# Rotational-Shearing-Interferometer Response for a Star-Planet System without Star Cancellation
## 1 Introduction  
In the last twenty years, the interest in exoplanet detection has been increased. More than 4000 planets have been discovered [1], The main interest of the scientific community are the earthlike planets [2]. The overwhelming majority of discovering planet was discovered by indirect techniques such as transit light curves [3], radial velocity [4], and gravitational microlensing [5]. These techniques sense the effect of the planet over the star radiation. They measure variations in the star radiation and determine the planet presence using statistical techniques. However, these variations may be produced by unknown processes in the star and not by the presence of a planet. Additionally, the measurements obtained by the microlensing technique are not repeatable because they require the alignment of two stars and the planet. Moreover, the time necessary to perform a measurement employing indirect techniques may last from days to years because the planet must complete at least one orbit. The direct detection of an exoplanet will confirm the currently available evidence for its existence, with shorter observation periods and incorporating the repeatability ofthe measurement. The principal challenges in the direct detection of exoplanets are the image resolution and the signal-to-noise ratio. The image resolution refers to the minimum angular distance between two discernible sources. This angular distance is conditioned by the diameter of the primary mirror of the telescope. The signal-to-noise ratio may be defined as the quotient between the planet radiance and the star radiance. At visible wavelengths, the planet radiance arises primarily from the radiation reflected from its parent star. The amount of reflected radiation depends on the planet's albedo, its radius, and its distance from the star. This ratio for a Jupiter-like planet and a Sun-like star is at least \(10^{-10}\)[6]. In the infrared (IR) spectral region, the planet radiance consists primarily of the planet thermally emitted radiation. Additionally, the radiance for a Sun-like star is lower in the IR than in the visible region. Under these conditions, the radiation signal-to-noise ratio increases up to \(10^{-5}\)[7]. However, this is still a very low signal-to-noise ratio. In order to improve the radiation ratio, a coronagraph and interferometric techniques are often implemented [8, 9]. The coronagraph technique consists of occulting the star using a mechanical aperture. This technique usually implements spatial transmission filters in the focal plane to remove the diffraction rings due to the hard stop edge [10; 11]. However, due to the limitations of the spatial resolution, this technique is applied primarily to the potential planets with large orbits. Most of interferometric techniques attenuate the star radiation by means of destructive interference. Consequently, these interferometers are called Nulling Interferometers. They interfere with the wavefronts with a delayed version of them. When the delay between the interferometer arms is \(\lambda/2\), the star radiation is canceled, with only the planet radiation remaining [12; 13; 14]. Previously, we proposed a Rotational Shearing Interferometer (RSI) for planet detection [15; 16; 17; 18]. This interferometer interferes with the wavefronts with a rotated and delayed version of them. Thus, this interferometer may cancel the star radiation in a similar way to the nulling interferometers. Additionally, we may discriminate against false-positive results rotating the wavefront. Unfortunately, when the star is canceled, the remaining amount of radiation is too small. In order to improve the signal magnitude, we propose using the RSI without the total-cancellation of the star radiation. When the star-radiation is not canceled, the fringe visibility is decreased, but the signal magnitude is further increased. In this work, we describe the response of the RSI to a star-planet system radiation in Section 2. In Section 3, we present computational simulations to validate the viability of the method.