Abstract

Bound states in the continuum (BICs), an emerging type of long-lived resonances different from the cavity-based ones, have been explored in several classical systems, including photonic crystals and surface acoustic waves. Here, we reveal symmetry-protected mechanical BICs in the structure of slab-on-substrate optomechanical crystals. Using a group theory approach, we identified all the mechanical BICs at the Γ point in optomechanical crystals with C4v and C6v symmetries as examples, and analyzed their coupling with the co-localized optical BICs and guided resonances due to both moving boundary and photo-elastic effects. We verified the theoretical analysis with numerical simulations of specific optomechanical crystals which support substantial optomechanical interactions between the mechanical BICs and optical resonances. Due to the unique features of high-Q, large-size mechanical BICs and substrate-enabled thermal dissipation, this architecture of slab-on-substrate optomechanical crystals might be useful for exploring macroscopic quantum mechanical physics and enabling new applications such as high-throughput sensing and free-space beam steering.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Light trapping is a key technique for enhancing light-matter interactions in physical architectures such as cavity-QED [1] and cavity-optomechanics [2]. One method to achieve light confinement at wavelength scales is by dispersion engineering of light to create localized optical resonances which are spectrally separated from the radiation continuum, as represented by the photonic crystal cavities. Recently, a different mechanism for light trapping has gained resurgent interest, where long-lived optical resonances can reside in the radiation continuum while still being confined in certain spatial dimensions [3]. In contrast to conventional optical cavities with hard “mirrors”, such bound states in the continuum (BICs) can exist simply due to the symmetry incompatibility between the confined modes and the free-space modes, decoupling the former from the radiation continuum. While for other non-symmetry-protected BICs, it generally requires tuning of system parameters to accidentally cancel all the radiation amplitudes.

Although the concept of BICs can be traced back to the early work by Wigner and von Neumann in quantum mechanics [4], they are now more commonly found in classical systems. For example, in photonic crystal slabs, both symmetry-protected and non-symmetry-protected optical BICs have been observed [5, 6], leading to applications including large-area lasing [7]. New types of optical BICs in other platforms are being proposed [8–10]. Certain types of surface acoustic waves in anisotropic elastic media or layered structures are also identified as BICs [11, 12]. However, acoustic BICs in artificial periodic structures, such as mechanically-compliant photonic crystals, have not been explored. In mechanically-compliant photonic crystals, acoustic modes can strongly interact with the optical modes via radiation-pressure force, leading to the paradigm of optomechanical crystals [13] which are widely explored for inertia sensing [14], information processing [15], and quantum science [16]. In contrast to most free-standing optomechanical structures, here we study slab-on-substrate optomechanical crystals (Fig. 1), in which mechanical BICs can exist due to symmetry incompatibility with acoustic waves in the substrate, and analyze their interactions with the co-localized optical resonances using a group theory approach. We verify the theoretical analysis using numerical simulations of specific types of optomechanical crystals which exhibit substantial coupling between the mechanical BICs and optical resonances.

This new optomechanical architecture, with long-lived mechanical modes in the continuum, might help resolve some long-standing problems in cavity-optomechanics. Conventional on-chip optomechanical structures are suspended to prevent leakage of resonance phonons into the substrate. However, such suspended structures also result in slow heat dissipation which limits the parametric pump power and causes excess noise, especially when they are operated at low temperature and near quantum regime [17, 18]. The structure of slab-on-substrate optomechanical crystals, while trapping phonons of specific frequencies in mechanical BICs, facilitates the dissipation of thermal phonons as well as other non-resonant phonons via direct contact with the substrate. This mechanism is expected to reduce optically induced thermal noise and nonlinear acoustic noise while accommodating optical pumps for parametrically-enhanced coupling between the macroscopic mechanical BICs and optical resonances, for optically cooling the mechanical modes and addressing single phonons [16].

In principle, the mechanical BICs in optomechanical crystal slabs are not limited in size in the planar dimensions, in contrast to the resonances in micro-cavities which have a mode size comparable to the wavelength. Thus, such mechanical BICs with large masses, as well as their frequencies in the gigahertz range, represent a unique system for studying macroscopic quantum physics. The large-area mechanical BICs coupled with optical fields might also enable new applications, such as high-throughput sensing and free-space beam steering.

 figure: Fig. 1

Fig. 1 Schematic of the structure of a slab-on-substrate optomechancial crystal.

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2. Symmetry-protected mechanical and optical BICs in slab-on-substrate optomechanical crystals

In photonic crystal slabs with a spatial symmetry represented by a point group G, the optical modes with Bloch momentum k belong to the irreducible representation of the k-group Gk, i.e., a sub-group of G whose elements keep vector k invariant [19]. When the representation of a photonic crystal mode is incompatible with that of far-field radiation modes with the same parallel momentum, this mode decouples from the radiation continuum and becomes trapped in the photonic crystal slab [20]. In mechanically-compliant photonic crystals, i.e., optomechanical crystals, the mechanical Bloch modes can also be classified by the irreducible representations of the point group, and symmetry-protected mechanical BICs exist when their representations are incompatible with the acoustic waves in the surrounding media. We are interested in the coupling between mechanical BICs and optical resonances, including BICs and guided resonances, in slab-on-substrate optomechanical crystals. In general, because of the phase matching condition, only mechanical modes with Bloch momentum k=0 can couple with an optical mode. For the same reason, normal bound states not in the continuum, e.g. those below sound lines, always have non-zero momentum and thus cannot couple with a singleoptical mode. As such, we will only consider mechanical BICs at the Γ point in this paper, whose k-group is simply G. However, mechanical BICs at other high-symmetric points in the Brillouin zone can be similarly analyzed, which are relevant to coupling two optical modes with different momenta.

As examples, we will analyze two types of slab-on-substrate optomechanical crystals, with spatial symmetry described by C4v and C6v groups, which are commonly seen in crystals with square and triangular lattices, respectively. In Table 1 and 2, we list all the irreducible representations of the C4v and C6v groups and their characters to facilitate the discussion below.

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Table 1. Character table for the C4v point group

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Table 2. Character table for the C6v point group

2.1. Mechanical BICs

The evolution of elastic waves in solids is governed by the following wave equation of elastodynamics:

ρ2Qt2=Y2(1+ν)2Q+Y2(1+ν)(12ν)Q,
where ρ, Y and ν are the density, Young’s modulus and Poisson’s ratio of the solid, respectively, and Q is the mechanical displacement. According to Bloch’s theorem, when the elastic parameters are periodically distributed in two dimensions, such as in optomechanical crystal slabs, the mechanical eigenmodes of Eq. (1) can be written as:
Qk(r)eiωt=uk(r)ei(kρωt),
where ω is the eigenfrequency, k is the Bloch wave vector in the plane of the slab, ρ is the in-plane component of the position vector r, and uk(r) is a periodic function in the two dimensions. We can further expand uk(r) using discrete Fourier transform:
uk(r)=j=0fj(z)eiGjρ,
where Gj are the reciprocal lattice vectors. For example, in an optomechanical crystal with a square lattice, the first few reciprocal lattice vectors are G0=(0,0)2πa, Grallelj=1,2,3,4=(±1,0)2πa and (0,±1)2πa, etc..

In contrast to electromagnetic fields, the coupling of high-frequency mechanical modes into the air can be ignored due to the substantial impendence mismatch between air and solids for the ultrasound. Thus, below we only consider the coupling into the substrate. In the far field, the solution of Qk(r) is merely a superposition of plane waves. To find it, we can decompose Q into the curl-free (QL) and divergence-free (boldsymbolQT) components and separate Eq. (1) into two independent equations:

2QT(L)t2=cT(L)22QT(L),
where cT=Yρ12(1+ν) and cL=Yρ1ν(12ν)(1+ν) are the speed of transverse and longitudinal elastic waves, respectively. Note the elastic parameters in these formula of speed of sound are those of the substrate. The mechanical mode in the far field is then given by
Qk,(r)=eikρj=0(AT,jeikT,zjz+AL,jeikL,zjz)eiGjρ,
where
kT(L),zj=ω2cT(L)2|k+Gj|2,
and AT,j and AL,j are the amplitudes of transverse and longitudinal waves, respectively.

Because of the phase-matching condition for non-vanishing optomechanical coupling between a mechanical mode and an optical mode, we only consider mechanical BICs at the Γ point, i.e., k=0. In most solids, Poisson’s ratio is less than one, leading to cT<cL. For k=0, and ω<cT2πa (for C4v structures) or ω<cT4π3a (for C6v structures), which we call the cut-off frequency, only the zeroth-order terms (j = 0) in Eq. (5) have real z-component of the wave vector, and they can be parameterized as

Q0=QT,0+QL,0=(uex+vey)eikT,z0z+wezeikL,z0z.

These are the only radiation channels that a mechanical slab mode at the Γ point below the cut-off frequency can couple to.

