Stable scalable control of soliton propagation in broadband nonlinear optical waveguides

Stable scalable control of soliton propagation in broadband nonlinear optical waveguides

Avner Peleg, Quan M. Nguyen, and Toan T. Huynh Department of Exact Sciences, Afeka College of Engineering, Tel Aviv 69988, Israel Department of Mathematics, International University, Vietnam National University-HCMC, Ho Chi Minh City, Vietnam Department of Mathematics, University of Medicine and Pharmacy-HCMC, Ho Chi Minh City, Vietnam
Abstract

We develop a method for achieving scalable transmission stabilization and switching of colliding soliton sequences in optical waveguides with broadband delayed Raman response and narrowband nonlinear gain-loss. We show that dynamics of soliton amplitudes in -sequence transmission is described by a generalized -dimensional predator-prey model. Stability and bifurcation analysis for the predator-prey model are used to obtain simple conditions on the physical parameters for robust transmission stabilization as well as on-off and off-on switching of out of soliton sequences. Numerical simulations for single-waveguide transmission with a system of coupled nonlinear Schrödinger equations with show excellent agreement with the predator-prey model’s predictions and stable propagation over significantly larger distances compared with other broadband nonlinear single-waveguide systems. Moreover, stable on-off and off-on switching of multiple soliton sequences and stable multiple transmission switching events are demonstrated by the simulations. We discuss the reasons for the robustness and scalability of transmission stabilization and switching in waveguides with broadband delayed Raman response and narrowband nonlinear gain-loss, and explain their advantages compared with other broadband nonlinear waveguides.

pacs:
42.65.Tg, 05.45.Yv, 42.65.Dr, 42.65.Sf

I Introduction

The rates of information transmission through broadband optical waveguide links can be significantly increased by transmitting many pulse sequences through the same waveguide Agrawal2001 (); Tkach97 (); Mollenauer2006 (); Gnauck2008 (); Essiambre2010 (). This is achieved by the wavelength-division-multiplexed (WDM) method, where each pulse sequence is characterized by the central frequency of its pulses, and is therefore called a frequency channel multisequence (). Applications of these WDM or multichannel systems include fiber optics transmission lines Tkach97 (); Mollenauer2006 (); Gnauck2008 (); Essiambre2010 (), data transfer between computer processors through silicon waveguides Agrawal2007a (); Dekker2007 (); Gaeta2008 (), and multiwavelength lasers Chow96 (); Shi97 (); Zhang2009 (); Liu2013 (). Since pulses from different frequency channels propagate with different group velocities, interchannel pulse collisions are very frequent, and can therefore lead to error generation and severe transmission degradation Agrawal2001 (); Tkach97 (); Mollenauer2006 (); Gnauck2008 (); Essiambre2010 (); Iannone98 (); MM98 (). On the other hand, the significant collision-induced effects can be used for controlling the propagation, for tuning of optical pulse parameters, such as amplitude, frequency, and phase, and for transmission switching, i.e., the turning on or off of transmission of one or more of the pulse sequences NP2010 (); PNC2010 (); PC2012 (); CPJ2013 (); NPT2015 (); PNT2015 (). A major advantage of multichannel waveguide systems compared with single-channel systems is that the former can simultaneously handle a large number of pulses using relatively low pulse energies. One of the most important challenges in multichannel transmission concerns the realization of stable scalable control of the transmission, which holds for an arbitrary number of frequency channels. In the current study we address this challenge, by showing that stable scalable transmission control can be achieved in multichannel optical waveguide systems with frequency dependent linear gain-loss, broadband delayed Raman response, and narrowband nonlinear gain-loss.

Interchannel crosstalk, which is the commonly used name for the energy exchange in interchannel collisions, is one of the main processes affecting pulse propagation in broadband waveguide systems. Two important crosstalk-inducing mechanisms are due to broadband delayed Raman response and broadband nonlinear gain-loss. Raman-induced interchannel crosstalk is an important impairment in WDM transmission lines employing silica glass fibers Chraplyvy84 (); Tkach95 (); Ho2000 (); Yamamoto2003 (); P2004 (); P2007 (); CP2008 (); Golovchenko2009 (); PC2012b (), but is also beneficially employed for amplification Islam2004 (); Agrawal2005 (). Interchannel crosstalk due to cubic loss was shown to be a major factor in error generation in multichannel silicon nanowaveguide transmission Gaeta2010 (). Additionally, crosstalk induced by quintic loss can lead to transmission degradation and loss of transmission scalability in multichannel optical waveguides due to the impact of three-pulse interaction on the crosstalk PC2012 (); PNG2014 (). On the other hand, nonlinear gain-loss crosstalk can be used for achieving energy equalization, transmission stabilization, and transmission switching PNC2010 (); PC2012 (); CPJ2013 (); NPT2015 ().

In several earlier studies NP2010 (); PNC2010 (); PC2012 (); CPJ2013 (); NPT2015 (); PNT2015 (), we provided a partial solution to the key problem of achieving stable transmission control in multichannel nonlinear waveguide systems, considering solitons as an example for the optical pulses. Our approach was based on showing that the dynamics of soliton amplitudes in -sequence transmission can be described by Lotka-Volterra (LV) models for species, where the specific form of the LV model depends on the nature of the dissipative processes in the waveguide. Stability and bifurcation analysis for the steady states of the LV models was used to guide a clever choice of the physical parameters, which in turn leads to transmission stabilization, i.e., the amplitudes of all propagating solitons approach desired predetermined values NP2010 (); PNC2010 (); PC2012 (); CPJ2013 (); NPT2015 (); PNT2015 (). Furthermore, on-off and off-on transmission switching were demonstrated in two-channel waveguide systems with broadband nonlinear gain-loss CPJ2013 (); NPT2015 (). The design of waveguide setups for transmission switching was also guided by stability and bifurcation analysis for the steady states of the LV models CPJ2013 (); NPT2015 ().

