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This work introduces a class of 1D spatial-frequency-modulated structures with transmittance T ( x ) , in which the period changes along the x axis so that the corresponding spatial frequency f ( x ) sinusoidally alternates between two values. It is shown that T ( x ) generally is an almost-periodic function and has an impulsive spatial spectrum. However, we find the condition under which T ( x ) is a periodic function and its spatial spectrum form a lattice of impulses. When the periodicity condition is fulfilled, we call these structures as 1D spatially chirped periodic structures. These structures are characterized by two natural numbers, named as n c and n a v , and a real parameter named as frequency modulation strength (FMS). As an important special case, we define a 1D spatially chirped amplitude sinusoidal grating (SCASG) based on the transmission function of a conventional amplitude sinusoidal grating, in which the phase of conventional amplitude sinusoidal grating is replaced by desired chirped phase. Then the spatial spectrum of a 1D SCASG is investigated in detail, and it is shown that the spatial spectrum can be managed by changing the value of FMS. In other words, the grating’s spectrum can be manipulated by adjusting the value of FMS. This feature might find applications in optical sharing of the incident power among different diffraction orders. Moreover, near-field diffraction from 1D SCASGs is studied by using the so-called angular (spatial) spectrum method, and Talbot distances for these gratings are determined and verified experimentally. It is shown that the intensity profiles at quartet- and octant-Talbot distances strongly depend on the values of the parameters n c and n a v . In comparison with the conventional gratings, we see some new and interesting aspects in the diffraction from 1D SCASGs. For instance, unlike the conventional gratings, in some propagation distances, the diffraction patterns possess sharp and smooth intensity bars at which the intensity is several times of the incident light beam’s intensity. It is shown that the maximum intensity of these bright bars over the diffraction patterns depends on the characteristic parameters of the grating, including n c , n a v , and FMS of the grating. These intensity bars might find applications for trapping and aggregation of particles along straight lines.

In this work, by providing comprehensive theoretical foundations, we revisit and improve a simple and efficient method that has been used for generation of 2D orthogonal arrays of optical vortices with components having different topological charges (TCs). This method has been implemented by the diffraction of a plane wave from 2D gratings where the gratings’ profiles are determined by iterative computational process. Here, based on the theoretical predictions, specifications of the diffraction gratings can be easily adjusted in a way to generate experimentally a heterogeneous vortex array with the desired power shares among different elements of the array. We use the diffraction of a Gaussian beam from a class of pure phase 2D orthogonal periodic structures having sinusoidal or binary profiles possessing a phase singularity, calling pure phase 2D fork-shaped gratings (FSGs). The transmittance of each of the introduced gratings is obtained by multiplying the transmittance of two pure phase 1D FSGs along x and y directions, having topological defect numbers l x and l y and phase variation amplitudes γ x and γ y , respectively. By solving the Fresnel integral, we show that the diffraction of a Gaussian beam from a pure phase 2D FSG leads to generation of a 2D array of vortex beams having different TCs and power shares. The power distribution among the generated optical vortices over the different diffraction orders can be adjusted by γ x and γ y , and it strongly depends on the profile of the grating. Meanwhile the TCs of the generated vortices depend on l x and l y and the corresponding diffraction orders, namely lm,n = −(ml x + nl y ) presents the TC of (m, n)th diffraction order. We recorded the intensity patterns of the experimentally generated vortex arrays which are fully consistent with the theoretically predicted results. Furthermore, the TCs of the experimentally generated vortices are measured individually by the diffraction of each of them through a pure amplitude quadratic curved-line (parabolic-line) grating. The absolute values and signs of the measured TCs are consistent with the theoretical prediction. The generated configuration of vortices with adjustable TC and power sharing features might find many applications such as non-homogeneous mixing of a solution consisting trapped particles.