It is easy to see that {ex,ey} forms the basis of the E representation of C4v point group and the E1 representation of C6v point group, and ez belongs to the A1 representation of C4v and C6v groups. Therefore, for a mechanical eigenmode with frequency ω<cT2πa in an optomechanical crystal with C4v symmetry, it can be a BIC only if its representation belongs to {A2,B1,B2}. And in a structure with C6v symmetry, a mechanical mode with frequency ω<cT4π3a can be a BIC only if its representation belongs to {A2,B1,B2,E2}.

Above the cut-off frequency, at least the first-order transverse terms in Eq. (5), i.e.,

Q1,T=lAT,leikT,zlzeiGlρ,
where {Gl,l=1,,4}={±2πaex,y} for C4v structures and {Gl,l=1,,6}={±4π3aex,±4π3a(±12ex+32ey)} for C6v structures, become viable radiation channels for the mechanical modes at the Γ point. We find the basis {exeiGlρ,eyeiGlρ} can be decomposed into A1+B1+A2+B2+2E representations of C4v group and A1+B1+A2+B2+E1+E2 representations of C6v group, respectively, which contain all the irreducible representations of the two groups. As a result, above the cut-off frequency, no mechanical BICs exist. We summarize these results in Table 3.

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Table 3. Mechanical and optical BICs at the Γ point

2.2. Optical BICs

The analysis of optical BICs largely follows from the analysis of mechanical BICs above. In order to have confined optical modes in the slab, we require the refractive index of the material of slab to be larger than that of the substrate. In the far field, the electric fields satisfy the Maxwell’s equation,

1c22Et2=1n22E,
where n is the refractive index of the substrate and c is the speed of light in vacuum. We can decompose the electric fields into plane waves:
Ek(r)=eikρj=0Ajeikzjz+iGjρ,
where
kzj=ω2(c/n)2|k+Gj|2.

The difference from elastic waves is that electromagnetic waves can only be transverse.

At the Γ point, when ω<cn2πa (C4v structures) or ω<cn4π3a (C6v structures), the only radiation component in Eq. (10) is E0=A0eikz0z. E0 has the same form as QT,0, so it belongs to the E representation of C4v or the E1 representation of C6v. Above the cut-off frequencies, similar to the mechanical case, the radiation terms contain all the irreducible representations of C4v or C6v group, and thus no optical BICs exist. The result of optical BICs is summarized in Table 3 as well.

3. Optomechanical couplings between mechanical BICs and optical modes

The optomechanical interaction between a mechanical mode and an optical mode can be modeled by the Hamiltonian H=g0aa(b+b), where a(b) and a(b) are the annihilation and creation operators of the optical(mechanical) mode, and g0 is the single-photon optomechanical coupling. The main contribution to the optomechanical coupling includes moving boundary effect g0,MB and photo-elastic effect g0,PE, with the total optomechanical coupling given by [21]:

g0=g0,MB+g0,PE2meffωm(gOM,MB+gOM,PE),
where ωm is the frequency of the mechanical mode and meff=ρ|Q|2dV is its effective mass with properly normalized Q. The moving boundary and photo-elastic components are given by
gOM,MB=ωo2(Qn^)(Δϵ|E|2Δϵ1|D|2)dSE*DdV,
gOM,PE=ωo2ϵ0n4Ei*EjpijklSkldVE*DdV,
where ωo is the frequency of the optical mode, n^ is the unit vector normal to the boundary, and denote the components parallel and perpendicular to the boundary, Δϵ=ϵintϵext and Δϵ1=ϵint1ϵext1 (ϵext is the permittivity of the media which n^ points to and ϵint is the permittivity of the media on the other side of the boundary.), ϵ0 is the vacuum permittivity, n is the refractive index, p is the rank-four photo-elastic tensor and S is the strain tensor.

For mechanical and optical Bloch modes in optomechanical crystals, the optomechanical coupling can be calculated using

g0=g¯0N1Nn,meik(na1+ma2),
where k is the Bloch wavevector of the mechanical mode, a1,2 are the lattice vectors, N is the number of unit cells in the optomechanical crystal slab, and g¯0 is the coupling in a single unit cell at (n,m)=(0,0), where all the integrations are performed, leading to the 1/N factor from meff. For large enough optomechanical crystals, from Eq. (15), g0 is zero unless k=0, which is equivalent to the phase-matching condition. Thus, for the mechanical BIC at the Γ point, its coupling with an optical mode is given by
g0=g¯0N.

Because of the scaling of 1/N, the single-photon optomechanical coupling between the delocalized mechanical BICs and optical modes is generally smaller than that in state-of-the-art optomechanical crystal cavities [21]. However, because of the better unit-area photon capacity thanks to the substrate-enabled heat dissipation, parametrically enhanced optomechanical coupling, i.e.,

G=g0Nnp=g¯0np,
where np is the number of pump photons in a single unit cell, might be comparable or even better than the state-of-the-art, given sufficient pump powers can be supplied.

Below, we will systematically study the optomechanical coupling between a mechanical BIC and an optical mode, based on the symmetry of the modes indicated by their representations. This general analysis helps to identify modes with desired symmetries for non-vanishing optomechanical couplings which can be further optimized through numerical simulations.

3.1. Moving boundary effect

The optomechanical coupling due to moving boundary effect is given by Eq. (13). We define fΔϵ|E|2Δϵ1|D|2 to simplify the discussion below. We first consider the coupling between a one-dimensional (1-d) representation mechanical BIC and a 1-d representation optical mode. For 1-d representation optical modes, f is even under mirror operations σα, i.e., {2σv,2σd} in the C4v group and {3σx,3σy} in the C6v group, since E and D are either even or odd under these mirror operations. Here we used subscript α to denote the angle between the x-z plane andthe mirror plane, e.g., σπ/2=σx. The integral in the numerator of Eq. (13) can be calculated in the two regions separated by the mirror plane,

(Qn^)fdS=αα+π(Qn^)fdS+ααπ(Qn^)fdS =θ=0π(Q(α+θ)+σαQ(αθ))n^(α+θ)f(α+θ)dS.

We see that, if the mechanical mode is odd under σα, the integration is zero. According to Tables 1 and 2, for both C4v and C6v groups, only A1 is even under all mirror reflections. However, mechanical modes with A1 representation cannot be a BIC as shown above. As a result, the interaction between a 1-d mechanical BIC and a 1-d optical mode due to moving boundary effect is always zero.

Next we consider the case when at least one of the mechanical and optical modes is a 2-d representation. For a 2-d representation, the matrix form of the group elements are generally not diagonal, except for C2. One can find C2=I for E (C4v) and E1 (C6v) representations and C2=I for E2 (C6v) representation. Therefore, all the modes, irrespective of the dimension of the representation, are either odd or even under C2. Making use of the unique property of C2, the integralin the numerator of Eq. (13) can be calculated as

(Qn^)fdS=0π(Qn^)fdS+π2π(Qn^)fdS =θ=0π(Q(θ)+C2Q(θ+π))n^(θ)f(θ)dS.

From Eq. (19), we see that the mechanical BIC needs to be even under C2 for non-vanishing moving boundary effect in this case.

The optomechanical coupling due to moving boundary effect between a mechanical BIC and an optical mode is summarized in Table 4 and Table 5 for structures with C4v and C6v symmetry, respectively. In these tables, when the coupling is indicated as 0, it only means the coupling is not constrained to be zero by symmetry; but could be zero due to other reasons, such as linear combination of degenerate modes.

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Table 4. Moving boundary effect gOM,MB, C4v

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Table 5. Moving boundary effect gOM,MB, C6v

3.2. Photo-elastic effect

The optomechanical coupling due to photo-elastic effect is given by Eq. (14). Unlike the moving boundary effect, the photo-elastic effect depends on the crystalline type of materials through the photo-elastic tensor. We will consider two types of crystalline: the hexagonal crystal, e.g., aluminum nitride, and the cubic crystal, e.g., silicon. Another subtlety of photo-elastic effect is the orientation of the material’s crystal lattice relative to the artificial optomechanical crystal. To take this into account, we use a general photo-elastic tensor with an in-plane rotation with respect to the optomechanical crystal slab,

pijkl(θ)=Riq(θ)Rjr(θ)Rks(θ)Rlt(θ)pqrst,
where
R(θ)=(cos(θ)sin(θ)0sin(θ)cos(θ)0001),
and θ is the rotation angle. We find that the photo-elastic tensor of hexagonal crystals is independent of θ, while that of cubic crystals is periodic in θ with a periodicity of π/2. The analysis of the photo-elastic effect is much more involved comparing to the moving boundary effect, due to a large number of terms present in Eq. (14). We summarize the result in Tables 6 and 7. In these tables, when the coupling is indicated as 00, it means that the coupling depends on the rotation angle and it is 0 when θ is a multiple of π/2.

