Nonlinear photonic crystals: I. Quadratic nonlinearity

Waves Random Media

Volumn 11, Issue 2, 2001, Pages

  Babin, A ^a   Figotin, A ^a  

^a UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

CRYSTALS; DIELECTRIC MATERIALS; MATHEMATICAL MODELS; MAXWELL EQUATIONS; NONLINEAR EQUATIONS; NONLINEAR OPTICS;

PHOTONIC CRYSTALS;

PHOTONS;

EID: 0035304043     PISSN: 09597174     EISSN: None     Source Type: Journal    
DOI: 10.1088/0959-7174/11/2/201     Document Type: Review

References (84)

1. Agrawal G 1995 Nonlinear Fiber Optics (New York: Academic)
2. Ashcroft N and Mermin N 1976 Solid State Physics (New York: Holt, Rinehart and Winston)
3. Agrawal G and Boyd R 1992 Contemporary Nonlinear Optics (New York: Academic)
4. Arnold V I 1989 Methods of Classical Mechanics (Graduate Texts in Mathematics, vol 60) (New York: Springer)
5. Arnold V I, Gusein-Zade S M and Varchenko A N 1985 Singularities of Differentiable Mappings vol 1 (Monographs in Mathematics vol 82) (Boston, MA: Birkhäuser)
6. Arnold V I, Gusein-Zade S M and Varchenko A N 1985 Singularities of Differentiable Mappings vol 1 (Monographs in Mathematics vol 82) (Boston, MA: Birkhäuser)
7. Arnold V I, Gusein-Zade S M and Varchenko A N 1988 Singularities of Differentiable Mappings vol 2 (Monographs in Mathematics vol 83) (Boston, MA: Birkhäuser)
8. Arnold V I, Gusein-Zade S M and Varchenko A N 1988 Singularities of Differentiable Mappings vol 2 (Monographs in Mathematics vol 83) (Boston, MA: Birkhäuser)
9. Aközbek N and John S 1998 Optical solitary waves in two- and three-dimensional nonlinear photonic band-gap structures Phys. Rev. E 57 2287-319
10. Axmann W and Kuchment P 1999 An efficient finite element method for computing spectra of photonic and acoustic band-gap materials J. Comput. Phys. 150 468-81
11. Bulgakov A, Bulgakov S and Vazquez L 1998 Second-harmonic resonant excitation in optical periodic structures with nonlinear anisotropic layers Phys. Rev. E 58 7887-98
12. Berger V 1999 Nonlinear photonic crystals Phys. Rev. Lett. 81 4136-9
13. Bleistein N and Handelsman R A 1986 Asymptotic Expansions of Integrals (New York: Dover)
14. Bogoliubov N N and Mitropolsky Y A 1961 Asymptotic Methods in the Theory of Non-Linear Oscillations (Delhi: Hindustan)
15. Babin A, Mahalov A and Nicolaenko B 1997 Regularity and integrability of 3D Euler and Navier-Stokes equations for rotating fluids Asymptotic Anal. 15 103-50
16. Belyakov V 1992 Diffraction Optics of Complex-Structured Periodic Medium (Berlin: Springer)
17. Butcher P and Cotter D 1990 The Elements of Nonlinear Optics (Cambridge: Cambridge University Press)
18. Boyd R 1992 Nonlinear Optics (New York: Academic)
19. Broderick N, Mongo T, Bennett P and Richardson D 1999 Nonlinearity in holey optical fibers: measurements and future opportunities Opt. Lett. 24 1395-7
20. Bennink R, Yoon Y, Boyd R and Sipe J 1999 Accessing the optical nonlinearity of metals with metal-dielectric photonic bandgap structures Opt. Lett. 24 1416-8
21. Centini M et al 1999 Despersive properties of finite, one-dimensional photonic bandgap structures: applications to nonlinear quadratic interactions Phys. Rev. E 60 4891-8
22. Dobson D 1999 An efficient method for band structure calcuations in 2D photonic crystals J. Comput. Phys. 149 363-76
23. Estrada R and Kanwal R 1994 Asymptotic Analysis: a Distributional Approach (Boston, MA: Birkhäuser)
24. Fedorjuk M V 1987 Asimptotika Integraly i Rjady (Moscow: Nauka)
