Acousto-optic coupling in phoxonic crystal nanobeam cavities with plasmonic behavior

Jin-Chen Hsu; Tsung-Yi Lu; Tzy-Rong Lin

doi:10.1364/OE.23.025814

1. Introduction

In recent years, interaction of photons and phonons in micro- and nanostructures using dual photonic and phononic band gaps has received considerable attention. These structures are made up of periodic composites and are also called phoxonic or optomechanical crystals; their dielectric and elastic properties are tailored simultaneously to achieve very efficient control of the optical and acoustic fields in the same small region of a system [1–5]. By means of the dual band gaps, photons and phonons of similar wavelengths can be tightly confined in a phoxonic crystal cavity to enhance their interaction as a consequence of increased density of states. Cavity configurations with high quality (Q) factors can generally be produced by introducing well-designed defects into the phoxonic crystals. In a realistic scheme, photons and phonons can be pumped into a phoxonic crystal cavity for acousto-optic (AO) coupling applications using several known mechanisms [6–8]. For example, photons can be injected into the cavity from an outside light source using optical fiber coupling. On the other hand, acoustic phonons can be generated with frequencies up to several gigahertz inside the cavity by, for example, laser-induced thermal fluctuations, electro-acoustic excitation, or opto-acoustic conversion. Strong AO coupling or photon–phonon interaction can lead to many applications, such as cooling of phoxonic resonators to the quantum mechanical ground state and wide-band optical signal processing capabilities in nanoscale photonics, including opto-acoustic sensing, light pulse storage, and phonon lasers [9–12].

In general, phoxonic crystals can be classified into three types according to their spatial periodicities, i.e., one-, two-, and three-dimensional (1D, 2D, and 3D) phoxonic crystals. Their spatial periodicities are directly relevant to the confinement of the optical and acoustic propagation directions with the dual band gaps they can provide. As a result, a wide variety of structural forms, such as beams and slabs, have been considered for building different types of phoxonic crystals and desired cavity designs. Among the many structural forms, nanobeams have been demonstrated to realize high-Q phoxonic crystal cavities [13–16].

The underlying physics of AO coupling can be explained on the microscopic level by a photon–phonon exchange mechanism. Alternatively, a recently developed approach using continuum theories has successfully been used to analyze and model AO coupling in phoxonic crystal structures and devices [17–19]. In this approach, the AO coupling can be categorized into two contributions: the bulk and the interface parts that arise from the photoelastic (PE) disturbances applied to bulk and interfaces, respectively. The two contributions are called the bulk and interface effects, respectively, here. The bulk effect describes how the acoustic field induces local stress inside the material to change its local refractive index. The interface effect describes how the mechanical displacement field of acoustic waves perturbs the structural boundaries from their equilibrium positions. As a result, the resonant optical field and resonance frequency are then modulated. Psarobas et al. reported strong AO coupling in a multilayer 1D phoxonic cavity. They showed that the strong coupling occurs when simultaneous acoustic and optical resonant modes are confined in the same cavity to trigger multiphonon exchange [18]. Rolland et al. investigated AO coupling in 2D phoxonic crystal cavities. They showed that the strength of AO coupling estimated by modulation of the optical resonance frequency depends on the choice of optical and acoustic mode combinations [20]. El-Jallal et al. analyzed AO coupling in 2D phoxonic crystal slab cavities and showed that the cavity geometry is also significant in achieving a higher AO coupling rate [21]. Davanço et al. proposed and analyzed the so-called slot-mode-coupled optomechanical crystals, showing that a large coupling strength can be achieved by squeezing an optical mode inside a slot [22].

Many studies have shown that AO coupling can be enhanced in phoxonic crystal cavities; however, thus far, most of them focused on phoxonic crystal and cavity designs in order to achieve dual band gaps and high-Q resonances or on different combinations of photonic–phononic mode pairs to obtain different or stronger coupling strengths. Only dielectrics and semiconductors were used as materials. Few studies have focused on the possibilities of tailoring the confined optical modes or even acoustic modes to obtain reduced mode volumes and high mode overlap. A small optical mode volume and high photonic–phononic mode overlap can enhance the contributions of the interface and/or bulk effects to AO coupling. These features may yield more efficient exchange of energy and momentum between the confined phonons and photons, which gives rise to stronger AO coupling [23, 24].