A comprehensive theoretical study on the generation of a 2D orthogonal lattice of optical beams based on the near-field diffraction and Talbot effect is presented. First we investigate the near-field diffraction of an optical beam with a finite lateral extension from an infinite 2D orthogonal grating. It is shown that the resulting diffraction patterns over the Talbot planes depend on the following parameters: the period and opening ratio (OR) of the grating, wavelength and spatial spectral bandwidth of the incident beam, and the propagation distance. In terms of these parameters, we find multiplication conditions: the certain conditions under which a 2D orthogonal lattice of the Fourier transform of the incident beam is generated on the Talbot planes. Therefore, if the incident beam is Fourier-invariant and all the established multiplication conditions are fulfilled, the intensity profile of each of the individual Talbot images resembles the intensity profile of the incident beam. We consider the Laguerre–Gaussian beams having zero radial index as an important class of the vortex beams. We explicitly show that these beams are Fourier-invariant and we calculate their spatial spectral bandwidth. As a result, in the illumination of a 2D orthogonal binary grating with this kind of vortex beam, a 2D orthogonal lattice of the incident optical vortex is generated at the Talbot planes. Considering the obtained multiplication conditions, for the first time, to our knowledge, we determine a multiplication interval. This interval covers the propagation distances at which the vortex beam multiplication occurs. Moreover, we obtain the maximum possible value of the grating’s OR for the realizations of the vortex multiplication. It is shown that both the multiplication interval and the maximum value of the OR depend on the topological charge (TC) of the incident beam. With the aid of some practical examples and defining a multiplication quality factor, the mentioned results are verified quantitatively. In addition to the vortex beam multiplication effect, we consider another interesting phenomenon that results from the interference of the grating’s first diffraction orders. We call this phenomenon the first diffraction orders interference (FDOI) effect. We show that both the multiplication and the FDOI effects occur simultaneously but at different propagation distances. It is also shown that the multiplication and FDOI intervals separate and distance from each other by increasing the TC of the incident beam.

In this work, the theory of self-imaging in the polar coordinates for azimuthally periodic Bessel-based structures (APBBSs) is presented. For the first time, to the best of our knowledge, we define single- and multi-frequency APBBSs and show that these structures have self-images under plane-wave illumination. We also define sinusoidal and binary-like single-frequency APBBSs and theoretically and experimentally investigate the near-field diffraction of these structures. The diffraction from these structures provides 2D arrays of optical traps that can be used in multi-trapping.

In this work, the diffraction-based discrimination of two-dimensional (2D) orthogonal separable and nonseparable periodic structures and prediction of the reduced Talbot distances for 2D orthogonal nonseparable periodic structures are presented. 2D orthogonal periodic structures are defined and classified into separable (multiplicative or additive) and nonseparable categories with the aid of a spatial spectrum lattice. For both the separable and nonseparable cases, the spatial spectra or far-field impulses are 2D orthogonal lattices. We prove that for a 2D orthogonal separable structure, in addition to the DC impulse, there are other impulses on the coordinate axes. As a result, if all the spectrum impulses of a structure on the coordinate axes, except for the DC impulse, vanish, we conclude that the structure is nonseparable. In the second part of this work, using a unified formulation, the near-field diffraction of the 2D orthogonal separable and nonseparable periodic structures is investigated. In general, the Talbot distance equals the least common multiple of the individual Talbot distances in the orthogonal directions, say, z t =z lcm . For the 2D orthogonal nonseparable periodic structures having Fourier coefficients only with odd indices, we have found surprising results. It is shown that for this kind of structure, the Talbot distance strongly depends on the number theoretic properties of the structure. Depending on the ratio of the structure’s periods in the orthogonal directions, pxpy, the Talbot distance reduces to z lcm 2, z lcm 4, or z lcm 8. In addition, for the 2D orthogonal nonseparable sinusoidal grating, we show that, regardless of the value of pxpy, self-images are formed at distances smaller than the conventional Talbot distances attributed to px and py that we name the reduced Talbot (RT) distances. Halfway between two adjacent RT distances, the formation of negative self-images with a complementary amplitude of the self-images is predicted. Halfway between two adjacent self-image and negative-elf-image, subimages are formed. As another interesting result, we show that the intensity patterns of the subimages are 2D multiplicatively separable with halved periods in both directions. Finally, we show that 2D almost periodic structures with impulses on zone-plate-like concentric circles have self-images under plane wave illumination.

We report a family of solutions of the homogeneous free-space scalar wave equation. These solutions are determined by linear combinations of the half-integer order Bessel functions. We call these beams ``combined half-integer Bessel-like beams.'' It is shown that, by selecting suitable combinations of the half-integer order Bessel functions, a wide set of beams can be produced in which they may carry the orbital angular momentum (OAM) or not. We show that this family of beams satisfies a ``radial structured'' boundary condition at $z=0$ plane, therefore they can be produced by the diffraction of a plane wave from suitable ``radial structures.'' Some specific examples of the half-integer Bessel-like beams are introduced. Especially, a set of spatially asymmetric beams, having half-integer OAM, is introduced that can be used to make the concentration of the absorbing and dielectric micro- or nanoparticles in a microsolution inhomogeneous. Also, by manipulating the Fourier series of the radial structures, three subfamily of beams can be produced including the radial carpet, petallike, and ringlike vortex beams. The intensity profile of the petallike beams forms two dimensional optical latices with polar symmetry at the transverse plane. The ringlike vortex beams carry OAM. Here, by solving the wave equation we present the full image of the radial carpet beams. All the presented beams have nondiffracting, accelerating, and self-healing features. The combined half-integer Bessel-like beams can be considered in other areas of wave phenomena, ranging from sound and elastic waves to many other kinds of classical waves. Therefore, this work has profound implications in many linear wave systems in nature.

