Gerchberg-Saxton Algorithm Research Articles

Wavefront correction of spatial light modulators using an optical vortex image

We present a fast and flexible non-interferometric method for the correction of small surface deviations on spatial light modulators, based on the Gerchberg-Saxton algorithm. The surface distortion information is extracted from the shape of a single optical vortex, which is created by the light modulator. The method can be implemented in optical tweezers systems for an optimization of trapping fields, or in an imaging system for an optimization of the point-spread-function of the entire image path.

3D interpolation of irregular data with a POCS algorithm

Seismic surveys generally have irregular areas where data cannot be acquired. These data should often be interpolated. A projection onto convex sets (POCS) algorithm using Fourier transforms allows interpolation of irregularly populated grids of seismic data with a simple iterative method that produces high-quality results. The original 2D image restoration method, the Gerchberg-Saxton algorithm, is extended easily to higher dimensions, and the 3D version of the process used here produces much better interpolations than typical 2D methods. The only parameter that makes a substantial difference in the results is the number of iterations used, and this number can be overestimated without degrading the quality of the results. This simplicity is a significant advantage because it relieves the user of extensive parameter testing. Although the cost of the algorithm is several times the cost of typical 2D methods, the method is easily parallelized and still completely practical.

Symmetrical iterative Fourier-transform algorithm using both phase and amplitude freedoms

A modified Gerchberg–Saxton algorithm, which we call symmetrical iterative Fourier transform algorithm, is presented and employed to a series of super-Gaussian beam shaping problems and arbitrary beam shaping problems. The main features of this algorithm include (1) using an amplitude-symmetric function of the failed result about the Fourier-domain constraint; (2) using both phase and amplitude freedoms in the iteration process, i.e. the noise produced outside the signal window by each iteration is kept. A formula is given to determine a critical constant in the search for the amplitude-symmetric function. The new method results in highly efficient solutions with highly precise beam profiles in the signal window for all the test problems.

Dynamic holographic optical tweezers using a twisted-nematic liquid crystal display

In this paper, we realize dynamic holographic optical tweezers (HOT) using a commercial twisted-nematic liquid crystal display (TN-LCD) removed from a video projector, and manipulate multiple particles simultaneously. After measuring the parameters of the TN-LCD, we find that it is not suited to work as a phase-only modulator for infrared optical tweezers because of the small range of its phase modulation. To overcome this shortcoming, we employ binary computer-generated holograms obtained with the Gerchberg–Saxton algorithm instead of kinoforms, and optimize the diffraction efficiency of the TN-LCD by the choice of the polarization angel of the input laser and the two grey level video signals driving the TN-LCD. Since TN-LCDs are more prevalent than other types of LC devices, such as parallel-aligned nematic and ferroelectric LC devices, our approach provides an easier and lower-cost access to the set-up of a HOT system.

Image recovery from double amplitudes in fractional Fourier domain

The classical Gerchberg–Saxton algorithm is introduced into the image recovery in fractional Fourier domain after adaptation. When this algorithm is applied directly, its performance is good for smoothed image, but bad for unsmoothed image. Based on the diversity of fractional Fourier transform on its orders, this paper suggests a novel iterative algorithm, which extracts the information of the original image from amplitudes of its fractional Fourier transform at two orders. This new algorithm consists of two independent Gerchberg–Saxton procedures and an averaging operation in each circle. Numerical simulations are carried out to show its validity for both smoothed and unsmoothed images with most pairs of orders in the interval [0, 1].

A novel algorithm for designing diffractive optical elements for two colour far-field pattern formation

One of the many uses for diffractive optical elements (DOEs) is in pattern formation for display purposes. These images can be made much more arresting to the eye if they are constructed from more than just one colour. To achieve this with DOEs is not simple, as they are inherently wavelength specific. We present a novel approach to designing quantized elements that produce distinct intensity patterns, in the far field, for two wavelengths. The technique uses a modification of the Gerchberg–Saxton algorithm to design the phase profiles, together with phase delays greater than 2π to give the required extra degree of freedom. We will further discuss methods of enhancing the design algorithm. Experimental examples are presented to illustrate the success of these techniques. Efficiencies of 50%–60% are typical for 16 level, 128 × 128 elements with a maximum phase delay of 8π.

