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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