1. INTRODUCTIONThe problem of the dielectric characterization of an objectby using an illuminating electromagnetic field is of greatinterest in many applications. The dielectric permittiv-ity in fact can be useful in testing or identifying materialssuch as glass, crystals, textiles, minerals, and the newmaterials employed in the aeronautic industry; in elec-tronics, the dielectric permittivity profile characterizesthe behavior of devices such as planar waveguides and op-tical fibers; in biomedical applications, it is useful becauseit is linked to parameters such as water content and tem-perature. Even if the various applications require inci-dent radiation that varies from the optical wavelengths1–3
to microwaves,4–6 the theoretical framework is the same,and one must investigate it carefully to formulate a reli-able reference model of the electromagnetic inverse scat-tering.
The reconstruction of the dielectric profile of an objectfrom external measurements of the scattered field is notan easy task. In fact, it is a nonlinear inverse problem.Moreover, it is ill posed.7 The commonly used approachis based on a drastic simplification of the problem, whichconsists in linearizing the operator that links the un-known permittivity and the scattered field outside theobject.1–4,6,8–10 However, the range of validity of a linearformulation is limited.11 Other approaches are based onthe iterative application of the linear approximation,12,13
but the conditions that guarantee convergence to the so-lution have not been well investigated.14
We have focused our attention on the theoretical andalgorithmical aspects of the problem. Here we apply aquadratic approximation of the operator equation that de-fines the problem.15–17 The quadraticity of the operatorto be inverted allows us to overcome a well-known prob-lem of nonlinear inversion: the presence of false solu-tions, or traps, of the inversion procedure.14 Moreover,this approximation allows us to obtain better results than
the linear approximation does. In particular, the class ofprofiles that can be reconstructed is enlarged.
The exploitation of a rotationally symmetric cylindricalgeometry has two main advantages. It allows us to in-vestigate with a certain simplicity the crucial points in-volved in the inversion, and for this reason it constitutesthe basis for the study of a complete two-dimensional ge-ometry. Moreover, it can be directly applied to many re-alistic cases that have the same symmetry, such as, thecharacterization of optic fibers.
This paper is organized as follows: in section 2 we in-troduce the scalar equations of the scattering, we approxi-mate them with a quadratic operator, and then we formu-late the problem as the minimization of the distancebetween the data point and the image of the operator; inSection 3 we discuss the presence of local minima in thefunctional to be minimized; and in Section 4 we show nu-merical examples of reconstruction. Conclusions follow.
2. FORMULATION OF THE PROBLEMOur aim in this paper is to recover the dielectric permit-tivity of a rotationally symmetric cylindrical object (Fig.1) from far-zone measurements. We assume that the ob-ject is embedded in a homogeneous medium of dielectricpermittivity eb and that it is illuminated by a plane elec-tromagnetic wave at fixed frequency with the electric fieldpolarized along the axis of the system. In these hypoth-eses the electromagnetic scattering can be described bythe following integral scalar equations18:
E~r, u! 5 Ei~r, u! 1 k2E0
2pE0
R
Gi~r, r8, u, u8!x~r8!
3 E~r8, u8!r8dr8du8 (1)
inside the object (r , R) and
1997 Optical Society of America
2778 J. Opt. Soc. Am. A/Vol. 14, No. 10 /October 1997 R. Pierri and A. Brancaccio
Es~r, u! 5 k2E0
2pE0
R
Ge~r, r8, u, u8!x~r8!
3 E~r8, u8!r8dr8du8 (2)
outside the object (r . R) where k is the wave number inthe external medium, E is the total internal electric field,Es is the scattered field, Ei is the incident field, Gi is thebidimensional Green’s function of the external medium19
evaluated inside the object, Ge is the bidimensionalGreen’s function of the external medium evaluated in thefar zone of the object,20 and
x~r ! 5er~r !
eb2 1 (3)
is the contrast function normalized with respect to the ex-ternal medium.
