# Rotational-Shearing-Interferometer Response for a Star-Planet System without Star Cancellation
## 2 Theory  
The closest star to Earth is Proxima Centauri, its distance to the Earth is 1.295 parsecs [19]. At this distance, the optical radiation from a massive light source like a star or a planet may be considered coherent. This is because these conditions satisfy the Van Cittert-Zernike theorem [20]. Then, the radiation from any planetary-system outside the solar system may interfere between them. Additionally, the star and the planet may be considered as point sources [21]. Figure 1 shows a diagram of the star-planet system viewed for the observer. The alignment of the planet and the star is characterized by means of their elevation angle (\(\theta\)) and their azimuth angle (\(\varphi\)). The wavefronts from a star or a planet in its periphery may be modeled as planes with uniform intensity due to the long distance from the observer. The Rotational Shearing Interferometer (RSI) was proposed for extra-solar planet detection because it is insensitive to rotational-symmetrically wavefronts. The RSI makes incident wavefront interfere with a rotated version of them. When a solitary star is aligned  
Figure 1: Star–planet system viewed for the observer. The star and the planet may be modeled as point sources with an elevation (\(\theta\)) and an azimuth (\(\varphi\)) angle with respect to the optical axis. on the optical axis of the interferometer, its wavefront is rotational-symmetrically, and the RSI does not produce a fringe pattern. Instead, when the wavefronts from the star and the planet are incident on the RSI, its interference produces a fringe pattern. The RSI consists of a Mach-Zehnder interferometer with a Dove prism in each arm as shown in Figure 2. When the Dove prism rotates around the optical axis, the propagated wavefront rotates double of the Dove-prism rotation angle. One of the Dove prism is rotated in order to generate the wavefront rotation. The other Dove prism remains static for compensation purposes. We use a mirror array as an Optical Path Modulator (OPM). This array is formed by the mirrors \(M_{4}\), \(M_{5}\), and \(M_{6}\). The mirrors \(M_{5}\) and \(M_{6}\) are collocated over a displacement platform to control the elongation of the optical Path. We add an additional mirror array in the other arm for compensation purposes. When a beam insides on the first beam splitter, it is divided in two. The first one is propagated trough the prism \(DP_{1}\), and it is rotated by \(\Delta\phi\) with respect to the other beam. The second beam is propagated through the OPM to adjust the OPD. Finally, both beams interfere in the observation plane (OP). When a beam coming from a star-planet system is incident on the RSI entrance, four wavefronts interfere between them on the observation plane. Two of these wavefronts correspond to the planet and the other two correspond to the star. The resulting interference may be divided into three terms: the first one for the interference between the star wavefronts (\(M_{SS}\)), the second one for the interference between the planet wavefronts (\(M_{PP}\)), and an additional term for the interference between the planet and the star wavefronts (\(M_{SP}\)). Then, the incidence (\(M\)) in the interference plane may be modeled as the sum of the terms:  
\[M=M_{SS}+M_{PP}+M_{SP}. \tag{1}\]  
Specifically, these terms may be written in terms of the incidence of the star (\(M_{P}\)), incidence of the planet (\(M_{P}\)) and phase (\(\Phi_{ij}\)) of each wavefront. The subscripts \(i=S,P\) and \(j=1,2\), represent the wavefronts, and the interferometer arms, respectively:  
\[M_{SS} =M_{S}[2+2\cos(\Phi_{S2}-\Phi_{S1})], \tag{2}\] \[M_{PP} =M_{P}[2+2\cos(\Phi_{P2}-\Phi_{P1})], \tag{3}\]  
Figure 2: The Rotational Shearing Interferometer consists of a modified Mach–Zehnder interferometer, with a Dove prism as Wavefront Rotator (WR), a mirror array as an Optical Path Modulator (OPM), and its respective compensation components in the opposite arm.