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本文目录如下：

目录

1 概述

2 运行结果

3 参考文献

4 Matlab代码实现

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1 概述

经环形孔径衍射的径向偏振光束的非近轴传播是一个复杂的问题，涉及到光的衍射、偏振和传播特性的综合考虑。在非近轴传播情况下，光束的传播方向和光束的强度分布会发生变化，需要使用适当的数值方法进行模拟和分析。

对于这个问题，可以考虑使用数值计算方法，如有限差分法（Finite Difference Method，FDM）、有限元法（Finite Element Method，FEM）或光线追迹法（Ray Tracing Method）等。这些方法可以用于求解光的传播方程和偏振方程，并模拟光束在非近轴传播过程中的衍射和偏振变化。

本文基于矢量瑞利衍射积分，推导了径向偏振拉盖尔-高斯光束在环形孔径衍射后的电磁场解析表达式。

2 运行结果

部分代码：

factorH1=factorial(n+1).*2.^(l-1)./(factorial(l+1).*factorial(l).*factorial(n-l).*w0.^(2.*l+1));
factorH2=((i.*k.*r-2)./r.^4).*(rho(s,j).*(k.*rho(s,j)./(2.*r)).^(2.*u+1)./(factorial(u).*factorial(u+1))-i.*(k.*rho(s,j)./(2.*r)).^(2.*u)./(factorial(u).*factorial(u))).*(mfun('GAMMA',l+u+2,-g.*b.^2)-factorial(l+u+1))./g.^(l+u+2);
factorH3=(2+(i.*k.*r-2).*z.^2./r.^2).*(k.*rho(s,j)./(2.*r)).^(2.*u+1)./(rho(s,j).*r.^2.*factorial(u).*factorial(u+1)).*(mfun('GAMMA',l+u+2,-g.*b.^2)-factorial(l+u+1))./g.^(l+u+2);
factorH4=k./(2.*r.^5).*((i.*(3.*rho(s,j)-2.*r.^2./rho(s,j)).*(k.*rho(s,j)./(2.*r)).^(2.*u+1)./(factorial(u).*factorial(u+1))+z.^2.*(k.*rho(s,j)./(2.*r)).^(2.*u+2)./(factorial(u).*factorial(u+2))+(k.*rho(s,j)./(2.*r)).^(2.*u)./(factorial(u).*factorial(u))).*(mfun('GAMMA',l+u+3,-g.*b.^2)-factorial(l+u+2))./g.^(l+u+3)+z.^2.*(k.*rho(s,j)./(2.*r)).^(2.*u)./(factorial(u).*factorial(u)).*(mfun('GAMMA',l+u+2,-g.*b.^2)-factorial(l+u+1))./g.^(l+u+2));
factorH5=-k.*z.^2./(2.*r.^5).*((-i.*(k.*rho(s,j)./(2.*r)).^(2.*u+1)./(rho(s,j).*factorial(u).*factorial(u+1))+(k.*rho(s,j)./(2.*r)).^(2.*u+2)./(factorial(u).*factorial(u+2))).*(mfun('GAMMA',l+u+3,-g.*b.^2)-factorial(l+u+2))./g.^(l+u+3)+(k.*rho(s,j)./(2.*r)).^(2.*u)./(factorial(u).*factorial(u)).*(mfun('GAMMA',l+u+2,-g.*b.^2)-factorial(l+u+1))./g.^(l+u+2));
factorH=factorH1.*(factorH2+factorH3+factorH4+factorH5);
Hxsum0b=Hxsum0b+factorH;
Hysum0b=Hysum0b+factorH;
l=1;
factorH1=factorial(n+1).*2.^(l-1)./(factorial(l+1).*factorial(l).*factorial(n-l).*w0.^(2.*l+1));
factorH2=((i.*k.*r-2)./r.^4).*(rho(s,j).*(k.*rho(s,j)./(2.*r)).^(2.*u+1)./(factorial(u).*factorial(u+1))-i.*(k.*rho(s,j)./(2.*r)).^(2.*u)./(factorial(u).*factorial(u))).*(mfun('GAMMA',l+u+2,-g.*b.^2)-factorial(l+u+1))./g.^(l+u+2);
factorH3=(2+(i.*k.*r-2).*z.^2./r.^2).*(k.*rho(s,j)./(2.*r)).^(2.*u+1)./(rho(s,j).*r.^2.*factorial(u).*factorial(u+1)).*(mfun('GAMMA',l+u+2,-g.*b.^2)-factorial(l+u+1))./g.^(l+u+2);
factorH4=k./(2.*r.^5).*((i.*(3.*rho(s,j)-2.*r.^2./rho(s,j)).*(k.*rho(s,j)./(2.*r)).^(2.*u+1)./(factorial(u).*factorial(u+1))+z.^2.*(k.*rho(s,j)./(2.*r)).^(2.*u+2)./(factorial(u).*factorial(u+2))+(k.*rho(s,j)./(2.*r)).^(2.*u)./(factorial(u).*factorial(u))).*(mfun('GAMMA',l+u+3,-g.*b.^2)-factorial(l+u+2))./g.^(l+u+3)+z.^2.*(k.*rho(s,j)./(2.*r)).^(2.*u)./(factorial(u).*factorial(u)).*(mfun('GAMMA',l+u+2,-g.*b.^2)-factorial(l+u+1))./g.^(l+u+2));
