Lecture/Lab 5:
Interaction of light with
particles Mie
particles.
Mie’ss solution
solution.
Scattering of light by spherical particles (Mie scattering).
The problem (Bohren and Huffman, 1983):
Given a particle of a specified size, shape and optical properties that is
illumin t d b
illuminated
by an
n arbitrarily
bit il p
polarized
l i dm
monochromatic
n ch m tic wave, determine
d t min the
th
electromagnetic field at all points in the particles and at all points of the
homogeneous medium in which it is embedded.
We will assume that the wave is plane harmonic wave and a spherical particles.
Define four fields:
v v
v v
v v v v
(E1 , H1 ), (E2 , H 2 ), (Ei , H i ), (Es , H s )
v v
(Es , H s )
v v
(Ei , H i )
Outside the particle:
2
1
v
v v v
v
v
E2 = Ei + Es , H 2 = H i + H s
Plane parallel harmonic wave:
((
((
))
))
v
v
v
Ei = E0 exp i k ⋅ x − ωt
v
v
v
H i = H 0 exp i k ⋅ x − ωt
Must satisfy Maxwell’s equation where material properties are constant:
v
∇⋅E = 0
v
∇⋅H = 0
v
v
∇ × E = iωμH
v
v
∇ × H = −iωεE
ε is the permittivity μ is the premeability.
define:
k 2 = εμω 2
The vector equation reduce to:
v
v
2
∇ E+k E =0
v 2v
2
∇ H +k H = 0
2
Boundary conditions: the tangential components of the electric and magnetic
fields must me continuous across the boundary of the particle (analogous to
energy conservation):
[
[
]
v
v
E2 − E1 × nˆ = 0
v
v
H 2 − H1 × nˆ = 0
]
The equations and BCs are linear Æ superposition of solutions is a solution.
An arbitrarily polarized light can be expressed as a supperposition of two
orthogonal polarization states:
Amplitude
p
scattering
g matrix
⎛ El , s ⎞ exp(ik (r − z )) ⎛ S 2 (θ , ϕ ) S3 (θ , ϕ )⎞⎛ El ,i ⎞
⎜
⎟=
⎜
⎟
⎜
⎟
⎜ S (θ , ϕ ) S (θ , ϕ ) ⎟⎜ E ⎟
⎜E ⎟
−
ik
ikr
1
⎝ 4
⎠⎝ r ,i ⎠
⎝ r ,s ⎠
For spheres:
0 ⎞⎛ El ,i ⎞
⎛ El , s ⎞ exp(ik (r − z )) ⎛ S 2 (θ )
⎜
⎟=
⎜
⎟
⎜
⎟
⎜
⎟
⎜E ⎟
⎜E ⎟
(
)
0
S
θ
−
ikr
1
⎝
⎠⎝ r ,i ⎠
⎝ r ,s ⎠
exp(ik (r − z ))
El , s =
S1 (θ )El ,i
− ikr
⇒
exp(ik (r − z ))
Er , s =
S 2 (θ )Er ,i
− ikr
Taking the real part of the squares of the electric fields we get the radiant
intensity [W Sr-1]:
⇒
I r ,s =
I l ,s =
S1 (θ )
2
2 2
I r ,i
2 2
I l ,i
k r
2
S 2 (θ )
k r
For unpolarized light:
{
}
1
2
2
S1 (θ ) + S 2 (θ )
Is = 2
Ii
2 2
k r
Polarization:
Describe the plane of propagation and phase of the EM radiation.