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Table 6. Photo-elastic effect gOM,PE, C4v

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Table 7. Photo-elastic effect gOM,PE, C6v

4. Cross-structure optomechanical crystals: an example

We designed and numerically simulated some slab-on-substrate optomechanical crystals with C4v and C6v symmetry to verify the theory above. For the structure with C4v symmetry, the unit cell of the optomechanical crystal is shown in Fig. 2(a). According to our simulation, such cross structures support more mechanical BICs than a unit cell with a regular square or circular hole. In our simulation, the slab is made of 600 nm thick aluminum nitride (AlN) and the substrate is silicon dioxide (SiO 2). For AlN, the Young’s modulus, Poisson’s ratio, density and refractive index are 325 GPa, 0.25, 3120 kg/m3 and 2.14, respectively; for SiO 2, these material parameters are 74.8 GPa, 0.19, 2650 kg/m3 and 1.53, respectively. Although the speed of sound in bulk AlN is greater than that of SiO 2, by making void optomechanical crystals in the slab, the effective speed of sound in the slab becomes smaller, which makes mechanical BICs below the cutoff frequency possible.

 figure: Fig. 2

Fig. 2 (a) Top view of the unit cell of an optomechanical crystal with C4v symmetry. a=1 μm. Red and white areas are AlN and air, respectively. (b) The phononic bandstructure. The red dots indicate three mechanical BICs at the Γ point and the green dot indicates a pair of degenerate mechanical guided resonances. The gray shaded region indicates the region below substrate sound line (ω=cTk). (c) The photonic bandstructure. The red dots indicate two optical BICs and the green dot indicates a pair of degenerate optical guided resonances. The gray shaded region indicates the region below substrate light line (ω=cnk).

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 figure: Fig. 3

Fig. 3 Mechanical and optical modes at the Γ point. (a-c) Total displacement (|Q|) (left) and z-component of the displacement (Qz) at the interface of the slab and substrate (right) of the three mechanical BICs with frequency 2.53 GHz (a), 2.77 GHz (b) and 2.93 GHz (c). (d-f) |E|2 (left) and Ez (right) of the two optical BICs with frequency 182 THz (d) and 194 THz (e), and one of the degenerate guided resonance with frequency 190 THz (f).

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The phononic and photonic band structures for the unit cell with a lattice constant a=1 μm is shown in Fig. 2(b) and 2(c), calculated using finite element method (COMSOL) and plane wave expansion method (MIT Photonic Bands), respectively. We find that the modes at the Γ point are consistent with the result of Table 3. For the mechanical modes, below the cut-off frequency cT,SiO2/a= 3.44 GHz, there are three mechanical BICs, indicated by the red dots in Fig. 2(b), with frequency and representation of (2.53 GHz, A2), (2.77 GHz, B1), and (2.93 GHz, B2), respectively. For the optical modes, below the cut-off frequency c/nSiO2a=196 THz, there are two optical BICs, indicated by the red dots in Fig. 2(c), with frequency and representation of (182 THz, B1) and (194 THz, A1), respectively. The mode profiles of the three mechanical BICs and two optical BICs as well as an optical guided resonance (i.e., leaky modes which nevertheless have significant energy confined in the slab [22]; the green dot in Fig. 2(c) with frequency 190 THz and representation E) are shown in Fig. 3. Due to the finite thickness (4 μm) of the substrate used in our simulations, the mechanical BICs have limited quality factors instead of being infinity in theory. However, as shown in Fig. 4, comparing to the mechanical guided resonances (e.g. the green dot in Fig. 2(b)), the quality factors of mechanical BICs are still significantly higher (106107).

 figure: Fig. 4

Fig. 4 Quality factor (Q) of the five mechanical modes (red and green dots in Fig. 2(b)) as the Bloch wavevector is scanned near the Γ point. The BICs have significantly higher quality factor than the guided resonances.

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We also calculated the optomechanical coupling between the mechanical BICs and optical modes. The photo-elastic tensor of hexagonal AlN is taken from Ref. [23]. We found that, in this C4v structure, the optomechanical coupling between the mechanical BICs and optical BICs are indeed zero, and nonzero couplings occur between the mechanical BICs and the optical guided resonance with E representation (Fig. 3(f)) as listed in Table 4, among which the largest total optomechanical coupling in a single unit cell is g¯0/2π=690 kHz for the mechanical BIC with frequency 2.93 GHz (Fig. 3(c)).

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Table 8. Optomechanical couplings between the optical guided resonance 190 THz (E) and mechanical BICs in a unit cell of the C4v cross-structure optomechanical crystal

We also simulated cross-structure optomechanical crystals with C6v symmetry in the same material system, and find that all the modes and optomechanical couplings are consistent with the theory (see Appendix). We note the optomechanical couplings obtained here are rather preliminary and can be further optimized with new structure design and parameter sweeping.

5. Summary

In summary, we have proposed a new paradigm for chip-scale optomechanics using long-lived mechanical bound states in the continuum in slab-on-substrate optomechanical crystals. Using a group theory approach, we systematically identified mechanical BICs at the Γ point in optomechanical crystals with C4v and C6v symmetry, and non-vanishing optomechanical couplings between the mechanical BICs and optical modes. We verified the theory with an example of slab-on-substrate optomechanical crystals that support a variety of mechanical BICs with large optomechanical couplings. The mechanical BICs in slab-on-substrate structures might be useful for exploring macroscopic quantum optomechanics and applications such as high-throughput sensing.

Appendix: Simulation of an optomechanical structure with C6v symmetry

Here we show the simulation results of an optomechanical crystal structure with C6v symmetry. The material system is the same as the structure with a square lattice. The top view of its unit cell is shown in Fig. 5(a). Thephononic and photonic bandstructures are shown in Fig. 5(b) and 5(c), respectively. For the triangular lattice with lattice constant a=1 μm, the cut-off frequencies for mechanical and optical modes are cT3a/2=3.98 GHz and cnSiO23a/2= 226 THz, respectively. Below the cut-off frequency, there are four mechanical BICs, with frequency and representation (2.69 GHz, B2), (3.90 GHz, B2), and (3.95 GHz (2), E2), respectively, where the last one is a degenerate pair. There are six optical BICs, with frequency and representation (198 THz, B1), (202 THz (2), E2), (209 THz, B2), and (222 THz (2), E2), respectively. The mode profiles of the mechanical BICs, optical BICs, and a pair of degenerate optical guided resonances are shown in Fig. 6. Table 8 shows the moving boundary contribution and photo-elastic contribution to the optomechanical coupling between one of the degenerate E2 mechanical BIC and the optical resonances. The coupling between the other E2 BIC and optical resonances happens to be zero because of the special linear combination of the degenerate modes COMSOL chooses. Optomechanical interactions between other modes are all zero as expected.

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Table 9. Optomechanical couplings between one mechanical BIC (3.95 GHz, E2) and optical modes in a unit cell of the C6v cross-structure optomechanical crystal

 figure: Fig. 5

Fig. 5 (a) Top view of the unit cell of an optomechanical crystal with C6v symmetry. a=1 μm. Red and white areas are AlN and air, respectively. (b) The phononic bandstructure. The red dots indicate three mechanical BICs at the Γ point. The gray shaded region indicates the region below substrate sound line (ω=cTk). (c) The photonic bandstructure. The red dots indicate two pairs of degenerate optical BICs and two non-degenerate BICs. The green dot indicates a pair of degenerate optical guided resonances. The gray shaded region indicates the region below substrate light line (ω=cnk).

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 figure: Fig. 6

Fig. 6 (a-d) Mechanical modes at Γ point. Total displacement (left) and zcomponent of the displacement at the interface between the slab and substrate (right) of the four mechanical BICs with frequency 2.69 GHz (a), 3.90 GHz (b) and 3.95 GHz (2) (c, d). (e-l) Optical modes at Γ point. |E|2 (left) and Ez (right) of the six optical BICs with frequency 198 THz (e), 202 THz (2) (f, g), 209 THz (h), 222 THz (2) (i, j) and guided resonances with frequency 226 THz (2) (k, l).

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16. R. Riedinger, A. Wallucks, I. Marinković, C. Löschnauer, M. Aspelmeyer, S. Hong, and S. Gröblacher, “Remote quantum entanglement between two micromechanical oscillators,” Nature 556, 473 (2018). [CrossRef]   [PubMed]  

17. J. D. Cohen, S. M. Meenehan, G. S. MacCabe, S. Gröblacher, A. H. Safavi-Naeini, F. Marsili, M. D. Shaw, and O. Painter, “Phonon counting and intensity interferometry of a nanomechanical resonator,” Nature 520, 522 (2015). [CrossRef]   [PubMed]  

18. S. M. Meenehan, J. D. Cohen, G. S. MacCabe, F. Marsili, M. D. Shaw, and O. Painter, “Pulsed excitation dynamics of an optomechanical crystal resonator near its quantum ground state of motion,” Phys. Rev. X 5, 041002 (2015).

19. T. Inui, Y. Tanabe, and Y. Onodera, Group theory and its applications in physics(Springer Science & Business Media, 2012).

