# Research on Optical System of Dim Target Simulator Based on Polarization Stray Light Suppression
## 2 Analysis and Design of Optical System of Dim Target Simulator
### Analysis and Design of Depolarization Optical Engine
#### 2.2.3 Analysis of Illumination Optical System Design Objectives
Parabola \(BB^{\prime}\) is shown in Equation (15):  
\[\left[2\sqrt{fk_{1}}x+(f-k_{1})y\right]^{2}+8 af^{2}x-8 ak_{2}\sqrt{fk_{1}}y=4a^ {2}f(a+k_{2}) \tag{15}\]  
where \(k_{1}=2a-f\), \(k_{2}=a+f\), \(f\) is the focal length, \(f=a(1+\sin\theta_{\text{max}})\), \(a\) is the radius of the LED, \(a=b\sin\theta_{\text{max}}\), \(b\) is the radius of the light outlet hole \(AB\), and \(\theta_{\text{max}}\) is the maximum beam absorption angle of the composite parabolic mirror. Figure 8: Structure of illumination optical system. The luminous intensity \(I(\theta_{i})_{z}\) at the exit of the composite paraboloid mirror includes two parts: the luminous intensity \(\theta_{i}\) directly emitted by the LED and the luminous intensity \(\theta_{i}\) reflected by the composite paraboloid mirror. If the luminous intensity in the normal direction of the LED is \(I_{0}\), the luminous intensity at the angle \(I_{0}\) is \(I_{0}\cos\theta_{i}\) and the luminous intensity at the exit of the composite paraboloid mirror can be shown by Equation (16):  
\[I(\theta)_{z}=I_{0}\cos\theta_{i}+kI_{0}\frac{\int_{0}^{L}\cos(F(\theta_{i},y) )dy}{\int_{0}^{\theta_{\max}}\int_{0}^{L}\cos(F(\theta_{i},y))dyd\theta_{i}} \tag{16}\]  
where \(F(\theta_{i},y)\) is the function of \(v\) related to \(\theta_{i}\) and \(y\), \(v\) is the outgoing light angle of the LED corresponding to \(\theta_{i}\), \(v=\pi-\theta_{i}-2\eta\), \(y\) is the ordinate of the intersection point of the light with the incident angle of \(v\) emitted by the LED and the CPC, \(k\) is the reflectance of the reflector surface of the composite parabolic mirror, \(L\) is the length of the composite parabolic mirror, and \(L=\frac{\cos\theta_{\max}}{\sin^{2}\theta_{\max}}\). When \(\theta_{\max}\) is larger, the range of \(\theta_{i}\) is larger, and the reflected light is reflected and compensated on both sides of the irradiation surface. The radius of the LED is \(a=1.88\) mm and the incident light angle is \(60^{\circ}\). In order to ensure that the CPC length is appropriate, \(\theta_{\max}=\pm 17^{\circ}\) and \(\theta_{i}=\theta_{\max}\) are used to calculate the CPC focal length \(f=2.042\) mm. CPC length \(L=22.844\) mm; CPC light output diameter \(2b=10.81\) mm. The design of the homogenizing rod  
We designed a homogenizing rod for high uniform lighting. The homogenizing rod uses the principle of total reflection to reflect the light incident inside the rod many times and then emit it. The aperture ratio of the homogenizing rod determines the total reflection times, and the incident irradiation is divided according to the incident angle, so as to improve the irradiation uniformity of the system [20]. The principle of the homogenizing rod is shown in Figure 10, in which 1 to 4 respectively represent the equivalent position of the edge light of \(\theta_{1}\) to \(\theta_{4}\) light from different incident angles. The homogenizing rod shall satisfy Equation (17):  
\[m=\frac{2}{n}\bigg{(}\frac{L_{z}}{L_{y}}\bigg{)}\tan\theta_{m},\theta_{m} \langle\theta_{\max} \tag{17}\]  
where \(m\) is the maximum number of total reflections in the homogenizing rod, \(n\) is the refractive index of the homogenizing rod, \(L_{z}\) is the length of the homogenizing rod, \(L_{y}\) is the cross-section width of the homogenizing rod, \(\theta_{m}\) is the division angle of the homogenizing rod, and \(\theta_{\max}\) is the incidence angle of the homogenizing rod. Figure 9: Working principle diagram of CPC. Since the CPC incident angle in front of the homogenizing rod is 17\({}^{\circ}\) and the outlet diameter is 10.81 mm, \(L_{Y}=10.81\) mm. Considering that the effective irradiation surface size is 25 mm, the maximum number of total reflections should be greater than or equal to 25/10.81 = 2.3. The maximum number of total reflections is 3, and since JGS3 heat-resistant quartz glass \(n=1.487\) is selected as the homogenizing rod, \(l_{z}=78\) mm can be calculated. 4.