a modified fiber dispersion model using continuous over wavelength for numerical simulation of...

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Optical Fiber Technology 11 (2005) 46–55

www.elsevier.com/locate/yofte

A modified fiber dispersion model using continuoβ2 over wavelength for numerical simulation of

optical pulse propagation through dispersivenonlinear fibers in WDM systems

Jaehoon Lee∗

Dong Suwon, P.O. Box 105, 416, Maetan-3dong, Gyeonggi-do 442-600, South KoreaDigital Home Laboratory, Telecommunication R&D Center, Samsung Electronics, South Korea

Received 7 November 2003; revised 30 July 2004

Available online 11 November 2004

Abstract

To accurately simulate transmissions of opticallywavelength division multiplexed signals withlarge spectral width through dispersive nonlinear fibers, a modified fiber dispersion model uscontinuous group velocity dispersion (GVD) over wavelength (= β2(ω)) has been proposed. Thmodel can provide an accurate simulation of the interaction between the dispersion and thnonlinearities by the extraction of the propagation constant (= β(ω)) from the continuous GVDwithout neglecting higher order dispersion terms. This method has been applied to dispersioaged 32 channel×40 Gbps WDM systems to simulate the transmission performance. Our simuresults suggest that the proposed method can accurately model the dispersion effect with theof fiber nonlinearities, compared with conventional method. 2004 Elsevier Inc. All rights reserved.

Keywords: Optical fiber communication; Optical fiber nonlinear effect; Dispersion; Simulation

* Fax: +82-31-279-5255.E-mail address: [email protected].

1068-5200/$ – see front matter 2004 Elsevier Inc. All rights reserved.doi:10.1016/j.yofte.2004.09.002

J. Lee / Optical Fiber Technology 11 (2005) 46–55 47

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1. Introduction

The demand of computer modeling and simulation has been increased for design ocomplicated optical transmission systems such as long haul amplified terrestrial and sumarine systems, and ultrahigh capacity (Tbit/s) systems. Especiallyfor ultrahigh capacityWDM (wavelength division multiplexing) systems and long haul amplified links, thealization of optical transmission systems through trial and error experiments has bunreasonable because of tremendous cost andtime of the implementation due to the icreased system complexities. So, complete modeling and simulation for fine-tunsystem parameters at the design stage should be performed before system impltions.

The efforts of simulation of optical signal transmissions over fibers have been pusince the early days of optical communications [1–4]. In general, the simulation of osignal transmissions is based on solving thenonlinear Schrödinger equation that includboth fiber dispersion and nonlinear effects such as self-phase modulation, cross-phaulation, and four-wave mixing [5,6].

The dispersion in fibers plays a critical role in optical signal propagations of oppulses through fibers. The dispersion caused by the nonlinear phase response oaround the optical carrier frequency makes opticalsignal distortions. In addition, the inteaction between the dispersion and the nonlineareffects of fibers largely affects on opticsignal propagations. Self-phase modulation, one of the nonlinear effects in fibermodify the pulse broadening effect due to the dispersion effect. The interaction ofvelocity dispersion and self-phase modulation can reduce the pulse broadening due todispersion in the anomalous regime of fibers. Especially, in WDM systems, the noneffects of fibers such as cross-phase modulation and four-wave mixing which mainterchannel crosstalk are largely dependent on the fiber dispersion. The walkoffcaused by the dispersion can mitigate cross-phase modulation, because this effect mathe optical pulse of each channel walk away from each other. In addition, four-wave mis dependent on the fiber dispersion because the dispersion is related to the phase matchcondition. So, it is required to model the fiberdispersion accurately to include linear anonlinear effects, and theirinteraction in fibers.

In the present work, we indicate the limit of the currently used dispersion model bon the nonlinear Schrödinger equation for the simulation of high-speed WDM syand then propose a new method including an accurate dispersion effect of fibers bason using the continuous group velocity dispersion (GVD) over wavelength. Moreovedemonstrate that the continuous GVD over wavelength can be used to model the diseffect accurately including high order terms offibers, completely, and the walkoff effecautomatically.

