On the independent significance of generalizations of the Wigner distribution function.

The Wigner distribution function (WDF) is a significant time-frequency analysis tool in, e.g., the theory of optical coherence and signal processing. Recently, various generalizations of the WDF associated with linear canonical transforms have been proposed to improve and broaden its applications. It is useful to identify which of these novel distributions have independent significance for further investigation. We plot these distributions for a test signal using symbolic integration to find which distributions are linear coordinate transforms of the WDF or have unique features. Five distributions are determined to be linear coordinate transforms of the WDF. Two distributions show unique characteristics. We focus on the mathematical interpretation, properties, and possible applications of those two distributions. We demonstrate how one of them can be used in the analysis of partially coherent systems.

Automated filter selection for suppression of Gibbs ringing artefacts in MRI.

Gibbs ringing creates artefacts in magnetic resonance images that can mislead clinicians. Reconstruction algorithms attempt to suppress Gibbs ringing, or an additional ringing suppression algorithm may be applied post reconstruction. Novel reconstruction algorithms are often compared with filtered Fourier reconstruction, but the choices of filters and filter parameters can be arbitrary and sub-optimal. Evaluation of different reconstruction and post-processing algorithms is difficult to automate or subjective: many metrics have been used in the literature. In this paper, we evaluate twelve of those metrics and demonstrate that none of them are fit for purpose. We propose a novel metric and demonstrate its efficacy in 1D and 2D simulations. We use our new metric to optimise and compare 17 smoothing filters for suppression of Gibbs artefacts. We examine the transfer functions of the optimised filters, with counter-intuitive results regarding the highest-performing filters. Our results will simplify and improve the comparison of novel MRI reconstruction and post-processing algorithms, and lead to the automation of ringing suppression in MRI. They also apply more generally to other applications in which data is captured in the Fourier domain.

Fast, sub-pixel accurate, displacement measurement method: optical and terahertz systems.

A novel (to the best of our knowledge), fast method to measure in-plane object motion in 1D with sub-pixel accuracy which complements the correlation technique is proposed. The method is verified experimentally using both visible and terahertz images. The absolute sum of grey level accumulated change is used to quantify object motion. The method requires calibration for each target, but only addition and subtraction operations. This results in a decrease of two orders of magnitude in the computation time.

Optical Trajectory Manipulations Using the Self-Written Waveguide Technique.

This study is novel for several reasons: We used a thin drop cast layer of dry photosensitive materials to study the behaviors of wet photopolymer media using microscopic distances during the Self-Written Waveguide (SWW) process; then, we examined the self-trajectories formed inside the solid material. The results provide a framework for theoretical and experimental examinations by handling the effects of manipulating the alignment of fibers. The other main advantage of these techniques is their lightweight, easy to process, highly flexible, and ultimately low-cost nature. First, the SWW process in wet photopolymer media (liquid solutions) was examined under three cases: single-, counter-, and co-fiber exposure. Then, the SWWs formed inside the solid material were examined along with the effects of manipulating the alignment of the fibers. In all cases, high precision measurements were used to position the fiber optic cables (FOCs) before exposure using a microscope. The self-writing process was indirectly monitored by observing (imaging) the light emerging from the side of the material sample during SWW formation. In this way, we examined the optical waveguide trajectories formed in Acrylamide/Polyvinyl Alcohol (AA/PVA), a photopolymer material (sensitized at 532 nm). First, the transmission of light by this material is characterized. Then, the bending and merging of the waveguides that occur are investigated. The predictions of our model are shown to qualitatively agree with the observed trajectories. The largest index changes taking place at any time during exposure, i.e., during SWW formation, are shown to take place at the positions where the largest exposure light intensity is present. Typically, such maxima exist close to the input face. The first maximum is referred to as the location of the Primary Eye. Other local maxima also appear further along the SWW and are referred to as Secondary Eyes, i.e., eyes deeper within the material.

Multi-layered full-field phase imaging using continuous-wave terahertz ptychography.

Due to the unique properties of terahertz (THz) waves, THz phase imaging has been widely investigated to retrieve the absorption and phase modulation of dielectric two-dimensional thin samples, as well as multiple stacked samples. In this Letter, we apply the three-dimensional ptychographic iterative engine algorithm for continuous-wave THz full-field multi-layered phase imaging. The complex-valued transmission function of two-layered polypropylene thin plates and the corresponding probe function are reconstructed, respectively, which are immune to crosstalk of different layers. The phenomenon of the field-of-view enlargement at the second object layer is observed. This lensless compact imaging method can be potentially used for THz three-dimensional imaging.

