Interplay between relaxation of nonlinear response and coupling coefficient dispersion in the instability spectra of dual core optical fiber

Optics Communications 303 (2013) 46–55

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Interplay between relaxation of nonlinear response and coupling coefficient dispersion in the instability spectra of dual core optical fiber K. Nithyanandan, K. Porsezian n Department of Physics, Pondicherry University, Puducherry 605 014, India

art ic l e i nf o

a b s t r a c t

Article history: Received 20 February 2013 Received in revised form 1 April 2013 Accepted 7 April 2013 Available online 28 April 2013

The modulational instability of a dual core optical fiber under the combined effect of coupling coefficient dispersion (CCD) and relaxation of nonlinear response is presented. The effect of wavelength dependence of coupling coefficient is effectively included in the pair of generalized linearly coupled nonlinear Schrödinger equation (CNLSE). In order to account for the finite response time, a time dependent nonlinear response is incorporated in the system of CNLSE using the Debye relaxation model. The results from the linear stability analysis show that in anomalous dispersion regime for any finite relaxation time, the nonlinearity becomes complex and there exist two unstable modes. In the normal dispersion regime, a single unstable band at higher detuning frequency is observed which is identified as the Raman band. In symmetric case the CCD is found to be immaterial, whereas in asymmetric case the CCD is found to be crucial and brings new spectral bands. A critical value of CCD is predicted where the dynamics evolve dramatically in different manners. Thus in this paper, the MI dynamics of a dual core optical fiber is fully explored and the interplay between CCD and relaxation is clearly highlighted. & 2013 Elsevier B.V. All rights reserved.

Keywords: Modulational instability Optical fibers Nonlinear Schrödinger equation Coupling coefficient dispersion Linear stability analysis Relaxing nonlinearity

1. Introduction In the classical optics, the intensity dependent process in the optical system seeds the emergence to the field of nonlinear optics [1], which deserves a considerable attention for more than five decades in terms of both fundamental and application perspectives. One such exciting prospect that earns serious scientific interest nearly for three decades is the so-called modulational instability (MI) [2–9]. In nonlinear optics, MI is a phenomena in which a weak perturbation imposed on a continuous or quasi continuous wave leads to the exponential growth of the weak perturbation as a consequence of the phase matching between linear and nonlinear effects. Such conservative interaction between dispersion and self phase modulation is closely related to the concept of soliton, thus the MI is recognized as the soliton precursor [2,4,10]. MI as a natural means of generating ultrashort soliton like pulses was proposed theoretically by Hasegawa in 1984 [6], and confirmed through experiments by Tai et al. in 1986 [7]. Later on, MI picks momentum and evolves with ages, despite new findings and modern innovations, MI still remains as the subject of high scientific curiosity. For instance, the sweeping interest in MI is powered by its potential applications such as generation of

n

Corresponding author. Tel./fax: +91 413 2655183. E-mail addresses: [email protected] (K. Nithyanandan), [email protected], [email protected] (K. Porsezian). 0030-4018/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.optcom.2013.04.018

ultrashort pulses, soliton aided ultra high bit-rate optical communication system, instability induced supercontinuum generation, loss compensation and optical amplification of weak signal, frequency comb for metrology, all-optical switching are some of the examples of enduring interest [11–21]. The dynamical equation governing the propagation of light beam in optical fiber is deeply connected with the nonlinear Schrödinger equation (NLSE). The perception of investigation takes different dimensions depending upon the nature of waveguide such as single or multimode [22], conventional or photonic crystal fiber, high or low birefringence accordingly scalar or vector MI [23–27], constant or varying dispersion such as dispersion decreasing (increasing) fibers, dispersion shifted and flattened fibers [28–31], single core or multicore (two, three core fibers) [2,32], type of dopants such as erbium or ytterbium in the fibers [33], etc. Also, the input profile of the injecting light, such as nature of the pulse (Gaussian, sech, etc.), continuous or discrete pulses, input pulse width (pico, subpico and femtosecond pulses), intensity, etc., will affect and significantly modifys the NLSE and the associated dynamics of the MI [2–4]. Among all, our present focus is on the dual core optical fiber, which acts as a nonlinear directional coupler [34,35]. The dual core fiber can be either formed by means of two parallel single mode optical fiber or through a photonic crystal fiber. Some of the useful devices based on dual core fibers that find wide applications are nonlinear directional couplers, ultrafast optical switching, power splitters, wavelength division multiplexing, multifrequency generation or supercontinuum

