Iterative Fourier Research Articles

Energy adjustment pulse shaping algorithm part I: accuracy improvement of phase retrieval IFTA.

Techniques to generate a targeted temporal waveform with high accuracy are desirable to extend the application range for pulse shapers. In this study, a target energy adjustment mechanism is applied to the input-output iterative Fourier transform algorithm (IFTA). It is numerically demonstrated that, considering multi-pulse temporal waveforms, the developed algorithm provides a suitable spectral phase modulation pattern and improves the shape of the temporal waveform compared to that of the input-output IFTA.

Phase-Only Beam Broadening of Contiguous Uniform Subarrayed Arrays

In modern antenna systems, beam broadening of subarrayed arrays provides continuous coverage of a wide angular extent in a cost-effective manner. While many methods have been published that address beam broadening of traditional (non-subarrayed) arrays, there is a knowledge gap in the published literature with respect to efficient beam broadening of contiguous uniform subarrayed arrays. This article presents efficient methods for beam broadening of contiguous uniform subarrayed arrays where the excitation of each element is not individually controlled, but the elements of the array are grouped together as subarrays to have the same element excitations. Particularly, this article focuses on phase-only optimization to preserve maximum power output. Three modilied iterative Fourier transform (IFT) methods and one genetic algorithm (GA) are presented to efliciently search the vast solution space of possible phase settings for a solution that satislies the desired broadened pattern. These methods are evaluated on idealized 1 x 40 and 1 x 80 linear arrays with live element subarrays and 40 x 40 and 80 x 80 element rectangular arrays with 5 x 5 element subarrays. The proposed modilied IFT methods are found to be faster than the GA approach, while the GA approach only offers a few percentage points of better effectiveness.

Rapid phase calibration of a spatial light modulator using novel phase masks and optimization of its efficiency using an iterative algorithm

ABSTRACT We develop an improved phase calibration method of a reflective SLM using interferometry by employing novel phase masks. In the process, we definitively determine the actual maximum phase throw of our SLM which provides a recipe for users to verify supplier specifications. We generate optimised phase masks by using Iterative Fourier Transform Algorithm (IFTA) and compare their performance with global linear corrections in the look-up table (LUT) and find that the former perform with around 20 better efficiency. Besides obtaining an array of 1D/2D spots having high uniformity (90) using IFTA, our result exemplifies the use of iterative algorithms for improving efficiency of phase limited SLMs. Finally, our improved phase calibration method enables threefold faster phase measurements, and to the best of our knowledge, is the first endeavour directed towards enabling rapid phase characterisation of an SLM using interferometric measurements. We believe that it can have very useful applications in settings which may require fast phase calibrations as well as for real-time, multi-wavelength spectroscopic applications.

Coherent diffraction imaging based on iterative fractional Fourier transform

This paper presents a phase retrieval method which is based on a fractional Fourier transform with estimating on-line the order of the transform during the iterative projections between space domain and space-frequency domain. This method overcomes the limitations of the same-kind methods which require a prior knowledge of the accurate order of fractional Fourier transform. Simulations and experiments demonstrated that the proposed method converges faster and achieves less error.

Development of a position–velocity–time-modulated two-dimensional ion beam figuring system for synchrotron x-ray mirror fabrication

With the rapid evolution of synchrotron x-ray sources, the demand for high-quality precision x-ray mirrors has greatly increased. Single nanometer shape accuracy is required to keep imaging capabilities at the diffraction limit. Ion beam figuring (IBF) has been used frequently for ultra-precision finishing of mirrors, but achieving the ultimate accuracy depends on three important points: careful alignment, accurate dwell time calculation and implementation, and accurate optical metrology. The Optical Metrology Group at National Synchrotron Light Source II has designed and built a position-velocity-time-modulated two-dimensional IBF system (PVT-IBF) with three novel characteristics: (1) a beam footprint on the mirror was used as a reference to align the coordinate systems between the metrology and the IBF hardware; (2) the robust iterative Fourier transform-based dwell time algorithm proposed by our group was applied to obtain an accurate dwell time map; and (3) the dwell time was then transformed to velocities and implemented with the PVT motion scheme. In this study, the technical aspects of the PVT-IBF systems are described in detail, followed by an experimental demonstration of the figuring results. In our first experiment, the 2D RMS in a $ 50\;{\rm mm} \times 5\;{\rm mm} $50mm×5mm clear aperture was reduced from 3.4 to 1.1 nm after one IBF run. In our second experiment, due to a 5 mm pinhole installed in front of the source, the 2D RMS in a $ 50\;{\rm mm} \times 5\;{\rm mm} $50mm×5mm clear aperture was reduced from 39.1 to 1.9 nm after three IBF runs, demonstrating that our PVT-IBF solution is an effective and deterministic figuring process.

