## 1. Introduction

[2] Solving radiation and scattering problems in the vicinity of a half-space or stratified medium with integral equation methods is complicated, and slowed by computation of expensive Sommerfeld integrals. In addition, large scattering problems often require millions of unknowns, thus motivating the fast algorithm. Several methods have been developed for this class of problem, and most rely on the fast multipole algorithm (FMA) [*Rokhlin*, 1990; *Coifman et al.*, 1993] and multilevel paradigm, as first proposed by *Brandt* [1991], and implemented for electromagnetics in *Lu and Chew* [1993] and the multilevel fast multipole algorithm (MLFMA) [*Lu and Chew*, 1994]. Recent work includes use of hierarchical basis functions JorgensenGRS2004, the discrete complex image method (DCIM) with MLFMA [*Geng et al.*, 2000; *Li et al.*, 2003; *Ginste et al.*, 2004], and steepest descent path integrals with the fast inhomogeneous plane wave algorithm (FIPWA) [*Hu and Chew*, 2001; *Hu et al.*, 2001].

[3] Much of this work stems from MLFMA and FIPWA; both are multipole-based algorithms that are beyond the scope of introductory discussion. Hence, efforts have been made throughout this article to introduce sufficient detail from the seminal papers on FMA [*Rokhlin*, 1990; *Coifman et al.*, 1993], MLFMA [*Brandt*, 1991; *Lu and Chew*, 1993, 1994] and FIPWA [*Hu and Chew*, 2001; *Hu et al.*, 2001, 1999; *Hu and Chew*, 2000], and it is likely that the interested reader will find them quite useful.

[4] In *Jorgensen et al.* [2004], Legendre polynomial basis functions help reduce the number of unknowns *N* to achieve a faster solution. Meshing the scatterer with curvilinear quadrilateral patches requires fewer unknowns than Rao, Wilton, Glisson (RWG) basis functions [*Rao et al.*, 1982], and the higher-order basis functions can span multiple media. Yet, the computational complexity is higher than MLFMA with RWG bases ((*N*^{2})), and the method is only beneficial to geometries that can be represented with few curvilinear patches. In *Geng et al.* [2000] and *Li et al.* [2003], MLFMA extends earlier work [*Geng et al.*, 1999] with the fast multipole algorithm and DCIM, where complex images were computed with approximate reflection coefficients. Using multipole expansions requires a prohibitive number of terms for sources that reside in highly lossy media. The method achieves (*N* log *N*) complexity of MLFMA. Whiles parallelization efforts enable novel integration of multiple scatterers in very large problems, the cost to compute the complex images is equal to the free space case, and increases memory requirements up to 50%. In addition, the multipole expansion is still needed.

[5] Yet, FIPWA established a simpler formulation for removing the multipole expansion [*Hu and Chew*, 2001; *Hu et al.*, 2001, 1999]. Hu et al. developed FIPWA [*Hu and Chew*, 2001; *Hu et al.*, 2001, 1999; *Hu and Chew*, 2000] as an alternative to the fast multipole algorithm (FMA) and multilevel fast multipole algorithm (MLFMA) (thoroughly discussed and cited in *Chew et al.* [2000]). FIPWA evolves from the fast steepest descent path algorithm (FASDPA) developed by *Michielssen and Chew* [1996], which was the first fast algorithm ((*N*^{4/3} ln *N*)) to use the spectral integral representation of the Green's function, i.e., propagating and evanescent plane waves.

[6] In FIPWA, the spectral dyadic Green's function for layered media is computed with high accuracy because it rigorously treats the reflection coefficients, and it also accelerates computation of the dyadic components using modified steepest descent path (SDP) integrals. Unlike the DCIM approaches of *Geng et al.* [2000] and *Li et al.* [2003], FIPWA only requires one image term, making the setup and computation of FIPWA faster than the DCIM approach. Following the MLFMA paradigm of factoring and diagonalizing the Green's function operator, FIPWA constructs a diagonal operator with extrapolation of evanescent waves from propagating waves. In doing so, FIPWA achieves (*N* log *N*) efficiency for various two- and three-dimensional scattering problems, and in the case of the layered media problem, FIPWA only costs up to 20% more in memory than the free space case.

[7] While FIPWA still relies on the multipole expansion, the formulation is highly adaptable to a completely multipole-free form. Previous work [*Saville and Chew*, 2006a, 2006b] laid the groundwork for the multipole-free fast algorithm, with the implication that one can simply substitute the 2-D FIPWA translator for the 2-D FMA translator. Hence, this paper presents the multipole-free expansion of the spectral layered media Green's function (LMGF) in a new fast algorithm called the Multipole-Free Fast Inhomogeneous Plane Wave Algorithm (MF-FIPWA). MF-FIWPA is validated by comparing the scattering solution of a moderately sized problem to the solutions from FIPWA and a full matrix solver. Finally, MF-FIPWA is demonstrated for a larger complex target and shown to retain (*N* log *N*) complexity in total memory cost and processing time of the matrix-vector product.

[8] This paper is organized as follows. First, the integral equation formulation is presented for the spectral LMGF. Then, the Green's function operator is factorized and diagonalized without using multipoles. The computational cost to constuct the 3-D translation matrix is shown to be (*N*) with controllable accuracy. Finally, numerical results demonstrate the accuracy, complexity, and stability of the new multipole-free algorithm.