We report the amplitude-equalized high order arbitrary numerator rational harmonic mode-locked (RHMLed) pulse generations based on the technique of nonlinear polarization rotation in the fiber-ring laser. The numerator effect on the amplitude uniformity of RHMLed pulse train is first investigated. It is demonstrated that the high order RHMLed pulse train with uniform amplitude can be achieved by choosing a proper numerator. The pulse amplitude equalization in up to the 10th order RHMLed pulse train is observed when the numerator is equal to 3 instead of 1. It is important that the technique may be useful to generate the high-speed pulse trains with uniform amplitude by using the high order RHMLing technique.
©2005 Optical Society of America
An optical short pulse sources with a high repetition rate is very important to realize future ultra-high-speed optical communications. Rational harmonic mode- locking (RHMLing), which is first demonstrated by Onodera et al , represents a potential technology to generate such high repetition rate optical pulse trains [2–7]. By applying RF (radio frequency) driving frequency slightly deviated from the multiple of the fundamental cavity frequency, the pulse trains with a multiplicated RF driving frequency can be produced. To date, Wu et al. have reported on the generation of up to the 22nd order RHMLed pulse , while Yoshida et al. have obtained the RHMLed pulse train with a high repetition rate of up to 200GHz .
However, the pulse amplitude becomes severely uneven except at the second order RHMLing. The amplitude uniformity usually degrades with an increase of the order number in RHMLing. Uneven amplitude will create difficulties on the application of the RHMLed pulses. To solve this problem, several methods to equalize the RHMLed pulse amplitude have been proposed and demonstrated [8–14]. Due to its simple configuration and the relative pulse stability, the technique of nonlinear polarization rotation (NPR)  is an attractive approach of them. To date, the amplitude equalization using the NPR has been demonstrated in up to the 4th order RHMLed pulse train . However, owing to the strong unevenness on amplitude distribution beyond the 6th order RHMLing, this technique cannot be applicable to realize the amplitude equalization in higher order RHMLed pulse trains.
Recently, we reported that the generation of the arbitrary numerator RHMLed pulse  when the RF driving frequency fm(n+m/p)fc, in which, the numerator m was an arbitrary integer, fc was the fundamental cavity frequency, and n and p were both integers. In this work, the numerator effect on the amplitude- distribution of RHMLed pulse was observed. It was found that the pulse-amplitude -distribution of non-unity numerator high order RHMLed pulse is strongly modulated by the numerator=1 low order RHMLing. The change of the pulse amplitude distribution by modifying the numerator is helpful to approach the amplitude equalization in the high-order RHMLing based on the NPR technique. In this paper, we demonstrated experimentally the generation of the high order arbitrary numerator RHMLed pulse with uniform amplitude based on the NPR technique. The effect of the numerator on the pulse-amplitude- equalization has been observed. Up to the 10th order RHMLed pulse trains with uniform amplitude are obtained when the numerator is equal to 3 instead of 1. The result is important, as it shows that the high order RHMLed pulse with uniform amplitude can be achieved using this simple technique.
2. Experiments and principle
The experimental configuration of our RHMLed fiber-ring laser to produce the equal-amplitude pulse train is shown in Fig. 1. In the configuration, a 1550/980nm wavelength division multiplexer (WDM) coupled the 180mW, 980nm laser into the 20m-long erbium-doped fiber (EDF) for pumping. Two isolators (ISO1 and ISO2) were used to propagate the lasing power along a single direction with a co-propagating pump. A 5-GHz bandwidth proton-exchanged lithium niobate intensity modulator with an insertion loss of ~7dB was used as the mode locker, which was driven by a frequency-tunable RF signal synthesizer. Finally, a 10:90 coupler was used to output 10% lasing power for measurement. In the configuration, the pulse-amplitude-equalization mechanism of NPR included the polarization controller (PC1), and a section of 5m long high- birefringence polarization-maintaining fiber (PMF).
