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We show characterization of stimulated Brillouin scattering (SBS) in a circular-core two-mode fiber (c-TMF) using Brillouin optical time-domain analysis (BOTDA) with a pulsed pump and a counter-propagating continuous wave probe. By using two free-space mode combiners (FSMCs), we can launch any combination of spatial modes into both ends of the c-TMF. Combined with coherent detection, measurement of distributed Brillouin gain spectra (BGS) is achieved for all possible counter-propagating spatial mode pairs with high spectral resolution and stability. Both intra- and inter-modal SBS are investigated for the c-TMF. The inter-modal SBS between two degenerate LP11 modes (LP11a/LP11b) is demonstrated for the first time. From the Brillouin frequency shift (BFS) measured in each intra-modal SBS, the distributed modal birefringence between non-degenerate modes (LP01/LP11) and degenerate LP11 modes is obtained. The proposed setup can potentially be used as a c-TMF based distributed Brillouin sensor.
©2013 Optical Society of America
1. Introduction
The exponential growth of Internet traffic has led to a huge bandwidth demand on the high-speed optical transport, from core to access networks. Much research has been focused on standard single-mode fiber (SSMF) to improve the spectral efficiency (SE) by various multiplexing techniques, e.g., polarization-division multiplexing [1], orthogonal-frequency division multiplexing (OFDM) [2], and multilevel modulation [3]. However, SSMF is fast approaching its so-called nonlinear Shannon limit [4,5]. To solve the capacity crunch problem, recently space-division multiplexing (SDM) based on either multi-core fiber (MCF) [6,7] or multi-mode fiber (MMF) [8–12] has been proposed. By utilizing another dimension – the spatial mode, the spectral efficiency (SE) can be increased several times than that of SSMF. Few-mode fiber (FMF) has recently attracted much interest due to many advantages such as high SE per core, cost and energy efficient amplification [13,14], ease of splicing and low-cost [15]. Compared with conventional MMF, FMF supports only a limit number of spatial modes (normally 3-6 modes), which greatly simplifies the system design. By utilizing the techniques of coherent detection and multi-input-multi-output (MIMO) digital signal processing (DSP), linear impairments of a FMF such as chromatic dispersion (CD), polarization-mode dispersion (PMD), differential-modal-delay (DMD) and mode coupling, can be completely rewound by the MIMO algorithm at the receiver, enabling high-speed mode-division multiplexed (MDM) transmission. To realize the full potential of MDM, the FMF needs be fully characterized. Previous experiments on MDM transmission [8–11] and fiber design [16,17] have explored a number of global parameters of a FMF including the loss, CD and DMD. However, distributed effects in a FMF are not well understood, such as the mode coupling and modal birefringence, because measurement of these effects is not straightforward.
Stimulated Brillouin scattering (SBS) is one of the major nonlinear effects in a silica fiber which can be utilized for many applications, such as fiber characterization, optical amplification, and fiber sensor. Measurement of the Brillouin gain spectrum (BGS) could provide important information along the fiber, such as the effective refractive index of the guided mode, which is very sensitive to the temperature and strain. Distributed measurement or sensing can also be achieved using techniques such as Brillouin optical time-domain reflectometer (BOTDR) [18] or Brillouin optical time-domain analysis (BOTDA) [19]. The SBS effects have been intensively studied on single-mode fibers (SMFs) including the standard single-mode fiber and polarization-maintaining fiber (PMF). For the SBS effects in FMF, however, only a few reports can be found. For example, an early work in [20] has demonstrated the forward SBS between co-propagating LP01 and LP11 modes on a dual-mode fiber. Very recently, Song, et al. [21] reported the inter-modal SBS between the counter-propagated LP01 and LP11 modes in an elliptical-core two-mode fiber (e-TMF). Compared with the most used circular-core fibers, these specially designed elliptical-core fibers are not suitable for use as a transmission fiber due to the reduced capacity and high cost compared with circular-core TMF fibers. Li, et al. [22] demonstrated all-optical generation of Brillouin dynamic grating (BDG) in a FMF. All these measurements touch the interesting topic of generating