The results of Refs. NP2010 (); PNC2010 (); PC2012 (); CPJ2013 (); NPT2015 (); PNT2015 () provide the first steps toward employing crosstalk induced by delayed Raman response or by nonlinear gain-loss for transmission control, stabilization, and switching. However, these results are still quite limited, since they do not enable scalable transmission stabilization and switching for pulse sequences with a general value in a single optical waveguide. To explain this, we first note that in waveguides with broadband delayed Raman response, such as optical fibers, and in waveguides with broadband cubic loss, such as silicon waveguides, some or all of the soliton sequences propagate in the presence of net linear gain NP2010 (); PNC2010 (); PNT2015 (). This leads to transmission destabilization at intermediate distances due to radiative instability and growth of small amplitude waves. As a result, the distances along which stable propagation is observed in these single-waveguide multichannel systems are relatively small even for small values of the Raman and cubic loss coefficients PNC2010 (); PNT2015 (). The radiative instability observed in optical fibers and silicon waveguides can be mitigated by employing waveguides with linear loss, cubic gain, and quintic loss, i.e., waveguides with a Ginzburg-Landau (GL) gain-loss profile PC2012 (); CPJ2013 (); NPT2015 (). However, the latter waveguides suffer from another serious limitation because of the broadband nature of the waveguides nonlinear gain-loss. More specifically, due to the presence of broadband quintic loss, three-pulse interaction gives an important contribution to collision-induced amplitude shifts PC2012 (); PNG2014 (). The complex nature of three-pulse interaction in generic three-soliton collisions in this case (see Ref. PNG2014 ()) leads to a major difficulty in extending the LV model for amplitude dynamics from to a general value in waveguides with broadband nonlinear gain-loss. In the absence of a general -dimensional LV model, it is unclear how to design setups for stable transmission stabilization and switching in -sequence systems with . For this reason, transmission stabilization and switching in waveguides with broadband nonlinear gain-loss were so far limited to two-sequence systems PC2012 (); CPJ2013 (); NPT2015 ().

In view of the limitations of the waveguides studied in Refs. NP2010 (); PNC2010 (); PC2012 (); CPJ2013 (); NPT2015 (); PNT2015 (), it is important to look for new routes for realizing scalable transmission stabilization and switching, which work for -sequence transmission with a general value. In the current paper we take on this task, by studying propagation of soliton sequences in nonlinear waveguides with frequency dependent linear gain-loss, broadband delayed Raman response, and narrowband nonlinear gain-loss. Due to the narrowband nature of the nonlinear gain-loss, it affects only single-pulse propagation and intrasequence interaction, but does not affect intersequence soliton collisions. We show that the combination of Raman-induced amplitude shifts in intersequence soliton collisions and single-pulse amplitude shifts due to gain-loss with properly chosen physical parameter values can be used to realize robust scalable trasmission stabilization and switching. For this purpose, we first obtain the generalized -dimensional predator-prey model for amplitude dynamics in an -sequence system. We then use stability and bifurcation analysis for the predator-prey model to obtain simple conditions on the values of the physical parameters, which lead to robust transmission stabilization as well as on-off and off-on switching of out of soliton sequences. The validity of the predator-prey model’s predictions is checked by carrying out numerical simulations with the full propagation model, which consists of a system of perturbed coupled nonlinear Schrödinger (NLS) equations. Our numerical simulations with soliton sequences show excellent agreement with the predator-prey model’s predictions and stable propagation over significantly larger distances compared with other broadband nonlinear single-waveguide systems. Moreover, stable on-off and off-on switching of multiple soliton sequences and stable multiple transmission switching events are demonstrated by the simulations. We discuss the reasons for the robustness and scalability of transmission stabilization and switching in waveguides with broadband delayed Raman response and narrowband nonlinear gain-loss, and explain their advantages compared with other broadband nonlinear waveguides.

The rest of the paper is organized as follows. In Section II, we present the coupled-NLS model for propagation of pulse sequences through waveguides with frequency dependent linear gain-loss, broadband delayed Raman response, and narrowband nonlinear gain-loss. In addition, we present the corresponding generalized -dimensional predator-prey model for amplitude dynamics. In Section III, we carry out stability and bifurcation analysis for the steady states of the predator-prey model, and use the results to derive conditions on the values of the physical parameters for achieving scalable transmission stabilization and switching. In Section IV, we present the results of numerical simulations with the coupled-NLS model for transmission stabilization, single switching events, and multiple transmission switching. We also analyze these results in comparison with the predictions of the predator-prey model. In Section V, we discuss the underlying reasons for the robustness and scalability of transmission stabilization and switching in waveguides with broadband delayed Raman response and narrowband nonlinear gain-loss. Section VI is reserved for conclusions.

