# Theory of light propagation in arbitrary two-dimensional curved space

Chenni Xu  Li-Gang Wang<EMAIL_ADDRESS>Zhejiang Province Key Laboratory of
Quantum Technology and Device, Department of Physics, Zhejiang University,
Hangzhou, 310027, Zhejiang, China

(July 13, 2021)

###### Abstract

As an analog model of general relativity, optics on some two-dimensional (2D)
curved surfaces has been increasingly paid attention to in the past decade.
Here, in light of 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”
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.

††preprint: APS/123-QED

##  I 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], gravitational lensing [2], scattering theory [3], as well as
photon rings [4] which are predicted to ensconce 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 analogue
models to study GR phenomena by table-top equipments in laboratory [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], mimicking gravitational lensing by a microstructured optical
waveguide [9]. Another analogue model is to abandon one spatial dimension and
fix 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, 11, 12, 13, 14, 15, 16, 17, 18] and experimentally
[19, 20, 21, 22, 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 et
al. [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 longitudinal direction, which is a special one 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 convariant 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
doesn’t 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 surface affect its divergence? Besides, it has also been revealed
in prior studies [10, 20, 15, 17] that the existed 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 Huygens-Fresnel
principle. We assume the light wave propagates along 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 us from
complicated mathematics. Using this approach, we take a Flamm’s paraboloid,
which is the 2D correspondence of Schwarzschild metric, as an example, and
demonstrate the behaviors of both collimated and highly divergent light in
such curved space. At last, we figure out the remaining enigma of artificial
singularities in the previous method, and suggest some possible schemes for
experimental verification.

##  II RESULTS AND DISCUSSION

###  II.1 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 $\left[x,y,z=H(x,y)\right]$, with $x,y$ being the
planar Cartesian coordinates and an arbitrary function $H$ marking the height
difference between the curved surface and the $x-y$ plane, as is sketched in
Fig. 1. The corresponding metric of the curved surface is

$\displaystyle ds^{2}=$ $\displaystyle\,g_{ij}dx^{i}dx^{j}$ $\displaystyle=$
$\displaystyle\left[1+\left(\frac{\partial H}{\partial
x}\right)^{2}\right]dx^{2}+\left[1+\left(\frac{\partial H}{\partial
y}\right)^{2}\right]dy^{2}$ $\displaystyle+2\frac{\partial H}{\partial
x}\frac{\partial H}{\partial y}dxdy,$ (1)

where the third term indicates that the coordinate system we choose to
parametrize the surface is not orthogonal on the curved surface. On curved
surfaces, light rays propagate along the so-called geodesics, which are the
counterparts of the straight lines in flat space. The geodesic equation is
given by
$\frac{d^{2}x^{\sigma}}{ds^{2}}+\Gamma_{\mu\nu}^{\sigma}\frac{dx^{\mu}}{ds}\frac{dx^{\nu}}{ds}=0$,
where $\Gamma_{\mu\nu}^{\sigma}=\frac{1}{2}g^{\sigma\rho}\left(\frac{\partial
g_{\rho\mu}}{\partial x^{\nu}}+\frac{\partial g_{\rho\nu}}{\partial
x^{\mu}}-\frac{\partial g_{\mu\nu}}{\partial x^{\rho}}\right)$ are Christoffel
connections, $g^{\sigma\rho}$ are the elements of the inverse of the metric
tensor $\mathbf{g}$, and Einstein summation convention is applied, with
$x^{\sigma},x^{\mu},x^{\nu}$ running through $x,y$. Therefore, one can track a
light ray on a curved surface by solving the following equations

$\frac{d^{2}x}{ds^{2}}+\Gamma_{xx}^{x}\left(\frac{dx}{ds}\right)^{2}+\Gamma_{yy}^{x}\left(\frac{dy}{ds}\right)^{2}+2\Gamma_{xy}^{x}\frac{dx}{ds}\frac{dy}{ds}=0,$
(2)
$\frac{d^{2}y}{ds^{2}}+\Gamma_{xx}^{y}\left(\frac{dx}{ds}\right)^{2}+\Gamma_{yy}^{y}\left(\frac{dy}{ds}\right)^{2}+2\Gamma_{xy}^{y}\frac{dx}{ds}\frac{dy}{ds}=0.$
(3)

Generally, it is difficult to acquire the analytical solution to this equation
set, unless some extra properties of surfaces, such as symmetries, are
present. Usually numerical methods, such as the Runge-Kutta method, can be
utilized to solve the above equations as long as the step length meets the
accuracy requirements.

