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We report the simultaneous existence of acoustic and optical asymmetric transmission by combining a periodic array of silicon rods in air with a periodic grating (grating-phoxonic crystals). Theoretically and numerically, we demonstrate that the grating-phoxonic crystals with a square lattice of silicon rods can exhibit simultaneous asymmetric propagation in the same reduced frequency range both for sound and light. The sound and light can transmit through the composite structure when incident from the grating interface but are completely reflected for the reverse direction incidence. Furthermore, when the silicon rods are tuned into a triangular lattice, reversed asymmetric transmissions are also achieved. Sound and light are blocked when incident from the grating interface while being fully able to pass through the composite structure for the reverse direction incidence. These two tunable dual acoustic and optical asymmetric transmissions show great potential for acoustic-optic devices by simultaneously manipulating sound and light.
© 2017 Optical Society of America
1. Introduction
Due to interesting phenomena arising from the interaction of light and sound with periodic materials, artificial structures called optomechanical or phoxonic crystals (PxCs) [1] have gained increasing attention and have been extensively investigated for more than ten years. PxCs are well known for phoxonic band gaps (the simultaneous existence of photonic and phononic band gaps in same reduced frequency range) which provide promising means to simultaneously control the propagation of electromagnetic and elastic waves. Several combinations of materials and structures have been studied in order to demonstrate the opening of simultaneous photonic crystals (PtCs) and phononic crystals (PnCs) bandgaps [2–4] and acoustic-optical interaction [5,6].
Maldovan and Thomas [1,7] theoretically demonstrated that triangle-latticed structures with air rods in a silicon matrix can obtain large, complete and simultaneous photonic and phononic band gaps. Pennec et al. [8] have demonstrated that PxCs exhibit band gaps for both electromagnetic and plate-mode waves. Hassouani et al. [9] theoretically demonstrated the dual phononic-photonic band gaps in a periodic array of silicon pillars deposited on a thin plate. The experimental evidence of PxC band gaps is also reported in 3D PnCs and PtCs of amorphous silica spheres [10]. Moreover, with different materials, the opening of simultaneous photonic and phononic band gaps have also been investigated in lithium niobate slabs [11,12]. Later, the confinement of both electromagnetic and elastic waves in a single area was realized by appropriately introducing point or liner defects into PxCs [13]. Meanwhile, the interaction of acoustic and optical has been proposed and studied. Eichenfield et al. [14] utilized silicon-on insulators to demonstrate that the PtCs structures can also be used to detect, generate and control mechanical vibrations (phonons) within the same planar, chip-scale architecture. Psarobas et al. [15] reported that the strong acousto-optic interaction occurs when simultaneous acoustic and optical resonant modes are simultaneously confined in a 1D PxC cavity with a single defect to motivate multiphonon exchange. Some optomechanical crystal devices have also been created for the new acoustic-optical effects, such as the slow light [16,17] and coherent optical wavelength conversion [18].
In recent years, unidirectional optical transmission devices which have received considerable attention, due to the capability of rectifying electromagnetic energy, have important potential applications in optical communication and quantum computers [19,20]. A variety of unidirectional optical transmission devices have been proposed and demonstrated based on PtCs including: a one-dimensional PtC structure with lossy metallic film [21], a two-dimensional PtC waveguide with an asymmetric array of nonlinear defect rods [22] and other unidirectional optical devices based on non-symmetric diffraction structures [23,24], left-handed [25], and magneto-optical [26] materials. In parallel, the unidirectional acoustic transmission [27] also has been a hot research topic in recent years for their promising applications, such as rectifier, ultrasonic medical imaging and noise insulation. The impressive contributions of unidirectional acoustic transmission have been made in non-reciprocal systems by breaking the time reversal symmetry with a nonlinear medium [28,29], or with active elements [30,31]. Even the large gaps between the acoustic bands are opened by strong opto-mechanical interaction in the non-reciprocal system [32]. Meanwhile, the asymmetric acoustic transmission in reciprocal systems has also been realized by breaking the spatial inversion symmetry with many artificial asymmetric structures [33–35]. Despite such extensive studies, the concept of the simultaneous asymmetric acoustic and optical transmission has not been exploited. From this viewpoint, co-localization of simultaneous asymmetric acoustic and optical transmission in same reduced frequency is unique and attractive, as it can provide the potential possibility of enhancing opto-mechanical interaction and simultaneously manipulating photons and phonons.
