The concept of chirality is important in many fields, from the study of chiral molecules to quantum optics [1–3]. Chiral phonons are much studied in solids, when the constituent atoms exhibit rotational motion, generating angular momentum in the direction of propagation [4]. As a new degree of freedom for the investigation of vibrational mechanical states, the chiral phonons play an important role for many fundamental fields such as phonon-driven topological states [5], electron-phonon coupling in solids [6], and energy-efficient information processing [3]. Recent works on chiral phonons, either magnetically driven or inherently present in topological materials [7–10] and helically chained biomolecules [11], have created a lot of interest and triggered a series of new explorations and developments in phonon-related physical processes.

In recent years, twisted photonic crystal fiber (PCF)—a versatile chiral material infinitely extended in the direction of the twist—has been shown to robustly preserve circularly polarized optical vortex modes with low loss, enabling novel investigations of linear and nonlinear optical effects in the presence of chirality [12–14]. Chiral PCF offers tight confinement of light in wavelength-scale glass cores, with close to 100% optoacoustic overlap [15], making it an ideal vehicle for investigating chiral forward Brillouin photon-phonon interaction. More importantly, the fibers have a special chiral structure that can stably preserve chiral flexural phonons (CFPs) and enable the efficient generation and stable propagation of CFP via the Brillouin scattering effect. Although in recent years there has been a series of studies of backward intervortex Brillouin scattering in twisted PCF [16–18], the excited phonons propagated as one-dimensional longitudinal waves that were non-chiral.

Here, we report the use of intervortex forward stimulated Brillouin scattering (FSBS) to optically excite and read out CFP in twisted PCF. The beat-note between co-propagating pump and Stokes modes during FSBS drives the creation of CFPs that robustly propagate as rotating single-spiral cork-crew flexural waves carrying vortices. Phase-matching and angular momentum conservation has the consequence that CFP created by FSBS in the twisted PCF can be used for intervortex frequency conversion at a widely separated wavelength in the backward direction. The intervortex FSBS opens a new path to advanced Brillouin scattering by controlling the chiral flexural modes for mode conversion. These modes represent a new version of acoustic phonons that can spatiotemporally modulate the optical vortices, with potential for use in vortex lasers or communication systems. The results offer a new perspective for topology photonics and Brillouin scattering, with potential applications in inter-band frequency conversion [19–21] and manipulation of vortex states in classical and quantum regimes [22,23].

Chiral PCFs with $N$-fold rotational symmetry support helical Bloch modes (hBMs) [24,25], whose $m$-th order azimuthal harmonic carries an optical vortex of order ${\ell }_A^{(m)}={\ell }_A^{(0)}+Nm$, where ${\ell }_A^{(m)}$ is the number of complete periods of phase progression around the azimuth for fields expressed in cylindrical components and ${\ell }_A^{(0)}$ is the principal azimuthal order. Note that ${\ell }_A^{(0)}$ is always an integer and is robustly conserved. In chiral PCF it is found, both experimentally and by numerical modeling, that the fields are almost perfectly circularly polarized, under which circumstances the topological charge ${\ell }^{(m)}$ (the number of on-axis discontinuities for fields evaluated in Cartesian coordinates) is linked to the azimuthal order by ${\ell }^{(m)}={\ell }_A^{(m)}-s$, where $s$ is the spin [$s=+1$ for left-circular (LCP) and $-1$ for right-circular (RCP) polarization states]. Here, we use the shorthand $[\ell ,s]$ to denote the parameters of an hBM, where for ease of notation the principal topological order is defined as ${\ell }^{(0)}\equiv \ell$. In chiral PCF, hBMs with equal and opposite values of $\ell$ are generally non-degenerate in index, i.e., topologically birefringent, while modes with opposite spin but the same value of $\ell$ typically display weak circular birefringence [12].

A three-dimensional sketch of the chiral PCF used in the experiments is shown in Fig. 1(a). The fiber has a three-fold rotationally symmetric structure with one on-axis and three satellite cores [see also scanning electron micrograph in Fig. 1(b)]. The twist pitch is 5 mm, the diameter $d$ of the hollow channels is 1.58 μm, and the distance $\mathrm{\Lambda }$ between adjacent channels is 1.79 μm, yielding $d/\mathrm{\Lambda }$ of 0.88, which results in tight confinement of both acoustic and optical fields and strong optoacoustic coupling (see Appendix B for more details). Finite element analysis of the chiral PCF in a helicoidal frame yields the near-field intensity patterns in Fig. 1(b), showing good agreement between experiment and theory. The measurements of polarization maintaining ability were made after propagation along an 8 m length of the fiber, where the modulus of the Stokes parameter $|S_3|$ of $[+1,+1]$ and $[-1,-1]$ modes was higher than 0.98 at the output, showing very good preservation of the circular polarization state. Such robust circular polarization transmission in the chiral PCF also means that the vortex modes stably propagate with low crosstalk, since the topological charges and spin orders are always associated together with azimuthal order by $\ell ={\ell }_A-s$, as mentioned above. Any deviations from integer values of spin orders are caused by the crosstalk among vortex modes, and vice versa. The fiber loss at 1550 nm, measured by a cut-back method, was 0.195 dB/m for the $\ell =0$ modes and 0.262 dB/m for the $\ell =\pm 1$ modes. More details about polarization maintaining and fiber loss measurements are shown in Appendix C.

