Law Of Refraction Research Articles

Comprehensive refractive manipulation of water waves using electrostriction

The control of wave propagation based on refraction principles offers unparalleled possibilities as shown by the striking example of optics. This approach is unfortunately limited for water waves as it relies mainly on variations of the liquid depth which, while controlling the wave velocity, also trigger nonlinearities and damping. In this article, we show experimentally that electrostriction allows to implement extensive refraction-based control of water waves in a precise and contactless manner. The setup consists of an electrode under high voltage placed above the grounded conductive water. The waves propagating under the electrode can be slowed down up to approximately half their speed compared to free propagation. We characterize the Snell-Descartes laws of refraction and the total internal reflection for the water waves. We implement emblematic refraction-based devices such as electrically tunable focusing lenses, waveguides without obstacles, and beam splitters based on frustrated internal reflection to perform interference experiments.

Active particles crossing sharp viscosity gradients

Active particles (living or synthetic) often move through inhomogeneous environments, such as gradients in light, heat or nutrient concentration, that can lead to directed motion (or taxis). Recent research has explored inhomogeneity in the rheological properties of a suspending fluid, in particular viscosity, as a mechanical (rather than biological) mechanism for taxis. Theoretical and experimental studies have shown that gradients in viscosity can lead to reorientation due to asymmetric viscous forces. In particular, recent experiments with Chlamydomonas Reinhardtii algae swimming across sharp viscosity gradients have observed that the microorganisms are redirected and scattered due to the viscosity change. Here we develop a simple theoretical model to explain these experiments. We model the swimmers as spherical squirmers and focus on small, but sharp, viscosity changes. We derive a law, analogous to Snell’s law of refraction, that governs the orientation of active particles in the presence of a viscosity interface. Theoretical predictions show good agreement with experiments and provide a mechanistic understanding of the observed reorientation process.

The MLS-based numerical manifold method for Darcy flow in heterogeneous porous media

The moving least squares (MLS) interpolation gets rid of the shackles of meshes, while the numerical manifold method (NMM) can solve both continuity and discontinuity problems in a unified way. The MLS-based NMM (abbreviated as MLS-NMM) inherits the individual merits of MLS and NMM. In the present work, the MLS-NMM is extended to solve groundwater flow problems in heterogeneous porous media. To accurately simulate the flow velocity field at and in the vicinity of the material interfaces, the refraction law is introduced into the proposed model as a posteriori condition. Several examples are employed to verify the correctness and accuracy of the proposed model. The results indicate that the MLS-NMM can model the Darcy flow in heterogeneous porous media with high accuracy, as well as satisfy the refraction law rigorously. In addition, the smooth flow velocity field can be obtained directly without the need for extra post-processing.

A Compact, Single-Layer, Index-Modulated Microstrip Antenna With Stable Customized Tilted Beam Over a Wide Bandwidth

A compact, single-layer, wideband, tilted-beam microstrip antenna with customized tilt angle is presented. The antenna is comprised by one subwavelength driven patch and several subwavelength parasitic patches. The tilt angle of the antenna is determined by the index distributions of those subwavelength patches. The detailed index-modulation principle is developed from the generalized law of refraction, wherein the reflection phase is mapped to the index by utilizing the equivalent transmission-line theory and the transverse resonance technique. As a result, the tilt angle could be customized by allocating patch cells following specific index distributions. On top of that, as a new degree of freedom in design, one could choose subwavelength patches with low dispersion characteristics as unit cells to further enhance the operational bandwidth and improve the stability of radiation pattern. For demonstration, two types of I-shaped patch units with different dispersion characteristics have been utilized to configure tilted-beam antennas that follow the same index distributions. In addition, the effectiveness of the developed tilted-beam antenna for array application has been proven. Experiments have been carried out to verify their performance.