K. Nithyanandan, K. Porsezian / Optics Communications 303 (2013) 46–55

generation, etc. [36–41]. In 1988 [21], Trillo et al. with an insight theoretical treatment analyzed the impact of various combination of physical effects in the MI dynamics of nonlinear directional coupler. Also, the authors studied the close relation between the MI in two core fiber and the birefringent glass fibers. The MI in the asymmetric dual core optical fiber was analyzed by Tasgal et al. [42]. The evolution of the optical beam in the dual core fiber is given by a pair of linearly coupled nonlinear Schrödinger equation [43–45]. The periodic power transfer between the two cores of the fiber is governed by the linear coupling coefficient. However, in the ultrashort pulse regime, the so-called coupling coefficient dispersion (CCD) arising due to the wavelength dependence of coupling coefficient is found to be crucial [46–49] especially in long fibers and can cause severe pulse distortion, which eventually leads to pulse break up [50,51]. The effects of CCD and its relative influence in various physical mechanisms were studied in detail both theoretically and also through experiments by Chiang and his group through various analysis [43–48,50,51]. Considering the importance of CCD, Li et al. extended the results of the Refs. [21,42] with the inclusion of CCD [52]. The authors have reported that CCD does not affect the symmetric/asymmetric CW state but dramatically changes the MI of the asymmetric CW state. However, the Ref. [52] has not taken into account the effects of delayed Raman response, which is found to be essential, especially in the ultrashort pulse regime. A comprehensive picture of the interplay between MI and SRS in the ultrashort regime can be seen from the Refs. [53–57]. Therefore, one must include, in addition to the instantaneous Kerr nonlinearity, the effects of relaxation of nonlinearity or Raman scattering. Thus, we extended further the result of Ref. [52] beyond the limiting case of instantaneous nonlinear response. We have suitably modeled the CNLSE to account for the noninstantaneous nonlinearity by including a time dependent nonlinear response through a model called as Debye relaxational model [21,58–61]. Therefore the nonlinearity is not only a pure function of the intensity of the optical beam but depends also on the finite response time of the medium. Thus, in the present study, we discuss the cumulative effect of the finite response time and the coupling coefficient dispersion in a two-core optical fiber. In order to provide a self-explanatory note of the above stated objective, we consider both the dispersion regime, anomalous and normal dispersion regime. The manuscript is organized as follows: following a detailed introduction in the Section 1. Section 2 features the theoretical model, followed by the linear stability analysis of both symmetric and asymmetric CW states in Section 3. Section 4 deals with the detailed MI analysis of different types of dispersion regime under subsection. The conclusion of the paper is presented in Section 5 with a detailed summary of the results.

2. Theoretical model The equation describing the evolution of the slowly varying electric field envelopes (Ej, j¼ 1,2) in a single mode dual core fiber is given by a pair of linearly coupled modified nonlinear Schrödinger equations (CNLSE) as follows [52,44,50]: i

∂Ej 1 ∂2 Ej ∂E3−j − β ¼0 þ γjEj j2 Ej þ κE3−j þ iκ1 ∂z 2 2 ∂t 2 ∂t

ð1Þ

where z and t are the longitudinal coordinate and time in the comoving frame of reference, respectively. β2 and γ are the group velocity dispersion and Kerr parameter, respectively. γ ¼ ðn2 ω0 Þ= ðcAeff Þ, where n2 is the nonlinear index coefficient, Aeff is the effective core area and ω0 is the carrier frequency. κ is the coupling coefficient and κ 1 ¼ dκ=dω is the coupling coefficient dispersion corresponding to the wavelength dependence of the coupling coefficient and is equivalent to the intermodal dispersion.

47

The above equation is similar to the Ref. [52], which can only describe the propagation of pulses in the limiting case of instantaneous response. In principle, when operating the system in the ultrashort regime, in addition to the instantaneous response due to electronic contribution, one must include the slow response of thermal origin or re-orientational nonlinearity, whose time scale ranges between tens of picosecond to hundreds of nanoseconds. The usual description to account for this delayed response is to include additional terms corresponding to the Taylor expansion of the delayed envelope amplitude in the nonlinear Schrödinger equation. Although the above assumption is suitable in the low frequency regime, but it notably fails in the high frequency regime [21,60]. In order to address the issue, a simple relaxational model known as Debye relaxational model is considered, which is found to be reliable regardless of the frequency regime [21,60,62]. The equation governing the dynamics of the pulse propagation in a noninstantaneous dual core fiber with a finite response time ðτÞ can be written as i

∂Ej 1 ∂2 Ej ∂E3−j − β þ γNj Ej ¼ 0; þ κE3−j þ iκ 1 ∂z 2 2 ∂t 2 ∂t

ð2aÞ


∂N j 1 ¼ ð−N j þ
Ej j2 Þ: τ ∂t

ð2bÞ

The parameter Nj ¼ Nj ðz; tÞ is the nonlinear index of the medium and can replace jE2j j in the original NLSE. The dynamics of N depends on the finite response time of the medium (τ) and the local intensity of the field. 3. Linear stability analysis 3.1. Symmetric/antisymmetric solutions The stability of the steady state solution against harmonic perturbation is studied using linear stability analysis (LSA). The symmetric/antisymmetric continuous wave (CW) steady state solution can be written as 0 ECW 1 ¼ E1 expðiϕðzÞÞ;

0 ECW 2 ¼ δE 2 expðiϕðzÞÞ:

The symmetric/antisymmetric CW corresponds to E0 ¼ NCW j

2

¼ jE0 j :

ð3aÞ E01

¼

E02 . ð3bÞ

where ϕðzÞ is the nonlinear phase shift with ϕðzÞ ¼ ðγN þ δκÞz. δ can take either +1 or −1 corresponding to symmetric or antisymmetric solutions. The stability of the steady state solution can be studied by perturbing the above fields of the form Ej ¼ ðE0 þ aj ðz; tÞÞ expðiϕðzÞÞ;

