An object or system is said to be chiral if it cannot be superimposed on its mirror reflection. Chirality is ubiquitous in nature, for example, in protein molecules and chiral phonons—acoustic waves carrying angular momentum—which are usually either intrinsically present or magnetically excited in suitable materials. Here, we report the use of intervortex forward Brillouin scattering to optically excite chiral flexural phonons in a twisted photonic crystal fiber, which is itself a chiral material capable of robustly preserving circularly polarized optical vortex states. The phonons induce a spatiotemporal rotating linear birefringence that acts back on the optical vortex modes, coupling them together. We demonstrate intervortex frequency conversion under the mediation of chiral flexural phonons and show that, for the same phonons, backward and forward intervortex conversion occurs at different wavelengths. The results open up, to our knowledge, new perspectives for Brillouin scattering and the chiral flexural phonons offer new opportunities for vortex-related information processing and multi-dimensional vectorial optical sensing.
【AIGC One Sentence Reading】:Chiral flexural phonons in twisted photonic crystal fibers enable intervortex frequency conversion, offering new avenues for optical sensing and information processing.
【AIGC Short Abstract】:We report the excitation of chiral flexural phonons in a twisted photonic crystal fiber using intervortex forward Brillouin scattering. These phonons induce a rotating linear birefringence, enabling intervortex frequency conversion at different wavelengths. This discovery offers new opportunities for vortex-related information processing and optical sensing.
Note: This section is automatically generated by AI . The website and platform operators shall not be liable for any commercial or legal consequences arising from your use of AI generated content on this website. Please be aware of this.
1. INTRODUCTION
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].
Sign up for Photonics Research TOC Get the latest issue of Advanced Photonics delivered right to you!Sign up now
2. TWISTED PCF AND VORTEX-CARRYING HELICAL BLOCH MODES
Chiral PCFs with -fold rotational symmetry support helical Bloch modes (hBMs) [24,25], whose -th order azimuthal harmonic carries an optical vortex of order , where is the number of complete periods of phase progression around the azimuth for fields expressed in cylindrical components and is the principal azimuthal order. Note that 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 (the number of on-axis discontinuities for fields evaluated in Cartesian coordinates) is linked to the azimuthal order by , where is the spin [ for left-circular (LCP) and for right-circular (RCP) polarization states]. Here, we use the shorthand to denote the parameters of an hBM, where for ease of notation the principal topological order is defined as . In chiral PCF, hBMs with equal and opposite values of are generally non-degenerate in index, i.e., topologically birefringent, while modes with opposite spin but the same value of 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 of the hollow channels is 1.58 μm, and the distance between adjacent channels is 1.79 μm, yielding 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 of and 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 , 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 modes and 0.262 dB/m for the modes. More details about polarization maintaining and fiber loss measurements are shown in Appendix C.
Figure 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 hBMs, and the spiral interference fringes that form between the modal far-fields and a divergent Gaussian beam. The intensity patterns for the and 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 . When superimposed with a 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 -out-of-phase flexural waves. When the waves have different frequencies (as in chiral FSBS), the shape rotates in time.
Figure 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.
3. CHIRAL PHOTON-PHONON INTERACTION VIA INTERVORTEX FSBS
A. Theory
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 and , 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] where is a slowly varying amplitude, is the azimuthally periodic field distribution of the mode hBM (, are cylindrical coordinates, is the twist rate (rad/m), the angular frequency, the spin, and the topological charge. Note that the hBM field pattern rotates with at a rate , locking it to the structure of the chiral fiber. can be expressed as the sum over azimuthal harmonics, with the -th harmonic having propagation constant and topological charge (to simplify the notation we have set and ). For more details see Appendix A.1.
In the absence of any perturbation ( and 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 in the form where , , is the acoustic wavevector at frequency , is the acoustic topological charge, and is an induced anisotropic change in dielectric constant that yields linear birefringence (needed to couple LCP and RCP light), the coupling rate from mode to mode will be proportional to , 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, out-of-phase, orthogonal flexural waves differing in Brillouin frequency by (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 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 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., , and collecting terms with the slowest rates of phase progression, we obtain (see Appendix A.2 for details) where the quantity is the dephasing rate, and are loss coefficients (dB/m) of pump and Stokes waves, is the phase index of mode , and , where is a dimensionless parameter that is proportional to the integral of over the fiber cross-section.
