Generalized nonlinear Schrödinger equations describing the Second Harmonic Generation of femtosecond pulse, containing a few cycles, and their integrals of motion

Autoři: Vyacheslav A. Trofimov aff001;  Svetlana Stepanenko aff001;  Alexander Razgulin aff001
Působiště autorů: Faculty of Computational Mathematics and Cybernetics, Lomonosov Moscow State University, Moscow, Russia aff001
Vyšlo v časopise: PLoS ONE 14(12)
Kategorie: Research Article


An interaction of laser pulse, containing a few cycles, with substance is a modern problem, attracting attention of many researches. The frequency conversion is a key problem for a generation of such pulses in various ranges of frequencies. Adequate description of such pulse interaction with a medium is based on a slowly evolving wave approximation (SEWA), which has been proposed earlier for a description of propagation of the laser pulse, containing a few cycles, in a medium with cubic nonlinear response. Despite widely applicability of the frequency conversion for various nonlinear optics problems solutions, SEWA has not been applied and developed for a theoretical investigation of the frequency doubling process until present time. In this study the set of generalized nonlinear Schrödinger equations describing a second harmonic generation of the super-short femtosecond pulse is derived. The equations set contains terms, describing the pulses self-steepening, and the second order dispersion (SOD) of the pulse, a diffraction of the beam as well as mixed derivatives. We propose the transform of the equations set to a type, which does not contain both the mixed derivatives and time derivatives of the nonlinear terms. This transform allows us to derive the integrals of motion of the problem: energy, spectral invariants and Hamiltonian. We show the existence of two specific frequencies (singularities in the Fourier space) inherent to the problem. They may cause an appearance of non-physical absolute instability of the problem solution if the spectral invariants are not taken into account. Moreover, we claim that the energy preservation at the laser pulses propagation may not occur if these invariants do not preserve. Developed conservation laws, in particular, have to be used for developing of the conservative finite-difference schemes, preserving the conservation laws difference analogues, and for developing of adequate theory of the modulation instability of the laser pulses, containing a few cycles.

Klíčová slova:

Computer modeling – Conservation of energy – Diffraction – Electric field – Laser beams – Lasers – Modulation – Optics


1. Franken PA, Hill AE, Peters CW, Weinreich G. Generation of optical harmonics. Phys. Rev. Lett. 1961; 7(4): 118–120. doi: 10.1103/PhysRevLett.7.118

2. Armstrong J, Bloembergen N, Ducuing J, Pershan P. Interactions between Light Waves in a Nonlinear Dielectric. Phys. Rev. 1962; 127(6): 1918–39. doi: 10.1103/PhysRev.127.1918

3. Linde D, Schulz H, Engers T, Schüler H. Second Harmonic Generation in Plasmas Produced by Intense Femtosecond Laser Pulses. IEEE Jour. of Quant. Electr. 1992; 28(10): 2388–97. doi: 10.1109/3.159545

4. Streeter M, Foster PS, Cameron FH, Borghesi M, Brenner C, Carroll DC, et. al. Relativistic plasma surfaces as an efficient second harmonic generator. New J. Phys. 2011; 13: 023041. doi: 10.1088/1367-2630/13/2/023041

5. Singh M, Gupta DN, Suk H. Efficient second- and third-harmonic radiation generation from relativistic laser-plasma interactions. Phys. of Plasm. 2015; 22: 063303. doi: 10.1063/1.4922435

6. Kim S, Jin J, Kim YJ, Park IY, Kim Y, Kim SW. High-harmonic generation by resonant plasmon field enhancement. Nature. 2008; 453: 757–760. doi: 10.1038/nature07012 18528390

7. Steingrube D, Schulz E, Binhammer T, Gaarde MB, Couairon A, Morgner U, Kovacev M. High-order harmonic generation directly from a filament. New J. Phys. 2011; 13: 043022. doi: 10.1088/1367-2630/13/4/043022

8. Zheng J, Qiu E, Lin Q. High harmonic generation with sub-cycle pulses. J. Opt. 2011; 13: 075206. doi: 10.1088/2040-8978/13/7/075206

9. Lucchini M, Calegari F, Kim K, Sansone G, Nisoli M. Nonadiabatic quantum path analysis of the high-order harmonic generation in a highly ionized medium. New J. Phys. 2012; 14(2): 023025.

