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Absorption and Scattering Peng Xi Changhui Li 北北北北北北北 北北北北北北北 2011/09/09

Absorption and Scattering

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Absorption and Scattering. Peng Xi Changhui 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. - PowerPoint PPT Presentation

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Page 1: Absorption and Scattering

Absorption and Scattering

Peng XiChanghui Li

北京大学工学院生物医学工程系2011/09/09

Page 2: Absorption and Scattering

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

Page 3: Absorption and Scattering

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.

Page 4: Absorption and Scattering

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.

Page 5: Absorption and Scattering

The intensity of the scattered light 44

1

or

os

oi

EELet is the incident amplitude,

is the scattered amplitude at a distance r from the scatterer.

V is the volume of the scatterer.

Example 4.1Establish the dependence of the percentage of light scattered in Rayleigh scattering.

Vr

EE oios

1

Assume

rEVK

rEVE oioi

os

4

Page 6: Absorption and Scattering

rEVKE oi

os

rKV

Must be unitless, and

K must has units of ( Length )2

4

2

1,

1

os

oioios

I

rEV

rEVKE

Page 7: Absorption and Scattering

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.

Page 8: Absorption and Scattering

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

Page 9: Absorption and Scattering

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.

Page 10: Absorption and Scattering

The Phases of the wavelets at P differ greatly

The Transmission of Light Through Dense

Media

Page 11: Absorption and Scattering

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!

Page 12: Absorption and Scattering

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 tE 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.

Page 13: Absorption and Scattering

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

Page 14: Absorption and Scattering

Light excites atoms, which emit light that adds (or subtracts) with the input light.When light of frequency excites an atom with resonant frequency 0:

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 ( = 0)

( )ex t( )E t

( )E t +

=

Incident light

Emitted light

Transmitted light

Page 15: Absorption and Scattering

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

Page 16: Absorption and Scattering

The relative phase of emitted light with respect to the input light depends on the frequency.

Below resonance << 0

Electric field at atom

Electron cloud

On resonance = 0

Above resonance >> 0

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.

Page 17: Absorption and Scattering

Refractive index and Absorption coefficient

2 20

2 2 2 20 0 0 0 0

/ 2 12 ( ) ( / 2) 4 ( ) ( / 2)e e

Ne Nenc m m

0

Absorption coefficient

Refractive index

0

n–1

Frequency, Frequency, 0

Page 18: Absorption and Scattering

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

Page 19: Absorption and Scattering

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.

Page 20: Absorption and Scattering

Light scattering regimes

Particle size/wavelength

Ref

ract

ive

inde

x

Mie Scattering

Ray

leig

h S

catte

ring

Totally reflecting objects

Geo

met

rical

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

Page 21: Absorption and Scattering

Mie Scattering.

                                                                                                          

Page 22: Absorption and 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(i-j)]. When the ’s are not random, they don’t cancel out!

If the phases are random, we add the irradiances:

Coherent

Incoherent

Etotal = E1 + E2 + … + En

Page 23: Absorption and Scattering

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

Page 24: Absorption and Scattering

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

Page 25: Absorption and Scattering

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

Page 26: Absorption and Scattering

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

dI Idx

dI dxI

I x I x I x l

IIx

l

Page 27: Absorption and Scattering

Ballistic photonsSnake photonsDiffused photons

Page 28: Absorption and Scattering

Trajectories of Optical Photons in Biological Tissue

Tissue

Laser beam

1 mm

Reflectometry

Photoacoustics

Page 29: Absorption and Scattering

Spectra of Major Biological Absorbers

102

103

104

105

10−4

10−2

100

102

104

106

Wavelength (nm )

Abs

orpt

ion

coef

ficie

nt (c

m−1

)

HbO2

Near IR window: ~700 nm

2.95 µm

~1 µm penetration

Page 30: Absorption and Scattering

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

Page 31: Absorption and Scattering

Spectrum of Fat (Lipids)

http://omlc.ogi.edu/spectra

Page 32: Absorption and Scattering

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

Page 33: Absorption and Scattering

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

Page 34: Absorption and Scattering

Molar Extinction Spectra of Hemoglobin

[nm]259.93339.54390.01422.05452.36500.11529.24545.26570.18584.09796.80

Isosbesticpoint

Page 35: Absorption and Scattering

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

Page 36: Absorption and Scattering

Extinction

• Extinction = absorption + scattering• Molar extinction coefficient = extinction

coefficient per Molar concentration per length• Molar = moles/L

sat

a

CC

tcoefficien extinction molar the is ion,concentrat the is where

,10ln

Page 37: Absorption and Scattering

Beer’s Law (Extinction)

path freemean :tcoefficien n)(extinction interactio total:

pathlength:intensity ballistic:

expexp 00

t

t

tt

t

l

xI

lxIxIxI

dxIdI

Page 38: Absorption and Scattering

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

Page 39: Absorption and Scattering

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 >>

Page 40: Absorption and Scattering

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

Page 41: Absorption and Scattering

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

Page 42: Absorption and Scattering

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

Page 43: Absorption and Scattering

Absorption: Beer’s lawBiological Scattering-Elastic:

Rayleigh scattering: 1/λ4

Mie scattering: weak relative to wavelength

Summary