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A simple and robust technique for measuring the loop gain and bandwidth of a phase-locking loop (PLL) for mode-locked laser is proposed. This technique can be used for the real-time measurement of the PLL’s real loop gain and bandwidth in a closed loop without breaking its locking state. The agreement of the experimental result and theoretical calculation proves the validity of the proposed technique for measuring the loop gain and bandwidth. This technique with a simple configuration can be easily expanded to other laser’s locking system whose loop gain and bandwidth should be measured in advance.
© 2016 Optical Society of America
1. Introduction
Mode-locked laser (MLL) [1] has received considerable attentions for years as it is widely used in metrology [2], frequency synthesis [3], optical atomic frequency standard [4], and timing distribution [5]. In these applications, one of the most important work is to lock the MLL to a highly-stable frequency reference, for maintaining its high-precise frequency and phase coherence. Generally, in order to obtain a high stability in locking a MLL, a phase-locked loop (PLL) should be built between the MLL and highly-stable frequency reference [6]. Many works have reported that highly-stable MLLs can be achieved by using the PLL with proportional-integral (PI) controller [7–19]. For a MLL-based PLL, its loop gain and bandwidth are two key parameters which shows how stable the laser is locked [16,17]. With the precise measurement of loop gain and bandwidth in a laser-based PLL, we can characterize the real timing jitter or phase noise of a femtosecond MLL [20–23]. We can also achieve the best photonic-microwave generator with ultra-low phase noise by adjusting the PLL’s loop gain and bandwidth [24,25]. Therefore, it is significant that the real loop gain and bandwidth of the laser-based PLL can be measured precisely. In principle, the loop gain and bandwidth can be calculated out based on the theoretical model of a laser-based PLL [16,17]. However, it is very difficult to directly measure the real loop gain and bandwidth of a laser-based PLL in an open loop, because a PLL system is usually a first or second order system, and in this case, the error signal in an open loop is very large and cannot be tracked.
In this paper, we propose a simple and robust technique to measure the loop gain and bandwidth of a MLL-based PLL. This technique can be used for real-time measurement of the loop gain of a phased-locked MLL without breaking its locking state.
2. Scheme for measuring the loop gain and bandwidth of PLL
A PLL [26,27] which is used to lock MLL is a control system allowing the repetition frequency of MLL to track with a frequency reference. It is possible to have a phase offset between MLL and frequency reference source, but when the MLL is locked, its frequency and phase must be exactly tracked. Its purpose is to force the MLL to replicate and track the frequency and phase of the input frequency reference when in locking.
Figure 1 shows the configuration of a PLL for locking a MLL. In this PLL, a phase-error detection circuit is used to extract the phase difference between a MLL and an electronic frequency reference source, and then lock the repetition frequency of the MLL to the reference source. A microwave signal from a fast photo detector is mixed with the reference source by a mixer or phase detector, and then a phase-error signal at the intermediate frequency (IF) is produced. A loop filter eliminates any unwanted higher frequency products, and the filtered phase-error signal continuously adjusts the voltage on the laser’s piezoelectric transducer (PZT) or the laser’s pump source to maintain quadrature between the MLL’s output signal and the reference source. In this case, the laser’s repetition frequency is locked to that of the reference source, and the relative phase fluctuations are reduced. This auto-adjusting locking technique realizes a PLL between MLL and reference source.
Fig. 1 Configuration of phase-locked loop for locking a MLL. VCO, voltage-controlled oscillator. PD, photo detector. PZT, piezoelectric transducer.
Three basic functional blocks in the configuration (shown in Fig. 1), phase detector, loop filter, and voltage-controlled oscillator (VCO) which consists of a PZT (or Pump) and MLL, form a PLL. Based on a theoretical analysis for this PLL (presented in [16, 17]), the functional blocks of the PLL is demonstrated as Fig. 2.
