Comment on "Quality of paraxial electromagnetic beams"

We comment on a recent paper by Seshadri [Appl. Opt.45, 5335 (2006)]APOPAI0003-693510.1364/AO.45.005335. In the cited work the author utilized the so-named Felsen method in order to derive the full wave expression of a scalar Bessel-Gauss field, propagating in free space, originated from an assigned input distribution on the source plane z = 0. As we show, the author's obtained results do not correctly reproduce the aforesaid input field distribution, as, on the contrary, they have to do.     

Angular momentum flux of counterpropagating paraxial beams

We study the angular momentum (AM) of the arbitrary superposition of counterpropagating paraxial beams that have the same magnitude of the wavenumber. We derive compact analytical expressions for the total AM in a transverse cross section (linear AM density) and the total AM flux through the cross section. We demonstrate that whereas for the time-averaged linear AM density its separation into the spin and orbital parts is not, generally, observed, the total time-averaged AM flux is separated into well-identifiable spin and orbital constituents. Moreover, we show that such a flux is also naturally separated into the fluxes of forward- and backward-propagating beams.     

Fundamental electromagnetic Gaussian beam beyond the paraxial approximation

The fundamental electromagnetic Gaussian beam is constructed from a single component of the electric vector potential oriented normal to the propagation direction. The potential is cylindrically symmetrical about the propagation direction. The paraxial beam and the first-order nonparaxial beam are obtained. In solving the inhomogeneous paraxial wave equation governing the evolution of the nonparaxial beam, both the particular integral and the complementary function are included. A procedure for deducing the proper asymptotic state of the nonparaxial beam is summarized. The amplitude coefficients of the cylindrically symmetric complex-argument Laguerre-Gauss beams, which constitute the complementary function are determined by requiring the potential to have the proper behavior asymptotically at infinity and near the input plane. From the potential function, the electromagnetic fields are developed and the electrodynamics of the fundamental electromagnetic Gaussian beam beyond the paraxial approximation is investigated. The role of the first-order nonparaxial beam in determining the average beam characteristics is examined.     

Propagation of Hermite-Gaussian and Laguerre-Gaussian beams beyond the paraxial approximation

On the basis of the vectorial Rayleigh-Sommerfeld formulas and by means of the relation between Hermite and Laguerre polynomials, the analytical expressions for the propagation of the Hermite-Gaussian (HG) and Laguerre-Gaussian (LG) beams beyond the paraxial approximation are derived, with the corresponding far-field propagation expressions and that for the Gaussian beams being given as special cases of the results. Some detailed comparisons of our results with the expansion series and paraxial expressions are made, which show the advantages of our results over the expansion series. With the results obtained, some typical intensity patterns of nonparaxial HG and LG beams are shown.     

Waveletlike basis function approach to the propagation of paraxial beams

A semianalytical method is described for calculating the diffraction integral for paraxial propagation through an optical system. The field at the input plane is represented by a linear superposition of nearly Gaussian basis functions that keep a simple analytical form under ABCD propagation. The coefficients of the superposition are obtained by a numerical fit. The flexibility of the basis functions makes the method well suited to dealing with sharp local variations of the input field.     

Recurring beams in hollow metal waveguides: paraxial approximation

For optical and near-optical applications in electromagnetics, the directed propagation of waves in free space and in lenslike media is often in the Cartesian form of Gaussian or more general Hermite-sinusoidal-Gaussian beams. It has been shown that recurring (rather than continuing) forms of such beams are possible in the paraxial approximation for certain hollow metal waveguides, in which multiple reflections from the waveguide walls may occur. Limitations on this recurrence behavior implicit in use of the paraxial approximation are considered here, and estimates are obtained for the maximum propagation distance before the onset of significant distortion of the recurring beams.     

Transformation properties of an indefinite media slab for paraxial beams

On the basis of paraxial beam propagation theory in indefinite media, the effective refraction index for both of the two types of polarized electromagnetic wave, TE and TM waves, are introduced. According to the definition of effective refraction index, we obtain the conditions that paraxial beams pass through the indefinite media. We propose to employ a slab of indefinite media to construct a polarizer or polarizing beam splitter by choosing appropriate anisotropic parameters. Under certain conditions, the indefinite media slab exhibits polarization-selective focusing effect.     

