Theoretical and Experimental Analysis of Optical Frequency Combs Synchronized to a Microwave Reference Achieving 10⁻¹⁹-Level Additional Stability

Abstract

This paper presents a combined theoretical and experimental method for noise suppression in the repetition frequency (f_r) locking of erbium-doped fiber optical frequency combs (OFCs). This study proposed a novel mathematical model to bridge the noise relationship of f_r between the free-running and locked modes, and analyzed this relationship from two perspectives: the additional phase noise and the frequency stability. In addition, to integrate theoretical modeling with experimental validation, this study designed f_r locking strategy that uses a phase-locked loop (PLL) with PFD + PIID (a phase frequency detector and a proportional, first-order integer, second-order integer, first-order differential controller). Under synchronization of the f_r with a microwave reference (REF), this study achieved OFC additional frequency stabilities of 2.81 × 10⁻¹⁵@1 s and 8.08 × 10⁻¹⁹@10,000 s at 200 MHz fundamental frequency locking and 4.25 × 10⁻¹⁶@1 s and 1.91 × 10⁻¹⁹@10,000 s at 1200 MHz harmonic locking. The simulated and experimental results are in good agreement, confirming the consistency of the theoretical model and experiment. This work provides a reliable theoretical model that can be used to predict stability for OFC locking and significantly improves the additional frequency stability of OFCs.

1. Introduction

Since their groundbreaking realization in 2000 [1], optical frequency combs (OFCs) have demonstrated significant potential in various applications [2], including precision frequency measurement [3,4], time-frequency transfer [5,6,7,8], and optical frequency conversion [9,10]. These applications exploit the distinctive comb-like spectral properties of OFC, where each radio frequency (RF) or optical spectral line follows the formula f_n = f_ceo ± nf_r ± f_beat (with n being the comb tooth order, f_ceo the carrier-envelope offset frequency, f_r the repetition frequency, and f_beat the beat frequency) [11,12]. This attribute facilitates precise conversion between the RF and optical frequencies, with the precision heavily contingent upon the frequency stability of the OFCs. Therefore, enhancing the frequency stability is essential for accurate and reliable measurements, transfers, and conversions, directly impacting the precision and dependability of these applications. Improving the frequency stability of OFCs remains a key focus in ongoing research and technological advancements. The frequency stability of OFCs is typically achieved by synchronization with external microwave or optical references through a phase-locked loop (PLL). Optical references offer stability at 1 s from the 10⁻¹⁵ to 10⁻¹⁸ level and long-term stability at 1000 s to 10,000 s from the 10⁻¹⁷ to 10⁻²⁰ level [10,11,13,14,15,16,17,18,19,20], whereas microwave references offer a stability range from the 10⁻¹² to 10⁻¹⁴ level and long-term stability at the 1000 s to 10,000 s from the 10⁻¹⁵ to 10⁻¹⁷ level [11,13,14,21,22]. The high stability of the optical reference is essentially achieved by amplifying the phase gain through harmonic locking, where the phase gain for harmonics is nu/φ (u is the output voltage of the phase detector, φ is the phase difference between the reference and the OFCs, and the phase gain for the fundamental frequency is u/φ). Similarly, the frequency stability of OFCs utilizing a microwave reference source (REF) can also be enhanced by employing higher-order harmonic locking [8,11,14,22,23]. However, higher-order harmonic locking causes the system’s phase control range to decrease by a factor of n, leading to a reduction in the system’s anti-interference capability. Therefore, despite existing methods, effective strategies to further improve OFC frequency stability are limited.

This paper applies “First-Principles Thinking” to deconstruct and redesign the OFC frequency locking system, identifying key factors influencing stability. To achieve this, the system is initially broken down into independent components: the OFCs, the phase frequency detector (PFD), the phase error controller, and the REF. Based on the fundamental characteristics of these components, this study developed mathematical models that describe their behavior. Through an analysis of phase noise suppression factor (PSF) and stability suppression factor (SSF) in the Laplace domain, this study proposes a comprehensive mathematical model that uses measurable system noise as input to simulate the additional phase noise and stability of the OFC repetition frequency. Additionally, the derived PSF and SSF characterize the OFC noise suppression performance. By utilizing experimental and simulated results, the model and system design were iteratively optimized, significantly improving the repetition frequency stability of the OFCs. To enhance the frequency stability and noise suppression capabilities of existing OFC systems, this study ultimately devises a PLL frequency locking system based on a PFD + PIID (proportional, first-order integral, second-order integral, first-order differential phase error controller). The PFD provides a wide phase detection range of (−2π, 2π), which is four times greater than that of multiplier (e.g., mixer) and twice that of phase detectors [13,14,19,24], significantly expanding the phase detection range and improving the signal-to-noise ratio (SNR). Moreover, the PFD has low noise and excellent frequency response linearity, enabling more accurate phase change detection and more effective suppression of OFC noise.

