Gerchberg-Saxton Research Articles

Experimental analysis of adaptive optics correction methods on the beam carrying orbital angular momentum mode through oceanic turbulence

In recent years, as the demand for underwater communication has increased, underwater wireless optical communication (UWOC) has attracted a lot of attentions. Meanwhile, orbital angular momentum (OAM) has been applied in UWOC system to increase communication link capacity. However, the aberrations caused by oceanic turbulence (OT) is unavoidable, which results in serious intermodal crosstalk. Usually, adaptive optics (AO) is used to compensate these distortion aberrations. In this work, we experimentally demonstrate the influence of the distortion caused by OT, and evaluate the performance of AO correction algorithms, including Shack-Hartmann (SH), Stochastic-Parallel-Gradient-Descent (SPGD) and Gerchberg-Saxton (GS) algorithms. During the experiment, OT is simulated by using improved random phase screen model. We discuss the influence of various parameters of OT, including relative strength of temperature and salinity, strength of OT Cn2, and propagation distance, on the compensation effect with and without correction algorithms. The results show that all the three AO algorithms have good compensation effect on the distortion caused by OT, and GS algorithm has a better capability than the other two algorithms. Additionally, when the iteration numbers is less than 100, the compensation effect of GS and SPGD algorithms becomes more obvious as the iteration numbers increases, and GS algorithm is superior to SPGD algorithm. This work is beneficial to aberration correction of OAM-based UWOC system.

Intelligent coding metasurface holograms by physics-assisted unsupervised generative adversarial network

Intelligent coding metasurface is a kind of information-carrying metasurface that can manipulate electromagnetic waves and associate digital information simultaneously in a smart way. One of its widely explored applications is to develop advanced schemes of dynamic holographic imaging. By now, the controlling coding sequences of the metasurface are usually designed by performing iterative approaches, including the Gerchberg–Saxton (GS) algorithm and stochastic optimization algorithm, which set a large barrier on the deployment of the intelligent coding metasurface in many practical scenarios with strong demands on high efficiency and capability. Here, we propose an efficient non-iterative algorithm for designing intelligent coding metasurface holograms in the context of unsupervised conditional generative adversarial networks (cGANs), which is referred to as physics-driven variational auto-encoder (VAE) cGAN (VAE-cGAN). Sharply different from the conventional cGAN with a harsh requirement on a large amount of manual-marked training data, the proposed VAE-cGAN behaves in a physics-driving way and thus can fundamentally remove the difficulties in the conventional cGAN. Specifically, the physical operation mechanism between the electric-field distribution and metasurface is introduced to model the VAE decoding module of the developed VAE-cGAN. Selected simulation and experimental results have been provided to demonstrate the state-of-the-art reliability and high efficiency of our VAE-cGAN. It could be faithfully expected that smart holograms could be developed by deploying our VAE-cGAN on neural network chips, finding more valuable applications in communication, microscopy, and so on.

Data-Aided Iterative Algorithms for Linearizing IM/DD Optical Transmission Systems

DSP-enhanced intensity-modulation direct-detection (IM/DD) systems can support up to 56 Gb/s over 100 km signal transmissions at C-band. To achieve higher data rates and longer transmission distances, we propose data-aided iterative algorithm (DIA) and decision-directed DIA (DD-DIA) to digitally mitigate signal-signal beating interference (SSBI) without requiring any modifications to physical layer structures. DIA utilizes pilot symbols with uniformly spaced insertions to relax the modified Gerchberg-Saxton (G-S) algorithms that suffer from the local optimum problem. To further improve the symbol error rate (SER) performance and convergence speed, DD-DIA introduces a decision process to generate pseudo-pilots. We numerically compare DIA with other algorithms and find that DIA can recover signals subject to large fiber dispersions corresponding to which conventional IA and Volterra filter (VF) fail, while DD-DIA significantly accelerates the convergence speed and improves the reconstruction performance compared with DIA, it can support 100-Gb/s PAM4 over 400-km IM/DD transmissions within just 50 iterations. Two orders of magnitude reductions in SER is observed for 100 Gb/s PAM4 signal transmission over 100-km SSMFs. Compared with conventional IA, the proposed techniques have higher convergence speeds, better global optimum features and large tolerances to physical model errors. In particular, DD-DIA results in larger optical signal-to-noise ratio (OSNR)/ received optical power (ROP) improvement, higher transmission capacities and less computational complexity. DD-DIA is a promising algorithm for efficiently reconstructing 2-dimensional optical field for conventional IM/DD optical transmission systems.

