Fresnel Diffraction Algorithm Research Articles

Applying multi-focus filtering plane in double-step Fresnel diffraction method for enhanced holographic projection region

Computer-generated holography has been anticipated in augmented reality (AR) since it can fully provide multi-depth 3D information to users, but speckle noises are often accompanied as one of the most significant degradation factors. Since complex amplitude encoding can suppress speckle noises, applying double-phase encoding to a lensless holographic projection system using a double-step Fresnel diffraction (DSF) algorithm has been reported to reduce those noises. However, a single filtering focus at a virtual focal plane might significantly reduce the projection region compared to the maximum diffraction bandwidth of the spatial light modulator. In this paper, we applied multiple foci for the focal plane of the DSF method with off-axis phase encoding, which can expand the projection region and is applicable to display multi-depth images with DSF. Numerical analysis conducted for image quality compared with conventional single focus DSF algorithm shows that the speckle performance does not degrade so much as the number of foci increase. We expect the proposed method to be useful for holographic projection optical systems such as AR head-up display.

Method of color holographic display with speckle noise suppression.

In this paper, a method of color holographic display with speckle noise suppression is proposed. Firstly, the intensity information of the object is extracted according to the red, green and blue (RGB) channels. The band-limited phase is calculated and used as the initial phase for each color channel. Secondly, the double-step Fresnel diffraction algorithm is used to calculate the computer-generated holograms (CGHs), and a filter plane that dynamically adjusts the position of the filter in the optical path is designed. Then, a divergent spherical phase factor is added to the CGHs. Finally, the time average method is used to further reduce the speckle noise. When the CGHs of the RGB channels are loaded on the digital micromirror device and illuminated by the RGB lights emitting in a temporal sequence, the color reconstructed images with speckle noise suppression can be displayed. The validity of the proposed method is verified.

37‐3: Student Paper: Method of Color Amplitude‐Only Hologram Generation for Speckle Noise Suppression

In this paper, a calculation method of color amplitude‐only computer‐generated hologram (CGH) that can suppress speckle noise is proposed. Firstly, the amplitude information of the red, green and blue channels of the object is extracted. Secondly, to reduce the speckle noise, the convolution result of the amplitude information with a constrained random phase is used as the complex amplitude distribution of the object. Then, the multi‐frame amplitude‐only CGHs for each color channel of the object are generated by using the modified double‐step Fresnel diffraction algorithm. Finally, the digital micromirror device is used to refresh the multi‐frame amplitude‐only CGHs, and the speckle noise is suppressed by using the time‐averaging method. The validity of the computational method is verified.

Curved hologram generation method for speckle noise suppression based on the stochastic gradient descent algorithm

In this paper, a curved hologram generation method with suppressed speckle noise is proposed. In the process of generating the curved hologram, the angle spectrum method is used to calculate the 3D object in layers. By analyzing the loss function relationship between the diffraction image of the curved hologram and the target light field, the loss function is calculated. The phase of the hologram is updated based on the stochastic gradient descent algorithm, thereby obtaining the optimal phase distribution of the curved hologram. When the curved hologram is reproduced at different bending center angles, the reconstructed image of the corresponding angle can be seen. The proposed method has obvious advantages compared with the error diffusion method and Fresnel diffraction algorithm. The feasibility of the proposed method is verified by the experiments.

Effective method for further magnifying the image in holographic projection under divergent light illumination.

Divergent light illumination can effectively increase the diffraction angle of holographic projection. However, the achieved maximum image size is limited to the geometric enlargement of the hologram size with the existing double-sampling Fresnel diffraction algorithm. In this paper, an effective method for further magnifying the image is proposed. The theoretical maximum size under divergent illumination is first analyzed. On this basis, a virtual intermediate plane is introduced between the source plane and the image plane. Then three-step diffraction calculation is performed to evaluate the Fresnel diffraction between the hologram plane and the image plane so that the sampling interval on the image plane is related to the position of the virtual plane and can be flexibly tuned. Consequently, the achieved image size can be further enlarged. The feasibility of the proposed method is demonstrated by simulations and experiments.

