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1
Diffraction decorating caustics in gravitational lensing
M V Berry
H H Wills Physics Laboratory, Tyndall Avenue, Bristol BS8 1TL,
UK
[email protected]
Abstract
The spacings of interference fringes in an observation plane,
and the
accompanying intensity amplifications, are calculated for the
gravitational
lensing of a distant source by an isolated star and near the
fold and cusp
singularities from a binary star lens. In the astronomically
natural transitional
approximations, the familiar wavelength scalings associated with
short-wave
asymptotics are accompanied by a variety of dependences on
disparate
astronomical lengths, such as the Schwartzschild radius,
separation of binary
stars, and distance to the lens. Also discussed briefly are the
images seen in a
telescope looking in the direction of the source; if the
aperture is modelled by
Gaussian apodization, the image is a complexification of the
wave in the
observation plane.
Keywords: asymptotics, diffraction catastrophes, scaling,
focusing,
apodization, cusps
Submitted to: J.Optics, January 2021
Orcid id: 0000-0001-7921-2468
“...it would not be surprising if the wave optical features of
gravitational lensing emerge observationally during the next
century.”[1]
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1. Introduction
With gravitational lensing based on the deflection of light rays
now established
as a valuable tool in astronomy [1-4], there is growing
anticipation of effects of
wave interference, promising to add interferometric precision to
inferences
about distant objects. It is well understood that lensing is
dominated by events
and places where focusing occurs, that is, by caustics, and that
the decoration of
these singularities by ‘diffraction catastrophes’ [5, 6] will
play a major role in
the envisaged wave effects [7]. On astronomically relevant
scales, the
associated diffraction integrals are highly oscillatory, raising
challenges for their
numerical evaluation; an important recent advance, especially
for
multidimensional integrals, has been the development and
application of
sophisticated contour deformation techniques [8, 9].
My aim here is to calculate the fringe spacings and
intensity
amplifications associated with several kinds of focusing, in an
observation plane
(Earth-based for example). For gravitational lensing,
geometrical optics almost
suffices, so the relevant regime is extreme short-wave
asymptotics: small
wavelengths l, i.e. large wavenumbers k=2p/l. The scalings with
k are well
understood [10, 11]: for example near a fold caustic the spacing
is O(l2/3) –
much larger than the familiar interference fringe spacing O(l)
away from
caustics – and the intensity amplification is O(l–1/3). (This
explains, for
example, why the rainbow – which is a fold caustic – is bright,
and why our
naked eyes can resolve supernumerary rainbows decorating the
main bow [12],
even though the wavelengths in sunlight are much smaller than
raindrops.) But
in addition to the wavelength dependence, the fringe spacings
and
amplifications are strongly influenced by the enormously varied
relevant
astronomical lengths: Schwartzschild radii, binary star
separations, lens-
observer distances. It is these additional dependences that will
be calculated.
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3
Section 2 revisits the basic wave theory for the wave in the
observation
plane. I will also briefly discuss the images in a telescope
looking in the
direction of the lensing objects; this is the intensity of the
Fourier transform of
this wave, windowed by the telecope aperture, and if this is
modelled by
Gaussian apodization the images are complexifications of those
in the
observation plane. The main results of the paper are explicit
calculations for
three simple lens models: the focal spot from a single star
(section 3), and
across the fold and cusp caustics from a binary star (section
4). The concluding
section 5 summarises the main results, in terms of physical
variables rather than
the scaled variables convenient for calculation.
In short-wave asymptotics there are four different regimes, and
it is worth
pinpointing the one relevant to the anticipated wave phenomena
in gravitational
lensing. Regime 1 is the coarsest scale, in which the wave
intensity is simply
the sum of intensities of contributing rays: inverse Jacobians
between source
and observer. This is the regime currently employed in
conventional
gravitational lensing: geometrical optics, with singularities at
the caustics. Next
is regime 2, in which the intensity is the square of the sum of
interfering wave
amplitudes, each with a phase (optical path length) decorating
the square root of
the inverse Jacobian, This is the ‘geometrical wave’; it gives
an accurate
description of the wave everywhere except very near the
caustics, where the
light is brightest and wave effects strongest – the very
features of interest here.
We need to describe the waves very close to the geometrical
caustics. This is
regime 3, where the lensed wave is described by canonical
special functions -
diffraction catastrophes [13, 14] – each representing a
different type of caustic.
