Diffraction of conic and Gaussian beams by a spiral phase plate

An analytical expression for the spatial spectrum of the conic wave diffracted by a spiral phase plate (SPP) with arbitrary integer singularity of order n is obtained. Conic wave diffraction by the SPP is equivalent to plane-wave diffraction by a helical axicon. A comparison of the conic wave and Gaussian beam diffraction on a SPP is made. It is shown that in both cases a light ring is formed, with the intensity function growing in proportion to rho(2n) at small values of radial variable rho and decreasing as n(2)rho(-4) at large rho. By use of direct e-beam writing on the resist, a 32 level SPP of the 2nd order and diameter 5 mm is manufactured. By use of this SPP, a He-Ne laser beam is transformed into a beam with phase singularity and ringlike intensity distribution. A four-order binary diffractive optical element (DOE) with its transmittance proportional to a linear superposition of four angular harmonics is also manufactured. With this DOE, simultaneous optical trapping of several polystyrene beads of diameter 5 microm is performed.     

Diffraction of a finite-radius plane wave and a Gaussian beam by a helical axicon and a spiral phase plate

We derive what we believe to be new analytical relations to describe the Fraunhofer diffraction of the finite-radius plane wave by a helical axicon (HA) and a spiral phase plate (SPP). The solutions are deduced in the form of a series of the Bessel functions for the HA and a finite sum of the Bessel functions for the SPP. The solution for the HA changes to that for the SPP if the axicon parameter is set equal to zero. We also derive what we believe to be new analytical relations to describe the Fresnel and Fraunhofer diffraction of the Gaussian beam by a HA are derived. The solutions are deduced in the form of a series of the hypergeometric functions. We have fabricated by photolithography a binary diffractive optical element (a HA with number n=10) able to produce in the focal plane of a spherical lens an optical vortex, which was then used to perform rotation of several polystyrene beads of diameter 5 microm.     

Gaussian laser beam transformation into an optical vortex beam by helical lens

In this article, we investigate the Fresnel diffraction characteristics of the hybrid optical element which is a combination of a spiral phase plate (SPP) with topological charge p and a thin lens with focal length f, named the helical lens (HL). As incident a Gaussian laser beam is treated, having its waist a distance ζ from the HL plane and its axis passing through the centre of the HL. It is shown that the SPP introduces a phase singularity of pth order to the incident beam, while the lens transforms the beam characteristic parameters. The output light beam is analyzed in detail: its characteristic parameters and focusing properties, amplitude and intensity distributions and the vortex rings profiles, and radii, at any z distance behind the HL plane, as well as in the near and far field.     

Fresnel and Fraunhofer diffraction of a Gaussian laser beam by fork-shaped gratings

Expressions describing the vortex beams that are generated by the process of Fresnel diffraction of a Gaussian beam incident out of waist on fork-shaped gratings of arbitrary integer charge p, and vortex spots in the case of Fraunhofer diffraction by these gratings, are deduced. The common general transmission function of the gratings is defined and specialized for the cases of amplitude holograms, binary amplitude gratings, and their phase versions. Optical vortex beams, or carriers of phase singularity with charges mp and -mp, are the higher negative and positive diffraction-order beams. The radial part of their wave amplitudes is described by the product of the mpth-order Gauss-doughnut function and a Kummer function, or by the first-order Gauss-doughnut function and the difference of two modified Bessel functions whose orders do not match the singularity charge value. The wave amplitude and the intensity distributions are discussed for the near and far fields in the focal plane of a convergent lens, as well as the specialization of the results when the grating charge p=0; i.e., the grating turns from forked into rectilinear. The analytical expressions for the vortex radii are also discussed.     

