Can 𝜆/4 Radiators Be Really Isotropic?

Abstract

In search for isotropic radiators with reasonable quality Factor (Q), bandwidth, and efficiency, one looks for practical radiators with a typical resonant length of 𝜆/4. We present here a Green's function analysis in Fourier of a 𝜆/4 microstrip element and a far-field integral method in configuration (real) space of single and dual U-shaped 𝜆/4 elements. Both solutions analytically prove that the power radiation patterns are isotropic in nature (while the thickness and the width tend to zero), although the polarizations are not symmetrical in all cuts. It is also shown that the power isotropic U-shaped radiator, for which the surface current density is infinite, can be replaced by another finite-size radiator, having finite-surface current density, such that its far-field is exactly the same as the far-field of the U-shaped isotropic radiator.

1. Introduction

Accumulated experience, empirical and theoretical, with 𝜆/4 microstrip elements proved their efficiency as elements of antenna arrays, as can be shown, for example, in [1]. The beamwidth of these elements is relatively broad, where the vertical part of the element contributes to the end-fire direction. The beamwidth becomes broader as the width and the height of the elements become narrower. This phenomenon has motivated us to reinvestigate the age-old problem of the unity gain antenna. Such an antenna would be considered as an achievement from the technological point of view: one is eager to have a multidirectional circularly polarized antenna for satellites and space vehicles in order to ensure communicability with Earth ground stations [2]. Indeed, [3] states that the need for isotropic antennas not academic in that a radar designed with such an antenna and so interconnected that the polarization is reversed on reception, would receive equal signals from most reflecting objects, no matter where they were placed on a sphere centered at the antenna. In [4] it is shown for the so-called “𝜆/4 transmission line antenna” that as the distance between the conductors becomes smaller, the pattern tends to be more isotropic. A relatively recent summary of the search for an isotropically radiating source is given in [5], from which we learn that an infinite long source current distribution based on parallel to 𝑥𝑦 “turnstiles” distributed uniformly over the 𝑧-axis radiated power isotropically. In this case the feed function is 𝐾₀(𝑘𝑧), where 𝑘=2𝜋/𝜆 is the wavenumber and 𝐾₀(⋅) is the zero order of the modified Bessel function of the second kind. The pattern of one turnstile is shown, for example, in [6].

The peculiarity of the class of isotropic power radiators offered here is that the source has a finite size and that the far-field is analytically solvable. Moreover, it is shown [7, 8] that the origin of their phenomenon is the magnetoelectric symmetry of the radiator. These radiators are based on an infinite-surface current density, but this is not a serious impediment, because a finite-size, finite-surface current distribution radiator can be constructed such that the fields outside this radiator are the same as for the basic infinite-current isotropic power radiator, as will be shown in what follows.

The structure of the paper is as follows: in Section 2 we present the far-field calculation of an idealized 𝜆/4 element and present its radiation cuts for various values of width and height above ground. It is shown that as the width and the height above ground of the element become smaller relative to the wavelength, the power pattern becomes more and more isotropic. A U-shaped power isotropic radiator is presented in Section 3. The proof of the isotropic behavior of the far-field radiation power density of this radiator is described, as well as its polarization patterns. In Section 4 we present a double U radiator, for which the polarization patterns are more symmetric. In Section 5 we show how to replace the U-shaped radiator, for which its surface current density is infinite, by a spherical surface having 𝜆/2 radius and finite-surface current density, for which its far-field is exactly the same as the far-field of the U-shaped radiator. Conclusions are given in Section 6.

2. Patterns of an Idealized 𝜆/4 Microstrip Element

Consider the geometry of a 𝜆/4 element above an infinite ground plane as shown in Figure 1. For simplicity there is no dielectric substrate, although the general analysis takes such substrate of thickness 𝐻 and relative dielectric constant 𝜀_𝑟 into consideration. The surface current density on the radiator is composed of horizontal and vertical component as follows:𝐉𝑥,𝑦,𝑧=𝐽_𝑥𝑥,𝑦𝐱+𝐽_𝑧𝑦,𝑧𝐳̂, (1) where bold letters are used to designate vectors hereafter. A detailed Fourier analysis of the radiated fields [9–11] can be summarized as follows.

