Defect modes of a two-dimensional photonic crystal in an optically ...

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280 J. Opt. Soc. Am. B/Vol. 16, No. 2/February 1999 Painter et al.lattice of air holes C 6 . The modes can be labeled by theirreducible representations (IRREP) of C 6 . 42 We plottedthe magnitude of the electric field of one of the modesof the defect band in Fig. 10(a). The defect mode transformslike the x component of a vector under the pointgroupoperations of C 6 . Thus we label it an x dipolemode. The character table of C 6 (Ref. 43) shows that thex dipole mode is a member of a doubly degenerate IRREP.This IRREP can be represented by a pair of basis functionsthat transform like x and y vector components. Weplotted the other defect mode, which we term the y dipolemode, in Fig. 10(b). To excite the x dipole mode we usedan initial field with xˆ symmetry. Similarly, for the y dipolemode we used an initial field with ŷ symmetry. Thedipole modes are mainly TE polarized in the slab andhave an antinode in the center of the defect region. Thisis important for maximizing the overlap of the defectmode and the semiconductor gain region, where we anticipatethat electron and hole densities will be much lowerFig. 11. Guided in-plane radiation losses of the x and y dipolemodes (degenerate case) are shown in (a) and (b), respectively.In (c) a cross section along the ŷ direction shows the radiation outthe top half of the cavity for the y dipole mode. In each plot theelectric-field amplitude has been enhanced to highlight thelosses.Fig. 10. 2D slice through the middle of the slab, showing theelectric-field amplitudes of the degenerate defect modes. (a) xdipole mode, (b) y dipole mode.near the surfaces of the etched holes. In Fig. 11 we showplots of the magnitude of the electric field for the x and ydipole modes, enhanced so as to highlight the radiation
Painter et al. Vol. 16, No. 2/February 1999/J. Opt. Soc. Am. B 281pattern in the in-plane directions. From the field plotsone can see that the guided in-plane radiation is highlydirectional and is well suited for coupling to waveguidesfor in-plane routing. The fundamental k component ofthe envelope of the defect modes is clearly near the Xpoint of the Brillouin zone. As discussed above, this isbecause the air band edge is at the X point. Also plottedin Fig. 11 is the radiation pattern in the vertical directionfor the y dipole mode. With a definition of modal volumesimilar to that of Foresi et al. 16 the volume of the x and ydipole modes is calculated to be approximately 2(/2n) 3 .The in-plane symmetry of the defect modes allows us toapply the appropriate mirror conditions to reduce thecomputation size by one fourth. Because we are interestedonly in TE-like modes, we can also reduce the meshsize by a factor of 2 in the ẑ direction by applying an evenmirror boundary condition at the middle of the slab. Usingthe reduced mesh, we calculated the quality factor ofthe y dipole mode for the defect refractive-index range1.4–3.4. As the defect refractive index is varied, the dipolemode frequency sweeps across the bandgap as shownin Fig. 9; thus we are able to obtain plots of the qualityfactor versus frequency.Fig. 14. Plot of the effective in-plane quality factor Q versusnormalized frequency.We used two different methods to calculate the qualityfactor of the cavity modes. The first method measuresthe slope of the exponential decay of the energy of a givencavity mode 44 :Ut U0expt/ ph U0exp o t/Q, (3)where U is the energy in the mode and we have used therelationQ o ph . (4)The photon lifetime of the cavity mode is ph , and o isthe angular frequency. This method is useful for relativelylow Q modes; however, for high Q modes, the slopeis nearly zero and an alternate method is employed. Thesecond method calculates the power absorbed in theboundary P(t) and divides it by the energy stored in thecavity mode, 44Fig. 12. Plot of the quality factor versus normalized frequencyof the y dipole mode.Fig. 13. Plot of the effective quality factor in the vertical directionQ versus normalized frequency.Q oUt. (5)PtFor the defect modes of the structure analyzed in this sectionwe found little discrepancy between the two methods.In the following plots we chose to display the quality factorof the defect modes calculated with the secondmethod, because this method allows one to separate outthe radiation losses into different directions.A plot of Q versus normalized frequency is given in Fig.12 for three different slab thicknesses: d 0.933a, d 0.533a, and d 0.4a. The quality factors of the defectmodes seem at first sight to be rather unspectacularand the cavity losses too high to support lasing in such asmall volume structure. To separate out radiation loss inthe vertical direction that is due to leaky modes, and thelosses that are due to the small number of photonic crystallayers that surround the defect region, we plot the effectivevertical Q and in-plane Q for the y dipole mode inFigs. 13 and 14, respectively. We define these effectivequality factors as follows:
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