# Single-Photon Vibrometry

Patrick Rehain  Department of Physics, Stevens Institute of Technology, 1
Castle Point Terrace, Hoboken, NJ, 07030, USA  Center for Quantum Science and
Engineering, Stevens Institute of Technology, 1 Castle Point Terrace, Hoboken,
NJ, 07030, USA  Jeevanandha Ramanathan  Department of Physics, Stevens
Institute of Technology, 1 Castle Point Terrace, Hoboken, NJ, 07030, USA
Center for Quantum Science and Engineering, Stevens Institute of Technology, 1
Castle Point Terrace, Hoboken, NJ, 07030, USA  Yong Meng Sua  Department of
Physics, Stevens Institute of Technology, 1 Castle Point Terrace, Hoboken, NJ,
07030, USA  Center for Quantum Science and Engineering, Stevens Institute of
Technology, 1 Castle Point Terrace, Hoboken, NJ, 07030, USA  Shenyu Zhu
Department of Physics, Stevens Institute of Technology, 1 Castle Point
Terrace, Hoboken, NJ, 07030, USA  Center for Quantum Science and Engineering,
Stevens Institute of Technology, 1 Castle Point Terrace, Hoboken, NJ, 07030,
USA  Daniel Tafone  Department of Physics, Stevens Institute of Technology, 1
Castle Point Terrace, Hoboken, NJ, 07030, USA  Center for Quantum Science and
Engineering, Stevens Institute of Technology, 1 Castle Point Terrace, Hoboken,
NJ, 07030, USA  Yu-Ping Huang  Department of Physics, Stevens Institute of
Technology, 1 Castle Point Terrace, Hoboken, NJ, 07030, USA  Center for
Quantum Science and Engineering, Stevens Institute of Technology, 1 Castle
Point Terrace, Hoboken, NJ, 07030, USA  Corresponding author:
<EMAIL_ADDRESS>
###### Abstract

We propose and demonstrate a single-photon sensitive technique for optical
vibrometry. It uses high speed photon counting to sample the modulated
backscattering from a vibrating target. Designed for remote vibration sensing
with ultralow photon flux, we show that this technique can detect small
displacements down to 110 nm and resolve vibration frequencies from DC up to
several kilohertz, with less than 0.01 detected photons per pulse. This
single-photon sensitive optical vibrometry may find important applications in
acousto-optic sensing and imaging, especially in photon-starved environments.

††journal: ol

Optical means have become the tool of choice for the measurement and profiling
of mechanical vibrations [1, 2, 3]. In optical vibrometry, a portion of the
returning optical probe is detected and the target’s displacement or vibration
information is extracted from the measured signal. To this end, interference
based detection mixes the returning signal with a reference beam and
demodulates the target motion from the beating between the two. This method
uses the phase coherence between the probe and reference to achieve
displacement sensitivity on the order of a fraction of the optical wavelength
used [3]. While such interferometry offers high performance in terms of
bandwidth and displacement sensitivity, it requires a laser with long
coherence time and strong and stable power in the returning beam for faithful
measurement. These requirements preclude the interference-based systems from
operating when the signal is very weak, and makes them sensitive to intensity
fluctuations arising from the dynamic speckle patterns returning from a rough
target.

The speckle noise present in interference-based vibrometry manifests as an
intensity modulation that distorts the desired phase-modulated signal.
Demodulating the two effects can be exceedingly difficult because they occur
at the same frequency. Speckle noise has been widely studied and various
techniques have been shown to mitigate its effects in certain applications [4,
5, 6]. For example, it can be detected using the kurtosis ratio of the
measured electronic signal and then reduced via a combination of both time and
frequency domain filtering [7]. Also, a scanning average technique can blur
the speckle-pattern by rapidly sweeping the beam across the target surface and
later dropping the scanning frequency [8]. In another method, a balanced
detection setup using two photodiodes is shown to be highly resilient against
intensity fluctuations by taking a differential measurement between two phase
angles [9]. These are just a few example of techniques that can mitigate the
speckle noise, and while they can be effective, they can be costly and
complicated to implement. The need for a strong optical signal and the
distortions caused by speckle noise remain the major limiting factors in the
interference-based vibrometry.

