g. barrera - ifunam€¦ · rubén g. barrera instituto de física, unam mexico rubén g. barrera....
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Rubén
G. BarreraInstituto de Física, UNAM
Mexico
Rubén
G. BarreraInstituto de Física, UNAM
Mexico
Rubén
G. BarreraInstituto de Física, UNAM
Mexico
Rubén
G. BarreraInstituto de Física, UNAM
Mexico
Recent Advances on the Effective Optical Properties of
Turbid Colloids
Recent Advances on the Effective Optical Properties of
Turbid Colloids
In collaboration with:In collaboration with:
Edahí
GutierrezAugusto
García Celia Sánchez Pérez
Felipe Pérez Luis Mochán Alejandro Reyes
The
problem
milk
blood optical
properties
index
of
refraction
of
complex
fluids
Refractive
index
blood1
1.71n
2n
refractometer
2
1
sin cnn
θ =
Abbe-type
′ ′′= +2 2 2n n in
refraction
reflectance
μ μ=2 0non-magnetic
ε ε ′ ′′= = +2 2 0 2 2/n n in
Fresnel´s
relationsFIT
actually
θ′ ′′22 2( , ; )ir n n
critical-angle
refractometry
Applications
states of aggregation2nδreal time
δ→ +2 2 2n n nchanges
in structure
1.3462 1.3489...3 410 10− −−
MODEL
1.3333
mn 2n 2nδ
Index
of
refraction
in continuum
electrodynamics
Electrodynamics
of
continuous
media
E E Eδ= +
10 20 30 40
-0.6-0.4
-0.2
0.20.4
0.6
Average
spatial average
average fluctuations
λ
macroscopiccoherent
E
Average power
( ) ( ) ( )δ δ δ⎡ ⎤= × = + × + = +⎣ ⎦
1 1Re * Re *2 2
S E H E E H H S S
S E H E Hδ δ= × + ×
Plane
waves
cohS
diffuseS
one neglects the power carried bythe diffuse field
coh diffuseS S
not
enough
S
cohS
diffuseS
∂= +∇×∂indPJ Mt
PJ MJ
Average induced
current
indρindJ
0indindJ
tρ∂
∇ ⋅ + =∂
material
Traditional
approach
ind indρ ρ→
ind indJ J→
( , )P M
AVERAGE
polarization
magnetization
P
M
MATERIAL FIELDS
ε= +0D E PDisplacementfield
εμ
scheme
0
BH M
μ= −H
field
Linear materials
Tradition
ε= ˆD E 1ˆH Bμ−=
Isotropic
and
homogeneous
“on the average”
Frequency
domain
3( , ) (| |; ) ( , )D r r r E r d rω ε ω ω′ ′ ′= −∫ 3 1( , ) (| |; ) ( , )H r d r r r B rω μ ω ω−′ ′ ′= −∫
( , )kε ω1
( , )kμ ω
non-local
spatial
dispersion
electric
permittivity magnetic
permeability
time dispersionNLa
Local optics
Index
of
refraction
0( ) ( ) /n ω ε ω ε=
( ) ( )n i nω ω′ ′′= +“continuum”
NLa λ
( 0, ) ( )kε ω ε ω→ = ( 0, ) ( )kμ ω μ ω→ =
ε ω ε ω ε ω′ ′′= +( ) ( ) ( )i
dissipation
0( )μ ω μ≈
long-wavelength
limit ( 0)k →
0( , )ε μ
How
about
inhomogeneous
materials?
Can one
extend
the
continuum
approach?
colloids
dispersed
phase
/ homogeneous
phase
colloidal
particles
/ matrix
EXAMPLES
“ordered”
colloids
turbiditytransparency
SIZE
Optical
properties
0
2 anka πλ
=
2a
size
parameter 1ka 1ka ∼
gold
colloids
diffuse cohS S diffuse coh
S S∼
Effective-medium
approach
small
particles 1ka
( , )eff effε μ
eff eff effn ε μ=
electrodynamicsof
continuous
media
optical
properties
always
possible…although
it
may be difficult
advantage
UNRESTRICTED
continuum
homogenization
0
0
i effeff z z
p i effeff z z
k krk k
ε εε ε
−=
+
Fresnel´s
relation
0 0eff
eff effk k n k ε= ≈
Reflection
& refraction
unrestricted
( , )eff effε μ
effk
effn1n1.71
measurement
critical-anglerefractometry
Big
colloidal
particles
1ka ∼
diffuse cohS S∼
turbidity diffraction
Effective
medium
Is
there
an
effective
medium?