Ii Coupled-NLS and predator-prey models

ii.1 A coupled-NLS model for pulse propagation

We consider sequences of optical pulses, each characterized by pulse frequency, propagating in an optical waveguide in the presence of second-order dispersion, Kerr nonlinearity, frequency dependent linear gain-loss, broadband delayed Raman response, and narrowband nonlinear gain-loss. We assume that the net linear gain-loss is the difference between amplifier gain and waveguide loss and that the frequency differences between all sequences are much larger than the spectral width of the pulses. Under these assumptions, the propagation is described by the following system of perturbed coupled-NLS equations:

(1)

where is proportional to the envelope of the electric field of the th sequence, , is propagation distance, and is time. In Eq. (1), is the linear gain-loss coefficient for the th sequence, is the Raman coefficient, and is a polynomial of , describing the waveguide’s nonlinear gain-loss profile. The values of the coefficients are determined by the -dimensional predator-prey model for amplitude dynamics, by looking for steady-state transmission with equal amplitudes for all sequences. The second term on the left-hand side of Eq. (1) is due to second-order dispersion, while the third and fourth terms represent intrasequence and intersequence interaction due to Kerr nonlinearity. The first term on the right-hand side of Eq. (1) is due to linear gain-loss, the second corresponds to intrasequence interaction due to nonlinear gain-loss, the third describes Raman-induced intrasequence interaction, while the fourth and fifth describe Raman-induced intersequence interaction. Since we consider waveguides with broadband delayed Raman response and narrowband nonlinear gain-loss, Raman-induced intersequence interaction is taken into account, while intersequence interaction due to nonlinear gain-loss is neglected. The polynomial in Eq. (1) can be quite general. In the current paper, we consider two central examples for waveguide systems with nonlinear gain-loss: (1) waveguides with a GL gain-loss profile, (2) waveguides with linear gain-loss and cubic loss. The expression for for waveguides with a GL gain-loss profile is

(2)

where and are the cubic gain and quintic loss coefficients. The expression for for waveguides with linear gain-loss and cubic loss is

(3)

where is the cubic loss coefficient. We emphasize, however, that our approach can be employed to treat a general form of the polynomial . Note that since some of the perturbation terms in the propagation model (1) are nonlinear gain or loss terms, the model can also be regarded as a coupled system of GL equations.

The dimensional and dimensionless physical quantities are related by the standard scaling laws for NLS solitons Agrawal2001 (). Exactly the same scaling relations were used in our previous works on soliton propagation in broadband nonlinear waveguide systems PNC2010 (); PC2012 (); CPJ2013 (); NPT2015 (); PNT2015 (). In these scaling relations, the dimensionless distance in Eq. (1) is , where is the dimensional distance, is the dimensional dispersion length, is the soliton width, and is the second-order dispersion coefficient. The dimensionless retarded time is , where is the retarded time. The solitons spectral width is and the frequency difference between adjacent channels is . , where is proportional to the electric field of the th pulse sequence and is the peak power. The dimensionless second order dispersion coefficient is , where is the Kerr nonlinearity coefficient. The dimensional linear gain-loss coefficient for the th sequence is related to the dimensionless coefficient via . The coefficients , , and are related to the dimensional cubic gain , cubic loss , and quintic loss , by , , and , respectively NPT2015 (). The dimensionless Raman coefficient is , where is a dimensional time constant, characterizing the waveguide’s delayed Raman response Agrawal2001 (); Chi89 (). The time constant can be determined from the slope of the Raman gain curve of the waveguide Agrawal2001 (); Chi89 ().

We note that for waveguides with linear gain-loss and cubic loss, some or all of the pulse sequences propagate in the presence of net linear gain. This leads to transmission destabilization due to radiation emission. The radiative instability can be partially mitigated by employing frequency dependent linear gain-loss . In this case, the first term on the right hand side of Eq. (1) is replaced by , where is the Fourier transform of with respect to time, and stands for the inverse Fourier transform. The form of is chosen such that existence of steady-state transmission with equal amplitudes for all sequences is retained, while radiation emission effects are minimized. More specifically, is equal to a value , required to balance amplitude shifts due to nonlinear gain-loss and Raman crosstalk, inside a frequency interval of width centered about the frequency of the th-channel solitons at distance , , and is equal to a negative value elsewhere g_omega_z (). Thus, is given by:

where . The width in Eq. (LABEL:global3a) satisfies , where for . The values of the coefficients are determined by the generalized predator-prey model for collision-induced amplitude dynamics, such that amplitude shifts due to Raman crosstalk and nonlinear gain-loss are compensated for by the linear gain-loss. The values of and are determined by carrying out numerical simulations with Eqs. (1), (3), and (LABEL:global3a), while looking for the set, which yields the longest stable propagation distance g_omega_z (). Figure 1 shows a typical example for the frequency dependent linear gain-loss function at for a three-channel system with , , , , , , , and . These parameter values are used in the numerical simulations, whose results are shown in Fig. 7.

Figure 1: An example for the frequency dependent linear gain-loss function of Eq. (LABEL:global3a) at in a three-channel system.

The optical pulses in the th sequence are fundamental solitons of the unperturbed NLS equation with central frequency . The envelopes of these solitons are given by , where , , and , , and are the soliton amplitude, position, and phase. Due to the large frequency differences between the pulse sequences, the solitons undergo a large number of fast intersequence collisions. The energy exchange in the collisions, induced by broadband delayed Raman response, can lead to significant amplitude shifts and to transmission degradation. On the other hand, the combination of Raman-induced amplitude shifts in intersequence collisions and single-pulse amplitude shifts due to frequency dependent linear gain-loss and narrowband nonlinear gain-loss with properly chosen coefficients can be used to realize robust scalable transmission stabilization and switching. In the current paper, we demonstrate that such stable scalable transmission control can indeed be achieved, even with the simple nonlinear gain-loss profiles (2) and (3).

