Caorsi et al. Vol. 13, No. 3 / March 1996 / J. Opt. Soc. Am. A 509 Rytov approximation: application to scattering by two-dimensional weakly nonlinear dielectrics Salvatore Caorsi* Department of Electronics, University of Pavia, Via Abbiategrasso 209, I-27100 Pavia, Italy Andrea Massa and Matteo Pastorino Department of Biophysical and Electronic Engineering, University of Genoa, Via Opera Pia 11, I-16145, Genoa, Italy Received May 30, 1995; revised manuscript received September 29, 1995; accepted October 5, 1995 The Rytov approximation is applied to devise an iterative numerical approach to the computation of electro- magnetic scattering by two-dimensional weakly nonlinear configurations. The scatterers are assumed to be isotropic, inhomogeneous, lossless, and of arbitrary bounded shapes. The nonlinearity is assumed to be of the Kerr type, and the illumination is given by a monochromatic plane wave. The effects of the application of the Rytov approximation to various configurations and the convergence of the approach are demonstrated by several examples. Comparisons with the iterative Born approach are also made. 1996 Optical Society of America 1. INTRODUCTION The Rytov 1 and Born approximations are among the most useful and most widely applied approximations to simplify electromagnetic (but not only electromagnetic) scattering problems. In the case of direct electromagnetic scatter- ing, the starting point is usually the wave equation that describes the history, throughout space and time, of the electromagnetic field inside a given medium. In a two- dimensional case (e.g., the case of an infinite isotropic cylindrical geometry under TM illumination conditions 2 ), the wave equation for the z component of the electric field vector (TM z wave) is given by = 2 E z sx, y , td 2 me d 2 E z sx, y , td d 2 t › 0. (1) Solution of Eq. (1) usually requires that the electric field at a given point be written in terms of the field inside the scatterer. Therefore, when possible, one tries to approxi- mate this unknown internal field by a known distribution in order to linearize the problem. To this end, the Rytov approximation requires that dielectric discontinuities (i.e., scatterers) be weak with respect to the propagation medium. The same holds for the Born approximation, but it has been shown that the Born approximation also requires that the inhomogeneities within propagation media exhibit gradual changes. This requirement had to be met, in particular, when this approximation was applied to solve inverse problems for imaging purposes (e.g., for computerized ultrasound tomography 3 and for electromagnetic imaging 4 ). In the case of electromag- netic imaging, the first-order Rytov approximation has been found to produce accurate dielectric reconstructions only for media with a contrast no higher than a factor of 2 but essentially without constraints on object dimensions. 4 On the contrary, as is well known, the first-order Born approximation exhibits severe limitations on the imaging of bodies that are large in terms of wavelength. 5 In the imaging field the Rytov approximation, when applicable, generally constitutes an improvement over the Born ap- proximation, but not in all cases. For example, when the Rytov approximation is used for the study of scattering in random media, it is not so useful as the Born approxi- mation, as it suffers more serious divergence problems. 6 In this paper we apply the Rytov approximation to develop an iterative numerical approach to computing electromagnetic scattering by two-dimensional weakly nonlinear objects of bounded shapes, where the nonlin- earity is expressed by the dependence of its dielectric permittivity on the internal electric field. The objects are assumed to be isotropic, nonmagnetic, lossless, and inhomogeneous, the inhomogeneity being due not only to the nonlinearity but also to the inhomogeneous nature of the linear part of the dielectric permittivity. 7 The use of the Rytov approximation in the context of nonlin- ear problems was recently considered by Lin and Fiddy in Ref. 6, where they extended the analysis previously developed in Ref. 8 (to assess the validities of the Born and Rytov approximations) to the case in which such ap- proximations were applied to one-dimensional nonlinear half-plane scattering and to random media. Previously we discussed the nonlinear scattering prob- lem from a different point of view. 9 In that paper we pro- posed a numerical approach based on an integral-equation formulation to the computation of the electromag- netic scattering by three-dimensional weakly nonlinear objects in free space. This approach took into account the possibility of harmonic generation in the case of a Kerr-like nonlinearity. 10 A formal solution was obtained in which each field component was expressed in terms of the Green’s function for free space, through coupling co- efficients that accounted for the effects of the other field components. In general, the nonlinearity is assumed to be weak, in the sense that the evolution of the non- 0740-3232/96/030509-08$06.00 1996 Optical Society of America
