Modified Gerchberg-Saxton Algorithm Research Articles

An adaptive optical technique for structured beam generation based on phase retrieval using modified Gerchberg–Saxton algorithm

Generation of high-purity structured light beams is challenging due to transverse phase aberrations in the optical setup consisting of components such as spatial light modulators (SLMs), lenses, and mirrors. Several phase compensation approaches have been explored in previous work, of which the Gerchberg–Saxton(GS) phase retrieval algorithm has been quite popular. In this paper, we have proposed and experimentally demonstrated a modified GS algorithm based on aperture optimization for phase compensation and compared its performance for a wide range of topological charges using inline, pre-, and post-phase compensation techniques. We find that the post-compensation configuration provides the best performance, and by employing such phase compensation we can achieve a 35% enhancement in modal purity of the generated orbital angular momentum (OAM) beam while reducing the nearest neighbor cross-talk by 9.37 dB.

Real-time hologram generation using a non-iterative modified Gerchberg-Saxton algorithm

Computer-generated holography (CGH) is a technique that aims to produce specific illumination patterns using coherent light. However, traditional CGH algorithms often struggle to balance computational speed with the accuracy of the generated hologram. To address this issue, a non-iterative algorithm named "DL-GSA" is proposed in this paper. DL-GSA combines unsupervised learning in machine learning with convolutional neural networks (CNN) to generate holograms with high accuracy and fixed computational complexity. Simulation experiments reveal that DL-GSA generates hologram patterns faster than the Modified Gerchberg-Saxton algorithm (MGSA) and double-phase retrieval algorithm (DPRA). Furthermore, the average accuracy of the generated holograms is higher than 95 %. These findings suggest that DL-GSA has the potential to significantly enhance the real-time hologram generation capabilities, making it a promising technique for future applications.

Research on multiple-image encryption method using modified Gerchberg–Saxton algorithm and chaotic systems

Research on multiple-image encryption method using modified Gerchberg–Saxton algorithm and chaotic systems

High-fidelity ghost diffraction through complex scattering media using a modified Gerchberg-Saxton algorithm.

In this paper, we propose a modified Gerchberg-Saxton (GS) algorithm to generate random amplitude-only patterns as information carriers in ghost diffraction. With the generated random patterns, high-fidelity ghost diffraction through complex scattering media can be realized with a single-pixel detector. The modified GS algorithm adopts a support constraint in the image plane, which is divided into a target region and a support region. In the Fourier plane, amplitude of the Fourier spectrum is scaled to regulate the sum of the image function. A random amplitude-only pattern can be generated to encode a pixel of the data to be transmitted using the modified GS algorithm. Optical experiments are conducted to verify the proposed method in complex scattering environments, e.g., dynamic and turbid water with non-line-of-sight (NLOS). Experimental results demonstrate that the proposed ghost diffraction is of high fidelity and high robustness against complex scattering media. It is expected that an avenue could be opened up for ghost diffraction and transmission in complex media.

What Binarization Method Is the Best for Amplitude Inline Fresnel Holograms Synthesized for Divergent Beams Using the Direct Search with Random Trajectory Technique?

Fast reconstruction of holographic and diffractive optical elements (DOE) can be implemented by binary digital micromirror devices (DMD). Since micromirrors of the DMD have two positions, the synthesized DOEs must be binary. This work studies the possibility of improving the method of synthesis of amplitude binary inline Fresnel holograms in divergent beams. The method consists of the modified Gerchberg-Saxton algorithm, Otsu binarization and direct search with random trajectory technique. To achieve a better quality of reconstruction, various binarization methods were compared. We performed numerical and optical experiments using the DMD. Holograms of halftone image with size up to 1024 × 1024 pixels were synthesized. It was determined that local and several global threshold methods provide the best quality. Compared to the Otsu binarization used in the original method of the synthesis, the reconstruction quality (MSE and SSIM values) is improved by 46% and the diffraction efficiency is increased by 27%.

Non-interferometric key recording applied to a joint transform cryptosystem.

In this Letter, we propose an opto-digital cryptosystem based on the joint transform correlator architecture without the need for a reference beam, phase-shifting techniques, or an additional window in the input plane. In this system, only two intensity recordings are necessary: the intensity of the key Fourier transform, and the joint power spectrum between the key and an arbitrary object in contact with a random phase mask. Combining them with the knowledge of their respective input modules, we implement a modified Gerchberg-Saxton algorithm to recover the phase associated with the encryption key. The validity of our approach is demonstrated by computer simulations and experimental results.

Design of a single-mode fiber coupling system based on the modified Gerchberg-Saxton algorithm.

