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Physical Review A

Laser Doppler velocimetry experiment with a water flow to measure the Fourier transform of the time-interval probability: Comparison between experimental and theoretical results

Jose Luis Tornos, M. Á. Rebolledo, J. M. Alvarez

Published Jul 1, 1989• 6 citations
Publisher

Abstract

In this paper a laser Doppler velocimetry (LDV) experiment with a water flow is performed to show that the experimental results for the Fourier transform of the two-photon time-interval probability ${Q}_{F}$(\ensuremath{\nu}) agree with the theoretical results derived in previous papers. The conclusions of these papers can therefore be applied to a real LDV experiment. It is concluded that by using this technique for small scattered intensities the velocities can be obtained with much more accuracy than by measuring the autocorrelation function ${g}^{(2)}$(\ensuremath{\tau}). The values of the intensity and of the Doppler frequencies that can be used are discussed.

DOI
10.1103/physreva.40.362

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In this paper a laser Doppler velocimetry (LDV) experiment with a water flow is performed to show that the experimental results for the Fourier transform of the two-photon time-interval probability QF{Q}_{F}QF​(\ensuremath{\nu}) agree with the theoretical results derived in previous papers. The conclusions of these papers can therefore be applied to a real LDV experiment. It is concluded that by using this technique for small scattered intensities the velocities can be obtained with much more accuracy than by measuring the autocorrelation function g(2){g}^{(2)}g(2)(\ensuremath{\tau}). The values of the intensity and of the Doppler frequencies that can be used are discussed.
If a beam of light is coherent, then the photons constituting the beam arrive at random moments in time. Consequently, the photon counting statistics in a given time interval below the coherence time, should in theory follow a Poissonian distribution. When the measurement interval exceeds the coherence time, the distribution should broaden and become a super-Poissonian distribution. The aim of this study is to experimentally demonstrate the Poissonian nature of light, as well as quantifying the transition to a super-Poissonian distribution. In the experiment, a low-intensity laser beam was measured with single-photon detectors, able to capture individual photons. The arrival time of the photons was detected with an oscilloscope, and the data further analysed with our own Python code. The probability distribution of the arrival time of photons was constructed and fitted with a Poissonian distribution for different time intervals. The results showed a clear Poissonian distribution for all intervals from 0.2 μs up to 2000 μs. We did however, observe tendencies of randomness for the longest measured intervals. A Poissonian distribution is defined by the variance being equal to the mean, whilst for a super-Poissonian distribution the variance is higher. The biggest measured ratio between the variance and the mean in our experiment was 1.06 for interval time 2000 μs, which is too small to infer super-Poissonian. Due to limited data memory, our dataset was restricted in length and resolution. Therefore, longer intervals could not be studied without statistical uncertainty. To cross-check the result we also calculated the probability distribution for the arrival time itself. The result perfectly fitted an exponential distribution which was in accordance with the theory of coherent light. Hence, within the given measurement accuracy and statistical uncertainty, it is safe to conclude that the laser beam is coherent for all tested time intervals. However, this does not allow us to conclude that it is a single-mode laser.
A photon-statistics technique that is based on the time-interval probability, W(θ), is studied. This technique consists of measuring the transform QR(X)=∫0∞W(θ)R(X,θ)dθ, where R(X, θ) is a square-wave function of frequency X. This function oscillates between one and zero as θ varies. The measurement of QR(X) can be carried out in a simple way. We study the error with which the period of a low-intensity square-wave-modulated light beam can be obtained by fitting a Lorentzian curve to the experimental values of QR(X). It is found that this technique is more accurate than the intensity-correlation technique when the intensity is small. The advantage increases as the intensity of the light beam decreases. This technique can be applied for those experiments in which oscillating-intensity light beams are obtained, the period providing the required information. For instance, it can be used in laser Doppler velocimetry and in quantum-beat spectroscopy.
Laser Doppler Vibrometry (LDV) is a high-precision, non-contact measurement technique that leverages optical interference between a reference beam and a Doppler-shifted measurement beam derived from coherent laser light. By detecting phase variations induced by surface vibrations, LDV has become a widely utilized tool in industrial environments for vibration analysis, structural diagnostics, and quality control. The detection sensitivity of the LDV is fundamentally constrained by the Poissonian statistics of photon arrivals. In heterodyne LDV systems, the enhancement of signal-to-noise ratio (SNR) and measurement resolution under shot-noise-limited conditions typically requires increasing the optical power of the measurement beam. However, this approach becomes infeasible in scenarios with strict constraints on thermal load or optical power. Recent studies have demonstrated that injecting squeezed vacuum states as non-classical light sources can suppress quantum noise while maintaining low optical power, thus surpassing the standard quantum limit. This thesis introduces the principle of the heterodyne LDV combined with squeezed light sources. The first scientific hypothesis proposed herein is that introducing laser light with squeezed quantum uncertainties into industrial heterodyne LDVs enables vibration measurements to exceed the physical resolution limits of traditional LDVs, achieving sub-shot-noise sensitivity. The thesis designed a squeezed-light-enhanced heterodyne LDV, achieving over 3 dB of squeezing within a 40 MHz measurement bandwidth and demonstrating a noise-equivalent displacement resolution reaching 8 fm/√Hz, thereby confirming the first hypothesis. Addressing the limitations imposed by the finite dynamic range of analog-to-digital converters (ADCs) on the SNR of digitally demodulated phase-modulated signals, the thesis proposes a clipping-based synchronous zero-crossing sampling technique and a corresponding second scientific hypothesis: that this sampling scheme enables heterodyne LDV to overcome the digital noise floor defined by the ADC's SNR. Experimental results demonstrate that this sampling and demodulation approach effectively reduces ADC quantization noise, allowing the digital resolution of the demodulated phase-modulated signal to surpass the ADC-imposed SNR limit while preserving the quantum noise-level squeezing, thereby validating the second hypothesis. Compared to previous work, this research constitutes the first experimental verification of squeezed-light-enhanced measurement feasibility in a classical heterodyne LDV operating at a 40 MHz difference frequency, advancing the application of quantum-enhanced precision metrology in industrial vibration testing.
An important feature of heterodyne laser Doppler vibrometer (LDV) is the possibility of measuring an optical path length oscillation at frequency f at a choosable frequency f_het-f, at which the photo-electric detector shows a sufficiently high dark noise clearance. The full squeezed light enhancement of a heterodyne LDV's signal to noise has not been achieved so far. Here we use a sideband spectrum that is squeezed around f_het=40 MHz and demonstrate the squeezing enhanced measurement of an optical path length vibration at f=1 MHz of about 3.5 dB while fully maintaining the signal power. The proof of principle we provide will enable the realization of ultra-precise LDVs over extended signal bandwidth for probes or environments that require low intensities.