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The concept of irreversible and reversible powers and that of real and reactive powers are developed with particular reference to the surface polariton. The characteristics of the surface polariton excited by a current source are obtained. The limitations of the various propagation velocities are examined. It is shown that in the frequency range where the surface polariton is transparent, a physically meaningful transport velocity, or equivalently wave packet velocity, does not exist.

The propagation characteristics of the fundamental Airy wave are obtained; the intensity distribution is the same as that for a point electric dipole situated at the origin and oriented normal to the propagation direction. The propagation characteristics of the modified fundamental Airy wave are determined. These characteristics are the same as those for the fundamental Gaussian wave provided that an equivalent waist is identified for the Airy wave. In general, the waves are localized spatially with the peak in the propagation direction.

In this communication, certain comments are made on the X-shaped field generated by a really superluminal (point) charge treated in Subsection 2.D in the paper by Zamboni-Rached et al., “Cherenkov radiation versus X-shaped localized waves,” [J. Opt. Soc. Am. A27, 928 (2010)].

The complex space source theory is used to derive a general integral expression for the vector potential that generates the extended full Gaussian wave in terms of the input value of the vector potential of the corresponding paraxial beam. The vector potential and the fields are assumed to fluctuate on a time scale that is large compared to the wave period. The Poynting vector in the propagation direction averaged over a wave period is expressed in terms of the cross-spectral density of the fluctuating vector potential across the input plane. The Schell model is assumed for the cross-spectral density. The radiation intensity distribution and the power radiated are determined. The effect of spatial coherence on the radiation intensity distribution and the radiated power are investigated for different values of the physical parameters. Illustrative numerical results are provided to bring out the effect of spatial coherence on the propagation characteristics of the fluctuating light wave.

The propagation characteristics of a spatially localized electromagnetic wave produced by a planar current source of different states of spatial coherence are analyzed by the use of a Gaussian Schell-model source. A linearly polarized fundamental electromagnetic Gaussian wave with the electric field perpendicular to the direction of propagation is treated. The effects of the degree of coherence of the source distribution on the radiation intensity distribution and the total radiated power are determined.

The linearly polarized real-argument Hermite–Gauss beam is investigated by the Fourier transform method. The complex power is obtained and the reactive power of the paraxial beam is found to be zero. The complex space source required for the full-wave generalization of the real-argument Hermite–Gauss beam is deduced. The resulting basic full real-argument Hermite–Gauss wave is determined. The real and the reactive powers of the full wave are evaluated. The reactive power of the basic full real-argument Hermite–Gauss wave is infinite, and the reasons for this singularity are described. The real power depends on kw_0, m, and n, where k is the wavenumber, w_0 is the e-folding distance of the Gaussian part of the input distribution, and m and n are the mode numbers. The variation in the real power with respect to changes in kw_0 for specified m and n as well as with respect to changes in m and n for a specified kw_0 is examined.

The basic full wave corresponding to the fundamental Gaussian beam was discovered for the outwardly propagating wave in a half-space by the introduction of a source in the complex space. There is a class of extended full waves all of which reduce to the same fundamental Gaussian beam in the appropriate limit. For the extended full Gaussian waves that include the basic full Gaussian wave as a special case, the sources are in the complex space on different planes transverse to the propagation direction. The sources are cylindrically symmetric Gaussian distributions centered at the origin of the transverse planes, the axis of symmetry being the propagation direction. For the special case of the basic full Gaussian wave, the source is a point source. The radiation intensity of the extended full Gaussian waves is determined and their characteristics are discussed and compared with those of the fundamental Gaussian beam. The extended full Gaussian waves are also obtained for the oppositely propagating outwardly directed waves in the second half-space. The radiation intensity distributions in the two half-spaces have reflection symmetry about the midplane. The radiation intensity distributions of the various extended full Gaussian waves are not significantly different. The power carried by the extended full Gaussian waves is evaluated and compared with that of the fundamental Gaussian beam.

