Wide-angle passive beam steering using 3D modified partial Maxwell fisheye lens

This study presents a broadband, 3D gradient index beam-steering lens, derived from an optimized modification of the partial Maxwell fisheye (PMFE) design, achieving a boresight gain of 23 dBi, -80° to 80° beam steering, and <10 dB gain roll-off. Utilizing fused filament fabrication (FFF) to realize its intricate geometry, the design employs a novel polar space-filling curve (PSFC) to establish a 3D varying, effective permittivity distribution. Rigorous simulations and experimental validation attest to its effectiveness, marking the first 3D implementation of a PMFE-type lens to our knowledge. This research underscores the feasibility and diverse applications of a low-cost, wide-angle passive beam-steering dielectric lens.

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 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.

Double-layer geodesic and gradient-index lenses

A double-layer lens consists of a first gradient-index/geodesic profile in an upper waveguide, partially surrounded by a mirror that reflects the wave into a lower guide where there is a second profile. Here, we derive a new family of rotational-symmetric inhomogeneous index profiles and equivalent geodesic lens shapes by solving an inverse problem of pre-specified focal points. We find an equivalence where single-layer lenses have a different functionality as double-layer lenses with the same profiles. As an example, we propose, manufacture, and experimentally validate a practical implementation of a geodesic double-layer lens that is engineered for a low-profile antenna with a compact footprint in the millimeter wave band. Its unique double-layer configuration allows for two-dimensional beam scanning using the same footprint as an extension of the presented design. These lenses may find applications in future wireless communication systems and sensing instruments in microwave, sub-terahertz, and optical domains.

A double-layer lens consists of a first gradient-index/geodesic profile in an upper waveguide, partially surrounded by a mirror that reflects the wave into a lower guide where there is a second profile. A family of such lens profiles are derived.

Transformation Metamaterials

Based on the form-invariance of Maxwell's equations under coordinate transformations, mathematically smooth deformation of space can be physically equivalent to inhomogeneous and anisotropic electromagnetic (EM) medium (called a transformation medium). It provides a geometric recipe to control EM waves at will. A series of examples of achieving transformation media by artificially structured units from conventional materials is summarized here. Such concepts are firstly implemented for EM waves, and then extended to other wave dynamics, such as elastic waves, acoustic waves, surface water waves, and even stationary fields. These shall be cataloged as transformation metamaterials. In addition, it might be conceptually attractive and practically useful to control diverse waves for multi-physics designs.

Flat lens design to rotate a cylindrical beam of a line source to an arbitrary angle

The theory of transformation optics is used to adjust the direction of emitted beams from a flat lens. In this paper, a planar lens is presented based on the transformation optics approach, which converts cylindrical beams emitted from a line source into a planar beam at the desired angle. The index profile of a planar inhomogeneous lens is considered as the refractive index of the original coordinate system. So, this yields a lens that produces a flat wave at an arbitrary angle. The performance of the structure is confirmed by COMSOL software.

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.

Observation of light rays on absolute geodesic lenses

Absolute optical instruments with rotation symmetry and corresponding absolute geodesic lenses have drawn considerable attention for their property of perfect imaging of light rays. In this paper, we systematically explore a series of absolute geodesic lenses which is mapped from generalized Maxwell's fish-eye lenses with a rational number index {p}. Moreover, we construct new types of duplex absolute geodesic lenses by splicing two different half absolute geodesic lenses, which is inspired by the work [Huiyan Peng, et al Phys. Rev. Applied13, 034050 (2020)]. Also, we fabricate some samples of absolute geodesic lenses based on the 3D printing technique and observe light rays on them. Our findings enlarge the family of absolute geodesic lenses and might find an application on classical imaging systems.

Transformation seismology: composite soil lenses for steering surface elastic Rayleigh waves

Metamaterials are artificially structured media that exibit properties beyond those usually encountered in nature. Typically they are developed for electromagnetic waves at millimetric down to nanometric scales, or for acoustics, at centimeter scales. By applying ideas from transformation optics we can steer Rayleigh-surface waves that are solutions of the vector Navier equations of elastodynamics. As a paradigm of the conformal geophysics that we are creating, we design a square arrangement of Luneburg lenses to reroute Rayleigh waves around a building with the dual aim of protection and minimizing the effect on the wavefront (cloaking). To show that this is practically realisable we deliberately choose to use material parameters readily available and this metalens consists of a composite soil structured with buried pillars made of softer material. The regular lattice of inclusions is homogenized to give an effective material with a radially varying velocity profile and hence varying the refractive index of the lens. We develop the theory and then use full 3D numerical simulations to conclusively demonstrate, at frequencies of seismological relevance 3–10 Hz, and for low-speed sedimentary soil (v_s: 300–500 m/s), that the vibration of a structure is reduced by up to 6 dB at its resonance frequency.

Multi-functional lens based on conformal mapping

Based on conformal mapping method, a two dimensional, multi-functional lens structure is proposed and designed in this work. The lens is an infinitely-long, gradient-index dielectric cylinder with a semi-elliptic cross-section. The lens can first be considered like a flattened Luneburg lens, which produces highly-directive electromagnetic waves by adjusting the feed position along the line connecting the two foci. It also functions like an Eaton lens. When an incoming beam impinges on the same line but outside the two foci, it will be guided through the lens structure and take a U-turn. Besides, if properly shaped, the structure can also be used as a waveguide bend. The lens can be realized using non-resonant metamaterials with inhomogeneous hole arrays. Simulation results demonstrate excellent performance of the lens and agree well with theoretical prediction. The designed lens can be used in the electromagnetic control. And it is especially useful in the real optical lens system.