Transcript
Caorsi et al. Vol. 13, No. 3 /March 1996/J. Opt. Soc. Am. A 509
Rytov approximation: application to scatteringby two-dimensional weakly nonlinear dielectrics
Salvatore Caorsi*
Department of Electronics, University of Pavia, Via Abbiategrasso 209, I-27100 Pavia, Italy
Andrea Massa and Matteo Pastorino
Department of Biophysical and Electronic Engineering, University of Genoa,Via Opera Pia 11, I-16145, Genoa, Italy
Received May 30, 1995; revised manuscript received September 29, 1995; accepted October 5, 1995
The Rytov approximation is applied to devise an iterative numerical approach to the computation of electro-magnetic scattering by two-dimensional weakly nonlinear configurations. The scatterers are assumed to beisotropic, inhomogeneous, lossless, and of arbitrary bounded shapes. The nonlinearity is assumed to be ofthe Kerr type, and the illumination is given by a monochromatic plane wave. The effects of the applicationof the Rytov approximation to various configurations and the convergence of the approach are demonstratedby several examples. Comparisons with the iterative Born approach are also made. 1996 Optical Societyof America
1. INTRODUCTIONThe Rytov1 and Born approximations are among the mostuseful and most widely applied approximations to simplifyelectromagnetic (but not only electromagnetic) scatteringproblems. In the case of direct electromagnetic scatter-ing, the starting point is usually the wave equation thatdescribes the history, throughout space and time, of theelectromagnetic field inside a given medium. In a two-dimensional case (e.g., the case of an infinite isotropiccylindrical geometry under TM illumination conditions2),the wave equation for the z component of the electric fieldvector (TMz wave) is given by
=2Ezsx, y, td 2 med2Ezsx, y, td
d2t 0 . (1)
Solution of Eq. (1) usually requires that the electric fieldat a given point be written in terms of the field inside thescatterer. Therefore, when possible, one tries to approxi-mate this unknown internal field by a known distributionin order to linearize the problem. To this end, the Rytovapproximation requires that dielectric discontinuities(i.e., scatterers) be weak with respect to the propagationmedium. The same holds for the Born approximation,but it has been shown that the Born approximation alsorequires that the inhomogeneities within propagationmedia exhibit gradual changes. This requirement hadto be met, in particular, when this approximation wasapplied to solve inverse problems for imaging purposes(e.g., for computerized ultrasound tomography3 and forelectromagnetic imaging4). In the case of electromag-netic imaging, the first-order Rytov approximation hasbeen found to produce accurate dielectric reconstructionsonly for media with a contrast no higher than a factor of 2but essentially without constraints on object dimensions.4
On the contrary, as is well known, the first-order Bornapproximation exhibits severe limitations on the imaging
0740-3232/96/030509-08$06.00
of bodies that are large in terms of wavelength.5 In theimaging field the Rytov approximation, when applicable,generally constitutes an improvement over the Born ap-proximation, but not in all cases. For example, when theRytov approximation is used for the study of scatteringin random media, it is not so useful as the Born approxi-mation, as it suffers more serious divergence problems.6
In this paper we apply the Rytov approximation todevelop an iterative numerical approach to computingelectromagnetic scattering by two-dimensional weaklynonlinear objects of bounded shapes, where the nonlin-earity is expressed by the dependence of its dielectricpermittivity on the internal electric field. The objectsare assumed to be isotropic, nonmagnetic, lossless, andinhomogeneous, the inhomogeneity being due not only tothe nonlinearity but also to the inhomogeneous natureof the linear part of the dielectric permittivity.7 Theuse of the Rytov approximation in the context of nonlin-ear problems was recently considered by Lin and Fiddyin Ref. 6, where they extended the analysis previouslydeveloped in Ref. 8 (to assess the validities of the Bornand Rytov approximations) to the case in which such ap-proximations were applied to one-dimensional nonlinearhalf-plane scattering and to random media.
