Gerchberg-Saxton Research Articles

Single-Shot Full-Field Characterization of Short Pulses by Using Temporal Annealing Modified Gerchberg–Saxton Algorithm

The dispersive Fourier transform is suitable for characterizing short pulses. However, the traditional temporal Gerchberg–Saxton (GS) algorithm suffers from the timing error of measured dispersed waveforms, which limits the measurement performance of the temporal phase. An annealing modified GS algorithm that can simultaneously retrieve the temporal profile and phase is proposed. The inevitable timing error in the measurement can be accurately recovered by the algorithm, which significantly improves the retrieval performance. Based on the annealing modified GS algorithm, an experimental structure is proposed to achieve single-shot full-field measurement. The algorithm is analyzed numerically and the temporal waveform and phase of short pulses can be retrieved successfully in the experiment. The results indicate that this single-shot full-field measurement method is free from timing error, which is important for real applications. The simple single-shot method promises future applications to measure the temporal profile and temporal phase of short pulses with high accuracy simultaneously.

Speeding up image quality improvement in random phase-free holograms using ringing artifact characteristics.

A holographic projector utilizes holography techniques. However, there are several barriers to realizing holographic projections. One is deterioration of hologram image quality caused by speckle noise and ringing artifacts. The combination of the random phase-free method and the Gerchberg-Saxton (GS) algorithm has improved the image quality of holograms. However, the GS algorithm requires significant computation time. We propose faster methods for image quality improvement of random phase-free holograms using the characteristics of ringing artifacts.

Gerchberg–Saxton algorithm based phase correction in optical wireless communication

Optical wireless communication (OWC) enables to establish the backhaul in the fifth generation (5G) wireless communication networks. The air turbulence, however, could distort the phases of optical signals, thus limit transmission capacity. In this paper, we study the correction of phase distortions by using the Gerchberg–Saxton (GS) algorithm. The air turbulence-induced phase is generated by the Monte-Carlo phase screen method, which characterizes the realistic air turbulence effect on the optical signals. The numerical results reveal that the GS algorithm is able to retrieve the phase information with fast convergence by adopting a proper initial condition. Also, the GS algorithm based phase correction in OWC is confirmed.

Optical double image security using random phase fractional Fourier domain encoding and phase-retrieval algorithm

We propose a novel security scheme based on the double random phase fractional domain encoding (DRPE) and modified Gerchberg-Saxton (G-S) phase retrieval algorithm for securing two images simultaneously. Any one of the images to be encrypted is converted into a phase-only image using modified G-S algorithm and this function is used as a key for encrypting another image. The original images are retrieved employing the concept of known-plaintext attack and following the DRPE decryption steps with all correct keys. The proposed scheme is also used for encryption of two color images with the help of convolution theorem and phase-truncated fractional Fourier transform. With some modification, the scheme is extended for simultaneous encryption of gray-scale and color images. As a proof-of-concept, simulation results have been presented for securing two gray-scale images, two color images, and simultaneous gray-scale and color images.

Phase-only stereoscopic hologram calculation based on Gerchberg–Saxton iterative algorithm**Project supported by the National Basic Research Program of China (Grant No. 2013CB328803) and the National High Technology Research and Development Program of China (Grant Nos. 2013AA013904 and 2015AA016301).

A phase-only computer-generated holography (CGH) calculation method for stereoscopic holography is proposed in this paper. The two-dimensional (2D) perspective projection views of the three-dimensional (3D) object are generated by the computer graphics rendering techniques. Based on these views, a phase-only hologram is calculated by using the Gerchberg–Saxton (GS) iterative algorithm. Comparing with the non-iterative algorithm in the conventional stereoscopic holography, the proposed method improves the holographic image quality, especially for the phase-only hologram encoded from the complex distribution. Both simulation and optical experiment results demonstrate that our proposed method can give higher quality reconstruction comparing with the traditional method.

Pre-turbulence compensation of orbital angular momentum beams based on a probe and the Gerchberg-Saxton algorithm.

We propose a scheme that uses a probe Gaussian beam and the Gerchberg-Saxton (GS) algorithm to realize the pre-turbulence compensation of beams carrying orbital angular momentum (OAM). In the experiment, spatial light modulators are utilized to simulate the turbulent atmosphere and upload the retrieval holograms. A probe Gaussian beam is used to detect the turbulence. Then, the retrieval holograms, which can correct the phase distortion of the OAM beams, are obtained by the GS algorithm. The experimental results show that single or multiplexed OAM beams can be compensated well. The compensation performances under different iterations are also analyzed.

19-4: Stereoscopic Hologram Calculation based on Gerchberg-Saxton (GS) Algorithm

We propose a phase-only CGH calculation method for stereoscopic holography. 2-D perspective projection views of the 3-D object are firstly generated. Based on these views, phaseonly hologram for hogel is calculated by using GS iterative algorithm instead of non-iterative algorithm in conventional stereoscopic holography. Both simulation and optical experiment results demonstrate that our proposed method can perform higher quality reconstruction comparing with the traditional method.

