# Gravitational lensing in the presence of plasma scattering with application
to Fast Radio Bursts

Pawan Kumar1 & Paz Beniamini2,3
1Department of Astronomy, University of Texas at Austin, Austin, TX 78712, USA
2Department of Natural Sciences, The Open University of Israel, P.O Box 808,
Ra’anana 4353701, Israel
3Astrophysics Research Center of the Open university (ARCO), The Open
University of Israel, P.O Box 808, Ra’anana 4353701, Israel
pk<EMAIL_ADDRESS>
###### Abstract

We describe how gravitational lensing of fast radio bursts (FRBs) is affected
by a plasma screen in the vicinity of the lens or somewhere between the source
and the observer. Wave passage through a turbulent medium affects
gravitational image magnification, lensing probability (particularly for
strong magnification events), and the time delay between images. The
magnification is suppressed because of the broadening of the angular size of
the source due to scattering by the plasma. The time delay between images is
modified as the result of different dispersion measure (DM) along photon
trajectories for different images. Each of the image lightcurve is also
broadened due to wave scattering so that the images could have distinct
temporal profiles. The first two effects are most severe for stellar and sub-
stellar mass lens, and the last one (scatter broadening) for lenses and plasma
screens at cosmological distances from the source/observer. This could limit
the use of FRBs to measure their cosmic abundance. On the other hand, when the
time delay between images is large, such that the lightcurve of a transient
source has two or more well separated peaks, the different DMs along the wave
paths of different images can probe density fluctuations in the IGM on scales
$\lesssim 10^{-6}$ rad and explore the patchy reionization history of the
universe using lensed FRBs at high redshifts. Different rotation measure (RM)
along two image paths can convert linearly polarized radiation from a source
to partial circular polarization.

###### keywords:

fast radio bursts – stars: neutron – radio continuum: transients –
gravitational lensing – ISM: structure

††pagerange: Gravitational lensing in the presence of plasma scattering with
application to Fast Radio Bursts–LABEL:lastpage

##  1 Introduction

The probability of strong lensing of a compact source at redshift larger than
one, to magnification $>\mu$, by an intervening galaxy is $p(>\mu)\sim
0.3\Omega_{\rm gal}/\mu^{2}$ for $z\gtrsim 2$ and $P(>\mu)\approx\Omega_{\rm
gal}z^{2}/(4\mu^{2})$ for $z\ll 1$ (Narayan & Bartelmann, 1996). ; where
$\Omega_{\rm gal}$ is the mean mass density in galaxies divided by the
critical mass density. Strong lensing occurs when a source has multiple images
and typically (depending on the mass density in the lens) requires $\mu\gtrsim
1$. For a galaxy of mass $\sim 10^{11}M_{\odot}$ at a Gpc distance from both
the source and the observer, the angular separation between the images is of
order a few arc-seconds, and the travel time difference of order $\sim 10$
days. For $\mu\lower
2.0pt\hbox{$\,\buildrel\scriptstyle>\over{\scriptstyle\sim}\,$}1$, the
corresponding lensing probability is $\sim 6{\rm x}10^{-3}(\Omega_{11}/0.02)$
(where $\Omega_{11}$ is $\Omega_{\rm gal}$ for galaxies of mass $\sim
10^{11}M_{\odot}$). Thus, in an FRB survey of 104 sources, we expect of order
60 lenses by intervening galaxies. In addition, considering that the
population of FRB-repeaters is about 20% of the entire FRB population, about
12 repeating FRBs in the survey should be lensed. The expected number of FRB
lenses is small. However, considering that the duration of FRBs is typically a
few milli-seconds, one can determine the travel time difference along multiple
paths to better than about a ms or about one part in a billion. This
remarkable accuracy, which is better than any other class of astronomical
object, has led a number of people e.g., (Eichler, 2017; Zitrin & Eichler,
2018; Li et al., 2018; Liu et al., 2019; Wucknitz et al., 2021; Connor & Ravi,
2022; Leung et al., 2022), to suggest that FRB lensing can be a good probe of
cosmology. In particular, it has been suggested that this type of lensing
delay from FRB-repeaters could be used to measure the Hubble constant (Zitrin
& Eichler, 2018; Li et al., 2018; Liu et al., 2019; Wucknitz et al., 2021), by
observing the rate of change in the lensing delay (the difference between the
arrival of signals from two images of the same burst) over a period of years.

FRBs can also be micro-lensed by a stellar object. The probability for that at
$z\lesssim 0.5$ is of order111This $\Omega_{M_{\odot}}$ estimate is from Madau
& Dickinson 2014., $\tau\sim 2.5\times 10^{-4}(\Omega_{M_{\odot}}/0.004)$ for
a solar mass object (where $\Omega_{M_{\odot}}$ is the mass density in $\sim
1M_{\odot}$ objects at $z\lesssim 0.5$ divided by the closure density). The
lensing probability is significantly larger (by a factor of up to $\sim 50$)
if all of dark matter were in $\sim M_{\odot}$ primordial black holes (Carr &
Hawking, 1974). Thus, observational limits on $\tau(M)$ can constrain the
density of primordial black holes with mass $\sim M$. One noteworthy advantage
of observing micro-lensing of FRBs by objects with stellar or smaller mass is
that it allows to probe the regime of physical, rather than geometrical,
optics. Gravitational lensing is typically considered in the latter regime.
This is in large part due to the fact that one needs a source whose size is
smaller than $\sim
10^{12}(M/M_{\odot})^{-1/2}(\nu/\mbox{GHz})^{-1}(D/\mbox{Gpc})^{-1/2}\mbox{
cm}$ 222This requirement ensures that the size of the source is less than
$d_{\rm SO}\pi\theta_{\rm F}^{2}/\theta_{\rm E}$ (where $d_{\rm SO}$ is the
source-observer distance, and $\theta_{\rm F},\theta_{\rm E}$ are
correspondingly the Fresnel and Einstein angular scales) and therefore that
the time-delay between the two images is shorter than a wave period. (Nakamura
& Deguchi 1999; where $D$ is the distance to the source and we considered for
clarity the case in which the lens is halfway between the source and the
observer). This condition is not easily satisfied by sources observable at
cosmological distances. It has been argued that the coherent nature of FRBs,
their detection up to cosmological distances and the potentially small sizes
of their sources, make it possible to use them to probe wave effects of plasma
and gravitational lensing (Grillo & Cordes, 2018; Cordes et al., 2017; Katz et
al., 2020; Jow et al., 2020). In the physical optics regime (unlike in
geometrical optics) micro-lensing events become frequency dependent. As a
result, if the dynamic spectrum of a micro-lensing event can be observed, it
allows to uniquely constrain the mass of the lens, $M$, and the combination
$\mu_{\rm rel}^{2}D$ (where $\mu_{\rm rel}$ is the angular velocity of the
lens relative to the source in the plane of the sky). This is as opposed to
geometrical micro-lensing where only the combination $M/\mu_{\rm rel}^{2}D$
can be measured.

A different application of micro-lensing of FRBs by stellar mass objects is to
constrain the possibility that dark matter is composed of massive compact halo
objects (MACHOS) in the range of several to hundreds of solar masses (Muñoz et
al., 2016). The idea suggested by these authors uses the fact that the time
difference between two (unresolved) ‘images’ is a direct measure of the lens
mass. Therefore, if this time difference is longer than the duration of an FRB
burst, micro-lensing will leave a measurable signature on the lightcurve. This
can then be used to constrain the optical depth of lenses with a given mass
range.

As apparent from the above description, many applications of FRB lensing rely
on the fact that cosmological FRBs can often be assumed to be effectively
point sources. However, the spectra of many FRBs is often seen to suffer from
spectral decoherence (e.g. Bannister et al., 2019), and their light-curves
show signatures of scatter broadening (e.g. Thornton et al., 2013). These
signatures indicate that the FRB wave suffers from scintillation as it passes
through a turbulent plasma screen (or multiple screens) on its path from the
source to us. Indeed, multi-path propagation induced by plasma scintillation
can lead to other notable effects on the observed FRB signals, such as induced
temporal variability (Beniamini & Kumar, 2020) and depolarization / induced
circular polarization (Beniamini et al., 2022). As plasma scintillation is
commonly inferred to affect FRB observations, it is natural to ask how it
affects gravitational lensing. This is the topic of the present work. As we
will show below, perhaps the most important effect of plasma scintillation on
lensing is that it effectively increases the size of the source (as mentioned
in Cordes & Chatterjee 2019), and therefore suppresses lensing effects for
lenses below a critical mass.

The effects of uniform and non-uniform plasma on gravitational lensing by a
point source was considered also by Bisnovatyi-Kogan & Tsupko (2010, 2017).
Eshleman (1979) explored how the magnification of a background source’s
brightness due to gravitational lensing by the Sun is limited by waves
traveling through the solar corona. These authors, however, did not consider
wave scattering in turbulent plasma, which is the topic of the current work.

The paper is organized as follows. We begin in §2 with a description of some
important timescales and length-scales related to gravitational lensing and
plasma scintillation. This sets the scene to understand when plasma
scintillation affects can not be ignored in the treatment of lensing. In §3 we
focus on a simple, but very useful test case of the effects of plasma
scattering on lensing from a point source. This case can be worked out in some
detail and allows to estimate the effects of a plasma screen on the lensing
magnification and on the lens delays as reflected in the FRB lightcurve. In
§4.2, we describe ways in which gravitational lensing in the presence of
plasma scattering can provide unique constraints on cosmology and in addition
show that gravitational lensing can enhance the degree of induced circular
polarization due to passage of the FRB wave through a magnetized plasma
screen. We conclude in §5.

##  2 Basic considerations

The travel time difference along two different paths corresponding to two
gravitational lens images of an FRB by a point lens of mass $M_{\rm l}$ is
approximately $R_{\rm s}/c=2GM_{\rm l}/c^{3}$; where $R_{\rm s}\equiv 2GM_{\rm
l}/c^{2}$ is the lens’ gravitational radius, and $c$ is the speed of light in
vacuum. For a stellar mass lens, this time difference is $\Delta t_{g}\sim
10\mu$s. Coherent FRB radio waves traveling along two different trajectories
could interfere and physical optics effects could influence the magnification
of the observed flux when the lens mass is small, e.g. Jow et al. (2020).

In addition, an EM wave propagating through a medium of non-zero electron
density moves at a speed slightly smaller than $c$ and that contributes to a
delay in photon arrival time that is different for different trajectories. We
show below that this plasma effect cannot be ignored for stellar mass lens.
The dispersion relation and group speed for EM waves in plasma are

$\omega^{2}=\omega_{\rm p}^{2}+c^{2}k^{2},\quad{\rm and}\quad v_{\rm
g}=\frac{d\omega}{dk}\approx c\left[1-\frac{\omega_{\rm
p}^{2}}{2\omega^{2}}\right]$ (1)

where the plasma frequency is

$\omega_{\rm p}=\left(\frac{4\pi q^{2}n_{\rm e}}{m}\right)^{1/2}$ (2)

and $n_{\rm e}$, $q$, and $m$ are electron density, charge and mass
respectively. Thus, the delay in photon arrival when it travels a distance
$d_{s}$ through plasma is

$t_{\rm p}=\int_{0}^{d_{s}}\frac{dr}{v_{\rm
g}}-\frac{d_{s}}{c}=\int_{0}^{d_{s}}\frac{dr}{c}\frac{2\pi q^{2}n_{\rm
e}}{m\omega^{2}}={\rm 4.2\,ms}\,\nu_{9}^{-2}\,{\rm DM},$ (3)

where

${\rm DM}\equiv\int_{0}^{d_{s}}\frac{dr}{1\,{\rm pc}}\,n_{\rm e}$ (4)

is the dispersion measure along the photons trajectory, with $n_{\rm e}$
measured in CGS units, and 1 pc = 3.14x1018 cm.