This ‘transitional asymptotics’ does not match smoothly onto the
geometrical
wave of regime 2. The two are unified in the ‘uniform
asymptotics’ of regime 4,
in which the variables in the diffraction catastrophes are
stretched to represent
the exact optical path lengths of the contributing rays so as to
be valid far from
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4
caustics as well as close to them [5, 6, 13, 15-17]. In
gravitational lensing, it is
only very close to caustics that wave effects are likely to be
observable, so the
additional sophistication of regime 4 is unnecessary: regime 3
suffices, and has
the advantage of leading to simple explicit formulas.
2. Reprise of basic wave theory
For present purposes, the simplest formulation suffices:
paraxial diffraction
from a distant point source, in the thin screen approximation
[7, 18], in which
the lenses are projected along the propagation direction,
leading to a phase
distribution W(r) in the lens plane r=(x,y) (figure 1). Mass
endows space with a
refractive index, for which an adequate approximation is
, (2.1)
where f(r,z) is the Newtonian gravitational potential. The lens
plane phase is
then
, (2.2)
in which ‘regularised’ refers to ignoring irrelevant infinite
constants.
n r, z( ) = 1− 2φ r, z( )c2
W r( ) = dz−∞
∞
∫ n r, z( )−1( )⎛
⎝⎜⎞
⎠⎟ regularised
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5
Figure 1. Lens and observation planes
Elementary Fresnel-Kirchhoff diffraction gives the wave in
the
observation plane R=(X,Y), distant Z from the phase screen,
normalised to unit
strength for the wave arriving at the screen:
. (2.3)
The intensity across the observation plane is |y|2. If the
source is at a finite
distance Zs from the screen, the only effect on the intensity is
that R and Z are
scaled by 1+Z/Zs: the wave beyond the screen is stretched and
further away.
Like all wave patterns, the interference detail will depend on
k; this
colour dependence will be a characteristic signal of wave
lensing, distinguishing
it from geometric gravitational lensing, the latter being
achromatic because the
index (2.1) is wavelength-independent. Spectral distortion of
the Airy pattern
observation plane R
r
Z
distant source
wavefront -W(r)
lens plane
ψ R,Z( ) = −ik2πZ
dr exp ik W r( )+ r − R( )2
2Z⎛
⎝⎜⎞
⎠⎟⎛
⎝⎜
⎞
⎠⎟
r plane
⌠
⌡⎮⎮⌠
⌡⎮⎮
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6
near a fold caustic was discussed elsewhere [19]. Simply put:
caustics arising
from polychromatic sources will be reddened on the dark
side.
It is not my purpose here to give a detailed discussion of the
wave
features of images seen when looking in directions W=(qX,qY)
from the point
R,Z with a telescope of finite aperture. Elsewhere [19] I have
elaborated some
of the peculiarities associated with looking at images when
there are caustics in
the telescope aperture, including the connection with Husimi
functions. But it is
worth drawing attention to one aspect.
The wave in direction W is the windowed Fourier transform of the
wave
at R,Z in the observation plane. If the telescope aperture with
radius L is
modelled by Gaussian apodization rather than the more realistic
sharp edges,
this is
. (2.4)
A straightforward calculation using (2.3) leads to
. (2.5)
Thus the wave corresponding to looking in direction W is a
complexification of
the wave in the observation plane. The corresponding
complexifications of the
Airy and Pearcey functions were discussed in [19].
An easy connection with the aperture-blurred geometrical image
is
obtained by approximating (2.5) by including only the terms that
do not vanish
for large k:
ψ Ω,R,Z( ) = 12π L2
d ′RR plane∫∫ exp −
′R − R( )22L2
⎛
⎝⎜⎞
⎠⎟exp ikΩ⋅ ′R( )ψ ′R ,Z( )
ψ Ω,R,Z( ) = exp − 12 k 2L2Ω2( )exp ikΩ⋅R( ) ψ R,Z( )⎡⎣
⎤⎦R→R+ikL2Ω,Z→Z−ikL2
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(2.6)
The saddle points r of the fast-varying first exponential factor
in the integral are
the lens plane points from which rays emerge in the viewing
direction W, and
with these r the second exponential factor gives the geometric
aperture blurring
(the diffraction blurring O(1/(kL)) requires considering both
factors).