20. T. Ochiai and K. Sakoda, “Dispersion relation and optical transmittance of a hexagonal photonic crystal slab,” Phys. Rev. B 63, 125107 (2001). [CrossRef]  

21. J. Chan, A. H. Safavi-Naeini, J. T. Hill, S. Meenehan, and O. Painter, “Optimized optomechanical crystal cavity with acoustic radiation shield,” Appl. Phys. Lett. 101, 081115 (2012). [CrossRef]  

22. S. Fan and J. D. Joannopoulos, “Analysis of guided resonances in photonic crystal slabs,” Phys. Rev. B 65, 235112 (2002). [CrossRef]  

23. S. Y. Davydov, “Evaluation of physical parameters for the group III nitrates: BN, AlN, GaN, and InN,” Semiconductors 36, 41–44 (2002). [CrossRef]  

References

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  1. H. Walther, B. T. Varcoe, B.-G. Englert, and T. Becker, “Cavity quantum electrodynamics,” Rep. Prog. Phys. 69, 1325 (2006).
    [Crossref]
  2. M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86, 1391 (2014).
    [Crossref]
  3. C. W. Hsu, B. Zhen, A. D. Stone, J. D. Joannopoulos, and M. Soljačić, “Bound states in the continuum,” Nat. Rev. Mater. 1, 16048 (2016).
    [Crossref]
  4. J. von Neumann and E. Wigner, “Über merkwürdige diskrete eigenwerte,” Phys. Z 30, 465–467 (1929).
  5. J. Lee, B. Zhen, S.-L. Chua, W. Qiu, J. D. Joannopoulos, M. Soljačić, and O. Shapira, “Observation and differentiation of unique high-Q optical resonances near zero wave vector in macroscopic photonic crystal slabs,” Phys. Rev. Lett. 109, 067401 (2012).
    [Crossref] [PubMed]
  6. C. W. Hsu, B. Zhen, J. Lee, S.-L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Observation of trapped light within the radiation continuum,” Nature 499, 188 (2013).
    [Crossref] [PubMed]
  7. A. Kodigala, T. Lepetit, Q. Gu, B. Bahari, Y. Fainman, and B. Kanté, “Lasing action from photonic bound states in continuum,” Nature 541, 196 (2017).
    [Crossref] [PubMed]
  8. M. Minkov, I. A. Williamson, M. Xiao, and S. Fan, “Zero-index bound states in the continuum,” Phys. Rev. Lett. 121, 263901 (2018).
    [Crossref]
  9. S. I. Azzam, V. M. Shalaev, A. Boltasseva, and A. V. Kildishev, “Formation of bound states in the continuum in hybrid plasmonic-photonic systems,” Phys. Rev. Lett. 121, 253901 (2018).
    [Crossref]
  10. A. Cerjan, C. W. Hsu, and M. C. Rechtsman, “Bound states in the continuum through environmental design,” arXiv e-prints arXiv:1901.07126 (2019).
  11. T. Lim and G. Farnell, “Character of pseudo surface waves on anisotropic crystals,” J. Acoust. Soc. Am. 45, 845–851 (1969).
    [Crossref]
  12. A. Maznev and A. Every, “Bound acoustic modes in the radiation continuum in isotropic layered systems without periodic structures,” Phys. Rev. B 97, 014108 (2018).
    [Crossref]
  13. M. Eichenfield, J. Chan, R. M. Camacho, K. J. Vahala, and O. Painter, “Optomechanical crystals,” Nature 462, 78 (2009).
    [Crossref] [PubMed]
  14. A. G. Krause, M. Winger, T. D. Blasius, Q. Lin, and O. Painter, “A high-resolution microchip optomechanical accelerometer,” Nat. Photons. 6, 768 (2012).
    [Crossref]
  15. K. Fang, M. H. Matheny, X. Luan, and O. Painter, “Optical transduction and routing of microwave phonons in cavity-optomechanical circuits,” Nat. Photons. 10, 489 (2016).
    [Crossref]
  16. R. Riedinger, A. Wallucks, I. Marinković, C. Löschnauer, M. Aspelmeyer, S. Hong, and S. Gröblacher, “Remote quantum entanglement between two micromechanical oscillators,” Nature 556, 473 (2018).
    [Crossref] [PubMed]
  17. J. D. Cohen, S. M. Meenehan, G. S. MacCabe, S. Gröblacher, A. H. Safavi-Naeini, F. Marsili, M. D. Shaw, and O. Painter, “Phonon counting and intensity interferometry of a nanomechanical resonator,” Nature 520, 522 (2015).
    [Crossref] [PubMed]
  18. S. M. Meenehan, J. D. Cohen, G. S. MacCabe, F. Marsili, M. D. Shaw, and O. Painter, “Pulsed excitation dynamics of an optomechanical crystal resonator near its quantum ground state of motion,” Phys. Rev. X 5, 041002 (2015).
  19. T. Inui, Y. Tanabe, and Y. Onodera, Group theory and its applications in physics(Springer Science & Business Media, 2012).
  20. T. Ochiai and K. Sakoda, “Dispersion relation and optical transmittance of a hexagonal photonic crystal slab,” Phys. Rev. B 63, 125107 (2001).
    [Crossref]
  21. J. Chan, A. H. Safavi-Naeini, J. T. Hill, S. Meenehan, and O. Painter, “Optimized optomechanical crystal cavity with acoustic radiation shield,” Appl. Phys. Lett. 101, 081115 (2012).
    [Crossref]
  22. S. Fan and J. D. Joannopoulos, “Analysis of guided resonances in photonic crystal slabs,” Phys. Rev. B 65, 235112 (2002).
    [Crossref]
  23. S. Y. Davydov, “Evaluation of physical parameters for the group III nitrates: BN, AlN, GaN, and InN,” Semiconductors 36, 41–44 (2002).
    [Crossref]

2018 (4)

M. Minkov, I. A. Williamson, M. Xiao, and S. Fan, “Zero-index bound states in the continuum,” Phys. Rev. Lett. 121, 263901 (2018).
[Crossref]

S. I. Azzam, V. M. Shalaev, A. Boltasseva, and A. V. Kildishev, “Formation of bound states in the continuum in hybrid plasmonic-photonic systems,” Phys. Rev. Lett. 121, 253901 (2018).
[Crossref]

A. Maznev and A. Every, “Bound acoustic modes in the radiation continuum in isotropic layered systems without periodic structures,” Phys. Rev. B 97, 014108 (2018).
[Crossref]

R. Riedinger, A. Wallucks, I. Marinković, C. Löschnauer, M. Aspelmeyer, S. Hong, and S. Gröblacher, “Remote quantum entanglement between two micromechanical oscillators,” Nature 556, 473 (2018).
[Crossref] [PubMed]

2017 (1)

A. Kodigala, T. Lepetit, Q. Gu, B. Bahari, Y. Fainman, and B. Kanté, “Lasing action from photonic bound states in continuum,” Nature 541, 196 (2017).
[Crossref] [PubMed]

2016 (2)

C. W. Hsu, B. Zhen, A. D. Stone, J. D. Joannopoulos, and M. Soljačić, “Bound states in the continuum,” Nat. Rev. Mater. 1, 16048 (2016).
[Crossref]

K. Fang, M. H. Matheny, X. Luan, and O. Painter, “Optical transduction and routing of microwave phonons in cavity-optomechanical circuits,” Nat. Photons. 10, 489 (2016).
[Crossref]

2015 (2)

J. D. Cohen, S. M. Meenehan, G. S. MacCabe, S. Gröblacher, A. H. Safavi-Naeini, F. Marsili, M. D. Shaw, and O. Painter, “Phonon counting and intensity interferometry of a nanomechanical resonator,” Nature 520, 522 (2015).
[Crossref] [PubMed]

S. M. Meenehan, J. D. Cohen, G. S. MacCabe, F. Marsili, M. D. Shaw, and O. Painter, “Pulsed excitation dynamics of an optomechanical crystal resonator near its quantum ground state of motion,” Phys. Rev. X 5, 041002 (2015).

