1. Introduction

Polarization-maintaining fibers (PMF) are optical waveguides specially designed to preserve the polarization of the input light, which have attracted much interest over many decades. There are countless applications, including distributed temperature and strain sensing [1,2], environment acoustic impedance detection [3], random fiber lasers [4], ultra-wideband communications [5], ultra-fast optics [6] and quantum communications [7,8]. In the fabrication process of PMF, the refractive index transverse profile is made radially asymmetric, defining two orthogonal polarization axis with refractive indexes $n_x$ and $n_y$. The difference between $n_x$ and $n_y$ is the fiber’s birefringence, and it is one of the key parameters of PMF. Standard single-mode fibers (SMF) also exhibit birefringence, but in a much smaller scale ($\sim$ two orders of magnitude), which is due to the imperfect symmetry of the core shape that is randomly distributed along the fiber length. Despite the strong intrinsic birefringence of PMF introduced in the manufacturing process, smaller random birefringence fluctuations are also present [9], resulting in a position- and time-dependent birefringence that can be written as $\Delta n(z,t) = \overline {\Delta n} + \delta n(z,t)$, where $\overline {\Delta n}$ is the mean intrinsic birefringence and $\delta n(z,t)$ corresponds to the birefringence fluctuation.

Some applications require an accurate measurement of distributed birefringence fluctuations over long distances for optimal performance. One example are fiber-optic gyroscopes (FOG), which exhibit enhanced performance when built with km-long polarization-maintaining fibers [10], but still undergo polarization non-reciprocal errors due to birefringence fluctuations. However, through distributed birefringence measurement one could compensate the amplitude error due to polarization non-reciprocity, thus enhancing the accuracy of FOG [11,12]. Another example is fiber-sensing: Brillouin dynamic gratings (BDG) in PMF were used for discrimination between temperature and strain at high accuracy (0.08 $^{\circ }$C and 3 $\mu \varepsilon$) in [1], and later shown to be capable of distributed discrimination in a 110 m-long PMF [13]. To extend the sensing capabilities to longer ranges, it is necessary to measure the distributed birefringence profile before setting the frequency separation between lights launched into orthogonal axis. Recently, we showed that random birefringence fluctuations along a PMF could be used for random number generation [14], where the frequency difference between pump and probe waves must precisely match a region of high birefringence fluctuation in the PMF, thus requiring pre-knowledge of the distributed birefringence profile. Therefore, there is an increasing demand to measure the distributed birefringence over long PMF.

Most methods for distributed birefringence measurement of PMF rely on the generation of BDG [9,15,16]. When two co-polarized and counter-propagating pump optical waves are launched into one axis of the PMF, if their frequency difference matches the fiber’s Brillouin frequency shift $\nu _{B}$, then a longitudinal acoustic wave of frequency $\nu _{B}$ is generated. The acoustic vibrations modulate the refractive index creating a dynamic grating, which scatters light through stimulated Brillouin scattering (SBS) for that axis – thus named as BDG. The BDG properties can be investigated if a probe light is launched at polarization orthogonal to that of pump waves. In this process, a fourth wave is generated when the frequency difference $\Delta \nu (z,t)$ between probe and pump waves is equal to $\nu _{p} \Delta n (z,t)/n_{x}$ [17], where $\nu _{p}$ is the optical frequency of higher-energy pump wave launched in y-axis, and $n_{x}$ is the group refractive index of x-axis. This generated fourth wave, here called idler, is coupled to the other three optical waves through the acoustic wave, characterizing a four-wave mixing (FWM) process enhanced by stimulated Brillouin scattering [18,19]. By pulsing one of the pump waves and scanning the frequency of the probe wave while detecting the intensity of the idler signal, the birefringence profile of an 8 m-long PMF was measured in [9]. Later, it was shown that the method could be used for distributed birefringence measurement over long fibers, extending the measurement range to 500 m in a Brillouin-gain configuration – the pump pulse has higher frequency than the CW pump [15]. However, as the pump pulse propagates along the fiber, it loses energy due to SBS depletion and fiber attenuation, thus reducing the strength of the BDG and hence hindering the signal-to-noise ratio (SNR), and ultimately limiting the measurement distance. To extend the range, the Brillouin-loss configuration [20] was recently employed in [16], where the pump pulse has lower frequency than the CW pump, enhancing the strength of the BDG as the pulse propagates and increasing the SNR over longer distances. Although the method shows an improvement over the Brillouin-gain scheme, suitable for over 3 km-long PMF, the BDG is still much stronger at the launching end of the CW pump, and exponentially decays as the CW pump propagates, limiting the birefringence measurement range to a few km. To extend the measurement range even further, the BDG must be sustained over long distances. Simply increasing the power of optical waves is not an alternative because of pump depletion – the higher frequency pump wave is highly consumed in the first meters when high powers are used [21]. To sustain strong BDGs over several km, we need to find a new amplification mechanism to compensate attenuation of the phonon field and pump depletion.

