Search

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, the far-field propagation of multi-vortex beams is investigated. We consider diffraction of a Gaussian wave from a spatial light modulator (SLM) in which a multi-fork grating is implemented on it at the waist plane of the Gaussian wave. In the first-order diffraction pattern a multi-vortex beam is produced, and we consider its evolution under propagation when different multi-fork gratings are implemented on the SLM. We consider two different schemes for the phase singularities of the implemented grating. A topological charge (TC) equal to l 1 is considered at the center of the grating, and four similar phase singularities all having a TC equal to l 2 = l 1 4 (or l 2 = − l 1 4 ) are located on the corners of a square where the l 1 singularity is located on the square center. Some cases with different values of l 1 , and consequently l 2 , are investigated. Experimental and simulation results show that if signs of the TCs at the corners and center of the square are the same, the radius of the central singularity on the first-order diffracted beam increases, and it convolves the other singularities. If their signs are opposite, the total TC value equals zero, and at the far-field, the light beam distribution becomes a Gaussian beam. For determining the TCs of the resulting far-field beams, we interfere experimentally and by simulation the resulting far-field beams with a plane wave and count the forked interference fringes. All the results are consistent.

We report on applications of moiré deflectometry in measurements of the anisotropy and scaling of the phase structure function (PSF), obtained after passing a laser beam through an indoor enclosure containing convective air turbulence. We combine the use of two telescopes, with a two-channel wavefront sensor based on moiré deflectometry, to attain high sensitivity and resolution to fluctuations in the wavefront phase, caused by turbulent fluctuations in the enclosure. The measurements of the wavefront PSF along two directions perpendicular to the direction of the light beam propagation at different heater temperatures show that the convective air turbulence is anisotropic turbulence, where the value of the anisotropy increases with increasing temperature gradient. Various models are fitted to the measured PSFs, and we find that the turbulent is also non-Kolmogorov, in which, for the separation distances of two points on the wavefront less than 10 cm, the von Kármán PSF is the best fit to the experimental data. For higher values of separations, the experimental data do not fit with existing models. By fitting the von Kármán PSF on the data, we estimate values of the refractive index structure constant, C n 2 , as well as the outer scale of the turbulence. The value of the outer scale decreases with increasing temperature of the heater up to approximately 50°C, where it saturates, while the value of C n 2 monotonically increases. Over the complete range of heater temperatures, from 40°C to 160°C, the Rayleigh number, Ra, for the enclosed air flow varied from 5.80 × 10 8 < R a < 5.89 × 10 9 so that all measurements were conducted in a state of developed convective turbulence.

This Letter reports a new, to the best of our knowledge, technique for the quality testing of steep optical samples by introducing an absolute interferometry method based on a double-slit interference experiment. We determine the quality of the sample with an ultrahigh-dynamic-range wavefront sensor by determining the deformation of the central fringe of the double-slit interferometer recorded for two different separations of the slits. The transmission function of the double slit is implemented on an amplitude spatial light modulator. Therefore, the slits’ location can be easily displaced over the entire area of the sample’s wavefront. We applied the proposed method on two samples: a microscope slide and a conventional ophthalmic lens, and maximum absolute phase variations of 0.33 and 26.7 rad were measured, respectively. Our estimation shows that an absolute phase variation of about 700 rad can be measured with this method.

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.

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.

In this paper, the anisotropy of optical convective underwater turbulence is investigated in terms of the variance of angle of arrival (AOA) fluctuations of a narrow laser beam propagating through it in different sections of the medium. The collimated laser beam with a wavelength 532 nm and a diameter 1 cm, which passes through a convective underwater turbulence. The turbulence is generated in a water tank with dimensions of 20 cm × 36 cm × 20 cm, which is installed on a flat surface electrical heater. During the experiments, the mean water temperature (MWT) can be changed from room temperature to 34 ∘C by increasing the heater temperature. The use of the heater also generates a temperature gradient in the medium. The laser beam propagates along a horizontal path with a length of 20 cm inside the tank at different altitudes from the heater source, as well as at different distances from one of the side walls of the turbulent medium. After passing the laser beam through the turbulent medium, the fluctuations of the AOA components in the vertical and horizontal directions are measured. From the time series of the measured AOA fluctuations, their variances are determined. The anisotropy of the medium is investigated by comparing the variance of AOA components measured in the vertical and horizontal directions. We show that the variances of both of vertical and horizontal components of the AOA fluctuations are increased with the MWT, and they are saturated at higher MWTs. In addition, different anisotropic behaviors are observed for the variances of the measured AOA fluctuations at the vicinity of the lateral wall and upper surface of the water. At the vicinity of the lateral wall the variances of the AOA fluctuations in the horizontal component are larger, but at the vicinity of the upper surface the variances of the AOA fluctuations in the vertical component are dominant. This behavior may be caused by the change of the convection motion direction in the turbulent fluid.

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.

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.

In this work, we investigate and compare propagation of Laguerre–Gaussian (LG) and the recently introduced radial carpet (RC) beams through an indoor convective air turbulence under weak to strong turbulence conditions. By determining the radial displacement of the intensity maxima over the donut ring of the LG beam, and tracing the radial coordinates of the main intensity spots of the RC beam, when the beams are propagating through the turbulence, we estimate the variance of the local displacements for both of the beams in the radial direction. It is shown that, under the same turbulence conditions, the RC beam experiences less disturbance and is more resilient to turbulence, especially when it has a complicated structure. The measured values of the image displacement variance for the RC beam are almost one order of magnitude smaller than the same parameter for the LG beam. In addition, for the LG beam, the symmetry of the beam is broken and in many places the intensity ring is even cut off. Although there are slight changes in the RC beams transverse intensity distribution, the number of the main intensity spots rarely changes. This feature makes the RC beam a good candidate for the transmission of information. We also show that a set of RC beams having different values of the main intensity spots (generated with different radial gratings having different spoke numbers) can be used as an orthogonal bases for free-space optical communication.

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.

The paper’s high-resolution figures and an additional figure that presents a schematic representation of the use of Talbot interferometer as a wavefront sensor.

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, measurement of thermally induced nonlinear refractive index of meso-tetraphenylporphyrin (H2TPP) at different concentrations in 1,2-dicoloroethane using a double-grating interferomete...

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 paper, anisotropy and inhomogeneity of an indoor convective air turbulence are studied using two-channel moire deflectometry by statistical analyzing of the angle of arrival (AA) fluctuations over about 1300 points on the wave front passing through the turbulent area. The beam of a laser source is expanded and collimated with a diameter of 18 cm by a telescopic system consisting of an aspheric lens and a telescope. This beam propagates horizontally through a turbulent area. A heater is used for generating a convection air turbulence. At the end of the turbulent medium, the distorted wave front enters the aperture of another telescope. By the aid of another telescopic system, the diameter of the beam is reduced to a value of 2 cm, and the beam enters to a two-channel moire deflectometer. This device is able to measure AA of the wave front being fluctuated due to the turbulence. By statistical analyzing of the AA fluctuations in two perpendicular directions on the transverse plane, the homogeneity and isotropy of the turbulent medium are investigated at different distances from the heater in the presence of different temperature gradients. Results show that the temperature gradient and the distance from the heat source have considerable impacts on the inhomogeneity and anisotropy of the convective air turbulence. With increasing the distance from the heat source and decreasing the temperature gradient, the homogeneity increases, while by increasing the temperature gradient and in high distances from the heater, it becomes more isotropic. Finally, we report a way for estimating the depth of boundary layer.

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.

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, 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.