PROPAGATION-PARAMETER MAPPINGMETHOD FOR EFFICIENT SIMULATIONOF IN-LINE AMPLIFIED TRANSMISSIONYongbum Kim,1 Yongwon Lee,1 Myungmoon Lee,1 Jintae Yu,1

and Jinwoo Park11 Optical Networks LaboratoryDepartment of Electronics EngineeringKorea UniversitySeoul 136-701, Korea

Recei�ed 1 March 2001

ABSTRACT: In this paper, a propagation parameter-mapping methodfor an efficient simulation to e�aluate the performance of in-line opticalamplified transmission line is proposed. The method helps to extract thepropagation parameters for simplified simulation from both a single-channel and a WDM transmission line. It has been shown in thesimulation results that the simplified simulation can describe the propa-gation beha�ior of optical pulses �ery accurately, with less than only 3%difference in the eye-opening ratio when compared with the normalsimulation method, but with a great reduction of calculation time.� 2001 John Wiley & Sons, Inc. Microwave Opt Technol Lett 30:364�368, 2001.

Key words: in-line amplified transmission; numerical analysis;single channel; WDM; parameter mapping

1. INTRODUCTION

Recently, there have been many research efforts to analyzethe performance of in-line optical amplified transmissionsystems. In such research, it is often required to solve thenonlinear Schrodinger equation to determine the propaga-¨tion phenomena of optical pulses in optical fiber. There havebeen two methods used to address the nonlinear Schrodinger¨

� �equation: numerical methods 1�4 , and analytical methods� �5�7 .

Analytical methods provide solutions with good accuracy.However, it has been known that the nonlinear Schrodinger¨equation does not generally lend itself to general analyticsolutions, except for limited cases with some a priori informa-tion and assumptions. Therefore, with the analytical methods,it may be difficult to obtain satisfactory results for the manypractical cases where the propagation conditions are chang-ing arbitrarily. In contrast, the numerical methods for solvingthe nonlinear Schrodinger equation are preferred since they¨provide solutions with fairly good accuracy, and the ability toexamine the optical transmission characteristics with the vari-ous design parameter sets. However, the numerical methodoften demands a tedious time-consuming effort to reach thefinal results, especially for a long optical transmission linewith many fiber sections and in-line amplifiers.

In this paper, we propose a propagation-parameter map-ping method that can simplify the calculation procedure andspeed up the numerical simulation for the in-line opticalamplified transmission line. In the proposed method, wemodel a long transmission line consisting of many fibersections and amplifiers by a transmission line having one fibersection with parameters which are to be found by the pro-posed mapping method. We demonstrate in this paper that,with the found propagation parameters, the propagation ofthe optical pulses over single-channel and WDM-multiplexedin-line amplified transmission lines can be described withgood accuracy and reduced computation time. It is found in

the simulations that the simulation run time is reduced to34.72% in single-channel transmission and 29.62% in WDMtransmission, respectively, with less than 3% difference be-tween the direct simulations and the simplified simulations.

2. PROPAGATION-PARAMETER MAPPING FORSINGLE-CHANNEL TRANSMISSION

An intensity-modulated optical signal undergoes a nonlinearphase shift in an optical fiber due to self-phase modulationŽ .SPM . SPM-induced spectral broadening �� is the conse-quence of the time dependence of the nonlinear phase shift

Ž . � ��� z, T , which is given by the following equation 8 :

2Ž .� U 0, TŽ . Ž . Ž .�� z , T � � P z 1

�T

where z is the distance, T is the time measured in a frame ofŽ .reference moving with the pulse at the group velocity, P z is

Ž .the peak power at z, U 0, T is the field amplitude at z � 0,and is the nonlinearity coefficient defined by 2� n �2Ž .� A , where n is the nonlinear index coefficient, � is0 eff 2 0the signal wavelength, and A is the effective core area.effThe combined effects of the SPM-induced frequency compo-

Ž .nents and the group velocity dispersion GVD cause nonlin-ear distortion of the original optical signal.

