Photonic crystal heterostructures: Waveguiding phenomena and methods of solution in an envelope function picture

We present an envelope approximation formalism to study three-dimensional photonic crystal heterostructures which only requires knowledge of the bulk crystal band structure and heterostructure design. Applying this method to photonic crystal waveguides, we predict within 1% accuracy the frequencies of guided modes and obtain the correct waveguided mode shapes. We show that guided modes are allowed for wave vectors where the curvature of a band in a direction perpendicular to the plane of the waveguide has the same sign as the refractive index contrast between the core and the cladding. We show that elementary waveguide theory can be employed to compute mode shapes and dispersion relations.