SBS effect in a FMF. However, the SBS in a circular-core FMF has not been fully characterized. The complication is that FMF supports not only the non-degenerate mode (such as LP01) but also degenerate mode (such as LP11a and LP11b). In this work, we demonstrate detailed measurement of the SBS in a circular-core two-mode fiber (c-TMF). Unlike the specially designed e-TMF in [21], a c-TMF can support three spatial modes: LP01 mode and two degenerate LP11 modes - LP11a and LP11b. The intra- and inter-modal SBS between the three supported spatial modes has been fully explored. To the best of our knowledge, the intermodal SBS between the two-degenerate LP11 modes are demonstrated and characterized for the first time. Measurement of the distributed SBS over the c-TMF has been achieved by the BOTDA technique using high power pulsed pump and a weak continuous wave (cw) probe. From the distributed Brillouin gain spectra (BGS), the mode birefringence between LP01/LP11 mode and between degenerate LP11 modes (LP11a/LP11b) is also obtained. The proposed setup can potentially be used as a c-TMF based distributed Brillouin sensor through characterization of the Brillouin dynamic grating (BDG) [22] generated by the intra- or inter-modal SBS effect. It is worth noting that the electrostrictively excited low-frequency cladding Brillouin scattering (CBS) may perturb the SBS dynamics in sufficiently long fibers [23]. For a stable measurement of SBS the CBS effect must be suppressed, which can be achieved by various methods such as removing the fiber coating, using short FMF or reducing the pump power.
2. Fundamentals of SBS and BOTDA
When light propagates through a fiber, acoustic waves (phonons) will be excited and a fraction of the light will be backscattered due to the interaction between photon and acoustic phonon, known as the spontaneous Brillouin scattering. The backscattered light will undergo a frequency downshift (stokes) or upshift (anti-stokes), and the frequency shift depends on the acoustic velocity and is given by
where is the Brillouin frequency shift (BFS), is the effective refractive index (ERI), is the effective velocity of the acoustic wave, and is the wavelength of light. If a weak probe light is also counter-propagating along the fiber and has the same frequency downshift, stimulated Brillouin scattering will happen and the probe wave will experience gain. Through time-domain analysis using a pulsed pump and a counter-propagating cw probe, distributed measurement or sensing along the fiber can be achieved. This technique is also known as the Brillouin optical time-domain analysis (BOTDA) [19]. For FMFs the SBS becomes more complicated as many spatial modes can be involved [21].3. Experimental setup
3.1 Fiber under test
The fiber under test (FUT) is a 4-km custom-designed circular-core TMF. The TMF is a Ge-doped circular-core step-index fiber that supports three spatial modes, LP01, LP11a and LP11b, the same as we used in [8]. The fiber parameters are summarized in Table 1.
3.2 Mode stripper and mode converter
A mode stripper (MS) is a device that can strip out higher order modes. The MS can be realized by tightly winding the bare TMF over an 8-mm post of about 20 turns. In so doing, the modal rejection ratio to the LP11 mode is above 30 dB, with very low loss (< 0.2 dB) for the LP01 mode and negligible polarization dependence. A mode converter (MC) can convert the fundamental mode in a FMF to the specific higher order mode, e.g., LP11 mode, or the other way round. Various known methods can realize mode conversion, such as long-period fiber grating (LFPG) [9], spatial light modulator (SLM) [10], free-space phase plate [11], and fused spatial mode coupler (SMC) [12]. In this experiment, we use LPFG based MC to realize mode conversion between LP01 and LP11 modes in a TMF. The advantage of LPFG based MC is that it is very compact and the conversion ratio is tunable. The LPFG is generated by pressing a metal grating to the TMF. The metal grating is fabricated with 20 evenly-spaced grooves on one polished surface. The groove pitch is 510 ± 5 μm, which equals to the beating length between LP01 and LP11 modes. The center wavelength and conversion ratio (or extinction ratio, ER) is controlled by the applied force and angle to the fiber [9]. The measured performance can be found in Fig. 4 of [9]. It is confirmed that the losses of the fabricated MCs are about 1.5 dB with polarization dependent loss (PDL) of 0.4 - 1.2 dB. The modal ER is maintained beyond 20 dB for a 13-nm wavelength range from 1545 to 1558 nm.