UAV trajectory planning based on an improved sparrow optimization algorithm with multi-strategy integration

Real-time monitoring of urban high-altitude data is an important goal in the construction and development of smart cities today. However, with the development of modern cities, the monitoring space becomes complicated and narrow because of the different building heights and no-fly zones, which makes UAV trajectory planning more difficult. In this paper, a multi-strategy sparrow search algorithm (MSSA) is proposed to solve the UAV trajectory planning problem in a three-dimensional environment. The algorithm aims to minimize the flight distance and maximize the use efficiency of the UAV. First, the improved algorithm employed a reverse-learning strategy based on the law of refraction to improve the search range and enhance the optimization performance. Second, we introduced a random step size generated by Levy flight into the position update strategy of the participant. The algorithm accuracy and speed of convergence were improved by the randomness feature. Finally, the algorithm incorporated the Cauchy mutation to improve the scout position, which enhanced its ability to jump out of the local optimum of the algorithm. Sixteen benchmark test functions, Wilcoxon rank sum test, and 30 CEC2014 test function optimization results demonstrated that MSSA had better optimization accuracy, convergence speed, and robustness than the comparison algorithms. In addition, the proposed algorithm was applied to the UAV trajectory planning problem in different complex 3D environments. The results confirmed that the MSSA outperformed the other algorithms in complex 3D trajectory planning problems.

The Beginnings of Optimization: Snell's Law and Fermat's Principle

The study provides a brief description of the basic concepts of the theory and its history. We considered Fermat's principle of least time, proposed to substantiate Snell's experimentally established law of refraction of light. This example can be applied as the beginning of the formation of any scientific discipline. The main purpose is to show the benefits of applying three components: experiments, bold hypotheses and mathematical models.

Optics of the metalens

A metalens is an ultra-thin lens made of a metasurface, with focusing and imaging functions, and with promising applications in miniature optical systems, such as mobile devices. This paper gives an introduction to the optics of metalenses. First, we examine the focusing and imaging characteristics of metalenses. And then, as an essential step, we derive the refraction law of metasurfaces from the Huygens’s principle. It is then that some basic formulas of the metalens, for focusing and imaging, are derived at a level appropriate for undergraduate students. We begin with the derivation of the phase profile of a metalens. Then we derive the lens equation of the metalens, which relates the object and image distance in a metalens. The paper closes with a discussion of the relationship between the metalens and the classical refractive lens.

Kepler in search of the “Anaclastic”

This article was written as a tribute to Enrique Tirapegui who passed away on 10 March 2020. We were working on these issues with him. He will always be remembered.We describe Kepler’s attempt to discover the law of refraction. Its very elegant analysis can be seen as a numerical search for the solution of a differential equation. Kepler’s arguments and the reasons for his failure are discussed. This analysis, which precedes Descartes’ solution of the inverse tangent problem posed by Florimond de Beaune, is an opportunity to briefly recall some aspects of the history of differential equations. It is also an opportunity to propose a pedagogical activity that consists of introducing the law of refraction to students in the wake of Kepler and Descartes. This article is intended to be pedagogical. It is self-consistent and all the properties of the hyperbola necessary to follow Kepler’ arguments are demonstrated using elementary geometry. It will also be the material for the chapter of a book intended to revisit the elementary laws of physics using Euclidean geometry. This project started with Enrique Tirapegui. This work was presented at the XVIII Instabilities and Nonequilibrium Structures Workshop in tribute to Enrique held in December 2021 in Valparaiso, Chile.

Three-dimensional reconstruction from a fringe projection system through a planar transparent medium.

A vision measurement system is placed in a protective cover made of a transparent medium to avoid environmental influences. Due to the deflection of light rays on the front and rear surfaces of the transparent medium, the imaging position of an object on the camera target plane is deviated, which makes the traditional vision detection methods based on the triangulation principle produce large measurement errors. This work introduces a three-dimensional (3D) reconstruction method by fringe projection system through a planar transparent medium. We derive the coordinate transformation relationship between a real-object point and a pseudo-object point caused by light refraction based on Snell's law of flat refraction. Based on the relationship, a modified fringe projection method is proposed for unbiased 3D reconstruction. Two experiments, including 3D shape measurement of a white plate with ring markers and 3D shape measurement of a regular spherical object are conducted. The results demonstrate the effectiveness of the proposed method in such measurement environment.

