Internal, Near-Surface, and Scattered Electromagnetic Fields for a Layered Spheroid with Arbitrary...

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Internal, near-surface, and scattered electromagneticfields for a layered spheroid with arbitrary illumination

John P. Barton

A spheroidal coordinate separation-of-variables solution has been developed for the determination ofinternal, near-surface, and scattered electromagnetic fields of a layered spheroid ~either prolate or oblate!with arbitrary monochromatic illumination ~e.g., plane wave or focused Gaussian beam!. Calculatedresults are presented for layered 2:1 axis ratio prolate and oblate spheroids with an equivalent sphere sizeparameter of 20. © 2001 Optical Society of America

OCIS codes: 290.4020, 350.3950, 280.1100, 260.2110.

1. Introduction

In an earlier paper,1 a spheroidal coordinateeparation-of-variables solution was developed forhe determination of internal and external electro-agnetic fields for a homogeneous spheroid, of either

rolate or oblate geometry, illuminated by monochro-atic light of arbitrary character ~plane wave, fo-

used beam, etc.!. In this paper, the proceduredeveloped in Ref. 1 is generalized so as to obtain acorresponding solution for a spheroid consisting of ahomogeneous core surrounded by a single homoge-neous layer. This development could be used, forexample, to determine the electromagnetic fields thatresult from the interaction of a focused laser beamwith an appreciably elongated ~prolatelike! or appre-ciably flattened ~oblatelike! solid particle coated with

layer of condensed liquid or with a spheroid-shapediological particle that has been approximately mod-led as a homogeneous nucleus surrounded by a ho-ogeneous layer of cytoplasm.The problem of plane-wave scattering by a lay-

red spheroid has been considered previously.ang and Barber2 solved the problem by applying

the extended boundary condition method in spher-ical coordinates, whereas Sebak and Sinha,3 Som-sikov,4 and Farafonov et al.5–7 have solved the

J. P. Barton ~jbarton@uniserve.unl.edu! is with the Departmentof Mechanical Engineering, College of Engineering and Technol-ogy, University of Nebraska–Lincoln, Lincoln, Nebraska 68588-0656.

Received 12 September 2000; revised manuscript received 23April 2001.

0003-6935y01y213598-10$15.00y0© 2001 Optical Society of America

3598 APPLIED OPTICS y Vol. 40, No. 21 y 20 July 2001

problem using a spheroidal coordinate separation-of-variables method. The research presented herediffers from these previous efforts, however, in thatthe incident illumination is not restricted to theplane wave, but can be of any arbitrary form includ-ing, for example, a focused Gaussian beam or scat-tered light from an adjacent particle or surface.8–10

In addition, in Refs. 2–7 only far-field scatteringresults are presented. In this paper, internal andnear-surface electromagnetic field distributions arealso presented. A study of the internal and near-surface electromagnetic field distributions can as-sist in the development of a physical interpretationof the origins of the resultant far-field scattering.For high-intensity laser applications, the internaland near-surface electromagnetic field distribu-tions can be used to determine the location withinthe particle where maximum heating occurs or thelocation where electrical breakdown would initiate.

Internal and near-surface electromagnetic fieldsfor a layered spheroid with arbitrary illuminationcould also be determined with the spherical coordi-nate boundary-matching method described in Ref. 11.As discussed in Ref. 11, however, this method is lim-ited to near-spherical geometries and could be usedonly for spheroids with axis ratios up to approxi-mately 1.40. The spheroidal coordinate method pre-sented here is applicable for high ~as well as low! axisratio spheroids.

2. Theoretical Development

A. General Theory

The electromagnetic interaction of a known incidentmonochromatic field with a layered spheroid consist-ing of a core surrounded by a single layer is consid-

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ered. A schematic of the assumed geometry isshown in Fig. 1. The spheroid core, spheroid layer,and the surrounding medium are each homogeneous,isotropic, and nonmagnetic ~m 5 1!; and the sur-rounding medium is assumed to be infinite and non-absorbing. An implicit time dependence ofexp~2ivt! is assumed, and the electromagnetic fieldis nondimensionalized relative to an electric field am-plitude E0 associated with the incident field.

A spheroid is created when an ellipse is rotatedabout its axis. If the ellipse is rotated about its ma-jor axis, a prolate spheroid is formed. If the ellipse isrotated about its minor axis, an oblate spheroid isformed. The spheroid has an associated semifocallength f. If a is the length of the semimajor axis and

is the length of the semiminor axis of the spheroid,hen

f 5 a@1 2 ~bya!2#1y2. (1)

In the following, the rectangular coordinates ~x, y, z!,the spatial position vector r, and the vector calculusoperators ~¹, ¹2! are all nondimensionalized relativeto the semifocal length f.

