Distributed analysis of nonlinear wave mixing in fiber due to forward Brillouin scattering and Kerr effects

Forward stimulated Brillouin scattering (F-SBS) is a third-order nonlinear-optical mechanism that couples between two co-propagating optical fields and a guided acoustic mode in a common medium. F-SBS gives rise to nonlinear wave mixing along optical fibers, which adds up with four-wave mixing induced by the Kerr effect. In this work, we report the distributed mapping of nonlinear wave mixing processes involving both mechanisms along standard single-mode fiber, in analysis, simulation and experiment. Measurements are based on a multi-tone, optical time-domain reflectometry setup, which is highly frequency-selective. The results show that F-SBS leads to nonlinear wave mixing processes that are more complex than those that are driven by the Kerr effect alone. The dynamics are strongly dependent on the exact frequency detuning between optical field components. When the detuning is chosen near an F-SBS resonance, the process becomes asymmetric. Power is coupled from an upper-frequency input pump wave to a lower-frequency one, and the amplification of Stokes-wave sidebands is more pronounced than that of anti-Stokes-wave sidebands. The results are applicable to a new class of distributed fiber-optic sensors, based on F-SBS.

3 nano-structured fiber, using vibrometric measurements from outside the fiber [15]. Spatiallycontinuous, nonlinear propagation involving F-SBS is addressed in a recent theoretical study by Wolff and coworkers [32]. However, a corresponding experimental characterization has not yet been reported.
In this work we report an experimental distributed analysis of nonlinear wave mixing due to both F-SBS and the Kerr effect along 8 km of standard single-mode fiber. The measurements are based on a multi-tone, optical time-domain reflectometry (multi-tone OTDR) setup: the monitoring of Rayleigh back-scatter of multiple spectral field components as a function of time [30]. The extension of this measurement principle to address F-SBS presents two main challenges.
First, the separation of back-scattered components that are detuned in optical frequency by only several hundreds of MHz is necessary. Such separation is obtained here using tunable, narrowband B-SBS amplification [33]. Second, the efficient stimulation of F-SBS requires input optical fields that are highly coherent, whereas Rayleigh back-scatter of coherent light is extremely noisy [34][35]. This difficulty is resolved in the experimental setup as well.
The results show that wave mixing involving F-SBS is highly sensitive to the exact frequency offset between incident optical fields. When the offset matches a resonance frequency of F-SBS, the process becomes asymmetrical: Power is coupled from a higher-frequency input pump wave to a lower-frequency one, and the Stokes-wave sidebands are amplified more efficiently than their anti-Stokes wave counterparts. This asymmetry is in marked contrast to the symmetric characteristics of FWM due to Kerr nonlinearity. In addition, detuning of the frequency offset between the incident waves above or below the F-SBS resonance gives rise to nonlinear mixing processes that are qualitatively different. Measurements are in agreement with analysis and simulations. 4 The observations contribute a more complete description of nonlinear wave mixing in fiber, for a more general case where the Kerr effect is not the exclusive underlying mechanism. On top of the interest to basic research, the results are also relevant to emerging new concepts of distributed fiber-optic sensing that are based on F-SBS [36][37]. These measurement schemes allow for the analysis of media outside the fiber, where light cannot reach [36][37]. However, the signal-to-noise ratio, range and resolution of at least one such protocol are currently limited by the onset of nonlinear amplification of modulation sidebands [36]. A more complete description of wave mixing involving F-SBS may be incorporated in the sensor arrangement, and improve its performance. The study is therefore of timely and practical interest.
A model for nonlinear wave mixing through the combination of F-SBS and the Kerr effect is presented in Section 2. Approximate analytic solutions are proposed, and a numerical analysis is reported. Experimental results are provided in Section 3, and a summary is given in Section 4.
Preliminary results were briefly reported in conference proceedings [38,39].

