# Regimes in astrophysical lensing: refractive optics, diffractive optics, and
the Fresnel scale

Dylan L. Jow,2,3 Ue-Li Pen,1,2,3,4,5,6 and Job Feldbrugge7,8
1Institute of Astronomy and Astrophysics, Academia Sinica, Astronomy-
Mathematics Building, No. 1, Section 4, Roosevelt Road, Taipei 10617, Taiwan
2Canadian Institute for Theoretical Astrophysics, University of Toronto, 60
St. George Street, Toronto, ON M5S 3H8, Canada
3Department of Physics, University of Toronto, 60 St. George Street, Toronto,
ON M5S 1A7, Canada
4Perimeter Institute for Theoretical Physics, 31 Caroline St. North, Waterloo,
ON, Canada N2L 2Y5
5Canadian Institute for Advanced Research, CIFAR program in Gravitation and
Cosmology
6Dunlap Institute for Astronomy & Astrophysics, University of Toronto, AB
120-50 St. George Street, Toronto, ON M5S 3H4, Canada
7Higgs Centre for Theoretical Physics, James Clerk Maxwell Building, Edinburgh
EH9 3FD, UK
8Department of Physics, Carnegie Mellon University, 5000 Forbes Ave,
Pittsburgh, PA 15217, USA E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

Astrophysical lensing has typically been studied in two regimes: diffractive
optics and refractive optics. Diffractive optics is characterized by a
perturbative expansion of the Kirchhoff-Fresnel diffraction integral, while
refractive optics is characterized by the stationary phase approximation.
Previously, it has been assumed that the Fresnel scale, $R_{F}$, is the
relevant physical scale that separates these two regimes. With the recent
introduction of Picard-Lefschetz theory to the field of lensing, it has become
possible to generalize the refractive description of discrete images to all
wave parameters, and, in particular, exactly evaluate the diffraction integral
at all frequencies. In this work, we assess the regimes of validity of
refractive and diffractive approximations for a simple one-dimensional lens
model through comparison with this exact evaluation. We find that, contrary to
previous assumptions, the true separation scale between these regimes is given
by $R_{F}/\sqrt{\kappa}$, where $\kappa$ is the convergence of the lens. Thus,
when the lens is strong, refractive optics can hold for arbitrarily small
scales. We also argue that intensity variations in diffractive optics are
generically small, which has implications for the study of strong diffractive
scintillation (DISS).

###### keywords:

waves – radio continuum: ISM – pulsars:general – fast radio bursts

††pubyear: 2021††pagerange: Regimes in astrophysical lensing: refractive
optics, diffractive optics, and the Fresnel scale–A

##  1 Introduction

When coherent sources, such as pulsars and fast radio bursts (FRBs), undergo
lensing, wave effects become important. Pulsar scintillation, for example, is
due to the interference of many coherent images formed due to scattering in
the interstellar medium (ISM). As more data of these coherent sources becomes
available from current and future observations, there has been increased
interest in the study study of wave optics in the context of astrophysical
lensing (see e.g. Grillo & Cordes, 2018; Main et al., 2018; Feldbrugge et al.,
2019; Feldbrugge & Turok, 2020; Feldbrugge, 2020; Jow et al., 2020; Shi & Xu,
2021).

In principle, wave optics is fully described by the Kirchhoff-Fresnel
diffraction integral, which is a path integral over all paths connecting the
source and observer through a refractive medium. In practice, however, the
full diffraction integral is rarely computed, as there exist few analytic
solutions for lenses of interest, and its highly oscillatory nature makes it
challenging to compute numerically. It has only been through the recent
application of the mathematical framework of Picard-Lefschetz theory that
computing the diffraction integral exactly has become numerically tractable
(see e.g. Feldbrugge et al., 2019; Feldbrugge & Turok, 2020; Feldbrugge, 2020;
Jow et al., 2021; Shi & Xu, 2021). Previously, astrophysical lensing has
typically been studied in the geometric regime, where the contribution to the
observed flux is isolated to a discrete set of well-defined images whose
individual fluxes can be easily computed. Wave effects are then added, to
first order, by computing the phase associated with each image and allowing
them to coherently interfere at the observer. This is also known as the
Eikonal limit, which is equivalent to the stationary-phase approximation of
the diffraction integral111Note that some authors make a distinction between
geometric optics and the Eikonal limit of wave optics, where geometric optics
is when the images are added incoherently at the observer and the Eikonal
limit is when they are added coherently. Here we will use the terms “geometric
optics” and “Eikonal limit” interchangeably to refer to the regime where
isolated images interfere coherently..

Geometric optics, or the Eikonal limit, is the high-frequency limit of the
Kirchhoff-Fresnel integral. The low-frequency limit of optics has also been
studied, as it is often possible to compute the Kirchhoff-Fresnel integral
through a perturbative expansion. In the plasma lensing literature, these two
limits of optics (geometric and perturbative) are sometimes referred to as the
refractive and diffractive regimes, respectively, as the geometric limit is
characterized by a discrete set of isolated images that are refracted into the
direction of the observer, whereas the perturbative regime is characterized by
contributions to the diffraction integral from an extended area on the sky
(see e.g. Rickett, 1990; Lorimer & Kramer, 2012).

It is of central importance to the study of wave optics to understand at what
frequency these two regimes become valid, and whether or not there is any
overlap. In the absence of an exact result for the Kirchhoff-Fresnel integral,
it has often been assumed in the literature that the Fresnel scale sets the
frequency which separates geometric optics from the diffractive regime (see
e.g. Cordes et al., 1986; Lorimer & Kramer, 2012; Pen & King, 2012; Johnson et
al., 2018; Er et al., 2022). In this work, however, we will show that the
Fresnel scale is, generically, the wrong scale. Rather, a combination of the
Fresnel scale and lens strength sets the correct separation scale between the
geometric and diffractive regimes.

It is by employing the mathematical framework of Picard-Lefschetz theory that
we are able to make a detailed study of the transition from refractive optics
to diffractive optics. Not only does Picard-Lefschetz theory allow for an
exact evaluation of the Kirchhoff-Fresnel integral at all frequencies that we
can quantitatively compare to the various approximations, it also allows for a
well-defined separation of the integral into contributions from a discrete set
of images at all frequencies, which, in the high-frequency limit, agrees with
the geometric images. As such, Picard-Lefschetz theory bridges the conceptual
gap between the image description of geometric optics and the perturbative
description of the diffractive regime.

This paper is structured in the following way. In Section 2, we will review
the stationary-phase approximation which is used to compute geometric optics,
as well as the perturbative expansion which can be used to approximate the
diffraction integral in the low-frequency limit. In Section 3, we will study a
simple 1D rational lens to argue that most of parameter space is well
described by either of these two approximations, and that the separation scale
between the two regimes is given by the Fresnel scale divided by the square-
root of the convergence. Contrary to what has previously been assumed in much
of the lensing literature, the separation scale is not the Fresnel scale
alone. Indeed, for either large or small lens strengths, the Fresnel scale may
be off by orders-of-magnitude. In Section 4, we will argue that lensing in the
diffractive regime is typically weak, and therefore an observation of strong
scintillation already suggests lensing in the refractive regime. In Section 5
we show how the two regimes behave differently in delay space. Our numerical
results are quite general and can be applied to scenarios in gravitational
lensing and plasma lensing, as well as potential applications to scattering
theory in quantum mechanics. However, in Section 6, we discuss in greater
detail the connection between our results and previous results in the
literature on turbulent plasma lensing. In Appendix A, we present the detailed
Picard-Lefschetz analysis of the rational lens.

We note that while our focus will be astrophysical lensing, our results may be
applicable to problems in scattering theory in quantum mechanics. In
particular, while we focus on single-plane lensing, the formalism for multi-
plane lensing is equivalent to the real-time Feynman path integral in the
limit of infinite planes using the time-slicing approximation. In this
setting, the diffractive limit corresponds to the application of perturbative
methods resulting in the Feynman rules, whereas the refractive limit
corresponds to non-perturbative methods such as the use of instantons and the
study of (complex) classical trajectories. The tools of Picard-Lefschetz
theory, which we use here to evaluate the Kirchoff-Fresnel integral, have been
recently used to evaluate the Feynman path integral to re-interpret phenomena
in quantum mechanics (Turok, 2014; Cherman & Unsal, 2014; Tanizaki & Koike,
2014; Gozzi et al., 2021).

##  2 The perturbative expansion and Eikonal approximation of wave optics

Figure 1: Diagram of thin-plasma lensing showing distances and co-ordinates
involved.

The flux at $y$ of a source lensed by a thin screen can be expressed as the
magnitude $|F(y)|^{2}$, where the amplitude $F$ is obtained from the
dimensionless Kirchhoff-Fresnel diffraction integral

$F(y)=\sqrt{\frac{\nu}{2\pi
i}}\int_{-\infty}^{\infty}e^{i\nu[\frac{(x-y)^{2}}{2}+\kappa\psi(x)]}dx\,,$
(1)

with the phase variation induced by the lens given by $\psi$ and where we
integrate over all possible paths from the source to the observer intersecting
the lens plane at $x$. For convenience, we use the dimensionless version of
the Kirchhoff-Fresnel integral with the dimensionless relative position of
source and observer, frequency, and lens strength $y$, $\nu$, and $\kappa$.
Fig. 1 shows a diagram of the lensing setup, with dimensionful parameters. The
dimensionless parameters in Eq. (1) are related to these by $x=\hat{x}/a$,
where $a$ is some length scale associated with the lens,
$y=\hat{y}/a=(\hat{x}_{s}d_{l}+\hat{x}_{o}d_{sl})/ad_{s}$, and $\nu=\omega
a^{2}/c\overline{d}$ where $\overline{d}=d_{sl}d_{l}/d_{s}$. In the following
sections, we will refer only to the dimensionless parameters so that our
analysis can be easily applied to a wide variety of contexts. We will further
discuss the underlying physical parameters in the context of plasma and
gravitational lensing in Sections 6 and 7. See also Schneider et al. (1992),
Nakamura & Deguchi (1999), Feldbrugge et al. (2019) and Jow et al. (2021) for
a more thorough discussion of the correspondence of these dimensionless
parameters to physical parameters in different contexts.

In this work, we restrict our attention to the case of a one-dimensional lens,
with a single peak, and we make a choice of co-ordinates $x$ and $y$ such that
the peak is centred at $x=0$ and has curvature $\psi^{\prime\prime}(0)=-2$.
Under this choice of coordinates, the parameter $\kappa$ is the convergence of
the lens (Pen & King, 2012; Schneider et al., 1992). A canonical example of
such a lens is the Gaussian lens, $\psi(x)=\exp\\{-x^{2}\\}$, which has been
used in many lensing studies (see, e.g., Clegg et al., 1998; Pen & King, 2012;
Cordes et al., 2017). One may also construct rational approximations of the
Gaussian lens, such as the simple rational lens, $\psi(x)=1/(1+x^{2})$, which
corresponds to the first-order Padé approximation of the Gaussian potential.
For the sake of simplicity, we will also restrict our attention to convergent
lenses, i.e. $\kappa>0$.