ð4aÞ

Nj ¼ nj ðz; tÞ þ jE0 j2

ð4bÞ

2

2

jaj j ⪡jE0 j . Here, aj is the small amplitude perturbation and nj is the deviation from the stationary solution of the nonlinear index of the medium. Using Eqs. (4) in Eqs. (2) and neglecting the higher order perturbation terms one will arrive into a linearized equation for the perturbations aj and nj as follows: i

∂aj 1 ∂2 aj ∂a3−j − β þ γnj E0 ¼ 0; þ κða3−j −δaj Þ þ iκ 1 ∂z 2 2 ∂t 2 ∂t

∂nj 1 ¼ ½−nj þ E0 ðaj þ anj Þ τ ∂t

ð5aÞ ð5bÞ

We assume the following ansatz for the perturbation with frequency detuning from the pump Ω, and K will be the wavenumber of the perturbation: aj ¼ U j exp½−iðKz−ΩtÞ þ V j exp½iðKz−ΩtÞ;

ð6aÞ

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K. Nithyanandan, K. Porsezian / Optics Communications 303 (2013) 46–55

nj ¼ U j exp½−iðKz−ΩtÞ þ V j exp½iðKz−ΩtÞ:

ð6bÞ

where Uj and Vj are the amplitudes of the perturbations representing the anti-Stokes and Stokes sidebands, respectively. Introducing Eqs. (6) in Eqs. (5) results in four homogenous equations for Uj and Vj. The compatibility condition of two equations in U1 and V1 will lead to the relation as follows: ! 1 4γE20 2 2 2 ðK−δκ 1 ΩÞ ¼ β2 Ω β2 Ω þ ð7Þ 4 ð1 þ iΩτÞ Similarly, the compatibility condition corresponding to U2 and V2 leads to the dispersion relation corresponding to the second wave and can be written as ! 1 4γE20 2 2 2 ðK þ δκ 1 ΩÞ ¼ ðβ2 Ω −4δκÞ β2 Ω −4δκ þ ð8Þ 4 ð1 þ iΩτÞ The general dispersion relation after some mathematical manipulations can be written as ½ðK−δκ1 ΩÞ2 −f 1 ½ðK þ δκ 1 ΩÞ2 −f 2 Þ ¼ 0:

ð9Þ

! 1 4γE20 2 2 f 1 ¼ β2 Ω β2 Ω þ ; 4 ð1 þ iΩτÞ 1 4γE20 f 2 ¼ ðβ2 Ω2 −4δκÞ β2 Ω2 −4δκ þ 4 ð1 þ iΩτÞ

ð10aÞ ! ð10bÞ

In principle, the MI is said to occur whenever Im[KðΩÞ]≠0. For the case of instantaneous nonlinear system, the delay time τ-0, then γ~ -γ. Therefore K becomes complex for f 1 o 0, which can be made possible only when group dispersion coefficient takes negative value. However, in relaxing nonlinear system, for any finite value of τ, γ-~γ . Thus the nonlinearity becomes complex which leads to nonzero imaginary part for K even for the case of normal dispersion regime. This leads to the understanding that the delay time ðτÞ extends the range of unstable frequencies and the sign of the group dispersion coefficient is found to be insignificant in the origin of the MI process. Furthermore, the system with finite delay time can be understood as a system possessing complex nonlinearity, which consists of both the real and imaginary parts. The imaginary part of the nonlinear response models the Raman like process and leads to Raman self-matched instability, while the real part accounts for the conventional parametric MI process through phase matching between linear and nonlinear effects. Thus, the instability band in the noninstantaneous nonlinear system comprises (i) parametric MI bands and (ii) the Raman band. Thus, in Eq. (7) when K becomes complex for real Ω, then the field intensity of the perturbation will be amplified down the fiber with an exponential growth rate, given by the gain GðΩÞ ¼ Im½KðΩÞ. This can be made possible only when the system is in the anomalous dispersion regime. On contrary, the presence of linear coupling coefficient in Eq. (8) plays crucial role and extends the MI domain to the normal dispersion regime even in the instantaneous limit. The complex nonlinearity coefficient simply leads to additional Raman band on the higher detuning frequencies. MI analysis in the absence of the CCD was already studied in detail in different contexts and therefore will not be discussed further. Interested readers can refer Ref. [21], where Trillo et al. with their extensive theoretical study bring to the light, the interplay of different mechanisms in MI, namely, dispersion, relaxation, parametric and Raman interactions and so on. Since, we eye on the role of CCD, we observed that κ1 (a measure of CCD) does not have any significant impact on MI dynamics (no change in the gain). Since, only one supermode corresponding to either symmetric (even supermode) or antisymmetric (odd supermode) exists, the intermodal dispersion does not have any impact [52].

Thus we conclude from the section that the role of CCD in the symmetric/antisymmetric solution is found to be insignificant and the inclusion of delay time leads to Raman band at higher detuning frequencies.

