linear and nonlinear transfer functions of single mode fiber for optical transmission systems

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1564 J. Opt. Soc. Am. A/Vol. 26, No. 7 /July 2009 Le Nguyen Binh

Linear and nonlinear transfer functions of singlemode fiber for optical transmission systems

Le Nguyen Binh

Department of Electrical and Computer Systems Engineering, Monash University, P.O. Box 35, Clayton,Victoria 3800, Australia ([email protected])

Received January 23, 2009; revised April 22, 2009; accepted May 4, 2009;posted May 8, 2009 (Doc. ID 106672); published June 11, 2009

The transmittance transfer function of single mode optical fibers operating in both linear and nonlinear re-gions is presented. For the linear domain, Fresnel sine and cosine integrals are obtained via the Fourier trans-form. In the nonlinear region dominated by self-phase-modulation effects, the Volterra series is essential toobtain the nonlinear transfer function. A convergence criterion for the Volterra series transfer function (VSTF)approach is described for solving the nonlinear Schrödinger wave propagation equation. Soliton transmissionover single fibers is demonstrated as a case study of the application of the VSTF and a modified VSTF with anumber of segmented steps whose distance is within the limit of the convergence of the VSTF. © 2009 OpticalSociety of America

OCIS codes: 060.0060, 190.0190.

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. INTRODUCTIONiber-optic communication is the fastest growing areas inodern communication systems in order to satisfy the in-

ensive demand for bandwidth by the Internet. Systemsf higher bandwidth and longer transmission distancend multiplexed optical carriers have been installed at aremendous pace. With the advent of erbium-doped fibermplifiers (EDFAs) since 1989, the major limitation ofber-optic communication systems has changed from loss-

imited to dispersion-limited. The increase in bandwidthnd bit rates by high-speed modulation and multiplexingf optical carriers in the S-, C- and L-bands and the in-rease in transmission distance further complicates theesign of optical communications systems. It is thus es-ential to develop a more efficient model to represent theommunications channel using optical fibers.

Further due to the increase of the total number of op-ical channels of multiplexed optical-information-odulated carriers, the fiber channel reaches its nonlin-

ar regime and hence a frequency domain approach inodeling fibers is normally avoided by employing theell-known split-step Fourier (SSF) procedures. There-

ore the principal difficulty in modeling a single-mode op-ical fiber as a transmission channel lies in an effectiveepresentation of the self-induced nonlinear effects. Non-inearities existing in fibers, optical amplifiers, and otherevices have long been recognized as one of the majorimitations for optical communication systems. Variousonlinear effects such as self-phase-modulation (SPM),ross-phase-modulation (CPM), stimulated Raman scat-ering (SRS), stimulated Brillouin scattering, and four-ave mixing have been known to cause serious problems

n long-haul high-speed optical networks. The widely usedSF method incorporates the fiber nonlinear effects in theime domain into the calculation of linear effects in therequency domain via an intermediate step where the op-

1084-7529/09/071564-12/$15.00 © 2

ical fields have been Fourier transformed into the fre-uency domain. Despite the high efficiency of the SSFethod, it is a recursive method and hence can be un-ieldy in designing optical communication systems.The Volterra series transfer function (VSTF) is a math-

matically tool commonly employed to approximate theonlinear effects in a system [1]. It has been used to over-ome the nonlinearity in several engineering nonlinearroblems for its intuitive representation of nonlinear ele-ents. Unlike other numerical methods, the VSTFethod is nonrecursive. An application of the Volterra se-

ies for optical communication system first appeared inhe literature in 1997 [2], and since then a number of pa-ers have been published on this topic reporting on sev-ral aspects of nonlinear effects on the channel capacityimit of single mode optical fibers [3–10]. These worksave, however, focused on the applications of the VSTFethod for solving the nonlinear Schrödinger wave

ropagation equation (NLSE) for optical communicationystems, and have compared the accuracy of the VSTFethod with the more traditionally used methods such as

he SSF approach. However, none of the papers has dis-ussed the issue of the validity of the VSTF method inepresenting the nonlinear effects in single mode opticalber channel. This means the limit of the convergence ofhe VSTF, besides which it is also important to estimateccurately the upper limit of the input optical power to bepplicable in a VSTF model. Indeed the radius of conver-ence of the VSTF model dictates the convergence of theeries and correlates to this limit.

In this paper, we present the effects of the convergencef VSTF and its applications in the modeling of opticalommunication systems, and most importantly the upperimit for convergence and its physical relationship withhe possible maximum power that can be transmittedhrough fiber for linear operation. In this linear operating

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Le Nguyen Binh Vol. 26, No. 7 /July 2009 /J. Opt. Soc. Am. A 1565

egion of the fiber, Fresnel sine and cosine functions arebtained and shown to represent the transfer function ofhe dispersive medium.

The paper is thus organized as follows. A brief repre-entation of the NLSE is given in Section 2, followed byts representation by the Fresnel functions in Section 3nd the VSTF in Section 4 for the nonlinear operating re-ion. Numerical results and their corresponding physicalnterpretation are given in Section 5. The impacts of suchndings of the convergence of the VSTF on optical com-unications systems are also described. Finally, some

oncluding remarks are given.

