Isotropic Radiators – Princeton University

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Isotropic Radiators
                                       Haim Matzner
                   Holon Academic Institute of Technology, Holon, Israel
                                     Kirk T. McDonald
            Joseph Henry Laboratories, Princeton University, Princeton, NJ 08544
                           (April 8, 2003; updated Jan. 2, 2013)

1     Introduction
Can the radiation pattern of an antenna be isotropic?
    A simple argument suggests that this is difficult.
    The intensity of radiation depends on the square of the electric (or magnetic) field. To
have isotropic radiation, it would seem that the magnitude of the electric field would have
to be uniform over any sphere in the far zone. However, the electromagnetic fields in the
far zone of an antenna are transverse, and it is well known that a vector field of constant
magnitude cannot be everywhere tangent to the surface of a sphere (Brouwer's "hairy-ball
theorem" [1]). Hence, it would appear that the transverse electric field in the far zone cannot
have the same magnitude in all directions, and that the radiation pattern cannot be isotropic,
IF the radiation is everywhere linearly polarized [2, 3].1
    However, electromagnetic waves can have two independent states of polarization, de-
scribed as elliptical polarization in the general case. While a transverse electric field with
a single, linearly polarized component cannot be uniform over a sphere in the far zone, it
may be possible that the sum of the squares of the electric fields with two polarizations is

2     The U-Shaped Antenna of Shtrikman
Shmuel Shtrikman has given an example of a U-shaped antenna that generates an isotropic
radiation pattern in the far zone [5] in the limit of zero intensity of the radiation.2 This
example shows that any desired degree of "isotropicity" can be achieved for a sufficiently
weak radiation pattern.
    Matzner [7] has also shown that the radiation pattern of the U-shaped antenna can in
principle be produced by specified currents of finite strength on the surface of a sphere (see
sec. 2.2).

2.1     The U-Shaped Antenna
The U-shaped antenna of Matzner et al. [5] is illustrated in Fig. 1. It consists of two vertical
arms of length L = /4 (kL = /2), separated by a short cross piece of length h           .
      Comay [4] has given a different argument based on the incorrect assumption that the radial components
of the electromagnetic fields E and B are "exactly" zero at finite distances from the source.
      The U-shaped antenna is a variant of the famous "rabbit ears" TV antenna. It is a limiting case of
antennas described in [6].

        Figure 1: The U-shaped antenna whose radiation pattern is isotropic in the
        limit that h  0, for which the intensity also goes to zero. The dashed lines
        indicate the spatial dependence of the current. From [5].

   Denoting the peak current in the antenna by I , the current density J can be written

                                            J(r, t) = J(r)e-it,                             (1)


                J(r) = I [(x - h/2) (y ) cos(kz + /4)^
                                                     z + (y )(z + /8)^
                       - (x + h/2) (y ) cos(kz + /4)^z] ,                                   (2)

and -/8  z  /8 on the vertical arms, -h/2  x  h/2 on the horizontal arm.
   The time-averaged, far-zone radiation pattern of an antenna with a specified, time-
harmonic current density can be calculated (in Gaussian units) according to [8]
                              dP    2 ^                                     2
                                 =        ^
                                       k k                J(r)eikr dVol       .           (3)
                              d    8c3
   For an observer at angles (, ) with respect to the z axis (in a spherical coordinate
system), the unit wave vector has rectangular components
                               ^ = sin  cos  x
                               k             ^ + sin  sin  y
                                                           ^ + cos  ^
                                                                    z.                      (4)

The integral transform Jk =          J(r)eikr dVol in eq. (3) has rectangular components
 Jk,x = I             dx eik sin  cos x   dy (y )eik sin  sin y   dz  (z + 1/8)eik cos z
              sin[(k/2)h sin  cos ] -i(/4) cos 
        = I                        e             Ihe-i(/4) cos  ,                           (5)
                 (k/2) sin  cos 

