Modulation instabilities in nonlinear two-core optical fibers with fourth order dispersion

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Original research article

Modulation instabilities in nonlinear two-core optical fibers with fourth order dispersion Jin Hua Li*, Ting Ting Sun, You Qiao Ma, Y.Y. Chen, Zhao Lou Cao, Feng Lin Xian, Dong Zhi Wang, Wan Wang School of Physics and Optoelectronic Engineering, Nanjing University of Information Science and Technology, Nanjing 210044, China

ARTICLE INFO

ABSTRACT

Keywords: Fourth-order dispersion Modulation instability Two-core optical fibers

We study in detail the modulation instabilities (MIs) in nonlinear two-core optical fibers (TCFs) in the presence of fourth-order dispersion (FOD) by considering the symmetric and antisymmetric CW states. The analytical instability gain spectra with the FOD are shown explicitly. For the symmetric CW state, in the normal (anomalous) dispersion regime, the positive (negative) FOD reduces the original MI band, and the negative (positive) FOD can generate a completely new dominate MI band in the very high frequency range even with quite small amplitude of FOD for typical TCF parameters. For the antisymmetric CW state, in the normal dispersion regime, there is no MI without FOD. With the negative FOD, MI is generated at any given total powers. In the anomalous dispersion regime, being similar with the behaviors for symmetric CW state, the positive (negative) FOD effect is to reduce the original MI band, and the negative (positive) FOD can generate a completely new dominate MI band in the very high frequency range. We also verify our MI analysis by solving the coupled-mode equations numerically.

1. Introduction As a universal phenomenon in diverse physical systems, MI is a fundamental way to understand the interactions between the nonlinearity and dispersion/diffraction of the nonlinear system involved [1–3], and is closely related to the soliton existence of the nonlinear system involved [4–6]. It manifests itself as the exponential growth of small perturbations imposed on the continuous waves (CWs). This exponential perturbation growth can not be sustained indefinitely due to the pump depletion, and instead, the CWs evolve into a train of periodic narrow pulses with the period of initial modulation frequency, at which the MI arises [1–3]. In optics, MI degrades the performance of optical communication system [7]. In the side of application, MI can be used for fiber lasers [8,9], ultrashort pulse generation [10], supercontinuum generation [11], fiber characterization [12], fiber-optic sensing [13,14], all-optical switching [15], frequency-conversion optimization [16] and so on. The information transmission capacity of traditional optical fiber communication systems based on the single mode and single core fibers is approaching its limit [17,18]. To overcome this limit, mode-division-multiplexing (MDM) technology has been recently proposed by employing the multimode/multicore fibers [19–21]. In the recent years, the wave propagation in multicore and multimode fibers has attracted much attention of scholars as the new information transmission medium [22–29]. The nonlinearity plays an important role in affecting the performance of MDM system [30–38]. As a fundamental study to understand the interactions between the nonlinearity and the dispersion of the system, MI in multicore and multimode fibers should be investigated

⁎

Corresponding author. E-mail address: [email protected] (J.H. Li).

https://doi.org/10.1016/j.ijleo.2019.164134 Received 24 September 2019; Received in revised form 22 December 2019; Accepted 23 December 2019 0030-4026/ © 2019 Elsevier GmbH. All rights reserved.

Please cite this article as: Jin Hua Li, et al., Optik - International Journal for Light and Electron Optics, https://doi.org/10.1016/j.ijleo.2019.164134

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J.H. Li, et al.

systematically. Currently, there are already several studies exploring MI in few-mode [39–42] and few-core fibers [43–53]. The frequency dependence of the propagation constant in optical fibers determines the existence of higher order dispersions, like the third-order dispersion (TOD) and fourth-order dispersion (FOD), which can distort the pulses both in the linear and nonlinear regimes, and have to be taken into account for the ultrashort optical pulses, or when the input wavelength approaches the zerodispersion wavelength [54]. In MI, TOD usually doesn’t affect the gain spectra of the nonlinear systems, but FOD can significantly change the gain spectra of the nonlinear systems [55–57]. However, as far as we know, the FOD effects on MI in TCFs with two identical cores have never been systematically explored. The primary goal of this paper is to study in detail the effects of FOD on MI in TCFs with the symmetric and antisymmetric CW states. In Section 2, we show the coupled nonlinear Schrödinger equations governing the pulse propagation in nonlinear TCFs with the FOD effects and discuss the physical meanings of each parameter. We then calculate the MI spectra analytically in the presence of FOD by the linear stability method and discuss the FOD effects on MI for the symmetric CW state in Section 3, and for the antisymmetric CW state in Section 4. Finally, we give a main conclusion of FOD effects on MI spectra in TCFs in Section 5. 2. Coupled-mode equations The generalized nonlinear pulse propagation in TCFs with two parallel identical cores are described by

i

a1 z

1 2

2

i

a2 z

1 2

2

2a 1 t2

1 i 6

3

2a 2 t2

1 i 6

3

3a 1 t3

+

1 24

4

3a 2 t3

+

1 24

4

4a 1 t4

+

|a1 |2 a1 + Ca2 + iC1

a2 = 0, t

(1)

