  According to a recent proposal [S. Takayama et al., Appl. Phys. Lett. 87,
061107 (2005)], the triangular lattice of triangular air holes may allow to
achieve a complete photonic band gap in two-dimensional photonic crystal slabs.
In this work we present a systematic theoretical study of this photonic lattice
in a high-index membrane, and a comparison with the conventional triangular
lattice of circular holes, by means of the guided-mode expansion method whose
detailed formulation is described here. Photonic mode dispersion below and
above the light line, gap maps, and intrinsic diffraction losses of
quasi-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 complete
photonic band gap for the fundamental guided mode, but the useful region is
generally limited by the presence of second-order waveguide modes; (ii) the
lattice may support the usual photonic band gap for even modes (quasi-TE
polarization) and several band gaps for odd modes (quasi-TM polarization),
which could be tuned in order to achieve doubly-resonant frequency conversion
between an even mode at the fundamental frequency and an odd mode at the
second-harmonic frequency; (iii) diffraction losses of quasi-guided modes in
the triangular lattices with circular and triangular holes, and in line-defect
waveguides or point-defect cavities based on these geometries, are comparable.
The results point to the interest of the triangular lattice of triangular holes
for nonlinear optics, and show the usefulness of the guided-mode expansion
method for calculating photonic band dispersion and diffraction losses,
especially for higher-lying photonic modes.
