Design and analysis of an isotropic two-dimensional planar Composite Right / Left-Handed waveguide structure

A two-dimensional isotropic Composite Right/Left-Handed (CRLH) waveguide structure is proposed which is designed for operation in X-band. The balanced structure possesses left-handed behaviour over a large bandwidth from 7.5 GHz up to its transition frequency at 10 GHz. Above this region, the unit cell behaves in a right-handed manner up to 13.5 GHz. Operating the structure within these bands yields a frequency dependent index of refraction ranging from −2.5 ≤ n ≤ 0.8. Isotropic characteristics are obtained between 8 .5GHz≤ f ≤ 12GHz resulting in−1.5 ≤ n ≤ 0.8. The planar CRLH structure is designed based on transmission line theory, implemented in microstrip technology and optimized using full-wave simulation software. An equivalent circuit model is determined describing the electromagnetic behaviour of the structure whose element values are obtained by even and odd mode analysis. The design of the unit cell requires an appropriate de-embedding process in order to enable an analysis in terms of dispersion characteristics and Bloch impedance, which are performed both.


Introduction
A structure possessing at least one frequency band in which the vectors E, H and k g form a left-handed triad while a right-handed one is found in the remaining frequency range is known as a Composite Right/Left-Handed structure.Operation of the guiding structure in the left-handed region results in wave propagation with negative phase velocity.In contrast, the group velocity associated with the flow of energy is positive in all regions.Since materials possessing such a behaviour have not been found in nature yet, they have to be produced artificially.One of the first structures Correspondence to: M. Eberspächer (mark.eberspaecher@tum.de)exhibiting CRLH behaviour was realised by split-ring resonators located in the proximity of a resonant wire arrangement (Shelby et al., 2001).Due to the loose coupling of the resonators, this configuration suffers from high losses and only a small operative bandwidth of approximately 10% is achieved.It was shown that these problems can be overcome by synthesizing CRLH structures utilizing the transmission line approach (Lay et al., 2004) which is based on periodically arranged unit cells emulating a homogeneous structure.If mutual coupling between the unit cells is not considered, the waveguiding characteristics of the entire cascade is solely governed by the characteristics of the constitutive unit cells.As a consequence, the most challenging part in the artificial material synthesis is the design of appropriate unit cells which fulfil predefined requirements.Besides meeting the electrical characteristics, the structures must be feasible with an acceptable complexity.It has been shown that cascading planar CRLH unit cells in a specific manner enables alternative concepts for a multitude of applications like power dividers, beam forming concepts and antennas (Eccleston, 2010), (Caloz and Itoh, 2006).

