1. Introduction

Photonic devices that exhibit polarization dependence (e.g., polarization dependent loss/gain and polarization mode dispersion) are considered detrimental in general, as they degrade the performance of photonic systems when employed. On the other hand, if the polarization dependence could be all-optically controlled with a large dynamic range at modest optical power levels, it would be highly attractive for active control and manipulation of the polarization state of light, which can be adopted in new types of photonic components and advanced all-optical signal processing. Nonlinear optical phenomena, e.g., the optical Kerr effect, stimulated Raman scattering and stimulated Brillouin scattering (SBS), have offered compelling nonlinear polarization effects that can be potentially adopted for the realization of all-optical polarization control. Recent examples include lossless polarizers based on the self-polarization effect [1], Kerr-based polarization scramblers [2], and polarization-switchable Brillouin fiber lasers [3]. Although SBS in conventional single-mode fibers has been long considered as one of the most significant nonlinear optical effects, the insufficient polarization dependence of SBS gain [4,5] has hindered its practical use for implementing all-optical polarization manipulation. Strongly polarization-selective SBS amplification of one particular polarization mode could also dramatically improve the performances of various SBS-based systems, e.g., tunable bandpass filters [6], Brillouin-based optical spectrometers [7], and optical vector network analyzers [8].

With the advancement of nano-fabrication techniques, it has been possible to realize micro/nano-scaled photonic systems that allow for strong confinement of light in small volumes to yield highly efficient nonlinear optical effects. In particular, novel kinds of nonlinear optomechanical phenomena can emerge at modest optical powers, when light and acoustic phonons are tightly trapped simultaneously in tiny spaces, as recently demonstrated in micron/submicron-thick fiber tapers [9–11], small-solid-core microstructured optical fibers with high air-filling fractions [12–14], and silicon on-chip suspended waveguides [15,16]. In these systems, tailored dispersions of trapped acoustic phonons yield forward SBS (FSBS) via the phase-matched nonlinear coupling between the co-propagating guided light and transverse acoustic resonances (ARs) [13,14]. In sharp contrast to conventional backward SBS (BSBS), the optically excited AR in the FSBS process can be even automatically phase-matched with the higher-order Stokes and anti-Stokes waves at the same time, which in some cases leads to the generation of equally-spaced optical frequency comb [13].

The ARs and guided light can be further tailored by manipulating the waveguide cross-sections, as they are strongly influenced by the waveguide boundaries. Although this possibility could be investigated for engineering of the properties of the resulting nonlinear photon-phonon interactions via FSBS, the relevant previous works have been limited to changing the cross-sectional waveguide dimension to adjust the Brillouin frequency shift and FSBS gain [9,15,17]. It has been recently reported that the backward Brillouin scattering mediated by the traveling surface acoustic waves (SAWs) can be eliminated [11,18] or even extremely polarization-sensitive [18] in suitably designed subwavelength silica glass waveguides. However, the inherently weak interactions between the SAWs and the guided light still greatly limit the BSBS gains, which seriously obstructs its practical application. In strong contrast, in the FSBS the intrinsically larger optomechanical overlaps between the transverse ARs and the guided light yield much greater FSBS gains. In particular, this makes the polarization-selective light amplification via FSBS remarkably advantageous over the backward counterpart for advanced all-optical signal processing, although such the possibility has not been explored yet.

In this paper, we report for the first time that the FSBS driven by the transverse ARs, not to be confused with the backward SAW Brillouin scattering [11,18], can be completely eliminated and even strongly polarization-selective in subwavelength waveguides with suitably designed core geometry. By carefully selecting both the aspect ratio (or ellipticity) and dimension of the waveguide core, we can make the FSBS mediated by a certain AR mode completely suppressed for only one optical polarization mode, while rather enhancing that for the other polarization mode, which provides a unique practical way of highly efficient all-optical dynamic polarization control. Furthermore, we also show that the qualitative behavior of polarization selectiveness is tailorable by choice of the sign of photoelastic coefficient (PEC) of the waveguide material. We perform full-vectorial simulations of optical and AR modes in the waveguides and investigate their nonlinear interactions to fully understand the polarization selectiveness.