Figure 1: Schematic of a 2D curved surface generated from the planar Cartesian
coordinates $(x,y)$, here as an example with the height of the surface
$z=H(x,y)=\sin{x}\cos{y}$. The red solid line denotes an arbitrary geodesic as
the propagation axis of a light beam on this curved surface. $\Sigma_{i}$ and
$\Sigma_{o}$ are the two geodesics locally vertical to the propagation axis as
the input and output interfaces, respectively. The gray-dark lines are the
shortest geodesics from the points on $\Sigma_{i}$ to the orange-color point
on $\Sigma_{o}$ and one of the angles between these shortest geodesics and the
propagation axis is denoted by $\theta$ in the inset figure.  Figure 2: (a)
(Left) Sketch of a Flamm’s paraboloid, and (Right) the output-field intensity
distribution at different propagation distance $d$ for the cases of a well-
collimated Gaussian beam launched from Point A (the incident end) to Points
B1, B2, and B3 (different output ends) along the curve (ii). The curves
(i)-(iv) with different colors on the surface show four typical geodesics
along which a light beam propagates. (b) Magnitudes of the obliquity factor
$K$ between the input and output ends under different values of $d=110$mm and
$280$mm. (c)(d) Changes of the beam width $\sigma(d)$ of light along the
geodesics (i) and (iii) in (a), respectively. The insets in (c) and (d) show
the intensity evolution along the geodesics (i) and (iii), respectively, and
the near-horizontal dot lines in (c) and (d) denote the changes of $\sigma(d)$
in flat space. (e)(f) The dependence of the output $\sigma(d)$ on the initial
$\sigma_{0}$ along the four geodesics in (a) with the fixed value of
$d=220$mm, when the input end is locating differently at (e) $r=200$ mm and
(f) $r=300$ mm. Other parameters are $r_{s}=20$ mm, $\lambda=7\times 10^{-5}$
m, $r(\Sigma_{i})=200$ mm, $\sigma_{0}=10$ mm.

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, for example, 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 $\Sigma_{i}$ and
$\Sigma_{o}$ on that surface. In light of Huygens-Fresnel principle, each
point $S_{i}$ on the input interface $\Sigma_{i}$ is a source of secondary
spherical wavelet. The secondary wavelets emanating from different points on
the initial interface interfere mutually, whose superposition forms the far-
field wavefront at $P_{o}$ on the output interface $\Sigma_{o}$. Put it
mathematically, the complex amplitude at $P_{o}$ on $\Sigma_{o}$ is described
as

$\Phi_{o}(P_{o})=\sqrt{\frac{1}{i\lambda}}\int\limits_{\Sigma_{i}}\Phi_{i}(S_{i})\frac{e^{ikL(S_{i},P_{o})}}{L(S_{i},P_{o})}K(S_{i},P_{o})d\mathbf{l},$
(4)

where $\Phi_{i}(S_{i})$ and $\Phi_{o}(P_{o})$ are, respectively, the incident
and output complex fields at the input and output ends, $k=2\pi/\lambda$ is
the wavenumber with wavelength $\lambda$, and $d\mathbf{l}$ denotes the
primary wave source on the incident end $\Sigma_{i}$, which now is essentially
1D. Here, $K=\cos\theta$ is the obliquity factor with $\theta$ being the angle
between the propagation axis and the geodesic connecting any input point
$S_{i}$ on $\Sigma_{i}$ and the output point $P_{o}$ on $\Sigma_{o}$, as is
shown in the inset of Fig. 1. In Fig. 2(b), it shows that this obliquity
factor can be reasonably taken as $1$ when the propagation distance $d$ is one
order of magnitude larger than the transverse dimension. Here, we emphasize
again that both the incident and output ends, $\Sigma_{i}$ and $\Sigma_{o}$,
should be taken along the geodesics which are locally orthogonal to the
propagation optical axis, thus the points on $\Sigma_{i}$ and $\Sigma_{o}$ are
$S_{i}=(x,y,z)$ and $P_{o}=(x^{\prime},y^{\prime},z^{\prime})$, obeying Eqs.
(2) and (3). The function $L(S_{i},P_{o})$ represents the eikonal function
which is essentially the length of the geodesic connecting $S_{i}$ on
$\Sigma_{i}$ and $P_{o}$ on $\Sigma_{o}$, obtained by $L=\int ds$.
Technically, except for few special surfaces, the analytical expression of $L$
is scarcely available, leading to the difficulty in the integral in Eq. (4).
Therefore in practice, for a certain point $P_{o}$ on $\Sigma_{o}$, we also
calculate the geodesic lengths directly, according to Eqs. (2) and (3),
connecting $P_{o}$ and hundreds of discrete points $S_{i}$ on $\Sigma_{i}$,
which are deviated from the optical propagation axis as shown in Fig. 1. Thus
one can obtain the exact field distribution of light along $\Sigma_{o}$.

