Recently, a method for synthesizing a hologram of three dimensional (3D) objects from captured light field array is demonstrated. The 3D objects can be captured under incoherent light illumination using a micro lens array and their orthographic projection view images are generated from the captured elemental images. The synthesized orthographic projection view images are then multiplied by the corresponding phase functions and combined to form a digital hologram. For the first time, we analysis the performance of synthesized hologram under photon counting (low light imaging) conditions. The feasibility of this technique is experimentally verified by recording the orthographic projection images using a micro lens array and the reconstructed photon counted hologram is presented with varying photoncounting measurements.
We calculate the forces of a single-beam optical trap, also known as an "optical tweezers," on micron-sized dielectric spheres based on the wave optics regime. These forces have previously been calculated using a geometrical ray optics approach by Ashkin. The relatively simple approach proposed by Ashkin is based on tracing a set of equal intensity rays from a uniformly sampled microscope objective lens aperture, all passing through a single point in space, the focal point of the lens. This model does not take into account the wave nature of light and is, therefore, it cannot account for the effects of diffraction Here we propose the use of the angular spectrum method, based on the fast Fourier transform algorithm, to calculate the scalar wavefield on the surface of a microsphere, from which ray amplitudes and directions can be estimated. This allows for the Ashkin’s method to calculate forces on trapped spherical particle to be amended to account for the effects of diffraction. Numerical results are presented for a laser power of 10mW and a microscope objective with a numerical aperture of 1.25, and compared with those obtained using the traditional geometrical approach.
The propagation of a light beam through a photo-sensitive photopolymer Polyvinyl Alcohol/Acrylamide (PVA/AA), and the creation of self-written waveguides (SWWs), has received much attention. Here we explore the manufacture and characterization of SWWs in PVA/AA for applications at near infrared communication wavelengths 850nm and 1300nm. The SWWs are fabricated using visible light at wavelength 532nm. The insertion and optical loss of the SWWs at different wavelengths will be interrogated. An optical loss and attenuation profile is to be built up for each of the three wavelengths as they propagate down the resulting SWWs.
The linear canonical transform (LCT) is used in modeling a coherent light-field propagation through first-order optical systems. Recently, a generic optical system, known as the quadratic phase encoding system (QPES), for encrypting a two-dimensional image has been reported. In such systems, two random phase keys and the individual LCT parameters (α,β,γ) serve as secret keys of the cryptosystem. It is important that such encryption systems also satisfy some dynamic security properties. We, therefore, examine such systems using two cryptographic evaluation methods, the avalanche effect and bit independence criterion, which indicate the degree of security of the cryptographic algorithms using QPES. We compared our simulation results with the conventional Fourier and the Fresnel transform-based double random phase encryption (DRPE) systems. The results show that the LCT-based DRPE has an excellent avalanche and bit independence characteristics compared to the conventional Fourier and Fresnel-based encryption systems.
Proc. SPIE. 10233, Holography: Advances and Modern Trends V
KEYWORDS: Analytics, Holograms, Digital signal processing, Solar energy, Digital holography, Data modeling, Fourier transforms, Data conversion, Electronics engineering, Signal analyzers, Radium, Systems modeling, Digital Light Processing
The 2D non-separable linear canonical transform (2D-NS-LCT) can model a range of various paraxial optical systems. Digital algorithms to evaluate the 2D-NS-LCTs are important in modeling the light field propagations and also of interest in many digital signal processing applications. In [Zhao 14] we have reported that a given 2D input image with rectangular shape/boundary, in general, results in a parallelogram output sampling grid (generally in an affine coordinates rather than in a Cartesian coordinates) thus limiting the further calculations, e.g. inverse transform. One possible solution is to use the interpolation techniques; however, it reduces the speed and accuracy of the numerical approximations. To alleviate this problem, in this paper, some constraints are derived under which the output samples are located in the Cartesian coordinates. Therefore, no interpolation operation is required and thus the calculation error can be significantly eliminated.
Digital holographic microscopy (DHM) is an optoelectronic technique that is made up of two parts: (i) the recording of the interference pattern of the diffraction pattern of an object and a known reference wavefield using a digital camera and (ii) the numerical reconstruction of the complex object wavefield using the recorded interferogram and a distance parameter as input. The latter is based on the simulation of optical propagation from the camera plane to a plane at any arbitrary distance from the camera. A key advantage of DHM over conventional microscopy is that both the phase and intensity information of the object can be recovered at any distance, using only one capture, and this facilitates the recording of scenes that may change dynamically and that may otherwise go in and out of focus. Autofocusing using traditional microscopy requires mechanical movement of the translation stage or the microscope objective, and multiple image captures that are then compared using some metric. Autofocusing in DHM is similar, except that the sequence of intensity images, to which the metric is applied, is generated numerically from a single capture. We recently investigated the application of a number of sparsity metrics for DHM autofocusing and in this paper we extend this work to include more such metrics, and apply them over a greater range of biological diatom cells and magnification/numerical apertures. We demonstrate for the first time that these metrics may be grouped together according to matching behavior following high pass filtering.
