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We report the properties of dual phononic-photonic band gaps and localized modes of eightfold lithium niobate (LiNbO3) phoxonic quasicrystals (PhXQCs). Complete and large phoxonic band gaps are easily achieved despite the low refractive index of LiNbO3 substrate. Point defect intentionally introduced can form localized modes within both forbidden and transparency bands over a wide range of geometric parameters. Further analysis indicates that the localized modes within transparency bands originate from the intrinsic high-order rotational symmetry of quasiperiodic structures, which resemble whispering gallery modes. LiNbO3 PhXQCs provide a good candidate to enhance phononic-photonic interaction and show considerable advantage over the periodic counterparts.
© 2016 Optical Society of America
1. Introduction
Quasicrystalline structures, which were discovered by Shechtman et al. [1], have been introduced to periodic artificial structure domains, including photonic crystals (PhTCs) [2, 3] and phononic crystals (PhNCs) [4–7]. The concept of photonic and phononic quasicrystals was presented by Chan et al. [8] and He et al. [9], respectively. Compared with their periodic counterparts, these new types of PhTCs and PhNCs with aperiodic structures exhibit many distinct characteristics, such as highly isotropic and low-index contrast band gaps, intrinsic localization in defect-free structures, and large fabrication tolerance [10–14]. High flexibility and tunability for localized modes are easily achieved in quasiperiodic structures.
In the past few years, periodic artificial structures called optomechanical or phoxonic crystals (PhXCs) have gained increasing attention and have been extensively investigated. PhXCs possess phoxonic band gaps (the simultaneous existence of photonic and phononic band gaps) and the confined phonon-photon interaction [15–21]. Recently, we have introduced the concept of PhXCs to quasiperiodic counterparts called phoxonic quasicrystals (PhXQCs) [14]. Silicon PhXQCs can easily form large and complete dual phonon-photonic band gaps, and they possess abundant defect-free localized modes. Defect-free localized modes can simultaneously confine electromagnetic and elastic fields in a large area, thereby enlarging the phonon-photonic interaction space [14].
LiNbO3 material has been widely used in acousto-optic and electro-optic components because of its excellent acousto-optic, electro-optic, and piezoelectric properties [22–27]. The linear photonic [23] and phononic [27] band gap effects have been demonstrated experimentally through the introduction of artificial crystal concept, such as air hole arrays in LiNbO3 material. Moreover, the phoxonic band gap effect can be achieved in LiNbO3-based PhXCs [25, 26], which provide the possibility for realization of ultracompact acousto-optic devices. However, the lattices and geometrical parameters are limited selectively in forming efficient photonic band gaps because of the low refractive index of LiNbO3 material [25, 26]. The lattice symmetries should be decreased to achieve large phoxonic band gaps [25], thereby increasing the fabrication difficulty. Furthermore, complete phoxonic band gaps cannot be realized despite the decrease in lattice symmetries [25]. Thus, the intrinsic characteristics of low refractive index limit the degree of freedom of the realization of LiNbO3 PhXCs. Fortunately, quasiperiodic structures can form phoxonic band gaps even in a low threshold of the refractive index contrast. In this paper, we consider a new type of LiNbO3-based PhXQCs with phoxonic band gaps. Complete and large phoxonic band gaps can be easily realized despite the low refractive index of the substrate materials. We also realize the simultaneous strong localization of photons and phonons within the transparency bands of LiNbO3 PhXQCs, which generally appear in the forbidden bands for traditional PhXCs. The localization mechanism is addressed in PhXQCs.
2. Complete and large phoxonic band gaps
Eightfold quasiperiodic arrays of air holes in two-dimensional infinite structure of LiNbO3 material are considered. These arrays are based on octagonal Ammann–Beenker tiling created from square and rhombus tiles (minimum angle of 45°) [28]. The models under study are finite structures composed of 109 scatters, as shown in Fig. 1. The parameter a0 is the side length of square and rhombus composition, and the lattice constant a is 2a0/(1 +). LiNbO3 is anisotropic material and the holes are aligned along the Z crystallographic axis (z axis in Fig. 1). Both electromagnetic and elastic wave propagates in the x-y plane. The material parameters are set according to Refs [26, 29, 30]. Correspondingly, the refractive index, mass density, transverse and longitudinal speeds of sound is 2.2, 4300 kg/m3, cT,LiNbO3 = 3590 m/s and cL,LiNbO3 = 6570 m/s, respectively. The finite element method, which has been proven efficient, is applied to calculate the band gaps and field distributions [6, 7, 14, 26, 31].
