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Showing papers in "Electromagnetics in 1995"


Journal ArticleDOI
TL;DR: In this paper, the authors combined the discrete complex image method (DCIM) with a hybrid mixed-potential integral equation (MPIE) formulation for microstrip patch antennas with vertical probe feeds.
Abstract: The discrete complex image method (DCIM) is combined with a hybrid mixed-potential integral equation (MPIE) formulation for microstrip patch antennas with vertical probe feeds. The properties and performance of the DCIM are investigated. To assess the accuracy and efficiency of the DCIM-MPIE approach, it is applied to a finite array of vertical probe fed microstrip patch antennas. The same structure is also analyzed by a well-established approach based on a rigorous numerical treatment of the Sommerfeld integrals and the results of the two methods are compared.

19 citations


Journal ArticleDOI
TL;DR: In this article, an expression for the singularity (or near-field) structure of the dyadic Green function of an anisotropic dielectric medium is presented by using an operational procedure.
Abstract: An expression for the singularity (or near-field) structure of the dyadic Green function (of the electric type) of an anisotropic dielectric medium is presented by using an operational procedure. The permittivity dyadic of the dielectric medium is assumed to be real symmetric biaxial that makes the result applicable to a wide variety of substances and materials which are, in one form or another, of interest in electromagnetic field problems. Subsequently we use the obtained delta-function singularity of the dyadic Green function as a key ingredient to obtain the effective properties of a discrete random composite made of spherical inclusions dispersed in an anisotropic host medium. By employing the long-wavelength scattering approximation we are able to establish Maxwell Garnett estimates of the effective constitutive dyadics of such a composite, which in our case is an essemble of small bianisotropic spheres embedded in a general dielectric host medium.

17 citations


Journal ArticleDOI
TL;DR: In this paper, a vectorial far field analysis of the Fresnel-zone plate antenna (FZPA) is presented using two different calculation methods: Physical Optics and Uniform Theory of Diffraction Expressions.
Abstract: A vectorial far-field, analysis of the Fresnel-zone plate antenna (FZPA) is presented using two different calculation methods The first method, Physical Optics, is based on Kirchhoff's diffraction theory and holds for apertures which are large in terms of the wavelength Because this is not the case for the width of all the radiating rings of the FZPA, a second method is used which accounts for the influence of the edges of the rings on the incident field This method is based on a combination of the Geometrical Theory of Diffraction and the Uniform Theory of Diffraction Expressions which describe the radiation patterns are derived using both methods, and the differences and similarities between them are examined The results of this analysis give good insight into the influence of the FZPA geometry on its radiation properties

13 citations


Journal ArticleDOI
TL;DR: In this paper, the conformal mapping solution to the rectangular slot aperture having depth is briefly reviewed and the inverse of the internal impedance per unit length is calculated by means of the usual method of using the perfectly conducting current or magnetic field distribution and the surface impedance.
Abstract: The conformal mapping solution to the rectangular slot aperture having depth is briefly reviewed The inverse of the internal impedance per unit length is calculated by means of the usual method of using the perfectly conducting current or magnetic field distribution and the surface impedance The connection between the local edge magnetic field strength and the global slot magnetic flux per unit length is determined The conformal mapping solution and internal impedance per unit length (determined by the usual method) for the single rectangular conductor are briefly reviewed The connection between the local edge magnetic field strength and the global conductor current is determined The results obtained in this paper are of direct interest, but are also used in future papers to obtain the internal impedance per unit length, including local impedance corrections for right angle corners of arbitrary ratios of internal-to-external magnetic permeabilities

12 citations


Journal ArticleDOI
TL;DR: The relationship between the representations of an RTM in different Cartesian coordinate systems art: developed is developed and three examples are given to demonstrate the versatility of the combined application.
Abstract: General coordinate system transformations and rotation transformations for antenna, applications are presented in a novel coherent and unified fashion. In coordinate system transformations, the Cartesian transformation matrix (CTM) is used to relate the representations of a vector in two Cartesian coordinate systems. Useful CTMs such as the Cartesian–spherical transformation and the Eulerian transformation are formulated based on three elementary CTMs. In rotation transformations, the rotation transformation matrix (RTM) is used to describe the rotation of an object, which may be a point, a vector, a coordinate system, or any physical entity. The relationship between the representations of an RTM in different Cartesian coordinate systems art: developed. Three examples: (i) Scanning dual–reflector antennas, (ii) Map generation for satellite antennas, and (iii) Coordinate systems for antenna measurement (EL–over–AZ and AZ–OVER–EL) are given to demonstrate the versatility of the combined application...

