Gdansk Lecture: materials for optics
P. Audebert
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Outline
Introduction
–
Basics on light and matter
Fluorescent molecules and materials.
–
–
–
–
–
What is fluorescence-theory
Fluorescent molecules
Fluorescent materials
Plasmon resonnance and sensing
Applications
Molecules and materials for NLO
–
–
–
–
–
Second order
Third order
Non-linear absorption
Molecules and materials for NLO
Figures of merit and influence of size.
–
Conclusion
INTRODUCTION:
Recalling what light is.
Wavelengths of “Light”
nm: for near UV, visible,
and near IR light
mm: for IR and far IR light
(sometimes wavenumbers
preferred, n = 10000/l if n
in cm-1 and l in mm)
Typical
range of
IR spectra
recording
Å: for x-ray. But in this
regime people usually use
photon energy in eV.
We have
eV 
1240
l (nm)
Light Wave
Plane electromagnetic wave
E x (x, t)  E0 cos (wt - kz  f0 )
 Re[ E0 exp( jf0 ) exp j (wt  kz)]
 Re[ Ec exp j (wt  kz)]
– k: propagation constant or wave
number
k  2
 w: angular frequency
– Phase of the wave (wt –kz+f0)
Wave front : A surface over
which the phase of a wave is
constant.
Optical field : refers to the
electrical field Ex.
Ec  E0 exp( jf0 )
/l
Ex
Direction of Propagation
k
x
z
z
y
By
An electromagnetic wave is a travelling wave which has time
varying electric and magnetic fields which are perpendicular to each
other and the direction of propagation, z.
© 1999 S .O. Kasap,Optoelectronics (Prentice Hall)
Traveling wave along Z
Propagation of Light
Light is a kind of electro-magnetic wave. In the
general case the field varies with all space
ordinates (in addition to time)
A: amplitude vector. f: phase.
Wave Vector and Wave number
Wave Vector, k : Use to indicate the direction of
propagation. The vector whose direction is normal to the
wavefront, and magnitude is k = 2/l.
For a plane wave, A is constant, and
k
f  k  r  wt
The magnitude of k, k = 2/l, is
also called the wave number.
Phase velocity
The relationship between time and space for a given
phase, f, that corresponds to a maximum field, can be
described by:
f  wt  kz  f0  const
So, during a time interval dt, this constant phase (max. field) moves
a distance dz. From the relation above it comes :
wdt  kdz  0
Therfore it defines the phase velocity of this wave as:
v  dz
where
dt
 w / k  l ,

is
frequency
(w  2)
Basics of fluorescence
What happens to molecules upon
photoexcitation?
Fluorescence deals with light reemission after absorption; It competes with
plenty of other phenomena that can also occur after a photon absorption.
Absorption is a linear process, which occurs when the incident photon energy
matches a molecule/atom orbital gap + some additionnal conditions…
Optical absorption basics: What are the
possible transitions in a simple molecule?
Not all transitions are allowed (there are symetry rules) and
some of them, eg the n* are associated to a partial charge
transfer (results in a increase of the transition dipole).
Singlet and triplet states
From Hund’s rule, the
triplet state lies always
below the singlet
state. Conversion is
sometimes possible,
but not always.
Transmittance and absorbance; the
Beer-Lambert law.
Experimentally, the efficiency of light absorption at a wavelength l by an
absorbing medium is characterized by the absorbance A(l) or the
transmittance T(l) , defined as
In a (very) large majority of case, the absorption of a solution is given by the
Beer-Lambert law below. The unit of e is therefore L.mol-1.cm-1
Absorption coefficients and cross section
We can define the decadic absorption coefficient:
And the Naperian absorption coefficient:
Which allows to introduce in turn the
molecular absorption cross-section:
Relation between s and e
This is exactly the Beer-Lambert law with
e = (1/2.3)Nas
Examples of e values
The molar absorption coefficient is a very widespread value to
estimate the absorption efficiency of a given compound. Here are
reported the values for classical organic chemicals and dyes (at
maximum).
Origin of emission from a molecule : The
Perrin-Jablonski diagram.
Emission (or non-emission) from a molecule :
The time scale for the processes.
Absorption and emission from a molecule :
The fine structure.
Molecules can be in different vibrational states; the relative proportion of
molecules in the different states is given by the Boltzmann law:
N0/N1 = exp[-(E1-E0)/kT]
This can induce a fine structure in the spectrum, if the vibrationnal levels
have enough spacing.