Adaptive weighted Gerchberg-Saxton algorithm for generation of phase-only hologram with artifacts suppression.

In the conventional weighted Gerchberg-Saxton (GS) algorithm, the feedback is used to accelerate the convergence. However, it will lead to the iteration divergence. To solve this issue, an adaptive weighted GS algorithm is proposed in this paper. By replacing the conventional feedback with our designed feedback, the convergence can be ensured in the proposed method. Compared with the traditional GS iteration method, the proposed method improves the peak signal-noise ratio of the reconstructed image with 4.8 dB on average. Moreover, an approximate quadratic phase is proposed to suppress the artifacts in optical reconstruction. Therefore, a high-quality image can be reconstructed without the artifacts in our designed Argument Reality device. Both numerical simulations and optical experiments have validated the effectiveness of the proposed method.

Generalizing the Gerchberg–Saxton algorithm for retrieving complex optical transmission matrices

The Gerchberg–Saxton (GS) algorithm, which retrieves phase information from the measured intensities on two related planes (the source plane and the target plane), has been widely adopted in a variety of applications when holographic methods are challenging to be implemented. In this work, we showed that the GS algorithm can be generalized to retrieve the unknown propagating function that connects these two planes. As a proof-of-concept, we employed the generalized GS (GGS) algorithm to retrieve the optical transmission matrix (TM) of a complex medium through the measured intensity distributions on the target plane. Numerical studies indicate that the GGS algorithm can efficiently retrieve the optical TM while maintaining accuracy. With the same training data set, the computational time cost by the GGS algorithm is orders of magnitude less than that consumed by other non-holographic methods reported in the literature. Besides numerical investigations, we also experimentally demonstrated retrieving the optical TMs of a stack of ground glasses and a 1-m-long multimode fiber using the GGS algorithm. The accuracy of the retrieved TM was evaluated by synthesizing high-quality single foci and multiple foci on the target plane through these complex media. These results indicate that the GGS algorithm can handle a large TM with high efficiency, showing great promise in a variety of applications in optics.

Modified Gerchberg–Saxton (G-S) Algorithm and Its Application

The Gerchberg–Saxton (G-S) algorithm is a phase retrieval algorithm that is widely used in beam shaping and optical information processing. However, the G-S algorithm has difficulty obtaining the exact solution after iterating, and an approximate solution is often obtained. In this paper, we propose a series of modified G-S algorithms based on the Fresnel transform domain, including the single-phase retrieval (SPR) algorithm, the double-phase retrieval (DPR) algorithm, and the multiple-phase retrieval (MPR) algorithm. The analysis results show that the convergence of the SPR algorithm is better than that of the G-S algorithm, but the exact solution is not obtained. The DPR and MPR algorithms have good convergence and can obtain exact solutions; that is, the information is recovered losslessly. We discuss the security advantages and verification reliability of the proposed algorithms in image encryption. A multiple-image encryption scheme is proposed, in which n plaintexts can be recovered from n ciphertexts, which greatly improves the efficiency of the system. Finally, the proposed algorithms are compared with the current phase retrieval algorithms, and future applications are discussed. We hope that our research can provide new ideas for the application of the G-S algorithm.

Design of beam shaping and focusing metasurface device based on G-S algorithm

We demonstrate a sub-diffraction flat-top beam shaping device based on sub-wavelength square silicon pillar array at operating wavelength of 10.6 μm, followed with improved Gerchberg-Saxton (G-S) algorithm to enable the advanced shaping processing capability. The theoretical simulation results show that the device generates a sub-diffraction flat-top optical needle with length of 10.6 cm along the optic axis, and its diffraction efficiency is higher than 90%. Moreover, the focal spot, with a transverse size of 3.5 λ, is smaller than the diffraction limit of 6.8 λ. This sub-diffraction beam shaping device with large focal depth has practical applications in laser machining technologies.