This paper is structured as follows. In Section 2, the conventional model of thedispersion for WDM systems based on the nonlinear Schrödinger equation is providthis section, the limit of the conventional method is described. In Section 3, the derivof a new method of the fiber dispersion based on using the continuous GVD is desIt is demonstrated that this method can includeall effects related to the fiber dispersioIn Section 4, we applied the proposed method to 32 channel× 40 Gbps WDM optical sig-

48 J. Lee / Optical Fiber Technology 11 (2005) 46–55

ntional

opti-therical

xer

s

dlar fre-

withals as

Fig. 1. Simulation schematic of the fiber dispersion in WDM systems.

nals, transmitted over fiber, and then compared the results obtained from the convemethod.

2. Conventional method considering dispersion in fibers for numerical analysis ofWDM systems using discrete β2 and β3

For the simulation of optical signals of WDM systems, it is required to combinecal signals of each channel into a concatenatedoptical signal after passing a wavelengdomain multiplexer, as shown in Fig. 1. To consider the fiber dispersion for the numanalysis of transmission characteristics of WDM signals, the complex electrical fieldA(t)

at the output of the wavelength domain multiplexer can be expressed as

A(t) =∑

i

Ai(t)exp[−j (ωi − ω0)t

], (1)

wherei is channel number,Ai(t) andωi are the field envelope after wavelength multipleand an optical frequency of transmitters of theith channel, respectively.ω0 is the opticalangular frequency of the center wavelength of WDM signals.

In general, the multiplexed complex electrical field,A(t), in the single mode fiber obeythe following nonlinear Schrödinger equation [5]:

∂A

∂z= −j

2β2

∂2A

∂T 2 + 1

6β3

∂3A

∂T 3 − α

2A + jγ |A|2A, (2)

whereβ2 is GVD, β3 is the third order dispersion,α is the absorption coefficient, anγ is the nonlinearity coefficient. These parameters are at the center optical anguquency,ω0. T is made by the time delay ofβ1z. We can replacet in Eq. (1) withT , becausethe time delay occurs identically in every channel. If we substitute Eq. (1) into Eq. (2)an approximation, we can get another nonlinear Schrödinger equation for WDM signfollows:


ds

e

cy

tutcy,s

teis

wholeticalz. Es-ectral

es,s ofbers ore pulsenonlin-

seriousms.dInhere-

on,

∑i

∂Ai

∂ze−j (ωi−ω0)T

= −∑

i

{β2(ω0)(ωi − ω0) + 1

2β3(ω0)(ωi − ω0)

2}

∂Ai

∂Te−j (ωi−ω0)T

−∑

i

j

2

{β2(ω0) + β3(ω0)(ωi − ω0)

}∂2Ai

∂T 2 e−j (ωi−ω0)T

+∑

i

1

6β3(ω0)

∂3Ai

∂T 3e−j (ωi−ω0)T −

∑i

α(ω0)

2Aie

−j (ωi−ω0)T + jγ |A|2A. (3)

This equation indicates that each optical signal experiences different time delay angroup velocity dispersion. It means that the optical signal of theith channel propagatethrough fibers with a new relativeβ1(= β2(ω0)(ωi −ω0)+ 1

2β3(ω0)(ωi −ω0)2) andβ2(=

β2(ω0) + β3(ω0)(ωi − ω0)). Precisely, the new relativeβ1 at ωi can be obtained from thdiscrete value ofβ2 andβ3 at the optical frequency of the center wavelength,ω0, multipliedby the difference of the radian frequency between theith channel and the center frequen(ωi −ω0). Besides,β2 atωi can be calculated from the discrete value ofβ2(ω0) andβ3(ω0)

multiplied by (ωi − ω0). This method can automatically include the walkoff effect thaaffects on the fiber nonlinearities, because a relativeβ1 of each channel is considered. Bit is expected that the approximatedβ1 andβ2 that are far away from the center frequenmake large errors, becauseβ1 andβ2 in real fibers have nonlinearcharacteristics versuwavelength.

