Law Of Refraction Research Articles

Refracting spacetime wave packets

A particular class of focused, pulsed light beams can propagate self-similarly in free space at a fixed group velocity. Now, scientists present a law of refraction that determines how the group velocity of these beams changes as they refract at an interface between two materials.

Anomalous refraction of optical spacetime wave packets

Refraction at the interface between two materials is fundamental to the interaction of light with photonic devices and to the propagation of light through the atmosphere at large. Underpinning the traditional rules for the refraction of an optical field is the tacit presumption of the separability of its spatial and temporal degrees-of-freedom. We show here that endowing a pulsed beam with precise spatio-temporal spectral correlations unveils remarkable refractory phenomena, such as group-velocity invariance with respect to the refractive index, group-delay cancellation, anomalous group-velocity increase in higher-index materials, and tunable group velocity by varying the angle of incidence. A law of refraction for `space-time' wave packets encompassing these effects is verified experimentally in a variety of optical materials. Space-time refraction defies our expectations derived from Fermat's principle and offers new opportunities for molding the flow of light and other wave phenomena.

Concept of a Generalized Law of Refraction: A Phenomenological Model

This paper presents investigations on the generalized laws of refraction and reflection for metasurfaces made of diffractive elements. It introduces a phenomenological model that reproduces all the features of the experiments dedicated to the generalized Snell–Descartes laws. Our main finding is that the generalized laws of refraction and reflection as previously stated have to be modified in order to describe the propagation of light through metasurfaces made of diffractive elements. We provide the appropriate laws that take a different form depending on the properties of the metasurface. Our models apply to both periodic and nonperiodic metasurfaces. We show that the generalized law of refraction strictly exists only for linear-phase profiles and sawtooth-wave phase profiles under constraints that we specify. It can be approximately defined for nonlinear phase profiles.

Light pulse propagation in a double prism layout close to the angle of total internal reflection

Both transmission and reflection of a short light pulse are considered for a double prism system in the geometry of in-gap refraction with an account of material dispersion. Within a frame of linear optics and Snell refraction law, we revealed the effect of a ‘negative dispersion’ regime close to the total internal reflection. A narrow area of parameters (wave number and angle of incidence) in the close vicinity of the total internal reflection is found to possess specific properties for the refracted field. A resonance excitation of leaky waveguide modes enables to extract the refracted field from the gap efficiently and inspect various transmission regimes. Analytical consideration and simulations demonstrate an appearance of advanced pulse in the transmitted and reflected waveforms, with the temporal shape depending on a number of modes enclosed in a spectral width of an incident pulse as well as the existence of elongated precursor within a gap.

The generalized vectorial laws of reflection and refraction leading to the development of novel equations involving the coordinates of the source point, point of observation, and the point of incidence

This paper considers the derivation of novel equations involving the coordinates of a pair of points lying respectively on the incident ray and the reflected (refracted) ray as well as those of the point of incidence by making use of the generalized vectorial laws of reflection and refraction developed by the author in 2005. The novel equations developed will be applicable irrespective of the positions of the pair of points lying on the incident ray and the reflected (refracted) ray respectively. If, however, one of the pair of points considered is supposed to be the source point and the other one is considered as the point of observation, then such novel equations will effectively represent the relation(s) between the coordinates of the source point, point of observation and the point of incidence. The problem addressed is novel and interesting and the development of such novel equations increases the range of applicability of the generalized vectorial laws of reflection and refraction, there by enhancing the traditional optical physics literature as well.

A generalized fractional-order elastodynamic theory for non-local attenuating media.

This study presents a generalized elastodynamic theory, based on fractional-order operators, capable of modelling the propagation of elastic waves in non-local attenuating solids and across complex non-local interfaces. Classical elastodynamics cannot capture hybrid field transport processes that are characterized by simultaneous propagation and diffusion. The proposed continuum mechanics formulation, which combines fractional operators in both time and space, offers unparalleled capabilities to predict the most diverse combinations of multiscale, non-local, dissipative and attenuating elastic energy transport mechanisms. Despite the many features of this theory and the broad range of applications, this work focuses on the behaviour and modelling capabilities of the space-fractional term and on its effect on the elastodynamics of solids. We also derive a generalized fractional-order version of Snell's Law of refraction and of the corresponding Fresnel's coefficients. This formulation allows predicting the behaviour of fully coupled elastic waves interacting with non-local interfaces. The theoretical results are validated via direct numerical simulations.

