Pulse-amplitude equalization in arational harmonic mode-locked

semiconductor fiber ring laser using adual-drive Mach-Zehnder modulator

Yun Jong Kim1, Chung Ghiu Lee2, Young Yun Chun3, and Chang-SooPark1

1Department of Information and Communications, Kwangju Institute of Science andTechnology (K-JIST), 1 Oryong-dong, Buk-gu, Gwangju, 500-712, South Korea

2Korea Photonics Technology Institute (KOPTI), 971-35 Wolchul-dong, Buk-gu, Gwangju,500-210, South Korea

3Department of Information and Communication, Wonkwang Health Science College, 344-2Shinruung-dong, Iksan-City, Chollabuk-do, 570-749, South Korea

csp@kjist.ac.kr

Abstract: We present and demonstrate a simple method of pulse-amplitude equalization in a rational harmonic mode-locked semiconductorring laser, using a dual-drive Mach-Zehnder (MZ) modulator. Pulse-amplitude equalization was achieved by adjusting the voltages applied toboth arms of the modulator, such that each mode-locked pulse experiencesthe same transmission coefficient in the transmission curve of the modulator.With this method, amplitude-equalized pulse trains with repetition ratesof ∼7.42GHz (third rational harmonic) and ∼12.34GHz (fifth rationalharmonic) were successfully obtained without any additional function tothe ring laser itself.

© 2004 Optical Society of America

OCIS codes: (060.2330) Fiber optics communications, (320.5550) Pulses, (140.3560) Lasers,ring, (140.4050) Mode-locked lasers, (250.5980) Semiconductor optical amplifiers.

References and links1. C. Wu and N. K. Dutta, “High-repetition-rate optical pulse generation using a rational harmonic mode-locking

fiber laser,” IEEE J. Quantum Electron. 36, 145–150 (2000).2. N. Onodera, A. J. Lowery, L. Zhai, Z. Ahmed, and R. S. Tucker, “Frequency multiplication in actively mode-

locked semiconductor lasers,” Appl. Phys. Lett. 62, 1329–1331 (1993).3. Z. Ahmed and N. Onodera, “High repetition rate optical pulse generation by frequency mudltiplication in actively

modelocked fibre ring lasers,” Electron. Lett. 32, 455–457 (1996).4. M. -Y. Jeon, H. K. Lee, J. T. Ahn, K. H. Kim, D. S. Lim, and E. -H. Lee, “Pulse-amplitude-equalized output from

a rational harmonic mode-locked fiber laser,” Opt. Lett. 23, 855–857 (1998).5. H. J. Lee, K. Kim, H. G.. Kim, “Pulse-amplitude equalization of rational harmonic mode-locked fiber laser using

a semiconductor optical amplifier loop mirror,” Opt. Commun. 160, 51–56 (1999).6. Z. Li, C. Lou, K. T. Chan, Y. Li, and Y. Gao, “Theoretical and Experimental Study of Pulse-Amplitude-

Equalization in a Rational Harmonic Mode-Locked Fiber Ring Laser,” IEEE J. Quantum Electron. 37, 33–37(2001).

7. C. G. Lee, Y. J. Kim, H. K. Choi, C. -S. Park, “Pulse-amplitude equalization in a rational harmonic mode-lockedsemiconductor ring laser using optical feedback,” Opt. Commun. 209, 417–425 (2002).

8. G. Zhu, H. Chen, and N. Dutta, “Time domain analysis of a rational harmonic mode locked ring fiber laser,” J.Appl. Phys. 90, 2143–2147 (1990).

9. A. Takada and H. Miyazawa, “30GHz picosecond pulse generation from actively mode-locked erbium-dopedfibre laser,” Electron. Lett. 26, 216–217 (1990).

(C) 2004 OSA 8 March 2004 / Vol. 12, No. 5 / OPTICS EXPRESS 907#3571 - $15.00 US Received 22 December 2003; revised 27 February 2004; accepted 1 March 2004

10. S. Walklin and J. Conradi, “Effect of Mach-Zehnder Modulator DC Extinction Ratio on Residual Chirp-InducedDispersion in 10-Gb/s Binary and AM-PSK Duobinary Lightwave Systems,” IEEE Photon. Technol. Lett. 9,1400–1402 (1997).