4a 2 t4

+

|a2 |2 a2 + Ca1 + iC1

a1 = 0, t

(2)

where aj (j = 1, 2) denotes the slowly varying electric-field envelope in the j-th core; z is the propagation distance, t is the retarded

time coordinate in the frame of reference moving at the group velocity;

m

=

dm d m

= 0

is the m-th order group velocity dispersion

(GVD) at the carrier wave frequency w0 with β as the propagation constant. The GVD effects in most cases of practical interest is dominated by β2, however, for pulse wavelength near the zero-dispersion wavelength (β2 ≈ 0), or for ultrashort pulses, we have to consider the TOD and FOD effects to justify the truncation of the propagation constant after the β2 term. γ is the cubic Kerr nonlinear coefficient with γ = 2πn2/(λAeff), where λ, n2, and Aeff are the free-space optical wavelength, nonlinear refractive index of the fiber material, and the effective area of each core respectively; C is the linear coupling coefficient being responsible for the periodic pulse transfer between the two cores; C1 = dC dw|w = w0 with w0 as the optical angular frequency of the carrier wave is called the linear coupling coefficient dispersion (CCD) denoting the frequency dependence of the linear coupling coefficient, or equivalently, the intermodal dispersion arising from the group-delay difference of the supermodes of the fiber. Eq. (1) and (2) permit the symmetric and antisymmetric CW states [49], which are respectively,

a1 = a2 =

P0 exp (ikz ), 2

(3)

with k = γP0 + C,

a1 =

P0 exp (ikz ), a2 = 2

P0 exp (ikz ), 2

(4)

with k = γP0 − C. The fields in the two cores are identical for the symmetric CW state, and have the same amplitude with the opposite sign for the antisymmetric CW state. Eq. (1) and (2) also permit the asymmetric CW state with unequal field amplitudes in the two cores. However, in this paper, we focus only on the symmetric and antisymmetric CW states, and study analytically the corresponding MI gain spectra as shown by Eq. (3) for the symmetric and Eq. (4) for the antisymmetric CW state. 3. MI analysis for the symmetric CW state According to the linear stability method, we first perturb the CW state of Eq. (3) slightly, i.e.

a1 =

P + u exp (ikz ), a2 = 2

P + v exp (ikz ), 2

(5)

with u = u(z, t) and v = v(z, t) as the small imposed perturbations in the form of

u = F1 exp (iKz

i t ) + G1 exp( iKz + i t ),

(6)

v = F2 exp (iKz

i t ) + G2 exp( iKz + i t ),

(7)

where K and Ω are the perturbation wavenumber and perturbation frequency respectively. We substitute Eq. (5) into Eqs. (1) and (2), linearize in u and v, and then insert Eq. (6) and (7) into the resulting linear equations. Finally, the nontrivial solutions of Fi and Gi (i = 1, 2) lead to 2

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K

C1

1 6

3

K+

C1

1 6

3

2

3

2

3

=

1 576

2(

=

1 ( 576

4

2

4

4

+ 12 2 )(

+ 12

2

2

4

4

+ 12

48C )(

2

4

4

2

+ 24 P ),

+ 12

2

2

(8)

48C + 24 P ).

(9)

Let the TOD parameter 3 = 0 and FOD parameter 4 = 0 , the dispersion relations of Eq. (8) and (9) essentially degenerate into the dispersion relation of Eq. (6) in Ref. [46]. We also note that Eq. (8) in the absence of TOD and FOD is also essentially the dispersion relation of the single-core optical fibers (SCFs) [51]. Therefore, as analyzed in the literature, the MI spectra in TCFs have contained that in SCFs [49]. As usual, we define the instability gain as (10)

g ( ) = |Im (K )| of which, Im denotes the maximum absolute value of the imaginary part of K solution. Now, we study the gain spectra from the dispersion relations of Eq. (8) and Eq. (9) respectively in the next sections. 3.1. Gain spectra from Eq. (8) For convenience, we first define the following parameters, namely, 2 0

12

=

2

2 1

,

4

6

=

2

+2

2 2

,

4

6

=

2

2

,

(11)

4

6 4 P . The defined parameter with = 9 (i = 0, 1, 2) in Eq. (11) can be either positive or negative depending on the sign of β2, β4 and the value of total power P, but all appeared i2 in the following texts of this paper are positive for the appropriate parameters of TCFs or by restricting the necessary parameter values of TCFs. Now, we first study the effects of FOD for the normal dispersion. There exists no MI in the presence of positive FOD, i.e. β4 > 0, and MI arises with β4 < 0 only. The corresponding gain spectrum is, 2 2

