Microwave Waveguide-Type Hyperbolic Metamaterials

Hyperbolic metamaterials (HMs) supporting hyperbolic isofrequency curves (IFCs) provide unprecedented control on wave propagation and light-matter interaction. However, in the microwave regime, ultrahigh wave vectors and customizable permittivity tensors cannot be supported simultaneously by current common HMs. Based on the waveguide principle, low-loss waveguide-type HMs (WHMs) are suggested as a new geometry format of microwave HMs, ﬂ exibly mixing positive and negative components in their effective permittivity tensors. Then, a deeply subwavelength WHM cavity, con ﬁ rming the existence of giant wave vectors, is demonstrated, whose transverse size has a potential to be reduced to three orders smaller than the working free-space wavelength. Such a WHM cavity is also demonstrated to follow the fascinating anomalous scaling law originating from the unique shape of hyperbolic IFCs. As a low-loss platform, WHMs are predicted to produce distinctive physical phenomena based on HMs and provide novel functional devices at low frequencies.

DOI: 10.1002/adpr.202000043 Hyperbolic metamaterials (HMs) supporting hyperbolic isofrequency curves (IFCs) provide unprecedented control on wave propagation and light-matter interaction. However, in the microwave regime, ultrahigh wave vectors and customizable permittivity tensors cannot be supported simultaneously by current common HMs. Based on the waveguide principle, low-loss waveguide-type HMs (WHMs) are suggested as a new geometry format of microwave HMs, flexibly mixing positive and negative components in their effective permittivity tensors. Then, a deeply subwavelength WHM cavity, confirming the existence of giant wave vectors, is demonstrated, whose transverse size has a potential to be reduced to three orders smaller than the working free-space wavelength. Such a WHM cavity is also demonstrated to follow the fascinating anomalous scaling law originating from the unique shape of hyperbolic IFCs. As a low-loss platform, WHMs are predicted to produce distinctive physical phenomena based on HMs and provide novel functional devices at low frequencies.

Constructing WHMs
A waveguide, consisting of two metal plates sandwiching a dielectric plate of permittivity ε b and height h, is investigated. Under the TE 10 mode, the waveguide mode behaves like a plane wave propagating in an effective electric medium. As treating the metal as a perfect electric conductor (PEC), the effective relative permittivity can be simply deduced as where λ 0 is the free-space wavelength. According to Equation (1), ε wg exhibits the Drude dispersion and switches its sign by adjusting ε b or h appropriately. Then, two types of subwavelength dielectric slices with thicknesses d 1 and d 2 , respectively, are alternately arranged in a period of p y ¼ d 1 þ d 2 along the y-direction between the top and bottom metal plates (Figure 1a), whose relative permittivities are ε b1 and ε b2 , respectively. The field behavior is equivalent to that of a plane wave propagating in an electric anisotropic medium. With the effective medium theory (EMT), [1] the effective relative permittivity tensor of the composite waveguide structure is found to be The wave propagation behavior follows the dispersion equation where k 0 is the free-space wave number. When the two filling materials are two types of ceramics (barium tetratitanate of relative permittivity ε b1 ¼ 38.50 with loss factor tan δ b1 ¼ 1.24 Â 10 À4 and anorthite of relative permittivity ε b2 ¼ 5.5 with tan δ b2 ¼ 1 Â 10 À4 ), and h ¼ 60 mm, the working frequency is around f ¼ 0.7 GHz, and the effective relative permittivities corresponding to the two waveguide segments are ε wg1 ¼ 25.76 þ 0.0049i and ε wg2 ¼ À7.24 þ 0.00055i according to Equation (1) (time harmonic factor exp(Àiωt) is assumed). By inserting the two values into Equation (2) with d 1 ¼ d 2 ¼ 1 mm, one has that ε whm,x ¼ 9.26 þ 0.0027i and ε whm,y ¼ À20.13 þ 0.0036i. According to Equation (3), it is known that the IFC at 0.7 GHz of the composite waveguide is hyperbolic; thus, a WHM is attained. The values of ε whm,x and ε whm,y can be varied in a large range by adjusting the parameters in Equation (2), including zero, infinite, and negative, which is an advantage of the WHM compared with wire HMs. The above-mentioned investigation is made under the assumption that there only exists waveguide mode TE 10 . An unwanted waveguide mode, TM 10 , can exist simultaneously along the parallel-plate waveguide. To prevent the cross coupling from TE 10 to TM 10 due to interface scattering, additional thin metal wires are added on each slice interface to short-circuit the electric component of E z and quench waveguide mode TM 10 . [29] Here, thin silver wires (conductivity σ Ag ¼ 6.30 Â 10 7 S m À1 ) are distributed in a period of p x with width d x , thickness d y , and  ). Full-wave simulation is conducted by the eigenfrequency solver in the wave optics module of COMSOL. The mesh-element size within the metal wires is set below 5 μm along the xand y-directions, and below 1.5 mm along the z-direction. The solver is required to find the eigenfrequencies around 0.7 GHz by iteration, and then, we check the electrical field patterns to determine the correct eigenmodes. For a given Bloch vector of k x and k y , a complex eigenfrequency, f eig , can be found, and Re( f eig ) represents the working frequency, and Im( f eig ) the attenuated rate. When p x ¼ 1 mm, d x ¼ 0.2 mm, and d y ¼ 0.02 mm for the silver wires, the simulated IFC at Re( f eig ) ¼ 0.7 GHz is shown in Figure 1b. The hyperbolic IFC, curved toward the k y -direction, ends at the Bloch boundary of π/p y . As the WHM is anisotropic, the total in-plane wave vector and the effective mode attenuation rate (or quality factor Q whm ) change with k x . The former one is k tot ¼ (k x 2 þ k y 2 ) 1/2 , whereas the latter one can be calculated by Q ¼ ÀRe( f eig )/2Im( f eig ) as a general definition. [31] k tot and Q whm corresponding to Figure 1b are shown in Figure 1c. Maximum k tot is up to 120 k 0 , whereas Q whm remains larger than 100.
If one further reduces p x and p y to weaken the nonlocal effect, larger wave vectors can be produced. With a set of optimized parameters (p , the IFC at 0.7 GHz is shown in Figure 1 d. The finite period of metal wire arrays may also limit the largest attainable wave vectors due to the nonlocal effect. The Bloch boundaries of K Bragg ¼ π/p x ¼ π/p y ¼ 2141.5 k 0 terminate the IFC, giving the maximum k tot of 3028.5 k 0 . This giant wave vector over 3000 k 0 is realized with Q whm larger than 10 (see Figure 1e). WHMs provide an effective way to break the limit of natural materials and achieve giant effective refractive index (n whm ¼ k tot /k 0 ) with relatively low dissipation, [32,33] which can simply scale to work at various frequencies across the microwave range. Some synthesized composite materials can possess the huge relative permittivities up to 10 4 , but work only at low frequencies below a few megahertz. [34] In the gigahertz range, the relative permittivity of typical high-permittivity natural material is only around 100. [35]

