Canonical Integrals for Diffraction Catastrophes

​
integral
Pearcey
swallowtail, x=-6
elliptic umbilic, z=4
hyperbolic umbilic, z=2
plot
2D
contour
3D
f
Re
Im
Abs
Arg
Catastrophe theory was developed in the late 1970s. A catastrophe is a discontinuous change in the behavior of a function that can occur even when its parameters are varied continuously. In diffraction theory, when higher-order catastrophes appear, rapidly oscillating diffraction integrals are required. These integrals represent the light intensity or the quantum-mechanical probability density. When catastrophes occur, classical intensity functions are no longer adequate. The diffraction integrals contain several control parameters (which determine the codimension
K
) and a set of two-state variables. The curves where intensities accumulate are called caustics.

Details

Examples for four of the seven possible types of catastrophe are:
Pearcey integral (cusp catastrophe)
P(x,y)=
Ψ
2
(X)=
∞
∫
-∞
expi
4
u
+x
2
u
+yudu
swallowtail
S(x,y,z)=
Ψ
3
(X)=
∞
∫
-∞
expi
5
u
+
3
u
x+
2
u
y+zudu
elliptic umbilic
(E)
Ψ
(x,y,z)=
∞
∫
-∞
∞
∫
-∞
expi
3
s
-3s
2
t
+z(
2
s
+
2
t
)+yt+xsdsdt
hyperbolic umbilic
(H)
Ψ
(x,y,z)=
∞
∫
-∞
∞
∫
-∞
expi
3
s
+
3
t
+zst+yt+xsdsdt

References

[1] I. S. Gradsteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 6th ed., San Diego: Academic Press, 2000.
[2] M. Berry, "Why Are Special Functions Special?," Physics Today, 54(4), 2001 pp. 11–12. scitation.aip.org/content/aip/magazine/physicstoday/article/54/4/10.1063/1.1372098.
[3] R. Gilmore, Catastrophe Theory for Scientists and Engineers, New York: Dover Publications, 1993.
[4] M. V. Berry and C. J. Howls. "Chapter 36: Integrals with Coalescing Saddles." Digital Library of Mathematical Functions. Version 1.0.9; Release date 2014-08-29. dlmf.nist.gov/36.
[5] Souichiro-Ikebe. "Special Functions for Catastrophe Theory." Graphics Library of Special Functions (in Japanese). (Nov 7, 2014) math-functions-1.watson.jp/sub1_spec_370.html.
[6] T. Pearcey,"The structure of an Electromagnetic Field in the Neighbourhood of a Cusp of a Caustic," Philosophical Magazine, 37(268), 1946 pp. 311–317. www.tandfonline.com/doi/abs/10.1080/14786444608561335#.VF0IB4e3ZM4.
[6] R. Borghi, "Evaluation of Cuspoid and Umbilic Diffraction Catastrophes of Codimension Four," Journal of the Optical Society of America A, 28(5), 2011 pp. 887–896. doi:10.1364/JOSAA.28.000887.
[7] J. N. L. Connor and C. A. Hobbs, "Numerical Evaluation of Cuspoid and Bessoid Oscillating Integrals for Applications in Chemical Physics," Khimicheskaya Fizika, 23(2), 2004 pp. 13–19.
​arxiv.org/ftp/physics/papers/0411/0411015.pdf.
[8] S. Scibelli. "Research Journal." (Nov 7, 2014) laser.physics.sunysb.edu/~samantha/journal.
[9] J. A. Lock and J. H. Andrews, "Optical Caustics in Natural Phenomena," American Journal of Physics, 60(5), 1992 p. 397. doi:10.1119/1.16891.

External Links

Caustic (Wolfram MathWorld)
Catastrophe (Wolfram MathWorld)
Elliptic Umbilic Catastrophe (Wolfram MathWorld)
Parabolic Umbilic Catastrophe (Wolfram MathWorld)
Hyperbolic Umbilic Catastrophe (Wolfram MathWorld)
Swallowtail Catastrophe (Wolfram MathWorld)
Codimension (Wolfram MathWorld)

Permanent Citation

Enrique Zeleny
​
​"Canonical Integrals for Diffraction Catastrophes"​
​http://demonstrations.wolfram.com/CanonicalIntegralsForDiffractionCatastrophes/​
​Wolfram Demonstrations Project​
​Published: November 11, 2014