Fig. 1 Eightfold quasiperiodic arrays of air holes in lithium niobate (LiNbO3) material. A point defect is created via removing a hole in the middle.
Figure 2 shows the calculated phoxonic band gaps. Figure 2(a) shows the normalized photonic frequencies ωa/2πc versus the dimensionless ratio r/a0, where c is the light speed in air and r is the radius of air holes. The results for r/a0>0.375 are not considered because of the overlap of the neighboring holes. Both transverse electric (TE, filled blue) and transverse magnetic (TM, filled brown) photonic band gaps begin to appear when the radius (r) is 0.13a0. The electric field is normal to x-y plane for TE polarization, while the magnetic field is normal to x-y plane for TM polarization. TE band gap width is increased with the increase in r. When r is 0.375a0, the mid-gap ratio (Δω/ωmid) for TE modes reaches the maximum value of 13.9%; 15.3% and 4.6% are the first and second TM band gap widths, respectively. The first TM band gap width reaches the maximum value of 19.1% when r is 0.34a0. Moreover, complete photonic band gaps (filled green) begin to exist when r is larger than 0.16a0. Whereas, both TE and TM band gaps are intrinsically narrow and independently exist over a narrow range of geometric parameters in periodic PhXCs with all the lattices, including square, hexagonal, and honeycomb lattices [25, 26]. Large photonic band gaps can be realized by decreasing the lattice symmetry. Nevertheless, the periodic PhXCs still do not possess complete photonic band gaps despite the decrease in lattice symmetry [25].
Fig. 2 Phoxonic band gaps and localized modes formed via a point defect in LiNbO3 phoxonic quasicrystal (PhXQCs). (a) Normalized photonic frequencies (ωa/2πc) for transverse electric (TE) and transverse magnetic (TM) band gaps (filled color) and the localized TE and TM modes (symbols) vs. r/a0 within and outside photonic band gaps, where c is the light speed in air. (b) Normalized phononic frequencies (ωa/2πcT,LiNbO3) for phononic band gaps (filled color) and localized in-plane and out-of-plane modes (symbols) vs. r/a0 within and outside phononic band gaps, where cT,LiNbO3 is the transverse sound speed in LiNbO3 material. r is the radius of air holes.
The polarization modes of elastic waves are decoupled into a transverse (out of plane) mode, corresponding to shear vibrations along z-axis, and mixed (in-plane) modes, corresponding to coupled shear and dilatational vibrations in the the x-y plane [16].The material anisotropy of LiNbO3 can induce the coupling of phononic polarizations in phononic crystals [32]. Rolland et al. [26] have investigated period phoxonic bandgaps for Z-cut LiNbO3 slabs, and compared these bandgaps with two-dimensional counterparts. In their calculation, out of plane and in-plane modes are approximately decoupled [26]. In this work, we also adopt a same approximation. Figure 2(b) shows the normalized phononic frequencies (ωa/2πcT,LiNbO3) versus r/a0, where cT,LiNbO3 is the transverse sound speed in LiNbO3 material. The out-of-plane (filled yellow) and in-plane (filled red) phononic band gaps begin to appear when the r values are 0.14a0 and 0.23a0, respectively. When r is 0.375a0, the ratio Δω/ωmid reaches the maximum values of 19.8% and 21.4% for the out-of-plane and in-plane modes, respectively. Complete phononic band gaps begin to appear when r is 0.29a0. The photonic and phononic band gaps occur when the r values are 0.13a0 and 0.14a0, respectively. Therefore, photonic and phononic band gaps almost simultaneously open for a small radius r. Furthermore, the phoxonic band gaps exist within a considerably large radius range of 0.14≤r/a0≤0.375. This result indicate that photonic and phononic band gaps coexist over a wide range of geometric parameters, which is impossibly realized in periodic counterparts [25, 26].
LiNbO3 PhQXCs also possess complete phoxonic band gaps when r≥0.29a0. The maximum Δω/ωmid values for complete photonic and phononic band gaps reach 6.24% and 10.1%, respectively, when r is 0.375a0. With the absence of complete photonic band gaps in periodic PhXCs, phononic band gaps coexist with photonic band gaps of either TE or TM modes. Consequently, complete phoxonic bands cannot be achieved. Thus, the PhQXCs exhibit significant advantage over the PhXCs for the potential ability of controlling and tailoring electromagnetic and elastic waves simultaneously.
