We utilize the Bloch-mode analysis of periodic composite structures to introduce an approach for retrieving effective parameters of homogenized metamaterials. In the case of single-mode propagation we can restore a complex effective refractive index with a high accuracy. By further employing surface or volume averaging of the electromagnetic fields of the dominating (fundamental) Bloch modes we are able to determine the Bloch and wave impedances, leading to wave and material effective parameters, respectively. The approach is demonstrated on several examples. We focus our discussion on the nature of microfields returned by Maxwell's solvers, showing that ignoring of difference between magnetic strength and induction lead to incorrect determination of the Poynting vector.

We introduce an approach for retrieving effective parameters of metamaterials based on the Bloch-mode analysis of quasiperiodic composite structures. We demonstrate that, in the case of single-mode propagation, a complex effective refractive index can be assigned to the structure, being restored by our method with a high accuracy. We employ both surface and volume averaging of the electromagnetic fields of the dominating (fundamental) Bloch modes to determine the Bloch and wave impedances, respectively. We discuss how this method works for several characteristic examples, and demonstrate that this approach can be useful for retrieval of both material and wave effective parameters of a broad range of metamaterials.

We experimentally study the fields close to an interface between two photonic crystal waveguides that have different dispersion properties. After the transition from a waveguide in which the group velocity of light is v(g) ~ c/10 to a waveguide in which it is v(g) ~ c/100, we observe a gradual increase in the field intensity and the lateral spreading of the mode. We attribute this evolution to the existence of a weakly evanescent mode that exponentially decays away from the interface. We compare this to the situation where the transition between the waveguides only leads to a minor change in group velocity and show that, in that case, the evolution is absent. Furthermore, we apply novel numerical mode extraction techniques to confirm experimental results.

The main bottleneck in the restoration of electromagnetic effective parameters is connected to the impedance retrieving. The S-parameters method gives the input (Bloch) impedance, which, being then used for permittivity and permeability determination, causes some fundamental physics principles violation, like antiresonance behaviour with Im(e) < 0, Im(μ) < 0. We employ the Bloch mode analysis of periodic metamaterials to extract the dominating (fundamental) Bloch mode. Then it is possible to determine the Bloch and wave impedances by the surface and volume averaging of the electromagnetic field of the Bloch mode, respectively. Case studies prove that our approach can determine material and wave effective parameters of lossy and lossless metamaterials. In some examples when the passivity is violated we made further analysis and showed that this is due to the failure of concept of impedance retrieving through the volume averaging.

We demonstrate the fabrication by direct laser writing and the operation of a directional coupler containing Bragg gratings in each waveguide. We achieve high-precision control over the longitudinal shift between the gratings, which feature first-order Bragg resonance at telecommunication wavelengths. We observe fundamental differences between light transmission characteristics in couplers with unshifted and shifted gratings in agreement with theoretical predictions.

We report the experimental observation of tunneling of slow and fast electromagnetic modes in coupled periodic waveguides shifted longitudinally by half of modulation period. According to the symmetry analysis, such a coupler supports two electromagnetic modes with exactly matched slow or fast group velocities but different phase velocities for frequencies close to the edge of the photonic band. We confirm the predicted properties of the modes by directly extracting their dispersion and group velocities from the near-field measurements using specialized Bloch-wave spectral analysis method.

We propose the method of effective parameters retrieval based on the Bloch mode analysis of periodic metamaterials. We perform the surface and volume averaging of the electromagnetic field of the dominating (fundamental) Bloch mode to determine the Bloch and wave impedances, respectively. We show that our method is able to retrieve both material and wave EPs for a wide range of materials, which can be lossy or lossless, dispersive, possess negative permittivity, permeability and refractive index values. It is simple and unambiguous, free of the "branch" problem, which is an issue for the reflection/transmission based method and has no limitations on a metamaterial slab thickness. The method does not require averaging different fields' components at various surfaces or contours. The retrieval of both wave and material EPs is performed within a single computational cycle, after exporting fields on the unit cells facets or in its volumes directly from Maxwell's equations solver.

Coupled modes of two nanobeams brought together are shown to have pronounced frequency dependence on a spatial displacement of the resonators, while the quality factor of the coupled modes preserves the same order of magnitude as that of a single nanobeam. Both a longitudinal shift and transversal displacement of nanobeams with respect to each other are under investigation. We show that the mode degeneracy occurs at the longitudinal shift of about a half a modified lattice constant in the center of the defect region. To perform eigenmode analysis of the coupled system we used a 3D finite-difference frequency-domain method and a 48 Gb station.

