Showing papers on "Paraxial approximation published in 2007"
TL;DR: It is shown numerically that vector antenna arrays can generate radio beams that exhibit spin and orbital angular momentum characteristics similar to those of helical Laguerre-Gauss laser beams in paraxial optics.
Abstract: We show numerically that vector antenna arrays can generate radio beams that exhibit spin and orbital angular momentum characteristics similar to those of helical Laguerre-Gauss laser beams in paraxial optics. For low frequencies (1 GHz), digital techniques can be used to coherently measure the instantaneous, local field vectors and to manipulate them in software. This enables new types of experiments that go beyond what is possible in optics. It allows information-rich radio astronomy and paves the way for novel wireless communication concepts.
903 citations
TL;DR: The least-squares Gaussian approximations of the diffraction-limited 2D-3D paraxial-nonparaxial point-spread functions (PSFs) of the wide field fluorescence microscope (WFFM), the laser scanning confocal microscope (LSCM), and the disk scanning conf focal microscope (DSCM) are studied.
Abstract: We comprehensively study the least-squares Gaussian approximations of the diffraction-limited 2D-3D paraxial-nonparaxial point-spread functions (PSFs) of the wide field fluorescence microscope (WFFM), the laser scanning confocal microscope (LSCM), and the disk scanning confocal microscope (DSCM). The PSFs are expressed using the Debye integral. Under an L(infinity) constraint imposing peak matching, optimal and near-optimal Gaussian parameters are derived for the PSFs. With an L1 constraint imposing energy conservation, an optimal Gaussian parameter is derived for the 2D paraxial WFFM PSF. We found that (1) the 2D approximations are all very accurate; (2) no accurate Gaussian approximation exists for 3D WFFM PSFs; and (3) with typical pinhole sizes, the 3D approximations are accurate for the DSCM and nearly perfect for the LSCM. All the Gaussian parameters derived in this study are in explicit analytical form, allowing their direct use in practical applications.
479 citations
TL;DR: A novel family of paraxial laser beams forming an overcomplete yet nonorthogonal set of modes that have a singular phase profile and are eigenfunctions of the photon orbital angular momentum are studied.
Abstract: We studied a novel family of paraxial laser beams forming an overcomplete yet nonorthogonal set of modes. These modes have a singular phase profile and are eigenfunctions of the photon orbital angular momentum. The intensity profile is characterized by a single brilliant ring with the singularity at its center, where the field amplitude vanishes. The complex amplitude is proportional to the degenerate (confluent) hypergeometric function, and therefore we term such beams hypergeometric-Gaussian (HyGG) modes. Unlike the recently introduced hypergeometric modes [Opt. Lett. 32, 742 (2007)], the HyGG modes carry a finite power and have been generated in this work with a liquid-crystal spatial light modulator. We briefly consider some subfamilies of the HyGG modes as the modified Bessel Gaussian modes, the modified exponential Gaussian modes, and the modified Laguerre-Gaussian modes.
240 citations
TL;DR: The generalized Airy-Gauss (AiG) beams are introduced and their propagation through optical systems described by ABCD matrices with complex elements in general is analyzed to describe in a more realistic way the propagation of the Airy wave packets.
Abstract: We introduce the generalized Airy-Gauss (AiG) beams and analyze their propagation through optical systems described by ABCD matrices with complex elements in general. The transverse mathematical structure of the AiG beams is form-invariant under paraxial transformations. The conditions for square integrability of the beams are studied in detail. The model of the AiG beam describes in a more realistic way the propagation of the Airy wave packets because AiG beams carry finite power, retain the nondiffracting propagation properties within a finite propagation distance, and can be realized experimentally to a very good approximation.
238 citations
TL;DR: The main properties of the gyrator operation which produces a rotation in the twisting phase planes are formulated and this transform can be easily performed in paraxial optics that underlines its possible application for image processing, holography, beam characterization, mode conversion and quantum information.
Abstract: In this work we formulate the main properties of the gyrator operation which produces a rotation in the twisting (position - spatial frequency) phase planes. This transform can be easily performed in paraxial optics that underlines its possible application for image processing, holography, beam characterization, mode conversion and quantum information.As an example, it is demonstrated the application of gyrator transform for the generation of a variety of stable modes.
