The linear canonical transforms (LCTs) are a Lie group of transforms including the Fresnel and Fourier transforms that describe scalar wave propagation in quadratic phase systems. As such, they are useful in system analysis and design, and their discretisations are important for opto-numerical systems, e.g. numerical reconstruction algorithms in digital holography. An important topic in the literature is therefore the generalization of Fourier transform properties for the LCTs. A number of authors have proposed convolution theorems for the linear canonical transform, with different goals in mind. In this paper, we compare those methods, with particular attention being paid to the consequences of discretization. In a similar way to how discrete convolution associated with the DFT differs from that associated with the Fourier transform, we must take the chirp-periodic nature of discrete LCTs into account when determining the discrete convolution associated with LCTs. This work is of significance for the simulation of VanderLugt correlators, which have been used for optical implementations of neural networks, and for optical filtering operations and coherent optical signal processing in general.
The two-dimensional non-separable linear canonical transform (2D-NS-LCT) can model a wide range of paraxial optical systems. Digital algorithms to calculate the 2D-NS-LCTs are of great interested in both light propagation modeling and digital signal processing. We have previously reported that the transform of a 2D image with rectangular sampling grid generally results in a parallelogram output sampling grid, thus complicating further calculations. One possible solution is to use interpolation techniques. However, it usually leads to poor calculation speed and reduced accuracy. To alleviate this problem, we previously proposed a unitary algorithm by choosing an advantageous sampling rate related to the system parameters. In this paper, a fast algorithm is further proposed based on a novel matrix decomposition, which can significantly improve the efficiency of the numerical approximations.
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.
Propagation and diffraction of a light beam through nonlinear materials are effectively compensated by the effect of selftrapping. The laser beam propagating through photo-sensitive polymer PVA/AA can generate a waveguide of higher refractive index in direction of the light propagation. In order to investigate this phenomenon occurring in light-sensitive photopolymer media, the behaviour of a single light beam focused on the front surface of photopolymer bulk is investigated. As part of this work the self-bending of parallel beams separated in spaces during self-writing waveguides are studied. It is shown that there is strong correlation between the intensity of the input beams and their separation distance and the resulting deformation of waveguide trajectory during channels formation. This self-channeling can be modelled numerically using a three-dimension model to describe what takes place inside the volume of a photopolymer media. Corresponding numerical simulations show good agreement with experimental observations, which confirm the validity of the numerical model that was used to simulate these experiments.
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 accurate measurement of optical phase has many applications in metrology. For biological samples, which appear transparent, the phase data provides information about the refractive index of the sample. In speckle metrology, the phase can be used to estimate stress and strains of a rough surface with high sensitivity. In this theoretical manuscript we compare and contrast the properties of two techniques for estimating the phase distribution of a wave field under the paraxial approximation: (I) A digital holographic system, and (II) An idealized phase retrieval system. Both systems use a CCD or CMOS array to measure the intensities of the wave fields that are reflected from or transmitted through the sample of interest. This introduces a numerical aspect to the problem. For the two systems above we examine how numerical calculations can limit the performance of these systems leading to a near-infinite number of possible solutions.
A practical technique is presented based on DH, for the reconstruction of a wavefront from three recorded intensity images. Combining the off-axis Fourier spatial filtering (OFSF) technique with iterative phase retrieval algorithms, it is shown how the twin image can be eliminated. The proposed method overcomes system geometry constraints and improves both the flexibility and resolution associated with OFSF-based DH. It also overcomes the cost problem associated with phase-shifting interferometry-based DH. In order to demonstrate the performance of the proposed DH method, both simulation and experiment results for objects having smooth and rough surfaces are presented.
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.
In this paper, a new practical technique is presented based on digital holography, for the reconstruction of a wave front from three intensity recordings. Combining the off-axis Fourier filtering technique with boundary detection and iterative phase retrieval algorithms, it is shown how problems such as elimination of the twin image can be overcome. The proposed methods deal with also the issues of feasibility and accuracy associated with off-axis Fourier spatial filtering (OFSF), and those of cost and alignment associated with phase shifting interferometry (PSI). Problems associated with working with diffuse objects are also overcome.
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 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.