3. Localized modes within the forbidden and transparency bands
Localized modes exist in defect-free PhXQCs, as demonstrated in [14]. Localized modes similar to those of periodic structures can still be achieved via intentionally introducing point or line defects into quasiperiodic crystals [10, 31, 33–35]. In the present work, we consider a defect created by removing a middle air hole, as shown in Fig. 1. Figure 2 also shows the calculated phoxonic localized modes. The frequencies (ωa/2πc) of photonic localized modes and ωa/2πcT,LiNbO3 of phononic modes versus r/a0 are shown as different symbols in Figs. 2(a) and 2(b), respectively. Three localized photonic modes simultaneously exist when the radius is r≥0.3a0. Two out-of-plane localized phononic modes simultaneously appear when r is 0.33a0. Subsequently, the third out-of-plane mode appears when r is increased to 0.35a0. An in-plane localized phononic mode can also be observed when r reaches 0.36a0.
One TE mode (black squares) and one TM mode indicated by TM 1 (red circles) are within the photonic forbidden bands in Fig. 2(a). Only one localized phononic mode, which is the out-of-plane mode (red squares) within the phononic forbidden bands in Fig. 2(b), exists. The field distribution of these localized modes within the forbidden bands is shown in Figs. 3(a)-3(c), where r is 0.375a0. Both the electromagnetic and displacement fields are strongly localized in the point defect of PhXQCs. The field of the TE mode is a monopole, the TM 1 mode is a dipole, and the out-of-plane phononic mode is a quadrapole. The localization for these modes originates from the rejection of electromagnetic waves in photonic band gaps or elastic waves in phononic band gaps, which is the same as the case of PhXCs with periodic structures [16, 18, 20].
Fig. 3 Field distribution of localized modes with frequencies within the forbidden bands of PhXQCs having a middle point defect; the radius r is 0.375a0. These localized modes are indicated in Fig. 2: (a) electric field profile for the TE mode, (b) magnetic field profile for the first TM mode indicated by TM 1, and (c) displacement uz for the first out-of-plane phononic mode indicated by OUT 1.
Notably, the defect properties of PhXQCs are significantly richer than those of PhXCs, which have been partly demonstrated as the appearance of abundant localized modes in defect-free PhXQCs [14]. Some localized phoxonic modes also exist outside the forbidden gaps, which can be observed in Fig. 2. Figure 2(a) shows that the second TM localized mode indicated by TM 2 (blue triangles) begins to appear within the transparency photonic band when r is 0.3a0. The frequencies of this localized mode are larger than those within the forbidden bands. In addition, two out-of-plane modes (purple circles and yellow triangles) exist within the transparency phononic band, as shown in Fig. 2(b). Moreover, one in-plane (green triangles) phononic modes can be achieved when r reaches 0.36a0. In-plane modes are not observed within the forbidden band, thereby indicating that this type of localized modes can exist irrespective of phoxonic band gaps. Therefore, complete confinement for all polarizations of both electromagnetic and displacement fields can be realized via using the defect modes outside band gaps. Figure 4(a) displays the magnetic field profile for the second TM mode indicated by TM 2. The displacement field distribution of the three phononic modes is shown in Figs. 4(b)-4(d). Both the electromagnetic and displacement fields can also be strongly confined in the point defect even if no band gap effect exists.
Fig. 4 Field distribution of the localized modes with frequencies within the transparency band of PhXQCs having a middle defect; the radius r is 0.375a0. These localized modes are indicated in Fig. 2: (a) magnetic field profile for the second TM mode indicated by TM 2; displacement uz of two out-of-plane phononic modes indicated by (b) OUT 2 and (c) OUT 3, respectively; and (d) total displacement u = [(ux)2 + (uy)2]1/2 for the in-plane phononic mode.
