Fresnel Diffraction Research Articles

Fresnel diffraction from a step in the general case.

Recently, Fresnel diffraction from phase steps with parallel plates has been studied in detail, and the subject has led to many interesting metrological applications. In this report we formulate Fresnel diffraction from a physical step with arbitrarily oriented plates in reflection mode. We simulate the diffraction patterns for different orientations of the plates and develop the required procedure for determining the involved angles by analysis of the diffraction pattern. In the experimental part of the report we arrange a setup to form diffraction patterns in different orientations of the step plates and test the derived formulations. Also, we briefly review the application potentials of the subject.

Theoretical conversion of the hypergeometric-Gaussian beams family into a high-order spiraling Bessel beams by a curved fork-shaped hologram

Based on the process of the Fresnel diffraction, the possibility of generating a new type of laser beams family by illuminating a curved fork-shaped hologram, with an input hypergeometric-Gaussian beams family of orders n and m is studied in this paper. The theoretical and the numerical results showed that, at the output plane, a high order spiraling Bessel vortex beam is produced. This vortex beam is divergence or non-divergence depending upon the waist position of the input hypergeometric-Gaussian beams, regarding the plane where the curved fork-shaped hologram is situated. Analytical expressions of the amplitude and the intensity distribution of the diffracted wave field are calculated and deduced using the stationary phase method. The actual work generalizes also the Fresnel diffraction study of some subfamilies of the hypergeometric-Gaussian beams family, such as: fundamental Gaussian, hollow Gaussian, modified quadratic Bessel–Gaussian and elegant Laguerre–Gaussian beams.

Image reconstruction in an electro-holographic display

The optical phenomena arising in the process of forming reconstructed images in a hologram are explained and shown visually with the use of light field images. The light fields at different distances from the hologram on a DMD reveal that the reconstructed image of each object point is formed by the corresponding Fresnel zone pattern, which is reconstructed from the hologram when it is illuminated by a reconstruction laser beam. The reconstructed image is a circle of least confusion laden with noise and distortion. It has a finite size and does not appear at the object distance from the hologram due to the presence of aberrations, especially that of a strong astigmatism. The astigmatism appears along the direction of the rotating axis of each pixel and its cross at right angles. The aberrations and noise are responsible for the distortion and deterioration of the resolution in the reconstructed image, the difference of the image position from that of the object, and a reduction in the depth resolution. The light field images also reveal intensity fluctuations due to the addition of the in- and out-phase of the rays from the hologram.

Displacements and evolution of optical vortices in edge-diffracted Laguerre–Gaussian beams

Based on the Kirchhoff–Fresnel approximation, we consider the behavior of optical vortices (OV) upon propagation of diffracted Laguerre–Gaussian (LG) beams with topological charge ∣m∣ = 1, 2. Under conditions of weak diffraction perturbation (i.e. the diffraction obstacle covers only the far transverse periphery of the incident LG beam), these OVs describe almost perfect 3D spirals within the diffracted beam body, which is an impressive demonstration of the helical nature of an OV beam. The far-field OV positions within the diffracted beam cross section depend on the wavefront curvature of the incident OV beam, so that the input wavefront curvature is transformed into the output azimuthal OV rotation. The results are expected to be useful in OV metrology and OV beam diagnostics.

Emergence of Fresnel diffraction zones in gravitational lensing by a cosmic string

The possibility to detect cosmic strings – topological defects of early Universe, by means of wave effects in gravitational lensing is discussed. To find the optimal observation conditions, we define the hyperbolic-shaped Fresnel observation zones associated with the diffraction maxima and analyse the frequency patterns of wave amplification corresponding to different alignments. In particular, we show that diffraction of gravitational waves by the string may lead to significant amplification at cosmological distances. The wave properties we found are quite different from what one would expect, for instance, from light scattered off a thin wire or slit, since a cosmic string, as a topological defect, gives no shadow at all.

