TETRA Project Team Members Publishes Groundbreaking Book on Metamaterial Anisotropy

We are pleased to announce that some of TETRA project team members published a new scientific book titled "Anisotropy of Metamaterials: Beyond Conventional Paradigms".

This comprehensive work, published in the Series in Optics and Optoelectronics, is authored by distinguished researchers including Prof. Vladimir Mityushev (TETRA Management Committee member), Prof. Tatjana Gric (TETRA project coordinator), Prof. Radoslaw Kycia, and Prof. Natalia Rylko.​

The book explores the anisotropic properties of metamaterials, challenging conventional paradigms and presenting innovative approaches to understanding these complex structures. This publication represents a significant contribution to the field of metamaterial science and aligns with the TETRA project's mission to advance metamaterial researches including cancer detection applications.​

Available in Kindle Edition format, this first edition book consolidates cutting-edge research and theoretical frameworks developed within the COST Action CA23125 - TETRA (The mETamaterial foRmalism approach to recognize cAncer) project network. The publication demonstrates the productive collaboration between TETRA project members and their commitment to disseminating advanced knowledge in metamaterial science to the broader research community. This work is expected to serve as a valuable reference for researchers, academics, and professionals working in the fields of optics, optoelectronics, and metamaterial applications. The book is available through digital platforms and represents another milestone in the TETRA project's ongoing efforts to push the boundaries of metamaterial research.​

LİNK

Chapter 1. Electromagnetics of metamaterials​

This chapter is devoted to the physical foundation of metamaterials and Maxwell's equations in the frequency domain. Maxwell’s equations are supplemented by constitutive relations on the electric field vector E and the electric flux density (displacement) vector D, the magnetic flux density (displacement) vector B, and the magnetic field vector H. The linear dependencies through the medium's permittivity and permeability are considered. Special attention is paid to dielectrics, materials described only by permittivity, and their behavior in the quasistatic limits. Temporal dispersion and the behavior of metamaterials and piezoelectric materials are outlined. The basic dispersion models are introduced, and the resonance notion is explained. The perfect contact condition (transmission condition) is written in terms of the boundary value problem for the potential.​

Chapter 2. Generalized alternating method of Schwarz

The generalized alternating method of Schwarz for composites and its various implementations are applied to determine the local fields in 2D dispersed composites. Following the homogenization theory, we consider a doubly periodic representative cell Q with an arbitrary number of inclusions per cell. The method of complex potentials and constructive results on the R-linear problem is systematically applied. The proposed method yields approximate and exact analytical formulas for the effective properties of dispersed composites with the strictly derived precision of their validity in concentration f and the contrast parameter ρ for two-phase composites. First, we developed formulas for real-valued permittivity. Next, we explore complex values of the normalized permittivity of components Ɛk for multi-phase composites. Consequently, the contrast parameters ρ k also become complex. Building upon the foundations laid out in this chapter, we extend these formulas to the complex domain through analytic continuation in terms of ρ k. To ensure the validity of this continuation, it is imperative that we establish a groundwork for understanding the interplay between complex permittivity and its matrix representation.​

Chapter 3. Effective Medium Approximation

As described in the previous chapters, Schwarz's method leads to new analytical exact and approximate formulas with explicitly written precision. Comparison of these new formulas with the previously derived formulas by other approaches (self-consistent method, effective medium approximations, differential scheme with Mori-Tanaka method, and other modifications) based on empirical arguments demonstrate the discrepancy for the approximations in f and ρ beginning from the second-order terms. It is shown that many general formulas are derived using the first-order approximation of Schwarz’s method, valid up to O(f2) for the implicit iterative scheme and up to O(| ρ |2) for the explicit scheme. This investigation explains plenty of illusory different formulas that are reduced to the classic Maxwell-Garnett (Clausius-Mossotti) estimation for dilute composites. Some self-consistent methods violate homogenization principles and lead to methodologically misleading approaches.​

Chapter 4. Circular and elliptic inclusions

The generalized alternating method of Schwarz for composites with circular inclusions transforms into the method of functional equations. By a functional equation, we mean an equation with a composition of the unknown function with a given function instead of the integral operator. Schwarz's iterative method includes the result in terms of multiple integrals. The composition of functions can be easily implemented for numerical and symbolic computations. In the present chapter, we develop the constructive results obtained for the real-valued permittivity to the complex permittivity. A method of structural sums is developed for dispersed random composites. Next, a simple rule is established to transform the higher-order formulas derived for real-valued permittivity to complex permittivity. The exact formula for one circular inclusion per periodicity unit cell Q is derived. An approximate analytical formula is derived for an arbitrary location of N non-overlapping disks per periodicity cell, where $N$ is given in symbolic form. These formulas are the most general formulas known for the effective properties of composites with circular, non-overlapping inclusions. They are extended to inclusions of elliptic shapes. The approach described above leads to the theory of the analytical Representative Volume Element aRVE.​

