Diffraction of electromagnetic waves by an extended gravitational lens

We continue our study of the optical properties of the solar gravitational lens. 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 of the lens, thus affecting its optical properties. The new wave-theoretical solution allows the study of gravitational lensing by a realistic lens that possesses an arbitrary number of gravitational multipoles. This angular eikonal method represents an improved treatment of realistic gravitational lensing. It may be used for a wave-optical description of many astrophysical lenses.

Figure 1

The different optical regions of the SGL (from [11]) with the strong interference region formed beyond 547.8 AU.

Figure 2

The basis vectors used to characterize the vector deflection angle. ${\mathbf{e}}_b$ and $\mathbf{k}$ are in the plane spanned by the incident light ray and the center of the lens; ${\mathbf{e}}_{{\varphi }_{\xi }}$ is normal to this plane. The multipole moments change the amount by which the incident ray is deflected compared to the effect of the monopole (dashed arrow), but also lift the ray out of the plane spanned by the incident ray and the center of the lens.

Figure 3

Even and odd caustics representing individual contributions of the multipoles of a gravitational field to the PSF of the extended axisymmetric gravitational lens, obtained through numerical integration of $\mathrm{PSF}=|B(\mathbf{x}){|}^2$ with $B(\mathbf{x})$ from (141). Images of a point source formed in the image plane of the lens. From top left, clockwise: (a) $J_2$, $J_4$, $J_6$ and $J_8$; (b) monopole, $J_3$, $J_5$ and $J_7$. For the odd-numbered caustics, a change in sign flips the image in the north-south direction.

Figure 4

Interaction between caustics and the effects of sign, calculated by numerically integrating $|B(\mathbf{x}){|}^2$ using (141). Top row depicts the effect of $J_3$, distorting the $J_2$ astroid starting with a negative value similar in magnitude to $J_2$, going through 0 and reaching a positive value. Bottom row depicts the effect of $J_4$ on $J_2$ in a similar fashion. These images demonstrate that the sign of the $J_3$ caustic reverses its vertical, “north-south” orientation, whereas the sign of the $J_4$ caustic determines if it is the astroid’s vertical or horizontal pair of cusps that are “split” as the astroid is stretched in the horizontal vs vertical direction.

Figure 5

The PSF of the SGL with multipole gravitational moments, obtained by integrating $|B(\mathbf{x}){|}^2$ using (141), using realistic solar parameters. These examples show light from a $\lambda =1\mu \mathrm{m}$ point source, projected by the SGL to 650 AU from the Sun. Left: $\sin {\beta }_s=0.1$ (i.e., ${\beta }_s\sim 5.74°$ from the solar axis of rotation) in an $8\times 8$ meter area; at a resolution of 4 mm, fine details due to diffraction are visible both inside and outside the caustic boundary. Right: $\sin {\beta }_s=0.387$ (${\beta }_s\sim 22.78°$) in a $120\times 120\mathrm{m}$ area, at 6 cm resolution.

Figure 6

The PSF of the SGL in color, evaluated in multiple wavelengths between 400 and 675 nm in 25 nm increments, each assigned an approximate RGB color. The parameters used are $\sin {\beta }_s=0.05$ (${\beta }_s\sim 2.87°$) at 650 AU from the Sun, ${\varphi }_s=30°$; a $2\times 2$ meter area is depicted at 1 mm resolution. Whereas a rainbow pattern is visible in many parts of the image, the cusps are achromatic white, indicating that their position and appearance is not wavelength dependent.