First a theoretical model was developed in order to understand whether RSA
properties were kept upon including RSA molecules in a solid host. In fact this is a
necessary step in order to realize photonic devices.
Then photophysical properties of fullerene C
60
were studied either theoretically or
by nanosecond time resolved spectroscopy. Principally triplet photophysics of these
systems was investigated, since several problems connected with the multi-molecular
processes active in solutions were still unresolved. Moreover phenomena connected to
the photochemistry of those systems needed to be clarified.
Finally CNs, in particular SWNTs, which are the simplest form of CNs, were
studied. Since the spectroscopic characterization of these materials is still lacking, not
only NL optical properties were investigated but also linear ones. The complexity of
these extended structures is very high compared to that of conventional molecular
systems. Therefore investigations covered a wide experimental range: from Raman
and fluorescence spectroscopy to OL and pump-probe measurements.
PART I
NONLINEAR OPTICS,
NONLINEAR ABSORPTION AND SCATTERING
In this section basic equations describing interactions between matter and optical field
are summarized. Moreover nonlinear phenomena, as nonlinear absorption and
scattering are introduced.
I.1 Nonlinear polarization
In nonlinear optics, the nonlinear optical response can be described by expressing the
polarization )t(
~
P as a power series in the field strength )t(
~
E as
1,2, 3
...
~~~~
:
~~
)t(
~
)3()2()1(
+χ+χ+χ= EEEEEEP M
(I-1)
where
)1(
χ is the linear susceptibility,
)i(
χ 3,... 2, i = are known as the first, second,
third order nonlinear optical susceptibilities, respectively. They are tensors of 1i + rank.
The steady state linear electric polarization response may be written as
EP
~~
)1(
L
χ=
(I-2)
while the terms of order higher than first in the field strength are included in the
nonlinear polarization term:
...)t(
~
)t(
~
...
~~~~
:
~
)t(
~
)3()2()3()2(
NL
++=+χ+χ= PPEEEEEP M (I-3)
The i-th nonlinear polarization can be written as a summation of its Fourier components
ti
n
n
)i()i(
n
e)(
~
)t(
~
ω−
ω=
∑
PP
(I-4)
So that for the second order polarization term we obtain:
)(E)(E),;(D)(
~
2k1j2
jk
13
)2(
3
)2(
ωωωωω−χ=ω
∑
P . (I-5)
For the third order:
)(E)(E)(E),,;(D)(
~
3l2k1j32
jkl
14
)3(
4
)3(
ωωωωωωω−χ=ω
∑
P (I-6)
where
)2(
D and
)3(
D are degeneracy factors.
Nonlinear susceptibilities can be derived either by a more approximate approach,
based on quantum-mechanical perturbation on the atomic or molecular wave functions,
or by a density matrix formalism leading to more general solutions.
1,4
I.2 Maxwell’s equations in nonlinear optical media
The response of a nonlinear material system to an intense laser field can be described
by Maxwell’s equations, which are able to account for the way the various frequency
components of the field become coupled by nonlinear interaction. New frequencies not
present in the incident radiation field can be developed, which act as new frequency
components of the electric field.
For most of optics the optical wave may be characterized by defining its electric
field. The electric field vector of the optical wave is expressed as the sum of a number
of frequency components, indicated by the subscript n , as:
()
.c.ce)t,()t,(
~
t
n
n
nn
+=
ω−⋅
∑
rk
rArE
(I-7)
where the superscripted tilde ~ implies that the fields are rapidly varying in time and
they are real quantities. The summation is taken over positive frequencies only. In this
equation,
n
k is the wave vector of propagation,
n
ω is the circular frequency of the
rapidly oscillating wave. The wave amplitude )t,(rA may have a slowly varying space
and time dependence, so that )t,(
nn
rAA = . This amplitude in general is complex.
For a large number of problems in linear and nonlinear optics the optical wave can
be considered an infinite plane wave. Within this approximation the field extent can be
assumed to be infinite and constant in amplitude and phase, moreover lying in a plane
transverse to the direction of propagation, taken as z direction. Thus the complex field
amplitude becomes a function of z and t only: )t,z(AA
nn
= .
