Propagation of confluent hypergeometric beam through uniaxial crystals orthogonal to the optical axis

Based on the theory of paraxial electromagnetic beams in anisotropic medium, propagation properties of a linearly polarized confluent hypergeometric (HyG) beam through uniaxial crystals orthogonal to the optical axis are investigated. Analytical expressions are derived for the resultant optical field components. Effects of separate beam parameters together with the ratio of refractive indices of crystals on transversal intensity distributions are revealed by numerical calculations, respectively. It is shown that the beam profile finally converts into an elliptical shape due to influences of anisotropic crystals. Moreover, it is also indicated that through suitable selection of parameters, types of the optical vortices of resultant field would change from a central dark spot to a slight dark ring or an axial shadow during the propagation. These numerical results may provide alternative advantages to the trapping of tiny particles by utilizing HyG beams in practical experiments.     

Hypergeometric-Gaussian modes

We studied a novel family of paraxial laser beams forming an overcomplete yet nonorthogonal set of modes. These modes have a singular phase profile and are eigenfunctions of the photon orbital angular momentum. The intensity profile is characterized by a single brilliant ring with the singularity at its center, where the field amplitude vanishes. The complex amplitude is proportional to the degenerate (confluent) hypergeometric function, and therefore we term such beams hypergeometric-Gaussian (HyGG) modes. Unlike the recently introduced hypergeometric modes [Opt. Lett. 32, 742 (2007)], the HyGG modes carry a finite power and have been generated in this work with a liquid-crystal spatial light modulator. We briefly consider some subfamilies of the HyGG modes as the modified Bessel Gaussian modes, the modified exponential Gaussian modes, and the modified Laguerre-Gaussian modes.     

Rotation of laser beams with zero of the orbital angular momentum

We show that among the multitude of rotating light beams whose complex amplitude can be represented as a linear superposition of the Laguerre-Gaussian (LG) modes with definite numbers there are light beams with zero orbital angular momentum (OAM) and vice versa, multi-mode LG beams that show no rotation and are lacking the radially symmetric intensity distribution can possess the non-zero OAM. Also, we give examples of the rotating light beams with zero OAM, represented as a superposition of the Bessel and new hypergeometric modes. Using an SLM, we generate a rotating Bessel beam with zero OAM for the first time.     

Circular beams

A very general beam solution of the paraxial wave equation in circular cylindrical coordinates is presented. We call such a field a circular beam (CiB). The complex amplitude of the CiB is described by either the Whittaker functions or the confluent hypergeometric functions and is characterized by three parameters that are complex in the most general situation. The propagation through complex ABCD optical systems and the conditions for square integrability are studied in detail. Special cases of the CiB are the standard, elegant, and generalized Laguerre-Gauss beams; Bessel-Gauss beams; hypergeometric beams; hypergeometric-Gaussian beams; fractional-order elegant Laguerre-Gauss beams; quadratic Bessel-Gauss beams; and optical vortex beams.     

Cartesian beams

A new and very general beam solution of the paraxial wave equation in Cartesian coordinates is presented. We call such a field a Cartesian beam. The complex amplitude of the Cartesian beams is described by either the parabolic cylinder functions or the confluent hypergeometric functions, and the beams are characterized by three parameters that are complex in the most general situation. The propagation through complex ABCD optical systems and the conditions for square integration are studied in detail. Applying the general expression of the Cartesian beams, we also derive two new and meaningful beam structures that, to our knowledge, have not yet been reported in the literature. Special cases of the Cartesian beams are the standard, elegant, and generalized Hermite-Gauss beams, the cosine-Gauss beams, the Lorentz beams, and the fractional order beams.     

Accessible solitons in three-dimensional parabolic cylindrical coordinates

A class of self-similar beams, named three-dimensional (3D) spatiotemporal parabolic accessible solitons, are introduced in the 3D highly nonlocal nonlinear media. We obtain exact solutions of the 3D spatiotemporal linear Schrödinger equation in parabolic cylindrical coordinates by using the method of separation of variables. The 3D localized structures are constructed with the help of the confluent hypergeometric Tricomi functions and the Hermite polynomials. Based on such an exact solution, we graphically display three different types of 3D beams: the Gaussian solitons, the ring necklace solitons, and the parabolic solitons, by choosing different mode parameters. We also perform direct numerical simulation to discuss the stability of local solutions. The procedure we follow provides a new method for the manipulation of spatiotemporal solitons.     

Rotation of multimodal Gauss–Laguerre light beams in free space and in a fiber

A gradient algorithm is proposed for design of phase diffractive optical elements (DOEs) that can form the light beams being effectively a superposition of a small number of nonradially symmetric Gauss–Laguerre modes with a preset energy contribution of each mode. We show that under certain conditions there is a rotation of the cross-section of such light beams. The rotation of multimodal Gauss–Laguerre light beams propagating in a fiber with quadratic dependence of the refractive index is discussed.     

