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A band separating device that uses an anisotropic metamaterial exhibiting negative refraction is designed and measured. The metamaterial has frequency dispersion in one component of the permeability tensor. A beam is incident onto the metamaterial and undergoes frequency dependent reflection and refraction directing different bandwidths towards three distinct measurement ports. Design issues are discussed, and measurement results are presented.
©2005 Optical Society of America
1. Introduction
Negative values of permeability and permittivity have been shown to be present in certain composite materials [1]. These metamaterials are produced synthetically by the repetition of a sub-wavelength structure consisting of metallic elements [2, 3], and can often be described by biaxial, anisotropic constitutive tensors. The appropriate choice and orientation of these repeated structures can result in the negative refraction of electromagnetic waves [4].
Metamaterials are necessarily frequency dispersive which can result in a frequency dependent sign of refraction. This feature could be used to make a three-port device of the T-junction setup studied in [5, 6], which used a metamaterial prism inside a waveguide to demonstrate negative refraction of power.
In this paper we extend this idea to a four-port device, a conceptual representation of which is shown in Fig. 1. The device consists of a metamaterial wedge at the junction of the four ports. An electromagnetic wave is fed into port 1 which is then incident upon the wedge. Frequency dependent refraction and reflection occur at the interface. The transmitted “LH frequency band” is negatively refracted towards port 4. Similarly the transmitted “RH frequency band” is positively refracted towards port 3. The reflected power is collected at port 2.
Fig. 1. Four-Port Concept Design: An electromagnetic wave excited at the incident port (1) is reflected (port 2), positively refracted (“RH” port 3), or negatively refracted (“LH” port 4) at the surface of a metamaterial prism as a function of frequency.
We define the passband at each port as the 3 dB band centered at the most highly transmitted frequency. Designing the device consists of selecting an appropriate metamaterial and defining a geometry for the wedge that will allow transmission to the LH and RH ports.
2. Metamaterial Selection
In order for the four-port device to work properly, a metamaterial that supports negative refraction is needed. We consider metamaterials that can be described by biaxial, anisotropic constitutive parameters as potential candidates in the design and write the frequency dependent permittivity and permeability tensors respectively as:
The free-space constants are represented by εo and µo such that εx,y,z and µx,y,z represent the relative values. These relative values are functions of the radial frequency, ω.
The device is built inside a parallel plate waveguide with a plate separation small enough such that the incident field consists only in the TEM mode. We take the plate separation in the ŷ direction, which gives a ŷ polarized electric field for the TEM mode. The dispersion relationship [7], for this field is:
where c represents the speed of light in vacuum and the wave-vector is k̄=x̂kx+ẑkz. Note that only {εy,µx,µz} are relevant for this mode. This dispersion relation has been studied in detail in [8], as a function of the sign of these parameters. When the ŷ polarized electric field is incident from air onto a boundary that coincides with a principle axis of the metamaterial, the sign of the refraction can be determined by the sign of the constitutive parameters, assuming phase matching is possible [7]. Table 1 summarizes the refraction properties when the physical boundary is parallel to the x principle axis of the material. The shape of the dispersion relation in the kx,kz plane is also given.
Table 1. Refraction Properties for Anisotropic Materials
Table 1 is not exhaustive since it does not cover any of the cases where the material axis and boundary are not aligned. However, most metamaterials are deficient in representing such boundaries well since the sub-wavelength structural unit cell the materials are composed of generally have a cross section in the x, z plane that is rectangular. Boundaries that do not fall along the unit cell grid must be approximated by a corrugated edge which may introduce un-intended diffraction [9]. This issue is avoided here by taking the boundary along a principle axis.
As negative refraction of the power is indispensable to the operation of the four-port device, cases ii, vi, and viii are possible solutions. Case ii is the simplest, as it only requires one dispersive component. The other cases require two or three simultaneously negative parameters.
3. Metamaterial Construction and Effective Parameters
We proceed by focusing on case ii in Table 1, which requires only negative permeability in one direction. To that affect, the symmetric split ring resonator (SRR) first introduced in [10], is used as the unit cell structure. The cell constructed for measurement is given in Fig. 2 and was designed in [11]. It consists of two metallic rings printed on a dielectric substrate.
Fig. 2. Symmetric SRR unit cell with superimposed photograph. The labeled dimensions are: a=0.24 mm, b=1.56 mm, c=1.2 mm, d=3.12 mm, e=2.25 mm, f=5 mm, and g=0.5 mm. The substrate has εr≈4. The remaining volume is air-filled.
