Gerchberg-Saxton Algorithm Research Articles

Triple-optical autocorrelation for direct optical pulse-shape measurement

Triple optical autocorrelation of femtosecond optical pulses was realized simply with third-harmonic-generation technique. This optical technique provides complete knowledge of transient pulse intensity variation directly in time domain. Only analytic calculation is needed to obtain the pulse-shape from data without direction-of-time ambiguity. Combined with a spectral measurement and the Gerchberg–Saxton algorithm, except for pulses with complete temporal and spectral symmetry that will cause a twofold ambiguity, exact phase variation in time can also be retrieved through an iterative calculation with an O(n) complexity.

Phase retrieving and extension by combination of TIE and Gerchberg-Saxton algorithm

Phase retrieving and extension by combination of TIE and Gerchberg-Saxton algorithm

HRTEM Resolution Extension for Interface by Gerchberg-Saxton Algorithm with Supported Constraint

HRTEM Resolution Extension for Interface by Gerchberg-Saxton Algorithm with Supported Constraint

Atomic and Electronic Structure of Interfaces at Sic Studied by Indirect Super Hrtem and Electron Spectroscopyn Imaging

AbstractObtaining electronic and atomic structure of material simultaneously is very important for developing the nano-technology. In this paper, we demonstrate that atomic and electronic structure of an interface can be extracted with combination of Gerchberg-Saxton indirect microscopy and electron spectroscopy imaging (ESI) technique. Basically, Gerchberg-Saxton algorithm includes two projections. Projection in the real space is a maximum entropy (ME) de-convolution process and in reciprocal space is an amplitude substitution process. It has been shown that Gerchberg-Saxton algorithm can extend the structural resolution to near 0.1nm. An application case of Gerchberg-Saxton algorithm to solve the atomic structure for 3C-polytypic SiC boundary is shown.ESI spectrum processed by FFT interpolation, maximum entropy de-convolution and wavelet transformation allow us to extract 2-dimensional map of the sp2/sp3 with a sub-nanometer resolution. Grain boundary and interface at SiC are good candidates for this study, since the bond distance of Si-C is slightly less than 0.1nm which is not routinely resolvable using a FEG TEM and Si-L (99eV) and C-K-edges (283 eV) locate in a reasonable energy range. The resultant electronic structure can be compared with that calculated using WIEN97. An example of quantitative analysis on 2-dimensional sp3/sp2 map deduced from the C K-edge of ESI spectra acquired from 6H-SiC is given.

Extension of HRTEM resolution by semi-blind deconvolution method and Gerchberg-Saxton algorithm: application to grain boundary and interface.

A generalized maximum entropy method coupled with Gerchberg-Saxton algorithm has been developed to extend the resolution from high-resolution TEM image(s) for weak objects. The Gerchberg-Saxton algorithm restores spatial resolution by operating real space and reciprocal space projections cyclically. In our methodology, a generalized maximum entropy method (Kullback-Leibler cross entropy) dealing with weak objects is used as a real space (P1) projection. After P1 projection, not only are the phases within the input spatial frequencies improved, but also the phases in the next higher frequencies are extrapolated. An example of semi-blind deconvolution (P1 project only) to improve the resolution in SiC twin boundary is shown. The nature of the bonding in this twin boundary is Si-C but it was rotated 180 degrees along the boundary normal. The optimum solution from P1 projection can be further improved by a P2 projection. The square roots of diffraction intensities from a diffraction pattern are then substituted to complete a cycle operation of the Gerchberg-Saxton algorithm. Application examples of Gerchberg-Saxton algorithm to solve the atomic structure of defects (2 x 1 interfacial reconstruction and dislocation) in NiSi2/Si interfaces will be shown also.

