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.

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 [

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 [

For profile reconstructions, one of the following two approaches is usually utilized: time-domain methods [

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 [

Among all the possible solution techniques, analytical approximation techniques [

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 [

Of all the existing global optimization methods, which are used in electromagnetic inversion problems, the neural network technique [

In the present work, an algorithm developed in [

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 [_{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

Let the plane electromagnetic harmonic wave of type

Geometry of the problem.

Measurements of

So,

The direct problem for a harmonic wave of frequency

The function

Thus, the diffraction problem is reduced to an ordinary differential equation

After

The function

The relative error of the determined refractive index

It can be noted that for a starting profile it is more convenient to choose a line connecting values closest to

The problem of reconstructing the

The layer thickness will be taken as

In case of refractive index changing continuously in the whole domain, transited energy has a unique and clearly pronounced minimum. For instance, for a parabolic profile the minimum is close to 24 THz (the dashed line in Figure

The layer’s transmittance

In reconstructing a profile of the dielectric layer’s refractive index, it is often quite difficult to take a wide frequency range because of a spread in values _{2} can be found in [

Profiles reconstructed at various frequency ranges are given in Figure

Reconstructed profiles. The solid curve corresponds to the initial profile

Dependence of an error of the profile reconstruction

The relative error of reconstruction of

The relative error of reconstruction of refractive index for the parabolic layer versus the number of iterations

The second interval for reconstructing the

Details of the profile reconstruction for the case of

The main conclusion regarding reconstruction of refractive index for ridge-shaped profiles is the following. The optimal frequency interval ensuring the highest accuracy and the minimum number of iterations corresponds to the global minimum of the transited energy. At small shifts from the optimal frequencies, the same error of the profile reconstruction can be reached but at a greater number of iterations.

It is also worth noting that the number of measured values in the optimal frequency range (minimum of the transited energy) is not a key issue at all. The case

In this section, reconstruction of more complex profiles will be considered. In Figure

Reconstructed profiles. The solid curve corresponds to the initial profile

The frequency interval 65–67 THz corresponds to the minimum of energy transited through the layer (Figure

The layer’s transmittance

The general behavior of dependence of the approximation error

The relative error of reconstruction of

Unlike the previous section, a case of discontinuous refractive index (

Just as in the case of simple profiles, the number of measurements in the optimal range for complex profiles is not a key issue at all. A measurement at a single frequency provides the same approximation as measurements carried out at several frequencies.

In this section, influence of the measurement error

It certainly makes more sense to perform reconstructions based on several measurements instead of using only one measurement. The measurements can be carried out both at one frequency and at several frequencies. If the measurements undergo perturbations, the inverse problem gets worsened. In this case, regularization parameters must be chosen as nonzero. In Figure

Reconstructed profiles for the case of perturbed data (

In reconstructing the layer profile using some perturbed data, the optimal interval for reconstruction remains to be the range corresponding to the minimum values of

For reconstruction of refractive index, the best results are obtained in cases in which measurements are carried out at frequencies close to frequencies of minimum transition of energy through the layer. The frequency interval can be very narrow such that dispersion effects of the substance filling the layer remain insignificant. The measurements can be also carried out at a single frequency. In this case, accuracy of the profile reconstruction practically remains the same.

Thus, the following strategy can be utilized. First, the flow of energy passed through the layer must be measured. Next, a frequency, at which the flow has a minimum, is selected. High precision measurements for the entire diffracted field at the selected frequency (or at a narrow frequency interval) are carried out at the next stage.

The functional used for the profile reconstruction is constructed with zero regularization parameters. However, if it is known a priori that the measurements were carried out with essential errors, the regularization parameters

The author declares that there is no conflict of interests regarding the publication of this paper.

This work was funded by the subsidy allocated to Kazan Federal University for the state assignment in the sphere of scientific activities.

_{2}-SiO

_{2}glass over a wide spectral range

_{2}and SiO

_{2}using an ITO hard mask