Then the coupled FSBS equations for complex slowly varying field amplitudes that scale with the square-root of the modal power () can be written in the form where is the dephasing parameter, is the speed of light in vacuum, and is a characteristic parameter (with units ) that depends on the electrostrictive parameters and the optoacoustic overlap.
With the good approximation (commonly used in SBS) that , Eq. (4) can be recast in the form where and we have used the very good approximation . Comparing Eqs. (6) and (3) we see that for , 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 in Eq. (5) where and are real-valued and extracting the imaginary parts, we obtain 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 and Eq. (5b) by and adding each equation to its complex conjugate, we obtain the coupled power equations
B. Writing of ℓph = ±1 Chiral Flexural Phonons
In the experiment of CFP writing/excitation, a fixed frequency pump signal is launched into the mode, along with a frequency-tunable Stokes signal in the mode, where . 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 [see Fig. 3(a)]. Photons are converted from pump to Stokes () 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.
Figure 3.(a) Dispersion curves for (from left to right) , and hBMs. The last two are separated by a small circular birefringence of , 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 pump to 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 has the highest overlap with the optical modes. The open circle marks the CFW that phase-matches the pump and 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 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)].
We start the experiment by demonstrating chiral FSBS between and modes and generation of CFP. Some simulations were preformed beforehand to estimate the acoustic frequency. Figure 3(a) shows the 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 and ). In addition, the simulations confirm that modes with the same topological charge but opposite spins display weak circular birefringence, of order . Any deviations from integer values of topological charge are caused by the polarization state not being perfectly circular [24]. Figure 3(b) shows the 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 pump to Stokes conversion. Efficient FSBS only occurs when phase matches 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 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 by approximately μ, 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 (or ) signal, which is down-shifted (or up-shifted) into an (or ) signal. For the detailed descriptions on the experimental setup see Appendices E and F. In the case of chiral FSBS between and modes, the CFW must supply a topological charge of , 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 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 pump and 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 . The green dotted curve shows the length dependence of the optoacoustic coupling, which is directly proportional to and takes the value at . Excellent agreement with theory is obtained for .
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 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 μ and the gain is half of its peak value, i.e., μ, corresponding to a 1/e acoustic power decay length of 70 μm. This in turn allows us to estimate . In the experiment, the phonon lifetime is 42 ns, which is calculated from the experimentally measured Brillouin linewidth with the formula . The acoustic power decay length is 86.52 μm, which can be calculated by and the flexural acoustic velocity . 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.
C. Read-Out of ℓph = ±1 Chiral Flexural Phonons at Different Wavelengths
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 [27], where and 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 , plotted as a function of the frequency shift , yield the curve in Fig. 4(b). The error falls to zero at , 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 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 depends on the dispersion of the two modes. Note that if the pump and Stokes modes are exchanged, the phonon propagates backwards and 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 for the backward process as a function of . The theory is based on FEM and predicts at , 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.
Figure 5(a) shows the read-out spectra recorded by a high resolution (20 MHz) OSA when a 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 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 . The Brillouin frequency shift 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 mode and Stokes read-out in the mode. When the pump signal is launched forwards, only very weak conversion to the backward 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 Stokes signal is launched backwards into the fiber at the phase-matching point and converted to the mode with a frequency up-shift of 98.5 MHz. When the pump signal is launched in forward direction, only very weak conversion to the 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 [27]. The and are the group indices of pump and Stokes modes. For conversion between and modes, the is calculated to be . 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 mode.
Figure 5.(a) Read-out spectra recorded by high resolution OSA when a 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 relative to the forward pump frequency. When the pump signal is launched forwards, only very weak conversion to the 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 pump and 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 Stokes to pump. The double peaks are caused by pixellation in the OSA. The recorded modal patterns confirm that the pump is in the mode. (c) Calculated evolution of power in backward-propagating pump and 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 , where . (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.
The read-out process (subscript ) can be conveniently modeled using Eq. (5): where and are the read-out amplitudes of pump and Stokes modes, [from Eq. (6)], and , the sign of being chosen to minimize . The distributions of and are obtained from solutions of Eqs. (5) and (11) and then numerically integrated to yield the length dependence of and , 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)].
