Optical Caustics Research Articles

Symmetrical optical caustics

The purpose of this paper is to revive interest in the problem of caustics with symmetry. Small imperfections in the point focus of a perfect lens are classified by the Seidel aberrations, while catastrophe theory deals satisfactorily with large ones. But the middle ground where an initial wave front is not perfectly spherical, and yet retains some symmetry, has proved more difficult to occupy. Caustics from water-drop lenses provide a motivation, but the analysis given is not restricted to such lenses. We start from a water-drop lens whose periphery has n-fold rotational symmetry combined with reflection symmetry (Dn). In 3D the unperturbed caustic is a flared and scalloped double cone with n rib lines. When n is even, the two cones are mutually rotated. Perturbing the drop by one of the terms suggested by catastrophe theory produces a sequence of caustics. It unfolds the original caustic. With parabolic umbilic singularities acting as off-stage organising centres, butterfly singularities and beak-to-beak events change the number of cusps in a cross-section until the caustic may eventually become n − 2 cusped triangles. We present such sequences for a number of low-order perturbations and find that identical sequences are produced if the perturbation is held fixed and the screen is moved. The two ways are related by power laws. For n = 6, a practical way of making a perturbation would be provided by the Bragg bubble model of a metal. Iluminating a single bubble in one of the hexagonally symmetric rafts would produce a hexagonally symmetric caustic that could be perturbed by distorting the raft elastically. The treatment can be extended to cover the chiral point groups Cn.

Microlensing as a possible probe of event-horizon structure in quasars

In quasars which are lensed by galaxies, the point-like images sometimes show sharp and uncorrelated brightness variations (microlensing). These brightness changes are associated with the innermost region of the quasar passing through a complicated pattern of caustics produced by the stars in the lensing galaxy. In this paper, we study whether the universal properties of optical caustics could enable extraction of shape information about the central engine of quasars. We present a toy model with a crescent-shaped source crossing a fold caustic. The silhouette of a black hole over an accretion disc tends to produce roughly crescent sources. When a crescent-shaped source crosses a fold caustic, the resulting light curve is noticeably different from the case of a circular luminosity profile or Gaussian source. With good enough monitoring data, the crescent parameters, apart from one degeneracy, can be recovered.

Scaling the abruptly autofocusing beams in the direct-space.

We propose a simple technique to scale the abruptly autofocusing beams in the direct space by introducing a scaling factor in the phase. Analytical formulas are deduced based on optical caustics, explicitly revealing how the scaling factor controls location, peak intensity, and size of the focal spot. We demonstrate that the multiplication of a scaling factor on the phase is equivalent to the axial-scaling transformation under the paraxial approximation. Further numerical and experimental results confirm theoretical predictions. In addition, amplitude modulation using phase-only holograms is used to maintain the peak intensity level of the focal spots.

Generation of Caustics and Rogue Waves from Nonlinear Instability.

Caustics are phenomena in which nature concentrates the energy of waves and may exhibit rogue-type behavior. Although they are known mostly in optics, caustics are intrinsic to all wave phenomena. As we demonstrate in this Letter, the formation of caustics and consequently rogue events in linear systems requires strong phase fluctuations. We show that nonlinear phase shifts can generate sharp caustics from even small fluctuations. Moreover, in that the wave amplitude increases dramatically in caustics, nonlinearity is usually inevitable. We perform an experiment in an optical system with Kerr nonlinearity, simulate the results based on the nonlinear Schrödinger equation, and achieve perfect agreement. As the same theoretical framework is used to describe other wave systems such as large-scale water waves, our results may also aid the understanding of ocean phenomena.

Electromagnetic scattering of a plane wave by a radially inhomogeneous sphere in the short wavelength limit

The formulas of ray theory and Airy theory for scattering of an electromagnetic plane wave by a radially inhomogeneous sphere are obtained from the generalization of exact Lorenz–Mie wave scattering theory using the WKB approximation to determine the radial portion of the partial wave scalar radiation potential, carrying out a Debye series expansion of it, approximating the sum over partial waves by an integral over an associated impact parameter, and approximately evaluating the integral using either the method of stationary phase or mapping it into one of the phase integrals that describe optical caustics. Various features of the results are commented on, and the procedure is extended to the merging of two rainbows in a given Debye series channel in the context of the longitudinal cusp caustic.

