In the previous chapter, the equations governing the three - dimensional scat-tering phenomena involving dielectric and conducting materials were derived. A preliminary distinction between direct and inverse scattering problems has been introduced.
Since this book is devoted to microwave imaging techniques, which are essentially short - range imaging approaches, the scattering equations consti-tute the foundation for the formulation and the development of the various reconstruction procedures.
The inverse scattering problem considered here belongs to the category of inverse problems (Colton and Kress 1998 ), which includes many very challenging problems encountered in several applications, including atmos-pheric sounding, seismology, heat conduction, quantum theory, and medical imaging.
From a strictly mathematical perspective, the defi nition of a problem as the inverse counterpart of a direct one is completely arbitrary. To this end, it is helpful to recall the following well - known sentence by J. B. Keller (Keller 1976 ) quoted by Bertero and Boccacci ( 1998 , pp. 1 – 2): “ We call two problems inverses of one another if the formulation of each involves all part of the solution of the other. Often, for historical reasons, one of the two problems has been studied extensively for some time, while the other has never been studied and is not so well understood. In such cases, the former is called a direct problem, while the latter is the inverse problem. ”
On the contrary, from a physical perspective, it is generally recognized that the differences between direct and inverse problems are related to the con-cepts of cause and effect. In the specifi c case considered in this book, the direct problem is related to computation of the fi eld that is scattered by a known object (where the interaction between the incident fi eld and the object is the cause of the scattering phenomena), whereas the inverse problem concerns the determination of the object starting from knowledge of the fi eld scattered (the effect of the interaction).
Nowadays, inverse problems, which in the past were considered diffi cult and “ strange ” problems, have been widely studied from a mathematical per-spective, and several books discuss such problems in depth (e.g., Colton and Kress 1998 , Bertero and Boccacci 1998 , Engl et al. 1996 , Tarantola 1987 , Chadan and Sabatier 1992 , Herman et al. 1987 , Tikhonov and Arsenin 1977 ).
The most critical aspect of an inverse problem is usually its ill - posedness . Following the defi nition by Hadamard (1902, 1923) , a problem is well posed if its solution exists, is unique, and depends continuously on the data. The last property essentially means that a small perturbation of the data results in a small perturbation of the solution. If one of these conditions is not satisfi ed, the problem is called ill - posed or improperly posed .
In imaging applications, one measures the scattered fi eld and tries to obtain information on the object subjected to the incident radiation. If, for a given set of measurement values (real data can be affected by noise or measurement errors), there is no object that produce the prescribed fi eld distribution, the problem lacks in existence. Moreover, if two or more different objects produce the same measurement data, the problem solution is not unique. Finally, if two very similar sets of measurements are generated by two signifi cantly different objects, the problem solution does not depend continuously on the data, since small errors in the measurements result in large errors in the solution.
The well - posedness of the very general problem of fi nding f ∈ X , given g ∈ Y , such that
Af g= , (3.1.1)
where A is an operator (potentially nonlinear) mapping elements of the normed space X into elements of the normed space Y , depends essentially on the properties of the operator itself.
In particular, the problem turns out to be well posed if the operator A is bijective (i.e., injective and surjective) and the inverse operator A − 1 such that
A g f− =1 (3.1.2)
is continuous. Injectivity ensures that for any g for which the solution exists, such a solution is unique (uniqueness), whereas surjectivity guarantees that there is a solution f for any g (existence). Finally, if A − 1 is continuous, the
22 THE ELECTROMAGNETIC INVERSE SCATTERING PROBLEM
solution depends continuously on the data (stability). Obviously, if A does not fulfi ll all the requirements, described above, the problem is ill - posed.
If A is a completely continuous operator (i.e., an operator that is compact and continuous), then the problem described by equation (3.1.1) can represent an important example of an ill - posed problem. In fact, if A is such an operator, then equation (3.1.1) is ill - posed unless X is of fi nite dimension.
The proof of this statement (Colton and Kress 1998 ) can be provided as follows. Assume that A − 1 exists and is continuous. As a result, I = A − 1 A (where I is the identity operator in the space X ) is compact, since the composition of a continuous and a compact operator is also compact.
Because the identity operator is compact only if the space in which they are defi ned is of fi nite dimension, the thesis follows. We note that such a result holds true even if A is a nonlinear operator.
Ill - posedness is a very common property of inverse problems that make them very diffi cult to solve. Regularization procedures are useful tools in controlling ill - posedness. Applying a regularization procedure means replac-ing the original ill - posed problem with another well - posed problem, in which some additional information can be added. From this new problem one expects to obtain an approximate solution of the original problem. However, adding further information requires some knowledge of the behavior of the solution to the original problem. This information is usually called a priori information and can be related, in imaging applications, to the physical nature of the body to be inspected, such as its spatial extension, and/or to the noise level of the measured data.
3.2 THREE - DIMENSIONAL INVERSE SCATTERING
Let us consider Figure 3.1 . According to the previous defi nitions, it is assumed that E can be measured for r ∉ V o . In particular, E can be available in an observation domain V m , which corresponds, in practical applications, to the region spanned by the measurement probes (resulting in V m ∩ V o = Ø ). Moreover, because of the limited information content of the data, equation (2.7.2) , for r ∈ V m (hereafter called the data equation ), is often solved together with the equation that provides the internal fi eld distribution (often called the state equation ). This relationship is still given by equation (2.7.2) , but in this case it is valid for r ∈ V o . So the problem solution is reduced to solving the following set of nonlinear integral equations:
E r E r r E r G r r r r( ) = ( ) + ′( ) ′( ) ⋅ ′ ′ ∈∫inc data equationj d Vb V mo
ωμ τ ( ) , (( ), (3.2.1)
E r E r r E r G r r r r( ) = ( ) + ′( ) ′( ) ⋅ ′ ′ ∈∫inc state equatioj d Vb V oo
ωμ τ ( ) , nn( ). (3.2.2)
THREE-DIMENSIONAL INVERSE SCATTERING 23
Scatterer
ε μ
Einc
Investigation domain
Measurement points
εb μb
FIGURE 3.1 Imaging confi guration for three - dimensional inverse scattering; investi-gation and observation domains.
Although formally similar, these equations are, of course, very different, since the left - hand side of equation (3.2.1) (which is a Fredholm equation of the fi rst kind) is a known quantity, whereas the corresponding term in equation (3.2.2) (which is a Fredholm equation of the second kind) is unknown.
Formulated in this way, one can view the resolution of the inverse problem as the search for the object that produces a prescribed scattered fi eld distribu-tion (equal to the one that has been measured in the observation domain), but, at the same time, produces an internal fi eld distribution consistent with the known incident fi eld inside the object itself. It is simply the presence of the known internal fi eld [and, consequently, the constraint imposed by the requirement of fulfi lling equation (3.2.2) ] that distinguishes the inverse scat-tering problem so sharply (also in terms of well - posedness issues) from the inverse source problem (i.e., the retrieval of a source from the fi eld that it radiates). This point is discussed further, for example, by Bleinsten and Cohen (1977) , Devaney (1978) , Hoenders (1978) , and Stone (1987) and in the books cited in Section 3.1 .
It should also be noted that the formulation given above is based entirely on electric fi eld integral equations (EFIEs). However, other formulations for describing the scattering phenomena are available. In microwave imaging, one of the most widely applied is the contrast source formulation (van den Berg and Kleinman 1997 , van den Berg and Abubakar 2001 , Abubakar et al. 2006 ). In such a framework, the inverse problem is still treated in its nonlinear form, but the problem unknowns are the object function and the equivalent current density [which is directly related to the internal fi eld through equation (2.6.10) ].
24 THE ELECTROMAGNETIC INVERSE SCATTERING PROBLEM
According to this alternative formulation, the following two integral equa-tions are considered:
E r E r J r G r r r r( ) = ( ) + ′( ) ⋅ ′ ′ ∈ ( )∫inc eq data equationj d Vb V mo
ωμ ( ) , , (3.2.3)
J r r E r r J r G r r r req inc eq state ( ) = ( ) ( ) + ( ) ′( ) ⋅ ′ ′ ∈∫τ ωμ τj d Vb V oo
( ) , eequation( ).
(3.2.4)
As mentioned, the problems unknowns are τ and J eq , which vanish outside V o .