Figure 1 

The geometry of a 𝜆/4 antenna element above an infinite ground plane. The height of the element above ground is 𝐻, its width is 𝑊, and its length is 𝜆/4.

The Fourier transform of the tangential electric field on the plane which includes the patch is given by𝐄_𝑡𝑘_𝑥,𝑘_𝑦=𝜂𝑘₀𝑘²₀𝐉⋅𝐤^⟂_𝑡𝐤^⟂_𝑡𝑗𝛾₂cot𝛾₂𝐻−𝛾₁+𝛾₁𝛾₂𝐉⋅𝐤^‖_𝑡𝐤^‖_𝑡𝑗𝜀_𝑟𝛾₁cot𝛾₂𝐻−𝛾₂, (2) where 𝜂=120𝜋Ω is the free space impedance, 𝑘₀=2𝜋/𝜆 is the wavenumber, 𝜀_𝑟 is the relative dielectric constant layer under the patch, and 𝐻 is the thickness of the dielectric layer, and 𝐉 is the surface current density on the patch in Fourier space. The other variables are𝛾₁=𝑘²₀−𝑘²_𝑥−𝑘²_𝑦,𝛾₂=𝜀_𝑟𝑘²₀−𝑘²_𝑥−𝑘²_𝑦,𝐤_𝑡=𝑘_𝑥𝐱+𝑘_𝑦𝐲,𝑘_𝑡=𝑘²_𝑥+𝑘²_𝑦,𝐤^⟂_𝑡𝑘=−^𝑦𝑘_𝑡𝑘𝐱+^𝑥𝑘_𝑡𝐤𝐲,_‖𝑡=𝑘_𝑥𝑘_𝑡𝑘𝐱+^𝑦𝑘_𝑡𝐲. (3) The electric field in spatial coordinates can be calculated by taking the inverse Fourier transform of 𝐄(𝑘_𝑥,𝑘_𝑦). However, this transform cannot be done analytically, so further results will be written in the Fourier domain.

The complex input power at the antenna terminals is𝑃_in1=−8𝜋^2∞_−∞𝐄_𝑡𝑘_𝑥,𝑘_𝑦⋅𝐉^∗𝑘_𝑥,𝑘_𝑦𝑑𝑘_𝑥𝑑𝑘_𝑦. (4)

This expression has real and imaginary parts. The contribution to the real part comes from the radiation into free space and from surface waves excitation effects. In the case of air as a dielectric substrate (𝜀_𝑟=1) surface waves do not exist. The contribution to free space radiation comes from integration in the last equation over the range𝑘²_𝑥+𝑘²_𝑦<𝑘²₀ (5) which is called the “visible range.”

Transforming the equation for 𝑃_in into spherical coordinates and restrict it to the visible range gives the following result:𝑘_𝑥=𝑘₀sin𝜃cos𝜑,𝑘_𝑦=𝑘₀sin𝜃sin𝜑,𝑃_𝑟𝜃,𝜑=₀^𝜋/2₀^2𝜋𝑓𝜃,𝜑sin𝜃𝑑𝜃𝑑𝜑, (6)(7) where 𝑓(𝜃,𝜑) is the power radiation of the antenna that is expressed in terms of the Fourier transforms of the surface current density components 𝐽_𝑥̃ and 𝐽_𝑦̃: 𝑓=𝜃,𝜑15𝑘²₀𝜋−𝐽_𝑥̃sin𝜑+𝐽_𝑦̃cos𝜑²cos²𝜃𝜀_𝑟−sin²𝜃cot²𝐻𝑘₀𝜀_𝑟−sin²𝜃+cos²𝜃+𝐽_𝑥̃cos𝜑+𝐽_𝑦̃sin𝜑²cos²𝜃𝜀_𝑟−sin²𝜃𝜀_𝑟−sin²𝜃+𝜀²_𝑟cos²𝜃cot²𝐻𝑘₀𝜀_𝑟−sin²𝜃. (8)

In our case 𝜀_𝑟=1, hence𝑓𝜃,𝜑=15𝑘²₀𝜋−𝐽_𝑥̃sin𝜑+𝐽_𝑦̃cos𝜑²cot²𝐻𝑘₀+𝐽̃cos𝜃+1_𝑥cos𝜑+𝐽_𝑦̃sin𝜑²cos²𝜃1+cot²𝐻𝑘₀=cos𝜃15𝑘²₀𝜋sin²𝐻𝑘₀×−𝐽̃cos𝜃_𝑥sin𝜑+𝐽_𝑦̃cos𝜑²+𝐽_𝑥̃cos𝜑+𝐽_𝑦̃sin𝜑²cos²𝜃. (9)