Recognizing these limitations, intensity-based vibrometry has been used to
probe target vibrations by directly measuring the speckle dynamics. This has
been demonstrated using a CCD-camera to track the motion of individual
speckles using post-processing algorithms [10, 11]. These systems are limited
by the camera frame rate, the distribution of the intensity across many
pixels, and the computational cost of processing all pixels. Recently, a
modified version of this setup was introduced where Newtonian telescope was
used to focus the 2D speckle pattern onto a line-scan CCD [12]. This increased
the maximum frame rate and the intensity at each pixel, while reducing the
dimensionality of post processing. Single-pixel systems have also been used by
directly reading the intensity-modulated output of a photodiode[10, 13]. An
aperture or mask is placed in front of the detector to tune the number of
collected speckles. The average intensity is increased by summing over more
speckles, but comes at the cost of signal contrast (modulation depth). The
single photodiode can achieve much higher frame rate and detection sensitivity
than a CCD camera, but is subject to harmonic distortion at large amplitudes.
While these intensity-based methods can increase the detection SNR, they
cannot operate under highly photon-starved conditions when the returning
signal is single-photon level (below the sensitivity of a photodiode).

Pushing the limits on remote sensing involves practical scenarios where the
returning optical signal is very weak. While in principle this can be overcome
by increasing the output laser power and detection dwell-time, this is not
always possible in practice. High output power can lead to eye-safety
concerns, meaning the laser power cannot be arbitrarily increased. Time
correlated single photon counting (TCSPC) techniques using pulsed laser source
synchronized to a Geiger-mode avalanche photodiode (APD) have proved
successful in 3D imaging systems with less than one detected photon per pulse
[14, 15]. These systems accumulate detections over long dwell-times and use
photon counting statistics to extract the target information. In vibration
sensing, high-speed data acquisition is a necessity as the detection bandwidth
is inversely related to the dwell-time.

In this letter, we demonstrate a single-photon sensitive technique for remote
displacement and vibration sensing, based on direct photon counting the ultra-
weak (single-photon level) backscattered optical signal by using a Geiger mode
APD. Our single-photon vibrometry (SPV) technique measures changes in the
photon flux that occur as the target surface is slightly displaced and tilted,
similar to other intensity based systems, but is suitable for low flux
applications where the number of returning photons is much less than one per
probe pulse. Using 50 MHz repetition rate pulses, the variation in photon
counts is captured by integrating the number of photon detections over a
preset dwell-time ($\Delta t$) using a field-programmable-gate-array (FPGA).
This process is repeated to form a time-series of photon counting measurements
with sampling rate limited by the programmable bandwidth set by the FPGA. The
digital nature of photon counting circumvents the need for a complex chain of
electronic amplifiers and filters commonly found in analog detection systems.
The time-series measurement generated by the FPGA is passed via ethernet to a
computer for analysis.

When a vibrating target is well sampled with high contrast to noise ratio, the
motion of the target is directly manifest in the time-series. The vibration
frequency can be resolved even when the signal contrast is below the shot
noise by identifying the peaks in the Fourier Transformed spectrum. SPV is
fundamentally limited by the Poisson noise that is intrinsic to any photon
counting measurement. Nonetheless, the statistically random nature of such
shot noise leads to unbiased baselines in both time and frequency domains,
which can be mitigated through post-processing. While we only show a simple
method of post-processing in this paper, the one-dimensional data offer an
easily accessible platform for more extensive digital signal processing
techniques such as signal smoothing, frequency filtering, and harmonic
analysis. We show SPV can sense mechanical displacements down to 110 nm, and
resolve vibration frequencies from DC up to 4 kHz with less than 0.01
detections per pulse.

