1. Introduction

Forward stimulated Brillouin scattering (FSBS) is an acousto-optic interaction between co-propagating pump and Stokes fields through an acoustic wave. Different from the conventional backward stimulated Brillouin scattering induced by the longitudinal acoustic mode [1], the transverse acoustic mode involves the interaction between the pump and Stokes in forward process [2]. It has been theoretically studied in a fiber with a nano-structured core [3] and demonstrated experimentally in photonic crystal fibers (PCFs) [4–6]. Moreover, PCFs with a multi-scale photonic crystal structure are numerically and experimentally studied to realize guided acoustic wave Brillouin scattering [7]. By the controllable acousto-optic interactions, FSBS arising from the transverse acoustic guided mode is widely studied [8,9]. Therefore, the realization of on-chip FSBS is essential and promising for the applications in integrated photonic platforms. Recent works have reported that in phoxonic crystal waveguides [10,11] the propagation of photons and phonons can be controlled effectively by introducing periodic holes in the medium, giving rise to the confinement of optical and acoustic wave simultaneously [12–14] and enhanced acousto-optic interaction in tiny volumes. This provides the possibilities for the realization of FSBS on chip. However, the phoxonic waveguide is constituted by the only one material with artificial periodic structure. It means that both the properties of optical and acoustic modes are determined by the same waveguide geometry. The optical and acoustic dispersion relationships cannot be independently engineered and the phase-matching condition is difficult to be satisfied, which limits the applications of phoxonic crystals. Hybrid phononic-photonic waveguides [15,16], which are studied for optical mode conversion [15] and control of coherent information [16], are favorable to solve this issue with flexible tunability of acoustic and optical modes. The compound-material device comprises two kinds of material and provides independent control of the optical and acoustic dispersion relationships. This separate control over the optical and acoustic modes improves the structural tunability. Moreover, FSBS can be enhanced greatly with the combination of electrostrictive force and radiation pressure in nanowire waveguides [17–21] and hybrid waveguides [15,22]. It provides an efficient approach to achieve FSBS on chip.

In this paper, we investigate FSBS in hybrid phononic-photonic waveguides which are designed to independently control the optical and acoustic guided modes. To analyze the influences of waveguide structures on the acoustic field, three different kinds of hybrid phononic-photonic waveguides are considered and studied, including the square, hexagonal and honeycomb cases. The honeycomb hybrid waveguide is fabricated on a silicon-on-insulator wafer with a total length of 1 cm. We experimentally demonstrate the FSBS in honeycomb phononic-photonic waveguides. A pump-probe measurement scheme is performed to measure the Brillouin frequency shift. It presents that FSBS operates at 2.425 GHz with the acoustic Q-factor 1100. Through the numerical simulation, the acousto-optic interaction based on FSBS is analyzed and discussed under different parameters in terms of pump power, acoustic loss, nonlinear optical loss and lattice constant.

2. Principle and theory

Figure 1 shows our designed hybrid phononic-photonic waveguides for generating FSBS. The z-direction rectangular silicon (Si) waveguide is embedded in the suspended silicon nitride (Si₃N₄) phononic crystal slab. The optical mode is determined by Si waveguide while the acoustic mode is determined by the Si₃N₄ phononic crystal slab. This separate control over the optical and acoustic modes significantly increases the tunability of the whole structure. In FSBS, the properties of acoustic guided wave extremely depend on the waveguide structure. Thus we concentrate on the study of acoustic field in these hybrid waveguides where three different kinds of Si₃N₄ slab, the square, hexagonal and honeycomb cases, are explored respectively (Fig. (1)). Meanwhile, the Si waveguides are in the same dimensions to ensure the uniformity of optical properties in each hybrid waveguide. The basic geometrical parameters of such a structure are the air radius $r$, the lattice constant $a$, the Si waveguide width $b$, the Si waveguide height $h$ and the central waveguide width $w$.

 

Fig. 1 Proposed hybrid phononic-photonic waveguides with different phononic crystal structures. (a) The hexagonal structure. (b) The square structure. (c) The honeycomb structure.

