Gerchberg-Saxton Iterative Algorithm Research Articles

Encoding terahertz holographic bits with a computer-generated 3D-printed phase plate

A sub-terahertz holographic image of a two-dimensional 576-bit data code is produced using a diffractive phase-plate element. The phase plate was designed using a modified Gerchberg-Saxton iterative algorithm to encode a focused image of the data code into a phase modulation profile. The complex phase plate structure is fabricated from polylactic acid using fused deposition modeling, a common three-dimensional-printing technique. The design achieves a significantly simplified optical setup, consisting of a 0.14 THz diverging source, the holographic phase plate and a scanning detector, without the need for additional optical elements. The information stored in the data code is an example of a cryptographic private key. Specifically, the private key for a Bitcoin wallet address. Successful retrieval of the encoded information demonstrates a potential use case for terahertz holographic memory, using a storage medium that can be fabricated with consumer-level three-dimensional-printing techniques.

Resolution-enhanced imaging by interferenceless coded aperture correlation holography with two competing methods of phase mask synthesis

Introducing a scattering phase mask between an object and an objective lens can improve the imaging resolution of a maskless system. The resolution enhancement is achieved using the technique of interferenceless coded aperture correlation holography with a point spread hologram (PSH) of N randomly distributed dots. For the same PSH type, there is more than one way to synthesize the phase mask. In this study, we suggest a new method of synthesizing masks and compare it with the previous method. The new method is random multiplexing of linear phase functions, each of which is responsible for a single dot on the camera plane. The previous method is based on the well-known iterative Gerchberg-Saxton algorithm with the constraints of a randomly chosen dot pattern on the camera plane and a pure phase function on the mask plane. In terms of the signal-to-noise ratio of the reconstructed image, the iterative algorithm is better for high N but worse for low N, as expected. This article has also answered the question of which method is preferred to synthesize the phase mask.

Fast non-iterative algorithm for 3D point-cloud holography.

Recently developed iterative and deep learning-based approaches to computer-generated holography (CGH) have been shown to achieve high-quality photorealistic 3D images with spatial light modulators. However, such approaches remain overly cumbersome for patterning sparse collections of target points across a photoresponsive volume in applications including biological microscopy and material processing. Specifically, in addition to requiring heavy computation that cannot accommodate real-time operation in mobile or hardware-light settings, existing sampling-dependent 3D CGH methods preclude the ability to place target points with arbitrary precision, limiting accessible depths to a handful of planes. Accordingly, we present a non-iterative point cloud holography algorithm that employs fast deterministic calculations in order to efficiently allocate patches of SLM pixels to different target points in the 3D volume and spread the patterning of all points across multiple time frames. Compared to a matched-performance implementation of the iterative Gerchberg-Saxton algorithm, our algorithm's relative computation speed advantage was found to increase with SLM pixel count, reaching >100,000x at 512 × 512 array format.

High-precision Fourier ptychographic microscopy based on Gaussian apodization coherent transfer function constraints.

Fourier ptychographic microscopy (FPM) combines the concepts of phase retrieval algorithms and synthetic apertures and can solve the problem in which it is difficult to combine a large field of view with high resolution. However, the use of the coherent transfer function in conventional calculations to describe the linear transfer process of an imaging system can lead to ringing artifacts. In addition, the Gerchberg-Saxton iterative algorithm can cause the phase retrieval part of the FPM algorithm to fall into a local optimum. In this paper, Gaussian apodization coherent transfer function is proposed to describe the imaging process and is combined with an iterative method based on amplitude weighting and phase gradient descent to reduce the presence of ringing artifacts while ensuring the accuracy of the reconstructed results. In simulated experiments, the proposed algorithm is shown to give a smaller mean square error and higher structural similarity, both in the presence and absence of noise. Finally, the proposed algorithm is validated in terms of giving reconstruction results with high accuracy and high resolution, using images acquired with a new microscope system and open-source images.

Modeling of the driver transverse profile for laser wakefield electron acceleration at APOLLON research facility

The quality of electron bunches accelerated by laser wakefields is highly dependant on the temporal and spatial features of the laser driver. Analysis of experiments performed at APOLLON PW-class laser facility shows that spatial instabilities of the focal spot, such as shot-to-shot pointing fluctuations or asymmetry of the transverse fluence, lead to charge and energy degradation of the accelerated electron bunch. It is shown that PIC simulations can reproduce experimental results with a significantly higher accuracy when the measured laser asymmetries are included in the simulated laser's transverse profile, compared to simulations with ideal, symmetric laser profile. A method based on a modified Gerchberg–Saxton iterative algorithm is used to retrieve the laser electric field from fluence measurements in vacuum in the focal volume, and accurately reproduce experimental results using PIC simulations, leading to simulated electron spectra in close agreement with experimental results, for the accelerated charge, energy distribution, and pointing of the electron beam at the exit of the plasma.