Gravitational images in the lens plane are separated by a distance of order
the Einstein radius which is defined by

$R_{\rm E}=\left(\frac{2R_{\rm s}d_{\rm SL}d_{\rm LO}}{d_{\rm
SO}}\right)^{1/2},$ (5)

where $d_{\rm SL}$ is the distance between the lens and source, $d_{\rm LO}$
is distance between the lens and observer, and $d_{\rm SO}$ is the distance
between the source and the observer. It is useful to define also the Einstein
angle

$\theta_{\rm E}={R_{E}\over d_{\rm LO}}=\left(\frac{2R_{\rm s}d_{\rm
SL}}{d_{\rm SO}d_{\rm LO}}\right)^{1/2},$ (6)

For a stellar mass lens at a cosmological distance, $R_{\rm E}\sim 10^{17}$cm.
Even if the difference in DM along the two photon trajectories is only
$\delta\mbox{DM}=10^{-3}$ cm-3 pc – due to fluctuations in the electron
density on a scale $R_{\rm E}$ – the travel time difference along the two
trajectories due to plasma effects is $\Delta t_{p}=4\mu$s at 1 GHz (eq. 3),
i.e. of the same order as the gravitational time delay. Density fluctuations
of this magnitude are present when the outer scale of turbulence for the IGM
is of order $10^{22}$cm or less and this is discussed further in §3.3.
Furthermore, even if $\Delta t_{p}\ll\Delta t_{g}$, plasma scattering may
significantly modify lensing. Indeed, it turns out that typically the limiting
condition for modification of gravitational lensing by plasma effects is
related to the ratio of the plasma scattering angle and $\theta_{\rm E}$
(§3.1).

For a lens of mass larger than $\sim$102 M⊙, the time delay between different
images is $\Delta t_{g}\lower
2.0pt\hbox{$\,\buildrel\scriptstyle>\over{\scriptstyle\sim}\,$}$1ms. In this
case (barring plasma effects) no interference between different lensed images
is expected for most FRBs as their durations are a few ms. The magnification
of each image is set by interference of the bundle of rays, the Fresnel
bundle, that have traveled along different trajectories with travel-times that
lie within a few wave periods of each other. The width of the Fresnel bundle,
$R^{2}_{F}\sim\lambda\min\\{d_{\rm SL},d_{\rm LO}\\}$; $\lambda$ is photon
wavelength. The deflection of rays across the Fresnel bundle does not change
much as long as $R_{F}\ll R_{\rm E}$. And that means that geometrical optics
is a good description of gravitational lensing as long as $R_{\rm
s}\gg\lambda$ and $\theta_{\rm S}\not\ll\theta_{E}$; where $\theta_{\rm S}$ is
the angle between the observer-source and observer-lens lines.

This can also be understood by considering the time delay between the images.
For a point mass lens, and a source angle position $\theta_{\rm
S}\lesssim\theta_{E}$, the time delay between the two images is $\Delta
t\sim(4R_{\rm s}/c)(\theta_{\rm S}/\theta_{E})$ (see §3.3 for the derivation
of this result; eq. 53). The delay is less than the wave period, or the images
interfere, when $\theta_{\rm S}<\theta_{\rm F}^{2}/\theta_{\rm E}$; where the
Fresnel angle is defined to be

$\theta_{F}=\left[\frac{\lambda d_{\rm SL}}{d_{\rm LO}d_{\rm
SO}}\right]^{1/2}.$ (7)

Since, the most likely lensing event is one where $\theta_{\rm
S}\sim\theta_{\rm E}$, we see that typically interference is important when
$\theta_{\rm F}<\theta_{\rm E}$ or equivalently $R_{\rm s}<\lambda$.
Therefore, geometrical optics is a good description of gravitational lensing
for most cases of interest333 The interference between bundles of rays
associated with the two images can modify the overall magnification as given
by the geometrical optics even when $R_{\rm s}\gg\lambda$. This can occur if
the difference in travel time between images is smaller than the coherence
time, $T_{c}$, of the transient source being lensed. This translates to the
condition that $R_{\rm s}\lower
2.0pt\hbox{$\,\buildrel\scriptstyle<\over{\scriptstyle\sim}\,$}cT_{c}$ for
physical optics to be important. However, it should be noted that the
interference between the two images can increase the total magnification by at
most a factor two, but destructive interference can reduce the total flux for
the two images to almost zero.. However, plasma effects can be important more
widely as discussed above, and that is the subject of this paper.

The effect of plasma on image magnification is important when the scattering
angle for waves traveling through the plasma screen ($\theta_{\rm scat}$) is
of order $\theta_{\rm S}$. The scattering broadens the angular size of the
source to $\theta_{\rm scat}$, and limits the magnification to $\theta_{\rm
E}/(2\theta_{\rm scat})$ as we show in §3.

##  3 Gravitational lensing due to a point mass in the presence of plasma
scattering screen

The flux observed from an astronomical source when photons travel through a
gravitational potential and plasma on their way to the observer is given by

$f(\omega,{\vec{\theta}_{\rm s}})=\frac{1}{i\theta_{F}^{2}}\int
d^{2}\theta\,\exp\left\\{\frac{i\pi|\vec{\theta}-\vec{\theta}_{s}|^{2}}{\theta_{F}^{2}}-i\omega\left[\psi(\vec{\theta})-\delta
t_{p}(\vec{\theta})\right]\right\\}$ (8)

where $\vec{\theta}_{s}$ is the angular location of the source wrt observer-
lens line, and $\vec{\theta}$ is angular position of a point in the lens plane
again wrt observer-lens line. The first term in the bracket,
$\pi|\vec{\theta}-\vec{\theta}_{s}|^{2}/\theta_{F}^{2}$ is the geometric phase
shift suffered by the wave as a result of its trajectory not being a straight
line from the source to the observer,

$\psi(\vec{\theta})=\int d\ell\,\frac{\Phi(\vec{x})}{c^{3}}(1+n_{r}^{2})$ (9)

is the time delay due to travel in the gravitational potential
$\Phi$444General relativistic effects have been neglected in the expression
for $\psi$, where the integration is done over the photon path, $n_{r}$ is the
radial component of unit vector that is tangent to the photon path, and
$\delta t_{p}$ is the time delay suffered by the wave due to propagation
through plasma. The expression for $\psi$ is valid only for spherically
symmetric gravitational potential, and the radial component is wrt the center
of the symmetry. This equation is a generalization of the result given in
(Nakamura & Deguchi, 1999) to include plasma effects. For a point lens of mass
$M$, as per Schneider et al. (1992)

$\psi(\vec{\theta})=\frac{4GM}{c^{3}}\log|\vec{\theta}|=\frac{2R_{\rm
s}}{c}\log|\vec{\theta}|.$ (10)

Much of the contribution to the integral in eq. (8) comes from the extrema of
the phase function in the exponent. In the absence of the turbulent plasma
screen, the phase has two extrema for a point lens corresponding to two
gravitational lens images. However, for waves passing through a turbulent
plasma screen $\delta t_{p}(\theta)\propto\theta^{5/6}$ and it fluctuates on
an eddy length scale of $\ell_{\phi}$. Thus, the exponent has large number of
extrema that correspond to waves being scattered from different segments of
the plasma screen to arrive at the observer. The area around each of these
extrema that contributes to the integral is of angular size
$\delta\theta_{\phi}\sim\ell_{\phi}/d_{\rm LO}$ such that the exponent changes
by $\sim\pi$ radian across the area. Expanding the exponent about one of the
image locations in the absence of the plasma, $\theta_{\rm I}$, we see that
when $\delta\theta\lower
2.0pt\hbox{$\,\buildrel\scriptstyle>\over{\scriptstyle\sim}\,$}\theta_{F}^{2}/\delta\theta_{\phi}$,
the phase is dominated by the geometrical path length term and grows as
$\delta\theta^{2}$, hence the integrand oscillates rapidly and does not
contribute much to the integral. This can be cast in the usual picture that
the radius of the screen from which photons arrive at the observer is
$\sim(\theta_{F}/\delta\theta_{\phi})^{2}\ell_{\phi}$, which is defined as the
refractive scale. Waves are scattered by different patches of size
$\ell_{\phi}$ within the refractive scale and add incoherently at the observer
location.

In the next sub-section we calculate the effect on magnification of
gravitationally lensed images due to the turbulent plasma screen. And in §3.3
we discuss how the time delay between different lensed images are modified by
the plasma screen.

###  3.1 Effect of plasma screen on lens magnification

Figure 1:  Schematic figure representing the geometry of the source - lens -
plasma - observer system. The plasma plane is assumed here to coincide with
the lens plane.

We refer to Fig. 1 for the derivation of magnification of gravitationally
lensed point-source in the presence of a scattering plasma screen. The
derivation assumes that the plasma lies in the lens plane to simplify the
algebra. However, the results are broadly applicable when the lens and plasma
planes are separate except for some geometrical factors which can be important
when the distance between the two planes becomes comparable to the smaller of
the distances between the source and the lens or lens and the observer (the
relevant modification in this case is described in §5).

The lens equation is

$\theta_{\rm I}d_{\rm SO}=\theta_{d}d_{\rm SL}+\theta_{\rm S}d_{\rm SO},$ (11)

where the deflection angle

$\theta_{d}=\frac{2R_{\rm s}}{\theta_{0}d_{\rm SL}}=\frac{2R_{\rm
s}}{\theta_{\rm I}d_{\rm LO}},$ (12)

$R_{\rm s}=2GM_{\rm l}/c^{2}$ is the gravitational radius of the lens.
Substitution of $\theta_{d}$ in the lens equation gives the standard result
for image location in terms of the angular position of the source wrt the
lens:

$\theta_{\rm I}=\frac{1}{2}\left[\theta_{\rm S}\pm\left(\theta_{\rm
S}^{2}+4\theta_{\rm E}^{2}\right)^{1/2}\right].$ (13)

We calculate the magnification of the image that is on the same side of the
lens-source axis as the observer. The magnification of the other image can be
worked out in an equivalent way by considering the image with the minus side
in eq. 13. One way to calculate the magnification by the lens, which can be
easily generalized to include the plasma scintillation, is to consider a
bundle of rays between two cones of angles $\theta_{0}$ and
$\theta_{0}+\delta\theta_{0}$, with the apex of these cones at the source. The
rays are bent by the lens so that the cross-sectional area of this bundle in
the observer plane is $2\pi x\delta x$ instead of
$2\pi\theta_{0}\delta\theta_{0}d_{\rm SO}^{2}$ as would occur with no lensing.
$x,\delta x$ are given by

$x=\theta_{0}(1-\theta_{\rm E}^{2}/\theta_{\rm I}^{2})d_{\rm
SO}\quad\&\quad\delta x=d_{\rm SO}\delta\theta_{0}\left[1+\theta_{\rm
E}^{2}/\theta_{\rm I}^{2}\right].$ (14)

The magnification in the absence of the scattering screen is
$\mu_{gl}=\theta_{0}\delta\theta_{0}d_{\rm SO}^{2}/(x\delta x)$, which reduces
to the standard expression, i.e.

$\mu_{gl}=\frac{1}{1-\theta_{\rm E}^{4}/\theta_{\rm I}^{4}}.$ (15)

The effect of the plasma screen is that a bundle of rays within a finite angle
between $\theta_{-}$ & $\theta_{+}$ are scattered and their intersection with
the observer plane has a larger area than it would in the absence of
scattering. We make use of the circular symmetry about the source–lens axis to
simplify the calculation of magnification.

The magnification is calculated approximately by making the simplifying
assumption that a bundle of rays from the point source with cross-section of
$\ell_{\phi}$ in the lens plane are scattered by the column of turbulent
eddies in the plasma screen into a cone of angle $\theta_{\rm scat}$, and
these rays are bent by an angle $\theta_{d}$ by the point lens as they travel
toward the observer. The scattered rays in the bundle lie between distances
$x(\theta_{0})\pm\theta_{\rm scat}d_{\rm LO}$ from the source-lens line in the
observer plane (Fig. 1 shows the geometry). If the source is at an angle
$\theta_{\rm S}$ and the observer is located at distance $x_{\rm o}$ wrt
source-lens line (see Fig. 1), then the observer will lie in the scattering
cone when the outer edge of the cone lies between $x_{\rm o}\pm 2\theta_{\rm
scat}d_{\rm LO}$. The source angle $\theta_{\rm S}$ and $x_{\rm o}$ are
related by (as the geometry in Fig. 1 makes clear)

$x_{\rm o}=\frac{\theta_{\rm S}d_{\rm SO}d_{\rm LO}}{d_{\rm SL}}.$ (16)

Thus, the range of angles $\theta_{0}$, $[\theta_{-},\theta_{+}]$, from which
rays arrive at the observer after crossing the lens and plasma screen is given
by

$\theta_{\pm}d_{\rm SO}\left[1-{\theta_{\rm E}^{2}d_{\rm
LO}^{2}\over\theta_{\pm}^{2}d_{\rm SL}^{2}}\right]=\max\Big{\\{}0,x_{\rm
o}\pm\theta_{\rm scat}d_{\rm LO}\Big{\\}}.$ (17)

Where we made use of equation 14 that expresses the distance from the source-
lens line that a ray that started out from the source at angle $\theta_{\pm}$
will be at in the observer plane in absence of the scattering plasma.

