## Abstract

Landgrave and Berriel-Valdos presented axial and radial sampling expansions for three-dimensional light amplitude distribution around the Gaussian focal point. [J. Opt. Soc. Am. A **14**, 2962 (1997)]. The expansions were obtained under the assumption that the pupil function was rotationally symmetric. We present a new derivation of the axial expansion that does not make use of arbitrary formal assumptions used by Landgrave and Berriel-Valdos and eliminates some faults of the derivation given by Arsenault and Boivin, who published this expansion in 1967 [J. Appl. Phys. **38**, 3988 (1967)]. We also discuss generalizations of the axial expansion to the case of pupils that exhibit no symmetry with respect to the axis considered.

© 2003 Optical Society of America

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### Equations (10)

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(1)
$$G(\delta ,v)=\sum _{m=-\infty}^{\infty}G(m,0)L(\delta -m,v),$$
(2)
$$G(y,z)=\sum _{n=-\infty}^{\infty}G(-4n\pi ,0)O(y+4n\pi ,z),$$
(3)
$${p}_{T}(r)=\left\{\begin{array}{l}1,\hspace{1em}\hspace{1em}-T/2<r<T/2\\ 0,\hspace{1em}\hspace{1em}\mathrm{otherwise}\end{array}\right.,$$
(4)
$$G(\delta ,v)\hspace{0.5em}=2{\int}_{0}^{1}F({\rho}^{2})exp(-i2\pi \delta {\rho}^{2}){J}_{0}(\rho v)\rho \mathrm{d}\rho .$$
(5)
$$exp(-i2\pi \delta {\rho}^{2}){J}_{0}(v\rho )=\sum _{m=-\infty}^{\infty}{f}_{m}^{(a)}(\delta ,v)exp(-i2\pi m{\rho}^{2}),$$
(6)
$$p(\rho )=\left\{\begin{array}{l}1,\hspace{1em}\hspace{1em}0<\rho <1\\ 0,\hspace{1em}\hspace{1em}\mathrm{otherwise}\end{array}\right.,$$
(7)
$$G(\delta ,v)={\int}_{0}^{\infty}p(\rho )F(\rho )exp(-i2\pi \delta \rho ){J}_{0}(v{\rho}^{1/2})\mathrm{d}\rho .$$
(8)
$${F}_{p}(\rho )=p(\rho )\sum _{m=-\infty}^{+\infty}{C}_{m}exp(i2\pi m\rho ),$$
(9)
$${C}_{m}={\int}_{0}^{1}F(\rho )exp(-2\pi \mathit{im}\rho )\mathrm{d}\rho =G(m,0).$$
(10)
$$G(\delta ,v)={\int}_{0}^{\infty}p(\rho )\left[\sum _{m=-\infty}^{\infty}G(m,0)exp(2\pi \mathit{im}\rho )\right]\times exp(-i2\pi \delta \rho ){J}_{0}(v{\rho}^{1/2})\mathrm{d}\rho =\sum _{m=-\infty}^{\infty}G(m,0){\int}_{0}^{1}exp[-2\pi i(\delta -m)\rho ]\times {J}_{0}(v{\rho}^{1/2})\mathrm{d}\rho =2\sum _{m=-\infty}^{\infty}G(m,0){\int}_{0}^{1}exp[-2\pi i(\delta -m){\rho}^{2}]\times {J}_{0}(v\rho )\rho \mathrm{d}\rho =\sum _{m=-\infty}^{\infty}G(m,0)L(\delta -m,v),$$