Transcript The architecture of diffraction catastrophes Michael Berry Physics Department University of Bristol United Kingdom http://www.phy.bris.ac.uk/staff/berry_mv.html Institute for Mathematics and its Applications, Minneapolis, 22 July 2002 forthcoming chapter with.

The architecture of
diffraction catastrophes
Michael Berry
Physics Department
University of Bristol
United Kingdom
http://www.phy.bris.ac.uk/staff/berry_mv.html
Institute for Mathematics and its Applications, Minneapolis, 22 July 2002
forthcoming chapter with Christopher Howls for the
Digital Library of Mathematical Functions (DLMF)
Integrals with coalescing saddles
mathematics, physics, philosophy...
diffraction catastrophes are oscillatory integrals, constructed
from the catastrophe polynomials of Thom and Arnold
catastrophe theory: stable bifurcations of critical points of
smooth functions as parameters vary
1 t; x   t 3  xt
variable
critical points: ∂1/∂t=0
smooth function
parameter
x<0
1
x=0
t
1
x>0
t
bifurcation (coalescence) at x=0
1
t
bifurcation set: locus of parameter(s) x where
critical points coalesce
2
 1
for 1:
x
2
t
x=0
0
stability: bifurcation (pattern of coalescence of critical points)
persists under perturbation, e.g. to
3
2
4
t  xt  x  x t  x t 
and can still be described locally by 1, by diffeomorphism
with more parameters x, y, z..., more critical points can collide,
and the stable bifurcations are more complicated
catastrophe theory classifies these by
codimension K: number of active parameters
codimension K=1: fold catastrophe
x
K=2: cusp catastrophe
y
x
1 t; x   t 3  xt
critical point locus in x, t:
t
x
4
2
 2 t; x, y  t  yt  xt
cusp bifurcation set:
2
3
27x  8y
K=3: swallowtail
3 t; x, y, z   t 5  zt 3  yt2  xt
polynomials represent equivalence classes
K=3: elliptic umbilic
3
E s,t; x, y, z   s  3st

2
z s t
2
K=3: hyperbolic umbilic
2
 yt  xs
variables s, t: corank 2
3
H s,t; x, y, z   s  t
3
 zst  yt  xs
critical points   s,t E,H  0
in physics, catastrophes are realised as caustics
caustics: envelopes (focal surfaces) of families of rays
rays: critical points of path length (Fermat-Hamilton)
field point
x,z
caustics
path length
rays
 t; x, z  
 z  ht 2   x  t 2
t
initial wavefront, h(t), e.g. from a wavy lens or mirror
rays: t  0
caustic:
t  0 and t2  0
sunsparkles on Lake Como: a many-to-one gradient map from
the lake to the eye at a given instant
images are places on the
water where the distance
sun-water-eye is stationary
singularities of the map:
images coalesce, when
a caustic in the space
above the water passes
the eye
rapid succession of singularities
sparkling
mathematical catastrophes: stable singularities: natural focusing
wavefront
ht   0.15t 6  0.4t 5  t 4  12 t 2
z
x
caustics are the singularities
of geometrical optics
h
(holistic property of family of rays)
geometric caustic singularities are decorated by waves when
the wavelength l is small, i.e. k=2p/l is large
then, short-wave (high-frequency, semiclassical…) asymptotics
gives, for the wave decoration of the catastrophe generated by
the polynomial K(t; x) (x=x, y, z…), the diffraction catastrophe

K x;k   k 
dt expik K t;x (corank 1)


U x;k   k




 ds  dt expikU s,t;x (corank 2, U=E,H)
ray physics: differentiate  with respect to t (and s)
wave physics: integrate exp(i with respect to t (and s) - superposition
asymptotics: O(1/k) corrections, atoms of uniform approximations
as k
∞, the integrals oscillate infinitely fast, and the geometric
singularity is characterized by wave exponents describing
k-scaling of the diffraction catastrophe:
K x;k   k
where
and

yk   k
K y 
g 1K
K
K yk 
x1,k
g 2K
x2


 dt expi K t;y

canonical integrals
k=1
wave exponents K, {gKi}
wave exponents
K
K 
, U  13 U  E, H 
2 K  2 
m
g mK  1
, g xU  g yU  23 , g zU  13
K2
singularity
fold
codimension K
1
cusp
2
swallowtail
3
E, H umbilics
3
meaning of exponents
as k
 K g 1K g 2K g 3K
1
6
1
4
3
10
1
3
K x;k   k
K
2
3
3
4
4
5
2
3
1
2
3
5
2
3

