According to a recent proposal [S. Takayama et al., Appl. Phys. Lett. 87,061107 (2005)], the triangular lattice of triangular air holes may allow toachieve a complete photonic band gap in two-dimensional photonic crystal slabs.In this work we present a systematic theoretical study of this photonic latticein a high-index membrane, and a comparison with the conventional triangularlattice of circular holes, by means of the guided-mode expansion method whosedetailed formulation is described here. Photonic mode dispersion below andabove the light line, gap maps, and intrinsic diffraction losses ofquasi-guided modes are calculated for the periodic lattice as well as for line-and point-defects defined therein. The main results are summarized as follows:(i) the triangular lattice of triangular holes does indeed have a completephotonic band gap for the fundamental guided mode, but the useful region isgenerally limited by the presence of second-order waveguide modes; (ii) thelattice may support the usual photonic band gap for even modes (quasi-TEpolarization) and several band gaps for odd modes (quasi-TM polarization),which could be tuned in order to achieve doubly-resonant frequency conversionbetween an even mode at the fundamental frequency and an odd mode at thesecond-harmonic frequency; (iii) diffraction losses of quasi-guided modes inthe triangular lattices with circular and triangular holes, and in line-defectwaveguides or point-defect cavities based on these geometries, are comparable.The results point to the interest of the triangular lattice of triangular holesfor nonlinear optics, and show the usefulness of the guided-mode expansionmethod for calculating photonic band dispersion and diffraction losses,especially for higher-lying photonic modes.
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