The electric current sources that are required for the excitation of the fundamental Gaussian beam and the corresponding full Gaussian light wave are determined. The current sources are situated on the secondary source plane that forms the boundary between the two half-spaces in which the waves are launched. The electromagnetic fields and the complex power generated by the current sources are evaluated. For the fundamental Gaussian beam, the reactive power vanishes, and the normalization is chosen such that the real power is 2 W. The various full Gaussian waves are identified by the length parameter b_t that lies in the range 0?b_t?b, where b is the Rayleigh distance. The other parameters are the wavenumber k, the free-space wavelength λ, and the beam waist w_0 at the input plane. The dependence of the real power of the full Gaussian light wave on b_t/b and w_0/λ is examined. For a specified w_0/λ, the reactive power, which can be positive or negative, increases as b_t/b is increased from 0 to 1 and becomes infinite for b_t/b=1. For a specified b_t/b, the reactive power approaches zero as kw_0 is increased and reaches the limiting value of zero of the paraxial beam.

A method is developed for deducing the required source in the complex space for the full-wave generalization of a specified paraxial beam. This method is applicable to all paraxial beams. However, the details of the method are presented for the anisotropic Gaussian beam. The required source in the complex space for the anisotropic Gaussian beam is a line source with a Gaussian distribution. An expression is derived for the anisotropic Gaussian full wave, which in the appropriate limit reproduces the anisotropic Gaussian paraxial beam. The radiation intensity of the anisotropic Gaussian light wave is determined; its characteristics are analyzed and compared with those of the limiting case of the anisotropic Gaussian paraxial beam.

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.

A simple electric multipole of finite size in the form of a spherical current sheet with a surface current density in the θ direction varying as sin 2θ is considered. The electromagnetic fields outside the spherical surface are of the same form as that of the corresponding point electric multipole situated at the origin and oriented in the z direction. The power is separated into the radiative and the reactive parts and compared with the separation made on the basis of propagating and evanescent waves. The evanescent waves contribute only to the reactive power, and the propagating waves contribute to both the radiative and the reactive powers. The power flux density is also separated into the real and the reactive parts, and the characteristics of the two parts of the power flux density are discussed.

An electric dipole of finite size in the form of a spherical current sheet with a surface current density in the θ direction varying as sin θ is introduced. External to the spherical surface, the electromagnetic fields are of the same form as that of a point electric dipole situated at the origin and oriented in the z direction. The power and the power flux density are separated into the radiative and the reactive parts. This type of separation of power is related to the usual separation made on the basis of propagating and evanescent waves. There is a natural relationship between the radiative and the reactive parts with the far-field and the near-field effects, respectively.

The transverse magnetic (TM) modified Bessel-Gauss beams and their full-wave generalizations are treated. Attention is paid to the spreading properties on propagation of the null in the radiation intensity pattern for the azimuthal mode numbers m=0 and 1. The rate of spreading of the null in the propagation direction is significantly less for the TM modified Bessel-Gauss waves than those for the corresponding TM Bessel-Gauss waves. The total power transported by the waves is determined and compared with that of the corresponding paraxial beam to estimate the quality of the paraxial beam approximation of the wave. The dependence of the quality of the paraxial beam approximation on the azimuthal mode number, the beam shape parameter, and the ratio of the beam waist to the wavelength has a regular pattern for the TM Bessel-Gauss wave and not for the TM modified Bessel-Gauss wave.

The exact full-wave generalization of the azimuthally varying scalar real-argument Laguerre-Gauss beam is obtained. The radiation intensity of the resulting azimuthally varying real-argument Laguerre-Gauss wave is determined. The main characteristics of the radiation intensity pattern are discussed. The total power P_n,m is determined, where n is the radial mode number and m is the azimuthal mode number. By the use of 1/P_n,m, the characteristics of the quality of the paraxial beam approximation to the real-argument Laguerre-Gauss wave are investigated.

For modified Bessel-Gauss beams, the modulating function for the Gaussian, instead of a Bessel function of real argument, is a Bessel function of imaginary argument. The modified Bessel-Gauss beams and their full wave generalizations are treated with particular attention to the spreading properties on propagation for the azimuthal mode numbers m=0 and m=1. The spreading on propagation of the peak and the null in the radiation pattern obtained in the propagation direction for m=0 and m=1, respectively, is substantially less for the modified Bessel-Gauss waves than that for the corresponding Bessel-Gauss waves. The total power transported by the waves is determined and compared with that of the corresponding paraxial beam to assess the quality of the paraxial beam approximation for the wave. The powers in the Bessel-Gauss wave and the modified Bessel-Gauss wave are finite in contrast to that in the Bessel wave. With respect to both the spreading properties and the quality of the paraxial beam approximation, the modified Bessel-Gauss beam is an improvement over the Bessel-Gauss beam.