10. Kemper A, Moritz B, Freericks JK, Devereaux TP. Theoretical description of high-order harmonic generation in solids. New J. Phys. 2013; 15: 023003. doi: 10.1088/1367-2630/15/2/023003

11. Vampa G, Hammond TJ, Thiré N, Schmidt BE, Légaé F, McDonald CR, Brabec T, Corkum PB. Linking high harmonics from gases and solids. Nature. 2015; 522: 462–464. doi: 10.1038/nature14517 26108855

12. Neyra E, Videla F, Ciappina MF, Pérez-Hernández JA, Roso L, Lewenstein M, Torchia GA. High-order harmonic generation driven by inhomogeneous plasmonics fields spatially bounded: influence on the cut-off law. J. Opt. 2018; 20(3): 034002. doi: 10.1088/2040-8986/aaa6f7

13. Zdanowicz M, Kujala S, Husu H, Kauranen M. Effective medium multipolar tensor analysis of second-harmonic generation from metal nanoparticles. New J. Phys. 2011; 13: 023025. doi: 10.1088/1367-2630/13/2/023025

14. Pavlyukh Y, Berakdar J, Hübner W. Semi-classical approximation for second-harmonic generation in nanoparticles. New J. Phys. 2012; 14: 093044. doi: 10.1088/1367-2630/14/9/093044

15. Kautek W, Sorg N, Krüger J. Femtosecond pulse laser second harmonic generation on semiconductor electrodes. Electrochimica Acta. 1994; 39(8/9): 1245–49. doi: 10.1016/0013-4686(94)E0043-Y

16. Mlejnek M, Wright E, Moloney J, Bloembergen N. Second Harmonic Generation of Femtosecond Pulses at the Boundary of a Nonlinear Dielectric. Phys. Rev. Lett. 1999; 83(15): 2934–37. doi: 10.1103/PhysRevLett.83.2934

17. Sidick E, Knoesen A, Dienes A. Ultrashort-pulse second-harmonic generation. I. Transform-limited fundamental pulses. J. Opt. Soc. Am. B. 1995; 12(9): 1704–12. doi: 10.1364/JOSAB.12.001704

18. Kim D-W, Xiao G-Y, Ma G-B. Temporal properties of the second-harmonic generation of a short pulse. Appl. Opt. 1997; 36(27): 6788–93. doi: 10.1364/ao.36.006788 18259545

19. Telegin LS, Chirkin AS. Interaction in frequency doubling of ultrashort laser pulses. Sov. J. Quantum Electron. 1982; 12: 1358. doi: 10.1070/QE1982v012n10ABEH006061

20. Razumikhina TB, Telegin LS, Kholodnykh AI, Chirkin AS. Three-frequency interactions of high-intensity light waves in media with quadratic and cubic nonlinearities. Sov. J. Quantum Electron. 1984; 14(10): 1358–63. doi: 10.1070/QE1984v014n10ABEH006408

21. Ditmire T, Rubenchik AM, Eimerl D, Perry MD. Effects of cubic nonlinearity on frequency doubling of high-power laser pulses. J. Opt. Soc. Am. B. 1996; 13(4): 649–652. doi: 10.1364/JOSAB.13.000649

22. Choe W, Banerjee PP, Caimi FC. Second-harmonic generation in an optical medium with second- and third-order nonlinear susceptibilities. J. Opt. Soc. Am. B. 1991; 8(5): 1013–22. doi: 10.1364/JOSAB.8.001013

23. Komissarova MV, Sukhorukov AP, Trofimov VA. Self-compression of the fundamental and second harmonic pulses in the media with quadratic and cubic nonlinearities. Bulletin of the Russian Academy of Sciences. Physics supplement. Physics of Vibration. 1993; 57: 189–192.

24. Lysak TM, Trofimov VA. The bistable mode of second harmonic generation by femtosecond pulses. Technical Physics. 2001; 46: 1401–06. doi: 10.1134/1.1418503

25. Lysak TM, Trofimov VA. Bistability and uniqueness of solutions in the problem of second harmonic generation of femtosecond pulses. Computational Mathematics and Mathematical Physics. 2001; 41: 1214–26.

