Anomalous refractive properties of photonic crystals at the band edges

S. Enoch, B. Gralak, G. Tayeb, and D. Maystre

Laboratoire d'Optique Électromagnétique, ESA 6079, Faculté des Sciences et Techniques, Centre de Saint-Jérôme, 13397 Marseille Cedex 20, France

Some experimental studies about ultra refraction, negative refraction, highly dispersive properties of structures based on photonic crystals have been reported recently [1,2]. Ultrarefractive properties of photonic crystals have been suggested previously to realize components such as lenses of ultra-short focal lengths or laser accelerator [3]. However, the electromagnetic basis which support these phenomena have not been clarified. To our knowledge, the most complete work concerns corrugated waveguides [4].

?Since we are interested with finite size photonic crystals illuminated by an incident field the problem is, a priori, different from the study of Bloch modes propagating in an infinite crystal. Bloch theory only consider propagating modes. In a finite crystal evanescent modes exist as well. It is well known in grating theory that the evanescent modes play an important role and cannot be neglected in a quantitative analysis.

?Our approach establish the link between the dispersion relation of Bloch modes for an infinite crystal, which describes the intrinsic properties of the photonic crystal in the absence of any incident field, and the diffraction problem of a grating (finite photonic crystal) illuminated by an incident field. We develop the relationship between the translation operator of the first problem and the transfer matrix of the second one and the links between the incident field, the eigensolutions of this matrix, and the Bloch modes of the associated infinite structure.

?It enables us to find suitable parameters to obtain ultrarefractive or negative refraction properties. For example, we determine the parameters that enable one to design an artificial material whose effective index is less than unity, and if possible close to zero, regardless to the direction of propagation.

Since the index contrast with other materials is very large, new optical elements can be imagined, for instance a microlens as shown on the figure. Rigorous computations add a quantitative aspect, and demonstrate the relevance of our approach.

Note that even if the crystal behaves as a homogeneous material, the physical situation is very different from that generally studied in homogenization studies where one considers quasi-static limits.

 

 

Of course, inside a bandgap the evanescent modes govern the behavior of the crystal. In this case, we show how the field decay is related to the eigenvalues of the elementary transfer matrix.