Hidden possibilities in controlling optical soliton in fiber guided doped resonant medium
aa r X i v : . [ n li n . S I] J un Hidden possibilities in controlling optical soliton in fiber guided dopedresonant medium
Anjan KunduTheory Group & CAMCS, Saha Institute of Nuclear Physics, Calcutta, INDIAOctober 2, 2018
Fiber guided optical signal propagating in a Erbium doped nonlinear resonant medium is known to producecleaner solitonic pulse, described by the self induced transparency (SIT) coupled to nonlinear Schr¨odingerequation. We discover two new possibilities hidden in its integrable structure, for amplification and control ofthe optical pulse. Using the variable soliton width permitted by the integrability of this model, the broadeningpulse can be regulated by adjusting the initial population inversion of the dopant atoms. The effect canbe enhanced by another innovative application of its constrained integrable hierarchy, proposing a system ofmultiple SIT media. These theoretical predictions are workable analytically in details, correcting a well knownresult.
I. INTRODUCTION
Optical communication through fiber has achieved phenomenal development over the last twodecades OptCom¸ . Dissipation and dispersion in the media, which are the main hindrances in signaltransmission, are usually attempted to be solved by the dispersion management techniques and de-vices OptCom,dismang¸ . On the other hand, in soliton based optical communication, mediated by thenonlinear Schr¨odinger (NLS) equation proposed much earlier nlssolit¸ , the group velocity dispersioncan be countered by the self phase modulation in the nonlinear fiber medium agarwal¸ . However, theexperiments revealed insufficiency of the model for its efficient practical application nlsdrab¸ . Anotherproposal with improved solitonic transmission was due to the self-induced transparency (SIT), pro-duced by the coherent response of the medium to an ultra short optical pulse SIT,lamb¸ . Finally,the benefits of both the NLS and the SIT systems, were combined in a coupled NLS-SIT modelMaimitsov¸ , by transmitting the optical soliton through an Erbium doped nonlinear resonant mediumnakazawa1,nakazawa2¸ .However the solitonic communication, in spite of its favorable features and theoretical advantagesdue to its underlying integrability, did not receive the needed response. Our aim here therefore, isto revisit the NLS-SIT model for exploring new possibilities hidden in its integrable structures anduse them for the control and amplification of the solitonic pulse. Though the soliton usually moveswith a constant velocity or speed, the integrable property of this coupled NLS-SIT system, as we findhere, allows the soliton speed to be a tunable function. And since the soliton width in this case isrelated to its speed, which in turn is linked here to the initial population inversion, the pulse width canbe regulated by manipulating the population inversion profile of the dopant atoms. This controllingeffect can be further enhanced by exploring another specialty of this integrable system, namely itsconstrained integrable hierarchy, through a novel use of coupled multiple SIT media, in place of the1onventional single doped medium. The details can be worked out exactly due to the integrability ofthe model, detecting the limitation of a well known result on NLS-SIT soliton nakazawa1,nakazawa2¸ .Propagation of a stable optical pulse through a fiber medium, serving as a dispersive and nonlin-ear wave guide with Kerr nonlinearity nlssolit¸ , can be described by the optical electric field E ( z, t )satisfying the well known NLS equation iE z − E tt − | E | E = 0 , (1)with space and time variables being interchanged as customary in nonlinear optics agarwal¸ . On theother hand, ultra short optical pulse producing SIT in the medium can be described by the Maxwell-Bloch equation SIT,lamb¸ iE z = 2 p, ip t = 2 N E, iN t = − ( Ep ∗ − E ∗ p ) , (2)with the induced polarization p and the population inversion N of the medium, contributed by theBloch equation. Fascinatingly, it is possible to combine these two effects by transmitting the stablenonlinear pulse produced in the fiber wave