Hollow-core photonic-bandgap fibers (HC-PBFs) are unique for their ability to guide light in air through Bragg reflection from a periodic silica matrix that forms the waveguide cladding [1–3] [Fig. 1(a)]. In comparison to conventional silica (step-index) fibers, Bragg guidance in HC-PBF drastically reduces the nonlinear interactions with silica and increases the power handling to permit new forms of high-power laser delivery [4], pulse compression [5], and light sources [6]. Conversely, the introduction of atomic vapors within these hollow-core fibers produces sustained photon–atom interactions over unprecedented length scales, which enables new light sources for both classical [7] and quantum [8–12] applications. While electronic nonlinearities are widely studied in such fibers [1–3], comparatively little is known of their acousto-optic (or optomechanical) interactions, and the dynamics and noise that they can produce. Only recently has coupling between megahertz (MHz) elastic waves within the silica matrix been quantified [13] and identified as a source of noise in quantum optics [13,14].

 

Fig. 1. (a) Hollow-core fiber cross section with the vibrating air in the core represented in the inset, (b) mode profiles in the fiber core and forward Brillouin scattering frequencies of interest, indicated as inputs and outputs, (c) single-mode optical dispersion curve with the pump (${\omega }_p$) and Stokes (${\omega }_s$) frequencies indicated, (d) the acoustic mode dispersion curves, and (e) energy-level diagram of the Stokes transition. The common green arrow represents the acoustic four-vector that satisfies phase matching, where $K$ is the wavevector projection.

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In this paper, we show that photon–phonon coupling mediated by air (gases) within the hollow core of the fiber constitutes a much larger and perhaps more tailorable form of optomechanical coupling. Through a combination of theory and experiment, we show that the hollow core of the fiber acts as a conduit for guided acoustic waves in air. Optical waves that are guided in this same region produce strong photo-acoustic coupling to these sound waves, yielding appreciable forward Brillouin coupling at MHz frequencies. Using precision spectroscopy methods, we identify a single air-mediated Brillouin resonance at 35 MHz with 20 times stronger photo-acoustic coupling than is produced by elastic waves in the silica cladding alone. We show that the strength, frequency, and character of this Brillouin resonance is explained by the properties of the gas filling the core and the dimension of the hollow-core fiber. Since this form of Brillouin coupling depends strongly on both the acoustic and dispersive optical properties of the gas within the fiber, this new type of optomechanical interaction is highly tailorable. More generally, gas optomechanics may lead to new forms of spectroscopy and may be valuable for sensing. In contrast, photon–phonon coupling in gases presents a power limitation in applications delivering high-power narrowband sources [6,7] and a noise limitation for noise-sensitive classical and quantum applications [8–12]. In what follows, we identify the optomechanical properties of air in HC-PBFs.

Through driven Brillouin scattering processes, interfering optical waves produce time-varying optical forces that drive the excitation of acoustic waves. These elastic waves scatter light to new frequencies through dynamical changes in the dielectric response. Depending on the type of deformation and the medium response, light can be scattered to optical modes with similar or dissimilar polarization states. These distinct scattering processes are termed intra-polarization and inter-polarization scattering, respectively. It is important to note that while elastic solids can mediate both intra- and inter-polarization scattering (e.g., see Ref. [15]), inter-polarization scattering is generally forbidden in gases. Both scattering processes occur in solids because elastic deformation of a solid produces optical birefringence (through the photoelastic effect) that readily changes the polarization of light. By contrast, pressure waves do not produce birefringence in typical gases, meaning that inter-polarization scattering does not occur in air [16]. It is for this reason, for example, that no Brillouin signature from air was measured previously, in Ref. [13]. In that work, measurements were made only of inter-polarization scattering, which can occur only in the glass.