2 i

1 24

g( ) =

2(

4

2

+ 12 2 )(

4

4

+ 12

2

2

+ 24 P )

(12)

with 2 0

2

<

(13)

2 2

<

and the maximum gain

gmax (

max )

=

1 P at 2

2 m

=

6

2

2 9

2 2

3 P

4

> 0,

(14)

4

where > 0, > 0, > 0 for β4 < 0 and β2 > 0. Eq. (13) shows that there is a single MI band with the gain increasing with the increment in the total power P as shown by Eq. (14), and the maximum gain is not affected by FOD (β4), which only imposes effect on the optimum modulation frequency (OMF) generating the dominate gain. The main MI characteristics from Eq. (12) – Eq. (14) are shown by Fig. 1 for typical TCF parameters of β2 = 0.02 ps2/m, β4 = −0.001 ps3/m, γ = 5 /(kW·m), C = 200 /m, and C1 = 0 ps/m. , lim 22 , which show us that the gain vanishes in the absence of FOD, in good We note that as β4 → 0, lim 02 2 2

2 0

2 max

0

4

0

4

agreement with the conclusion that there is no MI for the normal dispersion. In conclusion, β4 < 0 plays an important role, which can generate MI in the normal dispersion. We now consider the effects of FOD for the anomalous dispersion. For β4 > 0, the gain is the same with Eq. (12), but the modulation frequencies generating MI become 2 1 2

2

<

2 0,

<

2 0

<

2

and

<

2 2,

for P < Pcr 1,

for P > Pcr1,

(15)

where

Pcr1 =

3 2

2 2

,

(16)

4

and the maximum gain is

gmax (

max )

=

1 P at 2

2 m

=

6

2

±2 9

2 2

3 P

4

,for P < 2Pcr1 ,

4

3

(17)

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Fig. 1. The variation of MI spectrum with the total power (P) and perturbation frequency (Ω/2/π) for β2 = 0.02 ps2/m, β4 = −0.0001 ps4/m, γ = 5 /(kW·m), C = 200 /m, C1 = 0 ps/m.

gmax (

max )

3

=

4

2 2 2 4

( 3

2 2

+ 2 P 4)

2 m

at

=

6

2

,for P > 2Pcr1.

4

(18)

From Eq. (15), we see that there are two MI bands for P < Pcr1. As P increases, the two bands merge into a single MI band when P > Pcr1. As long as P < 2 Pcr1, the maximum gain occurs at two different perturbation frequencies as shown by Eq. (17). When P > 2 Pcr1, the maximum gain relies on β4, and the corresponding OMF depends on β2 and β4 only as shown by Eq. (18), which is dramatically different from that shown in Eq. (17). The main MI characteristics are shown by Fig. 2 with β2 = −0.02 ps2/m, β4 = 10−6 ps4/m, γ = 2.5 /(kW·m), C = 200 /m, and C1 = 0 ps/m. For β4 < 0, the gain is also the same with Eq. (12), but the modulation frequencies are 2

(19)

2 2,

<

with the maximum gain

gmax (

max )

=

1 P 2

2 m

at

6

=

2

2 9

2 2

3 P

4

.

(20)

4

Eq. (19) shows that there is a single MI band with β4 < 0, and from Eq. (20), we can also see that the maximum MI gain is not affected by FOD, which only alters the OMF. The main characteristics are shown by Fig. 3 with the identical parameters in Fig. 2 but β4 = −10−6 ps4/m. + , and the gain spectrum of Eq. (12), Eqs. (15)–(17), and Eqs. (19) and 20) basically degenerate into Let β4 → 0, Pcr1 respectively,

g( ) = 2

<

gmax (

1 4

2

2(

2

2

+ 2 P) ,

(21)

2 P , | 2| max )

(22)

=

1 P at 2

2 m

=

P , | 2|

(23) 4

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Fig. 2. The variation of MI spectrum with the total power (P) and perturbation frequency (Ω/2/π) for β2 = −0.02 ps2/m, β4 = 10−6 ps4/m, γ = 2.5 /(kW·m), C = 200 /m, C1 = 0 ps/m.

which are essentially identical with those in SCFs. Only one MI band is present with the gain increasing with the increment in the total power. The main characteristics are shown by Fig. 4 with the identical parameters in Fig. 2 but β4 = 0 ps4/m. Compared with Fig. 2, β4 > 0 can generate a completely new MI band in the high frequency, which can merge with the original MI band into a single one as P increases across Pcr1 as shown by Eqs. (15) and 16). Compared with Fig. 3, β4 < 0 reduces the modulation frequency band that generates MI for the typical TCF parameters selected. 3.2. Gain Spectrum from Eq. (9) For the sake of convenience, let’s define the following parameters, 2 3

=

2 5

=

1

=9

6

2

+2

1

,

4

6

2

+2

2

,

4

2 4

=

2 6

=

6

2

2

1

,

(24)

4

6

2

2

2

,

(25)

4

with 2 2

+ 12 4 C ,

2

=9

2 2

+ 12 4 C

6

4

P.