An interesting technique has been presented in the context of Brillouin optical time domain analysis (BOTDA) to extend the temperature/strain measurement range in standard SMF by using a third optical wave as a distributed pump, which is referred as distributed Brillouin amplification (DBA) pump [22]. In this so-called DBA configuration, the pump pulse loses energy to a lower-frequency CW light due to SBS depletion, similar to the Brillouin-gain configuration, but it is amplified by a higher frequency DBA pump through a second SBS process. As demonstrated, the Brillouin gain experienced by the lower-frequency CW light, which is proportional to the acoustic wave strength, was almost evenly distributed along 50 km. For an exact even distribution, the DBA pump bandwidth can be tailored, thus controlling the DBA pump energy density and the SBS gain, resulting in an acoustic wave evenly distributed along the fiber [23].

In this paper, we extend the distance of distributed birefringence measurement in PMF by making use of a DBA pump, which works as a distributed amplification for the pump pulses, sustaining strong BDG for longer distances. Furthermore, to increase the birefringence measurement accuracy, we make use of an optical frequency comb, generated from a narrow-linewidth laser, to precisely tune the frequency difference between pumps and probe in orthogonal axis. This approach completely removes birefringence measurement errors due to optical frequency fluctuations between different optical sources when two or more lasers are used as pumps and probe [9,15,16]. Ultimately, this allows us to achieved the lowest measurable birefringence fluctuation as $10^{-9}$ combined with the longest sensing length as reported so far (7 km) to the best of our knowledge.

2. Theoretical analysis

We start this section by analyzing the strength distribution of longitudinal acoustic waves generated in PMF from the interaction of optical waves in three different configurations: Brillouin-gain, Brillouin-loss and DBA. Next, we discuss the theory of ultra-long range distributed birefringence measurement assisted by DBA.

2.1 Brillouin-gain, Brillouin-loss and DBA

The principle of Brillouin-gain technique is shown in Fig. 1(a). An optical pulse with frequency $\nu _{P}$ is launched into an optical fiber of length $L$ at position $z=0$, while a counter-propagating CW light with lower frequency $\nu _{CW}$ is launched at $z=L$. The frequency difference between the optical waves is $\nu _{B}$, so that a longitudinal acoustic wave is excited and the CW light experiences SBS gain along the fiber, depleting the pulse power in the process. The power of optical waves and the strength of the generated acoustic wave are described by the following coupled-wave equations [21][15].

 

Fig. 1. Comparison between three SBS configurations: Brillouin-gain (top panel), Brillouin-loss (middle panel), and DBA (bottom panel). The basic principles of the three techniques are given in (a), (d) and (g). Power distribution of launched optical waves over a 10 km-long PMF is presented in (b), (e) and (h), while (c), (f) and (i) show the distribution of the acoustic wave generated from the interaction between CW and pulse waves.

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Optical powers of pulsed and CW lights are indicated by $P_{P}$ and $P_{CW}$, respectively, with slowly varying field amplitudes $A_{P}$ and $A_{CW}$. The fiber’s material density is expressed by $\rho$, representing the longitudinal acoustic wave propagating along the fiber with velocity $v_{A}$ and angular frequency $\Omega _{B} = 2\pi \nu _{B}$. $A_{\text {eff}}$, $g_{a}$ and $\tau _{ph}$ stand for the effective area, the acoustic coupling coefficient and the acoustic phonon lifetime, respectively [15]. Considering propagation in a PMF, the group velocity of optical waves must be written in terms of the polarization-dependent refractive index, i.e., $n_x$ or $n_y$. In this section, we assume that all optical waves are co-polarized, so the group velocity can be simply written as $v_g$. Fiber losses are expressed by $\alpha$, which is higher in PMF compared to standard SMF. SBS gain is frequency-dependent and represented by $g_{B}$, which is given by:

