Integral equation methods

The starting point are the Stratton-Chu equations. In [Holliday, 1987] the author first takes these equations (in the perfectly conducting case, in which the two equations decouple) to the surface. In this way, he obtains an equation for the surface currents:

$$\vec{J}(\vec{x})=\vec{J}_i(\vec{x})-2\hat{n}(\vec{x})\times \oint_S \vec{J}'\times \nabla' G \, da'.$$ (1.31)

The goal, then, is to solve this current equation. Note that knowing the currents (and charges, in general) at the surface is all that is needed to compute the scattered field. [Holliday, 1987] proceeds to do this iteratively. In the first iteration, the incident (induced) current is used, leading to the Kirchhoff approximation --note that this brings new light to the interpretation of the Kirchhoff approximation. The next iteration, in the small slope and small $k_z \sigma_\zeta$ limits, lead to the Small Perturbation Method result.

In [Fung et al., 1992,Fung, 1994] we find a discussion of the what is known as the Integral Equation Method (IEM). This method is based on the same philosophy as was just described. The current equations (or, rather, the equations for the tangential components of the fields) are to be solved for iteratively. In the first iteration, the Kirchhoff approximation fields are used . The tangential fields at the surface are represented as a sum of the Kirchhoff fields plus a complementary field. Then, integral equations are derived for the complementary fields at the surface.

Based on previous work by [Holliday, 1987], in [Elfouhaily et al., 1998b] the Stratton-equations are again studied in terms of the induced currents on the surface (which are well behaved) and are treated directly as a linear integral equation.

An integral equation  is defined by a Kernel K, and symbolically, it may be described in generality by

$$f=f_0+\lambda Kf,$$ (1.32)

where f₀ is the inhomogeneous (given) term, Kf stands for the convolution of the kernel with the unknown function f, and $\lambda$ is a constant. An iterative solution to this problem is given by a Newmann series  (see [Courant & Hilbert], p. 140)

$$f=f_0+\lambda Kf = f_0 +\lambda K( f_0 +\lambda K f)= ....,$$ (1.33)

and so on. For instance, to first order in $\lambda$, $f_1=f_0+\lambda K f_0$. This is the Kirchhoff solution described above.