In this paper, we analyze AO coupling in a subwavelength phoxonic crystal nanobeam cavity. The proposed phoxonic crystal nanobeam consists of dielectric and metal materials, producing a metal–dielectric interface to hybridize photons and surface plasmons. Therefore, the metallodielectric cavity of the crystal nanobeam provides stronger confinement of electric and/or magnetic fields near the interface and concentrates their energy in a smaller region of space through excitation of surface plasmon resonances. Furthermore, because the metal has a sound velocity smaller than the dielectric, the acoustic energy in the metallodielectric cavity is also localized around the interface. Using the phoxonic crystal nanobeam cavity we designed in this study, we also show that a large opto-acoustic eigenmode overlap exists in our structure, in which the energy field distributions of the photonic and phononic eigenmodes overlap highly at smaller local regions in the cavity. As a result, the interface and/or bulk effects can be more effective in boosting the AO coupling strength. In addition, another purpose of this study is to give insightful understanding into the AO mechanisms in the combination of phoxonic crystals and plasmons. This paper is organized as follows. In section 2, we describe the theoretical method and define the finite-element (FE) model we use for calculating the AO coupling. In section 3, for comparison, we first present the results of AO coupling in a dielectric phoxonic crystal nanobeam cavity (without metal) and discuss its effects in modulation of the photonic resonance frequency. Then the results of the phoxonic crystal nanobeam cavity with plasmonic behavior are presented and discussed in section 4. In section 5, we conclude this study.

2. Theory and numerical model definition

2.1 Theoretical description

Figure 1 shows the geometry of the phoxonic crystal nanobeam. The nanobeam is a two-layer structure that consists of a layer of gallium arsenide (GaAs) and a layer of silver (Ag). GaAs and Ag are chosen as the building materials because of their remarkable strain-optic coefficients and plasmonic performance in the near-infrared range, respectively, and their wide-ranging applications in electro-optic devices. Perforated semi-cylindrical holes are arranged periodically along the length direction (z direction) and symmetrically in the width direction (x direction) of the nanobeam. Figure 1(b) illustrates the unit cell of a perfect crystal nanobeam (without defects), where the lattice constant is denoted by a, the radius of the semi-cylindrical holes is denoted by r₀, the thicknesses of the GaAs and Ag layers are denoted by h_GaAs and h_Ag, respectively, and the beam width is w. We consider that a cavity with linear variations in the lattice constant and semi-cylindrical hole radius from the center is inserted into the nanobeam, as shown in Fig. 1(c).

Fig. 1 (a) Schematic of the GaAs/Ag phoxonic crystal nanobeam cavity and definition of the coordinates. (b) Geometry of the unit cell of the nanobeam outside the cavity region. The lattice constant a is 294 nm, beam width w is 294 nm, hole radius r₀ is 100 nm, GaAs thickness h_GaAs is 176 nm, and Ag thickness h_Ag is 75 nm. (c) Definition of the graded cavity obtained by varying the lattice constant and hole radii. The cavity consists of 15 varied lattice constants.

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Because the optical frequency (~200 THz) is five orders of magnitude larger than the acoustic frequency (in the gigahertz range), a quasi-static approximation is adopted. As a result, at any instant of time t, the acoustic field can be regarded as static when we calculate the optical field in the acoustically perturbed structure. Acoustic waves propagating in an elastic medium are governed by Cauchy’s equation of motion [25]. The time-harmonic acoustic displacement field u_i has the form

u_{i} (x, y, z, t) = U_{i} (x, y, z) \exp (i Ω t - i K z), i = x, y, z,

where Ω and K are the acoustic frequency and wavevector, respectively, and U_i is the acoustic displacement amplitude. The equation governing the acoustic displacement field is given by

ρ \frac{\partial^{2} u_{i}}{\partial t^{2}} - \frac{\partial}{\partial x_{j}} (c_{i j k l} \frac{\partial u_{k}}{\partial x_{l}}) = F_{i}, i, j, k, l = x, y, z,

where c_ijkl and ρ are the elastic stiffness tensor and mass density, respectively, and F_i is the body force. The strain and stress fields, S_ij and T_ij, induced by the acoustic displacement field u_i can be deduced from

S_{i j} = \frac{1}{2} (\frac{\partial u_{i}}{\partial x_{j}} + \frac{\partial u_{j}}{\partial x_{i}}) and T_{i j} = c_{i j k l} S_{k l} .

