results from the Mie series. But for higher values of �, thePMCHWT formulation gives incorrect results, while the presentformulation still agrees with the Mie series. For � � 105 S/m, ��r�� 1.8 � 1011 and the skin depth is about 16 mm. The presentformulation still works well, even when � is as high as 108 S/mand the corresponding skin depth is about 0.5 mm. So the presentformulation can work with much higher conductivities than thePMCHWT formulation, due to the effective suppression of theerror contribution from the internal conductive region.

To explore the generality of the present formulation, a high-frequency case using RWG basis is also calculated and the resultsare illustrated in Figure 4. The frequency here is 10 MHz. For �varying from 1 to 1000 S/m, the present formulation providesoverall satisfactory performance, while the PMCHWT formulationgives incorrect results when � is higher than 10 S/m.

For � � 104 S/m, the new formulation also fails to give correctresults, because the skin depth is too small to be accurate enoughfor the typical seven-point Gaussian quadrature integration, evenwhen the error has already been suppressed. To increase theintegration accuracy, we simply divide each triangular patch intofour subpatches and apply the seven-point Gaussian quadratureintegration to each subpatch, thus effectively increasing the inte-gration points per patch. Note that here the number of basisfunctions is not increased. Then the new formulation gives acorrect result again, as shown in Figure 4(f). Although usingsubpatches is not the most efficient way to improve the integrationaccuracy, we illustrate the fact that the number of basis functionsdoes not have to increase with the internal-region wavenumber.

The present formulation performs better in the low-frequencyband than in the higher frequencies because, for the same skindepth, ��r2� is larger at lower frequencies and thus the internal-region terms are suppressed by a larger factor. Hence, the accuracyof the internal-region terms will be less important for lower fre-quencies.

5. CONCLUSION

A new SIE formulation has been obtained by tuning the weightingcoefficients of the integral equations for the external and internalregions, respectively. Thus, the integration error over the Green’sfunction for the internal conductive region is suppressed and thepresent formulation provides satisfactory results in a significantlylarger range of skin depths than previous SIE formulations. Thepresent formulation can also work in tandem with techniques thatimprove the integration accuracy for Green’s functions in conduc-tive media in order to further enlarge the solvable range of con-ductivities.

REFERENCES

1. A. Sebak, L. Shafai, and Y. Das, Near-zone fields scattered by three-dimensional highly conducting permeable objects in the field of anarbitrary loop, IEEE Trans Geosci Remote Sensing 29 (1991), 9–15.

2. V.A. Fock, Electromagnetic diffraction and propagation problems,Pergamon Press, New York, 1965.

3. J.R. Wait, Theories for radio ground wave transmission over a multi-section patch, Radio Sci 15 (1980), 972–976.

4. D.-S. Wang, Limits and validity of the impedance boundary conditionon penetrable surfaces, IEEE Trans Antennas Propagat 35 (1987),453–457.

5. S.Y. Chen, J.S. Zhao, and W.C. Chew, Analyzing low-frequencyelectromagnetic scattering from a composite object, IEEE TransGeosci Remote Sensing 40 (2002), 426–433.

6. B.M. Kolundzija, Electromagnetic modeling of composite metallic anddielectric structures, IEEE Trans Microwave Theory Tech 47 (1999),1021–1032.

7. A.A. Kishk and L. Shafai, Different formulations for numerical solu-

tion of single or multibodies of revolution with mixed boundaryconditions, IEEE Trans Antennas Propagat 34 (1986), 666–673.

8. S. Chakraborty and Jandhyala, Accurate computation of vector poten-tials in lossy media, Microwave Opt Technol Lett 36 (2003), 359–363.

9. F. Shubitidze, K. O’Neil, S.A. Haider, K. Sun, and K.D. Paulsen,Application of the method of auxiliary sources to the wide-bandelectromagnetic induction problem, IEEE Trans Geosci Remote Sens-ing 40 (1991), 928–942.

10. Y. Chu, W.C. Chew, J.S. Zhao, and S.Y. Chen, A surface integralequation formulation for low-frequency scattering from a compositeobject, IEEE Trans Antennas Propagat 51 (2003), 2837–2844.

11. S. Rao, D. Wilton, and A. Glisson, Electromagnetic scattering bysurfaces of arbitrary shape, IEEE Trans Antennas Propagat 30 (1982),409–418.

© 2005 Wiley Periodicals, Inc.