The eigenfunctions of the separation-of-variablessolution of the scalar Helmholtz equation,

~¹2 1 h2!P 5 0, (2)

in spheroidal coordinates ~h, j, f! are of the form

Plm 5 Slm~h, h! Rlm~h, j!exp~imf!, (3)

here h 5 kf, k is the wave number, Slm is the sphe-roidal angle function, Rlm is the spheroidal radialfunction ~which may be of the first kind, Rlm

~1!; secondkind, Rlm

~2!; third kind, Rlm~3! 5 Rlm

~1! 1 iRlm~2!; or fourth

kind, Rlm~4! 5 Rlm

~1! 2 iRlm~2!!, and the azimuthal angle

ependence is contained in the complex exponentialxp~imf!. l and m are integer indices with 2` ,

, ` and, for a given m, umu # l , `. h is thepheroidal angle coordinate, j is the spheroidal radialoordinate, and f is the azimuthal coordinate. Thelm and Rlm functions are different depending on

Fig. 1. Schematic of the geometric arrangement. The boundaryof the spheroid rotates about the z axis. For the prolate spheroid~as shown!, the major axis of the spheroid is along the z axis. Forthe oblate spheroid, the major axis of the spheroid is along the xaxis. The outer surface of the spheroid ~the layer–external inter-face! is located at j 5 j0 and the core–layer interface is located at

5 jc , j0.

whether a prolate or oblate spheroidal coordinate sys-tem is chosen. A computer program for generatingthe associated eigenvalues, llm~h!, and subroutinesfor generating the Slm~h, h! and Rlm~h, j! functionswere written based on procedures described in Flam-mer.1,12,13

Within a source-free, homogeneous medium, boththe electric field E and the magnetic field H are so-lenoidal ~¹ z E 5 0, ¹ z H 5 0! and satisfy the vector

elmholtz equations

~¹2 1 h2!E 5 0, (4)

~¹2 1 h2!H 5 0. (5)

Following the approach of Stratton,14 independentsolenoidal vector eigenfunction solutions ~Mlm, Nlm!f the vector Helmholtz equation can be obtainedrom the corresponding scalar eigenfunctions by theollowing vector operations:

Mlm 5 ¹ 3 ~rPlm!, (6)

Nlm 51h

¹ 3 Mlm. (7)

In spheroidal coordinates, a spheroid is formed bya surface of constant radial coordinate ~j 5 constant!;thus a specified value of j0 is used to define the outerboundary of the spheroid, and a specified value of jc ,j0 is used to define the boundary of the core–layerinterface. The external particle ~j . j0! electromag-

etic field consists of a sum of the known incidenteld indicated by the superscript ~i! and the to-be-

determined scattered field indicated by the super-script ~s!. The electromagnetic field within theparticle core ~j , jc! indicated by the superscript ~c!and the electromagnetic field within the particlelayer ~jc , j , j0! indicated by the superscript ~l ! arealso to be determined. Expressions for the scatteredand internal fields are developed when we take sum-mations over the appropriate forms of the corre-sponding vector eigenfunctions as follows.

For the scattered field,

E~s! 5 (l,m

@almNlm~s! 1 blmMlm

~s!#, (8)

H~s! 5 2iÎeext (l,m

@almMlm~s! 1 blmNlm

~s!#, (9)

here

Plm~s! 5 Slm~hext, h! Rlm

~3!~hext, j!exp~imf!. (10)

or the core field,

E~c! 5 (l,m

@clmNlm~c! 1 dlmMlm

~c!#, (11)

H~c! 5 2iÎeext nc (l,m

@clmMlm~c! 1 dlmNlm

~c!#, (12)

here

Plm~c! 5 Slm~hc, h! Rlm

~1!~hc, j!exp~imf!. (13)

20 July 2001 y Vol. 40, No. 21 y APPLIED OPTICS 3599

w

t

e

c

um1s

d

t

tt

cet

c

s

l

l

l

l

l

l

l

l

fI

3

For the layer field,

E~l ! 5 (l,m

@elmNlm~l,1! 1 flmMlm

~l,1!

1 glmNlm~l,2! 1 hlmMlm

~l,2!#, (14)

H~l ! 5 2iÎeext nl (l,m

@elmMlm~l,1! 1 flmNlm

~l,1!

1 glmMlm~l,2! 1 hlmNlm

~l,2!#, (15)

here

Plm~l,1! 5 Slm~hl, h! Rlm

~1!~hl, j!exp~imf!, (16)

Plm~l,2! 5 Slm~hl, h! Rlm

~2!~hl, j!exp~imf!. (17)

In Eqs. ~8!–~17!, eext is the dielectric constant ofthe surrounding ~external! medium, nc 5 =ecyeext ishe relative index of refraction of the core, and nl 5

=elyeext is the relative index of refraction of the layer.Thus hext 5kextf 5 ~2pylext! f, hc 5 kcf 5 nchext, andhl 5 klf 5 nlhext, where lext is the wavelength in thesurrounding medium. For the scattered field, onlythe Rlm

