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Abstract
The dual beam guides for transverse-electric and transverse-magnetic polarizations of electromagnetic (EM) wave and elastic wave in defect-free phoxonic crystals are reported. The realization for phoxonic virtual waveguides relies on dual flat equifrequency contours (EFCs) enabling self-collimation for EM and elastic waves. As a possible application of our work, the enhanced acousto-optic (AO) interaction in this kind of defect-free phoxonic waveguide, just as it does in defect-based waveguides, is further studied. Results show that obvious shifts of the transmission peaks of EM waves exist for both polarizations during one period of the elastic wave, and single-phonon exchange dominates the AO interaction. This kind of phoxonic virtual waveguide provides an effective platform to enhance AO interaction and exhibits some advantage over defect-based waveguides by properly manipulating the photonic and phononic dispersion surfaces.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The strong enhancement of opto-mechanical and acousto-optic (AO) interactions at micro- and nanoscale cavities [1] and waveguides [2] has attracted considerable interest in recent years. Periodic or quasiperiodic structures called optomechanical crystals [3] or phoxonic crystals (PhXCs) [4–6], which originate from the concepts of photonic crystals (PhTCs) [7] and phononic crystals (PhNCs) [8,9], can easily form cavities and waveguides on a wavelength scale, thereby serving as a desirable and powerful planar circuit to couple photons and phonons in an unprecedented manner [5,10–12]. Electromagnetic (EM) and elastic waves can be confined or guided in the same defect area by appropriately introducing point or line defects into PhXCs due to the existence of phoxonic bandgaps (the coexistence of photonic and phononic bandgaps). The enhancement of the AO interaction via bandgap-based waveguides (BGWs) is realized and allows for the high performance of integrated circuits and devices [13–16]. However, a high threshold of refractive index contrast in BGWs is needed to form complete bandgaps, which are the key condition for ensuring the simultaneous implementation of the transverse-electric (TE) and transverse-magnetic (TM) polarizations [17–19]. The refractive indexes of some important acousto- and electro-optic materials are usually very low, thereby limiting the applications of nonlinear materials in BGWs. Furthermore, the introduction of defects results in difficulties in the technological process of actual fabrication.
Photonic or phononic virtual waveguides based on self-collimation effect have no such deficiencies [20–28]. Recently, we have reported virtual waveguides using self-collimation effect for EM and acoustic waves in defect-free phoxonic crystals made of silicon column array in air [25]. AO effect originates from the elastic strain caused by the elastic wave passing through the medium. However, the elastic strain of the acoustic wave in the air is rather weak, indicating that the AO effect in the air will be barely noticeable. Thus, this kind of structure is unsuitable for enhancing the AO interaction. In this letter, we consider a phoxonic structure that consists of an air hole array in a silicon matrix, which ensures that the elastic wave is propagated in the solid, possibly enhancing the AO effect. The coexistence of beam guides of EM wave and elastic wave in defect-free PhXCs based on self-collimation effect is first achieved by properly manipulating the dispersion surfaces. Without the existence of complete bandgaps, TE and TM polarizations of EM and elastic waves can be simultaneously guided. The application of the enhanced AO interaction in this kind of phoxonic virtual waveguides is further investigated.
2. Phoxonic virtual waveguides using self-collimation effect for both electromagnetic and elastic waves
The phoxonic structure that we considered is constituted by a square lattice of air holes in a silicon matrix. The material parameters of the silicon are as follows: refractive index n = 3.46, mass density ρ = 2331 kg/m3, and transverse and longitudinal speeds of sound ct = 5360 m/s and cl = 8950 m/s, respectively. The lattice constant is denoted by a, and the relative radius r/a is 0.40. In view of possible telecommunication applications, the lattice constant a is selected as 463 nm. Because the large density contrast between the solid and the air, the rigid boundary conditions are applied to the interface between the silicon and the air for elastic wave. Figure 1(a) shows the corresponding photonic dispersion curves, where the green and purple lines represent TM and TE polarization, respectively. We extracted the Brillouin zone of the EFCs of the TE and TM waves in the second band shown in Figs. 1(b) and 1(c). The normalized frequencies of 0.301 (blue line) and 0.317 (red line) own flat TE and TM EFCs along the ΓX direction, respectively. This means that TE and TM waves can exhibit self-collimation along the ΓX direction even if no complete bandgaps exist. Figure 2(a) shows the corresponding phononic dispersion curves of PhXCs for the in-plane modes. We extracted the Brillouin zone of the EFCs in the sixth band (orange lines) of the elastic waves shown in Fig. 2(b). Flat EFCs exist at the normalized frequency of 1.058 along the ΓX direction. Therefore, by manipulating the dispersion surface, we obtained the coexistence of flat photonic and phononic EFCs along the same direction, that is, ΓX direction. Note that the out-of-plane modes of elastic waves can also exhibit self-collimation effect. However, the shear vibrations of the out-of-plane mode is parallel to the holes along the z axis, which makes the dynamic motion of the silicon-air interfaces is negligible, thereby resulting in the interaction between the EM and the out-of-plane modes is very weak. Thus, we did not consider this kind of phononic modes.
