1. Introduction

Artificial periodic structures have been widely used to control the propagation of classic waves, including electromagnetic and elastic waves, because of the existence of photonic and phononic band gaps [1–5]. During the past few years, a growing interest toward periodic structures exhibiting simultaneous photonic and phononic band gaps has arisen; these structures are called optomechanical crystals [6,7] or phoxonic crystals (PhXCs) [8,9]. The confinement of both electromagnetic and elastic waves in a same area can be realized by appropriately introducing point or linear defects into PhXCs [8–14]. Thus, PhXCs provide a promising method for enhancing opto-mechanical [6,7,10,11] and acousto-optical interaction [11–14] on a wavelength scale.

Compared with their periodic counterparts, quasiperiodic structures lack long-range translational order, but have orientational order, higher order rotational symmetries, and many non-equivalent sites. Thus, these structures exhibit many interesting characteristics, such as highly isotropic and low index contrast band gaps and intrinsic localization in defect-free structures, along with large fabrication tolerance [15–29]. Researchers focused on investigating the control of propagation of electromagnetic wave in quasiperiodic photonic crystals [15–24] and elastic waves in quasiperiodic phononic crystals [24–29], respectively. To our knowledge, few studies have focused on dual photonic and phononic band gaps in quasiperiodic structures. In this paper, our aim is to demonstrate the large simultaneous band gaps for both photons and phonons in quasiperiodic phoxonic crystals called phoxonic quasicrystals (PhXQCs). We also study the localization of both electromagnetic and elastic waves in same area without intentionally creating defects.

2. Model and band gaps of phoxonic quasicrystals

We consider 8-fold quasiperiodic structures based on octagonal Ammann-Beenker tiling created from square and rhombus tiles [20]. The existence of band gaps in quasiperiodic composites shows that a long range periodic order is not a necessary condition for forming gaps, meaning that this kind of gaps originate from the quasiperiodic structure [16,19,25]. The transmissions are calculated to achieve the band gaps [5,16,19,23,25]. The models under study in this work are finite structures composed of 109 scatters, as shown in Fig. 1. We confirm that the two computed models are large enough to characterize the transmission properties by calculating gaps of larger structures. $a_0$ is the side length of square and rhombus (minimum angle of 45°) composition. Then lattice constant $a$ is $2a_0/(1+\sqrt{2})$ [25]. Solid silicon materials are considered. The physical parameters of silicon are the refractive index$n_{Si}=3.5$, mass density ${\rho }_{Si}=2331\text{ kg}/{\text{m}}^3,$ and transverse and longitudinal speeds of sound $c_{T,Si}=5360\text{m}/\text{s}$ and$c_{L,Si}=8950\text{m}/\text{s},$respectively. Finite element method (FEM) is extensively applied in the field of physics and engineering. The calculations performed in this work are based on FEM, which has been proven to be an efficient method for achieving band gaps and field distributions in photonic crystals and phononic crystals [5,12,23,30]. Two kinds of quasiperiodic structures, silicon rods in air (PhXQCs I), and air holes in silicon (PhXQCs II) are considered, as shown in Figs. 1(a) and 1(b), respectively.

 

Fig. 1 8-fold phoxonic quasicrystals (PhXQCs) models: (a) silicon rods in air (PhXQCs I) and (b) air holes in silicon (PhXQCs II).

Download Full Size | PDF

To calculate transmission, a line source with a width 10$a_0$ along y-direction is placed on the left side of these models. The line source generates propagating elastic or electromagnetic wave. A received port is placed on the right side of the models. The ratio of received power to source power is defined as transmission coefficient. To illustrate the way how to define the band gaps, the transmission spectra of transverse electric (TE) modes in PhXQCs I for different incidence angles are shown as examples in Fig. 2. Here the radius of rods is 0.375$a_0$ The largest angle of 22.5° is enough large to characterize the transmission properties due to eightfold symmetry of the calculated quasicrystals [16]. There appear three photonic band gaps, which correspond to the normalized frequency regions ranging from $0.161$to 0.211 for the first gap, 0.311 to 0.348 for the second gap, and 0.46 to 0.478 for the third gap, respectively. Furthermore, positions and widths of all the three gaps do not shift when the incidence angle varies from 0° to 22.5°. Sharp peaks at a same frequency 0.34427 within the second gaps for all the incidence angles are also found, indicating that a localized mode exists in this defect-free quasiperiodic crystals. Figure 2(b) shows the magnified view of the sharp peaks. The field distribution of this localized mode is shown in Fig. 5(a). Thus, highly isotropic band gaps [16,24] and intrinsic localized modes [19,21,29] in defect-free quasicrystal structures are naturally achieved.