Nongeneric polarization singularities in combined vortex beams

Nongeneric polarization structures of singular beams formed as a coherent coaxial mixture of weighted orthogonally polarized single-charged Laguerre-Gauss modes with different radial indices are analyzed. A general solution for the superposition of elliptically orthogonally polarized partial vortex beams is obtained; the limiting special cases in which the mixed modes are linearly or circularly polarized are investigated. It is established that unusual spatially stable polarization structures such as closed C contours and L contours with a fixed azimuth of linear polarization arise in such combined beams. The results are experimentally confirmed by using a new diffraction method for testing phase singularities in optical beams.     

Entangling different degrees of freedom by quadrature squeezing cylindrically polarized modes

Quantum systems such as, for example, photons, atoms, or Bose-Einstein condensates, prepared in complex states where entanglement between distinct degrees of freedom is present, may display several intriguing features. In this Letter we introduce the concept of such complex quantum states for intense beams of light by exploiting the properties of cylindrically polarized modes. We show that already in a classical picture the spatial and polarization field variables of these modes cannot be factorized. Theoretically it is proven that by quadrature squeezing cylindrically polarized modes one generates entanglement between these two different degrees of freedom. Experimentally we demonstrate amplitude squeezing of an azimuthally polarized mode by exploiting the nonlinear Kerr effect in a specially tailored photonic crystal fiber. These results display that such novel continuous-variable entangled systems can, in principle, be realized.     

Symmetric bifurcations and phenomena of polarization symmetry breaking in a single-mode gas laser

An analysis is carried out of the optical cavities whose natural modes have the form of Gaussian beams with complex astigmatism. In the case of coincident eigenvalues of the cavity ray matrix, the wave beams were found to be determined not only by the geometrical optics characteristics of a cavity, but also by additional cavity-independent parameters. A detailed analysis is given for the fundamental mode of such cavities. Concrete cavities possessing these properties are noted.     

Comparison of three kinds of polarized Bessel vortex beams propagating through uniaxial anisotropic media

A comparison of differently polarized Bessel vortex beams propagating through a uniaxial anisotropic slab is discussed in terms of the vector wave function expansions.The magnitude profiles of electric field components, the transformation of polarization modes, and the distributions of orbital angular momentum(OAM) states of the reflected and transmitted beams for different incident angles are numerically simulated.The results indicate that the magnitude profiles of electric field components for different polarization modes are distinct from each other and have a great dependence on the incident angle,thus the transformation of polarization modes which reflects the change of energy can be affected largely.As compared to the x and circular polarization incidences, the reflected and transmitted beams for the radial polarization incidence suffer the fewest transformation of polarization modes, showing a better energy invariance.The distributions of OAM states of the reflected and transmitted beams for different polarization modes are diverse as well, and the derived OAM states of the transmitted beam for radial polarization present a focusing effect, concentrating on the state between two predominant OAM states.     

Simple annular-beam generator with a laser-diode-pumped axially off-set power build-up cavity

We have developed a novel, compact-mode selective annular-beam generator that employs a laser-diode (LD)-pumped, passively locked power build-up cavity (PBC). The PBC functions as a flexible and compact light source that is capable of producing TEM_0n high-order Hermite-Gaussian (HG) beams. The generated HG beams can then be converted to LG modes, particularly to annular-beams, using two cylindrical lenses. The system was theoretically studied and a mode-matching condition was derived. The tailoring position of the active semiconductor laser medium relative to the external cavity of the PBC enabled the production of single high-order HG modes. The optimum position was calculated by model calculations of the mode-coupling efficiency. The desired TEM_0n modes of HG beams were generated by translating the axial position of the external cavity of the PBC in increments of ∼100 μm relative to the LD emission surface. High quality TEM₀₁ to TEM₀₁₅ of the HG beams and their conversion to the desired LG modes having powers of the order of 1 mW was successfully demonstrated. This system is suitable for optical tweezers.     

