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Abstract
With the developments of the tunable laser source (TLS), there are increasing demands for high-resolution dynamic wavelength calibration in recent years. Considering mutual constraints between wide measurement range and high calibration resolution, we propose a dynamic wavelength calibration method based on an auxiliary Mach-Zehnder interferometer (MZI) and the synchrosqueezed wavelet transform (SSWT). Our proposed method can achieve a calibration resolution of 5 fm and a tuning range of 10 nm. Moreover, the measurement range and spatial resolution of the optical frequency domain reflectometer (OFDR) system are improved to ∼80 m and ∼mm, respectively. Our proposed approach can substantially reduce the subtle spectrum distortion (tens of fm) in coherent optical spectrum analyzer (COSA) systems.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Tunable laser sources (TLS) have manifested the ability to extend coherent detections from single-point measurements to time-series analyses. With a foreknown time-wavelength tuning curve, the interferometric signals at different frequencies could map to particular moments. And the enhancement of the multiplexing capability envisages various applications such as the optical frequency domain reflectometer (OFDR) [1–4], coherent optical spectrum analyzer (COSA) [5,6], swept-source optical coherence tomography (SSOCT) [7], frequency-modulated continuous wave laser-radar (LADAR) [8], etc. It is clear that in these uses, the accuracy of the time series decoupling depends on the resolution of the wavelength calibration. But for most commercially available wavemeters, the resolution and measuring speed are limited, leading them insufficient to calibrate the time-wavelength curves of the TLS. As a result, the residual tuning nonlinearity will distort the responses to each detection variable, and the aliasing between adjacent variations will degrade the accuracy of the whole system [2,8–11]. The existing defects motivate the exploration of the dynamic wavelength calibration methods with high wavelength resolution.
A common method for dynamic wavelength calibration is to sample the interferometric signals at equal frequency intervals. In 2005, Brian J. Soller et al. designed an auxiliary interferometer whose zero-crossing points could serve as the markers for specific phases, so the instantaneous wavelength of the TLS could be linearized by monitoring the output pulses of the interferometer [9]. On that basis, Fabry–Perot etalon [10] and loop structure [11] were utilized to sharpen the transmission spectrum of the interferometer, improving the resolution to sub-pm. Nevertheless, these approaches still cannot resolve the phases between zero-crossing points, so the reachable resolution of the dynamic wavelength calibration is impeded. To address this problem, T.-J. Ahn et al. utilized the Hilbert transform to produce a π/2 phase shift in the interferometric signal, and estimated the phases between zero-crossing points from the ratio of the phase-shifted signal to the original signal [12]. Eric D. Moore et al. shifted the signal through digital filtering and Fourier transform, so the phases could be demodulated from the complex exponential of the inverse Fourier transform [13]. While improving resolution, these methods suffer from the value range of the arctangent function (varying from -π to π). Phase changes exceeding 2π need additional unwrapping algorithms to demodulate, deepening the systematic dependence on the high sampling rate and high signal-to-noise ratio (SNR).
After that, calibration methods based on envelope detection [14], deskew filter [15], and concatenate-generated phase [16] were successively proposed to avoid phase-unwrapping, but most of them could only support the calibration within a tuning range of several GHz (corresponding to tens of pm at 1550 nm). Zheyi Yao et al. reported a compact calibration setup based on the short-time Fourier transform (STFT) [17]. This approach avoided adaptive hardware and electronic filters. Therefore, its calibration range could be extended indefinitely (as long as the computation burden and the time consumption are affordable). Differing from STFT, which could only provide fixed time-frequency resolution, K. Feng et al. utilized continuous wavelet transform (CWT) to improve the time-resolution of the beating signal at a particular frequency [18]. Despite the success in improving spatial resolution and stability of OFDR systems, the method is still limited by the trade-off between the time- and frequency- resolution [19,20].
Aiming at the mutual constraints between wide measurement range and high calibration resolution, in this paper, we propose a dynamic wavelength calibration method by combining the auxiliary Mach-Zehnder interferometer (MZI) with the synchrosqueezed wavelet transform (SSWT). The analysis results of the MZI photocurrents demonstrate that the signals corresponding to distributed strong reflection points could be naturally decomposed into several intrinsic mode functions, enabling the reallocation and squeezing of the wavelet coefficients. The simulations and proof-of-concept experiments demonstrate that such reallocation can sharpen the time-frequency diagram of the interferometric signals (from tens of fm to several fm), and improve the calibration resolution by at least one order. The proposed method is also implemented to correct the tuning nonlinearity in OFDR and COSA systems.
