Optomechanic interaction in a corrugated phoxonic nanobeam cavity

The interaction between phonons and photons is investigated theoretically in a phoxonic cavity inside a corrugated nanobeam waveguide presenting band gaps for both electromagnetic and elastic waves. The structure is made by drilling periodic holes on a silicon nanobeam with lateral periodic stubs and the tapered cavity is constructed by changing gradually the geometrical parameters of both the holes and stubs. We show that this kind of cavity displays localized phonons and photons inside the gaps, which can enhance their interaction and also promotes the presence of many optical confined modes with high quality factor. Using the finite-element method, we demonstrate that with appropriate design of the tapering construction, one can control the cavity modes frequency without altering significantly the quality factor of the photonic modes. By changing the tapering rates over the lattice constants, we establish the possibility of shifting the phononic cavity modes frequency to place them inside the desired gap, which enhances their confinement and increases the mechanical quality factor while keeping the strength of the optomechanic coupling high. In our calculations, we take account of both mechanisms that contribute to the acousto-optic interaction, namely photoelastic and interface motion effects. We show that in our case, these two effects can contribute additively to give high coupling strength between phononic and photonic cavity modes. The calculations of the coupling coefficient which gives the phonon-photon coupling strength give values as high as 2 MHz while photonic cavity modes display quality factor of 10${}^5$ and even values up to 3.4 MHz but with smaller photonic quality factors.

Figure 1

(a) Periodic corrugated silicon nanobeam with parallelepiped stubs grafted on each side and circular holes drilled in the middle. The radius of the holes, the thickness of the nanobeam, the stubs’ length and width are respectively denoted $r$, $e$, $L$, and $d$ and fixed to $r$ = 0.3$a$, $e$ = 0.44$a$, $L=a$, and $d$ = 0.5$a$, where $a$ is the lattice parameter fixed to 500 nm. (b) Nanobeam's phononic band structure for wave propagation in $x$ direction, in which we highlighted the modes with the symmetry EO, EE, OO, and OE in green, red, purple, and blue colors respectively. The gray region indicates the absolute band gap.

Figure 2

(a) Schematic view of the tapered cavity created inside the phoxonic strip waveguide where the center is between two holes. (b) Unit-cell period profile from cell to cell outside and inside the cavity as a function of the spatial position. Two parabolic shapes called tapering α and β are plotted respectively with red circles and black squares.

Figure 3

(a) Distribution of the electric and magnetic fields of the OE cavity photonic mode selected for the calculation of the AO interaction. (b) Transmission coefficient through the structure showing the peak of the cavity mode in the two cases of tapering α and β.

Figure 4

(a) Phononic band structure of the perfect nanobeam where the red shaded regions named BG1, BG2, and BG3 are the gaps for EE symmetry modes. (b) AO coupling between the photonic cavity mode at 1550.4 nm and the phononic mode at 2.46 GHz, estimated by the wavelength shift modulation along half of the acoustic period, and by the calculation of the coupling coefficients $g_{\mathrm{PE}}$ and $g_{\mathrm{MI}}$.

Figure 5

(a) Schematic view of the cavity created inside the phoxonic strip by tapering the unit-cell period, the hole's radius, as well as the lateral stub's height. (b) The unit-cell period and lateral stub heights profiles from cell to cell indicated in red circles and green squares, respectively, outside and inside the cavity.

Figure 6

(a) Phononic band structure of the perfect nanobeam. (b) AO coupling between the photonic cavity mode at 1605.2 nm and the phononic modes at 2.08, 5.34, and 5.49 GHz, estimated by the calculation of the coupling coefficients $g_{\mathrm{PE}}$ and $g_{\mathrm{MI}}$.

Figure 7

AO coupling between the photonic cavity mode at 1605.2 nm as well as the one at 1596.7 nm and the phononic modes at 2.08 GHz, estimated by the calculation of the coupling coefficients $g_{\mathrm{PE}}$ and $g_{\mathrm{MI}}$ as well as their wavelength modulation by the acoustic field.

Figure 8

(a) Schematic view of the cavity created inside the phoxonic strip by tapering the unit-cell period, the hole's radius, as well as reducing the lateral stub's height. (b) The unit-cell period and lateral stub heights profiles from cell to cell are indicated in red circles and green squares, respectively, outside and inside the cavity.

Figure 9

(a) Phononic band structure of the perfect nanobeam. (b) AO coupling between the photonic cavity mode at 1538.2 nm and the phononic modes at 3.74, 3.78, and 3.94 GHz, estimated by the calculation of the coupling coefficients $g_{\mathrm{PE}}$ and $g_{\mathrm{MI}}$.