We examine additional symmetries of specific refraction index profiles that are used in the well-known phenomena of perfect imaging and cloaking.In the considered cases, the translation generator and the angular momentum are conserved. We express the ray trajectory parameters through the integrals of motion and observe the existence of a photon state with maximal angular momentum, which can be used as an optical resonator. Application in plasmons and the role of polarization are discussed, and the Spin Hall effect in an extended symmetry profile is predicted.
Optical tweezers have found widespread application in many fields, from physics to biology. Here, we explain in detail how optical forces and torques can be described within the geometrical optics approximation and we show that this approximation provides reliable results in agreement with experiments for particles whose characteristic dimensions are larger than the wavelength of the trapping light. Furthermore, we provide an object-oriented software package implemented in MatLab for the calculation of optical forces and torques in the geometrical optics regime: \texttt{OTGO - Optical Tweezers…
Geometrical optics is extended so as to provide a model for spinning light rays via the coadjoint orbits of the Euclidean group characterized by color and spin. This leads to a theory of ``geometrical spinoptics'' in refractive media. Symplectic scattering yields generalized Snell-Descartes laws that include the recently discovered optical Hall effect.
We investigate imaging by spherically symmetric absolute instruments that provide perfect imaging in the sense of geometrical optics. We derive a number of properties of such devices, present a general method for designing them and use this method to propose several new absolute instruments, in particular a lens providing a stigmatic image of an optically homogeneous region and having a moderate refractive index range.
The basic laws of geometrical optics can be deduced from energy-momentum conservation for electromagnetic waves, without other wave concepts. However, the concept of quanta is required; it arises naturally, hence such a hypothesis could have arisen earlier than it did historically. Measurements to determine the angles of reflection and refraction demand that each incident quantum be either reflected or refracted; such a separation is central to the experimental results.
This paper presents a general approach to designing the isotropic spherical media with complex spatial structure that provide different types of imaging for different light rays. It is based on equivalence of the spherical medium and the corresponding geodesic lens. We use this approach to design multi-focal gradient- index lenses embedded into an optically homogeneous region and multi-focal absolute instruments that provide perfect imaging of three-dimensional domains.
We demonstrate that laser beam collapse in highly nonlinear media can be described, for a large number of experimental conditions, by the geometrical optics approximation within high accuracy. Taking into account this fact we succeed in constructing analytical solutions of the eikonal equation, which are exact on the beam axis and provide: i) a first-principles determination of the self-focusing position, thus replacing the widely used empirical Marburger formula, ii) a benchmark solution for numerical simulations, and iii) a tool for the experimental determination of the high-order nonlinear …
Optical forces are often calculated by discretizing the trapping light beam into a set of rays and using geometrical optics to compute the exchange of momentum. However, the number of rays sets a trade-off between calculation speed and accuracy. Here, we show that using neural networks permits one to overcome this limitation, obtaining not only faster but also more accurate simulations. We demonstrate this using an optically trapped spherical particle for which we obtain an analytical solution to use as ground truth. Then, we take advantage of the acceleration provided by neural networks to st…
Geometrical optics (GO) is widely used in studies of electromagnetic materials because of its ease of use compared to full-wave numerical simulations. Exact solutions for waves can, however, differ significantly from the GO approximation. In particular, effects that are "perfect" for waves cannot usually be derived using GO. Here we give a method for designing materials in which GO is exact for some waves. This enables us to find interesting analytical solutions for exact wave propagation in inhomogeneous media. Two examples of the technique are given: a material in which two point sources do …
We present a compact two-lens HDPE f/1.6 refractor design that is capable of supporting a 28-deg diffraction-limited field of view at 1-mm wavelengths and contrast it to a similar two-lens refractor using silicon lenses. We compare the optical properties of these two systems as predicted by both geometrical and physical optics. The presented analysis suggests that by relaxing telecentricity requirements, a plastic two-lens refractor system can perform comparably to a similar silicon system across a wide field of view and wavelengths up to 1 mm. We show that for both telescope designs, cold sto…
As an approximate theory that is highly regarded for its computational efficiency, geometrical optics (GO) is widely used for modeling waves in various areas of physics. However, GO fails at caustics, which significantly limits its applicability. A new framework, called metaplectic geometrical optics (MGO), has recently been developed that allows caustics of certain types to be modeled accurately within the GO framework. Here, we extend MGO to the most general case. To illustrate our new theory, we also apply it to several sample problems, including calculations of two-dimensional wavefields n…
We proposed in a previous paper [Opt. Commun. 228, 33 (2003)] a modified radiative transfer equation to describe radiative transfer in a medium with a spatially varying refractive index. The present paper is devoted to the demonstration that this equation perfectly works in the non-absorbing / non-scattering limit, what was contested by L. Martí-López and coworkers [Opt. Commun. 266, 44 (2006)]. The assertion that this equation would imply a zero divergence of the rays is also commented.