Security-enhanced optical nonlinear cryptosystem based on modified Gerchberg–Saxton iterative algorithm

For any optical encryption and decryption systems, security is the major concern. In this paper, security of time varying nonlinear term embedded double random phase encoding (DRPE) scheme is enhanced by using Modified Gerchberg Saxton Iterative Algorithm (MGSIA). The encrypted image of DRPE is given as an amplitude input to MGSIA. This makes the cryptosystem asymmetric and endures the basic attacks such as known-plaintext attack, chosen-plaintext attack, and chosen-cipher text attack. Robustness of this model has been checked with various results.

Triple-level cryptosystem using deterministic masks and modified gerchberg-saxton iterative algorithm in fractional Hartley domain by positioning singular value decomposition

We present a novel method for image encryption by Modified Gerchberg–Saxton Iterative Algorithm (MGSIA) using deterministic masks (DMKs) and singular value decomposition (SVD) in fractional Hartley transform (FrHT) domain. This proposal delivers triple-level encryption (L1, L2 and L3) and all these three levels can be iterated number of times. First and third level of encryption routines MGSIA, DMK and FrHT, whereas mid-level encryption uses SVD. Since L1 delivers single cipher image and L3 yields double cipher images, this technique is also known as multiple image encryption system. Each MGSIA generates private keys and these keys are engendered in such a way that decryption is to be expected with traditional DRPE system. Because of the nature of these two asymmetric keys, the proposed system is non-linear and by the way, it boosts security. In the place of traditional random phase masks, the proposed cryptosystem uses statistically independent deterministic masks in L1 and L3. DMKs are nothing but the class of structured phase masks. DMKs are generated and used as the phase for MGSIA in L1 and L3. These DMKs are formed by the deviation from conventional rectangular function and limited range values which delivers key components with reduced size, better performance and lower complexity. Deployment of SVD on L2 decomposes the cipher image1 into three components and out of these three components, due to the nature of diagonality, S components plays a major role in proposed work. Numerical simulations have been achieved to prove the strength and feasibility of the proposed scheme.

Sub-Terahertz Computer Generated Hologram with Two Image Planes

An advanced optical structure such as a synthetic hologram (also called a computer-generated hologram) is designed for sub-terahertz radiation. The detailed design process is carried out using the ping-pong method, which is based on the modified iterative Gerchberg–Saxton algorithm. The novelty lies in designing and manufacturing a single hologram structure creating two different images at two distances. The hologram area is small in relation to the wavelength used (the largest hologram dimension is equivalent to around 57 wavelengths). Thus, it consists of a small amount of coded information, but despite this fact, the reconstruction is successful. Moreover, one of the reconstructed images is larger than the hologram area. Good accordance between numerical simulations and experimental evaluation was obtained.

Far-Field Antenna Pattern Measurement Using Near-Field Thermal Imaging

The Active Denial nonlethal weapon system utilizes a large millimeter-wave (mmW) antenna with over 66 dBi of gain operating at 95 GHz. In its final state, the active denial antenna cannot be characterized by conventional measurement techniques including direct far-field pattern characterization or near-field scanning. However, the high power transmitted from Active Denial (100 kW average) provides the means of rapidly recording near-field antenna beam patterns using infrared imaging. Although these thermal images provide a magnitude-only snapshot of the near-field antenna beam, phase can be retrieved from measurements at two different ranges using the iterative Gerchberg–Saxton algorithm. The resulting complex near fields can then be used to compute the antenna far-field pattern and the complex aperture fields. This method has been used to successfully characterize these mmW antennas in both indoor laboratory and outdoor field test environments.