1. Introduction

A stable pulse train with high repetition rate is very essential for a high-speed OTDM system.An actively mode-locked fiber laser is ideal for generating a short pulse train with various bitrates. Recently, rational harmonic mode-locking techniques that generate pulse trains with highrepetition rates have been reported [1, 2, 3]. When the repetition rate of the generated opticalpulse train becomes an integer multiple of the RF drive frequency, the amplitudes among thepulses become varied and characterized by large fluctuations. To overcome this unevenness,several methods have been reported, including the use of another fiber laser with a nonlinearoptical loop mirror (NOLM) [4] and an SOA loop mirror [5], nonlinear polarization rotation(NPR) [6], and optical feedback [7]. G. Zhu, et al. were able to show theoretically an equal-ized amplitude pulse train up to the fourth order rational harmonic mode-locking [8]. In asemiconductor-based ring laser, however, DC biasing the single electrode intensity modulatorof an actively mode-locked fiber ring laser is commonly known to generate optical pulses atrepetition rates that are twice as much as those of the RF drive frequency [9].

In this letter, We propose a simple method of equalizing pulse-amplitudes of rational har-monic mode-locked pulses in a semiconductor-based ring laser. Equalization can be easilyachieved by using a dual-drive LiNbO3 Mach-Zehnder (MZ) modulator placed inside the ringlaser itself and individually adjusting two voltages to the modulator. With this simple structure,amplitude-equalized pulse trains with more than two times the RF modulation frequency can berealized and will be demonstrated experimentally without introducing additional componentsor changes in structure.

2. Principle of pulse-amplitude equalization

The multiplication of repetition rate by rational harmonic mode-locking has been described inthe literatures [1, 2, 3]. If the RF drive frequency ( f mod) is equal to a harmonic of the fun-damental cavity frequency ( f cav), that is, fmod = n fcav (n is a positive integer), the nth-orderharmonic mode-locking occurs, and the pulse repetition rate ( f rep) is the same as the RF drivefrequency. That is frep = fmod = n fcav. To achieve rational harmonic mode-locking, the modu-lation frequency is slightly detuned from the harmonic mode-locking condition by f cav/p. Thusfmod = n fcav ± fcav/p, where p is a positive integer. The pulse train of the repetition rate at ptimes the RF drive frequency can then be obtained ( f rep = p fmod). Figures 1(a) and 1(d) showthe third (p = 3) rational harmonic mode-locked pulse train as an example. The pulse repetitionrate becomes three times the RF drive frequency ( f rep = 3 fmod , i.e., Trep = Tmod/3) [7]. Tmod isthe time interval of the modulation frequency for harmonic mode-locking. The pulses in solidline in Fig. 1(a) indicate the harmonic mode-locked pulses.

The amplitudes of the generated harmonic mode-locked pulses are determined by the trans-mission coefficients through the MZ modulator. Generally, the transmission curve of the modu-lator from input port to output port is defined as TMZM(t) = Iout(t)/Iin(t), where Iin = |Ein(t)|2,Iout = |Eout(t)|2, and Iin(t) denotes the shape of the input optical pulse train at time t to themodulator. Ein and Eout are the input and output optical fields, respectively. The output field ofthe dual drive MZ modulator is given by

Eout(t) =Ein(t)

2

[exp

(jπv1(t)

Vπ

)+ γ exp

(jπv2(t)

Vπ

)], (1)

(C) 2004 OSA 8 March 2004 / Vol. 12, No. 5 / OPTICS EXPRESS 908#3571 - $15.00 US Received 22 December 2003; revised 27 February 2004; accepted 1 March 2004

inI

(d)

3modT

modT

inI

(a)

t

MZMT

(b)t

outI

outI

(c)t

MZMT

(e)t

t

(f)t

πVVac < πVVac >

Fig. 1. Timing diagrams for pulse-amplitude equalization. (a), (d) The input pulse train ofthe MZ modulator, (b), (e) transmission curve of the MZ modulator, and (c), (f) the outputpulse train of the MZ modulator. [(a), (b), and (c)] are due to the small signal modulation(Vac < Vπ ) and [(d), (e), and (f)] large signal modulation (Vac > Vπ ).

