# Ultra-Efficient Wideband Multi-Layer Dielectric Resonator Antennas and Arrays

*A multi-layer dielectric resonator antenna (DRA) with a coaxial feed has a wide radiating and impedance bandwidth of 43 percent, from 8 to 12 GHz, thereby covering the entire X-Band. A single DRA exhibits a high gain of 8.2 dB at 10 GHz with an efficiency greater than 95 percent over the entire band. The single DRA includes a coaxial feed and a metal reflector cavity. The dielectric portion comprises an inner layer with a dielectric constant of 10.5 and an outer layer with a dielectric constant of 2.1. Prototypes of single DRAs are fabricated with unit cell sizes of 20 × 20 mm and 22 × 22 mm. RF measurements of single DRAs agree closely with simulations. Measurements of a 5 x 5 steerable array including a Rotman-Turner lens for steering closely matches simulated results over the entire 8 to 12 GHz frequency range for multiple steering angles from 0 to 30 degrees. The 5 x 5 array has a measured gain of 21.5 dB at 10 GHz.*

Antennas are critical parts of many electromagnetic (EM) systems since they couple near-field operation with far-field EM radiation. Most conventional antennas leverage current flow in metals to generate EM radiation, however metal antennas exhibit degraded efficiency at elevated frequencies due to the skin effect and metal roughness. In addition, PCB-based 2D metal antennas have limited bandwidth and gain.

DRAs are 3D antennas formed of low loss dielectric materials that provide efficient emission and reception of EM radiation. For DRAs, EM energy in the so called “near zone” is almost entirely supported by displacement currents, which in contrast to conductive currents, are not subject to skin effect. Also, in contrast to metal-based PCB antennas, DRAs operate based on the principle of exciting multiple resonant EM modes in the dielectric structure, which enables very efficient operation up to mmWave frequencies.

Furthermore, when properly designed, the extended 3D shape of DRAs enables inherent and efficient wideband radiation operation with consistently shaped radiation patterns over the entire operating band.

**RADIATING AND NON-RADIATING MODES OF A DIELECTRIC RESONATOR**

Dielectric resonator (DR) resonant modes represent different EM field distributions. Their mathematical representations are found as solutions to the Maxwell-Helmholtz equation under certain symmetries and boundary conditions. The symmetry defines the type of mathematical function and the boundary conditions identify the mode. For example, if the symmetry is spherical, cylindrical or rectangular, the solution to the Maxwell-Helmholtz equation will be spherical harmonics, Bessel functions or the usual sine and cosine functions, respectively.

The most common resonator is a cylindrical resonator, also referred to as a puck, operating in its fundamental transverse electric TE_{01δ} mode, or its lowest resonant frequency. Because of the cylindrical shape and symmetry, all eigenstates or resonant modes are represented by Bessel functions of the first and second kind, inside and outside the resonator, where the boundary conditions are reflected in the mode index.

For example, for the TE_{01δ} mode, the first index (0) is indicative of radial symmetry with no azimuthal cyclical structure. The second index (1) indicates that the electric field has one maximum, as represented by the Bessel function J_{1}(kr), where k is the wavenumber along the resonator radius. More intuitively, the second index represents a Bessel type function with a half wavelength fitting within the radius of the resonator. Finally, the third index (δ) is associated with the mode structure along the z-direction normal to the cylindrical resonator’s top and bottom surfaces and is associated with the mode’s functional form along the z-direction. In the case of a puck in a “sandwich” configuration, where the dielectric is placed between two metal plates, δ = 1, and in the most general case, 0 < δ < 1.

The field structure of DR higher order modes is more complex and there are various ways to represent them. The method introduced above, based on φ, r and z coordinates, was proposed by Kobayashi-Senju and Zaki-Atia for the microwave regime and Snitzer in the optical regime.^{1-3} They also introduced the EH or HE designation for hybrid modes depending on which field, electric or magnetic, contributes the 0most dominant component in the z-direction.

In contrast, we propose an intuitive method of mode designation based on fundamental TE_{01δ }and TM_{01δ }transverse magnetic (TM) modes. We omit the first and the third index, therefore referring to the fundamental modes (i.e., unit cells) as TE_{1} and TM_{1}, which represent magnetic and electric dipoles (MD and ED). **Figure 1** illustrates the electric and magnetic field distributions for the fundamental TE_{1 }and TM_{1} unit cells. For these fundamental modes, the field lines in the XY plane are parallel to the top and bottom surfaces of the cylindrical resonator and exhibit cylindrical symmetry, therefore they are very close to “pure” TE and TM modes.

Any higher order mode can be represented as a combination of multiple EDs or MDs. In our case though, TE or TM means “quasi” TE or TM. Examples of higher order modes are electric and magnetic octupole modes, HE_{22δ} and EH_{22δ }in the Kobayashi representation, which are shown schematically in **Figure 2**. Simulated fields corresponding to these modes are shown in **Figure 3**. In the proposed representation, they are referred to as 4TM_{1} and 4TE_{1}, or four ED and four MD.

