Optical Ray Tracing Based on Dijkstra Algorithm in Inhomogeneous Medium

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.

A millimeter-wave (mmW) fully metallic crossover with six intersecting channels based on Maxwell fish-eye (MFE) lens is proposed. The quasi-MFE lens consists of an almost-parallel plate waveguide with a curved top plate and an array of uniform metallic posts loaded on the bottom plate. Six pairs of rectangular waveguides are added that surround the lens to form a six-channel crossover. The results show a performance of broadband and efficient crossing transmissions, indicating the application potential of the proposed crossover in the multichannel mmW systems.

In this paper, we designed Maxwell's fish-eye lens in a square shape from a conformal mapping and constructed it into a waveguide and a photonic crystal. We studied the unidirectional self-imaging effect from both geometric optics and wave optics. Ray tracing and full-wave simulations were performed to validate this effect. Specifically, the designed waveguide is of a finite width while having the same imaging properties as the ideal infinite waveguide---the Mikaelian lens, the refractive index profile of which is 1/cosh ($x$) [Mikaelian and Prokhorov, Prog. Opt. 17, 279 (1980)].

It has recently been shown that passive lens designs can retain perfect lensing despite intrinsic loss in the comprising left-handed materials. Here we show that energy conservation is not at odds with the operation of such lossy perfect lenses: The irreversible transfer of electromagnetic power to the lossy material dictates that a smaller portion of the power emitted by the source arrives at the image, but image fidelity remains absolute. Moreover, the power that does arrive at the perfect image must drain from the system there. This is shown to apply to any layered perfect lens based on left-handed media, including the well-known Veselago lens. Combined with what is known about Maxwell's fisheye lens, our results suggest that power drainage is universal to all perfect lenses and is intrinsic to any perfect image. Properly addressing this unique power dynamics is therefore essential to any successful realization of practical perfect lenses with existing metamaterials.

The Maxwell fish eye lens has previously been reported as being capable of the much sought after phenomenon of subwavelength imaging. The inclusion of a drain in this system is considered crucial to the imaging ability, although its role is the topic of much debate. This paper provides a numerical investigation into a practical implementation of a drain in such systems, and analyzes the strong frequency dependence of both the Maxwell fish eye lens and an alternative, the Mi\~nano lens. The imaging capability of these types of lens is questioned, and it is supported by simulations involving various configurations of drain arrays. Finally, a discussion of the near-field and evanescent wave contribution is given.

Recent work suggesting that ‘perfect’ far-field imaging is possible using Maxwell's fish-eye lens (Leonhardt 2009 New J. Phys.11 093040) has raised a number of questions and controversies about the nature of imaging and field localization in inhomogeneous media. In this brief paper we present analogous results for a purely reflector-based imaging system—an elliptical cavity. With a source at one focus of the ellipse we show that sub-wavelength field localization can be achieved at the other focus when an active ‘drain’ is present there, but not without it. Does this show that far-field ‘perfect’ imaging is possible even without refraction (negative or positive)? Unfortunately not, giving further evidence that these are solely drain-induced effects.

The design and performance of a novel octave-bandwidth highly-directive half Maxwell fish-eye (HMFE) lens antenna are presented in superextended C band. The three-dimensional (3D) HMFE lens is implemented by gradient-refractive-index (GRIN) metamaterials and launched by an omnidirectional planar microstrip trapezoid printed monopole from the perspective of high integration, light weight and low profile. A new approach is proposed to design the GRIN metamaterial element in terms of a deep subwavelength feature by incorporating fractal geometry. Numerical and experimental results coincide well, showing that the lens enables a considerable gain enhancement of the monopole near 10 dB across a frequency range of 3 to 7.5 GHz while without significantly affecting the cross-polarization patterns and impedance matching. The near-field free-space measurement is also performed in an octave to afford a physical insight into the high gain, which is attributable to the accurate conversion of quasi-spherical waves to plane waves. Moreover, the truncation and homogenization effects of the lens on the antenna directivity are investigated to illustrate the fundamental mechanisms and afford the design guidelines.

A Luneburg lens is a fascinating gradient refractive index lens that can focus parallel light on a perfect point without aberration in geometrical optics. A three-dimensional (3D) Luneburg lens operating at optical frequencies is proposed by Zhao et al. as a first experimental demonstration. This gradient refractive index lens is realized by laser direct writing, and the refractive profile of simple cubic metamaterials is tailored by the fill factor. Experimentally, 3D ideal focus performance has been observed via near-field optical microscopy, which is a direct experimental evidence for 3D Luneburg lens to achieve the ideal focus of mid-IR light. (Picture: Yuan-Yuan Zhao et al., 10.1002/lpor.201600051 pp. 665–672, in this issue)

A technological process to control the foam dielectric constant, an important issue for the design of microwave components and antennas, is described. For that purpose, the use of different commercial foam materials has been considered. This kind of foam substrate is made of original material (Polyvinyl chloride, resin, and…) into which gas is injected. Therefore, the dielectric constant of such foam is close to one. It can be increased by expelling the gas out of the foam material. The authors are presenting the technological process used to expel the gas by pressing a foam slab at relatively low temperature (90 $^{\circ}{\rm C}$ ). Because of this technological process, the dielectric constant variation can be controlled by the ratio between the initial and final slab thickness. It holds a great interest for the design of microwave antennas and circuits. Indeed, the dielectric constant inside gradient index lenses (Luneburg, Maxwell fish-eye, and Fresnel lenses) must follow a particular law to obtain the desired radiation capabilities. The results of materials characterization are presented to validate the technological process. Foam-based antennas and components are also shown to illustrate the interest of the process.

