Ray and caustic structure of Ince-Gauss beams

R. Gutiérrez-Cuevas; Mark Dennis; Miguel A. Alonso

doi:10.1088/1367-2630/ad17dc

Ray and caustic structure of Ince-Gauss beams

R. Gutiérrez-Cuevas^*, Mark Dennis, Miguel A. Alonso

^*Corresponding author for this work

Physics and Astronomy

Research output: Contribution to journal › Article › peer-review

Abstract

The Ince-Gauss beams, separable in elliptic coordinates, are studied through a ray-optical approach. Their ray structure can be represented over a Poincaré sphere by generalized Viviani curves (intersections of a cylinder and a sphere). This representation shows two topologically different regimes, in which the curve is composed of one or two loops. The overall beam shape is described by the ray caustics that delimit the beams’ bright regions. These caustics are inferred from the generalized Viviani curve through a geometric procedure that reveals connections with other physical systems and geometrical constructions. Depending on the regime, the caustics are composed either of two confocal ellipses or of segments of an ellipse and a hyperbola that are confocal. The weighting of the rays is shown to follow the two-mode meanfield Gross–Pitaevskii equations, which can be mapped to the equation of a simple pendulum. Finally, it is shown that the wave field can be accurately estimated from the ray description.

Original language	English
Article number	013011
Number of pages	19
Journal	New Journal of Physics
Volume	26
Issue number	013011
DOIs	https://doi.org/10.1088/1367-2630/ad17dc
Publication status	Published - 8 Jan 2024

Bibliographical note

Acknowledgment
The authors dedicate this work to the memory of Kurt Bernardo Wolf and Willard Miller, Jr., both of whom passed recently. MRD and MAA first presented aspects of this work almost exactly a decade ago at a workshop to celebrate Wolf's 70th birthday and where Miller was also present. Discussions with both of them led us to a deeper appreciation of these results. The authors also acknowledge J M Fellows and D H J O'Dell for useful discussions. R G-C and M A A acknowledge the Excellence Initiative of Aix-Marseille University—A*MIDEX, a French 'Investissements d'Avenir' programme for funding this research. R G C also acknowledges funding from the Labex WIFI (ANR-10-LABX-24, ANR-10-IDEX-0001-02 PSL*). MRD acknowledges support from the EPSRC Centre for Doctoral Training in Topological Design (EP/S02297X/1).

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title = "Ray and caustic structure of Ince-Gauss beams",

abstract = "The Ince-Gauss beams, separable in elliptic coordinates, are studied through a ray-optical approach. Their ray structure can be represented over a Poincar{\'e} sphere by generalized Viviani curves (intersections of a cylinder and a sphere). This representation shows two topologically different regimes, in which the curve is composed of one or two loops. The overall beam shape is described by the ray caustics that delimit the beams{\textquoteright} bright regions. These caustics are inferred from the generalized Viviani curve through a geometric procedure that reveals connections with other physical systems and geometrical constructions. Depending on the regime, the caustics are composed either of two confocal ellipses or of segments of an ellipse and a hyperbola that are confocal. The weighting of the rays is shown to follow the two-mode meanfield Gross–Pitaevskii equations, which can be mapped to the equation of a simple pendulum. Finally, it is shown that the wave field can be accurately estimated from the ray description.",

author = "R. Guti{\'e}rrez-Cuevas and Mark Dennis and Alonso, {Miguel A.}",

note = "Acknowledgment The authors dedicate this work to the memory of Kurt Bernardo Wolf and Willard Miller, Jr., both of whom passed recently. MRD and MAA first presented aspects of this work almost exactly a decade ago at a workshop to celebrate Wolf's 70th birthday and where Miller was also present. Discussions with both of them led us to a deeper appreciation of these results. The authors also acknowledge J M Fellows and D H J O'Dell for useful discussions. R G-C and M A A acknowledge the Excellence Initiative of Aix-Marseille University—A*MIDEX, a French 'Investissements d'Avenir' programme for funding this research. R G C also acknowledges funding from the Labex WIFI (ANR-10-LABX-24, ANR-10-IDEX-0001-02 PSL*). MRD acknowledges support from the EPSRC Centre for Doctoral Training in Topological Design (EP/S02297X/1).",

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N2 - The Ince-Gauss beams, separable in elliptic coordinates, are studied through a ray-optical approach. Their ray structure can be represented over a Poincaré sphere by generalized Viviani curves (intersections of a cylinder and a sphere). This representation shows two topologically different regimes, in which the curve is composed of one or two loops. The overall beam shape is described by the ray caustics that delimit the beams’ bright regions. These caustics are inferred from the generalized Viviani curve through a geometric procedure that reveals connections with other physical systems and geometrical constructions. Depending on the regime, the caustics are composed either of two confocal ellipses or of segments of an ellipse and a hyperbola that are confocal. The weighting of the rays is shown to follow the two-mode meanfield Gross–Pitaevskii equations, which can be mapped to the equation of a simple pendulum. Finally, it is shown that the wave field can be accurately estimated from the ray description.

AB - The Ince-Gauss beams, separable in elliptic coordinates, are studied through a ray-optical approach. Their ray structure can be represented over a Poincaré sphere by generalized Viviani curves (intersections of a cylinder and a sphere). This representation shows two topologically different regimes, in which the curve is composed of one or two loops. The overall beam shape is described by the ray caustics that delimit the beams’ bright regions. These caustics are inferred from the generalized Viviani curve through a geometric procedure that reveals connections with other physical systems and geometrical constructions. Depending on the regime, the caustics are composed either of two confocal ellipses or of segments of an ellipse and a hyperbola that are confocal. The weighting of the rays is shown to follow the two-mode meanfield Gross–Pitaevskii equations, which can be mapped to the equation of a simple pendulum. Finally, it is shown that the wave field can be accurately estimated from the ray description.

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ER -

Ray and caustic structure of Ince-Gauss beams

Abstract

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