## Abstract

We demonstrate that it is possible to measure the local geometrical thickness and the refractive index of a transparent pellicle in air by combining the diffractive properties of a Gaussian beam with the analytical equations of the light that propagates through a thin layer. We show that our measurement technique is immune to inherent piston-like vibrations present in the pellicle. As our measurements are based on characterizing properly the Gaussian beam in a plane of detection, a homodyne technique for this purpose is devised and described. The feasibility of our proposal is confirmed by measuring local geometrical thicknesses and the refractive index of a commercially available stretch film.

© 2014 Optical Society of America

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### Equations (16)

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(1)
$${z}_{T}={z}_{1}+t+{z}_{2},$$
(2)
$$\mathrm{\Psi}({x}_{1},{y}_{1})=A\text{\hspace{0.17em}}\mathrm{exp}(-\frac{{x}_{1}^{2}+{y}_{1}^{2}}{{r}^{2}})\mathrm{exp}[i\beta ({x}_{1}^{2}+{y}_{1}^{2})].$$
(3)
$${r}_{A}=\frac{r}{\pi}\sqrt{{[\beta \lambda {z}_{T}+\pi ]}^{2}+\frac{{\lambda}^{2}}{{r}^{4}}{z}_{T}^{2}}.$$
(4)
$${r}_{B}=\frac{r}{\pi}\sqrt{{[\beta \lambda ({z}_{1}+\frac{t}{n}+{z}_{2})+\pi ]}^{2}+\frac{{\lambda}^{2}}{{r}^{4}}{({z}_{1}+\frac{t}{n}+{z}_{2})}^{2}}.$$
(5)
$${z}_{1}+\frac{t}{n}+{z}_{2}={z}_{1}^{\prime}+\frac{{t}^{\prime}}{{n}^{\prime}}+{z}_{2}^{\prime}={z}_{1}^{\prime \prime}+\frac{{t}^{\prime \prime}}{{n}^{\prime \prime}}+{z}_{2}^{\prime \prime}=\cdots .$$
(6)
$$t\frac{n-1}{n}={t}^{\prime}\frac{{n}^{\prime}-1}{{n}^{\prime}}={t}^{\prime \prime}\frac{{n}^{\prime \prime}-1}{{n}^{\prime \prime}}=\cdots .$$
(7)
$$F=t\frac{n-1}{n}.$$
(8)
$$P={\left(\frac{4{n}_{0}{n}_{1}}{M}\right)}^{2},$$
(9)
$$M=\left|\begin{array}{cccc}1& -1& -1& 0\\ -{n}_{0}& -{n}_{1}& {n}_{1}& 0\\ 0& \mathrm{exp}(-i\frac{2\pi}{\lambda}{n}_{0}{n}_{1}t)& \mathrm{exp}\left(i\frac{2\pi}{\lambda}{n}_{0}{n}_{1}t\right)& -1\\ 0& {n}_{1}\text{\hspace{0.17em}}\mathrm{exp}(-i\frac{2\pi}{\lambda}{n}_{0}{n}_{1}t)& -{n}_{1}\text{\hspace{0.17em}}\mathrm{exp}(-i\frac{2\pi}{\lambda}{n}_{0}{n}_{1}t)& -{n}_{0}\end{array}\right|.$$
(10)
$$P=A{\int}_{\alpha}^{\infty}\mathrm{exp}(-2\frac{{x}^{2}}{{r}_{0}^{2}})\mathrm{d}x.$$
(11)
$$\alpha ={x}_{0}+{\delta}_{0}\text{\hspace{0.17em}}\mathrm{cos}(2\pi ft),$$
(12)
$${P}_{\text{linear}}({x}_{0})=B\text{\hspace{0.17em}}\mathrm{exp}(-2\frac{{x}_{0}^{2}}{{r}_{0}^{2}})\mathrm{cos}(2\pi ft),$$
(13)
$${P}_{\text{rel}}=\frac{{P}_{\text{film}}}{{P}_{\text{air}}}.$$
(14)
$$F=t\frac{n-1}{n},$$
(15)
$${F}^{\prime}={t}^{\prime}\frac{n-1}{n},$$
(16)
$$\frac{F}{{F}^{\prime}}=\frac{t}{{t}^{\prime}}.$$