Image formation for extended sources with the solar gravitational lens

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.

Figure 1

The different optical regions of the SGL (adapted from [3]).

Figure 2

The geometry of imaging a point source with the SGL. A point source with coordinates $(x^{\prime },y^{\prime })$ is positioned in the source plane, at the distance $z_0$ from the Sun. The SGL image plane is at the heliocentric distance $\overline{z}$. Rays with different optical paths produce a diffraction pattern in the SGL image plane that is observed by an imaging telescope.

Figure 3

Imaging of extended resolved sources with the SGL. The SGL is a convex lens, producing inverted images of a source.

Figure 4

Imaging a point source with the SGL with a telescope. The telescope is positioned on the optical axis that connects the source and the Sun and it “sees” the Einstein ring. The telescope is represented by a convex lens with a diameter $d$ and a focal length $f$. Positions in the SGL image plane, $(x,y)$, and the optical telescope’s focal plane, $(x_i,y_i)$, are also shown.

Figure 5

Top row: Density plots simulating images that appear in the focal plane of the optical telescope. Left: The directly imaged region, in accordance with Eq. (31). The brightness of this image is exaggerated to ensure that the Einstein ring and diffraction artifacts remain visible. Center: Light from the rest of the source, in accordance with Eq. (50). This is the dominant light contribution, yielding a much brighter Einstein ring with less prominent diffraction artifacts. Right: Image contamination due to a nearby source of light, in accordance with Eq. (58), showing light from another uniformly illuminated disk of the same size, offset horizontally by ten radii. Bottom row: Corresponding dimensionless intensities depicted on a decimal logarithmic scale. The contribution from the directly imaged region is $\mathcal{O}({10}^3)$ less than the contribution from the rest of the source. Contribution from a nearby object is of similar intensity but confined to narrow sections of the Einstein ring.

Figure 6

Combined behavior of $\epsilon ({\rho }_0)$ (53), for $0\le {\rho }_0/r_{\oplus }\le 1$ (solid red curve) and $\beta ({\rho }_0)$ (60), for ${\rho }_0/r_{\oplus }\ge 1$ (dashed green curve). The horizontal axis is in units of ${\rho }_0/r_{\oplus }$. The dots represent values obtained from a numerical simulation of (16) with (25).

Figure 7

Density plot simulating the image seen by the optical telescope when it is positioned in the region of weak interference, ${\rho }_0\gtrsim R_{\odot }$ from the optical axis, with the resulting minor and major images shown in accordance with Eq. (85). The Sun is indicated with a dashed line, while the Einstein ring is shown as a solid line. Note that in the region of geometric optics only the major image remains, as described by Eq. (84).

Figure 8

Encircled energy and its normalized distribution for the directly imaged region (97) and the rest of the source (106). Horizontal axis is in seconds of arc, as seen by a telescope positioned at $\overline{z}=650\mathrm{A}\mathrm{.}\mathrm{U}\mathrm{.}$. The peak at $\sim {1.6}^{\prime \prime }$ corresponds to the location of the Einstein ring.

Figure 9

The annular coronagraph concept. The coronagraph blocks light from both within and outside the Einstein ring. The thickness of the exposed area is determined by the diffraction limit of the optical telescope at its typical observational wavelength.

Figure 10

Angular sizes of the Sun and the diffraction-limited view of the Einstein ring as functions of heliocentric distance (for $\lambda =0.6\mu \mathrm{m}$). As the heliocentric distance increases, the Einstein ring (together with the entire imaged region) further separates from the Sun. A coronagraph may have to be able to compensate for decreasing angular sizes.

Figure 11

A simulation of the effects of the monopole solar gravitational lens on an Earth-like exoplanet image. Top row, left: A monochrome image, sampled at $128\times 128\mathrm{pixels}$; center: blurred image; right: deconvolution at $\mathrm{SNR}\sim 4.5$. From [28]. Bottom row, left: Original RGB color image with a $1024\times 1024\mathrm{pixel}$ resolution; center: image blurred by the SGL; right: the result of image deconvolution at an SNR of $\sim 5.2$ per color channel, or combined SNR of $\sim 9$.

Figure 12

Solar corona brightness from [34]. As heliocentric distances increase, the Einstein ring further separates from the Sun. Positions of the Einstein ring for $z=600\mathrm{A}\mathrm{.}\mathrm{U}\mathrm{.}$ and $z=1500\mathrm{A}\mathrm{.}\mathrm{U}\mathrm{.}$ are shown, both corresponding to distances from the center of the Sun of $\le 2R_{\odot }$. For such solar separations, the K corona dominates.

Figure 13

Typical surface brightness of the solar corona (from [29]) as seen at 800 A.U., as given by (a16).

Figure 14

Behavior of the averaged PSF of the SGL. Left: Numerical integration of Eq. (b3). Evaluating the analytical expression for the averaged PSF given by the combination of Eqs. (b5) and (b7) yields an identical plot. Right: The PSF from Eq. (b12) given by $\epsilon (r)$ (blue line) and $\beta (r)$ (red, dashed line). The plots are nearly identical. Note that a minor oscillatory behavior evident on the left is absent on the right. The horizontal axis is the distance from the center of the aperture in meters.

Figure 15

Behavior of the averaged PSF of the SGL. Left: Numerical integration of Eq. (b3). Right: The piecewise defined PSF from Eqs. (b15) and the smoothed PSF from Eq. (b17). The horizontal axis is the distance from the center of the aperture in meters.