It is also worth noting that, in several applications, the external shape of the object is known and the integration domain is therefore known. In other applications, the shape of the unknown object is itself a problem unknown. In those cases, it is natural to defi ne an investigation domain (a test region ) V i , which by defi nition includes the support of the scatterer under test ( V o ⊂ V i ).
In order to retrieve the shape of the scatterer, specifi c methods can be used (see Chapter 5 ). However, as a general rule, it is possible to assume that the unknown object to be inspected coincides with the investigation domain; that is, the support of τ ( r ) is extended to all r ∈ V i and the integrals in equations (3.2.1) and (3.2.2) [or in equations (3.2.3) and (3.2.4) ] are defi ned over V i instead of V o . Clearly, a perfect reconstruction (i.e., a success-ful solution of the inverse scattering problem) would yield τ ( r ) = 0, for r ∈ ( V i \ V o ), and so it would be possible to precisely defi ne the actual object shape.
3.3 TWO - DIMENSIONAL INVERSE SCATTERING
The scattering formulation reported in the previous section concerns three - dimensional confi gurations. In fact, although microwave imaging techniques can in principle be applied to three - dimensional confi gurations without theo-retical limitations, most of the approaches proposed so far in the scientifi c literature are still related to two - dimensional problems. In fact, the imaging of two - dimensional structures can be simplifi ed by means of some assumptions regarding the scatterer under test and the illumination system considered. On the contrary, fully three - dimensional approaches can still be considered as preliminary proposals. Nevertheless, in the following chapters, general formu-lations will be discussed in a three - dimensional framework, whereas some approaches proposed for two - dimensional confi gurations will be described with reference to their specifi c imaging modalities.
TWO-DIMENSIONAL INVERSE SCATTERING 25
z
y
x
Tx
Scatterer cross section
Infinite cylinder
Rx
Observation domain
S0
FIGURE 3.2 Infi nite cylinder with an inhomogeneous cross section.
If the object to be inspected has an elongate shape with respect to the space region illuminated by the source, it can be approximated as an infi nite cylinder (see Fig. 3.2 ). This is an assumption that should be carefully verifi ed in each application. However, under this approximation, the cross section of the cylinder can be assumed to be independent of one of the spatial coordinates (in Fig. 3.2 , the z coordinate), and we obtain
ε ε εr r( ) = ( )0 r t , (3.3.1)
μ μ μr r( ) = ( )0 r t , (3.3.2)
where r t is the transversal component of r , such that (in Cartesian coordinates)
r x y z r z= + + = +x y z ztˆ ˆ ˆ ˆ . (3.3.3)
Moreover, if we assume that E r r zinc inc( ) = ( )E z t ˆ , that is, that the incident fi eld is z - polarized and uniform along z [ transverse magnetic incident fi eld (TM)], for symmetry reasons both the scattered electric fi eld and the total electric fi eld turn out to be independent of z and z - polarized fi elds [i.e., E r r zscat scat( ) = ( )E z t ˆ and E ( r ) = E z ( r t ) z ]. In this case, equation (2.7.2) can be rewritten as
E r E r r E r G r r rt t b t t t tSj dz d
o( ) = ( ) + ′( ) ′( )⋅ ′ ′ ′
−∞
∞∫∫inc ωμ τ ( ) , (3.3.4)
26 THE ELECTROMAGNETIC INVERSE SCATTERING PROBLEM
where S o is the cross section of the cylindrical object. By using (2.4.7) , we have
E r E r r E r Ir r
t t b t tb
jk
jk
e b t
( ) = ( ) + ′( ) ′( ) ⋅ + ∇∇⎛⎝⎜
⎞⎠⎟
− − − ′
inc ωμ τ 142 ππ r r
rt
tSdz d
o − ′′⎡
⎣⎢⎤⎦⎥
′−∞
∞∫∫ . (3.3.5)
Moreover, since the following relation holds (Balanis 1989 )
−− ′
= − ′( )− − ′
−∞
+∞( )∫
14 4 0π
edz
jH k
jk
tb t t
b tr r
r rr r2 , (3.3.6)
from equation (3.3.5) , one obtains
E E E H k dz t tb
t z t b t t tSz or r r r r r r( ) = ( ) − ′( ) ′( ) − ′( ) ′( )∫inc
2ωμ τ4 0 , (3.3.7)
or
E E j E G dz t t b t z t t t tSz or r r r r r r( ) = ( ) + ′( ) ′( ) ′ ′∫inc Dωμ τ 2 ( ) , (3.3.8)
where
Gj
H kt t b t t2 04D
2( )r r r r′ = − ′( )( ) (3.3.9)
is the free - space Green function for two - dimensional problems. By using the integral representation of the Hankel function, which will be useful in devel-opment of the diffraction tomography algorithm (Section 5.2 ), this can also be written as (Morse and Feshback 1953 )
Gj e
ke dt t
j k y y
b
j x xb
2
4
2 2 2
2
2 4
2 2 2
D ( )r r′ = −−
− − − ′− − ′( )
−∞
+∞
∫π λ
πλ
π λλ.. (3.3.10)
It is important to note that these assumptions result in a greatly simplifi ed formulation, since the inverse scattering problem turns out to be a two - dimensional scalar problem.
Furthermore, as in the three - dimensional case, the two - dimensional inverse scattering problem under transverse magnetic illumination conditions can be formulated in terms of the following two nonlinear scalar integral equations
E E j E G d Sz t t b t z t t t tS t mz or r r r r r r r( ) = ( ) + ′( ) ′( ) ′ ′ ∈∫inc D dωμ τ 2 ( ) , aata equation( ),
(3.3.11)
TWO-DIMENSIONAL INVERSE SCATTERING 27
E E j E G d Sz t t b t z t t t tS t oz or r r r r r r r( ) = ( ) + ′( ) ′( ) ′ ′ ∈∫inc D sωμ τ 2 ( ) , ttate equation( ),
(3.3.12)
where S m is the observation domain ( S m ∩ S o = Ø ) where the measurements are performed. This domain is usually a circle or a segment ( probing line ) along which the probes are located or moved (see Chapter 4 ). Clearly, (3.3.11) and (3.3.12) are the two - dimensional counterparts of (3.2.1) and (3.2.2) of the three - dimensional case. Analogously, if the contrast source approach is used, the two - dimensional counterparts of equations (3.2.3) and (3.2.4) are
E E j J G d Sz t t b t t t tS t mz zor r r r r r r( ) = ( ) + ′( ) ′ ′ ∈∫inc eq D data ωμ 2 ( ) , eequation( ),
is the z component of the equivalent current density. It should be mentioned that, when the incident fi eld is a transverse electric
(TE) fi eld [i.e., E r r x r yinc inc inct t tE Ex y( ) = ( ) + ( )ˆ ˆ ], the problem is still two - dimensional but the vector nature of the equations is conserved. The TE illumination has been considered by some authors (e.g., Chiu and Liu 1996 , Cho and Kiang 1999 , Ramananjoana et al. 2001 , Qing 2002 ). In certain cases this illumination condition can provide better results, due to the increased information contained in the measured samples of the scattered electric fi eld. However, the simplifi cation inherent to the transverse magnetic illumination is partially lost.
Moreover, in exactly the same way as described for the three - dimensional case in Section 3.2 , if the cross section of the object is unknown, an investiga-tion area (a test region ) S i can be defi ned, which includes the cross section of the scatterer under test ( S o ⊂ S i ). In this case, the integrals in (3.3.11) – (3.3.14) are now extended to S i instead of S o . In this case, too, a perfect reconstruction would yield τ ( r t ) = 0 for r t ∈ ( S i \ S o ), allowing the defi nition of the object cross section.
Finally, if the infi nite cylinder is a PEC one, then, under the transverse magnetic (TM) illumination condition, equation (2.8.1) can be rewritten as
E E j J G dz t t b S t t tGz zr r r r r r( ) = ( ) + ′( ) ′ ′∫inc Dωμ 2 ( ) ,� (3.3.16)
where G is a closed line that determines the object profi le in the transverse plane and is the contour of the object cross section in that plane. Moreover,
28 THE ELECTROMAGNETIC INVERSE SCATTERING PROBLEM
JSz is the z component of the surface current density J S (see Section 2.8 ) and
is defi ned on G .
