A wideband, high-resolution vector spectrum analyzer for integrated photonics

The analysis of optical spectra—emission or absorption—has been arguably the most powerful approach for discovering and understanding matter. The invention and development of many kinds of spectrometers have equipped us with versatile yet ultra-sensitive diagnostic tools for trace gas detection, isotope analysis, and resolving hyperfine structures of atoms and molecules. With proliferating data and information, urgent and demanding requirements have been placed today on spectrum analysis with ever-increasing spectral bandwidth and frequency resolution. These requirements are especially stringent for broadband laser sources that carry massive information and for dispersive devices used in information processing systems. In addition, spectrum analyzers are expected to probe the device’s phase response where extra information is encoded. Here we demonstrate a novel vector spectrum analyzer (VSA) that is capable of characterizing passive devices and active laser sources in one setup. Such a dual-mode VSA can measure loss, phase response, and dispersion properties of passive devices. It also can coherently map a broadband laser spectrum into the RF domain. The VSA features a bandwidth of 55.1 THz (1260–1640 nm), a frequency resolution of 471 kHz, and a dynamic range of 56 dB. Meanwhile, our fiber-based VSA is compact and robust. It requires neither high-speed modulators and photodetectors nor any active feedback control. Finally, we employ our VSA for applications including characterization of integrated dispersive waveguides, mapping frequency comb spectra, and coherent light detection and ranging (LiDAR). Our VSA presents an innovative approach for device analysis and laser spectroscopy, and can play a critical role in future photonic systems and applications for sensing, communication, imaging, and quantum information processing.

Introduction.The analysis of light and its propagation in media is fundamental in our information society.The discovery of light refraction and dispersion in media has resulted in the invention of prisms and gratings that are ubiquitously used in today's optical systems for imaging, sensing, and communication.Key enabling building blocks to these applications are dispersive elements that separate light components of different colors (i.e.frequencies) either spatially or temporally 1 , with precisely calibrated chromatic dispersion.With these elements, modern optical spectrum analyzers (OSA) and spectrometers can deliver unrivaled frequency resolution, large dynamic range, and wide spectral bandwidth of hundreds of nanometers.Time-stretched systems 2 can probe ultrafast and rare events in complex nonlinear systems.
Here we demonstrate a new paradigm of vector spectrum analysis that units OVNA for passive devices and OSA for active laser sources in one setup. .Principle and architecture of the vector spectrum analyzer (VSA).a.The principle of our VSA is based on a chirping CW laser that is sent to and transmits through a device under test (DUT).The DUT can be either a passive device or a broadband laser source.The transmission spectrum of the chirping laser through the DUT is a time-domain trace.For passive devices, this trace carries the information of the DUT's loss, phase and dispersion over the chirp bandwidth.For active laser sources, the chirping laser beats progressively with different frequency components of the optical spectrum, thus analyzing the beat signal in the RF domain allows extraction of the spectral information.In short, the chirping laser coherently maps the DUT's frequency-domain response into the time domain.Critical to this frequency-time mapping is precise and accurate calibration of the instantaneous laser frequency during chirping.This requires to refer the chirping laser to a "frequency ruler".b.Experimental setup.The frequency-calibration unit here is a phase-stable fiber cavity of 55.58 MHz FSR.The chirping laser unit can be a single laser, or multiple lasers that are bandwidth-cascaded together.The latter allows the extension of the full spectral bandwidth by seamless stitching of individual laser traces into one trace.PD, photodetector.OSC, oscilloscope.
sion property of passive devices, or coherently map an optical spectrum into the RF domain.

I. PRINCIPLE AND SETUP
The principle of our VSA is illustrated in Fig. 1a.A continuous-wave (CW), widely chirping laser is sent to and transmits through a device under test (DUT) that can be either a passive device or a laser source.During laser chirping, for passive devices, the frequencydependent LTF containing the DUT's loss and phase information is photodetected and recorded.For laser sources, the chirping laser beats progressively with different frequency components of the optical spectrum, and the beatnote signal is digitally recorded in the RF domain using a narrow-band-pass filter.In both cases, the VSA outputs a time-domain trace, with each data point corresponding to the DUT's instantaneous response at a particular frequency during laser chirping.In short, the chirping laser coherently maps the DUT's frequencydomain response into the time domain.Since the laser cannot chirp perfectly linearly, critical to this frequencytime mapping is precise and accurate calibration of the instantaneous laser frequency at any given time.This requires to refer the chirping laser to a calibrated "frequency ruler".