We assume as an unknown the contrast function de-fined by Eq. (3); the known term of the problem is thescattered far field. An important point that must bestressed is that the information content of the scatteredfield is finite. The number of independent measurementsof the field outside the object under test cannot be in-creased over a certain N that depends on the object di-mensions with respect to the external wavelength.21,22
This property is due to the fact that the operator thatyields Es [Eq. (2)] has an analytical kernel of the exponen-tial type; i.e., its eigenvalues decay abruptly to zero afteran index N 5 @kR# for sufficiently large N. As a directconsequence the scattered field can be expressed, outsidea circular cylinder, as a truncated series of exponentials:
Es~u! 5 (n52N
N
an exp~2jnu!. (4)
The coefficients an can be calculated by a discrete Fou-rier transform from a finite number of measurements per-formed around the cylinder. In the numerical examplesthat follow, we use as data such Fourier coefficients in-stead of the actual measurements of the scattered field.This choice permits the filtering of data eventually cor-rupted by noise and does not affect the whole discussion,because measurements and Fourier coefficients are lin-early linked.
The fact that the number of independent data is finitehas some important implications regarding the choice ofthe functional space in which the unknown can besearched. First, a uniqueness problem must be solved.If in principle the uniqueness is guaranteed if we searchfor the unknown in C0
3(R3) (Ref. 23) when the scattered
Fig. 1. Geometry of the problem.
far field is known for every direction and every polariza-tion of the incident plane wave,24 in practice the above-mentioned analytical properties of the scattered fieldlimit the search for the unknown within a functionalspace whose dimension is at most the same as the numberof independent data. Another point that must be made isthat, whereas for a linear problem the use of a number ofdata equal to the number of unknowns is sufficient notonly for the uniqueness of the solution but also for theachievement of it (without considering ill-conditioning),nonlinear inverse problems can present false solutions ordivergence from the solution, even if the number of un-knowns is less than or equal to the number of indepen-dent data.14
Keeping in mind these considerations, we assume thatthe unknown contrast is a function belonging to a finite-dimensional space. Then it can be expressed as a super-position of a finite number of basis functions:
x~r ! 5 (n
cnfn~kr !, n P I, (5)
where I is a finite interval of integer values. The choiceof the basis functions fn(kr) and of their number dependson some a priori knowledge of the object under test and onthe number of independent measurements that one hasavailable to perform the inversion.
Let us now come back to Eqs. (1) and (2). As can beseen, the relation between the contrast and the scatteredfield is nonlinear. In fact, the scattered field depends onthe product between the contrast and the total internalfield. The latter is unknown and depends on the contrastagain. A possible way to simplify the problem is to ap-proximate the internal field in Eq. (2) by substituting forit the incident field. This assumption, well known as theBorn approximation for weak and smooth scatterers,transforms our problem into a linear one, thus greatlysimplifying it. The resulting equation to be inverted is
Es 5 A~x!, (6)
where A is the linear operator:
A~x! 5 k2E0
2pE0
R
Ge~r, r8, u, u8!x~r8!
3 Ei~r8, u8!r8dr8du8. (7)
A better approximation consists of substituting for theinternal field the incident one in the integral in Eq. (1)and then substituting Eq. (1) into Eq. (2). This procedureleads to a quadratic problem. In this paper we refer tothis model that allows us to obtain better results with re-spect to the linearization, as we show in what follows.Then we assume as a model for the electromagnetic scat-tering the equation16
Es 5 A~x! 1 B~x, x!, (8)
where A has been defined above and B is the quadraticoperator:
R. Pierri and A. Brancaccio Vol. 14, No. 10 /October 1997 /J. Opt. Soc. Am. A 2779
B~x, x! 5 k2E0
2pE0
R
Ge~r, r8, u, u8!x~r8!
3 E0
2pE0
R
Gi~r8, r9, u8, u9!x~r9!