Return back to the 𝜆/4 radiator, one can take an educated model for the horizontal current (near resonance) as 𝐽_𝑥=𝐼𝑥,𝑦𝑊sin𝜋𝑥−𝑊2𝐿0≤𝑥≤𝐿,2𝑊≤𝑦≤20,𝑧=𝐻elsewhere, (10) and the vertical current𝐽_𝑧=−𝐼𝑦,𝑧𝑊𝑊𝑥=𝐿,−2𝑊≤𝑦≤20,0≤𝑧≤𝐻,elsewhere. (11)𝐼 is the current (in Amperes). Taking Fourier transform of 𝐽_𝑥(𝑥,𝑦)𝐽_𝑥̃𝑘_𝑥,𝑘_𝑦=𝐽_𝑥𝑥,𝑦exp𝑗𝑘_𝑥𝑥+𝑘_𝑦𝑦=𝐼𝑑𝑥𝑑𝑦𝑊^L₀sin𝜋𝑥2𝐿exp𝑗𝑘_𝑥𝑥𝑑𝑥×^𝑊/2_−𝑊/2exp𝑗𝑘_𝑦𝑦𝑗()⋅𝑘𝑑𝑦=𝐼𝑗exp𝜋/2_𝑥/𝑘₀𝑘_𝑥+𝑘₀𝑘²₀−𝑘²_𝑥sinc𝑘_𝑦𝑊2. (12)

The vertical current can be replaced by two horizontal contributions [11]. This replacement is exact for the relevant fields on and above the plane 𝑧=𝐻. The replacement is as follows:𝐉_ℎ𝑘_𝑥,𝑘_𝑦=𝐽_𝑣̃𝑘_𝑥,𝑘_𝑦𝑗𝐤_𝑡𝛾²₂, (13) where 𝐉_ℎ(𝑘_𝑥,𝑘_𝑦)=𝐽̃_ℎ𝑥(𝑘_𝑥,𝑘_𝑦)𝐱+𝐽̃_ℎ𝑦(𝑘_𝑥,𝑘_𝑦)𝐲 is the Fourier transform of the surface current density which replaces the vertical surface current density:𝐽̃_ℎ𝑥𝑘_𝑥,𝑘_𝑦𝑗()𝑘=−𝐼𝑗exp𝜋/2_𝑥/𝑘₀𝑘_𝑥𝑘²₀−𝑘²_𝑥−𝑘²_𝑦sinc𝑘_𝑦𝑊2,𝐽̃_ℎ𝑦𝑘_𝑥,𝑘_𝑦𝑗()𝑘=−𝐼𝑗exp𝜋/2_𝑥/𝑘₀𝑘_𝑦𝑘²₀−𝑘²_𝑥−𝑘²_𝑦sinc𝑘_𝑦𝑊2, (14)(15) where sinc(𝑥)=sin(𝑥)/𝑥.

Substituting the surface current densities in the pattern formula, we can calculate the far-field pattern of the idealized 𝜆/4 microstrip element as function of 𝐻 and 𝑊. Figure 2(a) presents the 𝐸-plane power pattern cut and Figure 2(b) shows the 𝐻-plane power pattern cut. It is clearly seen that as 𝑊 and 𝐻 tend to zero, the pattern cuts become more and more isotropic.

(a)

(b)

Figure 2 

(a) 𝐸-plane radiation pattern (𝜑 = 0° cut) of the ideal 𝜆/4 antenna element for 𝐻=𝑊, measured in wavelengths. The pattern becomes more isotropic as 𝑊 and 𝐻 become smaller. (b) 𝐻-plane radiation pattern (𝜑 = 90° cut) of the ideal 𝜆/4 antenna element for 𝐻=𝑊, measured in wavelengths. The pattern becomes more isotropic as 𝑊 and 𝐻 become smaller.