Following this principle, we construct the setup as shown in Fig. 1b.A widely tunable, mode-hop-free, external-cavity diode laser (ECDL, Santec TSL) is used as the chirping laser.Cascading multiple ECDLs covering different spectral ranges allows the extension of full spectral bandwidth, which is 1260 to 1640 nm (55.1 THz) in our VSA with three ECDLs (see Note 1 in Supplementary Materials).
The ECDL's CW output is split into two branches.One branch is sent to the DUT and the other is sent to a frequency-calibration unit.Such frequency-calibration involves relative-(i.e. the frequency change relative to the starting laser frequency) and absolute-frequency-  The experimental setup to calibrate f fsr is shown in Fig. 2a.The ECDL's CW output is phase-modulated by an RF signal generator to create a pair of sidebands.The carrier and both sidebands are together sent into the fiber cavity with maintained polarization.The transmitted signal through the fiber cavity is probed by a 125-MHz-bandwidth photodetector, analyzed by an oscillo-scope, and fed back to the RF signal generator.Based on the fiber cavity length, an initial value of the fiber cavity's FSR, ∆f 0 = 55.58MHz, is estimated.The RF driving frequency f mod of the phase modulator is set to f mod = N • ∆f 0 , where N is an integer (N = 3 in our case).Since ∆f 0 ̸ = f fsr , as shown in Fig. 2b, in the frequency domain, the carrier and both sidebands locate at different positions of the respective three resonances.Therefore, the three CW components experience different cavity responses, and together create an amplitude interference in the time domain at the fiber cavity's output.This interference can be completely eliminated when f mod is slightly varied such that This time-domain interference can be photodetected and observed by the oscilloscope.Figure 2c depicts the transmission spectrum of a cavity resonance.When f mod ̸ = N • f fsr , the resonance profile is modulated (red curves); When f mod = N • f fsr , the resonance profile is unaffected as a normal Lorentzian profile probed by a single CW laser (blue curves).We also simulate this modulation behavior (left red panels) which agrees with the experimental data (right blue panels).The modulation amplitude is extracted with fast Fourier transformation (FFT) as shown in Fig. 2d, where red curves represent We apply this method to measure the fiber cavity's f fsr from 1260 to 1640 nm wavelength (55.1 THz frequency range) with an interval of 10 nm, with ambient temperature of T 0 = 23.5 • C. The fiber cavity is made of phase-stable fibers (PSF, described later).Figure 2e shows that, plots and analysis of frequency-dependent f fsr enable extraction of the fiber dispersion using a cubic polynomial fit (see Note 2 in Supplementary Materials).This dispersion-calibrated fiber cavity's resonance grid is used as the frequency ruler in our VSA and following experiments.
We further characterize the temperature stability of f fsr .The fiber cavity is heated and its f fsr shift versus the relative temperature change at 1490 nm is measured, as shown in Fig. 2f.In addition, we compare two types of fibers to construct the cavity: the normal single-mode fiber (SMF, blue data) and phase-stable fiber (PSF, red data).The linear fit shows that the PSF-based fiber cavity features temperature-sensitivity of df fsr /dT = −262 Hz/K, in comparison to −676 Hz/K of the SMF.The lower df fsr /dT of PSF is the reason why we use PSF instead of SMF.Correspondingly, 1 K temperature change (the level of our ambient temperature stabilization and control) causes ∼ 240 MHz cumulative error of the PSF-based fiber cavity over the entire 55.1 THz bandwidth.
We also measure the fiber cavity's dispersion at elevated temperature T 0 + ∆T , where ∆T = 9.3 • C. Figure 2e shows that, the two measured fiber dispersion curves at different temperatures are nearly identical except with a global relative shift in the y-axis.This indicates that the temperature change only affects f fsr but not fiber dispersion.More details concerning the measurement are found in Note 3 in Supplementary Materials.Therefore, once the ambient temperature is known, the f fsr at 1490 nm can be calculated, as well as the f fsr variation over frequency.
Finally, to verify the measurement reproducibility, the f fsr value at 1490 nm is repeatedly measured 150 times.Figure 2g shows the occurrence histogram, with a standard deviation of 112.5 Hz.