Figure 1: Experimental setup. A Laser sends an optical pulse train to be
launched from a transceiver, and a synchronizing electronic signal to trigger
the time-gate of InGaAs APD. A probe beam is guided towards PZT using mirrors.
A fiber circulator passes the returning probe to APD for detection.

The current SPV setup is shown in Fig 1. A mode-lock laser is used to generate
the pulse train with 6 ps full width at half maximum (FWHM) and 50 MHz
repetition rate at 1554 nm. The laser sends a synchronized electronic pulse to
the InGaAs APD to trigger the time-gated detection. Collimated probe pulses
(Gaussian beam diameter: 2.2 mm) are transmitted toward the scene through an
optical transceiver and programmable scanning MEMS mirror. A fiber optics
circulator separates the outgoing signal pulses and the incoming backscattered
photons with a minimum isolation ratio of 55 dB. The transceiver is based on a
simple monostatic coaxial arrangement using off-the-shelf optical components.
It is comprised of an angle-polished single mode fiber (SMF) coupled to an
aspheric lens, providing diffraction limited collimation of the probe laser
beam. The MEMS mirror allows for easy beam steering towards the desired
target, and the use of a coaxial transceiver negates the need for careful
alignment between separate emitter and receiver when the distance to the
target is changed. The low numerical aperture of the SMF acts as a spatial
filter, with a small angle of acceptance, making the transceiver sensitive to
spatial and angular changes in the returning signal that occur as the target
moves. Intensity-based vibration sensors using the coupling sensitivity of SMF
have been demonstrated previously [16, 17, 18, 19], but these sensors are
built for very short working distances for localized displacement sensing and
do not achieve single-photon sensitivity.

We test the performance of SPV using a piezo-electric-transducer (PZT) placed
1 meter from the transceiver, for precise control over the target motion. The
beam was fixed at a nearly normal angle of incidence to the surface of the
PZT, and parallel to the axis of displacement. The average outgoing power was
0.15 mW and additional fiber attenuators were used to reduce the strength of
the returning signal prior to detection, thus imposing photon-starved
conditions. Direct detection of the attenuated signal is done using an InGaAs
APD with detection efficiency of 10 % within a 1 ns gate. The detector dark
count rate is about 50 kHz and the deadtime is 100 ns.

Figure 2: Top: Photon counting rate time-series with dwell-times of 1 ms
(blue), 10 ms (red), and 1 second (yellow). Bottom: Photon counting rate as a
function of the PZT displacement.

Data acquisition is done using a FPGA to repeatedly count the number of
detections (N) over a fixed dwell-time ($\Delta t$). The results are saved to
a file and then sent to a computer for processing. The time varying photon
count is given by

$N(t)=\Phi(t)\eta_{det}\Delta t,$ (1)

where $\Phi(t)$ is the collected photon flux (photons/second), and
$\eta_{det}$ is the detector efficiency. Note $\Phi(t)$ is directly
proportional to the repetition rate of the laser and Eq. (1) holds when the
photon detection rate is much less than 1 per gate. Considering the Poisson
nature of photon counting, the detection SNR is given by $\sqrt{N}$, and
scales as $\sqrt{\Delta t}$. Better SNR can be obtained with longer dwell-
time, but comes at the cost of detection bandwidth ($\Delta B=1/2\Delta t$).
Following the Nyquist theorem, the maximum resolvable frequency is half the
sampling rate. Target displacement is detected by a differential in photon
counting $\Delta N=|N_{1}-N_{2}|$, where $\Delta N$ is the signal contrast.
The contrast to noise ratio (CNR) is defined as $\Delta
N/(\sqrt{N_{1}+N_{2}})$ [20, 21, 22]. When the contrast is low, we can
approximate this using the mean ($\bar{N})$. The condition for discriminating
between two measurements amounts to having CNR greater than 1, or $\Delta
N>\sqrt{2\bar{N}}$.