Synthesis
In order to realise a two-dimensional waveguiding structure which behaves within a certain bandwidth isotropically, the unit cell is supposed to be a fully symmetric and reciprocal four-port with a pair of ports for each direction of space.Using scattering matrix representation, this can be written as Published by Copernicus Publications on behalf of the URSI Landesausschuss in der Bundesrepublik Deutschland e.V. Fig. 1.Picture of the fabricated structure which is produced on Rogers RO4350B substrate (thickness h = 0.762 mm and r = 3.66).The 2d CRLH unit cell is within the square marked by the dashed line (side length l = 4.54 mm ≈ λ/4).Parts located outside the square belong to either adjacent cells or the feeding network.
Fig. 2. Equivalent circuit model of the structure seen in figure 1 with the series elements C L , L R /2 and R S /2 and the shunt elements C R /2, 2L L and 2R P .The equivalent circuit model of the 2d CRLH unit cell is within the region marked by the dashed line.Elements located outside the marked region belong to either adjacent cells or the feeding network.
based on the simulated or measured network response which can be done by performing even and odd analysis.
Starting with the impedance matrix representation of the four-port network referred to as extended unit cell Zeuc , which comprises the unit cell and the feeding network, al-   based on the simulated or measured network response which can be done by performing even and odd analysis.
Starting with the impedance matrix representation of the four-port network referred to as extended unit cell Zeuc , which comprises the unit cell and the feeding network, al-Fig.2. Equivalent circuit model of the structure seen in figure 1 with the series elements C L , L R /2 and R S /2 and the shunt elements C R /2, 2L L and 2R P .The equivalent circuit model of the 2d CRLH unit cell is within the region marked by the dashed line.Elements located outside the marked region belong to either adjacent cells or the feeding network.
whereas the phase variation experienced by a wave travelling through the cell is described by the exponential function e −jφ .For the synthesis process the impedance matrix representation of S is of further interest, which can be determined by that at the resonant frequency of the shunt resonator the impedance of Z even is expected to peak since the susceptances of the resonator cancel each other out.Consequently, Fig. 3. Equivalent circuit model of the impedance Z even obtained by even mode excitation of the unit cell with the feeding network.4. Equivalent circuit model of the admittance Y odd obtained by odd mode excitation of the unit cell with the feeding network.
the product of L L and C R is immediately known.Due to the fact that the unit cell realises a band pass structure with only weak losses, the parallel resistance R P must be highly resistive whereas its series counterpart R S must be of low resistance.Hence, we can conclude that R P >> R S and tuning the element values manually by performing the steps as follows: Starting with the observation of Z even shows that at the resonant frequency of the shunt resonator the impedance of Z even is expected to peak since the susceptances of the resonator cancel each other out.Consequently, Fig. 3. Equivalent circuit model of the impedance Z even obtained by even mode excitation of the unit cell with the feeding network.the product of L L and C R is immediately known.Due to the fact that the unit cell realises a band pass structure with only weak losses, the parallel resistance R P must be highly resistive whereas its series counterpart R S must be of low resistance.Hence, we can conclude that R P >> R S and  with I being the identity matrix and Z 0 the reference impedance.Since the four-port network which is to be realised is assumed lossless its impedance matrix is purely imaginary.Thus, taking the imaginary part of Z S results in Now, the goal is given by finding a network comprising lumped elements which emulates Z S , at least within a certain bandwidth.In the simplest case this could be achieved by extending a T-shaped network with the series impedance Z and the shunt admittance Y to a four-port structure.The resulting network is given by with 1 being a matrix with all elements equal to 1. Consequently, the elements of Z T may be determined and yield and Since we want the synthesized structure to appear as a homogenous one, only small phase angles φ are considered.
Adv n.Due to ucture with st be highly t be of low > R S and approximation of the element values can require some further iterations.Figure 5 shows the comparison of the even impedance and the odd admittance obtained by EM simulations and circuit simulations after the element values have been determined correctly.The determined elements can be Y/Ω −1 f GHz Fig. 5. Comparison between even impedance and odd admittance obtained from EM simulation (solid lines) and circuit simulation (triangles), respectively.Imaginary parts are displayed in red colour, real parts are drawn in blue colour.considered either as the feeding network, in case of analysing a single section, or as a part of adjacent cells in case of a periodically arranged cascade.Thus, by finding these element values the equivalent circuit model of the CRLH unit cell is fully determined.

Results
A well balanced unit cell is achieved after optimizing the structure supported by EM full-wave simulation using CST Microwave Studio (CST) combined with the procedure in- itself, the isotropic behaviour and the power balance must be conserved anyway.As seen, the result is very close to the theoretical expectations of an isotropic structure which requires magnitudes of -6 dB, identical phase shifts of all transmission paths as well as a phase difference of 180 • between transmitted and reflected waves.The waveguiding properties of the de-embedded unit cell, possessing a Bloch impedance Z Blochx ≈ 60 Ω, are shown in figures 7, 8 and 9.In regions where an isotropic behaviour is exhibited, the wave vector must have an absolute value which is independent of the direction.In terms of figure 9 this is represented by cone shaped characteristics or relating to the isofrequency plots this results in circles around the point β x p = β y p = 0 which can be observed for 8.5 GHz ≤ f ≤ 12 GHz.It should be mentioned that the gap in figure 9 is an artefact of the visualization and does not represent a stop band as validated by the dispersion diagram depicted in figure 10.In the region Γ − X and M − X the index of refraction is determined and presented in figure 11 which also reveals the isotropy of the unit cell.A prototype, seen in figure 1, was fabricated and measured.The results of the measurements are presented in figure 12 and the corresponding dispersion diagram in figure 13.As seen from the increasing attenuation 7 .9 9 6 7 .9 9 6 7 .9 9 6 7 .9 9 6 8.5 8 .5 9 .00 4 9 .49 6 Fig. 7. Isofrequency contour plot (in GHz) of the left-handed mode with respect to two-dimensional wave propagation in the x and y directions.