 

Fig. 2 (a) The transmission spectra of transvers electric (TE) modes of PhXQCs I with the radius 0.375$a_0$of rods for six incidence angles, namely 0°, 4.5°, 9°, 13.5°, 18°, 22.5°. (b) Magnified view of the sharp peaks indicated by a dash rectangle within the second bands in (a).

Download Full Size | PDF

Figure 3 shows the calculated photonics and photonics band gaps. For the model of PhXQCs I, Figs. 3(a) and 3(b) show the normalized photonic frequencies $\omega a/2\pi c$ and phononic frequencies $\omega a/2\pi c_{air}$ versus the dimensionless ratio$r/a_0$, respectively, where c and $c_{air}$ are the light and sound speed in air, respectively. $r$ is the radius of the rods. The results for $r/a_0>0.375$ are not considered because of the overlap of the neighboring rods. Three band gaps exist for both TE photonic and phononic modes. The photonic and phononic band gaps begin to appear when the radius $r$ values are 0.06$a_0$and 0.07$a_0$, respectively, meaning that photonic and phononic band gaps almost simultaneously open up for a small radius$r$. When $r$ is 0.19$a_0$, the gap width to mid-gap ratios Δω/ω_mid reaches a maximum value of 47.4% for the first photonic band gaps with lower frequencies; 15.1% and 11.1% were the values for the first and second phononic band gaps, respectively. The band gap width of photonics is slightly reduced, but that of phononics is slightly increased. The maximum of Δω/ω_mid for phononic band gaps can be 24.6% when $r$ is increased to 0.375$a_0$, whereas that for photonic band gaps still is as large as 33.7%. Thus, the width of the gaps is simultaneously large within a very large radius range.$0.19<r/a_0<0.375$ In the periodic counterparts, large simultaneous photonic and phononic band gaps are not easily obtained [9]. To achieve large band gaps, excess cells have to be added to decrease the lattice symmetry [31].

 

Fig. 3 Photonic and phononic band gaps in 8-fold PhXQCs. For PhXQCs I made of silicon rods in air, the following are shown: (a) normalized photonic frequencies $\omega a/2\pi c$ for TE photonic band gaps versus the dimensionless ratio $r/a_0$, where c is the light speed in air. (b) Normalized phononic frequencies $\omega a/2\pi c_{air}$ versus $r/a_0$, where $c_{air}$ is the sound speed in air. For PhXQCs II made of air holes in silicon, the following are shown: (c) Normalized photonic frequencies $\omega a/2\pi c$ for TE and TM photonic band gaps versus $r/a_0$. (d) Normalized phononic frequencies $\omega a/2\pi c_{T,Si}$ for out-of-plane and in-plane phononic band gaps versus $r/a_0$, where $c_{T,Si}$ is the transverse sound speed in silicon. $r$ is the radius of the silicon rods or air holes.

Download Full Size | PDF

For the model of PhXQCs II, photonic frequencies $\omega a/2\pi c$ and phononic frequencies $2\pi c_{T,Si}$ versus$r/a_0$ are also shown, where $r$ is the radius of air holes, see Figs. 3(c) and 3(d), respectively. Multiple out-of-plane (yellow) and in-plane (red) modes phononic band gaps are observed. The second phononic band gaps for out-of-plane modes can be achieved when $r$ is 0.11$a_0$, and in this case, transverse magnetic (TM) photonic band gaps also appear. Both photonic and phononic band gaps coexist all from this radius, thereby indicating that simultaneous photonic and phononic band gaps is also easily obtained within a large radius range. With increasing radius r, the band gaps for TE, TM, out-of-plane, the first and second in-plane modes are increased in size, but that of highest frequency in-plane mode increases and then decreases in size. When $r$ is 0.375$a_0$, the gap width ratio simultaneously reaches maximum 33.8% and 15.5% for the TE and TM modes, 22.2% for the out-of-plane mode and 20% for the in-plane mode for the first phononic band gaps. Furthermore, complete photonic band gaps (green) exist when $r$ is larger than 0.16$a_0$. The maximum of Δω/ω_mid for complete photonic band gaps reaches 17.1% when $r$ is 0.34$a_0$. Complete phononic band gaps (brown) begin to appear when $r$ is larger than $0.33a_0$ and the maximum of Δω/ω_mid reaches 10.3% when $r$ is 0.375$a_0$. The Δω/ω_mid for complete photonic band gaps is 12.7%. These values can be comparable with those of periodic counterparts [32]. Thus, large complete photonic and phononic band gaps can be achieved, indicating that electromagnetic and elastic waves can simultaneously be completely prohibited or localized.