In the case of anthracene, the
spacing is around 1400 cm-1,
which comes to 2.8 10-20 J, and
has the consiquence that virtually
all molecules are in the ground
state (N0/N1 = 0.001). In this case
the spectrum has the shape
represented on the left.
In the general case, the levels are tighter spaced (quasi continuum) which
leads to overlap between absorption and fluorescence spectrum)
Summary of all the possibilities for
desactivation of a molecule.
Each process can be favoured according to the position of the
different energy levels and the molecular structure (presence of
heavy atoms favour intersystem crossing.
Fluorescence life-times
Once a molecule has been excited by absorption of a photon, to its excited state
that we will call A*, it has therefore several paths of deactivation, fluorescence
being one of them. This is quite well exemplified in the scheme below:
We can call knr the constant summarizing all the non radiative processes, against
kr which summarizes the radiative ones (mainly fluorescence). The disparition of
A* follows a classical 1rst order kinetics, and its life-time can be measured.
Fluorescence intensity
The fluorescence intensity is directly linked to the amount of excited
molecules still remaining inside the solution, and the radiative rate constant:
Most of the time the decay is monoexponential, and parallels what is
observed in radioactive decay, although with much faster decay rates!
Quantum yields
A very important property for a fluorescent molecule is the radiative quantum
yield, that is, the proportion of reemitted light against the absorbed light.
The fluorescence yield is therefore nothing else than the ratio of the radiative
rate constant against the sum of the deexcitation constants.
Or otherwise:
It is also possible, on the same basis, to define the yield for the
intersystem crossing (isc) and the phosphorescence, which are usually
lower than for fluorescence.
Some values for classical fluorophores
Aromatic hydrocarbons are usually good fluorophores, here are
some examples with life-times and quantum yields.
Emission spectra and Stokes shift
Since the quantum yield concerns all photons emitted from a molecule,
it can also be described from the integral of the emission spectrum.
The Stokes shift is a very important parameter, which describes the
energy gap (often expressed in nm) between the absorption and the
emission spectra.
Examples of Stokes shift
Examples of a large and small Stokes shift in two classical dyes,
a benzoazinone and a rhodamine.
Heavy atom effect
The presence of heavy atoms in fluorescent molecules has huge
effects on the intersystem crossing, and favors the phosphorescence
at the expense of fluoresence, especially with bromine and iodine, as
exemplified with the naphtalene derivatives below.
Fluorescence quenching
The excited state of a molecule can react with several type of
substrates, exchanging energy, electrons or chemical species
(mainly protons) leading to fluorescence quenching.
The kinetic analysis is very similar to deactivation processes,
except that it is now a bimolecular rate! (which can comes to a
1rst order kinetics in case of quencher excess)
Fluorescence quenching : Main paths
Summary of all possible deactivation paths:
Fluorescence quenching : Three main
situations, relatively to the process.
1) The excited state of a molecule can react immediately with quencher in large
excess (interactions already exist between the quencher and the fluorophore): We
have extinction of part of the fluorophores. Two life times can be distinguished
according to the association of the quencher with the fluorophore or not.
2) The quencher is not in larger excess, but the life-time of the fluorophore is short
enough and long-range interactions (eg energy transfer) can occur. Again, part of
the fluorophore that are in the vicinity of the quencher are extinct, while others are
not. This case is kinetically analogous to 1) for part of the fluorophores, and again
two life times can be distinguished according to the presence or not of the
quencher in the vicinity of the fluorophore. In the two above cases, the response
are concentration dependent.
These cases are called : Static quenching.
3) The quencher is not in large excess, and transport can occur during the quenching
process (long life-time and/or fast diffusion). Then the pseudo first order may not
applies any longer. This case may be more complex.
This last case is called « dynamic quenching » and the apparent rate constant
sometimes change with time.
