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Abstract
A scheme for improving the bandwidth of a phase-locked loop (PLL) for mode-locked laser is proposed. In the proposed scheme, a modified PLL with a multiple-differentials-based loop filter can be used to increases the upper limit bandwidth of the laser-based PLL. The mechanism of the bandwidth improvement is explained in detail, and the experimental results of a laser-based PLL with the proposed scheme show that the upper-limit bandwidth of the PLL has been increased about by one order at offset frequency from 3 kHz to 30 kHz. This scheme with the simple multiple-differential-based loop filter configuration can be easily used in another laser’s phase locking system whose bandwidth should be improved.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Passive mode-locked laser (MLL) [1] which can produce ultra-short optical pulse has widely used in many applications in recently years, including optical/radio frequency standard [2], ultra-low noise microwave synthesis [3], metrology [4], and time-frequency dissemination [5]. For these applications of MLL, it is required that the MLLs should be stabilized to a highly-stable optical/radio frequency reference source with a phase-locked loop (PLL) [6]. With this phase-locking scheme, the phase and repetition frequency of MLL are both synchronized to an external frequency reference by using a PLL. In the past years, many works demonstrate that ultra-stable MLLs can be achieved by using the PLL with the help of proportional-integral (PI) controller [7–17]. In general, the bandwidth of a PLL is a crucial parameter which determines the locking performance of a MLL [17,18]. By increasing the loop gain and bandwidth of a MLL-based PLL, we are able to improve the stability and timing fluctuation of a stabilized MLL [18–20], and further synthesize an ultra-low noise microwave signal in a photonic-microwave generator [21]. Therefore, it is very important that a laser-based PLL with high bandwidth is realized.
In our previous works, we have studied the phase-locking technique for MLL, including the theoretical analysis of the laser-based PLL and the scheme for measuring the loop gain and bandwidth of the PLL [22,23]. From the studies, we found that the locking bandwidth of a PLL for a MLL is mainly limited by the laser oscillator, because the piezoelectric transducer (PZT) and its high-voltage (HV) driver in laser system have a time delay and frequency resonance, which introduce an extra phase lag in the loop. In this case, increasing the locking bandwidth of the PLL will results in the oscillation or unlocking of the PLL, and the maximum value of the bandwidth in the locked state is called as the upper limit bandwidth. Therefore, to break the limit of the bandwidth of a laser-based PLL, the PLL should be modified to compensate the extra phase lag. A typical method for improving the bandwidth of a laser-based PLL is optical or mechanical scheme, where the fast PZT is fixed on an elaborately-designed mount [24,25]. Although this optical method can solve the problem of the bandwidth limit, it is also not very convenient for the laser designer. Because the laser cavity has to be reconstructed if its bandwidth is needed to be increased.
In this paper, we propose an electronic PLL scheme with an upgrade loop filter to improve the bandwidth of a MLL-based PLL. This scheme can be used to increase the upper limit bandwidth of the PLL for a mode-locked laser by using a loop filter with multiple differentials without replacing any other electronic and optic components. Furthermore, with this multiple-differentials-based loop filter, the stability and timing jitter of the PLL can also be improved since the phase lag has been compensated.
2. Scheme for improving the bandwidth of the PLL with multiple-differential-based loop filter
A PLL [26] is an important electronic tool which can stabilize a mode-locked laser to a frequency reference source, where the phase and repetition frequency of the laser are both synchronized to the reference source precisely. As we presented above, the locking bandwidth of the laser-based PLL is a crucial parameter, which determines the locking performance of the PLL, for example, the timing jitter and stability. However, the bandwidth of the PLL with the typical PI-based loop filter has a limit due to the extra phase lag introduced by the PZT and its HV driver. To break the limit, one of the most effective method is to modify the loop filter if the laser is not allowed to be rebuilt. Here, we will first explain the problem of the limit for the laser-based PLL with typical PI-based loop filter, and then demonstrate an upgraded PLL scheme with a multiple-differentials-based loop filter which can compensate the extra phase lag and improve the bandwidth of a laser-based PLL.
2.1 Limit of bandwidth of the PLL with typical PI loop filter for mode-locked laser
Figure 1(a) shows the schematic of the PLL with typical PI-based loop filter for MLL. In this PLL, a fast photo detector and band-pass filter are used to convert the laser pulses to a microwave signal. The detected microwave is mixed with the frequency reference signal by a phase detector, and then a phase-error signal at the intermediate frequency (IF) is produced. A PI-based loop filter eliminates any unwanted higher frequency products, and introduces a frequency-depended gain in the loop. The error signal generated from the loop gain continuously adjusts the voltage on the PZT of the laser via a HV driver to maintain quadrature between the laser output signal and the reference source. In this case, the phase and repetition frequency of the MLL is locked to that of the reference source, and hence the timing jitter of the laser is reduced. This auto-adjusting phase-locking technique realizes a PLL between a MLL and reference source.