Previously we discussed the nonlinear scattering prob-lem from a different point of view.9 In that paper we pro-posed a numerical approach based on an integral-equationformulation to the computation of the electromag-netic scattering by three-dimensional weakly nonlinearobjects in free space. This approach took into accountthe possibility of harmonic generation in the case of aKerr-like nonlinearity.10 A formal solution was obtainedin which each field component was expressed in terms ofthe Green’s function for free space, through coupling co-efficients that accounted for the effects of the other fieldcomponents. In general, the nonlinearity is assumedto be weak, in the sense that the evolution of the non-
1996 Optical Society of America
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linear process considered does not involve catastrophes,such as shock waves and wave breaking. In the caseof monochromatic illumination, the two main effects ofthe nonlinearity (i.e., variations in the field distributionat the fundamental frequency and harmonics generation)are worth considering. However, for very weak nonlin-earities, such as those of most actual nonlinear materials,the variations in the field distribution at the fundamen-tal frequency can be considered to be the only significanteffect, whereas the generation of higher-order harmonicscan be neglected. We made this assumption in Ref. 11,where we developed an iterative approach to the com-putation of the bistatic scattering width for nonlinearcylinders by using the distorted-wave Born approxima-tion and, in a simplified version of the approach, theclassic first-order Born approximation.
The iterative approach presented here is aimed at theapproximate computation of nonlinear scattering by useof the Rytov approximation and at developing a numeri-cal solution by application of the Richmond formulation,12
which has been proved to be effective in evaluating two-dimensional scalar scattering by linear dielectrics (i.e.,cylindrical geometries under transverse-magnetic illumi-nation only). In this paper the usefulness of applying theRytov approximation to a nonlinear scattering problem isassessed from a numerical point of view, on the basis ofthe results of several numerical simulations. Compari-sons with the results obtained by the iterative Born ap-proach are also made. In particular, the focus is on theconvergence of the approach, which is expected for veryweak nonlinearities only, whose effective (time-varying)relative dielectric permittivities correspond to very weakscattering dielectrics.
2. MATHEMATICAL FORMULATIONUnder stationary conditions and monochromatic illumina-tion [time dependence expsivtd], the following wave equa-tion holds for the scattered electric field inside a nonlinearmedium A (Fig. 1):
s=2 1 km2dCscattsx, yd fkp2
A sx, yd 2 km2gCtotsx, yd
sx, yd [ A , (2)
where km2 v2m0em, em being the dielectric permittiv-
ity of the background, Cscattsx, yd and Ctotsx, yd are thespace-dependent parts of the scattered and the total elec-tric fields, and kp2
A sx, yd is the propagation constant in-side A. Since the dielectric permittivity is assumed toexhibit a nonlinearity of the Kerr type, kp2
A sx, yd can beexpressed as
kp2A sx, yd vm0epsx, yd km
2fe1sx, yd 1 e2jCtotsx, ydj2g ,(3)
where epsx, yd is the nonlinear complex dielectric permit-tivity, e1sx, yd is the linear part of its relative value, ande2 is a nonlinear coefficient. The scatterer cross sectionis inhomogeneous, owing to both the nonlinearity andthe inhomogeneous linear part of the dielectric permit-tivity, which corresponds to the actual permittivity whenthe amplitude of the total electric field Ctotsx, yd is verysmall. Under such assumptions, we can obtain a solution
of Eq. (2) by applying the Rytov approximation, providedthat the related application conditions are fulfilled. Tothis end, we represent Ctotsx, yd as
