## Abstract

As an analog model of general relativity, optics on some two-dimensional (2D) curved surfaces has received increasing attention in the past decade. Here, in light of the Huygens–Fresnel principle, we propose a theoretical frame to study light propagation along arbitrary geodesics on any 2D curved surfaces. This theory not only enables us to solve the enigma of “infinite intensity” that existed previously at artificial singularities on surfaces of revolution but also makes it possible to study light propagation on arbitrary 2D curved surfaces. Based on this theory, we investigate the effects of light propagation on a typical surface of revolution, Flamm’s paraboloid, as an example, from which one can understand the behavior of light in the curved geometry of Schwarzschild black holes. Our theory provides a convenient and powerful tool for investigations of radiation in curved space.

© 2021 Chinese Laser Press

## 1. INTRODUCTION

In general relativity (GR), spacetime is distorted in the vicinity of massive celestial bodies. Dynamics of electromagnetic (EM) waves in the context of strong gravitational fields has attracted increasing attention, ranging from wave optics [1] to gravitational lensing [2] and scattering theory [3], as well as photon rings [4], which are predicted to be ensconced in the shadow of the M87* black hole image recently published by EHT Collaboration [5]. Despite the flourishing astrophysical explorations, investigations from the perspective of optics are still rare. Because of feeble gravitational effects, measurements and verification of GR phenomena are difficult to perform unless in an astronomical scale. Therefore, researchers have proposed various analog models to study GR phenomena using tabletop equipment in laboratories [6], such as observation of spontaneous Hawking radiation in a flowing Bose–Einstein condensate [7], emulation of Schwarzschild precession with a gradient index lens [8], and mimicking gravitational lensing by a microstructured optical waveguide [9]. Another analog model is to abandon one spatial dimension and fix the time coordinate of the four-dimensional (4D) curved spacetime. In this manner, the remaining 2D spatial metric tensor can be depicted as a 2D curved surface embedded in 3D space, and the interplay between EM waves and spatial curvature can be revealed by investigating light propagation on such appropriately fabricated surfaces. Ever since this notion was put forward by Batz and Peschel [10] in 2008, various optical phenomena have been reported both theoretically [10–18] and experimentally [19–23]. Besides optics and photonics, similar studies on curved surfaces have also been extended to surface plasmon polaritons [24], acoustic topological insulators [25], and quantum particles [26].

The theory of light propagation in 2D curved space was initiated by Batz and Peschel [10], by obtaining a nonlinear Schrödinger equation on surfaces of revolution (SORs) with constant Gaussian curvature. Owing to the rotational symmetry of SORs, the curvilinear coordinates on surfaces are conveniently taken along longitudes and latitudes. This paradigm ingeniously simplifies the calculation to a great extent. However, the solution applies exclusively to propagation along the longitudinal direction, which is special among innumerable geodesics. Indeed, considering light propagation along non-longitudinal directions is more challenging, not only because of the tedious calculation of analytically solving the covariant wave equation but also the ambiguous physical images that are beyond intuitive imagination. Due to the rotational symmetry of SORs, a light beam launched tangent to a longitude recognizes an axisymmetric distribution of spatial curvature, which guarantees its propagation right along the very longitude. However, such axisymmetry does not hold true for light beams with other initial directions, whose trajectories will therefore be bent somehow. Intriguing questions naturally arise; for instance, which pathway would the light beam take and how would the curvature of the surface affect its divergence? It has also been revealed in prior studies [10,15,17,20] that the existing method for calculating light fields on closed SORs collapsed at artificial singularities (such as both the north and south poles on spherical surfaces), leading to an artificial “infinite intensity” thereat.

In this paper, we propose an alternative approach to study light propagation along arbitrary geodesics on any curved surfaces, in light of the Huygens–Fresnel principle. We assume the light wave propagates along the geodesic, the natural path of its ray counterpart in curved space. This approach not only figures out the problems mentioned above but also in a manner that refrains from using complicated mathematics. With this approach, we take a Flamm’s paraboloid, which is the 2D correspondence of the Schwarzschild metric, as an example, and we demonstrate the behaviors of both collimated and highly divergent light in such curved space. We then figure out the remaining enigma of artificial singularities in the previous method and suggest some possible schemes for experimental verification.

