Resonant properties of composite structures consisting of several resonant diffraction gratings.

We theoretically and numerically investigate resonant optical properties of composite structures consisting of several subwavelength resonant diffraction gratings separated by homogeneous layers. Using the scattering matrix formalism, we demonstrate that the composite structure comprising N gratings has a multiple transmittance zero of the order N. We show that at the distance between the gratings satisfying the Fabry-Pérot resonance condition, an (N - 1)-degenerate bound state in the continuum (BIC) is formed. The results of rigorous numerical simulations fully confirm the theoretically predicted formation of multiple zeros and BICs in the composite structures. Near the BICs, an effect very similar to the electromagnetically induced transparency is observed. We show that by making the proper choice of the thicknesses of the layers separating the gratings, nearly rectangular reflectance or transmittance peaks with steep slopes and virtually no sidelobes can be obtained. In particular, one of the presented examples demonstrates the possibility of obtaining an approximately rectangular transmittance peak with a significantly subnanometer width. The presented results may find application in the design of optical filters, sensors and devices for optical differentiation and transformation of optical signals.

Spatial integration and differentiation of optical beams in a slab waveguide by a dielectric ridge supporting high-Q resonances.

We show that a very simple structure consisting of a single subwavelength dielectric ridge on the surface of a slab waveguide enables spatial integration and differentiation of the profile of optical beams propagating in the waveguide. The integration and differentiation operations are performed in reflection and in transmission, respectively, at oblique incidence of the beam impinging on the ridge. The implementation of these operations is associated with the resonant excitation of a cross-polarized eigenmode of the ridge. We demonstrate that the quality factor of the resonances strongly varies along the dispersion curves and allows one to achieve the required tradeoff between the integration (or differentiation) quality and the amplitude of the resulting beam. The presented rigorous numerical simulation results confirm high-quality integration and differentiation. The proposed integrated structure may find application in ultrafast all-optical analog computing and signal processing systems.

First-order optical spatial differentiator based on a guided-mode resonant grating.

We present an experimental demonstration of a subwavelength diffraction grating performing first-order differentiation of the transverse profile of an incident optical beam with respect to a spatial variable. The experimental results are in a good agreement with the presented analytical model suggesting that the differentiation is performed in transmission at oblique incidence and is associated with the guided-mode resonance of the grating. According to this model, the transfer function of the grating in the vicinity of the resonance is close to the transfer function of an exact differentiator. We confirm this by estimating the transfer function of the fabricated structure on the basis of the measured profiles of the incident and transmitted beams. The considered structure may find application in the design of new photonic devices for beam shaping, optical information processing, and analog optical computing.

Coupled-mode theory and Fano resonances in guided-mode resonant gratings: the conical diffraction mounting.

We study resonances of guided-mode resonant gratings in conical mounting. By developing 2D time-dependent coupled-mode theory we obtain simple approximations of the transmission and reflection coefficients. Being functions of the incident light's frequency and in-plane wave vector components, the obtained approximations can be considered as multi-variable generalizations of the Fano line shape. We show that the approximations are in good agreement with the rigorously calculated transmission and reflection spectra. We use the developed theory to investigate angular tolerances of the considered structures and to obtain mode excitation conditions. In particular, we obtain the cross-polarization mode excitation conditions in the case of conical mounting.

Planar two-groove optical differentiator in a slab waveguide.

We propose a simple planar optical differentiator consisting of two grooves on the surface of a slab waveguide. The studied differentiator operates in reflection and enables temporal and spatial differentiation of optical pulses and beams propagating in the waveguide. The differentiation is associated with the excitation of an eigenmode localized at the ridge located between the grooves. The presented numerical simulation results demonstrate high-quality spatial, temporal and the so-called spatiotemporal differentiation. The proposed differentiator may find application in ultrafast analog computing and signal processing systems.

Study of propagation of vortex beams in aerosol optical medium.

A theoretical and experimental study of the propagation of vortex laser beams in a random aerosol medium is presented. The theoretical study is based on the extended Huygens-Fresnel principle with the generation of a random field, using the fast Fourier transform. The simulation shows that the stability of vortex beams to fluctuations of an optical medium falls with rising order of optical vortices. Moreover, a coherence length (radius) of the random medium is of great importance. The coherence radius extension affects adversely the conservation of a beam structure in the random medium. During further free-space propagation, increasing coherence enables reduction of the negative effects of fluctuations for beams with high-value topological charges. Experimental studies in the random aerosol medium have shown that at small distances vortex beams mostly demonstrate lower stability than a Gaussian beam. However, at considerable distances, vortex beams start to demonstrate greater stability that may be explained by their capacity to be regenerated after they passed obstacles.