Experimental demonstration of holographic three-dimensional light shaping using a Gerchberg–Saxton algorithm

We use a three-dimensional Gerchberg–Saxton algorithm (Shabtay (2003) Opt. Commun. 226 33) to calculate the Fourier-space representation of physically realizable light beams with arbitrarily shaped three-dimensional intensity distributions. From this representation we extract a phase-hologram pattern that allows us to create such light beams experimentally. We show several examples of experimentally shaped light beams.

Reconstruction of In-Line Holograms Using Phase Retrieval Algorithms

In-line holography is a simple technique for high-resolution imaging, however, due to the fact that the reference and illuminating lights are overlapped during the hologram is recorded, the real image and its twin image are in the same position, which will bring difficulties to distinguish the real one. The Yang-Gu (YG) and Gerchberg-Saxton (GS) algorithms are used to reconstruct pure real object from a single in-line hologram. The differences between the YG and GS algorithms are shown. The modified GS algorithm is also used for phase-contrast imaging and optical metrology. This method can present quantitative phase-contrast image based on only one in-line hologram. The image reconstructed from experimentally obtained data shows the good agreement with the sample.

Assembly of 3-dimensional structures using programmable holographic optical tweezers

The micromanipulation of objects into 3-dimensional geometries within holographic optical tweezers is carried out using modified Gerchberg-Saxton (GS) and direct binary search (DBS) algorithms to produce the hologram designs. The algorithms calculate sequences of phase holograms, which are implemented using a spatial light modulator, to reconfigure the geometries of optical traps in many planes simultaneously. The GS algorithm is able to calculate holograms quickly from the initial, intermediate and final trap positions. In contrast, the DBS algorithm is slower and therefore used to pre-calculate the holograms, which are then displayed in sequence. Assembly of objects in a variety of 3-D configurations is semi-automated, once the traps in their initial positions are loaded.

Interactive application in holographic optical tweezers of a multi-plane Gerchberg-Saxton algorithm for three-dimensional light shaping

Phase-hologram patterns that can shape the intensity distribution of a light beam in several planes simultaneously can be calculated with an iterative Gerchberg-Saxton algorithm [T. Haist et al., Opt. Commun. 140, 299 (1997)]. We apply this algorithm in holographic optical tweezers. This allows us to simultaneously trap several objects in individually controllable arbitrary 3-dimensional positions. We demonstrate the interactive use of our approach by trapping microscopic spheres and moving them into an arbitrary 3-dimensional configuration.

Iterative algorithm for the design of free-space diffractive optical elements for fiber coupling.

We present a design method based on the Gerchberg-Saxton algorithm for the design of high-performance diffractive optical elements. Results from this algorithm are compared with results from simulated annealing and the iterative Fourier-transform algorithm. The element performance is comparable with those designed by simulated annealing, whereas the design time is similar to the iterative Fourier-transform method. Finally, we present results for a demanding beam-shaping task that was beyond the capabilities of either of the traditional algorithms. The element performances demonstrate greater than 85% efficiency and less than 2% uniformity error.

Wave-function reconstruction of complex fields obeying nonlinear parabolic equations.

We present a generalized Gerchberg-Saxton (GS) algorithm for reconstructing a (2+1)-dimensional complex scalar wave field which obeys a known nonlinear nondissipative parabolic differential equation, given knowledge of the wave-field modulus at three or more values of an evolution parameter such as time. This algorithm is used to recover the complex wave function of a (2+1)-dimensional Bose-Einstein condensate (BEC) from simulated modulus data. The Gross-Pitaveskii equation is used to model the dynamics of the BEC, with the modulus information being provided by a temporal sequence of simulated absorption images of the condensate. The efficacy of the generalized GS algorithm is examined for a wide range of simulation conditions, including strong nonlinearities, vortex states and Poisson noise. The general form of this algorithm, which allows one to reconstruct a time-dependent wave function, will be useful for studying the phase dynamics of topological defects in coherent quantum systems.

