## Abstract

A design for diffractive phase elements (DPE’s) that implement wavelength demultiplexing and annular focusing simultaneously is presented based on the general theory of amplitude phase retrieval. The optical system considered is illuminated by a beam of polychromatic light consisting of several wavelengths. With the use of an iterative algorithm the pattern of a surface-relief DPE can be determined by solving the relevant equations. Computer simulations are carried out for several model examples. The calculation results show that the designed DPE’s can satisfactorily achieve the desired functions.

© 1997 Optical Society of America

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### Equations (20)

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(1)
$${U}_{1\alpha}={U}_{1}({\mathbf{X}}_{1},{\mathrm{\lambda}}_{\alpha})={\rho}_{1}({\mathbf{X}}_{1},{\mathrm{\lambda}}_{\alpha})exp[i{\varphi}_{1}({\mathbf{X}}_{1},{\mathrm{\lambda}}_{\alpha})].$$
(2)
$${U}_{2\alpha}={U}_{2}({\mathbf{X}}_{2},{\mathrm{\lambda}}_{\alpha})={\rho}_{2}({\mathbf{X}}_{2},{\mathrm{\lambda}}_{\alpha})exp[i{\varphi}_{2}({\mathbf{X}}_{2},{\mathrm{\lambda}}_{\alpha})].$$
(3)
$${U}_{2}({\mathbf{X}}_{2},{\mathrm{\lambda}}_{\alpha})=\int G({\mathbf{X}}_{2},{\mathbf{X}}_{1},{\mathrm{\lambda}}_{\alpha}){U}_{1}({\mathbf{X}}_{1},{\mathrm{\lambda}}_{\alpha})\mathrm{d}{\mathbf{X}}_{1}.$$
(4)
$${U}_{2}({\mathbf{X}}_{2},{\mathrm{\lambda}}_{\alpha})=\stackrel{\u02c6}{G}({\mathrm{\lambda}}_{\alpha}){U}_{1}({\mathbf{X}}_{1},{\mathrm{\lambda}}_{\alpha}),$$
(5)
$${U}_{1j}({\mathrm{\lambda}}_{\alpha})={\rho}_{1j\alpha}exp[i2\pi {h}_{1j}({n}_{{s}_{\alpha}}-1)/{\mathrm{\lambda}}_{\alpha}],$$
(6)
$$j=1,2,3\dots {N}_{1},\hspace{1em}\hspace{1em}\alpha =1,2,3\dots {N}_{\mathrm{\lambda}},$$
(7)
$${U}_{2m\alpha}={\rho}_{2m\alpha}exp(i{\varphi}_{2m\alpha}),\hspace{1em}\hspace{1em}m=1,2,3\dots {N}_{2s},$$
(8)
$${U}_{2m\alpha}=\sum _{j=1}^{{N}_{1}}{G}_{\mathit{mj}}({\mathrm{\lambda}}_{\alpha}){U}_{1j}.$$
(9)
$${D}^{2}=\sum _{\alpha}\Vert [{U}_{2\alpha}-\stackrel{\u02c6}{G}({\mathrm{\lambda}}_{\alpha}){U}_{1\alpha}]{\Vert}^{2}=\sum _{\alpha}\mathrm{Tr}[{U}_{2\alpha}^{+}{U}_{2\alpha}-{U}_{2\alpha}^{+}\stackrel{\u02c6}{G}({\mathrm{\lambda}}_{\alpha}){U}_{1\alpha}-{U}_{1\alpha}^{+}{\stackrel{\u02c6}{G}}^{+}({\mathrm{\lambda}}_{\alpha}){U}_{2\alpha}+{U}_{1\alpha}^{+}{\stackrel{\u02c6}{G}}^{+}({\mathrm{\lambda}}_{\alpha})G({\mathrm{\lambda}}_{\alpha}){U}_{1\alpha}].$$
(10)
$$exp[i2\pi {h}_{1k}({n}_{0}-1)/{\mathrm{\lambda}}_{0}]=\frac{{\tilde{Q}}_{k}^{*}}{|{\tilde{Q}}_{k}|},\hspace{1em}\hspace{1em}k=1,2,3\dots {N}_{1},$$
(11)
