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\title{Signal Conversion in Radio Optics of Metamaterials}
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             \author[1]{A  Salakhitdinov}

             \author[2]{Kh  Mirzokulov}

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\date{\small \em Received: 12 December 2020 Accepted: 4 January 2021 Published: 15 January 2021}

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\begin{abstract}
        


In this paper, we propose a theory of RHM and LHM materials with the aim of their possible creation in the optical range by analogy with oscillatory processes with wave processes. The basis for signal conversion in radio optics of metamaterials is taken from radiation-induced color centers in potassium-aluminoborate glasses with paramagnetic additions of Fe3+ ions, interacting with color centers of the glass matrix of the 3x and 4x types of coordinated boron [ ] 3 e i Bo ? and [ ] 4 e i Bo + , respectively. A distributed parameter communication system with limited linear spatial dimensions is considered as a radio frequency analogy. In metamaterials located between the transmitter and receiver, the interaction of moving and backward waves is considered.

\end{abstract}


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\let\tabcellsep& 	 	 		 
\section[{Introduction}]{Introduction}\par
Author ? ?: Samarkand branch of Tashkent university of information technology named after Muhammad al-Khwarizmi, Samarkand, Uzbekistan. e-mail: liverpool\textunderscore 2592@mail.ru direction in the science of metamaterials and their applications in a wide field. This is due to the fact that Hyperboloid metamaterials make a scientific and practical contribution to optics as well as laser effects in their time. Nonlinear signal transfo-rmations using indefinite tensors ? ij (?) and µ ij (?) \hyperref[b12]{[13]} are practically implemented in all areas of optics: in the processes of scattering, absorption, reflection, diffraction of light, in holographic effects, wave and waveguide phenomena, in focusing light and others.\par
Along with hyperboloid metamaterials, an alternative direction has been developing in recent years the use of magneto-optical properties of borate glasses with paramagnetic additives ions of Cu 1+ , Cu 2+ , Fe 2+ , Fe 3+ and others \hyperref[b13]{[14]}\hyperref[b14]{[15]}\hyperref[b15]{[16]}\hyperref[b16]{[17]}\hyperref[b17]{[18]}. It should be noted the features of the radiation-optical and thermo-radiation properties of potassium-aluminoborate (KAB) glasses, activated by Fe 3+ ions, in which, on the one hand, the radiationoptical properties are well studied \hyperref[b17]{[18]}, and on the other hand, they exhibit peculiar transformations of paramagnetic radiation-induced color centers into oxygen-containing medium of the form [BO 3 ] and  {\ref [BO 4} ], meaning 3x and 4x coordinated boron. The latter arise under the influence of X-ray and gamma irradiation of 60Co and temperature \hyperref[b16]{[17]}\hyperref[b17]{[18]}. Radiation-induced color centers are [ ] and other complex ones with ?u 1+ , ?u 2+ ions \hyperref[b17]{[18]}.\par
It turned out that the simultaneous effect of the thermo radiation field causes a change in the coordination state of the activator ions in the medium in such a way that negative differential absorption ?D<0 is observed and, as a consequence, the medium becomes with a negative refractive index ?n<0  {\ref [19 -20]}.\par