3.2. Asymmetric solutions Following the symmetric/antisymmetric CW solution, the so-called asymmetric CW solution can be readily obtained from Eq. (3) by setting E01 ≠E02 : 0 ECW 1 ¼ E 1 expðiϕðzÞÞ;

0 ECW 2 ¼ E 2 expðiϕðzÞÞ

ð11Þ

The nonlinear phase shift ϕðzÞ can take the form ϕðzÞ ¼ γðN 1 þ N 2 Þz, with N2 ¼ κ2 =γ 2 N1 . It is worth noting that for change pffiffiffi in variables ðjE01 j- P j ; jE0j j2 -P j -N j Þ our calculation is identical to that of Ref. [52] at the instantaneous limit. The required minimum total power E20 ¼ jE01 j2 þ jE02 j2 to maintain CW solution for a given κ is found to be ðE20 Þmin ¼ 2κ=γ and it is identified to be same as the case for instantaneous system dealt in Ref. [52]. Thus the ðE20 Þmin required for CW is found to be independent of the delay time ðτÞ. Performing the similar procedure and using linear stability analysis, as described in the previous section, the dispersion relation with some mathematical manipulation is given by    κ1 Ω 2 κ1 Ω 2 K þ pffiffiffi Þ −h1 K þ pffiffiffi Þ −h2 ¼ h 2 2

h1 ¼

" # pffiffiffi 1 2 4 β 2 Ω4 γ~ E0 −4κ 2 þ ðκ 21 −2 2β2 κÞΩ2 þ 2 2 2

" # pffiffiffi 1 2 4 β22 Ω4 2 2 2 γ~ E0 −4κ þ ðκ 1 þ 2 2β2 κÞΩ þ h1 ¼ 2 2

ð12Þ

ð13aÞ

ð13bÞ

h ¼ 14ð~γ 2 E40 −4κ2 Þ2 −ð~γ 2 E40 −4κ 2 Þðβ2 γ~ E20 þ 2κ 21 ÞΩ2 −ð5β22 κ2 þκ 41 −β22 γ~ 2 E40 ÞΩ4 þ 12β22 κ21 Ω6 γ~ ¼ γ=ð1 þ iΩτÞ

ð13cÞ ð13dÞ

One can readily observe from the dispersion relation Eq. (12), that when the delay time ðτÞ is turned off, the solution for K becomes fourth order polynomial with real coefficients. Therefore, there exist four distinct solutions, with two being always real and the rest of the two are most probably complex conjugate. The two real solutions are insignificant as far as MI is concerned, and hence only the complex pair of solutions involves in MI dynamics. The imaginary part of the complex K leads to instability band. Thus the instantaneous system results in a single unstable mode (parametric band) leading to gain band. On the contrary the system with delayed response is interesting, where for any finite value of τ, Eq. (12) becomes fourth order polynomial with complex coefficients. Also, the complex roots do not appear in conjugate pairs, therefore there exist two unstable modes for any frequency, this is in contrast to the single instability band in the case of instantaneous system. In what follows, without loss of generality, we consider the dispersion coefficient and nonlinear parameter as β2 ¼ 7 −1 0:02 ps2 m−1 , γ ¼ 3:05 kW m−1 . The operating input power varies in the range 0–200 kW and the delayed response time varies between τ ¼ 0−10 ps. The coupling coefficient and the coupling coefficient dispersion can take values between κ ¼ 0−25 m−1 and κ1 ¼ 0−10 ps m−1 , respectively.

K. Nithyanandan, K. Porsezian / Optics Communications 303 (2013) 46–55

Here the system is in the anomalous dispersion regime with β2 taking negative value, and the MI is said to occur through a conservative balance between nonlinearity and the negative group dispersion coefficient. In order to provide a quantitative picture of the role of delay in MI spectrum, we consider the limiting case of instantaneous nonlinearity alongside the relaxation (delayed) nonlinear response; since the spectrum is symmetric across the center frequency (i.e.) GðΩÞ ¼ Gð−ΩÞ. For better illustration, we content with GðΩÞ, instead of complete spectrum. The eigenvectors corresponding to the dispersion relation for zero coupling coefficient dispersion can be written as  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1=2 1 K 1 ¼ pffiffiffi ξ þ 8π 4 β2 Ω4 − ðξ2 −16E20 π 2 βγ~ ξΩ2 þ 64π 4 β2 ðE40 γ~ 2 −3κ 2 ÞΩ4 Þ 2

ð14aÞ 

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1=2 1 K 1 ¼ − pffiffiffi ξ þ 8π 4 β2 Ω4 − ðξ2 −16E20 π 2 βγ~ ξΩ2 þ 64π 4 β2 ðE40 γ~ 2 −3κ 2 ÞΩ4 Þ 2

ð14bÞ  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1=2 1 K 3 ¼ pffiffiffi ξ þ 8π 4 β2 Ω4 þ ðξ2 −16E20 π 2 βγ~ ξΩ2 þ 64π 4 β2 ðE40 γ~ 2 −3κ 2 ÞΩ4 Þ 2

ð14cÞ  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1=2 1 K 4 ¼ − pffiffiffi ξ þ 8π 4 β2 Ω4 þ ðξ2 −16E20 π 2 βγ~ ξΩ2 þ 64π 4 β2 ðE40 γ~ 2 −3κ 2 ÞΩ4 Þ 2

