Opto-mechanical inter-core cross-talk in multi-core fibers

Optical fibers containing multiple cores are widely regarded as the leading solution to the optical communications capacity crunch. The most prevalent paradigm for the design and employment of multi-core fibers relies on the suppression of direct coupling of optical power among cores. The cores, however, remain mechanically coupled. Inter-core, opto-mechanical cross-talk, among cores that are otherwise optically isolated from one another, is shown in this work for the first time. Light in one core stimulates guided acoustic modes of the entire fiber cladding. These modes, in turn, induce refractive index perturbations that extend across to other cores. Unlike corresponding processes in standard fiber, light waves in off-axis cores stimulate general torsional-radial guided acoustic modes of the cylindrical cross-section. Hundreds of such modes give rise to inter-core cross-phase modulation, with broad spectra that are quasi-continuous up to 1 GHz frequency. Inter-core cross-talk in a commercial, seven-core fiber is studied in both analysis and experiment. Opto-mechanical cross-talk is quantified in terms of an equivalent nonlinear coefficient, per acoustic mode or per frequency. The nonlinear coefficient may reach 1.9 [W*km]-1, a value which is comparable with that of the intra-core Kerr effect in the same fiber.


Introduction
Multi-core fibers represent a major area of interest of the electro-optics community in recent years.
They hold promise for supporting a large number of parallel optical communication channels, in space division multiplexing architectures [1][2][3][4]. The most common approach to the design and employment of these fibers is to try and reduce the residual coupling of optical power among cores as much as possible, in attempt to simplify system operation. Coupling may be suppressed by large physical separation between cores [5], heterogeneous cores with trenches of depressed cladding [6], or air holes [7]. The extent of residual coupling among cores has been studied in many works [5], which also addressed the effects of bending [8] and Kerr nonlinearity [9,10]. However, these works did not take into account opto-mechanical considerations. Although optically separated, the multiple cores are nevertheless assembled as part of a single, unified mechanical structure. The implications of this mechanical coupling are examined in this work. The results demonstrate that inter-core, opto-mechanical cross-phase modulation (XPM) may take place in multi-core fibers, even where direct optical coupling is very weak.
Opto-mechanical XPM is driven by guided acoustic modes of the entire fiber cladding [11].
GAWBS has been studied since 1985 [12]. The interactions were mapped in standard fibers [12,24], highly nonlinear fibers [17], solid-core photonic-crystal fibers [16,19,20] and microstructured fibers [21][22][23], and their temperature and strain dependence was examined as well [25][26][27][28][29]. The effect was also observed in hollow-core fibers [30]. Recently, we have shown that GAWBS supports a new paradigm for the sensing of chemicals outside the cladding of an unmodified, standard optical fiber [24]. Measurements could be taken even though the guided light wave never came in contact with the substance under test [24].
The role of GAWBS in multi-core fibers may be understood as follows: the stimulated acoustic waves introduce perturbations to the local value of the refractive index, through the photo-elastic effect. These index variations extend across the entire cladding cross-section. When using standard single-mode fibers, the index perturbations may only be probed within the single core from which they are stimulated. In multi-core fibers, on the other hand, acoustically-induced index variations may affect light that is propagating in other cores as well. Opto-mechanical coupling among modes is extensively studied in various photonic devices [31,32], and was also addressed in few-ordermode, single-core fibers [33] and in dual nano-web, micro-structured fibers [21][22][23]. The phenomenon, however, was not yet investigated in multi-core fibers.
In the following we report a quantitative analytic and experimental study of inter-core, optomechanical coupling in a commercial, seven-core fiber. Our results show significant qualitative differences between GAWBS in a multi-core fiber, and corresponding processes in a standard, single-mode fiber. Guided acoustic waves scattering in standard single-mode fibers only involves radial modes, and one specific sub-category of torsional-radial modes, due to symmetry considerations [12,13]. The resulting spectra of probe wave modulation consist of discrete and sparse resonances, with comparatively large separations [12,17,24]. In contrast, light propagating in outer, off-axis cores of a multi-core fiber stimulates general, torsional-radial guided acoustic modes of the cylindrical cladding cross-section. Hundreds of such modes contribute to inter-core XPM, with spectra that are broad and quasi-continuous up to 1 GHz frequency. Broadband GAWBS spectra were previously observed in a hexagonal, single-core photonic crystal fiber [34].