Static fluorescence quenching : Illustration
Fluorescence quenching : Calculation of the two
cases of static quenching
In the first case (sphere of effective quenching) the quenching efficiency is related
to the number of quenchers, equal to Na Q Vq , where Q is the quencher
concentration, Vq the sphere volume, and Na the Avogadro number. It can be
shown that:
In the second case (preequilibrium) there is an equilibrium M + Q = [MQ]. MQ
does not fluoresce while the fluorescence of M is unaffected. Therefore:
And, at steady state:
Dynamic fluorescence quenching : Stern-Volmer
kinetics
This is what happens in cases 1) or 2) (for selected fluorophores),
let be M the fluorophore, and Q the quencher, we have:
It comes to:
Since the fluorescence intensity is proportionnal to the M*
concentration:
Fluorescence quenching : Stern-Volmer kinetics (2)
Since the fluorescence intensity decay is therefore a single exponential,
whose characteristic time comes from the factor inside the exponential:
And therefore we have the relation, known
as the Stern-Vomer law:
For quantum yields
we have :
Fluorescence quenching : Stern-Volmer kinetics (3)
Under steady-state illumination, we have:
Where I0 and I are the steady-state fluorescence intensities in the
absence and presence of quencher respectively, and KSV = kq t0 Q,
proportionnal to the quenching rate, is called the Stern-Volmer
constant. The relation is called the Stern-Volmer relation.
Fluorescence quenching : Summary, including lifetime dependance.
The table below shows the different I/Q and lg(I)/t curves that can be
expected from the various mechanisms previously detailed.
Examples of classical fluorophores and
their syntheses.
Very classical fluorescent laser dyes (1)
Rhodamine
Rhodamine 6G
Coumarines (coumarine and umbelliferone)
Tétracene
Malachite green
Classical laser dyes (2)
Acridine orange
Acridine yellow
Pyrilium dye
Cyanine
Fluorol
Phenoxazine dye
Cresyl violet
Other fluorescent dyes
Flavanthrone
quinophtalone
isoindolinone
isoindoline
Classical fluorescent dyes: Metal complexes and analogues.
Fluorophore
Anchoring
group
BODIPY TR-X
Iridium complex
Magnesium
tetraphenylporphyrin
Zinc octaethylporphyrin
Magnesium phtalocyanin
Fluorescein
(A. von Baeyer, 1871)
Fluorescein : synthesis
Fluorescein: pH sensing
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2 excitation lex1 lex2, 1 emission lem
Rhodamin synthesis and activation
Synthesis of the core
Functionnalisation
Synthesis of indolium dyes (1)
Fischer
indole
synthesis
Synthesis of indolium dyes (2)
Near
infrared
dye
Sensing with fluorescence
What kind of parameters may modulate
fluorescence ?
Electric potential
H bonding
ions
ions
pressure
Fluorescence
pH
viscosity
quenchers
temperature
polarity
pO2
pH
Ion sensing fluorophores
Calcium green
Sodium green
I
IFAH
pH sensing : principles
single wavelength measurement
AH
pH increase
IFAH = a.C0 with a eAH , FAH
l
I
IF
AH = H+ + Bl
I
IFB-
IF = a.[AH] + b.[B]
with C0 = [A] + [B]
B-
IFB = b.C0 with b eB , FB
I
IFAH
pH sensing : principles
single wavelength measurement
AH
I  I AH
= F F
[AH]
IB I
F
F
pH increase
[B ]
l
I

IF
l
I
IFB-


pH  pKa  log B  log [B ]

[AH]
AH
AH = H+ + B-

I  I AH
pH  pKapp  log F  F
IB I
F
F
Bl

Disadvantages of single wavelength
measurement
Measurements of all intensities MUST be done
in the very same conditions
Measurements of IFAH and IFB are difficult in-vivo
Calibration may depend upon [probe]
Needs for ratiometric measurements
(independent of [probe])
pH increase
I
pH sensing : principles
dual wavelength measurement
AH
I(l )
R 1
I(l )
2
I
I
l1 l2
l
AH = H+ + B-

RR
A
pH  pKapp  log
R R
B
+ log 2
b
2

l
RR
A
pH  pKapp  log
R R
B
B-

l
a
I (l )
+ log A 2
I (l )
B 2
Advantages of dual wavelength
measurement
Independant of source fluctuations intensity
Independant of instrument sensitivity
Independant of [probe]
Fluorescein
2 excitation lex1 lex2, 1 emission lem
Fluorescent polymers
Organic polymers model
vide
Energie
*
(LUMO)
*
(LUMO)
*
(LUMO)

(HOMO)

(HOMO)

(HOMO)
Ethylène
BC
Butadiène
Octatétraène
2
*
(LUMO)

(HOMO)
BV
(pleine)
Polyène
n
As much as the conjugation length increases, the levels get
closer and closer
Fluorescent polymers (of interest in emitting devices)
O
MEH-PPV
PPV family
n
O
R
R
Polyfluorene
family
n
PPV SYNTHESIS
The Wessling route
ClH2C
S
S
CH2Cl
CH3OH
T = 65°C
Cl
Cl
S
1) NaOH, MeOH/H2O, T = 0°C
or Bu4NOH, MeOH, T = 0°C
2) HCl
Cl
S
Cl
S
n
Quinodimethane
Soluble precursor
Precursor conversion :
S
Cl
THT
+
T = 180-300°C
n
+
HCl
S
vacuum 12h
n
Insoluble
BY PRODUCTS
PPV SYNTHESIS
The Wessling route
Mechanism of the polymerisation of the para quinodimethane has not been
completely elucidated yet
Cl
S
- radical polymerisation?
