## Abstract

A method using a wedge-plate shear shifting system with adjustable accuracy and measuring range is proposed for measuring the focus shifting caused by thermal distortion in a single-shot laser system. Two beam splitter groups are used in this method to precisely split a single beam into multiple beams with different optical path difference. The focus shifting is determined by position change of the minimum spot on the detector. This method is convenient and economic, especially as it powerfully solves the problem of catching focus shifting in an ultrashort time.

© 2013 Optical Society of America

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### Equations (9)

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(1)
$$\{\begin{array}{cc}{W}_{N,N}={R}_{1}^{2}& (N=1)\\ {W}_{N,N}={T}_{1}^{4}\xb7{R}_{2}^{2}& (N=2)\\ {W}_{N,N}={T}_{1}^{4}\xb7{T}_{2}^{4}\xb7{R}_{2}^{(N-2)\xb72}& (N>2)\end{array},$$
(2)
$$\{\begin{array}{cc}{K}_{1,1}=\frac{{W}_{1,1}}{{W}_{N,N}}=1& (N=1)\\ {K}_{1,N}=\frac{{W}_{1,1}}{{W}_{N,N}}=\frac{{R}_{1}^{2}}{{T}_{1}^{4}\xb7{T}_{2}^{4}\xb7{T}_{2}^{(N-2)\xb72}}& (N\ge 2)\end{array}\mathrm{.}$$
(3)
$$\{\begin{array}{cc}{K}_{2,2}=\frac{{W}_{2,2}}{{W}_{N,N}}=1& (N=2)\\ {K}_{2,N}=\frac{{W}_{2,2}}{{W}_{N,N}}=\frac{{R}_{2}^{2}}{{T}_{2}^{4}\xb7{R}_{2}^{(N-2)\xb72}}& (N>2)\end{array}.$$
(4)
$${R}_{2}=\frac{N-3}{N-1}\phantom{\rule[-0.0ex]{2em}{0.0ex}}(N>3).$$
(5)
$${c}^{\prime}=\mathrm{arcsin}[\mathrm{sin}(2\xb7\theta +\mathrm{arcsin}\left(\frac{\mathrm{sin}(a)}{n}\right))\xb7n].$$
(6)
$$\mathrm{\Delta}l=2w[\mathrm{sec}({b}^{\prime})+n\frac{\mathrm{sin}(\theta )\mathrm{tan}({b}^{\prime})}{\mathrm{cos}(c)}-\mathrm{sin}({c}^{\prime})\mathrm{tan}({b}^{\prime})(\mathrm{cos}(\theta )+\mathrm{sin}(\theta )\mathrm{tan}(c))].$$
(7)
$${\sigma}_{\mathrm{\Delta}l}^{2}={\left(\frac{\partial \mathrm{\Delta}l}{\partial w}\right)}^{2}\xb7{\sigma}_{w}^{2}+{\left(\frac{\partial \mathrm{\Delta}l}{\partial a}\right)}^{2}\xb7{\sigma}_{a}^{2}+{\left(\frac{\partial \mathrm{\Delta}l}{\partial \theta}\right)}^{2}\xb7{\sigma}_{\theta}^{2}+{\left(\frac{\partial \mathrm{\Delta}l}{\partial n}\right)}^{2}\xb7{\sigma}_{n}^{2}.$$
(8)
$$\{\begin{array}{l}\frac{\partial {a}^{\prime}}{\partial a}=\frac{\partial b}{\partial a}=\frac{\partial c}{\partial a}=\frac{\mathrm{cos}(a)}{\sqrt{{n}^{2}-{\mathrm{sin}}^{2}(a)}}\\ \frac{\partial {a}^{\prime}}{\partial n}=\frac{\partial b}{\partial n}=\frac{\partial c}{\partial n}=\frac{-\mathrm{sin}(a)}{n\sqrt{{n}^{2}-{\mathrm{sin}}^{2}(a)}}\\ \frac{\partial c}{\partial \theta}=2\frac{\partial b}{\partial \theta}=2\end{array}\{\begin{array}{l}\frac{\partial {b}^{\prime}}{\partial a}=\frac{n\text{\hspace{0.17em}}\mathrm{cos}(b)}{\sqrt{1-{(n\text{\hspace{0.17em}}\mathrm{sin}(b))}^{2}}}\frac{\partial b}{\partial a}\\ \frac{\partial {b}^{\prime}}{\partial n}=\frac{\mathrm{sin}(b)-n\text{\hspace{0.17em}}\mathrm{cos}(b)\frac{\partial b}{\partial n}}{\sqrt{1-{(n\text{\hspace{0.17em}}\mathrm{sin}(b))}^{2}}}\\ \frac{\partial {b}^{\prime}}{\partial \theta}=\frac{n\text{\hspace{0.17em}}\mathrm{cos}(b)}{\sqrt{1-{(n\text{\hspace{0.17em}}\mathrm{sin}(b))}^{2}}}\end{array}\{\begin{array}{l}\frac{\partial c}{\partial a}=\frac{n\text{\hspace{0.17em}}\mathrm{cos}(c)}{\sqrt{1-{(n\text{\hspace{0.17em}}\mathrm{sin}(c))}^{2}}}\frac{\partial b}{\partial a}\\ \frac{\partial {c}^{\prime}}{\partial n}=\frac{\mathrm{sin}(c)+n\text{\hspace{0.17em}}\mathrm{cos}(c)}{\sqrt{1-{(n\text{\hspace{0.17em}}\mathrm{sin}(c))}^{2}}}\frac{\partial b}{\partial n}\\ \frac{\partial {c}^{\prime}}{\partial \theta}=\frac{2n\text{\hspace{0.17em}}\mathrm{cos}(c)}{\sqrt{1-{(n\text{\hspace{0.17em}}\mathrm{sin}(c))}^{2}}}\end{array}.$$
(9)
$$\{\begin{array}{l}\frac{\partial \mathrm{\Delta}l}{\partial w}=2[\mathrm{sec}({b}^{\prime})+n\frac{\mathrm{sin}(\theta )\mathrm{tan}({b}^{\prime})}{\mathrm{cos}(c)}-\mathrm{sin}({c}^{\prime})\mathrm{tan}({b}^{\prime})(\mathrm{cos}(\theta )+\mathrm{sin}(\theta )\mathrm{tan}(c))]\\ \frac{\partial \mathrm{\Delta}l}{\partial a}=\frac{\partial \mathrm{\Delta}l}{\partial {b}^{\prime}}\frac{\partial {b}^{\prime}}{\partial a}+\frac{\partial \mathrm{\Delta}l}{\partial c}\frac{\partial c}{\partial a}+\frac{\partial \mathrm{\Delta}l}{\partial {c}^{\prime}}\frac{\partial {c}^{\prime}}{\partial a}\\ \frac{\partial \mathrm{\Delta}l}{\partial \theta}=\frac{\partial \mathrm{\Delta}l}{\partial {b}^{\prime}}\frac{\partial {b}^{\prime}}{\partial \theta}+\frac{\partial \mathrm{\Delta}l}{\partial c}\frac{\partial c}{\partial \theta}+\frac{\partial \mathrm{\Delta}l}{\partial {c}^{\prime}}\frac{\partial {c}^{\prime}}{\partial \theta}+n\text{\hspace{0.17em}}\mathrm{tan}({b}^{\prime})\frac{\mathrm{cos}(\theta )}{\mathrm{cos}(c)}-\mathrm{sin}({c}^{\prime})\mathrm{tan}({b}^{\prime})(\mathrm{cos}(\theta )\mathrm{tan}(c)-\mathrm{sin}(\theta ))\\ \frac{\partial \mathrm{\Delta}l}{\partial n}=\frac{\partial \mathrm{\Delta}l}{\partial {b}^{\prime}}\frac{\partial {b}^{\prime}}{\partial n}+\frac{\partial \mathrm{\Delta}l}{\partial c}\frac{\partial c}{\partial n}+\frac{\partial \mathrm{\Delta}l}{\partial {c}^{\prime}}\frac{\partial {c}^{\prime}}{\partial n}+\mathrm{sec}(c)\mathrm{sin}(\theta )\mathrm{tan}({b}^{\prime})\end{array}.$$