2 RADAR TOMOGRAPHY WITH TIME DOMAIN SIGNALS
2.1 Electromagnetic wave propagation in material
2.1.1 Maxwell’s equations
The electromagnetic phenomena are governed by the four fundamental Maxwell equations linking the physical quantities of the electric fieldE⃗, the magnetic field B⃗, the electric flux densityD⃗, the magnetic field intensityH⃗ , the electric current densityJ⃗, and the volumetric electric charge densityρ˜. In the differential form, these equations are
∇ × E⃗=−∂B⃗
∂t (2.1)
∇ · D⃗ =ρ˜ (2.2)
∇ × H⃗ =J⃗ +∂D⃗
∂t (2.3)
∇ · B⃗ =0. (2.4)
The first equation of Maxwell equations (Eq. 2.1) is the Faraday’s law of induc-tion which is based on the experimental fact that a time-changing magnetic flux in-duces electromotive force. Hence a spatially varying, and also time-varying, elec-tric field is always accompanied with a time-varying magnetic field. The equation 2.2, Gauss’s law, states that the electric flux per unit volume in space in equal to the volumetric electric charge density at that point. The volume charge densityρ˜ therefore represents the source from which electric fields originate. In a source-free
medium this equation takes the form∇ · D⃗ =0. A generalisation of Ampère’s law (Eq. 2.3) states that magnetic fields can be generated by an electric current and chang-ing electric fields, predictchang-ing that a changchang-ing magnetic field induces an electric field and vice versa. The fourth Maxwell equation (Eq. 2.4) states that there are no mag-netic charges and that magmag-netic field lines always close on themselves.
The medium-independent quantitiesDandH are related to the electric and mag-netic fields in a dielectric medium through an electric permittivityϵand a magnetic permeabilityµby
D⃗ =ϵE⃗ and H⃗ =B⃗
µ. (2.5)
The current densityJ⃗ in the equation 2.3 is given by J⃗ =J⃗
s+J⃗
c, (2.6)
whereJ⃗
s represents the source current such as that in a transmitting antenna, and J⃗
c the conduction current flowing in a medium which conductivityσ̸=0, when-ever there is an electric field present. The conduction currentJ⃗
c is given by
J⃗c=σE⃗. (2.7)
The parametersϵ,µ, andσdescribe the relationships between macroscopic field quantities. They are constants only for simple material media, which are linear, homogeneous, time-invariant and isotropic. For complex materials, these quantities my depend on the magnitudes of the field quantitiesE⃗andB⃗ (non-linear media), on spatial coordinates (inhomogeneous), on time (time-variant), or on the orientations ofE⃗ andH⃗ (anisotropic).
When substituting the medium-independent quantitiesD⃗andH⃗ with the medium-specific quantities given by the equation 2.5 to emphasise the contributions of the medium in the fields, the Maxwell equations 2.1-2.4 in a source-free medium take the form
∇ × E⃗=−∂B⃗
∂t (2.8)
∇ ·ϵE⃗=0 (2.9)
∇ × B⃗ =µJ⃗ +ϵµ∂E⃗
∂t (2.10)
∇ · B⃗ =0. (2.11)
Taking the curl of the equation 2.8 yields
∇ × ∇ × E⃗=∇(∇ · E⃗)− ∇2E⃗ =−∇ ×∂B⃗
∂t . (2.12)
Substituting equation 2.10 to the equation 2.12, using the equations 2.6 and 2.7 to expand the current density, and assuming a source-free medium (∇ · E⃗ = 0), the equation yields
ϵµ∂2E⃗
∂t2 +σµ∂E⃗
∂t − ∇2E⃗=µ∂J⃗
s
∂t . (2.13)
This equation can be recognised as a hyperbolic wave equation for the electric field and it will be used as the basis for building the forward model for the radar tomographic inverse problem.
2.1.2 Time-harmonic forms of Maxwell’s equations
In radar tomography performed in space, the radar antenna acts as a source of elec-tromagnetic waves. The target SSSB is the dielectric medium surrounded by free space. Hence, an amplitude-modulated radar operating at a carrier frequency fc can be treated as a source of steady-state sinusoidal waves, with its amplitude modulated within a narrow bandwidthBaround the carrier frequency. Assuming that the char-acteristics of the propagation medium do not vary significantly over the bandwidth, the propagation behaviour of the radar signal can be described by a single sinusoidal carrier wave.
