Fundamentals of Polarization and Polarizability

Seth R. MarderJoseph W. PerryDepartment of

Chemistry

+-

+-

t1t0 t2

CHARGEDISTRIBUTION

INDUCEDPOLARIZATION

F = qE (1)

Induced Polarization

Electric Field

Molecular polarization = µ = (2)

Polarizability: A Microscopic View

E = 0 E1 E2

Effect of Application of an Oscillating Electric Field such as Light

APPLIED FIELD

INDUCED POLARIZATION

Application of an oscillating electric field will induce an oscillating polarization in a material.

For linear polarizability, the polarization will have the same frequency as the applied electric field.

This induced polarization is a source of light will propagate through the material in the same direction as the light beam that created it.

E o

r P

0

Mechanisms of Polarization

The oscillating electric field of light affects all charges in the optical medium, not only the electrons.

Vibrational polarization and involves nuclear motion. Dipolar molecules can rotate to create molecular polarization. In ionic materials, the ionic motion can cause polarization.

IONIC MOTIONROTATIONALVIBRATIONALELECTRONIC

δ−

δ+

__

____

_ _

_

__

_

+

++

+

++

+

+

++

+

_

+

+ -+ -

+ -+ -+ -

+ -

+ -+ -+ -

+ -+ -+ -

+ -

No Fielδ

With Fielδ

O

M

O

M

Dipoles in Electric Fields

For materials that contain electric dipoles, such as water molecules, the dipoles themselves reorient in the applied field.

2a

θ

+F

–F

θ

P

t

Anisotropic Nature of Polarizability

The polarization of a molecules need not be identical in all directions.

μ

μ

E

E

+-+-+ -+ -+ - +-H3C CH3H3C CH3

dipole moment

xx

αxy

αxz

αyx

αyy

αyz

αzx

αzy

αzz

applied field

induced polarization

Polarizability is a tensor quantity as shown below:

Tensorial Nature of Polarization

Each entry of the tensor is a component of the polarizability

x

y

z

μx

μy

μz

eg. μx xx Ex + xy Ey + xz Ez

Polarizability: A Macroscopic View

In bulk materials, the linear polarization is given by:

Pi() = ij( Ej( (4) i,j

where ij() is the linear susceptibility of an ensemble of molecules

The total electric field (the "displaced" field, D) within the material becomes:

D = E + 4P = (1 + 4E (5)

Since P = E (Equation (4)), 4E is the internal electric field created by the induced displacement of charges (polarization).

The Dielectric Constant

The dielectric constant and the refractive index n) are two bulk parameters that characterize the susceptibility of a material.

e is defined as the ratio of the displaced internal field to the applied field (e = D/E) in that direction:

eij() = 1 + 4ij() (6)

The frequency dependence of the dielectric constant provides insight into the mechanisms of polarization.

e

RADIO VISIBLEMICROWAVE

v + e e

r + v + e

IR

i + r + v + e

Frequency

The Index of Refraction

The ratio of the speed of light in a vacuum, c, to the speed of light in a material, v, is called the index of refraction (n):

n = c/v. (7)

The dielectric constant equals the square of the refractive index:

e() = n2(). (8)

Consequently, we can relate the refractive index to the bulk linear (first-order) susceptibility:

n2() = 1+ (9)

Index of refraction depends therefore on chemical structure.

Harmonic and Anharmonic Potential Surfaces

Harmonic potentials gives rise to linear polarizability

Anharmonic potentials gives rise to nonlinear polarizability

POTENTIAL ENERGY

-X 0 +X

X DISTORTION COORDINATE

HARMONIC POT

+ CUBIC TERM

Asymmetric Polarization

With the addition of this anharmonicity the induced polarization depends on the direction of displacement.

For the covalent C=O bond in acetone, for example, one expects that the electron cloud would be more easily polarized towards the oxygen atom.

_+ _++

_

Applied Field +0 _ChargeDistribution

Pinduced

Induced polarization

Linear

NonlinearApplied field

Pind = +

The application of an oscillating electric field to the electrons in an anharmonic potential leads to an asymmetric polarization response.

This polarization wave has diminished maxima in one direction and accentuated maxima in the opposite direction.

Creation of an Asymmetric Polarization Wave

Fourier Analysis of Asymmetric Polarization Wave

This asymmetric polarization can be Fourier decomposed into a DC polarization component and components at the fundamental and second harmonic frequencies.

Since only the time averaged asymmetrically induced polarization leads to second-order NLO effects, only molecules and materials lacking a center of symmetry possess them.

