Influence of the short-range lateral disorder in the meta-atoms positioning on the effective parameters of the metamaterials is investigated theoretically using the multipole approach. Random variation of the near field quasi-static interaction between metaatoms in form of double wires is shown to be the reason for the effective permittivity and permeability changes. The obtained analytical results are compared with the known experimental ones.

Metamaterials are artificial media that allow tailoring the macroscopic properties of the light propagation by a careful choice of a design for the microscopic unit cell (called the meta-atom). By controlling the geometrical shape and the material dispersion of the meta-atom, novel effects such as negative refraction [

The interaction between the small particles, both dielectric and metallic, and propagation of an optical excitation in a regular chain of such particles have been intensively investigated [

Natural expansion of the developed models of the electromagnetic excitation transport on a chain of particles with randomly varying parameters revealed several interesting peculiarities. The problem of wave propagation through disordered systems attracts great attention in both quantum and classical physics [

An influence of various types of disorder on the effective properties of the metamaterials was intensively investigated as well. Light propagation and Anderson localization in superlattices were theoretically considered in [

In the presented work, attention is primarily devoted to the extension of the multipole approach to describe in-plane disorder in metamaterials, which means the randomness in positioning of the meta-atoms in the plane of the substrate; see Figure

(a) Regular and (b) laterally (along

The qualitative explanation of the influence of the spatial disorder on the effective parameters is in following. The positional disorder creates different conditions for the charged dynamics in the meta-atoms due to the interaction between them [

This qualitative hypothesis requires further development of the existing theoretical multipole model; in particular, the interaction between the meta-atoms [

Geometry for the probability function elaboration, the spheres show meta-atoms. The first row shows a regular arrangement of the meta-atoms, where each meta-atom occupies the center of a slot of length equal to the mean period. The second row depicts an arrangement of the meta-atoms exhibiting random uncorrelated positional disorder (denoted by

The quest to obtain such

The main difference of the approach here in comparison with the previous ones in the use of the multipole model is that the charge dynamics in meta-atoms is primarily considered and calculated taking into account the interaction between meta-atoms, which is expressed as a function of distance between them. Finally, averaging over all possible realization of the inter-metaatom distance gives the expression for the effective parameters. This paper is primarily devoted to the elaboration of the model and to the effective parameters calculation; further applications of the presented approach (disorder in propagation direction, transition “coherent-incoherent” states, influence of the Anderson localization on the effective parameters, etc.) will be done elsewhere. Interaction between meta-atoms is taken into account using a simplest way of dipole-dipole near field interaction in quasi-static limit; extrapolation of the model on the dynamic case is left for the future work. The interaction between quadrupoles is treated the same way, which makes the presented approach suitable for consideration of the magnetic properties of the metamaterials. In spite of the excessive simplification of the interaction, the presented model treats the effective parameters (especially magnetic response) in much more correct way than it was done before by just introduction of permeability and/or magnetic susceptibility, and is believed to provide a suitable platform for analytical or semianalytical treatment of the problems, appearing in the case of disordered metamaterials.

The problem of the positional disorder modeling can be tackled in several ways. The most general formulation of the problem requires a Markovian treatment. As an illustration, the one-dimensional equivalent of the problem is considered here. Supposing that the meta-atoms are introduced one by one on a line of given length, the probability that a particle will take up a certain position on the line, and hence the probability of a particular inter separation distance, depends not only on the last particle, but also on the history and existing configuration. This is the essence of the Markovian approach. Standard techniques exist for tackling such problems, formulating a rigorous treatment. Nevertheless, due to its complex nature, an alternative simpler math treatment is developed in the present paper.

The math treatment accepted in the presented work is stipulated by the real technological chains for the producing of the nanostructure. When performing control experiments, masks for random metamaterials are manufactured by e-beam lithography methods. The writing algorithms are modified so that instead of a periodic grid a randomized one is generated by the scanner. The extent of randomness can be specifically controlled, and statistically relevant parameters such as the mean period and the variance can be assigned to each mask. Translating the above approach to mathematics, one assumes that the meta-atoms are initially placed in well-defined slots of equal length (see Figure

Define the characteristic functions for

The mathematical procedure has to ensure that the perturbation does not become so large that the meta-atoms overlap each other. In the analysis, the particles are assumed to be placed in average in the center of the slots of a length equal to the mean spacing period. The particles can randomly move within their own slot, and the extent of the displacement from the center of the slot is given by

In this section, using the above mentioned principles, the effect of disorder in a chain of periodically placed dipoles is investigated. The geometry is given in Figure

Geometry of propagation for randomly arranged dipole ensemble.

