2 Electrodynamical Analysis of Open Lossy Metamaterial Waveguide and Scattering Structures

Large stream of articles devoted to the study of metamaterial waveguide and metamaterial scattering (reflecting) structures points that there is a need for development devices possessing unique characteristics, as multifunctionality, reconfigurability, certain frequency bandwidth, ability to operate at high-powers and high-radiation conditions. The importance of diffraction problems for scattering structures is based on their great practical utility for many applications, such as reflector antennas, the analysis of structures in open space, electromagnetic (EM) defence of structures, the scattering modeling for remote sensing purposes, high frequency telecommunications, computer network, invisibility cloaks technology and radar systems (Li et.al., 2011; Zhou et.al.; 2011, Zhu et al., 2010; Mirza et al., 2009; Abdalla & Hu, 2009; Engheta & Ziolkowski, 2005).


Introduction
Large stream of articles devoted to the study of metamaterial waveguide and metamaterial scattering (reflecting) structures points that there is a need for development devices possessing unique characteristics, as multifunctionality, reconfigurability, certain frequency bandwidth, ability to operate at high-powers and high-radiation conditions.The importance of diffraction problems for scattering structures is based on their great practical utility for many applications, such as reflector antennas, the analysis of structures in open space, electromagnetic (EM) defence of structures, the scattering modeling for remote sensing purposes, high frequency telecommunications, computer network, invisibility cloaks technology and radar systems (Li et.al., 2011;Zhou et.al.;2011, Zhu et al., 2010;;Mirza et al., 2009;Abdalla & Hu, 2009;Engheta & Ziolkowski, 2005).
The technological potential of metamaterials for developing novel devices offers a very promising alternative that could potentially overcome the limitations of current technology.The metamaterial waveguide and scattering structures can operate as different devices that possess different specific qualities as well.In order to create a new microwave device it is necessary to know the main electrodynamical characteristics of metamaterial structures on the basis of which the device is supposed to be created.
Here are presented electrical field distributions and dispersion characteristics of open metamaterial waveguides in subsections 1-3 and numerical analysis of the scattered and absorbed microwave powers of the layered metamaterial cylinders in the subsection 4.

Analyses of electrical field and dispersion characteristics of square metamaterial lossy waveguides by the SIE method
Here the open (without conductor screen) square metamaterial waveguides with sizes 5x5•mm 2 and 4x4• mm 2 are investigated by our algorithms that were created on the base of the Singular Integral equations' (SIE) method (Nickelson & Sugurov, 2005;Nickelson et al., InTech2011).Due to the fact that metamaterial is a substance with losses we have determined the complex roots of the dispersion equation by using of the Muller method in our researches (Nickelson et al., InTech2011).
Our computer programmes are written in MATLAB.They let us to investigate open absorptive waveguides with different shapes of cross-section (Gric & Nickelson, 2011).We have used the values of the complex relative permittivity met and the complex relative permeability met of metamaterial from (Penciu et al, 2006) in suctions 1 and 2.