A common form of a finite beam is the TEM
00
mode of a circular Gaussian beam.
The field of this type of wave has the form:
c.ce
)z(w
w
)t,z()t,(
~
t
z
z
tankz
)z(q2
kr
i
0
n
n
R
1
2
+=
ω−
−+
−
n
ArE (I-8)
The beam has a Gaussian cross-section with a variable radius )z(w , which is
defined as the half-width of the Gaussian curve at the point r where the curve is at 1/e
of its maximum. The diameter of the beam has a minimum
0
w at focus 0z = , where
0
w2 is called beam waist.
The plane of constant phase for a focused Gaussian beam is, in general, curved. At
the beam waist, however, the radius of curvature is infinite, like in a plane wave. The
quantity )z(q is called complex radius of curvature. Finally the quantity
R
z is called the
Rayleigh range and it is defined as:
λ
π
=
2
0
R
wn
z (I-9)
where n is the index of refraction of the medium and λ the optical wavelength in free
space.
The form of the wave equation for the propagation of light through a nonlinear
optical medium can be obtained by solving Maxwell’s equations (c.g.s. units):
J
D
H
B
E
B
D
~
c
4
t
~
c
1~
,
t
~
c
1~
,0
~
,ρ
~
4
~
π
+
∂
∂
=×∇
∂
∂
−=×∇
=⋅∇
π=⋅∇
(I-10)
Solutions are taken in the region of space that contain no charges i.e. 0=ρ and no
currents so that 0
~
=J . Moreover the material is nonmagnetic, i.e. HB
~~
= .
If the material is non linear, the fields D
~
and E
~
are related as
PED
~
4
~~
π+=
(I-11)
Therefore Maxwell’s equations can be derived in the usual manner as:
2
2
22
2
2
t
~
c
4
t
~
c
1~
∂
∂π
−=
∂
∂
+×∇×∇
PE
E
(I-12)
When the intensity of the light is sufficiently high the total polarization can be split into
its linear and nonlinear part as:
NLL
~~~
PPP += (I-13)
so that substituting (I-2) into (I-12) the wave equation becomes:
2
NL
2
22
2
2
)1(
t
~
c
4
t
~
c
~
∂
∂π
−=
∂
∂ε
+×∇×∇
PE
E
(I-14)
For the case of a dispersive medium each component of the field has to be considered
separately as in I-7 so that the nonlinear polarization, if representing a small
perturbation to the total polarization can also be expressed as:
.c.ce)(
~
)t,(
~
ti
n
n,NLNL
n
+=
ω−
∑
rPrP (I-15)
In that case (I-14) is valid for each frequency component separately.
I.3 Nonlinear Absorption
The optical properties of a material can undergo a profound change upon exposition to
an intense monochromatic laser radiation. If nonlinear absorption occurs, the
transmittance of a material changes as a function of intensity or fluence. As the
probability of a material to absorb more than one photon is greatly enhanced at
sufficiently high intensities of radiation, multiphoton absorption has been widely studied
since the invention of lasers. Moreover, intense laser fields may make population to
redistribute over rather inaccessible high energy levels of molecular systems, so that a
wealth of nonlinear phenomena can be induced, such as stimulated emission and
absorption, complicated energy transitions, which might lead also to charge separation
or to photochemical processes, or generation of free carriers in solids. Some of the
latter phenomena are manifested optically in reduced (saturable) or increased (reverse
saturable) absorption, named SA and RSA respectively.
Many effects arising from nonlinear absorption can be exploited in diverse areas
such as nonlinear spectroscopy and optical limiting, in fact nonlinear optical
phenomena are interesting for several applications either in science or in technology.
I.1.1 Multiphoton absorption
Multiphoton absorption refers to processes leading to the absorption of n photons. The
process is coherent if photons are absorbed simultaneously i.e. without loss of
coherence of the nonlinear polarization induced in the medium. The initial state
involved in the process is directly coupled to the final state without intermediate steps
leading to population in other excited states.
