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Register a new userOptical properties of an extended gravitational lens
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We continue to study the optical properties of the solar gravitational lens (SGL). The aim is prospective applications of the SGL for imaging purposes. We investigate the solution of Maxwells equations for the electromagnetic (EM) field, obtained on the background of a static gravitational field of the Sun. We now treat the Sun as an extended body with a gravitational field that can be described using an infinite series of gravitational multipole moments. Studying the propagation of monochromatic EM waves in this extended solar gravitational field, we develop a wave-optical treatment of the SGL that allows us to study the caustics formed in an image plane in the SGLs strong interference region. We investigate the EM field in several important regions, namely i) the area in the inner part of the caustic and close to the optical axis, ii) the region outside the caustic, and iii) the region in the immediate vicinity of the caustic, especially around its cusps and folds. We show that in the first two regions the physical behavior of the EM field may be understood using the method of stationary phase. However, in the immediate vicinity of the caustic the method of stationary phase is inadequate and a wave-optical treatment is necessary. Relying on the angular eikonal method, we develop a new approach to describe the EM field accurately in all regions, including the immediate vicinity of the caustics and especially near the cusps and folds. The method allows us to investigate the EM field in this important region, which is characterized by rapidly oscillating behavior. Our results are new and can be used to describe gravitational lensing by realistic astrophysical objects, such as stars, spiral and elliptical galaxies.
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We continue our study of the optical properties of the solar gravitational lens (SGL). Taking the next step beyond representing it as an idealized monopole, we now characterize the gravitational field of the Sun using an infinite series of multipole moments. We consider the propagation of electromagnetic (EM) waves in this gravitational field within the first post-Newtonian approximation of the general theory of relativity. The problem is formulated within the Mie diffraction theory. We solve Maxwells equations for the EM wave propagating in the background of a static gravitational field of an extended gravitating body, while accounting for multipole contributions. Using a wave-theoretical approach and the eikonal approximation, we find an exact closed form solution for the Debye potentials and determine the EM field at an image plane in the strong interference region of the lens. The resulting EM field is characterized by a new diffraction integral. We study this solution and show how the presence of multipoles affects the optical properties of the lens, resulting in distinct diffraction patterns. We identify the gravitational deflection angle with the individual contributions due to each of the multipoles. Treating the Sun as an extended, axisymmetric, rotating body, we show that each zonal harmonics causes light to diffract into an area whose boundary is a caustic of a particular shape. The appearance of the caustics modifies the point-spread function (PSF) of the lens, thus affecting its optical properties. The new wave-theoretical solution allows the study gravitational lensing by a realistic lens that possesses an arbitrary number of gravitational multipoles. This {em angular eikonal method} represents an improved treatment of realistic gravitational lensing. It may be used for a wave-optical description of many astrophysical lenses.
We study the optical properties of the solar gravitational lens (SGL) while treating the Sun as an extended, axisymmetric and rotating body. The gravitational field of the Sun is represented using a set of zonal harmonics. We develop an analytical description of the intensity of light that is observed in the image plane in the strong interference region of a realistic SGL. This formalism makes it possible to model not only the point-spread function of point sources, but also actual observables, images that form in the focal plane of an imaging telescope positioned in the image plane. Perturbations of the monopole gravitational field of the Sun are dominated by the solar quadrupole moment, which results in forming an astroid caustic on the image plane. Consequently, an imaging telescope placed inside the astroid caustic observes four bright spots, forming the well-known pattern of an Einstein cross. The relative intensities and positions of these spots change as the telescope is moved in the image plane, with spots merging into bright arcs when the telescope approaches the caustic boundary. Outside the astroid caustic, only two spots remain and the observed pattern eventually becomes indistinguishable from the imaging pattern of a monopole lens at greater distances from the optical axis. We present results from extensive numerical simulations, forming the basis of our ongoing study of prospective exoplanet imaging with the SGL. These results are also applicable to describe a large class of gravitational lensing scenarios involving axisymmetric lenses that can be represented using zonal harmonics.
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 the optical properties of an oblate gravitational lens, such as the solar gravitational lens, which, in addition to a monopole, is characterized by the presence of a small quadrupole zonal harmonic. We obtain a new type of diffraction integral using our recently developed angular eikonal method. We evaluate this integral using the method of stationary phase. The resulting quartic equation can be solved algebraically using the method first published by Cardano in 1545. We find that the resulting solution provides a good approximation to the electromagnetic field almost everywhere in the image plane, yielding the well-known astroid caustic of the quadrupole lens. The sole exception is the immediate vicinity of the caustic boundary, where a numerical treatment of the diffraction integral yields better results. We also convolve the quartic solution with the point-spread function of a thin-lens optical telescope. We explore the direct relationship between the algebraic properties of the quartic, the geometry of the astroid caustic, and the geometry and shape of the resulting Einstein-cross that appear on the image sensor of the model telescope. This leads to improvements in numerical simulations as the quartic solution is computationally far less expensive than numerical integration.
We continue our investigation of the optical properties of the solar gravitational lens (SGL). We treat the Sun as an extended axisymmetric body and model its gravitational field using zonal harmonics. We consider a point source that is positioned at a large but finite distance from the Sun and, using our new angular eikonal method, we established the electro-magnetic (EM) field on the image plane in the focal region behind the SGL and derive the SGLs impulse response in the form of its point-spread function (PSF). The expression that we derive describes the extended Sun in all regions of interest, including the regions of strong and weak interference and the region of geometric optics. The result is in the form of a single integral with respect to the azimuthal angle of the impact parameter, covering all lensing regimes of the SGL. The same expression can be used to describe gravitational lensing by a compact axisymmetric mass distribution, characterized by small deviations from spherical symmetry. It is valid in all lensing regimes. We also derive results that describe the intensity of light observed by an imaging telescope in the focal region. We present results of numerical simulations showing the view by a telescope that moves in the image plane toward the optical axis. We consider imaging of both point and extended sources. We show that while point sources yield a number of distinct images consistent with the caustics due to zonal harmonics of a particular order (e.g., Einstein cross), extended sources always result in the formation of an Einstein ring. These results represent the most comprehensive wave-theoretical treatment of gravitational lensing in the weak gravitational field of a compact axisymmetric gravitating object.
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