## 2. RESULTS AND DISCUSSION

#### A. Basic Theory

Consider an arbitrary 2D curved surface that can be fabricated by deforming a plane. The points on an arbitrary curved surface can be expressed by the 3D Cartesian coordinates as $[x,y,z=H(x,y)]$, with $x$, $y$ being the planar Cartesian coordinates and an arbitrary function $H$ marking the height difference between the curved surface and the $xy$ plane, as is sketched in Fig. 1. The corresponding metric of the curved surface is

Now let us consider the propagation of light on curved surfaces. First, we build up the coordinates on a curved surface, as is shown in Fig. 1. The optical propagation axis of a light beam is taken along the arbitrary geodesic that we are interested in, e.g., the red line on the surface. Vertical to the optical propagation axis, the input and output interfaces are taken along the orthogonal geodesics, respectively, denoted by ${\mathrm{\Sigma}}_{i}$ and ${\mathrm{\Sigma}}_{o}$ on that surface. In light of the Huygens–Fresnel principle, each point ${S}_{i}$ on the input interface ${\mathrm{\Sigma}}_{i}$ is a source of secondary spherical wavelet. The secondary wavelets emanating from different points on the initial interface interfere mutually; the superposition forms the far-field wavefront at ${P}_{o}$ on the output interface ${\mathrm{\Sigma}}_{o}$. Put mathematically, the complex amplitude at ${P}_{o}$ on ${\mathrm{\Sigma}}_{o}$ is described as

In practice, the rotational symmetry exists extensively in many celestial systems. Here we consider a special family of curved surfaces with rotational symmetry, universally known as SORs, whose metrics can be generally expressed as $\mathrm{d}{s}^{2}=[1+{(\frac{\mathrm{d}H}{\mathrm{d}r})}^{2}]\mathrm{d}{r}^{2}+{r}^{2}\mathrm{d}{\phi}^{2}$ in a polar coordinate system for convenience. Thanks to the orthogonality of the coordinates as well as the rotational symmetry, now one is able to solve Eqs. (2) and (3) analytically (for mathematical details, see Appendix A):

#### B. Light Propagation on Flamm’s Paraboloids

Next we will use the above approach to consider the light propagation on a specific SOR, the Flamm’s paraboloid, as is shown in Fig. 2(a). This interesting surface reveals the spatial curvature in the vicinity of a Schwarzschild black hole [28]. As known, the gravitational field outside the Schwarzschild radius ${r}_{s}$ of an uncharged irrotational spherical mass is described by the Schwarzschild metric:

Now we consider the propagation of a very narrow light beam with a large divergent angle launched directly toward the black hole, along the geodesic (iv) in Fig. 2(a). This is usually similar to the situation that a point-like light source is far away from a black hole. The beam is so divergent that instead of being entirely captured, only the central portion is absorbed by the event horizon, while the periphery grazes the black hole and escapes. As a result, signals can be detected at the opposite side of the light source (i.e., the forward scattering), or even be detected at the same side of the light source (i.e., the backward scattering). In practice, we calculate the distribution of the light field on the entire surface of Flamm’s paraboloid, rather than taking a specific far-field output end as we did above. Interestingly, there are always two possible geodesics from a certain point on the input interface to an arbitrary point on such Flamm’s paraboloid surface of the black hole. As an example, in Fig. 3(a), we can see that light rays can travel clockwise and anticlockwise along two different geodesics, respectively, to reach the cross and/or asterisk points in the forward and/or backward directions. The interplay of these two branches of light rays may lead to interference fringes on the forward hemi-surface, as is clearly revealed in Fig. 3(b). Such behaviors are similar to the interference characteristics shown in a recent work by Nambu *et al*. [1]. Furthermore, some portions of light rays may even return as the backward scattering, via the backward geodesics after circling around the black hole, to the vicinity of the incident source, resulting in the complicated interference patterns in Fig. 3(c). We believe such forward-scattering and backward-scattering phenomena, in spite of its feebleness, could possibly be used to obtain the structural information of black holes and detect the gravitational effects of invisible and small black holes. More detailed properties of the forward- and backward-scattering effects for light near a black hole will be further explored elsewhere. Indeed, the interaction between two branches of rays also exists in the collimated light beams investigated in the above, yet one branch dominates, and consequently no interference can be observed (see Appendix B).