3.4 DISCRETIZATION OF THE CONTINUOUS MODEL
In practical applications, the continuous model must be discretized. Several numerical methods can be applied to solve the equations involved in scattering problems. The most widely used are the method of moments (MoM), the fi nite - element method (FEM), and the fi nite - difference (FD) methods. These methods can be implemented with reference to several different schemes, and a plethora of modifi ed and hybrid techniques can be adopted. Detailed descrip-tion of these approaches is outside the scope of the present monograph, and the reader is referred to specialized books (e.g., Harrington 1968 , Moore and Pizer 1984 , Wang 1991 , Zienkiewicz 1977 , Chari and Silvester 1980 , Chew et al. 2001 , Mittra 1973 , Press et al. 1992 , Bossavit 1998 , Jin 2002 , Monk 2003 , Tafl ove and Hagness 2005 , Sullivan 2000 , Kunz and Luebbers 1993 , and refer-ences cited therein). However, for illustration purposes only, and in order to defi ne some quantities used in the following chapters, a straightforward dis-cretization, often used to obtain pixelated representations (images) of the original and reconstructed dielectric distributions, is presented, which is in principle based on application of MoM to a two - dimensional dielectric confi gu-ration under TM illumination. With this goal in mind, let us consider equation (3.3.8) and search for an approximate numerical solution. The problem unknowns can be expanded in a set of N basis function f n ( r t ), such that
τ τr rt n n tn
N
f( ) = ( )=∑
1 (3.4.1)
and
E E fz t n n tn
N
r r( ) = ( )=∑
1. (3.4.2)
In order to obtain the simplest partitioning of the investigation domain S i , piecewise constant representations of the unknowns can be used. To this end, one chooses f n ( r t ) such that f n ( r t ) = 1 if r t ∈ S n , where S n is the n th subdomain of the partitioned investigation domain (i.e., � n S n = S i ), and f n ( r t ) = 0, else-where. Moreover, an inner product must be introduced. Typically, for complex functions, the following product is considered (Harrington 1968 )
< >= ′ ′ ′∫u v u v dt t t t tS( ), ( ) ( ) *( ) ,r r r r r (3.4.3)
where u and v are two generic functions of r t having S as domain, and v * denotes the complex conjugate of v . Considering the discrete nature of the measurements in imaging applications, one can assume that the values of the scattered fi eld are available in a set of M measurement points where the
DISCRETIZATION OF THE CONTINUOUS MODEL 29
acquisition probes are located. These conditions suggest the use, as testing functions, of a set of Dirac delta functions, such that
wm t t mr r r( ) = −( )δ , (3.4.4)
where r m , m = 1, … , M , is the m th measurement point. By substituting (3.4.1) and (3.4.2) in equation (3.3.8) and sequentially multiplying [by means of the inner product of equation (3.4.3) ] the resulting equation by each testing func-tion, one obtains
h E E m Mmn n nn
N
msτ
=∑ = =
11, , , ,… (3.4.5)
where
h j G dmn b m t tSn= ′ ′∫ωμ 2D ( )r r r (3.4.6)
and
E E E E Ems
m mi
z m mz= − ≡ ( ) − ( )r rinc , (3.4.7)
where Ems , E m , and Em
i respectively are the z components of the scattered, total, and incident fi elds at the m th measurement point. In order to obtain equation (3.4.5) , it has been assumed that the integral in equation (3.4.3) is extended to an infi nite domain [this is clearly possible since τ ( r t ) = 0 for r t ∉ S ] and the well - known properties of the Dirac delta functions have been exploited.
As a result, the problem solution is reduced to the resolution of a set of M nonlinear algebraic equations given by (3.4.5) . Equation (3.4.5) can also be expressed in matrix form as
H T e e[ ][ ] = s, (3.4.8)
where e = [ E 1 , … , E N ] T , es sMs T
E E= [ ]1, ,… ; [ H ] is a rectangular M × N matrix, whose elements are the coeffi cients h mn , m = 1, … , M , n = 1, … , N ; and [ T ] is a square N × N diagonal matrix whose diagonal elements are given by τ n , n = 1, … , N .
It should be noted that the coeffi cients h mn can be numerically computed. A very simple expression, widely used in imaging approaches and suffi ciently accurate for two - dimensional TM scattering, is obtained by approximating the subdomains S n , n = 1, … , N , by circles of equivalent areas (Richmond 1965 ). In this case
hj
ka J k a H kmn n b n m n= ( ) −( )( )
21 0
2π r r , (3.4.9)
30 THE ELECTROMAGNETIC INVERSE SCATTERING PROBLEM
where a Sn n= Δ π is the radius of the equivalent circle, Δ S n is the area of S n , and r n , n = 1, … , N , is the barycenter of S n .
As mentioned previously, the discretization procedure presented above is very simple. Various different approaches can be followed. For example, several different basis and testing functions can be used. The objective is usually to minimize the number of problem unknowns, which has a direct impact on the computational load of the method. It is also worth noting that the selected basis and testing functions are required to allow a simple computation of the coeffi cients and the known terms of the discretized equations.
It is also evident that the various above mentioned numerical methods can be applied not only to the two - dimensional equations considered here but also to the three - dimensional formulation of Section 3.2 . A detailed example is given below. (This procedure has been applied for the numerical simulations presented thoroughout this book.) For three - dimensional scattering by isotropic dielectric bodies, the integral equation (2.7.2) can be immediately rewritten as follows (Zhang et al. 2003 ):
E r E r r E r r r r
r E r
( ) = ( ) + ′( ) ′( ) ′ ′
+ ∇∇ ⋅ ′( ) ′
∫inc j G d
jk
b V
b
b
oωμ τ
ωμ τ
( )
2(( ) ′ ′∫ G d
Vo( ) .r r r
(3.4.10)
This integrodifferential equation relating the electric fi eld and the scattering potential is often preferred to the EFIE since it avoids the problems concern-ing the singularities of Green ’ s tensor (Van Bladel 2007 ). For development of the numerical method, it is then useful to introduce the vector fi eld (Zhang et al. 2003 )
A r r E r r r r( ) = − ′( ) ′( ) ′ ′∫jk
G db
bVo
ωμ τ2
( ) , (3.4.11)
where V o = {( x , y , z ) ∈ � 3 : x 1 ≤ x ≤ x 2 , y 1 ≤ y ≤ y 2 , z 1 ≤ z ≤ z 2 } is assumed for convenience to be a parallelepiped containing the support of the scatterers. As a consequence, equation (3.4.10) can be written as
E r E r A r A r( ) = ( ) − ( ) −∇∇ ⋅ ( )inc kb2 . (3.4.12)
The fi rst step in the implementation of the method consists in discretizing region V 0 into N = I × J × K parallelepipeds with faces parallel to coordinate directions and centers located at
r x yijk x i x y j y z k= + −⎛⎝⎞⎠
⎡⎣⎢
⎤⎦⎥+ + −⎛
⎝⎞⎠
⎡⎣⎢
⎤⎦⎥+ + −⎛
⎝1 1 112
12
12
Δ Δˆ ˆ ⎞⎞⎠
⎡⎣⎢
⎤⎦⎥
Δz ˆ ,z (3.4.13)
DISCRETIZATION OF THE CONTINUOUS MODEL 31
for i = 1, … , I , j = 1, … , J , k = 1, … , K , where
Δ Δ Δxx x
Iy
y yJ
zz z
K=
−=
−=
−2 1 2 1 2 1, , , (3.4.14)
are the lengths of the sides of the cells. In such a way, a grid of points G = { r ijk ∈ � 3 , i = 1, … , I , j = 1, … , J , k = 1, … , K } is introduced.
