††thanks: These authors contributed equally to this work.††thanks: These
authors contributed equally to this work.

# A vector spectrum analyzer of 55.1 THz spectral bandwidth
and 99 kHz frequency resolution

Yi-Han Luo  International Quantum Academy, Shenzhen 518048, China  Shenzhen
Institute for Quantum Science and Engineering, Southern University of Science
and Technology, Shenzhen 518055, China  Baoqi Shi  International Quantum
Academy, Shenzhen 518048, China  Department of Optics and Optical Engineering,
University of Science and Technology of China, Hefei, Anhui 230026, China  Wei
Sun  International Quantum Academy, Shenzhen 518048, China  Ruiyang Chen
International Quantum Academy, Shenzhen 518048, China  Shenzhen Institute for
Quantum Science and Engineering, Southern University of Science and
Technology, Shenzhen 518055, China  Sanli Huang  International Quantum
Academy, Shenzhen 518048, China  Hefei National Laboratory, University of
Science and Technology of China, Hefei 230088, China  Zhongkai Wang
International Quantum Academy, Shenzhen 518048, China  Jinbao Long
International Quantum Academy, Shenzhen 518048, China  Chen Shen
International Quantum Academy, Shenzhen 518048, China  Haiyan Jia  Qaleido
Photonics, Hangzhou 310000, China  Zhichao Ye  Qaleido Photonics, Hangzhou
310000, China  Hairun Guo  Key Laboratory of Specialty Fiber Optics and
Optical Access Networks, Shanghai University, Shanghai 200444, China  Junqiu
Liu<EMAIL_ADDRESS>International Quantum Academy, Shenzhen 518048, China
Hefei National Laboratory, University of Science and Technology of China,
Hefei 230088, China

The analysis of optical spectra – emission or absorption – has been arguably
the most powerful approach for discovering and understanding matters. 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 to characterize passive devices and active
laser sources in one setup. Such a dual-mode VSA can measure loss, phase
response and dispersion property 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 to 1640 nm), frequency resolution of 99 kHz, and
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 successfully 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.

Figure 1:  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 extension of the full spectral bandwidth by seamless
stitching of individual laser traces into one trace. PD, photodetector. OSC,
oscilloscope.

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 Yang  _et al._ (2021), with
precisely calibrated chromatic dispersion. With these elements, modern optical
spectrum analyzers (OSA) and spectrometers can deliver unrivalled frequency
resolution, large dynamic range and wide spectral bandwidth of hundreds of
nanometers. Time-stretched systems Mahjoubfar  _et al._ (2017) can probe
ultrafast and rare events in complex nonlinear systems.

For spectrum analysis, precise and broadband frequency-calibration of
dispersive elements is pivotal. Due to the ultimate need of spectrometers with
reduced size, weight, cost, and power consumption, extensive effort have been
made to create miniaturized spectrometers Li  _et al._ (2022); Ryckeboer  _et
al._ (2013); Pohl  _et al._ (2020); Yang  _et al._ (2019a); Yoon  _et al._
(2022); Zhang  _et al._ (2022) and broadband laser sources Kippenberg  _et
al._ (2018); Gaeta  _et al._ (2019); Diddams  _et al._ (2020); Moss  _et al._
(2013); Lu  _et al._ (2020); Perez  _et al._ (2023); Li  _et al._ (2016); Lu
_et al._ (2019); Johnson  _et al._ (2015); Porcel  _et al._ (2017); Carlson
_et al._ (2017); Guo  _et al._ (2018); Pu  _et al._ (2018); Ye  _et al._
(2021); Riemensberger  _et al._ (2022) based on integrated waveguides. For
these devices, frequency-calibration is particularly crucial yet challenging
since the dispersion of integrated waveguides can be significantly altered by
the structures and sizes Foster  _et al._ (2004). Meanwhile, stationary phase
approximation for time-stretch dispersive Fourier transform Goda and Jalali
(2013) necessitates carefully frequency-calibrated elements that are strongly
dispersive. For these purposes, optical vector network analyzers (OVNA) are
viable tools. Analogue to an electrical VNA, an OVNA enables direct
characterization of the linear transfer function (LTF) of passive devices,
therefore allowing simultaneous measurement of transmission (i.e. loss), phase
response and dispersion Sandel  _et al._ (1998); Gifford  _et al._ (2005); Jin
_et al._ (2013). Previously demonstrated OVNAs are based on interferometry
Gifford  _et al._ (2005); Li  _et al._ (2012), optical channel estimation Yi
_et al._ (2012); Jin  _et al._ (2013), single-sideband modulation Tang  _et
al._ (2012); Pan and Xue (2017), and frequency-comb-assisted asymmetric double
sidebands Qing  _et al._ (2019). Despite, all these methods have limited
measurement bandwidth of sub-terahertz to a few terahertz. Therefore for
booming demands to understand and to engineer devices used for broadband laser
sources that span over tens of terahertz, including optical frequency combs
Kippenberg  _et al._ (2018); Gaeta  _et al._ (2019); Diddams  _et al._ (2020),
parametric oscillators Moss  _et al._ (2013); Lu  _et al._ (2020); Perez  _et
al._ (2023), quantum frequency translators Li  _et al._ (2016); Lu  _et al._
(2019), supercontinua Johnson  _et al._ (2015); Porcel  _et al._ (2017);
Carlson  _et al._ (2017); Guo  _et al._ (2018), and parametric amplifiers Pu
_et al._ (2018); Ye  _et al._ (2021); Riemensberger  _et al._ (2022), all
these methods fail.

