+ 1 Qk
,
where Qk stands for the degree of confinement in-plane and Q⊥ stands for that in vertical directions. This definition of Q is equivalent to that found by fitting parameters from U = U0e−ω0Qt.
The quality factor can be derived from the FDTD simulations. The excitation is the same as that described in the previous section. The defect mode is excited using point sources narrowly peaked at the resonant frequency of the mode and located according to the symmetry of the mode to excite the mode of interest. Then we record the electromag-netic energy stored in the cavity and the integral of the Poynting vectors over the surface of the domain as the radiated power. P⊥ is measured at ±2a above and below the slab, where a is the lattice constant. And Pk is measured on the other four sides of the domain.
The simulation results are shown in Table 4.4.
Table 4.4: The quality factors.
Type Q⊥ Qk Qtot Q0* Type A 282 443 172 170 Type B 276 1032 218 213 Type C 287 1325 236 231 Type D 303 1524 253 247 Type E 321 1671 269 264
*Q
0is derived from fitting parameters
We can observe that Qk increases as the radii increase per layer outward. It can be interpreted that this effect is originated from the increase of averaged ε1
r(r) as r goes away from the defect center, which acts as the character of potential function in quantum mechanics, providing the effect in analogy to quantum well structures.
Chapter 5 Summary
First the techniques for the FDTD method including Yee’s algorithm and various boundary conditions used for the simulation for photonic crystals are introduced. Then a brief introduction of symmetry analysis by group theory and design rules of defect modes of photonic crystal nano-cavities are investigated. The modes in single defect in 2D photonic crystal slab are doubly degenerate dipole modes. Numerical simulations of the band structures, mode profiles, and resonant frequencies are obtained by both the PWE method and the FDTD method with various boundary conditions. Some techniques like supercell scheme or domain containing two unit cells are employed.
For the defect modes, quality factors are also derived by the FDTD method. Proper excitation is needed. The excitation of point sources narrowly peaked around the fre-quency of the desired mode are placed according to the symmetry property of the mode.
The simulated mode profiles are consistent with the criterion of symmetry analysis.
For the design of the cavity for high quality factors, we also try to compare slightly different structures by gradually increasing or decreasing the hole radii per layer outward.
Numerical experiments showed that a gradually increase of radii of holes per layer out-ward may increase the in-plane quality factor of donor modes, just as the case of the enhancement of confining electrons by quantum wells.
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