Expression for Microscopic Nonlinear Polarizabilities

A common approximation is to expand the polarizability as a Taylor series:

μ = μ0 + E (∂μi/Ej)Eo+ (1/2) E·E (2μi/EjEk)Eo +

(1/6) E·E·E (3μi/EjEkEl)Eo + … (10)

μ = μ0 + ijE + (ßijk/2) E·E + ijkl /6) E·E·E + ... (11)

The terms beyond ijE are not linear in E and are therefore referred to as the nonlinear polarization and give rise to nonlinear optical effects.

Expression for Macroscopic Nonlinear Polarizabilities

The observed bulk polarization density is given by an expression analogous to (11):

P = Po + (1) ij ·E j + ((2)

ijk/2) ·· E jE k + (3) ijkl/6)··· E jE kE l + ... (12)

Where: the (i) susceptibility coefficients are tensors of order i+1 (e.g., (2)

ijk). Po is the intrinsic static dipole moment density of the sample.

Taylor Expansion for Bulk Polarization

Or when all the fields are identical:

P = Po + (1)·E + (1/2)(2)·· E2 + (1/6)(3)···E3+ … (13)

Just as a molecule can only have a if it is noncentrosymmetric, a material can only have a (2) if the material is noncentrosymmetric.

(i.e., a centrosymmetric arrangement of noncentrosymmetric molecules lead to zero (2)) .

Taylor Expansion with Oscillating Electric Fields-SHG

Recalling that the electric field of a plane light wave can be expressed as:

E = E0cos(t), (14)

equation (14) can be rewritten as:

P = P + (1)E0cos(t) + E02cos2(t) + (3) E0

3cos3(t) + ... (15)

Since cos2(t) equals 1/2 + (1/2) cos(2t), the first three terms of equation (13) become:

P = (P + (1/2) (2) E02) + (1)E0cos(t) + (2)E0

2cos(2t) + .. (16)

Second Harmonic Generation (SHG) P = (P + 1/2 (2) E0

2) + (1)E0cos(t) + (2)E02cos(2t) + … (16)

Physically, equation (16) states that the polarizationconsists of a:

Second-order DC field contribution to the static polarization (first term),

Frequency component corresponding to the light at the incident frequency (second term) and

A new frequency doubled component, 2 (third term)-- recall the asymmetric polarization wave and its Fourier analysis.

Sum and Difference Frequency Generation

In the more general case, NLO effects involves the interaction of NLO material with two distinct waves with electric fields E1 with the electrons of the NLO material.

Consider two laser beams E1 and E2, the second-order term of equation (4) becomes:

·E1cos(1t)E2cos(2t) (17)

From trigonometry we know that equation (17) is equivalent to:

(1/2)·E1E2cos [(1+2)t] + (1/2)·E1E2cos 1-)t] (18)

thus when two light beams of frequencies and 2 interact in an NLO material, polarization(light) is created at sum (1+2) and difference 1-) frequencies.

Changing the Propagation Characteristics of Light: The Pockels Effect

It is possible to change the amplitude, phase or path of light at a given frequency by using a static DC electric field to polarize the material and modify the refractive indices.

Consider the special case 2 = 0 [equation (17)] in which a DC electric field is applied to the material.

The optical frequency polarization (Popt) arising from the second-order susceptibility is:

(1/2) ·E1E2(cos 1t) (19)

where E2 is the magnitude of the electric field due to the voltage applied to the nonlinear material.

Pockels Effect

Recall that the refractive index is related to the linear susceptibility that is given by the second term of Equation (15):

·E1(cos t), (20)

so the total optical frequency polarization is:

Popt = ·E1(cos 1t) + (1/2) ·E1E2(cos1t) (21)

Popt = [ + (1/2) ·E2] E1(cos t) (22)

Changing the Propagation Characteristics of Light: The Pockels Effect (cont.)

The applied field in effect changes the linear susceptibility and thus the refractive index of the material.

This is, known as the linear electrooptic (LEO) orPockels effect, and is used to modulate light by changing the applied voltage.

At the microscopic level, the applied voltage anisotropically distorts the electron density within the material. Thus, application of a voltage to the material causes the optical beam to "see" a different material with a different polarizability and a different anisotropy of the polarizability than in the absence of the voltage.

Technological Applications Of The Pockels Effect

The Pockels effect has many important technological applications.

Light traveling through an electrooptic material can be phase of polarization modulated by refractive index changes induced by an applied electric field.

Devices exploiting this effect include optical switches, modulators, and wavelength filters.

