1. Introduction

Stimulated Raman scattering induces energy transfer between photons and lattice vibrations (phonons), and this effect can be quantitatively analyzed as a nonlinear optical scattering phenomenon with a medium possessing complex third order susceptibility. The growth of a Stokes wave can be written as [1]:

This is in contrast to well-known fiber parametric nonlinearities four-wave-mixing [1] (FWM), where phases of individual beams matter, and hence phase matching among the interacting photons determines the emission spectra [Fig. 1(a)]. The fundamental distinction is between elastic and inelastic scattering. Raman scattering is inelastic because a participating optical phonon mediates both energy and momentum conservation. Since the magnitude of momentum carried by lattice vibrations is orders of magnitude greater than that carried by photons, optical phonon wavevectors at almost any arbitrary orientations in relation to those of the incident and scattered photons are able to offer momentum conservation for the scattering process [Fig. 1(b)]. Hence the Raman scattering process automatically conserves momentum, and since momentum is intimately tied to the phase of a photon, Raman is typically known to be a self-phase-matched process. This imperviousness to phase is actually a key attribute of Raman scattering and its widespread utility in telecom amplification [9] or building fiber lasers [10], since spectral shapes of Raman gain are not influenced by properties such as pump-probe frequency separation or the dispersive properties of the fiber modes. But this also means that it is inevitably present in all systems and can be detrimental in applications such as fiber-based quantum source generation [11] or power-scaling of fiber lasers [12], where it either acts as a noise source or reduces device efficiency. Thus, reported schemes for suppressing Raman scattering have involved ‘brute-force’ methods such as increasing the mode area of fibers [13], inhibiting transmission at Stokes wavelengths [14] or cooling the material [15] to decrease the magnitude of ${g_R}(\mathrm{\Omega } )$. Even so, none of these techniques can significantly alter the spectral shape ${g_R}(\mathrm{\Omega } )$, defined solely by the material. Here, we show that light carrying orbital angular momentum (OAM) that experiences spin-orbit coupling, as readily occurring due to the confining potential of an optical fiber waveguide, can fundamentally alter Raman scattering by making it sensitive to the phases of the participating beams. This enables, for the first time to our knowledge, dispersive control of the strength as well as spectrum of Raman scattering simply by tailoring the topological charge of light.

 

Fig. 1. Phase-matched Raman scattering principle. (a) A parametric nonlinear process such as four-wave-mixing requires momentum conservation of all interacting photons for maximum gain. The colored arrows indicate wavevectors. The subscripts p₁, p₂, s and as represent two pumps, Stokes and anti-Stokes photons. (b) Raman scattering is a phase-insensitive process as the optical phonons compensate for any phase mismatch between the pump and Stokes photons. (c) Facet image of the ring-core fiber used in the experiment, schematic plot of effective index splitting ($\Delta {n_{eff}}$) between ${\hat{\sigma }^ + }$ and ${\hat{\sigma }^ - }$ of same L due to spin-orbit interaction. (d, e) Schematic plots of polarization evolution for optical activity modes of different topological charges at pump and Stokes wavelengths. Red: Pump light in ${L_p} = 21$, green: Stokes light in ${L_s} = 22$, blue: Stokes light in ${L_s} = 20$. (f) Beat length as a function of wavelength for different modes.

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2. Principle of phase-matching in Raman scattering