where γ is a scaling factor, between 0 to 1, that accounts for an asymmetric device, V π theswitching voltage, and v1(t) and v2(t) the voltages applied to the arms of the MZ modulator[10]. The scaling factor γ is less than one if the splitting/combining ratio of the MZ modulatoris not exactly 50/50. This parameter, γ , is related to the optical extinction ratio δ , as γ = (

√δ −

1)/(√

δ + 1) [10]. The applied voltages to the two arms can be expressed as

v1(t) = Vbias1 + |Vac|sin(2π fmodt + φ1)v2(t) = Vbias2 + |Vac|sin(2π fmodt + φ2),

(2)

where Vac is a modulation amplitude, Vbias1 and Vbias2 the bias voltages applied to arm1 andarm2, φ1 and φ2 the phase of applied voltage to arm1 and arm2, respectively. Figure 2 showsthe calculated values of the modulation amplitude, Vac, in order to achieve the pulse-amplitudeequalization for the third (p=3) and the fifth (p=5) rational harmonic mode-locking. When thephase difference between the applied voltages is π , |φ1 − φ2| = π , small amount of modula-tion amplitude is sufficient to acquire pulse-amplitude equalization for the third (p=3) and thefifth (p=5) rational harmonic mode-locking cases. Pulse-amplitude equalization can be easilyachieved in that condition.

Therefore, we choose the phase difference of π for pulse-amplitude equalization with small

(C) 2004 OSA 8 March 2004 / Vol. 12, No. 5 / OPTICS EXPRESS 909#3571 - $15.00 US Received 22 December 2003; revised 27 February 2004; accepted 1 March 2004

0.25 0.50 0.75 1.00 1.25 1.50 1.75

1.0

1.5

2.0

2.5

3.0

3.5

Mod

ulat

ion

ampl

itude

, |V ac

| (V π)

Phase difference (π)

Third (p=3) rational harmonic mode-locking Fifth (p=5) rational harmonic mode-locking

Fig. 2. Minimum modulation amplitude as a function of phase difference between the twoapplied signals for a pulse-amplitude equalization with mode-locking cases of p=3 andp=5.

modulation amplitude. In our case, φ1 = 0 and φ2 = π . Then, the voltages are described as

v1(t) = Vbias1 + |Vac|sin(2π fmodt) = Vbias1 + vac(t)v2(t) = Vbias2 + |Vac|sin(2π fmodt + π) = Vbias2 + vac(t),

(3)

where vac(t)= |Vac|sin(2π fmodt), vac(t)= |Vac|sin(2π fmodt +π). The output optical pulse trainderived from Eq. (1) is expressed as

Iout(t) =

∣∣∣Ein(t)∣∣∣2

4

[1+ γ2 + 2γ cos

(πv1(t)

Vπ− πv2(t)

Vπ

)]. (4)

Therefore, TMZM(t) is given as

TMZM(t) =14

[1+ γ2 + 2γ cos

(πv1(t)

Vπ− πv2(t)

Vπ

)]. (5)

Figures 1(b) and 1(e) represent the transmission curves of the MZ modulator given by Eq. (5).As shown in Fig. 1(b), the third rational harmonic mode-locking pulses experience differenttransmission coefficients in the MZ modulator. This causes the pulse-amplitude variation. Thispulse-amplitude fluctuation has been known to result from the interaction between circulatingpulses and cavity loss modulation [3]. Equalization of pulse-amplitudes occurs only when thepulse gets the same transmission coefficient in the modulator [8]. The transmission curve of themodulator depends on the voltages applied to each arm of the MZ modulator. By adjusting thevoltages applied to the modulator, i.e., bias voltages, Vbias1, Vbias2 and the modulation amplitude,Vac, the shape of transmission curve can be changed from Fig. 1(b) to Fig. 1(e). Conventionalrational harmonic mode-locked pulse is usually obtained by operating the modulator with thepeak-to-peak amplitude of the applied voltage (Vac) smaller than that of the switching voltage,Vπ as shown in Fig. 1(c). On the contrary, by providing voltages larger than the switchingvoltage of the modulator, more minimum and maximum values of transmission curve can begenerated in such a way that the rational harmonic mode-locked pulses experience almost the

(C) 2004 OSA 8 March 2004 / Vol. 12, No. 5 / OPTICS EXPRESS 910#3571 - $15.00 US Received 22 December 2003; revised 27 February 2004; accepted 1 March 2004

0.0 0.5 1.0

Tranmission curve of the modulator Output pulse waveform

Time (ns)

Inte

nsity

(a.u

.)

