First-order Optical Systems Research Articles

Estimation of dislocated phases and tunable orbital angular momentum using two cylindrical lenses.

A first-order optical system consisting of two cylindrical lenses separated by a distance is considered. It is found to be non-conserving of orbital angular momentum of the incoming paraxial light field. The first-order optical system is effectively demonstrated to estimate phases with dislocations using a Gerchberg-Saxton-type phase retrieval algorithm by making use of measured intensities. Tunable orbital angular momentum in the outgoing light field is experimentally demonstrated using the considered first-order optical system by varying the distance of separation between the two cylindrical lenses.

A Simplified Model for Optical Systems with Random Phase Screens

A first-order optical system with arbitrary multiple masks placed at arbitrary positions is the basic scheme of various optical systems. Generally, masks in optical systems have a non-shift invariant (SI) effect; thus, the individual effect of each mask on the output cannot be entirely separated. The goal of this paper is to develop a technique where complete separation might be achieved in the common case of random phase screens (RPSs) as masks. RPSs are commonly used to model light propagation through the atmosphere or through biological tissues. We demonstrate the utility of the technique on an optical system with multiple RPSs that model random scattering media.

Realization of general first-order optical systems using nine thin cylindrical lenses of arbitrary focal length and four units of free propagation distance.

General first-order optical systems-represented by a four-dimensional real symplectic group-can be realized using thin lenses and free propagation transformations. It is shown that such systems can be realized using four units of free propagation transformation and nine thin rotated cylindrical lenses (or equivalently, four thin rotated astigmatic lenses and a thin rotated cylindrical lens). If these nine thin lenses placed in five transverse planes can be realized using spatial light modulators (SLMs), then our gadget realizes any general first-order optical system using five SLMs. It is also outlined that any general first-order optical system with finite symplectic matrix entries can always be realized using at least any one of the identities presented here, when a particular decomposition demands thin lenses with impractical focal length.

Realization of general first-order optical systems using thin lenses of arbitrary focal length and fixed free propagation distance.

Any general first-order optical system can be represented using S∈Sp(4,R), where Sp(4,R) is the symplectic group with real entries in four dimensions. We prove that any S∈Sp(4,R) can be realized using not more than 18 thin lenses of arbitrary focal length and seven unit distance. New identities that realize S=S1⊕S2, where S1,S2∈Sp(2,R), are obtained. Also, it is proved any S of the form S1⊕S2 can be realized using a maximum of eight thin lenses of arbitrary focal length and three unit distance. Moreover, decompositions for examples such as differential magnifier, partial Fourier transform, and inverse partial Fourier transform are also provided. A "gadget" is proposed that can realize any S∈Sp(4,R) using thin lens transformations-which can be realized through the use of eight spatial light modulators (SLMs) and seven unit distance. Experimental limitation imposed by SLMs while realizing thin lens transformations is also outlined. The justification for the choice of unit distance according to the availability of thin lenses in a lab is given too.

Non-Euclidean symmetries of first-order optical systems.

We revisit the basic aspects of first-order optical systems from a geometrical viewpoint. In the paraxial regime, there is a wide family of beams for which the action of these systems can be represented as a Möbius transformation. We examine this action from the perspective of non-Euclidean hyperbolic geometry and resort to the isometric-circle method to decompose it as a reflection followed by an inversion in a circle. We elucidate the physical meaning of these geometrical operations for basic elements, such as free propagation and thin lenses, and link them with physical parameters of the system.