Axial multi-particle trapping and real-time direct observation

The optical tweezers with the special advantages of non-mechanical contact and the accurate measurement of positions of particles, are a powerful manipulating tool in numerous applications such as in colloidal physics and life science. However, the standard optical tweezers system uses a single objective lens for both trapping and imaging. As a result, the trapping and imaging regions are confined to the volume near the focal plane of the objective lens, making it difficult to track the trapped particles arranged in the axial direction. Therefore, multiple trapping along axial direction remains a challenge. The three-dimensional imaging technology can realize the monitoring of the axial plane, but neither the laser scanning microscopy nor the wide-field imaging technology can meet the requirement of the real-time imaging. To address this issue, we propose a modified axial-plane Gerchberg-Saxton (GS) iterative algorithm based on the Fourier transform in the axial plane. Compared with the direct algorithm such as the Fresnel lens method, the modified axial-plane GS iterative algorithm has a higher modulation efficiency, and the generated axial distribution has a sharper intensity. In theory, the traps generated each have an ideal Gaussian intensity distribution independently, which is proved by the simulation of reconstructed field. With such an iterative algorithm, we can directly create multiple point-trap array arranged along the axial direction. We also develop an axial-imaging scheme. In this scheme, the particles are trapped and a right-angled silver-coated 45 reflector is used to realize axial-plane imaging. The scheme is verified by imaging silica particles in an axial plane and a lateral plane simultaneously. Furthermore, we combine the axial-plane imaging technique with holographic optical tweezers, and demonstrate the simultaneous optical trapping in 22 trap array and the monitoring of multiple silica particles in the axial plane. The trap stiffness of traps array in axial plane is calibrated by measuring the Brownian motion of the trapped particles in the axial trap array with digital video microscopy. The proposed technique provides a new perspective for optical micromanipulation, and enriches the functionality of optical micromanipulation technology, and thus it will have many applications in biological and physical research.

New optical sensing technique of tissue viability and blood flow based on nanophotonic iterative multi-plane reflectance measurements.

Physiological substances pose a challenge for researchers since their optical properties change constantly according to their physiological state. Examination of those substances noninvasively can be achieved by different optical methods with high sensitivity. Our research suggests the application of a novel noninvasive nanophotonics technique, ie, iterative multi-plane optical property extraction (IMOPE) based on reflectance measurements, for tissue viability examination and gold nanorods (GNRs) and blood flow detection. The IMOPE model combines an experimental setup designed for recording light intensity images with the multi-plane iterative Gerchberg-Saxton algorithm for reconstructing the reemitted light phase and calculating its standard deviation (STD). Changes in tissue composition affect its optical properties which results in changes in the light phase that can be measured by its STD. We have demonstrated this new concept of correlating the light phase STD and the optical properties of a substance, using transmission measurements only. This paper presents, for the first time, reflectance based IMOPE tissue viability examination, producing a decrease in the computed STD for older tissues, as well as investigating their organic material absorption capability. Finally, differentiation of the femoral vein from adjacent tissues using GNRs and the detection of their presence within blood circulation and tissues are also presented with high sensitivity (better than computed tomography) to low quantities of GNRs (<3 mg).

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.

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.

Quantum enhanced phase retrieval

The retrieval of phases from intensity measurements is a key process in many fields in science, from optical microscopy to x-ray crystallography. Here we study phase retrieval of a one-dimensional multi-phase object that is illuminated by quantum states of light. We generalize the iterative Gerchberg-Saxton algorithm to photon correlation measurements on the output plane, rather than the standard intensity measurements. We report a numerical comparison of classical and quantum phase retrieval of a small one-dimensional object of discrete phases from its far-field diffraction. While the classical algorithm was ambiguous and often converged to wrong solutions, quantum light produced a unique reconstruction with smaller errors and faster convergence. We attribute these improvements to a larger Hilbert space that constrains the algorithm.

On the use of the supporting quadric method in the problem of the light field eikonal calculation.

A new iterative technique for calculating the eikonal function of a light field providing the focusing into a set of points is introduced. This technique is a modification of the supporting quadric method widely used for design of reflecting and refracting optical surfaces for generating prescribed illuminance distributions at given discrete set of points. As an example, we design a refractive optical element which focuses an incident beam into a set of points with energy pattern forming an image of a keyboard of a calculator. It is shown that the proposed technique is well-suited for the design of diffractive optical elements producing continuous intensity distributions within the scalar theory of diffraction. It is also shown that the calculated eikonal function is a good initial guess when designing diffractive optical elements using the iterative Gerchberg-Saxton algorithm.