The differential equation describing optical loss for the absorption of n photons is
given by:
∑
γ−α−=
n
nn
II
dz
dI
(I-16)
where α is the linear absorption coefficient,
n
γ (n > 1) is the n-photon absorption
coefficient and has units of [
1n)1n(2
W/m
−−
].
The coefficient
n
γ is a macroscopic parameter characterizing the material, however
it is related to the molecular n-photon absorption cross-section
)n(
σ as:
)n(
n
n
N
)(
σ
ω
=γ
h
. (I-17)
For n=1, the linear absorption coefficient is related to the absorption cross-section as
σ
ω
=α
N
h
(I-18)
The absorption cross-section
)n(
σ
for an n-photon coherent absorption process is
related to the imaginary part of (2n-1)-th order susceptibility
)1n2( −
χ
.
In case two photon are absorbed at the same frequency, this relation becomes
5
)(Im
Nc
32
)3(
2
22
)2(
ωχ
ωπ
=σ
h
(I-19)
The transition probability per molecule for a non resonant n-photon transition is
n21
n
nn2211
)n(
)n(
...
)(I)...(I)(I
W
ωωω
ωωωσ
=
h
(I-20)
Most commonly observed multiphoton absorption processes are those at lower
orders, as two-photon absorption (TPA), three-photon absorption (3PA), up to five-
photon absorption. Higher order absorption processes are rather weak, indeed very
high intensities are required to get them excited, so that avalanche ionization and
breakdown in condensed matter might occur too.
The n-th order absorption cross-section could be enhanced by many orders of
magnitude if the absorption process involves intermediate states, i.e. incoherent
multiphoton absorption.
In this case one photon absorption processes occur between populated
intermediate states, so that the whole process consists in several steps, at each step
the polarization loses its coherence. The system indeed, after each absorption step,
relaxes through non radiative processes (i.e. internal conversion or intersystem
crossing), so that coherence is lost before the further photon is absorbed.
Here the induced nonlinearity is cumulative, thus incoherent n-photon absorption
becomes fluence-driven rather than intensity-driven. Therefore time differential beam
loss equation (I-16) should be integrated over time, the fluence being defined as
∫
∞−
=
t
'dt)'t(I)t(F (I-21)
Moreover
)n(
σ is no longer related to the n-order susceptibility, but it is an effective
n-th order absorption cross-section, as it depends on linear absorption cross-sections
of intermediate processes.
Since intermediate resonances imply finite lifetimes of states, a population rate
equation approach must be employed to describe the whole stepwise process.
Populations should be computed including all decay rates of intermediate states, so
that the differential beam loss equation becomes time dependent as well.
I.1.2 Excited state absorption. SA, RSA
If the incident intensity is sufficient to deplete the ground state significantly, the excited
state becomes populated. In systems as polyatomic molecules or semiconductors
there is a high density of states close to the state of arrival after the first absorption, so
that a rapid jump into one of these states can occur before the excited electron relaxes
to the ground state. However there might be also a large number of higher lying
electronic states coupled with these intermediate levels by the incident laser radiation.
If a resonance condition is satisfied between the intermediate state and upper excited
states, a further photon can be absorbed and the electron can be promoted to a higher
lying electronic level. This process is called excited state absorption.
When the excited state cross-section is smaller than that of the ground state, the
transmission of the system is increased by higher excitation. This process is called
saturable absorption (SA). At the opposite when the absorption cross-section of the
excited state is larger than that of the ground state, the system becomes less
transparent if excited. This process thus is called reverse saturable absorption (RSA).
The RSA phenomenon was first observed by Giuliano and Hess, during investigations
on bleachable dyes.
6
In order to describe excited state absorption of polyatomic molecules, it is useful to
build simplified electronic diagrams including only the n electronic levels involved in the
whole process, this is usually called n-level model. The number of levels that should be
included depends on the system characteristics and on degree of approximation one
intends to introduce. Anyhow the simplest diagram suitable to describe essential
features of both SA and RSA is a five-level model shown in figure I-1. This diagram
consists of five distinct electronic states, within each of those there exist a manifold of
very dense vibrational-rotational states. Therefore in the absence of any radiation field
the electron is lying in a vibrational-rotational state of the ground state, which must
have a singlet spin multiplicity. As the electron is promoted to an excited state it gets
into one of these vibro-rotational sublevels, then, occurring a fast thermalization
process due to collisions, it drops to the lowest lying vibrational-rotational state of the
electronic manifold. The latter process is called internal conversion (IC). Afterward the
electron may either absorb another photon or decay to lower energy states.
