PREFACE ix
INTRODUCTION x
CHAPTER 1. ELECTROMAGNETIC WAVE SCATTERING FROM RANDOM ROUGHSURFACES: BASICS 1
1.1. Introduction 1
1.2. Generalities 2
1.2.1. Maxwell equations and boundary conditions 2
1.2.2. Propagation of a plane wave (Helmholtz equation and planewave) 5
1.2.3. Incident wave at an interface: polarization 7
1.3. Random rough surfaces: statistical description andelectromagnetic roughness 12
1.3.1. Statistical description of random rough surfaces 12
1.3.2. Specific case of sea surfaces 19
1.3.3. Electromagnetic roughness and Rayleigh roughnesscriterion 21
1.4. Scattering of electromagnetic waves from rough surfaces:basics 28
1.4.1. Presentation of the problem (2D/3D) 28
1.4.2. Huygens' principle and extinction theorem 30
1.4.3. Green functions (2D/3D) 33
1.4.4. Scattered powers and scattering coefficients 36
CHAPTER 2. DERIVATION OF THE SCATTERED FIELD UNDER ASYMPTOTICMODELS 41
2.1. Bibliography on existing models 42
2.1.1. Introduction 42
2.1.2. Rigorous models 43
2.1.3. Asymptotic models 44
2.1.4. General properties of scattering 52
2.1.5. A few details on the KA and the GO 55
2.2. Scattering in reflection and transmission under the KA withshadowing effect 58
2.2.1. KA in reflection and transmission with shadowing effectfor 2D problems 58
2.2.2. Extension of the KA model to 3D problems 62
2.3. Scattering in reflection for 3D problems under variousasymptotic models 69
2.3.1. Context and specific notations 69
2.3.2. The small perturbation model 71
2.3.3. The Kirchhoff approximation-high-frequency regime73
2.3.4. The weighted curvature approximation 75
2.3.5. The small slope approximation 75
2.3.6. The local curvature approximation 76
2.3.7. The resonant curvature approximation 76
2.3.8. Validation of the different asymptotic numerical modelsfor 2D problems 77
CHAPTER 3. DERIVATION OF THE NORMALIZED RADAR CROSS-SECTIONUNDER ASYMPTOTIC MODELS 81
3.1. Derivation of incoherent normalized radar cross-sectionunder the GO for 2D problems 82
3.1.1. Incoherent NRCS under the GO with shadowing effect for 2Dproblems 82
3.1.2. Calculation of the bistatic shadowing functions inreflection and transmission 86
3.2. General properties and energy conservation of the GO for 2Dproblems 97
3.2.1. General properties of the GO for 2D problems 97
3.2.2. Study of energy conservation under the GO for 2D problems99
3.3. Scattering coefficients under the GO with shadowing effectfor 3D problems 108
3.4. Energy conservation of the GO model for 3D problems 111
3.4.1. Case of a perfectly conducting lower medium 112
3.4.2. Case of a lossless dielectric lower medium 115
3.5. Scattering in reflection for 3D problems under variousasymptotic models 117
3.5.1. Expression of the NRCS under the SPM1 118
3.5.2. Expression of the NRCS under the GO 118
3.5.3. Expression of the NRCS under the SSA 119
3.5.4. Validation and comparison of the different asymptoticanalytical models for 2D problems 119
3.5.5. Comparison between numerical and analytical asymptoticmodels for 3D problems 121
APPENDIX 1. FAR-FIELD SCATTERED FIELDS UNDER THE METHOD OFSTATIONARY PHASE 125
APPENDIX 2. CALCULATION OF THE SCATTERING COEFFICIENTS UNDERTHE GO FOR 3D PROBLEMS 131
BIBLIOGRAPHY 137
INDEX 149
1
Electromagnetic Wave Scattering from Random Rough Surfaces: Basics
This chapter recalls the basic necessary concepts for dealing with electromagnetic wave scattering from random rough surfaces, by using integral equations. First, it recalls the notions of Maxwell equations, plane wave propagation, polarization, Snell–Descartes laws. Second, it gives a statistical description of the heights of random rough surfaces and defines the concept of electromagnetic roughness through the Rayleigh roughness parameter. Last, it introduces the integral equations describing the electromagnetic scattering, and the necessary Green functions, for both 2D and 3D problems, and defines the notion of a normalized radar cross section.
1.1. Introduction
In this book, the incident wave illuminating the surfaces will be considered as a plane wave. A wave can be called locally plane if it is located in the so-called Fraunhofer zone1 of the transmitter source, or far-field zone of the source. This assumes that the source is far enough from the surface such that the incident wave may appear as a plane on a distance greater than any dimension of the surface [LYN 70a]. The media are assumed to be linear, homogeneous and isotropic (LHI), stationary and non-magnetic. The incident medium is perfectly dielectric2, and can be assimilated to vacuum in general, although we will endeavor to write the equations in the general case of any lossless perfect dielectric medium.
The problem of electromagnetic (EM) wave scattering from non-flat surfaces, called rough surfaces, has been studied for decades. In particular, let us quote the works of Lord Rayleigh [RAY 45, RAY 07], who was the first to give a rigorous definition of the EM roughness of a surface (characterized by the so-called Rayleigh roughness criterion, which will be detailed further). Among rough surfaces, two main categories may be distinguished: periodic surfaces (such as square surfaces, triangular surfaces, sawtooth surfaces and sinusoidal surfaces), which are deterministic, and random surfaces for which only some statistical features are known. This latter category is discussed in this book.
