REFERENCES

1. K.S. Yee, Numerical solution of initial boundary value problems in-volving Maxwell’s equations in isotropic media, IEEE Trans AntennasPropagat 14 (1966), 302–307.

2. A. Taflove and S.C. Hagness, Computational electromagnetics: Thefinite-difference time-domain method, 2nd ed., Artech House, Norwood,MA, 2000.

3. T. Martin, An improved near- to far-zone transformation for the finite-difference time-domain method, IEEE Trans Antennas Propagat 46(1998), 1263–1271.

4. R.F. Harrington, Time-harmonic electromagnetic fields, 2nd ed., Wiley–IEEE Press, Hoboken and Piscataway, NJ, 2001.

5. W. Yu and R. Mittra, Conformal finite-difference time-domain Max-well’s equations solver: Software and user’s guide, Artech House,Norwood, MA, 2004.

© 2005 Wiley Periodicals, Inc.

GROUP-DELAY CHARACTERIZATIONOF CASCADED CHIRPED FIBER-BRAGG-GRATING DISPERSIONCOMPENSATORS

Yan Liu, Zhongwei Tan, Yong Chen, Jihong Cao, Kai Zheng,Tigang Ning, Ting Chen, and Shuisheng JianInstitute of Lightwave TechnologyBeijing Jiaotong UniversityBeijing (100044), P.R. China

Received 16 May 2005

ABSTRACT: Based on linear-system theory, a method to obtain thetotal group-delay characteristics of the cascaded chirped fiber Bragggrating (FBG) from the measured group delays of each grating is pro-posed. The method is then used to characterize cascaded chirped FBGsused as the dispersion compensators of an experimental eight-channel10-Gb/s NRZ system over 1500-km G.652 fiber. The inband total local-dispersion deviation curve is calculated and shows good consistencywith the tendency of the inband BER fluctuations, which indicates thatthe inband local dispersion deviation may be the main reason for thedependence of such a system performance on signal wavelength, and itsRMS value can be taken as the indicator of performance tolerance towavelength instability. Numerical simulation is also performed, bywhich a near linear scaling of the simulated EOP with total local-dispersion deviation is shown. © 2005 Wiley Periodicals, Inc.Microwave Opt Technol Lett 47: 343–346, 2005; Published online inWiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.21165

Key words: fiber Bragg grating; dispersion compensation; group delay;long-haul transmission

1. INTRODUCTION

As an effective solution for dispersion compensation in high-speedand long-haul optical-communication systems, chirped fiber Bragggratings (FBGs) have attracted more and more attention because oftheir low insertion loss, high dispersion figure of merit, compactdesign, capability to compensate higher-order dispersion, the ab-sence of nonlinear effects (especially), and so on. However, be-cause of their resonant nature and the imperfection of manufactureprocess, chirped FBGs inevitably represent ripples in both thegroup delay and reflective spectra, which has detrimental effectson the quality of the signal. Thus, the proper performance charac-terization of chirped FBGs is very important for their properapplication in communication systems.

Compared with reflection fluctuations, the extent and complex-ity of the influence of group-delay ripple on the final performanceof the chirped FBG dispersion compensators are much greater [1].So in this paper we focus on the study of the group-delay charac-teristics of the gratings. For long-haul applications, chirped FBGsare often cascaded as periodic in-line compensators for each fiberspan so as to offer a large dispersion-compensation capability; insuch case, the total dispersion characteristics need to be known.There have been several studies regarding the cascadability ofchirped gratings [2, 3], but their research has been limited to theanalysis of a small number cascaded gratings and all the analyseswere based on the directly measured group delay of the cascadedgratings. However, if the number of cascaded gratings is verylarge, the measurement of a large total dispersion will be limitedby many factors. For example, if many gratings are to be mea-sured, optical amplifiers must be inserted to compensate the lossthat each grating brings, which will increase the complexity of themeasurement settings and decrease the reliability of the measure-ment results. Furthermore, considering the widely used group-delay measurement method—the phase-shift method—the maxi-mum dispersion which can be measured is limited by the phaserange of the phase detector used in such device, the typical valueof which is 180°. Exceeding the range of the phase detector willcause a phase-wrapping error [4]. The wrap-free range is expressedby �� � 360 � D � fm; �� � 180, where D is the dispersion ofthe device under test, fm is the modulation frequency, and �� is thewavelength step. To avoid the problems above, in this paper, wepropose a method to calculate the total dispersion characteristicsbased on the measured group-delay data of separate gratings. Andgratings in three channels of an experimental 10-Gb/s NRZ trans-mission system over 1500-km G.652 fiber are analyzed using thismethod. The total inband local-dispersion deviation is obtainedusing the method. And a finding is that the tendency of local-dispersion deviation corresponds with the inband BER fluctuationvery well for all of the channels. The simulations of the transmis-sion performance of the three channels are also performed, andsome useful conclusions are drawn.