3 Ei~r9, u9!r9dr9du9r8dr8du8. (9)
Let us consider the physical meaning of this quadraticapproximation with respect to the linear one. The prod-uct of the internal field and the contrast can be inter-preted as a polarization current that radiates the scat-tered field. Every point of the object under testcontributes to it, but every point of the object also inter-acts with all the other points.25 Equation (1) takes intoaccount this fact. In fact, the internal field itself is cal-culated by integration over the entire object. Then, incalculating the induced currents that appear in Eq. (2) weshould take into account this interaction. The contribu-tion to the internal field in a given point is due mainly tothe same point and decreases (or interferes destructively)for far-away points. Linearizing Eq. (2) by using the in-cident field in place of the internal one [i.e., avoiding us-ing Eq. (1) at all] implies that we are considering eachpoint of the object as if it were standing alone in the ex-ternal homogeneous medium. However, with the qua-dratic approximation we take into account at least themutual interaction between any two points of the object.This behavior is shown schematically in Fig. 2. Thenwhat we can expect is that the second-order approxima-tion will permit us to reconstruct profiles that vary morerapidly in the space.
An explicit evaluation of the second-order scatteredfield is now in order. We obtain for the linear operator Athe following result (Appendix A):
Fig. 2. Physical interpretation of (a) the linear and (b) the qua-dratic approximations of the scattering operator.
A~x! 5 (n52N
N
exp~2jnu!(pPI
cpc1~n, p !, (10)
where
c1~n, p ! 5 2j2
~2p/kr !1/2 exp~ jp/4 2 jkr !
3 E0
kR
xfp~x !Jn2~x !dx, (11)
where Jn(x) is a Bessel function of the first kind, and forthe quadratic operator B (Appendix B):
B~x, x! 5 (n52N
N
exp~2jnu!(pPI
(qPI
cpcqc2~n, p, q !,
(12)
where
c2~n, p, q ! 5 2p exp~ jp/4 2 jkr !
4~2p/kr !1/2
3 E0
kR
xfp~x !Jn~x !
3 FHn~2 !~x !E
0
x
x8fq~x8!Jn2~x8!dx8
1 Jn~x !Ex
kR
x8fq~x8!
3 Hn~2 !~x8!Jn~x8!dx8Gdx, (13)
where Hn(2)(x) is a Hankel function.
Some interesting considerations can be made by ob-serving the functions c1(n, p), since we can look at theintegral [Eq. (11)] as a scalar product. Then, if the basisfunction fp(x) is orthogonal to Jn
2(x), the correspondingc1(n, p) is null. This can happen, for example, if fp(x)varies rapidly in the integration interval (i.e., if the con-trast function presents rapid variations inside the object).The result is that there certainly are profiles that, inde-pendently of their weakness, are not visible when the lin-ear Born approximation is used. The same conclusioncan be achieved for a cylinder that varies only with theangular coordinate,17 suggesting that this reasoning canbe applied to the general bidimensional case.
Let us now say a few words about the way to tackle theinversion. The problem of solving operator equation (8)is ill posed. In fact, a solution to it does not exist, be-cause of the approximation involved in the adopted modeland because of unavoidable measurement errors. Toovercome this ill-positioning, we introduce a quasi-solution as the global minimum (if it exists) of the func-tional:
f~x! 5 iF~x! 2 Esi2, (14)
where Es is the measured scattered field, possibly af-fected by noise, and F 5 A 1 B. In other words, we ac-cept as a solution the contrast that minimizes the dis-tance between the measured scattered field and the fieldcalculated by the quadratic model. It can easily be dem-
2780 J. Opt. Soc. Am. A/Vol. 14, No. 10 /October 1997 R. Pierri and A. Brancaccio
onstrated that this assumption is equivalent to projectingthe measured data onto the image of the operator F. Wesolve the question of the existence of the minimum of thefunctional [Eq. (14)] by searching for the solution in afinite-dimensional space. In this case, the compactnessof a limited sphere of the space in which we search for theminimum guarantees the existence of it.26
Thus the problem of the inversion of Eq. (8) has beenreduced to the minimization of f (x) in a limited sphere ofthe chosen finite-dimensional space. Since, because ofthe quadratic approximation, f (x) is a fourth-order func-tional, local minima (traps for the inversion procedure)can be encountered. However, their presence can beavoided by choosing a proper ratio between the number ofindependent data and the number of unknowns.27 InSection 3 this topic will be examined in a more detailedway.