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
$ds^{2}=\left[1+\left(\frac{dH}{dr}\right)^{2}\right]dr^{2}+r^{2}d\varphi^{2}$
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),

$d\varphi=\pm\frac{\kappa}{r\sqrt{r^{2}-\kappa^{2}}}\sqrt{1+\left(\frac{dH}{dr}\right)^{2}}dr,$
(5)

where
$\kappa=r_{\text{initial}}^{2}\left(\frac{d\varphi}{ds}\right)_{\text{initial}}$
is a constant determined by initial conditions, and the sign of $\pm$
corresponds to two different trajectories according to the sign of the initial
condition $\frac{dr}{ds}$.

###  II.2 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 [27]. As is known, the gravitational field outside
the Schwarzschild radius $r_{s}$ of an uncharged irrotational spherical mass
is described by the Schwarzschild metric

$\displaystyle ds^{2}=$
$\displaystyle-\left(1-\frac{r_{s}}{r}\right)c^{2}dt^{2}+\left(1-\frac{r_{s}}{r}\right)^{-1}dr^{2}$
$\displaystyle+r^{2}d\psi^{2}+r^{2}\sin^{2}\psi d\varphi^{2}.$ (6)

With its spherical symmetry, the equatorial slice is taken (i.e.,
$\psi=\pi/2$) without loss of generality, and the remnant of spatial part of
Eq. (6) establishes a SOR with $H(r)=\pm 2\sqrt{r_{s}(r-r_{s})}$ for
$r>r_{s}$. As is shown in Fig. 2(a), first we consider the propagation of a
well-collimated Gaussian wavepacket launched along four different geodesics.
The Gaussian profile is well maintained at an arbitrary output end when the
beam is a little bit away from the event horizon of the black hole. For
example, when the beam propagates along the curve (ii) (the purple curve), the
beam width changes little from A to B1 since the light is far away from the
black hole ($r>>r_{s}$), while it increases fast as the light approaches the
black hole, such as near the points B2 and B3. The intensity profiles at these
points are shown in the right side of Fig. 2(a). In this situation, we can
define the beam width $\sigma$ of light as the full width at its half-maximum
of the intensity profile and track its evolution along the propagation. The
changes of the beam width along the red and green geodesics in Fig. 2(a) are
demonstrated in Figs. 2(c) and (d), respectively, along with their intensity
evolution in the corresponding inset figures. The propagation distance $d$
along the propagation axis is normalized by the Rayleigh distance $z_{r}$ of
light in flat space. It is seen that the light beams in both of these cases
diverge rapidly even within a short distance compared to the cases in flat
space. This tells us that the divergence of light beams on such curved surface
is greatly amplified due to the large spatial curvature generated from the
strong gravitational field near the black hole. The diverging nature of light
beams on Flamm’s paraboloid is further revealed in Figs. 2(e) and 2(f), where
the variation of $\sigma(d)$ at a certain output end versus the initial beam
width $\sigma_{0}$ along the four geodesics in Fig. 2(a) are illustrated. We
can see that the Gaussian beams with small $\sigma_{0}$ are subject to strong
divergence, under the same $d$, due to the experience of the strong spatial
curvature, and the turning points in these four case are slightly different.
These effects are actually related to the strength of negative spatial
curvature [10, 14].

Figure 3: (a) Sketch of two geodesics from a certain point (red color) on the
input to an arbitrary point (black cross or black asterisk) on surface. (c-d)
Intensity distributions (denoted by the common logarithm of intensity
$I=\left|\Phi_{o}\right|^{2}$) in the green and blue regions of (a) are shown
in (b) and (c), respectively, with magnitude being normalized by the maximal
intensity of the incident light. Note that the intensity distribution in the
gray region of (a) is not shown in (c) in case the intense field near the
input end obscures the remaining region in (c). Other parameters are
$r_{s}=15$ mm, $\lambda=1.5$ cm, $r(\Sigma_{i})=150$ mm, $\sigma_{0}=1$ mm.

Now we consider the propagation of a very narrow light beam with a large
divergent angle launched directly towards 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 complicated interference patterns in Fig.
3(c). We believe such forward scattering and backward scattering phenomena, in
spite of their 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).

###  II.3 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. Take a hemispherical surface in Fig. 4(a)
as example, the longitudinal coordinate $u$ along the longitudinal arc
direction has the range of $u\in\left[0,\pi R/2\right]$ with $R$ being the
radius of hemisphere, while the latitudinal coordinate $v$, being the
rotational angle, is within the range $\left[0,2\pi\right]$. The arclength of
latitudes is thus $2\pi R\cos\left(u/R\right)$, 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

$\displaystyle I(u,v)$
$\displaystyle=\frac{1}{\cos\left(\frac{u}{R}\right)\left[1+\left.R^{2}\tan^{2}\left(\frac{u}{R}\right)\right/z_{r}^{2}\right]^{\frac{1}{2}}}$
$\displaystyle\times\exp\left[-\frac{kz_{r}R^{2}v^{2}}{R^{2}\tan^{2}\left(\frac{u}{R}\right)+z_{r}^{2}}\right],$
(7)

where $z_{r}=k\sigma_{0}^{2}/2$ is the Rayleigh distance in flat space, and
accordingly the beam width obeys