Vectorial discrete ray tracing approach for solving radiative transfer in irregular variable index media via discrete transfer method

A novel vectorial discrete ray tracing approach is applied for tracking the curve paths of radiative beams through the variable index media with complex geometries. In this approach, the irregular solution domain is discretized by a boundary-fitted grid, where the radiative properties, including the refractive index, are considered to be constant for each control volume and control surface. Radiative beams travel along straight lines through each volume element until they hit the interfaces between adjacent volume elements, where the direction of rays changes according to the Snell's law of refraction. The radiation heat transfer in absorbing, emitting, and isotropically scattering media is solved by the discrete transfer method. The problem is solved for one-, two-, and three-dimensional media. Axisymmetric media are simulated by detaching a sector with specular sidewalls. Verification is done by comparing the results with available benchmark solutions in literature. For the cases where the data is not available, the accuracy is proved by satisfaction the energy conservation criterion.

Descartes’s Deduction of the Law of Refraction and the Shape of the Anaclastic Lens in Rule 8

Descartes’s most extensive discussion of the law of refraction and the shape of the anaclastic lens is contained in Rule 8 of Rules for the Direction of the Mind. Few reconstructions of Descartes’s discovery of the law of refraction take Rule 8 as their basis. In Rule 8, Descartes denies that the law of refraction can be discovered by purely mathematical means, and he requires that the law of refraction be deduced from physical principles about natural power or force, the nature of the action of light, and the behavior of light rays in a variety of transparent media. For over a century, however, there has been broad agreement that Descartes discovered the law of refraction by purely mathematical means, and that he only later provided the relevant physical rationale (via comparisons or analogies) in Dioptrics II. I execute each step in Descartes’s proposed deduction of the law of refraction and the shape of the anaclastic lens in Rule 8 and concretely show how Descartes could have discovered the law of refraction and the shape of the anaclastic by its means. Rule 8, I argue, reflects Descartes’s actual path to the discovery of the law of refraction and the shape of the anaclastic lens.

Examining and explaining the "generalized laws of reflection and refraction" at metasurface gratings.

The widespread concept of "generalized laws of reflection and refraction" that is commonly applied to wave propagation through metasurfaces is thoroughly explained on the foundation of diffraction theory. This allows definition of strict constraints to the applicability of these generalized laws and highlights the underlying physical effects. A diffraction-based explanation of the reported phenomena is provided that yields a solid theoretical foundation for the prediction of experimental results and that clarifies many of the convoluted explanations found throughout the literature.

Nondispersive Space–Time Wave Packets Propagating in Dispersive Media

AbstractSpace–time wave packets can propagate invariantly in free space with arbitrary group velocity thanks to the spatio‐temporal correlation. Here it is proved that the space–time wave packets are stable in dispersive media as well and free from the spread in time caused by material dispersion. Furthermore, the law of anomalous refraction for space–time wave packets is generalized to the weakly dispersive situation. These results reveal new potential of space–time wave packets for the applications in real dispersive media.

Measuring the refractive index of a transparent sphere from the primary and secondary rainbows

We propose a method to obtain the refractive index of a transparent sphere from the fascinating phenomena of rainbows, which can be used as an interesting subject in the experimental courses of primary optics for undergraduate or high school students. The primary and secondary rainbows are reproduced on white walls and screens in the classroom with the sun and water drops replaced by the LED flashlight and K9 glass (one type of borosilicate glass) sphere respectively. The locations of the primary and secondary rainbows on the screen are measured, with the locations of the screen and sphere changed. By fitting the multiple measurements to the theoretical result using the least square method, the refractive index of the glass sphere, as an unknown parameter in the fitting process, is obtained. For both the primary and secondary rainbows, the locations of the red, yellow and purple bows are measured independently. Repetitive measurements give very stable and accurate output with the standard deviation ratio as low as 10−4 and the relative error all around 0.1% compared to the result obtained from special instruments in the literature. The fascinating phenomena and precise measurements enable this method to fully deepen the students’ understanding of the law of refraction and the application of the least square method.

Extended Snell’s law based on surface current radiation

The generalized laws of reflection and refraction have been proposed in many related reports, but they still have some imperfections. For example, the first principle lies in the Fermat principle, but the realization of surface phase gradient utilizes the scattering of superelements. The former belongs to geometric optics, and the latter is attributed to wave optics. Therefore, this paper avoids the Fermat principle and gives the generalized Snell’s law from a new perspective by controlling the conductivity or polarizability of materials. The findings indicate that the surface current can manipulate the reflected and refracted light. The key is to obtain the relationship between the different components of the spectrum and the wave vectors parallel to the surface of the reflected and refracted waves by using the Fourier transform of the polarizability. This model is applicable to the metamaterial at the nanoscale, making it possible to design flexible surface structures.

