Geometrical Optics Ray Research Articles

The geometrical description of electromagnetic radiation

The paper describes the relationship between the solutions of Maxwell’s equations which can be considered at least locally as plane waves and the curvilinear coordinates of geometrical optics; it generalizes the results achieved by Lüneburg, concerning the evolution of surfaces of electromagnetic fields discontinuities. If vectors and are orthogonal to each other and their directions do not change with time t, but may vary from point to point in the domain G, then under some conditions there is an orthogonal coordinate system in which -lines represent rays of geometrical optics, -lines point out -direction and, -lines point out -direction. This coordinate system will be called phase-ray coordinate system. In the article, it will be proved that field under study can be represented by two scalar functions. The article will also specify the necessary and sufficient conditions for the existence of a coordinate system, generated by the solution of Maxwell’s equations with the holonomic field of the Poynting vector. It is shown that the class of solutions of Maxwell’s equations, as described in this work, includes monochromatic polarized waves, and the Hilbert–Courant solutions and their generalizations.

Propagation of optical beams in two transverse gradient index media

We investigate the propagation of optical beams in two gradient index inhomogeneous media. The Green’s function for paraxial propagation in an inhomogeneous medium is derived as a Feynman path integral involving summation over real rays. We use a simple method based on discontinuous functions, which is similar to that used in General Relativity when studying metric discontinuities across an hypersurface, to study the propagation in two transverse refractive index gradients which are coupled in the propagation direction. We show that this method is consistent with the geometric-optics ray theory, and then with the definition of the coupled Green’s function. The propagation of Gaussian beams in two homogeneous media with different refractive indices is analyzed. We also study the propagation of Airy beams in two media with different linear transverse refractive index gradients. We demonstrate that by controlling the gradient strength of the media it is possible to reduce to zero their acceleration, yielding an Airy beam that propagates linearly.

Flat Optics for Surface Waves

The name flat optics (FO) has been introduced in a recent paper by Capasso’s group for denoting light-wave manipulations through a general type of penetrable or impenetrable metasurfaces (MTSs). There, the attention was focused on plane waves, whereas here we treat surface waves (SWs) excited on impenetrable impedance surfaces. Space variability of the boundary conditions imposes a deformation of the SW wavefront, which addresses the local wavector along not-rectilinear paths. The ray paths are subjected to an eikonal equation analogous to the one for geometrical optics (GO) rays in graded index materials. The basic relations among ray paths, ray velocity, and transport of energy for both isotropic and anisotropic boundary conditions are presented for the first time. This leads to an elegant formulation which allows for closed form analysis of flat operational devices (lenses or beam formers), giving a new guise to old concepts. It is shown that when an appropriate transformation is found, the ray paths can be conveniently controlled without the use of ray tracing, thus simplifying the problem and leading to a flat version of transformation optics, which is framed here in the general FO theory.

Mathematical simulation of propagation of frequency-modulated radio waves in ionospheric plasma

A numerical simulation of single-hop and double-hop propagation of frequency-modulated signals in anisotropic ionospheric plasma is performed on the basis of a numerical solution to a space–time Hamiltonian bicharacteristic system whose unknowns are the components of the wave vector, coordinates, and also the frequency and time. It is assumed that the radiation source is a point one and is located outside the magnetoactive plasma, the frequency modulation of decameter waves is linear, and the amplitude of the magnetic field is constant. A model of a two-layer ionospheric plasma with a wavelike disturbance is considered. The specific features of the departure of the ordinary and extraordinary waves from the plane of propagation, formation of caustics of space–time geometrical-optics rays, and the Doppler frequency shift are analyzed.

Controlling local temperature in water using femtosecond optical tweezer.

A novel method of directly observing the effect of temperature rise in water at the vicinity of optical trap center is presented. Our approach relies on changed values of corner frequency of the optical trap that, in turn, is realized from its power spectra. Our two color experiment is a unique combination of a non-heating femtosecond trapping laser at 780 nm, coupled to a femtosecond infrared heating laser at 1560 nm, which precisely controls temperature at focal volume of the trap center using low powers (100-800 µW) at high repetition rate. The geometric ray optics model quantitatively supports our experimental data.