Gouy phase shift of lens-generated quasi-nondiffractive beam

Gouy phase shift (GPS) along geometrical rays of quasi-nondiffractive beam is studied by Fresnel diffraction to understand its phase behavior. Two typical quasi-nondiffractive beams, quasi-Bessel beam generated by axicon and quasi-Airy beam generated by cubic phase mask, were studied. Results show that those two beams have different kinds of GPS. While quasi-Airy beam follows general phenomenon of converging light wave where its phase change across focal line is -π, quasi-Bessel beam has phase change of -π/2 although it is two-dimensional.

High focusing efficiency or high signal-to-noise ratio diffractive optical element for color separation and light focusing

Diffractive optical elements (DOEs) are designed by the simulated annealing method for simultaneously implementing color separation and light focusing (CSLF) functions. Through changing the maximum permitted phase, the quantization level number, the input pixel number or the target focusing region width, various CSLF DOEs are optimally designed. Optical performances of the designed DOEs are calculated by the Fresnel diffraction integral method. Simulation results reveal that the designed DOEs not only successfully realize the expected CSLF functions, but also exhibit an excellent performance of high focusing efficiency or high signal to noise ratio under specific parameter circumstances. It is expected that the designed DOEs should have practical applications in solar cell systems or optical interconnection systems.

Full-color holographic display with increased-viewing-angle [Invited

Among the important features of holographic displays are the wide viewing angles and the full color of the reconstructed images. The present work focuses on achievement of both features. We propose an increased-viewing-angle full-color holographic display using two tiled phase-only spatial light modulators (SLMs), a 4f concave mirrors system, and a temporal-spatial multiplexing method. The 4f optical system consists of two concave mirrors and serves to increase the viewing angle. A temporal-spatial multiplexing synchronization control (TSMSC) method is developed to achieve a full-color image and to remove the color crosstalk of the image. We calculate RGB phase-only holograms of a computer-generated color pyramid by using a slice-based Fresnel diffraction algorithm. The experimental results indicate that the proposed display system is feasible to reconstruct a full-color holographic 3D image with a viewing angle of 12.8°, which is about 3.8 times wider than the viewing angle formed by a single SLM.

Speckle reduced lensless holographic projection from phase-only computer-generated hologram.

This paper presents a method for the implementation of speckle reduced lensless holographic projection based on phase-only computer-generated hologram (CGH). The CGH is calculated from the image by double-step Fresnel diffraction. A virtual convergence light is imposed to the image to ensure the focusing of its wavefront to the virtual plane, which is established between the image and the hologram plane. The speckle noise is reduced due to the reconstruction of the complex amplitude of the image via a lensless optical filtering system. Both simulation and optical experiments are carried out to confirm the feasibility of the proposed method. Furthermore, the size of the projected image can reach to the maximum diffraction bandwidth of the spatial light modulator (SLM) at a given distance. The method is effective for improving the image quality as well as the image size at the same time in compact lensless holographic projection system.

Evaluation of noise limits to improve image processing in soft X-ray projection microscopy.

Soft X-ray microscopy has been developed for high resolution imaging of hydrated biological specimens due to the availability of water window region. In particular, a projection type microscopy has advantages in wide viewing area, easy zooming function and easy extensibility to computed tomography (CT). The blur of projection image due to the Fresnel diffraction of X-rays, which eventually reduces spatial resolution, could be corrected by an iteration procedure, i.e., repetition of Fresnel and inverse Fresnel transformations. However, it was found that the correction is not enough to be effective for all images, especially for images with low contrast. In order to improve the effectiveness of image correction by computer processing, we in this study evaluated the influence of background noise in the iteration procedure through a simulation study. In the study, images of model specimen with known morphology were used as a substitute for the chromosome images, one of the targets of our microscope. Under the condition that artificial noise was distributed on the images randomly, we introduced two different parameters to evaluate noise effects according to each situation where the iteration procedure was not successful, and proposed an upper limit of the noise within which the effective iteration procedure for the chromosome images was possible. The study indicated that applying the new simulation and noise evaluation method was useful for image processing where background noises cannot be ignored compared with specimen images.

