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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 reemits light in all other
directions.
Biomedical Optics
Tissue optical properties
E. Göran Salerud
Department Biomedical Engineering
Molecule
Light source
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.
Spherical waves
A spherical wave is also a solution to Maxwell's equations and is a good
model for the light scattered by a molecule.
•
where k is a scalar, and
•
r is the radial magnitude.

E( r ,t) ∝ ( E0 / r ) exp[i(kr − ω t)]
A spherical wave has spherical wave-fronts.
Unlike a plane wave, whose amplitude remains constant as it
propagates, a spherical wave weakens. Its irradiance goes as 1/r2.
Scattered spherical waves often combine to
form plane waves
A plane wave impinging on a surface will produce a reflected plane wave
because all the spherical wavelets interfere constructively along a flat
surface.
The interference is measured in one direction
at time and far away
This way we can approximate spherical waves by plane waves in that
direction, vastly simplifying the math.
The mathematics of scattering
The math of light scattering is analogous to that of light sources.
If the phases aren t random, we add the fields:
Coherent
Etotal = E1 + E2 + … + En
Far away,
spherical wavefronts are almost
flat…
Usually, coherent constructive interference will occur in one direction, and
destructive interference will occur in all others.
Itotal = I1 + I 2 + ... + I N + ce Re{E1E2* + E1E3* + ... + EN -1EN* }
I1, I2, … In are the irradiances of
the various beamlets. They re all
positive real numbers and add.
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:
To understand scattering in a given situation,
we compute phase delays
Because the phase is
constant along a wave-front,
we compute the phase delay
from one wave-front to
another potential wave-front.
If the phase delay for all
scattered waves is the same,
then the scattering is
constructive and coherent.
If it varies uniformly from 0
to 2p, then it s destructive
and coherent.
Incoherent
Coherent constructive scattering
A beam can only remain a plane wave if there s a direction for which
coherent constructive interference occurs.
Wave-fronts
Consider the different
phase delays for
different paths.
L1
qi qr
L2
L3
L4
Scatterer
Potential
wave-front
fi = k Li
Incident
wave-front
Potential
outgoing
wave-front
If it s random (perhaps due to random motion), then it s incoherent.
Coherent constructive interference occurs for a reflected beam if the angle of
incidence = the angle of reflection: qi = qr.
Coherent destructive scattering
Coherent scattering occurs in one direction, with
coherent destructive scattering occurring in all others
•Imagine that the reflection angle is too big.
•The symmetry is now gone, and the phases are now all different.
f = ka sin(qtoo big)
qi qtoo big
a
A smooth surface scatters light coherently and constructively only in the
direction whose angle of reflection equals the angle of incidence.
f = ka sin(qi)
Potential
wave front
Coherent destructive interference occurs for a reflected beam direction if the
angle of incidence ≠ the angle of reflection: qi ≠ qr.
Incoherent scattering:
reflection from a rough surface
No matter which direction we look
at it, each scattered wave from a
rough surface has a different
phase. So scattering is incoherent,
and we ll see weak light in all
directions.
Looking from any other direction, you ll see no light at all due to coherent
destructive interference.
Why can t we see a light beam?
Unless the light beam is propagating right into your eye or is scattered into it,
you won t see it. This is true for laser light and flashlights.
This is due to the facts that air is very sparse (N is relatively small), air is also
not a strong scatterer, and the scattering is incoherent.
This eye sees almost no light.
This eye is blinded
(don t try this at home…)
This is why rough surfaces look different from smooth surfaces and mirrors.
To photograph light beams in laser labs, you need to blow some smoke
into the beam…
On-axis vs. off-axis light scattering
Off-axis light scattering: scattered
wavelets have random relative
phases in the direction of interest
due to the often random place-ment
of molecular scatterers.
Forward (on-axis) light
scattering: scattered wavelets
have nonrandom (equal!)
relative phases in the forward
direction.
Randomly spaced scatterers in a plane
Incident
wave
Off-axis scattering is incoherent
when the scatterers are randomly
arranged in space.
Path lengths are random.
Light scattering regimes
Particle size/wavelength
~1
Large
Rayleigh-Gans Scattering
Mie Scattering
Totally reflecting objects
Geometrical optics
Air
Rayleigh Scattering
Relative refractive index
Large
~1
~0
For large particles, we must first consider the fine-scale scattering from the
surface microstructure and then integrate over the larger scale structure.
