基于FFT以及次谐波的两种相位屏仿真方法

基于FFT（快速傅里叶变换）和次谐波补偿的两种相位屏仿真方法的MATLAB源程序。这些方法常用于大气湍流相位屏的模拟，适用于自适应光学、自由空间光通信等领域。

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方法1：基于FFT的经典相位屏生成

function [phase_screen] = fft_phase_screen(N, delta, r0, L0)
% FFT-based phase screen simulation
% Inputs:
%   N     : Number of grid points (N x N)
%   delta : Grid spacing [m]
%   r0    : Fried parameter [m]
%   L0    : Outer scale [m]
% Output:
%   phase_screen : Simulated phase screen [rad]

% Generate frequency grid
k = 2*pi/(N*delta) * [0:N/2-1, -N/2:-1];  % Spatial frequencies [rad/m]
[Kx, Ky] = meshgrid(k, k);
K = sqrt(Kx.^2 + Ky.^2);  % Radial wave number

% Set zero frequency to small value to avoid singularity
K(1,1) = 1e-6;

% Von Karman power spectrum
P_k = 0.023 * r0^(-5/3) * (K.^2 + 1/L0^2).^(-11/6);

% Generate random complex field with Gaussian statistics
rand_phase = 2*pi*rand(N);
H = sqrt(P_k) .* (randn(N) + 1i*randn(N)) .* exp(1i*rand_phase);

% Ensure Hermitian symmetry for real output
H(1,1) = 0;
H(N/2+1, N/2+1) = 0;
H(N/2+1,1) = 0;
H(1,N/2+1) = 0;

% Inverse FFT to get phase screen
phase_screen = real(ifft2(ifftshift(H)));

% Normalize to zero mean
phase_screen = phase_screen - mean(phase_screen(:));
end

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方法2：次谐波补偿相位屏生成

function [phase_screen] = subharmonic_phase_screen(N, delta, r0, L0, M)
% Subharmonic-compensated phase screen simulation
% Inputs:
%   N     : Number of grid points (N x N)
%   delta : Grid spacing [m]
%   r0    : Fried parameter [m]
%   L0    : Outer scale [m]
%   M     : Number of subharmonic layers
% Output:
%   phase_screen : Simulated phase screen [rad]

% Base FFT phase screen (high frequencies)
phase_base = fft_phase_screen(N, delta, r0, L0);

% Initialize subharmonic compensation
phase_sub = zeros(N);

% Generate subharmonic layers
for m = 1:M
    % Determine number of subharmonics for this layer
    n = 3^m;
    delta_k = 2*pi/(n*N*delta);  % Frequency step
    
    % Generate random coefficients
    a = randn(n, n);
    b = randn(n, n);
    
    % Create spatial grid
    [X, Y] = meshgrid((0:N-1)*delta);
    
    % Add subharmonic components
    for p = 1:n
        for q = 1:n
            kx = (p - n/2 - 0.5) * delta_k;
            ky = (q - n/2 - 0.5) * delta_k;
            
            % Von Karman spectrum value
            K_mag = sqrt(kx^2 + ky^2);
            P_k = 0.023 * r0^(-5/3) * (K_mag^2 + 1/L0^2)^(-11/6);
            
            % Add component to subharmonic phase
            phase_sub = phase_sub + sqrt(P_k) * ...
                       (a(p,q)*cos(kx*X + ky*Y) + ...
                        b(p,q)*sin(kx*X + ky*Y));
        end
    end
end

% Combine base and subharmonic components
phase_screen = phase_base + phase_sub;

% Normalize to zero mean
phase_screen = phase_screen - mean(phase_screen(:));
end

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方法使用

% Simulation parameters
N = 512;        % Grid size (512x512)
delta = 0.01;   % Grid spacing [m]
r0 = 0.1;       % Fried parameter [m]
L0 = 10;        % Outer scale [m]
M = 3;          % Subharmonic layers

% Generate phase screens
fft_screen = fft_phase_screen(N, delta, r0, L0);
sub_screen = subharmonic_phase_screen(N, delta, r0, L0, M);

% Visualization
figure;
subplot(1,2,1);
imagesc(fft_screen);
title('FFT Method');
axis square;
colorbar;

subplot(1,2,2);
imagesc(sub_screen);
title('Subharmonic Method');
axis square;
colorbar;
colormap(jet);

参考代码 基于FFT以及次谐波的两种相位屏仿真方法源程序 youwenfan.com/contentcnb/78668.html

关键参数说明

1. Fried参数 (r0):

   • 描述大气湍流强度
   • 典型值: 0.05-0.2 m (强湍流), >0.2 m (弱湍流)
   • 计算: \(r_0 = \left[0.423k^2\sec z \int C_n^2 dh\right]^{-3/5}\)

2. 外尺度 (L0):

   • 湍流涡旋的最大尺寸
   • 典型值: 10-100 m (地面), >100 m (高空)

3. 次谐波层数 (M):

   • 控制低频补偿精度
   • 典型值: 3-5 层
   • 增加层数提高低频精度但增加计算量

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方法对比

特性	FFT方法	次谐波补偿方法
计算复杂度	O(N² log N)	O(M·N²)
低频分量表现	不足	精确
相位结构函数精度	高空间频率精确	全频谱精确
适用场景	快速生成、高空间频率主导的湍流	精确模拟、特别是大尺度湍流结构
参数依赖	r0, L0	r0, L0, M

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物理背景

相位屏模拟基于大气湍流的Kolmogorov理论：

• 功率谱密度：\(\Phi_n(\kappa) = 0.033C_n^2\kappa^{-11/3}\)
• 相位结构函数：\(D_\phi(r) = 6.88\left(\frac{r}{r_0}\right)^{5/3}\)
• 次谐波补偿解决FFT方法在低频（\(\kappa < 2\pi/L_0\)）的采样不足问题

这些代码提供了大气湍流相位屏仿真的基础实现，可根据具体应用需求调整参数和优化算法。