方形自聚焦透镜的折射率分布研究

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

为了获得方形自聚焦透镜的折射率分布,提出了一种求解其折射率分布的半经验方法。该方法利用圆形边界条件下获得的扩散方程的解去近似方形边界条件下扩散方程的解,该近似解中的4个待定系数用雅明干涉法测得的方形自聚焦透镜4个位置点上的折射率来确定。该方法避免了在方形边界条件下求解扩散方程的复杂过程,得到的半经验公式形式简单、计算方便,利用半经验公式计算得到的折射率与实验结果吻合得较好,二者之间的最大相对误差为0.94%,平均相对误差不超过0.3%。该公式为以后研究方形自聚焦透镜阵列成像问题提供了可供参考的理论依据。

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.