3. DISCUSSION OF THE PRESENCE OFLOCAL MINIMAEquation (14) is minimized by using an iterativegradient-based procedure (in this paper we use theBroyden–Fletcher-Goldfarb–Shanno method28) that, froma starting point x0 , follows the updating rule
xm11 5 xm 1 lTm¹fm , (15)
where m denotes the mth step, ¹fm is the gradient of thefunctional calculated in the current (mth) point, Tm is alinear operator that changes step by step from the iden-tity operator I to the inverse of the Hessian of the func-tional, and l is a parameter to be chosen at every step toguarantee the maximum decrease of the functional in thedirection jm 5 Tm¹fm . As is well known, the efficiencyand the precision of the method depend on the exactevaluation of l, which entails minimizing at every stepthe function f(xm 1 ljm) that is a fourth-order polyno-mial of the real variable l:
f~xm 1 ljm! 5 al4 1 bl3 1 cl2 1 dl 1 e, (16)
where
a 5 iB~jm , jm!i2, (17a)
b 5 2 Re^A~jm! 1 B~jm , xm!
1 B~xm , jm!, B~jm , jm!&, (17b)
c 5 2 Re^F~xm! 2 Es , B~jm , jm!& 1 iA~jm!
1 B~jm , xm! 1 B~xm , jm!i2, (17c)
d 5 2 Re^F~xm! 2 Es , A~jm! 1 B~jm , xm!
1 B~xm , jm!&, (17d)
e 5 f~xm!, (17e)
where ^ & denotes the usual dot product. Then at everystep the optimum value l makes the first derivative of Eq.(16) equal to zero; i.e., it is the solution of the third-degreeequation
4al3 1 3bl2 1 2cl 1 d 5 0. (18)
If Eq. (18) has only one real solution, then along the
current direction there are no points of local minimum.
Fig. 3. Functional along the direction txtrue 1 (1 2 t)x local for10 real unknowns and (a) 5 complex data, (b) 6 complex data, (c)10 complex data.
R. Pierri and A. Brancaccio Vol. 14, No. 10 /October 1997 /J. Opt. Soc. Am. A 2781
In contrast, if there is more than one real solution, theminimization procedure could terminate in a local mini-mum. Let us suppose that the coefficients [Eq. (17)] arecalculated in the global minimum of the functional andthat the data in that point coincide with the image of theoperator F (in the absence of model and measurement er-ror). In these conditions d 5 0 and the absence of a mul-tiple solution of Eq. (18) is guaranteed if along any direc-tion j the following inequality is verified:
b2
ac5
4@Re^A~j! 1 B~j, xm! 1 B~xm , j!,B~j, j!
iB~j, j!i2iA~j! 1 B~j, xm! 1 B~xm , j!i2
<32
9. (19)
As one can see from Eqs. (10) and (12) for the operatorsA and B, if the number of independent measurementchanges, the coefficients [Eqs. (17)] and then relation (19)change. Whereas the numerator in relation (19) is re-lated to a scalar product that can increase or decreasewhen the number of measurements increases, the de-nominator can increase only if the number of measure-ments increases. A higher number of independent mea-surements can lead to satisfaction of relation (19) andthen to the elimination of local minima. Therefore thefact that our functional is of fourth order makes it pos-sible to determine a criterion for the existence of localminima. A deeper and more comprehensive discussion ofthis topic can be found in Ref. 27. Here we validate theabove considerations by showing an example of how, byadding independent measurements, we can modify thefunctional and make the local minimum disappear. Tothis end we consider a contrast function expressed by thesuperposition of 10 rectangular basis functions fn(kr).The radius of the cylinder is 1.5 times the wavelength inthe external medium (i.e., kR 5 3p). First we performthe inversion by using only the first five Fourier coeffi-cients of the scattered field (the data are complex values,while the unknowns are assumed to be real). In Fig. 3(a)we show the functional along a direction joining the truesolution xtrue with the point in which the inversion proce-dure stopped, x local . The Hessian eigenvalues are calcu-lated in x local and are positive, so it is a local minimum.In Figs. 3(b) and 3(c) the functional is depicted along thesame direction when 6 and 10 coefficients, respectively,are used to perform the inversion. The minimum disap-pears, confirming that by increasing the number of datawe can positively influence the functional.