Computing Close to Optimal Weighted Shortest Paths in Practice

This paper proposes a new practical method for the weighted region problem (WRP). The objective of WRP is to find a minimum cost path between two vertices among different regions where each region incurs a traversal cost per unit distance. Currently, there is no practical algorithm that solves this problem exactly. Among the approximation methods that solve instances of WRP, there is a limited number of algorithms that compute paths whose lengths are close to optimal, which we call very-close optimum paths. However, they are considered as theoretical methods. On the other hand, algorithms for solving WRP that can be applied to practical data sets (using decomposition ideas or heuristics) are not guaranteed to find a very-close optimum path within an acceptable amount of time. In this paper, we consider an alternative method for solving WRP that exploits Snell's law of physical refraction. We compare the performance of our new algorithm with that of two existing algorithms, using at least 500 test cases for each such comparison. The experimental results show that our algorithm returns a very-close optimum weighted shortest path in reasonable time.

Laws of reflection and refraction of TE and TM polarization waves on the border of rhombic crystals

The article analytically solves the problem of reflection and refraction of electromagnetic plane waves of different polarization at the boundary of anisotropic half-spaces of rhombic symmetry. Based on the matrix method, the angles of refraction of electromagnetic waves of different polarization, the amplitudes of the reflected and refracted waves, the angles that determine the direction of the group velocities and vectors of the flow of electromagnetic energy, the magnitudes of the flows of electromagnetic energy and their components depending on the direction of the wave vector of the incident wave are determined. The determination of the angles of total internal reflection and the refractive index of electromagnetic waves of different polarization is considered. A significantly different dependence of the kinematic and energy characteristics of electromagnetic waves of different polarization on the anisotropy of the magnetic and dielectric tensors is shown. For electromagnetic waves, the polarization of which is determined by the component of the electric tension vector perpendicular to the wave propagation plane (TE wave), the characteristics of the reflected and refracted waves, the velocity indicatrix, the propagation angles, etc. determined primarily by the components of the magnetic permeability tensor. In the case of electromagnetic waves, when the polarization is determined by the component of the magnetic field (TM wave), perpendicular to the plane of wave propagation, all characteristics depend mainly on the components of the dielectric constant. The validity of the Fresnel formulas for determining the coefficients of reflected and refracted waves at the boundary of anisotropic media of rhombic symmetry is shown. However, in this case, the components of the wave vectors included in the Fresnel formulas are determined by their indicatrices. In addition, it is necessary to take into account the dependence of these components on the angle of refraction in the second medium.

Geometric aspects of negative refraction

Using Ptolemy’s theorem and inequality, it is shown that the law of negative refraction of an optical ray on the interface between an ordinary isotropic medium and an electromagnetic metamaterial with negative permittivity and negative permeability corresponds to the Fermat’s principle of least time. The optical path length of the ray between two corresponding points at positive and negative refraction is compared.

Mechanical refraction in action

Abstract We study the analog of Snell's law for particles moving across the interface of two regions with two different potential energies, from two different points of view. First, a simple demonstration involving marbles and a potential step is shown. Then, from a theoretical point of view, this law describing mechanical refraction is derived from the Maupertuis and Jacobi variational principles, in close analogy with the well known derivation of Snell's law for refraction from Fermat's least time principle. Finally, the relativistic version of mechanical refraction is obtained by the Maupertuis principle, by trading the Newtonian dispersion relation with the relativistic one. The pedagogical significance of this treatment is discussed.