3.3 Free-space mode combiner (FSMC)
To analyze the SBS between different spatial modes, mode multiplexer/de-multiplexer (MUX/DeMUX) is needed to combine or split the spatial mode. We use free-space mode combiner (FSMC) as the mode MUX/DeMUX. Configuration of the FSMC is shown in Fig. 1. The FSMC consists of three XY translation stages, two 50:50 non-polarizing beamsplitters (BS), and four collimating lenses with focal length f = 11.0 mm. For LP01 mode, the signal is directly launched into the free-space SMC from a single-mode fiber (SMF). For LP11 mode, the signal is mode converted from fundamental mode to LP11 mode by a LPFG based MC before being launched into the FSMC from a TMF. The two input TMFs and one input SMF that carry LP11a, LP11b or LP01 mode are mounted on the XY stages so that the fiber position can be manually aligned to maximum the coupling ratio to the output TMF. The collimated beams are subsequently combined together by the two BS’s through either transmission or reflection path, and finally focused and coupled to the output TMF by a packaged fiber collimator. The BS is polarization insensitive with less than 5% difference in transmission for s- and p-polarization at 1550 nm. The TMFs mounted on the XY stage are connectorized with specially designed FC-type connectors whose ferrule can be axially rotated. By adjusting the key thus the TMF, the orientation of the two LP11 modes can be manipulated to be orthogonal (90°) to each other. An infrared CCD camera can be placed in the unused path of either the first or second BS, so that the position, pattern and orientation of the individual or combined modes can be monitored in real time. The output TMF is connected with the 4-km FUT. The theoretical minimum loss is 3 dB for LP01 mode and 6 dB for the two LP11 modes, respectively. The FSMC is reciprocal and therefore it can also be used as a mode DeMUX.
Fig. 1 Schematic diagram of a 3 × 1 FSMC. The translation stages have two free axes, X and Y (Z is the light propagation direction). BS: 50:50 non-polarizing beamsplitter. The second BS can be removed in case only two modes need to be multiplexed so that it becomes a 2 × 1 FSMC.
3.4 Experimental setup
The experimental configuration for characterization of SBS in a c-TMF is shown in Fig. 2. We build a mode-multiplexed pump-probe setup to analyze the intra- and inter-modal SBS. By utilizing two FSMCs, the pump and probe wave can be launched in one or a combination of the three supported spatial modes, LP01, LP11a and/or LP11b. A tunable external-cavity laser (ECL) with nominal 100-kHz linewidth is used as a light source. The laser wavelength is set at 1550 nm in our experiment. The 1550-nm cw light from ECL is first divided with a 50:50 single-mode coupler. The upper path is further divided with a 50:50 single-mode coupler to provide the pump and probe wave. The lower path is first amplified by an erbium doped fiber amplifier (EDFA), and then divided again by a 1 × 3 coupler to generate 3 carriers as the local oscillators (LOs) for heterodyne coherent detection. The pump wave is modulated by an electro-optical modulator (EOM) driven by an arbitrary function generator (AFG). The EOM can generate 30 ns Gaussian pulse at 5-kHz repetition frequency in order to achieve high pump power. The pump pulse is then split to three paths by a 1 × 3 coupler and amplified by three EDFAs. For the upper two paths, MS and MC are used to generate LP11a and LP11b mode, and for the lower path the pump pulse simply passes through to provide LP01 mode. Finally, the three spatial modes are mode multiplexed by a FSMC and launched into the FUT. The probe wave is cw and modulated by a Mach-Zehnder modulator (MZM) driven by a RF synthesizer. The synthesizer can generate cosine wave up to 40 GHz. By biasing the MZM at null point with the main carrier much suppressed, a double-sideband (DSB) probe signal can be obtained. The DSB signal then passes through a tunable 10-GHz optical band-pass filter (OBPF) with center frequency set at the lower sideband (longer wavelength) signal. The upper sideband (shorter wavelength) is thus attenuated by > 20 dB, and the probe signal becomes single-sideband (SSB) with its frequency downshifted from the carrier. The spectrum of the probe wave before and after OBPF is shown in the insets (i) and (ii) of Fig. 2. The probe signal is then split into three paths by another 1 × 3 coupler to generate three spatial modes, and mode multiplexed with another FSMC. The mode combined probe wave is then fed into the FUT in opposite direction.