If the surface isn t smooth, the scattering is incoherent.
If the surfaces are smooth, then we
use Snell s Law and angle-of-incidenceequals-angle-of-reflection.
Incident
wave
Forward scattering is coherent—
even if the scatterers are randomly
arranged in space.
Path lengths are equal.
Scattering from particles is much stronger than
that from molecules
There are many
regimes of particle
scattering, depending
on the particle size,
the light wave-length,
and the refractive
index.
Rainbow
This plot considers only single scattering by spheres. Multiple scattering and
scattering by non-spherical objects can get really complex!
Then we add up all the waves resulting
from all the input waves, taking into
account their coherence, too.
Scattering
• The light scattered by a tissue has interacted
with the ultrastructure of the tissue.
• Tissue ultrastructure extends from
membranes to membrane aggregates to
collagen fibers to nuclei to cells.
• Photons are most strongly scattered by those
structures whose size matches the photon
wavelength.
Scattering possibilities
Scattering
• Scattering provides feedback during therapy.
10 µm
cells
nuclei
• during laser coagulation of tissues, the onset of
scattering is an observable endpoint
mitochondria
• Scattering has diagnostic value.
1 µm
lysosomes, vesicles
• the density of lipid membranes in the cells, the
size of nuclei, the presence of collagen fibers, the
status of hydration in the tissue.
0.1 µm
striations in collagen fibre
macromolecular aggregates
0.01 µm
Scattering coefficient
membranes
Scattering Coefficient
Scattering coefficient
Scattering cross-section
• The scattering coefficient (µs) describes a medium
containing many scattering particles and is defined
as:
• The scattering cross-section (ss) is defined as:
µs = rss s
Where,
ss is the scattering cross-section (cm2)
rs is a volume density (cm-3)
Where,
s s = Qs A s
Qs is the scattering efficiency (can be calculated from Mie
theory)
As is the area of the scatterer (cm2)
Scattering Coefficient
Scattering coefficient µs
Reduced scattering coefficient
efficiency
abs. crosssectional area
(µs’)
• Lumped property
incorporating the scattering
(µs) coefficient and the anisotropy factor (g):
σ s = Qs A
geometrical area
[cm 2 ] = [−][cm 2 ]
µs ' = µs (1 - g )
Where,
µs is the scattering coefficient (cm-1)
σ s = Qs A
A
geometrical
cross-section
effective
cross-section
ECE532 Biomedical Optics
©Steven L. Jacques, Scott A.
Prahl
Oregon Graduat Institute
Mean Free Path
Scattering coefficient µs
µs = ρ sσ s
ls =
scattering crosssectional area
density
The probability of transmission T of the photon without
redirection by scattering after a pathlength L is:
•
µs
•
ls
•
Most tissues
1
µs
100 cm-1
= 0.1 mm
µ s > 50 cm-1 (prostate)
< 1000 cm-1 (tooth emanel)
T = exp[−µsL]
24
Anisotropy g
Scattering Phase Function
azimuthal
angle y
deflection
angle q
Differential scattering cross section:
scattering in direction s from input direction s’
dσ s (ŝ, ŝ ')
dΩ
y
The angular dependence of scattering is
q
p(ŝ ⋅ ŝ ') =
4π dσ s (ŝ ⋅ ŝ ')
σ s +σ a
dΩ
cos(q)
photon
trajectory
scattering
event
•
Often the scattering phase function does not depend on input direction:
p(θ)
•
p(θ) describes the probability of a photon scattering into a unit solid
angle, relative to the original photon trajectory
•
p(θ) has historically been called the scattering phase function
26
Scattering Anisotropy
The proper definition of anisotropy (g) is the expectation value for cos (θ):
Effectiveness of Scattering
g≡
∫ p(ŝ ⋅ ŝ ')ŝ ⋅ ŝ 'dΩ
∫ p(ŝ ⋅ ŝ ')dΩ
∫
• The proper definition of anisotropy (g) is the
expectation value for cos (q):
1
π
p(θ )cosθ 2π sin θ dθ
0
=
∫ p(cosθ )cosθ d(cosθ )
-1
1
π
where
∫
0
p(θ )2π sin θ dθ = 1
Expression for anisotropy
Sometimes also written
in terms of cos(θ)
g =< cos (θ )
=
Scattering Coefficient
where
∫ p(cosθ )d(cosθ ) = 1
−1
g =< cosq >
Scattering Anisotropy
•
Anisotropy is a measure of forward direction retained after a
single scattering event (mean value of cos(θ))
" −1 Total backward scattering
$
g=# 0
$ 1
Total forward scattering
%
Scattering function
• The angular dependence of scattering is called the
scattering function, p(q) which has units of [sr-1] and
describes the probability of a photon scattering into a
unit solid angle oriented at an angle relative to the
photons original trajectory.
