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Abstract
In this paper, we present a real-time measurement technology for the free spectral range (FSR) of an ultrahigh-aspect-ratio silicon nitride (Si3N4) waveguide ring resonator (WRR). Two different correlated resonant modes were tracked by two optical single-sideband frequency-shifted lights to eliminate interference noise in the Pound–Drever–Hall error signals. A relative precision of 0.1474 ppm was achieved for a 35 mm WRR with FSR = 1,844,944.5 kHz and finesse (F) = 13.2. Furthermore, a cross-correlation of 0.913 between FSR-calculated and thermistor-measured temperatures indicated a high correlation between the real-time FSR and room temperature. We believe this technology is currently the best way to realize low-finesse (F < 50) real-time FSR measurements in the GHz range.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Optical dispersion information is widely used in many fields of photonics [1–3]. Dispersion properties are typical measured using standard white-light interferometric methods. WRR-based tests are employed to obtain more precise and definitive measurements of dispersion properties [4]. In addition to the Q-factor and mode volume, the free spectral range (FSR) is a parameter that affects intracavity dispersion. Precise knowledge of the FSR plays an important role in numerous photonics developments, such as passive resonant gyroscopes [5,6], optical frequency combs [7,8] refractive index and thermo-optic coefficient measurement [9], and sub-Doppler molecular spectroscopy [2].
The FSR can be measured using various methods. One common method is based on a tunable laser locking system, which incorporates a high-accuracy optical wavelength meter. In addition, acousto-/electro-optic modulations [10,11] and optical frequency combs [12] have been utilized for FSR measurements in the last decade. However, most studies have focused more on precision than on real-time requirements. Owing to the integration of the optical inertial method and microwave photonic systems, the demand for real-time FSR measurement has continued to increase [13]. For example, in a passive resonant gyroscope, the diameter of the WRR must be maintained at the centimeter-level to achieve a certain Sagnac effect sensitivity [14,15]. At such a WRR-diameter scale, the resonant frequency drift caused by temperature fluctuations cannot be ignored. To evaluate the effects of temperature fluctuations, the FSR drift must be detected in real-time with Hertz-level precision [16]. In addition, precise measurement of FSR drift can also improve the accuracy of modulation frequency, thereby improving the detection sensitivity of the gyroscope. However, the precision of FSR measurement heavily depends on the WRR finesse. Higher WRR finesse helps obtain better precision of FSR drifts. Limited by the intracavity high polarization-dependent loss requirement and the resulting propagation losses, the WRR finesse values are far from reaching the ideal value designed [16–18]. In the current study, we focused on obtaining real-time FSR information with high precision from such low-finesse WRR.
Apart from WRR finesse, the narrow laser linewidth, obtained by a narrow laser control bandwidth (LCB), facilitates high-precision FSR measurements. In [16], the master resonance-tracking loop (RTL) was developed, as a direct laser-locking method; however, this method requires a large LCB to ensure adequate resonance tracking bandwidth (RTB) and to track the resonant mode drift in real time. As such, we locked the laser directly to the WRR. In that case, the RTB was equal to the LCB. Expanding the LCB enlarges the laser linewidth and deteriorates the precision of resonance tracking. In a direct laser-locking method, achieving a perfect balance between narrow linewidth (narrow LCB required) and resonance tracking ability (the LCB should be larger than the required RTB) is difficult. The additional random walks, caused by the correlation between linewidth and resonance tracking ability, are more divergent than the FSR random walk, and hinder the improvement in the FSR-tracking performance. The optical single-sideband (OSSB) method provides a way to nullify the correlation between the LCB and the RTB [19], helps reduce the correlation between linewidth and resonance tracking ability, and enables the analysis of the LCB and the RTB separately.
