The circular grating around the center disc acts as a Bragg mirror and imposes a photonic band gap, i.e. stop band, which prevents lateral leakage of light from the cavity. The resonance frequency which has to lie inside the band gap is then mainly determined by the radius of the central defect. To achieve a small cavity volume, which is imperative for high switching speeds, we optimize the geometric parameters of the circular grating to maximize the band gap. A larger band gap leads to a larger reflectivity R of the circular grating for the same number of gratings. Therefore, the light decays faster inside the circular grating and decreases the spatial extension of the cavity mode. The band gap of the circular grating can be approximated by the corresponding linear grating which in principle is just the cross-section of the circular grating.
- The band gap of linear goating
The inset of Fig. 2 shows the cross section of the linear grating. The duty cycle dc = q=a is the length of the air section relative to the period of the circular gratings, i.e., the lattice constant a. Together with the height h of the Si structure, the duty-cyle dc, and the lattice constant a they are the three free parameters for the mirror structure. These parameters are varied to obtain the largest possible band gap centered around the telecom wavelength of 1.550 nm. Focussed only on the TM polarization the calculation results are shown in Fig. 2. There is a maximum band gap of 12% (band-gap width divided by center wavelength) for a thickness of approx. 320 nm. For thicknesses larger than 420 nm, even larger band gaps can be achieved. The linear grating which maximizes the band gap for a Si thickness of 340 nm has a duty cycle of 0.24 and a lattice constant of 386 nm. A CGR with a quality-factor Q of a few thousands achieves the target specification of switching on a picosecond timescale. This means a target reflectivity R of the circular gratings of approx. 85%, which corresponds to approx. 6 to 8 circular gratings around the defect.