Self-accelerating shape-invariant beams are attracting major attention, presenting applications in many areas such as laser manipulation and patterning, light-sheet microscopy, and plasma channels. Moreover, optical lattices are offering many applications, including quantum computation, quantum phase transition, spin-exchange interaction, and realization of magnetic fields. We report observation of a class of accelerating and self-healing beams which covers the features required by all the aforementioned applications. These beams are accelerating, shape invariant, and self-healing for more than several tens of meters, have numerous phase anomalies and unprecedented patterns, and can be feasibly tuned. Diffraction of a plane wave from radial phase gratings generates such beams, and due to their beauty and structural complexity we have called them ``carpet'' beams. By tuning the value of phase variations over the grating, the resulting carpet patterns are converted into two-dimensional optical lattices with polar symmetry. Furthermore, the number of spokes in the radial grating, phase variation amplitude, and wavelength of the impinging light beam can also be adjusted to obtain additional features. We believe that radial carpet beams and lattices might find more applications in optical micromanipulation, optical lithography, super-resolution imaging, lighting design, optical communication through atmosphere, etc.

We experimentally demonstrate and theoretically predict a new and unprecedented optical carpet that includes all the geometric shadow and far-field and near-field diffraction patterns at the transverse plane in the diffraction from a radial grating illuminated by a plane wavefront. The main feature of using radial grating is the continuous change of the spatial period along the radial direction. Therefore, the geometric shadow, and the near-field and far-field diffraction regimes are mixed at various propagation distances, and the traditional definitions for the different diffraction regimes would not apply here. We show that for a given propagation distance, at a certain radial distance the shadow regime changes to the near-field regime and at another certain radial distance, the diffraction pattern changes from a near-field to a far-field case.

In this work we theoretically and experimentally investigate the diffraction of spatially coherent and collimated white light beam from radial amplitude gratings. Theoretical part of the work is resolved with the Fresnel-Kirchhoff integral. In the experimental part, a collimated wave-front of a white light beam emitting from a halogen lamp is transmitted through a radial amplitude grating. We digitally record the diffraction pattern in various distances from the grating using a CCD camera. The resulted diffraction pattern that we call it "Colorful radial Talbot carpet at the transverse plane" has a shape-invariant form under propagation. The other significant aspects of this pattern are the existence of a quite patternless dark area located around the optical axis and an intense rainbow-like ring in the vicinity of the patternless area. The rainbow color changes radially from the violet in the vicinity of the patternless area to red by increasing the radius in which the purity of the colors in the inner side is dominant. We call this phenomena "diffraction-based rainbow". In addition, the transverse plane Talbot carpet pattern consists colorful self-images of the grating's spokes at the larger radii. The theoretical calculations and experimental results verify each other completely. The introduced diffraction-based rainbow can be utilized in spectrometry.

In this work, a comprehensive analytic study of the diffraction of vortex beams from structured apertures is presented. We formulate the near- and far-field diffraction of a vortex beam from an aperture having an arbitrary functionality in the Cartesian coordinates by two general and different approaches. We show that each of the resulting diffraction patterns can be determined by a number of successive derivatives of the 2D Fourier transform of the corresponding hypothetical aperture function or equally can be obtained by a summation of 2D Fourier transforms of the corresponding modified aperture function. We implement both introduced analytic approaches in predicting the diffraction of a vortex beam from an elliptic Gaussian aperture, an elliptic Gaussian phase mask, and a hyperbolic Gaussian phase mask in the near- and far-field regimes. It is shown that the predicted diffraction patterns by both these approaches are exactly the same. It is shown that the diffraction of a vortex beam from an elliptic Gaussian aperture at the far-field regime forms a light beam that belongs to a family of light beams we call elegant elliptical vortex Hermite-Gaussian beams. In addition, the diffractions of a vortex beam from a Fresnel zone plate in general form for the on- and off-axis situations are formulated, and sinusoidal and binary zone plates are investigated in detail. Our general analytic formula can be used for a large variety of apertures including off-center situations and asymmetrical cases.