12 citations


Journal ArticleDOI
TL;DR: In this article, a hybrid technique from the reciprocity theorem, method of moments, and triangular basis function is developed to investigate the scattering and radiation properties of microstrip line fed arbitrarily shaped aperture antennas.
Abstract: A hybrid technique from the reciprocity theorem, method of moments, and triangular basis function is developed to investigate the scattering and radiation properties of microstrip line fed arbitrarily shaped aperture antennas. The electric field distribution on the arbitrarily shaped aperture is expanded by a set of triangular basis functions. S parameters and the unknown magnetic current coefficients are obtained by matching the boundary conditions on the aperture and applying the reciprocity theorem simultaneously. The rectangular, exponentially-tapered, and dumbbell slots are investigated using this technique. Inclination and offset of the aperture are introduced to improve the antenna radiation characteristics. Good agreement between experimental and numerical results has been obtained. Lastly, the advantages and limitations of this technique are discussed.

12 citations


Journal ArticleDOI
TL;DR: In this article, the radiation characteristics of elementary current sources on cylindrical bianisotropic substrates are presented, where special constitutive relations are used to describe the bisotropic properties of a material, such that the Green's function can be formulated.
Abstract: Radiation characteristics of elementary current sources on cylindrical bianisotropic substrates are presented. Special constitutive relations are used to describe the bianisotropic properties of a material, such that the Green's function can be formulated. The material bianisotropic effects on the radiation properties of the antennas including beam scanning, directivity enhancement, dark-region illumination, and on horizon radiation, are discussed. Radiation patterns as a function of material parameters are also presented.

11 citations


Journal ArticleDOI
TL;DR: In this article, the existence of dual characteristics between gyroelectric and gyromagnetic planar structures with electric and magnetic walls interchanged is con finned directly from the characteristic equations of related planar circuits.
Abstract: Longitudinally magnetized gyroelectric waveguide with circular electric wall boundaries exhibit split limit-TE cut-off numbers but degenerate limit-TM cut-off numbers for non-symmetric modes. The magnetic wall waveguide displays degenerate limit-TE and limit-TM cut-off numbers. These results have been deduced directly from the characteristic equations of the related planar circuits. The existence of dual characteristics between gyroelectric and gyromagnetic planar structures with electric and magnetic walls interchanged are con finned. Measured resonant frequencies are compared with theoretical models for some gyromagnetic devices.

11 citations


Journal ArticleDOI
TL;DR: In this article, the electromagnetic scattering from elliptic impedance cylinders illuminated by an obliquely incident plane wave is presented, and the finite difference technique is used to solve this problem.
Abstract: The electromagnetic scattering from elliptic impedance cylinders illuminated by an obliquely incident plane wave is presented. The finite difference technique is used to solve this problem. Impedance Boundary Condition (IBC) is enforced on the cylinder surface and an Absorbing Boundary Condition (ABC) is applied on the outer boundary. Between the cylinder surface boundary and the outer boundary, the wave equation is solved with the finite difference technique to obtain the scattered field. Two methods are used to solve this problem and comparison between both methods is used to verify the solution. One method is based on transforming the elliptic cylinder into circular cylinder in polar coordinates. The other method is the direct implementation of the finite difference method in the elliptic coordinates. Consequently, proper transformation of IBC, ABC and wave equation is used. The solution obtained from both methods are found to be in good agreement with each other.

10 citations


Journal ArticleDOI
TL;DR: In this paper, it is shown how the modern theory of edge diffracted rays can be constructed as a natural generalization of the Rubinowicz results, which is carried out first for the scalar diffraction problem and further for the vector problem.
Abstract: In 1924 the Polish scientist Rubinowicz published a paper where he investigated the field penetrating through an aperture in an opaque screen using the Kirchhoff approximation. In that paper he established the ray properties of the diffraction field and predicted some basic elements of the modern asymptotic theories such as the Geometrical Theory of Diffraction, the Physical Theory of Diffraction, and Uniform Theories of Diffraction. In particular, he demonstrated that every stationary point on an aperture edge created an entire cone of diffracted rays which satisfy Fermat's priniciple. In the present paper it is shown how the modern theory of edge diffracted rays can be constructed as a natural generalization of the Rubinowicz results. This generalization is carried out first for the scalar diffraction problem and further for the vector problem.

