## Abstract

A method of measuring thin film thickness is described, based on a previous paper where the authors analyzed a graphical means for grating modulation calculation. Metallic or dielectric films on any substrate may be measured, and precision was shown to be comparable with that achieved ellipsometrically at least in the 300–1500-Å thickness range. The method is non-contact and destructive: the grating is recorded on the film. Measurements are simple, requiring a low-power He–Ne laser and a photodetector, and may be carried out at a distance from the sample. Experimental results are presented for three types of sample, including measurements by reflection and transmission.

© 1984 Optical Society of America

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### Equations (13)

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(1)
$${I}_{N}/{I}_{0}=\frac{\mid {\widehat{r}}_{a}-{\widehat{r}}_{b}{\mid}^{2}{(a/d)}^{2}}{\mid {\widehat{r}}_{b}+({\widehat{r}}_{a}-{\widehat{r}}_{b})(a/d){\mid}^{2}}\hspace{0.17em}{\text{sinc}}^{2}(Na/d),$$
(2)
$$\begin{array}{ll}{\widehat{r}}_{a}=\frac{{\widehat{r}}_{1}+{\widehat{r}}_{2}\hspace{0.17em}\text{exp}(i2\beta )}{1+{\widehat{r}}_{1}{\widehat{r}}_{2}\hspace{0.17em}\text{exp}(i2\beta )},\hfill & \text{where}\hspace{0.17em}2\beta \equiv 4\pi h{\widehat{n}}_{1}/\mathrm{\lambda},\hfill \\ {\widehat{r}}_{b}={\widehat{r}}_{3}\hspace{0.17em}\text{exp}(i2\gamma ),\hfill & \text{where}\hspace{0.17em}2\gamma \equiv 4\pi h{n}_{0}/\mathrm{\lambda}.\hfill \end{array}\}$$
(3)
$$\begin{array}{l}{\widehat{r}}_{1}=({n}_{0}-{\widehat{n}}_{1})/({n}_{0}+{\widehat{n}}_{1}),\hfill \\ {\widehat{r}}_{2}=({\widehat{n}}_{1}-{\widehat{n}}_{2})/({\widehat{n}}_{1}+{\widehat{n}}_{2}),\hfill \\ {\widehat{r}}_{3}=({n}_{0}-{\widehat{n}}_{2})/({n}_{0}+{\widehat{n}}_{2}).\hfill \end{array}\}$$
(4)
$$\begin{array}{ll}{\widehat{r}}_{a}={\widehat{r}}_{1}\hfill & \hspace{0.17em}\hfill \\ {\widehat{r}}_{b}={r}_{1}\hspace{0.17em}\text{exp}(i2\gamma )\hfill & \text{where}\hspace{0.17em}2\gamma =4\pi h/\mathrm{\lambda}\hfill \end{array}\},$$
(5)
$$\text{cos}(2\gamma )=1-\frac{{I}_{N}/{I}_{0}}{2{(a/d)}^{2}\hspace{0.17em}{\text{sinc}}^{2}(Na/d)+({I}_{N}/{I}_{0})2(1-a/d)a/d}.$$
(6)
$$\begin{array}{l}2\gamma =K2\pi \pm 2{\gamma}_{0}\hfill \\ \text{or}\hfill \\ h=K\mathrm{\lambda}/2\pm {h}_{0}\hfill \end{array}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\begin{array}{l}\text{with}\hspace{0.17em}K\hspace{0.17em}\text{an}\hspace{0.17em}\text{integer}\hfill \end{array}\}.$$
(7)
$${I}_{N}/{I}_{0}=\frac{\mid {\widehat{t}}_{a}-{\widehat{t}}_{b}{\mid}^{2}{(a/d)}^{2}}{\mid {\widehat{t}}_{b}+({\widehat{t}}_{a}-{\widehat{t}}_{b})a/d{\mid}^{2}}{\text{sinc}}^{2}(Na/d).$$
(8)
$$\begin{array}{ll}{\widehat{t}}_{a}=\frac{{t}_{01}{t}_{10}\hspace{0.17em}\text{exp}(i\beta )}{1+{r}_{01}{r}_{10}\hspace{0.17em}\text{exp}(i2\beta )}\hfill & \text{with}\hspace{0.17em}\beta \equiv 2\pi (H+{H}_{s}){n}_{1}/\mathrm{\lambda},\hfill \\ {\widehat{t}}_{b}=\frac{{t}_{01}{t}_{10}\hspace{0.17em}\text{exp}(i\gamma )}{1+{r}_{01}{r}_{10}\hspace{0.17em}\text{exp}(i2\gamma )}\hfill & \text{with}\hspace{0.17em}\gamma \equiv 2\pi ({H}_{s}{n}_{1}+H{n}_{0})/\mathrm{\lambda},\hfill \end{array}\}$$
(9)
$$\text{cos}(\beta -\gamma )=1-\frac{{I}_{N}/{I}_{0}}{2{(a/d)}^{2}\hspace{0.17em}{\text{sinc}}^{2}(Na/d)+2{I}_{N}/{I}_{0}(1-a/d)a/d}.$$
(10)
$$H=h2{n}_{0}/({n}_{1}-{n}_{0}).$$
(11)
$$\begin{array}{l}{r}_{1}=-0.1870,\hfill \\ {\widehat{r}}_{2}=-0.4501-i0.0021,\hfill \\ {\widehat{r}}_{3}=-0.5876-i\mathrm{0.0017.}\hfill \end{array}\}$$
(12)
$$\begin{array}{l}{\widehat{r}}_{a}=\frac{-0.1870-0.4501\hspace{0.17em}\text{exp}[i(2\beta +0.0047)]}{1+0.0842\hspace{0.17em}\text{exp}[i(2\beta +0.0046)]},\hfill \\ {\widehat{r}}_{b}=-0.5876\hspace{0.17em}\text{exp}[i(2\gamma +0.0029)].\hfill \end{array}\}$$
(13)
$$\begin{array}{l}{\widehat{r}}_{a}\simeq \frac{-0.1870-0.4501\hspace{0.17em}\text{exp}(i2\beta )}{1+0.0842\hspace{0.17em}\text{exp}(i2\beta )},\hfill \\ {\widehat{r}}_{b}\simeq -0.5876\hspace{0.17em}\text{exp}(i2\gamma ),\hfill \end{array}\}$$