SPECTRAL HOLE-BURNING AND SLOW
LIGHT IN EMERALD AND RUBY
IVANA CARCELLER
A THESIS SUBMITTED IN FULFILMENT OF THE
REQUIREMENTS FOR THE DEGREE OF MASTERS BY
RESEARCH
School of Physical, Environmental and Mathematical Sciences
University of New South Wales Canberra
March 2012
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‘I hereby grant the University of New South Wales or its agents the right to archive and to make available my thesis or dissertation in whole or part in the University libraries in all forms of media, now or here after known, subject to the provisions of the Copyright Act 1968. I retain all proprietary rights, such as patent rights. I also retain the right to use in future works (such as articles or books) all or part of this thesis or dissertation. I also authorise University Microfilms to use the 350 word abstract of my thesis in Dissertation Abstract International (this is applicable to doctoral theses only). I have either used no substantial portions of copyright material in my thesis or I have obtained permission to use copyright material; where permission has not been granted I have applied/will apply for a partial restriction of the digital copy of my thesis or dissertation.' Signed ……………………………………………........................... Date 31-03-2012
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‘I certify that the Library deposit digital copy is a direct equivalent of the final officially approved version of my thesis. No emendation of content has occurred and if there are any minor variations in formatting, they are the result of the conversion to digital format.’ Signed ……………………………………………........................... Date 31-03-2012
Originality Statement
I hereby declare that this submission is my own work and to the best of my
knowledge it contains no materials previously published or written by another
person, or substantial proportions of material which have been accepted for the
award of any other degree or diploma at UNSW or any other educational
institution, except where due acknowledgement is made in this thesis. Any
contribution made to the research by others, with whom I have worked at
UNSW or elsewhere, is explicitly acknowledged in the thesis. I also declare that
the intellectual content of the thesis is the product of my own work, except to the
extent that assistance from others in the project's design and conception or in
style, presentation and linguistic expression is acknowledged.
Ivana Carceller
Signed ..........................
Date 31/03/2012
Acknowledgements
Firstly, I would like to thank my supervisor Prof. Hans Riesen for his guidance,
patience, support and kindness in the development of this thesis. Without his
knowledge, encouragement and willingness, the completion of this thesis would
have been impossible. I am very grateful to him for his generous help,
tolerance, sense of humour and respect even during the hard times.
I also want to thank Dr. Wayne Hutchison for his proofreading despite the short
notice we gave him.
I want to express my gratitude to the University of New South Wales Canberra
for giving me the opportunity and offering me a Research Training Scholarship.
I wish to thank the other students and staff at the Australian Defence Force
Academy for all the good times and company in the last two years.
I would like to thank my Mum, Dad and the rest of family because without their
love and support, I would not have finished this thesis.
Special thanks go to my beloved friends Renata for her unconditional support
during hard times in Canberra, her wise advices and maturity and to Karina for
her sense of humour and for making me laugh even in the worst moments.
I would like to thank Greg for his patience, invaluable support and
understanding throughout the final stage of this project.
I want to mention my good friends Sarah, Lucia, Bea, Maria, Miriam and Alex
for their invaluable friendship throughout so many years regardless the
distance. I want to thank David and Tom for their help upon my settling into
Canberra. Also thanks to Federica, Bridget, Rhys, Michael and Rob for her
friendship and encouragement.
Also I would like to thank all the staff and students in the School of Physical,
Environmental and Mathematical Sciences at the University of New South
ii
Wales. In particular, I want to thank Emeritus Prof. Brian Lees, Prof. Warrick
Lawson, Ms Tessa Hodson, Dr Steve James, Dr Barry Gary, Mrs Annabelle
Boag and Mrs Nadia Seselja. My sincere thanks are also due to Ms Julie Kesby
who helped me when I was sick.
Also, thanks to Tracy Massil for her wise advices in my first months in Canberra
and her funny stories. I would like to thank Baran and Xianglei for their
proofreading of my thesis.
iii
Abstract
The temperature dependence of the R1 line of Chatham laboratory grown
emeralds (Be3Al2(SiO3)6) containing low concentrations of chromium (III) (0.04%
and 0.0017% per weight) is investigated in the temperature range of 2.5 - 260 K
by spectral hole-burning, luminescence line narrowing experiments and
luminescence spectroscopy. The contribution to the homogeneous linewidth at
6 K is 70 kHz and over 400 GHz at 260 K. The data is well described by the
two-phonon Raman process above 80 K. Below 80 K, the direct process
between the two 2E levels is dominant. However, at temperatures below 10 K a
low energy phonon is required to explain the temperature dependence.
The generation of slow light by transient hole-burning is reported for the first
time. A Gaussian probe pulse is propagated in an optically dense medium
provided by the R1(±3/2) line of 130 ppm Bagdasarov ruby. A delay of 10.8 ns is
observed, corresponding to a reduction of the group velocity of 213000 m/s
which is a reduction by a factor of 1400 compared to air.
iv
List of Publications
Publications that will result from this thesis:
1. Ivana Carceller, Wayne D. Hutchison, Hans Riesen; Temperature
dependence of the R1 linewidth in emerald; manuscript in preparation for
publication.
2. Hans Riesen, Aleksander Rebane, Ivana Carceller, Alex Szabo; Slow light in
pink ruby by transient spectral hole-burning; manuscript in preparation for
publication.
v
Table of contents
Chapter 1 Introduction.........................................................................................1
1.1. Theoretical background.....................................................................3
1.1.1. Electronic transition.............................................................3
1.1.2. Homogeneous linewidth......................................................4
1.1.3. Inhomogeneous broadening................................................8
1.2. Chromium (III) systems....................................................................10
1.2.1. General properties..............................................................10
1.3. Temperature dependence of the homogeneous linewidth...............14
1.4. Laser spectroscopy..........................................................................19
1.4.1. Spectral hole-burning (SHB)..............................................19
1.4.1.1. transient spectral hole-burning.............................20
1.4.1.2. Persistent hole-burning.........................................21
1.4.1.3. Potential applications of SHB...............................22
1.4.2. Fluorescence line narrowing (FLN)....................................23
1.5. Slow light..........................................................................................24
1.5.1. Kramers-Kronig relations....................................................28
1.5.2. Methods to create slow light...............................................29
1.5.2.1. Electromagnetically induced transparency (EIT)..29
1.5.2.2. Coherent population oscillation (CPO).................32
vi
1.5.2.3. Hole-burning .......................................................36
1.6. References......................................................................................38
Chapter 2 Experimental....................................................................................44
2.1. Sample preparation.........................................................................44
2.2. Room temperature spectroscopy....................................................44
2.3. Spectroscopy at Liquid temperatures..............................................44
2.3.1. Mounting of samples.........................................................45
2.3.2. Cooling method and temperature control..........................46
2.3.3. Absorption and Transmission spectroscopy......................46
2.3.4. Non-selective luminescence spectroscopy........................47
2.3.5. Spectral hole-burning.........................................................48
2.3.6. Fluorescence line narrowing..............................................49
2.3.7. Slow light experiments.......................................................50
2.4. Application of an external magnetic field.........................................57
2.5. Laser sources..................................................................................58
2.5.1. Nd:YAG laser ....................................................................59
2.5.2. He-Ne laser........................................................................60
2.5.3. Diode laser.........................................................................61
2.5.4. External cavity diode laser.................................................63
2.5.4.1. Frequency lock.....................................................64
2.6. Monochromator................................................................................65
2.7. Fabry-Pérot interferometer...............................................................67
vii
2.8. The Laue method.............................................................................69
2.9. Data acquisition and analysis...........................................................70
2.9.1. Data acquisition..................................................................70
2.9.2. Data analysis......................................................................71
2.10. References.....................................................................................73
Chapter 3...........................................................................................................75
3.1. Emerald............................................................................................75
3.1.1. Crystal structure and background......................................75
3.1.2.Results and discussion........................................................77
3.2. Slow light in ruby by transient spectral hole-burning........................90
3.2.1. Crystal structure and orientation........................................90
3.2.2. Results and discussion.......................................................92
3.2.2.1. Experiment...........................................................92
3.2.2.2. Simulations..........................................................97
3.3. References.....................................................................................100
Chapter 4................................................................................................. ........102
Appendix 1.......................................................................................................103
Appendix 2.......................................................................................................104
Appendix 3.......................................................................................................105
Appendix 4................................................................................................. ......107
Appendix 5.......................................................................................................109
Chapter 1
Introduction
The optical spectroscopy of transition metal ions in the solid state
has been of interest for decades. Electronic transitions in the solid
state are subject to inhomogeneous broadening as a consequence
of the presence of impurities or imperfections in the lattice. Many
laser-based techniques have been developed to overcome this
broadening; these techniques include hole-burning, fluorescence
line narrowing and photon echo measurements which can give us
information at very high resolution enabling a better understanding
of the most fundamental properties of electronic structure.
Therefore, understanding the dynamics and structure of crystalline
materials (as well as amorphous solids) is an important aim and a
significant effort has been made over the years in order to achieve
this goal.
Many potential applications of laser-based techniques are also
receiving significant attention such as frequency and time domain
optical data storage which are based on spectral hole-burning. This
implies that in the future new technologies may be applied for
advancement in the areas of data storage and quantum computing.
Parallel to this, researchers have made a great effort in order to slow
down the speed of light. In the telecommunications industry,
enormous amounts of data are transmitted through optical fibres at
very high transfer rates. But, unfortunately, information cannot be
processed at those rates. Due to limiting factors of existing
technology, light signals cannot be stored or processed without
being transformed into electrical signals, which only work on a much
slower timescale. The idea would be to control the light signals by
light in order to avoid the necessity to convert them into electrical
signals.
3
1.1. Theoretical background
Some theoretical concepts are presented in this section which are used
throughout this thesis in order to provide the reader with a better understanding
of the work undertaken. A rigorous quantum treatment of electronic excitations
and their linewidths can be found in [1, 2].
1.1.1. Electronic transition
The simplest electronic transition takes place in a two-level system where an
electron jumps from the ground state a to the excited state b (see Figure 1.1).
When this transition occurs, the electron gains energy which is subsequently
lost through radiative or non-radiative processes when it relaxes back to the
ground state.
