Conditions of near-zero dispersion of defect modes in one-dimensional photonic crystals containing negative-index materials

Abstract

It is shown that the defect modes inside the zero-n¯ gap can be sensitive to the incident angle, the same as those inside the Bragg gap. An expression of the dispersion of the defect modes (i.e., the incident angle dependence of defect modes) inside the Bragg gap or the zero-n¯ gap of one-dimensional photonic crystals containing negative-index materials is derived. It is found from the expression that the dispersion of the defect mode approaches zero when the phase change on reflection from periodic stacks of the photonic crystals can cancel out the change of the optical phase thickness of the defect layer. However, if they cannot cancel each other out, the dispersion will be of either the positive or the negative type. Practical designs for omnidirectional defect modes are given according to the conditions of the near-zero dispersion. In addition, a narrow frequency and sharp angular defect mode are designed by combining two photonic crystals with defect modes of the positive and the negative types of dispersion, respectively.