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We investigate image formations in gravitational lensing systems using wave optics. Applying the Fresnel-Kirchhoff diffraction formula to waves scattered by a gravitational potential of a lens object, we demonstrate how images of source objects are obtained directly from wave functions without using a lens equation for gravitational lensing. As an example of image formation in gravitational lensing, images of a point source by a point mass gravitational lens are presented. These images reduce to those obtained by a ray tracing method in the geometric optics limit.

Gravitational lensing is one of predictions of Einstein’s general theory of relativity and many samples of images caused by gravitational lensing have been obtained observationally [

Although interference and diffraction of waves by gravitational lensing has been discussed in connection with amplification of gravitational waves, a little was discussed about how images by gravitational lensing are obtained based on wave optics. For electromagnetic wave, E. Herlt and H. Stephani [

We review the basic formalism of gravitational lensing based on wave optics [

where is the gravitational potential of the lens object with the condition. The scalar wave propagation in this curved spacetime is described by the following wave equation:

and for a monochromatic wave with the angular frequency,

where is the flat space Laplacian.

We show the configuration of the gravitational lensing system considering here (

where is the effective path length (eikonal) along a path from the source position to the observer position via a point on the lens plane

and we assume that and. A constant represents the intensity of a point source. The two dimensional gravitational potential is introduced by

Then the wave amplitude at the observer can be written as [1,2]

where is the wave amplitude at the observer in the absence of the gravitational potential:

is the path length along a straight path from to. The amplification factor is given by the following form of a diffraction integral

where is the Fermat potential along a path from the source position to the observer position via a point on the lens plane. The first term in is the difference of the geometric time delay between a straight path from the source to the observer and a deflected path. The second term is the time delay due to the gravitational potential of the lens object. Now we introduce the following dimensionless variables:

where we choose as

is the mass of the gravitational source, and represent the Einstein radius and the Einstein angle, respectively. Using these dimensionless variables,

In the geometrical optics limit, the diffraction integral (13) can be evaluated around the stationary points of the phase function in the integrand. The stationary points are determined by the solution of the following equation:

This is the lens equation for gravitational lensing and determines the location of the image for given source position. As the specific model of gravitational lensing, we consider a point mass as a gravitational source. In this case, the two dimensional gravitational potential is

and the deflection angle is given by

For, the solution of the lens Equation (15) is

and represents the Einstein ring with the apparent angular radius defined by (12). We show an example of images obtained as solutions of the lens Equation (15) (

The wave property is obtained by evaluating the diffraction integral (13). For a point mass lens potential (16), the integral can be obtained exactly

On the observer plane, an interference pattern appears (

Using this formula, near, the distance between adjacent fringes of the interference pattern is

This fringe pattern is interpreted as interference between double images of a point source by the gravitational lensing. The question we raise in this paper is how the interference pattern on the observer plane is related to the images of gravitational lensing in the geometrical optics limit. The wave amplitude on the observer plane does not make the image of the source and we have to transform the wave function to extract images. To answer this question, we introduce a “telescope” in the gravitational lensing system and simulate observation of a star (a point source) using the telescope. With this setup, it is possible to understand how images of a source are formed in the framework of wave optics.

To establish relation between the interference pattern of the wave and the images of the source in the gravitational lensing system, we first consider an image formation system composed of a single convex lens and review how images of source objects appear in the framework of wave optics [

Let us is the incident wave from a point source in front of a thin convex lens and is the transmitted wave by the lens (

where is called a lens transformation function. The action of a convex lens is to modify the phase of the incident wave. For a point source placed at (front focal point), the incident wave and the transmitted wave are

where we have used assuming. Thus, a convex lens converts spherical wave fronts to plane wave fronts.

Using this action of a convex lens for the incident wave and the transmitted wave, we can demonstrate the image formation by a convex lens in the framework of wave optics. Let us consider the configuration of the lens system shown in

where is the path length from a point on the object plane to a point on the lens plane and we have assumed. The amplitude of the wave just behind the lens is given by the relation (22)

where is the aperture function of the lens defined by for and for. represents a radius of the lens. With the assumption, the amplitude of the wave on the plane behind the lens is

For a value of satisfying the following relation (the lens equation for a convex thin lens),

the wave amplitude becomes

For limit, the Bessel function in (25) becomes the delta function and we obtains the following wave amplitude on:

Thus, a magnified image of the source field appears on the plane. This reproduces the result of image formation in geometric optics; we have shown that an inverted images with magnification of a source object appears on satisfying the lens Equation (24).