The experimental results validate the mathematical models and system design. Over a 1 s measurement period, this study achieved OFC additional frequency stabilities of 2.81 × 10⁻¹⁵ at a fundamental frequency locking of 200 MHz and 4.25 × 10⁻¹⁶ at a harmonic locking of 1200 MHz. Over a 10,000 s measurement period, this study achieved additional frequency stabilities of 8.08 × 10⁻¹⁹ at 200 MHz and 1.91 × 10⁻¹⁹ at 1200 MHz. Compared with existing locking strategies, the stability is improved significantly, and the noise suppression effect was significantly enhanced. These experimental results are consistent with the theoretical predictions of the mathematical model, confirming the reliability of the proposed model in analyzing OFC locking. This research provides valuable guidance for optimizing OFC frequency locking strategies by reducing noise and improving stability, thereby enhancing the precision and reliability of OFCs and supporting broader and more precise applications in scientific and industrial fields, including but not limited to optical frequency measurement, time-frequency transfer, and optical frequency conversion.

2. Theoretical Model and Experimental Implement

2.1. Theoretical Model

As depicted in Figure 1a, the system in this study adopts a PFD + PIID locking strategy: The PFD converts the phase difference between f_r and the reference signal REF into a voltage signal representing the phase error. In the PIID phase error controller, the proportional component offers rapid phase error correction; the first-order integrator mitigates the residual steady-state error of the proportional component; the second-order integrator further reduces the residual error of the first-order integrator, enhancing the locking precision; and the first-order differential component enhances the dynamic response speed and suppresses high-frequency oscillation interference. As depicted in Figure 1b, the principle of OFCs can be referred to our previous work [20]. The PLL composed of PFD, PIID, and OFC forms the locking strategy of this paper. As shown in the Figure 2, it presents the experimental method for collecting data required for theoretical simulation.

2.1.1. Model of Additional Phase Noise and Frequency Stability

The background phase noise S_Free(f) of the OFCs and the equivalent background phase noise S_PDF(f) of the PFD in free-running mode can be obtained experimentally. Since S_Free(f) and S_PDF(f) are statistically independent, the result of the superposition of both phase noises is the direct sum. Therefore, according to Formulas (A7) and (A8) in Appendix A, the phase noise S_Locked(f) of OFCs in locked mode can be obtained as

$$S_{Locked}\left(f\right)=T^2\left(f\right)·\left(S_{Free}\right(f)+S_{PFD}(f)+S_{other}(f\left)\right)$$

(1)

The equivalent impact of the PFD background voltage noise on the phase noise of OFCs is denoted as S_PDF(f) (the equivalent background phase noise of the PFD), and S_other(f) is defined as the equivalent impact of other circuit noise. The derivation of T can be found in Appendix A.1.

After phase noise enters the locked loop, it is suppressed by PSF T²(f). Since the influences of the voltage noise of the PFD and other circuits are negligible compared with S_Free(f), as S_Free(f) >> (S_PDF(f) + S_other(f)), in the calculation of S_Locked (f), S_PDF(f) and S_other(f) can be neglected. Therefore, the following relationship holds

$$S_{Locked}\left(f\right)\approx T^2\left(f\right){·S}_{Free}\left(f\right)$$

(2)

Substitute Formula (2) into the relationship formula between Allan variance and phase noise [25]. The resulting relationship is obtained as

$${\sigma }_{Locked}^2=2{\int }_0^{\infty }{T}^2(f)·{\displaystyle \frac{f^2}{f_r^2}}{·S}_{Free}(f)·{\displaystyle \frac{{sin}^4\left(\pi f\tau \right)}{(\pi f\tau {)}^2}}df$$

(3)

where ${\sigma }_{Locked}^2$ is the Allan variance of OFCs in locked mode, and where τ is the sampling time interval.