Ž .Figure 1 a shows an in-line amplified transmission line, inŽ .which P n � 1, 2, . . . , N is the output peak power of ann

amplifier, and l is the distance from z � 0. The amount ofnSPM-induced frequency chirp varies, depending on the prop-agation distance z because the frequency is dependent on theoptical pulse power and the signal power attenuates exponen-tially as the optical pulses pass through the fiber. Therefore,the resulting frequency chirp should be counted as a signaldistortion process during propagation over a given fiberlength. The total nonlinear distortion � is the sum ofdistdistortions occurring over the whole fiber span of interest,which is produced by the interaction between the GVD andthe SPM-induced frequency components. The total nonlineardistortion of the whole transmission line can therefore bewritten by

L L LŽ . Ž . Ž . Ž . Ž .� T � �� z , T D r dr D r dr dz 2H H Hdist0 z 0

where L is the propagation distance, and D is the dispersionŽ .parameter of an optical fiber. In Eq. 2 , the term in the

brackets is a distance-dependent weighting function definedat distance z.

Ž .The optical power P z at the distance z should take intoaccount the amplifier’s gain at each stage and the attenuationof fiber. It now can be expressed as follows:

N

Ž . � Ž .�P z � P exp �� z � lÝ n n�1n�1

� Ž . Ž .� Ž .� u z � l � u z � l 3n�1 n

Ž .where � is the loss parameter of fiber and u z is the unitstep function. Assuming that the fiber’s dispersion parameteris constant over a fiber span, the total nonlinear distortion

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 30, No. 5, September 5 2001364

oa

P1

z=0oa

P2

l1

oa

P3

l2

oa

PN

lN-1

Rx

lN (z=L)

(a)

oa

Pm

z=0Rx

z=L

(b)

Ž . Ž .Figure 1 a In-line amplified transmission system. b Systemmodel

Ž . Ž .can be obtained by combining Eqs. 1 � 3 as follows:

2Ž . � U 0, TŽ .� T � �dist L �T

N L� � Ž .� �� P � exp �� l � l � 1Ý n n n�1�n�1

1� � Ž .� �� l exp �� l � l � ln n n�1 n�1�

1� � Ž .� � Ž .� exp �� l � l � 1 . 4n n�12�

Ž .To make Eq. 4 simpler, it is assumed that the outputpower of the amplifiers and the amplifier spacing is identical,which are denoted by P and l, respectively. Then, � can0 distbe rewritten as

2Ž . P � U 0, T0Ž .� T � �dist L �T

NL�� lŽ .� � e � 1

�

Ž . Ž .l N N � 1 N N � 1�� l� e �½ 5� 2 2

N�� lŽ . Ž .� e � 1 . 52�

Ž .Figure 1 b is a simplified transmission-line model consist-ing of a fiber section and an amplifier. Its nonlinear distortion� can be written asdist, m

2Ž . P � U 0, Tm mŽ .� T � �dist, m L �Tm

1 Ž . 4 Ž .� exp �� L � 1 � L � 6m m m m2�m

where , P , L , and � are the nonlinearity coefficient,m m m mthe output peak power of amplifier, the fiber length, and the

Ž .loss parameter, respectively, in the model of Figure 1 b .Directly comparing � with � , we can find the propa-dist, m distgation parameters of the transmission-line model of FigureŽ .1 b . The relationships between the propagation parameters

Ž . Ž .based on Eqs. 5 and 6 are

Ž .P � P 7m 0

Ž . � 8m

Ž .L � L 9m

L�� l 2 �� LmŽ . Ž .K e � 1 � � � L� � e � 1 � 0 10m m�

Ž . Ž Ž . . Ž 2.where K � NL�� � L N � 1 �2� � N�� . Now,Ž .what we have to do is to find � satisfying Eq. 10 . It can bem

said that, with the propagation parameters found from Eqs.Ž . Ž . Ž .7 � 10 , the in-line amplified transmission line in Figure 1 a

Ž .is simplified by the transmission line of Figure 1 b .

3. PROPAGATION-PARAMETER MAPPING FOR WDMCHANNEL TRANSMISSION

When we consider WDM-multiplexed transmission, the addi-tional spectral broadening due to the cross-phase modulationŽ .XPM must be taken into account. If k � 1 WDM-multi-plexed channels are transmitted over a fiber, the total fre-quency chirp occurring on the jth channel can be written by,combining the effects of SPM and cross-phase modulationŽ .XPM ,

Ž .�� z , TS, X , j

� 2Ž . Ž .� � P z U 0, Tj j j�T

k channels2Ž . Ž . Ž .�2 P z U 0, T . 11Ý x x

Ž .x x j

Ž .In Eq. 11 , the second term in the brackets is due to XPM.Assuming that the optical input powers to each channel arein the same level as in the most practical applications, we