- anionic propagation?
R. A. Wessling, J. Polym. Sci., Polym. Symp., 72, 55-66, (1985)
Molar mass determination :
S
Cl
T = 50°C
n
OCH3
CH3OH
n
SOLUBLE
Mn > 100 000 Da
PPV DERIVATIVES SYNTHESIS
MEH-PPV synthesis
O
MEH-PPV
O
n
hEL = 1%
ITO/MEH-PPV/Ca
D. Braun, A. J. Heeger, Appl. Phys. Lett., 58, 1982, (1991)
Synthesis : Gilch polymerisation
O
OH
O
tBuOK
Br
HCHO
ClH2C
HCl dioxane
KOH, EtOH re flux
O
O
O
CH2Cl
THF, T = 20°C
O
O
PPV DERIVATIVES SYNTHESIS
Dialkoxy-PPV derivatives
* A lot of polymers have been prepared following the previously described
synthetic route.
* Two homopolymers have emerged :
O
O
O
n
O
MEH-PPV
max
lEL =
610nm hPL = 15%
n
OC1C10-PPV
Philips Hoechst
max
max
lEL = 592nm lPL = 575nm
PPV DERIVATIVES SYNTHESIS
Side reaction in the GILCH polymerisation
OR2
ClH2C
OR2
1 eq tBuOK
H2C
CH2Cl
CH
Cl
R1O
R1O
Side reaction
Main reaction
OR2
OR2 Cl
OR2
Cl
OR 2
Cl
Cl
OR2
n
OR1
OR1
OR1
Cl
OR 1
OR1
1 eq tBuOK
1 eq tBuOK
Tolane bis benzyl moiety (TBB)
OR 2
OR 2
OR 2
OR2
OR 2
n
R1O
OR 1
R1O
R1O
R1O
For OC1C10-PPV, the defect concentration is in the range of 1.5 – 2.2%
H. Becker et al., Macromolecules, 32, 4925, (1999)
PPV DERIVATIVES SYNTHESIS : Cyano PPV
Synthesis of polymer with high electron affinity
The Wessling route is ineffective when e- withdrawing substituents are involved
Knoevenagel condensation
OC6H13
NC
CN
NaCN
C6H13O
OC6H13
OC6H13
ClH2C
CH2Cl
C6H13O
OC6H13
CN
t-BuOH/THF 50°C
t-BuOK or Bu4NOH
1) NaOAc
C6H13O
NC
C6H13O
OC6H13
2) KOH, EtOH
OHC
3) PCC
n
CN-PPV
CHO
C6H13O
The synthetic route is flexible
CN-PPV
MEH-CN-PPV
OC6H13
many cyano PPV derivatives
O
OC6H13
CN
C6H13O
NC
C6H13O
OC6H13
O
CN
CN
CH3O
n
NC
CH3O
C6H13O
n
NC
n
PPV DERIVATIVES SYNTHESIS : Heck coupling reaction
Preparation of alternating copolymers derived from PPV :
OR2
OR2
I
I
Et3N, Pd(OAc)2 , DMF
+
P
3
R1O
n
R1O
R. Heck, Org. React.,27, 345, (1982)
Precursor synthesis :
OR2
OR2
I2, HIO 3
I
I
H2SO4, AcOH, CCl4
R1O
R1O
* The Heck coupling reaction approach is versatile
* But the obtained molecular weights are limited
Z. Bao, Y. Cen, R. Cai, L. Yu, Macromolecules, 26, 5281-5286, (1993)
POLYFLUORENE DERIVATIVES
R
R
n
* Good opportunity for getting soluble blue emitting polymer
* Chemically and photochemically stable
* Good hole injecting materials
* Good electron transporting materials
First attempts for preparing poly(9,9-dihexylfluorene) (oxidative polymerisation)
R
2 n-BuLi T = -78°C
2 RBr
R
R
R
FeCl3
CHCl3
n
Not really suitable for application for the moment…
* very low molecular weight
* branching
* non conjugative linkages through other positions than 2 and 7
Y. Ohmori et al., Jpn. J. of Appl. Phys., 30(11B), L1941-L1943, (1991)
POLYFLUORENE DERIVATIVES
Yamamoto reaction (Dow Chemical Company) :
R
R
R
2 Br2
R
R
bis(1,5-cyclooctadienyl) Nickel (0)
Br
R
Br
Br
CHCl3
N
N
DMF
Br
n
T = 80°C
Polymers were end-capped with monobrominated aromatic derivatives