2. THEORETICAL BACKGROUND

H��� � R���exp� j����� (1)

y�t� � x�t� � h1�t� � h2�t�· · · � hn�t� (2)

Y��� � X��� � H1��� � H2���· · · Hn��� (3)

Y��� � X��� � R1 � R2· · · Rn � exp� j��1���

� �2��� � · · · � �n����� (4)

When FBGs are used as dispersion compensators of a systemoperating at low input fiber power, they are usually working inlinear style. According to linear-system theory, the total systemfunction can be obtained by the product of the subsystems in thefrequency domain. So, as in any linear system, if one FBG can becharacterized by its system function in the frequency domain, as inEq. (1), when many such devices are concatenated (see Fig. 1), thetotal system function can be given as the product of that for eachone, as in Eq. (4), and obviously the total phase response can beobtained by the summation of the individual grating’s phase re-sponse. In Eqs. (1)–(4), Rn(�) and �n(�) are the amplitude andphase response for the number n chirped grating, y(t), x(t), andhn(t) are, respectively, the input signal, output signal, and system

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 47, No. 4, November 20 2005 343

function of the number n grating in the time domain, X(�), Y(�),and Hn(�) are their corresponding functions in frequency domain.

From Eq. (4), it can be seen that the total phase response of thecascaded grating can be obtained by the addition of that of therespective grating. This method operates as follows: first, weobtain the phase response of each grating by the integration of itsmeasured group delay; second, we obtain the total phase by thesummation of the separate grating’s phase; and finally, we differ-entiate the total phase response so as to obtain the total group delayand calculate the dispersion by using the quadratic differential ofthe total phase. All the above calculations should observe therelation between phase, group delay, and dispersion, as follows:

�g��� � d����/d� � ��2

2�c

d����

d�,

D��� � d�g���/d�, (5)

where c denotes the velocity of light in vacuum, �g(�) denotes thegroup delay, and D(�) denotes dispersion.

Using this method, the total dispersion, group-delay ripple, orphase ripple can be easily obtained, and then the many cascadedgratings can be viewed as one grating, which will greatly benefitsthe anticipated performance of the system. In order to test thereliability of this method, the analysis of four chirped gratings wasperformed. The total group-delay ripple was obtained using themethod above and compared with the one obtained from thedirectly measured total group delay, as shown in Figure 2, fromwhich good similarity can be seen. Note that due to the integrationinvolved in the method, the calculated error is inevitable and canbe decreased to an extent by the adoption of a wavelength-mea-surement step that is as small as possible when the separate gratinggroup delay is measured and numerical integration is separatelyperformed in the subwavelength section.

3. EXPERIMENTS AND ANALYSIS

An eight-channel chirped FBG-compensated 10-Gb/s NRZ systemover 1500-km G.652 fiber was set up, as shown in Figure 3. All ofthe gratings used in this system were fabricated by the widely-usedscanning phase-mask method in hydrogen-loaded fiber with thereflection ripple being lower than 1 dB and group-delay ripplebeing lower than 30 ps. The input signal power of each channel isdesignated to be about 2 dBm, which is believed to make thesystem operate in a nearly linear style. The system can achievesteady zero-BER transmission. Unfortunately, the system perfor-mance exhibited dependence on the source wavelength. Threechannels were chosen so as to have an in-depth analysis. There arenine gratings in each channel. The group-delay characteristics ofthe gratings used in the three channels were measured separatelyby a chromatic dispersion analyzer CD400 from PerkinElmerCorp. The received optical power was adjusted to allow a slightdeviation from the optimal one, in order to offer nonzero BER. Thedependence of the system performance on the wavelength wasrecorded by the measurement of the averaged BER over half anhour at 21 operating wavelengths spaced at 0.008 nm across thecentral 0.16-nm bandwidth of the total reflective spectra of the ninegratings. The fluctuating errors of the total reflective spectra of thethree channels are all less than 2 dB, as measured by the opticalspectrum analyzer, because of the mutual compensation effect ofthe different gratings and the calculated amplitude of the group-delay ripple is about 100 ps, as calculated using the above method.According to [5, 6], the estimated maximum EOP induced by thereflectivity and group-delay fluctuations are about 0.6 and 4 dB,respectively. So compared with the group-delay ripple, the effectof nonideal characteristics of reflectivity is trivial.