3. Focal diffraction from a single star
For a single star with mass M, the phase W in (2.3) depends on
its
Schwartzschild radius RS:
. (3.1)
This case has been studied extensively, and the diffraction
integral evaluated
exactly [7, 20], but we need to extract the relevant
asymptotics. It is convenient
to scale observation plane distances R, Z and wavenumbers k with
RS, and lens
plane distances r differently:
. (3.2)
In astronomical lensing, the parameters k and z greatly exceed
unity. With these
scalings, (2.3) becomes
, (3.3)
ψ Ω,R,Z( ) = exp −ikZΩ2( )
2π L2
× drr plane∫∫ exp ik W r( )+Ω⋅ r( )( )exp − r − R −ΩZ( )
2
2L2⎛
⎝⎜⎞
⎠⎟.
W r( ) = −2RS logr, RS =2GMc2
k ≡ κRS, R ≡ Rsρ,θR( ), Z ≡ RSζ , r ≡ 2Zk t,φr
⎛
⎝⎜
⎞
⎠⎟
ψ ρ,ζ ,κ( ) = 2 dtt exp i −2κ log t + t2( )( ) J00
∞
∫ tρ2κζ
⎛
⎝⎜⎞
⎠⎟
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in which here and hereafter we neglect all external phase
factors irrelevant to
the wave intensity.
The integral can be expressed in terms of confluent
hypergeometric
functions [7, 20]:
. (3.4)
The intensity falls from the initial value, which for k>>1
is
, (3.5)
to the asymptotic value unity for large r.
Of the variety of asymptotic approximations for confluent
hypergeometric functions [13] (or equivalently Whittaker
functions [21]), the
transitional asymptotics (regime 3) relevant to astronomy is
obtained by
applying large k stationary-phase approximation to the
exponential factor in
(3.3) and using this saddle value t=√k in the Bessel factor.
Thus
. (3.6)
As the dashed curves in figure 2 show, the transitional
approximation accurately
describes more interference oscillations as k increases, that
is, as l decreases for
fixed Z.
ψ ρ,ζ ,κ( ) = 2πκ1− exp −2πκ( ) 1F1 iκ ,1,i
ρ2κ2ζ
⎛⎝⎜
⎞⎠⎟
ψ 0,ζ ,κ( ) 2 ≈ ψ 0 ζ ,κ( ) = 2πκ
ψ ρ,ζ ;κ( ) 2 ≈ 2πκ J02 ρκ2ζ
⎛
⎝⎜⎞
⎠⎟
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Figure 2. Focal spot intensity pattern for the indicated
parameter values. Full red curves: exact intensity (3.4); dashed
curves: Bessel transitional approximation (3.6) (regime 3); dotted
curves: geometric wave approximation (regime 2); the full line is
the asymptotic limiting intensity 1
The canonical star-lensing function is shown in figure 3a
(figures
3b,3c show the fold and cusp canonical functions to be
calculated in section 4).
Figure 3. Canonical functions representing diffraction
intensity, with indicated principal
fringe separations X for (a) focal spot: ; (b) fold caustic: ;
(c) axial section of cusp |P(c)|2
A measure of the fringe spacing is the distance Xstar between
the focus and the
nearest maximum. Thus, from the argument of J0, the spacing in
scaled units is
0 1 2 3 4 50
2
4
6
0 1 2 30
4
8
12
0 1 20
10
20
30
0.0 0.4 0.8 1.20
20
40
60
|ψ|2
κ=1 κ=2
κ=5 κ=10
ρ/√ζ
J02 c( )
-5 0 50.0
0.4
0.8
-6 -4 -2 0 2 40.0
0.1
0.2
0.3
-8 -6 -4 -2 0 2 4 60
2
4
6a b cXstar Xfold Xcusp
c
|ψ|2
J02 c( ) Ai c( )2
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. (3.7)
The intensity amplification factor is 2pk.
The transitional approximation (3.6) is different from the
geometrical
wave approximation; this is obtained from (3.3) by using the
asymptotic
approximation for J0 in the integrand, in the form of a cosine,
separated into two
exponentials, and then evaluating the integral by stationary
phase [20]. The
resulting intensity is shown as the dotted curves in figure 2.