The Kirchhoff-Fresnel integral is equivalent to a path integral through the
lens plane with phase
$S(x,y;\nu,\kappa)=\nu[\frac{(x-y)^{2}}{2}+\kappa\psi(x)]$. Geometric optics
is obtained in the high-frequency limit ($\nu\to\infty$), through the
stationary-phase approximation. That is, in geometric optics, the
contributions to the Kirchhoff-Fresnel integral reduce to a discrete set of
isolated points, or images, $\\{x_{j}\\}$, which have stationary phase,
$\partial_{x}S(x_{j})=0$. The condition of stationary phase gives rise to the
lens map,

$\xi(x)\equiv y=x+\kappa\psi^{\prime}(x),$ (2)

which maps points on the lens plane to a source position $y$. For a given
source position, the images, $\\{x_{j}\\}$, with stationary phase, are
solutions to the lens equation, $\xi(x)=y$. Note that while the map $\xi$ is
one-to-one, the inverse map $\xi^{-1}$ need not be, and a given source
position may correspond to multiple images.

In the geometric, or stationary-phase approximation, each individual image has
an associated wave-field given by

$F^{\mathrm{geom.}}_{j}=\frac{1}{|\Delta_{j}|^{1/2}}e^{iS(x_{j},y;\nu,\kappa)-i\frac{\pi}{2}n_{j}},$
(3)

where $\Delta_{j}=\xi^{\prime}(x_{j})$ is the Jacobian of the lens map, and
$n_{j}$ is the Morse index, which is $0$ or $1$ when $x_{j}$ is a maximum or
minimum of the lens map, respectively. The total wave-field at the observer in
this limit is given by $F^{\mathrm{geom.}}=\sum_{j}F^{\mathrm{geom.}}_{j}$
(Schneider et al., 1992). This is also sometimes referred to as the Eikonal
limit of wave optics. This approximation is valid in the limit $\nu\to\infty$,
except at caustics, which are points where the lens map is degenerate, and
$\Delta_{j}=0$. At caustics, the intensity of the degenerate images formally
goes to infinity. Despite this divergence, geometric optics is an extremely
useful approximation in the high-frequency limit as the regions around the
caustics at which the divergence occurs typically contribute very little
overall flux.

So far we have assumed the lens map is a real map. However, generically the
lens potential, $\psi$, and the associated lens map may be analytically
continued so that the lens map can admit complex solutions (i.e. the images,
$x_{j}$, may be complex). For such images, the wave-field is given by (Grillo
& Cordes, 2018)

$F^{\mathrm{geom.}}_{j}=\frac{1}{|\Delta_{j}|^{1/2}}e^{i[S(x_{j},y;\nu,\kappa)-\frac{\mathrm{arg}(\Delta_{j})}{2}]}.$
(4)

Such an analytic continuation of the lens map actually improves the geometric
approximation, and, indeed, is necessary to capture the behaviour of lensing
just outside of caustics (Wright, 1980; Grillo & Cordes, 2018; Jow et al.,
2020). Note, however, that not all complex solutions to the lens equation will
contribute to the total flux. In general, determining whether or not a complex
saddle point is relevant, i.e. whether or not it contributes to the total
integral, requires computing the contour of steepest ascent in the function
$h(x)=\mathrm{Re}\\{iS(x,y;\nu,\kappa)\\}$ in the complex plane, originating
at the saddle point. As we discuss in Appendix A, according to Picard-
Lefschetz theory, whether or not the saddle point is relevant depends on
whether its steepest ascent contour intersects the real axis. The saddle point
is relevant if and only if such an intersection exists. See, e.g., Jow et al.
(2021) for a more detailed discussion of imaginary images in lensing. Note
that real images are always relevant, because the steepest ascent contour
originating from a real saddle point always intersects the real axis at the
saddle point.

Now, the geometric or Eikonal limit is the high-frequency limit of the
Kirchhoff-Fresnel integral. In order to apply this limit, it is necessary to
know at what frequency it becomes a good approximation of the diffraction
integral. It has sometimes been assumed that this frequency is set by the
Fresnel scale (see e.g. Pen & King, 2012; Er et al., 2022),

$y_{\mathrm{Frnl.}}\equiv 1/\sqrt{\nu}.$ (5)

That is, if $y_{\mathrm{Frnl.}}\ll 1$, then one is assumed to be firmly in the
geometric regime. Note that $y_{\mathrm{Frnl.}}$ is the dimensionless version
of the Fresnel scale given our parametrization of the Kirchhoff-Fresnel
integral. Typically, the Fresnel scale is given as a distance scale,
$R_{F}=\sqrt{\lambda\overline{d}/2\pi}$, where $\lambda$ is the wavelength of
the light. In these units, the relevant limit is when the Fresnel scale is
much smaller than the angular scale of the lens. One of the purposes of this
work is to argue that the assumption that the Fresnel scale sets the frequency
above which geometric optics holds is generically false.

Another tool for approximating the Kirchhoff-Fresnel diffraction integral is
the perturbative expansion. In this approximation, one expands
$\exp\\{i\nu\kappa\psi(x)\\}=1+i\nu\kappa\psi(x)+...$ in the integrand of the
diffraction integral, giving

$F^{\mathrm{pert.}}(y)=1+i\kappa\nu\sqrt{\frac{\nu}{2\pi i}}\int
e^{i\nu\frac{(x-y)^{2}}{2}}\psi(x)dx+\mathscr{O}((\kappa\nu)^{2}).$ (6)

The small parameter that is being expanded in is given by

$\epsilon\equiv\kappa\nu,$ (7)

which we define here as it will later become of central importance.

The utility of performing this expansion is that for certain potentials, the
first term is easier to compute analytically or numerically. For example, for
the Gaussian lens, $\psi(x)=\exp\\{-x^{2}/2\\}$, the perturbative expansion
can be computed to first order analytically as

$F^{\mathrm{pert.}}(y)=1+i\epsilon\sqrt{\frac{\nu}{2i+\nu}}e^{-\frac{\nu
y^{2}}{2i+\nu}}\,.$ (8)

Similarly, for the rational lens, $\psi(x)=1/(1+x^{2})$, one can compute the
first-order expansion analytically as

$\displaystyle F^{\mathrm{pert.}}(y)$
$\displaystyle=1+i\frac{\epsilon}{2}\sqrt{\frac{-i\nu\pi}{2}}\Big{[}e^{i\frac{\nu}{2}(y+i)^{2}}\big{(}1+\mathrm{erf}\\{\sqrt{\frac{i\nu}{2}}(y+i)\\}\big{)}$
$\displaystyle+e^{i\frac{\nu}{2}(y-i)^{2}}\big{(}1+\mathrm{erfc}\\{\sqrt{\frac{-i\nu}{2}}(1+iy)\\}\big{)}\Big{]},$
(9) $\displaystyle\approx
1+i\epsilon\sqrt{\frac{-i\nu\pi}{2}}e^{-\nu|y|}e^{i\frac{\nu}{2}(y^{2}-1)},$
(10)

where the last approximation is valid for large $\nu|y|$. The first-order
integral can be computed by first taking the integral in Fourier space by
taking the Fourier transform of the potential, $\tilde{\psi}(k)=e^{-|k|}$.

When the perturbative expansion is valid, the flux at the observer can be
described as a small oscillatory modulation about an un-lensed image. That is,
when $\kappa=0$ (i.e. when there is no lens), $F(y)\equiv F_{0}(y)=1$,
exactly. Thus, in the perturbative limit, the flux is described by a small
modulation about the un-lensed flux, $F^{\mathrm{pert.}}(y)=F_{0}(y)+i\epsilon
g(y)$, for some function $g(y)$. This is what is sometimes referred to as the
diffractive regime (Rickett, 1990), and it is qualitatively very distinct from
geometric optics (a.k.a. the refractive regime). Whereas in geometric optics
multiple isolated images form and interfere at the observer, the diffractive
regime is effectively described by a single background image plus a small
modulation.

Fig. 2 shows a schematic of the kind of lensing we are considering. A source
emits radiation at some distance far from the lens and observer. The waves
from the source encounter some lensing potential, $\psi(x)$, localized to the
lens plane. This induces a phase variation in the wave front, which then
propagates un-interrupted towards the observer, forming an interference
pattern in the observer plane. The bottom panel shows the observed intensity
computed for the rational lens for a fixed $\kappa=5$ and different values of
$\nu$. For $\nu=0.01$, the lensing is firmly in the diffractive limit and the
intensity is well-described by a small oscillatory modulation about unity
(note that in Fig. 2 the intensity fluctuations for $\nu=0.01$ are multiplied
by a factor of ten so that all three curves can be plotted on the same axis).
For $\nu=2$, the lensing is firmly in the refractive regime and is described
by an interference pattern of multiple images. The curve for $\nu=0.1$ is in a
regime where neither the perturbative/diffractive or Eikonal/geometric
approximations are good approximations.

The middle panel of Fig. 2 shows the light rays that pass through a given
observer position, $y$. These rays correspond to the geometric images that an
observer at $y$ observes. Multiple rays may pass through an observer, and
regions in the observer plane with different numbers of images per source
position are separated by caustics (the caustics are easily visualized in the
middle panel of Fig. 2). Now the field at the observer plane can be computed
by assuming that each ray corresponds to a point source at the lens plane
emitting a spherical wave with a relative phase determined by the lens. The
field at the observer is then a sum of all of these spherical waves (this is
known as the Huygens-Fresnel principle). Geometric optics arises when the
frequencies are large, and so all the waves associated with rays that do not
pass through the observer position $y$ cancel out. Thus, the observed
intensity in geometric optics is completely determined by the rays passing
through a neighbourhood of the observer. In particular, the intensity of the
individual images is given by the density of the rays passing through a small
neighbourhood of the observer. Thus, in the geometric regime, the intensity
curve traces the density of the rays passing through the observer plane. As
$\nu$ decreases, the cross section of rays that contribute to the observed
intensity at a given source position $y$ becomes finite. As such, the
interference between multiple images that is easily visible in the geometric
regime gets washed out.

Figure 2: Schematic of thin-plane lensing. The source emits waves that scatter
off of the lens potential, $\psi(x)$, localized to a lens plane (top panel).
The lens induces a phase shift in the wave-front, which then propagates to the
observer. The middle panel shows the light rays propagating from lens to
observer, which correspond to the paths of stationary phase connecting source
to observer, and are always perpendicular to the wave-front. The observed
intensity as a function of observer position is shown in the bottom panel for
fixed $\kappa=5$ and different values of $\nu$ for the lens potential
$\psi(x)=1/(1+x^{2})$. Note that the intensity fluctuations for $\nu=0.01$ are
multiplied by a factor of ten so that all three curves can be visualized on
the same axis. The grey curve corresponds to the incoherent sum
($\sum_{j}|F_{j}|^{2}$) of the intensities of the images in the geometric
limit.