(26)

We first consider the normal dispersion. For β4 > 0, the MI is always present with the instability gain as

1 24

g( ) =

(

4

4

+ 12

2

2

48C )(

4

4

+ 12

2

2

48C + 24 P ) ,

(27)

of which, 2 5

0<

2

< 2

<

2 3,

< 2 3,

for for

P< P>

2C 2C

,

,

(28) 5

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Fig. 3. The variation of MI spectrum with the total power (P) and perturbation frequency (Ω/2/π) for β2 = −0.02 ps2/m, β4 = −0.001 ps4/m, γ = 2.5 /(kW·m), C = 200 /m, and C1 = 0 ps/m.

and the maximum gain

gmax ( ) =

1 P at 2

gmax ( ) =

2C ( P

2 m

6

=

2

+2 9

2 2

+ 12C

3 P

4

4

for P <

4C

,

(29)

4

2 m

2C ) at

= 0 for P >

4C

.

(30)

From Eqs. (27)–(30), we see that the instability gain increases as the increment in the total power P, and the MI band broadens at the same time. When P increases across the critical total power of 2C/γ, the MI band locates in the range of [0, 32 ], which doesn’t vary with P. The maximum MI gain is not influenced by β4, which affects the OMF only, as shown by Eq. (29) for P < 4C/γ. These main characteristics are shown by Fig. 5 for β2 = 0.02 ps2/m, β4 = 0.001 ps4/m, γ = 5 /(kW·m), C = 200 /m, and C1 = 0 ps/m. For β4 < 0, the instability gain is the same with Eq. (27), but the modulation frequencies generating MI depend strongly on the values of β4, i.e. 2 4

<

2

<

2 6

and

2 5

2 4

<

2

<

2 6

and

2

2 5

<

2

<

2 6,

2

<

2 6,

for P >

gmax ( ) =

1 P at 2

2

< <

2 3,

<

and

4

for P <

for P > 2C

for Pcr 2 < P < 2C

2 3,

<

2C

and 3

2 2

4C

4

2C

and <

and

4

>

3

2 2

4C

4

3

3

>

4C

2 2

4C

2 2

,

,

,

,

(31)

with 2 m

=

6

2

2 9

2 2

+ 12C

4

3 P

4

, for P >

4

6

4C

,

(32)

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Fig. 4. The variation of MI spectrum with the total power (P) and perturbation frequency (Ω/2/π) for β2 = −0.02 ps2/m, β4 = 0 ps4/m, γ = 2.5 /(kW·m), C = 200 /m, C1 = 0 ps/m.

1 P at 2

gmax ( ) =

2 m

6

=

2

±2 9

2 2

+ 12C

4

3 P

4

, for P <

4C

with

4

>

3

4

2 2

4C

, and for 2Pcr 2 < P <

4C

with

4

<

3

2 2

4C

,

(33)

3

gmax ( ) =

4

2 2

3

+C

4

2 2

+ 4C

2 P

at

2 m

=

6

2

4

4

, for P < 2Pcr 2 and

4

<

3

2 2

4C

,

(34)

where

Pcr2 =

3 2

2 2

+

2C

.

(35)

4

As shown by Eq. (32)–(34), the MI spectrum for β4 > −3β22/(4C) is quite different from the case for β4 < −3β22/(4C). There is only one MI band for the case of β4 < −3β22/(4C), and the existence of this MI band requires total power above the critical value of Pcr2 (Eq. (35)). While for the case of β4 > −3β22/(4C), two MI bands are always present for any input total power. These main MI characteristics are shown by Fig. 6(a) and (b) for β4 > −3β22/(4C) and Fig. 6(c) and (d) for β4 < −3β22/(4C) respectively. Ignoring the FOD effects, the gain of Eq. (27) degenerates into

1 (4C 4

g( ) =

2

2 )(

2

2

4C + 2 P ) ,

(36)

Ω3 and Ω5 become respectively,

lim 4

0

2 3

=

4C , lim | 2| 4 0

2 5

=

4C

2 P , | 2|

(37)

thus the Eq. (28) and the first two equations in Eq. (31) (the limit of β4 → 0 doesn’t apply to the last two equations in Eq. (31) as β4 → 0 > −3β22/(4C)) degenerate into 7

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Fig. 5. The variation of MI spectrum with the total power (P) and perturbation frequency (Ω/2/π) for β2 = 0.02 ps2/m, β4 = 0.001 ps4/m, γ = 5 /(kW·m), C = 200 /m, C1 = 0 ps/m.