2014 (1)

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86, 1391 (2014).
[Crossref]

2013 (1)

C. W. Hsu, B. Zhen, J. Lee, S.-L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Observation of trapped light within the radiation continuum,” Nature 499, 188 (2013).
[Crossref] [PubMed]

2012 (3)

J. Lee, B. Zhen, S.-L. Chua, W. Qiu, J. D. Joannopoulos, M. Soljačić, and O. Shapira, “Observation and differentiation of unique high-Q optical resonances near zero wave vector in macroscopic photonic crystal slabs,” Phys. Rev. Lett. 109, 067401 (2012).
[Crossref] [PubMed]

A. G. Krause, M. Winger, T. D. Blasius, Q. Lin, and O. Painter, “A high-resolution microchip optomechanical accelerometer,” Nat. Photons. 6, 768 (2012).
[Crossref]

J. Chan, A. H. Safavi-Naeini, J. T. Hill, S. Meenehan, and O. Painter, “Optimized optomechanical crystal cavity with acoustic radiation shield,” Appl. Phys. Lett. 101, 081115 (2012).
[Crossref]

2009 (1)

M. Eichenfield, J. Chan, R. M. Camacho, K. J. Vahala, and O. Painter, “Optomechanical crystals,” Nature 462, 78 (2009).
[Crossref] [PubMed]

2006 (1)

H. Walther, B. T. Varcoe, B.-G. Englert, and T. Becker, “Cavity quantum electrodynamics,” Rep. Prog. Phys. 69, 1325 (2006).
[Crossref]

2002 (2)

S. Fan and J. D. Joannopoulos, “Analysis of guided resonances in photonic crystal slabs,” Phys. Rev. B 65, 235112 (2002).
[Crossref]

S. Y. Davydov, “Evaluation of physical parameters for the group III nitrates: BN, AlN, GaN, and InN,” Semiconductors 36, 41–44 (2002).
[Crossref]

2001 (1)

T. Ochiai and K. Sakoda, “Dispersion relation and optical transmittance of a hexagonal photonic crystal slab,” Phys. Rev. B 63, 125107 (2001).
[Crossref]

1969 (1)

T. Lim and G. Farnell, “Character of pseudo surface waves on anisotropic crystals,” J. Acoust. Soc. Am. 45, 845–851 (1969).
[Crossref]

1929 (1)

J. von Neumann and E. Wigner, “Über merkwürdige diskrete eigenwerte,” Phys. Z 30, 465–467 (1929).

Aspelmeyer, M.

R. Riedinger, A. Wallucks, I. Marinković, C. Löschnauer, M. Aspelmeyer, S. Hong, and S. Gröblacher, “Remote quantum entanglement between two micromechanical oscillators,” Nature 556, 473 (2018).
[Crossref] [PubMed]

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86, 1391 (2014).
[Crossref]

Azzam, S. I.

S. I. Azzam, V. M. Shalaev, A. Boltasseva, and A. V. Kildishev, “Formation of bound states in the continuum in hybrid plasmonic-photonic systems,” Phys. Rev. Lett. 121, 253901 (2018).
[Crossref]

Bahari, B.

A. Kodigala, T. Lepetit, Q. Gu, B. Bahari, Y. Fainman, and B. Kanté, “Lasing action from photonic bound states in continuum,” Nature 541, 196 (2017).
[Crossref] [PubMed]

Becker, T.

H. Walther, B. T. Varcoe, B.-G. Englert, and T. Becker, “Cavity quantum electrodynamics,” Rep. Prog. Phys. 69, 1325 (2006).
[Crossref]

Blasius, T. D.

A. G. Krause, M. Winger, T. D. Blasius, Q. Lin, and O. Painter, “A high-resolution microchip optomechanical accelerometer,” Nat. Photons. 6, 768 (2012).
[Crossref]

Boltasseva, A.

S. I. Azzam, V. M. Shalaev, A. Boltasseva, and A. V. Kildishev, “Formation of bound states in the continuum in hybrid plasmonic-photonic systems,” Phys. Rev. Lett. 121, 253901 (2018).
[Crossref]

Camacho, R. M.

M. Eichenfield, J. Chan, R. M. Camacho, K. J. Vahala, and O. Painter, “Optomechanical crystals,” Nature 462, 78 (2009).
[Crossref] [PubMed]

Cerjan, A.

A. Cerjan, C. W. Hsu, and M. C. Rechtsman, “Bound states in the continuum through environmental design,” arXiv e-prints arXiv:1901.07126 (2019).

Chan, J.

J. Chan, A. H. Safavi-Naeini, J. T. Hill, S. Meenehan, and O. Painter, “Optimized optomechanical crystal cavity with acoustic radiation shield,” Appl. Phys. Lett. 101, 081115 (2012).
[Crossref]

M. Eichenfield, J. Chan, R. M. Camacho, K. J. Vahala, and O. Painter, “Optomechanical crystals,” Nature 462, 78 (2009).
[Crossref] [PubMed]

Chua, S.-L.

C. W. Hsu, B. Zhen, J. Lee, S.-L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Observation of trapped light within the radiation continuum,” Nature 499, 188 (2013).
[Crossref] [PubMed]

J. Lee, B. Zhen, S.-L. Chua, W. Qiu, J. D. Joannopoulos, M. Soljačić, and O. Shapira, “Observation and differentiation of unique high-Q optical resonances near zero wave vector in macroscopic photonic crystal slabs,” Phys. Rev. Lett. 109, 067401 (2012).
[Crossref] [PubMed]

Cohen, J. D.

J. D. Cohen, S. M. Meenehan, G. S. MacCabe, S. Gröblacher, A. H. Safavi-Naeini, F. Marsili, M. D. Shaw, and O. Painter, “Phonon counting and intensity interferometry of a nanomechanical resonator,” Nature 520, 522 (2015).
[Crossref] [PubMed]

S. M. Meenehan, J. D. Cohen, G. S. MacCabe, F. Marsili, M. D. Shaw, and O. Painter, “Pulsed excitation dynamics of an optomechanical crystal resonator near its quantum ground state of motion,” Phys. Rev. X 5, 041002 (2015).

Davydov, S. Y.

S. Y. Davydov, “Evaluation of physical parameters for the group III nitrates: BN, AlN, GaN, and InN,” Semiconductors 36, 41–44 (2002).
[Crossref]

Eichenfield, M.

M. Eichenfield, J. Chan, R. M. Camacho, K. J. Vahala, and O. Painter, “Optomechanical crystals,” Nature 462, 78 (2009).
[Crossref] [PubMed]

Englert, B.-G.

H. Walther, B. T. Varcoe, B.-G. Englert, and T. Becker, “Cavity quantum electrodynamics,” Rep. Prog. Phys. 69, 1325 (2006).
[Crossref]

Every, A.

A. Maznev and A. Every, “Bound acoustic modes in the radiation continuum in isotropic layered systems without periodic structures,” Phys. Rev. B 97, 014108 (2018).
[Crossref]

Fainman, Y.

A. Kodigala, T. Lepetit, Q. Gu, B. Bahari, Y. Fainman, and B. Kanté, “Lasing action from photonic bound states in continuum,” Nature 541, 196 (2017).
[Crossref] [PubMed]

Fan, S.

M. Minkov, I. A. Williamson, M. Xiao, and S. Fan, “Zero-index bound states in the continuum,” Phys. Rev. Lett. 121, 263901 (2018).
[Crossref]

S. Fan and J. D. Joannopoulos, “Analysis of guided resonances in photonic crystal slabs,” Phys. Rev. B 65, 235112 (2002).
[Crossref]

Fang, K.

K. Fang, M. H. Matheny, X. Luan, and O. Painter, “Optical transduction and routing of microwave phonons in cavity-optomechanical circuits,” Nat. Photons. 10, 489 (2016).
[Crossref]

Farnell, G.

T. Lim and G. Farnell, “Character of pseudo surface waves on anisotropic crystals,” J. Acoust. Soc. Am. 45, 845–851 (1969).
[Crossref]

Gröblacher, S.

R. Riedinger, A. Wallucks, I. Marinković, C. Löschnauer, M. Aspelmeyer, S. Hong, and S. Gröblacher, “Remote quantum entanglement between two micromechanical oscillators,” Nature 556, 473 (2018).
[Crossref] [PubMed]

J. D. Cohen, S. M. Meenehan, G. S. MacCabe, S. Gröblacher, A. H. Safavi-Naeini, F. Marsili, M. D. Shaw, and O. Painter, “Phonon counting and intensity interferometry of a nanomechanical resonator,” Nature 520, 522 (2015).
[Crossref] [PubMed]

Gu, Q.

A. Kodigala, T. Lepetit, Q. Gu, B. Bahari, Y. Fainman, and B. Kanté, “Lasing action from photonic bound states in continuum,” Nature 541, 196 (2017).
[Crossref] [PubMed]

Hill, J. T.

J. Chan, A. H. Safavi-Naeini, J. T. Hill, S. Meenehan, and O. Painter, “Optimized optomechanical crystal cavity with acoustic radiation shield,” Appl. Phys. Lett. 101, 081115 (2012).
[Crossref]

Hong, S.

R. Riedinger, A. Wallucks, I. Marinković, C. Löschnauer, M. Aspelmeyer, S. Hong, and S. Gröblacher, “Remote quantum entanglement between two micromechanical oscillators,” Nature 556, 473 (2018).
[Crossref] [PubMed]

Hsu, C. W.

C. W. Hsu, B. Zhen, A. D. Stone, J. D. Joannopoulos, and M. Soljačić, “Bound states in the continuum,” Nat. Rev. Mater. 1, 16048 (2016).
[Crossref]

C. W. Hsu, B. Zhen, J. Lee, S.-L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Observation of trapped light within the radiation continuum,” Nature 499, 188 (2013).
[Crossref] [PubMed]

A. Cerjan, C. W. Hsu, and M. C. Rechtsman, “Bound states in the continuum through environmental design,” arXiv e-prints arXiv:1901.07126 (2019).

Inui, T.

T. Inui, Y. Tanabe, and Y. Onodera, Group theory and its applications in physics(Springer Science & Business Media, 2012).

