Canonical Problems in Scattering and Potential Theory Part II: Acoustic and Electromagnetic Diffraction by Canonical Str

315 Pages·2002·2.54 MB·English

by  S.S. Vinogradov

194 4. Electromagnetic Di(cid:11)raction froma Metallic Spherical Cavity.
1 q (1)
qTM (2)
0.9 TM
(s) M
qT0.8
N ,
TIO0.7
U
B
RI0.6
T
N
O
C0.5
Y
G
ER0.4
N
E
C 0.3
NI
O
M0.2
R
A
H
0.1
0
0 2 4 6 8 10 12 14 16 18 20
ka
Æ
FIGURE 4.3. Excitation of the cavity ((cid:18) = 30 ) by a electric dipole located in
z<0 with q=d=a=0:9. First (solid) and second (dashed) harmonic terms.
4.1.2 The Vertical Magnetic Dip ole (TE Case)
Letusnowconsidertheverticalmagneticdipolereplacingtheverticalelec-
tric dipole described at the beginning of Section 4.1. Taking into account
the continuity condition and the far-(cid:12)eld behaviour of the scattered (cid:12)eld,
we decompose the total (cid:12)eld as the sum
(0) s
E(cid:30) =E(cid:30) +E(cid:30);
and seek the scattered (cid:12)eld in the form
2 1
s mk 1 (cid:16)n(ka) n(kr); r <a
E(cid:30) = ynPn(cos(cid:18)) (4.50)
(cid:0) r n=1 (cid:26) n(ka)(cid:16)n(kr); r >a
X
where the unknown coeÆcients yn are to be found. The (cid:12)nite energy con-
dition(4.6)e(cid:11)ectively de(cid:12)nedthesolutionclassforthecoeÆcient sequence
1
yn n=1: Takingthe integration region in (4.6) to be the sphere of radius
f g
a, it is easily deduced that
2 4
W =WTE =2(cid:25)m k a
(cid:2)
1
n(n+1) 2 2 02 n(n+1) 2
yn (cid:16)n(ka) n (ka) 2 1 n(ka)
n=1 2n+1 j j j j ( (cid:0)" (ka) (cid:0) # )
X
(4.51)
4.1 Electric or Magnetic Dipole Excitation. 195
mustbe(cid:12)nite.UsingtheasymptoticbehaviourofthesphericalBesselfunc-
tions and their derivatives (see equations [4.10]) it can be readily shown
that
1
2
WTE C2(ka) yn (4.52)
(cid:20) j j
n=1
X
where C2(ka) is a function of ka alone, with (cid:12)nite value. Thus the coeÆ-
1
cient sequence yn n=1 is square summableand belongs to the functional
f g
space l2.
Now enforce the boundary conditions (4.3){(4.4), bearing in mind the
explicitformofthe scattered (cid:12)eld (4.50),the incident (cid:12)eld(1.213)and the
relationsbetween the basic (cid:12)eld componentE’ and the other (cid:12)eld compo-
nents (1.90). As a result, one obtains the followingdual series equations.
1
1
ynPn(cos(cid:18))=0, (cid:18) (0;(cid:18)0) (4.53)
2
n=1
X
1
1
yn n(ka)(cid:16)n(ka)Pn(cos(cid:18))=
n=1
X 1
(cid:0)2 1
(kd) (2n+1) n(kd)(cid:16)n(ka)Pn(cos(cid:18)), (cid:18) ((cid:18)0;(cid:25)): (4.54)
(cid:0) 2
n=1
X
The (cid:12)rst step is to identify a suitable asymptotically small parameter.
Thus de(cid:12)ne
i 4n(n+1)
(cid:22)n =1 n(ka)(cid:16)n(ka): (4.55)
(cid:0) ka 2n+1
(cid:0)2
It is readily veri(cid:12)ed from the asymptotics (4.10) that (cid:22)n = O n , as
n : The dual series equations (4.53){(4.54)maybe rearranged as
!1 (cid:0) (cid:1)
1
1
ynPn(cos(cid:18))=0, (cid:18) (0;(cid:18)0) (4.56)
2
n=1
X
1
2n+1 1
(1 (cid:22)n)ynPn(cos(cid:18))
n(n+1) (cid:0)
n=1
X 1
2n+1 1
= (cid:12)nPn(cos(cid:18)), (cid:18) ((cid:18)0;(cid:25)) (4.57)
n(n+1) 2
n=1
X
196 4. Electromagnetic Di(cid:11)raction froma Metallic Spherical Cavity.
