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In this work, we report the characterization of a Laguerre-Gaussian (LG) beam with given values of topological charge (TC) and radial index in a simple, efficient, and robust experimental diffraction scheme. The beam diffracts from an amplitude parabolic-line linear grating and the resulting diffraction patterns at zero- and first-order reveals the values of the TC, l, and radial index p of the incident LG beam using a simple analysis. The zero-order diffraction pattern consists of p + 1 concentric intensity rings and the first-order diffraction pattern contains an (l + p + 1) by (p + 1) two-dimensional array of intensity spots. The experimental scheme is robust since it is not sensitive to the relative locations of the impinging beam axis and the grating center, and is efficient since most of the energy of the output beam is in the diffraction order of interest for LG beam characterization. The measurement is also simple since the intensity spots of the array are placed exactly over straight and parallel lines. Both experimental and simulation results are presented and are consistent with each other.

We investigate the reforming of a plane wave into a periodic waveform in its propagation through a structural asymmetry four-level quantum dot molecule (QDM) system that is induced by an inter-dot tunneling process and present the resulting tunneling-induced Talbot effect. The tunneling process between two neighborhood dots is provided with the aid of a gate voltage. Using a periodic coupling field the response of the medium to the propagating plane probe beam becomes periodic. The needed periodic coupling field is generated with the interference of two coherent plane waves having a small angle and propagating almost parallel to the probe beam direction. In the presence of the tunneling effect of an electron between two adjacent QDs, for the probe beam propagating through the QDM system, the medium becomes transparent where the coupling fields interfere constructively. As a result, the spatial periodicity of the coupling field modulates the passing plane probe beam. We determine the minimum length of the QDM system to generate a periodic intensity profile with a visibility value equal to 1 for the probe field at the exit plane of the medium. It is also shown that by increasing the propagation length of the probe beam through the QDM medium, the profile of the maximum intensity areas becomes sharper. This feature is quantified by considering a sharpness factor for the intensity profile of the probe beam at the transverse plane. Finally, we investigate free space propagation of the induced periodic field and present the Talbot images of the tunneling-induced periodic patterns at different propagation distances for different values of the QDM medium lengths. The presented dynamically designing method of the periodic coherent intensity patterns might find applications in science and technology. For instance, in optical lithography, the need to use micro/nanofabricated physical transmission diffraction gratings, in which preparation of them is expensive and time-consuming, can be eliminated.

Optical tweezers have become a powerful tool in the fields of biology, soft condensed matter physics, and nanotechnology. Here, we report the use of recently introduced radial carpet beams (RCBs) in the optical tweezers setup to trap multiple particles. An RCB is produced by diffraction of a plane or Gaussian beam from an amplitude radial grating. Because of the radial symmetry of the grating, all the diffraction orders are propagated along the optical axis and are used for trapping. Based on the number of grating spokes, the produced RCB has a definite number of high-intensity spots on the transverse plane located over a circular ring. These high-intensity spots of the beam provide multi-traps when it passes through an objective lens and have enough gradient force to trap polystyrene and silica particles. Moreover, the diffracted light from the grating has this property to transfer the angular momentum. We show that the multi-trapped birefringent particles could rotate in their own traps when polarization of the trapping RCB to be circular. In addition, the orbital rotation of the particles is simply executable by manually rotating the grating in its plane around the optical axis.

In this work, a three-point spatial phase shifting (SPS) method is implemented for chasing of the moving interference fringes in the homodyne laser Doppler vibrometry (HoLDV). By the use of SPS method, we remove disability of the HoLDV in the discrimination of the motion direction for long-range displacements. From the phase increments histogram, phase unwrapping tolerance value is selected, and adequacy of the data acquisition rate and required bandwidth limit are determined. Also in this paper, a detailed investigation on the effect of detectors positioning errors and influence of the Gaussian profile of the interfering beams on the measurements are presented. Performance of the method is verified by measuring a given harmonic vibration produced by a loudspeaker. Also, by the proposed method, vibration of mounting system of a disk laser gain medium is characterized.

We analytically, numerically, and experimentally determine a topological charge (TC) of the sum of two axisymmetric off-axis Laguerre-Gaussian (LG) beams with the indices (0, m) and (0, n). In particular, we find that at m = n, the combined beam has TC = n, which suggests that the sum of two identical off-axis LG beams has the TC of an individual constituent LG beam. At m < n, the TC of the sum is found to take one of the following four values: TC1 = (m + n)/2, TC2 = TC1 + 1, TC3 = TC1 + 1/2, and TC4 = TC1 – 1/2. We also establish rules for selecting one of the four feasible values of TC. For the sum of two on-axis LG beams, TC of the superposition equals the larger constituent TC, i.e. TC = n. Meanwhile following any infinitesimally small off-axis shift, TC of the sum either remains equal to the pre-shift TC or decreases by an even number. This can be explained by an even number of optical vortices (OV) with TC = –1 instantly ‘arriving’ from infinity that compensate for the same number of OV with TC = +1 born in the superposition. We also show that when two LG beams with different parity are swapped in the superposition, the topological charge of the superposition changes by 1. Interestingly, when superposing two off-axis LG beams tilted to the optical axis so that their superposition produces a structurally stable beam, an infinite number of screw dislocations with TC = +1 are arranged along a certain line, with the total TC of the superposition equal to infinity.

