Quasi-conformal Transformation Research Articles

A Miniaturized High-Gain Multi-Beam Lens Antenna Based on Quasi-Conformal Transformation Optics

A Miniaturized High-Gain Multi-Beam Lens Antenna Based on Quasi-Conformal Transformation Optics

All-dielectric carpet cloaks with three-dimensional anisotropy control

Abstract In this article, we propose all-dielectric carpet cloaks composed of jungle gym shaped dielectric unit cells and present a design strategy for three-dimensional (3-D) anisotropy control based on the transformation optics. The carpet cloaks are 3-D printable and operate with polarization independent incident waves in arbitrary incident angles due to the 3-D anisotropy control. Realizable anisotropic permittivities of cubic and rectangular unit cells are numerically studied based on the relative permittivity and loss tangent of ɛ r = 2.9 and tan δ = 0.02 of ultra-violet curing resin measured at the microwave frequency. It is shown that the unit cell has little frequency dependence even with the anisotropy in the low frequency range where the effective medium approximation is valid. A carpet cloak is designed based on the design method with a quasi-conformal coordinate transformation and implemented with the unit cells taking into account its realizable anisotropy. Polarization independent 3-D cloaking operations of the designed cloak are confirmed numerically. The designed cloak is fabricated by stereolithography 3-D printing technology and its cloaking performances are evaluated experimentally at 10 GHz. It is shown that non-specular reflections are well suppressed by the carpet cloak for both TE and TM incident waves with different incident angles of 30, 45, and 60°. Frequency independent cloaking operations are also shown experimentally in the X-band. The measured near-field distributions and bistatic radar cross sections are in good agreement with simulated predictions and the validity of the design method is confirmed.

Broadband flattened underwater acoustic Luneburg lens

Conventional Luneburg lenses are symmetric circular gradient-index lens with no aberration, but they are restricted by their circular focal surface. Here, we show the design, realization and measurement of an underwater acoustic Luneburg lens with flattened focal surface. The flattened lens is designed by using the quasi-conformal transformation technique and constructed by metamaterials based on air-filled photosensitive resin. Through numerical simulations and experimental studies, the flattened lens is demonstrated to have excellent focusing performance for the incident plane waves propagating at desired angles over a broad frequency band from 20 kHz to 35 kHz. The proposed flattened Luneburg lens can be potentially applied in the applications such as biomedical imaging, underwater acoustic sensing and communication.

Wide-Angle Ceramic Retroreflective Luneburg Lens Based on Quasi-Conformal Transformation Optics for Mm-Wave Indoor Localization

This paper presents a quasi-conformal transformation optics (QCTO) based three-dimensional (3D) retroreflective flattened Luneburg lens for wide-angle millimeter-wave radio-frequency indoor localization. The maximum detection angle and radar cross-section (RCS) are investigated, including an impedance matching layer (IML) between the lens antenna and the free-space environment. The 3D QCTO Luneburg lenses are fabricated in alumina by lithography-based ceramic manufacturing, a 3D printing process. The manufactured structures have a diameter of 29.9 mm ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$4 \lambda _{0}$ </tex-math></inline-formula> ), showing a maximum realized gain of 16.51 dBi and beam steering angle of ±70° at 40 GHz. The proposed QCTO Luneburg lens with a metallic reflective layer achieves a maximum RCS of −20.05 dBsqm at 40 GHz with a wide-angle response over ±37°, while the structure with an IML between the lens and air improves these values to a maximum RCS of −15.78 dBsqm and operating angular response between ±50°.

Field Decorrelation and Isolation Improvement in an MIMO Antenna Using an All-Dielectric Device Based on Transformation Electromagnetics.

This work presents a new technique for enhancing the performance of a multiple-input multiple-output (MIMO) antenna by improving its correlation coefficient ρ. A broadband dielectric structure is designed using the transformation electromagnetics (TE) concept to decorrelate the fields of closely placed radiating elements of an MIMO antenna, thereby decreasing ρ and mutual coupling. The desired properties of the broadband dielectric wave tilting structure (DWTS) are determined by using quasi-conformal transformation electromagnetics (QCTE). Next, the permittivity profile of the DWTS is realized by employing air-hole technology, which is based on the effective medium theory, and the DWTS is fabricated using the additive manufacturing (3D printing) technique. The effectiveness of the proposed technique is verified by designing two-element patch-based MIMO antenna prototypes operating at 3 GHz, 5 GHz, and 7 GHz, respectively. The proposed technique helped to reduce the correlation coefficient ρ in the range of 37% to 99% in the respective operating bandwidth of each MIMO antenna, thereby, in each case, improving the isolation between antenna elements by better than 3 dB, which is an excellent performance.