Assume a wave propagating in the z-direction, and the observer is at z=0.
http://instruct1.cit.cornell.edu/Courses/ece303/lecture15.pdf
Stokes notation and the scattering matrix:
Is =
k
2
El , s + Er , s
2
2ωμ
2
2
k
Qs =
El , s − Er , s
2ωμ
k
Us =
El , s E *r ,s + Er , s E *l ,s
2ωμ
k
*
*
Vs = i
El , s E r ,s − Er , s E l ,s
2ωμ
⎛ Is ⎞
⎛ S11 S12
⎜ ⎟
⎜
1 ⎜ S 21 S 22
⎜ Qs ⎟
⎜U ⎟ = k 2 r 2 ⎜ S
S32
s
31
⎜ ⎟
⎜
⎜V ⎟
⎜S
⎝ s⎠
⎝ 41 S 42
No polarizers
H i
Horizontal
t l - vertical
ti l polorizers
l i
45°
5 - ((-45°)
5 ) po
polorizers
or z rs
Right handed – left handed
circular polorizers
S13
S 23
S33
S 43
S14 ⎞⎛ I i ⎞
⎟⎜ ⎟
S 24 ⎟⎜ Qi ⎟
S34 ⎟⎜U i ⎟
⎟⎜ ⎟
S 44 ⎠⎟⎜⎝ Vi ⎟⎠
For a sphere:
⎛ Is ⎞
⎛ S11
⎜ ⎟
⎜
1 ⎜ S12
⎜ Qs ⎟
⎜U ⎟ = k 2 r 2 ⎜ 0
⎜ s⎟
⎜
⎜V ⎟
⎜ 0
⎝ s⎠
⎝
S12
S11
0
0
0
0
S33
− S34
Link to amplitude scattering matrix:
{
}
S11 = S1 + S 2 ,
2
{
2
}
{
0 ⎞⎛ El ,i ⎞
⎛ El , s ⎞ exp(ik (r − z )) ⎛ S 2 (θ )
⎜
⎟
⎟
⎜⎜
⎟⎟⎜⎜
⎜E ⎟ =
⎟
E
(
)
S
0
θ
ikr
−
1
⎝
⎠⎝ r ,i ⎠
⎝ r ,s ⎠
2
2
1
2
S12 = S
{
0 ⎞⎛ I i ⎞
⎟⎜ ⎟
0 ⎟⎜ Qi ⎟
S34 ⎟⎜U i ⎟
⎟⎜ ⎟
S33 ⎟⎠⎜⎝ Vi ⎟⎠
}
−S
S33 = S S + S S 2 , S34 = S S − S S 2
*
2 1
*
1
*
2 1
S12
S =S +S +S , P≡−
S11
2
11
2
12
2
33
2
34
*
1
}
Solution method:
Expand incident and scattered fields in spherical harmonic functions for
each polarization. Match solutions on boundary of particle and require them
to be finite at large distances.
Input to Mie code:
W
Wavelength
l
h in
i medium
di
(λ).
( )
Size (diameter, D) in the same units as wavelength.
Index of refraction relative to medium (n + in’).
S l ti d
Solution
depends
p ds on::
Size parameter: πD/λ
Index of refraction relative to medium
Output to Mie code:
Efficiency factors:
Qa, Qc (also called Qext)
Scattering matrix elements:
S1 and S2
From which we can calculate:
Qb=Qc-Qa
β ∝ S11=|S1|2+|S2|2
Other polarization scattering matrix elements:
S12 ∝ |S1|2-|S2|2
P=-S12/S11
S33=Real(S
Real(S2 x S1*)/S11
S34=Imag(S2 x S1*)/S11
Populations of particles:
Monodispersion- example, obtaining the scattering
coefficient:
b=NQbG,
G G=πD2/4.
/4 Watch for units!
Polydispersion: discrete bins:
b=ΣNiQb,iGi
When using continuous size distribtion:
N ( D , ΔD ) =
D + ΔD / 2
∫ f (D )dD
D − ΔD / 2
b=
π
4
Dmax
2
(
)
Q
D
D
f (D )dD
b
∫
Dmin
Similar manipulations are done to obtain the absorption and attenuation
coefficients, as well as the population’s volume scattering function.
Resources (among many others…):
Barber and Hill, 1990.
Bohren and Huffman, 1983.
Kerker, 1969.
Van de Hulst, 1981 (original edition, 1957).
Codes (among many others):
http://www iwt-bremen
http://www.iwt
bremen.de/vt/laser/wriedt/index_ns.html
de/vt/laser/wriedt/index ns html