## 2 Construction of the Defocus and Tilt Compensation Theoretical Model
### Dual-Prism Compensation Theoretical Model  
A Cartesian coordinate system, OXYZ, is established as shown in Figure 3, with its origin O positioned at the center of the first prism's flat surface. The \(Z\)-axis aligns with the mechanical axis (MCA). The two wedge prisms, prism 1 and prism 2, share an identical wedge angle \(\alpha\) and have respective thicknesses denoted as \(t_{1}\) and \(t_{2}\). Prism 1 is fixed on the MCA, while prism 2 allows for oblique movement, axial displacement, and independent rotation. The rotation angle is defined as \(\omega\). The direction of the optical axis is accurately determined using Snell's law:  
Figure 3: Schematic diagram of image plane deformation compensation model. (**a**,**b**) represent different adjustment states respectively. Figure 2: Schematic diagram of image plane deformation compensation model and adjustment method. (**a**) Schematic diagram of axial movement and oblique movement. (**b**) Schematic diagram of prism rotation. \[\overset{\rightarrow}{A_{i}}=\frac{n_{i-1}}{n_{i}}\overset{\rightarrow}{A_{i-1}}- \left\{\sqrt{1-\left(\frac{n_{i-1}}{n_{i}}\right)^{2}\left[1-\left(\overset{ \rightarrow}{A_{i-1}}\overset{\rightarrow}{N_{i}}\right)^{2}\right]}-\frac{n_ {i-1}}{n_{i}}\overset{\rightarrow}{A_{i-1}}\overset{\rightarrow}{N_{i}} \right\}\overset{\rightarrow}{N_{i}}\left(i=1,2,3,4\right) \tag{1}\]  
where \(n_{i-1}\) and \(n_{i}\) represent the refractive indices of the object and image sides, respectively. \(\overset{\rightarrow}{A_{i-1}}\) and \(\overset{\rightarrow}{A_{i}}\) denote the direction cosines of the incident and refracted light rays, while \(\overset{\rightarrow}{N_{i}}\) refers to the direction cosine of the surface normal. [PERSON]'s equations play a crucial role in the study of astigmatism by mathematically relating the vertex of a surface to the distances of the astigmatic lines. In this study, these equations are used to determine the position of the sagittal image plane, as expressed below:  
\[\frac{n^{\prime}}{S^{\prime}}-\frac{n}{S}=\frac{n^{\prime}\cos(I^{\prime})-n\cos (I)}{R} \tag{2}\]  
\[S_{i}={S_{i-1}}^{\prime}-\Delta S_{i} \tag{3}\]  
Since the curvature of the plane is zero, the equation simplifies to:  
\[R\rightarrow\infty,n_{i}^{\prime}S_{i}=n_{i}{S_{i}}^{\prime} \tag{4}\]  
By decomposing the wedge prism into a parallel plate and a wedge, the effects of system parameters on image plane adjustments can be more clearly understood. The wedge facilitates tilt adjustments of the image plane (rotation), calculated using [PERSON]'s law, while the parallel plate handles positional adjustments (translation), as expressed by the following equation:  
\[\left(\Delta x,\Delta y,\Delta z\right)^{T}=\left(0,0,\left(1-\frac{N}{\sqrt{n ^{2}+N^{2}-1}}\right)t\right)^{T} \tag{5}\]  
where \(t\) represents the thickness of the plate. The adjustment of the image plane by the dual-prism is expressed as follows:  
\[ \begin{pmatrix}\Delta x_{total}\\ \Delta y_{total}\\ \Delta z_{total}\end{pmatrix} = \begin{pmatrix}L_{4}\cdot S_{3}\\ M_{2}\cdot\Delta S+M_{4}\cdot S_{3}\\ N_{2}\cdot\Delta S+N_{4}\cdot S_{3}+\left(1-\frac{N}{\sqrt{n^{2}+N^{2}-1}} \right)\left(T+\Delta T\right)\end{pmatrix}  \tag{6}\]  
This method, by decomposing the wedge prism, provides a deeper understanding of its functionality. The prism moves along its inclined surface, adjusting the optical path by varying its thickness \(\Delta T\) within the optical path, thereby achieving axial movement of the image plane. Furthermore, since the air gap between the prisms remains constant during oblique movement, this adjustment does not introduce lateral displacement of the image plane. The lateral displacement is determined by the object distance \(S\), while the orientation of the image plane is defined by the prism's rotation angle. To visually illustrate the effects of different parameters on the image plane, a schematic diagram is provided (see Figure 4).