Thus, the picture is that a conical bundle of rays that leave the source
between angles $\theta_{-}$ and $\theta_{+}$ arrive at the observer after
crossing the lens-plasma plane. And this bundle of rays occupy a circular
annulus in the observer plane of radii

$x_{\pm}=\max\Big{\\{}0,x_{\rm o}\pm 2\theta_{\rm scat}d_{\rm LO}\Big{\\}}.$
(18)

with the observer lying in the middle of this ring. Since the energy in a
bundle of rays between the cones of angle $\theta_{\pm}$ flows through the
observer plane between $x_{\pm}$, the average magnification at the observer
location as a result of gravitational lensing and scattering by turbulent
plasma is given approximately by

$\mu\approx{d_{\rm
SO}^{2}\left[\theta_{+}^{2}-\theta_{-}^{2}\right]\over\left(x_{+}^{2}-x_{-}^{2}\right)/2},$
(19)

The factor 2 in the denominator is geometric in origin and accounts for the
fact that the flux in the middle of a ring formed by superposition of cones
rotated about the source-lens axis is larger than the average by a factor
$\sim 2$.

Equation (17) for $\theta_{\pm}$ can be rewritten using eq. 16 as

${d_{\rm SL}\theta_{\pm}\over d_{\rm LO}}\left[1-{\theta_{\rm E}^{2}d_{\rm
LO}^{2}\over\theta_{\pm}^{2}d_{\rm SL}^{2}}\right]=\max\Big{\\{}0,\theta_{\rm
S}\pm{\theta_{\rm scat}d_{\rm SL}\over d_{\rm SO}}\Big{\\}}.$ (20)

This equation is further simplified when expressed in terms of

$\theta^{I}_{\pm}\equiv\theta_{\pm}\left[{d_{\rm SL}\over d_{\rm
LO}}\right]\quad\&\quad\theta^{\prime}_{\rm scat}\equiv\theta_{\rm
scat}\left[{d_{\rm SL}\over d_{\rm SO}}\right]$ (21)

and it takes the following form

$\theta^{I}_{\pm}\left[1-{\theta_{\rm
E}^{2}\over{\theta_{\pm}^{I}}^{2}}\right]=\max\Big{\\{}0,\theta_{\rm
S}\pm\theta^{\prime}_{\rm scat}\Big{\\}}.$ (22)

This is the standard point mass lens equation with source located at
$\max\Big{\\{}0,\theta_{\rm S}\pm\theta^{\prime}_{\rm scat}\Big{\\}}$. The
physical interpretation of the result is straightforward, viz. the effect of a
scattering screen is to broaden a point source so that it has an effective
angular size of $\theta^{\prime}_{\rm scat}$.

The equation for magnification (19) simplifies to

$\mu\approx{2\left[{\theta^{I}_{+}}^{2}-{\theta^{I}_{-}}^{2}\right]\over\left(\theta_{\rm
S}+2\theta^{\prime}_{\rm scat}\right)^{2}-\left(\max\big{\\{}0,\theta_{\rm
S}-2\theta^{\prime}_{\rm scat}\big{\\}}\right)^{2}}.$ (23)

It is easy to see that this equation reduces to the standard form, i.e. eq.
15, in the absence of the plasma ($\theta_{\rm scat}=0$); the numerator in
that limit is $2(d\theta_{\rm I}^{2}/d\theta_{\rm
S})\delta\theta=4(\delta\theta)\theta_{\rm I}^{3}/(\theta_{\rm
I}^{2}+\theta_{\rm S}^{2})$, where $\delta\theta=2\theta_{\rm scat}^{\prime}$.
We also note that in the limit $\theta_{\rm scat}^{\prime}\gg\theta_{\rm
E},\theta_{\rm s}$, eq. 15 reduces to $\mu\to 1/2$. This might appear counter-
intuitive as in the presence of strong scattering we would expect no
magnification or $\mu\approx 1$. However, in this limit, the area in the lens
plane that is traversed by rays that compose image 1 (on the same side of the
lens-source axis as the observer) almost completely overlaps the area in the
lens plane of the rays forming image 2 (on the opposite side of the said
axis). This means that the images can no longer be separated in any physical
way (i.e. either in terms of their angular position on the sky or in terms of
the time delays associated with them), and one can no longer treat the two
images separately. The magnification for the two images together approaches
unity in this case as one would expect.

For the case where $\theta_{\rm E}>\theta_{\rm scat}^{\prime}\lower
2.0pt\hbox{$\,\buildrel\scriptstyle>\over{\scriptstyle\sim}\,$}\theta_{\rm
S}$, $\theta_{-}=\theta_{\rm E}$ as per equation (22), and the magnification
is

$\mu={(\theta_{\rm S}+\theta_{\rm scat}^{\prime})\over(\theta_{\rm
S}+2\theta_{\rm scat}^{\prime})^{2}}\left[\theta_{\rm S}\pm\sqrt{\theta_{\rm
S}^{2}+4\theta_{\rm E}^{2}}\right].$ (24)

The presence of the scattering screen decreases the magnification by a factor
$\sim 2\theta_{\rm scat}^{\prime}/\theta_{\rm S}$ in this case.

The magnification in the opposite case of $\theta_{\rm
scat}^{\prime}<\theta_{\rm S}$ is

$\mu={(\theta_{\rm S}+\theta_{\rm scat}^{\prime})\theta^{I}_{+}-(\theta_{\rm
S}-\theta_{\rm scat}^{\prime})\theta^{I}_{-}\over 4\theta_{\rm S}\theta_{\rm
scat}^{\prime}}.$ (25)

The total magnifications (summing over the two images) for a few values of
$\theta_{\rm S}/\theta_{\rm scat}^{\prime}$ are shown in Fig. 2. The
magnification in the presence of a scattering screen for the sum of the flux
for the two images is capped at $\theta_{\rm E}/2\theta^{\prime}_{\rm scat}$
(see. Fig. 2). Thus, there is a minimum lens mass, $M_{\rm min,\mu}(\mu_{\rm
max})$, below which the magnification cannot be larger than $\mu_{\rm max}$.
This mass is given by

$\displaystyle M_{\rm min,\mu}(\mu_{\rm max})$ $\displaystyle\sim{\mu_{\rm
max}^{2}\theta_{\rm scat}^{2}c^{2}\over G}{d_{\rm SL}d_{\rm LO}\over d_{\rm
SO}}$ (26) $\displaystyle\sim(7\,M_{\odot})\mu_{\rm max,1}^{2}\theta_{\rm
scat,-9}^{2}d_{\rm LO,22}(d_{\rm SL}/d_{\rm SO}).$

The scatter broadening timescale for a lensed burst scales as555The reason
that the scatter broadening time scales as $\theta^{\prime 2}_{\rm scat}$ and
not as $\theta_{E}\theta^{\prime}_{\rm scat}$ is that at the extrema points of
the path-integral in eq. 8, the first derivative of the geometrical +
gravitational contributions to the phase vanishes. In other words, their
respective contributions cancel each other to first order, and for small
deviations around $\theta_{\rm I}$, only the second order term contributes to
the time delay. $\theta^{\prime 2}_{\rm scat}$, and is given by

$t_{\rm sc}\approx\frac{(\theta_{\rm scat}^{\prime}d_{\rm
LO})^{2}}{c}\frac{d_{\rm SO}}{d_{\rm LO}d_{\rm SL}}=\frac{\theta_{\rm
scat}^{2}d_{\rm SL}d_{\rm LO}}{cd_{\rm SO}}=\frac{2R_{\rm
s}}{c}\left(\frac{\theta_{\rm scat}^{\prime}}{\theta_{\rm E}}\right)^{2}$ (27)

As will be shown in §3.3, the first term on the R.H.S. is of order the
gravitational+geometrical time delay between the images, $\Delta t_{g}$. For
$\theta_{\rm scat}^{\prime}\ll\theta_{\rm E}$, we see from eq. 27 that $t_{\rm
sc}\ll\Delta t_{g}$. This timescale is also shorter than $t_{\rm FRB}$, if

$\min(d_{\rm LO},d_{\rm SL})<d_{\rm sc}\equiv(3\times 10^{25}{\rm cm})\,t_{\rm
FRB,-3}\theta_{\rm scat,-9}^{-2}.$ (28)

For lens mass below $M_{\rm min,\mu}(\mu_{\rm max}=1)$, the temporal
broadening for each of the ‘image’ pulse due to plasma turbulence is larger
than the delay between their arrival times ($\Delta t_{g}$). As a result, the
presence of gravitational lensing would be very hard (if not impossible) to
infer from the lightcurve below this mass.

Even if the scatter broadening timescale is short compared to the time delay
between the images, it might still be larger than the intrinsic duration of
the FRB. In such a situation, a delay between the image arrival times would be
detectable, but the shape of the lightcurves for the two images would be
different. It is important to note that lensing could still be identifiable in
this case for non-repeating FRBs, as the later image lightcurve will have a
very similar DM and likely an indistinguishable angular position relative to
that of the first image. The situation is more complicated for a repeater, as
the different looking lightcurves of different images could be confused with
different bursts from the source. Lensing could still be identified by
consistent delays between pairs of bursts (for a point lens) from the source,
or perhaps by sophisticated data analysis that can deconvolve the different
scatter broadening of the two images.

The scattering angle $\theta_{\rm scat}$ in eqns. 26, 28 can be estimated as
$\theta_{\rm scat}\sim\lambda/\ell_{\phi}$, where $\ell_{\phi}$ is the
diffraction scale for the turbulent plasma screen, and $\lambda$ is the
wavelength of FRB radiation. We take a brief detour here to provide an
estimate for $\ell_{\phi}$, and thus $\theta_{\rm scat}$.

Let us consider that the electron density fluctuation in the turbulent medium
is a power-law function in the inertial sub-range between length scale
$\ell_{\rm min}$ and $\ell_{\rm max}$, and is given by

$\delta n_{\rm e}(\ell)=n_{\rm e}(\ell/\ell_{\rm max})^{\alpha}.$ (29)

The index $\alpha=1/3$ is for Kolmogoroff density fluctuations. The largest
eddy size, $\ell_{\rm max}$, is the scale at which energy is injected to
maintain the turbulence, and the smallest scale $\ell_{\rm min}$ is determined
by dissipation physics of turbulence. The fluctuation in the dispersion
measure (DM) for waves traveling through a plasma screen of width ${\rm L}$,
at two points separated by length scale $\ell$, is given by

$\delta{\rm DM}(\ell)\sim\delta n_{\rm e}(\ell)(\ell/L)^{1/2}\sim{\rm
DM_{L}}\,(\ell/\ell_{\rm max})^{\alpha}\,(\ell/\rm{L})^{1/2},$ (30)

where ${\rm DM_{L}}=n_{\rm e}{\rm L/(1pc)}$.

The diffraction scale ($\ell_{\phi}$) is defined to be the transverse length
in the plasma screen such that a wave suffers a differential phase shift of
$\sim\pi$ across $\ell_{\phi}$ after crossing the screen, and is given by

$\ell_{\phi}\sim\left({mc^{2}\over q^{2}n_{\rm e}\lambda}\right)^{{6\over
5}}\ell_{\rm max}^{{2\over 5}}L^{-{3\over 5}}\sim(2{\rm x}10^{13}{\rm
cm})\,n_{\rm e}^{-{6\over 5}}L^{-{1\over 5}}\nu_{9}^{6\over 5}\left({\ell_{\rm
max}\over L}\right)^{2\over 5}.$ (31)

The various exponents in the above expression for $\ell_{\phi}$ are for the
particular case of $\alpha=1/3$ or the Kolmogoroff spectrum. Strong
scattering, or diffractive scintillation, occurs when
$\ell_{\phi}<R_{F}\equiv\theta_{F}d_{\rm LO}$.