In what follows, we consider forward Brillouin scattering processes (FBS) [13,15,17–26], not to be confused with more widely studied backward Brillouin scattering processes [27]. Through FBS, co-propagating pump and Stokes waves of frequencies ${\omega }_p$ and ${\omega }_s$ are coupled through parametrically generated phonons of frequency $\mathrm{\Omega }={\omega }_p-{\omega }_s$, as indicated by Figs. 1(b)–1(e). Coupling is mediated by guided acoustic phonons that satisfy the phase-matching condition $K(\mathrm{\Omega })=k({\omega }_p)-k({\omega }_s)$, where $K(\mathrm{\Omega })$ and $k(\omega )$ are the acoustic and optical dispersion relations of the type sketched in Figs. 1(c) and 1(d). An analogous set of conditions apply for the anti-Stokes process.

In contrast to backward Brillouin processes, the phase-matching condition for FBS is not satisfied by elastic waves in bulk media, and typically requires wave guidance for both light and sound [13,15,17–26]. Due to the small wave-vector mismatch between the pump and Stokes fields, only guided acoustic phonons with nonzero frequencies for $K=0$ can mediate forward Brillouin coupling. Within the HC-PBF of Fig. 1, this phase-matching condition is satisfied only by guided acoustic waves that exhibit a frequency cutoff, labeled with mode indices $\mathrm{AR}1–\mathrm{AR}N$ in Fig. 1(d).

We will analyze this problem by using both (1) an approximate analytical model and (2) a full vectorial numerical model of Brillouin coupling. We begin with an analytical model. The Brillouin coupling strength ($G$) is determined by the overlap of the optical and acoustic fields, as well as by electrostrictive coupling parameters. By treating the inner glass boundary of the fiber as a hard reflecting boundary for sound, one finds a set of guided acoustic mode solutions (for further details, see Supplement 1). By using the formalism described in Refs. [13,28,29], one can show that the Brillouin gain coefficient associated with intra- and inter-polarization scattering, denoted as $G^{\parallel }$ and $G^{\perp }$, respectively, stems from the overlap between the dipole (or electrostrictive) forces that act to compress the gas and the acoustic modes within the hollow core.

Using an approximation for the acoustic profiles in terms of Bessel functions and a Gaussian approximation for the optical profile, one finds

The boundary condition of the fiber determines the acoustic mode spectrum, which ultimately shapes the optical Brillouin response. Specifically, this boundary condition constrains the radial displacement to zero at the radius of the hollow core and results in the Bessel zeros in Eq. (1). This condition determines the acoustic resonance frequencies as ${\mathrm{\Omega }}_i=X_{1i}v/R$; further details can be found in Supplement 1. Note that while the excited axial–radial modes $\mathrm{AR}1–\mathrm{AR}N$ satisfy the phase-matching conditions for forward Brillouin scattering, the fundamental longitudinal mode ($L0$) cannot (Fig. 1).

The approximate analytical model can now be applied to the hollow-core fiber to calculate the frequency and the strength of the interaction. The gain in air for the first five resonances is plotted in Fig. 2 along with the optical field and electrostrictive force distributions. The overlap integral of the first mode dominates over the higher order modes because this mode profile is well- matched to the optical forcing profile. The higher order modes have alternating directions of displacement, which, when forced optically in only one direction, results in a reduced coupling strength. Consequently, the gain for the first mode (AR1) is more than 200 times greater than it is for the next higher order mode. Therefore, the simple analytical model predicts a single high-gain resonance at the same polarization as the driving beams (intra-polarization) at a frequency of $\sim 35\mathrm{MHz}$, with a coupling strength of $G^{\mathrm{AR}1}\sim {10}^{-3}{\mathrm{W}}^{-1}{\mathrm{m}}^{-1}$ and a linewidth consistent with the acoustic loss of air at this frequency (e.g., $\sim 3\mathrm{MHz}$ [30]).

 

Fig. 2. Calculated forward Brillouin gain with the magnitude of the acoustic displacement field plotted by the respective gain peak. Inset: Gaussian electric field and resultant electrostrictive force distribution. Calculations assume an inner diameter of 5.65 μm, $Q=8.75$, $\lambda =1550\mathrm{nm}$, $n=1.0003$ for air, the air density is $1.2754\mathrm{kg}/{\mathrm{m}}^3$, and the pressure-wave speed is $343\mathrm{m}/\mathrm{s}$.

