International Journal of Optics

Volume 2014, Article ID 841960, 7 pages

http://dx.doi.org/10.1155/2014/841960

## On Optimal Frequencies for Reconstruction of a One-Dimensional Profile of Gradient Layer’s Refractive Index

Kazan Federal University, 18 Kremlyovskaya Street, Kazan 420008, Russia

Received 26 May 2014; Revised 27 November 2014; Accepted 30 November 2014; Published 16 December 2014

Academic Editor: Stefan Wabnitz

Copyright © 2014 Dmitrii Tumakov. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The problem of reconstruction of a one-dimensional profile of gradient layer’s refractive index is investigated. An algorithm for choosing a right frequency, at which a scattered field is measured, is proposed. It is concluded that at the correct choice of frequency one measurement must be sufficient. Moreover, in this case, regularization parameters of the residual functional are chosen as zero. It is shown that in case of measurements being carried out with errors, residual terms must be added to the functional.

#### 1. Introduction

For developing efficient antireflective coatings on the solid surfaces, the structure of coating layers often requires a change in physical properties of the layers across the direction of light propagation by setting a proper gradient in this direction. This engineering application has been under thorough investigation for quite a long time, for example, through studying evaporation of very thin alternating high/low refractive index films creating an effective refractive index gradient by varying thicknesses of the layers [1, 2].

For developing the right technology of manufacturing the coating layers and testing the manufactured layers as well as for other applications encountered in optics and electrodynamics, it is often required to reconstruct refractive indices of the layers. Thus, reconstruction of a one-dimensional profile of the layer’s refractive index, which is an inverse problem, is one of the major problems in this field. Reconstruction of an unknown profile can be carried out using various types of data such as data for reflection coefficients, input impedance, and scattered electric or magnetic fields. In those techniques, the reconstructed profile can be considered as either a complex profile or a continuous profile [3].

For profile reconstructions, one of the following two approaches is usually utilized: time-domain methods [4, 5] or frequency-domain methods. The time-domain methods require a rather sophisticated and high-precision equipment for generating and registering narrow pulses, which complicates a practical application of these methods. Often the use is made of terahertz pulsed spectroscopy [6, 7]. The frequency-domain methods are used for determining the desired parameters through measuring multiple frequencies, multiple angles, and different polarizations. The permittivity profile is either approximated by an expansion series involving a finite number of elements [8] or represented in the form of some discrete values [9].

The inverse problems are ill-posed problems, and regularization methods are often used to find a solution to these problems. Methods of this kind stabilize the solution but lead to certain losses of accuracy. Convergence of the frequency-domain methods depends on the choice of the frequency range used in the inverse problem. However, rigorous criteria for choosing the frequency range have not yet been proposed [10].

Among all the possible solution techniques, analytical approximation techniques [11–13] and layer stripping techniques [14, 15] can be especially emphasized. Most often, the profile reconstruction problem is solved by an optimization method that minimizes the error between the observed and calculated data. Besides, the problems, in which the data are either incomplete or contaminated with some noise, are also frequent [16].

The optimization methods can be further subdivided into local and global optimization methods. Examples of local optimization methods include gradient methods and quasi-Newton and Gauss-Newton techniques [17, 18]. These methods are fast but often converge to local minima due to the nonlinear nature of the problem. Therefore, these methods are recommended for being used only in case of availability of sufficient amount of priory information. Global optimization methods do not require any a priori information, but a large number of iterations are needed to reach convergence.

Of all the existing global optimization methods, which are used in electromagnetic inversion problems, the neural network technique [19], the genetic algorithms [20, 21], and the particle swarm optimization techniques [22–26] can be accentuated. Each of these methods has its own advantages and disadvantages [27–29]. In view of this, sometimes hybrid techniques combining different methods for their advantages are utilized [30, 31].

In the present work, an algorithm developed in [32] is developed further. The key component of the algorithm is determining an optimal frequencies range for the profile reconstruction. First, the flow of energy transited through the layer is measured. Next, a frequency, at which the flow has a minimum, is determined. Measurements are carried out both after the layer and before the layer. The first reason for that is that measurements at both ends provide more stability to the problem. Secondly, it must be kept in mind that in certain cases different profiles ensure identical fields at the layer exit [33], while reflected fields can differ. High precision measurements of the entire diffracted field at a chosen frequency (or a narrow frequency interval) are carried out at the next stage.

The reconstruction problem is reduced to the minimization of a certain functional. For minimizing the functional, the direct problem is solved iteratively. The algorithm for reconstruction of refractive index is validated through the experiments with several frequency domains. At first, the case in which refractive index of a layer monotonically increases and then monotonically decreases is considered. The layer having a complex profile is considered next. One of the sections is devoted to problems related to perturbed original measured data.

Gradient layers made up of different TiO_{2}/SiO_{2} ratios are chosen for numerical experiments. Refractive indices of TiO_{2} and SiO_{2} are approximately 2.5 and 1.5, respectively, and both materials are optically transparent in the visible and near-IR region [34]. By varying the TiO_{2}/SiO_{2} ratios, thin films of any desired refractive index lying between those of TiO_{2} and SiO_{2} can be obtained. Refractive index of the composite material is determined by the TiO_{2}/SiO_{2} ratio in a deposited dielectric film, defined as in [35].

#### 2. Problem Statement

Let the plane electromagnetic harmonic wave of type fall on a layer of thickness with unknown refractive index from a homogenous isotropic medium (see Figure 1), where is a wave amplitude, is the vacuum wave number, is the refractive index of medium 1, and is a wave frequency. Variable is changed by and it is supposed that the functions depend on variables and . It is necessary to reconstruct the refractive index by using the assumption that the reflected and transmitted waves are known for several frequencies . The profile of refractive index is supposed to vary continuously.