Propagation of partially coherent truncated polymorphic beams

The recently introduced concept of coherent polymorphic beam (PB), which is focused into a 2D light curve of arbitrary form with independently prescribed phase along it, is a fruitful generalization of the perfect ring vortex and opens up new perspectives in all-optical particle manipulation and light material processing. Its application for optical transport of micro/nano-particles has been experimentally demonstrated. However, the propagation of the PB has not been studied yet. In this Letter, we derive analytical expressions for the propagation of the truncated PB and its partially coherent counter-part through the first-order optical systems, in particular, the rotationally symmetric and twisting systems described by the fractional Fourier and Gyrator transforms, respectively. These expressions clarify the light-curve formation from a truncated PB and can be easily applied for the numerical simulation of the partially coherent PB propagation

Second-order moments of a twisted Gaussian Schell-model beam in anisotropic turbulence

Analytical formulas for the Wigner distribution functions and the second-order moments (SOMs) of a twisted Gaussian Schell-model (TGSM) beam propagating through anisotropic turbulence have been derived by means of a tensor method. It is found that the propagation law for the SOMs of a TGSM beam spreading in turbulence can be described as a first-order optical systems propagation law with an additional turbulence-induced effect. Based on the SOMs, second-order statistics in terms of the effective beam width, the M2-factor, the orbital angular momentum flux, and the effective radius curvature are analyzed in detail. One finds that the anisotropic turbulence leads to an anisotropic spreading of light beams, and a TGSM beam is less affected by turbulence than a Gaussian Schell-model beam without the twist phase. For short distance propagation, the light beam is more sensitive to the initial beam parameters than turbulence parameters, while for sufficiently long transmission distance, the beam characteristics are determined at most by turbulence statistics. The method can be extended to study the propagation characteristics of various coherent and partially coherent complex Gaussian beams, such as flat-topped beams, dark hollow beams, and array beams in turbulence, and promotes important supports in free-space optical communications and remote sensing.

Radial-angular entanglement in Laguerre-Gaussian mode superpositions.

Classical optic entanglement between the radial and angular degrees of freedom in Laguerre-Gaussian mode superpositions is explored within the framework of symmetric first-order optical systems. The Gouy phase picked by a Laguerre-Gaussian mode on free propagation is seen to be of consequence to the radial-angular entanglement in the mode superpositions. We illustrate examples of mode superpositions for which radial-angular entanglement is preserved on passage through symmetric first-order optical systems. An indicator of radial-angular entanglement in two-mode Laguerre-Gaussian superpositions is demonstrated to be a robust free space signaler in the presence of atmospheric turbulence, through examples.

Realization of first-order optical systems using thin lenses of positive focal length

An explicit decomposition of the most general first-order optical system characterized by an Sp(4,R) matrix is obtained in terms of free propagation, thin convex lenses, and thin cylindrical lenses of positive focal length. The Euler decomposition of an Sp(4,R) matrix is used in arriving at the decomposition. It is shown that not more than four convex lenses and 14 cylindrical lenses of positive focal length are required to realize the same. Explicit decompositions for the case of differential free and inverse free propagation, differential magnifiers, and differential fractional Fourier transform are provided.

Characterization of optical fields with quantized orbital angular momentum by invariants of higher order moments of radial coordinates

We show that the skewness and kurtosis parameters of optical fields with quantized orbital angular momenta (OAM) and integer topological charge, which depend on the propagation distance only through normalized transverse coordinates, remain invariant at propagation through axially symmetric first-order optical systems, if defined in terms of higher-order moments of the radial coordinate. The values of these parameters, which characterize the shape of optical fields, depend on the type of OAM beams (Gaussian, Laguerre–Gauss or spiral phase plates in far-field) and the topological charge. As a result, the skewness and kurtosis can be used to identify the type of OAM beam and the absolute value of the topological charge for Gaussian and Laguerre–Gauss vortices encountered in most applications.

Choice of optical system is critical for the security of double random phase encryption systems

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.

Evaluation of the influence of arbitrary masks on the output field of optical systems using ABCD matrices.

We propose a method to express the complex field amplitude (CFA) at the output of any optical system with arbitrary masks. The method provides a general equation for any first-order optical system with given parameters and arbitrary masks. The method is based on the ABCD matrices of the system components, ignoring the masks. The obtained output CFA is formulated in a way that helps recognize the influence of each mask on the output CFA by projecting its influence on the output with the aid of ABCD matrices. The method offers a more transparent approach for efficient organization and minimization of the calculation steps.

