Eaton Lens Research Articles

Controlling electromagnetic surface waves with conformal transformation optics

The application of transformation optics to the development of intriguing electromagnetic devices can produce weakly anisotropic or isotropic media with the assistance of quasi-conformal and/or conformal mapping, as opposed to the strongly anisotropic media produced by general mappings; however, it is typically limited to two-dimensional applications. By addressing the conformal mapping between two manifolds embedded in three-dimensional space, we demonstrate that electromagnetic surface waves can be controlled without introducing singularity and anisotropy into the device parameters. Using fruitful surface conformal parameterization methods, a near-perfect conformal mapping between smooth manifolds with arbitrary boundaries can be obtained. Illustrations of concealing and illusions, including surface Luneburg and Eaton lenses and black holes for surface waves, are provided. Our work brings the manipulation of surface waves at microwave and optical wavelengths one step closer.

540-degree deflecting lens and its general version.

We demonstrate an isotropic device called 540-degree deflecting lens, which has symmetric refractive index and can deflect parallel beam by 540 degrees. The expression of its gradient refractive index is obtained and generalized. We discover it's an optical absolute instrument with self-imaging characteristic. Using conformal mapping, we deduce its general version in one-dimensional space. We also introduce a combined lens called the generalized inside-out 540-degree deflecting lens similar to the inside-out Eaton lens. Ray tracing and wave simulations are used to demonstrate their characteristics. Our study expands the family of absolute instruments and provides new ideas to design optical systems.

Ray tracing and wave optics behavior of the Eaton lens and invisible sphere triangular lattice photonic crystal

The Eaton lens and invisible sphere are known as devices that create fixed deflection angles in the ray trajectories striking them. The wave optics behavior of these tools, however, can be different. Two triangular lattice photonic crystals (TLPCs) composed of the Eaton lens and invisible sphere are studied in this paper. The differences between ray and wave optics behaviors in the TLPCs are explained. We investigate the ray trajectories in the two TLPCs, separately. Then, for studying wave behavior in TLPCs, the band structures of the two photonic crystals are extracted numerically. Transmission diagrams for six-layer TLPCs are also plotted to approve the band calculations. It is observed that, in some particular frequency ranges, the wave behavior in the TLPCs composed of the Eaton lens and invisible sphere can be quite opposite to the expected behavior of the rays.

Bifunctional Luneburg-Eaton Lens Fabricated of 3-D-Printed Anisotropic Medium

Luneburg lens (LL) and Eaton lens (EL) are well known as optical and microwave components. This letter proposes a bifunctional Luneburg-Eaton lens (LEL) fabricated by three-dimensional (3-D)-printed anisotropic medium. The spherical LEL is designed as a LL for incident vertical polarization and as an EL for incident horizontal polarization through the use of anisotropic dielectric. In order to realize a good continuous refractive index, the proposed lens structure is divided into many cells, and the equivalent permittivity of each unit is calculated by using the equivalent media theory. The prototype of LEL is designed and excited by X-band waveguide WR-90, and the ideal results are achieved. The gain of the antenna is from 15.7 to 16.4 dBi in X-band with the LEL functioning as a LL, and the waveguide can have a beam deflection of set angle with the LEL functioning as an EL.

Ray tracing in concentric gradient-index media: optical Binet equation.

The Binet equation in mechanics describes the orbital geometry of a moving particle under a central force field. In this paper, as its counterpart in optics, we show this formula can be similarly utilized in ray tracing of a gradient-index (GRIN) medium with a concentric field. As an inference of Fermat's principle, this generalization is called the optical Binet equation (OBE). A remarkable advantage of OBE is that it can not only determine the ray trace or concentric GRIN field once one of them is given, but also derive the propagation time inside the medium. As examples, we apply OBE to rays passing through a Maxwell fish-eye lens, Luneburg lens, Eaton lens, concentrator, and hyperbolic deflector, the time delay of which can be calculated once the GRIN field or ray trace equation is solved. The results are well matched with simulations, proving it to be an effective tool in solving problems of the concentric GRIN field.