In this paper, we present a real-time FSR measurement system realized using a fiber-based, hybrid, OSSB Mach–Zehnder modulator (MZM) based on a correlated dual-OSSB resonance tracking method. This method isolates laser control and resonance tracking to ensure a narrow LCB and a wide RTB, and to reduce laser linewidth deterioration. The correlated dual-OSSB resonance tracking method utilizes two correlated OSSB frequency-shifted lights to track two resonant modes of a WRR by using the Pound-Drever-Hall (PDH) method. A correlated radio-frequency signal generator (CO-RF SG) is used to synthesize two phase-correlated microwave signals for driving the OSSB modulators. In addition to the reduction of laser linewidth deterioration, the tracking systematic errors reduce as the correlation between the tracking transient responses of the resonance mode and the FSR is reduced, thereby enabling real-time high-precision measurement. Two sources of optical interference noise affect the PDH error signals—: one is due to the phase difference between the original Path 1 and Path 2 carrier light waves, and the other is caused by the unutilized optical sideband located in the untracked resonant peak. The systematic errors of the resonant frequency judgment occur due to the demodulation of desired and undesired PDH error signals. These interference noise problems can be solved using the correlated dual-OSSB resonance tracking technology in which the DC accuracy and the OSSB-shifted frequencies are determined and analyzed.
2. Principle
The schematic of the proposed real-time FSR measurement system comprising a WRR, is illustrated in Fig. 1. The fiber-based, hybrid OSSB MZM is located on the light path next to the optical carrier generator (a tunable narrow-linewidth DFB fiber laser (T-DFB laser) with a central wavelength of 1550 nm and piezoelectric tuning capability). Each grandparent arm (i.e., Path 1 and Path 2) comprises an OSSB modulator with two parallel child MZIs located on each parent arm, a phase modulator (PM), and an erbium-doped fiber amplifier (EDFA) for light power adjustment. The OSSB frequency-shifted lights are generated using the method presented in the literature [16]. Light from the T-DFB laser is split equally into two waves at point ① in Fig. 1, where both waves are modulated as OSSB frequency-shifted signals by using the OSSB modulators. The relative optical phase between the parent arms in Path 1 (i.e., ${\varphi _{pa}}$ in Fig. 1) is set as $\pi /2$ by adjusting the bias voltage through DC control, as shown in Fig. 2(b). Similarly, ${\varphi _{pa}}$ of Path 2 is set as $- \pi /2$. Thus, the main power of Path 1 and Path 2 waves move to the lower and upper sidebands [20], respectively. Each light wave at point ② is modulated using different single-frequency signals (i.e., Ω1 and Ω2) to separate the WRR-dropped intensity of Path 2 wave from Path 1 wave, as shown in Fig. 2(c)-(d). Ω1 and Ω2 should be meticulously selected to avoid the beats formed by the image and higher-order frequencies. Next, the phase samples of modes $\ell $ and $\ell + \Delta \ell $ are reflected by different PDH error signals at Ω1 and Ω2, respectively. The EDFAs are used to ensure that the light power of Path 1 is close to that of Path 2 at point ③. Finally, the utilized sidebands of Path 1 and Path 2 waves are tuned and centered at the $\ell $-th and $\ell + \Delta \ell $-th resonant modes of the WRR, respectively, by using the PDH method. Then, the FSR is calculated in real-time as follows:
where ${\omega _{OSSB1}}$ and ${\omega _{OSSB2}}$ denote the angular frequency of the microwave signals and $\Delta \ell $ denotes the mode number difference between the tracked modes.
Fig. 1. Schematic diagram of the real-time FSR measurement system. Top: structure of the modulation path; Bottom: experimental setup. (MZI: Mach-Zehnder interferometer; PD: photodetector; C1 and C2: evanescent wave couplers; Demod: demodulation module; T-DFB Laser: tunable distributed feedback fiber laser; PM: phase modulator; EDFA: erbium doped fiber amplifier; WRR: waveguide ring resonator).
Fig. 2. (a) WRR resonant modes for propagating lasers, (b) the spectrum of the Path 1 wave at point ②, (c) and (d) show the simplified spectra of the Path1 and Path 2 waves at point ③ when the resonant modes are tracked, (e) the simplified spectra of the undesired double-sideband resonance state (OCSR, optical carrier to sideband power ratio).