Although considerably broader than GAWBS process in standard fiber, the bandwidth of the resulting inter-core XPM remains much narrower than the data rates of modern optical fiber communication. Hence the effect is unlikely to restrict the capacity of space-division multiplexing networks. On the other hand inter-core XPM may affect, and even serve for, other potential applications of multi-core fibers such as distributed chemical and shape sensors, opto-electronic oscillators, parametric amplifiers, or lasers operating at transverse super-modes across multiple cores. These prospects are addressed in the concluding discussion.

Analysis of Brillouin scattering by radial guided acoustic waves in multi-core fibers
The fiber used in this work consists of a central, inner core, and six outer cores that are equally spaced on a hexagonal grid. The centers of the outer cores are 35 µm away from the fiber center.
The mode field diameter of the optical modes in all cores is specified as 6.4 ± 0.2 µm at 1550 nm wavelength. The optical power coupling between any pair of cores, in the fiber itself and in the fan-out units at its both ends, was verified as lower than -40 dB.
The mathematical analysis of inter-core XPM induced by GAWBS in multi-core fibers is provided in detail in the Supplementary Material. Only final expressions are given briefly below.
We first address radial guided acoustic modes 0m R , where m is an integer. These modes are stimulated by a pump wave that is propagating, for the time being, at the central, inner core. Intercore XPM due to radial acoustic modes is characterized by a set of discrete resonances, in similarity to GAWBS in standard single-mode fiber. Radial modes are simpler to characterize quantitatively in experiment, hence their study also serves for the validation of our model.
Treatment is extended to the more general case of pump waves in outer cores in section 4.
Let us denote the instantaneous power of the pump as a function of time t by   Pt . The Fourier where  is a radio-frequency (RF) variable. XPM is introduced to a second, optical probe wave. The Fourier components of opto-mechanical phase modulation due to mode 0m R are given by [13]: Here m  is the cut-off frequency of mode 0m R , m (1) and summation over m .
Based on Eq. (1), and following [35], we define an equivalent nonlinear coefficient for optomechanical XPM: At the acoustic resonance frequencies we obtain the following relation between the power spectral density (PSD) of XPM and that of the pump power [35]: The coefficient is mode-specific and core-specific, depends on the geometry of the fiber, and may also depend on the state of polarization (SOP) of the probe wave (see Supplementary Material). It is independent of the pump wave. The opto-mechanical coefficient holds an analogous role to that of the nonlinear coefficient Kerr  associated with the intra-core Kerr effect, and the two might be compared.

Setup and measurement procedures
A schematic illustration of the experimental setup is shown in Fig. 1  Pulse modulation was carried out by a semiconductor optical amplifier (SOA) and an electrooptic modulator (EOM) connected in series [24]. The SOA provided a high modulation extinction ratio, which cannot be achieved in the EOM, but could not support short pulses. Pulses were first generated in the SOA with 5 ns duration and 1 µs period, and were then further modulated to 0.5 ns duration with the same period by the EOM. The pulse generators driving the SOA and EOM were synchronized. Pulses were amplified to peak power levels between 0.3-6.0 W by an erbiumdoped fiber amplifier (EDFA). Sine-wave modulation of the pump was carried out using the EOM only. The sine-wave pump was amplified by the EDFA to average optical power levels between 20-60 mW. The pump wave was launched into the inner core of a 30 m-long, seven-core fiber under test, in one direction. A polarization scrambler along the input path of the pump wave was used to suppress the stimulation of polarization-sensitive, torsional-radial 2m TR acoustic modes [11][12][13] (see also Supplementary Material).
