Research Article
Year 2024,
Volume: 53 Issue: 2, 405 - 416, 23.04.2024
Abstract
The Hermite-Gauss basis functions have been extensively employed in classical and quantum optics due to their convenient analytic properties. A class of multivariate Hermite-Gauss functions, the anisotropic Hermite-Gauss functions, arise by endowing the standard univariate Hermite-Gauss functions with a positive definite quadratic form. These multivariate functions admit useful applications in optics, signal analysis and probability theory, however they have received little attention in literature. In this paper, we examine the properties of these functions, with an emphasis on applications in computational optics.
Keywords
Hermite functions orthogonal basis computational optics linear canonical transform Fourier transform Wigner-Vile distribution eigenfunctions
References
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Year 2024,
Volume: 53 Issue: 2, 405 - 416, 23.04.2024
Abstract
References
- [1] D. Aguirre-Olivas, G. Mellado-Villaseñor, V. Arrizón, and S. Chávez-Cerda, Selfhealing of Hermite-Gauss and ince-Gauss beams, in A. Forbes and T. E. Lizotte editors, Laser Beam Shaping XVI, SPIE, 2015.
- [2] M. Allgaier, V. Ansari, J. M. Donohue, C. Eigner, V. Quiring, R. Ricken, B. Brecht and C. Silberhorn, Pulse shaping using dispersion-engineered difference frequency generation, Phys. Rev. A 101 (4), 2020.
- [3] S.-i. Amari and M. Kumon, Differential geometry of edgeworth expansions in curved exponential family, Ann. Instit. Stat. Math. 35 (1), 1–24, 1983.
- [4] S. Ast, S. Di Pace, J. Millo, M. Pichot, M. Turconi, N. Christensen and W. Chaibi, Higher-order Hermite-Gauss modes for gravitational waves detection, Phys. Rev. D, 103 (4), 2021.
- [5] S. Chabou and A. Bencheikh, Elegant Gaussian beams: nondiffracting nature and self-healing property, Appl. Opt. 59 (32), 2020.
- [6] M. A. Cox, L. Maqondo, R. Kara, G. Milione, L. Cheng and A. Forbes, The resilience of Hermite- and Laguerre-Gaussian modes in turbulence, J.Light. Technol. 37 (16), 3911–3917, 2019.
- [7] B. Holmquist, The d-variate vector Hermite polynomial of order k, Linear Algebra Appl. 237–238, 155–190, 1996.
- [8] M. E. H. Ismail and P. Simeonov, Multivariate holomorphic Hermite polynomials, Ramanujan J. 53 (2), 357–387, 2020.
- [9] J. C. T. Lee, S. J. Alexander, S. D. Kevan, S. Roy and B. J. McMorran, Laguerre- Gauss and Hermite-Gauss soft x-ray states generated using diffractive optics, Nat. Photonics 13 (3), 205–209, 2019.
- [10] J. J. Perkins, R. T. Newell, C. R. Schabacker and C. Richardson, Novel fiber-optic geometries for fast quantum communication, in M. Razeghi, E. Tournié and G. J. Brown editors, Quantum Sensing and Nanophotonic Devices XI, SPIE, 2013.
- [11] B. K. Singh, H. Nagar, Y. Roichman and A. Arie, Particle manipulation beyond the diffraction limit using structured super-oscillating light beams, Light: Sci. & Appl. 6 (9), e17050–e17050, 2017.
- [12] J. Stecha and V. Havlena, Unscented kalman filter revisited – Hermite-Gauss quadrature approach, 15th International Conference on Information Fusion in 2012, pages 495–502, 2012.
- [13] S. Steinberg and L.-Q. Yan, Physical light-matter interaction in hermite-gauss space, ACM Trans. Graph. 40 (6), 2021.
- [14] H. Stoof, K. Gubbels and D. Dickerscheid, Gaussian Integrals, pages 15–31, Springer Netherlands, Dordrecht, 2009.
- [15] A. Takemura and K. Takeuchi, Some results on univariate and multivariate cornishfisher expansion: Algebraic properties and validity, Sankhy: The Indian J. Stat., Series A (1961-2002) 50 (1), 111–136, 1988.
- [16] L. Tao, A. Green and P. Fulda, Higher-order Hermite-Gauss modes as a robust flat beam in interferometric gravitational wave detectors, Phys. Rev. D 102 (12), 2020.
- [17] S. P. Walborn and A. H. Pimentel, Generalized HermiteGauss decomposition of the two-photon state produced by spontaneous parametric down conversion, J. Phys. B: At., Mol. Opt. Phys, 45 (16), 165502, 2012.
- [18] W. Zhen and D. Deng, Gooshänchen shift for elegant HermiteGauss light beams impinging on dielectric surfaces coated with a monolayer of graphene, App. Phys. B 126 (3), 2020.
There are 18 citations in total.
Details
| Primary Language | English |
|---|---|
| Subjects | Mathematical Sciences |
| Journal Section | Mathematics |
| Authors | |
| Early Pub Date | August 15, 2023 |
| Publication Date | April 23, 2024 |
| Published in Issue | Year 2024 Volume: 53 Issue: 2 |