Showing papers on "Paraxial approximation published in 1986"
TL;DR: In this article, it was shown that the Laguerre-Gaussian and Hermite Gaussian beams with complex arguments arise naturally in correction terms of a perturbation expansion whose leading term is the fundamental paraxial Gaussian beam.
Abstract: Hermite–Gaussian and Laguerre–Gaussian beams with complex arguments of the type introduced by Siegman [ J. Opt. Soc. Am.63, 1093 ( 1973)] are shown to arise naturally in correction terms of a perturbation expansion whose leading term is the fundamental paraxial Gaussian beam. Additionally, they can all be expressed as derivatives of the fundamental Gaussian beam and as paraxial limits of multipole complex-source point solutions of the reduced-wave equation.
156 citations
TL;DR: In this paper, the analytical properties and computational implications of the Gabor representation are investigated within the context of aperture theory, where the radiation field in the pertinent half-space is represented by a discrete set of linearly shifted and spatially rotated elementary beams.
Abstract: The analytical properties and computational implications of the Gabor representation are investigated within the context of aperture theory. The radiation field in the pertinent half-space is represented by a discrete set of linearly shifted and spatially rotated elementary beams that fall into two distinct categories, the propagating (characterized by real rotation angles) and evanescent beams. The representation may be considered a generalization in the sense that both the classical plane wave and Kirchhoff’s spatial-convolution forms are directly recoverable as limiting cases. The choice of a specific window function [w(x)] and the corresponding characteristic width (L) are, expectedly, cardinal decisions affecting the analytical complexity and the convergence rate of the Gabor series. The significant spectral compression achievable by an appropriate selection of w(x) and L is demonstrated numerically, and simple selection guidelines are derived. Two specific window functions possessing opposite characteristics are considered, the uniformly pulsed and the Gaussian distributions. These are studied analytically and numerically, highlighting several outstanding advantages of the latter. Consequently, the primary attention is focused on Gaussian elementary beams in their paraxial and their far-field estimates. Although the main effort is devoted to aperture analysis, demonstrating the advantages and limitations of the proposed approach, reference is also made to its potential when applied to aperture-synthesis and spatial-filtering problems. The quantitative effects of basic filtering in the discrete Gabor space are depicted.
121 citations
TL;DR: To use separated telescopes to achieve diffraction-limited performance over a finite field of view, the following rule must be obeyed: as viewed from a point in the focal plane, beams from separated telescopes must be recombined so that they appear to be coming directly from a single large telescope which has been masked so as to reproduce exactly the ensemble of collecting telescopes.
Abstract: In the present paper, work is discussed which arose in the context of optical design studies for the proposed space-based COSMIC (Coherent Optical System of Modular Interferometric Collectors) telescope. The operation of COSMIC involves the combination of a number of parallel beams with a central beam-combining telescope which forms an image of the sky. The paper is mainly concerned with the aspects of beam combination. The paraxial constraint condition is considered along with tolerances regarding the individual optical elements, and imaging as a nonshearing recombination. A first-order requirement for achieving a finite field of view is discussed, taking into account specifically the case of a coherent imaging optical system.
91 citations
01 Jan 1986
TL;DR: In this paper, a Lie algebraic method for characterizing optical systems and computing their aberrations is presented, which represents the action of each separate element of a compound optical system, including all departures from paraxial optics.
Abstract: We present the foundations of a new Lie algebraic method of characterizing optical systems and computing their aberrations. This method represents the action of each separate element of a compound optical system —including all departures from paraxial optics— by a certain operator. The operators can then be concatenated in the same order as the optical elements and, following well-defined rules, we obtain a resultant operator that characterizes the entire system. These include standard aligned optical systems with spherical or aspherical lenses, models of fibers with polynomial z-dependent index profile, and also sharp interfaces between such elements. They are given explicitly to third aberration order.
73 citations
TL;DR: In this paper, a new exact representation for point sources is given in terms of complex point sources, where a point source is equivalent to a distribution of sources on the surface of a sphere in complex space.