$\sigma(u)=\sigma_{0}\sqrt{\cos^{2}\left(\frac{u}{R}\right)+\frac{R^{2}}{z_{r}^{2}}\sin^{2}\left(\frac{u}{R}\right)}$
(8)

where $\sigma_{0}$ is the initial beam width (for the detailed derivation, see
Appendix C). Apparently the intensity is infinite at $u=\pi R/2$, yet the beam
width is not vanishing at the singularity. Such singularity won’t occur when
the light beam propagates along the equator, which, however, should be exactly
identical to the propagation along a meridian, owing to the perfect symmetry
of a sphere. Clearly, such singular behaviors along a meridian are artificial.

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). Especially, on the north pole, the geodesic
perpendicular to the propagation axis is marked by the blue bold line, 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 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 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.

###  II.4 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, 20, 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 Nanoscribe 3D laser lithography technique [22,
28] in nanometric scale. Very recently, an intriguing work [29] demonstrates
light propagation on thin curved soap membranes (see its supplementary video
3). This scheme provides a novel promising platform, especially when the
varying thickness of membrane, acting as 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 azumuthally symmetric
distribution of refractive index [30]. Therefore, an alternative pathway is to
fabricate the predesigned refractive index profile on a planer surface, by,
for example, a microsphere-embedded variable-thickness polymethyl methacrylate
waveguide [9, 31], or through the optically-induced giant Kerr effect in
liquid crystal [32], with its landscape being provided by a spatial light
modulator working in reflection.

Figure 4: (a) The evolution of a light beam along a meridian of a hemisphere,
with radius $R$, from the equator. The intensity profile of the light beam at
the north pole along the geodesic (the blue line) is shown in the inset. (b)
The intensity evolution of a light beam circling around a spherical surface
along the direction (denoted by the orange arrow), starting from the geodesic
(denoted by the blue solid line), which starts from $r(\Sigma_{i})=0.6R$,
$\varphi(\Sigma_{i})=\pi$ on the bottom projected plane. (c) Changes of beam
width along the propagation on the spherical surface of (b). Other parameters
are $R=50$ mm, $\lambda=5\times 10^{-4}$ m, $\sigma_{0}=2$ mm.

##  III CONCLUSION

In conclusion, we develop heuristics to study the propagation of light in 2D
curved space, founded on Huygens-Fresnel principle. This method is feasible
when the direction of light propagation is along arbitrary geodesics on any
curved surfaces. 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
wavepackets propagating along different geodesics, and reveal the diverging
nature of light on such curved surface. 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 which greatly broaden the possibilities of investigations about light
propagation in curved space. Exotic geodesics [33] could be utilized 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], Hanbury Brown and Twiss effect
[20], geodesic lens [34], Talbot effect [35] and acceleration radiation [36].
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 surfaces of
revolution is

$ds^{2}=\left[1+\left(\frac{dH}{dr}\right)^{2}\right]dr^{2}+r^{2}d\varphi^{2}.$
(9)

Therefore, the geodesic equations take the form

$\frac{d^{2}r}{ds^{2}}+\frac{\frac{dH}{dr}}{1+\left(\frac{dH}{dr}\right)^{2}}\frac{d^{2}H}{dr^{2}}\left(\frac{dr}{ds}\right)^{2}-\frac{r}{1+\left(\frac{dH}{dr}\right)^{2}}\left(\frac{d\varphi}{ds}\right)^{2}=0,$
(10)
$\frac{d^{2}\varphi}{ds^{2}}+\frac{2}{r}\frac{dr}{ds}\frac{d\varphi}{ds}=0.$
(11)

Eq. (11) can be solved as

$\frac{d\varphi}{ds}=\frac{\kappa}{r^{2}},$ (12)

where
$\kappa=r_{\text{initial}}^{2}\left(\frac{d\varphi}{ds}\right)_{\text{initial}}$
is an integration constant whose value is determined by initial conditions of
the geodesic. Besides, since Eq. (9) is basically a spatial metric (i.e.,
$ds>0$), one can divide both sides by $ds^{2}$, and readily have

$\frac{dr}{ds}=\pm\frac{1}{\sqrt{1+\left(\frac{dH}{dr}\right)^{2}}}\sqrt{1-\frac{\kappa^{2}}{r^{2}}}.$
(13)

Here $+(-)$ is taken when
$\left(\frac{dr}{ds}\right)_{\text{initial}}>\left(<\right)0$, therefore $\pm$
corresponds to two different geodesics. Eq. (13) can also be obtained by
solving Eq. (10) with Eq. (12), by the method of constant variation. At last,
after dividing Eq. (12) by Eq. (13), we have Eq. (5) in the main text,