EFFICIENT CHARACTERIZATION OFHARMONIC AND INTERMODULATIONDISTORTION EFFECTS IN DISPERSIVERADIO OVER FIBER SYSTEMS WITHDIRECT LASER MODULATION

Giovanni Tartarini,1 and Pier Faccin2

1 Department of Electronics, Computer Science and SystemsUniversity of BolognaBologna, Italy2 TEKMAR Sistemi Srl–An Andrew CompanyVia De Crescenzi 40I 48018 Faenza, Italy

Received 5 January 2005

ABSTRACT: Theoretical and experimental radio over fiber (RoF) sys-tem performance is studied in the presence of fiber dispersion and di-rectly modulated DFB laser chirp. The measurements determine the dis-tortion terms of the links with G-652 fibers. Agreement with thecomputed values validates the model, used presently in RoF link designfor wireless-signal distribution. © 2005 Wiley Periodicals, Inc.Microwave Opt Technol Lett 46: 114–117, 2005; Published online inWiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.20917

Key words: microwave photonics; laser chirp; harmonic distortion;intermodulation distortion; dispersive channel

INTRODUCTION

The analysis of harmonic and intermodulation distortion is of greatconcern in the design of analog or subcarrier multiplexed (SCM)optical systems. These effects may negatively affect the perfor-mance of systems that exploit, for example, wavelength conver-sion via cross gain modulation (XGM) in semiconductor opticalamplifiers (SOAs) [1] or systems that exploit both direct andindirect modulations of a laser source to realize upconvertedmillimeter-wave fiber-optic links [2].

A typical cause of the appearance of these distortion terms isthe combined action of laser chirp and propagation in a dispersiveoptical channel [3]. This problem has been studied intensely (see,for example, [4]) with reference to systems exploiting externalelectrooptical modulators. These systems, however, have the draw-back of cost related to the use of expensive components such asexternal modulators. This indicates the need to investigate lessexpensive solutions, such as those based on standard direct inten-

e-mail: gtartarini@deis.unibo.it

114 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 46, No. 2, July 20 2005

sity modulation (IM) of the laser source [5]. Unfortunately, fromthe point of view of the chirp effect, these systems present evenlarger values of laser chirp.

Therefore, the overall spurious-free dynamic range (SFDR)performances of dispersive RoF systems should be evaluated inview of correct system design.

The effect of laser chirp in direct IM systems has been studied,as for example in [6], where a Gaussian SCM modulating signal isassumed. Moreover, in [7], both experimental and theoreticalstudies of the problem were performed in large-signal direct mod-ulation. In both cases, the models that relate the laser chirp to theIM of the laser source were very detailed. In the former, both staticand dynamic chirping was taken into account; in the latter, aprecise expression of a quasi-adiabatic chirp was used.

However, for preliminary evaluation of the system perfor-mances, it is useful to have at one’s disposal an easy-to-implementnumerical tool, which, although approximate, uses an assumptionthat can be regarded as acceptable in relation to the characteristicof the optical link considered. These characteristics are included inthe model described below, which present a great conceptualsimplicity, based on the extension of those discussed in [8, 9].

To usefully exploit this model, the characterization of the chirpof the laser source must be performed experimentally. The mea-sured values of the chirp must then be input to the numericalprogram in order to evaluate the system performances in thepresence of a dispersive optical channel. In the following, wedescribe the experimental characterization of the performance ofdifferent optical links. Then, after a presentation of the numericalmodel, we compare the theoretical results obtained with this modelwith the experimental ones. Finally, conclusions will be drawn.

EXPERIMENTAL RESULTS

The first step of the experimental part of this work was to measurethe value of the laser chirp to be inserted in the model equations.For this purpose, the self-homodyne technique was used to mea-sure the angular modulation index mf caused by the laser chirp,with frequency fm of the RF input signals ranging betweenfm,min � �m,min/(2�) � 100 MHz and fm,max � �m,max/(2�) �2.11 GHz and input powers Cin of the modulating signals rangingfrom �14 dBm to 10 dBm. We verified that for a fixed value of fm,the value of mf is proportional to (Cin)1/ 2 with a good degree ofapproximation.

The adiabatic chirp factor kf � mf fm/lj MHz/mA, where lj isthe peak value of the modulating current injected in the transmitter,was then determined. This evaluation was done in the hypothesis,which will be explained in the next section; its contribution to theoverall chirp effect is by far the predominating one.

