We deduce and study an analytical expression for Fresnel diffraction of a plane wave by a spiral phase plate (SPP) that imparts an arbitrary-order phase singularity on the light field. Estimates for the optical vortex radius that depends on the singularity’s integer order n (also termed topological charge, or order of the dislocation) have been derived. The near-zero vortex intensity is shown to be proportional to ${\rho }^{2n}$, where ρ is the radial coordinate. Also, an analytical expression for Fresnel diffraction of the Gaussian beam by a SPP with $n\mathrm{th}$-order singularity is analyzed. The far-field intensity distribution is derived. The radius of maximal intensity is shown to depend on the singularity number. The behavior of the Gaussian beam intensity after a SPP with second-order singularity $(n=2)$ is studied in more detail. The parameters of the light beams generated numerically with the Fresnel transform and via analytical formulas are in good agreement. In addition, the light fields with first- and second-order singularities were generated by a 32-level SPP fabricated on the resist by use of the electron-beam lithography technique.

We report creation by photolithography techniques of the phase rotor filter, an optical element whose complex transmittance depends in a linear fashion on the azimuth angle. Relationships are given that describe the scalar diffraction of coherent light by the rotor filter. The results of the numerical simulation and experiments are discussed.

Diffraction of a 3D optical beam on a multilayer phase-shifted Bragg grating (PSBG) is considered. It is shown that the PSBG enables optical computation of the spatial Laplace operator of the electromagnetic field components of the incident beam. The computation of the Laplacian is performed in reflection at normal incidence. As a special case, the parameters of the PSBG transforming the incident Gaussian beam into a Laguerre–Gaussian mode of order (1,0) are obtained. Presented numerical results demonstrate high quality of the Laplace operator computation and confirm the possibility of the formation of Laguerre–Gaussian mode. We expect the proposed applications to be useful for all-optical data processing.

We present a theoretical study of a new application of the phase-shifted Bragg grating (PSBG) as an optical spatial differentiator operating in reflection. We demonstrate that the PSBG allows to calculate the first-order spatial derivative at oblique incidence and the second-order derivative at normal incidence. As an example, the differentiator is numerically shown to be able to convert an input 2D Gaussian beam into a 2D Hermite–Gaussian mode. We expect the proposed application to be useful for all-optical data processing.

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 $n$th-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 $c\ge 0$: 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 $c\gg 1$. 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 $\hslash (n+c{I}_1(2c)/I_0(2c))$, where $\hslash$ is the Planck constant and $I_n(x)$ is a modified Bessel function.

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