Joannopoulos, J. D.

C. W. Hsu, B. Zhen, A. D. Stone, J. D. Joannopoulos, and M. Soljačić, “Bound states in the continuum,” Nat. Rev. Mater. 1, 16048 (2016).
[Crossref]

C. W. Hsu, B. Zhen, J. Lee, S.-L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Observation of trapped light within the radiation continuum,” Nature 499, 188 (2013).
[Crossref] [PubMed]

J. Lee, B. Zhen, S.-L. Chua, W. Qiu, J. D. Joannopoulos, M. Soljačić, and O. Shapira, “Observation and differentiation of unique high-Q optical resonances near zero wave vector in macroscopic photonic crystal slabs,” Phys. Rev. Lett. 109, 067401 (2012).
[Crossref] [PubMed]

S. Fan and J. D. Joannopoulos, “Analysis of guided resonances in photonic crystal slabs,” Phys. Rev. B 65, 235112 (2002).
[Crossref]

Johnson, S. G.

C. W. Hsu, B. Zhen, J. Lee, S.-L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Observation of trapped light within the radiation continuum,” Nature 499, 188 (2013).
[Crossref] [PubMed]

Kanté, B.

A. Kodigala, T. Lepetit, Q. Gu, B. Bahari, Y. Fainman, and B. Kanté, “Lasing action from photonic bound states in continuum,” Nature 541, 196 (2017).
[Crossref] [PubMed]

Kildishev, A. V.

S. I. Azzam, V. M. Shalaev, A. Boltasseva, and A. V. Kildishev, “Formation of bound states in the continuum in hybrid plasmonic-photonic systems,” Phys. Rev. Lett. 121, 253901 (2018).
[Crossref]

Kippenberg, T. J.

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86, 1391 (2014).
[Crossref]

Kodigala, A.

A. Kodigala, T. Lepetit, Q. Gu, B. Bahari, Y. Fainman, and B. Kanté, “Lasing action from photonic bound states in continuum,” Nature 541, 196 (2017).
[Crossref] [PubMed]

Krause, A. G.

A. G. Krause, M. Winger, T. D. Blasius, Q. Lin, and O. Painter, “A high-resolution microchip optomechanical accelerometer,” Nat. Photons. 6, 768 (2012).
[Crossref]

Lee, J.

C. W. Hsu, B. Zhen, J. Lee, S.-L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Observation of trapped light within the radiation continuum,” Nature 499, 188 (2013).
[Crossref] [PubMed]

J. Lee, B. Zhen, S.-L. Chua, W. Qiu, J. D. Joannopoulos, M. Soljačić, and O. Shapira, “Observation and differentiation of unique high-Q optical resonances near zero wave vector in macroscopic photonic crystal slabs,” Phys. Rev. Lett. 109, 067401 (2012).
[Crossref] [PubMed]

Lepetit, T.

A. Kodigala, T. Lepetit, Q. Gu, B. Bahari, Y. Fainman, and B. Kanté, “Lasing action from photonic bound states in continuum,” Nature 541, 196 (2017).
[Crossref] [PubMed]

Lim, T.

T. Lim and G. Farnell, “Character of pseudo surface waves on anisotropic crystals,” J. Acoust. Soc. Am. 45, 845–851 (1969).
[Crossref]

Lin, Q.

A. G. Krause, M. Winger, T. D. Blasius, Q. Lin, and O. Painter, “A high-resolution microchip optomechanical accelerometer,” Nat. Photons. 6, 768 (2012).
[Crossref]

Löschnauer, C.

R. Riedinger, A. Wallucks, I. Marinković, C. Löschnauer, M. Aspelmeyer, S. Hong, and S. Gröblacher, “Remote quantum entanglement between two micromechanical oscillators,” Nature 556, 473 (2018).
[Crossref] [PubMed]

Luan, X.

K. Fang, M. H. Matheny, X. Luan, and O. Painter, “Optical transduction and routing of microwave phonons in cavity-optomechanical circuits,” Nat. Photons. 10, 489 (2016).
[Crossref]

MacCabe, G. S.

S. M. Meenehan, J. D. Cohen, G. S. MacCabe, F. Marsili, M. D. Shaw, and O. Painter, “Pulsed excitation dynamics of an optomechanical crystal resonator near its quantum ground state of motion,” Phys. Rev. X 5, 041002 (2015).

J. D. Cohen, S. M. Meenehan, G. S. MacCabe, S. Gröblacher, A. H. Safavi-Naeini, F. Marsili, M. D. Shaw, and O. Painter, “Phonon counting and intensity interferometry of a nanomechanical resonator,” Nature 520, 522 (2015).
[Crossref] [PubMed]

Marinkovic, I.

R. Riedinger, A. Wallucks, I. Marinković, C. Löschnauer, M. Aspelmeyer, S. Hong, and S. Gröblacher, “Remote quantum entanglement between two micromechanical oscillators,” Nature 556, 473 (2018).
[Crossref] [PubMed]

Marquardt, F.

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86, 1391 (2014).
[Crossref]

Marsili, F.

J. D. Cohen, S. M. Meenehan, G. S. MacCabe, S. Gröblacher, A. H. Safavi-Naeini, F. Marsili, M. D. Shaw, and O. Painter, “Phonon counting and intensity interferometry of a nanomechanical resonator,” Nature 520, 522 (2015).
[Crossref] [PubMed]

S. M. Meenehan, J. D. Cohen, G. S. MacCabe, F. Marsili, M. D. Shaw, and O. Painter, “Pulsed excitation dynamics of an optomechanical crystal resonator near its quantum ground state of motion,” Phys. Rev. X 5, 041002 (2015).

Matheny, M. H.

K. Fang, M. H. Matheny, X. Luan, and O. Painter, “Optical transduction and routing of microwave phonons in cavity-optomechanical circuits,” Nat. Photons. 10, 489 (2016).
[Crossref]

Maznev, A.

A. Maznev and A. Every, “Bound acoustic modes in the radiation continuum in isotropic layered systems without periodic structures,” Phys. Rev. B 97, 014108 (2018).
[Crossref]

Meenehan, S.

J. Chan, A. H. Safavi-Naeini, J. T. Hill, S. Meenehan, and O. Painter, “Optimized optomechanical crystal cavity with acoustic radiation shield,” Appl. Phys. Lett. 101, 081115 (2012).
[Crossref]

Meenehan, S. M.

S. M. Meenehan, J. D. Cohen, G. S. MacCabe, F. Marsili, M. D. Shaw, and O. Painter, “Pulsed excitation dynamics of an optomechanical crystal resonator near its quantum ground state of motion,” Phys. Rev. X 5, 041002 (2015).

J. D. Cohen, S. M. Meenehan, G. S. MacCabe, S. Gröblacher, A. H. Safavi-Naeini, F. Marsili, M. D. Shaw, and O. Painter, “Phonon counting and intensity interferometry of a nanomechanical resonator,” Nature 520, 522 (2015).
[Crossref] [PubMed]

Minkov, M.

M. Minkov, I. A. Williamson, M. Xiao, and S. Fan, “Zero-index bound states in the continuum,” Phys. Rev. Lett. 121, 263901 (2018).
[Crossref]

Ochiai, T.

T. Ochiai and K. Sakoda, “Dispersion relation and optical transmittance of a hexagonal photonic crystal slab,” Phys. Rev. B 63, 125107 (2001).
[Crossref]

Onodera, Y.

T. Inui, Y. Tanabe, and Y. Onodera, Group theory and its applications in physics(Springer Science & Business Media, 2012).

Painter, O.

K. Fang, M. H. Matheny, X. Luan, and O. Painter, “Optical transduction and routing of microwave phonons in cavity-optomechanical circuits,” Nat. Photons. 10, 489 (2016).
[Crossref]

S. M. Meenehan, J. D. Cohen, G. S. MacCabe, F. Marsili, M. D. Shaw, and O. Painter, “Pulsed excitation dynamics of an optomechanical crystal resonator near its quantum ground state of motion,” Phys. Rev. X 5, 041002 (2015).

J. D. Cohen, S. M. Meenehan, G. S. MacCabe, S. Gröblacher, A. H. Safavi-Naeini, F. Marsili, M. D. Shaw, and O. Painter, “Phonon counting and intensity interferometry of a nanomechanical resonator,” Nature 520, 522 (2015).
[Crossref] [PubMed]

J. Chan, A. H. Safavi-Naeini, J. T. Hill, S. Meenehan, and O. Painter, “Optimized optomechanical crystal cavity with acoustic radiation shield,” Appl. Phys. Lett. 101, 081115 (2012).
[Crossref]

A. G. Krause, M. Winger, T. D. Blasius, Q. Lin, and O. Painter, “A high-resolution microchip optomechanical accelerometer,” Nat. Photons. 6, 768 (2012).
[Crossref]

M. Eichenfield, J. Chan, R. M. Camacho, K. J. Vahala, and O. Painter, “Optomechanical crystals,” Nature 462, 78 (2009).
[Crossref] [PubMed]

Qiu, W.

