Homogeneous
spectral line shape
Let us consider the absorption of
light by one single molecule embedded in an optically transparent solid. An example of
such solid could be a piece of plastic or polymer material, a chunk of a laser crystal
or a piece of simple window glass. The relevant quantum-mechanical system consists of the
electronic and vibrational degrees of freedom of the molecule and of
the vibrational motion of the surrounding solid. Figure 1 shows a molecule
surrounded by a solid. The molecule together
with its closest neighbors (region inside red circle) is called impurity center. The
absorption spectrum of one molecule is called homogeneous absorption spectrum or
homogeneous line shape. Suppose that absorption occurs because of transition from
the ground electronic state of the impurity center to it's excited electronic state.
For organic dye molecules the ground electronic sate is singlet S0 and the
excited electronic state is the lowest excited singlet S1. The absorption
spectrum gives the probability of transition from the ground state to the excited
state as a function of frequency n (or as a function of wavelength,
l=c/n). Experimentally, the absorption spectrum is
obtained by illuminating the crystal with a beam of light of frequency n, and by measuring the ratio,
I(n)/I0(n), where I0(n) and
I(n) is the intensity
at the input and at the output of the crystal, respectively.
In the first approximation, the
absorption spectrum has a sharp maximum, where the frequency n equals
the energy difference between the states, divided by Planck's constant,
n = (E(S1)-E(S0))/h. This is most true if the molecule is
free in gas phase. In the solid, however, the transition probability depends also on the
coordinates of the surrounding atoms. More precisely, the probability is a function of the
density and of the frequency of the vibrational states of the solid. Crystal lattice
vibrations, which propagate like waves are called phonons. Most critical is the
presence of such phonons, which are part of the ground state wave function. Because the
phonon state population varies largely with the temperature of the crystal, the whole
absorption spectrum depends strongly on the temperature. Figure
2 shows how the homogeneous absorption spectrum of the molecule changes if the
temperature is varied between room temperature (T=300 K ) and absolute zero temperature
(T=0). At higher temperatures T=100 - 300 K the spectrum is tens to hundreds of cm-1
broad. It hardly contains any sharp structure. Since typical phonons in a solid matrix
have an energy quanta of hnph
~10 - 1000 cm-1, the thermal motion at around room temperature
( kT~300 cm-1) has enough energy to excite a wealth of
phonons. If many phonons are present, then each time the electronic
transition occurs in the impurity center, it is impossible to predict what will be exactly
the energy difference between the ground state and the excited state. Therefore the
room temperature absorption spectrum appears to be broad and without sharp lines. It
consists mostly of what is called phonon side band. At lower temperatures, however,
the number of phonons is much reduced. Then there exists a real probability
for electronic transitions where the phonons do not participate at all. Such
transitions are called zero phonon transitions. Their important property is that they
have
a very well defined frequency. The corresponding spectral feature shows up as a narrow
zero phonon line (ZPL). The narrowest and most intense zero
phonon line is observed at absolute zero temperature. The width of the ZPL is then given
by the inverse value of the excited state's lifetime. The phonon side band reduces at low
temperatures to a relatively weak feature on the shorter wavelength side of the
ZPL. In
some special cases the zero phonon line can be detected already at liquid
nitrogen temperature (T=77oK), however more typical is that the sample has to
be first cooled to 10 - 20 K.
Inhomogeneous
line broadening
Now let's assume that the same piece of
solid contains not just one, but many molecules, which all have the same chemical
structure. Nevertheless, because no solid has a 100% perfect regular structure, different
molecules are going to find themselves in slightly different surroundings. Figure 3a shows five chemically indistinguishable
dye molecules in a randomly fluctuating environment. Figure
3b shows that this makes the ground and excited state energy to vary randomly
from molecule to molecule, which causes the transition frequency to change randomly
as well. The probability of finding the transition
frequency in a unit frequency interval {n,
n+dn} is given by inhomogeneous distribution function,
g(n). The absorption profile, which results from the
superposition of many homogeneous line shapes is called inhomogeneously broadened
absorption band. Mathematically, the inhomogeneous absorption band can be described as a
convolution of the inhomogeneous distribution function with the function describing the
homogeneous line shape. Figure 4 shows the
composition of the inhomogeneous absorption band at high (room) and at low
(cryogenic) temperatures.
At high temperatures the inhomogeneous spectrum is a superposition of many
broad phonon-induced bands. Usually the width of the phonon bands is on the same
order as the width of the inhomogeneous distribution function. In this case the absorption
spectrum of molecules within the inhomogeneous band can be hardly be distinguished
from each other. Such absorption band has practically no spectral selectivity. At low
temperatures each molecule has a sharp zero phonon line. This allows groups of
molecules to be addressed selectively based on their ZPL frequency. We say that in this
case the material has a high degree of spectral selectivity.
Spectral hole burning
In most cases, molecules and atoms always
return from the excited state back to the initial ground state. There are situation,
however, when this is not always the case. For example, some organic dye molecules can
undergo a photochemical reaction, which alters the whole chemical structure of the
molecule. If such photochemically active molecule absorbs light, then with a probability
of a few % it will not return to the initial state called educt, but rather switches over
to a new ground state called product. Often the homogeneous absorption spectrum of the
product is much different from the educt, so that the corresponding inhomogeneous bands do
not even overlap. Figure 5 shows one such
example, where photochemical tautomerization at liquid helium temperature results in the
shift of the S1 ←
S0 absorption band from 634nm to 570nm. In
fact, by illuminating the sample in the spectral interval around 634nm most of the
molecules can be transferred from the educt to the product sate. If the illumination is
terminated, then the initial absorption profile is not restored unless the sample is
heated up to about liquid nitrogen temperature. To illustrate this fact, Figure 5 shows
the absorption profile taken before (red) and after (green) such illumination. Since at low temperatures
the inhomogeneous absorption band of the educt consists of narrow zero phonon lines, it is
possible to produce such photochemical transformations only in a small group of molecules,
which are selected by their ZPL frequency. Selective bleaching of the inhomogeneously
broadened absorption band consisting of narrow homogeneous absorption lines is called
spectral hole burning (SHB). Besides the photochemical tautomerization reaction shown in
Fig.5, there are many different mechanism for spectral hole burning in organic as well as
in inorganic materials. In all cases the spectral hole burning relies on three basic
factors: existence of narrow homogeneous zero phonon lines; existence of inhomogeneous
broadening; existence of some kind of molecular of electronic mechanism, which alters the
homogeneous absorption spectrum upon absorption of light.
Frequency-Selective
Optical Storage
Time-Space
Holography
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