Prediction and characterization of multiple extremal paths in continuously monitored qubits

We examine most-likely paths between initial and final states for diffusive quantum trajectories in continuously monitored pure-state qubits, obtained as extrema of a stochastic path integral. We demonstrate the possibility of "multipaths" in the dynamics of continuously-monitored qubit systems, wherein multiple most-likely paths travel between the same pre- and post-selected states over the same time interval. Most-likely paths are expressed as solutions to a Hamiltonian dynamical system. The onset of multipaths may be determined by analyzing the evolution of a Lagrange manifold in this phase space, and is mathematically analogous to the formation of caustics in ray optics or semiclassical physics. Additionally, we develop methods for finding optimal traversal times between states, or optimal final states given an initial state and evolution time; both give insight into the measurement dynamics of continuously-monitored quantum states. We apply our methods in two systems: a qubit with two non-commuting observables measured simultaneously, and a qubit measured in one observable while subject to Rabi drive. In the two-observable case we find multipaths due to caustics, bounded by a diverging Van-Vleck determinant, and their onset time. We also find multipaths generated by paths with different "winding numbers" around the Bloch sphere in both systems.

Design of accelerating beams based on caustic method

Self-accelerating beam is a kind of light beam capable of self-bending in free space without any external potential, of which a typical one is the well-known Airy beam. Such a beam has gained great attention for its extraordinary properties, including nondiffracting, self-accelerating and self-healing, which may have versatile applications in the delivery and guiding of energy, information and objects using light, such as particle manipulation, micro-machining, optical routing, super-resolution imaging, etc. However, since Airy beam can only propagate along parabolic trajectory, which reduces the flexibility in practical applications, thus how to design accelerating beams propagating along arbitrary trajectory is still a crucial problem in this area. One scheme is to keep on finding other analytical solutions of the wave equation besides Airy beam, such as semi-Bessel accelerating beams, Mathius beams, and Weber beams, moving along circular, elliptical, or parabolic trajectories, but it becomes increasingly difficult to find out any more solutions. A more effective solution to this problem is based on the caustic method, which associates the predesigned trajectory with an optical caustics and then obtains the necessary initial field distribution by performing a light-ray analysis of the caustics. This method has been implemented in real space and Fourier space based on Fresnel diffraction integral and angular-spectrum integral, respectively. It has been found recently that they can be unified by constructing Wigner distribution function in phase space. Based on the caustic method, accelerating beams were constructed to propagate along arbitrary convex trajectories in two-dimensional space at first. With continuous development of this method, the types of accelerating beams available have been extending from convex trajectories to nonconvex trajectories, from two-dimensional trajectories to three-dimensional trajectories, and from one main lobe to multiple main lobes, which opens up more possibilities for emerging applications based on accelerating beams. In future, previous researches and applications based on Airy beams will certainly be generalized to all these new types of accelerating beams, and owing to the great flexibility in designing accelerating beams, more application scenarios may emerge in this process with huge development potential. Thus in this paper, we review the principle and progress of the caustic method in designing accelerating beams.

Intensity-symmetric accelerating caustic beams

We construct and generate symmetric accelerating caustic beams (ACBs) by using 3/2-order phase-only masks with elliptical contour based on optical caustics and diffraction theory. The symmetric ACBs are a type of bimodal accelerating caustic beam with two quasi-constant intensity peaks, very similar to the combination of two face-to-face Airy-like beams judging by appearance. Their fundamental optical morphology and force properties of particles in ACBs are subsequently provided. The unique optical properties of ACBs can be exploited for practical uses, such as accelerating electrons and clearing micrometer-sized particles as a laser micrometer-sized "water pump" instead of a laser micrometer-sized "snowblower" of accelerating Airy beams.

Simulation of optical caustics associated with the tertiary rainbow of oblate droplets.

In this paper, a vector ray tracing (VRT) model is used to simulate optical caustic structures, including rainbow and hyperbolic umbilic (HU) fringes, in the tertiary rainbow region of light scattering from oblate spheroidal droplets. In order to apply the optical caustic structures to particle diagnostics, the evolution of rainbow and HU fringes with an increase in the aspect ratio of oblate spheroidal droplets is investigated in detail, and the curvature of rainbow fringes are calculated. Next, on the basis of the VRT model, the location of cusp caustics is calculated and compared with theoretical prediction.