The scattering angle is approximately equal to the diffraction angle
corresponding to size $\ell_{\phi}$, and can be written using equation 31 as

$\theta_{\rm scat}\sim{\lambda\over\ell_{\phi}}=(1.5{\rm x}10^{-12}{\rm
rad})\,n_{\rm e}^{{6\over 5}}L^{{1\over 5}}\nu_{9}^{-{6\over
5}}\left({\ell_{\rm max}\over L}\right)^{-{2\over 5}}.$ (32)

As an example, a plasma screen at 1 pc of the FRB source with $n_{\rm e}\sim
1$ cm-3 and $\ell_{\rm max}/L\sim 10^{-4}$, has $\ell_{\phi}\sim
10^{8}\nu_{9}^{6/5}$ cm. The apparent source size due to scattering by the
plasma screen is $d_{\rm SP}\lambda/\ell_{\phi}\sim 10^{11}$cm whereas the
Einstein radius for a stellar mass lens within 100 kpc of the source is,
$R_{\rm E}\sim 10^{15}$cm. Thus, the plasma screen within a few pc of the
source has little effect on the magnification by a stellar mass lens 100 kpc
away.

We can now recast the expression for $M_{\rm min,\mu}$ by making use of
equation (32) for $\theta_{\rm scat}$:

$M_{\rm min,\mu}\sim 2{\rm x}10^{4}M_{\odot}\,{\mu_{\rm max,1}^{2}{\rm
DM}_{2}^{12\over 5}d_{\rm LO,22}\,d_{\rm SL}\over\nu_{9}^{12\over
5}L_{22}^{6\over 5}\ell_{\rm max,17}^{4\over 5}\,d_{\rm SO}}.$ (33)

Similarly, we recast eq. 27 as

$\min(d_{\rm LO},d_{\rm SL})<d_{\rm sc}\equiv(10^{26}\,{\rm cm})\,\frac{t_{\rm
FRB,-3}\mbox{DM}_{2}^{12/5}}{L_{22}^{6/5}\nu_{9}^{12/5}\ell_{\rm
max,17}^{4/5}}.$ (34)

Figure 2:  Effect of plasma scattering on gravitational lensing magnification.
Solid lines depict the total magnification (i.e. the sum of the two images) as
a function of $\theta_{\rm scat}^{\prime}/\theta_{\rm s}$ and for different
values of $\theta_{\rm E}/\theta_{\rm s}$ (shown by different colors).
Horizontal dotted lines show the values of the magnification for the same
parameter but without plasma scattering. A dashed (dot dashed) vertical line
represents $\theta_{\rm scat}^{\prime}=\theta_{\rm s}$ ($\theta_{\rm
scat}^{\prime}=\theta_{\rm E}$). Plasma scattering causes lensing
magnification to be suppressed by a factor $\sim 2\theta_{\rm
scat}^{\prime}/\theta_{\rm s}$ for $\theta_{\rm scat}^{\prime}>\theta_{\rm s}$
until there is virtually no magnification when $\theta_{\rm
scat}^{\prime}\approx\theta_{\rm E}$.

####  3.1.1 Interference between multiple images due to a lens

Photons from multiple images of a coherent source would interfere provided
that the time delay between images is less than the duration of the burst or
the source coherence time, whichever is smaller. Let us assume that there are
$N$ images and that the travel time difference wrt to the first image
(selected arbitrarily) is $t_{i}^{l}$ ($t_{1}^{l}=0$). If there is a plasma
screen somewhere between the source and the observer then that would cause an
additional time delay for these $N$ images, and affect their interference and
the magnification for the combined $N$ images. Let us take the transverse
separation between the $i$-th and the first image path in the plasma screen to
be $\ell_{\perp i}$. Combining equations (3) & (30) we find the time delay due
to plasma for image $i$ to be

$t_{i}^{p}\approx{\rm 4.2\,ms}\,\nu_{9}^{-2}\,{\rm DM}_{\rm
L}\left({\ell_{\perp i}\over L}\right)^{1/2}\left({\ell_{\perp
i}\over\ell_{\rm max}}\right)^{\alpha},$ (35)

where DM is the mean DM for the plasma screen. If the flux at the observer for
image $i$ is $f_{i}$ then in the Eikonal approximation, the combined flux for
the $N$ images is

$f=2\sum_{i,j}(f_{i}f_{j})^{1/2}\cos\left[\omega(t_{i}^{l}+t_{i}^{p}-t_{j}^{l}-t_{j}^{p})\right].$
(36)

The flux $f_{i}$ is also modified by the presence of the plasma screen when
$\ell_{\phi}$ is smaller than the Fresnel length, i.e. in the strong
scattering regime. The phase difference between waves along the different
image trajectories through the turbulent medium is random because the travel
time difference, $t_{i}^{p}-t_{j}^{p}$, is random for most astrophysical
systems of interest as discussed in §3.3.

###  3.2 Optical depth for lensing in the presence of plasma scattering

We showed in §3.1 that the magnification is capped at $\min(\theta_{\rm
E}/2\theta_{\rm S},\theta_{\rm E}/2\theta_{\rm scat}^{\prime})$ in the limit
of strong lensing. Therefore, the optical depth for lensing is modified in the
presence of plasma.

Consider a distribution of lenses, all with mass $M$ and with an optical depth
(uncorrected for scattering) $\tau(>\mu)=\tau_{0}\mu^{-2}$ 666$\tau_{0}$ is
directly proportional to the mass density of the lenses, see §1. For clarity,
we focus here on the case in which the sources under consideration have $z\ll
1$, so that cosmological redshift corrections can be neglected, and in which
the plasma screen location is in the vicinity of the gravitational lens. We
denote a dimensionless distance $x\equiv d_{\rm LO}/d_{\rm SO}$ such that
$0<x<1$. For a homogeneous distribution of lenses between the source and
observer, the fraction of lenses per dimensionless distance $x$ is $dP_{\rm
x}/dx=3x^{2}$. In the absence of plasma scattering, a magnification $\mu>1$ is
obtained for sources with $\theta_{\rm s}$ such that

$\theta_{\rm s}<\theta_{\rm s,\mu}\equiv\frac{\theta_{\rm
E}}{2\mu}=\frac{\theta_{\rm E,0}(1-x)^{1/2}}{2\mu x^{1/2}}$ (37)

where $\theta_{\rm E,0}\equiv\theta_{\rm E}(x=1/2)$. Assuming an isotropic
population of sources and lenses, the fraction of sources with an angle
$<\theta_{\rm s}$ is $P_{\rm s}(\theta_{\rm s})=(1-\cos\theta_{\rm s})/2$ or
$P_{\rm s}(\theta_{\rm s})=(1/4)\theta_{\rm s}^{2}$ for $\theta_{\rm s}\ll 1$.
In the presence of plasma, to obtain magnification $>\mu$, the scattering
angle of rays going through the plasma plane must satisfy

$\theta_{\rm scat}^{\prime}<\frac{\theta_{\rm E}}{4\mu}\to\theta_{\rm
scat}<\theta_{\rm scat,\mu}\equiv\frac{\theta_{\rm
E,0}}{(1-x)^{1/2}x^{1/2}4\mu}$ (38)

where we have used eq. 21 to write $\theta_{\rm scat}^{\prime}=\theta_{\rm
scat}(1-x)$. The distribution of scattering angles for plasma screens depends
on the parameters of their turbulence. For generality, we take that to be an
unspecified function $P_{\rm scat}(<\theta_{\rm scat})$. We explore specific
forms of $P_{\rm scat}$ below.

Combining the conditions given by eqns. 37, 38 we can write an expression for
the optical depth

$\displaystyle\tau(>\mu)=\frac{32\tau_{0}}{\theta_{\rm
E,0}^{2}}\int_{0}^{1}dx\frac{dP_{\rm x}}{dx}P_{\rm s}(\theta_{\rm
s,\mu})P_{\rm scat}(\theta_{\rm scat,\mu})$
$\displaystyle=\frac{6\tau_{0}}{\mu^{2}}\int_{0}^{1}dx\,x(1-x)P_{\rm
scat}\left(\frac{\theta_{\rm E,0}}{(1-x)^{1/2}x^{1/2}4\mu}\right).$ (39)

In particular we see that in the limit of no scattering, $P_{\rm
scat}(\theta\to 0)=1$, $\tau(>\mu)=\tau_{0}/\mu^{2}$ as required. The effects
of scattering can be understood by examining a couple of concrete examples for
$P_{\rm scat}$. First, consider that all plasma screens have the same
scattering angle, $\theta_{\rm scat,0}$. In this situation, $P_{\rm
scat}=\Theta(\theta_{\rm scat}-\theta_{\rm scat,0})$; where $\Theta$ is the
step function. $P_{\rm scat}$ will equal 1 as long as $x(1-x)<(\theta_{\rm
E,0}/4\mu\theta_{\rm scat,0})^{2}$ and 0 otherwise. Stated differently, there
is a critical magnification

$\mu_{\rm scat}=\frac{\theta_{\rm E,0}}{2\theta_{\rm scat,0}}$ (40)

such that for $\mu<\mu_{\rm scat}$ the optical depth is unaffected by
scattering. For $\mu\gg\mu_{\rm scat}$, the contribution to eq. 3.2 comes from
two separate regions of $x$ satisfying $0<x<x_{\rm-}$ and $x_{\rm+}<x<1$,
where $x_{-,+}=0.5\pm 0.5\sqrt{1-(\mu_{\rm scat}/\mu)^{2}}$. The integral in
eq. 3.2 is symmetric in $x$ around $x=0.5$, so it is sufficient to work out
the scaling for one of those regions to get $\tau(\mu\gg\mu_{\rm scat})$. For
$\mu\gg\mu_{\rm scat}$, $x_{-}\approx 0.5(\mu/\mu_{\rm scat})^{2}\ll 1$.
Plugging this back to 3.2, we see that
$\tau(>\mu)\propto\mu^{-2}x_{-}^{2}\propto\mu^{-6}$. Overall, we have

$\displaystyle\tau(>\mu)\approx\tau_{0}\left\\{\begin{array}[]{ll}\mu^{-2}&\mu<\mu_{\rm
scat},\\\ \\\ \mu_{\rm scat}^{4}\mu^{-6}&\mu>\mu_{\rm scat}\
.\end{array}\right.$ (44)

The implication is that even if all plasma screens provide the same scattering
angle, the magnification reduces only as a powerlaw function beyond the
critical magnification. This is because there is always a small region of
space where the lens is sufficiently close to the source or the observer, and
the effective scattering angle is small enough so that the magnification is
not suppressed by plasma. If, instead, the plasma screen scattering angles are
distributed as a powerlaw, $P_{\rm scat}=(\theta_{\rm scat}/\theta_{\rm
scat,0})^{a}\Theta(\theta_{\rm scat,0}-\theta_{\rm scat})$ with $a>0$, then
for $\mu\gg\mu_{\rm scat}$, there is a limiting value of $\tilde{\theta}_{\rm
scat}=\theta_{\rm E,0}/2\mu$. For $\theta<\tilde{\theta}_{\rm scat}$,
magnification is unsuppressed by plasma for any $x$. As a result,
$\tau(>\mu)\propto\mu^{-2}\tilde{\theta}_{\rm scat}^{a}\propto\mu^{-2-a}$.
This scaling is relevant as long as contributions from $\tilde{\theta}_{\rm
scat}$ are dominant over those from $\theta_{\rm scat,0}$. Overall,

$\displaystyle\tau(>\mu)\approx\tau_{0}\left\\{\begin{array}[]{ll}\mu^{-2}&\mu<\mu_{\rm
scat},\\\ \\\ \max(\mu_{\rm scat}^{-2}\left(\frac{\mu_{\rm
scat}}{\mu}\right)^{2+a},\mu_{\rm scat}^{4}\mu^{-6})&\mu>\mu_{\rm scat}\
,\end{array}\right.$ (48)

These results are presented in figure 3 where we show $\tau(>\mu)$ for
different $P_{\rm scat}$distributions.