Equations 1–3 can be simplified by assuming steady-state condition, and that acoustic phonons are strongly dumped, thus propagating only over short distances [24]. With these simplifications, we numerically calculate the power distribution of pulsed and CW waves for four different pulse powers in a PMF of length $L = 10$ km, and the result is shown in Fig. 1(b). The CW power was kept fixed at 0.1 mW, while the pulse power varied from 1 to 4 mW. PMF attenuation coefficient was set to 0.4 dB/km. As the pulse power increases, higher gain is provided to the CW light, which clearly increases. However, this power increase is mostly concentrated in the first few km, where the pulse power is more severely depleted at high powers, and nearly zero gain is added close to the end of the fiber. The acoustic wave strength, as shown in Fig. 1(c), is stronger in the first few km, and vanishes at longer distances at all input pump powers. Such a process limits long-range distributed birefringence measurements due to small modulation-depths of the refractive index from weak acoustic wave. Hence, according to Fig. 1(c), the idler wave would only be generated in the first few km of optical fiber, so it could not be used to measure the birefringence throughout 10 km.

An alternative configuration is the Brillouin-loss scheme, described in Fig. 1(d). In this case, the pulsed wave has a lower frequency than the CW light, while their frequency difference remains the same, i.e. $\nu _{CW}-\nu _{P}=\nu _{B}$. This configuration offers two important advantages compared to the Brillouin-gain scheme. First, the pulse experiences SBS gain as it propagates along the fiber, which often overcome the fiber losses and thus enhances the strength of the acoustic wave. Second, as the pulse wave propagates along the fiber it is always amplified by a non-depleted CW light, which can propagate over several km only attenuated by fiber losses before meeting the pulse for SBS amplification. The Brillouin-loss configuration was first analyzed in [20], where theoretical calculations were developed based on Eqs. (1) and (2), but with a negative gain parameter $g_{B}$ to account for the pulse power increase. However, authors did not provide an analysis of the strength of the acoustic wave with distance. Figures 1(e) and (f) show the power distribution for the pulse and CW waves, and acoustic wave strength, respectively, for four different CW powers with a fixed pulse power of 0.1 mW. The pulse power increases with CW power, reaching more than 2 mW at $z=10$ km for a CW power of 4 mW. Opposite to the Brillouin-gain configuration, the acoustic wave strength enhances with distance, as shown in Fig. 1(f). This behavior was also reported in [16], however authors neglected fiber attenuation, which can be quite severe for long PMF. In addition, although a stronger acoustic wave was obtained at $z=L$ compared to that of the Brillouin-gain case at $z=0$, the high attenuation along the PMF prevents the formation of strong acoustic waves over the entire fiber length, which are unevenly distributed along the PMF and concentrated in a few km.

DBA principle is summarized in Fig. 1(g). Similar to the Brillouin-gain configuration, $\nu _{P}>\nu _{CW}$, and their difference is equal to $\nu _{B}$. However, a DBA pump with frequency $\nu _{DBA}$ is added to the system to compensate for fiber attenuation and pulse depletion. This is achieved by means of a second SBS process, and thus a second acoustic wave is generated from the interaction between $\nu _{DBA}$ and $\nu _{P}$ satisfying $\nu _{DBA}-\nu _{P}=\nu _{B}$. The modified coupled-wave equations considering the three optical and two acoustic waves are given as follows.

In Eqs. (5)–(9), the two SBS processes are distinguished by the subscripts 1 or 2. Note that, although the two acoustic waves are longitudinal and oscillate with the same frequency, their wavevectors have opposite sign, so they propagate in opposite directions [22]. Despite the overall increase in complexity when a third optical wave is added to the system, it offers a new degree of freedom to control the acoustic wave distribution along the fiber. A new set of numerical simulations was performed based on Eqs. (5)–(9) in a steady state regime. The CW and DBA pump powers were kept fixed at 0.01 mW and 1.46 mW, respectively, while the pulse power was varied from 0.2 to 0.4 mW. Power distribution for the three optical waves is shown in Fig. 1(h), where powers were normalized with respect to the DBA pump power. Different from the Brillouin-gain case, the pulse power increases with distance, and the CW power remains almost constant along the fiber for the three pulse powers analyzed. A major difference from both Brillouin-gain and Brillouin-loss configurations is found when analyzing the distribution of the acoustic wave $\rho _{1}$ as shown in Fig. 1(i). For a pulse power of 0.3 mW, $\rho _{1}$ is almost evenly distributed along 10 km of PMF. This offers a great advantage compared to the two other configurations, which is the key to measure the distributed birefringence over long PMF. Nonetheless, it is relevant to mention that $\rho _{1}$ is slightly weaker in the central portion of the fiber as shown in Fig. 1(i). For fibers much longer than 10 km, this effect will be more significant, which cannot be compensated increasing the input optical power due to pump depletion, thus limiting the uniform distribution of the acoustic wave.