To solve the acoustic field using Eq. (2), mechanical boundary conditions must be specified. Two types of boundaries are included: 1) Along the interface between the nanobeam and vacuum, a traction-free boundary is required; 2) Along the interface between GaAs and Ag, the stress and displacement vectors must be continuous. The stress and displacement boundary conditions are summarized as

Traction (stress vector) free : T_{j k} m_{k} = 0;

Continuity of stress vector : T_{j k}^{(GaAs)} m_{k}^{(GaAs)} = T_{j k}^{(Ag)} m_{k}^{(Ag)};

Continuity of displacement vector : u_{j}^{(GaAs)} = u_{j}^{(Ag)};

where m_k is the outward unit normal vector.

The acoustic displacement and strain fields determine the acoustically perturbed boundary and refractive index, respectively. The optical impermeability tensor Δ(1/n²)_ij is calculated by [26]

Δ {(1 / n^{2})}_{i j} = p_{i j k l} S_{k l},

where p_ijkl is the strain-optic tensor. Assuming that the acoustic strain is infinitesimally small, the change in the refractive index Δn_ij can then be approximated after a simple limiting process, and the refractive index tensor n_ij in terms of the strain tensor components can be obtained. The time-harmonic electric field E_i has the form

E_{i} (x, y, z, t) = A_{i} (x, y, z) \exp (i ω t - i k z),

where ω and k are the optical frequency and wavevector, respectively, and A_i is the electric field amplitude. At a given instant of time in terms of the acoustic phase Ωt, the modulated optical field can then be determined using the perturbed nanobeam geometry and refractive index tensor n_ij. The time-harmonic solution of the electric field E_i is governed by Maxwell’s equation

b_{l m} b_{m t} e_{s k t} \frac{\partial}{\partial x_{s}} (e_{i j k} \frac{\partial E_{j}}{\partial x_{i}}) = {(\frac{ω}{c})}^{2} E_{l},

where e_ijk is the permutation symbol, c is the free-space light speed, and b_ikn_kj = δ_ij. Along an interface, the electromagnetic boundary conditions require continuity of the tangential component of the electric field and continuity of the normal component of the electric displacement field. We note that perfect conductors will not contribute the bulk part of the PE disturbances because they contain so many free electrons. Therefore, the acoustic strain field induces no microscopic local charge distribution (or dielectric polarization) in conductors, and we do not consider the contribution of the bulk effect from Ag.

To estimate the AO coupling strength, we define the total variation of the optical resonance wavelength Δλ_c during a time period (Ωt = 0–2π) of the acoustic perturbation as

Δ λ_{c} = λ_{\max} - λ_{\min},

where λ_max and λ_min are the longest and shortest optical resonance wavelengths, respectively, during the period Ωt = 0–2π. In addition, the instantaneous optical resonance wavelength shift Δλ_r due to the acoustic perturbation is defined as Δλ_r(t) = λ_r(t) – λ_0r, where λ_r(t) and λ_0r are the perturbed and unperturbed resonance wavelengths, respectively.

2.2 Numerical model definition

We implement and solve the governing equations for the phoxonic models with plasmonic behavior using the FE software COMSOL Multiphysics. The acoustic problems are solved using the structural mechanics module dealing with the 3D elastodynamic calculations for the frequency responses of a structure composed of anisotropic (dielectric) and isotropic (metal) materials. The calculated domains include GaAs and Ag by taking into account their mass densities and elastic stiffness constants. The optical problems are solved using the RF module dealing with the 3D electrodynamic calculations in the frequency domain for the acoustically perturbed and unperturbed structures. The calculated domains include GaAs, Ag, and the vacuum surroundings. Dielectric constants or refractive indices are required to define the optical material properties; Ag is defined to have frequency-dependent complex dielectric constants so that the metal loss is taken into account by the imaginary part [27]. The perturbed structure geometry used in the optical model is defined using the so-called moving-mesh technique by acquiring the displacement field at a specific acoustic phase from the calculated results of the acoustic problems. Because FE analysis deals only with finite domains, we apply the Bloch theorem and supercell approximation to address the infinite periods and infinite media, respectively. These approaches are justified when using eigenfrequency analysis and studying localized fields. The models in the acoustic and optical problems are discretized by a tetragonal element mesh, and second-order Lagrangian elements are employed. In addition, the used elements must be sufficiently small to obtain a convergent result, which means that in wave problems (optical and acoustic) at least five or six elements per wavelength have to be employed. Because the confined wavelengths in the Bragg band gaps can be estimated using λ_r ~2a, where a is the lattice constant, we used a general criterion for the calculations of the opto-acoustic coupled problem that the element sizes are smaller than a/10. The convergence is further justified using more refined elements than the mesh criterion.