If equation (3.4.12) is enforced at each point of such a grid [i.e., Dirac delta functions located at the center of each cell are used again as testing functions (Harrington 1968 )], it must hold that
E E A Aijk ijk b ijk ijkk= − − ∇∇⋅( )inc 2 , (3.4.15)
where E ijk = E ( r ijk ), E E rijk ijkinc
inc= ( ) , A ijk = A ( r ijk ), and ( ∇ ∇ · A ) ijk is the value of ∇ ∇ · A evaluated at point r ijk . In order to express ( ∇ ∇ · A ) ijk in terms of the values of A computed at r ijk , a fi nite - difference scheme is used to approxi-mate the vector differential operator ∇ ∇ · , which in a Cartesian frame can be represented as
∇∇⋅ = ∇∇⋅( ) + ∇∇⋅( ) + ∇∇⋅( )A A x A y A zx y zˆ ˆ ˆ , (3.4.16)
where
( ) ,∇∇⋅ =∂∂
+∂∂ ∂
+∂∂ ∂
A xx y zA
x
A
x yA
x z
2
2
2 2
(3.4.17)
( ) ,∇∇⋅ =∂∂ ∂
+∂∂
+∂∂ ∂
A yx y zA
y x
A
yA
y z
2 2
2
2
(3.4.18)
( ) ,∇∇⋅ =∂∂ ∂
+∂∂ ∂
+∂∂
A zx y zA
z x
A
z yAz
2 2 2
2 (3.4.19)
where A x , A y , and A z are the Cartesian components of the vector fi eld A . The adopted fi nite - difference scheme is based on two different approxima-
tions of the fi rst - order derivatives. Namely, for computation of the second order - derivatives ∂ 2 / ∂ p 2 , p = x , y , z , the following expressions are used twice
∂ ( )
∂≈
+⎛⎝
⎞⎠ − −⎛
⎝⎞⎠
=A x y z
x
A xx
y z A xx
y z
xp x y zp
p p, , , , , ,, , , ,
Δ Δ
Δ2 2 (3.4.20)
∂ ( )
∂≈
+⎛⎝
⎞⎠ − −⎛
⎝⎞⎠
=A x y z
y
A x yy
z A x yy
z
yp x y zp
p p, , , , , ,, , , ,
Δ Δ
Δ2 2 (3.4.21)
32 THE ELECTROMAGNETIC INVERSE SCATTERING PROBLEM
∂ ( )
∂≈
+⎛⎝
⎞⎠ − −⎛
⎝⎞⎠
=A x y z
z
A x y zz
A x y zz
zp x y zp
p p, , , , , ,, , , ,
Δ Δ
Δ2 2 (3.4.22)
whereas the approximations used for computing the second - order mixed derivatives are as follows:
∂ ( )
∂≈
+( ) − −( )=
A x y z
x
A x x y z A x x y z
xp x y zp p p, , , , , ,
, , , ,Δ Δ
Δ2 (3.4.23)
∂ ( )
∂≈
+( ) − −( )=
A x y z
y
A x y y z A x y y z
yp x y zp p p, , , , , ,
, , , ,Δ Δ
Δ2 (3.4.24)
∂ ( )
∂≈
+( ) − −( )=
A x y z
z
A x y z z A x y z z
zp x y zp p p, , , , , ,
. , , .Δ Δ
Δ2 (3.4.25)
As a result, one obtains
∂ ( )∂
≈
∂ +⎛⎝
⎞⎠
∂−∂ −⎛
⎝⎞⎠
∂
≈
2
2
2 2A x y z
x
A xx
y z
x
A xx
y z
xx
A x
p
p p
p
, ,
, , , ,Δ Δ
Δ++( ) − ( )
−( ) − −( )
=+
ΔΔ
ΔΔ
ΔΔ
x y z A x y zx
A x y z A x x y zx
x
A x x y
p p p
p
, , , , , , , ,
, ,, , , , ,, , , ,
z A x y z A x x y z
xp x y zp p( ) − ( ) + −( )=
22
ΔΔ
(3.4.26)
and
∂ ( )∂ ∂
≈
∂ +⎛⎝
⎞⎠
∂−∂ −⎛
⎝⎞⎠
∂
≈
22 2
A x y z
x y
A xx
y z
y
A xx
y z
yx
A
p
p p
p
, ,
, , , ,Δ Δ
Δxx x y y z A x x y y z
y
A x x y y z A x x y
p
p p
+ +( ) − + −( )
−− +( ) − −
Δ Δ Δ ΔΔ
Δ Δ Δ
, , , ,
, , ,
2
−−( )
=+ +( ) − + −( )
−−
ΔΔΔ
Δ Δ Δ ΔΔ Δ
Δ
y zy
x
A x x y y z A x x y y z
x y
A x x
p p
p
,
, , , ,
22
4
,, , , ,, , , ,
y y z A x x y y z
x yp x y zp+( ) − − −( )=
Δ Δ ΔΔ Δ4
(3.4.27)
DISCRETIZATION OF THE CONTINUOUS MODEL 33
and analogous relations for the other derivatives. Accordingly, the fi nite - difference approximation of the components of the differential operator ∇ ∇ · can be written as
It is worth noting that, as readily follows from (3.4.31) , the different fi nite - difference approximations (3.4.20) – (3.4.22) and (3.4.23) – (3.4.25) make it pos-sible to express ( ∇ ∇ · A ) ijk in terms of the values of the vector fi eld A computed on the grid points neighboring r ijk . Equation (3.4.31) also shows that, in order to compute ( ∇ ∇ · A ) ijk for i = 1, … , I , j = 1, … , J , i = 1, … , K , the vector fi eld A has to be known on a grid G ′ = { r ijk ∈ � 3 : i = 0, … , I + 1, j = 0, … , J + 1, k = 0, … , K + 1} containing G . If the cells are so small that the dielectric permittivity and the electric fi eld can be assumed to be constant over each cell, the vector fi eld A can then be written as
A r E r r rijkb
bi j k i j k
k
K
ijkV
jk
G di j k
( ) = − ′ ′′ ′ ′ ′ ′ ′′=′∑ ∫ ′ ′ ′
ωμ τ2
1( )
Δjj
J
i
I
=′=∑∑
11, (3.4.32)
for i = 0, … , I + 1, j = 0, … , J + 1, k = 0, … , K + 1, where Δ V i ′ , j ′ , k ′ = {( x , y , z ) ∈ � 3 : x 1 + ( i ′ − 1) Δ x ≤ x ≤ x 1 + i ′ Δ x , y 1 + ( j ′ − 1) Δ y ≤ y ≤ y 1 + j ′ Δ y , z 1 + ( k ′ − 1) Δ z ≤ z ≤ z 1 + k ′ Δ z }, and τ i ′ j ′ k ′ = τ ( r i ′ j ′ k ′ ).
It can be proved that (Zwamborn and van den Berg 1992 )
DISCRETIZATION OF THE CONTINUOUS MODEL 35
G d
kjk e
ijkV
bb
jk
i j k
b
( )r r r′ ′ ≈
+⎛⎝⎜
⎞⎠⎟
−⎡⎣⎢
⎤
′ ′ ′∫
−( )
Δ
ΔΔ81
12
12
1 2ξ ξ
⎦⎦⎥∈
−
−
′ ′ ′
− −
′ ′ ′
′ ′ ′
, ,
sinh
r
r r
r r
ijk i j k
jk
b ijk i j k
V
ek
b ijk i j k
Δ
Δ42
ξ
112
12
12
jk
jk
jk
b
b
b
i
Δ
Δ
Δ
ξ
ξ
ξ
⎛⎝⎜
⎞⎠⎟
− ⎛⎝⎜
⎞⎠⎟
⎡
⎣
⎢⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥⎥cosh
, r jjk i j kV∉
⎧
⎨
⎪⎪⎪⎪
⎩
⎪⎪⎪⎪
′ ′ ′Δ , (3.4.33)
where Δ ξ = min{ Δ x , Δ y , Δ z }. Since the ratio in equation (3.4.33) depends on
r rijk i j k i i x j j y k k z− = − ′( ) + − ′( ) + − ′( )′ ′ ′2 2 2 2 2 2Δ Δ Δ , (3.4.34)
it can be written as
g G di i j j k k ijkVi j k− ′( ) − ′( ) − ′( ) = ′ ′
′ ′ ′∫ ( ) .r r rΔ
(3.4.35)
As a consequence
A r Eijkb
bi j k i j k
k
K
i i j j k kj
jk
g( ) = − ′ ′ ′ ′ ′ ′′=
− ′( ) − ′( ) − ′( )′=∑ωμ τ
21111
J
i
I
∑∑′=
. (3.4.36)
It is noteworthy that equation (3.4.36) allows one to compute the vector fi eld A at the points of grid G ′ even if the electric fi eld and the contrast function are known only on grid G .
With the development of a computer code in mind, it is useful to introduce an array e of 3 N elements containing the values of the three Cartesian com-ponents of the electric fi eld E on grid G . Moreover, it is very easy to check that equations (3.4.15) , (3.4.31) , and (3.4.36) defi ne a linear system of equa-tions for the elements of e . If e inc denotes an array whose 3 N entries are the three Cartesian components of the electric fi eld E inc on grid G , then
L e e[ ] = inc, (3.4.37)
where [ L ] is a 3 N × 3 N matrix expressing the relationships given by equations (3.4.15) , (3.4.31) , and (3.4.36) .
