Mitigating stimulated Brillouin scattering in multimode fibers with focused output via wavefront shaping

We first demonstrate SBS mitigation by coherent multimode excitation. To this end, we vary the number of excited modes and measure the SBS threshold. The input signal (serving as the pump for SBS) is a linearly-polarized continuous wave from a narrowband (15-kHz) fiber laser at 1064 nm. It is launched into a 50-meter-long, Ge-doped, step-index MMF with $\sim$20-$\mu$m core diameter and 0.3 numerical aperture (NA), supporting $\sim$80 modes per polarization. The transmitted light from the distal fiber end is split to simultaneously measure the output power and near-/far-field intensity patterns. We also monitor the input and backscattered light at the proximal fiber end (Supplementary Information, sec. 1).

We start with exciting only the fundamental mode (FM) in the MMF by focusing the incident light to the proximal fiber facet using a lens of NA matching that of the FM [Fig. 2a]. The transmitted beam produces a single spot at the center of the far field. We gradually increase the input power, and the output power first increases linearly but then levels off [Fig. 2d]. Meanwhile, the reflected power grows rapidly, indicating the onset of backward SBS. The transmitted power at this reflection threshold is 1.8 W.

Next, we excite multiple modes in the MMF by switching to a lens of larger NA (close to the core NA) [Fig. 2b]. The input beam is focused onto a small spot at the proximal fiber facet. When the focal spot is at the core center, predominantly the FM and several purely radial modes are excited. The output far-field pattern displays concentric rings. Due to modal interference, the near-field pattern is smaller than that of FM-only excitation. Nevertheless, the SBS threshold power rises to 2.6 W, indicating that the SBS suppression is not caused by transverse energy spreading that lowers the intensity. Instead, it is due to spectral broadening and peak reduction of the SBS gain by multimode excitation (see sec. 2.2).

To excite even more modes, we move the focal spot away from the core center [Fig. 2c]. Additional higher-order modes (HOM) with nonzero azimuthal index are excited in addition to the FM and radial HOMs. Consequently, the output beam is highly speckled. The SBS threshold increases with the distance $d_{\rm in}$ of the focal spot from the fiber core center [Fig. 3a], as more modes are excited. Near the core edge, the threshold reaches 5.4 W, $\sim$3 times the FM-only threshold.

To understand the mechanism of SBS suppression with multimode excitation, we have developed a multimode SBS theory (detailed derivation in [42]). Consider Brillouin scattering of a forward-propagating photon (signal) in fiber mode $l$ to a backward photon (Stokes) in mode $m$ via emission of a forward acoustic phonon at frequency $\Omega$. The Stokes photon has a frequency shift of $\Omega$ from the signal photon. The scattering strength $g_{\rm B}^{(m,l)}(\Omega)$ is determined by the spatial overlap of optical modes $m$ and $l$ with acoustic modes in the fiber [42]. We order the modes by their propagation constants from high to low.

We refer $m=l$ to intramodal scattering and $m\neq l$ to intermodal scattering. Figure 3b shows that if the signal is in the FM ($l$ = 1), the SBS is the strongest for Stokes in the FM ($m$ = 1), with scattering strength equal to $g_{\rm B}^{(1,1)}(\Omega)$, where the peak Stokes frequency $\Omega$ equals the eigenfrequency of the lowest-order acoustic mode. The intramodal scattering for an HOM (e.g., HE₂₄) has a lower peak at the same $\Omega$, due to smaller overlap with the lowest-order acoustic mode, and an additional peak at higher $\Omega$ corresponding to a higher-order acoustic mode. Intermodal scattering is generically weaker than intramodal scattering, due to smaller acousto-optic overlap for non-identical signal and Stokes mode profiles. The intermodal peak is not only lower, but also spectrally shifted from the intramodal peak, because the acoustic mode having the largest spatial overlap with the signal and Stokes modes is a higher-order mode with higher eigenfrequency.