In a recent study, we have reported a simple, efficient, and robust method that is based on diffraction in an amplitude parabolic-line linear grating for determination of the topological charge (TC), l , of an optical vortex beam [J. Opt. Soc. Am. B 37, 2668 (2020)JOBPDE0740-322410.1364/JOSAB.398143]. Here, we present a demonstration of the application of that method for characterization of a pair of superposed vortex beams having different winding numbers. It is shown that, when two vortex beams, described by Laguerre–Gaussian beams with winding numbers l 1 and l 2 and radial index p = 0 , impinge on-axis and collinearly on a diffraction grating having a quadratic curvature on its lines, with a simple analysis of the resulted diffraction patterns at the zero and first order, the TCs and their signs can be determined. The zero-order diffraction pattern shows an interference pattern of the beams. For close values of l 1 and l 2 , it has a petal-like pattern in which the number of spots is equal to | l 1 − l 2 | . It is also found that the first-order diffraction pattern depending to the signs of the beams’ TCs shows two different forms. If l 1 and l 2 have the same signs, the first-order diffraction pattern is only a set of elongated intensity spots. When the signs of l 1 and l 2 are opposite, the resulted pattern is a ( l 1 + 1 ) by ( l 2 + 1 ) slanted checkered-like matrix of bright spots. In addition, in this work, we use a simple, novel, and initiative method to generate and combine on-axis and collinearly a pair of vortex beams. Finally, a supporting theoretical study is presented that fully confirms the experimental results and simulation of propagation.

In this work the anisotropy and inhomogeneity of real atmospheric turbulence have been investigated using image motion monitoring and differential image motion monitoring methods. For this purpose the light beam of a point source is propagated through the atmospheric turbulence layers in horizontal path and then impinged to a telescope aperture. The telescope and point source were 350 m apart. In front of the telescope's aperture a mask consisting of four subapertures was installed. Image of the point source was formed on a sensitive CCD camera located at the focal plane of the telescope. By displacing CCD camera along the axis of telescope, four distinct images were recorded. Angle of arrival (AA) of each spot was calculated by image processing. Air turbulence causes AA to fluctuate. By comparing AA fluctuation variances of different spots in two directions isotropy and homogeneity of turbulence were studied. Results have shown that atmospheric turbulence in near ground layers is treated as an anisotropic and inhomogeneous medium. In addition, the inhomogeneity and anisotropy of turbulence decreases with the distance from earth surface.

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.

Diffraction gratings are important optical components and are used in many areas of optics such as in spectroscopy. A diffraction grating is a periodic structure that splits and diffracts the impinging light beam into several beams travelling in different directions. The diffracted beams from a grating are commonly called diffraction orders. The directions of the diffraction orders depend on the grating period and the wavelength of the impinging light beam so that a grating can be used as a dispersive element. In the diffraction of a plane wave from a conventional grating, the intensities of diffracted beams decrease with increasing order of diffraction. Here, we introduce a new type of grating where in the diffraction of a plane wave, the intensity of a given higher order diffracted beam can be higher than the intensity of the lower orders. We construct these gratings by adding an azimuthal periodic dependency to the argument of the transmission function of a linear phase grating that has a sinusoidal profile and we call them azimuthally-modified linear phase gratings (AMLPGs). In this work, in addition to introducing AMLPGs, we present the generation of varied radial carpet beams over different diffraction orders of an AMLPG with controlled intensity sharing among the generated beams. A radial carpet beam is generated in the diffraction of a plane wave from a radial phase grating. We show that for a given value of the phase amplitude over the host linear phase grating, one of the diffraction orders is predominant and by increasing the value of the phase amplitude, the intensity sharing changes in favor of the higher orders. The theory of the work and experimental results are presented. In comparison with the diffraction of a plane wave from radial phase gratings, the use of AMLPGs provides high contrast diffraction patterns and presents varied radial carpet beams over the different diffraction orders of the host linear phase grating. The resulting patterns over different diffraction orders are specified and their differences are determined. The diffraction grating introduced with controlled intensity sharing among different diffraction orders might find wide applications in many areas of optics such as optical switches. We show that AMLPG-based radial carpet beams can be engineered in which they acquire sheet-like spokes. This feature nominates them for potential applications in light sheet microscopy. In addition, a detailed analysis of the multiplication of the diffraction pattern of an AMLPG by the 2D structure of a spatial light modulator is presented. The presented theory is confirmed by respective experiments.

