Acousto-optic coupling in 1-D phoxonic potential well nanobeam cavity using slow modes

Abstract

We propose a novel acousto-optic (AO) devise using suspended fishbone nanobeams based on phononic and photonic potential well (PnPW and PtPW). We aim to reduce the traditional device size and eliminate the need for mirror regions in 1-D nanobeam cavity. By manipulating the frequencies of slow sound (SS) and slow light (SL) modes through geometric parameters, we can create potential wells for phonons and photons. The potential wells concentrate the waves inside the resonant cavity simultaneously, leading to an increase in AO coupling rate. We demonstrate the capability of potential wells in improving the AO coupling rate and investigate the impact of both wells. Some distribution of the phonons may affect the photons with a negative contribution, leading to a decrease in the coupling rate. Careful selection of appropriate acoustic and optical modes can overcome this limitation, making this structure a promising candidate for designing AO devices in silicon photonics.

Keywords:

• Acousto-optic coupling
• phononic potential well
• photonic potential well
• silicon photonics
• slow sound
• slow light

1. Introduction

In modern society, various microelectromechanical computing components, composed of integrated circuits, have become an essential part of our daily lives. Nanoscale electronic sensors, filters, and resonators are integrated into chips for practical usage. As the demand for improved computational performance and component miniaturization grows, alternative methods, such as light circuits, are being explored to replace integrated circuits. Over the past decade, quantum computers have propelled such research further, and using light as a replacement for electricity in signal transmission has emerged as an effective approach.^[1–3] In the commonly employed near-infrared communication band, Si has emerged as an excellent optical waveguide material, owing to its extremely low absorption.^[4–6] Thanks to mature semiconductor processing technology, Si can be easily processed into components and integrated onto SiO₂ substrates. Furthermore, the substantial refractive index contrast between Si (n = 3.46) and the surrounding air or SiO₂ substrate (n = 1.46) makes it the preferred optical waveguide material.^[4–6]

In the field of designing integrated light circuits employing silicon photonics, the demand for a variety of circuit elements is also evident in optical circuits. For instance, while electrons can be stored in capacitors, photons can only prolong their lifetime through resonant cavities^[7–11] or decrease the speed of light in structures utilizing metamaterials.^[12–16] In contrast to electrons that transmit signals in conventional circuits, photons that transmit signals in optical circuits require bandwidth reduction and polarization manipulation through filters.^[17–21] Switching components, accomplished via transistors in computational circuits, can also be achieved in integrated light circuits using external electrical, thermal, or acoustic signals.^[22–25] A commonly used approach when designing light circuit components is controlling light waves using sound waves. By inputting elastic waves, conditions of light wave propagation in acousto-optic (AO) devices can be rapidly and effectively altered by inducing deformation and internal strain in the structure, thereby achieving swift control.^[26–31] Optomechanical devices can also convert optical signals into mechanical signals for output when reading optical signals, providing a more versatile way of reading signals in light circuits.^[32,33]

As early as 2009, Matt et al. proposed a one-dimensional optomechanical crystal that could be integrated into a planar lightwave circuit.^[26] They designed a 1-D photonic crystal by creating periodic rectangular holes in a long and narrow silicon waveguide, forming a resonant cavity by changing the lattice constant in the central region and fixing the lattice constant in the mirror region on the outside to form reflective area. Through experiments, they confirmed that photons at 200 THz and phonons at 2 GHz can be coupled in the structure. Since then, 1-D optomechanical waveguides have been extensively studied. Initially, most of the research used similar hole-digging methods to design optomechanical waveguides. However, they were limited by the difficulty of generating phonon band gaps for mirror regions. In 2011, Pennec et al. began adding wing-shaped structures extending outward to the outside of the hole-dug silicon photonic waveguide to make it easier to generate phonon band gaps in the optomechanical crystal.^[27] In 2012, Hsiao et al. changed the position of air holes originally in the center of the waveguide to two semicircles on both sides of the waveguide, thus designing a fishbone-shaped phoxonic crystal.^[28] This design greatly enhanced the AO coupling by using photons and phonons with slow group velocities generated in the structure. In 2017, Chiu et al. designed a defect with a gradually varying period constant in the fishbone-shaped structure, forming a high-efficiency AO coupling resonant cavity and analyzed the relationship between acoustic modes and AO coupling rate.^[29] Most recently, in 2022, Daniel et al. integrated the structure of surface acoustic waves and designed an on-chip electro-opto-mechanical system at room temperature. In the experiment, the microwave signal was successfully converted to an optical wave signal with high sensitivity.^[31]