Conclusions
A two-dimensional CLRH unit cell was presented which possesses an isotropic behaviour between 8.5 GHz ≤ f ≤ 12 GHz.Within this frequency region a frequency dependent of refraction ranging from −1.5 ≤ 0.8 is achieved.A systematic procedure based on even and odd mode analysis was presented which allows to determine the element values of the equivalent circuit model of the proposed structure as well as its feeding network.Hence, a first order Taylor series expansion in φ may be performed around φ = 0 resulting in and Composite Right/Left-Handed behaviour is achieved if at least one frequency band with ∂ ω φ(ω) < 0 and another with ∂ ω φ(ω) > 0 exists.This can be accomplished by choosing appropriate network elements for Y and Z, respectively.For example, a parallel resonant circuit is inserted for Y and a series resonant circuit for Z. isotropic behaviour and the power balance must be d anyway.As seen, the result is very close to the thexpectations of an isotropic structure which requires es of -6 dB, identical phase shifts of all transmiss as well as a phase difference of 180 • between ed and reflected waves.The waveguiding properties -embedded unit cell, possessing a Bloch impedance ≈ 60 Ω, are shown in figures 7, 8 and 9.In rehere an isotropic behaviour is exhibited, the wave ust have an absolute value which is independent of tion.In terms of figure 9 this is represented by cone haracteristics or relating to the isofrequency plots lts in circles around the point β x p = β y p = 0 which bserved for 8.5 GHz ≤ f ≤ 12 GHz.It should be d that the gap in figure 9 is an artefact of the visuand does not represent a stop band as validated by rsion diagram depicted in figure 10.In the region and M − X the index of refraction is determined ented in figure 11 which also reveals the isotropy nit cell.A prototype, seen in figure 1, was fabnd measured.The results of the measurements are d in figure 12 and the corresponding dispersion dia-7 .9 9 6 7 .9 9 6 7 .9 9 6 7 .9 9 6 8.5 8 .5 9 .00 4 9 .4 9 6 Fig. 7. Isofrequency contour plot (in GHz) of the left-handed mode with respect to two-dimensional wave propagation in the x and y directions.

Conclusions
A two-dimensional CLRH unit cell was presented which possesses an isotropic behaviour between 8.5 GHz ≤ f ≤ 12 GHz.Within this frequency region a frequency dependent index of refraction ranging from −1.5 ≤ 0.8 is achieved.A systematic procedure based on even and odd mode analysis was presented which allows to determine the element values of the equivalent circuit model of the proposed structure as well as its feeding network.isotropic behaviour and the power balance must be d anyway.As seen, the result is very close to the thexpectations of an isotropic structure which requires es of -6 dB, identical phase shifts of all transmiss as well as a phase difference of 180 • between ed and reflected waves.The waveguiding properties -embedded unit cell, possessing a Bloch impedance ≈ 60 Ω, are shown in figures 7, 8 and 9.In reere an isotropic behaviour is exhibited, the wave ust have an absolute value which is independent of tion.In terms of figure 9 this is represented by cone haracteristics or relating to the isofrequency plots ts in circles around the point β x p = β y p = 0 which bserved for 8.5 GHz ≤ f ≤ 12 GHz.It should be d that the gap in figure 9 is an artefact of the visuand does not represent a stop band as validated by rsion diagram depicted in figure 10.In the region nd M − X the index of refraction is determined ented in figure 11 which also reveals the isotropy it cell.A prototype, seen in figure 1, was fabnd measured.The results of the measurements are in figure 12 and the corresponding dispersion dia-7 .9 9 6 7 .9 9 6 7 .9 9 6 7 .9 9 6 8.5 8 .5 9 .00 4 9 .4 9 6 Fig. 7. Isofrequency contour plot (in GHz) of the left-handed mode with respect to two-dimensional wave propagation in the x and y directions.

Conclusions
A two-dimensional CLRH unit cell was presented which possesses an isotropic behaviour between 8.5 GHz ≤ f ≤ 12 GHz.Within this frequency region a frequency dependent index of refraction ranging from −1.5 ≤ 0.8 is achieved.A systematic procedure based on even and odd mode analysis was presented which allows to determine the element values of the equivalent circuit model of the proposed structure as well as its feeding network.