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. Our vector
spectrum analyzer (VSA) can measure LTF and dispersion property of passive
devices, or coherently map an optical spectrum into the RF domain.

##  I Principle and setup

Figure 2:  Frequency-calibration of the fiber cavity. a. Experimental setup.
PM, phase modulator. PC, polarization controller. PD, photodetector. OSC,
oscilloscope. FFT, fast Fourier Transformation. b-d. Principle of frequency-
calibration process of the fiber cavity’s FSR. Charts compare the differences
when $f_{\text{mod}}\neq N\cdot f_{\text{fsr}}$ (red curves) and
$f_{\text{mod}}=N\cdot f_{\text{fsr}}$ (blue curves). From the experimental
data (blue area) and simulation (red area), the differences are revealed in
the envelope modulation on the time-domain trace (panel b), fiber cavity’s
resonance profile (panel c), and Fourier peaks in the RF domain (panel d). e.
Measured fiber cavity’s FSR variation over the 55.1 THz frequency range with
fitted dispersion. We perform the measurement at two different temperatures
$T_{0}$ and $T_{0}+\Delta T$, where $T_{0}=23^{\circ}$C and $\Delta
T=9.3^{\circ}$C. f. For fiber cavities made of single-mode fibers (SMF) or
phase-stable fibers (PSF), the measured cavity FSR drifts versus relative
temperature change, as well as the linear fit. g. Totally 150 measurements of
the fiber cavity’s FSR shows a standard deviation (STD) of $112.5$ Hz.

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 frequency-dependent 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 particularly frequency during laser
chirping. In short, the chirping laser coherently maps the DUT’s frequency-
domain response into the time domain. Since the laser cannot chirp perfectly
linearly, critical to this frequency-time 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 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 frequency resolution of each individual time-domain trace is
determined by the chirp range divided by the oscilloscope’s memory depth
($2\times 10^{8}$). For the ECDL of widest spectral range from 1480 to 1640 nm
(19.8 THz), the estimated frequency resolution of our VSA is around 99 kHz.
However, this resolution can be compromised by the laser linewidth.

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 calibration (i.e. accurately
measured starting laser frequency), The absolute-frequency calibration is
performed by referring to a build-in wavelength meter with an accuracy of
$200$ MHz (see Note 1 in Supplementary Materials). The relative-frequency
calibration is described in the following.

Relative-frequency calibration. Figure 2 illustrates the principle of
relative-frequency calibration. We use a fiber cavity with equidistant grid of
resonances as the frequency ruler. By counting the number of resonances passed
by the chirping laser and multiplying the number with the fiber cavity’s free
spectral range (FSR, $f_{\text{fsr}}$), the laser frequency excursion (i.e.
the instantaneous laser frequency) is calculated. Therefore, critical to this
method is the measurement precision of $f_{\text{fsr}}$ and compensation of
fiber dispersion to account $f_{\text{fsr}}$ variation over the 55.1 THz
spectral range.

The experimental setup to calibrate $f_{\text{fsr}}$ 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
oscilloscope, and fed back to the RF signal generator. Based on the fiber
cavity length, an initial value of the fiber cavity’s FSR, $\Delta
f_{0}=55.58$ MHz, is estimated. The RF driving frequency $f_{\text{mod}}$ of
the phase modulator is set to $f_{\text{mod}}=N\cdot\Delta f_{0}$, where $N$
is an integer ($N=3$ in our case). Since $\Delta f_{0}\neq f_{\text{fsr}}$, 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 fiber cavity’s output.
This interference can be completely eliminated when $f_{\text{mod}}$ is
slightly varied such that $f_{\text{mod}}=N\cdot f_{\text{fsr}}$ is satisfied.

