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 susceptibility. Successful comparison with several experiments is presented.
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…
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.
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.
When an optical beam propagates through dielectric blocks, its optical phase is responsible for the path of the beam. In particular, the first order Taylor expansion of the geometrical part reproduces the path predicted by the Snell and reflection laws whereas the first order expansion of the Fresnel phase leads to the Goos-Haenchen shift. In this paper, we analyze the effects of the second order Taylor expansion of the geometrical phase on the shape of the optical beam and show how it affects the transversal symmetry of Hermite-Gaussian beams. From the analytical expression of the transmitted…
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 …
A centred system forms the coherent image of the optical field on a spherical cap, taken as an object, on another spherical cap, whose vertex and curvature center are the respective paraxial images of the vertex and center of the object cap. That ``double-conjugation'' law, usually obtained in the framework of a scalar theory of diffraction, is deduced from concepts of geometrical optics. A magnification law between radii of conjugate spherical-caps extends the notion of longitudinal magnification to finite distances, and generalizes the longitudinal Lagrange-Helmholtz formula. Applying the do…
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.
Light scattering inhibits high-resolution optical imaging, manipulation and therapy deep inside biological tissue by preventing focusing. To form deep foci, wavefront-shaping and time-reversal techniques that break the optical diffusion limit have been developed. For in vivo applications, such focusing must provide high gain, high speed, and a large number of spatial modes. However, none of the previous techniques meet these requirements simultaneously. Here, we overcome this challenge by rapidly measuring the perturbed optical field within a single camera exposure followed by adaptively time-…
We report on the experimental measurement of the backscattering matrix of a weakly scattering medium in optics, composed of a few dispersed gold nanobeads. The DORT method (Decomposition of the Time Reversal Operator) is applied to this matrix and we demonstrate selective and efficient focusing on individual scatterers, even through an aberrating layer. Moreover, we show that this approach provides the decomposition of the scattering pattern of a single nanoparticle. These results open important perspectives for optical imaging, characterization and selective excitation of nanoparticles.
Controlling the wavefront of an extreme ultraviolet (XUV) high-order harmonic beam during the generation process offers to focus the beam without resorting to any XUV optics. By characterizing the XUV intensity profile and wavefront, we quantitatively retrieve both the size and the position of the waist of each generated harmonics and show that optics-free focusing leads to focused XUV spot with micrometer size. We use this remarkable coherent effect to demonstrate efficient and adjustable spectral filtering of the XUV light, along with a strong rejection of the fundamental beam, without using…
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…
We present a novel family of paraxial optical beams having a confluent hypergeometric transverse profile, which we name hypergeometric Gauss modes of type-II (HyGG-II). These modes are eigenmodes of the photon orbital angular momentum and they have the lowest beam divergence at waist of HyGG-II among all known finite power families of paraxial modes. We propose to exploit this feature of HyGG-II modes for generating, after suitable focusing, a "light needle" having record properties in terms of size and aspect ratio, possibly useful for near-field optics applications.
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…
Focusing light into opaque random or scattering media such as biological tissue is a much sought-after goal for biomedical applications such as photodynamic therapy, optical manipulation, and photostimulation. However, focusing with conventional lenses is restricted to one transport mean free path in scattering media, limiting both optical penetration depth and resolution. Focusing deeper is possible by using optical phase conjugation or wavefront shaping to compensate for the scattering. For practical applications, wavefront shaping offers the advantage of a robust optical system that is less…
Conventional light focusing, i. e. concentration of an extended optical field within a small area around a point, is a frequently used process in Optics. An important extension to conventional focusing is the generation of the annular focal field of an optical beam. We discuss a simple optical setup that achieves this kind of focusing employing a phase plate as unique optical component. We first establish the class of beams that being transmitted through the phase plate can be focused into an annular field with topological charge of arbitrary integer order q. Then, for each beam in this class …
Focusing light inside scattering media in a freely addressable fashion is challenging, as the wavefront of the scattered light is highly disordered. Recently developed ultrasound-guided wavefront shaping methods are addressing this challenge, albeit with relatively low modulation efficiency and resolution limitations. In this paper, we present a new technique, time-reversed ultrasound microbubble encoded (TRUME) optical focusing, which is able to focus light with improved efficiency and sub-ultrasound wavelength resolution. This method ultrasonically destructs microbubbles, and measures the wa…
Non-invasively focusing light into strongly scattering media, such as biological tissue, is highly desirable but challenging. Recently, wavefront shaping technologies guided by ultrasonic encoding or photoacoustic sensing have been developed to address this limitation. So far, these methods provide only acoustic diffraction-limited optical focusing. Here, we introduce nonlinear photoacoustic wavefront shaping (PAWS), which achieves optical diffraction-limited (i.e. single-speckle-grain) focusing in scattering media. We develop an efficient dual-pulse excitation approach to generate strong nonl…
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,…
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.