Chapter 5. Fibrous magneto-electro-elastic composites

Magneto-electro-elastic (MEE) composites extend the physical notion of piezoelectric composites when the electric and elastic fields are coupled with the magnetic field. We develop Lekhnitskii’s formalism to 2D MEE problems following Filshtinsky’s method. The MEE fields are written in terms of complex potentials. It is worth noting that Filshtinsky’s extension of Lekhnitskii’s formalism does not reduce to a formal increase of analytic functions from two to four. We formally present Filshtinsky’s method for dielectrics, i.e., for a problem for one complex potential. A series of exact formulas known for the anti-plane statement of composites was automatically extended to piezoelectric composites by a matrix transformation of the R-problem. This result completes the constructive theory of the piezoelectric transversely isotropic fibrous composites. More precisely, any formula for the scalar permittivity of a 2D composite can be transformed by a matrix formalism to a formula for the effective tensor of a piezoelectric fibrous composite. ​

Chapter 6. Digital Image Processing and basic granulometry

This chapter is split into two logical parts. The introduction to selected aspects of DIP (Digital Image Processing) will be presented in the first one. In the remaining part, we will employ this knowledge to construct image processing pipelines for specific images. We will illustrate all concepts using the Python programming language. Therefore, we assume that the Reader has some experience with programming using this language. As a short introduction, we can recommend an appendix of our book. You can also study free Scientific Python Lectures [4]. Complete examples of the code from this chapter can be downloaded from [1]. You can run it and experiment. We will use the OpenCV (Open Computer Vision) library for image processing. We will learn the specific algorithms in the library that are useful for image processing of tissues and selecting parts suspected to be cancer cells. Moreover, the same methods can be used in materials engineering for analyzing defects and identifying different phases in materials, and in biology, where one has to count bacterial cells in a microscope image. We focus mainly on cells in medical or biological applications or grains/granules in materials engineering. Therefore, this part will also be for nonspecialists who want to learn granulometry, a specific topic of DIP that focuses on characterizing granules in the image, e.g., counting them, defining their shape, color, classifying them, etc. ​

Chapter 7. Cancer cells detection using Neural Networks

In this chapter, we construct a simple artificial neural network (NN) to classify cancer in histopathologic scans. It is more technical and demanding since we expect the reader to be familiar with the Digital Image Processing methods described in the previous chapter and have at least basic knowledge of the construction of Artificial Neural Networks. In the previous chapter, we used unsupervised learning, which allows an algorithm for clustering similar pixels. obtained some groups of cells, which experts should examine, since the algorithm𝑘-Means, which was used, has no internal knowledge of detecting cancer cells. To make the classification more automatic, we must invest additional knowledge in preparing data and then use supervised learning algorithms to learn these examples. The NN approach is one of the most universal approaches to supervised learning that can detect complex relations in the data. In the chapter, we show an example of how to construct and train a model and then use it for classification. This chapter contains two sections. In the first one, we describe the reparation of the classifier; we use it to make the classification. Complete examples of the code from this chapter can be downloaded from. You can run it and experiment. ​

Chapter 8. Applications

First, image analysis of one cancerous mouse brain biopsy microscope image is applied to detect the inclusions (in the terminology of composite media) or cancerous and noncancerous cells (in medical terminology). Next, the derived analytical formulas for three-phase composites are applied to determine the macroscopic permittivity for various frequencies. Different variants of shapes are considered, in particular, elliptic for glioma and circular for neurons. The contributions of the Maxwell-Garnett term, the J-shape term, and the interaction term ℘ into the macroscopic permittivity tensor are analyzed. The results for the Maxwell-Garnett and Bruggeman approximations, as well as derived higher-order approximations, coincide only for very low frequencies. Low frequency corresponds to a small absolute values of contrast parameters. The aRVE theory is applied to study the distribution of all the inclusions (cells) in the following way. A small sliding box 𝐵(x) centered at x with a side less than 10 times the side of the large square 𝑄 is considered. The structural sums are calculated over 𝐵(x) as functions of x. According to the decomposition theorem, these structural sums characterize the local geometrical distribution of inclusions in 𝑄, which impacts the macroscopic permittivity tensor. ​



Short description of the book by chapters