The efficiency of a hollow beam received by the Cassegrain antenna coupling into a single-mode fiber is low, and converting the hollow beam into a solid beam can remarkably improve the coupling efficiency. In this paper, shaping diffractive optical elements (DOEs) are designed through a modified Gerchberg-Saxton algorithm (MGS) with Fresnel diffraction. Further, the MGS algorithm can be applicable in the issue of circular symmetric beam shaping. The properties of the system with/without shaping DOEs are analyzed and compared. According to the simulation results, in consideration of the energy loss of the antenna, DOEs, and coupling lens, the total transmission efficiency of the receiving antenna system at 1550nm wavelength can reach 77.81%. In addition, the system with shaping DOEs can better adapt for coupling lenses with different focal lengths, and the variation of the maximum coupling efficiency of the DOEs shaping system at different focal lengths studied in this paper is within 2.00%, which is 6.73% lower than that of the lens shaping system. The research results provide an idea of reverse design for improving a coupling system, which can also provide inspiration for other optical system designs.

Modified Gerchberg-Saxton algorithm-based probe-free wavefront distortion compensation of an OAM beam

In the free space optical communication system, atmospheric turbulence (AT) will distort the helical phase of the orbital angular momentum (OAM) beam, resulting in inter-mode crosstalk and system performance degradation. Based on modified Gerchberg-Saxton (GS) algorithm, this paper proposes a wavefront sensorless adaptive optics system to compensate the wavefront distortion of an OAM beam. For the modified GS algorithm, at each iteration, the spatial phase is re-modulated to descend along a new phase gradient direction to improve algorithm convergence. At the same time, in the spatial domain, negative feedback is generated by nonlinear operation to optimize the beam amplitude, so as to speed up the convergence speed of the algorithm. Simulation results show that our modified GS algorithm can better compensate the wavefront distortion of an OAM beam caused by atmospheric turbulence than the traditional GS and hybrid input–output algorithms, which provides a new idea to solve the problem of atmospheric turbulence in free space optical communication system.

Modified Gerchberg-Saxton Algorithm Based Electrical Dispersion Pre-Compensation for Intensity-modulation and Direct-detection Systems

In intensity-modulation and direct-detection (IM-DD) communication systems, electrical chromatic dispersion (CD) compensation cannot be directly applied in the transceivers due to the absence of phase information. Gerchberg-Saxton (GS) algorithm has been utilized to iteratively solve for real-valued CD pre-compensated signal in an IM-DD system. In this scheme, the amplitude information at the transmitter and phase information before the direct detection receiver are treated as the degree of freedoms (DoFs) and jointly optimized. In this paper, a modified GS algorithm is proposed and experimentally demonstrated for CD pre-compensation in a 28-GBaud PAM-4 system over 80-km single mode fiber (SMF) transmission. Compared with the basic GS electrical dispersion pre-compensation (pre-EDC) scheme, the amplitude information of received samples at the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">T/2</i> position is regarded as DoFs as well, in order to reduce the error. Besides, to accelerate the convergence speed, two error reversing factors associated with the phase error at the transmitter and amplitude error at the receiver are introduced. The experimental results show that the bit error rate (BERs) of 56-Gbps PAM-4 signal over 80-km SMF transmission without CD compensation and with 10 iterations of basic GS pre-EDC scheme are 2.35×10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">−1</sup> and 5.13×10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">−2</sup> , respectively, when a 21-taps feedforward equalizer (FFE) is equipped at the receiver side. While the BER can be reduced to below the hard-decision forward error correction (HD-FEC) threshold of 3.8×10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">−3</sup> by the proposed scheme with the iterative number of 10. The results reveal that the proposed scheme is a promising solution for 50-Gbps class 80-km metro optical communication links based on the IM-DD structure.

Visual Cryptography Using Binary Amplitude-Only Holograms

Visual cryptography (VC) is developed to be a promising approach to encoding secret information using pixel expansion rules. The useful information can be directly rendered based on human vision without the usage of decryption algorithms. However, many VC schemes cannot withstand occlusion attacks. In this paper, a new VC scheme is proposed using binary amplitude-only holograms (AOHs) generated by a modified Gerchberg-Saxton algorithm (MGSA). During the encryption, a secret image is divided into a group of unrecognizable and mutually-unrelated shares, and then the generated shares are further converted to binary AOHs using the MGSA. During image extraction, binary AOHs are logically superimposed to form a stacked hologram, and then the secret image can be extracted from the stacked hologram. Different from conventional VC schemes, the proposed VC scheme converts a secret image into binary AOHs. Due to the redundancy of the generated binary AOHs, the proposed method is numerically and experimentally verified to be feasible and effective, and possesses high robustness against occlusion attacks.