coherent
diffuse
incident
…. for the coherent beamIf
there
is
one, it
should
be
If
there
is
one, the
theory
should
be…
incomplete
continuum
scattering… as... dissipation
0 2 40.0
0.2
0.4
0.6
0.8
1.0
diffuse
coherent
penetration
trans
mitt
ed p
ower
Energy
conservation
effective
properties… coherent
beam…
∝ = +2 2 2
AV flucPower E E E
van de Hulst
( )γ = 3
0
32
fk a
effnδ
1 (0)effn i Sγ= +
dilute
limit
complexLight scattering by smallparticles (1957)
1γ
Effective
index
of
refraction
first attempts
Scattering
matrix
θθ⊥ ⊥
⎛ ⎞ ⎛ ⎞⎛ ⎞=⎜ ⎟ ⎜ ⎟⎜ ⎟− ⎝ ⎠⎝ ⎠ ⎝ ⎠
2
1
( ) 00 ( )
s incikr
s inc
SE EeSikrE E
= =1 2(0) (0) (0)S S S
sphere
MIE
-1 0 1 2 3 4 5 6 7-2.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0 TiO2/resina = 0.10 μm
dCsca/dΩ
θ
0 2 4 6 8 100
0.5
1
1.5
2
2.5
0 2 4 6 8 10
-0.5
0
0.5
1
1.5
2
Effective
index
of
refraction
2.8Sn =
2.8Sn =
2 aπλ
2 aπλ
effnf
δ ′′
δ ′effnf
Is this effective-mediumtheory unrestricted?
Van de Hulst
compare with experiment
( )30
3 (0 )2e ff
Sn i fk a
δ =
Comparison
with
experiment
critical-angle
refractometer
( )iR θA García-Valenzuela, RG Barrera,C. Sánchez-Pérez, A. Reyes-Coronado,E Méndez, Optics Express, 13, 6723 (2005)
Results
TiO2
/ water
Pure
waterunrestricted
experiment
σ=
=0 112
1.23a nmLog-normal
2.73
1.3313p
m
n
n
=
=
0 6350λ = nm
•
Results
latex
/ water
2.73
1.3313p
m
n
n
=
=
0 6350λ = nm
1.47pn =
0 186.5a = nm
Pure
waterunrestricted
experiment•
Why?
Our
result
IN TURBID COLLOIDAL SYSTEMS AN EFFECTIVE MEDIUM EXISTS, BUT IT IS NONLOCAL
( , )
( , )
ε ω
μ ωeff
eff
k
k
magnetic
response
Nonlocal
nature
of
the
electrodynamic
response of
colloidal
systemsRubén G. Barrera, Alejandro Reyes-Coronado & Augusto García-
ValenzuelaPhysical Review B 75, 184202 (2007)
Fresnel´s
equations
are not
valid
spheres
Ag
/ vacuum μ= 0.1a m
ka
μ ω⎡ ⎤
−⎢ ⎥⎣ ⎦
1 1Re 1( , )k f
μ ω⎡ ⎤
−⎢ ⎥⎣ ⎦
1 1Re 1( , )k f μ= 0.1a m
ka
spheres
TiO2
/ vacuum
dispersion
relation
( , ) 0Leff kε ω =
0 ( , )Teffk k kε ω=
longitudinal
transverse
k k ik′ ′′= +
effective
indexof
refraction
( )Lk ω
( )Tk ω
0( ) ( )Teffk k nω ω=
Electromagnetic
modes
( , )
( , )
ε ω
μ ωeff
eff
k
k
( , )
( , )
ε ω
ε ω
Leff
Teff
k
k
NON-LOCAL
“generalized”
Approximations
Local (long wavelength)
0 0( 0, ) ( )Teff effk k k kε ω ε ω= → =
( ) ( )effn ω ε ω=
0 0( ) ( , )T Teffk k kω ε ω=
0( ) ( , ) 1 (0)Teffn k i Sω ε ω γ= → +
Light-cone
approximation
(LCA)
ωω =0
( )( )T
effkn
k
LOCAL NONLOCALOR
LCA is close to the exact…van de Hulst has a non-local ancestryThus it is restricted
How to measure the effective refractive index?
Critical-angle
refractometry
●
Reflectance from a medium with a non-local response
Reflectance
Although
there
are attempts…there is still not a reliable solution to the reflectance problem…
Reflectance
of
a half-space
with
an
effective
non-local response
( , )
( , )
ε ω
μ ωeff
eff
k
k
translation-invariance
Surface
region
(| |; )
(| |; )
ε ω
μ ω
′−
′−eff
eff
r r
r r
( , ; )
( , ; )
ε ω
μ ω
′
′eff
eff
r r
r r
“bulk like”
…but there will be one soon…
2a∼ANISOTROPIC
Practical
problems
Surface
sensitive
3
if 0( ) 0 if 0.