. SCHRÖDINGER EQUATION AND FIBERONLINEARITIESs in most nonlinear systems, nonlinear effects in opticalommunication systems increase with an increase in thenput power level, as well as with increases in the fiberength. The generalized NLSE is used to model all thenown linear and nonlinear effects in optical fibers [11]:

�A

�z+

�0

2A + �1

�A

�t+ j

�2

2

�2A

�t2 +�3

6

�3A

�t3

= j��A�2A − a1

��A�2A�

�t+ a2

�A

�t�A�2 + a3

��A�2

�tA + jQRA

��−�

�

sr�t − t1��A�t1,z��2dt1 �1�

here A=A�t ,z� represents the slowly varying complexnvelope of the input pulse, � is the linear attenuation co-fficient of the optical fiber; �1=1/vg is the inverse of theptical carrier group velocity vg; �2 is the second-orderispersion parameter and is usually termed the group ve-ocity dispersion (GVD); �3 is the third-order dispersionarameter; �=�0 /cAeffn2, a1=a0 /�0, a2 and a3 are theonlinear coefficients of the single mode fiber with c thepeed of light, Aeff the effective cross-section of the fiber,nd n2 the refractive index of the core section; QR�0 /cAeffGR, is the Raman constant with GR as the Ra-an gain coefficient factor; and sR�t� is the time-domain

epresentation of the Raman gain spectrum SR��. Weote here that the frequency component �o is assigned ashe central optical frequency of the carrier under consid-ration, while all other frequency components �n (as de-cribed later) are generalized, including all other adjacentarriers or components affecting the transmission chan-el. The NLSE is in fact evaluated for all the signal fre-uency components over the signal base-band, the opticalpectrum including the optical carrier and its lower andpper side passband information spectrum.The NLSE cannot be solved analytically due to its non-

inear nature. Normally, a complete numerical methoduch as the SSFM or the Runga–Kutta method are usedo reach the final solution, the output pulses at the end ofhe transmission fiber. Under these numerical methods,he fiber is divided into small segments and the waveformt each segment is computed sequentially. The length ofach segment has to be kept small to ensure accuracy ofhese numerical methods, and this can become very com-

utationally extensive when the length of the transmis-ion fiber increases. Furthermore numerical methods areifficult in designing a complete system by interconnec-ion of different transfer functions since they do not pro-ide sufficient information of the system’s characteristic.n important aspect and advantage of representing theptical transmission channel in the frequency domain ishat any optical component can be inserted into the opti-al transmission system and can be modeled and ana-yzed by inspection, hence reducing the analysis cycleime.

. FIBER TRANSMISSION TRANSFERUNCTION UNDER LINEAR DISPERSIONEGIMEhe treatment of the propagation of modulated light-aves through single mode fiber has been well docu-ented [4,8,9]. For completeness of the transfer function

f single mode optical fibers, in this section we restrict ourtudy the frequency transfer function and impulse re-ponses of the fiber to the linear region of the medium.urthermore, the delay term in the NLSE can be ignored,s it has no bearing on the size and shape of the pulses.rom the NLSE we can thus model the fiber simply as auadratic phase function. This is because the nonlinearerm of the NLSE can be removed and the Taylor seriespproximation around the operating frequency (centralavelength) can be obtained, along with the frequencynd impulse responses of the single mode fiber [12–14].he input–output relationship of the pulse can thereforee depicted. The time-domain impulse response h�t� andhe frequency domain transfer function H�� are ex-ressed as a Fourier transform pair:

h�t� =� 1

j4��2exp� jt2

4�2� ↔ H�� = exp�− j�2�2�, �2�

here �2 is well known as the GVD parameter. The inputunction f�t� is typically a rectangular pulse sequence and2 is proportional to the length of the fiber. The output

unction g�t� is the dispersed waveform of the pulse se-uence. The propagation transfer function in Eq. (2) is anxact analogy of diffraction in optical systems (see item 1,able 2-1, p. 14, [15]). Thus, the quadratic phase functionlso describes the diffraction mechanism in one-imensional optical systems, where distance x is analo-ous to time t. The establishment of this analogy lets usorrow many of the concepts and analytical results thatave been developed in diffraction theory. Thus, we mayxpress the step response s�t� of the system H�� in termsf Fresnel cosine and sine integrals by

s�t� =�0

t� 1

j4��2exp� jt2

4�2�dt =� 1

j4��2�C��1/4�2t�

+ jS��1/4�2t� �3�

ith

C�t� =�0

t

cos��

2�2�d�, �4�

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S�t� =�0

t

sin��

2�2�d�,

here C�t� and S�t� are the Fresnel cosine and sine inte-rals, respectively.

Using this analogy, one may argue that it is always pos-ible to restore the original pattern f�x� by refocusing thelurry image g�x� (e.g., see image formation, item 5, Table-1, [11]). In the electrical analogy, it implies that it isossible to compensate for the quadratic phase media per-ectly. This is not surprising. The quadratic phase func-ion H�� in Eq. (2) is an all-pass transfer function; thus its always possible to find an inverse function to recover�t�. One can express this differently in informationheory terminology by saying that the quadratic phasehannel has a theoretical bandwidth of infinity, hence itsnformation capacity is infinite. Shannon’s channel capac-ty theorem states that there is no limit on the reliableate of transmission through the quadratic phase chan-el.Figure 1 shows the pulse and impulse responses of the

ber under consideration. It is noted that only the enve-ope of the pulse is shown and the phase of the lightwavearrier is included as the complex values of the ampli-udes. As observed the chirp of the carrier is significant athe edges of the pulse. At the center of the pulse, the chirps almost negligible at some limited fiber length, thus therequency of the carrier remains nearly the same as at itsriginal starting value. One could obtain the impulse re-ponse quite easily but in this work we believe that theulse response is much more relevant in the investigationf the uncertainty in the pulse sequence detection. Thempulse response, on the other hand, is much more impor-ant in the process of equalization.