Forward stimulated Brillouin scattering
We begin with a brief introduction of F-SBS. Detailed analysis and discussion of the effect may be found in numerous earlier works [1][2][3][4][5][6]26]. In addition to the single optical mode, standard fibers also support a variety of guided mechanical (acoustic) modes that propagate along the fiber axis.
This study is restricted to radial modes, denoted by 0,m R where m is an integer, in which the transverse displacement of the fiber material is purely radial [1][2][3]. The propagation of each mode is characterized by a cut-off frequency m  [1][2][3]. Close to cut-off, the axial phase velocities of the guided acoustic modes approach infinity. Hence for each mode 0,m R there exists a frequency, very 5 near cut-off, for which the axial phase velocity matches that of the optical mode [1][2][3]. Two copropagating optical field components that are offset in frequency by m    may be coupled with the guided acoustic mode [1][2][3]. The optical fields may stimulate the acoustic wave via electrostriction. The acoustic wave, in turn, can scatter and modulate light through photo-elasticity.
Spontaneous scattering by acoustic modes is often denoted as guided acoustic waves Brillouin scattering [1]. The stimulated effect, in which the acoustic wave oscillations are driven by light, is referred to as F-SBS. The efficiency of stimulated scattering through a given acoustic mode is determined by spatial overlap considerations (see also below, [1][2][3]26] [27]. Note that the opto-mechanical nonlinear coefficient on resonance is comparable with typical values of Kerr  .

Nonlinear polarization terms, nonlinear wave equations and approximate solutions
Consider two continuous optical field components of frequencies   The optical waves may stimulate the oscillations of a guided acoustic mode 0,m R , at frequency  .
One may show that the complex magnitude of the material displacement of the stimulated acoustic wave is given by [36]: Here 0  is the density of silica, 0  is the vacuum permittivity, denotes the spatial overlap integral between the transverse profile of the electrostrictive driving force and that of the modal acoustic displacement [26]. The strain associated with the acoustic wave induces nonlinear polarization, through photo-elasticity. The vector magnitudes of the nonlinear polarization components at frequencies 1,0  are given by [36]: In Eq. (6) and Eq. (7), the opto-mechanical nonlinear coefficient   It is assumed herein that the length scale of nonlinear wave mixing processes is much longer than the beat length of residual linear birefringence in standard fiber [40][41][42]. The chromatic dispersion of standard fibers is not considered. It has negligible effect for the values of  and lengths of fiber used in this work.
Both F-SBS and the Kerr effect are third-order nonlinear-optical phenomena. As such, they both bring about nonlinear mixing between the two optical waves. However, there are several 8 differences between the two contributions. The Kerr nonlinearity is parametric: it preserves the overall energy of the optical field terms combined. F-SBS, on the other hand, is a dissipative effect, in which part of the optical energy is lost to the acoustic wave stimulation. The Kerr terms describe FWM behavior (which is degenerate when only two optical components are considered), with no additional waves involved in the medium. In contrast, mixing between optical waves through F-SBS is mediated by an additional idler, in the form of the acoustic wave. Hence we do not refer to the F-SBS terms strictly as those of a FWM process.
Further, the Kerr effect is independent of frequency changes on the scale of  , whereas F-SBS is extremely frequency-selective. As seen in Eq. (2) and Eq. (3), the nonlinear polarization terms due to F-SBS exhibit the resonance response of a second-order system. This response adds a 2  phase shift on resonance. The phase shift, in turn, manifests in Eq. (9) and Eq. (10) in nonlinear terms with non-zero real parts, as opposed to those of the Kerr effect that are purely imaginary.
Resonances associated with the Kerr effect occur at frequencies outside the optical range. The primary objective of this work is to study the interplay between Kerr and F-SBS terms in nonlinear wave mixing, through analysis, numerical calculations and experiments.
For two input fields that are co-polarized, analytic solutions for the power levels Pz exist in the following form [29]: is the effective length up to point z ,     10 00 M P P  denotes the ratio between input power levels, and the gain coefficient in units of m -1 is defined by: The results show that the stimulation of the acoustic wave is associated with the coupling of power from the higher-frequency optical component to the lower frequency one, as may be expected. In addition to 0,1  , however, the stimulated acoustic wave also induces nonlinear polarization components at additional frequencies, , with respective vector magnitudes: These nonlinear polarization terms add up with those due to the Kerr effect, and may contribute to the generation of anti-Stokes-wave and Stokes-wave sidebands at optical frequencies 2, 1   . The processes may be described in terms of the following nonlinear wave equations: Here Az  denote the complex Jones vector envelopes of the field components at frequencies 2, 1   , scaled as above. First-order approximations for the power levels of the two sidebands may be obtained in terms of equivalent overall nonlinear coefficients: Here   Eq. (12)), we may qualitatively predict that the nonlinear amplification of However this trend is not accounted for by the simplified solutions of Eq. (20) and Eq. (21). These first-order expressions are quantitatively relevant, therefore, only as long as coupling between the two input tones remains comparatively modest. In this regime, the sidebands power levels Pz  are also fairly weak. Note also that Eq. (11) and Eq. (12) themselves are only valid when no other field components exist. When the sidebands power levels  In the general case, nonlinear coupling among all four waves, involving both F-SBS and the Kerr effect, must be considered. Cascaded generation of higher-order sidebands may take place as well. Numerical simulations of the more general problem are described in the next sub-section.

