## 3 Theoretical Model and Computational Analysis of Aberrations in Double Prism Compensation Devices
### Aberration Calculation  
By extending the wavefront aberration function to include planar symmetric systems, the aberration field of planar symmetric system combinations is described. The aberration coefficient of a combined system depends on the orientation of a single aberration coefficient and a single plane symmetry component system. The aberration coefficient can then be used to calculate the magnitude of different types of aberrations. Conceptually, if the appropriate symmetry plane direction is selected, the given aberration can be eliminated by adding equal and opposite amounts of aberration. If an inappropriate symmetry plane is chosen, aberrations cannot be eliminated [22]. The planar symmetry aberration coefficient can be expressed as a function of the first-order system parameters. Due to the particularity of prisms (i.e., each surface having zero curvature), a prism can be seen as a combination of several planar symmetric systems. Each subsystem's wave function has its own symmetric direction. Therefore, the total aberration wave function of a rotating prism can be written as the sum wave aberration function of each unit, just like vector addition [20]. This work validates and supplements the combination of aberrations in optical systems. In this study, a rotating dual-prism system can be seen as a combination of four planar symmetric systems. To calculate the aberrations of the system, it is essential to know the aberrations and their orientations generated by each surface (or each planar symmetric system). [PERSON] provided a formula for calculating aberrations in composite systems; therefore, planar symmetric aberrations can be regarded as vectors with direction and amplitude. When the prism rotates around the mechanical axis, it alters the symmetry direction of each surface plane, meaning each surface has its own vectors before and after rotation. Consequently, the total aberration of the prism system can be determined by vector addition. \(\overrightarrow{\rho}\) is the normalized pupil vector, \(\overrightarrow{H}\) is the normalized field vector, \(W\) is the total or global aberration of the optical system, and j is the number of planar symmetric systems. The total or global aberration function of a number j of plane symmetric systems with relative orientation \(\overrightarrow{i}_{j}\) and with aberration coefficients \(W_{aberationtypej}\) is the sum of the individual aberration functions. This can be expressed as:  
\[W\bigg{(}\overrightarrow{H}\cdot\overrightarrow{\rho}\bigg{)}=\sum_{1}^{j} \left( \begin{array}{l}W_{caj}\left(\overrightarrow{i}_{j}\cdot \overrightarrow{\rho}\right)^{2}+W_{aj}\left(\overrightarrow{i}_{j}\cdot \overrightarrow{H}\right)\left(\overrightarrow{i}_{j}\cdot\overrightarrow{ \rho}\right)+W_{ccij}\left(\overrightarrow{i}_{j}\cdot\overrightarrow{\rho} \right)\left(\overrightarrow{\rho}\cdot\overrightarrow{\rho}\right)+\\ W_{laj}\left(\overrightarrow{i}_{j}\cdot\overrightarrow{\rho}\right) \left(\overrightarrow{H}\cdot\overrightarrow{\rho}\right)+W_{ffj}\left( \overrightarrow{i}_{j}\cdot\overrightarrow{H}\right)\left(\overrightarrow{ \rho}\cdot\overrightarrow{\rho}\right)+W_{qdlj}\left(\overrightarrow{i}_{j} \cdot\overrightarrow{\rho}\right)\left(\overrightarrow{H}\cdot \overrightarrow{H}\right)\\ +W_{qdllj}\left(\overrightarrow{i}_{j}\cdot\overrightarrow{H}\right)\left( \overrightarrow{H}\cdot\overrightarrow{\rho}\right)+W_{sqj}\left( \overrightarrow{\rho}\cdot\overrightarrow{\rho}\right)^{2}+W_{laj}\left( \overrightarrow{H}\cdot\overrightarrow{\rho}\right)\left(\overrightarrow{ \rho}\cdot\overrightarrow{\rho}\right)\\ +W_{qaj}\left(\overrightarrow{H}\cdot\overrightarrow{\rho}\right)^{2}+W_{fcj} \left(\overrightarrow{H}\cdot\overrightarrow{H}\right)\left( \overrightarrow{\rho}\cdot\overrightarrow{\rho}\right)+W_{cdj}\left( \overrightarrow{H}\cdot\overrightarrow{H}\right)\left(\overrightarrow{H} \cdot\overrightarrow{\rho}\right)\end{array}