Since the waveguide characteristic of proton-exchanged lithium niobate modulator is strongly dependent on the polarization , only TM mode can be waveguided to pass through the modulator. Therefore, the modulator also played the same role as the polarizor to realize additive pulse limiting. The generated pulse trains were measured using a 500MHz Hewlett- Packard 54610B oscilloscope after a photo-detector. As the bandwidth of the used photo-detector is 1 GHz, the measured pulse width is approximately 1ns. We observed the harmonic mode locked pulse generations in this fiber laser when a RF driving frequency fm in the range of 29.0–40.0 MHz was applied. The measured fundamental cavity frequency fc was ~6.72 MHz. In the experiment, the most narrow and stable pulses were measured by carefully tuning the RF frequency in the range of 33.6–40.0 MHz. Such a range was to make sure that the observed pulses were the RHMLed ones. The RF driving frequency could be read from the synthesizer with a precision of 0.05MHz.
In principle, the NPR effect mainly occurs in the high-birefringence PMF. Fig. 2 illustrates the working principle of NPR in the PMF , in which, α 1 is the angle between the polarization direction of the input pulse and the fast axis of PMF, and α 2 is the angle between the fast axis of PMF and the polarization direction of the polarizor. The transmittivity of the structure can be expressed as follows:
where, L is the length of the PMF, nx and ny are linear birefringence coefficients, γ is the nonlinear coefficient of the PMF, and P is the power of input signal.
From Eq. (1), the structure can function as a saturable transmitter under an appropriate condition, whereby a high peak power is suppressed and a low peak power is enhanced. Hence, the pulse-amplitude can be equalized using this manner.
3. Results and discussion
In the experiment, we at first observed the generation of the integer harmonic mode-locked pulse when the driving modulation frequency was tuned at 33.6 MHz, which was the 5 times of the fundamental cavity frequency. By tuning the RF modulation frequency from 33.6 MHz, the RHMLed pulse trains with different detuning of mfc/p were obtained. Figure 3 shows the oscilloscope traces of the recorded pulse train at different values of RF driving frequency of (a) 35.7 MHz, (b) 35.2 MHz, (c) 34.8 MHz, (d) 34.6 MHz. Figure 3(a) shows the 3rd order RHMLed pulse train, as the driving frequency fm≈(5+1/3)fc. Figure 3(b) shows the 4th order RHMLed pulse train, as the driving frequency fm≈(5+1/4)fc. Figure 3(c) shows the 5th order RHMLed pulse as the RF driving frequency fm≈(5+1/5)fc. Fig.3d shows the 6th order RHMLed pulse train, as the driving frequency fm≈(5+1/6)fc. In all of them, the numerator=1. Commonly, the amplitude of RHMLed pulse train is not uniform except the 2nd order RHMLing. It should be noted that, here, the 3rd order, the 4th order, the 5th order and the 6th order RHMLed pulse trains are almost amplitude-equalized due to the effect of NPR. The results indicate that the Kerr nonlinearity of the PMF generates a rotation of polarization state, which depends on the pulse intensity. Meanwhile, it is demonstrated that the proton-exchanged modulator can provide the mechanism of the polarizor for additive pulse limiting to achieve the pulse-amplitude-equalization.
Figure 4 shows the oscilloscope traces of the 7th order RHMLed pulse trains recorded at different values of RF driving frequency of (a) 34.5 MHz, (b) 35.4 MHz, (c) 36.4 MHz. By analyzing the RF driving frequency, the detuning in Fig.4a is fc/7, and the numerator is 1; the detuning in Fig. 4(b) is 2fc/7, and the numerator is 2, the detuning in Fig.4c is 3fc/7, and the numerator is 3. It should be noted that, the amplitude of the 7th RHMLed pulse in Fig. 4(a) is no longer uniform, in which, the numerator=1. The pulse train in Fig. 4(b) performs the best on the pulse- amplitude- equalization, where the numerator=2. As shown in Fig. 4(c), the pulse- amplitude- equalization is again degraded when the numerator=3. In other words, the amplitude- equalization using the NPR technique is strongly influenced by the numerator in the high order RHMLing. In the conventional numerator=1 RHMLing, the pulse- amplitude beyond the 6th order RHMling could not be equalized using NPR technique. When the numerator=3, the pulse also becomes strongly uneven, due to the effect of the second RHMLing. As indicated in our previous works , this is because the power difference between the weakest pulse and the strongest one become too large to achieve the pulse-equalization by utilizing the NPR technique. From Eq. (1), we can infer that using NPR technique, only the pulses, whose powers are in an alternative range, can be equalized. Since the effect of a proper numerator (for example m=2) improves the distribution of the high order RHMLed pulse amplitude, and then, the uniformity of pulse amplitude is further improved using the NPR technique.
