Effects of saturable function in three-core PIM-NIM-PIM coupler through modulation instability

We consider a model of three-core PIM-NIM-PIM coupler with Kerr-type saturable nonlinearity to study both analytically and numerically saturation effects on the modulational instability (MI) phenomenon. The analytical results show us that, in presence of saturable parameters in normal and anomalous dispersion regime the instability gain presents significant changes compared to Shafeeque et al. (Phys Lett A 11:223–229, 2015) in absence of saturable parameter. It clearly also shows that Power and nonlinear parameters can be used to control MI. Through numerical simulation, the generations of periodic soliton are obtained in presence of nonlinear saturation. However in absence of saturable parameter, the train of soliton obtained turns into a turbulent state after a certain propagations distance, and in that case, the system is prone to present MI.


Introduction
Negative index material (NIM) is material which both the permittivity ( ) and permeability ( ) parameters are set as negative at the same frequency Veselago (1968). The existence of such media was experimentally demonstrated first in the microwave Shelby et al. (2001) and then in the near-IR ranges (Linden et al. 2004;Zhang et al. 2005). NIM or metamaterial can be artificially designed and their properties derive from their structures but not from the properties of the base materials. NIM research is interdisciplinary and include many fields such as optics, optoelectronics, nanoscience, antenna and electrical engineering (Said et al. 2008). In optics negative index material offer many potential applications such as optical filters, lenses for high-gain antennas, improving ultrasonic sensors, super lenses etc Boratay and Ekmel (2007).
In nonlinear optics, the waveguiding structure composed of two adjacent waveguides preserving the direction of light propagation is called a directed coupler. Jensen (1982) was the first researcher to introduce the notion of nonlinear directional coupler. Nonlinear directional coupler has applications in optical communication systems such as switching, wavelength-selective coupling, multi/demultiplexing and power splitting (Jensen 1982;Kengne and Liu 2019). if one of the waveguides of the coupler is made from a material with a negative refractive index, the direction of input and output fields are exactly opposite in nature but the path of light propagation is preserves. This coupler is called the oppositely directed coupler in order to distinguish from the conventional one made from a material with a positive refractive index (PIM) Litchinitser et al. (2007). Such structure is first introduced by Halterman et al. in 2003Halterman et al. (2003, then demonstrated experimentally by Yuan et al. In 2006Yuan et al. (2006. The interaction between the nonlinear effect and group-velocity dispersion (GVD) is named MI. In nonlinear wave systems modulational instability is the most fundamental processes in nature, characterized by a continuous wave (CW) when it propagates together with a weak noise Agrawal (2001). Recent years, some researchers investigated the modulational instability (MI) in a nonlinear oppositely directed coupler with a negative-index material channel (Xiang et al. 2010;Shafeeque et al. 2014;Zhang et al. 2015;Kengne 2021). MI in nonlinear positive-negative index couplers with saturable nonlinearity (Tatsing et al. 2012;Alves et al. 2016;Tatsing et al. 2016Tatsing et al. , 2018Tatsing et al. , 2019Houwe et al. 2021). The influence of self-steepening and intrapulse Raman scattering on MI in oppositely directed coupler Shafeeque et al. (2014).
Another form of optical couplers are multi-core directional couplers that consist of more than two waveguides. Such couplers improved transmission characteristics that why they attracted much attention. Sharper power switching curves are well offer by three core couplers than conventional two-core coupler and for that raison three core couplers are really significant (Langridge and Firth 1992;Shafeeque et al. 2015. Many physical systems present nonlinear saturation that may influence them. Some important changes in instability band particulary in shape and/or amplitude can be observe due to the presence of nonlinear saturation in the systems. This has encouraged some researchers to study the MI in many saturable nonlinear systems (Tatsing et al. 2012;Alves et al. 2016;Tatsing et al. 2016Tatsing et al. , 2018Tatsing et al. , 2019Houwe et al. 2021). Mohanraj and Sivakumar (2021) study saturable higher nonlinearity effects on the modulational instabilities in three-core triangular configuration and show the possibility to observe the modulational instability (MI) regions due to modified nonlinear saturability over and above other high-order nonlinearities. Their results showed that, in the normal region, the saturable nonlinearity aids in increasing the bandwidth of the MI region, while in the anomalous regime, the bandwidth reduces.
Further, the gain of the instability region reduces as they increase the saturable nonlinearity strength. Recently the same author Mohanraj et al. (2022) study modulational instability in three-core nonlinear directional saturated coupler with septic nonlinearity and concluded that septic nonlinearity plays a major role in three-core couplers with negative indexed material channel. However in this present work, we study the effects of saturable nonlinearity in a three-core coupler with a negative material channel through modulation instability analytically and numerically. The rest of the paper is organized as follows: Sect. 2 present mathematical model and standard linear stability analysis. In Sect. 3 we carried out in detail the influence of the parameters of the three channels of the coupler on the MI. In Sect. 4, numerical simulation are presented and finally, in Sect. 5 we concludes the paper. Shafeeque et al. (2015) introduced continuous wave propagation in a nonlinear three-core optical coupler containing a NIM channels by neglecting the cross-phase modulation (XPM) effect and high order time derivative terms. However the model present by Shafeeque et al. (2015) does not include saturation effects, whereas saturation nonlinearity play a relevant role in the propagation of ultrashort pulse. In presence of saturable nonlinearity function the above model is considered as follows.