The scattering parameters of the two-port system, comprised ofan optical transmitter, an optical homodyne interferometer, and aphotodetector, were measured for each couple of values ( fm, Cin)to determine the true value of the modulating injection current.Three different optical transmitters were then characterized, ex-hibiting a slight dependence of kf on the values of fm and Cin [10].The average values of kf that were determined are 97, 128, and 210MHz/mA, respectively.

The second experimental step was the harmonic- and inter-modulation-distortion measurement of the links (see Fig. 1). Theoptical transmitters were modulated by one or two RF sinusoidalcarriers. The modulated signal was then coupled to the opticalfiber. An optical variable attenuator assured that the signal reach-ing the photodiode suffered the same overall loss, regardless offiber length, in order to allow direct comparison of the measuredresults. The optical back reflection of the link was continuously

monitored in order to guarantee that its value was always greaterthan 35 dB. An RF spectrum analyzer was used to determine thedesired quantities of the output signal.

The ith-order distortion terms C/Di can then be measured: C isthe received power of the RF carrier component at angular fre-quency �m and Di is the power of the terms at angular frequencyi�m. The intercept points of the 2nd order (IIP2) and 3rd order(IIP3) can also be measured. The transmitters were modulated withRF carriers of frequencies fm1 and fm2, given respectively by (890,940), (900, 930), (1720, 1850), (1710, 1860), (930, 1720), and(940, 1720) MHz. In all cases, the distortion terms were measuredfor Cin equal to 3, 6, and 9 dBm. Different fiber-spool lengths wereused, ranging between 25 and 50 km. The value of the dispersionof the fibers at the transmitter’s wavelength (� � 1550.9208 nm)was directly measured using an Agilent 86038A.

THEORETICAL MODEL AND COMPARISON

The optical field (whose angular frequency is �0) is assumed to beintensity modulated by different RF signals (angular frequencies�m1, �m2, . . .) with optical modulation indexes ma1, ma2, . . . .Each of these signals then causes a frequency modulation withangular index mfi of the optical carrier due to the chirping effect.This is taken into account when characterizing the main compo-nent of the field as follows:

E � �1 � �i

maicos��mit� � exp�j��0t � �i

sin(�mit)��, (1)

and subsequently expanding it as the product of the Fourier seriesof the IM term [square-root factor in Eq. (1)] and of the FrequencyModulated (FM) term [exponential factor in Eq. (1)], respectively.A more rigorous introduction of the chirping effect in Eq. (1) canbe made (see, for example, [6, 7, 11]). However, in the hypothesisof modulation frequencies well below the relaxation resonance ofthe laser sources, it is allowable to assume a frequency deviationproportional and in phase with the intensity modulation [3, 12].

The evaluation of the product at the second side of Eq. (1) is nota trivial task. First, the different terms of Eq. (1) exhibit differentperiodicities in time. Second, while the IM term can be expandedin a simple Fourier series, it must take into account that the FMterm results in the product of a number of Fourier series equal tothe number of RF modulating carriers.

A way to overcome the first problem (without loss of general-ity) is to determine a common angular frequency �c such that for

Figure 1 Sketch of the experimental setup used for measurement of thedistortion terms of the RoF systems analyzed

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 46, No. 2, July 20 2005 115

all the nc modulating carriers, it holds �mi � qi�c, where qi is aninteger. Once �c has been determined, Eq. (1) takes the followingform:

E � �k0

2� �

l���

�kl

2exp( jl�ct)�

� �i�1

nc � �pi���

�

Jpi(mfi)exp( jpiqi�ct)�exp� j�0t�. (2)

The second problem is overcome by developing an appropriatealgorithm of collection. Through this algorithm, for the termcorresponding to a certain angular frequency n�c, a multiple of�c, the contributions coming from all the possible sums anddifferences of the various terms of indexes l, p1, . . . pnc are takeninto account. The final form eventually obtained can be formallywritten as in [8]:

E � � �n���

�

Lnexp( jn�ct)�exp� j�0t�. (3)

The amount of computing time necessary to arrive at Eq. (3) canbe kept low by exploiting the recursive properties of the integralsleading to the determination of the kl terms of Eq. (2) and also byexploiting the fact that, depending on the values of mfi, a valuepi max can be determined such that Jpi(mfi) � 0 for �pi� � pi max.

