Light scattering from a sliced target through use ofthe internal field of infinite cylinders: comparisonbetween Mie theory and a sliced sphere

Ariel Cohen, Richard D. Haracz, and Leonard D. Cohen

Light scattering from a particle that can be sectioned into circular slices is calculated by performing acoherent integration of the internal field over the volume of the target. The internal field in each slice istaken to be the internal-field solution of an infinite cylinder of radius equal to the radius of the slice. It isshown that for a spherical scatterer with size parameters up to 1.4, the integration leads to results thatare in good agreement with those predicted by the Mie theory. Thus, we show the remarkable result thatthe internal field from an infinite cylinder can reproduce scattering intensities for such a radicallydifferent shape as a sphere. This being the case, a wide variety of target shapes between a sphere and acylinder should be open to evaluation by this technique. The approach also has the benefit of beingcomputationally efficient, requiring a double integration of the internal field over a disk and thencoherently adding these calculations. The computations demonstrated in this paper are performedrelatively quickly on a computer such as the Macintosh Centris 650, and this efficiency allows us to obtainthe scattered fields for many target shapes.

Introduction

An integral equation for solving the Maxwell equa-tions for the scattering of light from dielectric targetsis used to evaluate the scattered-field intensities.The targets are spherical particles of size parametersup to x = 1.4 and refractive index m = 1.33. Thisintegral equation uses the internal electric field toobtain the scattered field, and thus reduces theproblem to a search for the internal field. Thehistory of this approach begins with the work ofShifrin' and Acquista 2 -the first improves on theBorn approximation (internal field equal to the ambi-ent field) by introducing the polarizability of themedium, and the latter iterates Shifrin's first approxi-mation in a series involving the polarizability. Laterwork by our group extended Acquista's approach to awider range of shapes by dividing the target intoslices3 4 and then attempted to evaluate the internalfield by a direct evaluation of the relevant integralequations.5

A. Cohen is with the Department of Atmospheric Sciences, TheHebrew University of Jerusalem, Jerusalem, Israel. The otherauthors are with the Department of Physics and AtmosphericScience, Drexel University, Philadelphia, Pennsylvania 19104-0001.

Received 25 March 1993; revised manuscript received 15 Septem-ber 1993.

0003-6935/94/091776-04$06.00/0.© 1994 Optical Society of America.

The work described in Ref. 5, though guaranteeingsuccess if the numerical evaluation is precise, leads toextremely time-consuming computations, and applica-tions to realistic problems demand quick and accurateresults and the ability to deal with random orienta-tions and multiple-scattering effects.

This work thus focuses on an approach that takesadvantage of the fact that the internal electric field ofan infinite cylinder can be obtained to any degree ofaccuracy, and this field can be used for each of thedisks that form the wide range of targets consideredin the previous papers (including helixes). The as-sumption is that the internal field of a disk derivedfrom the infinite-cylinder theory is a better choice fordetermining the internal field than merely taking aconstant polarizability matrix, as is done by Shifrinand Acquista. Moreover, it is the coherent combina-tion of these disk contributions that produces theeffects of the target shape. A further work,6 usesthis technique-taking the polarization matrix froman infinite cylinder to form the internal field within adisk in the long-wavelength limit-to obtain thescattered field for a helical dielectric target. Theresults are compared with experiment in Ref. 6, andthe comparison is good. Thus, the internal fieldfrom an infinite cylinder used within the disks thatcompose small targets has been established to workwell in calculating the scattered field from thesetargets. This provides the motivation for the pres-ent work that uses the internal field from an infinite

1776 APPLIED OPTICS / Vol. 33, No. 9 / 20 March 1994

cylinder as a target in which the disk diameters andthe wavelengths are of comparable size.

We will use the internal fields from a series ofinfinite cylinders and compare the resulting far-fieldresults to those corresponding to Mie theory for thesame spheres. We will show that this approach is inremarkably good agreement with the exact theory forthe scattering matrix element SI,, and that it givesgood agreement for S12 for scattering angles up toapproximately 1200, being within 20% of the exactvalue even at 1800.