The material axes are taken as marked so that the dispersive permeability (normal to the surface of the rings) is in the ẑ direction. The structure in Fig. 2 is simulated using CST’s Microwave Studio and the dispersive component of the permeability is found using the retrieval method of [12]. The permeability, µz, can be approximated by the Lorentz model [1], with the characteristic resonance frequency fmo≈10.08 GHz and plasma frequency fmp≈10.56 GHz.
The components εy and µx are approximated as constants. Ignoring the metallic structure, εy and µx assume the values of the background material which is dominantly air leading us to use εy=µx=1.
Fabrication of the unit cell is done by printing rows of the SRRs onto large sheets of the dielectric substrate. The metamaterial is formed by cutting the sheet into strips which are lined up with the appropriate periodicity.
4. Geometric Design Issues
An appropriate geometry for the wedge needs to be specified in which the four parameters (shown in Fig. 3) can be optimized: the angle of incidence, θi, that the wave from port 1 impinges the material; the material rotation angles, ϕ3 and ϕ4, at the RH and LH ports (3 and 4); and the scale or size of the material, S. The rotation angle is defined as the counter-clockwise angle between the x axis of the material and the boundary.
Each geometric parameter is discussed in terms of the effect it has in transmitting power to the measurement ports. Consistent with our selection of case ii from Table 1, the x axis of the material is parallel to the boundary where waves from port 1 strike the metamaterial.
Having a transmission band at the LH port places restrictions primarily on θi and ϕ4. These angles should be chosen so that a propagating wave transmits to port 4, and the positively refracted band does not meet the port 4 material boundary. By studying eq. (3) in the region where µz(ω)<0, we find many restrictions on these two angles which can simultaneously be met only over a small portion of the LH band [13].
We choose to design the device for the band near µz=-0.5. Figure 4 shows the path of an incident wave from port 1 to port 4 using the final selected values of ϕ4=60°, and θi=15°. We see that if ϕ4 were made significantly larger, phase matching would no longer be possible at the second boundary. Even if a correspondingly smaller θi was used, the negative refraction may not be large enough for the wave to reach the boundary. Also, if ϕ 4 is made smaller, some of the RH band strikes the boundary at port 4 rather than at port 3, deteriorating the stop band measured at port 4. While the scale parameter, S, also plays a role in determining if a refracted wave reached the port 4 or port 3 boundary, the issue is differed by assuming that the incident wave can be adjusted relative to the corner of the wedge. The metamaterial then can look as large or small as necessary to achieve the desired operation.
Fig. 3. Metamaterial Design Parameters: θi - incidence angle; ϕ 3 - rotation angle at port 3; ϕ 4 - rotation angle at port 4; S - metamaterial size. The material axes are shown. Dashed lines indicate port boundaries as absorbing material has been suppressed for clarity. The gray lines forming the material indicate the orientation of the dielectric cards with SRRs for reference.
It is convenient to select the orientation of the RH boundary, ϕ 3, to be the same as the incident boundary. We can justify this selection by noting that if ϕ 3=0, the material appears as a slab to the positively refracted frequencies. By symmetry, incident waves that propagate into the medium can be phased matched at the second boundary. The net effect is a frequency dependent beam shift of the RH band [14, 15, 16].
In the frequency band just past the plasma frequency, where 0<µz≪1, the dispersion relation describes a narrow ellipse where kx≪k o for real kz. Since θi>0 implies kx≠0, there is a frequency band above the plasma frequency where phase matching cannot be satisfied and a wave cannot propagate into the material. We expect the power reflected to port 2 to be high in this bandwidth. This conclusion is supported by theory when the material is approximated as a 2D slab and the analytical formulations are used to calculate the reflected power for a 15° incidence [17, 18].
The scale, or thickness of the material, S, is a parameter that serves as a wild card in design. So far we have only defined angles for the structure, which has left the scale at large. The material must be ample enough so that the unit cells form an effective medium. It also must be thick enough to define a boundary at port 4. Nevertheless, the material cannot be arbitrarily large since the refracted angle of the RH band is less than ϕ 4 for frequencies sufficiently above the plasma frequency. If the material was too big, these waves would first reach the port 4 boundary rather than that of port 3. A thickness of 6 cm (twelve layers of unit cells) from the incident to the port 4 material boundaries has proven satisfactory in measurements.
5. Experimental Setup and Results
The material construction itself was consistent with the description in Sec. 3 but with one notable difference. In order to minimize corrugation at the LH port boundary, the centers of the unit cells were shifted to fall directly onto the boundary. This technique avoided uneven stair-stepping in favor of a constant shift of the cells from row to row insuring that diffraction effects are minimal [9].
