Standing Wave Solutions in Twisted Multicore Fibers
aa r X i v : . [ m a t h . D S ] F e b Standing Wave Solutions in Twisted Multicore Fibers
Ross Parker and Alejando Aceves
Department of Mathematics, Southern Methodist Univeristy, Dallas, TX 75275 ∗ In the present work, we consider the existence and spectral stability of standing wave solutionsto a model for light propagation in a twisted multi-core fiber with no gain or loss of energy. Nu-merical parameter continuation experiments demonstrate the existence of standing wave solutionsfor sufficiently small values of the coupling parameter. Furthermore, standing waves exhibiting op-tical Aharonov-Bohm suppression, where there is a single waveguide which remains unexcited, existwhen the twist parameter φ and the number of waveguides N is related by φ = π/N . Spectralcomputations and numerical timestepping simulations suggest that standing wave solutions wherethe energy is concentrated in a single site are neutrally stable. Solutions with asymmetric couplingand multi-pulse solutions are also briefly explored. I. INTRODUCTION
There has been much recent theoretical and exper-imental interest in light dynamics in twisted multi-core optical fibers. Early work on twisted fibers canbe found in [7, 8], in which the coupled mode equa-tions describing light propagation in a circular ar-rangement of helical waveguides is derived. The in-troduction of a fiber twist in a circular array allowsfor control of diffraction and light transfer, in a sim-ilar manner to axis bending in linear waveguide ar-rays [6]. The fiber twist introduces additional phaseterms to the model, which is known as the Peierlsphase [8, 17]. In [13], this system is considered asan optical analogue of topological Aharonov-Bohmsuppression of tunneling [10], where the fiber twistplays the role of the magnetic flux in the quantummechanical system. Fiber arrangements featuringparity-time ( PT ) symmetry with balanced gain andloss terms are considered in [2, 9]. More compli-cated fiber bundle geometries have since been stud-ied, which include Lieb lattices [12] and honeycomblattices [1, 11]. Experimental applications of twistedmulti-core fibers include the construction of sensorsfor shape, strain, and temperature [3, 18]. N (cid:1) FIG. 1. Schematic of N twisted fibers arranged in a ring. ∗ [email protected] In this paper, we consider a multi-core fiberconsisting of N waveguides arranged in a ring(Figure 1). The entire fiber is twisted in a uniformfashion along the propagation direction z . For thesystem with an optical Kerr nonlinearity, the dy-namics are given by the coupled system of equations i∂ z c n = k (cid:0) e − iφ c n +1 + e iφ c n − (cid:1) + iγ n c n + d | c n | c n (1)for n = 1 , . . . , N , where c = c N and c N +1 = c due to the circular geometry [2, 16]. The quanti-ties c n ( z ) are the complex-valued amplitudes of eachwaveguide, k is the strength of the nearest-neighborcoupling, γ n is the optical gain or loss at site n , and φ is a parameter representing the twist of the fibers.(See [2, section 2] for a description of the parametersin terms of the geometry of the optical waveguidesystem and [5, 7] for a derivation of this equation).If γ n = 0 for all n , i.e. there is no gain or loss ateach node, the system is conservative. Furthermore,upon normalizing the fields by taking c n → √ | d | c n ,equation (1) becomes i∂ z c n = k (cid:0) e − iφ c n +1 + e iφ c n − (cid:1) ± | c n | c n , (2)which is Hamiltonian with conserved energy givenby H = N X n =1 k ( c n +1 c ∗ n e − iφ + c n c ∗ n +1 e iφ ) ± | c n | . (3)In