Design of Molecules for Nonlinear Optics

MOLECULAR ASYMMETRY STATE-OF-ART ca. 1987

N

H3

C C H3

Electron withdrawing group

Conjugated bridge

Electron donating group

N O2

Molecules

75 x -3 esu

μ x-8 esu iμortnt FOM for oleδ

olyμers

Poleδ Polyμers

r33 5 μ V

LiNO3 r33 38 μV

A prototypical NLO chromophore was 4-(N,N-dimethylamino)-4'-nitrostilbene (DANS), which is shown below

Role for Materials ChemistsTAKE SUM-OVER-STATES EXPRESSION FOR :

ijk

+ βikj

= −

e

4 h2

rg ′ n

j

rn

′ n

i

rgn

k

+ rg

′ n

k

rn

′ n

i

rgn

j

( )

n ≠′

n

n ≠ g

′ n ≠ g

∑

1

ω′ n g

− ω( )

ωng

+ ω( )

+

1

ω′ n g

+ ω( )

ωng

− ω( )

⎛

⎝

⎜

⎜

⎜

⎞

⎠

⎟

⎟

⎟

+ rg

′ n

i

rn ′ n

j

rgn

k

+ rg

′ n

k

rn

′ n

i

rgn

j

( )

1

ω′ n g

+ 2 ω( )

ωng

+ ω( )

+

1

ω′ n g

− 2 ω( )

ωng

− ω( )

⎛

⎝

⎜

⎜

⎜

⎞

⎠

⎟

⎟

⎟

+

rg ′ n

j

rn ′ n

k

rgn

i

+ rg ′ n

k

rn ′ n

j

rgn

i

( )

1

ω′ n g

− ω( ) ωng

− 2 ω( )

+

1

ω′ n g

+ ω( ) ωng

+ 2 ω( )

⎛

⎝

⎜

⎜

⎜

⎞

⎠

⎟

⎟

⎟

+ 4 r

gn

j

rgn

k

Δ rn

i

ωng

2

− 4 ω

2

( )+ r

gn

i

rgn

k

Δ r

n

j

+ rgn

i

Δ rn

k

( )ω

ng

2

+ 2 ω

2

( )[ ]

n

∑

1

ωng

2

− ω

2

( )ω

ng

2

− 4 ω

2

( )

⎡

⎣

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

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⎤

⎦

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⇓

TRANSLATE INTO AN OPTIMIZED MOLECULE

⇓INCORPORATE IN AN OPTIMIZED MATERIAL

Bond Length AlternationBond-length alternation (BLA) is defined as the average of thedifference in the length between adjacent carbon-carbon bonds in apolymethine ((CH)n) chain.

SingleBond

DoubleBond

l (2)l (1)

Polyenes have alternating double (1.34 Å) and single bonds (1.45 Å)and thus show a high degree of BLA (+ 0.11 Å).

Resonance Structures and BLA

( C H3

)2

N

( C H3

)2

N OO

-( C H

3)

2N

+

+

N ( C H3

)2 N ( C H

3)

2( C H

3)

2N

+

-+

Decreasing Magnitude of BLA

Decreasing Energy Gap

Electric Field Perturbation of Structure

-0.1

-0.05

0

0.05

0.1

BLABOA

0.6

0.4

0.2

0

-0.2

-0.4

-0.6

0 2 4 6 8 10 12 14F (10 7 V/cm)

(+) (-)

δ+ δ-

N O

Me

Me

N O

Me

Me

N O

Me

Me

APPLIED FIELD

• An electric field can increase charge separation in the ground –state of molecules

• This in turn modifies the BLA, the Bond Order Alternation (BOA) and the dipole moment

Linear Polarizability and BOA

∝ (μ 2ge Ege

)

20

40

60

80

100

120

αxx (10 -24 esu)

(arb. units)

-0.6 -0.4 -0.2 0 0.2 0.4 0.6BOA

Legend:l : αxx

O: αmodel

∆: μ2ge

r : 1

 E  ge

• The linear polarizabilty is peaked at BOA = O called the cyanine limit

First Hyper-polarizability and BOA

∝ (μ 2ge(μee-μgg) E 2

ge )

Oudar, Chemla, Garito and Lalama

-800

-600

-400

-200

0

200

400

600

800

βxxx (10 -30 esu)

-0.6 -0.4 -0.2 0 0.2 0.4 0.6BOA

Legend:l : βxxx

O: βmodel

+: (μee -μgg )

• β is peaked between polyene and cyanine limit

• Beyond cyanine limit β is negative

• Most aromatic molecules tend to be near polyene limit with small β

Factors Affecting Charge Separation

(CH3

)2

N OO

-(CH

3)

2N

+

CHARGE SEPARATION

(CH3

)2

N+

O-

(CH3

)2

N O

CHARGE SEPARATION & LOSS OF AROMATICITY

N

O

CH3

O-

N

CH3

+

CHARGE SEPARATION & GAIN OF AROMATICITY

Manipulation of BLA Through Topology

+

( C H3

)2

N ( C H3

)2

N

O

N

O

N

O

P hP h

O

-

.. ..