Integral Equation for Scattering

A solution to the Maxwell equations1 for the scatter-ing of radiation from dielectric targets is

E(r) = Einj(r) + VxVx dV'(m2- 1)/47r

x exp(ik0Ir - r'l)/Ir - r'lE(r')+ (1 - m2 )E(r), (1)

where the integration is over the volume of thescattering target, m is the refractive index of thetarget, k0 = 2r/X, r is a field point and r' is a pointwithin the target. We introduce the effective field by

Eeff(r) = (M2 + 2)/3E(r). (2)

Next, we use a lemma presented by Born and Wolf,7page 760, to bring the operator VxVx inside theintegral:

VxVx dV'(m2 - 1)/4Tr exp(ik I r - r'1)/I r - r'l E(r')

= dV'(m2- 1)/4rVxVx exp(ik, I r - r' I)

/ I r - r'I E(r') + (8r/3)(m2 - 1)E(r')/4rr,

where f dV' is an integral over the target volume,excluding a small sphere surrounding the point r'r, if the field point r is within the target region.Using the above lemma, the identity VxVxA _ (graddiv - k0

2) and putting Eq. (2) into Eq. (1), we get thefinal form for the effective field:

Eeff(r) = Einc(r) + a J dV'(grad div - k 2)

x exp(iko0 r - r' I )/ I r - r' I Eeff(r'). (3)

The coefficient a is the polarizability

a = (3/4,m)( 2 - 1)/(m2 + 2).

A

Fig. 1. A slice within the divided sphere showing the internalcoordinates, the direction of incidence, and the direction of scatter-ing.

where = r/r and [. . .]per indicates that the compo-nent of the vector within the square brackets is to betaken in a direction transverse to the direction ofscattering (see Fig. 1). Thus, the far field deter-mined from Eq. (5) depends on the effective fieldwithin the target, which by Eq. (2), depends on theeffective internal electric field.

Model for the Evaluation of the Integral Equation

As in our previous work, we treat targets that have asymmetry axis and possess a circular cross-sectionalarea perpendicular to this axis. The target in thispaper will be a sphere because an exact theory existsto test the model, and we will take the scatteringplane (the plane formed by the directions of incidence

(4)

When the field point is far from the target, Eq. (3)reduces to

Eeff(r) = ak, 2[exp(ikor)]/r

x [f dV' exp(-ikoP . r')Eeff(r')]per

(5) Fig. 2. Sphere divided into slices, showing the directions ofincidence and scattering.

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0.6-

0.4-

0.20 90 180

Scattering Angle

Fig. 3. Scattering matrix element S11 for a sphere of index ofrefraction 1.33 and size parameter (a) x = 2Tra/X = 0.6, (b) x = 0.8,and (c) x = 1.0, where a is the radius of the sphere and is thewavelength of the light outside the sphere.

and scattering) to be parallel to the plane of the disksthat form the sphere, as shown in Fig. 2.

The scattered field is calculated with the internalfields obtained from the infinite-cylinder theory foreach of the disks that form the target (19 are usedhere). As each disk has a different radius, the corre-sponding internal fields are derived from infinitecylinders of the appropriate radius. The internalfield transverse to the scattering direction, [Eeff(r')]per,is taken at each point in a disk, the integration overthe disk is performed, and these contributions aresummed coherently for disks from the bottom to thetop of the sphere.

The internal field that is used in Eq. (5) when theincident electric field is linearly polarized parallel to

U,

0 90 180

Scattering Angle

Fig. 5. ScatteringmatrixelementS 1 lform= 1.33and(a)x= 1.2,(b)x = 1.4.

the axis of a disk is (TM wave):+x-

Eint(r') = i exp i(@t -dz') 2 n-~

x exp(inO')mk0 /r'Jn'(kmr')k, (6)

where for incidence perpendicular to the z axis8'9

dn = [Jn(koa)H(2)'(ka) - Jn'(ka)H(2)(ka)]

/[mJ(kma)H( 2)'(ka) - m2Jn'(kma)H( 2)(k~a)].

The internal field when the incident electric field islinearly polarized perpendicular to the axis of the diskis (TE wave):

+0 j

Eint(r') = exp i(wt - kz') j cn(-i) exp(inO')

x [in/r'Jn(k~r')r' - Jn'(kmr )O ],

S12 Slices (a) x=0.6

S12 Mie

S12 Slices (b) x=0.8

S12 Mie

S12 Slices (c) x=1.0

S12 Mie

180

Scattering Angle

Fig. 4. Scattering matrix element S12 for a sphere of index ofrefraction 1.33 and size parameter (a) x = 2'ra/X = 0.6, (b) x = 0.8,and (c) x = 1.0, where a is the radius of the sphere and is thewavelength of the light outside the sphere.

V'

0 90

Scattering Angle

Fig. 6. Scattering matrix element S12 for m(b)x = 1.4.

(7)

--- S12 Slices (a) x=1.2

- S12 Mle

--- S12 Slices (b) x=1.4

-0-- 512 Mie

180

= 1.33 and (a) x = 1.2,

1778 APPLIED OPTICS / Vol. 33, No. 9 / 20 March 1994

W'

0 90 180

Scattering Angle

Fig. 7. Convergence of the intensity Ip, = Sl1 + S1 2 for sphereswith 1, 9, and 19 slices.

where

Cn = [Jn(k0a)Hn(2)' (k0a) - Jn'(k0 a)Hn(2)(k0a)]

/[m2 Jn(k0ma)Hn(2 )'(k0 a) - mJ'(k 0ma)H( 2)(k a)].