3. AO coupling in graded phoxonic crystal nanobeam cavities

We first consider an all-dielectric phoxonic crystal nanobeam cavity and AO coupling in it. The structure has the geometry described in Fig. 1 except that the Ag layer is removed. The lattice constant a is 294 nm, the beam width w is 294 nm, the hole radius r₀ is 100 nm, and the thickness of the GaAs beam h_GaAs is 176 nm. The material properties used are listed in Table 1. Figure 2(a) shows the photonic band structure of the crystal nanobeam with no defects. Solid and dashed curves represent the transverse magnetic (TM) and transverse electric modes, respectively. The photonic band gap for the TM modes ranges from 317.2 to 327.1 THz below the light line. The ratio of the band-gap width to its mid-gap frequency, Δω/ω_m, is 3.07%. Figure 2(b) shows the electric field intensity (|E_y|²) distributions for the upper and lower band-edge TM modes at the Brillouin zone boundary. The |E_y|² distributions show that the modes are guided along the nanobeam by index guiding below the light line, and the |E_y|² fields are distributed over the cross section of the beam. For the phononic viewpoint, Fig. 2(c) shows the phononic band structure of the same GaAs nanobeam. An obvious phononic band gap for all phononic eigenmodes appears from 5.35 to 5.94 GHz. The relative band-gap width ΔΩ/Ω_mis 10.45%. The total displacement field |u_i| of the upper and lower phononic band-edge modes is shown in Fig. 2(d), which have twisting deformations.

Table 1. Material constants of GaAs and Ag used for the calculations

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Fig. 2 (a) Photonic band structure of the GaAs (all-dielectric) phoxonic crystal nanobeam with no Ag layer. Shaded region represents the inner region divided by the light line. A band gap exists from 317.2 to 327.1 THz. (b) Electric field intensity |E_y|² distributions of the lower and upper band-edge modes at the Brillouin zone boundary. (c) Corresponding phononic band structure. A phononic band gap for all phononic eigenmodes appears from 5.35 to 5.94 GHz. (d) Total displacement field |u_i| of the upper and lower phononic band-edge modes.

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Figure 3(a) shows the |E_i|² field distribution of the photonic cavity mode with the graded cavity inserted in the middle of the nanobeam. The mode is similar to the lower band-edge TM modes but highly confined to the graded cavity. The resonance frequency is 319.7 THz, and the corresponding resonance wavelength is 938.53 nm. The Q factor of the photonic cavity mode, estimated using FE method with excited frequency-domain analysis, is 3500. However, the photonic cavity mode apparently shows that it has a considerable mode volume V_m, according to a comparison of the cavity size (i.e., cavity length and beam thickness). The mode volume V_m can be quantified by [28]

V_{m} = \frac{∭_{} W (x, y, z) d x d y d z}{\max [W (x, y, z)]}, where W = \frac{1}{2} (Re (\frac{d (ω ε)}{d ω}) {| E_{i} |}^{2} + μ {| H_{i} |}^{2}) .

In the equation, μ is the permeability, and H_i is the associated magnetic field. Using Eq. (11), the mode volume of the photonic cavity mode is V_m = 2.37(λ/n_GaAs)³. On the other hand, a phononic cavity mode, as shown in Fig. 3(b), is supported by the graded cavity. In addition to this mode, several other phononic cavity modes can be supported within the phononic band gap as well. This phononic cavity mode is considered because it has the most concentrated acoustic energy in the central region of the cavity, which is highly colocalized with the photonic cavity mode shown in Fig. 3(a). Such colocalization can achieve better AO coupling than other phononic cavity modes do. The phononic cavity mode vibrates with extension-and-contraction displacement and up-and-down bending of the limbs along the x and y directions, which are concentrated in the central region of the graded cavity. The corresponding resonance frequency is 5.77 GHz. On the basis of the AO coupling mechanisms presented in section 2, the modulation of the photonic resonance wavelength by the phononic cavity mode is shown in Fig. 3(c) as a function of the acoustic phase Ωt. In the figure, the wavelength variations between Ωt = 0–π and π–2π are symmetric; therefore, only the range Ωt = 0–π is presented. The individual contributions from the bulk and interface effects and the overall contribution of their joint effect are presented separately. In the calculations, the displacement amplitude |U_i| is limited to 0.01a (i.e., 2.97 nm). This criterion is typically adopted in estimation of the AO modulation and is used throughout this study. As a result, the totalvariation of the resonance wavelength Δλ_c is 0.2 nm for the photonic–phononic mode pair combination supported by the all-dielectric crystal nanobeam cavity, where the dominant contribution is from the interface effect. The highly confined photonic–phononic mode pair with optical and acoustic energy distributions in the same (central) region of the phoxonic cavity results in an obvious photonic resonance wavelength shift due to AO coupling. In the following section, we show more enhanced AO coupling by simultaneously reducing the photonic mode volume and tailoring the acoustic energy distributions using a metal layer.

