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Absorption and Scattering
Peng XiChanghui Li
北京大学工学院生物医学工程系
2011/09/09
The Propagation of Light
The processes of transmission, reflection, and refraction are macroscopic manifestations of
scattering occurring on a submicroscopic level.
http://faculty.qu.edu.qa%2Fmalrabban%2FOptics%2FOptics_08Propagation.ppt
Elastic Scattering
• In elastic scattering, the energy of the incident photon is conserved and its propagating direction is changed by the potential of the target.
Rayleigh Scattering
When a photon penetrates into a medium composed of particles whose sizes are much smaller than the wavelength of the incident photon, the scattering process is elastic and is called Rayleigh scattering. In this scattering process, the energy (and therefore the wavelength) of the incident photon is conserved and only its direction is changed. In this case, the scattering intensity is proportional to the fourth power of the reciprocal wavelength of the incident photon.
The scattering of electromagnetic radiation by particles with
dimensions much smaller than the wavelength of the radiation,
resulting in angular separation of colors and responsible for the
reddish color of sunset and the blue of the sky.
The intensity of the scattered light 4
4
1
or
os
oi
E
ELet is the incident amplitude,
is the scattered amplitude at a distance r from the scatterer.
V is the volume of the scatterer.
Example 4.1
Establish the dependence of the percentage of light scattered in Rayleigh scattering.
Vr
EE oios
1
Assume
r
EVK
r
EVE oioi
os
4
r
EVKE oi
os
r
KV Must be unitless, and
K must has units of ( Length )2
4
2
1,
1
os
oioios
I
r
EV
r
EVKE
The Transmission of Light Through Dense Media
Interference produces a redistribution of energy, out of the regions where it’s destructive into the regions where it’s
constructive.
Little or no light ends up scattered laterally or backwards in a dense homogeneous medium.
This makes sense from the perspective of conservation of energy– we can’t have constructive interference in every direction.
Constructive vs. destructive interference;Coherent vs. incoherent interference
Waves that combine in phase add up to relatively high irradiance.
Waves that combine 180° out of phase cancel out and yield zero irradiance.
Waves that combine with lots of different phases nearly cancel out and yield very low irradiance.
=
=
=
Constructive interference(coherent)
Destructive interference(coherent)
Incoherent addition
Scattering from molecules and small particles
Scattering from an individual molecule or particle is weak, but many such scatterings can add up—especially if interference is coherent and constructive.
Huygens’ Principle says that waves propagate as if each point on a wave-front emits a spherical wave (whether or not there’s a molecule or particle involved).
A plane wave impinging on a molecule or particle scatters into a spherical wave.
The Phases of the wavelets at P differ greatly
The Transmission of Light Through Dense
Media
Waves using complex amplitudes
• The resulting "complex amplitude" is:
•
0 exp( ) E A i
(note the " ~ ")
0, expE x t E i kx t
As written, this entire field is complex!
Complex numbers simplify optics!
This isn't so obvious using trigonometric functions, but it's easywith complex exponentials:
1 2 3
1 2 3
( , ) exp ( ) exp ( ) exp ( )
( ) exp ( )totE x t E i kx t E i kx t E i kx t
E E E i kx t
where all initial phases are lumped into E1, E2, and E3.
Adding waves of the same frequency, but different initial phase, yields a wave of the same frequency.
When two waves add together with the same complex exponentials,
we add the complex amplitudes, E0 + E0'.
Adding complex amplitudes
Slower phase velocityLaser Absorption
+
=
time
1.0
-0.2
0.8
Destructive interference:
1.0
0.2
1.2
+
=
time
Constructive interference:
+
=
time
"Quadrature phase" ±90° interference:
1.0
-0.2i
1-0.2i
Light excites atoms, which emit light that adds (or subtracts) with the input light.
When light of frequency w excites an atom with resonant frequency w0:
An excited atom vibrates at the frequency of the light that excited it and re-emits the energy as light of that frequency.
The crucial issue is the relative phase of the incident light and this re-emitted light. For example, if these two waves are ~180° out of phase, the beam will be attenuated. We call this absorption.
Electric field at atom
Electron cloud
Emitted field
On resonance ( w = w0)
( )ex t
( )E t
( )E t +
=
Incident light
Emitted light
Transmitted light
The interaction of light and matterLight excites atoms, which then emit more light.
The crucial issue is the relative phase of the incident light and this re-emitted light. If these two waves are ~180° out of phase, destructive interference occurs, and the beam will be attenuated—absorption. If they’re ~±90° out of phase: the speed of light changes—refraction.
Electric field at atom
Electron cloud
Emitted electric field
On resonance (the light frequency is the same as that of the atom)
( )ex t
( )E t
( )E t +
=
Incident light
Emitted light
Transmitted light
The relative phase of emitted light with respect to the input light depends on the frequency.
