A. A. Kovalev

Diffraction of a plane, finite-radius wave by a spiral phase plate

We derive analytical expressions containing a hypergeometric function to describe the Fresnel and Fraunhofer diffraction of a plane wave of circular and ringlike cross section by a spiral phase plate (SPP) of an arbitrary integer order. Experimental diffraction patterns generated by an SPP fabricated in resist through direct e-beam writing are in good agreement with the theoretical intensity distribution.

Asymmetric Bessel modes

We propose a new, three-parameter family of diffraction-free asymmetric elegant Bessel modes (aB-modes) with an integer and fractional orbital angular momentum (OAM). The aB-modes are described by the nth-order Bessel function of the first kind with complex argument. The asymmetry degree of the nonparaxial aB-mode is shown to depend on a real parameter c≥0: when c=0, the aB-mode is identical to a conventional radially symmetric Bessel mode; with increasing c, the aB-mode starts to acquire a crescent form, getting stretched along the vertical axis and shifted along the horizontal axis for c≫1. On the horizontal axis, the aB-modes have a denumerable number of isolated intensity zeros that generate optical vortices with a unit topological charge of opposite sign on opposite sides of 0. At different values of the parameter c, the intensity zeros change their location on the horizontal axis, thus changing the beams OAM. An isolated intensity zero on the optical axis generates an optical vortex with topological charge n. The OAM per photon of an aB-mode depends near-linearly on c, being equal to ℏ(n+cI1(2c)/I0(2c)), where ℏ is the Planck constant and In(x) is a modified Bessel function.

Asymmetric Bessel–Gauss beams

We propose a three-parameter family of asymmetric Bessel-Gauss (aBG) beams with integer and fractional orbital angular momentum (OAM). The aBG beams are described by the product of a Gaussian function by the nth-order Bessel function of the first kind of complex argument, having finite energy. The aBG beams asymmetry degree depends on a real parameter c≥0: at c=0, the aBG beam is coincident with a conventional radially symmetric Bessel-Gauss (BG) beam; with increasing c, the aBG beam acquires a semicrescent shape, then becoming elongated along the y axis and shifting along the x axis for c≫1. In the initial plane, the intensity distribution of the aBG beams has a countable number of isolated optical nulls on the x axis, which result in optical vortices with unit topological charge and opposite signs on the different sides of the origin. As the aBG beam propagates, the vortex centers undergo a nonuniform rotation with the entire beam about the optical axis (c≫1), making a π/4 turn at the Rayleigh range and another π/4 turn after traveling the remaining distance. At different values of the c parameter, the optical nulls of the transverse intensity distribution change their position, thus changing the OAM that the beam carries. An isolated optical null on the optical axis generates an optical vortex with topological charge n. A vortex laser beam shaped as a rotating semicrescent has been generated using a spatial light modulator.

Hermite–Gaussian modal laser beams with orbital angular momentum

A relationship for the complex amplitude of generalized paraxial Hermite-Gaussian (HG) beams is deduced. We show that under certain parameters, these beams transform into the familiar HG modes and elegant HG beams. The orbital angular momentum (OAM) of a linear combination of two generalized HG beams with a phase shift of π/2, with their double indices composed of adjacent integer numbers taken in direct and inverse order, is calculated. The modulus of the OAM is shown to be an integer number for the combination of two HG modes, always equal to unity for the superposition of two elegant HG beams, and a fractional number for two hybrid HG beams. Interestingly, a linear combination of two such HG modes also presents a mode that is characterized by a nonzero OAM and the lack of radial symmetry but does not rotate during propagation.

Vortex Hermite–Gaussian laser beams

We study elliptical vortex Hermite-Gaussian (vHG) beams, which are described by the complex amplitude proportional to the nth-order Hermite polynomial whose argument is a function of a real parameter a. At |a|<1, on the vertical axis of the beam cross section, there are n isolated optical nulls that produce optical vortices with topological charge +1(a<0) or -1(a>0). At |a|>1, similar isolated optical nulls of the vHG beams are found on the horizontal axis. At a=0, the vHG beam becomes identical to the HG mode of the order (0,n). We derive the orbital angular momentum (OAM) of the vHG beams, which depends on the parameter a and an ellipticity parameter of the Gaussian beam. The derived equation allows the transverse intensity of the vHG-beam to be changed without changing its OAM. The experimental and theoretical results are in good agreement.