Firstly, we establish the theories of chiral FSBS and optically excited CFP. Note that throughout the paper quantities related to the pump and Stokes are subscripted $P$ and $S$, and we adopt the usual convention that the pump always has a higher frequency than the Stokes. In the Cartesian laboratory frame, the transverse field of a circularly polarized hBM can be written as [26] $$\begin{gathered} {\mathbf{e}}_k=a_k(z){\mathbf{e}}_{Bk}=a_k(z)\left(\begin{array}{c}{e}_{Bkx}\\ e_{Bky}\end{array}\right) \\ =a_k(z)\left(\begin{array}{c}1\\ s_{k}i\end{array}\right)\frac{1}{\sqrt{2}}{B}_k(\rho ,\varphi +\alpha z)e^{i({\beta }_{k}z+{\ell }_k\varphi -{\omega }_{k}t)}, \end{gathered}$$(1)where $a_k(z)$ is a slowly varying amplitude, $B_k(\rho ,\varphi +\alpha z)$ is the azimuthally periodic field distribution of the mode $k$ hBM ($k=P,S)$, $(\rho ,\varphi ,z)$ are cylindrical coordinates, $\alpha$ is the twist rate (rad/m), ${\omega }_k$ the angular frequency, $s_k=\pm 1$ the spin, and ${\ell }_k$ the topological charge. Note that the hBM field pattern $B_k$ rotates with $z$ at a rate $\partial \varphi /\partial z=-\alpha$, locking it to the structure of the chiral fiber. $B_k$ can be expressed as the sum over azimuthal harmonics, with the $m$-th harmonic having propagation constant ${\beta }_k^{(m)}={\beta }_k+mN\alpha$ and topological charge ${\ell }_k^{(m)}={\ell }_k+mN$ (to simplify the notation we have set ${\beta }_k^{(0)}={\beta }_k$ and ${\ell }_k^{(0)}={\ell }_k$). For more details see Appendix A.1.

In the absence of any perturbation ($a_P$ and $a_S$ constant), the expression in Eq. (1) is itself a solution of Maxwell’s equations in the chiral PCF. In the presence of a dielectric constant perturbation ${\epsilon }_{\mathrm{ph}}$ in the form $$[{\epsilon }_{\mathrm{ph}}]=\left[\begin{array}{cc}\cos \zeta & -\sin \zeta \\ \sin \zeta & \cos \zeta \end{array}\right]\left[\begin{array}{cc}{\epsilon }_1(\rho ,\varphi )/2& 0\\ 0& -{\epsilon }_1(\rho ,\varphi )/2\end{array}\right],$$(2)where $\zeta =Kz+{\ell }_{\mathrm{ph}}\varphi -\mathrm{\Omega }t$, $\mathrm{\Omega }={\omega }_P-{\omega }_S$, $K$ is the acoustic wavevector at frequency $\mathrm{\Omega }$, ${\ell }_{\mathrm{ph}}$ is the acoustic topological charge, and ${\epsilon }_1$ is an induced anisotropic change in dielectric constant that yields linear birefringence (needed to couple LCP and RCP light), the coupling rate from mode $k$ to mode $l$ will be proportional to $⟨{\mathbf{e}}_{Bl}^{†}\text{|}{\epsilon }_{\mathrm{ph}}\text{|}{\mathbf{e}}_{Bk}⟩$, where $†$ denotes the Hermitian conjugate. The rotation matrix in Eq. (2) ensures that the pattern of linear birefringence rotates with position, azimuthal angle, and time, which enables the generation of CFPs that stably propagate as chiral flexural waves (CFWs) resembling a rotating single-spiral corkscrew. A CFW with this geometry can be produced by a superposition of two, $\pi /2$ out-of-phase, orthogonal flexural waves differing in Brillouin frequency by $\mathrm{\Omega }$ (see next section). Finite element modeling of the untwisted PCF also reveals acoustic flexural modes that are strongly confined to the core region [Fig. 1(c)]. Degenerate orthogonal versions of these modes are found that, when superimposed with a $\pi /2$ phase difference, produce a wave with the topology of a rotating single-spiral corkscrew, as shown in Fig. 1(d) (see Visualization 1, Visualization 2, Visualization 3, Visualization 4, Visualization 5, Visualization 6, Visualization 7 and Visualization 8). It is clear that these acoustic phonons carry OAM and display chirality.

