Soliton-mode proliferation induced by cross-phase modulation of harmonic waves by a dark-soliton crystal in optical media
aa r X i v : . [ n li n . PS ] F e b Soliton-mode proliferation induced by cross-phase modulation of harmonic waves by adark-soliton crystal in optical media
E. Chenui Aban and Alain M. Dikand´e ∗ Laboratory of Research on Advanced Materials and Nonlinear Science (LaRAMaNS),Department of Physics, Faculty of Science, University of Buea P.O. Box 63 Buea, Cameroon. (Dated: March 2, 2021)The generation of high-intensity optical fields from harmonic-wave photons, interacting via a cross-phase modulation with dark solitons both propagating in a Kerr nonlinear medium, is examined.The focus is on a pump consisting of time-entangled dark-soliton patterns, forming a periodicwaveguide along the path of the harmonic-wave probe. It is shown that an increase of the strengthof cross-phase modulation respective to the self-phase modulation, favors soliton-mode proliferationin the bound-state spectrum of the trapped harmonic-wave probe. The induced soliton modes,which display the structures of periodic soliton lattices, are not just rich in numbers, they also forma great diversity of population of soliton crystals with a high degree of degeneracy.
I. INTRODUCTION
Optical wave trapping, cloning, reconfiguration, dupli-cation, parametric amplification and recompression [1–11] are physical processes associated with wave inter-actions [12, 13, 15] in nonlinear optical media. Theseprocesses are usually controlled by nonlinear phenom-ena originating from the intensity-dependent index of re-fraction of propagation media, which causes cross-phaseor induced-phase modulations [16, 17] determining shapeprofiles of the propagating optical fields. Such processesfind widespread applications in modern communicationtechnology, and particularly in the processings of rela-tively low-power fields (such as harmonic fields) interact-ing with fields of sufficiently high intensity (such as opti-cal solitons), leading to their cloning and reconfigurationinto optical fields of higher powers [10].Since soliton cloning and reconfiguration can involve asizable energy cost from the pump field, these processeshave most often been envisaged between two solitons ofslightly different powers. Thus in refs. [5, 18], a re-configuration scheme was proposed in which an intensepump beam with soliton features induces the focusing ofa weaker probe beam of different wavelength, but alsowith soliton features, via cross-phase modulation. Theunderlying mechanism is simply understood by recall-ing that an optical soliton propagating in a Kerr nonlin-ear medium, creates a local distortion of the refractiveindex that travels with the soliton down the nonlinearpropagation medium. As a result of this refractive indexdistortion a waveguide can be induced, that acts like alocal potential by trapping and reshaping another muchweaker pulse, different from the soliton pump in both fre-quency and polarization. However, in ref. [10], Steiglitzand Rand suggested the possibility to use optical soli-tons propogating for instance in an optical fiber, to trapan reshape continuous-wave photons by means of theircross-phase modulation with the optical solitons. Thus, ∗ Corresponding author: [email protected] by considering a localized bright soliton pump interact-ing with an harmonic photon field, they established thatthe probe field was reshaped into new modes the eigen-states of which were described by a linear eigenvalueequation with a reflectionless potential. Subsequent tothe study of Steiglitz and Rand [10], the phenomenonof harmonic-wave trapping and reconfiguration by brightsolitons was extended to the context of a waveguide cre-ated by a periodic train of bright solitons [12] forminga bright-soliton crystal. This later study led to a lineareigenvalue problem of the Lam´e type for the probe field[19], and its bound states were shown to form spectraof rich and abundant soliton modes. Namely in ref. [9]it was established that increasing the strength of cross-phase modulation relative to the self-phase modulation,favors an increase of the population and the degeneracyof soliton modes composing bound-state spectra of thetrapped probe.While the dynamics of bright solitons as well as theirstability under mutual collisions are relatively well un-derstood, dark solitons have remained a curiosity forsome reasons. Most importantly dark solitons are odd-symmetry structures, and for this reason they can onlypropagate in specific media [20]. Nevertheless it is wellestablished that dark solitons have simpler collision dy-namics than their bright counterparts [21], are generallymore stable against various perturbations [22], and hencemay offer some important advantages in optical field pro-cessing applications. Based on this later features Steiglitz[23] considered using a localized dark soliton pump totrap and reshape harmonic-wave probes. He found thatprobe modes induced by cross-phase modulation in thewaveguide of the single dark soliton, were determined bya linear eigenvalue equation with a reflecting scatteringpotential. Because of this the groundstate was not aGoldstone translation of the pump as in the case involv-ing a bright-soliton pump [10], but instead a localizedsech-type pulse soliton.Motivated by results of a previous study [9], in which wefound that a waveguide consisting of a crystal of brightsolitons favors relatively more diverse and abundant soli-ton modes in the probe spectrum, in the present studywe shall examine the problem of harmonic-wave recon-figuration by a waveguide consisting of a periodic trainof dark solitons. Below we start with the presentationof the model, and obtain the periodic dark-soliton solu-tion to the pump equation. With this solution we showthat the probe equation can be formulated in terms ofa Lam´e-type eigenvalue problem, and derive some exactbounded states to this eigenvalue problem under specificconditions.
