Use and abuse of the effective-refractive-index concept

in turbid colloidal systems

Use and abuse of the effective-refractive-index concept

in turbid colloidal systems

Rubén G. Barrera*Instituto de Física,

Benemérita UniversidadAutónoma de Puebla

Rubén G. Barrera*Instituto de Física,

Benemérita UniversidadAutónoma de Puebla

*Permanent address: Instituto de Física, UNAM

In collaboration with:In collaboration with:

Alejandro Reyes Augusto García

Felipe Pérez

and withand with

Edahí Gutierrez

Also,I acknowledge very interesting discussions with:

Luis Mochán Peter HaleviEugenio Méndez

MotivationMotivation

milk1

1.71n

2n

critical angle

2

1

sin cnn

θ =

states of aggregation 2nδ

internal-reflection configuration

Real time

QuestionQuestion

What is the index of refraction of milk?

…it is white…and turbid…

ColloidColloid

Inhomogeneous phasedispersed within ahomogeneous one

colloidal particleshomogeneous phase

DISORDER

fogliquid aereosolliquidgas

smoke, powdersolid aereosolsolidgas

porous media, opalssolid foamgassolid

milky quartz, …solid emulsionliquidsolid

composites, policrystals, rubys…

solid solsolidsolid

foam, whipped cream…foamgasliquid

oil/water, water/benzeneemulsionliquidliquid

milk, paints, blood, …solsolidliquid

examplesnamedispersephase

continuousphase

Photonic crystals and metamaterials: ordered colloids?

MeasurementsMeasurements

….inconsistencies…

…reflectance around the critical angle…

OPTICAL SPECTRUM

400 800λ≤ ≤ nm

coherent(average)

diffuse(fluctuating)

TurbidityTurbidity coherent

turbid

…light scattering…

Total fieldTotal field E E Eδ= +

3

T

d rV

“on the average”homogeneous and isotropic

ensemble

…probability…

0.12

a mλ µπ≈

Small particlesSmall particles

averagethe diffuse fieldcan be neglected

2 1aka πλ

=size parameter

i.e. macroscopic electrodynamics

nano…

effective medium

effective properties

ε

µ

eff

eff

eff

n

ContinuumElectrodynamics

Effective mediumEffective medium

“unrestricted“

Effective-medium theories

Homogenization theories

[ ] , effn optical structural

Extended effective mediumExtended effective medium

colloid

fluctuations

effective medium

εµ

eff

eff

effn

average

Continuum Electrodynamics

INCOMPLETE !INCOMPLETE !

?

λ µπ

≈ ≈ 0.12

a mBIG

δ∝ = +22 2Power E E E

Energy conservationEnergy conservation

~ 1ka

0 2 40.0

0.2

0.4

0.6

0.8

1.0

diffuse

coherent

penetration

trans

mitt

ed p

ower

AttemptsAttemptsMODEL: Random system of identical spheres

vacuum

0kcω

=

0

0

( )

( ) /

µ µε ε ω

ε ω ε

=

=

=

p

p p

a

n

identical

nonmagnetic

local

ε ω ε σ ωω

≡ +0( ) ( )p pi

van de Hulstvan de Hulst

( )γ = 3

0

32

fk a

⊥ ⊥

⎛ ⎞ ⎛ ⎞⎛ ⎞=⎜ ⎟ ⎜ ⎟⎜ ⎟− ⎝ ⎠⎝ ⎠ ⎝ ⎠

2

1

00

s incikr

s inc

SE EeSikrE E

scattering matrix

= =1 2(0) (0) (0)S S Ssphere

-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

1f dilute

Light scattering by smallparticles (1957)

effnδ

γ= +1 (0)effn i Scomplex

volumefilling

fraction

0 2 4 6 8 100

0.5

1

1.5

2

2.5

effnf

δ ′′

2 aπλ

van de Hulstvan de Hulst scattering

nS = 1.5

nS = 2.8

Craig BohrenCraig 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)

r µ εµ ε−

=+

( )

γ θµ θθ

ε θ γ θ θ θ

−

+ −

= +

= + −

(1)

2

(1) (1) 2

( )( ) 1cos

( ) 1 2 ( ) ( )tan

TE iieff

iTE

i i i ieff

i S

i S S

θ π θ+ = + −⎡ ⎤⎣ ⎦(1)

11( ) (0) ( 2 )2i iS S S

θ π θ− = − −(1)1( ) (0) ( 2 )i iS S S

MAGNETIC

BOHREN

1

1

1 (0) ( )1 (0) ( )

eff

eff

i S Si S S

ε γ π

µ γ π

= + +⎡ ⎤⎣ ⎦= + −⎡ ⎤⎣ ⎦

Normal incidence

1(0) ( 2 )iS S π θ= −

1TEeffµ = NON-MAGNETIC

Small particles

Comment:…a very unconfortable result…

RG Barrera & A García-ValenzuelaRG Barrera & A García-Valenzuela JOSA A 20, 296 (2003)