This time-domain interference can be photodetected and observed by the
oscilloscope. Figure 2c depicts the transmission spectrum of a cavity
resonance. When $f_{\text{mod}}\neq N\cdot f_{\text{fsr}}$, the resonance
profile is modulated (red curves); When $f_{\text{mod}}=N\cdot
f_{\text{fsr}}$, 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
$f_{\text{mod}}\neq N\cdot f_{\text{fsr}}$ and blue curves represent
$f_{\text{mod}}=N\cdot f_{\text{fsr}}$. When $f_{\text{mod}}\neq N\cdot
f_{\text{fsr}}$, a binary search to minimize $|f_{\text{mod}}-N\cdot\Delta
f_{0}|$ is performed, until the modulation peaks vanishes, signalling
$f_{\text{mod}}=N\cdot f_{\text{fsr}}$.

We apply this method to measure the fiber cavity’s $f_{\text{fsr}}$ from 1260
to 1640 nm wavelength (55.1 THz frequency range) with an interval of 10 nm,
with ambient temperature of $T_{0}=23.5\leavevmode\nobreak\ ^{\circ}$C. The
fiber cavity is made of phase-stable fibers (PSF, described later). Figure 2e
shows that, plots and analysis of frequency-dependent $f_{\text{fsr}}$ 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 $f_{\text{fsr}}$. The
fiber cavity is heated and its $f_{\text{fsr}}$ 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
$\text{d}f_{\text{fsr}}/\text{d}T=-262$ Hz/K, in comparison to $-676$ Hz/K of
the SMF. The lower $\text{d}f_{\text{fsr}}/\text{d}T$ 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 $\sim 240$ 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
$T_{0}+\Delta T$, where $\Delta T=9.3\leavevmode\nobreak\ ^{\circ}$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
$f_{\text{fsr}}$ but not fiber dispersion. More details concerning the
measurement is found in Note 3 in Supplementary Materials. Therefore, once the
ambient temperature is known, the $f_{\text{fsr}}$ at 1490 nm can be
calculated, as well as the $f_{\text{fsr}}$ variation over frequency.

Finally, to verify the measurement reproducibility, the $f_{\text{fsr}}$ value
at 1490 nm is repeatedly measured for 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 Diddams  _et
al._ (2020); Coddington  _et al._ (2016); Yang  _et al._ (2019b) with
precisely equidistant grid of frequency lines can also be used Del’Haye  _et
al._ (2009); Twayana  _et al._ (2021). While frequency combs have been a
proven technology for spectroscopy Picqué and Hänsch (2019) with unparalleled
accuracy, they have several limitations in 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 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 Giorgetta  _et al._ (2010). 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.

##  II Characterization of passive integrated devices

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
$5\times 5$ mm2 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
$n_{g}=2.087$ at 192.681 THz. The loss rate $\alpha=-3.0$ dB/m (physical
length) is calculated with a linear fit of power decrease over distance (red
line). f. Measured group index $n_{g}$ (blue dots) and loss $\alpha$ (red
dots) of the waveguide over the 55.1 THz spectral range.

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 meter-long spiral
waveguide. Both devices, fabricated on silicon nitride (Si3N4) Ye  _et al._
(2023), have been extensively used in integrated nonlinear photonics Moss  _et
al._ (2013); Gaeta  _et al._ (2019). For example, optical microresonators of
high quality ($Q$) factors are central building blocks for miniaturized
microresonator-soliton-based optical frequency combs (“soliton microcomb”)
Kippenberg  _et al._ (2018); Gaeta  _et al._ (2019); Diddams  _et al._ (2020),
ultralow-threshold optical parametric oscillators Moss  _et al._ (2013); Lu
_et al._ (2020); Perez  _et al._ (2023), and quantum frequency translators Li
_et al._ (2016); Lu  _et al._ (2019). Ultralow-loss, dispersion-flattened
waveguides are cornerstones for multi-octave supercontinua Johnson  _et al._
(2015); Porcel  _et al._ (2017); Carlson  _et al._ (2017); Guo  _et al._
(2018) and continuous-travelling-wave optical parametric amplifiers Pu  _et
al._ (2018); Ye  _et al._ (2021); Riemensberger  _et al._ (2022). All these
applications require precisely characterized properties of integrated devices,
such as loss, phase and dispersion over a bandwidth spanning more than 100 nm.

