Transfer-matrix methods adopting a plane-wave basis have been routinely used to calculate the scattering of electromagnetic waves by general multilayer gratings and photonic crystal slabs. In this paper we show that this technique, when combined with Bloch's theorem, can be extended to solve the photonic band structure for 2D and 3D photonic crystal structures. Three different eigensolution schemes to solve the traditional band diagrams along high-symmetry lines in the first Brillouin zone of the crystal are discussed. Optimal rules for the Fourier expansion over the dielectric function and electromagnetic fields with discontinuities occurring at the boundary of different material domains have been employed to accelerate the convergence of numerical computation. Application of this method to an important class of 3D layer-by-layer photonic crystals reveals the superior convergency of this different approach over the conventional plane-wave expansion method.

We systematically investigate lensing of electromagnetic waves by a negative refractive-index material slab constructed from a two-dimensional photonic crystal with properly designed equifrequency-surface configuration ͓Luo et al., Phys. Rev. B 65, 201104 ͑2002͔͒. We find that a point source placed in the vicinity of the slab can form a good-quality image in the opposite side of the slab. However, the image is strongly confined in the near-field region of the slab and gradually degrades and disappears when moved beyond the near-field domain. In addition, the image-slab distance has little dependence on the source-slab distance and the slab thickness. On the other hand, the image can also form by a slab with a positive effective refractive index. We have analyzed the equifrequency-surface contour configuration of this photonic crystal and found that the overall imaging properties of this photonic crystal slab are dominantly governed by the self-collimation effect and complex near-field wave scattering effect, rather than by the all-angle negative-refraction effect.

Quantum electrodynamics of atom spontaneous emission from a three-dimensional photonic crystal is studied in a full vectorial framework. The electromagnetic fields are quantized via solving the eigenproblem of photonic crystals with use of a plane-wave expansion method. It is found that the photon density of states and local density of states (LDOS) with a full band gap vary slowly near the edge of band gap, in significant contrast to the singular character predicted by the previous isotropic model. Therefore, the spontaneous emission can be solved by conventional Weisskopf-Wigner approximate theory, which yields a pure exponentially decaying behavior with a rate proportional to the LDOS. PACS numbers: 42.70.Qs, 42.50.Dv In recent years, there has been vast interest in fabrication of photonic crystals with a full band gap [1][2][3]. In the band gap, there is no electromagnetic (EM) mode that can propagate in the crystal, which results in some peculiar physical properties, such as inhibition of atom spontaneous emission and localization of light [2,3]. Recently, much progress has been made toward fabricating threedimensional (3D) photonic crystals with an absolute band gap in visible and infrared regimes by means of microlithography and inverse-opal technique [4][5][6].Another important subject that has attracted much attention concerns the understanding of quantum electrodynamics (QED) behaviors of atoms and molecules in photonic crystals. In the theoretical side, an isotropic model developed by John and co-workers [7] was almost exclusively adopted to treat the QED problems by this group and other groups. In that model, the dispersion relation of photon in one-dimensional periodic multilayers is extended directly to the 3D case. This results in a singular behavior of photon density of states (DOS) as proportional to ͑v 2 v c ͒ 21͞2 , where v c is the edge frequency of the band gap. As the photon DOS plays an important role in determining the QED behavior of atoms when their emission frequency lies near the gap edge, some peculiar properties were predicted by this model, such as the anomalous Lamb shift [7], oscillatory behavior of spontaneous emission [8], and enhanced quantum interference effects [9]. In addition, the vectorial nature of EM fields is completely omitted. Although an improved model employing more realistic photon dispersion as v v c 1 A͑k 2 k 0 ͒ 2 was discussed in these works [7,8,10], the extension of the dispersion formula near the gap edge to the whole momentum space and the neglect of vectorial nature of EM fields remain.On the other hand, the spontaneous emission and fluorescence of molecules in photonic crystals have been investigated by several groups [11][12][13]. However, there still is a lack of clear quantitative understanding of such QED characters observed in these experiments, as other effects such as complex chemical and electronic interactions may account for a major fraction of the change in the measured QED characters.All these QED subjects require more accurate solution of...