Diffractive phase element for shrinking focal spot diameter: Design, fabrication, and application to laser beam lithography

A diffractive phase element (DPE) capable of shrinking the main-lobe of the focal spot of an incident beam has been developed. The DPE is expected to be used in laser beam lithography to improve the resolution. Special constraints, which force the focal spot to be small while permitting increase of the side-lobes, are introduced in the design of the DPE using an iterative method based on the Gerchberg-Saxton algorithm. Two kinds of DPEs are fabricated by a laser beam lithography system of our own composition and their performances are experimentally measured. The minimum line width obtained by the system is improved from 1.2 μm to 1.0 μm with the fabricated DPE. Focal depth of the focusing system is discussed.

Efficient optimization of diffractive optical elements based on rigorous diffraction models.

An efficient optimization strategy for the design of diffractive optical elements that is based on rigorous diffraction theory is described. The optimization algorithm combines diffraction models of different degrees of accuracy and computational complexity. A fast design algorithm for diffractive optical elements is used to yield estimates of the optimum surface profile based on paraxial diffraction theory. These estimates are subsequently evaluated with a rigorous diffraction model. This scheme allows one to minimize the need to compute diffraction effects rigorously, while providing accurate design. We discuss potential applications of this scheme as well as details of an implementation based on a modified Gerchberg-Saxton algorithm and the finite-difference time-domain method. Illustrative examples are provided in which we use the algorithm to design Fourier array illuminators.

Algorithm based on rigorous coupled-wave analysis for diffractive optical element design.

Diffractive optical element design is an important problem for many applications and is usually achieved by the Gerchberg-Saxton or the Yang-Gu algorithm. These algorithms are formulated on the basis of monochromatic wave propagation and the far-field assumption, because the Fourier transform is used to model the wave propagation. We propose an iterative algorithm (based on rigorous coupled-wave analysis) for the design of a diffractive optical element. Since rigorous coupled-wave analysis (instead of Fourier transformation) is used to calculate the light-field distribution behind the optical element, the diffractive optical element can thus be better designed. Simulation results are provided to verify the proposed algorithm for designing a converging lens. Compared with the well-known Gerchberg-Saxton and Yang-Gu algorithms, our method provides 7.8% and 10.8%, respectively, improvement in converging the light amplitude when a microlens is desired. In addition, the proposed algorithm provides a solution that is very close to the solution obtained by the simulated annealing method (within 1.89% error).

Extension of HRTEM Resolution: Solving Structures of Grain Boundary and Interface Using Gerchberg-Saxton Algorithm and Blind Deconvolution

Oxides, nitrides, carbides and silicides contain more than one type of atoms that gives one extra degree of freedom in the grain boundary and interfacial structure. This degree of freedom is the choice of the boundary plane containing different type of atoms which results of several structural multiplicities in the boundary. Coexistence of the structural multiplicities (or boundary domains) in the interface and grain boundary were reported [1, 2]. The structural information in high frequency domain is usually blurred and lost in the lens system of a high resolution transmission electron microscopes (HRTEM) such that the HRTEM image may not truly represent the real structure. Conventionally, the structures of grain boundary and interface were solved using forward method. The forward method involves matching the simulated images from guessed possible models with the through focal series experimental images. Recently, rereconstruction methods such as focal variation holography [3, 4], tiltbeam holography [5] and Gerchberg-Saxton algorithm [6] provide a backward (direct) way to recover the structural information lost in a modern microscope.

Structure determination at the atomic level from dynamical electron diffraction data under systematic row conditions

We discuss a method to obtain structural information on crystals at the atomic level in high-resolution transmission electron microscopy from dynamical diffraction data under systematic row conditions. Working at a fixed incident energy and within an N-beam approximation, data is required at a well defined set of N incident beam orientations to determine the scattering matrix S , one orientation for each column in the matrix. At each orientation the corresponding column of the S -matrix is obtained by Fourier transformation of the exit surface wave function. Thus, in addition to each exit surface image, we must recover the phase of the wave function for that orientation in the image plane. We show that retrieval of the phase using algorithms based on conservation of flux, which assume continuity of the phase, can yield incorrect solutions for the phase. This is because singularities can occur in the phase of the wave field at points where the intensity is zero, which can lead to edge dislocations in the phase. We demonstrate, using a model example, how these edge dislocations arise. We will show that phase retrieval from a through focal series of measurements or using the Gerchberg–Saxton algorithm (starting from measurements of an image and the corresponding diffraction pattern), correctly retrieves the phase and hence the exit surface wave function for all the orientations required to obtain the S -matrix. The dynamical (multiple) scattering can then be inverted to uniquely obtain the projected potential.