Note, that the perturbative expansion that gives rise to diffractive optics
explicitly converges when the quantity $\epsilon\ll 1$. This is our first hint
that the Fresnel scale may not be the relevant quantity for assessing when
geometric optics holds. Given how qualitatively distinct the perturbative
(diffractive) and geometric (refractive) descriptions are, the expectation is
that there would not be a large region of parameter space where the two
regimes overlap. However, if, indeed, geometric optics is valid whenever
$y_{\mathrm{Frnl.}}=1/\sqrt{\nu}\ll 1$ (or, equivalently, when $\sqrt{\nu}\gg
1$), and the diffractive regime occurs whenever $\epsilon=\kappa\nu\ll 1$,
then these two regimes should overlap for any given $\nu$ greater than unity,
so long as $\kappa$ is chosen to be sufficiently small. Another consequence of
this would be that there would be large regions of parameter space that could
neither be described by geometric optics, nor by the perturbative/diffractive
expansion. As such, some alternative description of optics in this regime
would be called for.

We argue that this is not, in fact, the case. Rather, contrary to previous
assumptions, the Fresnel scale alone is not the appropriate quantity for
determining when geometric optics is valid. Instead, a combination of the
Fresnel scale and the convergence, $\epsilon=\kappa\nu$, sets both the scale
below which the perturbative expansion is valid ($\epsilon\ll 1$), as well as
the scale above which geometric optics holds ($\epsilon\gg 1$). In the next
section, we will study a specific lens model to make this argument more
concrete. In particular, by using Picard-Lefschetz theory to exactly evaluate
the Kirchhoff-Fresnel integral at all frequencies, we can quantitatively
determine when each of the two approximations we have described are valid.

##  3 A simple rational lens

We will study the simple rational lens with potential given by

$\psi(x)=\frac{1}{1+x^{2}}.$ (11)

The lens map is given by

$\displaystyle\xi(x)\equiv y=x-\frac{2\kappa x}{(1+x^{2})^{2}},$ (12)

and its Jacobian takes the form

$\displaystyle\xi^{\prime}(x)=1+\frac{2\kappa(3x^{2}-1)}{(1+x^{2})^{3}}.$ (13)

The images corresponding to a position of the observer $y$ – defined by the
identity $\xi(x)=y$ – are roots of the fifth-order polynomial

$x^{5}-yx^{4}+2x^{3}-2yx^{2}+(1-2\kappa)x-y=0\,.$ (14)

The quintic polynomial has five roots, either one or three of which being real
depending on the position $y$. The real images are always relevant to the
integral. The complex images emerge as conjugate pairs (since the polynomial
is real), of which one is potentially relevant. By evaluating the
intersections of the steepest ascent contour of the complex images and the
real line, we determine the relevance of each complex saddle point (see
Appendix A for a detailed discussion).

The perturbative/diffractive approximation of the lens integral is given by
Eq. (9). We have chosen to study the rational lens as it has an analytic
potential with a similar form to the more widely studied Gaussian lens;
indeed, the potential is the first-order Padé approximation of the Gaussian
potential $\psi(x)=\exp{(-x^{2}/2)}$. Thus, we expect the rational lens to
behave similarly to the Gaussian lens. However, since the lens equation
reduces to a simple fifth-order polynomial, there is only a small number of
potentially relevant images, in contrast with the Gaussian lens, for which a
potentially infinite number of images may be relevant, making the evaluation
of the Kirchhoff-Fresnel integral much simpler for the rational lens. Once we
have found the relevant images for the rational lens, we can evaluate the
geometric or Eikonal approximation of the Kirchhoff-Fresnel integral. Our goal
is to compare the regimes of validity of the geometric approximation and the
perturbative/diffractive approximation given by Eq. (9).

In order to make this comparison, we use Picard-Lefschetz theory to
numerically evaluate the Kirchhoff-Fresnel integral. As we discuss in Appendix
A, Picard-Lefschetz theory allows for an exact evaluation of the diffraction
integral at arbitrary frequencies. Thus, we are able to compare the various
approximations to the exact value of the integral. In particular, Fig. 3 shows
the average minimum residual over a range of source positions, $y$, between
the intensity computed by either the perturbative expansion or the
geometric/Eikonal approximation with the exact value of the intensity as a
function of $\nu$ and $\kappa$. That is, we define the residual to be

$\displaystyle\text{Res}=\frac{1}{2L}\text{min}$
$\displaystyle\bigg{[}\int_{-L}^{L}||F^{\rm
pert.}(y)|^{2}-|F(y)|^{2}|\mathrm{d}y,$ (15)
$\displaystyle\int_{-L}^{L}||F^{\rm
geom.}(y)|^{2}-|F(y)|^{2}|\mathrm{d}y\bigg{]}/\max_{-L\leq y\leq
L}(|F(y)|^{2}),$

where we normalize the residual by the maximum value of the intensity. The
exact intensity is computed using Picard-Lefschetz theory, as described in
Appendix A. The residual is small when one of the approximations is valid, and
becomes large when neither is valid. Note that the geometric/Eikonal
approximation diverges for a small region around caustics. To avoid this
divergence, we choose $L=\frac{3}{4}y_{c}$, where $y_{c}$ is the location of
the fold caustic. That is, for the rational lens, when $\kappa>\frac{1}{2}$,
the lens exhibits a caustic at $y=\pm y_{c}$ where
$y_{c}=\frac{3\sqrt{3}}{8}\kappa$ for large $\kappa$. Since most of the flux
is contained within the region $-y_{c}<y<y_{c}$, we choose $L$ such that we
are within this region, in order to avoid the caustics. For
$\kappa<=\frac{1}{2}$, we choose $L=2$.

Fig. 3 shows the residual computed in this way as a function of $\kappa$ and
$\epsilon=\kappa\nu$. From the residual, we can see that both approximations
fail at $\epsilon=1$, which is shown as the red, dashed line. As $\epsilon$
decreases, the diffractive/perturbative approximation becomes valid, whereas
when $\epsilon$ increases, the geometric approximation becomes valid. Thus,
the line $\epsilon=1$ forms a boundary between the regimes of validity of
these two approximations. This suggests that $\epsilon$ is the appropriate
scale that determines when geometric optics holds, as opposed to the Fresnel
scale which only depends on the frequency $\nu$. The $\nu=1$ line is shown as
the white, dashed line. If, indeed, the Fresnel scale were the appropriate
separation scale, one would expect the geometric approximation to always hold
above this line, and always fail below it. This is clearly not the case,
however, as for small $\kappa$, the geometric approximation appears to fail
even when $\nu>1$, and for large $\kappa$ it can remain valid for $\nu<1$.

Fig. 3 also highlights the boundary at $\kappa=1/2$, which is where the cusp
caustic occurs. The cusp occurs when the two fold caustics merge, and
$y_{c}\to 0$. The increase in the residual around this line is due to the fact
that as $y_{c}\to 0$, our selection of the range $-y_{c}<y<y_{c}$ no longer
avoids the caustics. Note, however, that as $\epsilon$ increases, the region
around the caustic for which the geometric/Eikonal approximation diverges
becomes smaller. We label the two regions to the left and right of
$\kappa=1/2$ weak and strong geometric optics, respectively, as it is only
when $\kappa>1/2$ that multiple real images are formed by the lens, as we
discuss further in Section. 4.

Figure 3: The minimum average residual between the exact value of the
intensity of the Kirchhoff-Fresnel integral (computed using Picard-Lefschetz
theory) with the perturbative expansion or geometric/Eikonal approximation
over a range of source positions, $y$. This residual is defined explicitly in
Eq. (15). For $\kappa>{1/2}$, we choose $L=y_{c}$, where $y_{c}$ is the
location of the fold caustic. For $\kappa<1/2$, we choose $L=2$. The residual
is shown as a function of $\kappa$ and $\epsilon=\kappa\nu$. The regions are
labelled according to the approximation which minimizes the residual. The
labels “strong" and “weak" refer to the strong and weak lensing regimes
discussed in Section 4 and corresponding to Table 1. The red, white, and black
dashed lines correspond to the lines $\epsilon=1$, $\nu=1$, and $\kappa=1/2$,
respectively.

Fig. 4 further examines the transition from the geometric regime to the
diffractive/perturbative regime by comparing the asymptotic behaviour of the
exact Kirchhoff-Fresnel integral, the perturbative expansion, and the
geometric/Eikonal approximation as a function of frequency for specific values
of $\kappa$ and $y$. Five values of $\kappa$ and $y$ were chosen,
corresponding to the five distinct regions discussed in Appendix A. In short,
these five regions correspond to the five distinct Lefschetz thimble
topologies for this lens, or, in other words, each region has a unique number
of real and relevant complex images. What is shown in Fig. 4 is not the total
intensity, $|F(y)|^{2}$, but rather the absolute value squared of the total
wave-field minus one, $|F(y)-1|^{2}$. This way we are comparing the value of
the asymptotic scaling of the perturbative part in the diffractive regime. The
exact value, computed using Picard-Lefschetz theory, is shown in blue, whereas
the perturbative expansion and geometric optics results are shown in purple
and green, respectively.

Figure 4: Comparison of the absolute value of the wave-field minus one,
$|F(y)-1|^{2}$, as a function of frequency for the perturbative/diffractive
expansion (dashed purple lines), geometric optics (dashed green lines), and
the exact evaluation of the Kirchhoff-Fresnel integral computed using Picard-
Lefschetz theory (blue). The curves are computed for five different values of
$\kappa$ and $y$, which correspond to values in the five topologically
distinct regions of the rational lens shown in Fig. 9.

As expected, regardless of the choice of source position and lens strength,
the geometric limit returns the correct asymptotic behaviour for large
frequencies (except at caustics), whereas the perturbative/diffractive limit
assumes the correct asymptotic behaviour for low frequencies. What is notable,
however, is that the frequency where this transition occurs depends on the
lens strength $\kappa$, and, in particular, is given to an order-of-magnitude
by $\epsilon=1$. In the region where both approximations agree with each other
(around $\epsilon=1$), neither approximation agrees with the true result, but
unlike with small frequencies, where geometric optics is off by orders-of-
magnitude, and large frequencies, where the perturbative expansion is off by
orders-of-magnitude, both are only off by a few factors of unity. It is
particularly striking that for large lens strengths, the geometric
approximation can hold for very low frequencies (for example, when
$\kappa=5,000$ and $y=1000$, geometric optics holds down to $\nu\sim
10^{-3}$). Similarly, when $\kappa$ is small, geometric optics can be made to
fail for large frequencies (when $\kappa=0.25$ and $y=0$, the geometric
approximation fails as early as $\nu\sim 10$).

We infer two things from these results. First, that, together, geometric
optics and the perturbative expansion can be used to accurately describe most
of parameter space, except for a small region where they overlap. Secondly,
the quantity that determines the separation between these two regimes is
$\epsilon=\kappa\nu$, as opposed to the Fresnel scale. Expressed in terms of
dimensionful quantities, $\epsilon=\kappa a^{2}/R_{F}^{2}$, where $a$ is the
size of the lens (see Section 6). Thus, the physical scale that separates the
diffractive and refractive regimes is given by $R_{F}/\sqrt{\kappa}$ (i.e.
when the size of the lens is much larger than this scale, refractive optics
holds, and when it is much smaller, diffractive optics holds). This means that
the convergence plays an important role in determining whether geometric
optics holds, rather than the frequency alone. Thus, the simple heuristic that
geometric optics is valid when the wavelength is small compared to the size of
the lens is generically false.