(a)

0.0 0.5 1.0

Transmission curve of the modulator Output pulse waveform

Time (ns)

Inte

nsity

(a.u

.)

(b)

Fig. 3. Simulated pulse trains (solid line) and transmission curves (dotted line) for thethird (p = 3) rational harmonic mode-locking (a) without pulse-amplitude equalization(Vbias1 = 1.51Vπ , Vbias2 = 1.51Vπ and Vac = 0.86Vπ ) and (b) with pulse-amplitude equal-ization (Vbias1 = 1.30Vπ , Vbias2 = 1.40Vπ and Vac = 1.01Vπ ).

0.0 0.5 1.0

Transmission curve of the modulator Output pulse waveform

Time (ns)

Inte

nsity

(a.u

.)

(a)

0.0 0.5 1.0

Transmission curve of the modulator Output pulse waveform

Time (ns)

Inte

nsity

(a.u

.)

(b)

Fig. 4. Simulated pulse trains (solid line) and transmission curves (dotted line) for thefifth (p = 5) rational harmonic mode-locking (a) without pulse-amplitude equalization(Vbias1 = 1.50Vπ , Vbias2 = 1.50Vπ and Vac = 0.83Vπ ) and (b) with pulse-amplitude equal-ization (Vbias1 = 1.30Vπ , Vbias2 = 1.72Vπ and Vac = 1.12Vπ ).

same transmission coefficient. This is to the case of similar in the single-drive modulator [8],but the freedom of adjusting is largely limited because only one voltage can be used to obtainthe phase variation. Figure 1(e) is the result of large modulation (Vac > Vπ) with a dual control.In this case, transmission coefficients are almost equal to the mode-locked pulses arriving to themodulator and, as a result, pulse-amplitude equalization of the third (p=3) rational harmonicmode-locked pulse train is obtained (see Fig. 1(f)). The dash-and-dotted horizontal line is drawnfor showing the same transmission coefficient met by the input optical pulses.

To validate the principle of the proposed pulse-amplitude equalization, amplitude equaliza-tion of rational harmonic mode-locked pulse was simulated (Fig. 3 and Fig. 4) using Eq. (4)

(C) 2004 OSA 8 March 2004 / Vol. 12, No. 5 / OPTICS EXPRESS 911#3571 - $15.00 US Received 22 December 2003; revised 27 February 2004; accepted 1 March 2004

SOA

PPGDifferential

DriverSampling

OscilloscopeRF Spectrum

Analyzer

O/E

Optical

Electrical

Clock Trig

Output

3dB Coupler

OTDLPC

Isolator

Dual-ElectrodeLiNbO3 Modulator

2biasV

)(tvac

)(tvac

1biasV

AAATT ATT

Bias-TBias-T

SOA

PPGDifferential

DriverSampling

OscilloscopeRF Spectrum

Analyzer

O/E

Optical

Electrical

Clock Trig

Output

3dB Coupler

OTDLPC

Isolator

Dual-ElectrodeLiNbO3 Modulator

2biasV

)(tvac

)(tvac

biasV

AAAAATT ATT

Bias-TBias-T

Fig. 5. Experimental setup. PC: polarization controller; PPG: pulse pattern generator; SOA:semiconductor optical amplifier; OTDL: optical tunable delay line; ATT: RF attenuator.

and (5). It was helpful in determining the optimum bias level and modulation depth for gettingthe equalized pulse train. In the next section, the theoretical analysis is experimentally verifiedup to the fifth (p=5) rational harmonic mode-locking.