The uncertainty of the detection depends on the modu-ation formats and detection process. The modulation cane implemented by manipulation of the amplitude, thehase, or the frequency of the carrier, or of both ampli-ude and phase of multi-subcarriers such as in orthogonalrequency division multiplexing (OFDM) [16]. The ampli-ude detection would be mostly affected by the ripples ofhe amplitudes of the edges of the pulse. The phase of thearrier is mostly affected near the edge due to the chirpffects. However, if differential phase detection is usedhen the phase change at the transition instant is theost important and the opening of the detected eye dia-

ram. For frequency modulation the uncertainty in theetection is not very critical provided that the chirpingoes not enter into the region of the neighborhood of theenter of the pulse in which the frequency of the carrieremains almost constant.

The picture changes completely if the detector/decoders allowed only a finite time window to decode each sym-ol. In the convolution coding scheme, for example, it ishe decoder’s constraint length that is manifested becausef the finite time window. In an adaptive equalizationcheme, it is the number of equalizer coefficients that de-ermines the decoder window length. Since the transmit-ed symbols have already been broadened by the qua-ratic phase channel, if they are next gated by a finiteime window, the information received could be severely

educed. The longer the fiber, the more the broadening ofhe pulses, hence the more uncertain it becomes in the de-oding. It is the interaction of the pulse broadening on oneand, and the restrictive detection time window on thether, that gives rise to the finite channel capacity.

It is observed that chirp occurs mainly near the edge ofhe pulses when it is in the near-field region—about a fewilometers for standard single mode fibers. In this near-eld distance the accumulation of nonlinear effects is stillery weak and thus this chirp effects dominate the behav-or of the single mode fiber. The nonlinear VSTF pre-ented in Section 4 would thus have minimum influence.his point is important for understanding the behavior of

ightwaves circulating in short-length fiber devices inhich the both linear and nonlinear effects are to be bal-nced, such as in active mode locked soliton and multi-ound soliton lasers [17,18]. In the far field the output ofhe fiber is a Gaussian like for the square pulse launched

ig. 1. (Color online) Rectangular pulse transmission through aingle mode fiber: (a) pulse response, (b) frequency spectrum, (c)tep response of the quadratic-phase transmittance function.

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t the input. In this region the nonlinear effects wouldominate over the linear dispersion effect as they haveeen accumulated over a long distance.

. FIBER TRANSMISSION MODEL BYOLTERRA SERIES TRANSFER FUNCTIONhe weakness of most of the recursive methods in solvinghe NLSE is that they do not provide much useful infor-ation to help the characterization of nonlinear effects.he Volterra series model provides an elegant way of de-cribing a system’s nonlinearities, and enables the design-rs to see clearly where and how the nonlinearity affectshe system performance. Although [2–4] have given anutline of the kernels of the transfer function using theolterra series, it is necessary for clarity and physical rep-esentation of these functions to give brief derivationsere on the nonlinear transfer functions of an optical fiberperating under nonlinear conditions. The Volterra seriesransfer function of a particular optical channel can be ob-ained in the frequency domain as a relationship betweenhe input spectrum X�� and the output spectrum Y��:

Y�� = n=1

� �−�

�

¯�−�

�

Hn��1, ¯ ,�n−1,� − �1 − ¯ − �n−1�

� X��1� ¯ X��n−1�X�� − �1 − ¯

− �n−1�d�1 ¯ d�n−1, �5�

here Hn��1 , ¯ ,�n� is the nth-order frequency domainolterra kernel including all signal frequencies of orders 1o n. The wave propagation inside a single mode fiber cane governed by a simplified version of the NLSE (1) wherenly the self-phase-modulation effect is included as

�A

�z= −

�0

2A − �1

�A

�t− j

�2

2

�2A

�t2 −�3

6

�3A

�t3 + j��A�2A, �6�

here A=A�t ,z�. The proposed solution of the NLSE cane written with respect to the VSTF model of up to fifth
rder as m
A��,z� = H1��,z�A�� +�−�

� �−�

�

H3��1,�2,� − �1 + �2,z�

�A��1�A*��2�A�� − �1 + �2�d�1d�2

+�−�

� �−�

� �−�

� �−�

�

H5��1,�2,�3,�4,� − �1 + �2

− �3

+ �4,z�A��1�A*��2�A��3�A*��4�

�A�� − �1 + �2 − �3 + �4�d�1d�2d�3d�4, �7�

here A��=A�� ,0�, that is, the amplitude envelope ofhe optical pulses at the input of the fiber. Taking theourier transform of Eq. (4) and assuming A�t ,z� is ofinusoidal form we have

�A��,z�

�z= G1��A��,z� +�

−�

� �−�

�

G3��1,�2,� − �1 + �2�

�A��1,z�A*��2,z�A�� − �1 + �2,z�d�1d�2, �8�

here

G1�� = −�0

2+ j�1� + j

�2

2�2 − j

�3

6�3, G3��1,�2,�3� = j�.