26. Lin R, Gao Y. Observation of the modulation instability and frequency-doubling in self-defocusing crystal. Phys. Lett. A. 2011; 375: 3228–31. doi: 10.1016/j.physleta.2011.07.007

27. Kasumova RJ, Safarova GA, Ahmadova AR, Kerimova NV. Influence of self- and cross-phase modulations on an optical frequency doubling process for metamaterials. Appl. Opt. 2018; 57(25): 7385–90. doi: 10.1364/AO.57.007385 30182960

28. Trofimov VA, Kharitonov DM, Fedotov MV. Theory of SHG in a medium with combined nonlinear response. J. Opt. Soc. Am. B. 2018; 35(12): 3069–87. doi: 10.1364/JOSAB.35.003069

29. Trofimov VA, Trofimov VV. High effective SHG of femtosecond pulse with ring profile of beam in bulk medium with cubic nonlinear response. Proceedings of SPIE. 2007; 66100: 66100R. doi: 10.1117/12.740023

30. Ashihara S, Nishina J, Shimura T, Kuroda K. Soliton compression of femtosecond pulses in quadratic media. J. Opt. Soc. Am. B. 2002; 19(10) 2505–10. doi: 10.1364/JOSAB.19.002505

31. Liu X, Qian L, Wise F. High-energy pulse compression by use of negative phase shifts produced by the cascade χ(2): χ(2) nonlinearity. Opt. Lett. 1999; 24(23): 1777–79. doi: 10.1364/ol.24.001777 18079931

32. Wyller J, Królikowski WZ, Bang O, Petersen DE, Rasmussen JJ. Modulational instability in the nonlocal χ(2)-model. Physica D. 2007; 227: 8–25. doi: 10.1016/j.physd.2007.01.002

33. Wang J, Ma Z, Li Y, Lu D, Guo Q, Hu W. Stable quadratic solitons consisting of fundamental waves and oscillatory second harmonics subject to boundary confinement. Phys. Rev. A. 2015; 91: 033801. doi: 10.1103/PhysRevA.91.033801

34. Karamzin YN, Sukhorukov AP. Nonlinear interaction of diffracted light beams in a medium with quadratic nonlinearity: mutual focusing of beams and limitation on the efficiency of optical frequency converters. JETP Lett. 1974; 20: 339.

35. Torruellas WE, Wang Z, Hagan DJ, VanStryland EW, Stegeman GI, Torner L, et. al. Observation of two-dimensional spatial solitary waves in a quadratic medium. Phys. Rev. Lett. 1995; 74: 5036. doi: 10.1103/PhysRevLett.74.5036 10058667

36. Schiek R, Baek Y, Stegeman GI. One-dimensional spatial solitary waves due to cascaded second-order nonlinearities in planar waveguides. Phys. Rev. E. 1996; 53: 1138. doi: 10.1103/PhysRevE.53.1138

37. Fuerst RA, Baboiu DM, Lawrence B, Torruellas WE, Stegeman GI, Trillo S, Spatial modulational instability and multisolitonlike generation in a quadratically nonlinear optical medium. Phys. Rev. Lett. 1997; 78: 2756. doi: 10.1103/PhysRevLett.78.2756

38. Costantini B, De Angelis C, Barthelemy A, Bourliaguet B, Kermene V. Collisions between type II two-dimensional quadratic solitons. Opt. Lett. 1998; 23: 424–426. doi: 10.1364/ol.23.000424 18084532

39. Di Trapani P, Caironi D, Valiulis G, Dubietis A, Danielius R, Piskarskas A. Observation of temporal solitons in second-harmonic generation with tilted pulses. Phys. Rev. Lett. 1998; 81: 570. doi: 10.1103/PhysRevLett.81.570

40. Liu X, Qian LJ, Wise FW. Generation of optical spatiotemporal solitons. Phys. Rev. Lett. 1999; 82: 23. doi: 10.1103/PhysRevLett.82.4631

41. Kim DH, Kang JU, Khurgin JB. Cascaded Raman self-frequency shifted soliton generation in an Er/Yb-doped fiber amplifier. Appl. Phys. Lett. 2002; 81: 2695. doi: 10.1063/1.1512823

