Radiation losses in planar photonic crystals: two-dimensional representation of hole depth and shape by an imaginary dielectric constant

Waveguide modes in two-dimensional (2-D) photonic crystals (PhCs) deeply etched through monomode slab waveguides, e.g., AlGaAs/GaAs, $\mathrm{GaAs}/{\mathrm{AlO}}_x,$ or InP/GaInAsP, suffer from radiation losses that are strongly affected by the air hole depth and shape. The issue of three-dimensional (3-D) out-of-plane losses is addressed analytically by means of an incoherent approximation. Assuming separability both for the dielectric map and for the electric field, this approach is valid for defects such as in-plane microcavities, PhC-based waveguides, bends and couplers. Out-of-plane scattering is translated into an effective imaginary index in the air holes, so that 3-D losses can be cast in a simple 2-D calculation. The case of cylindroconical holes is treated, and the validity of this approach is experimentally confirmed by transmission measurements through simple PhC slabs.