In these studies, the mirror regions that possess both photonic and phononic bandgaps play a crucial role in each structure. The number of periods in the mirror region should be at least five or more to effectively utilize the bandgap and form a reflective area on both sides of the cavity. To ensure smooth entry of photons and phonons into the cavity region through the mirror region and to reduce losses, the geometric parameters of the structure should be gradually changed over several periods. Therefore, most acousto-optic structures require a length of about 10 µm. This study aims to provide an alternative approach to designing the resonant cavity and overcoming the limitations of using mirror regions. The structure will utilize slow light (SL) and slow sound (SS) modes to enhance the AO coupling efficiency and phononic potential wells (PnPW) and photonic potential wells (PtPW) to confine photons and phonons within the resonant cavity.

In traditional quantum wells, electrons are confined to specific regions by exploiting the discrete energy levels available in the material. The energy of both electrons and photons can be calculated using the Planck-Einstein relationship, E = hν, with their energy levels represented by frequency. Similarly, phonons can be distinguished by frequency to indicate their relative energy levels.^[34] In our previous study, a 1-D waveguide with a PnPW that employs phonons with different frequencies was designed.^[35] When the frequency of the acoustic mode in the unit cell changes due to geometric parameters, it can be treated as having corresponding phonon energy levels, similar to the electron energy level in dielectric materials. By properly combining multiple unit cells with different geometric parameters, PnPW can be established using the phonon energy levels in each unit of the structure. Using a similar approach, PtPW can also be designed in the cavity by combining the photon energy levels in the structure.

In this study, we aim to design a phoxonic crystal waveguide device that replaces the traditional mirror region with PnPW and PtPW simultaneously, leading to the reduction in the size of the component. The paper’s structure is organized as follows: Section 1 introduces the unit cell, geometric parameters of the fishbone structure, as well as the SL and SS modes used in this study. In Section 2, we describe the quantification of the AO coupling rate g value used in the AO devices, which is determined by the frequency shift of the SL mode due to the moving boundary (MB) effect and the photo-elastic (PE) effect. Section 3 discusses the impact of PnPW on the g value in the structure without PtPW. In Section 4, we introduce the design with PtPW and further analyze its effect on the g value. Finally, we summarize the conclusions of this study.

2. Structure units and mode analysis

We utilized a 1-D nanobeam in a fishbone shape as the carrier for SS and SL transmission. By periodically excavating air holes on both sides of a silicon nanobeam and elongating the regions between the holes into wing-like structures, the fishbone shape was formed. Figure 1(a) displays the top view (top) and side view (bottom) of the unit cell in a periodic fishbone structure, and labels the corresponding geometric parameters. To control the wavelength of the optical wave mode to be near 1,550 nm, the lattice constant a was set to 440 nm, and the radius of the excavated air holes r was set to 0.32a to achieve the acoustic wave mode close to the optical wavelength. Additionally, the thickness of the nanobeam H was set to 220 nm, and the base width of the nanobeam w was set to 440 nm.

Figure 1. (a) The top view and side view of the fishbone unit cell with geometric parameters. The (b) phononic band structure and (c) photonic band structure are shown by blue circles with the SS mode and SL mode indicated by red circles and red arrow respectively in the corresponding figure. (d) The displacement field distribution of the SS mode and (e) the electric field in x-component of the SL mode are depicted below the band structures.

In order to facilitate the modulation of the wave modes of light and sound, we have further defined the length of the fishbone wing, dw, and the lattice constant, a′, as illustrated in Figure 1. Both parameters are adjusted based on the basic width, w, and the basic lattice constant, a. Figure 1(b,c) respectively depict the phononic and photonic band structures of the structure with a′= a and dw = 0.5w, with the operating SS mode and SL mode marked in red. By applying periodic boundary conditions on the upper and lower surfaces of the unit cell in the z-direction depicted in the top view of Figure 1(a), we can use the finite element method (FEM) to calculate the eigenmodes of the structure under different k-vector conditions and obtain the energy band structure.