Numerical investigations of square 5x5 mm 2 metamaterial waveguide
The dispersion characteristics and the 3D electric field distributions are presented here .The characteristics are shown in Fig. 2.1 (Gric et al., 2010).The main wave (mode) is denoted with black points; the first higher mode is denoted with circles.The values of met and met are different at every frequency.The real part of the permittivity is always negative at the all frequency range 75-115 GHz.The imaginary part of the permittivity is negative approximately when 90 ≤ f ≤ 100 GHz.The real part of the permeability is negative when 100 ≤ f ≤ 105 GHz.The imaginary part of the permeability is always equal to zero or a positive number at the all mentioned frequency range.In the Fig. 2.1 are presented dependencies of the complex propagation constant h = h'-i h" on the frequency.In Fig. 2.1(a) we see that dependencies of the normalized propagation constant h'/k on the frequency, where h'=2•π/ and is the wavelength of microwave in the metamaterial waveguide, k=2 π f/c, k is the wavenumber in vacuum, f is an operating frequency, c is the speed of light in vacuum.The curves of the main mode and the first higher mode are not smooth.The magnitudes h'/k <1 for both modes (Fig. 2.1(a)).It means that the main and the first higher modes are the fast waveguide waves.In Fig. 2. 1(b) are shown dependencies of the waveguide attenuation constant (losses) h" on the frequency.We see that the values of the main and higher mode losses are commensurate and the losses are not higher in the frequency range.The loss maximums of the main mode and the first higher mode are slightly shifted.The maximum of main mode losses is 0.32 dB/mm at f = 87.5 GHz and the wavelength of this mode is equal to 4.2•mm.The maximum of the first higher mode is 0.36 dB/mm at f = 90 GHz and the wavelength of this mode is equal to 4•mm (Fig. 2.1).It is important to remark that the losses of the main mode are very low at the frequency range from 105 GHz till 115 GHz.This feature could be used in practice for creation of feeder lines and specific devices that require low distortions in the signal transmission.We see that the waveguide losses are large at f = 95 GHz and they are low at f = 110 GHz (Fig. 2.1(b)).We present here the electric field line distribution at these two frequencies in order to compare how the losses influence on the field picture.The calculations of the electric fields in this section were fulfilled at the approximately 10000 points in every crosssection.
The metamaterial is an epsilon-negative media at f = 95 GHz and f = 110 GHz.In Fig. 2.2(a) we see the electric field is very small in the center of the waveguide cross-section.Such distribution can be explained by the large loss at f = 95 GHz and the EM wave does not deeply penetrate into the metamaterial.The refractive index of single negative metamaterial and the transverse propagation constants of waveguide made of the metamaterial are imaginary numbers.For this reason the electric field concentrates near the waveguide border.The tendency of the field to concentrate at the interface of the single negative metamaterial is the important feature of this kind metamaterial.
The behaviour of the EM field components when approaching to the apex of the waveguide cross-section contour (waveguide edge) are an important point of an electrodynamical solution.We would like to note that the condition on the waveguide edge is satisfied in our solutions of considered electrodynamical problems.
Examining the electric field lines near the upper right corner of the square waveguide we would like to note that the electric field lines are directed counter-clockwise on the right side of the corner and the lines are directed clockwise on the left side of the corner.The electric field lines near the left bottom corner of the square waveguide are distributed in similar way.The lines are directed counter-clockwise on the right side of the corner and they are directed clockwise on the left side of the corner.The electric field lines are diverging of the left bottom corner while the lines are converging to the right upper corner.For this reason, we believe that there is an increased density of electric lines in the upper corner and the weaker density in the bottom corner (Fig. 2.2(a)).
The same effect of no uniformity in the electric field distribution on the waveguide perimeter we can see for the circular waveguide (see below in section 2, Figs 2.2,2.3).
Table 2.2.The EM field components of the first higher mode at the point with coordinates x = 4 mm and y = 4 mm when f = 95 GHz and f = 110 GHz In tables 2.1 and 2.2 we demonstrate the values of complex EM field components of the main mode and the first higher mode at two frequencies.On the base of the table data we can say, that the both modes on these frequencies are hybrid modes.
We do not classified the waveguide modes here in the usual way, e.g. the hybrid magnetic HE mn or the hybrid electric EH mn modes because the kind of mode of strong lossy waveguides may change when we change the frequency [Nickelson et al., 2011;Asmontas et al., 2010].In Fig. 2.3(a) we see that the electric field at f = 95 GHz for the first higher modes is also concentrated near the metamaterial borders and the strongest field is at two diagonal corners of the cross-section, i.e. at the right upper corner and the left bottom corner.There is a strong asymmetry of the electric field distribution on the perimeter of waveguide.It happened probably by reason that the real and imaginary parts of permittivity are negative and relatively large at this frequency.
In Figs 2.2(b)-2.3(b) we see that the electric field distributions of the main and first higher modes have a more homogeneous picture in the waveguide cross-section at f = 110 GHz.This happened because the electric field penetrates deeper into the metamaterial at this frequency because the waveguide loss is small at f = 110 GHz (Fig. 2.1(b)).We can also see here some asymmetry of electric field lines on the waveguide cross-section.The projections of the vector electric fields on the waveguide sidewalls are depicted along the waveguide (Figs 2.2-2.3).

Numerical investigations of square 4x4 mm 2 metamaterial waveguide
The dispersion characteristics of the square metamaterial waveguide are presented in Fig. 2.4 (Gric et al., 2010).We used the values of r m and r m from (Penciu et al, 2006).In Fig. 2.4(a) the normalized propagation constant h'/k is shown.In Fig. 2.4(a) we see that the magnitude h' is less than the wave number k in the frequency range 85-108 GHz.It means that at these frequencies the main and the first higher modes are fast waves.The wavelengths of both modes differ slightly.The losses of both propagating modes in the metamaterial square waveguide change in very complicated way when the frequency increases.Mode losses have several maximums and minimums.We see that the losses of the main mode become a very small at the frequency 103.75 GHz.In Fig. 2.5 we see that the electric field is weak inside of square metamaterial waveguide of size 4x4 mm 2 at f=92.5 GHz.The electric field lines concentrate in the two diagonal waveguide corners in other way in the comparison with the waveguide of size 5x5 mm 2 at f=95 GHz.

Investigations of the circular waveguides by the partial area method
Here the open circular metamaterial waveguide is investigated by the partial area method (Nickelson et al., 2008).The presentation of longitudinal components of the electric E z m and magnetic H z m fields that satisfy to the Maxwell's equations in the metamaterial medium is in the form: where A 1 , B 1 are unknown arbitrary amplitudes, J m is the Bessel function of the m−th order, k + ⊥ is the transverse propagation constant of the metamaterial medium, r is the radius of the circular metamaterial waveguide, m is the azimuthal index characterizing azimuthal variations of the field, φ is the azimuthal angle.The presentation the electric field E z a and the magnetic field H z a components that satisfy to Maxwell's equations in air are: where A 2 , B 2 are unknown arbitrary amplitudes, H m is the Hankel function of the m−th order and the second kind, k − ⊥ is the transverse propagation constant of air medium.
As far as the circular waveguide is researched in the cylindrical coordinate system we have to satisfy the boundary conditions for two components of the electric field (E φ , E z ) and the magnetic field (H φ , H z ) on the cylindrical interface metamaterial-air.The condition at infinity also is satisfied.The sign of k − ⊥ changes on the opposite one for the metamaterial hollow-core waveguide, when a hole is surrounded by a metamaterial medium.After substitution of expressions ( 1) and ( 2) in the transverse components expressed in terms of the longitudinal components (Kong, 2008) we obtain the expressions of all transverse EM field components.The result of solution is the dispersion equation in the determinant form.We determine complex roots of the dispersion equation by using of the Muller method.