guide through a doped medium with coherent response,governed by a coupled NLS-SIT system given by a deformed NLS equation iE z − E tt − | E | E = 2 p , (3)together with the SIT equations ip t = 2( N E − w p ) , iN t = − ( Ep ∗ − E ∗ p ) , (4)representing a nonholonomic constraint kundu09¸ . In (4) , p = ν ˜ ν ∗ is the induced polarization and N = | ˜ ν | − | ν | , − ≤ N ≤ , is the population inversion of the two level dopant atoms with nor-malized wave functions ν, ˜ ν for the ground and the excited states, respectively and w is the naturalfrequency of these resonant ions. Assuming a homogeneous broadening of the frequency spread witha sharp resonance at ∆ w = w − w , we have taken the symmetric distribution g (∆ w ) = δ (∆ w ) andreplaced the average value < p > = R dwg (∆ w ) p ( z, t, w ) appearing in (3) by p = p ( z, t, w ) andnormalized the coupling constants to ensure the integrability of the model. II. INTEGRABILITY AND SOLITON SOLUTION
Recall that, an integrable nonlinear equation may be associated with a linear system Φ t = U Φ , Φ z = V Φ , defined through a Lax pair U ( λ ) , V ( λ ), which are matrices with their elements depend-ing on the basic fields and a parameter λ , known as the spectral parameter. Through compatibility ofthe linear Lax equations, inducing flatness condition: U z − V t + [ U, V ] = 0 , the Lax pair yield the givennonlinear equation and at the same time can be used to extract its exact solutions through the inversescattering method (ISM) soliton¸ . We remind again that, the space z and time t are interchanged herein the context of nonlinear optics. It is noteworthy that, while the set of coupled NLS-SIT equations(3-4) generalize both the NLS (1) and the SIT (2) equations, the associated Lax pair U nls : sit , V nls : sit contain these subsystems as the constituent parts: U nls : sit ( λ ) = U nls ( λ ) = U sit ( λ ) , V nls : sit ( λ ) = V nls ( λ ) + V sit ( λ ) (5)2here U nls , V nls and U sit , V sit are the Lax pairs related to the NLS and the SIT equations, respectively.The NLS Lax pair is well known as soliton¸ U nls ( λ ) = i ( σ λ + U (0) ) , U (0) = Eσ + + E ∗ σ − (6) V nls ( λ ) = V (0) + V (1) λ + V (2) λ ,V (0) = ( U (0) x − i ( U (0) ) ) σ , V (1) = 2 iU (0) , V (2) = 2 iσ , (7)where σ , σ ± = ( σ ± iσ ) are the 2 × time -Lax operator as that of the NLS: U sit ( λ ) = U nls ( λ ) , while the space -Lax operator V sit ( λ ) = i ( λ − w ) − G , G = N σ + p σ + + p ∗ σ − , (8)can be linked to the nonholonomic deformation kundu09¸ . We check easily that, the flatness conditionof (6,7) yields the NLS equation (1), while (6,8) the SIT equation (2) and similarly, the Lax pair (5)would yield the coupled NLS-SIT equations (3,4)The time -Lax operator U ( λ ) plays the central role in the ISM for finding the exact solutions of thenonlinear equation soliton¸ and therefore, since U ( λ ) is the same for NLS (1), SIT (2) and NLS-SIT(3,4) equations as seen from (5), the form of soliton solutions and the ISM procedure are remarkablysimilar for all the three equations. We therefore present soliton solutions for all of them in an unifiedway following the ISM soliton¸ , which though an involved method, gives the 1-soliton solution in anamazingly simple form: E = c g | g | , g = exp[2 i (u ∞ t + ˜v ∞ z + φ )] , (9)with c, φ = constants. Note that, the crucial elements u ∞ , ˜v ∞ = z R z dz v ∞ in (9), though linked tothe Lax pair of the given system, need information only about their asymptotic properties: u ∞ = σ U ( λ ) | t = −∞ , v ∞ = σ V ( λ ) | t = −∞ , at discrete spectral parameter λ . Therefore, fixing the initialcondition of the basic fields involved in the NLS-SIT equations as E ( z, t = −∞ ) → , p ( z, t = −∞ ) → , N ( z, t = −∞ ) = N ( z ) (10)with an arbitrary function N ( z ) , we can easily derive from the Lax pair (5-8): σ u ∞ nls : sit = U nls : sit ( λ ) | t = −∞ = U nls ( λ ) | t = −∞ = U sit ( λ ) | t = −∞ = 2 iσ λ ,σ v ∞ nls : sit = V nls : sit ( λ ) | t = −∞ = ( V nls ( λ ) + V nls ( λ )) | t = −∞ = iσ (2 λ + ( λ − w ) − N ( z )) , (11)which at the discrete spectral parameter with complex value: λ = k + iη, take the explicit formu ∞ nls : sit = iλ , v ∞ nls : sit = v ∞ nls + v ∞ sit , v ∞ nls = 2 iλ , v ∞ sit = i ( λ − w ) − N ( z ) . (12)Inserting the needed complex valued expressions (12) in (9) and grouping its real (Re) and imaginary(Im) parts we get the 1-soliton solution for the optical field in NLS-SIT equations (3,4) in the familiar sech - form E = − iη sech2 ζe iθ , ζ = η ( t − t − vz ) , θ = ωz + kt + φ , (13)where t and φ are constant time and phase shift. Inverse speed v and phase rotation ω for theNLS-SIT soliton (13) are given by the superposition v = v nls + v sit , ω = ω nls + ω sit , (14)3f the corresponding parameters from the NLS and the SIT subsystems, derived from (12) using λ = k + iη as v nls = − η Im [v ∞ nls ] = − k, ω nls = Re [˜v ∞ nls ] = 2( k − η ) , (15) v sit = − η Im [˜v ∞ sit ] = 1 ρ f ( z ) , ω sit = Re [˜v ∞ sit ] = ˜ kρ f ( z ) , (16)with ˜ k = k − w , ρ = ˜ k + η , and f ( z ) = z R z N ( z ′ ) dz ′ .It is intriguing to note that, since the z-evolution of the optical field E in the NLS-SIT modelfollows the superposition rule (14) contributed separately by the NLS and the SIT parts, the term iE z in equation (3), evolving according to solution (13), breaks up into two parts: one follows the NLScontribution with parameters (15) and satisfies the pure NLS part of the equation in the left handside , while the other part equates to the SIT deformation 2 p in the right hand side of (3) involvingthe related parameters (16). Using this dynamics we derive the soliton solution for the dipole p from(13), in the form p = ηρ N sech2 ζ ( i ˜ k − η tanh2 ζ ) e iθ , (17)with ζ, θ as expressed in (13). Inserting solutions (13,17) for E and p in (4) and integrating by t wederive further the solution for population inversion N = N (1 − η ρ sech ζ ) , (18)again in the solitonic form with arbitrary function N ( z ) = N ( t → −∞ ), adjusted by the integration constant . We obtain thus the complete set of exact soliton solutions to the NLS-SIT equations (3-4) as (13) for the optical field E , (17) for the dipole p and (18) for the population inversion N .A beautiful interaction pattern can be noticed in these solutions, manifested in the superpositionrelations: v = v nls + v sit , ω = ω nls + ω sit , for the solitonic parametersappearing in (13,14). Intriguingly,in the absence of the SIT system with p = N = 0, when the coupled NLS-SIT equations reduce tothe NLS equation (1) for the field E , one recovers from (13) the well known NLS soliton by simplyputting v sit = ω sit = 0 due to the vanishing of (8). Therefore, the NLS soliton takes exactly the sameform as (13), though the parameters are reduced to pure NLS case: v = v nls , ω = ω nls . Similarly,we can directly get the soliton solution for the pure SIT equations (2) in the same form (13,17,18),but with soliton parameters reducing to v = v sit , ω = ω sit , due to switching off the NLS influence: v nls = ω nls = 0. Thus our exact NLS-SIT soliton can reproduce the solutions for both the NLS andthe SIT equations in a unified way, consistent with the ISM. However this rich interaction pictureseems to have been missed in a well known earlier work nakazawa1,nakazawa2¸ , leading to wrongconclusions in the general case. In particular, the soliton solution for the NLS-SIT equation presentedin nakazawa1,nakazawa2¸ gives the expression for the pulse delay as δ = nc (1 + γ ) ((4.9) in nakazawa2¸ ),which is equivalent to the inverse soliton speed for the SIT ((2.22) in nakazawa2¸ ), i.e. v ≡ δ = v sit ,in our notation solNLSsit¸ . Similarly, the phase rotation in nakazawa1,nakazawa2¸ is given as α = 2 η ,meaning ω ≡ α = − ω nls , (at k = 0, see (15)) in our notation solNLSsit¸ . Both these results for thecoupled NLS-SIT equations appear to be incomplete, when compared with our exact result (14). Itis clear that, the solution of nakazawa1,nakazawa2¸ can be justified only in a very limited sense, when v nls = ω sit = 0 and therefore unlike our soliton solution can not interpolate between the solutions ofthe NLS and the SIT equations. 