated distribution function was in good agreement with the simulated polarization distribution. Experimental polarization distributions in a poorly poled sample of polyvinylidene fluoride (PVDF) and in a LiNbO₃ bimorph, and space charge in polyethylene were analyzed. The new techniques were applied to the analysis of 3-dimensional polarization distributions.