It is known that random group-delay ripples of the grating willcause degradation of the system performance, and this degradationis not only related to the magnitude of the ripple but also to theripple period. When the period of the group-delay ripple is lowerthan the signal bandwidth, the ripple will mainly cause intersymbolinterference. The ripples in such a case can be called high-fre-quency ripples and the peak-to-peak value of the correspondingphase ripple is a better performance indicator of such ripples thanthe peak-to-peak value of the group-delay ripple. However, whenthe period of group-delay ripple is larger than signal bandwidth,the dominant effect will be a modification of the gradient of thedelay curve [7], which is equal to dispersion fluctuations and theripple can then be called a low-frequency ripple. Which parameterwill be most suitable for its characterization is still under discus-sion. To characterize the low-frequency group-delay ripple, local-dispersion deviation is defined as the deviation of the slope of alinear fit on the group delay centered at a specified wavelengthover a sliding window with width being equal to the signal band-width based on a designated dispersion value from [1]. Adoptingthe method mentioned above, the total inband local-dispersiondeviation based on the measured group delay can be calculated atthe corresponding wavelengths at which the BER is measured, as

Figure 1 Schematic diagram of n pieces of cascaded gratings

Figure 2 Comparison of the ripple obtained by the directly measuredgroup delay of the cascaded gratings and that calculated by the separategroup delay of four gratings

Figure 3 Schematic diagram of the 1500-km transmission experiment

344 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 47, No. 4, November 20 2005

shown in Figure 4, in which the measured BER and the calculatedtotal local-dispersion deviation are given together for comparison.For 10-Gb/s systems, the width of the sliding window is set to 0.16nm, and the referenced dispersion is calculated from the real fiberlength in the experiment. Although high-frequency deviation of thetwo curves exists, good consistency has been observed between thetendency of in-band BER fluctuation and the local-dispersiondeviation for all of the three channels, which for the first timeconfirms that in-band dispersion fluctuation is a dominant sourceof the dependence of transmission degradation on source wave-length, even more than the effect of phase ripple by experiments.Furthermore, even though when the system is near linear theaccumulated residual dispersion can be compensated at the re-ceiver [8], large inband dispersion fluctuation will still lead to afirm requirement of source-wavelength stability, which is indicatedby the fact that the RMS of the BER increases with the RMS value

of local-dispersion deviation for the three channels, as can be seenfrom Table 1. So the RMS value of the inband local-dispersiondeviation may be taken as a figure of merit in specifications ofchirped FBGs in order to characterize the system-performancetolerance to the signal-wavelength instability.

The EOP and BER at the above wavelength series for the threechannels were also calculated using a numerical simulation pro-gram (ignoring the effect of reflection fluctuations), and the pa-rameters were designated to be almost the same as those of theexperimental system. The simulated BER and measured BER ofthe three channels are simultaneously shown in Figure 5, where thegood consistency further confirms the trivial effect of the reflectionripple. The simulated EOP is shown as a function of the localdispersion deviation calculated at the same wavelength in anascending order; the relations follow a linear scaling, as can beshown in Figure 6. Since the analysis of group-delay characteris-tics over the same wavelength span while centered at differentwavelength is equal to the analysis of many gratings because of therandomness of the group-delay ripple, the conclusions drawn overdifferent source wavelengths can be also extended to differentgratings.

4. CONCLUSION

In this paper, a method to characterize the dispersion characteris-tics of cascaded chirped FBGs based on linear system theory hasbeen presented. The method is then used to analyze cascadedgrating-dispersion compensators of an experimental 10-Gb/s NRZsystem over 1500-km G.652 fiber. The inband total local-disper-sion deviation curve is calculated and good consistency with thetendency of the inband BER fluctuations is obtained, so the fluc-tuation of system degradation with the source wavelength may bemainly attributed to the inband local-dispersion fluctuations. Andthe RMS of the inband local-dispersion deviation of the gratingshould be taken as a figure of merit in the specifications of chirpedFBGs in order to characterize the system performance tolerance tothe signal wavelength instability. Numerical simulation was alsoperformed, by which a near linear scaling of the simulated EOPFigure 4 Comparisons between the inband BER fluctuation and the

local-dispersion deviation for the three channels

TABLE 1 Comparison of the RMS of the BER and InbandLocal-Dispersion Deviation (LDD) for the Three Channels

CH1 CH2 CH3

RMS of BER 1.65e-8 8.02e-5 3.93e-4RMS of inband LDD (ps/nm) 296.86 708.92 1644.96

Figure 5 Comparison between the simulated BER (ignoring the effect ofreflection fluctuation) and the measured BER for the three channels

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 47, No. 4, November 20 2005 345

with total local-dispersion deviation is shown. This method willhelp enable better utilization of chirped gratings as dispersioncompensators in long-haul systems.

