## Abstract

Maxwell's equations formulated for media with gradually changing conductivity are reduced to Volterra integral equations. Analytical and numerical investigations of the equations are presented for the case of gradual splash-like change in conductivity. Splash-like change in medium parameters can model any discharge phenomena, growing plasma, charge injection, etc. Exact analytical solution for the resolvent is presented and different field behaviours are analysed for the incident field as a plane wave and as an impulse.

## 1. Introduction

Properties of many media where radiation and propagation of electromagnetic field is considered are often varying with time. Normally, those variations are considered by different material sciences, e.g. plasma physics. However, there are some fundamental properties of field transformation in time-varying media, which can be derived from Maxwell's equations with time-varying coefficients. Such a description can also be another way of considering nonlinear problems, if it is assumed that we know a character of the variations, like splash for a discharge. Different mechanisms such as sudden ionization of a gas or semiconductor crystal (Yablonovitch 1989), transience induced by laser pulse excitation (de Sande *et al*. 1994), electric arc discharge or ball lightning explosion (Vorgul & Vorgul 2001) have similarities in the way of media parameter temporal variations.

Investigation of wave behaviour in transient media started from theoretical works by Felsen (Felsen & Whitman 1970) and Fante (1971), where they studied electromagnetic waves under the influence of step-like time jump in media permittivity. Their main result, which is the key to understanding field behaviour under temporal variations in media, and which is often not obvious for those who do not deal with this subject, is in that after the abrupt change in permittivity the incident wave splits into direct and inverse ones. That is, we have a wave, reflected from time inhomogeneity, by analogy with reflection from spatial boundaries between media with different parameters.

Infinitely fast (step-like) temporal variations were investigated in detail for different incident fields, media and geometry, as in Nerukh & Shavorykina (Vorgul) (1992), Aberg *et al*. (1995), Vorgul & Marciniak (2001) and Trifkovic & Stanic (2006). Gradual change in medium parameters with time and its influence on propagating field usually were solved numerically or approximately (Harfoush & Taflov 1991; Vorgul & Nerukh 1998; Nerukh *et al*. 2001; Bakunov & Grachev 2002; Porti *et al*. 2003) and only a few types of continuous dependences were considered, as time-harmonic variations in Harfoush & Taflov (1991) or approximation of continuous change by a sequence of abrupt ones as in Vorgul & Nerukh (1998).

Plasma media are the most relevant ones for transient descriptions. Plasma frequency temporal variations for Lorentz plasmas and magnetoplasmas were investigated in Kalluri (1999) for two opposite cases of abrupt and slow changes. Several cases of temporal variations of plasma frequency in application to photon acceleration were considered in Mendonca (2001), including changing over a slow time scale where WKB solutions can be applied, front of ionization moving with constant velocity and harmonic time dependence corresponding to Fermi acceleration. Slow time-scale analysis in Mendonca & Guerreiro (2005) elaborates solutions for the case when continuous time dependence is substituted with a sequence of abrupt steps of infinitely short duration.

Temporal variations in media are also a focus of modern optics and optoelectronics (Nerukh *et al*. 2001; Shvartsburg & Petite 2002). Frequency shift effect in transient media is used in subwavelength optics where enhanced bistability of gradient nanofilm provides tunnelling of cut-off frequency waves (Shvartsburg & Petite 2002).

There are much fewer publications with rigorous analysis of the field transformation when medium parameters change continuously with time (Vorgul 1998; Bing-Kang 2006). Analytical solutions for the transformed field not only have the advantage of seeing and saving computational time but also can be used for further development in conjunction with other aspects of the problem, as, for example, being substituted into material equation to derive a self-content nonlinear equation with direct account of field impact on the matter.

Here the transient electromagnetic problem is solved exactly for a time splash of conductivity. Analysis of the solution shows new features of the field transformation.

## 2. Formulating transient electromagnetic problem for gradually time-varying conductivity

Consider a medium, occupying some region, inside which the parameters are time varying, and in the general case inhomogeneous (figure 1). Outside this region the medium is stationary. Since we consider an evolutionary problem, we introduce some initial time moment, assuming that this zero time moment is a beginning of the non-stationarity. The field before this moment is called the initial field and assumed to be known, and the transformed field after this moment is the one to be found.

The problem solution is based on Volterra integral equations for the transformed field, which can be obtained by the following way: from Maxwell's equation, the wave equation can be obtained,(2.1)with .

In (2.1), the l.h.s. is the same as for the stationary case, and all the non-stationarities are collected at the r.h.s. This is achieved by introducing the step function, *h*(*t*, *x*), which is equal to 1 inside the transient object and 0 outside it. Therefore, the electric polarization is mathematically uniformly described in the whole space. Then convolution of Green's function of this equation with its r.h.s. yields the integral equation (Nerukh & Khizhnjak 1991).