J. Lee, B. Zhen, S.-L. Chua, W. Qiu, J. D. Joannopoulos, M. Soljačić, and O. Shapira, “Observation and differentiation of unique high-Q optical resonances near zero wave vector in macroscopic photonic crystal slabs,” Phys. Rev. Lett. 109, 067401 (2012).
[Crossref] [PubMed]

Rechtsman, M. C.

A. Cerjan, C. W. Hsu, and M. C. Rechtsman, “Bound states in the continuum through environmental design,” arXiv e-prints arXiv:1901.07126 (2019).

Riedinger, R.

R. Riedinger, A. Wallucks, I. Marinković, C. Löschnauer, M. Aspelmeyer, S. Hong, and S. Gröblacher, “Remote quantum entanglement between two micromechanical oscillators,” Nature 556, 473 (2018).
[Crossref] [PubMed]

Safavi-Naeini, A. H.

J. D. Cohen, S. M. Meenehan, G. S. MacCabe, S. Gröblacher, A. H. Safavi-Naeini, F. Marsili, M. D. Shaw, and O. Painter, “Phonon counting and intensity interferometry of a nanomechanical resonator,” Nature 520, 522 (2015).
[Crossref] [PubMed]

J. Chan, A. H. Safavi-Naeini, J. T. Hill, S. Meenehan, and O. Painter, “Optimized optomechanical crystal cavity with acoustic radiation shield,” Appl. Phys. Lett. 101, 081115 (2012).
[Crossref]

Sakoda, K.

T. Ochiai and K. Sakoda, “Dispersion relation and optical transmittance of a hexagonal photonic crystal slab,” Phys. Rev. B 63, 125107 (2001).
[Crossref]

Shalaev, V. M.

S. I. Azzam, V. M. Shalaev, A. Boltasseva, and A. V. Kildishev, “Formation of bound states in the continuum in hybrid plasmonic-photonic systems,” Phys. Rev. Lett. 121, 253901 (2018).
[Crossref]

Shapira, O.

J. Lee, B. Zhen, S.-L. Chua, W. Qiu, J. D. Joannopoulos, M. Soljačić, and O. Shapira, “Observation and differentiation of unique high-Q optical resonances near zero wave vector in macroscopic photonic crystal slabs,” Phys. Rev. Lett. 109, 067401 (2012).
[Crossref] [PubMed]

Shaw, M. D.

J. D. Cohen, S. M. Meenehan, G. S. MacCabe, S. Gröblacher, A. H. Safavi-Naeini, F. Marsili, M. D. Shaw, and O. Painter, “Phonon counting and intensity interferometry of a nanomechanical resonator,” Nature 520, 522 (2015).
[Crossref] [PubMed]

S. M. Meenehan, J. D. Cohen, G. S. MacCabe, F. Marsili, M. D. Shaw, and O. Painter, “Pulsed excitation dynamics of an optomechanical crystal resonator near its quantum ground state of motion,” Phys. Rev. X 5, 041002 (2015).

Soljacic, M.

C. W. Hsu, B. Zhen, A. D. Stone, J. D. Joannopoulos, and M. Soljačić, “Bound states in the continuum,” Nat. Rev. Mater. 1, 16048 (2016).
[Crossref]

C. W. Hsu, B. Zhen, J. Lee, S.-L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Observation of trapped light within the radiation continuum,” Nature 499, 188 (2013).
[Crossref] [PubMed]

J. Lee, B. Zhen, S.-L. Chua, W. Qiu, J. D. Joannopoulos, M. Soljačić, and O. Shapira, “Observation and differentiation of unique high-Q optical resonances near zero wave vector in macroscopic photonic crystal slabs,” Phys. Rev. Lett. 109, 067401 (2012).
[Crossref] [PubMed]

Stone, A. D.

C. W. Hsu, B. Zhen, A. D. Stone, J. D. Joannopoulos, and M. Soljačić, “Bound states in the continuum,” Nat. Rev. Mater. 1, 16048 (2016).
[Crossref]

Tanabe, Y.

T. Inui, Y. Tanabe, and Y. Onodera, Group theory and its applications in physics(Springer Science & Business Media, 2012).

Vahala, K. J.

M. Eichenfield, J. Chan, R. M. Camacho, K. J. Vahala, and O. Painter, “Optomechanical crystals,” Nature 462, 78 (2009).
[Crossref] [PubMed]

Varcoe, B. T.

H. Walther, B. T. Varcoe, B.-G. Englert, and T. Becker, “Cavity quantum electrodynamics,” Rep. Prog. Phys. 69, 1325 (2006).
[Crossref]

von Neumann, J.

J. von Neumann and E. Wigner, “Über merkwürdige diskrete eigenwerte,” Phys. Z 30, 465–467 (1929).

Wallucks, A.

R. Riedinger, A. Wallucks, I. Marinković, C. Löschnauer, M. Aspelmeyer, S. Hong, and S. Gröblacher, “Remote quantum entanglement between two micromechanical oscillators,” Nature 556, 473 (2018).
[Crossref] [PubMed]

Walther, H.

H. Walther, B. T. Varcoe, B.-G. Englert, and T. Becker, “Cavity quantum electrodynamics,” Rep. Prog. Phys. 69, 1325 (2006).
[Crossref]

Wigner, E.

J. von Neumann and E. Wigner, “Über merkwürdige diskrete eigenwerte,” Phys. Z 30, 465–467 (1929).

Williamson, I. A.

M. Minkov, I. A. Williamson, M. Xiao, and S. Fan, “Zero-index bound states in the continuum,” Phys. Rev. Lett. 121, 263901 (2018).
[Crossref]

Winger, M.

A. G. Krause, M. Winger, T. D. Blasius, Q. Lin, and O. Painter, “A high-resolution microchip optomechanical accelerometer,” Nat. Photons. 6, 768 (2012).
[Crossref]

Xiao, M.

M. Minkov, I. A. Williamson, M. Xiao, and S. Fan, “Zero-index bound states in the continuum,” Phys. Rev. Lett. 121, 263901 (2018).
[Crossref]

Zhen, B.

C. W. Hsu, B. Zhen, A. D. Stone, J. D. Joannopoulos, and M. Soljačić, “Bound states in the continuum,” Nat. Rev. Mater. 1, 16048 (2016).
[Crossref]

C. W. Hsu, B. Zhen, J. Lee, S.-L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Observation of trapped light within the radiation continuum,” Nature 499, 188 (2013).
[Crossref] [PubMed]

J. Lee, B. Zhen, S.-L. Chua, W. Qiu, J. D. Joannopoulos, M. Soljačić, and O. Shapira, “Observation and differentiation of unique high-Q optical resonances near zero wave vector in macroscopic photonic crystal slabs,” Phys. Rev. Lett. 109, 067401 (2012).
[Crossref] [PubMed]

Appl. Phys. Lett. (1)

J. Chan, A. H. Safavi-Naeini, J. T. Hill, S. Meenehan, and O. Painter, “Optimized optomechanical crystal cavity with acoustic radiation shield,” Appl. Phys. Lett. 101, 081115 (2012).
[Crossref]

J. Acoust. Soc. Am. (1)

T. Lim and G. Farnell, “Character of pseudo surface waves on anisotropic crystals,” J. Acoust. Soc. Am. 45, 845–851 (1969).
[Crossref]

Nat. Photons. (2)

A. G. Krause, M. Winger, T. D. Blasius, Q. Lin, and O. Painter, “A high-resolution microchip optomechanical accelerometer,” Nat. Photons. 6, 768 (2012).
[Crossref]

K. Fang, M. H. Matheny, X. Luan, and O. Painter, “Optical transduction and routing of microwave phonons in cavity-optomechanical circuits,” Nat. Photons. 10, 489 (2016).
[Crossref]

Nat. Rev. Mater. (1)

C. W. Hsu, B. Zhen, A. D. Stone, J. D. Joannopoulos, and M. Soljačić, “Bound states in the continuum,” Nat. Rev. Mater. 1, 16048 (2016).
[Crossref]

Nature (5)

C. W. Hsu, B. Zhen, J. Lee, S.-L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Observation of trapped light within the radiation continuum,” Nature 499, 188 (2013).
[Crossref] [PubMed]

A. Kodigala, T. Lepetit, Q. Gu, B. Bahari, Y. Fainman, and B. Kanté, “Lasing action from photonic bound states in continuum,” Nature 541, 196 (2017).
[Crossref] [PubMed]

R. Riedinger, A. Wallucks, I. Marinković, C. Löschnauer, M. Aspelmeyer, S. Hong, and S. Gröblacher, “Remote quantum entanglement between two micromechanical oscillators,” Nature 556, 473 (2018).
[Crossref] [PubMed]

J. D. Cohen, S. M. Meenehan, G. S. MacCabe, S. Gröblacher, A. H. Safavi-Naeini, F. Marsili, M. D. Shaw, and O. Painter, “Phonon counting and intensity interferometry of a nanomechanical resonator,” Nature 520, 522 (2015).
[Crossref] [PubMed]

