## 3 Theoretical Model and Computational Analysis of Aberrations in Double Prism Compensation Devices
### Aberration Result Verification
\begin{split}\overset{\rightarrow}{\varepsilon}& =\frac{1}{mu^{\prime}}\
abla_{\rho}W(\overset{\rightarrow}{H}, \overset{\rightarrow}{\rho})\\ &=\frac{1}{mu^{\prime}}\
abla_{\rho}\left(\begin{array}{l} \frac{j}{1}\bigg{(}W_{mj}(\overset{\rightarrow}{H}\overset{\rightarrow}{ \rho})\bigg{)}+\frac{j}{1}\bigg{(}W_{aj}(\overset{\rightarrow}{i}_{j}\cdot \overset{\rightarrow}{H})\bigg{(}\overset{\rightarrow}{i}_{j}\cdot \overset{\rightarrow}{\rho}\bigg{)}\bigg{)}\\ +\frac{j}{1}\bigg{(}W_{qdlj}(\overset{\rightarrow}{i}_{j}\cdot \overset{\rightarrow}{\rho})\bigg{(}\overset{\rightarrow}{H}\overset{ \rightarrow}{H}\overset{\rightarrow}{H}\bigg{)}\bigg{)}+\frac{j}{1}\bigg{(}W _{qdlII}(\overset{\rightarrow}{i}_{j}\cdot\overset{\rightarrow}{H})\bigg{(} \overset{\rightarrow}{H}\cdot\overset{\rightarrow}{\rho}\bigg{)}\bigg{)}\\ +\frac{j}{1}\bigg{(}W_{cdj}(\overset{\rightarrow}{\rho}\cdot \overset{\rightarrow}{\rho})\bigg{(}\overset{\rightarrow}{H}\cdot\overset{ \rightarrow}{\rho}\bigg{)}\bigg{)}\end{array}\right)\\ &=\frac{1}{mu^{\prime}}\
abla_{\rho}\left(\begin{array}{l}W_{m}(\overset{ \rightarrow}{H}\overset{\rightarrow}{\rho})+W_{a}(\overset{i}{i}_{a}\cdot \overset{\rightarrow}{H})(\overset{i}{a}_{a}\cdot\overset{\rightarrow}{\rho} )+W_{qdl}(\overset{\rightarrow}{i}_{qdl}\cdot\overset{\rightarrow}{\rho})( \overset{\rightarrow}{H}\cdot\overset{\rightarrow}{H})\\ +W_{qdlI}(\overset{\rightarrow}{i}_{qdlI}\cdot\overset{\rightarrow}{H})( \overset{\rightarrow}{H}\cdot\overset{\rightarrow}{\rho})+W_{cd}( \overset{\rightarrow}{H}\cdot\overset{\rightarrow}{\rho})(\overset{ \rightarrow}{H}\cdot\overset{\rightarrow}{H})\\ \end{array}\right)\\ &=\frac{1}{mu^{\prime}}\left(\begin{array}{l}W_{m}(\overset{H}{H})+W_{a}( \overset{i}{i}_{a})(\overset{i}{i}_{a}\cdot\overset{\rightarrow}{H})+W_{qdlI} (\overset{\rightarrow}{i}_{qdl})(\overset{\rightarrow}{H}\cdot\overset{ \rightarrow}{H})\\ +W_{qdII}(\overset{\rightarrow}{H})(\overset{\rightarrow}{i}_{qdlI}\cdot \overset{\rightarrow}{H})+W_{cd}(\overset{\rightarrow}{H})(\overset{ \rightarrow}{H}\cdot\overset{\rightarrow}{H})\end{array}\right)\\ &\text{where }\overset{\rightarrow}{\epsilon}\text{ is the lateral aberration vector on the image plane, }n\text{ is the index of refraction, and }u^{\prime}\text{ is the marginal ray slope in image space, }W_{m}(\overset{\rightarrow}{H}\cdot\overset{\rightarrow}{\rho})\text{, }W_{a}(\overset{\rightarrow}{i}_{a}\cdot \overset{\rightarrow}{H})(\overset{\rightarrow}{i}_{a}\cdot\overset{ \rightarrow}{\rho})\text{, }W_{qdlI}(\overset{\rightarrow}{i}_{qdlI}\cdot \overset{\rightarrow}{H})\text{, }W_{qdlI}(\overset{\rightarrow}{i}_{qdlI}\cdot \overset{\rightarrow}{H})\text{(}\overset{\rightarrow}{H}\cdot\overset{ \rightarrow}{\rho})\text{, and }W_{cd}(\overset{\rightarrow}{H}\cdot\overset{\rightarrow}{\rho})( \overset{\rightarrow}{H}\cdot\overset{\rightarrow}{\rho})\text{ are the vector sums of }W_{mj}(\overset{\rightarrow}{H}\cdot\overset{\rightarrow}{\rho})\text{, }W_{aj}(\overset{\rightarrow}{i}_{j}\cdot \overset{\rightarrow}{H})\bigg{(}\overset{\rightarrow}{i}_{j}\cdot \overset{\rightarrow}{\rho}\bigg{)},W_{qdlI}(\overset{\rightarrow}{i}_{qdl} \cdot\overset{\rightarrow}{\rho})(\overset{\rightarrow}{H}\cdot\overset{ \rightarrow}{H})\text{, }W_{qdlIj}(\overset{\rightarrow}{i}_{j}\cdot \overset{\rightarrow}{H})\bigg{(}\overset{\rightarrow}{H}\cdot\overset{ \rightarrow}{\rho}\bigg{)}\text{, and }W_{cd}(\overset{\rightarrow}{H}\cdot\overset{\rightarrow}{\rho})( \overset{\rightarrow}{H}\cdot\overset{\rightarrow}{H})\text{, respectively.}\end{split} \]  
To demonstrate the consistency of the two methods, we calculate the aberrations generated by the prism using system and structural parameters that are identical to those in Section 3.1. The types and magnitudes of aberrations generated are observed and compared with theoretical values, as shown in Figure 6. It can be seen that the aberration magnitudes obtained by ray tracing are generally consistent with the theoretical values. This article focuses on the simpler and more common cases of rotating prisms.