A Hybrid Diffractive Optical Element Design Algorithm Combining Particle Swarm Optimization and a Simulated Annealing Algorithm

With the rapid development of computer hardware and the emergence of the parallel calculation of diffraction fields, a breakthrough has been made in terms of the limitation of the unacceptable amount of computational cost to design diffractive optical elements (DOEs), and more accurate global search algorithms can be introduced to the design of complex DOEs and holographic projections instead of traditional iterative algorithms. In this paper, a hybrid algorithm which combines particle swarm optimization (PSO) with a simulated annealing (SA) algorithm is proposed for the designing of DOEs and projecting holographic images with less noise. PSO is used to reduce the invalid disturbance in SA, and SA can jump out from local extreme points to find the global extreme points. Compared with the traditional Gerchberg–Saxton (GS) algorithm, the simulation and experimental results demonstrate that the proposed SA–PSO hybrid algorithm can improve uniformity by more than 10%.

Optimization of Phase-Only Computer-Generated Holograms Based on the Gradient Descent Method

The Gerchberg–Saxton (GS) algorithm is a Fourier iterative algorithm that can effectively optimize phase-only computer-generated holograms (CGHs). This study proposes a new optimization technique for phase-only CGHs based on the gradient descent method. The proposed technique evaluates the intensity distributions of reconstructed images to directly obtain the phase distributions of the CGHs, whereas the GS algorithm equivalently evaluates the amplitude distributions of reconstructed images and extracts phase distributions from complex-amplitude distributions of the holograms using a constant-amplitude constraint. The proposed technique can reduce the errors in the reconstructed images with fewer iterations than the GS algorithm.

Small-phase retrieval using single focused far-field intensity

We develop a method to retrieve small wave-front aberrations from a single far-field intensity. The solution specifies the even and odd parts of the unknown phase T in two separate equations. Both the odd part and the even part require only one focus intensity measurement with a given phase diversity produced by the deformable mirror or active surface system. We take the solution, the approximation of T, zero and random phase distributions as the initial guesses of GS (Gerchberg & Saxton) algorithm respectively, the final recovered TR is chosen from the one with the minimum amplitude Root Mean Squared Error (RMSE). Numerical simulation results show that the method works well when the root mean squared (RMS) of the phase error is less than 3.14 rad, and it is effective when the signal-to-noise ratio(SNR)>10. Phase ambiguity is solved by moving the deformable mirror twice according to the twin images TR and TR’ TR*, and the image with higher peak intensity is selected as the final estimation of T.

Noniterative multiplane holographic projection

In this paper, we introduce a mixed complex and phase-only constraint for noniterative computer generation of phase-only holograms from multiplane intensity distributions. We are able to reproduce three-dimensional intensity distributions with the same number of planes achieved with the Gerchberg-Saxton (GS) algorithm; at the same time, we maintain the fast computation time of a noniterative method. In this way, we enable the possibility of multiplane light field control in dynamic applications. We show numerical results for three- and eight-plane holograms, for different interplane distances-using either the same or different amplitude constraints in each plane. In all of these tests, our method results in a comparable or better reconstruction quality than the GS algorithm, while achieving a significant decrease in computing time. Finally, we experimentally demonstrate the capability of our proposal to achieve multiplane holographic projection.

Multiview holographic 3D display based on blazed Fresnel DOE

Glasses-free three-dimensional (3D) displays are regarded as a promising technology for the next-generation display industry. Multiview 3D displays based on diffractive optics can present high resolution and low crosstalk 3D images at a wide viewing angle. However, the limited viewing angle in the vertical direction and low diffraction efficiency hinder the development of the technology. In this work, a multiview holographic 3D display based on a phase-only blazed Fresnel-type diffractive optical element (FDOE) is proposed. The core part of the 3D display is a view modulator that is fully covered with an aperiodic arranged transmission FDOE array, and it serves to redistribute the diverging rays and to shape the vertical extended views. The corresponding FDOE for each view is optimized by a modified Gerchberg–Saxton (GS) algorithm. Optical experiments demonstrate that the proposed FDOE can reconstruct converged views and virtual 3D scenes with an extended field of view (FOV) of 17.8° in the vertical direction. This work may find applications in mobile electronics and head-mounted 3D displays.

Error tracking-control-reduction algorithm for designing diffractive optical element with high image reconstruction quality.