2.2 Principle of DBA-assisted birefringence measurement

The principle of long-range distributed birefringence measurement in PMF assisted by DBA pump is shown in Fig. 2. Four optical waves are launched into a long PMF: a pump pulse ($\nu _{P}$), a CW light ($\nu _{CW}$) and a DBA pump wave ($\nu _{DBA}$) are launched into the y-axis (slow), and a probe pulse ($\nu _{pr}$) is launched into the x-axis (fast). Pump and probe pulses are launched from one end of the PMF, while DBA pump and CW light are launched from the other end. As shown in the previous section, the powers of the pump pulse, CW light and DBA pump must be carefully tuned to allow an even distribution of the acoustic wave $\rho _1$, which is generated from the interaction between $\nu _{P}$ and $\nu _{CW}$. The probe wave is used to investigate the properties of the acoustic wave, and it is scattered by the BDG, which creates an idler wave of frequency $\nu _{id}$ provided that $\nu _{pr} - \nu _{P} = \overline {\nu }\Delta n (z,t)/\overline {n}$. By scanning the frequency of the probe wave over a region of interest and detecting the generated idler wave, the fiber’s birefringence is readily obtained. It is important to mention that, although pump and probe pulses propagate in the same direction, their group velocity is different due to the different refractive indexes in fast and slow axis of PMF, which leads to a severe walk-off over long PMF. For example, the time-delay between two pulses launched into orthogonal-axis of a 10 km-long PMF with birefringence of $6\times 10^{-4}$ is about $20$ ns. Such large walk-offs can prevent the scattering of the probe wave as the acoustic phonons lifetime is typically about 10 ns. To overcome this issue, the pump pulse is made longer than the probe pulse, and their widths shall be designed depending on the fiber length. Note that the birefringence measurement spatial resolution is defined by the probe pulse width, so the increasing of the pump pulse does not affect the spatial resolution.

 

Fig. 2. Principle of ultra-long distance distributed birefringence measurement assisted by DBA-pump. The inclusion of a third optical wave as a distributed amplification for pump pulses allows the generation of a uniformly distributed dynamic grating along the PMF, which is strong enough to scatter probe pulses launched into the orthogonal axis over ultra-long distances. This is analogous to a three-level laser, where the population density of the energy level $h\times \nu _{P}$ is stable because of the DBA-pump.

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The system described in Fig. 2 with three optical waves launched in the y-axis is analogous to a laser system with three energy levels. The DBA-pump acts as the highest energy level, continuously amplifying the pump pulses at a lower level, which should have a steady number of photons even while amplifying $\nu _{CW}$ at the lowest energy level. In the case of the three-level laser, if enough pumping is provided, then photons are continuously emitted when atoms decay from the first to the ground level. Here, provided that enough DBA-pump is present, then a continuous acoustic wave will be sustained, which is ideal to scatter the probe wave, launched in the orthogonal axis, for ultra-long distance distributed birefringence measurement.

Different from [15] and [16], where the birefringence measurement setup involved a system with five coupled-wave equations, here we must consider seven coupled-wave equations, which are described as follows.

Equations 10–16 are written in terms of the slowly varying field amplitudes, $A_{i,j}$, where $i$ corresponds to the wave identification label, and $j$ to the polarization axis at which the wave propagates ($x$ or $y$). The optical coupling coefficient $g_o$ and the acoustic coupling coefficient are related to the peak gain by $g_p = 2 g_o g_a \tau _{ph}$ [15]. The pump pulse, the CW light, the probe pulse and the idler wave interacts through Brillouin-enhanced FWM [19], which has highest efficiency when the phase-matching condition is satisfied: ${\Delta } k=0$, where $\Delta k = (k_{pr} - k_{id})-(k_{P} - k_{CW})$. Thus, high efficiency phase-matching is obtained at particular frequency differences $\nu _{pr} - \nu _{P}$ for each position, depending on the local birefringence. In addition, as shown in Eq. (15), the acoustic wave strength depends on the interactions between both $\nu _{P}$–$\nu _{CW}$ and $\nu _{pr}$–$\nu _{id}$. However, since no light is launched at $\nu _{id}$, the interaction between pump pulse and CW light is much stronger than that between probe pulse and idler wave, then we can assume that the acoustic wave strength (or BDG reflectivity) depends mainly on the interaction between $\nu _{P}$–$\nu _{CW}$. Hence, the acoustic wave strength distribution shown in Fig. 1(i) is a good approximation for this case. An interesting comparison can be traced with the well-known optical time-domain reflectometry (OTDR) technique. In OTDR, the input light is backscattered from Rayleigh scattering, which has a constant backscattering coefficient along the fiber, commonly expressed by the parameter $C$. Therefore, an evenly distributed BDG, with equal reflectivity along the fiber, is very similar to OTDR in this sense, with the probe pulse being equally scattered along the fiber when phase-matching is satisfied.