Abstract: A new exact representation for point sources is given in terms of complex point sources. In the simplest configuration, a point source is equivalent to a distribution of sources on the surface of a sphere in complex space. The representation can be used to consider the propagation of point disturbances through inhomogeneous media and across interfaces. In the high-frequency limit, these solutions may be obtained by the use of complex ray-tracing methods, which are just the analytic extension of ordinary ray methods. It is shown that the additional use of the paraxial approximation yields a procedure that is similar to the Gaussian beam summation method. The latter technique is normally based on a matched asymptotics argument. However, the complex point-source representation now offers an exact basis for this widely used method.
65 citations
TL;DR: Based on the images obtained in stereopairs as well as in photographs obtained before and after inclination of the specimen, a three-dimensional model of the paraxial rod is proposed.
Abstract: . The fine structure of the paraxial rod of Phytomonas davidi and Herpetomonas megaseliae was analyzed in thin sections of Triton X-100-extracted, tannic acid-glutaraldehyde-fixed cells and in replicas of quick-frozen, freeze-fractured, deeply etched and rotary shadowed cells. The paraxial rod is formed by a complex array of filaments. Two regions, designated as proximal and distal, are formed by two and at least 11 plates, respectively, composed of an association of 25-nm and 7.0-nm-thick filaments which are oriented at an angle of-50° in relation to the major axis of the axoneme. The intermediate region is less dense and is formed by thin filaments. Short single and Y-shaped filaments connect the proximal plate to doublets numbers 4 and 7 of the axoneme. Based on the images obtained in stereopairs as well as in photographs obtained before and after inclination of the specimen, a three-dimensional model of the paraxial rod is proposed.
55 citations
TL;DR: In this article, the paraxial Gaussian-beam reflection and transmission in the presence of a convex circular boundary separating two different dielectrics is analyzed by the complex ray method whereby rays are traced in a complex coordinate space from the complex source point to the complex extension of the interface and then to the real observation point.
Abstract: Gaussian-beam reflection and transmission in the presence of a convex circular boundary separating two different dielectrics is analyzed by the complex ray method whereby rays are traced in a complex coordinate space from the complex source point to the complex extension of the interface and then to the real observation point. The search for the complex parameters describing the ray connecting the source and the observer can be numerically time consuming. Much more efficient are paraxial approximations that express the field as perturbations about its values on the axis of maximum strength. Such paraxial approximations are explored here, using on-axis complex ray and even on-axis real ray solutions with corrections accounting for differences in phase, ray reflection and transmission coefficients, and ray divergence coefficients. Extensive numerical comparisons establish a reasonable range of validity of both paraxial approximations, except in regions characterizing the transition to total reflection, if that can occur.
50 citations
10 May 1986
TL;DR: In this paper, a general mathematical form for this beam was derived by applying the angular spectrum approach to the paraxialized wave equation, and it was compared to those derived by other researchers and it is found that the beam can be equivalently represented by a set of paraxials traced about the central beam ray.
Abstract: The propagation of a Gaussian beam through an asymmetric optical system results in a generally or "twisted"" astigmatic beam whose elliptical spot rotates as it propagates in free space. A general mathematical form for this beam is derived by applying the angular spectrum approach to the paraxialized wave equation. This result is compared to those derived by other researchers and it is found that the beam can be equivalently represented by a set of paraxial rays traced about the central beam ray. The application of general Gaussian beam propagation to the analysis of non-Gaussian beam propagation in arbitrary optical systems is also discussed.
47 citations
TL;DR: In this article, the complex source point method is extended to the time domain by considering the field due to an impulsive source point with complex space coordinates and a complex initiation time.
