## 2 Dark Current Measurement and Noise Correction Method Based on Focal-Plane Temperature
### Dark Current Noise Correction Method  
After the analysis in the previous section, the overall dark current of the QWIP focal-plane changes due to temperature fluctuations, which in turn causes changes in the pixel response and generates low-frequency time noise. The introduction of a method to eliminate this type of dark current time noise forms the main content of this section.  
Focal-plane devices usually set a target cooling temperature, \(T_{set}\). \(T_{set}\) is essentially the same physical quantity as the previously mentioned T. \(T_{set}\) is used for experiments and T is used for theoretical analysis. The performance of the refrigeration machine varies, and the rated operating mode of \(T_{set}\) generates temperature fluctuations of \(\Delta T_{set}\). After obtaining the parameters of Equation (12), this equation can be used to solve for the dark current value \(I_{dark}\) at multiple temperature points in the range of \(T_{set}\pm 1/2\Delta T_{set}\). Within a very small temperature difference \(\Delta T_{set}\), a good linear relationship exists between \(I_{dark}\) and temperature. The quantized values of temperature measurement, corresponding to \(T_{set}\) and \(\Delta T_{set}\), \(g_{tem}\), and \(\Delta g_{tem}\), can be obtained with Equations (9)-(11). Then, both temperature and response are transformed into the form of response values, and the linear relationship between the two is solved to obtain the change in response values under equal temperature changes.  
The design of the information acquisition circuit includes a module for the preprocessing and quantization of the detector signal. The preprocessed detector signal voltage is:  
\[U_{det\_done}=(U_{det}-U_{det\_sub})\times a_{det}, \tag{13}\]where \(U_{det\_done}\) is the preprocessed detector signal voltage, \(U_{det\_sub}\) is the bias subtraction voltage of the preprocessing circuit, and \(a_{det}\) is the gain of the preprocessing circuit.  
The quantized response value of the detector signal is:  
\[g_{det}=\frac{U_{det\_done}-U_{I2}}{U_{w2}}\times 2^{n_{det}}, \tag{14}\]  
where \(g_{det}\) is the quantized detector response value, \(U_{I2}\) is the lower limit of the input voltage interval of the ADC of the preprocessing module, \(U_{w2}\) is the input voltage range, and \(n_{det}\) is the number of quantization bits.  
The detector signal quantization value \(\Delta g_{det}\) corresponding to \(\Delta T_{set}\) can be obtained according to Equations (13) and (14).  
The coefficient of the system response value with respect to the temperature response value \(a_{dark}\) is obtained by solving:  
\[a_{dark}=\frac{\Delta g_{det}}{\Delta g_{tem}}, \tag{15}\]  
The corrected system response value \(g^{\prime}_{det}\) is:  
\[g^{\prime}_{det}=g_{det}+(g_{tem}-\overline{g}_{tem})\times a_{dark}, \tag{16}\]  
where \(\overline{g}_{tem}\) is the mean value of the response of the temperature measurement pin during the time of data acquisition. Since the dark current variation occurs within the pixels, independent of the spectroscopic system, the method can be applied to QWIP hyperspectral imagers and gaze-type cameras for the indistinguishable correction of the pixels for each spectral channel or face array spatial dimension.  
The direct use of temperature measurement pin response data for target data correction, without a corresponding theoretical basis, is a purely mathematical method. The present method incorporates a physical model with reliable theoretical support.