Of course the
approximation fails near the focus at r=0, but it agrees with
the exact intensity
very well everywhere else. In particular, it reaches the
necessary large r limit
unity (as an elementary exercise confirms) – unlike the
transitional
approximation (3.6), which by design works from r=0 out to a
finite value r(k)
that increases with k.
4. Caustic diffraction from a binary star
4.1. Diffraction integral, caustic, geometrical optics
For an equal-mass binary with separation 2a, the phase in (2.3)
is
. (4.1)
It is convenient to scale distances in the observation and lens
planes by a,
distances Z by a2/RS, and wavenumber k by RS:
. (4.2)
Astronomically, k>>1 always. We will often specialise
formulas for general z
to the asymptotic regime z>>1, because this gives
relatively simple formulas
Δρstar =X starκ
ζ2, X star = 3.832
W r( ) = −RS log x − a( )2 + y2( ) x + a( )2 + y2( )⎡⎣ ⎤⎦
k ≡ κRS, x, y( ) ≡ a u,v( ), X ,Y( ) ≡ a ξ,η( ), Z ≡ a
2
RSζ
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and corresponds to many cases of interest (for a binary
consisting of two solar-
mass stars (RS~3km) separated by 1au (i.e a=0.5au), z>1
corresponds to
Z>65pc). With the scaling (4.2), the binary star lensed wave
becomes
(4.3)
For large k, the wave is dominated by the caustics, where real
saddles of
the phase S coincide. As is known [1], for constant z sections
the caustic
consists of fold curves and cusp points forming closed astroids:
two for z
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12
Figure 4. Caustics from a binary-star lens, for the distances
from the observation plane (a) z =2.05, (b) z=4; the red dots
indicate points on the fold and cusp near which the local
approximations are calculated in sections 4.2 and 4.3.
The saddle-points of the integrand in (4.3) determine the ray
(x, h) in the
observation plane that originates from the point (u,v) in the
lens plane, and, by
inversion, the lens plane points from which rays reach given
observation points:
-1 0 1-4
-2
0
2
4
-1 0 1-8
0
8
4
-4
a b
ξ
fold
cusp1cusp2
cusp3
η
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13
(4.4)
There are five saddles, all real within the astroids and three
real and two
complex outside. The caustic is determined by their coalescence,
where the
determinant of the Hessian matrix
(4.5)
vanishes:
. (4.6)
Explicitly, the caustic is given parametrically in terms of w by
the real values of
(4.7)
The transitional approximations (regime 3) that will be
calculated in the
following sections are those associated with the singularities
illustrated in figure
4: the fold and cusp1 and cusp2 along the h axis, and cusp3
along the x axis.
Explicitly, these are located at
∂u S = 0⇒ ξ = u 1−4ζ w2 −1( )1+ w2( )2 − 4u2
⎛
⎝⎜⎜
⎞
⎠⎟⎟,
∂v S = 0⇒η = v 1−4ζ w2 +1( )1− w2( )2 + 4v2
⎛
⎝⎜⎜
⎞
⎠⎟⎟, w = u2 + v2
⇒ u = uj ξ,η,ζ( ),u = uj ξ,η,ζ( ){ }, j = 1…5( ).
M =∂uu S ∂uv S∂vu S ∂vv S
⎛
⎝⎜⎜
⎞
⎠⎟⎟
detM = 0⇒w2 −1( )2 + 4u2( )w2 +1( )2 − 4u2( )2
=w2 +1( )2 − 4v2( )w2 −1( )2 + 4v2( )2
= 116ζ 2
u± w,ζ( ) = ± ζ 32 1+ w4 + 2ζ 2( ) + 1+ w2( )2 + 8ζ 2 ,v± w,ζ( )
= ± ζ 32 1+ w4 + 2ζ 2( ) − 1− w2( )2 − 8ζ 2 .
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(4.8)
where in each case we also give the limiting forms: near where
the caustic is
born (z=1/4 for the fold, z=2 for cusp1 and cusp2, and z=0 for
cusp3), and far
way, i.e. z>>1. We will also need the points u, v in the
lens plane from which
these caustics originate:
(4.9)
In all cases, the procedure will be the same: locally expanding
the phase S in
(4.3) about the caustic points x, h, u, v.