10 citations


Journal ArticleDOI
TL;DR: This paper covers recent work on a novel concept for multibeam transmit reflector antennas, called semi-active antennas since the feed array elements are not directly connected to the power modules, as in an active antennas, but separated from them by several multiport hybrid couplers.
Abstract: This paper covers recent work on a novel concept for multibeam transmit reflector antennas, called semi-active antennas since the feed array elements are not directly connected to the power modules, as in an active antennas, but separated from them by several multiport hybrid couplers. By controlling the phase of the low level equi-amplitude signals at the inputs of the amplifiers, these can be directed to selected feeds in the focal area of the reflector to generate multiple fixed or movable pencil or shaped beams. More or less channels can be routed into the beams by low level switching at the beam forming network input ports. Under all beam pointing and beam loading conditions, all amplifiers operate at nominal power, therefore with optimum efficiency. The basic principle of this new antenna architecture is presented and discussed and designs are evaluated for regional and global coverage satellite applications.

Journal ArticleDOI
TL;DR: In this paper, a general framework is presented for the computer simulation of narrow band adaptive nulling in hybrid reflector antennas that takes account of the details of reflector scattering and polarization.
Abstract: Hybrid antennas, the combination of a radiating or receiving aperture with an array feed, have been the subject of considerable recent interest because they offer some of the versatility of phased array antennas at a much lower cost. Adaptive nulling in hybrid antennas has not received as much attention as other important aspects, such as compensation for reflector surface distortion, scanning, pattern calculation, and the design of multibeam antennas. In this report we investigate adaptive nulling in hybrid reflector antennas using the Generalized Sidelobe Canceller algorithm. A general framework is presented for the computer simulation of narrow band adaptive nulling in hybrid reflector antennas that takes account of the details of reflector scattering and polarization. The general framework is then illustrated by the results of calculations performed for an offset paraboloid with a planar feed located in the focal plane of the paraboloid.

Journal ArticleDOI
TL;DR: In this article, two high-frequency methods are used to compute the scattered fields from both square and triangular trihedral corner reflectors, which are dominated by single, double, and triple reflections.
Abstract: Two high-frequency methods are used to compute the scattered fields from both square and triangular trihedral corner reflectors. Scattering from a trihedral corner reflector is dominated by single, double, and triple reflections. Both methods use Physical Optics (PO) to calculate the single reflected fields. For the evaluation of the double and triple reflected fields, the first method uses Geometrical Optics (GO) for the initial reflections and PO only for the last reflection. On the other hand, the second method uses PO for the last two consecutive reflections. In addition to the reflected fields, the two methods use the Method of Equivalent Currents (MEC) to calculate the single diffracted fields from all the exterior edges. Higher-order terms, such as multiple diffractions, reflection-diffractions and diffraction-reflections, are not included in the analysis. The backscattered fields are computed for incident angles along both elevation and conical paths. The predicted results are compared wi...

Journal ArticleDOI
TL;DR: In this paper, a low-frequency computational procedure based on the Finite Difference Time Domain (FDTD) algorithm was proposed for numerical modeling of electromagnetic scattering by buried objects in sediment layers under sea water.
Abstract: In this paper we discuss a low-frequency computational procedure based on the Finite Difference Time Domain (FDTD) algorithm, for numerical modeling of electromagnetic scattering by buried objects in sediment layers under sea water. The FDTD algorithm is found to be accurate for modeling buried objects in sediment layers tens of meters away from a constant current dipole source of 1A in sea water. For validation of the low-frequency FDTD modeling, the computed FDTD results are compared with those calculated by using analytic expressions and integral,equation techniques. In this paper we also present a technique for detecting conductivity anomalies in sediments, e.g., a buried object in sedimentary layers under sea water, by using the neural network approach. The electric field values are used as the inputs to the neural network and the associated conductivities are used as the targets. The neural network is then trained to associate these conductivities and field values. It is shown in this paper...

