Simulating first order optical systems—algorithms for and composition of discrete linear canonical transforms

Linear canonical transforms (LCTs) are a family of integral transforms with wide application in optical, acoustical, electromagnetic, and other wave propagation problems. The Fourier and fractional Fourier transforms are special cases of LCTs. We present the exact relation between continuous and discrete LCTs (which generalizes the corresponding relation for Fourier transforms), and also express it in terms of a new definition of the discrete LCT (DLCT), which is independent of the sampling interval. This provides the foundation for approximately computing the samples of the LCT of a continuous signal with the DLCT. The DLCT in this letter is analogous to the DFT and approximates the continuous LCT in the same sense that the DFT approximates the continuous Fourier transform. We also define the bicanonical width product which is a generalization of the time-bandwidth product.

Linear canonical transform (LCT) is very useful and powerful in signal processing and optics. In this paper, discrete LCT (DLCT) is proposed to approximate LCT by utilizing the discrete dilated Hermite functions. The Wigner distribution function is also used to investigate DLCT performances in the time-frequency domain. Compared with the existing digital computation of LCT, our proposed DLCT possess additivity and reversibility properties with no oversampling involved. In addition, the length of input/output signals will not be changed before and after the DLCT transformations, which is consistent with the time-frequency area-preserving nature of LCT; meanwhile, the proposed DLCT has very good approximation of continuous LCT.

Multichannel sampling expansion in the linear canonical transform domain and its application to superresolution

The linear canonical transform (LCT) has been shown to be useful and powerful in signal processing, optics, etc. Many results of this transform are already known, including sampling theory. Most existing sampling theories of the LCT consider the class of bandlimited signals. However, in the real world, many analog signals arising in engineering applications are non-bandlimited. In this correspondence, we propose a sampling and reconstruction strategy for a class of function spaces associated with the LCT, which can provide a suitable and realistic model for real applications. First, we introduce definitions of semi- and fully-discrete convolutions for the LCT. Then, we derive necessary and sufficient conditions pertaining to the LCT, under which integer shifts of a chirp-modulated function generate a Riesz basis for the function spaces. By applying the results, we present a more comprehensive sampling theory for the LCT in the function spaces, and further, a sampling theorem which recovers a signal from its own samples in the function spaces is established. Moreover, some sampling theorems for shift-invariant spaces and some existing sampling theories for bandlimited signals associated with the Fourier transform (FT), the fractional FT, or the LCT are noted as special cases of the derived results. Finally, some potential applications of the derived theory are presented.

Linear canonical transform play an important role in many fields of optics and signal processing society. Well known transforms in these fields such as the Fourier transform, the fractional Fourier transform, and the Fresnel transform can be looked at as special cases of the linear canonical transform. In this paper we obtain new sampling formulae for reconstructing signals that are band-limited or time-limited in the linear canonical transform sense. In the new sampling formulae, we use samples from both the signal and its Hilbert transform, but each taken at half the Nyquist rate of linear canonical transform domain

It has been shown that complex paraxial optical systems, consisting of various lens and distances of free space propagation, can be described using the Linear Canonical Transform (LCT). Indeed it can be shown that many well know optical transforms such as the Optical Fourier Transform (OFT), Optical Fractional Fourier Transform (OFRT), the effect of a lens or Chirp Modulation Transform (CMT) are all subsets of the more general LCT. Using the <i>ABCD</i> Collins matrix formula it is possible to represent these integral transforms in a simpler form, which facilitates system analysis and design. Speckle Photography (SP) in combination with an optical LCT can be used to measure surface motion of an optically rough body. It has previously been shown that Optical FRT's (OFRT) can be used in speckle based metrology systems to vary the range and sensitivity of a metrology system and also to determine both, the magnitude and direction, of tilting (rotation) and translation motion simultaneously, provided that the motion is captured in two separate OFRT domains. In this paper we extend the OFRT analysis to more general LCT systems and demonstrate how simultaneous tilt and translation measurements can be discerned from the speckle images captured prior to, and after motion. A spherical wavefront can be conveniently described using the Collin's matrix notation. By changing the wavefront of the illuminating light we show that we effectively change the domain of the LCT system without changing the bulk elements in the optical system. Thus the complete motion (in-plane translation and small surface tilting) of a rigid body can be determined using one optical LCT system and illuminating fields of varying curvature.

