## Abstract

A three-dimensional model for beam propagation through optical interference filters is presented. The model predicts a wavelength-dependent lateral beam displacement of tens or hundreds of micrometers in narrowband filters at an angle of incidence of only 3° to 5°. The effects of filter bandwidth, wavelength offset, angle of incidence, and beam size are investigated. The effect is experimentally confirmed for a
$100\text{\hspace{0.17em} GHz}$ filter at a 3.5° angle of incidence.

© 2006 Optical Society of America

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### Equations (9)

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(1)
$$A\left({k}_{x},{k}_{y}\right)=\frac{1}{2\pi}\int \int E\left(x,y,0\right){\displaystyle \mathrm{exp}\left[i\left({k}_{x}x+{k}_{y}y\right)\right]\mathrm{d}x\mathrm{d}y}.$$
(2)
$$k\equiv \left|\mathbf{k}\right|{=\left({{k}_{x}}^{2}+{{k}_{y}}^{2}+{{k}_{z}}^{2}\right)}^{1/2}=\frac{n\omega}{c}=\frac{2\pi n}{\lambda},$$
(3)
$$E\left(x,y,z,t\right)={\displaystyle \int \int A\left({k}_{x},{k}_{y}\right)\mathrm{exp}\left[i\left(\omega t-{k}_{x}x-{k}_{y}y-{k}_{z}z\right)\right]\mathrm{d}{k}_{x}\mathrm{d}{k}_{y},\text{\hspace{0.17em}\hspace{0.17em}}z\ge 0.}$$
(4)
$$E\left(x,y,z,t\right)=\int \int t\left({k}_{x},{k}_{y}\right)A\left({k}_{x},{k}_{y}\right){\displaystyle \mathrm{exp}\left\{i[\omega t-{k}_{x}x-{k}_{y}y-{k}_{z}\left(z-d\right)]\right\}\mathrm{d}{k}_{x}\mathrm{d}{k}_{y}},\text{\hspace{1em}}$$
(6)
$$\phi \left({k}_{x}\right)=\phi \left({k}_{x0}\right)+\phi \prime \left({k}_{x0}\right)\left({k}_{x}-{k}_{x0}\right)+\cdots \mathrm{.}$$
(7)
$$E\left(x,t\right)=t\left({k}_{x0}\right)\mathrm{exp}\left\{i\left[\phi \left({k}_{x0}\right)-{k}_{x0}\phi \prime \left({k}_{x0}\right)\right]\right\}\times \int A\left({k}_{x}\right)\mathrm{exp}\left(i\left\{\omega t-{k}_{x}[x-\phi \prime \left({k}_{x0}\right)\right]\right\})\mathrm{d}{k}_{x}$$
(8)
$$=\text{\hspace{0.17em}}t\left({k}_{x0}\right)\mathrm{exp}\left(j{\mathrm{\Phi}}_{0}\right)E\left[x-\phi \prime \left({k}_{x0}\right),t\right].$$
(9)
$$\text{Substrate \hspace{0.17em}}\left|{\left(HL\right)}^{6}\text{\hspace{0.17em}}H\text{\hspace{0.17em} 6}L\text{\hspace{0.17em}}H\text{\hspace{0.17em}}{\left(LH\right)}^{6}\text{\hspace{0.17em}}L\text{\hspace{0.17em}}{\left(HL\right)}^{7}\text{\hspace{0.17em}}H\text{\hspace{0.17em} 4}LH\text{\hspace{0.17em}}{\left(LH\right)}^{7}\text{\hspace{0.17em}}L{\left(HL\right)}^{7}\text{\hspace{0.17em}}H\text{\hspace{0.17em} 6}L\text{\hspace{0.17em}}H\text{\hspace{0.17em}}{\left(LH\right)}^{7}\text{\hspace{0.17em}}L\text{\hspace{0.17em}}{\left(HL\right)}^{6}\text{\hspace{0.17em}}H\text{\hspace{0.17em} 8}L\text{\hspace{0.17em}}H\text{\hspace{0.17em}}{\left(LH\right)}^{7}\text{\hspace{0.17em}}L\right|\text{Air,}$$