## 1. Introduction

Electromagnetic wave reflection from dispersive media has been a subject of interest to researchers for many years. The advent of ultra wideband (UWB) short pulse sources has recently attracted renewed interest in this aspect. Accurate modeling and improved physical understanding of pulse reflection from dispersive media is crucial in a number of applications, including optical waveguides, UWB radar, ground penetrating radar, UWB biological effects, stealth technology and remote sensing. Numerous researchers have demonstrated that Lorentz, Debye and Cole-Cole models can be used to accurately predict dispersive properties of many media.

In the seminal work of Sommerfeld (Sommerfeld, 1914) and in the subsequent refinements of Oughstun and Sherman (Oughstun & Sherman, 1988) (Oughstun & Sherman, 1989) (Oughstun & Sherman, 1990), the investigations have focused on the Lorentz material, which is a good model for many materials encountered in optics and engineering. The reflection of a short pulse by a Lorentz medium has been considered for TE (transverse electric) polarization by Gray (Gray, 1980) and for TM (transverse magnetic) polarization by Stanic et al (Stanic et al., 1991). In each of these studies, the authors find the impulse response of the reflected field by calculating the inverse transform of the frequency domain reflection coefficient as an infinite series of fractional order Bessel functions. Although this gives a convenient analytical result, the series form provides little insight into the behavior of the reflected field waveform. Cossman et al have presented a compact form for both TE (Cossman et al., 2006) and TM (Cossman et al., 2007) reflection coefficients, which provide useful intuition about the response of a Lorentz medium half space. However, the mathematical derivations are lengthy and the solutions involve exponential and modified Bessel functions and require convolution operation to evaluate. Particularly, because of the greater complexity of the TM frequency domain reflection coefficient, a more involved process is needed, including the introduction of a term that does not appear in the TE case. Under some conditions this term is noncausal, although the final expression for the impulse response is causal. Moreover, when the incident angle is equal to 45^{0}, the general expressions cannot be used directly due to singularities, and a special form of TM time domain reflection coefficient is separately achieved.

The use of short pulses to probe materials has prompted the study of the reflection of transient waves from material half space of other types. The Debye model (Debye, 1945) is utilized to describe the frequency behavior of the permittivity of many type materials, especially polar liquids. This model has been extended to include conductivity (Kosmas et al., 2004) and several relaxation components (Oswald et al., 1998), and has been used to describe the behavior of such diverse materials as biological tissues (Ong et al., 2003), building materials (Ogunsola et al., 2006), circuit boards (Zhang et al., 2003) and ceramics (Guerra & Eiras, 2004). A standard technique for the measurement of material parameters is to interrogate the material, either in free space (Piesiewicz et al., 2005) or in a waveguide system (Jones et al., 2005) with an electromagnetic pulse. It is therefore important to have an efficient method analyzing the time-domain reflection properties of a Debye material. Rothwell (Rothwell, 2007) worked out the time domain reflection coefficients of a Debye half space for both horizontal and vertical polarizations that involve exponential and modified Bessel functions and require convolution operations to evaluate. Another model commonly used to capture the relaxation-based dispersive properties is the Cole–Cole model (Cole & Cole, 1941) that is more general than the Debye model. For many types of materials including biological tissues, the Cole–Cole models provided an excellent fit to experimental data over the entire measurement frequency range. However, to our knowledge, the time-domain reflection coefficient of a Cole–Cole half space for any polarization has been not available so far, perhaps due to the computational complexity of embedding a Cole–Cole dispersion model into numerical methods.

All materials are to some extent dispersive. If a field applied to a material undergoes a sufficient rapid change, there is a time lag in the response of the polarization or magnetization of the atoms. It has been found that such materials have complex, frequency dependent constitutive parameters. On the one hand, the lossy material is dispersive since it has a complex, frequency dependent permittivity. On the other hand, the Kronig−Kramers relations imply that if the constitutive parameters of a material are frequency dependent, they must have both real and imaginary parts (Rothwell & Cloud, 2001). Such a material, if isotropic, must be lossy. So dispersive materials are general lossy and must have both dissipative and energy storage characteristics. However, many materials have frequency range called transparency ranges over which the imaginary parts are smaller compared to real parts of constitutive parameters. If we restrict our interest to these ranges, we may approximate the material as lossless.