4C 2

2 P < | 2| <

4C , | 2|

2

<

2C 4C , , for P < | 2| 2C

for P >

,

(38)

and the maximum gain follows,

gmax ( ) =

1 P at 2

gmax ( ) =

2C ( P

2 m

=

4C P 4C for P < , | 2| 2 m

2C ) at

= 0 for P >

(40)

4C

.

(41)

With β4 → 0, we note that the maximum gain of Eqs. (29) and (30) for β4 > 0 directly degenerate into Eq. (40) and Eq. (41) respectively. While the maximum gain of Eq. (32) for β4 < 0 can not degenerate into Eq. (41) for P > 4C/γ due to the effect of β4, which generates a completely new instability band dominating the MI in the high frequency as shown by Fig. 6(a) and (b). Fig. 7 shows the variation of the MI spectrum with the total power in the absence of FOD for β2 = 0.02 ps2/m, β4 = 0 ps4/m, γ = 5 /(kW·m), C = 200 /m, C1 = 0 ps/m. Compared with Fig. 6(a), β4 < 0 can generate a completely new dominate MI band in the very high frequency range even with very small amplitude of FOD, i.e. (|β4|) as shown by Eq. (32) and (33). Compared with Fig. 5, β4 > 0 reduces the modulation frequency band that generates MI for the typical TCF parameters selected. We next consider the anomalous dispersion. For β4 > 0, the instability gain is also the same with Eq. (27), while the modulation frequencies are 2 5

<

2

<

2 3,

2 5

<

2

<

2 3 ,and

2

<

2 3,

for P <

for P >

2C

2

2C

2 6,

< +

,

3 2

2 2

for

2C


2C

+

3 2

2 2

,

4

,

(42)

4

8

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Fig. 6. The variation of the MI with the total power and the perturbation frequencies for (a) β4 = −5 × 10−7 ps4/m, (b) β4 = −0.0001 ps4/m, β2 = 0.02 ps2/m, γ = 5 /(kW·m), C = 200 /m, C1 = 0 ps/m.

with the maximum gain

gmax ( ) =

1 P at 2

gmax ( ) =

1 3 24

2 m

2 2

=

6

2

+2 9

2 2

+ 12C

4

3 P

4

, for P <

4C

+

4

+ 4C

2 P

4

3

2 2

+ 4C

2 2

3

,

(43)

4

2 m

at

6

=

2

for P >

4

4

4C

+

2 2

3 4

.

(44)

For P < 2C/γ, there is one single MI band in the high frequency initially. As the total power increases, a new MI band in the low frequency is generated for 2C/γ < P < 2C + 3 22 (2 4 ) . As P > 2C + 3 22 (2 4 ) , the two MI bands rapidly merge into a broad single MI band. The maximum gain keeps on increasing with the increment in the total power, as shown by Eqs. (43) and (44), and can be dramatically changed by β4 when P > 4C + 3 22 ( 4 ) as shown by Eq. (44). Besides, from Eqs. (43) and (44), we also see that FOD, i.e. β4 can significantly affect the OMF. The main characteristics are shown by Fig. 8 for β2 = −0.02 ps2/m, β4 = 0.001 ps4/m, γ = 2.5 /(kW·m), C = 200 /m, C1 = 0 ps/m. For β4 < 0, the instability gain is also the same with Eq. (27), and the modulation frequencies and the maximum gain are respectively, 2

<

2 6,

with P >

gmax ( ) =

2C ( P

gmax ( ) =

1 P at 2

2C

,

(45) 2 m

2C ) at

2 m

=

6

2

= 0, for P <

2 9

2 2

+ 12C

4C

4

,

3 P

(46) 4

,for P >

4

4C

.

(47)

From Eq. (45), we see that the MI can be only present for P > 2C/γ with the gain increasing with the increment in the total power as shown by Fig. 9 for β2 = −0.02 ps2/m, β4 = −0.001 ps4/m, γ = 2.5 /(kW·m), C = 200 /m, C1 = 0 ps/m. From Eqs. (46) and 9

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Fig. 7. The variation of MI spectrum with the total power (P) and perturbation frequency (Ω/2/π) for β2 = 0.02 ps2/m, β4 = 0 ps4/m, γ = 5 /(kW·m), C = 200 /m, C1 = 0 ps/m.

Fig. 8. The variation of MI spectrum with the total power (P) and perturbation frequency (Ω/2/π) for β2 = −0.02 ps2/m, β4 = 10−6 ps4/m, γ = 2.5 /(kW·m), C = 200 /m, C1 = 0 ps/m. 10

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Fig. 9. The variation of MI spectrum with the total power (P) and perturbation frequency (Ω/2/π) for β2 = −0.02 ps2/m, β4 = −0.001 ps4/m, γ = 2.5 /(kW·m), C = 200 /m, C1 = 0 ps/m.