Note that the band gaps of periodic structures depend on the direction, but band gaps of the two PhXQCs are all isotropic because of their high order rotational symmetry [16], thereby indicating that the positions and widths of both the photonic and phononic band gaps are insensitive to the incident direction. That has also been demonstrated in Fig. 2.

3. Localized mode in defect-free PhXQCs

In periodic structures, defect has to be intentionally introduced in order to achieve defect modes, allowing localize electromagnetic or elastic waves [8–14]. However, intrinsic localization exists in defect-free quasiperiodic structures [19–21,29], as also shown in Fig. 2. Figures 4(a) and 4(b) display the normalized photonic and phononic frequency of localized modes in defect-free PhXQCs I and II versus the ratio$r/a_0$, respectively. In PhXQCs I, see Fig. 4(a), phononic localized mode begin to appear when the radius $r$ is 0.21$a_0$. The phononic frequency $\omega a/2\pi c_{air}$ is around 0.55, and this frequency is insensitive to the variation of the radius. The photonic localized mode is also found when $r$ is larger than $0.25a_0$. The frequency of this localized mode is reduced when $r$ increases. Thus, both photonic and phononic localized modes simultaneously exist when $r$ is larger than $0.25a_0$, thereby presenting that both electromagnetic and elastic waves can be confined in defect-free PhXQCs I within a large radius range.

 

Fig. 4 The normalized photonic and phononic frequencies of localized modes versus the ratio$r/a_0$ in defect-free (a) PhXQCs I, (b) PhXQCs II.

Download Full Size | PDF

In PhXQCs II, an out-of-plane phononic localized mode whose frequency is almost the same as that of PhXQCs I is found when the radius $r$ is equal to or larger than 0.21$a_0$. An in-plane localized mode with a lower frequency appears when $r$ is increased to be $0.35a_0$. There appear two TE localized modes when $r$ is larger than $0.285a_0$, and a TE localized mode is found when $r$ is $0.315a_0$. Moreover, two TM localized modes with higher frequencies arise when $r$ is increased to be $0.325a_0$.. PhXQCs II possesses phononic and photonic localized modes simultaneously when $r$ is larger than $0.285a_0$. Furthermore, all polarizations for both photonic and phononic localized modes can be realized if $r$ is equal to or larger than $0.35a_0$_..

Strong acousto-optic interaction depends on the degree of simultaneous spatial localization of optical and acoustic waves. FEM is also used to calculate the distribution of defect modes. The rotational symmetry of the quasiperiodic structures induces average indexes of the surrounding area is lower (PhXQCs I) or higher (PhXQCs II) than that of the central zone, thereby producing that the central zone may behave as a defect. The nonperiodicity of the quasiperiodic structure introduces the localization of the field distributions, while the self-similarity leads to the extension of the field distributions, meaning that the competition between the nonperiodicity and self-similarity determines the occurrence of localized modes in PhXQCs [19]. Thus, the field distributions would be localized and extended around the center. The distributions of electric field for TE localized mode and pressure field for phononic defect modes in PhXQCs I with a radius 0.375$a_0$ are shown in Figs. 5(a) and 5(b), respectively. These localized modes have 8-fold symmetry, which is the same as the quasicrystal structure’s symmetry. The strong field localization and overlap of both photonic and phononic localized modes take place in the central region of PhXQCs. Meanwhile, the field distribution of localized modes extends around the center and decays in a length scale of one or two basic cells. However, the defect modes are tightly confined in the intentional introduced cavity within a very small region in periodic structures [9]. Thus, the overlapping region of photonic and phononic modes in PhXQCs is much larger than that in periodic counterparts, inducing that the interaction space between the optical and acoustic waves is highly increased. This feature possibly leads to new characteristics of acousto-optic effect in PhXQCs, which should be intensively studied.

 

Fig. 5 The field distribution of localized modes in defect-free PhXQCs I with a radius 0.375$a_0$: (a) electric field profile for the TE mode and (b) pressure field profile for the phononic mode.