In this paper, we report on the simultaneous asymmetric acoustic and optical transmission in a linear system which is composed of the silicon grating-phoxonic crystals (GPxCs). For GPxCs with a square lattice, the acoustic and optical waves in the same reduced frequency range can transmit through the composite structure when they are incident from the grating interface but are completely reflected when they are incident from the PxCs interface. This asymmetric transmission is attributed to high-order diffractions caused by the grating, which can lead to the transition of different spatial modes and overcome the barrier of the ГX direction band gap [33]. When square lattice silicon rods are adjusted into the triangular lattice in the GPxCs, the optical and acoustic waves are blocked when they are incident from the grating interface while able to fully pass through the structure when they are incident from the PxCs interface. This simultaneous asymmetric acoustic and optical transmission in the reverse direction is obtained due to the properties of the Dirac point K that allow the normal incident waves to pass but block the oblique incident waves. These simultaneous asymmetric acoustic and optical transmissions in two different directions can be tuned to each other, which will be particularly significant for opto-acoustic interactions and integrating the management of elastic and electromagnetic waves.
2. Models and method
The composite GPxCs structures are illustrated in Fig. 1. Figure 1(a) describes the first geometry consisting of rectangular grating and square lattice of circular silicon rods in air. The square lattice constant is a = 600 nm, and the radius of the rods are r1 = 220 nm. The grating constant, depth, thickness and the distance between the grating and PxCs are set to be ag1 = 1800 nm, h1 = 600 nm, t1 = 200 nm and d1 = 75 nm, respectively. Figure 1(b) shows the second geometry made of rectangular grating and a triangular lattice of circular rods in air. The triangular lattice constant remains the same value of a = 600 nm. The radius of the rods are adjusted to r2 = 150 nm. And the grating constant, depth, thickness and the distance between the grating and PxCs are also changed to be ag2 = 1800 nm, h2 = 1400 nm, t2 = 220 nm and d2 = 75 nm, respectively. The material parameters used in the calculations are, for silicon, a refractive index n = 3.6, mass density ρs = 2332 kg/m3 and longitudinal wave velocity cs = 8429.4 m/s, and for air, the density ρa = 1.25 kg/m3 and acoustic velocity ca = 343 m/s. The structures are considered finite in x-direction, infinite in z-direction and periodic in y-direction. We define the incidence of the acoustic wave and transverse electric (TE) polarized plane light with the magnetic field along z-axis from the left side as positive incidence (PI), while that from the right side as negative incidence (NI), denoted by green and blue arrows, respectively. Throughout this work, the PnCs and PtCs band gaps are calculated by using the plane wave expansion (PWE) method, and the transmission spectra of the composite GPxCs structures are obtained by employing the finite element (FE) method (COMSOL MULTIPHSICS). It is noted that the simultaneous asymmetric acoustic and optical transmission in the GPxCs can be achieved by using a large number of geometrical parameter adjustments. The specific procedures are as follows. First, we search the common optical and acoustic directional band gaps by adjusting the filling ratio of the silicon rods. Then, we combine the periodic grating and the phoxonic crystals to achieve the simultaneous asymmetric acoustic and optical transmission based on the diffraction effect. Finally, by adjusting the grating constant, depth and thickness of the periodic grating, the good results of simultaneous asymmetric acoustic and optical transmission can be obtained.
Fig. 1 Schematic geometry of composite GPxCs structure: (a) a square lattice array of circular silicon rods combined with rectangular grating in air and (b) a triangular lattice array of circular silicon rods combined with rectangular grating in air.
