1. INTRODUCTION

Orbital angular momentum (OAM), a photonic degree of freedom with inherent multiple dimensions shown by Allen and co-workers [1–3], has made rapid progress in light manipulation, enhanced imaging, high-capacity communication, and memory [4–12] due to its unique profile, ranging from light to radio waves, and even electrons and plasma [13–15]. Total photonic angular momentum can be divided into spin angular momentum (SAM) and OAM in paraxial approximation. SAM and OAM are separately conserved when a light beam propagates in vacuum or a homogeneous and isotropic medium [16,17], providing a basis for using OAM beams to communicate information over a long distance. In comparison, information processing usually requires some interactions between signals. For OAM beams, these interactions can be well implemented by nonlinear optical processes both in classical and quantum domains, such as parametric downconversion, second-harmonic generation, and four-wave mixing [18–23], which are more applicable for matching signals’ frequency interface [19,23]. Besides, light–atom interactions afford some key protocols for processing optical information in the quantum domain [12], but these interactions are difficult to be applied in the classical realm, especially for the on-chip level. In general, the signal-processing functions based on the above interactions are far from the demand of OAM multiplexing. Additionally, another nonlinear process for concern is Brillouin interaction, an inelastic scattering of light from sound, which has received increasing attention with the aim of harnessing reversible photon–phonon conversion to perform signal processing [24,25]. Compared with other nonlinear interactions, the Brillouin process affords more signal-processing functions due to the involvement of acoustic phonons with low velocities and long-coherence times, and has good compatibility with a silicon circuit. Some related demonstrations such as light storage, signal processing, and Brillouin scattering-induced transparency have been carried out in a silicon photon–phonon circuit or a silica resonator recently [24–28].

To our knowledge, phonons carry no SAM due to its nature of electrostatic oscillations, but it is worth noting that they can carry OAM by forming acoustic vortices in the propagation medium; in other words, phonons’ OAM coincides with their total angular momentum. In theory, the exchange of OAM between an electromagnetic and a plasmon/phonon wave has been studied in Ref. [29]. In experiment, OAM transfer from an acoustic wave to photons has been observed in a fiber via acousto-optic interaction [30]. More interestingly, we found that the photonic OAM conservation was violated in stimulated Brillouin amplification (SBA) recently, and we proposed an assumption of a phonon carrying OAM to explain this phenomenon [31]. This assumption indicates that an OAM beam involved in SBA may be a feasible method to realize the prediction of exciting a specific OAM phonon from outside in Ref. [29]. And on this basis, we believe that the excited OAM phonon, if it really exists, can be probed via photon–phonon conversion as well. Consequently, a new insight is naturally gained once deliberating on the above assumption and the requirement of signal processing in an OAM-multiplexing system, i.e., whether we can achieve a high-capacity and flexible information-processing technique by extending the reversible photon–phonon conversion scheme into an OAM-multiplexing area. Here, to demonstrate the physical mechanism of this proposal, we report on the observation of reversible OAM photon–phonon conversion via the backward-stimulated Brillouin scattering (SBS) process. The OAM conservation relations in SBA and Brillouin acoustic parametric amplification (BAPA) are confirmed. More specifically, a well-defined OAM state can be converted into a phonon beam via SBA, and reconverted back to the optical domain via BAPA within the decay time of coherent phonons. SAM and OAM are independently conserved in these processes due to phonons carrying only OAM. Beyond the fundamental significance, this demonstration reveals a great potential use of photon–phonon conversion for information processing in an OAM-multiplexing system.