Here we use a dispersion-calibrated, phase-stable fiber cavity for relative-frequency calibration.We note that frequency comb spectrometers 11,35,36 with a precisely equidistant grid of frequency lines can also be used 37,38 .While frequency combs have been a proven technology for spectroscopy 39 with unparalleled accuracy, they have several limitations in the characterization of passive devices.First, in addition to being bulky and expensive, commercial fiber laser combs as spectrometers suffer from limited frequency resolution due to the RF-rate comb line spacing (typically above 100 MHz).Second, the simultaneous injection of more than 10 5 comb lines can saturate or blind the photodetector, yielding a severely deteriorated signal-to-noise ratio (SNR) and dynamic range.
Different from frequency combs, CW lasers featuring high photon flux and ever-increasing frequency tunability and agility are particularly advantageous for sensing 40 .In our method, after frequency-calibration by the fiber cavity, the chirping CW laser behaves as a frequency comb with a "moving" narrow-band-pass filter, where the filter selects only one comb line each time and rejects other lines.Therefore the nearly constant laser power during chirping provides a flat power envelope over the entire spectral bandwidth.Therefore our method avoids photodetector saturation and device damage, and increases SNR and dynamic range.
To improve frequency resolution, the extrapolation of instantaneous laser frequency between two neighbouring fiber cavity's resonances is performed, which relies on the frequency linearity of the chirping laser.Such linearity is experimentally characterized in a parallel work 41 of ours, where the chirping ECDL (Santec TSL) is referenced to a commercial optical frequency comb.The result from Ref. 41 evidences that, using a fiber cavity with 55 MHz FSR and laser chirp rate of 50 nm/s, we experimentally achieve relative-frequency calibration with precision better than 200 kHz.The error is caused by the laser chirp nonlinearity.More details are elaborated in the Note 4 in Supplementary Materials.
The ultimate frequency resolution of each individual time-domain trace is determined by the chirp range divided by the oscilloscope's memory depth (2 × 10 8 ).For the ECDL of the widest spectral range from 1480 to 1640 nm (19.8 THz), we estimate that the ultimate frequency sample resolution of our VSA is around 99 kHz, i.e. the frequency interval between two recorded neighbouring data points.The actual resolution can be compromised further by the chirping laser linewidth.Therefore we experimentally measured the dynamic laser linewidth using a self-delayed heterodyne setup.Experimental details are elaborated in Note 5 in Supplementary Materials.Within 100 µs time scale, the ECDL's dynamic linewidth at 50 nm/s chirp rate is averaged as 471 kHz.This linewidth is due to multiple reasons including laser intrinsic linewidth, laser chirp nonlinearity, and the fiber delay-line's instability in the heterodyne setup.The measured laser dynamic linewidth of 471 kHz sets the lower bound of our VSA's frequency resolution.

II. CHARACTERIZATION OF PASSIVE INTEGRATED DEVICES
Next we demonstrate several applications using our VSA.We first use our VSA as an OVNA to characterize passive devices.We select two types of optical devices: an integrated optical microresonator and a meterlong spiral waveguide.Both devices, fabricated on silicon nitride (Si 3 N 4 ) 42 , have been extensively used in integrated nonlinear photonics 10,12 .For example, optical microresonators of high quality (Q) factors are central building blocks for miniaturized microresonator-solitonbased optical frequency combs ("soliton microcomb") [9][10][11] , ultralow-threshold optical parametric oscillators 12,14,15 , and quantum frequency translators 16,17 .
Characterization of integrated optical microresonators.Figure 3a shows an optical microscope image of a Si 3 N 4 optical microresonator.The resonance frequency ω/2π and linewidth κ/2π of each fundamentalmode resonances, ranging from 1260 nm (237.9THz) to 1640 nm (182.8THz) wavelength, are measured.The microresonator's integrated dispersion is defined as where ω µ /2π is the µ-th resonance frequency relative to the reference resonance frequency ω 0 /2π, D 1 /2π is microresonator FSR, D 2 /2π describes group velocity dispersion (GVD), and D 3 , D 4 , D 5 are higher-order dispersion terms.Figure 3c top plots the measured D int profile, with each parameter extracted from the fit using Eq. 1.
