Analysis of linear defect modes in two-dimensional photonic lattices by employing Brillouin zone spectroscopy

The formation of defect modes in optically induced photonic lattices with one-and two-dimensional defects are analyzed by combining the diffraction relationship of uniform lattices and the Brillouin zone spectra. The defect modes associated with different high symmetry points in Brillouin zone are obtained, including the positive and negative defect modes bifurcating from the X₁ point, which is ″embedded″ in the photonic bandgap. It is revealed that, for two-dimensional defects, the positive (negative) defect modes exist only in the regions corresponding to normal (anomalous) diffraction along both principal axes in k-space, whereas for one-dimensional defects, the positive (negative) defect modes can exist in the regions corresponding to normal (anomalous) diffraction along a single direction. The positive defect mode at X₁ point predicates the existence of in-band solitons under self-focusing nonlinearity. The conclusions obtained above could be helpful to understanding the formation of lattice solitons (especially in-band solitons) in photonic lattices.