Orbital angular momentum of Laguerre-Gaussian beams beyond the paraxial approximation

We derive a full field solution for Laguerre-Gaussian beams consistent with the Helmholtz equation using the angular spectrum method. Field components are presented as an order expansion in the ratio of the wavelength to the beam waist, f=λ/(2πw?), which is typically small. The result is then generalized to a beam of arbitrary polarization. This result is then used to reproduce the signature angular momentum properties of Laguerre-Gaussian beams in the paraxial limit. The subsequent higher-order term is similarly obtained, which does not display a clear separation of orbital and spin angular momentum components.     

Propagation of hollow Gaussian beams through apertured paraxial optical systems

On the basis of the generalized Collins formula and the expansion of the hard-aperture function into a finite sum of complex Gaussian functions, an approximate analytical formula for a hollow Gaussian beam propagating through an apertured paraxial stigmatic (ST) ABCD optical system is derived. Some numerical examples are given. Furthermore, by using a tensor method, we derive approximate analytical formulas for a hollow elliptical Gaussian beam propagating through an apertured paraxial general astigmatic ABCD optical system and an apertured paraxial misaligned ST ABCD optical system. Our results provide a convenient way for studying the propagation and transformation of a hollow Gaussian beam and a hollow elliptical Gaussian beam through an apertured general optical system.     

Intrinsic orbital angular momentum of paraxial beams with off-axis imprinted vortices

We investigate the orbital angular momentum (OAM) of paraxial beams containing off-axis phase dislocations and put forward a simple method to calculate the intrinsic orbital angular momentum of an arbitrary paraxial beam. Using this approach we find that the intrinsic OAM of a fundamental Gaussian beam with a vortex imprinted off axis has a Gaussian dependence on the vortex displacement, implying that the expectation value of the intrinsic OAM of a photon can take on a continuous range of values (i.e., integer and noninteger values in units of h). Finally, we investigate, both numerically and experimentally, the far-field profiles of beams carrying half-integer OAM per photon, these beams having been created by the method of imprinting off-axis vortices.     

Partially coherent J 0-correlated Schell-model beams beyond the paraxial approximation

The concept of partially coherent J ₀-correlated Schell-Model (JSM) beams is extended to the nonparaxial regime. By using the generalized Rayleigh-Sommerfeld diffraction integral, analytical propagation equations of nonparaxial JSM beams in free space are derived. Under the paraxial approximation our results reduce to the corresponding paraxial ones. The free-space propagation properties of nonparaxial JSM beams are illustrated with numerical examples. A comparison with paraxial JSM beams shows that the nonparaxial approach should be used if the parameters f and σ are not small.     

Response to "Comments on 'Quality of paraxial electromagnetic beams'"

The nonparaxial wave obtained by Seshadri [Appl. Opt.45, 5335 (2006)]APOPAI0003-693510.1364/AO.45.005335 is correct. The difference in the input field distributions of the nonparaxial wave and the corresponding paraxial beam is correct and is not caused by any error in the treatment. As is to be expected, in the appropriate limit, the nonparaxial wave reduces to the paraxial beam for 0相似文献   

Vectorial structure of nonparaxial electromagnetic beams.

A representation of the general solution of the Maxwell equations is proposed in terms of the plane-wave spectrum of the electromagnetic field. In this representation the electric field solution is written as a sum of two terms that are orthogonal to each other at the far field: One is transverse to the propagation axis, and the magnetic field associated with the other is also transverse. The concept of the so-called closest field to a given beam is introduced and applied to the well-known linearly polarized Gaussian beam.     

Average characteristics of partially coherent electromagnetic beams

Average characteristics of partially coherent electromagnetic beams are treated with the paraxial approximation. Azimuthally or radially polarized, azimuthally symmetric beams and linearly polarized dipolar beams are used as examples. The change in the mean squared width of the beam from its value at the location of the beam waist is found to be proportional to the square of the distance in the propagation direction. The proportionality constant is obtained in terms of the cross-spectral density as well as its spatial spectrum. The use of the cross-spectral density has advantages over the use of its spatial spectrum.     