Abstract: The complex source point method is extended to the time domain by considering the field due to an impulsive source point with complex space coordinates and a complex initiation time. The resulting solution, which is a particular analytic continuation of the free space time-dependent Green's function, describes a propagating pulsed beam that has a moving peak along the beam axis. Near the pulse maximum in the paraxial region, this new field type is the Fourier transform into the time domain of the time-harmonic paraxial Gaussian beam field but the method also accommodates, in closed form, observation points far from the beam axis and observation times long before and after the peak has passed. By corresponding analytic continuation of available space-time Green's functions for various propagation and diffraction environments, one may generate directly the response due to the pulsed beam incident on the environment.
46 citations
TL;DR: The optical caustics associated with a number of higher umbilic catastrophes have been studied experimentally by passing light through drops of water resting on a horizontal glass slide as discussed by the authors.
Abstract: The optical caustics associated with a number of higher umbilic catastrophes have been studied experimentally by passing light through drops of water resting on a horizontal glass slide. When the perimeters of the drops were constrained to have the symmetry of a square (4mm), the caustics were organized by X 9 , which is the lowest catastrophe of corank 2 to possess a modulus. Changing the shape of the perimeter of a thin drop of this symmetry that is large enough to be affected by gravity has the effect of sweeping the modulus K through its complete range -6 to +2; in particular, one can study the physical caustics as K passes through the special value -2, which is mathematically excluded from X 9 . Similar variations of K are produced by varying the size of thin drops with square outlines, or simply by adding water to a drop of fixed outline so that its profile becomes highly curved. Twofold (2mm) and fourfold (4mm) symmetries are the only ones that allow X 9 , with its full range of moduli, to participate. However, the range of caustics produced by drops with these symmetries cannot be understood in terms of X 9 alone; to explain the observed details it is necessary to take account of the fact that X 9 is embedded in the higher catastrophe Y 1 / 2 , 2 . A theory of the drop profile under the combined effects of surface tension and gravity, valid for small slopes, leads to a calculation of the caustics in the paraxial approximation. This is completely adequate to explain the observations, for thin drops, but, as a surprising example of structural stability, it also suffices when the drops are highly curved and the paraxial approximation breaks down. However, the paraxial theory does fail, as expected, when highly curved drops are inverted so that they hang from the lower surface of the glass slide. Then the X 9 patterns due to gravity are opposed to those due to non-paraxiality; for a drop of square outline, filled nearly to the point where it is about to detach itself, and illuminated from below, the two effects exactly cancel when the side is 7.3 mm. Analysis of these patterns of fourfold symmetry helps in understanding the caustic patterns produced by drops having other symmetries, or no symmetry at all.
TL;DR: In this article, the canonical operator theory of paraxial optics is generalized to address the case of misaligned optics, and the formal group structure is extended from the aligned case in terms of Heisenberg-Weil and inhomogeneous canonical transforms and the associated 3 × 3 augmented ray matrices.
Abstract: Canonical operator theory of paraxial optics is generalized to address the case of misaligned optics. The formal group structure is extended from the aligned case in terms of Heisenberg–Weil and inhomogeneous canonical transforms and the associated 3 × 3 augmented ray matrices. Certain misalignment phase shifts that are often mistreated and ignored have been derived and incorporated into the theory.
TL;DR: The Wei-Norman algebraic techniques for time-ordering problems are applied to the study of the evolution of quantum states ruled by a harmonic-oscillator Hamiltonian with a time-dependent frequency and the adiabatic theorem is rederived together with higher-order corrections.
Abstract: The Wei-Norman algebraic techniques for time-ordering problems are applied to the study of the evolution of quantum states ruled by a harmonic-oscillator Hamiltonian with a time-dependent frequency. The slowly varying frequency case is studied; the adiabatic theorem is rederived together with higher-order corrections. The analogy with the propagation of a paraxial beam through a self-focusing fiber is also pointed out.
TL;DR: In this article, a Lie algebraic analysis of a simple imaging system in which a magnetic sextupole is used to remove the spherical aberrations is also reviewed, and the solenoids and drifts in the system used are arranged in such a way that, in the paraxial approximation, a crossover (focus) occurs in the center of the sextubole.