ð14dÞ where ξ ¼ E40 γ~ 2 −4κ2 . For τ ¼ 0, the MI spectrum consists only the instantaneous band and identical to the Ref. [52], whereas for nonzero τ, there exist two unstable modes as it is evident from Fig. 1. The first unstable mode (represented by solid line) is given by the expression Abs½Im ½K 1  or Abs½Im ½K 2  that constitutes both the conventional parametric band and the Raman band. It is interesting to note that, any finite value of nonlinear delay time results in complex nonlinearity, the real part of which leads to parametric band as a consequence of the phase matching between the linear and nonlinear effects and the imaginary part of nonlinearity leads to Raman band by means of self phase matching. The second unstable mode (represented by dashed line) is given by the expression Abs½Im ½K 3  or Abs½Im ½K 4 , which is purely due to the finite value of the nonlinear response time and it is defined as the Raman band. Thus, the inclusion of finite value of nonlinear response time leads to two unstable modes, whereas the instantaneous case manages only the parametric band as it is evident from Fig. 1. Moreover, there exist wide range of unstable frequencies in the case of relaxing nonlinear system, this is in contrast to the finite frequency width ðΩ0 o Ω oΩCMF Þ of instantaneous system,

4.1.1. Variation of MI spectrum with power As in most cases, we first consider the variation of MI gain spectrum at κ1 ¼ 0 for some representative value of power as shown in Figs. 2 and 3. Fig. 2 is the plot showing the variation of parametric MI and the first unstable mode with power, and Fig. 3 portrays the MI spectrum of the second unstable mode for different powers. One can readily infer that the MI gain spectrum behaves very much in a perceptible way, such that any increase in power monotonously increases the gain of the unstable band. 4.1.2. Variation of MI spectrum with κ Figs. 4 and 5 show the MI spectrum for some representative value of κ at a constant power E20 ¼ 20 kW. As reported in Ref. [52], the maximum gain of the conventional parametric MI band is inversely proportional to κ and the highest value of GMAX register for uncoupled case with κ ¼ 0. The variation of GMAX of Raman band is interesting where unlike the instantaneous case here the GMAX increases with κ (similar to the variation of gain with power). This leads to the understanding that κ enhances the Raman process and thereby the gain of the Raman band. For an in-depth study, the variation of GMAX and ΩMAX as a function of κ is shown in Fig. 6. The opposite nature of variation of GMAX with respect to power between the conventional parametric MI band and the Raman band can be readily observed from Fig. 6a.

140 120 1

4.1. Anomalous dispersion regime

where CMF is the critical modulation frequency. It is interesting to note that our argument resembles with the paper by Canabarro et al. where the author performed similar study in the case of relaxing nonlinear system using the Debye relaxational model [62]. Thus the delayed system leads to new unstable bands and extends the instability domain of MI to a wide frequency range. With this detailed note, we begin to explore the interplay between various physical parameters of the problem in terms of the MI spectrum.

Gain m

4. Modulational instability analysis

49

100

50 40 30 20 10 6.6

80 60 40 20 0 0

50

100

150

Frequency

200

250

300

THz

Fig. 2. MI spectra at zero CCD for different input powers (kW) with κ ¼ 10 m−1 .

80

50 25

{Conventional & Intantaneous Band} Gain m

1

Gain m

40

60

30

1

20 15 I Unstable Mode

10

20 10

20

II Unstable Mode 5

40

6.6

Instantaneous Band

0

0 0

50

100 Frequency

150

200

THz

Fig. 1. The MI spectra at instantaneous and delayed nonlinear response ðτ ¼ 0:01 psÞ for E20 ¼ 10 kW, κ ¼ 10 m−1 .

0

100

200 Frequency

300

400

500

THz

Fig. 3. Variation of gain of the second unstable mode (Raman band) at different input powers (kW).

50

K. Nithyanandan, K. Porsezian / Optics Communications 303 (2013) 46–55

The instantaneous band starts from a maximum at κ ¼ 0 and decreases gradually with increase in κ, on the other hand GMAX of the Raman bands (two unstable modes) increases with increase in κ and at a higher value of κ, GMAX of Raman band dominates as shown in Fig. 6a. Fig. 6b shows the ΩMAX as a function of κ. It is observed that ΩMAX of both the parametric bands and the Raman bands increases with increase in κ. 4.1.3. Impact of nonlinear response time ðτÞ in the MI spectrum We now switch to investigate precisely the role of nonlinear response time ðτÞ in the MI system. To realize the objective, we consider a wide range of delay time varying between τ ¼ ð0−1Þ ps. The delayed nonlinear response regime is basically divided into fast ðτ ¼ 0:001−0:1 psÞ and slow responses ðτ ¼ 0:1−1 psÞ. One can straightforwardly observe from Fig. 7, as the power increases, the GMAX of both parametric and Raman bands increases monotonously. For a deeper understanding, we have further divided Fig. 7 into two subplots, where Fig. 7a depicts the overall variation of gain versus τ 60

1

40

Gain m

50

30

5 0

20

25 20 15 10

10 0 0

100

200 Frequency

300

400

500

THz

Fig. 4. MI gain spectra for different coupling coefficients (m−1).