According to the noise model of a mode-locked laser [25,26], the theoretical trend coincides with the measured trend, as illustrated by the curve “free comb” in Figure 3a. Thus, S_Free(f) decreases rapidly with increasing frequency offset f [25,26,27]. Moreover, because the period of f in sin⁴(πfτ) is 1/τ, it is a function that approximates a comb, thus, it can be approximated as sampling T² at intervals of Δf = 1/τ. Therefore, in Formula (3), the limits of integration can be approximated as (0, 1/τ). Meanwhile T² is a monotonically increasing function, as illustrated by the curves T² in Figure 3, therefore, according to the integral mean value theorem, the upper bound of ${\sigma }_{Locked}^2$ can be approximated as

$$\begin{array}{cc}\hfill {\sigma }_{Locked}^2& =2{\int }_0^{{\displaystyle \frac{1}{\tau }}}{T}^2\left({\displaystyle \frac{1}{\tau }}\right)·{\displaystyle \frac{f^2}{f_r^2}}{·S}_{Free}\left(f\right)·{\displaystyle \frac{{sin}^4\left(\pi f\tau \right)}{(\pi f\tau {)}^2}}df \hfill \\ & \le T^2\left({\displaystyle \frac{1}{\tau }}\right)·{\sigma }_{Free}^2\hfill \\ \hfill \mathrm{and}, {\sigma }_{Locked}^2& \approx T^2\left({\displaystyle \frac{1}{\tau }}\right)·{\sigma }_{Free}^2\hfill \end{array}$$

(4)

where ${\sigma }_{Locked}$ is the additional frequency stability of the OFCs in locked mode, and where ${\sigma }_{Free}$ is the frequency stability of the OFCs in free-running mode. As illustrated by the curve T in the Figure 3c, the SSF is defined as $T\left({\displaystyle \frac{1}{\tau }}\right)$, and SSF characterizes the ability of the locking system to suppress the noise represented by frequency stability of the OFCs in free-running mode.

The background voltage noise of the PFD also affects the phase noise and frequency stability of the OFCs in locked mode, meaning that the PFD’s voltage noise is converted into the corresponding equivalent background phase noise S_PDF(f) for the OFCs. Because the background noise of PFD is persistent, and not reduced by suppression of the locking system, the S_PDF(f) is directly measured by the DMM and FFT analyzer. Since S_Locked(f) and S_PDF(f) are statistically independent, the result of the superposition of the two phase noises is the direct sum. Therefore, the overall additional phase noise of the OFCs in locked mode is obtained as

$$S_{overall}\left(f\right)=T^2\left(f\right)·S_{Free}\left(f\right)+S_{PFD}\left(f\right)$$

(5)

By substituting Formula (5) into the relationship formula between the Allan variance and phase noise [25,28], the overall additional frequency stability ${\sigma }_{overall}$ of the OFCs in locked mode can be obtained as

$${\sigma }_{overall}=\sqrt{T^2({\displaystyle \frac{1}{\tau }})·{\sigma }_{Free}^2+{\sigma }_{PFD}^2}$$

(6)

where ${\sigma }_{PFD}$ is the equivalent background frequency stability of the PFD.

2.1.2. Model of the n-th Harmonic Locking

The frequency of the n-th harmonic is n times the fundamental frequency, and the phase variation is also amplified by n times. While G(s) >> 1, therefore, the phase error transfer function R_n(s) representing the output of the PFD can be approximated as [24]

$$\begin{array}{cc}\hfill {\displaystyle \frac{R_n\left(s\right)}{R\left(s\right)}}& ={\displaystyle \frac{1+G\left(s\right)}{1+nG\left(s\right)}}\hfill \\ & \approx {\displaystyle \frac{1}{n}}\hfill \end{array}$$

(7)

Thus, the additional phase noise in the n-th harmonic locking is obtained as

$$S_{n-overall}(f)={\displaystyle \frac{S_{overall}\left(f\right)}{n^2}}$$

(8)

The additional frequency stability of the n-th harmonic locking is obtained as

$${\sigma }_{n-overall}={\displaystyle \frac{{\sigma }_{overall}}{n}}$$

(9)

Finally, this section derives the theoretical model of OFCs in the free-running mode, that is, the mathematical relationships (Formulas (5) and (6)) between the background noise of OFC, and the PFD in the free-running mode and the additional phase noise or frequency stability of the f_r in locked mode. In addition, this model is extended to the noise model of OFCs in the harmonic locked mode (Formulas (8) and (9)). The theoretical model analysis in this section provides positive guidance for frequency locking strategies in experimental design. By iteratively optimizing the locking strategies of system through simulation and experimentation, the noise performance of OFCs has been improved, providing a theoretical basis for this improvement.