FREE VIBRATIONS OF SELF-STRAINED ASSEMBLIES OF BEAMS

This paper examines the natural frequencies and modes of transverse vibration of two simple redundant systems comprising straight uniform Euler-Bernoulli beams in which there are internal self-balancing axial loads (e.g., loads due to non-uniform thermal strains). The simplest system consists of two parallel beams joined at their ends and the other is a 6-beam rectangular plane frame. Symmetric mode vibration normal to the plane of the frame is studied. Transcendental frequency equations are established for the different systems. Computed frequencies and modes are presented which show the effect of (1) varying the axial loads over a wide range, up to and beyond the values which cause individual members to buckle (2) pinning or fixing the beam joints (3) varying the relative flexural stiffness of the component beams. When the internal axial loads first cause any one of the component beams to buckle, the fundamental frequency of the whole system vanishes. The critical axial loads required for this are determined. A simple criterion has been identified to predict whether a small increase from zero in the axial compressive load in any one member causes the natural frequencies of the whole system to rise or fall. It is shown that this depends on the relative flexural stiffnesses and buckling loads of the different members. Computed modes of vibration show that when the axial modes reach their critical values, the buckled beam(s) distort with large amplitudes while the unbuckled beam(s) move either as rigid bodies or with bending which decays rapidly from the ends to a near-rigid-body movement over the central part of the beam. The modes of the systems with fixed joints change very little (if at all) with changing axial load, except when the load is close to the value which maximizes or minimizes the frequency. In a narrow range around this load the mode changes rapidly. The results provide an explanation for some computed results (as yet unpublished) for the flexural modes and frequencies of flat plates with non-uniform thermal stress distributions.     

Free-space propagation of autofocusing Airy vortex beams with controllable intensity gradients

Vortex splitting is one of the main causes of instability in orbital angular momentum(OAM) modes transmission. Recent advances in OAM modes free-space propagation have demonstrated that abruptly autofocusing Airy vortex beams(AAVBs) can potentially mitigate the vortex splitting effect. However, different modes of vortex embedding will affect the intensity gradients of the background beams, leading to changes in the propagation characteristics of vortex beams. This study presents the unification of two common methods of coupling autofocusing Airy beams with vortices by introducing a parameter(m), which also controls the intensity gradients and focusing properties of the AAVBs. We demonstrate that vortex splitting can be effectively reduced by selecting an appropriate value of the parameter(m) according to different turbulence conditions. In this manner,the performance of OAM-based free-space optical systems can be improved.     

Intensity-based modal analysis of partially coherent beams with Hermite-Gaussian modes

Many partially coherent beams are made up of a superposition of mutually uncorrelated Hermite-Gaussian modes. We prove that knowledge of the transverse intensity profile of such a beam is sufficient for evaluating the weights of the modes in an exact way. Simulations indicate that the proposed method resists noise well.     

Lasing modes in a spiral-shaped dielectric microcavity

The resonance patterns and lasing modes in a spiral-shaped dielectric microcavity are investigated through passive and active medium calculations. We find that the high-Q resonance modes are whispering-gallery-like modes, and these resonance modes can be easily excited as lasing modes. We also find that the quasi-scarred resonance mode, which shows strong directional emission beams from the cavity boundary, can be excited with selectively applied external pumping. Through a spectral analysis of the time evolution of the light field, the competition between these lasing modes is discussed.     

Photorefractive two-beam coupling equations with multiple spatial–temporal features: an SVD approach

We analyze the evolution of multi-feature two-beam coupling, wherein each beam contains several spatial–temporal features (spatial patterns modulated by different signals), using a one-dimensional plane wave model to describe the evolution of paired components. This general scenario is of interest for analyzing signal-processing applications of photorefractives, such as source-separation by orthogonalization of source-modulated spatial patterns. We use singular-value decomposition (SVD) to express each beam as a simple superposition of modes that are both temporally uncorrelated and spatially orthogonal. We find a solution that is a natural matrix generalization of the scalar solution for simple two-beam coupling, and a test for its validity: the two operators that give the spatial overlap associated with the temporal basis signals in the two beams must commute. Equivalently, this means that the same set of signals must be modulating the SVD modes in the two beams. Then the SVD modes are preserved in the two-beam coupling evolution, with only their amplitudes changing. Dedicated to Prof. Dr. Eckard Kr?tzig on the occasion of his 60th birthday. Received: 10 November 1998 / Revised version: 13 January 1999 / Published online: 7 April 1999     

Coherent and partially coherent twisting beams

We present coherent and partially coherent beams that exhibit transverse pattern expansion and rotation upon free propagation. Coherent twisting beams may be generated by a suitable superposition of Laguerre-Gauss modes, while a model for partially coherent twisting beams is given by a continuous incoherent superposition of lowest order modes.     

Harmonic oscillator fiducials for Hermite-Gaussian laser beams

This paper exploits the identical functional forms of quantum mechanical harmonic oscillators and Hermite-Gaussian laser beams to investigate the probability densities of large-order modes. The classical limits of a two-dimensional harmonic oscillator provide corresponding integration limits for the photon probability densities of the laser-beam modes to determine the fraction of photons detected therein. The probabilities of detecting photons within the ‘classical limits’ of Hermite-Gaussian laser beams exhibit a power-law dependence in the limit of large-order modes and asymptotically approach unity, in agreement with the Correspondence Principle. The classical limits for large-order modes are shown to include all of the nodes and peaks for Hermite-Gaussian laser beams; Sturm's theorem provides a direct proof that is further illustrated by examination of the concavity and convexity of Hermite functions.