Refraction and light transportin turbid colloids:

Is the Poynting vector ill defined?

Refraction and light transportin turbid colloids:

Is the Poynting vector ill defined?

Rubén G. BarreraInstituto de Física, UNAM

Mexico

Rubén G. BarreraInstituto de Física, UNAM

Mexico

ETOPIM 8

In collaboration with:In collaboration with:

Felipe PérezLuis Mochán

EdahíGutierrez

Augusto García

Motivation 1Motivation 1

milk11.71n

2n

critical angle

2

1

sin cnn

θ =

refractive index of milk

…but is white…and turbid…

Absorption

k′

k′′

Inhomogeneous wave

1θ2Θ

0

BS Eµ

= × 0indE J⋅ >

0k k′ ′′⋅ >

Motivation 2Motivation 2

k k ik′ ′′= +

V.A. Markel, Optics Express 16 (2008) 19152

Q

S E H= ×

negative refractionimposible

BHµ

=

ColloidColloid

Inhomogeneous phasedispersed within ahomogeneous one

colloidal particleshomogeneous phase

continuousphase

dispersephase name examples

liquid solid sol Milk, paints, blood, tissues

liquid liquid emulsion oil in water

liquid gas foam foam, whipped cream

solid solid solid sol composites, policrystals, rubys

solid liquid solid emulsion milky quatz, opals

solid gas solid foam porous media

gas solid solid aereosol smoke, powder

gas liquid liquid aereosol fog

Colloidal systemsColloidal systems

“Ordered” colloids“Ordered” colloids

Photonic crystals Metamaterials

coherent

diffuse

Light scatteringLight scatteringcoherent

turbidity

Effective mediumEffective medium

colloid

fluctuations

effective medium ?

εµ

eff

eff

effn

average

Continuum Electrodynamics

Our resultOur result

IN TURBID COLLOIDAL SYSTEMSTHE EFFECTIVE MEDIUM EXISTSBUT IT IS NONLOCAL

3( ; ) (| |; ) ( , )i nd effJ r r r E r d rω σ ω ω= ′ ′ ′−∫the probability density Is homogeneous

ω σ ω ω= ⋅( , ) ( , ) ( , )ind

effJ k k E k

Spatial dispersion

. Phys. Rev. 75 (2007) 184202 [1-19].

total

LT schemeLT scheme

Probability density is homogeneous and isotropic

σ ω σ ω σ ω= + −ˆ ˆ ˆ ˆ( ; ) ( , ) ( , )(1 )L Teff eff effk k kk k kk

ε ω ε σ ωω

= +0( ; ) 1 ( ; )eff effik k

generalized effective nonlocal dielectric function

ε ω( , )Leff k ε ω( , )Teff k

2[0] [2]

20

( , ) ( ) ( ) ...L L kkk

ε ω ε ω ε ω= + +

Long wavelength2

[0] [2]20

( , ) ( ) ( ) ...T T kkk

ε ω ε ω ε ω= + +

( )0ka →

2 2/ cω

“local limit”

ˆ kkk

≡

ε µ

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

Ag (radius=0.1µm)

ε ω −Re[ ( ; )] 1T pf

Phys. Rev. 75 (2007) 184202 [1-19].

0

-5

-10

Ag (radius=0.1µm)

3.5

0 1 2 3 4 5 6pa

32.5

21.5

10.5

0.83 µm

0.62 µm

0.45 µm

0.30 µm

0.22 µm

λ0

ε ω −Im[ ( ; )] 1T pf

Phys. Rev. 75 (2007) 184202 [1-19].( )effn ω

Poynting TheoremPoynting Theorem

( )* *

*

0 0 0

1 1Re Re2 2

B B BE E Eµ µ µ

⎛ ⎞⎛ ⎞ ⎛ ⎞∇ ⋅ × = ⋅ ∇× − ⋅ ∇×⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠

( )* *

* *0

0

1Re2

ext indB B EE J J Et t

εµ

⎛ ⎞∂ ∂= − ⋅ − ⋅ + − ⋅⎜ ⎟

∂ ∂⎝ ⎠

( ) ( )εµ µ⎛ ⎞⎛ ⎞ ∂ ⎜ ⎟∇ ⋅ × + + = − ⋅ − ⋅⎜ ⎟ ⎜ ⎟∂⎝ ⎠ ⎝ ⎠

2*2 * *0

0 0

1 1 1Re Re Re Re2 2 2 2 2

ext indBBE E J E J Et

extW QPlane waveextW Q→

Math

Maxwell

WORKWORK

( ) ( )* 2 *01 1Re Re 12 2ind

indW J E i n E Eωε⎡ ⎤= ⋅ = − ⋅⎣ ⎦

220 Im2

n Eω ε= 2Im 0n >Q

k ′

k ′′

( )22 2 20k k k k ik k n′ ′′= ⋅ = + =

Inhomogeneous wave

2 2 2 20 Rek k k n′ ′′− =

′ ′′⋅ = 2 202 Imk k k n 0>

Negative refraction is impossible

( )20 1indJ i n Eωε= − free-propagating mode

dispersion relation

Negative Refraction at Visible FrequenciesHenri J. Lezec,1,2*† Jennifer A. Dionne,1* Harry A. Atwater1

Nanofabricated photonic materials offer opportunities for crafting the propagation anddispersion of light in matter. We demonstrate an experimental realization of a two-dimensional negative-index material in the blue-green region of the visible spectrum, substantiated by direct geometric visualization of negative refraction. Negative indices were achieved with the use of an ultrathin Au-Si3N4-Ag waveguide sustaining a surface plasmon polariton mode with antiparallel group and phase velocities. All-angle negative refraction was observed at the interface between this bimetal waveguide and a conventional Ag-Si3N4-Ag slot waveguide. The results may enable the development of practical negative index optical designs in the visible regime.

Science 316, 430 (2007)

Nonlocal calculationNonlocal calculation

*

2 20

0 0

1 1Re2 2 2 ext ind

BE E B W W

tε

µ µ

⎛ ⎞ ⎛ ⎞∂⎜ ⎟∇ ⋅ × + + = − −⎜ ⎟⎜ ⎟ ∂⎜ ⎟ ⎝ ⎠⎝ ⎠

( )*1 Re2indJ E⋅

3( ; ) ( ; ) ( ; )indJ r t dt d r r r t t E r tσ′ ′ ′ ′ ′ ′= − − ⋅∫ ∫quasi-monochromatic wave

( ) ( ) 00 0 0( , ) ( , ) ...∂′ ′ ′ ′= + − ⋅∇ + − +⎡ ⎤⎣ ⎦ ∂EE r t E r t r r E t tt

0( , ) Re ( , ) exp[ ]E r t E r t ik r tω= ⋅ − 0 0E kEα∇0

0E Etα ω∂

∂

ε ω ε σ ωω

= +0( ; ) 1 ( ; )eff effik k

( ){ 0 0( , ) exp[ ( )] ( , ) ( , )indJ r t i k r t i k E r tα αβ αβ βω ω ε ω ε δ= ⋅ − − −

0 00

( , ) ( , ) ( , ) ( , )( )

k E r t k E r tk x t

αβ β αβ βαβ αβ

ε ω ε ωω ε ε δ ω

γ γ ω

⎫⎡ ⎤∂ ∂ ∂ ∂ ⎪− + − +⎢ ⎥ ⎬∂ ∂ ∂ ∂⎢ ⎥ ⎪⎣ ⎦ ⎭

Induced currentInduced current

( )*1 Re2indJ E⋅

2[0] [2]

20

( , ) ( ) ( ) ...L L kkk

ε ω ε ω ε ω= + +2

[0] [2]20

( , ) ( ) ( ) ...T T kkk

ε ω ε ω ε ω= + +

Im ReL Lε ε Im ReT Tε ε

long wavelength 0ka → “local”

