Distributed acoustic and vibration sensing via optical fractional Fourier transform reflectometry

Lihi Shiloh; Avishay Eyal

doi:10.1364/OE.23.004296

1. Introduction

Fiber-optic sensing is a highly promising technology for a variety of applications, ranging from biomedical sensing to structural health monitoring. In the field of Distributed Acoustic Sensing (DAS) there is a growing demand for a fiber optic sensing system that can provide fast scan rates over long distances with high spatial resolution and high sensitivity [1]. Among the various approaches for implementing DAS or quasi-DAS, methods based on Rayleigh backscattering standout as practical and affordable solutions due to the use of a conventional communication fiber as the sensing element and the simplicity of the interrogation scheme. In such techniques an interrogating waveform is launched into the fiber and the Rayleigh backscattered light is detected and analyzed to extract spatiotemporal information. There are two main approaches: Optical Time Domain Reflectometry (OTDR) in which the interrogation waveform is a short pulse and the detected light is analyzed in the time domain [2–4]. The second approach is Optical Frequency Domain Reflectometry (OFDR) where the interrogation waveform is linearly chirped and the received reflections are mixed with a reference and analyzed in the frequency domain [5–7]. To facilitate sensitive detection and to avoid the need of averaging, it is required to increase the optical energy which is reflected from a given spatial resolution cell as much as possible. In OTDR, once the power of the source is set to the highest possible level, any additional increase in the returning energy can only be achieved by increasing the pulse duration and, thereby, reducing the spatial resolution. In OFDR this issue is alleviated as the spatial resolution is determined by the frequency scan range $, Δ F,$ and not by the duration of the interrogating waveform [8]. Hence, it is possible to use OFDR interrogating waveforms which are orders of magnitude more energetic than OTDR pulses and obtain higher resolution with similar or better signal to noise ratio.

Another central trade-off in reflectometry-based techniques is between the fiber length $, L,$ and the scan repetition rate $, f_{scan} .$ This is due to the requirement that the back-reflected signal, following an interrogation pulse, will arrive before the next interrogation pulse is transmitted. Accordingly, the scan repetition rate must satisfy $f_{scan} < v / (2 L) \equiv f_{scan_max}$ where $v$ is the speed of light in the fiber. In practice different technical considerations may limit the scan rate to significantly below this theoretical bound. In our recently introduced dynamical OFDR system [9] the frequency scan is performed by supplying periodic voltage signal to a piezoelectric element which directly controls the laser's instantaneous frequency. To prevent undesired transient responses the tuning element is derived with a sinusoidal rather than a saw-tooth waveform and only the linear part of the sinusoid is used. In this mode of operation the frequency scan rate must be kept low enough to ensure sufficient overlap between the linear sweep regions of the reference signal and the signal returning from the fiber end. This limits the scan rate to roughly $f_{scan} < 0.1 f_{scan_max} .$

Recently we introduced a new optical reflectometry method which relaxes this limitation and enables interrogation of distant sections of the fiber with a scan rate of $f_{scan} \approx 0.5 f_{scan_max}$ [10]. At this scan rate the back-reflected light from the fiber end arrives at the receiver with a delay of $τ = 1 / (2 f_{scan})$ with respect to the light at the reference arm. Accordingly, the time functions describing the instantaneous frequencies of the back-reflected light and the reference light have opposite phases. Hence, there will be an overlap between linear sweep segments of the signals but the slopes will have opposite signs (see Fig. 2(a)). Within this time segment the beat signal at the receiver output will have a component whose instantaneous frequency varies linearly in time. Using a coherent optical communications type receiver with I/Q detection this linearly chirped signal, which comprises both positive and negative beat frequencies, can be recorded without aliasing. There is a variety of methods to detect such a linearly chirped signal buried in measured data. Here we describe the use of the Fractional Fourier Transform (FrFT) to that end. Its main advantage in the present case is its compatibility with conventional OFDR. By adjusting a single parameter ( $α,$ the FrFT rotation angle) the FrFT will yield either the usual FFT data corresponding to conventional OFDR or detect the linearly chirped components which correspond to distant reflections. Moreover, as will be shown below, the use of FrFT enables mapping of the fiber reflection profile not only at the point $z = v / (2 f_{scan})$ but also at its vicinity. The length of the fiber section around the point $z$ where FrFT yields high spatial resolution reflection profile, is determined by the scan frequency and shown to be approximately 3.5km at $f_{scan} \approx 2 .5kHz .$ In addition, by tuning the scan frequency, it is possible to tune the measurement section and probe different regions along the fiber.