Hyperbolic IFCs in Broadband
The IFC behavior of the constructed WHMs at various frequencies can be qualitatively understood according to Equation (2). The WHM investigated in Figure 1b is taken as an example. When f approaches the topological transition frequency around 0.533 GHz, as Re(ε whm,x ) ¼ 0 and Re(ε whm,y ) ! ∞, the corresponding IFC should be rather flat, which can be used to realize the canalization effect. [5] Above 0.533 GHz, the hyperbolic IFCs should open along the AEk y -directions, because Re(ε whm,x ) > 0 and Re(ε whm,y ) < 0 (i.e., the central line of each continuous branch is along the k y axis). Below 0.533 GHz, the hyperbolic IFCs should open along the AEk x -directions, because Re(ε whm,x ) < 0 and Re(ε whm,y ) > 0. However, below 0.403 GHz, the WHM acts as an anisotropic plasmonic metal, and there are no bulk waveguide modes existing, because Re(ε whm,x ) and Re(ε whm,y ) are both negative. To verify the above-mentioned prediction, the IFCs at various frequencies are obtained by COMSOL ( Figure 2). The simulated result is consistent with the qualitative analysis. By the above-mentioned investigation, it is demonstrated well that the constructed WHMs possess hyperbolic IFCs in a broadband, because they do not depend on localized resonance.
Note that we use EMT just to show that a constructed wavetype composite material behaves like an HM as a simple illustration. The accurate description of the propagating-mode behavior, considering the nonlocal effect and the influence of the added metal wires, is given in Figure 1b-d and 2 in terms of IFCs based on rigorous full-wave simulation.