No matter how complicated the mode, it can be represented in terms of an even number of alternating electric or magnetic dipoles. The cyclic number, the first index in Kobayashi designation, which at the same time represents the azimuth structure of the Bessel functions, defines the number of alternating pairs in our representation. For example, for the HE_{22δ} mode, the first index (2) means that there are two pairs (four nodes in azimuth), therefore four dipoles in our notation. The number of alternating dipoles is twice the first Kobayashi index.

The proposed method lacks the mathematical rigor of the Kobayashi method, but it offers a lot of advantages from the point of view of providing an intuitive understanding, a field visualization and an enabler for recalling the mode configuration. This representation is based on dipoles and is in complete alignment with the fundamental structure of electromagnetism and Maxwell equations, particularly the first two, which in principle represent infinitesimal electric and magnetic dipole structures.

The representation also aids in recollection of the mode-frequency hierarchy. More electric or magnetic loops imply more alternating dipoles, and hence a higher corresponding mode frequency. The method also provides a good understanding of the mode Q (quality factor), since the higher the number of dipoles, the tighter the fields are bound to the resonator and the closer the Q is to the inverse of tan(δ). The method also helps in providing and improving ways to excite the modes.

Most importantly, the method clearly shows the mode radiating properties, since the alternating dipoles represent evanescent modes, where their far fields cancel systematically and form very inefficient overall radiators. Only modes that are represented by unpaired dipoles can radiate efficiently, providing the two families of TE and TM unit cell modes: TE_{01δ} (TE_{1}), TE_{02δ} (TE_{2}), TE_{03δ} (TE_{3}), … TE_{0nδ} (TE_{n}) and TM_{01δ} (TM_{1}), TM_{02δ} (TM_{2}), TM_{03δ} (TM_{3}), … TM_{0nδ} (TM_{n}).

From now on, we omit the last index δ and call the modes simply TE_{01}, TE_{02}, TE_{03}, … TE_{0n}.

As mentioned before, TE modes have radial symmetry, therefore the system of the field lines can be supported by a half resonator on a metal surface (i.e., an electric wall), which also plays the role of electric mirror.

**Figure 4a** shows one of the symmetry planes of the TE01 electric field lines. In **Figure 4b**, the plane is replaced by an electric wall, such as a metal, and half of the resonator is removed. Boundary conditions are still satisfied and the TE01 mode (together with all other members of TE family) is still supported. The half resonator can be a half cylinder, a half sphere (or dome) or a half ellipsoid.

This structure is used as the foundation of our designs because it offers an ideal way to excite and couple to DR TE modes and fully exploit their excellent radiation properties. The modifications in resonator shape and symmetry introduced by the electric wall suppresses DR modes with electric fields parallel to the ground. TM modes, as originally defined above, are also suppressed. However, the presence of the electric wall forming a ground surface creates favorable boundary conditions for a new family of modified TM modes. A strong electric field can be sourced from the ground and supported by the half resonator height, as shown in **Figure 5**.

In fact, the radiating bandwidths of DRAs is determined by an interplay between radiating TE and TM modes (see **Figure 6**). As explained above, only TE and TM modes of radial symmetry, which are represented by unpaired dipoles, are efficient radiators. They also have very different radiation patterns: TE modes radiate maximally at boresight and minimally sideways (see Figure 6a), where TM modes radiate in exactly the opposite manner, minimally at boresight and maximally sideways (see Figure 6b).

For a typical DRA with a uniform dielectric constant spatial distribution, such as half cylinder DRA on a metal ground plane, TE and TM modes are present in the frequency spectrum in an alternating sequence (see **Figure 7**). The TE_{01 }mode appears at the low end of the spectrum, followed by the TM_{01 }mode, the TE_{02} mode, the TM_{02 }mode and so on. Although such a DRA structure has a wide impedance bandwidth, the boresight radiating bandwidth is severely limited due to the alternation in the TE and TM modes in the spectrum.

**DESIGN AND FABRICATION**

Depending on the application and the required direction of the radiation, either TE or TM modes may be desired. In addition, in many applications, a wide radiating bandwidth formed by the same type of radiating modes is desirable. The DRA design presented herein has an air center (see **Figure 8**) to provide a wide radiating bandwidth formed by multiple TE modes. It accomplishes this by suppressing side-radiating TM modes within the radiating frequency band and shifting the spectral location of the TM modes to higher frequencies. **Figure 9** shows a representative gain and |S_{11}| of the DRA and depicts a wide radiating bandwidth of TE modes. Such DRAs can achieve TE mode radiating bandwidths of over 40 percent.