$\frac{d\varphi}{dr}=\pm\frac{\kappa}{r^{2}\sqrt{1-\frac{\kappa^{2}}{r^{2}}}}\sqrt{1+\left(\frac{dH}{dr}\right)^{2}}.$
(14)

## APPENDIX B: SUPERPOSITION OF CLOCKWISE (CW) AND ANTICLOCKWISE (ACW)
GEODESICS

Figure 5: (a) Sketch of a Flamm’s paraboloid. (b1)(b2) Intensity distribution
on the incident (b1) and opposite (b2) hemi-surface with wavelength
$\lambda=1.5\times 10^{-2}$ m. These two subfigures are same as Figs. 4(b) and
(c). (c1)(c2)(d1)(d2) Intensity of the field induced exclusively by CW
(c1)(d1) and ACW (c2)(d2) geodesics along a certain latitude [white dashed
lines in (b1) and (b2)] on the opposite hemi-surface (c1)(c2) and incident
hemi-surface (d1)(d2). Due to symmetry, only a quarter of surface is plotted,
with $\varphi$ labeled in (a). (c3)(d3) Intensity of the field after
superimposing the complex amplitudes induced by CW and ACW geodesics. (e1)(e2)
Intensity distribution of a light beam with same parameters except for
wavelength $\lambda=1\times 10^{-5}$ m, which is therfore less divergent.
Inset of (e2) is the zoom of the area in the blue dashed frame.
(f1)(f2)(f3)(g1)(g2)(g3) Intensity profiles along white dashed latitudes in
(e1) and (e2).

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 CW direction (i.e., along the blue solid line to the cross point
and along the yellow dashed line to the asterisk point) and another in 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 (for example, the ones in Fig. 2) is also a result of
the superposition of CW and ACW geodesics. In Figs. 5(e1) and (e2), we
illustrate a light beam with same initial beam width but much smaller
wavelength (that is, much longer Rayleigh distance and thus more collimated)
as that in Fig. 4 as well as Figs. 5(b1) and (b2). It is clearly seen that at
the incident-side half hemi-surface which 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. While at
another 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 that 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)-(g3), we find out that for collimated light beams, the field
induced by one branch (ACW in this case) is way smaller than another, or in
other words, the contribution from 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-c2) and Figs. 5(d1-d2), and therefore interference
fringes present in Figs. 5(c3) and (d3).

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

The electric field of light on curveds surface obeys the covariant Helmholtz
equation

$\frac{1}{\sqrt{g}}\partial_{i}\left(\sqrt{g}g^{ij}\partial_{j}\Phi\right)+k^{2}\Phi=0,$
(15)

where $k$ is the wave number, $g^{ij}$ is the element of the inverse matrix of
metric tensor $\mathbf{g}$, and $g$ is the determinant of $g_{ij}$. To
construct a spherical surface in Fig. 4, we take the longitudinal arc length
as the longitudinal coordinate $u$, and the rotational angle as the
latitudinal coordinate $v$, the spherical surface can thus be defined by
$r_{\text{ROR}}(u)=R\cos\left(\frac{u}{R}\right)$, with $r_{\text{ROR}}$ being
radius of revolution and $R$ being the radius of sphere. With the position
vector $\mathbf{\rho}=\left[R\cos\left(\frac{u}{R}\right)\cos
v,R\cos\left(\frac{u}{R}\right)\sin v,R\sin\left(\frac{u}{R}\right)\right]$,
one is able to calculate the metric by
$g_{ij}=\frac{\partial\mathbf{\rho}}{\partial
x^{i}}\cdot\frac{\partial\mathbf{\rho}}{\partial x^{j}}$, where $i,j=1,2$ and
$x^{1}=u$, $x^{2}=v$. Therefore, the wave equation, Eq. (15), on spherical
surfaces is expressed as

$\frac{\partial^{2}\Phi}{\partial
u^{2}}-\frac{\tan\left(\frac{u}{R}\right)}{R}\frac{\partial\Phi}{\partial
u}+\frac{1}{R^{2}\cos^{2}\left(\frac{u}{R}\right)}\frac{\partial^{2}\Phi}{\partial
v^{2}}+k^{2}\Phi=0,$ (16)

with metric $g_{11}=1$, $g_{22}=R^{2}\cos^{2}\left(\frac{u}{R}\right)$.