The use of the diffractive optical element (DOE) can often significantly reduce the size and enhance the performance of the optical system, but it is mostly prevented by low diffraction efficiency and serious speckle noise due to the quantization error. In this paper, an error tracking-control-reduction (ETCR) algorithm is proposed to suppress the quantization error, which adjusts the accumulative action, controls the current state and predicts the trend of the error. The simulation and experiment results verify that the ETCR algorithm has high diffraction efficiency which can be comparable with the Gerchberg-Saxton (GS) and Modified GS algorithms. Furthermore, the root-mean-square error (RMSE) of the proposed algorithm is significantly lower than that of the GS and MGS algorithms. Based on the proposed method, a 2-level DOE has been designed and fabricated to generate several grey images with only 0.05 RMSE.

Dual Polarization Full-Field Signal Waveform Reconstruction Using Intensity Only Measurements for Coherent Communications

Conventional optical coherent receivers capture the full electrical field, including amplitude and phase, of a signal waveform by measuring its interference against a stable continuous-wave local oscillator (LO). In optical coherent communications, powerful digital signal processing (DSP) techniques operating on the full electrical field can effectively undo transmission impairments such as chromatic dispersion (CD), and polarization mode dispersion (PMD). Simpler direct detection techniques do not have access to the full electrical field and therefore lack the ability to compensate for these impairments. We present a full-field measurement technique using only direct detection that does not require any beating with a strong carrier LO. Rather, phase retrieval algorithms based on alternating projections that makes use of dispersive elements are discussed, allowing to recover the optical phase from intensity-only measurements. In this demonstration, the phase retrieval algorithm is a modified Gerchberg Saxton (GS) algorithm that achieves a simulated optical signal-to-noise ratio (OSNR) penalty of less than 4dB compared to theory at a bit-error rate of 2 times 10-2. Based on the proposed phase retrieval scheme, we experimentally demonstrate signal detection and subsequent standard 2x2 multiple-input-multiple-output (MIMO) equalization of a polarization-multiplexed 30-Gbaud QPSK transmitted over a 520-km standard single-mode fiber (SMF) span.

Phase retrieval with fast convergence employing parallel alternative projections and phase reset for coherent communications.

Phase-retrieval (PR) receivers can reconstruct complex-valued signals using only direct detection without the use of any optical carriers. We propose and demonstrate two PR receiver solutions with faster and better convergence. First, we demonstrate a PR receiver based on parallel alternative projections that are produced by propagating the signal through an array of dispersive elements of increasing length followed by direct detection. Fast convergence and high retrieved phase accuracy are achieved using a modified Gerchberg-Saxton (GS) algorithm that uses each projection as an intensity constraint. Second, we achieve similar performances employing an enhanced single projection GS algorithm with selective phase reset using symbol-wise GS errors. We experimentally reconstruct a 30 Gbaud QPSK signal after 55 km single-mode fiber transmission using the proposed solutions with a reduced number of iterations.

Three-Dimensional Holographic Reconstruction of Brain Tissue Based on Convolution Propagation

In this study, a dynamic holographic projection system for brain tissue and its anatomical structures extracted from Magnetic Resonance (MR) plane slice is reported. Computer holograms are calculated using a modified Gerchberg-Saxton (GS) iterative algorithm where the projection is based on the plane wave decomposition. First, brain anatomy includes white matter (WM), grey matter (GM) and brain tissue are extracted. Then, phase holograms using the proposed method are generated. Finally, single-phase hologram for the whole brain anatomy is generated and is optically reconstructed by a phase-only spatial light modulator (SLM) at different depths. The obtained results revealed that the three-dimensional holographic projection of MR brain tissue can aid to provide better interpretation of brain anatomical structure to achieve better diagnostic results.

Method to suppress speckle noise using time multiplexing in phase‐only holographic display

AbstractIn holographic display, the reconstructed image suffers from speckle noise severely. In this paper, we propose a method to suppress speckle noise using time multiplexing in phase‐only holographic display. Adjacent pixels of the recorded object are separated into object point groups firstly. Particularly, the pixel interval of each object point group is larger compared with the conventional pixel separation method. And then, sub‐computer–generated holograms (sub‐CGHs) are calculated by the modified Gerchberg–Saxton (GS) algorithm with different initial random phases. Finally, the final integrated image is reconstructed with low speckle noise using time multiplexing technique. Both numerical and optical experimental results are presented to demonstrate the effectiveness and feasibility with our proposed method.