Investigations of the circular metamaterial waveguide (r=2.5 mm) by the partial area method
We discovered the particularity in the electric field distribution on the cross-section of the open circular metamaterial waveguide at the operating frequency 95 GHz.We find that this waveguide could be used as a narrowband filter at frequencies 102-102.5 GHz.
The circular metamaterial waveguide with r=2.5 mm was researched.The dispersion characteristics and the 3D electric and magnetic field distributions were calculated (see Figs 2.1-2.5).The dispersion characteristics are presented in Fig. 3.1.The main mode is denoted with black points and the first higher mode is denoted with circles.
In Fig. 3.1 (a) we see that the normalized propagation constants of the main and the first higher modes are fairly smooth except only one protrusion at frequencies between 97 GHz and 102 GHz.There is a large peak of the main mode losses at frequency f = 101.25 GHz.At this frequency metamaterial is double negative with r,met = -9.17-i0.83 and r,met = -0.75.We see that losses of the main mode are very small at the frequency ranges 75-100 GHz and 102.5-115GHz.While the losses of the first higher mode are significantly higher at this frequency ranges (Fig. 3.1(b)).Therefore the investigated circular metamaterial waveguide can be used as a filter at the frequencies 100-102.5 GHz and as a one mode lossless waveguide at the frequency ranges 75-100 GHz and 102.5-115GHz.As we can see the first higher mode is a quickly attenuated wave because it has very large losses in all frequency range.The 3D electric field distributions of the main mode were calculated at frequencies 95 GHz and 110 GHz (Fig. 3.2 and 3.3) as well as the first higher mode at the same frequencies (Fig. 3.4 and 3.5).The electric field inside the circular metamaterial waveguide is much smaller than outside of waveguide for this reason we have increased the electric field strength lines inside of the waveguide in order to see them ( Because the metamaterial has more losses at 95 GHz than at 110 GHz, so the electric field inside of waveguide is weaker at 95 GHz (see tables 3.1, 3.2) compare to the inner electric field at 110 GHz.On this reason we increased the electric field strength lines at 95 GHz in the 143 times and at 110 GHz in the 14 time.The calculations of the electric fields were fulfilled at the approximately 10000 points in every cross-section. -1 -i1.1109•10 -5  -3.3354•10 -6 -i3.3354•10 -6  7.1930•10 -5 -i7.1930•10 -5   Table 3.1.The electromagnetic field components of the main mode at the point with coordinates r = 2 mm, φ = 45° when f = 95 and f = 110 GHz   The electrical field inside of the waveguide is very small at all frequencies.However the observable electric field strength lines appear at the waveguide boundary.
We would like to note that the boundary conditions on the border of waveguide are satisfied, i.e. the tangential components of electric and magnetic fields are equal on the interface air-metamaterial.For this reason when the electric field lines has a tangential character outside the waveguide ( We would like to draw attention to the fact that the feature of the irregular distribution of electric field lines in the cross-section of square and circular metamaterial waveguides at 95 GHz is very similar. 3.2 Investigations of the circular metamaterial waveguide (r = 2 mm) by the partial area method

The metamaterial rod waveguide
We have investigated circular metamaterial waveguides by our algorithm that was created using the partial area method (Nickelson et al., 2008).
In Fig. 3.6 the dispersion characteristics of the metamaterial waveguide are presented.In Fig. 3.6 (a) the normalized propagation constant h'/k of the main mode and the first higher mode is shown.The main mode is denoted with black points and the first higher mode is denoted with circles.In Fig. 3.6 (b) losses of the main mode are shown.In Fig. 3.6 (c) losses of the first higher mode are presented.We show the losses of the modes in the different scales because the losses are not commensurate.We see (Fig. 3.6 (a)) that the main mode is a slow mode and the first higher mode can be a slow mode or a fast mode dependent on the frequency range.
There are two frequency ranges 75-97.5 GHz and 100-115 GHz when losses of the main mode are extremely small.On the other side there are frequency ranges, for example f=95 -101.25 GHz and f=107.5-115GHzwhen losses of the first higher mode are large.We have calculated the 3D vector electric field distributions of the main mode propagating in the open circular metamaterial waveguide.The calculation was fulfilled inside and outside the waveguide in 1500 points.The electric field distributions were calculated at frequency f=95 GHz.At this frequency the metamaterial is single-negative.At this frequency r,met = -23.75-i18.75 and r,met = 1.75+ i1.625.
In Fig. 3.7(a) an enlarged picture of the electric field lines inside the metamaterial waveguide is shown.We see that the strongest electric field is in the thin surface layer which is located at the interface metamaterial-air.In Fig. 3.7(b) the electric field lines inside and outside of the metamaterial waveguide are shown.In Fig. 3.7(b) we see that the electric field inside the waveguide is significantly weaker than outside it.We also clearly see that the electric field distributions are periodically repeated in the longitudinal direction.