Constructing WHM Cavities
The supported ultrahigh wave vectors enable one to construct subwavelength cavities based on WHMs. Such a cavity configuration is shown in Figure 3a, whose cross section is given in Figure 3b. It consists of five pairs of alternately arrayed anorthite slices and barium tetratitanate slices, bounded by two copper plates at the top and bottom. Each slice has a size of 10 Â 1 Â 60 mm 3 (along the x-, y-, and z-directions, respectively), www.advancedsciencenews.com www.adpr-journal.com and silver wires (0.2 Â 0.02 Â 60 mm 3 ) are distributed on the two large surfaces of each barium tetratitanate slice with a period of p x ¼ 1 mm along the x-direction. The total transverse size of the WHM cavity is small: The width along the x-direction is L x ¼ 10 mm, and the thickness along the y-direction is also L y ¼ 10 mm (the thickness of the silver wires is ignored here). The top and bottom copper plates are very thin with a size of 10 Â 10 Â 0.2 mm 3 . When some propagating mode, performing a round trip inside the WHM cavity, fulfills the Fabry-Pérot (FP) resonance condition, a cavity resonant mode can be formed. For analysis convenience, the WHM cavity is first assumed to have boundary conditions of PMC at AExand AEy-directions to eliminate the phase induced by the boundary reflection. A resonant mode is found to appear at 0.725 GHz by simulation. As shown by the electric mode pattern given in Figure 3c (normalized to some large value to saturate the illustrating color around the metal wires), there is one node in the electric field of E x along the y-direction, and also in E y along the x-direction. The normalized magnetic pattern of H z has only an antinode at the center; thus, one may call this mode as resonant mode (1,1). [10] By observing the resonant mode patterns, one can know that the propagation phases inside the cavity along the xand y-directions are both π/2. Thus, the wave vector of the propagating mode supported by the WHM constituting the cavity can be determined by k x ¼ π/2/L x and k y ¼ π/2/L y . Then, by referring to Figure 2b, it is found that the propagating mode should have a working frequency of 0.717 GHz, nearly equal to that of resonant mode (1,1), which illustrates that this resonant mode originates from the underlying WHM. When the cavity is put back into the air, resonant mode (1,1) shifts to 0.890 GHz (Figure 3 d). The electric and magnetic fields leak out of the cavity, and the cavity boundaries induce some reflection phases.
It should be pointed out that another kind of resonant modes can exist inside a WHM cavity besides the resonant modes depicted by the corresponding bulky propagation modes inside the WHM. For example, near resonant mode (1,1) in Figure 3 d, such a special resonant mode exists at 0.701 GHz (Figure 3e). For this mode, the components of E y and H z are mainly localized near the cavity boundaries, and the component of E x does not change its sign along the y-direction. This resonant mode origins from the SPL waveguide mode along a metal-dielectric multilayer, [36] which will be called a SPL resonant mode thereafter.
Due to the unique shapes of hyperbolic IFCs, the so-called anomalous scaling law predicts that narrowing a WHM cavity in one direction may decrease the resonance frequency counterintuitively. [10] To demonstrate this property, three other WHM cavities of different transverse sizes are investigated with the WHM cavity investigated in Figure 3 d acting as a reference. Cavity 1 has a smaller thickness of L y ¼ 6 mm compared with the reference cavity (the ceramic slice number is decreased). The propagating wave-vector component of k y is required to increase to maintain the FP resonance condition, and this enforces resonant mode (1,1) to blueshift according to Figure 2b. By simulation, it is found that the resonance frequency of cavity 1 does shift from 0.890 to 0.970 GHz, which is verified by Figure 4a. Cavity 2 has a smaller width of L x ¼ 6 mm compared with the reference cavity (the ceramic slices are narrowed, and the number of the metal wires is also decreased correspondingly while each individual metal wire keeps unchanged). The propagating wave-vector component of k x is also required to increase to maintain the FP resonance condition; thus, resonant mode (1,1) The left column is for E x , the middle column for E y , and the right column for H z . c,d) These are for resonant mode (1,1) when the cavity is surrounded by a perfect magnetic conductor (PMC) condition or air, respectively, and e) this is for a surface-plasmon-like (SPL) mode when the cavity is in the air.
www.advancedsciencenews.com www.adpr-journal.com should be redshifted according to Figure 2b. As confirmed by simulation, the resonance frequency of cavity 2 is redshifted from 0.890 to 0.777 GHz (Figure 4b).
The above-mentioned investigation shows that compressing the WHM cavity along the xand y-directions produce opposite effects, which confirms the anomalous scaling behavior. The anomalous scaling law indicates that the working frequency of resonant mode (1,1) may change little when one simultaneously shrinks the width and thickness of the WHM cavity in some proportion. A cavity with L x ¼ L y ¼ 6 mm is investigated in Figure 4c. The resonant frequency is 0.846 GHz, near that of the cavity in Figure 3 d. To achieve an ultrasmall cavity, the transverse size of the cavity in Figure 4c is further reduced to 3 mm Â 3 mm (one anorthite slice was sandwiched between two barium tetratitanate slices along the y-direction). For this extremely reduced structure, the pattern of resonant mode (1,1) still preserves (similar to the mode pattern of the 10 mm Â 10 mm cavity), as shown in Figure 4 d, but the resonance frequency is shifted from 0.89 to 0.722 GHz. Compared with the free-space resonant wavelength (λ 0 ¼ 415.2 mm), the present cavity is rather small, especially in the transverse size (λ 0 /138.4 Â λ 0 /138.4 Â λ 0 /6.9). The quality factor has a moderate value Q cav ¼ 182, and the mode dissipation mainly comes from the absorption loss of the metal wires.
In fact, a WHM cavity has a potential to possess a higher transverse confinement. As an example, two barium tetratitanate slices and one anorthite slice are alternately arranged (each slice is of 200 μm Â 50 μm Â 54.2 mm), and two metal wires (20 μm Â 20 μm Â 54.2 mm) are adopted on each large surface of the ceramic slices of high permittivity. This ultrasmall cavity is covered by two copper plates of 5 μm thickness at the top and bottom. Resonant mode (1,1) occurs at 0.823 GHz with Q cav ¼ 12.6. This cavity (λ 0 /1821 Â λ 0 /2143 Â λ 0 /6.7) is the smallest design in the transverse size as far as we know. It is further hoped to realize a 3D ultracompact cavity by reducing the height of the WHM cavity. Considering the FP resonance condition is fulfilled along the cavity height, this problem may be resolved using metamirrors supporting nonzero reflection phases or adopting higher permittivity dielectric materials.

