Chirped nonlinear resonant states in femtosecond fiber optics
Shailza Pathania, Amit Goyal, Thokala Soloman Raju, C. N. Kumar
aa r X i v : . [ n li n . PS ] N ov Chirped nonlinear resonant states in femtosecond fiberoptics
Shailza Pathania a , Amit Goyal b, ∗ , Thokala Soloman Raju c , C. N. Kumar a a Department of Physics, Panjab University, Chandigarh 160014, India b Department of Physics, GGDSD College, Chandigarh 160030, India c Indian Institute of Science Education and Research (IISER) Tirupati, Andhra Pradesh517507, India
Abstract
We show the existence of nonlinear resonant states in a higher-order nonlin-ear Schr¨odinger model that appertains to the wave propagation in femtosecondfiber optics, under certain parametric regime. These nonlinear resonant statesare analytically illustrated in terms of Gaussian beams, Airy beams, and peri-odic beams that resulted due to the presence of quadratic, linear, and constanttype of ‘smart’ potentials, respectively, of the ensuing model. Interestingly, thenonlinear chirp associated with each of these novel resonant states can be effi-ciently controlled, by varying the self-steepening term and self-frequency shift.Furthermore, we have conducted numerical experiments corroborative of ouranalytical predictions.
Keywords:
Gaussian and Airy beams, Frequency chirp, Higher-ordernonlinear Schr¨odinger equation, Smart potential
1. Introduction
The development of all-optical soliton transmission systems is consideredto be one of the hottest technologies of 21st century. In 1973, Hasegawa and ∗ Corresponding author
Email address: [email protected] (Amit Goyal)
Preprint submitted to Journal of L A TEX Templates December 1, 2020 appert [1] theoretically predicted optical solitons in fibers. Later in 1980, Mol-lenaeur, Stolen and Gordon from the Bell Telephone Laboratories confirmedthe existence of such phenomena experimentally [2]. It is a well-known factthat the optical solitons owe their existence due to the delicate balance betweendispersion and self-phase modulation (SPM) [1, 3] . Owing to the prevalenceof different phase sensitive nonlinear processes, only few nonlinear effects thatmay arise from the nonlinear susceptibility χ (3) will be present in the nonlinearfibers. Among them will be self-steepening effect and stimulated Raman scat-tering (SRS) effect, because of χ (3) in ultrashort pulses [3]. Additionally, theseultrashort pulses will also be subjected to group-velocity dispersion and thirdorder dispersion (TOD). Then the model equation that describes the pulse prop-agation through optical waveguide effectively, is the celebrated higher-order non-linear Schr¨odinger (HNLS) equation. Kodama and Hasegawa, obtained HNLSequation, for the first time, that includes higher order effects such as TOD,self-steepening, and SRS [4, 5]. In 1985, Dianov et al. [6] discovered the phe-nomena of Raman-induced frequency shift. A significant work has been doneon the analysis of HNLS equation by Potasek [7] and Serkin [8]. Grudinin etal. [9] report the first experimental study of the dynamics of the femtosecondpulses in optical fibers. The HNLS equation has been shown to support bright/ dark soliton [10] and rogue wave solutions [11] under different parametricconditions. Recently, dipole solitons [12], periodic soliton interactions [13] andself-similar solitons [14] has been studied in the presence of higher-order effects.For achieving ultra-high speed in long-haul telecommunication networks, soli-tons are transmitted at a high-pulse-repetition rate. In view of this, it is highlydemanded that the higher-order effects may be retained in the HNLS equationfor femtosecond pulse propagation.For femtosecond pulse propagation, TOD plays pivotal role in a situationwhen the value of GVD is near to zero. But for pulses whose width is about100fs, power of the order of 1 Watt, and GVD bit away from zero, the effectdue to TOD can be neglected [15]. Despite this, the significant