zernike的多项式有多种定义，如下图
而Zermax中的Standard多项式的定义为：
那么对应的matlab实现为
- function [z,zer]= zemaxZernike(res)
- % ref1 - Optical Shop Testing, Third Edition (2007) P517 TABLE 13.5.
- % ref2 - https://blog.csdn.net/xsz591541060/article/details/124275188
- % ref3 - Zemax中Zernike Standard多项式 https://zhuanlan.zhihu.com/p/667867516
- % 参数j： Zernike多项式的序号
- % 参数res：Zernike多项式的分辨率
- relative = 1;%设置为单位圆
- x = linspace(-relative,relative,res);
- [x,y] = meshgrid(x,x);
- [theta,rho] = cart2pol(x,y);% 由(x,y)换算(r,theta)
- z{1} = ones(size(x)).*1/sqrt(pi); %Piston
- z{2} = rho .* cos(theta).*2/sqrt(pi); %x tilt
- z{3} = rho .* sin(theta).*2/sqrt(pi); %y tilt
- z{4} = (2.*rho.^2 - 1) .* sqrt(3/pi); %Defocus
- z{5} = rho.^2 .* sin(2.*theta) .*sqrt(6/pi); %Primary astigmatism at 45°
- z{6} = rho.^2 .* cos(2.*theta) .* sqrt(6/pi); %Primary astigmatism at 0°
- z{7} = (3.*rho.^3 - 2.*rho) .* sin(theta) .* sqrt(8/pi); %Primary y coma
- z{8} = (3.*rho.^3 - 2.*rho) .* cos(theta) .* sqrt(8/pi); %Primary x coma
- z{9} = rho.^3 .* sin(3.*theta) .* sqrt(8/pi);
- z{10} = rho.^3 .* cos(3.*theta) .* sqrt(8/pi);
- z{11} = (6.*rho.^4 - 6.*rho.^2 + 1).*sqrt(5/pi); %Primary spherical
- z{12} = (4.*rho.^4 - 3.*rho.^2) .* cos(2.*theta) .* sqrt(10/pi); %Secondary astigmatism at 0°
- z{13} = (4.*rho.^4 - 3.*rho.^2) .* sin(2.*theta) .*sqrt(10/pi); %Secondary astigmatism at 45°
- z{14} = rho.^4 .* cos(4.*theta) .* sqrt(10/pi);
- z{15} = rho.^4 .* sin(4.*theta) .*sqrt(10/pi);
- z{16} = (10.*rho.^5 - 12.*rho.^3 + 3.*rho) .* cos(theta) .* sqrt(12/pi); %Secondary x coma
- z{17} = (10.*rho.^5 - 12.*rho.^3 + 3.*rho) .* sin(theta) .* sqrt(12/pi); %Secondary y coma
- z{18} = (5.*rho.^5 - 4.*rho.^3) .* cos(3.*theta) .* sqrt(12/pi);
- z{19} = (5.*rho.^5 - 4.*rho.^3) .* sin(3.*theta) .* sqrt(12/pi);
- z{20} = rho.^5 .* cos(5.*theta) .* sqrt(12/pi);
- z{21} = rho.^5 .* sin(5.*theta) .* sqrt(12/pi);
- z{22} = (20.*rho.^6 - 30.*rho.^4 + 12.*rho.^2 + 1).* sqrt(7/pi); %Secondary spherical
- z{23} = (15.*rho.^6 - 20.*rho.^4 + 6.*rho.^2) .* sin(2.*theta) .* sqrt(14/pi);%Tertiary astigmatism at 45
- z{24} = (15.*rho.^6 - 20.*rho.^4 + 6.*rho.^2) .* cos(2.*theta) .* sqrt(14/pi);%Tertiary astigmatism at 0
- z{25} = (6.*rho.^6 - 5.*rho.^4) .* sin(4.*theta) .* sqrt(14/pi);
- z{26} = (6.*rho.^6 - 5.*rho.^4) .* cos(4.*theta) .* sqrt(14/pi);
- z{27} = rho.^6 .* sin(6.*theta) .* sqrt(14/pi);
- z{28} = rho.^6 .* cos(6.*theta) .* sqrt(14/pi);
- z{29} = (35.*rho.^7 - 60.*rho.^5 + 30.*rho.^3 - 4.*rho) .* sin(theta) .* sqrt(16/pi);%Tertiary y coma
- z{30} = (35.*rho.^7 - 60.*rho.^5 + 30.*rho.^3 - 4.*rho) .* cos(theta) .* sqrt(16/pi);%Tertiary x coma
- z{31} = (21.*rho.^7 - 30.*rho.^5 + 10.*rho.^3) .* sin(3.*theta) .* sqrt(16/pi);
- z{32} = (21.*rho.^7 - 30.*rho.^5 + 10.*rho.^3) .* cos(3.*theta) .* sqrt(16/pi);
- z{33} = (7.*rho.^7 - 6.*rho.^5) .* sin(5.*theta) .* sqrt(16/pi);
- z{34} = (7.*rho.^7 - 6.*rho.^5) .* cos(5.*theta) .* sqrt(16/pi);
- z{35} = rho.^7 .* sin(7.*theta) .* sqrt(16/pi);
- z{36} = rho.^7 .* cos(7.*theta) .* sqrt(16/pi);
- z{37} = (70.*rho.^8 - 140.*rho.^6 + 90.*rho.^4 - 20.*rho.^2 + 1) .* sin(theta) .* sqrt(9/pi);%Tertiary spherical
- zer = zeros(res,res,37);
- for n = 1:37
- z{n}(rho>relative) = 0; % 只保留单位圆内的数据
- z{n} = z{n}/max(abs(z{n}(:)));
- zer(:,:,n) = z{n};
- %z{n} = z{n}/max(abs(z{n}(:)));归一化
- end
- end
- %%demo
- % res = 256;
- % z = zemaxZernike(res);
- % figure;
- % for n = 1:37
- % subplot(5,8,n);imshow(z{n},[]);colormap jet;colorbar;xlabel(['Z',num2str(n)]);
- % end
发表于 2024-2-20 15:28