Abstract
We study, for times of order \({1/\varepsilon}\), solutions of wave equations which are \({\fancyscript{O}(\varepsilon^2)}\) modulations of an \({\varepsilon}\) periodic wave equation. The solutions are of slowly varying amplitude type built on Bloch plane waves with wavelength of order \({\varepsilon}\). We construct accurate approximate solutions of the three scale WKB type. The leading profile is both transported at the group velocity and dispersed by a Schrödinger equation given by the quadratic approximation of the Bloch dispersion relation at the plane wave. A ray average hypothesis of the small divisor type guarantees stability. We introduce techniques related to those developed in nonlinear geometric optics which lead to new results even on time scales \({t=\fancyscript{O}(1)}\). A pair of asymptotic solutions yield accurate approximate solutions of oscillatory initial value problems. The leading term yields H 1 asymptotics when the envelopes are only H 1.
Similar content being viewed by others
References
Albert J.H.: Genericity of simple eigenvalues for elliptic PDE’s. Proc. A.M.S. 48, 413–418 (1975)
Allaire G.: Dispersive limits in the homogenization of the wave equation. Annales de la Faculté des Sciences de Toulouse XII, 415–431 (2003)
Allaire G., Palombaro M., Rauch J.: Diffractive behavior of the wave equation in periodic media: weak convergence analysis. Annali di Matematica Pura e Applicata 188, 561–590 (2009)
Altug H., Vuckovic J.: Experimental demonstration of the slow group velocity of light in two-dimensional coupled photonic crystal microcavity arrays. Appl. Phys. Lett. 86, 111102–11111023 (2004)
Bajcsy M., Zibrov A., Lukin M.: Stationary pulses of light in an atomic medium. Nature 426, 638–641 (2003)
Bamberger A., Engquist B., Halpern L., Joly P.: Parabolic wave equation approximations in heterogenous media. SIAM J. Appl. Math. 48(1), 99–128 (1988)
Barrailh K., Lannes D.: A general framework for diffractive optics and its applications to lasers with large spectrum and short pulses. SIAM J. Math. Anal. 34(3), 636–674 (2003)
Bensoussan, A., Lions, J.-L., Papanicolaou, G.: Asymptotic Analysis for Periodic Structures. North-Holland, Amsterdam, 1978
Bloch F.: Uber die Quantenmechanik der Electronen in Kristallgittern. Z. Phys. 52, 555–600 (1928)
Brahim-Otsmane S., Francfort G., Murat F.: Correctors for the homogenization of the wave and heat equations. J. Math. Pures Appl. 71(9), 197–231 (1992)
Brillouin L.: Propagation of Waves in Periodic Structures. Dover, New York (1953)
Conca C., Planchard J., Vanninathan M.: Fluids and Periodic Structures. RMA 38, Wiley Paris (1995)
Conca C., Orive R., Vanninathan M.: On Burnett coefficients in strongly periodic media. J. Math. Phys. 47(3), 032902 (2006)
Donnat, P., Joly, J.-L., Métivier, G., Rauch, J.: Diffractive nonlinear geometric optics. Séminaire sur les Equations aux Dérivées Partielles 1995–1996, Exp. No. XVII, 25 pp. Ecole Polytechnique, Palaiseau, 1996
Donnat, P., Rauch, J.: Modeling the Dispersion of Light. Singularities and Oscillations (Minneapolis, MN, 1994/1995), 17–35. IMA Vol. Math. Appl., 91. Springer, New York, 1997
Donnat P., Rauch J.: Dispersive nonlinear geometric optics. J. Math. Phys. 38(3), 1484–1523 (1997)
Dumas E.: Diffractive optics with curved phases: beam dispersion and transitions between light and shadow. Asymptot. Anal. 38(1), 47–91 (2004)
Francfort G., Murat F.: Oscillations and energy densities in the wave equation. Commun. Partial Differ. Equ. 17, 1785–1865 (1992)
Gérard P., Markowich P., Mauser N., Poupaud F.: Homogenization limits and Wigner transforms. Commun. Pure Appl. Math. 50(4), 323–379 (1997)
Gersen H., Karle T., Engelen R., Bogaerts W., Korterik J., van Hulst N., Krauss T., Kuipers L.: Real-space observation of ultraslow light in photonic crystal waveguides. Phys. Rev. Lett. 94, 073903–10739034 (2005)
Hau L.V., Harris S.E., Dutton Z., Behroozi C.: Light speed reduction to 17 meters per second in an ultracold atomic gas. Nature 397, 594–598 (1999)
Joly J.-L., Métiver G., Rauch J.: Generic rigorous asymptotic expansions for weakly nonlinear multidimensional oscillatory waves. Duke Math. J. 70, 373–404 (1993)
Joly J.-L., Métiver G., Rauch J.: Diffractive nonlinear geometric optics with rectification. Indiana U. Math. J. 47, 1167–1242 (1998)
Kato T.: Perturbation Theory for Linear Operators. Springer, Berlin (1966)
Kuchment P.: Floquet theory for partial differential equations. Operator Theory: Advances and Applications, Vol. 60. Birkhäuser Verlag, Basel (1993)
Kuchment, P.: The mathematics of photonic crystals. Mathematical Modeling in Optical Science, 207–272. Frontiers Appl. Math., Vol. 22. SIAM, Philadelphia, 2001
Lannes D.: Dispersive effects for nonlinear geometrical optics with rectification. Asymptot. Anal. 18(1–2), 111–146 (1998)
Lax P.D.: Asymptotic solutions of oscillatory initial value problems. Duke Math. J. 24, 627–646 (1957)
Leontovich M., Fock V.: Solution of the problem of propagation of electromagnetic waves along the earth’s surface by the method of parabolic equation. Acad. Sci. USSR. J. Phys. 10, 13–24 (1946)
Reed M., Simon B.: Methods of Modern Mathematical Physics. Academic Press, New York (1978)
Russell P.St.J.: Photonic crystal fibers. J. Lightwave Technol. 24(12), 4729–4749 (2006)
Santosa F., Symes W.: A dispersive effective medium for wave propagation in periodic composites. SIAM J. Appl. Math. 51, 984–1005 (1991)
Sipe J., Winful H.: Nonlinear Schrödinger equations in periodic structure. Optics Lett. 13, 132–133 (1988)
de Sterke C., Sipe J.: Envelope-function approach for the electrodynamics of nonlinear periodic structures. Phys. Rev. A 38, 5149–5165 (1988)
Tappert, F.: The parabolic approximation method. Wave Propagation and Underwater Acoustics (Workshop, Mystic, Conn., 1974), 224–287. Lecture Notes in Phys., Vol. 70. Springer, Berlin, 1977
Tartar L.: H-measures, a new approach for studying homogenization, oscillations and concentration effects in partial differential equations. Proc. R. Soc. Edinburgh 115, 193–230 (1990)
Vlasov Y.A., Petit S., Klein G., Hönerlage B., Hirlmann C.: Femtosecond measurements of the time of flight of photons in a three-dimensional photonic crystal. Phys. Rev. E 60, 1030–1035 (1999)
Wilcox C.: Theory of Bloch waves. J. Anal. Math. 33, 146–167 (1978)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by F. Otto
Rights and permissions
About this article
Cite this article
Allaire, G., Palombaro, M. & Rauch, J. Diffractive Geometric Optics for Bloch Wave Packets. Arch Rational Mech Anal 202, 373–426 (2011). https://doi.org/10.1007/s00205-011-0452-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00205-011-0452-9