In a quasicrystal structure with finite size, Villa et al. stated that the interactions between a limited number of cells induce the occurrence of localized modes within the transparency bands [28]. The properties of octagonal LiNbO3 ring are discussed using the physical and geometrical parameters, which are the same as the structures of PhXQCs above, to investigate the origin of the localized modes within the transparency bands in the present work. The octagonal ring is the middle part of PhXQCs (inset of Fig. 5(a)). All the localized modes outside the forbidden bands of the PhXQCs can also be observed in this single octagonal ring (frequencies of localized phoxonic modes as a function of r/a0, as shown in Fig. 5(a)). Furthermore, the frequencies and variation trend of the one photonic and three phononic modes are almost identical to those of PhXQCs as a whole. The field profiles of localized modes in octagonal rings are also shown in Figs. 5(b)-5(e). The symmetry and distributions of these modes are almost the same as those of PhXQCs. Strong confinement can still be realized. Therefore, this basic octagonal ring determines the properties of the defect modes of PhXQCs in general. Whispering gallery modes are formed around a concave surface, as demonstrated in PhNCs and quasiperiodic PhTCs [36, 37]. With the high-order rotational symmetry of quasicrystal structure, eight holes that are similar to a concave surface around the middle defect exist. The region of the eight holes displays a lower efficient index than the middle part, thereby inducing the localized mode occurrence via total internal reflection. The multiple maxima of the displacement fields exist at the edges of the eight holes, which demonstrate that these localized modes resemble the whispering gallery modes [36–38], as shown in Figs. 5(b)-5(e). When rings with higher-order rotational symmetry, such as 10-fold and 12-fold, are selected, localization can be easier to be achieved because the surrounding holes are more similar to a concave surface, as demonstrated by our calculations. Thus, the occurrence of these modes depends on the effective index contrast instead of the band gap effect [16, 18, 20], as well as on the interactions between the limited numbers of holes [28]. These defect modes are insensitive to the size of the quasicrystal structures maintaining a middle part, which is an octagonal ring, thus providing considerable application freedom.
Fig. 5 (a) Normalized frequencies of the localized photonic and phononic modes vs. r/a0 in octagonal rings, as shown in the inset. The field distribution of the localized modes indicated in Fig. 5(a), where the radius is 0.375a0: (b) magnetic field profile for the TM modes indicated by TM 2; displacement uz of two out-of-plane phononic modes indicated by (d) OUT 2 and (d) OUT 3, respectively; and (e) total displacement u = [(ux)2 + (uy)2]1/2 for the in-plane phononic mode.
Figures 3 and 4 show that the strong field localization and overlap of both photonic and phononic localized modes occur in the point defect of PhXQCs. Furthermore, these localized modes can be freely selected within and outside the band gaps. These defect modes are tightly confined in the intentionally introduced cavity within a considerably small region. Strong phononic-photonic interactions in the point defect can be expected despite whether the defect modes are within the band gaps or not. The localization of defect-free modes in PhXQCs is distributed within a large area [14]. Consequently, the distribution of these localized fields, which is similar to that of periodic structures [16], is different from that of defect-free localized modes. This result is attributed to the fact that the localization stems from the cavity intentionally introduced for the two former ones, and the nonperidicity and self-similarity for the latter one.
4. Conclusion
We have investigated the properties of photonic and phononic band gaps in eightfold quasiperiodic arrays of air holes in LiNbO3 material. Results show that PhXQCs provide a good candidate to realize complete and large phoxonic band gaps despite the low refractive index of the substrate. Localized modes can be easily achieved via introducing point defect into PhXQCs. These localized modes can also be within or outside band gaps maintaining a strong confinement around the defect within a small area. Further study shows that the localized modes within transparency bands resemble whispering gallery modes. High-order rotational symmetry leads to low-efficient surrounding index around the defect. Moreover, the higher-order rotational symmetry of PhXQCs results in stronger field confinement. Our results imply that PhXQCs provide another efficient method to confine and tailor all the polarizations of electromagnetic and elastic waves simultaneously, in addition to defect-free localized modes. The slab thickness plays an important role in determining the phoxonic bandgaps and modal localization in a PhXC slab. Especially, out of plane phononic modes are strongly dependent on the slab thickness [26]. However, the structures discussed in this work are infinite along z axis. Slab structures for applications will be considered in our subsequent works.
Funding
National Science Foundation of China (NSFC) (Grant Nos. 11664024, 11304144, 11264030, 11264029, 61367006), the Natural Science Foundation of Jiangxi Province (NSFJP) (Grant No. 20151BAB202015), the External science and technology cooperation program of Jiangxi Province (ESTCPJP) (Grant No. 20151BDH80030), the Research Project from the Department of Education of Jiangxi Province (RPDEJP) (Grant No. GJJ14159).
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