A continuous planar array of dipoles that are oriented in a particular direction and have an amplitude distribution that is Gaussian in the paraxial limit is introduced as a source for the fundamental Gaussian light wave. The radiation intensity of the Gaussian light wave is determined and its characteristics are analyzed. The universal Gaussian beam factor is deduced and identified as the radiation intensity of the scalar Gaussian wave. The total radiated power, the mean center of the localized wave, and the beam widths of the intensity distribution are obtained. The ratio of the power in the Gaussian wave to that in the corresponding paraxial Gaussian beam is used as a measure of the quality of the paraxial beam approximation. A limiting factor for the power ratio is introduced as an indicator for the acceptability of the paraxial beam approximation. The cross section and the beam widths of the localized light wave are investigated in the large and small kw_0 limits, where k is the wavenumber and w_0 is the beam waist at the input plane. The beam width of the full Gaussian wave is found to be less than that of the corresponding paraxial Gaussian beam both for the scalar Gaussian wave and for the Gaussian light wave.

Clarifications are provided for the proper choice of the branch of a square-root function that is connected with the singularity on the focal plane occurring in the treatment by Sheppard on "High-aperture beams" [J. Opt. Soc. Am. A18, 1579 (2001)]. The author's claim that the singularity is nonphysical is disputed.

In this communication, certain comments are made on the paper by Kostenbauder , "Eigenmode expansions using biorthogonal functions: complex-valued Hermite-Gaussians,"J. Opt. Soc. Am. A14, 1780 (1997).

The Gaussian beam propagation in the direction of the optic axis of a uniaxial crystal is treated by the complex-source-point technique. At the input plane the electric field is linearly polarized. A particular superposition of the ordinary-mode and the extraordinary-mode beams is generated. The electrodynamics of the composite beam has features that are different from those of the two constituent beams. As a result of the anisotropy, on propagation, the cross-polarized component of the electric field is generated except along the beam axis; the cross section of the beam, which is circular at the input plane, becomes elliptical; and the mean squared width of the beam departs from the usual quadratic dependence on the distance from the waist in the direction of propagation.

Electromagnetic beams in a uniaxial crystal are treated with emphasis on the extraordinary mode. A virtual source that generates a basic elliptical Gaussian wave propagating obliquely to the optic axis is identified. An exact expression is obtained for this basic elliptical Gaussian wave that simplifies to the corresponding basic elliptical Gaussian beam in the appropriate limit. In the direction of amplitude propagation, the paraxial result becomes identical to the exact result and the sum of all the nonparaxial contributions vanish. The characteristics of the basic elliptical Gaussian beam are illustrated with a numerical example. From the spectral representation of the basic Gaussian wave, the first three orders of nonparaxial corrections for the basic elliptical Gaussian beam are determined. The nonparaxial results reduce correctly to those of the fundamental Gaussian beam in an isotropic medium.

With reference to the analysis contained in J. Opt. Soc. Am. A9, 765 (1992), certain explanations of and remarks on the entire series of nonparaxial corrections for the fundamental Gaussian beam are presented.

A dielectric-film waveguide with a grating etched on its film–cover interface can be used as a polarization converter. The e wave can be converted into the h wave and vice versa if the guided wave is incident obliquely to the grating vector. The polarization converter is analyzed by use of a quasi-optic technique. A high degree of polarization-conversion efficiency is achieved by a suitable choice of the grating length. The Bragg angles of incidence and observation, the maximum polarization-conversion efficiency, and the frequency and the angular selectivity are all found to increase with an increase in the grating period. An illustrative numerical example is presented.

Particular higher-order sources give rise to electromagnetic Gaussian beams, which are linearly polarized and have their maximum in the propagation direction. For this dipolar beam the cross-sectional shape changes in the propagation direction. Nodal surfaces exist on which the tangential component of the electric field vanishes in the standing wave that is formed by the two oppositely directed dipolar, electromagnetic Gaussian beams. These surfaces are identified as the mirror shapes for an open resonator that supports this standing wave. For standing waves that have a particular cross-sectional shape at the waist the cross section of the beam near the mirror surfaces is circular. The resonant frequencies for the fundamental transverse mode of such a resonator have been determined as a function of the geometry and the axial mode number. By a perturbation technique the resonant frequency of an open resonator with spherical mirrors has been obtained. This result is valid in only the paraxial approximation. Illustrative numerical results are included.