D. Writing and Read-Out of ℓph = ±2 Chiral Flexural Phonons
Next, we show the results about writing and read-out of CFPs, with interaction between and optical modes. An expanded view of the calculated diagram for the four vortex-carrying hBMs is shown in Fig. 6(a), revealing a refractive index difference of between the and modes at 1550 nm, or a propagation constant difference of 0.00132 rad/μm. Figure 6(b) shows the diagram of the acoustic mode having the highest overlap with two optical modes (the inset shows the 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 for the forward writing process of 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 and writing the CFW must supply a topological charge of , which because of the three-fold symmetry is provided by the first harmonic of an 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 harmonic of the CFW.
Figure 6.(a) Dispersion curves for (from left to right) hBMs. Zero on the frequency axis corresponds to 1550 nm. The propagation constants of the and modes differ by 0.00132 rad/μm. (b) Calculated dispersion curves for the CFW in an untwisted PCF, with a cut-off frequency of . The open circle marks the point at which the CFW phase-matches the pump and 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 , which because of the three-fold symmetry is provided by the first harmonic of an CFW. (d) The transmitted power in the 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 Stokes signal during read-out by a weak backward pump signal, measured as a function of detuning from the phase-matching frequency of 187.2 THz (), which is shifted 6.21 THz () from the writing wavelength (1550 nm).
Figure 6(d) shows the transmitted power in the 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 Stokes signal during the read-out process by a backward pump signal, measured as a function of detuning from the phase-matching frequency of 187.2 THz (), which is shifted 6.21 THz () 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 flexural phonons, it was difficult to theoretically characterize the process by numerical calculation because the response of to 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 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.
4. DISCUSSION AND CONCLUSIONS
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 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 and core size. Generally, to improve the conversion efficiency of FSBS, the 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 and , theory predicts that backward conversion is possible not only between and modes (observed experimentally), but also between the 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 and 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 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
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.
APPENDIX A: THEORY OF CHIRAL INTERVORTEX FSBS
LCP to RCP Plane-Wave Coupling by Linear Birefringence Wave
Two orthogonal circularly polarized plane waves of different frequencies can be coupled together by a weak acoustic wave that induces a rotating linear birefringence. The dielectric constant of such a system can be written as where , is the average dielectric constant, causes circular birefringence and a weak linear birefringence, and is the acoustic wavevector [acoustic phase velocity ]. For , Maxwell’s equations yield LC and RC polarized fields with propagation constants , where is the refractive index. The resulting circular birefringence helps maintain circular polarization states against perturbations. Assuming a transverse field in the form where and are slowly varying pump and Stokes amplitudes, coupling between the two waves will be proportional to , resulting in the equations with coupling constants and dephasing rate , the sign being chosen to minimize . Coupling between LCP and RCP modes can only occur if there is some linear birefringence.
Optoacoustic Coupling between hBMs and Generation of Chiral Phonons
We now extend this analysis to coupling between circularly polarized helical Bloch modes (hBMs) carrying optical vortices and guided in an -fold rotationally symmetric chiral PCF. In the Cartesian laboratory frame, the transverse field of such an hBM can be written as a superposition of Bloch harmonics, each of which carries an optical vortex [26]: where is a slowly varying amplitude, is the azimuthally periodic field distribution of the mode hBM , are cylindrical coordinates, the angular frequency, the spin, and the -th orbital harmonic has propagation constant , topological charge , and amplitude (to simplify the notation we have set and ). The radial distribution is given by the summation , where is the -th zero of the Bessel function, and the effective edge of the photonic crystal region, where the fields vanish, is set at (this should be large enough to accommodate the guided modes). Note that each harmonic in contains the factor , i.e., so that the hBM field pattern rotates with at a rate , as required of the chiral fiber.
In the absence of any perturbation ( and constant), the expression in Eq. (A4) is itself a solution of Maxwell’s equations. As in Eq. (3) of the main text, the coupling rate from mode to mode will be proportional to where denotes the Hermitian conjugate and this time the perturbation has a transverse profile (caused by the guided acoustic wave) and , where is the acoustic topological charge and is the induced anisotropic change in dielectric constant that yields linear birefringence. Note that it is important to ensure that corresponds to the acoustic harmonic that carries topological charge , since adjacent harmonics differ in wavevector by the twist rate . Coupling between orthogonal circularly polarized vortex modes can only occur for , and the rotation matrix in Eq. (A1) ensures that the pattern of linear birefringence rotates with position and time, which enables the generation of chiral flexural phonons that resemble a rotating single-spiral corkscrew. A chiral flexural wave (CFW) with this topology can be produced by a superposition of two, out-of-phase, orthogonal flexural waves differing in frequency by (see next section).