Fig. 1 (a) Schematic of the PLL with typical PI-based loop filter for MLL. PZT: piezoelectric transducer. HV: high voltage. PI: proportional-integral. VCO: voltage-controlled oscillator. (b) Simulated loop gain of the PLL with different kp.
In the schematic shown in Fig. 1(a), the HV driver, PZT and MLL are be treated as a voltage-controlled oscillator (VCO). By adjusting the input voltage of the HV driver, the cavity length of the laser is changed, and hence the repetition frequency of the laser can be controlled. Based on the theoretical analysis for the standard PLL in literatures [13,22], the loop gain of the MLL-based PLL can be calculated out, and it is given by
where G(s) is the loop gain of the laser-based PLL, Θi(s) and Θo(s) is the phase arguments of the input reference signal and the output signal of the MLL, Θe(s) is the phase error between the reference signal and the laser signal, c is light velocity in fiber, n is the laser’s nth harmonic, ko is the regulation gain of the HV driver and PZT, kd is the gain of phase detector, To is the time-delay of the HV driver and PZT which limits the PLL bandwidth, Lo is the laser’s cavity length, and F(s) is the transfer function of the loop filter. Here, the loop filter has a proportional and an integral, and its transfer function F(s) is expressed as [27]where kp is the proportional gain (P gain) of the PI controller, and Ti is its integral time constant. By inserting Eq. (2) into Eq. (1), the final loop gain of the PI-based PLL is given bywhere k = 2πcnkokdkp. The loop gain presented in Eq. (3) shows that the PLL is a third-order controlling system. Based on Eq. (3), we calculated and simulated the loop gain and bandwidth of the PLL with this PI-based loop filter. In our simulation, we assume the repetition frequency of a MLL is 100 MHz, and its 10th harmonic (1 GHz) is locked to a reference signal. Therefore, n is 10. The frequency corner of the integral of the PI controller is about 100 Hz. Therefore, Ti is 10−2 s. The other parameters of the PLL are assumed as follow: ko is 10−5 m/V, kd is 10−1 V/rad, Lo is c/(100 × 106 Hz), To is 10−4 s.To simulate the PLL with different bandwidth, we changed the loop gain G(s) by adjust kp, and the simulated G(s) is shown as the Bode plot in Fig. 1(b). The Bode plot has two parts which show magnitude and phase lag, separately. Here, curve (i), (ii) and (iii) show the magnitudes of the loop gain, when the PI gain kp are 2, 8 and 20, respectively. For the phase plot, the phase lag of the PLL for different kp is same. As we known, the locking bandwidth of a closed loop system is determined by the frequency where the magnitude of the loop gain is zero. Therefore, the bandwidths of the PLL for curve (i), (ii) and (iii) in our simulation are about 300 Hz, 1 kHz, and 3 kHz, respectively. However, the bandwidth has an upper limit which is determined by the phase lag. This is because that, in engineering, it is required that phase lag is less than −135 degree when the amplitude plot crosses zero-dB at only one frequency (called upper limit bandwidth) [13,14,26]. If the locking bandwidth is above the upper limit bandwidth, the oscillation will be happened in the loop. In other words, the PLL will be in the unstable state, and any noise signal can break the loop. In our simulation, we found the upper limit bandwidth of the PLL is 3 kHz, because the phase lag is −135 degree at this frequency. Therefore, it is can be estimated that the PLLs with kp of 2 and 8 (shown as curves (i) and (ii)) are stable, and the PLL with kp of 20 (shown as curve (iii)) is unstable, because the bandwidth of curve (iii) is very close to its upper limit. To break the bandwidth limit of the laser-based PLL, one of the possible methods is to modify the PLL with an upgraded loop filter if the laser cavity is not allowed to be redesigned. The modified PLL with the upgraded loop filter will be introduced in the next subsection.