2.2. Experimental Implement

Figure 1 depicts the experimental setup for the f_r locking of OFC. As depicted in Figure 1b, this paper utilized an Er-fiber femtosecond laser, which is based on the mode-locking mechanism of a nonlinear amplification loop mirror (NALM). The EOM in Figure 1b is used for locking with the optical frequency reference, but it is not employed in the locking process with the microwave reference, which is why it is not connected in the diagram. More details on this can be found in our previous work [20]. The fundamental repetition frequency f_r of the OFCs is 200 MHz. Figure 1a illustrates the f_r locking system. The f_r from a PD of the OFCs and the REF were fed back into the PFD and LF, which generated phase error. The phase error is modulated by the PIID controller and then applied to the OFCs to adjust f_r. The f_r of the OFCs is fed back to the PFD through the BF, where it is phase-compared with the REF, forming a PLL. The DMM is the Truevolt 34461A(KEYSIGHT, Santa Rosa, CA, USA), which is employed to measure the PFD output voltage V_out. The calculations related to V_out and other experimental data are referred to Appendix A.2.

To analyze the phase noise and frequency stability of f_r, this study employs a combined experimental–computational approach based on the theoretical model established in preceding section, with the method illustrated in Figure 2, through the experimental and computational steps that follow:

Step 1: OFC background noise evaluation in the free-running mode

• Measured the OFC background phase noise S_Free(f) utilizing a phase noise meter and the corresponding background frequency stability ${\sigma }_{Free}$ utilizing a frequency meter (Figure 2a);
• Simulated phase noise S_Locked(f) and stability ${\sigma }_{Locked}$ using Formulas (2) and (4) in the locked mode.

Step 2: PFD background noise evaluation

• Connected both PFD inputs to REF and measured the output voltage with a DMM (Figure 2b);
• Simulated the equivalent PFD background phase noise S_PDF(f) and frequency stability ${\sigma }_{PFD}$ via Formulas (A10)–(A12).

Step 3: Overall System noise simulation

• Combine S_Free(f), ${\sigma }_{Free}$, S_PDF(f), and ${\sigma }_{PFD}$ in Formulas (5)–(6) to simulated overall additional phase noise S_overall(f) and frequency stability ${\sigma }_{overall}$;
• For n-th harmonic locking, simulated the overall additional phase noise S_n-overall(f) and frequency stability ${\sigma }_{n-overall}$ via Formulas (8)–(9).

According to the steps outlined above, this paper ultimately enables the simulation of the additional phase noise and frequency stability of the OFC, based on its noise data in the free-running mode.

3. Experimental Results and Discussion

First, following the methodology detailed in the preceding section, this paper measured the background phase noise and corresponding frequency stability of the OFCs (Figure 3 “free comb”) and the PFD (Figure 3 “PFD”) in the free-running mode. The phase noise S_Free(f) at the 1 Hz offset for the 200 MHz fundamental frequency (Figure 3a “free comb”) and the 1200 MHz harmonic (Figure 3b “free comb”) were measured to be −39.8 dBc/Hz and −60.3 dBc/Hz, respectively. The frequency stability ${\sigma }_{Free}$ was determined to be 4.50 × 10⁻¹⁰ at τ = 1 s (Figure 3c “free comb”). Uncertainties in the measurement results attributable to noise, the environment, equipment, and other interferences during the measurement process prevent the phase noise of the two systems from differing strictly by a factor of n². As illustrated by the curve “free comb” in Figure 3c, the frequency stability increases with τ, primarily because temperature variations induce frequency drift in the OFCs. As illustrated by the curves “locked comb” in Figure 3, the simulated results S_Locked(f) demonstrate the suppression of the OFC’s background noise S_Free(f) (Figure 3 “free comb”) by the locking system, and the additional phase noise was −169.7 dBc/Hz at the 1 Hz offset for the 200 MHz fundamental frequency and the additional frequency stability ${\sigma }_{Locked}$ was 1.01 × 10⁻¹⁵ at τ = 1 s.

As illustrated by the curves “PFD” in Figure 3, the equivalent background phase noise S_PFD(f) was −154.8 dBc/Hz at the 1 Hz offset for the 200 MHz fundamental frequency and the equivalent frequency stability ${\sigma }_{PFD}$ was 2.23 × 10⁻¹⁵ at τ = 1 s. This represents the equivalent impact of the PFD’s background noise on the phase noise and frequency stability when the input signal frequency is 200 MHz. The frequency distribution is relatively uniform, predominantly consisting of white noise.