Then we evaluate the $⟨{\mathbf{e}}_{Bl}^{†}\text{|}{\epsilon }_{\mathrm{ph}}\text{|}{\mathbf{e}}_{Bk}⟩$ in a coupled-mode description. Assuming slowly varying power-normalized amplitudes, a separately excited CFW (i.e., no FSBS), conservation of topological charge, i.e., ${\ell }_{\mathrm{ph}}={\ell }_S-{\ell }_P$, and collecting terms with the slowest rates of phase progression, we obtain (see Appendix A.2 for details) $$\frac{\partial a_P}{\partial z}+\frac{{\alpha }_P}{2}{a}_P=i{\kappa }_P{a}_S{e}^{i\vartheta z},\hspace{1em}\frac{\partial a_S}{\partial z}+\frac{{\alpha }_S}{2}{a}_S=i{\kappa }_S{a}_P{e}^{-i\vartheta z},$$(3)where the quantity $\vartheta ={\beta }_S({\omega }_S)-{\beta }_P({\omega }_P)-K(\mathrm{\Omega })$ is the dephasing rate, ${\alpha }_P$ and ${\alpha }_S$ are loss coefficients (dB/m) of pump and Stokes waves, $n_k$ is the phase index of mode $k$, and ${\kappa }_k={\omega }_{k}Q/c$, where $Q$ is a dimensionless parameter that is proportional to the integral of $⟨{\mathbf{e}}_{Bl}^{†}|{\epsilon }_{\mathrm{ph}}|{\mathbf{e}}_{Bk}⟩$ over the fiber cross-section.

Then the coupled FSBS equations for complex slowly varying field amplitudes $a_i$ that scale with the square-root of the modal power ($p_i={|a_i|}^2$) can be written in the form $$\begin{gathered} \frac{\partial a_P}{\partial z}+\frac{{\alpha }_P}{2}{a}_P=i\frac{{\omega }_P}{c}\sqrt{Q_B}{a}_S{a}_{\mathrm{ph}}, \\ \frac{\partial a_S}{\partial z}+\frac{{\alpha }_S}{2}{a}_S=i\frac{{\omega }_S}{c}\sqrt{Q_B}{a}_P{a}_{\mathrm{ph}}^{*}, \\ \frac{\partial a_{\mathrm{ph}}}{\partial z}+(\frac{{\alpha }_{\mathrm{ph}}}{2}+i\vartheta )a_{\mathrm{ph}}=i\frac{{\omega }_{\mathrm{ph}}}{c}\sqrt{Q_B}{a}_S^{*}{a}_P, \end{gathered}$$(4)where $\vartheta ={\beta }_P-{\beta }_S-{\beta }_{\mathrm{ph}}$ is the dephasing parameter, $c$ is the speed of light in vacuum, and $Q_B$ is a characteristic parameter (with units ${\mathrm{W}}^{-1}$) that depends on the electrostrictive parameters and the optoacoustic overlap.

With the good approximation (commonly used in SBS) that $|\partial /\partial z|\ll |{\alpha }_{\mathrm{ph}}/2+i\vartheta |$, Eq. (4) can be recast in the form $$\frac{\partial a_P}{\partial z}+\frac{{\alpha }_P}{2}{a}_P=-\frac{g_B}{2}(1-i\frac{2\vartheta }{{\alpha }_{\mathrm{ph}}})(a_P{a}_S^{*})a_S,$$(5a)$$\frac{\partial a_S}{\partial z}+\frac{{\alpha }_S}{2}{a}_S=\frac{g_B}{2}(1+i\frac{2\vartheta }{{\alpha }_{\mathrm{ph}}})(a_S{a}_P^{*})a_P,$$(5b)where $$g_B=\frac{g_{B0}}{1+4{\vartheta }^2/{\alpha }_{\mathrm{ph}}^2}=\frac{4\overline{\omega }{\omega }_{\mathrm{ph}}{Q}_B/(c^2{\alpha }_{\mathrm{ph}})}{1+4{\vartheta }^2/{\alpha }_{\mathrm{ph}}^2}{\mathrm{m}}^{-1}{\mathrm{W}}^{-1},$$(6)and we have used the very good approximation ${\omega }_P\cong {\omega }_S\cong ({\omega }_P+{\omega }_S)/2=\overline{\omega }$. Comparing Eqs. (6) and (3) we see that for $\vartheta =0$, $$Q=\frac{4{\omega }_{\mathrm{ph}}{Q}_B}{c{\alpha }_{\mathrm{ph}}}|a_P{a}_S^{*}|,$$(7)which will be useful later in analyzing the backward pump-to-Stokes coupling, which is essentially a linear scattering process. We also need to check the phase of the CFW at every point along the fiber. Setting $a_i=q_i(z)e^{i{\psi }_i(z)}$ in Eq. (5) where $q_i$ and ${\psi }_i$ are real-valued and extracting the imaginary parts, we obtain $$\frac{\partial {\psi }_P}{\partial z}=\vartheta \frac{g_B}{{\alpha }_{\mathrm{ph}}}{q}_S^2,\frac{\partial {\psi }_S}{\partial z}=\vartheta \frac{g_B}{{\alpha }_{\mathrm{ph}}}{q}_P^2,$$(8)which shows that in general the relative phase of pump and Stokes waves varies in a non-trivial manner along the fiber. In the experiment, however, the dephasing is zero in the writing process, so that the pump and Stokes phases are constant.

