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Brillouin optical correlation-domain analysis (B-OCDA) allows for distributed measurements of strain and temperature with sub-cm resolution. Time-multiplexing techniques have previously extended B-OCDA to the monitoring of many km of fiber and two million resolution points. Thus far, however, the number of scans of correlation peaks positions, necessary to cover the fiber under test, was restricted to the order of 100 or more. In this work we report a B-OCDA protocol that is able to address an entire fiber using only 11 pairs of position scans per choice of frequency. The measurements protocol relies on a merger between B-OCDA principles and double-pulse-pair analysis, previously incorporated in time-domain Brillouin sensors. Phase coding of the pump and signal waves with a repeating, short and high-rate code stimulates Brillouin interactions in a large number of narrow correlation peaks, with substantial temporal overlap. Unambiguous measurements are achieved by repeating each experiment twice, using a pair of pump pulses of different durations, and subtracting the two output traces. The principle is demonstrated in the analysis of a 43 m-long fiber with 2.7 cm resolution. Several local hot-spots are properly identified in the measurements. The experimental uncertainty in the measurement of the local Brillouin frequency shift is estimated as ± 1.9 MHz. The proposed method requires broader detection bandwidth and a larger number of averages than those of previous time-gated B-OCDA setups. Hence the overall number of measurements is similar to that of previous setups.

1. Introduction

Stimulated Brillouin scattering (SBS) in optical fibers is a nonlinear interaction between counter-propagating pump and signal waves [1]. The interaction is mediated by an acoustic wave that is stimulated by the two optical fields through electro-striction. SBS may lead to the amplification of signal waves at optical frequencies that are lower than that of the pump. Amplification, however, is highly narrow-band: it requires that the difference between the optical frequencies of pump and signal should closely match the Brillouin frequency shift (BFS) of the fiber, which is on the order of 11 GHz for standard fiber at 1550 nm wavelength. The Brillouin gain spectrum (BGS) is only 30 MHz-wide. Since the BFS varies with both temperature and mechanical strain, measurements of the local BGS serve in the distributed monitoring of both quantities for over 25 years [2,3]. State-of-the-art distributed Brillouin sensors can cover hundreds of km [4], perform sound and vibration monitoring at kHz rates [5], and recover the local BFS with better than 1 MHz precision [6].

One of the key metrics of distributed Brillouin analysis is its spatial resolution. The most commonly-employed measurement protocol, known as Brillouin optical time-domain analysis (B-OTDA), relies on the amplification of a continuous signal wave by a pump pulse [2,3]. The spatial resolution of the fundamental B-OTDA protocol is determined by the pulse duration, which must exceed in turn the lifetime associated with the stimulation of the acoustic wave: approximately 5-10 ns [7]. Hence the spatial resolution of basic B-OTDA is restricted to the order of 1-2 m.

Numerous solutions for improving B-OTDA resolution were proposed and demonstrated (for details see a recent review and references therein [6], as well as [8]). One of the most successful approaches is based on a double-pulse pair (DPP) [9]: in DPP B-OTDA each trace of the output signal is acquired twice. Both measurements involve pump pulses that are sufficiently longer than the acoustic lifetime, however their exact durations slightly differ. The subtraction of the two output traces provides distributed analysis with high spatial resolution that is governed by the difference between the durations of the two pulses, rather than their absolute durations [9]. DPP B-OTDA was successfully performed over 2 km with 2 cm resolution [9]. The main advantage of the protocol is the addressing of the entire fiber under test (FUT) with a single pair of measurements per choice of frequency offset between pump and signal. On the other hand, the method mandates a broad detection bandwidth which degrades the measurements signal-to-noise ratio (SNR).

An alternative paradigm for distributed Brillouin sensors is that of Brillouin optical correlation-domain analysis (B-OCDA), which was first developed by Hotate and associates [10]. B-OCDA relies on the close connection between the strength of the SBS interaction at a given location and the cross-correlation between the temporal complex envelopes of the pump and signal waves at that position [10,11]. The careful joint modulation of pump and signal, in frequency or phase, may largely confine their cross-correlation to discrete and narrow peaks in which the acoustic stimulation is stationary [10,11]. B-OCDA can retrieve the local BGS at the correlation peak with comparatively narrow-band detection and exceptional spatial resolution, down to mm-scale [12]. However, the main drawback of the concept is the need for scanning the correlation peak position along the entire FUT, one resolution point at a time.