We note that, due to our 55.1 THz measurement bandwidth and 471 kHz frequency resolution, our method can measure higher-order dispersion 43 up to the fifth-order D 5 term.This is validated in Fig. 3c  shows that, after further subtraction of the D 5 term, no prominent residual dispersion is observed.Some data points deviate from the fit due to avoided mode crossings in the microresonator 44 .
Here we split laser power into two branches as shown in Fig. 1b.In one branch the laser transmits through the DUT, while in the other the laser experiences a delay ∆τ .The delay ∆τ introduces a frequency difference ∆f = γ∆τ between the two branches, where γ is the laser chirp rate.Thus when the two branches recombine, a beat signal is photodetected.The extra phase shift ϕ introduced by the DUT also applies to the beat signal, which can be extracted with Hilbert transformation 47 (see Note 6 in Supplementary Materials).The measured and fitted phases are shown in Fig. 3d red curves.The continuous phase transition across the resonance in Fig. 3d top and bottom represents under-coupling, while the phase jump by 2π in Fig. 3d middle represents over-coupling.From top to bottom, the fitted loss values (κ 0 /2π, κ ex /2π) for each resonances are (23.8,14.0), (19.9, 42.4), and (24.7, 12.8) MHz.The complex coupling coefficient 48 in the bottom is κ c /2π = 29.1 + 2.25i MHz.
Characterization of single-pass waveguides.In addition to microresonators as well as other resonant structures, our method can also characterize singlepass waveguides.Figure 3b shows an optical microscope image of a Si 3 N 4 photonic chip containing a spiral waveguide of L 0 = 1.6394 meter physical length.We use our VSA as optical frequency-domain reflectometry (OFDR) 49 to characterize the waveguide loss and dispersion.Figure 3e plots the OFDR signal from the spiral waveguide.The prominent peak located at 1.6394 meter physical length (3.4214 meter optical length) is attributed to the light reflection at the rear facet of the chip, where the waveguide terminates.The difference in the physical and optical lengths indicates a group index of n g = 2.087 at 192.681 THz.
In the presence of waveguide dispersion, the optical path length L op varies due to the frequency-dependent n g .This dispersion-induced optical path variation leads to deteriorated spatial resolution in broadband measurement 50 .By dividing the broadband measurement data into narrow-band segments 51,52 , the optical path length at different optical frequencies can be obtained, and thus the frequency-dependent n g over the 55.1 THz spectral range can be extracted.With the extracted n g , the waveguide dispersion can be de-embedded with a re-sample algorithm 52,53 .
Light traveling in the waveguide experiences attenuation following the Lambert-Beer Law I(L) = I 0 •exp(αL).In Fig. 3e, the average linear loss α = −3.0dB/m (physical length) is extracted by applying a first-order polynomial fit of the power profile (red line) within the 19.8 THz bandwidth and centered at 192.681 THz. Figure 3f shows the frequency-dependent α (red dots) and n g (blue dots) extracted using segmented OFDR algorithm 51,54 .The n g is further fitted at 208.015 THz, and the dispersion parameters are extracted up to the fourth order as β 1 = 6955.0fs/mm, β 2 = −74.09fs 2 /mm, β 3 = 199 fs 3 /mm, and β 4 = 2.4 × 10 2 fs 4 /mm.The loss fluctuation with varying frequency is likely due to multi-mode interference in the spiral waveguide 55 .
In OFDR, the resolution δL op of optical path length is determined by the laser chirp bandwidth B as δL op = c/2B, with c being the speed of light in vacuum.Our VSA can provide a maximum B = 19.8THz in a single measurement, which enables δL op = 7.6 µm.As shown in Fig. 3e, such a fine resolution allows unambiguous discrimination of scattering points in the waveguide, which are revealed by small peaks.Thus our VSA is proved as a useful diagnostic tool for integrated waveguides.

III. CHARACTERIZATION OF SOLITON SPECTRA AND LIDAR APPLICATIONS
Coherent detection of frequency comb spectra.Next, we use our VSA as an OSA to characterize broadband laser spectra.While modern OSAs can achieve wide spectral bandwidth, they suffer from a limited frequency resolution ranging from sub-gigahertz to several gigahertz.This issue prohibits OSAs from resolving fine spectral features.For example, individual lines of modelocked lasers or supercontinua with repetition rates in the RF domain cannot be resolved by OSAs.Soliton microcombs with terahertz-rate repetition rate can be useful for low-noise terahertz generation 56,57 , but their precise comb line spacing can neither be measured by normal photodetectors nor OSAs.