Optical heterodyne detection of random electromagnetic beams

Abstract

In this paper we present a general analysis for the optical heterodyne detection of random electromagnetic beams. To describe the ensemble of quasimonochromatic beams which are partially polarized and partially coherent, we use a recently developed matrix treatment. We derive an expression for the signal-to-noise ratio (SNR) in terms of the beam coherence polarization matrices of the beams on the detector surface. Numerical examples are given for the SNR variation in the case of partially polarized Gaussian Schell model beams and the optimum detection is discussed in terms of beam parameters of the local oscillator.     

Relative phase shift in laguerre-gaussian beams propagating through an apertured paraxial ABCD system

Abstract

Based on the generalized Huygens-Fresnel diffraction integral, closed-form expressions for the relative phase shift of Laguerre-Gaussian beams propagating through an apertured paraxial optical ABCD system are derived. The dependence of the relative phase shift on the beam and system parameters, and the condition for eliminating phase shift are analysed and illustrated with numerical examples.     

Radiation efficiency of partially coherent electromagnetic beams.

We present a general definition of the radiation efficiency of stationary electromagnetic fields and prove that it is bounded between zero and unity for beams of any state of coherence and polarization. The radiation efficiency may be interpreted as a measure of how directed the radiated fields are, and therefore it can be used to assess the allowed spatial coherence and intensity variations across a beam. We consider a class of partially coherent electromagnetic fields that were recently introduced in the literature and evaluate the radiation efficiencies for two particular examples, namely, the azimuthally polarized symmetric beams and the dipolar beams that are nearly linearly polarized in the central region. The results show that the radiation efficiency is fairly insensitive to the state of polarization and that it differs appreciably from unity for only small values of source and correlation widths.     

Propagation characteristics of Gaussian beams through a paraxial ABCD optical system with an annular aperture

By introducing a hard aperture function into a finite sum of complex Gaussian functions, an approximate analytical expression for a Gaussian beam passing through a paraxial ABCD optical system with an annular aperture has been derived. The results could be reduced to the case of circular black screen or circular aperture. Some numerical simulations are also performed and illustrated for the propagation characteristics of a Gaussian beam through a paraxial ABCD optical system with an annular aperture, a circular black screen or a circular aperture.     

Stochastic electromagnetic beams focused by a bifocal lens

In this paper, we study the focusing of a stochastic electromagnetic beam by a bifocal lens. By taking the electromagnetic Gaussian Schell-model (EGSM) beam as an example, the changes in the spectral density, in the spectral degree of coherence, and in the spectral degree of polarization of the EGSM beam as the beam is focused by an unapertured bifocal lens are investigated. It is shown that the spectral density, the spectral degree of coherence, and the spectral degree of polarization of the focused electromagnetic EGSM beams depend upon the coherence lengths and focal lengths of the bifocal lens. The influence of the coherence lengths and the focal lengths on the focused spectral density, the spectral degree of coherence, and the spectral degree of polarization are investigated in great detail.     

Generalized Stokes parameters of a stochastic electromagnetic beam propagating through a paraxial ABCD optical system

On the basis of the generalized diffraction integral formula for an ABCD optical system in the spatial domain, a propagation law for the generalized Stokes parameters of a stochastic electromagnetic beam passing through an ABCD optical system is obtained. We describe the Stokes parameters of the source as linear combinations of the elements of the cross-spectral density matrix, and study the changes in the spectral degree of polarization and in the state of the polarization ellipse of a stochastic electromagnetic Gaussian Schell-model beam propagating through a gradient-index fiber with the help of generalized Stokes parameters and the cross-spectral density matrix. The medium has significant effect on the change of the spectral degree of polarization. However, when the correlation coefficients of the source satisfy the relation delta(xx)=delta(yy)=delta(xy)=delta(yx), the medium does not influence the spectral degree of polarization.