The idea underlying the numerical method described above is to solve the linear system (3.4.37) to compute the values of the electric fi eld inside V o and afterward to use these results to compute the electric fi eld at any point r outside V o . In principle, the linear system (3.4.37) can be solved by any numer-ical algorithm developed for linear systems, both direct and iterative. However,
36 THE ELECTROMAGNETIC INVERSE SCATTERING PROBLEM
because of the usually enormous number of unknowns, the iterative approach is much more convenient since the particular structure of the equations involved allows for a very effi cient computation of the products between matrix [ L ] and the solution vectors. In order to perform the simulation described, the so - called biconjugate stabilized gradient method has been applied (van der Vorst 1992 , Xu et al. 2002 ). Following the notation by Zhang et al. (2003) , the method consists in the following steps:
1. Choose a guess solution e 0 and a tolerance e > 0. 2. Set
r e L e0 0= −[ ]inc , (3.4.38)
ρ0 1= , (3.4.39)
α0 1= , (3.4.40)
ω0 1= , (3.4.41)
v0 0= , (3.4.42)
p0 0= . (3.4.43)
3. Choose r 0 such that
ˆ , ˆ ,r r r r0 0 0 01
3
3 0( ) = ( ) ( ) ≠=∑C n nn
N
N (3.4.44)
and set i = 0. 4. Compute
ρi i= −( , ),�r r0 1 (3.4.45)
β ρρ
αωi
i
i
i
i
=− −1 1
, (3.4.46)
p r p vi i i i i i= + −( )− − −β ω1 1 1 , (3.4.47)
v L pi i= [ ] , (3.4.48)
α ρi
i=( )�r v0,
, (3.4.49)
s s vi i i i= −−1 α , (3.4.50)
t L si i= [ ] , (3.4.51)
ω ii i C
i i C
N
N
=( )( )t st t
,
,,
3
3 (3.4.52)
DISCRETIZATION OF THE CONTINUOUS MODEL 37
e e p si i i i i i= − +−1 α ω , (3.4.53)
r s ti i i i= +ω . (3.4.54)
5. If
r r
e ei i C
C
N
N
e,
,,
( )( ) <
3
3inc inc
(3.4.55)
then terminate and the solution is e i . Otherwise, increment the counter i and go back to step 3. Once the array e has been determined, the electric fi eld is known at every point of grid G . In order to compute the electric fi eld for r ∉ V 0 , the integral relation (2.7.2) can be used, without any problem concern-ing the proper meaning of the involved integral operator.
As a result, by exploiting the previously used approximations, the p Cartesian component of the total electric fi eld vector, E p ( r ), p = x , y , z , can be written as (Livesay and Chen 1974 )
E E j E g dp b i j kq
q x y zk
K
pqp i j kr r r r( ) ≈ ( ) + ′ ′′ ′ ′
=′=′ ′ ′∑∑inc ωμ τ
, ,( )
1rr
ΔVj
J
i
I
i j k′ ′ ′∫∑∑′=′= 11
, (3.4.56)
where E Ei j k
qq i j k′ ′ ′
= ( )′ ′ ′r , E pinc is the p Cartesian component of the incident fi eld vector, and g pq , p , q = x , y , z is the pq component of Green ’ s dyadic tensor G .
Equation (3.4.56) can be simply implemented in a numerical code, since (Livesay and Chen 1974 )
g dj k V e
pq ijkV
b b i j kjk
i j k
b ijk i j k
( )r r rr r
′ ′ = −′ ′ ′
′ ′ ′
∫′ ′ ′
− −
Δ
Δωμπ4 kk
k jkb ijk i j k
b ijk i j k b ijk i j k pq
3
2 2 1
r rr r r r
−− − − −( )⎡⎣
+
′ ′ ′
′ ′ ′ ′ ′ ′ δpp p q q
k
ijk i j k ijk i j k
ijk i j k
b ijk i j
−( ) −( )−
− −
′ ′ ′ ′ ′ ′
′ ′ ′
′ ′ ′
r rr r
2
23 kk b ijk i j kjk2 3+ −( )⎤⎦′ ′ ′r r (3.4.57)
where p ijk and p i ′ j ′ k ′ are the p Cartesian components of r ijk , and r i ′ j ′ k ′ , respec-tively. It is worth remarking that the entries of the array e are suffi cient to compute the electric fi eld at any point outside V 0 , according to (3.4.56) and (3.4.57) .
Some simulation results on plane - wave scattering by a homogeneous sphere are illustrated in Figure 3.3 . The sphere has a radius equal to λ /2 and is char-acterized by ε r = 3.0 and σ = 0.0166 S/m. The incident fi eld is a unit plane wave [equation (2.4.24) ] with k = k 0 z and p = x . The amplitude and the phase of the
38 THE ELECTROMAGNETIC INVERSE SCATTERING PROBLEM
FIGURE 3.3 Scattered electric fi eld produced by the interaction of a plane wave with a sphere ( ε r = 3.0, σ = 0.0166 S/m, radius equal to λ /2), with comparison between numerical and analytical data; total electric fi eld (amplitude and phase): (a), (b) x component; (c), (d) y component; (e), (f) z component. (Simulations performed by G. Bozza, University of Genoa, Italy.)
(a)
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
0 20 40 60 80 100 120 140 160 180
|Ex| [V
/m]
m
analytic solutionFFT-BiCGStab solution
0
50
100
150
200
250
300
350
400
0 20 40 60 80 100 120 140 160 180
∠ E
x [
°]
m(b)
analytic solutionFFT-BiCGStab solution
DISCRETIZATION OF THE CONTINUOUS MODEL 39
(c)
0
2e-06
4e-06
6e-06
8e-06
1e-05
0 20 40 60 80 100 120 140 160 180
|Ey| [V
/m]
m
analytic solutionFFT-BiCGStab solution
(d)
0
50
100
150
200
250
300
350
400
0 20 40 60 80 100 120 140 160 180
∠ E
y [
°]
m
analytic solutionFFT-BiCGStab solution
FIGURE 3.3 Continued
40 THE ELECTROMAGNETIC INVERSE SCATTERING PROBLEM
(f)
0
50
100
150
200
250
300
350
400
0 20 40 60 80 100 120 140 160 180
∠ E
z [
°]
m
analytic solutionFFT-BiCGStab solution
0
5e-06
1e-05
1.5e-05
2e-05
2.5e-05
3e-05
3.5e-05
4e-05
0 20 40 60 80 100 120 140 160 180
|Ez| [V
/m]
m
analytic solutionFFT-BiCGStab solution
(e)
FIGURE 3.3 Continued
SCATTERING BY CANONICAL OBJECTS: THE CASE OF A MULTILAYER ELLIPTIC CYLINDER 41
total electric fi eld have been computed in a set of M = 180 points of Cartesian components given by
rm m m m mM
m M= ( ) = =0 2 22
1, sin , sin , , , , .λ ϕ λ ϕ ϕ π… (3.4.58)
Figure 3.3 plots the three Cartesian components of the total electric fi eld. The numerical values are compared, with excellent agreement, with analytical data computed by using the eigenfunction solution for the homogeneous sphere (30 modes are considered) (Stratton 1941 ).
3.5 SCATTERING BY CANONICAL OBJECTS: THE CASE OF A MULTILAYER ELLIPTIC CYLINDER
As explained above, computation of the electromagnetic fi eld scattered by arbitrarily shaped dielectric objects when they are illuminated by a generic incident fi eld is a complex task that seldom can be performed by using analytic techniques. However, for some canonical geometries of the scatterers and for particular incident fi elds (e.g., plane or cylindrical waves), series expansions of the solutions for the scattered fi elds with analytically computable coeffi -cients can be provided.
Among the various canonical objects reported in the literature (e.g., Stratton 1941 , Bowman et al. 1969 , Wait 1962 ), we consider here, for the sake of illustration, the scattering by a multilayer dielectric cylinder of elliptic cross section when the incident fi eld is a plane wave polarized along the cylinder axis, that is, under TM illumination conditions (Caorsi et al. 1997a ). Similar formulations can be obtained under different illumination conditions (e.g., line - current sources). A stratifi ed elliptic cylinder is suffi ciently complex to provide a good test for imaging procedures and, in general, for numerical algorithms. In fact, the assumed confi guration is inhomogeneous and may contain both dielectric and conducting materials. Moreover, it does not exhibit a circular symmetry, which is an important aspect in evaluating the capabilities of tomographic imaging confi gurations, which are essentially based on circular geometries. In addition, elliptic cylinders are often used to approximately model several real structures, such as aircraft fuselage and other cylindrical bodies (Uslenghi 1997 ).
