## Abstract

On the basis of the Rayleigh–Sommerfeld formulas, the analytical propagation equation of nonparaxial controllable dark-hollow beams (CDHBs) in free space is derived. The far-field approaches and the paraxial approximation are dealt with as special cases of our general results. By using the derived formulas, the nonparaxial propagation properties of CDHBs in free space are illustrated and are analyzed with numerical examples. Some detailed comparisons of the results obtained with the paraxial results are made, which show that the *f* parameter and the propagation distance play an important role in determining the nonparaxiality of the CDHBs.

© 2008 Optical Society of America

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### Equations (22)

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(1)
$$E({\rho}_{0},0)=\sum _{n=1}^{N}{a}_{n}[\mathrm{exp}(-\frac{n{\rho}_{0}^{2}}{{w}_{0}^{2}})-\mathrm{exp}(-\frac{n{\rho}_{0}^{2}}{{v}_{0}^{2}})],$$
(3)
$${a}_{n}=\frac{{(-1)}^{n-1}}{N}\left(\begin{array}{c}N\\ n\end{array}\right)$$
(4)
$$E(\rho ,\phi ,z)=-\frac{1}{2\pi}{\int}_{0}^{2\pi}{\int}_{0}^{\infty}E({\rho}_{0},{\phi}_{0},0)\frac{\partial}{\partial z}\left(\frac{{e}^{ikR}}{R}\right){\rho}_{0}\mathrm{d}{\rho}_{0}\mathrm{d}{\phi}_{0},$$
(5)
$$R=\sqrt{{\rho}_{0}^{2}+{\rho}^{2}-2{\rho}_{0}\rho \phantom{\rule{0.2em}{0ex}}\mathrm{cos}(\phi -{\phi}_{0})+{z}^{2}}.$$
(6)
$$R=r+\frac{{\rho}_{0}^{2}-2{\rho}_{0}\rho \phantom{\rule{0.2em}{0ex}}\mathrm{cos}(\phi -{\phi}_{0})}{2r},$$
(7)
$${\int}_{0}^{2\pi}\phantom{\rule{0.2em}{0ex}}\mathrm{exp}\left(ix\phantom{\rule{0.2em}{0ex}}\mathrm{cos}\phantom{\rule{0.2em}{0ex}}\theta \right)\mathrm{d}\theta =2\pi {J}_{0}\left(x\right),$$
(8)
$${\int}_{0}^{\infty}\phantom{\rule{0.2em}{0ex}}\mathrm{exp}(-pt){J}_{0}\left(2{\alpha}^{1\u22152}{t}^{1\u22152}\right)\mathrm{d}t={p}^{-1}\phantom{\rule{0.2em}{0ex}}\mathrm{exp}(-\alpha \u2215p),$$
(9)
$$E(\rho ,z)=-\frac{ikz}{2{r}^{2}}\phantom{\rule{0.2em}{0ex}}\mathrm{exp}\left(ikr\right)\sum _{n=1}^{N}{a}_{n}\{{(\frac{n}{{w}_{0}^{2}}-\frac{ik}{2r})}^{-1}\phantom{\rule{0.2em}{0ex}}\mathrm{exp}[-\frac{{(k\rho \u22152r)}^{2}}{n\u2215{w}_{0}^{2}-ik\u22152r}]-{(\frac{n}{{v}_{0}^{2}}-\frac{ik}{2r})}^{-1}\phantom{\rule{0.2em}{0ex}}\mathrm{exp}[-\frac{{(k\rho \u22152r)}^{2}}{n\u2215{v}_{0}^{2}-ik\u22152r}]\}.$$