Based on the definitions of the horizontal and vertical axes in the band structure, the group velocity of each mode can be calculated using the first derivative at each point. In Figure 1(b), the SS mode is marked with red circles and its displacement field is represented in Figure 1(d). Observing Figure 1(d), it is evident that the SS mode is primarily driven by the vibration of the wings, while the central region of the nanobeam experiences almost no displacement in comparison to the wing region. Thus, the resonance frequency of the SS mode can be adjusted by changing the length of the wings, i.e., by adjusting dw. On the other hand, Figure 1(c) shows the photonic band structure of the fishbone structure, where the gray area represents the light cone. Modes located within the light cone would result in speeds exceeding the speed of light in a vacuum when calculating the first derivative, so they are not considered in this study.

The mode indicated by the red arrow is the SL mode in the structure. Although the first derivative of the mode is not close to 0 for k < 0.5, it meets the desired slow-light condition at k = 0.5. This SL mode is primarily controlled by the electric field component in the x-direction, as shown in Figure 1(e). We can analyze it by studying the distribution of this component. Figure 1(e) displays the x component of the electric field of this mode. From the figure, we can observe that the electric field is distributed in the center of the region with holes, and the wavelength is approximately double that of a. Based on the electric field distribution, we can infer that changing the length of the wings will not affect the frequency of the light wave, and the frequency of the SL mode can be modulated by changing the lattice constant of the unit, which is a’.

To validate the capability of modulating SS and SL modes using dw and a′, we plot the variations of SS and SL modes as a function of these two parameters in Figure 2. Figure 2(a) depicts the change in SS and SL modes with dw/w in a structure with a fixed a′/a ratio. To ensure that the frequency shift of modes in different structures with different a′/a ratios exhibit similar trends, we show the frequency shift of SS and SL modes in two structures with a′/a = 1 (solid line) and a′/a = 0.5 (dotted line). As seen in Figure 2(a), the SS mode, indicated by the blue triangle, demonstrates a decrease in frequency with increasing dw in both structures, while the SL mode, indicated by the orange circle, remains almost unchanged. This indicates that adjusting dw (the length of the wings) affects only the SS mode.

Figure 2. The resonance frequency of the SS mode and the SL mode varying with (a) the wing length dw and (b) the lattice constant a’.

On the other hand, Figure 2(b) shows the trend of the SS and SL modes with respect to changes in a′/a in a structure with a fixed dw/w ratio. To ensure that the frequency shift of modes in structures with different dw/w ratios exhibit the same trend, the frequency shift of the SS and SL modes in structures with dw/w = 0 (solid line) and dw/w = 0.5 (dashed line) are displayed. From Figure 2(b), it can be seen that changes in a′ impact both the acoustic and optical wave modes. As the value of a′/a decreases, the frequency of both the SS and SL modes increases, with a larger impact on the optical wave mode. The results show that the modulation method based on the electric field distribution that was mentioned in the previous paragraph is workable. As the modes in the structure are closely related to the geometric structure parameters, we kept the radius of air holes in each unit maintain a ratio of 0.32 with the lattice constant when changing the lattice constant a′. The change in the lattice constant then affects the width of the wing and thus the resonance frequency of the acoustic wave. However, the SS mode is mainly dominated by the length of the wing, and the width of the wing has a relatively minor impact on the SS mode, so adjusting a′ can still serve as a means of modulating the frequency of the SL mode.

3. Acousto-optic coupling ratio

The coupling rate g represents the performance of the structure as an AO device, with a higher value indicating a better ability of sound waves to modulate light waves. In the AO coupling element, the calculation of the g value needs to consider two effects induced by sound waves: the moving boundary (MB) effect caused by the deformation of the structure and the photo-elastic (PE) effect caused by the strain generated in the structure during deformation that changes the local refractive index. When the frequency of the light wave is ω₀, the g value generated by the two effects can be calculated using the following formula and presented as a frequency shift of the light wave mode:^[29] (1) $$g_{MB}=-\frac{{\omega }_0}{2}\frac{\int \left(Q·\widehat{n}\right)\left(\Delta \epsilon E_{\parallel }^2-\Delta {\epsilon }^{-1}{D}_{\perp }^2\right)dS}{\int \mathit{E}·\mathit{D}dV}\sqrt{\mathrm{\hslash }/2m_{\mathit{eff}}{\mathrm{\Omega }}_m}$$(1) (2) $$g_{PE}=-\frac{{\omega }_0}{2}\frac{⟨E|\delta \epsilon |E^{\prime }⟩}{\int \mathit{E}·\mathit{D}dV}\sqrt{\mathrm{\hslash }/2m_{\mathit{eff}}{\mathrm{\Omega }}_m}.$$(2)

