Defect modes of a two-dimensional photonic crystal in an optically ...
Defect modes of a two-dimensional photonic crystal in an optically ...
Defect modes of a two-dimensional photonic crystal in an optically ...
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278 J. Opt. Soc. Am. B/Vol. 16, No. 2/February 1999 Pa<strong>in</strong>ter et al.Fig. 5. B<strong>an</strong>d diagram for TM-polarized light (E field polarized<strong>in</strong> the ẑ-direction). In this case the <strong>in</strong>dex contrast <strong>an</strong>d r/a arenot large enough to open a full 2D b<strong>an</strong>dgap between the first (dielectric)<strong>an</strong>d second (air) b<strong>an</strong>ds.with respect to the center <strong>of</strong> the waveguide. The TE<strong>modes</strong> have the electric field polarized <strong>in</strong> the pl<strong>an</strong>e <strong>of</strong> thewaveguide, <strong>an</strong>d the TM <strong>modes</strong> have the magnetic field polarized<strong>in</strong> the pl<strong>an</strong>e <strong>of</strong> the waveguide. The reson<strong>an</strong>t<strong>modes</strong> <strong>of</strong> a 2D patterned dielectric slab surrounded by air,however, are not purely TE or TM but rather what wedesignate TE-like <strong>an</strong>d TM-like. The TE-like <strong>an</strong>d TM-like<strong>modes</strong> are classified by how they tr<strong>an</strong>sform under thehorizontal mirror operation <strong>in</strong> the middle <strong>of</strong> the dielectricslab. TE-like <strong>modes</strong> are even under reflection, <strong>an</strong>d TMlike<strong>modes</strong> are odd. TE-like <strong>modes</strong> are composed <strong>of</strong> evenTE slab <strong>modes</strong> <strong>an</strong>d odd TM slab <strong>modes</strong>, whereas TM-like<strong>modes</strong> are formed from even TM slab <strong>modes</strong> <strong>an</strong>d odd TEslab <strong>modes</strong>. As mentioned <strong>in</strong> Section 2, we are <strong>in</strong>terestedprimarily <strong>in</strong> active regions with predom<strong>in</strong><strong>an</strong>tly TE ga<strong>in</strong>.Only the even TE slab <strong>modes</strong> will couple to such qu<strong>an</strong>tumwells placed at the center <strong>of</strong> the waveguide, or, <strong>in</strong> the 2Dpatterned waveguides, coupl<strong>in</strong>g will be limited to the TElike<strong>modes</strong>. For this reason we focus on design<strong>in</strong>g defectcavities that support TE-like localized <strong>modes</strong>. Also, depend<strong>in</strong>gon the thickness <strong>of</strong> the dielectric slab, there c<strong>an</strong>be higher-order guided <strong>modes</strong> supported by the patternedslab. 35 Here we consider dielectric slabs approximately ahalf-wavelength thick <strong>in</strong> which there is a b<strong>an</strong>dgap betweenthe fundamental (0th order) air <strong>an</strong>d dielectric b<strong>an</strong>dTE-like <strong>modes</strong>.A 3D FDTD simulation is used to model the fundamentalTE-like b<strong>an</strong>d structure <strong>of</strong> the <strong>optically</strong> th<strong>in</strong> patternedwaveguide. By apply<strong>in</strong>g appropriate Bloch boundaryconditions over a unit cell <strong>of</strong> the <strong>photonic</strong> <strong>crystal</strong>, one c<strong>an</strong>obta<strong>in</strong> the spectral response for a given <strong>in</strong>-pl<strong>an</strong>e wavevector. The peaks <strong>in</strong> the frequency spectrum give theeigen<strong>modes</strong> <strong>of</strong> the <strong>photonic</strong> <strong>crystal</strong> at the k vector determ<strong>in</strong>edby the boundary conditions. The <strong>in</strong>terestedreader is referred to the paper by Ch<strong>an</strong> et al. 36 for furtherdetails. In our case the unit cell consists <strong>of</strong> <strong>an</strong> <strong>in</strong>-pl<strong>an</strong>egeometry given by the 2D unit cell <strong>of</strong> the hexagonal lattice.In the ẑ direction we do not have periodicity, so