Let us consider the elliptic coordinates ( u , v , z ) shown in Figure 3.4 . In this coordinate system, u = constant represents a family of elliptic surfaces having the same foci (located at points x = ± d on the x axis), whereas v = constant represent a family of confocal hyperbolic surfaces. Accordingly, the i th layer of the N - layer (lossless and nonmagnetic) cylinder is bounded by the elliptic surfaces u = u i − 1 and u = u i , i = 1, … , N , with u 0 = 0. The external medium is assumed to be the free space and is denoted as the N + 1 layer. The semimajor and semiminor axes of the cylinders bounding the various layers are denoted by a i and b i .
42 THE ELECTROMAGNETIC INVERSE SCATTERING PROBLEM
As mentioned previously, the illuminating fi eld is a transverse magnetic uniform plane wave [equation (2.4.24) ] with p = z and
k x y= − +( )kb cos sin ,ϕ ϕinc inc� � (3.5.1)
where ϕ inc is the incident direction in the horizontal plane. The total electric fi eld in the i th layer is given by
E ziziu v E u v, , ,( ) = ( ) � (3.5.2)
H u v u v Eiui
vi
i
iu v H u v H u vj
h u vu v, , , ,( ) = ( ) + ( ) = −
∂∂+
∂∂
⎡⎣⎢
⎤⎦⎥
� � � �ωμ
(( ), (3.5.3)
with i = 1, … , N , where
h d u v= −cosh cos .2 2 (3.5.4)
According to Yeh ’ s (1963) solution, the fi eld in the i th layer is expressed in terms of Mathieu functions, which are the eigenfunctions for the elliptic cylinder (Stratton 1941 , Morse and Feshback 1953 , Blanch 1965 )
E u v e Mc q u e Mc q u ce q vzi
mi
m i mi
m i m im
, , , ,( ) = ( ) + ( )[ ]⋅ ( )( ) ( )
=
∞
11
22
0∑∑
∑+ ( )+ ( )[ ]⋅ ( )( ) ( )
=
∞o Ms q u o Ms q u se q vm
im i m
im i m i
m1
12
2
1, , , , (3.5.5)
F1 F2
x
y
d
u
v
u = ui
v = vj
FIGURE 3.4 Elliptic coordinates for multilayer elliptic cylinders .
SCATTERING BY CANONICAL OBJECTS 43
where ce m and se m denote even and odd angular Mathieu functions, Mcm1( ) and
Msm1( ) denote even and odd radial Mathieu functions of the fi rst kind, Mcm
2( ) and Msm
2( ) denote even and odd radial Mathieu functions of the second kind (Blanch 1965 ), and q i = 0.25( k i d ) 2 , where k i is the wavenumber of the i th layer. Finally, emj
i and omji , j = 1, 2 are the expansion coeffi cients in the i th layer (to
conserve notation, we set e om m21
21 0= = ).
In the external region, the electric fi eld is the sum of the incident and scat-tered waves. The scattered wave can be expressed in terms of Mathieu func-tions of the fourth kind, Mcm
4( ) and Msm4( ) , which are analogous to the Hankel
functions for the circular cylinder and can be expressed as linear combinations of the corresponding radial Mathieu functions of the fi rst and second kinds [i.e., Mc Mc jMcm m m
4 1 2( ) ( ) ( )= − and Ms Ms jMsm m m4 1 2( ) ( ) ( )= − ]. We then obtain
E u v e Mc q u ce q v
o Ms
z mN
m N m Nm
mN
m
scat , , ,( ) = ( )⋅ ( )
+
+ ( )+ +
=
∞
+
∑ 1 41 1
0
1 4(( )+ +
=
∞( )⋅ ( )∑ q u se q vN m N
m1 1
1, , , (3.5.6)
where the coeffi cients emN +1 and om
N +1 depend on both the amplitude and the direction of the horizontally directed incident plane wave [see equations (2.4.24) and (3.5.1) ], which, expanded in Mathieu functions, has the form
E u vj Mc q u ce q v ce q
ce qz
mm N m N m N
m
incinc,
, , ,( ) = ( ) ( ) ( )( )+ + +2
11 1 1π ϕ
NNm
mm N m N m N
v dv
j Ms q u se q v se q+=
∞
( )+ + +
( )[ ]
+ ( ) ( )∫
∑1
2
0
20
11 12
,
, ,
π
π 11
12
0
21
,
,,
ϕπ
inc( )( )[ ]+=
∞
∫∑
se q v dvm Nm (3.5.7)
where q N +1 = 0.25( k b d ) 2 . The unknown coeffi cients can be deduced by enforcing the continuity of the tangential components of the electric fi eld across the interfaces between layers. To this end, let us defi ne the following quantities:
C ce q v ce q v dvmnij
m i n j= ( ) ( )∫ , , ,0
2π (3.5.8)
S se q v se q v dvmnij
m i n j= ( ) ( )∫ , , ,0
2π (3.5.9)
CS ce q v se q v dvmnij
m i n j= ( ) ( )∫ , , .0
2π (3.5.10)
For the i th layer, by applying the Galerkin ’ s method, one obtains
C e Mc q u e Mc q u
C
mni i
mi
m i i mi
m i im
nni
+( ) ( ) ( )
=
∞
+
( ) + ( )[ ]
=
∑ 11
12
2
01
, ,
(( ) +( ) + ( )+
+ ( )+( ) + ( )[ ] =i
ni
n i i ni
n i ie Mc q u e Mc q u i111 1
1 21 2
1, , if 11 12 1
1 11 1
, , ,, ,
… Nj Mc q u ce q
C e
nn i i n i
nni i
n
−( ) ( )
+
( )+ +
+( ) +( )π ϕ inc
iin i iMc q u i N+ ( )
+( ) =
⎧⎨⎪
⎩⎪1 4
1, ,if (3.5.11)
44 THE ELECTROMAGNETIC INVERSE SCATTERING PROBLEM
where n = 0, 1, 2, … , and
S o Ms q u o Ms q u
S
mni i
mi
m i i mi
m i im
nni
+( ) ( ) ( )
=
∞
+
( ) + ( )[ ]
=
∑ 11
12
2
11
, ,
(( ) +( ) + ( )+
+ ( )+( ) + ( )[ ] =i
ni
n i i ni
n i io Ms q u o Ms q u i111 1
1 21 2
1, , if 11 12 1
1 11 1
, , ,, ,
… Nj Ms q u se q
S o
nn i i n i
nni i
n
−( ) ( )
+
( )+ +
+( ) +( )π ϕ inc
iin i iMs q u i N+ ( )
+( ) =
⎧⎨⎪
⎩⎪1 4
1, ,if (3.5.12)
where n = 1, 2, … . Analogously, the continuity of the H u vv
i ,( ) component of H i ( u , v ) gives
C e DMc q u e DMc q u
C
mni i
mi
m i i mi
m i im
nni
+( ) ( ) ( )
=
∞( ) + ( )[ ]
=
∑ 11
12
2
0, ,
++( ) +( ) + ( )+
+ ( )+( ) + ( )[ ]1 1
11 1
1 21 2
1i
ni
n i i ni
n i ie DMc q u e DMc q u, , iff
inc
i Nj DMc q u ce q
C
nn i i n i
nni i
= −( ) ( )
+
( )+ +
+( ) +
1 12 1
1 11
, , ,, ,
…π ϕ
11 1 41
( ) + ( )+( ) =
⎧⎨⎪
⎩⎪ e DMc q u i Nni
n i i, ,if
(3.5.13)
where n = 0, 1, 2, … , and
S o DMs q u o DMs q u
S
mni i
mi
m i i mi
m i im
nn
+( ) ( ) ( )
=
∞( ) + ( )[ ]
=
∑ 11
12
2
1, ,
(ii ini
n i i ni
n i io DMs q u o DMs q u+ + + ( )+
+ ( )+( ) + ( )[ ]1 1
11 1
1 21 2
1)( ) , , iff
inc
i Nj DMs q u se q
S
nn i i n i
nni i
= −( ) ( )
++ +
+
1 12 1
1 11
, , ,, ,( )
( )(
…π ϕ
