Study on the Acousto-Optic Coupling Effect of a One-Dimensional Hetero-Optomechanical Crystal Nanobeam Resonator

Abstract

The optomechanical crystal nanobeam resonator has attracted the attention of researchers due to its high optomechanical coupling rate and small modal volume. In this study, we propose a high-optomechanical-coupling-rate heterostructure with a gradient cavity, and the optomechanical rates of the single mirror and hetero-optomechanical crystal nanobeam resonators are calculated. The results demonstrate that the heterostructure based on the utilization of two mirror regions realizes better confinement of the optical and mechanical modes. In addition, the mechanical breathing mode at 9.75 GHz and optical mode with a working wavelength of 1.17 μm are demonstrated with an optomechanical coupling rate g₀ = 3.81 MHz between them, and the mechanical quality factor is increased to 3.18 × 10⁶.

1. Introduction

As artificial structures, photonic and phononic crystals are formed by the periodic arrangement of different optical and mechanical parameters with photonic and phononic bandgaps in space, respectively, which can modulate electromagnetic and elastic waves [1,2]. When electromagnetic and elastic waves are simultaneously localized in a micro-nano structure, the interaction between the two waves is significantly enhanced with the increase in state density. As a new type of cavity optomechanical system, optomechanical crystals have both photonic and phononic bandgaps. Furthermore, they have the characteristics of a high optomechanical coupling rate, high quality factor, small modal volume, etc. Recently, significant progress has been made in electromagnetically induced transparency [3], quantum information processing [4], and laser cooling [5,6] to explore the interaction between single optical cavity mode and single mechanical mode. Moreover, optomechanical crystal nanobeam resonant cavities can be used to control and manipulate light at the quantum mechanical level, reduce the propagation speed of light waves, and achieve the slow light effect, resulting in enhanced optical gain, absorption, and nonlinearity per unit length, and many optical devices, such as lasers, amplifiers, detectors, absorption modulators, and wavelength converters, can be miniaturized and have promising applications in the future [7].

For the first time, Maldovan and Thomas [8] proved theoretically that periodic structures can produce both photonic and phononic bandgaps. They analyzed two systems, in which air holes are periodically arranged in a silicon matrix and silicon pillars are periodically arranged in the air. Then, the localization of electromagnetic waves and elastic waves is achieved by introducing point defects. In 2009, Sadat-Saleh et al. [9] not only elaborated on the concept of phoxonic crystals but also analyzed the effects of different lattice types and unit cell sizes on the photonic and phononic bandgaps. In the same year, Eichenfield et al. [10,11] introduced a one-dimensional periodic structure with both phononic and photonic bandgaps into the field of cavity optomechanics for the first time. The concept of optomechanical crystals was proposed. Furthermore, the possibility of optomechanical crystal as an ultrahigh-precision mass sensor was verified by combining theory with experiment. In 2012, Chan et al. [12] used a combination of finite element simulation and numerical optimization to calculate the optomechanical coupling rate of optomechanical crystal nanobeam at first. The optomechanical coupling rate under the effects of the moving boundary effect and photoelastic effect was given on the basis of first-order electromagnetic perturbation theory. Recently, there has been a trend to consider how to significantly improve the optomechanical coupling rate through structural design and optimization, which has become a research hotspot. In recent years, single-mirror optomechanical crystal nanobeam structures such as slit-type [13], fishbone-type [14], and oscillator-type [15] have achieved a high optomechanical coupling rate. However, the photonic and phononic bandgaps are both affected by the structural parameters, and the synchronization optimization is difficult, which limits the further improvement of the optomechanical crystal quality factor and the optomechanical coupling rate.