In formula (1), Q is the normalized displacement field, $\widehat{n}$ is the normal vector of the structure surface; E is the electric field, D is the electric displacement field, and the subscripts $\parallel$ and $\perp$ denote the components parallel and perpendicular to the structure surface, respectively. Then take the volume integral of the inner product of the E field and D field in the AO cavity. $\Delta \epsilon$ represents the relative permittivity difference between the structural material and the surrounding medium, which is the difference between silicon and air in this paper, and $\Delta {\epsilon }^{-1}$ is the reciprocal difference of relative permittivity. In formula (2), $\delta {\epsilon }_{ij}=-{\epsilon }_0{n}^4{p}_{\mathit{ijkl}}{S}_{kl},$ where n is the refractive index of silicon, $p_{\mathit{ijkl}}$ is the photo-elastic tensor, and $S_{kl}$ is the strain tensor generated by sound waves. Finally, $\sqrt{\hslash /2m_{\mathit{eff}}{\mathrm{\Omega }}_m}$ is the zero-point momentum of the resonant structure, where $\hslash$ is the reduced Planck constant, $m_{\mathit{eff}}$ is the effective mass generated by the sound wave displacement field in the structure, and ${\mathrm{\Omega }}_m$ is the frequency of the sound wave.

The total AO coupling rate of the structure can be obtained by adding the coupling rate produced by the two effects: $$g=g_{MB}+g_{PE}.$$

It can be noticed that the calculated g value can be positive or negative, representing the frequency shift of the optical mode to a higher or lower frequency. When discussing the AO coupling rate in the structure, that is, the modulation ability of sound on light, the |g| value will be taken for discussion, representing the amount of frequency shift of the optical mode.

4. Phononic potential well nanobeam

Using the relationship between resonance frequency and geometric parameters, structures with PnPW and PtPW can be designed by combining units with resonance modes in different frequencies in a fishbone nanobeam. Figure 3(a) illustrates the structure of a fishbone nanobeam cavity comprising seven gradually varying unit cells, with two half-cells located on both sides and five waveguide units connected to the half-cells, respectively. The width of each waveguide unit is w, and the length is a, which is used to set up the waveguides before and after the structure in the simulation. To simulate the suspended nanobeam, the bottom surfaces of the outermost three waveguide units on both ends are fixed boundaries representing the connection surface fixed on the substrate. During the frequency response transmission simulation, the optical signal is introduced from the left of the waveguide and passes through the structure along the nanobeam towards the right side. The acoustic signal is applied from the top surface of the fourth unit on the left, and reflection is avoided by setting the loss factor in the outermost three waveguide units on both ends.

Figure 3. (a) The full sketch of the fishbone nanobeam cavity with parameter denotations. (b) The PnPW line shape of the Type α and β structures. (c) The displacement field of the SS mode at 5.744 GHz and (d) the Ex field distribution of the SL mode calculated by the eigenvalue solver in Type α structure.

To facilitate discussion, we have assigned labels to each unit of the nanobeam. With the center unit 0 as the reference, the units on the right are sequentially numbered from 1 to 3, while those on the left are numbered from −1 to −3, as shown in Figure 3(a). The geometric parameters of the center and right units are denoted as a′₀, a′₁, a′₂, a′₃, and dw₀, dw₁, dw₂, dw₃, respectively, based on their corresponding unit numbers. The geometric parameters of the left units are arranged in a mirror-image relationship with those on the right, as depicted in the figure.

In this section, we aim to investigate the influence of PnPW by maintaining constant lattice constants in the nanobeam (a’₀ = a’₁ = a’₂ = a’₃ = a) and varying only the wing lengths of each unit. We propose two PnPW fishbone structures, denoted as Type α and Type β. In Type α, the wing lengths are dw₀ = 0.3w, dw₁ = 0.2w, dw₂ = 0.1w, and dw₃ = 0. The center unit has the longest wing length, which gradually decreases to both sides until the −3 and 3 units with a base width of w, and then passes through an additional half unit to connect with the outer waveguide unit. In the Type β structure, the wing length of the central unit increases to 0.5w, and it decreases at intervals of 0.2w until the −3 and 3 units, and then connects to the waveguide unit (dw₀ = 0.5w, dw₁ = 0.3w, dw₂ = 0.1w, and dw₃ = 0) through additional half units. Table 1 presents the detailed parameters of both structures. By utilizing symmetrically varying geometric variables, we can regulate the frequency of the SS mode in each unit to exhibit the valley trend, as shown in Figure 3(b), and form the PnPW. The PnPW of Type α and Type β structures are represented by blue circles and orange triangles, respectively, in Figure 3(b). It is evident that the potential well of Type β has a greater depth than Type α within the same well width range. Our previous studies demonstrate that this design can enhance the Q value of the acoustic mode in the structure,^[35] which is expected to improve the efficiency of AO coupling.