25. Fedorjuk M V 1962 The method of stationary phase for multiple integrals J. Commun. Math. Math. Phys. 2 145-50
26. Fedorjuk M V 1977 The Saddle-Point Method (Moscow: Nauka)
27. Figotin A 1999 High-contrast photonic crystals Diffusive Waves in Complex Media ed J P Fouque (Dordrecht: Kluwer) pp 109-36
28. Figotin A and Godin Yu 1997 The computation of spectra of some 2D photonic crystals J. Comput. Phys. 136 585-98
29. Figotin A, Godin Yu and Vitebsky I 1998 2D tunable photonic crystals Phys. Rev. B 57 2841-8
30. Figotin A and Klein A 1997 Localization of classical waves II. Electromagnetic waves Commun. Math. Phys. 184 411-41
31. Figotin A and Klein A 1996 Localization of classical waves I. Acoustic waves Commun. Math. Phys. 180 439-82
32. Figotin A and Klein A 1998 Localization of light in lossless inhomogeneous dielectrics J. Opt. Soc. Am. A 15 1423-35
33. Figotin A and Kuchment P 1996 Band-gap structure of the spectrum of periodic dielectric and acoustic media. II. 2D photonic crystals SIAM J. Appl. Math. 56 1561-620
34. Figotin A and Kuchment P 1998 Spectral properties of classical waves in high contrast periodic media SIAM J. Appl. Math. 58 683-702
35. Felsen L and Marcuvits N 1994 Radiation and Scattering of Waves (Oxford: Oxford University Press, Piscataway, NJ: IEEE)
36. Fogel I et al 1998 Spontaneous emission and nonlinear effects in photonic bandgap materials Pure Appl. Opt. 7 393-407
37. Jackson J 1975 Classical Electrodynamics (New York: Wiley)
38. John S 1991 Localization of light Phys. Today (May)
39. Joly J-L, Metivier G and Rauch J 1998 Diffractive nonlinear geometric optics with rectification Indiana Univ. Math. J. 47 1167-241
40. Donnat P, Joly J-L, Metivier G and Rauch J 1996 Diffractive nonlinear geometric optics Séminaire sur les Équations aux Dérivées Partielles 1995-6 (Palaiseau: École Polytechnique) Exp. no XVII, 25 pp
41. Donnat P and Rauch J 1997 Dispersive nonlinear geometric optics J. Math. Phys. 38 1484-523
42. Georgieva A, Richerbauer K T and Venakides S 1999 Wave propagation and resonance in one-dimesional nonlinear discrete periodic medium SIAM J. Appl. Math. 60 272-94
43. Fejer M M, Magel G A, Jundt D H and Byer R L 1992 Quasi-phase-matched second harmonic generation: tuning and tolerances IEEE J. Quantum Electron. 28 2631
44. Harrington R 1961 Time-Harmonic Electromagnetic Fields (New York: McGraw-Hill)
45. Hattori T, Tsurumachi N and Nakatsuka H 1997 Analysis of optical nonlinearity by defect states in one-dimensional photonic crystals J. Opt. Soc. Am. B 14 348-55
46. Haus J, Viswanathan, Scalora M, Kalocsai A, Cole J and Theimer J 1998 Enhanced second-harmonic generation in media with a weak periodicity Phys. Rev. A 57 2120-8
47. Hunter J K and Keller J B 1983 Weakly nonlinear high frequency waves Commun. Pure Appl. Math. 36 547-69
48. Joly J-L, Métivier G and Rauch J 1996 Nonlinear oscillations beyond caustics Commun. Pure Appl. Math. 49 443-527
49. Joannopoulos J, Meade R and Winn J 1995 Photonic Crystals (Princeton, NJ: Princeton University Press)
50. Kappeler T 1988 Multiplicities of the eigenvalues of the Schrödinger equation in any dimension J. Funct. Anal. 77 346-51
51. Kato T 1995 Perturbation Theory for Linear Operators (Berlin: Springer)
52. Kevorkian J and Bosley D 1998 Multiple-scale homogenization for weakly nonlinear conservation laws with rapid spatial fluctuations Stud. Appl. Math. 101 127-83
53. Kevorkian J and Cole J 1966 Multiple Scale and Singular Perturbation Methods (Berlin: Springer)