Transformation optics (TO) has been used to propose various novel optical devices. With the help of metamaterials, several intriguing designs, such as invisibility cloaks, have been implemented. However, as the basic units should be much smaller than the working wavelengths to achieve the effective material parameters, and the sizes of devices should be much larger than the wavelengths of illumination to work within the light-ray approximation, it is a big challenge to implement an experimental system that works simultaneously for both geometric optics and wave optics. In this letter, by using…
The geometrical interpretation of electromagnetism in transparent media (transformation optics) is extended to include media with isotropic, inhomogeneous, chirality. It is found that such media may be described through introducing the non-Riemannian geometrical property of torsion into the Maxwell equations, and shown how such an interpretation may be applied to the design of optical devices.
Geometrical chirality is a property of objects that describes three-dimensional mirror-symmetry violation and therefore it requires a non-vanishing spatial extent. In contrary, optical chirality describes only the local handedness of electromagnetic fields and neglects the spatial geometrical structure of optical beams. In this manuscript, we put forward the physical significance of geometrical chirality of spatial structure of optical beams, which we term "Kelvin's chirality". Further, we report on an experiment revealing the coupling of Kelvin's chirality to optical chirality upon transmissi…
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…
Laser filamentation in transparent material has a wide range of applications, from three dimensional manufacturing to biological technologies. Various experimental results showed that femtosecond laser pulse filamentation in fused silica strongly depends on laser focusing conditions. However, the physical mechanism governing each regime has not been fully understood. For the first time, single and multiple re-focusing of the laser pulse in interaction of femtosecond laser pulse with fused silica, and consequent single and multiple damage zones (filaments) have been observed in our extensive th…
We design three-dimensional (3D) metallic nanowire media with different structures and numerically demonstrate that they can be homogeneous effective indefinite anisotropic media by showing that their dispersion relations are hyperbolic. For a finite slab, a nice fitting procedure is exploited to obtain the dispersion relations from which we retrieve the effective permittivities. The pseudo focusing for the real 3D wire medium agrees very well with the homogeneous medium having the effective permittivity tensor of the wire medium. Studies also show that in the long-wavelength limit, the hyperb…
The propagation of a light ray in thin layer (film) within geometrical optics is considered. It is assumed that the ray is captured inside the layer due to reflecting walls or total internal reflection (in the case of a dielectric layer). It has been found that for a very thin film (the length scale is imposed by the curvature of the surface at a given point) the equations describing the trajectory of the light beam are reduced to the equation of a geodesic on the limiting curved surface. There have also been found corrections to the trajectory equation resulting from the finite thickness of t…
We theoretically and experimentally investigate the mutual collapse dynamics of two spatially separated optical beams in a Kerr medium. Depending on the initial power, beam separation, and the relative phase, we observe repulsion or attraction, which in the latter case reveals a sharp transition to a single collapsing beam. This transition to fusion of the beams is accompanied by an increase in the collapse distance, indicating the effect of the nonlinear coupling on the individual collapse dynamics. Our results shed light on the basic nonlinear interaction between self-focused beams and provi…
We theoretically and experimentally investigate the design of an all-optical noiseless magnification and sampling function free from any active gain medium and associated high-power continuous wave pump source. The proposed technique is based on the co-propagation of an arbitrary shaped signal together with an orthogonally polarized intense fast sinusoidal beating within a normally dispersive optical fiber. Basically, the strong nonlinear phase shift induced by the sinusoidal pump beam on the orthogonal weak signal through cross-phase modulation turns the defocusing regime into localized tempo…
The use of individual multimode optical fibers in endoscopy applications has the potential to provide highly miniaturized and noninvasive probes for microscopy and optical micromanipulation. A few different strategies have been proposed recently, but they all suffer from intrinsically low resolution related to the low numerical aperture of multimode fibers. Here, we show that two-photon polymerization allows for direct fabrication of micro-optics components on the fiber end, resulting in an increase of the numerical aperture to a value that is close to 1. Coupling light into the fiber through …
The ability to steer light propagation inside scattering media has long been sought-after due to its potential widespread applications. To form optical foci inside scattering media, the only feasible strategy is to guide photons by using either implanted or virtual guide stars. However, all of these guide stars must be introduced extrinsically, either invasively or by physical contact, limiting the scope of their application. Here, we focus light inside scattering media by employing intrinsic dynamics as guide stars. By time-reversing the perturbed component of the scattered light adaptively, …
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.