For measurement, the metamaterial was placed inside a parallel plate waveguide like shown in [1]. A diagram of the setup is shown in Fig. 5. Microwave absorber was used as necessary to maintain plate separation, minimize direct coupling between ports, minimize diffraction, and minimize influences from the external environment. A WR90 waveguide to SMA adapter fed the microwave signal between the plates at port 1. The same type of adapter was used for measurement at ports 2, 3, and 4. The plate separation was approximately the height of the material, 1 cm.
In comparison to Fig. 1, port 1 and 2 are much closer due to the small incidence angle needed. In fact, there was no need to use absorber between these two ports to achieve the desired device operation. Also, the conceptual triangular shape is replaced with one more suited to insuring propagating waves at ports 3 and 4. Finally port 4 is closer to the edge of the material due to the generally large transmission angle anticipated at the port 4 material boundary.
An HP-8510 network analyzer was used for measurements in the X-band, (8.2 to 12.4 GHz), and the measurement results are presented in Fig. 6(a). The measurement at the RH port shows a strong stop band centered near 10.4 GHz which is inside the negatively refracted band. The left portion of the stop band can be attributed to power being refracted towards the LH port, and the right due to high reflection. Measurements at the LH and reflection ports corroborate this assessment. At the LH port a peak appears in a band corresponding to µz<0. The measured LH band is in good agreement with the retrieval results, peaking at 10.3 GHz. In the region near the plasma frequency, where |µz|≪1 the reflected power increases and peaks at 10.5 GHz. Above the plasma frequency, as µz→1, transmission to the RH port dominates.
Fig. 4. Refraction to Port 4 when µz=-0.5, with ϕ 4=60° and θi=15°. Refraction at the port 1 and port 4 boundaries are shown with the dispersion relations superimposed. The dashed line indicates the phase matching component of the suppressed k vector. The small and large arrows indicate the direction of energy flow in air and in the negative medium, respectively. The inset figures detail the dispersion relation in the area of interest. The thick line in the detail of the second interface indicates the possible k vectors in the negative medium that meet the transmission requirement.
Fig. 5. Top view of experimental setup with top plate removed. The bottom plate is shown in gray. Microwave absorbing material is represented by the black wedges. The material axes are shown. The final design angles are used: ϕ 4=60° and θi=5°. S=6 cm (12 unit cells).
Passbands have been shaded using the criteria outlined in Sec. 1. In the operating band of 10.1 to 11.1 GHz, we find that an LH band of 190 MHz is achieved at port 4. At port 2, a bandwidth of 187 MHz is measured and at port 3, the RH band is 192 MHz. The RH bandwidth however is superficially limited by the band of interest and could easily be increased by raising the upper limit of the frequency range. In the remainder of the X-band the signal at port 3 is dominant.
A setup similar to the experimental setup in Fig. 5 is constructed in Microwave Studio and the transmission results are shown in Fig. 6(b). A homogenous anisotropic metamaterial prism is used with the retrieved Lorentz dispersion. The simulation also demonstrates the predicted three band behavior. Performance differences are due to simplifications made in the simulated device, and the difficulty in accessing the precise nature of the losses in the metamaterial.
6. Conclusion
In this paper we have realized a four-port band select device using an anisotropic metamaterial with a permeability that is frequency dispersive and achieves negative values. The use of frequency dependent refraction to differentiate an incident signal into different bands has been verified experimentally. The proposed device could potentially be improved in the microwave regime, or extended to optical frequencies for applications such as an optical router.
Fig. 6. (a) Experimental measurement results. The 3 dB passbands are indicated by shading. (b) Simulation results. Magenta — Reflection Port 2. Green — RH Port 3. Red — LH Port 4.
While case ii of table 1 was studied in detail, the dispersion curves for two (case vi) or three (case viii) dispersive parameters suggest a device in which negatively refracted waves can more easily be phase matched over a broader band at the LH port. Many of the metamaterial structures thus far proposed (and those available at the time of measurement) use vertical rods to achieve plasmatic permittivity, which cases vi and viii both require. Good electrical contact between the rods and parallel plates is needed for the plasma effect, which is difficult to achieve experimentally. S-shaped structures which achieve both negative permittivity and permeability have been proposed to eliminate this issue [19, 20]. Negative permittivity could be skirted entirely if a denser dielectric was used outside the wedge in case ii, allowing for easier phase matching at the LH port due to the larger dispersion circle.
Acknowledgments
This material is based upon work supported under a National Science Foundation Graduate Research Fellowship. This work was sponsored by the Department of the Air Force under Air Force Contract F19628-00-C-0002, and the ONR under Contract N00014-01-1-0713. Opinions, interpretations, conclusions and recommendations are those of the author and are not necessarily endorsed by the United States Government.
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