K k
g mK
2
5
1
3
xm

∞: 1. intensity ||2 diverges as k 2 K
2. interference fringes in xm direction shrink as
k g mK
adapted from Beresford Parlett:
only wimps study only the general
case; real scientists understand
through examples
fold diffraction catastrophe
Airy 1838
2/3 

k x
1/ 6 2p
1 x;k   k
1/ 3 Ai 1/ 3 
3
 3 

Ai  X   21p



du exp i 13 u 3  Xu

interference oscillations
no (real) rays (evanescent wave)
2 rays
X
ray optics limit
1/l1/3
~l2/3
intensity
supernumerary rainbows
interference fringes near an angular caustic
for deflection D, droplet radius a,refractive index n,
1/ 3
a 

intensity  D   
l 





1 / 2 

2 / 3 n2  1
2pa 

2 

Ai Dmin  D


2 1/ 6

3
l

4 n




Descartes: the rainbow is a caustic of deflected rays
x
a
singularity (caustic) at Dmin
for wavelength l, intensity D
1/ 3
a 

  
l
D-Dmin




1/ 2 
2

2
/
3
n 1
2pa

2 

Ai Dmin  D 

2 1/ 6
3
l
4n




 1 4  n 2 3/ 2 
Dmin  2 cos 1  2 
 
3  

n 

 138o
platonic rainbow
for white illumination
Kelvin’s ship-wave pattern
gravity waves on deep water
ship speed v
‘Hamiltonian’ w(k)=√(gk)+v.k
distance scale a=v2/g
r=r/a
r

wave height
arcsin(1/3)
z r,  
p /2

p / 2
r cos   
d cos
 cos2  



 cosrA Ai r 2/3 B 
scaling
gravity’s rainbow - interfering falling neutrons
neutron mass m, gravity acceleration g, Planck constant h
caustic (bounding parabola)
x
for energy E,
neutron wavelength is
l=h/√2mE
x
2
fringe spacing
1/ 3
 h 
x  a 2 2 
p m g
=0.026mm
spacing of Airy maxima
scaling exponent h2/3
x independent of l!
cusp diffraction catastrophe (Pearcey 1946)
x
2 x, y 


|2|2

dt exp i t 4  yt 2  xt

y

x
y
Imt
contour
Ret
much ado about nothing: zeros of the cusp diffraction catastrophe
inside the cusp, 3-wave
interference:
y
x
ym   2p 4m 1,
xnm
2

2n  14 p

ym
yM   2p 4M  1,
x NM
2
1 p

2N

1



4
yM
outside, interference between one real wave and one complex
(evanescent, decaying) wave
(topological effect of a complex wave)
pairs of perfectly
black points of
complete destructive
interference -zeros of
intensity (cf. Young:
“on blackness”)
=1/4; g1=3/4; g2=1/2
zeros have codimension 2, and are phase singularities
(wave dislocations, vortices, topological charges…)
y=rexp(ic)=0, i.e. ReyImy=0
modulus phase
zeros are complementary to
caustics
singularities of ray physics
singularities of wave physics
phase contours of 2(x,y), at intervals of p/4
cusp diffraction catastrophe
modulus
phase
(colour-coded by hue)
diffraction catastrophes in white light (real colours)
theory
experiment
rainbow skittles:
colours deep inside the cusp
hyperbolic umbilic diffraction catastrophe
section of the three-dimensional elliptic umbilic diffraction
catastrophe, involving four interfering waves
caustic
E x, y, z  
wave pattern


3
2

2
dsdt exp i s  3st  z s  t
st plane
evaluated on slices with constant z
2
 yt  xs
z=0
unfolding the three dimensional wave pattern,
made by shining light on a triangular water-droplet ‘lens’
E x, y,0  