In the radiative relaxation process the system is deactivated to the ground state
by emitting a photon. The emitted photon has an energy equivalent to the
energy difference between the ground and the excited state. On the other hand,
in non-radiative relaxation the electronic excitation energy is dissipated into
phonons (vibrations) to the rest of the crystal lattice.
The three basic processes of absorption, spontaneous and stimulated emission
are not independent of each other in a system. At equilibrium with an external
electromagnetic radiation field the overall transition rate from level a to level b
must be the same as that from level b to level a [3, 4].
4
Figure 1.1. The three possible transitions between a ground and an excited state.
Two parameters determine the width of an electronic transition: T1, which is the
lifetime of the excited state and T2*, which is the pure dephasing time [1].
1.1.2. Homogeneous linewidth
A significant effort has been made in order to rigorously understand electronic
structures in crystalline and amorphous solids and many sophisticated
experimental techniques have been applied so far to achieve this goal.
As even the most perfect crystals contain either imperfections or impurities, an
important experimental technique can be the optical spectroscopy of these
impurities (or optical centres) in the host structure. Hence, optical spectroscopy
of impurities embedded in a host matrix is an important tool to unravel its
electronic structure and associated dynamics [5].
An electronic transition between the levels of a two-level system can be
represented by a one-dimensional damped harmonic oscillator as is illustrated
in Figure 1.2.
relaxation
excitation
Incident photon
hγ
ground state a
nonradiative relaxation
OR
radiative relaxation
hγ
excited state b
emitted photon
5
Figure 1.2. Dampened harmonic oscillator. Adapted from [6].
If monochromatic electromagnetic radiation disturbs a two-level system, then
after a time t, an electron can be excited. A harmonic oscillation then happens
in this system and it is damped by dissipation of energy (radiative and non-
radiative relaxation processes as explained above). The response of such a
system can be represented with the following equation:
Eq. (1.1)
The damping term can be expressed as the inverse of the excited state lifetime,
that is, γ=1/T1 which is determined by the radiative and non-radiative relaxation
processes. The angular frequency is given by:
Eq. (1.2)
x(t)
x0
Time t
6
where k is the restoring force constant and is the mass of the electron.
The solution of the Eq. (1.1) is
Eq. (1.3)
where is the amplitude of the undamped oscillator [8].
To understand this better, let us consider a system with just one impurity at very
low temperatures. This impurity has an absorption associated with the two
electronic levels responsible for the transition. When that transition is weakly
coupled to lattice vibrations of the host, the spectrum has a Lorentzian line
shape which is the Fourier transform of Eq. (1.3). The Fourier transform of Eq.
(1.3) has a Lorentzian profile that can be expressed in terms of frequency by
[8]
Eq. (1.4)
where is the resonance frequency and is the full width at half maximum in
Hz expressed by
Eq. (1.5)
If, now, we consider many impurities distributed randomly in a perfect crystal
with equivalent local geometry and environment, the resulting spectrum will still
have a Lorentzian lineshape as in the case of just one impurity.
7
At room temperature, the interactions between electrons and phonons and, to a
lesser extent, electron and nuclear fluctuations are the main contributions to the
homogeneous linewidth; this means that the pure dephasing time T2*
determines the linewidth at high temperatures. In contrast, at lowest
temperatures, the excited state lifetime, T1, can become the dominant
contributor. The homogeneous spectral linewidth is given by the effective optical
dephasing time, T2 [9, 10]. The full width at half maximum of the Lorentzian line
shape can be expressed as [11, 12]
Гhom=
=
+
Eq. (1.6)
where the relationship between the excited state lifetime T1, the pure dephasing
time T2* and the effective dephasing time T2 is given by
Eq. (1.7)
Figure 1.3. Perfect crystal and the Lorentzian lineshape of identical optical centres.
w0
γ=1/T1
Angular frequency w
8
1.1.3. Inhomogeneous broadening
Unfortunately, all crystals contain imperfections, impurities, dislocations, isotope
distribution, etc. As a consequence every single optical centre is in a different
local environment. Because the local field of an impurity does affect its
transition frequency, each single impurity will have a slightly shifted transition
frequency with respect to a central frequency. As a result, the lineshape will
most likely be Gaussian since it is the sum of shifted individual Lorentzian line
shapes [7]. This effect is called inhomogeneous broadening and its width is
usually referred to as Гinh.
The theory of inhomogeneous broadening in crystals is founded on the
statistical model of Stoneham [13]. He assumes that in crystals some point
defects are distributed randomly and that those defects interact with the impurity
transition [13]. If we consider r as the distance between the optical centre and
the defect then the perturbation of the optical centres transition frequency varies
as 1/r3 and a Lorentzian lineshape is predicted. When the defects are more
concentrated the lineshape changes from Lorentzian to Gaussian [14, 15].
Figure 1.4. Inhomogeneous broadening. The spectrum is the sum of a large number of much narrower individual homogeneously broadened lines that are each shifted in frequency with respect to each other. Gaussian line
shapes are often observed. Adapted from [5].
9
At liquid helium temperatures, that is at temperatures < 5 K, only the
inhomogeneous linewidth can be observed in non-selective spectroscopy as the
phonon-induced fluctuations are frozen out and hence, are negligible. In other
words, the homogeneous linewidth cannot be observed in conventional
spectroscopy at low temperatures [5]. On the other hand, at high temperatures,
the inhomogeneous broadening becomes less important and the homogeneous
linewidth may be directly observed and the lineshape becomes Lorentzian [5].
When MacFarlane and his collaborators [16] started using single frequency dye
lasers in the 1970s, many spectral lines of rare-earth impurities were shown to
have inhomogeneous linewidths of 1-10 GHz. In the late 1980s, the
development of reliable single frequency lasers enabled a significant increase of
the number of materials and linewidths that could be studied [17, 18].
The inhomogeneous linewidth of an electronic origin is mainly determined by its
nature. So far rare-earth doped crystals have provided the narrowest
inhomogeneous linewidths. To a great extent this is because their f-f transitions
are not strongly perturbed by the surrounding atoms. Also pure spin-flip
transitions in chromium (III) are not very sensitive to inhomogeneous
broadening and present narrower inhomogeneous linewidths than most other d-
d transitions [11].
An interesting example for rare earth impurity ions is the case of Nd3+ ions at
low concentration in Y7LiF4 showing inhomogeneous broadening as narrow as
10 MHz [19].
Another example of spectral lines with very low inhomogeneous broadening,
are the transitions of Pr3+ in CaF2, where six hyperfine lines of 600 MHz were
resolved [20].
On the other side, the largest inhomogeneous linewidths observed are for
charge-transfer transitions and for ligand-field transitions. These transitions are
also affected by strong coupling to vibrations.
10
Table 1.1 shows a comparison of d-d and f-f transitions in crystalline and
amorphous structures is presented summarizing the very different values for the
inhomogeneous linewidths [11].
Transition Crystalline host
(cm-1)
Amorphous host
(cm-1)
d – d 0.1 - 100 10 - 1000
f - f 0.01 – 10 1 - 100
Table 1.1. Typical inhomogeneous linewidths of d-d and f-f transitions in crystalline and amorphous hosts [11].
1.2. Chromium (III) systems
The 4A2 – 2E transition in chromium (III) systems is discussed in this section. A
brief overview of the structure and bonding characteristics of these systems are
presented to allow a better understanding of the spectroscopic observations
explained later.
1.2.1. General properties
The chromium (III) cation is a 3d3 system having 3d electrons on the outside of
the ion. These outer electrons are very sensitive to the environment. The
dominant ligand field component in chromium (III) complexes is generally of
octahedral symmetry. Apart from this contribution, lower symmetry fields can
exist, such as trigonal, rhombic or tetragonal components, and they have to be
taken into account in a rigorous description [1, 21].
11
Figure 1.5. Octahedral geometry. This is a representation of the central chromium ion surrounded by the six ligands in an octahedral environment. Adapted from [22].
As is shown in the Figure 1.5, due to the octahedral ligand field symmetry, the
five degenerate d orbitals are divided into two sets of levels, eg (twofold
degenerate) and t2g (threefold degenerate). This splitting is due to the
interaction between electron orbitals of the metal and electron configuration of
the ligand. The dxy, dxz and dyz orbitals are stabilised and their energy is
reduced. The dz2 and dx
2-y
2 orbitals are destabilised due to the fact that these
orbitals point in the same direction as the ligands. The energy difference
between these two orbitals eg and t2g is characterized as 10Dq, where Dq is a
ligand field parameter that measures the octahedral field strength [21, 23]. The
splitting is shown in Figure 1.6.
Figure 1.6. Octahedral ligand field splitting. Both eg and t2g levels exhibit degeneracy.
Energy
dyz
dx2-y
2
dxz
dz2
dxy
10 Dq
t2g
eg
12
In a perfect octahedral field, numerous energy levels are present in a chromium
(III) system. Two factors are important when it comes to determine the energy
of these levels: the octahedral field strength Dq and the Racah parameter B
which is a parameter for the description of electron-electron repulsion.
In the Tanabe-Sugano [1] diagram shown in Figure 1.7, the energy levels for
3d3 systems are represented.
Figure 1.7. Tanabe-Sugano energy level diagram for 3d3 electronic configuration in an octahedral field. The
octahedral field increases from left to right. The energy of the 4A2 ground state is used as a reference to measure
the energy of each electronic state. Adapted from [1].
Emerald Ruby
13
In addition to the lower symmetry fields that add more complexity to the
structure of chromium (III) systems, the effects of spin-orbit coupling must be
considered. In the Figure 1.8 a diagram for the 2E-4A2 transition is shown.
Figure 1.8. Schematic energy level diagram for ruby and emerald. a) Energy level diagram of Cr3+
in a an octahedral field. b) Fine structure of
4A2 and
2E. c) Luminescence spectrum of ruby at low temperature. d) Absorption spectrum
of ruby at low temperature. Two intense and broad spin-allowed transitions are shown.
