Dual-band composite right/left-handed metamaterial lines with dynamically controllable nonreciprocal phase shift proportional to operating frequency

3.1 Equivalent circuit model and conditions for formation of two Dirac points

In this section, we briefly introduce how to construct two Dirac points in the dispersion diagram of nonreciprocal CRLH metamaterial lines and specifically how to design dispersion of the phase-shifting nonreciprocity proportional to the operating frequencies. Figure 3a shows an equivalent circuit model for the unit cell of the nonreciprocal CRLH metamaterial line along the y axis, and Figure 3b shows schematic of the dispersion curves without any Dirac points for general cases [27].

Figure 3:

An equivalent circuit model and dispersion diagram for the nonreciprocal dual-band CRLH metamaterial lines.

(a) The equivalent circuit model for the unit cell. (b) Typical dispersion curves without any Dirac points.

In Figure 3a, the length of the unit cell is p, and variables C _L ^c, C _L ^d, and C _R ^d denote capacitances of the designated capacitive elements, and L _R ^d, L _L ^d, and L _L ^c represent the inductances. The superscript “c” on the right-hand side of the variables stands for conventional inductive or capacitive elements used to construct single-band CRLH metamaterial lines, whereas the superscript “d” denotes elements additionally inserted in the unit cell to realize dual-band operation. In the unit cell, two identical nonreciprocal transmission line sections having wavevectors β p and β n and characteristic impedances Z p and Z n in the positive and negative y directions have also been introduced. In the present paper, the nonreciprocal line section is composed of a normally magnetized ferrite microstrip line, and the circuit parameters characterizing the sections are specifically determined by the gyro-magnetic characteristics of the ferrite and the geometrical configuration. The parameters including β p and β n must be previously given before finding solutions to the eigen modes for waves propagating along the periodic CRLH structures, based on Floquet’s theorem. The phase constants for the periodic structures correspond to the pair of variables β + and β − in Figure 1. Under the assumption that the characteristic impedances of the nonreciprocal transmission line sections show almost identical characteristics with Z p = Z n = Z 0 = 1 / Y 0 , we have general dispersion curves with significantly large bandgaps, as shown in Figure 3b. To eliminate two bandgaps in the frequency regions from ω L 1 to ω U 1 and from ω L 2 to ω U 2 in Figure 3b and achieve the formation of two Dirac points at β = Δ β , as shown in Figure 1, the following two relations must be simultaneously satisfied in the same manner as reciprocal cases [9],

where

The frequency dependence of the nonreciprocity Δβ is expressed in terms of admittances of shunt stubs per unit cell, Y ₁ and Y ₂, that are asymmetrically inserted at left- and right-hand side strip edges, respectively, when looking at positive y direction, and is given by [23],

with

where it should be emphasized that the expression (5) explicitly shows the phase-shifting nonreciprocity originates from a combination of broken time-reversal and space-inversion symmetries, in that a factor κ in (5) is an off-diagonal component of the permeability tensor in ferrite and related to broken time-reversal symmetry, while another factor ( Y 1 − Y 2 ) indicates the geometrical asymmetry of the waveguides due to broken space-inversion symmetry. In (5), h denotes the thickness of the substrate for the ferrite microstrip line, and in (6) the quantity ω h represents the “magnetic resonance frequency” proportional to the internal dc magnetic field H 0 in the ferrite, γ  is the gyromagnetic ratio, and M S the saturation magnetization.

In order to specify the frequency dependence of Δβ in (5), we assign the inductive element L L c in the shunt branch in Figure 3a for Y ₁ and series LC resonant circuit including L L d and C R d for Y ₂. Then we have

For the weakly applied dc magnetic field to the ferrite where the operating frequency is selected at frequencies much higher than the magnetic resonance frequency with the relation ω ≫ ω h , we have dispersive nonreciprocity Δβ, whose frequency dependence becomes complicated and looks like Figure 1a [29]. On the other hand, for the strongly applied dc magnetic field to the ferrite where the operational frequency is much lower than the magnetic resonance frequency with ω ≪ ω h , the nonreciprocity Δβ (ω) in (5) reduces to

The above expression (8) clearly shows that the nonreciprocity is approximately proportional to the operational frequency ω both at lower and higher Dirac points that are located far below the magnetic resonance frequency ω h , as ideally illustrated in Figure 1c.