Fig. 3 (a) |E_i| field distributions of the GaAs photonic cavity mode on the x–z and y–z planes. The cavity mode is confined by the graded cavity with a high quality, Q = 3500. The mode volume V_m is 2.37(λ/n_GaAs)³. (b) The |u_i| field distribution of the phononic cavity modes, which is concentrated in the cavity central region. (c) Modulation of the optical resonance wavelength λ_r by the phononic cavity mode as a function of the acoustic phase Ωt.

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4. AO coupling in graded phoxonic crystal nanobeam cavities with plasmonic behavior

We now turn our attention to the two-layer phoxonic crystal nanobeam composed of GaAs and Ag.

4.1 Band structures of GaAs/Ag phoxonic crystal nanobeam

Figure 4 shows the photonic and phononic band structures and the band-edge eigenmodes of the GaAs/Ag heterogeneous nanobeam. Compared with the GaAs (all-dielectric) nanobeam in section 3, an additional layer of Ag is deposited on the bottom of the GaAs beam here. The thickness of the Ag layer is h_Ag = 75 nm, and the other geometrical parameters are the same as those used for the GaAs nanobeam. The Ag layer thickness is chosen to generate dual photonic and phononic band gaps and sufficiently prevent leakage of optical energy from the bottom. In the photonic band structure shown in Fig. 4(a), a photonic band gap below the light line appears from 182 to 198 THz. This photonic band gap has a much wider relative band- gap width, i.e., Δω/ω_m = 8.42%, than the GaAs nanobeam. Three bands of waveguide modes appear around the band gap and are labeled modes A, B, and C at the Brillouin zone boundary. These modes are TM modes with the magnetic field polarization primarily in the x direction, and their dominant electric field component is Ey, by which the photons and surface plasmons are hybridized. The electric field intensity (|Ey|²) distributions are shown in Fig. 4(b); in the side view (x–y plane), they exhibit concentrated energy above and close to the interface of the GaAs and Ag layers. In the top view (x–z plane), the |Ey|² distributions have also been tailored to be concentrated in smaller regions in the unit cell. The |Ey|² distributions of modes A and C are concentrated in the entire and ends of the limbs, whereas that of mode B is concentrated in the central region of the unit cell.

Fig. 4 (a) Photonic band structure of the GaAs/Ag phoxonic crystal nanobeam. A photonic band gap appears from 182 to 198 THz. Three bands of waveguide modes appear around the band gap and are labeled as modes A, B, and C at the Brillouin zone edge. With the graded cavity, two cavity modes A_d1 and A_d2 exist in the band gap. (b) Electric field intensity |E_y|² distributions of the lower and upper band-edge modes at the Brillouin zone boundary.

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In the phononic band structure in Fig. 5(a), two band gaps appear. Their ranges are 3.92–4.12 GHz and 4.54–4.76 GHz, and their relative band-gap widths ΔΩ/Ω_m are 4.98% and 4.73%, respectively. For the phononic modes, the band gap widths of the GaAs/Ag nanobeam are reduced compared with that of the GaAs nanobeam. The two-layer nanobeam with a metal layer of large mass density does not benefit opening wider phononic band gaps. Reducing the Ag layer thickness can increase the phononic band-gap widths. However, the Ag layer breaks the spatial symmetry in the thickness direction of the nanobeam and gives rise to asymmetric phononic eigenmodes with energy highly concentrated near the Ag layer because of its low sound velocity, which may be collocated in the photonic modes to enhance the interface effect for AO coupling. Figure 5(b) shows the upper and lower band-edge modes of the phononic band gaps, corresponding to the points labeled α, β, γ, and δ and indicated in the phononic band structure. These modes are slightly or significantly asymmetric along the y direction, where the α and γ modes are highly concentrated near the Ag layer at the limbs of the unit cell.

