Fourier reciprocity between generalized elliptical Gaussian and elegant elliptical Hermite-Gaussian beams carrying orbital angular momenta.

We explore two distinct families of orbital angular momentum carrying light beams, which we refer to as generalized elliptical Gaussian and elegant elliptical Hermite-Gaussian vortex beams, respectively. We show that the fields of the two vortex families are related via a Fourier transform. Hence, one family can be viewed as a source of the far-field intensity distribution of the other and vice versa. We also examine the orbital angular momentum evolution of both beam families on their free space propagation and establish a relationship between the orbital angular momentum, TC, and beam ellipticity factors. Our results may find applications to optical communications and imaging with structured light.

Voltage-controlled two-dimensional Fresnel diffraction pattern in quantum dot molecules.

This study explores the influence of inter-dot tunneling effects within a quantum dot molecule on the Fresnel diffraction phenomenon. Our findings indicate that the Fresnel diffraction of the output probe Gaussian field can be manipulated by adjusting the inter-dot tunneling parameter's strength and the characteristics of the coupling field. The inter-dot tunneling effect establishes a closed-loop system, setting conditions for the interference of the applied fields. We specifically examine a Laguerre-Gaussian (LG) coupling field, investigating how its properties-such as strength, value, and sign of the orbital angular momentum (OAM)-impact the Fresnel diffraction of the output probe field. Increasing the inter-dot tunneling parameter and the coupling LG field's strength allows for control over the spatial distribution of the Fresnel diffraction pattern. Notably, the inter-dot tunneling parameter can disturb the symmetry of the diffraction patterns. Additionally, considering a negative OAM for the coupling LG field transforms the diffraction pattern into its inverse shape. This suggests that, in the presence of the inter-dot tunneling effect, the Fresnel diffraction pattern is contingent on the direction of rotation of the helical phase front of the coupling LG field. Our results offer insights into quantum control of Fresnel diffraction patterns and the identification of OAM in LG beams, presenting potential applications in quantum technologies.

Gaussian beam diffraction from radial structures: detailed study on the diffraction from sinusoidal amplitude radial gratings.

In this work, we report a comprehensive theoretical investigation on the diffraction of a Gaussian beam from structured radial apertures. In particular, the study of near- and far-field diffraction of a Gaussian beam from an amplitude radial grating having a sinusoidal profile provides new theoretical insights and possible applications. We observe a high self-healing feature at far-field for the Gaussian beam in the diffraction from amplitude radial structures. It is also shown that by increasing the spokes number of the grating, the strength of the self-healing decreases, and reforming of the diffracted pattern into a Gaussian beam occurs at longer propagation distances. The energy flow towards the central lobe of the diffraction pattern and its dependence on the propagation distance are also investigated. In the near-field regime, the diffraction pattern is very similar to the intensity distribution in the central area of the radial carpet beams generated in the diffraction of a plane wave from the same grating. It is shown that by optimally choosing the waist radius of the Gaussian beam, in the near-field regime, it is possible to have a petal-like diffraction pattern, which has been experimentally used in multiple-particle trapping. Compared to radial carpet beams, since in this case there is no energy in the geometric shadow of the radial spokes of the grating, the main part of the power of the incident Gaussian beam is transferred to the main intensity spots of the petal-like pattern, which significantly increases the multi-particle trapping efficiency. We also show that regardless of the grating spokes number, at the far field, the diffraction pattern becomes a Gaussian beam, and its power share reaches 2/3 of the total power passed through the grating.

Transformation of Laguerre-Gaussian beams into 1D array of Hermite-Gaussian modes under the Talbot effect.

This work explains diffraction of Laguerre-Gaussian (LG) beams having non-zero radial indices from one dimensional (1D) periodic structures and their transformation into Hermite-Gaussian (HG) modes, theoretically, verifies using simulations and demonstrates the phenomenon experimentally. We first report a general theoretical formulation for such diffraction schemes, and then use it to investigate the near-field diffraction patterns from a binary grating having a small opening ratio (OR) by providing numerous examples. Results show that for OR≲ 0.1, at the Talbot planes, mainly at the first Talbot image, the images of individual lines of the grating obtain HG modes' intensity patterns. Therefore, the topological charge (TC) of the incident beam and its radial index can be determined from the observed HG mode. In this study, the effects of the OR of the grating and the number of Talbot plane on the quality of the generated 1D array of HG modes are also investigated. The optimum beam radius for a given grating is also determined. The theoretical predictions, are well confirmed by a number of simulations based on the free space transfer function and fast Fourier transform approach, and by experiments. The observed phenomenon, the transformation of LG beams into 1D array of HG modes under the Talbot effect, in addition of providing a way for characterization of LG beams with non-zero radial indices, itself is interesting and may be used in other fields of wave physics, especially for long-wavelength waves.