If we do not take limit, due to the diffraction effect, an image of a point source has finite size on the image plane called the Airy disk [

This value determines the resolving power of image formation system. For two point sources at , their separation on the image plane is. To resolve them, their separation must be larger than the size of the Airy disk:

The lefthand side of this inequality is the angular separation of the sources and determines the resolving power of the image formation system.

As we have observed that a convex lens can be a device for image formation in wave optics, we combine it with a gravitational lensing system and obtain images by gravitational lensing. We consider a configuration of the gravitational lens system shown in

This equation is the same as (7). After passing through the convex lens, the wave amplitude on the image plane is given by

where denotes the aperture of the convex lens. Using dimensionless variables, the wave amplitude on the image plane is

If we choose the location of the image plane $z_2$ to satisfy the following “lens equation” for a convex lens,

then the wave amplitude on the image plane becomes

where. Thus the wave amplitude on the image plane is the Fourier transform of the amplification factor that gives the interference fringe pattern. Under the geometrical optics limit, integral in the amplification factor (13) can be approximated by the WKB form

where is the solution of the lens equation

We have assumed that the aperture of the convex lens is sufficiently smaller than the size of the gravitational lensing system and holds. Then, the wave amplitude on the image plane is

For limit (large lens aperture limit or high frequency limit), we obtains

and the image of the point source appears at the following location on the image plane determined by the lens Equation (33):

Equations (35) and (36) reproduce the same result of image formation in the geometrical optics (ray tracing) in terms of the wave optics. This is what we aim to clarify in this paper. If the lens Equation (33) has multiple solutions, the wave amplitude on the image plane becomes

where are constants.

As an example of image formation in a gravitational lensing system using wave optics, we present the wave optical images of a point source by the gravitational lensing of a point mass (

In each image, we can observe concentric interference pattern which is caused by finite size of the lens aperture and this is not intrinsic feature of the gravitational lensing system. We can also observe radial non-concentric

patterns. They are caused by interference between double images and represent the intrinsic feature of the gravitational lensing system. For case that corresponds to the Einstein ring in the geometrical optics limit, we can observe a bright spot at the center of the ring, which is the result of constructive interference and does not appear in geometric optics. For sufficiently large values of, the wave amplitude at the observer coincides with the result obtained by geometric optics. It is possible to estimate analytically the intensity distribution of the Einstein ring using the formula (20):

The intensity of the image has a peak at (

We investigated image formation in gravitational lensing system based on wave optics. Instead of using the ray tracing method, we obtained images directly from wave functions at the observer without using the lens equation of gravitational lensing. For this purpose, we introduced a “telescope” with a single convex thin lens, which acts as a Fourier transformer for waves at the observer. The analysis in this paper relates the wave amplitude and images of the gravitational lensing directly. In the geometric optics limit of waves, images by lensing systems are

obtained by a lens equation that determines paths of each light rays. As light rays are trajectories of massless test particles (photon), expressing image in terms of wave is to express particle motion in terms of waves.

As an application and extension of analysis presented in this paper, we plan to investigate gravitational lensing by a black hole and obtain wave optical images of black holes. This subject is related to observation of black hole Shadows [5,6]. As the apparent angular sizes of black hole shadows are so small, the diffraction effect on images are crucial to resolve black hole shadows in observation using radio interferometer. For SgrA^{*}, which is the black hole candidate at Galactic center, the apparent angular size of its shadow is estimated to be arc seconds and this value is the largest among black hole candidates. For a sub-mm VLBI with a baseline length, using Equation (28), the condition to resolve the shadow becomes and this requirement shows the possibility to detect the black hole shadow of SgrA^{*} using the present day technology of VLBI telescope. Thus, analysis of black hole shadows based on wave optics is an important task to evaluate detectability of shadows and determination of black hole parameters via imaging of black holes.

The topic of wave optical image formation in black hole spacetimes belongs to a classical problem of wave scattering in black hole spacetimes [

This work was supported in part by the JSPS Grantin-Aid for Scientific Research (C) (23540297). The author thanks all member of “Black Hole Horizon Project Meeting” in which the preliminary version of this paper was presented.