Eigenmodes in a circularly symmetric waveguide such as an optical fiber [16] carry both orbital angular momentum, which is quantified by its topological charge L, and spin angular momentum ${\hat{\sigma }^ \pm }$ (left/right-handed circular polarization). Spin-orbit interaction (SOI), originating from the light-matter interaction at dielectric interfaces, splits the effective refractive index ($\Delta {n_{eff}}$) between ${\hat{\sigma }^ \pm }$ states of same L, inducing optical activity (OA) [17]. The ring-core fiber we use in our experiments [Fig. 1(c)] amplifies this SOI, enabling stable propagation of each {$L,{\hat{\sigma }^ \pm }$} state. In addition, since this splitting is geometrodynamic in nature, optically active superpositions of these states, such as linearly polarized OAM modes, are also stable during propagation [17]. As a result, the polarization orientation angle of linearly polarized light carrying OAM at the input of the fiber systematically rotates as it propagates, with the beat length ${Z_{beat}} = \lambda /\Delta {n_{eff}}$. Schematic plots of this polarization evolution are shown in Figs. 1(d) and 1(e), which also illustrate a key attribute of OA due to the SOI effect – Fig. 1(d) illustrates this rotation for two eigenmodes, and it is evident that ${Z_{beat}}$ varies strongly with topological charge, since SOI is known to be strongly dependent on topological charge (varies as ${L^3}$ in the ring-core fibers used in our experiments, see section 3 in Supplement 1). In addition, waveguide dispersion makes ${Z_{beat}}$ a strong function of wavelength too. Thus, a spatially complex but deterministic polarization rotation effect typically dependent primarily on the material (dispersive) properties of chiral media [18], is now dependent on light’s topological charge itself [Fig. 1(f)]. Now, considering the Raman interaction, we can quantify the spatial frequency ($k = 2\pi /{Z_{beat}}$) difference of the linear polarization rotation between a pump and Stokes photon for arbitrary mode combinations as:

3. Results

We probe this effect with a 650-ps, 1.7-kW-peak-power (${P_{peak}}$) pump source comprising a Nd:YAG laser and a fiber amplifier. The Guassian beam of the pump is converted to the desired OAM mode of a 9.8-m-long ring-core fiber [Fig. 1(c)] with a spatial light modulator (SLM), and another SLM is used to modally sort the output spectrum by the topological charge (see Section 1 and 2 in Supplement 1). We sweep ${L_p}$ from 7 to 24 in both ${\hat{\sigma }^ - }$ and $\hat{x}$ (OA) polarizations whose resultant measured mode intensity and phase profiles at the fiber output are shown in Fig. 2(b) (the details of the fiber and mode properties are discussed in Section 3 of Supplement 1). Similar ring shapes and sizes for all the modes result in similar nonlinear overlap integrals $\eta $ [see Eq. (2)] for all different combinations of ${L_p}$’s and ${L_s}$’s – which is what conventionally controls the Raman scattering strength [19]. To account for the slight differences, and hence to avoid any variations in Raman scattering strengths due to differing pump-Stokes overlaps, we adjust ${P_{peak}}$ slightly for each ${L_p}$ such that $({{P_{peak}}\cdot \eta } )$ remains constant throughout the experiments. For pumps in a pure circular polarization ${\hat{\sigma }^ - }$, we observe that the Stokes light is also in ${\hat{\sigma }^ - }$ (Stokes power in the orthogonal polarization ${\hat{\sigma }^ + }$ is lower by more than 40 dB, hence the cross-polarization Raman gain contribution is ignored in this case), and almost identical, conventionally co-polarized Raman spectra are obtained for all pump and Stokes mode combinations. Given their similarity, for illustrative clarity, we only plot one of them (${L_p} = 7,\; {\hat{\sigma }^ - }$;${L_s}\; = 8,\; {\hat{\sigma }^ - }$) as the red curves in Fig. 2(a). But when the pump is linearly polarized along $\hat{x}$, despite the fact that the spatial overlap, $\eta $, still remains similar across all mode combinations, the non-zero $\Delta k$ terms for Stokes and pump combinations satisfying (${L_s} \ne {L_p} - 1$) result in complete polarization walk-off between the pump and Stokes light [shown schematically in Fig. 1(d)], leading to significant Raman gain reduction. Again, since all (${L_s} \ne {L_p} - 1$) yield similar spectra, Fig. 2(a) plots the spectra obtained for only one such mode combination: ${L_p} = 7,\hat{x}\; $ and ${L_s}\; = 8$ [green curves in Fig. 2(a)]. Note that while their spectral shapes are similar to the conventional case where the pump is in ${\hat{\sigma }^ \pm }$ polarization, power in the Stokes band is dramatically suppressed by ∼20 dB for the same pump powers and mode sizes (intensity overlaps) [20]. Finally, the blue curves in Fig. 2(a) are Raman spectra for the Stokes mode in ${L_s} = {L_p} - 1$. Due to the perfect polarization overlap at every position along the fiber [Fig. 1(e)], a strong Raman gain peak with strength similar to that of co-polarized Raman scattering (red curves) is obtained at the beat length matching wavelength ${\lambda _b}$ where $\Delta k = 0$. Furthermore, since $\Delta k$ increases as Stokes wavelength deviates from ${\lambda _b}$, the Raman gain spectrum is re-shaped and narrowed (yellow shaded region under blue curve), and the polarization state of the Stokes light evolves from linearly polarized (at ${\lambda _b}$) to elliptically polarized and then circularly polarized due to differential gain in ${\hat{\sigma }^ + }$ and ${\hat{\sigma }^ - }$ as wavelength deviation increases (see Section 5C in Supplement 1). Additional sharp peaks evident on every spectral plot arise from the multitude of parametric FWM interactions that are also possible in these fibers [21] – as described in Section 4 in Supplement 1, these neither influence, nor change the conclusions evident from the Raman spectra. The modulated Raman peak wavelength ${\lambda _b}$ shifts as pump mode changes since the $\Delta k = 0$ condition is strongly dependent on the spectral dispersion of the modes, as well as, crucially, the topological charge that the pump mode carries. Hence, Fig. 2(a) illustrates the ability to tailor both the strength and wavelength of Raman gain by controlling the angular momentum content of light.