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In this section we outline a universal analytical model for the generation of intra-mode FSBS. The optical pump wave at frequency $w_p$ and Stokes wave at frequency $w_s$ are launched into the waveguide with the same direction. And then the beat pattern between the two optical waves will excite the acoustic eigenmode of hybrid waveguides with frequency $\Omega$. This produces a moving density wave, which changes the refractive index, leading to the scattering of pump wave. To keep sustainable translation for enhancing FSBS, the optical and acoustic waves should satisfy the energy and momentum conservations $\Omega ={\omega }_p-{\omega }_s$ and $q(\Omega )={\beta }_p({\omega }_p)-{\beta }_s({\omega }_s)$. Here $\beta (\omega )$ and $q(\Omega )$ are the optical and acoustic propagation constant respectively. The coupled-mode equations [23,24] that govern the acousto-optic interaction between propagating modes are as follows:

3. Optical and acoustic guided modes

A key point for the future applications based on the acousto-optic interaction is to reduce the propagation losses. From the photonic point of view, the Si waveguide is inserted in a Si₃N₄ phononic crystal slabs where total internal reflection between Si (n = 3.45) and Si₃N₄ (n = 2.05) tightly confines the optical modes in the Si waveguide core. We choose the same Si waveguide width $b=800$nm and height $h=220$nm in hybrid waveguides.

We compute optical properties by the finite element method (FEM). The software used for simulation is COMSOL Multiphysics. The computed $E_x$ field profile of quasi-transverse electric (TE) optical mode at the telecommunication wavelength of 1550 nm is shown in Fig. 2(a). The silicon waveguide with the lower contrast with SiN is beneficial to the excitation of the single mode propagation. The acousto-optic interaction will be enhanced thanks to the large overlap between the acoustic and optical field. For FSBS, the optical forces which are responsible for driving the acousto-optic interaction are almost entirely transverse [19,22]. Optical forces contain electrostrictive force [17] and radiation pressure [19], as shown in Figs. 2(b)-2(c). The electrostriction force is localized within the center of the Si waveguide and it induces the antisymmetric distribution with respect to the central line (y-axis) of the waveguide. Radiation pressure, resulting from the scattering of light at boundaries, localizes near the discontinuous dielectric boundary of the step-index waveguide. It is shown that both the two optical forces contribute to the generation of the acoustic wave and act to push the Si waveguide in opposite directions (x-axis), which drives the acoustic field to vary in x-axis. Electrostrictive force is dominant in the process of acousto-optic interaction. This is attributed to the decrease of radiation pressure as waveguide size increases [17]. The corresponding acoustic guided mode is excited and largely confined in hybrid waveguides. Figures 2(d)-2(f) show the displacement distributions (x-displacement) of transverse acoustic mode in three kinds of hybrid phononic-photonic waveguides with$a=1850\mathrm{nm}$ and $w=1700\mathrm{nm}$. The acoustic modes are computed using the COMSOL Multiphysics software. The structure within the simulation is the super-cell which consists of the central solid core and bilateral phononic crystal structure (Figs. 2(d)-2(f)). It is only repeated in the z direction and the periodic boundary conditions are applied along the z direction. This is different from the simulation of unit cell in which two dimensional periodic boundary conditions are applied at the lateral sides [27]. Here the local velocities for longitudinal acoustic wave are chosen to be $v_{\text{Si}}=8430\text{m/s}$ and $v_{{\text{Si}}_{\text{3}}{\text{N}}_{\text{4}}}=11000\text{m/s}$ [28], with material densities ${\rho }_{\text{Si}}=2329$kg/m³, ${\rho }_{{\text{Si}}_{\text{3}}{\text{N}}_{\text{4}}}=3440$kg/m³. It is noted that three kinds of structures all support the transverse acoustic modes in the Si waveguide. However, as shown in Fig. 2(d), the acoustic field leaks out from the central waveguide, indicating the weakly lateral confinement of the hexagonal phononic crystals for the acoustic wave. For two other structures the lateral leakage of acoustic field is reduced obviously (Figs. 2(e)-2(f)). Furthermore, compared to the square case the more acoustic energy is confined in the honeycomb case, meaning that acousto-optic interaction is more efficient. Therefore, we focus on the honeycomb case in the following study.

