Characteristic mode analysis
Characteristic modes (CM) form a set of functions which, under specific boundary conditions, diagonalizes operator relating field and induced sources. Under certain conditions, the set of the CM is unique and complete (at least theoretically) and thereby capable of describing the behavior of a studied object in full. This article deals with characteristic mode decomposition in electromagnetics, a domain in which the CM theory has originally been proposed. Background. CM decomposition was originally introduced as set of modes diagonalizing a scattering matrix. The theory has, subsequently, been generalized by Harrington and Mautz for antennas. Harrington, Mautz and  their students also successively developed several other extensions of the theory. Even though some precursors were published back in the late 1940s, the full potential of CM has remained unrecognized for an additional 40 years. The capabilities of CM were revisited in 2007 and, since then, interest in CM has dramatically increased. The subsequent boom of CM theory is reflected by the number of prominent publications and applications. Definition. For simplicity, only the original form of the CM — formulated for perfectly electrically conducting (PEC) bodies in free space — will be treated in this article. The electromagnetic quantities will solely be  represented as Fourier's images in frequency domain. Lorenz's gauge is used. The scattering of an electromagnetic wave on a PEC body is represented via a boundary condition on the PEC body, namely with formula_2 representing unitary normal to the PEC surface, formula_3 representing incident electric field intensity, and formula_4 representing scattered electric field intensity defined as with formula_6 being imaginary unit, formula_7 being angular frequency, formula_8 being vector potential formula_10 being vacuum permeability, formula_11 being scalar potential formula_13 being vacuum permittivity, formula_14 being scalar Green's function and formula_16 being wavenumber. The integro-differential operator formula_17 is the  one to be diagonalized via characteristic modes. The governing equation of the CM decomposition is with formula_19 and formula_20 being real and imaginary parts of impedance operator, respectively: formula_21 The operator, formula_22 is defined by The outcome of (1) is a set of characteristic modes formula_24, formula_25, accompanied by associated characteristic numbers formula_26. Clearly, (1) is a generalized eigenvalue problem, which, however, cannot be analytically solved (except for a few canonical bodies). Therefore, the numerical solution described in the following paragraph is commonly employed. Matrix formulation. Discretization formula_27 of the body of the scatterer formula_28 into formula_29 subdomains as  formula_30 and using a set of linearly independent piece-wise continuous functions formula_31, formula_32, allows current density formula_33 to be represented as and by applying the Galerkin method, the impedance operator (2) The eigenvalue problem (1) is then recast into its matrix form which can easily be solved using, e.g., the generalized Schur decomposition or the implicitly restarted Arnoldi method yielding a finite set of expansion coefficients formula_37 and associated characteristic numbers formula_26. The properties of the CM decomposition are investigated below. Properties. The properties of CM decomposition are demonstrated in its matrix form. First, recall that the bilinear forms  and where superscript formula_41 denotes the Hermitian transpose and where formula_42 represents an arbitrary surface current distribution, correspond to the radiated power and the reactive net power, respectively. The following properties can then be easily distilled: then spans the range of formula_46 and indicates whether the characteristic mode is capacitive (formula_47), inductive (formula_48), or in resonance (formula_49). In reality, the Rayleigh quotient is limited by the numerical dynamics of the machine precision used and the number of correctly found modes is limited. This last relation presents the ability of characteristic modes to diagonalize the impedance operator (2) and demonstrates far  field orthogonality, i.e., Modal quantities. The modal currents can be used to evaluate antenna parameters in their modal form, for example: These quantities can be utilized for analysis, feeding synthesis, radiator's shape optimization, or antenna characterization. Applications and further development. The number of potential applications is enormous and still growing: The prospective topics include Software. CM decomposition has recently been implemented in major electromagnetic simulators, namely in FEKO, CST-MWS, and WIPL-D. Other packages are about to support it soon, for example HFSS and CEM One. In addition, there is a plethora of in-house and academic packages  which are capable of evaluating CM and many associated parameters. Alternative bases. CM are useful to understand radiator's operation better. They have been used with great success for many practical purposes. However, it is important to stress that they are not perfect and it is often better to use other formulations such as energy modes, radiation modes, stored energy modes or radiation efficiency modes.