# 线性光学笔记（15）：角谱衍射理论（二）

$U(x,y,0) = \iint^\infty_{-\infty} A(f_X,f_Y;0)e^{i2\pi(f_X x+f_Y y)}\,\mathrm{d} f_X\,\mathrm{d} f_Y,$

$A(f_X,f_Y;0) = \iint^\infty_{-\infty} U(x,y,0)e^{-i2\pi(f_X x+f_Y y)}\,\mathrm{d} x\,\mathrm{d} y.$

$\alpha^2+\beta^2+\gamma^2 = 1.$

$\alpha = \lambda f_X, \quad \beta = \lambda f_Y, \quad \gamma = \sqrt{1-(\lambda f_X)^2-(\lambda f_Y)^2}.$

$A\left(\frac{\alpha}{\lambda},\frac{\beta}{\lambda};0\right) = \iint^\infty_{-\infty} U(x,y,0)e^{-i2\pi\left(\frac{\alpha}{\lambda} x+\frac{\beta}{\lambda} y\right)}\,\mathrm{d} x\,\mathrm{d} y,$

$A\left(\frac{\alpha}{\lambda},\frac{\beta}{\lambda};0\right)$ 被称作 $U(x,y,0)$ 的角谱（angular spectrum）。

$U(x,y,z) = \iint^\infty_{-\infty} A(f_X,f_Y;z)e^{i2\pi(f_X x+f_Y y)}\,\mathrm{d} f_X\,\mathrm{d} f_Y,$

$A(f_X,f_Y;z) = \iint^\infty_{-\infty} U(x,y,z)e^{-i2\pi(f_X x+f_Y y)}\,\mathrm{d} x\,\mathrm{d} y.$

$A(f_X,f_Y;z) = H(f_X, f_Y) A(f_X,f_Y;0).$

$\frac{\mathrm{d}^2}{\mathrm{d}z^2}A(f_X,f_Y;z) + k^2(1-\alpha^2-\beta^2)A(f_X,f_Y;z) = 0.$

$A(f_X,f_Y;z) = A(f_X,f_Y;0) e^{ik\sqrt{1-\alpha^2-\beta^2}z} \quad (\alpha^2+\beta^2<1).$

$H(f_X, f_Y) = \begin{cases} e^{i\frac{2\pi}{\lambda}z\sqrt{1-(\lambda f_X)^2-(\lambda f_Y)^2}}& \left(\sqrt{f_X^2+f_Y^2}<\frac{1}{\lambda}\right),\\ 0& \left(\sqrt{f_X^2+f_Y^2}>\frac{1}{\lambda}\right). \end{cases}$

http://blog.sciencenet.cn/blog-373392-727588.html

## 全部精选博文导读

GMT+8, 2019-9-19 20:37