## 1. Introduction

[2] Dense antenna arrays have been a subject of intense research for about half a century [*Hansen*, 1966; *Mailloux*, 1965; *Serracchioli and Levis*, 1959; *Stein*, 1962]. Such arrays have been primarily considered for radar systems but they see increasing applications in the fields of communications [*Oestges and Clerckx*, 2007], near-field sensing [*Gibbins et al.*, 2010], radiometry [*Camps et al.*, 1998; *Weissman and Le Vine*, 1998] and radio astronomy [*van Ardenne et al.*, 2009]. Also, it is interesting to notice that more recent fields of research also benefit from concepts and methods developed in the area of antenna arrays. This is the case of frequency-selective surfaces [*Munk*, 2000], as well as metamaterials (the Yagi antenna is sometimes regarded as the first metamaterial). Many applications consider very dense arrays, i.e., arrays with interelement distances smaller than the wavelength. For instance, to avoid grating lobes in regular antenna arrays, the interelement distance is generally kept smaller than half a wavelength. Irregular arrays [*Manica et al.*, 2009] may mitigate those grating lobes, but when such arrays are limited to very few elements, a proper control of the level of sidelobes may require quite small average spacings [*Bucci et al.*, 2010]. Consequently, in view of the scattering that takes place among the elements of the array, the behavior of an antenna in its array environment may be significantly different from that of the element taken in isolation.

[3] The goal of the present paper is to review concepts and methods developed for the analysis of mutual coupling. In view of the very wide literature on this subject, the task appears as a major challenge and an exhaustive treatment seems impossible. This is why we preferred to select a number of complementary aspects of mutual coupling, which either have known a fast development over the recent decade, or are key to the understanding of the limits of some models. Among those aspects, we shall note mutual coupling in irregular and regular arrays, beam coupling factors, fast finite-array analysis and noise coupling.

[4] The main goal being physical analysis tools, the vast subject of mutual coupling correction will be treated very succinctly. In the following, we will assume that the reader is familiar with fundamental electromagnetic concepts (equivalence principle, reciprocity, unicity, radiation integrals, …) as well as with fundamental array theory (embedded element pattern, array factor, grating lobes, visible region, array impedance matrix, etc.). Regarding the *N*-port coupling matrix, an impedance (rather than scattering) matrix approach will be adopted, because we believe this formulation is closer to field representations (this is mainly a matter of taste). Apart from a few comments on limit cases, we will avoid the single-mode approximation for current distributions on antennas. Indeed, although some aspects of mutual coupling can benefit from rich insights under that approximation [*Wasylkiwskyj and Kahn*, 1970; *Capolino and Albani*, 2009], (among which the asymptotic decay rate of mutual coupling [*Galindo and Wu*, 1968]), they can also be the source of some misunderstandings. Finally, when it comes to considering full-wave approaches, integral-equation methods will be preferred to differential-equation approaches, because physical interpretations are already possible at the level of periodic Green's functions and because those functions implicitly involve radiation conditions. Whenever possible, developments presented by different authors will be described under a common notation such that the advantages of different analysis methods can be clearly identified. Throughout the paper, a phasor notation will be adopted, with an implicit exp(*j ω t*) time dependence. Free-space impedance and wave number will be denoted by *η* and *k*_{°}, respectively. The materials composing the array will be assumed linear and reciprocal.

[5] This paper is structured as follows. In section 2, generalities are provided about simple interpretations of mutual coupling. Starting from the definition of open-circuit patterns, we explain why descriptions based only on mutual coupling matrices, as often considered in the communications community, are bound to single-mode approximations. In section 3, we show that exact relationships however exist between the array *N*-port impedance matrix and the beam coupling factors, which are receiving increasing attention among the astronomic and radiometric communities. An original extension to the case of the unmatched array is provided. Section 4 defines basic quantities for infinite regular arrays. This is done primarily for linear arrays, with the help of Fourier methods relying on distributions theory, and the array scanning method is introduced. In section 5, in the mathematical framework recalled in section 4, two important relationships are given regarding the embedded element pattern in an infinite terminated array. Numerical methods for array analysis are reviewed in section 6, with special emphasis on integral-equation approaches and on good approximations based on macro basis functions (MBF). This approach allows one to dramatically reduce the number of effective degrees of freedom per antenna when large arrays of identical and complex elements are considered. The special case of eigenmodes on regular structures is shortly reviewed in section 7, while section 8 reviews a number of increasingly finer methods for the analysis of finite arrays, that exploit quantities provided by infinite-array calculations. Since ultrawideband arrays form an often extreme study case in terms of mutual coupling, for which most approximations fail, we found it useful to devote a specific part (section 9) to such arrays. Finally, over the recent decade, the issue of noise coupling in high-sensitivity arrays has received great attention; the basic analysis tools for such arrays are reviewed in section 10. Conclusions and further prospects are drawn in section 11.