~3! spheroidal radial function is included in thexpression for Plm

~s! because, in the limit of large j, thisfunction corresponds to an outgoing traveling wave,appropriate for the scattered field solution. @In thesame limit, the Rlm

~4! spheroidal wave function wouldcorrespond to an incoming traveling wave.# For theore field, only the Rlm

~1! spheroidal radial function isincluded in the expression for Plm

~c! because Rlm~2! is

nbounded as j f 0. In practice, the double sum-ation indices are truncated such that 2M , m ,M and umu # l , L, where M and L are chosen

ufficiently large for convergence of the solution.The electromagnetic field components of the inci-

ent field E~i! and H~i! ~plane wave, focused beam,etc.! are assumed known. The unknown expansioncoefficients ~alm, blm, clm, dlm, elm, flm, glm, hlm! aredetermined when we apply the boundary conditionsof continuity of the tangential electromagnetic fieldover the outer surface of the particle ~j 5 j0! and overhe core–layer interface ~j 5 jc!. At j 5 j0,

Eh~l ! 2 Eh

~s! 5 Eh~i!, (18)

Ef~l ! 2 Ef

~s! 5 Ef~i!, (19)

Hh~l ! 2 Hh

~s! 5 Hh~i!, (20)

Hf~l ! 2 Hf

~s! 5 Hf~i!; (21)

and at j 5 jc,

Eh~l ! 2 Eh

~c! 5 0, (22)

Ef~l ! 2 Ef

~c! 5 0, (23)

Hh~l ! 2 Hh

~c! 5 0, (24)

Hf~l ! 2 Hf

~c! 5 0. (25)

Next, the series expansions of Eqs. ~8!–~17! are usedo substitute for the scattered, layer, and core elec-romagnetic field components in the eight boundary

600 APPLIED OPTICS y Vol. 40, No. 21 y 20 July 2001

ondition equations. The resultant series-xpansion forms of Eqs. ~18!–~21! are then each mul-iplied by Sl9m9~hext, h!exp~2im9f! and integrated

over the outer surface of the spheroid ~j 5 j0!. Sim-ilarly, the resultant series-expansion forms of Eqs.~22!–~25! are each multiplied by Sl9m9~hl,h!exp~2im9f! and integrated over the surface of theore–layer interface ~j 5 jc!.

For a maximum summation limit L and for a par-ticular value of m, we obtain the following set ofimultaneous linear algebraic equations:

(5umu

L

~2Ilml91 alm 2 Ilml9

2 blm 1 Ilml93 elm 1 Ilml9

4 flm

1 Ilml95 glm 1 Ilml9

6 hlm! 51

2pAl9m

h , (26)

(5umu

L

~2Ilml97 alm 2 Ilml9

8 blm 1 Ilml99 elm 1 Ilml9

10 flm

1 Ilml911 glm 1 Ilml9

12 hlm! 51

2pAl9m

f , (27)

(5umu

L

~2Ilml92 alm 2 Ilml9

1 blm 1 nl Ilml94 elm 1 nl Ilml9

3 flm

1 nl Ilml96 glm 1 nl Ilml9

5 hlm! 5i

2pÎeext

Bl9mh , (28)

(5umu

L

~2Ilml98 alm 2 Ilml9

7 blm 1 nl Ilml910 elm 1 nl Ilml9

9 flm

1 nl Ilml912 glm 1 nl Ilml9

11 hlm! 5i

2pÎeext

Bl9mf , (29)

(5umu

L

~2Ilml913 clm 2 Ilml9

14 dlm 1 Ilml915 elm 1 Ilml9

16 flm

1 Ilml917 glm 1 Ilml9

18 hlm! 5 0, (30)

(5umu

L

~2Ilml919 clm 2 Ilml9

20 dlm 1 Ilml921 elm 1 Ilml9

22 flm

1 Ilml923 glm 1 Ilml9

24 hlm! 5 0, (31)

(5umu

L

~2nc Ilml914 clm 2 nc Ilml9

13 dlm 1 nl Ilml916 elm

1 nl Ilml915 flm 1 nl Ilml9

18 glm 1 nl Ilml917 hlm! 5 0, (32)

(5umu

L

~2nc Ilml920 clm 2 nc Ilml9

19 dlm 1 nl Ilml922 elm 1 nl Ilml9

21 flm

1 nl Ilml924 glm 1 nl Ilml9

23 hlm! 5 0, (33)

or all l9 5 umu to L. The one-dimensional integralslml9

1224 are dependent solely on the geometry of theparticle ~j0, jc! and are defined in Appendix A. Thesurface integrals Al9m

h , Al9mf , Bl9m

h , and Bl9mh are depen-

s

~~

~ls~

Ts

cbsccE

dent on both the geometry of the particle ~j0! and theincident field @E~i!, H~i!# and are defined as follows:

Al9mh 5 *

0

2p

*21

1

Eh~i!~j0, h, f!Sl9m~hext, h!