A conversion method of two-dimensional image into three-dimensional holographic display

A method to convert two-dimensional(2D) image into three-dimensional(3D) holography display is proposed. The conjectured phase distribution is directly obtained by a polynomial equation based on the 2D image, and is superimposed over the object plane. The parameters of polynomial equation are automatically calculated by the gray value of 2D image. A special kinoform is designed, which contains the phase information generated by Fresnel diffraction algorithm. It can effectively suppress the speckle noise. The sequential kinoforms are reconstructed by the 3D dynamic holographic display system with phase-only liquid crystal spatial light modulator (LC-SLM) and the cylindrical fog screen. The results of simulation and optic experiment exhibit that the proposed method can reconstruct 3D scenes with a better vision effect. The display results show more advantages, such as high contrast, clear detail and low noise. The method presented in this paper can be applied to high performance 3D dynamic holographic display.

Calculation of computer-generated hologram (CGH) from 3D object of arbitrary size and viewing angle

We propose a method to calculate computer-generated hologram (CGH) from 3D object of arbitrary size and viewing angle. The spectrum relation between a 3D voxel object and the CGH wavefront is established. The CGH is generated via diffraction calculation from 3D voxel object based on the development of a scaled Fresnel diffraction algorithm. This CGH calculation method overcomes the limitations on sampling imposed by conventional Fourier transform based algorithm. The calculation and reconstruction of 3D object with arbitrary size and viewing angle is achieved. Both of simulation and optical experiments proves the validation of the proposed method.

54‐3: Lensless Holographic 3D Display based on Fast Calculated Computer‐Generated hologram

A method of lensless holographic 3D display is proposed based on the fast calculated computer‐generated hologram (CGH). The CGH is calculated by using a virtual nonuniform sampled wavefront recording plane (NS‐WRP) which is close to the 3D object. The calculation process consists of two steps: first, the 3D object is calculated to the NS‐WRP using nonuniform fast Fourier transform (NUFFT) based algorithm. Second, the NS‐WRP is propagated to the CGH using Fresnel diffraction algorithm. This method can provide fast and simple calculation of CGH which can reconstruct high quality 3D object with perceptual depth cue in space. Both of simulation and experiment demonstrate the proposed method.

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.

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.

Numerical study for the calculation of computer-generated hologram in color holographic 3D projection enabled by modified wavefront recording plane method

A method of calculating computer-generated hologram (CGH) for color holographic 3D projection is proposed. A color 3D object is decomposed into red, green and blue components. For each color component, a virtual wavefront recording plane (WRP) is established which is nonuniformly sampled according to the depth map of the 3D object. The hologram of each color component is calculated from the nonuniform sampled WRP using the shifted Fresnel diffraction algorithm. Finally three holograms of RGB components are encoded into one single CGH based on the multiplexing encoding method. The computational cost of CGH generation is reduced by converting diffraction calculation from huge 3D voxels to three 2D planar images. Numerical experimental results show that the CGH generated by our method is capable to project zoomable color 3D object with clear quality.

Simulation of Anderson localization in a random fiber using a fast Fresnel diffraction algorithm

Anderson localization has been previously demonstrated both theoretically and experimentally for transmission of a Gaussian beam through long distances in an optical fiber consisting of a random array of smaller fibers, each having either a higher or lower refractive index. However, the computational times were extremely long. We show how to simulate these results using a fast Fresnel diffraction algorithm. In each iteration of this approach, the light passes through a phase mask, undergoes Fresnel diffraction over a small distance, and then passes through the same phase mask. We also show results where we use a binary amplitude mask at the input that selectively illuminates either the higher or the lower index fibers. Additionally, we examine imaging of various sized objects through these fibers. In all cases, our results are consistent with other computational methods and experimental results, but with a much reduced computational time.