BiologicalTissues,0.65<g>0.95
29
Scattering function
Scattering functions
• Plotting p(q) indicates how photons will
scatter as a function of q in a single plane of
observation
• Plotting p(q) 2psinq indicates how photons
will scatter as a function of the deflection
angle q regardless of the azimuthal angle y,
in other words integrating over all possible y
in an azimuthal ring of width dq and
perimeter 2psinq at some given q .
ECE532 Biomedical Optics
©Steven L. Jacques, Scott A.
Prahl
Oregon Graduate Institute
Isotropic scattering
Albedo
π
1
p(θ ) =
, such that
4π
p(ŝ ⋅ ŝ ') =
∫ p(θ )2π sinθdθ = 1
4π dσ s (ŝ ⋅ ŝ ')
σ s +σ a
dΩ
Fraction of light energy incident on a
scatterer or absorber from direction s that
gets scattered into direction s prime.
0
g=0
∫ p(ŝ ⋅ ŝ ')dΩ = σ
4π
µs
=
+
σ
µ
s
a
s + µa
Albedo, in tissue can range from 0.3 to
0.99 depends on wavelength
Ratio of scattering versus total attenuation
Scattering in tissue is dominant!
34
Henyey-Greenstein scattering function
Henyey-Greenstein
Heyney Greenstein scattering phase function is an analytical expression
which mimics the angular dependence of light scattering by small particles
and is based on the anistropy factor g (used for Monte Carlo Simulations)
• The function allows the anisotropy factor, g to specify
p(θ) such that <cos(θ)> returns the same value of g
• This function is useful in approximating the angular
dependence of single scattering events in biological
tissue
π
1
1− g 2
p(θ ) =
, such that
2
4π (1+ g − 2g cosθ )3/2
∫ p(θ )2π sinθ dθ = 1
0
π
and
∫ p(cosθ )d(cosθ ) = 1
• The function does not represent true scattering
phase functions very well but it is a good average
approximation
0
p(θ ) =
1
1− g 2
, such that
2
2 (1+ g − 2g cosθ )3/2
π
∫ p(cosθ )d(cosθ ) = 1
0
π
and
∫ p(θ )2π sinθ dθ = 1
0
36
Different anisotropy values
Reduced scattering coefficient
• µs' = µs(1 - g) [cm-1]
• The purpose of µs' is to describe the diffusion of
photons in a random walk of step size of 1/µs' [cm]
where each step involves isotropic scattering.
• This occurs if there are many scattering events
before an absorption event, i.e., µa << µs'.
• This situation of scattering-dominated light transport
is called the diffusion regime and µs’ is useful in the
diffusion regime which is commonly encountered
when treating how visible and near-infrared light
propagates through biological tissues.
ECE532 Biomedical Optics
©Steven L. Jacques, Scott A.
Prahl
Oregon Graduate Institute
Example, reduced scattering coefficient
Importance of light scattering
• Important because
g = <cos(q) = 0.90
• light propagation is affected by the tissue optical properties,
26o
• the physiological condition or state of single cells or tissues
is expressed through (but not exclusively) changes in cell
size or refractive index,
<q> ≈
one mfp´
• changes in refractive index or cell size influence the optical
properties.
µs = µs(1-g) = 0.1
mfp = 1/µs
mfp = 1/µs
ten mfp
Therefore measurements or analysis of scattering provide
information about the tissue.