For a single channel transmission system, this method can provide a relatively accurafiber dispersion effect, because the spectral width of optical signals for a single channelsufficiently small. However, for WDM transmission systems (largeωi −ω0), it occurs largediscrepancy to use this method in order to consider the fiber dispersion effect overspectra of WDM signals into a numerical method. The spectral width of WDM opsignals with 100 GHz channel spacing and a few tens of channels reaches a few THpecially, to simulate WDM systems using S-, C-, and L-bands, simultaneously, the spwidth of optical signals becomes larger. As the spectral width of optical signals increasthe error of the relativeβ1, β2, andβ3 increases due to neglecting higher order termthe dispersion. The higher order terms should be considered in zero dispersion fidispersion flattened fibers. The error in the dispersion parameters affects not only thbroadening due to the dispersion but also the interaction between the dispersion andear effects of fibers [7]. For longer transmission systems, this effect becomes morebecause of the accumulation of the dispersion effect, even in dispersion-managed systeTo avoid this problem, Eq. (2) can be used for each channel, individually. But this methocannot accurately consider the interaction between channels such as the walkoff effect.addition, the larger the number of channels is, the longer the simulation time is. Tfore, it is required to modify the dispersion model in the nonlinear Schrödinger equatiEq. (2).


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allely,e

3. Proposed method considering dispersion in fibers for numerical analysis of WDMsystems using continuous β2

To solve the nonlinear Schrödinger equation (Eq. (2)), the optical field,A(z,T ), is prop-agated for a distance with the dispersion effect only or with the nonlinear effect only. Thdispersion effect of fibers can be solved in afrequency domain. In calculating with the dipersion effect only, we can use a measured or calculated continuousβ2 over the interestingwavelength instead of discrete values ofβ2 andβ3, to improve the conventional methodfollows:

∂A

∂z=

(−jβ(ω) − α(ω)

2

)A. (4)

In the conventional model,β(ω) is approximated to12β2(ω − ω0)

2 + 16β3(ω − ω0)

3.Therefore, as the spectral width of optical signals increases or the length of fibers increasthis approximation in the conventional model becomes inaccurate. If we obtainβ(ω) overthe interesting wavelength range, we can accurately include the dispersion effect of fiberEquation (4) can be applied to any fibers by extracting the information ofβ(ω) from acontinuousβ2 over the interesting wavelength.

From the continuousβ2 over the interesting wavelength, we can obtainβ1(ω):

β1(ω) = ∂β(ω)

∂ω=

ω∫ωc

β2(ω′′) dω′′ + C1, (5)

whereC1 is constant andωc is the initial angular frequency in the interesting range. FrEq. (5), we can obtainβ(ω) as follows:

β(ω) =ω∫

ωc

β1(ω′′) dω′′ + C2 =

ω∫ωc

ω′′∫ωc

β2(ω′) dω′ dω′′ + C1ω − C1ωc + C2. (6)

After substituting Eq. (6) into Eq. (4), the transfer function(H(ω) = A(z′ = z)/A(z′ =0)) of Eq. (4) becomes

H(ω) = exp

[{−jβ(ω) − α(ω)

2

}z

]

= exp

[{−j

( ω∫ωc

ω′′∫ωc

β2(ω′) dω′ dω′′ + C1ω − C1ωc + C2

)− α(ω)

2

}z

]. (7)

We can neglectC1ω representing on only the time delay in time domain overWDM channels and−C1ωc + C2 representing only the constant phase shift. Namexp(−jC1ωz) makes a time delay ofC1z and exp(jC1ωc − jC2) makes only the phas


lly,

pulseidth.

-

-

ion.in. Inion

tained

n, fourulation

ce

ctiondeth thatlectric

dreeristicss

shift, because it is just constant. These reasons make the unknown terms ignored. Finathe transfer function becomes

H(ω) = exp

[{−j

( ω∫ωc

ω′′∫ωc

β2(ω′) dω′ dω′′

)− α(ω)

2

}z

]. (8)

Because this method considers the fiber dispersion usingβ(ω) without neglectinghigher order group velocity dispersion terms, this method can simulate the opticalpropagation through fibers in WDM systems without any constraint of the spectral wIn addition, this method can consider the accurate walkoff effect in itself.

Continuousβ2(ω) can be used to extract the information ofβ(ω) to model the dispersion effect accurately in fibers using Eqs. (5) and (6), and continuousβ1(ω) can be used toreconstructβ(ω) through Eq. (6). A continuousβ3(ω) or higher order group velocity dispersion cannot be used to makeβ(ω), because it makes the terms ofω2 or higher order ofω while reconstructingβ(ω) through integration, which cannot be neglected in calculatIn Eq. (7), the terms ofω can be neglected because it is just a time delay in time domaaddition, other constant terms can be ignored, because they don’t affect the pulse evolutover the fiber.