2. Polarization-selective FSBS in silica subwavelength waveguides

We first consider silica-glass elliptical waveguides suspended in the air, as described in Fig. 1(a), which is feasible in the form of highly birefringent microstructured fibers [20] and elliptical microfibers [21]. The FSBS is dependent generally on both the ellipticity and dimension of the core. Here, we define the core ellipticity as e = (a–b)/a (0≤e≤1), where a and b are the semi-major and semi-minor axis, respectively, and use the ‘equivalent radius’ $R_{\text{eq}}=\sqrt{ab}$ to represent the core dimension [18]. By considering the symmetry of the optical modes, it can be verified that the TR₂₁-like (0°/90°) torsional-radial AR and the R₀₁-like radial AR (Fig. 1(b)) can drive intramodal FSBS significantly [13,14]. We note that rectangular waveguides could be an alternative model of realistic waveguides having the C2 symmetry, while the polarization sensitivity of FSBS has not been investigated in either the previous experiments [15,16] or theoretical analyses [17,19] with on-chip rectangular waveguides. Our analysis on the rectangular waveguides, not to be presented here, shows that their key qualitative features of polarization selectiveness are the same as those of elliptical ones, whereas the polarization selectiveness takes place at different core aspect ratios for the two cases. We emphasize that the polarization selectiveness could be thus practically exploited in the on-chip integrated air-suspended rectangular waveguides [15–17,19] as well.

 

Fig. 1 (a) Cross-section of a typical elliptical waveguide suspended in the air, together with the Cartesian coordinates. The equivalent radius R_eq represents the core dimension, in such a way that the cross-sectional area of the core (grey ellipse) equals that of the circular core of radius R_eq (dotted circle). (b) Transverse ARs in elliptical waveguides that mediate FSBS significantly. A color map is used to describe the profile of magnitude of total displacement, where blue and red correspond to the zero and maximum displacement, respectively. Green arrows indicate the direction of acoustic displacement.

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We calculate the intramodal ‘FSBS coupling’ (G), defined as a figure of merit for SBS [18], at the wavelength of 1550 nm for the two types of ARs, over the entire range of ellipticity and equivalent radius of the core (Fig. 2). The photon-phonon interactions via intramodal FSBS can be highly polarization-selective, in the sense that the FSBS coupling is eliminated for one polarization mode, while it is kept significant for the other. The strong polarization selectiveness is observed over a wide range of core dimension for both ARs. For instance, for the TR₂₁-like AR, the FSBS driven by the x-polarized mode is completely suppressed at certain core ellipticities, which appears as a zero-FSBS-coupling curve in Fig. 2(a). In sharp contrast, around the zero-FSBS-coupling conditions, the FSBS coupling for the y-polarized mode is rather enhanced, having a maximum value of G = 29 W⁻¹km⁻¹ at (R_eq, e) = (607 nm, 0.42), as can be seen in Fig. 2(b). For the R₀₁-like AR, on the contrary, the FSBS coupling is eliminated only when driven by the y-polarized mode, as shown in Figs. 2(c) and 2(d). We emphasize that this strong polarization selectiveness is markedly different from the conventional common sense of polarization sensitivity. The latter implies that the complete suppression of FSBS by a certain AR mode would take place at different core ellipticities and/or dimensions for the two polarization modes. For instance, this would then create a zero-FSBS-coupling curve in Fig. 2(b) as well in a similar fashion to Fig. 2(a) but along a different trajectory of core parameters compared to that in Fig. 2(a).

 

Fig. 2 Contour plots of the FSBS coupling in silica glass elliptical waveguides as functions of the equivalent radius (R_eq) and ellipticity of the core for each AR and polarization mode. The white dashed curves in (a) and (d) indicate the core parameters at which the FSBS coupling is eliminated. The pink dashed curve in (c) follows the acoustic anti-crossing points between the R₀₁-like and the flexural F₁₁ ARs. Notice that the plot in (a) is in the logarithmic scale.

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To figure out the strong suppression of FSBS, we take into account the contribution of electrostriction and that of radiation pressure separately to the total FSBS coupling. We define G^(es) and G^(rp) as the FSBS coupling components independently contributed by the electrostrictive force and radiation pressure, respectively, and calculate them at R_eq = 500 nm over the entire range of core ellipticity. At the core ellipticity for which FSBS is suppressed, the two FSBS coupling components have the same magnitude as each other as shown in Figs. 3(a) and 3(d), which implies that they are canceled out to yield the zero total FSBS coupling.