2. THEORY BACKGROUND

The backward-SBS process, used in this proof-of-principle study, can be classified into two quasi-parametric downconversions, i.e., SBA and BAPA. In SBA, a pump beam and a Stokes frequency-shift seed beam counterpropagate in a medium; the seed is amplified by pump and a coherent phonon beam is simultaneously generated as an idler field. The energies and the momenta of two optical fields and the phonon field in SBA must satisfy the conservation relations of ${\omega }_p-{\omega }_s=\mathrm{\Omega }$ and $k_p-k_s=q$, respectively. Here ${\omega }_p(k_p)$, ${\omega }_s(k_s)$, and $\mathrm{\Omega }(q)$ are the energy (momentum) of the pump, seed and phonon field, respectively. In BAPA, a probe beam co-propagates with and enhances a coherent phonon beam usually generated by SBA or focused-SBS. A major difference from SBA is that here the role of the phonon beam is of a seed field of downconversion, and a Stokes frequency-shift light beam is generated as an idler field. Indeed, the physical nature of the so-called Brillouin dynamic grating in the distributed fiber sensor domain is just a BAPA process, where the coherent phonon (dynamic grating) is generated by SBA [32]. The energy and momentum conservation required by BAPA are ${\omega }_b-{\omega }_k=\mathrm{\Omega }$ and $k_b-k_k=q$, respectively, where ${\omega }_b(k_b)$ and ${\omega }_k(k_k)$ are the energy (momentum) of the probe and the Stokes field, respectively. It is noted that the momentum conservation relations $k_p-k_s=k_b-k_k=q$ include the angular momentum as well, and considering the fact that the phonons carry no SAM, the SAM conservation for SBA and BAPA are $S_p=S_s$ and $S_k=S_b$, respectively. Here, $S_x$ is the SAM of corresponding optical fields. That is to say, the SAM conservation in photon–phonon coupling only involves optical fields. More specifically, the pump and seed should be set at the same linear or opposite circular polarizations in SBA, whereas in BAPA, any polarized probe beam can interact with coherent phonons and generate a same linearly or opposite circularly polarized Stokes beam.

Next, we first discuss the OAM conservation relation in SBA for deducing the phonon OAM state excited by SBA. Then, in order to intuitionally confirm the existence of SBA-excited OAM phonons, we discuss the OAM conservation relation in BAPA for deducing the Stokes beam’s OAM state extracted from the phonons. First, for obtaining the phonon OAM state excited by SBA, consider both the interaction picture Hamiltonian of SBA and the requirement of OAM conservation at $x$ axis, expressed, respectively, as

Second, for obtaining the OAM conservation relation in BAPA, consider both the interaction picture Hamiltonian of BAPA and the requirement of system OAM conservation expressed, respectively, as

The behavior of the photon OAM states shown in Eq. (6) looks like a four-photon process; this is why the connected two quasi-parametric downconversions were called Brillouin-enhanced four-wave mixing (BEFWM). It should be noted that the name BEFWM is not appropriate because it is not four-wave mixing but two independent three-wave mixing processes sharing the same phonon field, which is excited in SBA and enhanced in BAPA. By removing the operator in Eq. (6), we obtain an OAM relation between the output idler Stokes beam and the input optical beams (pump, seed, and probe), which is given by

Note that the quantity of OAM transfer from phonons to photons in BAPA is opposite to the increment of the phonon OAM, i.e., $-{\ell }_{\rho }$ per photon, and it is to be expected that the OAM selection rules shown in Eqs. (6) and (7) are an equivalent version of $k_p-k_s=k_b-k_k=q$.