 

Fig. 2 (a) The $E_x$ field profile. (b) The x-component of electrostrictive force. (c) The x-component of radiation pressure. (d-f) Acoutic displacement fields (x-displacement) of (d) the hexagonal hybrid waveguide with $r/a=0.25$ ($\Omega /2\pi =2.63\mathrm{GHz}$), (e) the square hybrid waveguide with $r/a=0.46$ ($\Omega /2\pi =2.43\mathrm{GHz}$) and (f) the honeycomb hybrid waveguide with $r/a=0.25$ ($\Omega /2\pi =2.38\mathrm{GHz}$).

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Figure 3(a) displays phononic dispersion relations of the defect-based honeycomb hybrid waveguide with $a=1850\mathrm{nm}$. The transverse acoustic guided mode (blue dots) locates inside the phononic bandgap (red region) around the frequency of 2.4 GHz. This is beneficial to reducing the acoustic loss resulting from the prohibition of coupling to other modes in fabrication defects [13]. Figures 3(b)-3(d) show the displacement fields (x- component) of three acoustic eigenmodes with the acoustic propagation constant $q=1.0\times {10}^4$ m⁻¹,$q=3.4\times {10}^5$ m⁻¹,$q=1.0\times {10}^6$ m⁻¹. It shows that the transverse acoustic mode tends to degrade gradually as the acoustic propagation constant increases. Therefore, FSBS is more likely to be produced with small acoustic propagation constant. The corresponding 3D acoustic displacement field is computed to further obtain the displacement distribution of the acoustic mode over the whole waveguide (Fig. 3(e)). It shows that although the structure of the hybrid waveguide is periodic in the z direction, the acoustic displacement field in the central waveguide is nearly invariant along the z direction. Therefore, Eqs. (1)-(3) are still applicable to this condition. Figure 3(f) shows distribution of elastic strain energy density. It is well confined in the Si waveguide where the optical mode is guided.

 

Fig. 3 (a) Phononic dispersion relation of the honeycomb hybrid waveguide for the acoustic guided mode. (b-d) Displacement field of three eigenmodes with acoustic propagation constant $q=1.0\times {10}^4$m⁻¹,$q=3.4\times {10}^5$m⁻¹,$q=1.0\times {10}^6$m⁻¹. (e) 3D acoustic displacement field. (f) Acoustic elastic strain energy density.

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4. Experimental results

Figures 4(a)-4(c) show the scanning electron micrograph (SEM) images of the hybrid phononic-photonic waveguide. Our device is fabricated on a standard silicon-on-insulator wafer with a silicon structure layer of 220 nm.

 

Fig. 4 (a) SEM image of the fabricated device. (b) Close-up SEM view of the hybrid phononic-photonic waveguide. (c) SEM image of the cross-section of waveguide (d) Experimental setup to measure FSBS in the hybrid phononic-photonic waveguide.

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The Si waveguide is patterned with electron-beam lithography (EBL) and etched with inductively coupled plasma (ICP) etching. A 250 nm Si₃N₄ layer is deposited on the whole sample to fill the etched grooves on the both sides of Si waveguide. Then the EBL and ICP etching are performed to thin the Si₃N₄ atop Si waveguide. Hot phosphoric acid etch is used to ensure that the remaining Si₃N₄ atop the Si waveguide is cleared completely. The phononic crystal structures are also patterned with the EBL and etched with ICP etching. Finally, the device is suspended by soaking it in hydrofluoric acid to remove the silica. Note that each suspended waveguide section is separated by a 5μm unsuspended region to avoid the collapse of suspended waveguide. The cross-sectional dimensions of the Si waveguide are 800 nm × 220 nm ($b$ $\times$ $h$). The total length of the hybrid waveguide is 1 cm with the central waveguide width of 1700 nm. The air radius is $r=0.25a$ where $a$ is 1850 nm.