 Jk,y = 0,                                                                                                 (6)
 Jk,z = I         dx  (x - h/2)eik sin  cos x       dy (y )eik sin  sin y          dz cos(kz + /4)eik cos z
             -I     dx(x + h/2)eik sin  cos x        dy (y )eik sin  sin y         dz cos(kz + /4)eik cos z
          sin[(k/2)h sin  cos ] i(/4) cos 
        = I                    (ie         + cos e-i(/4) cos  )
                (k/2) sin2 
            cos  i(/4) cos 
         Ih      (ie        + cos e-i(/4) cos  ).                                                          (7)
   ^ k
   k  ^  Jk            = | Jk | 2 - | k
                                       ^  Jk | 2
                               ^2 )|Jk,x|2 + (1 - k
                        = (1 - k                  ^2 )|Jk,z |2 - 2k
                                                                  ^x k
                                                                     ^z ReJk,x J
                                x                  z                             k,z
                        = (1 - sin2  cos2 )|Jk,x |2 + sin2 |Jk,z |2 - 2 sin  cos  cos ReJk,x Jk,z
                                sin2 [(k/2)h sin  cos ]
                        = I2                            (1 - sin2  cos2 )
                                  [(k/2) sin  cos ]2
                                   + cos2 (1 + cos2  - 2 cos  sin[(/2) cos ])
                                   -2 cos2 (cos2  - cos  sin[(/2) cos ])
                                sin2 [(k/2)h sin  cos ]
                        = I2                             I 2h2.                                            (8)
                                  [(k/2) sin  cos ]2
Thus, the radiation pattern is indeed isotropic in the limit that h  0. But in this limit,
the radiation vanishes, for a fixed peak current I .3
    For a finite separation h between the two vertical arms of the antenna, the deviation from
isotropicity is roughly 1 - sin2(kh/2)/(kh/2)2 . Thus the pattern will be isotropic to 1% for
h  0.05 . However, this uniformity is achieved at the expense of a substantial reduction in
the intensity of the radiation. For example, the case of a U-shaped antenna with h = 0.05
has an intensity only 1/40 of that of a basic half-wave, center-fed antenna.4
    As is to be expected, the polarization of the radiation of the U-shaped antenna is elliptical
in general. The far-zone electromagnetic fields are related to the integral transform Jk
according to
                                    ei(kr-t) ^                      ^.
                            B = ik           k  Jk ,    E = Bk                            (10)
The components of the far-zone electromagnetic fields in spherical coordinates are therefore,
                            ^  B = 0,
                  Er = Br = k                                                                             (11)
       Matzner et al. [5] tacitly assume that the product Ih = 1 as h  0. Their result then appears to have
a finite radiation intensity, but the current in their U-shaped antenna is infinite.
       Using eq. (14-55) of [8] for a center-fed linear antenna of length L = /2 (kL =  ), and peak current I ,
we have
                 ^ k  ^  Jk
                                 2    4I 2 cos[(kL/2) cos  ] - cos(kL/2)      I 2 cos2 [(/2) cos  ]
                 k                 = 2                                       = 2                    .      (9)
                                      k             sin  sin(kL/2)                       sin2 
for which the maximum intensity occurs at  = /2 where eq. (9) becomes 0.10I 2 .

               E = B = ik                (cos  cos Jk,x - sin Jk,z )
                                                 sin[(k/2)h sin  cos ] ei(kr-t)
                          =   Ik cos ei(/4) cos 
                                                      (k/2) sin  cos        r
                                                     i(kr -t)
                              Ihk cos ei(/4) cos                 ,                        (12)
             E = -B =         -ik          sin Jk,x
                                                       sin[(k/2)h sin  cos ] ei(kr-t)
                          =   -iIk sin e-i(/4) cos 
                                                            (k/2) sin  cos       r
                                                           i(kr -t)
                              -iIhk sin e-i(/4) cos                 .                     (13)
The magnitudes of the fields are

                                    Ik sin[(k/2)h sin  cos ]  Ihk
                         E=B=                                     ,                       (14)
                                     r    (k/2) sin  cos       r

which are isotropic in the limit of small h. Figure 2 from [7] illustrates the character of the
elliptical polarization of the fields (12)-(13) for various directions in the limit of small h.

2.2    Isotropic Radiation from Currents on a Spherical Shell
In sec. 6.6 of his Ph.D. thesis [7], Matzner shows how the far-zone radiation pattern of the
U-shaped antenna (in the limit h  0) can be reproduced by an appropriate distribution
of currents on a spherical shell of radius R = /4. For this, he first expands the far-zone
fields (12)-(13) in vector spherical harmonics, and then matches these to currents on a shell
of radius R and to an appropriate form for the fields inside the shell.
    Figures 3 and 4 illustrate this procedure. The key point is that the surface currents
are finite in magnitude, and hence an isotropic radiator is realizable in the laboratory (in
contrast to the U-shaped antenna, which requires an infinite current I to achieve perfectly
isotropic radiation).
    In principle, many other surfaces besides that of a sphere could support a pattern of
finite, oscillating currents whose far zone radiation pattern is isotropic.

3     A Linear Array of "Turnstile" Antennas
Saunders [3] has noted that a certain infinite array (a certain vertical stack) of so-called
"turnstile" antennas [9, 10] can also produce a far-zone radiation pattern that is isotropic
    A turnstile antenna consists of a pair of half-wave, center-fed linear dipole antennas
oriented at 90 to each other, and driven 90 out of phase, as shown in Fig. 5.
    If we approximate the half-wave dipoles by point dipoles, then the dipole moment of the
system can be written
                               p = p0 e-it = p0 (^    ^ )e-it ,
                                                 x + iy                                  (15)

Figure 2: The elliptical polarization of the fields (12)-(13) of the U-shaped
antenna in the limit of small h. From [7].

Figure 3: The spherical shell of radius R = /4 on which a set of currents can
be found that produces the same far-zone fields as does the U-shaped antenna.
From [7].

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