3. EXPERIMENT

Now we experimentally demonstrate the reversible OAM photon–phonon conversion based on the above discussion. Figure 1(a) shows the schematic diagram of the experimental setup, a simple method that can be easily duplicated in any optical laboratory. A P-polarized pump beam and a P-polarized Stokes-frequency-shift seed beam quasi-collinearly interact in a coupling cell to excite a coherent phonon beam via SBA. Then an S-polarized probe beam, collinearly counterpropagating with the seed, interacts with the SBA-excited phonons via BAPA, and meanwhile generates an S-polarized Stokes beam that collinearly counterpropagates with the pump. The Stokes beam carries the phonon’s information and outputs from PBS3 [see vector diagram at the right bottom in Fig. 1(a)]. Usually there are two methods to insulate the probe from interacting with the seed; one of them is called frequency decoupling, in which a frequency difference more than the linewidth of the coherent phonon between the probe and the pump is used for violating the phase matching for the probe and seed; the other is called polarization decoupling, which is used in this experimental demonstration. Here the frequencies of the probe and pump are the same for convenience, with the consequence of phase matching between the probe and seed. Hence, violating SAM conservation is a workable way to prevent their coupling, i.e., setting the probe and seed in mutually orthogonal polarizations. As a result, the probe can only interact with the SBA-excited coherent phonons, and no Stokes beam will be generated via BAPA if the pump or seed is shut down. Figures 1(b1) and 1(b2) show the timing of the beams, where pump arrives 1 ns after the seed at the coupling cell for suppressing interaction between the pump and the incoherent-phonon noise, and the probe is 1 ns later than the pump as well to ensure that the coherent phonon have been well generated by SBA. All beams’ powers are below the threshold of SBS to avoid self-SBS noise, where the energy of the pump, seed, and probe are set to 2, 0.5, and 2 mJ, respectively, and the Stokes beam of 1.05 mJ is obtained (for more experimental details, see Supplement 1).

 

Fig. 1. Schematic presentation of reversible OAM photon–phonon conversion. (a) Experimental setup. A $\lambda /2$ plate and a polarized beam splitter (PBS1) are used to vary the S and P contributions, and the P component is injected into the coupling cell as a pump in SBA. The transmitted S component from the BS is directed toward the SBS cell to generate a P-polarized Stokes-frequency-shift seed beam, whereas the reflected S component from the BS is used as a probe in BAPA. The pump and seed interact quasi-collinearly in the coupling cell via SBA to create a coherent phonon field, and then the probe interacts with the phonon via BAPA to create a Stokes beam and reflect from PBS3. The OAM beams are converted by spiral phase plates (SPP) or a q-plate from Gaussian beams. The vector diagrams at the right bottom describe the momentum and energy conservation relations of SBA and BAPA. (b1, b2) Timing of the beams (b1) before and (b2) after interactions in the experiments.