Improved compensation of turbulence-induced amplitude and phase distortions by means of multiple near-field phase adjustments.

An approach for compensation of turbulence-induced amplitude and phase distortions is described. Two deformable mirrors are placed optically conjugate to the collecting aperture and to a finite range from this aperture. Two control algorithms are presented. The first is a sequential generalized projection algorithm (SGPA) that is similar to the Gerchberg-Saxton phase retrieval algorithm. The second is a parallel generalized projection algorithm (PGPA) that introduces constraints that minimize the number of branch points in the control commands for the deformable mirrors. These approaches are compared with the approach of placing the second deformable mirror conjugate to the far field of the collecting aperture and using the Gerchberg-Saxton algorithm to determine the optimal mirror commands. Simulation results show that placing the second deformable mirror at a finite range can achieve near-unity Strehl ratio regardless of the strength of the scintillation induced by propagation through extended paths, while the maximum Strehl ratio of the far-field approach drops off with increasing scintillation. The feasibility of the solutions is evaluated by counting the branch points contained in the deformable mirror commands. There are large numbers of branch points contained in the control commands that are generated by the Gerchberg-Saxton SGPA-based algorithms, irrespective of where the second deformable mirror is located. However, the control commands generated by the PGPA with branch point constraints achieves excellent Strehl ratio and minimizes the number of branch points.

Strain profiles in epitaxial films from x-ray Bragg diffraction phases

We present a versatile and model-independent approach for the analysis of strain distributions in thin-film systems which is based on the x-ray phase retrieval by an iterative Gerchberg–Saxton algorithm. This scheme is used to determine the strain profile across a thin ordered Cu3Au(111) film grown epitaxially on Al2O3.

Characterization of nuclear wave packets describing molecular photodissociation

A bound-to-free transition initiated by femtosecond excitation of diatomic molecules results in photofragments with a distribution of kinetic energies. A measurement of the kinetic-energy distribution yields the modulus squared of the asymptotic momentum-space wave packet prepared in the laser excitation process. On the other hand, the coordinate-space density of the wave packet entering the interaction-free region can be determined from pump–probe integrated fluorescence spectroscopy. We provide several numerical examples to show that this information can be used to determine the phase of the asymptotic wave packet so that this particular quantum-mechanical wave function can be characterized completely. To achieve this aim we use an iteration scheme (Gerchberg–Saxton algorithm) which does not require any further information about the system or the laser pulses.

Experimental study of the laser and stimulated Raman scattering wave phases by a nonlinear imaging method

Applications of Stimulated Raman Scattering to spatial information transfer from a laser exciting beam (or a Raman seed beam) to an amplified Raman beam require the knowledge and the control of the Raman phase. A method using a Fourier optics set up and the Gerchberg-Saxton algorithm allows the measurements of the laser and Raman phases. The results show the importance of the propagation in the Raman medium. Links between the Raman phase and the laser intensity distribution are observed, opening thus perspectives concerning the Raman spatial structure control by the laser structure.