Below resonance
w << w0
Electric field at atom
Electron cloud
On resonance w = w0
Above resonance
w >> w0
The emitted light is 90° phase-shifted with respect to the atom’s motion.
Emitted field
Weak emission.90° out of phase.
Strong emission.180° out of phase.
Weak emission.-90° out of phase.
Refractive index and Absorption coefficient
2 20
2 2 2 20 0 0 0 0
/ 2 1
2 ( ) ( / 2) 4 ( ) ( / 2)e e
Ne Nen
c m m
0
Absorption coefficient
Refractive index
0
n–1
Frequency, Frequency, ww0
Variation of the refractive index with wavelength (dispersion) causes the
beautiful prismatic effects we know and love.
Prisms disperse white light into its various colors.
Prism
Input white beam
Dispersed beam
Light Scattering
When light encounters matter, matter not only re-emits light in the forward direction (leading to absorption and refractive index), but it also re-emits light in all other directions.
This is called scattering.
Light scattering is everywhere. All molecules scatter light. Surfaces scatter light. Scattering causes milk and clouds to be white and water to be blue. It is the basis of nearly all optical phenomena.
Scattering can be coherent or incoherent.
Light scattering regimes
Particle size/wavelength
Re
frac
tive
inde
x
Mie Scattering
Ra
yle
igh
Sca
tteri
ng
Totally reflecting objects
Ge
om
etr
ica
l opt
ics
Rayleigh-Gans Scattering
Larg
e
~1.
5
~0 ~0 ~1 Large
There are many regimes of particle scattering, depending on the particle size, the light wavelength, and the refractive index. You can read an entire book on the subject:
Rainbow
Air
Mie Scattering.
The mathematics of scattering
Itotal = I1 + I2 + … + In
I1, I2, … In are the irradiances of the various beamlets. They’re all positive real numbers and add.
* * *1 2 1 2 1 3 1... Re ...total N N NI I I I c E E E E E E
If the phases aren’t random, we add the fields:
Ei Ej* are cross terms, which have the
phase factors: exp[i(qi-qj)]. When the q’s are not random, they don’t cancel out!
If the phases are random, we add the irradiances:
Coherent
Incoherent
Etotal = E1 + E2 + … + En
The Biological Origin of Light Scattering
Structure name Refractive index SizeNucleus 1.38-1.41 ~ 4-10μmMitochondrion 1.38-1.41 ~ 1μmCytoplasm 1.36-1.375
College of Engineering, Peking University 生物医学光学 II 23
The absorption coefficient (μa) is defined as the probability of photon absorption in a medium per unit path length (strictly speaking, per unit infinitesimal path length).
The absorption coefficient can be considered as the total cross-sectional area for absorption per unit volume.
Absorption
: absorption coefficient of the medium
: number density of absorbers
: absorption cross-section of an absorber
a a a
a
a
a
N
N
The Beer-Lambert law (or Beer's law) is the linear relationship between absorbance and concentration of an absorbing species. The general Beer-Lambert law is usually written as:
A = a(λ) * b * cwhere A is the measured absorbance, a(λ) is a wavelength-dependent absorptivity coefficient, b is the path length, and c is the analyte concentration. When working in concentration units of molarity, the Beer-
Lambert law is written as:A = ε * b * c
where ε is the wavelength-dependent molar absorptivity coefficient with units of M-1 cm-1.
Beer-Lambert Law
http://elchem.kaist.ac.kr/vt/chem-ed/spec/beerslaw.htm
Beer’s Law (Absorption)
0 0
0
exp exp
: light intensity
: incident light intensity
: pathlength
: absorption coefficient (decay rate)
: absorption length (decay constant)
a
a
a a
a
a
dII
dxdI
dxI
I x I x I x l
I
I
x
l
Ballistic photonsSnake photonsDiffused photons
Trajectories of Optical Photons in Biological Tissue
Tissue
Laser beam
1 mm
Reflectometry
Photoacoustics
Spectra of Major Biological Absorbers
102
103
104
105
10−4
10−2
100
102
104
106
Wavelength (nm )
Abs
orpt
ion
coef
fici
ent (
cm−
1 )
HbO2
Near IR window: ~700 nm
2.95 µm
~1 µm penetration
Spectrum of Melanosome• Melanin:
– Eumelanin: A black-to-dark-brown insoluble material found in human black hair and in the retina of the eye.
– Pheomelanin: A yellow-to-reddish-brown alkali-soluble material found in red hair and red feathers.