194 citations
TL;DR: A solution to the problem of partial reflection and refraction of a polarized paraxial Gaussian beam at the interface between two transparent media and a diffraction effect of angular transverse shifts of the reflected and refracted beams is described.
Abstract: We present a solution to the problem of partial reflection and refraction of a polarized paraxial Gaussian beam at the interface between two transparent media. The Fedorov-Imbert transverse shifts of the centers of gravity of the reflected and refracted beams are calculated. Our results differ in the general case from those derived previously by other authors. In particular, they obey general conservation law for the beams' total angular momentum but do not obey one-particle conservation laws for individual photons, which have been proposed by [Onoda et al. Phys. Rev. Lett. 93, 083901 (2004)]. We ascertain that these circumstances relate to the artificial model accepted in the literature for the polarized beam; this model does not fit to real beams. The present paper resolves the recent controversy and confirms the results of our previous paper [Bliokh et al. Phys. Rev. Lett. 96, 073903 (2006)]. In addition, a diffraction effect of angular transverse shifts of the reflected and refracted beams is described.
179 citations
TL;DR: In this article, simple explicit localized solutions are systematized over the whole space of a linear wave equation, which models the propagation of optical radiation in a linear approximation, and a similarity between these exact solutions and harmonic in time fields obtained in the paraxial approximation based on the Leontovich-Fock parabolic equation has been studied.
Abstract: Simple explicit localized solutions are systematized over the whole space of a linear wave equation, which models the propagation of optical radiation in a linear approximation. Much attention has been paid to exact solutions (which date back to the Bateman findings) that describe wave beams (including Bessel-Gauss beams) and wave packets with a Gaussian localization with respect to the spatial variables and time. Their asymptotics with respect to free parameters and at large distances are presented. A similarity between these exact solutions and harmonic in time fields obtained in the paraxial approximation based on the Leontovich-Fock parabolic equation has been studied. Higher-order modes are considered systematically using the separation of variables method. The application of the Bateman solutions of the wave equation to the construction of solutions to equations with dispersion and nonlinearity and their use in wavelet analysis, as well as the summation of Gaussian beams, are discussed. In addition, solutions localized at infinity known as the Moses-Prosser “acoustic bullets”, as well as their harmonic in time counterparts, “X waves”, waves from complex sources, etc., have been considered. Everywhere possible, the most elementary mathematical formalism is used.
158 citations
TL;DR: It is shown, specifically, that the interpretation of this beam as accelerating, i.e., one characterized by a nonlinear lateral shift, depends significantly on the parameter a entering into the solution.
Abstract: A recently derived Airy beam solution to the (1+1)D paraxial equation is shown to obey two salient properties characterizing arbitrary finite energy solutions associated with second-order diffraction; the centroid of the beam is a linear function of the range and its variance varies quadratically in range. Some insight is provided regarding the local acceleration dynamics of the beam. It is shown, specifically, that the interpretation of this beam as accelerating, i.e., one characterized by a nonlinear lateral shift, depends significantly on the parameter a entering into the solution.
154 citations
TL;DR: In this article, a general study of transverse energy flows (TEF) as physically meaningful and informative characteristics of paraxial light beams' spatial structure is presented, where the total TEF can be decomposed into the spin and orbital contributions giving rise to the spin angular momentums, correspondingly.
126 citations
TL;DR: In this paper, Belsky and Khapalyuk's exact paraxial theory of conical diffraction is explored with an emphasis on pinhole and Gaussian beams.
Abstract: Publisher Summary This chapter describes how the current understanding of conical refraction has been acquired after nearly two centuries of theoretical and experimental studies. A systematic use of the simplifying approximation of paraxiality is made in the chapter. This is justified by the small angles involved in conical diffraction. Conical diffraction exemplifies a fundamental feature of crystal optics––namely, the diabolical point. Belsky and Khapalyuk's exact paraxial theory of conical diffraction is explored in the chapter with an emphasis on pinhole and Gaussian beams. The chapter focuses on chiral and biaxially birefringent materials. Chirality has been found to destroy the diabolical point by separating the two sheets of the wave surface, reflecting the change of the dielectric matrix from real symmetric to complex Hermitian. The chapter demonstrates that each diabolical point splits into two branch-points, reflecting the fact that the dielectric matrix is non-Hermitian. The effects of absorption on the pattern of emerging light, as well as the combined effects of dichroism and chirality, are described.