In these equations, Jn and H, are Bessel and Hankelfunctions of integral order n, the prime denotesdifferentiation with respect to the argument, and a isthe radius of the slice or disk.

Here, r' and ' are cylindrical coordinates in theplane of a disk, as shown in Fig. 1.

Discussion and Conclusions

The scattering intensities are defined by

I = k 2r2I Esc 12 / E. 12, (8)

where Esc is the scattered electric field. The scatter-ing matrix elements

Sll = (par + Iper)/2, S12 = (par - Iper)/2,

are then calculated, where "par" and "per" refer tothe direction of incident polarization in and normal tothe scattering plane, respectively. The fact that thedepolarization of the sphere is zero imposes an essen-tial normalization condition:

Ipar(O) = Iper(0), (9)

where the forward scattering direction is implied byzero. Furthermore, we have taken as an overallnormalization condition S11(0) = 1.

The results are given in Figs. 3 through 6, wherethese calculations are compared with those of the Mietheory. The index of refraction is 1.33, and the sizeparameters are 0.6, 0.8, 1.0, 1.2, and 1.4. In each ofthese calculations the sphere is sliced into 19 disks.Figure 7 demonstrates the convergence of the ap-proach by comparing the intensity Ipar = S11 + S12, for

1, 9, and 19 slices. The difference between 9 and 19slices is seen to be small.

The scattering matrix element S,1 is remarkablywell produced in every case, with a very slight devia-tion occurring at large scattering angles. The ma-trix element S12 is also well produced 'to scatteringangles up to approximately 1200, with deviation occur-ring for the higher size parameters, above 120.

Thus, we have shown that the internal-field charac-teristics produced by light scattering from an infinitecylinder seem to have relevance to a radically differ-ent target shape. The only additional requirementtakes into account the fact that the sphere has zerodepolarization. Therefore, this symmetry must beimposed as a normalization of the cylindrical results.If the cylinder can then mimic the sphere for thesephysical conditions, it should be able to do even betterin reproducing the scattering patterns of such targetshapes as spheroids and shapes that are closer to acylinder than to a sphere. For such shapes, thesymmetry condition in Eq. (9) would undoubtedlyhave to be modified, perhaps with an adjustableparameter.

The results of this approach suggest that theangular scattered intensities can be calculated with ahigh degree of accuracy for particles of arbitraryshape and size comparable to the wavelength of theincident light that can be sliced into circular sections.For example, the scattering of radar waves by tear-drop-shaped raindrops can be studied with the abovetheoretical tools.

This work was supported by the Technical Re-search, Development and Engineering Center (JanonEmburg) under the auspices of the U.S. Army Re-search Office Scientific Services Program adminis-tered by Battelle Memorial Institute.

References

1. K. S. Shifrin, "Scattering of light in a turbid medium," NASAdoc. TT F-477, (NASA, Washington, D.C., 1968).

2. C. Acquista, "Light scatteringby tenuous particles, a generaliza-tion of the Rayleigh-Gans-Rocard approach," Appl. Opt. 15,2932-2940 (1976).

3. R. D. Haracz, L. D. Cohen, A. Cohen, and C. Acquista, "Lightscattering from dielectric targets composed of a continuousassembly of circular disks," Appl. Opt. 25, 4386-4394 (1986).

4. A. Cohen, R. D. Haracz, and L. D. Cohen, "Scattering from ahelix using the exact cylinder theory," J. Wave Mater. Interact.3, 219-225 (1988).

5. R. D. Haracz, L. D. Cohen, A. R. W. Presley, and A. Cohen,"Scattering of linearly polarized microwave radiation from adielectric target including self-interaction," Appl. Opt. 28,1338-1344(1989).

6. R. D. Haracz, L. D. Cohen, A. Cohen, and R. Wang, "Thescattering of linearly polarized light from a dielectric helix,"Appl. Opt. 26, 4632-4638 (1987).

7. M. Born and E. Wolf, Principles of Optics, 6th. ed. (Pergamon,Oxford, 1980) App. 5, pp. 760-761.

8. H. C. van de Hulst, Light Scattering by Small Particles (Dover,1957, 1981) Chap. 15, pp. 297-322.

9. M. Kerker, The Scattering of Light and Other ElectromagneticRadiation (Academic, New York, 1969) Chap. 6, pp. 255-310.

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