Prof. David R. Jackson
ECE Dept.
Spring 2009
Notes 27
ECE 6341
ECE 6341
1
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Scattering by a Sphere: Mie Series Solution and Backscattered Field - Prof. David Jackson, Study notes of Electrical and Electronics Engineering
An in-depth analysis of scattering by a sphere using the mie series solution. It covers the theory behind the solution, the backscattered field representation, and the behavior of the legendre functions. The document also includes formulas for evaluating the bessel functions and the total field.
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Prof. David R. Jackson
Spring 2009 ECE Dept.^ Notes 27
ECE 6341ECE 6341
Scattering by SphereScattering by Sphere
z
y
x
a
i H
PEC
PW
i E
^
^
0
0
i^
i
jkz^
jkz
E
E^
x E e
H^
y^
e
η
−^
−
=^
Scattering by Sphere (cont.)Scattering by Sphere (cont.)
Let
=^
=^
i^ +
s
r^
r^
r i^
s
r^
r^
r
A^
A^
A
F^
F^
F
l
(^ )
(^
)^
(^
)
l^
(^
)^
(^
)
2
1
0
0
(2)^
1
0
0
cos
cos
sin
cos
s^
n
r^
n^
n
n
s^
n
r^
n^
n
n
A^
E^
d H
kr
P
E
F^
e H
kr
P
φ
θ
φ
θ
η
∞ = ∞ =
∑ ∑
where Examine the field components to determine the boundary conditions. 4
Scattering by Sphere (cont.)Scattering by Sphere (cont.)
For example,
2
sin
θ^
ε^
θ^
φ^
ωμε
θ
∂^
=^
∂^
r^
r
F^
A
E^
r^
j^
r^
r
=^
r ∂
r^
S^
S
A
F^
r
Scattering by Sphere (cont.)Scattering by Sphere (cont.)
The exact solution to scattering by spherical particles is called theMie †^ Gustav Adolf Feodor Wilhelm Ludwig Mie (Sept. 29, 1869 – Feb. 13, 1957) was aGerman physicist.He was the first to publish the solution toscattering by a dielectric sphere.
†^ series solution (first published in 1908).
Backscattered FieldBackscattered Field
z
(^
)^
^
(^
)
,^
,^
,^
,
s^
s x
E^
r^
x E
r
π^
φ
π φ
=
Fromsymmetry,
y
x
a
PEC
θ^
scattered
incident
Backscattered Field (cont.)Backscattered Field (cont.)
( l
)^
(^
( l )
(^
)(^
)^
(^
2
1
0
0
2
1
0
0
cos
cos
sin 1
cos
sin
cos
s^
n n^
n
n
n n^
n
n
E
E^
e H
kr^
P
r
E^
d k H
kr^
P
j^
r
θ
φ
θ
ε^
θ η
φ
θ
θ
ωμε
∞ = ∞ =
′^
+^
∑ ∑
2
sin
θ^
ε^
θ^
φ^
ωμε
θ
∂^
=^
∂^
r^
r
F^
A
E^
r^
j^
r^
r
From the handout:
Low-Low
-Frequency ApproximationFrequency Approximation
( l
)^
(^
( l )
(^
)(^
)^
(^
2
1
0
0
2
1
0
0
cos
cos
sin 1
cos
sin
cos
s^
n n^
n
n
n n^
n
n
E
E^
e H
kr^
P
r
E^
d k H
kr^
P
j^
r
θ
φ
θ
ε^
θ η
φ
θ
θ
ωμε
∞ = ∞ =
′^
+^
∑ ∑
0
0
λ^
→ a Let
Keep the dominant term in theseries, which is the
n^
= 1 term
(^
) (^
)
1 0
0
P^
x^
=
See the formulas for
d^ n^
and
e^ n^
to verify that lower terms are more dominant.
Backscattered Field (cont.)Backscattered Field (cont.)