3. The 𝜆/4 U-Shaped Power Isotropic Radiator

We present here another current source which radiated its power isotropically. This time the calculations are done in configuration (or real) space. The geometry of the current source is shown in Figure 3, where the element is placed in a different position for convenience. The relevant current density is given by 𝐉=𝛿(𝑥)𝛿𝑦𝛿𝐿𝑥,𝑦,𝑧𝑧+2𝐱−𝛿^′(𝑥)𝛿𝑦𝜋cos4+𝜋𝑧̂𝐿2𝐿𝐳−2𝐿≤𝑧≤2,0elsewhere. (16) The relation between the wavenumber 𝑘 and the length is given by𝜆𝐿=4=𝜋.2𝑘 (17) Following Elliott’s procedure [12], the electric far-field is given by 𝐄=−𝑗𝜔𝐀_𝑡,𝐀_𝑡=𝑗(𝑟,𝜃,𝜑,𝑡exp𝜔𝑡−𝑘𝑟)4𝜋𝜇₀⁻¹𝑟𝒜𝜃,𝜑, (18)(19) where 𝒜(𝜃,𝜑) is the directional weighting function given by 𝒜𝜃,𝜑=𝐉𝑥^′,𝑦^′,𝑧^′×exp𝑗𝑘𝑥^′sin𝜃cos𝜑+𝑦^′sin𝜃sin𝜑+𝑧^′cos𝜃𝑑𝑥^′𝑑𝑦^′𝑑𝑧^′. (20) The 𝒜_𝜃 and 𝒜_𝜑 components are given in our case by 𝒜_𝜃𝜃,𝜑=cos𝜃cos𝜑𝒜_𝑥𝜃,𝜑−sin𝜃𝒜_𝑧𝜃,𝜑,𝒜_𝜑𝜃,𝜑=−sin𝜑𝒜_𝑥𝜃,𝜑,𝒜_𝑥𝜃,𝜑=𝛿𝑥^′𝛿𝑦^′𝛿𝜋4𝑘+𝑧^′×exp𝑗𝑘𝑥^′sin𝜃cos𝜑+𝑦^′sin𝜃sin𝜑+𝑧^′cos𝜃𝑑𝑥^′𝑑𝑦^′𝑑𝑧^′𝜋=exp−𝑗4cos𝜃,𝒜_𝑧=𝜋𝜃,𝜑cos4+𝑧^′𝑘𝛿^′𝑥^′𝛿𝑦^′×exp𝑗𝑘𝑥^′sin𝜃cos𝜑+𝑦^′sin𝜃sin𝜑+𝑧^′cos𝜃𝑑𝑥^′𝑑𝑦^′𝑑𝑧^′=𝑗cos𝜑𝜋sin𝜃exp−𝑗4×𝑗𝜋cos𝜃exp2cos𝜃−𝑗cos𝜃, (21)(22)(23)(24) hence, we have for the polar components 𝒜_𝜃, 𝒜_𝜑𝒜_𝜃𝑗𝜋𝜃,𝜑=−𝑗cos𝜑exp4cos𝜃∼𝐸_𝜃𝜃,𝜑,𝒜_𝜑𝜋𝜃,𝜑=−sin𝜑exp−𝑗4cos𝜃∼𝐸_𝜑𝜃,𝜑 (25)(26) and finally𝒜_𝜃𝜃,𝜑²+𝒜_𝜑𝜃,𝜑²=1. (27)

Figure 3 

The geometry of a current source which radiates power isotropically. We put 𝐿=𝜆/4 and take the limit 𝐻 → 0 and 𝑊 → 0, keeping 𝐼⋅𝐻=1, where 𝐼 is the maximum value of the current.

Hence, the U-shaped 𝜆/4 radiator whose thickness 𝐻 and width 𝑊 tend to zero radiates its power isotropically in the far-field.

The polarization patterns of the radiator are presented is what. Figures 4(a), 4(b), and 4(c) present a view from directions Z, −Y, and X, respectively. The radiator is drawn at the center of the plots. It is shown that the polarization of the radiator strongly dependents on the direction.