Temporal decorrelation of short laser pulses

We describe a unique approach for extracting the temporal profile of ultrashort laser pulses from typical autocorrelation measurements. The use of the constraint that intensity is a nonnegative quantity enables an iterative numerical algorithm to reconstruct pulse shapes in a one-dimensional procedure. With the reconstruction of the intensity profile, the Gerchberg–Saxton algorithm can be used to retrieve the phase of the electric field from a spectral measurement. Because these procedures are carried out in one dimension, they are numerically much faster than two-dimensional techniques such as frequency-resolved optical gating. Their high computational efficiency can save substantial time by constructing good trial solutions for the more accurate but slower procedure of frequency-resolved optical gating.

Numerical investigation of phase retrieval in a fractional Fourier transform

Recently the combination of the Gerchberg–Saxton (GS) algorithm and a fractional Fourier transform was proposed to implement beam shaping in the fractional Fourier domain [ Zalevsky , Opt. Lett.21, 842 (1996)]. We generalize this idea to deal with the problem of phase retrieval from two intensity measurements in a fractional Fourier transform system. The relevant equations for determining the unknown phases are derived, based on the general theory of amplitude–phase retrieval in an optical system. The unitarity condition of the fractional Fourier transform in a practical optical system with finite aperture is discussed. For different fractional orders P, the phase retrieval of several typical model images is studied in detail. A comparison of the GS and our algorithms is given, based on numerical simulations. It follows that our algorithm can offer the desired phase in all cases considered. However, the GS algorithm may fail when the transform system is nonunitary.

Computer-generated holograms from 3D-objects written on twisted-nematic liquid crystal displays

In order to optimize computer-generated holograms (CGHs) for three-dimensional objects, an extension of the well known Gerchberg-Saxton algorithm is used. Optical reconstructions using an electrically addressed twisted-nematic liquid crystal display are presented. For arbitrary CGHs, reconstructing 3D-objects, we will discuss the problems and solutions associated with periodic replication of holograms.

Methods for reconstruction of 2-D sequences from Fourier transform magnitude

The Gerchberg-Saxton (GS) algorithm and its generalizations have been the main tools for phase retrieval. Unfortunately, it has been observed that the reconstruction using these algorithms does not always converge to the correct result even if the desired solution satisfies the uniqueness condition. In this paper, we propose a new deautocorrelation algorithm and a few auxiliary techniques. We recommend that a combination of the iterative Fourier transform (IFT) algorithm with our new algorithm and techniques can improve the probability of success of phase retrieval. A pragmatic procedure is illustrated. Different reconstruction examples that are difficult to reconstructed using the single IFT algorithm are used to show the robustness and effectiveness of the new combination of algorithms. If the given Fourier modulus data contain no noise, it is sometimes possible to get a perfect reconstruction. Even when the signal-to-noise ratio (SNR) of the Fourier modulus data is only 10 dB, a meaningful result remains reachable for our examples. A concept concerning the intrinsic ambiguity of phase retrieval is suggested. We emphasize the necessity of verification of the solution, since the available phase retrieval algorithms are incompetent for distinguishing between an intrinsically ambiguous solution and the true solution.

High-efficiency arbitrary array generator

Implementation of a beam-shaping system, whereby an arbitrary array of spots is generated, is proposed. The suggested beam-shaping generator, which is based on the Gerchberg-Saxton algorithm, contains two phase-only filters and thus yields a very-high-power throughput. The flexibility of the suggested approach is demonstrated with some computer simulations.

Phase retrieval of recorded kinoform transmittance and estimation of the phase-recording curve characteristic

Methods are presented for recovering, by the use of the basic Gerchberg-Saxton algorithm, the complex-valued transmittance of a recorded kinoform and for estimating the phase-recording curve characteristic of a kinoform-fabrication system by comparison of the phase to be recorded with that actually recorded. Related issues considered here are the shift of the central sampling point from the optical axis in the Fourier-image domain, a kinoform phase sequence suitable for estimating the curve characteristic, the initial estimate of the iterative algorithm, and the influence of errors in the measurement of the kinoform or of the Fourier-image amplitude. Results of an optical experiment with a fabrication system including a photographic process are discussed and shown to confirm the effectiveness of the proposed methods.