For a long time it was believed there was no reason to include the geometrical phase in studying the propagation of gaussian optical beams through dielectric blocks. This can be justified by the fact that the first order term in the Taylor expansion of this phase is responsible for the lateral shift of the optical beam which is also predicted by ray optics. From this point of view, the geometrical phase can be seen as a purely auxiliary concept. In this paper, we show how the second order term in the Taylor expansion accounts for the symmetry breaking of the transversal spatial distribution an…
A nonlinear self-focusing material can amplify random small-amplitude phase modulations present in an optical beam, leading to the formation of amplitude singularities commonly referred to as optical caustics. By imposing polarization structuring on the beam, we demonstrate the suppression of amplitude singularities caused by nonlinear self-phase modulation. Our results are the first to indicate that polarization-structured beams can suppress nonlinear caustic formation in a saturable self-focusing medium and add to the growing understanding of catastrophic self-focusing effects in beams conta…
Non-Euclidean geometry combined with transformation optics has recently led to the proposal of an invisibility cloak that avoids optical singularities and therefore can work, in principle, in a broad band of the spectrum [U. Leonhardt and T. Tyc, Science 323, 110 (2009)]. Such a cloak is perfect in the limit of geometrical optics, but not in wave optics. Here we analyze, both analytically and numerically, full wave propagation in non-Euclidean cloaking. We show that the cloaking device performs remarkably well even in a regime beyond geometrical optics where the device is comparable in size wi…
We study the interaction of focused radially-polarized light with metal nanospheres. By expanding the electromagnetic field in terms of multipoles, we gain insight on the excitation of localized surface plasmon-polariton resonances in the nanoparticle. We show that focused radially-polarized beams offer more opportunities than a focused plane wave or a Gaussian beam for tuning the near- and far-field system response. These results find applications in nano-optics, optical tweezers, and optical data storage.
