Improved differential-transfer-matrix method for inhomogeneous one-dimensional photonic crystals

A band structure of one-dimensional periodic media having an arbitrary inhomogeneous refractive index profile is extracted based on the modification committed on the differential-transfer-matrix method (DTMM). Furthermore, the most recently modified differential transfer matrix is improved by reshaping the formulation in terms of which the electromagnetic fields are expanded and closed form formulas, providing the allowed values of Bloch wavenumbers, which were not available before. Although the frequency gaps in previously published results that we derived by using the conventional DTMM were not in agreement with the well-known Bragg law at the edge of each Brillouin zone, the new results obtained by the proposed method are now matched with the Bragg condition. The final results are also justified by either employing conventional transfer-matrix method or comparing it with exact analytical solutions, wherever such exact solutions were available.