The fiber under test was placed within a Sagnac loop [16,24] [16,24]. In contrast, the counter-clockwise (CCW) propagating probe was subject to negligible nonlinear phase perturbation [16,24]. The acoustic waves therefore introduced non-reciprocal phase modulation of the probe wave. A second PC inside the loop was used to adjust the SOPs of the CW and CCW probe waves and the bias value of non-reciprocal phase delay B  . Fig. 1. Schematic illustrations of the experimental setup used to study opto-mechanical inter-core cross-phase modulation in a commercial seven-core fiber [16,24]. The pump wave propagated in the central core of the fiber under test, in one direction only. In some experiments the pump wave was modulated by pulses of 0.5 ns duration and 1 µs period, using a semiconductor optical amplifier (SOA) and an electro-optic modulator (EOM) connected in series. Pulses were amplified to peak power levels between 0.3-6.0 W by an erbium-doped fiber amplifier (EDFA). In other experiments, the pump wave was modulated by continuous RF sine waves using the EOM only, and amplified by the EDFA to average power levels of 20-60 mW. A continuous probe wave propagated in either the inner core (panel (a)) or an outer core (panel (b)) of the same fiber, in both directions, in a Sagnac loop configuration. When the probe propagated in the inner core alongside the intense pump (panel (a)), optical bandpass filters (BPFs) were used to block the pump wave from reaching the loop output. GAWBS driven by the pump pulses induced non-reciprocal phase delay perturbation to the probe wave.
The non-reciprocal phase delay was converted to an intensity signal upon detection of the probe wave at the loop output. When the probe wave propagated at the inner core alongside the intense pump wave, optical bandpass filters were used to block the pump from reaching the detector (see Fig. 1(a)). The bandpass filters were not needed when the probe wave propagated in an outer core.
The detector output voltage   Vt  was either observed with an RF spectrum analyzer, or sampled by a real-time digitizing oscilloscope of 6 GHz bandwidth. Traces recorded by the oscilloscope were averaged over 1,024 to 4,096 repetitions and analyzed using offline signal processing.
We may relate the instantaneous detector voltage to the opto-mechanical XPM of the probe wave as follows: Here max V denotes the maximum output voltage of the detector, which is obtained when the CW and CCW probe waves interfere constructively. Eq. (4)

Results
Measurements of probe waves at the inner core were used for calibration and validation purposes.  with the probe wave moved to an outer core, obtained using short pulses. Opto-mechanical XPM due to pump pulses in the inner core is evident. The temporal trace retains its fundamental periodicity of 0 t . However, the separation between adjacent impulses is no longer fixed, since the distance from the outer cores to the cladding boundary is not exactly equal to the distance from the outer cores to the fiber axis. A filtered trace of shown in Fig. 3(b).   noted above, as could be expected. Experimental error is primarily due to uncertainty in the pump power that was actually coupled into the multi-core fiber. The endto-end coupling losses from standard fiber to the multi-core fiber and back were measured as 2.5 dB, however the division of these losses between the two interfaces is unknown. Hence the pump power could not be determined with better precision.
Comprehensive numerical analysis of the probe wave propagation in the Sagnac loop, subject to opto-mechanical XPM and many realizations of arbitrary fiber birefringence [36],

Brillouin scattering by general torsional-radial guided acoustic waves
Light in a central, axis-symmetric core of a fiber with a cylindrical cross-section may only stimulate two categories of guided acoustic modes: the radial modes 0m R addressed thus far, and torsional-radial modes with two-fold azimuthal symmetry, noted as 2m TR [11][12][13]. The stimulation of the latter category was deliberately suppressed in the experiments of section 3, by scrambling the polarization of the pump wave [11][12][13]. The fiber cross-section supports a broad variety of general torsional-radial guided acoustic modes, denoted as pm TR , where 0 p  is any integer [37][38][39]. These modes cannot be addressed by GAWBS processes in standard, single-mode fibers.
However, they may be stimulated by pump light which propagates in an outer, off-axis core of a multi-core fiber.