50

25 20

Gain m

1

40

15 10

30

5

20

0 10 0 0

100

200 Frequency

300

400

500

THz

Fig. 5. Variation of gain of the second unstable mode (Raman band) for different coupling coefficients (m−1).

and Fig. 7b shows the variation of Raman GMAX in the fast response range. It is observed that for fast response the gain of the conventional band and Raman band remains constant and well separated. Also, the gain of the parametric band is found to be nearly twice the gain of the Raman band, as it is evident from Fig. 7a and b. As the delay increases further, the parametric MI gain decreases and coalesced with the Raman band when the delay becomes of the order of the inverse of the critical modulation frequency (ΩCMF Þ at the instantaneous limit. Further increase in delay results in the slow response regime, where the gain of the coalesced band substantially decreases at the rate of 1=τ2=3 [59]. Thus, one can infer that the slowly responding media inevitably suppresses the gain of the instability band. The variation of OMF as a function of τ is shown in Fig. 8. For better illustration, Fig. 8 is divided into subplots showing ΩMAX as a function of time for the coalesced (Fig. 8a) and Raman bands (Fig. 8b). In the fast responding limit the OMF of the instantaneous band is found to get anchored at a particular frequency, which is identified as the ΩMAX corresponding to the instantaneous band. Thus at fast responding media the conventional parametric MI band behaves identical to the instantaneous band. In contrast, the Raman band is found to be highly sensitive to the response time and the ΩMAX of the Raman band decreases at the rate of 1=τ, quite similar to the Ref. [59]. With increasing delay time τ the Raman band downshifts towards the center frequency and gets coalesced with the conventional parametric band to form a single band. The ΩMAX of coalesced band continues the downshift in the slow response regime at a rate 1=τ1=3 as shown in Fig. 8a. The interplay between coupling coefficient and delay time is shown in Fig. 9, where the OMF and GMAX as a function of τ is shown for different values of κ at an input power of 20 kW. Unlike the earlier case, where impact of power is identified to be straightforward, such that any increase in power monotonously increases both the gain of the conventional and Raman bands. Here κ is found to be robust, in the sense that strong coupling although decreasing the parametric gain substantially enhances the Raman band and thereby increases the gain of the Raman band. Thus emphasizing the strong affinity of the coupling coefficient towards the Raman band. Also, the relation Gainconv ≈2  GainRaman readily fails for higher value of κ. It is evident from Fig. 9a and b that the increasing κ decreases (increases) the parametric band (Raman band). For κ o 15 both conventional and Raman bands have different gains (Gainconv 4 GainRaman Þ and are well distinguishable at fast response. As τ increases both coalesced bands and results in a single coalesced band, whose gain decreases at the rate of 1=τ2=3 . However, for κ 4 15, the gain of the Raman band dominates ðGainconv o GainRaman Þ. Thus the resulting coalesced band possesses higher gain than the conventional band. For higher value of τ in the slow responding regime the GMAX of the coalesced band decreases as discussed earlier.

Fig. 6. Variation of OMF and GMAX of the instantaneous ðτ ¼ 0Þ and relaxing system (conventional+Raman bands) as a function of κ.

K. Nithyanandan, K. Porsezian / Optics Communications 303 (2013) 46–55

51

Fig. 7. Variation of GMAX of MI and Raman bands as a function of τ for different values of power with κ ¼ 10 m−1 .

Fig. 8. Variation of OMG of MI and Raman bands as a function of τ for different values of power.

Fig. 9. Variation of OMF and GMAX of MI and Raman bands as a function of τ for different values of κ at an input power E20 ¼ 20 kW.

The variation of OMF with τ is found to be a sensitive function of κ as depicted in Figs. 9c and d. At higher value of κ (typically κ 4 15), the OMF of the coalesced band is found to be higher than the conventional parametric band. The behavior of ΩMAX at lower values of κ is obvious and very much similar to Fig. 8 and hence will not be discussed.

4.1.4. Interplay between delayed nonlinear response ðτÞ and coupling coefficient dispersion ðκ 1 Þ Here in this section, we intent to analyze exclusively the impact of CCD in the MI spectrum of the delayed nonlinear system. Before that we fix κ ¼ 10 m−1 , which makes the power threshold for CW limit as ðE20 Þmin ¼ 6:56 kW. Similar to Ref. [52], we consider both

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K. Nithyanandan, K. Porsezian / Optics Communications 303 (2013) 46–55

low and high power regimes, since the MI gain spectrum significantly differs in the two input power regimes. At low power limit ðE20 ¼ 7 kWÞ, when κ 1 ¼ 0, there exist a primary low frequency band (conventional band) and Raman band on the higher frequency side. It should be noted that the Raman band on the higher frequency side behaves quite similar to Fig. 3 and hence we focus 14

B

12

A

C

10 Gain m

1

{D, E, F

8 6

D

4

F

E

2 0 0

5

10 15 Frequency

20

25

30

THz

Fig. 10. The MI gain spectra at low input power ðE20 ¼ 7 kWÞ with CCD for different values of κ 1 (ps m−1). Curves κ 1 (A ¼ 0; B¼ −0.2; B¼ −0.4; D¼ −0.6; E¼ −0.8; F ¼−1) ps m−1.