II. THE PUMP-PROBE EQUATIONS ANDDARK-SOLITON-CRYSTAL SOLUTION TO THEPUMP EQUATION
The propagation equations for the system composed ofa nonlinear pump field, coupled to an harmonic-wave fieldvia a cross-phase modulation anf propagating together ina Kerr nonlinear optical medium, are given by: i ∂v∂z + ∂ v∂t − ζ | v | v = 0 , (1) i ∂u∂z + k ∂ u∂t − k | v | u = 0 . (2)In the first equation, which is precisely the cubic nonlin-ear Schr¨odinger equation with self-defocusing nonlinear-ity, the quantity v is the pump envelope, z is the prop-agation distance, t is the propagation time and ζ is thecoefficient of self-phase modulation. The quantity u inthe second equation is the probe envelope, k is the co-efficient of group-velocity-dispersion and k is the coeffi-cient of cross-phase modulation.In eq. (1) the nonlinear coefficient is effectively nega-tive i.e. − ζ with ζ >
0, corresponding to a Kerr opticalmedium with self-defocusing nonlinearity. In ref. [23] theproblem of harmonic-wave trapping and reshaping by awaveguide created by a single dark soliton, was consid-ered. Here we are interested in the context when thewaveguide is created by a period train of single dark soli-tons, forming a dark-soliton crystal. In this purpose weconsider a solution to eq. (1) describing a stationarywave, v ( z, t ) = A ( t ) exp [ − i ( kz − ωt )], where k is the wavenumber and ω is the frequency. Substituting this in eq.(1), we find that the wave amplitude A ( t ) must obey thefirst-integral equation: ∂A∂t = ζ (cid:18)r A − sζ A + ρ (cid:19) , (3)with ρ an energy contant determining shape profile of A ( t ). Solving eq. (3 with periodic boundary conditions[24–27], the pump amplitude A ( t ) is found to be the fol-lowing nonlocalized periodic pattern of time-entangleddark solitons: A ( t ) = Q √ ζ sn [ Q ( t − t )] , (4) where sn () is a Jacobi elliptic function of modulus κ (with0 ≤ κ ≤ Q = r s (1 + κ ) , s = k − ω . (5)The amplitude A ( t ) of the periodic dark soliton (4) isrepresented in fig. 1, for κ = 0 .
98 (left graph) and κ = 1(right graph). Note that when κ → sn () → tanh (), corresponding to the darksoliton pump obtained in [23]. III. PUMP-INDUCED TRAPPING,RESHAPING AND PROBE-MODEPROLIFERATION
Using the periodic dark soliton solution to the pumpequation obtained in eq. (4), we will now seek solutionsto the probe equation (2). Proceeding with it is use-ful to start by the important remark that eq. (2) is alinear chr¨odinger equation, but with a time-dependent”external” potential represented by the norm squared ofthe pump envelope q ( z, t ). Substituting (4) in the probeequation given by (2, and expressing the probe envelopeas a stationary wave i.e. A ( t ) = u ( t ) exp ( − iqz ), where u ( t ) is the core of the probe envelope and q is its wavenumber, we obtain the Lam´e equation [19]: ∂ u∂τ + P ( q ) − l ( l + 1) κ sn ( τ ) ! u = 0 , (6) P ( q ) = qk Q , τ = Q ( t − t ) , l ( l + 1) = 2 k κ k ζ . (7)It is worth stressing that when κ = 1, the Lam´e equation(7) becomes the Associated Legendre equation obtainedin ref. [23].The Lam´e equation possesses a rich spectrum with agreat variety of eigenmodes [19]. However the most rel-evant to us are its eigenmodes that display a permanentprofile typical of solitons. Precisely these later modesare bound states of the Lam´e equation, and becausetheir formation through the cross-phase modulation withthe dark-soliton crystal involves energy cost (momen-tum transfer) from the pump, they can be looked outas low-energy states of the probe spectrum created bythe pump-induced periodic potential (4). Discrete statesof Lam´e’s equation form a spectrum of finite orthogonalmodes, whose population depends on the integer quan-tum number l [19]. According to equation (7), valuesof the integer quantum number l will be determined bythe competition between the self-phase modulation re-sponsible for the fiber nonlinearity, and the cross-phasemodulation exerted by the pump field on the harmonicprobe. For a given value of l , the discrete spectrum ofLam´e equation possesses 2 l + 1 modes some of which canbe degenerate [9, 12, 19]. - -
50 0 50 100 - - A κ = - -
50 0 50 100 - - A κ = FIG. 1. (Color online) Amplitude of the pump field A ( t ) given by eq. (4) versus time, for κ = 0 .