Fast phase retrieval with a combined method between interferometry and non-interferometry in the holographic data storage

Non-interferometric phase retrieval is a fundamental technique for phase-modulated holographic data storage due to its advantages of easy implementation, simple system setup, and robust noise tolerance. Usually, the iterative algorithm of non-interferometry needs hundreds of iteration numbers to retrieve phase accurately, which decreased the data transfer rate severely. Strong constraint conditions, such as embedded data, can be used on the phase data page to reduce the iteration numbers. However, introducing embedded data will reduce the code rate of the system. We proposed a method that combined the single-shot interferometric method with the non-interferometric iterative Fourier transform algorithm method. We used the phase decoding result by single-shot interferometry as the embedded data in the process of non-interferometry. Therefore, no extra embedded data are needed in the signal code. We realized the code rate improvement as well as keeping fast data transfer rate. In the demonstration, we recorded a four-level phase pattern and retrieved the phase correctly. The bit error rate of phase retrieval is less than 1% within 20 iterations, which proves our approach is practical. In our case, the code rate is increased by two times.

Iterative local Fourier transform-based high-accuracy wavelength calibration for Fourier transform imaging spectrometer.

An iterative local Fourier transform (ILFT)-based high-accuracy wavelength calibration for Fourier transform imaging spectrometer (FTIS) is proposed. The wavelength calibration for FTIS is to determine the relation between the wavelength and the wavenumber position. However, the wavenumber position solved by conventional method is only accurate up to integers restricted by the picket-fence effect of discrete Fourier transform. While the proposed ILFT can increase the accuracy of calculating the wavenumber position by combining the local Fourier transform and a few iterations. In this paper, the method is investigated in theory and then by simulations and experiments. The simulations show that the accuracy of the wavenumber position calculated by the ILFT is increased by 100 times than conventional method with noise, phase error, and non-uniform sampling of optical path difference. And the experimental results indicate that the ILFT decreases the absolute error of wavelength calibration from about 2.03 nm to 0.16 nm. Therefore, the method provides theoretical and technical support for FTIS and promotes the development of superior resolutions therein.

Alleviating the nonlinear effect by narrowing down the differences between the modulus values of the constellation points in all-optical OFDM system

We use the iterative split-step Fourier method to simulate the eight quadrature amplitude modulation all-optical orthogonal frequency division multiplexing (OFDM) system employing different constellation design schemes. It is found that properly narrowing down the modulus difference between constellation points has an impact on the whole system’s ability to resist nonlinear effects. The results show that by means of changing the modulus values of the constellation points, the system bit error rate will be significantly reduced when the optical signal-to-noise ratio is high. Furthermore, by comparing four different constellation diagrams, we propose an alternative solution to reduce the distortion brought by the nonlinear effects on all-optical OFDM systems, that is, the constellation points can be appropriately redesigned to improve the performance of high nonlinear effect systems.

Light shaping from a physical-optics point of view

In the design of optical element for light shaping, a geometric-optics assumption is usually used, where the validity of the assumption is rarely discussed in literature. In this work, the field tracing techniques for modeling light-shaping systems are presented, which reveals the optical element resulted from those geometric-base algorithm is not always accurate enough for the design task. An example is demonstrated with the functional embodiment of the element. The simulation result shows that diffraction effect may occur, especially in paraxial situation. However, the designed result start with the assumption is well-introduced initial guess for further optimization with the iterative Fourier transform algorithm (IFTA).

Security-enhanced optical encryption based on JTC architecture with confused ciphertext

Security is the key issue of an optical cryptosystem. In this work, a security-enhanced optical encryption scheme based on JTC architecture with confused ciphertext is demonstrated. Firstly, the original image is decomposed into two phase-only masks. Then one of the masks is transformed into joint power spectrum (JPS) by using a JTC architecture. Finally, the JPS is confused by operations of rotation and superposition, and the obtained data is employed as the final ciphertext. As a consequence, the proposed scheme removes the linear character of the conventional JTC-based cryptosystem. Therefore, the proposed scheme achieve a high robustness against attacks based on linearity of the cryptosystem, especially the attack based on iterative Fourier transform (IFT). Furthermore, good performances of simplicity and low cost are remained at the same time. Some numerical simulations are presented to verify the validity and performance of the security-enhanced scheme.