Multiplying Eq. (5a) by $a_P^{*}$ and Eq. (5b) by $a_S^{*}$ and adding each equation to its complex conjugate, we obtain the coupled power equations $$\begin{gathered} \frac{\partial p_P}{\partial z}+{\alpha }_P{p}_P=-g_B{p}_P{p}_S, \\ \frac{\partial p_S}{\partial z}+{\alpha }_S{p}_S=g_B{p}_P{p}_S. \end{gathered}$$(9)

In the experiment of CFP writing/excitation, a fixed frequency pump signal is launched into the $[{\ell }_P,\mathrm{+s}]$ mode, along with a frequency-tunable Stokes signal in the $[{\ell }_S,\mathrm{-s}]$ mode, where ${\ell }_P\ne {\ell }_S$. Interference between these two co-propagating signals creates a spiraling interference pattern that, through electrostriction at the phase-match frequency, excites CFP with topological charge of ${\ell }_{\mathrm{ph}}={\ell }_P-{\ell }_S$ [see Fig. 3(a)]. Photons are converted from pump to Stokes (${\omega }_P>{\omega }_S$) via chiral FSBS, with the energy defect going to excite phonons. This CFW induces a spiraling three-fold rotationally symmetric pattern of linear birefringence, which acts back on the optical fields, coupling together two orthogonal circularly polarized vortex lights as explained above and in Appendix A.

We start the experiment by demonstrating chiral FSBS between $[0,+1]$ and $[-1,+1]$ modes and generation of ${\ell }_{\mathrm{ph}}=\pm 1$ CFP. Some simulations were preformed beforehand to estimate the acoustic frequency. Figure 3(a) shows the $\omega -\beta$ diagram for the six hBMs considered in this paper, calculated by the finite element method (FEM) and plotted over the spectral regions of interest. There is strong topological birefringence between modes with different topological charges (the calculated refractive index differences at 1550 nm are $n_{[0,+1]}-n_{[+1,+1]}=0.034$ and $n_{[-1,-1]}-n_{[+1,+1]}=0.00047$). In addition, the simulations confirm that modes with the same topological charge but opposite spins display weak circular birefringence, of order $8\times {10}^{-6}$. Any deviations from integer values of topological charge are caused by the polarization state not being perfectly circular [24]. Figure 3(b) shows the $\omega -\beta$ diagram for three acoustic flexural modes in an untwisted fiber over the frequency range where the maximum SBS gain is seen in the experiments for $[0,+1]$ pump to $[-1,-1]$ Stokes conversion. Efficient FSBS only occurs when phase matches $\vartheta =0$ and the optoacoustic overlap is high. Theory predicts that these conditions are fulfilled at the marked points in Figs. 3(a) and 3(b), corresponding to a frequency difference of $\sim 95.8\mathrm{MHz}$ and an acoustic propagation constant of 0.0465 rad/μm, i.e., an acoustic wavelength of 139 μm. The acoustic dispersion in the vicinity of this point is 0.0037 rad/μm per MHz, while the optical dispersion is negligible. In the chiral PCF the acoustic wavevector will differ for ${\ell }_{\mathrm{ph}}=\pm 1$ by approximately $2\alpha =4\pi /0.005-0.002{\mathrm{\mu m}}^{-1}$, which only slightly alters the phase-matching condition and is neglected.

The experimental setup is sketched in Fig. 2. The acoustic phonons are first written (i.e., excited) by FSBS between the forward pump and Stokes modes, and then read out at a different wavelength by a backward $P$ (or $S$) signal, which is down-shifted (or up-shifted) into an $S$ (or $P$) signal. For the detailed descriptions on the experimental setup see Appendices E and F. In the case of chiral FSBS between $[0,+1]$ and $[-1,-1]$ modes, the CFW must supply a topological charge of ${\ell }_{\mathrm{ph}}=\pm 1$, depending on which mode the pump is in. Transmitted pump and Stokes signals were separated at a circularly polarizing beam splitter and monitored using a power meter or OSA. Peak conversion is seen at 98.5 MHz (in good agreement with a theoretical value of 95.8 MHz), at which point the transmitted Stokes and pump powers were 1.89 and 0.71 W, respectively, as shown by the red circles in Fig. 3(c). In the absence of SBS, after taking the optical loss into account, these powers are 1.39 and 1.23 W.