ii.2 A generalized -dimensional predator-prey model for amplitude dynamics

The design of waveguide setups for transmission stabilization and switching is based on the derivation of LV models for dynamics of soliton amplitudes. For this purpose, we consider propagation of soliton sequences in a waveguide loop, and assume that the frequency spacing between the sequences is a large constant, i.e., for . Similar to Refs. NP2010 (); PNC2010 (), we can show that amplitude dynamics of the sequences is approximately described by a generalized -dimensional predator-prey model. The derivation of the predator-prey model is based on the following assumptions. (1) The temporal separation between adjacent solitons in each sequence satisfies: . In addition, the amplitudes are equal for all solitons from the same sequence, but are not necessarily equal for solitons from different sequences. This setup corresponds, for example, to phase-shift-keyed soliton transmission. (2) As , intrasequence interaction is exponentially small and is neglected. (3) Delayed Raman response and gain-loss are assumed to be weak perturbations. As a result, high-order effects due to radiation emission are neglected, in accordance with single-collision analysis.

Since the pulse sequences are periodic, the amplitudes of all solitons in a given sequence undergo the same dynamics. Taking into account collision-induced amplitude shifts due to broadband delayed Raman response and single-pulse amplitude changes induced by gain and loss, we obtain the following equation for amplitude dynamics of the th-sequence solitons (see Refs. NP2010 (); PNC2010 () for similar derivations):

(5)

where , and . The function on the right hand side of Eq. (5) is a polynomial in , whose form is determined by the form of . For and given by Eqs. (2) and (3), we obtain and , respectively. The coefficients on the right hand side of Eq. (5), which describe the strength of Raman interaction between th- and th-sequence solitons, are determined by the frequency dependence of the Raman gain. For the widely used triangular approximation for the Raman gain curve Agrawal2001 (); Chraplyvy84 (), in which the gain is a piecewise linear function of the frequency, for and NP2010 ().

In order to demonstrate stable scalable control of soliton propagation, we look for an equilibrium state of the system (5) in the form for . Such equilibrium state corresponds to steady-state transmission with equal amplitudes for all sequences. This requirement leads to:

(6)

Consequently, Eq. (5) takes the form

(7)

which is a generalized predator-prey model for species Lotka25 (); Volterra28 (). Notice that and are equilibrium states of the model for any positive values of , , , , and .

We point out that the derivation of an -dimensional predator-prey model with a general value is enabled by the narrow bandwidth of the waveguide’s nonlinear gain-loss. Indeed, due to this property, the gain-loss does not contribute to amplitude shifts in interchannel collisions, and therefore, three-pulse interaction can be ignored. This makes the extension of the predator-prey model from to a general value straightforward. As a result, extending waveguide setup design from to a general value for waveguides with broadband delayed Raman response and narrowband nonlinear gain-loss is also straightforward. This situation is very different from the one encountered for waveguides with broadband nonlinear gain-loss. In the latter case, interchannel collisions are strongly affected by the nonlinear gain-loss, and three-pulse interaction gives an important contribution to the collision-induced amplitude shift PC2012 (); PNG2014 (). Due to the complex nature of three-pulse interaction in generic three-soliton collisions in waveguides with broadband nonlinear gain or loss (see Ref. PNG2014 ()), it is very difficult to extend the LV model for amplitude dynamics from to a generic value for these waveguides. In the absence of an -dimensional LV model, it is unclear how to design setups for stable transmission stabilization and switching in -sequence systems with . As a result, transmission stabilization and switching in waveguides with broadband nonlinear gain-loss have been so far limited to two-sequence systems PC2012 (); CPJ2013 (); NPT2015 ().

Iii Stability analysis for the predator-prey model (7)

iii.1 Introduction

Transmission stabilization and switching are guided by stability analysis of the equilibrium states of the predator-prey model (7). In particular, in transmission stabilization, we require that the equilibrium state is asymptotically stable, so that soliton amplitude values tend to with increasing propagation distance for all sequences. Furthermore, transmission switching is based on bifurcations of the equilibrium state . To explain this, we denote by the value of the decision level, distinguishing between on and off transmission states, and consider transmission switching of sequences, for example. In off-on switching of sequences, the values of one or more of the physical parameters are changed at the switching distance , such that turns from unstable to asymptotically stable. As a result, before switching, soliton amplitudes tend to values smaller than in sequences and to values larger than in sequences, while after the switching, soliton amplitudes in all sequences tend to , where . This means that transmission of sequences is turned on at . On-off switching of sequences is realized by changing the physical parameters at , such that turns from asymptotically stable to unstable, while another equilibrium state with components smaller than is asymptotically stable. Therefore, before switching, soliton amplitudes in all sequences tend to , where , while after the switching, soliton amplitudes tend to values smaller than in sequences and to values larger than in sequences. Thus, transmission of sequences is turned off at in this case. In both transmission stabilization and switching we require that the equilibrium state at the origin is asymptotically stable. This requirement is necessary in order to suppress radiative instability due to growth of small amplitude waves PC2012 (); CPJ2013 (); NPT2015 ().