At very low temperatures only the transitions from the lowest excited states can
be observed in luminescence of chromium (III) ions. This is because higher
energy states are not populated at these temperatures as they relax non-
radiatively to the lowest excited state. The 4A2 – 2E transition is a spin-forbidden
transition (because ∆S≠0). Despite being forbidden, it still happens due to lower
symmetry fields and spin-orbit coupling. In chromium (III) systems with strong
ligands, the 2E spin-flip level is the lowest excited state. Luminescence is
possible from the two levels of 2E as seen in Figure 1.8. As a result the four R-
lines due to ground and excited state splitting can be observed [24, 25].
4A2
2E
d) Absorption c) Luminescence b) Octahedral +
trigonal ligand
field + spin-orbit
coupling
R1 R2
a) Octahedral field
No
n-r
adia
tive
rel
axat
ion
Ab
sorp
tio
ns
4T1
4T2
Emis
sio
ns
(flu
ore
sce
nce
)
14
Transition metal complexes are very sensitive to the metal-ligand distances,
which affect the ligand field and can cause large shifts in the electronic
potentials of the energy levels. For instance, the displacement between ground
and excited state potentials in d-d spin-allowed transitions can be very
significant and lead to very pronounced vibrational sidebands. Also the
electronic origin accounts for a fraction of the total intensity of the spectrum
only. Hence, the intensity and energy of the electronic origin are very
susceptible to changes of the metal-ligand distances for spin-allowed d-d
transitions.
When transitions occur in the same electronic configuration, they are to a great
extent independent of the ligand field. As is seen in the Tanabe-Sugano energy
diagram of Figure 1.7, the 2E energy remains almost constant when increasing
the octahedral field strength. However, the energy separation of other levels
such as 4T1 and 4T2 grow linearly as the octahedral field is increased (these last
energy levels are interconfigurational d-d spin-allowed transitions). Hence,
spin-flip transitions such as 4A2 → 2E present weak vibrational sidebands and
most intensity is in the narrow electronic origin.
Additional broadening of an optical transition is observed when other factors
affect the symmetry of the system. Spin allowed transitions usually appear
without a specific structure due to a high degree of inhomogeneous broadening.
4A2 – 2E spin-flip transitions which happen in the same electronic configuration
are less sensitive to these changes owing to their first order independence from
the ligand field strength [8, 26].
1.3. Temperature dependence of the homogeneous linewidth
The temperature dependence of the homogeneous linewidth in crystalline
structures is due to different mechanisms which are based on electron and
nuclear spin-spin interactions and electron-phonon interactions [8, 11].
15
Figure 1.9. Schematic representation of important mechanisms of electron-phonon interaction. From left to right: direct process, Orbach process and Raman process.
The first case is the direct process. This process occurs very rapidly when the
energy gap is close to the maximum density of phonon states. Due to the high
density of phonon states in coordination compounds, it is a very important
source of homogeneous line broadening. In coordination compounds the energy
gaps are typically between 10 and 3000 cm-1. As a consequence the direct
process is very likely to happen because of the high density of phonon states is
in the same range [1].
The Raman and Orbach processes are non-radiative transitions mechanisms
between close-lying levels through two phonons as depicted in Figure 1.9. In
the Orbach process one phonon is absorbed from the lower to the upper state
and then emitted back to another sublevel of the lower state. The Raman
process is basically the same as the Orbach mechanism but the difference is
that the upper state is virtual only. There is rapid broadening with temperature
of the homogeneous linewidth when one of these two processes takes place
according to the Debye approximation for the density of phonon states [1, 8,
25].
A highly influential paper in the field of thermally induced broadening and shift of
optical transitions of impurity centres in solids was published by McCumber and
Sturge in 1963 [27]. They calculated the contributions of one-phonon (“direct”)
a
b
c
16
and two-phonon (“Raman”) processes to the broadening and shift for optical
transitions for systems with level splitting of the order of kT. By means of non-
selective spectroscopy they analysed the R-lines of ruby [27]. They used a
perturbation approach in the weak coupling limit resulting in the Eq. (1.8.a):
Eq. (1.8.a)
where is the Debye temperature. The linewidth contribution expressed in Eq.
(1.8.a) is due to two phonon Raman scattering of Debye phonons and it
describes the temperature dependence of the homogeneous linewidths of
impurity centres in solids.
In the 1980s, Hsu and Skinner [28] published a series of papers using a non-
perturbative approach for the temperature dependence of the homogeneous
linewidth within the weak-coupling regime and the Debye approximation for the
density of phonon states. They obtained the following temperature dependent
contribution to the homogeneous linewidth for the Debye model of acoustic
phonons:
Eq. (1.8.b)
In this equation W is the quadratic electron-phonon coupling constant,
where is the Boltzmann constant. In the weak coupling limit (W<<1) Eq.
(1.8.b) simplifies to Eq. (1.8.a) (original equation obtained by McCumber and
Sturge) [27]. They suggested that the direct one-phonon process (a phonon is
absorbed or emitted between the 2Ā and Ē levels of the 2E state separated by
∆) is not relevant for temperatures > 90 K.
The direct one-phonon process is expressed by the well known Eq. (1.9):
17
Eq. (1.9)
where is the 2E splitting (29 cm
-1 in ruby) and is the R2 linewidth
at 2.5 K ( expresses the zero-point lifetime of the 2Ā level).
Blume et al. [29] calculated the relaxation lifetime of 2Ā(2E) to Ē(2E) level due to
spontaneous phonon emission in ruby and it was found to be 0.3 ns. This
allowed calculating the direct one-phonon process contribution. More recently a
study [30, 31] was undertaken to measure the temperature dependence of the
R lines in ruby. The temperature dependence of the R2 linewidth was found to
be dominated by a direct one-phonon process up to 50 K and by a two-phonon
Raman scattering process above that temperature. In Figure 1.10 a re-
examination of the R1(±3/2) linewidth in ruby by doing transient hole-burning
measurements in a low magnetic field is shown. One can observe the direct
one-phonon process between the split levels of the 2E excited state up to 50 K
and the two-phonon Raman scattering process at higher temperatures.
18
Figure 1.10. Temperature dependent contribution to the homogeneous linewidth of the R1( ) in ruby. Adapted from [30].
Some data on the temperature dependent contribution of the linewidths at low
temperature has also been measured by Muramoto et al. [32]. They obtained
constant values below 10 K.
Extensive measurements of the temperature dependence of the linewidth and
lineshift of the electronic origins of many materials have been made [33-38].
Raman
Direct
Direct + Raman
Temperature (K)
Co
ntr
ibu
tio
n t
o h
om
oge
neo
us
linew
idth
(M
Hz)
19
1.4. Laser spectroscopy
With the development of tunable lasers over the last decades there have been a
variety of techniques for increasing spectral resolution and extracting
information which is obscured by inhomogeneous broadening. The existing
techniques can be classified into two categories:
- Time-domain techniques, such as photon echoes and optical free
induction decay.
- Frequency-domain techniques, such as fluorescence line-narrowing
and spectral hole-burning.
Spectral hole-burning and fluorescence line-narrowing, which have been used
in the experiments of the present thesis, are explained in the following
paragraphs.
1.4.1. Spectral hole-burning
There are two types of hole-burning: persistent and transient hole-burning.
Within persistent hole-burning, one must distinguish between photochemical
and non-photochemical hole-burning.
- Photochemical HB: the so-called “photoproduct” absorption band is
well separated from the original absorption band. This type of hole
can persist infinitely at low temperatures.
- Non-photochemical HB (NPHB): this kind of hole is typical for
amorphous systems. In this case the “photoproduct” is within the
inhomogeneous absorption line.
- Transient HB (THB): in this case, a fraction of the population is stored
in any excited level. A narrow-band laser of width Гl and frequency fl
excites molecules within the inhomogeneous broadening that absorb
20
resonantly with the laser. A hole is created in the original absorption
band. The hole width can approach the homogeneous width Гhom in
ideal cases [10].
1.4.1.1. Transient spectral hole-burning
Some of the optical centres lying in the inhomogeneously broadened transition
are excited. The lifetime of the transient hole is given by the lifetime of the state
in which the population is stored.
When high powered lasers are needed, care must be taken in order avoid
excessive heating of the sample. A larger broadening of the transition is
obtained otherwise. Obviously, stimulated emission takes part in the readout
process if the population is stored in the excited state [8, 11].
Transient hole-burning experiments can be performed by using one or two
single-frequency lasers. If only one laser is used, this has to be scanned by fast
modulation, e.g. current. When two lasers are used, the first burns the hole at a
particular frequency and the second laser scans the spectrum of the hole [10,
39, 40]. In this project, transient hole-burning has been conducted by using a
single laser.
The first transient hole-burning experiments in the solid state were carried out
by Szabo in the 1970’s [39]. However, hole-burning was observed for the first
time in condensed matter in the 1960’s [41].
The value of Гhom can be estimated as Гhom~ 0.5 Гhole– Гl, where Гl is the laser
linewidth.
21
Figure 1.11. Representation of mechanisms for transient hole-burning. From left to right: direct depletion of ground state, relaxation to metastable excited state and relaxation to metastable sublevel of the ground state. Adapted
from [11].
1.4.1.2. Persistent hole-burning
There are two possible mechanisms to create persistent hole-burning. Their
names are photochemical hole-burning (PHB) and non-photochemical hole-
burning (NPHB).
In PHB optical centres in the excited state can be subject to a photochemical
reaction and a shift in their transitional frequency results leading to a decrease
of the absorbance at that particular frequency. As a result the spectrum
changes and there is a photoproduct in the absorption spectra outside of the
inhomogeneous linewidth [5, 8, 11].
Ground state
Read
laser
Bu
rn laser
Excited state
22
Figure 1.12. Persistent hole-burning. Appearance of photoproduct and difference between spectra before and after burning. Adapted from [11].
On the other hand, NPHB is more likely to happen in amorphous structures
rather than in crystalline systems. The idea is that the appearance of anti-holes
occurs in proximity to the burned hole due to rearrangements of host-guest
interactions [42-44].
1.4.1.3. Potential applications of spectral hole-burning
Hole-burning is not only important for being a tool to unravel details of the
electronic structures of ions. It has potential applications in quantum computing,
laser stabilisation and in frequency and time domain optical storage (particularly
persistent spectral hole-burning).