An extended analysis of GAWBS in multi-core fibers, which accounts for pump waves in outer cores, is reported in detail in the Supplementary Material. The analysis suggests that several hundreds of guided torsional-radial acoustic modes contribute to GAWBS in the seven-core fiber under test. Figure 5 shows an example of the normalized transverse profile of the GAWBS-induced perturbation  ε to the local dielectric tensor of the fiber, due to mode 18,14 TR . The acoustic cut-off frequency of the particular mode is 18,14  = 2•526 MHz. Local maxima of the dielectric tensor perturbation are in spatial overlap with the outer cores, suggesting effective inter-core XPM through that mode. The resonant spectra of many torsional-radial modes are in significant overlap. Due to that overlap, the definition of mode-specific nonlinear coefficients for the inter-core GAWBS process (as in Eq. (2)) is less convenient for pm TR modes. Instead, we may define two frequency-dependent opto-mechanical nonlinear coefficients, which take into account the combined effects of all guided acoustic modes: Here      Figure 6 shows an example of the calculated position-averaged coefficient MHz and 800 MHz. Similar spectra were calculated for probe waves at other outer cores. XPM in the opposite outer core remains significant even up to 1.5 GHz. Measurements of GAWBS were carried out using the setup of Fig. 1(b), with the short pump pulses moved to an outer core and the probe wave propagated in an adjacent outer core. Figure   7(a) shows an example of the instantaneous detector output   Vt  . Unlike Fig. 2(a) and Fig. 3(a), the temporal trace is noise-like and irregular. Figure 7(b) shows the measured normalized PSD of the output probe wave Inter-core, opto-mechanical XPM is observed up to 750 MHz frequency, limited by the bandwidth of the pump pulses. One of the strongest spectral peaks matches 18,14  . Similar spectra were obtained when the probe wave was moved to other outer cores. The qualitative prediction for broad inter-core cross-talk is therefore corroborated by experiment. Measured spectra are markedly different from those obtained through 0m R modes only (see Fig. 2(c) and Fig. 3(d)), which consist of discrete, sparse resonances with comparatively large separations. The results provide a first demonstration of GAWBS involving general pm TR modes in fibers with cylindrically-symmetric mechanical structure. Broadband GAWBS spectra due to a large number of modes were previously demonstrated in a single-core, photonic crystal fiber of hexagonal structure [34].
Unlike the case of sparse radial modes addressed earlier, the details of the calculated and measured XPM spectra due to hundreds of overlapping pm TR modes do not fully agree. A possible source for discrepancy is depolarization. The probe wave modulation is polarization-dependent, with principal axes that vary with frequency. Hence the visibility of interference between CW and CCW probe waves may become frequency-dependent, and distort the measured spectra. In addition, torsional-radial modes of azimuthal orders p > 36 may also contribute to inter-core cross-talk. Last, the doping profiles of the cores may modify their mechanical properties, and thereby affect the exact resonance frequencies and transverse profiles of high-order torsional-radial modes. Acoustic guiding in the core is known, for example, in backwards stimulated Brillouin scattering. This effect was not included in the analysis.

Discussion
In this work, we have studied inter-core GAWBS processes in a commercial, seven-core fiber. The results provide a first demonstration of cross-talk among cores that are otherwise optically isolated from one another. This cross-talk mechanism was not considered before. The specific case of a polarization-scrambled pump wave at the central core was examined first. Cross-talk in this case takes place through radial guided acoustic modes. Analysis showed that opto-mechanical XPM of probe waves propagating in all cores may be expected. The modulation spectra consist of a series of narrowband resonances. The magnitude of the effect is quantified in terms of an equivalent nonlinear coefficient per each acoustic mode [35]. Calculations suggest that the nonlinear coefficient can be on the same order of magnitude as that of the intra-core, Kerr nonlinearity. Intercore XPM subject to these conditions was observed experimentally. Excellent quantitative agreement was obtained between model and measurement, in both magnitude and PSD of the intercore cross-talk.