Considerable reduction in the necessary number of position scans has been achieved using hybrid B-OTDA / B-OCDA (or time-multiplexed B-OCDA) approaches [13–18]. In these schemes the pump and signal are jointly and repeatedly phase-modulated by a comparatively short, high-rate code, and the amplitude of the pump wave is also modulated by a single pulse. The propagation of the pump pulse sequentially stimulates SBS at multiple correlation peaks in a single flight along the FUT. The interaction in each peak is restricted to the duration of the pulse, and therefore multiple Brillouin gain events may be resolved by direct temporal analysis of the output signal wave [14,15,17,18].

In one realization of the method, an entire set of 440,000 high-resolution points was successfully addressed using only 211 scans of correlation peaks positions [17]. A different experiment achieved a record-high number of over 2 million potential resolution points [18]. However, the minimum separation between adjacent correlation peaks in these measurements equaled the spatial extent of the pump pulse, which was again limited to 2 meters or longer. Consequently, thus far B-OCDA with cm-scale resolution required about 100 or more scans of correlation peaks positions, in order to avoid temporal overlap between gain events at neighboring peaks and ambiguous analysis.

In this work, we combine between the principles of DPP acquisition and time-multiplexing in order to reduce the number of correlation peak positions scans in B-OCDA by an order of magnitude. The number of scans does not depend on the length of the FUT. The pump and signal waves are jointly phase-modulated by a very short phase code: the sequence contains only 11 bits, and its period is much shorter than the duration of the overlaying pump pulses and even shorter than the acoustic lifetime. Therefore, the SBS amplification events taking place at individual peaks cannot be directly separated using time-domain analysis of a single trace of the output signal. However, unambiguous measurements are successfully restored by acquiring each trace twice, using pump pulse durations that are somewhat different, as in DPP setups. The subtraction of the two traces recovers individual SBS gain events. The measurement protocol may also be regarded as DPP B-OTDA with underlying phase coding.

The principle is supported by direct numerical integration of the SBS equations subject to the appropriate boundary conditions. The proposed method is experimentally demonstrated in the analysis of a 43 m-long fiber with 2.7 cm resolution. The entire set of 1,600 resolution points is addressed using only 11 pairs of traces per frequency, the smallest number of necessary position scans of any B-OCDA experiment to-date. Multiple local hot-spots are properly identified in the measurements. The experimental uncertainty in the measurement of the local BFS is estimated ± 1.9 MHz.

The remainder of this paper is organized as follows: the principle of operation and numerical simulations are described in section 2. Experiments are reported in section 3, and a concluding discussion is given in section 4.

2. Principle of operation and simulations

Let us denote the complex envelopes of the pump and signal waves as $A_{p, s} (z, t)$ where $t$ stands for time and $z$ represents position along the FUT. The pump wave is launched at $z = 0$ in the positive $z$ direction, whereas the signal propagates from $z = L$ in the negative $z$ direction, with $L$ the length of the FUT. The group velocity of light in the FUT is $v_{g}$ . The central optical frequencies of the pump and signal are $ω_{p, s}$ , and their difference equals $Ω = 2 π ν \approx Ω_{B} (z)$ where $Ω_{B} (z)$ is the BFS of the FUT at point $z$ . The Brillouin gain linewidth of the FUT is denoted by $Γ_{B} \approx 2 π \cdot 30 MHz$ .

The modulation of pump and signal follows that of time-multiplexed B-OCDA setups [14,15,17,18]. The signal wave is phase-modulated by a binary sequence with symbol duration $T_{p h a s e}$ , which is on the order of hundreds of ps:

(1)

A_{s} (z = L, t) \equiv {\tilde{A}}_{s} (t) = A_{s 0} \sum_{n} c_{n} rect (\frac{t - n T_{p h a s e}}{T_{p h a s e}}) .

In Eq. (1) $A_{s 0}$ is a constant magnitude, $rect (ξ) = 1$ for $| ξ | \leq 0.5$ and equals zero elsewhere, and ${c_{n}}$ are the elements of a perfect Golomb code which is repeated with a short period $N_{p h a s e}$ = 11 bits [14,19,20]. The magnitude of all code elements is unity. The phases of elements {1,2,3,5,6,8} equal zero, whereas the phases of elements {4,7,9,10,11} are given by $φ = \cos^{- 1} [- (N_{p h a s e} - 1) / (N_{p h a s e} + 1)]$ ~2.556 rad. Off-peak values of the cyclic auto-correlation function of perfect Golomb codes are exactly zero [19]. This property helps reduce noise due to off-peak Brillouin interactions in B-OCDA setups [14,15,17,20]. The pump wave is phase-modulated by the same code. In addition, the amplitude of the pump is modulated by a single pulse of duration $T_{p u l s e}$ , which is several times longer than the acoustic lifetime:

(2)

A_{p} (z = 0, t) \equiv {\tilde{A}}_{p} (t) = A_{p 0} rect (\frac{t}{T_{p u l s e}}) \sum_{n} c_{n} rect (\frac{t - n T_{p h a s e}}{T_{p h a s e}}) .