In this chapter, the time domain technique based on the numerical inversion of Laplace transform is developed and extended to the modeling of ultra wideband pulse reflection from Lorentz, Debye and Cole−Cole media. All these three dispersive models satisfy the Kronig−Kramers relations required for a causal material (Rothwell & Cloud, 2001). Firstly, for readers’ convenience, the numerical inversion of Laplace transform is presented. Next, the time domain reflection coefficients, viz impulse responses, of Lorentz, Debye and Cole−Cole half spaces are achieved for both TE and TM cases. Then, the transient reflections of an arbitray pulse from these media are determined by convolving the incident pulse with the impulse responses of these media, instead of using Prony’s method to decompose the incident pulse into a series of finite attenuating exponential signals as in our previous work (Zeng & Delisle, 2006). Based on the time domain analysis of reflected pulses from these dispersive half spaces, some waveform parameters are estimated and the material diagnosis is carried out. Lastly, the work on transient wave reflection from dispersive media is summarized with some meaningful conclusions. Our results show excellent agreement with those in the literature, validating the correctness and effectiveness of our technique.

## 2. Numerical inversion of Laplace transform

The Laplace transform (image function in the complex frequency domain)

and the inverse transformation

In general, it is straightforward to take the Laplace transform of a function. However, the inverse transformation is often difficult. In many cases, the method using simple rules and a table of transforms, and the method using the Bromwich integral and Cauchy integral theorem do not work well, hence some numerical technique must be utilized. To implement the numerical inversion method, the following conditions must be satisfied (Zeng, 2010). Based on the properties of

The most distinctive feature of this method lies in the approximation for

The Bromwich integral is transformed to the integral around the poles of

Then

where

Equation (5) shows that the function

where

In this method, the upper bound for the truncation errors is given by

while the upper bound for the approximation errors is given by

If

As indicated in (10), the relative approximation errors are less than

## 3. Pulse reflection from a Lorentz medium half space

Consider a sinusoidal stead-state plane wave of frequency

where the wave impedance of the incident wave is

Here

Here

where

where

To compare our results with those in (Cossman et al, 2006), the same incident angle,

The next set of parameters is

The third choice of parameters is

For the case of TM polarization (magnetic field perpendicular to the plane of incidence), the originally defined reflection coefficient is still given by (11), the impedance of a plane wave in the free space (ratio of tangential electric field to tangential magnetic field at the interface) is

The incident angle

where

Here

As indicated in the introduction, the solution of transient reflection coefficient in TM case is more complicated and involved than that in TE case. In (Cossman et al., 2007), a term that does not appear in the TE case needs to be introduced and a special form of the time domain reflection coefficient needs to be separately solved when the incident angle is equal to 45^{0}, which is however not needed with numerical Laplace transform. The image function

To compare our results with those in (Cossman et al, 2007), the same incident angles and the same sets of material parameters are chosen as in (Cossman et al., 2007). The first case uses

In the second case, the material parameters are chosen as

In the third case, the choice of parameters is

In the fourth case, the material parameters are the same as those in the first case, but

The final case examines the special case of

Figure 1 and Figure 2 demonstrate that our results perfectly agree with those in (Cossman et al., 2006) and in (Cossman et al., 2007), respectively, for all the different cases.

## 4. Pulse reflection from a Debye and Cole–Cole medium half space

The knowledge of material properties is required in various technological fields, such as geophysics, material science and biomedical engineering. The characterization of bulk materials would be the most direct way to acquire this knowledge and greatly helpful to understand the underlying physics at the microscopic level, which is much more complicated in comparison with the existing formulations of the bulk effects. A typical approach to bulk material characterization is to examine reflected electromagnetic pulses from the interface between free space and the investigated material. Many kinds of materials show the relaxation-based dispersive properties that are commonly captured by the Debye (Debye, 1945) and Cole–Cole (Cole & Cole, 1941) models. Rothwell (Rothwell, 2007) worked out the time domain reflection coefficients of a Debye half space for both horizontal and vertical polarizations that involve exponential and modified Bessel functions and require convolution operations to evaluate. To our knowledge, the time domain reflection coefficient of a Cole–Cole half space for any polarization has been not available so far. It is the purpose of this section to develop a new technique for transient analysis of pulse reflection from Debye and Cole–Cole media, and apply this technique to waveform parameter estimation and material characterization.