98 (left graph) and κ = 1(right graph). We start with the lowest value of l ; l = 1 correspondingto the case when the cross-phase modulation and the self-phase modulation coefficients are related by k = k ζκ .In this case the Lam´e equation possesses three distinctlocalized modes, namely: u ( τ ) = u (11) cn ( τ ) , q = q = k Q κ ζ , (8) u ( τ ) = u (12) dn ( τ ) , q = q = k Q ζ , (9) u ( τ ) = u (13) sn ( τ ) , q = q = (1 + κ ) k Q κ ζ , (10)where u (1 i ) are normalization constants. The threemodes are represented in fig.2, for κ = 0 .
98 (left column)and κ = 1 (right column).Taking l = 2, or equivalently k = 3 k ζκ , leads to fivedistinct localized modes for the probe which are listedbelow: u ( τ ) = u (21) cn ( τ ) dn ( τ ) , q = q = (cid:18) κ κ (cid:19) k Q ζ , (11) u ( τ ) = u (22) sn ( τ ) dn ( τ ) , q = q = (cid:18) κ κ (cid:19) k Q ζ , (12) u ( τ ) = u (23) sn ( τ ) cn ( τ ) , q = q = (cid:18) κ κ (cid:19) k Q ζ , (13) u ( τ ) = u (24) " sn ( τ ) − κ + p − κ (1 − κ )3 κ ,q = q = " κ ) − p − κ (1 − κ )3 κ k Q ζ , (14) u ( τ ) = u (25) " sn ( τ ) − κ − p − κ (1 − κ )3 κ , q = q = " κ ) + p − κ (1 − κ )3 κ k Q ζ . (15)The five bounded modes are plotted versus time in fig.3,for κ = 0 .
98 (left column) and κ = 1 (right column).The third and last case considered is l = 3, corre-sponding to k = 6 k ζκ . In this case the bound-statespectrum of the probe comprises seven distinct localizedmodes i.e.: u ( τ ) = u (31) sn ( τ ) cn ( τ ) dn ( τ ) ,q = q = (cid:18) κ )3 κ (cid:19) k Q ζ , (16) u ( τ ) = u (32) " sn ( τ ) − κ ) − √ − κ + 4 κ κ sn ( τ ) ,q = q = " κ ) + 2 √ − κ + 4 κ κ k Q ζ , (17) u ( τ ) = u (33) " sn ( τ ) − κ ) + √ − κ + 4 κ κ sn ( τ ) ,q = q = " κ ) − √ − κ + 4 κ κ k Q ζ , (18) u ( τ ) = u (34) cn ( τ ) " sn ( τ ) − κ − p − κ (1 − κ )5 κ ,q = q = " κ + 2 p − κ (1 − κ )6 κ k Q ζ , (19) u ( τ ) = u (35) cn ( τ ) " sn ( τ ) − κ + p − κ (1 − κ )5 κ ,q = q = " κ − p − κ (1 − κ )6 κ k Q ζ , (20) - -
50 0 50 100 - - u1 κ = - -
50 0 50 1000.00.20.40.60.81.01.2 time u1 κ = - -
50 0 50 1000.00.20.40.60.81.0 time u2 κ = - -
50 0 50 1000.00.20.40.60.81.01.2 time u2 κ = - -
50 0 50 100 - - u3 κ = - -
50 0 50 100 - - u3 κ = FIG. 2. (Color online) Amplitudes of the three probe modes corresponding to l = 1 versus time, for κ = 0 .
98 (left graphs) and κ = 1 (right graphs). u i in the graphs mean u i in eqs. (8)-(10), with i = 1 , , u ( τ ) = u (36) dn ( τ ) " sn ( τ ) − κ − √ − κ + 4 κ κ ,q = q = " κ + 2 √ − κ + 4 κ κ k Q ζ , (21) u ( τ ) = u (37) dn ( τ ) " sn ( τ ) − κ + √ − κ + 4 κ κ ,q = q = " κ − √ − κ + 4 κ κ k Q ζ . (22)Although the analytical expressions of the seven modesseem to suggest complex combinations of Jacobi ellip- tic functions, we can convince ourselves of the contraryby examining their expressions for κ = 1, a value forwhich these analytical expressions are fundamental com-ponents composing the soliton trains in the seven distinctmodes. This remark also holds for the cases l = 1 and l = 2. To gain a better understanding of shape profilesof these fundamental components, in tables I, II and IIItheir eigenfunctions u ( τ ) are listed together with corre-sponding eigenvalues q .As it is apparently, the three tables feature very richand varied spectra of bounded states for the three valuesof l considered. Also remarkable, table I suggests thatthe l = 1 spectrum possesses two modes which are nearlydegenerate, whereas the l = 2 spectrum has three nearly - -
50 0 50 100 - - u1 κ = - -
50 0 50 1000.00.20.40.60.81.01.2 time u1 κ = - -
50 0 50 100 - - u2 κ = - -
50 0 50 100 - - - u2 κ = - -
50 0 50 100 - - u3 κ = - -
50 0 50 100 - - - u3 κ = - -
50 0 50 100 - - - - - u4 κ = - -
50 0 50 100 - - - - - - u4 κ = - -
50 0 50 100 - u5 κ = - -
50 0 50 100 - u5 κ = FIG. 3. (Color online) Temporal profiles of amplitudes of the five bounded modes given by eqs. (11)-(15), for κ = 0 .
