## 1. Introduction

[2] Electromagnetic waves can be guided in space by a hollow waveguide, where the walls are electric conductors. This can be used to guide waves traveling from one point to another, and provides a controlled environment in which measurements can be made. In order to be able to interpret these measurements, we need to investigate what influence the waveguide structure and material or filling have on the wave propagation.

[3] The waves can be decomposed in modes, which can be defined as the eigenfunctions of a transversal differential operator. These modes are orthogonal, and the wave equation for each mode decouples completely from the other modes for a linear, homogeneous filling. However, in practice there may still be a mode coupling due to irregularities or finite conductivity in the walls, particularly when degenerate modes are present. When the waveguide is filled with a nonlinear material an additional coupling is introduced, and the equations do not decouple even when assuming perfectly conducting walls. However, it is still expedient to use the standard waveguide modes due to the possibility of using the resulting equations with a mode-matching algorithm in direct and inverse scattering problems [*Petit*, 1980; *Roberts*, 1988].

[4] In this paper we study the propagation of transient waves in nonlinear waveguides, i.e., waves generated by an arbitrary signal. The theory of linear waveguides is well established since the major efforts during the second world war. The analysis is often made in the frequency domain, but since the nonlinear filling not only couples the modes but also induces a coupling between the different frequency components, we choose to treat the problem entirely in the time domain. The propagation of transient waves in waveguides has been treated for linear materials by, among others, *Bernekorn et al.* [1996], *Kristensson* [1995], and *Cotte* [1954], and some general references are *Collin* [1991], *Jackson* [1999, chapter 8], and *Olyslager* [1999].

[5] There has been a number of papers on nonlinear waveguides. Some recent contributions consider the problem of self focusing, where the field energy inside the waveguide moves closer to the center as the wave propagates. A few early studies are found in the works of *Kelley* [1965] and *Chiao et al.* [1964] and a more recent one is [*Polstyanko et al.*, 1997]. In the work of *Rozzi and Zappelli* [1996] a modal field expansion is applied for a dielectric slab waveguide at a fixed frequency. The resulting equations are mainly used to determine where the energy will be localized.

[6] This paper is organized as follows. In section 2 we introduce Maxwell's equations and the instantaneous constitutive relations. The waveguide geometry is presented in section 3, together with the relevant expansion functions. We use these expansion functions to obtain wave equations for each mode in section 4, and the explicit results for a parallel-plate waveguide are calculated in section 5. Since even this simple example proves very challenging, we make the reasonable assumption that almost all the energy is contained in one mode in section 6, which enables us to derive a system of quasi-linear, homogeneous, hyperbolic differential equations for the dominant mode, and a system of inhomogeneous hyperbolic equations with source terms for the minor modes. Some energy relations are derived for both the dominant and the minor modes, allowing an estimate of the growth of the minor modes. The final conclusions and discussions are given in section 7.