++ + ( )+( ) =
⎧⎨⎪
⎩⎪1 1 4
1) , ,o DMs q u i Nn
in i i if
(3.5.14)
where n = 1, 2, … , and D denotes the derivatives of the Mathieu functions with respect to the u variable. To obtain these relations, we consider the fact that CSmn
i i+( ) =1 0 for any m , n , and i . Moreover, from (3.5.11) – (3.5.14) , the following matrix equations can be obtained by imposing series truncations
B w A wi i i i+ + +[ ] = [ ]1 1 1 , (3.5.15)
where
w j j j j j j jMj T
e e o o e e e= ⎡⎣ ⎤⎦−( )01 02 11 12 11 12 1 2, , , , , , , ,� (3.5.16)
and the matrices involved in (3.5.15) are block matrices given by
B
BeBo
Be
Be
j
j
j
j
Mj
[ ] =
⎡⎣ ⎤⎦⎡⎣ ⎤⎦
⎡⎣ ⎤⎦
⎡⎣ ⎤⎦
⎡
⎣
⎢⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥⎥−
0
1
1
1
0
0�
,, (3.5.17)
SCATTERING BY CANONICAL OBJECTS 45
A
AC ASC AC AC
ACS Aj
j j jM
j
j
[ ] =
⎡⎣ ⎤⎦ ⎡⎣ ⎤⎦ ⎡⎣ ⎤⎦ ⎡⎣ ⎤⎦⎡⎣ ⎤⎦
−( )00 01 01 0 1
10
�SS ACS ACS
AC ASC AC11 11 1 1
10 11 11
j jM
j
j j j
⎡⎣ ⎤⎦ ⎡⎣ ⎤⎦ ⎡⎣ ⎤⎦⎡⎣ ⎤⎦ ⎡⎣ ⎤⎦
−( )�⎡⎡⎣ ⎤⎦ ⎡⎣ ⎤⎦
⎡⎣ ⎤⎦ ⎡⎣ ⎤⎦
−( )
−( ) −( ) −( )
�� � � �
AC
AC ACS AC
1 1
1 0 1 1 1 1
Mj
Mj
Mj
Mjj
M Mj⎡⎣ ⎤⎦ ⎡⎣ ⎤⎦
⎡
⎣
⎢⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥⎥−( ) −( )� AC 1 1
, (3.5.18)
where
Bekj
kkjj k j j k j j
k j j
CMc q u Mc q u
DMc q u⎡⎣ ⎤⎦ =
( ) ( )( )−
( )−
( )−
11
21
11
, ,,(( ) ( )
⎡⎣⎢
⎤⎦⎥
= −( )−DMc q u
k Mk j j2
1
0 1 1,
, , , , ,… (3.5.19)
Bokj
kkjj k j j k j j
k j j
SMs q u Ms q u
DMs q u⎡⎣ ⎤⎦ =
( ) ( )( )−
( )−
( )−
11
21
11
, ,,(( ) ( )
⎡⎣⎢
⎤⎦⎥
= −( )−DMs q u
k Mk j j2
1
1 1,
, , , ,… (3.5.20)
ACkhj
hkj j h j j h j j
h
CMc q u Mc q u
DMc⎡⎣ ⎤⎦ =
( ) ( )−( )( )
− −( )
− −11
1 12
1 11
, ,(( )
− − − −( ) ( )⎡⎣⎢
⎤⎦⎥
= − = −q u DMc q u
k M h Mj j h j j1 1
21 1
0 1 0 1, ,
,
, , , , ,
( )
… … ,, (3.5.21)
ACSkhj
hkj j h j j h j jCS
Mc q u Mc q uDMc
⎡⎣ ⎤⎦ =( ) ( )−( )
( )− −
( )− −1
11 1
21 1, ,
hh j j h j jq u DMc q uk M h M
11 1
21 1
1 1 0
( )− −
( )− −( ) ( )
⎡⎣⎢
⎤⎦⎥
= − =, ,
,
, , , , ,… … −− 1, (3.5.22)
ASCkhj
khj j h j j h j jCS
Ms q u Ms q uDMs
⎡⎣ ⎤⎦ =( ) ( )−( )
( )− −
( )− −1
11 1
21 1, ,
hh j j h j jq u DMs q uk M h M
11 1
21 1
0 1 1
( )− −
( )− −( ) ( )
⎡⎣⎢
⎤⎦⎥
= − =, ,
,
, , , , ,… … −− 1, (3.5.23)
ASkhj
hkj j h j j h j j
h
SMs q u Ms q u
DMs⎡⎣ ⎤⎦ =
( ) ( )−( )( )
− −( )
− −11
1 12
1 11
, ,(( )
− −( )
− −( ) ( )⎡⎣⎢
⎤⎦⎥
= − = −q u DMs q u
k M h Mj j h j j1 1
21 1
1 1 1 1, ,
,
, , , , ,… … ,, (3.5.24)
From equation (3.5.15) , the coeffi cients of the fi eld expansion in the ( i + 1)th layer can be explicitly expressed in terms of the coeffi cients of the i th layer, yielding
w D wi i i+ = [ ]1 , (3.5.25)
where
D
DC DSC DC DC
DCS Dj
j j jM
j
j
[ ] =
⎡⎣ ⎤⎦ ⎡⎣ ⎤⎦ ⎡⎣ ⎤⎦ ⎡⎣ ⎤⎦⎡⎣ ⎤⎦
−( )00 01 01 0 1
10
�SS DCS DCS
DC DSC DC11 11 1 1
10 11 11
j jM
j
j j j
⎡⎣ ⎤⎦ ⎡⎣ ⎤⎦ ⎡⎣ ⎤⎦⎡⎣ ⎤⎦ ⎡⎣ ⎤⎦
−( )�⎡⎡⎣ ⎤⎦ ⎡⎣ ⎤⎦
⎡⎣ ⎤⎦ ⎡⎣ ⎤⎦
−( )
−( ) −( ) −( )
�� � � �
DC
DC DSC DC
1 1
1 0 1 1 1 1
Mj
Mj
Mj
Mjj
M Mj⎡⎣ ⎤⎦ ⎡⎣ ⎤⎦
⎡
⎣
⎢⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥⎥−( ) −( )� DC 1 1
, (3.5.26)
46 THE ELECTROMAGNETIC INVERSE SCATTERING PROBLEM
where
DCkhj
kj
kj
kj
pj
kj
k
be be be be
be ac
⎡⎣ ⎤⎦ = −( )+ + + + −
+11
122
112
1211 1
221
hhj
kj
khi
kj
khj
kj
khi
ki
be ac be ac be ac
be11 12
121 22
112 12
122
11
− −+ + +
++ + + +− −⎡
121 21
111 11
122 21
112ac be ac be ac be ackh
ikj
khj
ki
khi
kj
khj⎣⎣
⎢⎤
⎦⎥ , (3.5.27)
where k = 0, 1, … , M − 1, h = 0, 1, … , M − 1, and
DSCkhj
kj
kj
kj
pj
kj
be be be be
be as
⎡⎣ ⎤⎦ = −( )+ + + + −
+11
122
112
1211 1
221 cc be asc be asc be asc
bkhj
kj
khi
kj
khj
kj
khi
11 121
21 221
12 121
22− −+ + +
ee asc be asc be asc beki
khi
kj
khj
ki
khi
kj
111
21 211
11 111
22 211+ + + +− − aasckh
j12
⎡
⎣⎢
⎤
⎦⎥ , (3.5.28)
where k = 0, 1, … , M − 1, h = 1, … , M − 1, and
DCSkhj
kj
kj
kj
pj
kj
bo bo bo bo
bo ac
⎡⎣ ⎤⎦ = −( )+ + + + −
+11
122
112
1211 1
221 ss bo acs bo acs bo acs
bkhj
kj
khi
kj
khj
kj
khi
11 121
21 221
12 121
22− −+ + +
oo acs bo acs bo acs boeki
khi
kj
khj
ki
khi
kj
111
21 211
11 111
22 21+ + + +− − 11
12acskhj
⎡
⎣⎢
⎤
⎦⎥ , (3.5.29)
where k = 1, … , M − 1, h = 0, 1, … , M − 1, and
DSkhj
kj
kj
kj
pj
kj
k
bo bo bo bo
bo as
⎡⎣ ⎤⎦ = −( )+ + + + −
+11
122
112
1211 1
221
hhj
kj
khi
kj
khj
kj
khi
ki
bo as bo as bo as
bo11 12
121 22
112 12
122
11
− −+ + +
++ + + +− −⎡
121 21
111 11
122 21
112as bo as bo as bo askh
ikj
khj
ki
khi
kj
khj⎣⎣
⎢⎤
⎦⎥ , (3.5.30)
where k = 1, … , M − 1, h = 1, … , M − 1. In equations (3.5.27) – (3.5.30) , beklp
j , boklpj , ackhlp
j , as acskhlpj
khlpj , asckhlp
j , l , p = 1, 2 indicates the elements of the l th row and p th column of matrices [ Bek
j ], [ Bokj ],
[ ACkhj ], [ ASkh
j ], [ ACSkhj ], and [ ASCkh
j ], respectively. If equation (3.5.25) is applied recursively, then, for i = 1, … , N − 1, we can obtain
w F wi i+ = [ ]1 1, (3.5.31)
where
F F D Di i i p
p
i
[ ] = [ ][ ] = [ ]−
=∏1
1, (3.5.32)
with [ D 0 ] = [ I ]. At the external boundaries, using the same procedure, we obtain analogous
relations. In particular
B u A w K[ ] = [ ] + [ ]+N N N1 , (3.5.33)
where
SCATTERING BY CANONICAL OBJECTS 47
uN N N NMN T
e o e e+ + + +−+= [ ]1
01
01
11
11, , , , ,… (3.5.34)
B
BeBo
Be
Be
[ ] =
[ ][ ]
[ ]
[ ]
⎡
⎣
⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥
−
0
1
1
1
0
0�
M
, (3.5.35)
K[ ] =
− ( ) ( )− (
( )+ +
+
22
001
1 0 10
01
1
π ϕπ
j Mc q u ce qj DMc q u
N N N
N N
, ,,( )
inc
)) ( )− ( ) ( )−
+( )
+ +
ce qj Ms q u se q
j DMs
N
N N N
0 11
11
1 1 11
22
,, ,
ϕπ ϕπ
inc
inc
111
1 1 1( )
+ +( ) ( )
⎡
⎣
⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥
q u se qN N N, ,
,
ϕ inc
�
(3.5.36)