2. Principles
Figure 1(a) demonstrates the schematic diagram of the proposed method. The optical coupler 1(OC1), optical circular (OCir), optical coupler 3(OC3), and balanced photodiode (BPD) form a basic MZI employed to sample the heterodyne signal. The splice 1, splice 2 and fiber Bragg grating (FBG) serve as the distributed strong reflection points S(n). Assuming that the reflectivity of S(n) is R(n), the response current of the BPD can be expressed as:
Fig. 1. Schematic diagram of the dynamic wavelength calibration method, a. the connection diagram of the hardware, the abbreviations TLS, OC, Ocir, H, PD, BPD, and DAQ represent tunable laser source, optical coupler, optical circular, fiber holder, photodiode, balanced photodiode, and digital acquisition device, respectively, S(n) are the distributed strong reflection points provided by the splice 1, splice 2 and fiber Bragg grating (FBG); b. the flow chart of the calibration process.
Equation (2) indicates that the instantaneous beating frequency $\omega _\textrm{b}^{(n )}$ implies the information of the tuning rate $\gamma$. Therefore, we could calculate the wavelength shifts of the TLS within discrete temporal bins $[{p{\sigma_t},\textrm{ }({p + 1} ){\sigma_t}} ]$, where p is a non-negative integer, ${\sigma _t}$ represents the time-resolution of $\omega _\textrm{b}^{(n )}$. Then, the dynamic wavelength of the TLS could be evaluated by
As the tuning rate $\gamma$ varies slowly, Eq. (2) could be decomposed into several intrinsic mode functions (IMFs) which supports the synchrosqueezed wavelet transform (SSWT). In this case, the CWT coefficients of $i(t )$ could be expressed as
To visualize the flow chart demonstrated in Fig. 1(b). We investigate the CWT and the SSWT of an arbitrarily-constructed signal $i(t )= 0.5\cos ({2\pi {t / 5}} )+ \cos [{2\pi ({{t / {20}} + 2{{{t^2}} / 5}} )} ]$, wherein the time- and frequency- scales of i(t) are normalized by the tuning period and the Nyquist frequency (fN) for comparison purposes (the sampling rate, the number of wavelet layers, and the mother function are 104, 384, and Morlet wavelet, respectively). The overall distributions of Figs. 2(a) and 2(c) are similar, apart from the linewidths of the white lines in Fig. 2(c) are much narrower than that in Fig. 2(a). The intuitive comparison between Figs. 2(a) and 2(c) illustrates that the map $({b,a} )\to ({b,\omega_\textrm{b}^{(n )}} )$ could centralize the time-frequency distributions of the CWT coefficients, so the resolution of the dynamic wavelength calibration could be enhanced accordingly.
Fig. 2. Simulation of the synchrosqueezing operation, a. the frequency- and time- normalized CWT of i(t); b. the mapping relationship between the scaling factor and frequency; c. the frequency- and time- normalized SSWT of i(t), the sampling rate, the maximum decomposition level, and the mother function remain unchanged for comparison purposes; d., e., and f. are the extractions of a., b., and c. when the normalized time equals 0.45.
Figures 2(d)∼2(f) exhibit the extractions of Figs. 2(a)∼2(c) when the normalized time is 0.45, in which the bule and orange circles correspond to the sinusoidal component $0.5\cos ({2\pi {t / 5}} )$ and linearly chirped component $\cos [{2\pi ({{t / {20}} + 2{{{t^2}} / 5}} )} ]$, respectively. Figure 2(e) reflects the step-shaped mapping relationship between the normalized scaling factor and the normalized frequency, the blue and orange circles are relocated to 0.2 and 0.46, consistent with the theoretical instant frequency of the sinusoidal and linearly chirped components, respectively. Finally, Fig. 2(f) shows that the wavelet coefficients are synchrosqueezed to the peaks at 0.2 and 0.46, the likelihood ratio of these peaks is 1:2, also matching the initial settings.