Fig. 5 (a) Phononic band structure and defect bands of the GaAs/Ag phoxonic crystal nanobeam. Two phononic band gaps appear from 3.92 to 4.12 and 4.54 to 4.76 GHz. The lower and upper band-edge modes of the two band gaps are labeled as modes α, β, γ, and δ. (b) The |u_i| field distributions of the four band-edge modes, which exhibit asymmetric distributions along the beam thickness.

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4.2 Cavity modes and mode volume

Consider that the graded cavity is inserted into the GaAs/Ag crystal nanobeam. Figure 5 shows the photonic and phononic cavity modes in the photonic and phononic band gaps, respectively. Their frequencies are shown by the horizontal dashed lines in Figs. 4(a) and 5(a), respectively. In the photonic band gap, two cavity modes with resonance frequencies of 193.9 and 186.6 THz (corresponding to resonance wavelengths of 1547.9 and 1608.4 nm, respectively) are supported by the cavity, and their |E_i|² field distributions are shown in Fig. 6(a). They are labeled A_d1 and A_d2 because of their similarity to the TM mode A, in which the energy of modes A_d1 is concentrated at a single spot in the central cavity, whereas that of mode A_d2 is concentrated separately at double spots located symmetrically with respect to the cavity central line. As a result of using the GaAs/Ag combination, the cavity modes A_d1 and A_d2 are tailored to have reduced mode volumes, which are equal to V_m = 0.0657(λ/n_GaAs)³ and 0.11(λ/n_GaAs)³, respectively. We note that because of metal loss introduced by the Ag layer, the Q factors of the A_d1 and A_d2 cavity modes are reduced to 176 and 191, respectively. For the phononic part, the cavity supports seven and eight cavity modes in the first and second phononic band gaps, respectively. The enlarged plot with the frequency range around the phononic band gaps in Fig. 5(a) shows the defect bands. By observing the vibration of these phononic cavity modes, we classify them into three types according to their dominant displacement component: 1) symmetric stretching (SS_i) modes, 2) asymmetric stretching (AS_i) modes, and 3) twisting (T_i) modes, where the subscript i denotes a mode number. In addition, another cavity mode (i.e., SS₃ mode) with a frequency slightly above the upper edge of the first phononic band gap is also highly confined. This kind of confinement is known as deaf-band confinement and is due to weak coupling between the cavity mode and phononic eigenmodes of the same frequency. Figures 6(b) and 6(c) show the total displacement and deformation fields for several phononic cavity modes in the first and second band gaps, respectively.

Fig. 6 (a) |E_i| field distributions of the GaAs/Ag photonic cavity modes A_d1 and A_d2 on the x–z and y–z planes, respectively. The two cavity modes are confined by the graded cavity with reduced mode volumes V_m = 0.0657(λ/n_GaAs)³ and 0.11(λ/n_GaAs)³. (b), (c) Illustration of the |u_i| field distributions of several phononic cavity modes labeled in Fig. 5(a). They are classified into symmetric stretching (SS_i) modes, asymmetric stretching (AS_i) modes, and twisting (T_i) modes. The field distributions of the SS₃ and SS₂ modes are highly matched to those of the A_d1 and A_d2 modes, respectively.

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To generate photonic–phononic mode pair combinations for efficient AO coupling, in the following we focus on the SS_i and AS_i modes because their spatial energy distributions exhibit better correlation with those of the two photonic cavity modes.