Derived Characteristic properties
•
Four important properties
•
Cross section
•
•
absorption
•
scattering
•
extinction
Cross sections
Scattering cross section
σs =
4π
σt
4π
∫
Albedo
p(s,i)dω
4π
W0 =
Absorption cross section
σa
Angular dependence
•
∫ σ d dω =
scattering phase function
σs
σ
= s
σs + σa σt
Note: W0 is close to
zero for most tissues
Back-scatter cross section
σ b = 4 πσ d (−i,i)
Extinction cross section
σt = σs + σa
σ b = differential scattering cross section
Scattering cases
•
Size parameter x = 2πa /( l/nmed)
•
Refractive index ratio nr = np/nmed
•
Rayleigh approximation
•
the particle a dipole, strength related to its volume
•
valid if a<<1 or until a is 5% of l
•
proportional to l-4
Rayleigh scattering
•
Scattered intensity
I s ( R,θ ) = (1+ cos 2 θ )
•
k4
2
α Ii
2
2R
where k=angular wavenumber and nr=refractive index ratio
k=
2π nmed
λ
nr =
np
nmed
Rayleigh scattering
•
•
Scattering cross-section
σs =
Rayleigh scattering
8π k 4 2
α
3
Polarizability of a sphere
n2 − 1 3
α = 2r
a
nr + 2
Scattering cross-section
8π a 2 x 4 nr2 − 1
σs =
, x = ka
3
nr2 + 2
2
•
•
•
Size parameter
x = ka
Scattering efficiency
8x 4 nr2 − 1
Qs =
3 nr2 + 2
2
•
Diameter of sphere: 2a = 20 nm
•
Wavelength in vacuum: l = 400 nm
•
Refractive index of sphere: ns = 1.57
•
Refractive index of background: nb = 1.33
•
Specific weight of sphere: rs = 1.05 g/cm3
•
Specific weight of background: rb = 1.00 g/cm3
•
Concentration of spheres in background by weight: Cwt = 10-5
•
ss = 2.15x10-20 m2
•
Qs = 6.83x10-5
Mie theory model for tissue optical properties
•
Mie theory describes the scattering of light by particles, with
refractive index (np) that differs from the refractive index of its
surroundings (nmed).
•
The dipole re-radiation pattern from oscillating electrons in the
molecules of such particles superimpose to yield a strong net
source of scattered radiation.
•
•
Also, the re-radiation patterns from all the dipoles do not cancel
in all but the forward direction of the incident light as is true for
homogeneous medium, but rather interfere both constructively
and destructively in a radiation pattern.
Hence, particles "scatter" light in various directions with varying
efficiency.
Mie
• Mie's classical solution is described in terms of two
parameters, nr and x:
•
the magnitude of refractive index mismatch between particle and
medium expressed as the ratio of the n for particle and medium,
nr = np/nmed
•
a
the size of the surface of refractive index mismatch which is the
"antenna" for re-radiation of electro-magnetic energy, expressed as
a size parameter (x) which is the ratio of the meridional
circumference of the sphere (2pa, where radius = a) to the
wavelength (l/nmed) of light in the medium,
x = 2pa /( l/nmed)
Mie calculations
MIE scattering
• A Mie theory calculation will yield the efficiency of
scattering which relates the cross-sectional area of
scattering, ss to the true geometrical cross-sectional
area of the particle, A = pa2
{Q s,p(θ )} = Mie(
np
2πa
,
)
n med λ /n med
σ s = Q s πa 2
• ss = QsA
• Finally, the scattering coefficient is related to the
product of scatterer number density rs, and the
cross-sectional area of scattering, ss
µs = rsss
π
g=
∫ p(θ )cosθ ⋅ 2π sinθdθ
0
π
∫ p(θ ) ⋅ 2π sinθdθ
0
The math of Mie scattering
Scattering matrix
•
Describes the relationship between incident and scattered
electric field components perpendicular and parallel to the
scattering plane as observed in the "far-field”
⎡E lls ⎤ exp(−ik(r − z)) ⎡S2 S3 ⎤⎡E lli ⎤
⎢ ⎥=
⎢
⎥⎢ ⎥
ikr
⎣E ⊥s⎦
⎣S4 S1 ⎦⎣E ⊥i ⎦
•
S is known as the "amplitude scattering matrix”
•
The total field (Etot) depends on the incident field (Ei), the
scattered field (Es) , and the interaction of these fields (Eint). If
one observes the scattering from a position which avoids Ei,
then both Ei and Eint are zero and only Es is observed.
Scattering matrix
• For the "far-field” observation of Es at a distance r
and particle of diameter d such that:
kr >> nc2
2π
k=
λ
d
nc =
λ
• Then the scattering elements S3 and S4 are equal to
zero.