R
R
R
R
Yamamoto coupling
Br
Br
n
2
Br
n
Ni-catalysed
Oxydative coupling
DPn
48
14
Mw/Mn
2.4
6.8
Tg (°C)
95
55
Mesomorphism
193 N 249
none
Fluorescence (nm)
424, 448, 475
425, 495
M. Bernius, et al., Proc. SPIE, 3797, 129-137, (1999)
POLYFLUORENE DERIVATIVES
Drawback of the Yamamoto route : low solubility of the polymer in DMF
 Adaptation of the Suzuki reaction (Dow Chemical + others)
R
R
R
1) 2.1 eq n-BuLi THF -78°C
Br
Br
O
R
O
O
B
O
B
O
R
B
2)
R
Pd(0)[(PPh3)4]
O
R
R
Toluene, Na2CO3
Br
n
Reflux 48h, Ar
Br
Possibility of making fluorene based copolymers with a wide variety of comonomers :
N
N
N
N
N
R
R
R
R
R
OR
S
N
RO
S
N
Possibility of finely tuning properties of the EL polymers
COPOLYMERISATION
Fluorene based copolymers
Poly(9,9-dialkylfluorene)s tend to aggregate upon annealing or during operation
Use of a lower band gap comonomer
PL spectra of PDHF
C6H13
C6H13
OC10H21
C10H21O
n
PDHFDDOP
hPL = 40%
PL spectra of PDHFDDOP
W. L. Yu, et al. Chem. Commun., 1837-1838, (1999)
Quantum confinement – Perovskite
layers and Q-dots
Luminescence from quantum confinement
When a wave lenght can be held into a small size
environment, this is called quantum confinement
Exemple 1 : Plasmon resonnance into a gold nanoparticle of a
Q-dot
Exemple 2 : Confinement into a bidimensionnal layer of an
hybrid organic-inorganic perovskite
Résolution in the harmonic case
Avec:
Hypotheses
Solutions
A second order developpment shows the fonctions are paraboles
Exemple : Organic-inorganic perovskites – (2 d-Qwell)
Alternance of infinite
organic/inorganic plans
Schematic electronic
structure
[2]
Photoluminescence
Optical Density
Eliaison≈ 220 meV
2,2 2,3 2,4 2,5 2,6 2,7 2,8 2,9 3,0 3,1 3,2
Energie (eV)
• Luminescence at room temperature
« strong » excitons
observables at 300 K
Q-dots - Size effects
The Q-dots are very small nanoparticles of chalcogenides, where the
wavelength is confined and the emission is size-dependant.
Chemical synthesis
Inorganic materials where optical properties are
linked with electron confinement
Type of materials: Various metal chalcogenides
Composition effects
Blinking as a consequence of single photon
emission!
Second part of the course:
Basics of
Nonlinear Optics (NLO)
Basics of Nonlinear Optics
At the molecular scale, molecules are influenced by
electromagnetic fields without need of absorption.
The first effect is simply diffraction (linear index
change) as a result of the « slowing down » of the
propagation in matter vs vacuum. Simple one-photon
absorpation is also a linear effect.
There are higher order effects, whose intensity is
much smaller (not observable at standard intensities)
Macroscopic situation: Intense light modifiy
matter polarization.
Upon application of an electric field, induced dipolar moments
appear in the matter, which cause induced polarization P.