Anisotropic Kepler problem in a non–rotationally-symmetric Eaton lens

In this paper, we prove that the anisotropic Kepler problem in two-body motion can be analogous to optical non--rotationally-symmetric Eaton lens. According to the numerical calculation and simulation results, when the mass ratio of anisotropic Kepler problem $\ensuremath{\mu}>1$, the corresponding non--rotationally-symmetric Eaton lens can simulate and reproduce the periodic orbits and chaotic characteristics of anisotropic Kepler problem in geometric optics and wave optics, respectively. In the extended study, when a beam is incident to the non--rotationally-symmetric Eaton lens from air, different from the light properties in traditional Eaton lens, the deflected beam is no longer along 180\ifmmode^\circ\else\textdegree\fi{}, and may have scatterings in other directions, which are influenced by the incident direction and mass ratio \ensuremath{\mu}. The results provide a successful reference for the analogy between optical and mechanical phenomena.

2-D Beam-Steerable Generalized Mikaelian Lens With Unique Flat-Shape Characteristic

A novel two-dimensional (2-D) beam-steerable generalized Mikaelian lens is presented in this letter. The main difference of proposed lens with respect to conventional Luneburg, Maxwell fish-eye and Eaton lens is its unique flat-shape characteristic, which provides great advantages in designing the low-profile flat-shape lens antenna, avoiding using extremely complex conformal mapping methods. The all-dielectric low-cost lens prototype presenting desired graded permittivity profile is fabricated by 3-D printing. The measurement results at 10 GHz show 2-D beam-steering capabilities from −20° to 20° in both the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">E</i> -and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> -plane with around 18.9 dBi of realized gain with less than ∼1.2 dB variation and low-side lobe can be achieved, revealing great potentials for developing the low-profile and cost-effective lens antenna for microwave applications.

Partial Maxwell fish-eye lens inspired by the Gutman lens and Eaton lens for wide-angle beam scanning.

This paper presents a novel two-dimensional (2-D) partial Maxwell fish-eye (PMFE) lens with the capability of wide-angle beam scanning inspired by the Gutman lens and Eaton lens, which is obtained by cutting a part from the 2-D Maxwell fish-eye (MFE) lens along a straight line. In terms of the refractive index profile, the MFE lens is similar to the Gutman lens near the center and the Eaton lens near the edge, respectively. We demonstrate the potential of the PMFE lens in wide-angle beam scanning based on its Gutman-like focusing and Eaton-like rotating characteristics corresponding to different feed points. As an example, a fully metallic PMFE lens antenna in the Ka-band composed of a bed of nails and a series of linearly arranged waveguide feeders is designed and experimentally verified. The measured results reveal wide-angle scanning ranges, especially about ±90° at 36 GHz, low reflections and low mutual couplings. The frequency scanning due to the dispersion of the lens is also discussed.

Singular Lenses for Flexural Waves on Elastic Thin Curved Plates

Transformation optics, which is generically applicable to other classical waves such as acoustic and elastic waves, provides an emerging design paradigm to manipulate waves. However, some lenses and optical-transformation devices require a singular refractive index; meeting this requirement is a significant challenge. A method called transmutation can relax some types of index singularity into finite anisotropy around the singularity. Here, we show that such lenses with a singularity for flexural waves can be obtained by approaching a near-zero thickness of the plate precisely at the location of the singularity. As examples, we demonstrate a series of Eaton lenses theoretically and experimentally by projecting the refractive index in space onto the thickness in plates and by working in a broad frequency range in which impedance mismatch is negligible. This framework offers an insight into feasible methods that can be used to develop singular devices such as cloaking devices on thin flexible curved plates and can be further extended to a general methodology for shaping elastic waves. We hope that this elastic platform can also be a test bed to indirectly study unprecedented phenomena enabled by gravitational and quantum fields in terms of analog models.

Analytic solution of the generalized Eaton lens

ABSTRACT Eaton lens is a typical gradient index lens which has many noble properties. The refractive index of the lens has a singularity at the center of the lens. The singularity makes the generalized Eaton lens rotate waves at arbitrary deflecting angles. The analytic solutions of the generalized Eaton lens were studied herein. The solutions were obtained from a nonlinear equation of various order. Only four deflecting angles produced analytic solutions: 2π, π, π/2, and π/3. The solutions of the other angles are obtained by numerical calculations only. The refractive indices and trajectories at the four angles were plotted. The possible application areas of the lens were discussed.