To obtain transmitted optical fields after fiber transmission with a length ofz, Eq. (8) canbe solved using the Fourier transform. Finally, the transmitted optical field can be obfrom

A(z,T ) = 1

2π

∞∫−∞

A(0,ω)H(ω)exp(−iωT ) dω, (9)

whereA(0,ω) is the Fourier transform ofA(z = 0, T ).Other nonlinear effects, such as, self phase modulation, cross-phase modulatio

wave mixing, and stimulated Raman scattering can be considered, through the calcwith the nonlinear effect only.

4. Application of the proposed method to 32 channel × 40 Gbps WDM systems

To see a difference between the conventional method using the discreteβ2 andβ3, andthe proposed method using the continuousβ2, we simulated the transmission performanof 32 channel× 40 Gbps WDM systems. For the conventional method, theD and D′parameters atω0 were applied to Eq. (2). For the proposed method, the transfer funin Eq. (8) was calculated fromβ2(ω). The continuous GVD of the standard single mofiber can be obtained from the measurement or the refractive index over wavelengis well approximated by the Sellmeier equation and waveguide dispersion due to diewaveguides [8].

Figure 2 showsβ2(ω) calculated from two methods.β2(ω) of the conventional methois linear with the slope ofβ3, because the higher order terms of dispersion parameters aneglected in Eq. (3). But that of the proposed method is not linear due to real charactof the standard single mode fiber [8]. We can confirm that the discrepancy increases a


of

s de-ede wave-the bited by aithmode

spans

plified

Fig. 2. Comparison of GVD (= β2(ω)) between the conventional method and the proposed method.

Fig. 3. Simulation configuration for 32 channel× 40 Gbps WDM systems.

the wavelength is far away from the center wavelength because the higher order termsdispersion parameters are ignored in the conventional method.

The simulation configuration is shown in Fig. 3. The simulation configuration wasigned as follows. The center wavelength and the first channel wavelength were placat 1.550116 and 1.525662 µm, respectively, and other channels were located at thlengths according to the ITU-T standard with the channel spacing of 200 GHz, andrate of each channel was 40 Gbps. Each optical transmitter was externally modulatLiNbO3 modulator with a bit stream of 27 length and generated optical NRZ signals wthe extinction ratio of 12 dB. The optical power launching into the standard singlefiber and the dispersion compensating fiber were set to 6 and−3 dB m, respectively. Thelength of the standard single mode fiber for one span was set to be 80 km and twowere used in this paper. A receiver used in oursimulation was optically preamplified andconsisted of an EDFA as a preamplifier, a Gaussian-shaped optical filter to block am


sed

h-ordere PINed to

ludingreebsolute32,ictsation.nnel 1

sitivities

pt fore mod-ficient

eng

Fig. 4. Calculated receiver sensitivities using the standard single mode fiber of the conventional and propomethods considering the only dispersion effect.

spontaneous emission noise, a PIN photodetector, and a power amplifier. The fourtBessel–Thomson electrical filter represented the frequency characteristics form thphotodetector to power amplifier in the receiver. The EDFA preamplifier was assumhave a flat gain of 20 dB and a noise figure of 4 dB.

Figure 4 shows receiver sensitivities of 32 channels after 160 km transmission incthe dispersion effect only. In the case of the conventional method, receiver sensitivities asymmetrical around the center wavelength, becauseβ2(ω) was modeled to be linear in thstandard single mode fiber and the dispersion compensating fiber. It means that the avalues of the residual dispersion of two pair channels between channel 1 and channelchannel 2 and channel 31, etc., are the same. So, this model cannot accurately predthe impairment due to the residual dispersion by the incomplete dispersion compenOn the other hand, in the case of the proposed method, receiver sensitivities of chaand channel 32 are different, because of the nonlinearβ2(ω) of the single mode fiber. Athe channel is away from the center wavelength, the discrepancy of receiver sensbetween two models increases.

