Femtosecond Quasi-Bright Soliton Solution and Its Properties Under Influence of Higher-Order Effects in Metamaterials
【Abstract】Based on the higher-order nonlinear Schrödinger equation describing ultrashort pulse transmission in metamaterials, this study presents an exact femtosecond quasi-bright soliton solution and determines its existence conditions by using the traveling wave method. When the group speed dispersion, third-order dispersion, cubic-quintic nonlinearities, self-steepening, and second-order nonlinear dispersion effects are properly balanced, the femtosecond quasi-soliton can exist in nonlinear metamaterials. Without the third-order dispersion and second-order nonlinear dispersion, the soliton in metamaterials can not occur. Based on the Drude model, the existence index regions of the femtosecond quasi-bright soliton are discussed in different nonlinear metamaterials. The results show that femtosecond quasi-soliton can exist in the negative index region of self-defocusing nonlinear metamaterials, and in the positive index region of self-focusing nonlinear metamaterials. Moreover, the intensities and widths of the solitons differ in different regions of the metamaterials, implying that the properties of the formed solitons can be adjusted by choosing different nonlinear metamaterials and different frequencies of the incident wave, making them in the corresponding existence areas.
【Keywords】 nonlinear optics; femtosecond quasi-bright soliton; self-defocusing; self-focusing; nonlinear metamaterials; negative refraction;
(Translated by CHENG H)
 Hasegawa A, Tappert F. Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion [J]. Applied Physics Letters, 1973, 23 (3): 142–144.
 Agrawal G P. Nonlinear fiber optics&application of nonlinear fiber optics [M]. Jia D F, Yu Z H, Tan B, et al. , Transl. Beijing: Electronic Industry Press, 2002: 94–104.
 Kruglov V I, Peacock A C, Harvey J D. Exact solutions of the generalized nonlinear Schrödinger equation with distributed coefficients [J]. Physical Review E, 2005, 71 (5): 056619.
 Yang J W, Gao Y T, Feng Y J, et al. Solitons and dromion-like structures in an inhomogeneous optical fiber [J]. Nonlinear Dynamics, 2017, 87 (2): 851–862.
 Kong D F, Jia D F, Feng D J, et al. Soliton self-frequency shift in optical fibers [J]. Laser & Optoelectronics Progress, 2018, 55 (10): 101902 (in Chinese)．
 Wang L, Yang R C, Jia H P, et al. Periodically lumped amplification and recovery of soliton in dispersion-decreasing optic fiber link [J]. Acta Optica Sinica, 2017, 37 (6): 0619001 (in Chinese)．
 Veselago V G. The electrodynamics of substances with simultaneously negative values ofεandμ [J]. Soviet Physics Uspekhi, 1968, 10 (4): 509–514.
 Lee S H, Park C M, Seo Y M, et al. Reversed Doppler effect in double negative metamaterials [J]. Physical Review B, 2010, 81 (24): 241102.
 Shalaev V M. Optical negative-index metamaterials [J]. Nature Photonics, 2007, 1 (1): 41–48.
 Yang R C, Shadrivov I V. Double-nonlinear metamaterials [J]. Applied Physics Letters, 2010, 97 (23): 231114.
 Fan K B, Hwang H Y, Liu M K, et al. Nonlinear terahertz metamaterials via field-enhanced carrier dynamics in GaAs [J]. Physical Review Letters, 2013, 110 (21): 217404.
 Gao F, Yang R C, Jia H P, et al. Exact solution of spatial chirped dark soliton in metamaterials and its propagation characteristics [J]. Acta Optica Sinica, 2019, 39 (2): 0219001 (in Chinese)．
 Kozyrev A B, Shadrivov I V, Kivshar Y S. Soliton generation in active nonlinear metamaterials [J]. Applied Physics Letters, 2014, 104 (8): 084105.
 English L Q, Wheeler S G, Shen Y, et al. Backward-wave propagation and discrete solitons in a left-handed electrical lattice [J]. Physics Letters A, 2011, 375 (9): 1242–1248.
 Lazarides N, Tsironis G P. Publisher′s note: coupled nonlinear Schrödinger field equations for electromagnetic wave propagation in nonlinear left-handed materials[Phys. Rev. E71, 036614 (2005)] [J]. Physical Review E, 2005, 71 (4): 049903.
 Scalora M, Syrchin M S, Akozbek N, et al. Generalized nonlinear Schrödinger equation for dispersive susceptibility and permeability: application to negative index materials [J]. Physical Review Letters, 2005, 95 (1): 013902.
 Wen S C, Xiang Y J, Dai X Y, et al. Theoretical models for ultrashort electromagnetic pulse propagation in nonlinear metamaterials [J]. Physical Review A, 2007, 75 (3): 033815.
 Li P G, Yang R C, Xu Z Y. Gray solitary-wave solutions in nonlinear negative-index materials [J]. Physical Review E, 2010, 82 (4): 046603.
 Sharma V K, Goyal A, Raju T S, et al. Periodic and solitary wave solutions for ultrashort pulses in negative-index materials [J]. Journal of Modern Optics, 2013, 60 (10): 836–840.
 Yang R C, Min X M, Tian J P, et al. New types of exact quasi-soliton solutions in metamaterials [J]. Physica Scripta, 2016, 91 (2): 025201.
 Yang R C, Zhang Y. Exact combined solitary wave solutions in nonlinear metamaterials [J]. Journal of the Optical Society of America B, 2011, 28 (1): 123–127.
 Triki H, Zhou Q, Moshokoa S P, et al. Novel singular solitons in optical metamaterials for self-steepening effect [J]. Optik, 2018, 154: 545–550.
 Biswas A, Ekici M, Sonmezoglu A, et al. Chirped solitons in optical metamaterials with parabolic law nonlinearity by extended trial function method [J]. Optik, 2018, 160: 92–99.
 Daoui A K, Triki H, Biswas A, et al. Chirped bright and double-kinked quasi-solitons in optical metamaterials with self-steepening nonlinearity [J]. Journal of Modern Optics, 2019, 66 (2): 192–199.
 Daoui A K, Azzouzi F, Triki H, et al. Propagation of chirped gray optical dips in nonlinear metamaterials [J]. Optics Communications, 2019, 430: 461–466.
 Li X Z, Wang M L. A sub-ODE method for finding exact solutions of a generalized KdV-mKdV equation with high-order nonlinear terms [J]. Physics Letters A, 2007, 361 (1/2): 115–118.
 Serge D Y, Justin M, Betchewe G, et al. Optical chirped soliton in metamaterials [J]. Nonlinear Dynamics, 2017, 90 (1): 13–18.
 Dong J, Jin Y J, Chen C, et al. The transmission properties of exact bright-like and dark-like solitons in metamaterials [J]. Acta Sinica Quantum Optica, 2013, 19 (2): 162–170 (in Chinese)．
 Ferrando A. Nonlinear plasmonic amplification via dissipative soliton-plasmon resonances [J]. Physical Review A, 2017, 95 (1): 013816.
 Zhang Q, Tan C H, Huang G X. Lossless Airy surface polaritons in a metamaterial via active Raman gain [J]. Scientific Reports, 2016, 6: 21143.