4his partial result unfortunately led to wrong conclusions, for the NLS-SIT system in general, stat-ing that (sect. IV nakazawa2¸ ), the normalized speed (i.e. δ − ) of the NLS-SIT soliton is determinedonly by the SIT effect and similarly, the z dependence of the phase of the dipole (i.e. α ) is determinedsolely by the nonlinear phase change due to the NLS soliton . Our exact solutions for the optical field E (13) and the dipole p (17) with correct expressions (14), conclude on the other hand that, only a part(i.e. v sit ) in the normalized speed v − = ( v nls + v sit ) − of the NLS-SIT soliton is determined by the SITeffect, while there is an additional contribution coming from the NLS part v nls . Similarly, the z depen-dence of the phase of the dipole and the input optical field gets contribution from both the NLS andthe SIT parts as ω = ω nls + ω sit , consistent with the interaction picture in the coupled NLS-SIT system. III. CONTROLLING OPTICAL SOLITON EXPLOITING INTEGRABLE STRUC-TURES
Based on the integrable structures underlying the NLS-SIT system describing the propagationof optical soliton in fiber guided doped medium, we propose two possible ways for controlling theamplitude and width of the optical pulses.
A. Soliton control by regulating initial population inversion
It is commonly believed that, the exact soliton solution of a homogeneous equation always moves witha constant speed, width and frequency, as in the case of the NLS soliton (15) with constant values for v nls , ω nls . However, it is crucial to note that, for the NLS-SIT soliton the parameters ( v, ω ), as evidentfrom (14,16) can become variable functions, depending on the initial population inversion N ( z ) (10).Due to this peculiarity of integrable structure of the NLS-SIT system, hidden in the expressions like(5,8,12,16), the soliton speed : v − and width : ( ηv ) − , as defined from the soliton argument ζ (13), canbe variable and linked to a controllable arbitrary function N ( z ).We show that, this important observation embedded in the integrability of the NLS-SIT systemcan open up a new avenue for controlling the optical soliton propagating through the doped medium,by regulating its initial population inversion profile N ( z ). This fact however remained unexploredin earlier investigations nakazawa1,nakazawa2,kakei,porsezian¸ , due to the restriction to a fixed initialprofile N ( z ) = −
1. Note that, at this particular value giving f ( z ) = −
1, our more general result (16)reduces to the simplified expressions obtained earlier: v sit = − ρ , ω sit = − ˜ kρ , (19)The choice for the initial population inversion in the NLS-SIT model as an arbitrary function N ( z ) > −
1, that we propose here, gives us the needed freedom for obtaining the excited and theground state occupancies at the initial moment as | ˜ ν | = (1 + N ( z )) and | ν | = (1 − N ( z )),respectively. Therefore, for N > −
1, giving | ˜ ν | >
0, we can prepare the dopant atoms initially inan excited state by optical prepumping, resulting to the creation of a laser-active amplifying mediumwith its intensity determined by N . Note that, only in such a case when more active dopant atomsare in the excited state, the optical soliton can gain net energy OptCom¸ .In addition to the soliton pulse amplification, variable initial profile N ( z ) > −
1, permitted by theintegrability of the NLS-SIT system, can play a crucial role in controlling the shape and dynamics of theoptical soliton. It is possible, as we see below, to address the important problem of pulse broadeningby regulating the initial profile of the dopant atoms. For example, a solitonic pulse governed by theNLS equation under small perturbation by a term − i Γ2 E with Γ <<
1, would suffer broadening5y a factor (4 kη ( z )) − , which can be worked out through the variational perturbation method as η ( z ) = η e − Γ z agarwal¸ , which is valid however upto the range Γ z ≈ z . Beyond this range with z >>
