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

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Pa<strong>in</strong>ter et al. Vol. 16, No. 2/February 1999/J. Opt. Soc. Am. B 279Fig. 7. Plot <strong>of</strong> the air <strong>an</strong>d dielectric b<strong>an</strong>d edges as a function <strong>of</strong>slab thickness. The midgap frequency is also plotted, as adashed curve.Fig. 8. Plot <strong>of</strong> the b<strong>an</strong>dgap between the fundamental guided airb<strong>an</strong>d <strong>an</strong>d dielectric b<strong>an</strong>d versus slab thickness.the slab thickness is reduced, ow<strong>in</strong>g to <strong>in</strong>creas<strong>in</strong>g photonmomentum <strong>in</strong> the ẑ direction. However, the <strong>in</strong>crease <strong>of</strong>the b<strong>an</strong>dgap width with th<strong>in</strong>ner dielectric slabs 38 seemsat first to be counter<strong>in</strong>tuitive. A simple argument mightbe made to expla<strong>in</strong> this result by <strong>an</strong>alogy with the frequencydispersion relation for a photon <strong>in</strong> a const<strong>an</strong>t <strong>in</strong>dexmaterial, 2 1 c n 2 k 2 k 2 , (2)where n is the material refractive <strong>in</strong>dex, c is the speed <strong>of</strong>light <strong>in</strong> vacuum, k is the photon momentum normal tothe slab, <strong>an</strong>d k is the <strong>in</strong>-pl<strong>an</strong>e momentum (<strong>in</strong> this discussionfixed at the X or J po<strong>in</strong>t). The b<strong>an</strong>dgap <strong>in</strong>crease resultsfrom the concentration <strong>of</strong> the air b<strong>an</strong>d mode <strong>in</strong> theair holes <strong>an</strong>d the dielectric b<strong>an</strong>d <strong>in</strong> the high dielectric region.The energy required for a given photon momentumis larger <strong>in</strong> a lower-<strong>in</strong>dex material. Thus, as the slab becomesth<strong>in</strong>ner <strong>an</strong>d k <strong>in</strong>creases, the air b<strong>an</strong>d frequency<strong>in</strong>creases faster th<strong>an</strong> the dielectric b<strong>an</strong>d, <strong>an</strong>d the b<strong>an</strong>dgap<strong>in</strong>creases. The variation <strong>of</strong> b<strong>an</strong>dgap with slab thicknesswill have nonnegligible effects on the defect cavities <strong>an</strong>alyzedbelow.4. THREE-DIMENSIONAL FINITE-DIFFERENCE TIME-DOMAIN ANALYSIS OFDEFECT MODESWe now <strong>in</strong>troduce a defect <strong>in</strong>to the <strong>photonic</strong> <strong>crystal</strong> <strong>an</strong>d<strong>an</strong>alyze the properties <strong>of</strong> the localized <strong>modes</strong> that develop.By <strong>an</strong>alyz<strong>in</strong>g the defect <strong>modes</strong> <strong>in</strong> three dimensions,we are able to evaluate the effectiveness <strong>of</strong> the verticalconf<strong>in</strong>ement <strong>of</strong> the defect <strong>modes</strong> for a f<strong>in</strong>ite-thicknessdielectric slab. The FDTD calculations <strong>in</strong> this section areperformed for three different thicknesses <strong>of</strong> the slab d:(i) d 0.933a, (ii) d 0.533a, <strong>an</strong>d (iii) d 0.4a. Forfrequencies <strong>in</strong> the b<strong>an</strong>dgap <strong>of</strong> the <strong>photonic</strong> <strong>crystal</strong>, thesethicknesses correspond to approximately , /2, <strong>an</strong>d 3/8,where is the wavelength <strong>of</strong> light <strong>in</strong> the slab.To simulate the defect cavity we use the FDTD method,where we truncate the computational mesh by plac<strong>in</strong>g <strong>an</strong>onreflect<strong>in</strong>g absorber at all boundaries. 39,40 Our computationalmesh has a resolution <strong>of</strong> 15 po<strong>in</strong>ts per <strong>in</strong>terholespac<strong>in</strong>g, or equivalently 15 po<strong>in</strong>ts per wavelength forfrequencies with<strong>in</strong> the <strong>photonic</strong> <strong>crystal</strong> b<strong>an</strong>dgap. As <strong>in</strong>the previous calculations the radius <strong>of</strong> the air holes areequal to 0.3a. The number <strong>of</strong> layers <strong>of</strong> air holes that surroundthe defect is three, as shown <strong>in</strong> Fig. 3.An <strong>in</strong>itial TE-polarized electric field is used to excitethe TE-like <strong>modes</strong> <strong>of</strong> the defect structure. Then the <strong>in</strong>itialfield is evolved <strong>in</strong> time with the FDTD method. Afast Fourier tr<strong>an</strong>sform is applied to the result<strong>in</strong>g time series<strong>of</strong> the field at a po<strong>in</strong>t <strong>of</strong> low symmetry <strong>in</strong> the cavity, topick out the reson<strong>an</strong>ce peaks <strong>of</strong> the defect mode. Thefield is then convolved <strong>in</strong> time with a b<strong>an</strong>dpass filter toselect out the defect <strong>modes</strong>. 41 In the follow<strong>in</strong>g calculationthe radius <strong>of</strong> the defect hole is kept const<strong>an</strong>t at 0.3a,whereas its refractive <strong>in</strong>dex is varied between 1.4 <strong>an</strong>d 3.4.With the chosen r/a <strong>an</strong>d n slab , <strong>an</strong>d for the defectrefractive-<strong>in</strong>dex r<strong>an</strong>ge 1.4–3.4, only one defect b<strong>an</strong>d isformed. A plot <strong>of</strong> the defect b<strong>an</strong>d frequency versus defectrefractive <strong>in</strong>dex is given <strong>in</strong> Fig. 9 for the three differentslab thicknesses.Because we chose to use a circular defect, the symmetry<strong>of</strong> the defect cavity is the po<strong>in</strong>t group <strong>of</strong> the hexagonalFig. 9. Plot <strong>of</strong> the normalized frequency versus defect refractive<strong>in</strong>dex <strong>of</strong> the degenerate defect mode. The radius <strong>of</strong> the defecthole was kept const<strong>an</strong>t at 0.3a, whereas the refractive <strong>in</strong>dex wasvaried.

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