##  4 Strong scintillation in the refractive and diffractive regimes

The result of the previous section suggests that it cannot always be taken for
granted that short wavelengths automatically mean geometric optics holds.
Rather, one must also know the lens strength, or convergence, in order to
assess whether one is in the geometric or diffractive regime. At first glance,
this seems like a challenge, as the convergence is a much more difficult
quantity to measure. However, in this section, we will make a simple scaling
argument that an observation of large flux variations implies that the lensing
is taking place in the geometric regime. In other words, diffractive lensing
is typically weak. Thus, for many practical purposes (e.g. pulsar
scintillation and extreme-scattering events), any lensing observations are
likely to occur in the geometric optics regime. It is possible to define
strong versus weak lensing in many ways; however, in order to make a simple
scaling argument, here we will consider lensing to be strong when most of the
flux is localized to some finite region in the observer plane (as opposed to
being diffuse across the observer plane). We will again study the simple
rational lens to make our argument, and we will consider four limiting cases:
when $\kappa\gg 1$ and $\kappa\ll 1$, with $\epsilon\gg 1$ or $\epsilon\ll 1$.

As we have shown, when $\epsilon\gg 1$ we are in the geometric regime and when
$\epsilon\ll 1$ we are in the diffractive or perturbative regime. In either
case, outside of a certain region centred around the origin, the intensity
above unity, $|F|^{2}-1$, falls off exponentially. This is easy to see for the
perturbative/diffractive regime from Eq. (10). This is a little more difficult
to see for geometric optics, but as we show in Appendix A, for any $\kappa$
there exists a $y$ beyond which the only relevant images are a single real
image and one or two imaginary images. The real image is effectively
unperturbed and has an intensity of order unity, and the imaginary images have
an intensity that has an exponential dependence on the impact parameter that
goes like $~{}e^{-i\nu y}$. Thus, in both the geometric and diffractive
regimes there is a region in the observer plane (with an area we will call
$\sigma$) outside of which the intensity decays exponentially. We will
consider the lensing to be strong if the total rms intensity variations in
this region is of order unity or larger. If this is not the case, then most of
the displaced flux is in the exponentially damped region and we consider this
to be weak lensing. Fig. 5 shows an illustrative example computed for
$\kappa=5$ and $\nu=2$ in the geometric regime. Outside of a region of width
$\sigma$, the intensity peaks decay exponentially, and the rms intensity
fluctuations within the region are given by $\mu$. The total rms fluctuations
are then given by $m=\sigma\mu$, which is sometimes referred to as the
scintillation index. The lensing is considered weak when $m\ll 1$.

Figure 5: Intensity fluctuations as a function of $y$ for the rational lens
with $\kappa=5$ and $\nu=2$. Outside of a region of width $\sigma$, the
intensity fluctuations decay exponentially. Within the region, the intensity
fluctuations have an rms of $\mu$.

First, let us consider $\kappa\gg 1$. If $\epsilon\gg 1$, then we are in the
geometric regime. Since $\kappa>0.5$, the value of $\sigma$ is set by the
region between the two caustics (see Fig. 9). For $\kappa\gg 1$, the area
between the caustics scales as $\sigma\sim\kappa$. Now, when $\kappa$ is
large, using Eq. (13) and Eq. (4), the intensity fluctuations, which we will
call $\mu$, are of order $\sim\kappa^{-1}$. Thus, the total rms fluctuations
are $m=\sigma\mu\sim 1$, which we consider strong lensing. When $\epsilon\ll
1$, we are in the perturbative/diffractive regime. From Eq. (10), the cross
section is given by $\sigma\sim\nu^{-1}$, and the intensity fluctuations are
$\mu\sim\epsilon^{2}\nu$. Thus, in this regime,
$m=\sigma\mu\sim\epsilon^{2}\ll 1$. So, when $\kappa$ is large, strong lensing
is never described by the perturbative or diffractive regime.

Now, if $\kappa\ll 1$, there are no caustics. Thus, for $\epsilon\gg 1$ (the
geometric regime), the cross section is given by the width of the exponential
decay of the intensity of the imaginary images, which is $\sigma\sim\nu^{-1}$.
Expanding the intensity fluctuations in the geometric limit in terms of
$\kappa$, we obtain that the intensity fluctuations are $\mu\sim\kappa$. Thus,
the total rms fluctuations are $m=\sigma\mu\sim\kappa/\nu\ll\kappa^{2}\ll 1$,
since $\epsilon=\kappa\nu\gg 1$. When $\epsilon\ll 1$ (the perturbative
regime), we have again that $\sigma\sim\nu^{-1}$ and $\mu\sim\epsilon\nu$, so
$\sigma\mu\sim\epsilon$. Thus, when $\kappa\ll 1$, regardless of whether we
are in the refractive or diffractive regime, the lensing is weak.

Table 1 summarizes these results. We have argued that strong lensing generally
occurs when $\kappa\gg 1$ and $\epsilon\gg 1$, but not when either $\kappa\ll
1$ or $\epsilon\ll 1$. In other words, any lensing described by the
perturbative or diffractive regime is weak. Another way to make this argument
is to look at the peak intensity value for the perturbative expansion, which
occurs when $y=0$; in particular,
$|F^{\mathrm{pert.}}(0)-1|^{2}\sim\epsilon^{2}\nu=\kappa^{2}\nu^{3}$. In order
for this to be of order unity or larger and for the perturbative expansion to
hold, we require $\kappa^{2}\nu^{3}\gtrsim 1$ and $\kappa\nu\ll 1$, which
together imply that $\nu\gtrsim\kappa^{2/3}$ and $\kappa^{1/3}\ll 1$. Thus,
the maximum intensity fluctuations are only ever larger than unity in the
perturbative regime in a small region of parameter space when the lens
strength $\kappa$ is small. This has implications for plasma lensing
observations, as it suggests that strong diffractive interstellar
scintillation (sometimes referred to as DISS in the plasma lensing literature)
may never be a good description. In other words, extrapolating the
perturbative expansion to strong lensing regimes may generically lead to an
incorrect description of the lensing phenomenon. We discuss the implications
for plasma lensing further in Section 6.

$\kappa$ | $\epsilon=\kappa\nu$ | regime | cross section ($\sigma$) | intensity fluctuations ($\mu$) | $m=\sigma\mu$ | strong/weak
---|---|---|---|---|---|---
$\gg 1$ | $\gg 1$ | geometric / refractive | $\sim\kappa$ | $\sim\kappa^{-1}$ | $\sim 1$ | strong
$\gg 1$ | $\ll 1$ | perturbative / diffractive | $\sim\nu^{-1}$ | $\sim\epsilon^{2}\nu$ | $\sim\epsilon$ | weak
$\ll 1$ | $\gg 1$ | geometric / refractive | $\sim\nu^{-1}$ | $\sim\kappa$ | $\ll\kappa^{2}$ | weak
$\ll 1$ | $\ll 1$ | perturbative / diffractive | $\sim\nu^{-1}$ | $\sim\epsilon^{2}\nu$ | $\sim\epsilon$ | weak
Table 1: Summary of the scaling arguments for the strength of lensing in the
different regimes for the simple rational lens. Strong lensing never occurs
when $\kappa\ll 1$, or when the perturbative expansion holds.

##  5 Measuring time delays

In the previous section, we argued that strong scintillation generically
occurs in the geometric regime, whereas diffractive optics generically
produces weak scintillation, as the total rms intensity fluctuations only
becomes of order unity in the geometric regime. However, while the total rms
fluctuations may increase, even if the intensity fluctuations are large
(especially near caustics), if the resolution of an observation is too small,
the actual observed intensity will be an average over many oscillations, and
any large intensity fluctuations will be washed out. In particular, in
geometric optics, when $\kappa\gg 1$, the average fluctuations are
$\mu\sim\kappa^{-1}\ll 1$. Thus, even when the lensing is described by
multiple geometric images, it may still appear to be weak. How then are we to
distinguish geometric optics from diffractive/perturbative optics in such a
scenario? Fortunately, the two regimes behave qualitatively very distinctly in
delay space.

Observationally, the quantity one typically observes is the wave-field as a
function of frequency and time, $F(\nu,t)$, where the time dependence comes
from the relative motion of the source and lens as a function of time, $y(t)$.
Here we will focus on the wave-field as a function of frequency. Typically,
one actually observes the dynamic spectrum, which is the intensity, $|F|^{2}$,
as a function of frequency and time; however, novel phase retrieval techniques
have made observations of the underlying wave-field possible (Sprenger et al.,
2021).

Consider lensing in the geometric regime when multiple images are present.
Then the wave-field takes the form

$F^{\mathrm{geom.}}(\nu)=\sum_{j}\frac{1}{|\Delta_{j}|^{1/2}}e^{iS(x_{j},y;\nu,\kappa)-i\frac{{\rm
arg}(\Delta_{j})}{2}},$ (16)

as in Eq. (4). Now, consider taking a Fourier transform of the wave-field as a
function of frequency. In the geometric regime, the result is

$\hat{F}_{\nu}\propto\sum_{j}\frac{1}{|\Delta_{j}|^{1/2}}\delta(\tau-T(x_{j},y,\kappa)),$
(17)

where $\hat{F}_{\nu}$ denotes the Fourier transform of $F$ as a function of
$\nu$, $\tau$ is the conjugate variable of $\nu$, and
$T(x,y,\kappa)\equiv\partial_{\nu}S=\frac{1}{2}(x-y)^{2}+\kappa\psi(x)$ is the
dimensionless time-delay associated with an image at $x$. Thus, in the
geometric regime, the wave-field in delay space is a sum of delta functions
centred at well-defined time delays associated with each image. Conversely, as
we argue in Appendix A, in the low-frequency limit, the geometric images
become mutually coherent, so that the diffractive/perturbative wave-field,
rather than being characterized by an interference pattern of multiple
incoherent images, effectively becomes a single image with a small modulation
on top. Thus, one would not expect the diffractive/perturbative wave-field to
exhibit distinct images in delay space.

Figure 6: (Left column) The intensity $|F(\nu)|^{2}$ of the rational lens
computed for $\kappa=1000$ and $y=162$, for different ranges of frequency,
$\nu$. The frequency ranges are chosen such that the top panel occurs firmly
in the geometric regime ($\epsilon=\kappa\nu\gg 1$), the bottom panel occurs
in the perturbative regime ($\epsilon<1$), and the middle panel occurs in an
intermediate regime. (Right column) The Fourier transform of the wave-field as
a function of $\nu$, denoted by $\hat{F}_{\nu}(\tau)$, where $\tau$ is the
conjugate variable of $\nu$. The vertical axis shows
$|\hat{F}_{\nu}/\Delta\nu|^{2}$ where $\Delta\nu$ is the frequency bandwidth
for which the wave-field was computed. Under this normalization, the conjugate
wave-field in the geometric optics regime is a sum of isolated peaks centred
at the time delay for each image, with a height given by the individual
intensity of the image in the geometric optics approximation. The black X’s in
the right column correspond to each of the images’ individual intensities and
time delays. Picard-Lefschetz theory makes it possible to assign individual
intensities to the relevant images even at small frequencies. The horizontal
axis of the left column is shown in two units: units of the fundamental delay
resolution given by $(\Delta\nu)^{-1}$, and units of the relative time delay
$\Delta\tau$, which is the distance between the two peaks shown in the top
right panel.