Fig. 1. (a) Photonic dispersion curves for a PhXC of air hole array in a silicon matrix. The TE and TM polarizations are represented by the purple and green lines, respectively. The blue and red dashed lines correspond to the normalized frequency of 0.301 and 0.317, respectively. (b) The Brillouin zone of the EFCs of the second band and the blue EFC indicates the case of TE waves at the frequency 0.301 c/a along the ΓX direction. (c) The Brillouin zone of the EFCs of the second band and the red EFC indicates the case of TM waves at the frequency 0.317 c/a along the ΓX direction. (d) Electric field of TE wave distribution at the frequency 0.301 c/a. (e) Magnetic field of TM wave distribution at the frequency 0.317 c/a.
Fig. 2. (a) Phononic dispersion curves for a PhXC of air hole array in a silicon matrix. The orange dashed lines correspond to the normalized self-collimation frequency of 1.058. (b) The Brillouin zone of the EFCs of the sixth band and the orange contours represent the case of the elastic waves at the normalized frequency 1.058 along the ΓX direction. (c) Displacement field distributions of the longitudinal elastic wave at the frequency 1.058 ct/a.
To calculate the field distributions in the phoxonic virtual waveguides, the excitation source with a width 4a along the y-direction is placed on the left side of the PhXCs. Figures 1(d) and 1(e) show the distributions of the electric field at the frequency 0.301 c/a and magnetic field at the frequency 0.317 c/a, respectively, where c is the speed of EM waves in air. The phononic propagation modes within the passing band along the ΓX direction are of longitudinal character, and only longitudinal waves can be strongly transmitted [29]. Thus, we only considered the case of longitudinal elastic wave (P waves, normal harmonic vibration). The distribution of the longitudinal elastic wave field at the frequency 1.058 ct/a is shown in Fig. 2(c). As shown in the figure, the EM and longitudinal elastic waves are well propagated along the ΓX direction without diffraction in the defect-free structure. The fields of EM and elastic waves overlap effectively due to their non-diffraction propagation in the same area, as shown in Figs. 1(d), 1(e), and 2(c), which can possibly enhance the interaction of EM and elastic waves. Thus, this kind of virtual waveguides, which requires neither defects nor bandgaps, can control and guide EM and elastic waves simultaneously, thereby providing an effective platform to realize enhanced AO coupling.
3. Application to the enhanced AO interaction based on phoxonic virtual waveguides
To further calculate the AO interaction, the three independent stiffness coefficients of the silicon material are given as follows [13,30]: C11 = 16.57 × 1010 N m−2, C12 = 6.39 × 1010 N m−2, and C44 = 7.962 × 1010 N m−2. We simulated the phoxonic self-collimation in PhXCs along the ΓX direction considering the stiffness coefficient of the silicon. The displacement field distributions of longitudinal elastic wave at the frequency 1.0148 ct/a, electric field of TE wave at the frequency 0.29968 c/a, and magnetic field of TM wave at the frequency 0.31424 c/a are shown in Figs. 3(a), 3(b), and 3(c), respectively. These field distributions verify that phoxonic self-collimation still exists after considering the stiffness coefficient of the silicon besides the fact that the transmission peak frequencies have a slight departure from the original values.
Fig. 3. Phoxonic self-collimation in PhXCs along the ΓX direction considering the stiffness coefficient of the silicon: (a) displacement field distributions of the longitudinal elastic wave at the frequency 1.0148 ct/a. (b) electric field distribution of the TE wave at the frequency 0.29968 c/a. (c) magnetic field distribution of the TM wave at the frequency 0.31424 c/a.
