# Kerr spacetime geometric optics for vortex beams

Fabrizio Tamburini<EMAIL_ADDRESS>ZKM – Zentrum für Kunst und
Medien,
Lorenzstraße 19, D-76135 Karlsruhe, Germany  Fabiano Feleppa
<EMAIL_ADDRESS>Institute for Theoretical Physics, Utrecht
University, Princetonplein 5, 3584 CC Utrecht, The Netherlands  Ignazio Licata
<EMAIL_ADDRESS>Institute for Scientific Methodology (ISEM), I-90146
Palermo, Italy  School of Advanced International Studies on Theoretical and
Nonlinear Methodologies of Physics, I-70124 Bari, Italy  International
Institute for Applicable Mathematics and Information Sciences (IIAMIS), B.M.
Birla Science Centre, Adarsh Nagar, Hyderabad-500 463, India  Bo Thidé
<EMAIL_ADDRESS>Swedish Institute of Space Physics, Ångström Laboratory,
P.O. Box 537, SE-75121 Uppsala, Sweden

###### Abstract

We apply the analogy between gravitational fields and optical media in the
general relativistic geometric optics framework to describe how light can
acquire orbital angular momentum (OAM) when it traverses the gravitational
field of a massive rotating compact object and the interplay between OAM and
polarization. Kerr spacetimes are known not only to impose a gravitational
Faraday rotation on the polarization of a light beam, but also to set a
characteristic fingerprint in the orbital angular momentum distribution of the
radiation passing nearby a rotating black hole (BH). Kerr spacetime behaves
like an inhomogeneous and anisotropic medium, in which light can acquire
orbital angular momentum and spin-to-orbital angular momentum conversion can
occur, acting as a polarization and phase changing medium for the
gravitationally lensed light, as confirmed by the data analysis of M87* black
hole.

###### pacs:

97.60.Lf, 04.70.-s, 98.35.Jk, 98.62.Js)

##  I Introduction

Electromagnetic (EM) radiation is still the main carrier of information
exploited to understand our Universe, to communicate and share information
encoded in the EM conserved quantities and written in the observables of the
EM field. The classical observables of the EM field are related to the set of
the ten conserved quantities concomitant with the $10$-dimensional Poincaré
group $P_{10}(EM)$ of Noether invariants: $3$ components of the linear
momentum vector, $3$ of the pseudovector total angular momentum, $3$
components of the vector center of energy and the scalar energy. EM radiation
transports both energy and momentum. The linear momentum vector is connected
with force action, while the pseudovector total angular momentum
$\mathbf{J}=\mathbf{S}+\mathbf{L}$ is connected with torque action. The spin-
like form $\mathbf{S}$ is related to the spin angular momentum (SAM),
associated with helicity, and hence with the polarization of light. The second
form, $\mathbf{L}$, is associated with the orbital phase profile of the beam,
measured in the direction orthogonal to the propagation axis, and is also
known as orbital angular momentum (OAM). The transfer of SAM from light to a
mechanical body was first demonstrated experimentally by Beth in 1935 Beth
(1935, 1936), while the corresponding transfer of OAM has been verified only
recently Garcés-Chávez et al. (2004); Babiker et al. (1994); Andersen et al.
(2006). When a beam of light is propagating in inhomogeneous and anisotropic
media, the conserved quantity $\mathbf{J}$ can evolve in time, together with
$\mathbf{S}$ and $\mathbf{L}$, with an interchange between SAM and OAM
Marrucci et al. (2011); Puentes (2019). Only in the paraxial approximation,
when the beam is propagating in vacuum or in a homogeneous medium,
$\mathbf{S}$ and $\mathbf{L}$ can be handled as two separate variables. These
properties remain valid down to the single photon level: each individual
photon carries a quantized amount of SAM, $S=\sigma\hbar$, where $\sigma=\pm
1$, and can additionally carry OAM, whose $z$ component is quantized as well,
$L_{z}=m\hbar$, where $m=0,\pm 1,\pm 2,\ldots,\pm{}N$ Berestetskii et al.
(1982); Calvo et al. (2006); Tamburini and Vicino (2008) as confirmed
experimentally Mair et al. (2001); O’Neil et al. (2002); Leach et al. (2002,
2004).

Laguerre-Gaussian (LG) helical beams are classical examples of beams with
axial symmetry Vallone et al. (2016), characterized by inclined wavefronts
that carry a well-defined value $L_{z}=m\hbar$ of OAM per photon Schmitz et
al. (2006); Molina-Terriza et al. (2007) and harbor an EM or optical vortex
(OV) in their center of symmetry. OVs are singularities of the EM field where
the phase is not defined and the field amplitude drops to zero because of
interference, forming in the plane transverse to the beam’s propagation
direction a doughnut-shaped structure with a central dark core where the
vortex is found.

OAM of light finds practical applications in many fields of research. To give
a few examples, we cite nanotechnology Grier (2003); Shen et al. (2019),
quantum cryptography Vaziri et al. (2002); Yuan Qian and Zhao Sheng-mei
(2010); Mirhosseini et al. (2015) and telecommunication Gibson et al. (2004);
Thidé et al. (2007); Torres and Torner (2011); Tamburini et al. (2011a);
Tamburini et al. (2012) that has a vast literature with real-world
demonstrations of efficient information transfer in the radio Spinello et al.
(2015); Tamburini et al. (2015); Spinello et al. (2016); Oldoni et al. (2015)
and optical Huang et al. (2014) domains.

