Geodesic
Exact solutions of the $m$-dimensional wave equation from paraxial ones. Further generalization of the Bateman solution
Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 41, Tome 393 (2011), pp. 167-177 
A. P. Kiselev; A. B. Plachenov. Exact solutions of the $m$-dimensional wave equation from paraxial ones. Further generalization of the Bateman solution. Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 41, Tome 393 (2011), pp. 167-177. http://geodesic.mathdoc.fr/item/ZNSL_2011_393_a11/
@article{ZNSL_2011_393_a11,
     author = {A. P. Kiselev and A. B. Plachenov},
     title = {Exact solutions of the $m$-dimensional wave equation from paraxial ones. {Further} generalization of the {Bateman} solution},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {167--177},
     year = {2011},
     volume = {393},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2011_393_a11/}
}
TY  - JOUR
AU  - A. P. Kiselev
AU  - A. B. Plachenov
TI  - Exact solutions of the $m$-dimensional wave equation from paraxial ones. Further generalization of the Bateman solution
JO  - Zapiski Nauchnykh Seminarov POMI
PY  - 2011
SP  - 167
EP  - 177
VL  - 393
UR  - http://geodesic.mathdoc.fr/item/ZNSL_2011_393_a11/
LA  - ru
ID  - ZNSL_2011_393_a11
ER  - 
%0 Journal Article
%A A. P. Kiselev
%A A. B. Plachenov
%T Exact solutions of the $m$-dimensional wave equation from paraxial ones. Further generalization of the Bateman solution
%J Zapiski Nauchnykh Seminarov POMI
%D 2011
%P 167-177
%V 393
%U http://geodesic.mathdoc.fr/item/ZNSL_2011_393_a11/
%G ru
%F ZNSL_2011_393_a11

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

A review of earlier generalizations of the classical Bateman solution, involving an arbitrary function, is presented. Its further generalization, described by $m(m-1)$ real parameters characterizing the phase, is given. Under a proper choice of the arbitrary function, it may describe Gaussian beam or Gaussian packet.

[1] H. Bateman, The mathematical analysis of electrical and optical wave-motion on the basis of Maxwell's equations, Cambridge University Press, Cambridge, 1915 | Zbl

[2] V. A. Fock, Electromagnetic diffraction and propagation problems, Pergamon Press, Oxford, 1965 | MR

[3] L. A. Vainshtein, Otkrytye rezonatory i otkrytye volnovody, Nauka, M., 1966

[4] J. A. Arnaud, H. Kogelnik, “Gaussian light beams with general astigmatism”, Appl. Optics, 8 (1969), 1687–1693 | DOI

[5] V. M. Babich, V. S. Buldyrev, Asimptoticheskie metody v zadachakh difraktsii korotkikh voln, Nauka, M., 1972 | MR

[6] J. N. Brittingham, “Focus waves modes in homogeneous Maxwell's equations: Transverse electric mode”, J. Appl. Phys., 54 (1983), 1179–1189 | DOI

[7] A. P. Kiselev, “Modulirovannye gaussovy puchki”, Izv. vyssh. uch. zaved. Radiofizika, 26:5 (1983), 1014–1020 | MR

[8] P. A. Bélanger, “Packetlike solutions of the homogeneous-wave equation”, J. Opt. Soc. Am. A, 1:7 (1984), 723–724 | DOI

[9] R. W. Ziolkowski, “Exact solutions of the wave equation with complex source locations”, J. Math. Phys., 26 (1985), 861–863 | DOI | MR

[10] I. M. Besieris, A. M. Shaarawi, R. W. Ziolkowski, “A bidirectional traveling plane wave representation of exact solutions of the wave equation”, J. Math. Phys., 30 (1989), 1254–1269 | DOI | MR | Zbl

[11] R. W. Ziolkowski, “Localized transmission of electromagnetic energy”, Phys. Rev. A, 39 (1989), 2005–2033 | DOI | MR

[12] A. P. Kiselev, M. V. Perel, “Highly localized solutions of the wave equation”, J. Math. Phys., 41:4 (2000), 1934–1955 | DOI | MR | Zbl

[13] A. P. Kiselev, M. V. Perel, “Otnositelno neiskazhayuschiesya resheniya $m$-mernogo volnovogo uravneniya”, Diff. uravneniya, 38:8 (2002), 1128–1129 | MR | Zbl

[14] I. M. Besieris, A. M. Shaarawi, A. M. Attiya, “Bateman conformal transformations within the framework of the bidirectional spectral representation”, Progress in Electromagnetics Research, 48 (2004), 201–231 | DOI

