## Abstract

We show that the transmission of a periodic grating with circular symmetry can be composed of the sum of the transmissions of positive and negative axicons of different orders. The properties of the light beams generated by the gratings can be understood from the well-known properties of the light beam of an axicon, which is approximately a Bessel beam of zeroth order. The method is applied to a circular binary phase grating, which is investigated both theoretically and experimentally in the spatial domain and the Fourier domain. A simple and accurate method for determining the focal length of a lens with use of Bessel light beams is presented.

© 1997 Optical Society of America

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### Equations (21)

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(1)
$${T}_{\mathrm{ax}}\left(r\right)=exp\left(-\frac{2\pi i}{{\rho}_{\mathrm{ax}}}r\right).$$
(2)
$${\rho}_{\mathrm{ax}}=\frac{2\pi {R}_{\mathrm{ax}}}{{k}_{L}{H}_{\mathrm{ax}}(n-1)}.$$
(3)
$$\frac{E\left(r,z\right)}{{E}_{0}}=-\frac{{\mathit{ik}}_{L}}{z}exp\left[{\mathit{ik}}_{L}\left(z+\frac{{r}^{2}}{2z}\right)\right]\times {\int}_{0}^{R}T\left({r}^{\prime}\right)exp\left(\frac{{\mathit{ik}}_{L}}{2z}{r}^{\prime 2}\right){J}_{0}\left(\frac{{k}_{L}r}{z}{r}^{\prime}\right){r}^{\prime}\mathrm{d}{r}^{\prime}.$$
(4)
$$\frac{{I}_{\mathrm{ax}}(r,z)}{{I}_{0}}=\frac{2\pi}{{k}_{L}}{\alpha}^{2}z{J}_{0}^{2}(\alpha r).$$
(5)
$${z}_{max}=\frac{{k}_{L}}{\alpha}{R}_{\mathrm{ax}}$$
(6)
$$\alpha =\frac{2\pi}{{\rho}_{\mathrm{ax}}}.$$
(7)
$${d}_{0}=2{r}_{0}\approx 4.81/\alpha \approx 0.77{\rho}_{\mathrm{ax}}.$$
(8)
$${T}_{\mathrm{na}}\left(r\right)=exp\left(+\frac{2\pi i}{{\rho}_{\mathrm{na}}}r\right),$$
(9)
$$A(u)=2\pi {\int}_{0}^{\infty}a(r){J}_{0}(2\pi \mathit{ru})r\mathrm{d}r.$$
(10)
$$u=\frac{{k}_{L}{r}_{F}}{2\pi f}.$$
(11)
$${d}_{\mathrm{ax}}=\frac{4\pi f}{{k}_{L}{\rho}_{\mathrm{ax}}}.$$
(12)
$${A}_{\mathrm{na}}(u)=[{A}_{\mathrm{ax}}(u){]}^{*}$$
(13)
$$f(r)=\sum _{m=-\infty}^{\infty}{c}_{m}exp(-2\pi i\frac{m}{\rho}r)$$
(14)
$${c}_{m}=\frac{1}{\rho}{\int}_{0}^{\rho}f(r)exp(2\pi i\frac{m}{\rho}r)\mathrm{d}r.$$
(15)
$$T\left(r\right)=\left\{\begin{array}{cc}\sum _{m=-\infty}^{\infty}{c}_{m}exp\left(-2\pi i\frac{m}{\rho}r\right)& \mathrm{for}\hspace{0.5em}0\u2a7dr\u2a7dR\\ 0& \mathrm{elsewhere}\end{array}\right..$$
(16)
$${\rho}_{m}=\rho /m,$$
(17)
$$T\left(r\right)={c}_{0}+\sum _{m=1}^{\infty}{c}_{m}exp\left(-\frac{2\pi i}{{\rho}_{m}}r\right)+\sum _{m=1}^{\infty}{c}_{-m}exp\left(\frac{2\pi i}{{\rho}_{m}}r\right).$$
(18)
$$f(r)=\left\{\begin{array}{ll}1& \mathrm{for}\hspace{0.5em}l\rho <r<(l+1/2)\rho ,\hspace{1em}l\in \mathbf{Z}\\ {e}^{-i\pi}=-1& \mathrm{elsewhere}\end{array}\right..$$
(19)
$${c}_{m}=\left\{\begin{array}{cc}\frac{2i}{\pi m}& \hspace{1em}\mathrm{for}\hspace{0.5em}\mathrm{odd}\hspace{0.5em}m\\ 0& \hspace{1em}\mathrm{for}\hspace{0.5em}\mathrm{even}\hspace{0.5em}m\end{array}\right..$$
(20)
$${d}_{m}=m\frac{4\pi f}{{k}_{L}\rho}\hspace{0.5em}\hspace{1em}\mathrm{for}\hspace{0.5em}\mathrm{odd}\hspace{0.5em}m.$$
(21)
$$f=s\frac{{k}_{L}\rho}{4\pi}.$$