In this work, we present an analysis of the influence of the geometrical parameters on the sensitivity and linear range of the fiber optic angular displacement sensor, through computational simulations and experiments. The geometrical parameters analyzed were the lens focal length, the gap between fibers, the fibers cladding radii, the emitting fiber critical angle (or, equivalently, the emitting fiber numerical aperture), and the standoff distance (distance between the lens and the reflective surface). Besides, we analyzed the sensor sensitivity regarding any spurious linear displacement. The…
Geometrical optics (GO) is often used to model wave propagation in weakly inhomogeneous media and quantum-particle motion in the semiclassical limit. However, GO predicts spurious singularities of the wavefield near reflection points and, more generally, at caustics. We present a new formulation of GO, called metaplectic geometrical optics (MGO), that is free from these singularities and can be applied to any linear wave equation. MGO uses sequenced metaplectic transforms of the wavefield, corresponding to symplectic transformations of the ray phase space, such that caustics disappear in the n…
In this paper we present a modification of the geometrical optics method, which allows one to properly separate the complex amplitude and the phase of the wave solution. Appling this modification to a smoothly inhomogeneous isotropic medium, we show that in the first geometrical optics approximation the medium is weakly anisotropic. The refractive index, being dependent on the direction of the wave vector, contains the correction, which is proportional to the Berry geometric phase. Two independent eigenmodes of right-hand and left-hand circular polarizations exist in the medium. Their group ve…
We consider the scattering of light in participating media composed of sparsely and randomly distributed discrete particles. The particle size is expected to range from the scale of the wavelength to the scale several orders of magnitude greater than the wavelength, and the appearance shows distinct graininess as opposed to the smooth appearance of continuous media. One fundamental issue in physically-based synthesizing this appearance is to determine necessary optical properties in every local region. Since these optical properties vary spatially, we resort to geometrical optics approximation…
Geometrical optics describes, with good accuracy, the propagation of high-frequency plane waves through an electromagnetic medium. Under such approximation, the behaviour of the electromagnetic fields is characterised by just three quantities: the temporal frequency $ω$, the spatial wave (co)vector $k$, and the polarisation (co)vector $a$. Numerous key properties of a given optical medium are determined by the Fresnel surface, which is the visual counterpart of the equation relating $ω$ and $k$. For instance, the propagation of electromagnetic waves in a uniaxial crystal, such as calcite, is r…
Lens design uses a calculation of the lens' surfaces that permit to obtain an image from a given object. A set of general rules and laws permits to calculate the essential points of the optical system such as distances, thickness, pupils, and focal distance among others. Now, the theory on which the classical lens design is based changes radically as our theoretical foundations do not rely on the classical ray tracing rules. We show that with the rules expressed in a reduced vector analytical solution set of equations, we can take into account all optical elements, i.e. refractive, reflective,…
We study the geometrical optics generated by a refractive index of the form $n(x,y)=1/y$ $(y>0)$, where $y$ is the coordinate of the vertical axis in an orthogonal reference frame in $\R^2$. We thus obtain what we call "hyperbolic geometrical optics" since the ray trajectories are geodesics in the Poincaré-Lobachevsky half--plane $\HH^2$. Then we prove that the constant phase surface are horocycles and obtain the \emph{horocyclic waves}, which are closely related to the classical Poisson kernel and are the analogs of the Euclidean plane waves. By studying the transport equation in the Beltrami…
We assess the accuracy and relevance of the numerical algorithms based on the principles of Geometrical Optics (GO) and Physical Optics (PO) in the analysis of reduced-size homogeneous dielectric lenses prone to behave as open resonators. As a benchmark solution, we use the Muller boundary integral equations discretized with trigonometric Galerkin scheme that has guaranteed and fast convergence as well as controllable accuracy. The lens cross-section is chosen typical for practical applications, namely an extended hemiellipse whose eccentricity satisfies the GO focusing condition. The analysis…
Symmetries and their associated selection rules are extremely useful in all fields of science. In particular, for system that include electromagnetic (EM) fields interacting with matter, it has been shown that both of symmetries of matter and EM field's time-dependent polarization play a crucial role in determining the properties of linear and nonlinear responses. The relationship between the system's symmetry and the properties of its excitations facilitate precise control over light emission and enable ultrafast symmetry-breaking spectroscopy of variety of properties. Here. we formulate the …
The paper presents an ab initio account of the paraxial complex geometrical optics (CGO) in application to a scalar Gaussian beam propagation and diffraction in a 3D smoothly inhomogeneous medium. The paraxial CGO deals with quadratic expansion of the complex eikonal and reduces the wave problem to the solution of ordinary differential equations of Riccati type. This substantially simplifies description of Gaussian beams diffraction as compared to full wave or parabolic (quasi-optics) equations. For a Gaussian beam propagating in a homogeneous medium or along the symmetry axis in a lens-like m…