Near-field anomalous spectral behavior in diffraction of a Gaussian pulsed beam from an annular aperture

Based on the Fresnel diffraction integral and by introducing a hard-aperture function into a finite sum of complex Gaussian functions, the approximate analytical expression for the near-field spectral intensity distribution of a space-time-dependent Gaussian pulsed beam passing through an annular aperture is derived, which permits us to study the on- and off-axis spectral anomalies that are near phase singularities of the diffracted Gaussian pulsed beam in the near-field. The expressions for a circular black screen and a circular aperture are given as special cases of the general results. The relative spectral shift of a space-time-dependent Gaussian pulsed beam versus the different values of the truncation parameters and the position parameters of observation points are also studied and illustrated with numerical calculations. It is shown that the spectral switch appears near phase singularities in the near-field, and the near-field spectral behavior depends on the truncation parameters, the pulse duration tau, and the position parameter. The results of this work have potential applications in free-space information encoding and transmission.     

Hollow Gaussian beams with the power-exponent-phase vortex

A kind of hollow Gaussian beams with the power-exponent-phase vortex is introduced. Based on the Collins integral, an analytical expression of a hollow Gaussian beam with the power-exponent-phase vortex passing through a paraxial optical system described by the ABCD matrix approach is derived. The analytical expressions for the beam propagation factors and the orbital angular momentum density of such hollow vortex Gaussian beam passing through a paraxial optical system described by the ABCD matrix approach are also derived, respectively. As a numerical example, the propagation properties of a hollow Gaussian beam with the power-exponent-phase vortex are demonstrated in free space. The evolutions of the normalized intensity, the phase and the orbital angular momentum density distributions are investigated, respectively. The influences of the power order and the topological charge on the beam propagation factors in the x- and y-directions are analysed. The introduced hollow Gaussian beam has potential applications in the atom manipulation and the optical trapping.     

Focusing Properties of the Gaussian Beam Generated by Optical Resonators with Gaussian Reflectivity Mirrors

Abstract

This paper presents a parametric study concerning the focusing properties of the light beam generated by optical resonators with Gaussian reflectivity mirrors. The diffraction integral has been evaluated and analytical expressions are derived for intensity patterns near the focus in systems of different Fresnel numbers. Universal curves have been plotted against dimensionless parameters to show the variations in the intensity patterns due to the change in focusing geometry and the incident beam intensity profile. Modification to the well known diffraction formula are considered in order to extend the conventional Gaussian formalism to the case of Gaussian mirror resonators.     

Effective radius of curvature of partially coherent Hermite–Gaussian beams propagating through non-Kolmogorov turbulence

Based on the extended Huygens–Fresnel principle and non-Kolmogorov spectrum, the analytical expression for the effective radius of curvature of partially coherent Hermite–Gaussian (PCHG) beams propagating through non-Kolmogorov turbulence is derived, and the relative effective radius of curvature is used to describe the effect of turbulence on the effective radius of curvature. It is shown that the effective radius of curvature of PCHG beams depends on the beam and non-Kolmogorov turbulence parameters and on the propagation distance. The variation of relative effective radius of curvature with increasing generalized exponent parameter α of non-Kolmogorov turbulence is non-monotonic. The longer the propagation distance is, the larger the effect of turbulence on the effective radius of curvature of PCHG beams is. The effective radius of curvature of PCHG beams with shorter wavelength, smaller beam order, larger beam waist width or better spatial coherence is more affected by the non-Kolmogorov turbulence. The results are interpreted physically.     

Spectral shifts and spectral switches of a pulsed Gaussian beam from a rectangular aperture in the far field

The far-field anomalous spectral behaviours of a space–time-dependent Gaussian pulsed beam passing through a rectangular aperture are studied. By expanding a hard aperture function into a finite sum of complex Gaussian functions and starting from the Fresnel diffraction integral, the approximate analytical expression for the spectral intensity of a space–time-dependent Gaussian pulsed beam passing through a rectangular aperture is derived. Meanwhile, the corresponding closed-forms for the slit and the unapertured cases are also given as special cases of the general results. The red and blue shifts and the spectral intensity distribution are studied and illustrated with numerical calculations. Specifically, it is shown that the spectral switch takes place when the truncation parameter is equal to particular values or the observation position is at the critical diffraction angle. The possibility of tunable spectral switching in the far field with an apertured pulsed beam by varying the size of the rectangular aperture is highlighted.     