Iterative phase retrieval for digital holography: tutorial.

This paper provides a tutorial of iterative phase retrieval algorithms based on the Gerchberg-Saxton (GS) algorithm applied in digital holography. In addition, a novel GS-based algorithm that allows reconstruction of 3D samples is demonstrated. The GS-based algorithms recover a complex-valued wavefront using wavefront back-and-forth propagation between two planes with constraints superimposed in these two planes. Iterative phase retrieval allows quantitatively correct and twin-image-free reconstructions of object amplitude and phase distributions from its in-line hologram. The present work derives the quantitative criteria on how many holograms are required to reconstruct a complex-valued object distribution, be it a 2D or 3D sample. It is shown that for a sample that can be approximated as a 2D sample, a single-shot in-line hologram is sufficient to reconstruct the absorption and phase distributions of the sample. Previously, the GS-based algorithms have been successfully employed to reconstruct samples that are limited to a 2D plane. However, realistic physical objects always have some finite thickness and therefore are 3D rather than 2D objects. This study demonstrates that 3D samples, including 3D phase objects, can be reconstructed from two or more holograms. It is shown that in principle, two holograms are sufficient to recover the entire wavefront diffracted by a 3D sample distribution. In this method, the reconstruction is performed by applying iterative phase retrieval between the planes where intensity was measured. The recovered complex-valued wavefront is then propagated back to the sample planes, thus reconstructing the 3D distribution of the sample. This method can be applied for 3D samples such as 3D distribution of particles, thick biological samples, and other 3D phase objects. Examples of reconstructions of 3D objects, including phase objects, are provided. Resolution enhancement obtained by iterative extrapolation of holograms is also discussed.

Iterative Algorithm for Electronic Dispersion Compensation in IM/DD Systems

A novel iterative algorithm for electronic dispersion compensation (EDC) at the transmitter is described and demonstrated for intensity modulation and direct detection (IM/DD) systems. The proposed algorithm is based on the Gerchberg-Saxton (GS) algorithm, where the link dispersion performs time-to-frequency mapping, while the unconstrained phase at the direct detection receiver acts as a degree of freedom. Simulation results for a transmitter employing a single drive Mach-Zehnder modulator (MZM) operating at 28 Gb/s non-return-to-zero on-off keying (NRZ-OOK), show pre-compensation of upto 100 km of single mode fiber (SMF), with 1 dBQ margin above the forward error correction (FEC) bit error rate (BER) limit of $3.8 \times 10^{-3}$ . It is shown that with five iterations of the proposed EDC algorithm in digital-signal processing (DSP) coupled with an 8-bit 56 GSa/s digital-to-analog converter (DAC), the transmission reach is extended by a factor of 5 in unamplified SMF links. Simulations for a 56 Gb/s four-level pulse-amplitude-modulation (PAM-4) show the compensation of upto 10 km with five iterations of the EDC algorithm at 1 dBQ above the FEC limit.

OAM mode recognition based on joint scheme of combining the Gerchberg–Saxton (GS) algorithm and convolutional neural network (CNN)

Optical vortex carrying orbital angular momentum (OAM) has attracted a lot of attentions in the field of free-space optical (FSO) communication. Generally, after transmitting in atmospheric turbulence, the helical phase-front of Laguerre-Gaussian (LG) beam carrying OAMs will be severely distorted, thus result in intermodal crosstalk, which is a critical challenge to the effective recognition of the OAM modes. In this paper, we have proposed a joint scheme of combining Gerchberg–Saxton (GS) algorithm and convolutional neural network (CNN) (GS-CNN) to achieve the efficient recognition of the multiplexing LG beams under turbulence environment. The feasibility of the scheme is verified by investigating the recognition performances under various turbulence-strength levels and mode spacings. Moreover, we also discuss the performances of recognition LG beam with different radial indices. The numerical simulation shows that after initial restoration of the beams’ intensity distribution by the GS algorithm, the following CNN can effectively extract the features of retrieved intensity distribution and achieve higher accuracy on recognizing multiplexing LG beams in severe turbulence environment. Besides, the good performance can also be achieved in recognizing LG beams with different radial indices. The results demonstrate great potentials of the GS-CNN for the implementation of actual OAM-based communication system.