Analytical description of 3D optical pulse diffraction by a phase-shifted Bragg grating.

Diffraction of a three-dimensional (3D) spatiotemporal optical pulse by a phase-shifted Bragg grating (PSBG) is considered. The pulse diffraction is described in terms of signal transmission through a linear system with a transfer function determined by the reflection or transmission coefficient of the PSBG. Resonant approximations of the reflection and transmission coefficients of the PSBG as functions of the angular frequency and the in-plane component of the wave vector are obtained. Using these approximations, a hyperbolic partial differential equation (Klein-Gordon equation) describing a general class of transformations of the incident 3D pulse envelope is derived. A solution to this equation is found in the form of a convolution integral. The presented rigorous simulation results fully confirm the proposed theoretical description. The obtained results may find application in the design of new devices for spatiotemporal pulse shaping and for optical information processing and analog optical computing.

Near-wavelength diffraction gratings for surface plasmon polaritons.

In this Letter, we study numerically diffraction gratings for surface plasmon polaritons. Investigated plasmonic gratings consist of a periodic set of dielectric ridges located on the metal surface. The grating period is comparable with the wavelength of the incident wave. The simulation results obtained within the rigorous coupled-wave analysis framework have shown that the SPP diffraction on a plasmonic grating with parasitic scattering suppression is remarkably close to the diffraction of a TE-polarized plane wave on a conventional grating. A compact and efficient reflecting plasmonic grating is considered as an example. The presented results can be used for the design of high-efficiency two-dimensional optical elements for steering surface plasmon propagation.

Optical computation of the Laplace operator using phase-shifted Bragg grating.

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.

Spatial differentiation of optical beams using phase-shifted Bragg grating.

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.

Graded photonic quasicrystals.

We introduce graded photonic quasicrystals and investigate properties of such structures on the example of a Luneburg lens based on a dodecagonal photonic quasicrystal. It is shown that the graded photonic quasicrystal lens has better focusing properties as compared with the graded photonic crystal lens in a frequency range suitable for experimental realization. The proposed graded photonic quasicrystals can be used in optical systems where compact and powerful focusing elements are required.

Single-resonance diffraction gratings for time-domain pulse transformations: integration of optical signals.

A general transformation of the optical pulse envelope implemented by a single-resonance diffraction grating is studied. The particular cases considered include optical pulse integration and differentiation implemented by the grating in the Wood anomalies and the fractional integration and differentiation of order 1/2 implemented in the Rayleigh-Wood anomalies. The extraordinary-optical-transmission plasmonic gratings are shown to be well suited for the integration in the transmission. Diffraction gratings to perform the integration and semi-integration of optical pulses with temporal features in the picosecond range are designed. Numerical simulations based on the rigorous coupled-wave analysis of Maxwell's equations are in good agreement with presented theoretical analysis.

Hankel-Bessel laser beams.

An analytical solution of the scalar Helmholtz equation to describe the propagation of a laser light beam in the positive direction of the optical axis is derived. The complex amplitude of such a beam is found to be in direct proportion to the product of two linearly independent solutions of Kummer's differential equation. Relationships for a particular case of such beams-namely, the Hankel-Bessel (HB) beams-are deduced. The focusing of the HB beams is studied.

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.

Photonic crystal lens for coupling two waveguides.

We report the design, fabrication, and characterization of a new nanophotonic device comprising a two-dimensional photonic crystal (PhC) lens of size 3x4 microm fabricated in silicon-on-insulator. The PhC lens is put at the output of a planar waveguide of width 4.5 microm to couple light into a planar waveguide of width 1 microm, with two waveguides being of length 5 mm. A 1 microm off-axis displacement of the smaller waveguide leads to an 8-fold reduction of output light intensity, which means that the focal spot size at output of the PhC lens in silicon is less than 1 microm. The simulation has shown that the PhC lens has maximal transmittance at 1.55 microm, with the coupling efficiency being 73%. The focal spot size of the lens in air calculated at the FWHM is 0.32lambda (where lambda is the wavelength).

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.

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 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.

Generation of phase singularity through diffracting a plane or Gaussian beam by a spiral phase plate.

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 rho2n, where p is the radial coordinate. Also, an analytical expression for Fresnel diffraction of the Gaussian beam by a SPP with nth-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.