This work presents a detailed analytic approach to the diffraction of vortex beams from 2D orthogonal periodic structures. Using the presented formulation, the diffraction of vortex beams from 2D sinusoidal and Ronchi gratings is investigated. For these gratings, the Talbot self-healing effect under vortex beam illumination is examined. In the illumination of a 2D grating with a vortex beam, we refer the Talbot self-healing effect to the filling of the null area of the incident beam under propagation. We show that, for an incident vortex beam having odd value of topological charge, the generated Talbot self-images over the self-healing area are a 2D array of optical vortices, in which each of the individual self-images gets the form of an optical vortex with a topological charge of l=1 regardless of the topological charge of the incident beam. Both the Talbot self-healing effect and generation of the 2D array of optical vortices occur optimally between a definite interval of propagation distance. Using an intuitive approach based on the interference of the diffracted orders of the grating, we determine the self-healing interval. We show that a 2D array of optical vortices can be generated directly in the interference of eight copies of a vortex beam having proper lateral shifts and relative tilts. Easy tuning and energy preservation are two main advantages that the interference-based method has over the above-mentioned diffraction-based method for generating a 2D array of optical vortices. However, setup and implementation of the diffraction-based method are very simple. We believe that both the diffraction-based and interference-based methods for creating vortex beam arrays might find applications in optical tweezers, micromanipulations, and microfluidics.

For convenient optical communications by the aid of vortex beams, topological charge (TC) alterations should be translated to the change in intensity of the output light. In this paper, we formulate and experimentally investigate diffraction of vortex beams from amplitude radial gratings having sinusoidal profiles. We show that the diffraction pattern simply renders both TC and twist direction of the impinging vortex beam. When the TC of the vortex beam and the radial grating spoke number are equal, intensity on the optical axis of the Fraunhofer pattern gets a maximum value. Otherwise, its value on the optical axis remains zero. We examined the method on different vortex beams; the measured TCs of generated beams are in excellent agreement with the expected values. We show that an alteration between two vortex beams, in which one has a TC equal to the grating spoke number, is translated to a binary change in intensity of the output light on the optical axis. This feature might find wide applications in optical communications.

In this paper, first a detailed investigation of the various behaviors of the near- and far-field diffractions from multiplicatively separable (MS) structures in the x and y directions is presented. It is shown that in near-field propagation, the diffraction pattern of a 2D MS structure is same as the product of the corresponding near-field diffractions of the 1D individual components of the structure. For the far-field diffraction, although the resulting diffraction pattern is not equal to the product of the individual 1D structures' diffraction patterns, we show that it is still a 2D MS pattern. Second, a detailed theoretical investigation of the contrast enhancement effect for the multiplication of two 1D orthogonal intensity patterns (not necessarily periodic) is presented. By merging the above-mentioned facts, we deduced that in near-field diffraction, the contrast of the diffraction pattern of a 2D MS structure is always larger than the contrast of each of the diffraction patterns of the corresponding individual 1D structures. In the second part of this paper, we implement the intensity contrast as a quantity to describe self-images of the 2D MS periodic structures. For the first time, two very important results are obtained based on the contrast enhancement effect. We show that the depth of focus of the self-images increases for the 2D periodic structures in comparing to their corresponding 1D structures. We also predict the existence of additional self-images in addition to the Talbot images located at the least common multiple of each of the individual 1D Talbot distances for the 2D MS periodic structures. In addition, in this work we present a very straightforward manner for the prediction of the 1D or 2D forms of the diffraction pattern and the direction of the 1D pattern strips at given propagation distances from the 2D structure by introducing another intensity-contrast-based parameter. Furthermore, we show that the diffraction pattern of a 2D MS periodic structure depends strongly on the square of the ratio of the 1D structure periods in the x and y directions. These theoretical achievements are verified by some experimental works.

Using analytical modeling and detailed numerical simulations, we investigate the input-output transmission regimes in one-dimensional (1D) nonlinear photonic crystal including array defect layers. A coupled-mode system is derived from the Maxwell equations and analyzed for the stationary-transmission regime in the new proposed structure. Using the idea about introducing defect layers into 1D nonlinear photonic crystals, a new method for creating and controlling optical bistability is proposed. The periodic optical structures with array defect layers can be used as all optical switches between lower- and higher-transmissive states, whereas it possesses one jumping from a low-transmissive state to a transparent state.