B.L.G. Jonsson: A bandwidth limitation for certain array antennas and its connection to sum-rules

To carry information in electromagnetic waves has become a backbone of our society. There is currently a demand on higher data rates which in an essential way depend on the bandwidth of the electromagnetic waves carrying the information. To meet the demand of more data more frequency bands are reserved for communication, which in turn makes it interesting with antennas that can radiate power over a larger bandwidth. Another side of the issue is that there is no particular societal desire to mount many antennas, densely, in a populated area, neither from esthetic nor from a cost based (rent) perspective.

Wide-band array antennas with a ground plane are a class of antennas that have interesting possibilities in this setting. The construction of such wide-band arrays are very challenging and quite a complex antenna engineering task. It is about limitations and properties of this type of antennas that this talk is a about. One essential quantity in the design of antennas is the input reflection coefficient. This quantity measure how much of our input electrical signal (carrying voice, emails or music) that will be reflected back towards the signal generator. Ideally we would like that there is no reflection of the input signal, this is however not realistic. It was somewhat of a surprise, when it was shown that a combined product of reflection level (return loss) and bandwidth measured in a certain way can be limited by the antenna thickness, a material parameter and something called the scan angle. Complex shaped antennas can hence be bandwidth limited by three physical parameters.

This result is based on a sum-rule for the reflection coefficient which relates its dynamical (frequency) behavior to static properties of the device. The result is connected to Fano’s result (1950) on bandwidth matching for electrical circuits and it is also connected to Rozanov (2000) limitation for absorbers. One of the underlying physical properties that enables the result is that the antennas reflection coefficient is scattering passive, which implies that it is holomorphic in the upper half plane and, furthermore that its absolute value is limited by 1. The reflection coefficient is directly related to a Herglotz function from which the sum-rule follows, see e.g. the PhD thesis of Bernland 2012. The above outlined result is based on passivity in the time-variable, i.e. the one-dimensional concept of passivity. However, passivity is also well defined in a multi-dimensional setting and the talk ends with reflections the applicability of this concept to array limitations.