We next calculated the AO interaction. The moving interfaces (MIs) and photoelastic (PE) effects are considered in AO interaction. The MI effect is due to the dynamic motion of the silicon–air interfaces, which is the deformation of the structure induced by the elastic waves. The PE effect is related to the change of the refractive index by the generation of the strain field in the structure. The relations of the refraction index and elastic strain tensor are given by [31]
The optical transmission spectrum of TE and TM waves varies periodically in time with the period of the elastic wave [33]. Figure 4 shows the optical transmission peak frequency modulations of the TE and TM waves during one period of the elastic wave and the corresponding Fourier transforms. As shown in the figure, the MI and PE effects are in phase leading to a strengthened AO interaction. Figure 4(a) shows an almost sinusoidal modulation of the full AO coupling. The transmission peak frequency shifts of the TE wave reach ${\Delta}fa/c$=2.5${\times} 10$−4. The analysis on the corresponding Fourier spectrum indicates that the first-order Fourier component is predominant, whereas the contribution of the higher-order components is relatively little. This means that the single-phonon exchange process (the photon absorbs and/or emits one phonon) dominates the AO interaction [32]. On the other hand, we found a steepening sinusoidal modulation for TM wave in Fig. 4(b). This process can also be explained by the corresponding Fourier spectrum. The frequencies of the TM wave have a first-order perturbation, and at the same time, the second-order effect is not negligible, thereby forming the superposition of the sine and square sine function of Ωt. The maximum transmission peak frequency shift is $\Delta fa/c$=3.1${\times} 10$−4 by the full AO coupling for the TM wave. Of note, the contribution of the MI effect is predominant in both cases by comparing the strength of PE and MI effects. The EM and elastic waves propagate without diffraction in the same area. Thus, an efficient overlap field is formed. Then, all those air holes located in the overlapping region move dynamically due to the disturbance of elastic waves. Therefore, the larger variation of the displacement of the silicon–air interfaces in the overlapping region derived by the elastic wave helps to improve the strength of the MI effect, allowing larger frequency shifts to be achieved. Comparing the strength of AO interaction between the TE and TM waves during one period of the elastic wave, we found that the frequency shift of the TM wave is greater than that of the TE wave in the same deformation. One reason is that the group velocity of the self-collimated TM wave is lower than that of the TE wave from Fig. 1(a). This lower group velocity increases the AO interaction life-time so that the frequency shift of the TM wave becomes greater. Comparing the AO interaction in BGWs with a line defect, the strength of the AO interaction for TE and TM waves during one period of the elastic wave can be comparable to the strength of AO interaction based on defect-based silicon phoxonic waveguide [34].
Fig. 4. Frequency modulations of the photonic mode in one period of the phononic mode and the corresponding Fourier transforms. The black and red dashed–dotted lines stand for MI and PE effects, respectively. The blue dashed–dotted lines represent the full AO coupling. (a) AO coupling between TE and elastic waves. (b) AO coupling between TM and elastic waves.
The finite thickness is an important factor in determining the phoxonic self-collimation effect in a PhXC slab. Slab structures can confine waves in the vertical direction by total internal reflection. It has been shown that the 2-D and slab structures have similarities in the band structures and EFCs [35]. Furthermore, the first and second bands of the photonic mode can be easily below the light cone [35], and the elastic waves cannot radiate into the air region. Thus, it can be predicted that the phoxonic self-collimation and enhanced AO effect can be achieved in a real PhXC slab.
4. Conclusions
By manipulating dispersion surfaces properly, we have achieved flat phoxonic EFCs that can guide and control both polarizations of EM and elastic waves due to self-collimation effect in defect-free PhXC along the ΓΧ direction without diffraction. The application to the enhanced AO interaction in this kind of virtual waveguides is further investigated, and the results show that the strength of the AO interaction can be comparable to that based on defect-based phoxonic waveguide using the bandgap effect. Furthermore, compared with BGWs, this new platform realizing enhanced AO interaction has a relatively simple technological process in actual fabrication because no defects are needed and there is no limitation for some nonlinear materials with low refractive index.
Funding
National Natural Science Foundation of China (11604136, 11664024, 11704175); Natural Science Foundation of Jiangxi Province (20171ACB21020).
Disclosures
The authors declare that there are no conflicts of interest related to this article.
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