where
4
(cid:12)n = i 2n(n+1) n(kd)(cid:16)n(ka): (4.58)
(cid:0) ka(kd)
Because the unknowncoeÆcient sequence issquare summable,itcan be
readily estimated that the general term of series (4.56) and (4.57) decay
1 3
(cid:0)2 (cid:0)2
at rates O n and O n ; respectively, as n : Thus the (cid:12)rst
! 1
series conve(cid:16)rges u(cid:17)niformly(cid:16),whil(cid:17)st the second converges nonuniformly.The
diÆculty is circumvented in the same way as for the VED case. Set
1 d
Pn(cos(cid:18))= Pn(cos(cid:18))
(cid:0)d(cid:18)
1
and then validlyintegrate term by term (because yn n=1 l2) to obtain
f g 2
thefollowinguniformlyconvergingseries towhichAbelintegraltransforms
maybe applied,
1
ynPn(cos(cid:18))=C2; (cid:18) (0;(cid:18)0) (4.59)
2
n=1
X
where C2 is a constant of integration.
We now use the integral representations (4.19) of Mehler-Dirichlet kind
to obtain the equivalent pre-regularised form
1 1 C12cos12(cid:18); (cid:18) 2(0;(cid:18)0)
yncos n+ (cid:18) = 1
nX=1 (cid:18) 2(cid:19) 8< n=1((cid:12)n+(cid:22)nyn)cos n+ 2 (cid:18); (cid:18) 2((cid:18)0;(cid:25)):
P (cid:0) (cid:1) (4.60)
:
The constant C2 formally arising as the result of integration has an
interpretation that parallels that given to the constant C1 in the previ-
ous section. The original dual series equations were series expansions in
1
the complete orthogonal family of functions Pn(cos(cid:18)) (n=1;2;:::); the
transformed equations are series expansions in another complete orthog-
1 1
onal family of functions, the trigonometric functions cos n+ 2 (cid:18) n=0;
the unknown coeÆcients yn are associated with an incomplete subset of
(cid:8) (cid:0) (cid:1) (cid:9)
this family (indexed by n = 1;2;:::); and the constant C2 is associated
1
with the zero index element (cos 2(cid:18)) that makes this family of functions
complete.
Makinguse oforthogonality,the equations (4.60)are easily transformed
to the i.s.l.a.e.of the second kind
1 1
(1) (1)
(1 (cid:22)m)ym+ yn(cid:22)nQnm((cid:18)0)=(cid:12)m (cid:12)nQnm((cid:18)0); (4.61)
(cid:0) (cid:0)
n=1 n=1
X X
4.1 Electric or Magnetic Dipole Excitation. 197
(1)
where m=1;2;::: and Qnm((cid:18)0) was de(cid:12)ned in the previous section.
Introducing the angle (cid:18)1 = (cid:25) (cid:18)0 leads to the alternative and more
(cid:0)
compact form(recall [4.24])
1 1
(1) (1)
Ym Yn(cid:22)nRnm((cid:18)1)= BnRnm((cid:18)1); (4.62)
(cid:0)
n=1 n=1
X X
where m=1;2;::: and
m m
Ym =( 1) ym; Bm =( 1) (cid:12)m: (4.63)
(cid:0) (cid:0)
Equations (4.61) or (4.62) remain valid when d < a and the dipole is
locatedinupper half-spacez >0,providedthe coeÆcients (cid:12)n arereplaced
by
4
(cid:12)n = i 2n(n+1)(cid:16)n(kd) n(ka): (4.64)
(cid:0) ka(kd)
If the dipole is located in lower half-space (z < 0) then the values of (cid:12)n,
de(cid:12)ned by (4.58) or (4.64), are modi(cid:12)ed by a simple multiplication with
n(cid:0)1
the factor ( 1) .
(cid:0)
Letuscollectthenear-(cid:12)eldandfar-(cid:12)eldcharacteristics. Thesurface cur-
rent density has one component and is equal to the jump in the magnetic
(cid:12)eld intensity across the shell,
2 1
mk 1
j(cid:30) = H(cid:18)(a 0;(cid:18)) H(cid:18)(a+0;(cid:18)) = ynPn(cos(cid:18)): (4.65)
(cid:0)f (cid:0) (cid:0) g (cid:0) a
n=1
X
v ikr
The radiation pattern S2 de(cid:12)ned by E(cid:18);H(cid:30) S2((cid:18))e =r as r
! 1
equals
1
2 n(kd) n(ka) n(cid:0)1 1
S2((cid:18))=mk (2n+1) 2 + yn ( 1) Pn(cos(cid:18)):
n=1 ( (kd) 2n+1 ) (cid:0)
X
(4.66)
The radiation resistance equals
1 2
3 n(kd) n(ka)
R= n(n+1)(2n+1) 2 + yn : (4.67)
2n=1 (cid:12) (kd) 2n+1 (cid:12)
X (cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
198 4. Electromagnetic Di(cid:11)raction froma Metallic Spherical Cavity.
