## Abstract

Line-shape distortions caused by the misalignment of the moving cube mirror in Fourier transform spectrometers have been described. A method of studying and correcting these distortions is presented. By using this method we can estimate the accuracy of the line position, which is especially important in high-resolution Fourier transform spectroscopy. The method is verified in simulations, and in practice it has been used to align the Oulu Fourier transform spectrometer.

© 1992 Optical Society of America

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### Equations (16)

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(1)
$$\Delta \nu =\frac{{\nu}_{0}\Omega}{2\pi},$$
(2)
$${L}_{0}\approx \frac{1.21\pi}{{\nu}_{0}\Omega}.$$
(3)
$$\begin{array}{ll}E(\alpha )=1,\beta <\theta ,\alpha \le \theta -\beta ,\hfill & \hfill \\ E(\alpha )=\frac{1}{\pi}\text{arccos}\left(\frac{{\alpha}^{2}+{\beta}^{2}-{\theta}^{2}}{2\alpha \beta}\right)\hfill & \alpha \in \left(\left|\theta -\beta \right|,\theta +\beta \right),\hfill \\ E(\alpha )=0\hfill & \text{elsewhere},\hfill \end{array}$$
(4)
$$m\left(0\right)=\frac{2{J}_{1}\left(q\right)}{q},$$
(5)
$$q=4\pi {\nu}_{0}d\left(0\right)\theta \approx 4\pi {\nu}_{0}d\left(0\right){\left(\frac{\Omega}{\pi}\right)}^{1/2},$$
(6)
$$d\left(0\right)<\frac{1}{8\pi {\left(\Delta \nu {\nu}_{0}\right)}^{1/2}},$$
(7)
$$\beta <\frac{\theta}{{\left(\mathrm{S}/\mathrm{N}\right)}^{1/2}},$$
(8)
$$\begin{array}{ll}{x}_{i0}=\frac{j-1/2}{2{\nu}_{R}},\hfill & j=1,2,3,\dots .\hfill \end{array}$$
(9)
$${\u220a}_{j}={x}_{j}-{x}_{j0}={x}_{j}-\frac{j-1/2}{2{\nu}_{R}}.$$
(10)
$${\nu}_{\text{true}}={\nu}_{R}-\frac{d\u220a\left(x\right)}{dx}{\nu}_{R},$$
(11)
$${\nu}_{\text{true}}x={\nu}_{R}{x}_{R}={\nu}_{R}\left[x-\u220a\left(x\right)\right]$$
(12)
$${\nu}_{\text{true}}={\nu}_{R}\left[1-\frac{\u220a\left(x\right)}{x}\right].$$
(13)
$${\mathcal{F}}^{-1}\left[E\left(\nu -{\nu}_{R}\right)\right]=\text{Re}\left(x\right)+i\phantom{\rule{0.2em}{0ex}}\text{Im}\left(x\right)$$
(14)
$$\varphi \left(x\right)=\text{arctan}\left[\frac{\text{Im}\left(x\right)}{\text{Re}\left(x\right)}\right]=2\pi {\nu}_{R}\u220a\left(x\right).$$
(15)
$${I}^{\prime}\left(j\Delta x\right)={\displaystyle \sum _{k=j-N}^{j+N}{I}_{k}{g}_{j-k},}$$
(16)
$${g}_{j}=\text{sinc}\left[\frac{\pi}{2}\left(j-{\u220a}_{j}\right)\right]\text{cos}\left[\frac{\pi}{2}\left(j-{\u220a}_{j}\right)\right],$$