Geodesic
On the explicit parametric description of waves in periodic media
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 49 (2009) no. 6, pp. 1119-1130 
A. V. Vinogradov; A. V. Popov; D. V. Prokopovich. On the explicit parametric description of waves in periodic media. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 49 (2009) no. 6, pp. 1119-1130. http://geodesic.mathdoc.fr/item/ZVMMF_2009_49_6_a13/
@article{ZVMMF_2009_49_6_a13,
     author = {A. V. Vinogradov and A. V. Popov and D. V. Prokopovich},
     title = {On the explicit parametric description of waves in periodic media},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {1119--1130},
     year = {2009},
     volume = {49},
     number = {6},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2009_49_6_a13/}
}
TY  - JOUR
AU  - A. V. Vinogradov
AU  - A. V. Popov
AU  - D. V. Prokopovich
TI  - On the explicit parametric description of waves in periodic media
JO  - Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki
PY  - 2009
SP  - 1119
EP  - 1130
VL  - 49
IS  - 6
UR  - http://geodesic.mathdoc.fr/item/ZVMMF_2009_49_6_a13/
LA  - ru
ID  - ZVMMF_2009_49_6_a13
ER  - 
%0 Journal Article
%A A. V. Vinogradov
%A A. V. Popov
%A D. V. Prokopovich
%T On the explicit parametric description of waves in periodic media
%J Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki
%D 2009
%P 1119-1130
%V 49
%N 6
%U http://geodesic.mathdoc.fr/item/ZVMMF_2009_49_6_a13/
%G ru
%F ZVMMF_2009_49_6_a13

Voir la notice de l'article provenant de la source Math-Net.Ru

A method for the parameterization of the one-dimensional wave equation is proposed that makes it possible to find its solution by quadratures under an arbitrary dependence of the refraction index on the current wave phase. The form of the solution found is used to investigate the structure of the wave function for a periodic refraction index. Explicit expressions for the fundamental system of solutions and for the Floquet index are obtained. Examples of applying the proposed method to the optimal synthesis of multilayer interference mirrors and Bragg waveguides are discussed.

[1] Joannopulos J. D., Johnson S. G., Winn J. N., Meade R. D., Photonic crystals, Princeton Univ. Press, New York, 2008 | MR

[2] Yariv A., Yukh P., Opticheskie volny v kristallakh, Mir, M., 1987

[3] Sveshnikov A. G., Tikhonravov A. B., “Matematicheskie metody v teorii sinteza opticheskikh tonkosloinykh sistem”, Nekorrektnye zadachi estestvoznaniya, Izd-vo MGU, M., 1987, 254–274

[4] Tikhonravov A. B., Trubetskov M. K., “Novye zadachi mnogosloinoi optiki”, Radiotekhn. i elektronika, 50:26 (2005), 265–272

[5] Brechet F., Roy P., Marcou J., Pagnoux D., “Singlemode propagation into depressed-core-index photonic-bandgap fibre designed for zero-dispersion propagation at short wavelengths”, Electronics Letts., 36:6 (2000), 514–515 | DOI

[6] Uitteker E. T., Vatson Dzh. N., Kurs sovremennogo analiza, Ch. 2, URSS, M., 2002

[7] Beitmen G., Erdeii A., Vysshie transtsendentnye funktsii, Ch. 3, Nauka, M., 1967 | MR

[8] Brekhovskikh L. M., Volny v sloistykh sredakh, Izd-vo AN SSSR, M., 1957 | MR

[9] Babikov V. V., Metod fazovykh funktsii v kvantovoi mekhanike, Nauka, M., 1988 | MR

[10] Landau L. D.,Lifshits E. M., Teoreticheskaya fizika. T. 1. Mekhanika, Nauka, M., 1988 | MR

[11] Born M., Volf E., Osnovy optiki, Nauka, M., 1973

[12] Prokopovich D. V., Popov A. B., Vinogradov A. B., “Skalyarnaya teoriya slabokontrastnykh breggovskikh volnovodov”, Kvantovaya elektronika, 37:9 (2007), 873–880