1

Diffraction decorating caustics in gravitational lensing

M V Berry

H H Wills Physics Laboratory, Tyndall Avenue, Bristol BS8 1TL, UK

asymptotico@bristol.ac.uk

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]

2

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.

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

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

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

⌠

⌡⎮⎮⌠

⌡⎮⎮

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

7

(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

⎛

⎝⎜⎞

⎠⎟

8

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

⎛

⎝⎜⎞

⎠⎟

9

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

10

. (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

11

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

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

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 .

14

(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 .

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

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

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

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

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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