Surface wave direction control on curved surfaces

Transformation optics (TO) has been recognized as an established scheme that can precisely manipulate electromagnetic surface wave propagations. Utilising the advantages of the TO technique, we proposed a TO device which can control the propagation direction of surface waves. The device is based on a generalized Eaton lens and can be realized on curved surfaces. Such a device can be composed of stacked ring layers, where the refractive index of each ring segment can be individually engineered. By engineering the refractive index, the stacked ring structure can guide the propagation direction of the surface wave. It may open a new door to manipulate surface wave propagation on demand.

Multimode T-junctions based on truncated Eaton lens

Abstract Mode-division multiplexing (MDM) in silicon-on-insulator platform is an emerging technology to increase the channel number of a single wavelength carrier by the number of modes and consequently increase the transmission capacity of on-chip optical interconnects. We propose and theoretically demonstrate a multimode branching structure based on the truncated Eaton lens. The proposed T-junctions efficiently convert the higher-order modes into fundamental modes; therefore, they can be potentially employed to manipulate modes in MDM systems. The designed T-junctions are implemented by varying the guiding layer’s thickness on a silicon-on-insulator platform. The three-dimensional simulations verify that the proposed structures can split the TE2 (TE1) mode into the fundamental modes with an average transmitted power of 32% (47%) in a 1550–1600 nm bandwidth.

Ultra-thin silicon-on-insulator waveguide bend based on truncated Eaton lens implemented by varying the guiding layer thickness

Silicon-on-insulator (SOI) waveguides with different geometries have been employed to design various integrated optical components. Reducing the bending radius of the SOI waveguides with low bending loss is essential in minimizing the footprint of light-wave circuits. The propagating mode is less confined in the core of the ultra-thin SOI waveguide and penetrates to substrate and cladding, leading to higher bending loss compared to conventional SOI waveguides with a thicker guiding layer. Although various bending mechanisms have been utilized to reduce the bending loss of conventional SOI waveguides, the ultra-thin SOI waveguide bends have not been studied in detail. In this paper, we present a 60-nm-thick SOI waveguide bend based on the truncated Eaton lens implemented by varying thickness of the guiding layer. The three-dimensional full-wave simulations reveal that the designed waveguide bend, with a radius of 3.9 μm, reduces the bending loss from 3.3 to 0.42 dB at the wavelength of 1550 nm. Moreover, the bending loss for the wavelength range of 1260–1675 nm is lower than 0.67 dB while the bending loss in the C-band is lower than 0.45 dB.

Morse lens [Invited

In this Letter, we find that Morse potential (proposed about 90 years ago) could be connected to Coulomb potential (or Newton potential) and harmonic potential (or Hooke potential) by conformal mappings. We thereby design a new conformal lens from Morse potential, Eaton lens, and Luneburg lens and propose a series of generalized Eaton/Luneburg lenses. We find that this Morse lens is a perfect self-focusing asymmetric lens that differs from a Mikaelian lens. Our theory provides a new insight to Morse potential and other traditional potentials, and revisits their classical applications on designing lenses.

Polymer multimode waveguide bend based on a multilayered Eaton lens.

Reducing the bending radius of low-index-contrast waveguides is essential in reducing the size of the integrated optical components. A polymeric multimode waveguide bend is presented based on the Eaton lens. Ray-tracing calculations are utilized to truncate the Eaton lens in order to improve the performance of the bend. The truncation of the lens decreases the footprint of the bend as well. A designed waveguide bend with a radius of 18.4μm is implemented by a concentric cylindrical multilayer structure. Average bend losses of 0.69 and 0.87dB are achieved for the TM0 and TM1 modes, respectively, in the C-band of optical communication. The bend loss is lower than 1dB in a bandwidth of 1520-1675nm for both modes.