Measured scattering matrix
• Practical experiments measure intensity,
I =< EE * >= (1/2)a 2 ,where E = aexp(−iδ )
⎡ Ills
⎢
⎢⎣ I⊥ s
⎡ S
⎤
2
⎥ = constant ⎢
⎢ 0
⎥⎦
⎢⎣
2
0
S1
2
Stokes vector
Angular scattering pattern of polarized light
⎤⎡
⎥ ⎢ Illi
⎥ ⎢ I⊥ i
⎥⎦ ⎣
Angular patterns of Mie scattering
Click here for Mie calculator
http://omlc.ogi.edu/calc/mie_calc.html
⎤
⎥
⎥⎦
Scattered light intensity for random polarised
Mie theory calculation of anisotropy (g)
π
2
2
1
1
S11 (θ ) = S1 (θ ) + S2 (θ )
2
2
where S11 (θ ) = first element in Mueller matrix
g=
∫ S (θ ) cos(θ )2π sin(θ )dθ
11
0
π
∫ S (θ )2π sin(θ ) dθ
11
I s = S11 * I i
0
p(θ ) =
Mueller Matrix, a 4x4 matrix which relates an input vector of Stokes
parameters (Ii, Qi, Ui, Vi) describing a complex light source and the
output vector (Is, Qs, Us, Vs) describing the nature of the transmitted
light.
S11 (θ )
π
∫ S (θ )2π sin(θ ) dθ
11
0
Scattering vs wavelength
Mie scattering from cellular
structures
•
Soft tissue optics are dominated by the lipid content of the tissues.
•
Mie theory provides a simple first approximation to the scattering of soft tissues
•
The approximation involves a few assumptions:
•
Assume the refractive index of the lipid membranes of cells is 1.46, based
on the reported
•
Assume the refractive index of the cytosol of cells is 1.35, based on the reported value
for cellular cytoplasms
•
Assume the lipid content of soft tissue is about 1-10% (fv = 0.02-0.10).
•
Let's choose fv = 0.02 for this example to match the value for several typical soft
tissues such as lungs, spleen, prostate, ovary, intestine, liver, arteries, to name a few.
•
Assume all the lipid is packaged as small spheres of various sizes whose number
density maintains a constant volume fraction fv.
•
Ignore the interference of scattered fields from particles which can alter the apparent
scattering properties based on isolated particles.
Scattering vs wavelength
Scattering Assumptions
•
The optical properties of the particle are completely described
by the refractive index
•
The medium in which the particles are embedded is considered
to be homogeneous
•
Only the interactions of a single particle with light of arbitrary
wavelength are considered
•
The total scattered field is merely the sum of the fields scattered
by each particle (i.e. the particles do not affect each other).
•
For a collection of particles, the number of particles is large and
their separations are random
ECE532 Biomedical Optics
©Steven L. Jacques, Scott A.
Prahl
Oregon Graduate Institute
62
Scattering vs wavelength
Scattering vs wavelength
ECE532 Biomedical Optics
©Steven L. Jacques, Scott A.
Prahl
Oregon Graduate Institute
ECE532 Biomedical Optics
©Steven L. Jacques, Scott A.
Prahl
Oregon Graduate Institute
Summary optical properties
Summary
• Optically tissue may be characterized by its
•
•
• scattering, refractive index, and absorption.
• The scattering arises from
• cell membranes, cell nuclei, capillary walls, hair follicles...
• The absorption arises from
• visible and NIR wavelengths (400 nm - 800 nm);
•
» hemoglobin and melanin,
• IR wavelengths;
•
» water and molecular vibrational/rotational states.
A survey of all tissue types has not been completed.
A survey would only be a guideline since subject to subject
variation and normal vs diseased variation apparently are
sufficiently significant to warrant ad hoc measurements on any
particular tissue site of interest.Other tissues may present different
types of scattering than just the simple "soft tissue" and "dermis"
models.
•
Muscles have myoglobin and actin-myosin fibers.
•
Brain has myelin sheaths around nerves.
•
Fatty tissues are distinct in their properties. The scattering by
the nucleus has been ignored here, but is a potential scattering
site of clinical importance.
Wavelength dependence in tissue
Wavelength dependence in tissue
• Absorption and scattering decreases as a function
of wavelength
• Ratio of scattering to absorption coefficient
increases with the wavelength
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