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1: 36.2 Catastrophes and Canonical Integrals
Diffraction Catastrophes
36.2.10 Ψ K ( x ; k ) = k - exp ( i k Φ K ( t ; x ) ) d t , k > 0 .
36.2.11 Ψ ( U ) ( x ; k ) = k - - exp ( i k Φ ( U ) ( s , t ; x ) ) d s d t , U = E , H ; k > 0 .
2: 36.1 Special Notation
§36.1 Special Notation
The main functions covered in this chapter are cuspoid catastrophes Φ K ( t ; x ) ; umbilic catastrophes with codimension three Φ ( E ) ( s , t ; x ) , Φ ( H ) ( s , t ; x ) ; canonical integrals Ψ K ( x ) , Ψ ( E ) ( x ) , Ψ ( H ) ( x ) ; diffraction catastrophes Ψ K ( x ; k ) , Ψ ( E ) ( x ; k ) , Ψ ( H ) ( x ; k ) generated by the catastrophes. …
3: 36.6 Scaling Relations
§36.6 Scaling Relations
Diffraction Catastrophe Scaling
Ψ K ( x ; k ) = k β K Ψ K ( y ( k ) ) ,
Ψ ( U ) ( x ; k ) = k β ( U ) Ψ ( U ) ( y ( U ) ( k ) ) ,
4: 36.14 Other Physical Applications
Diffraction catastrophes describe the (linear) wave amplitudes that smooth the geometrical caustic singularities and decorate them with interference patterns. … Diffraction catastrophes describe the connection between ray optics and wave optics. … Diffraction catastrophes describe the “semiclassical” connections between classical orbits and quantum wavefunctions, for integrable (non-chaotic) systems. …
5: 36.5 Stokes Sets
Stokes sets are surfaces (codimension one) in x space, across which Ψ K ( x ; k ) or Ψ ( U ) ( x ; k ) acquires an exponentially-small asymptotic contribution (in k ), associated with a complex critical point of Φ K or Φ ( U ) . …
6: 36.12 Uniform Approximation of Integrals
36.12.1 I ( y , k ) = - exp ( i k f ( u ; y ) ) g ( u , y ) d u ,
This technique can be applied to generate a hierarchy of approximations for the diffraction catastrophes Ψ K ( x ; k ) in (36.2.10) away from x = 0 , in terms of canonical integrals Ψ J ( ξ ( x ; k ) ) for J < K . For example, the diffraction catastrophe Ψ 2 ( x , y ; k ) defined by (36.2.10), and corresponding to the Pearcey integral (36.2.14), can be approximated by the Airy function Ψ 1 ( ξ ( x , y ; k ) ) when k is large, provided that x and y are not small. …
7: Bibliography N
  • J. F. Nye (2006) Dislocation lines in the hyperbolic umbilic diffraction catastrophe. Proc. Roy. Soc. Lond. Ser. A 462, pp. 2299–2313.
  • J. F. Nye (2007) Dislocation lines in the swallowtail diffraction catastrophe. Proc. Roy. Soc. Lond. Ser. A 463, pp. 343–355.
  • 8: Bibliography B
  • M. V. Berry and C. J. Howls (1990) Stokes surfaces of diffraction catastrophes with codimension three. Nonlinearity 3 (2), pp. 281–291.
  • M. V. Berry and C. J. Howls (2010) Axial and focal-plane diffraction catastrophe integrals. J. Phys. A 43 (37), pp. 375206, 13.
  • M. V. Berry, J. F. Nye, and F. J. Wright (1979) The elliptic umbilic diffraction catastrophe. Phil. Trans. Roy. Soc. Ser. A 291 (1382), pp. 453–484.
  • M. V. Berry and F. J. Wright (1980) Phase-space projection identities for diffraction catastrophes. J. Phys. A 13 (1), pp. 149–160.
  • M. V. Berry (1980) Some Geometric Aspects of Wave Motion: Wavefront Dislocations, Diffraction Catastrophes, Diffractals. In Geometry of the Laplace Operator (Proc. Sympos. Pure Math., Univ. Hawaii, Honolulu, Hawaii, 1979), Vol. 36, pp. 13–28.
  • 9: 36.7 Zeros
    Table 36.7.1: Zeros of cusp diffraction catastrophe to 5D. …
    Zeros { x y } inside, and zeros [ x y ] outside, the cusp x 2 = 8 27 | y | 3 .
    10: Bibliography T
  • H. Trinkaus and F. Drepper (1977) On the analysis of diffraction catastrophes. J. Phys. A 10, pp. L11–L16.