4.3 AO coupling

To demonstrate the enhancement of AO coupling in the GaAs/Ag phoxonic crystal nanobeam cavity, the two photonic cavity modes (A_d1 and A_d2 modes) and several phononic cavity modes (SS_i, i = 2, 3, 5, and 7, and AS₁ modes) are paired, and the acoustically perturbed resonance wavelengths of the two photonic cavity modes are calculated. We first consider the phononic cavity modes AS₁, SS₂, and SS₃, which are related to the first phononic band gap. Figure 7 shows the modulations in the photonic resonance wavelength by these modes. To observe the influence of different photonic–phononic mode pair combinations on the AO coupling mechanisms, modulations contributed by the joint (bulk + interface), bulk, and interface effects are calculated separately. In Figs. 7(a) and 7(d), significant differences appear in the total variation of the resonance wavelength Δλ_c among different photonic–phononic mode pairs. The mode pairs A_d1–SS₃ and A_d2–SS₂ exhibit the strongest AO coupling, with Δλ_c equal to 3.88 and 6.03 nm, respectively. This shows that tailoring the photonic and phononic cavity mode distributions to overlap and reducing the photonic mode volumes using the metal (Ag) layer is effective and efficient in enhancing the AO coupling. To further examine the underlying contributions to this enhancement, Figs. 7(b) and 7(c) and Figs. 7(e) and 7(f) compare the contributions from the bulk and interface effects. Figures 7(b) and 7(e) show that the bulk effect results in only small modulations for any photonic–phononic mode pair combinations, and Figs. 7(c) and 7(f) show that the interface effect dominates the enhancement of the AO coupling. This can be understood as both the photonic and phononic cavity modes being squeezed into the small regions near the GaAs/Ag interface.

Fig. 7 AO modulation of optical resonance wavelength in the GaAs/Ag phoxonic crystal nanobeam cavity as a function of acoustic phase Ωt from 0 to π for different photonic–phononic mode pair combinations, taking into account the contributions of (a), (d) joint (bulk + interface) effect, (b), (e) bulk effect, and (c), (f) interface effect. The used phononic cavity modes, the AS₁, SS₂, and SS₃ modes, are in or near the first phononic band gap. (a–c) A_d1 mode. (d–f) A_d2 mode.

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Figure 8 shows the modulations in the photonic resonance wavelength obtained by pairing the A_d1 and A_d2 modes with the phononic SS₅ and SS₇ modes. These phononic cavity modes have an obvious z component of the displacement field compared with those cavity modes in the first phononic band gap. The joint, bulk, and interface effects are considered separately.

Fig. 8 AO modulation of optical resonance wavelength in the GaAs/Ag phoxonic crystal nanobeam cavity as a function of acoustic phase Ωt from 0 to π for different photonic–phononic mode pair combinations, taking into account the contributions of (a), (d) joint (bulk + interface) effect, (b), (e) bulk effect, and (c), (f) interface effect. The used phononic cavity modes, the SS₅ and SS₇ modes, are in the second phononic band gap. (a–c) A_d1 mode. (d–f) A_d2 mode.

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Among these pairings, A_d1–SS₅, A_d1–SS₇, A_d2–SS₅, and A_d2–SS₇ exhibit the strongest AO coupling, with Δλ_c equal to 1.16, 1.29, 0.52, and 1.36 nm, respectively, where the Ag layer also contributes to AO coupling by an enhanced interface effect. However, because the SS₅ and SS₇ modes in the second band gap are not as concentrated as the SS₂ and SS₃ modes in the first band gap, the AO coupling is not as strong as that of the mode pairs A_d1–SS₃ and A_d2–SS₂. Similarly, the interface effect again dominates the AO coupling, whereas the bulk effect contributes much less. Table 2 summarizes the mode volume V_m, quality factor Q, and variation in photonic resonance wavelength Δλ_c for the two photonic cavity modes in relation to the properties of the colocalized phononic cavity modes discussed in this study. Note that by examining the modulation of the optical resonance wavelength over an acoustic period, the multi-phonon process dominates the photon-phonon exchange of AO coupling in the most mode combinations (except the A_d1–AS₁ combination where single-phonon process dominates) [18, 29]. Using the GaAs/Ag phoxonic crystal nanobeam cavity with plasmonic behavior, AO coupling is stronger as a result of stronger multi-phonon process. According to the estimation with optical wavelength shift, several to tens phonon frequencies can sustain the AO coupling, depending on the choice of the photonic–phononic mode combination (see Table 2). However, the modulation degree also depends on acoustic displacement amplitude. Generating smaller or larger amplitude would decrease or increase the modulation degree. We also conclude that the small mode volume and high mode overlap significantly enhance the AO coupling through the interface effect in the GaAs/Ag phoxonic crystal structures.

Table 2. Properties of photonic resonant modes and AO coupling (displacement amplitude |U_i| = 2.97 nm).