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.

The characteristics of highly directional electromagnetic beams generated by planar sources of different states of spatial coherence are investigated by using variations of Gaussian Schell-model beams. Azimuthally symmetric beams with the electric vector polarized in the azimuthal direction and dipolar beams with linearly polarized electric fields are treated. The effect on the shape of the beam of decreasing the coherence length of the source distribution is discussed.

A planar dielectric film waveguide with a substrate of a nearly-free-electron metal has an ordinary diffraction grating etched on its surface. The reflection of a light beam incident on the grating from the cover is investigated as a function of the angle of incidence. A perturbation procedure is used for the determination of the resulting absorption spectra. The angular location and the depth of the antiresonance in the angular response of the reflection coefficient are used to obtain estimates of the average thickness of the dielectric film and the relative amplitude of the grating.

Average characteristics of a partially coherent Bessel-Gauss optical beam are treated by using the paraxial approximation. The intensity distribution, as well as the coherence properties of the source, is assumed to vary in the azimuthal direction. The transport of the mean-squared width of the beam in the propagation direction is determined. The variation of the mean squared width of the beam in the azimuthal direction is investigated. The anisotropy of the intensity distribution can be balanced against the anisotropic coherence properties to obtain an optical beam whose mean-squared width in the far zone is a constant in the azimuthal direction. S0740-3232(99)00712-7

Average characteristics of a partially coherent Bessel–Gauss optical beam are treated by using the paraxial approximation. The intensity distribution, as well as the coherence properties of the source, is assumed to vary in the azimuthal direction. The transport of the mean-squared width of the beam in the propagation direction is determined. The variation of the mean squared width of the beam in the azimuthal direction is investigated. The anisotropy of the intensity distribution can be balanced against the anisotropic coherence properties to obtain an optical beam whose mean-squared width in the far zone is a constant in the azimuthal direction.

A grating coupler is investigated for the case when the incident direction of a guided wave in a planar waveguide is inclined, within the guide plane, to the direction of a grating vector. The coupling and conversion between a TE, or TM, guided mode and the TE and TM radiative and nonradiative modes are considered in a weakly corrugated guide. The amplitude transport equation and the expressions for the leakage parameter, and the phase correction to a wave number of an unperturbed waveguide, are deduced using a quasi-optical technique. The power-conservation relation of a grating coupler for obliquely incident waves is established, and the coupling and conversion efficiencies are examined. It has been found that significant differences exist for excitation, or scattering, of waves at the normal and oblique incidences.

A ray model for the treatment of a planar anisotropic dielectric waveguide is presented. The attenuation coefficient of the guided wave predicted by this model is consistent with the energy-conservation requirement.

The governing equations for the paraxial approximation are deduced from Maxwell’s equations. The plane wave, in the paraxial approximation, is found to be transverse electromagnetic. For a field distribution at the input plane having a general azimuthal variation and a radial variation in the form of a Bessel function of an integer order with a Gaussian envelope of a given waist size, the spreading due to the Fresnel diffraction is determined as the paraxial beam is transported in the axial direction. The effects of Fresnel diffraction are illustrated with examples for a beam transporting unit power. Diffraction patterns of azimuthally symmetrical and dipolar modes are presented.

The coupled-mode equations governing the interaction of two guided modes in a pair of parallel, identical dielectric waveguides are deduced by a quasi-optical technique that also yields an elegant expression for the attenuation coefficient of the guided wave owing to losses in the various spatial regions.

A perturbation analysis of a reflection grating for waves at oblique incidence is given. Three canonical equations governing the behavior of the reflection grating are systematically deduced, and from them the power-conservation principle and reciprocity are established. The characteristics of the output coupler are obtained with particular reference to both the direction and the polarization of the radiated waves. The important features of an input coupler are investigated with particular emphasis on the direction and the polarization of the incident wave and the length of the grating that lead to the maximum of the input-coupling efficiency. The guided wave can have its electric or magnetic field polarized normal to its sagittal plane, and the grating can be incorporated either on the conducting surface or on the film-cover interface. The relative merits of these four different schemes of operation of the reflection grating are discussed with reference to their applications as output and input couplers.