jj
j
j
d rzp r Vz
⎧>⎪= ⎨
⎪ <⎩
0z =
Probability
interface
+
++++
+
++
+
surface
region
An
alternative
experimental set
up to
measurethe
effective
refractive
index
Non-local refraction
=i effk ksin ( )sinθ ω θ=i eff tn
local or
non-local ancestry
absorption
( , , )θ ′ ′′→eff eff in N n n
weak dependence
“generalized”
Snell’s Law
“surface regions”
“Probability interfaces”
Generalized
Snell’s Law for complex
refractive indices
“bulk”z
2a
“surface region”
“probability interface”
effk
iθ
Experimental setup for refraction and extinction measurements
A. Reyes-Coronado, A García-Valenzuela,C. Sánchez-Pérez, RG BarreraNew
Journal
of
Physics
7 (2005) 89 [1-22]
Latex particles in solution
f = 1.2%
1.2 ml H2 O incial
Diameter: 0.31 μm
1.2 ml of water + 0.15 ml in solution + 0.25 ml in solution
+ 0.35 ml in solution + 0.45 ml in solution + 0.55 ml in solution
Particle
sizing
0 ( , ) ( , )effi k k T k kωμ σ ′ ′→
Effective-medium
& multiple-scattering
3( , ) ( , ; ) ( , )ind effJ k k k E k d kω σ ω ω′ ′ ′= ⋅∫
“generalized”
conductivity
TOTAL
Journal
of
Quantitative
Spectroscopy
&Radiative
Transfer
79–80 (2003) 627–
647
J. Opt. Soc. Am. A/Vol. 20, No. 2/February
2003
coherent-scattering model
. 1
. 2pol s mpol p m
== ( )
γ = 30
32
fk a
dilute
regime
BULK ( , ; )eff k kσ ω
effective
medium scattering
theory
2
2
( 2 )( ) (0)
m ii eff iz z z
k Sri k k k k S
γ π θγ
−=
+ −
Results
TiO2
/ water
Pure
waterunrestricted
experiment
σ=
=0 112
1.23a nmLog-normal
2.73
1.3313p
m
n
n
=
=
0 6350λ = nm
•CSM
CONCLUSIONS
We
have
provided
arguments
for
the
use of
refraction
as a secureway
for
the
experimental determination
of
the
effective
index
ofrefraction
in turbid
colloids.
0.0 0.2 0.4 0.6 0.8 1.0
neff
for Ag (radius=0.1μm & f =0.02)Im
[nef
f]0.04
0.03
0.02
0.01
λ0
[μm]
Exact
QA
LCA
LWA
LCA is close to the exact…
van de Hulst has a non-local ancestryThus it is restricted
Craig
Bohren
1 (0)effn i Sγ= +
11 ( )effn i Sγ π= +
transmission
reflection
ε γ π
μ γ π
= + +⎡ ⎤⎣ ⎦= + −⎡ ⎤⎣ ⎦
1
1
1 (0) ( )1 (0) ( )
eff
eff
i S Si S S MAGNETIC ?
Proposition
J. Atmos Sci. 43, 468 (85)
μ ε
μ ε
−=
+eff eff
eff eff
r
multiple-scattering
theory Effective-medium
approach
Half-space
dilute
EFA excpE E≈
2
2
( 2 )( ) (0)
m ii eff iz z z
k Sri k k k k S
γ π θγ
−=
+ −
0Z =
ik
effkrk
Journal
of
Quantitative
Spectroscopy
&Radiative
Transfer
79–80 (2003) 627–
647
J. Opt. Soc. Am. A/Vol. 20, No. 2/February
2003
coherent-scattering model
. 1
. 2pol s mpol p m
==
multiple-scattering
theory
( )γ = 3
0
32
fk a
Effective
medium
1 (0)effn i Sγ= +
1effμ =
eff effnε =
assume
to
be unrestricted
0.1f =
Fresnel’s relations
assume to be unrestricted
Comparison
with
experiment
critical-angle
refractometer
( )iR θA García-Valenzuela, RG Barrera,C. Sánchez-Pérez, A. Reyes-Coronado,E Méndez, Optics Express, 13, 6723 (2005)
A García-Valenzuela, RG Barrera,C. Sánchez-Pérez, A. Reyes-Coronado,E Méndez, Optics Express, 13, 6723 (2005)
Comparison
with
experiment
critical-angle
refractometer
A García-Valenzuela, RG Barrera,C. Sánchez-Pérez, A. Reyes-Coronado,E Méndez, Optics Express, 13, 6723 (2005)
Internal reflectionconfiguration
great
sensitivity
TiO2
/ water
( )iR θ
( ; , 1)Fresnel vdHi eff effR nθ μ =
Comparison
with
experiment
assume to beunrestricted
22
2
( 2 )( ) (0)
m ii eff iz z z
k SRi k k k k S
γ π θγ
−=
+ −
Coherent
scattering
model
“surface regions”
“Probability interfaces”
(b)Generalized
Snell’s Law for
complex refractive indices.
2a 2ad
“bulk”
z
“probability interfaces”
“surface regions”
(a)
“bulk”
z
2a
“surface region”
“probability interface”
“bulk like”
ik
effktθ
iθ