42. Kharenko DS, Bednyakova AE, Podivilov EV, Fedoruk MP, Apolonski A, Babin SA. Cascaded generation of coherent Raman dissipative solitons. Opt. Lett. 2016; 41: 175. doi: 10.1364/OL.41.000175 26696187

43. Buryak AV, Di Trapani P, Skryabin DV, Trillo S. Optical solitons due to quadratic nonlinearities: from basic physics to futuristic applications. Phys. Rep. 2002; 370: 63. doi: 10.1016/S0370-1573(02)00196-5

44. Cheng Z, Fu HY, Li Q. Cascaded photonic crystal fibers for three stage non-integer order soliton compression. Opt. Comm. and Net. 2017; 1–2.

45. Bache M, Wise FW. Type-I cascaded quadratic soliton compression in lithium niobate: Compressing femtosecond pulses from high-power fiber lasers. Phys. Rev. A. 2010; 81: 053815. doi: 10.1103/PhysRevA.81.053815

46. Zeng X, Ashihara S, Shimura T, Kuroda K. Adiabatic femtosecond pulse compression and control by using quadratic cascading nonlinearity. Nonlinear Optics: Technologies and Applications.—International Society for Optics and Photonics. 2008; 6839: 68390B.

47. Li Q, Kutz J, Wai P. Cascaded higher-order soliton for non-adiabatic pulse compression. J. Opt. Soc. B. 2010; 27: 2180. doi: 10.1364/JOSAB.27.002180

48. Bache M, Królikowski WZ, Moses J, Wise FW. Limits to compression with cascaded quadratic soliton compressors. Opt. Expr. 2008; 16: 3273. doi: 10.1364/OE.16.003273

49. Moses J, Wise FW. Soliton compression in quadratic media: high-energy few-cycle pulses with a frequency-doubling crystal. Opt. Lett. 2006; 31: 1881. doi: 10.1364/ol.31.001881 16729102

50. Šuminas R, Tamošauskas G, Valiulis G, Dubietis A. Spatiotemporal light bullets and supercontinuum generation in β-BBO crystal with competing quadratic and cubic nonlinearities. Opt. Lett. 2016; 41: 2097. doi: 10.1364/OL.41.002097 27128083

51. Conti C, Trillo S, Di Trapani P, Kilius J, Bramati A, Minardi S, et. al. Effective lensing effects in parametric frequency conversion. J. Opt. Soc. Am. B. 2002; 19(4): 852–859. doi: 10.1364/JOSAB.19.000852

52. Di Trapani P, Bramati A, Minardi S, Chinaglia W, Trillo S, Conti C, et. al. Focusing versus defocusing nonlinearities in self-trapping due to parametric frequency conversion. Phys. Rev. Lett. 2001; 87: 183902. doi: 10.1103/PhysRevLett.87.183902

53. Trofimov V, Lysak T. Catastrophic self-focusing of axially symmetric laser beams due to cascading SHG. Proceedings of SPIE. 2011; 7822: 78220E–78220E-11. doi: 10.1117/12.891265

54. Lysak TM, Trofimov VA. Achieving high-efficiency second harmonic generation in a sequence of laser pulses with random peak intensity. Part I. Efficient generation in optical fibers. Comp. Math. and Modeling. 2008; 19: 333–342. doi: 10.1007/s10598-008-9012-z

55. Lysak TM, Trofimov VA. Achieving high-efficiency second harmonic generation in a sequence of laser pulses with random peak intensity Part II. Suppression of intensity fluctuations in a quadratic-nonlinearity medium. Comp. Math. and Modeling. 2009; 20: 1–25. doi: 10.1007/s10598-009-9015-4

56. Lysak TM, Trofimov VA. Achieving high-efficiency second harmonic generation in a sequence of laser pulses with random peak intensity Part III. Propagation of pulses in a bulk medium. Comp. Math. and Modeling. 2009; 20: 101–112. doi: 10.1007/s10598-009-9030-5

57. Trofimov VA, Lysak TM. Highly efficient SHG of a sequence of laser pulses with a random peak intensity and duration. Opt. Spectrosc. 2009; 107: 399–406. doi: 10.1134/S0030400X0909015X

58. Huttunen M, Mäkitalo J, Bautista G, Kauranen M. Multipolar second-harmonic emission with focused Gaussian beams. New J. Phys. 2012; 14: 113005.