Fig. 2 Experimental setup for the BOTDA measurement of BGS of a 4-km c-TMF. ECL: external-cavity laser; MZM: Mach-Zehnder modulator; EOM: electro-optic modulator; AFG: arbitrary function generator; EDFA: Erbium-doped fiber amplifier; OBPF: optical band-pass filter; PC: polarization controller; OC: optical circulator; MS: mode stripper; MC: mode converter; BR: balanced receiver; TDS: time-domain (sampling) scope; FUT: fiber under test. Insets in the bottom: spectrum of probe signal (i) before OBPF, and (ii) after OBPF.
To detect the probe signal, three optical circulators (OCs) are inserted before the FSMC in the pump path. The counter-propagating probe is first mode de-multiplexed to three spatial modes by the same FSMC in the pump path. The optical signal in each mode is then directed to the coherent receiver which is coupled with a 1550 nm local oscillator (LO) from the same ECL through a 90° optical hybrid for coherent detection. The outputs of the hybrids are subsequently detected by 6 balanced receivers (BRs). The photo detectors (PDs) in each BR have a 3-dB bandwidth of 15 GHz. Finally, the electrical signals comprising the in-phase (I) and quadrature (Q) components of all 3 modes are sampled by a 25-GSa/s time-domain sampling scope (TDS). The TDS is synchronized and triggered by the AFG with the same frequency of the pump pulse, so that distributed measurement of BGS is enabled. The state of polarization (SOP) of LOs is aligned with the pump and probe wave at the input end of pump (point A in Fig. 2), so that the electrical signal power after coherent detection is maximized. Since a circular-core FMF is used as the FUT which is normally a randomly birefringent fiber, the state of polarization (SOP) of pump/probe wave is not maintained and will scramble over distance. Therefore, even though the SOP of pump and probe is well aligned at the input end of pump, it will deviate during propagation because the pump-probe wave is counter-propagating. In extreme cases the SOP can be orthogonal, which will lead to zero SBS gain. This problem can be solved by various means such as using a polarization maintained fiber, fast scrambling the pump or probe polarization, and nonlinear polarization pulling (NLPP) [24]. Nevertheless, in our case the fiber is tightly spooled and placed on an vibration isolated optical table to avoid perturbation, and also a short distance of FMF (<1-km) is measured. From the time trace and spectra (Fig. 3) we did not observe fast fluctuation in SBS gain over distance, therefore we think that the polarization scrambling in our FMF is slow and the measurement can be stable. Even if the SOP evolves very fast over distance in the fiber, theoretically we can partition the fiber into many short segments where polarization can be considered constant. In this way we can align the polarization state at each fiber segment and measure the SBS gain for each segment separately (assume the space resolution is smaller than the length of each segment). A polarization-maintaining (PM) FMF will be an ideal solution where no polarization adjusting is needed, and this will be investigated in the future.
Fig. 3 Received probe signal using heterodyne coherent detection. (a) Time-domain trace, (b) probe spectrum measured before pump pulse (w/o. SBS), and (c) probe spectrum measured after pump pulse (w. SBS). The frequency of probe is offset from pump by −10.512 GHz. Pump power = 4.5 mW.