The experimental setup to measure FSBS in the phononic-photonic waveguide is shown in Fig. 4(d). In the upper arm, a tunable continuous wave (CW) laser is amplified by an erbium-doped fiber amplifier (EDFA) to serve as a pump. In the lower arm, the Stokes is generated by the multi-longitudinal-mode erbium-doped fiber laser (MLM-EDFL) which is constructed by a ring cavity configuration in the dotted box (Fig. 4(d)). In the ring cavity the length of erbium-doped fiber (EDF) is 3.8 m. The MLM-EDFL will oscillate at the center wavelength of the tunable optical filter (TOF). The wavelength division multiplexer (WDM) is used to couple the 980 nm pump light and the MLM-EDFL output is extracted from the 3 dB coupler. The polarization controller positioned in the experimental setup is utilized to couple both the pump and the Stokes simultaneously to the certain polarization state that aligns to the polarization state required by the hybrid phononic-photonic waveguide to realize effective forward stimulated Brillouin scattering. The polarization state of the input light is adjusted to be x-polarized to optimize the polarization state. The radio frequency (RF) signal is detected using photodetector (PD) and observed by the electrical spectrum analyzer (ESA). The optical spectrum analyzer (OSA) is utilized to measure the optical response. The optical power is measured by the optical power meter (OPM). The optical waves are coupled into and out of the waveguide by the vertically coupling method through the focusing gratings integrated on the chip. The total loss including the fiber-to-chip coupling loss and the waveguide propagation loss is measured to be 19.3 dB at the wavelength of 1550 nm. The fiber-to-chip coupling loss of input and output is measure to be 16.1 dB by the reference straight waveguide though cutback measurements. Taking into account the total waveguide length of 1 cm, the waveguide propagation loss of $3.2dB/cm$ is obtained.

The RF spectrum of MLM-EDFL before being injected into the hybrid waveguide is shown in Fig. 5(a). It will serve as the Stokes to perform the measurement of FSBS. The frequency of RF spectrum (MLM-EDFL) ranges from 0 GHz to 3 GHz which covers the phononic bandgap to ensure the generation of interaction between pump and Stokes. As the Stokes light from the MLM-EDFL experiences the waveguide loss, the RF self-beating signal is weakened, and RF self-beating signals within low-frequency region are still observed through the electrical spectrum analyzer (Fig. 5(b)). Then the pump wave (25 mW) is coupled into the waveguide together with the MLM-EDFL. It is noteworthy that with the satisfaction of phase-matching condition the optical energy transfers from pump to Stokes while the acoustic guide mode is excited. The Brillouin frequency shift is measured by monitoring the heterodyne interference between the pump and Stokes waves. It generates the strong beat signal at 2.425 GHz which corresponds to the Brillouin frequency shift and locates in the phononic bandgap (Fig. 5(c)). The frequency of the strongest peak is in good agreement with the frequency of transverse acoustic mode in previous simulation. Figure 5(d) displays the enlarged RF spectrum at 2.42 GHz with a full-width at half-maximum of 2.2 MHz. The acoustic Q-factor of the transverse acoustic mode is 1100. The acoustic loss is extracted as $\alpha =\Gamma /v_b$~$2.0\times {10}^7{m}^{\text{-1}}$.

 

Fig. 5 (a) Measured RF spectrum of MLM-EDFL before being injected into the hybrid waveguide. (b) Measured RF spectrum of MLM-EDFL after propagating through the hybrid waveguide without the pump. (c) Measured RF spectrum of the hybrid waveguide with the pump. (d) Zoomed-in RF spectrum at 2.42 GHz. The dot is the experimental value and the solid curve is the fitted.