The Fractional Fourier transform was first introduced by Namias [11] and was applied to problems in quantum mechanics. Since then the transform was used in numerous applications. Mendlovich and Ozaktas [12] have shown that the FrFT is a natural tool for describing light propagation in quadratic graded index media. Other examples for its application span a wide variety of fields such as cryptography [13], image watermarks [14], chirp parameter estimation [15] and advanced bat's chirps research [15]. A significant contribution was made by Ozaktas [16] who implemented a fast discrete FrFT with computational complexity of $Ο (NlogN)$ and made FrFT a practical and useful tool in modern signal processing.

In the previous paper [10] the Optical FrFT Reflectometry (OFFR) method was introduced and preliminary experimental results were described. This paper presents, for the first time, a comprehensive study of OFFR which includes: theoretical formulation of the method and its limitations, numerical simulation of its performance in comparison with OFDR and a static, as well as dynamic, experimental results. It is shown that the method provides detection of external perturbations to the fiber at scan rates of ~2.5kHz and at a distance of more than 20km. The spatial resolution at this range and scan-rate is characterized by a measurement of the backscatter profile from a FBG's-array and is found to be ~2.8m.

2. Theory

This section outlines the theoretical background of OFFR, introduction to the Fractional Fourier transform can be found in Appendix I. The measurement system, which is similar to a typical OFDR setup, is described in section 4. The source is an ultra-narrowband laser whose frequency can be directly controlled by deriving a piezoelectric tuning element. For a sinusoidal driving voltage the laser output can be expressed as:

E (t) = E_{0} \exp {j [ω_{0} t - \frac{A}{f_{scan}} \cos (ω_{scan} t)]}

where

ω_{0} = 2 π f_{0}

is the center (angular) optical frequency

, f_{scan}

is the scan frequency and

A

is proportional to the amplitude of the driving voltage. The instantaneous frequency deviation from

f_{0}

is given by

: δ f (t) = A \sin (ω_{scan} t) .

In this case the signal at the coherent I/Q receiver output corresponding to a discrete reflector with a delay of

τ = 2 z / v,

can be expressed as:

s (t, z) = a r (z) {| E_{0} |}^{2} \exp {j [\frac{2 A}{f_{scan}} \sin (ω_{scan} (t - \frac{z}{v})) \sin (ω_{scan} \frac{z}{v}) - θ (z)]}

where

θ (z)

is the roundtrip phase associated with point

z, r (z)

is the reflection coefficient and

a

describes the responsivity of the receiver, the launched power etc. The instantaneous frequency of this signal is a function of time and depends on the distance of the reflector. It can be expressed as:

f_{inst} (t) = 2 A \sin (ω_{scan} \frac{z}{v}) \cos (ω_{scan} (t - \frac{z}{v}))

For close reflectors that satisfy

ω_{scan} z / v ≪ 1

there is a time window

| ω_{scan} t | ≪ 1

for which

f_{inst} (t) \approx 2 A ω_{scan} z / v .

Namely, by properly choosing the time window, the signal of a close reflector is sinusoidal with a constant frequency and therefore the reflection profile of close reflectors can be obtained by a simple Fourier transform. This analysis of the received data is the common practice in conventional OFDR. For distant reflectors at the vicinity of

z = v / (4 f_{scan})

the instantaneous frequency at the observation window becomes:

f_{inst} (t) \approx 2 A ω_{scan} t .

Denoting the deviation from

z = v / (4 f_{scan})

as

Δ z,

the signal at the observation window takes the form of a linear chirp (quadratic phase dependence) as follows:

s (t, z) \approx E_{1} \exp {j [\frac{1}{2} χ t^{2} + ζ t + \tilde{θ}]}

where

E_{1} = a r (z) {| E_{0} |}^{2}, χ = 4 π A ω_{scan}, ζ = - 4 π A ω_{scan} Δ z / v

and

\tilde{θ}

is time independent phase. As described in Appendix II, when FrFT is applied to the output of the detector, all reflectors near

z = v / (4 f_{scan})

yield peaks for the same rotation angle

, α_{opt} [rad]

but at different positions along the FrFT axis. Moreover, as shown in Appendix II, the separation in the FrFT domain between these reflectors will be proportional to the delay difference between them. Hence, the use of FrFT yields the reflection profile of the fiber in a certain section of the fiber whose center can be tuned by varying

f_{scan} .

