Matlab：拉盖尔高斯光束的干涉
代码：
clc
clear all
close all
%% 拉盖尔-高斯光束与平面光波干涉
N = 300; %取样点数
lambda = 632e-9; %波长632nm
k = 2*pi/lambda; %波数
w0 = 3;
x = linspace(-1e-4,1e-4,N); y = x;
[X,Y] = meshgrid(x,y);
[theta,r] = cart2pol(X,Y);
p = 0;
Z_R = pi*w0^2/lambda; %瑞利长度
z = 0;
w_z = w0*sqrt(1+(z/Z_R)^2);%光束在z位置的半径
figure;
for m = -4:4
E1 = exp(-1i*k*X); %平面波
E2 = sqrt(2*factorial(p)/pi/(p+factorial(abs(m))))*(1/w_z)*(sqrt(2)*r/w_z).^abs(m)...
.*exp(-r.^2/w_z^2).*laguerre(p,abs(m),2*r.^2/w_z^2).*exp(-1i*m*theta).*exp(-1i*k*z)...
.*exp(-1i*k*r.^2*z/2/(z^2+Z_R^2))*exp(-1i*(2*p+abs(m)+1)*atan(z/Z_R));
c1 = E1+E2;
E_1 = c1.*conj(c1);
subplot(3,3,m+5)
h1 = pcolor(X,Y,E_1);
colorbar;
set(h1,'edgecolor','none','facecolor','interp');
title(['m = ',num2str(m)]);
colormap(gray);
axis square;
end
suptitle('拉盖尔-高斯光束与平面光波干涉') %为图一添加总标题
%% 涡旋光束与球面光波干涉
N = 200; %取样点数
lambda = 632e-9; %波长632nm
k = 2*pi/lambda; %波数
x = linspace(-2e-3,2e-3,N); y = x;
[X,Y] = meshgrid(x,y);
[theta,r] = cart2pol(X,Y);
Z = 1;
Z_R = pi*w0^2/lambda; %瑞利长度
z = 0;
w_z = w0*sqrt(1+(z/Z_R)^2);%光束在z位置的半径
figure;
for m = -4:4
E3 = exp(-1i*k*Z*(1+0.5*X.^2/Z^2+0.5*Y.^2/Z^2)); %球面波
E4 = sqrt(2*factorial(p)/pi/(p+factorial(abs(m))))*(1/w_z)*(sqrt(2)*r/w_z).^abs(m)...
.*exp(-r.^2/w_z^2).*laguerre(p,abs(m),2*r.^2/w_z^2).*exp(-1i*m*theta).*exp(-1i*k*z)...
.*exp(-1i*k*r.^2*z/2/(z^2+Z_R^2))*exp(-1i*(2*p+abs(m)+1)*atan(z/Z_R));
c2 = E3+E4;
E_2 = c2.*conj(c2);
subplot(3,3,m+5)
h2 = pcolor(X,Y,E_2);
colorbar;
set(h2,'edgecolor','none','facecolor','interp');
title(['m = ',num2str(m)]);
colormap(gray);
axis square;
end
suptitle('拉盖尔高斯光束与球面光波干涉') %为图二添加总标题
%% 拓扑荷数相反的拉盖尔高斯光束互相干涉
N = 200; %取样点数
lambda = 632e-9; %波长632nm
k = 2*pi/lambda; %波数
x = linspace(-8,8,N); y = x;
[X,Y] = meshgrid(x,y);
[theta,r] = cart2pol(X,Y);
Z_R = pi*w0^2/lambda; %瑞利长度
z = 0;
w_z = w0*sqrt(1+(z/Z_R)^2);%光束在z位置的半径
figure;
for m = 1:9
E5 = sqrt(2*factorial(p)/pi/(p+factorial(abs(-m))))*(1/w_z)*(sqrt(2)*r/w_z).^abs(-m)...
.*exp(-r.^2/w_z^2).*laguerre(p,abs(-m),2*r.^2/w_z^2).*exp(-1i*(-m)*theta).*exp(-1i*k*z)...
.*exp(-1i*k*r.^2*z/2/(z^2+Z_R^2))*exp(-1i*(2*p+abs(-m)+1)*atan(z/Z_R));
E6 = sqrt(2*factorial(p)/pi/(p+factorial(abs(m))))*(1/w_z)*(sqrt(2)*r/w_z).^abs(m)...
.*exp(-r.^2/w_z^2).*laguerre(p,abs(m),2*r.^2/w_z^2).*exp(-1i*m*theta).*exp(-1i*k*z)...
.*exp(-1i*k*r.^2*z/2/(z^2+Z_R^2))*exp(-1i*(2*p+abs(m)+1)*atan(z/Z_R));
c2 = E5+E6;
E_3 = c2.*conj(c2);
subplot(3,3,m)
h3 = pcolor(X,Y,E_3);
colorbar;
set(h3,'edgecolor','none','facecolor','interp');
title(['m = ',num2str(m)]);
colormap(gray);
axis square;
end
suptitle('拓扑荷数相反的拉盖尔高斯光束互相干涉') %为图三添加总标题
%% 拉盖尔多项式(文献3中的公式)
function result = laguerre(p,l,x)
result = 0;
if p == 0
result = 1;
elseif p == 1
result = 1+abs(l)-x;
else
result = (1/p)*((2*p+l-1-x).*laguerre(p-1,abs(l),x)-(p+l-1)*laguerre(p-2,abs(l),x));
end
end
运行结果：
参考文献：
[1]徐丽娟. 涡旋光束的产生及特性研究[D]. 浙江大学, 2014.
[2]赵麒, 白忠臣, 周骅,等. 拉盖尔-高斯光束作用下熔石英温度及应力研究%Research of temperature and thermal stress of fused silica irradiated by Laguerre-Gaussian beam[J]. 激光技术, 2018, 042(001):121-126.
[3]石业娇. 面向Fredholm微分方程的广义拉盖尔多项式求解方法[J]. 湘潭大学自然科学学报, 2018, 040(001):31-35.