Nowadays OAM is attracting more and more the attention of the astronomical
community. Already in 2003, Harwit Harwit (2003) suggested to better exploit
the properties of EM waves in order to optimally extract the information there
encoded, and posed a particular attention to OAM. Only recently OAM has become
an efficient tool for astronomy as the first applications essentially
consisted in the use of the optical properties of OVs to build efficient
optical coronagraphs that can go beyond the Rayleigh limit: OVs obtained by
artificially imposing OAM to the light coming from celestial bodies can enable
the detection and imaging of extrasolar planets Swartzlander (2001, 2005); Foo
et al. (2005); Lee et al. (2006); Anzolin et al. (2008), and improve the
resolving power of any optical instrument by up to one order of magnitude
Tamburini et al. (2006); Anzolin et al. (2009).

The first remarkable result that demonstrated a clear advantage in the use of
the OAM naturally emitted by celestial bodies with respect to the standard
techniques used in astronomy is the measure of the rotation of the
supermassive black hole (BH) in the center of the galaxy Messier 87 yielding a
rotation parameter $a=0.90\pm 0.05$ with $\sim 95\%$ confidence level
Tamburini et al. (2019). This result was obtained from the analysis of the
data collected by the Event Horizon Telescope collaboration (EHT) EHT
Collaboration (2019a, b, c, d, e, f) with standard OAM techniques Torner et
al. (2005), confirming that light propagating near a rotating black hole
(Kerr) becomes twisted Tamburini et al. (2011b). The OAM analysis of the EM
waves lensed by the BH is now a valid alternative and complementary method of
investigation to the study of the shape of the BH shadow.

Theoretical studies indicate that EM waves can acquire OAM when propagating in
a structured astrophysical plasma Tamburini, F. et al. (2010), providing
information on the spatial structure and turbulence of the plasma Elias
(2008); Thidé et al. (2011); in this scenario, photons acquire an effective
Proca mass through the Anderson-Higgs mechanism Anderson (1963); Higgs (1964),
becoming Majorana quasiparticles Tamburini and Thidé (2011); Tamburini et al.
(2021). This idea was applied to relativistic astrophysics to estimate the
level of turbulence inside homogeneously distributed ultra-bright gamma ray
burst (GRB) progenitors Tamburini et al. (2017a); Tamburini, F. (2010). OAM
applied to the BH information paradox demonstrated, in agreement with the
latest Hawking’s studies on soft-hair implants that the event horizon actually
acts as a hologram, with a holographic memory that preserves the information
carried by OAM states HAWKING (1974); Hawking et al. (2017); Tamburini et al.
(2017b); Haco et al. (2018).

In this work we describe the evolution of the SAM and OAM components of a
light beam propagating in a Kerr spacetime through general relativistic
geometric optics. In certain conditions the two components $\mathbf{S}$ and
$\mathbf{L}$ of the total angular momentum invariant $\mathbf{J}$,
“disentangle” their initial superposition, modifying and transforming SAM and
OAM into each others as occurs in an inhomogeneous anisotropic medium.

##  II SAM and OAM evolution in Kerr spacetime

To describe the evolution of the angular momentum of a light beam propagating
in the spacetime of a rotating BH Kerr (1963), we use the geometric optics-
analogy of general relativity Plebanski (1960); de Felice (1971); Roy and Sen
(2019). Geometric optics in GR suggests that light, when traversing regions of
spacetime, behaves as it were propagating inside an optical medium. Following
this analogy, we can characterize the SAM component of the beam with the
gravitational Faraday effect Dehnen (1973) and study the evolution of the OAM
component of $\mathbf{J}$. Different scenarios of rotating BHs due e.g. to
modifications of the gravitational potential or modified theories of gravity
go beyond the purpose of the present work.

Theory and the results from numerical simulations suggest that spacetimes of
rotating BHs have geometries that are analogous to inhomogeneous and
anisotropic media, as these beams acquire OAM during their propagation
Tamburini et al. (2011b); Yang and Casals (2014) and change their polarization
Su and Mallett (1980). The analogies with particular optical media have been
studied in deep and this analogy was tested in laboratory experiments Galvez
and Freedman (2021) and can be in principle measurable with quantum
information techniques that go down to the quantum level Racorean (2018).

Light beams and, more in general, EM waves emitted from an accretion disk (AD)
around the rotating BH are characterized by specific fingerprints in the OAM
spectrum Torner et al. (2005) that depend on the source (the physical
properties of the AD) and on the GR effects i.e. the spacetime dragging and
gravitational lensing.

