## Abstract

The design of diffractive phase elements (DPE’s) for generating pseudo-nondiffracting beams (PNDB’s) is carried out by employing the conjugate-gradient method. By selecting a trapezoid profile as the preset axial intensity distribution to guide the design and by using an error function with a weighting factor to evaluate the performance of the DPE, we develop a model design and obtain a satisfactory diffractive pattern of a single-segment PNDB. We demonstrate the effectiveness of the conjugate-gradient method in the design of DPE’s that produce multiple-segment PNDB’s in which two consecutive segments are separated by a dark region along the optical axis. We also evaluate the degree of nonuniformity of the axial intensity over a given axial region. The design results match the requirements quite well.

© 1998 Optical Society of America

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### Equations (20)

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(1)
$${U}_{1}({r}_{1})={\rho}_{1}({r}_{1})exp[i{\varphi}_{1}({r}_{1})].$$
(2)
$${U}_{2}({r}_{2\alpha},{z}_{\alpha})={\rho}_{2}({r}_{2\alpha},{z}_{\alpha})exp[i{\varphi}_{2}({r}_{2\alpha},{z}_{\alpha})].$$
(3)
$${U}_{2}({r}_{2\alpha},{z}_{\alpha})=\int G({r}_{2\alpha},{r}_{1},{z}_{\alpha}){U}_{1}({r}_{1})\mathrm{d}{r}_{1},$$
(4)
$$G({r}_{2\alpha},{r}_{1},{z}_{\alpha})=\frac{2\pi}{i\lambda {z}_{\alpha}}exp(i2\pi {z}_{\alpha}/\lambda )\times exp[i\pi ({r}_{2\alpha}^{2}+{r}_{1}^{2})/\lambda {z}_{\alpha}]\times {J}_{0}\left(\frac{2\pi {r}_{2\alpha}{r}_{1}}{\lambda {z}_{\alpha}}\right){r}_{1},$$
(5)
$${U}_{2}({r}_{2\alpha},{z}_{\alpha})=\stackrel{\u02c6}{G}({z}_{\alpha}){U}_{1}({r}_{1}),$$
(6)
$${U}_{2m\alpha}=\sum _{n=1}^{{N}_{1}}{G}_{\mathit{mn}}({z}_{\alpha}){U}_{1n},$$
(7)
$${U}_{1n}={\rho}_{1n}exp(i{\varphi}_{1n}),\hspace{1em}{U}_{2m\alpha}={\rho}_{2m\alpha}exp(i{\varphi}_{2m\alpha}),$$
(8)
$$n=1,\hspace{0.5em}2,\hspace{0.5em}3,\hspace{0.5em}\dots ,\hspace{0.5em}{N}_{1},\hspace{1em}\hspace{0.5em}m=1,\hspace{0.5em}2,\hspace{0.5em}3,\hspace{0.5em}\dots ,\hspace{0.5em}{N}_{2\alpha},$$
(9)
$$\alpha =1,\hspace{0.5em}2,\hspace{0.5em}3,\hspace{0.5em}\dots ,\hspace{0.5em}{N}_{z}.$$
(10)
$$E=\sum _{\alpha =1}^{{N}_{z}}W(\alpha )\left\{\sum _{m=1}^{{N}_{2\alpha}}{\left[{\tilde{\rho}}_{2m\alpha}-\left|\sum _{n=1}^{{N}_{1}}{G}_{\mathit{mn}}({z}_{\alpha}){\rho}_{1n}exp(i{\varphi}_{1n})\right|\right]}^{2}\right\},$$
(11)
$${\mathbf{\Phi}}_{1}^{(k+1)}={\mathbf{\Phi}}_{1}^{(k)}+{\tau}^{(k)}{\mathbf{d}}^{(k)},\hspace{1em}\hspace{1em}\mathrm{for}\hspace{0.5em}k=0,1,2,3,\dots ,$$
(12)
$${\mathbf{d}}^{(k)}=-\nabla E({\mathbf{\Phi}}_{1}^{(k)})+{\beta}^{(k-1)}{\mathbf{d}}^{(k-1)},\hspace{1em}\hspace{1em}k=1,2,3,\dots ,$$
(13)
$${\beta}^{(k-1)}=\frac{|\nabla E({\mathbf{\Phi}}_{1}^{(k)}){|}^{2}}{|\nabla E({\mathbf{\Phi}}_{1}^{(k-1)}){|}^{2}},\hspace{1em}\hspace{1em}k=1,2,3,\dots $$
(14)
$$E({\mathbf{\Phi}}_{1}^{(k+1)})<E({\mathbf{\Phi}}_{1}^{(k)})$$
(15)
$$\frac{\partial E({\mathbf{\Phi}}_{1})}{\partial {\varphi}_{1n}}=\sum _{\alpha =1}^{{N}_{z}}W(\alpha )\left[\sum _{m=1}^{{N}_{2\alpha}}2(|{S}_{m\alpha}|-{\tilde{\rho}}_{2m\alpha})\frac{\partial |{S}_{m\alpha}|}{\partial {\varphi}_{1n}}\right]=\sum _{\alpha =1}^{{N}_{z}}W(\alpha )\left[\sum _{m=1}^{{N}_{2\alpha}}\left(1-\frac{{\tilde{\rho}}_{2m\alpha}}{|{S}_{m\alpha}|}\right){S}_{m\alpha}^{*}\frac{\partial {S}_{m\alpha}}{\partial {\varphi}_{1n}}+\mathrm{c}.\mathrm{c}.\right]=-2\mathrm{Im}\left[{\rho}_{1n}exp(i{\varphi}_{1n})\sum _{\alpha =1}^{{N}_{z}}W(\alpha )\sum _{m=1}^{{N}_{2\alpha}}\left(1-\frac{{\tilde{\rho}}_{2m\alpha}}{|{S}_{m\alpha}|}\right)\times {G}_{\mathit{mn}}({z}_{\alpha}){S}_{m\alpha}^{*}\right],$$
(16)
$${S}_{m\alpha}=\sum _{k=1}^{{N}_{1}}{G}_{\mathit{mk}}({z}_{\alpha}){\rho}_{1k}exp(i{\varphi}_{1k}).$$
(17)
$$G({r}_{2\alpha}=0,{r}_{1},{z}_{\alpha})=\frac{2\pi}{i\lambda {z}_{\alpha}}exp(i2\pi {z}_{\alpha}/\lambda )exp(i\pi {r}_{1}^{2}/\lambda {z}_{\alpha}){r}_{1}.$$
(18)
$$w(\alpha )=\left\{\begin{array}{ll}\frac{0.98}{{N}_{p}}& \mathrm{in}\hspace{0.5em}\mathrm{the}\hspace{0.5em}\mathrm{signal}\hspace{0.5em}\mathrm{region}\\ \frac{0.02}{{N}_{z}-{N}_{p}}& \mathrm{in}\hspace{0.5em}\mathrm{other}\hspace{0.5em}\mathrm{regions}\end{array}\right.,$$
(19)
$$\mathrm{nonuniformity}=\frac{(\u3008{I}^{2}\u3009-\u3008I{\u3009}^{2}{)}^{1/2}}{\u3008I\u3009},$$
(20)
$$\mathit{dz}\u2a7d\frac{2\pi {z}^{2}}{{\mathit{kR}}_{1m}^{2}-2\pi z},$$