Chemical modification
Photoproduct
23
When a narrow bandwidth laser excites those centres resonant with the laser
frequency, multiple spectral holes can be burned within the absorption line. The
absence or presence of spectral holes at given frequencies can be used to
represent binary data. The ratio of inhomogeneous to homogeneous linewidth,
determines the storage capacity in the frequency domain and is the figure
of merit (FOM) for this application [45, 46].
One of the major challenges is to find materials with a high storage capacity at
room temperature, i.e. a high FOM.
1.4.2 Fluorescence line narrowing (FLN)
In the 60’s Denisov et al. [47] were able to narrow the luminescence spectrum
of Eu3+ in a glass by means of mercury lines. FLN was also demonstrated in a
crystal by Szabo in 1970 [48, 49]. He observed a narrowing of the R1 line in
ruby when using resonant ruby-laser excitation. By means of a narrow band
laser, a subset of optical centres within an inhomogeneously broadened
absorption line is excited and the fluorescence is dispersed with a
monochromator or a Fabry-Pérot interferometer. The subset of optical centres
in the inhomogeneous broadened transition has the same transition energy. As
the energy levels in crystalline systems are very correlated, this is a site
selective technique in these systems [8]. It is another method of approaching
the homogeneous width of an optical transition.
The ions or molecules that are in resonance with the laser are excited and only
these molecules or ions will emit light. As a consequence much narrower lines
will appear in the fluorescence spectrum in comparison with non-selective
excitation [3, 5]. When the luminescence is observed resonantly with the laser
frequency, this luminescence must be separated from the laser light by means
of gating techniques [8]. Mechanical choppers can be used when the excited
24
state lifetime is longer than 10 µs. However for shorter excited state lifetimes
acoustic-optic or electro-optic modulators are recommended [1].
FLN processes in resonant experiments involve two photons so the linewidth
observed is twice the homogeneous linewidth. The optical centre is excited by
one photon and another photon is observed in emission. In non-resonant
experiments the lifetime and correlation of the third energy level affect the
measured linewidth [1, 8].
Figure 1.13. FLN technique, illustrating both resonant and non-resonant experiments. Adapted from [8].
1.5. Slow light
The concept of dispersive wave propagation and the group velocity has been
interrelated historically. In 1839, Sir William R. Hamilton [50] introduced the
concepts of phase and group velocities when he explained dispersive wave
Reso
nan
t FLN Laser excitation
No
n-reso
nan
t FLN
25
propagation as a coherent superposition of monochromatic wave disturbances.
Rayleigh [51] attributed the original definition of group velocity to Stokes [52]
claiming that: 'when a group of waves advances into still water, the velocity of
the group is less than that of the individual waves of which it is composed; the
waves appear to advance through the group, dying away as they approach its
anterior limit. This phenomenon was, I believe, first explained by Stokes, who
regarded the group as formed by the superposition of two infinite trains of
waves, of equal amplitudes and of nearly equal wavelengths, advancing in the
same direction'. Rayleigh [53] explained the difference between phase and
group velocity by using those results.
The understanding of both phase and group velocity is crucial to interpret the
results described in chapter 3. Let us start explaining some concepts related to
wave propagation [54]. The total electric field E(t) of a non-monochromatic wave
formed by two waves with amplitude A(t), angular frequencies and , wave
numbers and propagating along the z axis, is [55]:
Eq. (1.10)
By applying a trigonometric transformation, we obtain:
Eq. (1.11)
where , , and . If
both frequencies and are of the same order, Eq. (1.11) can be considered
as a non-monochromatic wave expressed by the product of a carrier wave (high
wave number and frequency ) and a modulation wave (low wave number
and frequency
). The carrier wave propagates at phase velocity and
the modulation wave propagates at [56].
The intensity of the field can be considered as
Eq. (1.12)
26
Eq. 1.12 the first term represents the carrier term oscillating at a frequency
and the second term represents the envelope travelling at a low frequency
, which is the frequency of the intensity modulation and modulates the
first term.
The group velocity of a wave is the velocity with which the intensity modulation
of the wave propagates though space whilst the phase velocity is the velocity at
which the carrier term displaces.
Figure 1.14. Phase and group velocity of a wave. Adapted from [54].
From the phase velocity equation
, it can be observed that each
monochromatic wave propagates at different phase velocities. In the case that
the monochromatic waves have close frequencies, the group velocity can be
expressed as follows [57]:
Eq. (1.13)
where is the real part of the refractive index of a medium and is frequency
dependent (because each wave comprises many frequencies travelling at
27
different phase velocities). From Eq. (1.13) it follows that the group velocity can
be smaller (slow light) or larger (superluminal) depending on the value of
.
Slow light can be defined as a reduction in the group velocity of the light (vg<<c)
and fast light as an increase in the group velocity (vg>c or vg is negative) [58].
When it comes to an absorbing medium, the group velocity can be influenced
by the refractive index and the real part of the refractive index can be
frequency dependent in a narrow optical resonance of the medium. It is in that
narrow optical region is where slow light takes place [57].
Figure 1.15. Horizontal axis represents optical frequency. Rapid change of the index of refraction (blue line) in a region of rapid change of absorption (gray line). The steep and linear region of the refractive index in the center
gives rise to slow light. Adapted form [59].
There are several methods to create slow light. The common factor is to create
a narrow spectral region with high dispersion. The methods are usually
classified in two groups [60]:
- Material dispersion, such as Electromagnetically Induced
Transparency (EIT), Coherent Population Oscillation (CPO), hole-
burning (HB) and Four Wave Mixing (FWM). They all produce a rapid
change in refractive index as a function of the frequency. A nonlinear
effect is used to modify the dipole response of a medium to a “probe”
field,
Rapid change of
absorption
Index of
refraction
28
- and waveguide dispersion, such as photonic crystals or other
resonator structures which modify the k-component of a wave
(structural dispersion).
1.5.1. Kramers- Kronig relations
In order to produce slow light in a system it is necessary that the refractive
index changes rapidly as a function of frequency. This is possible when the
material is in resonance (or very close) with the applied optical field. The
Kramers- Kronig relations are presented below. These equations relate the real
part of the refractive index and the absorption in a material [61]:
Eq. (1.14.a)
Eq. (1.14.b)
By analysing Eqs. (1.14.a) and (1.14.b) it can be observed that a narrow dip in
an absorption spectrum will lead to strong normal dispersion (dn/dw>>0) and
that a gain will lead to anomalous dispersion (dn/dw<<0). Figure 1.16 illustrates
the first case.
Figure 1.16. Absorption and refractive index are related though the Kramers-Kronig relations. A narrow spectral dip leads to strong normal dispersion. Adapted from [62].
29
The group index of a pulse can be expressed by Eq. (1.15).
Eq. (1.15)
It is obvious that if the dispersion is large, the group index can be made large as
well. Hence, the conclusion is that in order to produce slow light in a material, a
process that leads to narrow and strong features must be undertaken. By
Kramers-Kronig relations, this characteristic will lead to the necessary large
dispersion for slow light.
1.5.2 Methods to create slow light
The most relevant methods to create slow light are discussed in the following
sections.
1.5.2.1. Electromagnetically induced transparency (EIT)
It is well-known that a rapid change in the refractive index near the resonance of
a material can lead to large values of the group index. Low group velocities are
accompanied by strong absorption so the observation of this effect is very
difficult. In order to reduce this strong absorption the technique of
electromagnetically induced transparency has been used to render the material
medium very transparent and retain the strong absorption necessary to observe
slow light propagation [63-66]. In the 1990s Harris et al. observed low group
velocities for the first time in strontium [67-68].
Using this technique Kasapi et al. observed a group velocity of vg=c/165 in a 10
cm long Pb vapour cell [69]. The lowest group velocity observed so far in an EIT
medium was by Budher et al in 1999. Group velocities as low as 8 m/s were
observed in a warm thermal rubidium vapour [70]. In the same year, Hau and
30
his collaborators [71], used this method in order to produce slow light. The
group velocity of light was as slow as 17 m/s in an ultracold atomic gas at 450
nK. In the first experiments of ultra-slow light experiments in a solid, Turukhin et
al. observed a velocity of 45 m/s corresponding to a group delay of 66 μs in a
praseodymium doped Y2SiO5 crystal [72].
EIT is a coherent optical nonlinearity which presents a transparent medium over
a narrow spectral range within an absorption line. Dispersion is created within
the transparency window leading to slow light [73].
In order to observe slow light by means of this procedure, two optical fields
must be used (for example, two lasers). These optical fields must interact with
three quantum states of a material. The probe field (wp) and a much stronger
coupling field (also called pump or control, wc) are tuned near resonance at
different transitions. Hence, EIT can be achieved in a three-level atom with a
coupling and a probe field to produce a two-quantum coherence with a long
lifetime. There is population trapping in a dark, non-absorbing state and this
produces the required coherence. The atoms are trapped in this state and
cannot be excited due to destructive interference between transition paths to
the excited state. As a consequence of the existence of the dark state
coherence, there is a narrow "hole" in the absorption line for the probe. As the
Rabi frequency induced by the coupling field is larger than the holewidth, a
narrow EIT hole will be produced [74].
Figure 1.17. in order to produce EIT a control beam, wc, ia applied between I1> and I2>. This splits level I2> and a probe beam sees less absorption over a narrow spectral range.
strong
control
beam probe beam
I3>
I1>
I2>
31
In a configuration of three states at least two out of the three possible transitions
between the states must be dipole allowed and the other dipole forbidden. One
of these three states is connected to the other states by means of the probe and
coupling field.
As is described in the literature coherent preparation of the medium produces
remarkable changes in the optical properties of a molecular medium or an
atomic gas [75, 76]. The laser-induced coherence of atomic states is the cause
of the modification of the optical response of an atomic medium. This will lead to
quantum interference between the excitation paths which control the optical
response. When quantum interference between the excitation pathways
happens, linear susceptibility can be removed (absorption and refraction) at the
resonant frequency of the transition [77, 78].
