The combinatorial frequency generation by the periodic stacks of binary layers of anisotropic nonlinear dielectrics is examined. The products of nonlinear scattering are characterised in terms of the three-wave mixing processes. It is shown that the intensity of the scattered waves of combinatorial frequencies is strongly influenced by the constitutive and geometrical parameters of the anisotropic layers, and the frequency ratio and angles of incidence of pump waves. The enhanced efficiency of the frequency conversion at Wolf-Bragg resonances has been demonstrated for the lossless and lossy-layered structures.

A new generation of artificial electromagnetic materials has opened up new opportunities for engineering the media with the specified properties. The latest advancements in this field have prompted a surge of research in the new phenomenology, which could extend a range of functional capabilities and enable the development of innovative devices in the millimeter, terahertz (THz), and optical ranges.

Frequency conversion in dielectrics with nonlinearities of the second and third order has been investigated in optics, particularly, in the context of the second (SHG) and third (THG) harmonic generation. The recent studies have indicated that nonlinear photonic crystals (PhCs) and metamaterials (MMs) have significant potential for enhancement of the nonlinear activity associated with the mechanisms of field confinement, dispersion management and resonant intensification of the interacting waves. For example, it has been demonstrated in [

Combinatorial frequency generation by mixing pump waves of two different frequencies provides alternative means for frequency conversion. The efficiency of mixing process can be dramatically increased in the layered structures, for example, at the higher order Wolf-Bragg resonances of the combinatorial frequencies generated in the anisotropic nonlinear dielectric slabs. As shown in [

The aim of this paper is to explore the mechanisms of the combinatorial frequency generation in the PhC composed of a periodic stack of binary anisotropic nonlinear dielectric layers illuminated by two-tone pump waves that allows us to combine the effects of the resonance mixing with the dispersion control provided by the structure periodicity. Here the properties of the combinatorial frequencies generated by the nonlinear anisotropic dielectric PhC illuminated by plane waves of two tones are investigated. A generic approach, based on the transfer matrix method (TMM) [

Wave propagation and scattering in linear stratified media are usually modelled by TMM, which sequentially relates the fields at the layer interfaces, see, for example, [

In order to examine the three-wave mixing process in the 1D

Geometry of the problem.

Each layer has 6 mm class of anisotropy and is described by the linear dielectric permittivity tensor

In the approximation of weak nonlinearity, the scattering characteristics of the TM waves can be obtained separately at each frequency by the harmonic balance method. Thus at the combinatorial frequency

The full solution of inhomogeneous equation (

To satisfy the boundary conditions at the interfaces of the nonlinear layers at the combinatorial frequency

Finally, by combining (

Thus the modified TMM approach presented in this section gives the closed-form expressions for the nonlinear scattering coefficients of the finite PhC composed of the binary nonlinear layers. The obtained analytical formulations not only provide a qualitative insight in the formation of the nonlinear response and the properties of the scattered fields but also enable fast quantitative analysis of the specific PhC configurations.

The results of numerical simulations based upon the analytical solutions obtained here are presented in the next section to illustrate the effects of structure and materials parameters on the properties TM waves of combinatorial frequencies generated by nonlinear PhC in the three-wave mixing process.

The analytical solutions for the coefficients

To illustrate the features of the frequency mixing in the 1D nonlinear anisotropic PhCs, the characteristics of the combinatorial frequency waves are discussed here with the examples of periodic stacks of binary anisotropic dielectric layers of CdS and ZnO described by the tensors

CdS:

ZnO:

The constituent layer thicknesses are

PhCs are known to be instrumental in enhancing the SHG and THG efficiency by choosing the pump wave frequency close to the PhC band edge. Therefore, it was interesting to explore whether similar facility could be exploited for the combinatorial frequencies generated in the three-wave mixing process. The spectral bands of a periodic stack of binary linear anisotropic dielectric layers have been inferred first from the reflectance

Reflectance of plane TM wave incident at

The field intensities

The field intensity at frequency

Figure

As indicated in the preceding section, the number ^{2} has maxima at ^{2}. However, at the higher frequency

Intensities ^{2}: black dash-dot line) directions of the

Harmonic generation in 1D PhCs are usually analysed at normal incidence of pump wave on the stacked layers. In the case of combinatorial frequency generation by a pair of pump waves, incident at different angles, an additional degree of freedom exists in realising the phase synchronism and controlling the whole frequency mixing process. To gain insight in the effect of the incidence angle on the combinatorial frequency field intensities,

Examination of

The field intensity at frequency ^{13} s^{−1} and

Both the reflectance/transmittance of pump waves and the phase synchronism in the mixing process are essentially dependent on the permittivities and anisotropy of the constituent binary layers. Therefore the effect of the constituent layer parameters has been assessed first to discriminate contributions of the aforementioned mechanisms to the combinatorial frequency generation. In order to evaluate the effect of the layer anisotropy, the intensities

The field intensity at frequency

The stack overall thickness may have profound influence on the frequency mixing efficiency. This can be the result of the increased number of unit cells in the stack as illustrated in Figure

Intensities

The analytical study of nonlinear scattering by an isolated anisotropic dielectric slab in [

Intensities

The combinatorial frequency generation in the periodic stacks of binary layers discussed so far has been based upon the analysis of the lossless structures. To estimate the effect of dissipation on

Reflectance of TM wave incident at

The field intensity at frequency

The effect of loss on the intensity

Intensities

The properties and mechanisms of the combinatorial frequency generation by periodic stacks of binary nonlinear anisotropic dielectric layers have been analysed. The closed-form solutions for the nonlinear scattering coefficients have been obtained in the approximation of the three-wave mixing process in the presence of weak polarisation nonlinearity. The effects of the structure parameters and the incident pump wave characteristics on the efficiency of the combinatorial frequency generation have been investigated in detail. The performed parametric study has shown that in contrast to SHG and THG in the PhCs, the spectral band edges of the binary layer stacks do not improve the combinatorial frequency generation efficiency for the refracted waves. Alternatively, it is shown that the frequency conversion efficiency can be significantly enhanced at Wolf-Bragg resonances occurring at the appropriate combinations of the pump wave frequencies, incidence angles, and the layers’ constitutive parameters. The effects of the individual parameters on the frequency mixing efficiency have been discussed in detail for the lossless and lossy constitutive layers in the periodic stacks. It has been demonstrated that the combinatorial frequency generation efficiency can be dramatically increased at the higher order Wolf-Bragg resonances in the stacks with thick constitutive layers. The performed analysis provides insight in the main features of the combinatorial frequency generation by the periodic stacks of binary nonlinear anisotropic dielectric layers.

This work has been performed in the framework of the Project PEARL supported by the FP7 Marie Curie IIF Grant 255110.