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5. Discussion

The field distributions of optical and acoustic guided modes and corresponding loss have been obtained. The acousto-optic interaction based on FSBS can be numerically simulated by solving the Eqs. (1)-(3). During the simulation, the nonlinear optical loss factors are $\beta =38{\text{m}}^{\text{-1}}{\text{W}}^{\text{-1}}$ and $\gamma =3352{\text{m}}^{\text{-1}}{\text{W}}^{\text{-1}}$ [16] and the Stokes power is fixed to 0.1 mW.

Figures 6(a)-6(c) show the normalized evolution of optical and acoustic powers in the honeycomb hybrid waveguide with $a=1850\mathrm{nm}$. In Fig. 6(a), the results show that the acoustic power reaches maximum at the length of 4.3 mm where the energy conversion from pump to Stokes is particularly efficient. When high pump power (250 mW) is injected into the waveguide, the conversion efficiency from pump to Stokes is lower than the small pump power (25 mW) case due to the large nonlinear optical loss (Fig. 6(b)). It means that most of the optical energy is expended by the nonlinear optical absorption instead of being used to produce the acoustic wave. When the acoustic loss is 6000 m⁻¹, the variation of the acoustic power is the damped oscillation process with the pump power 25 mW (Fig. 6(c)). This can be ascribed to the reason that the acoustic loss is so small resulting in the reversible conversion between pump and Stokes [15]. Figure 6(d) shows the relationship between the effective length $L$ and pump powers. The effective length $L$ is defined as the waveguide length when the acoustic power first reaches to the maximum value. When the input pump power is lower than 75 mW, the $L$ decreases exponentially with the increase of pump power. It is obvious that the small acoustic loss will effectively reduce the $L$ under the same pump power.

 

Fig. 6 (a-c) Normalized evolution of optical and acoustic powers in the honeycomb hybrid waveguide with (a) pump power 25 mW and acoustic loss 2.0$\times$10⁷ m⁻¹, (b) pump power 250 mW and acoustic loss 2.0$\times$10⁷ m⁻¹, (c) pump power 25 mW and acoustic loss 6000 m⁻¹. (d) Effective length $L$ as a function of the pump power for honeycomb hybrid waveguide under different acoustic losses.

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Figure 7(a) shows that the maximum acoustic powers vary with the optical power in different hybrid waveguides with the same loss parameters and lattice constant ($a=1.85\mu m$). The value of acoustic power will tend to remain constant when the pump power exceeds a certain threshold. The sustained increase of pump power plays little role for the increase of acoustic power. This is attributed to the large nonlinear optical loss which has been discussed in the above. Moreover compared with the square and hexagonal cases, the higher acoustic power is achieved at the same pump power for the honeycomb case due to the stronger confinement of the acoustic wave in honeycomb hybrid waveguide. Figure 7(b) shows the maximum acoustic power varies with $a$ ranging from 1.25 $\mu m$ to 2.05 $\mu m$ under a given pump power 75 mW. For the square case, we can see that the acoustic power is maximum at $a=1.35\mu m$. As $a$ increases for the honeycomb case, the acoustic power reaches to the maximum at $a=1.85\mu m$. The larger lattice constant is more acceptable in the technological fabrication.

 

Fig. 7 The maximum acoustic powers which is normalized vary with (a) the optical powers and (b) the lattice constants.

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6. Conclusion

We have designed hybrid phononic-photonic waveguides for the generation of FSBS. It is presented that the honeycomb hybrid waveguide can support the highly confined optical mode at optical communication wavelength and transverse acoustic mode at the frequency of a few GHz. We experimentally demonstrate that the frequency of FSBS reaches to 2.425 GHz with the acoustic Q-factor 1100. By analyzing the influence of three structures on the acoustic field, we find that the honeycomb hybrid waveguide is a relatively optimum structure for the enhancement of acousto-optic interaction in FSBS with certain structural parameters. Moreover, the acousto-optic interaction is closely related to the pump power due to the nonlinear optical absorption. By optimizing the design of phononic crystal structure and improving the fabrication techniques, we believe that the higher acoustic Q-factor and FSBS frequency shift can be achieved. The proposed hybrid waveguides appear as alternative and promising acousto-optic devices for CMOS signal-processing technologies on chip.