Figure I-1 Schematic diagram of a five level system.
Selection rules allow only stimulated transitions which conserve spin angular
momentum, so that the first absorbed photon must lead the system to a singlet excited
state. The same is true for fluorescence or absorption from higher lying excited states.
However the first excited electronic state can also make a radiationless transition to a
lower lying triplet state, i.e. intersystem crossing (ISC), the spin flip being induced by
internal processes such as a strong spin-orbit coupling or by external processes such
as collision with paramagnetic ions. The energy of triplet is lower than that of singlet
since electron with the same spin occupy farther spatial regions, however singlet- triplet
energy split decreases in larger molecules where delocalisation of electrons is
enhanced. Again from excited triplet states radiation induced transitions are allowed
only to upper triplet states.
In short, allowed pathways for an electron in a five–level system are as follows.
Absorption of an incident photon promotes an electron into an excited singlet state,
hence leading to three possibilities. First the electron may decay to the ground state
either by radiative (fluorescence) or radiationless transitions, the total rate constant for
these processes being k
f.
Another possibility is that the
electron can drop non-
radiatively into a triplet state by intersystem crossing. The third possibility is that the
molecule may absorb another photon, promoting the electron to a higher lying singlet
(singlet-singlet absorption), from which a fast relaxation to the first exited singlet state
occurs.
If the second of those possibilities is realized, the electron has two accessible ways.
The first consists in a decay to the ground state implying a spin flip, which can occur
either radiatively (phosphorescence) or radiationless, the rate constant accounting for
those processes being k
ph
. The second possibility is that the triplet excited electron
would experience absorption of another photon (triplet-triplet absorption), then
decaying very rapidly back to the first triplet state.
Relaxation rates from higher lying excite states either singlets (S
n
) or triplets (T
n
)
are so fast that populations of these states can be neglected in respect to those of
other electronic states. Moreover stimulated emission from excited states may also be
considered irrelevant since the relaxation to the bottom of the electronic manifold is
very fast.
With all these assumptions, rate equations should account for the population
densities of only the most important electronic states, i.e. the ground singlet S
0
, the first
excited singlet S
1
and the first excited triplet T
1
state. The population density for an
electronic state is defined as the ratio of molecules occupying that electronic state to
total number of molecules N. Therefore the following equations apply:
()
Tk
t
T
SkkS
I
t
S
TkSkS
I
t
S
ph
1
1iscf0
01
1ph1f0
00
−=
∂
∂
+−
ω
σ
+=
∂
∂
++
ω
σ
−=
∂
∂
h
h
(I-22)
Then the equation for beam attenuation is:
NTNISNIS
z
I
121100
σ−σ−σ−=
∂
∂
(I-23)
Here the radiation intensity is )t,z(II = ,
0
σ ,
1
σ and
2
σ are cross-sections accounting
for absorption from ground, singlet excited and triplet excited states respectively.
Moreover population densities have to be conserved so that
1TSS
110
=++
(I-24)
I.4 Nonlinear scattering
Scattering of light is a complex phenomenon, including several different processes.
Here only phenomena connected to the nonlinear behavior of the studied systems are
reported.
Scattering processes can be spontaneous or stimulated. Spontaneous scattering
occurs under conditions such that the optical properties of the material system are
unmodified by the incident light beam, while stimulated scattering takes place
whenever the incident light of is sufficiently intense to modify the optical properties of
the material system. The main types of stimulated scattering are Raman scattering,
Brillouin scattering and Rayleigh scattering. Raman scattering can be equivalently
described as scattering by optical phonons, Brillouin scattering on the other side can be
considered scattering of light from acoustic phonons. Both of those processes make
the scattered radiation to be frequency-shifted with respect to the incident radiation.