The 2D non-separable linear canonical transform (2D-NS-LCT) can describe a variety of paraxial optical systems. Digital algorithms to numerically evaluate the 2D-NS-LCTs are not only important in modeling the light field propagations but also of interest in various signal processing based applications, for instance optical encryption. Therefore, in this paper, for the first time, a 2D-NS-LCT based optical Double-random- Phase-Encryption (DRPE) system is proposed which offers encrypting information in multiple degrees of freedom. Compared with the traditional systems, i.e. (i) Fourier transform (FT); (ii) Fresnel transform (FST); (iii) Fractional Fourier transform (FRT); and (iv) Linear Canonical transform (LCT), based DRPE systems, the proposed system is more secure and robust as it encrypts the data with more degrees of freedom with an augmented key-space.
Terahertz radiation lies between the microwave and infrared regions in the electromagnetic spectrum. Emitted frequencies range from 0.1 to 10 THz with corresponding wavelengths ranging from 30 μm to 3 mm. In this paper, a continuous-wave Terahertz off-axis digital holographic system is described. A Gaussian fitting method and image normalisation techniques were employed on the recorded hologram to improve the image resolution. A synthesised contrast enhanced hologram is then digitally constructed. Numerical reconstruction is achieved using the angular spectrum method of the filtered off-axis hologram. A sparsity based compression technique is introduced before numerical data reconstruction in order to reduce the dataset required for hologram reconstruction. Results prove that a tiny amount of sparse dataset is sufficient in order to reconstruct the hologram with good image quality.
The linear canonical transform (LCT) is essential in modeling a coherent light field propagation through first-order optical systems. Recently, a generic optical system, known as a Quadratic Phase Encoding System (QPES), for encrypting a two-dimensional (2D) image has been reported. It has been reported together with two phase keys the individual LCT parameters serve as keys of the cryptosystem. However, it is important that such the encryption systems also satisfies some dynamic security properties. Therefore, in this work, we examine some cryptographic evaluation methods, such as Avalanche Criterion and Bit Independence, which indicates the degree of security of the cryptographic algorithms on QPES. We compare our simulation results with the conventional Fourier and the Fresnel transform based DRPE systems. The results show that the LCT based DRPE has an excellent avalanche and bit independence characteristics than that of using the conventional Fourier and Fresnel based encryption systems.
Digital holographic microscopy is an optic-electronic technique that enables the numerical reconstruction of the complex wave-field reflected from, or transmitted through, a target. Together with phase unwrapping, this method permits a height profile, a thickness profile, and/or a refractive index profile, to be extracted, in addition to the reconstruction of the image intensity. Digital holographic microscopy is unlike classical imaging systems in that one can obtain the focused image without situating the camera in the focal plane; indeed, it is possible to recover the complex wave-field at any distance from the camera plane. In order to reconstruct the image, the captured interference pattern is first processed to remove the virtual image and DC component, and then back-propagated using a numerical implementation of the Fresnel transform. A necessary input parameter to this algorithm is the distance from the camera to the image plane, which may be measured independently, estimated by eye following reconstruction at multiple distances, or estimated automatically using a focus metric. Autofocus algorithms are commonly used in microscopy in order to estimate the depth at which the image comes into focus by manually adjusting the microscope stage; in digital holographic microscopy the hologram can be reconstructed at multiple depths, and the autofocus metric can be evaluated for each reconstructed image intensity. In this paper, fifteen sparsity metrics are investigated as potential focus metrics for digital holographic microscopy, whereby the metrics are applied to a series of reconstructed intensities. These metrics are tested on the hologram of a biological cell. The results demonstrate that many of the metrics produce similar profiles, and groupings of the metrics are proposed.
Proc. SPIE. 9599, Applications of Digital Image Processing XXXVIII
KEYWORDS: Modeling, Digital signal processing, Digital image processing, Digital holography, Matrices, Fourier transforms, Wave propagation, Electronics engineering, Direct methods, Communication engineering
The continuous linear canonical transforms (LCT) can describe a wide variety of wave field propagations through paraxial (first order) optical systems. Digital algorithms to numerically calculate the LCT are therefore important in modelling scalar wave field propagations and are also of interest for many digital signal processing applications. The continuous LCT is additive, but discretization can remove this property. In this paper we discuss three special cases of the LCT for which constraints can be identified to ensure the DLCT is additive.
