Fig. 1. (a) Sketch of a short length of the three-fold rotationally symmetric PCF. The small circles mark the positions of the hollow channels, embedded in fused silica. (b) Left to right: scanning electron micrograph of the PCF structure. Upper: measured near-field intensity patterns of the [ℓ,s]=[0,+1],[+1,+1] hBMs, and the spiral interference fringes that form between the modal far-fields and a divergent Gaussian beam. The intensity patterns for the [−1,−1] and [+1,+1] modes are very similar. Lower: numerical simulations of the modal intensity distribution and the azimuthal phase variation of the modes. (c) Left two panels: numerical simulations of the transverse displacements of two orthogonal degenerate flexural modes at ∼96  MHz. When superimposed with a π/2 phase shift these modes generate a chiral flexural wave (CFW). Right-hand panel: axial displacement of the acoustic mode, confirming the presence of a flexural wave. (d) 3D sketch of a CFW formed by the superposition of two orthogonal π/2-out-of-phase flexural waves. When the waves have different frequencies (as in chiral FSBS), the shape rotates in time.

Fig. 2. Experimental setup for FSBS and frequency conversion between vortex modes. SSBM, single-sideband modulator; IM, intensity modulator; EDFA, erbium-doped fiber amplifier; FPC, fiber polarization controller; VGM, vortex generation module; CPBS, circular-polarizing beam splitter; BS, beam splitter; LIA, lock-in amplifier; OSA, optical spectrum analyzer; PM, power meter; NBA, near-field scanning Brillouin analyzer. A coherent population of CFP is written in the twisted PCF by forward-propagating pump and Stokes modes, and read out in the backward direction at a different wavelength, determined by a special phase-matching condition.

Fig. 3. (a) Dispersion curves for (from left to right) [ℓ,s]=[+1,+1],[−1,−1],[−1,+1],[+1,−1], and [0,±1] hBMs. The last two are separated by a small circular birefringence of 8×10−6, so are indistinguishable on the scale of the plot. Zero on the frequency axis corresponds to 1550 nm. In the experiments the peak gain is seen for [0,+1] pump to [−1,−1] Stokes conversion. (b) Dispersion curves for three CFWs in an untwisted PCF, with cut-offs in the frequency range from 85 to 115 MHz, calculated by FEM. The mode with cut-off at ∼89  MHz has the highest overlap with the optical modes. The open circle marks the CFW that phase-matches the [0,+1] pump and [−1,−1] Stokes modes in (a). In the vicinity of this point the acoustic dispersion is 0.0037 rad/μm per MHz. (c) Solutions of the coupled power equations for the parameters in the experiment (full curves). The red circles mark the experimental measurement. The Stokes mode experiences slight gain over the first 3 m, when there is substantial conversion from pump to Stokes. Beyond this point the FSBS conversion gradually weakens, and the signals converge to the base level exponential loss. The strength of the acoustic wave is proportional to the product of pump and Stokes, so also falls off in strength with distance. (d) The measured Stokes power as a function of frequency difference (red open circles) together with a fit to numerical solutions of Eq. (9), showing good agreement, with a full width at half-maximum of ∼3.8  MHz and a peak at 98.5 MHz. The extended shoulder on the high frequency side is attributed to excitation of acoustic modes with lower overlap and higher cut-off frequencies [the upper curves in (b)].

Fig. 4. (a) Schematic illustrating the principle of phase-matched backward read-out of CFW excited by chiral FSBS, at a different wavelength band. The frequency shift ΔΩ depends on the dispersion of the two modes. Note that if the pump and Stokes modes are exchanged, the phonon propagates backwards and ΔΩ changes sign. The blue arrow illustrates the phase-mismatching of the forward conversion and non-reciprocity of the read-out process. (b) Plot of the dephasing parameter Δβ for the backward process as a function of ΔΩ. The theory is based on FEM and predicts Δβ=0 at ΔΩ/2π=15.5  GHz, in quite good agreement with the experimental value. (c) Conservation of absolute angular momentum in the forward and backward SBS processes. The black arrows represent the phonons. Note that in (a) the topological charge and spin are conventionally defined relative to the beam propagation direction; for correct comparison with the forward process, however, the same frame of reference must be used for both directions, so the backward signs must be reversed, as in the figure.