On the relative intensity of Poisson’s spot

The Fresnel diffraction phenomenon referred to as Poisson’s spot or spot of Arago has, beside its historical significance, become relevant in a number of fields. Among them are for example fundamental tests of the super-position principle in the transition from quantum to classical physics and the search for extra-solar planets using star shades. Poisson’s spot refers to the positive on-axis wave interference in the shadow of any spherical or circular obstacle. While the spot’s intensity is equal to the undisturbed field in the plane wave picture, its intensity in general depends on a number of factors, namely the size and wavelength of the source, the size and surface corrugation of the diffraction obstacle, and the distances between source, obstacle and detector. The intensity can be calculated by solving the Fresnel–Kirchhoff diffraction integral numerically, which however tends to be computationally expensive. We have therefore devised an analytical model for the on-axis intensity of Poisson’s spot relative to the intensity of the undisturbed wave field and successfully validated it both using a simple light diffraction setup and numerical methods. The model will be useful for optimizing future Poisson-spot matter-wave diffraction experiments and determining under what experimental conditions the spot can be observed.

Compact, cost-effective and field-portable microscope prototype based on MISHELF microscopy

We report on a reduced cost, portable and compact prototype design of lensless holographic microscope with an illumination/detection scheme based on wavelength multiplexing, working with single hologram acquisition and using a fast convergence algorithm for image processing. All together, MISHELF (initials coming from Multi-Illumination Single-Holographic-Exposure Lensless Fresnel) microscopy allows the recording of three Fresnel domain diffraction patterns in a single camera snap-shot incoming from illuminating the sample with three coherent lights at once. Previous implementations have proposed an illumination/detection procedure based on a tuned (illumination wavelengths centered at the maximum sensitivity of the camera detection channels) configuration but here we report on a detuned (non-centered ones) scheme resulting in prototype miniaturization and cost reduction. Thus, MISHELF microscopy in combination with a novel and fast iterative algorithm allows high-resolution (μm range) phase-retrieved (twin image elimination) quantitative phase imaging of dynamic events (video rate recording speed). The performance of this microscope prototype is validated through experiments using both amplitude (USAF resolution test) and complex (live swine sperm cells and flowing microbeads) samples. The proposed method becomes in an alternative instrument improving some capabilities of existing lensless microscopes.

Performance of the hybrid externally occultedLyotsolar coronagraph

Context. High-contrast hybrid coronagraphs, which combine an external occulter and a Lyot-style coronagraph became a reality in recent years, despite the lack of analytic and numerical end-to-end performance studies. The solar coronagraph ASPIICS which will fly on the future ESA Formation Flying mission Proba-3 is a good example of such a hybrid coronograph. Aims. We aim to provide a numerical model to compute theoretical performance of the hybrid externally occulted Lyot-style coronagraph, which we then aim to compare to the performance of the classical Lyot coronagraph and the externally occulted solar coronagraph. We will provide the level and intensity distribution of the stray light, when the Sun is considered as an extended source. We also investigate the effect of different sizes for the internal occulter and Lyot stop. Methods. First, we have built on a recently published approach, to express the diffracted wave front from Fresnel diffraction produced by an external occulter at the entrance aperture of the coronagraph. Second, we computed the coherent propagation of the wave front coming from a given point of the Sun through the instrument. This is performed in three steps: from the aperture to the image of the external occulter, where the internal occulter is set, from this plane to the image of the entrance aperture, where the Lyot stop is set, and from there to the final image plane. Making use of the axis-symmetry, we considered wave fronts originating from one radius of the Sun and we circularly average the intensities. Our numerical computation used the parameters of ASPIICS. Results. The hybrid externally occulted Lyot coronagraph rejects sunlight below 10E-8Mean Solar Brightness from 1,3 solar radius - in the particular configuration of ASPIICS. The Lyot coronagraph effectively complements the external occultation. We show that reducing the Lyot stop allows a clear gain in rejection, being even better than oversizing the internal occulter, that tends to exclude observations very close to the solar limb. As an illustration, we provide a graph that allows us to estimate performance as a function of the internal occulter and Lyot stop sizes. Conclusions. Our work consists of a methodological approach to compute the end-to-end performance for solar coronagraph.

Generation of phase edge singularities by coplanar three-beam interference and their detection.