2.1 Interference noise
To obtain a high-precision PDH error, two types of interference noises should be suppressed: (1) interference noise, as a function of the phase difference ${\varphi _{GP}}$ between the two paths, formed by the original carrier light waves, as shown in Fig. 2; and (2) interference noise caused due to the unutilized sideband power located in the untracked resonant peak, that is, the undesired double-sideband resonance (DSR) state, as shown in Fig. 2(e). In the DSR state, the final PDH error of Path 1 is the sum of the two PDH errors caused by the tracked and untracked resonant modes. The signal obtained from $\omega + {\omega _{OSSB1}} - {\omega _{\ell + \Delta \ell - 1}}$ might be mistaken for the difference between $\omega - {\omega _{OSSB1}}$ and ${\omega _\ell }$; thus, the tracking result of Path 1 (and Path 2) would not be the expected one.
Because both types of interference noises are determined by the energy of the carrier and unutilized sideband, the stabilization and improvement of the optical carrier-to-sideband power ratio (OCSR) [21] in Fig. 2(b) effectively suppress the interference phenomena.
The maximum deviation of PDH error at Ω1 influenced by the two aforementioned types of interference noise when $\Delta {\omega _{OSSB1}} \equiv {\omega _\ell } - ({\omega - {\omega_{OSSB1}}} )= 0$ is shown in Fig. 3. As the accuracy of the DC control increases, and a higher OCSR is achieved, the suppression of the interference noise gets enhanced. ${\omega _{OSSB1}}$, ${\omega _{OSSB2}}$ and $|{{\omega_{OSSB1}} - {\omega_{OSSB2}}} |$ are selected as fractional multiples of $2\pi \cdot FSR$ to achieve a stable suppression. The effect of ${\varphi _{GP}}$ on PDH error reduces greatly when ${\varphi _{GP}}$ is in the selectable range shown in Fig. 3. Consequently, for an OSCR of around -30 dB, the resonant frequency tracking error caused by interference noise is less than ${10^{ - 6}} \times FWHM$, and so is the FSR tracking error. Fortunately, we require a DC accuracy of only 1% (i.e., ∼0.1 V accuracy) to achieve an OSCR of -30 dB, which is not a difficult requirement.
Fig. 3. Maximum deviation of the master PDH error when ΔωOSSB1 = 0 (Normalized to full width at half maximum (FWHM)), (a) 10% DC accuracy, (b) 1% DC accuracy.
2.2 Correlated RF signal generator
Another important key to ensuring the precision of the proposed system shown in Fig. 1 is the correlated microwave signals at ${\omega _{OSSB1}}$ and ${\omega _{OSSB2}}$, which are synthesized by the CO-RF SG as shown in Fig. 3(a). The CO-RF SG separates the LCB and the RTB. In addition, it is used to expand the tracking bandwidth, stabilize phase noises, and repress additional random walks. We used a low-phase noise translation loop (LPNTL) synthesizer as the core of the CO-RF SG to reduce the amplification of phase noise by the multipliers and to ensure that both microwave signals have a similar phase noise spectrum. The compromised phase-locked loop (PLL) bandwidth is determined by a trade-off between additive noise and phase noise to minimize the total phase jitter. The continuous-phase signals, ${\hat{\omega }_{ref1}}$ and ${\hat{\omega }_{ref2}}$ in Fig. 4, formed by the direct digital synthesizer, are set as the reference signals of the LPNTL synthesizer.
Fig. 4. Schematic diagram of the CO-RF SG (VCO: voltage control oscillator; DDS: direct digital synthesizer; TCXO: temperature compensated crystal oscillator; LF: loop filter; LPF: low pass filter; 1/Nx: divide-by-Nx dividers; ×Mx: times-Mx multipliers).
If a frequency divider exists in the closed loop of a PLL, the PLL is equivalent to a frequency multiplier. As such, the phase noise at the input end gets amplified by the PLL. Therefore, the noise source model of the CO-RF SG system can be obtained, as shown in Fig. 5.