The metamaterial hollow-core waveguide
The dispersion characteristics of the metamaterial hollow-core waveguide with the radius of the hole (in the metamaterial medium) equal to 2 mm are presented in Fig. 3.8.
In Fig. 3.8 (a) dispersion characteristics of the main and the first higher modes are presented.Both modes are the fast ones.Their electromagnetic energy concentrates in the hollow-core air area.There are the frequency ranges where modes propagate with very small losses.We can see that losses of the main mode in the frequency range 75-90 GHz and 104-115 are very low and they can be large in the frequency range 91.25-103.75GHz.The first high mode' losses change abruptly.The losses can be very low at some frequencies approximately 75-87 GHz, 106 and 111 GHz.
Losses of the first high mode are low and this mode can easily propagate and it can modulate the amplitude of the main mode in devices that were created on the base of the metamaterial waveguide with r=2 mm.There is also good possibility to create a devise on the base of the first high mode in the range f=95-105 GHz.
We have calculated the 3D vector electric field distributions of the main mode propagating in the hollow-core metamaterial waveguide.The electric field distributions were calculated at frequency f = 95 GHz.The calculations of the electric fields were fulfilled at the approximately 10000 points in every cross-section.In Fig. 3.9 we see that the electric field inside the air hole is significantly weaker than outside in the metamaterial media.We also clearly see that the electric field distributions are periodically repeated in the longitudinal direction.As the waveguide losses exist only in the metamaterial and the most part of EM energy propagates into air hole then we see no decrease in the amplitude of the electric field with a change in coordinate z (please, compare with Fig. 3.9.The 3D vector electric field distribution of the main mode of the hollow-core metamaterial waveguide with r=2 mm at frequency 95 GHz.

Conclusions on sections 2&3
1.The open lossy metamaterial waveguides with different shapes of the cross-section were investigated by using of our computer programs that have written in MATLAB language.The computer codes were based on the method of singular integral equations and the partial area method.2. We have calculated the dispersion characteristics (the propagation and attenuation constants) at the frequency range 75-115 GHz as well as the 2D and 3D electromagnetic field distributions.We took the electromagnetic parameters of metamaterial close to practice.Our computer algorithms can be useful working out microwave devices on the base of waveguides made of strong lossy materials.3. We discovered the special feature of the open lossy metamaterial waveguides.
Propagation and attenuation constants depend on the waveguide sizes in an unpredictable complex manner.E.g., the circular metamaterial waveguide with r=2.5 mm (Fig. 3.1) and r=2 mm (Fig. 5. Investigated EM fields of square and circular waveguides have irregular distributions on the waveguide perimeter.Investigated EM field lines for the main and first higher modes of the squire waveguides are concentrated near the metamaterial borders and the strongest field is at two diagonal corners of the cross-section.The electric field lines are diverging of the one corner while the lines are converging at the other corner.

Dispersion characteristic analysis of circular anisotropic metamaterial
waveguide with the effective metamaterial permittivity and permeability near to zero

Introduction
In the past several years many specialist focused on the experimental and theoretical investigations of the zero-refractive index (or zero-index) metamaterials.These metamaterials attracted researches due to their unconventional constitutive parameters and anomalous effects to work out novel electromagnetic devices.Zero-index metamaterial may have the epsilon-near-zero (ENZ) or mu-near-zero (MNZ) properties simultaneously or in turn, one after another at different frequencies.
Zero-index metamaterials are dispersive (electromagnetic parameters dependence on a frequency) media.The constitutive parameters of anisotropic dispersive metamaterials can be described by expressions that involve the plasma frequencies.The metamaterial on frequencies near to plasma resonances is called a plasmonic metamaterial.
Zero-index metamaterials are used in different devices as a transformer to achieve the perfect impedance match between two waveguides with a negligible reflection or to improve the transmission through a waveguide bend as well as for the matching of waveguide structure impedance with the free space impedance (when the metamaterial epsilon and mu are simultaneously very close to zero).Plasmonic metamaterial provides manipulating of the antenna phase fronts and enhancing the antenna radiation directivity.