Characterizing an Ultrasmall WHM Cavity
For experimental demonstration, the WHM cavity investigated in Figure 4c is correspondingly fabricated following the same geometry and material parameters, and the sample is shown in Figure 5a. In measurement, a coaxial cable connected with a vector network analyzer (Rohde & Schwarz ZVA40) acts as an electric-dipole antenna to stimulate and detect the cavity resonances, of which the cable core's front part (about 120 mm in length) is made nude. Because the cavity modes are highly localized, their direct stimulation is difficult. Thus, the cavity Figure 4. Anomalous scaling law followed by WHM cavities. a,d) Normalized E y distributions of resonant mode (1,1) for four cavities of different transverse sizes. Compared with the reference cavity in Figure 3, their transverse sizes are reduced to 10 Â 6, 6 Â 10, 6 Â 6, and 3 Â 3 mm 2 , respectively, and the resonance frequencies are 0.970, 0.777, 0.846, and 0.722 GHz, respectively. www.advancedsciencenews.com www.adpr-journal.com is laid down upon a large copper substrate with a tiny gap of 0.1 mm (spaced by a layer of sticky tape) between them. The left cavity surface parallel to the y-z plane (see Figure 3a,b) faces the substrate. The copper substrate can force a larger part of mode pattern to reside in the surrounding environment. The corresponding measured result of S 11 is shown in Figure 5b. A resonant mode with Q cav ¼ 95 is observed at 0.709 GHz. This mode corresponds to resonant mode (1,1) shown in Figure 3 d. In simulation, the introduction of the copper substrate shifts the resonance frequency from 0.722 to 0.73 GHz with Q cav nearly unchanged (the dashed curve in Figure 5b represents the numerical result of S 11 obtained by the full-wave software of CST Microwave Studio). The deviation between the measured and numerical results is induced by the fabrication error. In Figure 5b, one can also observe another resonant mode at 0.652 GHz with Q cav ¼ 55. This corresponds to the SPL resonant mode shown in Figure 3e, whose simulated resonance frequency is 0.665 GHz with Q cav ¼ 111.

Conclusion
In conclusion, the waveguide approach allows for the construction of new-type microwave HMs. Ultrahigh wave vectors have been confirmed by the construction of ultrasmall WHM cavities. The anomalous scaling law has also been examined for WHM cavities. Our suggestion enables flexible variation of HMs, eliminating the limit of wire-type HMs in the microwave regime. As a low-loss platform, WHMs are predicted to flexibly enable distinctive phenomena in the microwave regime, and find their potential roles in many aspects, including microwave imaging and sensing, microwave communicating and integrating, controlling the behavior of microwave transition, and Cerenkov electron radiation. The proposed WHMs are potentially scalable across the electromagnetic spectrum from radio to terahertz to extend their significance.