Ž . Ž . Ž . Ž .know that P z � P z and U 0, T � U 0, T . Therefore,j x j xthe resulting spectral broadening can be rewritten as

2Ž .� U 0, TjŽ . Ž . Ž . Ž .�� z , T � � 1 � 2k P z . 12S, X , j j j �T

Through the same procedure applied to the case of single-channel transmission, the total nonlinear distortion of the jthchannel over the whole transmission line � can be ob-dist, jtained as

2Ž . � U 0, Tj jŽ . Ž .� T � � 1 � 2k Pdist, j j , 0L �TN

L �� z � Žn�1. l� e eH Ý0 n�1

� Ž Ž . . Ž .�Ž .� u z � n � 1 l � u z � nl L � z dz .Ž .13

The total nonlinear distortion of the WDM-multiplexed sig-nal occurring over the simplified transmission line shown in

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 30, No. 5, September 5 2001 365

Ž . Ž . Ž . ŽFigure 2 Pulse patterns after propagation. a Normal method NZ�DSF, N � 20, P � 0 dBm . b Mapping method NZ�DSF,0. Ž . Ž . Ž . Ž .N � 20, P � 0 dBm . c Normal method NZ�DSF, N � 7, P � 5 dBm . d Mapping method NZ�DSF, N � 7, P � 5 dBm .0 0 0

Ž . Ž . Ž . Ž . Ž . Že Normal method SMF, N � 4, P � 0 dBm . f Mapping method SMF, N � 4, P � 0 dBm . g Normal method SMF,0 0. Ž . Ž .N � 5, P � 5 dBm . h Mapping method SMF, N � 5, P � 5 dBm0 0

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 30, No. 5, September 5 2001366

TABLE 1 Simulation Parameters

Ž .Parameter Unit Value SMF, NZ�DSF

17PRBS pattern � 2 � 1Wavelength nm 1555Noise figure dB 4.5

Ž . Ž .Dispersion D ps�nm�km �16, �3.0Ž . Ž .Loss � 1�km 0.0510, 0.0472

Ž . Ž .Nonlinearity 1� W � km 2, 2.3Ž .coefficient

Ž .Input pulseform � RZ Gaussian FWHM � 40 ps

Ž .Figure 1 b , � , can be writtendist, j, m

2Ž . � U 0, Tj, m jŽ . Ž .� T � � 1 � 2k Pdist, j , m j , mL �Tm

L �� zm Ž . Ž .� e L � z dz . 14H0

The propagation parameters of the simplified transmissionŽ . Ž .line can be found by directly comparing Eqs. 13 and 14 .

Following the same procedure for single-channel transmis-sion, we can easily find the necessary propagation parameters

Ž .for the WDM channel transmission line in Figure 1 b .

4. SIMULATION RESULTS

We first applied the proposed propagation-parameter map-ping method to the simulation of 10 Gbits�s single-channelin-line amplified transmission. The amplifier spacing is as-sumed to be 20 km, and either standard single-mode fiberŽ . Ž .SMF or nonzero dispersion-shifted fiber NZ�DSF is used.The typical physical parameters used for the simulations areshown in Table 1.

The loss parameter � calculated by the parameter-map-mping method is shown in Table 2, where N is the number offiber sections in the transmission line. It is noted that thecase of N � 1 is equivalent to that of a transmission line witha fiber section and an amplifier. Therefore, the mapped lossparameter of the case N � 1 has the same value as theattenuation coefficient of the used fiber. It should be notedthat the effective loss parameter � decreases as N in-mcreases. Figure 2 shows the simulation results confirming thesimilarity between the pulse patterns after propagation, whichare obtained by the normal simulation method, and theproposed parameter-mapping method, where normal simula-tion means that the calculation procedures to describe thepropagation are performed over the first fiber section, andthen the next ones with the results of the previous sections

Ž .repeatedly, until the final section is reached. Figure 2 a andŽ .b shows the pulse patterns for NZ�DSF 20 � 20 km trans-

Ž . Ž .mission with P � 0 dBm; Figure 2 c and d is for NZ�DSF0Ž . Ž .7 � 20 km transmission with P � 5 dBm. Figure 2 e , f and0