role played byself-steepening and self-frequency shift terms can not be simply undermined in2ny way, and they should be retained in the model under consideration. Thus,keeping all these effects, the HNLS model equation in dimensionless units takesthe form iQ z + 12 Q xx + | Q | Q + iǫ [ AQ xxx + B ( | Q | Q ) x + CQ ( | Q | ) x ] − V ( x, z ) Q = 0 . (1)In Eq. (1), Q ( z, x ) indicates the complex envelope of the electric field, ǫ indicates perturbation in which A refers to TOD, B refers to self-steepeningand C signifies self-frequency shift. And V ( z, x ) signifies the ‘smart’ potential[16].We know that the cubic nonlinear Schr¨odinger equation (NLSE) or Gross-Pitaevskii eqution (GPE) with appropriate trapping potential aptly describesthe dynamics of dilute-gas Bose–Einstein condensate (BEC) [17]. The fact thatdifferent traps used to arrest BEC has paved the way for finding new solutionsof NLSE or GPE with new potentials [18–24]. We emphasize here that, ourmotivation to consider these resonant states in the presence of ‘smart’ potentialsessentially, stems from this fact.Now-a-days, there is renewed interest in studying the chirped pulses, as theyfind useful applications [25–28]. In particular, Hmurcik and Kaup [29] studiedlinearly chirped pulses with a hyperbolic-secant-amplitude profile, numerically.Following this, many authors have published their works showing the existenceof chirped soliton-like solutions [30–35]. Recently, the propagation of chirpedoptical pulses has been studied in the context of nonlinear metamaterials [36].One of the present authors solved Eq. (1) in the absence of ‘smart’ potential andobtained soliton-like solutions with nonlinear chirp [37]. In the present work,we show the existence of nonlinear resonant states in a higher-order nonlin-ear Schr¨odinger model that appertains to the wave propagation in femtosecondfiber optics, under certain parametric regime. These nonlinear resonant statesare analytically illustrated in terms of Gaussian beams, Airy beams, and peri-odic beams that resulted due to the presence of quadratic, linear, and constanttype of ‘smart’ potentials, of the ensuing model. Furthermore, we have con-ducted numerical experiments corroborative of our analytical predictions. In3ur context, these resonant states appear to be the stationary states in an opti-cal waveguide that exhibit perfect transmission, akin to the appropriate scenarioin Bose-Einstein condensates [38].
2. Chirped resonant states
For the purpose of obtaining chirped resonant states, we begin our analysisby assuming solution to Eq. (1) in the form of Q ( x, z ) = ρ ( ξ ) e i ( ψ ( ξ ) − ωz ) , (2)where ξ = ( x − vz ) , v , ω are real parameters, ψ ( ξ ) is the phase function and ρ ( ξ ) is the amplitude function. Substituting Eq. (2) into Eq. (1) and separatingthe real and imaginary parts, we obtain vψ ′ ρ + ωρ + 12 ( ρ ′′ − ψ ′ ρ ) + ρ + ǫ ( − Aψ ′′ ρ ′ − Aψ ′ ρ ′′ − Aψ ′′′ ρ + Aψ ′ ρ − Bρ ψ ′ ) − V ( x, z ) ρ = 0 , (3) − vρ ′ + ρ ′ ψ ′ + 12 ψ ′′ ρ + ǫ ( Aρ ′′′ − Aψ ′ ψ ′′ ρ − Aψ ′ ρ ′ + (2 C + B ) ρ ρ ′ ) = 0 . (4)For the femtosecond pulses, far away from zero GVD, we put A = 0 in Eq. (1).After integrating Eq. (4), we have an expression for ψ ′ written as ψ ′ = Iρ ( ξ ) + v + αρ ( ξ ) , (5)where I is a constant of integration and α = − ǫ (2 C + B ) . Also we put I = 0,thus ensuring that the phase does not diverge. By substituting the expressionof ψ ′ into Eq. (3), we obtain ρ ′′ + ( v + 2 ω − V ( ξ )) ρ + 2(1 − Bǫv ) ρ − ǫ C + B )(2 C − B ) ρ = 0 . (6)The frequency change across the pulse at any distance z is known as frequencychirp which is given by δω ( z, x ) = − ∂∂x [ ψ ( ξ ) − ωz ] = − ψ ′ ( ξ ). Here, the fre-quency chirp depends considerably on the exact pulse shape through the rela-tion δω ( z, x ) = − ψ ′ ( ξ ) = − ( v + αρ ( ξ )), where v and α are the constant and4onlinear chirp parameters, respectively. We note here that the parameter ‘ α ’depends on the model