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Phoxonic crystal is a promising material for manipulating sound and light simultaneously. In this paper, we theoretically demonstrate the propagation of acoustic and optical waves along the truncated surface of a two-dimensional square-latticed phoxonic crystal. Further, a phoxonic crystal hetero-structure cavity is proposed, which can simultaneously confine surface acoustic and optical waves. The interface motion and photoelastic effects are taken into account in the acousto-optical coupling. The results show obvious shifts in eigenfrequencies of the photonic cavity modes induced by different phononic cavity modes. The symmetry of the phononic cavity modes plays a more important role in the single-phonon exchange process than in the case of the multi-phonon exchange. Under the same deformation, the frequency shift of the photonic transverse electric mode is larger than that of the transverse magnetic mode.
© 2014 Optical Society of America
1. Introduction
Photonic crystals (PTCs) and phononic crystals (PNCs) are artificial structures with periodic changes in their material parameters. These structures can exhibit photonic bandgaps (PTBGs) or phononic bandgaps (PNBGs) in which the propagation of photons or phonons is prohibited, respectively. PTCs and PNCs are promising materials for optical or acoustic devices such as waveguides, resonators, sensors and super prisms, etc. Recently, periodic structures possessing simultaneous PTBGs and PNBGs are of increasing interest. These structures with the abilities of controlling, guiding and confining optical and acoustic energies simultaneously are termed as phoxonic crystals (PXCs) [1] or optomechanical crystals [2]. Furthermore, PXCs can enhance acousto-optical (AO) or optomechanical interactions. Some efforts have been devoted to the generation and tuning of simultaneous photonic and phononic bandgaps [1,3–9]. The interaction between phonons and photons in PXC cavities and waveguides has also been studied [2,10–18]. PXC sensors for simultaneous determination of optical and acoustic properties of analytes with high sensitivity in ultra small volume have been reported [19,20].
Compared with bulk acoustic waves, the investigations of surface acoustic waves (SAWs) in PNCs are rather limited, although SAW devices have been in commercial use for over decades. However, in recent years more and more theoretical research works focused on SAW bandgaps and PNC SAW devices have been reported [21–27]. Besides, several experimental investigations on SAWs in PNCs have been carried out [28–31]. Actually, SAWs can be conveniently excited and detected by making use of piezoelectric materials or an optical excitation. The introduction of a SAW PNC to SAW devices may increase the performances of conventional SAW devices.
Surface optical waves (SOWs) in PTCs are non-radiative waves localized at the dielectric-air interface, decaying into both media and propagating along the interface. Unlike surface plasmonic waves at a metallic surface, they are almost absorption-free because dielectrics exhibit much lower extinction coefficients than metals. The electromagnetic field of SOWs extends more into the environment, which is very beneficial for optical index sensing in chemical and biological analysis owing to the strong interaction between the surface waves and environmental analytes. Moreover, engineering of SOWs can be useful for the manipulation of light at the surface of PTCs, such as beam shaping and focusing. In recent years, many designs based on SOWs in PTCs have been demonstrated, including cavities [32], waveguides [33], collimated emissions [34], sensors [35,36], etc.
In the present paper, we demonstrate the existence of simultaneous surface acoustic and optical waves in a two-dimensional (2D) PXC. By tuning the surface geometry properly, we propose a hetero-structure cavity which can confine both acoustic and optical energies at the surface of the cavity. The AO interaction between different phononic and photonic cavity modes is investigated. The effect of the symmetry of phononic modes on the coupling strength is also discussed. The numerical results are calculated by the finite element method (FEM) using COMSOL Multiphysics.
2. Design of hetero-structure cavity
Figure 1(a) shows the schematic diagram of the PXC. The PXC is formed by drilling complex holes with a square lattice in silicon matrix. Alternatively it can be also considered as a square lattice of circular scatterers connected by thin connectors. This design of the PXC insures simultaneous large complete photonic and phononic bandgaps [8]. The PXC is truncated at both upper and lower surfaces to generate surface waves. Initially, it has a finite length of 9 unit-cells along the y-axis. For the purpose of modifying the surface bands, we add a silicon layer at the upper surface and change its thickness d, i.e. varying the truncation position at the upper surface. In Figs. 1(b)-1(d) we illustrate the calculated dispersion curves of the surface bands for the geometrical parameters r = 0.35a, w = 0.9a and d = 0.12a with a being the lattice constant. The phononic and photonic eigenfrequencies are normalized by 2πct/a and 2πc/a, respectively, with ct being the elastic transverse wave velocity of silicon and c the velocity of light in air. The corresponding material parameters are the refractive index n = 3.5, mass density ρ = 2331 kg/m3, elastic moduli C11 = 16.57 × 1010 N/m2, C12 = 6.39 × 1010 N/m2 and C44 = 7.962 × 1010 N/m2, respectively [15]. For the phononic case one can observe four surface bands in the PNBG, see Fig. 1(b). The phononic surface modes 1 and 4 are the upper surface modes while modes 2 and 3 are the lower ones. It is noteworthy that the SAWs corresponding to the bands above the sound line (the black dashed line in Fig. 1(b)) can exist. Unlike the optical waves, the elastic waves (acoustic waves in solids) cannot radiate into the air region from the free surfaces; and due to the existence of the PNBG, the elastic waves cannot propagate from the surface into the interior space. For the optical waves shown in Figs. 1(c)-1(d), one transverse electric (TE) upper surface mode, one TE lower surface mode and one transverse magnetic (TM) upper surface mode can be found. The generation of the SOWs is owing to the PTBG and total internal reflection effects. For the surface mode bands, both acoustic and optical waves can be highly confined near the surface and attenuate exponentially away from the surface, as the modal distribution confirms.