It has been shown that complex paraxial optical systems, consisting of various lens and distances of free space propagation, can be described using the linear canonical transform (LCT). Indeed it can be shown that many well know optical transforms such as the optical fourier transform (OFT), optical fractional fourier transform (OFRT), the effect of a lens or chirp modulation transform (CMT) are all subsets of the more general LCT. Using the ABCD Collins matrix formula it is possible to represent these integral transforms in a simpler form, which facilitates system analysis and design. Speckle Photography (SP) in combination with an optical LCT can be used to measure surface motion of an optically rough body. It has previously been shown that Optical FRT&apos;s (OFRT) can be used in speckle based metrology systems to vary the range and sensitivity of a metrology system and also to determine both, the magnitude and direction, of tilting (rotation) and translation motion simultaneously, provided that the motion is captured in two separate OFRT domains. In this paper we extend the OFRT analysis to more general LCT systems and demonstrate how simultaneous tilt and translation measurements can be discerned from the speckle images captured prior to, and after motion. A spherical wavefront can be conveniently described using the Collin&apos;s matrix notation. By changing the wavefront of the illuminating light we show that we effectively change the domain of the LCT system without changing the bulk elements in the optical system. Thus the complete motion (in-plane translation and small surface tilting) of a rigid body can be determined using one optical LCT system and illuminating holds of varying curvature.

The linear canonical transform (LCT), which is a generalized form of the classical Fourier transform (FT), the fractional Fourier transform (FRFT), and other transforms, has been shown to be a powerful tool in optics and signal processing. Many results of this transform are already known, including its convolution theorem. However, the formulation of the convolution theorem for the LCT has been developed differently and is still not having a widely accepted closed-form expression. In this paper, we first propose a generalized convolution theorem for the LCT and then derive a corresponding product theorem associated with the LCT. The ordinary convolution theorem for the FT, the fractional convolution theorem for the FRFT, and some existing convolution theorems for the LCT are shown to be special cases of the derived results. Moreover, some applications of the derived results are presented.

In this work, we present a new optical double-image encryption method based on a joint transform correlator (JTC) and the linear canonical domain for the simultaneous authentication of two images. This new extension of the encryption system overcame the vulnerability of the method based on the JTC and the conventional 4f-optical processor in the Fourier domain. Although the simultaneous authentication process is satisfied in the Fourier domain, the data content is partially disclosed in false validation. Therefore, we introduce a quadratic phase encryption system of the linear canonical transform (LCT) domain in this method. The linear canonical transform domain adds more degrees of freedom to the security method due to the six LCT orders. In addition, the double-image encryption scheme became secure against intruder attacks, and it was difficult to recognize confidential information during the negative validation process. A cryptanalysis is performed in terms of a chosen-plaintext attack (CPA) and chosen-ciphertext attack (CCA). Numerical simulations demonstrate the feasibility, security, and effectiveness of the proposed system.

Digital holography is an electronic means of measuring the complex amplitude of an optical wavefield using CCD or CMOS arrays. With an appropriate reconstruction algorithm the intensity values registered by the pixels produce images of a particular (object) scene. These holographic systems are very sensitive to even small vibrations or deformations of an object, mainly due to the phase information that is also recovered by the measurement. Hence these systems are useful in a wide array of different metrology problems. It is important that we somehow quantify the information that can be recovered with such a detection scheme; better if we can provide a theoretical framework to optimize an optical system design for a given metrology problem. In this manuscript we show how the Linear Canonical Transform can fulfill this role and can optimally match the space-bandwidth product (SBP) of a signal to be measured with the SBP of a CCD/CMOS detector array. We provide formulae that determine the performance of generalized holographic optical systems (containing lenses and sections of free space), taking into account the finite extent of the CCD array, the size of the pixels, and the spacing between them. Some illustrative examples are presented, with associated numerical simulations.