Wigner distribution function of a Lorentz–Gauss vortex beam: alternative approach

The Wigner distribution function of a Lorentz–Gauss vortex beam with one topological charge is considered. The topics have been the object of a recent publication. However, we present an alternative compact expression that could complement the analysis developed in the aforementioned publication. The deduced expression can be usefully exploited to fully account for the Wigner dynamics of the Lorentz–Gauss vortex beam as far as the paraxial propagation through first-order optical systems is concerned.

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.

Realization of first-order optical systems using thin convex lenses of fixed focal length.

A general axially symmetric first-order optical system is realized using free propagation and thin convex lenses of fixed focal length. It is shown that not more than five convex lenses of fixed focal length are required to realize the most general first-order optical system, with the required number of lenses depending on the situation. The free propagation distances are evaluated explicitly in each situation. The optimality of the decomposition obtained in each situation is brought out. Decompositions for some familiar subgroups of SL2(R) are also worked out. Convex or concave lenses of arbitrary focal length are realized using three or two convex lenses of fixed focal length, respectively. It is further shown that three convex lenses of arbitrary focal length are sufficient to realize the most general first-order optical system.

Approximate analytical method to evaluate diffraction crosstalk in free-space optical interconnects systems that use circular microlenses with finite uniform apertures

We propose an accurate analytical model to calculate the optical crosstalk of a first-order free space optical interconnects system that uses microlenses with circular apertures. The proposed model is derived by evaluating the resulted finite integral in terms of an infinite series of Bessel functions. Compared to the model that uses complex Gaussian functions to expand the aperture function, it is shown that the proposed model is superior in estimating the crosstalk and provides more accurate results. Moreover, it is shown that the proposed model gives results close to that of the numerical model with superior computational efficiency.

Multichannel sampling theorem for bandpass signals in the linear canonical transform domain

The linear canonical transform (LCT) describes the effect of first-order quadratic phase optical system on a wave field. The classical multichannel sampling theorem for common bandlimited signals has been extended differently to bandlimited signals associated with LCT. However, a practical issue associated with the reconstruction of the original bandpass signal from multichannel samples in LCT domain still remains unresolved. The purpose of this paper is to introduce a practical multichannel sampling theorem for bandpass signals in LCT domain. The sampling expansion which is constructed by the ordinary convolution in the time domain can reduce the effect of spectral leakage and is easy to implement. The classical multichannel sampling theorem and the well-known sampling theorems for the LCT are shown to be special cases of it. Some potential applications of the multichannel sampling are also presented to show the advantage of the theory.

Propagation Characteristics of Airy-Gaussian Beams Passing through a Misaligned Optical System with Finite Aperture

Propagation characteristics of finite Airy-Gaussian beams through an apertured misaligned first-order ABCD optical system are studied. In this work, the generalized Huygens-Fresnel diffraction integral and the expansion of the hard aperture function into a finite sum of complex Gaussian functions are used. The propagation of Airy-Gaussian beam passing through: an unapertured misaligned optical system, an apertured aligned ABCD optical system and an unapertured aligned ABCD optical system are derived here as particular cases of the main finding. Some numerical simulations are performed in the paper.

Linear canonical Wigner distribution and its application

Linear canonical transform (LCT) form a three-parameter family of intergral transforms with wide application in optics. In this paper, we investigate the linear canonical Wigner distribution (LCWD) which is based on the LCT and the classical Wigner distribution (WD). Firstly, the definition of LCWD is discussed. Moreover, the transformation law for the LCWD through a first-order optical system is derived. This new phase-space distribution provides analysis of signals in both space and LCT domains simultaneously. Then, the main properties of LCWD are investigated in detail. Finally, the application of the LCWD is presented. The LCWD is found to be the appropriate phase-space distribution function for light-beam characterization in first-order optical system. Moreover, the moment matrix formalism for beam characterization is studied.

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.