In this PLL configuration, MLL can be treated as a VCO. This is because the repetition frequency of the MLL is determined by the laser cavity length and pump power. We can use the PZT inserted into the cavity to change the total cavity length or use a tunable current source to change the pump power, for finely adjusting the laser’s repetition frequency. With the modeling of PZT-tuning VCO, its transfer function in complex domain can be expressed as [16],
where c is light velocity, n is the laser’s nth harmonic, ko is the PZT’s regulation gain, To is the PZT’s time-delay which limits the PZT’s bandwidth, and Lo is the laser’s cavity length. This equation presents the mathematic model of a PZT-tuning VCO, which is a second-order time-delay system. Note that for a fast PZT, its time-delay parameter To is small and can be omitted. Therefore, in the case of omitting To, the model is degraded to a first-order time-delay system. Generally, the fast PZT is used in a high-frequency harmonic-based PLL system, because it can provide a higher locking bandwidth.With the modeling of pump-tuning VCO, its transfer function in complex domain can be expressed as [17],
where c is light velocity, n is the laser’s nth harmonic, km is the linear coefficient between the refractive index and the current source, gm is defined as the coefficient between the control voltage of the current source and its output current, Lo is the laser’s cavity length, and nr is the average refractive index of the cavity. This equation presents the mathematic model of a pump-tuning VCO, which is a first-order time-delay system. Note that for a pump-tuning VCO, generally, its bandwidth is larger than a PZT-tuning VCO. This is because the speed of pump modulation is faster than a PZT.A phase detector in a PLL can generate a voltage signal which represents the difference in phase between the MLL and reference source. A loop filter with a PI controller is used to determine the loop dynamics [28]. It can eliminate the high order harmonics from the phase detector and limit the amount of reference frequency energy (ripple) appearing at the phase detector output. The transfer functions of the phase detector and loop filter are given respectively by,
where kd is the gain of phase detector, and F(s) is the transfer function of loop filter. In this paper, to simplify the analysis of our technique, we will just discuss the PZT-tuning PLL, and the analysis of PLL with tunable pump source is almost same [17]. Therefore, Based on Eq. (1) and (3), the loop gain of the PZT-tuning PLL can be obtained, which is shown in Fig. 2, and given by,where G(s) is the loop gain of the PLL for a locked MLL. The loop gain is an important parameter to determine the PLL’s locking ability for a locked MLL. In principle, G(s) can be calculated out theoretically. However, it is difficult to directly measure the real loop gain in an open PLL, because the open loop is a second order system (described in Eq. (4), and the loop gain is very large in low offset frequency, which increases the difficulty to detect any error signal in the open loop. In order to measure a real loop gain, there is only a chance if we can detect the error signal in a closed loop. Here, based on Eq. (4), we rewrite the loop gain G(s) = F(s)∙T(s), where T(s) = 2πcnKdKo/(s(Tos + 1)Lo2), which is called transfer function of MLL-based VCO and phase detector. Our scheme for measuring the loop gain in this paper is to first measure F(s) and T(s) individually, and then achieve the final G(s) by multiplying them. F(s) which is the transfer function of loop filter, can be measured directly. For measuring T(s), a simple technique is proposed here.The technique for measuring T(s) is shown in Fig. 3. Two new components, a low-pass filter and a 1:1 electronic adder, are inserted into the original PLL. The low-pass filter after the loop filer is used to limit the PLL’s locking bandwidth, and the adder is used to introduce a modulation signal which is from a network analyzer into the loop for gain measurement. When the PLL is in locking state, the modulation signal will be delivered to the adder, and the error signal after the phase detector will be monitored by the same network analyzer. With a Bode analysis between the modulation and error signal, a real loop gain (as a Bode plot) of the modulation signal after passing through the MLL-based VCO and phase detector can be achieved. Based on the measured gain plot of modulation signal, T(s) can be fitted out precisely. In the next section, we will theoretically analyze the mechanism of our technique for measuring T(s) and show the simulation result.
