# diffraction catastrophes

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##### 1: 36.2 Catastrophes and Canonical Integrals
###### DiffractionCatastrophes
36.2.11 $\Psi^{(\mathrm{U})}(\mathbf{x};k)=k\int_{-\infty}^{\infty}\int_{-\infty}^{% \infty}\exp\left(ik\Phi^{(\mathrm{U})}\left(s,t;\mathbf{x}\right)\right)% \mathrm{d}s\mathrm{d}t,$ $\mathrm{U=E,H}$; $k>0$.
##### 2: 36.1 Special Notation
###### §36.1 Special Notation
The main functions covered in this chapter are cuspoid catastrophes $\Phi_{K}\left(t;\mathbf{x}\right)$; umbilic catastrophes with codimension three $\Phi^{(\mathrm{E})}\left(s,t;\mathbf{x}\right)$, $\Phi^{(\mathrm{H})}\left(s,t;\mathbf{x}\right)$; canonical integrals $\Psi_{K}\left(\mathbf{x}\right)$, $\Psi^{(\mathrm{E})}\left(\mathbf{x}\right)$, $\Psi^{(\mathrm{H})}\left(\mathbf{x}\right)$; diffraction catastrophes $\Psi_{K}(\mathbf{x};k)$, $\Psi^{(\mathrm{E})}(\mathbf{x};k)$, $\Psi^{(\mathrm{H})}(\mathbf{x};k)$ generated by the catastrophes. …
##### 3: 36.6 Scaling Relations
###### DiffractionCatastrophe Scaling
$\Psi_{K}(\mathbf{x};k)=k^{\beta_{K}}\Psi_{K}\left(\mathbf{y}(k)\right),$
$\Psi^{(\mathrm{U})}(\mathbf{x};k)=k^{\beta^{(\mathrm{U})}}\Psi^{(\mathrm{U})}% \left(\mathbf{y}^{(\mathrm{U})}(k)\right),$
##### 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 $\mathbf{x}$ space, across which $\Psi_{K}(\mathbf{x};k)$ or $\Psi^{(\mathrm{U})}(\mathbf{x};k)$ acquires an exponentially-small asymptotic contribution (in $k$), associated with a complex critical point of $\Phi_{K}$ or $\Phi^{(\mathrm{U})}$. …
##### 6: 36.12 Uniform Approximation of Integrals
This technique can be applied to generate a hierarchy of approximations for the diffraction catastrophes $\Psi_{K}(\mathbf{x};k)$ in (36.2.10) away from $\mathbf{x}=\boldsymbol{{0}}$, in terms of canonical integrals $\Psi_{J}\left(\xi(\mathbf{x};k)\right)$ for $J. For example, the diffraction catastrophe $\Psi_{2}(x,y;k)$ defined by (36.2.10), and corresponding to the Pearcey integral (36.2.14), can be approximated by the Airy function $\Psi_{1}\left(\xi(x,y;k)\right)$ 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.
##### 10: Bibliography T
• H. Trinkaus and F. Drepper (1977) On the analysis of diffraction catastrophes. J. Phys. A 10, pp. L11–L16.