2.2 PLL with multiple-differentials-based loop filter for bandwidth improvement
In the last subsection, it is shown that the bandwidth of the PLL is limited by the extra phase lag which is introduced by the PZT and its HV driver in the high offset frequency. In this paper, a modified PLL with an upgraded loop filter is proposed to improve the bandwidth. With this proposed PLL, the extra phase lag can be compensated partially by a multiple-differentials-based loop filter, and consequently, the upper limit bandwidth of the loop can be increased significantly. The schematic of the modified PLL with the upgraded loop filter is shown in Fig. 2(a). The most of parts in the modified PLL is same as the standard PLL shown in Fig. 1(a), except the upgraded loop filter. In this new loop filter, we add two differentials in the previous PI controller, which can provide a phase compensation in a specific frequency section. In practice, the specific frequency section is determined by the cut-off frequencies of the two differentials which is designed elaborately. In addition, to evaluate the bandwidth improvement, a loop gain and bandwidth measurement technique reported in literature [23] is used to real-timely characterize the bandwidth of the modified loop.
Fig. 2 (a) Schematic of the modified PLL with the multiple-differentials-based loop filter for bandwidth improvement. (b) Function block of the modified PLL.
The function block of the modified PLL with the multiple-differentials-based loop filter is shown in Fig. 2(b). Here, the loop filter has a proportional, an integral and two differentials, and its transfer function F(s) is expressed as
where kp is the proportional gain of the loop filter, Ti is its integral time constant, TD1 and TD2 are the time constant of the two differentials. By inserting Eq. (4) into Eq. (1), the final loop gain of the PLL with the differentials-based loop filter is given bywhere k = 2πcnkokdkp. The loop gain presented in Eq. (5) shows that the PLL is also a third-order controlling system. Compared to Eq. (3), the two terms TD1 and TD2 are involved, which will benefit for the compensation of the phase lag introduced by the PZT and HV driver. Based on the Eq. (5), we simulated the loop gain G(s) of the PLL with the differentials-based loop filter. The parameters of the PLL in the simulation are same as above, and the cut-off frequencies of the two differentials are 10 kHz and 1 MHz, respectively. In this case, TD1 and TD2 are 10−4 s and 10−6 s.The simulation results as Bode plots are shown in Fig. 3. The curve in Fig. 3(a) is the loop gain of the PLL with typical PI-based loop filter, which has been demonstrated in Fig. 1(b), where the P gain of the loop filter kp is 20. From the magnitude plot, we found that the locking bandwidth of the PLL is about 3 kHz. However, the upper limit bandwidth of the PLL is also 3 kHz since the phase lag is less than −135 degree above 3 kHz. In this case, the loop is unstable, and there is a risk that the oscillation would be happened. The curve in Fig. 3 (b) is the loop gain of the PLL with differentials-based loop filter, where the P gain of the loop filter is also 20. Here, the magnitude plot is slightly different from the curve in Fig. 3 (a) at high offset frequency, and the locking bandwidth of the loop is still 3 kHz. From the phase plot, we found the upper limit bandwidth of the loop is almost above 100 kHz since the phase lag is always less than −135 degree from 100 Hz to 100 kHz. Therefore, it can be estimated that the loop with differentials-based loop filter is stable. From the simulation results, it demonstrates that the upper limit bandwidth of the PLL with the differentials-based loop filter has been increased significantly. However, in practice, the upper limit must be restricted by the resonance of the PZT, which is not shown in Fig. 3(b). This is because the resonance is determined by the physical structure of the PZT. Although it is difficult to model the structure precisely, the resonance of a PZT can be characterized by an actual test. In the next section, the upper limits bandwidths of a real PLL with and without differentials-based loop filter have been demonstrated.
Fig. 3 Simulation results of loop gain of the laser-based PLL with different loop filters. (a) For the typical PI loop filter without differentials. (b) For the modified loop filter with differentials
3. Experimental results
An experiment was conducted to measure the real loop gains and bandwidths of the modified laser-based PLL with and without differentials, respectively. Here, the laser is a passively polarization additive-pulse mode-locked (P-APM) erbium-doped fiber laser, and its P-APM configuration is similar to that presented in the works [23,28]. The repetition frequency of the MLL is 100 MHz, and we locked its 10th harmonic (1 GHz) to a highly-stable microwave generator (Agilent, E8257D). In our modified PLL, the loop filter (shown in Fig. 2) has a proportional, an integral, and two differentials. By switching S1, the loop filter can be selected as a typical PI controller or an upgraded controller with two differentials, and the two cut-off frequencies of the differentials of the loop filter are 10 kHz and 1 MHz, respectively. The parameters of the actual PLL are same as the simulation, which is as follow: n is 10, ko is 10−5 m/V, kd is 10−1 V/rad, Lo is c/(100 × 106 Hz), To is 10−4 s, Ti is 10−2 s, TD1 is 10−4 s, TD2 is 10−6 s and kp is 20. When the PLL was closed, the loop gain had been measured as Bode plot by the technique of loop gain measurement (S2 is active, shown in Fig. 2), which is presented in the literature [23].