This chapter aims at introducing the main necessary concepts for understanding the tools used in the following chapters. In section 1.2, first, we will recall some generalities on EM waves and their propagation in LHI media. The case of dielectric media will be discussed in general, these media being potentially lossy dielectric3. Then, the interaction of these EM waves with a flat interface will be studied by detailing the reflection and transmission of a plane wave at a flat (perfectly conducting, lossless or lossy dielectric) interface of infinite length. In section 1.3, a description of random rough surfaces, with either spatial or spatiotemporal variations, will be given. However, we will focus here only on the cases where spatiotemporal varying surfaces are equivalent to spatial varying surfaces (ergodicity). An application in the maritime domain will be given. Also, the so-called Rayleigh roughness EM criterion will be described for making a distinction among a slightly rough, a moderately rough and a very rough surface. Finally, in section 1.4, the general problem of EM wave scattering from random rough surfaces will be presented, in order to calculate the EM power scattered by such surfaces. In the rough surface scattering community, this quantity is generally called scattering coefficient as a general describer. The more specific terms used in radar and optics will also be given.
1.2. Generalities
1.2.1. Maxwell equations and boundary conditions
In their local form, the Maxwell equations in dielectric media are given by [BOR 80]:
[1.1]
[1.2]
[1.3]
[1.4]
Usually, in Cartesian coordinates, the operator div is replaced by · and the operator rot is replaced by ∧. The first two equations give the relationships of the fields’ structure, and are valid irrespective of the medium. The last two equations depend on the considered medium. Here, E and H refer to the electric and magnetic field vectors, respectively, which compose the EM field. They are expressed in V/m and A/m, respectively. It is important to note that, throughout the book, the vectors will be denoted in bold, and the unitary vectors will be denoted in bold and with a hat. D and B refer to the electric displacement and the magnetic induction, respectively, and describe the action of the EM field on the matter. They are expressed in C/m2 and Tesla, respectively. Finally, ρ and j refer to the densities of charge or current. They are expressed in C/m3 and A/m2, respectively. These quantities act as sources for the EM field. They check the charge conservation equation:
For an LHI medium4 (which is the case that we will always consider in the following), the quantities D, B and j are related to E and H by the following constitutive relations:
[1.5]
[1.6]
[1.7]
where, ε, μ and σ are, respectively, the permittivity, the permeability and the conductivity of considered matter, with ε0 and μ0, as their constants in vacuum, which are equal to:
[1.8]
[1.9]
These two quantities check the relation:
with c as the celerity of light in vacuum. εr and μr are the relative electric permittivity and magnetic permeability, respectively: they are equal to 1 in vacuum. Let us recall that in the following, only non-magnetic media will be considered; consequently, the relative magnetic permeability μr = 1. Moreover, propagation media will be assumed to be free of charge, ρ = 0, and most of the time free of current as well, j = 0. A medium that is free of charge is then qualified as a dielectric medium; a distinction will be made between a dielectric medium free of current, which will be called perfect dielectric medium or lossless dielectric medium, and a dielectric medium not free of current, which will be called lossy dielectric medium.
1.2.1.1. Boundary conditions
Figure 1.1. Interface between two semi-infinite LHI media Ω1 (incident medium) and Ω2
The Maxwell equations are applicable to infinite media, which does not reflect reality as every medium has boundaries. For practical applications of electromagnetics, it is essential to know how to deal with the problem of the boundary between two media of different EM properties. Let us assume that an arbitrary interface S12 separates two semi-infinite media (LHI) denoted by Ω1 for the incident (upper) medium and Ω2 for the transmission (lower) medium, respectively, and is a unitary vector that is orthogonal (normal) to the interface and oriented towards the incident (upper) medium Ω1. The boundary conditions [KON 90, FAR 98, PÉR 01] may be written in the local form as follows:
[1.10]
[1.11]
[1.12]
[1.13]
where ρs and js represent the superficial (or surface) density of charge and the vector of superficial (or surface) density of current, respectively, which may exist at the boundary between the two media (ρs = 0 for dielectric media, ρs = 0 and js = 0 for perfect dielectric media). Equations [1.10] and [1.12], called continuity relations, describe the continuity of the normal component of B and of the tangential component of E at the interface, respectively. The other two equations [1.11] and [1.13] describe the discontinuity of the normal component of D in the presence of superficial charges of density ρs and the discontinuity of the tangential component of H on a layer of current, respectively.
For the case where the lower medium is a perfectly conducting metal5, the equations take the form:
[1.14]
[1.15]
[1.16]
[1.17]
Condition [1.16] is usually called the Dirichlet boundary condition and condition [1.17], in the absence of current, is usually called the Neumann boundary condition.
Using the same method, for the case when the two LHI media are perfect dielectric, the equations take the form:
[1.18]
[1.19]
[1.20]
[1.21]
1.2.2. Propagation of a plane wave (Helmholtz equation and plane wave)
The propagation equations of fields are obtained from the Maxwell equations by using the property rot rot = grad div – 2, where 2 is the vector Laplacian6. Then, in a general way, we obtain:
[1.22]
[1.23]
For a perfect dielectric medium (ρ = 0, j = 0), the equations reduce to:
[1.24]
[1.25]
A wave...
