Stimulated Raman scattering crosstalk in massive WDM systems under the action of group velocity dispersion

15 July 2001

Optics Communications 194 (2001) 319±323

www.elsevier.com/locate/optcom

Stimulated Raman scattering crosstalk in massive WDM systems under the action of group velocity dispersion A.G. Grandpierre a, D.N. Christodoulides a,*, W.E. Schiesser b, C.M. McIntosh c, J. Toulouse c a

Department of Electrical Engineering and Computer Science, Lehigh University, Bethlehem, PA 18015, USA b Department of Chemical Engineering, Lehigh University, Bethlehem, PA 18015, USA c Department of Physics, Lehigh University, Bethlehem, PA 18015, USA Received 7 February 2001; accepted 21 May 2001

Abstract We provide a theoretical model that can accurately describe the evolution of stimulated Raman scattering crosstalk in high-speed massive wavelength-division-multiplexed (WDM) systems. Our model takes into account channel walko€ e€ects due to group-velocity dispersion and directly involves the bit-1 probability and the duty cycle of the modulation sequence. Our analytical predictions are found to be in excellent agreement with numerical simulations carried out using pseudo-random bit patterns in a return-to-zero format. These results are also applicable even in the case of ampli®ed WDM systems. Pertinent examples are provided. Ó 2001 Published by Elsevier Science B.V. Keywords: Stimulated raman scattering; Wavelength division multiplexed systems (WDM); Group velocity dispersion; Crosstalk; Optical ®bers

It is widely recognized that wavelength-divisionmultiplexed (WDM) technology provides a viable solution in terms of exploiting the large bandwidth o€ered by today's low-loss optical ®bers [1]. In the last few years or so, both the number of WDM wavelengths as well as the bit rate of each channel have kept increasing at an unprecedented pace. Fiber WDM transmission systems with more than 100 channels at aggregate rates greater than 1 Tb s 1 have been recently demonstrated in labo-

* Corresponding author. Tel.: +1-610-758-4069; fax: +1-610758-6279. E-mail address: [email protected] (D.N. Christodoulides).

ratory experiments [2,3]. It is by now well understood that in order to harness this bandwidth several technical issues must be ®rst addressed. Among them are crosstalk e€ects due to ®ber nonlinearities such as stimulated Raman scattering (SRS), four-wave mixing, self- and cross-phase modulation [1,4,5]. Ultimately however, as the total power in the WDM channels increases, this crosstalk process is eventually dominated by SRS [4,6]. Yet, limited information exists as to how exactly SRS impacts WDM systems. In general, WDM channels are in¯uenced by SRS in the following two ways. First, the average power level of a bit ``1'' in a particular channel is modi®ed (ampli®ed/attenuated) as a result of SRS. Second, as a result of this same process, the actual

0030-4018/01/$ - see front matter Ó 2001 Published by Elsevier Science B.V. PII: S 0 0 3 0 - 4 0 1 8 ( 0 1 ) 0 1 3 0 5 - 0

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level of a particular ``1'' bit varies statistically (around the ``1'' bit average power level) since it depends on the bit sequences encountered in the remaining channels [7±9]. Both these aspects are important since they directly a€ect the overall system performance. We would like to emphasize, that at this point, the inter-channel SRS power exchange in a WDM cluster has been solved analytically only in the CW regime [10,11]. This was done by employing the so-called triangular approximation for the Raman gain pro®le (of silica glass), as ®rst suggested by Chraplyvy [12]. On the other hand, in the case of high-speed (time modulated) signals, we know very little (at the analytical level) regarding WDM inter-channel SRS power transfer. This problem is of considerable complexity, especially when one considers groupvelocity dispersion e€ects [13,14]. In this case, every channel or wavelength travels at a di€erent group velocity (walks-o€) and, as a result, every bit slot may interact with hundreds of di€erent bits (from the other channels) during propagation. Even though it has been long speculated [14, 15] that the WDM SRS crosstalk under walk-o€ conditions can be described within an ``e€ective'' CW picture (using mean time-averaged values), so far no systematic study has been undertaken to support this view. Even more importantly, it is only recently that full analytical results have been developed (for the CW regime) to allow for such comparisons [10]. In this letter, we provide a theoretical model that can accurately predict how the average power per bit evolves under the action of stimulated Raman scattering crosstalk in high-speed massive WDM systems. Our model, which is based on the triangular approximation, takes into account dispersive channel walk-o€ e€ects by appropriately scaling the ``neighboring'' channel powers. This is done by considering the bit-1 probability, the duty cycle of the modulation sequence, and by assuming that e€ectively this process takes place under ``CW conditions''. Our analytical predictions are found to be in excellent agreement with numerical simulations carried out using pseudo-random bit patterns in a return-to-zero (RZ) format. These results are applicable even in the case of ampli®ed WDM systems. Pertinent examples are provided.

