## Abstract

In the mode-expansion method for modeling propagation of a diffracted beam, the beam at the aperture can be expanded as a weighted set of orthogonal modes. The parameters of the expansion modes are chosen to maximize the weighting coefficient of the lowest-order mode. As the beam propagates, its field distribution can be reconstructed from the set of weighting coefficients and the Gouy phase of the lowest-order mode. We have developed a simple procedure to implement the mode-expansion method for propagation through an arbitrary *ABCD* matrix, and we have demonstrated that it is accurate in comparison with direct calculations of diffraction integrals and much faster.

© 2007 Optical Society of America

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### Equations (23)

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(1)
$$\psi \left(r,z\right)=\frac{1}{w\left(z\right)}\sqrt{\frac{2}{\pi}}\text{\hspace{0.17em}}\mathrm{exp}\left[-j\text{\hspace{0.17em}}\frac{\pi {r}^{2}}{\lambda R\left(z\right)}\right]\mathrm{exp}\left[j\gamma \left(z\right)\right]\times \mathrm{exp}\left[-j\text{\hspace{0.17em}}\frac{2\pi \left(z-{z}_{0}\right)}{\lambda}\right]\mathrm{exp}\left[-\frac{{r}^{2}}{w{\left(z\right)}^{2}}\right]\text{,}$$
(2)
$$\gamma \left(z\right)=\mathrm{arctan}\left[\frac{\lambda \left(z-{z}_{0}\right)}{\pi {{w}_{0}}^{2}}\right]$$
(3)
$$w\left(z\right)={w}_{0}\sqrt{1+{\left[\frac{\lambda \left(z-{z}_{0}\right)}{\pi {{w}_{0}}^{2}}\right]}^{2}}\text{,}$$
(4)
$$R\left(z\right)=\left(z-{z}_{0}\right)\left\{1+\left[\frac{\pi {{w}_{0}}^{2}}{\lambda \left(z-{z}_{0}\right)}\right]\right\}.$$
(5)
$$U\left(r,z\right)={\displaystyle \sum _{n=0}^{\infty}{C}_{n}\left(z\right){\Psi}_{n}\left(r,z\right)},$$
(6)
$${C}_{n}\left(z\right)={\displaystyle {\int}_{0}^{2\pi}{\displaystyle {\int}_{0}^{\infty}U\left(r,z\right)\Psi *\left(r,z\right)r\mathrm{d}r\mathrm{d}\theta =2\pi {\displaystyle {\int}_{0}^{\infty}U\left(r,z\right)\Psi *\left(r,z\right)r\mathrm{d}r}}}.$$
(7)
$${\Psi}_{n}\left(r,z\right)=\frac{1}{\widehat{w}\left(z\right)}\sqrt{\frac{2}{\pi}}\text{\hspace{0.17em}}{L}_{n}\left[\frac{2{r}^{2}}{\widehat{w}{\left(z\right)}^{2}}\right]\mathrm{exp}\left[-j\text{\hspace{0.17em}}\frac{\pi {r}^{2}}{\lambda \widehat{R}\left(z\right)}\right]\times \mathrm{exp}\left[j\left(2n+1\right)\hat{\gamma}\left(z\right)\right]\mathrm{exp}\left[-j\text{\hspace{0.17em}}\frac{2\pi \left(z-{\widehat{z}}_{0}\right)}{\lambda}-j\phi \right]\times \mathrm{exp}\left[-\frac{{r}^{2}}{\widehat{w}{\left(z\right)}^{2}}\right]\text{,}$$
(8)
$${C}_{n}=2\pi {\displaystyle {\int}_{0}^{a}\psi \left(r,z=0\right){\mathrm{\Psi}}_{n}*\left(r,z=0\right)r\mathrm{d}r},$$
(9)
$$U\left(r,z\right)\approx {\displaystyle \sum _{n=0}^{N}{C}_{n}{\Psi}_{n}\left(r,z\right)}.$$
(10)
