Imaging extended sources with the solar gravitational lens

Abstract

We study the image formation process with the solar gravitational lens (SGL) in the case of an extended, resolved source. An imaging telescope, modeled as a convex lens, is positioned within the image cylinder formed by the light received from the source. In the strong interference region of the SGL, this light is greatly amplified, forming the Einstein ring around the Sun, representing a distorted image of the extended source. We study the intensity distribution within the Einstein ring observed in the focal plane of the convex lens. For any particular telescope position in the image plane, we model light received from the resolved source as a combination of two signals: light received from the directly imaged region of the source and light from the rest of the source. We also consider the case when the telescope points away from the extended source or, equivalently, it observes light from sources in sky positions that are some distance away from the extended source, but still in its proximity. At even larger distances from the optical axis, in the weak interference or geometric optics regions, our approach recovers known models related to microlensing, but now obtained via the wave-optical treatment. We then derive the power of the signal and related photon fluxes within the annulus that contains the Einstein ring of the extended source, as seen by the imaging telescope. We discuss the properties of the deconvolution process, especially its effects on noise in the recovered image. We compare anticipated signals from realistic exoplanetary targets against estimates of noise from the solar corona and estimate integration times needed for the recovery of high-quality images of faint sources. The results demonstrate that the SGL offers a unique, realistic capability to obtain resolved images of exoplanets in our galactic neighborhood. FIG. 1: The different optical regions of the SGL (adapted from [3]).SNRs, required integration times for a given observing scenario, to evaluate the quality of reconstructed images and, by doing so, to move the concept of imaging with the SGL from a domain of theoretical physics to the mainstream of astronomy and astrophysics.Our paper is organized as follows: Section II introduces the SGL and the solution for the EM field in the image plane in the strong interference region behind the Sun. Section III discusses the modeling of the intensity distribution observed in the focal plane behind the convex lens. We present the total signal received from the extended source as consisting of two parts: the signal from the directly imaged region of the source and the blur received from the rest of the source. Although our basic results are generic, to allow for the analytic evaluation of realistic observing scenarios, we model the source as a uniformly illuminated disk. This approach allows us to develop analytical expressions to estimate the total photon flux received by the telescope. In Section IV we study image formation in the geometric optics and weak interference regions, thus extending ...

Image formation process with the solar gravitational lens

“…Expression (15) is always finite, reaching its maximum at the Einstein ring for which η i = α, see discussion in [8]. Fig.…”

“…The quantity A(x i , x 0 ) in (4) is the complex amplitude of the EM wave as it observed on the focal plane of the optical telescope for a particular telescope position of x 0 . If the amplitude at the entrance of the telescope, A(x, x 0 ), is known for any location x on the image plane (in the case of imaging with the SGL, this amplitude is well-known, e.g., see [1,2,8]), then the wave's amplitude at the focal plane of the telescope is determined by the Fresnel-Kirchhoff diffraction formula [10]:…”

“…A. EM field in the interference region of the SGL In [8], we considered light from an extended source at a finite distance, r 0 from the Sun. We parameterize the problem using a spherical coordinate system (r, θ, φ) that is aligned with a preferred axis: a line connecting a preselected (e.g., central) point in the source to the center of the Sun.…”

“…For that, we consider light, that is to say, a high-frequency EM wave (i.e., neglecting terms ∝ (kr) −1 where k is the wavenumber) and for r ≫ r g (where r g = 2GM ⊙ /c 2 is the Sun's Schwarzschild radius) and derive the components of the EM field near the optical axis. Following [8], we have that for a point source located on the optical axis at the distance of z 0 from the Sun, up to terms of O(ρ 2 /z 2 , 2r g z/z 0 ), the z-component of the EM field which moves in the z-direction ( Fig. 1) behaves as (E z , H z ) = O(ρ/z, 2r g z/z 0 ), while the other components are given as…”

“…In Ref. [8], we considered a high-frequency EM wave (i.e., neglecting terms ∝ (kr) −1 ) and for r ≫ r g and derived the components of the EM field in the geometric optics and weak interference regions. As light amplification in these regions is rather weak, it is sufficient to derive the solution to the highest leading order term in the image field amplitude.…”