Automated spatiotemporal diffraction of ultrashort laser pulses

We exploit the close similarities between time-frequency and position-wave-vector correspondences to control the spatiotemporal diffraction pattern of ultrashort laser pulses. This approach permits novel, automated generation of sophisticated two-dimensional femtosecond waveforms. A two-dimensional space-time version of a Gerchberg-Saxton algorithm is used to iteratively determine the phase pattern in position-frequency space that produces a user-defined intensity profile in wave-vector-time space.

Phase extension in crystallography using the iterative Fienup-Gerchberg-Saxton algorithm and Hilbert transforms.

A procedure for phase extension in electron crystallography is proposed based on the iterative Fienup-Gerchberg-Saxton algorithm in combination with the use of discrete Hilbert transforms. This transform is used to provide oversampling in reciprocal space, thus satisfying the Shannon sampling requirement and introducing reflections with fractional indices. When the procedure is combined with the knowledge of a small set of strong phased Bragg reflections from electron-microscope images (or direct methods), the magnitudes of many non-Bragg reflections can be calculated with useful accuracy, thus enhancing the performance of the iterative algorithm for phase extension. The effects of various constraints used in the iterative algorithm are discussed. In this way, it is shown that the iterative algorithm conventionally used for phasing diffuse scattering from non-periodic objects can also be applied to problems in conventional crystallography to find the phases of high-order (resolution) beams from a known set of low-order (resolution) phases.

Terahertz waveform synthesis via optical rectification of shaped ultrafast laser pulses

We demonstrate the use of optical pulse-shaping technique in conjunction with difference frequency generation in a non-linear optoelectronic crystal for generating synthesized waveforms at terahertz frequencies. Spectral phase modulations, programmed using Gerchberg-Saxton algorithm and prepared in a spatial light Fourier filter, produce tailored terahertz pulses, including chirped pulses, zero-area pulses, and trains of multiple pulses for tunable narrow-band terahertz radiation up to 2.0 THz.

The specular component of light scattering and the characterization of the height distribution for random surfaces

The relations between the specular reflection component of the intensity scattered by random surfaces and the height distributions of the surfaces are analyzed theoretically. In the extraction of the height distribution, both the phase and the amplitude of the specular wave are required. The measured specular intensity data versus the perpendicular component of the wave vector are used for the retrieval of the phase distribution of the specular wave, in which the Gerchberg-Saxton iterative algorithm is employed, and the characterization of the height distribution of random surfaces is accomplished. In the experiment, two samples with Gaussian and quasi-two level height distributions, respectively, are practically measured and the results of the height probability density function obtained by light scattering method are in good accordance with those by atomic force microscopy. The method of this paper is of important significance for the characterizations and studies of random surfaces.

Characterization of ultrashort optical pulses with third-harmonic-generation based triple autocorrelation

We present a method to obtain complete information of femtosecond pulses. By measuring triple-optical autocorrelation directly with third-harmonic generation, without spectral information, a temporal pulse shape can be obtained by analytical calculation without direction-of-time ambiguity. Combining the resulting optical pulse shape with its corresponding optical spectrum, the exact phase and color variations in time can all be recovered with a Gerchberg-Saxton algorithm through an iterative calculation with an O(n) complexity.

Pulse shaping with the Gerchberg-Saxton algorithm

Generation of arbitrarily complex intensity profiles by using phase-only Fourier-domain pulse shaping has thus far been performed by using optimization algorithms to find the optimal phase profile. We present an alternative method based on the Gerchberg–Saxton (GS) algorithm that converges at least several hundred times faster and is independent of the number of phase points and gray levels. The numerical and experimental performance of the GS algorithm is characterized and compared against a genetic algorithm. An application of amplified GS-synthesized waveforms to large-amplitude coherent phonon generation and destruction is demonstrated.

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