For the sake of simplicity, let us focus on the Kerr solution. In Boyer-
Lindquist coordinates the line element of the Kerr spacetime with coordinates
$(t,r,\theta,\phi)$ is given by the following quadratic form (for $G=c=1$)
Ishihara et al. (1988a)

$\displaystyle ds^{2}$
$\displaystyle=\frac{\rho^{2}}{\Delta}dr^{2}+\rho^{2}d\theta^{2}+\frac{\sin^{2}\theta}{\rho^{2}}[adt-(r^{2}+a^{2})d\phi]^{2}$
$\displaystyle\hskip 11.38092pt-\frac{\Delta}{\rho^{2}}(dt-a\sin^{2}\theta
d\phi)^{2},$ (1)

where the coefficients

$\displaystyle\rho^{2}$ $\displaystyle=r^{2}+a^{2}\cos^{2}\theta,$ (2)
$\displaystyle\Delta$ $\displaystyle=r^{2}-2Mr+a^{2}$ (3)

depend on the mass $M$ and on the angular momentum per unit mass $a\leq{}M$ of
the KBH. Because of spacetime dragging, the effects of the gravitational
lensing caused by a Kerr BH consist of image deformation and rotation of the
lensed object; these effects are characterized by a particular anamorphic
transformation, including additional features due to wavefront warping and
gravitational Berry phase Carini et al. (1992); Chandrasekhar (1992); Beckwith
and Done (2005); Feng and Lee (2001); Yang and Casals (2014).

The propagation of the polarization vector (SAM) along a null geodesic shows
that the gravitational lensing produces a gravitational Faraday rotation
Dehnen (1973) of the plane of polarization with an angle proportional to the
mass and the line-of-sight component of the angular momentum of the BH
Ishihara et al. (1988b); in addition to that, the BH can impose helical modes
to the light which depend only on the rotation parameter $a$ of the BH and not
on its mass $M$ Tamburini et al. (2011b). A deep study and analysis of the
spatial distribution of these two quantities, SAM and OAM, allows a better
characterization of the phenomenology occurring in the regions surrounding the
BH. Of course, the distribution of the polarization vector and OAM in the
observer’s plane depends also on the intrinsic properties of the source before
being lensed by the BH. The different evolution of the polarization state
(SAM) and OAM, the latter depending on the spatial distribution of the phase
of the EM wavefront, can be put in evidence with some examples. Indeed,
different polarization transformations can occur in a gravitational field that
have the same initial and final polarization states but involves geometrical
optical phase differences that are described by the Pancharatnam-Berry phase
Feng and Lee (2001) and instead induce OAM. This effect can be traced and
mathematically characterized by the analysis of the EM field of the lensed
light with a multipole expansion Mashhoon (1973). This procedure allows to
identify and characterize the OAM through its projection on a specific
orthogonal polynomial basis like Laguerre-Gaussian polynomials Molina-Terriza
(2008).

To study the transfer of OAM with the interplay between SAM and OAM we
consider first a linearly polarized electromagnetic radiation propagating in a
curved spacetime and then a circularly polarized one. In the first case, this
wave experiences a rotation of the polarization plane due to the gravitational
Faraday rotation effect. On the other hand, when circularly polarized, light
passes through a Faraday rotator and it experiences a gravitationally induced
phase shift $\Delta\phi_{G}$ that depends upon the direction of propagation
through the rotator. In this way, clockwise and counterclockwise polarized
beams experience different phase shifts.

If the source of the gravitational field is rotating, the spacetime behavior
changes dramatically. In this case, the norm of the timelike Killing field is
positive in a region, called ergosphere, that extends outside the black-hole
(BH) horizon; the asymptotic time translation Killing field
$\xi^{\alpha}=(\partial/\partial t)^{\alpha}$ becomes spacelike and an
observer in the ergosphere cannot remain stationary even if it is orbiting
outside the BH; observers in the ergosphere are forced to move in the rotation
direction of the Kerr BH (KBH).

###  II.1 Evolution of SAM and OAM in Kerr spacetime

OAM is obtained from the spatial phase distribution imprinted by the
gravitational lensing Carini et al. (1992); Tamburini et al. (2011b); Yang and
Casals (2014); Tamburini et al. (2019). OAM can be generated or modified also
by the SAM-to-OAM interchange that can occur in inhomogeneous and anisotropic
media Marrucci et al. (2011); Torres and Torner (2011).

Consider a typical case of gravitational lens where the source (s) and the
observer (o) are located far away from the rotating BH Bray (1986) and the
light passes close to the BH itself. We denote the coordinates of the light
source by $(r_{s},\theta_{s},\phi_{s})$ and the observer’s coordinates by
$(r_{o},\theta_{o},\phi_{o})$; the deflection of a light ray from $s$ to $o$
occurs in a plane that includes the KBH too. We then introduce two local
orthogonal reference frames defined by the scattering plane and the
propagation directions of the light beam from (s) to the (o), $\mathbf{k_{s}}$
and $\mathbf{k_{o}}$, respectively.