In this work, in contrast to \hyperref[b0]{[1]}\hyperref[b1]{[2]}, a method is proposed for studying RHM (Right handed materials) and LHM materials in order to create materials in the optical range, taking into account the analogies of oscillatory processes with wave processes \hyperref[b21]{[21]}\hyperref[b22]{[22]}\hyperref[b23]{[23]}. In the calculations, we will use computer programs \hyperref[b24]{[24]}\hyperref[b25]{[25]}. Some questions of the theory of an effective medium in metamaterials can be found in \hyperref[b26]{[26]}\hyperref[b27]{[27]}\hyperref[b28]{[28]} {\ref [29]}. Year 2021 e have considered a method for creating a metamaterial in the radio frequency spectrum, where two parallel communication line elements are used as a cell of a metamaterial, one of which is a dielectric, the second is series-periodically connected odd single-wire open communication lines \hyperref[b0]{[1]}. From this pair of a single element, a multilayer metastructure is created taking into account the phase of the backward waves. This method of creating Left handed materials (LHM) material \hyperref[b1]{[2]} is broadband (nonresonant) and more versatile as applied to antenna technology than the second method of realizing a metamaterial in the form of a metastructure with a cell size much smaller than the wavelength of the transmitting signal, containing thin conducting rods and open frames \hyperref[b2]{[3]}\hyperref[b3]{[4]}.\par
As an example of resonator unit cells, we present the results \hyperref[b4]{[5]}\hyperref[b5]{[6]}\hyperref[b6]{[7]}, where a variety of cell shapes are considered in the form of broken and inserted into each other triangles, quadrangles, oval and other types, the sizes of which actually determine the narrowband characteristics, radiation pattern and gain of antennas. The external dimensions of such antenna cells reach 5-6 mm, with a gap of the order of 0.1-0.4 mm. The limited size of the unit cell does not allow mastering the creation of metamaterials in this way in the infrared (IR) and optical ranges.\par
Next, we will consider the possibilities of the first method \hyperref[b0]{[1]}\hyperref[b1]{[2]} for creating metamaterials in the IR and optical ranges \hyperref[b7]{[8]}\hyperref[b8]{[9]}\hyperref[b9]{[10]}\hyperref[b10]{[11]}\hyperref[b11]{[12]}\hyperref[b12]{[13]}. These works develop a new W II. Features of the Analogy of an Optical Wave Field with Radio Oscillations in a Communication System with Distributed Parameters\par
As a radio frequency analogy, let us consider an equivalent circuit of a communication system with distributed parameters \hyperref[b0]{[1]}\hyperref[b1]{[2]} with limited linear spatial dimensions. In this case, moving and backward waves will exist in the metamaterial located between the transmitter and the receiver, similar to the phenomena in a moving wave lamp or a backward klystron used on microwave waves. For simplicity of calculations, consider a plane scalar monochromatic wave of the form( ) ( ) ( ) 0 2 cos exp exp E A t kr A i i t ikr A i i t ikr ? ? ? ? ? ? = ? ? = + ? + ? + .\textbf{(1)}\par
If we take both terms in expression (1), then we can consider nonlinear processes, if we take only the second term of equation ( \hyperref[formula_0]{1}), then we can consider linear processes. In this case, it is necessary to add a complex conjugate term. Formula (1) can be expressed as follows:\par
(2) Where ,\par
Here i A e ? is the complex vibration amplitude.\par
The value is determined from the following relation ,\par
Which determines the projection of the direction of wave propagation ( )\par
, ,x y z k k k k = .\par
It can be shown that in a rectangular coordinate system the components of the vector are as follows:\par
sin , sin cos , cos cosx y z k k k k k k ? ? ? ? ? = = =\textbf{(5)}\par
Where the angle ? is chosen between k r and the plane OYZ, ? between z k and the projection k r on the plane OYZ. It can be seen from ( \hyperref[formula_4]{5}) that there are only two independent angular variables:1 2 sin , sin cos x y k u k k u k ? ? ? = = = = . 6)\par