1, as shown by some other method, the broadening of the pulse width follows a different rule,by increasing linearly with z at a rate slower than the linear medium agarwal¸ . Though an attenuationwith intensity loss would also occur simultaneously, the broadening leads to more serious problem ofinformation loss and bandwidth limitation. Therefore we concentrate here only on the broadeningproblem of the perturbed NLS soliton, due to the increasing solitonic width kη e Γ z along z , as shownin Fig 1. As stated above for z >> N ( z ). Fig 2a shows this controlling effect, where the broadening of the solitonic pulse suffered in Fig1, is countered by the narrowing of the pulse due to variable width V ( z ) = ( v nls + N ( z ) / ( ρ Γ)) − , bytaking N ( z ) ∼ η ( z ) − . Note that the profile N ( z ) has to be adjusted differently at different ranges,as mentioned above, to control the broadening in the respective regions for a wide range of z . Thesoliton dynamics would also change to a variable speed V ( z ), possible due to the energy supplied byoptical prepumping.This potential opportunity for controlling the pulse width, hidden in the integrable property of theNLS-SIT system, as explained above, was missed in earlier investigations nakazawa1,nakazawa2,kakei,porsezian¸ ,since the initial atoms are usually taken in their ground state: | ν | = 1 , | ˜ ν | = 0 , by restricting to N ( z ) = − | E ( z, t ) | along the fiber, moving with a constantspeed with parameter choice k = 0 . , η = 0 . , Γ = 0 . B. Enhanced soliton control through multiple doping
Another promising opportunity in managing optical soliton in fiber communication, emerging alsofrom the integrability of the NLS-SIT model, is overlooked completely in earlier investigations. Thisis the proposal of enhancing the effect of amplification and control of the optical soliton by replacingthe conventional single SIT system, the only case considered in the literature, by a coupled multipleSIT system, using recursively the constrained integrable hierarchy in the NLS-SIT model (see Fig. 3).The physical meaning of coupling the NLS equation to such multi SIT system can be given througha novel proposal of using coupled multiple doped resonant media, in place of a single doped medium.For generating the governing hierarchal equations and showing their integrability, we extend Lax6 a) (b)
Figure 2: a) Broadening NLS soliton pulse is controlled by a coupled NLS-SIT system with N ( z ) = − . e Γ z and w = 0 .
3. Variable speed of the soliton is evident from its bending in the ( z, t )- plane.b) Additional control is achieved by coupling to a second SIT system with N (2)0 ( z ) = 0 . e Γ z , showingan efficient restoration of the soliton width.operator V ( λ ) (5,8) by adding more deforming terms V sitM = i P Mj ( λ − w ) − j G j , linked to the M -thconstrained hierarchy kundu09¸ in the NLS-SIT system, fixed at level M from the possible infinitesequence : j = 1 , , . . . . In analogy with G (8) we can express the deforming matrices G j , throughdipole moment p j and population inversion N j of the j -th doped resonant medium. For explicitdemonstration we restrict to the next higher level M = 2 in the constrained hierarchy, by consideringonly an additional SIT system to the original NLS-SIT set. Compatibility of the Lax pair thus definedwould generate an extended set of equations given by the same deformed NLS (3) coupled however toa double SIT system ip t = 2( N E − w p − p ) , iN t = − ( Ep ∗ − E ∗ p ) ,ip t = 2( N E − w p ) , iN t = − ( Ep ∗ − E ∗ p ) , (20)with induced polarization p and population inversion N , linked to the additional doped mediumdescribed by the second SIT system. We find intriguingly that, the exact soliton solution for theoptical pulse E in this extended NLS-SIT model (3,20), can be expressed again in the same form (13),where the soliton parameters are to be modified with contributions from all its interacting parts, i.e.from the NLS as well as from the multiple SIT system as v = v nls + v sit + v sit , ω = ω nls + ω sit + ω sit . Parameters v nls , ω nls and v sit , ω sit have the same expressions as found already in (15,16), while theadditional SIT contribution v sit , ω sit , can be derived following a similar argument as (16) in the form v sit = − η Im [( λ − w ) − ] f ( z ) = 2 ˜ kρ f ( z ) , sit = Re [( λ − w ) − ] f ( z ) = 2 ˜ k − η ρ f ( z ) , (21)with f ( z ) = z R z N (2)0 ( z ′ ) dz ′ , involving an additional arbitrary function N (2)0 ( z ) = N ( z, t = −∞ ) . Itopens up therefore another novel