54. Kuchment P 1993 Floquet Theory for Partial Differential Operators (Basel: Birkhäuser)
55. Kuchment P 1999 The mathematics of photonic crystals Mathematical Modeling in Optical Sciences (Philadelphia, PA: SIAM)
56. Klein A 1999 Localization of light in randomized periodic media Diffusive Waves in Complex Media ed J P Fouque (Dordrecht: Kluwer) pp 73-92
57. Koroteev N, Magnitskii S, Tarasishin A and Zheltikov A 1999 Compression of ultrashort light pulses in photonic crystals: when envelopes cease to be slow Opt. Commun. 159 191-202
58. Kryloff N and Bogoliuboff N 1947 Introduction to Nonlinear Mechanics (Princeton, NJ: Princeton University Press) (a free translation by Solomon Lefschetz of excerpts from two Russian monographs)
59. Kuchment P and Kunyansky L 1999 Spectral properties of high contrast band-gap materials and operators on graphs Exp. Math. 8 1-28
60. Mills D 1991 Nonlinear Optics (Berlin: Springer)
61. Mitropolskii Yu A and Nguen Van Dao 1997 Applied Asymptotic Methods in Nonlinear Oscillations (Dordrecht: Kluwer)
62. Martorell J, Vilaceca R and Corbalan R 1997 Second harmonic generation in a photonic crystal Appl. Phys. Lett. 70 702-4
63. Nayfeh A 1973 Perturbation Methods (New York: Wiley)
64. Nelson R and Boyd R 1999 Enhanced third-order nonlinear optical response of photonic bandgap materials J. Mod. Opt. 46 1061-9
65. Nayfeh A H and Mook D T 1976 Nonlinear Oscillations (New York: Wiley)
66. Newell A C and Moloney J V 1992 Nonlinear Optics (Reading, MA: Addison-Wesley)
67. Rarity J and Weisbuch C 1996 Microcavities and Photonic Bandgaps: Physics and Applications (NATO ASI Series E, vol 324) (Dordrecht: Kluwer)
68. Reed M and Simon B 1978 Analysis of Operators vol 4 (New York: Academic)
69. Scalora M et al 1997 Pulsed second-harmonic generation in nonlinear, one-dimensional, periodic structures Phys. Rev. A 56 3166-74
70. Soukoulis C 1996 Photonic B and Gap Materials (NATO ASI Series E. vol 315) (Dordrecht: Kluwer)
71. Saleh B and Teich M 1991 Fundamentals of Photonics (New York: Wiley)
72. Sauter E G 1996 Nonlinear Optics (New York: Wiley)
73. Stein E 1993 Harmonic Analysis. Real-Variable Methods, Orthogonality and Oscillatory Integrals (Princeton, NJ: Princeton University Press)
74. Shubin M 1994 Partial Differential Equations, VII. Spectral Theory of Differential Operators (Berlin: Springer)
75. Tarasishin A, Zheltikov A and Magnitskii S 1999 Matched second-harmonic generation of ultrasoft laser pulses in photonic crystals JETP Lett. 70 819-25
76. Teytel M 1999 How rare are multiple eigenvalues? Commun. Pure Appl. Math. 52 917-34
77. Tran P 1997 Optical limiting and switching of short pulses by use of a nonlinear photonic bandgap structures with a defect J. Opt. Soc. Soc. B 14 2589-95
78. Uhlenbeck K 1976 Generic properties of eigenfunctions Am. J. Math. 98 1059-78
79. Villeneuve P and Piché M 1994 Photonic B and gaps in periodic dielectric structures Prog. Quantum Electron. 18 153-200
80. Whitham G 1974 Linear and Nonlinear Waves (New York: Wiley)
81. Winn J, Fan S and Joannopoulos J 1999 Interband transitions in photonic crystals Phys. Rev. B 59 1551-41
82. Wang Z et al 1997 Nonlinear transmission resonance in a two-dimensional periodic structure with photonic B and gap Phys. Rev. B 56 9185-8
83. Yariv A and Yeh P 1984 Optical Waves in Crystals (New York: Wiley)
84. Zhu Y and Ming N 1999 Dielectric superlattices for nonlinear optical effects Opt. Quantum Electron. 31 1093-128