A Natural Exposition and Concise Derivation of Geometric Optics Laws by Fermat Principle

With Fermat Principle, that is, the most fundamental principle in geometric optics, it is easy to obtain the three laws of geometric optics as well as the paraxial refraction and catoptric imaging laws. Making use of the principle and a simple mathematical relation, this paper expounds and derives the rectilinear propagation law of rays in the homogeneous medium, the optical reflection and refraction laws as well as the spherical reflection and refraction laws under the paraxial condition. Ulteriorly, with given character rules, the spherical dioptric and catoptric imaging formulas can be generalized into a unified formula separately. The elaboration and derivation in this paper reflect the simplicity of physics and reduce the error probability of complex trigonometric function operation, thus making it more easy for high school students or junior undergraduate and junior college students to accept and understand.

Propagation of polarization in transmission problems

For geometric systems of real principal type, we define a subprincipal symbol and derive a transport equation for polarizations which, in the scalar case, is a well-known equation of Duistermaat and H\"ormander. We apply the transport equation to propagation of polarization in transmission problems of elastodynamics, to interior bulk waves as well as to free (Rayleigh) surface waves. Using spectral factorizations of matrix polynomials having real spectrum, we establish reflection and refraction laws of polarizations at the boundary and at interior interfaces. The results are not limited to isotropic elasticity.

Addressing a novel problem on the basis of the generalized vectorial laws of reflection and refraction along with offering novel treatments of derivation of some results in geometrical optics

The generalized vectorial laws of reflection and refraction have been applied in this paper to develop novel determinantal equations involving the coordinates of the object point, image point, and point of incidence in cases of reflection and refraction of light when reflection (refraction) takes place from a plane reflecting surface (plane surface of discontinuity) or from a concave (convex) spherical reflecting surface (spherical surface of discontinuity). Establishment of the theoretical proof of the validity of the principle of reversibility of light has also been accomplished by making use of the novel determinantal equations developed in an earlier work by the author. Furthermore, looking at the interest of the readership, novel treatments for the derivation of the relationship between object distance and image distance in case of reflection of light at a plane reflecting surface as well as for the derivation of the relationships between real depth (height) and apparent depth (height) when refraction of light takes place at a plane surface of discontinuity are also offered in this paper.

Reflection and Refraction of a Spin at the Edge of a Quasi-Two-Dimensional Semiconductor Layer (Quantum Well) and a Topological Insulator

We derive the reflection and refraction laws for an electron spin incident from a quasi-two-dimensional semiconductor region (with no spin–orbit interaction) on the metallic surface of a topological insulator (TI) when the two media are in contact edge to edge. For a given incident angle, there can generally be two different refraction angles for refraction into the two spin eigenstates in the TI surface, resulting in two different ‘spin refractive indices’ (birefringence) and the possibility of two different critical angles for total internal reflection. We derive expressions for the spin refractive indices and the critical angles, which depend on the incident electron’s energy for given effective masses in the two regions and a given potential discontinuity at the TI/semiconductor interface. For some incident electron energies, there is only one critical angle, in which case 100% spin polarized injection can occur into the TI surface from the semiconductor if the angle of incidence exceeds that critical angle. The amplitudes of reflection of the incident spin with and without spin flip at the interface, as well as the refraction (transmission) amplitudes into the two spin eigenstates in the TI, are derived as functions of the angle of incidence.

Analytic solution for double optical metasurface beam scanners

Optical metasurfaces are researched more and more intensively for the possible realization of lightweight and compact optical devices with novel functionalities. In this paper, a new beam-steering system based on double metasurface lenses (metalenses) is proposed and developed. The proposed system is lightweight, small volume, low cost, and easy to integrate. The exact close-form forward and numerical inverse solutions are derived respectively using the generalized Snell’s law of refraction. Given the orientations of the double metalenses, the pointing position can be accurately determined. If the desired pointing position is given, the required metalenses’ orientations can be obtained by applied global optimization algorithms to solve nonlinear equations related to the inverse problem. The relationships of the scan region and blind zone with the system parameters are derived. The method to eliminate the blind zone is given. Comparison with double Risley-prism systems is also conducted. This work provides a new approach to control light beams.