This is actually an integral equation only for the internal field (the field inside the transient region), whereas the external region describes the external field determined after the internal one. Unlike the stationary case, this is a Volterra integral equation of the second type. For the one-dimensional case considered here, it takes the following form:(2.2)inside the transient region, *x*>0, andfor the external field (*x*<0), where for transient conductive medium; *θ* is the Heaviside step function; and .

Its solution can be presented in terms of a resolvent that is an inverse integral operator for the kernel of equation (2.2)(2.3)where function *M* satisfies the differential equation(2.4)whereand initial conditions at *t*=*t*′ are(2.5)This is an ordinary linear second-order differential equation with varying coefficients. Introducing a new function, *u*(*τ*), by(2.6)where unidimensional variables and parameters are used, with and *τ*=*ωt*, (2.4) can be transformed into the Riccati equation, which is the first-order nonlinear one as follows:(2.7)changing the initial conditions (2.5) correspondingly.

### (a) Solution of the equation

We model the splash time dependence by the following formula for *f*(*t*):(2.8)

This flexible formula describes accurately a variety of splash-like dependence shapes, when adjusting the constants *C* and *c*.

Looking for a similar to (2.8) form of solution for the function *u*(*τ*) but with different coefficients for different terms, the partial solution is found to be(2.9)

This partial solution allows one to obtain a general solution for the second-order differential equation (2.4) that has the following form for the function :(2.10)whereand *φ*(*t*) is determined by the conductivity time dependence.

This solution being substituted into the formula (2.3) represents an exact expression for electromagnetic field in a medium with time-splashing conductivity. The initial field *E*_{1} can have any dependence on time and space.

To reveal the features of electromagnetic field transformation, two fundamental varieties of the initial field, particularly a plane harmonic wave and a rectangular pulse, are analysed.

## 3. Plane wave transformation

For the plane wave as , the exact expression for transformed field after the transient behaviour started can be derived as follows:(3.1)The above integrals can be solved analytically when the value of the conductivity splash is significantly large for small frequencies. Approximate formulae then have the form(3.2)However, in that case the initial field amplitude decreases very rapidly and vanishes completely before the peak point of the conductivity.

In the opposite case of small medium variations, the field is slightly damping during the splash, regaining its initial amplitude after medium stabilizing.

For the intermediate splash amplitude, the field was calculated by the exact formula (3.1). Results presented in figure 2 show a very short splash of the field amplitude. Its maximum value is more than four times as high as the initial field amplitude at *x*=0, i.e. at the points where the initial field had its maximum at the moment when the media started changing. At *x*=*π*/2, the field behaviour looks similar but with its amplitude a hundred times less than at *x*=0. Similar field behaviour but with intermediate amplitude values relating to that mentioned above is at *x*=*π*/4.

## 4. Rectangular pulse reflection from a half-space with time splash of conductivity

Transformation of pulses is different from that of a harmonic wave due to their transient character. We consider here as a rectangular pulse being incident on a boundary of a half-space *x*>0 with changing conductivity. No limits are put on its duration. After finding the transformed field from (2.3) with determined by (2.10), the external (reflected) field was calculated using its integral representation (2.2) in terms of the internal field. The results of the calculations are shown in figure 3.

During the earlier stage of the pulse interaction with the half-space, when the whole pulse is not involved in it (for the time moments less than the pulse duration), the reflected pulse front that is presented in figure 3*b* then will have its shape and values conserved, moving with the medium light velocity (circle inclusion, figure 3*c*). As time passes, this pulse of a small amplitude moving away from the boundary leaves a field trace of a high amplitude (figure 3*c*). After the end of the conductivity splash, the trace amplitude decreases forming a splash-like pulse. The front of this pulse moves with a velocity lower than the medium light velocity. This trace evolution with time leads to its transformation into a short pulse of a very high amplitude, which is like a Dirac delta function as shown in figure 3*d*.

## 5. Conclusions

Exact solution for the transient electromagnetic problem of field interaction with a medium whose conductivity has a splash-like time dependence is presented in a form allowing analysis of different incident fields as well as considering spatially inhomogeneous medium. Detailed analysis for the plane wave and rectangular pulse incidences reveals features of the field transformation.

Thus, the field of a harmonic plane wave under the influence of intermediate time splash of conductivity is focused on the planes where *x* is divisible by *π*, i.e. where the initial field had its maximum values at the initial moment of the non-stationarity. An unusual thing about a reflected rectangular pulse from a half-space with time-splashing conductivity is that it leaves a trace of its field in the region it passed through, even if this is a region of non-dispersive medium. This field trace appears to be more stable than the main reflected pulse body.

## Acknowledgments

The author would like to greatly thank the organizers of the Maxwell Meeting in Aberdeen (September 2006), especially Charles Wang and John Reid, for providing the opportunity to discuss the results within a wide spectrum of expertise of the participants' relevant fields, as well as to thank Alan Cairns and Bob Bingham for discussing possible applications of the results and methodology to nonlinear analysis of field–plasma interaction.

## Footnotes

One contribution of 20 to a Theme Issue ‘James Clerk Maxwell 150 years on’.

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