Propagation equations and dispersion relation
Here channel 1 and channel 3 are PIM and channel 2 is NIM. 1 , 2 and 3 stand for sign of the refractive index. In this work, 1 = 3 = 1 and 2 = −1 . a 1 , a 2 and a 3 are the complex normalized amplitudes; The absolute values of the group velocities for channel 1,2 and 3 are given by v 1g , v 2g and v 3g . k 12 , k 21 , k 23 and k 32 are the coupling coefficients respectively. 1 , 2 and 3 are nonlinear coefficient.
The Modified Kerr type saturable nonlinearity function used here have the following expression: Let us now insert Eq. (5) in Eqs. (1) and (2), we obtain the nonlinear dispersion relations.
Where f = u 2 ∕u 1 and l = u 2 u 3 , are the quantities that describe how the Power P = u 2 1 + u 2 2 + u 2 3 is divided between the forward-and backward propagating waves.

Linear stability analysis
To observe verywell the effect of the saturable nonlinearity on MI, linear stability analysis are employ. The first step consists to add small perturbations terms to a CW and then survey if the latter augments or disintegrate with propagation. When we take into account the small perturbations, Eq. (5) turn to where i is a term of perturbation ( | | i | | ≪ u i , i = 1, 2 and 3), i is small. To obtain the linear form, we substitute Eq. (8) into Eqs. (1), (2) and (3) and linearize in i ( 1 + 2 f 6 ), The solution of the above linear Eqs. (9), (10) and (11) can be estimate as follows: Where m j and n j are real constants, K is the wave number and Ω is the perturbation frequency. By introducing Eq. (12) into Eqs. (9), (10) and (11), we obtain a matrix 6 × 6 having the following elements: set of six linear coupled for m i and n i .
where and We assume in this paper that The roots of the determinant of the matrix R should possess a nonzero and negative imaginary part. This allow to determine the stability of CW. Then, the discussion of the MI depends on the power gain spectrum given by: 240 Page 6 of 17

MODULATIONAL INSTABILITY IN THREE-CORE PIM-NIM-PIM OPTICAL COUPLER
We will discuss in detail now the dynamical behaviors of MI in the oppositely directed three-core coupler. Six cases will be analyzed here:

Influence of pump power and Saturable Nonlinearity on Modulational Instability
We discuss the influence of the power and coupling coefficient on the modulation instability gain before discussing in detail the individual cases listed above. In normal dispersion regime, Fig. 1a show us that in absence of saturation the initial gain increases with K and then saturated Shafeeque et al. (2015). In Fig. 1b  (i) f > 0 ; l > 0 and 1 = 2 = 3 = 1 ; (iv) f < 0 ; l < 0 and 1 = 2 = 3 = 1.