Dispersion characteristics
As shown Eqs.
(3) and (4) the impedance matrix of the ideal unit cell comprises only two different types of elements.
These are on the one hand the input impedances Z I = Z nn for n ∈ 1,2,3,4 and on the other hand all the remaining entries denoted by Z T .However, we want to take possible anisotropic behaviour into account for the calculation of the dispersion characteristics.Thus, a third element is introduced which allows to distinguish between the impedance          of two ports which are arranged oppositely and perpendicularly, respectively.Consequently, the impedance matrix for such a four-port network reads It is assumed here that the two ports 1, 2 and 3, 4 are arranged along one Cartesian coordinate axis.Performing the Floquet ansatz based on Z r yields the dispersion characteristics For wave propagation along a principle axis, i.e. the region −X in the dispersion diagram, this expression simplifies to cosh(γ x p) = cosh(γ y p) = Z I /Z T . (11) In this case all nodal points of the periodic arrangement being on a line perpendicular to the propagation direction are        equipotential points.Thus, no current is flowing across nodal points in transverse direction and the four-port matrix can be reduced to a two-port impedance matrix given by the upper left submatrix of Z r comprising the elements Z I and Z T .Consequently, the Bloch impedance reads 3 Design The goal is to realise an isotropic two-dimensional CRLH unit cell which exhibits a left-handed mode for f < 10GHz and a right-handed mode for f > 10GHz fulfilling Eq. (3) around the transition frequency of f T = 10GHz.Due to the given operation frequencies the use of discrete elements, for example surface mounted devices, may lead to difficulties.Hence, the design of the 2d unit cell is based on the 1d structure presented in (Eberspächer et al., 2009)    have inductively.Thus, this configuration realises a shunt inductor and a series capacitor simultaneously.Both elements are completed by the occurring parasitics to a series resonant circuit and a parallel resonant circuit, respectively.Adjusting the resonant frequencies as well as the Bloch impedance can be achieved easily by modifying either the stublength or the fingers of the arrangement allowing to meet given specifications and balancing the structure.Ensuring the artificial structure to behave homogeneously, the length of the unit cell must not exceed the quarter wave length limit (Lay et al., 2004).Dealing with 2d structures, this requirement must be fulfilled in both directions.Along with the desired isotropic characteristic this results in a quadratic footprint of the unit cell possessing a sidelength which is smaller or equal to a quarter of the wavelength.Although the absence of interlayer connections makes the initial design easy to fabricate and hence preferable, it cannot be extended to a two-dimensional unit cell in a straightforward manner, since the open-ended stubs cannot be reduced to fit into the footprint without losing its inductive behaviour.Therefore, the open-ended stubs are substituted by short-circuited ones which require to be reduced in length to maintain their electromagnetic properties.This modification influences strongly the field distribution within the interdigital capacitor.Whereas in the original case the maximum of capacitance was realised close to the open ends, in the modified version this appears close to the hostline.In order to reduce the amount of interlayer connections, these are placed at the corners of the unit cell so that they may be used by adjacent cells simultaneously, as seen in Fig. 1.