3 exp~2imf!dhdf, (34)

Al9mf 5 *

0

2p

*21

1

Ef~i!~j0, h, f!Sl9m~hext, h!

3 exp~2imf!dhdf, (35)

Bl9mh 5 *

0

2p

*21

1

Hh~i!~j0, h, f!Sl9m~hext, h!

3 exp~2imf!dhdf, (36)

Bl9mf 5 *

0

2p

*21

1

Hf~i!~j0, h, f!Sl9m~hext, h!

3 exp~2imf!dhdf, (37)

For a particular value of m, Eqs. ~26!–~33! thus pro-vide 8 ~L 2 umu 1 1! linear algebraic equations toolve for the 8 ~L 2 umu 1 1! values of ~alm, blm, clm,

dlm, elm, flm, glm, hlm!. Note that, for the special caseof m 5 0, the one-dimensional integrals I2, I4, I6, I7,I9, I11, I14, and I16 are zero, so Eqs. ~26!, ~29!, ~30!, and33! can be used to solve for the 4 ~L 1 1! values ofal0, cl0, el0, gl0!; and Eqs. ~27!, ~28!, ~31!, and ~32! can

be used to solve for the 4 ~L 1 1! values of ~bl0, dl0, fl0,hl0!.

In summary, the procedure for determining theinternal and external electromagnetic fields for aknown field incident on a layered spheroid is as fol-lows. For a given set of input parameters ~j0, jc,hext, nl, nc, incident field parameters, etc.!, the Ilml9one-dimensional integrals of Appendix A and theAl9m

h , Al9mf , Bl9m

h , Bl9mh surface integrals of Eqs. ~34!–

37! are determined by numerical integration. Theinear algebraic equations of Eqs. ~26!–~33! are thenolved for the internal and scattered field coefficientsalm, blm, clm, dlm, elm, flm, glm, hlm!. Once the ex-

pansion coefficients are known, Eqs. ~8!–~17! can thenbe used to determine the electromagnetic field any-where inside or outside the spheroid.

B. Prolate Geometry

In prolate spheroidal coordinates, the rectangular co-ordinates ~x, y, z! are related to the spheroidal coor-dinates ~21 , h , 11, 1 , j , `, 0 , f , 2p! by

x 5 @~j2 2 1!~1 2 h2!#1y2 cos f, (38)

y 5 @~j2 2 1!~1 2 h2!#1y2 sin f, (39)

z 5 jh. (40)

he semimajor to semiminor axis ratio of the prolatepheroid ~ayb! is related to the choice of j0 by

~ayb! 5j0

~j02 2 1!1y2 . (41)

So, for example, j0 5 1.15470 f ~ayb! 5 2, j0 51.06066 f ~ayb! 5 3, etc.

The components of the eigenfunctions of the vectorHelmholtz equation for a prolate spheroidal coordi-nate system are given in Ref. 1.

C. Oblate Geometry

In oblate spheroidal coordinates, the rectangular co-ordinates ~x, y, z! are related to the spheroidal coor-dinates ~21 , h , 11, 0 , j , `, 0 , f , 2p! by

x 5 @~j2 1 1!~1 2 h2!#1y2 cos f, (42)

y 5 @~j2 1 1!~1 2 h2!#1y2 sin f, (43)

z 5 jh. (44)

The semimajor to semiminor axis ratio of the oblatespheroid ~ayb! is related to the choice of j0 by

~ayb! 5 ~1 1 1yj02!1y2. (45)

So, for example, j0 5 0.57735 f ~ayb! 5 2, j0 50.35355 f ~ayb! 5 3, etc.

The components of the eigenfunctions of the vectorHelmholtz equation for an oblate spheroidal coordi-nate system are given in Ref. 1.

3. Verification Calculations

Computer programs were written based on the theo-retical development discussed in Section 2. I veri-fied these computer programs ~and the inherenttheoretical development! by performing several tests.Electromagnetic field distributions were generatedfor layered prolate and oblate spheroids of a 1.3 axisratio, and the results were compared point by pointwith corresponding results obtained by use of thespherical coordinate boundary-matching method ofRef. 11. For the limited axis ratio of 1.3, theboundary-matching method is rigorously valid, andthe results of the spheroidal separation-of-variablessolution and the spherical coordinate boundary-matching method solution were found to be identicalto the order of six significant digits. It was alsoverified that the homogeneous spheroid solution ~asgiven in Ref. 1! could be recovered from the layeredspheroid solution when either the relative index ofrefraction of the layer was set equal to the relativeindex of refraction of the core ~nl 5 nc! or when therelative index of refraction of the layer was set equalto unity ~nl 5 1!.

