Asymptotic variational approach to study light propagation in a nonlocal nonlinear medium

Results in Physics 27 (2021) 104536 Available online 12 July 2021 2211-3797/© 2021 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Contents lists available at ScienceDirect Results in Physics journal homepage: www.elsevier.com/locate/rinp Asymptotic variational approach to study light propagation in a nonlocal nonlinear medium Artorix de la Cruza,∗, Michael Cada a, Jaromir Pistora b, Tamara Diaz-Chang c aDepartment of Electrical and Computer Engineering, Dalhousie University, Halifax, Canada bNanotechnology Centre and IT4I, VSB - Technical University of Ostrava, 708 33 Ostrava-Poruba, Czech Republic cInstituto de Ciencias Fisicas y Matematicas, Universidad Austral de Chile, Chile ARTICLE INFO Keywords: Variational approach Nonlocal Nonlinear, optics Asymptotics Euler equations ABSTRACT We propose and demonstrate analytically, within the framework of a hydrodynamic model, a novel and simpler variational approach to study the asymptotic behavior of a continuous wave (cw) laser beam propagating in a nonlinear nonlocal medium. The starting point in the analysis is the light propagation in a weakly nonlocal nonlinear defocusing medium described by normalized NLSE 𝑖 𝜓𝑧+1 2𝜓𝑥,𝑥 −𝜑 𝜓 = 0 ,(1) where the dimensionless 𝑧and 𝑥are the spatial evolutionary variable and the transverse coordinates, respectively. Also, 𝜓is the complex electric field envelop, 𝜑is a real function that denotes the nonlinear nonlocal change of the refractive index depending on the intensity 𝐼=|𝜓|2. Finally 0< 𝜖 ≪ 1is a small quantity that deal with the weakly diffracting regime (see [1] for more details). Other examples of light propagating in different media are [2–9]. The above expression is coupled to a diffusion-like equation for the response of the nonlocal medium −𝜎2𝜑𝑥,𝑥 +𝜑=|𝜓|2,(2) where the parameter 𝜎is a spatial scale (setting the diffusion length) that measures the degree of nonlocality. We consider small amplitude slowly varying modulations of the steady state given by a continuous wave 𝜓=𝜓0exp(−𝑖|𝜓0|2𝑧), where 𝜓0is an arbitrary complex constant, |𝜓0|2= 1 and the constant 𝜑= |𝜓0|2. Applying the Mandelung transformation 𝜓(𝑧, 𝑥) = 𝜌1∕2(𝑧, 𝑥) exp[ 𝑖 ℎ(𝑧, 𝑥) ] and retaining leading orders in 𝜖, it is possible to obtain the following equations 𝜌𝑧+(𝜌 ℎ𝑥)𝑥= 0,(3a) ℎ𝑧+1 2ℎ2 𝑥+1 2𝜌−1∕2 𝜌1∕2 𝑥,𝑥 +𝜑= 0,(3b) ∗Corresponding author. E-mail address: [email protected] (A. de la Cruz). −𝜎2𝜑𝑥,𝑥 +𝜑=𝜌, (3c) The above system of equations can be derived from the appropriate Lagrangian density 𝐿=𝜌[ℎ2 𝑥 2+ℎ𝑧+𝜑− 1]+1 2(𝜌1∕2 𝑥)2−1 2[𝜑2+(𝜎 𝜑𝑥)2− 1].(4) Euler–Lagrange variation with respect to ℎyields (3a) whereas the 𝜌and 𝜑variations yield (3b) and (3c), respectively. To discuss the wave envelop dynamics in this long-wavelength limit due to weak nonlinear and weak dispersive effects, we introduce the stretched variables 𝜉=𝜖1∕2 (𝑥−𝑧)and 𝜏=𝜖3∕2 𝑧, where 𝜉allows us to study the system on different, slowly, moving frames and by 𝜏, longer propagation distance 𝑧. Also, 𝜖is a measure of the deviation from the background 𝜓0. Using the perturbation expansions 𝜌(𝜉, 𝜏) = 𝜌0+ ∞ ∑ 𝑗=1 𝜖𝑗𝜌(𝑗)(𝜉, 𝜏),(5a) 𝜑(𝜉, 𝜏) = 𝜑0+ ∞ ∑ 𝑗=1 𝜖𝑗𝜑(𝑗)(𝜉, 𝜏),(5b) ℎ(𝜉, 𝜏) = ∞ ∑ 𝑗=0 𝜖𝑗+1∕2 ℎ(𝑗+1)(𝜉, 𝜏).(5c) https://doi.org/10.1016/j.rinp.2021.104536 Received 19 June 2021; Received in revised form 5 July 2021; Accepted 6 July 2021 Results in Physics 27 (2021) 104536 2 A. de la Cruz et al. where 𝜌0= 1,𝜑0=|𝜓0|2. Therefore we can expand the Lagrangian density for small amplitudes following the method in [10,11]. 