2p 2 23
2
3
 x  iy  x  iy 
Re Ai  1 Bi  1 
  12 3   12 3 
z=3.85
z=4


4 z 2  1 xz  p 
E x, y, z   2 p3 exp i 27
3
4
u
2 
  6
y
4
2
2
du expiu  2zu  z  x u  12u2 
C


p/12
C
complicated
pattern of zeros nodal lines in
three dimensions
z=6
for |z|>>1, central region of pattern
periodic in z as well as x and y
‘crystallography’ - R3m
‘crystallographic’ region
|z|>>1, |x,y| small
z/z=0
y/x
z/z=1/36
z/z=1/6
nearly plane nodal rings
max
x/x
min
z
Nmax rings in Mth row,
before transition to hairpin
Nmax
256M 269

 int 

13
52 
rings
hairpins
N=14
M=1
N=1
cusp
fold
M=1, Nmax=14
M=2, Nmax=34
projections of nodal rings onto xz plane
theory reduction
nodal lines outside the caustic
curly antelope
horns
James Clerk Maxwell
The dimmed outlines of phenomenal things
all merge into one another
unless we put on the focusing-glass of theory,
and screw it up first to one pitch of definition and then to another
so as to see down into different depths
through the great millstone of the world.
projection identities - nonlinear integral identities
for diffraction catastrophes
prototype:
2
2
23
Ai  X   p




2
3
duAi 2 u 2  X

physical interpretation in terms of phase space,
leading to two infinite families of identities
two ways to get the square of a diffraction catastrophe |y(x)|2
two ways to get |y(x)|2
p
y(p)
A
C
B
y(x)
B
C
y x 
W(x,p)
Lagrangian manifold
x=x(p)
D
|y(x)|2
1
2p

x
-ipx
dp
y

p

e

A
p
y p  Ae
i  dp x  p 
p0
(Maslov,diffraction catastrophe)


W  x, p   p1  dPe 2ixPy p  P y* p  P  (Wigner)


D
y  x  2   dpW x, p

(projection)
dpdP or dPdp
examples

cusp


2
du
2 x, y   1 cos2xu1 2u 3 y  2u2
u 3
2

0
hyperbolic umbilic
H x, y, z   8p 2 29
1
3
2




 du  dv 
  x  zv  3u Ai  y  zu  3v 
Ai
4
3
1
3
2
4
3
1
3
2
the quietly beating heart of asymptotics: Stokes’ phenomenon
large-k asymptotics for fixed x far from caustic in K(x)

C
1
fold  X   Ai  X  
2p
X<0: two real rays
Ai  X  
cos
23 X 3/ 2 
p X 1/ 4
X  0
bright side


dt exp i 13 t 3  Xt

X>0: one complex ray
Ai  X  

exp  23 X 3 / 2
2 p X1/ 4
X  0
dark side


how can one exponential (dark side) become two (bright side)?
Stokes 1847: the second exponential switches on in the
complex plane where the two exponentials are maximally
disparate, that is where they have the same imaginary parts
ImX
Stokes line
one exponential
2p/3
two exponentials
ReX
formally, on the dark side,


exp  23 X 3/ 2  ar
4 X 3/ 2
Ai X  
,
F



3
2 p X1/ 4 r0 F r
a0  1, ar
r  16 !r  65 !



2pr!
r  1!
r 2p
divergent series
the second exponential is born from the resummed tail of the
divergent series, by a universal mechanism:
 a
ar F 1 ar
 r   r   rr
r0 F
r0 F
r F F

F 1 a
  rr
F 1 r0 F
terms get smaller, then bigger,
r=IFI is close to the least term
 iS  exp F 
Stokes multiplier S is an error
function:
ImX
Im F

2 Re F
S   
1
p
,ImF

 dt expt 
2

ReX
theory
|F|=2
exact
two contrasting general phenomena
bifurcation (caustic, catastrophe) set: real saddles collide
two real waves
one evanescent wave
(complex saddle)
violent birth/death of real waves
∂t=0
∂t2=0
stokes set
evanescent wave
+ dominant wave
nothing
+ dominant wave
gentle birth/death of evanescent waves
Re(1-2)=0
nonlocal bifurcation
ImX
so, for Ai(X) (Stokes 1857)
for higher-K diffraction
catastrophes, Stokes’
phenomenon can occur
for real parameters
2 complex
waves
complex plane of
parameter X
2 real waves
1 complex
wave
ReX
2 complex
waves
Stokes set
cusp diffraction catastrophe (Wright 1980)
x
bifurcation set
y   32 x 2 / 3
3 real
Stokes set
1/ 3 1 2 / 3
2x
y  33 3  5
1 real, 1 complex
1 real
y
1 real, 1 complex
K=3
elliptic umbilic
swallowtail
hyperbolic umbilic
Emotional asymptotics
Passions rise.
We cross the bifurcation;
something intense and complex is born.
But it is evanescent from the start,
and soon decays.
Much later, we meet again.
Now there is another dominant
exponential in her life.
As we cross our Stokes
line, all passion vanishes.
The whole thing was an error (function).