To obviously observe the pulse-amplitude change with the numerator, we present the oscilloscope traces of the 9th order RHMLed pulse trains in Fig. 5. In Fig. 5, the RF driving frequency and the numerator are the following: (a) 34.3 MHz, m=1; (b) fm=35.0 MHz, m=2; (c) fm=36.4 MHz, m=4. From Fig.5, the amplitude of the pulse train in Fig.5b is almost even, in which, the numerator m=2. However, the pulse trains when the numerator=1 cannot be equalized. This is because the power of the weakest pulse as shown in Ref.  is too small to utilize the NPR effect. When the numerator=4, the pulse amplitude distribution is strongly uneven, as the detuning is close to the half of the fundamental cavity frequency . We also present the oscilloscope traces of the 10th order RHMLed pulse trains in Fig. 6, in which, the RF driving frequency and the numerator are the following: (a) 34.2 MHz, m=1; (b) fm=35.5 MHz, m=3. From Fig. 6, the numerator=3 RHMLed pulse train exhibits the best performance on the pulse amplitude uniformity. This indicates that the amplitude in the high order RHMLed pulse train can be equalized using the NPR technique if choosing a proper numerator.
Therefore, it is experimentally demonstrated that the arbitrary numerator RHMLing technique is helpful to obtain the amplitude-equalized high order RHMLed pulse trains. Based on the experimental results shown in Figs. 4, 5 and 6, we find that the amplitude equalization of RHLMed pulse performs best when the detuning is around fc/3 or fc/4. This is to take advantage of the improvement on the amplitude distribution of the high order RHMLed pulse trains in such a detuning range. Analyzing the NPR technique, the pulse train can be amplitude-equalization when the power of each pulse in the train is in a certain range. As our previously reported results without the NPR mechanism, the amplitude of the each pulse in the high order RHMLed train was relatively well-confined in such a certain range when the detuning is around fc/3 or fc/4 . This also indicated that our results are in accordance with the previous experimental results on the pulse amplitude distribution without using NPR technique. When the detuning tends to the zero or fc/2, the NPR technique is not an efficient way to equalize the pulse amplitude. However, by using an alternative numerator, the amplitude equalization of the high order RHMLed pulses can be achieved using this technique.
Due to the limitation of the responsive bandwidth of 500MHz oscilloscope, the repetition frequency of the generated pulse train is limited at ~355MHz. However, as a way, this provides a significant physical approach to realize the equal amplitude pulse generation. This method may be useful for the generation of the ultra-high repetition frequency pulse train with uniform amplitude using the high order RHMLing technique.
In conclusion, we demonstrated experimentally the numerator effect on the pulse- amplitude-equalization in arbitrary numerator RHMLed fiber laser. Using NPR technique, we observed the generation of up to the 10th order RHMLed pulse train with equal amplitude when the numerator is 3. Due to its simple configuration, without requiring any additional devices or components, this technology may be applicable on the generation of amplitude-equalized high order RHMLed pulse train for ultra-high bit rate optical communication systems.
The authors acknowledge the financial support from the Science and Technology Committee of Shanghai Municipal, China under contracts 04DZ14001, and 022261003, and the National Natural Science Foundation of China under the grant 10474064.
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