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5 Coated Dielectric Spheroid In this chapter we consider the scattering of a linearly polarized plane mono- chromatic wave by a homogeneous lossy/lossless dielectric spheroid with a co

5

Coated Dielectric

Spheroid

In this chapter we consider the scattering of a linearly polarized plane mono- chromatic wave by a homogeneous lossy/lossless dielectric spheroid with a confocal lossy/lossless dielectric coating immersed in a homogeneous isotropic medium It is assumed that the surrounding medium is nonconducting and nonmagnetic Results are presented only for the prolate spheroids, as the re- sults for the oblate cases can be obtained by the transformations < + i<, c + -4~ The media of both the spheroid and the coating layer are both assumed

to be linear, homogeneous, and isotropic with permittivities ~1 and ~2 (in gen- eral, complex quantities), and nonmagnetic in nature The permittivity of the surrounding medium is ~0 The semiaxial lengths of the spheroidal core are a2 and b2, and those of the spheroid formed by the confocal outer layer are

al and br The thickness of the coating is defined as t = al - a2 The inner and outer spheroidal surfaces are defined by 5 = 52 and 5 = (1, respectively (Fig 5.1)

The relationships governing the parameters c and { and the spheroidal dimensions are the same as those in Chapter 4 In addition, the following relations hold:

Spheroidal Wave Functions in Electromagnetic Theory

Le-Wei Li, Xiao-Kang Kang, Mook-Seng Leong

ISBNs: 0-471-03170-4 (Hardback); 0-471-22157-0 (Electronic)

116 EM SCATTERING BY A COATED DIELECTRIC SPHEROID

Wave

Fig 5.1 Scattering geometry of a coated spheroid

INCIDENT, TRANSMITTED AND SCATTERED FIELDS 117

62 c2 =

lr

-co*

CO

(5.lc)

Unlike the case of the perfectly conducting spheroid, the existence of fields inside the dielectric spheroid makes the problem more complicated, and it is necessary to use the magnetic fields in the boundary conditions The magnetic field of a propagating wave is related to the electric field via

where k is the wave number and 2 is the characteristic impedance of the medium The incident electric fields for TE and TM polarizations are the same as those of the conductor case, and the corresponding magnetic fields are

(5.4a)

I (5*4b)

for the TE case; and

&TM = aM mn M+t4) *,m+l,ntCO) + PmM+l,n+l M$Z+l,n+l

00 +>:

118 EM SCATTERUVG BY A COATED DIELECTRIC SPHEROID

for the TM case

The transmitted fields inside the core (t < &) can be expressed as

oM

* +%

For the region & < c < &, the transmitted waves are represented as

mission coefficients Note that in the coating region both first and second kinds of spheroidal functions are used, because down here, the wave consists

of two components, one propagating inward

reflections at both boundary surfaces

and one outward This is due to

FIELDS

5.3.1 Boundary Conditions

The unknown scattering and transmission coefficients can be determined by applying the boundary conditions (i.e., continuity of the tangential compo- nents of the electric and magnetic fields across each of the spheroidal surfaces

at < = 51 and 5 = &) Thus, we have

where the suffixes q and 4 denote, respectively, the r)- and +-components of the fields These equations must hold for all allowed values of 0 < 4 < 27r - - and -1 < q < 1 - -

5.3.2 TE Polarization for Nonaxial Incidence

The same method is applied as for the conductor case in Chapter 4 This means that making use of the orthogonality of the trigonometric fu

and spheroidal angular functions, and integrating over q, we will get

nctions

and

120 EM SCATTERING BY A COATED DIELECTRIC SPHEROID

where

(R > a =

CR ) b =

’ -X$(Q)) -x:,, (co) -xC4) m2 (co)

RELAT/ONSH/P BETWEEN /NC/DENT AND SCATTERED FIELDS 121

122 EM SCATTERING BY A COATED DIELECTRIC SPHEROID

(R > 9 =

CR > h =

f -q&2, 1 -V$(c2) -x(‘) (c2) 1 -v(l) (c2) ml ml

I

RElATIOfUSHlP BETWEEN lNClDENT AND SCATTERED FIELDS 123

(Qa) = km _ _ i - _

y~~o(co)

Y i% (co)

B (1) cmow

B (1) cm1 hd

y(l) a10 (co 7 Sl)

(1) ya,,bh)

cY,,,+l -

P m+l,m+l E’

P m+l,m+2 E’

- YEI E’

Yon,m+l

bEf m+l,m+l liEI m+l,m+2

124 EM SCATTERING BY A COATED DIELECTRIC SPHEROID

SE - m2 -

XE m,m+l F”

- PiX’+l.,m+l Pit’+l,m+2

-

oE’ mm E’

cTm,m+l

- E’

7,+l,m+l E’

7 PoEd

POE;

- E’