Ctotsx, yd expfFsx, ydg . (4)
As =Ctotsx, yd expfFsx, ydg=Fsx, yd, from =2Ctotsx, yd = ? f=Ctotsx, ydg we derive
=2Ctotsx, yd = ? hexpfFsx, ydg=Fsx, ydj . (5)
Since, given a scalar and a vector field, c and F , respec-tively, the following relation holds,
The Rytov approximation allows the variations in thecomplex phase Fssx, yd inside A to be neglected. Con-sequently, the following approximate equation holds:
s=2 1 km2dCincsx, ydFssx, yd Cincsx, ydfkp2
A sx, yd 2 km2g
sx, yd [ A , (20)
whose solution can be given in terms of the Green’s func-tion for free space13:
Fssx, yd 1
Cincsx, yd
ZA
Cincsx0, y 0 dfkp2A sx0, y 0 d 2 km
2g
3 Gsx, yyx0, y 0 ddx0dy 0, (21)
where Gsx, yyx0, y 0 d s2jy4dH s2d0 skmrd, H s2d
0 skmrd is theHankel function of the second kind and zeroth order,and r fsx 2 x0 d 1 sy 2 y 0 dg1/2. Then the electric fielddistributions inside and outside A can be obtained byEqs. (12) and (4). At this point, the iterative computa-tion of Ctotsx, yd can be performed in a way analogous tothat used for the iterative process based on the distorted-wave Born approximation developed in Ref. 11. The ap-proach consists in expressing, at the step sk 1 1d, the totalelectric field C
sk11dtot sx, yd as the sum of the incident field
plus the scattered field expressed as the integral of quan-tities computed at the previous step k:
Csk11dtot sx, yd Cincsx, yd 2 s jy4d
ZA
hfkpskdA sx0, y 0 dg2 2 km
2j
3 Cskdtotsx0, y 0 dH s2d
0 skmrddx0dy 0, (22)
where kpskdA sx, yd indicates that kp
Asx, yd is computedthrough Eq. (3) in terms of C
skdtotsx, yd. Using the Rytov
approximation, we can start the iterative process by set-ting, at step k 0,
Cs0dtotsx, yd expfF0sx, yd 1 Fs0d
s sx, ydg , (23)
where Fs0ds sx, yd is given by Eq. (21) and, in that equation,
kpAsx, yd is computed through Eq. (3) in terms of Cincsx, yd.
In order to evaluate the convergence of the proposed itera-tive approach, we also define the following residual error:
Qhkj A21Z
A
(C
skdtotsx0, y 0d 2 Cincsx0, y 0 d
1 s jy4dZ
Ahfkpskd
A sj, hdg2 2 km2jCskd
totsj, hd
3 H s2d0 skmxddjdh
)dx0dy 0, (24)
where x fsx0 2 jd2 1 sy 0 2 hd2g1/2. It is worth notingthat this definition results in a convenient numerical com-putation, as will be clear in the following.
Since the formulation has to be applied to an arbitrarilyshaped inhomogeneous cross section, a discretization ofEq. (22) is obtained and the coefficients are computedby use of the Richmond formulation.12 To this end, wepartition A into P subdomains and denote the centerand the area of the pth subdomain by sxp, ypd and Dsp,respectively. Equation (22) can now be approximated as
Csk11dtot sx, yd ø Cincsx, yd 2 s jy4d
PPp1
hfkpskdA sxp, ypdg2 2 km
2j
3 Cskdtotsxp, ypdH s2d
0 skmrpdDsp , (25)
where rp fsx 2 xpd2 1 s y 2 ypd2g1/2. In the same man-ner we can easily compute the residual error in discretizedform:
Qhkj ø A21PP
p1sssCskd
totsxp, ypd 2 Cincsxp, ypd
1 s jy4dPP
q1hkpskd
A fsxq, yqdg2 2 km2jCskd
totsxq, yqd
3 H s2d0 skmxpqdDsqdddDsp . (26)
Roughly speaking, this error indicates whether the com-puted electric field satisfied the formal solution of Eq. (2)at each step and then whether it can be regarded as acorrect nonlinear field (under the theoretical hypothesismade in Section 2).
Finally, the convergence issue should be addressed,which is of major importance for an iterative approach.The approach will be said to converge if Qhkj decreasesas k increases. As discussed in Ref. 11, in the nonlin-ear case convergence depends on many more factors thanin the linear case: the linear part of the dielectric per-mittivity, the nonlinear coefficients, the incident electricfield, and the ratio between the wavelength and the scat-
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terer dimensions. At present, a convergence criterionis not available. Nevertheless, some interesting conclu-sions can be drawn from the results reported in Section 3.