Low-index-contrast waveguide bend based on truncated Eaton lens implemented by graded photonic crystals

Low-index contrast waveguides such as silica waveguides are indispensable part of passive integrated optical components. Reducing the bending radius of silica waveguides with low bend loss is vital in miniaturizing silica planar lightwave circuits. The Eaton lens is a gradient index lens that can bend parallel light rays by 90{\deg}, 180{\deg}, or 360{\deg}. We present a low-loss and compact 90{\deg} waveguide bend with a radius of 9 $\mu$m designed by truncating the Eaton lens. The performance of the designed bend, implemented by graded photonic crystal, was evaluated by full-wave two-dimensional finite element method. The average bend loss in the C-band is 0.9dB while the average bend loss for the wavelength range of 1260-1675nm is 1.05dB. The proposed design strategy can be applied to other low-index contrast waveguides.

Demonstration of steering acoustic waves by generalized Eaton lens

We demonstrate an acoustic generalized Eaton lens that steers an acoustic wave to a desired angle, to propose a design method for gradient-index devices using the sub-wavelength structure to shape an acoustic wave. Based on investigations on the effective parameter with several methods: the S-parameter retrieval method, Floquet-Bloch calculations, and multiple scattering theory (MST) for the cylindrical rigid rod structure, we speculate the design process to realize inhomogeneous refractive index distribution. For realization of a high effective index (∼2), the MST design inevitably fails, whereas the Floquet-Bloch calculations and S-parameter retrieval give identical results. By appropriately designing a two-dimensional array of cylindrical PMMA rods, we experimentally verify the acoustic generalized Eaton lenses for steering angles of 15° and 45°. Furthermore, the use of non-resonant metamaterials enables the proposed devices to work in the broad frequency range from 4 to 8 kHz.

On Ergodicity of Foliations on {mathbb{Z}^d}-Covers of Half-Translation Surfaces and Some Applications to Periodic Systems of Eaton Lenses

We consider the geodesic flow defined by periodic Eaton lens patterns in the plane and discover ergodic ones among those. The ergodicity result on Eaton lenses is derived from a result for quadratic differentials on the plane that are pull backs of quadratic differentials on tori. Ergodicity itself is concluded for {mathbb{Z}^d}-covers of quadratic differentials on compact surfaces with vanishing Lyapunov exponents.

The 90° Rotating Eaton Lens Synthesized by Metasurfaces

In this letter, the two-dimensional 90° rotating Eaton lens is designed and realized by flat metasurfaces. Two kinds of unit-cell metallization are employed, namely the square patch and the mushroom structure, the latter adopted to achieve the higher values of equivalent refractive indices required by the regions near the center of the circular lens. Designs by computer simulations have demonstrated the successful bending of impingent waves by right angles, via field distributions over the metasurface as well as by simulated gains towards the rotated direction. A prototype of the design was also manufactured, the measured surface field distribution and far-field radiation gain patterns of which corroborate the theoretical prediction of the lens' ability to rotate waves by 90°.

Genericity on curves and applications: pseudo-integrable billiards, Eaton lenses and gap distributions

In this paper we prove results on Birkhoff and Oseledets genericity along certain curves in the space of affine lattices and in moduli spaces of translation surfaces. In the space of affine lattices \begin{document}$ASL_2( \mathbb{R})/ASL_2( \mathbb{Z})$\end{document} , we prove that almost every point on a curve with some non-degeneracy assumptions is Birkhoff generic for the geodesic flow. This implies almost everywhere genericity for some curves in the locus of branched covers of the torus inside the stratum \begin{document}$\mathscr{H}(1,1)$\end{document} of translation surfaces. For these curves we also prove that almost every point is Oseledets generic for the Kontsevitch-Zorich cocycle, generalizing a recent result by Chaika and Eskin. As applications, we first consider a class of pseudo-integrable billiards, billiards in ellipses with barriers, and prove that for almost every parameter, the billiard flow is uniquely ergodic within the region of phase space in which it is trapped. We then consider any periodic array of Eaton retroreflector lenses, placed on vertices of a lattice, and prove that in almost every direction light rays are each confined to a band of finite width. Finally, a result on the gap distribution of fractional parts of the sequence of square roots of positive integers is also obtained.