A relationship exists between the optical properties of molecular and atomic
gases and their energy-level structures. The first order susceptibility χ(1) can
describe an atom that responds linearly to resonant light. This susceptibility can
be divided in two factors: Re(χ(1)) and Im(χ(1)). The real part determines the
refractive index and the imaginary part represents the absorption. As we can
see in Figure 1.18 the imaginary part as a function of frequency can be
represented by a Lorentzian function while the real part can be approximated by
a dispersion profile.
Figure 1.18. The picture to the left is the imaginary part of χ(1)
and to the right is the real part of χ(1)
.
(wp-w31)/γ31 (wp-w31)/γ31
Im(χ
(1) )
Re(χ
(1) )
32
Figure 1.18 shows the susceptibility as a function of the probe field wp with
respect to the atomic resonance frequency w31. The dashed line is for a
radiatively broadened two-level system with width γ31 while the solid line is an
EIT system with resonant coupling field.
In conclusion, EIT creates a narrow transparency window in the absorption line
and when a probe pulse propagates through that window it may be delayed.
This results in slow light [73, 79]. As dictated by the Kramers-Krӧnig (see
section 1.5.1.) relations a change in absorption in a narrow spectral range yields
a rapidly changing refractive index in that region. This rapid change produces a
very low group velocity.
1.5.2.2. Coherent population oscillation (CPO).
Years after EIT was developed, several groups investigated different ways to
apply slow light to more realistic fields and to overcome the hole bandwidth
problem in EIT experiments. In this section some aspects of slow light by
means of coherent population oscillations are analysed [58].
CPO involves the creation of a "spectral hole" due to population oscillations.
The "hole" is due to the fact that the ground state population is modulated at the
beat frequency between the probe and pump field applied to the material.
"Spectral holes" as narrow as 36 Hz lead to group velocities of 57 m/s [58].
In 1967 Schwartz predicted the appearance of these "holes" [80]. In 1983,
Hillman observed for the first time a CPO "hole" in ruby [81]. An argon-ion laser
was used in these experiments to pump population from the ground state to the
4T2 absorption band. The decay from this level to the metastable 2Ā and Ē
levels of 2E is fast (on the order of picoseconds) and from these levels to the
ground state is on the order of a few miliseconds (T’1). A "hole" at the laser
frequency is created as a consequence of that long lifetime. The width of that
hole was 37 Hz which is the inverse of the population relaxation time.
33
More recently Bigelow [82] carried similar experiments for the observation of
slow light in ruby at room temperature. In Figure 1.19 the 4A2 ground state is
denoted as a, the levels 2Ā and Ē are denoted as c and the 4T2 absorption band
is b. As said above level b decays very rapidly so the system is reduced to a
two-level system. In the picture T1 is the ground state recovery time and it is
twice the lifetime of level c T’1. T2 is the dipole moment dephasing time.
Figure 1.19. Schematic diagram of energy levels in ruby. Adapted from [82].
Bigelow et al. claim that when light with a frequency w1 is in resonance with the
transition a-b (with the intensity modulated at interacting with a system
as shown in Figure 1.19 there will be a decrease in the ground state population
Ng because of absorption. The ground state re-populates by transition c-a. Its
state population will be modulated at the modulation frequency and will be
delayed with respect to the incident intensity. The metastable level lifetime 1/T1’
and the recovery time of the pumping cycle are of the same order. The
modulation depth of the ground state population Ng is larger when the pumping
intensity has a modulation period similar to that of the metastable level
lifetime. When that resonance happens a narrow overall resonant behaviour
occurs and a decrease in absorption is observed. By Kramers-Kronig relations,
the real part of the refractive index nr depends on the ground state population
Ng. As Ng is frequency dependent, that dependence will give large values of
near resonance leading to small values of group velocities.
Bigelow [82] suggested that a beam with amplitude modulation experiences
large group index due to the spectral dip observed. In Figure 1.20 the delay as
a
b
a
b
c
W1
Rapid decay
34
a function of the modulation frequency for different pump powers is shown. It
was found that the largest delay corresponded to the deepest and narrowest
hole. Bigelow et al. reported a group velocity of 57 m/s and they claimed that
slow light can be observed only by applying a single intense pulse of light to
induce the saturation needed for the slow light, i.e. no separation of pump and
probe waves is required.
Figure 1.20. Time delay versus modulation frequency for different imput powers. The inset shows the normalized 60 Hz input and output signal at 0.25 W. There is a delay of 612 µs which corresponds to 118 m/s. Adapted from [82].
Different Gaussian pulses were applied to observe the delay when propagating
through the ruby. The longer pulses experienced the longer delays.
35
Figure 1.21. Normalized input and output intensities of different pulse lengths with the corresponding group velocities. The inset shows a close-up of the 20 ms pulse. Adapted from [82].
However, other researchers countered that the above CPO investigations are
the consequence of lack of physical understanding of these effects. Zapasskii et
al. claim that the discussed results presented above can be regarded as a
consequence of the model of a saturable absorber [83].
A saturable absorber is a layer of an optical medium whose absorption shows
saturation with increasing light intensity. Various groups [83-86] show how the
pulse delay is the result of any medium with nonlinear intensity and there is no
evidence of any change in the light group velocity. They say that when the light
interacts with a nonlinear medium a change in the intensity spectrum is
produced. The interpretation of these effects does not need the introduction of
concepts such as slow light or group velocity reduction [84-86]. In particular,
Selden claims that there is no demonstration of group velocity reduction since
the results are compatible with saturable absorption theory [86].
Separation of the probe and pump beams seems to be a minimum requirement
for making the difference between hole-burning and saturable absorption in
slow light experiments. In this way, the depth of the hole would be reduced,
36
which should produce a reduction in the slow light. Only when a very narrow
source is used to scan the absorption spectrum in the homogeneous linewidth a
hole can be created in the absorption line [87].
CPO experiments on slow light are not well interpreted because the observed
phase shift is treated as a signal transit time. This is used to calculate a group
velocity that can be done very small by reducing the length of the sample
(vg=L/Td≥L/Ts) where Td is the observed delay and Ts is the metastable lifetime.
A correct interpretation is that the transit time depends on sample length and it
should be larger than the relaxation time of the absorber in a long sample [88].
As a conclusion, the results of slow light by means of CPO do not demonstrate
unequivocally group velocity reduction.
1.5.2.3. Hole-burning.
Rebane et al. proposed to produce slow light by means of persistent spectral
hole-burning [89, 90]. A burn and a probe laser are applied to the system. The
burn laser creates a transparency window for the probe, creating narrow
spectral filters with very high contrast and, as a consequence, with high
dispersion.
Let us consider a weak probe pulse of amplitude launched into a medium at
the input end. In the case of a weak probe, the molecules behave as classical
damped harmonic oscillators and the interaction between the spectral hole-
burning medium and the probe pulse can be interpreted within classical pulse
propagation in a linear absorbing medium. The change in the phase and
amplitude can be expressed as
Eq. (1.16)
where is the complex frequency domain amplitude response function
expressed as
37
Eq. (1.17)
is the intensity transmission through the sample in the longitudinal
direction at frequency expressed as where is the
absorption coefficient and the length of the medium. is the phase of the
response function calculated by employing the Hilbert transform:
Eq. (1.18)
In Eq. (1.19) represents the frequency domain amplitude of the probe,
Eq. (1.19)
Eq. (1.20) expresses the time domain intensity at the output of the medium,
Eq. (1.20)
which is calculated for a Gaussian probe pulse.
Slow light takes place when all frequency components experience the same
attenuation and when the phase changes linearly with the frequency.
In 2004, spectral hole-burning was used to create slow light. Fan et al. [91]
slowed down light in a solid to approximately 43 m/s through the change in the
refractive index of the medium at room temperature.
In this thesis, this theory is applied to transient hole-burning in the R1-line of
chromium (III) in ruby.
38
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44
Chapter 2
Experimental
2.1. Sample preparation
Ruby and emerald crystals were cut with a water cooled diamond blade in a
Shell-Lap Gemmasta saw. The samples were polished using 1 µm, 0.25 µm
and 0.1 µm diamond paste on a Pedemin wheel. All the crystals were carefully
cleaned before they were used in optical experiments.
2.2. Room temperature spectroscopy
Non-polarised absorption spectra at room temperature were measured by a
Varian Cary 50 UV-Vis spectrometer. PC software facilitated the calibration and
baseline corrections.
2.3. Spectroscopy at Liquid Helium Temperatures
Very low temperatures (< 10 K) are necessary to observe details of electronic
structure that are obscured at higher temperatures. In this section an
explanation is provided about the steps required for correct mounting and
cooling of the samples. Also a brief description of the instrumentation used in
this project is given.
45
2.3.1. Mounting of samples
The sample was cleaned with hexane and was mounted over a hole in a brass
disc. The sample was imbedded in heat conducting cry-con copper grease. The
grease is used to ensure maximum thermal conductivity between the cold finger
of the cryostat and the sample. The sample holder was then attached to the
cold finger of the closed-cycle refrigerator (CCR).
Figure 2.1. Schematic of sample holder.
Crystal
sample
Brass plate
Cry-con
grease
46
2.3.2. Cooling method and temperature control
A Janis/Sumitomo SHI-4.5 closed-cycle refrigerator (CCR) was used in order to
cool down the samples. Temperatures as low as 2.5 K can be reached with the
use of this CCR without the need of liquid nitrogen and liquid helium. The CCR
uses a two-stage Gifford-McMahon cycle to cool down the cold finger. The cold
head is separated from the compressor unit by means of two high pressure
hoses. These hoses carry the expanded and compressed helium out and into
the cold head. The cold finger of the CCR is cooled to low temperature when
the helium gas is expanded [1].
In order to isolate the cold finger from the external environment, it is enclosed in
a radiation shield and a vacuum jacket. In this way convective and radiative
energy transfer between the environment and the CCR is minimized.
The vacuum jacket was evacuated using a turbomolecular pump to pressures
<8x10-4 mBar before cooling the system. The cooling process to 2.5 K takes at
least two hours. When the lowest temperature was reached a pressure of
around 10-6 mBar was measured. To measure and control the temperature at
the sample a Lakeshore Model 330 temperature controller and a 50 Ohm heater
was employed.