Figure 3: Optical depth for lensing in the presence of plasma scattering
(assumed here to take place in the lens plane). Plasma scattering leads to a
suppression of the optical depth for magnifications $\mu\gtrsim\mu_{\rm
scat}=\theta_{\rm E,0}/2\theta_{\rm scat,0}$. For $\mu\gg\mu_{\rm scat}$, the
suppression is by a factor of up to $(\mu_{\rm scat}/\mu)^{4}$. Results are
shown for a homogeneous and isotropic distribution of lenses and sources and a
distribution of plasma scattering angles that is either a powerlaw (PL)
function $P_{\rm scat}=(\theta_{\rm scat}/\theta_{\rm
scat,0})^{a}\Theta(\theta_{\rm scat,0}-\theta_{\rm scat})$ or a log-normal
distribution $P_{\rm scat}=\mbox{Lognormal}(\log(\theta_{\rm scat,0}),0.5)$.

###  3.3 Effect of plasma screen on lens delays and FRB lightcurve

For the calculation of arrival time of photons along different trajectories we
go back to equation 8. The exponent in that equation is the phase of the
photon at the observer location or the product of its arrival time and
frequency. The image location is given by stationary points of the phase, i.e.

${2\pi(\vec{\theta}_{I}-\vec{\theta}_{s})\over\theta_{F}^{2}}-{2\omega R_{\rm
s}\vec{\theta}_{I}\over c\theta_{\rm I}^{2}}+\omega{\partial\delta
t_{p}\over\partial\vec{\theta}}=0,$ (49)

and the unscattered image location by

${\vec{\theta}_{I}-\vec{\theta}_{s}\over\theta_{F}^{2}}={2\nu R_{\rm
s}\vec{\theta}_{I}\over c\theta_{\rm I}^{2}}\;\;{\rm
or}\;\;{|\vec{\theta}_{I}-\vec{\theta}_{s}|\over\theta_{\rm
E}^{2}}={1\over\theta_{\rm I}}.$ (50)

Thus, the arrival time of photons associated with the image is

$t={\pi|\vec{\theta}_{\rm I}-\vec{\theta}_{\rm
S}|^{2}\over\omega\theta_{F}^{2}}-\psi(\vec{\theta}_{\rm I})+\delta
t_{p}(\vec{\theta}_{\rm I})={2R_{\rm s}\over c}\left[{\theta_{\rm E}^{2}\over
2\theta_{\rm I}^{2}}-\ln\theta_{\rm I}\right]+\delta t_{p}(\theta_{\rm I}),$
(51)

where eqns. 10 & 50 were used to obtain the second equality and we have
assumed that the plasma causes a small correction to the image location.

Finally, the time difference between the arrival of photons for the two images
at angles $\theta_{I1}$ & $\theta_{I2}$ is

$\displaystyle\Delta t$ $\displaystyle={2R_{\rm s}\over c}\left[{\theta_{\rm
E}^{2}\over 2\theta_{I1}^{2}}-{\theta_{\rm E}^{2}\over
2\theta_{I2}^{2}}+\ln\left({\theta_{I2}\over\theta_{I1}}\right)\right]+\delta
t_{p}(\theta_{I1})-\delta t_{p}(\theta_{I2})$ (52) $\displaystyle={2R_{\rm
s}\over c}\left[{\theta_{\rm S}(\theta_{\rm S}^{2}+4\theta_{\rm
E}^{2})^{1/2}\over 2\theta_{\rm E}^{2}}+\ln\left({\sqrt{1+4\theta_{\rm
E}^{2}/\theta_{\rm S}^{2}}+1\over\sqrt{1+4\theta_{\rm E}^{2}/\theta_{\rm
S}^{2}}-1}\right)\right]$ $\displaystyle{\hskip 128.0374pt}+\delta
t_{p}(\theta_{I1})-\delta t_{p}(\theta_{I2})$

For $\theta_{\rm S}<\theta_{\rm E}$, this reduces to

$\Delta t\approx{4R_{\rm s}\theta_{\rm S}\over c\theta_{\rm E}}+\delta
t_{p}(\theta_{I1})-\delta t_{p}(\theta_{I2})$ (53)

When $\theta_{\rm S}=2\theta_{\rm E}$, the magnification factor in the absence
of plasma for one of the lens images is 1.03 whereas the other image is
demagnified by a factor 33; for a source at $3\theta_{\rm E}$, the second
image is demagnified by a factor 102. Thus, identifying superposition of two
lens images in the lightcurve of an FRB would be difficult when $\theta_{\rm
S}\lower
2.0pt\hbox{$\,\buildrel\scriptstyle>\over{\scriptstyle\sim}\,$}2\theta_{\rm
E}$. We, therefore, provide estimates for the particular case of $\theta_{\rm
S}\sim\theta_{\rm E}$, which has the highest probability of occurrence. The
geometrical plus the gravitational time difference between the two images in
this particular case is $4.16R_{\rm s}/c$ which is close to the value in eq.
(53). Next we consider the time difference due to wave propagation through
turbulent plasma.

The extra time it takes for radio waves of frequency $\omega$ to travel
through an eddy of size $\ell$ is $(\ell/c)\omega_{p}^{2}/\omega^{2}$. For
uncorrelated eddies, the total extra time to travel through a plasma screen of
thickness $L$ is

$\Delta t_{p}(\ell)={\ell\over c}{\delta n_{\rm e}(\ell)\over
n_{0}}\left({L\over\ell}\right)^{1\over 2}{4\pi q^{2}n_{0}\over
m\omega^{2}}={(\rm 4.3\,ms)\,DM\over\nu_{9}^{2}}\left({\ell\over
L}\right)^{1\over 2}\left({\ell\over\ell_{\rm max}}\right)^{\alpha}$ (54)

where $\mbox{DM}=Ln_{0}$ is the mean dispersion measure (DM) of the plasma
screen.

The eddy that contributes most to the travel time difference between the two
image trajectories has a size of order the distance between the two
trajectories in the plasma-plane, i.e.

$\ell_{12}\sim d_{\rm LO}(\theta^{I}_{+}-\theta^{I}_{-})\sim d_{\rm
LO}\left(\theta_{\rm S}^{2}+4\theta_{\rm E}^{2}\right)^{1/2}\sim 2\theta_{\rm
E}d_{\rm LO}.$ (55)

Substituting $\ell=\ell_{12}$ into 54 gives the arrival time difference for
the two lensed images due to the presence of turbulent plasma,

$\Delta t_{p}\sim({\rm 2\,s)\,DM}\,\nu_{9}^{-2}\,M_{\odot}^{5/12}{[d_{\rm
LO}(d_{\rm SO}-d_{\rm LO})]^{5/12}\over d_{\rm SO}^{5/12}L^{1/2}\ell_{\rm
max}^{1/3}}$ (56)

Taking $\ell_{\rm max}\sim 10^{17}$cm, $L=10^{2}$ pc, and $d_{\rm LO}=1$ Mpc,
we find from the above equation that $\Delta t_{p}\sim 0.4$ ms
DM${}_{2}M_{\odot}^{5/12}$. This should be compared with the geometric +
gravitational delay for the two images, $\Delta t_{g}\sim 4R_{\rm s}/c$ when
$\theta_{\rm S}\sim\theta_{\rm E}$ (see eq. 53). For $\Delta t_{p}\lower
2.0pt\hbox{$\,\buildrel\scriptstyle>\over{\scriptstyle\sim}\,$}\Delta t_{g}$,
the plasma delay dominates the delay between arrivals of radio signals for the
two lens images. In this regime, (provided that pulse broadening by scattering
is sub-dominant, see §3.1 for the appropriate condition) an observer would see
two pulses with similar spectro-temporal evolution (after de-dispersing the
signal) and with slightly different DM values (note that
$\Delta\mbox{DM}(\ell)/\mbox{DM}\sim\Delta t_{p}(\ell)/t_{p}<1$). This will
allow to establish that the lightcurve is a superposition of two lens images.
However, the time delay in this regime is no longer a direct proxy for the
lens mass. In particular, the connection between magnification and time-delay
that one calculates for gravitational lensing from a point source 777This
connection is found by relating the gravitational + geometric components of
$\Delta t$ in eq. 52 to the relative magnification of the two images (eq. 15).
The result (when plasma scattering can be ignored) is $\Delta t=2GM_{\rm
l}\left(\frac{\gamma-1}{\gamma}+\log\gamma\right)/c^{3}$ where $\gamma$ is the
ratio of magnifications of the two images (see e.g. Yang et al. 2021). will be
modified. This consideration places a lower limit on lens masses above which
plasma-delay is insignificant,

$M_{\rm min,t}\sim 0.05\,M_{\odot}\,{\mbox{DM}_{2}^{12\over 7}{\rm
min}\big{\\{}d_{\rm LO,22},d_{\rm SL,22}\big{\\}}^{5\over
7}\over\nu_{9}^{24\over 7}L_{22}^{6\over 7}\ell_{\rm max,17}^{4\over 7}}.$
(57)

Surveys at high frequencies can explore smaller lens masses as $M_{\rm
min,t}\propto\nu^{-24/7}$.

The scenario we have described thus far considers the lens to be located
inside a plasma screen. However, the calculation can be easily extended to a
wider set of possibilities. For instance, when the plasma density in the lens-
plane is small but the photon trajectories corresponding to the two images
pass through a turbulent plasma screen somewhere between the source and the
observer, equation (54) can be used for calculating the travel time difference
along different photon trajectories by substituting $\ell\sim 2\theta_{\rm
E}d_{\rm LO}\times f_{d}$. The factor $f_{d}=d_{\rm PO}/d_{\rm LO}$ when the
plasma screen is between the lens and the observer, and $f_{d}=d_{\rm
SP}/d_{\rm SL}$ when the plasma lies between the source and the lens; where
$d_{\rm PO}$ ($d_{\rm SP})$ is the distance between observer (source) and the
plasma screen. The factor $f_{d}$ accounts for the fact that the distance
between the photon trajectories for the two images in the plasma plane is
smaller than $\sim 2R_{\rm E}=2\theta_{\rm E}d_{\rm LO}$ by the factor
$f_{d}^{-1}$.

The effect of plasma on micro-lensing in the FRB host galaxy is similar to the
effect we have discussed above.

####  3.3.1 IGM turbulence

The size of the largest eddy in the IGM, $\ell_{\rm max}$, is highly uncertain
by several orders of magnitude. It could be as large as 1024cm – the scale for
energy deposition into the IGM by AGN jets and outflows from galaxy clusters –
or as small as a 1020cm. If the Mach number of the IGM turbulence were to be
$\xi_{m}$ then $\ell_{\rm max}\lower
2.0pt\hbox{$\,\buildrel\scriptstyle>\over{\scriptstyle\sim}\,$}10^{24}\xi_{m}^{3}$,
otherwise the heating of the IGM by dissipation of turbulent energy will raise
its temperature on a time scale smaller than the Hubble time, which is
contradicted by observations; the IGM data show that the mean temperature is
$\sim 10^{4}$K and is not increasing with decreasing redshift.

Therefore, taking $\ell_{\rm max}\sim 10^{22}$cm, $L\sim d_{\rm LO}$, $d_{\rm
SL}\sim 10^{28}$ cm and ${\rm DM}\sim 10^{3}$ pc cm-3 for the IGM turbulence,
we find from eq. (57) that the time delay between two images of an object due
to IGM turbulence is larger than the gravity+geometry effects for lens mass
$\lower
2.0pt\hbox{$\,\buildrel\scriptstyle<\over{\scriptstyle\sim}\,$}10^{-3}M_{\odot}$.
We note that according to our estimates, scattering of radio waves by IGM
turbulence is much weaker than expected for FRB host galaxy and the Milky Way
ISM, which is consistent with Cordes et al. (2022) & Ocker et al. (2022).

##  4 Synergistic effects of plasma scintillation and gravitational lensing

So far we have focused on ways in which plasma scattering suppresses the
magnification, smears out lightcurves of the two images, and changes time
delays of gravitational lensing events. These ultimately limit the range of
parameter space in which FRB lensing can be observed and used for cosmology.
In this section, we show that plasma scattering combined with gravitational
lensing can also, under certain circumstances, lead to unique constraints on
cosmology or on the nature of the environment of the FRB source.

