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 di idi

i dsds

s + dss

di i ds

(experimental observation)or

dii

ds

: extinction coefficient, units of reciprocal length [cm-1]

( , , , )iT P C

di i ds

a

a : absorption coefficient

: scattering coefficient

0 0

( )* *

( )( )

i s s

i

dis ds

i

,di

ids

00* *( )

ln ( )( )

si ss ds

i

00 * *( ) ( )exp ( )

si s i s ds Bouguer’s la

w

Radiation mean penetration distance energy absorbed and scattered

at s( ) ( )i s i s ds

fraction of energy absorbed and scattered at s

1

0( )i ds

i

0

* *exp ( )s

s ds ds

( ) ( )di di

i s i s ds dsds ds

1

0( )

dids

i ds

0

( )

( )s

sd

i

i

Mean penetration depth

0

* *( )exp ( )s

s s ds ds

when = constant

0exp( )ml s s ds

lm : average penetration distance

before absorption or scattering

s

0 0

* *( )exp ( )s

ml s s s ds ds

1

Optical thickness (Opacity)

0

* *( ) ( )s

s s ds

0( ) ( )exp ( )i s i s

for a uniform gas

s m

s

l

the number of mean penetration distances

00 * *( ) ( )exp ( )

si s i s ds

1 : optically thick (photon continuum)• A volume element within the

material is only influenced by surrounding neighbors.• diffusion approximation (Rosseland approximation)

1 : optically thin• interact directly with medium

boundary• Radiation emitted within the material is not reabsorbed by the material.• negligible self-absorption

Limiting cases

0 : transparent medium

: completely opaque

0rq

0rq

The absorption and Scattering Coefficients Absorption coefficient

if no scattering,

0( ), a

00 * *( ) ( )exp ( )

si s i a s ds

in the case of a uniform medium0( ) ( )exp( )i s i a s

in the electromagnetic theory of radiant energy propagation in conducting media4

0( ) ( )exp ,s

i s i

4a

True absorption coefficientordinary or spontaneous emissionstimulated 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 T P

larger than a calculated by using observed attenuation media

for a gas with reflective index n = 1

01 exphc

a ak T

except at a large value of T a a

1% deviation T < 3120 m

within 5% deviation T < 4800 m

Mie Solution1) Definitions

Cs : scattering cross-section

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

Ca : absorption cross-section

Ce : extinction cross-section

e a sC C C

Q : efficiency factor

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

e a sQ Q Q

r : radius of the sphere

: absorption efficiency factor

2a

a

CQ

r

2s

s

CQ

r : scattering efficiency

factor

2) Mie’s solution

Re : real part

x : size parameter,

an, bn: Mie coefficient (Riccati-Bessel function)

2 2

21

22 1 ,s n n

n

Q n a bx

21

22 1 Ree n n

n

Q n a bx

D

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

( ) ( ) / ( ) ( )

( ) ( ) / ( ) ( )n n n n

n

n n n n

x y y m xa

x y y m x

( ) ( ) / ( ) ( )

( ) ( ) / ( ) ( )n n n n

n

n n n n

m x y y xb

m x y y x

1 2

1

22

/

( ) ( ),nn

zz J z

1 2

1 1

2 2

12

/

( ) ( ) ( )n

nn n

zz J z iJ z

Basic parameters

1)index of refraction m = n - i relative to the surrounding medium

( = 0 : pure scatter)

3) scattering angle : phase function For a medium containing N particles per unit volume of the same composition and uniform size

2) size parameter:

D

2a aa C N r Q N

2s sC N r Q N

For a cloud of particles of different sizes

2

0 0( ) ( )a aa C dN r r Q N r dr

2

0 0( ) ( )s sC dN r r Q N r dr

volume element dV

dss = ds

s = 0

dAs

dA

dib

Increase of Intensity by Emission

dAs: a projected area normal to i(0)

R

spherical blackenclosure at T

nonparticipatingmedium

R

dss = ds

s = 0

dAs

Volume element dV

dA

d

i b

spherical blackenclosure at T

Nonparticipatingmedium

▪ spectral intensity incident at a location of dAs on dV, from element dA on the surface of the black enclosure: 0( ) ( , )bi i T

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

0( ) ( , )bdi a i ds a i T ds

▪ the energy absorbed by the differential subvolume dsdAs :

R

dss = ds

s = 0

dAs

Volume element dV

dA

d

i b

spherical blackenclosure at T

Nonparticipatingmedium

( , )b sa i T dsdA d d

2sdA

dR

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

( , )b sdVa i T d d dA ds

( , ) ( , )b s ba i T d d dA ds a i T dVd d

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

R

dss = ds

s = 0

dAs

Volume element dV

dA

d

i b

spherical blackenclosure at T

Nonparticipatingmedium

4 ( , )ba i T dVd

4 4

0 0( , ) ( , )b ba i T dVd d a i T dVd d

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

R

dss = ds

s = 0

dAs

Volume element dV

dA

d

i b

spherical blackenclosure at T

Nonparticipatingmedium

4 4( , ) ( , )b ba i T dVd a e T dVd

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

44,e p bdi dA d d a e dVd

4 4,e p bdi dA d a e dVd ,e bdi a i ds

,eb

dia i

ds

or

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

p

dVds

dA : the mean thickness of dV parallel

to the direction of emission

Increase of Intensity by Incoming Scattering

ds

Incoming scattering

( , )i r

d d

2, ( , , )sd i r

scattered intensity in the direction2

, ( , , )sd i r

( , ) ( , )i r di r

d

ds( , )i r

d

( , )i r ds

phase function: directional distribution of scattered radiation

2 1

4, ,( , , ) ( , ) ( , )s sd i r di r P

,( , ) ( , )si r di r

where 2

4, ,( , ) ( , , )s sdi r d i r d

4

11

4( , )P d

4

1

4 , ( , ) ( , )sdi r P d

2

4 , ( , , )sd i r d

isotropic scattering 4

11

4P d P

physical interpretation4

1

4( , )P d

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

Augmentation by incoming scattering

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

2 1

4, ( , , ) ( , ) ( , )sd i r i r ds P

ds( , )i r

d d

2, ( , , )sd i r

All scattered intensity in the direction

44, ( , ) ( , ) ( , )s

dsdi r i r P d

44, ( , ) ( , )sdi

i r P dds

or

ds( , )i r

d d

2, ( , , )sd i r

2 1

4, ( , , ) ( , ) ( , )sd i r i r ds P

( )ba i r

44( , ) ( , )i r P d

, ,a sdii

ds

,eb

dia i

ds

44, ( , ) ( , )isdi

i r P dds

Radiative Transfer Equation (RTE)

( , )di r

ds

( , )i r