700 E’

701

k m = 2 for nz = 0 and km = 1 for nz > 0, and

The column matrices Am and A+ are the same as those defined for the conductor case in Chapter 4 0 is the zero matrix Besides the row matrices used in Chapter 4, the other row matrices are defined as follows:

126 EM SCATTERING BY A COATED DIELECTRIC SPHEROID

x( )

B 0 sN i = (,(i) NO & SE; ) 7 (5.20)

& - - 2(I es N n- &~sNn)

Here CO = <I and/or 52 The integrals used are defined as follows:

RELATIONSHlP BETWEEN INCIDENT AND SCATTERED FIELDS 127

128 EM SCATTERING BY A COATED DIELECTRIC SPHEROID

The functions Bm+2 m+n+2, Cm+2 m+n+2, and Gm+2 m+n+2 are obtained from their counterparts (with subscripts m and nz + A) by substituting m bym+Zandm+nbym+n+Z

Equation (5.11) can be written as

SE m = (G$)IE, m=O,l,Z ,

where (GE) = (RE)-l(QE) re resents the transformation matrix to trans- p form the incident vector 1: = Am(cos 00)-l into the column vector of the unknown scattering and transmitting coefficients We also note that (GE)

is dependent only on the scattering body, which is the same as that of the conducting body

5.3.3 TM Polarization for Nonaxial Incidence

Using the same principles as before, we will get for the TM case,

RELATIONSHIP BETWEEN INCIDENT AND SCATTERED FIELDS 129

matrices which are required are

cyll M-’

Q12 - M-’

711 M-’

12

M-’

Oil M-’

O12

#)(c) iN = (p) (c) g(c) NO @Jc) ) 7 where

g&(c) - -

(5.25)

130 EM SCATTERlNG BY A COATED DIELECTRIC SPHEROlD

f (({’ - 1)‘/2$Rj:),,, IJllVn

c=to R(i)

1,1+n )I IJllVn,

t=t0

(0 = 51, and/or <2 and Xg, (c) is a row matrix defined in Chapter 4

As in the TE case, we can write Eq (5.23) in the form

SM - m - (G,M)I,M, m=0,1,2 , , where (GE) = (Rg)-‘(Q,M) and 1: = Am To obtain (GE), we note that (Q,M) is readily obtained from (QE), and the inverted matrices (RE)-l are the same as those for (GE) Thus, the matrices obtained for (GE) will readily yield (G,M) an vice versa, which provides great convenience in computation d

of scattered fields at both polarizations This is the same situation that we encountered previously in the conducting case

5.3.4 Fields at Axial Incidence

In the case where the propagating field is incident along the axis of 80 = 0, the solution becomes much simpler by virtue of Eq (4.6) Assuming that the electric field is polarized along the positive y-axis and magnetic field along the positive z-axis, the following would be obtained:

NUMERICAL COMPUTATION AND MATHEMATICA SOURCE CODE 131

Unlike the conducting case, the truncation number required to obtain a given accuracy in the computed cross sections depends on a number of factors: for example, the electrical size, the properties of the coating and the core material, and the thickness of the coating It is very difficult to come up with a rule that can take all these factors into account As such, for the sizes and permittivities of the spheroids and coatings considered here, it was found accurate enough to consider only using m = 0, 1, 2 and 72 = m, m + 1,

nz + 5 The reason is that due to the dimensions of the matrices involved, it takes much longer to compute the system for a given m than in the conducting case Moreover, the convergence has been verified by computing until m = 4 for most cases In all instances, the results matched at least to four significant figures and in most cases to five significant figures This ensures that the scheme of truncation above is proper

Three Mathematics packages were written for the coated dielectric spher- oid The first is Coatedback.nb, which contains the user module Coat- edback [a2-, ratio2-, thickness-, epsilonrl,, epsilonr2J for computing and plotting the normalized backscattering cross section The second pack- age, Coatedbistatic.nb, contains the module Coatedbistatic[a2-, ratio2,, thickness-, epsilonr l-, epsilonr2,, thetaO-, phi-] for the computing and plotting the normalized bistatic cross section at nonaxial incidence The last

tiofl,, thickness,, epsilonrl-, epsilonr2-1, is developed for the bistatic cross section at axial incidence Note that the formulas for the various cross sections are the same as those given for the conductor case

To compute the cross sections, the user needs to load the relevant packages and call any of the three modules The arguments a2 and ratio2 denote the semimajor axis length (in terms of X0) and the axial ratio of the core, respectively The argument thickness denotes the thickness of the dielec- tric coating (also in terms of &), while epsilonrl and epsilonr2 are the relative permittivities of the two regions (see Fig 5.1) In addition, Coat- edbistatic.m computes the bistatic cross section for a user-specified angle of incidence (thetao-) at a user-specified azimuthal angle of observation (phi-) For all three cases, the output is formatted in the same way as the conducting case