 

Fig. 3 Explanation of the polarization-selective FSBS in silica glass elliptical waveguides. (a-d) Contribution of electrostriction (blue curves) and radiation pressure (red curves) to the total FSBS coupling (black curves) at R_eq = 500 nm over a range of core ellipticity for each AR and polarization mode, which correspond respectively to Figs. 2(a)–2(d). The green vertical dashed sections indicated by ‘A’ in (a) and ‘B’ in (d) point to FSBS suppression. At the purple vertical dashed sections in (c) and (d), acoustic anti-crossing emerges between the R₀₁-like and the flexural F₁₁ ARs. (e-n) The optical field profiles, optical force distributions, and the resulting optomechanical work densities on the waveguide cross-section, in the condition where FSBS mediated by the TR₂₁-like AR is suppressed for the x-polarized mode (e = 0.63). The electrostrictive force (f^(es)) and the radiation pressure (f^(rp)) are shown in blue and red, respectively. The bulk and boundary forces are plotted in the same scale for comparison. The positive and negative work densities are displayed as red and blue, respectively.

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The strong polarization selectiveness of FSBS can be intuitively understood in terms of the time-averaged optomechanical work density $W\propto \mathrm{Re}\left(f^{*}\cdot {\text{u}}_{\text{m}}\right)$ done on the waveguide by the optical forces, as described in Figs. 3(e)–3(n), where f and u_m are the optical force distribution and the displacement profile of AR, respectively. The optical forces are almost transverse in the FSBS process, and the dominant electrostriction stress tensor components are then σ_xx and σ_yy for both polarization modes, which are expressed for the electric field profile E in isotropic media by [22]

We also obtain the spectra of FSBS for each polarization mode by calculating the acoustic frequency and FSBS coupling for several ARs (including the TR₂₁-like and the R₀₁-like ones) over the entire range of core ellipticity. Figure 4 shows the result, which we obtain as keeping the core dimension at R_eq = 500 nm and assuming an arbitrary but realistic value of AR quality factor of 1000 for silica glass [23]. It is noteworthy that a number of anti-crossings emerge at some core ellipticities, which we attribute to the simultaneous resonances of two acoustic modes satisfying the free-boundary conditions for the acoustic displacement at the waveguide interface [24]. For instance, the frequency of the R₀₁-like AR increases with the core ellipticity and intersects that of the higher-order flexural F₁₁ AR at e = 0.63. The amount of frequency splitting at the resulting anti-crossing depends on the coupling strength between the two ARs. As the core ellipticity increases further, the R₀₁-like AR branch forms a series of anti-crossings with other types of ARs. We collect these R₀₁-like ARs and designate them here as a ‘family of R₀₁-like ARs’. We note that for the x-polarized mode the FSBS coupling decreases significantly nearby the acoustic anti-crossings, as can be seen in Fig. 2(c). In addition, for the y-polarized mode, the flexural ARs dominate the FSBS coupling over the R₀₁-like ones at the core ellipticities above the first anti-crossing, which results in negligible FSBS couplings of the R₀₁-like ARs at high core ellipticities (Fig. 2(d)).

 

Fig. 4 (a) FSBS spectra of silica glass elliptical waveguides of R_eq = 500 nm over a range of core ellipticity, for each polarization mode. The black dashed curves indicate a family of R₀₁-like ARs in virtue of the strong coupling between torsional and radial displacements. The green dashed squares show FSBS suppression. (b) Zoomed-in FSBS spectra corresponding to the light blue dashed rectangle in (a). The white and black in the color map correspond to the zero and maximum value of FSBS coupling, respectively. Two dashed curves represent the R₀₁-like (red) and the flexural F₁₁ (yellow) ARs, and the inset shows their displacement profiles with the exaggerated deformation for clarity, where the color map represents the y-displacement (u_y) for the R₀₁-like AR and the x-displacement (u_x) for the F₁₁ AR, blue, white and red corresponding to negative, zero and positive values, respectively.

