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We discuss the optical properties of the solar gravitational lens (SGL). We estimate the power of the EM field received by an imaging telescope. Studying the behavior of the EM field at the photometric detector, we develop expressions that describe the received power from a point source as well as from an extended resolved source. We model the source as a disk with uniform surface brightness and study the contribution of blur to a particular image pixel. To describe this process, we develop expressions describing the power received from the directly imaged region of the exoplanet, from the rest of the exoplanet and also the power for off-image pointing. We study the SGLs amplification and its angular resolution in the case of observing an extended source with a modest size telescope. The results can be applied to direct imaging of exoplanets using the SGL.
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The remarkable optical properties of the solar gravitational lens (SGL) include major brightness amplification (~1e11 at wavelength of 1 um) and extreme angular resolution (~1e-10 arcsec) in a narrow field of view. A mission to the SGL carrying a modest telescope and coronagraph opens up a possibility for direct megapixel imaging and high-resolution spectroscopy of a habitable Earth-like exoplanet at a distance of up to 100 light years. The entire image of such a planet is compressed by the SGL into a region with a diameter of ~1.3 km in the vicinity of the focal line. The telescope, acting as a single pixel detector while traversing this region, can build an image of the exoplanet with kilometer-scale resolution of its surface, enough to see its surface features and signs of habitability. We report here on the results of our initial study of a mission to the deep outer regions of our solar system, with the primary mission objective of conducting direct megapixel high-resolution imaging and spectroscopy of a potentially habitable exoplanet by exploiting the remarkable optical properties of the SGL. Our main goal was to investigate what it takes to operate spacecraft at such enormous distances with the needed precision. Specifically, we studied i) how a space mission to the focal region of the SGL may be used to obtain high-resolution direct imaging and spectroscopy of an exoplanet by detecting, tracking, and studying the Einstein ring around the Sun, and ii) how such information could be used to detect signs of life on another planet. Our results indicate that a mission to the SGL with an objective of direct imaging and spectroscopy of a distant exoplanet is challenging, but possible. We composed a list of recommendations on the mission architectures with risk and return tradeoffs and discuss an enabling technology development program.
We report on the initial results obtained with an image convolution/deconvolution computer code that we developed and used to study the image formation capabilities of the solar gravitational lens (SGL). Although the SGL of a spherical Sun creates a greatly blurred image, knowledge of the SGLs point-spread function (PSF) makes it possible to reconstruct the original image and remove the blur by way of deconvolution. We discuss the deconvolution process, which can be implemented either with direct matrix inversion or with the Fourier quotient method. We observe that the process introduces a ``penalty in the form of a reduction in the signal-to-noise ratio (SNR) of a recovered image, compared to the SNR at which the blurred image data is collected. We estimate the magnitude of this penalty using an analytical approach and confirm the results with a series of numerical simulations. We find that the penalty is substantially reduced when the spacing between image samples is large compared to the telescope aperture. The penalty can be further reduced with suitable noise filtering, which can yield ${cal O}(10)$ or better improvement for low-quality imaging data. Our results confirm that it is possible to use the SGL for imaging purposes. We offer insights on the data collection and image processing strategies that could yield a detailed image of an exoplanet within image data collection times that are consistent with the duration of a realistic space mission.
We investigate the optical properties of the solar gravitational lens (SGL) with respect to an extended source located at a large but finite distance from the Sun. The static, spherically symmetric gravitational field of the Sun is modeled within the first post-Newtonian approximation of the general theory of relativity. We consider the propagation of monochromatic electromagnetic (EM) waves near the Sun. We develop, based on a Mie theory, a vector theory of diffraction that accounts for the refractive properties of the solar gravitational field. The finite distance to a point source can be accounted for using a rotation of the coordinate system to align its polar axis with the axis directed from the point source to the center of the Sun, which we call the optical axis. We determine the EM field and study the key optical properties of the SGL in all four regions formed behind the Sun by an EM wave diffracted by the solar gravity field: the shadow, geometric optics, and weak and strong interference regions. Extended sources can then be represented as collections of point sources. We present the power density of the signal received by a telescope in the image plane. Our discussion concludes with considering the implications for imaging with the SGL.
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.
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.
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