The study was then extended to the more general case of pump waves that propagate in outer, off-axis cores. Due to the removal of radial symmetry, GAWBS in this case becomes a fundamentally different process. Hundreds of general torsional-radial acoustic modes contribute to inter-core cross-talk, with a broad spectrum that reaches 1 GHz. Broad modulation spectra were also observed experimentally. Although general torsional-radial guided acoustic waves in cylindrical fiber are long known [37][38][39], they cannot be observed through Brillouin scattering in On the other hand, our analysis suggests that opto-mechanical cross-talk due to a single pair of cores is of comparable magnitude to the intra-core Kerr effect for frequencies up to the order of 1 GHz. GAWBS might therefore become the dominant nonlinearity when the transmission bandwidth is on that order. This could be the case, for example, in sub-carrier-multiplexed radioover-fiber transmission [40] or in microwave-photonic applications of multi-core fibers, as proposed in [41]. While multi-core fibers are currently pursued primarily for the purpose of highrate communications, their use may diversify as they become more widely available.
One application of highly coherent light in multi-core fibers is in fiber lasers [42][43][44][45]. Multicore fiber lasers exhibit self-organization of spatial super-modes across several cores. The formation and stability of these modes depend on the extent of coupling among cores [42][43][44][45].
Coupling might be modified by inter-core GAWBS processes. Opto-mechanical crosstalk might impede, or perhaps assist, the propagation of transverse super-modes in high-power fiber lasers.
GAWBS is also identified as a source of noise in parametric fiber amplifiers [46]. Parametric gain is being considered for phase-sensitive amplification in coherent optical communication [47]. In some cases, parametric amplifiers employ carefully designed doping profiles to suppress the guiding of acoustic modes in the core, and elevate the threshold of backwards stimulated Brillouin scattering at 10-11 GHz [47]. However, this solution path is inapplicable to GAWBS. Parametric amplification in space-division-multiplexing networks over multi-core fibers may have to take inter-core GAWBS into consideration.
While inter-core GAWBS can be detrimental from the standpoints of certain applications, the same effect could become very useful in other contexts. One such example is the recently-proposed protocol for chemical sensing outside the cladding [24]. In that approach, changes in the modal linewidth of guided acoustic waves due to the acoustic impedance at the cladding boundary are monitored. The method works around a long-time difficulty of fiber sensors: the lack of spatial overlap between light that is confined to the core and substances outside the cladding of unmodified fibers. Thus far, the method has been demonstrated over standard single-mode fiber [24], however its extension to multi-core fibers would bring about significant added values. First, spatial separation between pump and probe would improve the measurement signal-to-noise ratio and precision. Second, use of fibers consisted of heterogeneous, dissimilar cores would restrict the pump-probe interactions to the walk-off distances among different optical modes, and may give rise to distributed analysis. This prospect is highly sought-after by the optical fiber sensors community. Last but not least, simultaneous opto-mechanical measurements in multiple cores can also support advanced shape sensing [48,49]. Shape-sensing in multi-core fibers relies on narrowband optical signals.
Another potential application is in opto-electronic oscillators [50]. In such oscillators light is amplitude-modulated by an RF waveform, and propagates along a section of fiber. The waveform is detected at the output of the fiber, and the recovered RF signal is fed back to modulate the optical input. When the feedback gain is sufficiently high, stable self-sustained RF oscillations may be achieved [50]. Opto-electronic oscillators provide RF tones with extremely low phase noise, and they are pursued for applications in coherent communication, radars and precision metrology. We have recently demonstrated an opto-electronic oscillator which makes use of GAWBS in standard single-mode fiber as the sole mechanism for frequency selectivity [51]. Here too, the employment of multi-core fibers would reduce phase noise due to separation between pump and probe, and provide larger freedom in the choice of radio-frequencies.
In conclusion, multi-core optical fibers provide a rich playground for the study of optomechanics. The applications and implications of the principles demonstrated in this work are the subjects of ongoing research.