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.

In this work, we introduce for the first time the use of a pair of telescopes facing each other in conjunction with the use of a pair of moire deflectometers for the investigation of the inhomogeneity of atmospheric turbulence. In the experiment, a laser beam enters a telescope from its back focal point by virtue of a focusing lens and is expanded and re-collimated by it before passing through the turbulent ground level atmosphere. It then enters the aperture of a second telescope, where it is again re-collimated behind the focal point, finally entering a pair of moire deflectometers. We use the instrument for measuring the fluctuations of two components of the angle of arrival (AA) across the second telescope's aperture. Calculation of the structure functions of the vertical and horizontal components of the AA fluctuations on the second telescope's aperture (over typical vertical and horizontal moire fringes) at different altitudes and at different latitudes allows a quantitative measure of the inhomogeneity at the day and night times in the atmospheric surface layer. Experimental results show that on sunny days the difference between the structure functions of the horizontal component of the AA fluctuations calculated at two different altitudes over two horizontal moire fringes is changed as a function of the time and its value meaningfully correlated to the mean value of the temperature of the Earth's surface. We did not find any interpretative correlation for the difference between the structure functions of the vertical component of the AA fluctuations calculated at two different latitudes. In addition, for the data recorded on windy days the observed correlation between the inhomogeneity of the atmospheric turbulence and the Earth's temperature almost disappeared.

The diffraction from a linear grating having a quadratic curvature on its lines is proposed for vortex beam characterization. Three types of transmission functions are considered for the gratings, including pure amplitude, pure phase, and hybrid amplitude and phase profiles. The first-order diffraction of the vortex beam through such gratings is only a set of elongated intensity spots. The number of spots determines the value of the topological charge (TC), and the sign of the TC can be distinguished with the elongation (and rotation) direction of the spots. This method is effective because over a given diffraction order, all of the energy of the beam transfers into the bright bar-like spots. Another advantage of the method is its ease of use, because it is not sensitive to the relative location of the beam axis and the grating center. Using a spatial light modulator, we provide different hybrid amplitude and phase linear gratings with a quadratic curvature on their lines and demonstrate the effectiveness of the proposed method experimentally. Also, the same justification was done in the diffraction of vortex beams from printed pure amplitude gratings. We have also investigated the effect of lateral shearing between the amplitude and phase parts of the hybrid grating on the resulted diffraction pattern. It is shown that for given values of lateral shear, one of the first-order diffraction patterns is eliminated, and the intensity of the other one is maximized. Finally, we present the key results of the diffraction of optical vortices from annular amplitude and phase gratings and from phase objects having linearly increasing phase functions along the radial direction. It is shown that the diffraction of optical vortices only from the parabolic-line linear gratings is insensitive to the off-axis value of the beam and grating centers.

We measure wavefront (WF) distortions in a high-power thin-disk laser induced simultaneously by the gas-lens/wedge and disk front-surface deformation using a two-channel moiré-based WF sensor. Thermal lensing and tilting effects are characterized versus time, their pressure-dependent fluctuations are quantified, and finally the corresponding pure disk front-surface effects are estimated for zero pressure. A divergent probe beam with a WF mean curvature similar to the curvature of the disk is reflected off the disk front surface. The temporal evolution of the WF at laser start-up is characterized using the WF sensor. A camera records both temporal evolution of the moiré fringe patterns and the intensity profile of the laser beam to have simultaneously both phase and intensity profiles of the laser beam. Successive WF data quantify the temporal evolutions of the dioptric power and induced wedges in two directions and their fluctuations during laser operation. We investigate the effect of air pressure on the produced gas-lens/wedge. The method is not sensitive to translational vibrations and is very low cost with adjustable sensitivity.

In this work, we propose a method for designing an adjustable amplitude-phase hybrid grating in which by relative lateral shearing of its amplitude and phase parts, the intensity share between different diffraction orders can be controlled. The method is based on superimposing two pure-amplitude and pure-phase gratings with sinusoidal or binary profiles having the same periods and lines’ orientations. It is shown that, in the diffraction of a Gaussian beam from such hybrid gratings, the intensity share of each of the diffraction orders is related to the shear value and the amplitude of the transmission functions of the superimposed pure-amplitude and pure-phase gratings. For instance, when both of the amplitude and phase profiles are sinusoidal and the shear value between them is a quarter of the period, for given values of the transmissions amplitudes of the profiles, all positive (or negative) diffraction orders are removed. We also show that, by changing the values of the transmission's amplitudes, the intensity share for the higher orders can be increased. This kind of grating might find application in optical switching and in devices requiring power sharing between different channels such as in beam-steering devices, in optical interconnects, and in optical fiber communication.