Journal ArticleDOI
TL;DR: In this article, the analysis of modern integrated circuits for microwave, millimeter-wave and optical wave frequencies is described using the method of lines (MoL) in an alternative unified approach.
Abstract: The analysis of modern integrated circuits for microwave, millimeter-wave and optical wave frequencies is described using the method of lines (MoL) in an alternative unified approach The cross-section of the waveguides consist of inhomogeneous layers The permittivity is varying in lateral direction and is different from one layer to the other Metallic strips of finite thickness, or which are infinitely thin, may be embedded Loss can be taken into account by using complex permittivities Results will be given for a microstrip line on substrate of finite width, and for a rib waveguide in the optical frequency range

Journal ArticleDOI
TL;DR: In this article, a coaxial circular cylindrical transmission line at high frequencies is used as a wave launcher for the TEM mode, where the conical region is filled with a uniform, isotropic, dielectric with frequency-independent, constant (lossless and dispersionless).
Abstract: In launching the TEM mode on a coaxial circular cylindrical transmission line at high frequencies, one can use coaxial circular cones as a wave launcher. Characteristic impedances are matched at the boundary of the conical waveguide and the cylindrical waveguide, and in the usual lens sense rays on the conical structure travel with equal time to an aperture plane perpendicular to the axis of the system. To accomplish this the conical region is filled with a uniform, isotropic, dielectric with frequency-independent dielectric constant (lossless and dispersionless). While the lens is not perfect in that there are small reflections at the lens surface, the high-frequency performance can be quite good for a large range of lens parameters.

Journal ArticleDOI
TL;DR: In this article, a cylindrical can with end caps subjected to symmetric axial lightning current is treated by solving the problem of a current filament terminating normally in an infinite planar slab and the integral transform solution of both these problems is given and reduced to a convergent series expansion which is summed numerically.
Abstract: Wall voltages interior to a conducting enclosure subjected to lightning currents are estimated The results are applied to the specific problem of a cylindrical can with end caps subjected to symmetric axial lightning current The voltage on the side wall of the can is approximated by the solution of a planar conducting slab excited by a plane magnetic field The end caps are treated by solving the problem of a current filament terminating normally in an infinite planar slab The integral transform solution of both these problems is given and reduced to a convergent series expansion which is summed numerically Both early and late time approximations are given The end cap problem is shown to be well approximated by the much simpler planar solution when applied incrementally from the effective static radius of the filament to the end cap radius A simple circuit model is also presented, which very closely models the planar diffusion problem Thus the complete cylindrical enclosure problem is appr

Journal ArticleDOI
TL;DR: In this paper, a planar arrays of ring patches embedded in a stratified dielectric medium are analyzed and the solution of the scattering problem is carried out by the method of moments directly in the spectral domain.
Abstract: Frequency selective surfaces composed of planar arrays of ring patches embedded in a stratified dielectric medium are analyzed. The solution of the scattering problem is carried out by the method of moments directly in the spectral domain. In particular, a new set of basis functions with divergent behaviour has been used, and the appropriate truncation criterion for a stable numerical solution is discussed. Design curves of the resonance frequency and of the reflection bandwidth are shown. Finally, the design of a dichroic plate for a diplexer application in the millimeter wave range is presented.

Journal ArticleDOI
TL;DR: In this article, the radiation and scattering properties of resonant region reflectors with diameters in the range of 1 to 5 wavelengths were examined, and it was found that the method of moments (MM) and the physical optics approximation (PO) are in close agreement in the radiation case, even for reflector diameters as small as lλ.
Abstract: The radiation and scattering properties of resonant region reflectors with diameters in the range of 1 to 5 wavelengths are examined. It is found that the method of moments (MM) and the physical optics approximation (PO) are in close agreement in the radiation case, even for reflector diameters as small as lλ. This assumed that the reflector is in the far-field of an ideal feed (no blockage). Additional gain calculations using the feed near-field terms were not substantially different than the physical optics result, at least for feed diameters as large as the reflector radius. For plane wave scattering the agreement is not as good, but the physical optics result is still within a few dB of the method of moments. Neither the MM or PO data revealed significant resonances for these small reflectors when they were illuminated by ideal focal point feeds.