this paper, we will only be concerned with theHamiltonian case (2), with defocusing (minus) non-linearity. The case with symmetric gain-loss terms( PT symmetry) is considered in [2]. Asymptoticanalysis of the system (2) for N = 6 fibers where thepeak intensity is contained in the first fiber ( n = 1)shows that the opposite fiber in the ring ( n = 4)has, to leading order, zero intensity when the twistparameter is given by φ = π/ N = 4 and φ = π/ φ = π/N , both for N even and N odd. Wethen investigate the stability of these solutions insection IV. We conclude with a brief discussion ofasymmetric variants and multi-modal solutions andsuggest some directions for future research. II. STANDING WAVE SOLUTIONS
Standing wave solutions to (2) are bound states ofthe form c n = a n e i ( ωz + θ n ) , (4)where a n ∈ R , θ n ∈ ( − π/ , π/ ω is the fre-quency of oscillation. (Since we allow a n to be nega-tive, we can restrict θ n to that interval). Making thissubstitution and simplifying, equation (2) becomes k (cid:16) a n +1 e i (( θ n +1 − θ n ) − φ ) + a n − e − i (( θ n − θ n − ) − φ ) (cid:17) + ωa n − a n = 0 , (5)where we have taken the defocusing (minus) nonlin-earity. Equation (5) can be written as the system of2 n equations k (cid:0) a n +1 cos( θ n +1 − θ n − φ )+ a n − cos( θ n − θ n − − φ ) (cid:1) + ωa n − a n = 0 a n +1 sin( θ n +1 − θ n − φ ) − a n − sin( θ n − θ n − − φ ) = 0 (6)by separating real and imaginary parts. We notethat the the exponential terms in (5) depend onlyon the phase differences θ n +1 − θ n between adjacentsites. Due to the gauge invariance of (2), if c n issolution, so is e iθ c n , thus we may without loss ofgenerality take θ = 0. If φ = 0, i.e. the fibers arenot twisted, we can take θ n = 0 for all n , and so (5)reduces to the untwisted case with periodic bound-ary conditions. Similarly, if we take φ = 2 π/N and θ n = ( n − φ for all n , the exponential terms do not
11 22 33 44
FIG. 2. Schematic of symmetry relationship betweennodes for N = 6 and N = 7. For nodes connected witharrows, the amplitudes a k are the same and the phases θ k are opposite. contribute, and (5) once again reduces to untwistedcase. The interesting cases, therefore, occur when0 < θ < π/N .In the anti-continuum (AC) limit ( k = 0), thelattice sites are decoupled. Each a n can take on thevalues { , ±√ ω } , the phases θ n are arbitrary, and φ does not contribute. The amplitudes √ ω are real if ω >
0. We construct solutions to (6) by parametercontinuation from the AC limit with no twist usingthe standard continuation software package AUTO.As an initial condition, we choose a single excitedsite at node 1, i.e. a = √ ω and a n = 0 for all other n . (We can start with more than once excited state,but, in general, these solutions will not be stable.)In addition, we take θ n = 0 for all n and φ = 0. Wefirst continue in the coupling parameter k , and then,for fixed k , we continue in the twist parameter φ . Indoing this, we observe that the solutions have thefollowing symmetry: a k = a N − k +2 k = 2 , . . . , M − θ k = − θ N − k +2 k = 2 , . . . , M − , (7)where M = ( N/ N even and M = ( N +1) / N odd. For N even, node M is the node di-rectly across the ring from node 1, and θ M = 0. Forall N , θ = 0. See Figure 2 for an illustration ofthese symmetry relations for N = 6 and N = 7.Figure 3 shows an example of a standing wave solu-tion produced by numerical parameter continuationfor N = 6. The symmetry relations (7) among theamplitudes a k can be seen in the right panel. FIG. 3. Standing wave solution for N = 6, ω = 1, k =0 .
25, and φ = 0 .
