## Abstract

Frequency Domain Analysis of measurement results from coherence scanning interferometers provides an estimate of the topography of the surface scattering the light, with claims of nm levels of accuracy being achieved. In the following work we use simulations of the measurement result to show that the limited set of spatial frequencies passed by the instrument can lead to errors in excess of 200 nm for surfaces with curvature. In addition we present a method that takes the uncertainty in the amplitude and phase of each element of the transfer function and provides the upper and lower limit on the location of the surface provided by the Frequency Domain Analysis method. This provides an idea of the level of accuracy that the spatial frequency components must be known to in order to reproduce curved surfaces well.

© 2015 Optical Society of America

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### Equations (12)

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(1)
$$\tilde{O}(\mathbf{k})=2\tilde{\mathrm{\Delta}}(\mathbf{k})\tilde{H}(\mathbf{k})$$
(2)
$$F(\mathbf{k})=8\pi {r}_{0}^{3}{\displaystyle {\int}_{-\infty}^{\infty}\text{sinc}}\left({r}_{0}\sqrt{{\kappa}^{2}+{k}_{x}^{2}+{k}_{y}^{2}}\right)\text{sinc}\left({r}_{0}\left({k}_{z}-\kappa \right)\right)\mathrm{exp}\left(i{r}_{0}\left[1+\mathrm{cos}\mathrm{\Theta}\right]\left({k}_{z}-\kappa \right)\right)d\kappa $$
(3)
$$f(\mathbf{r})={\displaystyle \iiint F\left(\mathbf{k}\right)\mathrm{exp}\left(-i\mathbf{k}.\mathbf{r}\right)d{k}_{x}d{k}_{y}d{k}_{z}}$$
(4)
$$F{T}_{plane}=\mathrm{exp}\left(i\mathbf{k}.{\mathbf{r}}_{0}\right){|}_{\theta ={\theta}_{0},\varphi ={\varphi}_{0}}$$
(5)
$$Signa{l}_{\mathbf{k},\mathbf{r}}={A}_{\mathbf{k}}\mathrm{exp}(-i[{k}_{x}x+{k}_{y}y])\mathrm{exp}(-i{k}_{z}z)$$
(6)
$${B}_{\mathbf{k}}{}_{,x,y}\mathrm{exp}\left(-i{k}_{z}z\right)={A}_{\mathbf{k}}\mathrm{exp}(-i[{k}_{x}x+{k}_{y}y])\mathrm{exp}(-i{k}_{z}z)$$
(7)
$${\mathrm{F}}_{z}({k}_{z}){|}_{(x,y)}={\displaystyle \sum _{{k}_{x}}{\displaystyle \sum _{{k}_{y}}{B}_{\mathbf{k},x,y}{|}_{{k}_{z}}}}$$
(8)
$${U}_{0}=\sqrt{{|{B}_{\mathbf{k},x,y}|}^{2}+{\left(|{B}_{\mathbf{k},x,y}|+|\mathrm{\Delta}{B}_{\mathbf{k},x,y}|\right)}^{2}-2|{B}_{\mathbf{k},x,y}|(|{B}_{\mathbf{k},x,y}|+|\mathrm{\Delta}{B}_{\mathbf{k},x,y}|)\mathrm{cos}(\mathrm{\Delta}{\theta}_{\mathbf{k},x,y})}{|}_{{k}_{z}}$$
(9)
$${U}_{\mathrm{max}}={\displaystyle \sum _{{k}_{x}}{\displaystyle \sum _{{k}_{y}}\sqrt{{|{B}_{\mathbf{k},x,y}|}^{2}+{\left(|{B}_{\mathbf{k},x,y}|+|\mathrm{\Delta}{B}_{\mathbf{k},x,y}|\right)}^{2}-2|{B}_{\mathbf{k},x,y}|(|{B}_{\mathbf{k},x,y}|+|\mathrm{\Delta}{B}_{\mathbf{k},x,y}|)\mathrm{cos}(\mathrm{\Delta}{\theta}_{\mathbf{k},x,y})}}{|}_{{k}_{z}}}$$
(10)
$$\mathrm{\Delta}\theta =asin\left(\frac{{U}_{\mathrm{max}}}{\sqrt{{x}_{0}^{2}+{y}_{0}^{2}}}\right)$$
(11)
$${\beta}_{1}=\frac{m{\sum}_{n=0}^{m-1}{p}_{n}{w}_{n}+m{\sum}_{n=0}^{m-1}{w}_{n}{\epsilon}_{n}-{\sum}_{n=0}^{m-1}{w}_{n}{\sum}_{n=0}^{m-1}{p}_{n}-{\sum}_{n=0}^{m-1}{w}_{n}{\sum}_{n=0}^{m-1}{\epsilon}_{n}}{m{\sum}_{n=0}^{m-1}{w}_{n}^{2}-{\left({\sum}_{n=0}^{m-1}{w}_{n}\right)}^{2}}$$
(12)
$$\frac{\partial {\beta}_{1}}{\partial {\epsilon}_{n}}=\frac{m{w}_{n}-{\sum}_{r=0}^{m-1}{w}_{r}}{m{\sum}_{r=0}^{m-1}{w}_{r}^{2}-{\left({\sum}_{r=0}^{m-1}{w}_{r}\right)}^{2}}$$