 

Fig. 2. Raman spectra and pump mode images. (a) Raman spectra plots for pump with different topological charges, arranged, top to bottom as ${L_p} = 7,\; 8,\; 9,\; 11,\; 14,\; 17,\; 19,\; 21,\; 24$; Raman spectra shown as red curves represent {Stokes in ${L_s} = 8,{\hat{\sigma }^ - }$; Pump in ${L_p}\; = 7,\; {\hat{\sigma }^ - }$} combinations; green curves represent {${L_s} = 8$; ${L_p}\; = 7,\; \hat{x}$} combinations; and blue curves denote {${L_s} = {L_p} - 1$; ${L_p},\; \hat{x}$} combinations. The yellow shaded area represents contributions from the $\Delta k = 0\; $ phase matching condition. (b) Left and right columns: Intensity distribution of pump modes in $\hat{x}$ and ${\hat{\sigma }^ - }$ polarizations. Middle: Phase pattern acquired by interference with an expanded Gaussian beam, the number of spiral arms being equal to ${L_p}$, the order of OAM of the pump beam.

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Figure 3(a) shows the integrated Raman power for the three functionally distinct mode combinations described in Fig. 2. The red curve represents the co-polarized Raman scattering process where both pump and Stokes light are in the ${\hat{\sigma }^ - }$ polarization, with each data point and error bar denoting the average and deviation of Raman strength in all Stokes modes for a given ${L_p}$. Despite the fact that the error bar for the ${L_p} = 11$ case is large, due mainly to superimposed spectra from parasitic FWM (see Section 5B in Supplement 1), the flat trend with respect to ${L_p}$ confirms conventional wisdom, that Raman strength does not depend on the phases of the participating waves (note that topological charges essentially manifest in the spatial phase of a mode). In contrast, when the pump light is an optically active state arising from SOIs, all but one of the Stokes modes experience phase mismatches due to non-zero $\Delta k$, and Stokes power is suppressed by ∼ 20 dB, as illustrated with the green curve of Fig. 3(a) (again, large measurement errors for ${L_p} = 9,\; 12$ arise from parasitic FWM, but do not, otherwise, influence the observed trend). For the Stokes mode corresponding to ${L_s} = {L_p} - 1$, we observe a systematic tunability of strength of Stokes power by over 15 dB since the Raman spectra are now tailored by the $\Delta k = 0$ condition [blue curve in Fig. 3(a)]. The spectral position of peak Raman gain now depends not only on the conventional Raman scattering strength of the material, but also its relation to the beat-length-matched wavelength ${\lambda _b}$. Figure 3(b) shows this new degree of freedom – the ability to tune the Raman gain peak wavelength ${\lambda _b}$, simply by tuning the topological charge of the pump mode, by over 30 nm (∼ 8 THz), which is greater than half of the material’s (Silica’s) conventional Raman Stokes shift (∼13 THz).