From ( 4) we find ,\par
For the propagation of a plane wave in free space from the cutting plane of the transmitting antenna, the following condition must be met: .  \hyperref[formula_7]{9}) is not satisfied, then we get an inhomogeneous wave, which exponentially decays, at least along one of the coordinates . Such a wave is also a solution to the wave equation.( ) t i e z y x p P ? ? = , ,\textbf{(9)}? ? ? i e A r k i i A r k i i A z y x p = ? + = ? = ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? exp exp exp , , 2 k 2 2 2 2 2 2 2 4 2 z y x k k k c x k + + = = = ? ? 2 2 2 1 2 u u k k z ? ? ± = ( ) ( ) [ ] x u x u i u u k iz iy A z y x p 2 1 2 2 2 1 2 exp exp ) exp( , , + ? ? ? ± ? = ? ? ? ? ? ? ? ? 2 2 2\textbf{( ) ( )}\par
Note that equations and expressions ( \hyperref[formula_4]{5}) -(9) can be used to determine the radiation pattern of the transmitting antenna in a linear approximation.\par
Since the electromagnetic field in the form of a plane wave \hyperref[b7]{(8)} with different parameters is a solution to the wave equation, the solution will also be in the form of a sum (integral) of fields of the form (8) for a threedimensional system: \hyperref[b9]{(10)} where is a complex function that describes the amplitude and phase of an individual plane wave with the direction of propagation, which determines the set of real variables , that is, all possible plane waves, including inhomogeneous ones.\par
Equation (  {\ref 10}) is a generalization of the solution of the wave equation to the case of a nonplanar monochromatic wave, for example, for a spherical wave. From expression \hyperref[b9]{(10)}, one can pass to the real field by multiplying by exp(-j?t) and adding the complex conjugate term to the expression.\par
Let's solve the following problem: the values of the wave equation are given on the plane z=0 (the initial plane of the antenna location) of the directional pattern. It is required to find a solution of the wave equation for z?0, which turns into a given function on the plane z=0.\par
From the conditions of Kirchhoff radiation on an infinite sphere of the wave field, this function should be equal to zero. From \hyperref[b9]{(10)} we obtain the following , (11)??? g( )=\textbf{(12)}\par
Having determined the frequency spectrum g (u 1 ,u 2 ) from ( \hyperref[formula_9]{12}) and p (x,y,0,t), we find the boundary conditions at z = 0. It follows from (4.15) that for heterogeneous moving waves and for z>0 , \hyperref[b12]{(13)} For z <0, we obtain a solution corresponding to backward waves . \hyperref[b13]{(14)} Consider special cases (  {\ref 13}) and (  {\ref 14}): at z=0, y=0z y x k k k , ,\textbf{( ) ( ) ( )}2 1 2 1 2 2 2 1 2 2 1 , 4 1 , , 2 u du y u x u i e e u u g z y x p u u k iz + ? ? = ? ? ± ? + ? ? ? ( ) 2 1 ,u u g 2 1 , u u ( ) ( ) ( ) 2 1 2 1 2 1 , 4 1 0 , , 2 u du y u x u i e u u g z y x p + ? ? = = ? + ? ? ? 2 1 ,u u ( ) (\par
) dxdy y u x u i e y x p 2 1 0 , , + ? ? ? ? + ? ? 2 2 2 2 1 k u u > + ( ) ( ) ( ) 2 1 2 1 2 2 2 1 2 2 1 , 4 1 , , 2 u du e e u u g z y x p y u x u i u u k iz + ? ? ? ? = ? + ? ? ? ( ) ( ) ( )\textbf{2}\par
. (  {\ref 18})\par
Formula (  {\ref 18}) is valid for any value of z. The coordinates of points in the space x, y have the dimension of length l. The variables u1, u2 have the dimension of inverse wavelength cm-1 and correspond to spatial frequencies. Formulas (  {\ref 13}) and (  {\ref 14}) correspond to two-dimensional (D2) Fourier integrals  {\ref [30]}; formulas \hyperref[b14]{(15)}\hyperref[b15]{(16)}\hyperref[b16]{(17)}\hyperref[b17]{(18)} correspond to the usual one-dimensional (D1) Fourier integrals. 