As it is well-known, Kerr geometry belongs to the class of Petrov type D
spacetimes Petrov (2000). The Walker-Penrose conserved quantity (along null
geodesics) Walker and Penrose (1970); Chandrasekhar (1992), is used to
describe the evolution of the polarization vector. The conserved complex
quantity is

$K_{\mathrm{WP}}=(A+\mathrm{i}B)(r-\mathrm{i}a\cos\theta),$ (4)

where, in turn, $A$ and $B$ are defined by

$\displaystyle A$
$\displaystyle=(k^{t}f^{r}-k^{r}f^{t})+a\sin^{2}\theta(k^{r}f^{\phi}-k^{\phi}f^{r}),$
(5) $\displaystyle B$
$\displaystyle=(r^{2}+a^{2})\sin\theta(k^{\phi}f^{\theta}-k^{\theta}f^{\phi})$
$\displaystyle\hskip 11.38092pt-a\sin\theta(k^{t}f^{\theta}-k^{\theta}f^{t}).$

In the above formulae, $k^{i}$ is the four-vector tangent to the geodesic and
$f^{i}$ the polarization four vector that can be re-defined as
$\hat{f}^{r}\equiv{}f^{r}$, $\hat{f}^{\theta}\equiv{}rf^{\theta}$, and
$\hat{f}^{\phi}\equiv{}r\sin\theta{}f^{\phi}$. In the region far away from the
KBH, the real $\Re$ and imaginary $\Im$ parts of the invariant can be
approximated as follows:

$\displaystyle\Re(K_{\mathrm{WP}})$
$\displaystyle=r(1-au)\hat{f}^{r}+\gamma\hat{f}^{\theta}-\beta\hat{f}f^{\phi},$
(6) $\displaystyle\Im(K_{\mathrm{WP}})$
$\displaystyle=-r(1-au)\hat{f}^{r}-(\beta\hat{f}^{\theta}+\gamma\hat{f}^{\phi})\frac{k^{r}}{|k^{r}|},$

respectively. Here, $u$ is a constant of motion that relates the asymptotic
behavior of $k^{i}$ near the position of the source or of the observer,
$k^{\phi}\rightarrow\ u/(r^{2}\sin^{2}\theta)$, while
$\beta=(\eta-u^{2}\cot^{2}\theta+a^{2}\cos^{2}\theta)^{1/2}k^{\theta}/|k^{\theta}|$
contains another constant of the motion, $\eta$. Finally,
$\gamma=u\csc\theta-a\sin\theta$. While the radial term $\hat{f}^{r}$ is
characterized by a constant of the motion, the quantity $r(1-au)$, depends
directly on the radial coordinate and the transformation of the other two
spatial components of the polarization vector transform from the source $(s)$
to the observer $(o)$ obtained with a linear rotation matrix $R$ whose
coefficients depend on the geometry:

$\left(\begin{array}[]{c}\hat{f}^{\theta}\\\
\hat{f}^{\phi}\end{array}\right)_{o}=R\left(\begin{array}[]{c}\hat{f}^{\theta}\\\
\hat{f}^{\phi}\end{array}\right)_{s}.$ (7)

Following Ref. Ishihara et al. (1988a), we describe the evolution of the
polarization vector $\mathbf{f}=f_{\parallel}\mathbf{n}+f_{\perp}\mathbf{h}$
in terms of the perpendicular and projected components on the plane by
transforming the polarization vector, expressed in the $(r,\theta,\phi)$
coordinates

$N=\left(\begin{array}[]{cc}h^{\theta}&h^{\phi}\\\
n^{\theta}&n^{\phi}\end{array}\right).$ (8)

The final transformation is then

$\left(\begin{array}[]{c}f_{\parallel}\\\
f_{\perp}\end{array}\right)_{o}=N_{o}RN_{s}^{-1}\left(\begin{array}[]{c}f_{\parallel}\\\
f_{\perp}\end{array}\right)_{s},$ (9)

giving the rotation of the polarization vector. If the KBH spin is not lying
in the propagation plane of the light beam, the rotation matrix of the
polarization vector is reduced to the simple form

$N_{o}RN_{s}^{-1}=\left(\begin{array}[]{cc}\cos\xi&-\sin\xi\\\
\sin\xi&\cos\xi\end{array}\right).$ (10)

Let us now consider the two reference frames associated to $(s)$ and $(o)$.
The evolution of the polarization vector is fully described in the plane
orthogonal to the propagation axis of the photons, identified by the vector
components $f_{\perp}$ and $f_{\parallel}$. For a left-circular polarized beam
($f_{\perp}=1$ and $f_{\parallel}=\mathrm{i}$) of coherent light that
traverses the Kerr geometry of the rotating gravitational lens, we similarly
find that the lensed light acquires a phase factor related to the OAM

$N_{o}RN_{s}^{-1}\left(\begin{array}[]{c}1\\\
\mathrm{i}\end{array}\right)_{s}=e^{-i\xi}\left(\begin{array}[]{c}1\\\
-\mathrm{i}\end{array}\right)_{o}.$ (11)

In analogy with SAM-to-OAM conversion process Marucci et al. (2006), the OAM
acquired by a light beam due to lensing is in our case $q=\xi/2$, which
clearly depends on the phase retardation caused by the gravitational lens
geometry as occurs in a spiral phase plate Torres and Torner (2011).