Theory and generation of heterogeneous 2D arrays of optical vortices by using 2D fork-shaped gratings: topological charge and power sharing management.

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 lx and ly and phase variation amplitudes γx and γy, 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 γx and γy, and it strongly depends on the profile of the grating. Meanwhile the TCs of the generated vortices depend on lx and ly and the corresponding diffraction orders, namely lm,n = -(mlx + nly) 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.

Three-dimensional optical multiple trapping using pure amplitude octagonal almost periodic structures.

We demonstrate a novel method for three-dimensional optical multiple trapping using pure amplitude octagonal almost periodic structures (PAOAPSs). We use a Gaussian beam to diffract through these structures and create a three-dimensional array of trapping spots with the aid of an objective lens. Our device is simple, cost-effective, and easy to fabricate, and it has several advantages over conventional methods for trapping multiple particles. By adjusting the rotation of the PAOAPS and the polarization of the beam, we can simultaneously rotate the trapped particles in both axial and orbital directions. We show that our device achieves an ∼19-fold increase in trapping efficiency compared to a recently introduced method based on an amplitude radial grating. Furthermore, our device transfers about 1/70 of the transmitted beam power to each optical trap, which is much more efficient than a spatial light modulator (SLM).

1D spatially chirped periodic structures: managing their spatial spectrum and investigating their near-field diffraction.

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.

Propagation of a multi-vortex beam: far-field diffraction of a Gaussian beam from a multi-fork phase grating.

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 l1 is considered at the center of the grating, and four similar phase singularities all having a TC equal to l2=l14 (or l2=-l14) are located on the corners of a square where the l1 singularity is located on the square center. Some cases with different values of l1, and consequently l2, 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.

Investigation of the anisotropy and scaling of the phase structure function of a spatially coherent light beam propagating through convective air turbulence.

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, Cn2, 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 Cn2 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 5.80×108

Theoretical study on the diffraction-based generation of a 2D orthogonal lattice of optical beams: physical bases and application for a vortex beam multiplication.

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.

Ultrahigh-dynamic-range wavefront sensor based on absolute double-slit interferometry.

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.

Simple, efficient and reliable characterization of Laguerre-Gaussian beams with non-zero radial indices in diffraction from an amplitude parabolic-line linear grating.

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.

Tunneling-induced Talbot effect.

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.

Investigating the dynamic behavior of thermal distortions of the wavefront in a high-power thin-disk laser using the moiré technique.

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.

Gear-like rotatable optical trapping with radial carpet beams.

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.

Rotation-sensitive and direction-resolved homodyne laser-Doppler vibrometry method for simultaneous measurement of rotational and translational vibrations of a rigid object using a 1D array detector.

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.

Atmospheric coherence time measurement by four-aperture DIMM defocus velocity technique.

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.

Direction-resolved homodyne laser-Doppler vibrometry by analyzing space-time fringes created by the successive 1D intensity profiles of the interference fringes.

In this Letter, we introduce a simple direction-resolved homodyne Laser-Doppler vibrometry method by sewing successive one-dimensional images of the interference pattern recorded by a linear array detector and creating a two-dimensional space-time fringe pattern. A space-time fringe pattern visualizes the vibration form, and it can be used for characterizing the vibration of the object. We measure the vibration of a harmonically driven loudspeaker as a known source to demonstrate the capability of the method. We also employ the method to characterize the vibrational properties of the resonator elements of a thin-disk laser. The method reveals the environmental and instrumental sources of the vibration. The use of an array detector in the detection system simplifies the fringe chasing procedure and optical setup and, by the aid of a space-time image, the vibration waveform is directly determined with no requirement for a time-consuming SPS algorithm.

An azimuthally-modified linear phase grating: Generation of varied radial carpet beams over different diffraction orders with controlled intensity sharing among the generated beams.

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.

Talbot effect of azimuthally periodic Bessel-based structures.

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.