Design of refractive/diffractive hybrid optical elements for beam shaping with large diffraction pattern

Diffractive optics is an important technique for beam shaping with high light efficiency and strong diffraction pattern flexibility. Since the diffraction angle is limited by the unit size of the diffractive optical element (DOE), the size of the required diffraction pattern is always rather small. In this Letter, refractive/diffractive hybrid optical elements (RDHOEs) consisting of a DOE and a lens are used to realize beam shaping for a large diffraction pattern. The lens, as the component of the RDHOEs, can not only be concave but also convex, and the double sampling Fresnel diffraction algorithm is developed for the design of these two types of RDHOEs. The simulation and experimental results provide solid evidence to demonstrate the proposed method with the pure phase spatial light modulator.

Image magnification in lensless holographic projection using double-sampling Fresnel diffraction.

Since the diffraction angle is limited by the spatial resolution of the spatial light modulator (SLM), the size of the optical image of the lensless holographic projection with a SLM is very small. Using a divergent spherical beam to illuminate a SLM is an effective method to physically increase the projection angle; nevertheless, the sampling ranges of the existing Fresnel diffraction algorithms with fast Fourier transform keep unchanged. In this paper, a double-sampling Fresnel diffraction algorithm to enlarge the sampling range is proposed when using a divergent spherical beam to illuminate a SLM, and the magnification of the optical image is realized in lensless holographic projection. The hologram can be easily optimized by the Gerschberg-Saxton algorithm. Simulation and experimental results with enlarged optical image are presented successfully.

Experimental implementation of a virtual optical beam propagator system based on a Fresnel diffraction algorithm

Abstract. Wedescribetheexperimentalimplementationofavirtualopticalbeampropagatorsystem.Thisvirtualpropagator system allows the experimental study of beam propagation without physically moving any element.The approach uses a Fresnel diffraction algorithm (usually called the angular spectrum method) and its imple-mentation on a spatial light modulator. We discuss the limits of the technique and provide a detailed descriptionof the experimental procedures. Experimental results are included where we design a hologram capable of pro-ducingtwopatternsattwodifferentdistances,andwecanchangetheeffectiveplaneofobservationbychangingthe encodedpropagationinsteadof bymovinganyelement on the experimentalsystem. ©2015SocietyofPhoto-OpticalInstrumentation Engineers (SPIE) [DOI: 10.1117/1.OE.54.10.103101] Keywords: Fresnel propagation; angular spectrum method; spatial light modulators.Paper 150322 received Mar. 12, 2015; accepted for publication Aug. 31, 2015; published online Oct. 5, 2015. 1 IntroductionWe discuss the experimental implementation of a virtualoptical beam propagator system for encoding the Fresneldiffraction that a wavefront encounters as it propagates.The approach uses a Fresnel diffraction algorithm (usuallycalled the angular spectrum method or convolution method)encoded onto a spatial light modulator (SLM). In this way,the SLM can be used not only to implement a given opticalelement but also to implement the diffracted pattern thatis created at a certain distance. This encoding has beenextremely valuable in many applications.While parts of this idea have been known for some time,we analyze here the limits imposed by the rasterized struc-ture of the SLM and give the details for its experimentalimplementation. We expect that this discussion will be valu-able for new applications of this procedure.The idea of experimentally encoding the Fresnel-dif-fracted version of an optical element using an SLM wasfirst used (to our knowledge) in the design of a space-variantFresnel-transform optical correlator.

Dynamic holographic three-dimensional projection based on liquid crystal spatial light modulator and cylindrical fog screen

A dynamic holographic three-dimensional (3D) projection based on phase-only liquid crystal spatial light modulator (LC-SLM) and cylindrical fog 3D screen is introduced. Sequential kinoforms of a 3D real existing object are calculated from sixty viewing angles using the slice-based fresnel diffraction algorithm. To suppress speckle noise of reconstructed images, sub-kinoforms for each viewing angle are calculated by adding dynamic pseudorandom initial phase factor into each object plane. The sequential kinoforms are reconstructed by a holographic reconstruction system based on phase-only LC-SLM. A specially designed cylindrical fog 3D screen is used as the scattered carrying medium to project the dynamic 3D images. Through our holographic 3D projection system, a vivid dynamic holographic reconstructed projection image can be observed by some observers at the same time.