Broadband digital coding metasurface holography

Digital coding metasurfaces composed of subwavelength meta-atoms can flexibly control electromagnetic waves to achieve holography, which has great potential in millimeter-wave imaging systems and data storage. In this paper, we propose a 3-bit reflective digital coding metasurface. The incident linearly polarized waves can be transformed into cross-polarized components with distinct phase responses by adjusting the rotational and open angles of the coding elements. The 3-bit phase performance can be retained over a wide bandwidth from 12 to 18 GHz by simultaneously changing the rotational and open angles. Based on the proposed broadband metasurface, broadband holography is successfully demonstrated with the optimization of a modified Gerchberg–Saxton algorithm. As a proof of concept, five schemes with different holograms integrating the letters “S,” “E,” “U,” “X,” and “Z” are simulated from 12 to 18 GHz. Good simulation results validate the performance of the proposed broadband holography, showing a relative bandwidth of 40%. Two prototypes superposing the holograms of letters “U” and “X” are fabricated and measured in a near-field microwave anechoic chamber. The experimental results corroborate well with simulated results, further supporting the demonstration. We believe that the proposed broadband holography based on the digital coding metasurface paves a way to wideband applications for microwave imaging, information processing, and holographic data storage.

Transmission of a 56 Gbit/s PAM4 signal with low-resolution DAC and pre-equalization only over 80 km fiber in C-band IM/DD systems for optical interconnects.

A low-cost intensity modulation and direct detection scheme using a 4-bit digital-to-analog converter for 56 Gbit/s pre-equalized PAM-4 generation at the transmitter and without receiver side post-equalization over 80 km single-mode fiber transmission in a C-band is experimentally demonstrated in this Letter. Pre-equalization includes pre-emphasis for channel response, a modified Gerchberg-Saxton algorithm with five iterations for chromatic dispersion compensation, and a noise shaping technique implemented by a five-tap finite impulse response filter for quantization noise suppression. Enabled by the pre-equalization techniques, the bit error rate of the received signal without post-equalization can reach the hard-decision forward error correction threshold with -2dBm receiver optical power. The experimental results indicate that our proposed scheme is a promising candidate for the standardization of 400 GbE wavelength division multiplexing-based long-haul distance data center interconnects.

Simple phase retrieval method based on two intensity measurements on a single plane.

In this work, a simple phase retrieval method is proposed by observing two intensity patterns on a single plane, which are generated with and without a lens. Rigorous theoretical derivations show that the two fields constitute the Fourier transform pairs, and a modified Gerchberg-Saxton algorithm is proposed to recover the phase patterns from the Fourier pairs. The proposed method does not require the intensity patterns to be measured on two different planes along the propagation distance, and this is quite beneficial in a system with a phase tuning element like a spatial light modulator, which can form a virtual lens by creating a parabola-like phase distribution. Experiments are conducted to demonstrate the effectiveness of the proposed phase retrieval method.

Computer-Generated Hologram Based on Reference Light Multiplexing for Holographic Display

In this article, we propose a reference light wave multiplexing scheme to increase the information capacity of computer-generated holograms. The holograms were generated by different reference light waves and superimposed together as a multiplexed hologram. A modified Gerchberg–Saxton algorithm was used to improve image quality, and different images could be reconstructed when the multiplexed hologram was illuminated by corresponding reference light waves. We performed both numerical simulations and optical experiments to demonstrate the feasibility of the proposed scheme. Numerical simulations showed that the proposed method could reconstruct multiple images successfully by a single multiplexed hologram and optical experiments are consistently good with numerical simulations. It is expected that the proposed method has great potential to be widely applied in holographic displays in the future.

Optical hiding based on single-input multiple-output and binary amplitude-only holograms via the modified Gerchberg-Saxton algorithm.

Optics has provided a promising means for the development of information hiding in recent years. However, conventional optical information hiding systems can only hide a limited number of images, and optical implementation complexity is usually high in conventional methods. In this paper, we propose a new scheme to implement optical information hiding based on single-input multiple-output (SIMO) and binary amplitude-only holograms (AOHs) using the modified Gerchberg-Saxton algorithm (MGSA). Different from conventional optical hiding methods with the limited multiplexing capacity, the proposed scheme can retrieve a large number of different secret images from one single host image during optical retrieval. In addition, it is also illustrated that optical implementation complexity is reduced in the proposed method. Simulations and optical experiments are conducted to verify feasibility, security and robustness of the proposed method. It is expected that the proposed method could open up a different research perspective for optical multiple-image hiding.

