| Thanks to the intriguing electromagnetic properties of metamaterials, introducing metamaterials into photonic crystals (PC) will surely bring about lots of new phenomena and new rules. In this thesis, we study the properties of both one-dimensional and quasi-one-dimensional PCs in sub-wavelength limit.Based on the works of former researchers, we analyze the band structure properties of two different one-dimensional PCs. Two gaps, the zero-average-index gap in one-dimensional PC made of alternating positive- and negative-index materials and the zero-effective-phase gap in one-dimensional PC made of two kinds of single-negative materials, have similar properties and both of them can be considered as "spatial-averaged-single-negative" (SASN) gaps. SASN gaps are invariant to the geometrical scaling and insensitive to the incident angle and disorder.The quasi-one-dimensional PC we study is a comb-like structure composed of a negative-index backbone wave-guide along which finite size positive-index side branches are grafted equidistantly. This study is conducted within the frame of the Interface Response Theory. Through calculation, we obtain analytically transmission coefficient, band structure, field etc. The result shows the existence of a special band gap, which is very similar with the zero-average-index gap. This gap possesses general properties of SASN gap, e.g., it is invariant to the change of scaling. Compared to the zero-average-index gap, this gap is obviously deeper and broader, given the same parameters. In addition, the behavior of the band-edge of this gap is also interesting. One gap-edge coincides with the corresponding gap-edge of the zero-average-index gap, and the other is only subject to the changing of the permeability of the back bone.Furthermore, we also give simulation and experimental results for the part of work of quasi-one-dimensional PC. The results show good coincidence with theoretical prediction.
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