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泽尼克多项式

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泽尼克多项式是一组正交多项式，它们出现在具有圆形光瞳的光学系统的波前函数展开中。奇数和偶数泽尼克多项式由下式给出

^oU_n^m(rho,phi); ^eU_n^m(rho,phi)=R_n^m(rho)sin; cos(mphi)
(1)

其中径向函数 R_n^m(rho) 定义为 n 和 m 整数，且满足 n>=m>=0，由下式给出

R_n^m(rho)={sum_(l=0)^((n-m)/2)((-1)^l(n-l)!)/(l![1/2(n+m)-l]![1/2(n-m)-l]!)rho^(n-2l) for n-m even; 0 for n-m odd.
(2)

这里，phi 是方位角，满足 0<=phi<2pi，rho 是径向距离，满足 0<=rho<=1 (Prata 和 Rusch 1989)。偶数和奇数多项式有时也表示为

Z_n^(-m)(rho,phi)=^oU_n^m(rho,phi)=R_n^m(rho)sin(mphi)
(3)
Z_n^m(rho,phi)=^eU_n^m(rho,phi)=R_n^m(rho)cos(mphi).
(4)

泽尼克多项式在 Wolfram 语言中实现为ZernikeR[n, m, rho].

R_n^m(rho) 的其他闭合形式包括

R_n^m(rho)=(Gamma(n+1)_2F_1(-1/2(m+n),1/2(m-n);-n;rho^(-2)))/(Gamma(1/2(2+n-m))Gamma(1/2(2+n+m)))rho^n
(5)

对于 n-m 奇数且 m!=n，其中 Gamma(z) 是 伽玛函数，_2F_1(a,b;c;z) 是 超几何函数。这也可以用 雅可比多项式 P_n^((alpha,beta))(x) 表示为

R_n^m(rho)=(-1)^((n-m)/2)rho^mP_((n-m)/2)^((m,0))(1-2rho^2).
(6)

前几个非零径向多项式为

R_0^0(rho)=1
(7)
R_1^1(rho)=rho
(8)
R_2^0(rho)=2rho^2-1
(9)
R_2^2(rho)=rho^2
(10)
R_3^1(rho)=3rho^3-2rho
(11)
R_3^3(rho)=rho^3
(12)
R_4^0(rho)=6rho^4-6rho^2+1
(13)
R_4^2(rho)=4rho^4-3rho^2
(14)
R_4^4(rho)=rho^4
(15)

(Born 和 Wolf 1989, p. 465)。

径向函数满足正交关系

int_0^1R_n^m(rho)R_(n^')^m(rho)rhodrho=1/(2(n+1))delta_(nn^')R_n^m(1),
(16)

其中 delta_(ij) 是 克罗内克 delta，并且与第一类 贝塞尔函数相关，关系如下

int_0^1R_n^m(rho)J_m(vrho)rhodrho=(-1)^((n-m)/2)(J_(n+1)(v))/v
(17)

(Born 和 Wolf 1989, p. 466)。径向泽尼克多项式具有生成函数

([1+z-sqrt(1+2z(1-2rho^2)+z^2)]^m)/((2zrho)^msqrt(1+2z(1-2rho^2)+z^2))=sum_(s=0)^inftyz^sR_(m+2s)^(+/-m)(rho)
(18)

(更正了 Born 和 Wolf 的排版错误) 并且被归一化，使得

R_n^m(1)=1
(19)

(Born 和 Wolf 1989, p. 465)。

泽尼克多项式也满足递推关系

rhoR_n^m(rho)=1/(2(n+1))[(n+m+2)R_(n+1)^(m+1)(rho)+(n-m)R_(n-1)^(m+1)(rho)] R_(n+2)^m(rho)=(n+2)/((n+2)^2-m^2){[4(n+1)rho^2-((n+m)^2)/n-((n-m+2)^2)/(n+2)]R_n^m(rho)-(n^2-m^2)/nR_(n-2)^m(rho)} R_n^m(rho)+R_n^(m+2)(rho)=1/(n+1)(d[R_(n+1)^(m+1)(rho)-R_(n-1)^(m+1)(rho)])/(drho)
(20)