4. NUMERICAL RESULTSIn this section we show some numerical results obtainedby the application of the above-described quadratic inver-sion method. To obtain the data of the problem weimplemented a numerical code based on the solution ofthe direct scattering problem when the dielectric cylinderis a multilayer one.29
Once we have chosen the basis functions that representthe contrast, we can evaluate (partly in closed form andpartly by numerical computation) the functions c1(n, p)and c2(n, p, q) that appear in the expressions for A and
B. Then the problem is discretized and reduces to mini-mize the following functional:
f~c! 5 (n52N
N Uan 2 (pPI
cpc1~n, p !
2 (pPI
(qPI
cpcqc2~n, p, q !U2
, (20)
where c represents the vector of the unknown coefficientsof the sum [Eq. (5)] and an represents the nth Fourier co-efficient of the scattered field, possibly affected by noise.
We choose as basis functions to represent the dielectricprofile the rectangular windows
R. Pierri and A. Brancaccio Vol. 14, No. 10 /October 1997 /J. Opt. Soc. Am. A 2783
where the common factor A2p/(kr) exp( jp/4 2 jkr) isomitted for brevity. Note that c2(n, p, q)5 c2(n, q, p), so in this case the bilinear operatorB(x, j) is symmetric. The integrals of the squared Besselfunctions times the integration variable can be calculatedin closed form,30 whereas the remaining integrals must becalculated numerically.
We define two model errors, i.e., the rms errors be-tween the data and the scattered field calculated by thelinear operator A and by the quadratic operator A 1 B:
e12 5
(m
Uam 2 (p50
P21
cpc1~m, p !U2
(m
uamu2, (24)
e22 5
(m
U am 2 (p50
P21
cpc1~m, p ! 2 (p50
P21
(q50
P21
cpcqc2~m, p, q !U2
(m
uamu2,
(25)
where c is the vector of the actual coefficients of the pro-file. To evaluate the quality of the reconstruction we alsodefine two reconstruction errors, as follows:
re12 5
(p50
P21
u cp 2 cp~1 !u2
(p50
P21
u cpu2
, (26)
re22 5
(p50
P21
u cp 2 cp~2 !u2
(p50
P21
u cpu2
, (27)
Fig. 8. Second-order reconstructed profile for the example inFig. 4(a) in presence of noisy data.
where c (1) and c (2) are the vectors that contain the coeffi-cients of the profile reconstructed by inverting the linearoperator A and the quadratic operator A 1 B, respec-tively.
A first example refers to the reconstruction of a profilerepresented by the superposition of P 5 10 basis func-tions for a cylinder with kR 5 3p. Because of the rota-tional symmetry, at most 10 independent complex mea-surements are available to search for 10 real unknowns.The model error in the linear case is e1 5 5.5%, whereasin the quadratic case it is e2 5 1%. The starting point ofthe inversion procedure is chosen randomly in the rangeof the model validity (we used a uniform probability-density distribution between the zero value and the maxi-mum contrast 0.04). In Fig. 4(a) the actual profile to bereconstructed is depicted; in Figs. 4(b) and 4(c) the first-order and the quadratic reconstructions, respectively, areshown. As can be seen, the first-order reconstruction isnot acceptable, while the quadratic reconstruction showsgood agreement with the actual profile. In Figs. 5 and 6a more detailed comparison between the reconstructedand the actual contrasts as functions of the cylinder ra-dius makes it possible to appreciate the quantitative dif-ferences: while the linear model allows a reconstructionerror re1 5 73%, the quadratic reconstruction error isre2 5 1.5%.