In theory, the frequency resolution of SSWT could increase infinitely with the subdividing of the decomposition levels, achieving a higher calibration accuracy. But in an actual application with noises (e.g., the distributed Rayleigh scattering along the fiber axis, and the 1/f noise of the photodiodes), especially when the maximum decomposition level is sufficiently high, further increasing its value would become uneconomical because of the heavier computation requirements. To quantify this, we refer to the works led by W. J. Williams et al., i.e., using Rényi entropy to characterize the concentrations of the time-frequency graph [21,22]. For comparison convenience, we normalize the Rényi entropy by interpolating the SSWT coefficients into the same length:
When the spectral function of each moment is taken as a probability density function, the larger Rényi entropy is, the less concentrated the time-frequency distribution would be. Figure 3 demonstrates the tendencies of the Rényi entropy as the maximum decomposition levels and the signal-to-noise ratios (SNRs) alter, wherein the solid lines and the patches represent the expectation and the confidence interval (probability = 95%) of the Rényi entropy, respectively. We could see the concentration of the time-frequency pattern rises first and then slows down with the increment of the decomposition levels. Combining this trend with the characteristics of the FFT butterfly calculation, we finally found that ${2^8} \cdot \lfloor{{{\log }_2}{{{N_\textrm{s}}} / 2}} \rfloor$ is the most suitable value, where $\left\lfloor x \right\rfloor$ represents the round down of x, ${N_\textrm{s}}$ is the number of sampling points.
Fig. 3. Simulation of the Rényi entropy as the maximum decomposition levels and the SNR alter, the horizontal coordinates are normalized by $\lfloor{{{\log }_2}{{{N_\textrm{s}}} / 2}} \rfloor$.
3. Experiments and discussions
3.1 Proof-of-concept experiments
To verify the feasibility of the dynamic wavelength calibration method, this paper employed an arbitrary waveform generator (AWG, Keysight 81160A) to modify the output frequency of the tunable laser source (TLS, Yenista Tunics Reference). Figure 4 demonstrates the obtained photocurrent and its SSWT when the TLS tunes linearly, where three ridges (the white lines) correspond to the reflectance at the Splice 1 (S(1)), fiber Bragg grating (FBG), and Splice 2 (S(2)), respectively. These ridges were overall horizontal, but they converged into a straight line when the time was between 0.42s and 0.45s. This should attribute to the intensive reflectance when the TLS frequency satisfied the Bragg condition of the FBG. And its falling edge could serve as the trigging signal to ensure ${\lambda _0}$ is constant. Within the sampling range (around 10 nm width), it is clear that line S(2) shows a higher peak-valley contrast. Considering that S(2) corresponds to a longer time delay (${\tau _\textrm{d}}$) than FBG and S(1), this phenomenon is consistent with the expectation of Eq. (2) and will support a higher calibration resolution according to Eq. (3).
Fig. 4. Measurement of the beating frequency when the TLS tunes linearly (the expectation of the tuning rate is 2.5 GHz/ms), a. the photocurrent acquired by MZI and PD; b. the SSWT of the photocurrent (the sampling rate and maximum decomposition levels are 2 × 105 and 8192, respectively); three bright lines correspond to the reflectance at the trip points of refractive indexes, which is higher than the background (resulting from the Rayleigh scattering in quasi-homogeneous fibers).
The jitters of the ridges also indicate the existence of the tuning nonlinearity. To quantitatively test the proposed method in tuning nonlinearity compensation, the TLS is tuned in sine-shape (the tuning rate is around $70\cos ({2\pi \cdot 2.5\textrm{Hz} \cdot t} )\textrm{MHz/ms}$ beyond most variations introduced by the tuning nonlinearities). Figures 5(a)∼5(c) demonstrate the 50000-point STFT, 5000-point CWT, and 5000-point SSWT of the MZI photocurrent. For comparison purposes, the Fourier transform length of the STFT (${2^{12}} = 4096$) is approximately twice the maximum decomposition level of the CWT and SSWT (${2^8} \cdot \lfloor{{{\log }_2}{{5000} / 2}} \rfloor = 2816$), dividing the frequency range $[{0,{f_\textrm{N}}} ]$ into sub-intervals of similar length. The total length (4000 points) and overlap length (3990 points) of the STFT windows discretizes the time axis into 4601 segments, which is similar to the cases of 5000-point CWT and SSWT.
Fig. 5. Comparison among the beating frequencies demodulated by STFT, CWT, and SSWT (the TLS tunes in sine shape), a. STFT of the MZI photocurrent; b. CWT of the MZI photocurrent; c. SSWT of the MZI photocurrent; d. the frequency-likelihood curves of the STFT, CWT, and SSWT coefficients at t = 0.01 s.