Multiplexing Vectorial Holographic Images with Arbitrary Metaholograms

AbstractMetasurfaces achieving arbitrary phase profiles within ultrathin thickness, emerge as miniaturized, ultracompact, and kaleidoscopic nanophotonic platforms. However, it is often required to segment or interleave independent subarray metasurfaces to multiplex holograms in a single nanodevice, which in turn affects the device's compactness and channel capacity. Here, a flexible strategy is proposed for multiplexing vectorial holographic images by controlling the phase distributions of holographic images in far field. Benefitting from precisely controlling the phase difference of reconstructed images through the modified Gerchberg–Saxton algorithm, two different holographic images are independently designed for the circular light by two interleaved metasurfaces and an extra vectorial hologram is flexibly encrypted in far field without additional set of structures on the metasurface plane. An unlimited number of polarization can be achieved in the holographic image and additional information can be decrypted when different polarization‐dependent holographic images overlap. By continually varying phase difference between the incident right and left circular polarized light, the image within the overlap area can be modulated. The silicon dielectric metahologram with record absolute multiplexed efficiency (&gt;25%) is achieved in the experiment. This technique, as far as it is known, promises an enormous data capacity as well as a high level of information security.

Reconfigurable time-frequency division multiplexing metasurface: giving a helping hand to high signal-to-noise ratio and high data holographic imaging.

A holographic imaging system based on reconfigurable time-frequency division multiplexing metasurface (RTFDMM) has been designed, in an attempt to address the problems of insufficient flexibility, low signal-to-noise ratio (SNR), and low data of traditional metasurface imaging devices. The modified Gerchberg-Saxton (MGS) algorithm in the front end of this system has been proposed to calculate the phase distribution required by the diffraction metasurface at the back end. The 1-bit low loss reconfigurable metasurface has been desiged in X-band. To obtain the best imaging results, the appropriate ranges of imazging distance, pixel size and resolution ratio are given by adjusting the phase distribution of metasurface. Based on the MGS algorithm and the 1-bit metasurface, the high-quality imaging system has been successfully established through the time-frequency division multiplexing (TFDM) technology. The high SNR imaging can be generated with the expansion of the time domain, while the high data imaging can be yielded with the quantization of the frequency domain. The proof-of-concept has demonstrated that the high data, high SNR, and high flexibility of the RTFDMM are critical technologies for the advancement of detection imaging equipment. Therefore, there are great possibilities for RTFDMM holographic imaging system to be applied in the intelligent detection platform in the future.

Unequal modulus decomposition and modified Gerchberg Saxton algorithm based asymmetric cryptosystem in Chirp-Z transform domain

In this paper, we have presented an image encryption method in Chirp-Z transform domain using unequal modulus decomposition (UMD) and modified Gerchberg–Saxton (GS) algorithm. The proposed encryption scheme is highly sensitive to the encryption keys and the modified GS algorithm introduces an additional layer of security. The validity of the proposed method is tested with various grayscale and binary images and the numerical simulation results are demonstrated for ‘Cameraman’, ‘Medical’ and Binary ‘CUH’ images. The presented results confirm the robustness of the proposed method against various existing attacks such as, the noise attack, special attack, statistical attack, and brute force attack. A comparative analysis with existing similar methods is also performed and the enhanced security and efficiency of the proposed method is verified.

Polarization Multiplexing Hologram Realized by Anisotropic Digital Metasurface

AbstractMetasurface holograms have the potential to be applied in a variety of practical fields. Here an anisotropic digital metasurface is proposed that can project polarization multiplexing patterns at the microwave frequency, in which a modified Gerchberg–Saxton algorithm is presented to calculate the phase masks. A 3‐bit coding metamaterial unit is also proposed, which is capable of independently manipulating the transmission phases of orthogonal linearly polarized electromagnetic waves. The influence of wave‐front has been analyzed by comparing two holographic results with distinct source optimization methods. A prototype of the anisotropic digital coding metasurface optimized with horn wave‐front is designed, manufactured, and tested. The experimental measurements present the expected holographic fields and have a good match with the simulation results.

Spin-Switched Three-Dimensional Full-Color Scenes Based on a Dielectric Meta-hologram

The possibility to manipulate light fields with metasurfaces can facilitate more compact and efficient devices for encoding computer-generated holograms. However, most of the reconstructed holographic images for reported metasurfaces are either monochromatic or two-dimensional, limiting their applicability. Here, we design and fabricate a silicon metasurface, whose constituent meta-molecules are composed of three different kinds of meta-atoms capable of modulating red, green, and blue light independently. A modified Gerchberg–Saxton algorithm is utilized to retrieve the wavefront for the three-dimensional (3D) hologram. The reconstructed 3D full-color images can be switched by changing the helicity of the illuminating circularly polarized light. The spin-switched 3D full-color scenes greatly increase the information capacity of the device, which may find use in 3D dynamic displays, virtual reality, and data storage applications.