3.2 Designed and fabricated circuit configuration

In Figure 4, schematic of the designed CRLH metamaterial line, and photograph of the fabricated circuit are shown. As a commercial software, ANSYS HFSS ver. 19 was utilized in the numerical simulations for 3-D full-vector electromagnetic field analysis based on finite element method. Configuration parameters designed in the numerical simulations are given in the following: The heights of the ferrite and dielectric substrates are both h = 0.8 mm, their dielectric constants are ε _f = 15 and ε _d = 2.62. The length of the unit cell p = 6.9 mm, the width of the ferrite rod w = 4.0 mm, the length and width of the three parallel thin metallic strips in the center microstrip line are l _t = 4.5 mm and w _t = 0.3 mm. The length and width of the shorter inductive stub for Y ₁ on the left-hand side of Figure 4b, l _s ^c = 3.0 mm, w _s ^c = 1.0 mm. The length and width of longer inductive stubs made of the series LC tank for Y ₂ on the right-hand side, l _s ^d = 16.0 mm, and w _s ^d = 1.0 mm. The lumped-element capacitances C _L ^c = 2.1 pF, C _L ^d = 1.1 pF, and C _R ^d = 0.5 pF. We assume the saturation magnetization of the ferrite μ ₀ M _S = 175 mT, and the magnetic loss μ ₀ΔH = 5 mT. Since the Bloch impedance of the designed CRLH line was much smaller than 50 Ω, tapered feeding microstrip lines with the linearly varying widths were inserted between the designed CRLH line and 50-Ω input/output ports for impedance matching. Based on the designed configuration, we fabricated a prototype CRLH metamaterial line, as shown in Figure 4c. We utilized polycrystalline yttrium iron garnet (YIG) for the ferrite substrate, and a PTFE substrate (NPC-F260, Nippon Pillar Packing Co., Ltd.) was used for the dielectric substrate. In the measurement setup, the dc magnetic field was externally applied to the test circuits by using a neodymium magnet with the dimension of the length of 100 mm, width of 9.0 mm, and height of 50 mm that was placed just below a cupper plate for a ground plane, and the magnet was mechanically moved to adjust the desired magnitude of the external dc field.

Figure 4:

Schematic of the designed CRLH line and a photograph of the fabricated circuit for demonstration.

(a) Top view of the designed CRLH line. (b) Perspective view of the unit cell. (c) Photo of the fabricated line.

3.3 Numerical simulation and experimental demonstration

Dispersion diagrams extracted from the simulated and measured phases of the transmission coefficients S ₂₁ and S ₁₂ as well as the phase shifting nonreciprocity Δβ are illustrated in Figure 5. Figure 5a corresponds to nonreciprocal cases with the phase shifting nonreciprocity Δ β ≠ 0 , whereas Figure 5b corresponds to almost reciprocal cases with Δ β ≈ 0 . In the numerical simulation set-up, the magnetic resonance frequency was selected at f _h =  ω h /2π = 3.92 GHz under the internal dc magnetic field of μ ₀ H ₀ = 140 mT for the nonreciprocal case in Figure 5a, while the magnetic resonance frequency was f _h = 11.2 GHz under the dc field of μ ₀ H ₀ = 400 mT for the almost reciprocal case in Figure 5b. In the measurement setup, the external dc magnetic field of μ ₀ H _ext = 300 mT was applied to the ferrite for Figure 5a, and μ ₀ H _ext = 520 mT for Figure 5b. Figure 5a and b clearly shows that dual passbands of the CRLH line are well designed in the numerical simulation and successfully reproduced in the measurement. It is found from Figure 5a that there are no significant bandgaps in the two CRLH passbands, and two Dirac cones are constructed with the intersection of the dispersion curves at f _D1 = 1.64 GHz, Δβp/π = 0.0188 in the lower passband and at f _D2 = 2.47 GHz, Δβp/π = 0.0339 in the higher passband. In addition, these intersections of the curves are found to be located in the vicinity of the same linear line with the slope of the refractive index of 0.27. It is noted that there exists a singularity in the nonreciprocity Δβ around at 2.2 GHz within a band gap between lower and higher CRLH passbands, as seen in Figure 5a. The singularity is likely to disturb the formation of linearity over the wideband region in the phase-shifting nonreciprocity. Figure 5b shows a considerably small nonreciprocities Δβ both in the numerical simulation and measurement due to the magnetic resonance frequency f _h much higher than the operational frequency.

Figure 5:

Dispersion diagrams and the phase-shifting nonreciprocities that are extracted from the transmission coefficients for the designed and fabricated dual-band CRLH metamaterial lines.