Fig. 6 shows the intensity fluctuations and the conjugate wave-field,
$\hat{F}_{\nu}$, for the rational lens with $\kappa=1000$ and $y=162$, which
is halfway to the caustic ($y=y_{c}/2$). The lensing is computed for three
different frequency ranges, $\nu\in[0,0.002]$, $[0,0.01]$, and $[0,20]$, where
the ranges are chosen so that the lensing occurs firmly in the
diffractive/perturbative regime for the smallest range, firmly in the
geometric/Eikonal regime for the largest range, and neither for the
intermediate range.

For the choice of $\kappa$ and $y$, there are four relevant images: three
real, and one imaginary. In other words, the sum in Eq. (16) contains four
terms, where $x_{j}$ is purely real for three of those terms. As a result, the
conjugate wave-field for the geometric regime (the top panel) is, in
principle, a sum of four delta functions. Due to the finite bandwidth,
$\Delta\nu$, for which we calculate the wave-field, the conjugate wave-field
for the geometric regime is actually given by
$|F_{\nu}|^{2}\sim\sum_{j}(\tau-T(x_{j}))^{-2}$. The right column of Fig. 6
shows the amplitude of the conjugate wave-field normalized by the frequency
bandwidth, $|F_{\nu}/\Delta\nu|^{2}$. Under this normalization, the amplitude
of the individual peaks (in the geometric regime) are exactly given by
$|\Delta_{j}|^{-1}$. Thus, under this normalization, the height of the peaks
in delay space give the amplitude of the geometric images, and the location of
the peaks give the time delay associated with each image. Note that the top-
right panel of Fig. 6 only shows the peaks associated with two of the four
images. In geometric optics, the imaginary part of $x_{j}$ for the complex
image causes the amplitude to be exponentially suppressed, and so its
contribution to the total power is negligible compared to the other images.
The other real image is the “unperturbed" image centred at $\tau=0$, with
$|\Delta_{j}|^{-1}=1$. The top-right panel of Fig. 6 zooms in on the lensed
images so that they are easily resolvable by eye. Both of these images have
amplitudes of $|\Delta_{j}|^{-1}\sim\kappa^{-1}\sim 10^{-3}$. Thus, while the
intensity fluctuations shown in the top-left panel may be much larger,
$\sim\,0.2$, when the oscillations are averaged over (as might occur for an
observation with limited frequency resolution), the observed fluctuations
could potentially be as small as $10^{-3}$.

As we describe in Appendix A, one of the main conceptual advantages of using
Picard-Lefschetz theory to evaluate the Kirchhoff-Fresnel integral is that
even for low-frequencies, one can separate the integral into contributions
from a discrete set of images that correspond to the geometric limit as one
increases the frequency. The height of the black X’s in the right column of
Fig. 6 correspond to the amplitude of each of the relevant images computed
using Picard-Lefschetz theory, shown at the corresponding time delay. In
particular, one can see that while the imaginary image is exponentially
suppressed in geometric optics, and thus not shown on the top-right panel, as
the frequency decreases, the imaginary image becomes significant, and indeed
has an amplitude comparable to the real images in the bottom panel.

Now, while separating the wave-field into distinct contributions from the
relevant images is a mathematically well-defined procedure under Picard-
Lefschetz theory, we can see that the form of the conjugate wave-field in
delay space is qualitatively very different between the geometric and
diffractive/perturbative regimes. Specifically, the conjugate wave-field for
the perturbative regime is no longer a simple sum of delta functions. While,
regardless of frequency, there is always a peak centred at $\tau=0$ and with
height of unity corresponding to the unperturbed image (which always must be
present even in the absence of a lens), the lensed images cease to form well-
defined peaks in delay space, and instead merge into a single peak with some
width. The width of this merged peak increases as the frequency decreases.

Another way to think of this is by considering two fundamental time-scales in
delay space. First, there is the resolution scale given by $(\Delta\nu)^{-1}$.
That is, the maximum resolution in the variable $\tau$ for which one can
compute the conjugate wave-field is given by the inverse of the bandwidth over
which one observes the wave-field itself. The second time-scale one may
consider is the relative delay between the lensed images. Consider the two
real images (aside from the un-perturbed image), which we will call $x_{1}$
and $x_{2}$. Then this relative delay is
$\Delta\tau\equiv|T(x_{1})-T(x_{2})|$. This delay between the lensed images
can be thought of as the effective geometric width of the lens in delay space.

The conjugate variable $\tau$, scaled by both of these time-scales, is shown
in Fig. 6, with the values of $\tau\Delta\nu$ being shown on the bottom
horizontal axis, and the values of $\tau/\Delta\tau$ shown on the top
horizontal axis. Note that geometric optics corresponds to the regime when
$\Delta\tau\ll(\Delta\nu)^{-1}$. Conversely, the perturbative regime occurs
when $\Delta\tau\gg(\Delta\nu)^{-1}$. This can be easily made sense of when we
realize that for large $\kappa$, we have $\Delta\tau\sim\kappa$. Thus,
$\Delta\tau\sim(\Delta\nu)^{-1}$ when $\epsilon\sim\kappa\Delta\nu\sim 1$. In
other words, geometric optics is obtained in the limit where the fundamental
resolution limit in delay space is much smaller than the relative time delays
of the lensed images. Diffractive/perturbative optics occurs when the relative
time delays become smaller become un-resolvable in the conjugate wave-field.
Observationally, this leads to a rather straightforward way of diagnosing
whether or not an observation occurs in the geometric or diffractive regime,
since it is often possible to measure the relative time delays of scattered
images.

##  6 Implications for turbulent plasma lensing

In this section we will discuss the implications of our results for the study
of plasma lensing. Following observations of extreme scattering events (ESEs)
in pulsars (Fiedler et al., 1987) and the discovery of pulsar scintillation
(Scheuer, 1968; Rickett, 1977), both of which are thought to be due to
scattering in the interstellar medium (ISM), there has been significant effort
to understand wave effects in plasma lensing. The expectation that fast radio
bursts (FRBs) will be lensed in a similar way to pulsars has increased the
need for a robust understanding of wave optics in lensing.

Due to the difficulties in computing the Kirchhoff-Fresnel integral, studies
of strong plasma lenses typically restrict their attention to geometric
optics, especially in the case of studies attempting to model ESEs (see e.g.
Romani et al., 1987; Clegg et al., 1998; Pushkarev et al., 2013; Bannister et
al., 2016, 2019). Studies of pulsar scintillation have also included studies
of the diffractive interstellar scintillation phenomenon (DISS), for which the
perturbative expansion of the Kirchhoff-Fresnel integral is assumed to hold
(see e.g. Wang et al., 2005; Lewandowski et al., 2011; Dai et al., 2016). Much
of our understanding of when the different regimes of wave optics in plasma
lensing comes from an earlier effort to develop the theory of turbulent plasma
lensing (Lee & Jokipii, 1975; Prokhorov et al., 1975; Blandford & Narayan,
1985; Goodman & Narayan, 1985; Cordes et al., 1986; Codona et al., 1986;
Cordes & Rickett, 1998). Rickett (1990) reviews much of this earlier
theoretical work, the results of which are also presented in the textbook, The
Handbook of Pulsar Astronomy (Lorimer & Kramer, 2012), which is widely
referred to in the pulsar astronomy community. These early works on turbulent
plasma lensing relied on approximation schemes for the Kirchhoff-Fresnel
integral, which, in the absence of a robust method for evaluating oscillatory
integrals exactly, were assumed to hold based on heuristic arguments, rather
than explicit quantitative validation. The results we have presented here for
a simple lens model suggest that these previous works may have overestimated
the applicability of these approximations in certain regimes.

In order to compare our results with the turbulent plasma lensing literature,
we will first re-write the Kirchhoff-Fresnel integral using the dimensionful
parameters that appear in plasma lensing. Eq. (1) becomes

$F(y)\propto\int\exp\Big{\\{}i\frac{\omega}{\overline{d}c}\Big{[}\frac{(\hat{x}-\hat{y})^{2}}{2}+\frac{\overline{d}ce^{2}\Sigma_{e}(\hat{x})}{2m_{e}\epsilon_{0}\omega^{2}}\Big{]}\Big{\\}}d\hat{x},$
(18)

where $\omega$ is the angular frequency of the source radiation,
$\overline{d}=d_{sl}d_{l}/d_{s}$ is the reduced distance,
$\hat{y}=(\hat{x}_{s}d_{l}+\hat{x}_{o}d_{sl})/d_{s}$ is a weighted average of
the transverse displacement of the source and observer, and $\Sigma_{e}$ is
the projected electron column density of the plasma in the lens plane. Fig. 1
shows the distances and the coordinates involved in this parametrization.

Now, in order to compare this with our dimensionless analysis, we re-scale the
coordinates by $a$, which we take to be the size of the lens. That is, we take
$\hat{x}\to x=\hat{x}/a$, $\hat{y}\to y=\hat{y}/a$, and
$\Sigma_{e}(\hat{x})\equiv\Sigma_{0}\psi(x)$, where $a$ is defined so that
$\psi^{\prime\prime}(0)\approx-2$. Defined in this way, $\Sigma_{0}$ is the
amplitude of the plasma surface density variation. Thus, Eq. (18) reduces to
Eq. (1), with

$\displaystyle\nu$ $\displaystyle=\frac{\omega
a^{2}}{c\overline{d}}=\frac{a^{2}}{R_{F}^{2}},$ (19) $\displaystyle\kappa$
$\displaystyle=\frac{\overline{d}e^{2}\Sigma_{0}}{2a^{2}m_{e}\epsilon_{0}\omega^{2}},$
(20)

where $R_{F}\equiv(\lambda\overline{d}/2\pi)^{1/2}$ is the dimensionful
Fresnel scale. Thus, as we stated before, the condition that $\nu$ be large is
equivalent to the Fresnel scale being small relative to the size of the lens.
For plasma lensing, we obtain

$\epsilon=\frac{e^{2}\Sigma_{0}}{2m_{e}c\epsilon_{0}\omega}.$ (21)

Note that since the convergence scales with the inverse frequency squared in
plasma lensing, we have $\epsilon\sim\omega^{-1}$. Thus, geometric optics
($\epsilon\gg 1$) is obtained in the low frequency limit, and diffractive
optics ($\epsilon\ll 1$) in the large frequency limit.

The primary difficulty in comparing our results to the previous literature is
that much of the study of wave optics in plasma lensing was performed for
turbulent lenses. That is, whereas we consider a smooth $\psi(x)$ with a
single peak, studies of turbulent plasma lensing consider a Gaussian random
field as the lensing potential. As a result, the primary goal of these
calculations is not to compute the observed light curve for a particular lens,
but to compute the statistical properties of the electric field at the
observer given some power spectrum defining the lens. For example, Rickett
(1990) reviews the study of turbulent plasma lenses with power spectra of the
form

$P(k)\propto(k^{2}+k^{2}_{\mathrm{outer}})^{-\beta/2}\exp(-k^{2}/k^{2}_{\mathrm{inner}}).$
(22)

Between the inner and outer scales set by $k_{\mathrm{inner}}$ and
$k_{\mathrm{outer}}$, this power spectrum is a simple power law, $P(k)\propto
k^{-\beta}$, and, in particular, is Kolmogorov for $\beta=11/3$. Outside of
these scales, the power spectrum falls rapidly to zero.