As illustrated by the curves “simulation” in Figure 3, for the 200 MHz fundamental frequency locking, the simulated overall additional phase noise S_overall(f) at the 1 Hz offset was −154.8 dBc/Hz, and the simulated overall additional frequency stability ${\sigma }_{overall}$ was 3.23 × 10⁻¹⁵ at τ = 1 s and 2.73 × 10⁻¹⁸ at τ = 10,000 s. For the 1200 MHz harmonic locking, the simulated additional phase noise S_n-overall(f) at the 1 Hz offset was −170.4 dBc/Hz, and the simulated additional frequency stability ${\sigma }_{n-overall}$ was 5.39 × 10⁻¹⁶ at τ = 1 s and 4.55 × 10⁻¹⁹ at τ = 10,000 s. From the simulated results, for the system in this paper, it can be concluded that the impact of PFD noise on the measurement results is more significant.

Finally, utilizing the experimental setup depicted in Figure 1 and the methodology detailed in the preceding section, the experimental results of the curves “experiment” are presented in Figure 3. For the 200 MHz fundamental frequency locking, the overall additional phase noise at the 1 Hz offset was measured to be −168.7 dBc/Hz, and the overall additional frequency stability was 2.81 × 10⁻¹⁵ at τ = 1 s and 8.08 × 10⁻¹⁹ at τ = 10,000 s. For the 1200 MHz harmonic locking, the overall additional phase noise at the 1 Hz offset was measured to be −182.3 dBc/Hz, and the overall additional frequency stability was 4.25 × 10⁻¹⁶ at τ = 1 s and 1.91 × 10⁻¹⁹ at τ = 10,000 s. As shown in

Table 1. Comparison of phase noise improvement.

Locking Strategy	1 Hz	1 Hz (Harmonic)
PFD + PIID	−168.7 dBc/Hz	−182.3 dBc/Hz Δ
Multiplier + PI [22]	−96 dBc/Hz	−138 dBc/Hz *
Improvement ratio	72.7 dB	44.3 dB

^Δ 1200 MHz harmonic locking; * 3600 MHz harmonic locking.

and

Table 2. Comparison of frequency stability improvement.

Locking Strategy	τ = 1 s	τ = 8000 s	τ = 1 s (Harmonic)	τ = 8000 s (Harmonic)
PFD + PIID	2.81 × 10−15	2.72 × 10−19	4.25 × 10−16 Δ	2.80 × 10−19 Δ
Multiplier + PI [22]	4.54 × 10−14	1.67 × 10−16	1.46 × 10−14 *	4.85 × 10−17 *
Improvement ratio	16.2	614.0	34.4	173.2

^Δ 1200 MHz harmonic locking; * 3600 MHz harmonic locking.

, compared with the commonly used Multiplier + PI (proportional–integral) locking strategies, employing a PFD + PIID achieves lower additional phase noise and improving frequency stability. Furthermore, for the n-th harmonic locking, the experimental results approximate a factor of 1/n compared to the fundamental frequency locking described in the preceding section.

To validate the theoretical model and experimental methods presented in this paper, the simulated results were compared with the experimental results. As illustrated in Figure 3, the theoretical and experimental results are generally consistent. However, due to measurement noise introduced by factors such as the testing environment, equipment, and power supply, a slight discrepancy in the phase noise results remains. In contrast, the measurement of frequency stability benefits from the averaging of a large amount of data during the calculation process, which helps suppress measurement noise, resulting in better consistency between the simulated and experimental results.

4. Conclusions

In summary, this paper introduces a low-noise, high-stability frequency locking method for OFCs by employing a PFD + PIID as the phase error comparator and controller. This approach effectively suppresses the OFC noise and ensures synchronization with the REF, particularly in harmonic locking scenarios. This paper develops a theoretical model based on Laplace transforms and PLL feedback control principles to derive PSF and SSF, utilizing OFC noise in free-running mode as input to predict experimental results. A comparative analysis between the simulated and experimental results reveals good agreement, validating the proposed method and model. The analysis indicates that beyond higher harmonic locking, the keys to reducing OFC noise and improving frequency stability lie in the use of low-noise phase detection components and the design of the controller. Moreover, minimizing environmental interference and system noise through temperature control, vibration control, and other methods is crucial for reducing noise in free-running OFCs. Future work will focus on continuously optimizing the OFCs and locking system, further enhancing the frequency stability of OFCs by improving noise suppression capabilities, reducing background noise of OFCs, vibration isolation, and enhancing environmental stability. These advancements will broaden the applicability of OFC technology in precision frequency measurement, time–frequency transfer, and optical frequency conversion.