Finally the stored external energy (see the discussion in the previous
section) is
1
(e) 2 4 n(n+1) 2 2
WTE =2(cid:25)m k a yn n(ka)
2n+1 j j (cid:2)
n=1
X
n(n+1) 2 0 2
2+ 2 1 (cid:16)n(ka) (cid:16)n(ka) :
( " (ka) (cid:0) #j j (cid:0) )
(cid:12) (cid:12)
(cid:12) (cid:12)
The totalstored energy isthe sumofscattered energy thatis stored inthe
internal (r<a) and external (r>a) regions
i (e)
WTE =WTE +WTE, (4.68)
i
where the value of WTE is de(cid:12)ned by formula(4.51).
InanalysinghighQ-factoroscillationsthatdevelopinthisopenspherical
resonator it we maydecompose, as in the TM case,
1
(n)
WTE = WTE (4.69)
n=1
X
intoasumoftermsrepresentingthenormalisedenergycontributionofeach
harmonic,and to examine the ratios
(n)
(n) WTE
qTE = . (4.70)
WTE
Before presenting numericalresults let us consider analyticalfeatures of
the solution obtained above. First, a careful look at the expressions for
surface current density(4.65)reveals thatgeneral termsofthe series decay
1
(cid:0)2
at the rate O n as n ,i.e.,the series is slowlyconverging,and at
!1
a rate that is(cid:16)rathe(cid:17)r slower than the corresponding series for the VED.
Acceleration ofconvergence is achieved byrearrangement ofthe i.s.l.a.e.
(4.62) in the form
1
(1)
Yn = (Ys(cid:22)s+Bs)Rsn ((cid:18)1): (4.71)
s=1
X
n
Setting Yn = ( 1) yn into formula (4.65) and interchanging the order of
(cid:0)
summationproduces thecomputationalformulaforsurface current density
2 1
mk
j’ = (Ys(cid:22)s+Bs)Ls(#;(cid:18)1); (4.72)
a
s=1
X
4.1 Electric or Magnetic Dipole Excitation. 199
where #=(cid:25) (cid:18) and
(cid:0)
1
(1) 1
Ls(#;(cid:18)1)= Rsn ((cid:18)1)Pn(cos#): (4.73)
n=1
X
This function maybe expressed in the form
Rs0((cid:18)1)
Fs(#;(cid:18)1) 1(cid:0)R00((cid:18)1)F0(#;(cid:18)1);# (0;(cid:18)1)
Ls(#;(cid:18)1)= (cid:0) (cid:0) 2 (4.74)
( 0; # ((cid:18)1;(cid:25))
2
where
@
Fs(#;(cid:18)1)= Gs(#;(cid:18)1) (4.75)
@#
and Gs was de(cid:12)ned by (2.178).The function Fs(#;(cid:18)1) maybe easily com-
puted fromthe recurrence formula
1
2 cos s 2 (cid:18)1 2s 1
Fs(#;(cid:18)1)= sin# (cid:0) (cid:0) sin#Gs(cid:0)1(#;(cid:18)1)
(cid:25)s 2(co(cid:0)s# c(cid:1)os(cid:18)1) (cid:0) s
(cid:0)
2s 1 s 1
+ p(cid:0) cos#Fs(cid:0)1((cid:23);(cid:18)1) (cid:0) Fs(cid:0)2(#;(cid:18)1); (4.76)
s (cid:0) s
validfor s=1;2;:::, initialised by the elementary expression for F0
1
2 tan 2#
F0(#;(cid:18)1)=
(cid:0)(cid:25) 2(cos#(cid:0) c(cid:1)os(cid:18)1) (cid:2)
(cid:0)
1 1
sin(cid:18)1 sin2(cid:18)1+p 2(cos# cos(cid:18)1) +cos 2(cid:18)1(cos# cos(cid:18)1)
(cid:0) (cid:0)
: (4.77)
n1+cos# p2cos(cid:18)1+2sin12(cid:18)1o 2(cos# cos(cid:18)1)
(cid:0) (cid:0)
p (cid:0)25
Theconvergence rate ofthe modi(cid:12)edseries (4.72)isO s ass .
