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