The characteristics of the second-order reflection grating are investigated for the transverse electric mode with particular reference to such applications as frequency-selective filter and the output, input, and conjugate couplers. Three canonical equations that are valid for a distributed coupler with two guided waves, an incident wave, and a radiated wave are derived systematically. General relations among the coupling coefficients are deduced, and these lead to the principle of power conservation and reciprocity. The analyses of the devices, based on the solution of the canonical equations, lead to the determination of the various design parameters, such as the optimum operating frequencies and the optimum grating length. The reflection grating with the corrugation on the film-cover interface is compared with the grating having the corrugation on the conducting surface. An illustrative numerical example is included.

A metal half-space is situated parallel to a planar dielectric film waveguide. As a consequence of the wave interaction between the surface polariton supported by the metal surface and the transverse-magnetic guided mode of the dielectric film, either the latter can be completely extinguished or its power can be substantially reduced in a small interaction length. The insertion loss in the same interaction length for the transverse-electric guided mode of the dielectric film is negligibly small. A metal cladding therefore serves as a polarizer for a planar dielectric film waveguide. The theory of this surface-polariton polarizer for a planar dielectric film optical waveguide is developed, and representative numerical results are presented.

A quasi-optic model for the treatment of a planar dielectric waveguide having a substrate with material dispersion is presented. An application of the lowest-order waveguide mode to the development of an optical sensor for monitoring the thermal degradation of a film thickness is indicated.

The two governing equations for the evanescent wave excitation of planar dielectric waveguides are deduced by a quasi-optical technique. These governing equations, which are in a canonical form that is appropriate for any coupler with a guided mode and a radiated wave, satisfy reciprocity and power-conservation requirements.

A planar thin isotropic film sandwiched between an isotropic cover and an electro-optic substrate crystal can support transverse electric and transverse magnetic guided modes. When an external electric field is applied to the substrate and is harmonically modulated suitably in the propagation direction, the two modes have a phase-matched interaction and exchange energy in the propagation direction. The governing equations of this interaction are deduced, using a quasi-optical technique for two orientations of the substrate crystal, and are analyzed to yield the requirement for the complete polarization conversion. Representative numerical results are obtained to show that for practically realizable film thicknesses, complete polarization conversion can be achieved in acceptable values of the interaction length and with acceptable frequency selectivity.

The guided modes supported by two parallel, identical planar dielectric waveguides are investigated exactly by a quasi-optical technique and approximately by using the improved coupled-mode theory. The wave numbers obtained by the two methods are expanded into asymptotic series in terms of a suitable small parameter. By comparing the asymptotic series obtained by the two methods term by term, it is shown that the improved coupled-mode theory is asymptotically exact in the limit of weak coupling and not so in the strong-coupling regime. The merits and the defects of the improved coupled-mode theory are discussed.

The results deduced for a waveguide geometry consisting of four planar dielectric layers in a different context [ J. Opt. Soc. Am. A4, 2141 ( 1987)] are adapted to obtain the attenuation coefficient of the guided mode supported by a metal-clad planar dielectric waveguide for both the transverse electric and the transverse magnetic modes.

With reference to the treatment contained in a recent paper [J. Opt. Soc. Am. A 3, 550 (1986)], the regions of validity of the energy method of Renard for obtaining the Goos-HAnchen lateral shift are discussed, and the proper procedure for implementation of the energy method, including the interaction power flow introduced by Yasumoto and Öishi [J. Appl. Phys. 54, 2170 (1983)], is described.

With reference to the treatment contained in a recent paper [ J. Opt. Soc. Am. A3, 550 ( 1986)], the regions of validity of the energy method of Renard for obtaining the Goos–Hänchen lateral shift are discussed, and the proper procedure for implementation of the energy method, including the interaction power flow introduced by Yasumoto and Ōishi [ J. Appl. Phys.54, 2170 ( 1983)], is described.

With reference to the analysis contained in J. Opt. Soc. Am.73, 920 (1983), certain comments on and clarifications of the paraxial wave equation pertaining to the extraordinary-mode beam propagating obliquely to the optic axis in a uniform uniaxial crystal are presented.