(a)

(b)

(c)

Figure 4 

(a) Schematic view of an observer, of the far-field polarization of the U-shaped power isotropic radiator at the angles 𝜃=0,𝜋/4,𝜋/2, and at multiples of 𝜋/8 of the angle 𝜑. The observer looks into the direction of the negative Z-axis. The polarizations have been drawn by rotating them so that they appear perpendicular to the line of sight. The green drawing represents the direction of the current of the radiator at a given time. The red ellipses represent the electric field, while the blue ellipses the magnetic field. The tip of the arrow (in black) indicates the phase for a given time, and the arrow points in the direction of the rotation of the fields. (b) Similarly as in Figure 4(a), but here the observer looks into the direction of the Y-axis, and the polarizations are drawn for 𝜃=0,𝜋/4,𝜋/2,3𝜋/4,𝜋, and 𝜑 is between −𝜋/8 and −7𝜋/8 in multiples of 𝜋/8. (c) Similarly as in Figure 4(a), but here the observer looks into the direction of the negative X-axis, 𝜃=0,𝜋/4,𝜋/2,3𝜋/4,𝜋 and 𝜑=0,±𝜋/8,±𝜋/4,±3𝜋/8.

4. The 𝜆/4 Double U-Shaped Power Isotropic Radiator

We have found the electric far-fields 𝐸_𝜃,𝐸_𝜑 of the U-shaped 𝜆/4 radiator and have proved that the total power radiated is isotropic. However, this element shows different polarizations (linear, elliptical, and circular) at different cuts. In case that much more “regular” polarizations are required, we propose to use a double U-shaped 𝜆/4 radiator whose geometry is shown in Figure 5. The current density of this radiator is given by𝐉(𝑥)𝛿𝑦𝛿𝜆=𝛿𝑧+8𝐲−𝛿𝐱+𝑗_′(𝑥)𝛿𝑦+𝑗𝛿(𝑥)𝛿^′𝑦𝜋cos4+2𝜋𝜆𝑧𝐳. (28)

Figure 5 

The double U-shaped radiator. This radiator is composed of two U-shaped radiators, one radiator is rotated by 𝜋/2 radians in the azimuth relative to the second U-shaped radiator, and the phase difference between the currents of the two radiators is 𝜋/2.

The electric far-field components are given by𝐸_𝜃𝑗𝜋=−sin𝜑+𝑗cos𝜑exp4cos𝜃,𝐸_𝜑𝜋=−sin𝜑+𝑗cos𝜑exp−𝑗4cos𝜃. (29)(30)

Figures 6(a) and 6(b) describe the view from directions Z and X, respectively. It is shown clearly that the polarization patterns are more symmetric than in the case of the U-shaped radiator. It can be shown that the symmetry of the radiation patterns can be described by known symmetry groups [7].

(a)

(b)

Figure 6 

(a) Schematic view of an observer, of the far-field polarization of the double U-shaped power isotropic radiator at the angles 𝜃=0,𝜋/4,𝜋/2, and at multiples of 𝜋/8 of the angle 𝜑. The observer looks into the direction of the negative Z-axis. The polarizations have been drawn by rotating them so that they appear perpendicular to the line of sight. The green drawing represents the direction of the current of the radiator at a given time. The red ellipses represent the electric field, while the blue ellipses the magnetic field.  The tip of the arrow (in black) indicates the phase for a given time, and the arrow points in the direction of the rotation of the fields. (b) Similarly as in Figure 6(a), but here the observer looks into the direction of the Y-axis, and the polarizations are drawn for 𝜃=0,𝜋/4,𝜋/2,3𝜋/4,𝜋 and 𝜑=0,±𝜋/8,±𝜋/4,±3𝜋/8.