– Polymers– ~10 nm in diameter– Studding the inner walls of
melanosomes (~1 micron diameter organelle)
• Volume fraction of melanosome in epidermis: – Light skinned Caucasions: 1-3%– Dark pigmented Africans: 18-43%
http://omlc.ogi.edu/spectra
Spectrum of Fat (Lipids)
http://omlc.ogi.edu/spectra
Spectrum of Methylene Blue Dye:Contrast Agent Used in Sentinel Lymph Node Mapping
http://omlc.ogi.edu/spectra
C16H18ClN3S, MW 319.85. Also called Swiss blue. One gram dissolves in about 25 ml of water, or in 65 ml alcohol. Peak absorption at 668 and 609 nm. --- Merck Index
Spectrum of Indocyanine Green (ICG):Cardiac Output and Hepatic Function Measurements
http://omlc.ogi.edu/spectra
C43H47N2O6S2Na, Molecular weight 775. A tricarbocyanine type of dye with infrared absorbing properties; peak absorption at about 800nm. Has little or no absorption in the visible. It is used in infrared photography and in the preparation of Wratten filters. It is also used as a diagnostic aid for blood volume determination, cardiac output, or hepatic function. --- Merck Index
Molar Extinction Spectra of Hemoglobin
[nm]259.93339.54390.01422.05452.36500.11529.24545.26570.18584.09796.80
Isosbesticpoint
Scattering
• The scattering coefficient (mus) is defined as the probability of photon scattering in a medium per unit path length (strictly speaking, per unit infinitesimal path length).
• The scattering coefficient can be considered as the total cross-sectional area for scattering per unit volume.
scatterer a ofsection cross scattering:
scatterers ofdensity number :
medium theoft coefficien scattering:
:media packedloosely In
s
s
s
sss
N
N
Extinction
• Extinction = absorption + scattering• Molar extinction coefficient = extinction
coefficient per Molar concentration per length• Molar = moles/L
sat
a
C
C
tcoefficien extinction molar the is ion,concentrat the is where
,10ln
Beer’s Law (Extinction)
path freemean :
tcoefficien n)(extinction interactio total:
pathlength:
intensity ballistic:
expexp 00
t
t
tt
t
l
x
I
lxIxIxI
dx
IdI
Pat Arnott, ATMS 749 Atmospheric Radiation TransferW. P. Arnott, AAAR tutorial, Sept. 2007 38
The Distribution of Scattered Radiation (Phase Function)
D D D
Rayleigh Resonance Geometrical OpticsIncoming light direction
Adapted fromhttp://hyperphysics.phy-astr.gsu.edu/hbase/atmos/blusky.html
Pat Arnott, ATMS 749 Atmospheric Radiation TransferW. P. Arnott, AAAR tutorial, Sept. 2007 39
Example of a morning when the Mexico City Plume Goes South to Popocatepetl Volcano.
near forward scattering by particles sca = 30 degrees
r << r ~ r >>
Optical Properties of Biological Tissue
• Basic properties• n [–]: index of refraction; e.g., 1.37• µa [cm–1]: absorption coefficient; e.g., 0.1
• µs [cm–1]: scattering coefficient; e.g., 100• g [–]: scattering anisotropy, <cosq>; e.g., 0.9
• Derived properties• µt [cm–1]: total interaction (extinction) coefficient, µa +
µs
• lt [cm]: mean free path, 1/ µt; e.g., 0.1 mm
• µs’ [cm–1]: reduced scattering coefficient, µs(1 – g)
• µt’ [cm–1]: transport interaction coefficient, µa + µs’
• lt’ [cm]: transport mean free path, 1/ µt’; e.g., 1 mm
• µeff [cm–1]: effective attenuation coefficient, (3µa µt’)1/2
• δ [cm]: penetration depth, 1/(3µa µt’)1/2; e.g., 5 mm
Major Challenge in Optical Tomography: High Resolution Beyond the Quasiballistic (~1-mm Depth) Regime
CFM: Confocal microscopy 2PM: Two-photon microscopyOCT: Optical coherence tomography DOT: Diffuse optical tomographyPAT: Photoacoustic tomographylt’: Optical transport mean free path ~ 1 mm
(mean free path ~ 0.1 mm)δ: Effective penetration depth
1 m
m
DOT, PAT
OCT
* Applied Optics 38, 4951 (1999). Simulation software MCML
available on the web
Soft limit*~ lt’
CFM & 2PM
Hard limit ~ 10δ ~ 5-7 cm (20,000X or 43 dB one-way attenuation)
Laser
Elastic Rayleigh scatteringMie scattering
InelasticRaman scattering
The difference in energy generates a vibrational excitation in the molecule
Brillouin scattering The difference in energy generates acoustic phonons.
Scattering
Absorption: Beer’s lawBiological Scattering-Elastic:
Rayleigh scattering: 1/λ4
Mie scattering: weak relative to wavelength
Summary