Table 1. Geometry parameter ratio of all the six types of fishbone structures.

	Type α		Type β		Type A		Type B		Type C		Type D
Unit number	a‘/a	dw/w	a‘/a	dw/w	a‘/a	dw/w	a‘/a	dw/w	a‘/a	dw/w	a‘/a	dw/w
0	1	0.3	1	0.5	0.9	0.3	0.9	0.5	1.1	0.3	1.1	0.5
−1/1	1	0.2	1	0.3	0.8	0.2	0.8	0.3	0.9	0.2	0.9	0.3
−2/2	1	0.1	1	0.1	0.7	0.1	0.7	0.1	0.7	0.1	0.7	0.1
−3/3	1	0	1	0	0.6	0	0.6	0	0.6	0	0.6	0

Through the frequency response analysis, we identified the SS mode in Type α structure at 5.744 GHz in the acoustic spectrum and the SL mode at 203.54 THz in the optical spectrum, which exhibit field distributions similar to those illustrated in Figure 1(d,e), respectively. The displacement field distribution of the SS mode and the E_x field distribution of the SL mode are shown in Figure 3(c,d), respectively. Figure 3(c) illustrates that the SS mode is primarily localized in the center units (−1 ∼ 1) of the fishbone structure. The displacement field reveals that the maximum displacement is observed at the end of the wings, while there is a relatively small amount of displacement at the center of the fishbone. In Figure 3(d), the E_x field is observed to be uniformly distributed throughout the fishbone structure. By utilizing Eqs. (1)(1) $$g_{MB}=-\frac{{\omega }_0}{2}\frac{\int \left(Q·\widehat{n}\right)\left(\Delta \epsilon E_{\parallel }^2-\Delta {\epsilon }^{-1}{D}_{\perp }^2\right)dS}{\int \mathit{E}·\mathit{D}dV}\sqrt{\mathrm{\hslash }/2m_{\mathit{eff}}{\mathrm{\Omega }}_m}$$(1) and (2)(2) $$g_{PE}=-\frac{{\omega }_0}{2}\frac{⟨E|\delta \epsilon |E^{\prime }⟩}{\int \mathit{E}·\mathit{D}dV}\sqrt{\mathrm{\hslash }/2m_{\mathit{eff}}{\mathrm{\Omega }}_m}.$$(2) , we calculated g_MB = −0.1039 MHz, g_PE = −0.0057 MHz, and a total coupling rate of |g| = 0.1096 MHz for this structure. Based on the obtained g values, we found that the modulation effect produced by the MB effect is greater than the PE effect when the SS mode and the SL mode in Figure 3(c,d) are coupled, indicating that the structural displacement caused by the acoustic mode in Figure 3(c) exhibits better modulation ability than the refractive index change induced by the strain distribution in the structure.

In Figure 4, the SS and SL modes in the Type β fishbone are observed at the frequency of 5.137 GHz in the acoustic spectrum and the frequency of 204.3 THz in the optical spectrum, respectively. Similar to Type α, Figure 4(a) shows that the displacement field of the SS mode is concentrated within the central −1 ∼ 1 unit, while the E_x field in Figure 4(b) is evenly distributed within the fishbone structure. In the Type β structure, the values of g_MB, g_PE, and the total coupling efficiency |g| are −0.5774 MHz, −0.0141 MHz, and 0.5633 MHz, respectively. Numerically, it is evident that optimizing the acoustic mode with a deeper PnPW can effectively enhance the efficiency of AO coupling without optimizing the optical modes. Based on this idea, we further designed for the optimization of the optical modes.

Figure 4. (a) The displacement field of the SS mode at 5.137 GHz and (b) the Ex field distribution of the SL mode calculated by the eigenvalue solver in Type β structure.