GENERALIZ GENERALIZATION OF THE H TION OF THE HARDY CLASS FOR AN ASS FOR ANALYTIC FUNCTIONS

Introduction. Quoting from a well-known American mathematician Lipman Bers [1] “It would be tempting to rewrite history and to claim that quasiconformal transformations have been discovered in connection with gas-dynamical problems. As a matter of fact, however, the concept of quasiconformality was arrived at by Grotzsch [2] and Ahlfors [3] from the point of view of function theory”. The present work is devoted to the theory of analytic solutions of the Beltrami equation which directly related to the quasi-conformal mappings. The function is, in general, assumed to be measurable with almost everywhere in the domain under consideration. Solutions of equation (1) are often referred to as analytic functions in the literature. Research methods.

Broadband wide-angle terahertz antenna based on the application of transformation optics to a Luneburg lens

The design of antennas for terahertz systems remains a significant challenge. These antennas must provide very high gain to overcome significant free-space path loss, which limits their ability to broadcast or receive a beam over a wide angular range. To circumvent this limitation, here we describe a new device concept, based on the application of quasi-conformal transformation optics to the traditional Luneburg lens. This device offers the possibility for wide-angle beam steering and beam reception over a broad bandwidth, scalable to any frequency band in the THz range.

Synthesis of a Luneburg lens with a flat surface using quasi-conformal transformation optics

Annotation In modern systems of radiolocation, navigation and communication, the requirements for antennas are becoming higher requirements every year, namely: operation in a wide frequency range, the ability to change of direction of the main lobe of the radiation pattern. Antenna systems with similar characteristics can be built using dielectric antenna beamforming structures. One of these structures is the Luneberg lens, the peculiarity of which is its spherical symmetry. However, the curved surface of this lens significantly complicates the placement of transmitting and receiving elements along it, which increases the complexity of constructing the entire antenna system. This paper proposes an algorithm for constructing a Luneberg lens with a flat surface. The lens was synthesized using the method of quasi-conformal optical transformations, the mathematical algorithm of which is also described in this work. The paper also presents the results of mathematical modeling of the antenna system using a Luneberg lens with a flat surface at different positions of the emitter relative to the center of the lens, as well as different cut angles. The simulation results show that the synthesized lens can be used to construct a multi-beam antenna system that allows the direction of the main lobe of the antenna radiation pattern to be rearranged over a wide range of angles. However, the scanning angles of this system are limited by the lens geometry, the larger the maximum scanning angle we choose, the more significant the influence of the side lobes on the radiation pattern becomes.

Numerical quasiconformal transformations for electron dynamics on strained graphene surfaces.

The dynamics of low-energy electrons in general static strained graphene surface is modelled mathematically by the Dirac equation in curved space-time. In Cartesian coordinates, a parametrization of the surface can be straightforwardly obtained, but the resulting Dirac equation is intricate for general surface deformations. Two different strategies are introduced to simplify this problem: the diagonal metric approximation and the change of variables to isothermal coordinates. These coordinates are obtained from quasiconformal transformations characterized by the Beltrami equation, whose solution gives the mapping between both coordinate systems. To implement this second strategy, a least-squares finite-element numerical scheme is introduced to solve the Beltrami equation. The Dirac equation is then solved via an accurate pseudospectral numerical method in the pseudo-Hermitian representation that is endowed with explicit unitary evolution and conservation of the norm. The two approaches are compared and applied to the scattering of electrons on Gaussian shaped graphene surface deformations. It is demonstrated that electron wave packets can be focused by these local strained regions.

Hemispherical acoustic Luneburg lens with the acoustic Goos\u2013H\xe4nchen shift and Fresnel filtering effect

A two-dimensional (2D) slice of a 3D hemispherical acoustic Luneburg lens using a quasi-conformal transformation and face-centred-orifice-cubic (FCOC) unit cells is designed and fabricated. With the system, the focusing characteristics of acoustic waves with frequencies that satisfy the homogeneous medium condition of the metamaterial are observed, such as focusing of acoustic plane waves at the antipodal point on the transformed surface of the opposite side for the incident direction and focus spreading due to total internal reflection at the focus point. The attenuation losses of the system are measured and compared with those of an untransformed system with respect to frequency. The value of the acoustic Goos–Hänchen shift is determined by comparing the experimental and theoretical and simulated values of the focus points with respect to the incident angle. The effect of acoustic Fresnel filtering due to the angular distribution of the incident waves at the flat surface boundary is verified by comparing the results of the experiment and a simulation.