The setups of transmission switching that we develop and study in the current paper are different from the single-pulse switching setups that are commonly considered in nonlinear optics (see, e.g., Ref. Agrawal2001 () for a description of the latter setups). We therefore point out the main differences between the two approaches to switching. First, in the common approach, the amplitude value in the off state is close to zero. In contrast, in our approach, the amplitude value in the off state only needs to be smaller than , although the possibility to extend the switching to very small amplitude values does increase switching robustness. Second, in the common approach, the switching is based on a single collision or on a small number of collisions, and as a result, it often requires high-energy pulses for its implementation. In contrast, in our approach, the switching occurs as a result of the cumulative amplitude shift in a large number of fast interchannel collisions. Therefore, in this case pulse energies need not be high. Third and most important, in the common approach, the switching is carried out on a single pulse or on a few pulses. In contrast, in our approach, the switching is carried out on all pulses in the waveguide loop (or within a given waveguide span). As a result, the switching can be implemented with an arbitrary number of pulses. Because of this property, we can refer to transmission switching in our approach as channel switching. Since channel switching is carried out for all pulses inside the waveguide loop (or inside a given waveguide span), it can be much faster than conventional single-pulse switching. More specifically, channel switching can be faster by a factor of compared with single-pulse switching, where is the number of channels, whose transmission is switched, and is the number of pulses per channel in the waveguide loop. For example, in a 100-channel system with pulses per channel, channel switching can be faster by a factor of compared with single-pulse switching.

Our channel switching approach can be used in any application, in which the same “processing” of all pulses within the same channel is required, where here processing can mean amplification, filtering, routing, signal processing, etc. A simple and widely known example for channel switching is provided by transmission recovery, i.e., the amplification of a sequence of pulses from small amplitudes values below to a desired final value above it. However, our channel switching approach can actually be used in a much more general and sophisticated manner. More specifically, let represent the transmission state of the th channel, i.e., if the th channel is off and if the th channel is on. Then, the -component vector , where , represents the transmission state of the entire -channel system. One can then use this -component vector to encode information about the processing to be carried out on different channels in the next “processing station” in the transmission line. After this processing has been carried out, the transmission state of the system can be switched to a new state, , which represents the type of processing to be carried out in the next processing station. Note that the channel switching approach is especially suitable for phase-shift-keyed transmission. Indeed, in this case, the phase is used for encoding the information, and therefore, no information is lost by operating with amplitude values smaller than AK_transmission ().

iii.2 Stability analysis for transmission stabilization and off-on switching

Let us obtain the conditions on the values of the physical parameters for transmission stabilization and off-on switching. As explained above, in this case we require that both and the origin are asymptotically stable equilibrium states of the predator-prey model (7).

We first analyze stability of the equilibrium state in a waveguide with a narrowband GL gain-loss profile, where . For this purpose, we show that

(8)

where , is a Lyapunov function for Eq. (7) Lyapunov (). Indeed, we observe that for any with for , where equality holds only at the equilibrium point. Furthermore, the derivative of along trajectories of Eq. (7) satisfies:

(9)

where and . For asymptotic stability, we require . This condition is satisfied in a domain containing if . Thus, is a Lyapunov function for Eq. (7), and the equilibrium point is asymptotically stable, if linear_stab (). When , is globally asymptotically stable, since in this case, for any initial condition with nonzero amplitude values. When , for amplitude values satisfying for . Thus, in this case the basin of attraction of can be estimated by for . For instability, we require along trajectories of (7). This inequality is satisfied in a domain containing if . Therefore, is unstable for linear_stab ().

Consider now the stability properties of the origin for . Linear stability analysis shows that is asymptotically stable (a stable node) when for , i.e., when all pulse sequences propagate in the presence of net linear loss. To slightly simplify the discussion, we now employ the widely accepted triangular approximation for the Raman gain curve Agrawal2001 (); Chraplyvy84 (). In this case, for and NP2010 (), and therefore the net linear gain-loss coefficients take the form

(10)

Since is increasing with increasing , it is sufficient to require . Substituting Eq. (10) into this inequality, we find that the origin is asymptotically stable, provided that

(11)

The same simple condition is obtained by showing that is a Lyapunov function for Eq. (7).

Let us discuss the implications of stability analysis for and the origin for transmission stabilization and off-on switching. Combining the requirements for asymptotic stability of both and the origin, we expect to observe stable long-distance propagation, for which soliton amplitudes in all sequences tend to their steady-state value , provided the physical parameters satisfy

(12)

The same condition is required for realizing stable off-on transmission switching. Using inequality (12), we find that the smallest value of , required for transmission stabilization and off-on switching, satisfies the simple condition

(13)

As a result, the ratio should be a small parameter in -sequence transmission with . The independence of the stability condition for on and and the simple scaling properties of the stability condition for the origin are essential ingredients in enabling robust scalable transmission stabilization and switching.

Similar stability analysis can be carried out for waveguides with other forms of the nonlinear gain-loss Lyapunov (). Consider the central example of a waveguide with narrowband cubic loss, where . One can show that in this case , given by Eq. (8), is a Lyapunov function for the predator-prey model (7), and that

(14)

for any trajectory with for . Thus, is globally asymptotically stable, regardless of the values of , , , and . However, linear stability analysis shows that the origin is a saddle in this case, i.e., it is unstable. This instability is related to the fact that in waveguides with cubic loss, soliton sequences with values satisfying propagate under net linear gain, and are thus subject to radiative instability. The instability of the origin for uniform waveguides with cubic loss makes these waveguides unsuitable for long-distance transmission stabilization. On the other hand, the global stability of and its independence on the physical parameters, make waveguide spans with narrowband cubic loss very suitable for realizing robust scalable off-on switching in hybrid waveguides. To demonstrate this, consider a hybrid waveguide consisting of spans with linear gain-loss and cubic loss [] and spans with a GL gain-loss profile []. In this case, the global stability of for spans with linear gain-loss and cubic loss can be used to bring amplitude values close to from small initial amplitude values, while the local stability of the origin for spans with a GL gain-loss profile can be employed to stabilize the propagation against radiation emission.