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In experiments, first an OAM state of $\ell =1$, shown in Figs. 2(a1) and 2(a1i), is carried by the seed and pump, respectively, for exciting the OAM phonons via SBA. According to Eq. (3), the SBA-excited OAM phonon states should be $|0+1⟩=|1⟩$ and $|1+0⟩=|1⟩$, if they really exist. Then, to confirm these OAM phonons, a Gaussian probe beam is used to reconvert them from the acoustic domain to the optical domain via BAPA. According to Eq. (7), the output idler Stokes signals should be $|0+1-0⟩=|1⟩$ and $|1+0-0⟩=|1⟩$, and if using an OAM probe beam of $\ell =1$ to extract the phonons generated by the Gaussian pump and seed, the output Stokes signal should be $|0+0-1⟩=|-1⟩$. It can be easily seen that the experimental results shown in Figs. 2(a2/a2i)–2(a4/a4i) are all in good agreement with the above analysis. Therefore, the reversible OAM transfer in photon–phonon conversion and the corresponding OAM selection rules shown in Eqs. (3) and (7) are verified. Particularly, the nonsymmetrical uniformity of the intensity patterns in Figs. 2(a2) and 2(a4) is due to a noncollinear parametric process in this experiment. More specifically, the optical axis of the output beam is slightly changed compared with that of the input beam, i.e., carrying an off-axis vortex [33]. Second, a fractional OAM state of $\ell =1.5$ shown in Fig. 2(b1) is input from the seed, pump, and probe, where the SBA-excited OAM phonon states should be $\ell =1.5$, $\ell =1.5$, and $\ell =0$, respectively, according to Eq. (3). Note that the azimuthal index $\ell$ of the Laguerre–Gauss mode can only have integer values; a noninteger value in fact describes a superposition state, and here a ring gap facing bottom right arises from the phase discontinuity associated with a string of alternating charge vortices [34,35]. It can be seen that the ring gaps of the corresponding output Stokes states shown in Figs. 2(b2), 2(b3), and 2(b4) face bottom right ($\ell =1.5$), bottom right ($\ell =1.5$), and left ($\ell =-1.5$), respectively, which means still complying with the OAM selection rule in Eq. (7). Third, for demonstrating the OAM selection rule shown in Eq. (7) more intuitively, two beams of $\ell =p=1$ and $\ell =p=0$ are used as pump and seed, respectively, to excite an expected phonon state of $\ell =p=1$. Then, an OAM beam of $\ell =1$ is used as probe to generate a Stokes beam, as shown in Figs. 2(c1)–2(c4). The Stokes beam of $\ell =0$ and $p=1$ indicates that not only the azimuthal index $\ell$ but also the radial index $p$ is conserved in our experimental setup. It should be noted that the index $p$ is not always conserved in the nonlinear process [19,22], and a more general case about this point will be further discussed in the future. The generation method of beams of $\ell =1.5$ and $\ell =p=1$ can be seen in Refs. [36,37] and in the experimental details in Supplement 1. Fourth, a crucial point in an OAM-multiplexing system is whether the OAM-multiplexing channels can be reversibly converted between the photonic and phononic domains. Therefore, we implement a reversible conversion for OAM-multiplexed channels. Here, superposing OAM beams of $\ell =\pm 2$ and $\ell =\pm 3$ are used for simulating the OAM-multiplexed channels. It can be seen that the input seed states and the output Stokes states for the superposing beams, shown in Figs. 2(d1) and 2(d2) and Figs. 2(d3) and 2(d4), still preserve the rule in Eq. (7). This result indicates that the phononic OAM could be multi- and de-multiplexed as its photonic analog, a crucial property for an OAM-multiplexing system, and this is to be expected as phonons are quasi-Boson. Note that according to Eqs. (3) and (7), if a probe is a Gaussian beam, i.e., ${\ell }_b=0$, then the phonon signals will be reconverted back to the optical domain without any channel change, or else, the so-called channel switching will be performed. For example, a data exchange can be achieved by using an OAM probe of $\ell =6$ to reconvert OAM-multiplexed phonon channels of $\ell =2$, 4 back and implementing one mirror reflection for the output Stokes beam.

 

Fig. 2. Experimental results of reversible OAM photon–phonon conversion. (a1–a4) Observed intensity profile/interferogram (a1/a1i) of the input OAM beam of $\ell =1$ from the seed, pump, and probe, and the intensity profiles/interferograms (a2–a4/a2i–a4i) of the corresponding output Stokes beams. (b1–b4) Observed intensity profiles of the input OAM beam of $\ell =1.5$ (b1) from the seed, pump and probe, respectively, and (b2–b4) corresponding output Stokes beams. (c1–c4) Observed intensity profiles of the input pump ($\ell =p=1$), seed ($\ell =p=0$), probe ($\ell =1$, $p=0$), and output Stokes ($\ell =0$, $p=1$) beams, respectively. (d1–d4): (d1, d2) Observed intensity profiles of the input seed beams of $\ell =\pm 2/\pm 3$ and (d3, d4) the corresponding output Stokes beams.