118 citations
TL;DR: In this article, the authors derived analytical expressions for the fields of a tightly focused Gaussian laser beam and found that using the derived fields, the calculated power can be about 25% more accurate than when calculated using the paraxial approximation for a beam focused down to a waist radius w 0∼0.4λ, where λ is the wavelength.
Abstract: Analytic expressions for the fields of a tightly focused Gaussian laser beam are derived, accurate to e11, where e is the diffraction angle. It is found that, for example, using the derived fields, the calculated power can be about 25% more accurate than when calculated using the paraxial approximation for a beam focused down to a waist radius w0∼0.4λ, where λ is the wavelength.
TL;DR: Based on the Collins integral formula, an analytical propagation formula for the anomalous hollow beam passing through a paraxial ABCD optical system is derived and the propagation properties in free space are studied graphically.
Abstract: A theoretical model is proposed to approximately describe an anomalous hollow beam of elliptical symmetry with an elliptical solid core, which was observed in experiment recently [Phys. Rev. Lett.94, 134802 (2005)]. Expressions for the propagation factor and effective beam spot size for the anomalous hollow beam are derived. Based on the Collins integral formula, an analytical propagation formula for the anomalous hollow beam passing through a paraxial ABCD optical system is derived. The propagation properties in free space are studied graphically.
TL;DR: In this paper, a large family of new two-component composite screening photovoltaic spatial solitons in photorifractive crystals was identified and a wide parameter space involving spatial width and power was identified.
TL;DR: In this article, the most important results of the paraxial complex geometrical optics (CGO) in respect to Gaussian beams diffraction in the smooth inhomogeneous media and discusses interrelations between CGO and other asymptotic methods, which reduce the problem of Gaussian beam diffraction to the solution of ordinary differential equations.
Abstract: The paper outlines the most important results of the paraxial complex geometrical optics (CGO) in respect to Gaussian beams diffraction in the smooth inhomogeneous media and discusses interrelations between CGO and other asymptotic methods, which reduce the problem of Gaussian beam diffraction to the solution of ordinary differential equations, namely: (i) Babich’s method, which deals with the abridged parabolic equation and describes diffraction of the Gaussian beams; (ii) complex form of the dynamic ray tracing method, which generalizes paraxial ray approximation on Gaussian beams and (iii) paraxial WKB approximation by Pereverzev, which gives the results, quite close to those of Babich’s method. For Gaussian beams all the methods under consideration lead to the similar ordinary differential equations, which are complex-valued nonlinear Riccati equation and related system of complex-valued linear equations of paraxial ray approximation. It is pointed out that Babich’s method provides diffraction substantiation both for the paraxial CGO and for complex-valued dynamic ray tracing method. It is emphasized also that the latter two methods are conceptually equivalent to each other, operate with the equivalent equations and in fact are twins, though they differ by names. The paper illustrates abilities of the paraxial CGO method by two available analytical solutions: Gaussian beam diffraction in the homogeneous and in the lens-like media, and by the numerical example: Gaussian beam reflection from a plane-layered medium.
TL;DR: In this paper, an algebraic method to obtain the complete set of the paraxial eigenmodes of an astigmatic optical resonator is presented, where the relation between the fundamental mode and the higher-order modes is expressed in terms of raising operators in the spirit of the ladder operators of the quantum harmonic oscillator.
Abstract: An astigmatic optical resonator consists of two astigmatic mirrors facing each other. The resonator is twisted when the symmetry axes of the mirrors are nonparallel. We present an algebraic method to obtain the complete set of the paraxial eigenmodes of such a resonator. Basic ingredients are the complex eigenvectors of the four-dimensional transfer matrix that describes the transformation of a ray of light over a roundtrip of the resonator. The relation between the fundamental mode and the higher-order modes is expressed in terms of raising operators in the spirit of the ladder operators of the quantum harmonic oscillator.
TL;DR: In this article, a modified hollow Gaussian beam (HGB) model is proposed to describe a dark hollow beam with adjustable beam spot size, central dark size and darkness factor.