( C H3

)2

N

S

O

ON C

N C

( C H3

)2

N

S

O

ON C

N C

+

–

S

SN

C N

N C

C N

S

SN

C N

N C

C N

_

+

Tradeoff on Aromatic Stabilization Energy

Between Donor and Acceptor in Neutral

and CT VB Structure

Decreasse Aromatic Stabilization

Energy in Neutral VB Strucutre

Increasse Effective Conjugation Length

in CT VB Strucutre

Good, But Not Good Enough

(CH3

CH2

CH2

CH2

)2

N

SO

O

NC

CN

μ = ~13,500 x 10-48 esu at 1.907 μm; μβ(0) = ~4,000 x 10-48 esu.

Material n reff.(pm/V)

n3reff.(pm/V)

n3reff./ε(pm/V)

LiNbO3 2.2 31 330 12

SandozPolymer

1.7 55 270 ~45

However: sub-optimal thermal stability: 60%decomposition after 20 min. @ 150 C.

Second Hyper-polarizability and BOA

∝ – (μ 4ge E 3

ge ) + ∑

e'

( μ 2ge E 2

ge

μ 2ee' E  ge'

) + (μ 2ge(μee-μgg)2 E 3

ge )

Pierce, Garito, Kuzyk and Dirk

-2.500 104

-2.000 104

-1.500 104

-1.000 104

-5000

0

5000

1.000 104

1.500 104

γxxxx (10 -36 esu)

-0.6 -0.4 -0.2 0BOA

0.2 0.4 0.6

Legend:

l : γxxxx

O: γmodel

X: ∑e'

(μ2ge E2ge

μ2ee' E  ge'

)

+: (μ2ge(μee-μgg)2 E3ge

)◊ : – (μ4ge

 E3ge )

• γ has a rich structure as a function of BOA• Imaginary part of γ gives rise to two-photon absorption

Third-order Nonlinear Optical Properties of Polarized Polyenes

Chemical Stucturelμxnμ

-3esu

6

O 464 1.9

CN

CN

540 2.5

N

N

O

O S

590 8.1

NC

CN

O

680 19

S

O

O

NC

CN

740 45

Molecular Two-Photon Absorption--an Imaginary Side of Third Order

Nonlinearity

hnF

Two-photonAbsorptivity

δ FluorescenceFfl

hnA

S0

S2

S1

Sn

hnA Electron Transfer

Energy TransferPhotochemistry

hnA

Two-Photon Excited Processes

Two-Photon Processes Provide 3-D Resolution

z

TPA I2

TPA z-4

I -2

Excitation by two photons is confined to a volume very close to focus where intensity is highest , giving rise to pinpoint 3D resolution

Excitation by one photon results in absorption along the entire path of the laser beam in the cuvette.

TPA Provides Improved Penetration of Light Into Absorbing Materials

Excitation by one photon results in absorption by surrounding medium before beam reaches sample

Excitation by two photons of half the energy allows for penetration through the material, and then two photons can be absorbed by the sample

Effect of bis-Donor Substitution

δ ≈ 10 x 10-50 cm4 s photon-1 δ ≈ 200 x 10-50 cm4 s photon-1

NN

E

1Ag

1Bu

2Ag

7.4 D

8.9 D

3.9 eV4.8 eV

Δ5 eV7 Δ

3 Δ5 eV5 eV

Δ8 eV

δS0 →S2

∝ M012 M12

2

(E1 −E0 −hω )Γ

Proposed Model to Enhance TPA in Symmetrical Molecules

-0.15

-0.1

-0.05

0

0.05

0.1

N Phenyl Vinyl Phenyl N

Group

-C

harg

e D

iffer

ence

N

N

BDAS has large and symmetrical charge transfer from nitrogens to central vinyl group that is associated with large transition moment between S(1) and S(2).

These results suggest that a large change in quadrupole moment between S(0) and S(1) is leads to enhanced δ

Strategies for the Design of New Materials

D--D

DD

n

D

D

A

A

Increase conjugation length

Also:

D----D

A--D--A

Add electron acceptors to the backbone

Chain-Length Dependence

With increasing chain length: δ increases l(2)

max red-shifts

Method: Two-photon induced fluorescence (TPF)

Pulse duration: ≈ 5 ns

I

II

III

IV

NN

BuBuBu

Bu

N

NBu

BuBu

Bu

OMe

MeO

NBuBu

NBu

Bu

OMe

MeO

NBuBu

NBu

Bu

OMe

MeO

OR

RO

OMe

MeOR=C12H25

Design of TPA Chromophores

NN

N

N

N

NMeO

OMe

D-A-D D--D A-D-A

N

NCN

NC

N

N CN

NCBu

Bu

Bu

Bu

Hex

Hex

Hex

Hex

C12H25O

OC12H25

S

CN

NCO

O

S

NC

CNO

O

OMe

MeO

OMe

MeO

12

53

210

995

1250

1940

2300

4700

δ in 10-50 cm4 s/photon Albota et al., Science 1998