Numerically solving the coupled power equations [Eq. (9)] for $\vartheta =0$ and adjusting the gain until the results agree with the experiment, we are able to plot the power in each mode as a function of position along the PCF [Fig. 3(c)]. The loss coefficients of $[0,+1]$ pump and $[-1,-1]$ Stokes used in the calculation are 0.195 and 0.262 dB/m, which are obtained from experiments. The red circles in Fig. 3(c) mark the experimental measurements, and the dashed curves show the behavior with $g_B=0$. The green dotted curve shows the length dependence of the optoacoustic coupling, which is directly proportional to $g_{B0}|a_P{a}_S^{*}|/2$ and takes the value $g_{B0}\times 1\mathrm{W}$ at $z=0$. Excellent agreement with theory is obtained for $g_{B0}=0.043{\mathrm{m}}^{-1}{\mathrm{W}}^{-1}$.

Figure 3(d) shows the measured Stokes power (red circles) as a function of frequency difference, together with a theoretical fitting from Eq. (9). The calculated pump power as a function of frequency difference is also shown. The pump wave is depleted while the Stokes wave is amplified and both reach their minimum/maximum at the frequency of 98.5 MHz. The measured half-width at half-maximum bandwidth is 1.9 MHz [Fig. 3(d)], and the lineshape fits quite well to a Lorentzian, as predicted by Eq. (6). For strong dephasing, the transmission reverts to the dashed curves in Fig. 3(c), as expected. The pronounced shoulder on the high frequency side is attributed to excitation of acoustic modes with higher frequency cut-offs [Fig. 3(b)], which have lower overlap with the optical modes. Detuning from $\vartheta =0$ is dominated by the acoustic dispersion, which at the phase-matching frequency is 0.0037 rad/μm per MHz [Fig. 3(b)]. At 1.9 MHz detuning the dephasing rate is $\vartheta =0.0037\times 1.9=0.007\mathrm{rad}/\mathrm{\mu m}$ and the gain is half of its peak value, i.e., ${\alpha }_{\mathrm{ph}}=2\vartheta =0.014{\mathrm{\mu m}}^{-1}$, corresponding to a 1/e acoustic power decay length of 70 μm. This in turn allows us to estimate $Q_B=1.8\times {10}^{-5}{\mathrm{W}}^{-1}$. In the experiment, the phonon lifetime $\tau$ is 42 ns, which is calculated from the experimentally measured Brillouin linewidth $\mathrm{\Gamma }$ with the formula $\tau =1/(2\pi \mathrm{\Gamma })$. The acoustic power decay length is 86.52 μm, which can be calculated by $\tau \times V$ and the flexural acoustic velocity $V=\mathrm{\Omega }/K=2060\text{m/s}$. The experimentally estimated acoustic power decay length is very close to 70 μm, which verifies the assumptions regarding the phase-matching conditions in the theoretical model.

Once the CFPs are excited by a forward pump and Stokes, modal dispersion means that they can be used for phase-matched conversion in the backward direction at a shifted wavelength, as illustrated in Fig. 4(a), enabling frequency conversion of vortex states. The black arrows represent the phonons. The experimental setup of the reading process is also shown in Fig. 2 and detailed in Appendix E. In the absence of group velocity dispersion, the frequency shift at which this occurs is $\mathrm{\Delta }\mathrm{\Omega }\hspace{0.17em}\approx \mathrm{\Omega }(n_{\mathrm{gP}}+n_{\mathrm{gS}})/(n_{\mathrm{gP}}-n_{\mathrm{gS}})$ [27], where $n_{\mathrm{g}\mathrm{P}}$ and $n_{\mathrm{g}\mathrm{S}}$ are the group index of the mode P and mode S (note that if the pump and Stokes modes are swapped, the frequency shift will change sign). In our case, however, there is significantly higher order dispersion, so that this simple condition is not accurate. Numerical calculations of the error $\mathrm{\Delta }\beta$, $$\mathrm{\Delta }\beta (\mathrm{\Delta }\mathrm{\Omega })={\beta }_{[0,+1]}(0)-{\beta }_{[-1,-1]}(-\mathrm{\Omega })-({\beta }_{[-1,-1]}(-\mathrm{\Delta }\mathrm{\Omega })-{\beta }_{[0,+1]}(-\mathrm{\Delta }\mathrm{\Omega }-\mathrm{\Omega })),$$(10)plotted as a function of the frequency shift $\mathrm{\Delta }\mathrm{\Omega }$, yield the curve in Fig. 4(b). The error falls to zero at $\mathrm{\Delta }\mathrm{\Omega }/2\pi =15.5\mathrm{GHz}$, which is in reasonable agreement with the experimental value of 18.5 GHz. Conservation of absolute angular momentum in the forward and backward SBS processes is illustrated in Fig. 4(c). The black arrows represent the angular momentum of the phonons, which is the same in both writing and read-out. The topological charge and spin of the optical modes are, however, conventionally defined relative to the beam propagation direction, as in Fig. 4(a); for correct comparison with the forward process, the same frame of reference must be used for both directions, so the backward signs must be reversed, as shown in the figure.