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

$\frac{\partial^{2}\Psi}{\partial
u^{2}}+\frac{1}{R^{2}\cos^{2}\left(\frac{u}{R}\right)}\frac{\partial^{2}\Psi}{\partial
v^{2}}+k_{\text{eff}}^{2}\Psi=0,$ (17)

with $k_{\text{eff}}^{2}=k^{2}+\Delta$ and
$\Delta=\frac{1}{4R^{2}}\left[1+\cos^{-2}\left(\frac{u}{R}\right)\right]$. We
assume $\Psi(u,v)=\Xi(u,v)\exp\left(ik_{\text{eff}}u\right)$, with paraxial
approximation

$\frac{\partial^{2}\Xi}{\partial u^{2}}\ll 2k\frac{\partial\Xi}{\partial
u}\text{,}$ (18)

being carried out, and subsequently let
$\Xi(u,v)=\phi(u,v)\exp\left[\frac{i}{2k}\int\Delta(u^{\prime})du^{\prime}\right]$,
we eventually reach

$2ik\frac{\partial\phi}{\partial
u}+\frac{1}{R^{2}\cos^{2}\left(\frac{u}{R}\right)}\frac{\partial^{2}\phi}{\partial
v^{2}}=0.$ (19)

For Eq. (19), suppose the solution takes the form

$\phi(u,v)=\exp\left[i\alpha(u)+\frac{ikv^{2}}{2\beta(u)}\right],$ (20)

where $\alpha(u)$ and $\beta(u)$ are functions which satisfy

$-2k\frac{\partial\alpha(u)}{\partial
u}+\frac{1}{R^{2}\cos^{2}\left(\frac{u}{R}\right)}\frac{ik}{\beta(u)}=0,$ (21)
$\frac{\partial\beta(u)}{\partial
u}-\frac{1}{R^{2}\cos^{2}\left(\frac{u}{R}\right)}=0.$ (22)

Here we suppose that the light launches at the equator and is in a Gaussian
profile with the initial beam width $\sigma_{0}$, i.e.,
$\phi(u=0,v)=\exp\left[-\frac{R^{2}v^{2}}{\sigma_{0}^{2}}\right]$, Eqs. (21)
and (22) can be readily solved as

$\beta(u)=\frac{1}{R}\tan\left(\frac{u}{R}\right)-\frac{iz_{r}}{R^{2}},$ (23)
$\alpha(u)=i\left\\{\frac{\ln\left[\frac{z_{r}^{2}+R^{2}\tan^{2}\left(\frac{u}{R}\right)}{z_{r}^{2}}\right]}{4}+\frac{i}{2}\arctan\left[\frac{R\tan\left(\frac{u}{R}\right)}{z_{r}}\right]\right\\},$
(24)

with $z_{r}=k\sigma_{0}^{2}/2$. Therefore we eventually have the solution of
Eq. (15)

$\displaystyle\Phi(u,v)$ $\displaystyle=$
$\displaystyle\left[\cos\left(\frac{u}{R}\right)\right]^{-\frac{1}{2}}\left[\frac{z_{r}^{2}+R^{2}\tan^{2}\left(\frac{u}{R}\right)}{z_{r}^{2}}\right]^{-\frac{1}{4}}$
(25)
$\displaystyle\times\exp\left\\{-\frac{kz_{r}R^{2}v^{2}}{2\left[z_{r}^{2}+R^{2}\tan^{2}\left(\frac{u}{R}\right)\right]}\right\\}$
$\displaystyle\times\exp\left\\{-\frac{i}{2}\arctan\left[\frac{R\tan\left(\frac{u}{R}\right)}{z_{r}}\right]\right\\}$
$\displaystyle\times\exp\left[\frac{ikR^{2}v^{2}}{2}\frac{R\tan\left(\frac{u}{R}\right)}{R^{2}\tan^{2}\left(\frac{u}{R}\right)+z_{r}^{2}}\right]$
$\displaystyle\times\exp\left[\frac{i}{2k}\int\Delta(u^{\prime})du^{\prime}\right]\exp\left[iku\right],$

from which we further have the intensity

$\displaystyle I(u,v)$ $\displaystyle=$
$\displaystyle\Phi^{\ast}(u,v)\Phi(u,v)$ (26) $\displaystyle=$
$\displaystyle\left[\cos\left(\frac{u}{R}\right)\right]^{-1}\left[1+\frac{R^{2}\tan^{2}\left(\frac{u}{R}\right)}{z_{r}^{2}}\right]^{-\frac{1}{2}}$
$\displaystyle\times\exp\left[-\frac{kz_{r}R^{2}v^{2}}{z_{r}^{2}+R^{2}\tan^{2}\left(\frac{u}{R}\right)}\right]$

and beam width

$\sigma(u)=\sigma_{0}\sqrt{\cos^{2}\left(\frac{u}{R}\right)+\frac{R^{2}}{z_{r}^{2}}\sin^{2}\left(\frac{u}{R}\right)}.$
(27)

###### Acknowledgements.

Zhejiang Provincial Natural Science Foundation of China (No. LD18A040001), the
National Natural Science Foundation of China (NSFC) (No. 11974309 and
11674284), and National Key Research and Development Program of China (No.
2017YFA0304202).