TL;DR: The paraxiality of higher-order Hermite, Laguerre, and Bessel-Gaussian beams was completely determined and this method could be extended to nonlinear optics and Bose condensates.
Abstract: The validity of the paraxial approximation for laser beams in free space is studied via an integral criterion based on the propagation invariants of Helmholtz and paraxial wave equations. This approach allows one to determine the paraxial limit for beams with nondefined spot size and for beams described by more parameters in addition to typical longitudinal wavelength and transverse waist. As examples, the paraxiality of higher-order Hermite, Laguerre, and Bessel-Gaussian beams was completely determined. This method could be extended to nonlinear optics and Bose condensates.
TL;DR: In this paper, El Gawhary et al. presented a new and wide class of scalar, rectangular symmetrical optical fields, the free-space propagation of which can be given in a closed-form in the paraxial approximation.
TL;DR: The Texas time-domain code was extended to solve for the propagation of nonlinear beams for the case where all medium properties vary in space, and good agreement between the code and measurements in capturing the shift in the pressure distribution of both the fundamental and second harmonic due to the gel layer.
Abstract: A three-dimensional model of the forward propagation of nonlinear sound beams in inhomogeneous media, a generalized Khokhlov-Zabolotskaya-Kuznetsov equation, is described. The Texas time-domain code (which accounts for paraxial diffraction, nonlinearity, thermoviscous absorption, and absorption and dispersion associated with multiple relaxation processes) was extended to solve for the propagation of nonlinear beams for the case where all medium properties vary in space. The code was validated with measurements of the nonlinear acoustic field generated by a phased array transducer operating at 2.5 MHz in water. A nonuniform layer of gel was employed to create an inhomogeneous medium. There was good agreement between the code and measurements in capturing the shift in the pressure distribution of both the fundamental and second harmonic due to the gel layer. The results indicate that the numerical tool described here is appropriate for propagation of nonlinear sound beams through weakly inhomogeneous media.
TL;DR: In this article, a theoretical model is proposed to describe a partially coherent dark hollow beam (DHB) of circular or elliptical symmetry, which can be easily controlled by properly choosing the beam parameters.
TL;DR: In this article, the authors studied a family of paraxial laser beams forming an overcomplete yet nonorthogonal set of modes called hypergeometric gaussian (HyGG) modes.
Abstract: We studied a novel family of paraxial laser beams forming an overcomplete yet nonorthogonal set of modes. These modes have a singular phase profile and are eigenfunctions of the photon orbital angular momentum. The intensity profile is characterized by a single brilliant ring with the singularity at its center, where the field amplitude vanishes. The complex amplitude is proportional to the degenerate (confluent) hypergeometric function, and therefore we term such beams hypergeometric gaussian (HyGG) modes. Unlike the recently introduced hypergeometric modes (Opt. Lett. {\textbf 32}, 742 (2007)), the HyGG modes carry a finite power and have been generated in this work with a liquid-crystal spatial light modulator. We briefly consider some sub-families of the HyGG modes as the modified Bessel Gaussian modes, the modified exponential Gaussian modes and the modified Laguerre-Gaussian modes.
TL;DR: Numerical simulations show that, when there is only mismatch in the linear refractive index, Helmholtz solitons behave according to Snell's law.
Abstract: Reflection and refraction of spatial solitons at dielectric interfaces, accommodating arbitrarily angles of incidence, is studied. Analysis is based on Helmholtz soliton theory, which eliminates the angular restriction associated with the paraxial approximation. A novel generalization of Snell's law is discovered that is valid for collimated light beams and the entire angular domain. Our new theoretical predictions are shown to be in excellent agreement with full numerical simulations. New qualitative features of soliton refraction and limitations of previous paraxial analyses are highlighted.
TL;DR: From this expression, theParaxial approximation and the nonparaxial corrections of all orders for the corresponding paraxial cosh-Gaussian beam are determined.
Abstract: On the basis of superposition of beams, a group of virtual sources that generate a cosh-Gaussian wave is identified. A closed-form expression is derived for this cosh-Gaussian wave, which, in the appropriate limit, yields the paraxial approximation for the cosh-Gaussian beam. From this expression, the paraxial approximation and the nonparaxial corrections of all orders for the corresponding paraxial cosh-Gaussian beam are determined.
