Solar Gravitational Lens Research Articles

Lensing Point-spread Function of Coherent Astrophysical Sources and Nontrivial Wave Effects

Most research on astrophysical lensing has been conducted using the geometric optics framework, where there exists a clear concept of lensing images. However, wave optics effects can be important for coherent sources, e.g., pulsars, fast radio bursts, and gravitational waves observed at long wavelengths. There, the concept of lensing images needs an extension. We introduce the concept of the “lensing point-spread function” (LPSF), the smoothed flux density distribution of a coherent point source after being lensed, as a generalization of the lensing image concept at finite frequencies. The frequency-dependent LPSF captures the gradual change of the flux density distribution of the source from discrete geometric images at high frequencies to a smooth distribution at low frequencies. It complements other generalizations of lensing images, notably the imaginary images and the Lefschetz thimbles. Being a footprint of a lensing system, the LPSF is useful for theoretical studies of lensing. Using the LPSF, we identify a frequency range with nontrivial wave effects, where both geometric optics and perturbative wave optics fail, and determine this range to be ∣κ∣−1 ≲ ν ≲ 10, with κ and ν being the dimensionless lens amplitude and the reduced observing frequency, respectively. Observation of LPSFs with nontrivial wave effects requires either very close-by lenses or very large observing wavelengths. The potential possibilities are the lensing of gravitational waves, the plasma lensing of Milky Way pulsars, and lensing by the solar gravitational lens.

Imaging with a gravitational lens: the geometric view

ABSTRACT We investigate imaging point sources with a monopole gravitational lens, such as the Solar Gravitational Lens in the geometric optics limit. We compute the light amplification of the lens used in conjunction with a telescope featuring a circular aperture that is placed in the focal region of the lens, compared to the amount of light collected by the same telescope unaided by a gravitational lens. We recover an averaged point spread function that is in robust agreement with a wave-theoretical description of the lens, and can be used in practical calculations or simulations.

Imaging rotating and orbiting exoplanets with the solar gravitational lens

ABSTRACT We utilize the well-established properties of the solar gravitational lens (SGL) to consider more realistic observational scenarios. Actual exoplanets, which may be the target of an SGL observational campaign, are not stationary. Their appearance also changes in a variety of ways, including changes due to their diurnal rotation and varying illumination due to their orbital motion around their host star. The nature of the SGL is such that imaging with one telescope is accomplished with a cadence of one pixel at a time, with substantial per-pixel integration times. Therefore, capturing a single snapshot of the target planet with a realistically sized telescope is not possible. Instead, the planetary surface must be reconstructed by inverting the combined effect of the SGL’s point-spread function and temporal changes induced by the planetary dynamics. Using the Earth as a stand-in, we demonstrate practical feasibility of this approach, by simulating a dynamical system and then recovering topographic images of acceptable quality. The dynamics-induced temporal variability of the exoplanet represents an added challenge, but even in the presence of such dynamics, use of the SGL for exoplanet imaging remains feasible.

Imaging faint sources with the extended solar gravitational lens

We consider resolved imaging of faint sources with the solar gravitational lens (SGL) while treating the Sun as an extended gravitating body. We use our new diffraction integral that describes how a spherical electromagnetic wave is modified by the static gravitational field of an extended body, represented by series of multipole moments characterizing its interior mass distribution. Dominated by the solar quadrupole moment, these deviations from spherical symmetry significantly perturb the image that is projected by the Sun into its focal region, especially at solar equatorial latitudes. To study the optical properties of the quadrupole SGL, we develop an approximate solution for the point spread function of such an extended lens. We also derive semianalytical expressions to estimate signal levels from extended targets. With these tools, we study the impact of solar oblateness on imaging with the SGL. Given the small value of the solar quadrupole moment, the majority of the signal photons arriving from an extended target still appear within the image area projected by the monopole lens. However, these photons are scrambled, thus reducing the achievable signal-to-noise ratio during the image recovery process (i.e., deconvolution). We also evaluate the spectral sensitivity for high-resolution remote sensing of exoplanets with the extended SGL. We assess the impact on image quality and demonstrate that despite the adverse effects of the quadrupole moment, the SGL remains uniquely capable of delivering high-resolution imaging and spectroscopy of faint, small, and distant targets, notably terrestrial exoplanets within $\ensuremath{\sim}30--100\text{ }\text{ }\mathrm{parsec}$ from us.

