# Research on Performance Evaluation and Optimization Theory for Thermal Microscope Imaging Systems
## 2 Materials and Methods
### Spectral Radiant Flux of the TMISs  
In the TMISs, the light radiated by the microfacet of the object to be measured is received by the optical system in a divergent form, and then passes through the optical system focus on the imaging detector. The imaging optical path diagram of the TMISs is shown in Figure 5. Among them, \(ds_{0}\) and \(ds_{1}\) represent the microfacets of the object to be measured and the detector respectively, \(n_{0}\) and \(n_{1}\) represent the refractive index of the object space and image space respectively, \(2u_{0}\) and \(2u_{1}\) respectively represent the opening angle of the micro-unit of the object to be measured and the micro-unit of the detector of the optical system, \(l\) and \(l^{\prime}\) represent object distance and image distance, respectively, \(\tau_{0}\) represents the spectral transmittance of the optical system, \(d_{0}\) and \(d^{\prime}{}_{0}\) represent the diameter of the optical system in object space and image space, respectively. The radiant flux emitted by the micro-space element \(ds_{0}\) in object space to a lens with an aperture \(d_{0}\) can be expressed as [30]  
\[d\Phi(\lambda,T)=\varepsilon(\lambda,T)M_{0}(\lambda,T)\tau_{a}(\lambda)ds_{0} sin^{2}u_{0} \tag{1}\]  
where \(\varepsilon(\lambda,T)\) is the spectral emissivity; \(2u_{0}\) is the objective aperture angle of the microscope objective; \(\tau_{a}(\lambda)\) is the spectral transmittance of the atmosphere (assuming that \(\tau_{a}=1\)); \(M_{0}(\lambda,T)\) is the blackbody spectral emittance at temperature T ([PERSON]'s radiation law) [31]:  
\[M_{0}(\lambda,T)=\frac{c_{1}}{\lambda^{5}}\frac{1}{exp(c_{2}/\lambda T)-1} \tag{2}\]  
Figure 4: Block diagram of the research method in this paper.  
After \(d\Phi(\lambda,T)\) passes through a microscope objective with a spectral transmittance \(\tau_{0}(\lambda)\), the irradiance on the detector microunit \(ds_{1}\) is  
\[E(\lambda,T)=\varepsilon(\lambda,T)M_{0}(\lambda,T)\tau_{0}(\lambda)\frac{ds_{0} }{ds_{1}}sin^{2}u_{0} \tag{3}\]  
according to the theorem of Helmholtz-Lagrange: \(n_{0}\cdot r_{0}\cdot sinu_{0}=n_{1}\cdot r_{1}\cdot sinu_{1}\)[32], and Equation (3) can be rewritten as  
\[E(\lambda,T)=\varepsilon(\lambda,T)M_{0}(\lambda,T)\tau_{0}(\lambda)\frac{n_{ 1}^{2}}{n_{0}^{2}}sin^{2}u_{1} \tag{4}\]  
where \(\tau_{0}(\lambda)\) is spectral transmittance of the optical system, \(n_{0}\) and \(n_{1}\) are the refractive index of the microscope objective in the object space, and image space, and \(2u_{1}\) is the opening angle of the detector microunit to the system; it can be shown that  
\[sinu_{1}=\left.(\frac{d_{0}}{2})\right/\sqrt{[\left(\frac{d_{0}}{2}\right)^{2 }+l^{\prime 2}]} \tag{5}\]  
and so the spectral radiant flux of the object received by the detector unit of area \(a\times b\) can be calculated as  
\[\Phi(\lambda,T)=E(\lambda,T)ab=ab\cdot\varepsilon(\lambda,T)M_{0}(\lambda,T )\tau_{0}(\lambda)(\left.\frac{d_{0}}{2}\right)^{2}\left/[\left(\frac{d_{0}}{ 2}\right)^{2}+l^{\prime 2}]\right. \tag{6}\]  
where \(l^{\prime}\) is the image distance.