In this paper, we introduce a new rotation-sensitive and direction-resolved homodyne laser-Doppler vibrometry method for rigid body vibration study that is based on the discrete Fourier-transform of successive 1D profiles of the moving interference fringes recorded with a 1D array detector. By investigating the temporal evolution of the spatial phase distribution of the 1D profiles of the interference fringes, the out-of-plane translational and rotational vibrations of the vibrating object are simultaneously determined. We use a direction-cosine-based approach to establish the theory of the measurements. The merits and limitations of the proposed method is described. We show that the proposed method can be used for detection of both tip and tilt changes and out-of-plane displacement measurements of a rigid body using a couple of parallel 1D array detectors. In addition, we show that the presented method can be also used on optical diffused surfaces by adding three lenses in a corner-like arrangement to the detecting system.

This work presents a very simple and comprehensive approach for classification of the combinational spatial frequencies of the superimposed periodic or quasi-periodic structures. The reciprocal vectors of the structures are used to express their respective spectral components, and a unique reciprocal vectors equation is introduced for presenting the corresponding combinational frequencies. By the aid of the reciprocal vectors equation we classify moire patterns of combinational frequencies into four classes: the conventional moire pattern, moire fringes of higher-order harmonics, higher-order moire patterns, and pseudo-moire patterns. The difference between the moire fringes of higher-order harmonics and higher-order moire patterns is expressed in the formulas. By some typical examples, conditions for simultaneous formation of moire patterns of different harmonics of the superimposed gratings are investigated. We show that in the superimposition of two gratings, where at least one has a varying period and another has a non-sinusoidal profile, different moire patterns are formed over different parts of the superimposed area, where a distinct pair of spatial frequencies of the superimposed structures contributes to the formation of each of the patterns. We use the same procedure in the analysis of simultaneously produced defected moire patterns in the superimposition of a linear grating and a zone plate, where one or both consist of some topological defects at specific locations and at least one of the gratings has a non-sinusoidal profile. The topological defects of resulting moire fringes are similar to those appearing in the interference patterns of optical vortices. It is shown that the defect number of resulting moire fringes depends on the defect numbers and order of frequency harmonics of the gratings. The dependency of the defect number of the moire fringes and its sign to the defect numbers of the gratings and their contributed frequency harmonics is derived for both additive and subtractive terms of moire fringes, and the results are verified with several examples based on computational simulations.

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, by using a local reciprocal vector concept, we introduce a considerable number of quasi-periodic structures where each of them is constructed by a set of curved lines with varying period. By using a simple and very comprehensive recently proposed method, formulations and characterizations of the various moire patterns of different pairs of curved line quasi-periodic structures are presented. Some advantages of the concept of local reciprocal vector in the area of curved line quasi-periodic structures are clarified.

In this work, we estimate the atmospheric Fried parameter $ {r_0} $r0, average wind speed $ \bar v $v¯, and subsequently, atmospheric coherence time $ {\tau _0} $τ0 by experimental measurement via a four-aperture differential image motion monitor (DIMM) instrument at the Iranian National Observatory (INO) site. The experimental approach is based on the four-aperture DIMM defocus velocity theory, which uses the angle-of-arrival fluctuation measurement of starlight propagation through atmospheric turbulence in the form of a four-spot configuration provided by the four-aperture DIMM telescope. Here, we measure the defocus variance $ \sigma _{{C_4}}^2 $σC42 and its velocity variance $ \sigma _{\partial {C_4}/\partial t}^2 $σ∂C4/∂t2 and use the preceding theory to estimate the atmospheric turbulence parameters. We have implemented the data sampling at the INO site at an altitude of 3600 m above sea level with a 12-inch Meade Cassegrain telescope consisting of a four-aperture mask at its entrance pupil and a fast CCD camera recording short-exposure images with frame rates in the range of 480 fps to 620 fps from the Capella star. The experimental recorded data sets are analyzed and the results compared to those of our simulation and other methods and demonstrate good agreement.

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.

In this paper, we introduce a self-referencing microsphere-assisted transmission digital holographic microscopy (DHM) system in real-image operating mode. Self-referencing DHM arrangements provide a compact geometry that are temporally stable against environmental vibrations and suitable for the measurement of dynamic specimens such as cells. These advantages may be more pronounced for microsphere-assisted DHM systems. In the real-image mode, unlike the virtual-image mode, the working distance is increased. This, in turn, provides flexibility for insertion of additional elements for enhancement of the image quality, or for other required tasks. The lateral resolution enhancement is similar to the virtual image arrangement and the axial resolution is decoupled from the lateral one. The methodology is discussed theoretically and validated experimentally by conducting DHM experiments on standard micro-objects and aggregation of micro-particles. Our results show that, by the assistance of a rather big size 550 μm silica microsphere, a 10× microscope objective can resolve the 3D structure of a compact disk. The arrangement may be useful toward having a compact and inexpensive bench-top high-resolution three-dimensional imaging apparatus.