Generalized M2 factor of a partially coherent beam propagating through a circular hard-edged aperture

The second-order intensity moments and the beam-propagation M2 factor of partially coherent beams that propagate through a circular-symmetry hard-edged aperture are in the cylindrical coordinate system. AJo-correlated Schell-model beam with a Gaussian intensity distribution is an example. The analytical expression for the generalized M2 factor is derived. The numerical calculation results are analyzed.     

Spiral spectrum of anomalous vortex beams propagating in a weakly turbulent atmosphere

ABSTRACT

We study the spiral spectrum of anomalous vortex beams propagating through a turbulent atmosphere. Based on the Huygens–Fresnel integral and the Rytov approximation, the integral expression and then the analytical expression for the spiral spectrum of anomalous vortex beams in the weakly turbulent atmosphere are derived. The capacity of wireless optical links using the anomalous vortex beam is obtained. It is found that the spiral spectrum of the anomalous vortex beam is less affected by turbulence than that of the Laguerre-Gaussian beam. And thus, the information capacity of wireless optical links using the anomalous vortex beam is larger than that using the Laguerre-Gaussian beam. The influence of beam order, wavelength, topological charge, propagation distance, refractive index structure constant and the radius of receiver aperture on spiral spectrum is investigated. These results contribute to reduce the disturbing effects of atmospheric turbulence on the orbital angular momentum of the vortex beam.     

A comparative study of standard and elegant Hermite–Gaussian beams propagating through turbulent atmosphere

Based on the extended Huygens–Fresnel principle and the definition of the second-order moments of the Wigner distribution function (WDF), the analytical expressions for the root-mean-square (rms) beam width and far-field divergence angle, curvature radius and M ²-factor of standard Hermite–Gaussian (SHG) and elegant Hermite–Gaussian (EHG) beams passing through turbulent atmosphere are derived and compared. It is shown that in turbulent atmosphere the far-field divergence angle of SHG and EHG beams is equal under the same conditions, but the rms beam width, curvature radius and the M ²-factor of SHG and EHG beams are different except for beam orders m?=?0 and m?=?1. The relative rms beam width, relative curvature radius and relative M ²-factor of SHG beams are less than those of EHG beams. Therefore, the conclusion that SHG beams are less influenced by turbulence than EHG beams can be drawn if we examine one of the above three relative beam parameters.     

Vortex-trap-induced fusion of femtoliter-volume aqueous droplets

This paper describes the use of an optical vortex trap for the transport and fusion of single femtoliter-volume aqueous droplets. Individual droplets were generated by emulsifying water in acetophenone with SPAN 80 surfactant. We demonstrate the ability of optical vortex traps to position trapped droplets precisely while excluding surrounding aqueous droplets from entering the trap, thereby preventing unwanted cross contamination by other nearby droplets. Additionally, the limitation of optical vortex traps for inducing droplet fusion is illustrated, and a remedy is provided through modulation of the spatial intensity profile of the optical vortex beam. Spatial modulation was achieved by translating the computer-generated hologram (CGH) with respect to the input Gaussian beam, thereby shifting the location of the embedded phase singularity (dark core) within the optical vortex beam. We present both simulated and experimentally measured intensity profiles of the vortex beam caused by translation of the CGH. We further describe the use of this technique to achieve controlled and facile fusion of two aqueous droplets.     

Elliptic Laguerre-Gaussian beams

An analytical expression for the diffraction of an elliptic Laguerre-Gaussian (LG) beam is derived and analyzed. We show that a beam with even singularity order has nonzero axial intensity for any degree of ellipticity and at any finite distance z from the initial plane, whereas at z = 0 and z = infinity the axial intensity is zero. We show that for a beam with a small degree of ellipticity and even order of singularity, two isolated intensity zeroes appear in the Fresnel zone on a straight line at an angle of 45 deg or -45 deg, depending whether the beam's spin is right or left. The theoretical conclusions are confirmed by numerical simulation and physical experiments.     

