dorsal/arxiv
View SchemaDynamical diffraction in sinusoidal potentials: uniform approximations for Mathieu functions
| Authors | Duncan H. J. O'Dell |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0011105 |
| URL | https://arxiv.org/abs/quant-ph/0011105 |
| DOI | 10.1088/0305-4470/34/18/316 |
| Journal | J. Phys. A: Math. Gen. 34, 3897 (2001) |
Abstract
Eigenvalues and eigenfunctions of Mathieu's equation are found in the short wavelength limit using a uniform approximation (method of comparison with a `known' equation having the same classical turning point structure) applied in Fourier space. The uniform approximation used here relies upon the fact that by passing into Fourier space the Mathieu equation can be mapped onto the simpler problem of a double well potential. The resulting eigenfunctions (Bloch waves), which are uniformly valid for all angles, are then used to describe the semiclassical scattering of waves by potentials varying sinusoidally in one direction. In such situations, for instance in the diffraction of atoms by gratings made of light, it is common to make the Raman-Nath approximation which ignores the motion of the atoms inside the grating. When using the eigenfunctions no such approximation is made so that the dynamical diffraction regime (long interaction time) can be explored.
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"abstract": "Eigenvalues and eigenfunctions of Mathieu\u0027s equation are found in the short\nwavelength limit using a uniform approximation (method of comparison with a\n`known\u0027 equation having the same classical turning point structure) applied in\nFourier space. The uniform approximation used here relies upon the fact that by\npassing into Fourier space the Mathieu equation can be mapped onto the simpler\nproblem of a double well potential. The resulting eigenfunctions (Bloch waves),\nwhich are uniformly valid for all angles, are then used to describe the\nsemiclassical scattering of waves by potentials varying sinusoidally in one\ndirection. In such situations, for instance in the diffraction of atoms by\ngratings made of light, it is common to make the Raman-Nath approximation which\nignores the motion of the atoms inside the grating. When using the\neigenfunctions no such approximation is made so that the dynamical diffraction\nregime (long interaction time) can be explored.",
"arxiv_id": "quant-ph/0011105",
"authors": [
"Duncan H. J. O\u0027Dell"
],
"categories": [
"quant-ph"
],
"doi": "10.1088/0305-4470/34/18/316",
"journal_ref": "J. Phys. A: Math. Gen. 34, 3897 (2001)",
"title": "Dynamical diffraction in sinusoidal potentials: uniform approximations for Mathieu functions",
"url": "https://arxiv.org/abs/quant-ph/0011105"
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