Contributions to the Qualitative Theory of Scattering

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.