Here $A_{p 0}$ is a constant magnitude. Note that unlike previous implementations of hybrid B-OCDA / B-OTDA [14], the temporal period of the phase code $N_{p h a s e} T_{p h a s e}$ is deliberately chosen to be much shorter than the pump pulse duration $T_{p u l s e}$ .

It is assumed that the SBS interaction between pump and signal is confined to narrow segments, so that the changes in intensity of both waves are small. In this condition, the magnitude of the stimulated acoustic wave at position $z$ and time $t$ may be approximated by [1,11]:

(3)

Q (z, t) = j g_{1} \int_{0}^{t} e^{- Γ_{A} (t - t')} {\tilde{A}}_{p} (t' - \frac{z}{v_{g}}) {\tilde{A}}_{s}^{*} [t' - \frac{z}{v_{g}} + θ (z)] d t' .

In Eq. (3) $g_{1}$ is an electro-strictive parameter, $Γ_{A}$ denotes a complex linewidth: $Γ_{A} (Ω, z) \equiv j [Ω_{B}^{2} (z) - Ω^{2} - j Ω Γ_{B}] / (2 Ω)$ , and $θ (z) \equiv (2 z - L) / v_{g}$ is a position-dependent temporal offset.

Figure 1(left) shows the simulated $| Q (z, t) |$ , calculated using Eq. (3) subject to the boundary conditions of Eq. (1) and Eq. (2). The phase code symbol duration was chosen to be 267 ps, and the amplitude pulse duration was 30 ns. The BFS was taken to be uniform along most of the simulated, 9 m long FUT, and $ν$ was chosen to match that value. The local BFS was offset by 30 MHz within a 5.6 cm-wide region, located at $z_{h o t}$ = 3 m. Only part of the simulated FUT is presented, for better clarity. In similarity with previous works using phase-coded B-OCDA, the stimulation of the acoustic field and the SBS interaction are restricted to discrete and narrow peaks [14]. The spatial extent of the correlation peaks is $Δ z = \frac{1}{2} v_{g} T_{p h a s e}$ = 2.76 cm, and their spatial period is $N_{p h a s e} Δ z$ . The acoustic wave magnitude at $z_{h o t}$ is much weaker than those at other locations, as expected. The duration of the interaction in each peak location is $T_{p u l s e}$ . Unlike previous realizations of B-OCDA, the correlation peaks are closely-spaced with significant temporal overlap among the SBS interactions at neighboring locations.

figure: Fig. 1

Fig. 1 Left – calculated normalized magnitude of the stimulated acoustic field as a function of position and time along a 9 m long fiber under test. Only part of the fiber is shown, for better clarity. The BFS of the fiber was taken to be uniform, and the frequency offset between pump and signal was chosen to match that value. The BFS was modified by 30 MHz within a 5.6 cm-wide segment located 3 m from the input end of the pump wave. The signal and pump waves are jointly phase-modulated by a repeating perfect Golomb code with a period of 11 bits and symbol duration of 267 ps. The pump wave is also amplitude-modulated by a single pulse of 30 ns duration. The acoustic field is confined to multiple, discrete and closely-spaced correlation peaks. Due to the short period of the phase code, SBS interactions at neighboring peaks take place with substantial temporal overlap. The acoustic field at the peak which is in spatial overlap with the modified region is considerably weaker than all others. Right – calculated power of the signal wave at the output of the FUT. Red and blue traces correspond to pump pulse durations of 30 ns and 29 ns, respectively (see legend).

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The propagation of the signal wave is governed by the following equation:

(4)

\frac{\partial A_{s} (z, t)}{\partial z} + \frac{1}{v_{g}} \frac{\partial A_{s} (z, t)}{\partial t} = - g_{2} Q^{*} (z, t) {\tilde{A}}_{p} (t - \frac{z}{v_{g}}) .