Holographic display based on liquid crystal spatial light modulator

In conventional phase-only holographic display, the phase-only computer generated hologram is usually calculated based on the fast Fourier transform (FFT) algorithm, in which the Nyquist theory should be satisfied. However, due to the pixel structure of the liquid crystal spatial light modulator and a fixed spatial sampling rate, the size of the reconstructed image is limited by the space-bandwidth product of the liquid crystal phase modulator. The traditional solution is to use convolution algorithm or double-step Fresnel diffraction algorithm to calculate the Fresnel hologram, but FFT has to be calculated many times in both of the methods, thereby increasing the burden of hologram computation. Therefore, in this paper we propose a method to calculate the phase-only hologram based on setting a virtual hologram plane. This virtual hologram plane is set based on the principle of lens imaging. So the calculation of the hologram can be divided into two steps: the first step is to calculate the Fresnel diffraction from the object plane to the virtual hologram plane, and the second step is to calculate the hologram from the virtual hologram plane by being multiplied with a quadratic phase term. In this way, the hologram can be calculated from the original object with any sampling rate we need by adjusting the corresponding parameters of distance. By this method one can calculate the Fresnel diffraction between hologram plane and object plane with variable sampling rates, without considering the space-bandwidth product of the liquid crystal phase modulator, and this algorithm uses only one FFT calculation, which can speed up the calculation of hologram compared with the convolution based method (using three FFTs in calculation) and the double-step Fresnel method (using two FFTs in calculation). Both the computer simulation and the optical experiments demonstrate that the object can be reconstructed with different sizes in the holographic display system. In the optical experiment, the zero-order diffraction can be removed by placing a filter on the back focal plane of the imaging lens and the speckle noise can also be eliminated in order to improve the reconstruction quality by displaying multiple phase-only holograms at a high speed. The proposed method in this paper shows a potential application in zoom-able liquid crystal spatial light modulator based holographic display system.

Analysis of the propagation dynamics and Gouy phase of Airy beams using the fast Fresnel transform algorithm.

There is great interest in Airy beams because they appear to propagate in a curved path. These beams are usually generated by inserting a cubic phase mask onto the input plane of a Fourier transform system. Here, we utilize a fast Fresnel diffraction algorithm to easily derive both the propagation dynamics and the Gouy phase shift for these beams. The trajectories of these beams can be modified by adding additional linear and quadratic phase terms onto the cubic phase mask. Finally, we have rewritten the equations regarding the propagating Airy beams completely in laboratory coordinates for use by experimentalists. Experimental results are included. We expect that these results will be of great importance in applications of Airy beams.

Fresnel diffraction method with object wave rotation for numerical reconstruction of digital hologram on tilted plane

We present a new method for numerically reconstructing holography image on tilted planes within the Fresnel domain. The method is based on the Fresnel diffraction algorithm and object wave rotation transform. By suitable rotation the object wave on recording plane and once fast Fourier transformation this method can be implemented. The proposed method can render the real image of the hologram not only on parallel plane but also on the tilted. Experimental results are presented to demonstrate the method for a single axis rotation.

Coaxial waveguide mode reconstruction and analysis with THz digital holography

Terahertz (THz) digital holography is employed to investigate the properties of waveguides. By using a THz digital holographic imaging system, the propagation modes of a metallic coaxial waveguide are measured and the mode patterns are restored with the inverse Fresnel diffraction algorithm. The experimental results show that the THz propagation mode inside the waveguide is a combination of four modes TE₁₁, TE₁₂, TM₁₁, and TM₁₂, which are in good agreement with the simulation results. In this work, THz digital holography presents its strong potential as a platform for waveguide mode charactering. The experimental findings provide a valuable reference for the design of THz waveguides.