Although we are aiming for the transitional approximations,
in
subsequent sections it will be convenient also to use the
geometrical wave
(regime 2), as an useful check on the calculations. This is
ηfold ζ( ) = 2 ζ ζ + 2( ) −1− 2ζ ζ ζ + 2( ) −1+ζ( ) = 163 3 ζ
−
14( )3/2 +O ζ − 14( )5/2( ) = 2ζ −1+ 14ζ −1 +O ζ −2( ),
ηcusp1 ζ( ) = −2 ζ ζ − 2( ) −1+ 2ζ ζ ζ − 2( ) −1+ζ( ) = 3 + 2 23
ζ − 2 +O ζ − 2( ) = 2ζ −1− 34ζ −1 +O ζ −2( ),ηcusp2 ζ( ) = 2 ζ ζ −
2( ) −1+ 2ζ − ζ ζ − 2( ) −1+ζ( ) = 3 − 2 23 ζ − 2 +O ζ − 2( ) = 1/
ζ +O ζ −3/2( ),ξcusp3 ζ( ) = 2 ζ ζ + 2( ) +1+ 2ζ − ζ ζ + 2( ) +1+ζ(
) = 1−ζ +O ζ 3/2( ) = 1/ ζ +O ζ −3/2( ),
ufold = 0, vfold ζ( )= − 2 ζ ζ + 2( ) −1− 2ζ ,
ucusp1 = 0, vcusp1 ζ( )= − −2 ζ ζ − 2( ) −1+ 2ζ ,
ucusp2 = 0, vcusp2 ζ( )= − 2 ζ ζ − 2( ) −1+ 2ζ ,
vcusp3 = 0, ucusp3 ζ( )= 2 ζ ζ + 2( ) +1+ 2ζ .
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15
, (4.10)
in which
(4.11)
Even though the geometric wave fails on the caustics, it gives
an easy global
picture of the interference structure, as figure 5
illustrates.
Figure 5. Geometrical wave (regime 2) (4.10) for k=200, z=5,
showing the central astroid
It is also interesting to consider the geometric optics limit,
i.e. regime 1.
Here there is a slight subtlety, arising from the fact that
along symmetry axes
two or more rays may contribute with the same phase and so their
contributions
ψ geom ξ,η,ζ ,κ( ) =exp iκ S j + 14 iπµ j( )
ζ detM jj=1,real rays
5
∑
S j = S uj η,ζ( ),v j η,ζ( ),ξ,η,ζ( ), M j = M u=u j ξ ,η,ζ(
),v=v j ξ ,η,ζ( )( ),µ j = sgn eigenvalues M j( )( )
1,2∑ .
-0.6 0.60-0.6
0.6
0
ξ
η
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16
must be added coherently, with each intensity multiplied by a
degeneracy factor
dj (1 or 2 in our examples. The geometric intensity is thus
. (4.12)
Figure 6 illustrates the accuracy of the geometric wave
approximation (regime
2), and how geometric optics (regime 1) describes the mean
intensity,
everywhere except near the caustics, to the precise description
of which we now
turn.
Figure 6. Wave intensity for h=0, z=1, k=20, including two cusp
caustics (cf. figure 5). Red
curve: exact, calculated numerically from (4.3); full black
curve: geometric optics
approximation (4.10) (regime 1); dashed black curve: geometric
wave approximation (4.8)
(regime 2)
4.2. Fold diffraction
The calculation will correspond to the red dot with the largest
value of h on the
fold caustic in figure 4a. As z increases, this caustic
originates at z=1/4 and then
Igeom ξ,η,ζ ,κ( ) =d j
ζ 2 detM jj=1,real rays
5
∑
-2 -1 0 1 20
10
20
30
40
ξ
|ψ|2
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17
recedes (figure 4b). To get the transitional approximation we
expand the phase
in (4.3) about the caustic points in the lens and observation
planes in (4.8) and
(4.9):
(4.13)
in which K is an irrelevant constant and
. (4.14)
The u integral is Gaussian, and the contribution to the
diffraction integral close
to the fold at hfold, from the dv integral in the neighbourhood
of ufold=0, v=vfold,
is the anticipated Airy function [13, 22]
, (4.15)
where
(4.16)
and
, (4.17)
with k1/3Efold(z)2 giving the intensity amplification at the
fold caustic.
The separation of the first two intensity maxima (cf. figure 3b)
is thus
. (4.18)
S u,vfold ζ( )+δv,0,ηfold ζ( )+δη,ζ( ) = K − δvδηζ −13δv
3A ζ( )+ u2
ζ!