Characterization of integrated optical microresonators. Figure 3a shows an
optical microscope image of a Si3N4 optical microresonator. The resonance
frequency $\omega/2\pi$ and linewidth $\kappa/2\pi$ of each fundamental-mode
resonances, ranging from 1260 nm (237.9 THz) to 1640 nm (182.8 THz)
wavelength, are measured. The microresonator’s integrated dispersion is
defined as

$D_{\text{int}}(\mu)=\omega_{\mu}-\omega_{0}-D_{1}\mu=\sum_{n=2}^{\cdots}\frac{D_{n}\mu^{n}}{n!}$
(1)

where $\omega_{\mu}/2\pi$ is the $\mu$-th resonance frequency relative to the
reference resonance frequency $\omega_{0}/2\pi$, $D_{1}/2\pi$ is
microresonator FSR, $D_{2}/2\pi$ describes group velocity dispersion (GVD),
and $D_{3}$, $D_{4}$, $D_{5}$ are higher-order dispersion terms. Figure 3c top
plots the measured $D_{\text{int}}$ profile, with each parameter extracted
from the fit using Eq. 1. We note that, due to our 55.1 THz measurement
bandwidth and 99 kHz frequency resolution, our method can measure higher-order
dispersion Yang  _et al._ (2016) up to the fifth-order $D_{5}$ term. This is
validated in Fig. 3c middle, where $D_{2}$, $D_{3}$ and $D_{4}$ terms are
subtracted from $D_{\text{int}}$, and the residual dispersion is fitted with
$D_{5}\mu^{5}/120$. Figure 3c bottom 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 Herr  _et al._ (2014).

For each resonance fit Li  _et al._ (2013), the intrinsic loss
$\kappa_{0}/2\pi$, external coupling strength $\kappa_{\text{ex}}/2\pi$, and
the total (loaded) linewidth
$\kappa/2\pi=(\kappa_{0}+\kappa_{\text{ex}})/2\pi$, are extracted. Figure 3d
shows three typical resonances with fit curves (blue), including one with
visible mode split (bottom). Conventionally, based on a single resonance
profile, it is impossible to judge whether the resonance is over-coupled
($\kappa_{\text{ex}}>\kappa_{0}$) or under-coupled
($\kappa_{\text{ex}}<\kappa_{0}$) Cai  _et al._ (2000). The coupling condition
can only be revealed by phase (vector) measurement.

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 $\Delta\tau$. The delay $\Delta\tau$ introduces a frequency difference
$\Delta f=\gamma\Delta\tau$ between the two branches, where $\gamma$ is the
laser chirp rate. Thus when the two branches recombine, a beat signal is
photodetected. The extra phase shift $\phi$ introduced by the DUT also applies
on the beat signal, which can be extracted with Hilbert transformation Marple
(1999) (see Note 4 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 2$\pi$ in Fig. 3d middle represents over-coupling. From top to bottom,
the fitted loss values $(\kappa_{0}/2\pi,\kappa_{\text{ex}}/2\pi)$ for each
resonances are $(23.8,14.0)$, $(19.9,42.4)$, and $(24.7,12.8)$ MHz. The
complex coupling coefficient Pfeiffer  _et al._ (2018) in the bottom is
$\kappa_{c}/2\pi=29.1+2.25i$ MHz.

Characterization of single-pass waveguides. In addition to microresonators as
well as other resonant structures, our method can also characterize single-
pass waveguides. Figure 3b shows an optical microscope image of a Si3N4
photonic chip containing a spiral waveguide of $L_{0}=1.6394$ meter physical
length. We use our VSA as an optical frequency-domain reflectometry (OFDR)
Soller  _et al._ (2005) 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
$L_{\text{op}}$ varies due to the frequency-dependent $n_{g}$. This
dispersion-induced optical path variation leads to deteriorated spatial
resolution in broadband measurement Glombitza and Brinkmeyer (1993). By
dividing the broadband measurement data into narrow-band segments Bauters  _et
al._ (2011); Zhao  _et al._ (2017), the optical path length at different
optical frequency 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 Kohlhaas  _et al._ (1991); Zhao  _et al._ (2017).