A hybrid Gerchberg–Saxton-like algorithm for DOE and CGH calculation

The Gerchberg–Saxton (GS) algorithm is widely used in various disciplines of modern sciences and technologies where phase retrieval is required. However, this legendary algorithm most likely stagnates after a few iterations. Many efforts have been taken to improve this situation. Here we propose to introduce the strategy of gradient descent and weighting technique to the GS algorithm, and demonstrate it using two examples: design of a diffractive optical element (DOE) to achieve off-axis illumination in lithographic tools, and design of a computer generated hologram (CGH) for holographic display. Both numerical simulation and optical experiments are carried out for demonstration.

The Matrix Completion Method for Phase Retrieval from Fractional Fourier Transform Magnitudes

Inspired by the implementation of the fractional Fourier transform (FRFT) and its applications in optics, we address the problem of reconstructing a signal from its several FRFT magnitudes (or intensities). The matrix completion method is adopted here. Through numerical tests, the matrix completion method is proven effective in both noisy and noise-free situations. We also compare our method with the Gerchberg-Saxton (GS) algorithm based on FRFT. Numerical tests show that the matrix completion method gains a certain advantage in recovering uniqueness and convergence over the GS algorithm in the noise-free case. Furthermore, in terms of noisy signals, the matrix completion method performs robustly and adding more measurements can generally increase accuracy of recovered signals.

Broadband Gerchberg-Saxton algorithm for freeform diffractive spectral filter design.

A multi-wavelength expansion of the Gerchberg-Saxton (GS) algorithm is developed to design and optimize a surface relief Diffractive Optical Element (DOE). The DOE simultaneously diffracts distinct wavelength bands into separate target regions. A description of the algorithm is provided, and parameters that affect filter performance are examined. Performance is based on the spectral power collected within specified regions on a receiver plane. The modified GS algorithm is used to design spectrum splitting optics for CdSe and Si photovoltaic (PV) cells. The DOE has average optical efficiency of 87.5% over the spectral bands of interest (400-710 nm and 710-1100 nm). Simulated PV conversion efficiency is 37.7%, which is 29.3% higher than the efficiency of the better performing PV cell without spectrum splitting optics.

Speckle-suppressed phase-only holographic three-dimensional display based on double-constraint Gerchberg-Saxton algorithm.

The Gerchberg-Saxton (GS) algorithm is widely used to calculate the phase-only computer-generated hologram (CGH) for holographic three-dimensional (3D) display. However, speckle noise exists in the reconstruction of the CGH due to the uncontrolled phase distribution. In this paper, we propose a method to suppress the speckle noise by simultaneously reconstructing the desired amplitude and phase distribution. The phase-only CGH is calculated by using a double-constraint GS algorithm, in which both the desired amplitude and phase information are constrained in the image plane in each iteration. The calculated phase-only CGH can reconstruct the 3D object on multiple planes with a desired amplitude distribution and uniform phase distribution. Thus the speckle noise caused by the phase fluctuation between adjacent pixels is suppressed. Both simulations and experiments are presented to demonstrate the effective speckle noise suppression by our algorithm.

Overcoming the limitation of phase retrieval using Gerchberg-Saxton-like algorithm in optical fiber time-stretch systems.

We investigate the fundamental limitation of the full-field retrieval of optical pulses based on a time-equivalent Gerchberg-Saxton (GS)-like algorithm, in which the Fourier transformation of the temporal signal is performed by the group velocity dispersion (GVD) of optical fibers. The insufficient knowledge of the input pulse intensity fundamentally limited by the bandwidth of the photodetector influences the accuracy of the retrieved phase using the algorithm. To this end, we propose a modified GS algorithm, which involves multi-step intensity-only measurements, to mitigate this limitation. By studying the influence of spectral phase and amplitude modulations on the retrieval error, we show that the modified GS algorithm demonstrates a significant improvement in phase retrieval accuracy. More importantly, we anticipate the present study will provide valuable insight in understanding how the spectral variation generally influences the recovery accuracy in time-stretch-based GS algorithm, and thus identify its potential applications, particularly for those requiring ultrafast measurements.

Iterative phase-retrieval method for generating stereo array of polarization-controlled focal spots.

This Letter introduces an iterative phase-retrieval method based on the Gerchberg-Saxton (G-S) algorithm for generating any arbitrary 3D pattern in image space, while simultaneously controlling the polarization orientation at each pixel. For proof-of-principle, we generate a stereo focal spot array with distinct polarization orientation for each spot. This method is universal for controlling the output polarization; the only requirement is that the input polarization should be spatially inhomogeneous. This work has the potential to impact coherent imaging techniques and spectroscopy.

