# Dual-Gratings Imaging Spectrometer
## 2 Dual-Gratings Imaging Spectrometer
### Double Images Offset Problem  
Compared with the conventional imaging spectrometers, the dual gratings provide a wider spectral coverage and more free imaging methods. Nevertheless, the assembly errors might cause some unique issues, which are collectively referred to as the double images offset problem. By increasing or decreasing the angle between the dual gratings relative to the original design, the double image offset results in a relative change of the position and rotation angle of the two images. Severe double-image misalignment might make it impossible to image the double-image on the same detector, which will affect the spatial and spectral ranges of the image. The angle between the double gratings is represented by the rotation angle \(\alpha\) around the \(x\)-axis, the rotation angle \(\beta\) around the \(y\)-axis, and the rotation angle \(\gamma\) around the \(z\)-axis(see Figure 10). The impact of the double images offset problem can be analyzed according to the increase and decrease of the three angles relative to the original design (see, Table 1 and Figure 11). Figure 9: The dual-gratings imaging spectrometer. Figure 11 is a schematic diagram of the double images offset problem. The red square and blue square each represent the image of a waveband on the image plane. The two images are parallel in the normal state, but due to the change of angle between the double gratings, misalignment and tilt appear between the two images. The difference between the dual image offset problem and the tolerance lies in the difference in the measurement standards. The measurement standard for the tolerance is MTF (Modulation Transfer Function), and the dual images offset mainly affects the image position and tilt, which has only a slight effect on the MTF. Therefore, this issue needs to be considered based on system components, such as the adopted detector. For the given values of \(\alpha\), and \(\beta\), the number of pixels should be considered that are dislocated up, down, left, and right between the two images, which is acceptable at the detector. The value of \(\gamma\), is also corrected as much as possible during the assembly and adjustment. First, we derive the relationship between \(\alpha\) rotation and dual image offset. From the above, the edge wavelengths \(\lambda_{1}\) and \(\lambda_{3}\) of the two bands are imaged at one point. By calculating the distance changed between \(\lambda_{1}\) and \(\lambda_{3}\) caused by \(\alpha\) rotation, the relationship between the dual images offset and \(\alpha\) can be obtained. \begin{table}
\begin{tabular}{c c c} \hline
**3D Angle** & **Change** & **Influence** \\ \hline \(\alpha\) & Increase & Distance between the two images expand \\ \(\alpha\) & Decrease & Distance between the two images shrink \\ \(\beta\) & Increase & Dislocation in the spatial dimension. \\ \(\beta\) & Decrease & Opposite dislocation in the spatial dimension. \\ \(\gamma\) & Increase & Tilt between the two images \\ \(\gamma\) & Decrease & Opposite tilt between the two images \\ \hline \end{tabular}
\end{table} 
Table 1: The Impact of the 3D angle variations. Figure 11: The impact of the double image offset (blue square and red square both represent one image). Figure 10: The direction of \(\alpha\), \(\beta\), and \(\gamma\) angle. Assuming that the rotation angle between the dual gratings is increased by \(t1\) relative to the original design, the distance between the dual images is increased by \(h1\). The influence caused by the rotation \(t1\) of the grating can be decomposed into two parts. The first part is the change of the incident angle caused by the rotation of the grating, which causes the change of the exit angle, and the second part is \(t1\) which causes the reflected light to rotate as a whole by \(t1\). Here, the grating rotation angle change is recorded as \(t1\), the exit angle change is recorded as \(t2\), and the output light rotation angle is recorded as \(t3\), which can be obtained from the Equation (7):  
\[\left\{