A second example [Fig. 7(a)] refers to a contrast fourtimes greater than the preceding one. In this case the di-rect errors are e1 5 21% and e2 5 3%; thus even thequadratic model is out of its range of validity. However,the corresponding reconstruction is still acceptable, sincere2 5 17% [see Fig. 7(c)] and re1 5 116% [see Fig. 7(b)].
As the above examples show, the linear reconstructionis still worse than the quadratic one (as in the directmodel). However, we want to stress that the quadraticmodel allows us not only to obtain more-precise recon-structions (i.e., smaller reconstruction errors) than thelinear one but above all to enlarge the class of recon-structible profiles. For the case of real permittivity pro-files assumed in this paper, the use of the quadratic ap-proximation permits us to deal with a number ofunknowns for which the Born-type inversion fails becauseit is not stable owing to the eigenvalue properties of thelinear model. A further example that confirms the un-stable behavior of the linear model is obtained by addinga noise of 3% to the data of the first example [Fig. 4(a)].The linear reconstruction gives a meaningless result(re1 5 4000%), whereas the quadratic reconstruction re-mains acceptable (re2 5 19%; see Fig. 8).
5. CONCLUSIONSThe validity of a quadratic approach to the inverse prob-lem of the reconstruction of a dielectric profile has beenshown by means of theoretical considerations and nu-merical examples. The presented approach, which we in-troduced by adding a term of the Neumann series to theclassical Born approximation, permits us to improve theresults that can be obtained by using the Born linearmodel, since it enlarges the class of contrast functionsthat can be reliably reconstructed. The same approach
2784 J. Opt. Soc. Am. A/Vol. 14, No. 10 /October 1997 R. Pierri and A. Brancaccio
could be applied to a modified series expansion of thescattered field31 or to expansion around a proper refer-ence contrast function (distorted Born) without changingthe theoretical analysis (and thus the expected results)presented in this paper. The model that we developedalso constitutes a solid basis for extension to the more-general case of nonsymmetric cylinders (fully bidimen-sional case).
APPENDIX AIn this appendix we derive Eq. (10). Using Eq. (5) andthe expansion of the Green’s function and of the incidentplane-wave field in a series of Bessel functions20 in Eq.(7), we have
A~x! 5 2jk2
4 S 2krp D 1/2
expF jS p
42 kr D G
3 E0
2pE0
R
exp@ jkr8 cos~u 2 u8!#x~r8!
3 exp~2jkr8 cos u8!r8dr8du8
5 2j4 S 2
krp D 1/2
expF jS p
42 kr D G
3 E0
2pE0
kR
(m
~ j !mJm~x8!
3 exp@2jm~u 2 u8!#(pPI
cpfp~x8!
3 (n
~2j !nJn~x8!exp~2jnu8!x8dx8du8
5 2j4 S 2
krp D 1/2
expF jS p
42 kr D G
3 (m
~ j !m exp~2jmu!(pPI
cp(n
~2j !n
3 E0
2p
exp@ j~m 2 n !u8#du8
3 E0
kR
Jm~x8!fp~x8!Jn~x8!x8dx8
5 2j2 S 2p
kr D 1/2
expF jS p
42 kr D G
3 (m
exp~2jmu!(p
cp
3 E0
kR
fp~x8!Jm2~x8!x8dx8.
APPENDIX BIn this appendix we derive Eq. (12). Using Eq. (5), theexpansion of the Green’s function and of the incidentplane-wave field in a series of Bessel functions, and theaddition theorem for the Hankel function,20 we have
B~x, x! 5 2116 S 2
krp D 1/2
expF jS p
42 kr D G
3 E0
2pE0
kR
(m
~ j !mJm~x8!