A ζ( ) = 12ζ 2
2 ζ 2 +ζ( ) −1− 2ζ ζ 2 +ζ( ) 1+ζ( )+ζ 2 +ζ( )( )
ψ fold δη,ζ ,κ( ) =κ 1/6Efold ζ( )Ai δηκ 2/3Ffold ζ( )( )
Ffold ζ( ) =1
ζ A ζ( )1/3= 23 ζ − 14( )( )1/6
1+O ζ − 14( )( ) = 1ζ 1+O1ζ
⎛⎝⎜
⎞⎠⎟
⎛⎝⎜
⎞⎠⎟
Efold ζ( ) =πζ
1A ζ( )1/3
= π3 ζ − 14( )( )1/6
1+O ζ − 14( )( ) = πζ 1+O1ζ
⎛⎝⎜
⎞⎠⎟
⎛⎝⎜
⎞⎠⎟
Δηfold =X foldκ
−2/3
D ζ( ) ≈ζ >>1X foldκ−2/3ζ , X fold = 2.2294
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18
This shows that in the short-wave limit the fold intensity grows
as k1/3, and the
fringes get smaller as k–2/3 , as anticipated from catastrophe
optics [5]; the
fringes also expand linearly as the lens distance z increases.
For z>2 this fold
lies on an astroid (figure 4) that gets smaller as z increases:
its size (distance
from the fold down to cusp 1) is close to 1/z (cf. (4.8). The
local approximation
(4.15) and (4.16) is valid only while the fold and cusp are well
separated, i.e. the
distance between these singularities exceeds the fringe
separation. This requires
. (4.19)
These are the main results of this section. The Airy function
emerges
clearly only for very large k. This is because there is an
additional and non-
negligible contamination from the three geometrical rays that
are separate from
the two that coalesce on the fold. This contamination gets
slowly weaker as
k increases, but is still visible as very fast oscillations even
for k=108 (figure
7a). In any practical situation these are likely to be averaged
away by
decoherence, leaving the Airy approximation emerging clearly;
numerical
calculations (not shown) confirm this. This phenomenon, of
diffraction
catastrophes being contaminated by fast oscillations that are
often only
imperfectly resolved in practice, is familiar in several
contexts, including the
quantum scattering of atoms and molecules [23, 24] and optical
rainbows [25,
26].
ζ < κ1/3
X fold
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19
Figure 7. Wave intensity close to (a) fold; (b) cusp1; (c)
cusp2; (d) cusp3, corresponding to red dots in figure 4a for
z=2.05, k=108; the red curves are the transitional approximations
(regime 3) of (a) sections 4.1, and (b,c,d) section 4.2; the black
curves show the geometric
wave approximation (regime 2). In (a) the inset shows the finest
two interference scales of the rays contaminating the fold
caustic.
4.3. Cusp diffraction
The procedure for all three cusps is the same; we give the
details for cusp3, i.e.
the cusp on the x axis (figure 4a), and then state the results
for cusp1 and cusp2.
To get the transitional approximation for cusp3, we expand the
phase in (4.3)
about its points x=xcusp,h=0, in the observation plane (equation
(4.8)), and
u=ucusp3, v=0 in the lens plane (equation (4.9)):
(4.20)
in which the second equality follows after integrating over du,
and
a
3.2003 3.2005 3.20070
50
100
2.140 2.150 2.1600
4000
8000
1.400 1.405 1.4100
10000
20000
0.5548 0.5552 0.5556 0.55600
40000
80000
2.5x10-6b
c d
|ψ|2
η
η
η
ξ
S ucusp3 ζ( )+δu,v,ξcusp3 ζ( )+δξ,0,ζ( ) = K + A3v
4 +δu −δξζ
+C3v2⎛
⎝⎜⎞⎠⎟+ δu
2
ζ+!
= K + A3 − 14C32ζ( )v4 + 12δξC3v2 +!
-
20
(4.21)
Thus the contribution to the diffraction integral close to
xcusp3, from the
neighbourhood of v=0, u=ucusp is
. (4.22)
P is the quartic-phase diffraction catastrophe integral (Pearcey
function [13,
27]) describing waves along the symmetry axis of the cusp and
illustrated in
figure 3c:
(4.23)
Here the z-dependent functions in the argument of P, and the
prefactor, are
(4.24)
and
A3 =18ζ 2
1−ζ −ζ 2 +ζ ζ 2 +ζ( )( ),C3 =
12ζ 2
1+ 2ζ + 2 ζ (2 +ζ ζ 2 +ζ( ) ζ +1( )−ζ ζ + 2( )( ).