Then, taking into account that chromatic dispersion is thedominant effect in G-652 fibers operating in the third opticalwindow, the propagation in the optical fiber is computed bymultiplying the field determined above by the approximate fibertransfer function, given by

H��� � exp��z� � exp��j��0

��

���0(� � �0) �1

2

�2

��2�0

(� � �0)2�z�, (4)

where and are the attenuation and propagation constant of thefundamental mode propagating in the fiber, respectively. Thesquare-law detection of the photodiode is finally taken into accountby computing the square module of the propagated field.

This modeling program is quite general and can be used tocompute all the harmonic and/or intermodulation distortion termsof the different orders, assuming different modulation indexes forthe various RF carriers. In particular, it can be observed that, sincethe IM term is expressed through an appropriately expanded Fou-rier series, no approximation on the value of the optical modulationindexes (OMIs) is necessary.

In addition to the harmonic-distortion terms C/Di, we werealso interested in evaluating the 2nd- and 3rd-order intermodulationterms, in the case of two RF carriers. To make a comparison withthe experimentally characterized links, attention was therefore alsofocused on the computation of IIP2 and IIP3. To reproduce theexperimental conditions, the two RF carriers were fed each timewith the same input power: Cin1 � Cin2 � Cin.

The behavior of the links was modeled for different values ofCin, for different fiber lengths and for different values of �m1 and�m2. Using the set of data obtained in the first part of the exper-imental work, we can insert the true values of mf 1, mf 2 in themodeling program for each laser in relation to the values of Cin1,fm1, and fm2.

Figures 2–4 report typical results chosen from those obtained.They compare, respectively, the measurements of the 2nd harmonicdistortion term C/D2 and the intercept points of orders two (IIP2)and three (IIP3) with the corresponding theoretical predictions.The differences between the theoretical and measured values arealways lower than 1 dB for the different fiber lengths, thus allow-ing us to conclude that the agreement is very good.

CONCLUSION

An experimental and theoretical comparison has been carried outin order to evaluate the 2nd- and 3rd-order distortion terms causedby laser chirp in RoF systems. Very good agreement was foundbetween the theoretical and measured values of the distortionterms. The simulation program derived from the theoretical modelis, therefore, a reliable and efficient tool for the design of RoFsystems that use direct-intensity modulation of DFB laser diodesand direct detection.

Figure 2 Values of the term C/D2 (in dB) for different lengths of thefiber span. The optical transmitter used features an average chirp factor ofkf � 128 MHz/mA and is modulated with an RF carrier with fm � 900MHz. The value of the input RF power is Cin � 3 dBm. Here and in thefollowing figures, the continuous dotted line represents computed values,while the broken line represents measured values

Figure 3 Values of IIP2 (in dB) vs. fiber length corresponding to anoptical transmitter with an average chirp factor of kf � 97 MHz/mA andmodulated with two RF carriers with fm1 � 940 MHz and fm2 � 1720MHz. The value of the input RF power is Cin � 6 dBm

116 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 46, No. 2, July 20 2005

ACKNOWLEDGMENT

Part of this work was funded by the Italian Ministry for Education,University and Research (MIUR).

REFERENCES

1. J. Capmany, E. Peral, and D. Pastor, Formula for two-carrier inter-modulation distortion in wavelength converted subcarrier multiplexedsignals via cross gain modulation, IEEE Photon Technol Lett 12(2000), 278–280.

2. K. Kojucharow, M. Sauer, and C. Schaffer, Millimeter-wave signalproperties resulting from electro-optical upconversion, IEEE TransMicrowave Theory Tech 49 (2001), 1977–1985.

3. M.R. Phillips, T.E. Darcie, D. Marcuse, G.E. Bodeep, and N.J. Frigo,Nonlinear distortion generated by dispersive transmission of chirpedintensity-modulated signals, IEEE Photon Technol Lett 3 (1991),481–483.

4. S. Betti, E. Bravi, and M. Giaconi, Analysis of distortion effects insubcarrier-multiplexed (SCM) externally modulated lightwave sys-tems: a generalized approach, IEEE Photon Technol Lett 9 (1997),118–120.

5. Application note, Agere Systems, April 2001.6. E. Bravi, V. Moeyaert, S. Betti, M. Giaconi, J.C. Froidure, L. Ghislain,

and M. Blondel, Experimental and theoretical evaluation of distortioneffects of SCM optical transmission due to the joint action of staticchirping, dynamic chirping, and dispersive channel, Photon NetworkCommun 2 (2000), 393–401.