M. Eichenfield, J. Chan, R. M. Camacho, K. J. Vahala, and O. Painter, “Optomechanical crystals,” Nature 462, 78 (2009).
[Crossref] [PubMed]

Phys. Rev. B (3)

A. Maznev and A. Every, “Bound acoustic modes in the radiation continuum in isotropic layered systems without periodic structures,” Phys. Rev. B 97, 014108 (2018).
[Crossref]

S. Fan and J. D. Joannopoulos, “Analysis of guided resonances in photonic crystal slabs,” Phys. Rev. B 65, 235112 (2002).
[Crossref]

T. Ochiai and K. Sakoda, “Dispersion relation and optical transmittance of a hexagonal photonic crystal slab,” Phys. Rev. B 63, 125107 (2001).
[Crossref]

Phys. Rev. Lett. (3)

J. Lee, B. Zhen, S.-L. Chua, W. Qiu, J. D. Joannopoulos, M. Soljačić, and O. Shapira, “Observation and differentiation of unique high-Q optical resonances near zero wave vector in macroscopic photonic crystal slabs,” Phys. Rev. Lett. 109, 067401 (2012).
[Crossref] [PubMed]

M. Minkov, I. A. Williamson, M. Xiao, and S. Fan, “Zero-index bound states in the continuum,” Phys. Rev. Lett. 121, 263901 (2018).
[Crossref]

S. I. Azzam, V. M. Shalaev, A. Boltasseva, and A. V. Kildishev, “Formation of bound states in the continuum in hybrid plasmonic-photonic systems,” Phys. Rev. Lett. 121, 253901 (2018).
[Crossref]

Phys. Rev. X (1)

S. M. Meenehan, J. D. Cohen, G. S. MacCabe, F. Marsili, M. D. Shaw, and O. Painter, “Pulsed excitation dynamics of an optomechanical crystal resonator near its quantum ground state of motion,” Phys. Rev. X 5, 041002 (2015).

Phys. Z (1)

J. von Neumann and E. Wigner, “Über merkwürdige diskrete eigenwerte,” Phys. Z 30, 465–467 (1929).

Rep. Prog. Phys. (1)

H. Walther, B. T. Varcoe, B.-G. Englert, and T. Becker, “Cavity quantum electrodynamics,” Rep. Prog. Phys. 69, 1325 (2006).
[Crossref]

Rev. Mod. Phys. (1)

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86, 1391 (2014).
[Crossref]

Semiconductors (1)

S. Y. Davydov, “Evaluation of physical parameters for the group III nitrates: BN, AlN, GaN, and InN,” Semiconductors 36, 41–44 (2002).
[Crossref]

Other (2)

A. Cerjan, C. W. Hsu, and M. C. Rechtsman, “Bound states in the continuum through environmental design,” arXiv e-prints arXiv:1901.07126 (2019).

T. Inui, Y. Tanabe, and Y. Onodera, Group theory and its applications in physics(Springer Science & Business Media, 2012).

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Figures (6)

Fig. 1
Fig. 1 Schematic of the structure of a slab-on-substrate optomechancial crystal.
Fig. 2
Fig. 2 (a) Top view of the unit cell of an optomechanical crystal with C4v symmetry. a = 1   μm. Red and white areas are AlN and air, respectively. (b) The phononic bandstructure. The red dots indicate three mechanical BICs at the Γ point and the green dot indicates a pair of degenerate mechanical guided resonances. The gray shaded region indicates the region below substrate sound line ( ω = c T k ). (c) The photonic bandstructure. The red dots indicate two optical BICs and the green dot indicates a pair of degenerate optical guided resonances. The gray shaded region indicates the region below substrate light line ( ω = c n k ).
Fig. 3
Fig. 3 Mechanical and optical modes at the Γ point. (a-c) Total displacement ( | Q |) (left) and z-component of the displacement (Qz) at the interface of the slab and substrate (right) of the three mechanical BICs with frequency 2.53 GHz (a), 2.77 GHz (b) and 2.93 GHz (c). (d-f) | E | 2 (left) and Ez (right) of the two optical BICs with frequency 182 THz (d) and 194 THz (e), and one of the degenerate guided resonance with frequency 190 THz (f).
Fig. 4
Fig. 4 Quality factor (Q) of the five mechanical modes (red and green dots in Fig. 2(b)) as the Bloch wavevector is scanned near the Γ point. The BICs have significantly higher quality factor than the guided resonances.
Fig. 5
Fig. 5 (a) Top view of the unit cell of an optomechanical crystal with C6v symmetry. a = 1   μm. Red and white areas are AlN and air, respectively. (b) The phononic bandstructure. The red dots indicate three mechanical BICs at the Γ point. The gray shaded region indicates the region below substrate sound line ( ω = c T k ). (c) The photonic bandstructure. The red dots indicate two pairs of degenerate optical BICs and two non-degenerate BICs. The green dot indicates a pair of degenerate optical guided resonances. The gray shaded region indicates the region below substrate light line ( ω = c n k ).
Fig. 6
Fig. 6 (a-d) Mechanical modes at Γ point. Total displacement (left) and z component of the displacement at the interface between the slab and substrate (right) of the four mechanical BICs with frequency 2.69 GHz (a), 3.90 GHz (b) and 3.95 GHz (2) (c, d). (e-l) Optical modes at Γ point. | E | 2 (left) and Ez (right) of the six optical BICs with frequency 198 THz (e), 202 THz (2) (f, g), 209 THz (h), 222 THz (2) (i, j) and guided resonances with frequency 226 THz (2) (k, l).

Tables (9)

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Table 1 Character table for the C 4 v point group

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Table 2 Character table for the C 6 v point group

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Table 3 Mechanical and optical BICs at the Γ point

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Table 4 Moving boundary effect g OM , MB, C4v

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Table 5 Moving boundary effect g OM , MB, C 6 v

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Table 6 Photo-elastic effect g OM , PE, C 4 v

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Table 7 Photo-elastic effect g OM , PE, C 6 v

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Table 8 Optomechanical couplings between the optical guided resonance 190 THz (E) and mechanical BICs in a unit cell of the C4v cross-structure optomechanical crystal

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Table 9 Optomechanical couplings between one mechanical BIC (3.95 GHz, E 2) and optical modes in a unit cell of the C 6 v cross-structure optomechanical crystal

Equations (21)

Equations on this page are rendered with MathJax. Learn more.

ρ 2 Q t 2 = Y 2 ( 1 + ν ) 2 Q + Y 2 ( 1 + ν ) ( 1 2 ν ) Q ,
Q k ( r ) e i ω t = u k ( r ) e i ( k ρ ω t ) ,
u k ( r ) = j = 0 f j ( z ) e i G j ρ ,
2 Q T ( L ) t 2 = c T ( L ) 2 2 Q T ( L ) ,
Q k , ( r ) = e i k ρ j = 0 ( A T , j e i k T , z j z + A L , j e i k L , z j z ) e i G j ρ ,
k T ( L ) , z j = ω 2 c T ( L ) 2 | k + G j | 2 ,
Q 0 = Q T , 0 + Q L , 0 = ( u e x + v e y ) e i k T , z 0 z + w e z e i k L , z 0 z .
Q 1 , T = l A T , l e i k T , z l z e i G l ρ ,
1 c 2 2 E t 2 = 1 n 2 2 E ,
E k ( r ) = e i k ρ j = 0 A j e i k z j z + i G j ρ ,
k z j = ω 2 ( c / n ) 2 | k + G j | 2 .
g 0 = g 0 , MB + g 0 , PE 2 m eff ω m ( g OM , MB + g OM , PE ) ,
g OM , MB = ω o 2 ( Q n ^ ) ( Δ ϵ | E | 2 Δ ϵ 1 | D | 2 ) d S E * D d V ,
g OM , PE = ω o 2 ϵ 0 n 4 E i * E j p i j k l S k l d V E * D d V ,
g 0 = g ¯ 0 N 1 N n , m e i k ( n a 1 + m a 2 ) ,
g 0 = g ¯ 0 N .
G = g 0 N n p = g ¯ 0 n p ,
( Q n ^ ) f d S = α α + π ( Q n ^ ) f d S + α α π ( Q n ^ ) f d S   = θ = 0 π ( Q ( α + θ ) + σ α Q ( α θ ) ) n ^ ( α + θ ) f ( α + θ ) d S .
( Q n ^ ) f d S = 0 π ( Q n ^ ) f d S + π 2 π ( Q n ^ ) f d S   = θ = 0 π ( Q ( θ ) + C 2 Q ( θ + π ) ) n ^ ( θ ) f ( θ ) d S .
p i j k l ( θ ) = R i q ( θ ) R j r ( θ ) R k s ( θ ) R l t ( θ ) p q r s t ,
R ( θ ) = ( cos ( θ ) sin ( θ ) 0 sin ( θ ) cos ( θ ) 0 0 0 1 ) ,

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