Lagrangian geometrical optics of nonadiabatic vector waves and spin particles

Linear vector waves, both quantum and classical, experience polarization-driven bending of ray trajectories and polarization dynamics that can be interpreted as the precession of the “wave spin”. Both phenomena are governed by an effective gauge Hamiltonian vanishing in leading-order geometrical optics. This gauge Hamiltonian can be recognized as a generalization of the Stern–Gerlach Hamiltonian that is commonly known for spin-1/2 quantum particles. The corresponding reduced Lagrangians for continuous nondissipative waves and their geometrical-optics rays are derived from the fundamental wave Lagrangian. The resulting Euler–Lagrange equations can describe simultaneous interactions of N resonant modes, where N is arbitrary, and lead to equations for the wave spin, which happens to be an (N2−1)-dimensional spin vector. As a special case, classical equations for a Dirac particle (N=2) are deduced formally, without introducing additional postulates or interpretations, from the Dirac quantum Lagrangian with the Pauli term. The model reproduces the Bargmann–Michel–Telegdi equations with added Stern–Gerlach force.

An accurate projection model for diffraction image formation and inversion using a polychromatic cone beam

Recently, the concept of X-ray diffraction contrast tomography (DCT) has been extended to the case of more widely available laboratory source CT systems. Using well known concepts from geometrical ray optics, an exact formulation is derived for the forward and backward projection geometry encountered under polychromatic cone beam illumination, and it is shown how this projection model can be efficiently implemented in practice. The new projection model is subsequently used for iterative tomographic reconstruction of the three-dimensional shape of a grain from a set of experimentally observed cone beam projections and shows a clear improvement compared to the simplified projection model used previously.

Scattering of ECRF waves by edge density fluctuations and blobs

The scattering of electron cyclotron waves by density blobs embedded in the edge region of a fusion plasma is studied using a full-wave model. The full-wave theory is a generalization of the usual approach of geometric optics ray scattering by blobs. While the latter allows for only refraction of waves, the former, more general formulation, includes refraction, reflection, and diffraction of waves. Furthermore, the geometric optics, ray tracing, model is limited to blob densities that are slightly different from the background plasma density. Observations in tokamak experiments show that the fluctuating density differs from the background plasma density by 20% or more. Thus, the geometric optics model is not a physically realistic model of scattering of electron cyclotron waves by plasma blobs. The differences between the ray tracing approach and the full-wave approach to scattering are illustrated in this paper.

Optical Effects at projection measurements for Terahertz tomography

Abstract Optical effects like refraction, diffraction and edge effects have an influence on Terahertz measurements. They can result in image artifacts which makes it difficult to detect and resolve material defects inside the samples. We used a geometrical optical ray tracing approach to analyze the optical effects at Terahertz projection measurements which can be used to perform 2D or 3D THz images. We measured rectangular and cylindrical samples made of PEEK (Polyetheretherketon), POM (Polyoxymethylen), and PMMA (Polymethylmethacrylat) and compared the results to simulations that are realized with the software ZEMAX. We were able to simulate the measured Fresnel refraction and transmission behavior for rectangular cuboids with a length of 25 mm and cylinders with diameter of 25 mm. We showed the influence of diffraction and edge effects at samples with different sizes made of PMMA. Thus, the optical effect of refraction was significant and observable for cylinders with diameters greater than 1.5 mm and holes with diameter greater than 2.5 mm.

Large-scale structure and asymptotic regularities of scattering phase function for water drops in the visible spectrum range

The mechanisms of radiation scattering by a sphere in the range of angles θ of 0–180° is analyzed by comparing the exact scattering phase functions calculated with the Mie theory and the interference phase functions with the use of diffraction and partial rays of geometric optics (GO). It is found that the large-scale oscillation regularities of the exact scattering phase function at the high Mie parameters x correspond to an interference pattern of two or three rays with corrected amplitude of the diffraction ray and phase shifts of GO rays. For integral characteristics, the error of computation using the interference formulae in the range of angles θ = 0–10° does not exceed units of percent for x > 10 and tends to zero as x increases. For other ranges, depending on the combination of parity of the integer parts in the π intervals of the total scattering angle, the tendency of oscillation periods over θ toward zero is seen according to the x−1, x−2/3 (rainbow), and x−1/2 laws, while the Mie parameter increases, and the oscillation period over x begins to depend only on θ. The results of the calculations of the exact scattering phase functions averaged over the intervals Δθ = 10–15° for the refraction index m = 4/3 are presented in the form of approximate relations with the asymptotic tendency to the GO scattering phase function.