Label-free color staining of quantitative phase images of biological cells by simulated Rheinberg illumination.

Modern microscopes are designed with functionalities that are tailored to enhance image contrast. Dark-field imaging, phase contrast, differential interference contrast, and other optical techniques enable biological cells and other phase-only objects to be visualized. Quantitative phase imaging refers to an emerging set of techniques that allow for the complex transmission function of the sample to be measured. With this quantitative phase image available, any optical technique can then be simulated; it is trivial to generate a phase contrast image or a differential interference contrast image. Rheinberg illumination, proposed almost a century ago, is an optical technique that applies color contrast to images of phase-only objects by introducing a type of optical staining via an amplitude filter placed in the illumination path that consists of two or more colors. In this paper, the complete theory of Rheinberg illumination is derived, from which an algorithm is proposed that can digitally simulate the technique. Results are shown for a number of quantitative phase images of diatom cells obtained via digital holographic microscopy. The results clearly demonstrate the potential of the technique for label-free color staining of subcellular features.

Orthographic projection images-based photon-counted integral Fourier holography.

Unlike coherent imaging techniques, light field imaging uses incoherent (white light) illumination to generate a digital hologram of three-dimensional (3D) objects in real time. Multiple projections (or elemental images) of a 3D object are captured using a microlens array attached to a digital camera. Orthographic projection images (OPIs) can be synthesized from the recorded elemental images. The synthesized intensity-based OPIs are then multiplied by the corresponding phase functions and combined to form a digital hologram (also known as an integral hologram) of a 3D object under illumination. In this study, we analyze the performance of a synthesized integral hologram under low light imaging (photon-counting) conditions. The feasibility of this technique is verified experimentally by capturing the elemental images and subsequently generating orthographic projection images and by varying photon counts to reconstruct the digital holograms.

Self-written waveguides in photopolymer.

Self-written waveguide (SWW) trajectories fabricated inside a dry photopolymer bulk material, acrylamide/polyvinyl alcohol (AA/PVA), are studied. Their production using both Gaussian and Laguerre-Gauss exposing (writing) light beams, output from optical fibers, is explored. The formation of the primary and secondary eyes is also discussed. Furthermore, the interactions that take place when two counterpropagating beams pass through the photopolymer material (both Gaussian and Laguerre-Gauss) are examined. In all cases experimental and theoretical results are presented. Good agreement between the predictions of the proposed model and experimental observations are demonstrated.

Low photon count based digital holography for quadratic phase cryptography.

Recently, the vulnerability of the linear canonical transform-based double random phase encryption system to attack has been demonstrated. To alleviate this, we present for the first time, to the best of our knowledge, a method for securing a two-dimensional scene using a quadratic phase encoding system operating in the photon-counted imaging (PCI) regime. Position-phase-shifting digital holography is applied to record the photon-limited encrypted complex samples. The reconstruction of the complex wavefront involves four sparse (undersampled) dataset intensity measurements (interferograms) at two different positions. Computer simulations validate that the photon-limited sparse-encrypted data has adequate information to authenticate the original data set. Finally, security analysis, employing iterative phase retrieval attacks, has been performed.

A Review of Hologram Storage and Self-Written Waveguides Formation in Photopolymer Media.

Photopolymer materials have received a great deal of attention because they are inexpensive, self-processing materials that are extremely versatile, offering many advantages over more traditional materials. To achieve their full potential, there is significant value in understanding the photophysical and photochemical processes taking place within such materials. This paper includes a brief review of recent attempts to more fully understand what is needed to optimize the performance of photopolymer materials for Holographic Data Storage (HDS) and Self-Written Waveguides (SWWs) applications. Specifically, we aim to discuss the evolution of our understanding of what takes place inside these materials and what happens during photopolymerization process, with the objective of further improving the performance of such materials. Starting with a review of the photosensitizer absorptivity, a dye model combining the associated electromagnetics and photochemical kinetics is presented. Thereafter, the optimization of photopolymer materials for HDS and SWWs applications is reviewed. It is clear that many promising materials are being developed for the next generation optical applications media.

Constraints on additivity of the 1D discrete linear canonical transform.

The continuous linear canonical transform (LCT) can describe a wide variety of paraxial (quadratic phase) first-order optical systems. Digital algorithms to numerically calculate the LCT are therefore important in modeling the field propagations and are also of interest for many digital signal-processing applications. The continuous LCT is additive (and unitary), but discretization can destroy additivity. In this paper, the general constraint sufficient to ensure the discrete LCTs are additive is derived. Often, we wish to decompose the transform into a series of more computationally efficient steps. Having previously discussed the unitarity of such algorithms, in this paper we consider how our additivity constraint applies to the direct method (DM) and spectral method (SM) algorithms. Examples are presented showing how to correct nonadditive calculations and to appropriately choose parameters.

