Magnetic metamaterial design and fabrication

Our aim is to design and fabricate an isotropic metamaterial (MM) exhibiting Re{μ} < 0 at 13.56 MHz with minimal losses. Assuming operational frequency close to 13.56 MHz, the free-space wavelength is λ o ≈ 22 m; a conventional metamaterial whose elements are roughly λ o /10 in size would be far too large for practical implementations. Instead, we must demonstrate the desired behavior with elements whose size is only several centimeters, on the order of λ o /1000. To achieve this, we significantly increase the metamaterial unit cell inductance by utilizing the double-sided rotated coil design shown in Figure 5, which sandwiches a substrate between two via-connected multi-turn coils. The coils are rotated with respect to one another to form a composite circuit in which the inductances of two individual coils are added in series, resulting in the total inductance improved by a factor of four relative to the configuration with inductance in parallel.

Figure 5 (A) Schematic of the metamaterial unit cell suitable for Printed Circuit Board fabrication process. The coils on the opposite sides of the substrate are rotated with respect to each other and connected by vias. (B) Each resonator has 17 turns on each side of a 10 mil substrate. Metal line widths are 200 microns and the gap between neighboring cells is 1 millimeter. (C) The complete unit-cell consists of three mutually orthogonal resonators. (D) The 3-layer slab is composed of the unit-cells stacked along z. Full size image

We form the complete metamaterial unit cell by positioning three identical resonators perpendicular to each other, as shown in Figure 5C and iteratively tweak the design in CST microwave studio using the standard S-parameters retrievals20,21 to obtain Re{μ} = −1 in the desired ISM frequency band.

The figure of merit (FOM) for our design is the inverse loss-tangent ratio at the frequency where Re{μ} = −1:

To minimize the loss tangent we choose a low-loss Rogers 4350 substrate and construct the metamaterial using a 1-ounce (34 μm thick) copper clad; the skin depth in Cu at 10 MHz is about 20 μm. The S-parameter retrieval method lets us compute the transverse permeability components μ x and μ y by setting periodic boundary conditions along the x and y directions and enforcing a normally (z-) incident, transversely polarized plane wave. The final design, whose retrieved permeability is shown in Figure 6, utilizes coils with 17 turns on each side of the substrate. Each turn is 200 μm wide and the gap between turns is set to 200 μm as well. To reduce the design's sensitivity to fabrication errors, we insert three vias 200 μm in diameter into the outer-most leg in each coil and increase the width of that leg to 500 μm such that there are 150 um between each via and the metal's edge. The total unit-cell size, including a 1 mm gap between adjacent unit-cells, is 1.894 cm.

Figure 6 (A) Transverse component of the effective permeability fora 1-layer MM slab, calculated using the conventional S-parameter retrieval methods (circles) and the field averaging method (solid line). With the latter method, we can also calculate the normal component (crosses). Blue and orange represent the real and imaginary parts of μ, respectively. (B) The transverse and normal component of a 3-layer slab, retrieved using the Field Averaging method. Full size image

Although all three orthogonal coils in the unit-cell are identical, this does not mean the MM's permeability is isotropic because the normal (z) and transverse (x,y) boundary conditions observed by fields propagating through the slab are significantly different from each other.

Field-averaging homogenization method for finite-thickness, anisotropic metamaterial layers

While the standard S-parameter retrieval techniques20,21 allow one to compute the components of effective permeability and permittivity tangential to the surface of a MM layer, the normal components are difficult to retrieve since they are not excited by a normally incident, transversely polarized plane wave. Here we present a quasi-magnetostatic field averaging retrieval method and outline how we used it to compute both the transverse and normal components of our MM. Quasi-electrostatic field-averaging homogenization was described in detail in Ref. 22., where effective ε was expressed through the capacitance of a unit cell submerged into a curl-free electric field. Here, we extend this method to quasistatic permeability retrieval using the electric-magnetic duality theorem. For brevity we assume the medium to be uniaxial with permeabilities μ T and μ N ; the method is applicable to a general orthotropic medium with an orthorhombic lattice.