The frequency of the RF synthesizer is set at the vicinity of the Brillouin frequency shift νB of the c-TMF for all modes (~10.5 GHz), which is roughly estimated using the spontaneous Brillouin scattering. In each analysis, a ± 250 MHz frequency range is swept to obtain the BGS. The step size is 2 MHz per scan. The received time-domain signal trace is recorded twice for each frequency and later processed offline using Matlab program. The total length of the signal acquired is 40 µs each (1 × 106 points @ 25GSa/s sampling rate), which is equivalent to the roundtrip delay of a 4-km fiber. The spectrum of received signal is calculated using fast Fourier transform, smoothed over 10 points (1 MHz) and averaged 2 times to reduce the noise. We first analyze the intra-modal SBS by selectively launching the LP01, LP11a or LP11b for both the pump and probe waves. It is known that due to the imperfection of the MDM system such as limited ER of the MC and/or mode coupling in the mode MUX/DeMUX, even if only one mode is launched into the c-TMF, the received signal may still experience mode beating. The mode beating could result in many fringes in the received spectra due to DMD, which could contaminate the BGS information. However, this mode beating is almost static within a few milliseconds. Therefore it can be removed by comparing the received probe spectra with and without pump within one time frame. Because the set time interval between two pump pulses (200µs) is much longer than the roundtrip delay of a 4-km fiber (40µs), we are able to record the time trace with and without SBS consecutively in the same time frame, as shown in Fig. 3. The initial sharp peak observed is the reflected pump pulse in the FSMC due to Fresnel reflection, which happens before coupling into the c-TMF. The first part of trace (before pump) is the probe signal without Brillouin gain, and the second part of trace (after pump) is the probe signal with Brillouin gain. Consequently, we can accurately measure the distributed BGS by calculating the differential gain spectrum. The proposed method also makes our analysis immune to most system variations from either the channel or free-space components as long as it is slow varying. The digital signal processing (DSP) in our Matlab program has low computation complexity that includes the following procedures: (1) IQ imbalance compensation; (2) Discrete Fourier transform for spectrum; (3) peak search in the vicinity of νB; and (4) Lorentzian curve fitting.
4. Result and discussion
There are four possible scenarios for pump and probe pairing that can generate SBS in a c-TMF: (1) intra-modal SBS between LP01 modes, (2) intermodal SBS between LP01 (pump) and LP11 (probe), (3) intermodal SBS between LP11 (pump) and LP01 (probe), and (4) intra- and inter-modal between the two degenerate LP11 modes. We first analyze the intra-modal SBS of LP01 mode (scenario 1). We selectively launch LP01 mode at both pump and probe. For analysis of distributed SBS, the received time-domain signal is divided into 40 segments with equal length of 1 µs (100 m roundtrip delay), and the spectrum is analyzed in each segment using fast Fourier transform (FFT). Due to the gain limitation of our EDFA, the average pump power launched into the c-TMF is 4.5 mW (maximum 9 mW if using 1 BS and 4.5 mW if using 2 BS in the FSMC), after counting all the losses of the subsequent components including the optical circulator (~0.5 dB) and FSMC (~3 or 6 dB). The probe power launched into the c-TMF is 0.8 mW. We measure the received power profile from 0 to 1 km along the c-TMF, and the results are shown in Fig. 4. As can be seen, the small fringes that exist in all spectra are the mode beating pattern between LP01 and LP11 modes, which is confirmed to have a constant frequency spacing of around 82 MHz (12 ns) and is equivalent to the DMD of the 4-km c-TMF (3.0 ns/km). By calculating the differential spectra with and without SBS, the distributed BGS can be obtained, as shown in Fig. 5. A clear Lorentzian-shape like gain profile centered around 10.513 GHz is seen without any mode beating pattern. The gain profile is broadened and split into two peaks after 500 m, which could be due to the self-phase modulation (SPM) of the pump. The evolution of Brillouin gain along the fiber at the center frequency is measured for 0-3 km from the input end of pump, shown in Fig. 6. We find that the SBS gain decreases very fast along the fiber, due to the power depletion of the pump. For the first 1 km of c-TMF, the attenuation is −12.9 dB/km (−6.3 dB/km) at pump power of 9 mW (4.5 mW), respectively. Therefore, in the following figures we will only show the result for 0-500 m along the fiber, where the Brillouin gain is not much attenuated thus can be accurately measured.
Fig. 4 Measured power spectral profile of probe by scanning a frequency range of ± 250MHz around 10.5 GHz: (a) spectra without (w/o.) and with SBS for 0-500 m length, and (b) spectra with SBS for 600-1000 m length. Pump: LP01 mode, Probe: LP01 mode. Pump power = 4.5 mW.
Fig. 5 Measured BGS for LP01-LP01 modes in a c-TMF at (a) 0-500 m length, and (b) 600-1000 m length. Pump: LP01 mode, Probe: LP01 mode. Pump power = 4.5 mW.
Fig. 6 Measured Brillouin gain for LP01-LP01 modes at the center frequency of BGS (10.513 GHz). Analyzed from 0 to 3 km along the fiber. (a) Pump power = 9 mW, (b) pump power = 4.5 mW. The Brillouin gain decays faster with higher pump power.