3. NUMERICAL RESULTSIn all the simulations described in this section, we as-sume the incident electric field to be given by a uniformplane wave of the TM kind. For this wave, in Eq. (11),F0sx, yd fR 2 jfI sx, yd, where fR is a real constantsuch that the amplitude of the plane wave is given byC0 jCincsx, ydj efR , and fI sx, yd kmx (without lossof generality, the propagation direction is assumed to co-incide with the x axis).
The aim of the first simulation is to study the effects ofthe application of the Rytov approximation to a scattererfor different nonlinearity values that correspond to differ-ent values of the effective dielectric permittivity. To thisend, we assume a circular cylinder (for which, in the lin-ear case, the electric field can be computed analytically14).The radius a of the cylinder cross section is such thatkma 1.5p. The other assumed parameters are C0 1sVymd, e1sx, yd e1 1.1 (the cylinder is homogeneousin its linear part) and P 225 (this value is large enoughto produce an accurate discretization of the circularcross section, as it fully satisfies the Hagmann–Gandhi–Durney criterion for the choice of the cell size15 and, atthe same time, requires a reasonable computational load).
In this example, first we set e2 0.01, corresponding toa very weak nonlinearity, and then e2 0.1. A compari-son with the results obtained by use of the iterative Bornapproximation is made in terms of the residual error com-puted by relation (26). Figure 2 gives the residual errorsQhkj for various values of the number of iterations, k. Inboth cases (e2 0.01 and e2 0.1), the two iterative ap-proaches converge (in the sense that Qhkj decreases ask increases). Although the Rytov approximation alwaysinvolves smaller errors, for e2 0.01 the results providedby the two approaches are similar, in that the residualerrors decrease in the two cases in a similar way and be-come very small after a few steps. Such results are notsurprising, as it is well known that, if the scattered elec-tric field Cscattsx, yd is small, if one starts from the Rytovsolution,
Ctotsx, yd Cincsx, ydexpfFssx, ydg
ø Cincsx, ydexphexpf2F0sx, ydCscattsx, ydgj ,
(27)
the total electric field, Ctotsx, yd Cincsx, yd 1 Cscattsx, yd,can be approximated by use of an expansion in powerseries, as follows:
which shows that, for very small values of the amplitudeof Cscattsx, yd, the results obtained by the Rytov and Bornapproximations tend to coincide.
For stronger nonlinearities the behaviors of the twoapproximations are different, even though the range ofvalues of e2 for which the iterative Rytov approach isconvergent is slightly wider than that for which the itera-
tive Born approach converges. For example, for e2 , 1.6,both approaches converge (and, for these values of e2, theRytov approximation always yields results with smallerQhkj values). For e2 0.16 the iterative Born approachdiverges, whereas the iterative Rytov approach converges(although very slowly). This is shown in Fig. 3, whichgives the residual errors calculated for the first 100 itera-tions. For a slightly larger value of e2 (i.e., e2 0.18), theiterative Rytov approach exhibits an oscillating behaviorand the residual errors remain quite large compared withthe errors related to the previous e2 values (Fig. 4). Ac-tually, we are unable to provide a complete insight intothe causes of this oscillation. Since it cannot be ascribedto errors associated with the discretization, which is goodenough for the assumed values of the dimensions and per-mittivities, it may be due to significant errors in the com-putation of the starting values of the distribution of thephase Fssx, yd, which turns out to depend essentially onthe incident electric field.
Fig. 2. Scattering by a nonlinear circular cylinder. C0 1sVymd, e1 1.1, kma 1.5p; P 225. Residual error wascalculated with the Born and the Rytov iterative approaches.Nonlinearity: e2 0.01 and e2 0.1.
Fig. 3. Scattering by a nonlinear circular cylinder. C0 1sVymd, e1 1.1, kma 1.5p; P 225. Residual error wascalculated with the Born and the Rytov iterative approaches.Nonlinearity: e2 0.16.