In the observer’s rest frame, the optical axis orientation is specified by an
angle $\alpha(r,\phi)\propto q\phi$, where $q$ finds a relationship to the OAM
value acquired by the circularly polarized light beam $m$ through $m=2q$.
Similarly as in Eq. (10), the Jones matrix

$N(-\alpha)\left(\begin{array}[]{cc}1&0\\\
0&-1\end{array}\right)N(\alpha)=\left(\begin{array}[]{cc}\cos 2\alpha&\sin
2\alpha\\\ \sin 2\alpha&-\cos 2\alpha\end{array}\right)$ (12)

describes the change of the polarization vector due to the presence of the
medium centered at $r=0$. A left-circular polarized EM wave
$E_{s}=E_{0}\times[1,\mathrm{i}]^{T}$, emerging from the medium, becomes
right-circularly polarized with an additional phase factor
$\exp(\mathrm{i}m\phi)$. The beam has thus been transformed into a helical
beam carrying a quantity of OAM $m=2q$. This suggests that, while a linearly
polarized light experiences gravitational Faraday rotation, a circularly
polarized beam instead can exhibit a behavior analogous to that experienced in
an inhomogeneous anisotropic medium where spin-to-orbital angular momentum
conversion occurs, a phenomenon related to the Pancharatnam-Berry geometrical
phase Marucci et al. (2006) in a q-plate.

The phase evolution of a photon in a Kerr spacetime can also be described by
using the photon wavefunction approach Tamburini and Vicino (2008); different
geodesics impose different real and imaginary values of the phase retardation,
contributing to the gravitational Faraday rotation for linearly polarized
light and for the light intensity amplification. The real phase factor depends
on the coupling of the rotation of the BH with the photon SAM Carini et al.
(1992). Furthermore, the phase evolution of photons can be also described in
terms of Wigner rotation, tetradic representations of the curved spacetime
Alsing and Stephenson (2009).

Thus, each photon has a finite and known probability of being converted from
left-circular to right-circular, inducing an asymmetry in the helicity that
depends on the mass and angular momentum of the rotating body, an effect
related to the Pancharatnam-Berry geometrical phase for the evolution of the
polarization vector. This asymmetry, which exhibits the gravitationally-
induced spin transfer from the rotating body to the scattered photons, is the
key of the spin-to-OAM conversion in a Kerr metric as no birefringence is
present in GR Mohanty and Prasanna (1995).

With the same assumptions made in Ref. Bray (1986); Ishihara et al. (1988a),
including terms up to the third order in $M/r_{\mathrm{min}}$ and
$a/r_{\mathrm{min}}$, we obtain the approximated solution to the equations of
motion of a light beam that includes the multi-imaging aspect of the
gravitational lens and the variation of the polarization. In the small-
deviations regime obtained for large values of $r$, the spatial coordinates
used to describe the Kerr geometry can be approximated by a set of spherical
polar coordinates $(r,\theta,\phi)$ in an Euclidean geometric support. For
each different light beam that draws the lensed image, the parameters $\nu$
and $\eta$ of the equations of motion can be expressed in terms of the so-
called “celestial coordinates”, denoted by $(\alpha,\beta)$, where $\alpha$ is
the apparent perpendicular distance of the image from the axis of symmetry and
$\beta$ the apparent perpendicular distance of the image from its projection
on the equatorial plane Chandrasekhar (1992). For the $i$-th light ray of the
image, the constants of motion become

$\nu\approx\frac{-r_{0}\sin\theta_{0}\alpha_{i}}{\sqrt{\alpha^{2}_{i}+\beta^{2}_{i}+(r_{0}-\beta_{i}\cot\theta_{0})^{2}}},$
(13)

and

$\sqrt{\eta}\approx\frac{\mp
r_{0}\sqrt{\beta_{i}^{2}+\alpha^{2}_{i}\cos^{2}\theta_{0}}}{\sqrt{\alpha^{2}_{i}+\beta^{2}_{i}+(r_{0}-\beta_{i}\cot\theta_{0})^{2}}}.$
(14)

The sign $\mp$ depends on the sign of the derivative
$\frac{d\theta}{dr}|_{r=r_{0}}$ and this approximation excludes first- and
higher- order terms in $Mr_{\mathrm{min}}^{2}/r_{0}^{3}$. The expression for
the distance of the closest approach $r_{\mathrm{min}}$ can then be written as
a function of the celestial coordinates.

$\displaystyle r_{\mathrm{min}}$
$\displaystyle\approx\sqrt{\nu^{2}+\eta}\left[1-\frac{M}{\sqrt{\nu^{2}+\eta}}-\frac{3M^{2}}{2(\nu^{2}+\eta)}\right.$
$\displaystyle\left.\hskip
11.38092pt-\frac{a^{2}}{2(\nu^{2}+\eta)}+\frac{a^{2}\eta}{2(\nu^{2}+\eta)^{2}}+\frac{2aM\nu}{(\nu^{2}+\eta)^{3/2}}\right].$
(15)

By converting the celestial coordinates of the $i$th light beam
$(\alpha_{i},\beta_{i})$ in polar coordinates in the sky $(l,\varphi)$
centered on the KBH position, for the i-th beam, $(l_{i},\varphi)$. Here
$\varphi$ is the angle between the vector position of the $i$th image with the
$\alpha$ axis, the real part of the phase factor acquired by photons $\xi$,
expanded at the first order in $\varphi$, becomes