In this paper, a simple and accurate method based on Z-scan and parallel moire deflectometry for measuring the focal length of microlenses is reported. A laser beam is focused by one lens and is re-collimated by another lens, and then strikes a parallel moire deflectometer. In the presence of a microlens near the focal point of the first lens, the radius of curvature of the beam is changed; the parallel moire fringes are formed only due to the beam divergence or convergence. The focal length of the microlens is obtained from the moire fringe period graph without the need to know the position of the principal planes. This method is simple, more reliable, and completely automated. The implementation of the method is straightforward. Since a focused laser beam and Z-scan in free space are used, it can be employed for determining small focal lengths of small size microlenses without serious limitation on their size.

In this report is presented a simple and accurate method based on moire deflectometry for studying of nanofluids' thermal conductivity, containing gold nanoparticles suspended in a liquid medium. A collimated laser beam passes through a cell of nanofluids and then enters a moire deflectometer. Due to a temperature gradient in the nanofluids, a prism-like medium is formed. The propagation direction of the beam is changed and the moire fringes are displaced. Finally, thermal conductivity of the nano-fluids may be determined from the moire fringe displacement. The temperature gradient is then calculated using a simple model based on the Fourier law of heat transfer. This model is expected to reasonably predict the temporal evolution of nanofluid temperature gradient.

We have built a new strain gauge based on the moire technique. This strain gauge mainly consists of two frames that can move with respect to each other. Displacements are recorded by using the moire technique. We use a pair of similar gratings attached to the frames. The gratings are installed in parallel without physical contact and their lines making a small angle with one another. A moire pattern is formed due to superimposing of the gratings. A diode laser light passes through the moire pattern and a narrow slit, and hits on a light sensor. In response to external stress, one of the gratings is displaced and, as a result, the moire fringes move in front of the slit. Due to the fringes movements, the light intensity on the detector varies and is recorded as voltage. The voltage output can be used to measure the strain. This instrument can detect displacements of the order of micron. In this paper we show the experimental results of our instrument.

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.

A novel approach based on the two-channel moir\'e deflectometry has been used to measure both wave-front and transverse component of the Poynting vector of an optical vortex beam. Generated vortex beam by the q-plate, an inhomogeneous liquid crystal cell, has been analyzed with such technique. The measured topological charge of generated beams are in an excellent agreement with theoretical prediction., Comment: 3 pages, 2 figures

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.

In this work, for the first time to the best of our knowledge, an investigation on the moiré patterns of superimpositions of two radial or two circular gratings consisting of topological defects and their mutual superimpositions with each other, or with linear forked gratings or defected zone plates, is presented. For characterization of the resulting moiré patterns, we use the reciprocal vectors approach. In this approach, by considering local spatial frequencies for the superimposed structures, their reciprocal vectors are determined from the transmission function of the structures. The local reciprocal vector of the resulting moiré pattern at a given point is determined in terms of the local reciprocal vectors of the superimposed structures defined at the same point. In this approach, the topological singularities of the superimposed structures are described by the azimuthal component of the reciprocal vectors. This formulation is very simple, uniform, and comprehensive. In this work, we offer a detailed discussion on the different resulting moiré patterns for the above-mentioned superimpositions and some potential applications of the proposed superimpositions are introduced. In addition, different resulting moiré patterns are simulated.

When a light beam propagates through the turbulent atmosphere, the wavefront of the beam is distorted, which affect the image quality of ground based telescopes. Adaptive optics is a means for real time compensation of the wavefront distortions. In an adaptive optics system, wavefront distortions are measured by a wavefront sensor, and then using an active optical element such as a deformable mirror the instantaneous wavefront distortions are corrected. On the other hand, three physical effects are observed when a light beam propagates through a turbulent atmosphere: optical scintillation, beam wandering, and fluctuations in the angle-of-arrival (AA). These effects are used for measuring turbulence characteristic parameters. Fluctuations of light propagation direction, referred to as the fluctuations of AA, are measured by various methods. In wavefront sensing applications the AA fluctuations measurement is a basic step.

In this paper, the application of “parallel” moire deflectometry in measuring the nonlinear refractive index of materials is reported. In “parallel” moire deflectometry the grating vectors are parallel, and the resulting moire fringes are also parallel to the grating lines. Compared to “rotational” moire deflectometry and the Z-scan technique, which cannot easily determine the moire fringe’s angle of rotation and is sensitive to power fluctuations, respectively, “parallel” moire deflectometry is more reliable, which allows one to measure the radius of curvature of the light beam by measuring the moire fringe spacing. The nonlinear refractive index of the sample, including the sense of the change, is obtained from the moire fringe spacing curve. The method is applied for measuring the nonlinear refractive index of ferrofluids.