\section[{III.}]{III.} 
\section[{Calculation of the Signal Transmission System by Wave Analog for RHM Material}]{Calculation of the Signal Transmission System by Wave Analog for RHM Material}\par
Consider a signal transmission system over a wave channel. For calculations, we will use formulas (  {\ref 15}) -(  {\ref 18}) in a simplified one-dimensional (D1) version.\par
Recall that to calculate the signal spectrum \hyperref[b16]{(17)}, you need to know the specific form of the input function p(x,z x =0). As mentioned earlier, as p(x,z x =0) we take the spectral dependence of KAB -glass \hyperref[b13]{[14]}\hyperref[b14]{[15]}\hyperref[b17]{18]}, and the shape of the absorption spectrum for a separate spectrum of the color center in the form of a Gaussian curve:max 2 0 0 1 2 exp G G Ð?" I I ? ? ? ? ? ? ? ? ? ? ? = ? ? ? ? ? ? ? ? ? ? ? ? ? ,\textbf{(19)}\par
Where max 0 G I maximum amplitude of the Gaussian curve; \par
Where for simplicity of calculations for the RHM material the following designations are adopted:( ) 2 0 max 0 exp bx I A Ð?" ? = ; ( ) 2 2 / 1 2 ln 2 ? = b ; 0 0 ? = x ; ? = x . (\textbf{21})\par
After transformations and calculations using the theory of residues [30], we obtain ( ) ( ) 0 , 0 , , x p y x p = ( ) ( ) ( ) 2 1 2 1 2 , u g u g u u g ? = ( ) ( ) ? ? + ? ? ? ? ? = = x d x i e x p u g g ? ? ) 0 , ( x d e g z x p x i ? ? ? + ? ? ? ? = ? ? ? ) ( 2 1 ) , ( 1 0 ? dx e A g u g iux xx b x b e ? + ? ? ? ? + = = 0 0 2 2 2 2 ) ( ) ( ? © 2021? ? ? ? ? ? ? ? ? ? ? ? + ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? = ,\textbf{(22)}\par
( ) ? ? ? ? ? ? ? +? +? ? ? ?? ?? = ? + ? ? ? , (\textbf{24})\par
where; . dx ux d u ? ? = =\par
To determine the amount of deductions, change the following parameters2 2\textbf{2 2 2 1 2 ;}\par
; cos ; sin 1 1 1du u u tgu d u u u ? ? ? ? ? = = = = + + + .\par
After the appropriate calculations, we arrive at the following equation ( ) ( )0 9 0 4 1 2 2 bx RHM x p u j j e x ? ? ? ? ? ? = + ? ? ? ? ? ? ,\textbf{(25)}\par
where through0 2 x x ? ? ? ? ? ? ? indicated 2 0 exp 2x ? ? ? ? ? ? ? .\par
IV.\par
Calculation of the Signal Transmission System over the Wave Channel Taking into Account the Presence of lhm Material\par
Calculations carried out similarly to the previous paragraph, taking into account the LHM of the material in the III-square (? (?) <0, µ (?) <0), lead to the following formulas:( ) ( ) 2 0 0 2 cos sin bx bx x LHM g u A e ux j ux dx ? ? ?? = ? ? ? (26)\par
For spatial spectrum and for field distribution( ) ( )( ) 0 1 cos sin 2 LHM p u g u ux j ux du ? ?? = ? ? ? ,\textbf{(27)}\par
Where the physical meaning of the variable u is reflected: on the one hand, u -has the meaning of "spatial frequencies", having the dimension of the wavelength, on the other hand, it determines the propagation of plane waves, on which we mark the extended wave field. By analogy with (  {\ref 23}) -(  {\ref 26}) for a medium with a metamaterial (LHM), taking into account the change of variables and the application of the theory of residues [30], we obtain: ( ) ( )\textbf{2}? ? + ? ? ? ? ? ? ? = + ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?\textbf{(28)}\par
from which ? ? = ? ? ? ? ? ? ?\textbf{(30)}\par
Imagine part.\par
The following formulas were obtained for a metamaterial with a negative absorption coefficient:\par
( )2 2 0 0 298 1194 24 1 2 2 x x p u j x x ? ? ? ? ? ? ? ? = + ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? .\textbf{(31)}\par
V. 