$${C}_{0}=\frac{4}{w\widehat{w}}{\displaystyle {\int}_{0}^{a}\mathrm{exp}\left\{-j\left[\frac{\pi {r}^{2}}{\lambda R\left(0\right)}-\frac{\pi {r}^{2}}{\lambda \widehat{R}\left(0\right)}\right]\right\}}\times \mathrm{exp}\left[-j\text{\hspace{0.17em}}\mathrm{arctan}\left(\frac{\lambda {z}_{0}}{\pi {{w}_{0}}^{2}}\right)+j\text{\hspace{0.17em}}\mathrm{arctan}\left(\frac{\lambda {\widehat{z}}_{0}}{\pi {\widehat{w}}_{0}}\right)+j\text{\hspace{0.17em}}\frac{2\pi}{\lambda}\left({z}_{0}-{\widehat{z}}_{0}\right)+j\phi \right]\mathrm{exp}\left[-\left(\frac{{r}^{2}}{{w}^{2}}+\frac{{r}^{2}}{{\widehat{w}}^{2}}\right)\right]r\mathrm{d}r,$$
(11)
$${C}_{0}=\frac{4}{w\widehat{w}}\text{\hspace{0.17em}}{\displaystyle {\int}_{0}^{a}\mathrm{exp}\left[-\left(\frac{{r}^{2}}{{w}^{2}}+\frac{{r}^{2}}{{\widehat{w}}^{2}}\right)\right]r\mathrm{d}r}=\frac{2{w}^{2}{\widehat{w}}^{2}}{{w}^{2}+{\widehat{w}}^{2}}\left[1-\mathrm{exp}\left(-\frac{{a}^{2}}{{w}^{2}}-\frac{{a}^{2}}{{\widehat{w}}^{2}}\right)\right].$$
(12)
$$\frac{1}{{R}_{2}}=\frac{1}{{R}_{1}}-\frac{1}{f}.$$
(13)
$${\widehat{w}}_{0}=\widehat{w}/\sqrt{1+{\left(\frac{\pi {\widehat{w}}^{2}}{\lambda \widehat{R}}\right)}^{2}}\text{,}$$
(14)
$${\widehat{z}}_{0}=\widehat{R}/\left[1+{\left(\frac{\lambda \widehat{R}}{\pi {\widehat{w}}^{2}}\right)}^{2}\right]\text{,}$$
(15)
$$\Delta \gamma \left(z\right)=\mathrm{arctan}\left[\frac{\lambda \left(z-{\widehat{z}}_{0}\right)}{\pi {{\widehat{w}}_{0}}^{2}}\right]-\mathrm{arctan}\left[\frac{\lambda \left(-{\widehat{z}}_{0}\right)}{\pi {{\widehat{w}}_{0}}^{2}}\right].$$
(16)
$$\Delta \gamma ={\mathrm{tan}}^{-1}\left[\frac{B\lambda}{\pi {\widehat{w}}^{2}\left(A+B/\widehat{R}\right)}\right],$$
(17)
$$\left|\begin{array}{cc}A& B\\ C& D\end{array}\right|=\left|\begin{array}{cc}1& 0\\ -1/f& 1\end{array}\right|\left|\begin{array}{cc}1& d\\ 0& 1\end{array}\right|=\left|\begin{array}{cc}1-d/f& d\\ -1/f& 1\end{array}\right|,$$
(18)
$${C}_{n}=\frac{4}{w\widehat{w}}{\displaystyle {\int}_{0}^{a}{L}_{n}\left(\frac{2{r}^{2}}{{\widehat{w}}^{2}}\right)}\mathrm{exp}\left[-\left(\frac{{r}^{2}}{{w}^{2}}+\frac{{r}^{2}}{{\widehat{w}}^{2}}\right)\right]r\mathrm{d}r.$$
(19)
$$U\left(r,{z}_{m}\right)\approx \frac{1}{{\widehat{w}}_{m}}\sqrt{\frac{2}{\pi}}\text{\hspace{0.17em}}\mathrm{exp}\left(-\frac{{r}^{2}}{{{\widehat{w}}_{m}}^{2}}\right){\displaystyle \sum _{n=0}^{N}{C}_{n}{L}_{n}\left(\frac{2{r}^{2}}{{{\widehat{w}}_{m}}^{2}}\right)}\times \mathrm{exp}\left[j\left(2n+1\right)\Delta \gamma \left({z}_{m}-{z}_{0}\right)\right],$$
(20)
$${\Psi}_{n}\left(r,{z}_{m}\right)=\frac{1}{{\widehat{w}}_{m}}\sqrt{\frac{2}{\pi}}\text{\hspace{0.17em}}{L}_{n}\left(\frac{2{r}^{2}}{{{\widehat{w}}_{m}}^{2}}\right)\mathrm{exp}\left[j\left(2n+1\right)\Delta \gamma \left({z}_{m}-{z}_{0}\right)\right]\times \mathrm{exp}\left(-\frac{{r}^{2}}{{{\widehat{w}}_{m}}^{2}}\right),$$
(21)
$$F\equiv \frac{{a}^{2}}{\lambda f},$$
(22)
$${C}_{0}=\frac{2\sqrt{2\pi}}{{\widehat{w}}_{0}}{\displaystyle {\int}_{0}^{a}\mathrm{exp}\left(-\frac{{r}^{2}}{{{\widehat{w}}_{0}}^{2}}\right)r\mathrm{d}r=\sqrt{2\pi}}\text{\hspace{0.17em}}{\widehat{w}}_{0}\left[1-\mathrm{exp}(-\frac{{a}^{2}}{{\widehat{w}}^{2}})\right]\text{,}$$
(23)
$${C}_{n}=\frac{2\sqrt{2\pi}}{{\widehat{w}}_{0}}{\displaystyle {\int}_{0}^{a}{L}_{n}\left(\frac{2{r}^{2}}{{\widehat{w}}^{2}}\right)}\mathrm{exp}\left(-\frac{{r}^{2}}{{{\widehat{w}}_{0}}^{2}}\right)r\mathrm{d}r$$