Analysis
As seen in Figs. 1 and 2, the boundary of the unit cell is located right between the interdigital capacitor.
In terms of an equivalent circuit model this can be considered as two capacitors connected in series representing together the capacitance of the interdigital capacitor separated by the cell boundary.However, excitation of the structure may only be performed with a quasi TEM mode bound to the microstrip lines.Consequently, the reference plane for determining the scattering parameters cannot be set correctly.As a result of this, the determinable scattering parameters always contain contributions of adjacent cells, in case of a periodically arranged cascade, or contributions of the feeding network, in case of analysing a single section.Thus, to obtain the properties of the unit cell itself the reference plane for the scattering matrix analysis must be placed between two inseparable capacitors which is not possible in practice.This difficulty can be overcome by modelling the unwanted contributions with an equivalent circuit model which is used to de-embed the unit cell parameters.For this, the values of the elements L R /2, 2L L , C R /2, 2R P , R S /2 and 2C L must be determined based on the simulated or measured network response which can be done by performing even and odd analysis.
Starting with the impedance matrix representation of the four-port network referred to as extended unit cell Z euc , which comprises the unit cell and the feeding network, allows to calculate and It is assumed here that Z euc can be described by a matrix possessing the same characteristics as the matrix shown in Eg. (9).Z even is the impedance which appears at the terminals if the network is excited simultaneously at all ports with an identical signal.In this case no current is flowing across the nodal point and the equivalent circuit model simplifies to the one seen in Fig. 3. Likewise, exciting the structure at each two ports with signals possessing a phase shift of 180 • leads to the odd admittance whose equivalent circuit model is seen in Fig. 4. In order to determine the values of the containing elements both equivalent networks are simulated with the circuit simulation software AWR Microwave Office (AWR).The results obtained from the EM structure are then compared with the results obtained by circuit analyses and in an iterative process the elements are modified until both responses match.This fitting sequence can either be performed with an automatic optimizer or by tuning the element values manually by performing the steps as follows: Starting with the observation of Z even shows that at the resonant frequency of the shunt resonator the impedance of Z even is expected to peak since the susceptances of the resonator cancel each other out.
www.adv-radio-sci.net/9/73/2011/Adv.Radio Sci., 9, 73-78, 2011 Consequently, the product of L L and C R is immediately known.Due to the fact that the unit cell realises a band pass structure with only weak losses, the parallel resistance R P must be highly resistive whereas its series counterpart R S must be of low resistance.Hence, we can conclude that R P >> R S and for this reason {Z even } ≈ R P at the parallel resonant frequency.Now the quality factor of the shunt resonator can be determined which allows a first estimation of L L and C R .As seen in the equivalent circuit, depicted in Fig. 3, the series resonance frequency occurs at ω rs = √ 2/ √ L R C L which is, if a balanced structure is assumed, far above the parallel resonant frequency.Thus, the series resonant circuit behaves capacitively, depending on the elements C L and L R /2, since it is operated below its resonance frequency.By modifying these values, the imaginary part of Z even of the equivalent circuit model may be adjusted until it fits to the desired one.
After these first steps we can evaluate the odd admittance Y odd at the shunt resonance frequency.Obviously, the parallel resonator appears as an open-circuit and the remaining network is almost the series resonator, as it can be seen in Fig. 4. Again, utilizing the quality factor and the resonance frequency allows to estimate the remaining element values.Since both adjustment steps are not decoupled the approximation of the element values can require some further iterations.Figure 5 shows the comparison of the even impedance and the odd admittance obtained by EM simulations and circuit simulations after the element values have been determined correctly.
The determined elements can be considered either as the feeding network, in case of analysing a single section, or as a part of adjacent cells in case of a periodically arranged cascade.Thus, by finding these element values the equivalent circuit model of the CRLH unit cell is fully determined.

Results
A well balanced unit cell is achieved after optimizing the structure supported by EM full-wave simulation using CST Microwave Studio (CST) combined with the procedure introduced in the previous section.The S-parameters of the resulting unit cell including the feeding network are shown in Fig. 6.
Although this is not the response of the unit cell itself, the isotropic behaviour and the power balance must be conserved anyway.As seen, the result is very close to the theoretical expectations of an isotropic structure which requires magnitudes of -6 dB, identical phase shifts of all transmission paths as well as a phase difference of 180 • between transmitted and reflected waves.The waveguiding properties of the de-embedded unit cell, possessing a Bloch impedance Z Bloch x ≈ 60 , are shown in Figs. 7, 8 and 9.In regions where an isotropic behaviour is exhibited, the wave vector must have an absolute value which is independent of the di-rection.In terms of Fig. 9 this is represented by cone shaped characteristics or relating to the isofrequency plots this results in circles around the point β x p = β y p = 0 which can be observed for 8.5GHz ≤ f ≤ 12GHz.It should be mentioned that the gap in Fig. 9 is an artefact of the visualization and does not represent a stop band as validated by the dispersion diagram depicted in Fig. 10.In the region − X and M − X the index of refraction is determined and presented in Fig. 11 which also reveals the isotropy of the unit cell.
A prototype, seen in Fig. 1, was fabricated and measured.The results of the measurements are presented in Fig. 12 and the corresponding dispersion diagram in Fig. 13.As seen from the increasing attenuation constant between 9.5GHz ≤ f ≤ 10.5GHz the fabricated prototype is not balanced correctly.Further analysis reveals that the series resonant frequency is f rs = 10.57GHz and the parallel resonant frequency is f rp = 9.67GHz.This can be explained by tolerances within the fabrication process.

Conclusions
A two-dimensional CLRH unit cell was presented which possesses an isotropic behaviour between 8.5GHz ≤ f ≤ 12GHz.Within this frequency region a frequency dependent index of refraction ranging from −1.5 ≤ 0.8 is achieved.A systematic procedure based on even and odd mode analysis was presented which allows to determine the element values of the equivalent circuit model of the proposed structure as well as its feeding network.