!1
More importantly,the vanishing of Ls(#;(cid:18)1) when # ((cid:16)(cid:18)1;(cid:25))(cid:17)shows that
2
the boundary condition (4.3) is satis(cid:12)ed term by term; using (4.76) and
(4.77), analysis of the form (4.74) of Ls(#;(cid:18)1) in the interval # (0;(cid:18)1)
2
reveals that the surface current j’ has the expected singular behaviour as
(cid:18) (cid:18)0 (or # (cid:18)1): each term in (4.72) has the correct behaviour (a sin-
! ! (cid:0)12
gularityof form(cos# cos(cid:18)1) ) in the vicinity ofthe sharp edge. Thus
(cid:0)
accurate calculationofsurface current density distributionisfacilitatedby
this transformationto a muchmore rapidly convergent series.
Now consider the quasi-eigenoscillationsthat develop inthe open spher-
ical cavity with a small aperture ((cid:18)0 1) when the excitation frequency
(cid:28)
coincides with one of the quasi-eigenvalues of spectrum (cid:23)sl. The spectrum
200 4. Electromagnetic Di(cid:11)raction froma Metallic Spherical Cavity.
s=1 2 3 4 5 6 7
l=1 4:493 5:763 6:988 8:183 9:356 10:513 11:657
2 7:725 9:095 10:417 11:705 12:967 14:207 15:431
3 10:90 12:323 13:698 15:040 16:355 17:648 18:923
(0)
TABLE4.2. TEls0-oscillations, spectral values (cid:23)sl
ofeigenvaluesofquasi-eigenoscillationsinthe closed cavity(ofTEls0-type)
(0)
is determined by the roots (cid:23)sl (l =1;2;:::) ofthe characteristic equation
(0)
s (cid:23)sl =0; (4.78)
(cid:16) (cid:17)
where s=1;2;:::; foreach s, the roots are indexed inincreasing order by
l. The indices s and l designate the numberof(cid:12)eld variationsinthe radial
and angular variables (r and (cid:18)), respectively. Some lower order roots are
shown in Table 4.2.
Following the same idea used in Chapters 2 and 3, the perturbation to
(0)
the spectral value (cid:23)sl caused by a small aperture (with (cid:18)0 1) may be
(cid:28)
analyticallycalculated:
(0) 2s+1 (1)
(cid:23)sl=(cid:23)sl =1 Qss ((cid:18)0)
(cid:0) 4s(s+1) (cid:0)
1 (1) (1)
2s+1 (0) 2n+1 Qns ((cid:18)0)Qsn ((cid:18)0)
i (cid:23)sl 2 : (4.79)
16s(s+1) n(n+1) (0)
nnX=6=1s (cid:16)n (cid:23)sl
(cid:12) (cid:16) (cid:17)(cid:12)
(cid:12) (cid:12)
(0) (cid:12) (cid:12)
Thus Re((cid:23)sl) < (cid:23)sl and Im((cid:23)sl) < 0, indicating radiative losses with
Q-factor
TE Re((cid:23)sl)
Qls0 = 2 (4.80)
(cid:0) Im((cid:23)sl)
of (cid:12)nite value.
Providing that (2n+1)(cid:18)0 1 and (2s+1)(cid:18)0 1, the angle functions
(1) (1) (cid:28) (cid:28)
Rns ((cid:18)0) and Qns ((cid:18)0) may be approximated by formulae (4.41){(4.41),
and so for small apertures ((cid:18)0 1), the followingseries in (4.79) may be
(cid:28)
approximated
1 (1) (1) 2 2 1
2n+1 Qns ((cid:18)0)Qsn ((cid:18)0) 4s (s+1) n(n+1)(2n+1) 10
2 2 2 (cid:18)0
n(n+1) (0) ’ 2025(cid:25) (0)
nnX=6=1s (cid:16)n (cid:23)sl nnX=6=1s (cid:16)n (cid:23)sl
(cid:12) (cid:16) (cid:17)(cid:12) (cid:12) (cid:16) (cid:17)(cid:12)
(cid:12) (cid:12) (cid:12) (cid:12) (4.81)
(cid:12) (cid:12) (cid:12) (cid:12)
4.2PlaneWaveDi(cid:11)ractionfromaCircularHoleinaThinMetallicSphere. 201
and hence
TE (cid:0)10
Qls0 (cid:18)0 : (4.82)
(cid:24)
The above formula is of restricted interest. For accurate values of the
Q-factor for larger apertures, it is necessary to (cid:12)nd the complex roots of
the full characteristic equation, that may by approximated by the matrix
equation of order N (N must be chosen suÆciently large),
det IN M +B =0 (4.83)
f (cid:0) g
where IN is the identity matrix of order N, M is the diagonal matrix