25. Left panel is real part of solution c n versus z for nodes 1-4 over a full period (2 π ). Thesolution for the remaining nodes can be found from thesymmetry relations (7). Right panel is amplitude a n solution at each node. III. OPTICAL AHARONOV-BOHMSUPPRESSION
Numerical parameter continuation, starting froma single excited node at node 1, suggests that opticalAharonov-Bohm suppression occurs when the twistparameter is φ = π/N . For N even, the node op-posite node 1 in the ring is completely dark, whichagrees with [2, 16]. We now show this occurs forstanding wave solutions. We consider the cases of N even and N odd separately, since the symmetrypatterns are different. In both cases, we find thatthere is a single dark node when φ = π/N . A. N even Taking a M = 0, where M = ( N/
2) + 1, we use thesymmetries (7) to reduce the system (6) to2 ka cos( θ − φ ) + ωa − a = 0 k ( a n +1 cos( θ n +1 − θ n − φ )+ a n − cos( θ n − θ n − − φ ))+ ωa n − a n = 0 a n +1 sin( θ n +1 − θ n − φ ) − a n − sin( θ n − θ n − − φ ) = 0 n = 2 , . . . , M − ka M − cos( θ M − + φ ) = 0 θ = θ M = 0 . It follows that a n = 0 for all n unlesscos( θ M − + φ ) = 0sin( θ n − θ n − − φ ) = 0 n = 3 , . . . , M − θ − φ ) = 0 . FIG. 4. Standing wave solution for N = 6 and φ = π/ ω = 1, k = 0 . One solution to this is θ M − + φ = π/ θ n − θ n − − φ = 0 n = 3 , . . . , M − θ − φ = 0 , from which it follows that we can have a single darknode at site M when φ = π/N . If this is the case,the system of equations above reduces to the simplersystem2 ka + ωa − a = 0 k ( a n +1 + a n − ) + ωa n − a n = 0 n = 2 , . . . , M − ka M − + ωa M − − a M − = 0 . (8)This system is of the form F ( a, k ) = 0,where a = ( a , . . . , a M − ). F (˜ a,
0) = 0,where ˜ a = ( √ ω, , . . . , D F (˜ a,
0) =diag( − ω, ω, . . . , ω ), which is invertible for ω = 0,the system (8) has a solution for sufficiently small k by the implicit function theorem. Once (8) has beensolved numerically, the full solution to (6) is givenby a M = 0 a M + k = a M − k k = 1 , . . . , M − θ = 0 θ k = ( k − φ k = 2 , . . . , M − θ M = 0 θ M + k = − θ M − k k = 1 , . . . , M − . Figure 4 shows this solution for N = 6. This ob-servation of a dark node for N = 6 when φ = π/ | k | ≤ k , where k depends on N and ω (Figure 5, left panel). Although -0.5 0 0.500.20.40.60.81 FIG. 5. Left panel shows l norm of solution vs k , for N = 50 with dark node at node 4 and ω = 1. For thisvalue of ω , k = 0 .
5. Right panel is a plot of k vs ω together with least squares linear regression line for N = 50. it is possible that there are standing wave solutionsfor | k | > k , they cannot be reached by parametercontinuation from this branch of solutions. The de-pendence of k on ω is shown in the right panel ofFigure 5, which suggests that k approaches ω/ N gets large. As k approaches k in the parametercontinuation, the l norm of the solution approaches0, thus the solution approaches the zero solution. B. N odd We can also obtain a dark node when N is odd.For simplicity, we take node 1 to be the dark node;in this case, the dark node will be opposite a pair ofbright nodes at a M and a M +1 with the same ampli-tude, where M = ( N + 1) /
2. Using the symmetries(7), when a = 0, the system (6) reduces to2 ka cos( θ − φ ) = 0 ka cos( θ − θ − φ ) + ωa − a = 0 a sin( θ − θ − φ ) = 0 k ( a n +1 cos( θ n +1 − θ n − φ )+ a n − cos( θ n − θ n − − φ ))+ ωa n − a n = 0 a n +1 sin( θ n +1 − θ n − φ ) − a n − sin( θ n − θ n − − φ ) n = 3 , . . . , M − k ( a M cos( − θ M − φ ) + a M − cos( θ M − θ M − − φ ))+ ωa M − a M = 0 a M sin( − θ M − φ ) − a M − sin( θ M − θ M − − φ ) = 0 . It follows that a n = 0 for all n unlesscos( θ − φ ) = 0sin( θ n − θ n − − φ ) = 0 n = 3 , . . . , M − θ M + φ ) = 0 . FIG. 6. Standing wave solution for N = 7 and φ = π/ ω = 1, k = 0 . One solution to this is θ − φ = − π/ θ n − θ n − − φ = 0 n = 3 , . . . , M − θ M + φ = 0 , (9)from which it follows that we can have a single darknode at a when φ = π/N . This condition for a darknode is the same as when N is even. For this case,the system of equations above reduces to the simplersystem of equations ka + ωa − a = 0 k ( a n +1 + a n − ) + ωa n − a n = 0 n = 3 , . . . , M − k ( a M + a M − ) + ωa M − a M = 0 . (10)This system of equations is again of the form F ( a, k ) = 0, where a = ( a , . . . , a M ). F (˜ a,