Let us consider a WDM system of N channels co-propagating in an optical ®ber. We assume that the lowest propagating wavelength is represented by channel #1, whereas channel #N corresponds to the highest wavelength. Our theory takes into account ®ber group-velocity dispersion e€ects, i.e. a di€erent group velocity vn is assigned to each wavelength. Since the purpose of this study is to isolate the SRS power exchange process, we ignore any pulse distortion that may arise from four-wave mixing, cross- and self-phase modulation, and pulse dispersive e€ects. These latter assumptions can be justi®ed in the case where a nonzero dispersion shifted ®ber (NZDSF) is used and if the length of the ®ber is less than the characteristic dispersion distance. Following Ref. [5], the power distribution in every channel evolves according to the following system of partial di€erential equations: N X oPn 1 oPn 1 Pn ‡ ‡ a…z†Pn ‡ GR …xn 2A mˆ1 oz vn ot

ˆ0

xm †Pm …1†

In Eq. (1), Pn …z; t† represents the power in the nth channel, z is the propagation distance and t is the time variable. The loss/gain pro®le in the ®ber is described through a…z†, A is the e€ective crosssectional area of the ®ber, GR is the Raman gain pro®le of silica glass, and x the angular frequency. The dispersive properties of the group velocity vn with respect to channel #1, are given by 2 vn 1 ˆ b00 ‡ b000 X ‡ b000 x1 , is 0 X =2, where X ˆ xn the angular frequency di€erence. In deriving Eq. (1), we neglected any vibrational energy loss effects. By introducing new normalized travelling coordinates, s ˆ …t z=v1 †=s0 and n ˆ z=lw in Eq. (1), we obtain: N oPn lw oPn lw X ‡ ‡ a…n†lw Pn ‡ Pn GR …xn on lwn os 2A mˆ1

ˆ0

xm †Pm …2†

where lw ˆ s0 v1 v2 =…v1 v2 † is the walk-o€ length between channel #1 and #2, s0 is a characteristic pulse duration, and lwn ˆ s0 v1 vn =…v1 vn † is the walk-o€ distance between the nth channel and #1. Note that all walk-o€ lengths have been taken with

A.G. Grandpierre et al. / Optics Communications 194 (2001) 319±323

respect to channel #1. If, for example, the dispersion is anomalous [5] (i.e., vn < v1 ), then channel #1 will be travelling ahead of the WDM cluster. By assuming that the channels are equally separated in the frequency domain by an amount Df , 2 then vn 1 ˆ b00 ‡ …n 1†b000 DX ‡ …n 1†2 b000 0 DX =2, where X ˆ …n 1†DX and DX ˆ 2p Df . Thus, the term lw =lwn is given by: lw ˆ …n lwn