In the past few years, pioneering work has shown that a heterostructure can increase the degree of freedom of bandgap adjustment, which is beneficial to improve the optomechanical coupling rate. A two-dimensional hetero-optomechanical crystal surface mode resonator cavity capable of confining both acoustic and optical waves was proposed [16], and the acousto-optic coupling in the resonator was investigated. Results showed that the symmetry of the phononic cavity mode was an important factor affecting its strength. A hetero-optomechanical structure was proposed, which utilized two mirror regions to confine the optical and mechanical modes [17]. In addition, the even-symmetric structure could enhance the overlap between electric and strain fields in optomechanical crystals, and the optomechanical coupling rate of 1.31 MHz was obtained. Three different cavity structures including a heterostructured OMC cavity, an optomechanical cavity without heterostructures, and an optomechanical cavity with acoustic radiation shielding were compared. The experimentally obtained optical spectra showed that the second mirror region in the heterostructure could achieve a confinement effect similar to the acoustic radiation shielding and successfully confined the optical and mechanical modes, respectively [18]. Furthermore, a zipper-type optomechanical crystal cavity composed of double hetero-optomechanical crystal nanobeams was proposed [19], and the optomechanical coupling rate of the structure was measured to be 0.73 MHz through experiments. The above studies demonstrated that, by using two kinds of periodic structures, the photonic and phononic bandgaps can be regulated, which eliminates the constraint that only one periodic structure cannot optimize the photonic and phononic bandgaps simultaneously.

Consequently, in this study, a high-optomechanical-coupling-rate structure based on a one-dimensional hetero-optomechanical crystal nanobeam resonator with gradient cavities is proposed. The phononic and photonic band structures of the single-mirror and hetero-optomechanical crystal nanobeam are investigated using the finite element method. The influence of heterostructure on the optomechanical coupling rate is discussed using the calculation method of the optomechanical coupling coefficient. As a result, when the mechanical frequency is 9.75 GHz, the proposed structure has a working wavelength of 1.17 μm, and the optomechanical coupling rate is as high as 3.81 MHz.

2. Model and Theory

2.1. Model Design

A one-dimensional hetero-optomechanical crystal nanobeam structure is proposed, as depicted in Figure 1. Accordingly, two mirrors and a defect region can be characterized. The first and second mirrors have different lattice constants and radii of the central holes and act as optical and acoustic mirrors, respectively. The defective region consists of eight unit cells. The geometric parameters of the model are listed in

Table 1. Geometric parameters of the optomechanical crystal.

Geometry Parameters (nm)
r1	r2	aI	aII	RI	RII	t
91	45	390	331	83	71	230

. In addition, the material of this structure is silicon, the density is 2329 kg/m³, and the elastic matrix is as follows:

$$\mathit{C}=\left(\begin{array}{cccccc}{C}_{11}& C_{12}& C_{12}& 0& 0& 0\\ C_{12}& C_{11}& C_{12}& 0& 0& 0\\ C_{12}& C_{12}& C_{11}& 0& 0& 0\\ 0& 0& 0& C_{44}& 0& 0\\ 0& 0& 0& 0& C_{44}& 0\\ 0& 0& 0& 0& 0& C_{44}\end{array}\right).$$

(1)

Here, C₁₁, C₁₂, and C₄₄ represent three independent elastic constants, which are 165.6 GPa, 63.9 GPa, and 79.5 GPa, respectively [20].

2.2. Theory

The optomechanical coupling of the strong interaction between electromagnetic and elastic waves can be characterized by the frequency shift of the optical resonance mode, which is caused by the zero-point motion of the mechanical field generated by the elastic wave [12]. It is generally recorded as the optomechanical coupling rate g₀, and is defined as

$$g_0={\chi }_{\mathrm{zpf}}\frac{\mathrm{d}{f}_0}{\mathrm{d}\alpha },$$

(2)

where f₀ is the frequency of the optical resonance mode, $\alpha$ is the amplitude of the mechanical displacement, ${\chi }_{\mathrm{zpf}}=\sqrt{\hslash /2m_{eff}{\omega }_m}$ is the zero-point fluctuation displacement, $\hslash$ is the reduced Planck constant, ${\omega }_m$ is the intrinsic angular frequency of the phononic mode, and $m_{eff}$ is the effective mass of the cavity of the phononic mode.

In the optomechanical crystal nanobeam resonator, the mechanical vibration changes the shape of the resonator, such that the frequency of the optical resonant mode changes with the mechanical amplitude. Since the matrix material does not include piezoelectric materials, it is usually necessary to consider the moving boundary effect and the photoelastic effect on the optomechanical coupling rate [21]. The moving boundary effect is a surface effect, where mechanical mode displacements affect the change of the dielectric constant matrix at different interfaces in an environment consisting of materials with different boundaries [22]. The photoelastic effect is a bulk effect, where the photoelasticity causes a change of the dielectric constant due to the change in material strain [23].