3 exp@2jm~u 2 u8!#(pPI
cpfp~x8!
3 E0
2pE0
kR
(s
Js@min~x8, x9!#
3 Hs~2 !@max~x8, x9!#(
qPIcqfq~x9!
3 (n
~2j !nJn~x9!exp~2jnu9!
3 x9dx9du9x8dx8du8
5 214
pS 2p
kr D 1/2
expF jS p
42 kr D G
3 (m
exp~2jmu!(pPI
cp(qPI
cq
3 E0
kR
Jm~x8!fp~x8!E0
kR
Jm@min~x8, x9!#
3 Hm~2 !@max~x8, x9!#fq~x8!Jm~x9!x9dx9x8dx8
5 214
pS 2p
kr D 1/2
expF jS p
42 kr D G
3 (m
exp~2jmu!(p
cp(q
cq
3 E0
kR
Jm~x8!fp~x8!E0
kR
Jm@min~x8, x9!#
3 Hm~2 !@max~x8, x9!#fq~x9!
3 Jm~x9!x9dx9x8dx8,
where min( ) is the minimum between the arguments andmax( ) is the maximum between the arguments.
ACKNOWLEDGMENTSWe thank Giovanni Leone for his assistance during thewriting of this paper and for help during its revision.
The authors can be reached as follows: tel: 39-81-501-0222; fax: 39-81-503-7370; e-mail: [email protected].
REFERENCES AND NOTES1. M. H. Maleki, A. J. Devaney, and A. Schatzberg, ‘‘Tomogra-
phic reconstruction from optical scattered data,’’ J. Opt.Soc. Am. A 9, 1356–1363 (1992).
2. T. C. Wedberg and J. J. Stamnes, ‘‘Quantitative imaging byoptical diffraction tomography,’’ Opt. Rev. 2, 28–31 (1995).
3. T. C. Wedberg and W. C. Wedberg, ‘‘Tomographic recon-struction of the cross-sectional refractive index distribution
R. Pierri and A. Brancaccio Vol. 14, No. 10 /October 1997 /J. Opt. Soc. Am. A 2785
in semi-transparent, birifrangent fibres,’’ J. Microsc. 177,53–67 (1995).
4. W. Tabbara, B. Duchene, Ch. Pichot, D. Lesselier, L. Chom-meloux, and N. Joachimowitz, ‘‘Diffraction tomography:contribution to the analysis of some applications in micro-waves and ultrasonics,’’ Inverse Probl. 4, 305–331 (1988).
5. J. Ch. Bolomey, ‘‘New concepts for microwave sensing’’, inAdvanced Microwave and Millimeter-Wave Detectors,S. S. Udpa and H. C. Han, eds., Proc. SPIE 2275, 2–10(1994).
6. A. J. Devaney, ‘‘A computer simulation study of diffractiontomography,’’ IEEE Trans. Biomed. Eng. BME-30, 377–386(1983).
7. D. Colton, R. Kress, Inverse Acoustic and ElectromagneticScattering Theory (Springer-Verlag, Berlin, 1992).
8. I. Akduman, ‘‘An inverse scattering problem related to bur-ied cylindrical bodies illuminated by Gaussian beams,’’ In-verse Probl. 10, 213–226 (1994).
9. M. H. Maleki and A. J. Devaney, ‘‘Noniterative reconstruc-tion of complex-valued objects from two intensity measure-ments,’’ Opt. Eng. 33, 3243–3253 (1994).
10. T. C. Wedberg and J. J. Stamnes, ‘‘Experimental examina-tion of the quantitative imaging properties of optical dif-fraction tomography,’’ J. Opt. Soc. Am. A 12, 493–500(1995).
11. M. Slaney, A. C. Kak, and L. E. Larsen, ‘‘Limitation of im-aging with first order diffraction tomography,’’ IEEE Trans.Microwave Theory Tech. MTT-32, 860–874 (1984).