coefficients—self-steepening ‘ B ’ and self-frequency shift‘ C ’. Thus, the frequency chirp can be efficiently controlled by varying these co-efficients. In Eq. (5), for 2 C = − B , α comes out to be zero and correspondingsolutions of Eq. (6) have trivial phase or also known as unchirped solutions.But, in this work, we report the chirped solutions with non-trivial phase mod-ulation for α = 0. Now, for 2 C = 3 B and B = ǫv , Eq. (6) reduces to the formexpressed as ρ ′′ − V ( ξ ) ρ + (2 ω + v ) ρ = 0 . (7)For different choices of ‘smart’ potentials V ( ξ )—quadratic, linear and constantpotential, Eq. (7) yields different types of solutions such as Gaussian, Airyand periodic solutions, respectively. Interestingly, Eq. (7) can be observed asquantum mechanical Schr¨odinger equation, by identifying V ( ξ ) as potential and(2 ω n + v ) as energy eigenvalue for corresponding ρ n . Like quantum mechanicalSchr¨odinger equation, for different value of n , various solutions can be generatedfor Eq. (7). In order to obtain the resonant Gaussian beams as exact solutions of thismodel, we take the quadratic potential, V ( ξ ) = k ξ where k is a real positiveconstant. In this case, Eq. (7) becomes the quantum Schr¨odinger equation forharmonic oscillator and the well-known solutions are given in terms of Hermitepolynomials. Thus, we have a class of solutions ρ n for Eq. (7), given as ρ n ( ξ ) = s (2 k ) n n ! π e − q k ξ H n h (2 k ) ξ i , (8)for different values of n and such that ω n = (2 n + 1) r k − v . (9)In Table 1, we have shown the expressions for ρ n ( ξ ) and corresponding ω n ,for n = 0 , able 1: Expression of ρ n and ω n for different values of n n ρ n ω n n = 0 (2 k ) π − e − q k ξ q k − v n = 1 √ k ) π − ξ e − q k ξ q k − v n = 6 (2 k ) π − √ (16 √ k ξ − k ξ + 90 √ k ξ − e − q k ξ q k − v - - Ξ Q ¤ (a) - - Ξ Q ¤ (b) - - Ξ Q ¤ (c)Figure 1: Intensity profile of chirped resonant Gaussian beams for (a) n = 0, (b) n = 1 and(c) n = 6. The other parameter used is k = 1. Hermite modes of fixed velocity. The complete complex wave solution for Eq.61) can be written as Q n ( x, z ) = s (2 k ) n n ! π e − q k ξ H n h (2 k ) ξ i e i ( ψ ( ξ ) − ωz ) , (10)where phase profile ψ ( ξ ) is given as ψ ( ξ ) = vξ − erf (cid:2) (2 k ) / ξ (cid:3) v . (11)Here erf( x ) is known as error function. We have depicted the intensity distri-bution, | Q ( x, z ) | , of Gaussian beams for n = 0, n = 1 and n = 6, respectively,with k = 1 in Fig. 1. One can observe from these plots that there is change inthe intensity and the number of peaks for different modes. For ω = − v , Eq.(7) reduces to ρ ′′ − V ( ξ ) ρ = 0 . (12)For the choice of linear potential, V ( ξ ) = k ξ where k is a real positive constant,Eq. (12) turns out to be Airy differential equation or Stokes equation, and itssolutions can be expressed as [39] ρ ( ξ ) = Ai h (2 k ) ξ i , (13)where Ai( x ) is the Airy function. The complex wave solution for Eq. (1) reads Q ( x, z ) = Ai h (2 k ) ξ i e i ( ψ ( ξ ) − ωz ) , (14)and the phase profile is given as ψ ( ξ ) = vξ − ξv Ai h (2 k ) / ξ i + 2 / k / v Ai ′ h (2 k ) / ξ i , (15)where Ai ′ denotes the derivative of the Airy function. We have depicted theintensity distribution of Airy beams for different values of parameter k , inFig. 2. One can observe from these plots that the number of resonant modesincrease as we increase the value of the homogeneous parameter k of the ‘smart’potential. 7 - - - Ξ Q ¤ (a) - - - - Ξ Q ¤ (b)Figure 2: Intensity profile of Airy beams for different values of parameter k , (a) k = 1 and(b) k = 5. For constant potential V ( ξ ) = k , where k is a real constant, Eq. (7)reduces to a homogeneous second order differential equation ρ ′′ + δρ = 0 , (16)where δ = 2 ω + v − k . Eq. (16) possesses periodic solution given by ρ ( ξ ) = P cos( √ δξ ) + Q sin( √ δξ ) , (17)where P , Q are arbitrary constants and δ should be greater than zero whichimposes a constraint condition on the wave parameter ω , ω > k − v . Wewould like to emphasize here that the periodic solutions are exact solutions ofHNLS equation, unlike the canonical free NLSE, with parametric restrictions.The complex wave solution for Eq. (1) can be written as Q ( x, z ) = (cid:16) P cos( √ δξ ) + Q sin( √ δξ ) (cid:17) e i ( ψ ( ξ ) − ωz ) , (18)and the corresponding phase profile can be obtained integrating Eq. (5). InFig. 