Abstract: Publisher Summary This chapter presents new methods employing Lie algebraic tools for characterizing charged-particle optical systems and computing aberrations. These new methods represent the action of each separate element of a compound optical system, including all departures from the paraxial optics, by a certain operator. The use of Lie algebraic tools offers several advantages. First, the calculation of high-order aberrations is facilitated. Second, new insight into the origin and possible correction of aberrations is provided. The spherical aberration is a major factor affecting the optical performance of electron microscopes and makes it impossible to form a point image of a point object. The total A integral is always negative for an imaging system. A Lie algebraic analysis of a simple imaging system in which a magnetic sextupole is used to remove the spherical aberrations is also reviewed. The solenoids and drifts in the system used are arranged in such a way that, in the paraxial approximation, a crossover (focus) occurs in the center of the sextupole.
TL;DR: Wave description and ray matrix methods have been used to study the paraxial Fourier transforming and imaging properties of GRIN lenses with revolution symmetry, and the GRIN lens law and the Fourier transforms condition are derived for a simple optical system.
Abstract: Wave description and ray matrix methods have been used to study the paraxial Fourier transforming and imaging properties of GRIN lenses with revolution symmetry. Using these properties, the GRIN lens law and the Fourier transforming condition are derived for a simple optical system.
TL;DR: In this article, Gaussian brackets are derived as the denominator of the nth convergent of a continued fraction, whose every partial numerator is equal to unity, whose elements consist of constitutional parameters of an optical system.
Abstract: Publisher Summary This chapter reviews Gaussian brackets defined on the basis of the theory of continued fractions, and summarizes the paraxial theory formulated with these Gaussian brackets for both homogeneous and inhomogeneous optical systems and also for the Gaussian beam optical system. Some examples of the application of the Gaussian brackets formulation to the analysis and synthesis of the optical system are also presented. A summary of one of the useful methods for the analysis or synthesis of an optical system in lens design is presented. The method is based on the concept named “Gaussian brackets”. Gaussian brackets are derived as the denominator of the nth convergent of a continued fraction, whose every partial numerator is equal to unity. The Generalized Gaussian Constants (GGC's) are written with the Gaussian brackets, whose elements consist of constitutional parameters of an optical system. The GGC's have a clear physical meaning, and are useful to formulate paraxial theory. The chapter explores that the Gaussian brackets' formulation can be applied not only to other types of optical systems such as a decentered optical system, but also to the aberration theory.
01 Jan 1986
TL;DR: In this article, the authors apply the theory of canonical integral transforms built in quantum mechanics to wave optics and translate the treatament of coherent states and other wave packets to lens and pupil systems.
Abstract: Paraxial geometric optics in N dimensions is well known to be described by the inhomogeneous symplectic group I2N ∧ Sp(2N, ℜ). This applies to wave optics when we choose a particular (ray) representation of this group, corresponding to a true representation of its central extension and twofold cover \(\tilde \Gamma _N = W_N ^ \wedge Mp(2N,\Re )\). for wave optics, the representation distinguished by Nature is the oscillator one. There applies the theory of canonical integral transforms built in quantum mechanics. We translate the treatament of coherent states and other wave packets to lens and pupil systems. Some remarks are added on various topics, including a fundamental euclidean algebra and group for metaxial optics.
TL;DR: From a set of previously derived equations for conic surfaces, another set of equations from which the conic constant k and the paraxial radius of curvature r can be obtained are derived, if at least three values of the longitudinal aberration X and their corresponding angles θ of the normals to the surface are measured.
Abstract: From a set of previously derived equations for conic surfaces, we derived another set of equations from which the conic constant k and the paraxial radius of curvature r can be obtained, if at least three values of the longitudinal aberration X and their corresponding angles θ of the normals to the surface are measured. The procedure is useful when the area around the vertex surface cannot be used. Some experimental results are presented.