3. Simulation

To check the proposed method, a computer simulation was performed. The fiber's impulse response was used to characterize the back reflected light received at the detector. It was calculated by dividing the fiber to small sections and assigning to each one of them a random backscatter coefficient with complex Gaussian distribution as well as an appropriate delay and loss. The simulated sensing fiber comprised 10 equally spaced discrete reflectors (10m separation) in addition to the Rayleigh scatterers, similar to the one used in the experiment. The first reflector was positioned at $z = 20.4 km$ and the total fiber length was approximately 21km. For this fiber length, implementation of OFDR in our system necessitated limiting the scan frequency to 500Hz while for OFFR the scan frequency could be increased to 2445Hz. The simulated signal was analyzed using both FFT and FrFT, where in both cases a Hanning window was used prior to the transformation to reduce truncation effects.

In the case of OFDR two window sizes were used: one with duration equal to the window used in OFFR $, Δ t,$ and the other with a longer duration but with the same scan range $, Δ F .$ The instantaneous frequency of the reference arm and the sensing arm for the conventional interrogation $(f_{scan} = 500 Hz)$ are shown in Fig. 1(a). The red and green segments are the sampled windows for equal- $Δ t$ window and equal- $Δ F$ respectively. The simulated signal plotted in Fig. 1(b) shows poor resolution for both windows. In the case of the equal- $Δ t$ window the low resolution (~50m) stems from the narrow frequency scan range. In the equal- $Δ F$ case the frequency scan range is sufficient for resolving the reflectors but the resolution is hindered by the non-linearity of the sweep for distant reflectors. In contrast, for OFFR the instantaneous frequency variations of the reference and sensing arms, for a reflector at 20.4km, at the analysis window, show linear time-dependence but are crossed (Fig. 2(a)). The reflection profile as obtained by OFFR is shown in Fig. 2(b). The superior resolution is evident. Clearly, the resolution of OFFR can be increased further by increasing $, Δ F$ (see for example Fig. 2(b) where we used twice the scan range) but this was not feasible with the experimental setup that was used.

Fig. 1 Simulation of OFDR-type interrogation of a 21km FBG fiber with scan frequency of 500Hz. (a) Instantaneous frequency in reference (solid black) and sensing (dash black) arms and sampled signal segment in solid red/green, (b) The normalized reflection profile.

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Fig. 2 Simulation of OFFR interrogation of a 21km FBG fiber with scan frequency of 2445Hz. (a) Instantaneous frequency in reference (solid black) and sensing (dash black) arms and sampled signal segment in solid red. (b) Normalized reflection profile (red) and a higher resolution profile obtained with a scan range of $~ 2 Δ F$ (gray).

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3. Experiment

To test the new method the setup in Fig. 3 was constructed. It consisted of a 1550nm ultra-coherent tunable laser source whose instantaneous frequency was varied sinusoidally with time. The laser output was split between a reference arm and a sensing arm. The reference arm was connected to a variable attenuator and then to the Local Oscillator (LO) port of a dual-polarization $90^{o}$ optical hybrid. The other output of the splitter was connected to the sensing fiber via an optical circulator. The output of the circulator (port 3) was fed into the signal port of the optical hybrid. The optical hybrid enabled the detection of all four quadrature components of the signal, real and imaginary part of each polarization but only one polarization component was used. Two cascades of fibers were used for characterizing the performance of the method: the first comprised a 20km fiber (on a spool) connected to a 0.94km fiber with 10 weak fiber Bragg gratings (FBGs) inscribed in its middle. The reflectivities of the FBG's were all ~0.5% and they were equally spaced with separations of 10m. The purpose of this setup was to test the spatial resolution of the measurement method. The second cascade comprised a 20km SMF spool followed by a 0.1km SMF spool connected to a 0.9km SMF spool. This arrangement was used to test the ability of the method to sense dynamical excitations such as vibrations and acoustical signals. The receiver output was acquired using an Infiniium 9000A Agilent scope with sampling frequency of 1GHz, and the results were processed in Matlab.

Fig. 3 The experiment setup. (a) used for dynamic characterization, (b) used for spatial resolution characterization.