(47), we see that the maximum gain is not affected by FOD, which only influences the OMF when P > 4C/γ. Ignoring the FOD effects, Ω3 in Eq. (24), Ω5 and Ω6 in Eq. (25) degenerate into

lim

0

4

2 3

=+

, lim 4

0

2 5

=+

, lim 4

0

2 6

=

2 P 4C , | 2|

(48)

both Eqs. (42) and (45) essentially become 2

>

2 P 4C 2C with P > | 2|

(49)

and the maximum gain

gmax ( ) =

2C ( P

gmax ( ) =

1 P at 2

2C ) at 2 m

=

P

2 m

= 0 for P <

4C

,

(50)

4C 4C , for P > . | 2|

(51)

With β4 → 0, the maximum gain of Eq. (46) is identical with Eq. (50), and Eq. (47) for β4 > 0 degenerate into Eq. (51). While the maximum gain of Eq. (43) for β4 < 0 can not degenerate into Eqs. (50) and (51) due to the effect of β4, which generates the dominate gain in the high frequency as shown by Fig. 8. From Eqs. (49) and (51), we see that there exists a single MI band for P > 2C/γ, and the band broadens with gain increasing as the total power increases. The main characteristics are shown by Fig. 10 for β2 = −0.02 ps2/m, γ = 2.5 /(kW·m), C = 200 /m and C1 = 0 ps/m. In contrast with the situation for the normal dispersion, β4 > 0 can generate a completely new dominate MI band in the high frequency, which can merge with the original MI band with the increment in the total power as shown by Eq. (42) and Fig. 8. β4 < 0 reduces the modulation frequency band that generates MI for the typical TCF parameters selected, in comparison with Fig. 9. We remark that the dominate MI gain for the normal dispersion with β4 > 0 arises from the dispersion relation of Eq. (9) only as shown by Eqs. (27)–(30). While, for normal dispersion with β4 < 0 and anomalous dispersion with either sign of β4, the MI arises for both dispersion relations of Eqs. (8) and (9), and the dominate MI gain and the OMF generating the dominate gain can occur from either dispersion relation of Eq. (8) or Eq. (9) depending on the specific fiber parameters. For typical TCF parameters selected in the above figures, there is no big difference between the dominate gain values from Eqs. (8) and (9), however, the OMF generating the 11

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Fig. 10. The variation of MI spectrum with the total power (P) and perturbation frequency (Ω/2/π) for β2 = −0.02 ps2/m, β4 = 0 ps4/m, γ = 2.5 /(kW·m), C = 200 /m, C1 = 0 ps/m.

corresponding dominate gain from Eq. (8) can be quite different from that from Eq. (9). 4. MI analysis of the anti-symmetric CW In terms of the same technique in Section 3, the dispersion relations for the anti-symmetric CW follow,

K+

C1

1 6

3

K

C1

1 6

3

3

3

2

2

=

1 576

2(

=

1 ( 576

4

4

4

2

+ 12 2 )(

+ 12

2

2

4

2

+ 12

+ 48C )(

4

2

2

2

+ 24 P ),

+ 12

2

2

+ 48C + 24 P ),

(52) (53)

of which, the gain spectra from Eq. (52) are essentially identical with those from Eq. (8) in terms of the definition of the instability gain of Eq. (10). Therefore, we only need to calculate the gain spectra from Eq. (53) in the following. For the sake of convenience, we define 2 7

=

2 9

=

6

2

+2

3

,

4

6

2

+2

4

,

4

2 8

2 10

6

=

2

2

3

,

(54)

4

6

=

2

2

4

,

(55)

4

12 4 C 6 4 P . 12 4 C , 4 = 9 with 3 = 9 We first consider the normal dispersion. There exists no MI with β4 > 0, and MI arises for β4 < 0 only with 2 2

2 2

g( ) =

1 24

(

4

4

+ 12

2

2

+ 48C )(

4

4

+ 12

2

2

+ 48C + 24 P ) ,

(56)

where the modulation frequencies locate in the range of 2 8

<

2

<

(57)

2 10 ,

and the maximum gain is 12

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Fig. 11. The variation of MI spectrum with the FOD (β4) and perturbation frequency (Ω/2/π) for β2 = 0.02 ps2/m, β4 = −0.001 ps4/m, γ = 5 /(kW·m), C = 200 /m, and C1 = 0 ps/m.

gmax ( ) =

1 P at 2

2

6

=

2 2

2 9

2

12C

4

3 P

4

.