Let us analyze $G_{\rm B}^{(m)}(\Omega)$ for a few cases. When only the FM is excited by the signal in a MMF, the Brillouin gain is strongest for the Stokes in the FM [$G_{\rm B}^{(1)}(\Omega)=g_{\rm B}^{(1,1)}(\Omega)P_{1}$]. Since $g_{\rm B}^{(1,1)}(\Omega)$ has the highest peak among all $g_{\rm B}^{(m,l)}(\Omega)$ [Fig. 3b], it results in the lowest SBS threshold. Instead, if a single HOM ($l\neq 1$) is excited, Stokes growth in that HOM ($m=l$) is strongest. Since intramodal scattering for an HOM is weaker than that for the FM, the SBS threshold is slightly higher. A more dramatic increase of the SBS threshold occurs when multiple modes are excited by the signal. Since the Brillouin gain $G_{\rm B}^{(m)}(\Omega)$ is a power-weighted sum of intramodal and intermodal scattering, it is spectrally broadened for Stokes in individual mode $m$ due to frequency-offset peaks of $g_{\rm B}^{(m,l)}$ among different mode pairs ($m$,$l$). Moreover, intermodal peaks are lower than intramodal ones [Fig. 3b]. Consequently, the peak growth rate of Stokes power is greatly reduced, leading to significant threshold enhancement over single-mode excitation.

To compare with experimental data [Fig. 3a], we calculate the modes excited by a signal focused on the proximal facet and obtain the Brillouin gain spectrum and the SBS threshold (Supplementary Information, sec. 2.1). A tight focus on the core center excites multiple modes, leading to a broader Brillouin gain spectrum than FM-only excitation and thus a lower peak [Fig. 3c]. Shifting the focus transversely to excite even more modes, the spectrum is further broadened with a much lower peak. The SBS threshold, predicted by our theory, increases monotonically with $d_{\rm in}$, and agrees well with the experimental data [Fig. 3a]. To obtain such agreement, we experimentally characterize the mode-dependent loss, linear mode coupling, and polarization mixing of our MMF, and include these effects in the SBS theory (Supplementary Information, sec. 2.2). Slope variation in the data is reproduced and explained by the theory (Supplementary Information, sec. 2.1).

The experiment with input focusing validates our multimode scheme for SBS suppression, but is restricted to considering a focused Gaussian input. To examine how robust this method is, we conduct a statistical study of the SBS threshold for arbitrary input wavefronts. We imprint random phase patterns on the signal beam (CW at 1064 nm) with a spatial light modulator (SLM) [Fig. 4a]. Strikingly, for all random input wavefronts, the threshold enhancement over FM-only excitation is consistently $>2$ [Fig. 4b]. The histogram shows a mean enhancement of $\sim$2.7 and a small standard deviation $\sigma\approx 0.14$. The prediction from simulations based on our theory is consistent with the experimental data [Fig. 4b]. It confirms that the random input wavefront distributes the signal power rather uniformly among all fiber modes [upper panel, Fig. 4c]. The calculated Brillouin gain spectrum [inset, Fig. 4d] has a full-width-at-half-maximum (FWHM) of 72 MHz, wider than the FM-only FWHM of 40 MHz. Consequently, the gain peak is notably lower, leading to threshold increase.

Figure 4d reveals the correlation between the threshold enhancement and Brillouin gain spectrum width. Different random input wavefronts cause variations in the excited mode content, leading to slightly different widths of the Brillouin gain spectrum. Larger widths correspond to higher SBS thresholds, with a Pearson correlation coefficient of 0.78.

To demonstrate the generality of our approach for SBS suppression, we switch from CW to pulsed (200-ns) signal at a different wavelength (1550 nm) and repeat the above experiments on the same type of MMF [Fig. 4e]. Under random multimode excitation, the mean SBS threshold enhancement is 2.1, slightly lower than that at 1064 nm. This is expected at a longer wavelength, where the fiber supports fewer optical modes ($\sim$37 per polarization).