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As in the simulation, the scan frequency was limited to $f_{scan} = 500Hz$ when the system was used to implement OFDR and was increased to 2445Hz in the case of OFFR. The amplitude of the driving signal to the PZT was 20Vpp for all experiments and the processed time windows were $~32 .7μsec$ and $~160 .5μsec$ for the equal- $Δ F$ case (which yielded $~46 .7MHz$ ).

4. Experimental results

OFDR spectrogram and reflection profile of the sensing fiber with the discrete reflectors are plotted in Figs. 4(a) and 4(b) respectively. The negative frequencies visible in Fig. 4(a) are residuals of the imperfect cancelation of the real and imaginary components associated with the FBG's. In contrast with OFFR, the conventional OFDR approach makes no use with the negative frequencies. Despite the reduced scan frequency, the achieved spatial resolutions for both window widths, are rather poor and the discrete reflectors are practically unresolved, as predicted in the simulation.

Fig. 4 Experimental results, OFDR interrogation of 10 discrete FBG reflectors at 20.4km, at a scan frequency of 500Hz: (a) Spectrogram, (b) A single reflection profile (one horizontal cross-section of the spectrogram).

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To implement OFFR, the scan frequency was increased to 2445Hz. This scan frequency was chosen so that the reflections from the distant fiber end and from the reflector array in its vicinity would arrive at the detector with a delay of $~ 0 .5 / f_{scan}$ with respect to the reference. The measured reflection signal was processed using FrFT. The optimal FrFT-rotation angle was determined using a transient feature extraction criterion [17], but can also be theoretically calculated if the system parameters are known [18]. The spectrogram of the measured response and the FrFT domain spectrum, obtained by rotation in $α_{opt},$ are plotted in Figs. 5(a) and 5(b) respectively. The discrete reflectors as well as the fiber end are clearly observed and well resolved. Comparing with the simulation results (Fig. 2), it can be seen that the experimental peaks to floor ratio is lower. This can be attributed to the presence of an undesired reflector close to the input of the sensing fiber whose wide FrFT response adversely affected the profile at ~20.5km. Similar effect could be observed in a simulated response with the addition of a 1.6% reflector at 24.5m (Fig. 5(b)). To estimate the spatial resolution, the FBG's peaks were fitted to Gaussians. The inset of Fig. 5(b) shows a zoom on the peak of one of the FBGs (red) together with a fit to a Gaussian whose full width half max is 2.8m. The rest of the peaks had similar widths.

Fig. 5 Experimental results of OFFR measurment of 10 discrete FBG reflectors at 20.4[km] and scan frequency of 2445[Hz]. (a) Time-frequency Spectrogram, (b) the signal in FrFT domain, averaged over 35 consecutive measurements (red), simulated signal with an undesired close reflector (grey). Inset: zoom on the peak of a single FBG (red) and its Gaussian fit (blue).

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Dynamic measurements were taken by gently “tapping” the 100m spool which was connected between the 20km fiber and the 900m fiber (see Fig. 3). The gentle “tapping” on the fiber was performed at irregular rate and induced local phase variations at random positions in the 100m fiber. As in phase-sensitive OTDR [3] these phase variations led to correlated variations in the Rayleigh backscatter profile. OFFR-generated spectrograms of the excited and un-excited measurements are shown in Fig. 6. The excited fiber section is indicated in Fig. 6(b) in the FrFT domain. For better visualization, zoom-in on the excited section is provided in Fig. 7(a). While the temporal variations in the reflection profile clearly reveal the excited section, to obtain even better contrast of the perturbations it is possible to employ image processing tools. For example, in the current experiment it was found that the following steps yield enhanced contrast: first, the spectrogram is processed via an edge detection 2D-filter, next, it is low-pass filtered in both dimensions $(f_{time, cut-off} \approx 305 [Hz], f_{distance, cut-off} \approx 0.24 [m^{- 1}])$ and normalized. Then, the result is subtracted from the normalized original spectrogram and plotted in dB scale (Fig. 7(b)). This result suggests that the method can be used to produce the input for detection algorithms. For comparison, similar measurements were made using OFDR. While the effect of the excitation was evident, the spatial resolution was ~3 times worse than in the OFFR measurement and the scan frequency was ~5 times slower.

Fig. 6 OFFR spectrograms (a) un-excited measurement (b) “tapping” on the the 100m spool.

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Fig. 7 Zoom on the perturbed section: (a) original OFFR spectrogram (b) following image enhancement.