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# IRF: Interactive Recommendation through Dialogue

Oznur Alkan, Massimiliano Mattetti, Elizabeth M. Daly, Adi Botea, Inge
Vejsbjerg  oalkan2, massimiliano.mattetti, elizabeth.daly, adibotea,
<EMAIL_ADDRESS>IBM ResearchDublinIreland

(2019)

###### Abstract.

Recent research focuses beyond recommendation accuracy, towards human factors
that influence the acceptance of recommendations, such as user satisfaction,
trust, transparency and sense of control. We present a generic interactive
recommender framework that can add interaction functionalities to non-
interactive recommender systems. We take advantage of dialogue systems to
interact with the user and we design a middleware layer to provide the
interaction functions, such as providing explanations for the recommendations,
managing users’ preferences learnt from dialogue, preference elicitation and
refining recommendations based on learnt preferences.

Recommender System, Interaction Mechanism, Pref. Elicitation

††copyright: acmcopyright††journalyear: 2019††conference: Thirteenth ACM
Conference on Recommender Systems; September 16–20, 2019; Copenhagen,
Denmark††booktitle: Thirteenth ACM Conference on Recommender Systems (RecSys
’19), September 16–20, 2019, Copenhagen, Denmark††doi:
10.1145/3298689.3346966††isbn: 978-1-4503-6243-6/19/09††ccs: Human-centered
computing HCI design and evaluation methods††ccs: Human-centered computing
Natural language interfaces††ccs: Information systems Recommender systems

##  1\. Introduction

Most recommender systems offer very limited or no means to inform the
recommender that its assumptions (e.g., about user preferences) are incorrect
or outdated. However, users appreciate being more actively involved in the
recommendation process, and their feedback can lead to better recommendations.

We present a generic Interactive Recommender Framework (IRF) that: 1\. can
leverage an existing non-interactive recommender and turn it into an
interactive recommender 2. implements the modes of interaction with the user.
To tackle the first problem, IRF is designed to be easily deployed on top of
any existing non-interactive recommender, with a minimal configuration.
Specifically, IRF can consume any recommender that aligns to the required API
specifications, and the endpoints are designed based on (García and Bellogín,
2018). To handle the second problem, IRF implements different types of
interactions to acquire preferences from the user in a context where the user
is motivated to give them, as well as to facilitate the exploration of the
domain and the elicitation of the user’s preferences. To achieve this, IRF
implements the following interaction mechanisms:  presenting recommendations;
explaining why an item is recommended; presenting the user profile; presenting
item details; allowing users to provide their preferences on the feature or
item level; and asking a preference elicitation question.

##  2\. System Overview

Data. Data stored in the backend have 4 main types: user_profile,
item_profile, rec_list, and user_preferences. The first three are at the core
of any recommender system and allow the configuration of required generic API
endpoints. user_preferences is used within IRF to store the preferences for
features learnt from the conversation. user_profile is required to contain a
history field, which is a list of ¡item, score, timestamp¿ triples where score
and timestamp are optional. It can contain further optional information
depending on the underlying recommender service. If provided, they are sent to
the recommender as part of the request. item_profile includes a list of
features for the categories of the underlying domain.

External Services. IRF relies on the following external services: 1\.
Recommender Service 2. Item Data Service 3. User Data Service. We minimize the
assumptions about the external services. IRF expects the endpoints specified
in Table 1. These endpoints are designed following REST principles (García and
Bellogín, 2018). Some endpoints are optional, such as $user/update$, called if
a notification should be sent to the user data service when the user provides
a new preference.

Table 1. External Services - API endpoint details URL |  Required |  Parameters |  Returns |  Description
---|---|---|---|---
recommend/get/ |  required |  user_profile |  rec_list |  returns recommendations
user/get/uid |  required |  user_id |  user_profile |  returns a user_profile
user/update |  optional |  user_profile |  - |  updates a user_profile when a new item preference is mentioned
item/get/iid |  required |  item_id |  item_profile |  returns an item_profile
item/desc/iid |  optional |  item_id |  item description text |  returns a description text for the item_profile

Solution Components. IRF has 2 main components: Dialogue Manager (DM) and
Middleware (MW).

DM is responsible for coordinating and managing the conversation with the user
through 1. analyzing the user’s utterances, 2. calling the external services
or MW based on the type of the interaction, 3. presenting the final response
to the user. DM manages the conversation with a dialogue plan created with AI
planning as described in (Botea et al., 2019). It uses Watson Assistant
(WA),111https://www.ibm.com/watson/developercloud/conversation.html an
existing service that assigns to every user utterance an intent and zero or
more entity–value pairs. The workspace file, which includes all the
entity–value pairs should be provided to IRF to configure the WA. For response
generation, a configurable messages file is kept. The messages file saves
sentence structures, which can possibly have place-holders for the inputs that
will be set by the DM before presenting that response to the user.

