# Non-Rotationally Symmetric Field Mapping for Back-Scanned Step/Stare Imaging System
## 3 Optical System Design  
The mathematical model of non-rotationally symmetric field mapping between object space and image space is established in Section 2. In this section, we want to design optical system which can implement these field mappings. The design specifications are shown in Table 1. The spectral band is 3.7-4.8 \(\upmu\)m (medium-wave infrared). The focal length of the optical system is 500 mm. The F-number of the optical system is 4. The detector is a two-dimensional FPA having 1280 \(\times\) 1024 pixels. The pixel pitch of the FPA is 15 \(\upmu\)m. The magnification of the afocal telescope is 5\(\times\). The FOV of the frame is 2.2 \(\times\) 1.76 degrees. The scanning speed of optical system is 40 deg/s and the normal integration time is 10 ms. Consequently, the back-scan is over \(\pm\)0.4 degrees in object space, and then the FOV of the afocal telescope is 2.2 \(\times\) 2.56 degrees. It should be noted that the main purpose of this manuscript is to explain the image points wander during back-scanning. In order to simplify the design, the imager is set as an ideal lens, and the practical problems of infrared optical system design such as secondary imaging and cold stop matching are not considered. The optical system designed according to the field mapping condition \(h_{x}=f^{\prime}\theta_{x},h_{y}=f^{\prime}\theta_{y}\) is shown in Figure 5. The afocal telescope is consisted of six lenses. From left to right, the lens 1, the lens 3, the lens 4, and the lens 5 are made of silicon and have a positive power. The lens 2 and the lens 6 are made of germanium and have a negative power. The entrance pupil is placed on the front surface of the lens 1. The rear surface of the lens 2 is a quadratic aspherical surface. The lenses 3 and 4 are field lenses, and have a small power. The rear surface of the lens 3 and the front surface of the lens 4 are freeform surfaces. The remaining surfaces are spherical. The FSM is placed on the exit pupil of the \begin{table}
\begin{tabular}{c c} \hline \hline
**Characteristics** & **Performance** \\ \hline Spectrum & 3.7–4.8 \(\upmu\)m \\ Focal length & 500 mm \\ F/\# & 4 \\ FPA array format & 1280 \(\times\) 1024 \\ Pixel pitch & 15 \(\upmu\)m \\ Magnification of the afocal telescope & 5\(\times\) \\ Scanning speed of optical system & 40 deg/s \\ Normal integration time & 10 ms \\ FOV of the afocal telescope & 2.2 \(\times\) 2.56 degrees \\ FOV of the frame & 2.2 \(\times\) 1.76 degrees \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Design specifications. a focal telescope, and the optical axis is turned by 90 degrees. The ideal lens imager focuses the scene onto the FPA. The freeform surfaces used in this example are extended XY polynomial. The sagittal deviation \(\Delta z\) between the freeform surface and the standard spherical surface is expressed by Equation (18):  
\[\Delta z=a_{1}x^{2}+a_{2}y^{2}+a_{3}x^{4}+a_{4}x^{2}y^{2}+a_{5}y^{4}+a_{6}x^{ 6}+a_{7}x^{4}y^{2}+a_{8}x^{2}y^{4}+a_{9}y^{6} \tag{18}\]  
The above non-rotationally symmetric freeform surfaces are used to control the tilt of the chief ray in the X and Y directions, which can produce an anamorphic aberration to meet the Equation (11). A special macro is coding to constrain the relationship among the image height, the focal length and the incident angle of the imaging system to meet Equation (11). At the same time, a multi-configuration is designed to include five different angles of the FSM. In the process of optimization, the image heights of the same FOV are limited to be equal. If this new field mapping is strictly restricted, the image quality of the system will be degraded, and the image distortion will be very serious. For the actual optical systems design, it is necessary to balance aberrations to meet the special specifications, such as the maximum allowable distortion when the FSM does not move, the maximum image wander during back-scanning and the minimum MTF under the Nyquist frequency.