60

A

50

B 40 1

Gain m

C

{D, E, F

D

30

E

on the lower detuning frequencies where the CCD plays a crucial role. When κ 1 takes nonzero value new spectral bands emerge and the MI spectrum changes dramatically. For instance, it is evident from Fig. 10 that any small finite value of CCD substantially decreases the gain of the first unstable mode until κ1 reaches a critical limit κ1cr . At κ 1cr the primary spectral band gets divided into a secondary band at higher frequency. With further increase in κ 1 the gain of the primary band saturates and the secondary band upshifts and moves towards the higher frequency side with decreasing gain. Fig. 11 depicts the MI spectrum with the effect of CCD at high power E20 ¼ 20 kW. Similar to the previous case of low input power for small values of κ1 the gain of the primary band decreases. At κ1cr a new secondary band emerges at higher frequency whose gain is found to be larger than the low frequency primary band. With further increase in κ 1 the gain of the primary band slightly increases and finally saturates. The secondary band on the other hand moves towards the higher detuning frequency by gradually decreasing the gain. A brief illustration of the variation of κ1cr as a function of system parameter is shown in Fig. 12. As it is evident from Fig. 12, the CCD corresponding to the splitting of spectral band varies nearly linear. For instance, with increase in system parameters such as E20 , γ; and jβ2 j, the magnitude of κ 1cr ðjκ 1cr jÞ increases, whereas κ decreases jκ1cr j as shown in Fig. 12. It should be noted that our results are in complete agreement with the results of the Ref. [52] for τ ¼ 0 and also we conclude that κ 1cr in the delayed nonlinear system does not bring any new spectral band in the lower frequency but the two unstable modes due to the delayed nonlinear response are observed at higher frequency.

F

4.2. Normal dispersion regime

20 10 0 0

10

20

30

Frequency

40

50

60

THz

Fig. 11. The MI gain spectra at high input power ðE20 ¼ 20 kWÞ with CCD for different values of κ 1 (ps m−1). Curves κ 1 (A ¼ 0; B¼ −1; B¼ −1.5; D ¼−2; E¼ −2.5; F¼ −3) ps m−1.

This section deals with the propagation of pulse below the zero dispersion wavelength, such that the group dispersion coefficient takes positive value. Thus the parametric instability is not possible since there is no possibility of phase matching between the linear dispersive and nonlinear terms. However, the presence of coupling between the two cores will extend the domain of MI to the normal dispersion regime as reported in Refs. [21,3]. It is obvious from the

Fig. 12. Variation of critical CCD ðκ 1cr Þ with system parameters as depicted in the inset of the figure.

K. Nithyanandan, K. Porsezian / Optics Communications 303 (2013) 46–55

previous section that the delay in the nonlinear response dramatically changes the MI spectrum, a similar feature can also be expected in the normal dispersive case, as we will be discussing in the proceeding section. We follow in similar lines with our earlier discussion, such that the role of different system parameters on the MI spectrum will be emphasized followed by an insight analysis of the interplay between delay and CCD. To give a qualitative picture of the role of power in the MI spectrum without CCD, we have shown in Fig. 13 the gain spectrum for different values of E20 at τ ¼ 0:01 ps and κ ¼ 10 m−1 . At low input powers (close to power threshold for CW) two spectral bands of smaller gain are observed in addition to the inevitable Raman band on the higher frequency side due to the delayed nonlinear response. As P increases the shorter gain band near the center frequency vanishes and a single instability band survives for higher powers. Similar to the case of negative group dispersion coefficient the gain of the Raman band is found to increase with power.

50

1

15

Gain m

20

10

40 30 20 10 6.6

5 0 0

2

4

6

8

Frequency

10

12

14

THz

Fig. 13. MI gain spectra at zero CCD for different input powers ðE20 kWÞ.

20

25

Gain m

1

15

20 10

15 10

5

0

5

0 0

2

4

6 Frequency

8

10

12

14

THz

Fig. 14. The MI gain spectra in the absence of CCD at different coupling coefficients ðκ m−1 Þ.

53

Fig. 14 shows the variation of MI gain spectrum at fixed power ðE20 ¼ 20 kWÞ for different values of κ. At κ ¼ 0, the instability band is still observed without the aid of the coupling coefficient at higher frequency which is recognized to be Raman band. Thus the instability exists in the normal dispersion regime even for vanishing coupling coefficient, which we speculate as the unique feature of the delayed nonlinear response. For increasing κ two unstable bands of different gains emerge at lower frequencies, whose gain is proportional to the value of κ. The variation of GMAX and OMF of Raman band as a function of power and coupling coefficient is shown in Fig. 15. It is obvious from Fig. 15a that as power increases the ΩMAX decreases gradually and the gain constantly increases with power. Unlike, as κ increases the OMF also increases, this is found to be a unique feature of the delayed system since coupling coefficient enhances the Raman band. The gain of the Raman band behaves very much in a perceptible manner such that any increase in κ enhances the gain as shown in Fig. 15b.