ψ cusp3 δξ,ζ ,κ( ) = E3 ζ( )κ 1/4P δξκ 1/2F3 ζ( )( )
P c( ) = dt exp i t 4 + ct2( )( )−∞
∞
∫
= π c8exp − 18 ic
2( ) exp 18 iπ( ) J− 14 18 c2( )− sgn c( )exp − 18 iπ( ) J 14 18
c2( )( ).
F3 ζ( ) =C3
2 A3 − 14C32ζ
= 1ζ 3/2
−ζ ζ + 2( )+ ζ +1( ) ζ ζ + 2( )( )× 1+14ζ + 22ζ 2 + 8ζ 3 + 2 2 +
7ζ + 4ζ 2( ) ζ ζ + 2( )= 2
ζ1+O ζ( )( ) = 2 1+O ζ −1( )( ).
-
21
(4.25)
The intensity amplification at cusp3 is k1/2E3(z)2. The
separation of the
first two intensity maxima (cf. figure 3c) is
. (4.26)
The k-1/2 dependence was anticipated from catastrophe optics,
but the fact the
for large z the fringe separation is independent of z was
unexpected. An
analogous phenomenon occurs in optical ‘diffractionless beams’
[28-31].
For z>2, cusp3 lies on an astroid (figure 4) that gets
smaller as z
increases: its size (distance across to the corresponding cusp
on the other side of
the astroid, i.e. for x>112 X cuspκ
−1/2 , X cusp = 3.3770
2 / ζ
ζ < 16κX cusp2
-
22
Figure 8. Wave intensity across cusp3 for z=1 and (a) k=20, (b)
k=200, (c) k=1000, (d) k=106, showing emergence of the transitional
approximation (4.23) (red curves) as k
increases; the full black curves show the geometric optics
approximation (regime 1), and the dashed black curves show the
geometric wave approximation (regime 2)
For cusp1 and cusp2, the procedure is the same, with h and x
replacing
x and h, and v and u replacing u and v. For the inward-pointing
cusp1, the
counterparts of (4.23)-(4.29) are
(4.28)
where
(4.29)
and
0.0 0.2 0.4 0.6 0.8 1.00
1
2
3
4
0.5 0.6 0.70
0.05
0.10
0.60 0.62 0.64 0.66 0.680
0.005
0.010
0.679 0.680 0.681 0.682 0.68300.678
1x10-7
2x10-7
3x10-7
0.70
0.8
|ψ|
ξ
b
c d
a
ψ cusp1 δη,ζ ,κ( ) = E1 ζ( )κ 1/4P −δηκ 1/2F1 ζ( )( ),
F1 ζ( ) =1
ζ 3/2ζ ζ − 2( )+ ζ −1( ) ζ ζ − 2( )( )×
−1+14ζ − 22ζ 2 + 8ζ 3 − 2 2 − 7ζ + 4ζ 2( ) ζ ζ − 2( )= 12 3 ζ −
2( ) 1+O ζ − 2( )( ) = 1ζ 1+O ζ −1( )( ),
-
23
(4.30)
The intensity amplification at cusp1 is k1/2E1(z)2. The
separation of the
first two intensity maxima (cf. figure 3c) is
. (4.31)
This increases with z, and gets much bigger than the separation
(4.26) for
cusp3. The validity condition, analogous to (4.19) for the fold
and (4.27) for
cusp3, is that cusp1 is well separated from the fold on the same
astroid:
. (4.32)
For the outward-pointing cusp2,
, (4.33)
where
(4.34)
and
E1 ζ( ) =1
π ζ 1− 4ζ + 2ζ 2 + 2 ζ −1( ) ζ ζ − 2( )( )( )1/4
= 121/4 π
1+O ζ − 2( )( ) = 12πζ −3/4 1+O ζ −1( )( ).