Ž . Ž .g , h shows the pulse patterns after SMF transmission forthe cases of 4 � 20 km and P � 0 dBm, and 5 � 20 km and0P � 5 dBm, respectively. It can be noticed from Figure 20

0 2 4 6 8 10 12 14 16 18 200.820.840.860.880.900.920.940.960.981.001.021.041.061.081.101.121.141.161.18

NZ-DSF 0dBm NZ-DSF 5dBm SMF 0dBm SMF 5dBm NZ-DSF WDM -7dBm NZ-DSF WDM 0dBm SMF WDM -7dBm SMF WDM 0dBm

Eye

-ope

ning

ratio

Number of fiber sections

Figure 3 Eye-opening ratios for NZ�DSF and SMF

that the pulse patterns obtained by the two methods coincidevery closely, showing that each corresponding curve tracesalmost identically. In order to compare the results quantita-

Ž .tively, we calculated the eye-opening EO ratio of the simu-lation results, which is the ratio of the average eye opening ofthe pulse patterns by the proposed parameter-mappingmethod to the average eye-opening by the normal simulationmethod. The results are depicted in Figure 3 for variousconditions. The EO ratio remains less than 3% for all of thecases. The simulation time reduction is also measured, and isdepicted in Figure 4. As can be expected, the effect of thesimulation time reduction can be manifested clearly when thenumber of fiber sections is large. For an example, in the caseof single-channel transmission with N � 20 and P � 0 dBm,0the total simulation time, including the parameter calculationtime required in the proposed method, can be reduced to37.15% of the normal simulation method for NZ�DSF and

0 2 4 6 8 10 12 14 16 18 20 22

30

40

50

60

70

80

90

NZ-DSF 0dBm NZ-DSF 5dBm

NZ-DSF WDM-7dBm NZ-DSF WDM-7dBm

SMF 0dBm

SMF WDM 0dBm SMF WDM -7dBm

SMF 5dBm

Tim

e ra

tio (

%)

Number of fiber sections

Figure 4 Simulation time reduction ratio by the proposed mappingmethod

( )TABLE 2 Examples of the Mapping 10 Gbits ///// s, l = 20 km

N

Ž .� 1�km 1 2 3 4 5 6 7 8 9 10m

SMF 0.0510 0.0323 0.0231 0.0180 0.0147 0.0124 0.0108 0.0095 0.0085 0.0077NZ�DSF 0.0472 0.0298 0.0214 0.0166 0.0136 0.0115 0.0099 0.0088 0.0078 0.0071

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 30, No. 5, September 5 2001 367

Ž .Figure 5 Eye diagrams of WDM transmission. a Normal methodŽ . Ž . ŽNZ�DSF, N � 14, P � 0 dBm . b Mapping method NZ�DSF,0

. Ž . ŽN � 14, P � 0 dBm . c Normal method SMF, N � 8, P � 00 0. Ž . Ž .dBm . d Mapping method SMF, N � 8, P � 0 dBm0

tion time is based on the simplicity of the mapping transmis-sion scheme because the more elements there are in thetransmission line, the more time the simulation takes.

Figure 5 shows the pulse patterns after propagationby WDM transmission over NZ�DSF and SMF with P �00 dBm, where we have assumed that three channels areWDM multiplexed at 1554, 1555, and 1556 nm, and eachchannel has a data rate of 10 Gbits�s. It is observed that thegeneral pulse shapes are similar, and the EO ratios are lessthan 3% for the various cases, as depicted in Figure 3. Themerit of simulation time reduction in the WDM transmissioncases is also described in Figure 4. The simulation time for athree-channel WDM transmission line with 20 fiber sectionsand P � 0.2 mW is 29.62% of the normal simulation case0for NZ�DSF and 30.80% for SMF, respectively.

5. CONCLUSION

In this paper, a propagation-parameter mapping method foran in-line amplified transmission line was proposed, whichcan simplify and speed up the numerical simulation. It wasshown that there is a close coincidence between the pulsepatterns after propagation obtained by normal simulationand the simulation adopting the proposed mapping methodfor both a single-channel and a WDM in-line amplifiedtransmission line. The merit of the simulation time reductionhas been measured to be 34.72% for the single-channeltransmission line and 29.62% for the three-channel WDMtransmission line with 20 fiber sections, respectively. Theproposed method can be used regardless of bit rate, fibertype, amplifier spacing, and transmission distance.

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