## Abstract

We describe the generation of plasmonic modes that propagate in a curved trajectory inducing magnetic properties. This is performed by masking a metal surface with two screens containing a randomly distributed set of holes that follow a Gaussian statistic. The diameter of the holes is less than the wavelength of the illuminating plane wave. By implementing scaling and rotations on each screen, we control the correlation trajectory and generate long-range curved plasmonic modes. Using the evanescent character of the electric field, the study is implemented for the transmission of a plasmonic mode propagating in a tandem array of thin metal films offering the possibility to generate localization effects.

### Keywords

- plasmon mode
- surface plasmon field
- speckle
- thin films
- curved correlation trajectory

## 1. Introduction

During the last decade, the scientific community has shown an increasing interest in the models of plasmon fields due to their potential applications, which occur practically in all branches of science and technology. In the present study, we emphasize the analysis of correlation trajectories on a metal surface with random structure. The resulting model offers applications to development of nano-antennas having the possibility of a tunable bandwidth [1]. This type of structure has applications in the synthesis of new light sources and the control of magnetic effects [2]. The tunable effects are controlled with the curvature parameter having applications in surface-enhanced Raman spectroscopy (SERS), also as the local excitation of quantum dots. Implementing the evanescent behavior of the plasmon field, the analysis is extended to the propagation of plasmon fields through a tandem array of metal films similar to photonic crystal structures [3, 4].

As a starting point, we describe the study of the electric field in the neighborhood of a nanoparticle using the electrostatic approximation [2]. The electric field corresponds to the plasmon particle. This model allows the description of the interaction between two plasmon particles. The interaction is extended to describe the plasmon fields propagating on a surface generating a wave behavior satisfying the Helmholtz equation where the wave number must have complex values in order to recover the traditional surface plasmon models. Controlling the random distribution of nanoparticles, we analyze the correlation effects leading us to induce localization effects. This last statement is obtained by masking thin metal surface with two independent random array hole distributions. Controlling the scale factors, we modify the curvature of the correlation trajectory. The model is related with a speckle pattern emerging from a rough surface [5]. This configuration is similar to the configuration proposed by Reather for the coupling of plasmon fields. Experimental results are shown.

## 2. Analysis of plasmon particle

A nanoparticle is generated by a set of atoms; the plasmon particle corresponds with the surface current distribution of the atoms. The analysis is implemented applying the electrostatic approximation given by

where

Proposing the solution as

we obtain the equation system

where the coupling constant α is a complex number having the form

and the solution for

Then, the complete solution

with

### 2.1. Description for the interaction between plasmon particles

The model is extended to describe the propagation of the electric field. For this, we propose that the electrostatic approximation is no longer fulfilled, acquiring the form of the Helmholtz equation having the form

Looking for propagation along the x-coordinate, the equation acquires the form

where

whose solution acquires the form

this equation must recover the structure of the electrostatic approximation for a single nanoparticle.

From the previous solution, it is easy to identify its behavior. Along the

In Figure 1a, the electrostatic approximation is valid for a single nanoparticle; the wave behavior is generated by another set of particles interacting shown in Figure 1c.

Until this point we have described the generation of a wave propagating in the

## 3. Description statistics of correlation trajectories

In the present section, we describe the transfer of the statistical properties of an anisotropic two-dimensional random walk model to generate wave propagation on a metal surface, thus generating a curved surface plasmon mode. The model is conceptually simple. We describe a trajectory in a two-dimensional array, starting from a point

When the number of cells

where

This result can be explained as follows: the correlation function of two scaled and rotated surfaces have the form

Analyzing the argument of the exponential function as a quadratic form, it can be shown that the curves of constant correlation are ellipses, presenting a reference system where they acquire the canonical form

The probability flows through the orthogonal trajectories between the two regions of constant probability, whose differential equation is given by

Further, the corresponding solution is given by

where c is an arbitrary constant and

### 3.1. Graphical description and experimental implementation of the correlation trajectory

A fundamental part of the chapter consists of describing a method to generate surface plasmon fields propagating along predetermined trajectories. This can be obtained analyzing the correlation function between two screens where each one has a random hole distribution following a predetermined probability density function. This method has the characteristic that the correlation trajectory geometry presents a tunable curvature which allows the possibility to generate long-range surface plasmon.

An alternative model to generate the curved correlation trajectories is performed using a speckle pattern as it is shown in Figure 4.