3, we depicted the intensity profile of periodic beams for different valueof parameter k . It is to be noted that for k = 0, Eq. (1) returns to the8 - Ξ Q ¤ (a) - - Ξ Q ¤ (b) - - Ξ Q ¤ (c)Figure 3: Intensity profile of periodic beams for different values of k (a) k = 0, (b) k = 6and (c) k = 8. The value of other parameters used are P = 1, Q = 1, v = 2 . ω = 5. standard HNLS equation which has been considered earlier [37]. Like Airybeams, maximum intensity of the periodic beams is also same for different valuesof free parameter and it only affects the frequency of beams.
3. Numerical simulations
In order to corroborate our analytical results with the numerical simulations,we have numerically solved Eq. (1), with a constraint that 2 C = 3 B and inthe presence of quadratic ‘smart’ potential, using split-step Fourier method.The initial condition has been taken as the first excited resonant state. As9videnced from the surface plot (Fig. 4), the numerical evolution of the firstexcited resonant state almost complements our analytical predictions. Figure 4: Plot depicting the numerical evolution of first excited resonant state for the quadratic‘smart’ potential.
4. Conclusion
In conclusion, we have elucidated the mechanism to generate resonant statesin a higher-order NLSE that appertains to the wave propagation in femtosecondfiber optics, under certain parametric regime. We report the existence of Gaus-sian beams, Airy beams, and periodic beams that resulted due to the presenceof quadratic, linear, and constant type of ‘smart’ potentials, respectively. It isobserved that the free parameter in smart potentials imposes significant effectson the intensity of optical beams. Interestingly, the chirping associated withthese optical beams can be efficiently controlled through self-steepening andself-frequency shift parameters. We have performed numerical simulations cor-roborative of the analytical results, using split-step Fourier method, and foundthat both analytical and numerical results are nearly complement to each other.
5. Future work
Recently, our group has elaborated a theoretical method, relying on isospec-tral deformation of Hamiltonian in supersymmetric quantum mechanics [40],10o modulate the dynamics of self-similar waves in inhomogeneous graded-indexwaveguide [41, 42]. This approach helps to construct a one-parameter depen-dent family of potentials and corresponding expression of wave functions for agiven potential. Before the advent of supersymmetric quantum mechanics, thisapproach was first introduced by Infeld and Hull [43] and Mielnik [44], and wasfound to be useful in various physical systems [45, 46]. Here, Eq. (7) can bemapped to Schr¨odinger equation of quantum mechanics which enables one togenerate a class of dynamic potentials by invoking the concept of isospectralHamiltonian approach. This way one can control the dynamical behavior ofGaussian beams. Study of dynamics of resonant states and their control usingisospectral Hamiltonian approach can be well illustrated for an another inter-esting case of ‘ n ( n + 1) sech ξ ’ ‘smart’ potential, which is analytically tractable.This work is presently under progress and will be reported elsewhere.
6. Acknowledgment
S.P. would like to thank DST Inspire, India, for financial support throughJunior Research Fellow [IF170725]. A.G. gratefully acknowledges Science andEngineering Research Board (SERB), Government of India for the award ofSERB Start-Up Research Grant (Young Scientists), under the sanction no:YSS/2015/001803, during the course of this work.