12. M. M. Moghaddam and W. C. Chew, ‘‘Study of some practi-cal issues in inversion with the Born iterative method usingtime-domain data,’’ IEEE Trans. Antennas Propag. 41,177–184 (1993).
13. S. Gutman and M. Klibanov, ‘‘Iterative method for multi-dimensional inverse scattering problems at fixed frequen-cies,’’ Inverse Probl. 10, 573–599 (1994).
14. A. Roger and F. Chapel, ‘‘Iterative methods for inverseproblems’’, in Application of Conjugate Gradient Method toElectromagnetics and Signal Analysis, T. K. Sarkar, ed.(Elsevier, Amsterdam, 1991).
15. A. Brancaccio, V. Pascazio, and R. Pierri, ‘‘Reconstruction ofdielectric profiles: a new approach as a quadratic inverseproblem,’’ presented at the Institute of Electrical and Elec-tronics Engineers–International Union of Radio ScienceMeeting on Microwaves in Medicine, Rome, October 1993.
16. A. Brancaccio, V. Pascazio, and R. Pierri, ‘‘A quadratic
model for inverse profiling: the one dimensional case,’’ J.Electromagn. Waves Appl. 9, 673–696 (1995).
17. A. Addivinola, A Brancaccio, G. Leone, R. Pierri, ‘‘Micro-wave tomography with a quadratic model: numerical re-sults for the cylindrical case,’’ presented at the Interna-tional Conference on Electromagnetic for AdvancedApplications, Torino, Italy, September 1995.
18. J. H. Richmond, ‘‘Scattering by a dielectric cylinder of arbi-trary cross section shape,’’ IEEE Trans. Antennas Propag.AP-13, 334–341 (1965).
19. J. A. Stratton, Electromagnetic Theory (McGraw Hill, NewYork, 1941).
20. M. Abramowitz and I. Stegun, Handbook of MathematicalFunctions (Dover, New York, 1965).
21. O. M. Bucci and G. Franceschetti, ‘‘On the degrees of free-dom of scattered fields,’’ IEEE Trans. Antennas Propag. 37,918–926 (1989).
22. R. P. Porter and A. J. Devaney, ‘‘Generalized holographyand computational solutions to inverse source problems,’’ J.Opt. Soc. Am. A 7, 1707–1713 (1990).
23. I.e., space of the compact support continuous functions,with continuous derivatives until the third order.
24. D. Colton and L. Paivarinta, ‘‘The uniqueness of a solutionto an inverse scattering problem for electromagneticwaves,’’ Arch. Ration. Mech. Anal. 119, 59–70 (1992).
25. M. Azimi and K. C. Kak, ‘‘Distortion in diffraction tomogra-phy caused by multiple scattering,’’ IEEE Trans. Med. Im-aging MI-2, 176–195 (1983).
26. L. V. Kantarovic and G. P. Akilov, Functional Analysis(Nauka, Moscow, 1977) [Analisi Funzionale (Editori Ri-uniti, Rome, 1980)].
27. T. Isernia, G. Leone, and R. Pierri, ‘‘Phase retrieval of radi-ated fields,’’ Inverse Probl. 11, 183–203 (1995).
28. W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vet-terling, Numerical Recipes (Cambridge U. Press, Cam-bridge, 1987).
29. H. E. Bussey and J. H. Richmond, ‘‘Scattering by a lossy di-electric circular cylinder multilayer, numerical values,’’IEEE Trans. Antennas Propag. 723–725, 1975.
30. I. S. Gradshsteyn and I. M. Ryzhik, Tables of Integrals, Se-ries and Products (Academic, New York, 1965).
31. R. E. Kleinman, G. F. Roach, and P. M. van den Berg, ‘‘Con-vergent Born series for large refractive indices,’’ J. Opt. Soc.Am. A 7, 890–897 (1990).
