{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,20]],"date-time":"2026-08-20T12:28:48Z","timestamp":1787228928194,"version":"build-2736575974"},"reference-count":32,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Multiscale Model. Simul."],"published-print":{"date-parts":[[2013,1]]},"abstract":"<jats:p>Out-of-plane gap solitons in two-dimensional photonic crystals are optical beams localized in the plane of periodicity of the medium and delocalized in the orthogonal direction, in which they propagate with a nonzero velocity. We study such gap solitons as described by the Kerr nonlinear Maxwell system. Using a model of the nonlinear polarization, which does not generate higher harmonics, we obtain a closed curl-curl problem for the fundamental harmonic of the gap soliton. For gap solitons with frequencies inside spectral gaps and in an asymptotic vicinity of a gap edge we use a slowly varying envelope approximation based on the linear Bloch waves at the edge and slowly varying envelopes. We carry out a systematic derivation of the coupled mode equations (CMEs) which govern the envelopes. This derivation needs to be carried out in Bloch variables. The CMEs are a system of coupled nonlinear stationary Schr\u00f6dinger equations with an additional cross derivative term. Examples of gap soliton approximations are numerically computed for a photonic crystal with a hexagonal periodicity cell and an annulus material structure in the cell.<\/jats:p>","DOI":"10.1137\/120865914","type":"journal-article","created":{"date-parts":[[2013,1,15]],"date-time":"2013-01-15T12:50:13Z","timestamp":1358254213000},"page":"162-191","source":"Crossref","is-referenced-by-count":4,"title":["Coupled Mode Equation Modeling for Out-of-Plane Gap Solitons in 2D Photonic Crystals"],"prefix":"10.1137","volume":"11","author":[{"given":"Tom\u00e1\u0161","family":"Dohnal","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Willy","family":"D\u00f6rfler","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2013,1,15]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1137\/11082662X"},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1137\/040606053"},{"key":"atypb3","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.57.2287"},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1016\/0022-0396(92)90145-D"},{"key":"atypb5","first-page":"335","volume":"49","author":"Arriaga J.","year":"2003","journal-title":"Rev. Mexicana Fis."},{"key":"atypb6","doi-asserted-by":"crossref","unstructured":"N. W. Ashcroft and D. N. Mermin,\n                      Solid State Physics\n                      , 1st ed., Thomson Learning, Toronto, 1976.","DOI":"10.1063\/1.3037370"},{"key":"atypb7","doi-asserted-by":"publisher","DOI":"10.1080\/17455030500196929"},{"key":"atypb8","doi-asserted-by":"publisher","DOI":"10.1145\/1268776.1268779"},{"key":"atypb9","doi-asserted-by":"crossref","unstructured":"G. Bao, L. Cowsar, and W. Masters, eds.\n                      Mathematical Modeling in Optical Science\n                      , Frontiers Appl. 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Wieners,\n                      Photonic Crystals: Mathematical Analysis and Numerical Approximation\n                      , Oberwolfach Semin. 42, Birkha\u0308user Verlag, Basel, 2011.","DOI":"10.1007\/978-3-0348-0113-3"},{"key":"atypb17","doi-asserted-by":"publisher","DOI":"10.1090\/S0894-0347-2012-00745-0"},{"key":"atypb18","doi-asserted-by":"publisher","DOI":"10.1007\/s00332-001-0002-y"},{"key":"atypb19","doi-asserted-by":"publisher","DOI":"10.1016\/0022-0396(92)90118-7"},{"key":"atypb20","doi-asserted-by":"publisher","DOI":"10.1137\/090769417"},{"key":"atypb21","doi-asserted-by":"crossref","unstructured":"T. Kato\u0304,\n                      Perturbation Theory for Linear Operators\n                      , Classics Math., Springer, Berlin, 1995.","DOI":"10.1007\/978-3-642-66282-9"},{"key":"atypb22","doi-asserted-by":"publisher","DOI":"10.1364\/OPN.13.2.000059"},{"key":"atypb23","doi-asserted-by":"crossref","unstructured":"P. Kuchment,\n                      Floquet Theory for Partial Differential Equations\n                      , Oper. Theory Adv. Appl. 60, Birkha\u0308user Verlag, Basel, 1993.","DOI":"10.1007\/978-3-0348-8573-7"},{"key":"atypb24","first-page":"301","volume":"23","author":"Markowich P. A.","year":"1996","journal-title":"Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4)"},{"key":"atypb25","unstructured":"J. V. Moloney and A. C. Newell,\n                      Nonlinear Optics\n                      , Westview Press, Oxford, UK, 2004."},{"key":"atypb26","doi-asserted-by":"crossref","unstructured":"P. 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