Figure 5(a) shows the read-out spectra recorded by a high resolution (20 MHz) OSA when a $[-1,-1]$ pump signal was launched backwards into the fiber at the phase-match frequency, in the presence of strong intervortex chiral FSBS. The horizontal scale corresponds to the frequency shift $\mathrm{\Delta }\mathrm{\Omega }$ of the backward pump signal relative to the forward pump signal. Strong intervortex conversion between backward pump and Stokes frequencies is observed (blue curve) when $\mathrm{\Delta }\mathrm{\Omega }=-18.5\mathrm{GHz}$. The Brillouin frequency shift $\mathrm{\Omega }=98.5\mathrm{MHz}$ is identical in both directions. On the right are the modal intensity profiles of backward pump signal recorded by a CCD camera and backward Stokes signal by the NBA, confirming that the launched pump is in the $[-1,-1]$ mode and Stokes read-out in the $[0,+1]$ mode. When the pump signal is launched forwards, only very weak conversion to the backward $[0,+1]$ Stokes is seen (magenta curve), 42 dB weaker than the previous case. We attribute this signal to Rayleigh scattering and weak reflections at the fiber output face. Figure 5(b) shows the reverse process, when a $[0,+1]$ Stokes signal is launched backwards into the fiber at the phase-matching point and converted to the $[-1,-1]$ mode with a frequency up-shift of 98.5 MHz. When the $[-1,-1]$ pump signal is launched in forward direction, only very weak conversion to the $[0,+1]$ Stokes is seen: 42 dB weaker than in the backward case (magenta curve), showing strong non-reciprocity. This is due to the phase-mismatching of the read-out process in the forward direction, as illustrated by the blue arrow in Fig. 4(a). The acoustic wave couples the backward pump and Stokes of different frequencies during the read-out process, while the coupling with the same frequency is prohibited in the forward direction, due to wavenumber mismatching $\mathrm{\Delta }k=(n_{g,1}-n_{g,2})\mathrm{\Delta }\mathrm{\Omega }/c$ [27]. The $n_{g,1}$ and $n_{g,2}$ are the group indices of pump and Stokes modes. For conversion between $[0,+1]$ and $[-1,-1]$ modes, the $\mathrm{\Delta }k$ is calculated to be $\sim 26.3\text{rad}/\mathrm{m}$. In the experiment, the Brillouin frequency between pump and Stokes is so small that each signal cannot be well-resolved by OSA, which from time to time caused double peaks around the signal central frequencies due to pixelation. The recorded modal patterns confirm that the backward pump signal is in the $[-1,-1]$ mode.

The read-out process (subscript $r$) can be conveniently modeled using Eq. (5): $$\begin{gathered} \frac{\partial b_{\mathrm{Pr}}}{\partial z}+\frac{{\alpha }_{\mathrm{Pr}}}{2}{b}_{\mathrm{Pr}}=-\left(\frac{g_{B0}}{2}{a}_P{a}_S^{*}\right)b_{\mathrm{Sr}}{e}^{i{\vartheta }_{r}z}, \\ \frac{\partial b_{\mathrm{Sr}}}{\partial z}+\frac{{\alpha }_{\mathrm{Sr}}}{2}{b}_{\mathrm{Sr}}=\left(\frac{g_{B0}}{2}{a}_S{a}_P^{*}\right)b_{\mathrm{Pr}}{e}^{-i{\vartheta }_{r}z}, \end{gathered}$$(11)where $b_{\mathrm{Pr}}$ and $b_{\mathrm{Sr}}$ are the read-out amplitudes of pump and Stokes modes, $g_{B0}=0.043{\mathrm{m}}^{-1}{\mathrm{W}}^{-1}$ [from Eq. (6)], and ${\vartheta }_r={\beta }_{\mathrm{Sr}}-{\beta }_{\mathrm{Pr}}\pm K$, the sign of $K$ being chosen to minimize $|{\vartheta }_r|$. The distributions of $a_S$ and $a_P$ are obtained from solutions of Eqs. (5) and (11) and then numerically integrated to yield the length dependence of $b_P$ and $b_S$, as shown in Fig. 5(c). The backward conversion efficiency is strongly dependent on the frequency of the backward pump and from FEM calculations; the backward dephasing rate for deviations from a perfect phase-matching backward pump frequency is 0.4 rad/(μm GHz), for which solutions of Eq. (11) yield a spectral conversion efficiency that is in good agreement with experiment [Fig. 5(d)].

