## Abstract

Figure measuring interferometers generally work in the null condition, i.e., the reference rays share the same optical path with the test rays through the imaging system. In this case, except field distortion error, effect of other aberrations cancels out and doesn’t result in measureable systematic error. However, for spatial carrier interferometry and non-null aspheric test cases, null condition cannot be achieved typically, and there is excessive measurement error that is referenced as retrace error. Previous studies about retrace error can be generally distinguished into two categories: one based on 4th-order aberration formalism, the other based on ray tracing through interferometer model. In this paper, point characteristic function (PCF) is used to analyze retrace error in a Fizeau interferometer working in high spatial carrier condition. We present the process of reconstructing retrace error with and without element error in detail. Our results are in contrast with those obtained by ray tracing through interferometer model. The small difference between them (less than 3%) shows that our method is effective.

© 2015 Optical Society of America

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### Equations (10)

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(1)
$$V({x}_{0},{y}_{0},{z}_{0};{x}_{1},{y}_{1},{z}_{1})={\displaystyle \underset{({x}_{0},{y}_{0},{z}_{0})}{\overset{({x}_{1},{y}_{1},{z}_{1})}{\int}}nds},$$
(2)
$$V({x}_{0},{y}_{0},{z}_{0};{x}_{1},{y}_{1},{z}_{1})=S({x}_{1},{y}_{1},{z}_{1})-S({x}_{0},{y}_{0},{z}_{0}),$$
(3)
$$n\frac{d\overrightarrow{r}}{ds}=grad(S),$$
(4)
$$\begin{array}{c}gra{d}^{0}(V)=-{n}_{0}\overrightarrow{{e}_{0}}\\ gra{d}^{1}(V)={n}_{1}\overrightarrow{{e}_{1}}\end{array}\},$$
(5)
$$\begin{array}{c}u={h}_{x}{}^{2}+{h}_{y}{}^{2}\\ v={h}_{x}{p}_{xi}+{h}_{y}{p}_{yi}\\ w={p}_{xi}{}^{2}+{p}_{yi}{}^{2}\end{array}\}.$$
(6)
$$V={\displaystyle \sum _{ijk}{f}_{ijk}{u}^{i}{v}^{j}{w}^{k}}.$$
(7)
$$\begin{array}{c}\mathrm{cos}({\xi}_{0})=\frac{\partial V}{{r}_{h}\partial {h}_{x}}\\ \mathrm{cos}({\eta}_{0})=\frac{\partial V}{{r}_{h}\partial {h}_{y}}\\ \mathrm{cos}({\xi}_{1})=\frac{\partial V}{{r}_{pi}\partial {p}_{xi}}\\ \mathrm{cos}({\eta}_{1})=\frac{\partial V}{{r}_{pi}\partial {p}_{yi}}\end{array}\},$$
(8)
$${f}_{001}=-\frac{{r}_{pi}{}^{2}}{2l}.$$
(9)
$$\begin{array}{c}({p}_{x},{p}_{y})=(\frac{\mathrm{cos}({\xi}_{0})}{{r}_{p}},\frac{\mathrm{cos}({\eta}_{0})}{{r}_{p}})\\ {h}_{xi}=({r}_{pi}{p}_{xi}+l\frac{\mathrm{cos}({\xi}_{1})}{\sqrt{1-\mathrm{cos}{({\xi}_{1})}^{2}-\mathrm{cos}{({\eta}_{1})}^{2}}})/{r}_{hi}\\ {h}_{yi}=({r}_{pi}{p}_{yi}+l\frac{\mathrm{cos}({\eta}_{1})}{\sqrt{1-\mathrm{cos}{({\xi}_{1})}^{2}-\mathrm{cos}{({\eta}_{1})}^{2}}})/{r}_{hi}\end{array}\}.$$
(10)
$$\begin{array}{c}{h}_{xi}={\displaystyle \sum _{ijkm}{a}_{ijkm}{h}_{x}{}^{i}{h}_{y}{}^{j}{p}_{x}{}^{k}{p}_{y}{}^{m}}\\ {h}_{yi}={\displaystyle \sum _{ijkm}{b}_{ijkm}{h}_{x}{}^{i}{h}_{y}{}^{j}{p}_{x}{}^{k}{p}_{y}{}^{m}}\\ {p}_{xi}={\displaystyle \sum _{ijkm}{c}_{ijkm}{h}_{x}{}^{i}{h}_{y}{}^{j}{p}_{x}{}^{k}{p}_{y}{}^{m}}\\ {p}_{yi}={\displaystyle \sum _{ijkm}{d}_{ijkm}{h}_{x}{}^{i}{h}_{y}{}^{j}{p}_{x}{}^{k}{p}_{y}{}^{m}}\end{array}\}.$$