$${\tilde{Q}}_{k}=\sum _{\alpha}\left[2\pi \left({n}_{{s}_{\alpha}}-1\right)/{\mathrm{\lambda}}_{\alpha}\right]\left\{\sum _{j\ne k}^{{}^{\prime}}{\rho}_{1j\alpha}\times exp\left[-i2\pi {h}_{1j}\left({n}_{{s}_{\alpha}}-1\right)/{\mathrm{\lambda}}_{\alpha}\right]{A}_{\mathit{jk}}\left({\mathrm{\lambda}}_{\alpha}\right)-\sum _{j}{\rho}_{2j\alpha}\times exp\left(-i{\varphi}_{2j\alpha}\right){G}_{\mathit{jk}}\left({\mathrm{\lambda}}_{\alpha}\right)\right\}{\rho}_{1k\alpha}\times exp\left\{i\left[2\pi {h}_{1k}\left({n}_{0}-1\right)/{\mathrm{\lambda}}_{0}\right]\left[\frac{{\mathrm{\lambda}}_{0}\left({n}_{{s}_{\alpha}}-1\right)}{{\mathrm{\lambda}}_{\alpha}\left({n}_{0}-1\right)}-1\right]\right\},$$
(12)
$$exp(i{\varphi}_{2k\gamma})=\frac{\sum _{j}{G}_{\mathit{kj}}({\mathrm{\lambda}}_{\gamma}){\rho}_{1j\gamma}exp[i2\pi {h}_{1j}({n}_{{s}_{\alpha}}-1)/{\mathrm{\lambda}}_{\gamma}]}{\left|\sum _{j}{G}_{\mathit{kj}}({\mathrm{\lambda}}_{\gamma}){\rho}_{1j\gamma}exp[i2\pi {h}_{1j}({n}_{{s}_{\alpha}}-1)/{\mathrm{\lambda}}_{\gamma ]}\right|},$$
(13)
$$k=1,2,3\dots {N}_{2s},\hspace{1em}\hspace{1em}\gamma =1,2,3\dots {N}_{\mathrm{\lambda}},$$
(14)
$$G\left({\mathbf{X}}_{2},{\mathbf{X}}_{1};l,{\mathrm{\lambda}}_{\alpha}\right)=\frac{exp\left(i2\pi l/{\mathrm{\lambda}}_{\alpha}\right)}{i{\mathrm{\lambda}}_{\alpha}l}exp\left\{\frac{i\pi}{{\mathrm{\lambda}}_{\alpha}l}\left[{\left({x}_{2}-{x}_{1}\right)}^{2}+{\left({y}_{2}-{y}_{1}\right)}^{2}\right]\right\}.$$
(15)
$${U}_{2}({r}_{2})={\int}_{0}^{{R}_{1m}}G({r}_{2},{r}_{1}){U}_{1}({r}_{1})\mathrm{d}{r}_{1},$$
(16)
$$G\left({r}_{2},{r}_{1};l,{\mathrm{\lambda}}_{\alpha}\right)=\frac{2\pi}{i{\mathrm{\lambda}}_{\alpha}l}exp\left(i2\pi l/{\mathrm{\lambda}}_{\alpha}\right)exp\left[\frac{i\pi \left({r}_{2}^{2}+{r}_{1}^{2}\right)}{{\mathrm{\lambda}}_{\alpha}l}\right]\times {J}_{0}\left(\frac{2\pi {r}_{2}{r}_{1}}{{\mathrm{\lambda}}_{\alpha}l}\right){r}_{1},$$
(17)
$${J}_{0}(x)=\frac{1}{2\pi}{\int}_{0}^{2\pi}exp[\mathit{ix}cos({\theta}_{1}-{\theta}_{2})]\mathrm{d}{\theta}_{1}.$$
(18)
$$I({r}_{2},{\mathrm{\lambda}}_{\alpha})={\left|{\int}_{0}^{{R}_{1m}}G({r}_{2},{r}_{1};l,{\mathrm{\lambda}}_{\alpha})exp[i2\pi {h}_{1}({r}_{1})\times ({n}_{{s}_{\alpha}}-1)/{\mathrm{\lambda}}_{\alpha}]\mathrm{d}{r}_{1}\right|}^{2}.$$
(19)
$$\eta =\frac{1}{{N}_{\mathrm{\lambda}}}\sum _{\alpha}\frac{I({\mathrm{\lambda}}_{\alpha},{r}_{2{f}_{\alpha}})}{\sum _{m}I({\mathrm{\lambda}}_{\alpha},{r}_{2m})},$$
(20)
$$T=\frac{1}{{N}_{\mathrm{\lambda}}}\sum _{\alpha}\frac{\sum _{\beta \ne \alpha}I({\mathrm{\lambda}}_{\beta},{r}_{2{f}_{\alpha}})}{I({\mathrm{\lambda}}_{\beta},{r}_{2{f}_{\alpha}})},$$