Diffraction of a cylindrical wave by a perfectly conducting cylinder enclosed in a metamaterial shell with a negative refractive index

The problem of diffraction of a plane (cylindrical) wave by a perfectly conducting cylinder enclosed in a metamaterial shell is rigorously solved. The influence of the geometric dimensions of the shell, the value of the negative refractive index of the metamaterial medium, and the position of a cylindrical wave source on the field structure in the near zone of the scatterer is investigated. It is found that, in the quasi-optical range of the problem parameters, this structure does not exhibit ideal focusing. It is shown that, there are two types of caustics inside the shell. The first type is related with rays reflected by the surface of the interior cylinder and has one cusp point, and the second type is formed by the geometric-optics rays that are refracted by the outer boundary of the shell and do not fall on the surface of the interior perfectly conducting cylinder. The spatial distribution of the total field amplitude and of the equal-amplitude lines in the near zone of the scatterer is reported. The obtained numerical results are correctly interpreted from the physical viewpoint.

Numerical Eulerian method for linearized gas dynamics in the high frequency regime

We study the propagation of an acoustic wave in a moving fluid in the high frequency regime. We calculate the asymptotic approximation of the solution, around a mean flow, of this problem using an Eulerian method. By introducing the stretching matrix (deformation tensor for the geometrical optics rays) of the linearized Euler system, we deduce the geometrical spreading. This quantity is the key tool for computing the leading order term of the asymptotic expansion thanks to a conservation equation along the group velocity. The main contribution is to construct and implement a numerical scheme in the Eulerian framework for the eikonal equation and for the transport equation on the stretching matrix. We present numerical results for several test cases to study the convergence and validate our approach.

Computational Tools and Approaches for Design and Control of Coating and Composite Color, Appearance, and Electromagnetic Signature

The transport behavior of electromagnetic radiation through a polymeric coating or composite is the basis for the material color, appearance, and overall electromagnetic signature. As multifunctional materials become more advanced and next generation in-service applications become more demanding, a need for predictive design of electromagnetic signature is desired. This paper presents various components developed and used in a computational suite for the study and design of electromagnetic radiation transport properties in polymeric coatings and composites. Focus is given to the treatment of the forward or direct scattering problem on surfaces and in bulk matrices of polymeric materials. The suite consists of surface and bulk light scattering simulation modules that may be coupled together to produce a multiscale model for predicting the electromagnetic signature of various material systems. Geometric optics ray tracing is used to predict surface scattering behavior of realistically rough surfaces, while a coupled ray tracing-finite element approach is used to predict bulk scattering behavior of material matrices consisting of microscale and nanoscale fillers, pigments, fibers, air voids, and other inclusions. Extension of the suite to color change and appearance metamerism is addressed, as well as the differences between discrete versus statistical material modeling.

Angular tolerance and sensitivity for designing Fresnel-like prism structures with arbitrary surface profiles

Fresnel-like prism arrays are designed using geometric ray optics and fabricated on a curved surface according to specific focal length requirements. In this study, the tolerance of Fresnel-like prism structures on arbitrary surfaces was investigated. Compact equations (iterative and non-iterative) based on analytical formulation can be helpful in exploring the sensitivity associated with the fabrication of Fresnel-like structures. The curved surface in this study revealed a linear relationship between maximum angular tolerance and the angle of incident rays. An understanding of such linearity is a key issue in the fabrication of lenses as well as in the sensitivity formulae used for optical structures in the development of next-generation optics.