iii.3 Stability analysis for on-off switching

We now describe stability analysis for on-off switching in waveguides with a GL gain-loss profile, considering the general case of switching off of out of soliton sequences. As explained in Subsection 3.1, in switching off of sequences, we require that is unstable, the origin is asymptotically stable, and another equilibrium state with components smaller than is also asymptotically stable. The requirement for instability of and asymptotic stability of the origin leads to the following condition on the physical parameter values:

(15)

In order to obtain guiding rules for choosing the on-off transmission switching setups, it is useful to consider first the case of switching off of N-1 out of N sequences. Suppose that we switch off the sequences and . To realize such switching, we require that is a stable equilibrium point of Eq. (7). The value of is determined by the equation

(16)

Since the origin is a stable equilibrium point, transmission switching of sequences can be realized by requiring that Eq. (16) has two distinct roots on the positive half of the -axis (the largest of which corresponds to ). This requirement is satisfied, provided Negative_g_j ():

(17)

Assuming that , we see that the switching off of the low-frequency sequences is the least restrictive, since it can be realized with smaller values. For this reason, we choose to adopt the switching setup, in which sequences are switched off. Employing inequality (17) and the triangular-approximation-based expression (10) for , we find that Eq. (16) has two distinct roots on the positive half of the -axis, provided that

(18)

Therefore, the switching off of sequences can be realized when conditions (15) and (18) are satisfied Eta_s_N ().

We now turn to discuss the general case, where transmission of out of sequences is switched off. Based on the discussion in the previous paragraph, one might expect that switching off of sequences can be most conveniently realized by turning off transmission of the low-frequency sequences, . This expectation is confirmed by numerical solution of the predator-prey model (7) and the coupled-NLS model (1). For this reason, we choose to employ switching off of sequences, in which transmission in the lowest frequency channels is turned off. Thus, we require that is an asymptotically stable equilibrium point of Eq. (7). The values of are determined by the following system of equations

(19)

where . Employing the triangular approximation for the Raman gain curve and using Eq. (10), we can rewrite the system as:

(20)

Stability of is determined by calculating the eigenvalues of the Jacobian matrix at this point. The calculation yields for and ,

(21)

and

(23)

Note that the Raman triangular approximation was used to slightly simplify the form of Eqs. (21)-(23). Since for and , the first eigenvalues of the Jacobian matrix are , where the coefficients are given by Eq. (21). Furthermore, since is either monotonically increasing or monotonically decreasing with increasing , to establish stability, it is sufficient to check that either or . To find the other eigenvalues of the Jacobian matrix, one needs to calculate the determinant of the matrix, whose elements are , where . The latter calculation can also be significantly simplified by noting that for , the diagonal elements are of order , while the off-diagonal elements are of order at most. Thus, the leading term in the expression for the determinant is of order . The next term in the expansion is the sum of terms, each of which is of order at most. Therefore, the next term in the expansion of the determinant is of order at most. Comparing the first and second terms, we see that the correction term can be neglected, provided that . We observe that the last condition is automatically satisfied by our on-off transmission switching setup for , since stability of the origin requires [see inequality (15)]. It follows that the other eigenvalues of the Jacobian matrix are well approximated by the diagonal elements for . Therefore, for , stability analysis of only requires the calculation of diagonal elements of the Jacobian matrix.

We point out that the preference for the turning off of transmission of low-frequency sequences in on-off switching is a consequence of the nature of the Raman-induced energy exchange in soliton collisions. Indeed, Raman crosstalk leads to energy transfer from high-frequency solitons to low-frequency ones Chi89 (); Malomed91 (); Kumar98 (); Kaup99 (); P2004 (); CP2005 (); NP2010b (). To compensate for this energy loss or gain, high-frequency sequences should be overamplified while low-frequency sequences should be underamplified compared to mid-frequency sequences NP2010 (); PNT2015 (). As a result, the magnitude of the net linear loss is largest for the low-frequency sequences, and therefore, on-off switching is easiest to realize for these sequences. It follows that the presence of broadband delayed Raman response introduces a preference for turning off the transmission of the low-frequency sequences, and by this, enables systematic scalable on-off switching in -sequence systems.

Iv Numerical simulations with the coupled-NLS model

The predator-prey model (7) is based on several simplifying assumptions, which might break down with increasing number of channels or at large propagation distances. In particular, Eq. (7) neglects the effects of pulse distortion, radiation emission, and intrasequence interaction that are incorporated in the full coupled-NLS model (1). These effects can lead to transmission destabilization and to the breakdown of the predator-prey model description PNC2010 (); PC2012 (); CPJ2013 (); NPT2015 (); PNT2015 (). In addition, during transmission switching, soliton amplitudes can become small, and as a result, the magnitude of the linear gain-loss term in Eq. (1) might become comparable to the magnitude of the Kerr nonlinearity terms. This can in turn lead to the breakdown of the perturbation theory, which is the basis for the derivation of the predator-prey model. It is therefore essential to test the validity of the predator-prey model’s predictions by carrying out numerical simulations with the full coupled-NLS model (1).

The coupled-NLS system (1) is numerically integrated using the split-step method with periodic boundary conditions Agrawal2001 (). Due to the usage of periodic boundary conditions, the simulations describe pulse propagation in a closed waveguide loop. The initial condition for the simulations consists of periodic sequences of solitons with amplitudes , frequencies , and zero phases:

(24)

where the frequency differences satisfy , for . The coefficients represent the initial position shift of the th sequence solitons relative to pulses located at for . To maximize propagation distance in the presence of delayed Raman response, we use for . As a concrete example, we present the results of numerical simulations for the following set of physical parameters: , , and . In addition, we employ the triangular approximation for the Raman gain curve, so that the coefficients satisfy for NP2010 (); PNT2015 (). We emphasize, however, that similar results are obtained with other choices of the physical parameter values, satisfying the stability conditions discussed in Section III.