The white lines in each figure denote the S(2)-ridges of the STFT, CWT, and SSWT, reflecting the estimations of $\omega _\textrm{b}^{(2 )}$. The similarity among these ridges could prove that the $\omega _\textrm{b}^{(2 )}$ estimated by SSWT is close to that estimated by STFT and CWT. Besides, Fig. 5(d) extracts the frequency-likelihood curves of Figs. 5(a)∼5(c) at t = 0.01s. The noise floor of the SSWT (-50 dB) is much lower than that of the STFT (-12 dB, mainly caused by Gibbs effect) and CWT (-16 dB, mainly caused by low-resolution), indicating that SSWT has promoted the anti-noise performance of the dynamic wavelength calibration. Furthermore, the widths of these S(2)-ridges are around 250 Hz, 400 Hz, and 10 Hz, respectively. According to Eq. (3), the calibration resolution of the SSWT-based method could reach ∼0.5 fm (when ${\sigma _t}$ = 0.1 ms), which is at least one magnitude higher than that achieved by STFT (∼12 fm) and CWT (∼20 fm). Note that the resolution improvement above is analyzed under the premise that the sampling rate of SSWT are much lower than that of STFT, proving that the proposed method doesn’t need digital acquisition device (DAQ) with an extra higher sampling rate.
3.2 Applications in OFDR
As shown in Fig. 6, an OFDR system was constructed by connecting the fiber under test (FUT) to the S(2) splice of the dynamic wavelength calibration setup (blue dashed box). The scattering along the FUT, as well as the reflection of S(n), were demodulated simultaneously through the interferometer composed of OC2, Ocir1, and OC4, avoiding the usage of the external clock interferometer. Therefore, this design realized a sampling rate close to the internal clock of the DAQ. The frequency of the internal clock is much higher than that of the zero-crossing points of the auxiliary interferometer (which serves as the external clock), allowing for an extended measurable range
Fig. 6. Typical application of the SSWT-based calibration in an OFDR system, the abbreviations TLS, OC, Ocir, BPD, PD, FUT and DAQ represent tunable laser source, optical coupler, optical circular, balanced photodiode, photodiode, fiber under test, and digital acquisition device, respectively.
Besides, the proof-of-concept experiments illustrate that SSWT has two advantages over STFT. Firstly, SSWT enhances the calibration resolution, so its combination with optical fiber sensing technologies can achieve more precise measurement. Secondly, the proposed method realizes a higher utilization rate of the sampling points. Therefore, in case the sampling rate and sampling points of the DAQ are limited, SSWT could break the trade-off between the measurement range and resolution. To preliminarily verify the feasibility of this method, we employed a multi-core fiber (MCF) as the FUT, of which cores #1 and #3 were linked to two MZIs for OFDR decoupling. Since TLS tuning nonlinearity exerts the same effect on Rayleigh scattering in cores #1∼4, the spectral distortion of the interference signal in core #3 could be corrected by taking the interference in core #1 as prior knowledge. To confirm this, Figs. 7(a) and 7(b) demonstrate the SSWT of the photocurrent corresponding to the backward scatterings of cores #1 and #3. Wherein Ridges 1#(1 or 3) and 2#(1 or 3) represent the scatterings on the front and rear ends of cores #1 and #3; the region below ridges 1#1 (related to the S(1), S(2), and FBG) could serve as a finger spectrum identifying the tuning nonlinearities of the TLS. Figures 7(c) and 7(d) further depict the Fourier spectra of the photocurrents before (blue line) and after (orange line) calibration. We could see that the widths of the photocurrents’ Fourier spectrum are narrowed by two orders of magnitude, corresponding to an improvement of the spatial resolution from ∼100 mm to ∼ 1 mm (better than the 2 cm reported by [17]).
Fig. 7. OFDR sensing results in MCF, a. and b. are the SSWT distributions of the photocurrents corresponding to the backward scatterings of cores #1 and #3; c. and d. are the estimated reflectivity along cores #1 and #3, wherein the blue and orange lines represent the results before and after wavelength correction.
