matlab求解光学布洛赫方程

光学布洛赫方程表述了光在周期性介质中的传播，其公式如下：

$$i\frac{\partial \psi}{\partial z} = -\frac{\hbar^2}{2n_0}\frac{\partial^2 \psi}{\partial x^2} + \left[V(x) + \frac{\hbar^2 k^2}{2n_0} \right]\psi(x)$$

其中，$\psi(x)$表示波函数，$z$是传播距离，$x$是坐标，$n_0$是介质的折射率，$k$是波矢，$V(x)$是周期性势场。

Matlab可以通过数值方法求解光学布洛赫方程。例如，使用有限差分法（Finite Difference Method，FDM）可以将微分方程离散化，将空间和时间分为若干个网格点，并通过迭代方法求解方程。下面是一个简单的Matlab程序：

% Parameters 
n0 = 1.5;   % Refractive index 
k = 2*pi/633e-9/n0;   % Wavevector 
V0 = 5*k^2*n0;   % Potential strength 
L = 1e-5;   % Period length 
sigma = 0.1e-5;   % Pulse width 
zmax = 1e-3;   % Propagation distance 
Nz = 1000;   % Number of z steps 
Nx = 256;   % Number of x steps 
dx = L/Nx;   % Spatial step 
dt = 1e-17;   % Time step 
Tmax = 1e-14;   % Maximum time 
Nt = round(Tmax/dt);   % Number of time steps 
x = linspace(-L/2, L/2, Nx);   % x coordinates 

% Initialize wavefunction 
nx0 = round(Nx/2);   % Center index 
psi0 = exp(-x.^2/(2*sigma^2));   % Gaussian wavepacket 
psi = zeros(Nx,Nz);   % Wavefunction 

% Solve PDE using FDM 
for j = 1:Nt 
    % Finite difference scheme 
    psi(nx0,1) = exp(-1i*k*x(nx0)*j*dt)*psi0(nx0); 
    for n = 1:Nz-1 
        psi(1,n+1) = psi(1,n)*(1+1i*dt/2/dx^2*(1i*k+n0*sqrt(V0/2/n0-L/2/n0*exp(-1i*2*pi/L*x(1))))) - 1i*dt/2/dx^2*(1i*k+n0*sqrt(V0/2/n0-L/2/n0*exp(-1i*2*pi/L*x(2))))*psi(2,n); 
        for k = 2:Nx-1 
            psi(k,n+1) = psi(k,n)*(1+1i*dt/dx^2*(1i*k+n0*sqrt(V0/2/n0-L/2/n0*exp(-1i*2*pi/L*x(k)))))-1i*dt/2/dx^2*(1i*k+n0*sqrt(V0/2/n0-L/2/n0*exp(-1i*2*pi/L*x(k+1))))*psi(k+1,n)-1i*dt/2/dx^2*(1i*k+n0*sqrt(V0/2/n0-L/2/n0*exp(-1i*2*pi/L*x(k-1))))*psi(k-1,n); 
        end 
        psi(Nx,n+1) = psi(Nx,n)*(1+1i*dt/2/dx^2*(1i*k+n0*sqrt(V0/2/n0-L/2/n0*exp(-1i*2*pi/L*x(Nx))))) - 1i*dt/2/dx^2*(1i*k+n0*sqrt(V0/2/n0-L/2/n0*exp(-1i*2*pi/L*x(Nx-1))))*psi(Nx-1,n); 
    end 
end 

% Plot wavefunction 
figure; 
plot(x,real(psi(:,end))); 
xlabel('x (m)'); ylabel('Real(psi)'); 

此程序通过FDM解决该方程，并产生波函数的实部的数字表示。程序允许改变各个参数，并可用于研究不同情况下的解决方案。