The Invention of Glass, Microscopes and Telescopes - The Advancement of Microbiology and Astronomy

The ultimate cause of much historical, social and cultural change is the gradual accumulation of knowledge of the human environment. The invention of glass and microscopes and telescopes, substantially increased human knowledge of the human environment. Microscopes and telescopes would not have been possible without the prior discovery of glass making and some knowledge of the law of refraction. Without the microscope we would have no knowledge of micro organisms and the causes of many diseases. Without the telescope we would probably still believe the sun and planets orbited a stationary earth. The change from an earth centered universe, the common sense theory for societies with unassisted vision, to a sun centered theory such as the Newtonian system, and then to General Relativity was inevitable. This was because the knowledge provided by unassisted vision, naturally led to an earth centered universe, the knowledge available from 17th to 19th century telescopes and mathematics led naturally to a sun centered system, such as the Newtonian system and better telescopes led naturally to a theory such as General Relativity. The human environment has a particular structure so that human knowledge of the environment is acquired in a particular order. The simplest knowledge is acquired first and more complex knowledge is acquired later. The order of discovery determines the course of human social and cultural history as knowledge of new and more efficient means of meeting human needs, results in new technology, which results in the development of new social and ideological systems. This means human social and cultural history, has to follow a particular course, a course that is determined by the structure of the human environment.

Reflections in a moving mirror

Euclid's law of reflection, r = i, rests on three assumptions: Fermat's principle of least time, constancy of speed of light, and that the mirror is stationary. For a moving mirror, the corresponding law is known as the relativistic reflection law; however, it was first inferred prior to relativity. While modern derivations rely on Lorentz transformations or Maxwell's equations, the original derivation was an elegant application of geometric optics. In this paper, we develop a new geometric approach which shows that the relativistic reflection from a plane mirror is equivalent to the ordinary reflection from a certain hyperbolic mirror, and that the incident and reflected rays are the focal rays of the hyperbola. We demonstrate that incident concentric spherical waves reflect as non-concentric spherical waves, and thereby derive the Doppler effect formula. We conclude with a class project for students to derive a generalization of Snell's law of refraction for a particular type of a moving interface.

Formulation of the Snell–Descartes Laws in Terms of Geometric Algebra

A formulation of the laws of reflection and refraction of optical rays by the interface of two isotropic media using the mathematical apparatus of geometric algebra is given. The cases of specular reflection, positive and negative refraction, and refraction in the presence of a metasurface are considered.

The Rationale for the Method of Calculating the Angles of Arrival of Short Radio Waves Taking Into Account the Influence of Regular and Random Inhomogeneities of the Ionosphere

The article presents a reasonable method of calculating the angles of elevation of short radio waves with the influence of regular and random inhomogeneities of the ionosphere. The method calculates short radio waves (HF) through the horizontal inhomogeneous scattering ionosphere. The technique is based on application of the law of refraction of Snell’s for well-known models of the ionosphere and its evolution in the irregular parts. The description of the program to calculate the angles of radiation and reception HF is presented. The results of calculating the angles of arrival are compared with measurements of angles of arrival of HF. We obtained the best agreement between the experimental and the calculated resultsl of arrival angles of HF than for the regular ionosphere. The use of techniques is discussed in the article

Optimal navigation strategies for active particles

The quest for the optimal navigation strategy in a complex environment is at the heart of microswimmer applications like cargo carriage or drug targeting to cancer cells. Here, we formulate a variational Fermat's principle for microswimmers determining the optimal path towards a given target regarding travelling time, energy dissipation or fuel consumption. For piecewise constant forces (or flow fields), the principle leads to Snell's law, showing that the optimal path is piecewise linear, as for light rays, but with a generalized refraction law. For complex environments, like general 1D, shear or vortex fields, we obtain exact analytical expressions for the optimal path, showing, for example, that microswimmers sometimes have to temporarily navigate away from their target to reach it fastest. Our results apply to idealized microswimmers which can instantaneously steer, are fast enough so that translational noise is unimportant and might be useful, e.g., to benchmark algorithmic schemes for optimal navigation.