MW uses the following components to convert a non-interactive recommender into
an interactive one.

Post Processor (PP). The preferences learnt during conversation should be
reflected back to the recommendations immediately. To achieve this, whenever a
recommendation request is received from the user, the following actions are
executed: 1. DM calls MW with $user\\_profile$ and $user\\_preferences$. 2. MW
calls external recommender service with $user\\_profile$ to receive an initial
set of recommendations, 3. MW calls the post processor to rerank this initial
list using the $user\\_preferences$. Step 3 is performed by calculating a
weighted similarity score between the $user\\_preferences$ and each item
within the recommendation list using cosine similarity measure. Final score is
calculated by taking a weighted average of the initial recommender score and
the preference similarity score.

User Profiling and Preferences Manager (UPM). UPM differentiates between a
temporary and a permanent user profile. The temporary profile amplifies the
preferences learnt during the conversation and is created at the start of the
session based on the permanent profile. It gets updated whenever the user
states new preferences for features or preference for an item during a
conversation session. At the end of a session the temporary profile is merged
with the permanent profile by replaying the individual preferences expressed
in the dialogue so the newly learnt information still impacts the permanent
profile but to a lesser extent controlled by configured weights. The next time
the permanent profile is retrieved, all values are decayed based on how long
it has been since the user profile was updated to reflect changes in the
user’s interests over time.

Explanation Generation (EG). EG generates a justification text for every
recommended item for which an explanation is not already provided. It builds a
TF-IDF model (Lops et al., 2011) on top of the recommendation domain when the
IRF is first initialized. Furthermore, it builds an explanation through
calculating the similarity between the $item\\_profile$, $user\\_profile$ and
$user\\_preferences$ vectors that are built based on the TF-IDF model. As a
result, explanation generation returns a list of common features between the
item and the user profile and their associated similarity scores.

Similarly, for presenting the user profile, IRF uses the union of the
preferences from user_preferences and the user_profile objects.

Question Generation (QG). QG formulates a preference elicitation question to
rank and filter the list of candidates. We use a feature selection method
based on Information Gain (Yang and Pedersen, 1997). Assume that $k$ is the
number of distinct features in the domain, and each item is seen as a
$k$-tuple. We compute the information gain for each feature in the space of
the recommended items, and return the one with the highest value.

To set up IRF, the following files should be updated: 1\. configuration file,
2. workspace file that contains the entity types and values and 3. messages
file that includes the specific utterances to use while responding the end-
user. The configuration file configures the system properties such as the
details of the external services, the number of recommendations to present to
the user, etc. Workspace and the messages files are used by DM, as described
in previous paragraphs.

##  3\. System Demonstration

Figure 1. Sample Screenshots of the IRF prototype interface.

To validate IRF’s compatibility with different recommender algorithms and
domains,we build two prototypes. One uses the MovieLens
100k222https://grouplens.org/datasets/movielens/latest/ dataset and a state of
art content based recommender algorithm (Lops et al., 2011). The second uses
the Career Builder job recommendation333https://www.kaggle.com/c/job-
recommendation dataset with a sequential pattern mining based recommender
algorithm (Yap et al., 2012). From IRF’s perspective, the only difference
between the two systems are the setup files described earlier. Demonstration
of the demo for interactive
movie444https://www.youtube.com/watch?v=GwHyzLNVhNQ and
job555https://www.youtube.com/watch?v=wcgWAy5xoRw recommenders are available
on YouTube. See Figure 1 for sample screenshots.

##  4\. Conclusion and Future Work

We presented a framework for adding interaction capabilities to any existing
non-interactive recommender solution which is independent of the underlying
domain. A promising future direction is to design and test different
preference elicitation strategies and allow more complex interactions with the
user.

## References

  * (1)
  * Botea et al. (2019) Adi Botea, Christian Muise, Shubham Agarwal, Oznur Alkan, Ondrej Bajgar, Elizabeth Daly, Akihiro Kishimoto, Luis Lastras, Radu Marinescu, Josef Ondrej, Pablo Pedemonte, and Miroslav Vodolan. 2019\.  Generating Dialogue Agents via Automated Planning. In _The Second AAAI Workshop on Reasoning and Learning for Human-Machine Dialogues (DEEP-DIAL 2019)_.
  * García and Bellogín (2018) Iván García and Alejandro Bellogín. 2018.  Towards an Open, Collaborative REST API for Recommender Systems. In _Proceedings of the 12th ACM Conference on Recommender Systems_ _(RecSys ’18)_. ACM, 504–505.
  * Lops et al. (2011) Pasquale Lops, Marco de Gemmis, and Giovanni Semeraro. 2011\.  _Content-based Recommender Systems: State of the Art and Trends_.  73–105.
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