Geodesic
Parabolic equation method and Fresnel approximation in Weinstein's problems
Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 44, Tome 426 (2014), pp. 87-118 
A. I. Korol'kov; A. V. Shanin. Parabolic equation method and Fresnel approximation in Weinstein's problems. Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 44, Tome 426 (2014), pp. 87-118. http://geodesic.mathdoc.fr/item/ZNSL_2014_426_a7/
@article{ZNSL_2014_426_a7,
     author = {A. I. Korol'kov and A. V. Shanin},
     title = {Parabolic equation method and {Fresnel} approximation in {Weinstein's} problems},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {87--118},
     year = {2014},
     volume = {426},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2014_426_a7/}
}
TY  - JOUR
AU  - A. I. Korol'kov
AU  - A. V. Shanin
TI  - Parabolic equation method and Fresnel approximation in Weinstein's problems
JO  - Zapiski Nauchnykh Seminarov POMI
PY  - 2014
SP  - 87
EP  - 118
VL  - 426
UR  - http://geodesic.mathdoc.fr/item/ZNSL_2014_426_a7/
LA  - ru
ID  - ZNSL_2014_426_a7
ER  - 
%0 Journal Article
%A A. I. Korol'kov
%A A. V. Shanin
%T Parabolic equation method and Fresnel approximation in Weinstein's problems
%J Zapiski Nauchnykh Seminarov POMI
%D 2014
%P 87-118
%V 426
%U http://geodesic.mathdoc.fr/item/ZNSL_2014_426_a7/
%G ru
%F ZNSL_2014_426_a7

Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

A problem of diffraction of a high frequency plane wave by a grating, consisting of absorbing screens is studied. Difficulties of a correct mathematical formulation of the problem are addressed. It is shown how this problem is connected with the classical Weinstein's problem of scattering by an open end of a planar waveguide. All results are derived by two different approaches: by the parabolic equation approach and by the method of Fresnel integrals. The equivalence of these approaches allows one to use Fresnel integrals for rigorous reasoning keeping the parabolic equation method for clear physical understanding of the results obtained.

[1] A. V. Shanin, “Weinstein's diffraction problem: embedding formula and spectral equation in parabolic approximation”, SIAM J. Appl. Math., 70 (2009), 1201–1218 | DOI | MR | Zbl

[2] A. V. Shanin, “Difraktsiya vysokochastotnoi volny na reshetke so slozhnym periodom pri skolzyaschem padenii”, Zap. nauchn. semin. POMI, 409, 2012, 212–239 | MR

[3] A. I. Korolkov, A. V. Shanin, “Difraktsiya na reshetke iz pogloschayuschikh ekranov raznoi vysoty. Novye uravneniya”, Zap. nauchn. semin. POMI, 422, 2014, 62–89

[4] L. A. Vainshtein, Otkrytye rezonatory i otkrytye volnovody, Sov. radio, M., 1966

[5] L. A. Vainshtein, Teoriya difraktsii i metod faktorizatsii, Sov. radio, M., 1966

[6] A. Zommerfeld, Optika, In. lit., M., 1953

[7] M. B. Vinogradova, O. V. Rudenko, A. P. Sukhorukov, Teoriya voln, Nauka, M., 1979 | MR

[8] A. G. Fox, T. Li, “Resonant modes in a Maser interferometer”, Bell Syst. Tech. J., 40 (1961), 453–458 | DOI