Study on refractive index distribution of the square self-focusing lens

In order to acquire the refractive index distribution of a square self-focusing lens, a semi-empirical method was put forward.The diffusion equation's solution under the square boundary conditions was approximated by the solution under the circular boundary conditions. The four unknown coefficients of the approximate solution were acquired by the known refractive index of the square self-focusing lens, and the refractive index was obtained by Jamin interference. This method avoids the complex process of solving the diffusion equation under the square boundary conditions, and the obtained semi-empirical formula is very simple and easy calculation.The calculation result agrees well with the experimental data, the maximal relative error is 0.94%,and the average is less than 0.3%,this formula provides a basis for the theoretical analysis of imaging of the square self-mixing lenses arrays.