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 [1]. Light rays obey null geodesics in curved spacetime and they are deflected by gravitational potential of lens objects. In weak gravitational field with thin lens approximation, a path of a light ray obeys so called lens equation for gravitational lensing and many analysis concerning the gravitational lensing effects are carried out based on this equation. Especially, we can obtain images of source objects by solving the lens equation using a ray tracing method. As a path of light ray is derived as the high frequency limit of electromagnetic wave, wave effects of gravitational lensing become important when the wavelength is not so much smaller than the size of lens objects and in such a situation, we must take into account of wave effects. For example, when we consider gravitational wave is scattered by gravitational lens objects, the wave effect gives significant impact on the amplification factor of intensity for waves [2-4]. Another example that wave effects must be taken into account is direct detection of black holes via imaging their shadows [5,6]. The apparent angular size of black hole shadows are so small that their detectability depends crucially on the angular resolution of telescopes, that is determined by diffraction limit of image formation system. Thus, for successful detection of black hole shadows, it is important to investigate wave effects on images of black holes.

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 [7] discussed the position of images by a spherical gravitational lens evaluating the Poynting flux of scattered wave at an observer. They claimed that there is a disagreement between wave optics and geometrical optics concerning the position of double images of a point source. But they have not presented complete understanding of image formation. In wave optics, image formations are understood as a diffraction effect by image forming devices such as a convex lens. The process of image formations can be expressed as a Fourier transformation of incident waves by imaging devices. In this paper, we consider image formation in gravitational lensing using wave optics and aim to understand how images by gravitational lensing are obtained in terms of waves. For this purpose, we adopt the diffraction theory of image formation in wave optics [8], which explains image formation in optical systems in terms of diffraction of waves. This paper is organized as follows. In Section 2, we review gravitational lensing using the Fresnel-Kirchhoff diffraction formula. In Section 3, we introduce a convex lens as an image formation device and apply it to the gravitational lensing system. Section 4 is devoted to summary. We use units in which in this paper.

We review the basic formalism of gravitational lensing based on wave optics [1]. In this paper, we do not consider polarization of waves and treat scalar waves as a model for electromagnetic waves. Let us consider waves propagating under the influence of the gravitational potential of a lens object. The background metric is assumed to be

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 (Figure 1). The wave is emitted by a point source, scattered by the gravitational potential of the lens object and reaches the observer. We assume the wave scattering occurs in a small spatial region around the lens object and outside of this region, the wave propagates in a flat space. With the assumptions of the eikonal and the thin lens approximation, the Fresnel-Kirchhoff diffraction formula provides the following amplitude of the wave at the observer [1,3]

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) (Figure 2). To produce these images, we have assumed an extended source with Gaussian distribution of intensity.

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 (Figure 3). For, the asymptotic formula for the cofluent geometric function yields

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 [8].

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 [9]. As is well known, waves incident to a rotating black hole are amplified by the superradiance [10] due to dragging of spacetimes. This effect enables waves to extract the rotation energy of black holes. On the other hand, it is known that particles can also extract the rotation energy of black holes via so called Penrose process. By investigating images of scattered waves by a rotating black hole, we expect to find out new aspect or interpretation of phenomena associated with superradiance in connection with the Penrose process.

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.