Mission Architecture to Reach and Operate at the Focal Region of the Solar Gravitational Lens

We present initial results of an engineering study on the feasibility of a space mission to the focal region of the solar gravitational lens (SGL). The mission goal is to conduct exoplanet imaging operations at heliocentric distances in the range astronomical units (AU). Starting at 547.6 AU from the sun, light from an exoplanet located behind the sun is greatly amplified by the SGL. The objective is to capture this light and use it for multipixel imaging of an exoplanet up to 100 light years distant. Using a meter-class telescope, one can capture the data needed to produce images of the exoplanet with a surface resolution measured in tens of kilometers and to identify signs of habitability. The data are acquired pixel by pixel while moving an imaging spacecraft within the image. Given the long duration of the mission, decades to reach 900 AU, we address an architecture for the fastest possible transit time while reducing mission risk and overall cost. The mission architecture implements solar sailing technologies and in-space aggregation of modularized functional units to form mission capable spacecraft. The study reveals elements of such a challenging mission, but it is nevertheless found to be feasible with technologies that are either extant or in active development.

A Search for Radio Technosignatures at the Solar Gravitational Lens Targeting Alpha Centauri

Stars provide an enormous gain for interstellar communications at their gravitational focus, perhaps as part of an interstellar network. If the Sun is part of such a network, there should be probes at the gravitational foci of nearby stars. If there are probes within the solar system connected to such a network, we might detect them by intercepting transmissions from relays at these foci. Here, we demonstrate a search across a wide bandwidth for interstellar communication relays beyond the Sun’s innermost gravitational focus at 550 au using the Green Bank Telescope (GBT) and Breakthrough Listen (BL) backend. As a first target, we searched for a relay at the focus of the Alpha Centauri AB system while correcting for the parallax due to Earth’s orbit around the Sun. We searched for radio signals directed at the inner solar system from such a source in the L and S bands. Our analysis, utilizing the turboSETI software developed by BL, did not detect any signal indicative of a non-human-made artificial origin. Further analysis excluded false negatives and signals from the nearby target HD 13908. Assuming a conservative gain of 103 in the L band and roughly 4 times that in the S band, a ∼1 m directed transmitter would be detectable by our search above 7 W at 550 au or 23 W at 1000 au in the L band, and above 2 W at 550 au or 7 W at 1000 au in the S band. Finally, we discuss the application of this method to other frequencies and targets.

Spectrally resolved imaging with the solar gravitational lens

We consider the optical properties of the solar gravitational lens (SGL) treating the Sun as a massive compact body. Using our previously developed wave-optical treatment of the SGL, we convolve it with a thin-lens representing an optical telescope, and estimate the power spectral density and associated photon flux at individual pixel locations on the image sensor at the focal plane of the telescope. We also consider the solar corona, which is the dominant noise source when imaging faint objects with the SGL. We evaluate the signal-to-noise ratio at individual pixels as a function of wavelength. To block out the solar light, we contrast the use of a conventional internal coronagraph with a Lyot-stop to an external occulter (i.e., starshade). An external occulter, not being a subject to the diffraction limit of the observing telescope, makes it possible to use small telescopes (e.g., $\sim 40$~cm) for spatially and spectrally resolved imaging with the SGL in a broad range of wavelengths from optical to mid-infrared (IR) and without the substantial loss of optical throughput that is characteristic to internal devices. Mid-IR observations are especially interesting as planets are self-luminous at these wavelengths, producing a strong signal, while there is significantly less noise from the solar corona. This part of the spectrum contains numerous features of interest for exobiology and biosignature detection. We develop tools that may be used to estimate instrument requirements and devise optimal observing strategies to use the SGL for high-resolution, spectrally resolved imaging, ultimately improving our ability to confirm and study the presence of life on a distant world.