Figure 5 shows receiver sensitivities with the same configuration of Fig. 4 exceincluding the nonlinear effects of fibers, such as, self-phase modulation, cross-phasulation, and four-wave mixing. For the standard single mode fiber, the nonlinear coef(n2) and the effective core area (Aeff) were set to 1.3 × 10−20 m2/W and 78 µm2, re-spectively. For the dispersion compensating fiber, the nonlinear coefficient(n2) and theeffective core are (Aeff) were set to 2.69× 10−20 m2/W and 35 µm2, respectively. Thediscrepancy of two models increases becausethe nonlinear effects of fibers affect by thdispersion. Because the degradation due to cross-phase modulation and four-wave mixiis inversely proportional to the absolute value of the dispersion parameterD which is re-


sed

sed

nnels

pt fores sig-to the

Fig. 5. Calculated receiver sensitivities using the standard single mode fiber of the conventional and propomethods considering the dispersion and the nonlinear effects.

Fig. 6. Calculated receiver sensitivities using the standard single mode fiber of the conventional and propomethods considering the dispersion, the nonlinearities, and the stimulated Raman scattering.

lated to the walkoff effect [9,10], the receiver sensitivities of lower wavelength chaare more degraded than upper wavelength channels.

Figure 6 shows receiver sensitivities with the same configuration of Fig. 5 exceincluding the stimulated Raman scattering. The stimulated Raman scattering maknals at longer wavelength amplified by shorter wavelength signals. This effect leads


els are

wave-errorsecomeof the

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ersioncylethod.

issultss wellalkoffrs.

di-ve

and331.

nol. 14

emas

single

18

ptical

ty, in:

degradation of shorter wavelength signals. So, receiver sensitivities of lower channdegraded, compared to Fig. 5.

These results indicate that the proposed model using the continuous GVD overlength is required to simulate optical signal transmissions in WDM systems. Theof the transmission performance due to the dispersion in the conventional model blarger, as the channel is away from the center wavelength. Moreover, the errorstransmission performance become larger withconsidering the nonlinear effects of fibecompared to the results from considering the dispersion effect only, because of the intetion between the group velocity dispersion and fiber nonlinearities.

5. Conclusions

In this paper, we proposed a new method which models the group velocity dispeffect of fibers accurately with the presence of fiber nonlinearities. To confirm the accuraof the proposed model, we applied this model to the dispersion managed 32 channe×40 Gbps WDM systems, and their results were compared with the conventional mWe demonstrated that the new method is more accurate and the conventional methodnot suitable for the simulation of optical signals with a large spectral width. Our reproved that the proposed model can accurately model the group velocity effect aas the nonlinear effect including high order terms of the fiber dispersion and the weffects, which play important roles in the dispersion and the nonlinear effects in fibe

References

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[2] M.R.C. Caputo, M.E. Gouvea, Dispersion slope effects of the compensation dispersion fiber for broadbdispersion compensation in the presence of self-phase modulation, Opt. Commun. 178 (2000) 323–

[3] X.Y. Zou, M.I. Hayee, S.-M. Hwang, A.E. Willner, Limitations in 10 Gb/s WDM optical-fiber transmissionwhen using a variety of fiber types to manage dispersion and nonlinearities, IEEE J. Lightwave Tech(1996) 1144–1152.

[4] A. Carena, V. Curri, R. Gaudino, P. Poggiolini, S. Benedetto, A time-domain optical transmission systsimulation package accounting for nonlinear and polarization-related effects in fiber, IEEE J. Select. AreCommun. 15 (1997) 751–765.

[5] G.P. Agrawal, Nonlinear Fiber Optics, Academic Press, San Diego, 1995.[6] J.J. Yu, K. Guan, Z. Xu, B. Yang, The effects of different dispersion compensation ratios on nonlinear

channel and WDM systems, Opt. Commun. 150 (1998) 85–90.[7] M. Miyagi, S. Nishida, Pulse spreading in a single-mode fiber due to third-order dispersion, Appl. Opt.

(1979) 678–682.[8] Ch. Hentschel, Fiber Optics Handbook, second ed., Hewlett Packard, Germany, 1988.[9] A. Mecozzi, C.B. Clausen, M. Shtaif, Analysis of intrachannel nonlinear effects in highly dispersed o

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