# Rotational-Shearing-Interferometer Response for a Star-Planet System without Star Cancellation
## 3 Computational Simulation  
In order to verify the advantages of the proposed technique, we perform a computational simulation of the RSI and its response to a star-planet system. We use an exact ray trace over the RSI to determine the wavefront modification. The wavefront was simulated using three rays whose sources are located over a plain. The rays are propagated in parallel to the propagation vector of the wavefront. On each surface, a new ray set is calculated according to the reflection or refraction laws as appropriate. When the rays insides over the observation plane, their optical path length is calculated. Using this information, we determine the wavefront transformation after they have been propagated by the RSI. The process is repeated for each wavefront and each interferometer arm. The incidence at each point of the observation plane is calculated using the incidence and phase of each incident wavefront. Finally, the resultant interferogram is determined by mapping the resultant incidence with a grayscale value. This computational simulation technique was explained with more detail in [23]. We simulate a star-planet system where the angular distance between the star and the planet is 0.5 arcsec, and the star radiation is \(10^{5}\) times the planet radiation at 1 \(\upmu\)m. The star is perfectly aligned with the optical axis. The azimuth angle of the planet is 0. The observation plane dimensions are 1 m x 1 m. We use these characteristics to probe the improvement in the planet detection performed by the proposed technique. However, equivalent advantages may be achieved with any star-planet system. Figure 5 shows six interferograms obtained by computational simulation. The interferograms were generated by adjusting the OPD to \(\lambda/2\) and changing the rotation angle from \(0^{\circ}\) to \(180^{\circ}\) with the star aligned with the optical axis. These interferograms are composed of straight fringes for which density and orientation change when the interferometer rotation-angle is changed, according to Equation (8). The fringes are produced by the interference of the planet wavefront with itself; this confirms the planet presence. The rotation of the fringe allows for discarding several false-positives by alignment errors. The image grayscale range is adjusted to coincide its saturation level with the maximum incidence, produced by \(M_{pp}\) term. Figure 6 shows interferograms generated adjusting the OPD to \(\lambda/2+1\) nm with the star aligned to the optical axis. These interferograms are composed by two straight fringes superposed. The fringes density and orientation changes with the interferometer rotation-angle as predicted in Equation (8). These fringes correspond to the interference between the star and the planet. The fringes produced by the interference of the planet with itself are present; however, they are eclipsed by the brightness of star-planet fringes. The first image shows the background incidence produced by the interference of the star wavefront with its rotated version. The image grayscale is adjusted to coincide with the maximum incidence, produced by the \(M_{SP}\) term. In this case, we obtain a signal gain of 4 with respect to the previous case. The gain is calculated as the difference between the minimum and the maximum incidence level of the resultant interferogram compared to the difference between the minimum and the maximum of the \(M_{pp}\) term. The increment in the fringe incidence in accordance with the OPD variation is showed by Figure 7. It shows interferograms with different grayscale ranges: in the first row, the image saturation-level is 8 times the \(M_{PP}\) maximum, in the second row 64 times, in the third row 128 times, and 256 times in the fourth row. The OPD is changed in each column: in the first column, the OPD is \(\lambda/2\), in the second column, the OPD is \(\lambda/2+2\) nm, in the third column, the OPD is \(\lambda/2+5\) nm, in the fourth column, the OPD is \(\lambda/2+10\) nm, and, finally, in the last column, the OPD is \(\lambda/2+15\) nm.