Fig. 3 Configuration of technique for measuring the transfer function of MLL-based VCO and phase detector.
3. Theoretical analysis and simulation
We simplify the configuration of this technique to a functional block (shown in Fig. 4). Assuming the new low-pass filer has a transfer function L(s), adder has a 1 by 1 input gain ratio, and network analyzer outputs a modulation signal Um(s), then we have,
where Ud(s) is the voltage signal to drive the PZT, and Ue(s) is the error voltage signal from the phase detector. Here, in order to calculate out the transfer function of VCO and phase detector, the gain of the modulation signal Um(s) passing though the VCO and phase detector in a closed loop should be obtained, which is described as M(s) = Ue(s)/Um(s) in Fig. 4.In a closed loop, the modulation signal is treated as a noise signal, because a PLL will stabilize the voltage at any points to a constant, for synchronizing the VCO’s phase to that of frequency reference. In our analysis, without loss of generality, we assume the fixed phase of frequency reference is zero, which is also convenient for us to calculate out the gain of modulation signal in the loop. With the assumption, we get
Based on Eq. (1), (5) and (6), the gain of modulation signal passing through VCO and phase detector (described as M(s)) can be calculated out as below,
where F(s) and L(s) is the transfer function of loop filter and transfer function of inserted low-pass filter respectively, which can be expressed as F(s) = kp(1/Tis + 1), and L(s) = 1/(τs + 1) [28,29], where kp is the gain of the loop filter, Ti is the integration time of the loop filter, and τ is the timing-delay of the low-pass filter. We put the two expressions F(s) and L(s) into Eq. (7), and then the final expression of M(s) is given by,Based on the theoretical M(s) demonstrated in Eq. (8), a simulation for M(s) is performed with the actual PLL parameters as follow: n is 35, ko is 10−5, kd is 10−2, Lo is c/(100 × 106 Hz), To is 10−3 s, kP is 1, Ti is 1 × 10−3 s, and τ is 5 × 10−3 s. The simulation result is shown as Curve (i) in Fig. 5. It shows that the simulated Bode plot Curve (i) has two different sections with different characteristics due to the bandwidth of the closed loop. For in the bandwidth, the Bode plot indicates how much the modulation signal is suppressed by the closed loop gain of PLL, because the PLL treats the modulation signal as a noise in the locking bandwidth. For out of bandwidth (When frequency is large), the Bode plot presents the gain of the modulation signal when it passes through the VCO and phase detector without the suppression of the gain of PLL. In this case, therefore, the gain of modulation is equal to the gain provided by the VCO and phase detector, which means M(s) in this section (out of bandwidth) is the same as T(s). Furthermore, this analysis can also be explained mathematically by simplifying the expression of M(s) in Eq. (7), when the offset frequency is too large to omit the term F(s)L(s). In this case (out of bandwidth), M(s) is degenerated to T(s) due to the approximate zero term F(s)L(s). Therefore, with the analysis above, T(s) can be fitted out via the Bode plot of M(s) out of bandwidth. In the next section, we will build an actual PLL for a mode-locked laser with the same parameters in the simulation, to experimentally verify our technique for measuring the loop gain.
Fig. 5 Simulation and measurement results of transfer function of MLL and phase detector. Curve (i): theoretical gain of modulation signal passed through the MLL and phase detector. Curve (ii): measured gain of modulation signal passed through the MLL and phase detector. Curve (ii): T(s) by fitting slope.
4. Experimental results and discussion
A real loop gain of an actual PLL for a MLL was measured with the proposed technique. 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 [30, 31]. The repetition frequency of the MLL is 100 MHz, and we locked its 35th harmonic (3.5 GHz) to a high-stable microwave generator (Agilent, E8257D). A PI controller in the loop filter (shown in Fig. 1) has one integration. When the loop was closed, we measured T(s) with our scheme, and the Bode plot measurement results are shown in Fig. 5.