way, hidden again in the integrable structure of the NLS-SIT system,for an enhanced control of the soliton width and dynamics, by adding a coupled second SIT system,as shown in Fig. 2b.This process of coupling the NLS equation to the set of multiple SIT equations can be continuedwithin the framework of the integrable system, as mentioned above, creating a form of directionalconnected network with feedback, as shown in Fig. 3. In particular, as evident from the coupledequations (3,20), the input optical pulse E would influence the dipole field p j and the populationinversion N j in all the resonant SIT media with j = 1 , , . . . , M , while only p from the first mediumgives feed back to the field E . On the other hand, p j +1 are coupled sequentially to p j , across the media,while N j are mutually coupled only with p j from the same medium, in the multiple SIT system with j ∈ [1 , M ]. This network, would exhibit more and more manipulative power for control over widthand amplification of the optical pulse, enhanced sequentially by choosing a set of initial condition N ( j )0 ( z ) = N j ( z, t = −∞ ) , j = 1 , , . . . M and is based on the notion of constrained hierarchy of theintegrable NLS-SIT system (see Fig. 3).Figure 3: Connected network of the NLS and the multiple SIT system with E as the input opticalfield, p j as the induced polarization and N j as the population inversion of the j -th doped resonantmedium with j = 1 , , , . . . , M . The arrows show the directions of coupling with equations (3,20)describing this network in the particular case of M = 2. Sequential enhancement of the controlof width and amplification are predicted by this network, which is consistent with the constrainedintegrable hierarchy of the NLS-SIT system.This theoretical prediction, as presented schematically in Fig 3, is an experimental challenge toincorporate the contribution of coupled multiple SIT systems. Repeating the idea of available exper-imental realization of single doped fiber medium, either to a series of doped media coupled throughinduced polarization, or to multiple doping with parallel coupling in a single medium, such experi-8ental set up is likely to be organized. IV. CONCLUDING REMARKS
Exploring the integrability of the coupled NLS-SIT system we have given novel proposals forcontrolling its solitonic pulse. The broadening problem of the optical pulse can be addressed byadjusting initial population inversion of the dopant atoms, linked to the soliton width, by choosingmore general function N ( z ) > − N ( z ) = −
1. This also allows amplification of the signal through initial excitation by prepumpingenergy. The controlling effect can be refined further by using another integrable property of the coupledNLS-SIT model given by its constrained hierarchy. The idea is to replace the conventional single SITsystem by a network of sequentially coupled multiple SIT media with doping. Each additional SITmedium can bring in a new tunable function N ( j )0 ( z ) > − , j = 2 , , . . . in the form of initial populationinversion of additional dopant atoms, providing more manipulative power for controlling the shape anddynamics of the optical soliton. One set of dopant atoms in the resonant medium is coupled to anotherset by induced polarization, with all SIT media interacting in turn with the input optical field. Thisnetwork of interacting systems described by the constrained hierarchy of the integrable NLS-multiSITequations is predicted to have enhanced control over solitonic width and amplitude, which can increasesequentially with the number of coupled SIT media. In such a multi-doped media requiring higherthreshold intensity for the formation of solitonic pulse, one could possibly use a multi-level dopant likeneodymium (Nd ), where with more than two available levels the energy can be pumped throughoutthe process, unlike in two levels, resulting to a higher gain OptCom¸ .Both of our theoretical proposals with applicable potentials can be worked out analytically inminute details through ISM, due to the underlying integrability of the system. References [1] G. P. Agarwal,
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