The statistical properties of speckles in paraxial optical systems depend on the system parameters. In particular, the speckle orientation and the lateral dependence (x and y) of the longitudinal speckle size can vary significantly. For example, the off-axis longitudinal correlation length remains equal to the on-axis size for speckles in a Fourier transform system, while it decreases dramatically as the observation position moves off axis in a Fresnel system. In this paper, we review the speckle correlation function in general linear canonical transform (LCT) systems, clearly demonstrating that speckle properties can be controlled by introducing different optical components, i.e., lenses and sections of free space. Using a series of numerical simulations, we examine how the correlation function changes for some typical LCT systems. The integrating effect of the camera pixel and the impact this has on the measured first- and second-order statistics of the speckle intensities is also examined theoretically. A series of experimental results are then presented to confirm several of these predictions. First, the effect the pixel size has on the measured first-order speckle statistics is demonstrated, and second, the orientation of speckles in a Fourier transform system is measured, showing that the speckles lie parallel to the optical axis.

We present a method to calculate the spatiotemporal electric field distribution of ultrashort pulses focused by an aberration-free lens through a planar interface between free space and a nonlinear medium. The method combines the Fresnel diffraction integral, which is used to model the propagation of the focused pulse in free space, and the angular spectrum propagation method, used to propagate the focused pulse within the nonlinear medium by introducing the irradiance-dependent nonlinear refractive index in the angular spectrum propagator. We have modeled the propagation of ultrashort mildly focused pulses through a Ti:sapphire crystal, characterized only by its linear and nonlinear refractive indices, for pulses with different powers and durations, finding that the proposed method is able to reproduce the self-focusing phenomenon observed in nonlinear media. Our results show that the focal spot within the nonlinear medium is closer to the interface, and it is slightly wider for pulses with higher incident power. However, despite the dependence of the effective refractive index of the nonlinear medium on irradiance, which is the power per unit area, and assuming that the group velocity dispersion and the propagation time difference are suitably corrected, the focused pulse duration is essentially unaffected by the incident power and remains virtually constant during propagation in the nonlinear medium. Finally, the proposed method also reproduces the spatiotemporal coupling arising from the intrinsic correlation between spatial and temporal properties of the focused pulse.

A discrete linear canonical transform would facilitate numerical calculations in many applications in signal processing, scalar wave optics, and nuclear physics. The question is how to define a discrete transform so that it not only approximates the continuous transform well, but also constitutes a discrete transform in its own right, being complete, unitary, etc. The key idea is that the LCT of a discrete signal consists of modulated replicas. Based on that result, it is possible to define a discrete transform that has many desirable properties. This discrete transform is compatible with certain algorithms more than others.

In this paper, we proposed a novel optical geometry for direct visualization of spatial coherence function. The full-field distribution of the coherence function from an incoherent circular aperture source is experimentally investigated. Meanwhile, the experiments of non-diffracting coherence function from an incoherent circular slit source have also been done. The experimental results show that the coherence function generated by an incoherent circular aperture source is in the form of Fresnel diffraction integral which diffracting during the propagation and the coherence function generated by an incoherent circular slit source is unaltered by propagating in free space.

The Tolman-Oppenheimer-Volkov (TOV) equation describes the interior properties of spherical static perfect fluid object as a relationship between two physical observables - pressure and density. For a fluid sphere object, which contains electric charge, magnetic field, and scalar field, the pressure becomes anisotropic. In the previous article [Phys. Rev. D 76 (2007) 044024; gr-qc/0607001], we deformed TOV in terms of δρC and δpC, and we found a new physical and mathematical interpretation for the TOV equation. In this work, we cannot use the perfect fluid constrains because of the electromagnetic field and the massless scalar field within this object. The TOV equation was thus generalized to involve the electromagnetic and the scalar fields. This model is close to the realistic objects in our universe such as a neutron star. In this paper, we consider the modified TOV equation for Schwarzschild coordinates in a special case. The density is considered as a constant and the scalar field is considered absent. On the general model of the TOV equation, the pressure is expressed in terms of radius. However, this model shows that pressure is affected by electric charge. Moreover, we also calculate the rigorous bound on the transmission probability for the Tolman-Bayin type of charged fluid sphere.