FUNDAMENTALS AND PROPERTIES FOR RADIATION IN ABSORBING, EMITTING, AND SCATTERING MEDIA

FUNDAMENTALS AND PROPERTIES FOR RADIATION IN ABSORBING, EMITTING, AND SCATTERING MEDIA

• Attenuation by absorption and scattering

• Absorption and scattering coefficients

• Augmentation of intensity by emission

• Augmentation of intensity by incoming scattering

Attenuation by Absorption and Scattering

i_λ + di_λ i_λ

i di ds ds

λ λ

= + s + ds

s

κ

di_λ = −

κ

(experimental observation) or di

ds i

λ = −

κ

κ

( , , ,T P C_i )

λ λ

κ

κ λ

di_λ ∝ i ds_λ

λ aλ λ

κ

σ

a_λ : absorption coefficient

σ

0 0

( ) * *

( ) ( )

i s s

i

di s ds

i

λ λ

λ λ

λ

κ

∫

∫

di , ds i

λ = −

κ

0 0

* *

ln ( ) ( )

( ) i s s

i s ds

λ λ

λ

κ

= −

∫

0 0 ^{* *}

( ) ( )exp ^s ( )

i s_λ = i_λ ^⎡⎢⎣−

∫ κ

Radiation mean penetration distance energy absorbed and scattered at s

( ) ( )

i s_λ − i s_λ + ds =

fraction of energy absorbed and scattered at s

( )

1 0

( ) i ds

i_λ

κ

= − −

0

* *

exp ^s _λ ( )s ds ds

λ

κ

κ

= ⎢⎣−

∫

( ) ( ) di di

i s i s ds ds

ds ds

λ λ

λ − ⎡⎢⎣ λ + ⎤⎥⎦ = −

1 0 ( )

di ds i ds

λ λ

−

0 ( )

( )s s i d

i

λ λ

κ

=

Mean penetration depth

0

* *

( )exps ^s ( )s ds ds

λ λ

κ

∫ κ

when

κ

0 exp( )

lm =

κ

∫

κ

l_m : average penetration distance before absorption or scattering s ⋅

0 0

* *

( )exp ^s ( )

lm =

∫

κ

∫ κ

1

κ

=

Optical thickness (Opacity)

0

* *

( )s ^s ( )s ds

λ λ

τ

∫ κ

[ ]

0

( ) ( )exp ( )

i s_λ = i_λ −

τ

λ λ s

τ

κ

m

s

= l

the number of mean penetration distances

0 0 ^{* *}

( ) ( )exp ^s ( )

i s_λ = i_λ ^⎡⎢⎣−

∫ κ

1

τ

• A volume element within the material is only influenced by surrounding

neighbors.

• diffusion approximation (Rosseland approximation)

1

τ

• interact directly with medium boundary

• Radiation emitted within the material is not reabsorbed by the material.

• negligible self-absorption

Limiting cases

λ 0

τ

∇ ⋅ =

: transparent medium

τ

q′′ =r

The absorption and Scattering Coefficients Absorption coefficient

if no scattering, (

σ

κ

0 0 ^{* *}

( ) ( )exp ^s ( )

i s_λ = i_λ ⎡⎢⎣−

∫

in the case of a uniform medium 0

( ) ( )exp( )

i s_λ = i_λ −a s_λ

in the electromagnetic theory of radiant energy propagation in conducting media

0 4

( ) ( )exp s ,

i s_λ i_λ

πκ λ

⎛ ⎞

= ⎜⎝ − ⎟⎠

a_λ 4

πκ

=

λ

True absorption coefficient

ordinary or spontaneous emission stimulated or induced emission

The observed emerging intensity is the result of the actual absorption modified by the addition of

induced emission along the path

(negative absorption): resulting from the presence of the radiation field. At the same frequency and in the same direction as the incident photon

actual absorbed energy :

true absorption coefficient a_λ⁺( , , )

λ

observed attenuation media

for a gas with reflective index n = 1 1 exp hc0

a a

λ k T

λ

⎡ ⎛ ⎞⎤ +

= −⎢⎣ ⎜⎝ − ⎟⎠⎥⎦

except at a large value of

λ

1% deviation

λ

μ

within 5% deviation

λ

μ

Mie Solution 1) Definitions

C_s : scattering cross-section

the ratio of the rate of scattered energy by a spherical particle to the incident energy rate per unit area