Fig. 5. (a) Read-out spectra recorded by high resolution OSA when a [−1,−1] pump signal is launched backwards into the fiber in the vicinity of the phase-matching frequency, in the presence of strong chiral FSBS. The horizontal scale is the frequency shift of backward read-out signals over the forward writing signals. Backward FSBS is phase-matched when the backward pump frequency is shifted −18.5  GHz relative to the forward pump frequency. When the [−1,−1] pump signal is launched forwards, only very weak conversion to the [0,+1] Stokes is seen, 42 dB weaker than in the backward case (magenta curve), showing strong nonreciprocity. On the right are the recorded near-field distributions of the backward [−1,−1] pump and [0,+1] Stokes. The slanted fringe pattern measured at the focus of a cylindrical lens [28] confirms the presence of an optical vortex. (b) The same as (a), but the backward intervortex conversion is from [0,+1] Stokes to [−1,−1] pump. The double peaks are caused by pixellation in the OSA. The recorded modal patterns confirm that the pump is in the [−1,−1] mode. (c) Calculated evolution of power in backward-propagating [−1,−1] pump and [0,+1] Stokes read-out signals along an 8-m-long PCF in the presence of the strong acoustic wave created by intervortex FSBS (Fig. 3), with coupling constant κ(z)=(gB0/2)|aSaP*|, where κ0=κ(0)=0.043  m−1. (d) Backward intervortex conversion efficiency to the Stokes with frequency detuning from perfect phase-matching, measured (red circles) and theoretical (gray curve) based on a dephasing rate of 0.4 (rad/μm)/GHz, calculated by FEM. Theory and experiment agree well in both peak conversion efficiency and bandwidth.

Fig. 6. (a) Dispersion curves for (from left to right) [ℓ,s]=[+1,+1],[−1,−1],[−1,+1],[+1,−1] hBMs. Zero on the frequency axis corresponds to 1550 nm. The propagation constants of the [−1,−1] and [+1,+1] modes differ by 0.00132 rad/μm. (b) Calculated dispersion curves for the CFW in an untwisted PCF, with a cut-off frequency of ∼1.292  GHz. The open circle marks the point at which the CFW phase-matches the [−1,−1] pump and [+1,+1] Stokes modes in (a). In the vicinity of this point the acoustic dispersion is 0.136 rad/μm per MHz. The inset shows dispersion curves for neighboring CFWs that have high overlap with the optical modes, with cut-off frequencies in the range 1.1–1.6 GHz. (c) Transverse and axial displacements of the CFW at 1.292 GHz. The CFW has topological charge of ℓph=±2, which because of the three-fold symmetry is provided by the first harmonic of an ℓph=∓1 CFW. (d) The transmitted power in the [+1,+1] Stokes mode during writing, measured by the LIA, plotted as a function of frequency difference between pump and Stokes. (e) The power in the backward [−1,−1] Stokes signal during read-out by a weak backward [+1,+1] pump signal, measured as a function of detuning from the phase-matching frequency of 187.2 THz (∼1601.5  nm), which is shifted 6.21 THz (∼51.5  nm) from the writing wavelength (1550 nm).

Fig. 7. (a) Fiber cross-section used in the calculation of ℓph=±1 acoustic mode. (b) Fiber cross-section used in the calculation of ℓph=±2 acoustic mode. (c) Zoom-in of fiber central part with finite element meshes. (d) Numerically calculated overlap coefficients (QRP in red, QES in blue) for all the acoustic modes that satisfy phase-matching in the range 0–0.3 GHz, for [0,+1] pump and [−1,−1] Stokes modes. (e) Same as (d) but for [−1,−1] pump and [−1,−1] Stokes modes, and acoustic modes in the range 0.8–1.8 GHz.

Fig. 8. Total displacements of three acoustic modes at 95.8, 101.3, and 109.4 MHz. All of them have similar displacement distributions in the core region, but the one at 98.5 MHz is the most tightly confined (35.8%) to the core and has highest overlap with the optical modes. The other two modes are less tightly confined (25.4% and 13.6%) to the core and are distributed over the whole fiber cross-section.

Fig. 9. The strain energy density and transverse and axial displacements of all six labeled CFWs in Fig. 7(b). They all have relatively high optoacoustic overlaps with the [−1,−1] pump and [+1,+1] Stokes modes. The acoustic mode at 1.292 GHz has the highest overlap and dominates FSBS.

Fig. 10. (a) The measurement of polarization |S3| Stokes parameters of [+1,+1] (blue) and [−1,−1] (red) after 8 m chiral PCF, from 1530 to 1630 nm. (b) Loss measurement of [0,+1] and [−1,−1] modes from 1530 to 1630 nm. The losses of other vortex modes ([+1,−1],[+1,+1],[−1,+1]) are almost equal to that of [−1,−1], and therefore not shown here.