Finally, I obtained an additional verification byhecking the rigor with which the electromagneticoundary conditions are maintained over the outerurface of the spheroid and over the surface of theore–layer interface. These boundary conditions in-lude not only the imposed tangential conditions ofqs. ~18!–~25!, but also the following supplementary

20 July 2001 y Vol. 40, No. 21 y APPLIED OPTICS 3601

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conditions on the normal components of the electro-magnetic field. At j 5 j0,

Ej~i! 1 Ej

~s! 5 nl2Ej

~l !, (46)

Hj~i! 1 Hj

~s! 5 Hj~l !; (47)

nd at j 5 jc,

nl2Ej

~l ! 5 nc2Ej

~c!, (48)

Hj~l ! 5 Hj

~c!. (49)

The absolute values of the deviations of the left- andright-hand sides of Eqs. ~46! and ~47!, averaged overthe outer surface of the spheroid, and the absolutevalues of the deviations of the left- and right-handsides of Eqs. ~48! and ~49!, averaged over the surfaceof the core–layer interface, provided sensitive criteriafor convergence. Normal convergence was observed,and the two convergence criteria could be made neg-ligibly small when the summation index limits M and

were chosen to be sufficiently large.

4. Demonstration Calculations

A set of calculations demonstrating the effects of thepresence of the layer on the internal, near-surface,and scattered electromagnetic fields of the spheroidwas performed. For the calculations that follow, theexternal dielectric constant ~εext! was set equal to 1.0,nd the values of 1.33 and 1.50 were used for theelative index of refractions of the core ~nc! and theayer ~nl!. A 2:1 axis ratio spheroid ~j0 5 1.154701

for prolate, j0 5 0.577350 for oblate! with a 2.5:1 axisratio core ~jc 5 1.091089 for prolate, j0 5 0.436436 foroblate! was assumed. For these conditions, 54.0% ofhe total volume of the prolate spheroid is in theayer, whereas for the oblate spheroid 67.5% of theotal volume is in the layer. The spheroid size pa-ameter ~hext! was set to a value ~hext 5 27.494593 for

prolate, hext 5 21.822473 for oblate! corresponding ton equivalent sphere ~i.e., equal volume sphere! sizearameter of 20.The incident electromagnetic field, either planeave or focused Gaussian beam, was assumed to be

inearly polarized. The direction of propagation andhe direction of electric field polarization are specifiedith two beam direction angles, ubd and fbd. ubd is

the direction of propagation relative to the y–z plane~see Fig. 1!, and fbd is the direction of the incidentlectric field polarization relative to the x–z plane.Figure 2 shows a surface grid plot of the electric

field magnitude ~uEu! in the x–z plane for a plane waveincident upon a 2:1 axis ratio layered prolate spher-oid with a core relative index of refraction nc of 1.33and a layer relative index of refraction nl of 1.50.The angle of incident propagation ubd is 30°, and theangle of incident electric field polarization fbd is 90°perpendicular to the x–z plane!. ~Note: In the

plane perpendicular to the direction of the incidentelectric field polarization, the j component of the elec-tric field is zero, and thus the electric field is contin-uous across both the outer surface of the spheroid and

602 APPLIED OPTICS y Vol. 40, No. 21 y 20 July 2001

the surface of the core–layer interface, as shown inFig. 2.! The plot extends from 21.5 to 11.5 in boththe x and z directions and includes the electric field ofthe core, layer, and near surface.

As shown in Fig. 2, part of the incident plane waveapparently reflects and defracts off the front surfaceof the prolate spheroid creating lines of constructiveand destructive interference in the near field. Theremaining part of the incident plane wave is trans-mitted into the spheroid where it is refracted at thelayer–external interface and then subsequently in-teracts with the core–layer interface. There are in-ternal regions of constructive and destructiveinterference with several peaks on the shadow side ofthe spheroid ~perhaps created by a focusing of theelectromagnetic wave as it passes through the spher-oid!. Part of the electromagnetic energy is appar-ently transmitted out the shadow-side end of thespheroid in the form of a relatively well-definedbeam. Figure 3 gives contour and gray-level ~whiteimplies low, dark implies high! plots of the same datahown in Fig. 2. Both plots in Fig. 3 include lines

Fig. 2. Surface grid plot of the electric field magnitude in the x–zplane for a plane wave incident upon a 2:1 axis ratio prolate spher-oid with a 2.5:1 axis ratio core. Incident propagation angle ubd 530°, incident electric field polarization angle fbd 5 90°, spheroidize parameter hext 5 27.494593, relative index of refraction of core

nc 5 1.33, and relative index of refraction of layer nl 5 1.50.

Fig. 3. Contour and gray-level ~white implies low, black implieshigh! plots of the electric field magnitude in the x–z plane for a

lane wave incident upon a 2:1 axis ratio layered prolate spheroidith a 2.5:1 axis ratio core. Same conditions as Fig. 2.

fi

Fasosns

fil

osaspmwaTpshttpa

2

n

a1

showing the locations of the spheroid outer surfaceand core–layer interface boundaries.

