Effect of random linear mode coupling on intermodal ... - OSA Publishing

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Effect of random linear mode coupling on intermodal four-wave mixing in few-mode fibers Yuzhe Xiao1,2 , Sami Mumtaz1 , Ren´e-Jean Essiambre2 , and Govind P. Agrawal1 1 The

Institute of Optics, University of Rochester, Rochester, NY 14627 2 Bell Laboratories, Alcatel-Lucent, Holmdel, NJ 07733 [email protected]

Abstract: We study numerically intermodal four-wave mixing (IM-FWM) in few-mode fibers including both birefringence fluctuations and random linear coupling. We find that linear mode coupling reduces idler power by 3.5 dB for non-degenerate IM-FWM. OCIS codes: 060.4370, 060.2330, 190.0190

1.

Introduction

Four-wave mixing (FWM) is an important nonlinear process that can strongly impact transmission. Although it was studied in few-mode fibers (FMFs) as early as 1975 [1], until recently most studies focused on single-mode fibers [2]. From the recent advances in space-division multiplexing (SDM), few-mode, multimode and multicore fibers are seen as viable candidates for increasing the capacity of telecommunication systems [3]. In recent years, considerable work has been done, both experimental and theoretical, to study the impact of fiber nonlinearity on SDM systems [4–9]. In particular, Essiambre et. al have carried out an experimental investigation of non-degenerate intermodal FWM (IM-FWM) [7], where two types of IM-FWM processes were observed. To fully understand the IM-FWM processes in FMFs, a numerical model is needed that incorporates the role of random linear coupling that is virtually unavoidable between groups of modes in FMFs. In this paper, we present a suitable theoretical model and perform detailed numerical simulations to understand the impact of linear mode coupling on the IM-FWM process. 2.

Nonlinear propagation in multimode fiber

The equation that we use for this purpose was developed earlier as Eq. (6) in Ref. [10], which is given by ∂A B2 ∂ 2 A B3 ∂ 3 A γ ∂A = iB0 A − B1 −ı − +ı ∂z ∂t 2 ∂t 2 6 ∂t 3 3

Z Z

[(AT F(2) A)F(2)∗ A∗ + 2(AH F(1) A)F(1) A]dxdy + QA,

(1)

where “*” refers to conjugate transpose, and we refer to [10] for the definitions of various parameters. To account for random linear coupling between modes [11, 12] we included a random coupling matrix Q, whose elements qi j , where i and j each run from 1 . . . N with N the number of modes, can be calculated from the coupled-mode theory [10, Eq. (40)]. We focus on the FMF used in the 2013 experiment [7]. It supports 3 spatial modes: LP01, LP11a, and LP11b, each with two orthogonally polarized states, resulting in a vector A with N = 6 components. The dispersion properties of LP11a and LP11b are assumed to be identical but different from those of the LP01 mode. More specifically, D01 = 19.3, and D11 = 18.3 ps/(nm-km) at λ0 = 1540 nm. The intermodal group delay at λ0 is β1,01 − β1,11 ≈ 300 ps/km. All spatial modes are assumed to have the same value of dispersion slope S = 0.055 ps/(nm2 -km). The nonlinear parameter γ = 1.77(Wkm)−1 , and the fiber linear loss coefficient α = 0.226 dB/km. All nonlinear effects including IM-FWM are governed by RR the nonlinear overlap factors f plmn ∝ Fl Fm Fn∗ Fp∗ dx dy, where Fp (x, y) is the spatial profile of the pth mode [10]. For our 6-mode FMF, the values of f plmn were obtained using the designed modal profiles. Symmetry considerations lead to [13, Tab. II]:  1 when all 4 fields are LP01     when all 4 fields are either LP11a or LP11b  0.747 0.496 when 2 fields are LP01 and 2 are LP11 (a or b) flmnp = (2)   0.249 when 2 fields are LP11a and 2 are LP11b modes    0 in all other cases 3.