\section[{Calculation Results and their Discussion}]{Calculation Results and their Discussion}\par
Using the previously obtained formulas ( \hyperref[formula_19]{22}), ( \hyperref[formula_30]{28}) -(30), we find the ratio of the spatial frequency spectra , there is an increase harmonics.\par
Calculations for k=u=+1 depending on as well as for k=+1. This is due to the fact that in (33) 2 u ? and 2 cos ? even functions. However, if we take into account the phase of oscillations according to \hyperref[b22]{(22)}, \hyperref[b26]{(26)}\hyperref[b27]{(27)}\hyperref[b28]{(28)} {\ref (29)} {\ref (30)} = ? ? ? ? ? ? ? ? ? + ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? (34) When u=k=-1, x 0 =1 ( ) ( ) ( ) 1,31 1,51cos 1 17, 77 u ux U ? ? ? ? ? ? =? + ? ? ? ? . (\textbf{35})\par
Calculations by formula (35) are presented in Fig. 1.b, from the phase dependence of the spatial spectrum, it can be seen that the phase of the "moving wave" and "backward wave" changes its sign at   For this amplitude-frequency characteristic (Table  {\ref 1}), broadband and amplification is manifested among metamaterials (Fig. 2). ? ? ? ? ? ? + ? ? ? ? ? ? ? ? ? ? ? ? ? = ? ? + ? ? ? ? ? ? . (\textbf{36} 
\section[{Conclusions}]{Conclusions}\par
It should be noted that small gain is not a problem, since in practice, both series connected and parallel elements are created , which can provide significant gain in comparison with the considered unidirectional linear system.\par
Thus, the problem of obtaining a metamaterial from amorphous glass has been theoretically solved. As samples, it is necessary to take amorphous films made of magneto-optical potassium-aluminoborate glass with additions of iron oxide, which provides the necessary metamaterial parameters during radiation processing, at sufficiently high temperature irradiation.\begin{figure}[htbp]
\noindent\textbf{}\includegraphics[]{image-2.png}
\caption{\label{fig_0}}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{}\includegraphics[]{image-3.png}
\caption{\label{fig_1}}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{}\includegraphics[]{image-4.png}
\caption{\label{fig_2}}\end{figure}
      \begin{figure}[htbp]
\noindent\textbf{} \par 
\begin{longtable}{P{0.35619047619047617\textwidth}P{0.008095238095238096\textwidth}P{0.2388095238095238\textwidth}P{0.004047619047619048\textwidth}P{0.03642857142857143\textwidth}P{0.012142857142857143\textwidth}P{0.04452380952380952\textwidth}P{0.012142857142857143\textwidth}P{0.008095238095238096\textwidth}P{0.004047619047619048\textwidth}P{0.028333333333333332\textwidth}P{0.004047619047619048\textwidth}P{0.03642857142857143\textwidth}P{0.012142857142857143\textwidth}P{0.04452380952380952\textwidth}}
\tabcellsep \tabcellsep \multicolumn{11}{l}{Signal Conversion in Radio Optics of Metamaterials}\tabcellsep \tabcellsep \\
\tabcellsep \tabcellsep 2 0 bx 2\tabcellsep \tabcellsep 2 2 bu x\tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep 2 0 bx\tabcellsep \tabcellsep 2 2 bu x\tabcellsep \tabcellsep \\
( ) g u\tabcellsep 32\tabcellsep e\tabcellsep e\tabcellsep 2\tabcellsep cos\tabcellsep ( ) 0 bx ux\tabcellsep j e\tabcellsep 32\tabcellsep e\tabcellsep 2\tabcellsep e\tabcellsep 2\tabcellsep cos\tabcellsep ( ) 0 bx ux\\
( ) Volume Xx XI Is sue III Version I J\tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \\
lobal Journal of Researches in Engineering\tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \end{longtable} \par
 
\caption{\label{tab_1}}\end{figure}
 		 		\backmatter   			 
\subsection[{Acknowledgments}]{Acknowledgments}\par
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