Evaluating in a coupled-mode description, assuming slowly varying power-normalized amplitudes, a separately excited acoustic wave (i.e., no SBS), conservation of topological charge, i.e., , and collecting terms with the slowest rates of phase progression, we obtain
The quantity is the dephasing rate, is the propagation constant and the phase index of mode , and , where is a dimensionless parameter that is proportional to the amplitude overlap integral and the induced linear birefringence. Note that the optoacoustic overlap integral is calculated using the entire field of each helical Bloch mode, i.e., the sum over all harmonics.
In solving Eq. (A5) it is often convenient to make the substitution , resulting in
Equations of Chiral FSBS: Writing Process
The coupled SBS equations for power-normalized field amplitudes can be written in the form where is the dephasing parameter, is the speed of light in vacuum, and is a characteristic parameter (with units ) that depends on the electrostrictive parameters and the optoacoustic overlap. Noting that the power in the mode is proportional to , power conservation can be checked by multiplying the first equation by , adding the result to its complex conjugate, and doing the same for the other three equations. Adding all the equations together we obtain since , showing that power is conserved in the absence of loss.
With the good approximation (commonly used in SBS) that , Eq. (A7) can be recast in the form
Multiplying the first by and the second by and adding each equation to its complex conjugate, we obtain the coupled power equations
Although, because energy is lost to phonons, these equations no longer exactly conserve power for , they do (as required) conserve photon flux:
Equation (A10) can be cast in a simplified form by noting that : where
Equation (A12) has an instructive analytical solution if the loss in both pump and Stokes modes is identical, i.e., : where , , is the total input power, and .
Phase of Excited Chiral Acoustic Wave
To analyze backward coupling, which is essentially a linear scattering process, we need to know the strength and phase of the CFW at every point along the fiber. This requires a solution of the SBS amplitude equations [Eq. (A9)]. Setting , where and are real-valued, substituting into Eq. (A9), and extracting the imaginary parts, we obtain which show that in general the relative phase of pump and Stokes varies in a non-trivial manner along the fiber. As a result, the phase of the excited acoustic wave is not constant, which will affect the conversion efficiency of backward conversion. However, in the experiment the dephasing is zero in the writing process, so that the pump and Stokes phases are constant.
Electrostrictive Driving Term
Given the complexity of the PCF structure, standard approaches to electrostriction are not valid and numerical techniques must be used [44]. We can nevertheless make some general observations, starting with an electric field created by a superposition of two vortex modes with different topological charges, spins, and frequencies: where . Collecting and combining the two vector components we obtain where and . Taking the case when for illustrative purposes, Eq. (17) yields which is a fast-oscillating linearly polarized field whose average direction rotates slowly with time, azimuthal angle, and distance. Electrostriction cannot follow the fast oscillation , but it is able to respond to the much slower (up to ) rotation , which we propose induces an anisotropic rotating strain field that when phase-matched can drive chiral acoustic phonons, and cause an anisotropic change in the dielectric constant similar to Eq. (A1). Understanding this mechanism fully will require further study on the analytical model.
Intervortex Backward Conversion by Chiral Phonons: Read-Out Process
The read-out process can be analyzed by extracting the coupling term in Eq. (A9) and writing coupled equations for the read-out amplitudes : where , the sign being chosen to minimize , and is set in the writing step, as is the acoustic frequency . In deriving Eq. (A19) we have assumed that the writing process is perfectly phase-matched, i.e., , and made the approximation . The terms in the large brackets are proportional to the strength of the already-written acoustic wave and play the same role as and in Eq. (3) of the main text; note that the amplitude gain is one half of the power gain . Since the acoustic frequency and wavevector are fixed by the writing step and the optical dispersion is much weaker than the acoustic, the dephasing changes much more slowly with the optical frequency than in the writing step.