Motion detection retains a fascinating amount of scope for new ideas. Digital holography enables three-dimensional images to be recorded with high resolution and large depth of field. Appropriate design of a holographic target has the potential to unlock significant resolution gains from holographic measurement systems. In this paper, we explore the potential of proposed three-dimensional analogues of Costas arrays for the design of such targets. These innately sparse constructions have the potential to enable the use of additional compressive sensing techniques to unlock further gains in resolution.
Proc. SPIE. 9216, Optics and Photonics for Information Processing VIII
KEYWORDS: Optical design, Digital holography, Fourier transforms, Wave propagation, Signal processing, Fractional fourier transform, System on a chip, Wigner distribution functions, Americium, Gyrators
The continuous linear canonical transforms is known to describes wave field propagation through paraxial (quadratic phase) optical systems. Digital algorithms to numerically calculate the LCT are therefore important in modelling field propagation through first order optical systems and are also of interest for many purely digital signal processing applications. Significantly the continuous LCTs are unitary, but discretization can destroy this property resulting in a loss of conservative properties. Previously we presented a sufficient condition on the sampling rates chosen during discretization to ensure that digital implementations of the 1D and 2D separable LCTs were unitary. In this paper we extend our analysis to discuss the cases of the 2D non-separable LCT which are used to describe non-orthogonal, nonaxially symmetric and anamorphic systems. We also examine the consequences of ours results.
The Wigner distribution function (WDF) has been used as a tool in wave optics for more than forty years. It is desirable to numerically simulate the WDF for a variety of situations; we argue that the reasons this is not more commonly done are the difficulty in defining the discrete transform appropriately and the size of the computation. In this paper, we examine a number of software packages freely available online, each purporting to calculate the WDF. We present results on their speed and accuracy. Optical engineers desiring to make use of the WDF in optical analysis and design will find our results useful in choosing which package to use in their simulations.
Subpixel methods increase the accuracy and efficiency of image detectors, processing units, and algorithms and provide very cost-effective systems for object tracking. A recently proposed method permits micropixel and submicropixel accuracies providing certain design constraints on the target are met. In this paper, we explore the use of Costas arrays - permutation matrices with ideal auto-ambiguity properties - for the design of such targets.
Digital holographic microscopy is suitable for the detection of microbial particles in a rapidly flowing fluid since in this technique the focusing can be carried out as post-processing of a single captured image. This image, known as a digital hologram, contains the full complex wave front information emanating from the object which forms an interference pattern with a known reference beam. Post-processing is computationally intense and it constitutes a bottleneck for real time inspection of fast moving scenes. In the current work, GPU computation is used to accelerate the post-processing of the holographic images captured by digital holographic microscopy. Efficiency and reliability of a pre-processing step in order to eliminate low information content holographic images is also investigated.
The two-dimensional non-separable linear canonical transform (2D-NS-LCT) involves a significant generalization of the separable LCT (S-LCT), since it can represent orthogonal and non-orthogonal first order optical systems. Thus the availability of a discrete numerical approximation of the 2D-NS-LCT is important as it permits the modelling of a broad class of optical systems. The continuous 2D-NS-LCTs are unitary, but discretization can destroy this property. In this paper, we discuss the condition on the sampling chosen in the discretization, under which some special cases of the discrete 2D-NS-LCTs are unitary. The results presented here provide a basis for the discussion of the general condition for the discrete 2D-NS-LCT to be unitarity.
Luneburg's first order optical systems consist of sections of free space, lenses, and all possible combinations of these.
The linear canonical transform (LCT), a parameterised linear integral transform, may be used to model the paraxial
propagation of scalar optical fields through such systems. We consider the propagation of quasi-monochromatic,
coherent wave fields, though more general calculations are possible. Numerical approximation of such systems is an
active area of research, of interest for system design and analysis. We consider methods for the determination of the
sampling requirements for the wave fields at the input and output of such calculations, in conjunction with the
discretisation of the transform. We illustrate these considerations using phase space diagrams (PSDs), making use of the
LCT's simple co-ordinate transforming effect on such diagrams. We discuss the implications of the cross-terms present
in the Wigner distribution function, which are ignored in such PSD-based analyses, for the accuracy of the simulations
and for the selection of sampling schemes. We examine the available algorithms for performing the transformations in
O(N log N) time. In particular, we consider the relative merits of algorithms which decompose the optical system into
special cases for which fast algorithms are better developed and also algorithms which decompose the calculations into
smaller ones iteratively.