Analysis and simulation of phase constant dependencies
Here we presented the phase constant (real part of the waveguide longitudinal propagation constant) of propagating modes on the circular anisotropic metamaterial waveguide when the metamaterial permittivity and permeability may take values close to zero at certain frequencies.Further we call a plasmonic waveguide.
The solution of Maxwell's equations for the circular anisotropic metamaterial waveguide was carried out by the partial area method (Nickelson et al., 2009).The computer program for the dispersion characteristic calculations has created in MATLAB language.Computer program allows take into account a very large material attenuation (Nickelson et al., 2011;Asmontas et al., 2010).In this section constitutive parameters of the uniaxial electrically and magnetically anisotropic metamaterial were taken from the article (Liu et al., 2007).In the mentioned article was considered an anisotropic dispersive lossless metamaterial slab.For this reason there were given only the real parts of the permittivity r,ij = ( xx , xx , zz) and permeability r, ij =( xx , xx , zz ) tensor components.
The tensor components of the relative permittivity and the relative permeability are described by following formulae (Liu et al., 2007): here =2πf is the angular frequency of microwaves, f is the operating frequency.The electric plasma frequencies of metamaterial are epxx = 2πf epxx , f epxx =(12) 1/2 GHz, epzz =2πf epzz , f epzz =2.5 GHz.The magnetic plasma frequencies of metamaterial are mpxx = 2πf mpxx GHz, f mpxx = (6) 1/2 GHz and mpzz =2πf mpzz GHz, f mpzz =2 GHz.The values of angular frequencies are taken from (Liu et al., 2007).The magnitudes of tensor components xx , zz , xx , zz are real numbers.In We see that the permittivity components xx and zz have negative values from 1.5 to ~3.5 GHz and from 1.5 to ~2.5 GHz, respectively.The permeability components xx and zz have negative values from 1.5 to~2.5 GHz and from 1.5 to ~2 GHz, respectively.We realize that all tensor components are negative at the frequency range from 1.5 GHz to ~2 GHz.Absolute values of tensor components are less than 1 at the frequency range from ~2.5 GHz to 4 GHz.
The values of tensor components become equal to zero at the operating frequency f equal to the metamaterial electric f epxx =3.46 GHz, f epzz = 2.5 GHz or magnetic f mpxx =2.45 GHz, f mpzz =2 GHz plasma frequencies.
In Figs.5.2-5.7 are shown dispersion characteristics (phase constants) of open circular waveguide made of the uniaxial electrically and magnetically anisotropic metamaterial.The calculations are performed for the left-handed (extraordinary) circularly polarized microwaves when exp(+imφ), m=0, 1, 2,…is the wave (mode) azimuthal periodicity index, φ is the azimuthal coordinate.We investigated only modes with the index m=1, because it is known the main mode of open circular dielectric waveguide specify by m=1 and the main mode of a dielectric waveguide is a hybrid mode HE 11 (Nickelson et al., 2009).
Here is presented the phase constant h' (the real part of longitudinal propagation constant) dependencies of plasmonic metamaterial waveguides with radii r equal to 0.1 mm, 1 mm, 3 mm, 5 mm, 7 mm, 10 mm and 15 mm.The phase constant h' is equal to 2π/ w , where w is the wavelength of certain mode.Our aim is to investigate how an increase in the plasmonic waveguide radius affects on the eigenmode numbers, mode cutoff frequencies and a shape of dispersion characteristics.
The analysis of Figs 5.2-5.5 shows that there are three main frequency areas where localize dispersion curves.
A shape all dispersion characteristics Figs 5.2-5.5 are unusual in the comparison with traditional dispersion characteristics of open cylindrical waveguides made of dielectrics, semiconductors or magnetoactive semiconductor plasma (Nickelson et al., 2011;Asmontas et al., 2009;Nickelson et al., 2009).Because the dispersion characteristic branches of analyzed plasmonic waveguides are vertical.We see that there is a single mode with the cutoff frequency close to f=1.5 GHz.The cutoff frequency of this mode shifted in the direction of lower frequencies with increasing of the waveguide radius.This first single mode is special one because the mode does not match any of plasma f epxx , f epzz , f mpxx , f mpzz frequencies.We can observe how a shape of the dispersion characteristic changes in the vicinity of the cutoff frequency.
We would like to draw your attention to the fact that the anisotropic metamaterial is described by the negative tensor components xx , zz , xx , zz in the frequencies less than 2 GHz (see formulae 5.1 and 5.2).It is mean that the first mode propagates in the waveguide when the metamaterial is double negative (DN).This wave is particularly important because small changes in frequency produce large changes in phase.
We can watch a package of dispersion branches closed to cutoff frequency 2.5 GHz.We see that the left lateral dispersion branch of the package is a special eigenmode, i.e. this one is separated by a larger distance from other eigenmodes.The vertical branch of the left lateral mode is located at the magnetic plasma f mpzz frequency equal to 2 GHz.We can distinguish also the right lateral dispersion branch of the package.The mode with this dispersion characteristic is also more specific one.i.e. this mode is separated by a larger distance from other modes.The vertical branch of this mode is located about 2.7 GHz and shifted on the higher frequency side with increasing of a radius.A dense bunch of dispersion curves located between the extreme left and right curves that were previously described.The number of curves increases rapidly at increasing of waveguide radius.It is interesting to note that all dispersion branches of the dense bunch are within the frequency band of 2-2.5 GHz.Apparently the dense bunch of dispersion characteristics related to plasma f mpzz and f epzz frequencies.The cutoff frequencies of all dispersion characteristics of the dense bunch are the same and equal to f~2.46 GHz.The dispersion curves fan out from a point with a value equal to f mpxx .
Second dense bunch of dispersion curves is at the electric plasma frequency f epxx ~3.46 GHz.The number of curves increases rapidly at increasing of waveguide radius.All dispersion characteristics are within the frequency band of 2.5 GHz and 3.46 GHz.The dispersion characteristics on the right side of the bunch are more vertical.The greatest number of modes can be excited at the electric plasma frequency f epxx ~3.46 GHz in the comparison with other plasma frequencies.The cutoff frequencies of dispersion characteristics of this dense bunch are the same and equal to f~3.46 GHz.We did not find the plasmonic metamaterial waveguide eigenmodes in the frequency range from 3.5 GHz till 2000 GHz.
We expanded the searching of eigenmodes on frequencies below f=1 GHz for the plasmonic waveguide with r= 15 mm.In Fig. 5.5 (b) is shown eigenmodes dispersion characteristics of plasmonic metamaterial waveguide in the frequency range from 5 MHz till 3.5 GHz.We present here more detailed calculations of the dispersion characteristics of the plasmonic waveguide eigenmodes with the radius equal to 15 mm (Figs.5.6 and 5.7).Here are presented the new dispersion characteristic branches of waveguide eigenmodes in the band of frequency from 5 MHz till 600 MHz (Fig. 5.6 (a)).We see that the dispersion curves have the clear expressed cutoff frequencies and they have an opposite slope in the comparison with dispersion curves of open ordinary waveguides (Nickelson et al., 2011;Asmontas et al., 2010).
It should be stressed that these very low frequency dispersion characteristics have obtained by solving of Maxwell's equations with certain boundary conditions.These low frequency modes are also the metamaterial waveguide eigenmodes.
In the Fig. 5.6 (b) is shown the dispersion characteristic of a singular mode on a larger scale.We see that this mode from 1.35 till ~1.45 GHz is a static mode, because no dependence on the frequency.We can observe the anomaly dispersion closed to f~ 1.45 GHz and the very strong dispersion in the frequency band over ~1.47 GHz.We can see the dispersion curves in the area of cutoff frequencies on a larger scale 1.9-2.5 GHz in Figs 5.7(a) and (b).We can note the anomalous dispersion hook of eigenmodes in the band f=1.9-2GHz.