Fig. 1 (a) Schematic sketch of the 2D PXC and enlarged diagram of the upper surface region. The structural geometry is described by the lattice constant a, circle radius r, hole width w and thickness of the additional silicon layer d. (b) Phononic dispersion curves and displacement field distributions of the surface bands. The red solid and blue dotted lines denote the upper and lower surface bands, respectively. The black dashed line is the sound line. (c) Photonic TE mode dispersion curves and magnetic field (Hz) distributions of the surface bands. (d) Photonic TM mode dispersion curves and electric field (Ez) distributions of the surface bands. The red solid and blue dotted lines denote the upper and lower surface bands, respectively. The black dashed line and the grey region are the light line and light cone, respectively.
The design of the surface mode cavity is based on the mode gap effect [37]. We change the structural parameter d to modulate the upper surface bands. The dispersion curves for two different structural parameters, d = 0.12a and d = 0.16a, are shown in Fig. 2. The hetero-structure cavity is formed at the upper surface. So only the upper surface bands are plotted in the band structures. Here we note that the upper and lower surface modes cannot couple to each other because there are enough periods of the PXCs in the interior region, i.e. with the parameter d changing the upper bands move while the lower ones keep invariant. Mode gaps (the frequency range marked by yellow shadow) can be seen for all the three modes discussed in this work, meaning that the surface acoustic and optical waves exist for d = 0.12a but disappear for d = 0.16a. We take the PXCs with d = 0.12a as the cavity region, and those with d = 0.16a as the mirror region. In order to obtain more gently variation of the modal fields, the tapered regions with d linearly increasing are introduced between the cavity and mirror regions. The lengths of the cavity and tapered regions are 4a and 3a, respectively. The schematic view of the cavity is displayed in Fig. 3(a). Four phononic cavity modes can be observed in Fig. 3(b). The semi-circular scatterers at the upper surface can be considered as oscillators. For mode α, two oscillators next to the central one swing and the others follow them. The ruling motion of mode β is the swing of the central oscillator. Mode γ shows almost the swing of all oscillators in the cavity region. The connectors vibrate while the scatterers are almost still in mode δ. Modes α and δ are even symmetry while modes β and γ are odd symmetry with respect to the plane x = 0. The photonic TE and TM cavity modes are shown in Figs. 3(c) and 3(d), respectively. The TE cavity mode appears as the dielectric mode while the TM cavity mode is the air mode. The two modes are both even symmetry with respect to the plane x = 0.
Fig. 2 Dispersion curves of the surface waves propagating at the upper surface of the PXC structure. Red and blue bands are the results for d = 0.12a and d = 0.16a, respectively. The yellow shadow represents the mode gap formed by different structural parameter d.
Fig. 3 (a) Schematic diagram of the 2D PXC hetero-structure cavity and variation of the structural parameter d along the x-axis. The hetero-structure cavity is formed by changing the thickness of the additional layer d, i.e. the distance between the upper surface and the holes. Displacement (b), magnetic (c) and electric (d) field distributions of different cavity modes.