 

Fig. 3. (a) Integrated Raman power within 20 nm wavelength range for different pump-Stokes mode combinations, red: Raman power for {Stokes in ${L_s},{\hat{\sigma }^ - }$; Pump in ${L_p},{\hat{\sigma }^ - }\} $; green: Raman power for $\{ {L_s} \ne {L_p} - 1$; ${L_p},\hat{x}\} $ combination, i.e. $\Delta k \ne 0$; blue: Raman power for {${L_s} = {L_p} - 1$; ${L_p},\hat{x}\} $ combination, i.e. $\Delta k = 0$. Relatively larger error bars for some data points in red and green curves represent cases where a coincident, parasitic FWM peak existed, corrupting measurement of Raman power alone. (b) Peak wavelengths shift for Stokes mode in ${L_s} = {L_p} - 1$ as a function of pump mode $\{{{L_p},\hat{x}} \}$. (c) Simulated plot of phase mismatch multiplied by fiber length as a function of wavelength for $\{{{L_s} = 14;{L_p} = 15,\hat{x}} \}$ mode combination. (d) Experimentally measured Raman spectral shape for $\{{{L_s} = 14;{L_p} = 15,\hat{x}} \}$ mode combination in both 9.8- and 20-m-long ring-core fibers.

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Further affirmation of this phase-matched behavior is evident from recording the bandwidth of the modified Raman spectra [yellow shaded regions of Fig. 2(a)] as a function of fiber length. The spectra of self-phase matched, conventional Raman scattering should display no inherent dependence on interaction length, at least in the spontaneous scattering regime. In contrast, processes that have strict phase matching requirements, such as second harmonic generation, have a response that is proportional to both phase mismatch $\Delta k$ and propagation distance $z$ [1]:

Since $\Delta k$ maps to frequency, the bandwidth of a parametric process is inversely proportional to z. Figure 3(c) plots the simulated $\Delta k\cdot z/2$ values and Fig. 3(d) shows the corresponding experimentally recorded Raman spectra of ${L_p} = 15$; ${L_S} = 14$, for two different fiber lengths – 9.8 m and 20 m – shown in blue and red curves, respectively. The green band in the simulations indicate the spectral limits of $\Delta k\cdot z/2$ at which the nonlinear response is halved [per Eq. (4)]. Remarkably, not only do these intersection points correspond well with the measured Raman spectra for the two lengths, the measured bandwidth ratio (1.42 nm/2.64 nm) also closely corresponds to the fiber length ratio (9.8 m/20 m). This length dependence of an otherwise self-momentum conserved scattering process illustrates the centrality of the role of phase matching arising from SOIs.

4. Summary and conclusions

The ability to modulate Raman scattering strength by ∼20 dB, and to spectrally tune it by over ½ the Raman gain bandwidth by controlling light’s topological charge, yields a design degree of freedom to enhance, spectrally shape, or avoid Raman scattering in optical fibers, as diverse applications may demand. The fact that the phase of light plays a role opens this important nonlinear process to control via dispersion engineering – an established design tool for fibers and waveguides. Finally, since the underlying effect arises from SOI of topologically complex light, systems other than waveguides, such as tightly focused light in bulk media, or OAM beams in chiral media, which would experience SOI or optical activity, may yield similarly anomalous Raman scattering dependencies, enabling control of Raman scattering in media other than waveguides too.