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Through experiments that follow, we quantify the Brillouin coupling within the HC-PBF system using a two-color pump–probe method. While this pump–probe method relies on a driven Brillouin interaction, it is important to note that both stimulated and spontaneous Brillouin processes originate from the same physical interaction. Hence, the coupling coefficients obtained through these studies are readily used to predict both spontaneous and stimulated Brillouin dynamics within hollow-core fibers. For further discussion and calculation of spontaneous scattering rates, see Supplement 1.

We determine the strength and character of the Brillouin interaction by using a two-color pump–probe technique, which can also be viewed as a four-wave mixing process. For further details, see Refs. [13,25,31,32] and Supplement 1. The measurement concept is illustrated in Fig. 3; three laser fields $({\omega }_1,{\omega }_2,{\omega }_3)$ are injected into the fiber, and the Brillouin interaction produces a new scattered field (${\omega }_4$) at the output. Pump waves (red) of frequencies ${\omega }_1$ and ${\omega }_2$ are synthesized from a monochromatic laser ($\lambda =1535\mathrm{nm}$) field using an intensity modulation scheme. A probe wave (blue) of a disparate wavelength ($\lambda =1546\mathrm{nm}$) at frequency ${\omega }_3$ is simultaneously injected into the fiber. When a phonon is resonantly driven by the pump, these phonons scatter energy from the probe wave to produce a new scattered light field at frequency ${\omega }_4$. One process through which this scattered wave can be generated is as illustrated in Fig. 3.

 

Fig. 3. (a) Hollow-core fiber with input and output fields indicated and (b) energy-level diagram for the phonon-mediated four-wave mixing process described in this work (both inter- and intra-polarized versions). Each frequency is indicated with a number and a polarization in parentheses. Note that the energy-level diagram represents one specific anti-Stokes interaction; the corresponding Stokes process occurs simultaneously.

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The Brillouin spectral response is obtained by sweeping the pump-modulation frequency through the Brillouin-active resonances of the HC-PBF. Through this process, the magnitude and frequency of the scattered wave (${\omega }_4$) exiting the fiber is measured with heterodyne detection and microwave signal analysis. By polarization resolving the four frequencies, we can separately measure $G^{\parallel }$ and $G^{\perp }$ as functions of acoustic frequency. In other words, we can quantify inter- and intra-polarization coupling strengths and compare with theory. In addition, a frequency-independent background due to Kerr four-wave mixing is expected.

Brillouin spectra obtained from a 1.61 m segment of HC-1550-02 fiber filled with air at atmospheric pressure are seen in Fig. 4, revealing a pronounced resonance that is consistent with the Brillouin-active air mode. Detection of the co-polarized scattered light field versus drive frequency reveals a single 35 MHz Brillouin-active phonon mode with a 4 MHz linewidth [Fig. 4(a)]. This Brillouin signature is absent from the orthogonal polarization [Fig. 4(b)]. Therefore, the intra-polarization scattering from this resonance is consistent with Brillouin coupling to air. Multiple narrower peaks are also observed at various frequencies in both polarizations, which are consistent with Brillouin scattering from the elastic modes supported by the fiber’s silica microstructure, as described in Ref. [13]. Moreover, the inter-polarization spectra [Fig. 4(b)] are consistent with a comparable measurement from Ref. [13], in which the silica response is observed at multiple frequencies and the air response is absent. All of the Brillouin signatures sit atop a nonzero background that results from Kerr nonlinearities.

 

Fig. 4. Forward Brillouin gain measurement for (a) intra-polarization and (b) inter-polarization scattering (solid black line) along with the theoretical prediction for air coupling, neglecting Kerr nonlinearity and the narrow resonances from the elastic modes in the silica (dashed red line). The acoustic signature is absent from the inter-polarization measurement, as expected for acousto-optic coupling with gases.

