Poynting theorem constraints on the signs of the imaginary parts of the electromagnetic constitutive parameters: comment.

Contrary to an unphysical conclusion obtained by Alavikia et al. [J. Opt. Soc. Am. A32, 522 (2015)JOAOD60740-323210.1364/JOSAA.32.000522], the imaginary parts of both the permittivity and the permeability of any linear, homogeneous, passive, spatially local, dielectric-magnetic material are always positive, independent of the signs of their real parts.

Scattering of light from metamaterial gratings with finite length.

Using an integral equation approach based on the Rayleigh hypothesis, we investigate the scattering of a plane wave at the rough surface of a metamaterial with a finite number of sinusoidal grooves. To show the adequacy of the model, we present results that are in agreement with the predictions of physical optics and that quantitatively reproduce the polarization and angular dependences predicted by the C-formalism for metamaterial gratings with an infinite number of grooves.

Properties of TM resonances on metallic slit gratings.

Electromagnetic resonances on metallic slit gratings induced by TM polarized incident light have been investigated and physically interpreted. We have developed an electromagnetic model imposing surface impedance boundary conditions on the metallic grating surface from which we derive simple formulas explaining all physical properties of these resonances. It is demonstrated that Fabry-Perot (or cavity) resonances are generated by the zeroth slit mode yielding extraordinary transmission. For very narrow slits, the resonant H-field is squeezed to the slit walls and causes enhanced power losses. The excitation of surface plasmon polaritons (SPPs), however, is generated by two mode coupling. SPPs are linked to sharp absorption peaks and dips in transmittance. It is shown that these phenomena are primarily caused by the interaction of the electromagnetic fields with the finite conducting slit walls. These findings have been confirmed by measured transmittance data of gold gratings with periods of 0.5 µm, 1 µm, and 2 µm.

Anomalous group velocity at the high energy range of a 3D photonic nanostructure.

We report on a study of electromagnetic waves propagation in thin periodically ordered photonic nanostructures in the spectral range where the light wavelength is on the order of the lattice parameter. The vector KKR method we use allows us to determine the group index from finite photonic structures including extinction providing confirmation of recently emerged results. We show that for certain frequencies the group velocity of opal slabs can either be superluminal or approach zero depending on the crystal thickness and the unavoidable presence of losses. In some cases, group velocity can be negative. Such behavior can be clearly attributed to the finite character of the three-dimensional structure and reproduces previously reported experimental observations. Calculations show that contrary to the predictions of extraordinary group velocity reductions for infinite periodic structures, the group velocity of real opals may exhibit strong fluctuations at the high energy range. Hence, a direct identification between the calculated anomalous group velocities, for an actual opal film, and the predicted propagating low dispersion modes for an ideal infinite ordered structure seems difficult to establish.

Radiation characteristics of electromagnetic eigenmodes at the corrugated interface of a left-handed material.

We study the radiation characteristics of electromagnetic surface waves at a periodically corrugated interface between a conventional and a negatively refracting (or left-handed) material. In this case, and contrary to the surface plasmon polariton in a metallic grating, surface plasmon polaritons may radiate on both sides of the rough interface along which they propagate. We find novel radiation regimes which provide an indirect demonstration of other unusual phenomena characteristic of electromagnetic wave propagation in left-handed materials, such as negative refraction or backward wave propagation.

Light generation at the anomalous dispersion high energy range of a nonlinear opal film.

We study experimentally and theoretically light propagation and generation at the high energy range of a close-packed fcc photonic crystal of polystyrene spheres coated with a nonlinear material. We observe an enhancement of the second harmonic generation of light that may be explained on the basis of amplification effects arising from propagation at anomalous group velocities. Theoretical calculations are performed to support this assumption. The vector KKR method we use allows us to determine, from the linear response of the crystal, the behavior of the group velocity in our finite photonic structures when losses introduced by absorption or scattering by defects are taken into account assuming a nonzero imaginary part for the dielectric constant. In such structures, we predict large variations of the group velocity for wavelengths on the order or smaller than the lattice constant of the structure, where an anomalous group velocity behavior is associated with the flat bands of the photonic band structure. We find that a direct relation may be established between the group velocity reduction and the enhancement of a light generation processes such as the second harmonic generation we consider. However, frequencies for which the enhancement is found, in the finite photonic crystals we use, do not necessarily coincide with the frequencies of flat high energy bands.