REFERENCES

1. K.-M. Feng, S. Lee, R. Khosvarani, S.A. Havstadand, and J.E. Rothen-bery, Multichannel dispersion compensation using cascaded FBGs for10-Gb/s transmission, NFOEC, paper F8.3, Orlando, FL, 2003.

2. L.-S. Yan, T. Luo, Q. Yu, Y. Xie, and A.E. Willner, System impact ofgroup-delay ripple in single and cascaded chirped FBGs, ProcOFC2002, 2002, pp. 700–702.

3. S. Jamal and J.C. Cartledge, Variation in the performance of multispan10-Gb/s systems due to the group delay ripple of dispersion compen-sating fiber Bragg gratings, J Lightwave Technol 20 (2002), 28–35.

4. P. Elmer, Optoelectronics Corp., Instruction manual of CD400 fiberoptic measurement system, 2000.

5. C. Scheerer, C. Glingener, and G. Fischer, System impact of ripples ingrating group delay, IC-TON’99, 1999, pp. 33–36.

6. K. Ennser, M. Ibsen, M. Durkin, M.N. Zervas, and R.I. Laming,Influence of nonideal chirped fiber grating characteristics on dispersioncancellation, IEEE Photon Technol Lett 10 (1998), 1476–1478.

7. D. Garthe, G. Milner, and Y. Cai, System performance of broadbanddispersion compensating gratings, Electron Lett 34 (1998), 582–583.

8. M.H. Eiselt, C.B. Clausen, and R.W. Tkach, Performance characteriza-tion of components with group delay fluctuations, IEEE Photon TechnolLett 15 (2003), 1076–1078.

© 2005 Wiley Periodicals, Inc.

ON THE PARTIAL-CAPACITANCESCHEME FOR MULTIPLE CONDUCTORSYSTEMS

J.A. Brandao FariaCentro de Electrotecnia Teorica e Medidas ElectricasDEEC–Instituto Superior Tecnico1049-001 Lisboa, Portugal

Received 12 May 2005

ABSTRACT: In the analysis of electrostatic-field problems for multi-conductor systems with n � 1 conducting bodies immersed in dielectricmedia, the capacitive coupling effects among conductors can be de-scribed by means of a partial-capacitance scheme consisting of a net-work of n(n � 1)/ 2 capacitors, each one linking each conductor pair.From a physical viewpoint, each capacitor is usually associated with theexistence of lines of force of the displacement field vector D from oneconductor to another. In this paper, we show that such a physical interpre-tation not only cannot be taken literally, but also, above all, can be a mis-leading one. If care is not taken, computation procedures based on D-fieldintegration may lead to a wrong evaluation of the partial-capacitancescheme. For illustration purposes, a very simple example of a symmetricthree-conductor system with odd-mode excitation is analyzed. © 2005Wiley Periodicals, Inc. Microwave Opt Technol Lett 47: 346–349, 2005;Published online in Wiley InterScience (www.interscience.wiley.com).DOI 10.1002/mop.21166

Key words: multiconductor systems; capacitive coupling; partial-capac-itances

1. INTRODUCTION

The determination of the capacitive effects among conductors insimple or multiple conductor systems is of outmost importance ina variety of electrical engineering problems. Examples includethose concerned with electrostatic discharges, capacitive crosstalk,distributed capacitive coupling in transmission-line systems, mod-elling of interconnect capacitances in 3D structures, and so forth.Such topics have received much attention [1–4]; the developmentof methods for capacitance computation remains an active field ofinvestigation.

The procedure for the evaluation of system capacitances isusually formulated in terms of a Dirichlet’s problem in potentialtheory [5, 6]. Boundary conditions for the potential are prescribedon each conductor of the system V0, V1, V2, . . . Vn. Then, as-suming the dielectric medium is charge-free, the potential functionis obtained in the surrounding space by solving Laplace’s equation,�2V � 0. Once V has been found, the evaluation of its normalderivative on each conductor surface provides the necessary infor-mation on the charge densities � ��V/�n. Integration of thesedensities over each conductor surface finally gives the systemcharges Q0, Q1, Q2, . . . Qn. System capacitances are derived byexamining the relationship between conductor charges and theprescribed potential values.

Figure 6 Comparisons of the EOP and local-dispersion deviations forthe three channels

346 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 47, No. 4, November 20 2005