Microwave scattering and absorption by layered metamaterial-glass cylinders
In this section we are going to give our calculation results for a single-layered cylinder and a twelve layered cylinder.The single-layered cylinder consists of a metamaterial core which is coated with an acrylic-glass layer.The twelve-layered cylinder consists of a conductor core which is covered with 12 metamaterial and acrylic-glass alternately layers.The acrylic-glass material is an external layer of each cylinder.The calculation of the scattered (reflected) and absorbed powers are based on the rigorous solution of scattering boundary problem (Nickelson & Bucinskas, 2011).The solution of mentioned electrodynamical problems and expressions of absorbed and scattered powers are given in article (Bucinskas et al., 2010).
The number and thickness of layers is not limited in the presented algorithm.The central core of multilayered cylinder can be made of different isotropic materials as a metamaterial, a ceramic matter or a semiconductor as well as of a perfect conductor.The isotropic coated layers can be of strongly lossy (absorbed) materials.
The signs of the complex permittivity and the complex permeability can be negative or positive in different combinations.
Here are presented the scattered and absorbed power of layered cylinder dependent on the hypothetic metamaterial permittivity and permeability signs and losses.We used for calculations our computer programs which are written in FORTRAN language.
The extern radius of both (single-and twelve-layered) cylinders is the same and equal to 2 mm.We show here our results only in the frequency range from 1 till 120 GHz.We present dependencies of the scattered and absorbed powers by the cylinders at the incident perpendicularly (the angle = 0 o , Fig.We admitted that acrylic-glass material is a non-dispersive and weakly lossy one with the complex permittivity ε g = g '-i g " = │ε g │exp(-i g ) = 3.8i 0.0005, i.e. the phase of the complex glass permittivity is g =arctan(ε g '/ε g ") =1.3•10 -4 [rad] and the glass permeability is equal to g =1.
The complex metamaterial permittivity is ε met = ' met -i " met = │ε met │exp(-i met ).The complex permeability is met = ' met -i " met = │ε met │exp(-i met ).The metamaterial met and met are the same for the single-layer and twelve-layer cylinders.The module of the metamaterial permittivity is | met |=20 and the metamaterial permeability is | met |=2.The phase of metamaterial permittivity is ,met =0.7854 [rad] and the phase of permeability is ,met =0.6981 [rad] for the single-layer and twelve layer cylinders.
We present here the absorbed power by the acrylic-glass layer (Fig. 7.3) and the absorbed power by the metamaterial (Fig. 7.4) for the single-layer cylinder.The total absorbed power by single-layered cylinder is equal to the sum of absorbed powers by metamaterial and acrylic-glass.And we show here the total absorbed power by all layers of the twelve-layered cylinder.
In Figs 7.2-7.6 are presented the averaged scattered or absorbed power values per oscillation period for the unit length of the metamaterial-glass cylinder.
The absorbed and scattered power calculations were fulfilled using by formula (28) in (Bucinskas et al., 2010)].The integral of the formula has a positive sign when we calculate the scattered power.And this integral has a negative sign when we calculate the absorbed power.For this reason the scattered power has a positive value and the absorbed power has a negative one (see ).
We presented here the dependencies of scattered and absorbed powers of the single-layered cylinder on signs of ' met , " met , ' met and " met , i.e. when metamaterial is double positive (DP), εsingle negative (SN) orsingle negative (SN), double negative (DN).