TL;DR: In this paper, Bessel and Bessel-Gaussian beam propagation through an unapertured or apertured misaligned paraxial optical system is investigated, and analytical formulas are derived based on the generalized diffraction integral formula for treating the propagation of a laser beam through a misaligned POMO system in the cylindrical coordinate system.
TL;DR: In this article, the Stokes vector is used to describe the evolution of a paraxial electromagnetic wave propagating in a weakly anisotropic medium and characterizing by a non-uniform polarization distribution with polarization singularities.
Abstract: We describe evolution of a paraxial electromagnetic wave propagating in a weakly anisotropic medium and characterizing by a non-uniform polarization distribution with polarization singularities. Our approach is based on the evolution equation for the Stokes vector, well-approved in classical polarimetry, but supplied now with a non-uniform initial distribution of the polarization field. In the case of homogeneous anisotropic medium, the equation is integrated analytically, which yields a 3-dimensional distribution of the polarization parameters, including singularities, i.e. C-lines of circular polarization and L-surfaces of linear polarization. The general theory is illustrated by specific examples of the unfolding of a vectorial vortex in birefringent and dichroic media.
TL;DR: Comparisons of the obtained results with the paraxial results are made, which show the propagation ofParaxial and nonparaxial LBG beams is all instable in the near field and the f parameter plays the important role in determining the non Paraxiality of vectorialLBG beams.
Abstract: The concept of vectorial Laguerre-Bessel-Gaussian (LBG) beams is proposed. On the basis of vectorial Rayleigh-Sommerfeld formulas, the analytical formulas for the nonparaxial propagation of vectorial LBG beams are derived and applied to study the nonparaxial propagation properties of vectorial LBG beams. The far field and paraxial approximation are dealt with as special cases of our general results. Some detailed comparisons of the obtained results with the paraxial results are made, which show the propagation of paraxial and nonparaxial LBG beams is all instable in the near field and the f parameter plays the important role in determining the nonparaxiality of vectorial LBG beams. The beam parameter alpha also affects the propagation behavior of nonparaxial LBG beams. Under certain conditions, the obtained results can be reduced to those of the cases for vectorial Laguerre-Gaussian and Bessel Gaussian beams.
TL;DR: A model of a non-modulated pyramid wavefront sensor (P-WFS) based on Fourier optics has been presented and it was observed that in poor visibility the new calibration is better than the conventional.
Abstract: A model of a nonmodulated pyramid wavefront sensor (P-WFS) based on Fourier optics has been presented. Linearizations of the model represented as Jacobian matrices are used to improve the P-WFS phase estimates. It has been shown in simulations that a linear approximation of the P-WFS is sufficient in closed-loop adaptive optics. Also a method to compute model-based synthetic P-WFS command matrices is shown, and its performance is compared to the conventional calibration. It was observed that in poor visibility the new calibration is better than the conventional.
Book•
28 Nov 2007
TL;DR: In this paper, the physics of waves and photons are discussed and a discussion of the effect of reflection and reflection and refraction on nonlinear optical image processing is presented, with a focus on polarization and nonlinear optics.
Abstract: 1. The Physics of Waves 2. Electromagnetic Waves and Photons 3. Reflection And Refraction 4. Geometric Optics 5.Superposition and Interference 6. Diffraction 7. Lasers 8. Optical Imaging 9. Polarization and Nonlinear Optics
TL;DR: In this paper, a spin-controlled change in the orbital angular momentum of light beams propagating in patterned space-variant optical axis phase plates has been studied, which can be exploited to produce a strong modulation in angular momentum change upon variation of the optical path through the phase plates.
Abstract: When light is transmitted through optically inhomogeneous and anisotropic media the spatial distribution of light can be modified according to its input polarization state. A complete analysis of this process, based on the paraxial approximation, is presented, and we show how it can be exploited to produce a spin-controlled change in the orbital angular momentum of light beams propagating in patterned space-variant optical axis phase plates. We also unveil a new effect: the development of a strong modulation in the angular momentum change upon variation of the optical path through the phase plates.
TL;DR: The paraxial approximation offers an efficient method for computing the matrix system for finite-frequency inversions in global tomography, though care should be taken near reflection points, and alternative methods are needed to compute sensitivity near the antipode.