0) = 0,where ˜ a = (0 , . . . , , p − ω/d, D F (˜ a,
0) =diag( ω, . . . , ω, − ω ), which is invertible for ω = 0,the system (10) has a solution for sufficiently small k by the implicit function theorem. Once (10) hasbeen solved numerically, we obtain the full solutionto (6) using a = 0 a M + k = a M − k +1 k = 1 , . . . , M − θ = 0 θ k = ( k − φ − π/ k = 2 , . . . , Mθ M + k = − θ M − k +1 k = 1 , . . . , M − N = 7. The resultsof parameter continuation simulations are similar tothat of the N even case. IV. STABILITY
We now look at the stability of the standing wavesolutions we constructed in the previous section. Asa first step in stability analysis, the linearization ofequation (2) about a standing wave solution c n = a n e i ( ωz + θ n ) = ( v n + iw n ) e iωz is the 2 N × N blockmatrix A ( c n ) = k (cid:18) S C − C S (cid:19) + ω (cid:18) I − I (cid:19) − (cid:18) diag(2 v n w n ) diag( v n + 3 w n ) − diag(3 v n + w n ) − diag(2 v n w n ) (cid:19) (11)where each block is a N × N matrix, C is the peri-odic banded matrix with cos φ on the first upper andlower diagonals, and S is the periodic banded matrixwith sin φ on the first lower diagonal and − sin φ onthe first upper diagonal, i.e. C = φ . . . cos φ cos φ φ . . . . . .cos φ . . . cos φ S = − sin φ . . . sin φ sin φ − sin φ . . . . . . − sin φ . . . sin φ . Since (11) is a finite dimensional matrix, the spec-trum is purely point spectrum. Due to the gaugeinvariance, there is an eigenvalue at 0 with algebraicmultiplicity 2 and geometric multiplicity 1. Follow-ing the analysis in [4, Section 2.1.1.1], there are planewave eigenfunctions which are, to leading order, ofthe form e ± ( iqn + λz ) and satisfy the dispersion rela-tion λ = ± i ( ω + 2 k cos( q + φ )) . (12)The corresponding eigenvalues are thus purely imag-inary and are contained in the bounded intervals ± i [ ω − k, ω + 2 k ]. As N increases, these eigenval-ues fill out this interval. For | k | < k = ω/
2, theseeigenvalues do not interact with the kernel eigen-values. Figure 7 illustrates these results numericallyfor ω = 1 and k = 0 .
25 for the case of N even and φ = π/N , i.e. a single dark node opposite a singlebright node. Similar results are obtained for othervalues of ω and k in which there is a single brightnode as well as the solutions from Section III B withodd N and a single dark node.Since the spectrum of these solutions is purelyimaginary, we expect that they will be neutrally sta-ble. Figure 8 shows the results of timestepping for -1 -0.5 0 0.5 110 -12 -2-1012 -1 -0.5 0 0.5 110 -12 -2-1012 FIG. 7. Spectrum of linearization of (2) about solutionfor even N with a single dark node opposite a singlebright node. N = 6 (left panel) and N = 50 (left panel). k = 0 . ω = 1, φ = π/N . n=1n=2n=3n=4 n=1n=2n=3n=4 FIG. 8. Amplitude | c n | for first four nodes versus z forsolution with N = 6, φ = π/ N = 7, φ = π/ k = 0 . a small perturbation of the standing wave solutionwhen N = 6 and N = 7. The solutions show smallamplitude oscillations but no growth, which providesevidence for neutral stability. Similar results are ob-tained for other values of N , ω , and k . In addition,we can start with a neutrally stable standing wavesolution and perturb the system by a small change in k or φ . Figure 9 shows the results of perturbationsin k . In particular, note that in the right panel ofFigure 9, the system is evolved for a value of the cou-pling parameter k which is greater than k , where k is defined in Section III A. In both cases, the solu-tions show oscillations, indicating this to be robustdynamics. The simulation suggests the period of os-cillations has a strong dependence on k . Additionaltimestepping results can be found in [2]. In partic-ular, see [2, Figure 4] for timestepping results whenthe fiber is initially excited at a single site. V. ASYMMETRIC COUPLING
As an additional variant, if the strength of thenearest-neighbor coupling is allowed to differ be- n=1n=2n=3n=4 n=1n=2n=3n=4
FIG. 9. Amplitude | c n | for first four nodes versus z forsolution with N = 10 and φ = π/