2.3.3. Absorption and Transmission spectroscopy
Low temperature transmission spectra were obtained using an Osram HLX
Xenophot 64840 bulb powered by a GW GPC-1850 dual-tracking power supply
to provide white light. The white light was passed through a polarising filter to
measure the polarised spectrum. In order to reduce luminescence effects in the
sample an OD filter was used to control the light power reaching the sample. A
200 mm lens was used to focus the light onto the sample and another 75 mm
47
lens was used to collimate the transmitted light after the cryostat. A third 200
mm lens was used to re-focus the light through a Thorlabs MC1000 opto-
mechanical chopper onto the entrance slit of a monochromator. The light was
then dispersed by a 1 m single monochromator (Spex 1404 equipped with a
1200 grooves/mm holographic grating). The light was detected using a side-
window photomultiplier tube (Hamamatsu R928). A current-to-voltage
preamplifier (Femto DLPCA-200) and digital lock-in amplifier (Stanford
Research Systems Model SR810 DSP) processed the signal from the
photomultiplier. Finally the signal was collected by a PC (see Figure 2.4). The
PC processed the transmission spectra in order to obtain absorption spectra
(dividing the baseline by the spectra obtained).
Eq. (2.1)
Io is the intensity of the light before the sample and I is the intensity after the
sample. The absorption spectra was obtained using the transmission spectra
by means of the relationship:
Eq. (2.2)
Similar experiments were conducted using a laser diode to obtain spectra with
much higher resolution.
2.3.4. Non-selective luminescence spectroscopy
For non-selectively excited luminescence spectra a Nd:YAG was used with a
wavelength of 532 nm to excite the sample. A polarising filter was used to find
out the polarisation dependence of the luminescence and also an OD filter to
reduce sample heating. 75 nm and 200 mm lenses were used to collimate the
emitted light. The luminescent light passed through a chopper, monochromator
48
and detected by a photomultiplier. A pre-amplifier and lock-in amplifier
processed the signal from the photomultiplier. Finally signals were collected on
a PC (see Figure 2.5).
2.3.5. Spectral hole-burning
The crystals were cut perpendicular and parallel to the c axis. Spectral holes
were burnt and read out by a stabilised laser diode (Thorlabs TEC2000
temperature controller with a ultra-low noise current source ILX LightWave LDX-
3620). After a short burn period at constant wavelength, the laser frequency
was tuned by modulating the injection current with a triangular waveform
generated by a waveform synthesizer (Stanford Research Systems SRS
DS345).
Figure 2.2 shows the ramp used to modulate the injection current using a diode
laser.
Figure 2.2. Waveform used to modulate the laser current. Typical frequency of ramp and burst are 2500 Hz and 830 Hz, respectively.
2500 Hz
Burst 830 Hz
49
Scans were calibrated by passing the modulated laser diode light through a
Fabry-Pérot etalon (1 mA increase in current corresponds to a frequency shift of
approximately 2.5 GHz).
The laser light was focused onto the sample after attenuation by using
polarising films and density filters. The laser light was measured in transmission
by a Si photodiode (Thorlabs PDA55). The signal was averaged on a digital
storage oscilloscope (Tektronik TDS210) and subsequently on a PC. Typically
1280 waveforms were averaged (see Figure 2.6).
Very narrow holes were measured at very low temperatures (< 14 K). For these
experiments an external cavity diode laser (Toptica DL110) was used. The
external cavity diode laser was locked to a 1.5 GHz Fabry-Pérot etalon and the
latter was scanned by a triangular ramp (see Figure 2.7).
2.3.6. Fluorescence line narrowing
Measurements were carried in two different ways. A Fabry-Pérot interferometer
(Burleigh RC-110) was used for intermediate temperatures (see Figure 2.9) up
to 80 K and the Spex 1704 monochromator for temperatures > 80 K (see Figure
2.8).
A Thorlabs TEC2000 temperature controller and a Thorlabs LDC500 current
controller controlled a Thorlabs TCLDM9 thermoelectric mount. Two opto-
mechanical choppers (Thorlabs MC 1000) operated with a phase shift of 1800
and a duty cycle of 30% were utilized to eliminate the laser light from reaching
the detector for measurements with the monochromator. Only one chopper was
necessary for FLN with Fabry-Pérot interferometer as the laser light was passed
though the chopper wheel as the FLN signal. The signal was processed as is
discussed in 2.3.3 and 2.3.4.
50
2.3.7. Slow light experiments
We have investigated the creation of slow light by transient spectral hole-
burning in a ruby crystal.
An external cavity diode laser (ECDL) was used. The amplitude modulation was
undertaken by an acoustico-optic modulator (Isomet 1205C-1; modulator driver
222A-1). The laser light was measured in transmission by a Si photodiode
(Thorlabs fast PDA10A). The signal was averaged on a digital storage
oscilloscope (LeCroy Wavesurfer 422) (see Figure 2.10).
Burn pulses with a period from 0.25 to 2 ms and a 50 ns Gaussian probe pulses
were separated by an interval of 0.01 to 0.25 ms as shown in Figure 2.3.
Figure 2.3. Burn and probe pulse.
0.01-0.25 ms
Burn pulse
0.25 - 2 ms
Probe pulse
50 ns
51
Figure 2.4. Schematic diagram for transmission measurement at liquid helium temperatures.
Photomultiplier
tube
CCR
sample Temperature
controller
24 V halogen
lamp
Power supply
Polarising
filter
Chopper
OD filter
200 mm lens
75 mm lens
200 mm lens
Chopper driver
Lock-in Amplifier
Computer
Pre-amplifier PMT
Monochromator
52
Temperature
controller
Figure 2.5. Schematic diagram for non-selective luminescence measurement at liquid helium temperatures.
Chopper driver
Lock-in Amplifier
Computer
Polarising
filter
Pre-amplifier
Chopper
OD filter
PMT
Monochromator
Nd:YAG
CCR
sample
200 mm lens
75 mm lens
200 mm lens
53
Figure 2.6. Schematic diagram for spectral hole-burning measurement at liquid helium temperatures.
Waveform generator
200 mm lens
200 mm lens
75 mm lens
Temperature
controller
Polarising
filter
OD filter
Current
Controller
Temperature
controller
Laser diode
CCR
sample
Mirror
Computer
Photo diode Digital oscilloscope
54
Figure 2.7. Schematic diagram for high resolution spectral hole-burning measurement at liquid helium temperatures.
Computer
Photo diode
Temperature
controller
Polarising
filter
OD filter
Current
Controller
Temperature
controller
external
cavity diode
laser
CCR
sample
Mirror
Shutter Waveform generator 2
optical
isolator
Fabry-
Perot
Photo-
detector
Waveform generator 1
Digital oscilloscope
DigiLock
110
55
Figure 2.8. Schematic diagram for fluorescence line narrowing measurement with a monochromator at liquid helium temperatures.
luminescence
Chopper driver 2
Lock-in Amplifier
Computer
Pre-amplifier PMT
Monochromator
Chopper 2
CCR
sample Chopper driver 1
Chopper 1
Temperature
controller
Current controller
Laser diode
Excitation laser
200 mm lens
200 mm lens
75 mm lens
sync
56
Figure 2.9. Schematic diagram for fluorescence line narrowing measurement with Fabry-Pérot interferometer at liquid helium temperatures.
prism
CCR
sample
Temperature
controller
Current controller
Fabry- Pérot
interferometer
Chopper driver Chopper
200 mm lens
75 mm lens
200 mm lens
Laser diode
57
Figure 2.10. Schematic diagram for slow light experiments.
2.4. Application of an external magnetic field
Uniform low strength magnetic fields were applied to the sample via external
Helmholtz coils. The coils are formed by two water-cooled copper wire coils
(250 mm diameter) mounted around the CCR and separated by a distance of
125 mm by plastic spacers (125 mm corresponds to the radius of the coil).
Currents in the range of 0 to 9.5 A were supplied by two power supplies
(Hewlett-Packard 6269B DC power supply). The magnetic field is a
Oscilloscope PC
ECDL Toptica
DL110 AOM
2 mm aperture
CCR
sample
150 mm lens
750 mm lens
Waveform generator
Photo
diode
58
superposition of the fields created by two sets of current loops. These external
Helmholtz coils behave according to the Biort-Savart law given by the following
Eq. (2.3) [2]:
Eq. (2.3)
In Eq. (2.3) B is the magnetic field strength, N is the number of turns in a coil, I
is the current, µ0 is the permittivity of vacuum (4π∙10-7 T∙m/A) and R is the
radius of the coil. The separation between the two coils must be equal to the
radius of the coils in order to have an optimally uniform magnetic field at the
centre position.
2.5. Laser sources
In this section the laser sources used in this project are presented. All the laser
sources used are potentially dangerous and without any precaution can cause
permanent eye injure. Keeping this in mind, all the OHS measures were taken
into account in order to reduce this risk. Moreover, the power of these lasers
was high enough to damage the photomultiplier used in this project, so care
was taken when working with these lasers.
59
2.5.1. Nd:YAG laser
Nd:YAG is a solid state laser that uses a neodymium doped yttrium aluminium
garnet crystal (Nd:Y3Al5O12). These lasers are optically pumped by a flashtube
or a diode laser. They lase at a wavelength of 1064 nm in the infrared [3].
Nd:YAG lasers can operate in both continuous and pulsed mode. In pulsed
mode an optical switch is incorporated into the laser cavity until a maximum
population inversion occurs in the neodymium ions before it opens. After that,
light can go through the cavity provoking a depopulation of the excited laser
medium. The high-intensity pulses can be frequency doubled to generate laser
light at 532 nm within the green range of the spectrum [4].
As shown in Figure 2.11 below, an external laser diode excites neodymium ions
to a higher lying excited state. There is a decay from 4F3/2 to 4I11/2 (a photon is
emitted) after a non-radiative decay from the higher excited level. Finally, non-
radiative decay processes take place to go back to the ground state.
Figure 2.11. The four level laser system Nd:YAG.
Non-Radiative decay
Non-Radiative decay
Photon emission
Excitation
1064 nm
60
This laser was used to perform non-selective luminescence experiments. It is a
class 3b laser product.