In Eq. (4) $g_{2}$ denotes a second electro-strictive constant of the fiber medium. The acoustic wave magnitude is taken from Eq. (3). We neglect pump depletion, and the effect of small SBS amplification of the signal magnitude on $Q (z, t)$ .

Figure 1(right, red curve) shows the calculated power of the signal wave at the output of the FUT, ${| A_{s} (z = 0, t) |}^{2}$ . The trace consists of a series of gain events. Each event builds-up with an exponential rise time $τ = Γ_{B}^{- 1}$ , and terminates abruptly when the pump pulse leaves the location of the particular correlation peak. Since $T_{p u l s e} > T_{p h a s e} N_{p h a s e}$ , multiple gain events are in temporal overlap. Following an initial transient, the output signal power at any given instance is affected by SBS interactions taking place at multiple peaks. Hence, individual amplification events cannot be directly resolved without ambiguity. Unambiguous measurements can be recovered by repeating the experiment using a different duration of amplitude pulses: $T_{p u l s e} - Δ T_{p u l s e}$ . The difference between the two durations is taken to be shorter than the phase code period: $Δ T_{p u l s e} < T_{p h a s e} N_{p h a s e}$ . Figure 1(right, blue curve) shows the calculated output signal power ${| A_{s} (z = 0, t) |}^{2}$ with $Δ T_{p u l s e}$ = 1 ns.

Figure 2(blue trace) shows the result of the subtraction between the two calculations of the output signal power. The difference trace consists of a series of amplification events, separated by $T_{p h a s e} N_{p h a s e}$ = 2.9 ns. Each event is unambiguously related to the SBS interaction taking place at a specific correlation peak of known location. The gain event corresponding to the correlation peak at $z_{h o t}$ is much weaker than all others. The red and black traces of Fig. 2 show the subtraction between the two output signal calculations with $ν$ taken to be 15 MHz and 30 MHz above the nominal BFS of the modelled fiber, respectively. The relative magnitudes of the gain events, corresponding to correlation peaks that are outside or within the modified region, change with the different values of $ν$ in agreement with expectations.

figure: Fig. 2

Fig. 2 Blue trace: The result of subtraction between the two calculated traces of the output signal power of Fig. 1(right). The frequency offset between pump and signal was chosen to match the BFS of the fiber outside the modified region. The difference trace recovers the magnitude of individual SBS gain events without ambiguity. The weaker gain event at 105 ns corresponds to SBS in a correlation peak which is in overlap with the modified region. Red and black traces were obtained using the same process, with the frequency offset between pump and signal taken to be 15 MHz and 30 MHz above the BFS of fiber outside the modified region, respectively.

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The subset of addressed resolution points may be scanned along the FUT using proper adjustments to the relative delays in the pump and signal paths [14,21]. The entire FUT may be addressed using only $N_{p h a s e}$ pairs of traces per choice of $ν$ , regardless of $L$ . This number of scans may be much smaller than that required using previous time-multiplexed B-OCDA schemes, in which multiple events were separated directly by temporal analysis of a single output trace.

3. Experimental setup and results

Figure 3 shows an illustration of the experimental setup used in DPP B-OCDA measurements [14,15,17]. Light from a laser diode at 1550 nm wavelength was used as a common source for both SBS pump and signal waves. The output of the laser diode was connected to an electro-optic phase modulator, driven by the output voltage of an arbitrary waveform generator (AWG). The AWG was programmed to repeatedly generate the 11 bits-long perfect Golomb code. The symbol duration of the phase code matched that of the simulations: $T_{p h a s e}$ = 267 ps, corresponding to a B-OCDA spatial resolution $Δ z$ of 2.76 cm. The output voltage of the AWG was adjusted to match the phase values of the code.

figure: Fig. 3

Fig. 3 Schematic illustration of the experimental setup. SOA: semiconductor optical amplifier; EDFA: erbium-doped fiber amplifier; SSB Mod.: singles side-band electro-optic modulator; Phase mod.: electro-optic phase modulator.

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The phase-modulated light was split into pump and signal branches. Light in the pump branch was offset in frequency using a single sideband (SSB) electro-optic modulator, driven by a sine wave at frequency $ν \approx Ω_{B} / (2 π)$ from the output of a microwave generator. The pump light was amplitude-modulated into pulses of $T_{p u l s e}$ = 30 ns or $T_{p u l s e} - Δ T_{p u l s e}$ = 29 ns duration, using a semiconductor optical amplifier (SOA) that was connected to a second AWG. The modulation period was 2 µs. Use of the SOA provided a high modulation extinction ratio. The pump wave was amplified by an erbium-doped fiber amplifier (EDFA) to an average power of 0.4 W (estimated pulse peak power of about 10 W), and connected to one end of a 43 m-long standard single-mode FUT through a fiber-optic circulator.