3. Simultaneous acoustic and optical asymmetric propagation for GPxCs with square lattice
We first analyze the band gaps of the general PxCs with a square lattice, which is shown in Fig. 2. The frequencies of the PnCs and PtCs band structures are normalized by the dimensionless frequency Ω = ωa/2πc, where c is the longitudinal wave velocity in air for elastic waves or the light velocity for electromagnetic waves. From Fig. 2(a), it is clearly seen that there exist a broad band gap for the acoustic wave from Ω = 0.3376 to 0.6210 in the ГX direction and a narrow band gap for the acoustic wave from Ω = 0.5419 to 0.6221 in the ГM direction. One can notice a broad frequency region from Ω = 0.3376 to 0.5419 where the acoustic waves are in the band gap along the ГX direction but in the pass band along the ГM direction. Similarly, as shown in Fig. 2(b), the band structures of TE polarized light also exhibit a similar behavior with a broad band gap from Ω = 0.338 to 0.444 in the ГX direction but with a pass band in the ГM direction. This means that if there is a way to overcome the ГX direction band gap in one side, asymmetric transmission will be achieved. Thus, we use a periodic rectangular grating to implement this functionality that can diffract the normal incident waves (acoustic and TE polarization light) to higher orders and let it fall in the pass band of the PnCs or PtCs. Then, the simultaneous asymmetric acoustic and optical transmission can be successfully obtained. Figures 2(c) and (d) exhibit the corresponding transmission spectra of the GPxCs for acoustic waves and TE polarization light waves, respectively. From Fig. 2(c), within the reduced frequency range from Ω = 0.340 to 0.460 (corresponding to shaded region and the natural frequency of acoustic wave from 1.944 × 108 to 2.63 × 108 Hz), PI is associated with high transmission coefficient but the transmission is forbidden for NI, which shows a broadband asymmetric acoustic transmission. When the transmission spectra of TE polarization light are calculated by using the same geometrical parameters, one can find that an asymmetric transmission frequency region from Ω = 0.418 to 0.444 (corresponding to shaded region and the natural frequency of optical wave from 2.09 × 1014 to 2.22 × 1014 Hz) is obtained as shown in Fig. 2(d). Although the light in 0.338 < Ω < 0.372 shows a good performance of asymmetric transmission, this frequency region will not be considered for the low positive transmittance being less than 0.2. Interestingly, it is noticed that a common reduced frequency region from Ω = 0.418 to 0.444 (with a bandwidth of 0.026) corresponding to the simultaneous asymmetric acoustic and optical transmission is gained. In this common normalized frequency range, both the acoustic waves and TE polarization light waves are transmitted through the GPxCs for PI, but completely reflected for NI.
Fig. 2 (a) Calculated band structures of PnCs for a square lattice of silicon rods in air. (b) Calculated band structure of PtCs (TE polarized light) for a square lattice of silicon rods in air. And the transmission spectra of acoustic waves (c) and TE polarized light (d) in composite GPxCs structure for PI and NI, respectively.
In order to further illustrate the simultaneous asymmetric transmission more clearly, the spatial intensity distributions of the acoustic pressure field and magnetic field intensity distributions of TE polarization light for PI and NI are mapped out as shown in Fig. 3 at the same reduced frequency Ω = 0.428, respectively. It is worthwhile to note that Ω = 0.428 is chosen arbitrarily from the simultaneous asymmetric acoustic and optical transmission frequency region from Ω = 0.418 to 0.444. From Figs. 3(b) and (d), for NI, it is obvious that the acoustic and light waves are completely reflected because of the simultaneous band gaps of PnCs and PtCs along the ГX direction. However, in the case of PI, Figs. 3(a) and (c) show the strong acoustic and magnetic fields in the output region, respectively, which indicates that the normal incident waves (acoustic and light) can transmit through the composite structure by appropriately placing the periodic grating in front of the PxCs. Moreover, one can see that the transmitted waves are not parallel to the incident waves. It means that the incident waves are converted to other spatial modes that are outside of the ГX direction band gap [33]. Moreover, due to the diffraction properties of the rectangle grating [36], the incident waves are diffracted in different directions representing the different diffraction orders when they first arrive at the grating interface. Not all the frequencies within ГX direction band gap can pass through the PnCs and PtCs where some diffraction orders may be extremely weak. Therefore, the output pattern after the grating is the interference of different diffraction orders and the whole transmittance of the output area is the sum of these diffraction orders, which further verify our results in Figs. 2(c) and (d).
Fig. 3 Simulated acoustic pressure field intensity distributions at Ω = 0.428 for PI (a), NI (b), respectively. And simulated magnetic field intensity distributions of TE waves at Ω = 0.428 for PI (c), NI (d), respectively.