On using a supporting quadric method to design diffractive optical elements

Application of a modified supporting quadric method in diffractive optical element design is described. As an example, we compute the eikonal function to focus into a calculator's image represented by a set of points. Properties of the produced image are analyzed using the scalar diffraction theory. It is shown that the proposed method can be used for the design of diffractive optical elements focusing into a continuous required illuminance distribution. Numerical examples also show that the computed eikonal function is a good initial estimate for the iterative Gerchberg-Saxton algorithm.

Optimization of ultra-fast interactions using laser pulse temporal shaping controlled by a deterministic algorithm

Femtosecond laser pulse temporal shaping techniques have led to important advances in different research fields like photochemistry, laser physics, non-linear optics, biology, or materials processing. This success is partly related to the use of optimal control algorithms. Due to the high dimensionality of the solution and control spaces, evolutionary algorithms are extensively applied and, among them, genetic ones have reached the status of a standard adaptive strategy. Still, their use is normally accompanied by a reduction of the problem complexity by different modalities of parameterization of the spectral phase. Exploiting Rabitz and co-authors’ ideas about the topology of quantum landscapes, in this work we analyze the optimization of two different problems under a deterministic approach, using a multiple one-dimensional search (MODS) algorithm. In the first case we explore the determination of the optimal phase mask required for generating arbitrary temporal pulse shapes and compare the performance of the MODS algorithm to the standard iterative Gerchberg–Saxton algorithm. Based on the good performance achieved, the same method has been applied for optimizing two-photon absorption starting from temporally broadened laser pulses, or from laser pulses temporally and spectrally distorted by non-linear absorption in air, obtaining similarly good results which confirm the validity of the deterministic search approach.

Ultrasound-Mediated Targeted Drug Delivery Generated by Multifocal Beam Patterns: An In vitro Study

Ultrasound-mediated targeted drug delivery has been a subject for a dedicated research activity for several decades. Nevertheless, in vitro studies in this field of research are characterized by their inconsistencies. To improve the repeatability of such experiments, a novel approach of multifocal spot generation was investigated. A multifocal pattern of 16 spots was utilized using an iterative Gerchberg–Saxton algorithm. The pattern was applied to insonate a 96-well Petri dish plate using a clinically available planar-phased array transducer with approximately 1000 elements with a central frequency of 0.55 MHz. The pattern was acoustically characterized and applied to a monolayer of human breast cancer cell line in the 96-well plate. With the help of ultrasonic contrast agents, the intracellular drug uptake was increased by an average factor of 3.5 compared with the control group.

Bypassing absorbing objects in focused ultrasound using computer generated holographic technique

Focused ultrasound (FUS) technology is based on heating a small volume of tissue, while keeping the temperature outside the focus region with minimal heating only. Several FUS applications, such as brain and liver, suffer from the existence of ultrasound absorbers in the acoustic path between the transducer and the focus. These absorbers are a potential risk for the FUS therapy since they might cause to unwanted heating outside the focus region. An acoustic simulation based solution for reducing absorbers' heating is proposed, demonstrated, and compared to the standard geometrical solution. The proposed solution uses 3D continuous acoustic holograms, generated by the Gerchberg-Saxton (GS) algorithm, which are described and demonstrated for the first time using ultrasound planar phased-array transducer. Holograms were generated using the iterative GS algorithm and fast Fourier transform (FFT) acoustic simulation. The performances of the holograms are demonstrated by temperature elevation images of the absorber, acquired by GE 1.5T MRI scanner equipped with InSightec FUS planar phased-array transducer built out of 986 transmitting elements. The acoustic holographic technology is demonstrated numerically and experimentally using the three letters patterns, "T," "A," and "U," which were manually built into 1 × 1 cm masks to represent the requested target fields. 3D holograms of a focused ultrasound field with a hole in intensity at the absorber region were generated and compared to the standard geometrical solution. The proposed holographic solution results in 76% reduction of heating on absorber, while keeping similar heating at the focus. In the present work we show for the first time the generation of efficient and uniform continuous ultrasound holograms in 3D. We use the holographic technology to generate a FUS beams that bypasses an absorber in the acoustic path to reduce unnecessary heating and potential clinical risk. The developed technique is superior in performance and flexibility compared to the intuitive geometrical technique that is being used in clinical practice.

Formation of vortex light fields of specified intensity for laser micromanipulation

method for calculating phase diffractive elements is proposed to form light fields with intensity described by flat curves. This method is based on the optics of spiral light beams and Gerchberg-Saxton iterative algorithm. The experimental data on the formation of fields and their use in manipulation with latex microparticles are reported.