## 3 Theory
### Incidence Function  
No coherence between the star and the planet radiation is assumed. Then, the incidence in the interference plane, \(M_{SP}\), may be modeled as the sum of the star and the planet contributions. \[M_{SP}(\varphi_{S},\theta_{S};\varphi_{P},\theta_{P})=M_{SS}(\varphi_{S},\theta _{S})+M_{PP}(\varphi_{P},\theta_{P}) \tag{1}\]  
\[M_{SS}(\varphi_{S},\theta_{S})=M_{S}[1+cos(\Phi_{S2}-\Phi_{S1})] \tag{2}\]  
\[M_{PP}(\varphi_{P},\theta_{P})=M_{P}[1+cos(\Phi_{P2}-\Phi_{P1})] \tag{3}\]  
Equations (2) and (3) are substituted into Equation (1):  
\[M_{SP}(\varphi_{S},\theta_{S};\varphi_{P},\theta_{P})=M_{S}[1+cos(\Phi_{S2}- \Phi_{S1})]+M_{P}[1+cos(\Phi_{P2}-\Phi_{P1})]. \tag{4}\]  
The phase terms, \(\Phi\)ij(\(\rho\), \(\theta\), \(\varphi\)), in cylindrical coordinates for each wavefront correspond to the waves originating at the star, \(S\), and the planet, \(P\). The incident wavefront is divided in two using a beam splitter. Each half is travelling through one of the interferometer arms, 1 and 2. The \(\omega t\)- terms are left out: they are subtracted out in the next step anyway. \[\Phi_{S1}(\varphi_{S},\theta_{S})=k\rho\ sin\theta_{S}\ cos(\varphi-\Delta \varphi-\varphi_{S})+L_{1}\cos\theta_{S} \tag{5}\]  
\[\Phi_{S2}(\varphi_{S},\theta_{S})=k\rho\ sin\theta_{S}\ cos(\varphi+\Delta \varphi-\varphi_{S})+L_{2}\cos\theta_{S} \tag{6}\]  
\[\Phi_{P1}(\varphi_{P},\theta_{P})=k\rho\ sin\theta_{P}\ cos(\varphi-\Delta \varphi-\varphi_{P})+L_{1}\cos\theta_{P} \tag{7}\]  
\[\Phi_{P2}(\varphi_{P},\ \theta_{P})=k\rho\ sin\theta_{P}\ cos(\varphi+\Delta \varphi-\varphi_{P})+L_{2}\cos\theta_{P} \tag{8}\]  
Here, \(r^{2}=x^{2}+y^{2}\). In addition, \(\rho_{i}=z_{0}\ sin\theta_{i}\), where \(i=S\) and \(P\), and \(z_{0}\) is the distance of the planetary system from the interferometer plane. The wavenumber is \(k=2\pi/\lambda\), where \(\lambda\) is the wavelength in [m]. Instead of modeling the rotation of the wavefronts by \(2\Delta\varphi\) using the Dove prism in one arm, in the theoretical development, the prism is rotated in each arm  
Figure 4: Experimentally obtained interference patterns indeed change when the angle of orientation of the Dove prism increases from 0 to 20\({}^{\circ}\) for an off-axis point source. The nearly straight interference fringes increase in density and increase in the inclination angle when the shear angle is increased in increments of 5\({}^{\circ}\). This is one of the characteristics of the RSI in measuring the tilted wave fronts, such as those originating at an off-axis planet. This experimental data confirms that an off-axis source can be detected in an RSI by generating straight fringes. Furthermore, the fact that the fringes arise from an off-axis source, rather than because of an artifact, may be confirmed by changing the shear angle and observing how their density and angle of inclination change. by \(+/-\Delta\varphi/2\). The prism rotation in one arm by \(\Delta\varphi/2\) results in the wavefront rotation by \(\Delta\varphi\). The symbol \(L_{i}\) is used to denote the distance that the wavefront travels inside the interferometer. The symbol \(\theta_{i}\) indicates the elevation angles of the star, \(S\), and the planet, \(P\), measured from the (negative, in our diagram) optical axis. Similarly, \(\varphi_{i}\) are the azimuthal angles of the star and the planet, measured from the \(x^{\prime}\)- axis. After some trigonometric manipulation, the difference in phase is found in the observation plane: please change indent below  
\[\Phi_{S2}(\varphi_{S},\theta_{S})-\Phi_{S1}(\varphi_{S},\theta_{S})=k\rho\ sin \theta_{S}\left[ \begin{array}{c}cos(\varphi+\Delta\varphi-\varphi_{S})\\ -cos(\varphi-\Delta\varphi-\varphi_{S})\end{array} \right]+(L_{2}-L_{1})\cos \theta_{S}, \tag{9}\]  
and  
\[\Phi_{P2}(\varphi_{P},\theta_{P})-\Phi_{P1}(\varphi_{P},\theta_{P})=k\rho\ sin \theta_{P}\left[ \begin{array}{c}cos(\varphi+\Delta\varphi-\varphi_{P})\\ -cos(\varphi-\Delta\varphi-\varphi_{P})\end{array} \right]+(L_{2}-L_{1})\cos \theta_{P}.