59. O’Donnell K, Torre R. Characterization of the second-harmonic response of a silver-air interface. New J. Phys. 2005; 7: 154.

60. Rodrigo S, Laliena V, Martin-Moreno L. Second-harmonic generation from metallic arrays of rectangular holes. J. Opt. Soc. Am. B. 2015; 32: 15–25. doi: 10.1364/JOSAB.32.000015

61. Luo M, Liu Q. Extraordinary enhancement of second harmonic generation in a periodically patterned distributed Bragg reflector. J. Opt. Soc. Am. B. 2015; 32: 1193–1201. doi: 10.1364/JOSAB.32.001193

62. Yudovich S, Shwartz S. Second-harmonic generation of focused ultrashort x-ray pulses. J. Opt. Soc. Am. B. 2015; 32: 1894–1900. doi: 10.1364/JOSAB.32.001894

63. Butet J, Gallinet B, Thyagarajan K, Martin OJF. Second-harmonic generation from periodic arrays of arbitrary shape plasmonic nanostructures: a surface integral approach. J. Opt. Soc. Am. B. 2013; 30: 2970–79. doi: 10.1364/JOSAB.30.002970

64. Kolmychek IA, Krutyanskiy VL, Murzina TV, Sapozhnikov MV, Karashtin EA, Rogov VV, et. al. First and second order in magnetization effects in optical second-harmonic generation from a trilayer magnetic structure. J. Opt. Soc. Am. B. 2015; 32: 331–338. doi: 10.1364/JOSAB.32.000331

65. Samim M, Krouglov S, Barzda V. Double Stokes Mueller polarimetry of second-harmonic generation in ordered molecular structures. J. Opt. Soc. Am. B. 2015; 32: 451–461. doi: 10.1364/JOSAB.32.000451

66. Arjmand A, Abolghasem P, Han J, Helmy AS. Interface modes for monolithic nonlinear photonics. J. Opt. Soc. Am. B. 2015; 32: 577–587. doi: 10.1364/JOSAB.32.000577

67. Hardhienata H, Alejo-Molina A, Reitböck C, Prylepa A, Stifter D, Hingerl K. Bulk dipolar contribution to second-harmonic generation in zincblende. J. Opt. Soc. Am. B. 2016; 33: 195–201. doi: 10.1364/JOSAB.33.000195

68. Zhang S, Zhang X. Strong second-harmonic generation from bilayer-graphene embedded in one-dimensional photonic crystals. J. Opt. Soc. Am. B. 2016; 33: 452–460. doi: 10.1364/JOSAB.33.000452

69. Sabouri SG, Khorsandi A. Thermal dephasing compensation in high-power and high-repetition-rate second-harmonic generation using spillover loss. J. Opt. Soc. Am. B. 2016; 33: 1640–48. doi: 10.1364/JOSAB.33.001640

70. Tang D, Wang J, Zhou B, Xie G, Ma J, Yuan P, et. al. Temperature-insensitive second-harmonic generation based on noncollinear phase matching in a lithium triborate crystal. J. Opt. Soc. Am. B. 2017; 34: 1659–68. doi: 10.1364/JOSAB.34.001659

71. Guo S, Ge Y, Han Y, He J, Wang J. Investigation of optical inhomogeneity of MgO:PPLN crystals for frequency doubling of 1560 nm laser. Optics Communications. 2014; 326: 114–120. doi: 10.1016/j.optcom.2014.04.008

72. Yuan J.-H, Zhang Y, Mo H, Chen N, Zhang Z. The second-harmonic generation susceptibility in semiparabolic quantum wells with applied electric field. Optics Communications. 2015; 356: 405–410. doi: 10.1016/j.optcom.2015.08.030

73. Kanseri B, Bouillard M, Tualle-Brouri R. Efficient frequency doubling of femtosecond pulses with BIBO in an external synchronized cavity. Optics Communications. 2016; 380: 148–153. doi: 10.1016/j.optcom.2016.05.067