The linear c(1) term is a second
order tensor.
For low powers, P = c(1) E
This is the linear response of matter to light, the
polarisation has the same direction than the
incident light, and can be related to the linear
refraction index.
We have the simple relation e = 1 + 4c(1) = n2
Simplified situation: Only one light beam, and
only one direction counts
If the light is polarized, only the field direction counts, the
equation becomes scalar, and the powers 2 and 3 in the
trigonometric equations can be linearized.
With:
We have:
It comes out that, in this very classical situation, second
order NLO will give rise to generation of frequency doubling
and an additionnal constant electric field, while third order
NLO will give rise to frequency tripling + generation of an
harmonic at the same wavelength.
Second order NLO
The non-linear c(2)
The non-linear response is a third order tensor and
therefore can mix responses to two different incident
beams.
c
P2 = (2) E1E2 and therefore implies the possibility of
frequency mixing.
c(2) different from zero implies noncentrosymetry in both the material, and the
active component, otherwise the effects cancel.
The non-linear c(2) term is a third
order tensor.
Privileged direction (z)
For most NLO effects,the response of a material is
usually much higher in a privileged direction,
usually chosen for z axis (sometimes called x).
The P2 tensor comes down to a scalar, like
c(2) .We have:
Pzzz = c(2) zzz E2z.
Second order NLO, the general case:
w1
w1
w2
w2
2nd order NLO active material
w1  w2
w1  w2
2w1
2w2
Calculation (of the simplest case)
If beams have same direction and polarization, and
phase matching, we come back to the directionnal case,
with simple wave mixing. We have:
Etotal = E1cos(w1t) + E2cos(w2t)
c(2) Etotal2 ,
= c(2) {1/2[ E
P2 =
P2
1
therefore:
2
+ E22 + E12 cos(2w1t)+ E22cos(2w2t)]
+ E1E2cos(w1+w2)t + E1E2cos(w1+w2)t
The first term (in white) corresponds to the induced static polarization, the
second (in red) to the second harmonic generation (frequency doubling) the
third (in yellow) to the frequency sum generation and the last one (in green)
to the frequency difference generation.
The frequency doubling
This is by far the largest application of second order
NLO
c
P2 = (2) EE and therefore generates a wave with
doubled frequency.
NLO active material
Red
Blue (sum frequency)
The main application is the UV laser (of weak
power) for eg information storage..
Other possibilities
1) The non-linear response can be used to generate
sum and difference frequencies to detect eg IR beams.
IR
Yellow
Orange (frequency difference)
Green (frequency sum)
2) The non-linear response can be used to generate
modulation of the linear refraction index through
application of a constant field, this is the electrooptic effect
(or Pockels effect).
The electrooptic effect
Applying an external field comes to submit
the medium to a sollicitation E = E0 + E(w).
E0 is usually large compared to E.
If the field is aligned with the light direction the
tensor solves to the scalar, along this direction.
c(2) EE therefore P = c(1) (E +E) + c(2) (E +E)
P = c(1) E + c(2) E + (2c(2)E + c(1)) E + c(2) E
P2 =
0
2
0
0
Static term
0
2
0
« new » refraction ≈ n2
Small
Therefore we have the « new » refraction index given by:
c(2)E
n2 – n02 = 2
0
2
Third order NLO
The 3rd order NLO
One wave is generated at triple frequency, and one at
the same frequency.
NLO 3rd order active material
UV (frequency tripling)
Red
red (harmonic generation
at same frequency)
Applications in optical gates (Kerr effect) and
UV lasers.
Privileged direction (z)
Again the z direction can be privileged along the
field direction.
The P3 tensor is reduced, but not to a
simple scalar, we have:
Pzzzz = c(3) zzzz E3z.
However, this time, the c(3) term contains two
components, because not only frequency tripling
can occur, but also generation at the same
frequency, by simultaneous frequency addition and
soustraction
The non-linear c(3)
The third order non-linear response is a fourth order
tensor and therefore can mix responses to three
different incident beams.
c
P2 = (3) E1  E2  E3 and therefore implies many
possibilities of frequency mixing. The general case is
extremely complicated…
c(3) different from zero also for centrosymetric
molecules and materials  always observed!
The 3rd order NLO
Generation of triple frequency occurs just like frequency doubling,
only the
c(3) zzzz values are usually smaller than c(2) zzz values .