### 4.1. Time domain reflection coefficients

Without losing generality and for the comparison with the results in (Rothwell, 2007), the one-order model with zero ionic conductivity is utilized in this work. Introduce the Laplace variable

where

A plane wave is obliquely incident onto a dispersive half space from free space, at an incidence angle

and

for horizontal and vertical polarizations, respectively. Substituting

and

where

and

Either

and

So

and

Both

Before applying this technique to waveform parameter estimation and material characterization, its correctness and effectiveness are verified by comparing the reduced transient reflection coefficients with those in (Rothwell, 2007). Several different cases are considered, each following the Debye model (

Figure 3 (a) illustrates the reduced reflection coefficients of water (at standard temperature and pressure) calculated using our technique, and compares them to the results in (Rothwell, 2007) with an excellent agreement. The reduced reflection coefficients do not

include any impulsive component with the amplitude of

Figure 3 (b) compares the reduced reflection coefficients of a Martian soil stimulant found using our technique to the results in (Rothwell, 2007). Our results agree with those in (Rothwell, 2007) very well. Since the static and optical permittivities for the soil are comparable, the relaxation effect is less dramatic than that for water while the durations of transient reflection coefficients are longer than that for water due to the longer relaxation time.

Figure 3 (c) shows the reduced reflection coefficients of a Lanthanum modified

Consider a Gaussian waveform incident upon a water half-space at

where

### 4.2. Waveform parameter estimation and material characterization

Based on the above transient analysis, this technique can be utilized for the estimation of waveform parameters of reflected pulses. As an example, consider a mixture of water and ethanol with a volume fraction

For water the Cole–Cole parameter

One of the most important waveform parameters is the correlation between two waveforms. It indicates the degree to which two waveforms resemble and is defined by

Let

Assume that a mixture with

Assume that a mixture with

versus

versus

## 5. Conclusion

In general, two approaches are employed to analyze transient reflection and propagation. One is to approximate an arbitrary incident signal with a finite number of attenuating exponential signals using Prony’s method and to apply numerical inversion of Laplace

transform (NILT) to the final image function, which is the product of the frequency domain reflection or tramsimission coefficient and the image function of the approximating incident signal. The accuracy of this approach is limited by the numerical errors from both NILT and the decomposition of the original incident signal into of a series of finite attenuating exponential signals (Zeng & Delisle, 2006). As shown in sections 4.1 and 4.2, another approach is to use NILT for determinating the transient reflection or transmission coefficient and to convolve the incident signal with the time domain reflection or transmission coefficient. The accuracy of this approach depends on both NILT and numerical convolution, which normally incurs smaller errors than decomposing an arbitrary signal into a series of finite attenuating exponential signals. Section 4.1 presents time domain reflection coefficients for both TE- and TM-polarized plane waves incident on a Lorentz medium half space using NILT. Three possible cases are discussed, each of which is determined by a different relationship between the damping coefficient, oscillation and plasma frequencies. The result is an exponentially damped waveform that oscillates based on the conditions of each case.

In section 4.2, the properties of a half space are described in frequency domain by the Debye and Cole–Cole models, respectively, which are commonly used to capture the relaxation-based dispersive properties. First, transient reflected pulses are analyzed and waveform parameters are estimated. Then, based on the estimation, the relationships between the waveform parameters of reflected pulses and the properties of dispersive material as well as incident angles are discussed. Meanwhile, the results obtained with the Debye model are compared to those obtained with the Cole–Cole model. The application of these results to material characterization and diagnosis is explored. It is shown that using the reduced time domain reflection coefficients often brings more physical insights and leads to an efficient algorithm and a robust scheme for dispersive material diagnosis. There is excellent agreement between our results and those in (Rothwell, 2007), which validates the correctness and effectiveness of this work.