7. E. Peral and A. Yariv, Large-signal theory of the effect of dispersivepropagation on the intensity modulation response of semiconductorlasers, IEEE J Lightwave Technol 18 (2000), 84–89.

8. G.J. Meslener, Chromatic dispersion induced distortion of modulatedmonochromatic light employing direct detection, IEEE J QuantumElectron 20 (1984), 1208–1216.

9. C.S. Ih and W. Gu, Fiber induced distortions in a subcarrier multi-plexed lightwave system, IEEE J Sel Areas Commun 8 (1990), 1296–1303.

10. M. Sauer, K. Kojucharow, H. Kaluzni, and M. Otto, Impact of laserchirp on carrier and IMD power in electro-optical upconverted milli-metre wave fibre optic links, Electron Lett 35 (1999), 834–836.

11. I. Kaminov and T.L. Koch, Optical fiber telecommunications III B,Academic Press, New York, 1997, pp. 117–120.

12. A. Yariv, Optical electronics, Harcourt Brace Jovanovich, Orlando,FL, 1991, pp. 576–584.

© 2005 Wiley Periodicals, Inc.

MESHED MICROSTRIP PATCHANTENNAS WITH LOW RCS

Xiulian He, Shuxi Gong, Yicai Ji, and Qizhong LiuNational Laboratory of Antennas and Microwave TechnologyXidian UniversityXi’an 710071, P. R. China

Received 4 January 2005

ABSTRACT: The application of meshed microstrip patch antennas to(RCS) reduction is analyzed. The effect of mesh line-width, mesh line-spacing, and meshing style on the resonant frequency and gain of theantenna is studied. A meshed microstrip patch antenna with low RCS isdesigned. The gain suffers by 0.6 dBi, and the mean RCS over 2–10 GHz is4-dBsm lower compared to the standard patch antenna with the same reso-nant frequency. The maximum RCS reduction is 10.6 dB at 3.2 GHz.© 2005 Wiley Periodicals, Inc. Microwave Opt Technol Lett 46: 117–120,2005; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.20918

Key words: meshed microstrip antenna; radar cross section (RCS) re-duction

1. INTRODUCTION

Microstrip antennas possess numerous advantages, including con-formability, light weight and low cost, which make them attractivecandidates for use in aerospace applications. The radar crosssection (RCS) of aircraft determines whether aircraft will be de-tected by radar stations directly. Techniques such as geometricalshaping and the use of radar-absorbing materials have beenadopted to reduce the RCS of aircraft; thus it has become increas-ingly important to reduce the RCS of microstrip antennas mountedon aircraft, since such scattering may be the most significantcontribution to the overall RCS.

Several techniques have been employed to reduce the RCS ofmicrostrip antennas. A resistive skirt placed on the edge of thepatch has been reported to suppress the RCS, while having aminimal effect on antenna performance at the operating frequency[1]. In addition, by controlling the bias voltage across a varactordiode that is mounted between the conduction patch and groundplane, the scattering response of the antenna can be tuned tominimize the RCS at threatened frequencies [2]. The RCS peaks ofa microstrip patch antenna printed on a ferrite substrate can befrequency shifted by changing the magnetic bias field with littleeffect upon the radiation property of microstrip antennas [3]. Theapplication of meshed patch antennas to the RCS reduction ofmicrostrip antennas is investigated in this paper. The effects ofvarying the mesh line-width, mesh line-spacing, and the meshingstyle upon the resonant frequency and gain of the antenna arestudied. A meshed microstrip patch antenna with low RCS isdesigned.

2. ANALYSIS MODEL FOR RCS OF MICROSTRIPANTENNAS

The total scattering field of antennas constitutes structural scatter-ing and antenna-mode scattering. When the feed port is matchloaded, the scattering is structural scattering. If not, part of thereceived energy is reradiated, which is antenna-mode scattering.

The geometry of a microstrip patch antenna fed by a coaxialprobe is shown in Figure 1. The scattering field of the microstrippatch antenna with arbitrary load can be obtained using the scat-tering matrices [4, 5]:

Figure 4 Values of IIP3 (in dB) vs. fiber length corresponding to thecase where kf � 210 MHz/mA, fm1 � 1720 MHz, fm2 � 1850 MHz, andCin � 9 dBm

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 46, No. 2, July 20 2005 117