APPENDIX B: CALCULATION OF CHIRAL ACOUSTIC MODES AND OPTOACOUSTIC OVERLAPS
In the simulation, the full-vectorial wave equations were independently solved by the finite element method (FEM) in COMSOL Multiphysics software. We used the Electromagnetic Waves module to solve the optical modes and Solid Mechanics module to solve the acoustic modes. The simulations were carried out at a fixed optical wavelength of 1550 nm, and other parameters, such as fiber geometry, refractive index, and photoelastic tensors, are shown in Table 1. Considering the difficulty of calculating the acoustic mode in twisted fiber, we first calculate the normal flexural modes in non-twisted PCF and then produce the chiral ones by a superposition of two orthogonal normal modes by phase difference. For the calculation of acoustic modes of lower frequency, the cladding diameter is chosen to be 230 μm, which is the same as the parameter of real fiber, as shown in Fig. 7(a). However, for the simulation of acoustic modes of higher frequency, the number of parasitic modes was extremely large (more than 1000 in frequency span of 5 MHz around 1.29 GHz) if the cladding size is still 230 μm. Therefore, the cladding diameter in the latter case is set to 20 μm, as shown in Fig. 7(b). Additionally, to ensure the accuracy of calculation, we set the mesh size to “Extremely fine” in the COMSOL and refine the central region with “2nd-order refinements”. The maximum element size is 0.1 μm and the minimum element size is μ. The zoom-in picture of the fiber central part with finite element meshes is shown in Fig. 7(c).
Fiber and Material Parameters in Simulations
Air holes diameter
1.61 μm
Hole-to-hole pitch
1.828 μm
Core diameter
1.8 μm
Cladding diameter (for acoustic simulation)
230 μm
Cladding diameter (for acoustic simulation)
20 μm
Refractive index
1.444
Fiber twist rate
628 rad/m
Density ()
2203
Young’s modulus (GPa)
73
Poisson ratio
0.17
Viscosity tensor
Viscosity tensor
Photoelastic tensor
(0.121, 0.27, −0.0745)
Figure 7.(a) Fiber cross-section used in the calculation of acoustic mode. (b) Fiber cross-section used in the calculation of acoustic mode. (c) Zoom-in of fiber central part with finite element meshes. (d) Numerically calculated overlap coefficients ( in red, in blue) for all the acoustic modes that satisfy phase-matching in the range 0–0.3 GHz, for pump and Stokes modes. (e) Same as (d) but for pump and Stokes modes, and acoustic modes in the range 0.8–1.8 GHz.
Figure 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.
Figure 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 pump and Stokes modes. The acoustic mode at 1.292 GHz has the highest overlap and dominates FSBS.
Note that those calculations are only valid for uniaxial flexural acoustic modes in an untwisted PCF. Chiral flexural acoustic modes can then be formed by the superposition of two orthogonal, -out-of-phase, uniaxial flexural modes with different frequencies. In the chiral PCF the acoustic modes are expected to display some topological birefringence, which although not discussed here will have some small effect on the phase-matching conditions.
APPENDIX C: MEASUREMENTS OF CIRCULAR POLARIZATION MAINTAINING AND FIBER LOSS
Figure 10(a) shows the measurement of Stokes parameters of (blue) and (red) after 8 m chiral PCF, from 1530 nm to 1630 nm. Figure 10(b) shows the loss measurement of and modes from 1530 nm to 1630 nm.
Figure 10.(a) The measurement of polarization Stokes parameters of (blue) and (red) after 8 m chiral PCF, from 1530 to 1630 nm. (b) Loss measurement of and modes from 1530 to 1630 nm. The losses of other vortex modes () are almost equal to that of , and therefore not shown here.
The chiral PCF was fabricated by spinning the preform during fiber drawing [12]. To do so, a high speed spinning motor with a maximum speed of 2000 revolutions per minute (RPM) has been installed at the top of the fiber drawing tower. This makes it possible to draw and twist the fibers at the same time. The twisting pitch can be determined by , where is the rotation rate in Hz and is the fiber drawing speed in m/s. To apply both vacuum and pressure to the spinning preform a two-channel rotary pressure joint was installed.
The first step before starting to rotate the preform is to obtain the desired fiber structure. Once this has been achieved, the rotation speed is gradually increased while adjusting all the parameters to keep the structure consistent. The microstructure can be fine-tuned by the pressure system while the preform is rotating. In order to minimize the problem of preform vibration, it is important to keep the preform mounted tightly and straight in the center of the motor axis. At high rotational speeds, even a slight deviation from the center can damage the preform due to strong centrifugal forces.