C_a : absorption cross-section C_e : extinction cross-section

e a s

C = C + C

Q : efficiency factor

the ratio of the cross-section to the geometric cross-section

e a s

Q = Q + Q

r : radius of the sphere

: absorption efficiency factor

2 a a

Q C

π

=

2 s s

Q C

π

= : scattering efficiency factor

2) Mie’s solution

Re : real part

x : size parameter,

a_n, b_n: Mie coefficient

(Riccati-Bessel function)

( ) ( 2 2 )

2

1

2 2 1 ,

s n n

n

Q n a b

x

∞

=

=

∑

( ) { }

2

1

2 2 1 Re

e n n

n

Q n a b

x

∞

=

=

∑

π

λ

z = x or y, y = mx, m = n - i

κ

[ ]

[ ]

( ) ( ) / ( ) ( )

( ) ( ) / ( ) ( )

n n n n

n

n n n n

x y y m x

a x y y m x

ψ ψ ψ ψ

ξ ψ ψ ξ

′ − ′

= ′ − ′

[ ]

[ ]

( ) ( ) / ( ) ( ) ( ) ( ) / ( ) ( )

n n n n

n

n n n n

m x y y x

b m x y y x

ψ ψ ψ ψ

ξ ψ ψ ξ

′ − ′

= ′ − ′

1 2

1

2 2

/

( ) ( ),

n n

z

π

ψ

⎛ ⎞

= ⎜ ⎟

⎝ ⎠

1 2

( )

1 1

2 2

2 1

/

( ) ( ) ⁿ ( )

n n n

z

π

ξ

⎛ ⎞

= ⎜ ⎟ + −

⎝ ⎠

Basic parameters

1) index of refraction m = n - i

κ

(

κ

3) scattering angle

θ

For a medium containing N particles per unit volume of the same composition and uniform size

2) size parameter:

π

λ

2

a a

a_λ = C N =

π

2

s s

C N r Q N

σ

π

For a cloud of particles of different sizes

2

0 _a ( ) 0 _a ( )

a_λ =

∫

∫

π

2

0 C dN r_s ( ) 0 r Q N r dr_s ( )

σ

∫

∫

π

volume

element dV ds

s = ds s = 0

dA_s

dA

dω i_λb

Increase of Intensity by Emission

dA_s: a projected area normal to i_λ(0)

R

spherical black

enclosure at T

nonparticipating medium

R

ds

s = ds s = 0

dA_s

Volume element dV

dA

dω i_λb

spherical black

enclosure at T

Nonparticipating medium

▪ spectral intensity incident at a location of dA_s on dV, from element dA on the surface of the black enclosure: i_λ ( )0 = i_λb( , )

λ

▪ the change of this intensity in dV as a result of absorption

0

( ) _b ( , )

di_λ = −a i_{λ λ} ds = −a i_{λ λ}

λ

▪ the energy absorbed by the differential subvolume dsdA_s :

R

ds

s = ds s = 0

dA_s

Volume element dV

dA

dω i_λb

spherical black

enclosure at T

Nonparticipating medium

( , )

b s

a i_{λ λ}

λ

ω λ

2

dAs

d

ω

▪ the energy emitted by dA and absorbed by all of dV, dV =

∫

( , )

b s

dV a i_{λ λ}

λ

λ ω

∫

( , ) ( , )

b s b

a i_{λ λ}

λ

λ ω

λ

λ ω

=

∫

▪ for all energy incident upon dV from the entire spherical enclosure

R

ds

s = ds s = 0

dA_s

Volume element dV

dA

dω i_λb

spherical black

enclosure at T

Nonparticipating medium

4

π

λ

λ

=

4 4

0a i _b( , )T dVd d a i _b( , )T dVd 0d

π π

λ λ λ λ

ω

λ λ ω λ λ

ω

= = =

∫ ∫

▪ in equilibrium the energy spontaneously emitted by an isothermal volume element