74. Kato K, Umemura N, Petrov V. Sellmeier and thermo-optic dispersion formulas for CdGa2S4 and their application to the nonlinear optics of Hg1−xCdxGa2S4. Optics Communications. 2017; 386: 49–52. doi: 10.1016/j.optcom.2016.10.054

75. Zhang Y, Hyodo M, Okada-Shudo Y, Zhu Y, Wang X. Characteristics of pulse width for an enhanced second harmonic generation. Optics Communications. 2017; 387: 241–244. doi: 10.1016/j.optcom.2016.11.058

76. Cai L, Wang Y, Hu H. Efficient second harmonic generation in χ(2) profile reconfigured lithium niobate thin film. Optics Communications. 2017; 387: 405–408. doi: 10.1016/j.optcom.2016.10.064

77. Leo N, Meier D, Becker P, Bohatý L, Fiebig M. Magnetically driven second-harmonic generation with phase matching in MnWO4. Optics Express. 2015; 23: 27700. doi: 10.1364/OE.23.027700 26480432

78. Zeng J, Li J, Li H, Dai Q, Tie S, Lan S. Effects of substrates on the nonlinear optical responses of two-dimensional materials. Optics Express. 2015; 23: 31817. doi: 10.1364/OE.23.031817 26698973

79. Kim S, Qi M. Broadband second-harmonic phase-matching in dispersion engineered slot waveguides. Optics Express. 2016; 24(2): 773. doi: 10.1364/OE.24.000773 26832462

80. Stegeman GI, Schiek R, Fang H, Malendevich R, Jankovic L, Torner L, et. al. Beam evolution in quadratically nonlinear one-dimensional media: LiNbO3 slab waveguides. Laser Phys. 2003; 13: 137–147.

81. Su W, Qian L, Luo H, Fu X, Zhu H, Wang T, et. al. Induced group-velocity dispersion in phase-mismatched second-harmonic generation. J. Opt. Soc. Am. B. 2006; 23(1): 51–55. doi: 10.1364/JOSAB.23.000051

82. Bache M, Bang O, Zhou BB, Moses J, Wise FW. Optical Cherenkov radiation by cascaded nonlinear interaction: an efficient source of few-cycle energetic near- to mid-IR pulses. Opt. Expr. 2011; 19: 22557–62. doi: 10.1364/OE.19.022557

83. Cai Y, Xu S, Zeng X, Zou D, Li J. High-efficiency intracavity second-harmonic enhancement for a few-cycle laser pulse train. J. Opt. 2012; 14: 105202.

84. Ota Y, Watanabe K, Iwamoto S, Arakawa Y. Measuring the second-order coherence of a nanolaser by intracavity frequency doubling. Phys. Rev. A. 2014; 89: 023824. doi: 10.1103/PhysRevA.89.023824

85. Iaconis C, Walmsley IA. Self-Referencing Spectral Interferometry for Measuring Ultrashort Optical Pulses. IEEE J. Quantum Electron. 1999; 35: 501–509. doi: 10.1109/3.753654

86. Kane DJ, Trebino R. Characterization of arbitrary femtosecond pulses using frequency-resolved optical gating. IEEE J. Quantum Electron. 1993; 29: 571–579. doi: 10.1109/3.199311

87. Baltuska A, Pshenichnikov M, Wiersma D. Second-harmonic generation frequency-resolved optical gating in the single-cycle regime. IEEE J. Quant. El. 1999; 35(4): 459–478. doi: 10.1109/3.753651

88. Bragheri F, Faccio D, Bonaretti F, Lotti A, Clerici M, Jedrkiewicz O, et. al. Complete retrieval of the field of ultrashort optical pulses using the angle-frequency spectrum. Opt. Lett. 2008; 33(24): 2952–54. doi: 10.1364/ol.33.002952 19079503

89. Hause A, Kraft S, Rohrmann P, Mitschke F. Reliable multiple-pulse reconstruction from second-harmonic-generation frequency-resolved optical gating spectrograms. J. Opt. Soc. Am. B. 2015; 32: 868–877. doi: 10.1364/JOSAB.32.000868

90. Pirogova I, Sukhorukov A. Effect of nonlinear-wave coupling dispersion on the frequency doubling of subpicosecond light-pulses. Opt. and Spect. 1985; 59(3): 694–696.