COHERENT SCATTERING MODEL

Our new resultOur new result

IN TURBID COLLOIDAL SYSTEMS THE EFFECTIVE MEDIUM EXISTSBUT ITS ELECTROMAGNETIC RESPONSE IS NONLOCAL

Electromagnetic response

Our new resultOur new result

ˆind effJ Eσ=

GENERALIZED EFFECTIVE CONDUCTIVITY

TOTAL

LINEAR OPERATOR

PRB 75, 184202 (2007)

3( ; ) ( , ; ) ( ; )i n d effJ r r r E r d rω σ ω ω′ ′ ′= ⋅∫

Nonlocal

local vs nonlocallocal vs nonlocal

ISOLATED SPHERE

( )( ; )

0S

SS

S

r Vr

r Vσ ω

σ ω∈

=∉

⎧⎨⎩

extE

IE

SE

3

( ; ) ( ; ) ( ; )

( , ; ) ( ; )

ω σ ω ω

σ ω ω

=

′ ′ ′= ⋅∫i n d S I

ex tNLS

J r r E r

r r E r d r

NONLOCAL

LOCAL

30

( , ; )

( ; ) ( ) ( , ; ) ( , ; )

NLS

NLS

r r

U r r r I G r r r r d r

σ ω

ω δ ω σ ω

′ =

⎡ ⎤′ ′ ′′ ′ ′′− + ⋅⎢ ⎥

⎣ ⎦∫

Generalized NL conductivityGeneralized NL conductivity

0 ( ; )ωµ σ ωSi r

( , ; )T r r ω′0 ( , ; )ωµ σ ω′NL

Si r r

T matrix

0 ( , ; )ωµ σ ω′NLSi p p ( , ; )ω′T p p

Total induced currentTotal induced current

EXCITINGFIELD

,

3, 1 2 1 1

( ; ) ( ; )

; ) ( ; ; ), , ,... , ,...

i nd i nd ii

i e xc ii

NLS i i i N

J r J r

r r E r r r r r r d rr r

ω ω

ω ωσ − +

=

′ ′ ′= − ⋅−

∑

∑∫NONLOCAL

iextE

+

( , )ω≈ ′E r

Effective-Field ApproximationEffective-Field Approximation

…valid in the dilute regime…

3( ; ) ; ) ( , )( ,i nd ii

NLS iJ r r r E rr r d rω ω ωσ ′ ′= − ′− ⋅∑∫

3( ; ) ; )( , ( , )i nd iNLS i

iJ r r rr r E r d rω ωσ ω′= − ′ ′−∑∫

(| |; )eff r rσ ω′−

GENERALIZED NONLOCAL OHM’S LAW

GENERALIZED NONLOCALCONDUCTIVITY

ω σ ω ω= ⋅( , ) ( , ) ( , )ind

effJ p p E p

=0NnV

Momentum representationMomentum representation

( , ; )NLS r rσ ω′

FT

( , ; )NLS p pσ ω′ ( , ; )NL

S p p pσ ω′ =

0( , ) ( , ; )NLeff Sp n p p pσ ω σ ω′= =

3 X 3 = 9

3

... i

T

d rV

→ ∫

INTEGRAL EQUATION

NONLOCAL

LT schemeLT schemehomogeneous and isotropic “on the average”

σ ω σ ω σ ω= + −ˆ ˆ ˆ ˆ( ; ) ( , ) ( , )(1 )L Teff eff effp p pp p pp

ε ω ε σ ωω

= +0( ; ) 1 ( ; )eff effip p

generalized effective nonlocal dielectric function

ε ω( , )Leff p ε ω( , )T

eff p

2

“tradition”

0pa →

( )ε ω ε ω ε ω= + +2( ) [0] ( )[2]( , ) ( ) ( ) ...L T L T

eff eff effp pa εεε

≡0

Small paSmall pa

→ 0p “LOCAL LIMIT”

NONLOCAL DEPENDENCE

Calculation procedure

Phys. Rev. B, 75, 184202 (2007)

ResultsResults

( , ) 1T p fε ω = + ∆

( , ) 1T pf

ε ω −= ∆

4

5

0

-15

-5

-10

0 1 2 3 4 5 6pa

0.83 µm

0.62 µm

0.45 µm

0.30 µm

0.22 µm

λ0

Re[ ( , )] 1 ( 0.1 )Teff p for Ag radius m

fε ω

µ−

=

(εT(p,ω)/ ε0-1)/f for Ag (radius=0.1µm)

0.03

4

0.2 0.4 0.6 0.8 1.0

3

2

1

0

-1

1.51

0.50.1

0.01pa

λ0[µm]