2.5.2. He-Ne laser
This type of laser was used for alignment of optical axis. The mechanism to
produce population inversion and light amplification in a He-Ne laser comes
from many collisions of electrons with ground state helium atoms in the gas
mixture.
As shown in Figure 2.12 helium atoms are excited from the ground state to the
excited state (23S1 and 21S0 metastable states) by electron collisions. As a
consequence of a coincidence between these two metastable states and the 3s
and 2s levels of neon, collisions between atoms lying in these next levels
provoke a transfer of energy from helium to neon [5]. The population in 3s and
2s levels of neon is increased. When the population of these two levels is larger
than that of the ground state in neon, then there is a population inversion. The
light can be amplified by the medium in the 2s to 2p transition and the 3s to 2p
transition. Very fast radiative decay from 2p level to the 1s state is produced. A
final decay makes the population reach the ground state.
61
Figure 2.12. Energy transfer between Helium and Neon ions. Adapted from [6].
An optical oscillator is created by placing highly reflecting mirrors at each side of
the amplifying media so that the wave is reflected back and forth gaining more
power. When this happens, there is a stable laser beam output.
2.5.3. Diode laser
A range of current and temperature stabilised single-frequency laser diodes
were used in this project in high-resolution laser experiments. Conventionally,
they are used in devices such as DVD players, laser pointers, optical fiber
communications, etc. The most important advantage is the small size, high
power efficiency and low maintenance requirements [7].
Energy
He-Ne collision
Fast radiative transitions
Ground
state
Diffusion to walls
Excitation by
electron collision
62
The active medium in laser diodes is a semiconductor. The simplest kind of a
laser consists of a p-n junction (homojunction laser) and is powered by an
injection of electric current. When it comes to p-i-n structure (heterostructure)
electrons and holes can recombine and photons are released as energy
quantas. This process can be either spontaneous or stimulated by incident
photons leading to optical amplification, and with optical feedback in a laser
resonator to laser oscillation. When an electric potential is applied through the
p-n junction, electrons relax from the conduction band to the valence band of
the material. In this process there is emission of radiation. The semiconductor
material determines the output wavelength. One of the most common
semiconductors used in laser diodes is GaAlAs, commonly found in CD players
[8-10].
Laser diode sources were the laser source in experiments such as spectral
hole-burning or fluorescence line narrowing. They were installed in a Thorlabs
TCLDM9 temperature controller mount which was controlled by a Thorlabs
TEC2000 thermoelectric temperature controller and a ILX ultra low noise
current supply. By changing the supplied current and the temperature of the
laser diode, the frequency of the emitted light can be controlled.
The wavelength was measured using a wavemeter (Coherent Scientific
Wavemate). A calibration of this instrument can be found in [11].
In our experiments the laser light from the laser diode was attenuated with a
density filter and focused onto the sample by means of a lens. The injection
current of the laser diode was modulated using a waveform generator (Stanford
Research Systems DS345) yielding the desired frequency modulation.
63
2.5.4. External Cavity Diode Laser
In order to perform high resolution laser spectroscopy of very dilute samples,
where homogeneous linewidths are very narrow, a laser with a very small
linewidth must be used. In many diluted systems the spectral hole widths
observed are mainly due to the linewidth of the laser. A Toptica DL100 External
Cavity Diode Laser was used.
In an External Cavity Diode Laser a lens and a diffraction grating are placed at
the front of a standard laser diode as shown in the picture below allowing a
decrease in the linewidth of the laser output.
Figure 2.13. External cavity diode laser diagram. Adapted from [10].
The two most common configurations are Littrow and Littman-Metcalf
geometries. In both configurations a collimating lens and a diffraction grating
are placed in front of the laser diode. In the first configuration, the diffraction
grating is rotated so that the wavelength output is tuned. The inconvenience is
that when tuning the direction of the grating, the direction of the laser beam is
changed. The second configuration resolves this problem. In this geometry, a
mirror reflects the first-order beam back to the laser diode. Then the tuning of
the frequency is made by rotating that mirror. A high degree of selectivity can be
done, so narrower linewidths can be obtained [10].
Output
LD Lens Diffraction grating
64
Figure 2.14. External cavity diode laser configurations: Littrow to the left and Littman-Metcalf to the right. Adapted from [10].
.
The ECDL laser used in this thesis was based on the Littrow geometry with
optics that compensate for the beam walk.
2.5.4.1. Frequency lock
The output frequency of a diode laser depends on the injection current and the
temperature. Lasers with well-defined frequency are needed in many
spectroscopy experiments. But unfortunately after a time, the central frequency
of a diode laser with grating feedback will drift. This drift is the consequence of
fluctuations in temperature and injection current and mechanical
fluctuations. Hence, it is very important to stabilize the laser by locking it to an
external reference in order to reduce that drift.
The external cavity diode laser used in this project is a Toptica DL110 controlled
by a DigiLock 110 Feedback controlyzer.
The DigiLock PID controller was used to control the laser current in order to lock
the frequency of the laser. The laser was locked employing the locking
electronics to one of the fringes of a Thorlabs 1.5 GHz Fabry-Pérot cavity,
which has higher frequency stability. An error signal was created to measure
the deviation in frequency to the reference. The error signal is filtered (with the
locking electronics) and fed back to the laser. This feedback makes the laser
lens lens
output
diffraction
grating
mirror
65
frequency follow the reference point. In our case, one side of the fringe was
selected by scanning the laser. The laser was locked to a single optical
frequency when excursions of the laser frequency < 200 MHz were required.
The parameters of the locking electronics (like gain) were optimised for
scanning and locking the laser.
2.6. Monochromator
A 1m Spex 1404 Czerny-Turner monochromator has been used for
luminescence experiments. The main goal of a monochromator is to separate
and transmit a narrow portion of an optical signal from a broader range of
wavelengths available at the input.
In the simplest case a monochromator has two slits (entrance and exit) and a
dispersion element (prism or diffraction grating). In these two elements, the
dependence of the refraction angle (prism) or the reflection angle (grating) on
the radiation wavelength is very important. The dispersion or difraction in the
monochromator can be controlled only if the light is collimated, that is if all the
rays of light are parallel [12].
Monochromators use photoelectric recording of a selected small spectral
interval. An exit slit allows only limited intervals of wavelengths to go through to
the photoelectric detector [7]. The spectrum is scanned by rotating the grating.
This moves the grating normal with respect to the incident and diffracted
beams. The diffracted wavelength is reflected towards the second mirror. The
spectrum is still focused at the exit slit for each wavelength because the light
incident in the grating is collimated.
66
Figure 2.15. Czerny-Turner monochromator configuration. Light goes through a slit and is reflected by a collimating mirror to a diffraction grating. A focusing mirror reflects the diffracted light to the exit slit. Adapted from [13].
The spectral resolution of a monochromator is closely related to its spectral
dispersion. The dispersion governs how far apart two wavelengths are. The
resolution determines whether the separation of those two wavelengths can be
resolved. The Rayleigh criterion says that two wavelengths λ1 and λ2 are
resolved when the central maximum of one line falls on a diffraction minimum of
the other (see Figure 2.16). Hence, the spectral resolution can be defined by
Eq. (2.4):
Δλ =
Eq. (2.4)
where is the average wavelength between the two lines (λ1 and λ2), is the
angular dispersion of the system and is the slit width [7, 12].
Entrance slit
Exit slit
Collimating mirror
Focusing mirror
Focal
plane
67
Figure 2.16. Monochromator resolution.
2.7. Fabry- Pérot interferometer
A scanning plane-parallel Fabry-Pérot interferometer (Burleigh Instruments RC-
110) was used for fluorescence line narrowing experiments. It consists of two
parallel mirrors. There are multiple reflections between the two spaced mirrored
surfaces.
The condition for producing a maximum in transmission is given by the
condition
Nλ = 2nd cos θ Eq. (2.5)
where λ is the wavelength of the radiation in air, N is an integer, n is the index of
refraction of the air within the interferometer, d is the separation of the reflecting
surfaces and θ is the angle of incidence.
λ1 λ2
68
Figure 2.17 shows the transmission as a function of wavelength for a given
angle of incidence. If a particular N0 and λ0 satisfy the above equation, the
interference between the transmitted rays will be constructive and a maximum
of transmission will be produced. Another maximum of transmission will be
observed when other combinations such as N0+1 or λ0 - Δλ satisfy the
mentioned equation. The spacing Δλ between maxima of transmission is the
free spectral range (FSR). The equation for the FSR is given by [7]
Eq. (2.6)
Figure 2.17 shows how the reflectivity of the mirrors affects the transmission. It
follows that the maximum in transmission becomes broader when the reflectivity
is low and when the reflectivity is high the maxima of transmission observed will
be narrower.
Figure 2.17. Transmission as a function of wavelength through a Fabry-Perot interferometer. High reflectivity surfaces show sharper peaks and lower transmission minima than lower reflectivity surfaces . Adapted from [8].
High reflectivity surfaces
Wavelength
Low reflectivity surfaces
Transmission
69
The concept of finesse of the interferometer is very important as it measure the
ability of the interferometer to resolve separated spectral lines.
The total finesse of an interferometer can be expresses as the ratio of the free
spectral range to Δ, where Δ is the FWHM (full width half maximum) of the
response function of the system, also known as minimum resolvable bandwidth.
Eq. (2.7)
Two lines separated by Δ are resolvable when the sum of the two individual
lines at the midway point is at most equal to the intensity of one of the original
lines.
The reflectivity finesse Fr can be expresses as:
Eq. (2.8)
where R is the reflectivity of the surfaces. For high reflectivity, the transmission
of maxima of slightly different wavelengths can be resolved because the
transmission maxima are narrow.
2.8. The Laue method
White radiation is transmitted through or reflected from a fixed crystal. Arrays of
spots are formed by the diffracted beams lying on curves on a film. The Bragg
angle is fixed for each set of planes in the crystal. Every single wavelength from
the white radiation that accomplishes the Bragg law is diffracted by each set of
planes. The reflection from planes which belong to one zone are the spots on
one curve. There are two kinds of the Laue method, the transmission and the
back-reflection Laue method.