4.2.1. Competence of delay and CCD in the normal dispersion regime We now switch to investigate the interplay between τ and κ1 in the MI spectrum. As in the case of negative group dispersion coefficient, two different choices of input power are chosen, one near and other far from ðE20 Þmin . It is observed that for κ ¼ 10 m−1 , the minimum power required for the sustained CW is found to be ðE20 Þmin ¼ 6:56 kW. The MI spectrum at E20 ¼ 7 kW is shown in Fig. 16. At κ 1 ¼ 0, there exist a single primary band near the center frequency and an inevitable Raman band at far detuning frequencies. With increase in CCD, a new spectral band emerges near to the primary band, with further increase in κ 1 the gain of the primary band increases abruptly and saturates. The gain of the new secondary spectral band increases with κ 1 and reaches a maximum at a critical CCD, which is defined again as κ 1cr . At jκj 4jκ 1cr j the gain of the secondary band substantially decreases. Fig. 17 shows the MI spectrum with the effect of CCD at higher input power ðE20 ¼ 20 kWÞ. For any fractional increase in κ 1cr , two instability bands emerge and the gain of the primary band near the center frequency increases rapidly and saturates at higher value of κ. The secondary band on the higher frequency side upshifts with increase in κ 1 and the gain reaches a maximum at κ1cr . Further increase in κ 1 depletes the gain and shifts the instability band towards higher frequency side. However, the instability band at far detuning frequencies due to the delayed Raman response does not vary significantly with the fractional increases in κ 1 . A brief illustration of κ1cr as a function of system parameters is shown in the Fig. 18. It is observed that with increase in E20 , γ and jβ2 j, the magnitude of the critical CCD ðjκ 1cr jÞ increases. It is also observed that any increase in κ nearly keeps jκ1cr j as constant.

Fig. 15. Variation of OMF and GMAX of Raman band as a function of power E20 (kW) and κ ðm−1 Þ.

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K. Nithyanandan, K. Porsezian / Optics Communications 303 (2013) 46–55

5. Summary and conclusion In summary, we have investigated the modulational instability in dual core fiber with the effect of coupling coefficient dispersion and the relaxation of the nonlinear response. From the dispersion relation using linear stability analysis, it is observed for any finite

{B, C, D, E, F

10 8 Gain m

1

A

6

D

4

B

2

E

C

F

0 0

20

40 Frequency

60

80

100

THz

Fig. 16. MI gain spectra at low input power ðE20 ¼ 7 kWÞ with CCD for different values of κ1 (ps m−1). Curves κ 1 (A ¼ 0; B¼−0.5; B¼ −1; D¼ −2; E¼ −4; F¼ −6) ps m−1.

30

Gain m

1

25

D

20

E

C

F

15

AB

10 5 0 0

20

40 Frequency

60

80

100

THz

Fig. 17. MI gain spectra at high input power ðE20 ¼ 20 kWÞ with CCD for different values of κ1 (ps m−1). Curves κ 1 (A ¼ 0; B¼−0.5; B¼ −1; D¼ −2; E¼ −4; F¼ −6) ps m−1.

value of delay, there exist two unstable modes in addition to the parametric instability band, this is in contrast to the single parametric instability band observed in instantaneous Kerr case. Firstly, we studied the impact of various system parameters in the MI spectrum, and it is found that any increase in power monotonously increases the gain of the instability bands. However, the coupling coefficient behaves differently than the earlier reports, such that although κ decreases the gain of the parametric band as in the instantaneous case, but κ interestingly increases the gain of the Raman band. This leads to the understanding that coupling coefficient enhances the Raman band. To emphasize the role of delay ðτÞ in the MI spectrum, two typical delay regimes are considered based on the time scale, namely, fast and slow response regimes. In fast response, the parametric MI band dominates whose gain is twice that of the Raman band. As τ increases at a particular frequency both the parametric and Raman bands collide and lead to a single coalesced band. With further increase in τ the gain of the coalesced band decreases by a factor 1=τ2=3 . Thus, one can infer that the slow response leads to the overall supersession of MI. On the other hand, the OMF at fast response remains constant for parametric band, and the Raman band decreases by a factor 1=τ. At the slow response regime, the OMF of the coalesced band decreases at the rate of 1=τ1=3 . The role of CCD in the MI spectrum is interesting, since CCD leads to the emergence of new spectral bands near the center frequency, and the Raman band at the far detuning frequency does not vary significantly. There exist a critical CCD (measured as κ1cr ), where the dynamics of MI dramatically differs. To emphasize the role of critical CCD, κ1cr as a function of system parameters is shown. Our discussion also features the normal dispersion regime, and the MI is still achieved even without the aid of the coupling coefficient (see Ref. [52]). The variation of gain with power and coupling coefficient is observed to be same as in the anomalous dispersion regime. CCD, on the other hand, leads to new spectral bands and at threshold value ðκ 1cr Þ, MI spectrum behaves qualitatively in different manners.

Fig. 18. Variation of critical CCD ðκ1cr Þ with variation in system parameters such as E20 kW, κ m−1 , γ kW m−1 and β ps2 m−1 .

K. Nithyanandan, K. Porsezian / Optics Communications 303 (2013) 46–55

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