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From these spectra, the nonlinear Brillouin gain, $G^{\mathrm{AR}1}$, can be determined by using the known optical powers, fiber length, and coupling factor (for further details, see Supplement 1). The peak power of the 35 MHz peak is given by 15 nW. The gain after subtracting the power contribution from Kerr four-wave mixing is then given by $G^{\mathrm{AR}1}=9\times {10}^{-4}{\mathrm{W}}^{-1}{\mathrm{m}}^{-1}$. Therefore, the polarization, frequency, gain, and linewidth are all consistent with guided Brillouin scattering in air. Quantitatively, the forward Brillouin scattering gain given by experiment is 3 times larger than the prediction from the simple theoretical model presented above, which neglects the experimental complexity of the air and silica cladding.

To determine whether the discrepancies between the predicted frequencies and coupling strengths derived from our analytical formulation and experiment are due to our simplification of the fiber geometry, numerical simulations were performed. This full-vectorial model contains the complete (HC-1550-02) hollow-core fiber geometry. The air filling all of the voids in the silica fiber matrix is taken to have a mass density of $1.2754{\mathrm{kg}/\mathrm{m}}^3$ and a sound speed of $343\mathrm{m}/\mathrm{s}$. Using the parameters of Ref. [33] and with additional discussions with these authors, we accurately specify the hollow-core fiber geometry. Using COMSOL Multiphysics solver, we solve for the fundamental optical mode [Fig. 5(a)] as well as for the first excited axial–radial (AR1) acoustic mode of the 2D hollow-core geometry [Fig. 5(b)]. The simulated frequency is 36 MHz and the forward Brillouin gain is calculated and given by $G^{\mathrm{AR}1}=7\times {10}^{-4}{\mathrm{W}}^{-1}{\mathrm{m}}^{-1}$. This simulated gain agrees well with the measured value, which indicates that the silica/air cladding and the exact non-circular shape of the core contributes to the coupling strength. Nonetheless, the analytical model is useful for simple estimates of the coupling strengths and frequencies for arbitrary gases.

 

Fig. 5. Numerical simulations of the (a) near-field intensity profile of the fundamental optical mode and the (b) first excited axial–radial acoustic mode of the air in the core of the hollow-core fiber.

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Comparison with the detailed studies of Ref. [13] reveals that the characteristics of the Brillouin-active air modes are quite different from those produced by the surrounding nanomechanical silica matrix. In general, for glass, the photoelastic coupling is larger, the acoustic dissipation is lower, and the electric field only very weakly overlaps the silica structure. In contrast, the air photoelastic response is weak and air has a large dissipation, but the acousto-optical overlap is almost ideal in the core of the fiber. The net result is a 20 times larger integrated gain for air over that of silica. Therefore, it is only coincidence that in this case that the peak Brillouin gain for air is comparable to that from the silica cladding alone. In addition, the gas response can trivially be changed by modifying the properties of the gas itself.

Building on these studies, it may be possible to shape gas-mediated Brillouin interactions in a variety of ways. To appreciate the potential for enhancement of forward Brillouin scattering in gas, it is important to note that the gain is proportional to the index of refraction of the Brillouin-active gas. For instance, the refractive index increases sharply for wavelengths near one or more absorption resonances of an atomic vapor. While high dispersion is typically accompanied by high absorption, this is not always the case. For instance, strong interactions with atomic resonances can be achieved with low optical absorption in the case of electromagnetically induced transparency (EIT) [34]. Since EIT has been demonstrated in hollow-core fibers filled with rubidium [9,35] as well as with cesium [36] vapors, the study of gas-mediated Brillouin coupling could give rise to some intriguing new dynamics. By combining these methods with the gas-mediated Brillouin interactions described here, it may be possible to dramatically enhance and engineer new forms of forward Brillouin scattering in gas-filled hollow-core fibers.

In conclusion, we have observed a new form of optomechanical coupling in air-filled hollow-core fibers. Forward Brillouin scattering is identified and is accurately predicted by analytical and numerical models. This new optomechanical nonlinearity may be valuable for sensing and spectroscopy and presents a power- and noise-limitation that requires further consideration.