## Abstract

We present a design of the diffractive phase elements (DPE’s) that produce nondiffracting beams according to the beam-shaping scheme, in which the incident Gaussian-profile beam is converted into a Bessel-function ${J}_{0}$ beam. An optimization method is applied to solving this special beam-shaping problem. Numerical investigation of the generating ${J}_{0}$ Bessel beam shows that the designed DPE can satisfactorily produce the ${J}_{0}$ Bessel beam.

© 1998 Optical Society of America

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### Equations (13)

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(1)
$$B(\rho ,z)=exp(i\beta z){J}_{0}(\alpha \rho ),$$
(2)
$$exp(i\beta l){J}_{0}(\alpha \rho )=exp[-i\psi (\rho )]\times {\int}_{0}^{R}G(r,\rho )u(r)exp[i\varphi (r)]\mathrm{d}r,$$
(3)
$$G(r,\rho )=\frac{2\pi}{i\lambda l}exp\left(i\frac{2\pi l}{\lambda}\right)exp\left[\frac{i\pi ({r}^{2}+{\rho}^{2})}{\lambda l}\right]{J}_{0}\left(\frac{2\pi r\rho}{\lambda l}\right)r,$$
(4)
$${B}_{n}exp(i{\psi}_{n})=\sum _{m=1}^{M}{G}_{\mathit{mn}}{u}_{m}exp(i{\varphi}_{m})$$
(5)
$$\hspace{1em}\mathrm{for}\hspace{0.5em}n=1,2,\dots ,N.$$
(6)
$$minD({\mathbf{\Phi}}_{1})=\sum _{n=1}^{N}{\left[\left|\sum _{m=1}^{M}{G}_{\mathit{mn}}{u}_{m}exp(i{\varphi}_{m})\right|-{\tilde{B}}_{n}\right]}^{2},$$
(7)
$$\frac{\partial D({\mathbf{\Phi}}_{1})}{\partial {\varphi}_{m}}={\mathit{iu}}_{m}exp(-i{\varphi}_{m})\sum _{n=1}^{N}{G}_{\mathit{mn}}^{*}({\tilde{B}}_{n}-{B}_{n})\times exp(i{\psi}_{n})+\mathrm{c}.\mathrm{c}.\hspace{1em}\mathrm{for}\hspace{0.5em}m=1,2,\dots ,M,$$
(8)
$${\mathrm{grad}}^{(k)}=\left[\frac{\partial D(\mathbf{\Phi}_{1}{}^{(k)})}{\partial {\varphi}_{1}^{(k)}},\frac{\partial D(\mathbf{\Phi}_{1}{}^{(k)})}{\partial {\varphi}_{2}^{(k)}},\dots ,\frac{\partial D(\mathbf{\Phi}_{1}{}^{(k)})}{\partial {\varphi}_{M}^{(k)}}\right];$$
(9)
$$D(\mathbf{\Phi}_{1}{}^{(k+1)})=minD(\mathbf{\Phi}_{1}{}^{(k)}+\lambda {\mathbf{p}}^{(k)}).$$
(10)
$${H}^{(k+1)}={H}^{(k)}+\left(1+\frac{{\mathbf{W}}^{T}\xb7{H}^{(k)}\xb7\mathbf{W}}{\mathrm{\Delta}{q}^{T}\xb7\mathbf{W}}\right)\frac{\mathrm{\Delta}q\xb7\mathrm{\Delta}{q}^{T}}{\mathrm{\Delta}{q}^{T}\xb7\mathbf{W}}-\frac{\mathrm{\Delta}q\xb7{\mathbf{W}}^{T}\xb7{H}^{(k)}}{\mathrm{\Delta}{q}^{T}\xb7\mathbf{W}}-\frac{{H}^{(k)}\xb7\mathbf{W}\xb7\mathrm{\Delta}{q}^{T}}{\mathrm{\Delta}{q}^{T}\xb7\mathbf{W}},$$
(11)
$${w}_{m}=\frac{\partial D(\mathbf{\Phi}_{1}{}^{(k+1)})}{\partial {\varphi}_{m}^{(k+1)}}-\frac{\partial D(\mathbf{\Phi}_{1}{}^{(k)})}{\partial {\varphi}_{m}^{(k)}},\hspace{1em}\mathrm{for}\hspace{0.5em}m=1,2,\dots ,M,$$
(12)
$$\mathrm{\Delta}q=\mathbf{\Phi}_{1}{}^{(k+1)}-\mathbf{\Phi}_{1}{}^{(k)}.$$
(13)
$$\mathrm{MSE}=\frac{D(\mathbf{\Phi}_{1}{}^{(k+1)})}{{\displaystyle \sum _{m}}{u}_{m}^{2}}<\u220a,$$