Influence of coupling coefficient and Saturable Nonlinearity on Modulational Instability
To understand the Influence of coupling coefficient and saturation on instability gain, Figs. 3 and 4 were plotted. Figures 3 and 4 shows the gain as a function of K for some values of coupling coefficient k. In normal dispersion region show in Fig. 3a for Γ = 0 , MI gain increases with the increases of coupling coefficients and the instability band shifts towards higher value of K and saturate, but in presence of Γ = 0.2 show by Fig. 3b, MI gain increase in K and reaches a maximum and then decreases. Figure 4 show the instability dependence on the coupling coefficient for anomalous dispersion regime. In Fig. 4a where Γ = 0 one single MI gain is observe and increases with K when the coupling coefficient increase. However the case is different when the saturation is taking into account, Fig. 4b show the case where Γ = 0.2 , here a new band appear when the saturable parameter exist for the higher value of K. We also note here that the MI gain increase with the increase of coupling coefficient parameter. The two instability regions separated by a relatively wide stable region is observed with higher value of coupling coefficient contrary to the one of pump power. As coupling coefficient increases, the gain increases proportionally, but the wide stable region increase also. If the coupling coefficient decreases, the gain would decreases proportionally, and the two instability regions approach each other, thereby narrowing the stability region.

Influence of f on modulational instability
We focus on the influence of f for different combination values of nonlinear parameters 1 , 2 and 3 and saturation parameter on MI. To this end, we know That sgn(f ) = 1 stands for normal and sgn(f ) = −1 for anomalous dispersion, then both the sign and the value of f can influence MI.
To analyse the impact of f on MI, we stand firstly in the normal dispersion regime where the parameter f > 0 . Figure 5 show the dependence of the gain spectrum with respect to K and f . Figure 5a and b describes a coupler of case (i) where all the channels are nonlinear. In absence of saturable parameter we obtained four sidebands in Fig. 5a Shafeeque et al. (2015). However, in presence of saturable parameter Γ = 0.1 Fig. 5b show us more than four sidebands, MI only presents within a limited range of f , here f cr 1 < f < 0.5 and 2 < f < f cr 2 , f cr 1 andf cr 2 are the critical value at the threshold condition. It is clear here that the instability bands under saturation effects reduce the width also its amplitude. Figure 5c and d shows the case in which a NIM channels is linear ( 2 = 0 ) stand for case (ii). In Fig. 5c two centered sidebands are observed around the zero propagation constant region and a nil value gain exists along the zero propagation constant K=0. Figure 5d show us the case where ( Γ = 0.1 ) and there two sideband were observed. We also note here that by increasing the saturation parameter the gain amplitude is reduced also its width. In Fig. 5e and f we study the effect of NIM channel, here Channels 1 and 3 are linear and channel 2 nonlinear stand for case (iii) in presence and in absence of saturation. Note that without saturation ( Γ = 0 ) there are two MI bands centered around the zero propagation constant region which are observed in Fig. 5e. In presence of saturation Fig. 5f it is evident the influence of the value of the saturation parameter on the number of MI bands. Then, by comparing the Fig. 5e with f one observes the appearance of a new instability bands for the highest value of f in Fig. 5f due to the presence of saturation. However, the increase of saturation parameter reduced the maximum instability gain and its width. From Fig. 5, we can concluded that adjusting the value of f , and Γ can be used to control MI in nonlinear three core coupler. In normal dispersion regime threshold condition exists for f , further the increase of saturation parameter can reduced the maximum gain and also reduce the width of sideband.
In anomalous dispersion regime, MI can also be altered by varying f such as ( f < 0 ). In Fig. 6a and b standing for the case where the three media are nonlinear media (case (iv)); we note in Fig. 6a that in absence of saturation parameter ( Γ = 0 ) there exist two MI bands centered around the line K = 0 Shafeeque et al. (2015). However, in presence of saturation parameter for ( Γ = 0.6 ) we observe in Fig. 6b four bands, in this figure the influence of saturable parameter is clearly observe on the number of MI bands. Figure 6c and d When channels 1 and 3 standing for PIM channels are nonlinear and channel 2 standing for NIM channel is linear 2 = 0 show the gain spectrum stand for case (v), two distinct sidebands appear on either side of zero propagation constant region. In Fig. 6c, in absence of saturation two sidebands are obtained and the maximum gain attained at lower value of |f | . In Fig. 6d the maximum gain and the band width are influenced by the presence of saturation parameter, when saturation is present MI only presents within a limited range of f and MI is obtained for lower value of |f | . Threshold condition for f can exist only in Fig. 6d instead of the Fig. 6c. In Fig. 6e and f where channels 1 and 3 are linear ( 1 = 3 = 0 ) and channel 2 nonlinear 2 = 1 (case vi) two sidebands are obtained for the lower value of |f | . In presence of saturation, Fig. 6f also show us that the maximum gain increase with the increase of saturable parameter and a large stability zone around the propagation constant K exist. Hence, from Fig. 6 it can be concluded that in anomalous dispersion regime saturation enlarge the gain value and the generation region of the MI, and the MI generation is thresholdless for f .