Here we demonstrate that our VSA can act as an OSA which features a 55.1 THz spectral range and megahertz frequency resolution.As an example, we measure the repetition rate (line spacing) of a 100-GHz-rate soliton microcomb generated.The schematic is depicted in Fig. 4b, where the laser chirps across the entire soliton spectrum.Every time the laser passes through a comb line, it generates a moving beatnote.Using a finite impulse response (FIR) band-pass filter of 10 MHz center frequency and 3 MHz bandwidth, the beatnote creates a pair of marker signals when the laser frequency is ±10 MHz distant from the comb line.The polarization of the soliton spectrum is measured by varying the laser polarization until the beat signal with maximum intensity is observed.This search procedure of polarization can essentially be programmed and automated.Since the instantaneous laser frequency is precisely calibrated, the comb line spacing is extracted by calculating the frequency distance from two adjacent pairs of marker signals.With the known laser power and measured marker signals' intensity, the absolute power of each comb line can be calculated.
Figure 4 compares the measured soliton microcomb spectra using our VSA and a commercial OSA.Both spectra are nearly identical, particularly in that the left and right y-axes have identical power scales, which validates our VSA measurement.The dynamic range of our VSA is found as 56 dB, which is on par with modern commercial OSAs with the finest resolution (e.g. 45 to 60 dB at 0.02 nm resolution for Yokogawa OSAs). Figure 4a inset evidences that our VSA indeed provides significantly finer frequency resolution than the OSA.The soliton repetition rate measured by the VSA is (100.307± 0.002) GHz.
We emphasize that, here the frequency resolution of our VSA as an OSA is limited by the bandwidth of FIR band-pass filters.In digital data processing, we find that 3 MHz FIR bandwidth yields the optimal resolution bandwidth of 3 MHz.Experimentally, we verify the resolution bandwidth by phase-modulating a low-noise fiber laser (NKT Koheras) to generate a pair of sidebands of 3 MHz difference to the carrier.The carrier and the sidebands are unambiguously resolved using our VSA (See Note 5 in Supplementary Materials).The 3 MHz resolution bandwidth is also consistent with the uncertainty of measured soliton repetition rate of 100.307GHz.
Light detection and ranging.Finally, we note that the broadband, chirping, and interferometric nature of our VSA also enables coherent LiDAR.Frequency-modulated continuous-wave (FMCW) LiDAR is a ranging technique based on frequency-modulated interferometry 58 , as depicted in Fig 4d .The chirping laser is split into two arms, with one arm to the reference and the other to the target with a path difference of d.When the reflected signals from both arms recombine at the photodetector, the detected beat frequency is determined as ∆f = 2dγ/c, where c is the speed of light in air and γ is the chirp rate.Thus the measurement of ∆f in the RF domain allows distance measurement of d.The ranging resolution δd, i.e. the minimum distance that the LiDAR can distinguish two nearby objects, is limited by the chirp bandwidth B as δd = c/2B.One advantage of our VSA as a FMCW LiDAR is that, our laser can provide maximum B = 19.8THz that enables δd = 7.6 µm.
In our LiDAR experiment, we set the linear chirp rate of γ = 6.25 THz/s and duration of T = 0.4 s.The experimental setup and data analysis procedure of Li-DAR are found in Note 7 in Supplementary Materials.As a demonstration, we monitor the thermal expansion of our optical table due to ambient temperature drift, as shown in Fig. 4e.The distance difference between the target mirror and the reference mirror on the table Characterization of broadband laser spectra and coherent LiDAR applications.a. Single soliton spectra measured by our VSA (red) and a commercial OSA (blue).The spectral envelope of VSA data is fitted with a sech 2 function (green).Inset: Zoom-in of the comb line resolved by our VSA and the OSA, demonstrating the significant resolution enhancement by the VSA.b.Principle of coherent detection of broadband laser spectra using a chirping laser.The laser beats progressively with different frequency components of the optical spectrum, which allows frequency detection in the RF domain and continuous information output in the time domain.c.Histogram showing the deviations of 4625 LiDAR measurements from their mean values.The LiDAR precision is revealed by the standard deviation of 20.3 nm.d.Principle of coherent LiDAR using a frequency-calibrated chirping laser.With known chirp rate γ, the heterodyne measurement of frequency beat in the RF domain ∆f = 2dγ/c allows calculation of the time delay ∆t = 2d/c and thus to calculate the distance d. f.LiDAR Measurement of thermal expansion of our optical table using our VSA, in comparison with data from a digital ambient thermometer.is d = 137.63128mm.The measured distance change ∆d within 500 nm range agrees with the temperature decrease that causes contraction of the optical table.After subtracting the global trend, Figure 4c shows the histogram of the deviations of 4625 measurements from their mean values.Our LiDAR precision is revealed by the standard deviation of 20.3 nm.Such a precision is provided by the careful relative-frequency calibration and long-term stability of our VSA.