Light in the signal branch propagated through a polarization scrambler, used to eliminate polarization-induced fading of the SBS interaction [22]. A 2.2 km-long fiber path imbalance was used to allow for the fine-tuning of correlation peaks positions (see [14,21] for details). Peak positions were adjusted through slight changes in $T_{p h a s e}$ [14, 21]. The signal wave was launched into the opposite end of the FUT through a fiber-optic isolator. The signal wave at the FUT output was amplified by a second EDFA, and detected by a broadband photo-receiver. The detector output was sampled by a real-time digitizing oscilloscope, digitally filtered to 0.7 GHz bandwidth, and processed offline. Measurements were repeated for 67 values of $ν$ in 1.5 MHz increments, and $N_{p h a s e}$ = 11 positions of correlation peaks. A pair of traces were acquired for each set of positions and choice of $ν$ , using pump pulses of $T_{p h a s e}$ and $T_{p u l s e} - Δ T_{p u l s e}$ durations, as described in the previous section. Each trace was averaged over $N_{a v}$ = 512 acquisitions. Three 8 cm-wide hot-spots were placed along the FUT.

Figure 4(left) shows an example of a single pair of output signal traces taken at $ν$ = 10.765 GHz, a value that is close to the BFS of the FUT at room temperature. Only parts of the complete traces are shown for better clarity. The displayed section includes a single hot-spot. Both traces consist of series of overlapping gain events. Note the similarity between the experimental Fig. 4(left) and the simulated Fig. 1(right). Figure 4(right) shows the result of the subtraction between the two traces. The difference trace consisted of a series of individual gain events that are separated by $N_{p h a s e} T_{p h a s e}$ = 2.9 ns. The duration of each event is $Δ T_{p u l s e}$ = 1 ns, and it represents the SBS amplification in a single correlation peak of width $Δ z$ . Measurements are resolved even though the temporal separation between adjacent peaks is shorter than the Brillouin lifetime, and much shorter than the pump pulses durations. A single gain event, which corresponds to a correlation peak that is in overlap with the location of a hot-spot, appears much weaker than all others.

figure: Fig. 4

Fig. 4 Left – example of a pair of measurements of the output signal wave, taken for $ν$ = 10.765 GHz which is close to the BFS of the FUT at room temperature. Only parts of the traces are displayed for better clarity. The sections of the traces displayed are in overlap with a single hot-spot. The pump pulse durations were 30 ns (red) and 29 ns (blue). Both traces consist of series of overlapping SBS gain events. Right – result of the subtraction between the two traces of the left panel. The difference trace consists of a series of amplification events that are unambiguously associated with individual correlation peaks. A single gain event, which corresponds to a correlation peak that is in overlap with the hot-spot, is much weaker.

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The magnitude of the SBS gain at each peak was estimated as follows: the instantaneous maximum reading of each gain event in the difference trace of Fig. 4(right) was identified, and the trace was integrated over a 1 ns-wide window centered at that instance. Figure 5 shows the SBS amplification as a function of $z$ and $ν$ . The entire set of 1,600 resolution points was addressed using only 11 pairs of traces per choice of $ν$ . Lastly, Fig. 6(left) shows the retrieved $ν_{B} = Ω_{B} (z) / (2 π)$ . Magnified view of the fiber region which contains the three hot-spots is provided in Fig. 6(right). The three hot-spots are clearly identified. The experimental uncertainty in the measurement of the BFS was estimated based on the variations between adjacent resolution points: $σ_{ν}^{} \approx \sqrt{\frac{1}{2} {〈 {[ν_{B} (z + Δ z) - ν_{B} (z)]}^{2} 〉}_{z}}$ = 1.9 MHz [14]. Here ${〈〉}_{z}$ denotes averaging over position $z$ . It is assumed that changes in the retrieved BFS over short segments of $Δ z$ length, outside the hot-spots, represent measurement noise rather than physically meaningful variations. The estimate for the experimental error therefore represents a pessimistic upper bound.

figure: Fig. 5

Fig. 5 Measured normalized SBS gain (arbitrary units), as a function of correlation peak position and frequency offset between pump and signal waves.