Rayleigh scattering, on the other side, is known as quasi-elastic scattering since it
induces no frequency shift, in the latter case scattering occurs from non-propagating
density fluctuations.
1
Rayleigh scattering was firstly observed in 1871 by Sir. Rayleigh, since he noticed
that polarization of light scattered by small particles was maximum at 90°, so that
particles were acting as they had a polarizing angle of 45°.
There is a formal analogy between a particle and a slab undergoing an incident
wave. A slab can transmit or reflect an incident wave, whereas a particle can just
scatter it. The internal refracted waves in a slab are analogous to internal field
generated in a particle.
If one wants to calculate the scattered field by a particle of specified size, shape
and optical properties, it is necessary to solve Maxwell equation imposing boundary
conditions requiring that tangential components of E and H are continuous across the
boundary separating media with different properties.
In order to describe scattering phenomenon by a particle a scattering matrix should
be defined.
If the propagation direction of incident light is defined as z, being
x
e
y
e and
z
e
positive directions of Cartesian axes, the scattering direction
r
e and the forward
direction
z
e define a plane called scattering plane. It is convenient to resolve the
electric field lying in the scattering plane into its vertical and horizontal polarization
components with respect to an ideal polarizer (i.e. to define its Stokes parameters) as
shown in figure I-2. Scattering matrices can be written, so that polarization components
of incident light are related to those of scattered light as:
=
⊥⊥
i
i
||
14
32
s
s
||
E
E
SS
SS
E
E
(I-25)
where the elements )4,3,2,1j(S
J
= of scattering amplitude matrix depend on the
scattering angle θ and on the azimuthal angle φ . In order that scattering amplitude
matrix elements to be known, a measure of amplitude and phase components of the
light scattered in all directions for two incident orthogonal polarizations is required.
7
Figure I-2 Scattering by an arbitrary particle.
7
It should be remarked that the extinction cross-section
ext
σ is the sum of absorption
and scattering contributions as
σ+σ=σ
Sext
(I-26)
where σ denotes absorption cross-section. Scattering contribution may be
compared to absorption ones by defining a scattering cross-section
S
σ as
φθθ=σ
∫∫
ππ
ddsin
k
2
00
2
2
S
X
(I-27)
where X denotes the scattering vector accounting for x-polarized incident light, X is
defined as follows:
7
s14s||32
e)sinScosS(e)sinScosS(
⊥
φ+φ+φ+φ=X (I-28)
Analytical expressions for scattering matrices can be obtained by solving Maxwell’s
equations for particles of spherical, cylindrical or other simple geometrical structures. In
particular the scattering theory for spherical particles is named Mie Theory since it was
developed by Gustav Mie in 1908.
These results lead to rather complex relations between scattering cross-sections
and incident light frequencies, which cannot be shortly summarized here. However
scattering cross-sections of particles whose dimensions were in the micron scale were
found to depend on the inverse fourth power of the radiation wavelength.
Moreover Rayleigh scattering may be considered also from a thermodynamic point
of view, choosing as thermodynamic variables the temperature T and the density ρ .
This analysis might describe in particular thermally stimulated Rayleigh scattering: i.e.
the scattering of light from isobaric density fluctuations that are driven by the process of
optical absorption. This model includes the three equations of hydrodynamics, the first
being the equation of continuity. The second equation is that of momentum transfer,
while the third accounts for heat transfer. Solutions allow one to determine how
nonlinear polarization is affected by temperature variations and density fluctuations in
the medium as EP
~~
NL
χ∆= where χ∆ comes from solutions of thermodynamic
equations and depends on density of the medium, on heat transfer efficiency, on linear
absorption cross-sections, and on medium viscosity.
1
I.5 Optical Limiting (OL)
Optical limiters display a decreasing transmittance as a function of intensity or fluence.
Possible applications of optical limiters are related to their uses in optical pulse shaping
and smoothing and pulse compression. Moreover they are widely used as optical
sensor protection. However in recent years a great interest in optical limiters is grown
concerning their more general application in photonics as optical switches.
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