IV. CONCLUSION
In summary, we have demonstrated a dual-mode VSA featuring 55.1 THz spectral bandwidth, 471 kHz frequency resolution, and 56 dB dynamic range.The VSA can operate either as an OVNA to characterize the LTF and dispersion property of passive devices, or as an OSA to characterize broadband frequency comb spectra.A comparison of our VSA with other state-of-the-art OSAs and OVNAs is shown in Note 5 of Supplementary Mate-rial.Our VSA can also perform LiDAR with a distance resolution of 7.6 µm and precision of 20.3 nm.Meanwhile, our VSA is fiber-based, and neither requires highspeed modulators and photodetectors, nor any active feedback control.Therefore the system is compact, robust, and transportable for field-deployable applications.
There are several aspects to further improve the performance and reduce the complexity of our VSA.First, the frequency resolution can be improved by increasing oscilloscope's memory depth, or by sacrificing chirp bandwidth, until the laser noise dominates.Second, the frequency accuracy can be improved by adding a highly stable reference laser in the system.When the ECDL scans through the reference laser, the two lasers beat and create a marker in the time-domain trace.The marker marks the point where the chirping ECDL has an instantaneous frequency as the reference laser's frequency.Third, more ECDLs can be added into the system, allowing further extension of the spectral bandwidth and operation in other wavelength ranges such as the visible and mid-infrared bands.Meanwhile, even ECDLs with mode hopping can be used in our VSA.The selfcalibration and compensation of mode hopping can be realized by adding a calibrated, large-FSR cavity (e.g. a Si 3 N 4 microresonator of terahertz-rate FSR), in addition to the fine-tooth fiber cavity.By measuring the resonance-to-resonance frequency and referring to previously calibrated local FSR of the microresonator, the exact mode hopping range and location can be inferred.Adding more calibrated cavities of different FSR values to form a Vernier structure can further enhance the precision and accuracy.
Besides characterization of passive elements and broadband laser sources for integrated photonics, our VSA can also be applied for time-stretched systems 2 , optimized optical coherent tomography (OCT) 59 , linearization of FMCW LiDAR 60 , and resolving fine structures in Doppler-free spectroscopy 61 .Therefore our VSA presents an innovative approach for device analysis and laser spectroscopy, and can play a crucial role in future photonic systems and applications for sensing, communication, imaging, and quantum information processing.and recorded with the wavelength meter.The key question to answer is, whether the laser frequencies measured by the wavelength meter are precisely equal to the frequency value we set to the lasers.The measurement results for Lasers #2 and #3 are shown in Fig. S1(b, c).For each marker, a total of 60 traces are measured, and each trace is continuously measured for 60 s.In each trace, the data from 30 s to 60 s are used to calculate the mean frequency.The mean frequency values of Marker #1, #2 and #3, measured by the wavelength meter, are 220.92340(2)THz, 202.28896(2)THz and 202.28901(1)THz, respectively.The differences between the measured frequency values from the set values are 0.44 GHz, 0.15 GHz, and 0.10 GHz. Figure S1(b, c) shows that, for each marker, the deviations of measured frequency values to their mean value are within the accuracy of the wavelength meter (200 MHz).Therefore, in the experiment, we use the measured frequency values of 220.92340THz, 202.28896THz, and 202.28901THz as the markers' frequency values.