Light travelling in the waveguide experiences attenuation following the
Lambert-Beer Law $I(L)=I_{0}\cdot\text{exp}(\alpha L)$. In Fig. 3e, the
average linear loss $\alpha=-3.0$ dB/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 $\alpha$ (red dots) and $n_{g}$ (blue dots) extracted
using segmented OFDR algorithm Bauters  _et al._ (2011); Zhao  _et al._
(2020). The $n_{g}$ is further fitted at 208.015 THz, and the dispersion
parameters are extracted up to the fourth order as $\beta_{1}=6955.0$ fs/mm,
$\beta_{2}=-74.09$ fs2/mm, $\beta_{3}=199$ fs3/mm, and $\beta_{4}=2.4\times
10^{2}$ fs4/mm. The loss fluctuation with varying frequency is likely due to
multi-mode interference in the spiral waveguide Ji  _et al._ (2022).

In OFDR, the resolution $\delta L_{\text{op}}$ of optical path length is
determined by the laser chirping bandwidth $B$ as $\delta L_{\text{op}}=c/2B$,
with $c$ being the speed of light in vacuum. Our VSA can provide maximum
$B=19.8$ THz in a single measurement, which enables $\delta L_{\text{op}}=7.6$
$\mu$m. 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.

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
sech2 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 $\gamma$, the heterodyne measurement of frequency beat in the RF domain
$\Delta f=2d\gamma/c$ allows calculation of the time delay $\Delta t=2d/c$ 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.

##  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 gigaherz. This issue prohibits OSAs for
resolving fine spectral features. For example, individual lines of mode-locked
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 Tetsumoto  _et al._ (2021); Wang
_et al._ (2021), 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 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 2 MHz bandwidth, the beatnote
creates a pair of marker signals when the laser frequency is $\pm 10$ MHz
distant to 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 4a compares the measured soliton microcomb spectra using our VSA and a
commercial OSA. Both spectra are nearly identical, particularly that the left
and right y-axes have identical power scale, 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 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 $(100.307\pm 0.002)$ GHz. The
megahertz frequency resolution of our VSA is mainly determined by the FIR
filter’s design, and the optical linewidth of the laser and the comb lines.

Light detection and ranging. Finally, we note that, the broadband, chirping,
inteferometric nature of our VSA also enables coherent LiDAR. Frequency-
modulated continuous-wave (FMCW) LiDAR is a ranging technique based on
frequency-modulated interferometry Amann  _et al._ (2001), 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 $\Delta f=2d\gamma/c$, where $c$ is the speed of
light in air and $\gamma$ is the chirp rate. Thus the measurement of $\Delta
f$ in the RF domain allows distance measurement of $d$. The ranging resolution
$\delta d$, i.e. the minimum distance that the LiDAR can distinguish two
nearby objects, is limited by the chirp bandwidth $B$ as $\delta d=c/2B$. One
advantages of our VSA as a FMCW LiDAR is that, our laser can provide maximum
$B=19.8$ THz that enables $\delta d=7.6$ $\mu$m.

In our LiDAR experiment, we set the linear chirp rate of $\gamma=6.25$ THz/s
and duration of $T=0.4$ s. The experimental setup and data analysis procedure
of LiDAR are found in Note 5 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 is $d=137.63128$ mm. The
measured distance change $\Delta d$ 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, 99 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. Our VSA can also perform LiDAR with a distance resolution of 7.6
$\mu$m and precision of 20.3 nm. Meanwhile, our VSA is fiber-based, and
neither requires high-speed 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.
Ultimately, the resolution is limited by the laser linewidth. 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 self-calibration and compensation of
mode hopping can be realized by adding a calibrated, large-FSR cavity (e.g. a
Si3N4 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
Mahjoubfar  _et al._ (2017), optimized optical coherent tomography (OCT) Adler
_et al._ (2007), linearization of FMCW LiDAR Roos  _et al._ (2009), and
resolving fine structures in Doppler-free spectroscopy Meek  _et al._ (2018).
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.

Funding Information: J. Liu acknowledges support from the National Natural
Science Foundation of China (Grant No.12261131503), Shenzhen-Hong Kong
Cooperation Zone for Technology and Innovation (HZQB-KCZYB2020050), and from
the Guangdong Provincial Key Laboratory (2019B121203002). Y.-H L. acknowledges
support from the China Postdoctoral Science Foundation (Grant No.
2022M721482).

Acknowledgments: We thank Ting Qing and Jijun He for the fruitful discussion
on OVNA, Yuan Chen for assistance in the experiment, and Lan Gao for taking
the sample photos. J. Liu is indebted to Dapeng Yu who provided critical
support to this project.