Resolved imaging of exoplanets with the solar gravitational lens

ABSTRACT We discuss the feasibility of direct multipixel imaging of exoplanets with the solar gravitational lens (SGL) in the context of a realistic deep space mission. For this, we consider an optical telescope, placed in the image plane that forms in the strong interference region of the SGL. We consider an Earth-like exoplanet located in our immediate stellar neighbourhood and model its characteristics using our own Earth. We estimate photon fluxes from such a compact, extended, resolved exoplanet. This light appears in the form of an Einstein ring around the Sun, seen through the solar corona. The solar corona background contributes a significant amount of stochastic noise and represents the main noise source for observations utilizing the SGL. We estimate the magnitude of this noise. We compute the resulting signal-to-noise ratios (SNRs) and related integration times that are needed to perform imaging measurements under realistic conditions. It is known that deconvolution, removing the blur due to the SGL’s spherical aberration substantially decreases the SNR. Our key finding is that this ‘penalty’ is significantly mitigated when sampling locations in the image plane (image pixels) remain widely spaced. Consequently, we conclude that an imaging mission is challenging but feasible, using technologies that are either already available or in active development. Under realistic conditions, high-resolution imaging of Earth-like exoplanets in our galactic neighbourhood requires only weeks or months of integration time, not years as previously thought: a high quality 1000 × 1000 pixel image of an Earth-like planet at Proxima Centauri could be obtained with SNR > 10 using approximately 14 months of integration time.

A search for monochromatic light towards the Galactic Centre

ABSTRACT A region 140 square degrees towards the Galactic Centre was searched for monochromatic optical light, both pulses shorter than 1 s and continuous emission. A novel instrument was constructed that obtains optical spectra of every point within 6 square deg every second, able to distinguish lasers from astrophysical sources. The system consists of a modified Schmidt telescope, a wedge prism over the 0.28-m aperture, and a fast CMOS camera with 9500 × 6300 pixels. During 2021, a total of 34 800 exposures were obtained and analysed for monochromatic sources, both subsecond pulses and continuous in time. No monochromatic light was found. A benchmark laser with a 10-m aperture and located 100 light years (ly) away would be detected if it had a power more than ∼60 megawatt (MW) during 1 s, and from 1000 ly away, 6000 MW is required. This non-detection of optical lasers adds to previous optical SETI non-detections from more than 5000 nearby stars of all masses, from the Solar gravitational lens focal points of Alpha Centauri, and from all-sky searches for broadband optical pulses. These non-detections, along with those of broadband pulses, constitute a growing SETI desert in the optical domain.

Integral Field Spectroscopy with the Solar Gravitational Lens

The prospect of combining integral field spectroscopy with the solar gravitational lens (SGL) to spectrally and spatially resolve the surfaces and atmospheres of extrasolar planets is investigated. The properties of hyperbolic orbits visiting the focal region of the SGL are calculated analytically, demonstrating trade-offs between departure velocity and time of arrival, as well as gravity assist maneuvers and heliocentric angular velocity. Numerical integration of the solar barycentric motion demonstrates that navigational acceleration of is needed to obtain and maintain alignment. Obtaining target ephemerides of sufficient precision is an open problem. The optical properties of an oblate gravitational lens are reviewed, including calculations of the magnification and the point-spread function that forms inside a telescope. Image formation for extended, incoherent sources is discussed when the projected image is smaller than, approximately equal to, and larger than the critical caustic. Sources of contamination that limit observational signal-to-noise ratio (S/N) are considered in detail, including the Sun, the solar corona, the host star, and potential background objects. A noise mitigation strategy of spectrally and spatially separating the light using integral field spectroscopy is emphasized. A pseudo-inverse-based image reconstruction scheme demonstrates that direct reconstruction of an Earth-like source from single measurements of the Einstein ring is possible when the critical caustic and observed S/N are sufficiently large. In this arrangement, a mission would not require multiple telescopes or navigational symmetry breaking, enabling continuous monitoring of the atmospheric composition and dynamics on other planets.