We then measure the inter-modal SBS between LP01 and LP11 mode (scenario 2 and 3). Three possible pump and probe combinations are included in the measurement: (1) the pump signal is in LP01 mode (probe in LP11a or LP11b mode); (2) the pump signal is in LP11a mode (probe in LP01 mode); and (3) the pump signal is in LP11b mode (probe in LP01 mode). In the first measurement, we launch only one LP11 mode at probe but rotating the fiber key thus the mode orientation until the received signal has equal power in LP11a and LP11b, so that the SBS between LP01-LP11a and LP01-LP11b can be measured simultaneously. For the following two measurements, LP11a or LP11b mode is separately launched into the pump, which is for the reason of achieving high pump power. Since a circular-core TMF is used, the orientation of degenerate LP11 mode is not maintained and will evolve along the fiber. Therefore, the principle axes for LP11 mode only have local meaning. In the following section of this paper, the notion of LP11a and LP11b will be referred to the principle axes at the pump input end (point A). The result is depicted in Fig. 7. Two BSs are used in the FSMC to support all three modes.
Fig. 7 Measured BGS for LP01-LP11 modes in a c-TMF for varying distances. Pump-probe mode pairs: (a) LP01-LP11a, (b) LP01-LP11b, (c) LP11a-LP11a, and (d) LP11b-LP01. Pump power: (a) 4.5 mW, (b) 4.5 mW, (c) 2.3 mW, and (d) 2.3 mW. The reduced SBS gain in (c) and (d) is due to the 3-dB loss of pump power, which originates from the MS and MC in the LP11 path for mode conversion.
We next measure the intra- and inter-modal BGS between LP11 modes (scenario 4). It is known that in a c-TMF the LP11 mode in fact has two degenerate components, LP11a and LP11b, with similar characteristics such as propagation constant. The spatial orientation of the LP11 mode is not maintained over long distance in the c-TMF due to random mode coupling between the degenerate LP11 modes. We therefore choose to selectively launch LP11a or LP11b mode at the pump, and launch one LP11 mode in probe while adjusting the mode orientation until equal power in LP11a and LP11b is received. In so doing, the intra- and inter-modal SBS between degenerate LP11 modes can be measured simultaneously. In order to reduce the loss, we removed the second BS in the FSMC in the pump path, so that the system can only multiplex or de-multiplex the two LP11 modes, which is sufficient for the LP11 mode measurement. The results are shown in Fig. 8. For a fair comparison with other pump-probe scenarios, the pump power launched into the TMF is also 4.5 mW.
Fig. 8 Measured BGS for LP11-LP11 modes in a c-TMF for varying distances. Pump-probe mode pairs: (a) LP11a-LP11a, (b) LP11a-LP11b, (c) LP11b-LP11a, and (d) LP11b-LP11b. Pump power: 4.5 mW.
We also measure the SBS efficiency by varying the pump power and measure the Brillouin gain at the specific Brillouin frequency shift νB for each mode pair. The results are shown in Fig. 9. The markers and the solid lines respectively show the measured data and the linear curve fitting of the data in each mode pair. For the first two scenarios using LP01 pump and LP01 or LP11 probe, maximum pump power is reached (9 mW) by using only one 1 BS in the FSMC, as shown in Fig. 9(a) and 9(b). For the third scenario using LP11 mode pump and LP01 probe, the pump power is reduced by 6-dB due to the loss from the mode stripper, mode converter and additional BS in the FSMC, as shown in Fig. 9(c). For the last scenario using LP11 mode pump and LP11 probe, the second BS in FSMC is again removed so the pump power is only reduced by 3 dB from the input, as shown in Fig. 9(d). By linear fitting the SBS gain curve, we find out that the intra-modal SBS between LP01 modes has the highest gain slope of 2.38 dB/mW. This slope decreases slightly at 9 mW, which implies that we are reaching the maximum allowable pump power for SBS due to the nonlinear SBS pump depletion. For the inter-modal SBS between LP01 and LP11 modes, if the pump-probe configuration is LP01-LP11, the gain spectrum is almost the same for either LP11a or LP11b mode, which is about 