$\xi(\varphi)\simeq\frac{5\pi}{4}M^{2}a\cos\theta_{0}\left[C_{1}+C_{2}\varphi\right]+O[\varphi]^{2},$
(16)

where $C_{1}$ and $C_{2}$ are constants that depend on the parameters of the
KBH and on the position of the observer,

$\displaystyle C_{1}$
$\displaystyle=\left[\frac{a^{2}r_{0}^{2}}{2}-M+\frac{l_{i}^{2}\sin^{2}\theta_{0}\sqrt{l_{i}^{2}+r_{0}^{2}}}{l_{i}^{3}r_{0}^{3}}+\frac{l_{i}r_{0}}{\sqrt{l_{i}^{2}+r_{0}^{2}}}\right.$
$\displaystyle\left.\hskip
11.38092pt-\frac{(a^{2}+3M^{2}+2am\sin\theta_{0})\sqrt{l_{i}^{2}+r_{0}^{2}}}{l_{i}r_{0}}\right]^{-3},$
(17)

and

$\displaystyle C_{2}$
$\displaystyle=3\left[\frac{(a^{2}+3M^{2})\cot\theta_{0}+2aM\cos\theta_{0}}{\sqrt{l^{2}_{i}+r^{2}_{0}}}\right.$
$\displaystyle\left.\hskip
11.38092pt+\frac{l^{2}_{i}r^{2}_{0}\cot\theta_{0}}{(l_{i}^{2}+r_{0}^{2})^{3/2}}-\frac{l^{2}_{i}\sin\theta_{0}\cos\theta_{0}}{l^{2}_{i}r^{2}_{0}\sqrt{l^{2}_{i}+r^{2}_{0}}}\right]C_{1}^{-4/3},$
(18)

where $C_{1}=O[l]^{0}$ and $C_{2}=O[l]^{-1}$, respectively. Higher order terms
in $\varphi$ have constants $C_{j}$ that decrease more rapidly with increasing
$m$, being $O[C_{j}]\ll O[l]^{-2}$; thus, they can be neglected for our
purposes. The emission from the neighborhoods of any BH is limited by the
photon sphere and by the size of the shadow Feng and Lü (2020). With simple
algebra, if one rescales the distances with respect to the photon orbit radius
$r_{g}=3M$, it is easy to find that the OAM content of the lensed light
depends on the rotation parameter $a$ only and not on the mass of the BH as
numerically and experimentally proved in Ref Tamburini et al. (2011b);
Tamburini et al. (2019).

In the weak gravitational regime the fraction of photons that experienced the
spin-to-OAM conversion is fairly small. The first term, $C_{1}$, has the same
order of magnitude as the classical gravitational Faraday rotation, while the
term that describes the OAM component, is $C_{2}=O\left[10^{-11}\right]$.

In the framework of the analogy with geometric optics in GR, a good weak
lensing approximation useful to estimate the content of OAM induced by the
rotation of the KBH can be obtained from the geometrical considerations of the
simplified Kerr lensing where the KBH clearly behaves as a holographic $q$
plate that imparts OAM onto the impinging light,

$\langle
m\rangle\simeq\frac{5\pi}{4}\frac{M^{2}a\cos\theta_{0}}{r^{3}_{\mathrm{min}}},$
(19)

where $r_{\mathrm{min}}$ is the shortest distance of the lensed null geodesic
from the BH and $\theta_{0}$ is the projection angle on the line of sight of
the component of the angular momentum of the KBH. The reason we indicate the
OAM content in Eq. (19) with $\langle m\rangle$ is because OAM has a discrete
spectrum of values. A non-integer value of OAM obtained by the gravitational
lens is given by the superposition of the discrete OAM components present in
the spiral spectrum. As a rule of thumb, from the known results in Ref.
Tamburini et al. (2011c); Tamburini et al. (2019), the OAM content, namely the
components different from the untwisted light ($m=0$) can be at the first
order given by the ratio of the $m=1$ component with that of $m=0$, viz.,

$\langle m\rangle\simeq\mathrm{height}(m=1)/\mathrm{height}(m=0).$ (20)

The same order of magnitude is obtained with the gravitomagnetic approach
Guadagnini (2002); Barbieri and Guadagnini (2004).

Fig. 1 shows the OAM value $\langle m\rangle$ acquired by a circularly
polarized light beam lensed by three different KBHs with the rotation axis
pointing to the observer’s direction ($\theta_{0}=0$) Feng and Lü (2020)
having $a=0.1$, $a=0.5$ and $a=1$, respectively and finding $\langle
m\rangle_{a=0.1}=0.014$ , $\langle m\rangle_{a=0.5}=0.015$ and $\langle
m\rangle_{a=1}=0.036$. In the abscissa is reported the distance from the event
horizon expressed in natural units ($c=G=1$) and the radius in terms of the
photon orbit radius $r_{g}=3M$. The starting point of the x coordinate is at
$r=2$, the ending point at $r=6.2$. The vertical asymptote (not displayed in
the figure) corresponds to $r=0$ where the singularity is found and the OAM
value acquired by light would tend to infinity. This is what could be observed
with a naked singularity.