Figure 3: (a) Time-series of 150 Hz signal with 6.5 micron amplitude, where
the contrast created is high enough that we can clearly trace the motion in
the time-domain. (b) The same setup but with 0.15 micron amplitude, where the
Poisson noise distorts time-domain signal. (c) FFT of (a), showing peaks at
first three harmonics. (d) FFT of (b), still resolving driving frequency over
noise, but not showing 2nd and 3rd harmonics. (e) Blue solid line is the ratio
of the measured standard deviation and shot noise in the time-domain. Orange
dashed line is the peak SNR in the frequency domain.

We first demonstrate the performance of this system in detecting small
displacements with low signal flux. A singular displacement leads to a shift
in the mean counting rate, this amounts to a change in the DC level of the
signal. Figure 2 (top) shows the time-series acquired with dwell-times of 1 ms
(blue), 10 ms (red), and 1 second (yellow). In each case the PZT is displaced
0.55 $\mu m$, in steps of 0.11 $\mu m$. The average counting rate was 275 kHz,
meaning only 0.0055 detections per pulse. The signals in the figure are
normalized by the dwell-time, making the contrast independent of the dwell-
time, and allowing for easier visualization. As seen, the PZT stepping shifts
the mean counting rate, and the shorter dwell-times lead to more overlap
between the signal levels, making them harder to distinguish. Figure 2
(bottom) shows the linear change in the counting rate, with the contrast equal
to about 15 kHz per 0.11 $\mu$m step. The counting rates are obtained from the
1 second dwell-time results, where there is almost no deviation in the signal
at each level. The detection noise level of the 1 ms dwell-time (blue) is 22
kHz, measured by taking the standard deviation of a one second segment. Given
the mean counting rate of 275 kHz, the detection noise level is about 5 kHz
higher than pure shot noise. Using these values we calculate a CNR of $0.48$
for the 1 ms data. From here we determine a dwell-time of 10 ms is sufficient
to achieve CNR > 1 for the 110 nm displacements. This is confirmed in Fig 2
(red) where there is almost no overlap between the signal levels at different
displacements.

We next consider the case of a harmonically vibrating target, where the motion
is characterized by repeated displacements with a defined amplitude and
frequency. The sensitivity of SPV to these displacements leads to modulation
in the photon counting at the vibration frequency, and with contrast
proportional to the vibration amplitude. For a small amplitude vibration, the
modulated photon counting time-series can be expressed as a sine wave of
amplitude $A$ oscillating about a mean value $\bar{N}$

$N(t)=Asin(2\pi ft)+\bar{N},$ (2)

where $f$ is the target vibration frequency. The ability to resolve the time-
domain dynamics depends on the CNR of each successive measurement, and how
well the signal is sampled. When the frequency is near the Nyquist limit, or
the CNR is low, the signal amplitude cannot be directly measured from the
time-series. In this case, a statistical approach can be used to characterize
the time-domain signal. The standard deviation of the measured signal
$\sigma_{m}$ is the sum of the root-mean-square of the sine wave signal
$\sigma_{s}$ and the photon counting noise $\sqrt{\bar{N}}$.

$\sigma_{m}=\sqrt{\sigma_{s}^{2}+\bar{N}}$ (3)

Recognizing this, a viable metric for characterizing the time-domain signal
amplitude is to divide the measured standard deviation by the intrinsic photon
counting noise to arrive at $\sigma_{m}/\sqrt{\bar{N}}$. This scales linearly
with the vibrating signal amplitude, and is equal to 1 when the measured
standard deviation is only from the shot noise. In many cases, it is not
necessary to precisely resolve the dynamics of the time-domain signal. The
vibration frequency and amplitude information can be obtained by identifying a
peak in the frequency-domain representation of the signal, obtained by
applying fast-fourier-transform (FFT) to the photon counting time-series.