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The output of the system can also be represented in the form of a multisensor seismogram. The deviations of the backscatter amplitudes from their temporal means, at equally spaced locations along the fiber (12m separation), as a function of time, are plotted in Fig. 8. The graphs were low-pass filtered (same low-pass filter as above) and shifted vertically for clarity. The green lines represent the levels of zero deviation from the mean. The effect of the tapping on the 100m spool is seen in the plot as increased fluctuations of the graphs which correspond to the excited spool.

Fig. 8 The backscatter profile at discrete locations along the fiber (~12m spacing) as a function of time. The responses are shifted vertically for clarity. The green lines represent the level of zero deviation from the mean.

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5. Discussion and conclusions

OFDR is known for its excellent sensitivity, dynamic range, maximum fiber length and spatial resolution. There are many demonstrations of its effectiveness in characterizing the reflection profile of fiber optic systems at various lengths and resolutions. However, the use of OFDR for dynamical measurements is much rarer. This is because the implementation of high scan-repetition-rate OFDR systems, while maintaining its other merits, is challenging. The method proposed in this paper, OFFR, achieves ~2.8m spatial resolution, at a distance of ~20km and at a scan repetition rate of ~2.5kHz. It is implemented with a very simple and affordable configuration and uses a trivial narrowband RF waveform for generation of its optical interrogation signal. The same setup can be used for implementing conventional OFDR and both methods can be used alternately, with minor modifications, to measure the same sensing system.

6. Appendix 1. Fractional Fourier transform

The Fractional Fourier transform is a linear operator defined as:

X_{α} (u) = F^{α} {x (t)} \equiv \int_{- \infty}^{\infty} x (t) κ_{α} (t, u) d t

where

κ_{α} (t, u),

the operator kernel, is given by:

κ_{α} (t, u) = {\begin{matrix} \sqrt{\frac{1 - j \cot α}{2 π}} e^{j \frac{(u^{2} + t^{2}) \cot α}{2}} e^{- j \frac{u t \csc α}{2}} & α \neq 2 π k, k \in ℕ \\ δ (t - u) & α = 2 π k, k \in ℕ \\ δ (t + u) & α \pm π = 2 π k, k \in ℕ \end{matrix}

Here

δ (t)

is Dirac's delta function,

α

is the FrFT rotation angle and

t

and

u

are the time and FrFT-domain parameters respectively. It can be seen that the FrFT becomes the Fourier transform (FT) for

α = π / 2 + 2 πk,

or the inverse FT for

α = - π / 2 + 2 πk .

For

α = 2 πk

the FrFT becomes the identity operator. By changing the value of

α,

the representation of the signal can be continuously changed between its time-domain representation

, x (t),

to its frequency-domain representation

, X (ω),

and back to its time domain representation (for

α = 2 π) .

FrFT can be viewed as an operation which produces a transformed signal whose spectrogram is rotated with respect to the spectrogram of the original signal at an angle of

α .

Many different properties of FrFT can be found in the literature [19].

For a linear chirp, which appears as a section from a straight line in the time-frequency space, an optimal angle $, α_{opt},$ can be found such that the transformed signal in the FrFT domain is a delta function. Intuitively, the FrFT can be thought of as an operation which transforms a linear chirp into a delta function in the same manner as the FT transforms a sinusoidal function into a delta function. In order to automatically find the optimal rotation angle $, α_{opt},$ the following procedure can be taken: the FrFT is calculated for a range of relevant rotation angles. To each transform the kurtosis parameter is calculated [17]. The kurtosis is a parameter which describes how peaked a function is, namely, it quantifies the degree of its resemblance to a delta function. The optimal rotation angle is chosen as the angle for which the kurtosis is maximized. Alternatively, it is possible to calculate $α_{opt}$ analytically using simple trigonometry when the system parameters are known [18].

6. Appendix 2. FrFT reflectometry - mathematical formulation

In this appendix the mathematical formulation behind OFFR is described. Consider a signal with sinusoidally varying instantaneous frequency:

