Matlab：拉盖尔-高斯光束单缝衍射

代码：

function main()
clc;
clear;
close all;
%% 参数设置
%   /* 公共参数 */
Nxy = 200;          %取样点数
lambda = 632e-9;    %波长632nm
k = 2*pi/lambda;    %波数
p = 0;
z = 0;
%% 不同拓扑荷数的单缝衍射图样
w0 = 1e-3;          %高斯项的束腰 1mm
d = 1e-3;           %缝宽 1mm
zf = 0.5;           %传输距离0.5m
Z_R = pi*w0^2/lambda;      %瑞利长度
w_z = w0*sqrt(1+(z/Z_R)^2);%光束在z位置的半径
%   /* 坐标设置 */
[x,y] = meshgrid(linspace(-1.5*w0,1.5*w0,Nxy));   %近场坐标
[theta,r] = cart2pol(x,y);
[xf,yf] = meshgrid(linspace(-2e-3,2e-3,Nxy)); %积分坐标
Ef = zeros(Nxy);
for m = -1:1              %拓扑荷数-1,0,1
    tic
    %   /* 光源部分 z=0时的L-G光束 */
    E0 = 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));
    I0 = E0.*conj(E0);      I0 = I0/max(max(I0));
    figure(1);subplot(1,3,m+2);
    mesh(x*1e2,y*1e2,I0);
    view(2);
    axis square;
    set(gca,'fontname','times new roman','fontsize',16);    %设置图形对象属性
    title(['m = ',num2str(m)],'fontname','华文中宋','fontsize',16);
    xlabel('{\itx}/cm','fontname','times new roman','fontsize',16);
    ylabel('{\ity}/cm','fontname','times new roman','fontsize',16);
    zlabel('归一化强度','fontname','华文中宋','fontsize',16);
    if m == 1
        suptitle('不同拓扑荷数对应的光源光场');
    end
    %   /* 单缝衍射 */
    for a = 1:Nxy
        for b = 1:Nxy
            Ef(a,b) = (-1i/lambda/zf)*exp(1i*k*zf)*sum(sum(E0.*SingleSeam(E0,d,3*w0).*exp(1i*k/2/zf*((xf(a,b)-x).^2+(yf(a,b)-y).^2))));
        end
    end
    If = Ef.*conj(Ef);      If = If/max(max(If));
    figure(2);subplot(1,3,m+2);
    mesh(xf*1e2,yf*1e2,If);
    colormap gray;
    view(2);
    axis square;
    set(gca,'fontname','times new roman','fontsize',16);    %设置图形对象属性
    title(['m = ',num2str(m)],'fontname','华文中宋','fontsize',16);
    xlabel('{\itx}/cm','fontname','times new roman','fontsize',16);
    ylabel('{\ity}/cm','fontname','times new roman','fontsize',16);
    zlabel('归一化强度','fontname','华文中宋','fontsize',16);
    if m == 1
        suptitle(['{\it\sigma}=',num2str(w0*1e3),'mm,{\ita}=',num2str(d*1e3),'mm,{\itz}= ',num2str(zf),'m时,衍射图样随拓扑荷数的变化']);
    end
    toc
end
%% 不同缝宽时的衍射图样
w0 = 2e-3;          %高斯项的束腰 2mm
m = 1;              %拓扑荷数 1
zf = 0.5;           %传输距离0.5
Z_R = pi*w0^2/lambda;      %瑞利长度
w_z = w0*sqrt(1+(z/Z_R)^2);%光束在z位置的半径
%   /* 坐标设置 */
[x,y] = meshgrid(linspace(-1.5*w0,1.5*w0,Nxy));   %近场坐标
[theta,r] = cart2pol(x,y);
[xf,yf] = meshgrid(linspace(-3e-3,3e-3,Nxy)); %积分坐标
E0 = 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));
I0 = E0.*conj(E0);      I0 = I0/max(max(I0));
figure(3);mesh(x*1e2,y*1e2,I0);
view(2);
set(gca,'fontname','times new roman','fontsize',16);    %设置图形对象属性
title(['m = ',num2str(m)],'fontname','华文中宋','fontsize',16);
xlabel('{\itx}/cm','fontname','times new roman','fontsize',16);
ylabel('{\ity}/cm','fontname','times new roman','fontsize',16);
zlabel('归一化强度','fontname','华文中宋','fontsize',16);
for d = 0.2e-3:0.2e-3:0.6e-3        %d = 0.2,0.4,0.6mm
    tic
    %   /* 单缝衍射 */
    for a = 1:Nxy
        for b = 1:Nxy
            Ef(a,b) = (-1i/lambda/zf)*exp(1i*k*zf)*sum(sum(E0.*SingleSeam(E0,d,3*w0).*exp(1i*k/2/zf*((xf(a,b)-x).^2+(yf(a,b)-y).^2))));