Maxwell’s equations 2.1-2.4 allow the field vectorsE⃗, D⃗, H⃗ , andB⃗ be time-variant. To obtain the time-harmonic (sinusoidal steady-state) forms of the
equa-tions, these fields and the current density J⃗ are replaced with the time-invariant complex phasorsE⃗,D⃗,H⃗,B⃗, and⃗J. The former can be obtained from the latter by multiplying byejωt, and taking the real part thereof. For example,E⃗(x,y,z,t) = Re{E⃗(x,y,z)ejωt}. Substituting the field vectors in the equations 2.1-2.4 with the corresponding phasors leads to the time-harmonic forms of Maxwell’s equations:
∇ ×E⃗=−jωB⃗ (2.14)
∇ ·D⃗ =ρ (2.15)
∇ ×H⃗ =⃗J+jωD⃗ (2.16)
∇ ·B⃗=0. (2.17)
Using the phasor equivalents of the quantities in the equation 2.5, the above equa-tions 2.14-2.17 can be written as
∇ ×E⃗=−jωB⃗ (2.18)
∇ ·ϵE⃗=ρ (2.19)
∇ ×B⃗=µ⃗J
s+σE⃗+jωϵE⃗
(2.20)
∇ ·B⃗=0. (2.21)
The phasor current⃗J=⃗Js+⃗Jc =⃗Js+σE⃗(Eq. 2.20) is the sum of the antenna source current⃗Js and material conduction current⃗Jc as in the equations 2.6 and 2.7. In a conducting media, the equation 2.20 can be rearranged to
∇ ×B⃗=µ
⃗J
s+σE⃗+jωϵE⃗
=µh
⃗J
s+jω( σ
jω+ϵ)E⃗i
=µh
⃗Js+jω(ϵ− jσ ω )E⃗i
, (2.22)
where the term(ϵ−jσ/ω)can be interpreted as the effective complex permittivity ϵc of the medium.
Permittivityϵis a measure of the electric polarisability of a material. Together with the magnetic permeabilityµthey determine the phase velocityv of
electro-magnetic waves in the medium by
v= 1
⎷ϵµ. (2.23)
In free space, the electric permittivity and magnetic permeability have constant val-ues ofϵ0=8.854·10−12F/m, andµ0=4π·10−7H/m, respectively.
In lossy media, permittivity of a material is often expressed by the relative per-mittivityϵrwhich is the ratio of the absolute permittivity of the materialϵand the permittivity of the free spaceϵ0:
ϵr= ϵ
ϵ0. (2.24)
With these notations, the effective complex permittivity in the equation 2.22 can be formulated as:
ϵc =ϵ− jσ ω =ϵ0
ϵr−j σ ωϵ0
=ϵ0ϵc r, (2.25) whereϵc r is the complex relative permittivity of the medium. The complex relative permittivity is often also expressed as
ϵc r=ϵ′r−jϵ′′r. (2.26) This notation has been used in Publication IV of this thesis. The equation 2.22 can hence be written more concisely as
∇ ×B⃗=µ(J⃗
s+jωϵcE⃗). (2.27)
To obtain the time-harmonic formulation of the equation 2.13, the procedure again starts by taking the curl of the equation 2.20:
∇ × ∇ ×E⃗=−jω∇ ×B⃗=−jω(µJ⃗
s+jωµϵcE⃗)
=−jωµJ⃗s+ω2µϵcE⃗. (2.28) Using the vector identity∇×∇×E⃗=∇(∇·E⃗)−∇2E⃗and assuming a source-free
medium (∇ ·E⃗=0), the equation 2.28 yields
−∇2E⃗=−jωµJ⃗
s+ω2µϵcE⃗
ω2µϵcE⃗+∇2E⃗= jωJ⃗c. (2.29) The equation 2.29 is recognized as the inhomogeneous Helmholtz equation which can be solved by specifying a suitable boundary condition at infinity, such as the Sommerfeld radiation condition, and calculating the convolution with the Green’s function of the equation.