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To prove the feasibility of using a photon–phonon conversion-based signal-processing scheme in an OAM-multiplexing system, besides the above demonstrations, another important question is how long the OAM state could be maintained in the phonon signals, and, more important, what it will be like over time. So, finally, we further demonstrate the stability of the OAM state in the phononic domain, i.e., the stability of the OAM-channel labels. Here, an OAM state of $\ell =1$ is converted into a phonon beam and then reconverted back to a light beam after successively increasing the delay times between SBA and BAPA. Figure 3(a) shows the time sequences of the input beams; the presence of a delay time (1–26 ns) between the pump and probe can ensure that the pump and seed have exited from the coupling cell when the probe enters, and here the maximum reconversion efficiency (output Stokes energy/input probe energy) and retrieved Stokes energy are about 21% and 0.424 mJ, respectively, when the delay time is 1 ns, as shown in Figs. 3(b) and 3(c1). In contrast, in the case of no additional delay discussed above, as shown in Figs. 1(b1) and 1(b2), the photon–phonon coupling is enhanced due to SBA and BAPA simultaneously occurring in the cell; the reconversion efficiency is about 52%. An exponentially decaying reconversion efficiency shown in Fig. 3(c1) indicates that the SBA-excited phonon signal begins to dampen rapidly as soon as it is generated, as reported previously in Refs. [25,26]. Beyond that, here the phonon signal will remain in its initial OAM state before disappearing completely, as shown in Figs. 3(c2)–3(c5). This implies that no model crosstalk will be introduced when the phonon signals propagate in a homogeneous and isotropic medium, just like optical OAM channels. In addition, although the phonon channels can only be sustained up to 26 ns in this proof-of-principle experiment, the decay time of the phonon signals can be remarkably extended to tens of microseconds by handling low-frequency long-life phonons in a silicon photon–phonon circuit or a microcavity via the forward-SBS process [24–27]. Besides, a weak continuous probe light is able to sustain and enhance the signals in the phonon domain via BAPA as required. However, it is worth noting that a phonon is not a good candidate for storage, but an intermediary for mediating interactions between photons, i.e., a data cache. Therefore, most of the time we hope that its decay time is as short as possible for an applied pursuit of a high refresh rate. According to this requirement, the data in the phonon cache could also be quickly wiped by performing anti-Stokes-type photon–phonon coupling, i.e., an upconversion between the probe and phonons, with generating anti-Stokes photons.

 

Fig. 3. Experimental results of the OAM phonon signal’s time stability. (a) Timing of the delay experiment. The delay time between the pump and probe is from 1 to 26 ns; the position of the dotted line represents the case of 1 ns delay. (b) Time-domain waveform of the output Stokes beam versus delay time. (c1) Reconversion efficiency (output Stokes energy/input probe energy) and retrieved Stokes pulse energy versus delay time; the solid line is the exponential-decay fit. (c2–c5) Observed intensity profiles of output Stokes beams (the phonon state is prepared in $\ell =1$) with different delays.

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4. CONCLUSION

In conclusion, we have introduced the concept of OAM into photon–phonon coupling and used a simple experimental setup to demonstrate this proposal. A well-defined OAM state can be interconverted between the photonic and phononic domains via controllable OAM transfer in SBA and BAPA. Our finding reveals the potential of using phonons to modulate data in photonic OAM channels. Owing to the reversible OAM photon–phonon conversion, the long-coherence-time and low-velocity phononic OAM signals can provide many important signal-processing functions and protocols required by an OAM-multiplexing system, such as data cache, routing, switching, exchange, and correction. Particularly noteworthy is that the interconversion in this proof-of-principle study is demonstrated in a homogeneous and isotropic medium, which means that we just take the first step of this proposal. At present, the optical OAM waveguide is still in an initial stage; for realizing the proposed signal-processing photon–phonon circuit in this paper, a waveguide only supporting OAM photons is obviously not enough. Nevertheless, a hybrid OAM photonic–phononic waveguide is worth expecting. More remarkably, the OAM selection rules shown in Eqs. (3) and (7) largely depend on the fact that the phonons carry no SAM; or in other words, that SAM and OAM are independently conserved in the photon–phonon conversion. This property, separating SAM and OAM while converting, can provide unique protocols for processing signals carried by cylindrical vector beams as well. And furthermore, to achieve the total angular momentum’s state transfer between light and collective excitation, a reversible photon–magnon conversion in magnetic media is worth exploring, where magnons can also be manipulated by a microwave signal and a magnetic field; and during the preparation of this paper, some works about photon–magnon conversion have been reported on arXiv [38,39].