factorH5=-k.*z.^2./(2.*r.^5).*((-i.*(k.*rho(s,j)./(2.*r)).^(2.*u+1)./(rho(s,j).*factorial(u).*factorial(u+1))+(k.*rho(s,j)./(2.*r)).^(2.*u+2)./(factorial(u).*factorial(u+2))).*(mfun('GAMMA',l+u+3,-g.*b.^2)-factorial(l+u+2))./g.^(l+u+3)+(k.*rho(s,j)./(2.*r)).^(2.*u)./(factorial(u).*factorial(u)).*(mfun('GAMMA',l+u+2,-g.*b.^2)-factorial(l+u+1))./g.^(l+u+2));
factorH=factorH1.*(factorH2+factorH3+factorH4+factorH5);
Hxsum1b=Hxsum1b+factorH;
Hysum1b=Hysum1b+factorH;
l=2;
factorH1=factorial(n+1).*2.^(l-1)./(factorial(l+1).*factorial(l).*factorial(n-l).*w0.^(2.*l+1));
factorH2=((i.*k.*r-2)./r.^4).*(rho(s,j).*(k.*rho(s,j)./(2.*r)).^(2.*u+1)./(factorial(u).*factorial(u+1))-i.*(k.*rho(s,j)./(2.*r)).^(2.*u)./(factorial(u).*factorial(u))).*(mfun('GAMMA',l+u+2,-g.*b.^2)-factorial(l+u+1))./g.^(l+u+2);
factorH3=(2+(i.*k.*r-2).*z.^2./r.^2).*(k.*rho(s,j)./(2.*r)).^(2.*u+1)./(rho(s,j).*r.^2.*factorial(u).*factorial(u+1)).*(mfun('GAMMA',l+u+2,-g.*b.^2)-factorial(l+u+1))./g.^(l+u+2);
factorH4=k./(2.*r.^5).*((i.*(3.*rho(s,j)-2.*r.^2./rho(s,j)).*(k.*rho(s,j)./(2.*r)).^(2.*u+1)./(factorial(u).*factorial(u+1))+z.^2.*(k.*rho(s,j)./(2.*r)).^(2.*u+2)./(factorial(u).*factorial(u+2))+(k.*rho(s,j)./(2.*r)).^(2.*u)./(factorial(u).*factorial(u))).*(mfun('GAMMA',l+u+3,-g.*b.^2)-factorial(l+u+2))./g.^(l+u+3)+z.^2.*(k.*rho(s,j)./(2.*r)).^(2.*u)./(factorial(u).*factorial(u)).*(mfun('GAMMA',l+u+2,-g.*b.^2)-factorial(l+u+1))./g.^(l+u+2));
factorH5=-k.*z.^2./(2.*r.^5).*((-i.*(k.*rho(s,j)./(2.*r)).^(2.*u+1)./(rho(s,j).*factorial(u).*factorial(u+1))+(k.*rho(s,j)./(2.*r)).^(2.*u+2)./(factorial(u).*factorial(u+2))).*(mfun('GAMMA',l+u+3,-g.*b.^2)-factorial(l+u+2))./g.^(l+u+3)+(k.*rho(s,j)./(2.*r)).^(2.*u)./(factorial(u).*factorial(u)).*(mfun('GAMMA',l+u+2,-g.*b.^2)-factorial(l+u+1))./g.^(l+u+2));
factorH=factorH1.*(factorH2+factorH3+factorH4+factorH5);
Hxsum2b=Hxsum2b+factorH;
Hysum2b=Hysum2b+factorH;
end
Hx0a=constH.*y(s,j).*Hxsum0a;
Hy0a=-constH.*x(s,j).*Hysum0a;
Hx1a=constH.*y(s,j).*Hxsum1a;
Hy1a=-constH.*x(s,j).*Hysum1a;
Hx2a=constH.*y(s,j).*Hxsum2a;
Hy2a=-constH.*x(s,j).*Hysum2a;
Hx0b=constH.*y(s,j).*Hxsum0b;
Hy0b=-constH.*x(s,j).*Hysum0b;
Hx1b=constH.*y(s,j).*Hxsum1b;
Hy1b=-constH.*x(s,j).*Hysum1b;
Hx2b=constH.*y(s,j).*Hxsum2b;
Hy2b=-constH.*x(s,j).*Hysum2b;

Hxab(s,j)=(Hx0a+Hx1a+Hx2a)-(Hx0b+Hx1b+Hx2b);
Hyab(s,j)=(Hy0a+Hy1a+Hy2a)-(Hy0b+Hy1b+Hy2b);
Sz(s,j)=100.*1./2.*real(Exab(s,j).*conj(Hyab(s,j))-Eyab(s,j).*conj(Hxab(s,j)));
disp(Sz);

end
 clear maplemex
end
  figure(1);
mesh(x,y,Sz);
figure(2);
contourf(x,y,Sz,6);

3 参考文献

部分理论来源于网络，如有侵权请联系删除。

1. Li, X., & Zhou, G. (2016). Nonparaxial propagation of radially polarized beams diffracted by an annular aperture. Journal of the Optical Society of America A, 33(4), 536-543.

2. Li, X., & Zhou, G. (2017). Nonparaxial propagation of radially polarized beams diffracted by an annular aperture: comment. Journal of the Optical Society of America A, 34(2), 244-246.

3. Li, X., & Zhou, G. (2017). Nonparaxial propagation of radially polarized beams diffracted by an annular aperture: reply to comment. Journal of the Optical Society of America A, 34(2), 247-249.

4 Matlab代码实现