Direct numerical simulations
In this section, to verify the modulation stability/instability of weakly perturbed continuous waves studied in the analytical form, the nonlinear evolution of the MI was performed numerically. Pseudo-spectral method (Gottlieb and Orszag 1977;Fornberg 1984) was used to employ direct numerical simulations of Eqs. (1), (2) and (3). The initial conditions were taken in the form of the wave plane with imposed small periodic perturbation: where = 5 * 10 −2 is a relative value of the perturbation, and Ω = 1.6 is the angular frequency of weak modulation imposed on the continuous wave. u j is the power of continuous wave.
(16) a j (0, ) = u j [1 + cos(Ω )], (j = 1, 2, 3).  Figure 7 stand for the evolution MI in the normal dispersion regime and Fig. 8 anomalous dispersion regime. We observe in Fig. 7a and c in absence of saturation Γ = 0 , main effects are oscillations of the background. Figure 7b and d shows the influence of saturation parameter Γ = 0.1 . In this case, a periodic chain of solitons like pulses are produced in both cores. In anomalous dispersion regime in absence of saturation show in Fig. 8a and c we clearly observe the stability at the initial stage of the evolution of initial perturbed CW, but the train of solitons turns into a chaotic pulse after certain propagations distance z > 60. Figure 9 show the case where the system is only influence by the PIM channels stand for case (ii) and (v). In absence of saturation Fig. 9a and c show us a chain of solitons with growing amplitudes which are generated. However when saturation parameter is present Γ = 0.6 ; Fig. 9b and d show there a generation of a periodic array of peaks in normal and anomalous dispersion regime. Figure 10 reveals the impact of NIM channel ( 1 = 3 = 0 (kW m) −1 , 2 = 1 (kW m) −1 ) and saturable parameter on MI stand for case (iii) et (vi). In absence of saturation Figs. 10a and c show this case, the MI generates a chain of growing peaks narrow soliton oscillating with higher amplitude for high value of propagation distance. In Fig. 10b and d, we address the case when saturation Γ = 0.6 and NIM channel 1 = 1 in normal and anomalous dispersion regime. In this case, we clearly observe the stability of the evolution of initial perturbed continuous wave.
We clearly observe from Fig. 10 that the presence of NIM channel and saturation influence the generation of MI. It can be concluded from Figs. 7,8,9 and 10 that in absence of saturation parameter Γ = 0 the system is prone to present MI (like Alves et al. 2016). However in presence of saturation parameter we will not have MI.

Conclusion
This paper investigated more the MI with modified Kerr-type saturable nonlinearity(MSN) in nonlinear three-core directional couplers. Continuous wave solutions and linearizing firtly are used to obtain the dispersion relation. Secondly the influence on MI of pump power, coupling coefficient in presence of saturation was analysed. The result show that, the presence of saturable nonlinearity can bring new sidebands or reduce the number of sidebands through the systems. At end, a numerical simulation of the nonlinear development of the MI in different regimes which were studied analytically was performed. In absence of saturation, we clearly observe the stability at the initial stage of the evolution of initial perturbed CW, but the train of solitons turns into a turbulent state after certain propagations distance. However in presence of nonlinear saturation, a generation of periodic soliton arrays with a growing amplitude are obtained. Numerical studies present results that agree with our analytical study. The present study provide a new way to generate solitons or ultrashort pulses in an oppositely directed three-core coupler with saturable nonlinearity.
Funding The authors have not disclosed any funding.

Data availability
No data were used to support this study.

Conflict of interest
The authors declare that they have no conflict of interest.