Author contributions: Y.-H. L., B. S., W. S., Z. W., and J. Long built the
experimental setup. C. S. and H. G. provided assistance and advice. Y.-H. L.,
B. S., W. S., and R. C. performed the frequency-calibration. B. S., Y.-H. L.,
W. S., and S. H. performed the experiments on VSA applications. H. J. and Z.
Y. provided the silicon nitride chips. Y.-H. L, B. S. , and J. Liu analysed
the data and prepared the manuscript with input from others. J. Liu supervised
the project.

Conflict of interest: Y. -H. L, B. S. , W. S. and J. Liu are inventors on a
patent application related to this work. Others declare no conflicts of
interest.

Data Availability Statement: The code and data used to produce the plots
within this work will be released on the repository Zenodo upon publication of
this preprint.

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# Public Discourse in the Web Does Not Exhibit Group Polarization

Fang Wu and Bernardo A. Huberman
HP Laboratories
Palo Alto, CA 94304

###### Abstract

We performed a massive study of the dynamics of group deliberation among
several websites containing millions of opinions on topics ranging from books
to media. Contrary to the common phenomenon of group polarization observed
offline, we measured a strong tendency towards moderate views in the course of
time. This phenomenon possibly operates through a self-selection bias whereby
previous comments and ratings elicit contrarian views that soften the previous
opinions.

No aspect of the massive participation in content creation that the web
enables is more evident than in the countless number of opinions, news and
product reviews that are constantly posted on the Internet. Since these
opinions play such an important role in trust building and the creation of
consensus about many issues and products, there have been a number of recent
of studies focused on the design, evaluation and utilization of online opinion
systems [5, 6, 10, 11] (for a survey, see [7]). Given the importance of group
opinions to collective social processes such as group polarization and
information cascades [2, 3, 4, 15] it is surprising that with the exception of
one study [13], little research has been done on the dynamic aspects of online
opinion formation. It remains unclear, for example, whether the opinions about
books, movies or societal views fluctuate a long time before reaching a final
consensus, or they undergo any systematic changes as time goes on. Thus the
need to understand how online opinions are created and evolve in time in order
to draw accurate conclusions from that data.

Within this context we studied the dynamics of online opinion expression by
analyzing the temporal evolution of a very large set of user views, ranging
from millions of online reviews of the best selling books at Amazon.com, to
thousands of movie reviews at the Internet Movie Database IMDB.com.
Surprisingly, our analysis revealed a trend that runs counter to the well
known herding effect studied under information cascades, and in the smaller
instance of group polarization. Online, a self selection mechanism is at play
whereby previous comments and ratings elicit contrarian views that soften the
previous opinions.

It is well known that in the case of group polarization, members of a
discussion group tend to advocate more extreme positions and call for riskier
courses of action than individuals who did not participate in any such
discussion [1, 17]. However, on the massive scale that the web offers, we
observed that later opinions in the course of time tend to show a large
difference with previous ones, thus softening the overall discourse. This is a
robust and quantitative observation for which we can only offer a tentative
explanation in terms of the cost of expressing an opinion to the group at
large.

In order to perform this study we first analyzed book ratings posted on
Amazon.com. Our sample consisted of the book ratings of the top 4,000 best-
selling titles of Amazon in each of the following 12 categories, as of July 1,
2007: arts & photography, biographies & memoirs, history, literature &
fiction, mystery & thrillers, reference, religion & spirituality, sports,
travel, nonfiction, science, and entertainment. For each of the 48,000 books,
a series of user ratings was collected in time order, where each rating is an
integer between 1 and 5. Among the 48,000 books, 16,454 books have no less
than 20 ratings, and 11,920 have an average rating above 4.

We first checked the average rating of the 16,454 books as a function of the
index of the rating ($n=1,\dots,20$). As can be seen from Fig. 1(a), $EX_{n}$
decreases almost linearly with $n$, so there is a clear dynamical trend in the
ratings, which corroborates the observation reported in [13]. Later users tend
to write different reviews from those of earlier users. Like in the
experimental setup of group polarization, an Amazon user observes the existing
average rating of that book before she leaves her own (usually shown at the
top of the book page, right under the title). However, as opposed to group
polarization, the overall opinion on Amazon tends to decrease away from the
extreme ones.

Figure 1: (a) The average rating of 16,454 books on Amazon.com with more than
20 reviews. $EX_{n}$ is the sample average rating of all the 16,454 $n$’th
ratings. As one can see from the figure, $EX_{n}$ decreases by 0.4 stars in 20
steps. We did not obtain enough data from low selling books to show the
opposite trend.