Journal ArticleDOI
TL;DR: In this article, the radiation properties of a dielectric sphere with biaxial anisotropy were investigated and the primary field was defined and computed in terms of the convenient spherical vector wave functions, by solving the wave equation in the phase space.
Abstract: The radiation properties of a dielectric sphere with biaxial anisotropy are investigated. The excitation source is assumed to be an elementary dipole of arbitrary orientation illuminated at the centre of the anisotropic sphere. The primary field is defined and computed in terms of the convenient spherical vector wave functions, by solving the wave equation in the phase space. The boundary conditions at the sphere surface are satisfied and following the method of moments, Fourier series and spherical vector wave functions series are employed to express the unknown reflected and scattered fields at the sphere inner and outer region respectively. Although main emphasis is given to theoretical analysis, some numerical results are computed and presented.

Journal ArticleDOI
TL;DR: In this paper, a class of modified trigonometric functions that are related to repeated integrals of the cosine function is presented, which can be expressed as alternating power series and can be computed to any required accuracy.
Abstract: We present a class of modified trigonometric functions that are related to repeated integrals of the cosine function. In their closed form, these functions have a removable singularity at the origin: they are not defined there but their limit exists and is finite. The presence of the singularity makes them computationally unstable in an interval about the origin. In this interval, however, we can represent them as alternating power series and, hence, we can compute them to any required accuracy. We use these functions to stabilize the computation of two expressions in the physical theory of diffraction. One is related to the wedge-diffraction coefficient and the other is Kirchhoff’s integral over a polygon. Both of them are computationally unstable due to removable singularities. By re–writing them in terms of the new functions, we show that we can control their computation to any accuracy desired.

Journal ArticleDOI
TL;DR: In this paper, an analysis of a microstrip antenna over a finite ground plane is carried out with one disk patch over another, in which the bottom patch serves as a ground plane.
Abstract: An analysis of a microstrip antenna over a finite ground plane is carried out. This analysis is done with one disk patch over another, in which the bottom patch serves as a ground plane. The resulting Green's function for a patch in the Fourier domain is presented and the surface current density is calculated using the method of moments technique, with the inclusion of an attachment current mode and entire domain edge mode expansion functions to properly model the current in the vicinity of the probe–disk junction and to accelerate the convergence of the moment method solution. Finally, the radiation patterns and the ratio between the power above and below the ground plane are shown as a function of the ratio between the patch area and the ground area.

Journal ArticleDOI
TL;DR: In this article, the authors derived diffraction coefficients for an impedance wedge at skew incidence, which are based on the solution of a new set of difference equations, yielding a single expression which recovers the known skew incidence impedance wedge diffraction coefficient.
Abstract: New diffraction coefficients are derived for an impedance wedge at skew incidence. These are based on the solution of a new set of difference equations, yielding a single expression which recovers ail known skew incidence impedance wedge diffraction coefficients. An. important aspect of the derived expression is that a single computer code can be written for all wedge angles (with arbitrary face impedances) for which an exact solution is possible and furthermore it opens the possibility of deriving approximate solutions for other wedge angles.

Journal ArticleDOI
Frank Olyslager1
TL;DR: In this article, the authors studied the behaviour of the surface charge density and the potential at the edge of a perfect conductor taking into account the presence of nearby edges, and obtained an expression for the singular behavior of the charge density resulting from the interaction of the comers of the edges.
Abstract: In this paper we study the behaviour of the surface charge density and the potential at the edge of a perfect conductor taking into account the presence of nearby edges. Expressions for the singular behaviour of the surface charge density are obtained resulting from the interaction of the comers of the edges. The knowledge of the singular behaviour is useful for the choice of basis functions in method of moments techniques. The presented results allow also the determination of the extent of the disturbance of the potential distribution due to ripples on conductors.

Journal ArticleDOI
TL;DR: In this paper, the authors deal with an extension of PO by fringe currents at the reflector rim and at slots in the reflective surface, and find that they are small and primarily influence the crosspolarized radiation.
Abstract: The paper deals with an extension of PO by fringe currents at the reflector rim and at slots in the reflector surface. Such slots occur, for instance, in unfurlable reflectors composed of solid panels or in hinged solid reflectors with no electrical contacts between the panels. The fringe-current effects on the antenna far field are evaluated numerically for representative cases. It is found that they are small and primarily influence the crosspolarized radiation. They may in the case of circularly polarized offset reflector antenna systems be the dominant crosspolar contribution due to the reflector alone.