Figure 1: Plot of the OAM value $\langle m\rangle$ of the radiation emitted by
a particle falling into a KBH along the direction of the rotation axis versus
the distance $2<r<6.2$ in units of $r_{g}$. The BH spin is pointing in the
direction of the observer ($\theta=0$). The quantity $\langle m\rangle$ is
given by the ratio between the $m=1$ and the $m=0$ components in the spiral
spectrum. Here are reported the values for three different angular momenta of
the Kerr black hole: $a=0.1$ with $\langle m\rangle=0.014$ (dot-dashed),
$a=0.5$ with $\langle m\rangle=0.015$ (dashed) and the extremal black hole,
$a=1$ with $\langle m\rangle=0.036$ (continuous plot).

We now verify this model with the OAM content observed in the twisted light
observed in M87*; let us assume as rotation parameter $a=0.90\pm 0.05$ and
inclination $i=17^{\circ}\pm 2^{\circ}$ (or equivalently $i=163^{\circ}\pm
2^{\circ}$ with a magnetic arrested disk) Tamburini et al. (2011c). The shadow
of M87* is $\sim 2.6$ times larger than the black hole’s Schwarzschild radius
EHT Collaboration (2019a). By using Eq. (19), we find that the averaged OAM
value $\langle m\rangle=0.0683$ observed in the experimental data from EHT
from the analysis with the TIE method for epoch 1 (days April $5$–April $6$)
corresponds to a radius $r\simeq 3.5$. The value in epoch 2 corresponds to
$r\simeq 3.4$ Tamburini et al. (2019), confirming that the OAM radiation is
originated outside the BH shadow with null geodesics traveling in the
neighborhoods of the BH. The result agrees with that obtained by computing the
phase factor of Eq. (16).

###  II.2 OAM from image rotation

We now demonstrate, with another approach, the presence of OAM from the known
phenomenon of image rotation induced by the Kerr geometry. When the source is
in very close proximity of the BH, like in accretion disks, the value of OAM
is quite significant and, for some particular cases, $m$ becomes larger than
unity. This Ansatz is based on the concept of equivalence of the curved
spacetime to a medium with a precise index of refraction that affects the
propagation of a light beam Evangelidis (1980). The effect is analogous to the
case in which the image rotation is obtained after the pencil of light has
traversed a rotating medium. More generally, rotating bodies and/or rotating
observers experience a rotational coupling of the total angular momentum that
splits into their extrinsic orbital and intrinsic spin parts. In the case of
electromagnetic radiation, the orbital coupling is known in its limiting
Doppler form, namely the Sagnac effect.

The general phenomenon of spin-rotation coupling for photons has been the
subject of investigations Courtial et al. (1998). The OAM image rotation
effect can be explained as being due to the phase shifts between the different
constituent orbital angular momentum components of the image. The azimuthal
phase term $\exp(-\mathrm{i}m\theta)$, which describes the helical phase
fronts associated with orbital angular momentum (OAM), indicates that these
modes have an $m$-fold rotational symmetry about the beam axis. We can
describe this effect in analogy to the case in which the image rotation is
obtained through a rotating medium: a rotation of the mode about this axis
through an angle $\Delta\theta$ changes the phase by
$\Delta\phi=m\Delta\theta$ and, although the image rotation is independent of
$m$, the associated phase shift will be proportional to $m$. The rotation of
the plane of linear polarization is equivalent to a phase shift
$\Delta\phi=(n_{g}-1/n_{\phi})\Omega L$, where $\Omega$ is the angular
velocity of the medium, like a glass rod, $n_{g}$ and $n_{\phi}$ are the
refractive indices corresponding to the group and phase velocities of the
light in the medium and $L$ the thickness of the “rod” Padgett et al. (2006);
Leach et al. (2008).

As the Laguerre-Gaussian modes form a complete basis set, any arbitrary image
can be formed by an appropriately weighted superposition of these modes. The
rotation of the image observed when the light is traversing a rotating medium
is related to the OAM acquired by the photons Padgett et al. (2006); Leach et
al. (2008). Similar results could be obtained when the light is transversing
the distorted geometry in the close proximity of a Kerr black hole. In
addition to the distortion caused by the gravitational lensing, calculations
show that the lensed image is also rotated by the KBH, and in particular
situations even up to a factor of $\pi/3$, when the source is directly behind
or in front of the BH Pineault and Roeder (1977a, b).

In our case, from simple geometrical considerations both for an accretion disk
and for a lensed beam by a Kerr BH, the thickness of the ideal rod can vary in
the interval $0<L<2r$. A reasonable approximation to estimate the order of
magnitude of the OAM emitted is to assume $L\simeq r/2$.

In a particular geometrical formulation of the Kerr metric with cylindrical
coordinates Su and Mallett (1980), it was demonstrated that the light behaves
like traversing an anisotropic medium. There, the direction of energy
propagation given by the Poynting vector S does not coincide with that of the
wave vector k. This means that the group velocity $v_{g}$, associated to S,
and the phase velocity $v_{\phi}$, associated with k do not point in the same
direction. The fields E, D and H are never collinear in this medium. Letting
$g_{\alpha}=-g_{0\alpha}/g_{00}$ and $h=g_{00}$, the three-dimensional spatial
metric tensor becomes
$\gamma_{\alpha\beta}=-g_{\alpha\beta}+hg_{\alpha}g_{\beta}$ Landau and Lifšic
(1980). The fields can be written as
$D_{\alpha}=E_{\alpha}/\sqrt{h}+g^{\beta}H_{\alpha\beta}$,
$B^{\alpha\beta}=H^{\alpha\beta}/\sqrt{h}+g^{\beta}E^{\alpha}-g^{\alpha}E^{\beta}$,
and we have also
$B^{\alpha}=-1/(2\sqrt{\gamma})e^{\alpha\beta\gamma}B_{\beta\gamma}$; the
relations between the fields and the Poyinting and wave vectors are, as usual,
$\textbf{k}=\textbf{D}\times\textbf{B}$ and
$\textbf{S}=\textbf{E}\times\textbf{H}$.