𝐿=𝜖 𝐿(1) +𝜖2𝐿(2) +𝜖3𝐿(3) +(𝜖4).(6) For 𝜖: 𝐿(1) = −ℎ(1) 𝜉, from where no relevant information is obtained. For 𝜖2∶ 𝐿(2) =1 2ℎ(1)2 𝜉− 2𝜌(1)ℎ(1) 𝜉+𝜌(1)𝜑(1) +ℎ(1) 𝜏−ℎ(2) 𝜉−1 2𝜑(1)2(7) from where we have obtained the following expression as Euler– Lagrange equations 𝛿𝜌(1) ∶ℎ(1) 𝜉=𝜑(1),(8a) 𝛿𝜑(1) ∶𝜌(1) =𝜑(1),(8b) 𝛿ℎ(1) ∶𝜌(1) =ℎ(1) 𝜉,(8c) and the relation ℎ(2) 𝜉= −𝜌(1) ℎ(1) 𝜉.(9) The 𝜖3final Lagrangian is obtained with help of (8) and (9) as 𝐿(3) =1 2ℎ(1)3 𝜉−𝜌(2)ℎ(1) 𝜉+ 2ℎ(1) 𝜏ℎ(1) 𝜉+𝛾 8ℎ(1)2 𝜉𝜉 +ℎ(2) 𝜏(10) providing the condition ℎ(2) 𝜏= −𝜌(2) ℎ(1) 𝜉.(11) where 𝛾=(1−4𝜎2)is the optical analogue to surface tension [1]. Second-approximation terms ℎ(2) and 𝜌(2) could be obtained and studied [12] using the expressions (9)–(11). Assuming 𝑢=ℎ𝑥, the preceding equation yields, as its Euler– Lagrange equation, a KdV type [1,13] 𝑢𝜏+3 2𝑢 𝑢𝜉−𝛾 8𝑢𝜉𝜉𝜉 = 0.(12) The solution of (12) is given by 𝜌(𝜉, 𝜏)≡𝑢=𝑁 𝑠𝑒𝑐ℎ2[√𝑁 2𝛾(𝜉−𝑁 4𝜏)].(13) where 𝑁is the soliton amplitude. In original coordinates 𝑢(𝑧, 𝑥) = 𝑁 𝑠𝑒𝑐ℎ2{1 4√𝜖 𝑁 𝛾[𝑥−(1 + 𝜖𝑁 8)𝑧]},(14) and ℎ(𝑧, 𝑥)can be obtained readily from (8c), ℎ= − 4𝛾 𝜖√𝜖𝑁 𝛾tanh {1 4√𝜖𝑁 𝛾[𝑥−(1 + 𝜖𝑁 8)𝑧]}.(15) In the original (dimensionless) 𝑥and 𝑧, one may write down an approximate [up to order (𝜖)] solution for the macroscopic wavefunction 𝜓 𝜓=𝜓0√𝜌0+𝜖 𝜌1exp [−𝑖|𝜓0|2𝑧+𝑖 ℎ(𝑧, 𝑥)],(16) 𝜑=|𝜓0|2+𝜖 𝜑1,(17) where 𝜑1is written as (8b) and (14). Conclusions We have explored theoretically light propagation in a nonlocal nonlinear defocusing media through a proposed alternative simpler method, the asymptotic variational multiscale approach. The obtained KdV equation is similar to the one derived using reductive multiscale technique. Our results advance the understanding of nonlinear phenomena. CRediT authorship contribution statement Artorix de la Cruz: Conceptualization, Methodology, Software, Formal analysis, Investigation, Writing – original draft, Writing – review & editing, Visualization. Michael Cada: Project administration, Supervision. Jaromir Pistora: Supervision. Tamara Diaz-Chang: Review & editing. Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgments Artorix de la Cruz thanks the financial support from Killam Trust Predoctoral and Nova Scotia Research scholarships. Funding This work was supported by NSERC of Canada and by the IT4Innovations National Supercomputing Center - Path to exascale project (EF16-013/0001791) References [1] Horikis TP, Frantzeskakis DJ. Phys Rev Lett 2017;118:243903. [2] Triki H, Biswas A, Moshokoa SP, Belic M. Optik 2017;128:63. [3] Biswas H, Mirzazadeh m, Eslami M, Zhou Q, Bhrawy A, Belic M. Optik 2016;127:72450. [4] Zhou Q, Mirzazadeh M, Zerrad E, Biswas A, Belic M. J Modern Opt 2016;63:950. [5] Biswas A, Belic M. Optik 2018;171:217. [6] Zhang LW, Triki HY, et al. Nonlinear Dyn 2019;95:557–63. [7] Mirzazadeh M, Eslami M, Biswas Anjan. Optik 2014;125:6874. [8] Yildirim Y, Biswas A, Kara AH, Ekici M, Zayed EME, Alzahrani AK, Belic MR. J Opt 2020;49:580. [9] Asma M, Biswas A, Ekici M, Gonzalez-Gaxiola O, Alzahrani AK, Belic MR. Semicond Phys Quantum Electron Optoelectron 2021;24:64. [10] Infeld E, Rowlands G. Nonlinear waves, solitons and chaos. second ed.. Cambridge: Cambridge University Press; 2000. [11] Infeld E. Phys Rev B 1999;60:9302. [12] Karczewska A, Rozmej P, Infeld E. Phys. Rev. E. 2014;90:012907. [13] Baronio F, Wabnitz S, Kodama Yuji. Phys Rev Lett 2016;116:173901.