The workflow of our experiment procedure is detailed in Table S1.Before the selected ECDL starts chirping, its reference laser is set to the specified frequency and waits 60 s for stabilization.The actual emission frequency of the reference laser is measured by the wavelength meter.After the 60 s, the laser starts chirping.The photodetector probes the laser chirping, and the data is recorded by the oscilloscope.For the other two ECDLs, the operation procedure is similar.Meanwhile, during the 60 s waiting time for laser frequency stabilization, the recorded data by the oscilloscope can be processed by the computer.Finally, the three individually measured and calibrated data traces of each ECDL are stitched, forming the full 55.1 THz spectral bandwidth measurement.
Figure1.Principle and architecture of the vector spectrum analyzer (VSA).a.The principle of our VSA is based on a chirping CW laser that is sent to and transmits through a device under test (DUT).The DUT can be either a passive device or a broadband laser source.The transmission spectrum of the chirping laser through the DUT is a time-domain trace.For passive devices, this trace carries the information of the DUT's loss, phase and dispersion over the chirp bandwidth.For active laser sources, the chirping laser beats progressively with different frequency components of the optical spectrum, thus analyzing the beat signal in the RF domain allows extraction of the spectral information.In short, the chirping laser coherently maps the DUT's frequency-domain response into the time domain.Critical to this frequency-time mapping is precise and accurate calibration of the instantaneous laser frequency during chirping.This requires to refer the chirping laser to a "frequency ruler".b.Experimental setup.The frequency-calibration unit here is a phase-stable fiber cavity of 55.58 MHz FSR.The chirping laser unit can be a single laser, or multiple lasers that are bandwidth-cascaded together.The latter allows the extension of the full spectral bandwidth by seamless stitching of individual laser traces into one trace.PD, photodetector.OSC, oscilloscope.
middle, where D 2 , D 3 and D 4 terms are subtracted from D int , and the residual dispersion is fitted with D 5 µ 5 /120.Figure 3c bottom

2 Figure 3 .
Figure 3.Characterization of passive Si3N4 integrated devices.a, b.Optical microscope images showing a microresonator coupled with a bus waveguide (panel a), and a 1.6394-meter-long spiral waveguide contained in a photonic chip of 5 × 5 mm 2 size (panel b).The zoom-in shows the densely coiled waveguide.c.Measured integrated microresonator dispersion profile and fit up to the fifth order.d.Measured transmission and phase profiles of three resonances that are under-coupled (top), over-coupled (middle), or feature mode split (bottom, also under-coupled).e. Measured OFDR data of the spiral waveguide.The major peak at 1.6394 meter physical length (3.4214 meter optical length) is attributed to the light reflection at the rear chip facet, where the waveguide terminates.This length difference indicates a group index ng = 2.087 at 192.681 THz.The loss rate α = −3.0dB/m (physical length) is calculated with a linear fit of power decrease over distance (red line).f.Measured group index ng (blue dots) and loss α (red dots) of the waveguide over the 55.1 THz spectral range.
Figure 4.Characterization of broadband laser spectra and coherent LiDAR applications.a. Single soliton spectra measured by our VSA (red) and a commercial OSA (blue).The spectral envelope of VSA data is fitted with a sech 2 function (green).Inset: Zoom-in of the comb line resolved by our VSA and the OSA, demonstrating the significant resolution enhancement by the VSA.b.Principle of coherent detection of broadband laser spectra using a chirping laser.The laser beats progressively with different frequency components of the optical spectrum, which allows frequency detection in the RF domain and continuous information output in the time domain.c.Histogram showing the deviations of 4625 LiDAR measurements from their mean values.The LiDAR precision is revealed by the standard deviation of 20.3 nm.d.Principle of coherent LiDAR using a frequency-calibrated chirping laser.With known chirp rate γ, the heterodyne measurement of frequency beat in the RF domain ∆f = 2dγ/c allows calculation of the time delay ∆t = 2d/c and thus to calculate the distance d. f.LiDAR Measurement of thermal expansion of our optical table using our VSA, in comparison with data from a digital ambient thermometer.

c
Supplementary FigureS2: Characterization of the fiber cavity's temperature stability.a.The fit residual of the fiber cavity's FSR using a quadratic polynomial fit, evidencing the necessity to use a cubic polynomial fit.b.The fiber cavity's FSR drifts with ∆T = 9.3 • C temperature change.c.Measurement of the ambient temperature in our laboratory for 24 hours.

TABLE S1 :
Experimental workflow of the VSA.