We simulate a unit cell of dimensions a x × a y × a z in COMSOL Multiphysics's RF module. Air surrounded the cell along z while periodic boundary conditions (BCs) are along x and y such that unit-cell is part of a slab. Across the faces normal to the z-axis we assign an Electric Field which varies with z, . By using , we can be sure that the electric field has a linear variation in z and thus its curl is virtually uniform in the entire domain. From Faraday's law, this E-field leads to , uniform magnetic field H y . By using the duality theorem together with the definition of capacitance, C = εA/d = Q/V, we can replace ε with μ, electric charge Q with magnetic charge Q m and electric voltage V with magnetic potential V m and extract effective permeability according to

With H polarized along , we obtain μ y , one of the permeability principal values, by substituting d = a y , A = a x × a z and B = B y into (3).

To compute μ x , the remaining transverse component, one replaces the incident field E x (z) with E y (z) and utilizes the appropriate fields and dimensions in (3). To compute the normal component μ z , however, an additional subtle change has to be made. We begin by exciting an incident field E y (x) of the form , which gives rise to an almost uniform H-field H z . However, such a field still violates the periodic condition along x slightly; therefore on the x = const faces we use Floquet (phase-shifted periodic) boundary condition with the phase shift given by k x a x . We then compute μ z from (3), substituting d = a z , A = a x × a y .

We perform field-averaging retrievals across the 8 MHz–16 MHz frequency range and compute the transverse and normal components of μ for both the 1- and 3-layers configurations. We then fit each retrieved permeability to a Lorentzian curve defined as

where F is a constant representing the oscillator's strength, ω 0 = 2πf is the angular resonance frequency and y = ω 0 /2Q. The resulting fitted parameters of the 1-layer MM are

and the parameters fitted from the 3-Layer MM retrieval were calculated to be

Here we have used the superscripts 1 and 3 to distinguish between the 1- and 3-layer slabs, respectively and the subscripts N and T to distinguish between the normal and transverse permeability components, respectively. Not surprisingly, comparing the fitted parameters for the 1- and 3-layer MM suggests that as more layers are added the metamaterial behaves in a more isotropic fashion.

We note that the Lorentzian parameters provide a quality fit for the complex permeability curve only in the vicinity of the fundamental magnetic resonance studied. Good quality of fit is maintained at least through the frequency where Re(μ) crosses zero (roughly 16 MHz), that is, in the entire frequency band of interest.

Maximum transducer power gain calculations

Before we conduct WPT measurements with the non-resonant coils setup, we perform a calibration that moves reference planes of a VNA to the end of the cables that are connected to the coils (see Figure 7). This enables us to retrieve the direct coil-to-coil transmission efficiency. In Ref. 23. Pozar describes a suitable metric called the Maximum Transducer Power Gain, , which is the gain that would be achieved if a lossless matching network was inserted between the NA's reference planes and the non-resonant loops. Pozar defines in terms of only S parameters; here we summarize the calculations outlined in Ref. 23.

Figure 7 Pre-measurement calibration moves the VNA reference planes to the end of the cables that are connected to the non-resonant coils. Full size image

Transducer Power Gain G T is the ratio of power delivered to the load, P L , to the power available from the source, p s :

where Γ L = (Z L − Z 0 )/(Z L + Z 0 ) is the reflection coefficient seen looking toward the load, Γ S = (Z S − Z 0 )/(Z S + Z 0 ) is the reflection coefficient seen looking toward the source and Γ in is the reflection coefficient seen looking toward the input of the two port network

where Z in is the impedance seen looking into port 1 of the terminated network. Similarly, Γ out is the reflection coefficient seen looking into port 2 of the network when port 1 is terminated by Z S :

The Maximum Transducer Power Gain, , occurs when and . In the general case with a bilateral two port network, Γ in is affected by Γ out and vice versa, so that the input and output must be matched simultaneously. Equating and with the RHS of (8) and (9), respectively, yields

where Δ = S 11 S 22 − S 12 S 21 . Substituting (10) into (9) and rearranging the terms results in the quadratic equation

yielding the solutions