Chapter 4 夫琅禾费衍射-Matlab仿真
此内容理论部分出自书《Numerical Simulation Of Optical Wave Propagation With Example In Matlab》第四章 "Fraunhofer Diffraction and Lenses"
1、Fraunhofer Diffraction （夫琅禾费衍射）理论：
当光从一个很远的source aperture传播过来，在探测平面其光场分布可以近似为夫琅禾费衍射的积分：
(4.1)
其中，这个“很远”的距离满足公式：
(4.2)
其中，
为传播距离，D为source aperture的直径，
为光波长，
为源平面的坐标，
为观测平面坐标。
则公式（4.1）可以用傅里叶变换的方式表示：
(4.3)
因此可以利用傅里叶变换来计算这个夫琅禾费衍射积分
公式（4.3）中，
为轴上相位（on-axis phase）（在下文Matlab程序中，会暂时被忽略掉。）
Note：
为了获得一个准确的仿真结果，在程序当中应注意采样参数的设置，即grid spacing δ，spacial frequency的设置。 从上述公式推导可见，观测平面的坐标与源的空间频率存在以下关系：
(4.4a)
(4.4b)
则当观测平面的尺寸确定，则便会确定可观测到的来自源平面的最大空间频率：
(4.5a)
(4.5b)
2、Matlab code:
例程仿真的是一个单色平面波从一个圆形孔阑出射，传播到观测平面上，描绘其光场分布情况。
- 主程序：
% example_fraunhofer_circ.m
% Simulating a Fraunhofer diffraction pattern with comparison to the
% analtic result
close all; clear all; clc;
N = 512; % number of grid points per side
L = 7.5e-3; % total size of the grid [m]
delta = L/N; % source-plane grid spacing [m]
D = 1e-3; % diameter of the aperture [m]
lambda = 1e-6; % optical wavelength [m]
k = 2*pi/lambda;
Dz = 20; % propagation distance [m]
x = (-N/2:N/2-1)*delta;
[x1,y1] = meshgrid(x);
Uin = circ(x1,y1,D);
[Uout, x2, y2] = fraunhofer_prop(Uin, lambda, delta, Dz);
Iout = Uout .* conj(Uout); % Numerical intensity
% analytic result
Uout_th = exp(1i*k/(2*Dz)*(x2.^2+y2.^2))...
/ (1i*lambda*Dz)*D^2*pi/4 ...
.*jinc(D*sqrt(x2.^2+y2.^2)/(lambda*Dz));
Iout_th = Uout_th .* conj(Uout_th); % analytic intensity
% line profile
U_y0 = Uout(:,find(x==0)); % Amplitude
U_y0_th = Uout_th(:,find(x==0));
I_y0 = Iout(:,find(x==0)); % Intensity
I_y0_th = Iout_th(:,find(x==0));
%
figure,
imagesc(x2(1,:),y2(:,1),Iout)
axis([-0.1 0.1 -0.1 0.1]);axis square;
title('Numerical Intensity On detection plane')
figure,
imagesc(x2(1,:),y2(:,1),Iout_th),
axis([-0.1 0.1 -0.1 0.1]);axis square;
title('Analytic Intensity On detection plane')
figure,
plot(x2(1,:),I_y0,'x',x2(1,:),I_y0_th,'-s','linewidth',1.2)
xlim([-0.2 0.2]); xlabel('x [m]'); ylabel('Intensity');
legend('Numerical', 'Analytic')
grid on
figure,
plot(x2(1,:),abs(U_y0),'x',x2(1,:),abs(U_y0_th),'-s','linewidth',1.2)
xlim([-0.2 0.2]); xlabel('x [m]'); ylabel('Irradiance');
legend('Numerical', 'Analytic')
grid on- 调用子函数：
function [Uout, x2, y2] = fraunhofer_prop(Uin, lambda, delta, Dz)
% function [Uout, x2, y2] = fraunhofer_prop(Uin, wave, delta, Dz)
% performing a Fraunhofer propagation
% Uin : Source-plane optical field
% lambda : wavelength of the source
% delta : grid spacing in source plane
% Dz : the distance between object and lens
% Uout : optical field of observation plane
% x2, y2 : the coordinate in the observation plane
N = size(Uin,1); % assume square grid
k = 2*pi/lambda; % optical wavevector
fx = (-N/2:N/2-1)/(N*delta);
% observation plane coordinates
[x2, y2] = meshgrid(lambda * Dz * fx);
clear('fx');
Uout = exp(1i*k/(2*Dz)*(x2.^2+y2.^2))...
/ (1i*lambda*Dz) .* ft2(Uin,delta);
function z = circ(x,y,D)
% matlab code for evaluating the circ function
r = sqrt(x.^2+y.^2);
z = double(rfunction y = jinc(x)
% function y = jinc(x)
% code for evaluating the jinc function
y = ones(size(x));
idx = x ~= 0;
y(idx) = 2.0*besselj(1, pi*x(idx)) ./ (pi*x(idx));function G = ft2(g, delta)
% function G = ft2(g, delta)
G = fftshift(fft2(fftshift(g))) * delta^2;3、运行结果：