5. AO potential well nanobeam

In order to achieve both PtPW and PnPW in the fishbone structure, we manipulated the frequency of the SL mode by adjusting the lattice constant a′ of each unit in the fishbone structure. Similar to the approach used to discuss PnPW, we created two types of PtPWs with the following sets of geometric parameters: a′₀ = 0.9a, a′₁ = 0.8a, a′₂ = 0.7a, a′₃ = 0.6a and a′₀ = 1.1a, a′₁ = 0.9a, a′₂ = 0.7a, a′₃ = 0.6a. It can be noticed that the lattice constant of the outer two unit cells are keep in 0.7a and 0.6a in these two types of design to keep the width of the PtPW. By combining these with the wing length parameters from the last section, we designed a total of four fishbone structures, denoted as Type A, Type B, Type C, and Type D, listed in Table 1. These structures can be classified into two groups based on their PnPW as Type A, C and Type B, D, and two groups based on their PtPW as Type A, B and Type C, D.

Figure 5 shows the PnPW and PtPW of the four structures. Since changing the lattice constant also slightly affects the SS mode, it can be observed that Type A, C and Type B, D, which have the same wing length configuration, do not have exactly the same PnPW in Figure 5(a). It can be seen that structures with a larger lattice constant configuration, such as Type C and D, exhibit relatively deeper wells within the group with the same wing length configuration. On the contrary, the PtPW has shown in Figure 5(b) clearly presents two different depths of potential wells, confirming that the variation of wing length has little influence on the SL mode. The insets show slight differences between the PtPW in the Type A and B structures and the Type C and D structures, respectively.

Figure 5. (a) The PnPW line shape and (b) The PtPW line shape of the Type A to D structures. The inset show the tiny difference between Type A, B and Type C, D.

When deeper PnPW with the same well width have been found to effectively improve the Q value of acoustic modes in our previous studies, a similar effect is expected in PtPW structures, where higher optical quality should be induced by deepening the potential well depth and increasing the coupling rate. In Figure 6, we first analyzed Type A and Type B structures with shallower PtPW. Figure 6 displays the displacement field of the SS mode and the Ex field of the SL mode of Type A and Type B, and their corresponding frequencies are listed in Table 2. It can be observed from Figure 6(a,c) that when a′ is adjusted to control the optical wave, the two PnPWs still concentrate the SS mode within the central three units. In contrast, Figure 6(b,d) illustrates that the PtPW concentrates the SL mode in the central unit (on both sides of the dashed line), while higher-order modes also appear in the outer units. Unlike Type α and β without designed PtPW, the spatial structure along the periodic direction of the PtPW fishbone is no longer uniform, resulting in more resonant conditions that enable higher-order modes to exist in the potential well under the corresponding resonant condition.

Figure 6. (a) The displacement field of the SS mode at 6.283 GHz and (b) the Ex field distribution of the SL mode calculated by the eigenvalue solver in Type A structure. (c) The displacement field of the SS mode at 5.717 GHz and (d) the Ex field distribution of the SL mode calculated by the eigenvalue solver in Type B structure.

Table 2. Resonant frequencies of the SS modes and SL modes and the coupling rates of the six types fishbone structures.

Type	fSS (GHz)	fSL (THz)	gPE (MHz)	gMB (MHz)	g (MHz)
α	5.7436	203.54	−0.0057	−0.1039	−0.1096
β	5.1373	204.30	0.0141	−0.5774	−0.5633
A	6.2833	216.68	0.0104	−0.8789	−0.8685
B	5.7173	216.92	0.0132	−0.9118	−0.8986
C	5.7717	201.76	0.0764	−1.2850	−1.2086
		214.76	−0.0239	−1.2050	−1.2289
D	5.2173	201.80	0.0406	−0.5917	−0.5511
		214.88	−0.0083	−0.6713	−0.6796