The early work on turbulent plasma involved approximating the Kirchhoff-
Fresnel integral using perturbative methods, in addition to the Eikonal or
stationary phase approximation (see e.g. Prokhorov et al., 1975), as exact
methods for computing the diffraction integral were not generally available.
Naturally, these works were concerned with the delineation of the regimes of
applicability of these approximation schemes. One of the main results of this
early work, summarized in Rickett (1990) and Lorimer & Kramer (2012), was the
perturbative computation of the observed intensity fluctuations for a power
spectrum of the form of Eq. (22). It was found that the intensity fluctuations
were much less than unity when the Fresnel scale was larger than the coherence
scale, $s_{0}$, which is defined as the average lateral separation in the
observing plane across which there is a one-radian difference in phase
calculated along a straight line path from source to observer. Conversely,
when the coherence scale is larger than the Fresnel scale, the intensity
fluctuations predicted by the perturbative calculation exceed unity. Thus, a
quantity can be defined

$u=\frac{R_{F}}{s_{0}},$ (23)

such that when $u<1$ the amplitude of the intensity fluctuations predicted by
the perturbative expansion are weak and when $u>1$ they are strong (Rickett,
1990).

For $u<1$, which was taken to be the weak lensing regime, the perturbative
expansion was assumed to hold. The strong lensing regime ($u>1$) was further
split into the diffractive and refractive regimes, where diffraction occurs on
small scales and refraction occurs on large scales. The scales at which these
regimes occurred were determined by the following heuristic arguments, which
can be found in Rickett (1990) and Lorimer & Kramer (2012).

From the definition of the coherence scale, $s_{0}$, one can define the
diffractive scattering angle, $\theta_{d}\sim\lambda/s_{0}$, which is roughly
the average bending that light rays undergo as they pass through the plasma
(Salpeter, 1967; Rickett, 1990). If we consider the electric field at the
observer to be a sum of point sources located along the lens (the Huygens-
Fresnel principle), then the observed diffractive intensity variations come
only from the waves within the angular spectrum of width $\theta_{d}$. When
the observer is laterally displaced by $s_{0}$, the relative phase of the
incoming waves are substantially shifted from the original observer position
(by roughly one radian), resulting in significant differences in the observed
intensity across this scale. Thus, the diffractive scale, $s_{d}$, is taken to
be roughly equal to the coherence scale, $s_{d}\approx s_{0}$. Now, intensity
variations due to refraction (i.e. due to the bending of multiple light rays
into the line of sight of the observer) must occur on scales larger than
$s_{r}=\overline{d}\theta_{d}$. This is because two observers separated by a
distance less than $\overline{d}\theta_{d}$ will have overlapping scattering
disks. Thus, any significant intensity variations on scales larger than this
must be due to multiple images being refracted into the line of the sight of
the observer. Fig. 7 shows a diagram to help visualize this picture of
turbulent scattering.

Figure 7: Diagram of the turbulent plasma scattering picture. Diffractive
intensity variations at an observer come from waves emanating from within a
scattering angle of $\theta_{d}$. The diffractive scale, $s_{d}$, is taken to
be roughly equal to the coherence scale, $s_{0}$, which is the average lateral
distance over which the phase along a straight line path from source to
observer varies by one radian. The refractive scale, $s_{r}$, is taken to be
$\overline{d}\theta_{d}$, as observers separated by larger than this have no
overlap between their scattering disks.

Altogether, in terms of the Fresnel scale and $u$, one can write the
diffractive and refractive scales as (Rickett, 1990; Lorimer & Kramer, 2012)

$\displaystyle s_{r}$ $\displaystyle=uR_{F},$ (24) $\displaystyle s_{d}$
$\displaystyle=\frac{R_{F}}{u}.$ (25)

Fig. 8 summarizes this picture of turbulent plasma lensing that arises from
this description. When $u<1$, the intensity fluctuations are weak and the
perturbative expansion holds. When $u>1$, the intensity fluctuations are split
into distinct diffractive and refractive regimes. The Fresnel scale sets the
boundary between diffractive and refractive lensing, but as $u$ gets large,
the region around the Fresnel scale for which neither approximations hold gets
larger. Fig. 1 of Rickett (1990), which shows a schematic of the power
spectrum of the observed intensity fluctuations as a function of scale, also
presents this picture of lensing where the Fresnel scale sets the boundary
between two distinct regimes.

Figure 8: Summary of the regimes described in early turbulent plasma lensing
literature. The statistics of the observed electric field is taken to be
described by diffractive interstellar scattering (DISS) for spatial scales
less than $s_{d}=R_{F}/u$ and by refractive interstellar scattering (RISS) for
spatial scales greater than $s_{r}=uR_{F}$. The quantity $u$ plotted on the
$y$-axis sets the predicted strength of the diffractive intensity variations,
where they are much less than unity for $u<1$ and greater than unity for
$u>1$. The $x$-axis is the spatial scale of intensity variations $k^{-1}$
normalized by the Fresnel scale. Note that this description is in contrast
with our lens model, for which the refractive regime can be made to hold at
arbitrarily small scales when the lens strength is large.

Our results, however, are at odds with this picture. Firstly, we find that,
for the rational lens, rather than the Fresnel scale setting the transition
point between diffractive (perturbative) and refractive (geometric) optics,
the dimensionless parameter $\epsilon=\kappa\nu$ does. In particular, if the
lens strength is large, geometric optics may hold well below the Fresnel
scale, and likewise if it is small, it may fail at arbitrarily large scales.
Secondly, we find that even when the intensity variations are small, the
perturbative expansion may still fail. See, for example, panel (b) of Fig. 4,
for which the perturbative expansion fails even while the intensity variations
are of order $\sim 10^{-3}$. Likewise, large fluctuations predicted by the
perturbative expansion do not necessarily indicate large fluctuations in the
true value of the diffraction integral (see panel (c) of Fig. 4).

We also note that in Section 4, we argue that the condition for strong lensing
to occur is that both $\kappa>1$ and $\epsilon>1$. In other words, whenever
the perturbative expansion holds, the lensing is weak (though it need not be
the case that weak lensing implies the perturbative expansion holds). This is
in stark contrast with the definition of diffractive interstellar
scintillation (DISS), which is taken to be the regime where lensing is both
strong and described by a perturbative expansion. The study of DISS has led to
attempts to infer physical properties from observations of pulsar
scintillation, such as the spectral index of the ISM, the magnitude of
electron density fluctuations, and the relative velocity of the observed
pulsars (see e.g. Cordes & Rickett, 1998; Wang et al., 2005; Lewandowski et
al., 2011; Dai et al., 2016). If, as we have argued, perturbative calculations
of the lensing phenomena are inappropriate for regimes of strong
scintillation, then attempts to infer physical parameters using the framework
of DISS may lead to biased results.

One prediction of this framework is that for a turbulent plasma with a lensing
potential with Gaussian statistics, in the geometric optics limit, the
intensity fluctuations of a scatter-broadened image are Gaussian, in the sense
that as a function of the observation baseline, $b$, the intensity
fluctuations scale as $\sim\exp\\{-b^{2}\\}$. Conversely, in diffractive
optics, the intensity fluctuations are predicted to scale as
$\sim\exp\\{-b^{\beta-2}\\}$, where $\beta$ is the power-law index in Eq.
(22). These predicted scalings of the intensity fluctuations have been used,
for example, to separate diffractive and refractive regimes in interstellar
scattering (see e.g. Johnson et al., 2018); however, the predicted Gaussianity
of the refractive intensity fluctuations is based on the assumption that the
refractive fluctuations only occur on scales where the lens potential is
effectively smooth. Below this scale, it is assumed that
diffractive/perturbative optics holds. However, we have argued that, depending
on the amplitude of the lens, geometric optics may hold on scales much smaller
than have previously been assumed. Thus, the particular scaling of intensity
fluctuations may not be a strong indicator of the regime in which the lensing
is occurring. Indeed, the Gaussian scaling of the intensity fluctuations and
the regime of lensing may be entirely separate issues, though they have been
conflated in the literature.

Of course, as mentioned previously, a straightforward application of our
results to turbulent plasma lensing is challenging, as we have restricted our
attention to a simple lens model. Due to the highly non-linear nature of the
Kirchhoff-Fresnel integral, it is not immediately obvious that results for
such a simple lens would be applicable to the more complicated case of
turbulent lenses. Nevertheless, we feel that our description of the simple
rational lens is in sufficient contrast to the description of lensing in the
turbulent plasma lensing literature to warrant revisiting the theoretical
basis of diffractive scintillation now that more robust tools for exactly
evaluating the diffraction integral are available.

##  7 Typical values of dimensionless parameters

In this section, we compute the dimensionless parameters $\nu$, $\kappa$, and
$\epsilon$ for typical physical parameters that may occur in plasma lensing
and gravitational lensing.

###  7.1 Plasma lensing

The study of wave optics in astrophysical plasma lensing is particularly
relevant to pulsar observations, as pulsars are effectively coherent point
sources. Pulsars are radio sources with frequencies of order $~{}{\rm GHz}$,
and are found within the galaxy. Thus, we can expect typical distances to be
$\overline{d}\sim 1\,{\rm kpc}$. Pulsars have been observed to undergo extreme
scattering events (ESEs) due to lensing by roughly AU-scale features in the
ISM, with excess electron column densities of order $\sim 0.01\,{\rm
pc\,cm}^{-3}$ (Coles et al., 2015; Kerr et al., 2018). Thus, using Eqns. 19
and 20, we obtain

$\displaystyle\nu$ $\displaystyle=1.5\times
10^{4}\,\Big{(}\frac{f}{1\,\mathrm{GHz}}\Big{)}\Big{(}\frac{a}{1\,\mathrm{AU}}\Big{)}^{2}\Big{(}\frac{\overline{d}}{1\,\mathrm{kpc}}\Big{)}^{-1},$
(26) $\displaystyle\kappa$
$\displaystyle=100\,\Big{(}\frac{\Sigma_{0}}{0.01\,\mathrm{pc\,cm}^{-3}}\Big{)}\Big{(}\frac{\overline{d}}{1\,\mathrm{kpc}}\Big{)}\Big{(}\frac{a}{10\,\mathrm{km}}\Big{)}^{-2}\Big{(}\frac{f}{1\,\mathrm{GHz}}\Big{)}^{-2},$
(27) $\displaystyle\epsilon$ $\displaystyle=1.5\times
10^{6}\,\Big{(}\frac{\Sigma_{0}}{0.01\,\mathrm{pc\,cm}^{-3}}\Big{)}\Big{(}\frac{f}{1\,\mathrm{GHz}}\Big{)}^{-1}.$
(28)

Consistent with ESE observations, $\kappa\gg 1$, resulting in multiple real
images and large intensity fluctuations. Moreover, we note that $\epsilon\gg
1$, suggesting that ESEs occur firmly in the geometric regime.

