## Abstract

We compare the iterative angular spectrum (IAS) and the optimal rotation angle (ORA) methods in designing two-dimensional finite aperture diffractive optical elements (FADOEs) used as beamfanners. The transfer functions of both methods are compared analytically in the spatial frequency domain. We have designed several structures of 1-to-4 and 1-to-6 beamfanners to investigate the differences in the performance of the beamfanners designed by ORA method for near field operation. Using the three-dimensional finite difference time-domain (3-D FDTD) method with perfect matched layer (PML) absorbing boundary condition (ABC), the diffraction efficiency is calculated for each designed FADOE and the corresponding values are compared.

© 2009 OSA

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### Equations (10)

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(1)
$$\text{t(x,y)}=\tau \text{exp[j}\varphi (x,y)]\Pi (x/{\text{L}}_{x},y/{\text{L}}_{y}),$$
(2)
$${\mathbf{U}}_{0}({f}_{x},{f}_{y})={\displaystyle \colorbox[rgb]{}{$\underset{{P}_{0}}{\iint}{\mathbf{u}}_{0}(x,y)\mathrm{exp}[-j2\pi ({f}_{x}x+{f}_{y}y)]dxdy$}}.$$
(3)
$${\mathbf{u}}_{1}(x,y)={\displaystyle \colorbox[rgb]{}{$\int {\displaystyle \colorbox[rgb]{}{$\underset{-\infty}{\overset{\infty}{\int}}{\mathbf{U}}_{0}\mathrm{exp}[j{k}_{2}{z}_{obs}\sqrt{1-{({\lambda}_{2}{f}_{x})}^{2}-{({\lambda}_{2}{f}_{y})}^{2}}]\mathrm{exp}[j2\pi ({f}_{x}x+{f}_{y}y)]d{f}_{x}d{f}_{y}$}}$}},$$
(4)
$$\text{H(}{f}_{x},{f}_{y})=\mathrm{exp}[j{k}_{2}{z}_{obs}\sqrt{1-{({\lambda}_{2}{f}_{x})}^{2}-{({\lambda}_{2}{f}_{y})}^{2}}]$$
(5)
$${\mathbf{u}}_{\text{0,}k}={\mathbf{A}}_{k}^{\text{inc}}\mathrm{exp}[j({\varphi}_{k}^{\text{inc}}+{\varphi}_{k})],$$
(6)
$${\mathbf{u}}_{\text{1,}m}={\displaystyle \colorbox[rgb]{}{$\sum _{k}{\mathbf{u}}_{\text{0,}k}{\text{h}}_{k,m}$}},$$
(7)
$${\text{h}}_{k,m}=\genfrac{}{}{0.1ex}{}{1}{4\pi}\left[-j{k}_{2}-{z}_{obs}(j{k}_{2}-1/{r}_{km}^{\text{c}})/{r}_{km}^{\text{c}}\right]\genfrac{}{}{0.1ex}{}{4\mathrm{exp}(j{k}_{2}{r}_{km}^{\text{c}})}{{r}_{km}^{\text{c}}}\genfrac{}{}{0.1ex}{}{\mathrm{sin}({\tilde{k}}_{x}{\scriptscriptstyle \colorbox[rgb]{}{$\genfrac{}{}{0.1ex}{}{a}{2}$}})\mathrm{sin}({\tilde{k}}_{y}{\scriptscriptstyle \colorbox[rgb]{}{$\genfrac{}{}{0.1ex}{}{b}{2}$}})}{{\tilde{k}}_{x}{\tilde{k}}_{y}},$$
(8)
$$\text{h}(x,y;{z}_{obs})=\genfrac{}{}{0.1ex}{}{1}{4\pi}[-j{k}_{2}-\genfrac{}{}{0.1ex}{}{{z}_{obs}}{r}(j{k}_{2}-\genfrac{}{}{0.1ex}{}{1}{r})]\genfrac{}{}{0.1ex}{}{\mathrm{exp}(j{k}_{2}r)}{r},$$
(9)
$${\text{u}}_{\text{1}}(x,y)={\displaystyle \colorbox[rgb]{}{$\underset{{P}_{0}}{\iint}{\text{u}}_{\text{0}}(\eta ,\xi )\text{h}(x-\eta ,y-\xi ;{z}_{obs})d\eta d\xi $}}.$$
(10)
$$\text{DE}=\genfrac{}{}{0.1ex}{}{{\displaystyle \colorbox[rgb]{}{$\underset{W}{\iint}\text{I(x,y)}dxdy$}}}{{\displaystyle \colorbox[rgb]{}{$\underset{{P}_{1}}{\iint}\text{I(x,y)}dxdy$}}}$$