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 P with coordinates ð0; 0Þ. The random walk is characterized by a set of points randomly distributed, and the trajectory can be obtained from the correlation function corresponding to the flows of current probability. The statistical properties of a random distribution of points can be transferred to induce and control important physical effects. For example, it is known that the amplitude distribution of a speckle pattern follows Gaussian statistics [6, 7]. The statistic of the speckle pattern is matched with a random hole distribution, and it is transferred on a metal surface. The analysis is obtained by masking the surface metal which is considered to be formed by a set of square cells. The probability of a hole being present at the center of each cell is P; therefore, the probability of the absence of a hole is ð1 � PÞ. The surface contains N cells, and the probability of the surface contains n-holes, assuming that a Bernoulli distribution is

$$P(n) = \binom{N}{n} P^n (1-P)^{N-n}.\tag{12}$$

When the number of cells N increases, the Bernoulli distribution tends to a Gaussian distribution of the form

$$\rho(\mathbf{x}, \mathbf{y}) = \frac{1}{\sqrt{2\pi\sigma^2}} e^{-\frac{\mathbf{x}^2 + \mathbf{y}^2}{2\sigma^2}},\tag{13}$$

where σ<sup>2</sup> is the variance. Interesting features can be identified by describing the self-correlation in this type of distribution. The simplest case occurs when two screens are superposed and, subsequently, one of them is rotated by a small angle. In order to understand the generation of the self-correlation trajectory, we focus on a single hole. In this case, it is evident that the hole follows a circular arc by joining all the points of constant probability and the complete correlation trajectory is a circle. The result in this case is shown in Figure 2a. The correlation trajectory can be controlled by inducing a scale factor in the distribution of random points. By

### Figure 2.

(a) Set of points following a Gaussian distribution. (b) Correlation function between two Gaussian sets of points where one mask was rotated by a small angle. (c) Probability flow trajectories between two mask Gaussian points, one of them is scaled by approximately 95%, without rotation. (d) Same as in (c) but with a rotation of approximately 5° .

Synthesis of Curved Surface Plasmon Fields through Thin Metal Films in a Tandem Array DOI: http://dx.doi.org/10.5772/intechopen.81931

superposing the two screens again, it is evident that the scale factor shifts the point along a linear trajectory perpendicular to the regions of constant probability, which are sets of circles, as deduced from the argument of the Gaussian distribution. The analysis is presented in an equivalent way for a speckle pattern using the fact that both of them have the same probability distribution. In Figure 2b, we show these correlation trajectories. Finally, by introducing a small rotation, the linear trajectories are curved, as shown in Figure 2c.

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

$$\begin{split} \rho\_1(\mathbf{x}, \mathbf{y}) \cdot \rho\_2(\mathbf{x}^\*, \mathbf{y}^\*) &= \frac{1}{\sqrt{2\pi}\sigma\_1\sigma\_2} \exp\left\{-\frac{\mathbf{x}^2 + \mathbf{y}^2}{2\sigma\_1^2}\right\} \\ &\times \exp\left\{-\left[\frac{d(\mathbf{x}\cos\theta + y\sin\theta)^2 + \left[d(-\mathbf{x}\sin\theta + y\cos\theta)\right]^2}{2\sigma\_2^2}\right] \end{split} \tag{14}$$

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

$$\frac{\mu^2}{a^2} + \frac{\mathcal{V}^2}{b^2} = \mathbf{1}.\tag{15}$$

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

$$y' = \frac{b^2}{a^2} \frac{y}{x}.\tag{16}$$

Further, the corresponding solution is given by

$$
\mathcal{Y} = c\mathfrak{x}^a,\tag{17}
$$

where <sup>c</sup> is an arbitrary constant and <sup>α</sup> <sup>¼</sup> <sup>b</sup> a 2 2, which carries the information about the scale between the two probabilistic processes.