3. Experiments and results

3.1. Experimental setup

To demonstrate pulse-amplitude equalization in a rational harmonic mode-locked semiconduc-tor fiber ring laser, the experimental setup depicted in Fig. 5 was built. The rational harmonicmode-locked semiconductor fiber ring laser was composed of a semiconductor optical amplifier(SOA), an optical tunable delay line (OTDL), a polarization controller (PC), an isolator, anda dual-electrode LiNbO3 MZ type modulator. The modulator, which has 10Gb/s bandwidth,switching voltage of ∼5V, and insertion loss of 6dB, is connected to a high-speed differentialdriver. An RF clock from the pulse pattern generator (PPG) was applied to the differential driver(Anritsu A3HD2106), which gives the differential outputs with π phase difference to each otherbut the same amplitude. Then, Vbias1 and Vbias2 are controlled individually. The SOA (Alcatel1901) is a polarization-insensitive type ( 0.6dB typically) with low tensile bulk separate confine-ment heterostructure, and its gain and cavity length are 25dB and 1000µm, respectively. It hasa carrier lifetime of ∼320ps. It was driven at the operating bias current of 95.7mA. The cavitylength was estimated to be 22.31m, corresponding to a fundamental frequency of ∼8.96MHz.The output pulse train from the rational harmonic mode-locked fiber laser was measured us-ing a 3dB coupler. It was electrically converted by a high-speed photodiode with a 26GHzbandwidth and a 12ps full-width half maximum (FWHM) impulse response. The convertedelectrical signal was measured directly using an RF spectrum analyzer without a low-noise RFpreamplifier.

3.2. Third rational harmonic mode-locking (p = 3)

Conventional active mode-locking was obtained at 1560nm at a modulation frequency of2.48832GHz (∼277th harmonics), driven by the pulse pattern generator. By slightly detun-ing the RF drive frequency from the mode-locking frequency by f cav/p with an integer p,the pth rational harmonic mode-locked optical pulse train was obtained from the rational har-

(C) 2004 OSA 8 March 2004 / Vol. 12, No. 5 / OPTICS EXPRESS 912#3571 - $15.00 US Received 22 December 2003; revised 27 February 2004; accepted 1 March 2004

0.0 1.0 2.0 3.0

Inte

nsity

(a.u

.)

Time (ns)

(a)

0.0 1.0 2.0 3.0

Inte

nsity

(a.u

.)

Time (ns)

(b)

Fig. 6. Measured optical pulse trains from the third (p = 3) rational harmonic mode-lockedsemiconductor fiber ring laser (a) without and (b) with pulse-amplitude equalization.

monic mode-locked semiconductor fiber laser [7]. Based on this relationship, the third rationalharmonic mode-locking (p=3) was observed at the modulation frequency of 2.490840GHz de-tuned from 2.48832GHz by f cav/3. This produces a pulse train with a repetition rate that isthree times the RF drive frequency (∼7.42GHz) as shown in Fig. 6(a). Two arms of the dual-electrode modulator were equally biased at 7.558V (Vbias1,Vbias2

∼= 1.51Vπ) from the conditionacquired analytically in Section 2. Modulation amplitude of 4.335V (V ac = 0.86V) was alsodetermined from the analytical condition, Vac = 0.86V . The RF spectrum for Fig. 6(a) is shownin Fig. 7(a). The frequency scale is 1GHz/div, and the amplitude scale is 2dB/div. The mainpeak indicates the frequency component at the pulse repetition rate, i.e., ∼7.42GHz. There arealso other frequency components in the RF spectrum. When the rational harmonic mode-lockedpulse has a triple repetition rate, the peak amplitude repeats every three pulses. By controllingthe applied voltages to each arm of the dual-electrode modulator, pulse-amplitude equalizationwas obtained. From the amplitude equalization conditions described in Section 2, the appliedbias voltages and the modulation peak were extracted as Vbias1 = 1.30Vπ , Vbias2 = 1.40Vπ andVac = 1.01Vπ . The bias voltages and the modulation amplitude were then tuned around thosevalues. One of the arms was biased with a voltage of 6.240V (Vbias1

∼= 1.25Vπ) and the other at7.128V (Vbias2

∼= 1.43Vπ). In addition, the amplitude of the drive signals applied to two armswas increased from 4.335V (Vac

∼= 0.86Vπ) to 5.097V (Vac∼= 1.02Vπ). A slight deviation of

the experimental voltage values from the simulation results are mainly resulted from the minortuning by fiber polarization control in order to obtain the best equalized pulse amplitude. Figure6(b) shows that the pulse-amplitudes can be equalized using the unbalanced dual-drive Mach-Zehnder modulator. Figure 7(b) is the RF spectrum for Fig. 6(b), showing that all other fre-quency components except the 7.42GHz component were effectively suppressed. This verifiesthat the amplitude-equalized pulse train has a pure 7.42GHz frequency component that corre-sponds to the pulse repetition frequency. The suppression ratio of the signal power at 7.42GHzto the noise power level (other frequency components) appeared to be about 16dB without us-ing any low-noise amplifiers after the high-speed photo detector. As shown in Fig. 6 and Fig. 7,the proposed method equalizes the uneven pulse-amplitude effectively while keeping the pulserepetition rate at three times the RF drive frequency.