Navigating stellar wobbles for imaging with the solar gravitational lens

The solar gravitational lens (SGL) offers unique capabilities for direct high-resolution imaging of faint, distant objects, such as exoplanets. For that purpose, in the near future, a spacecraft carrying a meter-class telescope with a solar coronagraph would be placed in the focal region of the SGL. That region begins at $\ensuremath{\sim}547$ astronomical units from the Sun and occupies the immediate vicinity of the target-specific primary optical axis---the line that connects the center of the target and that of the Sun. Clearly, this axis is not at rest. It undergoes complex motion as the exoplanet orbits its host star, as that star moves with respect to the Sun, and even as the Sun itself moves with respect to the Solar System's barycenter due to the gravitational pull of planets in our Solar System. Although less prominent, other motions exist. An image of an extended object is projected by the SGL into an image plane and moves within that plane, responding to the motion of the optical axis. To sample the image, a telescope must always be on the move, following the projection, with precise knowledge of its own position with respect to the image. We consider the dominant motions that determine the position of the focal line as a function of time. We evaluate the needed navigational capability for the telescope to conduct a multiyear exoplanet imaging mission in the focal region for the SGL. We show that even in a rather conservative case, when an Earth-like exoplanet is in our immediate stellar neighborhood at $\ensuremath{\sim}10$ light years, the motion of the image is characterized by a small total acceleration that is driven primarily by the orbital motion of the exoplanet (its effect on the projected image estimated to be at the level of $\ensuremath{\sim}6\text{ }\text{ }\ensuremath{\mu}{\mathrm{m}/\mathrm{s}}^{2}$, decreasing inversely with distance to a target) and by the reflex motion of our Sun (target independent, contributing at $\ensuremath{\sim}0.2\text{ }\text{ }\ensuremath{\mu}{\mathrm{m}/\mathrm{s}}^{2}$). We discuss how the amplified light of the host star allows us to establish a local reference frame that significantly relaxes navigational requirements for the imaging operations. We conclude that the required navigation in the SGL's focal region, although complex, can be accurately modeled, and a $\ensuremath{\sim}10$-year prospective imaging mission is achievable with the already available propulsion technology.

A fast response mission to rendezvous with an interstellar object

A solar sail propelled small satellite mission concept to intercept and potentially rendezvous with newly discovered transient interstellar objects (ISOs) is described. The mission concept derives from the proposal for a technology demonstration mission for exiting the solar system at high velocity, eventually to reach the focal region of the solar gravitational lens. The ISO mission concept is to fly a solar sail toward a holding orbit around the Sun and when the ISO orbit is confirmed, target the sailcraft to reach an escape velocity of over 6 AU/year. This would permit rapid response to a new ISO discovery and an intercept within 10 AU from the Sun. Two new proven interplanetary technologies are utilized to enable such a mission: i) interplanetary smallsats, such as those demonstrated by the MarCO mission, and ii) solar sails, such as demonstrated by LightSail and IKAROS missions and developed for NEA Scout and Solar Cruiser missions. Current technology work suggests that already within this decade such a mission could fly and reach an ISO moving through the solar system. It might enable the first encounter with an ISO to allow for imaging and spectroscopy, measurements of size and mass, potentially giving a unique information about the object’s origin and composition. A similar approach may be used to allow for a sample return.

