Influence of optical "dipoles" on the topological charge of a field with a fractional initial charge.

In this work, using the Rayleigh-Sommerfeld integral and Berry formula, a topological charge (TC) of a Gaussian optical vortex with an initial fractional TC in the far field was calculated. It was found that, for diverse fractional parts of the TC, the beam contained different numbers of screw dislocations, which determined the TC of the entire beam. If a fractional part of the TC was small, the beam consisted of the main optical vortex centered on the optical axis, with the TC equal to the nearest integer (say n>0) and two edge dislocations located on the vertical axis (one above and the other below the center). When the fractional part of the initial TC increased, a "dipole" was formed from the upper edge dislocation, consisting of two vortices with TCs equal to +1 and -1. With a further increase in the fractional part, the additional vortex with TC=+1 moved to the center of the beam, and the vortex with TC=-1 moved to the periphery. When the fractional part of the TC increased further, another "dipole" was formed from the lower edge dislocation, in which, on the contrary, the vortex with TC=-1 was displaced to the optical axis (to the center of the beam) and the vortex with TC=+1 moved to the beam periphery. When the fractional part of the TC became equal to 1/2, the lower vortex with a TC=-1, which was earlier displaced to the center of the beam, began to shift to the periphery, and the upper vortex with a TC=+1 moved closer and closer to the center of the beam, eventually merging with the main vortex when the fractional part approached 1. Such dynamics of additional vortices with TCs above +1 and below -1 determined which whole TC the beam would have (n or n+1) for different values of the fractional part from the segment [n,n+1]. Our analysis has shown that, for any value of the fractional part of the initial topological charge, the TC of the beam in the far field will not be determined. Since, with an increase in the radius of the circle in the beam section on which the TC is calculated, more optical "dipoles" will appear, and the TC will be either n or n+1.

Vortex astigmatic Fourier-invariant Gaussian beams.

We find a two-parameter family of astigmatic elliptical Gaussian (AEG) optical vortices, which are free space modes up to scale and rotation. We calculate total normalized orbital angular momentum of AEG vortices, which can be an integer, fractional and zero, and which is equal to the algebraic sum of two terms reflecting the contribution of the vortex and astigmatic components of the light field. In any transverse plane, such a beam has an isolated n-fold degenerate intensity null on the optical axis (an optical vortex) embedded into an elliptical Gaussian beam. In addition to the quadratic elliptical phase, a beam has the phase of a cylindrical lens rotated by an angle of 45 degrees with respect to the principal axes of the ellipse of the Gaussian beam intensity distribution. The degenerated central intensity null in these beams does not split it into n spatially separated intensity nulls, as is usually assumed for elliptical astigmatic beams.

Tight focusing of laser light using a chromium Fresnel zone plate.

Using near-field scanning microscopy, we demonstrate that a 15-µm zone plate fabricated in a 70-nm chromium film sputtered on a glass substrate and having a focal length and outermost zone's width equal to the incident wavelength λ = 532 nm, focuses a circularly polarized Gaussian beam into a circular subwavelength focal spot whose diameter at the full-width of half-maximum intensity is FWHM = 0.47λ. This value is in near-accurate agreement with the FDTD-aided numerical estimate of FWHM = 0.46λ. When focusing a Gaussian beam linearly polarized along the y-axis, an elliptic subwavelength focal spot is experimentally found to measure FWHMx = 0.42λ (estimated value FWHMx = 0.40λ) and FWHMy = 0.64λ. The subwavelength focal spots presented here are the tightest among all attained so far for homogeneously polarized beams by use of non-immersion amplitude zone plates.

Thin high numerical aperture metalens.

We designed, fabricated, and characterized a thin metalens in an amorphous silicon film of diameter 30 µm, focal length equal to the incident wavelength 633 nm. The lens is capable of simultaneously manipulating the state of polarization and phase of incident light. The lens converts a linearly polarized beam into radially polarized light, producing a subwavelength focus. When illuminated with a linearly polarized Gaussian beam, the lens produces a focal spot whose size at full-width half-maximum intensity is 0.49λ and 0.55λ (λ is incident wavelength). The experimental results are in good agreement with the numerical simulation, with the simulated focal spot measuring 0.46λ and 0.52λ. This focal spot is less than all other focal spots obtained using metalenses.

Microlens-aided focusing of linearly and azimuthally polarized laser light.

We have investigated a four-sector transmission polarization converter (4-SPC) for a wavelength of 633 nm, that enables the conversion of a linearly polarized incident beam into a mixture of linearly and azimuthally polarized beams. It was numerically shown that by placing a Fresnel zone plate of focal length 532 nm immediately after the 4-SPC, the incident light can be focused into an oblong subwavelength focal spot whose size is smaller than the diffraction limit (with width and breadth, respectively, measuring FWHM = 0.28λ and FWHM = 0.45λ, where λ is the incident wavelength and FWHM stands for full-width at half maximum of the intensity). After passing through the 4-SPC, light propagates in free space over a distance of 300 µm before being focused by a Fresnel zone plate (ZP), resulting in focal spot measuring 0.42λ and 0.81λ. The focal spot was measured by a near-field microscope SNOM, and the transverse E-field component of the focal spot was calculated to be 0.42λ and 0.59λ. This numerical result was verified experimentally, giving a focal spot of smaller and larger size, respectively, measuring 0.46λ and 0.57λ. To our knowledge, this is the first implementation of polarization conversion and subwavelength focusing of light using a pair of transmission micro-optic elements.