Download Full Size | PDF

From Figs. 4(a) and 4(b), PhXQCs II have abundant localized modes compared with PhXQCs I. These localized modes exhibit different characteristics of field distribution. Thus, these two kinds of localized modes are discussed separately. The field distribution properties of the first kind of localized modes is similar to that of PhXQCs I, as shown in Fig. 6, thereby indicating that both electromagnetic and elastic fields are distributed in the central region of PhXQCs. TE mode is the third localized mode with the highest frequencies in the band gaps. The distributions of magnetic field for two TM localized modes are similar except that much field is also distributed in the eight holes neighboring the central hole for the second TM mode with higher frequency, as shown in Figs. 6(b) and 6(c). From Figs. 5(b) and 6(d), the distribution as well as frequency for the out-of-plane phononic mode in PhXQCs II is almost the same to that in PhXQCs I. The overlap region of photonic TM and phononic localized mode is larger than that of photonic TE and phononic modes.

 

Fig. 6 The field distribution of localized modes in defect-free PhXQCs II with a radius 0.375$a_0$: (a) electric field profile for the third TE mode with highest frequencies, (b) magnetic field profile for the second TM mode with higher frequency, (c) magnetic field profile for the first TM mode and (d) displacement u_z for the out-of-plane phononic mode.

Download Full Size | PDF

The field distribution scope of another kind of localized modes in defect-free PhXQCs II is wider than that of the first kind of localized modes. Figure 7(a) displays the electric field for the first and the second TE localized mode. The total displacement $u=\sqrt{[{\left(u_x\right)}^2+{(u_y)}^2]}$ for the in-plane phononic localized mode is shown in Fig. 7(b). Their normalized frequencies are lower than that of the first kind of localized modes in the PhXQCs II discussed above. Partial field for both photonic and phononic modes is distributed near the central hole. At the same time, much field is distributed in the circle region with a radius of about 0. _325a. Although the distribution for this kind of defect modes becomes more extended, the overlap between the photonic and phononic modes is still large because of their similar profiles. Thus, the acousto-optic interaction within a larger area can also be expected.

 

Fig. 7 The extended field distribution of the localized modes in defect-free PhXQCs II with a radius 0.375$a_0$: (a) electric field profile for the first TE modes with lower frequencies, (b) total displacement $u=\sqrt{[{\left(u_x\right)}^2+{(u_y)}^2]}$ for the in-plane phononic mode.

Download Full Size | PDF

Note that defect modes similar to that of periodic structures can still be achieved by intentionally introducing defects into quasiperiodic structures [21,22]. Thus, both the frequencies and field distributions of the photonic and phononic localized modes in PhXQCs can easily be adjusted to suit simultaneously confining and tailoring electromagnetic and elastic waves. Obviously, the quasiperiodic structures made of both rods in air (PhXQCs I) and air holes in solid material (PhXQCs II) can simultaneously confine electromagnetic and elastic waves in space.

Considering telecommunication applications in practice, the operating wavelength of optical wave is selected as about 1550nm. The structures with radii r of 0.375$a_0$ are chosen. For the photonic TE localized mode in PhXQCs I, the corresponding lattice constant a is about 534nm. Accordingly, the operating frequency of phononic localized mode is 349MHz. In PhXQCs II, the lattice constant is chosen as about 612nm on the basis of the third photonic TE localized mode. The frequency of the out-of-plane phononic localized mode is 5.81GHz. Both theorectical and experimental results based on optomechanical crystals or PhXCs of air holes in solid [11–14,33] indicate that PhXQCs II can match the practical technological feasibility. It should be pointed out that there exists strong decay of energy for high-frequency sonic signals propagated in air due to air absorption [34,35]. Thus, the telecommunication applications feasibility for PhXQCs I strongly depends on propagation technology of a few hundred MHz ultrasonic waves in air.

4. Conclusion

In conclusion, we numerically demonstrate quasiperiodic phoxonic crystal (PhXQCs) that can provide simultaneous large photonic and phononic band gaps. Furthermore, complete band gaps for both photons and phonons are achieved, thereby indicating that all polarizations of both the electromagnetic and elastic waves can simultaneously be controlled. The simultaneous localization of electromagnetic and elastic waves for defect-free localized modes are also investigated. Abundant photonic and phononic localized modes can coexist in a same large area, which can enlarge the interaction space of electromagnetic and elastic waves. Compared with the periodic counterparts, PhXQCs show some interesting features, such as large and complete simultaneous band gaps with isotropy, abundant localized modes, and localization in a same larger area. Thus, the proposed quasiperiodic structures provide a promising method for simultaneously confining and tailoring electromagnetic and elastic waves with potential applications for highly controllable photon-phonon interactions and acousto-optical devices. Acousto-optic coupling in this kind of phoxonic crystals will be investigated in our subsequent works.