Rectangular Maxwell’s fisheye lens via transformation optics as a crossing medium for dissimilar waveguides

Different material platforms, each offering a set of unique advantages, have been introduced for photonic integrated circuits. These platforms may be used in the same photonic chip to explore their advantages. Consequently, the in-plane crossing of dissimilar waveguides may be inevitable. In this paper, we investigate, for the first time to our knowledge, the crossing of silicon nitride ( S i 3 N 4 ) and silicon (Si) strip waveguides. Recently, the imaging property of the Maxwell’s fisheye lens has been exploited to design waveguide crossings. However, the refractive indices of S i 3 N 4 and Si are different by about 1.45, so the crossing medium should have different refractive indices at its edges to minimize reflection from the interface of the waveguide and the crossing medium. We utilize quasi-conformal transformation optics to design a rectangular Maxwell’s fisheye lens as a crossing medium of S i 3 N 4 and Si strip waveguides. We implement the rectangular lens with graded photonic crystal and numerically evaluate the performance of the designed waveguide crossing.

High gain, wide-angle QCTO-enabled modified Luneburg lens antenna with broadband anti-reflective layer

The gradient-index (GRIN) Luneburg lens antenna offers significant benefits, e.g. high aperture efficiency, low-power, minimal cost, wide beam scanning angle and broad bandwidth, over phased array antennas and reflector antennas. However, the spherical shape of the Luneburg lens geometry complicates the integration of standard planar feed sources and poses significant implementation challenge. To eliminate the feed mismatch problem, the quasi-conformal transformation optics (QCTO) method can be adopted to modify the lens’ spherical feed surface into a planar one. However, Luneburg lenses designed with QCTO method are limited to poor performance due to the presence of the reflections and beam broadening arising from the quasi-conformal mapping. In this paper, we present a new method of implementing QCTO-enabled modified Luneburg lens antenna by designing a broadband anti-reflective layer along with the modified lens’s planar excitation surface. The proposed anti-reflector layer is inherently broadband in nature, has a continuously tapered inhomogeneous dielectric permittivity profile along its thickness, and ensures broadband impedance matching. To show the new QCTO modified Luneburg lens antenna, an example lens antenna was designed at Ka-band (26–40 GHz) and fabricated using fused deposition modeling (FDM) based additive manufacturing technique. Electromagnetic performance of the lens antenna was experimentally demonstrated.

Flattened structural Luneburg lens for broadband beamforming.

A conventional structural Luneburg lens is a symmetric circular gradient-index lens with refractive indices decreasing from the centre along the radial direction. In this paper, a flattened structural Luneburg lens (FSLL) based on structural thickness variations is designed by using the quasi-conformal transformation technique. Through numerical simulations and experimental studies, the FSLL is demonstrated to have excellent beam steering performance for the manipulation of flexural wave propagation at desired angles.

Silicon nitride waveguide devices based on gradient-index lenses implemented by subwavelength silicon grating metamaterials.

The rapid development of photonic integrated circuits demands the design of efficient and compact waveguide devices such as waveguide tapers and crossings. Some components in the silicon nitride (SiN) waveguide platform are superior to their counterparts in the silicon waveguide platform. Designing a compact SiN waveguide taper and crossing is crucial to reduce the size of SiN photonic components. In this paper, we utilize the focusing property of the Luneburg lens to design an SiN taper connecting a 10-µm-wide waveguide to a 1-µm-wide waveguide. Three-dimensional full-wave simulations indicate that the designed 13-µm-long taper has an average transmission efficiency of 92% in the wavelength range of 1500-1600 nm. We also present an in-plane SiN waveguide crossing based on the imaging property of the square Maxwell's fisheye lens designed with quasi-conformal transformation optics. The designed waveguide crossing occupies a compact footprint of 5.65µm×5.65µm, while its average insertion loss is 0.46 dB in the bandwidth of 1500-1600 nm. To the best our knowledge, the designed SiN waveguide taper and crossing have the smallest footprints to date.

X Wave Radiator Implemented With 3-D Printed Metamaterials

A radiator is presented, capable of generating paraxial X waves over a 50% fractional bandwidth (7.5–12.5 GHz) in its radiative near-field. X waves are localized pulses formed by the superposition of Bessel beams with a common cone angle. Quasiconformal transformation optics, a 2-D variant of transformation optics, is employed to find the inhomogeneous, isotropic dielectric constant profile needed to convert the radiation of a monopole into a paraxial Bessel beam. An impedance matching layer is applied to reduce reflections at the air-dielectric interface of the device. The transformation and impedance matching regions are implemented using rotationally symmetric metamaterial unit cells that yield a spatially varying effective dielectric constant. The resulting design is fabricated through 3-D printing, by combining parts made from three low-loss dielectric filaments. The device is experimentally measured, and shows good agreement with simulations. Its ability to generate paraxial X waves when excited by a broadband pulse is verified.