57.5% of that of the SBS between LP01 modes. On the contrary, if the pump-probe configuration is LP11-LP01, the SBS efficiency for LP11a-LP01 or LP11b-LP01 is about 76% or 57.5% of that of the SBS between LP01 modes. The reason why the SBS efficiency is higher for LP11a-LP01 than that of LP11b-LP01 is because our fiber is not perfectly circular-core, and anti-symmetric higher-order acoustic mode (e.g., L11 mode) [25,26] is excited. Therefore, the more overlap integral between the acoustic mode and the LP11 mode could be larger for one orientation over the other. For the scenario of LP11 mode pump and probe, The SBS efficiency are respectively 73.9%, 48.7%, 39.0% and 47.0% for the pump-probe mode pairs of LP11a- LP11a, LP11a- LP11b, LP11b- LP11a, and LP11b- LP11b. The intra-modal SBS between LP11b modes is also smaller than that of the SBS between LP11a modes, and the possible cause could be that the anti-symmetric acoustic mode is generated. The absolute value of SBS efficiency we measured on c-TMF is much higher than the value measured in [21], which we think is due to the difference in pump pulse configuration. Nevertheless, the relative efficiency of LP01-LP11 mode we measured is almost the same as measured in [21] (57.5% vs. 58%), and the relative efficiency of LP11a-LP11b or LP11b-LP11b is also very close to the LP11-LP11 result in [21](48.7%/47.0% vs. 47%), which implies that these factors could be similar in both c-TMF and e-TMF.
Fig. 9 Brillouin gain as a function of pump power at 100 m of c-TMF. Pump-probe mode pairs: (a) LP01-LP01, (b) LP01-LP11, (c) LP11-LP01, and (d) LP11-LP11 (including all possible combinations of LP11a and LP11b). The measured gain slope is: 2.38 (LP01-LP01), 1.37 (LP01-LP11a), 1.35 (LP01-LP11b), 1.81 (LP11a-LP01), 1.37 (LP11b-LP01), 1.76 (LP11a-LP11a), 1.16 (LP11a-LP11b), 0.93 (LP11b-LP11a), and 1.12 (LP11b-LP11b), unit: dB/mW.
Finally, we analyze the Brillouin frequency shift νB along the fiber by Lorentzian curve fitting of the experimental data. The results are shown in Fig. 10. The center frequency and spectral width of the BGS for all mode pairs at 100 m fiber length is summarized in Table 2. According to Eq. (1), Brillouin frequency shift νB is linearly proportional to the ERI of the light propagating in the very spatial mode. Therefore if and does not change, the relative difference in effective refractive index (ERI) should be the same as the relative difference in νB. The simulation result shows a 2.052% (2.055%) decrease in ERI for LP11a (LP11b) mode with respective to the ERI of LP01 mode. As comparison, the measured νB is decreased by 2.015% (1.634%) for LP11a (LP11b) mode with respect to the LP01 mode, which also suggests that our c-TMF is not perfectly circular-core but has some modal birefringence between LP11a and LP11b modes. Since the propagation constant is also proportional to ERI, the modal birefringence can be represented by the difference in ERI (Δn), and the result is shown in Fig. 11.
Fig. 10 Measured Brillouin frequency shift νB in a c-TMF from 0 to 500 m after Lorentizan curve fitting. Pump-probe mode pairs: (a) LP01-LP01, (b) LP01-LP11, (c) LP11-LP01, and (d) LP11-LP11 (including all possible combinations of LP11a and LP11b).
Table 2. Characteristics of BGS at 100 m fiber length
Fig. 11 Measured distributed modal birefringence in a c-TMF from 0 to 500 m. The ERI for LP01 mode nLP01 = 1.449788 is calculated from the fiber parameter.
5. Conclusion
SBS in a custom-designed c-TMF is characterized by counter-propagating a pulsed pump and a cw probe wave. We build a 3 × 3 MDM BOTDA system so that the full mode diversity in a c-TMF can be exploited. The two FSMCs in the setup enable selective launching of different mode pairs into the c-TMF in counter-propagation configuration. Intra- and inter-modal SBS between all possible combinations of mode pairs are demonstrated. Using coherent detection and time-domain analysis, the distributed BGS is measured. By analyzing the Brillouin frequency shift νB, the distributed modal birefringence Δn between all three modes is obtained.
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