## Abstract

Interferometry in grazing incidence can be used to test cylindrical mantle
surfaces. The absolute accuracy of the resulting surface profiles is limited by systematic wavefront
aberrations caused in the interferometer, in particular due to an inversion of the test wavefront
in an interferometer using diffractive beam splitters. For cylindrical specimens, a
calibration method using four positions has therefore been investigated. This test is combined
with another method of optical metrology: the rotational averaging procedure. The
implementation for grazing incidence is described and measurement results for hollow
cylinders are presented. The gain in accuracy is demonstrated.

© 2006 Optical Society of America

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### Equations (14)

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(1)
$$h\left(z,\phi \right)={h}_{\text{circ}}\left(z\right)+{h}_{\text{noncirc}}\left(z,\phi \right)=\left[{h}_{\text{con}}\left(z\right)+{\stackrel{\u0303}{h}}_{\text{circ}}\left(z\right)\right]+{h}_{\text{noncirc}}\left(z,\phi \right)\mathrm{.}$$
(2)
$$\u3008{h}_{\text{noncirc}}\left(z,\phi \right)\u3009=0.$$
(3)
$${\varphi}_{\text{norm}}-\u3008{\varphi}_{\text{norm}}\u3009=\frac{4\pi}{{\lambda}_{\text{eff}}}{h}_{\text{noncirc}}\left(z,\phi \right),$$
(4)
$$\u3008{\varphi}_{\text{shift}}\u3009-\u3008{\varphi}_{\text{norm}}\u3009=\frac{4\pi}{{\lambda}_{\text{eff}}}\text{\hspace{0.17em}}\left[{\tilde{h}}_{\text{circ}}\left(z+\delta z\right)-{\tilde{h}}_{\text{circ}}\left(z\right)\right]+\frac{4\pi}{{\lambda}_{\text{eff}}}\left(a+b\right)\delta z,$$
(5)
$$\u3008{\varphi}_{\text{flip}}\u3009-\u3008{\varphi}_{\text{norm}}\u3009=\frac{4\pi}{{\lambda}_{\text{eff}}}\text{\hspace{0.17em}}\left[{\tilde{h}}_{\text{circ}}\left(-z\right)-{\tilde{h}}_{\text{circ}}\left(z\right)\right]-\frac{8\pi}{{\lambda}_{\text{eff}}}\text{\hspace{0.17em}}az,$$
(6)
$$\begin{array}{c}{\tilde{h}}_{\text{circ}}\left(z+\delta z\right)={\displaystyle \sum _{i=1}^{M}{m}_{i}}{T}_{i}\left(z+\delta z\right),\end{array}$$
(7)
$$\begin{array}{c}{\tilde{h}}_{\text{circ}}\left(z\right)={\displaystyle \sum _{i=1}^{M}{m}_{i}{T}_{i}\left(z\right)},\end{array}$$
(8)
$${T}_{i}\left(z\right)=\mathrm{cos}\left(i\text{\hspace{0.17em}}\mathrm{arccos}\text{\hspace{0.17em}}z\right),$$
(9)
$$\u3008\varphi \u3009=\frac{1}{N}{\displaystyle \sum _{n=0}^{N-1}\varphi \left({R}_{n\Delta \Theta}^{\text{\hspace{0.17em} \hspace{0.17em} \hspace{0.17em} \hspace{0.17em} \hspace{0.17em} \hspace{0.17em} \hspace{0.17em} \hspace{0.17em}}-1}x\right)},$$
(10)
$${\u3008\varphi \u3009}_{\mathbf{a}}=\frac{1}{N}{\displaystyle \sum _{n=0}^{N-1}\varphi \left({R}_{n\Delta \Theta}^{\text{\hspace{0.17em} \hspace{0.17em} \hspace{0.17em} \hspace{0.17em} \hspace{0.17em} \hspace{0.17em} \hspace{0.17em} \hspace{0.17em}}-1}x-a\right)}\mathrm{.}$$
(11)
$${\u3008\varphi \u3009}_{a}\approx \frac{1}{N}{\displaystyle \sum _{n=0}^{N-1}\left[\varphi \left({R}_{n\Delta \Theta}^{\text{\hspace{0.17em} \hspace{0.17em} \hspace{0.17em} \hspace{0.17em} \hspace{0.17em} \hspace{0.17em} \hspace{0.17em} \hspace{0.17em}}-1}x\right)-a\frac{\partial \varphi \left({R}_{n\Delta \Theta}^{\text{\hspace{0.17em} \hspace{0.17em} \hspace{0.17em} \hspace{0.17em} \hspace{0.17em} \hspace{0.17em} \hspace{0.17em}}-1}x\right)}{\partial \left({R}_{n\Delta \Theta}^{\text{\hspace{0.17em} \hspace{0.17em} \hspace{0.17em} \hspace{0.17em} \hspace{0.17em} \hspace{0.17em} \hspace{0.17em} \hspace{0.17em}}-1}x\right)}\right]}\mathrm{.}$$
(12)
$${\u3008\varphi \u3009}_{a}\approx \frac{1}{N}{\displaystyle \sum _{n=0}^{N-1}\varphi \left({R}_{n\mathrm{\Delta \Theta}}^{\text{\hspace{0.17em} \hspace{0.17em} \hspace{0.17em} \hspace{0.17em} \hspace{0.17em} \hspace{0.17em} \hspace{0.17em} \hspace{0.17em}}-1}x\right)}-\frac{a}{N}{\displaystyle \sum _{n=0}^{N-1}\frac{\partial \varphi \left({R}_{n\Delta \Theta}^{\text{\hspace{0.17em} \hspace{0.17em} \hspace{0.17em} \hspace{0.17em} \hspace{0.17em} \hspace{0.17em} \hspace{0.17em} \hspace{0.17em}}-1}x\right)}{\partial \left({R}_{n\Delta \Theta}^{\text{\hspace{0.17em} \hspace{0.17em} \hspace{0.17em} \hspace{0.17em} \hspace{0.17em} \hspace{0.17em} \hspace{0.17em} \hspace{0.17em}}-1}x\right)}}=\u3008\varphi \u3009-\mathrm{\Delta \varphi}.$$
(13)
$$\mathrm{\Delta \varphi}\approx a\frac{\partial \varphi \left(x\right)}{\partial \left(x\right)}\mathrm{.}$$
(14)
$$\mathrm{\Delta \varphi}\approx \frac{2\pi}{100}\mathrm{.}$$