10. Initial condition issolution to (8) with k = 0 .
45. Timestepping performedwith k = 0 .
35 (left) and k = 0 .
55 (right) using a fourthorder Runge-Kutta scheme.
FIG. 10. Standing wave solution to (13) for N = 6. ω = 1, k = 0 .
4, and k n = 0 . n . Left isreal part of solution c n versus z for nodes 1-4 over a fullperiod (2 π ), right is amplitude a n solution at each node. φ = 0 . tween pairs of nodes, equation (2) becomes i∂ z c n = k n +1 e − iφ c n +1 + k n − e iφ c n − + d | c n | c n . (13)This allows for asymmetric solutions, as shown inFigure 10. (Contrast to the symmetric solutions foruniform k in Figure 3). These asymmetric solutionsare also neutrally stable. VI. MULTI-PULSES
Another broad class of solutions is multi-pulses,which are solutions in which the energy is concen-trated at multiple nodes (see Figure 11 for two ex-amples). These nodes are typically well-separated inthe ring. Multi-pulses can be generated by parame-ter continuation from the AC limit, similar to whatwas done in section II. Although a systematic studyof the existence and stability of multi-pulses is be-yond the scope of this paper (see, for example, [14]for results on multi-pulses in the discrete NLS equa-tion), we present one example of a symmetric double
FIG. 11. Amplitudes a n for double pulse solutions withtwo bright nodes in opposite positions of the ring. N =8, φ = π/ N = 12, φ = π/ k = 0 . -1 -0.5 0 0.5 110 -12 -2-1012 n=1n=2n=3n=4 FIG. 12. Spectrum of linearization of (2) about symmet-ric double pulse solution with N = 12 and φ = π/ k = 0 . pulse solution for even N in which the two excitedsites are opposite each other in the ring (Figure 11).If N is a multiple of 4 and φ = 2 π/N , thereis a pair of dark nodes halfway between the twobright nodes (in both directions), as can be seen inFigure 11. Numerical spectral computations as wellas timestepping results of perturbations of these so-lutions suggest that these double pulse solutions areneutrally stable (Figure 12). VII. CONCLUSIONS
In this paper, we have demonstrated the existenceof standing wave solutions to a system of equationsmodeling light propagation in a twisted multi-corefiber in the setting of no gain or loss at the indi-vidual sites. If the twist parameter φ and the num-ber of waveguides N are related by φ = π/N , thenstanding wave solutions exist which exhibit opticalAharonov-Bohm suppression, i.e. there is a nodewhich is completely dark for all time. These solu-tions exist for both N even and N odd, and are allneutrally stable. For future research, it would be in-teresting to investigate whether such standing wavesexist for twisted optical fibers in more complicatedgeometries such as multiple concentric rings or Lieblattices. We could also systematically study multi-pulse solutions, as well as investigate the existenceand stability of breathers, which are localized, pe-riodic structures that are not standing waves. (See[11] for examples of breather solutions in honeycomblattices). We could also apply the techniques usedhere to the PT -symmetric system with symmetricgain and loss, which is studied in [2]. Finally, sincethese standing wave solutions are neutrally stable, itwould be interesting to see if they could be createdexperimentally in twisted multi-core fibers. ACKNOWLEDGMENTS
This material is based upon work supported by theU.S. National Science Foundation under the RTGgrant DMS-1840260 (R.P. and A.A.) and DMS-1909559 (AA). The authors would also like to thankP.G. Kevrekidis for his helpful comments and sug-gestions for numerical simulations. [1] Mark J. Ablowitz, Christopher W. Curtis, and Yi-Ping Ma,
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