Figure 2: The dependence of soliton amplitudes during transmission stabilization in waveguides with broadband delayed Raman response and narrowband GL gain-loss for two-sequence [(a) and (b)], three-sequence [(c) and (d)], and four-sequence [(e) and (f)] transmission. Graphs (b), (d), and (f) show magnified versions of the curves in graphs (a), (c), and (e) at short distances. The red circles, green squares, blue up-pointing triangles, and magenta down-pointing triangles represent , , , and , obtained by numerical simulations with Eqs. (1) and (2). The solid brown, dashed gray, dashed-dotted black, and solid-starred orange curves correspond to , , , and , obtained by the predator-prey model (7).

We first describe numerical simulations for transmission stabilization in waveguides with broadband delayed Raman response and a narrowband GL gain-loss profile [] for , , and sequences. We choose so that the desired steady state of the system is . The Raman coefficient is , while the quintic loss coefficient is for , for and , and for . In addition, we choose and initial amplitudes satisfying for , so that the initial amplitudes belong to the basin of attraction of . The numerical simulations with Eqs. (1) and (2) are carried out up to the final distances , , and , for , , and , respectively. At these distances, the onset of transmission destabilization due to radiation emission and pulse distortion is observed. The dependence of soliton amplitudes obtained by the simulations is shown in Figs. 2(a), 2(c), and 2(e) together with the prediction of the predator-prey model (7). Figures 2(b), 2(d), and 2(f) show the amplitude dynamics at short distances. Figures 3(a), 3(c), and 3(e) show the pulse patterns at a distance before the onset of transmission instability, where for N=2, for N=3, and for N=4. Figures 3(b), 3(d), and 3(f) show the pulse patterns at , i.e., at the onset of transmission instability. As seen in Fig. 2, the soliton amplitudes tend to the equilibrium value with increasing distance for , 3, and 4, i.e., the transmission is stable up to the distance in all three cases. The approach to the equilibrium state takes place along distances that are much shorter compared with the distances along which stable transmission is observed. Furthermore, the agreement between the predictions of the predator-prey model and the coupled-NLS simulations is excellent for . Additionally, as seen in Figs. 3(a), 3(c), and 3(e), the solitons retain their shape at despite the large number of intersequence collisions. The distances , along which stable propagation is observed, are significantly larger compared with those observed in other multisequence nonlinear waveguide systems. For example, the value for is larger by a factor of compared with the value obtained in waveguides with linear gain and broadband cubic loss PNC2010 (). Moreover, the stable propagation distances observed in the current work for , , and are larger by factors of 37.9, 34.3, and 10.6 compared with the distances obtained in single-waveguide transmission in the presence of delayed Raman response and in the absence of nonlinear gain-loss PNT2015 (). The latter increase in the stable transmission distances is quite remarkable, considering the fact that in Ref. PNT2015 (), intrasequence frequency-dependent linear gain-loss was employed to further stabilize the transmission, whereas in the current work, the gain-loss experienced by each sequence is uniform. We also point out that the results of our numerical simulations provide the first example for stable long-distance propagation of soliton sequences with in systems described by coupled GL models.

We note that at the onset of transmission instability, the pulse patterns become distorted, where the distortion appears as fast oscillations of that are most pronounced at the solitons’ tails [see Figs. 3(b), 3(d), and 3(f)]. The degree of pulse distortion is different for different pulse sequences. Indeed, for , the sequence is significantly distorted at , while no significant distortion is observed for the sequence. For , the sequence is significantly distorted, the sequence is slightly distorted, while the sequence is still not distorted at . For , the and sequences are both significantly distorted at , while no significant distortion is observed for the and sequences at this distance.

Figure 3: The pulse patterns near the onset of transmission instability for the two-sequence [(a) and (b)], three-sequence [(c) and (d)], and four-sequence [(e) and (f)] transmission setups considered in Fig. 2. Graphs (a), (c), and (e) show before the onset of instability, while graphs (b), (d), and (f) show at the onset of instability. The solid red curve, dashed-dotted green curve, blue crosses, and dashed magenta curve represent with , obtained by numerical solution of Eqs. (1) and (2). The propagation distances are (a), (b), (c), (d), (e), and (f).

The distortion of the pulse patterns and the associated transmission destabilization can be explained by examination of the Fourier transforms of the pulse patterns . Figure 4 shows the Fourier transforms at (before the onset of transmission instability) and at (at the onset of transmission instability). Figure 5 shows magnified versions of the graphs in Fig. 4 for small values. It is seen that the Fourier transforms of some of the pulse sequences develop pronounced radiative sidebands at . Furthermore, the frequencies at which the radiative sidebands attain their maxima are related to the central frequencies of the soliton sequences or to the frequency spacing . The latter observation indicates that the processes leading to radiative sideband generation are resonant in nature (see also Refs. PNT2015 (); CPN2016 ()).