Next, we show the results about writing and read-out of ${\ell }_{\mathrm{ph}}=\pm 2$ CFPs, with interaction between $[-1,-1]$ and $[+1,+1]$ optical modes. An expanded view of the calculated $\omega -\beta$ diagram for the four vortex-carrying hBMs is shown in Fig. 6(a), revealing a refractive index difference of $4.7\times {10}^{-4}$ between the $[-1,-1]$ and $[+1,+1]$ modes at 1550 nm, or a propagation constant difference of 0.00132 rad/μm. Figure 6(b) shows the $\omega -\beta$ diagram of the acoustic mode having the highest overlap with two optical modes (the inset shows the $\omega -\beta$ curves of six other acoustic modes that have moderately high overlap with the optical modes, with cut-off frequencies in the range of 1.1–1.6 GHz). The open circles in Figs. 6(a) and 6(b) mark the points where phase-matching occurs, predicting maximum gain at a Brillouin frequency shift of $\sim 1.292\mathrm{GHz}$ for the forward writing process of ${\ell }_{\mathrm{ph}}=\pm 2$ CFP. The mode in Fig. 6(b) comes in two degenerate orthogonal forms, which when superimposed with a frequency shift produce the rotating displacement patterns as shown in Fig. 6(c) (see Visualization 9 and Visualization 10). Note that these calculations are for an untwisted PCF; see Appendix B. In the case of $[-1,-1]$ and $[+1,+1],$ writing the CFW must supply a topological charge of ${\ell }_{\mathrm{ph}}=\pm 2$, which because of the three-fold symmetry is provided by the first harmonic of an ${\ell }_{\mathrm{ph}}=\mp 1$ CFW. As previously reported for optical hBMs [26], higher order hBM harmonics can sometimes be stronger than those inside the first Brillouin zone, so that it is not unexpected that the SBS gain is higher for the ${\ell }_{\mathrm{ph}}=\pm 2$ harmonic of the CFW.

Figure 6(d) shows the transmitted power in the $[+1,+1]$ Stokes mode during the writing process, measured by the LIA, plotted as a function of frequency difference between pump and Stokes. Figure 6(e) shows the power in the backward $[-1,-1]$ Stokes signal during the read-out process by a backward $[+1,+1]$ pump signal, measured as a function of detuning from the phase-matching frequency of 187.2 THz ($\sim 1601.5\mathrm{nm}$), which is shifted 6.21 THz ($\sim 51.5\mathrm{nm}$) from the writing wavelength (1550 nm). This is the first time, as far as we know, to observe such large frequency shifting with the mediation of acoustic phonons in the Brillouin scattering.

Although a clear signal was observed at this wavelength in the experiment, confirming that it is possible to create and scatter off ${\ell }_{\mathrm{ph}}=\pm 2$ flexural phonons, it was difficult to theoretically characterize the process by numerical calculation because the response of $[-1,-1]$ to $[+1,+1]$ scattering was very weak. This is because the optoacoustic overlap of the optical mode with the higher order phonon modes is smaller, which results in a smaller phonon population, when compared with the cases of intervortex FSBS with ${\ell }_{\mathrm{ph}}=\pm 1$ acoustic modes. The parasitic phonon modes that exist around the Brillouin peak frequency [Fig. 6(b)] are also detrimental for efficient scattering and therefore decrease the peak conversion efficiency. For detailed descriptions, please see Appendix B.

Chiral intervortex FSBS between circularly polarized optical vortex modes in a three-fold rotationally symmetric twisted PCF enables robust all-optical excitation of CFPs, which are vibrational mechanical states carrying angular momentum. Gain is seen when co-propagating pump and Stokes modes are orthogonally polarized, under which circumstances a rotating pattern of linear birefringence is induced, which has the correct symmetry to couple the pump and Stokes signals. In the case studied, the excited CFW has a strain field that resembles a rotating single-spiral corkscrew and may be viewed as the superposition of two orthogonal flexural waves with a frequency difference equal to the Brillouin shift. A combination of finite-element modeling and analytical theory produces results that agree very well with experiment, yielding a Brillouin gain of $g_{B0}=0.043{\mathrm{m}}^{-1}{\mathrm{W}}^{-1}$ in the twisted PCF studied.

The CFPs created by FSBS in twisted PCF helically change the optical properties of the material and induce dynamic vortex gratings that spatiotemporally back-act on the optical waves, enabling frequency conversion of vortex states. The phase-matching condition and annular momentum conservation control the whole process, while the CFPs provide necessary OAM to make this happen. Once the phonons have been excited by the forward-propagating “writing” signals, they can be used, within their coherence time, for non-reciprocal backward conversion of vortex modes at a wavelength that can be spectrally distant (51.5 nm in the experiments) from the “writing” wavelength. This large spectral separation far exceeds what has been achieved in micro-resonators [29] and conventional linear-polarization maintaining fibers [30], using non-chiral optical and acoustic modes. The equivalent read-out process in the forward direction is strongly dephased and 42 dB weaker. The launched backward signals can be up-shifted or down-shifted by the Brillouin frequency, corresponding respectively to anti-Stokes or Stokes scattering.