One point to be stressed is that these results do not necessarily imply that
as time goes on the average opinion of the whole population changes, for the
late reviewers might come from a different group than the earlier ones and
need not be representative of the whole population. This is seen when plotting
the average “helpful ratio” as a function of star rating in Fig. 2 for users
of Amazon. As can be seen, the whole population finds high ratings in general
more helpful than low ratings, implying that the majority of the population
does not necessarily agree with the low ratings. This additional data suggests
that rather than indicating a real opinion shift in the whole population, the
observed dynamic trend is more of an expression bias.

(a)

(b)

Figure 2: (a) The average helpful ratio of five different star ratings. (b)
The average review length of five different star ratings in the number of
characters. The data is calculated for 4,000 bestselling mystery books. By
comparing the two figures it is clear that people find high ratings more
helpful not just because they are longer. For instance, 5-star reviews are on
average shorter than 4-star and 3-star reviews but are nevertheless more
helpful.

On reflection, it is rather surprising that people contribute opinions and
reviews of topics which have already been extensively covered by others. While
posting views is easy to understand when it involves no effort, like clicking
on a button of a website, it is more puzzling in situations where it is
costly, such as composing a review.111When a user of Amazon decides to review
a book, she is required to write a short paragraph of review in addition to a
simple star rating. The average word count of Amazon reviews is 181.5 words
[12], so the cost of opinion expression is indeed high. If the opportunity to
affect the overall opinion or rating diminishes with the number of published
ones, why does anyone bother to incur the cost of contributing yet another
review? From a rational choice theory point of view, if the utility to be
gained does not outweigh the cost, people would refrain from expressing their
views. And yet they do. This is reminiscent of the well analyzed voter’s
paradox [9, 14, 16], where a rational calculation of their success probability
at determining the outcome of an election would make people stay home rather
than vote, and yet they show up at the polls with high turnout rates. In
contrast to a political election, there is no concept of winning in online
opinion systems. Rather, by contributing her own opinion to an existing
opinion pool, a person affects the average or the distribution of opinions by
a marginal amount that diminishes with the size of that pool.

One possible explanation for these results is that in cases like Amazon,
people will derive more utility the more they can influence the overall
rating, as in the voter’s paradox. To be precise, in cases where users’
opinions can be quantified and aggregated into an average value, the influence
of an online opinion can be measured by how much its expression will change
the average opinion. Suppose that $n$ users have expressed their opinions,
$X_{1},\dots,X_{n}$, on a given topic at a website, with $X_{i}$ denoting the
quantified value of the $i$’th opinion. If the $(n+1)$’th person expresses a
new opinion $X_{n+1}$, it will move the average rating to

$\bar{X}_{n+1}=\frac{n\bar{X}_{n}+X_{n+1}}{n+1},$ (1)

and the absolute change in the average rating is given by

$|\bar{X}_{n+1}-\bar{X}_{n}|=\frac{|X_{n+1}-\bar{X}_{n}|}{n+1}.$ (2)

Thus a person is more likely to express her opinion whenever
$|X_{n+1}-\bar{X}_{n}|$ is large — an opinion is likely to be expressed if it
deviates by a significant amount from those already stated. Indeed, what is
the point of leaving another 5-star review after one hundred people have
already done so?222This point has also been made within the “brag-and-moan”
model [8, 11] which assumes that consumers only choose to write reviews when
they are very satisfied with the products they purchased (brag), or very
disgruntled (moan). Note however, that the brag-and-moan model is static and
thus predicts that $\bar{X}_{n}$ is constant over time, in contradiction with
the observed dynamical trends.

In order to test this hypothesis, we measured directly how much one’s rating
deviates from the observed average rating. We plot the expected _deviation_
$Ed_{n}=E|X_{n}-\bar{X}_{n-1}|$ as a function of $n$ in Fig. 3, where $X_{n}$
is the rating left by the $n$’th user, and $\bar{X}_{n-1}$ is the average
rating the $n$’th user observes. As can be seen, $Ed_{n}$ increases with $n$.
Since the expected deviation $Ed_{n}$ of an i.i.d. sequence normally
_decreases_ with $n$, this increasing trend is indeed significant. This again
supports our conjecture that those users who disagree from the public opinion
will be more willing to express themselves and thus soften the overall opinion
of a given book.

Figure 3: The average deviation of Amazon ratings increases with the number of
people.