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Fig. 4. Scattering by a nonlinear circular cylinder. C0 1sVymd, e1 1.1, kma 1.5p; P 225. Residual error wascalculated with the Rytov approach. Nonlinearity: e2 0.18.
Fig. 5. Amplitudes of the total electric field for different valuesof e2. C0 1 sVymd, e1 1.1, kma 1.5p; P 225.
Figure 5 shows the amplitudes of the total electric fieldcomputed (inside and outside the cylinder) along the xaxis s y 0.0d by the iterative Rytov approach for theabove examples. The same figure also gives the linearfield (analytically computed14 for e2 0.0). Note thatin this figure the plot for e2 0.1 refers to results ob-tained for k 10, whereas the plots for e2 0.16 ande2 0.18 refer to results obtained for k 100. Fig-ure 5 also gives the field distributions obtained for e2 0.2 and e2 0.3 and for k 1 (i.e., using the Rytovapproximation without iterations). For these values ofthe nonlinearity the iterative Rytov approach (as well asthe Born one) diverges, but for k 1 the related errorsturn out to be equal to Qh1j ø 3.54% for e2 0.2 and toQh1j ø 6.83% for e2 0.3. These values can be consid-ered small enough to make such results of some interest.For comparison, by using the first-order Born approxima-tion (without iterations), we obtained Qh1j ø 52.6% fore2 0.2 and Qh1j ø 93.5% for e2 0.3.
Finally, to give an idea of the variations in the fielddistributions of the nonlinear field at the various iteration
steps, in Fig. 6 we present the amplitudes of the totalelectric field computed inside and outside the cylinder fore2 0.16 (for which the behavior of the residual erroris shown in Fig. 3) and for k 1, 2, 3, 4, 20, and 100.The values for the linear field are also given. As canbe seen, the plots for k 20 and k 100 are almostindistinguishable, and some differences can be noticedonly near the local maxima of the field profile inside thecylinder.
Other simulations are devoted to evaluating the effec-tiveness of the Rytov iterative approach for different lin-ear dimensions of the scatterer. As already mentionedin Section 1, for imaging applications the Rytov approxi-mation has been found to work reasonably well, almostindependently of the linear dimensions, if the dielectricpermittivity changes gradually.4 On the contrary, it iswell known that the Born approximation is effective onlyfor small objects in terms of the incident wavelength.
Fig. 6. Amplitudes of the total electric field for different num-bers of iterations with the Rytov approach: C0 1 sVymd,e1 1.1, e2 0.16, kma 1.5p; P 225.
Fig. 7. Scatterer cross section: distribution of the linear partof the relative dielectric permittivity se1d
514 J. Opt. Soc. Am. A/Vol. 13, No. 3 /March 1996 Caorsi et al.
(a)
(b)Fig. 8. Scattering by a nonlinear square cylinder. C0 1sVymd, e1 (see Fig. 7), e2 0.1, kmd 0.32p; P 144. (a) Re-sidual error calculated with the Born and the Rytov iterativeapproaches; (b) bistatic scattering width.
A similar conclusion holds for the direct scattering prob-lem considered here. In particular, we assume a layeredinhomogeneous dielectric cylinder with a square crosssection of side d, for which the distribution of the lin-ear part of the dielectric permittivity e1 increases fromthe outermost layer to the innermost one (Fig. 7). Thenonlinearity is assumed to be given by e2 0.1, and dis expressed in terms of the free-space wavelength of theincident monochromatic field. For d such that kmd 0.32p (small object), both the Rytov and the Born itera-tive approaches converge very fast and yield similar re-sults, even at the first iteration steps. Figure 8(a) givesthe histograms of the residual error Qhkj, and Fig. 8(b)gives the plots of the bistatic scattering width for k 1,k 2, and k 10. The bistatic scattering width (i.e., thescattering cross section per unit length) is expressed as16
ssud limr !`
2prjCscattsx, ydj2
jCincsx, ydj2, (29)
where r and u denote cylindrical coordinates.