Algorithmically Optimized Hemispherical Dome as a Secondary Optical Element for the Fresnel Lens Solar Concentrator

The significance of this work lies in the development of a novel code-based, detailed, and deterministic geometrical approach that couples the optimization of the Fresnel lens primary optical element (POE) and the dome-shaped secondary optical element (SOE). The objective was to maximize the concentration acceptance product (CAP), while using the minimum SOE and receiver geometry at a given f-number and incidence angle (also referred to as the tracking error angle). The laws of polychromatic light refraction along with trigonometry and spherical geometry were utilized to optimize the POE grooves, SOE radius, receiver size, and SOE–receiver spacing. Two literature case studies were analyzed to verify this work’s optimization, both with a spot Fresnel lens POE and a spherical dome SOE. Case 1 had a 625 cm2 POE at an f-number of 1.5, and Case 2 had a 314.2 cm2 POE at an f-number of 1.34. The equivalent POE designed by this work, with optimized SOE radiuses of 13.6 and 11.4 mm, respectively, enhanced the CAP value of Case 1 by 52% to 0.426 and that of Case 2 by 32.4% to 0.45. The SOE’s analytical optimization of Case 1 was checked by a simulated comparative analysis to ensure the validity of the results. Fine-tuning this design for thermal applications and concentrated photovoltaics is also discussed in this paper. The algorithm can be further improved for more optimization parameters and other SOE shapes.

Refraction and reflexion according to the principle of general covariance

We show how the principle of general covariance introduced by Souriau in smoothly uniform contexts, can be extended to singular situations, considering the group of diffeomorphisms preserving the singular locus. As a proof of concept, we shall see how we get again, this way, the laws of reflection and refraction in geometric optics, by applying an extended general covariance principle to Riemannian metrics, discontinuous along a hypersurface.

An underwater stereo matching method based on ORB features

Since the traditional algorithm may cause problems such as slow running speed and more mismatching points when perform stereo matching on underwater environment, the ORB characteristics detection and curve restriction has been applied in this paper. Firstly the image should be detected so as to find out the characteristics, generate the descriptor, and match the feature points. Then the underwater curve restriction can be deduced according to the law of refraction combining internal and external parameters of camera. Finally the mismatching points can be decreased by means of underwater curve restriction. The experimental results have shown that in the case of effectively controlling mismatches, the speed of this algorithm are faster than traditional SIFT algorithm combined with curve restriction. As a result, it is of practical significance to improve the speed of underwater binocular vision system.

A Monge\u2013Amp\xe8re Problem with Non-quadratic Cost Function to Compute Freeform Lens Surfaces

In this article, we present a least-squares method to compute freeform surfaces of a lens with parallel incoming and outgoing light rays, which is a transport problem corresponding to a non-quadratic cost function. The lens can transfer a given emittance of the source into a desired illuminance at the target. The freeform lens design problem can be formulated as a Monge–Ampère type differential equation with transport boundary condition, expressing conservation of energy combined with the law of refraction. Our least-squares algorithm is capable to handle a non-quadratic cost function, and provides two solutions corresponding to either convex or concave lens surfaces.

Optimizing the Fresnel-Lens Solar-Concentrator Design for Tracking Error Mitigation in Thermal Applications, Using a Statistical Algorithm

The driving motive of this work is to enhance the tolerability of a flat Fresnel-lens concentrator to tracking errors, without the use of secondary optics or sophisticated, and normally costly, meticulous tracking equipment. This is achieved by optimizing the prismatic geometry of the Fresnel lens through a statistical approach that incorporates laws of light refraction and trigonometry. The proposed design produces high focal heat flux that is more suitable for thermal applications. A comparative simulation analysis was conducted for two case studies from literature, each with a different design method for the Fresnel lens. On basis of equivalent geometric concentration ratio and f-number, the flat-spot lenses, designed by this study, returned higher values of the concentration acceptance product ( $$CAP$$ ). In case #1 of a 212× cylindrical lens with an f-number of unity, an equivalent flat lens by this work almost doubled the $$CAP$$ from 0.165 to 0.34. For the design case #2 of a 559× aspheric Fresnel lens made by simple geometrical optics, a 60.7% enhanced $$CAP$$ was achieved by this work’s lens, from 0.206 to 0.331, at an f-number of 0.83.