###  4.1 Using FRB lensing delays to constrain cosmology

Consider a situation in which $\theta_{\rm scat}^{\prime}\ll\theta_{\rm E}$
and $t_{\rm sc}\sim(2R_{\rm s}/c)(\theta_{\rm scat}^{\prime}/\theta_{\rm
E})^{2}<t_{\rm FRB}<\Delta t_{g}+\Delta t_{p}$. As described in §3.1, §3.3, in
this situation the FRB lightcurve will appear as consisting of two distinct
and similarly shaped components due to the lensing as the time delay between
the images is long compared to the intrinsic FRB duration which in turn is
longer than the scintillation broadening timescale (we have assumed
$\theta_{\rm s}\approx\theta_{\rm E}$ in the expression above for clarity and
since this is the most relevant case of interest). The plasma broadening for
each component of the lightcurve corresponding to an image may still be
detectable. Furthermore, the temporal broadening of each component will be
slightly different due to the different columns of turbulent eddies along the
photon trajectories for the two images and the slightly different DMs along
the two paths. Indeed, estimating the DM separately for the two components of
the FRB lightcurve will provide us with an estimate of $\Delta\mbox{DM}(\ell)$
along the two photon trajectories that are separated by a distance $\ell$. The
physical separation between the two paths, in the lens plane, is typically of
order $\sim R_{\rm E}$. As an example, for a $10^{11}M_{\odot}$ lens at a
cosmological distance, we can measure $\Delta\mbox{DM}$ on a scale of $5\mbox{
kpc}$. This corresponds to a tiny angular separation, $\Delta\theta\sim
10^{-6}$ rad. So, lensing could be used to probe density fluctuations on this
small length scale, which cannot be done by two unrelated FRBs (because of
their large typical angular separation) or any other means currently available
to us. Lensing, therefore, provides us with a unique way to constrain the
fluctuations of $\mbox{DM}_{\rm IGM}$, and explore the patchy reionization at
$z>6$. Moreover, lensing of a repeating FRB might be useful for probing the
time dependence of density fluctuation. It should be noted that the size of
ionizing bubbles at the end of the reionization epoch is much larger, $\sim
10$ Mpc (Wyithe & Loeb, 2004), so only minor fluctuations in $\mbox{DM}_{\rm
IGM}(z)$ are expected on the angular scales that are probed by lensing. Still,
given that at large redshift, $\mbox{DM}(z\gtrsim 6)\approx 6000\mbox{pc
cm}^{-3}$ Beniamini et al. (2021), even very small fractional changes in
$\mbox{DM}_{\rm IGM}$ may still be detectable.

Figure 4:  Schematic figure demonstrating how multi-path propagation by
gravitational micro-lensing leads to different parts of the FRB wave
accumulating different phases ($\phi_{i}$) and rotation ($\chi_{i}$) of their
electric wave vector. In this setup, the resulting signal, which is a
superposition of the images, will generally be elliptically polarized, even if
the magnetic field changes on a spatial scale ($\ell_{\chi}$) that is large
compared to the visible size of the plasma screen ($R_{\rm sc}$) in the
absence of the gravitational lens.

Figure 5: Minimum lens masses that can be probed with FRB lensing. A
detectable time delay between the separate images of a lensed FRB and a
magnification above unity, can only be seen above the diagonal solid line.
Larger magnifications require greater lens mass to overcome the blurring
effect of turbulence ($M_{\rm min,\mu}\propto\mu_{\rm max}^{2}$, see eq. 33).
$M_{\rm min,\mu}(\mu_{\rm max}=10)$ is shown by a dashed line. A vertical line
denotes the value for $\min(d_{\rm SL},d_{\rm LO})$ above which the lensed
images become scatter broadened. The temporal delay between images of a lensed
FRB is dominated by the geometrical + gravitational delay ($\Delta t_{g}$),
rather than the plasma delay, above the dot-dashed line. $\Delta t_{g}$ is
larger than the intrinsic FRB duration ($t_{\rm FRB}$) above the dotted line.
These plots demonstrate that the limiting criteria for detection of FRB
lensing are typically given by $M_{\rm min,\mu}$. Results are plotted as a
function of $d_{\rm min}=\min(d_{\rm LO},d_{\rm SL})$ and assuming that the
plasma screen is in the vicinity of the gravitational lens. Other parameters
assumed for plotting these figures are: $d_{\rm SO}=2\cdot
10^{28}\mbox{cm},\mbox{DM}=10^{2}\mbox{pc cm}^{-3},\mu_{\rm
max}=1,\nu=1\mbox{GHz},t_{\rm FRB}=1\mbox{ ms}$. Different panels represent
different values of $L,L/\ell_{\rm max}$ (corresponding to different values of
$\theta_{\rm scat}$ listed in each panel title); where $L$ is the width of the
plasma screen & $\ell_{\rm max}$ is the size of largest eddies in the
turbulent medium.

###  4.2 Induced circular polarization in the presence of gravitational
lensing

A radio burst passing through a plasma with inhomogeneous density and magnetic
field strength, can become partially circularly polarized and possibly also
depolarized when it reaches the observer (Beniamini et al., 2022). The
circular polarization stems from the fact that rays reaching the observer at a
given time, have propagated through different segments of the plasma and by
doing so have accumulated different phases and different degrees of Faraday
rotation. For a plasma screen, this effect is important when
$\ell_{\chi}<R_{\rm sc}$, where $\ell_{\chi}$ is the separation (along the
plane of the plasma screen) over which the difference in rotation between two
waves going through the screen is of order unity and $R_{\rm sc}$ is the
scattering radius, which is the visible size of the screen.

Gravitational micro-lensing can enhance the effect of an inhomogeneous plasma
screen if the projection of the Einstein radius on the plasma screen is,
$R_{\rm E}f_{d}>R_{\rm sc}$ (where we have assumed here that the distance
between the images in the lens plane is $\sim R_{\rm E}$ for the case in which
lensing causes significant magnification). In this case, assuming Kolmogoroff
turbulence, the condition for induced circular polarization by multi-path
propagation becomes

$\displaystyle f_{d}\frac{R_{\rm E}}{\ell_{\chi}}>1\rightarrow$ $\displaystyle
110\frac{f_{d}\mbox{RM}_{4}^{6/5}(M/100M_{\odot})^{1/2}\min\\{d_{\rm
LO,28},d_{\rm SL,28}\\}^{1/2}}{\nu_{9}^{12/5}\ell_{\rm
max,18}^{2/5}L_{20}^{3/5}}>1$ (58)

where RM is measured in units of $\mbox{rad m}^{-2}$. Eq. 4.2 shows that if
(for example) the plasma screen is in the vicinity of the gravitational lens,
then gravitational micro-lensing can enhance the induced circular
polarization. An additional requirement for this to occur is that the lens
should cause significant magnification of the source, as described in §3.1 and
figure 2 (such that there are multiple images that contribute significantly to
the measured flux). The degree of induced circular polarization is of order
unity in case this inequality is satisfied, and of order the L.H.S. otherwise.

Having multiple paths due to gravitational (rather than plasma) lensing means
that it is not essential for the magnetic field in the different paths to be
originating from a single, strongly turbulent plasma screen. Instead,
different rays may be intersecting plasma with different properties. In order
for the degree of rotation to significantly change between two such paths, we
require that

$\Delta\chi=0.07\nu_{9}^{-2}\left(B_{\mu G}\Delta\mbox{DM}+\Delta B_{\mu
G}\mbox{DM}\right)>1$ (59)

At the same time, light from the two paths must interfere, meaning that

$\Delta t_{p}+\Delta t_{g}<t_{\rm FRB}\sim 1\mbox{
ms}\rightarrow\Delta\mbox{DM}<0.23\nu_{9}^{2}\left(1-\Delta t_{g}/\mbox{
ms}\right)$ (60)

It is possible to simultaneously satisfy both eqs. 59 and 60 with
astrophysically plausible parameters. For example, if the lens is a stellar
mass black hole binary losing mass through wind at the Eddington rate, its
wind may extend up to $\sim 0.1$ pc. A ray intersecting this wind at $\sim
10^{16}$ cm (slightly less than the typical Einstein radius for such a black
hole), would acquire an excess DM of order $\Delta\mbox{DM}\sim
10^{-3}\mbox{pc cm}^{-3}$ (for a $10M_{\odot}$ black hole). At the same
distance, the wind has a magnetic field strength of order $B\sim 10\mbox{
mG}\sigma_{-2}^{1/2}$ where $\sigma$ is the magnetization parameter. For these
parameters we have $\Delta\chi\sim 1\nu_{9}^{-2}$, demonstrating that the
conditions 59 and 60 are satisfied at $\nu\lesssim 10^{9}$ Hz and that this
setup would result in strong circular polarization.

##  5 Conclusions and discussion

We have investigated in this work how gravitational lensing of point radio
sources, such as fast radio bursts (FRBs), by a point mass is affected by a
plasma screen between the source and the observer. The main results we found
are summarized and discussed below.

Much of the analysis in this work was presented for the case in which the
plasma screen is co-located with the gravitational lens. However,
gravitational lensing by stars is likely to preferentially occur in old and
massive elliptical galaxies. These galaxies have a more tenuous ISM, which
reduces the chance for the radio waves encountering a plasma scattering screen
within the same galaxies. This motivates us to consider the extent to which
plasma scattering in the FRB host galaxy or the Milky Way can affect
gravitational lensing. When the plasma screen is at a general location
relative to the gravitational lens, and the scattering angle for photons
through the plasma is $\theta_{\rm scat}$, the image magnification is capped
at $\theta_{\rm E}/2\theta_{\rm scat}^{\prime}$; where $\theta_{\rm
scat}^{\prime}=\theta_{\rm scat}f_{d}d_{\rm SL}/d_{\rm SO}$, S,L,O,P stand for
the source, lens, observer, and plasma screen, $d_{\rm XY}$ is the distance
between $X$ and $Y$ and $f_{d}=d_{\rm PO}/d_{\rm LO}$ ($f_{d}=d_{\rm
SP}/d_{\rm SL}$) when the plasma is between the observer and the lens (source
and lens). This effectively translates to a lower limit on the lens mass,
$M_{\rm min,\mu}$ (eq. 33) that can be probed with FRB lensing. For masses
below $M_{\rm min,\mu}(\mu_{\rm max}=1)$, the presence of gravitational
lensing cannot be inferred from either the magnification or by spotting a
duplicate copy of the signal in the lightcurve that is delayed w.r.t. the
first component. In particular, we note that $M_{\rm min,\mu}\propto
f_{d}^{-2}$, and that means that plasma effects on lensing magnification and
pulse broadening are strongly suppressed for gravitational lenses at
cosmological distances when plasma scattering takes place far away from the
lens either in the FRB host galaxy or in the Milky Way.

As a specific example for the role of plasma suppressed gravitational lensing,
consider the case of FRB 20191221A. This bright FRB is particularly remarkable
due to its long duration ($\sim 3$ s), and 217 ms periodicity (Chime/Frb
Collaboration et al., 2022). One possibility explored by the authors to
explain the uniqueness of this burst is that it could be an ordinary extra-
galactic pulsar that has been micro-lensed by its binary companion and the
flux amplified by a factor of $\sim 10^{11}$. Chime/Frb Collaboration et al.
(2022) pointed out the highly unlikely geometry required for this tremendous
magnification. The problem is made much worse by non-zero scintillation in the
host galaxy, which limits the lens magnification ($\mu$) to $\sim 10^{5}$. The
lensing model requires $d_{\rm SL}\sim 10\mbox{pc},d_{\rm LO}\sim
1\,\mbox{Gpc},{\rm and}\,M_{\rm l}>10^{6}M_{\odot}$. The Einstein angle for
these parameters is $\theta_{E}\sim 10^{-12}$rad. If the scattering angle for
the plasma in the host galaxy were to be similar to the Milky Way galaxy, i.e.
$\theta_{\rm scat}\sim 10^{-9}$rad or $\theta^{\prime}_{\rm scat}\sim
10^{-17}$rad, then the maximum magnification is limited to
$\theta_{E}/2\theta^{\prime}_{\rm scat}\sim 10^{5}$ due to broadening of the
angular size of the source. This is much smaller than the required $\mu\sim
10^{11}$. The required magnification can be achieved provided that $M_{\rm
l}\lower 2.0pt\hbox{$\,\buildrel\scriptstyle>\over{\scriptstyle\sim}\,$}7\cdot
10^{17}\theta_{\rm scat,-9}^{2}$ (see eq. 26), which is unphysical even when
we take into account the uncertainty in the value of $\theta_{\rm scat}$.