Journal ArticleDOI
TL;DR: In this article, a dynamic source reversal method based on potential theory is developed to determine the parameters of the equivalent circuit for capacitive gaps in microstrip lines with anisotropic substrates.
Abstract: A dynamic source reversal method, based on potential theory, is developed to determine the parameters of the equivalent circuit for capacitive gaps in microstrip lines with anisotropic substrates. The open circuit end capacitance is found as a special case by letting the gap spacing go to infinity. It is also shown that since the discontinuity is essentially characterized by the excess charge in the vicinity of the gap an equivalent problem with an isotropic substrate and an effective substrate thickness can be formulated that gives accurate results for the equivalent circuit capacitances.

Journal ArticleDOI
TL;DR: In this paper, the authors used the Method of Moments to study the transmission properties of a number of frequency selective surfaces (FSS) with 3D elements, such as the tri-dipole, the dipole-loop, the triloop, and the bi-loop.
Abstract: In this paper we use the Method of Moments to study the transmission properties of a number of frequency selective surfaces (FSS) with 3D elements, such as the tri-dipole, the dipole-loop, the tri-loop, and the bi-loop. The frequency responses of these 3D-FSSs and those of the planar frequency selective surfaces with 2D elements, such as the loop and the window, are presented graphically for normally as well as obliquely incident plane wave illuminations. In general, the 3D-FSSs degrade the wide-angle performance of the original planar loop structure. The non-planar elements do offer the flexibility of widening the bandwidth of the frequency response and introducing multi-band capabilities when the plane wave is incident from a wide angle. However, insofar as the wide-angle response is concerned, the planar loop is still the most robust of all the structures investigated in this study.

Journal ArticleDOI
TL;DR: In this article, a variety of multilayer planar transmission lines are analyzed by the generalized spectral-domain immitance approach using the three-directional electromagnetic-field Green's functions excited by horizontal electric currents/fields.
Abstract: A variety of multilayer planar transmission lines are analyzed by the generalized spectral-domain immitance approach The three-directional electromagnetic-field Green's functions excited by horizontal electric currents/fields are analytically derived for a multilayer planar structure With these explicit field expressions, the planar transmission lines, such as microstrip lines, coplanar waveguides (CPW's) with/without backed conductor, are characterized Rigorous Poynting vector analysis is utilized to calculate the propagated powers of the fundamental and higher-order modes According to the propagated power distribution, design considerations for planar transmission lines are discussed A slightly lossy conductor-backed CPW is proposed to attenuate the unwanted even-mode propagation and to maintain a low loss of the desired CPW odd-mode propagation

Journal ArticleDOI
TL;DR: In this article, Ewald's method is extended to frequency domain integral formulations for fields governed by the Helmholtz equation for formulations in Cartesian coordinates, and it is shown how frequency dependent potential integrals can be resolved into superpositions of short-range, standing wave fields plus radiated fields having low-pass spectral amplitudes.
Abstract: Ewald's method is extended to frequency domain integral formulations for fields governed by the Helmholtz equation For formulations in Cartesian coordinates, it is shown how frequency dependent potential integrals can be resolved into superpositions of short-range, standing wave fields plus radiated fields having low-pass spectral amplitudes The radiated component is shown to be a spectrum of plane waves given by the Fourier transform of a source density transform, spectral Green's function, and low-pass filter function This low-pass, spatial filtering property of fields from extended sources dominates far-field behavior in Ewald's framework Localized or concentrated representations are realized through a careful selection of Ewald's splitting parameter

Journal ArticleDOI
TL;DR: In this paper, the power of the reflected radiation waves excited from an abruptly terminated single-mode-fiber waveguide is treated in detail, and the relationship between the reflected guided and reflected radiation power in proportion with the numerical aperture Δ, the core radius a and the free-space diffractive index n0 is investigated.
Abstract: The power of the reflected radiation waves excited from an abruptly terminated single-mode-fiber waveguide is treated in detail. The field inside the fiber and free-space regions is described in terms of the mixed spectrum vector eigenwaves, while a Neumann-series iterative solution is developed in order to solve the basic integral equation. In addition, the expansion coefficient of the reflected radiation modes is calculated as well as their power. Finally, through numerical computations, the relationship between the reflected guided and reflected radiation power in proportion with the numerical aperture Δ, the core radius a and the free-space diffractive index n0 is investigated.