光瞳半径对纯位相调制激光束整形系统的影响

光瞳截断作用对纯位相调制激光束整形系统的系统性能有重要影响. 本文提出了一种关于光瞳半径对近场调制位相和远场系统评价函数影响的定量分析方法. 通过拉格朗日乘数法等方法分析光瞳半径对近场调制位相的影响, 结果表明: 近场调制位相随光瞳半径近似线性增加. 通过建立数学模型, 拟合分析光瞳半径对系统评价函数的影响. 结果表明: 对方形目标光强, 系统评价函数拟合精度确定系数达到99%左右, 光瞳半径为2.5倍高斯光束束腰半径时, 相关系数达到0.997, 光强偏离残差平方均值达到0.0004左右, 光瞳截断作用趋于最小; 对圆形目标光强, 系统评价函数拟合精度确定系数达到97%, 光瞳半径为3倍高斯光束束腰半径时, 相关系数与光强偏离残差平方均值变化幅度均在10^-3量级, 系统评价函数趋于收敛, 光瞳截断作用趋于最小.

Influence of pupil on the laser beam shaping system by pure phase modulation

In this paper, we propose a quantitative approach to analyze the influence of pupil truncation on the phase-only modulation laser beam shaping system, based on the near-field phase and the far-field metric functions. First, the relationship between near-field phase and pupil radius is studied by Lagrange multiplier method. Result indicates that both the peak-to-valley and the root-mean-square of the near-field phase increase approximately linearly with the pupil radius. Second, the influence of pupil radius on a beam shaping system is investigated. To quantify the performance of the beam shaping system, the correlation coefficient (C) and the mean square difference (MSD) are introduced as the metric functions. Then, by comparing the metric functions at different pupil radius, it is shown that the pupil radius influences the performance of focal beam shaping distinctly at the lower pupil radius, whereas the influence trails off, and both the C and the MSD get close to the theoretical limit as the pupil radius continuously increases. Third, the mathematical models of the C and the MSD are proposed to reveal the relationship among the metric functions, pupil radius and target intensity's size, as it is difficult to obtain the explicit expressions on the basis of metric functions' definition. And the three coefficients in each model are ascertained by surface fitting method based on the sampling data. In addition, SSE (sum of square due to error), RMSE (root mean square error) and R-square (coefficient of determination) are adopted to determine the fitting precision. For both the metric functions, the precision of SSE and RMSE can reach 10^-2 and the R-square is shown to be more than 97%. The SSE, RMSE and R-square verify the proposed mathematical models. Finally, according to the models, we analyze when the influence of pupil truncation becomes negligible for the rectangle or circle target intensity. In practice, the size of target intensity is determined first. Sequentially, by combining the mathematical models and their first-order partial differentials, the changing regularity of metric functions with respect to pupil radius is studied. Meanwhile, the regularity helps us to find the beginning points for rectangle target and circle target intensities respectively. For the rectangle target intensity, when the pupil radius is 2.5 times that of the Gaussian waist radius, the metric functions become stable. The C with a value of 0.997 and the MSD with a value of 4×10^-4 are both close to the theoretical limit. In the meantime, the influence of pupil truncation tends to be minimal as expected. For circle target intensity, when the pupil radius is 3 times that of the Gaussian waist radius, the first-order partial differentials of the C and the MSD decrease to about 10^-3. This means that the metric functions begin to converge and that the influence of pupil truncation tends to be minimal at this point. Consequently, it is effective and meaningful to determine the best pupil radius using the proposed models in the article when designing a beam shaping system. Moreover, the models can also be used to evaluate the performance of a laser beam shaping system.