Note that the distance parameters $\overline{d}$ and $a$ both cancel out in
the expression for $\epsilon$. Thus, the factor which determines whether or
not lensing occurs in the geometric or diffractive/perturbative regime depends
only on the excess electron surface density and the frequency. In particular,
we can use Eq. (28) to argue that pulsar scintillation generically occurs in
the geometric regime, as $\epsilon\sim 1$ only occurs when the excess electron
surface density is $\Sigma_{0}\sim 10^{-8}\,{\rm pc\,cm}^{-3}$.

###  7.2 Gravitational lensing

In gravitational lensing, the surface mass density, as opposed to the electron
column density, sets the amplitude of the lens potential. Moreover, the
amplitude only depends on the surface mass density, and does not depend on
frequency. In gravitational lensing, one is also often concerned with
cosmological distances, and so the distances in
$\overline{d}=d_{sl}d_{l}/d_{s}$ must be replaced by angular diameter
distances, and the frequency $\omega$ must include include a redshift
correction, $\omega\to(1+z)\omega$, where $z$ is the redshift of the lens
(Schneider et al., 1992). Otherwise, the formalism is exactly the same as
plasma lensing.

We may wish to know the values of the dimensionless parameters in the case of
lensing by a galaxy at cosmological distances. Thus, the lens has a size of
order $\sim\,10\,{\rm kpc}$ and has a distance of order $\sim 1\,{\rm Gpc}$.
Since fast radio bursts (FRBs) are the only coherent sources at cosmological
distances, we will also be primarily concerned with radio frequencies. Given
these typical parameters, the dimensionless frequency $\nu$ can be computed:

$\nu\sim 6.5\times
10^{16}\,(1+z)\Big{(}\frac{f}{1\,\mathrm{GHz}}\Big{)}\Big{(}\frac{a}{10\,\mathrm{kpc}}\Big{)}^{2}\Big{(}\frac{\overline{d}}{1\,\mathrm{Gpc}}\Big{)}^{-1}.$
(29)

Many lens profiles have been used to model lensing by galaxies. One simple
model is the Plummer model for which it is straightforward to re-parametrize
the convergence in terms of the total lens mass. One obtains that the
convergence is given by $\kappa=2GM\overline{d}/a^{2}c^{2}$ (Werner & Evans,
2006).

$\kappa\sim
1\,(1+z)^{-1}\Big{(}\frac{M}{10^{12}\,M_{\odot}}\Big{)}\Big{(}\frac{\overline{d}}{1\,\mathrm{Gpc}}\Big{)}\Big{(}\frac{a}{10\,\mathrm{kpc}}\Big{)}^{-2}.$
(30)

Combining Eqs. 29 and 30, we obtain

$\epsilon=6.5\times
10^{16}\Big{(}\frac{M}{10^{12}\,M_{\odot}}\Big{)}\Big{(}\frac{f}{1\,\mathrm{GHz}}\Big{)},$
(31)

which is generically much larger than unity.

##  8 Conclusion

Through the introduction of the mathematical framework of Picard-Lefschetz
theory to the field of astrophysical lensing, it has now become possible to
evaluate the Kirchhoff-Fresnel diffraction integral for a wide variety of
situations. Previously, theoretical treatments of the study of wave optics in
astrophysical contexts relied on various approximation schemes. Exact methods
of computing diffraction integrals allow us to quantitatively evaluate
longstanding assumptions in the field.

In this work, we have studied a simple lens model to assess when the
perturbative approximation (diffractive optics) holds, and when the geometric
limit of optics (refractive optics) holds. We find that, for the lens model we
consider, there is no overlap in the regime of applicability of these two
methods, and that the region of parameter space where neither method is valid
is limited. That is optics is generically well described by either diffractive
or refractive optics. We find that the spatial scale that separates these two
regimes is given by the Fresnel scale divided by the square-root of the
convergence, $R_{F}/\sqrt{\kappa}$. That is, for a lens with convergence
$\kappa$ and size $a$, refractive optics holds when $a$ is larger than
$R_{F}/\sqrt{\kappa}$, and diffractive optics holds when it is smaller.
Previously, it has been assumed that the Fresnel scale alone sets the scale at
which diffractive optics is valid. We have also argued that large intensity
variations do not generically occur in the diffractive regime. This has
important implications for the field of pulsar scintillometry.

In this work, we have focused on a simple, one-dimensional lens model. The
highly non-linear nature of the Kirchhoff-Fresnel integral means that it is
not straightforward to apply these results to more complicated lensing
systems. Further detailed study of systems of interest, for example, turbulent
plasma lensing, is needed. In particular, a similar analysis of the Gaussian
random lens is a natural next step.

## Data Availability

No new data were generated or analysed in support of this research.

## Acknowledgements

We thank Fang Xi Lin for useful discussions. We receive support from Ontario
Research Fund—research Excellence Program (ORF-RE), Natural Sciences and
Engineering Research Council of Canada (NSERC) [funding reference number
RGPIN-2019-067, CRD 523638-18, 555585-20], Canadian Institute for Advanced
Research (CIFAR), Canadian Foundation for Innovation (CFI), the National
Science Foundation of China (Grants No. 11929301), Thoth Technology Inc,
Alexander von Humboldt Foundation, and the Ministry of Science and
Technology(MOST) of Taiwan(110-2112-M-001-071-MY3). Computations were
performed on the SOSCIP Consortium’s [Blue Gene/Q, Cloud Data Analytics, Agile
and/or Large Memory System] computing platform(s). SOSCIP is funded by the
Federal Economic Development Agency of Southern Ontario, the Province of
Ontario, IBM Canada Ltd., Ontario Centres of Excellence, Mitacs and 15 Ontario
academic member institutions. Cette recherche a été financée par le Conseil de
recherches en sciences naturelles et en génie du Canada (CRSNG), [numéro de
référence 523638-18,555585-20 RGPIN-2019-067].

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##  Appendix A Picard-Lefschetz analysis of the rational lens

The purpose of this work has been to evaluate the regimes of validity of the
refractive and diffractive regimes of optics through explicit comparison with
the exact result of the Kirchhoff-Fresnel integral. In particular, we studied
the simple rational lens, $\psi=1/(1+x^{2})$, as an example. Our ability to
exactly compute the Kirchhoff-Fresnel integral for this lens is enabled by the
tools of the mathematical framework of Picard-Lefschetz theory, which was re-
introduced to the study of oscillatory integrals in physics by Witten (2010),
and has subsequently been applied to the study of lensing (Feldbrugge et al.,
2019; Jow et al., 2021; Shi & Xu, 2021). Witten (2010) and Feldbrugge et al.
(2019) give a detailed description of the theoretical and numerical
application of Picard-Lefschetz theory to oscillatory integrals, but we will
give a brief summary here which follows the description of the theory in Jow
et al. (2021).

The goal is to compute integrals of the form

$I=\int_{\mathbb{R}}dxe^{iS(x;\mathbf{\xi})},$ (32)

where $\mathbf{\xi}$ is a set of parameters that fix the phase function
$S(x;\mathbf{\xi})$. Picard-Lefschetz theory is an application of Cauchy’s
theorem for the deformation of integration domains and guarantees the
existence of a surface in the complex plane, $\mathbb{C}$, that is a
continuous transformation of the original integration domain, $\mathbb{R}$,
such that the integrand is localized and non-oscillatory.

To see how this works, we define the function $h(z)=\text{Re}\\{iS(z)\\}$. We
assume that $S(z)$ is analytic, and we have replaced the integration variable
$x$ with the variable $z$ to denote the analytic continuation of the integrand
to the complex plane. It follows from the Cauchy-Riemann equations that the
critical points of the function $h(z)$ are also the critical points of $S(z)$,
and that the non-degenerate critical points are saddle points of the function
$h$. Thus, every critical point $z_{i}$ of $h(z)$ has associated with it a
contour of steepest descent, $\mathscr{J}_{i}$, with a dimension of one, and a
contour of steepest ascent, $\mathscr{U}_{i}$, of equivalent dimension.

The contours of steepest descent are determined by the flow equations

$\frac{dz}{dt}=\mp\frac{d(i\bar{S})}{d\bar{z}},\quad\frac{d\bar{z}}{dt}=\pm\frac{d(iS)}{dz},$
(33)

When the sign in front of the right-hand side of the first of Eq. (33) is
negative (and the sign of the second is positive), the equations determine the
trajectories of steepest descent. When the signs are reversed, the equation
determines the trajectories of steepest ascent. The surface $\mathscr{J}_{i}$
associated with $z_{i}$ is a manifold determined by the set of initial points
for which solutions to the steepest ascent equation terminate at $z_{i}$ as
$t\to\infty$. Hence, the surface $\mathscr{J}_{i}$ is a contour of descent,
since the direction pointing away from the critical point $z_{i}$ along the
surface $\mathscr{J}_{i}$ is a direction of decreasing $h$.

One can show that along each of the contours of steepest descent,
$\mathscr{J}_{i}$, the quantity $\mathrm{Im}\\{iS(z)\\}$ is constant:

$\displaystyle\frac{\mathrm{d}\text{Im}[iS]}{\mathrm{d}t}=\frac{1}{2i}\frac{\mathrm{d}(iS+i\bar{S})}{\mathrm{d}t}=\frac{1}{2i}\left(\frac{\partial(iS)}{\partial
z}\frac{\mathrm{d}z}{\mathrm{d}t}+\frac{\partial(i\bar{S})}{\partial\bar{z}}\frac{\mathrm{d}\bar{z}}{\mathrm{d}t}\right)=0.$
(34)

Moreover, the total integrand exponentially approaches $0$ away from the
critical point. This means that the integrand is localized and non-oscillatory
along these surfaces, which we call the “Lefschetz thimbles". Picard-Lefschetz
theory guarantees that the integral along the original domain, $\mathbb{R}$,
is equal to a sum of integrals along the Lefschetz thimbles. Explicitly,

$\displaystyle I=\sum_{i}n_{i}\int_{\mathscr{J}_{i}}dze^{iS(z;\mathbf{\xi})},$
(35)

where $n_{i}=\langle\mathbb{R},\mathscr{U}_{i}\rangle$ is the intersection
number, which counts the number of times the steepest ascent contour,
$\mathscr{U}_{i}$, intersects the original integration domain, $\mathbb{R}$,
with sign given by the relative orientations of the two curves. Thus, a
thimble, $\mathscr{J}_{i}$, contributes to the integral if and only if the
contour of steepest ascent, $\mathscr{U}_{i}$, for the corresponding critical
point, $z_{i}$, intersects the original integration domain. In order to
compute the integral for a given set of parameters, $\xi$, one must find the
complex critical points of $z_{i}$, as well as the surfaces of steepest ascent
and descent associated with each critical point, which can be achieved by
solving the flow equations. Note that here we have given a brief description
of Picard-Lefschetz theory for one-dimensional oscillatory integrals, but the
theory may be extended to arbitrary dimension (see Witten, 2010; Feldbrugge et
al., 2019)

The topology of the Lefschetz thimbles in the complex domain varies smoothly
with the parameters $\xi$ except at caustics and Stokes lines. Caustics are
points in parameter space where a set of real critical points become
degenerate. Lensing near caustics has been widely studied in both the
geometric and diffractive regimes (Nye, 1999). Stokes lines are points at
which the number of relevant complex critical points changes, and occur when
the Lefschetz thimbles of multiple critical points overlap (see Witten (2010)
and Jow et al. (2021) for a more detailed description of Stokes lines).
Together, the caustics and Stokes lines form separate regions in parameter
space where the Lefschetz thimble topology is distinct. Fig. 9 shows the
different thimble topologies for the rational lens. The black and purple lines
are the caustics and Stokes lines, respectively, which separate the parameter
space into five distinct regions. The encircled diagrams for each region show
the relevant thimbles in the complex domain. The red dots correspond to the
relevant critical points, and the black and grey curves correspond to the
surfaces of steepest descent and ascent associated with each critical point.
The blue dots correspond to the poles of the analytically continued phase
function. As defined in Eq. (1), the integral would be performed over the real
axis and is conditionally convergent. However, Picard-Lefschetz theory
guarantees that the integral over the relevant Lefschetz thimbles (the black
curves in the encircled diagrams) is not only equal to the original integral
over the real axis, but is absolutely convergent.

