1
Scattering in the long wavelength limit
~ inc = ~"0 E 0 eik^n 0¢~x is incident on a small scattering
Suppose a plane wave with E
object with dimension d ¿ ¸: The incoming wave induces electric and magnetic
dipole monents in the scattering object, which then radiates.
The scattered ¤elds are
2 ikr
~ sc = k e
E
[(^n £ ~p ) £ n
^¡n
^ £ m]
~
r
and
~ sc = n
~s c
B
^£E
The power radiated into direction n
^ with polarization ~" per unit solid angle is
dP
c ¯¯ ¤ ~ ¯¯ 2
= r2
¯~" ¢ E sc ¯
d4¼
and the di¤erential scattering cross section is
d¾
(^
n; ~"; n
^ 0 ;~"0 ) =
d=
=
¯
¯2
c ¯ ¤ ~ ¯
r 2 4¼
¯~" ¢ E sc ¯
¯
¯2
¯
c ¯ ¤ ~
~
"
¢
E
¯
¯
in
c
0
8¼
k4 ¤
2
j~" ¢ [(^
n £ p~) £ ^n ¡ ^n £ m
~ ]j
E 02
k4 ¤
2
j~" ¢ p~ + (^n £ ~"¤ ) ¢ m
~j
E 02
1
(1)
¤
where we used the fact that ~" ¢ n
^ = 0; and we rearranged the triple scalar
product on the right.
The result shows that the di¤erential scattering cross section is proportional
to ¸ 4 : This is Rayleighzs law. It applies to all scattering in the long-wavelength
limit.
Example: scattering by a small conducting sphere, radius a.
Since the sphere is small (a ¿ ¸) it sees the incident ¤eld as slowly varying.
The sphere can adjust to the electric ¤eld in a time t » a=c ¿ ¸=c = T ; the
wave period. Thus we may use the results for the static ¤elds from chapter 3.
With polar axis along ~"0 ; the potential is
µ
¶
a3
Á = ¡E 0 r ¡
cos µ
r
The ¤rst term is the incident ¤eld and the second, dipole term is the ¤eld due
to the charge distribution on the surface of the sphere. The dipole moment is
~ 0 a3 : We may also ¤nd the scalar magnetic potential. The boundary
~p = E
condition is Br = 0 at r = a; giving
µ
¶
a3
ÁM = ¡B0 r +
cos µ
2r
~ 0 ; and hence
where here the polar axis is along B
µ
¶
a3
Br = B 0 1 ¡ 3 cos µ
r
which is clearly zero at r = a: Again the ¤rst term is the incident ¤eld, and the
seo cnd term is a dipole ¤eld. Here the magnetic moment is
m
~ =¡
~ 0 a3
B
E0 a3
=¡
(^
n0 £ ~"0 )
2
2
We may now put these moments into the general result (1):
d¾
d-
=
=
¯
¯2
¯
k4 2 6 ¯¯ ¤
1
¤
¯
E
a
~
"
¢
~
"
¡
(^
n
£
~
"
)
¢
(^
n
£
~
"
)
0
0
0
2 0
¯
¯
E0
2
¯
¯2
¯
¯
1
(ka) 4 a2 ¯¯~" ¤ ¢ ~"0 ¡ (^n £ ~"¤ ) ¢ (^
n0 £ ~" 0 )¯¯
2
The vectors n
^ 0 and ^n de¤ne the plane of scattering. ^" is perpendicular to ^n
and ~"0 is perpendicular to ^n0 : We choose polarization vectors ~"0;1 and ~"1 in the
plane of scattering. Thus~" 1 makes angle µ with ~"0;1 : Similarly ~"0;2 and ~" 2
are perpendicular to the plane of scattering and parallel to each other. For
scattering of unplarized incident radiation into polarization ~" 1; in the plane of
2
scattering, we have
@¾ k
d-
(¯
¯2 ¯
¯2 )
¯ ¤
¯
¯
¯
1
1
¯~"1 ¢ ~"01 ¡ (^n £ ~"¤1 ) ¢ (^
n0 £ ~"01 )¯¯ + ¯¯~"¤1 ¢ ~"02 ¡ (^n £ ~"¤1 ) ¢ (^
n0 £ ~"02 )¯¯
¯
2
2
(¯
)
¯2
(ka)4 a2 ¯¯
1 ¯¯
¯ cos µ ¡ 2 ¯ + 0
2
4
=
=
(ka) a2
2
For scattering intopolarization ~" 2 ;
(¯
¯2 ¯
¯2 )
¯
¯ ¤
¯
@¾ ?
(ka) 4 a2 ¯¯ ¤
1
1
¤
¤
=
~"2 ¢ ~"01 ¡ (^
n £ ~"2 ) ¢ (^
n0 £ ~"01) ¯¯ + ¯¯~"2 ¢ ~"02 ¡ (^
n £ ~"2 ) ¢ (^
n0 £ ~"02 )¯¯
¯
d2
2
2
(¯
)
¯
¯
¯
4
¯2 ¯
¯2
(ka) a2 ¯¯
1 ¤
¯ + ¯ 1 ¡ 1 (¡~"¤ ) ¢ (¡~"01 )¯
=
0
¡
~
"
¢
~
"
02
1
¯
¯
¯
¯
2
2 1
2
(
)
¯
¯2
4
¯
¯
(ka) a2
1
=
0 + ¯¯ 1 ¡ cos µ ¯¯
2
2
The sum of the two gives the di¤erential scattering cross section
d¾
d-
4
=
=
The polarization is
(ka) a2
2
µ
1
cos 2 µ
+ 1 ¡ cos µ +
4
4
·
¸
¡
¢
5
(ka)4 a2
cos 2 µ + 1 ¡ cos µ
8
¦ (µ) =
=
=
=
d¾?
dd¾?
d-
³
cos 2 µ ¡ cos µ +
¡
+
1 ¡ cos µ +
¡
3
4
5
4
@ ¾k
d@ ¾k
dcos2 µ
4
5
4
´
¡
¢
¡ cos2 µ ¡ cos µ + 14
(cos2 µ + 1) ¡ 2 cos µ
¢
1 ¡ cos2 µ
(cos2 µ + 1) ¡ 2 cos µ
3 sin2 µ
5 (cos2 µ + 1) ¡ 8 cos µ
3
¶
2
1.5
1
0.5
0
0.5
1
1.5
2
2.5
3
The dashed line is the di¤erential scattering cross section, the solid line is the
polarization. The polarization peaks at µ = ¼=3: The scattering peaks at µ = ¼
(backward scattering) and is minimum at µ = ¼=5:
4