Ultrasound beam steering with flattened acoustic metamaterial Luneburg lens

We report ultrasound beam steering based on 2D and 3D flattened acoustic metamaterial Luneburg lenses at 40 kHz. The effective properties of the lenses are obtained by using the quasi-conformal transformation technique and solving the Laplace equation with Dirichlet and Neumann boundary conditions. A 2D lens and a 3D lens were designed and fabricated. The numerical and experimental results with these lenses demonstrate excellent beam steering performance of ultrasonic waves in both near field and far field.

Active control of sound waves via three-dimensional quasi-conformal mapping

We describe use of an optimization algorithm to produce three-dimensional, quasi-conformal transformation acoustics. The results indicate that the anisotropy of the transformed material can be made arbitrarily small by increasing the auxiliary function degrees of freedom. A boundary function is defined to prevent the algorithm from affecting the original device function. Bent waveguides with isotropic non-resonant phononic crystals were fabricated. The transmission performances of the optimized and initial transformed waveguides were studied to validate the proposed design method. Numerical simulations predicted the waveguide transmission modes. These were then demonstrated experimentally by measuring the sound pressure components at the outlets.

Coupling Si3N4 waveguide to SOI waveguide using transformation optics

Silicon nitride (Si3N4) planar waveguide platform combined with silicon-on-insulator (SOI) devices offer a whole new generation of system-on-chip applications. Therefore, efficient coupling of an Si3N4 waveguide to an SOI waveguide is essential. We present a coupler to interface a 1.8μm-wide Si3N4 waveguide to a 0.5μm-wide SOI waveguide based on the focusing property of the Luneburg lens. In order to match the refractive indices of the waveguides with the edges of the lens, one side of the lens is flattened with quasi-conformal transformation optics. The designed coupler is implemented by graded photonic crystals. The three-dimensional numerical simulations indicate that the 1.93μm-long coupler has an average coupling loss of 0.13 dB in the C-band.

Ultrashort waveguide tapers based on Luneburg lens

In integrated photonic circuits, silicon-on-insulator waveguides with different geometries have been employed to realize a variety of components. Therefore, efficient coupling of two different waveguides is crucial. In this paper, focusing property of the Luneburg lens is exploited to design waveguide tapers. The Luneburg lens, truncated in a shape of a parabolic taper with reduced footprint, is utilized to connect a 10 μm wide waveguide to a 0.5 μm one with the same thickness with an average coupling loss of 0.35 dB in the entire O, E, S, C, L, and U bands of optical communications. The proposed compact taper with the length of 11 μm is implemented by varying the thickness of the guiding layer and compared with three conventional tapers with the same length. However, designing a coupler to connect waveguides with different thicknesses and widths is more challenging. By applying quasi-conformal transformation optics, we flatten the Luneburg lens and consequently increase the refractive index on the flattened side. As a result, we are able to couple two waveguides with different thicknesses and widths. The numerical simulations are used to evaluate the theoretically designed tapers. To our knowledge, this is the first study presenting ultrashort tapers based on truncated Luneburg lens.

Multimode waveguide crossing based on a square Maxwell's fisheye lens.

Mode-division multiplexing (MDM) is an emerging large-capacity data communication technology utilizing orthogonal guiding modes as independent data streams. One of the challenges of multimode waveguide routing in MDM systems is decreasing the mode leakage of waveguide crossings. In this article, a square Maxwell's fish-eye lens as a waveguide crossing medium based on quasiconformal transformation optics is designed and implemented on a silicon-on-insulator platform. Two approaches were taken to realize the designed lens: graded photonic crystal and varying the thickness of the silicon slab waveguide. Three-dimensional numerical simulations show that the designed multimode waveguide crossing has an ultrawide bandwidth from 1260 to 1675nm with a compact footprint of only 3.77×3.77 μm2. For the first three transverse electric modes (TE0, TE1, and TE2), the designed waveguide crossing exhibits an average insertion loss of 0.24, 0.55, and 0.45dB; a crosstalk of less than -72, -61, and -27 dB; and a maximum return loss of 54, 53, and 30dB, respectively. The designed waveguide crossing supports low-distortion pulse transmission with a high fidelity factor of 0.9857. Furthermore, the proposed method can be expanded to design waveguide crossings with an even higher number of supporting modes by increasing the size of the lens.