Using Eqs. (1)(1) $$g_{MB}=-\frac{{\omega }_0}{2}\frac{\int \left(Q·\widehat{n}\right)\left(\Delta \epsilon E_{\parallel }^2-\Delta {\epsilon }^{-1}{D}_{\perp }^2\right)dS}{\int \mathit{E}·\mathit{D}dV}\sqrt{\mathrm{\hslash }/2m_{\mathit{eff}}{\mathrm{\Omega }}_m}$$(1) and (2)(2) $$g_{PE}=-\frac{{\omega }_0}{2}\frac{⟨E|\delta \epsilon |E^{\prime }⟩}{\int \mathit{E}·\mathit{D}dV}\sqrt{\mathrm{\hslash }/2m_{\mathit{eff}}{\mathrm{\Omega }}_m}.$$(2) , we can calculate the AO coupling rate of Type A and Type B. In Type A structure, g_MB = −0.8789 MHz, g_PE = 0.0104 MHz, and the total coupling efficiency |g| = 0.8685 MHz. In the Type B structure, the values of g_MB, g_PE, and the total coupling rate |g| are −0.9118 MHz, 0.0132 MHz, and 0.8986 MHz, respectively, as presented in Table 2. By comparing the two structures, we can observe that the deeper potential well of the PnPW design can still effectively increase the g value of Type B. Additionally, comparing Type A and B with Type α and β, we can infer that the incorporation of PtPW to the original PnPW design can significantly enhance the g value. However, we can also observe that the enhancement of PnPW to the g value in the Type A and B group is diluted under the influence of PtPW compared to the Type α and β group.

Next, we increased the depth of PtPW and designed two fishbone structures, Type C and Type D. Figure 7 displays the displacement field of the SS mode and the E_x field of the SL mode of the Type C structure. With the deeper potential well providing more space to accommodate modes, we obtained two modes that are concentrated in the cavity by the potential well through the frequency response solver, which are the first-order mode at 201.76 THz in Figure 7(b) and the second-order mode at 214.76 THz in Figure 7(c). As the PtPW in the Type C and Type D structures is deeper than in the Type A and Type B structures and the ±1 units in Type C and Type D have the same a′ as the 0 units in Type A and Type B, which means the same optical resonance condition, the E_x field distribution shows that the first-order mode mainly exists on both sides of the central 0 unit (both sides of the dashed lines), and the second-order mode mainly exists on both sides of the ±1 unit, as indicated by the arrows in Figure 7(c). By coupling the two optical modes with the acoustic modes, the respective coupling rates |g₂₀₁| = 1.2086 MHz and |g₂₁₄| = 1.2289 MHz are obtained, both of which show considerably high coupling rates. This result preliminarily demonstrates that increasing the depth of PtPW indeed has a positive effect on the overall g value and can significantly enhance it, leading to better optimization results.

Figure 7. (a) The displacement field of the SS mode in Type C structure at 5.772 GHz. The E_x field distribution of (b) the first-order SL mode and (c) the second-order SL mode calculated by the eigenvalue solver in Type C structure are shown. The dashed line in (b) represents the symmetry axis of the cavity, and the arrows in (c) indicate the center of the −1 and 1 unit cells.

The displacement field of the SS mode and the E_x field of the SL mode for the Type D structure are presented in Figure 8. As in the Type C structure, the deeper PtPW in the Type D structure results in two SL modes, located at 201.8 THz and 214.88 THz, respectively, as shown in Figure 8(b,c). The first-order mode in Figure 8(b) primarily exists on both sides of the dashed line in the central 0 unit, while the second-order mode in Figure 8(c) shows an over concentrated distribution. In the C and D groups with the deeper PtPW, the Type D structure also features a deeper PnPW than Type C and is expected to have the highest coupling rate. However, the coupling rate induced by the two SL modes is only |g₂₀₁| = 0.5511 MHz and |g₂₁₄| = 0.6796 MHz, respectively. Table 2 shows that the g value of Type D is not only lower than that of Type C but also lower than that of Type A and B, with only higher values than that of Type α and β. This may imply that there are some limitations to confining acoustic and optical waves solely by potential wells.

Figure 8. (a) The displacement field of the SS mode in Type D structure at 5.2173 GHz. The E_x field distribution of (b) the first-order SL mode and (c) the second-order SL mode calculated by the eigenvalue solver in Type D structure are shown. The dashed line in (b) represents the symmetry axis of the cavity.