(10)
$$R=r-\frac{{\rho}_{0}\rho \phantom{\rule{0.2em}{0ex}}\mathrm{cos}(\phi -{\phi}_{0})}{r}.$$
(11)
$$E(\rho ,z)=-\frac{ikz}{2{r}^{2}}\phantom{\rule{0.2em}{0ex}}\mathrm{exp}\left(ikr\right)\sum _{n=1}^{N}{a}_{n}\{{\left(\frac{n}{{w}_{0}^{2}}\right)}^{-1}\phantom{\rule{0.2em}{0ex}}\mathrm{exp}[-\frac{{k}^{2}{\rho}^{2}{w}_{0}^{2}}{4n{r}^{2}}]-{\left(\frac{n}{{v}_{0}^{2}}\right)}^{-1}\phantom{\rule{0.2em}{0ex}}\mathrm{exp}[-\frac{{k}^{2}{\rho}^{2}{v}_{0}^{2}}{4n{r}^{2}}]\}.$$
(12)
$$E(\rho ,z)=-\frac{i(z\u2215{z}_{R})}{4{f}^{2}{\alpha}_{1}^{2}}\phantom{\rule{0.2em}{0ex}}\mathrm{exp}\left(i{\alpha}_{1}\right)\sum _{n=1}^{N}{a}_{n}[{\beta}_{11}^{-1}\phantom{\rule{0.2em}{0ex}}\mathrm{exp}(-\frac{{\pi}^{2}{\eta}^{2}}{{\alpha}_{1}^{2}{\beta}_{11}})-{\beta}_{12}^{-1}\phantom{\rule{0.2em}{0ex}}\mathrm{exp}(-\frac{{\pi}^{2}{\eta}^{2}}{{\alpha}_{1}^{2}{\beta}_{12}})],$$
(13)
$${\alpha}_{1}={[4{\pi}^{2}{\eta}^{2}+\frac{{(z\u2215{z}_{R})}^{2}}{4{f}^{4}}]}^{1\u22152},$$
(14)
$${\beta}_{11}=n{f}^{2}-i\u22152{\alpha}_{1},$$
(15)
$${\beta}_{12}=n{f}^{2}\u2215{\u03f5}^{2}-i\u22152{\alpha}_{1},$$
(16)
$$\eta =\rho \u2215\lambda .$$
(17)
$$E(\rho ,z)=-\frac{i(z\u2215{z}_{R})}{4{f}^{4}{\alpha}_{1}^{2}}\phantom{\rule{0.2em}{0ex}}\mathrm{exp}\left(i{\alpha}_{1}\right)\sum _{n=1}^{N}{a}_{n}[\frac{1}{n}\phantom{\rule{0.2em}{0ex}}\mathrm{exp}(-\frac{{\pi}^{2}{\eta}^{2}}{n{\alpha}_{1}^{2}{f}^{2}})-\frac{{\u03f5}^{2}}{n}\phantom{\rule{0.2em}{0ex}}\mathrm{exp}(-\frac{{\u03f5}^{2}{\pi}^{2}{\eta}^{2}}{n{\alpha}_{1}^{2}{f}^{2}})]$$
(18)
$$r\approx z+{\rho}^{2}\u22152z.$$
(19)
$$E(\rho ,z)=-\frac{i}{2{\alpha}_{2}}\phantom{\rule{0.2em}{0ex}}\mathrm{exp}\left(i{\alpha}_{2}\right)\mathrm{exp}\left(\frac{2i{\pi}^{2}{\eta}^{2}}{{\alpha}_{2}}\right)\sum _{n=1}^{N}{a}_{n}\{{\beta}_{21}^{-1}\phantom{\rule{0.2em}{0ex}}\mathrm{exp}[-\frac{{\pi}^{2}{\eta}^{2}}{{\alpha}_{2}^{2}{\beta}_{21}}]-{\beta}_{22}^{-1}\phantom{\rule{0.2em}{0ex}}\mathrm{exp}[-\frac{{\pi}^{2}{\eta}^{2}}{{\alpha}_{2}^{2}{\beta}_{22}}]\},$$
(20)
$${\alpha}_{2}=\frac{z\u2215{z}_{R}}{2{f}^{2}},$$
(21)
$${\beta}_{21}=n{f}^{2}-i\u22152{\alpha}_{2},$$
(22)
$${\beta}_{22}=n{f}^{2}\u2215{\u03f5}^{2}-i\u22152{\alpha}_{2}.$$