Valid regimes for Rayleigh, Mie and

geometric optics

• Rayleigh: Scattering D < !/15, Absorption/extinction D < !/50

• Geometric optics: D > 100!

Mie Theory (1908)Ishimaru, 2-8;

Bohern and Huffman (1983)

• Exact solution of

plane wave scattering

by a sphere

• Accurate calculationof radar reflectivity

and bistatic scattering

pattern

• Determine valid

regime for Rayleigthscattering

approximation.

Boundary problem

• Incident fields

• Boundary problem

• Component form

!

r E

i= e

ikzˆ x

!

ˆ r "r E

int r= a= ˆ r " [

r E

i+

r E

s]

r= a

!

ˆ r "r H

int r= a= ˆ r " [

r H

i+

r H

s]

r= a

!

Eint" r= a = [E

i"+ E

s"]r= a

!

Eint" r= a = [E

i"+ E

s"]r= a

!

Hint" r= a = [H

i"+ H

s"]r= a

!

Hint" r= a = [H

i"+ H

s"]r= a

!

r H i =

1

"e

ikzˆ y

Two approaches to represent

wave fields

• In terms of vector spherical harmonics Bohern and

Huffman (1983), Ch. 4

Born and Wolf (2001), Ch. 14

• By radial component of Hertz vectors for incident,

scattered and internal wave fields

!

r E =" #" # ($

1

r r ) + i%µ

0" # ($

2

r r )

!

r H = "i#$% & ('

1

r r ) +% &% & ('

2

r r )

Solution of wave equation

• Wave equation

• In spherical coordinate

• Assuming a function form

!

"2# + k

2# = 0

!

1

r

" 2(r#)

"r2+

1

r2sin$

"

"$sin$

"#

"$

%

& '

(

) * +

1

r2sin

2$

" 2#

"+ 2+ k

2# = 0

!

" = R(r)#($)%(&)

!

"mn(kr,#,$) =%

n(kr)P

n

m(cos#)eim$

Expressions of wave fields

• Expansion

• Solve for an, bn by matching the boundary condition

!

an

="

n(ka) # "

n(kma) $m"

n(kma) # "

n(ka)

%n(ka) # "

n(kma) $m"

n(kma) # %

n(ka)

!

r"1s

=#1

k2

in#1(2n +1)

n(n +1)an$n

n=1

%

& (kr)Pn

1(cos')cos(

!

r"2s =#1

$k 2in#1(2n +1)

n(n +1)bn%n

n=1

&

' (kr)Pn

1(cos()sin)

!

bn

=m"

n(ka) # "

n(kma) $"

n(kma) # "

n(ka)

m%n(ka) # "

n(kma) $"

n(kma) # %

n(ka)

Scattered wave fields

• Expression

• Angle dependence

!

S1(") =

2n +1

n(n +1)n=1

#

$ an%ncos"( ) + b

n&ncos"( )[ ]

!

Es"

Es||

#

$ %

&

' ( =

eikr

)ikr

S1

0

0 S2

#

$ %

&

' ( Ei"

Ei||

#

$ %

&

' (

!

S2(") =

2n +1

n(n +1)n=1

#

$ an%ncos"( ) + b

n&ncos"( )[ ]

!

"ncos#( ) =

Pn

1cos#( )sin#

!

"ncos#( ) =

d

d#Pn

1cos#( )

!

r E

s=

eikr

"ikrS2(#)cos$ ˆ # " S1(#)sin$ ˆ $ [ ]

Pattern functions

From Bohren & Huffman, 1983

X-band scattering pattern

differ from Rayleigh: |sin"|

D = 2 mm: |S(#)|

D = 2 cm: |S(#)|

• Extinction cross section

• Scattering cross section

• Radar cross section

Cross sections

!

"t=4#

kIm

S(0)

$ik

%

& '

(

) * =2#

k2

n=1

+

, (2n +1)Re[an

+ bn]

!

"s

=1

k2

S1(#)2sin

2 $ + S2(#)2cos

2 $( )4%& d' =

2%

k2

n=1

(

) (2n +1) | an|2 + |b

n|2( )

!

"b

=#

k2

n=1

$

% (2n +1)(&1)n an& b

n( )

2