takes the values over the signal bandwidth and beyondn overlapping the signal spectrum of other optically

odulated carriers, while �1 . . .�3 also take values over aimilar range as that of �. For general expression theimit of integration is indicted over the entire range to in-nity.Substituting Eq. (7) into Eq. (8) and equating both

ides, the kernels can be obtained after some algebraic
anipulations:
�

�z�H1��,z�A�� + �−�

� �−�i

�

H3��1,�2,� − �1 + �2,z�A��1�A*��2�A�� − �1 + �2�d�1d�2

+ �−�

� �−�

� �−�

� �−�

�

H5��1,�2,�3,�4,� − �1 + �2 − �3 + �4,z�A��1�A*��2�

�A��3�A*��4�A�� − �1 + �2 − �3 + �4�d�1d�2d�3d�4�= G1��H1��,z�A�� + �

−�

� �−�i

�

H3��1,�2,� − �1 + �2,z�A��1�A*��2�A�� − �1 + �2�d�1d�2

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+ �−�

� �−�

� �−�

� �−�

�

H5��1,�2,�3,�4,� − �1 + �2 − �3 + �4,z�A��1�A*��2�A��3�A*��4�

�A�� − �1 + �2 − �3 + �4�d�1d�2d�3d�4� + �−�

� �−�

�

G3��1,�2,� − �1 + �2��H1��1,z�A��1�

+ �−�

� �−�i

�

H3��11,�12,�1 − �11 + �12,z�A��11�A*��12�A��1 − �11 + �12�d�11d�12

+ �−�

� �−�

� �−�

� �−�

�

H5��11,�12,�13,�14,�1 − �11 + �12 − �13 + �14,z�A��11�A*��12�A��13�

�A*��14�A��1 − �11 + �12 − �13 + �14�d�11d�12d�13d�14��H1��2,z�A��2� + �

−�

� �−�i

�

H3��21,�22,�2 − �21 + �22,z�A��21�A*��22�A��2 − �21 + �22�d�21d�22

+ �−�

� �−�

� �−�

� �−�

�

H5��21,�22,�23,�24,�2 − �21 + �22 − �23 + �24,z�A��21�A*��22�A��23�

�A*��24�A��2 − �21 + �22 − �23 + �24�d�21d�22d�23d�24�*�H1�� − �1 + �2,z�A�� − �1 + �2�

+ �−�

� �−�i

�

H3��31,�32,� − �1 + �2 − �31 + �32,z�A��31�A*��32�A�� − �1 + �2 − �31 + �32�d�31d�32

+ �−�

� �−�

� �−�

� �−�

�

H5��31,�32,�33,�34,� − �1 + �2 − �31 + �32 − �33 + �34,z�A��31�A*��32�A��33�

�A*��34�A�� − �1 + �2 − �31 + �32 − �33 + �34�d�31d�32d�33d�34� . �9�

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T

quating the first-order terms on both sides we obtain

�

�zH1��,z� = G1��H1��,z�. �10�

hus the solution for the first-order transfer function (8)s then given by

H1��,z� = eG1��z = e�−�0/2+j�1�+j�2/2�2−j�3/6�3�z. �11�

his is in fact the linear transfer function of an optical fi-er with the dispersion factors �2 and �3. Similarly forhe third-order terms we have

�

�z�−�

� �−�

�

H3��1,�2,� − �1 + �2,z�A��1�A*��2�

�A�� − �1 + �2�d�1d�2

=�−�

� �−�

�

G3��1,�2,� − �1 + �2�H1��1,z�A��1�

�H2*��2,z�A��2�H1�� − �1 + �2�

�A�� − �1 + �2�d�1d�2. �12�

ow letting �3=�−�1+�2, then it follows

�H3��1,�2,�3,z�

�z= G1��1 − �2 + �3�H3��1,�2,�3,z�

+ G3��1,�2,�3�H1��1,z�

�H1*��2,z�H1��3,z�. �13�

he third kernel transfer function can be obtained as

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H3��1,�2,�3,z� = G3��1,�2,�3�e�G1��1�+G1

*��2�+G1��3��z − eG1��1−�2+�3�z

G1��1� + G1*��2� + G1��3� − G1��1 − �2 + �3�

. �14�

he fifth-order kernel can similarly be obtained as

H5��1,�2,�3,�4,�5,z� =H1��1,z�H1

*��2,z�H1��3,z�H1*��4,z�H1��5,z� − H1��1 − �2 + �3 − �4 + �5,z�

G1��1� + G1*��2� + G1��3� + G1

*��4� + G1��5� − G1��1 − �2 + �3 − �4 + �5�

� � G3��1,�2,�3 − �4 + �5�G3��3,�4,�5�

G1��3� + G1*��4� + G1��5� − G1��3 − �4 + �5�

+G3��1,�2 − �3 + �4,�5�G3

*��2,�3,�4�

G1*��2� + G1��3� + G1

*��4� − G1*��2 − �3 + �4�

+G3��1 − �2 + �3,�4,�5�G3��1,�2,�3�

G1��1� + G1*��2� + G1��3� − G1��1 − �2 + �3��

−G3��1,�2,�3 − �4 + �5�G3��3,�4,�5�

G1��3� + G1*��4� + G1��5� − G1��3 − �4 + �5�

�H1��1,z�H1

*��2,z�H1��1 − �2 + �3,z� − H1��1 − �2 + �3 − �4 + �5,z�

G1��1� + G1*��2� + G1��3 − �4 + �5� − G1��1 − �2 + �3 − �4 + �5�

−G3��1,�2 − �3 + �4,�5�G3

*��2,�3,�4�

G1*��2� + G1��3� + G1

*��4� − G1*��2 − �3 + �4�

�H1��1,z�H1

*��2 − �3 + �4,z�H1��5,z� − H1��1 − �2 + �3 − �4 + �5,z�

G1��1� + G1*��2 − �3 + �4� + G1��5� − G1��1 − �2 + �3 − �4 + �5�

−G3��1 − �2 + �3,�4,�5�G3��1,�2,�3�

G1��1� + G1*��2� + G1��3� − G1��1 − �2 + �3�

�H1��1 − �2 + �3,z�H1

*��4,z�H1��5,z� − H1��1 − �2 + �3 − �4 + �5,z�

G1��1 − �2 + �3� + G1*��4� + G1��5� − G1��1 − �2 + �3 − �4 + �5�

. �15�

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Higher-order terms can be derived with ease if higherccuracy is required. However in practice such higher or-er would not exceed the fifth rank. We can understandhat for a length of a uniform optical fiber the first- toth-order frequency spectrum transfer can be evaluated