A new sampling criterion for the linear canonical transform (LCT) was recently proposed by the authors. This criterion
was based on an analysis of the consequences of sampling the LCT of a discrete signal. Previous LCT sampling work in
the literature considered only sampling the LCT of continuous, band limited signals. This analysis has great significance
for numerical simulations of first order optical systems using the LCT, as these simulations must necessarily consider a
sampled input function, and evaluate only a sampled output function. It is also significant for the analysis of periodic
structures such as gratings. We present a review of this new sampling criterion and the associated analysis. We clarifying
its meaning and consider its applications. In particular, we consider the consequences of the analysis for fast LCT
algorithms, and for the use of the discrete Fresnel transform in digital holography. We conclude that the spectral method
of calculating Fresnel transforms may benefit greatly from recent advances to LCT sampling theory.
The linear canonical transform (also known as the quadratic phase integral and the special affine Fourier transform, among others) is an important tool for the modeling of quadratic phase systems for coherent optical signal processing, as it is a generalization of a number of important and widely used transforms such as the Fresnel transform, the Fourier transform and the fractional Fourier transform. We consider properties of the linear canonical transform which are important for numerical approximation of the integral transform, and thus for simulation of the related paraxial optical
systems. Some of these properties have been previously developed in the literature, but are analyzed here in the context of linear canonical transform simulations, others are developed here for the first time. We examine these properties analytically, including how the support and bandwidth of the signal are related to transform parameters, a review of sampling issues and some new proposals in this area. Finally, we examine the effect of the linear canonical transform on the sparsity of signals, which is useful for efficient transmission or storage or to aid certain signal processing tools such
as blind source separation.
KEYWORDS: Holography, Imaging systems, Signal attenuation, Fourier transforms, Linear filtering, Wave propagation, Fractional fourier transform, Solar energy systems, Systems modeling, Wigner distribution functions
In the paraxial limit, optical systems can be well described as ABCD systems, which are linear, lossless systems and can
be well modeled using the Linear Canonical Transform (LCT). In theory, their effects are perfectly reversible. In
practice, finite component sizes mean that a system designer must be aware that a waveform passing through a practical
implementation of such a system may lose information due to walk-off and apertures, and make allowances accordingly.
Such limitations also place restrictions on the bandwidth of a waveform which may be propagated through a system.
These considerations are also very important from the point of view of attempting to simulate such systems for design or
analysis. We consider the parameters of a system which result in low loss due to this factor, demonstrating the
parameters which result in lossless systems for particular types of signal. We offer mathematical proof that certain
classes of two-parameter LCTs preserve bandwidth or compact support, or transform one into the other. We propose a
matrix-based methodology for minimizing aperture effects in ABCD systems.
KEYWORDS: Lenses, Fourier transforms, Free space, Numerical analysis, Wave propagation, Free space optics, Fractional fourier transform, Solar energy systems, Systems modeling, Wigner distribution functions
The numerical simulation of linear, lossless, paraxial, quadratic phase systems using the linear canonical transform
(LCT) presents significant challenges due to the high sampling rate associated with the chirp function in the kernel.
However, such simulations are significant for the design and analysis of optical systems and other signal processing
purposes. Furthermore, such systems can be optically implemented using only lenses and free space. In this paper, we
implement a recently proposed technique for calculating the discrete LCT. We review the existing literature on sampling
the linear canonical transform. We apply a space-spatial frequency representation of the signal, the Wigner-Ville
distribution function (WDF), and the space bandwidth product (SBP) of the signal (or system) to identify suitable
sampling rates for simulation. We apply this method to simulate the effect on a 1D rectangular waveform of the
fractional Fourier transform (FRT). The numerical results are compared with analytical expressions for the same system.
Paraxial optical systems implemented entirely with thin lenses and propagation through free space and/or Graded Index (GRIN) media are quadratic phase systems (QPS). The effect of any arbitrary QPS on an input wavefield can be described using Linear Canonical Transform (LCT). In this paper, we examine a novel numerical implementation of Fast Linear Canonical Transform (FLCT). We then apply the results in various optical signal processing applications.
Any paraxial optical system which can be implemented using only thin lenses and propagation through free space or through sections of graded index (GRIN) media, belongs to the class of systems known as Quadratic Phase Systems (QPS). Given some input optical wave field, the output of any QPS can be described using the linear canonical
transform (LCT), a unitary, additive, three-parameter class of linear integral transform first discovered in the 1970s. The terminology used in relation to the LCT is not at all consistent across the literature, and it is frequently called by other names, such as Quadratic-phase Integral and Generalized Fresnel Transform. In this paper, we examine a new, more flexible numerical implementation of the FLCT. This algorithm is similar to the Sande-Tukey FFT algorithm, and is of general radix. We demonstrate the savings possible in terms of required samples with the flexibility inherent in a general radix algorithm.