In recent years singular optics has gained considerable attention in science and technology. Up to now optical vortices (phase point dislocations) have been of main interest. This paper presents the first general analysis of formation of phase edge singularities by coplanar three-beam interference. They can be generated, for example, by three-slit interference or self-imaging in the Fresnel diffraction field of a sinusoidal grating. We derive a general condition for the ratio of amplitudes of interfering beams resulting in phase edge dislocations, lateral separation of dislocations depends on this ratio as well. Analytically derived properties are corroborated by numerical and experimental studies. We develop a simple, robust, common path optical self-imaging configuration aided by a coherent tilted reference wave and spatial filtering. Finally, we propose an automatic fringe pattern analysis technique for detecting phase edge dislocations, based on the continuous wavelet transform. Presented studies open new possibilities for developing grating based sensing techniques for precision metrology of very small phase differences.

Off-axis phase-only holograms of 3D objects using accelerated point-based Fresnel diffraction algorithm

A method for calculating off-axis phase-only holograms of three-dimensional (3D) object using accelerated point-based Fresnel diffraction algorithm (PB-FDA) is proposed. The complex amplitude of the object points on the z-axis in hologram plane is calculated using Fresnel diffraction formula, called principal complex amplitudes (PCAs). The complex amplitudes of those off-axis object points of the same depth can be obtained by 2D shifting of PCAs. In order to improve the calculating speed of the PB-FDA, the convolution operation based on fast Fourier transform (FFT) is used to calculate the holograms rather than using the point-by-point spatial 2D shifting of the PCAs. The shortest recording distance of the PB-FDA is analyzed in order to remove the influence of multiple-order images in reconstructed images. The optimal recording distance of the PB-FDA is also analyzed to improve the quality of reconstructed images. Numerical reconstructions and optical reconstructions with a phase-only spatial light modulator (SLM) show that holographic 3D display is feasible with the proposed algorithm. The proposed PB-FDA can also avoid the influence of the zero-order image introduced by SLM in optical reconstructed images.

Perturbative High Harmonic Wave Front Control.

We pattern the wave front of a high harmonic beam by intersecting the intense driving laser pulse that generates the high harmonic with a weak control pulse. To illustrate the potential of wave-front control, we imprint a Fresnel zone plate pattern on a harmonic beam, causing the harmonics to focus and defocus. The quality of the focus that we achieve is measured using the spectral wave-front optical reconstruction by diffraction method. We will show that it is possible to enhance the peak intensity by orders of magnitude without a physical optical element in the path of the extreme ultraviolet (XUV) beam. Through perturbative wave-front control, XUV beams can be created with a flexibility approaching what technology allows for visible and infrared light.

Propagation factor of electromagnetic concentric rings Schell-model beams in non-Kolmogorov turbulence**Project supported by the National Basic Research Program of China (Grant No. 2013CBA01702), the National Natural Science Foundation of China (Grant Nos. 61377016, 11104049, 10974039, 61575055, and 61575053), the Fundamental Research Funds for the Central Universities, China (Grant No. HIT.BRETIII.201406), and the Program for New Century Excellent Talents in University,

We derive an analytical expression for the propagation factor (known as M2-factor) of electromagnetic concentric rings Schell-model (EM CRSM) beams in non-Kolmogorov turbulence by utilizing the extended Huygens–Fresnel diffraction integral formula and the second-order moments of the Wigner distribution function (WDF). Our results show that the EM CRSM beam has advantage over the scalar CRSM beam for reducing the turbulence-induced degradation under suitable conditions. The EM CRSM beam with multi-rings far-fields in free space is less affected by the turbulence than the one with dark-hollow far-fields or the electromagnetic Gaussian Schell-model (EGSM) beam. The dependence of the M2-factor on the beam parameters and the turbulence are investigated in detail.

