Matlab：厄米-高斯光束合成拉盖尔-高斯光束

代码：

clc
clear all
close all
%%
N = 300;            %采样点数
lambda = 632e-9;    %波长632nm
k = 2*pi/lambda;    %波数
w0 = 3e-3;          %束腰半径3mm
Z_R = pi*w0^2/lambda;      %瑞利长度
row = linspace(-3*w0,3*w0,N);   col = row;
[x,y] = meshgrid(row,col);
[theta,r] = cart2pol(x,y);
z = 1e-9;           %z=0时w_z会无穷大，暂时不知道如何解决，只能用一个相当小的z来等效z=0
w_z = w0*sqrt(1+(z/Z_R)^2);%光束在z位置的半径
p = 0;      l = 1;
E_01 = Hermite(p,sqrt(2)*x/w_z).*Hermite(l,sqrt(2)*x/w_z).*exp(-1i*(p+l+1)/tan(z/Z_R)).*exp(1i*k*r.^2/2/(z-1i*Z_R));
I_01 = E_01.*conj(E_01);      I_01 = I_01/max(max(I_01));

figure;
subplot(2,3,1)
h_01 = pcolor(x,y,I_01);
set(h_01,'edgecolor','none','facecolor','interp');
title('HG_0_1');
axis square;

subplot(2,3,4)
E_01_phase = angle(E_01);
h_01_phase = pcolor(x,y,E_01_phase);
set(h_01_phase,'edgecolor','none','facecolor','interp');
title('HG_0_1  phase');
axis square;

p = 1;      l = 0;
E_10 = Hermite(p,sqrt(2)*y/w_z).*Hermite(l,sqrt(2)*y/w_z).*exp(-1i*(p+l+1)/tan(z/Z_R)).*exp(1i*k*r.^2/2/(z-1i*Z_R));
I_10 = E_10.*conj(E_10);      I_10 = I_10/max(max(I_10));

subplot(2,3,2)
h_10 = pcolor(x,y,I_10);
set(h_10,'edgecolor','none','facecolor','interp');
title('HG_1_0');
axis square;

subplot(2,3,5)
E_10_phase = angle(E_10);
h_10_phase = pcolor(x,y,E_10_phase);
set(h_10_phase,'edgecolor','none','facecolor','interp');
title('HG_1_0  phase');
axis square;

E = E_01+1i*E_10;
I = E.*conj(E);     I = I/max(max(I));
subplot(2,3,3)
h = pcolor(x,y,I);
set(h,'edgecolor','none','facecolor','interp');
title('LG_0_1');
axis square;

subplot(2,3,6)
E_phase = angle(E);
h_phase = pcolor(x,y,E_phase);
set(h_phase,'edgecolor','none','facecolor','interp');
title('LG_0_1  phase');
axis square;
%% 厄米特多项式
function result = Hermite(n,x)
if n <= 0
    result = 1;
elseif n == 1
    result = 2*x;
else
    result = 2*x*Hermite(n-1,x)-2*(n-1)*Hermite(n-2,x);
end
end

运行结果：

参考文献：
[1]葛甜. 利用光纤产生涡旋光束的理论与实验研究[D].
[2]Yao A M , Padgett M J . Orbital angular momentum: origins, behavior and applications[J]. Advances in Optics and Photonics, 2011, 3(2):161-204.