We demonstrate, theoretically and experimentally, that the modes of coupled cavities created in periodic waveguides can depend critically on the lateral shift between the cavities. In the absence of such shift, the modes feature symmetric or antisymmetric profiles, and their frequency splitting generally increases as the cavities are brought closer. We show that the longitudinal shift enables flexible control over the fundamental modes, which frequency detuning can be reduced down to zero. Our coupled-mode theory analysis reveals an intrinsic link between the mode tuning and the transformation of slow-light dispersion at the photonic band-edge. We illustrate our approach through direct numerical modelling of cavities created in arrays of dielectric rods, nanobeam structures, and two-dimensional photonic-crystal waveguides. We also present experimental results for coupled rod cavities confirming our predictions.

In this work, we demonstrate that the compound mode properties of coupled photonic-crystal cavities can depend critically on the interplay of distance between cavities and their longitudinal shifts. Thus the robust control over the cavity modes can be imposed. The simple coupled-mode theory employed for such systems predicts a peculiar behavior of band dispersion in the slow light regime at the photonic band-edge. In particular, it reveals an interesting effect that the frequency detuning of the fundamental supermodes in the coupled cavities can be reduced down to zero. We anticipate that this property will be generic for side-coupled cavity systems irrespectively of the individual cavity design, e. g. point-defect cavities in a photonic crystal or linear cavities in one-dimensional arrays of elements (rods or holes). We report here about the finite-difference frequency-domain method (FDFD) developed by us to analyze nanocavities with a very high Q-factor. The method is utilized to confirm by simulations the coupled-mode theory predictions. As an example we choose coupled cavities in one-dimensional periodic arrays of holes in dielectric nanowires known also as nanobeams.

We report the measurement of slowly propagating and weakly evanescent modes close to dispersion inflection points of photonic crystal waveguides. Evanescent modes play a key role in coupling light to slow modes of photonic crystals

We investigate the modes of double heterostructure cavities where the underlying photonic crystal waveguide has been dispersion engineered to have two band-edges inside the Brillouin zone. By deriving and using a perturbative method, we show that these structures possess two modes. For unapodized cavities, the relative detuning of the two modes can be controlled by changing the cavity length, and for particular lengths, a resonant-like effect makes the modes degenerate. For apodized cavities no such resonances exist and the modes are always non-degenerate.

We demonstrate that the modes of coupled cavities created in periodic waveguides can depend critically on the longitudinal shift between the cavities. In the absence of such shift, the modes feature symmetric or antisymmetric profiles, and their frequency splitting generally increases as the cavities are brought closer. We show that the longitudinal shift enables flexible control over the fundamental modes, whose frequency detuning can be reduced down to zero. Our coupled-mode theory analysis reveals an intrinsic link between the mode tuning and the transformation of slow-light dispersion at the photonic band-edge. We illustrate our approach through numerical modeling of cavities created in arrays of dielectric rods, and confirm our predictions with experimental observations.

We propose a method for retrieval of effective parameters of metamaterials based on the Bloch-mode analysis of periodic composite structures. We employ both surface and volume averaging of the electromagnetic fields of the dominating (fundamental) Bloch mode to determine the Bloch and wave impedances, respectively. We discuss how this method works for several characteristic examples and demonstrate that this approach can be useful to unambiguously determine both the material and wave effective parameters of lossy and lossless metamaterials with local response.

We reveal that the reduction of the group velocity of light in periodic waveguides is generically associated with the presence of vortex energy flows. We show that the energy flows are gradually frozen for slow-light at the Brillouin zone edge, whereas vortices persist for slow-light states having non-vanishing phase velocity inside the Brillouin zone. We also demonstrate that presence of vortices can be linked to the absence of slow-light at the zone edge, and present calculations illustrating these general results.

We demonstrate that the modes of coupled cavities created in periodic waveguides can depend critically on the lateral shift between the cavities. In the absence of such shift, the modes feature symmetric or antisymmetric profiles, and their frequency splitting generally increases as the cavities are brought closer. We show that the longitudinal shift enables flexible control over the fundamental modes, which frequency detuning can be reduced down to zero. Our coupled‐mode theory analysis reveals an intrinsic link between the mode tuning and the transformation of slow‐light dispersion at the photonic band‐edge. We illustrate our approach through direct numerical modelling of cavities created in arrays of dielectric rods, nanobeam structures, and two‐dimensional photonic‐crystal waveguides.

We demonstrate that the spatial profiles of both propagating and evanescent Bloch modes in a periodic structure can be extracted from a single measurement of an electric field at the specified optical wavelength. We develop a systematic extraction procedure by extending the concepts of high-resolution spectral methods previously developed for temporal data series to take into account the symmetry properties of Bloch modes simultaneously at all spatial locations. We illustrate the application of our method to a photonic crystal waveguide interface and confirm its robustness in the presence of noise.