Then, Fig. 8 depicts the reflectivity along a single-mode fiber (SMF) nearly 80 meters long. From that, we see the proposed method could significantly narrow the reflection peaks along the SMF (150 mm to 2 mm at the front end, and 20 m to 0.2 m at the rear end), proving the proposed method could improve the measurement accuracy within the entire range. Currently, the reachable resolution is mainly restricted to the total number of sampling points. And sub-mm resolution can be reached by sampling over 106 points.
Fig. 8. OFDR sensing results in SMF, the blue and orange lines represent the results before and after wavelength correction.
3.3 Applications in COSA
Following the principles of Ref. [18], we built the COSA setup as shown in Fig. 9(a). The dynamic wavelength calibration setup was applied to correct the tuning nonlinearities of the local oscillator (LO) beam. And the transmission spectra of the Fabry-Pérot interferometer (FPI, Micron Optics Inc.) and electro-optic intensity modulator (EOIM, Thorlabs LN81S-FC) served as the spectrum under test (SUT). The heterodyne interference between LO and SUT transferred the optical spectral information into the radio frequency range, therefore, the subsequent electrical filter could abstract the components of the SUT whose wavelength is close to the instantaneous wavelength of the LO. Then the SUT could be recovered through the wavelength-tuning process of the LO.
Fig. 9. Typical application of the SSWT-based calibration in a COSA system, a. the experimental setup, the abbreviations LO, SUT, ASE, FPI, NLL, EOIM, and PC represent local oscillator, spectrum under test, amplified spontaneous emission, Fabry-Pérot interferometer, narrow linewidth laser, electro-optic intensity modulator, and polarization controller, respectively; b. the recovered FPI transmission spectra before (blue) and after (orange) wavelength correction; c. the recovered EOIM modulation spectra before (blue) and after (orange) wavelength correction
Figure 9(b) demonstrates the transmission spectra of the FPI recovered by the COSA system. The recovered spectra became more uniform within the measurable range after the dynamic wavelength calibration, indicating the proposed method could correct the spectral distortion in the GHz scale (corresponding to tens of picometers at 1550 nm). The data hints marked in the inset are the intervals among the absorption peaks. They would become closer to the nominal free spectral range of the FPI through the calibration. On the other hand, Fig. 9(c) depicts the results of the EOIM. When driven by a sinusoidal voltage, the lithium niobate crystal inside the EOIM could separate the spectrum of the narrow linewidth laser (NLL, Connect CoSF-D-ER-B-HP1) into combs, and the interval between two adjacent combs is equal to the modulation frequency. We could see that, after the wavelength calibration, the estimated comb intervals are closer to the set value (10MHz, corresponding to 80 fm at 1550 nm), this further proves the feasibility of this method to correct the fm-level fine spectrum.
4. Conclusions
In conclusion, we have proposed a dynamic wavelength calibration method based on the SSWT algorithm and MZI structure. Compared to the conventional approaches based on STFT and CWT, this SSWT-based method could enhance the time- and frequency- resolutions simultaneously. And as a result, the resolution of the wavelength calibration can reach 5 fm (when the sampling points are more than 5000 and the integral time is less than 1 ms), satisfying the demand for high-resolution coherent detections. The experimental results verify that the calibration range of the proposed method could reach tens of nanometers, making it possible to correct the tuning nonlinearity of the TLS in numerous potential applications, such as OFDR and COSA. The functional validation experiments show that the proposed method can optimize the trade-off between the measurement range (near 80m) and spatial resolution (several millimeters) of the OFDR system. For COSA systems, it can also correct the subtle spectrum distortion (tens of femtometers) within tens of picometers.
Funding
National Natural Science Foundation of China (62105144, 62205139); Stable Support Program for Higher Education Institutions from Shenzhen Municipal Science and Technology Innovation Council (20200925162216001); China Postdoctoral Science Foundation (2022M711498); Young Elite Scientist Sponsorship Program by CAST (YESS20200235); Special Funds for the Major Projects of Guangdong Education Department for Foundation Research and Applied Research (2021ZDZX1023); Basic and Applied Basic Research Foundation of Guangdong Province (2021B1515120013); Open Fund of State Key Laboratory of Information Photonics and Optical Communications, Beijing University of Posts and Telecommunications (IPOC2020A002); Natural Science Foundation of Guangdong Province (2022A1515011434); The Open Projects Foundation of State Key Laboratory of Optical Fiber and Cable Manufacture Technology (SKLD2105); Science, Technology and Innovation Commission of Shenzhen Municipality (JCYJ20220530113811026).
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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