Exceeding Hannan's Limit by a 3-D Antenna Array

By making clear the electromagnetic fundamentals of information theory, electromagnetic information theory (EIT) is expected to be used for approaching or even exceeding the performance limit of existing communication systems. According to EIT, the performance limits (e.g., directivity, gain, communication degree of freedom, and efficiency) of a two-dimensional (2-D) antenna array are generally bounded by the physical aperture of the array. For example, the maximum realized gain for an aperture larger than a wavelength is
$g_{\gamma}(\theta, (\beta)=41\mathrm{t}A/\lambda^{2}\cos 8(1-\vert R(\alpha,\ \beta)\vert ^{2})$
, where
$A$
is the physical area of the aperture,
$\theta,\phi$
indicate the beamforming direction,
$\lambda$
is the free-space wavelength,
$\vert R(\alpha,\ \beta)\vert ^{2}$
is the ratio of returned power to incident power, and
$\alpha_{r}\beta$
are the phases of excited elements. Additionally, the communication degree of freedom of the 2-D antenna array for multiple-input-multiple-output (MIMO) systems can be obtained with a similar formula
$\pi A/\lambda^{2}$
, a factor of
$\cos\theta$
can also be multiplied for angular-limited scattering environment. Particularly, according to Hannan's limit (P. Hannan, “The element-gain paradox for a phased-array antenna,” in IEEE Trans. Ant. Propagat., vol. 12, no. 4, pp. 423–433, 1964.), the maximum embedded radiation efficiency (or element efficiency) of a single element is
$\pi A_{e}/\lambda^{2}$
, where
$A_{e}$
is the area of a single element. It can be found that these limits, which are directly connected to Shannon's capacity, are closely related to and dependent on the projection area of the array towards wanted directions, i.e.
$A \cos \theta$
.