Fig. 1 .
Fig. 1.Picture of the fabricated structure which is produced on Rogers RO4350B substrate (thickness h = 0.762mm and r = 3.66).The 2d CRLH unit cell is within the square marked by the dashed line (side length l = 4.54mm ≈ λ/4).Parts located outside the square belong to either adjacent cells or the feeding network.

Fig. 1 .Fig. 2 .
Fig. 1.Picture of the fabricated structure which is produced on Rogers RO4350B substrate (thickness h = 0.762 mm and r = 3.66).The 2d CRLH unit cell is within the square marked by the dashed line (side length l = 4.54 mm ≈ λ/4).Parts located outside the square belong to either adjacent cells or the feeding network.

Fig. 3 .
Fig. 3. Equivalent circuit model of the impedance Z even obtained by even mode excitation of the unit cell with the feeding network.

Fig. 4 .
Fig. 4. Equivalent circuit model of the admittance Y odd obtained by odd mode excitation of the unit cell with the feeding network.

Fig. 4 .
Fig. 4. Equivalent circuit model of the admittance Y odd obtained by odd mode excitation of the unit cell with the feeding network.

Fig. 5 .Fig. 6 .
Fig. 5. Comparison between even impedance and odd admittance obtained from EM simulation (solid lines) and circuit simulation (triangles), respectively.Imaginary parts are displayed in red colour, real parts are drawn in blue colour.M. Eberspächer et al.: Design and Analysis of an Isotropic 2D Planar CRLH Waveguide Structure 5

Fig. 6 .
Fig. 6.Simulated scattering parameters of the structure seen in figure 1: S 11 red line, 21 green line, S 31 orange line, S 41 blue line.
pächer et al.: Design and Analysis of an Isotropic 2D Planar CRLH Waveguide Structure 5 in the previous section.The S-parameters of the unit cell including the feeding network are shown 6.Although this is not the response of the unit cell GHz mulated scattering parameters of the structure seen in figred line, S 21 green line, S 31 orange line, S 41 blue line.

Fig. 7 .
Fig. 7. Isofrequency contour plot (in GHz) of the left-handed mode with respect to two-dimensional wave propagation in the x and y directions.

Fig. 8 .
Fig. 8. Isofrequency contour plot (in GHz) of the right-handed mode with respect to two-dimensional wave propagation in the x and y directions.

Fig. 9 .
Fig. 9. Surface plot of the two-dimensional wave vector β exhibited by the CRLH structure.

Fig. 13 .
Fig. 13.Measured wave propagation characteristics.Dispersion diagram for the region Γ − X (left) and the corresponding attenuation constant (right).

Fig. 13 .
Fig. 13.Measured wave propagation characteristics.Dispersion diagram for the region Γ − X (left) and the corresponding attenuation constant (right).

Fig. 10 .
Fig. 10.Simulated wave propagation characteristics.Dispersion diagram for the region − X (left) and the corresponding attenuation constant (right).

Fig. 9 .
Fig. 9. Surface plot of the two-dimensional wave vector β exhibited by the CRLH structure.

Fig. 13 .
Fig. 13.Measured wave propagation characteristics.Dispersion diagram for the region Γ − X (left) and the corresponding attenuation constant (right).

Fig. 11 .
Fig. 11.Index of refraction obtained by EM simulations for the regions − X (red line) and M − (blue line).

Fig. 9 .
Fig. 9. Surface plot of the two-dimensional wave vector β exhibited by the CRLH structure.

Fig. 13 .
Fig. 13.Measured wave propagation characteristics.Dispersion diagram for the region Γ − X (left) and the corresponding attenuation constant (right).

Fig. 12 .
Fig. 12. Measured scattering parameters of the fabricated structure seen in figure 1: S 11 red line, S 21 green line, S 31 orange line, S 41 blue line.

Fig. 13 .
Fig. 13.Measured wave propagation characteristics.Dispersion diagram for the region Γ − X (left) and the corresponding attenuation constant (right).

Fig. 13 .
Fig. 13.Measured wave propagation characteristics.Dispersion diagram for the region − X (left) and the corresponding attenuation constant (right).
which is fabricated in an entirely printed circuit technology based on transmission line theory.The original design can be considered as an interdigital capacitor whose arms supporting the individual fingers are extended to open-ended stubs in order to be-Fig.12. Measured scattering parameters of the fabricated structure seen in figure 1: S11 red line, S21 green line, S31 orange line, S41 blue line.