1†

b000 ‡ …n 1†e b000 ‡ e

…3†

1†ej  jb000 j, where e ˆ b000 0 DX=2. Note that if j…n then Eq. (3) gives lw =lwn  n 1. In our examples, we will include the e€ects arising from the slope of the dispersion curve and thus the e term will be retained. Before we proceed any further, perhaps it would be useful to take a closer look at this process, i.e. SRS crosstalk under dispersion. Let us assume that a ``1'' bit or pulse (in channel k) propagates in the ®ber at some velocity vk . Depending on the frequency of the channel that carries this bit, after several kilometers, the pulse will pass (or be overtaken) tens or hundreds of other bits from the rest of the channels. The number of bits that this pulse will ``see'' depends on the characteristic walk-o€ lengths involved. For example, the walko€ distance between channel #1 and #n is approximately lwn  s0 =‰2p…n 1†jb000 Df jŠ (in the case that b000 can be neglected). Thus the walk-o€ length 0 decreases with pulse duration, or as the channel number increases. As a result, the greater the separation between two channels (in the frequency domain), the faster they pass each other. Incidentally, this same group of channels (spectrally apart) is the one most a€ected by SRS crosstalk. Because of this almost random succession of bits seen by say channel k, one may then conclude that on average this is equivalent to an interaction with (N 1) CW waves. The ``average'' CW amplitude P of the mth wave, as observed from channel k, is given by P m ˆ Pm0 pd, where Pm0 is the peak power of the ``1'' bit, p is the probability that ``1'' will occur in the data sequence, and d is the duty cycle. If the bit period is T and if the power distribution of ``1'' (in time) is given by Pm0 h…t=~ s† (where h…x† is a normalized function of peak amplitude 1) then

321

R

d ˆ …1=T † dt h…t=~ s†. Therefore, the SRS crosstalk problem of time modulated WDM signals under the action of dispersion can be treated by ``mapping'' it on previously obtained CW results [10]. More speci®cally, to ®nd out how channel k is a€ected by this process, we assume that the power in this channel at z ˆ 0 is the actual power Pk0 initially launched at the input, whereas the ``CW'' levels of the other channels (m 6ˆ k) is P ˆ Pm0 pd where again Pm0 are the actual launch powers. By following this procedure, and by employing the results of Ref. [10], one can easily show that within the triangular approximation, the power level in channel k evolves according to:  Z z  0 0 Pk …z† ˆ Qk …g…z†† exp a…z † dz …4† 0

where Qk …g† ˆ Pk0 J0 exp…G0 J0 …k 1†g† 2 N X Pm0 eG0 J0 …m 1†g ‡ Pk0 eG0 J0 …k  4 dp

3

1

1†g 5

mˆ1 m6ˆk

…5† and Z g…z† ˆ

z 0

" exp

J0 ˆ Pk0 ‡ dp

N X

Z

z0 0

Pm0 ;

# a…z00 †dz00 dz0 ; G0 ˆ g0 Df =…2A†

m6ˆk

In this latter expression, g0 represents the slope of the Raman gain. In the special case where all the input power levels are the same (Pm0 ˆ P0 ), Eq. (5) takes the form:  sinh…NG0 J0 g=2† Qk …g† ˆ J0 …1 dp† ‡ dp sinh…G0 J0 g=2†  1 …6†  exp…G0 J0 g…N ‡ 1 2k†=2† where in Eq. (6) J0 ˆ P0 ‰1 ‡ …N 1† dpŠ. Note that the introduction of the g…z† transformation [16], makes our results applicable irrespective of the loss/gain pro®le of the ®ber system. When the loss is constant, g ˆ Ze ˆ …1 exp… az††=a.