Therefore, the total optomechanical coupling rate is the superposition of the optomechanical coupling rate caused by the moving boundary effect and the photoelastic effect, which is recorded as

$$g_0=g_{\mathrm{mb}}+g_{\mathrm{pe}}=\left({\frac{\mathrm{d}{f}_0}{\mathrm{d}\alpha }|}_{\mathrm{mb}}+{\frac{\mathrm{d}{f}_0}{\mathrm{d}\alpha }|}_{\mathrm{pe}}\right){\chi }_{zpf}.$$

(3)

In this formula, g_mb is the optomechanical coupling rate of the moving boundary effect, and g_pe represents the optomechanical coupling rate of the photoelastic effect.

The g_mb is obtained by solving the Maxwell equations of the moving boundary effect via using the perturbation theory, which is denoted as

$$g_{\mathrm{mb}}=-\frac{f_0}{2}\frac{\langle \mathit{E}|\frac{d\epsilon }{d\alpha }|\mathit{E}\rangle }{{\displaystyle \int dV·\epsilon |{\mathit{E}|}^2}}{\chi }_{\mathrm{zpf}}==-\frac{f_0}{2}\frac{{\displaystyle \int \mathrm{d}A\cdot \mathit{q}\cdot \mathit{n}\left(\Delta \epsilon {\left|{\mathit{E}}_{\left|\right|}\right|}^2-\Delta {\epsilon }^{-1}{\left|{\mathit{D}}_{\perp }\right|}^2\right)}}{{\displaystyle \int \mathrm{d}V\cdot \epsilon {|\mathit{E}|}^2}}{\chi }_{\mathrm{zpf}},$$

(4)

where $\langle a|b|c\rangle$ represents the volume fraction on the spatial domain, i.e.,$\langle a|b|c\rangle ={\displaystyle \int {\displaystyle \int {\displaystyle \int a(x)·b(x)·c(x)dx}}}dydz$; q expresses the normalized displacement field, n is the unit normal vector of the unperturbed cavity surface, ${\mathit{E}}_{\left|\right|}$ describes the electric field component parallel to the interface, ${\mathit{D}}_{\perp }$ represents the electrical displacement field component perpendicular to the interface, $\Delta \epsilon ={\epsilon }_1-{\epsilon }_2$ is defined as the difference between the dielectric constants of the dielectric and air, and $\Delta {\epsilon }^{-1}={\epsilon }_1{}^{-1}-{\epsilon }_2{}^{-1}$ is the difference between the reciprocal of the dielectric constants of the dielectric and the air; ${\displaystyle \int \mathrm{d}A}$ indicates all interfaces of the optomechanical crystal nanobeam resonator.

For the same reason, g_pe is obtained by solving the Maxwell equations of the photoelastic effect using the perturbation theory, which is denoted as

$$g_{\mathrm{pe}}=-\frac{f_0}{2}\frac{\langle \mathit{E}|\delta \epsilon |\mathit{E}\rangle }{{\displaystyle \int dV·\epsilon |{\mathit{E}|}^2}}{\chi }_{\mathrm{zpf}}={\chi }_{\mathrm{zpf}}\frac{f_0{\epsilon }_0{n}_0{}^4}{2}{\displaystyle \int \mathrm{d}V\left\{\begin{array}{l}2\mathrm{Re}\left\{E_x^{*}{E}_y\right\}{p}_{44}{S}_{xy}\\ +2\mathrm{Re}\left\{E_x^{*}{E}_z\right\}{p}_{44}{S}_{xz}\\ +2\mathrm{Re}\left\{E_y^{*}{E}_z\right\}{p}_{44}{S}_{yz}\\ +{\left|E_x\right|}^2\left[p_{11}{S}_{xx}+p_{12}\left(S_{yy}+S_{zz}\right)\right]\\ +{\left|E_y\right|}^2\left[p_{11}{S}_{yy}+p_{12}\left(S_{xx}+S_{zz}\right)\right]\\ +{\left|E_z\right|}^2\left[p_{11}{S}_{zz}+p_{12}\left(S_{xx}+S_{yy}\right)\right]\end{array}\right\}}/{\displaystyle \int \epsilon {\left|E\right|}^2\mathrm{d}V},$$