ndicating the linear to nonlinear effects of the optical sig-als transmitting through it. Indeed the third- and fifth-rder kernel transfer functions based on the Volterra se-ies indicate the optical filed amplitude of the frequencyomponents which contribute to the distortion of theropagated pulses. An inverse of these higher-order func-ions would give the signal distortion in the time domain.hus the VSTFs allow us to conduct distortion analysis of

ptical pulses and hence an evaluation of the bit-error-ate of optical fiber communications systems.

The superiority of such VSTF expressions allows us tovaluate each effect individually, especially the nonlinearffects, so that we can design and manage the opticalommunications systems under linear or nonlinear opera-ions. Currently this linear–nonlinear boundary of opera-ions is critical for system implementation, especially forptical systems operating at 40 Gbps where linear opera-ion and carrier-suppressed return-to-zero format is em-loyed. As a norm in series expansion, the series need toe converged to a final solution. It is this convergence thatould allow us to evaluate the limit of nonlinearity in a

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ystem. The issues involved in the convergence and its re-ationship with linearity and nonlinearity are thereforeresented in Section 5.

. CONVERGENCE PROPERTY OF VSTFs we can see from Section 4 that the VSTF takes the

orm of a power series whose convergence can be exam-ned with a number of well-established tests. The ratioest is chosen in this paper to test the convergence of theSTF, as it would lead to the best estimation of the con-ergence of the series. An infinite Volterra series can beepresented by a function of all frequency components as

n=1

�

Yn��1, ¯ ,�n� = n=1

�

Hn��1, ¯ ,�n�U��1� ¯ U��n�

n=1

�

�Hn��1, ¯ ,�n��Umax�n, �16�

here Umax=max U��; Yn��1 , ¯ ,�n� is the output fromhe nth-order kernel, which forms a Hilbert space. Eachf the output terms Y1 ,Y2 , . . . ,Yn take on different dimen-ions, which consist of different numbers of dependentariables. The higher-order terms can be converted intoheir one-dimensional equivalents using dimensional con-raction technique by taking a multidimensional convolu-ion across all variables. Thus Eq. (16) can be rewritten as

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bfi

Y�� = Y1�� +�−�

�

Y2��1,� − �1�d�1 +�−�

� �−�

�

Y3��1,�2,�

− �1 − �2�d�1d�2 + ¯ = H1��U�� +�−�

�

H2��1,�

− �1�U��1�U�� − �1�d�1 +�−�

� �−�

�

H3��1,�2,� − �1

− �2� � U��1�U��2�U�� − �1 − �2�d�1d�2 + . . . .

�17�

tilizing the triangular inequality property of the Hilbertpace, the upper bound for the output can be obtained as

�Y�� H1��,z��Umax� + ��−�

�

H2��1,� − �1�d�1��Umax�2

+ ��−�

� �−�

�

H3��1,�2,� − �1 − �2�d�1d�2��Umax�3

+ . . . . �18�

he convergence for the Volterra series can be guaranteedf and only if the infinite power-series on the right handide of the inequality is convergent. Accordingly Eq. (18)eads to

��−�� . . . �−�

�

n+1

Hn+1��1, . . . ,� − �1 − . . . − �n�d�1 . . . d�n�

��−�� . . . �−�

�

n

Hn��1, . . . ,� − �1 − . . . − �n−1�d�1 . . . d�n−1��Umax� � 1.

�19�

ince the even-order kernels are null in a single-mode fiber, the convergence criterion (19) can be written as

��−�� . . . �−�

�

n+2

Hn+2��1, . . . ,� − �1 − . . . − �n+1�d�1 . . . d�n+1�

��−�� . . . �−�

�

n

Hn��1, . . . ,� − �1 − . . . − �n+1�d�1 . . . d�n−1��Umax�2 � 1,

�20�

here n is an odd positive integer. Hence the upperounds for the inputs to each kernel of different ordersan be expressed in terms of the integration of lower-rder kernels. This expression can be simplified—underhe case of n=1 or the convergence of the third-order ker-el with respect to the linear kernel—as

��−�� −�

� H3��1,�2,� − �1 − �2�d�1d�2�

�H1��Umax�2 1,

�21�

eading to

�H1�� −�

� �−�

�

H3��1,�2,� − �1 − �2�d�1d�2��Umax�2.

�22�

ndeed expression (22) indicates the relationship betweenhe linear transfer function, or the effective bandwidth ofn optical fiber operating in the linear dispersion region,nd the dispersion effect due to the self-phase-modulationue to the intense optical pulse power contributing viahe third-order kernel.