(58)

4

The Eqs. (57) and (58) show that there exists a single MI band with the gain increasing with the increment in the total power (P). The FOD doesn’t affect the maximum MI gain, which only influences the OMF that generates the maximum gain. The main MI characteristics are shown by Fig. 11 for β2 = 0.02 ps2/m, β4 = −0.001 ps3/m, γ = 5 /(kW·m), C = 200 /m, and C1 = 0 ps/m. We now consider the anomalous dispersion. For β4 > 0, the instability gain is also Eq. (56), but with 2 8

<

2

2 8

<

2

<

2 10

and

2 9

<

2 7,

for P > Pcr 3,

2

<

4

2 7,

< <

for P < Pcr 3,

2 2

3

4C

4

<

2 2

3

4C

,

,

(59)

where

Pcr3 =

3 2

2 2

2C

.

(60)

4

The corresponding maximum gain is

gmax ( ) =

gmax ( ) =

1 P at 2

(3

2 2

2

=

6

2

±2 9

2 2

12C

3 P

4

4

,for P < 2Pcr 3,

4

4C 4 )(4C 4

4 2 4

3

2 2

+ 2 P 4)

As shown by Eq. (59), MI is only present for

at

2

=

6 4

β4 < 3β22/(4C). 13

2

,for P > 2Pcr 3.

(61)

(62)

There are two MI bands for P < Pcr3 locating in two different

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J.H. Li, et al.

Fig. 12. The variation of MI spectrum with the FOD (β4) and perturbation frequency (Ω/2/π) for β2 = −0.02 ps2/m, β4 = 10−6 ps4/m, γ = 2.5 /(kW·m), C = 200 /m, and C1 = 0 ps/m.

modulation frequency ranges sharing the same maximum gain as shown by Eq. (61). As P increases across Pcr3, these two bands merge into a single band. From Eq. (62), we see that FOD exerts both important effects on the dominate MI gain and the corresponding OMF. The main MI characteristics are shown by Fig. 12 for β2 = −0.02 ps2/m, β4 = 10−6 ps4/m, γ = 2.5 /(kW·m), C = 200 /m, and C1 = 0 ps/m. For β4 < 0, the instability gain is also Eq. (56) but with 2 8

2

<

gmax ( ) =

(63)

2 10 ,

<

1 P at 2

2

6

=

2 9

2

2 2

12C

3 P

4

4

.

4

(64)

Being different with the effects of β4 > 0, MI occurs for any values of β4 < 0 with a single instability band as shown by Fig. 13. In the absence of FOD, 2 8

4C , | 2|

4C + 2 P , | 2|

2 10

7

+

,

9

+

,

(65)

the instability gain of Eq. (56) degenerates into

1 ( 4

g( ) =

2

2

+ 4C )(

2

2

+ 4C + 2 P ) ,

(66)

both modulation frequencies of Eq. (59) for β4 > 0 and Eq. (63) for β4 < 0 essentially become

4C < | 2|

2

<

4C + 2 P , | 2|

(67)

and the maximum gain of Eq. (61) for β4 > 0 and Eq. (64) for β4 < 0 become

gmax ( ) =

1 P at 2

2

=

4C + P . | 2|

(68)

There is a single MI band, and the instability gain increases with the increment in the total powers as shown by Fig. 14 for β2 = −0.02 ps2/m, β4 = 0 ps3/m, γ = 2.5 /(kW·m), C = 200 /m, and C1 = 0 ps/m. Compared with Fig. 12, β4 > 0 can generate a 14

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Fig. 13. The variation of MI spectrum with the FOD (β4) and perturbation frequency (Ω/2/π) for β2 = −0.02 ps2/m, β4 = −10−6 ps4/m, γ = 2.5 /(kW·m), C = 200 /m, and C1 = 0 ps/m.

Fig. 14. The variation of MI spectrum with the FOD (β4) and perturbation frequency (Ω/2/π) for β2 = −0.02 ps2/m, β4 = 0 ps4/m, γ = 2.5 /(kW·m), C = 200 /m, and C1 = 0 ps/m.

15

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Fig. 15. The symmetric CWs evolution with white noise imposed for β2 = 0.02 ps2/m, β4 = 0.001 ps4/m, γ = 5 /(kW·m), C = 200 /m, C1 = 0 ps/ m, and P = 120 kW.