The analogy with a rotating glass rod, of course different for each null-
geodesic, follows from the Hamilton-Jacobi formalism and Fermat’s principle.
The value of the refractive index $n$ for the Kerr metric, as a function of
the rotation parameter $a$, obtained with a post-Newtonian approximation Iorio
(2003), is

$\displaystyle n$
$\displaystyle\simeq\frac{1}{\sqrt{1-\frac{\tilde{r}}{\rho^{2}}}}\left[1+\frac{5a^{2}r\sin^{2}\theta_{0}}{\tilde{r}\left(\rho^{2}-\tilde{r}\right)\left(1+\sqrt{1-\frac{4a^{2}}{r^{2}_{g}}}\right)^{2}}\right],$
(21)

with $\tilde{r}=2M$. To obtain just an order of magnitude of the phase change
$\Delta\phi$, we approximate the region nearby the rotating BH by a rod with
the thickness of the order of the gravitational radius, rotating with the
angular velocity of the KBH at a radius $r^{\prime}$ (expressed in units of
the gravitational radius), namely

$\Delta\phi=am\left(n-\frac{1}{n}\right)r^{\prime},$ (22)

for an external observer $r^{\prime}>r_{sh}$, considering that the main
contributions of OAM is generated in the regions close to the BH; in certain
cases $r^{\prime}$ can be even smaller than the radii of rotating and
counterrotating photon circular orbits (see e.g. ref. Bardeen et al. (1972)).

We can see, from the results obtained with M87*, that this very simple model
of a “rotating rod” can give a good description of the Kerr spacetime as an
optic medium, confirming the validity of the GR geometric optics approach also
with twisted light. For M87*, from Eq. (21), the refraction index for $r=3.5$
is $n=1.16$ and at the shadow’s radius, $r_{sh}=2.6$, one finds $n_{sh}=1.28$.
From this one can easily calculate the “thickness” of the ideal rotating rod,
finding $L\sim r/2$ and confirming our initial assumptions.

##  III conclusions

In this work we discussed the evolution and conversion of the angular momentum
of light beams and photons from the spin to the OAM component when they
traverse the region of a rotating black hole. As in optical inhomogeneous and
anisotropic media, Kerr lensing can generate helical modes of light that
depend on the geometry and dynamics of the lensing body. Moreover, different
polarization transformations with the same initial and final polarization
states involve geometrical optical phase differences described by the
Pancharatnam-Berry phases, a principle that can be extended to a more general
form of wave fronts shaped by polarization transformations where OAM and
optical vorticity are included.

With this simplified approach we found that the orbital angular momentum
quantity $m$ acquired by a light beam lensed by a Kerr black hole can exceed
unity, carrying a quantity $m\hbar$ of orbital angular momentum per each
photon present in the beam. This result finds agreement with the values of $m$
found with numerical simulations and confirmed experimentally with the
analysis of the electromagnetic waves received from the neighborhood of the
black hole in M87* with radio telescopes.

According to Ref. Zhang et al. (2015), one obtains the OAM $m$ from the
intensity, the gradient of the spiral spatial phase distribution (obtained
from the Stokes parameters $V$ and $U$) and the variation of the state of the
polarization in space at each point in the observational plane of the
asymptotic observer (o).

For the sake of completeness only, one can in principle obtain higher values
of Faraday rotation on the order of $\Delta\phi_{G}\sim\pi$ with the approach
described in Ref. Su and Mallett (1980), implying a relevant contribution of
the components in the OAM spectrum with $m>1$ following the analogy with the
phase modifying device in Ref. Marucci et al. (2006). This occurs when the
sources are located in the neighborhoods of the event horizon or when light
rays, lensed from infinity, travel very close to the BH event horizon, a
scenario expected when the BH is fast-rotating with $a\sim 1$ or when the BH
is a fast spinner. The results obtained so far suggest that a relevant
presence of very high order components in the OAM spectrum could reveal the
presence of naked singularities.

Of course, the evolution of SAM and OAM and SAM-to-OAM transfer is in
principle valid also for other fields crossing the gravitational field of the
KBH. In particular, neutrino fields and gravitational waves may reveal the
hidden latest evolution stages of a supernova or a GRB during its collapse
when the forming KBH is setting its rotation, in an attempt to preserve the
total angular momentum while interacting with the surrounding matter and
fields. Helicity asymmetry and OAM can in fact be induced in both these fields
Barbieri and Guadagnini (2005).

## Acknowledgements

The authors thank Dennis Durairaj for his suggestions.

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