# Freeform Wide Field-of-View Spaceborne Imaging Telescope: From Design to Demonstrator
## 2 Methods: Optical Design  
### Optical System Design  
The optical design and analysis were performed using Zemax OpticsStudio(r). The design comprises two freeform mirrors, mounted off-axis, including an aperture stop located between the two mirrors at the focal point of the secondary mirror (Figure 1). The design is optimized within the 1100 nm and 1700 nm wavelength range, for a full FOV of 120\({}^{\circ}\), and features an f-number of 7.23. Each of the fields is imaged on a point, resulting in a line on the detector plane. Additionally, telecentricity is achieved at this detector plane. The design has dimensions of 96 mm \(\times\) 15 mm \(\times\) 95 mm, fitting within 1 CubeSat Unit. The final design is achieved using a two-step design process. First, an on-axis telescope without tilts, but including two mirrors and the slit after the secondary mirror was considered for optimization of the freeform terms, after which the tilts were introduced to avoid vignetting, and re-optimization was performed. This two-step approach enabled the decorrelation of the shape and geometry of the mirrors, avoiding possible confusion. The freeform mirrors are described and optimized using XY polynomials, as the telescope is off-axis with respect to the Y-axis and features a symmetrical FOV with respect to the X-axis. The surface sag (z) comprises a base conic aspherical surface upon which the polynomial aspheric terms are added, as described by equation 1, in which \(c\) is the reciprocal of the radius, \(r\) is the radial coordinate in lens units, \(k\) is the conic constant, \(N\) is the number of polynomial coefficients and \(A_{i}\) is the coefficient of the \(i\)th extended polynomial term. The polynomials \(E_{i}(x,y)\) are a power series in \(x\) and \(y\), where the position values \(x\) and \(y\) are divided by a normalization radius, implying the polynomial coefficients are dimensionless [18]. \[z=\frac{c\ r^{2}}{1+\sqrt{1-(1+k)c^{2}r^{2}}}+\sum_{i=1}^{N}A_{i}E_{i}(x,y) \tag{1}\]  
We optimize the shape of the freeform mirrors for 11 fields between +60\({}^{\circ}\) and \(-\)60\({}^{\circ}\), including their radius of curvature and the following XY polynomial terms: X2Y0, X0Y2, X0Y3, X4Y0, X2Y2, X0Y4, X2Y3, X0Y5, X6Y0, and X4Y2. Since we have an orthogonal symmetry along the optical axis (i.e., +X and \(-\)X coefficients are symmetric, as are the fields +\(\times^{\circ}\) and \(-\)x\({}^{\circ}\)), we only need even orders in X, such as X0, X2, X4, and X6. Therefore, we do not consider odd orders in X, such as X1, X3, and X5. We first considered terms up to the 4 th order, while after introducing the tilts in the design, we add the 5 th order term X0Y5 and 6 th order term X6Y0, to further correct for aberrations. An iterative optimization was performed, considering the spot size and modulation transfer function (MTF) in the merit function. The total track length, f-number, entrance pupil diameter, and the ray positions given telecentricity were constrained in the merit function to fulfill the pursued specifications. Considering the final application and integration, the freeform lenses will be mounted monolithtically to ensure fixed positioning, while the imaging detector or entrance slit of the spectrometer can be included at the specified detector plane in Figure 1. However, for the purpose of our laboratory demonstrator, the optics needed to be mounted using bulky optomechanics featuring micrometer screws to enable an evaluation of the mounting constraints, while a commercial off-the-shelf (COTS) camera was used (Xenics Bobcat 320 GigE 100 camera) to assess the image quality, consuming more space (55 \(\times\) 55 \(\times\) 82 mm\({}^{3}\)) than an OEM image sensor mounted on a printed circuit board would. Taking the larger dimension of the mounting optomechanics and the camera into account, a dummy thin fold mirror is added to the design, in a way that does not affect the image quality but relaxes the  
Figure 1: Optical system layout of the freeform two-mirror imaging telescope design.