## Abstract

In this paper, a novel method is proposed to design the freeform off-axis reflective imaging systems. A special algorithm is demonstrated to calculate the data points on the unknown freeform surface using the rays from multiple fields and different pupil coordinates. These points are used to construct the multiple three-dimensional freeform surfaces in an imaging system which works for a certain object size and a certain width of light beam. An unobscured design with freeform surfaces can be obtained directly with this method, and it can be taken as a good starting point for further optimization. The benefit of this design method is demonstrated by designing a freeform off-axis three-mirror imaging system with high performance and system specifications. The final system operates at F/1.49 with a 64mm entrance pupil diameter and an 8° × 9° field-of-view (FOV). The performance of the system is diffraction limited at LWIR (long-wavelength-infrared).

© 2014 Optical Society of America

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### Equations (5)

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(1)
$${r}_{i}\text{'}\times {N}_{i}={r}_{i}\times {N}_{i},$$
(2)
$$T(K)={\displaystyle \sum _{i=1}^{K-1}i(K-i})=\frac{1}{6}{K}^{3}-\frac{1}{6}K=O({K}^{3}).$$
(3)
$${N}_{i}=\frac{{r}_{i}\text{'}-{r}_{i}}{\left|{r}_{i}\text{'}-{r}_{i}\right|}.$$
(4)
$$T(K)={\displaystyle \sum _{i=1}^{K-1}K-i+i}-1={(K-1)}^{2}=O({K}^{2}),$$
(5)
$$\begin{array}{l}z(x,y)=\frac{c({x}^{2}+{y}^{2})}{1+\sqrt{1-(1+k){c}^{2}({x}^{2}+{y}^{2})}}+{A}_{2}y+{A}_{3}{x}^{2}+{A}_{5}{y}^{2}+{A}_{7}{x}^{2}y\\ \text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}+{A}_{9}{y}^{3}+{A}_{10}{x}^{4}+{A}_{12}{x}^{2}{y}^{2}+{A}_{14}{y}^{4}+{A}_{16}{x}^{4}y+{A}_{18}{x}^{2}{y}^{3}+{A}_{20}{y}^{5}.\end{array}$$