Generation of same frequency through 3rd order NLO effects leads
to a light induced apparent refraction index change called optical
Kerr effect. The effect looks like the previously presented Pockels
effects, except that there is no applied permanent electric field.
However, if a permanent electric field is applied on a 3rd order
optical material, it has also an effect on the apparent refraction
index (analogous to Pockels effect) which is called static Kerr
effect.
Static Kerr effect
For a nonlinear material, the electric polarization field P will depend on the electric field E:
where ε0 is the vacuum permittivity and χ(n) is the n-th order component of the electric susceptibility of the
medium. The ":" symbol represents the scalar product between matrices. We can write that relationship
explicitly; the i-th component for the vector P can be expressed as:
where i = 1,2,3. It is often assumed that P1 = Px, i.e. the component parallel to x of the polarization field; E2 = Ey
and so on.
For a linear medium, only the first term of this equation is significant and the polarization varies linearly with the
electric field. For materials exhibiting a non-negligible Kerr effect, the third, χ(3) term is significant, with the evenorder terms typically dropping out due to inversion symmetry of the Kerr medium. Consider the net electric field E
produced by a light wave of frequency ω together with an external electric field E0:
where Eω is the vector amplitude of the wave.
Combining these two equations produces a complex expression for P. For the DC Kerr effect (E°
surimposed) , we can neglect all except the linear terms and those in
Therefore :
Optical (or AC) Kerr effect
In the optical or AC Kerr effect, an intense beam of light in a medium can itself provide the modulating electric field,
without the need for an external field to be applied. In this case, the electric field is given by:
where Eω is the amplitude of the wave as before.
Combining this with the equation for the polarization, and taking only linear terms and those in χ(3)|Eω|3:
As before, this looks like a linear susceptibility with an additional non-linear term:
and since:
where n0=(1+χLIN)1/2 is the linear refractive index. Using a Taylor approximation, since χNL
<< n02, this gives an intensity dependent refractive index (IDRI) of:
where n2 is the second-order nonlinear refractive index, and I is the intensity of the wave. The refractive index change is thus
proportional to the intensity of the light travelling through the medium.
The values of n2 are relatively small for most materials, on the order of 10 -20 m2 W-1 for typical glasses. Therefore beam
intensities in the GW cm-2 range are necessary to produce significant variations in refractive index via the AC Kerr effect.
Two-photon adsorption
This process corresponds to the simultaneous absorption of two photons,
which is possible if there is of course phase matching. This is a 3rd order
process. This is a resonnant process involving the imaginary part of c(3).
2-photon absorption may generate classical fluorescence !
Two-photon adsorption : Very localized fluorescence
Molecular scale: What happens
Molecules for NLO
At the molecular scale, molecules are influenced by
electromagnetic fields without need of absorption.
A molecule which is sensitive to an electric field is a
molecule with electronic delocalization
For 2nd order NLO, it has to be non-centrosymetric,
and it is good to exhibit a high dipole moment
For 3rd order NLO, it is enough to have flexible
delocalized electrons.
Molecules for NLO
At the molecular scale, the molecules are individually
polarized. We have:
P = P(0) + a E(w) + b E(w1)E(w2) +  E(w1)E(w2)E(w3) +…
Where P(0) represents the permanent dipole moment of the
molecule, and the other terms the induced dipoles, through
interaction with light.
Despite a molecule may have a distorted dipole moment, most
of the molecules, and especially the one used in NLO, have a
main axis through which the largest component appears, and
which will be choses as z.
Therefore, the best molecules will present a high dipole
moment in a privileged direction.
Molecules for 2nd order NLO
Molecules with high dipole moment are prefered.
Typical example: The paranitroaniline (one of the
first molecules studied)
Unidirectional molecule:
bzzz = bTC
Molecules for 2nd order NLO: The
two levels model for SHG
Calculations show that:
bzzz = bCT (-2w, w, w) = (3 e2h/4m) F(w)fDm
Where m and e are respecitvely the mass and the charge of the electron,
h the Planck constant, F(w) a frequence dependant factor and Dm the
difference between the dipole moments in the fundamental and first
excited state (Dm = me-mg), and f the oscillator strength. E represents
the energy of the incident light (hn) and E° the energy of the electronic
transition between the two levels.