        end
    end
    If = Ef.*conj(Ef);      If = If/max(max(If));
    figure(4);subplot(1,3,round(d*1e3/0.2));
    mesh(xf*1e2,yf*1e2,If);
    colormap gray;
    view(2);
    axis square;
    set(gca,'fontname','times new roman','fontsize',16);    %设置图形对象属性
    title(['d = ',num2str(d*1e3),'mm'],'fontname','华文中宋','fontsize',16);
    xlabel('{\itx}/cm','fontname','times new roman','fontsize',16);
    ylabel('{\ity}/cm','fontname','times new roman','fontsize',16);
    zlabel('归一化强度','fontname','华文中宋','fontsize',16);
    if round(d*1e3/0.2) == 3
        suptitle(['{\it\sigma}=',num2str(w0*1e3),'mm,{\itl}=',num2str(m),',{\itz}= ',num2str(zf),'m时,衍射图样随缝宽的变化']);
    end
    toc
end
%% 不同光斑半径时的衍射图样
d = 1e-3;           %缝宽1mm
m = 1;              %拓扑荷数 1
zf = 0.5;           %传输距离0.5m
for w0 = 1e-3:1e-3:3e-3        %d = 1,2,3mm
    tic
    Z_R = pi*w0^2/lambda;      %瑞利长度
    w_z = w0*sqrt(1+(z/Z_R)^2);%光束在z位置的半径
    %   /* 坐标设置 */
    [x,y] = meshgrid(linspace(-1.5*w0,1.5*w0,Nxy));   %近场坐标
    [theta,r] = cart2pol(x,y);
    [xf,yf] = meshgrid(linspace(-4e-3,4e-3,Nxy)); %积分坐标
    E0 = 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));
    I0 = E0.*conj(E0);      I0 = I0/max(max(I0));
    figure(5);subplot(1,3,w0*1e3);
    mesh(x*1e2,y*1e2,I0);
    view(2);
    axis square;
    set(gca,'fontname','times new roman','fontsize',16);    %设置图形对象属性
    title(['\it\sigma = ',num2str(w0*1e3),'mm'],'fontname','华文中宋','fontsize',16);
    xlabel('{\itx}/cm','fontname','times new roman','fontsize',16);
    ylabel('{\ity}/cm','fontname','times new roman','fontsize',16);
    zlabel('归一化强度','fontname','华文中宋','fontsize',16);
    if m == 3e-3
        suptitle('不同光斑半径对应的光源光场');
    end
    %   /* 单缝衍射 */
    for a = 1:Nxy
        for b = 1:Nxy
            Ef(a,b) = (-1i/lambda/zf)*exp(1i*k*zf)*sum(sum(E0.*SingleSeam(E0,d,3*w0).*exp(1i*k/2/zf*((xf(a,b)-x).^2+(yf(a,b)-y).^2))));
        end
    end
    If = Ef.*conj(Ef);      If = If/max(max(If));
    figure(6);subplot(1,3,w0*1e3);
    mesh(xf*1e2,yf*1e2,If);
    colormap gray;
    view(2);
    axis square;
    set(gca,'fontname','times new roman','fontsize',16);    %设置图形对象属性
    title(['d = ',num2str(d*1e3),'mm'],'fontname','华文中宋','fontsize',16);
    xlabel('{\itx}/cm','fontname','times new roman','fontsize',16);
    ylabel('{\ity}/cm','fontname','times new roman','fontsize',16);
    zlabel('归一化强度','fontname','华文中宋','fontsize',16);
    if w0 == 3e-3
        suptitle(['{\ita}=',num2str(d*1e3),'mm,{\itl}=',num2str(m),',{\itz}= ',num2str(zf),'m时,衍射图样随光斑半径的变化']);
    end
    toc
end

end
%% 拉盖尔多项式
function result = laguerre(p,l,x)
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
%% 单缝透射函数
function [result] = SingleSeam(E0,d,L)
% d,L分别是缝宽以及x方向的取值长度
[Nx,Ny] = size(E0);
result = zeros(Nx,Ny);
X_begin = ceil((1-d/L)*Nx/2);
X_end = ceil((1+d/L)*Nx/2);
for a = X_begin:X_end
    result(:,a) = 1;
end

end

结果：



参考文献：
[1]高福海, 陈宝算, 蒲继雄,等. 拉盖尔-高斯光束经单缝后的光强分布和螺旋谱[J]. 激光与光电子学进展, 2011, 000(009):40-46.
[2]王涛, 蒲继雄. 涡旋光束单缝衍射的理论和实验研究[J]. 中国激光, 2009(11):136-141.