#### C. Deciphering the Leftover Singularity Puzzle

In Ref. [10], an ingenious expression about the propagation of a light beam on the special SORs with constant Gaussian curvature is analytically given by solving the covariant Helmholtz equation in the longitude–latitude coordinate system under paraxial approximation. Taking the hemispherical surface in Fig. 4(a) as an example, the longitudinal coordinate $u$ along the longitudinal arc direction has the range of $u\in [0,\pi R/2]$, with $R$ being the radius of the hemisphere, while the latitudinal coordinate $v$, being the rotational angle, is within the range $[0,2\pi ]$. The arc length of latitudes is thus $2\pi R\text{\hspace{0.17em}}\mathrm{cos}(u/R)$, which varies with $u$ and vanishes at the north pole $u=\pi R/2$, resulting in a mathematical singularity. The intensity evolution of a Gaussian beam propagating along the meridian and starting from the equator follows

Fundamentally, these puzzles are consequences of mischoice of the far-field output interface and can be well solved by the method mentioned in this work. In the longitude–latitude coordinate system, both the incident and far-field interfaces are supposed to be taken along the latitudinal lines, which are actually not geodesics (except the equator). In practice, when an observer stands on the curved surface, the coordinates should be taken locally along the two orthogonal geodesics. In this situation, the propagation of a light beam is illustrated in Fig. 4(a). In particular, on the north pole, the geodesic perpendicular to the propagation axis is marked by the blue bold line, and thus the north pole is naturally a regular point instead of an artificial singularity. Furthermore, we also inspect the propagation of a light beam circling a sphere in Fig. 4(b) and find that the evolution of the beam width oscillates after each half circumference on the spherical surface. This interesting property was discovered under the condition that the propagation is along the equator via the coordinate transformation [10]. By our method, this periodic oscillation of the beam width can also be obtained when the incident Gaussian beam starts from arbitrary positions on a sphere, which is unavailable in the previous method. These results not only solve the artificial singularity enigma but are also more pragmatic in real experiments, since it is more appropriate to take an arbitrary geodesic as the incident interface when a laser beam is coupled onto a curved surface.

#### D. Possible Experimental Schemes

Our theory can be experimentally implemented both macroscopically and microscopically. In pioneering experiments, constraining light propagation on curved surfaces was realized either by total internal reflection in curved crown glass [19–21] or by a thin liquid waveguide covered on a 3D solid object [19,20]. The 3D objects with prescribed shapes can be fabricated by state-of-the-art technologies, such as high-precision diamond turning for macroscopic structures [19] and the Nanoscribe 3D laser lithography technique [22,29] in nanometric scale. Very recently, an intriguing work [30] demonstrates light propagation on thin curved soap membranes (see its supplementary video 3). This scheme provides a promising novel platform, especially when the varying thickness of the membrane, acting as an effective refractive index, could be an extra dimension for modulation. Moreover, it is proved that a curved surface of revolution is equivalent to a plane with azimuthally symmetric distribution of refractive index [31]. Therefore, an alternative pathway is to fabricate the predesigned refractive index profile on a planar surface, by, e.g., a microsphere-embedded variable-thickness polymethyl methacrylate waveguide [9,32], or through the optically induced giant Kerr effect in liquid crystal [33], with its landscape being provided by a spatial light modulator working in reflection.