Diffraction of a Gaussian beam by a logarithmic axicon

We derive an explicit analytical relationship to describe the axial light intensity when a Gaussian beam is diffracted by the logarithmic axicon (LA). An evaluation formula for the effective radius of the diffraction pattern that we deduce shows the said radius to be in inverse proportion to the LA "force" parameter. The finite-difference time-domain-based simulation has shown that using the LA makes it possible to go beyond the diffraction limit: in the LA vicinity, the FWHM of the light beam can be as small as one fifth of the illumination wavelength.     

A dynamic plasmonic manipulation technique assisted by phase modulation of an incident optical vortex beam

A novel phase modulation method for dynamic manipulation of surface plasmon polaritons?(SPPs) with a phase engineered optical vortex?(OV) beam illuminating on nanoslits is experimentally demonstrated. Because of the unique helical phase carried by an OV beam, dynamic control of SPP multiple focusing and standing wave generation is realized by changing the OV beam's topological charge constituent with the help of a liquid-crystal spatial light modulator. Measurement of SPP distributions with near-field scanning optical microscopy showed an excellent agreement with numerical predictions. The proposed phase modulation technique for manipulating SPPs features has seemingly dynamic and reconfigurable advantages, with profound potential for development of SPP coupling, routing, multiplexing and high-resolution imaging devices on plasmonic chips.     

An intuitive model for on-axis pulse evolution of ultrashort pulsed Gaussian beams diffracted from a circular aperture

The diffraction of ultrashort pulsed Gaussian beams from a circular aperture is studied by means of Fresnel diffraction integral and Fourier transform method. A uniform analytical expression is derived for temporal pulse form of ultrashort pulsed Gaussian beams in two cases, i.e. with constant beam waist and with constant diffraction length. It is shown that the on-axis pulse can be formulated as a superposition of an unapertured pulse and an aperture-induced pulse. The superposition of these two pulses leads to an enhanced pulse intensity for small truncation parameters at certain distances in the near field. Our results may find applications in high-intensity laser waveform control.     

3D-vortex labyrinths in the near field of solid-state microchip laser

The usage of vortex-labyrinth fields and Talbot lattices as optical dipole traps for neutral atoms is considered for the wavelength of trapping radiation in the range 0.98–2.79 µm. The square vortex lattices generated in high Fresnel number solid-state microchip lasers are studied as a possible realization. The distribution of light field is obtained via a nonstationary computational model based on Maxwell–Bloch equations for a class-B laser, discrete Fox–Lee map with relaxation of inversion and a static model based on superposition of copropagating Gaussian beams. The spatial patterns obtained numerically and observed experimentally previously are interpreted as nonlinear superposition of vortices with helicoidal phase dislocations. The distribution of light field is approximated analytically by a sum of array of vortex lines and an additional parabolic subtrap. The separable optical trapping potential is proposed with similar intensity distribution. The factorization of the macroscopic wavefunction has led to the solution of the Gross–Pitaevsky equation for an ensemble of quantum particles trapped in a vortex labyrinth formed by a spatially periodic array of Laguerre–Gaussian beams.     

Simplified intracavity phase plates for increasing laser-mode discrimination

Recently, a new class of laser resonators was introduced that utilizes diffractive mirrors and an additional intracavity diffractive phase element. High modal discrimination and low fundamental-mode loss were achieved simultaneously by use of sinusoidal and pseudorandom diffractive phase elements. An intracavity phase element consisting of a simple single-step phase modulation is approximated by a Gaussian with a small radius. Explicit expressions are obtained for the modal-discrimination factor as a function of resonator parameters with a Gaussian output mirror. Numerical simulations are performed for a phase element with a step singularity in the phase function, the fundamental mode of this cavity being super-Gaussian. The modal discrimination of the cavity is studied for different radii of the single-step phase modulation, the position of the phase plate, and the cavity Fresnel number. Optimum solutions are found for a plane output mirror with either a striped or a circular shape.