TL;DR: Numerical computations show that for the majority of the microlenses whose truncations and aberrations are small, the truncated or aberrated Gaussian beam can be assumed as a Gaussianbeam.
Abstract: Focusing characteristics of hemispherical microlenses formed on the end of single-mode fibers are investigated. The Gaussian beam passing through a hemispherical microlens formed on the end of a single-mode fiber is always aberrated and truncated due to its spherical aberration and aperturing. However, numerical computations show that for the majority of the microlenses whose truncations and aberrations are small, the truncated or aberrated Gaussian beam can be assumed as a Gaussian beam. The shifts of the size and position of minimum 1/e spot size and the shift of the maximum intensity position due to the spherical aberration and the finite size of a microlens are also discussed. To analyze these shifts, Fresnel diffraction integrals are used in the intermediate field region to the hemispherical microlens. Minimum 1/e spot sizes for hemispherical microlenses are measured and compared with the theoretical values of minimum 1/e spot sizes derived with diffraction theory as well as with those derived using the paraxial theory.
TL;DR: It is demonstrated that with nonspecular optical systems it is more convenient to use, in place of the generalized radiance, a closely related physical-optics quantity known as the Wigner (distribution) function.
Abstract: The propagation of a generalized radiance through optical systems is considered under the customary assumptions of physical optics (paraxial propagation in a uniform medium and amplitude transmission functions associated with optical elements). It is shown that in specular optical systems that consist of homogeneous media separated by spherical surfaces, the generalized radiance remains invariant along each paraxial ray regardless of the state of coherence of the wavefield. It is further demonstrated that with nonspecular optical systems it is more convenient to use, in place of the generalized radiance, a closely related physical-optics quantity known as the Wigner (distribution) function. As an application, we analyze the energy transport through an optical device that has been employed to produce highly directional, partially coherent beams.
TL;DR: In this article, the fully relativistic quantum mechanical treatment of paraxial electron-optical image formation initiated in the previous issue is worked out and leads to a rigorous foundation of the linear transfer theory.
Abstract: The fully relativistic quantum mechanical treatment of paraxial electron-optical image formation initiated in the previous paper (this issue) is worked out and leads to a rigorous foundation of the linear transfer theory. Moreover, the status of the relativistic scaling laws for mass and wavelength, as advocated in the literature, is elucidated.
TL;DR: A ray tracing scheme using paraxial approximations is described to allow oblique rays to be traced through lenses or optical systems in which some or all of the surfaces may be astigmatic, even at different axes.
14 Feb 1986
TL;DR: Polarization aberrations are variations of intensity and polarization of an optical beam in the exit pupil of a symmetric optical system, and the dependence of these variations on the position of the object in the field as discussed by the authors.
Abstract: Polarization aberrations are variations of intensity and polarization of an optical beam in the exit pupil of an optical system, and the dependence of these variations on the position of the object in the field. A set of functions has been derived to describe the low-order polarization aberrations of a symmetric optical system. A method is given for calculating polarization aberration coefficients for a refracting system from a paraxial raytrace. A model of these aberrations as weak polarizers that vary across the pupil will be discussed.
01 Jan 1986
TL;DR: The aim of this chapter is to review recent results in the quantum mechanics of nonstationary systems, and to demonstrate how they are applied in paraxial fiber optics.
Abstract: The aim of this chapter is to review recent results in the quantum mechanics of nonstationary systems, and to demonstrate how they are applied in paraxial fiber optics.
TL;DR: The problem of finding the resistance between two circular contacts on an infinite conducting plane is related to that of the capacity between paraxial cylinders as mentioned in this paper, and the general formula for the capacity of two such contacts is obtained, resulting in a solution for the resistance of any two contacts, including the case in which one is inside the circle formed by another.
