## Abstract

An empirical procedure based on optical-density-bandwidth products was recently
proposed for thickness estimation of dielectric thin film reflectors. A parallel is established
with new results derived from the Fourier transform thin film
synthesis technique. Two Fourier-transform approaches are proposed and justified by
numerical examples.

© 2006 Optical Society of America

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### Equations (12)

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(1)
$$N\approx \frac{{\displaystyle {\int}_{{\sigma}_{\mathrm{MIN}}}^{{\sigma}_{\mathrm{MAX}}}\text{\hspace{0.17em}}\frac{\text{OD}}{\sigma}\text{\hspace{0.17em}}\mathrm{d}\sigma}}{\text{2 \hspace{0.17em} ln}\left(\frac{{n}_{H}}{{n}_{L}}\right){\mathrm{sin}}^{-1}\left(\frac{{n}_{H}-{n}_{L}}{{n}_{H}+{n}_{L}}\right)},$$
(2)
$$\mathrm{ln}\left[\frac{n\left(x\right)}{{n}_{0}}\right]\text{\hspace{0.17em}}\stackrel{\text{FT}}{\leftrightarrow}\text{\hspace{0.17em}}\frac{i}{\pi}\frac{\tilde{Q}\left(T,\sigma \right)}{\sigma}$$
(3)
$$\tilde{Q}\left(T,\sigma \right)=Q\left(T\right){e}^{i\varphi \left(\sigma \right)}$$
(4)
$${{\displaystyle {\int}_{-\infty}^{\infty}{\mathrm{ln}}^{2}\left[\frac{n\left(x\right)}{{n}_{0}}\right]\mathrm{d}x=\frac{1}{{\pi}^{2}}\text{\hspace{0.17em}}{\displaystyle {\int}_{-\infty}^{\infty}\left[\frac{Q\left(T\right)}{\sigma}\right]}}}^{2}\mathrm{d}\sigma .$$
(5)
$$\int {\mathrm{ln}}^{2}\left[\frac{n\left(x\right)}{{n}_{0}}\right]\mathrm{d}x\approx \frac{1}{2}\text{\hspace{0.17em}}}\left(\mathrm{\Sigma}nt\right){\mathrm{ln}}^{2}\left[\frac{{n}_{H}}{{n}_{L}}\right]\mathrm{.$$
(6)
$$Q\left(T\right)=\left|\tilde{Q}\left(T,\sigma \right)\right|=\sqrt{\mathrm{ln}\left(\frac{1}{T}\right)}=\sqrt{\mathrm{ln}\left(10\right)\text{OD}}.$$
(7)
$$\Sigma nt\approx 0.93\text{\hspace{0.17em}}\frac{{\displaystyle {\int}_{0}^{\infty}\frac{\text{OD}}{{\sigma}^{\text{2}}}\text{\hspace{0.17em}d}\sigma}}{{\mathrm{ln}}^{2}\left[\frac{{n}_{H}}{{n}_{L}}\right]}.$$
(8)
$$\Sigma nt\approx 2\text{\hspace{0.17em}}\frac{{\displaystyle \int {\mathrm{ln}}^{2}\left[\frac{{n}_{C}\left(x\right)}{{n}_{0}}\right]\mathrm{d}x}}{{\mathrm{ln}}^{2}\left[\frac{{n}_{H}}{{n}_{L}}\right]},$$
(9)
$$1+\frac{1}{{3}^{2}}+\frac{1}{{5}^{2}}+\text{\hspace{0.17em}}\cdots \text{\hspace{0.17em}}=\frac{{\pi}^{2}}{8}=1.23,$$
(10)
$$\begin{array}{cc}\mathrm{ln}\left[\frac{n\left(x\right)}{{n}_{0}}\right]=\frac{2}{p}{\displaystyle \text{\hspace{0.17em}}\sum _{m=-\infty}^{l}{F}_{Q}\left(\frac{m}{p}\right),}& \frac{l}{p}\text{\hspace{0.17em}}\le \text{\hspace{0.17em}}x\text{\hspace{0.17em}}\le \text{\hspace{0.17em}}\frac{l+1}{p}\end{array},$$
(11)
$${\left(\frac{{n}_{H}}{{n}_{L}}\right)}^{\sqrt{\sum nt}}=\text{const .}$$
(12)
$${\left(\frac{{n}_{H}}{{n}_{L}}\right)}^{\sum nt}=\text{const.,}$$