afull description <strong>of</strong> the slab <strong>an</strong>d surround<strong>in</strong>g air must begiven. The Bloch boundary condition is applied to allfour sides normal to pl<strong>an</strong>e <strong>of</strong> the slab, <strong>an</strong>d Mur’s absorb<strong>in</strong>gboundary condition 37 is applied to the top boundary.At the bottom boundary we apply <strong>an</strong> even mirror reflectionpositioned at the middle <strong>of</strong> the slab to select out onlythe TE-like <strong>modes</strong> <strong>of</strong> the structure. A uniform spatialresolution <strong>of</strong> 20 po<strong>in</strong>ts per <strong>in</strong>terhole spac<strong>in</strong>g is used to discretizethe unit cell, <strong>an</strong>d the <strong>in</strong>itial field is evolved for 2 14time steps. This gives a normalized frequency resolution<strong>of</strong> 0.0012 <strong>an</strong>d a spatial resolution <strong>of</strong> approximately 20po<strong>in</strong>ts per wavelength <strong>in</strong> the high-<strong>in</strong>dex slab for frequencieswith<strong>in</strong> the b<strong>an</strong>dgap. In these 3D calculations thereal slab refractive <strong>in</strong>dex <strong>of</strong> 3.4 is used as opposed to theeffective <strong>in</strong>dex from Subsection 3.A.The TE-like b<strong>an</strong>d structure for a 2D patterned slabwaveguide with thickness d 0.4a is plotted <strong>in</strong> Fig. 6.The light l<strong>in</strong>e, <strong>in</strong>dicated by a solid l<strong>in</strong>e <strong>in</strong> Fig. 6 <strong>an</strong>d adashed l<strong>in</strong>e <strong>in</strong> Fig. 4, is the divid<strong>in</strong>g l<strong>in</strong>e between guided<strong>an</strong>d leaky <strong>modes</strong> <strong>of</strong> the perforated dielectric slab. Thelight l<strong>in</strong>e is simply the l<strong>in</strong>ear dispersion curve <strong>of</strong> a photon<strong>in</strong> air (as the th<strong>in</strong> slab is surrounded by air <strong>in</strong> this structure).The region above the light l<strong>in</strong>e corresponds toleaky <strong>modes</strong> <strong>in</strong> which the optical mode leaks energy <strong>in</strong>tothe surround<strong>in</strong>g air as it propagates down the waveguide.The parts <strong>of</strong> the frequency b<strong>an</strong>ds that are below the lightl<strong>in</strong>e are guided <strong>an</strong>d do not leak energy as they propagate.Only the regions <strong>of</strong> the frequency b<strong>an</strong>ds that are guidedare displayed <strong>in</strong> Fig. 6. The air b<strong>an</strong>d is guided near theb<strong>an</strong>d edge at the X po<strong>in</strong>t but eventually becomes leaky asit moves toward the po<strong>in</strong>t. The dielectric b<strong>an</strong>d, however,is guided throughout k space. In the approximateTE b<strong>an</strong>d structure <strong>of</strong> Fig. 4 the frequency b<strong>an</strong>ds are plottedthroughout k space, show<strong>in</strong>g both the guided <strong>an</strong>d theleaky regions. The approximate 2D model <strong>of</strong> Subsection3.A <strong>an</strong>d the 3D model presented here compare well for thefirst few lower-ly<strong>in</strong>g b<strong>an</strong>ds.A plot <strong>of</strong> the air b<strong>an</strong>d edge at the X po<strong>in</strong>t <strong>an</strong>d the dielectricb<strong>an</strong>d edge at the J po<strong>in</strong>t for vary<strong>in</strong>g slab thicknessesis given <strong>in</strong> Fig. 7. Also shown <strong>in</strong> this figure is themidgap frequency. A plot <strong>of</strong> the b<strong>an</strong>dgap width is given<strong>in</strong> Fig. 8. As expected the midgap frequency <strong>in</strong>creases asFig. 6. B<strong>an</strong>d structure <strong>of</strong> the TE-like <strong>modes</strong> <strong>of</strong> the 2D patternedslab waveguide surrounded by air (d 0.4a). The solid l<strong>in</strong>erepresents the light l<strong>in</strong>e. Only the guided <strong>modes</strong> are plotted.