Consider first the Fourier transforms of the pulse patterns for . As seen in Figs. 4(b) and 5(b), in this case the sequence develops radiative sidebands at frequencies and at . In contrast, no significant sidebands are observed for the sequence at this distance. These findings explain the significant pulse pattern distortion of the sequence and the absence of pulse pattern distortion for the sequence at . In addition, the radiative sideband frequencies satisfy the simple relations: and . For , the sequence develops significant sidebands at frequencies and , the sequence develops a weak sideband at frequency , and the sequence does not have any significant sidebands at [see Figs. 4(d) and 5(d)]. These results coincide with the significant pulse pattern distortion of the sequence, the weak pulse pattern distortion of the sequence, and the absence of pulse pattern distortion for the sequence at . Additionally, the sideband frequencies satisfy the simple relations: , , and . For , the and sequences develop significant sidebands, while no significant sidebands are observed for the and sequences at [see Figs. 4(f) and 5(f)]. These findings explain the significant pulse pattern distortion of the and sequences and the absence of significant pulse pattern distortion for the and sequences at . The sideband frequencies of the sequence satisfy the relations: , and . Note that the values of and for are very close to the values found for . Finally, the sideband frequencies of the sequence satisfy the relations: , and .

Figure 4: The Fourier transforms of the pulse patterns near the onset of transmission instability for the two-sequence [(a) and (b)], three-sequence [(c) and (d)], and four-sequence [(e) and (f)] transmission setups considered in Figs. 2 and 3. Graphs (a), (c), and (e) show before the onset of instability, while graphs (b), (d), and (f) show at the onset of instability. The red circles, green squares, blue up-pointing triangles, and magenta down-pointing triangles represent with , obtained by numerical solution of Eqs. (1) and (2). The propagation distances are (a), (b), (c), (d), (e), and (f).
Figure 5: Magnified versions of the graphs in Fig. 4 for small values. The symbols and distances are the same as in Fig. 4.

We now turn to describe numerical simulations for a single transmission switching event in waveguides with broadband delayed Raman response and a narrowband GL gain-loss profile. As described in Section III, on-off switching of out of pulse sequences at a distance is realized by changing the value of one or more of the physical parameters, such that the steady state turns from asymptotically stable to unstable, while another steady state at is asymptotically stable. We denote the on-off switching setups by A1-A2, where A1 and A2 denote the sets of physical parameters used at and , respectively.

Off-on switching of out of soliton sequences at is realized by changing the physical parameter values such that turns from unstable to asymptotically stable. As explained in Section III, to achieve stable long-distance transmission after the switching, one needs to require that the origin is an asymptotically stable steady state as well. Under this requirement, must satisfy inequality (12), and as a result, the basin of attraction of is limited to for . This leads to limitations on the turning on of the sequences, especially for and . To overcome this difficulty, we consider a hybrid waveguide consisting of a span with a GL gain-loss profile, a span with linear gain-loss and cubic loss, and a second span with a GL gain-loss profile. The introduction of the intermediate waveguide span with linear gain-loss and cubic loss enables the turning on of the sequences from low amplitude values due to the global stability of the steady state for the corresponding predator-prey model. However, due to the presence of linear gain and the instability of the origin for the same predator-prey model, propagation in the waveguide span with linear gain-loss and cubic loss is unstable against emission of small amplitude waves. For this reason, we introduce the frequency dependent linear gain-loss of Eq. (LABEL:global3a) when simulating propagation in the second span. More importantly, propagation in the second span with a GL gain-loss profile leads to mitigation of radiative instability due to the presence of linear loss in all channels for this waveguide span. This enables stable long-distance propagation of the soliton sequences after the switching. We denote the off-on switching setups by A2-B-A1, where A2, B, and A1 denote the sets of physical parameters used in the first, second, and third spans of the hybrid waveguide. The first span is located at , the second at , and the third at , where is the final propagation distance. Thus, off-on switching of the soliton sequences occurs at , while final transmission stabilization takes place at .

We present here the results of numerical simulations for on-off and off-on switching of two and three soliton sequences in four-sequence transmission. As discussed in the preceding paragraphs, on-off switching setups are denoted by A1-A2 and off-on switching setups are denoted by A2-B-A1. The following values of the physical parameters are used. The Raman coefficient is , which is the same value used in transmission stabilization. The other parameter values used in setup A1 in both on-off and off-on switching are , , and . The parameter values used in setup A2 in on-off switching are , , and for and , , and for . The on-off switching distance is for both and . The parameter values used in setup A2 in off-on switching are , , and for and , , and for . The parameter values used in off-on switching in setup B are and for both and . To suppress radiative instability during propagation in waveguide spans with linear gain-loss and cubic loss (setup B), the frequency dependent linear gain-loss of Eq. (LABEL:global3a) with and is employed. The switching and final stabilization distances in off-on transmission switching are and for , and and for . We point out that similar results were obtained with other choices of the physical parameter values, satisfying the stability conditions discussed in Section III.

The results of numerical simulations with Eqs. (1) and (2) for on-off switching of two and three soliton sequences in four-sequence transmission in setup A1-A2 are shown in Figs. 6(a) and 6(b). The results of simulations with Eqs. (1)-(LABEL:global3a) for off-on switching of two and three sequences in four-sequence transmission in setup A2-B-A1 are shown in Figs. 6(c) and 6(d). A comparison with the predictions of the predator-prey model (7) is also presented. The agreement between the coupled-NLS simulations and the LV model’s predictions is excellent in all four cases. More specifically, in on-off transmission of sequences with and , the amplitudes of the solitons in the lowest frequency channels tend to zero, while the amplitudes of the solitons in the high frequency channels tend to new values , where . The values of the new amplitudes are and in on-off switching of two sequences, and in on-off switching of three sequences. As can be seen from Figs. 6(a) and 6(b), these values are in excellent agreement with the predictions of the predator-prey model (7). In off-on switching of soliton sequences, the amplitudes of the solitons in the low frequency channels tend to zero for , while the amplitudes of the solitons in the high frequency channels increase with for . After the switching, i.e., for distances