(C) 2004 OSA 8 March 2004 / Vol. 12, No. 5 / OPTICS EXPRESS 913#3571 - $15.00 US Received 22 December 2003; revised 27 February 2004; accepted 1 March 2004

(a) (b)

Fig. 7. RF spectra of the optical pulse trains from the third (p = 3) rational harmonic mode-locked semiconductor fiber ring laser. (a) Without and (b) with pulse-amplitude equaliza-tion.

0.0 1.0 2.0 3.0

Inte

nsity

(a.u

.)

Time (ns)

(a)

0.0 1.0 2.0 3.0

Inte

nsity

(a.u

.)

Time (ns)

(b)

Fig. 8. Measured optical pulse trains from the fifth (p = 5) rational harmonic mode-lockedsemiconductor fiber ring laser (a) without and (b) with pulse-amplitude equalization.

3.3. Fifth rational harmonic mode-locking (p = 5)

The fifth rational harmonic mode-locking (p = 5) was derived from the modulation frequencyat 2.491609GHz by detuning from 2.48832GHz by f cav/5. Figure 8 shows the amplitude-equalized and -unequalized pulse trains in the fifth rational harmonic mode-locked semicon-ductor fiber laser. Figure 8(a) was obtained when the dual arms of the modulator were biasedat the same voltage of 7.509V (Vbias1,Vbias2

∼= 1.50Vπ) and equipped with the same amplitudesignals of 4.14V (Vac

∼= 0.83Vπ). The bias voltages of the two arms of the modulator werechanged separately. One of the arms was biased at 6.855V (Vbias1

∼= 1.37Vπ) and the other at7.998V (Vbias2

∼= 1.60Vπ). With an increase in the amplitudes of the drive signals applied toeach arm from 4.14V (Vac

∼= 0.83Vπ) to 5.263V (Vac∼= 1.05Vπ), the pulse-amplitudes were

(C) 2004 OSA 8 March 2004 / Vol. 12, No. 5 / OPTICS EXPRESS 914#3571 - $15.00 US Received 22 December 2003; revised 27 February 2004; accepted 1 March 2004

(a) (b)

Fig. 9. RF spectra of the optical pulse trains from the fifth (p = 5) rational harmonic mode-locked semiconductor fiber ring laser. (a) Without and (b) with pulse-amplitude equaliza-tion.

equalized. Figure 9 represents the RF spectrum of Fig. 8. The frequency scale is 1.5GHz/div,and the amplitude scale is 2dB/div. The main peak indicates the pulse repetition rate at about12.34GHz. Three frequency components exist in Fig. 9(a). As shown in Fig. 9(b), the proposedpulse-amplitude equalization method suppressed the unwanted frequency components in thefrequency domain. The suppression ratio of the 12.34GHz component power level to the noisepower level is about 12dB. This result verifies that the proposed method effectively equalizesthe uneven amplitudes of the fifth rational harmonic mode-locked pulse.

4. Conclusion

We have proposed and successfully demonstrated a novel pulse-amplitude equalization methodusing an unbalanced dual-drive LiNbO3 Mach-Zehnder modulator. The dual-drive modulatorwas used as a mode-locker and a pulse-amplitude equalizer in a rational harmonic mode-lockedsemiconductor fiber ring laser. The experimental results were consistent with the theoreticalanalysis. This is believed to be the simplest one in the pulse-amplitude equalization methodsof higher order (> 2) rational harmonic mode-locking. Only, the equalized rational harmonicorder can be limited by the available amplitudes of the applying voltages. This can be utilized asa short optical pulse source with high repetition rate as required in high-speed OTDM systems.

Acknowledgement

This work was partially supported by grant No. R01-2001-000-00327-0 from the Basic Programof the Korea Science & Engineering Foundation and also by the Brain Korea 21 project inKorea.

(C) 2004 OSA 8 March 2004 / Vol. 12, No. 5 / OPTICS EXPRESS 915#3571 - $15.00 US Received 22 December 2003; revised 27 February 2004; accepted 1 March 2004