Two-dimensional nonseparable linear canonical transform: sampling theorem and unitary discretization.

The two-dimensional (2D) nonseparable linear canonical transform (NS-LCT) is a unitary, linear integral transform that relates the input and output monochromatic, paraxial scalar wave fields of optical systems characterized by a 4×4 ray tracing matrix. In addition to the obvious generalizations of the 1D LCT (which are referred to as separable), the 2D-NS-LCT can represent a variety of nonaxially symmetric optical systems including the gyrator transform and image rotation. Unlike the 1D LCT, the numerical approximation of the 2D-NS-LCT has not yet received extensive attention in the literature. In this paper, (1) we develop a sampling theorem for the general 2D-NS-LCT which generalizes previously published sampling theorems for the 1D case and (2) we determine which sampling rates may be chosen to ensure that the obvious discrete transform is unitary.

Unitary discrete linear canonical transform: analysis and application.

The numerical approximation of the linear canonical transforms (LCTs) is important in modeling coherent wave field propagation through first-order optical systems and in many digital signal processing applications. The continuous LCTs are unitary, but discretization can destroy this property. We present a sufficient condition on the sampling rates chosen in the discretization to ensure unitarity. We discuss the various subsets of the unitary matrices examined in this paper that have been proposed elsewhere. We offer a proof of the existence of all of the unitary matrices we discuss. We examine the consequences of these results, particularly in relation to the use of discrete transforms in iterative phase retrieval applications.

Digital computation of the complex linear canonical transform.

An efficient algorithm for the accurate computation of the linear canonical transform with complex transform parameters and with complex output variable is presented. Sampling issues are discussed and the requirements for different cases given. Simulations are provided to validate the results.

Space-bandwidth ratio as a means of choosing between Fresnel and other linear canonical transform algorithms.

The product of the spatial and spatial frequency extents of a wave field has proven useful in the analysis of the sampling requirements of numerical simulations. We propose that the ratio of these quantities is also illuminating. We have shown that the distance at which the so-called "direct method" becomes more efficient than the so-called "spectral method" for simulations of Fresnel transforms may be written in terms of this space-bandwidth ratio. We have proposed generalizations of these algorithms for numerical simulations of general ABCD systems and derived expressions for the "transition space-bandwidth ratio," above which the generalization of the spectral method is the more efficient algorithm and below which the generalization of the direct method is preferable.

Cross terms of the Wigner distribution function and aliasing in numerical simulations of paraxial optical systems.

Sampling a function periodically replicates its spectrum. As a bilinear function of the signal, the associated Wigner distribution function contains cross terms between the replicas. Often neglected, these cross terms affect numerical simulations of paraxial optical systems. We develop expressions for these cross terms and show their effect on an example calculation.

Reevaluation of the direct method of calculating Fresnel and other linear canonical transforms.

The linear canonical transform may be used to simulate the effect of paraxial optical systems on wave fields. Using a recent definition of the discrete linear canonical transform, phase space diagram analyses of the sampling requirements of the direct method of calculating the Fresnel and other linear canonical transforms are more favorable than previously thought. Thus the direct method of calculating these transforms may be used with fewer samples than previously reported simply by making use of an appropriate reconstruction filter on the samples output by the algorithm.

Fast linear canonical transforms.

The linear canonical transform provides a mathematical model of paraxial propagation though quadratic phase systems. We review the literature on numerical approximation of this transform, including discretization, sampling, and fast algorithms, and identify key results. We then propose a frequency-division fast linear canonical transform algorithm comparable to the Sande-Tukey fast Fourier transform. Results calculated with an implementation of this algorithm are presented and compared with the corresponding analytic functions.

Additional sampling criterion for the linear canonical transform.

The linear canonical transform describes the effect of first-order quadratic phase optical systems on a wave field. Several recent papers have developed sampling rules for the numerical approximation of the transform. However, sampling an analog function according to existing rules will not generally permit the reconstruction of the analog linear canonical transform of that function from its samples. To achieve this, an additional sampling criterion has been developed for sampling both the input and the output wave fields.

Cases where the linear canonical transform of a signal has compact support or is band-limited.

A signal may have compact support, be band-limited (i.e., its Fourier transform has compact support), or neither ("unbounded"). We determine conditions for the linear canonical transform of a signal having these properties. We examine the significance of these conditions for special cases of the linear canonical transform and consider the physical significance of our results.