Rigorous formulation for electromagnetic plane-wave scattering from a general-shaped groove in a perfectly conducting plane: comment.

We show that the problem of scattering of an obliquely incident plane wave by a general-shaped groove engraved on a perfectly conducting plane, which was recently studied by Basha et al. [J. Opt. Soc. Am. A24, 1647 (2007)], was solved 11 years ago using the same formulation. This method was further extended to deal with a finite number of grooves and also with complex apertures including several nonlossy and lossy dielectrics, as well as real metals.

Control of the diffracted response of wire arrays with double periods.

The possibility of controlling the diffracted response of a periodic structure is investigated by using dual-period arrays, i.e., periodic arrays with a compound unit cell. We consider wire gratings in which each period comprises several cylinders with circular cross sections and all the cylinder axes are contained in the same plane. It is shown that this kind of structure permits one to control the diffracted response, regardless of the cylinder material and the incident polarization. Our numerical results suggest that the effect produced by wire gratings with dual-period characteristics is basically a geometric effect, and it can be present for other shapes of individual scatterers within each subarray.

Diffraction by dual-period gratings.

The dynamical characteristics of dual-period perfectly conducting gratings are explored. Gratings with several grooves (reflection) or slits (transmission) within each period are considered. A scalar approach is proposed to derive the general characteristics of the diffracted response. It was found that compound gratings can be designed to cancel as well as to intensify a given diffraction order. These preliminary estimations for finite gratings are validated by numerical examples for infinitely periodic reflection and transmission gratings with finite thickness, performed using an extension of the rigorous modal method to compound gratings, for both polarization cases.

Diffraction by fractal metallic supergratings.

The reflectance of corrugated surfaces with a fractal distribution of grooves is investigated. Triadic and polyadic Cantor fractal distributions are considered, and the reflected intensity is compared with that of the corresponding periodic structure. The self-similarity property of the response is analyzed when varying the depth of the grooves and the lacunarity parameter. The results confirm that the response is self-similar for the whole range of depths considered, and this property is also maintained for all values of the lacunarity parameter.

Physical origin of the high energy optical response of three dimensional photonic crystals.

The physical origin of the optical response observed in three-dimensional photonic crystals when the photon wavelength is equal or lower than the lattice parameter still remains unsatisfactorily explained and is the subject of an intense and interesting debate. Herein we demonstrate for the first time that all optical spectra features in this high energy region of photonic crystals arise from electromagnetic resonances within the ordered array, modified by the interplay between these resonances with the opening of diffraction channels, the presence of imperfections and finite size effects. All these four phenomena are taken into account in our theoretical approach to the problem, which allows us to provide a full description of the observed optical response based on fundamental phenomena as well as to attain fair fittings of experimental results.

Narrow gaps for transmission through metallic structured gratings with subwavelength slits.

Transmission dips in the response of metallic compound gratings formed by several wires and slits in each period have been recently reported for normal illumination. These anomalies are generated by a particular arrangement of the magnetic field phases inside the subwavelength slits, and they are characterized by a significant enhancement of the interior field. We investigate the microwave response of such systems under non-normal illumination and show that new phase modes appear in this configuration. Contrary to the effect produced by a defect in a photonic crystal, these systems exhibit forbidden channels within a permitted band. We also found that the appearance of these resonances is not highly dependent on the slits' width and thickness, even though these parameters modify the overall transmittance.

Classification of dispersion equations for homogeneous, dielectric-magnetic, uniaxial materials.