An adjustable, high sensitivity, wide dynamic range two channel wave-front sensor based on moire deflectometry has been constructed for measuring distortions of light wave-front transmitted through the atmosphere. In this approach, a slightly divergent laser beam is passed through the turbulent ground level atmosphere and then a beam-splitter divides it into two beams. The beams pass through a pair of moire deflectometers which are installed parallel and close together. From deviations in the moire fringes we calculate the two orthogonal components of angle of arrival at each location across the wave-front. The deviations have been deduced in successive frames which allows evolution of the wave-front shape to be determined. The dynamic range and sensitivity of detection are adjustable by merely changing the separation of the gratings and the angle between the rulings of the gratings in both of channels. The spatial resolution of the method is also adjustable by means of bright, dark, and virtual traces for given moire fringes without paying a toll in the measurement precision.

A wavefront sensor which takes advantage of the moire deflectometry has been constructed for measuring atmosphere induced wavefront distortions. In this sensor a collimated laser beam propagates through turbulent atmosphere, then a beam splitter splits it into two beams and the beams pass through a pair of moire deflectometers. Directions of the grating's rulings are parallel in each moire deflectometer but are perpendicular in the two beams. Using a suitable array of lenses and mirrors two sets of moire patterns are projected on a CCD camera. A suitable spatial filter removes the unwanted frequencies. Recording the successive moire patterns by the CCD camera and feeding them to a computer, allow temporal fluctuations of the laser beam wavefront phase to be measured highly accurately. Displacements of the moire fringes in the recorded patterns correspond to the fluctuations of two orthogonal components of the angle of arrival (AA) across the wavefront. The fluctuations have been deduced in successive frames, and then evolution of the wavefront shape is determined. The implementation of the technique is straightforward and it overcomes some of the technical difficulties of the Schlieren and Shack-Hartmann techniques. The sensitivity of detection is adjustable by merely changing the distance between two gratings in both moire deflectometers and relative grating ruling orientation. This overcomes the deficiency of the Shack-Hartman sensors in that these require expensive retrofitting to change sensitivity. Besides, in the moire deflectometry, the measurement is relatively insensitive to the alignment of the beam into the device. Hence this setup has a very good potential for adaptive optics applications in astronomy. Since tilts are measured in the Shack-Hartmann method at discrete locations, it cannot detect discontinuous steps in the wavefront. By this method discontinuous steps in the wavefront are detectable, because AA fluctuations are measured across the wavefront.

When a slightly divergent laser beam passes through a turbulent ground level atmosphere and strikes a linear grating, fluctuating self-images are formed at Talbot distances. By superimposing a similar grating on one of the self-images, even for the case of parallel gratings' lines, fluctuating moire fringes are formed owing to the beam divergence. Recording the successive moire patterns by a CCD camera and feeding them to a computer, after filtering the higher spatial frequencies, produces highly magnified fluctuations of the laser beam. Using moire fringe fluctuations we have calculated the fluctuations of the angle of arrival and the atmospheric refractive index structure constant. The implementation of the technique is straightforward, a telescope is not required, fluctuations can be magnified more than ten times, and the precision of the technique is similar to that reported in our previous work.

The use of moire pattern of superimposition of linear forked gratings (LFGs) and Fresnel zone plates (ZPs) has already been reported for study of different physical effects. In spite of a considerable number of applications, there is no comprehensive formulation for this kind of moire pattern. In this work, we introduce a new family of ZPs containing topological defects that we named defected ZP (DZP) and we present a very simple, uniform, and comprehensive formulation for the moire pattern of superimposition of two LFGs, two DZPs, and superimposition of an LFG on a DZP, using the reciprocal vector approach. For the case of the two LFGs superimposition, we show that the resulting moire pattern has a starlike shape or is a large-scale LFG pattern. In the case in which two DZPs are superimposed, we show that the resulting moire pattern has three general forms: large-scale DZP pattern, starlike pattern, and large-scale LFG pattern. In the superimposition of an LFG on a DZP, in special conditions a new spiral ZP having a topological defect is produced in which its defect number related to the superimposed gratings structures. The presented formulation has potential applications in singular optics measurements.

The turbulence of the atmosphere puts an upper limit on the quality of the image of a ground object obtained by long-exposure photography from low or high altitudes in the atmosphere or in the space. By using good optics and high resolution film or CCD and a stable platform, this limit could be approached but not exceeded. A useful quantity for indicating the magnitude of this limit is the integral of the modulation transfer function (MTF) associated with the turbulence. In this work, we introduce a new method for measuring the MTF of the atmosphere in the surface layer, based on moire technique. In this technique, from a low frequency Ronchi grating, installed at a certain distance from a digital camera equipped with a tele lens, successive images are recorded and then transferred to a PC. By rotating each image by θ/2 and -θ/2, say ±3°, and superimposing them, a large number moire patterns are produced. The average transmittance function of the superimposed image gratings is measured in a moire fringe interval. The latter function is measured by scanning the moire pattern by a slit parallel to moire fringes. It is shown theoretically that from the Fourier transform of the latter function the MTF of the atmosphere can be deduced, if the MTFs of the imaging system and the grating are given or their effects are negligible. The atmospheric MTFs have been measured at different turbulence conditions. Also, we have studied the behavior of the atmospheric MTF respect to exposure time.