Orbital angular momentum of superposition of identical shifted vortex beams.

We have formulated and proven the following theorem: the superposition of an arbitrary number of arbitrarily off-axis, identical nonparaxial optical vortex beams of arbitrary radially symmetric shape, integer topological charge n, and arbitrary real weight coefficients has the normalized orbital angular momentum (OAM) equal to that of individual constituent identical beams. This theorem enables generating vortex laser beams with different (not necessarily radially symmetric) intensity profiles but identical OAM. Superpositions of Bessel, Hankel-Bessel, Bessel-Gaussian, and Laguerre-Gaussian beams with the same OAM are discussed.

Superpositions of asymmetrical Bessel beams.

We considered nonparaxial asymmetrical Bessel modes of the first and second types, which differ from a conventional symmetrical Bessel mode by a real-valued shift along one Cartesian coordinate and an imaginary shift along another (both shifts are equal in modulus). The first- and second-type Bessel modes differ only in signs of the shift and, therefore, have different orbital angular momentum (OAM) (integer or fractional). Addition and subtraction of complex amplitudes of two identical asymmetrical Bessel modes of the first and second type lead to light beams with the same integer OAM equal to the topological charge n of the original mode, but with different transverse intensity distributions, which depend on the shift magnitude. This proposed method allows controlling of the OAM of the beam with simultaneous changing of its shape, i.e., for matching with the object being trapped.

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.

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 beam's 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.

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 beam's 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.

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.

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.

Hyperbolic secant slit lens for subwavelength focusing of light.

Using the finite-difference time-domain simulation, we show that if a gradient-index or binary planar dielectric microlens that focuses light at the output surface has a near-focus subwavelength slit the focal spot width is determined by the slit width. Notably, the slit allows the output light proportion to be increased due to the surface wave scattering, thus forming a focal spot nearly devoid of side lobes. In this work, the focal spot width of λ/23 and the diffraction efficiency of focusing of 44% are achieved.

Tight focusing with a binary microaxicon.

Using a near-field scanning microscope (NT-MDT) with a 100 nm aperture cantilever held 1 µm apart from a microaxicon of diameter 14 µm and period 800 nm, we measure a focal spot resulting from the illumination by a linearly polarized laser light of wavelength λ=532 nm, with its FWHM being equal to 0.58λ, and the depth of focus being 5.6λ. The rms deviation of the focal spot intensity from the calculated value is 6%. The focus intensity is five times larger than the maximal illumination beam intensity.

Generating hypergeometric laser beams with a diffractive optical element.

We derive explicit analytical relations to describe paraxial light beams that represent a particular case of the hypergeometric (HyG) laser beams [J. Opt. Soc. Am. A25, 262-270 (2008)JOAOD60740-323210.1364/JOSAA.25.000262]. Among these are modified quadratic Bessel-Gaussian beams, hollow Gaussian optical vortices, modified elegant Laguerre-Gaussian beams, and gamma-HyG beams. Using e-beam microlithography, a binary diffractive optical element capable of producing near-HyG beams is synthesized. Theory and experiment are in sufficient agreement. We experimentally demonstrate the ability to rotate dielectric microparticles using the bright diffraction ring of a HyG beam.

Hypergeometric modes.

A new family of paraxial laser beams that form an orthogonal basis is discussed. When propagated in uniform space, these beams preserve their structure to scale. The intensity distribution profile for such beams is similar to that for the Bessel modes, representing a set of alternating bright and dark concentric rings. The complex amplitude of these beams is proportional to the degenerate (confluent) hypergeometric function, and therefore we term such beams hypergeometric (HyG) modes. The HyG modes are generated with a liquid-crystal microdisplay.

Sidelobe contrast reduction for optical vortex beams using a helical axicon.

We present a new approach for generating an optical vortex pattern with reduced sidelobes without increasing the radius of the vortex and without excessive energy loss. Our technique combines the spiral phase plate with a weak axicon to form a helical axicon. Experimental results using a liquid crystal display agree with theory.

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.

Analytical expression for radiation forces on a dielectric cylinder illuminated by a cylindrical Gaussian beam.

Radiation forces exerted upon a dielectric, circular-shaped cylinder of infinite length illuminated by a non-paraxial cylindrical Gaussian beam are considered. Vectorial projections of the radiation pressure force on a dielectric, arbitrary- and circular-shaped cylinder are expressed analytically. In particular, the radiation force is expressed through coefficients of the decomposition of the non-paraxial Gaussian beam into the cylindrical functions. Using numerical examples, a possibility to optically trap a circular-shaped cylinder in a non-paraxial cylindrical Gaussian beam is demonstrated.