Whenever a problem is difficult from the topological, algebraic or functional analytical points of view, a “qualitative theory” arises. The example of choice is the qualitative theory of dynamical systems, founded by Henri Poincar ́e in 1881–1882. In the context of electromagnetics and optics there exist comprehensive accounts on group theoretical methods [1] and on symmetries [2, 3]. This work aims at contributing results in two directions, limited to fixed wavenumber k. 1 - Symmetries and ε-false symmetries in scalar potential scattering at fixed wavenumber. 2 - Approximate propagators in scalar diffraction at fixed wavenumber and related symmetries. The second part of the presentation will focus on group properties and related symmetries of the Helmholtz equation and of its approximations. Aperture diffraction in the halfspace of a scalar wave at fixed k is described by a propagator. A suitable asymptotic approximation yields the Fresnel (≡ paraxial) propagator. Further approximation leads to the Fourier propagator. The related group properties and information contents of both approximation stages will be compared by taking some available results [4–8] into account. REFERENCES 1. Sanchez-Mondragon, J. and K. B. Wolf, Eds., Lie Methods in Optics, Springer, Berlin, 1985. 2. Ramm, A. G., “Symmetry properties of scattering amplitudes and applications to inverse problems,” J. Math. Anal. Appl., Vol. 156, No. 2, 333–340, 1991. 3. Baum, C. E. and H. N. Kritikos, Eds., Electromagnetic Symmetry, Taylor & Francis, Washington, DC, 1995. 4. Grella, R., “Fresnel propagation, diffraction and paraxial wave equation,” J. Optics (Paris), Vol. 13, No. 6, 367–374, 1982. 5. Sudarshan, E. C. G., R. Simon, and N. Mukunda, “Paraxial-wave optics and relativistic front description. I. The scalar theory,” Phys. Rev. A, Vol. 28, 2921–2932, 1983. 6. Crosta, G. F., “On approximations of Helmholtz equation in the halfspace: Their relevance to inverse diffraction,” Wave Motion, Vol. 6, No. 3, 237–246, 1984. 7. Bandres, M. A. and M. Guizar-Sicairos, “Paraxial group,” Opt. Lett., Vol. 34, No. 1, 13–15, 2009. 8. Rodr ́ıguez-Lara, B. M., R. El-Ganainy, and J. Guerrero, “Symmetry in optics and photonics: A group theory approach,” Science Bulletin, Vol. 63, 244–251, 2018, arXiv:1803.00121.

Crosta, G. (2019). Contributions to the Qualitative Theory of Scattering. In 2019 PhotonIcs & Electromagnetics Research Symposium also known as 2019 Progress In Electromagnetics Research Symposium (PIERS — Rome). Abstracts (pp.326-327). Cambridge, MA 02138 : The Electromagnetics Academy.

Contributions to the Qualitative Theory of Scattering

Crosta, GF
2019

Abstract

Whenever a problem is difficult from the topological, algebraic or functional analytical points of view, a “qualitative theory” arises. The example of choice is the qualitative theory of dynamical systems, founded by Henri Poincar ́e in 1881–1882. In the context of electromagnetics and optics there exist comprehensive accounts on group theoretical methods [1] and on symmetries [2, 3]. This work aims at contributing results in two directions, limited to fixed wavenumber k. 1 - Symmetries and ε-false symmetries in scalar potential scattering at fixed wavenumber. 2 - Approximate propagators in scalar diffraction at fixed wavenumber and related symmetries. The second part of the presentation will focus on group properties and related symmetries of the Helmholtz equation and of its approximations. Aperture diffraction in the halfspace of a scalar wave at fixed k is described by a propagator. A suitable asymptotic approximation yields the Fresnel (≡ paraxial) propagator. Further approximation leads to the Fourier propagator. The related group properties and information contents of both approximation stages will be compared by taking some available results [4–8] into account. REFERENCES 1. Sanchez-Mondragon, J. and K. B. Wolf, Eds., Lie Methods in Optics, Springer, Berlin, 1985. 2. Ramm, A. G., “Symmetry properties of scattering amplitudes and applications to inverse problems,” J. Math. Anal. Appl., Vol. 156, No. 2, 333–340, 1991. 3. Baum, C. E. and H. N. Kritikos, Eds., Electromagnetic Symmetry, Taylor & Francis, Washington, DC, 1995. 4. Grella, R., “Fresnel propagation, diffraction and paraxial wave equation,” J. Optics (Paris), Vol. 13, No. 6, 367–374, 1982. 5. Sudarshan, E. C. G., R. Simon, and N. Mukunda, “Paraxial-wave optics and relativistic front description. I. The scalar theory,” Phys. Rev. A, Vol. 28, 2921–2932, 1983. 6. Crosta, G. F., “On approximations of Helmholtz equation in the halfspace: Their relevance to inverse diffraction,” Wave Motion, Vol. 6, No. 3, 237–246, 1984. 7. Bandres, M. A. and M. Guizar-Sicairos, “Paraxial group,” Opt. Lett., Vol. 34, No. 1, 13–15, 2009. 8. Rodr ́ıguez-Lara, B. M., R. El-Ganainy, and J. Guerrero, “Symmetry in optics and photonics: A group theory approach,” Science Bulletin, Vol. 63, 244–251, 2018, arXiv:1803.00121.
abstract + slide
Helmholtz equation; asymptotic approximations; paraxial propagator; Fresnel propagator
English
PhotonIcs & Electromagnetics Research Symposium - PIERS 2019 Rome
2019
Abad, M.; Abaie, Behnam; Abajo, F. Javier Garcia De; ... ; Zwiller Val; Zyatkov Denis Olegovich; Zyss Joseph
Asquini, R; Chew, WC; He, S; Khoo, IC; Simoni, F
2019 PhotonIcs & Electromagnetics Research Symposium also known as 2019 Progress In Electromagnetics Research Symposium (PIERS — Rome). Abstracts
17-giu-2019
2019
41
326
327
1P4-05
http://piers.org/piers2019Rome/
partially_open
Crosta, G. (2019). Contributions to the Qualitative Theory of Scattering. In 2019 PhotonIcs & Electromagnetics Research Symposium also known as 2019 Progress In Electromagnetics Research Symposium (PIERS — Rome). Abstracts (pp.326-327). Cambridge, MA 02138 : The Electromagnetics Academy.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10281/240132
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