Figure 4 provides a surface grid plot of the electricfield magnitude for the same layered prolate spheroidconditions as in Fig. 2, except that the relative indexof refractions of the layer and core were interchangedso that now nc 5 1.50 and nl 5 1.33. As can be seenby a comparison with Figs. 2 and 4, the internalelectric field distribution is significantly affected bythe interchange of the two index of refraction values.Figure 5 is a surface grid plot of a correspondinghomogeneous spheroid with a uniform relative indexof refraction of 1.33 ~nc 5 nl 5 1.33!. The electricfield distribution for the homogeneous spheroid withn 5 1.33 is similar to ~but not identical to! the electric

eld distribution of the nc 5 1.33, nl 5 1.50 layeredspheroid given in Fig. 2. ~The electric field distribu-tion of a homogeneous spheroid with n 5 1.50, notshown, was found to be similar to that of the nc 51.50, nl 5 1.33 layered spheroid given in Fig. 4.!

Fig. 4. Surface grid plot of the electric field magnitude in the x–zplane for a plane wave incident upon a 2:1 axis ratio prolate spher-oid with a 2.5:1 axis ratio core. ubd 5 30°, fbd 5 90°, hext 57.494593, nc 5 1.50, nl 5 1.33.

Fig. 5. Surface grid plot of the electric field magnitude in the x–zplane for a plane wave incident upon a 2:1 axis ratio homogeneousprolate spheroid. ubd 5 30°, fbd 5 90°, hext 5 27.494593, nc 5

l 5 1.33.

Apparently, at least for the conditions consideredhere, even though more than 50% of the spheroidvolume is in the layer, the internal and near-surfaceelectric field distribution is most strongly affected bythe optical properties of the core, as opposed to theoptical properties of the layer.

We can determine the far-field scattering intensityby evaluating the scattered field in the limit of jf `.

or the spheroidal coordinate system—in the limit oflarge spheroidal radial coordinate ~jf `!—the con-

tant j surface becomes spherical, and spheroidal co-rdinates can be directly related to the correspondingpherical coordinates: jf f r and h f cos~u!. Aondimensionalized, time-averaged scattering inten-ity in the far field can then be defined as follows:

Sr~u, f! 5 limr3`

r2^S&r

c8p

E02~pf 2!

5 limj3`

j2

pRe@Ef

~s!Hh~s!* 2 Eh

~s!Hf~s!*#. (50)

Figure 6 shows a polar plot ~log scale! of the far-eld scattering in the x–z plane for the layered pro-

ate spheroid. The plot on the left is for nc 5 1.33and nl 5 1.50 ~corresponding to Fig. 2!, and the plotn the right is for nc 5 1.50 and nl 5 1.33 ~corre-ponding to Fig. 4!. Both scattering patterns exhibitstrong forward scattering lobe at 30°, which corre-

ponds with the propagation direction of the incidentlane wave. The scattering pattern is nonsym-etrical about the direction of incident propagation,ith relatively strong scattering in the 260° regionnd relatively weak scattering in the 120° region.he far-field scattering patterns for homogeneousrolate spheroids with n 5 1.33 and n 5 1.50 arehown in Fig. 7, and the scattering patterns of theomogeneous spheroids exhibit similar general fea-ures to those of the layered prolate spheroids. Al-hough the scattering patterns of the four types ofrolate spheroid have similar general features, therere differences, and an analysis of these differences

Fig. 6. Polar plots ~log scale, range from 0.001 to 10.0! of thefar-field scattering in the x–z plane for a plane wave incident upona 2:1 axis ratio prolate spheroid with a 2.5:1 axis ratio core. ubd 530°, fbd 5 90°, hext 5 27.494593. For the plot on the left, nc 5 1.33nd nl 5 1.50 ~Sr,max 5 18.77495!; for the plot on the right, nc 5.50 and nl 5 1.33 ~Sr,max 5 14.79815!.

20 July 2001 y Vol. 40, No. 21 y APPLIED OPTICS 3603

c

a

2

pw

2

3

might be necessary, for example, to assist the devel-opment of a light-scattering measurement instru-ment capable of distinguishing one type of prolatespheroid from another.

All results so far have been for the prolate geome-try. Figure 8 shows a surface grid plot of the electricfield magnitude ~uEu! in the x–z plane for a plane waveincident upon a 2:1 axis ratio layered oblate spheroidwith a core relative index of refraction nc of 1.33 anda layer relative index of refraction nl of 1.50. As wasthe case for the prolate calculations, the angle ofincident propagation ubd is 30°, and the angle of in-ident electric field polarization fbd is 90° ~perpendic-

ular to the x–z plane!. The electric field distributionfor the oblate spheroid is quite different from that ofthe prolate spheroid. The most salient feature is thestrong focusing peak that occurs on the shadow sideof the spheroid. Contour and gray-level plots of thesame data presented in Fig. 8 are given in Fig. 9.Figure 10 provides the electric field distribution forthe same layered oblate spheroid, but with the values

Fig. 7. Polar plots ~log scale, range from 0.001 to 10.0! of thefar-field scattering in the x–z plane for a plane wave incident upon

2:1 axis ratio homogeneous prolate spheroid. ubd 5 30°, fbd 590°, hext 5 27.494593. For the plot on the left, nc 5 nl 5 1.33~Sr,max 5 12.18744!; for the plot on the right, nc 5 nl 5 1.50~Sr,max 5 14.48922!.