Numerical analysis of the scattered and absorbed microwave power of the singlelayer cylinder
In this subsection we investigate the scattered and absorbed powers of the incident microwave by the cylinder consists a core of metamaterial which is covered with a singlelayer of acrylic-glass.The external cylinder radius is R 1 =2 mm and the cylinder core has radius R 2 =1.8 mm, so the thickness of the glass layer is 0.2 mm.Nowadays there is a huge interest to the composite materials with untraditional values of the complex permittivity ε met and the complex permeability μ met .We analyzed here the scattered and absorbed powers for four versions of hypothetic metamaterial parameter signs.The complex metamaterial permittivity ε met =s 1 │ε met │exp(-s 2 i met, ) was taken for two combinations of signs, when s 1 =s 2 =±1.It is known that each metamaterial is intended for use in a specific frequency range and has a specific value of the effective permittivity and permeability at the certain frequency.For this reason we took the absolute values of real and imaginary parts of permittivity ε met and permeability μ met constant at all frequencies in our calculations.And the major impact makes the sizes' relation of wavelength and cylinder layers.
In figures 7.2-7.4 of this section are analyzed how the signs of the complex metamaterial permittivity and permeability influence on the scattered and absorbed powers when the plane perpendicularly or parallel polarized microwave impinges on the metamaterial-glass cylinder (Fig. 7.1).
The scattered and absorbed dependences of the metamaterial-glass cylinder when the incident microwave has the perpendicular or parallel polarization are shown in Figs 7.2-7.4.In Fig. 7.2 is presented the dependence of total scattered power W S on the microwave frequency f at two polarizations.We see that the character of curves for all metamaterial sign versions (curves 1-4) is the same at the perpendicular polarization.A comparison of curves 1-4 (Fig. 7.2(a)) shows that only curve 3 that describes by the metamaterial permittivity and permeability signs s 1 =s 2 =-1, s 3 =s 4 =+1 is the most different in comparison with other three cases.At the beginning the scattered power grows till the maximum value after that decreases till the minimum and later increases again with increasing of frequency.The total scattered power maximums of all curves are in the frequency range about 44-53 GHz.Curves 1 and 4 practically coincide with each other.The lowest scattered power is for the curve 3 at the frequencies about 1-35 GHz and the total scattered power minimum exists for the curve 2 approximately at the frequency 80 GHz.GHz and the second position of the maxima is in the interval 85-100 GHz.The largest scattering is at the lower frequencies.The maximum scattering is higher for the incident parallel polarized microwave in comparison with the incident perpendicularly polarized one.While the total scattered power curves for the incident perpendicularly polarized microwave (Fig. 7.2(a)) have only one maximum.
In Fig. 7.3 is presented the absorbed microwave power W a,glass by the coated glass layer at two microwave polarizations.We see that the behavior of the microwave power absorption strongly depends on the signs s 1 , s 2 , s 3 , s 4 of the permittivity and the permeability.The absorption is especially different at the higher frequencies.We see that the absorption power of the glass layer is larger at higher frequencies.
In Fig. 7.3(b) is presented the absorbed power by the coated glass layer for the incident parallel polarized microwave.There are some small distortion "hooks" of the absorbed power at the low frequencies.The absorbed power by glass layer increases with increasing of frequencies (curves 1, 3, 4) for frequencies that are larger than 20 GHz.The absorbed power is approximately constant when the SN metamaterial core permittivity and permeability have signs s 1 =s 2 =+1, s 3 =s 4 =-1.In Fig. 7.4(a) is given the absorbed power by the metamaterial core of the cylinder for the perpendicular microwave polarization.We see that the W a,met magnitudes have the pronounced wave-like nature dependent on the frequency.The metamaterial core absorption has the largest absolute value when the metamaterial permittivity has some negative values and the permeability has some positive values (curve 3).In Fig. 7.4(b) is given the metamaterial core absorbed power W a,met for the incident parallel polarized microwave.We see that the absorbed powers have maximum values at about 5 GHz and their values vary slightly after 20 GHz with increasing of frequency.The absorption by the metamaterial is the largest at the low frequencies (curves 1 and 2) when the metamaterial has the positive permittivity.The comparison of absorbed powers in figures 7. 3 and 7.4 shows that dependencies are absolutely different.

Numerical analysis of the scattered and absorbed microwave power of twelvelayer cylinder
In this subsection we present the total scattered power and the total absorbed power of incident microwave by the cylinder that consists of conductor core covered with 12 metamaterial and glass alternately layers.Designations in Figs 7.5 and 7.6 correspond: the curve 1 is for a DP material when s 1 =s 2 =s 3 = s 4 =+1 (line with black squares) and the curve 2 is for a DN material when s 1 =s 2 =s 3 = s 4 =-1 (line with empty triangulars).
In Figs 7.5 and 7.6 the permittivities and permeabilities metamaterial and acrylic-glass are the same as in the previous subsection.We see that the total scattered and absorbed powers strongly dependent on the polarization of incident wave.The characteristics of perpendicular polarized wave (  Comparing curves for single-and twelve-layered cylinders (Fig. 7.4(a), curve 4 and Fig. 7.6 (a), curve 2) when the metamaterial is SN we can note that their lineaments are different at the low frequencies.
The dependencies of absorbed parallel polarized microwave power for single-layered and twelve-layered cylinders in general are alike.The first minimum of absorbed power is shifted from ~5 GHz (Fig. 7.4(b)) till 10 GHz (Fig. 7.6 (b)) i.e. with growing of the number of layers the minimum shifted to the side of higher frequencies.The absorbed power dependency of twelve layered cylinder has a wavy behavior in the frequency range from 20 GHz till 120 GHz in a comparison with the single-layered one.

Fig. 2 .
Fig. 2.4.The dispersion characteristics of the square 4x4 mm 2 metamaterial waveguide (a) -the dependence of the normalized propagation constant, (b) -the dependence of the attenuation constant on frequencies.

Fig. 2 .
Fig. 2.5.The 3D vector electric field distribution of the main mode propagating in the open square 4x4 mm 2 metamaterial waveguide at f=92.5 GHz.The 3D electric field distribution of the main mode at the frequency 92.5 GHz is depicted in Fig. 2.5.The permittivity of the metamaterial at f=92.5 GHz is -35-i2.5 and the permeability is 2.25+i0.25.The metamaterial is a single negative matter.The calculations of the electric fields were fulfilled at the approximately 160 points in every cross-section (Fig 2.5).

Fig. 3 . 1 .
Fig. 3.1.The dispersion characteristics of the circular metamaterial waveguide with r=2.5 mm (a) -the dependence of the normalized propagation constant, (b) -the dependence of the attenuation constant on frequencies.