4. Simultaneous acoustic and optical asymmetric propagation for GPxCs with triangular lattice
In this section, we further study the tunability of simultaneous asymmetric acoustic and optical transmission in the GPxCs. When the square lattice silicon rods in the GPxCs are tuned into the triangular lattice as shown in Fig. 1(b), the reversed asymmetric transmission of acoustic and optical waves is simultaneously realized. Namely, the acoustic and TE polarization light are blocked for PI while able to fully pass through the structure for NI. This unique simultaneous asymmetric acoustic and optical transmission phenomenon stems from the PxCs structure and is designed to have a Dirac point at the K point that allows the normal incident waves to pass but block the oblique incident waves. In the following work, we first illustrate this mechanism by discussing the transmission of acoustic and light waves in different incident angles, and then design a periodic grating to change the incidence direction; the simultaneous asymmetric acoustic and optical transmission is then obtained.
Figures 4(a) and (b) give the band structures of general PnCs and PtCs with triangular lattices for acoustic and TE polarization light, respectively, obtained by the PWE method. It can be seen clearly that both the acoustic and TE polarization light exhibit a Dirac point at the K point. For acoustic waves, as shown in Fig. 4(a), there is a pass band from Ω = 0 to 0.7 in the ГK direction but a band gap from Ω = 0.444 to 0.640 in the ГM direction. Similarly, for TE polarization light, Fig. 4(b) also displays a pass band from Ω = 0.448 to 0.632 in the ГK direction but a band gap from Ω = 0.448 to 0.625 in the ГM direction. This indicates that different incident directions will lead to different band structures both for sound and light. In order to further verify this property, we calculate the transmission spectra of normal incidence (incident angle θ = 0°) and oblique incidence (incident angle θ = 15°) plotted in Figs. 4(c) and (d), respectively. Obviously, for oblique incidence, the transmittances (Ω = 0.489 to 0.616 for acoustic and Ω = 0.464 to 0.564 for light) are forbidden but they exhibit high transmittance for normal incidence. That is because the Dirac point will be split and the band gap will occur in case of oblique incidence [37]. The splitting degree of Dirac point relies on the oblique incidence angle θ [38]. Therefore, it is known that the transmission characteristics of waves (acoustic and light) are determined by the incident directions.
Fig. 4 Calculated band structures of PnCs (a) and PtCs (b) for a triangular lattice of silicon rods in air. The transmission spectra of acoustic (c) and TE polarized light (d) for a triangular lattice of silicon rods in air under normal incidence (θ = 0°) and oblique incidence (θ = 15°), respectively.
To achieve the simultaneous asymmetric acoustic and optical transmission, we select a periodic rectangular grating combined with PxCs. Different from the first case, the period and thickness of the grating are appropriately designed according to the diffraction properties of the phase grating [36]. For PI, when the normally incident waves are adjusted to oblique incidence through the diffraction properties of grating, transmission is forbidden; but for NI, the incident waves pass through the GPxCs completely. Figure 5 describes the mechanism of asymmetric acoustic and optical transmission in GPxCs structure. We can see that for NI, the waves pass through the PxCs first and arrive at the grating normally, then they are diffracted to other orders to pass through the GPxCs. While for PI, the waves are first diffracted to many directions in which the waves of oblique directions will be mostly reflected, and only a little energy of the zero order can transmit the PxCs. Thus, compared with the preceding case, the simultaneous asymmetric transmission effect in reverse direction can be obtained.
Fig. 5 The Geometric viewpoint of GPxCs for asymmetrical propagation, the indigo rectangles represent the 2D PxCs layers, the blue arrays are the periodic grating. (a) The waves (acoustic and TE) are mostly blocked for PI and (b) the waves (acoustic and TE polarized light) can pass through the whole composite structure for NI.