To explain the abrupt reduction in g, we can observe the dominant factor g_MB in Eq. (1)(1) $$g_{MB}=-\frac{{\omega }_0}{2}\frac{\int \left(Q·\widehat{n}\right)\left(\Delta \epsilon E_{\parallel }^2-\Delta {\epsilon }^{-1}{D}_{\perp }^2\right)dS}{\int \mathit{E}·\mathit{D}dV}\sqrt{\mathrm{\hslash }/2m_{\mathit{eff}}{\mathrm{\Omega }}_m}$$(1) that affects the coupling rate. The main factors affecting g_MB are the electromagnetic (EM) energy and the displacement perpendicular to the surface. Since the displacement fields in Types A, B, C, and D are similar, we can focus on the EM energy in each structure. To visualize the energy distribution of the modes, we utilized the eigenmode solver to compute the existence form of each mode in every structure, as illustrated in Figure 9. We normalized the energy of each structure by that of Type D and restricted the color bar to the range of 0 to 1. Consequently, areas with values above 1 will appear darker as the numerical values increase, enabling us to compare the structures more easily. It is evident that in Type α and β, without the PtPW, the energy is evenly distributed in the nanobeam structure, whereas in Types A, B, C, and D, the energy is more focused on unit 0.

Figure 9. The EM energy of the SL modes in all the six types of fishbone structures. The field distributions represent the corresponding structure that denoted at the left bottom corner of each figure, which are (a) Type α, (b) Type β, (c) Type A, (d) Type B, (e) Type C in 201 THz, (f) Type D in 201 THz, (g) Type C in 214 THz, and (h) Type D in 214 THz.

Examining the six SL modes in structures A, B, C, and D, it is apparent that the EM energy in the Type C structure is stronger than in the other structures, consistent with its highest coupling rate. Comparing the two Type C modes in Figure 9(e,g) and the two Type D modes in Figure 9(f,h), it is also evident that the 201 THz mode has a stronger electric field energy in the central unit, indicating that the 201 THz SL mode is the ground state in the PtPW of Type C and D structures, while the 214 THz mode is the second state. Observing Figure 9(c–f), it can be noted that the EM energy in Type A and B is weaker than that in Type C but greater than that in Type D, explaining why the coupling rate in the D structure is only slightly higher than that in α and β.

From another perspective, we can also analyze the displacement field. Since the electromagnetic energy is distributed in the center of the nanobeam, while the displacement field is mainly distributed at the end of the wings, it can have a significant impact on the AO coupling. Among the four structures, Type B and Type D with longer wings are more susceptible to the influence of the coupling due to the displacement field being distributed further from the center of the nanobeam. The maximum displacement values for the four structures are as follows: Type A: $2.26\times {10}^{-10} m,$ Type B: $2.49\times {10}^{-10} m,$ Type C: $2.4\times {10}^{-10} m,$ Type D: $2.39\times {10}^{-10} m.$ From the numerical values, the maximum displacement caused by acoustic waves is very close. Type B, with a relatively slender wing, has the largest displacement and effectively compensates for the disadvantage of the dispersed distribution of the displacement field and EM field. Conversely, the relatively small displacement in Type D cannot provide sufficient contribution to the coupling rate, resulting in a low coupling rate.

Therefore, although both PnPW and PtPW can concentrate the corresponding sound field and light field in the resonant cavity, the distribution of the field will have a considerable impact on the coupling rate. Due to the coupling rate is not only affected by the strength and the concentrating rate of the acoustic and optical fields but also the overlap area between the fields, a too-small mode volume of one of the fields may give a negative contribution instead. In other words, though the acoustic mode and optical mode used in this study that can be manipulated respectively benefit from the separated distribution, the less overlapping area between the two fields should reduce the coupling rate. Thus, improving the coupling rate by deepening the potential well also has certain limitations, and the design of the potential well structure in the resonant cavity needs to be carefully considered.

6. Conclusion

In this study, we present a novel approach for designing AO waveguide devices using PnPW and PtPW. By adjusting the geometric parameters, PnPW and PtPW can be designed in a suspended nanobeam based on the different resonant conditions of phonons and photons. The use of PnPW and PtPW allows for the concentration of acoustic and optical waves inside the resonant cavity, eliminating the need for mirror regions and reducing device size. Our numerical calculations show that the PnPW structure has a significant impact on improving the AO coupling rate, and combining it with PtPW has the potential to further enhance the coupling rate. However, our results also reveal that when the phonons are too concentrated by the PnPW structure, it may affect the concentration of photons by the PtPW, leading to a decrease in efficiency in certain structures. Furthermore, differences in the distribution areas of phonon and photon modes can also result in decreased coupling rate. By carefully selecting appropriate acoustic and optical modes, we believe that this structure has tremendous potential for AO device design in silicon photonics.

Disclosure statement

The authors report there are no competing interests to declare.