Graphene is used as an ideal platform for plasmonic optical devices due to its unique electrical and optical characteristics. We proposed a unique effective index method to evaluate a graphene-based half Maxwell fish-eye (HMFE) plasmonic lens on a single flake of graphene, which can focus a surface plasmon polariton plane wave and transform into a spherical wave. Similar to traditional optics, the proposed method can be used to investigate the focusing performance of an HMFE lens versus different parameters, including the incident frequencies, the discrete semiring numbers, the chemical potentials, and the size of the proposed lens. In addition, we use two proposed lenses to build an optical-coupling element, which can transform the beam of a plane wave into a different beam width. The proposed effective index method can be used as a reference in designing plasmonic optical devices with a variable effective index profile on a flake of graphene. The finite element method is employed to realize the numerical simulation, which demonstrates results almost consistent with our design methodology.

Ultrasonic inspection in anisotropic and inhomogeneous media has long presented a challenge because of the complex steering of ultrasonic paths. An approach is presented in which the true geometry of a previously used austenitic steel weld model is distorted so that, from one viewing location, all ultrasound travels in straight lines with a constant isotropic velocity. The mapping from the real space to this distorted space is accomplished using Fermat’s theorem of least travel time applied through ray tracing. Applications specific to inspection design and data interpretation for manual ultrasonic inspection of welds in austenitic steel plates are given. Validation of some intermediate results is provided using finite element analysis.

We discuss (linear) geometrical optics and its scalar wave counterpart, Fourier optics, as a natural outgrowth of the group theoretical machinery emanating from the symplectic structure of (optical) phase space, as it derives from the Fermat principle. We do, in fact, quantize geometrical optics. The symplectic structure is preserved by the (symplectic) group of linear canonical transformations. Every element of the symplectic group can be realized as a part of an image system in geometrical optics. On the other hand, one may associate a (nilpotent) Lie algebra to the (symplectic) phase space on which the symplectic group Sp (4, ℜ) acts in the natural way. Looking further on the irreducible unitary representations of the underlying simply connected Lie group, the Heisenberg group, one finds that the symplectic action on the Heisenberg group induces a group of unitary transformations in the space of each (infinite-dimensional) irreducible representation. This group, called the metaplectic group, describes the evolution of the (scalar) wave field (Fourier optics) corresponding to the evolution of rays (geometrical optics), and thereby connects the two views on optics. We put emphasis on the interplay of concepts, and for this reason, we also invoke intuition and terminology from classical and quantum mechanics.

: Design configurations of nonoscillatory feed, line-scan, parallel- plate radar antennas were explored. Some historically important designs are described as a basis for understanding several new, proposed antenna geometries. It is shown that the continuous-motion scanner developed by Lewis can be combined with many parallel-plate lenses and reflectors in various geometric layouts to obtain useful microwave antennas. Designs that would normally produce good radiation by combining the continuous-motion scanner and the cut, rolled, and folded sectoral horn-lens assemblies. The continuous-motion scanner can be combined with dielectric, metal-plate, and geodesic lenses. Multiple linear and nonlinear reflectors may be installed in the sectoral horn and planar reflectors may be used in some geodesic lenses to achieve compactness. Some of these 'oblique' layouts are shown to be analogous to optical telescopes. Methods of reducing antenna size by folding over or rolling up flat portions of the sectoral horn are explained. Optical laws are shown to apply to the geometrical design of parallel-plate antennas and lenses. Coma and spherical aberration are defined and procedures for designing parallel-plate radar antennas are proposed. Choice of optical and radar system parameters and the interrelationship of such parameters as scan angle, focal length, beamwidth, scan rate, and physical dimensions are discussed. Fermat's extremum principle is defined and some fundamental criteria for designing geodesic lenses are outlined.

To provide an example of circularly polarized absolute‐value rays in a nondispersive absorbing medium, the theory of ray tracing is presented in the case in which a plane wave is incident on a planar layered medium. Specific tracing methods are presented for the loci of the real phase plane normal, the loci of the projection ray related to the flow line of the normalized Poynting flux, the loci of the circularly polarized absolute‐value rays, and the intensities of these rays. The relationships to Poynting flux and their tracing method are described. Further, in the planar layered medium, an anisotropic projection ray index of refraction is introduced to the projection ray. It is shown that Fermat's principle holds under this index of refraction. On the other hand, Fermat's principle also holds in the tracing of the trajectory of the real‐phase plane normal. In Fermat's principle of the projection ray, its relationship to the energy flux density is new.

MOFFAT, BRAD A. PhD; ATCHISON, DAVID A. PhD, FAAO; POPE, JAMES M. DPhil, FAIP Author Information