Iterative Phase-Only Hologram Generation Based on the Perceived Image Quality

Image quality metrics are a critical element in the iterative Fourier transform algorithms (IFTAs) for the computer generation of phase-only holograms. Conventional image quality metrics such as root-mean-squared error (RMSE) are sensitive to image content and unable to reflect the perceived image quality accurately. This would have a significant impact on the calculation speed and the quality of the generated hologram. In this work, the structure similarity (SSIM) was proposed as an image quality metric in IFTAs due to its ability to predict the perceived image quality in the presence of the white Gaussian noise and its independence on the image content. This would enable IFTAs to terminate when further iterations would no longer lead to improvement in the perceived image quality. As a result, up to 75% of unnecessary iterations were eliminated by the use of SSIM as the image quality metric.

Outgassing-induced acceleration of comet 67P/Churyumov-Gerasimenko

Context. Cometary activity affects the orbital motion and rotation state through sublimation-induced forces. The availability of precise rotation-axis orientation and position data from the Rosetta mission allows us to accurately determine the outgassing of comet Churyumov-Gerasimenko/67P (67P). Aims. We derived the observed non-gravitational acceleration of 67P directly from the trajectory of the Rosetta spacecraft. From the non-gravitational acceleration, we recovered the diurnal outgassing variations and study a possible delay of the sublimation response with respect to the peak of the solar illumination. This allowed us to compare the non-gravitational acceleration of 67P with expectations based on empirical models and common assumptions about the sublimation process. Methods. We used an iterative orbit refinement and Fourier decomposition of the diurnal activity to derive the outgassing-induced non-gravitational acceleration. The uncertainties of the data reduction were established by a sensitivity analysis of an ensemble of best-fit orbits for comet 67P. Results. We find that the Marsden non-gravitational acceleration parameters reproduce part of the non-gravitational acceleration, but need to be augmented by an analysis of the nucleus geometry and surface illumination to draw conclusions about the sublimation process on the surface. The non-gravitational acceleration closely follows the subsolar latitude (seasonal illumination), with a small lag angle with respect to local noon around perihelion. The observed minor changes of the rotation axis do not favor forced precession models for the non-gravitational acceleration. Conclusions. In contrast to the sublimation-induced torques, the non-gravitational acceleration does not place strong constraints on localized active areas on the nucleus. We find a close agreement of the orbit-deduced non-gravitational acceleration and the water production that is independently derived from Rosetta in situ measurements.

Linear Array Thinning Using Probability Density Tapering Approach

In this letter, a novel synthesis algorithm is presented by using a probability density tapering approach in determining element distributions in a thinned array. The probability density tapering is carried out with the iterative Fourier technique (IFT) in array thinning, since the IFT method is proved effective in large array thinning, but the traditional IFT often easily gets trapped in local optima. In the proposed method, namely the probability tapering IFT (PTIFT), the element locations are determined by probability estimation. Then, a probability density taper is used to model the element distributions across the aperture. With iterations, the probability model taper is optimized to approach the global optimal solution. The efficiency of the PTIFT method was validated by some representative examples of array thinning with minimum peak sidelobe level. In addition, null placement in the uniformly illuminated thinned array with low sidelobes was investigated by utilizing the PTIFT.

Synthesis of Monopulse Antenna Patterns for Elliptical Phased Array Antennas With Different Peak Sidelobes Along the Principal Planes

This paper deals with the synthesis of monopulse antenna patterns, sum and difference ones, for planar phased array antennas with an elliptical aperture boundary featuring unequal ultra-low peak sidelobe levels (PSLLs) for the two principal planes of its far field. The used synthesis method is the iterative Fourier transform (IFT) approach. The elliptical planar phased array antenna involved in this synthesis contains 3656 open-ended waveguide radiating elements arranged in a triangular element grid and has an aperture width and height of 50λ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> and 25λ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> , respectively, where λ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> is the wavelength at X-band center frequency 1.0f <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> . The simulated monopulse pattern results will be compared with those of a 4680-element rectangular planar phased array antenna having the same width, height, radiator type and triangular grid as the elliptical array antenna. The monopulse patterns of the rectangular array antenna meet the same unequal ultra-low PSLL requirements as those of the elliptical array antenna. It will be shown that the elliptical array antenna with a slightly enlarged aperture containing 4170 elements features on receive comparable or even better far-field performance characteristics as that of the rectangular array antenna. All reported far-field pattern simulations take mutual coupling between the array elements into account.