(^ )
(^
)
(^ )
(^
)
(^ )
(^
)
1
2
1
1
1
2
2
1 1 1
2
2
1 1
1
1
2 2 1
1 d
P^
x^
x^
P^
x dx d
P^
x^
x^
x^
x
dx x^
x
P^
x^
x^
x
= −
− = −
−^
= −
−
−^
−
′^
=^
= −^
− (^
)
1 1
2 cos
cos
cot
1
cos
P
′^
=^
=
−
(^
)
1 cos 1
sin
P
( = − )
(^
)^
(^
)^
(^
)
/ 2 2
1
1
m m
m
m n^
n d m
P^
x^
x^
P^
x
dx
=^
−^
−
Next, evaluate the Legendre functions:
so
Hence 13
Backscattered Field (cont.)Backscattered Field (cont.)
Hence,
(^
(^
0
1
0
1
~^
cos
sin
sin 1
cos
sin
cot
s^
jkr jkr
E
E^
e^
e
r
E^
d k
je
j^
r
θ
ε^
θ^
θ^
− −
−^
−^
+^
(^
0
1 0
1
~^
cos
cos
cos
s^
jkr
jkr
E
E^
e^
e
r
E d k
je
j^
r
θ
φ
ε^
η^
θ^
φ
ωμε
−
−
+^
or
Backscattered field (cont.)Backscattered field (cont.)
Next, examine the Bessel function terms as
ka
→
0 :
l^
(^
)^
(^
)
(^
)
1
1
3/ 2
3/ 2
3/ 2
~^
⎡^
⎢^
⎢^
⎛^
⎢^
⎜^
⎢^
⎝^
⎣^
J^
x^
x j
x x^
J^
x
x
x
x^
x
Backscattered field (cont.)Backscattered field (cont.)
Also l^
(^ )
(^ )^ (
(^ )
(^ )
(^
)^
(^ )
(^
(2)^
2
1
1
(2)3/ 2 3/ 2
3/ 2
2 ~^
(^22)
2
cos sin
2
H^
x^
x h^
x x^
H^
x x x^
jY^
x
x x^
j^
J^
x
x
x^
j
x^
j^
x
x^
x^
x^
x
π π π π^
− π
= =
−⎡
⎤
⎣^
⎦
=^
−^
⎡^
⎤
⎣^
⎦
⎡^
⎤
⎛^
⎞
=^
−^
−^
⎢^
⎥
⎜^
⎟
⎝^
⎠
⎣^
∼ ⎦
(^
)^
(^
)
3/ 2^
3/ 2
Y^
x^
J^
x − =
(^ )
(^ )
(^ )
cossin J^
x^
J^
x
Y^
x
υ
υ
υ
υπ υπ
− −
=
17
Backscattered field (cont.)Backscattered field (cont.)
so
3
3
1
3
3
!^
!
2
2
2
2
2
4
π^
⎛^
⎞
⎛^
⎞
=^
=^
=
⎜^
⎟
⎜^
⎟^
⎜^
⎟
⎝^
⎠^
⎝^
⎠
3
1
3 2
~^
e^
x
ω^
⎛^
⎜^
⎝^
⎝^
or
(^
3
1
1 1 ~
2 ω
=
e^
x^
x^
ka
Backscattered field (cont.)Backscattered field (cont.)
Therefore
Next, considerthe
d^1
term:
(^
(^3) )
,^
0
1
~^
(^
,^
jkr
S TE
r
E^
e
E^
ka
F^
e
r
θ
ω
με
− ⎛^
⎞⎛^
−^
⎜^
⎟⎜^
⎝^
⎝^
due to
or
l^
(^
)
l^
(^
)
l^
(^
) l^
(^
) 1
1
1
(2) 1 1
1
1
J^ (2)^1
x
d^
c
H^
x J^
x
e^
c
H^
x ′
= −
′
= −
Compare with
0
1
0
1
1
~ θ
με
με
−^
−
⎛^
⎞^
⎛^
⎞
−^
⎜^
⎟^
⎜^
⎟
⎝^
⎠^
⎝^
⎠
jkr^
jkr
s^
E^
e^
e
E^
e^
E^
d
r^
r
Recall that 20