(a) Nonreciprocal cases with the internal dc field μ ₀ H ₀ = 140 mT in the numerical simulation, and the externally applied dc magnetic field μ ₀ H _ext = 300 mT in the measurement. (b) Almost reciprocal cases with μ ₀ H ₀ = 400 mT in the numerical simulation and μ ₀ H _ext = 520 mT in the measurement.

In Figure 6, magnitudes of the simulated and measured transmission and reflection coefficients are plotted. Figure 6a and b shows the numerical simulation results and Figure 6c and d does the measurement results. Figure 6a and c denotes the nonreciprocal cases corresponding to Figure 5a, and Figure 6b and d illustrates almost reciprocal cases corresponding to Figure 5b. It should be emphasized that simulated magnitudes of the transmission coefficients S ₂₁ and S ₁₂ in Figure 6a and b show almost identical characteristics, regardless of whether the designed CRLH lines are nonreciprocal or not. In Figure 6c, the transmission coefficients S ₂₁ and S ₁₂ at higher frequencies were found to be seriously attenuated because the dc magnetic field applied was nonuniform in the ferrite. We have confirmed that by replacing the permanent magnet with an electromagnet in order to apply a uniform dc field to the ferrite, both the measured transmission coefficients S ₂₁ and S ₁₂ successfully recovered with greater than −3 dB and almost identical transmission in the frequency region from 2.4 to 3.6 GHz. In addition, a big notch is found around at 2.7 GHz in Figure 6b and d for almost reciprocal cases with sufficiently large dc field applied to the ferrite. This is because the Dirac point in the higher CRLH passband collapsed, and a bandgap appears due to the impedance-mismatching in which effective permeability of the CRLH line has been considerably changed by enhancing the applied dc magnetic field.

Figure 6:

Magnitudes of transmission and reflection coefficients of the designed and fabricated CRLH lines.

(a) and (b) denote numerical simulation results for nonreciprocal and almost reciprocal cases, respectively. (c) and (d) represent the measurement results corresponding to (a) and (b), respectively. (a) The internal dc field μ ₀ H ₀ = 140 mT. (b) μ ₀ H ₀ = 400 mT. (c) The external dc magnetic field μ ₀ H _ext = 300 mT. (d) μ ₀ H _ext = 520 mT.

To confirm the validity of the simulated dispersion curves in Figure 5a, simulated magnitude and phase profiles of the electric field component E _z along the transmission line are observed around at the Dirac points in the lower and higher CRLH passbands and plotted in Figures 7 and 8, respectively. Figures 7 and 8 represent the field profiles measured at 1.65 and 2.47 GHz. The phase distribution in Figure 7c clearly shows the dominant forward wave mode propagation with wavenumber vector parallel to the transmitted power direction, whereas that in Figure 7d represents the backward wave mode propagation with the wavenumber vector antiparallel to the transmitted power direction. It is found from Figure 7c and d that their phase gradients in the longitudinal y direction are found almost the same. Figure 8 is also the case similar to Figure 7. Therefore, the observed field profiles in Figures 7 and 8 clearly show the wave propagation around at Dirac points with nonzero phase constants. We can also confirm the validity of the simulation results from the rough estimate of phase gradients along the line in Figures 7 and 8. Figure 7c and d shows that the phase difference over the 8-cell CRLH line was 0.4 rad and then Δβp = 0.05 rad/cell, which corresponds to beam angle of θ = sin⁻¹(Δβ/β ₀) = 12.5° that was measured with respect to the normal to the planar line, where β ₀ is the phase constant in vacuum. On the other hand, Figure 8c and d denotes the phase difference over the line was 0.7 rad and then Δβp = 0.0875 rad/cell, corresponding to the beam angle of 14.6°. Therefore, we have confirmed that the beam directions estimated from the transmission characteristics of the designed nonreciprocal CRLH metamaterial lines at two different frequencies are found to be almost the same. Thus, we have successfully demonstrated the transmission of dual band nonreciprocal CRLH metamaterial lines having two Dirac points with phase-shifting nonreciprocity approximately proportional to the operating frequency.

Figure 7:

Magnitude and phase distribution of the electric field component E _z observed on a plane a half the substrate thickness above the ground plane at 1.65 GHz.

(a), (b) Magnitude. (c), (d) Phase distribution. Signal incidence (a), (c) from the left port 1 and (b), (d) from the right port 2.

Figure 8:

Magnitude and phase distribution of the electric field component E _z observed on a plane a half the substrate thickness above the ground plane at 2.47 GHz.

(a), (b) Magnitude. (c), (d) Phase distribution. (a), (c) Signal incidence from left port 1 and (b), (d) from right port 2.