2 2

0 0Im ( , ) Im ( , )2L L T T

indW k E k Eω ε ω ε ω⎡ ⎤= +⎢ ⎥⎣ ⎦

( ) ( )[2] 2* [2]* *

0 0 0 0 00 0 0 0

1 1 ( )Re ( )2

LT L L TE B k E kE Eε ωε ω

ε µ ωε µ⎡ ⎤

−∇⋅ × + +⎢ ⎥⎣ ⎦

[0] 2 [2] [2]2 22 20 0 0 0 0

0 0

1 ( ) ( ) ( )Re2

L TL TkE E E E

tωε ω ε ω ε ω ε

ω ε µ ω ω ω ω⎡ ⎤⎛ ⎞∂ ∂ ∂ ∂

+ + + −⎢ ⎥⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠⎣ ⎦

Work on the induced currentsWork on the induced currents

E

*

2 20

0 0

1 1Re2 2 2 ext ind

BE E B W W

tε

µ µ

⎛ ⎞ ⎛ ⎞∂⎜ ⎟∇ ⋅ × + + = − −⎜ ⎟⎜ ⎟ ∂⎜ ⎟ ⎝ ⎠⎝ ⎠

Energy theoremEnergy theorem

Transverse waves

( )( )[0]

2 2* [2]*0 0 0 0 0

0 0

1 1 1 1 ( )Re 1 ( ) / Re2 2

TE B B Et

ωε ωε ω εµ µ ω⎡ ⎤ ⎡∂ ∂

∇ ⋅ × − + +⎢ ⎥ ⎢∂ ∂⎣ ⎦ ⎣

2 [2] 2 [2]2 2[0]

0 020 0 0 0

( ) ( )Im ( )2

T T

extk kE W Eε ω ω ε ωε ωε µ ω ω ω ε µ

⎤ ⎡ ⎤∂+ = − − +⎥ ⎢ ⎥∂ ⎦ ⎣ ⎦

( )( )* [2]*0 0 00

1 1Re 1 ( ) /2

TtransS E B ε ω εµ

⎡ ⎤= × −⎢ ⎥

⎣ ⎦

e m schemee m scheme

( , )( , ) ( , )ind P r tJ r t M r tt

∂= +∇×

∂

( )0( , ) ( , ) expP r t P r t i k r tω⎡ ⎤= ⋅ −⎣ ⎦

( )0( , ) ( , ) expM r t M r t i k r tω⎡ ⎤= ⋅ −⎣ ⎦

( )0( , ) ( , ) expind indJ r t J r t i k r tω⎡ ⎤= ⋅ −⎣ ⎦

quasi-monochromatic

NOT UNIQUE

( ) ( )0 0 0 0( , ) ( , ) ( , ) ( , )l L t TP r t k P k P E r tε ω ε ε ω ε⎡ ⎤= − + − ⋅⎣ ⎦

0 00

1 1( , ) ( , )( , )

M r t B r tkµ µ ω

⎡ ⎤= −⎢ ⎥⎣ ⎦

ResponseResponse

CHOICE I

( , ) (0, )t tkε ω ε ω=

CHOICE II

( , ) ( , )t lk kε ω ε ω=

( , )( , ) ( , )ind P r tJ r t M r tt

∂= +∇×

∂

( , )I kµ ω ( , )II kµ ω

* * [2]**0 0

0 0 00 0 0

1 ( )Re2

t

transB BS E M E ε ωµ µ ε

⎡ ⎤⎛ ⎞= × − − ×⎢ ⎥⎜ ⎟

⎢ ⎥⎝ ⎠⎣ ⎦

Poynting vector in emPoynting vector in em

0HCHOICE I

[2]*( ) 0tε ω = [2]* [2]* [2]( ) ( ) ( )t l Lε ω ε ω ε ω= =

CHOICE II

long wavelength limit 0 00 0

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

B k Eµ µ ω µ µ ω ω⎡ ⎤ ⎡ ⎤

− = − ×⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

(0, )Iµ ω (0, )IIµ ω

( )( )* [2]*0 0 00

1 1Re 1 ( ) /2

TtransS E B ε ω εµ

⎡ ⎤= × −⎢ ⎥

⎣ ⎦

Poynting vector in LTPoynting vector in LT

The correct expression of the Poynting vector

[2]*( )Tε ω is NOT a non-local correction

This is the LOCAL limit

ConclusionsConclusions

(i) We used the formalism developed for the treatment of the non-local effective medium associated to turbid colloids to find a general expression for the energy theorem, in the long wavelength limit, in terms of parameters associated to the non-local response of the system.

(ii) We found an explicit expression for the Poynting vector in terms of these non-local parameters and show that there are many ways of writing itin terms of the electric permittivity e and the magnetic permeability m, depending on the choice taken to define them.

(iii) This approach can be extended to finite wave vectors and non negligible dissipation as well as energy transport for free-propagating modes