Wideband DRAs described in this work are multi-layer with an air center and a dielectric portion containing an inner layer having a dielectric constant of 10.5 and an outer layer having a dielectric constant of 2.1. The inner layer is a thermoplastic filled with filler particles, and the outer layer is unfilled PTFE. All dielectric portions are CNC machined and the inner layer is pressure fit to the outer layer. The DRA includes a coaxial feed and an aluminum reflector cavity to provide isolation between adjacent DRAs in an array. **Figure 10** shows the fabricated components and the assembled DRA. Two types of DRAs are fabricated, one with a metal reflector base size of 20 × 20 mm (referred to as unit 1), and the other unit with a metal reflector base size of 22 × 22 mm (referred to as unit 2).

**RESULTS AND DISCUSSION**

**Individual Antenna Elements**

The DRA units are each measured with a vector network analyzer (VNA); |S_{11}| versus frequency shows good agreement with simulation (see **Figure 11**). |S_{11}| is less than -10 dB from slightly under 8 to slightly over 12 GHz, providing a measured impedance bandwidth of 43 percent. Furthermore, |S_{11}| is insensitive to the size of the metal reflector, which is indicative of high mode confinement to the dielectric structure and the cavity. Also, |S_{11}| of the measured units exhibits distinct minima in the matched band from 8 to 12 GHz, which is attributed to the various TE modes supported by the DRA.

**Figure 12** shows the measured gain on boresight of the DRA units, which corresponds closely with the simulation and is consistently high from about 8.75 to 12 GHz due to the presence of adjacent TE modes in the frequency spectrum. A single DRA unit is simulated and measured boresight gain is about 8.2 dB at 10 GHz. Furthermore, RF simulations show that the DRAs have radiation efficiencies greater than 95 percent over the entire 8 to 12 GHz band.

To understand the benefits of the 3D nature of the DRA design, the aperture efficiency of a 20 × 20 mm aperture, with no additional ground, is calculated using the well-known formula:

where G is the gain of the antenna, A is the base area of the antenna aperture and λ is the wavelength.

At 10 GHz, the measured gain is 8.2 dB for the 20 × 20 mm DRA, and hence the aperture efficiency based on the equation is calculated to be 118 percent. It is greater than 100 percent because the formula provides the aperture efficiency for a 2D aperture, where the true aperture for a DRA is the entire curved radiating surface encompassing the DRA element.

**Figure 13** shows E-plane and H-plane radiation pattern measurements at 10 GHz for both DRA units. The measurements demonstrate that the reflector size beyond 20 × 20 mm has little influence on boresight gain, indicative of the radiation mode source being dominantly controlled by the EM mode in the dielectric portion of the DRA.

**Array Antenna**

To demonstrate the array performance of multiple DRAs in a typical steerable configuration used for various applications, a 5 × 5 DRA array is created. **Figure 14a** is a photograph of the 5 × 5 array of DRAs, where each DRA is fed with separate coaxial feeds. The assembly is contained within a frame that holds all the DRAs. **Figure 14b** shows the 5 × 5 DRA array fed with a Rotman-Turner lens for steering the resultant radiated beam up to angles of 30 degrees.

**Figure 15** shows simulated and measured radiation patterns of the 5 × 5 DRA array at 10 GHz along the steering direction plane for the Rotman-Turner lens beam ports associated with 0-, 10-, 20- and 30-degree steering angles. The results show good agreement between measured and simulated radiation patterns.

**CONCLUSION**

A very efficient, wideband and steerable DRA at 10 GHz is based on the radiation properties of a multi-layer dielectric resonator. The radiating bandwidth is maximized via an interplay between TE and TM radiating modes. The resonator shape, multi-layer structure and unit cell architecture favor multiple TE radiating modes in spectral proximity. Furthermore, radiating TM modes are suppressed and shifted to higher frequencies utilizing an air center. The technique provides a flat and high boresight gain over a wide frequency band. DRA designs are scalable over a broad frequency range from C-Band to automotive radar frequencies and beyond. We strongly believe that the technology provides valuable tradeoffs that cannot be readily achieved with conventional antenna technologies.

**ACKNOWLEDGMENTS**

We thank Stephen O’Connor at Rogers Corporation for material formulation. We also thank Shawn Williams, Karl Sprentall and Bob Daigle at Rogers Corporation for valuable discussions and feedback.

**References**

- Y. Kobayashi and T. Senju, “Resonant Modes in Shielded Uniaxial-Anisotropic Dielectric Rod Resonators,”
*IEEE Transactions on Microwave Theory and Techniques*, Vol. 41, No. 12, December 1993, pp. 2198–2205. - K. A. Zaki and A. E. Atia, “Modes in Dielectric-Loaded Waveguides and Resonators,”
*IEEE Transactions on Microwave Theory and Techniques*, Vol. 31, No. 12, December 1983, pp. 1039–1045. - E. Snitzer, “Cylindrical Dielectric Waveguide Modes,”
*Journal of the Optical Society of America*, Vol. 51, No. 5, May 1961, pp. 491–498.