Fabricating a chiral PCF requires some special attention. It is very important to keep the preform mounted tightly and straight in the center of the motor axis so that the rotational symmetry can be maintained during twisting; otherwise the fibers are likely to break during drawing and the fabricated fibers will have stress-induced linear birefringence, which will make the eigen-polarization states elliptical rather than purely circular [24]. Also, some crosstalk between the vortex modes will occur. During the fiber drawing, before start of the twisting, we need to draw some non-twisted fiber and check the fiber cross-section images segment by segment while fine-tuning the pressure and temperature (of course, we also need to check the fiber cross-section during twisting the fiber). This is to make sure that the fiber structure is really symmetrical and the distortion of air-holes is as little as possible; otherwise the circular birefringence will again be destroyed.
In our case, the fiber shows robust maintenance ability on circular polarization states and vortex modes, with the Stokes parameter after 8-m-long fiber higher than 0.98. Nevertheless, the fabrication process still needs to be constantly optimized, as the fiber is not always perfect and tiny structural distortions happen from time to time.
APPENDIX E: EXPERIMENTAL SETUP
A custom-built setup was used to investigate chiral FSBS (Fig. 2). In the CFP writing step, the output from a CW laser at 1550 nm was split into pump and Stokes signals. The pump was frequency up-shifted by 10 GHz in a single sideband modulator (SSBM) and the Stokes by GHz. Brillouin gain spectra were measured by sweeping Ω over the peak of the Brillouin gain. Both pump and Stokes were boosted to few-watt levels in erbium-doped fiber amplifiers (EDFAs). Fiber polarization controllers (FPCs) and vortex generation modules (VGMs) were optionally used to synthesize vortex beams with different topological charges. The VGMs consisted of a sequence of polarizer, plate, Q-plate, and plate. A circular-polarizing beam splitter (CPBS) was used to set orthogonal circular pump and Stokes polarization states. At the output of the chiral PCF, a plate and a PBS were adjusted to transmit the pump and reflect the Stokes signals. For the measurement of intervortex chiral FSBS between and modes, the amplification of Stokes was measured by a power meter (PM), while for the case of to scattering, a double-lock-in detection scheme [30] was used. In this scheme, two low frequency sinusoidal signals ( and ), generated in a lock-in amplifier (LIA), were superimposed separately onto the pump and Stokes signals using intensity modulators (IMs). The LIA measures the Brillouin response from the Stokes signal at the output, with a lock-in frequency of 5 kHz (= 20–15 kHz). This double-lock-in technique allows not only measurement of the weak FSBS signal, but also rapid location of the Brillouin peak before measurement of the gain spectrum.
In the chiral phonons read-out step, the output from a frequency tunable CW laser was launched into the fiber in the backward direction, keeping the forward pump and Stokes switched on with their frequency difference set to the peak of the Brillouin gain. A VGM was optionally used to create circularly polarized vortex states. The backward pump/Stokes was frequency-tuned until conversion to a frequency-shifted Stokes/pump was observed, due to the presence of the already-written acoustic wave. The power and spectrum of the backward pump and Stokes signals were measured during read-out by a power meter and a high resolution OSA. Again, the lock-in detection scheme was optionally used to measure backward intervortex conversion if the signal response was too weak, as was the case in to conversion. A near-field scanning Brillouin analyzer (NBA) [18] was used to measure the near-field distribution and topological charge of the modes of the newly generated signals. See Appendix D for more details on NBA.
APPENDIX F: MODE PROFILES MEASUREMENT
As Fresnel reflections and Rayleigh scattering made direct measurement of the near-field mode profiles of the read-out signals difficult, we employed a near-field scanning Brillouin analyzer (NBA). This consists of an objective lens, a fiber raster scanning stage controlled by a computer with closed-loop feedback, a narrow-band filter, and signal detection equipment (e.g., OSA or PM). Light from the fiber was collected pixel by pixel by the fiber raster scanning stage. The signal was then filtered to remove unwanted light, detected, and analyzed. The NBA was also used to measure the topological charge, requiring only the addition of a cylindrical lens in front of it to create the patterns characteristic of OAM.