Next we examined whether this dynamical trend is still prominent at the level
of each individual book. We defined $d=\bar{X}_{20}-\bar{X}_{10}$ as a measure
of the change in a book’s rating over time. The histogram of 16,454 $d$’s is
shown in Fig. 4. As can be seen, most of the changes are negative. A $t$-test
of the alternative hypothesis “$d<0$” yields a $p$-value less than $0.0001$,
which further confirms the declining trend.

Figure 4: Histogram of the change in average book ratings
($d=\bar{X}_{20}-\bar{X}_{10}$) on Amazon.com. Most of the changes are
negative, testifying a declining trend in the average ratings.

While our hypothesis of a costly expression bias seems to explain the
softening of opinions observed in Amazon, it would be more conclusive if one
could conduct a test that directly compares people’s opinions expressed at
different cost levels. In order to address this issue we conducted a study of
IMDB.com (The Internet Movie Database). Unlike users of Amazon who are
required to write a review when rating a book, users of IMDB are free to
_choose_ the effort level when reviewing a movie. Specifically, after
observing the current average rating of a movie, a user can either submit a
quick rating by clicking on a scale of 10 stars, or can make the extra effort
involved in writing a comment between 10 and 1000 words.

Our study focused on two sets of movie titles. The first consists of the 50
most top-rated movies released after year 2000, which we call the “good
movies”, and the second consists of the 50 most low-rated, which we call the
“bad movies”. For each movie we know its average rating (taken among all
ratings with or without a comment), as well as the value and date-stamp of its
each commented rating, but we do not have any specific information about each
uncommented rating.

(a) Good movies

(b) Bad movies

Figure 5: Average rating associated with a comment of the (a) good and (b) bad
movies, as a function of the number of existing ratings. It can be seen that
good movies tend to receive lower ratings as time goes on, and bad movies tend
to receive higher ratings.

(a)

(b)

Figure 6: Histogram of $d=\bar{X}_{10}-\bar{X}_{5}$ for the good movies and
bad movies.

The trend of the ratings associated with comments of the two sets of movies is
shown in Fig. 5. Similar to Amazon, a softening of the expressed view is once
again observed for both sets. Two histograms of $d=\bar{X}_{10}-\bar{X}_{5}$
for the good movies and the bad movies are shown in Fig. 6. A $t$-test of the
alternative hypothesis $d<0$ for the good movies yields a $p$-value 0.44. A
$t$-test of $d>0$ for the bad movies yields a $p$-value 0.018. While it is not
too reliable to conclude that good movies tend to receive lower ratings over
time, it is safer to conclude that bad movies accumulate higher ratings as
time goes on.

We also examined the difference between the overall average rating (with or
without a comment) and the average rating associated with a comment for each
movie, and the result is shown in Fig. 7. It can be seen that those who decide
to spend the time to write a comment tend to speak differently from the
majority users, who simply leave a star rating without any justification. Fig.
7 is thus a direct verification of our hypothesis that high cost induces
expression bias.

Figure 7: Expression bias of commented ratings. Each point in this figure
corresponds to one movie title. The horizontal coordinate represents the
movie’s overall average rating ($\bar{r}$) taken over both commented and
uncommented ratings. The vertical coordinate represents the movie’s average
rating taken over only commented ratings ($\bar{r}_{c}$). Good and bad movies
are represented by circles and crosses, respectively. Clearly, those users who
spend the additional cost to write a comment tend to speak oppositely to the
majority. A $t$-test of the alternative hypothesis that $\bar{r}_{c}<\bar{r}$
for good movies and a similar $t$-test of $\bar{r}_{c}>\bar{r}$ for bad movies
both yield a $p$-value less than $0.001$.

These results show that in the process of articulating and expressing their
views online, people tend to follow a different pattern from that observed in
information cascades or group polarization. What is observed is an anti
polarization effect, whereby previous comments and ratings elicit contrarian
views that soften the previous opinions. This is in contrast to the phenomenon
of herding and opinion polarization observed in both group dynamics and online
sites.333We point out that in a website like Jyte.com, where it takes only one
click to agree or disagree with an arbitrary claim, we did see a strong group
polarization [15]. It is possible that the latter is due to the fact that such
a vote is costless compared to the opinions on Amazon and IMDB.

In closing, besides their intrinsic novelty, these results throw a cautionary
note on the interpretation of online public opinion. This is because a simple
change in the order or frequency of given sets of views can change the ongoing
expression in the community, and thus the perceived collective wisdom that new
users will find when accessing that information.

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