Numerical simulations of nonlinear wave mixing
In this sub-section, the above model is extended to describe nonlinear wave mixing among a larger set of optical field components. Let  (22) In principle, optical field components that are detuned by 2 might also be coupled through the radial guided mode 0, 14 R , which has a cut-off frequency 14 7 2.035    [1]. However,   0,14 OM  in standard single-mode fiber is considerably smaller than   0,7 OM  due to spatial overlap considerations [27]. We verified that the addition of F-SBS through mode 0,14 R to the above model has marginal effect on the calculated outcome.

Parameter
Value Source The local power levels of the optical tones     2 nn P z A z  were calculated using numerical integration of the above set of equations. The parameters used in the simulations are listed in Table   1. The input phases of the two optical fields were assumed to be equal, and their input states of polarization were aligned. Results are presented for the local power levels of six optical field 13 components as a function of position:   k Pz, 2, 1, 0,1, 2, 3 k    . The notation of frequency components used throughout this work is illustrated in Fig. 2, for better clarity. Due to nonlinear coupling, the power levels   k Pz are also affected by higher-order sidebands. Therefore, an overall number of 2N = 10 spectral components were used in the numerical calculations. We verified that the addition of further terms beyond 10 has negligible effect on the six   k Pz traces subject to the specific boundary conditions.   Az . The calculations support the predicted general trends discussed earlier, and suggest that nonlinear wave mixing involving F-SBS is qualitatively different from a corresponding FWM process due to the Kerr effect alone.  is tuned to the F-SBS resonance 7  of guided acoustic mode 0,7 R , 0  . A transfer of power takes place, from the upper-frequency pump wave to the lower frequency one. The nonlinear wave mixing amplification of sidebands becomes asymmetric.    (Fig. 6(b)), since the amplification of sidebands is weaker in that case. The predictions of numerical simulations are tested against experimental results in the next section.

Experimental setup and procedures
A schematic illustration of the multi-tone OTDR experimental setup used in the distributed mapping of FWM processes is shown in Fig. 7. Light from a tunable laser diode source of 17 frequency  at the 1550 nm wavelength range and 100 kHz linewidth was split in two paths. Light in one branch was used to generate the input wave components. It passed first through a doublesideband electro-optic modulator (EOM), which was biased for carrier suppression and driven by the output voltage of a sine-wave generator of variable radio-frequency 1 2  . The modulation generated two symmetric primary input tones at frequencies 1 2   . The two field components represent pump waves at frequencies 1,0  , respectively. The optical power of all higher-order modulation sidebands was at least 20 dB below those of the primary tones.
The two tones were amplified by an erbium-doped fiber amplifier (EDFA), and then amplitude-  The magnitude of Rayleigh back-scatter of coherent optical fields is extremely noisy [34,35].
For that reason, commercial OTDR instruments launch incoherent light pulses. However, the effective stimulation of guided acoustic waves requires the use of a narrowband, coherent light source. To work around this difficulty, experiments were repeated over 512 choices of the central optical frequency  , between wavelengths of 1558 nm and 1560 nm, and traces were averaged with respect to  [36]. The ensemble average over many central optical frequencies helps reduce noise due to coherent Rayleigh back-scatter, and provides a mapping of local power levels along the fiber under test, similar to that of incoherent OTDR. As a final processing stage, the experimental traces were digitally filtered by a moving average window of duration  . The spatial resolution of the analysis is given by