The conversion efficiency can be potentially increased by raising the writing beam power and optimizing the fiber structure, such as $d/\mathrm{\Lambda }$ and core size. Generally, to improve the conversion efficiency of FSBS, the $d/\mathrm{\Lambda }$ should be increased so that the phonon modes can be confined into the core region as much as possible. But the large air hole size will induce more loss due to the Rayleigh scattering effect, which is difficult to be estimated due to the uncertainties in fiber fabrication. Increasing the twist rate and reducing the core size will also improve the conversion efficiency, but this will in turn introduce more difficulties on the fiber fabrication and may further increase fiber loss. Therefore, the fabrication technique of chiral PCF is essential in practical experiments and is still under optimization.

The backward conversion efficiency depends not only on the frequency of the read-out mode [Fig. 5(d)], but also on the frequency difference between the forward pump and Stokes seed signals, which must lie within the Brillouin gain band. In addition, the topological charge and polarization state of the backward read-out mode also affect the conversion efficiency, as dictated by angular momentum conservation. For example, if the forward pump and Stokes modes are set as $[0,+1]$ and $[-1,-1]$, theory predicts that backward conversion is possible not only between $[-1,-1]$ and $[0,+1]$ modes (observed experimentally), but also between the $[-1,+1]$ and [0,1] modes.

It is possible to observe chiral photon-phonon interactions in any medium that is able to support stable optical vortex states, such as longitudinal homogeneous ring-core fibers (RCFs) [31,32]. However, the generation of chiral flexural phonons in such fibers has not been reported, due to the fact that the acoustic modes therein are distributed over the entire cross-section of the fiber rods, while the optical modes are only in the core, which results in a very low optoacoustic overlap. Recently, as the first step towards chiral Brillouin interaction, the inter-circular-polarization coupling through backward SBS in nano-tapers was demonstrated [33], with the mediation of spin-orbit interaction [34] and excitation of chiral longitudinal acoustic phonons. However, our work is the demonstration of intervortex coupling and fundamentally different from the above studies in terms of Brillouin coupling mechanisms and phonon properties. Firstly, the work in Ref. [33] with a nano-taper is based on backward Brillouin scattering with the mediation of longitudinal acoustic modes, while our work is about forward Brillouin scattering with flexural acoustic waves. They are physically different and the realization of the latter one is in general substantially more difficult because of the low interaction efficiency with mostly transverse acoustic waves. Secondly, the nano-taper is incapable of preserving circular polarization states. While inter-circular-polarization coupling can be observed in their case, the polarization states of both pump and Stokes have to be controlled precisely during the experiment. In contrast, the chiral PCF has great ability of mode maintenance and acoustic confinement, enabling stable and efficient chiral Brillouin inter-modal scattering (either backward or forward scattering) that is almost impossible to achieve in other platforms.

The intervortex FSBS and chiral phonons mediated frequency conversion can also be scaled for higher order vortex modes, by using some other specially designed chiral PCFs, such as six-fold rotationally symmetric chiral PCFs [18]. The three-fold rotationally symmetric chiral PCFs used here can only preserve $\ell =0$ and $\ell =\pm 1$ modes, and the loss of higher order modes is at least two orders of magnitude larger. This means that there is very little possibility that the injected light couples to $\ell \ge \pm 2$ modes. Lastly, the pure circular polarization states (Appendix C) at the fiber output indicate that there is almost no mode crosstalk in the experiment.

Intervortex FSBS incorporating spatial optical and acoustic vortices stands for a further step (compared with inter-polarization FSBS [30]) towards multi-dimensional Brillouin scattering, with potential to be used in new applications. For example, the unidirectional intervortex scattering can be used in the design of vortex isolators or narrowband vortex lasers, which are essential in multimode fiber communications [22]. The chiral flexural waves can be used in the large capacity light storage [35–39], where the buffers are multiple vortex-carrying acoustic modes of different topological charges, each one mediating a specific write-read process; they can also be employed for three-dimensional vectorial Brillouin sensing with orbital angular momentum of light, enabling structural torsion sensing and ambient acoustic vortex sensing [40–42]. The results of light-driven CFPs provide another perspective for fundamental research in chiral physics and might be of interest in the manipulation of vortex states in both classical and quantum regimes [22,23], and the lifetime of chiral flexural phonons should be sufficient for the latter application. Lastly, the theory of optoacoustic coupling between vortex modes also provides new insight into recent work on circular-polarization-sensitive SBS [33,43].

Acknowledgment. The authors thank Michael H. Frosz for providing the chiral PCF, and Dr. Hagai Diamandi and Dr. Yosef London for helpful discussions concerning the experiments.