Note that the thimble topologies for the rational lens only depend on the
parameters $\kappa$ and $y$, as $\nu$ is simply an overall factor multiplying
the exponent of the integrand in the Kirchhoff-Fresnel integral (Eq. (1)).
Thus, $\nu$ does not affect the solution to the flow equations, and thereby
does not affect the thimble topology. Positive values of $y$ are shown on the
horizontal axis of Fig. 9 as the result of the Kirchhoff-Fresnel integral is
symmetric about $y=0$ for the rational lens. The vertical axis is taken to be
$2\kappa$, as the cusp catastrophe occurs at $2\kappa=1$.

Figure 9: Lefschetz thimble topologies for the rational lens,
$\psi=1/(1+x^{2})$. The black and purple lines show the position of the
caustics and Stokes lines in lens parameter space, which separate the regions
(labeled I, II, III, IV, and V) with distinct thimble topology. The encircled
diagrams show the topology of the relevant thimbles in the complex domain for
each of these regions. The red dots correspond to the relevant critical
points, and the black and grey curves correspond to the surfaces of steepest
descent and ascent associated with each critical point. The blue dots
correspond to the poles of the analytically continued phase function.

One of the most powerful aspects of Picard-Lefschetz theory as applied to
lensing is that it allows for the separation of the total integral into
contributions associated with a discrete set of critical points,
$\\{{\bm{z}}_{i}\\}$, by performing the integral along the associated
Lefschetz thimbles separately. Moreover, these critical points are precisely
the images of geometric optics. Indeed, when $\nu\to\infty$, the value of the
integral associated with each of the Lefschetz thimbles converges to the
geometric optics value of the field for each image given by Eq. (4). Since
Picard-Lefschetz theory gives an exact value of the integral for all values of
$\nu$, this essentially allows us to extend the image analysis of geometric
optics to low frequencies.

As an example, consider when $\kappa=5000$ and $y=1000$, which corresponds to
Region II in Fig. 9. The total intensity fluctuations above unity for these
parameters as a function of frequency is shown in the top middle panel of Fig.
4 (along with similar plots for the other regions). For these parameters,
there are four relevant critical points, one complex and three real. We will
label the complex critical point $z_{c}$ and the real critical points $x_{0}$,
$x_{1}$, and $x_{2}$, with descending value. Fig. 10 shows the critical points
and their corresponding Lefschetz thimbles in the complex domain. The
background colour corresponds to the value of the function, $h(z)$. In
general, Lefschetz thimbles terminate at poles of $h(z)$ or infinity. The
thimble for $x_{2}$ goes from the pole at infinity to the pole at $+i$; the
thimble for $x_{1}$ goes from the pole at $+i$ to $-i$; the thimble for
$z_{c}$ goes from $-i$ to infinity; and the thimble for $x_{0}$ goes from
infinity in the lower half-plane to infinity in the upper-half plane.

Figure 10: The Lefschetz thimbles in the complex plane for the rational lens,
$\psi=1/(1+x^{2})$, with $\kappa=5000$ and $y=1000$. For these parameters
there are three relevant real images, which have been labeled $x_{0}$,
$x_{1}$, $x_{2}$, and a single relevant complex image, $z_{c}$. The bottom
panel shows a zoomed out view of the complex plane. The background colour is
determined by the value of the function $h(z)$, evaluated for $\nu=10^{-5}$.
The Lefschetz thimbles are the surfaces of steepest descent in the function
$h$ from the critical points. The steepest ascent contours are shown as grey
dashed lines, and the original integration contour, the real line, is shown as
a solid grey line.

By examining the integrand along each of these thimbles, we can see exactly
how geometric optics fails at low frequencies. Fig. 11 shows the magnitude of
the integrand evaluated along the thimbles for different values of $\nu$. For
large $\nu$, the integrand generically takes the form of a single peak at the
critical point, which decays exponentially away from the peak. As $\nu$
increases, the width of this peak decreases, and the integrand effectively
becomes a delta function at the critical point. Thus, in the limit of
$\nu\to\infty$, the contribution of each thimble is entirely determined by a
small neighbourhood of the critical point, and geometric optics is recovered.
However, as $\nu$ decreases, the effective region that contributes to the
integral increases. This has two main effects. Firstly, since some of the
thimbles terminate at poles, rather than infinity, at low frequencies the
width of the integrand has become large enough that it effectively becomes cut
off by the presence of the pole before it can decay to zero (see the
intergands for $z_{c}$, $x_{1}$, and $x_{2}$ for low frequencies in Fig. 11).
Secondly, as the effective region that contributes to the integral increases,
the curvature of the thimble becomes important. In the geometric limit, when
the integrand is isolated to a small region about the critical point, only the
local curvature of the thimble matters. In particular, the measure of the
integral, $dz$, becomes
$dz\propto\exp\big{\\{}i\big{[}\frac{\pi}{4}-\frac{\mathrm{arg}(\Delta_{j})}{2}\big{]}\big{\\}}$
at the critical point. However, as the $\nu$ decreases, the curvature of the
integration contour away from the critical point becomes relevant. While the
location of the poles is independent of the lens parameters, the effective
width of the integrand along the thimbles is determined by both $\nu$ and
$\kappa$. In particular, it is when $\epsilon=\kappa\nu\gg 1$ that the
integrand becomes localized to a small region on the thimble about the
critical point with constant curvature. Thus, it is in this way that the
parameter $\epsilon$ sets when geometric optics holds. Note that since the
thimble for $x_{0}$ goes from minus infinity to positive infinity in a
straight line (i.e. with constant curvature), neither of these effects is
relevant, and, indeed, the contribution to the total integral computed along
its Lefschetz thimble is always equal to the geometric optics prediction for
all $\nu$.

As an aside, the fact that near the critical point, the measure along the
Lefschetz thimble becomes
$dz\propto\exp\big{\\{}i\big{[}\frac{\pi}{4}-\frac{\mathrm{arg}(\Delta_{j})}{2}\big{]}\big{\\}}$
is the origin of the extra phase factors in Eqs. 3 and 4. When the critical
points are real, $\mathrm{arg}(\Delta_{j})=0$ or $\pi$. This binary choice of
angle for real images is usually expressed as the statement that each real
image has a Morse index, $n_{j}$ associated with it, which is an integer of
either $0$ or $1$. However, when the formalism is extended into the complex
plane, we see that for complex images, the angle associated with the Lefschetz
thimble is arbitrary, and so images, in general, do not have an associated
integer Morse index.

Figure 11: The absolute value of the integrand of the Kirchhoff-Fresnel
integral, $\exp\\{i\nu[\frac{(x-y)^{2}}{2}+\frac{\kappa}{1+x^{2}}]\\}$,
evaluated along the Lefschetz thimbles shown in Fig. 10 for $\kappa=5000$ and
$y=1000$. From left to right, and top to bottom, the integrand is shown for
the thimbles associated with $x_{0}$, $x_{1}$, $x_{2}$, and $z_{c}$. The
thimbles are parametrized by the imaginary component of the complex variable,
$\mathrm{Im}(z)$. The integrand is shown for multiple values of $\nu$, such
that $\epsilon\equiv\kappa\nu=0.1,1,$ and 1.

Now, examining how the integrand behaves along the Lefschetz thimbles allows
us to understand how geometric optics systematically fails as $\nu$ decreases.
We can also look at how the integral evaluated along the thimbles separately
behaves as a function of frequency. Fig. 12 shows the flux associated with
each image as a function of frequency. That is, we compute the integral along
each thimble separately and plot the absolute value squared of the result. In
the high-$\nu$ limit, these values converge to the intensities $|F_{j}|^{2}$
for each of the images predicted by geometric optics. The grey dashed line in
Fig. 12 shows the value of the intensity fluctuations in the perturbative
expansion, $|F^{\mathrm{pert.}}-1|^{2}$. We know from Fig. 4 that the total
intensity computed using Picard-Lefschetz theory begins to agree with the
perturbative below $\nu\sim 10^{-3}$; however, in Fig. 12 we find that the
intensities of the individual images are, asymptotically, orders of magnitude
larger than the perturbative intensity fluctuations. While this may seem
strange at first glance, this is not contradictory. As $\nu\to 0$, we can see
that the intensities of the images at $z_{c}$ and $x_{2}$ become equal to each
other. Their phases, however, become almost exactly opposite. This is easy to
see from the bottom panel of Fig. 10. At large $|z|$, the thimbles associated
with $z_{c}$ and $x_{2}$ become anti-parallel. When $\nu\to 0$, the relevant
region of integration along these thimbles gets larger, and so the values of
the integral for these two images become roughly equal in magnitude but
opposite in sign. Thus, while individually their intensities may be large, the
magnitude of their sum is small. It turns out that this sum is close to
opposite in phase with the image at $x_{1}$, leading to a further
cancellation. As a result, the intensity of the sum of the three images
$x_{1}$, $x_{2}$, and $z_{c}$ becomes very small, and, indeed, agrees with the
perturbative intensity fluctuations (the intensity for the image $x_{0}$ is
unity, independent of $\nu$).

This gives us insight into how geometric optics becomes diffractive optics as
$\nu$ decreases. In this case, a subset of the images become mutually coherent
with each other, in such a way that they almost exactly cancel each other out.
What is left is an unperturbed image ($x_{0}$ in this case) with a small
modulation on top. We can perform the same exercise for the different regions
shown in Fig. 9, and a similar picture emerges: the multiple images in the
geometric picture attain the same or opposite phases in the limit as $\nu\to
0$, and their sum yields the perturbative expansion.

Figure 12: The magnitude squared of the contribution to the total Kirchhoff-
Fresnel integral associated with the Lefschetz thimbles of $x_{0}$ (red),
$x_{1}$ (orange), $x_{2}$ (blue), and $z_{c}$ (green), for the rational lens
with $\kappa=5000$ and $y=1000$. The thimbles and images are shown in Fig. 10.
The grey dashed line shows the perturbative intensity fluctuations,
$|F^{\mathrm{pert.}}-1|^{2}$.