Therefore for the case of n=3, the expression for upperounds can now be obtained as a relationship between thefth-order nonlinear coefficients of the fiber, such as the

fm

T

c

H

I

A

taintipcetp

6PLItpo�dsvt

gtcfit


our-wave-mixing effects, and that of the self-phase-
odulation, the third-order kernel function of the VSTF. b
cabcfasn

tottlvv

Ffi

he upper bounds for inputs to higher-order kernels can
e derived in the same way:
��−�� −�

� �−�� −�

� H5��1,�2,�3,�4,� − �1 − �2 − �3 − �4�d�1d�2d�3d�4�

��−�� −�

� H3��1,�2,� − �1 − �2�d�1d�2��Umax�2 1 �23�

an be replaced by

��−�� −�

� �−�� −�

� H5��1,�2,�3,�4,� − �1 − �2 − �3 − �4�d�1d�2d�3d�4��Umax�4

�H1�� 1. �24�

ere we denote the norm of the nth-order kernel by

�n = sup� ��−�� . . . �−�

� Hn��1, . . . ,� − �1 − . . . − �n−1�d�1 . . . d�n−1�

�H1�� . �25�

n general, the radius of convergence as a function of the total input optical field amplitude can therefore be expressed as

Umax = � 1

�n�1/n−1

= inf�� H1��

��−�� . . . �−�

� Hn��1, . . . ,� − �1 − . . . − �n−1�d�1 . . . d�n−1��1/n−1� . �26�

ccordingly, the peak power of the input pulse so that the VSTF is convergent and hence computable is given by

Ppeak = Umax2 = � 1

�n�2/n−1

= inf�� H1��1��

��−�� . . . �−�

� Hn��1, . . . ,� − �1 − . . . − �n−1�d�1 . . . d�n−1��2/n−1� . �27�

This is the most important result and can be used inhe determination of the linear and nonlinear operation ofn optical fiber communications system under high andntense optical input pulses, especially when there existsumerous optical channels in a single fiber. Furthermorehe radius of convergence of the higher order of the VSTFndicates the level of manageability of the input opticalulse power so that the linear dispersion effect can beompensated for by the nonlinear effects in the fiber. Oth-rwise the series would be divergent, meaning the radia-ion or complete depletion of the optical pulses due to dis-ersion.

. ESTIMATION OF OPTICAL INPUTOWER FOR GUIDING OVER A FIBERENGTH

n this section, numerical results of the kernels of theransfer functions expressed via the Volterra series areresented for a fiber with a linear attenuation coefficientf �0=0.2 dB/km at the operating wavelength of 1.55 m,2=−1.2746�10−27 corresponding to a group velocityispersion=2.0 ps/km nm, and �3=0, i.e., zero dispersionlope. The nonlinear refractive index assumes the normalalue for silica fiber n2=2.31�10−20 m2/W with an effec-ive cross-sectional core area of Aeff=80 m2.

As described in previous Section 5, the radius of conver-ence (ROC) of the nth-order kernels of the VSTF governshe upper bounds for the maximum input power and isorrespondent to their norm �. The norm �n for third-,fth-, and seventh-order kernels are thus calculated withhe fiber length as a parameter. The radius of convergence

ould then be computed under these conditions, leading torelationship between the optical peak power and the fi-

er length. The limits of integration of the kernels in theomputation of the respective norms are restricted to therequency range of the spectrum that can be appreciablyffected by the two adjacent optical carrier channels in atandard ITU grid of 100 GHz spacing for 10 Gbps chan-els.Figures 2–4 show the dependence of the input pulse op-

ical peak power evaluated using the radii of convergencef the third-, fifth-, and seventh-order kernels, respec-ively, of the Volterra series as a function of the totalransmission length of the fiber. A similar pattern for re-ationship between the fiber length and the radius of con-ergence for the input power has been observed. The ROCaries inversely proportionally to the fiber length over a

ig. 2. (Color online) Input pulse peak power as a function ofber length of the third-order transfer function.

sig5ltst

ATcslpT

wmf

vts

tt

BTTVoomtstb

1Hcanavo1mmtwac

Ft

Ffi

Ft

Fp


ignificant distance as displayed in the log-log scale. Annteresting phenomenon is that the decrease of ROC be-ins to slow down when the fiber length approaches0 km and remains roughly constant after the fiberength is beyond 100 km. This result agrees with [1] inhat with input power of 10 nW and 1 mW, the errors aremall, while with input power of 30 mW, which is close tohe upper bound, the error is several magnitudes larger.

. Comparison between VSTF and SSF Methodshe proposal of the computation of radius of convergencean be further supported by the comparison between re-ults obtained with VSTF and SSF method. In this simu-ation, input pulses of Gaussian shape with different peakower are propagated through a fiber length of 100 km.he difference is calculated by setting a criterion

Deviation in % =�−�

� �U2�� − U1��2d�

�−�� U2��2d�

, �28�

here U1�� is the output spectrum from the VSTFethod, while U2�� is the output spectrum obtained

rom the SSF method.Figures 5 and 6 show the relationship between the de-

iation and the input peak optical power of pules afterransmission over a fiber length of 50 km and 100 km, re-pectively. As the peak power increases and approaches

ig. 3. (Color online) Input pulse peak power as a function ofhe fiber length of the fifth-order transfer function.

ig. 4. (Color online) Input pulse peak power as a function ofber length of the seventh-order transfer function.

he upper-bound ROC, the deviation between results ob-ained VSTF and SSF increases quickly.