The small correction volume for conventional wavefront shaping methods limits their applications in biological imaging through scattering media. We demonstrate large volume wavefront shaping through a scattering layer with a single correction by conjugate adaptive optics and remote focusing (CAORF). The remote focusing module can keep the conjugation between the AO and scattering layer during three-dimensional scanning. This new configuration provides a wider correction volume by the best utilization of the memory effect in a fast three-dimensional laser scanning microscope. Our results show t…
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…
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…
Optical focusing at depths in tissue is the Holy Grail of biomedical optics that may bring revolutionary advancement to the field. Wavefront shaping is a widely accepted approach to solve this problem, but most implementations thus far have only operated with stationary media which, however, are scarcely existent in practice. In this article, we propose to apply a deep convolutional neural network named as ReFocusing-Optical-Transformation-Net (RFOTNet), which is a Multi-input Single-output network, to tackle the grand challenge of light focusing in nonstationary scattering media. As known, de…
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 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 dipole-traps are used in various scientific fields, including classical optics, quantum optics and biophysics. Here, we propose and implement a dipole-trap for nanoparticles that is based on focusing from the full solid angle with a deep parabolic mirror. The key aspect is the generation of a linear-dipole mode which is predicted to provide a tight trapping potential. We demonstrate the trapping of rod-shaped nanoparticles and validate the trapping frequencies to be on the order of the expected ones. The described realization of an optical trap is applicable for various other kinds of …
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…
Manipulating and focusing light deep inside biological tissue and tissue-like complex media has been desired for long yet considered challenging. One feasible strategy is through optical wavefront engineering, where the optical scattering-induced phase distortions are time reversed or pre-compensated so that photons travel along different optical paths interfere constructively at the targeted position within a scattering medium. To define the targeted position, an internal guidestar is needed to guide or provide a feedback for wavefront engineering. It could be injected or embedded probes such…
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…
We present an experimental study of Z-scan measurements of the nonlinear response of cold Cs atoms in a magneto-optical trap. Numerical simulations of the Z-scan signal accounting for the nonuniform atomic density agree with experimental results at large probe beam detunings. At small probe detunings a spatially varying radiation force modifies the Z-scan signal. We also show that the measured scan is sensitive to the atomic polarization distribution.
On-chip optical interconnect has been widely accepted as a promising technology to realize future large-scale multiprocessors. Mode-division multiplexing (MDM) provides a new degree of freedom for optical interconnects to dramatically increase the link capacity. Present on-chip multimode devices are based on traditional wave-optics. Although large amount of computation and optimization are adopted to support more modes, mode-independent manipulation is still hard to be achieved due to severe mode dispersion. Here, we propose a universal solution to standardize the design of fundamental multimo…
The focusing properties of three aspheric lenses with numerical aperture (NA) between 0.53 and 0.68 were directly measured using an interferometrically referenced scanning knife-edge beam profiler with sub-micron resolution. The results obtained for two of the three lenses tested were in agreement with paraxial gaussian beam theory. It was also found that the highest NA aspheric lens which was designed for 830nm was not diffraction limited at 633nm. This process was automated using motorized translation stages and provides a direct method for testing the design specifications of high numerical…
The critical power of self-focusing in the air for ultrashort UV laser pulses stretched due to dispersion from 90 up to 730 fs was experimentally measured. It was shown that the pulse duration enhancement due to its propagation in condensed media within above mentioned time-range leads to almost linear decrease of self-focusing critical power. It was also observed that when the pulse peak power exceeds the critical one, the maximum of linear plasma distribution along the UV laser filament does not shifted in the direction opposite to the laser pulse propagation as in the case of IR laser filam…
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…
Ray optics is an intuitive and computationally efficient model for wave propagation through nonuniform media. However, the underlying geometrical-optics (GO) approximation of ray optics breaks down at caustics, erroneously predicting the wave intensity to be infinite and thereby limiting the predictive capabilities of GO-based codes. Full-wave modeling can be used instead, but the added computational cost brings its own set of tradeoffs. Developing more efficient caustic remedies has therefore been an active area of research for the past few decades. In this thesis, I present a new ray-based a…
We consider propagation of a paraxial beam carrying the spin angular momentum (polarization) and intrinsic orbital angular momentum (IOAM) in a smoothly inhomogeneous isotropic medium. It is shown that the presence of IOAM can dramatically enhance and rearrange the topological phenomena that previously were considered solely in connection to the polarization of transverse waves. In particular, the appearance of a new-type Berry phase that describes the parallel transport of the beam structure along a curved ray is predicted. We derive the ray equations demonstrating the splitting of beams with…
Optical random scattering is generally considered to be a nuisance of microscopy that limits imaging depth and spatial resolution. Wavefront shaping techniques have recently enabled optical imaging at unprecedented depth, but a remaining problem is also to attain super-resolution within complex media. To address this challenge, we introduce a new technique to focus inside of complex media by the use of a quantum reference beacon (QRB), consisting of solid-state quantum emitters with spin-dependent fluorescence. This QRB provides subwavelength guidestar feedback for wavefront shaping to achieve…