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5. Conclusion

We studied AO coupling in a heterogeneous GaAs/Ag phoxonic crystal nanobeam cavity. The Ag layer enables the structure to hybridize photons and surface plasmons to squeeze the optical energy into much smaller regions near the GaAs/Ag interface, and the phononic cavity modes can be tailored to highly overlap the squeezed photonic cavity modes. As a result, enhanced AO coupling is achieved. Our results show that the enhancement benefits from great boosting of the interface effect. The simultaneous small mode volume of the photonic cavity mode and high spatial mode overlap between the photonic and phononic cavity modes give rise to a one order of magnitude enhancement of the photonic resonance wavelength modulation compared with that in the all-dielectric GaAs phoxonic crystal nanobeam cavity. This study introduces the possibility of achieving strong AO interaction in subwavelength nanostructures.

Acknowledgments

Authors thank Jheng-Hong Shih at National Taiwan Ocean University for technical support. This work is supported by Ministry of Science and Technology (MOST), Taiwan, under the Grant Nos. MOST 103-2221-E-224-002-MY3 and MOST 103-2221-E-019-028-MY3.

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Material	Mass Density (kg/m³)	Elastic Constants (10¹⁰ N/m²)			Strain-Optic Coefficients			Refractive Index
	ρ	c₁₁	c₁₂	c₄₄	p₁₁	p₁₂	p₄₄	n
GaAs	5300	11.8	5.37	5.94	–0.165	–0.14	–0.072	3.4
Ag	10635	12.4	9.37	4.61				Ref [27].

Photonic-Phononic Mode Pair	Photonic Cavity Mode			Total Optical Resonance Wavelength Variation Δλ_c (nm)
	V_m (λ/n_GaAs)³	Q	λ_0r (nm)	bulk	interface	bulk + interface
Dielectric	2.37	3500	938.53	< 0.01	0.20	0.20
A_d1–AS₁				< 0.01	0.05	0.04
A_d1–SS₂				0.16	0.41	0.25
A_d1–SS₃	0.0657	176	1547.9	0.25	4.14	3.88
A_d1–SS₅				0.28	1.57	1.29
A_d1–SS₆				0.27	1.64	1.35
A_d2–AS₁				< 0.01	0.31	0.28
A_d2–SS₂				0.39	6.42	6.03
A_d2–SS₃	0.11	190	1608.4	0.01	0.80	0.79
A_d2–SS₅				0.14	0.74	0.60
A_d2–SS₇				0.28	1.64	1.36

Material	Mass Density (kg/m³)	Elastic Constants (10¹⁰ N/m²)			Strain-Optic Coefficients			Refractive Index
	ρ	c₁₁	c₁₂	c₄₄	p₁₁	p₁₂	p₄₄	n
GaAs	5300	11.8	5.37	5.94	–0.165	–0.14	–0.072	3.4
Ag	10635	12.4	9.37	4.61				Ref [27].

Photonic-Phononic Mode Pair	Photonic Cavity Mode			Total Optical Resonance Wavelength Variation Δλ_c (nm)
	V_m (λ/n_GaAs)³	Q	λ_0r (nm)	bulk	interface	bulk + interface
Dielectric	2.37	3500	938.53	< 0.01	0.20	0.20
A_d1–AS₁				< 0.01	0.05	0.04
A_d1–SS₂				0.16	0.41	0.25
A_d1–SS₃	0.0657	176	1547.9	0.25	4.14	3.88
A_d1–SS₅				0.28	1.57	1.29
A_d1–SS₆				0.27	1.64	1.35
A_d2–AS₁				< 0.01	0.31	0.28
A_d2–SS₂				0.39	6.42	6.03
A_d2–SS₃	0.11	190	1608.4	0.01	0.80	0.79
A_d2–SS₅				0.14	0.74	0.60
A_d2–SS₇				0.28	1.64	1.36

Acousto-optic coupling in phoxonic crystal nanobeam cavities with plasmonic behavior

Abstract

1. Introduction

2. Theory and numerical model definition

2.1 Theoretical description

2.2 Numerical model definition

3. AO coupling in graded phoxonic crystal nanobeam cavities

4. AO coupling in graded phoxonic crystal nanobeam cavities with plasmonic behavior

4.1 Band structures of GaAs/Ag phoxonic crystal nanobeam

4.2 Cavity modes and mode volume

4.3 AO coupling

5. Conclusion

Acknowledgments

References and links

Cited By

Figures (8)

Tables (2)

Equations (11)

Optics Express