We test low flux vibration sensing, and the amplitude response of SPV, by
driving the PZT with a 150 Hz near sine wave signal at displacement amplitudes
ranging from 0.15 $\mu m$ to 6.5 $\mu m$. The mean detection rate was fixed at
about 350 kHz (0.007 detection per pulse). Data was collected for two seconds
with a 1 ms dwell-time. The 150 Hz is well below the 500 Hz Nyquist frequency,
allowing the time-domain waveform to be resolved and higher order harmonics to
be visible in the spectrum. A zoom-in of the time-series for the largest and
smallest displacements are shown in Fig 3 (a) and (c), their FFT results in
Fig 3 (b) and (d). Larger amplitude vibration leads to high CNR between each
successive measurement, directly revealing the target motion in the time-
series. When the vibration amplitude is small, the signal contrast is obscured
by the relative noise. In the time-domain, this noise is completely overlapped
with the signal. However, after taking the FFT, the white noise is spread
across the frequency domain and the signal is condensed to a narrow region,
creating better contrast between the two. The ability to distinguish the
signal from the white noise is given by the peak SNR, defined by the ratio
between the FFT amplitude at the signal frequency and the standard deviation
across other frequencies. Figure 3 (e) shows the linear change in SNR in both
frequency and time domains. The two trends are nearly identical because the
total signal and noise is the same in both representations.

We demonstrate the limits of the current SPV bandwidth by resolving a 4 kHz
vibration, with an amplitude of 2.3 $\mu$m. Here the dwell-time was shortened
to 0.1 ms (10 kHz sampling rate), and the mean counting rate was 640 kHz.
Higher bandwidth measurements require the use of shorter dwell-time, which
naturally leads to higher photon counting noise. Additionally, the 4 kHz
signal is near the Nyquist frequency and does not benefit from oversampling.
Figure 4 (a) shows the raw FFT results for a 10 second integration time. The
signal peak cannot be identified due to the interference from the noise floor.
However, the noise floor does reduce with a longer integration time, as shown
in Fig 4 (b). Similar to the normalized time-domain noise, the frequency
domain noise level is inversely proportional to the square-root of the total
integration time. This is because the photon counting shot noise is purely
white noise. Figure 4 (c) shows the results when a 10 Hz moving-mean filter is
applied to the raw FFT results of the 10 second integration. The moving-mean
is a common technique for smoothing data, and is sufficient post-processing to
resolve the peak at 4 kHz. It is useful here to further reduce the randomly
distributed noise, but comes at the cost of frequency resolution. Extending
the windowing length of the filter can lead to oversmoothing and the loss of
signal contrast altogether. Figure 4 (d) shows the diminishing improvement in
peak SNR as the window length is extended beyond 10 Hz, which indicates the
signal linewidth to be around 10 Hz.

Figure 4: (a) The raw FFT results of the 4 kHz vibrating signal sampled at 10
kHz integrated over 10 seconds. (b) The frequency domain noise level (standard
deviation) as a function of the total integration time. (c) FFT results in (a)
after applying a 10 Hz moving mean filter, revealing the 4 kHz vibrating
signal peak. (d) The peak SNR vs. the artificial frequency resolution set by
the moving mean filter.

In summary, we have demonstrated single-photon sensitive optical vibrometry
based on photon counting of very faint returning signals. It samples the
photon detection rate to directly measure the intensity modulated signal
backscattered from a target to retrieve the vibration information. Capable of
operating with less than 0.01 average detected photons per pulse, the SPV
technique has shown to detect vibration displacements as small as 110 nm and
resolve frequencies up to 4 kHz. The measurement bandwidth of SPV can be
increased using higher repetition pulses or a more efficient detector,
possibly allowing for single-photon sensitive vibrometry up to ultrasonic
frequency, useful for optical elastography [23], acousto-optic sensing [24]
and single photon imaging [25].

## Disclosures.

The authors declare no conflicts of interest.

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