On the analysis of diffraction catastrophes

Trinkaus, H.; Drepper, Friedhelm R.

doi:10.1088/0305-4470/10/1/003

Cited by 65 publications

(18 citation statements)

References 7 publications

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“…The topology near the beam axis has many intricate features, and it is considerably more complicated than the approximate ray topology of the 3D PE solution in [7] or the 2D case in [11]. Specifically, we show that the main lobe of the AiB is described by a cusped double-layered caustic, which is of the hyperbolic umbilic type [22][23][24][25][26] (see . We also identify the range where the isolated hyperbolic umbilic caustic that describes the AiB field near its propagation axis evolves into the more complicated parabolic umbilic caustic considered in Section 6, so that the AiB loses its form.…”

Section: Aibmentioning

confidence: 92%

Nonparaxial wave analysis of three-dimensional Airy beams

Kaganovsky

Heyman

2012

J. Opt. Soc. Am. A

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The three-dimensional Airy beam (AiB) is thoroughly explored from a wave-theory point of view. We utilize the exact spectral integral for the AiB to derive local ray-based solutions that do not suffer from the limitations of the conventional parabolic equation (PE) solution and are valid far beyond the paraxial zone and for longer ranges. The ray topology near the main lobe of the AiB delineates a hyperbolic umbilic catastrophe, consisting of a cusped double-layered caustic. In the far zone this caustic is deformed and the field loses its beam shape. The field in the vicinity of this caustic is described uniformly by a hyperbolic umbilic canonical integral, which is structured explicitly on the local geometry of the caustic. In order to accommodate the finite-energy AiB, we also modify the conventional canonical integral by adding a complex loss parameter. The canonical integral is calculated using a series expansion, and the results are used to identify the validity zone of the conventional PE solution. The analysis is performed within the framework of the nondispersive AiB where the aperture field is scaled with frequency such that the ray skeleton is frequency independent. This scaling enables an extension of the theory to the ultrawideband regime and ensures that the pulsed field propagates along the curved beam trajectory without dispersion, as will be demonstrated in a subsequent publication.

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Section: Aibmentioning

confidence: 92%

Nonparaxial wave analysis of three-dimensional Airy beams

Kaganovsky

Heyman

2012

J. Opt. Soc. Am. A

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“…The sequence in Fig. 1a-d shows, for the first time in a matter wave system, a progression from diffractive caus- tics [30,31,32,33] toward classical caustics: the diffractive structure is manifest for the shortest period lattice and nearly invisible in the longest.…”

Section: Classical Analysismentioning

confidence: 96%

Quantum and classical dynamics of a Bose-Einstein condensate in a large-period optical lattice

et al. 2009

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We experimentally investigate diffraction of a 87 Rb Bose-Einstein condensate from a 1D optical lattice. We use a range of lattice periods and timescales, including those beyond the Raman-Nath limit. We compare the results to quantum mechanical and classical simulations, with quantitative and qualitative agreement, respectively. The classical simulation predicts that the envelope of the time-evolving diffraction pattern is shaped by caustics: singularities in the phase space density of classical trajectories. This behavior becomes increasingly clear as the lattice period grows.

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“…Literature provides additional discussion on this type of catastrophe function as well as plots in different spaces. [21][22][23] To ease the upcoming transformations a term having the form of Eq. ͑6͒ is inserted for k in Eq.…”

Section: Fully Lit Regionmentioning

confidence: 99%

Doubly focused backscattering from finite targets in an Airy caustic formed by a curved reflecting surface

Dzikowicz

Marston

2005

The Journal of the Acoustical Society of America

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Caustics can be formed in the water column when sound scatters off a curved-reflecting surface such as the ocean floor or surface. The simplest caustic is an Airy caustic formed by the merging of two rays. Small targets lying in or near Airy caustics have backscattered echoes that can be focused both to the target and upon return. For a point target, the doubly focused backscattering amplitude is proportional to the square of an Airy function whose argument depends on the target location through the changes in relative return times of contributing rays. For a finite sized target, the symmetry is broken and the amplitude unfolds into a hyperbolic umbilic catastrophe. The arguments for the hyperbolic umbilic function are calculated using the relative return times of transient echoes. These doubly focused echoes can lead to amplitudes larger than that of direct or singly focused echoes ͑echoes which focus once, either to the target or upon return͒. Experiments using a cylindrical half-pipe as a reflecting surface confirm these predictions.

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On the analysis of diffraction catastrophes

Cited by 65 publications

References 7 publications

Nonparaxial wave analysis of three-dimensional Airy beams

Nonparaxial wave analysis of three-dimensional Airy beams

Quantum and classical dynamics of a Bose-Einstein condensate in a large-period optical lattice

Doubly focused backscattering from finite targets in an Airy caustic formed by a curved reflecting surface

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