Fig. 8. Surface grid plot of the electric field magnitude in the x–zplane for a plane wave incident upon a 2:1 axis ratio oblate spher-oid with a 2.5:1 axis ratio core. ubd 5 30°, fbd 5 90°, hext 51.822473, nc 5 1.33, nl 5 1.50.

604 APPLIED OPTICS y Vol. 40, No. 21 y 20 July 2001

of the core and layer relative index of refractionsinterchanged ~nc 5 1.50, nl 5 1.33!; and Fig. 11 givesthe electric field distribution for the correspondinghomogeneous oblate spheroid ~nc 5 nl 5 1.33!. In acomparison of Figs. 8, 10, and 11, it can be observedthat, although the general features of the electricfield distributions are similar, they are not identical.The corresponding far-field scattering patterns of thelayered and homogeneous oblate spheroids areshown, respectively, in Figs. 12 and 13. The scat-tering patterns are quite distinct from those of theprolate spheroid, and, although once again exhibitingsimilar general features, each scattering pattern hasdistinguishing features that could be used to identifyone type of oblate spheroid from the other.

Solutions can be obtained for any known incidentfield, not just the plane wave. As an example, Fig.14 shows a surface grid plot of the electric field dis-tribution for the same layered oblate spheroid con-sidered in Fig. 8, but now with focused beamillumination instead of plane-wave illumination.The corresponding contour and gray-level plots are

Fig. 9. Contour and gray-level ~white implies low, black implieshigh! plots of the electric field magnitude in the x–z plane for a

lane wave incident upon a 2:1 axis ratio layered oblate spheroidith a 2.5:1 axis ratio core. Same conditions as Fig. 8.

Fig. 10. Surface grid plot of the electric field magnitude in the x–zplane for a plane wave incident upon a 2:1 axis ratio oblate spher-oid with a 2.5:1 axis ratio core. ubd 5 30°, fbd 5 90°, hext 51.822473, nc 5 1.50, nl 5 1.33.

l

fa

a1

9

0

fw

given in Fig. 15. The beam propagates in the z-axisdirection ~ubd 5 0°! with electric field polarization

Fig. 11. Surface grid plot of the electric field magnitude in the x–zplane for a plane wave incident upon a 2:1 axis ratio homogeneousoblate spheroid. ubd 5 30°, fbd 5 90°, hext 5 21.822473, nc 5 nl 51.33.

Fig. 12. Polar plots ~log scale, range from 0.001 to 10.0! of thear-field scattering in the x–z plane for a plane wave incident upon2:1 axis ratio oblate spheroid with a 2.5:1 axis ratio core. ubd 5

30°, fbd 5 90°, hext 5 21.822473. For the plot on the left, nc 5 1.33nd nl 5 1.50 ~Sr,max 5 84.95962!; for the plot on the right, nc 5.50 and nl 5 1.33 ~Sr,max 5 64.08979!.

Fig. 13. Polar plots ~log scale, range from 0.001 to 10.00! of thefar-field scattering in the x–z plane for a plane wave incident upona 2:1 axis ratio homogeneous oblate spheroid. ubd 5 30°, fbd 50°, hext 5 21.822473. For the plot on the left, nc 5 nl 5 1.33

~Sr,max 5 60.93952!; for the plot on the right, nc 5 nl 5 1.50~Sr,max 5 51.01663!.

perpendicular to the x–z plane ~fbd 5 90°!. Thebeam waist is equal to one half of the semifocal lengthof the spheroid ~w0 5 0.5!, and the focal point isocated on the x axis at z0 5 0.75. The electromag-

netic field components of the incident beam wereevaluated by use of the fifth-order corrected funda-mental Gaussian ~i.e., TEM00 mode! beam model de-scribed in Ref. 15. As can be seen in Figs. 14 and 15,the layered oblate spheroid acts as a lens, with thebeam entering the illuminated side and then exitingthe shadow side at an angle of approximately 30 degrelative to the original direction of propagation.

5. Summary

A spheroidal coordinate separation-of-variables solu-tion has been developed, verified, and demonstratedfor determination of the internal, near-surface, andscattered fields of a layered spheroid ~either prolateor oblate! with arbitrary monochromatic illumina-tion. The demonstration calculations illustrate howthe presence of the layer can affect the internal andnear fields, as well as affect the resultant far-fieldscattering pattern. A possible future application ofthis research would be to perform additional system-atic calculations to develop a better understanding ofthe combined effects of the layered spheroid geome-

Fig. 14. Surface grid plot of the electric field magnitude in the x–zplane for a focused beam incident upon a 2:1 axis ratio oblatespheroid with a 2.5:1 axis ratio core. ubd 5 30°, fbd 5 90°, hext 521.822473, nc 5 1.33, nl 5 1.50, w0 5 0.50, ~x0, y0, z0! 5 ~0.0, 0.0,.75!.