The lensing probability, $\tau(>\mu)$, is modified by scintillating plasma as
discussed in §3.2. The probability is suppressed for high magnification events
as waves scattered by the turbulent medium between the source and the observer
increase the angular size of the source to $\sim\theta^{\prime}_{\rm scat}$,
and this reduces the magnification. A corollary of this is that $\tau(>\mu)$
is no longer proportional to $\mu^{-2}$ for $\mu\gg 1$, but falls off more
steeply.

Different dispersion measure (DM) along photon trajectories for the two
different images introduces an extra time delay between the two images of a
transient source. The distance between the photon trajectories in the lens
plane is of order the Einstein radius ($R_{E}$), and the extra time delay
($\Delta t_{p}$) is proportional to $R_{E}^{5/6}$ due to waves traveling
through a turbulent medium with Kolmogoroff spectrum for density fluctuation.
The effect is most severe for stellar and sub-stellar mass lens when $\Delta
t_{p}$ is comparable to the gravitational time delay or the duration of FRBs.
The implication of this plasma introduced extra time delay between the two
copies of an FRB is that the observed delay is no longer a direct proxy for
the lens mass, and that the plasma delay must be corrected for determining the
cosmic abundance of stellar mass dark matter. At the same time, the fact that
$|\Delta t_{p}|>0$, can under certain circumstances, also be an advantage.
When $\theta_{\rm scat}^{\prime}\ll\theta_{\rm E}\mbox{ \& }\Delta
t_{g}+\Delta t_{p}>t_{\rm FRB}$, the existence of lensing can be inferred from
the lightcurve, while the different DM along the different image trajectories
allow us to measure $\Delta\mbox{DM}_{\rm IGM}$ on tiny angular scales,
$\lesssim 10^{-6}$ rad, which cannot be explored by any other observational
means. This technique might also turn out to be useful in the study of patchy
reionization history of the universe.

Gravitational lensing can convert a linearly polarized source to partial
circular polarization when the time delay between the images is less than the
coherence time for the source. The reason for this is that photons traveling
through a magnetized plasma suffer different amounts of rotation of the
electric vector (different rotation measures, RM), and different phase shifts,
along the two image paths, thereby resulting in some degree of circular
polarization.

There is a lens mass above which image magnification and time delays are not
affected much by the turbulent plasma between the source and observer. This
minimum lens mass is shown in figure 5 for a few different combinations of
length scales associated with the plasma screen and the distance of the plasma
screen and the lens from the source/observer. Lensing by $\lesssim
10^{2}M_{\odot}$ objects may be strongly suppressed by plasma scattering and
the pulses of the lensed images become temporally overlapping. That makes it
hard to spot gravitational lensing of FRBs without resorting to specialized
analysis that can accurately remove different scatter-broadening of the two
images in the lightcurve. An additional consideration that the relative plasma
delay for the two images should be short compared with the geometrical +
gravitational delay turns out to be easier to satisfy, as it typically kicks
in at lower lens mass than quoted above. The third constraint is that the
scatter broadening timescale, $t_{\rm sc}$, should be short compared to
$t_{\rm FRB}$ in order that the two lensed images have a similar temporal
profile so that observers could identify the lensing event888It might be
possible to identify an FRB lensing event even when the temporal profiles of
the images look very different due to scattering by turbulent plasma along the
different photon trajectories of the images by e.g. a cross-correlation
analysis that deconvolves the different scatter broadening for different
images.. This is satisfied when $\min(d_{\rm SL},d_{\rm LO})$ is smaller than
a critical distance, $d_{\rm sc}$ (when the plasma plane is separated from the
lens plane, the condition becomes $\min(d_{\rm SP},d_{\rm PO})<d_{\rm sc}$);
where $d_{\rm sc}$ is given by eqns. 28 & 34. The plasma scattering reduces
rapidly with frequency (see eqns. 33 and 57). As a result, lensing of FRBs is
easier to observe at higher frequencies.

The effect of scintillation on gravitational lensing was considered also by
Katz et al. (2020). They focused on a particular lensing scenario where the
lightcurves of the images overlap and produce interference fringes. Much of
their work was devoted to the analysis of the effect of this interference on
the observed spectrum. By contrast, the present work describes how
scintillation affects the magnification, lensing probability, image
lightcurves and their time delays. Katz et al. (2020) analysis is for the
regime where $\Delta t_{g}^{-1}$ is larger than the spectral resolution of the
detector, but smaller than the coherence bandwidth of the source, and that
translates to lens masses in the range $10^{-4}M_{\odot}<M_{\rm
l}<10^{-1}M_{\odot}$ as per these authors. The “Time domain" lensing effects
discussed in this paper should be applicable to a larger range of lens mass
and wider parameter space of scintillating plasma screen. Furthermore, the
condition $\Delta t_{p}>\Delta t_{g}$ is expected to apply for
$10^{-4}M_{\odot}<M_{\rm l}<10^{-1}M_{\odot}$ (see figure 5), and that means
that even for the parameter space explored by Katz et al. (2020), the
determination of lens mass from the lensing time delay signal is affected by
the physical effects described in this work.

## Acknowledgments

This work has been funded in part by an NSF grant AST-2009619. PB’s research
was supported by a grant (no. 2020747) from the United States-Israel
Binational Science Foundation (BSF), Jerusalem, Israel. Some of the work
presented here was carried out while PK was visiting the Yukawa Institute,
Kyoto. He is grateful to Kunihito Ioka for his hospitality, for organizing an
FRB workshop during that visit, and for many stimulating science discussions.
He would like to acknowledge many excellent discussions with Kohta Murase who
was also a visitor at the Yukawa Institute. He would like to thank YITP for
the financial support provided under the Visitors’ Program of FY2022, and
YITP-W-22-18 for funds for the worshop. We are indebted to James Cordes for
numerous comments on the draft and for pointing out several previous work on
this topic that we were not familiar with. We would also like to thank Bing
Zhang, Ue-Li Pen, Liam Connor, Casey Law, Julian Muñoz and Marc Kamionkowski
for helpful comments on the manuscript.

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# Luminance-Guided Chrominance Image Enhancement for HEVC Intra Coding

Hewei Liu∗, Renwei Yang∗, Shuyuan Zhu∗, Xing Wen† and Bing Zeng∗ This work was
supported by the National Natural Science Foundation of China under Grant
U20A20184 and Grant 62031009. The first and second authors contributed equally
to this work. ∗School of Information and Communication Engineering
University of Electronic Science and Technology of China, Chengdu, China
†Kuaishou, Beijing, China

###### Abstract

In this paper, we propose a luminance-guided chrominance image enhancement
convolutional neural network for HEVC intra coding. Specifically, we firstly
develop a gated recursive asymmetric-convolution block to restore each
degraded chrominance image, which generates an intermediate output. Then,
guided by the luminance image, the quality of this intermediate output is
further improved, which finally produces the high-quality chrominance image.
When our proposed method is adopted in the compression of color images with
HEVC intra coding, it achieves 28.96% and 16.74% BD-rate gains over HEVC for
the U and V images, respectively, which accordingly demonstrate its
superiority. The code is available online:
https://github.com/Nickyang4900/Luminance-Guided-Chrominance-Enhancement-for-
HEVC-Intra-Coding.

###### Index Terms:

Luminance, chrominance, enhancement, HEVC, intra coding.

##  I Introduction

With the rapid development of multimedia technology, there has been a huge
demand for high-definition content. Currently, high efficiency video coding
(HEVC) [1] is widely used in the compression of high-definition content.
However, still images compressed by HEVC intra coding often suffer from the
compression artifacts, especially at the low bit-rate. Recently, the new
generation of video coding standard, i.e., versatile video coding (VVC) [2]
has been proposed. VVC offers more excellent rate-distortion performance
compared with HEVC. However, the high computational complexity of VVC limits
its application in practice. Improving the performance of HEVC is still very
necessary and challenged. Enhancing the quality of compressed image offers a
simple but effective way to obtain a high coding efficiency. Over the past few
years, the convolutional neural network (CNN) based solution is becoming more
and more popular to achieve this goal.

The CNN models are normally used to compose the end-to-end schemes to reduce
the compression artifacts [3, 4]. In [5], the CNN-based in-loop filter was
designed to replace the sample adaptive offset (SAO) [6] in HEVC. In [7], a
multi-channel long-short-term dependency residual network was proposed to
enhance the quality of video frames based on the temporal correlation of
frames. The CNN-based post-processing methods were proposed in [8] and [9] for
the HEVC-coded images. More specifically, a variable-filter-size residue-
learning was designed in [8] to implement the quality enhancement and a deep
CNN-based auto decoder was proposed in [9] to achieve the same goal. In [10],
several enhancement networks were designed to improve the quality of different
coding tree units which finally achieves a overall quality gain for the whole
image. Meanwhile, the coding information was also adopted in the CNN-based
quality enhancement solution. For example, in [11], the motion residue was
utilized to improve the video quality. The coding unit mask was defined in
[12] as the prior information for the CNN-based post-processing. Also, the
transform unit partition map was employed in [13] to improve the quality of
video frames.

Besides the coding information, the channel correlation between the luminance
image (Y) and the chrominance images (U and V) may also be utilized to improve
the quality of compressed images. However, most of the existing CNN-based
methods process each channel independently, which ignores the channel
correlation and cannot achieve a high efficiency. In this work, we propose a
luminance-guided chrominance image enhancement method for HEVC intra coding.
More specifically, the degraded chrominance image is firstly fed into the
recursive asymmetric-convolution block (GRAB) to obtain an intermediate
output. Then, the feature of luminance image is obtained via the feature
extraction block (FEB). Finally, the intermediate output and the luminance
feature are combined together to reconstruct the final chrominance images. The
pipeline of our proposed method is illustrated in Fig. 1.

The rest of this paper is organized as follows. The motivation of our proposed
method is given in Section II. Our proposed method is introduced in Section
III. The experimental results are presented in Section IV and the conclusions
are drawn in Section V.

Figure 1: Pipeline of our proposed method. Figure 2: An example of YUV-420
format coding by HEVC intra coding, where the quantization parameter is 22.
Figure 3: Architectures of GRAB.

##  II Motivation

In color image compression, the input RGB image is normally converted into the
YUV format, including one luminance image (Y) and two chrominance images,
i.e., U and V. Both the U and V images are then spatially down-sampled to form
the YUV-420 format. The RGB-YUV conversion is implemented as

$\\!\begin{bmatrix}Y\\\ U\\\
V\end{bmatrix}\\!=\\!\begin{bmatrix}\\!0.2126&0.7152&0.0722\\\
-0.1146&-0.3854&0.5000\\\
0.5000&-0.4542&-0.0458\\!\end{bmatrix}\\!\\!\begin{bmatrix}\\!R\\\ G\\\
B\\!\end{bmatrix}\\!+\begin{bmatrix}\\!16\\\ 128\\\ 128\\!\end{bmatrix}\\!.$

(1)

According to Eq. (1), although the luminance and chrominance images are
obtained by combining the red (R), green (G) and blue (B) channels with
different weighting factors, they are still highly correlated. We present some
YUV-420 coding results of HEVC intra coding in Fig. 2. It is found from Fig. 2
that the compressed luminance image may potentially offer some detailed
textures to enhance the compressed chrominance image. More specifically, the
content of the luminance image is much clear but the texture of the
chrominance image is blurred due to the spatial down-sampling and compression.
The clear luminance textures may potentially provide prior information to
enhance the quality of the chrominance image, which will finally offer an
overall quality gain for the whole image.