The refractive index profile of optical fiber preform is measured by a nondestructive technique based on Talbot interferometry. In this technique the preform is placed between two ronchi ruling gratings of 10 lines/mm and the system is illuminated by an expanded and collimated beam of He-Ne laser. In this arrangement the 2nd grating is positioned in the Talbot image of the 1st grating and the preform axis is parallel to the gratings planes. To eliminate the effect of clad on the light beam deflection during the measurements, the preform is immersed in an index matching liquid. The phase front of the laser light over the 2nd grating can be monitored by analysis of the moire pattern which is formed over there. The analysis is done by means of 4-step phase shift technique. In this technique the second grating is moved in four steps of 1/4 of the grating vector and in each step the intensity profile of the moire pattern is recorded. The phasefront can be specified by using the recorded intensities. The refractive index profile of the preform can be calculated from the changes on phasefront while the preform is placed between the gratings respect to the case when it is absent. The whole procedure is automated and computer controlled by using a CCD camera to record the moire fringes, a stepper motor for linear translation of the 2nd grating and a code in MATLAB to control the system and measurements.

In glass casting or shaping processes, some times wavy and distortion defects are developed in glass sheets. Specifying the locations of these defects and measuring the deflections they cause are significant parameters in many applications. There are several techniques for observing and measuring these defects, but the technique we introduce in this work is more simple and flexible and can be easy installed in production line for quality controlling purposes. The presented deflectometer functions in the following way. The imaging system of the device forms an image of large-scale low frequency linear periodical pattern painted on a vertical plane on a transmission grating. The distance between the object grating (painted pattern) and the second grating is so that the frequencies of the image grating and the second (probe) grating are practically the same. There is a small angle between the lines of the latter gratings to form moire fringes. A CCD camera transfers the moire fringes to a PC. By applying phase shifting technique, which is realized by shifting the probe grating in definite steps in its plane in a direction perpendicular to its lines, the phase distribution due to imprecations in optical system and gratings, are specified, At this stage the defected sheet glass is installed between the object grating and the imaging system and the distorted moire pattern is processed as described before. The difference between two-phase distributions is the phase distribution caused by the defects in the glass, which can be easily converted into the required ray deflections. All the processes are carried out automatically in a time less than 30 seconds and the accuracy of the measurement is of the order of 20 arc seconds.

Specification of vibrational modes and amplitudes of structures are crucial issues in civil and mechanical engineering. Several techniques have been used for this kind of studies, including holographic interferometry, speckle interferometry and moire technique. But, for large-scale structures modal analysis technique is usually used. In this work we have used time averaging moire technique to study in plane vibrations of large structures. The study includes specification of vibrating modes and amplitudes of structures. The technique is applied by painting a suitable size linear sinusoidal reflectance pattern on the lateral surface of the structure. As the structure is put into vibration, using a wide-angle high-resolution digital camera, the image of vibrating pattern is recorded in an exposure time much larger than the vibration period. The visibilities of the image along a line parallel to the painted pattern line are derived by processing the reflectance distribution. By dividing the resulted visibilities by the visibility of the image of the static pattern we get the normalized visibility curve. The number of normalized visibilities equal to 1 provides the number of vibrational modes and the magnitudes of the visibility minima or the locations of the zero visibility give the amplitudes of vibration.

Using a novel set up, experimental study of the statistical properties of a light beam propagating horizontally through indoor convective air turbulence in the presence of a 2D temperature gradient (TG) is presented. A laser beam enters a telescope from its back focal point by virtue of an optical fiber and is expanded and recollimated by it and then passes through the turbulent area. Then the beam enters another telescope’s aperture. A mask consisting of four similar widely separated small subapertures was installed in front of the second telescope’s aperture. The subapertures were equidistant from the optical axis of the telescope and located at the corners of a square. A flat plane heater is used to produce a vertical TG in the medium. Due to the limited width of the heater, a horizontal component for the TG appeared. Near the focal plane of the second telescope, four distinct images of the source are formed and recorded by a CCD camera. Due to the turbulence all the images (spots) in the successive frames fluctuate. Using the four spot displacements we have calculated the fluctuations of the angle of arrival (AA) over the subapertures. The statistical properties of the optical turbulence are investigated using variance analysis of the AA component fluctuations at horizontal and vertical directions over the subapertures for different temperatures of the heater at different heights of the beam path from the heater. Experimental results show that when the heater is turned off, the variances of horizontal and vertical components of the AA fluctuations are approximately equal to zero over all the subapertures. When it is turned on, the variance of the horizontal component of the AA fluctuations over all of the subapertures are larger than those from the vertical one. In addition, in this case, we find a significant dependence of the variance of the AA components on the height from the heater.