Δηcusp1 =X cusp
F1 ζ( )κ 1/2≈
ζ >>1ζ 1/2X cuspκ
−1/2 , X cusp = 3.3770
ζ < κ1/3
X cusp2/3
ψ cusp2 δη,ζ ,κ( ) = E2κ 1/4P δηκ 1/2F2 ζ( )( )
F2 ζ( ) =1
ζ 3/2−ζ ζ − 2( )+ ζ −1( ) ζ ζ − 2( )( )×
−1+14ζ − 22ζ 2 + 8ζ 3 + 2 2 − 7ζ + 4ζ 2( ) ζ ζ − 2( )= 12 3 ζ −
2( ) 1+O ζ − 2( )( ) = 2 1+O ζ −1( )( ),
-
24
(4.35)
The intensity amplification at cusp2 is k1/2E2(z)2. The
separation of the
first two intensity maxima (cf. figure 3c) is
. (4.31)
This is asymptotically independent of z, like the separation
(4.26) for cusp3.
The validity condition for these oscillations is the same as for
cusp3, on whose
astroid it lies, namely
. (4.32)
Figures 7b and 7c illustrate the accuracy of the transitional
approximation
near cusp1 and cusp2. For all three cusps, the fast-oscillating
contamination by
interfering waves not associated with the caustic is much weaker
than for the
fold (figure 7a). This because there are only two such
contaminating rays, rather
than three.
5. Concluding remarks
The calculations reported here are intended to indicate the
fringe separations
and intensity amplifications that could occur in gravitational
lensing. For
convenience, Table I presents the results in terms of physical
variables. The
wavelength dependences are those anticipated from singularity
theory, but the
dependences on astronomical lengths could not have been
anticipated without
E2 ζ( ) =1
π ζ 1− 4ζ + 2ζ 2 − 2 ζ −1( ) ζ ζ − 2( )( )( )1/4
= 121/4 π
1+O ζ − 2( )( ) = 2πζ 1/4 1+O ζ −1( )( ).
Δηcusp2 =X cusp
F2 ζ( )κ 1/2≈
ζ >>112 X cuspκ
−1/2 , X cusp = 3.3770
ζ < 16κX cusp2
-
25
detailed calculation. In particular the dependences of fringe
spacing on
Schwartzschild radius, binary separation, and lens distance, are
very different
for the inward-pointing cusp1 and the outward-pointing cusp2 and
cusp3.
Further similar calculations can be envisaged, for example of
the fringe
spacings across a cusp, rather than along its symmetry axis;
these spacings are
much smaller: O(l3/4) rather than O(l1/2) [5] (an example of
this diffraction
anisotropy is the longitudinal stretching of fringes in cusps
seen by people
wearing eyeglasses when looking at street lights on rainy
nights; the images are
distorted by raindrop ‘lenses’ on the glass lenses [11, 14, 32];
the apparent
oxymoron that these fringes appear black-and-white, i.e. not
coloured, is a
psychovisual effect [33]).
Table 1. Fringe separations, intensity amplifications and
validity conditions, expressed in
physical variables, for Z>>RS for star and
ZRS/a2>>1for fold and cusps; Xfold=2.2294,
Xcusp=3.3770,
Two main effects threaten the detection of wave effects such as
those
anticipated here. The fringes in the observation plane could be
blurred by the
finite size of the distant source; its angular size, seen from
the lens plane, should
be smaller than the angle subtended at the lens plane by these
fringes, i.e.
fringe size amplification validity
star X starλ23/2π
ZRS
4π 2 RSλ
–
fold X foldλ2/3RS
1/3Z2π( )2/3 a
π Aimax2 a2
Z2πλRS
2
⎛⎝⎜
⎞⎠⎟
1/3
Z < 2πa2
X fold λRS2( )1/3
cusp 1 X cuspZλ2π
Pmax2 a3
RS 2πλZ3
Z < a2 2πX cusp
2 RS2λ
⎛
⎝⎜⎞
⎠⎟
1/3
cusps 2,3X cuspa23/2
λπRS
2 Pmax2 RSa
2Zπλ
Z < 32πa2
λ
Aimax2 = 0.2869, Pmax
2 = 6.9438
-
26
(spacings in Table I)/Z. Of course, the size of the aperture of
the telescope
directed at the source is also a limiting factor; I have not
discussed this here,
apart from briefly at the end of section 2 (see also [19]).
Acknowledgements. My research is supported by a Leverhulme Trust
Emeritus
Fellowship.
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