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3. Polarization-selective FSBS in silicon subwavelength waveguides

It is worth comparing the results so far for silica waveguides with those for silicon counterparts, as the latter has recently attracted rapidly growing attention, being anticipated to exhibit ultrahigh optomechanical interaction efficiencies [17]. Silicon has the PECs of p₁₁ = –0.09 and p₁₂ = 0.017 under the [100] orientation. On the contrary to fused silica, since |p₁₁| > |p₁₂| for silicon, the electrostriction stress tensor component σ_xx[σ_yy] is dominant over σ_yy[σ_xx] for the x[y]-polarized mode (Eqs. (1) and (2)), and the direction of electrostrictive bulk force is then mostly parallel to the optical polarization (Figs. 5(g) and 5(h)). In addition, the negative value of p₁₁ makes the electrostrictive bulk force point outward (Figs. 5(g) and 5(h)), which combines constructively with the radiation pressure regardless of the optical polarization (Figs. 5(i) and 5(j)), though near the waveguide boundary relatively smaller electrostrictive forces exist that tend to pull inward against the radiation pressure (Figs. 5(i) and 5(j)).

The constructive combination of electrostrictive bulk force and radiation pressure for both polarization modes has eluded the observation of polarization selective FSBS in silicon waveguides. Indeed, the polarization-selective behavior does not exist for the photon-phonon interactions by the TR₂₁-like AR (Figs. 5(a) and 5(b)). For the R₀₁-like AR, however, strong polarization selectiveness is observed at certain core ellipticities (Figs. 5(c) and 5(d)). For the y-polarized mode, the electrostrictive bulk force and radiation pressure cooperate in the y-direction, which in turn excite efficiently the R₀₁-like AR having the dominant strain component S_yy (Figs. 5(h) and 5(j)). The time-averaged optomechanical works done on the waveguide bulk and boundary then combine in phase, which gives rise to non-zero FSBS couplings (Figs. 5(l) and 5(n)). On the other hand, when driven by the x-polarized mode, the bulk and boundary work densities are distributed in such a way that the net work done on the waveguide vanishes, yielding the FSBS suppression (Figs. 5(k) and 5(m)). We emphasize that the polarization-selective FSBS in silicon waveguides can be potentially a key phenomenon for implementing polarization devices in on-chip photonic integrated circuits.

 

Fig. 5 (a-d) FSBS coupling in silicon elliptical waveguides as a function of the core ellipticity at R_eq = 200 nm, for each AR and polarization mode. The green vertical dashed section indicated by ‘A’ in (c) points to FSBS suppression. (e-n) The optical field profiles, optical force distributions, and the resulting optomechanical work densities on the waveguide cross-section, in the condition where the FSBS mediated by the R₀₁-like AR is suppressed for the x-polarized mode (e = 0.55).

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4. Conclusion

In conclusion, we have shown that strongly polarization-selective nonlinear optomechanical interactions emerge in subwavelength waveguides with carefully designed core geometry. At certain core aspect ratios (or ellipticities), FSBS mediated by a specific AR mode is eliminated for only one polarization mode, while that for the other polarization mode is enhanced. Thanks to the sufficient FSBS coupling (on the order of 10 W⁻¹km⁻¹ for silica waveguides) compared to particularly the weak backward SBS mediated by traveling SAWs [18], the strongly polarization-selective FSBS offers a unique practical way of highly efficient all-optical reconfigurable polarization control with a huge dynamic range. This remarkable feature could be applied to, e.g., dynamic nonlinear polarizers, polarization switching, and wavelength-polarization-division multiplexing, mode-locked lasers with ultrahigh degrees of polarization, as well as the above-mentioned applications [6–8]. In these applications, the co-propagating pump and Stokes waves can be separated by using a commercially available ultra-narrow-band wavelength filter with the bandwidth far below the AR frequency (a few GHz), such as a phase-shifted fiber Bragg grating [25] and an optical cavity. The intriguing phenomenon of polarization-selective suppression of FSBS can be explained by the counterbalance between electrostriction and radiation pressure and turn out to be strongly affected by the PECs of waveguide materials. Our study provides a new opportunity of engineering boundary-enhanced optical forces and nonlinear photon-phonon interactions.