## 2 The Designed Infrared Optical Imaging System  
We designed a wide band infrared optical imaging system with large relative aperture. The ratio F of focal length to aperture of the optical system reaches 0.9. The parameters of broadband infrared optical system are shown in Table 1. The focal length \(f\) of the system is 30 mm, and the pixel size D of the detector is 15 \(\upmu\)m. The resolution is \(W\times H=640\times 512\). The field of view can be calculated by the following formula:  
\[2w=\frac{d\times\sqrt{W^{2}+H^{2}}}{f}=23.2^{\circ} \tag{1}\]  
The infrared optical material has a high absorption coefficient. In order to improve the transmittance of the infrared optical system, the lens should be simplified as much as possible, thus an even aspheric surface shape is used to correct the aberration caused by large aperture and large field of view. Even degree aspherical surface can be expressed by the following formula:  
\[z=\frac{cr^{2}}{1+\sqrt{1-(k+1)c^{2}r^{2}}}+\sum_{i=1}^{j}a_{i}r^{2i} \tag{2}\]  
where: \(r=\sqrt{x^{2}+y^{2}}\) is the incident height of light on aspheric surface, \(c\) is the radius of curvature at the vertex, \(k\) is the cone constant, and \(a_{i}\) is the high-order aspheric coefficient. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Index** \\ \hline wavelength & 3–12 \(\upmu\)m \\ focal length & 30 mm \\ F\(\#\) & 0.9 \\ Total length of optical system & 35 mm \\ detector resolution & 640 \(\times\) 512 \\ pixel size of detector & 15 \(\upmu\)m \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of wide band infrared optical system. In the actual manufacturing process, the aspheric surface of infrared material can be machined by single point diamond turning. We choose a three-piece structure, in which the first piece of lens material is AMTIR1, the second piece is ZnS, and the third piece is Ge. AMTIR1 is a kind of synthetic material that can transmit light from near-infrared to long-wavelength infrared, and it can achieve very good achromatic effect in the infrared band. The second and sixth surfaces are even-degree aspherical surfaces. The first lens has a large aperture and large off-axis aberration. Aspherical surfaces are introduced on the back surface of the first lens. The fourth and sixth order coefficients are 8.0502 \(\times\) 10\({}^{-7}\) and 1.0185 \(\times\) 10\({}^{-9}\) respectively. Due to achromatic aberration, the third lens has a large power. In order to avoid excessive high-order aberration, an aspheric surface is introduced behind it. The fourth and sixth order coefficients are 3.1083 \(\times\) 10\({}^{-6}\) and -5.128 \(\times\) 10\({}^{-12}\) respectively. When designing the initial structure, a series of aberration equations are established according to the linear relationship between the primary aberration and the structure parameters, and then the linear equations are solved to obtain the initial structure giving the aberration requirements. The aberration calculation formulas used in this design are as follows:  
\[\sum_{i=1}^{n_{1}}h_{i}\sum_{j=1}^{n_{2}}\varphi_{j}=1 \tag{3}\]  
\[\sum C_{I}=\sum_{i=1}^{n_{1}}h_{i}^{2}\sum_{j=1}^{n_{2}}\frac{\varphi_{j}}{v_{ j}}=0 \tag{4}\]  
\[\sum S_{I}=\sum_{3}h^{4}\varphi^{3}P=0 \tag{5}\]  
\[\sum S_{II}=\sum_{3}h^{3}h_{z}\varphi^{3}P+J\sum_{3}h^{2}\varphi^{3}W=0 \tag{6}\]  
wherein, \(P=\sum\left(\frac{\Delta u_{j}}{\Delta\left(\frac{1}{h_{j}}\right)}\right)^{2} \Delta\frac{u_{j}}{n_{j}}\), \(W=\sum\left(\frac{\Delta u_{j}}{\Delta\left(\frac{1}{h_{j}}\right)}\right) \Delta\frac{u_{j}}{n_{j}}\), \(\Delta u_{j}=u_{j}^{\prime}-u_{j}\), \(\Delta\frac{u_{j}}{n_{j}}=\frac{u_{j}^{\prime}}{n_{j}^{\prime}}-\frac{u_{j}}{ n_{j}}\).