## Abstract

We present a method to calculate wave propagation between arbitrary curved surfaces using a staircase approximation approach. The entire curved surface is divided into multiple subregions and each curved subregion is approximated by a piecewise flat subplane allowing the application of conventional diffraction theory. In addition, in order to reflect the local curvature of each subregion, we apply the phase compensation technique. Analytical expressions are derived based on the angular spectrum method and numerical studies are conducted to validate our method.

© 2014 Optical Society of America

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### Equations (20)

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(1)
$${A}_{n}({f}_{x};z=g(n\Delta x))={\displaystyle \int {u}_{i}(x){g}_{p}(n\Delta x,x)\text{rect}\left(\frac{x-n\Delta x}{\Delta x}\right)\mathrm{exp}(-i2\pi {f}_{x}x)dx},$$
(2)
$${g}_{p}(n\Delta x,x)=\mathrm{exp}\left[ik\left\{g(n\Delta x)-g(x)\right\}\right],$$
(3)
$$\text{rect}(x)=\{\begin{array}{l}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}1,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{}\left|x\right|1/2\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}1/2,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left|x\right|=1/2\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0.\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{otherwise}\end{array}$$
(4)
$${u}_{o,n}(x)={\displaystyle \int {A}_{n}({f}_{x};z=g(n\Delta x))\mathrm{exp}\left[ik\left\{{z}_{0}-g(n\Delta x)\right\}\sqrt{1-{\lambda}^{2}{f}_{x}^{2}}\right]\mathrm{exp}(i2\pi {f}_{x}x)d{f}_{x}}.$$
(5)
$${u}_{o}(x)={\displaystyle \sum _{n}{u}_{o,n}(x)}.$$
(6)
$${u}_{o}(x)={\displaystyle \sum _{n}{u}_{o,n}(x){g}_{p}(x,n\Delta x)\text{rect}\left(\frac{x-n\Delta x}{\Delta x}\right)},$$
(7)
$${u}_{o,n}(x)={\displaystyle \int A({f}_{x};z={z}_{0})\mathrm{exp}\left[ik\left\{g(n\Delta x)-{z}_{0}\right\}\sqrt{1-{\lambda}^{2}{f}_{x}^{2}}\right]\mathrm{exp}(i2\pi {f}_{x}x)d{f}_{x}},$$
(8)
$$A({f}_{x};z={z}_{0})={\displaystyle \int {u}_{i}(x)\mathrm{exp}(-i2\pi {f}_{x}x)dx}.$$
(9)
$${w}_{n}(x)=\mathrm{exp}\left[-\pi {\left(\frac{x-n\Delta x}{\Delta x}\right)}^{2}\right].$$
(10)
$${u}_{o}(x)={\displaystyle \int A({f}_{x};z={z}_{0})\mathrm{exp}(i2\pi {f}_{x}x)d{f}_{x}},$$
(11)
$$A({f}_{x};z={z}_{0})={\displaystyle \int {u}_{i}(x)\mathrm{exp}\left[ik\left\{{z}_{0}-g(x)\right\}\sqrt{1-{\lambda}^{2}{f}_{x}^{2}}\right]\mathrm{exp}(-i2\pi {f}_{x}x)dx},$$
(12)
$${u}_{o}(x)={\displaystyle \int A({f}_{x};z={z}_{0})\mathrm{exp}\left[ik\left\{g(x)-{z}_{0}\right\}\sqrt{1-{\lambda}^{2}{f}_{x}^{2}}\right]\mathrm{exp}(i2\pi {f}_{x}x)d{f}_{x}},$$
(13)
$$A({f}_{x};z={z}_{0})={\displaystyle \int {u}_{i}(x)\mathrm{exp}(-i2\pi {f}_{x}x)dx}.$$
(14)
$${A}_{n}({f}_{x};z=n\Delta z)={\displaystyle \int {u}_{i}(x)\mathrm{exp}\left[ik\left\{n\Delta z-g(x)\right\}\right]\text{rect}\left(\frac{n\Delta z-g(x)}{\Delta z}\right)\mathrm{exp}(-i2\pi {f}_{x}x)dx}.$$
(15)
$${u}_{o}(x;z={z}_{0})={\displaystyle \sum _{n}{u}_{o,n}(x;z={z}_{0})},$$
(16)
$${u}_{o,n}(x;z={z}_{0})={\displaystyle \int {A}_{n}({f}_{x};z=n\Delta z)\mathrm{exp}\left[ik\left\{{z}_{0}-n\Delta z\right\}\sqrt{1-{\lambda}^{2}{f}_{x}^{2}}\right]\mathrm{exp}(i2\pi {f}_{x}x)d{f}_{x}}.$$
(17)
$${u}_{o}(x;z=g(x))={\displaystyle \sum _{n}{u}_{o,n}(x;z=n\Delta z)\mathrm{exp}\left[ik\left\{g(x)-n\Delta z\right\}\right]\text{rect}\left(\frac{n\Delta z-g(x)}{\Delta z}\right)},$$
(18)
$${u}_{o,n}(x;z=n\Delta z)={\displaystyle \int {A}_{n}({f}_{x};z={z}_{0})\mathrm{exp}\left[ik\left\{n\Delta z-{z}_{0}\right\}\sqrt{1-{\lambda}^{2}{f}_{x}^{2}}\right]\mathrm{exp}(i2\pi {f}_{x}x)d{f}_{x}},$$
(19)
$${A}_{n}({f}_{x};z={z}_{0})={\displaystyle \int {u}_{i}(x)\mathrm{exp}(-i2\pi {f}_{x}x)dx}.$$
(20)
$${w}_{n}(x)=\mathrm{exp}\left[-\pi {\left(\frac{n\Delta z-g(x)}{\Delta z}\right)}^{2}\right].$$