E (t) = E_{0} \exp {j [ω_{0} t - \frac{A}{f_{scan}} \cos (ω_{scan} t)]}

The signal is used for interrogation in an OFDR-like measurement system. In this case the sampled signal at the I/Q receiver output, after interference with a backscatter signal from a discrete reflector at

z,

can be expressed as:

s (t, z) = a r (z) {| E_{0} |}^{2} \exp {j [\frac{2 A}{f_{scan}} \sin (ω_{scan} (t - \frac{z}{v})) \sin (ω_{scan} \frac{z}{v}) - θ (z)]}

where

θ (z)

is the roundtrip phase associated with point

z, r (z)

is the reflection coefficient and

a

describes the responsivity of the receiver, the launched power etc. The signal corresponding to the measured fiber section in OFFR is obtained by substituting

z = v / (4 f_{scan}) + Δ z :

s (t, z) \approx a r (z) {| E_{0} |}^{2} \exp {j [2 π A ω_{scan} {(t - \frac{Δ z}{v})}^{2} - \frac{2 A}{f_{scan}} - θ]} = E_{1} \exp {j (\frac{1}{2} χ t^{2} + ζ t + \tilde{θ})}

Therefore we obtain the chirp of Eq. (4) with the following parameters:

E_{1} = a r (z) {| E_{0} |}^{2}, χ = 4 π A ω_{scan}, ζ = - 4 π A ω_{scan} Δ z / v, \tilde{θ}

is time independent phase. In order to calculate the fractional FT of

s (t, z),

the following transform properties are used [19]:

FrFT {\exp (j \frac{b t^{2}}{2})} = \sqrt{\frac{1 + j \tan α}{1 + b \tan α}} \exp {j \frac{u^{2} (b - \tan α)}{2 (1 + b \tan α)}}

FrFT {x (t) \exp (j b t)} = X^{α} (u - b \sin α) \exp {j [u b \cos α - \frac{b^{2} \sin α \cos α}{2}]}

Using Eq. (10) and Eq. (11) the fractional Fourier transform of

s (t, z)

becomes:

S^{α} (u) = {\tilde{E}}_{1} \sqrt{\frac{1 + j \tan α}{1 + χ \tan α}} \exp {j [\frac{{(u - ζ \sin α)}^{2} (χ - \tan α)}{2 (1 + χ \tan α)} + ζ u \cos α - \frac{ζ^{2} \sin α \cos α}{2}]}

where

{\tilde{E}}_{1} = E_{1} \exp {j \tilde{θ}} .

Substituting the optimal FrFT rotation angle of a linear chirp,

α_{opt} = arc \tan (χ)

[18], the transform simplifies to:

S^{α_{opt}} (u) = E_{2} \exp {j \frac{u ζ}{\sqrt{χ^{2} + 1}}}

where

E_{2} = {\tilde{E}}_{1} \sqrt{(1 + j χ) / (1 + χ^{2})} \exp {- j 0.5 ζ^{2} χ / (χ^{2} + 1)} .

Eq. (13) is the FrFT of

s (t, z)

in the

u

-FrFT domain. The corresponding signal in the orthogonal

w

-FrFT domain can be obtained by taking the FT of Eq. (13) (equivalently, it is possible to use

α'_{opt} = α_{opt} + π / 2

in Eq. (12) to obtain the representation of

s (t, z)

in the

w

-FrFT domain in one step, however, the mathematical steps are somewhat more complex):

{\bar{S}}^{α_{opt}} (w) = ℱ {S^{α_{opt}} (u)} = {\tilde{E}}_{1} \sqrt{\frac{1 + j χ}{2 π (1 + χ^{2})}} \exp {- j \frac{ζ^{2} χ}{2 (χ^{2} + 1)}} δ (w - w_{1})

where

w_{1} \equiv 2ζ / \sqrt{χ^{2} + 1}

is, therefore, the theoretical position of the peak, which represents the chirp in the

w

-FrFT domain. The meaning of this result is that a discrete reflector at

z = v / (4 f_{scan}) + Δ z

will show up as a peak in the

w

-FrFT domain in the point:

w_{1} = \frac{- 4 π A ω_{scan} 2 Δ z}{v \sqrt{{(4 π A ω_{scan})}^{2} + 1}} \approx - τ

Hence, the position of a peak in the FrFT domain is linearly dependent on

Δ z

and for high gamma chirps it is approximately

τ .

Moreover, equally spaced array of reflectors will show up as an array of equally spaced peaks. This result is verified in the static experiment presented in section 4, Fig. 5(b).

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Distributed acoustic and vibration sensing via optical fractional Fourier transform reflectometry

Abstract

1. Introduction

2. Theory

3. Simulation

3. Experiment

4. Experimental results

5. Discussion and conclusions

6. Appendix 1. Fractional Fourier transform

6. Appendix 2. FrFT reflectometry - mathematical formulation

References and links

Cited By

Figures (8)

Equations (15)

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