R

ds

s = ds s = 0

dA_s

Volume element dV

dA

dω i_λb

spherical black

enclosure at T

Nonparticipating medium

4

π

λ

λ

λ

λ

▪ the radiation intensity emitted by a volume element into any direction

4 di dA d d_{λ,e p} 4a e dVd_{λ λb}

π

λ ω

λ

∫

4 4

,e p b

di_λ ⋅

π

λ

λ

,e

b

di a i ds

λ

= λ λ

or

dA_p: the projected area of dV normal to the direction of emission

p

ds dV

= dA : the mean thickness of dV parallel to the direction of emission

Increase of Intensity by Incoming Scattering

ds

Incoming scattering

( , ˆ ) i r_λ Ω ^′

dω^{′ d}ω

Ωˆ

Ωˆ ^′

2

,_s( , ˆ , ˆ )

d i_λ r

Ω Ω

scattered intensity in the direction

2

,_s( , , )

d i_λ r

Ω Ω

Ωˆ

ˆ ˆ

( , ) ( , )

i r_λ

Ω

Ω

( , ˆ ) i r ds

σ

Ω

=

phase function: directional distribution of scattered radiation

2 1

,_s( , ˆ , ˆ ) ,_s( , ˆ ) 4 ( ˆ , ˆ )

d i_λ r

Ω Ω

Ω

Ω Ω

′ ≡ ′

π

ˆ , ˆ

( , ) _s( , ) i r_λ

Ω

Ω

where

2

,_s( , ˆ ) 4 ,_s( , ˆ , ˆ )

di_λ r d i_λ r d

Ω

Ω Ω ω

′ =

∫

ˆ ds

( , ) i r_λ Ω^′

dω^′ dω

Ωˆ

Ωˆ ^′

2

,_s( , ˆ , ˆ )

d i_λ r Ω Ω^′

4

1 1

4

ˆ ˆ

( , )

P_λ d

ω π

Ω Ω ω

π ∫

4

1

4 ^,

ˆ ˆ ˆ

( , ) ( , )

di_λ s r P_λ d

Ω

Ω Ω ω

π

=

∫

2

4 d i_λ,_s( ,r ˆ , ˆ )d

ω π

Ω Ω ω

= ′

∫

isotropic scattering

4

1 1

4 P_λ d P_λ

ω π

ω

π ∫

physical interpretation

4

1 4

ˆ ˆ

( , )

P_λ d

ω π

Ω Ω ω π ∫

: probability that the incident radiation at will be scattered into the element of solid angle d

ω

Ω

Ω

Augmentation by incoming scattering

2 1

,_s( , ˆ , ˆ ) ( , ˆ ) 4 ( ˆ , ˆ )

d i_λ r

Ω Ω σ

Ω

Ω Ω

⎡ ⎤

π

′ = ⎣ ′ ⎦ ′

ˆ ds ( , ) i r_λ Ω ^′

dω^{′ d}ω

Ωˆ

Ωˆ ^′

2

,_s( , ˆ , ˆ )

d i_λ r

Ω Ω

scattering of the incident radiation per unit time, per unit wavelength, per unit volume, into an element of solid angle d

ω

4 4

,_s( , ˆ ) ds ( , ˆ ) ( ˆ , ˆ )

di_λ r _λ i r_λ P_λ d

ω π

Ω σ Ω Ω Ω ω

π

=

∫

4 4

,^s ( , ˆ ) ( ˆ , ˆ )

di i r P d

ds

λ λ

λ λ

ω π

σ Ω Ω Ω ω

π

=

∫

or

ˆ ds ( , ) i r_λ Ω ^′

dω^{′ d}ω

Ωˆ

Ωˆ ^′

2

,_s( , ˆ , ˆ )

d i_λ r

Ω Ω

2 1

,_s( , ˆ , ˆ ) ( , ˆ ) 4 ( ˆ , ˆ )

d i_λ r

Ω Ω σ

Ω

Ω Ω

⎡ ⎤

π

′ = ⎣ ′ ⎦ ′

All scattered intensity in the direction ^Ωˆ

b( ) a i_{λ λ} r +

4 4

ˆ ˆ ˆ

( , ) ( , )

i r P d

λ ω π λ λ

σ Ω Ω Ω ω

π

+

∫

,^{a s} ,

di i

ds

λ ⁺ = −

κ

λ = λ λ

4 4

,^is ( , ˆ ) ( ˆ , ˆ )

di i r P d

ds

λ λ

λ λ

ω π

σ Ω Ω Ω ω

π

=

∫

Radiative Transfer Equation (RTE)

( , ˆ ) di r

ds

λ

Ω

= −

κ

Ω