Effective refractive index and group velocity determination of three-dimensional photonic crystals by means of white light interferometry

Abstract

White-light interferometry measurements over a wide spectral range in the optical region have been performed on three-dimensional ͑3D͒ opal-based photonic crystals that have permitted extracting the optical phase delay introduced by samples with an increasing number of layers. The absolute phase that corresponds to the wave vector inside the samples has been obtained by a proper normalization procedure. From the absolute phase and the transmittance, we have determined the complex effective refractive index of the 3D photonic crystals, whose real part shows normal dispersion outside the pseudogap and anomalous ͑negative͒ dispersion across the pseudogap. By a numerical derivative of the measured phase, the group velocity is directly obtained, which displays slowing down at the band edge and superluminal behavior inside the photonic gap. The evolution of the measured quantities with sample thickness and their convergence toward the infinite crystal behavior are successfully compared to theoretical calculations of the optical properties for the finite system as well as of the energy bands. The role of structural disorder on the measured quantities is also discussed.

Towards a full understanding of the growth dynamics and optical response of self-assembled photonic colloidal crystal films

“…These features have been repeatedly observed for a large number of samples, hence confirming the relation between film growth speed, crystal thickness and density of intrinsic defects in colloidal crystal films. Other groups have recently reported thorough analyses of the variations of crystal thickness along the growth direction 21,25 or of the relation between crystal thickness and defect density or domain size, 23,26 such observations being in fair agreement with those reported in ref. 18 and 19. A complete picture of the film growth process still lacks a description of the ordering of microspheres in a facecentred-cubic lattice within a receding meniscus.…”

“…2(a) (dashed line), it is necessary to introduce certain extinction in the structure through an imaginary part of the refractive index of the spheres Im(n). 22,23 In fact, it can be rigorously demonstrated that losses in the form of diffusely scattered light, introduced, for instance, by imperfections such as particle size dispersion or vacancies, can be approximated by such imaginary part of the refractive index of the spheres for weakly disordered systems. 24 Thus, in our case Im(n) is an extinction coefficient directly related to the density of intrinsic defects in the structure.…”

Relation between growth dynamics and the spatial distribution of intrinsic defects in self-assembled colloidal crystal films

“…It should be noted that the amount of disorder present in colloidal crystal films has previously been approximated by assuming an optical extinction factor that accounts for imperfections in the lattice. 18,19 The most remarkable result of these two precedents was that the amount of disorder is smaller for thicker sample. Our study indicates that the thicker regions of a particular film are indeed expected to be less defective since they grow more slowly.…”

Effect of extinction on the high-energy optical response of photonic crystals

“…As expected, in the low-energy region ͑a / ഛ0.9͒, the effect of such losses is weak, although it has to be taken into account if a precise description of the measurements is needed. 16,17 For a / ഛ0.9, the band dispersion is almost identical to that of the crystal without extinction: the wave vector may be considered real and linear with frequency, Im͑k͒ becoming significantly larger within the pseudogap. However, well above it, the band structure becomes completely different from that of a crystal without losses ͑see Fig.…”