[15] A. P. Kiselev, “Lokalizovannye svetovye volny: paraksialnye i tochnye resheniya volnovogo uravnenya (Obzor)”, Optika i Spektroskopiya, 102:4 (2007), 661–681 | MR

[16] V. M. Babich, “Sobstvennye funktsii, sosredotochennye v okrestnosti zamknutoi geodezicheskoi”, Zap. nauchn. semin. LOMI, 9, 1968, 15—63 | MR | Zbl

[17] M. M. Popov, “Rezonatory dlya lazerov s razvërnutymi napravleniyami glavnykh krivizn. Metod parabolicheskogo uravneniya”, Optika i Spektroskopiya, 25:2 (1968), 314–316

[18] M. M. Popov, “Sobstvennye kolebaniya mnogozerkalnykh rezonatorov”, Vestnik Leningradskogo universiteta, ser. fiz.-khim., 1969, no. 22(4), 42–54 | MR

[19] E. G. Abramochkin, V. G. Volostnikov, “Generalized Hermite–Laguerre–Gauss beams”, Physics of Wave Phenomena, 18:1 (2010), 14–22 | DOI

[20] C. F. R. Caron, R. M. Potvliege, “Bessel-modulated Gaussian beams with quadratic radial dependence”, Optics Communications, 164 (1999), 83–93 | DOI

[21] V. P. Bykov, O. O. Silichev, Lazernye rezonatory, FIZMATLIT, M., 2004

[22] A. P. Kiselev, A. B. Plachenov, P. Chamorro-Posada, “General astigmatic nonparaxial wave beams and packets”, Proc. 11th International Conference on Laser Fiber-Optical Networks Modeling, Kharkov, 2011

[23] R. Kurant, D. Gilbert, Metody matematicheskoi fiziki, Gostekhizdat, M., 1941

[24] V. Smirnoff, S. Soboleff, “Sur le problème plan de vibrations élastiques”, Comptes Rendus, 194 (1932), 1437–1439 | Zbl

[25] S. L. Sobolev, “Nekotorye voprosy teorii rasprostraneniya kolebanii”: F. Frank, R. Mizes, Differentsialnye i integralnye uravneniya matematicheskoi fiziki, ONTI, M., 1937, 468–617

[26] V. I. Smirnov, Kurs vysshei matematiki, v. 3, Ch. 2, Nauka, M., 1974 | MR

[27] A. P. Kiselev, “Otnositelno neiskazhayuschiesya volny, zavisyaschie ot trekh prostranstvennykh peremennykh”, Mat. Zametki, 79:4 (2006), 635–636 | DOI | MR | Zbl

[28] H. Bateman, “The conformal transformations of four dimensions and their applications to geometrical optics”, Proc. London Math. Soc., 7 (1909), 70–89 | DOI | MR | Zbl

[29] P. Hillion, “Generalized phases and nondispersive waves”, Acta Appl. Math., 30:1 (1993), 35–45 | DOI | MR | Zbl

[30] E. A. Polyanskii, “O svyazi mezhdu resheniyami uravnenii Gelmgoltsa i tipa Shrëdingera”, ZhVMMF, 12:1 (1972), 241–249 | Zbl

[31] A. Wünsche, “Transition from the paraxial approximation to exact solutions of the wave equation and application to Gaussian beams”, J. Opt. Soc. Amer. A, 9 (1992), 765–774 | DOI | MR

[32] A. Torre, “Separable-variable solutions of the wave equation from a general type of solutions of paraxial wave equation”, Proc. Int. Conf. “Days on Diffraction 2009”, St. Petersburg, 2009, 178–184

[33] F. Gori, C. Guattari, C. Padovani, “Bessel–Gauss beams”, Opt. Commun., 64:6 (1987), 491–495 | DOI

[34] P. L. Overfelt, “Bessel–Gauss pulses”, Phys. Rev. A, 44:6 (1991), 3941–3947 | DOI | MR

[35] M. V. Perel, I. V. Fialkovskii, “Eksponentsialno lokalizovannye resheniya uravneniya Kleina–Gordona”, Zap. nauchn. semin. POMI, 275, 2001, 187–198 | MR | Zbl

[36] F. G. Friedlander, “Simple progressive solutions of the wave equation”, Proc. Cambridge Philos. Soc., 43 (1947), 360–373 | DOI | MR | Zbl

[37] A. B. Plachenov, “Naklonnye neparaksialnye puchki i pakety dlya volnovogo uravneniya s dvumya prostranstvennymi peremennymi”, Zap. nauchn. semin. POMI, 393, 2011, 224–233 | MR