Oscillator noise is dominant in the generator. Assuming that the other elements are ideal, only the phase noise of the oscillator is extracted separately. Accordingly, the output of the RF signal generator with phase noise can be obtained as follows:
2.3 Tracking analysis
A basic block diagram of the FSR measurement system is illustrated in Fig. 6. In the FSR ramp loop, a second-order integrator (F2(s) in Fig. 6) was used to track the FSR ramp. Owing to this design feature, FSR measurement is independent of the laser control and has a weak correlation with the laser transient response. The simulation results of the transient response of the real-time FSR tracking system in the case of an out-of-phase sideband are shown in Fig. 7. The bandwidth of the system is designed as 100 kHz. As can be seen from Fig. 7, the stability time of the system is less than 20 µs with the FSR step at 1 MHz and the FSR ramp signal input at 100 MHz/s. Furthermore, when the laser frequency has the same variation, the system remains very stable within 20 µs.
Fig. 6. Basic block diagram of the real-time FSR measurement system (EFD, equivalent frequency detector; PD, phase detector; ε, frequency; θ, phase noise).
Fig. 7. Transient response analysis. (a) FSR offset; (b) FSR ramp; (c) Laser frequency offset; (d) Laser frequency ramp.
3. Experiment results
3.1 Phase noise and stability
In order to evaluate the effect of the RF signal output by the relevant RF signal generator on the laser phase noise, the line width and phase noise of the DFB fiber laser were tested, as shown in Fig. 8. The phase noises of $\omega $, ${\omega _{OSSB1}}$ and ${\omega _{OSSB2}}$ are consistent within the tracking bandwidth. Therefore, we can obtain two utilized OSSB sidebands with a similar phase noise spectrum and no phase noise deterioration. Owing to this feature, the signal to noise ratio of the PDH signals would not deteriorate with the OSSB frequency shift. The laser linewidth is measured to be less than 1kHz and is plotted in the inset of Fig. 8.
Phase noise is generally used to assess the short-term noise of the oscillator. Because the frequency of the oscillator under less than 10 Hz frequency drifts, the phase noise spectrum cannot meet the requirements. In this case, for long-term frequency stability, time-domain evaluation methods are used; Allan variance is the most commonly used method for time-domain evaluation. As can be seen from the long-term stability of the RF signal generator shown in Fig. 9, when the integral time was approximately 50s, the level value of Allan standard deviation appeared (∼120 Hz). Furthermore, the stability results of ${\omega _{OSSB1}}$ and ${\omega _{OSSB2}}$ revealed that the optimal theoretical precision of FSR measurement is approximately 14 Hz.
Fig. 9. Stability measurement of ωOSSB1 and ωOSSB2. (a) Waterfall Plots (RBW = 100 Hz), (b) Allan deviation.
3.2 Real-time FSR measurement considering temperature fluctuation
The OSSB modulators (FTM7962EP) have an optical bandwidth of 22.5 GHz and yield the maximum Vπ at the 3.5-V RF port and 14-V DC ports. A chip-based, single-mode, ultrahigh-aspect-ratio [22–24] silicon nitride (Si3N4) WRR with a high polarization-dependent loss and a bend radius of 17.5 mm served as the optical device-under-test. To evaluate the Q factor of the WRR, the OSSB light frequency was swept as the laser frequency was linearly changed using a certain microwave frequency, as shown in Fig. 10. The finesse and Q factor of the Si3N4 WRR were measured as 13.2 and 1.36 × 106, respectively. To avoid spectrum aliasing, the frequencies of the phase modulation tone, Ω1 and Ω2, were selected as 17.77 MHz and 14.44 MHz, respectively [16]. The FSR of the sample was measured in real time by tracking the resonance at approximately 16.6 GHz (i.e., $\Delta \ell = 9$), and the initial frequencies of ${\omega _{OSSB1}}$ and ${\omega _{OSSB2}}$ were set as 6.9 GHz and 9.7 GHz, respectively.
Fig. 10. (a) Resonance plot of the Si3N4 WRR; (b) The time trace of the measured FSR (VBW = 100 Hz); (c) Short-term tracking performances (VBW = 1 Hz).