. Application of the VSTF to Soliton Opticalransmissionhis subsection demonstrates the effectiveness of theSTF modeling for soliton transmission over single modeptical fiber. The VSTF is applied to the transfer functionf a whole single section. However when the fiber is seg-ented into a number of subsections, the improvement of

he accuracy can be significant [this method is termed theegmented steps VSTF(SS-VSTF)]. This is due to the facthat the convergent region of the SS-VSTF is preservedetter compared with that of the single-section VSTF.

. Modeling Using Single-Step VSTFistorically, optical communication development can be

lassified into different generations according to the oper-ting regions of the transmission channel in the linear oronlinear regime with the total power the effective powers seen by the fiber with the superposition of all indi-idual optical channels. Current state-of-the-art systemsperate over a very wide wavelength range from480 to 1620 nm, utilizing wavelength-division-ultiplexing (WDM), with dispersion compensated oranaged-composite transmission fibers. Designers of op-

ical fiber communication systems need to be concernedith the balancing of the fiber linear dispersion (GVD)nd the self-phase variation due to nonlinear effect. Opti-al solitons preserve their shape during propagation in a

ig. 5. (Color online) Deviation factor as a function of input op-ical power for a fiber length of 50 km.

ig. 6. (Color online) Deviation factor as a function of inputower for a fiber length of 100 km.

ltemlp

elkptlwpGhqesdsk9mcs

a4

2SItmtc

opaga

Fm


ossless fiber by counteracting the effect of dispersion viahe shaping of the fiber nonlinearity. We demonstrate theffectiveness of the Volterra series approach in the trans-ission of a soliton of fundamental order through a 50 km

ong fiber. Further, both VSTF and SSF methods are com-ared for the evolution of the propagated pulses.It is well known that the power requirement for nonlin-

ar optical pulses under complete compensation betweeninear dispersion and self-phase modulation, usuallynown as solitons, is governed by the soliton order, itsulse width, the pulse peak power, and the dispersion fac-or of the transmission fiber. In this simulation, the fol-owing parameters are used: soliton order=1, pulseidth=10 ps �Tfwhm�, GVD=1 ps/km nm. Accordingly, aower P0 of 16.923 mW is required for soliton formation.enerally, the power required by soliton systems is muchigher than by conventional ones, since the power re-uired to generate nonlinearity to counter the dispersionffect is substantial. Figures 7 and 8 show the transmis-ion of such first-order solitonic pulses of fundamental or-er by using the SSF method and the VSTF method, re-pectively, under the convergence condition for the thirdernel transfer function as described in Section 5. Figureshows the discrepancies between the SSF and VSTFethods. With this difference we would expect a signifi-

ant penalty of more than 0.1 dB with a residual disper-ion of a few kilometers over a 100 km span of optically

Fig. 7. (Color online) Solitonic pulse pr

Fig. 8. (Color online) Solitonic pulse pro

mplified dispersion compensated transmission span at0 Gb/s. This is about a 1 dB penalty over 10 spans.

. Model of Solitonic Transmission Systems Usingegmented Steps VSTFt is well known that solitonic systems utilize the interac-ion between the SPM and GVD effects of the fiber toaintain the pulse shape over a long transmission dis-

ance when the optical pulse shape is of the hyperbolic se-ant form and the intensity is strong enough [11]. Under

tion via the simulation by SSF method.

ion via the simulation by VSTF method.

ig. 9. (Color online) Deviation factor between VSTF and SSFethods.

pagat

testhhiictiVfi

t

ocp

sectio

FS

F

F(


he soliton condition, the GVD and SPM effects balanceach other out, and hence the pulse shape can be pre-erved over a very long distance. In this condition and inhe far-field region, the system is strongly nonlinear,ence the conventional VSTF model cannot be appliedere. Thus we propose here a SS-VSTF model to further

mprove the accuracy of modeling the evolution of solitonsn such system and provide the solution in an analyticlosed form. The segmentation of the propagation dis-ance is shown in Fig. 10, in which the fiber is segmentednto a number of sections and represented with theirSTFs. The length of each section is longer than the near-eld region; 5 km length is sufficient.In the simulation, the following settings are used for

he solitonic transmission system: input pulses of soliton

Fig. 10. Representing a fiber

ig. 11. Pulse evolution simulation of a single soliton using (a)SF method, (b) SS-VSTF method.

rder one are used; the FWHM width of the hyperbolic se-ant pulse is 12.5 ps; loss in the fibre is assumed to beerfectly compensated for by an idealized distributed Ra-

n with eight VSTF segments.

ig. 12. NSD for soliton simulation with 500-segment SS-VSTF.

ig. 13. Pulse evolution for 1000-segment fiber using (a) SSF,b) SS-VSTF method.

mGtfe

tmitVtt1F

dmFtoVts

stcttnapt

7Tpeopbritlcrv

plcau

vnstp

R

1

1

1

1

1

1

1

1

1

FS


an amplifier (DRA) and hence is assumed 0 dB/km; theVD is 3 ps/nm km; total transmission distance 200 km;

ransmission bit rate is 4 Gb/s. The number of segmentsor the SS-VSTF model is 500. The following optical pulsevolution is obtained in the simulation.