Re[ ( , )] 1 ( 0.1 )Teff p for Ag radius m

fε ω

µ−

=

Electromagnetic modesElectromagnetic modes

dispersion relation

ε ω =( , ) 0Leff p

ε ω= 0 ( , )Teffp k p

longitudinal

transverse

′ ′′= +p p ip

effective index of refraction

ω ω= 0( ) ( )Teffp k n

ω( )Lp

ω( )Tp

ε ω= 0 ( , )Teffp k p

0 ( 0; )Teffp k pε ω= →

nonlocal

local

GENEALOGY

ComparisonsComparisons

Long wavelength approximation

[0]0 ( )effp k ε ω= ε ω= [0]( )effn

0 ( , )Teffp k pε ω=

Exact

local

0

( )( )T

effpn

kωω =

Quadratic approximation

ε ω ε ω= + +[0] [2] 20 ( ) ( )( ) ...Tp k pa

ε ωε ω

=−

[0]

2 [2]0

( )1 ( ) ( )eff Tn

k a

Light-cone approximation

ε ω= =2 20 0( , )Tp k p k γ= +1 (0)effn i S

nonlocal

nonlocal van de Hulst

ComparisonsComparisons

Exact

QA

LCA

LWA

0.0 0.2 0.4 0.6 0.8 1.0

1.04

1.03

1.02

1.01

1.0

0.99

λ0[µm]

Re[ ] ( 0.1 & 0.02)effn for Ag radius m fµ= =

0.0 0.2 0.4 0.6 0.8 1.0

neff for Ag (radius=0.1µm & f =0.02)0.04

0.03

0.02

0.01

λ0[µm]

Exact

QA

LCA

LWA

Im[ ] ( 0.1 & 0.02)effn for Ag radius m fµ= =

Reflection problemReflection problemnonlocal nature

( , )

( , )

Teff

Leff

p

p

ε ω

ε ω

( )effn ω

( )effn ω cannot be used in local CE (Fresnel’s relations)

translational invariance

( )r rε ′−( ),r rε ′

AbuseAbuse

( ; )Fresnel vdHi effR nθ

IEMM

0( , )Teff p kε ω=

( )vdHeffn ω

iθLCA

nonlocal nature

Isotropic effective-medium model

Internal Reflection configurationInternal Reflection configuration

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

( ; )Fresnel vdHi effR nθ

IEMM

milk?

ComparisonComparison TiO2 / water

Pure water

IEMM

CSM

0 1121.33

aσ

=

=nm

A. Reyes-Coronado, A García-Valenzuela,C. Sánchez-Pérez, RG BarreraNew Journal of Physics 7 (2005) 89 [1-22]

How to measure neff?How to measure neff?

Use refraction (propagation)

Latex spheres / water

ComparisonComparison van de HulsteffFit with n

NEXT STEP CBS

e m schemee m scheme

ε ω ε ω=( , ) ( , )Leff p p

( )µ ω

ε ω ε ω=

− −202

1( , )1 ( , ) ( , )

effT Leff eff

pk p pp

magnetic response !

ind P MJ J J= +

ResultsResults

( )202

1( , )1 ( , ) ( , )

effT Leff eff

pk p pp

µ ωε ω ε ω

=− −

( )202

1 1 ( , ) ( , )( , )

T Leff eff

eff

k p pp p

ε ω ε ωµ ω

− = − −

0 1 2 4 5 63pa

0.2

0.4

0.6

0.8

1.0

0.83 µm0.62 µm0.45 µm0.30 µm0.22 µm

λ0

1 1Re 1 ( 0.1 )( , )eff

for Ag radius mf p

µµ ω⎡ ⎤

− =⎢ ⎥⎣ ⎦

OPTICAL MAGNETISM

ConclusionsConclusions

In turbid colloidal systems the effective index of refraction, due to its nonlocal character, is able to describe the propagation of light, but it cannot describe its reflection

This is important because the naïve use of the effective index of refraction in the calculation of reflection amplitudes has been done many times without too much (intellectual) reflection

There is a nonlocal magnetic response in turbid colloidal systems even when its components are non magnetic (optical magnetism)

We have developed an effective-medium approach to describe the optical properties of turbid colloids in the bulk, that is useful and complimentary to the multiple-scattering approach

PerspectivesPerspectives

ENERGY TRANSFER

1ˆS E H E Bµ−= × = ×

0 ˆ( ) ( )effp E k n p Bω ω× = ×

ˆ ?S p⋅ =

LEFTHANDED ?

LONGITUDINAL MODES

( , ) 0L pε ω = DO THEY EXIST ?

BULK

PROPAGATING MODES

Coherent-beam specroscopyCoherent-beam specroscopy

0( )effn λnonlocal

effective-mediumapproach

particle-size distribution

optical properties of thecolloidal particles

ReflectionReflection

extJ

Coherent beam spectroscopy