After modes $\ell $ and $\ell + \Delta \ell $ were locked by the utilized sidebands, we recorded the multiplier factors N0 and M0 in Fig. 4 and used them to calculate the real-time FSR. The real-time FSR results are presented in the Fig. 10(b). As can be seen from a short-term comparison between this work and the previous one shown in the Fig. 10(c) [16], the stability of this work is better. The short-term measurement standard deviation reduced from 6.3 kHz to 1.37 kHz, and the short-term random walk frequency modulation noise decreased from 6.6 × 103$\textrm{H}{\textrm{z}^2}\textrm{/Hz}$ to 162 $\textrm{H}{\textrm{z}^2}\textrm{/Hz}$. We plotted the overlapping Allan deviation and noise analysis by using lag1 autocorrelation (Fig. 11(a)). The FSR was experimentally measured at 1,844,944.5 ± 0.27 kHz, hence with a relative precision of 0.1474 ppm.
Fig. 11. (a) Stability and noise type of the real-time FSR measurement system. (PM: phase modulation; FM: frequency modulation; W-: white; F-: flicker; RW-: random walk; FW-: flicker walk); (b) and (c) The time traces of the measured air and FSR-calculated temperatures with different averaging times.
In addition, parallel lines with logarithmic scale data proportional to ${\tau ^{\textrm{ - }1}}$ were used to evaluate F-PM noise representing the short-term uncertainty of FSR tracking when $\tau \textrm{ = }1s$, F-PM noise was less than 500 Hz/Hz1/2. Furthermore, to characterize the effect of temperature fluctuation on F-PM noise, two mean time lengths were selected as test samples according to Allan standard deviation. The FSR data were compared with the recorded data of the PT-100 thermistor thermometer on the selected mean time length, and the accuracy of FSR measurement in the low temperature range was evaluated. The FSR temperature expansion coefficient used in the calculation was -16.735 kHz/°C [16]. As can be seen, from the test results displayed in Fig. 11(b), F-PM noise dominated the FSR data. The normalized cross-correlation between the FSR calculated data and PT-100 data was 0.43, showing a positive correlation between them. When the average time was 100s, the FPM noise in the calculated FSR reduced (Fig. 11(c)), thus, demonstrating the effects of FFM noise. The correlation between the calculated data of FSR and the measured data of PT-100 improved significantly, and the normalized correlation coefficient reached 0.9. At this time, the two groups of data were highly correlated. This shows that the Allan standard deviation minimum is due to the introduction of FSR’s flicker noise and that the measurement accuracy of the system is reliable.
We compared the relative precision achieved using the proposed method with that reported in the literature [25–28]. As can be seen from the state-of-the-art relative precision of several FSR measurement systems (real-time and offline) by using WRRs with different finesses displayed in Fig. 7, the proposed method yielded the best real-time performance in the GHz range, which is currently the best way to realize low-finesse (F < 50) real-time FSR measurements in the GHz range. The relative precision for FSR and FWHM were at least 1–2 orders of magnitude smaller than those of other real-time methods [29,30].
4. Conclusion
In conclusion, we presented the correlated dual-OSSB resonance tracking method for real-time FSR measurement in a low-finesse (13.2) ultra-high aspect ratio Si3N4 WRR. A theoretical model of the system was established, and the two types of interference features were carefully analyzed. The correlated OSSB frequency-shifted lights were produced by the correlated RF signals synthesized via the CO-RF SG. A relative precision of 0.1474 ppm was achieved in the real-time FSR measurement, thus, demonstrating that the proposed technology is suitable for optical resonance cavities with a microwave-rate FSR. Furthermore, we found that the power of the interference noise and the stability of the correlated microwave signals limit the measurement precision. A more accurate DC voltage and a more stable microwave frequency will enable more precise real-time FSR measurements in the future. This technology will influence interdisciplinary applications requiring real-time FSR measurement.
In addition, we should also note that high propagation loss is the most important reason why the finesse of the resonator cannot reach its ideal value. This requires optimizing the manufacturing process of the waveguide, designing the optimal process with the use of thermal oxidation, low pressure chemical vapor deposition (LPCVD) and electron beam lithography (EBL) technology and so on [31,32], improving the optical properties of the material, reducing the scattering and absorption loss of the waveguide, and ultimately improving the finesse of the resonator.
Funding
the National key research and development program (2021YFB3201600).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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