According to the above mechanism, to gain a better asymmetric transmission, the period, thickness and depth of the grating are set to be ag2 = 6a = 1800nm, t2 = 220 nm and h2 = 1400 nm, respectively. We calculate the transmission coefficients of waves (acoustic and light) by employing the FE method for PI and NI as shown in Fig. 6. From Fig. 6(a), it can be seen clearly that the acoustic transmittances for the two opposite direction are quite different for frequencies ranging from Ω = 0.441 to 0.582 (corresponding to the natural frequency of acoustic wave from 2.521 × 108 to 3.327 × 108 Hz). The transmittance of NI is fluctuating at higher level while that of PI is relatively low with the value near to 0 at Ω = 0.516. On the optical waves side, Fig. 6(b) shows the same asymmetric transmission in broad frequency range from Ω = 0.450 to 0.595 (corresponding to the natural frequency of optical wave from 2.25 × 1014 to 2.975 × 1014 Hz). Moreover, it can be seen that there exists a narrow range (green region) where the positive transmittance is high whereas the negative transmittance is almost zero, which is determined by the band structure in Fig. 4(b) that exhibits a pass band in the ГM direction but a band gap in the ГK direction from Ω = 0.428 to 0.447. Due to the diffraction of the grating, the light can transmit through the composite GPxCs structure for PI but are reflected back for NI. By comparing the asymmetric transmission regions of the acoustic and optical waves, it is noticeable that they occupy a common reduced frequency range from Ω = 0.45 to 0.582 (with a bandwidth of 0.132). Therefore, the simultaneous asymmetric acoustic and optical transmission is well realized. Besides, it is noted that differences of the acoustic and optical transmittances for PI and NI are relatively small, which is due to the fact that the zero order diffraction wave of PI can be transmitted. In order to reduce the transmittances of PI, the method of small oblique incidence is attempted to suppress the zero order diffraction. The insets of Figs. 6(a) and (b) show the acoustic and optical transmittances of five degrees oblique incidence for PI and NI, respectively. By comparison, it is seen that the acoustic and optical transmittances of PI can be decreased by small angled incidence, but transmittances of NI are reduced. So, the angled incidence is helpful but it may be not the best way to address this problem. The optimization method may be a better choice for finding an optimal simultaneous asymmetric acoustic and optical transmission.
Fig. 6 The transmission spectra of normal incident acoustic (a) and TE polarized light (b) for a triangular lattice of silicon rods combined with periodic grating in air under the PI and NI, respectively. The inset in Fig. 6(a) shows the acoustic transmission spectra of five degree oblique incidence. And the inset in Fig. 6(b) shows the optical transmission spectra of five degree oblique incidence. Simulated acoustic pressure field intensity distributions at Ω = 0.516 for PI (c), and NI (d), respectively. And simulated magnetic field intensity distributions of TE polarized light at Ω = 0.516 for PI (e), and NI (f), respectively.
To further illustrate the simultaneous asymmetric acoustic and optical transmission, the Figs. 6(c),(d),(e) and (f) show the spatial intensity distributions of the acoustic pressure field and magnetic field intensity distributions of TE polarization light at the same reduced frequency Ω = 0.516 for PI and NI, respectively. It is worth noticing that 0.516 is selected since the positive transmittances both for acoustic and light are lowest near to zero while the negative transmittances are quite high at this frequency. Obviously, different from the previous case of Fig. 3, the acoustic and light are blocked for PI as shown in Figs. 6(c) and (e), respectively. However, for NI, the acoustic and light are transmitted by the GPxCs presented in Figs. 6(d) and (f), respectively. Lastly, it is worth mentioning that the difference between these two simultaneous asymmetric acoustic and optical transmissions only lies in two different lattice types and geometry sizes. Thus, by adjusting the lattice types and geometry sizes, the simultaneous asymmetric acoustic and optical transmissions in two different directions can be tuned to each other.
5. Conclusion
We have theoretically demonstrated the simultaneous asymmetric propagation of acoustic and optical waves in the silicon GPxCs. First, a simultaneous asymmetric acoustic and optical transmission in the GPxCs with square lattice is realized in the same reduced frequency range. Namely, for PI, the acoustic and optical waves can pass through the composite structure, but for NI, they are completely reflected. Furthermore, by adjusting the square lattice silicon rods into triangular lattice arrays and geometry sizes in the GPxCs, the acoustic waves and optical waves are blocked for PI while fully able to pass through the structure for NI. This completely reverse behavior is due to the properties of the Dirac point K that allow the normal incident waves to transmit but block the oblique incident waves. Besides, these two simultaneous asymmetric acoustic and optical transmissions in different directions can be tuned to each other by changing lattice types and geometry sizes. Compared to the previously proposed asymmetric acoustic transmission or asymmetric optical transmission separately, the simultaneous asymmetric acoustic and optical transmission that we have proposed shows great potential for use in acoustic-optic devices that can integrate the manipulation of sound and light.
Funding
National Natural Science Foundation of China (No.11374093); the Program of the State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body (71375006); and Young Scholar Fund sponsored by Common University and College of the Province in Hunan.
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