For a larger object, for which kmd 1.6p, the iterativeRytov approach is more effective. Both approaches areconvergent, but the Born approximation produces resultssimilar to those produced by the Rytov approximation forslightly larger values of k, whereas notable errors are in-curred for small values of k when the Born approximationis used. These conclusions can be drawn from Fig. 9(a),which gives the residual errors, and from Fig. 9(b), whichshows the plots of ssud.
For d such that kmd 3.2p, the body becomes ratherlarge in terms of free-space wavelength. As can be de-duced from Fig. 10(a), which gives the residual errors,and from Fig. 10(b), which shows the plots of ssud,the iterative Born approach becomes rapidly divergent,whereas the iterative Rytov approach converges, eventhough rather slowly. It should be noted that, in non-linear direct scattering, unlike in the case of microwaveimaging mentioned above, because the effective dielectricpermittivity depends on the internal field through Eq. (3),when the dimensions of the scatterer increase in terms of
(a)
(b)Fig. 9. Scattering by a square nonlinear cylinder. C0 1sVymd, e1 (see Fig. 7), e2 0.1, kmd 1.6p; P 144. (a) Re-sidual error calculated with the Born and the Rytov iterativeapproaches; (b) bistatic scattering width.
Caorsi et al. Vol. 13, No. 3 /March 1996/J. Opt. Soc. Am. A 515
(a)
(b)Fig. 10. Scattering by a nonlinear square cylinder. C0 1sVymd, e1 (see Fig. 7), e2 0.1, kmd 3.2p; P 144. (a) Re-sidual error calculated with the Born and the Rytov iterativeapproaches; (b) bistatic scattering width.
wavelength the distribution of the dielectric permittivityvaries more rapidly.
It is interesting to note the nonsymmetric nature ofsome curves in Figs. 9(b) and 10(b). This asymmetry oc-curs for small k values and is due to the application of theRytov approximation to the discretized problem. As al-ready mentioned for Fig. 4, at the first iteration the phaseFssx, yd essentially depends on the incident electric field,which is symmetric with respect to the x axis (propa-gation axis). In contrast, the center points of the dis-cretization cells are asymmetric with respect to this axis.Of course, in the numerical computation this results inthe shown nonsymmetric nature, as the computed contri-bution of each partition is only an approximation of the ac-tual contribution. Such asymmetry is not physical, andthe fact that it tends to disappear as k increases seems toshow the effectiveness of the numerical approach.
Finally, it is worth noting that the proposed iterativemethod is quite efficient from a computational point ofview. Since the Richmond formulation is used in con-
junction with the Rytov approximation, the computationis reduced at each iterative step to a sum of P termsgiven by the product of known quantities (subroutinesfor the Bessel functions are required). Specifically, forthe examples considered in this section and the relateddiscretizations (which are, as discussed above within therange needed for accurate solutions), the whole compu-tation (including the image formation) required only afew minutes (for approximately ten iterations) on an HP9000/825 workstation (with use of an unoptimized code).
4. CONCLUSIONSIn this paper we have applied the Rytov approximationto develop an iterative approach to the approximate com-putation of the fields inside and outside weakly nonlineardielectric objects of arbitrary shapes. Two-dimensionalconfigurations have been considered, under the assump-tion of a Kerr-like nonlinearity. The results obtainedby the Rytov approximation have been compared withthose yielded by the iterative Born approach, for severaldifferent scatterers. The bistatic scattering width andthe internal and external field distributions have beencalculated. The convergence of the proposed iterativeapproach has been demonstrated by means of somenumerical simulations, in which the effects of variousparameters on the convergence have been taken into ac-count. For some configurations, corresponding to rela-tively strong scatterers, the iterative approach did notconverge. Nevertheless, the results from these diver-gent cases can be considered significant, because at thefirst iteration the residual errors turned out to be rathersmall.
We believe that it may be interesting to explore theextension of the use of the iterative approach developedhere (after proper modifications in light of the formulationpresented in Ref. 9) to nonlinearities (if any) that aresufficiently weak to ensure convergence but for which theharmonics generation cannot be completely neglected.
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