5. A Finite-Size Finite-Current Density Source Which Radiates Power Isotropically

We have seen that the U-shaped and the double U-shaped power isotropic radiators have infinite-surface current density. In order to remove this infinity, it is possible to replace the infinite-current density source by a finite-equivalent-current density source. A way to do this is to calculate an equivalent spherical surface current density, where the radius of the sphere is 𝑅>𝜆/4 (see Figure 7), where the fields outside the sphere are the same as for the infinite-current density radiator. For convenience, this calculation is performed using Gaussian units. Following [13], we apply the multipole procedure. Let I be the region inside the spherical surface of radius 𝑅, and II be the region outside the spherical surface, the fields in region II are given by𝐁^II=_𝑙,𝑚𝑎_𝐸II(𝑙,𝑚)ℎ_𝑙⁽¹⁾(𝑘𝑟)𝐗_𝑙𝑚−𝑗𝑘𝑎_𝑀II(𝑙,𝑚)𝛁×ℎ_𝑙⁽¹⁾(𝑘𝑟)𝐗_𝑙𝑚,𝑟>𝑅, (31)𝐄^II=_𝑙,𝑚𝑗𝑘𝑎_𝐸II(𝑙,𝑚)𝛁×ℎ_𝑙⁽¹⁾(𝑘𝑟)𝐗_𝑙𝑚+𝑎_𝑀II(𝑙,𝑚)ℎ_𝑙⁽¹⁾(𝑘𝑟)𝐗_𝑙𝑚,𝑟>𝑅, (32) whereℎ_𝑙⁽¹⁾=𝑗_𝑙(𝑘𝑟)+𝑗𝑛_𝑙(𝑘𝑟), (33)𝑗_𝑙(𝑘𝑟),𝑛_𝑙(𝑘𝑟) are the spherical Bessel and Neumann functions [14]. The coefficients 𝑎_𝐸II(𝑙,𝑚),𝑎_𝑀II(𝑙,𝑚) can be calculated for a given source current by𝑎_𝐸II(𝑙,𝑚)=4𝜋𝑘²𝑗𝑙()×𝑌𝑙+1_∗𝑙𝑚𝛁⋅𝐉𝜕𝑗𝜔𝜕𝑟𝑟𝑗_𝑙(𝑘𝑟)+𝑗𝑘𝑐(𝐫⋅𝐉)𝑗_𝑙(𝑘𝑟)𝑑³𝑥,𝑎_𝑀II(𝑙,𝑚)=4𝜋𝑘²𝑗𝑐𝑙()𝑌𝑙+1^∗𝑙𝑚𝛁⋅(𝐫×𝐉)𝑗_𝑙(𝑘𝑟)𝑑³𝑥. (34)(35)

Figure 7 

A finite-size spherical radiator carrying a finite-surface current density, for which its fields outside the sphere are exactly the same as the fields of the U-shaped𝜆/4 power isotropic radiator. The radius of the sphere is 𝑅=𝜆/4.

Inserting the current density of the U-shaped power isotropic radiator𝐉=𝛿(𝑥)𝛿𝑦𝛿()𝐱𝑥,𝑦,𝑧𝑧+𝜆−𝛿^′(𝑥)𝛿𝑦𝜋cos4+2𝜋𝑧𝜆̂𝜆𝐳−8𝜆≤𝑧≤80,elsewhere,, (36) we have the expressions for the coefficients𝑎_𝐸II()=−𝑙,𝑚2𝜋(2𝑙+1)𝑐𝑘²B(𝑚)×𝐼_𝑐(𝑙)+√22𝑗_𝑙𝜋4,𝑙=odd,−2𝜋(2𝑙+1)𝑐𝑘²B(𝑚)×𝐼_𝑠(𝑙)+√22𝑗_𝑙𝜋4,𝑙=even,𝑎_𝑀II()=𝑙,𝑚=±12𝜋(2𝑙+1)𝑐𝑘²𝐼_𝑐(𝑙)+√22𝑗_𝑙𝜋4,𝑙=odd,−2𝜋(2𝑙+1)𝑐𝑘²𝐼_𝑠(𝑙)+√22𝑗_𝑙𝜋4,𝑙=even, (37)(38) and 𝑎_𝑀II(𝑙,𝑚≠±1)=0, whereB(𝑚)=1,𝑚=1,−1,𝑚=−1,0,else,𝐼_𝑐(𝑙)=√22𝑙𝜋/4𝑗𝑙+1^𝑙𝜋4−𝑗_𝑙+1𝜋4+𝑗_𝑙𝜋4,𝐼_𝑠(𝑙)=√22()×𝑙+1𝜋/4𝑙4𝜋−1𝑗_𝑙𝜋4−𝑗_𝑙+1𝜋4+𝑗_𝑙𝜋4. (39)(40)(41)