3. Experimental setup

The experimental setup for high-accuracy DBA-assisted distributed birefringence measurement is shown in Fig. 3. To suppress birefringence measurement errors due to laser frequency instability when multiple lasers are used to generate the three optical waves launched into the PMF as discussed in the previous section, all optical waves involved in the FWM process are generated from a single narrow linewidth distributed-feedback (DFB) laser with 5 kHz-linewidth. For that purpose, a simple optical frequency comb (OFC) was developed to generate multiple laser lines with high frequency stability as shown in the box of Fig. 3. The comb was produced from an electro-optic modulator (EOM) connected in series with a phase modulator (PM), both modulated by a tone with frequency matching that of the Brillouin frequency shift $\nu _{B}$ ($\sim$10.5 GHz) [25]. Two adjacent comb lines are filtered and used as pump pulse and CW light, and another distant comb line is selected as the probe pulse and then fine tuned for accurate measurement as explained in the following.

 

Fig. 3. Experimental setup of DBA-assisted distributed birefringence measurement. The box on the right shows the implementation of the optical frequency comb employed for ultra-high measurement accuracy. AWG: arbitrary waveform generator; DFB: distributed feedback laser; EOM: electro-optic modulator; FBG: fiber Bragg grating; OBPF: optical bandpass filter; PBS: polarization beam splitter; PC: polarization controller; PD: photodetector; PM: phase modulator; PMF: polarization-maintaining fiber.

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The OFC output is amplified and split into two branches. The upper branch concerns the generation of the probe pulse. An optical band-pass filter (OBPF1) is used to select a comb line that is $6\nu _{B}$ apart from the pump to be the basis for the probe pulse as shown in the graph in Fig. 3. This comb line is intensity modulated by EOM1 with a frequency that can be accurately scanned from 0–40 GHz with an electronic radio-frequency synthesizer. The higher frequency sideband from EOM1 modulation is filtered (OBPF2) and then pulse-modulated (EOM2) to serve as the probe pulse light. Note that the frequency separation between probe and pump lights ranges from $[6\nu _{B}+0]$ to $[6\nu _{B}+40]$ GHz, where the OFC frequency separation works as a coarse tuning, while EOM1 modulation frequency acts as a fine tuning. A fiber amplifier compensates the insertion losses of components in the upper branch, and a polarization controller (PC) aligns the polarization to the x-axis before sending the probe pulses to the PMF under test through an optical circulator and a polarization beam splitter (PBS).

In the bottom branch, OBPF3 filters two adjacent comb lines to act as pump and CW light as shown in Fig. 3. A circulator directs the filtered signal to a fiber Bragg grating (FBG1), which is narrow enough to reflect the pump light, while transmitting the CW light. The transmitted CW light is amplified, aligned to the y-axis, and combined with a y-polarized DBA pump. The DBA pump is obtained from a second DFB laser of 7 MHz-linewidth, which is directly modulated with a triangular waveform of 4 Vpp-amplitude and 1 MHz-frequency by means of an arbitrary waveform generator (AWG). This modulation format, also used in [22], induces a frequency chirp that provides a flat gain distribution over nearly 1 GHz in the vicinity of $\nu _{P}$, covering not only the spectral width of pump pulses but also any deviations of $\nu _{B}$ from stress/temperature fluctuations. One could also select the higher frequency comb line adjacent to $\nu _{P}$ for the DBA pump at the expense of adding to the setup a beam splitter, a narrow optical filter, an EOM and amplifiers. However, it is simpler to control the DBA pump chirp by direct modulation of an independent laser, which is the method we selected for this experiment and was also chosen in similar works [22,23]. Note that the usage of a second laser does not deteriorate the birefringence measurement accuracy compared to the case where a comb line is selected as the DBA pump. The reason is that the DBA pump does not participate directly in the Brillouin-enhanced FWM process, acting essentially as a distributed amplifier for pump pulses – birefringence measurement noise is investigate in the next section. The combined CW light and DBA pump are launched into the PMF in the opposite end of probe pulses. The pump light reflected at FBG1 is modulated with square pulses (EOM3) and amplified by an Er-doped fiber amplifier. A circulator and FBG2 are used to remove the amplified spontaneous emission noise (ASE) from the amplified pump signal, which is then aligned to the y-axis and launched into the PMF through the same PBS used to launch probe pulses. Finally, the backscattered x-polarized signal is collected through an optical circulator and amplified before detection. The x-polarized backscattered signal includes the idler wave and the Rayleigh scattered probe signal. The latter is filtered out with OBPF4, such that the idler wave is isolated and detected at a photodetector (PD).