Fig. 3 . 2 .
Fig. 3.2.The 3D electric field distributions of the main mode of circular metamaterial waveguide with r=2.5 mm at f = 95 GHz.(a) -the electric field strength lines outside the waveguide (b) -the 143 times were increased electric field strength lines inside the waveguide.In In Figs 3.2-3.5 we see that the electric field is irregular on the waveguide perimeter of the cross-section while the cross-section of the waveguide is a circle.We see that and the most part of the electric field localizes on the border and outside of the waveguide.The electric field is strongest outside the waveguide when φ is 0 or π radians.The electric field has the minimum values and the electric field lines are directed clockwise or counter-clockwise to the right and left of the points with φ equal to π/2 or 3π/2 radians (Figs 3.2(a)-3.5(a)).We see that at the points when the electric field outside of the metamaterial waveguide has the maximum value the field inside of the waveguide is minimal.In Figs 3.2(b)-3.5(b)we see that the maximum electric field inside the metamaterial waveguide is when φ is equal to π/2 or 3π/2 radians.The length of the circular waveguide in z-direction (the Figs 3.2-3.5) is three times longer than wavelength of microwave in the waveguide in our calculations.
Fig. 3.3.The 3D electric field distributions of the main of circular metamaterial waveguide with r=2.5 mm at f = 110 GHz.(a) -the electric field strength lines outside the waveguide (b) -the 14 times were increased electric field strength lines inside the waveguide Fig. 3.5.The 3D electric field distributions of the first higher mode of circular metamaterial waveguide with r=2.5 mm at f = 110 GHz (a) -the electric field strength lines inside the waveguide (b) -the 14 times were increased electric field strength lines outside the waveguide.Comparing dispersion characteristics of square and circular waveguides (Figs 2.1 and 3.1) we see that they are different due to the boundary conditions, which have a strong influence on the dispersion characteristics in our frequency range.As an additional example, the dispersion characteristic (2πf• SQRT( r,met r,met ) of plane EM wave propagating in the same metamaterial only when the one has the infinite dimensions is strongly different in comparison with the waveguide dispersion characteristics.The relatively small losses of the EM wave in the infinite metamaterial are only at the frequencies between 102.5 GHz and 105 GHz.

Fig. 3 .
Fig. 3.6.The dispersion characteristics of the circular metamaterial waveguide with r=2 mm (a) -the dependence of the normalized propagation constant, (b), (c) -the dependence of the attenuation constant on frequencies.

Fig. 3 .
Fig. 3.7.The 3D vector electric field distributions of the main mode of circular metamaterial waveguide with r=2mm (a) -inside; (b) -inside and outside it.

Fig. 3 .
Fig. 3.8.The dispersion characteristics of the hollow-core metamaterial waveguide with r=2 mm (a) -the dependence of the normalized propagation constant, (b)& (c) -the dependence of the attenuation constant on frequencies.
3.6).4. The microwave signals propagating on the open metamaterial waveguides (r=2.5 mm) are absorbed at the narrow 2.5 GHz frequency range.This waveguide can be used as a band-stop filter at f 1 =100-102.5GHz when the losses in the passing frequencies 75-100, 102.5-115GHz are 100 times less than losses at f 1 .

1 .
Fig. 5.1.Dependences of the relative (a) permittivity and (b) permeability tensor components of the metamaterial on the frequency.

2 .
Fig. 7.2.Scattered power of metamaterial-glass cylinder on the frequency of incident (a)perpendicular and (b) -parallel polarized microwaves.The total scattered power of the incident perpendicularly and parallel polarized microwaves (Fig.7.2(a,b)) are different.The scattered microwave power curves for the incident parallel polarized microwave have two maximums in the frequency range 1-120 GHz.The first position of the scattered power maxima are in the narrow frequency interval about 10-15 GHz and the second position of the maxima is in the interval 85-100 GHz.The largest scattering is at the lower frequencies.The maximum scattering is higher for the incident parallel polarized microwave in comparison with the incident perpendicularly polarized one.While the total scattered power curves for the incident perpendicularly polarized microwave (Fig.7.2(a)) have only one maximum.

3 .
Fig. 7.3.Absorbed power by the acrylic-glass layer on the frequency of incident (a)perpendicular and (b) -parallel polarized microwaves.
Figs 7.5(a) and 7.6(a)) and the parallel polarized wave (Figs 7.5(b) and 7.6(b)) are completely different.Particularly noticeable the correlation between the total scattered and absorbed powers for the parallel polarized wave of the twelvelayered cylinder.We see that extremums of the total scattered and absorbed powers coincide in the frequency scale.For example, when the scattered power of cylinder has a maximum (Fig.7.5(b)) then the absorbed power has a minimum (Fig.7.6(b)) at f~10 GHz.
Fig. 7.5.Total scattered power of twelve layer metamaterial-glass cylinder on the frequency of incident (a) -perpendicular and (b) -parallel polarized microwaves

Table 2 .
1.The EM field components of the main mode at the point with coordinates x = 4 mm and y = 4 mm when f=95 GHz and f=110 GHz f = 95 GHz, square waveguide 5x5 mm 2

Table 3 .
2. The EM components of the first higher mode with coordinates at the point with coordinates r = 2 mm , φ = 45° when f=95 and f=110 GHz