Rashba spin-orbit interaction and birefringent electron optics in graphene

Electron optics exploits the analogies between rays in geometrical optics and electron trajectories, leading to interesting insights and potential applications. Graphene, with its two-dimensionality and photon-like behavior of its charge carriers, is the perfect candidate for the exploitation of electron optics. We show that a circular gate-controlled region in the presence of Rashba spin-orbit interaction in graphene may indeed behave as a Veselago electronic lens but with two different indices of refraction. We demonstrate that this birefringence results in complex caustics patterns for a circular gate, selective focusing of different spins, and the possible direct measurement of the Rashba coupling strength in scanning probe experiments.

Propagation and annihilation of vortex pairs in a smoothly inhomogeneous Bose-Einstein condensate

An asymptotic theory has been developed to describe the propagation and annihilation of vortex pairs in a smoothly inhomogeneous Bose-Einstein condensate with the repulsive interaction between atoms. It has been shown that the trajectories of vortex pairs coincide with geometric-optical rays in an equivalent isotropic medium with the refractive index depending both on the density of the unperturbed inhomogeneous condensate and on the energy of structures under study. The transformation of vortex pairs to vortex-free quasisolitons and back has been described.

Aberration-like cusped focusing in the post-paraxial Talbot effect

We present an analysis of the Talbot effect beyond the paraxial regime, where deviation from Fresnel propagation destroys perfect, periodic self-imaging. The resulting interference structures are examples of aberration without geometric optical rays, which we describe analytically using post-paraxial theory. They are similar to, but do not precisely replicate, a standard integral representation of a diffraction cusp (the Pearcey function). Beyond the Talbot effect, this result illustrates that aberration—as the replacement of a perfect focus with a cusp-like pattern—can occur as a consequence of improving the paraxial approximation, rather than due to imperfections in the optical system.

Physically-based simulation of rainbows

In this article, we derive a physically-based model for simulating rainbows. Previous techniques for simulating rainbows have used either geometric optics (ray tracing) or Lorenz-Mie theory. Lorenz-Mie theory is by far the most accurate technique as it takes into account optical effects such as dispersion, polarization, interference, and diffraction. These effects are critical for simulating rainbows accurately. However, as Lorenz-Mie theory is restricted to scattering by spherical particles, it cannot be applied to real raindrops which are nonspherical, especially for larger raindrops. We present the first comprehensive technique for simulating the interaction of a wavefront of light with a physically-based water drop shape. Our technique is based on ray tracing extended to account for dispersion, polarization, interference, and diffraction. Our model matches Lorenz-Mie theory for spherical particles, but it also enables the accurate simulation of nonspherical particles. It can simulate many different rainbow phenomena including double rainbows and supernumerary bows. We show how the nonspherical raindrops influence the shape of the rainbows, and we provide a simulation of the rare twinned rainbow, which is believed to be caused by nonspherical water drops.

High Frequency Scattering by a Classically Invisible Body

We consider a polyhedron with zero classical resistance, i.e., a polyhedron invisible to an observer viewing only the paths of geometrical optics rays. The corresponding problem of scattering of plane waves by the polyhedron is studied. The quasi-classical approximation is obtained and justified in the case of impedance boundary conditions with nonzero absorbtion. It is shown that the total momentum transmitted to the obstacle vanishes as the frequency k tends to infinity and that the total cross section oscillates at high frequencies. When the impedance $\lambda_0$ is real (i.e., there is no absorption), it is shown that there exists a sequence of frequencies $k_n$ such that the average of the total cross section over shrinking intervals around $\lambda_0 $ tends to zero as $k_n \to \infty$.

Optical approach to gravitational redshift

An optical approach begins by interpreting the gravitational redshift resulting to a change in the relative velocity of light due to the medium of propagation in the gravitational field. The discussion continues by pointing out an agreement in structure between the equation for rays in geometrical optics and the geodesic equation of general relativity. From their comparison we learn that the path of rays should be given by the relation $ds^2=n^2(r)dr^2+r^2d\theta^2$, not by $ds^2=dr^2+r^2d\theta^2$, in a medium with spherical symmetry of refractive index $n(r)$. The development of an optical analogy suggests introducing $n^2(r)$ in place of $g_{rr}$ as an optical version of the Schwarzschild metric. In form and content, $n^2(r)$ is different from $g_{rr}$. The optical point of view replaces the general-relativity explanations in terms of time and gravitation.