Nonparaxial hypergeometric beams

We derive an analytical expression to describe the exact solution of the Helmholtz equation in cylindrical coordinates as a product of two Kummer functions. The solution is presented as a sum of two terms that describe the nonparaxial hypergeometric light beams propagated along the optical axis in the positive and negative directions. With the distance from the initial plane becoming much larger than the wavelength of light, the expression derived for the nonparaxial hypergeometric beam coincides with that for a paraxial hypergeometric mode.

Half Pearcey laser beams

We obtain a new solution of the paraxial Helmholtz equation that describes a family of three-dimensional and two-dimensional form-invariant half-Pearcey beams (HP-beams). HP-beams generalize Pearcey beams obtained in Ring et al (2012) Opt. Express 20 18955, since these Pearcey beams can be considered as the sum of two first-order HP-beams. Three-dimensional HP-beams have angular spectra of plane waves, which are non-zero at a half parabola. For functions of HP-beam complex amplitudes, the orthogonality properties have been revealed. Using a spatial phase modulator, we generated superposition of HP-beams. For two-dimensional HP-beam acceleration and deceleration of trajectory has been shown for areas before and beyond the focal plane respectively.

Propagation of hypergeometric laser beams in a medium with a parabolic refractive index

An expression to describe the complex amplitude of a family of paraxial hypergeometric laser beams propagating in a parabolic-index fiber is proposed. A particular case of a Gaussian optical vortex propagating in a parabolic-index fiber is studied. Under definite parameters, the Gaussian optical vortices become the modes of the medium. This is a new family of paraxial modes derived for the parabolic-index medium. A wide class of solutions of nonparaxial Helmholtz equations that describe modes in a parabolic refractive index medium is derived in the cylindrical coordinate system. As the solutions derived are proportional to Kummer?s functions, only those of them which are coincident with the nonparaxial Laguerre?Gaussian modes possess a finite energy, meaning that they are physically implementable. A definite length of the graded-index fiber is treated as a parabolic lens, and expressions for the numerical aperture and the focal spot size are deduced. An explicit expression for the radii of the rings of a binary lens approximating a parabolic-index lens is derived. Finite-difference time-domain simulation has shown that using a binary parabolic-index microlens with a refractive index of 1.5, a linearly polarized Gaussian beam can be focused into an elliptic focal spot which is almost devoid of side-lobes and has a smaller full width at half maximum diameter of 0.45 of the incident wavelength.

Subwavelength focusing with a Mikaelian planar lens

We show that an arbitrary TE-polarized light field propagating in a Mikaelian secant (MS) planar lens can be decomposed into modes described by the Jacobi polynomials. This light field will be periodically repeated at the Talbot length and focused with a half-Talbot length period. An analytical expression for the width of the focal spot has been obtained. The MS lens allows obtaining a focal spot of width equal to the diffraction limit in the medium. The MS lens has been fabricated as a planar photonic crystal lens in a silicon film for wavelength 1.55 μm, and its focusing properties have been demonstrated by visible light (532 nm) interference fringes.

Diffraction integral and propagation of Hermite–Gaussian modes in a linear refractive index medium

We derive a diffraction integral to describe the paraxial propagation of an optical beam in a graded index medium with the permittivity linearly varying with the transverse coordinate. This integral transformation is irreducible to the familiar ABCD transformation. The form of the integral transformation suggests that, unlike a straight path in a homogeneous space, any paraxial optical beam will travel on a parabola bent toward the denser medium. By way of illustration, an explicit expression for the complex amplitude of a Hermite-Gaussian beam in the linear index medium is derived.