(Prata 和 Rusch 1989)。任意径向函数 F(rho,phi) 按照泽尼克多项式展开的系数 A_n^m 和 B_n^m

F(rho,phi)=sum_(m=0)^inftysum_(n=m)^infty[A_n^m^oU_n^m(rho,phi)+B_n^m^eU_n^m(rho,phi)]
(21)

由下式给出

A_n^m; B_n^m=((n+1))/(epsilon_(mn)^2pi)int_0^1int_0^(2pi)F(rho,phi)^oU_n^m(rho,phi); ^eU_n^m(rho,phi)rhodphidrho,
(22)

其中

epsilon_(mn)={epsilon=1/(sqrt(2)) for m=0, n!=0; 1 otherwise
(23)

设“初级”像差由下式给出

Phi=a_(lmn)^'Y^__1^(2l+m)(theta,phi)rho^ncos^mtheta
(24)

其中 2l+m+n=4，并且 Y^_ 是 复共轭 Y，定义

A_(lmn)^'=a_(lmn)^'Y^__1^(2l+m)(theta,phi),
(25)

得到

Phi=1/(epsilon_(nm)^2)A_(lmn)R_n^m(rho)cos(mtheta).
(26)

然后，初级像差的类型在下表中给出（Born 和 Wolf 1989, p. 470）。

像差lnmAA^'
球差040A_(040)^'rho^4epsilonA_(040)R_4^0(rho)
彗差031A_(031)^'rho^3costhetaA_(031)R_3^1(rho)costheta
像散022A_(022)^'rho^2cos^2thetaA_(022)R_2^2(rho)cos(2theta)
场曲120A_(120)^'rho^2epsilonA_(120)R_2^0(rho)
畸变111A_(111)^'rhocosthetaA_(111)R_1^1(rho)costheta

另请参阅

雅可比多项式

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参考文献

Bezdidko, S. N. "The Use of Zernike Polynomials in Optics." Sov. J. Opt. Techn. 41, 425, 1974.Bhatia, A. B. and Wolf, E. "On the Circle Polynomials of Zernike and Related Orthogonal Sets." Proc. Cambridge Phil. Soc. 50, 40, 1954.Born, M. and Wolf, E. "The Diffraction Theory of Aberrations." Ch. 9 in Principles of Optics: Electromagnetic Theory of Propagation, Interference, and Diffraction of Light, 6th ed. New York: Pergamon Press, pp. 459-490, 1989.Mahajan, V. N. "Zernike Circle Polynomials and Optical Aberrations of Systems with Circular Pupils." In Engineering and Lab. Notes 17 (Ed. R. R. Shannon), p. S-21, Aug. 1994.Prata, A. and Rusch, W. V. T. "Algorithm for Computation of Zernike Polynomials Expansion Coefficients." Appl. Opt. 28, 749-754, 1989.Wang, J. Y. and Silva, D. E. "Wave-Front Interpretation with Zernike Polynomials." Appl. Opt. 19, 1510-1518, 1980.Wyant, J. C. "Zernike Polynomials." http://wyant.optics.arizona.edu/zernikes/zernikes.htm.Zernike, F. "Beugungstheorie des Schneidenverfahrens und seiner verbesserten Form, der Phasenkontrastmethode." Physica 1, 689-704, 1934.Zhang, S. and Shannon, R. R. "Catalog of Spot Diagrams." Ch. 4 in Applied Optics and Optical Engineering, Vol. 11. New York: Academic Press, p. 201, 1992.

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泽尼克多项式

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Weisstein, Eric W. "泽尼克多项式。" 来自 Web 资源。 https://mathworld.net.cn/ZernikePolynomial.html

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