completely new MI band possessing the same maximum gain with the original band when P < Pcr3 as shown by Eq. (61). Compared with Fig. 14, β4 < 0 reduces the modulation frequency band that generates the MI. The effects of FOD here are quite similar with the behaviors for the symmetric CW state in the anomalous dispersion. We draw the conclusion that MI generation for the normal dispersion requires β4 < 0, being different from the conclusion for symmetric CW state, where MI can occur for either sign of β4. The dominate gains for dispersion relation of Eq. (8), or equivalently Eq. (52), and that for Eq. (53) are equal, however, the OMF from Eq. (8) differs from that from Eq. (53) for the C effect. For anomalous dispersion with both β4 > 0 and β4 < 0, MI is always present. The dominate MI gain and the OMF generating the dominate gain can arise from either dispersion relations of Eq. (8) or Eq. (53) depending on the specific fiber parameters. 5. Wave propagation To verify the above MI analysis, we solve Eqs. (1) and (2) numerically by directly launching CWs into each core with weak white noise imposed. The numerical method we employed is the pseudospectral method in the time domain and fourth-order Runge-Kutta method with an adaptive step control in the spatial domain. The white noise is a series of random numbers uniformly distributed in the range of [−1, 1] generated by Matlab with the amplitude as 0.01 % of that of CWs. In terms of MI, the CWs should evolve into a series of periodic pulses at the OMF generating the dominate gain. As further propagation, the periodicity of the pulses is broken, and eventually, the pulses evolve into a series of irregularly spiky pulses, due to the interactions among different modulation frequency components that generate MI along propagation. Fig. 15 shows the perturbed symmetric CW state evolution for β2 = 0.02 ps2/m, β4 = 0.001 ps4/m, γ = 5 /(kW·m), C = 200 /m, C1 = 0 ps/m, and P = 120 kW. As predicted, the CWs first evolve into a train of ultrashort optical pulses, and the period of the initial ultrashort optical pulses is 161.2 fs measured at z = 4.5 LC, being equivalent to the frequency of 6.2 THz, which fits well with the MI predication of 6.0 THz generating the dominate gain as shown in Fig. 5. As the propagation distance increases, the waveform rapidly loses its periodicity and becomes increasingly irregular and spiky, which implies the presence of a range of modulation frequency component, and the interactions between the different frequency components lead to the break down of the periodic pulses. Fig. 16 shows the perturbed antisymmetric CW state evolution for β2 = −0.02 ps2/m, β4 = 10−6 ps4/m, γ = 2.5 /(kW·m), C = 200 /m, C1 = 0 ps/m, and P = 500 kW. Being similar with Fig. 15 for the symmetric CW state, the CWs first evolve into a series of ultrashort optical pulses, and then rapidly evolve into a series of quite irregular and spiky pulses, which just implies the presence of more modulation frequency components. The period of the initial ultrashort optical pulses is 19.9 fs measured at z = 2 LC, being equivalent to the frequency of 50.3 THz, which fits very well with the MI predication of 55.1 THz generating the dominate gain as shown in Fig. 2. As the gain in this case (624.5 /m) is much larger than that (300.0 /m) in the previous case, it needs a much shorter propagation distance to observe the MI effect in this case. 16

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Fig. 16. The symmetric CWs evolution with white noise imposed for β2 = −0.02 ps2/m, β4 = 10−6 ps4/m, γ = 2.5 /(kW·m), C = 200 /m, C1 = 0 ps/m, and P = 500 kW.

6. Conclusion The MI gain spectra in TCFs with the FOD effects for the symmetric and antisymmetric CW states are studied analytically. There are two dispersion relations for each CW state, and the MI spectra from both dispersion relations are shown analytically. The dominate gain from which dispersion relation depends on the specific TCF parameters and values of FOD. According to our study, we find that the MI spectra with the FOD effects are significantly different from those without FOD. We also verify our MI analysis by solving Eqs. (1) and (2) numerically, and the results fit well with the MI analysis. For the symmetric CW state, in the normal dispersion regime, β4 > 0 reduces the original MI band, and β4 < 0 can generate a completely new dominate MI band in the very high frequency range at any given total powers with quite small amplitude of FOD for typical TCF parameters, which can never merge with the original MI band. In contrast to the normal dispersion, in the anomalous dispersion, β4 < 0 reduces the original MI band, and β4 > 0 can generate a completely new dominate MI band in the high frequency range, which can merge with the original MI band with the increment in the total power. For the antisymmetric CW state, in the normal dispersion regime, there is no MI without FOD. With the negative FOD effect, i.e. β4 < 0, MI is generated at any given total powers. In the anomalous dispersion regime, being similar with the symmetric CW state, β4 < 0 reduces the original MI band, and β4 > 0 can generate a completely new dominate MI band in the high frequency range, which can merge with the original MI band as the total power increases. The results of this paper should provide helps in understanding of the interactions between the nonlinearity and dispersion in multicore fibers. The investigation of FOD effects for the asymmetric CW state, and the expansion to the multicore (> 3 cores) fibers will be reported in our future work. Declaration of Competing Interest The authors declare that they have no known competing financialinterestsor personal relationships that could have appeared to influence the work reported in this paper. Acknowledgements This work is supported by National Natural Science Foundation of China (11605090, 11447113), the Innovation and Entrepreneurship Training Program for College Students in NUIST (1211131901002). 17

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