The experimental results are shown in Fig. 4. Curves in Fig. 4(a) are the magnitude and phase lag plots of the PLL with the typical PI-based loop filter, where the differentials is not active and the P gain of the loop filter kp is 20. From the curves, we found the bandwidth of the PLL is 3 kHz, because the magnitude is less than zero when the offset frequency is above3 kHz. Here, the upper limit bandwidth is also 3 kHz, where the phase lag is less than −135 degree above 3 kHz, which is determined by the timing delay of the PZT and HV driver. In this case, the loop is not stable, and it has a high risk of being in the unlocked state if we increase the P gain of the loop filter. Curves in Fig. 4(b) are the magnitude and phase lag plots of the PLL with the upgraded loop filter, where the differentials is active and the P gain of the loop filter kp are 20 and 60, respectively. From the curves, we found, with the P gain of loop filter kp increasing from 20 to 60, the PLL is always stable, and the locking bandwidths of the PLL with the kp of 20 and 60 are 3 kHz and 10 kHz, respectively. This is attributed by the improvement of the upper limit bandwidth. From the phase plot, it shows that the upper limit bandwidth has been increased to be about 30 kHz, because the phase lag has been compensated by the differentials and is always larger than −135 degree below 30 kHz. It is estimated that the PLL will be stable, if we increase the locking bandwidth within 30 kHz by changing the P gain of loop filter continuously. Compared with Fig. 4(a) and (b), the experimental results prove that the locking bandwidth of the laser-based PLL with the multiple-differentials-based loop filter has been improved significantly. Especially, in our case, the upper limit bandwidth has been increased by about one order at offset frequency, from 3 kHz to 30 kHz. However, note that, although the differentials-based loop filter can compensate the extra phase lag introduced by the PZT and HV driver to improve the bandwidth, the upper limit is also restricted by the resonance of the PZT which is determined by its physical structure. Figure 4 shows there are strong hikes around 50 kHz in both figures due to the PZT resonance, which also results in the rapid drop of the phase lags. Therefore, it is hard to break the restriction of the PZT resonance and improve the bandwidth above resonance frequency. Nevertheless, the proposed scheme of PLL with multiple-differentials-based loop filter in this paper is an effective tool which can improve the bandwidth of the PLL within the resonance frequency.
Fig. 4 The experimental results for measurement of loop gain and bandwidth. (a) With typical loop filter without differentials; (b) With upgraded loop filter with multiple differentials.
Although the loop filter with differentials has been used in a lot of electronic feedback loops to optimize their stability, it is still a pioneer approach to increase the locking bandwidth in a laser-based PLL. Because the currently-used techniques for improving the bandwidth of the laser-based PLL focus on the optical methods. The scheme proposed in this paper can simplify the design of the bandwidth improvement without reconstruction of the laser, which can benefit for the rapid design of a highly-stable MLL. In addition, a feedback loop in a MLL is generally affected by the parameter mismatch, dynamical instability and amplitude phase coupling factor in a long-range control parameter [29,30]. However, in our laser-based PLL, the cavity adjustment is in a short-range control regime, and the locking bandwidth is just limited by the HV driver and PZT, not by the laser. Therefore, the electronic method demonstrated in this paper is more suitable to our case, compared to the optical/mechanical methods.
4. Conclusions
We demonstrate a scheme for improving the locking bandwidth of PLL for mode-locked laser. In this scheme, the standard PLL with PI-based loop filter has been modified to a new PLL with multiple-differentials-based loop filter. With the modified PLL, the upper limit bandwidth of the PLL is able to be increased significantly. The theoretical analysis and simulation of the modified PLL is presented in detail, and an experiment was also conducted to measure the bandwidths of a laser-based PLL with and without differentials-based loop filter. The experimental results show that the upper limit bandwidth of the modified PLL has been increased by about one order at offset frequency from 3 kHz to 30 kHz, which can benefit for the improvement of the locking bandwidth of the PLL in practice. Furthermore, there are other lasers system whose frequency should also be stabilized. For example, a CW laser or pulsed laser are needed to be stabilized by locking their optical frequencies to the ultra-stable optical or radio frequency reference. In these application, PLL is the most common technique to achieve the stabilization. Therefore, the scheme demonstrated in this paper can be easily applied to these laser-based PLL whose bandwidth should be improved.
Funding
National Natural Science Foundation of China (NSFC) (61871084 and 61601084).
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