CAJ | 학술논문

Zernike多项式拟合是一种在光学领域中广泛应用的分析技术。由于现代光学工程中采集数据的离散性和非圆孔径系统的大量使用，Zernike多项式拟合不能完全满足分析需要。提出了一种基于Zernike多项式的非圆孔径离散采样点的正交多项式。通过矩阵的QR分解方法得到在离散采样点上的正交多项式基底。分别使用Zernike多项式和正交多项式对150 mm×90 mm的矩形光栅反射波前进行拟合，结果表明两种方法残差波前的PV和RMS值分别相差0.013波长和小于0.001波长。对比不同项数拟合的正交多项式和Zernike多项式系数表明，正交多项式系数之间彼此独立，并由正交多项式系数计算得到了对应的Seidel像差。正交多项式各项系数可以逐项求解，该方法可以显著提高求解速度。
Zernike다항식의합시일충재광학영역중엄범응용적분석기술。유우현대광학공정중채집수거적리산성화비원공경계통적대량사용，Zernike다항식의합불능완전만족분석수요。제출료일충기우Zernike다항식적비원공경리산채양점적정교다항식。통과구진적QR분해방법득도재리산채양점상적정교다항식기저。분별사용Zernike다항식화정교다항식대150 mm×90 mm적구형광책반사파전진행의합，결과표명량충방법잔차파전적PV화RMS치분별상차0.013파장화소우0.001파장。대비불동항수의합적정교다항식화Zernike다항식계수표명，정교다항식계수지간피차독립，병유정교다항식계수계산득도료대응적Seidel상차。정교다항식각항계수가이축항구해，해방법가이현저제고구해속도。
The Zernike polynomial is a widely used analytical technique in optics. Because of the discrete sampled measurement data and widely used non-circle aperture system in modern optical engineering, Zernike polynomial fitting can not satisfy a requirement completely. A kind of non-circle aperture discrete sampled orthogonal polynomial based on Zernike polynomial was proposed. The orthogonal basis was obtained using matrix QR decomposition method for discrete samples. Zernike polynomial and orthogonal polynomial were used for fitting 150 mm×90 mm rectangular grating wave-front. The differences of PV and RMS between two methods are 0.013 waves and less than 0.001 waves respectively for the residual wave-front. Comparison of different order fitting of the orthogonal polynomial and Zernike polynomial coefficients, indicate that the orthogonal polynomial coefficients are independent of each other. And the corresponding Seidel aberrations were calculated by the orthogonal polynomial coefficients. Orthogonal polynomial coefficients can be solved one by one. This method can significantly improve the solution speed.