IM-FWM in the absence of linear mode coupling

We first ignore linear mode coupling (by setting Q = 0) and perform numerical simulations for the two IM-FWM processes termed “PROC1” and “PROC2” in Refs. [7, 13]. The two processes differ by which wave is conjugated in one of the

M3F.5.pdf 20 Pump1 (LP11a) P =16dBm in P =14.8dBm

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Idler (LP11a)

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−30

Pin=6dBm Pout=5.49dBm

−40 −50

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Pout=−3.78dBm

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OFC 2014 © OSA 2014

(a)

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Pin=18dBm Pout=16.97dBm

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1 0.8 0.6 0.4

Linear phase matching Numerical, Eq. (1) Pin =(P1,P2,Pprobe)= (4,6,−6) dBm

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Fig. 1. Input (solid green) and output (dotted red) spectra for PROC1 (a) and PROC2 (b). FWM efficiency of PROC2 versus (c) probe and pump2 wavelengths using linear phase analysis, and (d) at a fixed pump wavelengths obtained using linear phase analysis and Eq. (1). All fields are polarized in the same direction at the input, and no polarization rotation or linear coupling is applied during transmission.

conjugated pairs of the FWM process. The input parameters for the two pumps and probe waves are given in Fig. 1. We propagate these fields over a distance of 4.7 km by solving Eq. (1) using the split-step Fourier method [2], extended to deal with multiple spatial modes. The output spectra are plotted in Fig. 1 with dotted red curves; solid green lines show input spectra for comparison. As seen in Fig. 1(a), an idler wave is generated by IM-FWM at the wavelength corresponding to PROC1 while the strongest idler is generated by PROC2 in Fig. 1(b). A second idler wave of small amplitude is also present in Fig. 1(b). It is attributed to PROC1 which has low efficiency for the input wave configuration. In our simulations, the reduced power of the two pumps is mainly due to linear fiber loss, and the peak power of the idler can be estimated using the undepleted-pump approximation [2]: dAidler α ≈ iγ( f2121 + f2211 )Ap1 (0)Ap2 (0)A∗probe (0)e−3αz/2 − Aidler . (3) dz 2 Solving Eq. (3) returns Pidler (L) = −4 dBm, quite close to the numerical value of −3.78 dB. The reason that our estimate is slightly lower is that Eq. (3) does not consider amplification of the probe through IM-FWM. These results demonstrate that IM-FWM can achieve full efficiency in FMFs. As found in Ref. [7], PROC1 and PROC2 share an identical phase-matching condition. However, our results show that these two IM-FWM processes exhibit totally different bandwidths. The bandwidth can be estimated from a linear phase analysis, since the power of the idler is proportional to η = sinc2 (∆β L/2) under the undepleted-pump approximation, where ∆β is the linear phase mismatch. In Fig. 1(c) we plot the IM-FWM efficiency η as a function of λprobe and λpump2 for PROC2, with λpump1 = 1530 nm. The position of the crest in this figure varies with both λprobe and λpump2 . In particular, the efficiency η is extremely sensitive to λpump2 with a bandwidth < 0.02 nm. If we fix λpump2 , we obtain an efficiency curve as a function of λprobe plotted in Fig. 1(d). It shows that PROC2 has a relatively large bandwidth of 2 nm. We also plot the numerical results using dotted-red curve in Fig. 1(d), which shows excellent agreement to the linear phase analysis. The phase matching map for PROC1 is similar to that in Fig. 1(c), only with λprobe and λpump2 interchanged. At a fix λpump2 , the bandwidth is only about 0.01 nm, 200 times narrower than PROC2. The physical reason behind this difference is very simple. For the wave configuration in PROC2, the idler wave moves in the same direction as the probe moves, maintaining the difference between the averaged wavelengths (that determine the group velocities) in two spatial modes, and keeping the IM-FWM process phase-matched. On the other hand, the opposite happens for PROC1. 4.