4. Experimental results and discussions

To experimentally verify the theory described in section 2. and test it for long PMF, we make use of two cascaded panda PMF as our fiber under test: one with 2.1 km and the other with 5 km, thus comprising 7.1 km of PMF. The BFS of the two fibers are slightly different, $\nu _{B}^{2km} = 10.34$ GHz, and $\nu _{B}^{5km} = 10.27$ GHz. So when scanning the fine tuning frequency of EOM1 to measure the birefringence for the 2.1 km and 5 km fibers, we set the OFC frequency separation to $\nu _{B}^{2km}$ and to $\nu _{B}^{5km}$, respectively. This shows the flexibility of the setup, as it can be easily adjusted to match the BFS of any PMF while maintaining high frequency stability between optical waves. Note that, when setting the OFC separation to $\nu _{B}^{5km}$, less SBS-depletion is experienced by the pump pulses in the first 2.1 km, which still undergo strong Rayleigh-scattering losses in this section. Pump and probe pulses were designed with widths of 200 ns and 50 ns, respectively, while probe pulses were delayed by 150 ns with respect to the pump pulses to pre-compensate the walk-off effect. Thus, the measurement spatial resolution is about 5 m, limited by probe-pulse width. Optical powers of pump pulses, CW light and DBA pump were tuned according to the theoretical analysis, and set to 158.5, 0.5 and 316.3 mW at the input of the PMF, respectively. Note that, these power levels are higher than those used in the simulations for two main reasons: 1) the measured attenuation coefficient for the available fibers is 0.8 dB/km, which is higher than the typical value of 0.4 dB/km used in simulations; and 2) to measure the reflection of probe light over long distances, the BDG must not only be uniformly distributed, but also strong enough to reflect the probe light to detectable power levels, which could only be obtained at higher powers. Still, the optical powers used in this work are comparable to those employed in [16]. We kept low probe pulse power at 125.9 mW to prevent the generation of stimulated Brillouin scattering, which coincides in frequency to the idler wave and adds noise to the detected signal if present. The EOM1 modulation frequency was varied from 13 to 31 GHz in steps of 1 MHz, corresponding to a frequency separation range between $\nu _{pr}$ and $\nu _{P}$ of 75–93 GHz. For each frequency separation step, the photodetector output signal was measured in an oscilloscope, and twenty averages were applied to mitigate birefringence fluctuations from environmental changes. The data acquisition time for each frequency separation was of 22 ms, so to cover a 1 GHz range in 1 MHz steps took 22 s, and the complete frequency separation range investigated took about 7 min.

The measurement result is shown in Fig. 4(a), where the distributed birefringence of cascaded PMF is observed over 7.1 km, the longest distance measured to the best of our knowledge. The result shows a wide birefringence measurement range of $1.4\times 10^{-4}$ (or 18 GHz) with ultra-high accuracy of $7.5\times 10^{-9}$ (or 1 MHz). This accuracy comes from generating pump and probe from the same laser source and precisely controlling their frequency separation in steps of 1 MHz. Also, by investigating the standard deviation of the peak birefringence calculated for a specific location from fifty measured spectra, we found it to be 4 MHz, which could be attributed to environmental fluctuations over time between different measurements. Nonetheless, this indicates that the frequency step of 1 MHz is adequate for high accuracy and highly repeatable measurements while allowing reasonable measurement times. Regarding the birefringence measurement range, even wider ranges could be obtained by selecting different comb lines as base-probes by properly tuning OBPF1. For instance, we selected the sixth comb line to build the probe frequency before fine tuning with EOM1, but by selecting the seventh, or eight comb line, the birefringence measurement range could be improved by $\sim$10 or 20 GHz, making the measurement system suitable for the majority of PMFs. In addition, since the 40 GHz frequency range of the RF synthesizer encompasses three comb lines, there is no gap between measurements obtained from the selection of different comb lines as base-probes. Hence, the birefringence measurement range is only limited by the number of comb lines generated. The birefringence accuracy could be further improved by reducing the scanning step of EOM1 below 1 MHz, at the expense of increased scanning time. As shown in the inset of Fig. 4(a), a peak birefringence with full-width half-maximum (FWHM) of 12 MHz (or $9\times 10^{-8}$ birefringence width) could be measured with the high-accuracy method. The limitation for birefringence accuracy comes from the laser source used as seed for the OFC: the uncertainty of instantaneous laser frequency, manifesting as a broad linewidth light, reduces the accuracy of frequency separation between probe and pump, thus limiting the birefringence accuracy. We will soon see that using scanning steps smaller than 1 MHz does not bring benefits to the measurement accuracy because the birefringence itself fluctuates with time, which manifests as a spectral broadening much wider than 1 MHz ($7.5\times 10^{-9}$). Even though our proposed frequency comb and DBG are capable of higher accuracy, the response time of our method is limited by the acoustic wave response time, typically in nanoseconds scale, while our detection speed is limited by the product of the bandwidth of sampling rate and that of the probe pulse.