Ghost imaging with non-negative exponential speckle patterns

The speckle patterns with non-negative exponential statistics are generated through introducing an iterative algorithm: the Gerchberg-Saxton (GS) algorithm. Furthermore, ghost imaging (GI) with the non-negative exponential speckle patterns is investigated, and the effect of the non-negative exponential speckle patterns on the image quality is analyzed.

30.4: Multi‐Plane Holographic Display with a Uniform 3D Gerchberg‐Saxton Algorithm

A modified three‐dimensional (3D) Gerchberg‐Saxton (GS) algorithm for phase‐only holograms in holographic displays is proposed. Numerical and experimental results indicate that compared with the traditional 3D GS algorithm, image‐quality difference in the proposed method is reduced by four orders of magnitude, while the average image‐quality is increased by 28.7%.

Iterative phase retrieval algorithms. I: optimization.

Two modified Gerchberg-Saxton (GS) iterative phase retrieval algorithms are proposed. The first we refer to as the spatial phase perturbation GS algorithm (SPP GSA). The second is a combined GS hybrid input-output algorithm (GS/HIOA). In this paper (Part I), it is demonstrated that the SPP GS and GS/HIO algorithms are both much better at avoiding stagnation during phase retrieval, allowing them to successfully locate superior solutions compared with either the GS or the HIO algorithms. The performances of the SPP GS and GS/HIO algorithms are also compared. Then, the error reduction (ER) algorithm is combined with the HIO algorithm (ER/HIOA) to retrieve the input object image and the phase, given only some knowledge of its extent and the amplitude in the Fourier domain. In Part II, the algorithms developed here are applied to carry out known plaintext and ciphertext attacks on amplitude encoding and phase encoding double random phase encryption systems. Significantly, ER/HIOA is then used to carry out a ciphertext-only attack on AE DRPE systems.

Three-dimensional polarization marked multiple-QR code encryption by optimizing a single vectorial beam

We demonstrate the feasibility of three dimensional (3D) polarization multiplexing by optimizing a single vectorial beam using a multiple-signal window multiple-plane (MSW-MP) phase retrieval algorithm. Original messages represented with multiple quick response (QR) codes are first partitioned into a series of subblocks. Then, each subblock is marked with a specific polarization state and randomly distributed in 3D space with both longitudinal and transversal adjustable freedoms. A generalized 3D polarization mapping protocol is established to generate a 3D polarization key. Finally, multiple-QR code is encrypted into one phase only mask and one polarization only mask based on the modified Gerchberg–Saxton (GS) algorithm. We take the polarization mask as the cyphertext and the phase only mask as additional dimension of key. Only when both the phase key and 3D polarization key are correct, original messages can be recovered. We verify our proposal with both simulation and experiment evidences.

Design and Analysis of Flat-Top Beam Produced by Diffractive Optical Element

The modified Gerchberg-Saxton (G-S) methods of diffractive optical elements are presented for converting a Gaussian beam into a flat-top beam. The sample points of the input and output planes are discussed based on the sampling principle. The relationship between weighting factor, empirical constant, diffractive efficiency, and intensity uniformity is investigated. The simulation results of different modified methods are calculated when the incident beams are the same. The results show that the intensity uniformity of the output flat-top beam is the best when using the ST modified method.

Detecting concentrations of milk components by an iterative optical technique.

This paper introduces a theoretical and practical model for reconstructing the scattering properties of a participating media. Our theory is based on a robust generalization of the Gerchberg-Saxton (G-S) algorithm. At the end of this algorithm the reduced scattering coefficient μ's of a given substance, can be estimated from the standard deviation (STD) of the retrieved phase of the remitted light. We use the theory to compute the phase's STD that directly correlated to the optical properties for different types of milk components, and we derive a novel appearance model for milk parameterized by the lactose and protein contents. Our results show that we are able to detect the possibility of lactose and milk proteins' quantitative signature by the G-S optical tool, en route to the design of a novel milk-content-monitoring tool. Sketch of the experimental setup for light intensity measurements and reduced scattering coefficient reconstruction. The samples were prepared from various milk components: whey protein, sodium casienate and lactose, at different concentrations.

Detecting nanoparticles in tissue using an optical iterative technique.

Determining the physical penetration depth of nanoparticles (NPs) into tissues is a challenge that many researchers have been facing in recent years. This paper presents a new noninvasive method for detecting NPs in tissue using an optical iterative technique based on the Gerchberg-Saxton (G-S) algorithm. At the end of this algorithm the reduced scattering coefficient (µs'), of a given substance, can be estimated from the standard deviation (STD) of the retrieved phase of the remitted light. Presented in this paper are the results of a tissue simulation which indicate a linear ratio between the STD and the scattering components. A linear ratio was also observed in the tissue-like phantoms and in ex vivo experiments with and without NPs (Gold nanorods and nano Methylene Blue). The proposed technique is the first step towards determining the physical penetration depth of NPs.