F(w) =
E0
( E02  E 2 )( E02  4 E 2 )
Molecules for 2nd order NLO: The
two levels model for SHG (2)
When E  0, the b term does tend towards 0, but towards a value that is
really representing the polarizability of the molecules, ie its nonlinear reaction
to an electric field sollicitation. This term is called b(0) and we have :
b(0)CT = (3 e2h/4m) E0-3 fDm
Then, the b at any frequency can be expressed as a
function of b(0) and the frequency. It comes:
b = b(0)
E04
( E02  E 2 )( E02  4 E 2 )
The first term is an intrinsic characteristics of the molecule, and the second a
frequency factor that rises when one gets close to the one photon or the twophoton transition (but also absorption !!).
Molecules for 2nd order NLO: The
two levels model for SHG (2)
The b at any frequency can be also expressed as a
function of b(0) and the pulsations. It comes:
Molecules for 2nd order NLO: The
relation between c and b.
Basically, the macroscopic polarisability is the integrated sum of the
microscopic ones.
When all molecules are oriented, with an a angle, with the incident field:
f is the local electric field correction factor, which depends on the polarity
of the solvent/medium.
Non polar
( ew = n2 )
Polar
Ideal molecules for NLO response:
Donor-acceptor conjugated molecules
-conjugated
Donor
Bridge
Acceptor
Most prepared and studied families
PNA family
Oligophenylenes family
Stilbene family
Azo dyes family
Oligothienylenes family
Ground and excited state
In the 2-level approximation,
people consider that the first
excited state corresponds to
the complete charge transfer
between the donor and the
acceptor group, as
represented for the classical
NLO-phores shown left.
Examples of molecules and b values
Examples of molecules and b values (2)
Examples of molecules and b values (3)
Examples of molecules and b values: Organometallics
Examples of molecules with other
geometries (distorted and V-shaped)
Examples of molecules with other
geometries (calixarenes)
Examples of molecules with other
geometries (octupoles)
Synthesis of azodyes: Para red
Molecules for 3rd order NLO
We recall:
P = P(0) + a E(w) + b E(w1)E(w2) +  E(w1)E(w2)E(w3) +…
Therefore, the best third order molecules will also need to
present a high electronic flexibility, but a high dipole moment in
the ground state is no longer mandatory.
However, we have, for a 2-states molecule:
1
0
D
A
1 , µ1
µ01, w0
D
xxxx (-w, w, -w, w) 
A
0 , µ0
 m 401
m 201 (m1  m 0 ) 2
 2
2
2 3
(w 0  w )
(w 0  w 2 ) 2 (w 20  4w 2 )
Molecules for 3rd order NLO
D
A
1 , µ1
µ01, w0
D
Quadrupoles are OK,
despite the
momentum of the
ground tate is zero!
A
0 , µ0
D---A---D or
D-spacer-D or
A-spacer-A
Examples of molecules only for 3rd order NLO
Optical limitation
(2-photon absorption)
N
N
X
X
Fe
X = -, Ph ,
N
,
N
Optical Kerr effect
Fe
Figures of merit for Kerr effect
So the important parameter is n2, nut it may be considered
relatively, compared to the 2-photon absorption b and the
standard absorption a.
W = n2/al and T = bl/n2 , where a and b are as defined above,
are the figures of merit to consider sonce they represent respectively
the relative efficiency and transparency of the molecule
Figures of merit for Kerr effect: Comparison
between two record molecules
DO3 is a classical dye while PYDO3 is non classical annd belongs to
the pull-push-pull group.
One and two photons absorption, along with  of DO3
and its pyrrole counterpart PYDO3
1 photon
NH 2
N
2 photons
N
N
N
N
NO 2
NO 2
DO3
PYDO3
A la fois les absorptions à un et deux photons sont décalées vers le bleu du DO3 au Pyrrole-DO3
Gammas at 1280 nm, out of resonnance for the two molecules
PYDO3 : (+137±70)  10-36 esu,
DO3:
(+256±94)  10-36 esu.
The  are almost the same, despite the large offset from resonnance in the pyrrole !
P. AUDEBERT, K. OHTA, K. KAMADA and M. ANDO Chem.Phys. Lett, 2000.
Bibliography:
1)Fluorescence:
2)NLO Personnal and Zyss’s group data +
T. Verbiest, S. Houbrechts, M. Kauranen, K. Clays and A.
PersoonsJ. Mater. Chem., 1997, 7(11), 2175–2189
Dzekuje bardzo !