diag((cid:22)1;(cid:22)2;(cid:22)3;:::;(cid:22)N); and B is the square matrix
(1) (1) (1) (1)
(cid:22)1Q11 (cid:22)2Q21 (cid:22)3Q31 ::: (cid:22)NQN1
(1) (1) (1) (1)
0 (cid:22)1Q12 (cid:22)2Q22 (cid:22)3Q32 ::: (cid:22)NQN2 1
(1) (1) (1) (1)
B = (cid:22)1Q13 (cid:22)2Q23 (cid:22)3Q33 ::: (cid:22)NQN3 : (4.84)
B C
B ::: ::: ::: ::: ::: C
B (1) (1) (1) (1) C
B (cid:22)1Q1N (cid:22)2Q2N (cid:22)3Q3N ::: (cid:22)NQNN C
B C
@ A
Finally,inparallelwithequation(4.49),theloadedQ-factorofaspherical
open resonator is given by
TE 3
Qls0 = ka WTE=R: (4.85)
2 (cid:1)
By way of illustration, the frequency dependence of the stored internal
(i) (e)
and external energies (WTE and WTE), respectively, is presented in Fig-
Æ
ure 4.4 for the open spherical resonator with parameter (cid:18)0 = 30 excited
by a vertical magnetic dipole located in the lower half-space (z < 0) with
q = d=a = 0:9. As we have already seen in the electric dipole case, the
energy stored outside the resonator is very smallcompared to that stored
(s)
internally.The value of qTE, (s=1;2),is displayed in Figure 4.5,showing
that at least for the (cid:12)rst three resonances (solid line) the energy contribu-
tion of the (cid:12)rst harmonics exceeds 95%.
4.2 Plane Wave Di(cid:11)raction from a Circular Hole in
a Thin Metallic Sphere.
The previous section considered vertical dipole excitation of the spherical
cavity with a circular hole. If the dipole is located along the vertical axis
of symmetry, only TM or TE waves are excited depending upon whether
202 4. Electromagnetic Di(cid:11)raction froma Metallic Spherical Cavity.
104
W
i
W
e
103
W
Y , 102
G
R
E
N
E
ED 101
R
O
T
S
100
10−1
2 4 6 8 10 12 14 16 18 20
ka
FIGURE 4.4. Internal (solid) and external (dashed) stored energy for cavity
Æ
((cid:18)=30 )excited by a magnetic dipole located in z<0 with q =d=a=0:9.
1
q (1)
qTE (2)
0.9 TE
(s) E
qT0.8
N ,
TIO0.7
U
B
RI0.6
T
N
O
C0.5
Y
G
ER0.4
N
E
C 0.3
NI
O
M0.2
R
A
H
0.1
0
0 2 4 6 8 10 12 14 16 18 20
ka
FIGURE 4.5.Harmonic energy fractions for dipole excited cavity of Figure 4.4.
4.2PlaneWaveDi(cid:11)ractionfromaCircularHoleinaThinMetallicSphere. 203
a) z b) z
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)
y y
o a o
x (cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1) x
n n
H H
E E
FIGURE 4.6. The spherical cavity illuminated by a plane wave a) at normal
incidence (cid:11)=0 and b) with (cid:11)=(cid:25).
the dipole is of electric or magnetic type. However, when the cavity is
illuminated by an electromagnetic plane wave the TM and TE waves are
coupled even if the wave propagates normallyto the aperture plane. This
more complex cavity scattering scenario is addressed in this section (see
also [99], [111]).
Consider then a thin metallic sphere with a circular hole irradiated by
a plane electromagnetic wave, as shown in Figure 4.6a. Thez-axis is nor-
malto the aperture plane and the direction of propagation coincides with
positive z-axis. The incident (cid:12)eld is described by
0 0
Ex = Hy =exp(ikz)=exp(ikrcos(cid:18)) (4.86)
(cid:0)
where we have employed the symmetrised form of Maxw ell’s equations
(1.58){(1.59).
0 0
According to (1.243) the associated electric (U ) and magnetic (V )
Debye potentials are
1
0
U cos’ 1 1
V0 = sin’ ik2r An n(kr)Pn(cos(cid:18)), (4.87)
(cid:26) (cid:27) (cid:26) (cid:27) n=1
X
where
n 2n+1
An =i :
n(n+1)
s s s s
Debye potentials (U , V ) associated with the scattered (cid:12)eld (cid:0)E!;(cid:0)H!
must satisfy the Helmholtz equation. Furthermore, the total (cid:12)eld must