Erratum: Spectral transfer from phase to intensity in Fresnel diffraction [Phys. Rev. A 93 , 053834 (2016)

Received 20 December 2016DOI:https://doi.org/10.1103/PhysRevA.95.019901©2017 American Physical SocietyPhysics Subject Headings (PhySH)Research AreasLight propagation, transmission & absorptionOptical coherenceAtomic, Molecular & Optical

Correction of axial distance error in ptychography based on image information entropy

Ptychography provides an extremely robust and highly convergent algorithm to reconstruct the specimen phase with a wide field of view. The resolution and accuracy of ptychography are severely restricted by the uncertainty of the position error that includes the scanning position and axial distance error. In fact, it is difficult to accurately measure the distance between the target plane and entrance pupil of charge-coupled device (CCD) or complementary metal oxide semiconductor, which results in the axial distance error. The axial distance error can blur the reconstructed image, degrade the reconstruction quality and reduce the resolution. In order to analyze the effect of the axial distance error, the model for axial distance error is derived based on the amplitude constraint in CCD and Fresnel diffraction integral. This model indicates that the axial distance error can cause a stretching deformation of the retrieved image, which is similar to the defocusing effect caused by different axial distances in holography. In this paper, we propose a method of correcting the axial distance error by using the image information entropy in an iterative way to obtain the accurate axial distance and retrieve the distinct image. The correction method based on the image information entropy is composed of four parts:the initial calculation, the determination of the direction search, the axial error correction and the reconstruction of the distinct image. The initial calculation part is to ensure that the intensity of the reconstructed object tends to be stable before entering into the other processing parts. The search direction portion is to indicate that the initial axial distance is greater than the actual axial distance, or less than the actual axial distance. The axial error correction section is to calculate the sharpness values of the image at different axial distance, and find the peak position of the sharpness distribution that corresponds to the position of the clearest image. The axial distance can be taken from the peak position. The obtained axial distance is again taken into account in the ptychography algorithm to eliminate the axial distance error and obtain the distinct reconstructed image. In this paper, some simulations are conducted to verify the feasibility of the proposed method. The effect of the axial distance error is analyzed. The image energy variation, the Tamura coefficient and the image information entropy are selected as the image definition evaluation functions in our paper. We compare the distributions of three image definition evaluation functions in the correction process of the axial distance error. It is found that the image information entropy has higher sensitivity than the other image definition evaluation functions. Finally, both simulation and experiment have proved the feasibility of axial distance error correction based on image information entropy.

Design of accelerating beams based on caustic method

Self-accelerating beam is a kind of light beam capable of self-bending in free space without any external potential, of which a typical one is the well-known Airy beam. Such a beam has gained great attention for its extraordinary properties, including nondiffracting, self-accelerating and self-healing, which may have versatile applications in the delivery and guiding of energy, information and objects using light, such as particle manipulation, micro-machining, optical routing, super-resolution imaging, etc. However, since Airy beam can only propagate along parabolic trajectory, which reduces the flexibility in practical applications, thus how to design accelerating beams propagating along arbitrary trajectory is still a crucial problem in this area. One scheme is to keep on finding other analytical solutions of the wave equation besides Airy beam, such as semi-Bessel accelerating beams, Mathius beams, and Weber beams, moving along circular, elliptical, or parabolic trajectories, but it becomes increasingly difficult to find out any more solutions. A more effective solution to this problem is based on the caustic method, which associates the predesigned trajectory with an optical caustics and then obtains the necessary initial field distribution by performing a light-ray analysis of the caustics. This method has been implemented in real space and Fourier space based on Fresnel diffraction integral and angular-spectrum integral, respectively. It has been found recently that they can be unified by constructing Wigner distribution function in phase space. Based on the caustic method, accelerating beams were constructed to propagate along arbitrary convex trajectories in two-dimensional space at first. With continuous development of this method, the types of accelerating beams available have been extending from convex trajectories to nonconvex trajectories, from two-dimensional trajectories to three-dimensional trajectories, and from one main lobe to multiple main lobes, which opens up more possibilities for emerging applications based on accelerating beams. In future, previous researches and applications based on Airy beams will certainly be generalized to all these new types of accelerating beams, and owing to the great flexibility in designing accelerating beams, more application scenarios may emerge in this process with huge development potential. Thus in this paper, we review the principle and progress of the caustic method in designing accelerating beams.