91. Tzoar N, Jain M. Self-phase modulation in long-geometry optical waveguides. Phys. Rev. A. 1981; 23(3): 1266–70. doi: 10.1103/PhysRevA.23.1266

92. Anderson D, Lisak M. Nonlinear asymmetric self-phase modulation and self-steepening of pulses in long optical waveguides. Phys. Rev. A. 1983; 27(3): 1393–98. doi: 10.1103/PhysRevA.27.1393

93. Agrawal G. Nonlinear Fiber Optics. (4th ed. Academic Press). 2007.

94. Brabec T, Krausz F. Nonlinear optical pulse propagation in the single-cycle regime. Phys. Rev. Lett. 1997; 78(17): 3282–85. doi: 10.1103/PhysRevLett.78.3282

95. Dong M-J, Tian S-F, Yan X-W, Zhang T-T. Nonlocal symmetries, conservation laws and interaction solutions for the classical Boussinesq–Burgers equation. Nonlinear Dynamics. 2019; 95(1): 273–291. doi: 10.1007/s11071-018-4563-9

96. Yan X-W, Tian S-F, Dong M-J, Wang X-B, Zhang T-T. Nonlocal symmetries, conservation laws and interaction solutions of the generalised dispersive modified Benjamin–Bona–Mahony equation. Zeitschrift für Naturforschung A. 2018; 73(5): 399–405. doi: 10.1515/zna-2017-0436

97. Wang X-B, Tian S-F, Qin C-Y, Zhang T-T. Lie symmetry analysis, conservation laws and analytical solutions of a time-fractional generalized KdV-type equation. J. Nonlinear Math. Phys. 2017; 24(4): 516–530. doi: 10.1080/14029251.2017.1375688

98. Peng W-Q, Tian S-F, Zhang T-T. Dynamics of breather waves and higher-order rogue waves in a coupled nonlinear Schrödinger equation. Europhysics Letters. 2018; 123(5): 50005.

99. Wang X-B, Tian S-F, Qin C-Y, Zhang T-T. Lie symmetry analysis, analytical solutions, and conservation laws of the generalised Whitham–Broer–Kaup–Like equations. Zeitschrift für Naturforschung A. 2017; 72(3): 269–279. doi: 10.1515/zna-2016-0389

100. Varentsova SA, Trofimov VA. Lagrangian of the process of generation of the second harmonic of femtosecond light pulses. Diff. Eq. 1998; 34(7): 997–999.

101. Varentsova SA, Trofimov VA. Invariants in the process of second harmonic generation by femtosecond light pulses. Moscow University Comput. Math. and Cybern. 1998; 4: 45–47.

102. Hovhannisyan D, Stepanyan K, Avagyan R. Computational modelling of second-harmonic generation by a femtosecond laser pulse of a few optical cycles. J. Mod. Opt. 2005; 52(1): 97–107. doi: 10.1080/09500340410001703832

103. Szarvas T, Kis Z. Numerical simulation of nonlinear second harmonic wave generation by the finite difference frequency domain method. J. Opt. Soc. Am. B. 2018; 35(4): 731–740. doi: 10.1364/JOSAB.35.000731

104. Xiao Y, Maywar DN, Agrawal GP. Propagation of few-cycle pulses in nonlinear Kerr media: harmonic generation. Opt. Lett. 2013; 38(5): 724–726. doi: 10.1364/OL.38.000724 23455278

105. Shcherbakov AA, Tishchenko AV. New fast and memory-sparing method for rigorous electromagnetic analysis of 2D periodic dielectric structures. J. Quant. Spectrosc. Radiat. Transfer. 2012; 113: 158–171. doi: 10.1016/j.jqsrt.2011.09.019

106. Weismann M, Gallagher D, Panoiu N. Nonlinear generalized source method for modeling second-harmonic generation in diffraction gratings. J. Opt. Soc. Am. B. 2015; 32: 523–533. doi: 10.1364/JOSAB.32.000523

107. Karamzin YN, Sukhorukov AP, Trofimov VA. Mathematical modeling in nonlinear optics. Moscow: Publishing house of Moscow University [in Russian]; 1989.

Článek vyšel v časopise


2019 Číslo 12
Nejčtenější tento týden