Direct stigmatic imaging with curved surfaces

We study the possibilities of direct (using one intersection with each light ray) stigmatic imaging with a curved surface that can change ray directions in an arbitrary way. By purely geometric arguments we show that the only possible case of such imaging is the trivial one where the image of any point is identical to the point itself and the surface does not perform any change of the ray direction at all. We also discuss an example of a curved surface which performs indirect stigmatic imaging after twice intersecting each light ray.

Removing singular refractive indices with sculpted surfaces

The advent of Transformation Optics established the link between geometry and material properties, and has resulted in a degree of control over electromagnetic fields that was previously impossible. For waves confined to a surface it is known that there is a simpler, but related, geometrical equivalence between the surface shape and the refractive index, and here we demonstrate that conventional devices possessing a singularity - that is, the requirement of an infinite refractive index - can be realised for waves confined to an appropriately sculpted surface. In particular, we redesign three singular omnidirectional devices: the Eaton lens, the generalized Maxwell Fish-Eye, and the invisible sphere. Our designs perfectly reproduce the behaviour of these singular devices, and can be achieved with simple isotropic media of low refractive index contrast.

Transmutation of singularities and zeros in graded index optical instruments: a methodology for designing practical devices

We describe a design methodology for modifying the refractive index profile of graded-index optical instruments that incorporate singularities or zeros in their refractive index. The process maintains the device performance whilst resulting in graded profiles that are all-dielectric, do not require materials with unrealistic values, and that are impedance matched to the bounding medium. This is achieved by transmuting the singularities (or zeros) using the formalism of transformation optics, but with an additional boundary condition requiring the gradient of the co-ordinate transformation be continuous. This additional boundary condition ensures that the device is impedance matched to the bounding medium when the spatially varying permittivity and permeability profiles are scaled to realizable values. We demonstrate the method in some detail for an Eaton lens, before describing the profiles for an "invisible disc" and "multipole" lenses.

Perfect imaging with planar interfaces

We describe the most general homogenous, planar, light-ray-direction-changing sheet that performs one-to-one imaging between object space and image space. This is a nontrivial special case (of the sheet being homogenous) of an earlier result [Opt. Commun.282, 2480 (2009)]. Such a sheet can be realized, approximately, with generalized confocal lenslet arrays.

Exploiting design freedom in biaxial dielectrics to enable spatially overlapping optical instruments

The optical behavior of gradient biaxial dielectrics has not been widely explored in the literature due to their complicated nature, but the extra degrees of freedom in the index tensor have the potential of yielding useful optical instruments which are otherwise unachievable. In this work, a design method is described in detail which allows one to combine the behavior of up to four totally independent isotropic optical instruments in an overlapping region of space. This is non-trivial because of the mixing of the index tensor elements in the Hamiltonians; previously known methods only handled uniaxial dielectrics (where only two independent isotropic optical functions could overlap). The biaxial method introduced also allows three-dimensional multi-faced Janus devices to be designed; these are worked out in an example of what is possible to design with the method.

Two-dimensional inside-out Eaton lens: design technique and TM-polarized wave properties

In this paper we perform a theoretical and numerical study of two-dimensional inside-out Eaton lenses under transverse-magnetic-polarized excitation. We present one example design and test its performance by utilizing full-wave Maxwell solvers. With the help of the WKB approximation, we further investigate the finite-wavelength effect analytically and demonstrate one necessary condition for perfect imaging at the level of wave optics, i.e. imaging with unlimited resolution, by the lens.

Plasmonic Luneburg and Eaton lenses

Plasmonics takes advantage of the properties of surface plasmon polaritons, which are localized or propagating quasiparticles in which photons are coupled to the quasi-free electrons in metals. In particular, plasmonic devices can confine light in regions with dimensions that are smaller than the wavelength of the photons in free space, and this makes it possible to match the different length scales associated with photonics and electronics in a single nanoscale device. Broad applications of plasmonics that have been demonstrated to date include biological sensing, sub-diffraction-limit imaging, focusing and lithography and nano-optical circuitry. Plasmonics-based optical elements such as waveguides, lenses, beamsplitters and reflectors have been implemented by structuring metal surfaces or placing dielectric structures on metals to manipulate the two-dimensional surface plasmon waves. However, the abrupt discontinuities in the material properties or geometries of these elements lead to increased scattering of surface plasmon polaritons, which significantly reduces the efficiency of these components. Transformation optics provides an alternative approach to controlling the propagation of light by spatially varying the optical properties of a material. Here, motivated by this approach, we use grey-scale lithography to adiabatically tailor the topology of a dielectric layer adjacent to a metal surface to demonstrate a plasmonic Luneburg lens that can focus surface plasmon polaritons. We also make a plasmonic Eaton lens that can bend surface plasmon polaritons. Because the optical properties are changed gradually rather than abruptly in these lenses, losses due to scattering can be significantly reduced in comparison with previously reported plasmonic elements.

Generalization of ray tracing in a linear inhomogeneous anisotropic medium: a coordinate-free approach

The Hamiltonian of an optical medium is important in both the design and the description of optical devices in the geometrical optics limit. The results calculated in this article show in detail how ray tracing in anisotropic materials in arbitrary coordinate systems and curved spaces can be carried out. Writing Maxwell's equations in the most general form, we derive a coordinate-free form for the eikonal equation and hence the Hamiltonian of a general purpose medium. The expression works for both orthogonal and non-orthogonal coordinate systems, and we show how it can be simplified for biaxial and uniaxial media in orthogonal coordinate systems. In order to show the utility of the equations in a real case, we study both the isotropic and the uniaxially transmuted birefringent Eaton lens and derive the ray trajectories in spherical coordinates for each case.