The input-focusing data and the spread of SBS thresholds under random input wavefronts suggest that some multimode combinations are more efficient than others in mitigating SBS. These results prompt us to optimize the input wavefront for further enhancement of the SBS threshold. We sequentially optimize the phases of 8$\times$8 SLM macropixels, to increase the transmitted signal power and/or decrease the backscattered Stokes power (Supplementary Information, sec. 1.4). The threshold is increased to 3.6 times the FM-only threshold, higher than that with off-axis input-focusing, and $\sim$$7\sigma$ higher than the mean enhancement with random wavefronts [Fig. 4b]. To better understand the mechanism underlying the threshold enhancement, we combine our theory with a numerical simulation of the SLM phase optimization to maximize the SBS threshold. The threshold enhancement reaches 3.5, closely matching the experimental result (Supplementary Information, sec. 2.3).

As shown in Fig. 4c, the optimized mode content (from simulation) is not uniform; instead the power is concentrated in several groups that are widely spaced in propagation constant. Such selective mode excitation broadens the Brillouin gain spectrum more effectively than uniform mode excitation, and the FWHM reaches 85 MHz [inset, Fig. 4d]. Starting from different input wavefronts, the optimization procedure takes different routes, and the final SBS thresholds vary slightly [Fig. 4d]. Although the optimized mode contents differ from one another, they all consist of well-separated groups for maximal broadening of the Brillouin gain spectrum.

Finally, we show that wavefront shaping of a coherent signal not only can mitigate the SBS in an MMF, but also can shape the transmitted beam profile. Since our theory finds that the SBS suppression depends only on how the signal power is distributed among modes [Eq. 1], the $phase$ of the input field to individual modes can be adjusted to control the modal interference at the fiber output. Below we present an example of focusing the output light to a diffraction-limited spot. Perfect focusing requires full control of the input field amplitude and phase [43], whereas we have phase-only control with an SLM. Nevertheless, we find that this incomplete control can still direct most transmitted power to a designated location and simultaneously increase the SBS threshold.

As illustrated in Fig. 5a, a linearly-polarized Gaussian beam is phase-modulated by an SLM at the conjugate plane of the fiber input facet, so that the transmitted beam is focused to a spot near the output facet. We optimize the phases of 16$\times$16 macropixels sequentially to maximize the output intensity at the focus [44] (Supplementary Information, sec. 1.4). The focal spot can be placed anywhere within the field of view defined by the fiber core size and NA. We choose the focus to be 20 $\mu$m outside the output facet (in the air) to avoid damage to the fiber. Figure 5b shows that, upon optimization, the focal spot size ($D\approx 3.4$ $\mu$m at $e^{-2}$ of the maximum intensity) is almost at the diffraction limit ($D\approx 3.3$ $\mu$m) that is determined by the fiber core NA $\approx 0.3$. We also measure the phase of the transmitted field, and the flat phase across the focal spot confirms the diffraction-limited focusing. The power within the focus is $\sim$0.7 of the total output power, close to the limit for phase-only modulation of a Gaussian beam (Supplementary Information, sec. 2.4).

Since the formation of a tight focus comes from the interference of many fiber modes, multimode excitation leads to SBS suppression. We measure the SBS thresholds when focusing the output light at different distances $d_{\rm out}$ from the fiber axis. On-axis focusing ($d_{\rm out}=0$) results in a 1.8$\times$ increase of the threshold over FM-only excitation [Fig. 5c]. Moving the focus away from the fiber axis further increases the threshold, up to 3.1$\times$ enhancement at $d_{\rm out}\approx 9.5$ $\mu$m. This is because forming an off-axis focus needs an increased number of participating modes, e.g., $>70$ modes for focusing at $d_{\rm out}\approx 9.5$ $\mu$m. We also demonstrate this with 200-ns laser pulses at 1550 nm, confirming the broad applicability of our method for simultaneously achieving a high SBS threshold and controlling the output-beam profile.