Curve (i) in Fig. 5 is the theoretical gain of modulation signal passing through the VCO and phase detector, which is described as M(s) in Eq. (8) in last section. This curve has a corner, which indicates the locking bandwidth of the PLL after inserting the low-pass filter (shown in Fig. 3). In bandwidth, the modulation signal will be treated as a noise signal, and suppressed by the loop. The less frequency is, the stronger suppression is. This is because the loop has a larger gain in low offset frequency. For out of bandwidth, the loop will not act on the modulation signal, because the loop gain is not enough to affect the modulation signal in the loop. Therefore, curve (i) should coincide with T(s) out of locking bandwidth. Curve (ii) is the measured gain of modulation signal M(s) passing through the VCO and phase detector. Compared with Curve (i) and (ii), it demonstrates that both curves are nearly same, except a servo hike appeared around corner frequency in curve (ii). This obvious servo hike is mostly due to a big servo bump which introduced by our loop. We can estimate, in this offset frequency with servo pump, the gain of modulation signal is larger than 1 and the absolute value of phase lag is larger than to 135 degree simultaneously, which increases the gain of positive feedback in our measuring system slightly. This servo hikes, usually, is hard to be eliminated, because a feedback loop system always has a variable phase lag in its bandwidth. With fitting curve (ii) out of locking bandwidth, we got a real T(s) which is shown in Fig. 5 as curve (iii). Please note that the slope of Curve (iii) is −20 dB/Decade. This is also in agreement with the theoretical expression of T(s) = 2πcKdKo/s(Tos + 1)Lo2. In addition, curve (ii) has a strong fluctuation above 100 kHz due to the resonant frequency of PZT, which is determined by the PZT’s time-delay To.
After measuring T(s), we measured the transfer function of loop filter F(s) directly. By multiplying them, we got the final loop gain G(s) of the actual MLL-based PLL.
Figure 6 shows the measured final loop gain G(s) and bandwidth of the PLL. Curve (i) is the measured T(s) which is also shown in Fig. 5. Curve (ii) is the measured transfer function of the loop filter F(s). Curve (iii) is the loop gain G(s) by multiplying T(s) and F(s), and this curve indicates the bandwidth of the PLL is 1 kHz, because the loop gain is less than 1 above 1 kHz. For verifying the measured loop gain, a calculated loop gain is also demonstrated in Fig. 6 as curve (iv). In our case, the actual PLL has the following parameters: n is 35, ko is 10−5, kd is 10−2, Lo is c/(100 × 106 Hz), To is 10−3 s, kP is 1, Ti is 1 × 10−3 s, and τ is 5 × 10−3 s, which is same as that we used in simulation above. Here, the gain of loop filter is set to 1 and the integration cut-off frequency of the loop filter is ~200 Hz. It can be seen from Fig. 6 that the measured loop gain shown as curve (iii) is in agreement with the calculated plot shown as curve (iv). Therefore, the agreement of the two curves proves that our technique for measuring the loop gain is correct and reasonable.
5. Conclusions
In summary, a simple and robust technique in measuring the loop gain and bandwidth of a PLL for MLL is proposed in this paper. With this technique, a real transfer function of MLL-based VCO and phase detector T(s) can be measured precisely. By multiplying T(s) and transfer function of loop filter F(s), the real loop gain G(s) and bandwidth of the PLL can be achieved. The theoretical analysis and simulation of our technique for measuring loop gain is presented in detail, and an experiment is also carried out to measure the real loop gain. The agreement of measurement results and theoretical calculation proves that our proposed technique is correct and reliable. The scheme of this technique can be easily applied to other MLL-based locking systems whose loop gains should be measured in advance.
Funding
This work was supported in part by the Startup Foundation for Distinguished Scholars from the University of Electronic Science and Technology of China (No. ZYGX2016KYQD117), and the Basic Research Foundation from Institute of Electronic Engineering, Chinese Academy of Engineering Physics.
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