Image formation process with the solar gravitational lens

We study image formation with the solar gravitational lens (SGL). We consider a point source that is positioned at a large but finite distance from the Sun. We assume that an optical telescope is positioned in the image plane, in the focal region of the SGL. We model the telescope as a convex lens and evaluate the intensity distribution produced by the electromagnetic field that forms the image in the focal plane of the convex lens. We first investigate the case when the telescope is located on the optical axis of the SGL or in its immediate vicinity. This is the region of strong interference where the SGL forms an image of a distant source, which is our primary interest. We derive analytic expressions that describe the progression of the image from an Einstein ring corresponding to an on-axis telescope position, to the case of two bright spots when the telescope is positioned some distance away from the optical axis. At greater distances from the optical axis, in the region of weak interference and that of geometric optics, we recover expressions that are familiar from models of gravitational microlensing, but developed here using a wave-optical treatment. We discuss applications of the results for imaging and spectroscopy of exoplanets with the SGL.

Figure 1

The different optical regions of the SGL (adapted from [8]).

Figure 2

Imaging a point source with the SGL. The source is positioned on the optical axis at the distance $z_0$ from the Sun. The image plane is at the heliocentric distance $\overline{z}$. Rays with different optical paths produce a diffraction pattern in the image plane that is observed by an imaging telescope.

Figure 3

Imaging a point source with the SGL as seen by a telescope. The telescope is represented by a convex lens with aperture $d$ and a focal length $f$. The telescope optics projects an image of the Einstein ring around the Sun in its focal plane. Positions in the SGL image plane, $(x,y)$, and the telescope’s focal plane, $(x_i,y_i)$, are also shown.

Figure 4

Normalized density plots simulating images that appear on the image sensor of a diffraction-limited 1-m optical telescope, at the wavelength $\lambda =1\mu \mathrm{m}$, at 1000 AU from the Sun, at different positions with respect to the SGL optical axis. (Left panel) The telescope is positioned on the optical axis, ${\rho }_0=0$, in accordance with Eq. (15). (Center panel) The telescope is positioned several meters away from the optical axis in the ${\varphi }_0=-\pi /4$ direction, but still in the region of strong interference, in accordance with Eq. (33). (Right panel) The telescope is positioned in the region of weak interference, ${\rho }_0\gtrsim R_{\odot }$ from the optical axis, with the resulting minor and major images (their positions swapped, as expected, by the optical telescope) shown in accordance with Eq. (63). The Sun is indicated with a dashed (yellow) line, while the Einstein ring is shown as a solid line.