## Abstract

A deep-etched polarization-independent binary fused-silica phase grating as a three-port beam splitter is designed and manufactured. The grating profile is optimized by use of the rigorous coupled-wave analysis around the $785\text{\hspace{0.17em}}\mathrm{nm}$ wavelength. The physical explanation of the grating is illustrated by the modal method. Simple analytical expressions of the diffraction efficiencies and modal guidelines for the three-port beam splitter grating design are given. Holographic recording technology and inductively coupled plasma etching are used to manufacture the fused-silica grating. Experimental results are in good agreement with the theoretical values.

© 2008 Optical Society of America

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### Equations (15)

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(1)
$${n}_{\text{out}}\mathrm{sin}({\phi}_{m})={n}_{\text{in}}\mathrm{sin}({\phi}_{\text{in}})+m\lambda /\mathrm{\Lambda},$$
(2)
$$\mathrm{cos}[{k}_{1}(\mathrm{\Lambda}-b)]\mathrm{cos}({k}_{2}b)-\frac{{k}_{1}^{2}+{k}_{2}^{2}}{2{k}_{1}{k}_{2}}\mathrm{sin}[{k}_{1}(\mathrm{\Lambda}-b)]\mathrm{sin}({k}_{2}b)=\mathrm{cos}(\alpha \mathrm{\Lambda}),$$
(3)
$$\u3008{E}_{ym}(x)\leftrightarrow {u}_{q}(x)\u3009=\frac{|{\int}_{0}^{\mathrm{\Lambda}}{E}_{ym}(x){u}_{q}(x)\mathrm{d}x{|}^{2}}{{\int}_{0}^{\mathrm{\Lambda}}|{E}_{ym}(x){|}^{2}\mathrm{d}x{\int}_{0}^{\mathrm{\Lambda}}|{u}_{q}(x){|}^{2}\mathrm{d}x},$$
(4)
$${E}_{y\text{in}}={t}_{\text{in}0}{u}_{0}(x)+{t}_{\text{in}2}{u}_{2}(x),$$
(5)
$${t}_{\text{in}q}=\frac{{\int}_{0}^{\mathrm{\Lambda}}{E}_{y\text{in}}(x){u}_{q}(x)\mathrm{d}x}{\sqrt{{\int}_{0}^{\mathrm{\Lambda}}|{E}_{y\text{in}}(x){|}^{2}\mathrm{d}x{\int}_{0}^{\mathrm{\Lambda}}|{u}_{q}(x){|}^{2}\mathrm{d}x}}\mathrm{.}$$
(6)
$${E}_{y\text{out}}(x,h)={E}_{1}{e}^{-i{k}_{x}x}+{E}_{0}+{E}_{-1}{e}^{i{k}_{x}x}={t}_{\text{in}0}{u}_{0}(x){e}^{-i{k}_{z}{n}_{0\text{eff}}h}+{t}_{\text{in}2}{u}_{2}(x){e}^{-i{k}_{z}{n}_{2\text{eff}}h}.$$
(7)
$$\frac{1}{\mathrm{\Lambda}}{\int}_{0}^{\mathrm{\Lambda}}[{t}_{\text{in}0}{u}_{0}(x)+{t}_{\text{in}2}{u}_{2}(x)]\mathrm{d}x=1,$$
(8)
$${E}_{0}=\frac{1}{\mathrm{\Lambda}}{\int}_{0}^{\mathrm{\Lambda}}[{t}_{\text{in}0}{u}_{0}(x){e}^{-i{k}_{z}{n}_{0\text{eff}}h}+{t}_{\text{in}2}{u}_{2}(x){e}^{-i{k}_{z}{n}_{2\text{eff}}h}]\mathrm{d}x.$$
(9)
$$\frac{1}{\mathrm{\Lambda}}{\int}_{0}^{\mathrm{\Lambda}}[{t}_{\text{in}0}{u}_{0}(x)+{t}_{\text{in}2}{u}_{2}(x)]\mathrm{cos}({k}_{x}x)\mathrm{d}x=0,$$
(10)
$${E}_{1}={E}_{-1}=\frac{1}{\mathrm{\Lambda}}{\int}_{0}^{\mathrm{\Lambda}}[{t}_{\text{in}0}{u}_{0}(x){e}^{-i{k}_{z}{n}_{0\text{eff}}h}+{t}_{\text{in}2}{u}_{2}(x){e}^{-i{k}_{z}{n}_{2\text{eff}}h}]\mathrm{cos}({k}_{x}x)\mathrm{d}x.$$
(11)
$${\eta}_{0}^{\mathrm{TE}}=|{E}_{0}{|}^{2}=|A{e}^{-i{k}_{z}{n}_{0\text{eff}}h}+(1-A){e}^{-i{k}_{z}{n}_{2\text{eff}}h}{|}^{2},$$
(12)
$${\eta}_{1}^{\mathrm{TE}}={\eta}_{-1}^{\mathrm{TE}}=|{E}_{1}{|}^{2}=4{B}^{2}{\mathrm{sin}}^{2}\frac{\mathrm{\Delta}\phi}{2},$$
(13)
$$A=\frac{1}{\mathrm{\Lambda}}{\int}_{0}^{\mathrm{\Lambda}}{t}_{\text{in}0}{u}_{0}(x)\mathrm{d}x,$$
(14)
$$B=|\frac{1}{\mathrm{\Lambda}}{\int}_{0}^{\mathrm{\Lambda}}({t}_{\text{in}0}{u}_{0}(x)\mathrm{cos}{k}_{x}x)\mathrm{d}x|,$$
(15)
$$4[{B}^{2}+A(1-A)]{\mathrm{sin}}^{2}\frac{\mathrm{\Delta}\phi}{2}=1.$$