The packages mentioned above are very similar in their structures, except for the last part, when they compute the different scattering cross sections

So they shall be described as one entity The program flow in the packages can be summarized as follows:

132 EM SCATTERING BY A COATED DlELECTRlC SPHEROID

3 The spheroidal radial functions of the first, second and fourth kinds and their first and second order derivatives are evaluated next All of these are done with the following supporting modules:

5 The matrices for the TE and TM cases are then computed using the

“ Table” command

6 Once the scattering coefficients are obtained, they are substituted into the relevant expressions for the scattering cross sections, and a plot is generated with the corresponding title and label

One advantage of the source code for coated dielectric spheroids is that it allows us to model the scattering by homogeneous dielectric spheroids This

is done by letting the relative permittivity of the coated layer be unity Fig- ure 5.2 shows the variation of the bistatic cross sections (at an axial incidence

of 60 = 0) of two dielectric spheroids of the same axial ratios and permittivities but different semimajor axis lengths The solid circles represent the results from Asano and Yamamoto [24] The close agreement once again verifies the accuracy of the Mathematics source code used

In Fig 5.2, for the smaller spheroid (top), the scattering intensity is al- most a constant in the plane perpendicular to the plane of polarization of the incident wave (H-plane), whereas the scattering in the plane parallel to the plane of polarization (E-plane) is approximately proportional to cos2 0 This behavior is similar to that of the Rayleigh limit for the scattering by very small spheres, whose radii are much smaller than for the incident wave- length In such cases (Rayleigh scattering), the intensity is a constant in the plane perpendicular to the plane of polarization and varies as cos2 0 in the parallel plane [132] With the increase in size (bottom), the magnitude of the scattered intensity increases, and the forward scattering amplitude becomes much greater than the backscattering amplitude Again, this is similar to the Mie effect observed in scattering by spheres [133] In addition, the patterns

in power intensity distributions become more and more complicated, showing oscillating fluctuations with 0

RESULTS AND DISCUSSION 133

134 EM SCATTERING BY A COATED DIELECTRIC SPHEROID

Figures 5.3 and 5.4 show the variation of the bistatic cross sections with different angles of incidence for a dielectric spheroid For these cases, the plane of observation is taken to be the X, z-plane The cross sections for the

TE mode do not change dramatically as the incident angle varies, but the minimum in the TM scattering continues to move toward the other end of the spheroid as the incident angle increases and very strong backscattering is observed The maximum in the TM scattering also increases, as the surface area available for scattering reaches a maximum at t90 = 90’ This is similar

to what was observed for a conducting spheroid in Chapter 4

Figure 5.5 illustrates the difference between a lossy dielectric and lossless dielectric on the backscattering cross sections Notice that the shape of the cross sections depends mainly on the dimensions of the spheroid Since both spheroids are of the same axial ratio and semimajor axis, they have roughly the same shape A smaller amplitude is observed from the lossy one as expected

Figure 5.6 depicts the Mie scattering effect and the effect of dimensions on the bistatic cross sections of dielectric spheres The results were compared with those (in bullets) by Van de Hulst [134] In both cases, the forward- scattering cross section amplitude is larger than the backscattering amplitude For a bigger sphere, the scattered intensity increases, and the pattern exhibits more oscillations [ 1331

Figure 5.7 shows the bistatic cross sections of a single dielectric spheroid with a confocal coating for two values of the axial ratio The results were compared with those (in bullets) by Cooray and Ciric [33] Here we observe that the variation of both the E- and H-plane patterns is almost the same for the two cases There are signs of Rayleigh scattering, but the Mie scattering effect is also coming into play The difference between the two cases is that the magnitudes of scattering cross sections are smaller for the “thinner” spheroid (axial ratio of 5) than for the “fatter” spheroid This is because the area available for scattering is less when the axial ratio is 5 than that when it is 2

Figure 5.8 illustrates two different backscattering cross sections of two coated spheroids of the same type of material but of totally different dimen- sions The one at the bottom has a much thicker coating relative to the inner core, and it is also fatter (axial ratio of 2) than the one at the top (axial ratio of 5) The one with the thinner coating exhibits a TM backscatter- ing that is always higher than the TE backscattering For the one with the thicker coating, the TM backscattering starts off initially larger than the TE backscattering, but the TE backscattering increases steadily after 00 = 45” until it is largest at broadside incidence (00 = 900)

Here, the bistatic cross sections for dielectric-coated spheroids of oblique angles of incidence were also computed and discussed Figures 5.9 and 5.10 show the results for a coated dielectric spheroid, which is basically the product

of adding a confocal coating of thickness 0.02& to the dielectric spheroid of Figs 5.3 and 5.4 As such, the cross sections for both cases are almost identical