3. Acousto-optical interaction
To calculate the AO interaction, we excite different phononic cavity modes, and then the frequencies of the photonic cavity modes undergo a modulation around their values for the unperturbed cavity. Two effects of the AO interaction, the moving interfaces (MI) and the photoelastic (PE) effects, are taken into account in the investigation. The MI effect is directly related to the deformation of the structure induced by the acoustic waves, i.e. the dynamic motion of the silicon-air interfaces. The PE effect results in an inhomogeneous modulation of the refractive index due to the generation of the strain field in the structure. The relations between the changes of the refractive indices and the elastic strain tensor are given by [15]
where Δnij (i, j = x, y), Sij (i, j = x, y) and pij (i, j = 1, 2, 4) denote the change of the refractive indices, elastic strain tensor and photoelastic constants of the material, respectively. For silicon, we use the following values of the photoelastic constants: p11 = –0.1, p12 = 0.01 and p44 = –0.051 [15]. Firstly we calculate different acoustic cavity modes by applying a prescribed displacement in the cavity. As the acoustic energy is highly confined in the cavity region, the right and left ends of the structure are fixed for the PNC case. In the calculation, the maximum displacement does not exceed 1% of the lattice constant a, which is well below the material strength limit. Since the frequency of the optical waves is several orders higher in the magnitude than that of the acoustic waves, the acoustic wave period is divided into 51 time-steps. And then we re-calculate the optical frequencies of the deformed cavity at these time-steps. This method has been demonstrated for the estimation of AO coupling strength in PXC structures [10–15]. Moreover, multi-phonon exchange process, which we concern in this paper, is involved in this method.The frequency modulations of the photonic TE cavity mode during one period of different phononic cavity modes and the corresponding Fourier transforms of the optical frequency are shown in Fig. 4. The amplitude of the modulation indicates the strength of the AO interaction. The shifts of the optical frequency Δfa/c induced by phononic modes α, β, γ and δ are 3.9 × 10−4, 6.7 × 10−5, 1.0 × 10−4 and 7.6 × 10−4, respectively. For the photonic TE mode, the frequency modulation by the phononic modes with even symmetry (modes α and δ) is larger than those with odd symmetry (modes β and γ). For mode α and δ, the first-order Fourier component dominates the Fourier spectrum with small contributions from the higher-order components, meaning that the single-phonon exchange process (linear effect), i.e. the photon absorbs and/or emits one phonon, dominates the AO interaction [10]. However, for modes β and γ, the second-order term contributes significantly to the Fourier spectrum, implying that the two-phonon process (nonlinear effect) dominates. These phenomena can be inferred from the variation of the optical frequency. During one period of acoustic waves, the optical frequency tuned by modes α and δ displays a sinusoidal pattern while that tuned by modes β and γ displays a square-sine function behavior. Here we note that although the two-phonon exchange process dominates in the coupling of the TE-β and TE-γ modes, the amplitude of the second-order Fourier coefficient in the TE-β and TE-γ interactions is in the same order of the magnitude as that in the TE-α and TE-δ interactions. In other words, for four phononic cavity modes the coupling strengths of the two-phonon process are similar. Normally the strength of the linear effect is stronger than that of the nonlinear effect, see Figs. 4(a) and 4(d). However the single-phonon exchange is more sensitive to the symmetry of the phononic cavity mode, thus with the odd-symmetry phononic modes the single-phonon exchange weakens and then the two-phonon exchange becomes dominant in the AO interaction, as shown in Figs. 4(b) and 4(c). We also investigate the interaction between the phononic and photonic TM modes. The behaviors of the AO interaction for the TM mode are similar to that for the TE mode. So the results are not presented and repeated here. One difference between the TE and TM modes is the strength of the modulation. The shifts of the photonic TM mode Δfa/c induced by phononic modes α, β, γ and δ are 2.4 × 10−4, 5.3 × 10−5, 7.9 × 10−5 and 2.5 × 10−4, respectively. With the same deformation, the frequency shift of the TE mode is larger than that of the TM mode.
Fig. 4 Frequency modulations and their Fourier transform of the photonic TE mode in one period of the phononic cavity modes α, β, γ and δ.
In view of possible telecommunication applications, the photonic operating wavelength λ is about 1550 nm. For the practical case of the photonic TE (TM) mode, the corresponding geometrical parameters are a = 537 (605) nm, r = 187.95 (211.75) nm, w = 483.3 (544.5) nm and d = 64.44 (72.6) nm (with d = 0.12a), respectively. And the corresponding optical wavelength is 1550.4 (1550.1) nm. In turn, the frequencies of phononic cavity modes α and δ are 3.61 (3.21) GHz and 6.99 (6.20) GHz, respectively. The modulations of the optical wavelength Δλ by phononic modes α, β, γ and δ are 1.75 (0.97) nm, 0.30 (0.21) nm, 0.46 (0.30) nm and 3.4 (0.97) nm, respectively. These geometrical parameters match the technological feasibility very well, such as the experimental data on the PTC nanobeam cavity [38], the “snowflake” [17] and L3-nanobeam [39] optomechanical cavities.
4. Summary
In summary, we demonstrate the bandgaps for simultaneous surface acoustic and optical waves in 2D PXCs. A PXC hetero-structure cavity is proposed, which can confine the phononic and photonic surface modes. We also investigate the AO interaction in the cavity. For the AO interaction, the symmetry of the phononic cavity modes plays a more important role in the single-phonon exchange process than in the case of the multi-phonon exchange. Modulated by the same phononic mode, the frequency shift of the photonic TE mode is larger than that of the TM mode. This structure can be further designed as surface wave sensors for simultaneous high-sensitive detection of the optical and acoustic properties of analytes. As Giga-Hertz (GHz) phonons can tune PTCs efficiently [40,41], different phononic cavity modes can be used for modulating the photonic cavity modes in the PXC structure. The analysis of this paper is based on the 2D assumption. It is well known that the results of a PXC slab depend on the slab thickness and are different from the corresponding 2D structure. Our future works will be focused on the investigation of surface modes in 2D PXC slabs.
Acknowledgments
This work was supported by the National Natural Science Foundation (11372031). The first author is grateful to China Scholarship Council (CSC) for the financial support of the research at the Chair of Structural Mechanics, University of Siegen, Germany. The third author is also grateful to the support of the National High Technology Research and Development Program of China (863 Program) under Grant No. 2013AA030901.
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