Fig. 15. Contour and gray-level ~white implies low, black implieshigh! plots of the electric field magnitude in the x–z plane for aocused beam incident upon a 2:1 axis ratio layered oblate spheroidith a 2.5:1 axis ratio core. Same conditions as Fig. 14.

20 July 2001 y Vol. 40, No. 21 y APPLIED OPTICS 3605

0

3

try, layered spheroid optical properties, and incidentfield properties on far-field scattering.

Appendix A: One-Dimensional Integrals

Ilml91 5 2 *

0

1

Nlm,h~s! ~j0, h, 0!Sl9,m~hext, h!dh, (A1)

Ilml92 5 2 *

0

1

Mlm,h~s! ~j0, h, 0!Sl9,m~hext, h!dh, (A2)

Ilml93 5 2 *

0

1

Nlm,h~l,1! ~j0, h, 0!Sl9,m~hext, h!dh, (A3)

Ilml94 5 2 *

0

1

Mlm,h~l,1! ~j0, h, 0!Sl9,m~hext, h!dh, (A4)

Ilml95 5 2 *

0

1

Nlm,h~l,2! ~j0, h, 0!Sl9,m~hext, h!dh, (A5)

Ilml96 5 2 *

0

1

Mlm,h~l,2! ~j0, h, 0!Sl9,m~hext, h!dh, (A6)

Ilml97 5 2 *

0

1

Nlm,f~s! ~j0, h, 0!Sl9,m~hext, h!dh, (A7)

Ilml98 5 2 *

0

1

Mlm,f~s! ~j0, h, 0!Sl9,m~hext, h!dh, (A8)

Ilml99 5 2 *

0

1

Nlm,f~l,1! ~j0, h, 0!Sl9,m~hext, h!dh, (A9)

Ilml910 5 2 *

0

1

Mlm,f~l,1! ~j0, h, 0!Sl9,m~hext, h!dh, (A10)

Ilml911 5 2 *

0

1

Nlm,f~l,2! ~j0, h, 0!Sl9,m~hext, h!dh, (A11)

Ilml912 5 2 *

0

1

Mlm,f~l,2! ~j0, h, 0!Sl9,m~hext, h!dh, (A12)

Ilml913 5 2 *

0

1

Nlm,h~c! ~jc, h, 0!Sl9,m~hl, h!dh, (A13)

Ilml914 5 2 *

0

1

Mlm,h~c! ~jc, h, 0!Sl9,m~hl, h!dh, (A14)

Ilml915 5 2 *

0

1

Nlm,h~l,1! ~jc, h, 0!Sl9,m~hl, h!dh, (A15)

606 APPLIED OPTICS y Vol. 40, No. 21 y 20 July 2001

Ilml916 5 2 *

0

1

Mlm,h~l,1! ~jc, h, 0!Sl9,m~hl, h!dh, (A16)

Ilml917 5 2 *

0

1

Nlm,h~l,2! ~jc, h, 0!Sl9,m~hl, h!dh, (A17)

Ilml918 5 2 *

0

1

Mlm,h~l,2! ~jc, h, 0!Sl9,m~hl, h!dh, (A18)

Ilml919 5 2 *

0

1

Nlm,f~c! ~jc, h, 0!Sl9,m~hl, h!dh, (A19)

Ilml920 5 2 *

0

1

Mlm,f~c! ~jc, h, 0!Sl9,m~hl, h!dh, (A20)

Ilml921 5 2 *

0

1

Nlm,f~l,1! ~jc, h, 0!Sl9,m~hl, h!dh, (A21)

Ilml922 5 2 *

0

1

Mlm,f~l,1! ~jc, h, 0!Sl9,m~hl, h!dh, (A22)

Ilml923 5 2 *

0

1

Nlm,f~l,2! ~jc, h, 0!Sl9,m~hl, h!dh, (A23)

Ilml924 5 2 *

0

1

Mlm,f~l,2! ~jc, h, 0!Sl9,m~hl, h!dh, (A24)

This research was supported, in part, by the Edge-wood Chemical Biological Center ~Merrill E. Milham!under the auspices of the U.S. Army Research OfficeScientific Services Program administrated by Bat-telle ~delivery order 258, contract DAAH04-96-C-086!.

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1

for an absorbing spheroidal particle with arbitrary illumina-

0. J. P. Barton, W. Ma, S. A. Schaub, and D. R. Alexander, “Elec-tromagnetic field for a beam incident on two adjacent sphericalparticles,” Appl. Opt. 30, 4706–4715 ~1991!.

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tromagnetic field components for a fundamental Gaussianbeam,” J. Appl. Phys. 66, 2800–2802 ~1989!.

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