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A.G. Grandpierre et al. / Optics Communications 194 (2001) 319±323

We will now demonstrate the validity of our theoretical results by employing computer simulations. More speci®cally, Eq. (1) is solved numerically. The accuracy of our numerical code was checked (in special cases) against known analytical solutions [13]. For our simulations, let us consider a NZDSF ®ber system operating around 1.55 lm. At that wavelength the ®ber loss is 0.2 dB/km and its e€ective core area is A ˆ 50 lm2 . The WDM system involves 32 channels operating at 10 Gb/s (T ˆ 100 ps), equally spaced in the frequency domain with Df ˆ 100 GHz. Channel #1 is located at 1.55 lm whereas the last one at 1.575 lm. The ®ber dispersion taken at 1.55 lm is D ˆ 3 ps km 1 nm 1 or b000 ˆ 3:8 ps2 km 1 , and b000 0 ˆ 0:11 ps3 km 1 . The zero-dispersion wavelength is 1.5 lm. From our previous discussion, the walk-o€ length between channel 1 and 32 happens to be lw  1:05 km. This implies that a ``1'' bit from channel 1 will ``see'' up to 78 bits from channel 32 over a propagation distance of 80 km. Eqs. (1) and (2) are solved within the triangular approximation with a Raman gain slope of g0 ˆ 6:7  10 18 m W 1 GHz 1 at 1.55 lm [5,6]. In all cases we take p ˆ 1=2 and T ˆ s0 . Every channel is loaded with a pseudo-random sequence of 27 -1 bits in a RZ modulation format and with a peak power of P0 ˆ 4 mW per channel. In the ®rst example, the bits are simulated with rectangular-like superGaussian pulses [5] of duty cycle d ˆ 1=2. Thus, the total ``e€ective'' power at the origin is J0 ˆ 35 mW. The power distribution can be computed from a sample in time where bits from all channels interact via SRS. Fig. 1 shows the power distribution after propagation through 80 km of ®ber, over which the loss is constant. The solid bars give the power distribution per channel after solving Eq. (1). This latter distribution was obtained after averaging out the bit power in every channel. On the other hand, the dashed line exhibits the expected theoretical result as obtained from Eqs. (4) and (6). Clearly, the theoretical results are in excellent agreement with the numerical simulations. The distribution in dB is linear [6,10,11], and in this case the maximum crosstalk is PN =P1 ˆ 0:61 dB. In fact, one can show that Eqs. (4)±(6) lead to the following approximate result: …PN …g†= P1 …g†† dB  4:34G0 P0 dpN …N 1†g.

Fig. 1. Output power distribution (per channel) of a 32 channel WDM system, as obtained from numerical simulations (solid bars) and from theory (dashed line, i.e. Eqs. (4)±(6)).

As a second example, let us consider the same ®ber system and WDM parameters. This time however, a discrete ampli®er of gain 15.2 dB is inserted at 40 km. Moreover, the signal bits are s† with a soliton-like pulses, i.e., h…t=~ s† ˆ sech2 …t=~ duty cycle of d ˆ 1=5. The peak power of each pulse is P0 ˆ 4 mW and thus J0 ˆ 16:4 mW. The power distributions corresponding to this example are given in Fig. 2. Again, the numerical results (after solving Eq. (1) ± solid curve) are in excellent agreement with the theoretical ones of Eqs. (4) and (6) ± dashed curve. The maximum crosstalk in this

Fig. 2. Power distribution in a 32 channel WDM system when soliton-like pulses are used. Power distribution at the input (dotted line). Output power distributions as obtained from numerical simulations (solid line), and from theory (dashed line, i.e. Eqs. (4)±(6)). These last two curves almost coincide in the ®gure.

A.G. Grandpierre et al. / Optics Communications 194 (2001) 319±323

case is PN =P1 ˆ 1:32 dB. This illustrates the fact that our results can also be used when the loss/gain pro®le of the system is not necessarily constant. Finally we would like to note that similar results are expected when other transmission formats (such as NRZ) are used. In conclusion, we have developed a theoretical model that can accurately describe the evolution of stimulated Raman scattering crosstalk in highspeed massive wavelength division multiplexed systems. Our model takes into account channel walk-o€ e€ects due to group-velocity dispersion and directly involves the bit-1 probability and the duty cycle of the modulation sequence. Our analytical predictions are found to be in excellent agreement with numerical simulations carried out using pseudo-random bit patterns and a returnto-zero format.

Acknowledgements This project was supported in part by NSF and by ARO-MURI.

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