(5)

where n₀ is the refractive index of silicon, S is the strain tensor, and P represents the fourth-order photoelasticity tensor. In the cubic crystal system, it is expressed as

$$\mathit{P}=\left(\begin{array}{cccccc}{p}_{11}& p_{12}& p_{11}& 0& 0& 0\\ p_{12}& p_{11}& p_{12}& 0& 0& 0\\ p_{12}& p_{12}& p_{11}& 0& 0& 0\\ 0& 0& 0& p_{44}& 0& 0\\ 0& 0& 0& 0& p_{44}& 0\\ 0& 0& 0& 0& 0& p_{44}\end{array}\right),$$

(6)

for material silicon, where p₁₁ = −0.094, p₁₂ = 0.017, and p₄₄ = −0.051 [23]. When the photoelastic effect is dominant, the height of the optical field is localized in the dielectric region, the change in refractive index inside the dielectric body has a great influence on the optical mode, and the movement near the slit in the resonator has a relatively small influence on the optical field inside the slit.

3. Results and Discussion

3.1. Band Structure

In this work, the finite element method was used to calculate the photonic and phononic band structures of the first and second mirrors. Figure 2a is the photonic band structure of the one-dimensional hetero-optomechanical crystal nanobeam. The light line is defined as $c\mathit{k}/\omega =1$, c is the speed of light in a vacuum, k is the wave vector, and $\omega$ is the angular frequency. The photonic bandgap of the first mirror is 213.37 THz to 274.36 THz, and that of the second mirror is 226.25 THz to 253.16 THz. The frequency of the optical defect mode is 256.98 THz. From the photonic band structure, the optical defect mode is well localized by the first mirror. Therefore, the constraint of the optical mode can be achieved using the first mirror.

Figure 2b is the phononic band structure of the one-dimensional hetero-optomechanical crystal nanobeam. The phononic bandgap of the first mirror is 6.92 GHz to 8.63 GHz, and that of the second is 7.77 GHz to 10.01 GHz. The mechanical frequency of the breathing mode [11] is 9.75 GHz. In summary, the mechanical breathing mode is localized by the second mirror. Accordingly, the constraint of the mechanical mode can be achieved using the second mirror.

In conclusion, two mirrors of the one-dimensional hetero-optomechanical crystal nanobeam achieve better constraints on the optical mode and mechanical mode.

3.2. Modes and Optomechanical Coupling Rate

To analyze the effect of the heterostructure, the electric field and displacement field distributions of the single-mirror and the hetero-optomechanical crystal nanobeam resonators are compared. Figure 3a shows the electric field distribution of the single-mirror optomechanical crystal nanobeam resonator. The optical frequency of this mode is 257.02 THz. In this optical mode, the electromagnetic wave energy is well localized in the optical resonator and less dissipated in the mirror. Figure 3b is the electric field distribution of the hetero-optomechanical crystal nanobeam resonator. The optical frequency of this mode is 256.98 THz. The two optical modes indicate that the electric field energy is locally concentrated at the defect, and the maximum energy is concentrated at the center of the structure. It can be seen that the single-mirror optomechanical crystal nanobeam resonator is sufficient to localize the electric field energy in an optical resonator.

The mechanical properties of the proposed structures are mainly evaluated by the mechanical quality factor and effective mass of the optomechanical crystal nanobeam resonator. Among them, the mechanical quality factor represents the mechanical energy loss of the optomechanical crystal nanobeam resonator [24]. The effective mass is proportional to the mechanical modal volume [25]. According to Equations (2)–(5), the optomechanical coupling rate of the single-mirror and the hetero-optomechanical crystal nanobeam resonators are calculated. In addition, we analyze the mechanical mode with the largest optomechanical coupling rate in the two structures, as shown in Figure 4.

Figure 4a displays the breathing mode of the single-mirror optomechanical crystal nanobeam resonator. The mechanical frequency is 9.75 GHz, the effective mass is 42.75 fg, and the mechanical quality factor is 1.25 × 10⁵. The energy of the displacement field is well localized in the phononic cavity. Figure 4b illustrates the breathing mode of the hetero-optomechanical crystal nanobeam resonator. The mechanical frequency is 9.75 GHz, the effective mass is 43.29 fg, and the mechanical quality factor is 3.18 × 10⁶. It can be seen from the comparison that the two mechanical vibration modes are the stretching vibration of the resonator cavity to both sides of the beam, and the mode shapes are the same.

Table 2. Mechanical quality factor and optomechanical coupling rate of the single-mirror and the hetero-optomechanical crystal nanobeam resonators.