The normalized standard deviation (NSD) for the solu-ions obtained by the SSF method and the SS-VSTFodel for the simulation is calculated using Eq. (28) and

s shown in Fig. 11. It can be observed from Fig. 12 thathe discrepancy between the SSF method and the SS-STF model increases dramatically after 150 km of

ransmission distance. In order to improve the accuracy ofhe model, the number of segments is further increased to000, and the pulse evolution plot is obtained as shown inig. 13(a).As can be observed from the pulse evolution plot, the

iscrepancy between the SSF method and the SS-VSTFodel is greatly reduced as compared with that shown inig. 13(b). This can be verified by the NSD plot as illus-

rated in Fig. 14. The discrepancy between the solutionsbtained from the SSF method and the 1000-segment SS-STF model is less than 4%. The result demonstrates

hat the SS-VSTF model is applicable for modeling thetrongly nonlinear solitonic transmission systems.

In terms of the computational time required, the 500-egment SS-VSTF model took 26.3 s to complete, whilehe 1000-segment SS-VSTF model took 52.4 seconds. Theomputational time required is significantly greater thanhat of the SSF method. In terms of an efficient simula-ion method, the SSF method is a better choice in stronglyonlinear systems. However, the SS-VSTF model has thedvantage of the analytical expression, which can be im-ortant in the design and analysis process of opticalransmission systems.

. CONCLUDING REMARKShe VSTF has been effectively employed to model waveropagation in single mode fibers. Both linear and nonlin-ar effects can practically be approximated up to seventhrder, depending on the accuracy required. This paperrovides a conservative criterion for calculating the upperounds of the input power under which the VSTF modelemains convergent. The relationship between ROC fornput power and the fiber length is studied. It is shownhat the ROC varies inverse proportionate to the fiberength logarithmically. However, as the fiber length in-reases beyond 100 km, the ROC for input power remainsoughly unchanged. Therefore the VSTF model remains

ig. 14. NSD between the SSF method and the 1000-segmentS-VSTF model.

alid even for fiber of very long length, provided the input

ower is below the upper bound of the ROC. Since theaunched power in optical communication systems is typi-ally below 10 mW, and the distances between amplifiersre less than 100 km, the VSTF model is quite adequatender normal operating condition.Practically, the VSTF method is more suitable for in-

estigating the frequency response of an optical commu-ication system, and hence is appropriate for system de-ign. However, it is not as efficient as the SSF method inhe simulation of pulse evolution in the fibers unless theropagation distance is relatively long.

EFERENCES1. M. Schetzen, The Volterra and Wiener Theories of

Nonlinear Systems (Wiley, 1989).2. K. V. Peddanarappagari and M. Brandt-Pearce, “Volterra

series transfer function of single-mode fibers,” J. LightwaveTechnol. 15, 2046–2055 (1997).

3. K. V. Peddanarappagari and M. Brandt-Pearce, “Study offiber nonlinearities in communication systems using aVolterra series transfer function approach,” in Proceedingsof the 31st Annual Conference on Information Sciences andSystems (1997), pp. 752–757.

4. K. V. Peddanarappagari and M. Brandt-Pearce, “Volterraseries approach for optimizing fiber-optic communicationssystem designs,” J. Lightwave Technol. 16, 2046–2055(1998).

5. M. B. Brilliant, “Theory of the analysis of nonlinearsystems,” MIT Research Laboratory of Electronics, Tech.Rep. 345, March 3, 1958.

6. A. Samelis, D. R. Pehlke, and D. Pavlidis, “Volterra seriesbased nonlinear simulation of HBTs using analyticallyextracted models,” Electron. Lett. 30, 1098–1100 (1994).

7. J. Tang, “The channel capacity of a multispan DWDMsystem employing dispersive nonlinear optical fibers andan ideal coherent optical receiver,” J. Lightwave Technol.20, 1095–1101 (2002).

8. J. Tang, “A comparison study of the Shannon channelcapacity of various nonlinear optical fibers,” J. LightwaveTechnol. 24, 2070–2075 (2006).

9. J. Tang, “The Shannon channel capacity of dispersion-freenonlinear optical fiber transmission,” J. Lightwave Technol.19, 1104–1109 (2001).

0. B. Xu and M. Brandt-Pearce, “Comparison of FWM- andXPM-induced crosstalk using the Volterra series transferfunction method,” J. Lightwave Technol. 21, 40–54 (2003).

1. G. P. Agrawal, Optical Communication Systems (Academic,1997).

2. A. F. Elrefaie, R. E. Wagner, D. A. Atlas, and D. G. Daut,“Chromatic dispersion limitations in coherent lightwavetransmission systems,” J. Lightwave Technol. 6, 704–709(1998).

3. R. C. Srinivasan and J. C. Cartledge, “On using fibertransfer functions to characterize laser chirp and fiberdispersion,” IEEE Photon. Technol. Lett. 7, 1327–1329(1995).

4. A. V. T. Cartaxo, B. Wedding, and W. Idler, “Influence offiber nonlinearity on the fiber transfer function: theoreticaland experimental analysis,” J. Lightwave Technol. 17,1806–1813 (1999).

5. A. Papoulis, Systems and Transforms with Applications inOptics (McGraw-Hill, 1968) p. 14.

6. J. G. Proakis, Digital Communications, 4th ed. (McGraw-Hill, 2001), pp. 185–213.

7. L. N. Binh and N. Nguyen, “Generation of high-ordermulti-bound-solitons and propagation in optical fibers,”Opt. Commun. 282, 2394–2406 (2009).

8. W. J. Lai, P. Shum, L. N. Binh, and M. Karlsson, “Phase-plane analysis of rational harmonic mode-locking in anerbium-doped fiber ring laser,” IEEE J. Quantum Electron.
41, 426–433 (2005).

linear and nonlinear transfer functions of single mode fiber for optical transmission systems

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