The geometric representation at a fixed frequency of the wave vector (or dispersion) surface omega(k) for lossless, homogeneous, dielectric-magnetic uniaxial materials is explored for the case when the elements of the relative permittivity and permeability tensors of the material can have any sign. Electromagnetic plane waves propagating inside the material can exhibit dispersion surfaces in the form of ellipsoids of revolution, hyperboloids of one sheet, or hyperboloids of two sheets. Furthermore, depending on the relative orientation of the optic axis, the intersections of these surfaces with fixed planes of propagation can be circles, ellipses, hyperbolas, or straight lines. The understanding obtained is used to study the reflection and refraction of electromagnetic plane waves due to a planar interface with an isotropic medium.

Interaction between non-Bragg band gaps in 1D metamaterial photonic crystals.

We consider periodic multilayers combining ordinary positive index materials and dispersive metamaterials with negative index in some frequency range. These structures can exhibit photonic band gaps which, in contrast with the usual Bragg gaps, are not based on interference mechanisms. We focus on effects produced by the interaction between non-Bragg gaps of different nature: a) the zero averaged refractive index, b) the zero permeability and c) the zero permittivity gaps. Our analysis highlights the role played by the unavoidable dispersive character of metamaterials. We show that the degree of overlap between these bands can be varied by a proper selection of the constructive parameters, a feature that introduces novel degrees of freedom for the design of photonic band gap structures. The numerical examples illustrate the evolution of the dispersion diagrams of a periodic multilayer with the filling fraction of the ordinary material constituent and show a range of filling fractions where propagation in the multilayer is forbidden for any propagation angle and polarization.

Transmission resonances of metallic compound gratings with subwavelength slits.

Transmission metallic gratings with subwavelength slits are known to produce enhanced transmitted intensity for certain resonant wavelengths. One of the mechanisms that produce these resonances is the excitation of waveguide modes inside the slits. We show that by adding slits to the period, the transmission maxima are widened and, simultaneously, this generates phase resonances that appear as sharp dips in the transmission response. These resonances are characterized by a significant enhancement of the interior field.

Comment I on "Resonant and antiresonant frequency dependence of the effective parameters of metamaterials".

Noting that the imaginary parts of the permittivity and permeability of any linear, homogeneous, passive, dielectric-magnetic material are always positive, independent of the signs of their real parts, we conclude that recent claims by Koschny et al. [Phys. Rev. E 68, 065602 (2003)] are unphysical.

Direct visualization of surface-plasmon bandgaps in the diffuse background of metallic gratings.

When a surface plasmon propagates along a microrough grating, it interacts with the periodic plus the random roughness and emits light into the diffuse background, which can present intensity maxima called diffuse light bands. We reexamine previous studies on these bands within the framework of recent studies on photonic surfaces and show that the phenomenon of diffuse light provides an experimental technique for directly imaging the dispersion relation of surface plasmons, including the gap that, under appropriate circumstances, opens in the reciprocal grating space.

Plane-wave diffraction at the periodically corrugated boundary of vacuum and a negative-phase-velocity material.

Considering the diffraction of a plane wave by a periodically corrugated half-space, we show that the transformation of the refracting medium from positive (negative) phase velocity to negative (positive) phase velocity type has an influence on the diffraction efficiencies. This effect increases with increasing corrugation depth, owing to the presence of evanescent waves in the troughs of the corrugated interface.

Polarization conversion from highly conducting, asymmetric trapezoidal gratings.

The differences in the curves of the zeroth-order cross-polarization reflection coeffients (p --> s and s --> p) versus angle of incidence have remarkable potential for application in scatterometry because, if the differences are larger than the measurement error, they could contribute to a reliable nondestructive technique for detecting asymmetries in grating profiles. The cross-polarization efficiencies of highly conducting metallic gratings with asymmetric trapezoidal profiles are investigated theoretically by means of a rigorous electromagnetic code. The results show that the differences between p --> s and s --> p conversion tend to be undetectable for highly conducting materials, a fact that limits, in principle, the application of this potential detection technique.