Hybrid optimisation method of improved genetic algorithm and IFT for linear thinned array

The array thinning technique can greatly reduce the number of the array elements while keeping the performance of the array almost the same. However, the existing algorithms have slow convergence rates and are easy to fall into local optimum. To improve the optimisation performance, a hybrid method based on improved genetic algorithm (GA) and iterative Fourier transform (IFT) technique for linear thinned array is proposed in this study. The population is divided into improved GA group and IFT group according to the convergence of the population and different operations can be done paralleled to generate offspring in each iteration. In the improved GA processing, adaptive crossover rate and mutation rate are used. The mechanism that keeps the fill factor stable is removed for larger search range. The IFT processing is executed paralleled for fast convergence velocity. The proposed hybrid method can obtain the fast convergence velocity and avoid being trapped into the local optimum by the combination of the two approaches. Several examples are simulated to validate the performance of the proposed method.

Holographic projection based on programmable axilens

In a traditional holographic projection system, a clear reconstructed image is formed only in the focal plane of the Fourier lens. In this paper, We propose a novel method to form simultaneously holographic reconstructed images in good focus located at different planes. The large focal depth characteristic of the programmable axilens is applied to the holographic projection. First, the iterative Fourier algorithm is employed to generate a phase hologram. A new phase distribution is formed by superimposing the axilens phase on the phase hologram, and loading onto the phase-only liquid crystal on silicon (LCoS); With the programmability of the LCoS, the different focal-depth parameters of the axilens are tuned. We could obtain the desired holographic reconstructed images when the collimated laser is used to incident the LCoS. The experimental results show that compared with the holographic projection system based on Fresnel lens, the holographic reconstructed images in good focus can be observed simultaneously on a multi-plane in a range by dynamically changing the focal depth of the axilens.

Phase and amplitude modulation with acoustic holograms

Acoustic holograms are a low cost method for generating arbitrary diffraction limited pressure distributions in 3 dimensions. However, at present, the creation of complex fields using this approach is limited by the inability of these holograms to independently modulate both the phase and amplitude of an incident wave. In this work, it is shown that this limitation can be circumvented by using two phase holograms, designed using an iterative Fourier transform algorithm, to form the phase conjugate of a back-propagated target pattern over a predefined surface. An experimental test-case, designed to generate the letters “UCL” with the uniform amplitude and phase, is prepared to demonstrate the feasibility of this technique. Field measurements from this sample show that the modulation of both the phase and amplitude of the acoustic field can be achieved with this approach.

New design strategy for acoustic holograms demonstrates potential for biomedical applications

The inability of acoustic holograms to independently modulate both phase and amplitude of incident waves is circumvented by multiple phase holograms designed using an iterative Fourier transform algorithm.

Encoding of arbitrary micrometric complex illumination patterns with reduced speckle.

In nonlinear microscopy, phase-only spatial light modulators (SLMs) allow achieving simultaneous two-photon excitation and fluorescence emission from specific region-of-interests (ROIs). However, as iterative Fourier transform algorithms (IFTAs) can only approximate the illumination of selected ROIs, both image formation and/or signal acquisition can be largely affected by the spatial irregularities of the illumination patterns and the speckle noise. To overcome these limitations, we propose an alternative complex illumination method (CIM) able to generate simultaneous excitation of large-area ROIs with full control over the amplitude and phase of light and reduced speckle. As a proof-of-concept we experimentally demonstrate single-photon and second harmonic generation (SHG) with structured illumination over large-area ROIs.

Faceted Fresnel DOEs creating the perception of a floating 3D virtual object under divergent illumination

An approach for the optimization and fabrication of a phase-only faceted Fresnel type diffractive optical element (FDOE) creating 3D virtual object is proposed. The FDOE is a transmissive Fresnel type DOE array, which produces the perception of a customized floating 3D virtual object behind the FDOE when illuminated with a divergent monochromatic Light Emitter Diode (LED) source. Each DOE unit of the FDOE is optimized by a modified iterative Fourier transform algorithm (M-IFTA). Every unit of the FDOE locally deflects the incident light to the same position to form a designated view in the target plane. The FDOE is fabricated using our home-built parallel writing photo-lithography machine. Numerical simulations and optical experiments are performed to verify the proposed design method. This work may find important applications in the advanced design of optical security hologram and anti-counterfeiting component.