Experimental study on interaction between matrix crack and fiber bundles using optical caustic method

The stress singularity of the matrix crack perpendicular to the fiber bundles was studied by the method of optical caustics. First, the strain fields at the matrix crack tip in the neighbor of the fiber bundles were derived based on transformation toughening theory and Eshelby inclusion method. Then the caustic equation at the matrix crack tip in the neighbor of the fiber bundles was established. Finally, an optical caustic experiment for the interaction between the matrix crack and the fiber bundles was conducted to visualize the stress singularity at the matrix crack tip in the form of caustic spots.

Experimental investigation of the fiber bundle shielding effect on the dynamic matrix crack using optical caustic method

The fiber bundle shielding effect on the dynamic matrix cracking in fiber reinforced composite under dynamic loading was investigated using an optical caustics method. First, edge cracked epoxy beams, with a symmetrically positioned fiber bundle ahead of the crack tip, were preformed to simulate matrix crack shielding by the fiber bundle in fiber reinforced composites. Then, the effect of fiber bundle width on the dynamic matrix crack propagation behavior was analyzed using the caustics method, where the caustic spot shape, the dynamic stress intensity factor and crack propagation velocity were obtained to characterize the shielding effect of the fiber bundle. Finally, the shielding effect of a tilted fiber bundle with an angle of 30°on the dynamic matrix crack was analyzed using the optical caustics method, where the different fiber bundle width would cause different failure modes and failure mechanisms.

ƒ(R) Gravity, Relic Coherent Gravitons and Optical Chaos

We discuss the production of massive relic coherent gravitons in a particular class of ƒ(R) gravity, which arises from string theory, and their possible imprint in the Cosmic Microwave Background. In fact, in the very early Universe, these relic gravitons could have acted as slow gravity waves. They may have then acted to focus the geodesics of radiation and matter. Therefore, their imprint on the later evolution of the Universe could appear as filaments and a domain wall in the Universe today. In that case, the effect on the Cosmic Microwave Background should be analogous to the effect of water waves, which, in focusing light, create optical caustics, which are commonly seen on the bottom of swimming pools. We analyze this important issue by showing how relic massive gravity waves (GWs) perturb the trajectories of the Cosmic Microwave Background photons (gravitational lensing by relic GWs). The consequence of the type of physics discussed is outlined by illustrating an amplification of what might be called optical chaos.

Simulation of optical caustics associated with the secondary rainbow of oblate droplets

A vector-ray tracing (VRT) model is employed to simulate optical caustic structures [in terms of rainbow and hyperbolic umbilic (HU) fringes] in the secondary rainbow region of light scattering from oblate droplets. The changes of the rainbow fringe and HU fringe in response to shape deformation of oblate droplets are investigated and the curvature of the rainbow fringe is calculated. Then the location of cusp caustics is calculated from VRT simulations and compared with analytic solutions.

Optical caustics associated with the primary rainbow of oblate droplets: simulation and application in non-sphericity measurement

A vector ray tracing (VRT) model is developed to simulate optical caustic structures in the primary rainbow region of light scattering from oblate droplets. The changes of the optical caustic structures in response to shape deformation of oblate droplets are investigated. Then the curvature calculated from the simulated rainbow fringes is compared with that from the measured rainbow fringes and good agreement is found. Furthermore, according to the generalized rainbow patterns and the relation between aspect ratio and curvature of the rainbow fringe, non-sphericities in terms of aspect ratio of an oblate water droplet is measured with high measurement accuracy.

Arbitrary nonparaxial accelerating periodic beams and spherical shaping of light

We report the observation of arbitrary accelerating beams (ABs) designed using a nonparaxial description of optical caustics. We use a spatial light modulator-based setup and techniques of Fourier optics to generate circular and Weber beams subtending over 95deg of arc. Applying a complementary binary mask also allows the generation of periodic ABs taking the forms of snake-like trajectories, and the application of a rotation to the caustic allows the first experimental synthesis of optical ABs upon the surface of a sphere in three dimensions.

From Fabrication to Characterization of 3D Organic Microresonators: A Complementary Alliance of Microfluidics and Optics

This paper introduces a preview of targeted current research on organic optofluidic materials and devices devoted to 3D photonics microresonators (MRs). First, such an approach takes advantage from a significant know-how on optical simulations of 3D spherical MRs by complementary and coupled ways based on electromagnetism and quantum mechanics principle. Such simulations have allowed to preset the quantization of whispering gallery modes (WGMs) and to define a new formulation of optical caustics in global 3D MRs cavities. Leaning from these simulations, an interdisciplinary approach has been achieved by combining microfluidics techniques and thin layer processes that allowed with flow rates control the realization of 3D droplets MRs of several tens of micrometers in radii. Finally, free-space optical characterizations have been performed on such 3D polymeric MRs by judicious protocols based, respectively, on a modified Raman spectroscopy laser excitation and an adequate direct beam waist optical coupling. Spectral analysis on such 3D MRs of various sizes confirms the excitation of the expected WGMs revealing free spectral range (FSR) and caustics values close to the analytical ones.