98 (leftcolumn) and κ = 1 (right column). u i in the graphs correspond to u i in eqs. (11)-(15), with i = 1 , , , , κ = 1)of the l = 1 eigenmodes.Eigenfunction Eigenvalue u ( τ ) = u ( τ ) ∝ sech ( τ ) q = q = k Q /ζu ( τ ) ∝ tanh( τ ) q = 2 k Q /ζ degenerate modes (see table II) and the spectrum for l = 3 possesses three distinct double-degenerate modesas one can see in table III, where only u out of theseven bounded states is non degenerate. TABLE II. Fundamental components (solutions with κ = 1)of the l = 2 eigenmodes.Eigenfunction Eigenvalue u ( τ ) ∝ sech ( τ ) q = 2 k Q / (3 ζ ) u ( τ ) = u ( τ ) ∝ sech ( τ ) tanh( τ ) q = q = 5 k Q / (3 ζ ) u ( τ ) ∝ − sech ( τ ) q = k Q /ζu ( τ ) ∝ − sech ( τ ) q = 5 k Q / (3 ζ ) TABLE III. Fundamental components (solutions with κ = 1)of the l = 3 eigenmodes.Eigenfunction Eigenvalue u ( τ ) ∝ sech ( τ ) tanh( τ ) q = 4 k Q / (3 ζ ) u ( τ ) ∝ (cid:16) − sech ( τ ) (cid:17) tanh( τ ) q = 2 k Q /ζu ( τ ) ∝ − sech ( τ ) tanh( τ ) q = 4 k Q / (3 ζ ) u ( τ ) ∝ (cid:16) − sech ( τ ) (cid:17) sech ( τ ) q = 11 k Q / (6 ζ ) u ( τ ) ∝ − sech ( τ ) q = k Q / (2 ζ ) u ( τ ) ∝ sech ( τ ) tanh ( τ ) q = 11 k Q / (6 ζ ) u ( τ ) ∝ (cid:16) − sech ( τ ) (cid:17) sech ( τ ) q = k Q / (2 ζ ) IV. CONCLUSION
We examined profiles of an harmonic wave propagatingin the waveguide structure created by a periodic latticeof single dark solitons, obtained as a periodic solution tothe self-defocusing cubic nonlinear Schr¨odinger equationdescribing the propagation of a pump field along an op-tical fibre. In ref. [23] the same problem was discussedassuming that the harmonic probe is coupled to a singledark soliton. We obtained that due to the coupling ofthe pump and probe via the cross-phase modulation, theprobe equation can be transformed into a general familyof eigenvalue equation called Lam´e equation. Consider-ing bounded states of this eigenvalue problem, we foundthat the population of the associated discrete spectrumwas determined by an integer quantum number which inturn was determined by the competition between the self-phase moduation and the cross-phase modulation. We obtained analytical expressions of these bounded modesof the trapped probe for l = 1 , ,
3, and observed that as l increases the number of bounded modes was higher andhigher and the spectra more and more degenerate.Our study demonstrates unambiguously that waveg-uides induced by a periodic train of dark solitons, can beused to control weak probes in the same way bright soli-tons can. Dark solitons offer real advantages over brightsoliton collisions in controlling light waves: Firstly darksolitons are well known to be more stable in the pres-ence of noise [20–22], and are generally more robust thanbright solitons. Secondly the probe, which is of muchlower intensity and here an harmonic wave, is expectedto peak at the dip in the intensities of its host single-dark soliton components, thus increasing the signal-to-noise ratio and making it easier, in principle, to detect.The present study finds its most important application inquantum communication and cryptographic systems, wethink in particular of experimental verification of photoncapture and transport in an optical fiber. The problemof detecting the probe in the presence of a pump, andtailoring physical parameters to realize virtual devices,is also of current interest in quantum computing appli-cations involving entangled photon emitters and photonqubits [28–31]. ACKNOWLEDGMENTS
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