## 3. CONCLUSION

In conclusion, we develop heuristics to study the propagation of light in 2D curved space, founded on the Huygens–Fresnel principle. This method is feasible when the direction of light propagation is along arbitrary geodesics on any curved surface. By this method, we study the behaviors of light beams on a Flamm’s paraboloid, which is the 2D correspondence of spatial curvature outside a Schwarzschild black hole. We investigate the evolution of Gaussian wave packets propagating along different geodesics and reveal the diverging nature of light on such curved surfaces. We also illustrate the interference patterns induced by highly divergent light sources. Finally, we point out that this method works out the remaining puzzles about the coordinate singularities in the previous theory. Our work provides a powerful tool and refreshing insights that greatly broaden the possibilities of investigations about light propagation in curved space. Exotic geodesics [34] could be used to realize lights with special properties. With the help of the proposed method, further studies could be extended to other optical effects, such as spectral properties [14,15], phase information [17], the Hanbury Brown and Twiss effect [20], geodesic lens [35], the Talbot effect [36], and acceleration radiation [37]. Moreover, the investigations about optics on Flamm’s paraboloids open up a new perspective on the radiations in the vicinity of Schwarzschild black holes and contribute to the interdisciplinary explorations of cosmology and optics.

## APPENDIX A: DERIVATION OF EQ. (5)

When written in polar coordinates, the metric of general SORs is

## APPENDIX B: SUPERPOSITION OF CLOCKWISE AND ANTICLOCKWISE GEODESICS

In Fig. 3(a), we have illustrated that there are always two geodesics from the point on the input end to an arbitrary point on Flamm’s paraboloid, one traveling in the clockwise (CW) direction (i.e., along the blue solid line to the cross point and along the yellow dashed line to the asterisk point) and the other in the anticlockwise (ACW) direction (i.e., along the purple solid line to the cross point and along the green dashed line to the asterisk point). The interaction of these two branches of geodesics leads to interference patterns on the entire surface for highly divergent light. In essence, the existence of these two branches of geodesics is a geometrical property of surface *per se*, regardless of the incident light beams. Therefore, theoretically, the intensity distribution of a collimated light beam (e.g., the ones in Fig. 2) is also a result of the superposition of CW and ACW geodesics. In Figs. 5(e1) and 5(e2), we illustrate a light beam with same initial beam width but much smaller wavelength (i.e., much longer Rayleigh distance and thus more collimated) as that in Fig. 4 as well as Figs. 5(b1) and 5(b2). It is clearly seen that at the incident-side half hemi-surface that contains the incident field, the Gaussian profile of intensity almost remains intact during propagation and the light beam barely diverges, with no interference patterns observed. At the other half hemi-surface in the forward direction, the intensity is extremely feeble, indicating that no signal exists up there, since the black hole blocks or absorbs almost all light energy. These results accord with those shown in Fig. 2, where only one geodesic is taken into consideration. As a matter of fact, when we further inspect the contributions of CW and ACW geodesics in Figs. 5(g1)–5(g3), we find that for collimated light beams, the field induced by one branch (ACW in this case) is much smaller than the other, or in other words, the contribution from the CW geodesic dominates, and thus no interference phenomenon occurs. In contrast, for highly diverging light beams, the fields induced by both CW and ACW geodesics are at a comparable level as is shown in Figs. 5(c1), 5(c2), 5(d1), and 5(d2), and therefore interference fringes present in Figs. 5(c3) and 5(d3).

## APPENDIX C: DERIVATION OF EQS. (7) AND (8)

The electric field of light on a curved surface obeys the covariant Helmholtz equation:

Taking the ansatz $\mathrm{\Phi}(u,v)={[\mathrm{cos}(\frac{u}{R})]}^{-\frac{1}{2}}\mathrm{\Psi}(u,v)$, after tedious mathematics, one has

For Eq. (C5), suppose the solution takes the form

where $\alpha (u)$ and $\beta (u)$ are functions that satisfy## Funding

Natural Science Foundation of Zhejiang Province, China (LD18A040001); National Natural Science Foundation of China (11974309, 11674284); National Key Research and Development Program of China (2017YFA0304202).

## Disclosures

The authors declare no conflicts of interest.

## Data Availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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