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figure: Fig. 6

Fig. 6 Left - Measured Brillouin frequency shift as a function of position. Right – magnified view of the region containing the three local hot-spots.

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4. Summary

In this work, we have proposed, simulated and demonstrated experimentally a DPP B-OCDA protocol. The measurement procedure addresses the entire FUT using a small number of scans of the correlation peak positions. The number of positions scans is independent of the fiber length. In a proof-of-concept experiment, a 43 m-long FUT was analyzed with a spatial resolution of 2.7 cm. All 1,600 resolution points were addressed using only 11 scans of the correlation peaks positions, the smallest number of any B-OCDA experiment. The spacing between adjacent peaks positions was only 30.4 cm. Such close spacing cannot be applied in direct, single-pulse B-OCDA due to acoustic lifetime limitations.

A comparison between DPP B-OTDA, previous B-OCDA arrangements, and the setup proposed in this work is given in Table 1 below. The proposed DPP B-OCDA protocol represents a middle-ground between earlier, time-gated B-OCDA setups [14,15,17,18] and DPP B-OTDA [9], in terms of the tradeoff between the number of position scans and the detection bandwidth. Previous B-OCDA protocols of cm-scale resolution required at least 100 scans of correlation peaks positions to cover the entire FUT, but on the other hand typically involved detection at 100 MHz bandwidth or lower. The measurements reported in this work required only 11 pairs of position scans, with detection at 0.7 GHz bandwidth. The bandwidth requirement is set by the difference $Δ T_{p u l s e}$ between the durations of the pump pulses in the two measurements. On the other extreme, DPP B-OTDA covers the entire FUT with a single pair of scans, but requires detection at several GHz bandwidth [9]. A similar trend may be expected with respect to pump depletion. The pump pulse is depleted along the entire length of the FUT in DPP B-OTDA. Depletion is restricted to a shorter effective length $L / N_{p h a s e}$ in DPP B-OCDA, and to a length that is shorter still in previous time-gated B-OCDA setups.

Table 1. Comparison between this work, previous B-OCDA setups and DPP B-OTDA

The uncertainty in the BFS estimate of all protocols in Table 1 is governed by the SNR of the SBS gain measurement [23]. In systems that are limited by shot noise, additive detector noise or optical amplifier noise, the SNR scales inversely with the detection bandwidth. The degradation in SNR must be compensated by a larger number of repetitions $N_{a v}$ in the averaging of each trace. Therefore, subject to equal power levels of pump and signal, similar noise characteristics and the same spectral scanning, the value of $N_{a v}$ scales with the measurement bandwidth. DPP B-OCDA should require fewer averages than DPP B-OTDA, but more averages than previous time-multiplexed B-OCDA setups. The overall number of traces pre $ν$ that is required to achieve a given precision: namely the product of $N_{a v}$ and the number of positions scans, should be similar in time-gated B-OCDA, DPP B-OCDA and DPP B-OTDA. Theoretically, the measurement time should be comparable as well. In a research laboratory environment, however, it is often more convenient to perform fewer position scans at the expense of a larger $N_{a v}$ .

In some specific realizations other considerations might mandate a certain minimum number of $N_{a v}$ . For example, in our experiment the averaging over 512 repeating acquisitions was required due to the operation rate of the polarization scrambler. Use of polarization switching [24,25], or polarization diversity schemes [5], is expected to reduce this number. Noise due to residual off-peak SBS interactions is expected to be lower in the DPP B-OCDA measurements than in previous B-OCDA protocols, due to the use of shorter perfect codes.

Both time-multiplexed B-OCDA and DPP B-OTDA have reached a measurement range of several km or more with cm-scale resolution, addressing hundreds of thousands of points [9,14–18]. There is no fundamental limitation that should prevent the currently proposed DPP B-OCDA principle from reaching comparable range. However, these efforts are beyond the scope of the present work. Future studies would attempt to demonstrate the analysis of longer fibers, and reduce the necessary number of repetitions in the averaging of each trace.

Funding

D. Ba wishes to thank the Chinese Scholarships Council (CSC) and the Kort Fellowships Program of Bar-Ilan University, Israel, for supporting his stay at Bar-Ilan University.

Acknowledgments

This work is dedicated to the memory of Prof. Solomon W. Golomb of the University of Southern California (1932-2016). Among his many career achievements, Prof. Golomb developed phase codes with perfect periodic auto-correlation properties. His codes found numerous applications, including in B-OCDA.

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