##  III Our Proposed Method

###  III-A Overview

Our proposed method is designed for the YUV-420 format coding after the RGB-
YUV conversion. Its pipeline is illustrated in Fig. 1, where the decoded
chrominance image is firstly enhanced by GRAB, and then is further enhanced
with the guidance of the decoded luminance image. We stack four feature
extraction blocks (FEB) to extract the prior information from the luminance
image. In our work, the intermediate chrominance output is passed through a
$3\times 3$ convolution kernel with stride = 1. Therefore, we adopt the
$3\times 3$ convolution kernel with stride = 2 after four FEBs to guarantee
the luminance features and the chrominance features with the same size. To
fuse the guided features offered by the luminance image, the features of the
intermediate chrominance output and the ones of the luminance image are
combined together via the element-wise adding. Moreover, we employ six
convolution layers with the $3\times 3$ kernel size and stride = 1 in the last
reconstruction procedure to yield the final chrominance output.

###  III-B Gated Recursive Asymmetric-convolution Block

The proposed GRAB is used to produce the intermediate output for the
chrominance image, and it is composed by six recursive units (RU). The
recursive design adopted in our work aims at avoiding the dramatic increasing
of model parameters when the network depth increases [14]. The architecture of
GRAB is given in Fig. 3. In GRAB, each RU consists of four asymmetric units
(AU) [16], including a shallow one and three deep ones, a convolution layer,
and a gate coefficient (GC) [19]. Due to the adoption of skip connection [17],
RU only needs to learn the residual between the source high-quality image and
the compressed low-quality input image, which leads to a simple training[18].
Except for the given values of GC, the six RUs share the same architecture and
weighting parameters. In each RU, AU is designed with various branches, thus
it is able to strengthen the network capability to obtain diverse features
from input [15, 16]. Each AU consists of three $1\times 1$ convolution layers,
two $3\times 3$ convolution layers, one average pooling and Leaky ReLU. The
structure of AU is given in Fig. 4, and it is implemented as

$z=c_{1}(x)+c_{3}(c_{1}(x))+AvgPool_{3}(c_{1}(x))+c_{3}(x)$ (2)

where $c_{i\in\\{1,3\\}}(\cdot)$ represents the convolution operation with the
$i\times i$ kernel size, and $AvgPool_{3}$ denotes the average pooling with
the 3$\times$3 slide window. In our work, the shallow and deep AUs share the
same structures, but employ different weighting parameters, which can
effectively reduce the parameters.

The gate coefficient was initially proposed in [19] to solve the image super
resolution problem. We adopt it in this work to construct the depth-wise
attention mechanism so that the network can be trained to assign different
weights to the features of different depths, which potentially leads to a good
reconstruction result. Moreover, we do not employ the channel attention
methods [20, 21] in our work to avoid the high computational cost. The gate
coefficient $\beta$ adopted in GRAB is used as the weighting factor for each
RU and it is achieved via multiplication as

$\displaystyle z=\beta\cdot x.$ (3)

Let us use $s(\cdot)$ and $d(\cdot)$ to represent the operation of the shallow
and deep AUs, respectively. Then, in the $i$-th RU, we can obtain the output
of the last AU

$AU_{4}(y_{i-1})=d(d(d(s(y_{i-1}))+s(y_{i-1}))+s(y_{i-1}))$ (4)

where $i\in\\{1,\ldots,6\\}$, and $y_{i}$ and $\beta_{i}$ denote the output
and GC of the $i$-th RU, respectively. Note that $y_{0}$ is the output
obtained by inputting the compressed chrominance image to a convolution layer
whose kernel size is $3\times 3$. Finally, the procedure of passing through
the $i$-th RU is formulated as

$\displaystyle\begin{array}[]{l}\mathop{y}_{i}=\beta_{i}\cdot
conv(AU(y_{i-1}))+y_{i-1}.\\\ \end{array}$ (6)

Figure 4: Architectures of AU.

###  III-C Feature Extraction Block

FEU is designed to extract features of the luminance image and it is designed
with the skip connection structure, allowing the combination of shallow and
deep convolution layer outputs. This structure not only strengthens the
ability to extract features, but also makes the training more easily [14]. The
architecture of the FEB is illustrated in Fig. 5. In this unit, the feature
maps of all the depths directly connect to the last layer, which combines the
shallow and deep features from the input image. In our work, FEB is
implemented as

$f_{FEB}=\sum_{i=1}^{4}{\gamma}_{i}$ (7)

where ${\gamma}_{i}$ is the output of the $i$-th convolution layer, i.e.,
${\gamma}_{i}=ReLU(W_{i}\ast{\gamma}_{i-1})$, $\ast$ denotes the convolution
operation and $W_{i}$ is the weighting parameter of the convolution layer. In
this work, the ReLU activation function $ReLU$ is employed to obtain
${\gamma}_{i}$. Moreover, ${\gamma}_{0}$ should be the input image for the
first FEB in in our proposed network.

Figure 5: Architectures of FEB. Figure 6: Classical image dataset. TABLE I: Comparison of BD-rate. | ASN-1 | Ours-1 | ASN-2 | Ours-2
---|---|---|---|---
| U | V | U | V | U | V | U | V
| Classical | -9.06% | -8.91% | -9.60% | -9.68% | -22.38% | -11.39% | -21.92% | -14.33%
| McMater | -9.77% | -12.01% | -10.23% | -12.28% | -30.02% | -15.51% | -30.99% | -17.56%
| Kodak | -14.67% | -15.07% | -15.23% | -15.47% | -33.21% | -17.51% | -33.98% | -18.33%
| Average | -11.17% | -12.00% | -11.68% | -12.47% | -28.54% | -14.80% | -28.96% | -16.74%
|  |  |  |  |  |  |  |  |

##  IV Experimental Results

###  IV-A Experiment Setup

We verify the effectiveness of our proposed method by employing it to process
the HEVC-compressed color images, where the HM 16.7 [23] is adopted in this
work. Moreover, we compare the performance of our proposed method with the
other existing methods. The first 800 color images of Flickr2K dataset [24]
are selected as the training dataset. Moreover, these images are converted
into YUV-420 format and compressed by HEVC intra coding with the quantization
parameters (QP) 22, 27, 32 and 37. The compressed chrominance images,
including both the U and V images, are cropped into the $32\times 32$ non-
overlapping sub-images, and the compressed luminance images are cropped into
the $64\times 64$ non-overlapping sub-images. We train all the CNN models by
using these generated sub-images.

TABLE II: Comparison of $\Delta$PSNR (dB) at QP 22. | ASN-1 | Ours-1 | ASN-2 | Ours-2
---|---|---|---|---
| U | V | U | V | U | V | U | V
Classical | 0.191 | 0.164 | 0.187 | 0.164 | 0.466 | 0.133 | 0.491 | 0.203
McMater | 0.244 | 0.364 | 0.237 | 0.349 | 0.860 | 0.322 | 0.940 | 0.427
Kodak | 0.371 | 0.460 | 0.365 | 0.441 | 0.919 | 0.455 | 1.035 | 0.498
Average | 0.269 | 0.329 | 0.263 | 0.318 | 0.748 | 0.303 | 0.822 | 0.376
|  |  |  |  |  |  |  |
TABLE III: Comparison of $\Delta$PSNR (dB) at QP 27. | ASN-1 | Ours-1 | ASN-2 | Ours-2
---|---|---|---|---
| U | V | U | V | U | V | U | V
Classical | 0.304 | 0.269 | 0.319 | 0.297 | 0.791 | 0.325 | 0.781 | 0.444
McMater | 0.388 | 0.516 | 0.401 | 0.527 | 1.369 | 0.653 | 1.430 | 0.752
Kodak | 0.515 | 0.567 | 0.530 | 0.582 | 1.304 | 0.632 | 1.367 | 0.688
Average | 0.402 | 0.450 | 0.417 | 0.469 | 1.155 | 0.537 | 1.193 | 0.628
|  |  |  |  |  |  |  |
TABLE IV: Comparison of $\Delta$PSNR (dB) at QP 32. | ASN-1 | Ours-1 | ASN-2 | Ours-2
---|---|---|---|---
| U | V | U | V | U | V | U | V
Classical | 0.337 | 0.360 | 0.360 | 0.384 | 0.948 | 0.509 | 0.909 | 0.623
McMater | 0.469 | 0.568 | 0.503 | 0.589 | 1.641 | 0.802 | 1.692 | 0.908
Kodak | 0.571 | 0.619 | 0.606 | 0.645 | 1.506 | 0.784 | 1.528 | 0.797
Average | 0.459 | 0.515 | 0.490 | 0.539 | 1.365 | 0.699 | 1.376 | 0.776
|  |  |  |  |  |  |  |
TABLE V: Comparison of $\Delta$PSNR (dB) at QP 37. | ASN-1 | Ours-1 | ASN-2 | Ours-2
---|---|---|---|---
| U | V | U | V | U | V | U | V
Classical | 0.349 | 0.392 | 0.404 | 0.467 | 0.946 | 0.620 | 0.947 | 0.718
McMater | 0.493 | 0.589 | 0.522 | 0.595 | 1.576 | 0.828 | 1.628 | 0.925
Kodak | 0.585 | 0.674 | 0.598 | 0.685 | 1.401 | 0.775 | 1.453 | 0.861
Average | 0.476 | 0.552 | 0.508 | 0.582 | 1.308 | 0.741 | 1.343 | 0.835
|  |  |  |  |  |  |  |

###  IV-B Implementation Detail

In the network training, the batch size is set to 32 and the total epoch is
set to 40. We start with a learning rate of 10-4 and decay the learning rate
with 0.1 at the 20th epochs. We first train the whole model at QP = 37 from
scratch and the models at QP = 22, 27 and 32 are fine-tuned based on this
trained model. Our proposed network is implemented based on Pytorch and
optimized by the Adam optimizer via minimizing the $l_{1}$ loss. All the
models are trained with the NVIDIA GTX 2080TI GPU. Moreover, all models are
only trained to enhance the quality of the U-image and they are also applied
to improve the quality of the V-image.

###  IV-C Performance Comparison

To make a comprehensive comparison, three popular image datesets are utilized
in our experiments, including 1) the classical image dataset, containing 9
color images as shown in Fig. 6, 2) McMater dataset [25], containing 18 color
images, and 3) Kodak dataset [26], containing 24 color images. These test
datasets are compressed by HEVC at QP 22, 27, 32 and 37. The default HM is set
as benchmark, and we test the quality enhancement performance in terms of BD-
Rate and $\Delta$PSNR. The proposed approach is compared with adaptive-
switching neural network (ASN) [12] which also introduces the guidance image
to enhance the quality of the HEVC-compressed images.

We present the average BD-rate and average $\Delta$PSNR results over all the
test images of each dataset in Tables I-V, respectively, where Ours-1 and
Ours-2 represent our proposed method without and with the guidance of the
Y-image, respectively. Similarly, we use ASN-1 and ASN-2 to represent the ASN
method without and with the guidance of the Y-image, respectively.

Firstly, one can see from Table I that our proposed method achieves 28.96% (on
average) BD-rate gain over HEVC for the U-images and 16.74% (on average) gain
for the V-images. Moreover, for most test image datasets, our approach offers
higher BD-rate gain than ASN, not only for the processing of the U-images, but
also for the processing of the V-images. Secondly, according to the
$\Delta$PSNR results presented in Tables II-V, our proposed method does not
perform as well as ASN when it is applied to process the chrominance images
without the guidance image, especially at QP = 22. After the guidance image is
introduced in our method, the significant performance gain is achieved.
Meanwhile, when the images are compressed with larger QP, our method achieves
higher $\Delta$PSNR (on average). Thirdly, all the results given in Tables I-V
demonstrate that employing the luminance image to guide the enhancement of the
chrominance images can provide remarkable gain for our proposed method.

##  V Concluding Remarks

In this paper, we propose a luminance-guided chrominance image enhancement
method for HEVC intra coding. We design the recursive asymmetric-convolution
block to initially restore each degraded chrominance image. Then, the
compressed luminance image is introduced as the guidance to further improve
the quality of the chrominance image. The experimental results demonstrate
that our proposed approach achieves 28.96% and 16.74% BD-rate gains over HEVC
for the U- and V-images, respectively, when it is adopted in the compression
of color images.

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