and [ A N ] is as given by (3.5.18) . In equation (3.5.35) the involved matrices are given by
Bek kkN N k N N
k N N
CMc q u
DMc q uk[ ] = ( )
( )⎡⎣⎢
⎤⎦⎥
+( ) +( ) +( )
+
1 14
14
1
( ) ,,
, == −0 1 1, , , ,… M (3.5.37)
Bok kkN N k N N
k N N
SMs q u
DMs q uk[ ] = ( )
( )⎡⎣⎢
⎤⎦⎥
+( ) +( ) +( )
+
1 14
14
1
( ) ,,
, == −1 1, , ,… M (3.5.38)
By substituting (3.5.31) in (3.5.33) , we obtain
B u A F w K[ ] − [ ][ ] = [ ]+ −N N N1 1 1 , (3.5.39)
and fi nally
uw
BA F
KN
N N
+
−
−⎡⎣⎢
⎤⎦⎥=[ ]
−[ ][ ]⎡⎣⎢
⎤⎦⎥[ ]
1
1 1
100
, (3.5.40)
which allows us to derive the coeffi cients of the external and the innermost layers simultaneously. From u N +1 we can also deduce the far - fi eld properties of the scattered fi eld. In particular, the scattering width is defi ned as
W W v
E
E
ke ce q v e
z
z
bMN
p Nj p
ϕ πρρ
π
( ) = ( ) =
≈ ( )
→∞
++
( )
lim
,
2
4
2
11
2
scat
inc
++ ( )=
++
( )
=∑ ∑p
M
MN
p Nj p
p
M
o se q v e0
11
2
1
2
, ,π
(3.5.41)
in which the following asymptotic expression, valid for large values of u , has been applied to equation (3.5.6) :
48 THE ELECTROMAGNETIC INVERSE SCATTERING PROBLEM
Mc q uk d u
e Ms q upb
j k d u pp
b4 2 1 4 42( ) − − +( )[ ]{ } ( )( ) ≈ ≈ ( ),cosh
,cosh
ππ .. (3.5.42)
Moreover, once w 1 has been obtained from equation (3.5.33) , the expansion coeffi cients in the other internal layers ( i = 2, … , N ) can be immediately obtained by using equation (3.5.25) recursively.
It should be mentioned that the recursive method described above requires the solution of only one matrix equation, exactly as does the Yeh method for a single homogeneous elliptic cylinder (Yeh 1963 ). Therefore, it is computa-tionally effi cient.
This procedure has been checked for several cases. Some examples are described as follows (Caorsi et al. 2000 ):
1. A Three - Layer Elliptic Cylinder. In this case the semimajor axes of the ellipses constituting boundaries in the transversal plane are given by a 1 = 0.1 m, a 2 = 0.16 m, and a 3 = 0.2 m (external boundary). The semifocal distance is d = 0.02 m, and the dielectric properties of the three nonmag-netic layers are εr1 2 0= . , εr2 1 3= . , and εr3 2 5= . , respectively. The back-ground is vacuum ( ε b = ε 0 ), and the cross section center coincides with the origin of the coordinate system. The incident fi eld is produced by a line - current source ( f = 600 MHz) with unit amplitude and places on the x axis at point x = 0.505 m, y = 0.0. Figure 3.5 shows the computed total electric fi eld (amplitude and phase) along the x and y axes obtained by using M = 10 modes for each layer. Since d is very small, the ellipses almost degenerate into circular cylinders. Consequently, the produced fi eld can be compared with the one obtained by the eigenfunction expan-sion for the multilayer circular cylinder (Bussey and Richmond 1975 ). As can be seen, the agreement is quite good. Moreover, the same fi elds have been compared with the one numerically obtained by using a FEM code [with a perfectly matched layer (PML) for truncation of the domain]. Numerical codes can be used to evaluate the reliability of the procedure when the circular cylinder is not a good approximation of the scatterer under test. For the simulation reported in Figure 3.5 , a square domain (including the cylinder cross section) has been consid-ered. The side of this domain is 3 m, and the discretization mesh consists of 300 × 300 equal squares, each of them divided by the positive slope diagonal. The PML used is that described by Caorsi and Raffetto (1998) .
2. A Four - Layer Cylinder. In this case the simulation parameters are a 1 = 0.14 m, a 2 = 0.16 m, a 3 = 0.9 m, a 4 = 0.2 m, d = 0.12 m, εr1 2 1= . , εr2 2 4= . , εr3 1 8= . , and εr4 1 4= . . The illuminating source is placed at points x = 0.505 m and y = 0.305. The working frequency is again f = 600 MHz, and M = 12 modes are used. Figure 3.6 provides the results for this simu-lation and comparison with FEM/PML values (obtained using the same discretization as in case 1). As can be seen, the agreement is quite good,
although some differences are present between the two solutions, probably due to the quite coarse mesh used for the FEM/PML computation.
3. A Three - Layer Cylinder with a 1 = 0.14 m, a 2 = 0.2 m, and a 3 = 0.25 m, d = 0.12 m, εr1 1 5= . , εr2 2 6= . , and εr3 1 9= . . The line current is located at point x = y = 0.505 m, the working frequency in this case is f = 400 MHz, and M = 10 modes are used. Figure 3.7 provides the total electric fi eld (amplitude and phase) computed along a circle of radius R = 0.4 m and centered at the origin of the coordinate system (coinciding with the center of the elliptic cross section). In this case, too, there is a very good agreement between the semianalytical data and the data obtained by using the FEM/PML numerical code.
It should be mentioned that other semianalytical solutions involving elliptic cylinders have been considered. Some examples are the simple case in which a PEC core is present (Richmond 1988 , Caorsi et al. 1997b ), and cases in which the multilayer cylinder consists of isorefractive materials (Caorsi and Pastorino 2004 ) or metamaterials (see Section 11.5 ) (Pastorino et al. 2005 ). Moreover, other similar solutions are available for other geometries, including elliptic cylinders [e.g., the case in which a PEC elliptic cylinder is coated by a circular dielectric layer (Kakogiannos and Roumeliotis 1990 )].
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