The optical system that rotates the image can be a prism-type Dove. Modifying the illumination configuration using a convergent beam and changing the relative distance between the two speckle patterns obtained by shifting one mirror a scale factor are introduced. The irradiance superposition between the two speckle patterns generates the desired correlation trajectories. The speckle pattern is shown in Figure 3.

It is known that the irradiance function for the speckle pattern has associated a probability density function-type exponential decreasing function. The decreasing term can be matched with the decaying ratio of the plasmon mode. This configuration allows improving the generation of plasmon field avoiding the masking of the metal surface which must be made with lithography techniques. These comments represent novel applications of the speckle pattern.

The correlation trajectories generated will be implemented in the following section to describe the surface plasmon. By the fact that the correlation occurs in a curved trajectory, we expect the surface plasmon to present a magnetic behavior.

### 3.2. Generation of curved surface plasmon modes

The previous statistical description will be employed for the synthesis of surface plasmonic modes. The expression for the electric field of an elementary surface plasmonic mode propagating along the

where

Using the functional relation given by Eq. (17), the expression for the curved plasmonic mode is given by

By means of the Maxwell equations, we can obtain the expression for the magnetic field and the energy flux given by the Poynting vector.

For the experimental setup, we propose to illuminate a thin flat Au film (thickness

The analysis presented can be extended to other plasmonic configurations which are presented in the following section.

## 4. Propagation in a tandem array of thin metal films

The natural extension of the analysis presented is the transfer of the plasmonic mode to a tandem array of thin metal surface, shown in Figure 6. This is possible using the evanescent behavior along the

The transmission of the plasmonic mode satisfies the following system of differential equations:

where

The simplest case occurs when the system is formed by two thin metal films separated by a dielectric medium whose thickness must be less than 50 nm. The evanescent decay depends on the modulus of the permittivity quotient [13], and at this thickness is possible to generate tunneling effects [11]. Subsequently, the system of Eq. (21a) acquires the simple form

Rewriting it in matrix form, we obtain

It can be deduced that, as a consequence of the energy conservation, the matrix structure must be symmetric. This indicates that

where

Moreover, it is known that the eigenvectors must be complex [14]. Subsequently, without loss of generality, the solution can be rewritten as

which indicates that the shift generated between each plasmon mode presents similar features as the coupling mode theory [12]. This analysis leads to the expression for the plasmonic mode as

where *s* is the arc length on the same curve; we remark that the correlation trajectory is given by Eq. (20).

Eq. (24) describes the evanescent coupling through a tandem array of thin metal films. Notably, the boundary conditions of the electric field indicate that the geometry of the plasmon field generated in the first thin metal film must be preserved in all the surfaces. This shows that the transmission of the curved plasmonic mode allows inducing magnetic properties in the system [15, 16, 17, 18].

## 5. Conclusions

The statistical properties of the distribution of random holes or equivalently the speckle pattern were transferred to a metal surface to stablish the conditions to generate long-range curved plasmonic modes. In the case of hole distribution, this can be implemented by masking a thin metal film with two screens that allows controlling the correlation trajectory whose geometry corresponds to a curved long-range surface plasmonic mode. Another possibility was illuminating the metal thin film with two correlated speckle patterns. An important consequence of these configurations is that the set of curved surface plasmonic modes presents a vortex structure that allows to induce magnetic properties [17]. Using the evanescent character of the plasmon modes, the electric field was transferred to the propagation in a tandem array of thin metal films offering applications to design photonic crystals with tunable and localized magnetic properties.

The theoretical point of view presented in this study allows incorporating other effects such as percolation effects which consist in propagating the electric field through random structures. The main characteristic is that the plasmon field presents fractal properties which are the origin of interesting magnetic properties implicit in the curved trajectory of the set of plasmonic modes; more details can be found in [18]. The model presented can be extended by implementing different hole distribution geometries which modify the plasmonic resonance effects. Notably, the curved trajectories have associated focusing regions, and, subsequently, the corresponding magnetic singularity offers the possibility of implementation in the generation of plasmonic magnetic mirrors.

Finally, we remark that the analysis presented offers applications to photonic crystal as a metamaterial design [19, 20, 21, 22, 23] since breaking the periodicity or incorporating another type of metal on a selected region is similar to doping the structure and then is possible to induce localization effects. The excitation of plasmon fields using a speckle patterns offers the possibility to incorporate the tunable behavior of the correlation trajectory offering interesting applications in the development of plasmonic antennas and synthesis of accelerating plasmon modes [21], extending the plasmonic optical models.

## Acknowledgments

The authors MATR and MVM are very grateful to CONACyT for their support.