Theoretical investigations on optical caustics of spherical microresonators: Analytical expressions of caustics and their asymptotic behaviors, computational simulations

This paper introduces analytical expressions of optical caustics r1 and r2 in micro-resonators (respectively called inner and outer caustics in micro-optics) in terms of radial and azimutal modal numbers n and l mutually. Such a theoretical study focuses on spherical micro-resonators and is carried out according to two complementary ways: Electromagnetic (E.M) approaches based respectively on an eigenvector problem resolution called modal field in integrated optics and an analogy with Quantum Mechanics (Q.M) using the notions of potential and geometric eikonal principle. Thus, using quantization and phase considerations, fundamental new formulation of caustics, that depend on the two previous quantum modal numbers simultaneously, have been shaped and confronted with numerical and graphical simulations. Both ways lead to results in good agreement (correlation coefficient superior to 0.99) and allow us to specify the evolution of such caustics according to each modal number of the quantification; such an analytical caustics formulation involves a specific function f whose study is expanded for various values of n ranging from 1 to 500. Moreover a whole study of asymptotic behaviors of f-function and its inverse f−1 on specific n and l values, directly related to the previous caustics expressions, is conducted, allowing the expansion of new analytical expressions of caustics in various limit cases and asymptotic behaviors. Then, the evolution of the positions of caustics in terms of modal numbers is interpreted, and the physical meanings of optical properties such as quasi-resonance, position and confinement of field with resonance, and associated electromagnetic fields are discussed.

Faraday caustics

We describe singularities in the distribution of polarized intensity as a function of Faraday depth (i.e. the Faraday spectrum) caused by line-of-sight (LOS) magnetic field reversals. We call these features Faraday caustics because of their similarity to optical caustics. They appear as sharply peaked and asymmetric profiles in the Faraday spectrum, that have a tail that extends to one side. The direction in which the tail extends depends on the way in which the LOS magnetic field reversal occurs (either changing from oncoming to retreating or vice versa). We describe how Faraday caustics will form three-dimensional surfaces that relate to boundaries between regions where the LOS magnetic field has opposite polarity. We present examples from simulations of the predicted polarized synchrotron emission from the Milky Way. We derive either the probability or luminosity distribution of Faraday caustics produced in a Gaussian magnetic field distribution as a function of their strength, F, and find that for strong Faraday caustics P(F)\proptoF^{-3} . If fully resolved, this distribution is also shown to depend on the Taylor microscale, which relates to the largest scale over which dissipation is important in a turbulent flow.

Arbitrary accelerating micron-scale caustic beams in two and three dimensions

We generate arbitrary convex accelerating beams by direct application of an appropriate spatial phase profile on an incident Gaussian beam. The spatial phase calculation exploits the geometrical properties of optical caustics and the Legendre transform. Using this technique, accelerating sheet caustic beams with parabolic profiles (i.e. Airy beams), as well as quartic and logarithmic profiles are experimentally synthesized from an incident Gaussian beam, and we show compatibility with material processing applications using an imaging system to reduce the main intensity lobe at the caustic to sub-10 micron transverse dimension. By applying additional and rotational spatial phase, we generate caustic-bounded sheet and volume beams, which both show evidence of the recently predicted effect of abrupt autofocussing. In addition, an engineered accelerating profile with femtosecond pulses is applied to generate a curved zone of refractive index modification in glass. These latter results provide proof of principle demonstration of how this technique may yield new degrees of freedom in both nonlinear optics and femtosecond micromachining.

Axial and focal-plane diffraction catastrophe integrals

Exact expressions in terms of Bessel functions are found for some of the diffraction catastrophe integrals that decorate caustics in optics and mechanics. These are the axial and focal-plane sections of the elliptic and hyperbolic umbilic diffraction catastrophes, and symmetric elliptic and hyperbolic unfoldings of the X9 diffraction catastrophes. These representations reveal unexpected relations between the integrals.