## Abstract

We proposed an analytical method to design optical minus filters by the thickness modulation of discrete, homogeneous thin-film layers of a two-material multilayer coating. The main stack provides the narrow, second-order rejection band, and the correct thickness–modulation apodization and match layers can effectively suppress the sidelobes of the passband. Using this approach, we can design minus filters with layer thicknesses close to half-wave of the rejection wavelength, making this method well suited for accurate monitoring during the deposition.

© 2013 Optical Society of America

Full Article |

PDF Article
### Equations (7)

Equations on this page are rendered with MathJax. Learn more.

(1)
$$\mathrm{\Delta}{\lambda}_{0}=\frac{2}{\pi}\frac{\sqrt{{C}^{2}-1}\text{\hspace{0.17em}}\mathrm{sin}(\pi p)}{[(C+1)-{(1-4p)}^{2}(C-1)\mathrm{cos}(2\pi p)]}{\lambda}_{0},$$
(2)
$$R(\lambda )=\frac{{a}_{11}^{2}{({n}_{a}-{n}_{s})}^{2}+{a}^{2}{({n}_{a}{n}_{s}\alpha +1)}^{2}{a}_{12}^{2}}{{a}_{11}^{2}{({n}_{a}+{n}_{s})}^{2}+{a}^{2}{(1-{n}_{a}{n}_{s}\alpha )}^{2}{a}_{12}^{2}}\phantom{\rule[-0.0ex]{1em}{0.0ex}}\text{and}$$
(3)
$${a}_{11}=0.5[{(1+\sqrt{\alpha}a)}^{-n}+{(1-\sqrt{\alpha}a)}^{-n}],{a}_{12}=0.5[{(1-\sqrt{\alpha}a)}^{-n}-{(1+\sqrt{\alpha}a)}^{-n}]/\left(\sqrt{\alpha}a\right),$$
(4)
$${\lambda}_{1}=\lambda /3+2/9\lambda \text{\hspace{0.17em}}{\mathrm{sin}}^{-1}(({n}_{h}-{n}_{l})/({n}_{h}+{n}_{l}))/\pi \text{and}{\lambda}_{2}=\lambda -2\lambda \text{\hspace{0.17em}}{\mathrm{sin}}^{-1}(({n}_{h}-{n}_{l})/({n}_{h}+{n}_{l}))/\pi .$$
(5)
$$T={t}_{a}+0.5*{t}_{p}*A(l)*\mathrm{sin}(\pi *l+\varphi ),$$
(6)
$$A(l)=10{t}^{3}-15{t}^{4}+6{t}^{5},$$
(7)
$$t=2l/s\phantom{\rule[-0.0ex]{1em}{0.0ex}}\text{for}\text{\hspace{0.17em}}\text{\hspace{0.17em}}l\le s/2,=2(s-l)/s\phantom{\rule[-0.0ex]{1em}{0.0ex}}\text{for}\text{\hspace{0.17em}}\text{\hspace{0.17em}}l>s/2.$$