Abstract: The problem of finding the resistance between two circular contacts on an infinite conducting plane is related to that of the capacity between paraxial cylinders. The general formula for the capacity between two such cylinders is obtained, resulting in a solution for the resistance between any two circular contacts, including the case in which one is inside the circle formed by the other. This approach suggests a new experiment for the undergraduate laboratory using the electrodes and electrolytic tank often found in such laboratories intended for the usual field plotting experiment. The setup for the resistance experiment is described and some results presented.
TL;DR: A complete family of refractive-index profiles for GRIN rods with noncylindrical surfaces of constant index is derived that provides exact solutions of the paraxial wave equation that allow exact evaluation of the image and transform planes as well as the transmittance function and equivalent focal length of the rod.
Abstract: A complete family of refractive-index profiles for GRIN rods with noncylindrical surfaces of constant index is derived that provides exact solutions of the paraxial wave equation. These solutions allow exact evaluation of the image and transform planes as well as the transmittance function and equivalent focal length of the rod.
25 Nov 1986
TL;DR: In this paper, an extension of the physical optics codes GLAD and GLAD V to include a global coordinate system and exact ray aberration calculations is described. But this method requires a major advance in analytical capability.
Abstract: Historically, wave optics computer codes have been paraxial in nature. Folded systems could be modeled by ''unfolding'' the optical system. Calculation of optical aberrations is, in general, left for the analyst to do with off-line codes. While such paraxial codes were adequate for the simpler systems being studied 10 years ago, current problems such as phased arrays, ring resonators, coupled resonators, and grazing incidents optics require a major advance in analytical capability. This paper describes extension of the physical optics codes GLAD and GLAD V to include a global coordinate system and exact ray aberration calculations. The global coordinate system allows components to be positioned and rotated arbitrarily. Exact aberrations are calculated for components in aligned or misaligned configurations by using ray tracing to compute optical path differences and diffraction propagation. Optical path lengths between components and beam rotations in complex mirror systems are calculated accurately so that coherent interactions in phased arrays and coupled devices may be treated correctly.
Patent•
05 Sep 1986
TL;DR: In this paper, the authors considered the problem of correcting an aberration even if a guided plane wave is made incident diagonally to an optical axis by the constitution in which the radius of curvature of the inscribed circle of a single lens in an optical waveguide at the face vertex thereof and the respective effective refractive indices of the optical waveguarantee and lens part satisfy the specific equations.
Abstract: PURPOSE:To correct an aberration even if a guided plane wave is made incident diagonally to an optical axis by the constitution in which the radius of curvature of the inscribed circle of a single lens in an optical waveguide at the face vertex thereof and the respective effective refractive indices of the optical waveguide and lens part satisfy the specific equations. CONSTITUTION:Equation I is satisfied in the x-z coordinate system with the optical axis z of the single lens 3A in the optical waveguide 2 and the vertexes 8a, 8b of the 1st and 2nd faces 4a, 4b as the origin where the curvature cn of the inscribed circle at the vertexes, circular cone constant (k) and expansion coeffts. A-D respectively designated. Equation II is satisfied where the radius r1 of curvature of the inscribed circle at the vertex 8a and the effective refractive indices n0, n1 of the optical waveguide and lens part are respectively designated. The guided plane wave 6 refracts on the 1st and 2nd faces and is converted by a converted cylindrical wave 7 which is condensed to a focus F. Since the radius r1 of curvature of the lens 3 made into an aspherical shape is maintained within a set region, the aberration generated in the paraxial region is corrected and the sharply condensed spot is obtd. The condensed spot is thus kept within the diffraction limit.
TL;DR: In this paper, an iterative geophysical tomography algorithm is presented, in which the medium is approximated smoothly by means of B-splines, and the tww-point problem of ray computation is solved with the aid of paraxial approximation.
Abstract: The algorithm of iterative geophysical tomography is presented. The medium is approximated smoothly by means of B-splines. The tww-point problem of ray computation is solved with the aid of paraxial approximation. The parameters of the medium are obtained from the iterative algorithm of minimizing the quadratic form. Two numerical 2-D examples are given.