Recently, an adjustable, high-sensitivity, wide dynamic range, two-channel wavefront sensor based on moire deflectometry was proposed by Rasouli et al (2010 Opt. Express 18 23906). In this work we have used this sensor on a telescope for measuring turbulence-induced wavefront distortions. A slightly divergent laser beam passes through turbulent ground level atmosphere and enters the telescope’s aperture. The laser beam is collimated behind the telescope’s focal point by means of a collimator and the beam enters the wavefront sensor. First, from deviations in the moire fringes we calculate the two orthogonal components of the angle of arrival at each location across the wavefront. The deviations have been deduced in successive frames which allows evolution of the wavefront shape and Fried’s seeing parameter r0 to be determined. Mainly, statistical analysis of the reconstructed wavefront distortions are presented. The achieved accuracy in the measurements and comparison between the measurements and the theoretical models are presented. Owing to the use of the sensor on a telescope, and using sub-pixel accuracy for the measurement of the moire fringe displacements, the sensitivity of the measurements is improved by more than one order of magnitude. In this work we have achieved a minimum measurable angle of arrival fluctuations equal to 3.7 × 10−7 rad or 0.07 arc s. Besides, because of the large area of the telescope’s aperture, a high spatial resolution is achieved in detecting the spatial perturbations of the atmospheric turbulence.

Several mathematical approaches have been used to explore the moire pattern. All of them have been considered for superimposing the static structures. In this paper, we have presented a theoretical approach to take into account the relative motion in superimposed gratings and its effect on the moire fringe patterns. We have used a reciprocal vector approach and consequently obtained a comprehensive description of the dynamic behavior of the moire patterns in a cinematic viewpoint. Formulations of the rotation and parallel moire patterns of superposition of static and dynamic periodic structures have been derived in an unified form. Besides, some applications of moire technique that have been already carried out in dynamical phenomena will be briefly reviewed.

In this paper, we have presented a simple, stable, highly sensitive and timesaving method based on a double-grating interferometer in conjunction with a pump–probe technique for measuring the nonlinear refractive index. A pump laser beam is aligned collinearly with an expanded plane parallel probe beam by a dichroic mirror. These beams pass through the sample, while right behind the sample using a suitable bandpass filter the pump beam is intercepted. The distorted probe beam then passes through a double-grating interferometer. One of the lateral shearing interference patterns is recorded by use of a CCD camera and, after digitization, has been stored in a computer. The interference pattern is analyzed by means of a Fourier transform algorithm. The refractive index changes have been obtained from phase distribution of the recorded fringe patterns. The implementation of the technique is straightforward and the arrangement is very simple and stable yet its sensitivity is comparable with other interferometry methods. It is also not a time consuming method. The method is applied for measuring the thermal nonlinear refractive index n2 of colloidal gold nanoparticles in water solution.

An instrument has been built for the study of the atmospheric turbulence by measuring the fluctuation of the angle of arrival across a telescope aperture using moiré deflectometry. A slightly divergent laser beam passes through a turbulent ground level atmosphere and enters the telescope aperture. The laser beam is recollimated behind the telescope's focal point by means of a collimator. The collimated beam passes through a moiré deflectometer. The fluctuating self-image of the first grating is formed on the second grating of the moiré deflectometer and fluctuating moiré fringes are formed. Using moiré fringe fluctuations we have calculated the fluctuations of the angle of arrival, the Fried's parameter r(0), and the atmospheric refractive index structure constant. Because of the magnifications of the telescope and moiré deflectometry, the precision of the technique can potentially be 1 order of magnitude more precise than previous methods.

There are several methods for measuring ground-level atmospheric turbulence parameters, such as the refractive index profile and its fluctuations, correlations of the fluctuations in space and time, and the atmospheric refractive-index structure constant. These methods are based mainly on the measurement of fluctuations in intensity and location of an image formed by light propagating in the turbulent atmosphere or the fluctuations in impinging points of narrow light beams traversing the ground-level atmosphere. Exploiting the moiré technique, we suggest a high-precision approach for determining fluctuaions in the angle of arrival. When a low-frequency grating (carrier grating) is installed at a suitable distance from a telescope, its image, practically, forms on the focal plane of the telescope objective. Superimposing a physical grating (probe grating) of the same pitch as the image grating on the image forms the moiré pattern. The atmospheric turbulence distorts the image grating. Processing the fluctuations of successive moiré fringes can yield the mentioned parameters across a rather large cross section of the atmosphere with high accuracy, because of the moiré technique's magnifying character and access to a large volume of data, and does so in a comparatively simple and reliable manner.