The system can be mathematically modeled as follows. Considering the coupling dynamics between two equal adjacent oscillators, one can write the equation describing their dynamics as

The response of the system can thus be obtained by monitoring the susceptibility of the medium. The polarization of the system can be written as

The extension of the above model to metamaterials (i.e., taking into account the magnetic response) firstly requires that the interaction between the adjacent meta-atoms is taken into consideration. The system is taken to be similar as the one shown in Figure

Geometry of propagation for randomly arranged quadrupole ensemble.

Relationship between the positional disorder function (a) and the interseparation probability distribution function (b). As the positional PDF (b) deviates from the Gaussian form for higher values of disorder (due to restrained excursion), the interseparation PDF approaches a triangular form.

Assuming that a plane electromagnetic wave now propagates through the ensemble along the

The first step is to find the solution of the above set of equations. They can be transferred to the Fourier domain by using ansatz

The effect of disorder can then be taken into account by carrying out the extra averaging integration (

where

With the effective susceptibility as defined above, one may now consider a planar metamaterial, which is formed by the identical rows of the randomly positioned meta-atoms. The effect of randomness is taken into account by the averaging procedure, and hence the dispersion relation and the effective material parameters can be written in analogy to [

The previous expressions can be easily carried over to a numerical code to obtain the material parameters of interest. The following section presents the results and compares them with the experimental observations.

For convenience, the integrations and other expressions have been converted to their normalized versions. The frequencies are normalized with respect to the resonant frequency

where

All the constants used in the analysis were taken from [^{−1}. The damping coefficient was taken to be ^{−1}. The mean periodic spacing

To verify the correct functioning of the code, the results for a very small disorder were compared with the result for a perfectly ordered system (with neighboring meta-atoms interacting with each other); see Figure

Verification of the code—typical values from [

The results of the analysis for dipoles are presented in Figure

Effective material parameter curves for dipole ensembles exhibiting positional disorder. The effective permittivity and permeability curves for disordered dipole ensembles are presented for different values of disorder. The first column pertains to values obtained for a mean period of

Dispersion and effective material parameter curves for quadrupole ensemble with

Dispersion and effective material parameter curves for quadrupole ensemble with

The analysis was carried out for two values of the spacing period

The following features are clearly noted.

For the disordered dipole ensemble, the fall in the permittivity with increasing disorder is clearly visible (Figures

In the case of the quadrupole ensemble, a decrease in the value of the electric permittivity is observed as

Generally speaking, the final expressions for the permittivity and the permeability were derived under several approximations, associated with (

In the light of the above arguments, it is concluded that as the observed positions of the resonances and the relative magnitudes of the parameters are within the limits of approximation, the analysis is valid and can be used to roughly predict the properties of metamaterials with incorporated randomness.

In the preceding analysis, the effect of positional disorder (arising due to aperiodicity) on the averaged material parameters was considered. In a random metamaterial other forms of disorder can exist as well. A particular case of interest is positional disorder along the cut-wire axis; see Figure

Disorder along the cut-wire axis direction—the figure shows a one-dimensional disorder arrangement of meta-atoms. The extent of disorder can be quantified in terms of the angle, the total range of variation being limited to

If this form of positional disorder is taken into consideration along with the aperiodicity, the model would then be a step closer to emulate a true self-organized random metamaterial [

The curves in Figures

Two forms of the distribution function were used in the analysis (Figure

(a) Rectangular and (b) Gaussian forms of distribution function used for governing the positional disorder of the metamaterials along the lateral direction. The positional disorder is expressed in terms of the relative angle between two neighboring meta-atoms. The effective material parameters of the ensemble are derived and presented in Figure

Effective material parameters for metamaterials exhibiting positional disorder along the lateral direction (

Effective material parameters for metamaterials exhibiting positional disorder along the lateral direction (

In extending the multipole approach [