Impact of linear mode coupling on IM-FWM

So far, we have neglected any linear mode coupling. In reality, such coupling invariably occurs because of various fiber imperfections. The impact of random birefringence fluctuations on nonlinear propagation is included when using the set of Manakov equations in the case of a single-mode fiber [14]. Recently, generalized Manakov equations were derived in the context of multi-mode fibers [6,10]. The strength of linear coupling depends on not only the linear mode coupling (Q), but also the propagation constant difference ∆β0 between modes. To consider the impact of linear coupling, we introduce

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a transfer matrix: T = exp[i(B0 + Q)∆z], where ∆z is the step size, and we adopt the following three simple models. Model 1 considers the coupling between orthogonally polarized modes of the same spatial mode but neglect linear mode coupling between any pair of distinct spatial modes. The corresponding transfer matrix T1 is given by     r11p r12p 0 0 0 0 r11p r12p 0 0 0 0 r r r22p 0 0 0 0  r22p 0 0 0 0   21p    21p    0 0 0 r11l r12l r13l r14l  0 r11l r12l 0 0   . (4) , T = T1 =  2  0  0 0 r21l r22l r23l r24l  0 r21l r22l 0 0       0  0 0 r31l r32l r33l r34l  0 0 0 r11m r12m  0 0 r41l r42l r43l r44l 0 0 0 0 r21m r22m It takes a block-diagonal form with 2 × 2 sub-matrices (unitary random matrices) for each spatial mode. For the FMF considered, one must allow for the fact that LP11a and LP11b modes are nearly degenerate in constant of propagation and can strongly linearly couple. Model 2 considers this case and uses T2 given in Eq. (4). This form ignores linear coupling between LP01 and LP11 modes (a or b), which is expected to be weak for a large proportion of FMFs that are designed with a relatively large difference ∆β0 in their propagation constants. This transfer matrix contains a unitary 4 × 4 random rotation sub-matrix representing random coupling among the 4 modes: LP11ax , LP11ay , LP11bx and LP11by . Model 3 assumes that there is an equal level of random linear mode coupling among all spatial modes. It can be represented by a matrix T3 , a 6 × 6 random unitary matrix (not displayed here). We performed numerical simulations for PROC2-type IM-FWM using these three linear mode coupling models. The transfer matrix Ti , with i = 1, 2 or 3, is applied to the fields at each step of the split-step Fourier method. We choose identical βi , with i = 1, 2 or 3, amongst the LP01x and the LP01y modes and amongst the LP11ax , LP11ay , LP11bx and LP11by modes. We propagate the fields over 4.7 km using a step size of 7.8 m (shorter step sizes show convergence). Numerical simulations are repeated 5 times in each case with different random seeds for the random matrix generator. Table 1 shows how the power of idler wave changes for three different linear coupling models. Model 1 Model 2 Model 3

-7.41 -7.38 -49.28

-7.34 -7.40 -52.61

-7.37 -7.37 -50.87

-7.35 -7.38 -52.14

-7.40 -7.41 -51.25

Table 1. Idler power (dBm) for three different linear coupling models. As seen in Table 1, the power of the idler wave generated is very close to −7.4 dBm (with variations ±0.1 dBm) in models 1 and 2 but it becomes negligibly small in model 3. These results indicate the coupling between the LP11a and LP11b modes does not affect much the IM-FWM process because these two modes have identical dispersion parameters. In contrast, any coupling between the LP01 and LP11 (a or b) reduces the FWM efficiency drastically because the phase matching condition can no longer be satisfied. We also solved the generalized Manakov equations in the weak coupling region [10] and obtained an idler power of −7.36 dBm, a value quite close to the full simulations for models 1 and 2. Compared to the value in Fig. 1(b), mode coupling reduces the idler power by about 3.5 dB, which is expected. The idler wave is generated through a process similar to cross phase modulation. Averaging over random polarization rotations reduces the nonlinear coefficient 2 to 4/3 (Eq. (26) in Ref. [10]), which changes the idler power by a factor of 4/9 or 3.5 dB. In conclusion, we present a suitable theoretical model and perform detailed numerical simulations to understand the impact of linear mode coupling on IM-FWM process. We find that one process has a much larger bandwidth than the other, and random linear mode coupling reduces idler power by only 3.5 dB. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.

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