 

Fig. 4. Distributed birefringence measurement of two cascaded PMF with DBA-pump (a) and without it (d). A zoom-in over the birefringence fluctuation range of the two fibers is provided in (b)-(c) for the configuration with DBA-pump, and in (e)-(f) when the DBA-pump is removed.

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From the result in Fig. 4(a), we observe that the two cascaded PMF have largely different birefringences, which are shown in Fig. 4(b) and (c). The birefringence fluctuation $\delta n$ of the 2.1 km fiber ranges from 91 to 93.2 GHz, while that of the 5 km fiber ranges from 75 to 75.9 GHz, a range more than two times lower than the former. This difference is attributed to the fibers’ manufacturing processes and the induced stresses in the fiber spools. To evaluate the advantages of the proposed method, we performed a measurement without DBA pump; results are shown in Fig. 4(d)-(f). In this case, the configuration is similar to the Brillouin-gain scheme explored in [15], where a strong idler signal is detected in the first few hundred meters of fiber, but its intensity exponentially decays with distance, agreeing well with theoretical expectations shown in Fig. 1(c). The idler intensity becomes weak after $\sim$1 km, and vanishes beyond of 2 km, indicating the limitation of Brillouin-gain method and the benefits of our technique.

The addition of the DBA-pump softened the requirement of high pulse power (for Brillouin-gain configuration) or high CW light power (for Brillouin-loss configuration) over long range birefringence measurement. Therefore, the technique is robust enough to prevent unwanted non-linear effects when high powers are used, such as modulation instability or stimulated Raman scattering. To assess the limitations of our method, we performed an integration of the result displayed in Fig. 4(a) for each position, and the calculation outcome is presented in Fig. 5(a). It shows the variation of idler’s intensity with position, which overall decays with distance. As mentioned in section 2, an even distribution of the acoustic wave gives a constant backscattering coefficient just like in an OTDR. Hence, as the probe pulse propagates it loses energy from fiber losses before being scattered and generating an idler signal, which in turn back-propagates and also experiences fiber losses. Thus, the measured 0.8 dB/km losses are the main limitation for long distance measurements. The red lines in Fig. 5(a) correspond to a power decay of 0.8 dB/km, matching the average power decay of the first PMF. This is an indication that the acoustic wave is indeed uniformly distributed across the first 2.1 km, which is shown in Fig. 5(b). A power increase is observed at the connection of the two cascaded fibers, which is probably due to the usage of a different frequency separation at the OFC: when setting the OFC separation to $\nu _{B}^{5km}$, the pump pulse power suffers less SBS-depletion in the first 2.1 km, reaching such distance with higher power and contributing to a stronger acoustic wave. The average power decay from 2.1 to 7.1 km is about 0.25 dB/km, a total 0.55 dB lower than the fiber losses. This means that the acoustic wave strength must be exponentially increasing with distance due to a strong DBA pump, and it ends up compensating part of the fiber losses and allowing longer distance measurements. It is important to highlight that there is trade-off between such compensation and DBA-pump depletion. Increasing the DBA-pump power arbitrarily would result in an acoustic wave concentrated in the final portion of the PMF as shown in the solid line in Fig. 1(i). Nonetheless, a fine tuning of the DBA-pump power can help increasing the distributed birefringence measurement distance for fibers with length longer than 7.1 km.