Structures	gmb (MHz)	gpe (MHz)	g0 (MHz)	Q	meff (fg)
Single-mirror structure	−1.04	−2.64	−3.68	1.25 × 105	42.75
Heterostructure	−1.08	−2.73	−3.81	3.18 × 106	43.29

lists the coupling results of the optical and the breathing mechanical modes of the single-mirror and the hetero-optomechanical crystal nanobeam resonators. The results include the moving boundary effect coupling rate g_mb, the photoelastic effect coupling rate g_pe, the total coupling rate g₀, the mechanical quality factor Q, and the effective mass m_eff. It is obvious that the optomechanical coupling rate of both structures is negative, and the photoelastic effect strongly contributes toward the optomechanical coupling rate, considering its contribution $g_{pe}\infty {\displaystyle \int p_{11}}{S}_{yy}{\left|E_y\right|}^{2}dV$ in this condition, where p₁₁ denotes one of the photoelastic coefficients. The major electric field component E_y and the major displacement field component S_yy are calculated. The electric field energy of the single-mirror and the hetero-optomechanical crystal nanobeam resonators is mainly distributed in the center of the defect, and its dominant component E_y is shown in Figure 5a. The electric field in the adjacent unit cell is out of phase. The displacement field energy is mainly distributed in the defect region, and its main component S_yy is shown in Figure 5b. The displacement field in adjacent single cells is in phase and overlaps with the dominant component E_y. Comparing the displacement field distribution of the two structures, the elastic wave energy dissipation occurs in the single-mirror structure, leading to the occurrence of phonon leakage. However, the introduction of the second mirror makes the displacement field energy dissipation in the first mirror smaller, and there is almost no dissipation in the second mirror. The acoustic wave energy leaking decreases and makes the strain field better confined in the resonant cavity, which improves the mechanical quality factor. Ultimately, the simulation results indicate that a strong optomechanical coupling rate g₀ = 3.68 MHz was obtained in the single structure with a contribution of 2.64 MHz from the photoelastic effect and 1.04 MHz from the moving boundary effect. Due to the high flexibility of the heterostructure, the electric field energy and displacement field energy are mainly distributed at the center of the photonic and phononic resonant cavity. Therefore, the optomechanical coupling rate g₀ = 3.81 MHz was obtained in the heterostructure with a contribution of 2.73 MHz from the photoelastic effect and 1.08 MHz from the moving boundary effect.

The results related to the optomechanical coupling rate are listed in

Table 3. The optomechanical coupling rate with different structures.

Structures	g0 (MHz)	Structures	g0 (MHz)
Rectangular holes [11]	0.22	Circular holes [26]	1.16
Oval holes [12]	1.10	Round hole heterostructure [17]	1.31
Oscillator type [15]	0.54	Fishbone type [14]	1.89
Slit type [13]	2.80	The structure proposed in this paper	3.81

. Compared with the single-mirror structure, the introduction of the second mirror region can realize the separate constraints on the optical mode and the mechanical mode, such that the electric field energy and the displacement field energy can be better localized in the resonator. The coupling effect between sound and light can be enhanced such that the optomechanical coupling rate can be improved. Furthermore, the structure proposed in this paper evolved from the regular hexagon structure, and it is easier to obtain wider bandgaps of phononic and photonic with high symmetry, thus better enhancing the acoustic–optical interaction. Therefore, the proposed structure has a more significant increase in the optomechanical coupling rate.

4. Conclusions and Perspectives

In this study, a hetero-optomechanical crystal nanobeam structure with a high optomechanical coupling rate was proposed. In addition, the electric field and displacement field distribution of the optical and mechanical modes of the structure were calculated. Simultaneously, the influence of the heterostructure on the optomechanical coupling rate and quality factor was analyzed. Due to the more flexible design of the heterostructure, it was possible to use two mirrors to better constrain the optical and mechanical modes, resulting in a high mechanical quality factor and optomechanical coupling rate. Therefore, the optomechanical coupling rate of the proposed structure could reach 3.81 MHz, the mechanical quality factor was 3.18 × 10⁶, and the effective mass was 43.29 fg. It can be applied to the fields of integrated circuits, quantum manipulation, and quantum information processing. Furthermore, the mode separation confinement method used in this work enables the independent design of the phononic bandgap and the photonic bandgap regulation of the optomechanical crystal. In addition, it provides a feasible idea for the design of two-dimensional cavities and waveguides.