The details of the calculations are given in [7]. In order to find the surface current density on the surface of the sphere of radius 𝑅 such that the fields outside the sphere will be the same as the fields of the isotropic radiator, we expand the fields inside the sphere:𝐁^I=_𝑙,𝑚𝑎_𝐸I(𝑙,𝑚)𝑗_𝑙(𝑘𝑟)𝐗_𝑙𝑚−𝑗𝑘𝑎_𝑀I(𝑙,𝑚)𝛁×𝑗_𝑙(𝑘𝑟)𝐗_𝑙𝑚,𝑟<𝑅,𝐄^I=_𝑙,𝑚𝑗𝑘𝑎_𝐸I(𝑙,𝑚)𝛁×𝑗_𝑙(𝑘𝑟)𝐗_𝑙𝑚+𝑎_𝑀I(𝑙,𝑚)𝑗_𝑙(𝑘𝑟)𝐗_𝑙𝑚,𝑟<𝑅. (42)(43)

We can find the coefficients 𝑎_𝐸I(𝑙,𝑚),𝑎_𝑀I(𝑙,𝑚) from the equation𝐄^I_𝑡=𝐄^II_𝑡,𝑟=𝑅, (44) then 𝐁^I is known, and the surface current density on the sphere is given by𝐉=𝐁𝑅,𝜃,𝜑𝐧×^II_𝑡−𝐁^I_𝑡=_𝑙,𝑚𝑎_𝐸𝐼𝐼()×ℎ𝑙,𝑚_𝑙(1)()−()ℎ𝑘𝑟𝑙+1_𝑙⁽¹⁾(𝑘𝑟)−𝑘𝑅ℎ⁽¹⁾_𝑙+1(𝑘𝑟)(𝑙+1)𝑗_𝑙⁽¹⁾(𝑘𝑟)−𝑘𝑅𝑗_𝑙+1(𝑘𝑟)𝐧×𝐗_𝑙𝑚+𝑗𝑎𝑘𝑅^𝑀II()×)ℎ𝑙,𝑚(𝑙+1_𝑙⁽¹⁾(𝑘𝑟)−𝑘𝑅ℎ⁽¹⁾_𝑙+1()−ℎ𝑘𝑟^(1)l(𝑘𝑟)𝑗_𝑙()×)𝑗𝑘𝑅(𝑙+1_𝑙(𝑘𝑟)−𝑘𝑅𝑗_𝑙+1(𝑘𝑟)𝐗_𝑙𝑚, (45) where the coefficient in region I has been expressed in terms of the coefficients in region II [7].

Figure 8 describes constant surface current density curves on the sphere surface.

(a)

(b)

Figure 8 

Constant amplitude curves of the surface current density on the surface of the spherical radiator drawn in Figure 7.

The far-field can be expressed in terms of the multipole expansion by𝑑𝑃=𝑐𝑑Ω4𝜋𝑘²×_𝑙.𝑚−𝑗^𝑙+1𝑎_𝐸II(𝑙,𝑚)𝐗_𝑙𝑚×𝐧+𝑎_𝑀II(𝑙,𝑚)𝐗_𝑙𝑚². (46)

Figure 9 shows the first 6 terms in the former expansion are enough for an excellent convergence. In the graphs we see the far-field power density as function of 𝜃 for 𝜑=𝜋/4 as an example.

(a)

(b)

Figure 9 

Convergence of the multipole expansion of the far-field power for the isotropic spherical radiator. It is shown in two scales: in a coarse scale (a) and in a fine scale (b), for 0<𝜃<𝜋 and 𝜑=𝜋/4. It is shown that 6 multipole terms are enough for excellent convergence of the far-field power.

6. Conclusions

The 𝜆/4 microstrip element has been investigated in the limit case where its width and its height above ground tend to zero. We have shown that in this case the far-field power density of the radiator tends to be isotropic in half space. Adding the image of this element, we have the U-shaped power isotropic radiator. The polarization pattern of the U-shaped radiator has low symmetry; hence, we have presented the double U-shaped power isotropic radiator, for which the polarization pattern is better. In order to remove the infinity in the current density of the radiator, we change the U-shaped radiator by a power isotropic spherical surface, having finite-surface current. However, we have to learn how to control, in practice, the details of the current density of the antenna.

Acknowledgment

This paper is devoted to the memory of our great teacher Professore Shmuel Shtrikman of the Department of Complex Systems, Weizmann Institute of Science, Rehovot, Israel, Life Fellow IEEE, who passed away on November 2003.