 

Fig. 5. Integration of idler power with distance (a), and calculated distribution of corresponding acoustic wave strength (b).

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Last, we analyze the noise impacts in the birefringence measurement when adding a DBA-pump. We first compared the measured spectra with and without DBA-pump at several positions within the first 500 m of the fiber under test, where the measurement without DBA maintains high SNR. One example, at position 231 m, is shown in Fig. 6(a). The traces with and without DBA remarkably match, and both measurements exhibit the same standard deviation when measuring the peak birefringence for a fixed location ($\sim$4 MHz in both cases), indicating that no significant noise is added with the DBA pump. On average, the FWHM of measured BDG spectra was about 15 MHz (or $\sim$1.13$\times 10^{-7}$), which is mainly due to the fiber’s non-uniformity [26]. Narrower BDG spectral widths have been observed when using SMF [27], which have better spatial uniformity than PMF, however it is experimentally harder to work with BDGs in SMF since pump and probe waves must have approximately the same wavelength. Next, we take the peak birefringence value for every position over 500 m with DBA-pump and compare with the case without it. As shown in Fig. 6(b), despite some minor differences probably due to small birefringence fluctuations in time, the peak birefringence nearly matches for the two cases, leading to the conclusion that no additional noise is transferred to the birefringence result when the DBA-pump is included in the system. There are two main reasons for this result. First, according to Eqs. (15) and (16), the two acoustic waves propagate in opposite directions, such that the noise from one acoustic wave does not affect much the other. Second, although SBS amplification from DBA-pump has a large noise figure, the frequency separation between the optical waves involved in Brillouin-enhanced FWM requires phase-matching, which is guaranteed from the usage of a single narrow-linewidth laser. Therefore, no significant noise is transferred to the measured birefringence with the addition of the DBA-pump due to FWM. A similar result was reported in [22], where no significant noise was observed in the probe wave in the context of BOTDA.

 

Fig. 6. (a) Birefringence measured spectra at 231 m for the configurations with and without DBA-pump. (b) Comparison of the peak birefringence measured over the first 500 m for the two different configurations. (c) Distributed birefringence measurement with DBA for reference.

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Besides the evaluation of noise taken from Figs. 6(a) and (b), these results lead to an important conclusion regarding the fiber’s birefringence fluctuation with position. In Fig. 6(a) we observe a dominant birefringence peak, but also multiple side peaks. These can be observed in Fig. 6(c) at multiple positions, which shows the distribution of birefringence measured with the DBA technique as a reference (a similar result is found without DBA). The presence of multiple peaks has been investigated before in [26] and attributed to PMF sections with high non-uniformity. By enhancing the fiber’s uniformity, the side peaks were significantly reduced. In this work, where long fibers were used without uniformity control, side peaks can eventually become dominant, resulting in abrupt jumps in the peak birefringence observed in Fig. 6(b).

5. Conclusions

We have investigated the distribution of longitudinal acoustic waves from SBS process in long PMF for three different optical configurations, Brillouin-gain, Brillouin-loss and DBA, showing that uniformly distributed acoustic waves can only be obtained with the aid of a DBA-pump. We explored this aspect to extend the distance of distributed birefringence measurement in PMF, however, one could benefit from evenly distributed acoustic waves for multiple applications, such as high-sensitive discrimination of strain and temperature over long PMF [1]. Experimental results validate the theoretical expectations, and we were able to measure the distributed birefringence of two cascaded PMF covering 7.1 km, the longest measurement reported to the best of our knowledge. In addition, high-accuracy was obtained when interrogating the BDG with a probe light generated from the same laser used for the pump. Given the wide frequency separation required between pump and probe lights for BDG interrogation, we made use of a simple optical frequency comb to generate both lights from the same laser, avoiding the usage of complex frequency locking loops and multiple laser sources. The high accuracy over long distances indicates that the technique has the potential to detect the action of an eavesdropper: small changes in the local birefringence caused by the intervention of an eavesdropper in the physical layer could be detected by the method, which is much more sensitive than BOTDA as small shifts in $\nu _{B}$ translate into large birefringence changes [26]. This requires further validation in a future work. We also showed that by fine-tuning the power of the DBA-pump, the strength of the acoustic wave could be slightly increased at the end of the fiber, thus compensating propagation losses experienced by the probe pulse. Finally, we verified that no significant noise is added to the birefringence measurement when DBA pump is added, therefore proving to be a highly efficient way to extend the distance of distributed birefringence measurements with super-high accuracy.