Multipole decomposition of gravitational lensing

We study gravitational lensing by a generic extended mass distribution. For that, we consider the diffraction of electromagnetic (EM) waves by an extended, weakly aspherical, gravitating object. We account for the static gravitational field of such a lens by representing its exterior potential in the most generic form, expressed via an infinite set of symmetric trace free (STF) tensor multipole mass moments. This yields the most general form of the gravitational phase shift, which allows for a comprehensive description of the optical properties of a generic gravitational lens. We found that at each order of the STF moments, the gravitational phase shift is characterized by only two parameters: a magnitude and a rotation angle that characterize the corresponding caustics, which form in the point spread function (PSF) of the lens. Both of these parameters are uniquely expressed in terms of the transverse-trace free (TT) projections of the multipole moments on the lens plane. Not only does this result simplify the development of physically consistent models of realistic lenses, it also drastically reduces the number of required parameters in the ultimate model. To help with the interpretation of the results, we established the correspondence of the gravitational phase shift expressed via the TT-projected STF multipole mass moments and its representation via spherical harmonics. For axisymmetric mass distributions, the new results are consistent with those that we obtained in previous studies. For arbitrary mass distributions, our results are novel and offer new insight into gravitational lensing by realistic astrophysical systems. These findings are discussed in the context of ongoing astrophysical gravitational lensing investigations as well as observations that are planned with the solar gravitational lens (SGL).

Wave-optical study of the Einstein cross formed by a quadrupole gravitational lens

We study imaging of point sources with a quadrupole gravitational lens while focusing on the formation and evolution of the Einstein cross formed on the image sensor of an imaging telescope. We use a new type of a diffraction integral that we developed to study generic, opaque, weakly aspherical gravitational lenses. To evaluate this integral, we use the method of stationary phase that yields a quartic equation with respect to a Cartesian projection of the observer's position vector with respect to the vector of the impact parameter. The resulting quartic equation can be solved analytically using the method first published by Cardano in 1545. We find that the resulting solution provides a good approximation of the electromagnetic (EM) 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 for the EM field on the image plane with the point-spread function of a thin lens imaging telescope. By doing so, we are able to explore the direct relationship between the algebraic properties of the quartic solution for the EM field, the geometry of the astroid caustic, and the geometry and shape of the resulting Einstein-cross that appear on the image plane of the thin lens telescope. The new quartic solution leads to significant improvements in numerical modeling as evaluation of this solution is computationally far less expensive than a direct numerical treatment of the new diffraction integral. In the case of the solar gravitational lens (SGL), the new results drastically improve the speed of numerical simulations related to sensitivity analysis performed in the context of high-resolution imaging of exoplanets.

Laser communication with Proxima and Alpha Centauri using the solar gravitational lens

ABSTRACT A search was conducted for laser signals, both sub-second pulses and continuous emission, from the regions of the sky opposite Proxima and Alpha Centauri. These regions are located at the foci of the gravitational lensing caused by the Sun, ideal for amplifying transmissions between our Solar system and those two nearest stellar neighbours. The search was conducted using two objective prism telescopes operating with exposure times of 0.25 s, enabling detection of sub-second laser pulses coming from the solar gravitational foci. During 6 months in 2020 and 2021, 88 000 exposures for Proxima Cen and 47 000 exposures for Alpha Cen were obtained. No evidence was detected of light pulses or continuous laser emission in the wavelength range of 380–950 nm. We would have detected a laser having a power of just 100 W, for a benchmark 1-m laser launcher that was diffraction-limited and located at the Sun’s gravitational focus 550 au away. To be detected, that beam must intercept Earth either by intention or, by accident, or if intended for a probe near Earth that is communicating with another one at the solar gravitational lens. These non-detections augment a previous non-detection of laser light coming directly from Proxima Centauri conducted with the HARPS spectrometer on the ESO 3.6-m telescope.

Imaging point sources with the gravitational lens of an extended sun

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.

Gravitational lensing by an extended mass distribution

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 electromagnetic field on the image plane in the focal region behind the SGL and derive the SGL 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.

Optical properties of an extended gravitational lens

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 Maxwell's 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 SGL's 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.

Image recovery with the solar gravitational lens

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 SGL's 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.

Diffraction of electromagnetic waves by an extended gravitational lens

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 Maxwell's 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.