Experimental results
Figure 8(a) shows multi-tone OTDR measurements of the local power levels of the six spectral components   k Pz, with  adjusted to 2π×340 MHz. This radio-frequency is detuned by 2π×20 MHz from the nearest resonance of F-SBS due to radial acoustic modes. Corresponding calculated traces are shown again for comparison (see Fig. 3(a)). Symmetric nonlinear generation of the Stokes-wave and anti-Stokes-wave sidebands is observed as anticipated. Figure 8   In both cases, asymmetry is observed between the two pump waves, as well as between the Stoke-wave and anti-Stokes-wave sidebands. The amplification of sidebands is more effective with  below the F-SBS resonance, as predicted.
Here too, the agreement between measurements and simulations is very good.

Summary
Distributed mapping of nonlinear wave mixing processes involving F-SBS along standard singlemode fiber has been carried out in analysis, simulation and experiment. The results show that the combined effects of F-SBS and Kerr nonlinearity give rise to wave mixing processes that are markedly different from FWM that is driven by the Kerr effect alone. The process dynamics become strongly dependent on the exact frequency separation between co-propagating field components. When that separation is adjusted near a resonance of F-SBS, symmetry is removed.
The stimulation of the guided acoustic wave is associated with coupling of power from a highfrequency input tone to a lower-frequency one. Consequently, Stokes-wave sidebands are amplified more efficiently than respective anti-Stokes-wave components. The detuning of the frequency separation below or above the F-SBS resonance leads to different wave mixing dynamics. The trends observed in the experiments are fully supported by corresponding numerical simulations. Approximate, analytic solutions for the regime of weak sidebands were derived and compared against simulations results.

22
The measurement protocol requires that the duration of pulses  should be several times longer than the lifetime 1 m   of the guided acoustic mode. That lifetime is on the order of 100 ns in standard coated fibers [22]. This limitation restricts the spatial resolution z  of the analysis to the order of tens of meters or longer. Lifetime considerations also restrict the maximum input power of each pump wave, since the nonlinear length The input power should also be kept below the threshold of amplified spontaneous backwards Brillouin scattering along the spatial extent of the pulse 2 z  . This restriction is on the order of few W as well. Higher spatial resolution may be achieved using double-pulse acquisition schemes, as applied in distributed Brillouin sensing [44,45].
The mapping of F-SBS using Rayleigh back-scatter is at the basis of a new concept for distributed fiber-optic sensing, which we refer to as opto-mechanical time-domain reflectometry [36]. The protocol is based on Eq. (6) and Eq. procedure is similar to the one used here. However, as discussed in Section 2, the analysis of dualtone traces is only valid when no other spectral field components exist in the fiber. The nonlinear wave mixing generation of spectral sidebands currently restricts the input power levels and the measurement signal-to-noise ratio, range and resolution [36].
On the other hand, the simultaneous OTDR analysis of four, six and perhaps more field components, as demonstrated in this work, might allow for the recovery of local F-SBS spectra 23 even where nonlinear wave mixing takes place among multiple optical tones. The quantitative agreement between the coupled equations model (Eq. (22)) and experimental results suggests that such extensions of the sensing protocol may be feasible. The analysis of multiple sidebands might lead to higher resolution, longer range, and better precision. Sensing applications of multi-tone opto-mechanical time-domain reflectometry will be examined in future work.