70
Figure 2.18. Transmission to the left and back-reflection Laue method to the right. Adapted
from [14].
In the transmission Laue method, the crystal is placed before a film to record
the beams transmitted through the crystal. In the back-reflection method, the
film is located between the crystal and the x-ray source. The beams recorded
are diffracted in a backward direction.
2.9. Data acquisition and analysis
2.9.1. Data acquisition
The acquisition of the data were undertaken by a PC (500 MHz Intel Pentium III
processor, Intel MMX Technology computer with 128 MB RAM, 13 GB HDD and
Windows 98 OS). Motion controllers, software and National Instruments NI-
488.2 General Purpose Interface Bus (GPIB) hardware devices were used.
Several programs written by Riesen in Microsoft Visual Basic 6.0 for 32-bit
windows were used to enable remote operation of the instruments and
acquisition of processed data. The text format (.txt) in ASCII format was used to
save all the collected data so that standard software could be used to analyse
the data.
71
2.9.2. Data analysis
All the data were analysed using a PC (EliteBook 8530p, Intel® Core™ 2 Duo
Processor T9600, 120 GB Hard Disk Drive, OS Windows Vista). Both Microsoft
Office 2007 and Wavemetrics IGOR Pro 6.02A were used to analyse the data.
The sophistication of IGOR enabled us to graph, analyse and work up the data
presented in this thesis. Some fit functions such as Lorentzian or Gaussian
were used in order to measure very narrow linewidths or spectral holes in the
collected spectra. In addition to built-in functions some user-defined functions
were created in order to study temperature dependences and other data
presented in the chapter 3. This characteristic of IGOR to allow the creation of
user-defined functions makes this program very useful in comparison to other
analysis software [15].
In order to analyse lineshapes obtained in this project it was necessary to use
user-defined functions. For instance, the Voigt profile was used to analyse hole-
burning measurements and luminescence data. The Voigt profile is the
convolution of a Lorentzian with a Gaussian.
A Gaussian profile is represented by the equation:
Eq. (2.9)
and a Lorentzian profile is expressed by:
Eq. (2.10)
The convolution of both profiles gives us the equation:
Eq. (2.11)
By using Igor it was possible to obtain a mathematical model to fit our data. The
curves were fitted by choosing the optimal coefficients which make the function
match the data as closely as possible. Voigt fits were used in order to obtain the
Lorentzian contribution at higher temperatures in the presence of the fixed
Gaussian low temperature contribution.
72
In order to analyse the data obtained from FLN (intermediate range of
temperatures) the inhomogeneous distribution was taken into account. The
equation used is as follows
Eq. (2.12)
where is the Gaussian distribution of optical centres (inhomogeneous
distribution). For resonant FLN is equal to , being the Lorentzian
distribution of the emission and absorption functions respectively. In this case,
the inhomogeneous linewidth was fixed at 2.5 K and the Lorentzian contribution
at higher temperatures was obtained.
For the analysis of the final data a non-perturbative function was built in order to
obtain a good fit for the resulting graph (see Appendix 1, 2, 3 and 4).
73
2.10. References
1. Jirmanus, M.N.; Introduction to laboratory cryogenics; Janis Research
Company, Wilmington (1990).
2. Young, H.D.; Freedman, R.A.; University Physics 9th Edition; Adison-
Wesley Reading (1996).
3. Yariv, A.; Quantum Electronics 3rd Edition; Wiley p. 208–211 (1989).
4. Koechner, W.; Solid-state laser engineering, Springer-Verlag p. 507
(1965).
5. Javan, A., Bennett, W. R.; Herriott, D. R.; Phys. Rev. Lett. 6 (3), 106–110
(1961).
6. http://laser.physics.sunysb.edu/~dli/hnwork.html
7. Demtröder, W.; Laser Spectroscopy, Basic concepts and instrumentation
2nd Edition; Springer (1995).
8. Atkins, P.W.; Phys. Chem. 8th Edition; Oxford University Press (2000).
9. Hall, R. N.; Phys. Rev. Lett. 9 (9), 366 (1962).
10. http://www.rp-photonics.com/ Encyclopaedia of Laser Physics and
Technology.
11. Hayward, B. F.; BSc (Hons) Thesis: UNSW (2004).
12. Domanchin, J. L., Gilchrist, J. R.; Size and spectrum, Spectrometers;
Photonics spectra (2011).
13. http://gratings.newport.com/library/handbook/toc.asp 'Plane gratings and
their mounts'.
74
14. http://webpages.iust.ac.ir/panahib/Cryst_files/frame.htm#slide0013.htm. X-
ray application in crystal.
15. www.wavemetrics.com Igor Pro 6.02A manual, Wavemetrics, Inc.
75
Chapter 3
Results and discussion
The results obtained for the temperature dependence of the homogeneous
linewidth in emerald and the creation of slow light in ruby by transient spectral
hole-burning are presented and discussed in this chapter.
3. 1. Emerald
3.1.1. Crystal structure and background
Emerald is the green variety of the mineral beryl and its idealized formulae is
Be3Al2(SiO3)6. The bright clear green variety is emerald while a bluish green
type is called aquamarine.
In emerald a small percentage of the aluminium (III) sites are replaced by
chromium(III). Beryl comprises of hexagonal rings formed of six Si-O irregular
tetrahedra. These Si-O distances may vary from 1.54 to 1.68 Å. Each Al is
octahedrally coordinated by group of six oxygen atoms and each Be atom is
coordinated by four oxygen atoms [1, 2].
Alkalis, such as Na, Li, K, Cs and Rb can be found in many samples of natural
beryl. The pink colour in beryl is associated with alkali traces, especially Li.
Also, other impurities that can be found are H2O+ [3].
Beryl crystallises in the hexagonal space group P6/mcc. The hexagonal Si6O18
RRrings form channels parallel to the c axis. As it is seen in the picture below
76
tetrahedrally coordinated beryllium and octahedrally coordinated aluminium are
responsible to keep these channels together [3].
Several optical studies show that iron(II) and iron(III) can also substitute for
aluminium ions [4, 5].
Figure 3.1. The crystal structure of beryl. Adapted from [3].
Emerald has been the subject of many spectroscopic studies because of the
presence and interactions of many impurities in its structure [4-9, 10]. Rigby et
al. [9] observed, for the first time, persistent hole-burning in 1992. In those
experiments it was reported that the presence of other impurities such as
titanium may be responsible for persistent holes [9]. Recently, it was found that
persistent spectral hole-burning is present only in natural emerald but not in
laboratory created samples [6]. In addition, Riesen demonstrated that natural
emerald exhibits much broader inhomogeneous broadening in comparison with
laboratory created emerald [11].
77
There are two spin-allowed transitions from the 4A2 ground state to 4T2 and 4T1
[8]. The band with a better defined structure is the one from the ground state to
2E leading to very sharp zero-phonon lines at 679.5 and 682.6 nm.
The 4A2 ground state and 2E excited state are split as a consequence of the
trigonal crystal field plus spin-orbit coupling. As a result splittings of
approximately 1.7 cm-1 in the ground state and of 63 cm-1 in the excited state
occur [8].
Emerald created Chatham laboratory with Cr3+ 0.04% (pale green) and
0.0017% (nominally chromium free) per weight were employed in order to
obtain the temperature dependent contribution to the homogeneous linewidth of
the R1(±3/2) line in emerald.
3.1.2. Results and discussion
Polarised transient spectral hole-burning experiments in the R1-line of a
laboratory created emerald (0.0017% Cr(III) per weight) from 2.5 K to 14 K were
obtained. The sample was excited with a diode laser operating at 682.47 nm.
The experiments are reported as a function of low magnetic fields BIIc and
temperature.
A contribution to the homogeneous broadening of less than 1 MHz at 60 K was
measured in this sample. The hole width at 13 K is 46 MHz as is illustrated in
Figure 3.2.
78
Figure 3.2. High resolution transient spectral hole-burning in a laboratory created emerald (0.0017% Cr(III) per weight)at different temperatures. The blue lines represent Voigt fits.
When a magnetic field B is applied parallel to the c axis the 2E lower excited
state level ±1/2 (2Ā) and the 4A2 lower ground state level ±3/2 (2Ā) are split by
gexµBBIIc and 3ggsµBBIIc (where gex PPandPP
Pggs
PPare the g-factors of the excited state
and ground state, respectively, B is the magnetic field strength and µB RRis the
Bohr magneton). The trigonal field and spin-orbit coupling splits the ground and
the excited states into two Kramers doublets each. The splitting of the 4A2
ground state and 2PE excited state is 1.79 cm-1 and 63 cm-1, respectively, as
depicted in Figure 3.3.
79
Figure 3.3. Schematic energy level diagram for spectral hole-burning experiments in the R1(±3/2)-line in an external magnetic field, BIIc. The laser is in resonance with the -1/2 -3/2 and +1/2 +3/2 transitions yielding side holes at
+3/2 +1/2 and -3/2 -1/2, respectively.
Transient spectral holes of the R1-line in π polarization (E||c) are illustrated as a
function of low magnetic fields in Figure 3.4. The separation between the two
side holes and the main hole is ±(3ggs
-gex
)µBBIIc. Taking the gex
factor as 1.04
and ggs=1.97 as it was established in Ref. [6] we conclude that the scan is
approximately 300 MHz.
+1/2
-1/2
+3/2
-3/2
+1/2
-1/2
+3/2
-3/2
Laser
1.7
9 c
m-1
63
cm
-1
2E Ē
4A2
2Ā (±3/2)
Ē
+(1/2)gexμBB
-(1/2)gexμBB
-(3/2)ggsμBB
+(3/2)ggsμBB
2Ā (±1/2)
80
Figure 3. 4. Transient spectral hole spectra in π polarization of the R1(±3/2)-line in a laboratory created emerald (Cr(III) 0.0017% per weight) in low magnetic fields B||c. The inset shows the relative shifts of the side holes as a
function of the magnetic field strength.
Figure 3.5 shows transient spectral hole-burning measurements of the R1(±3/2)-
line between 10 and 30 K in the same sample (0.0017% Cr(III) per weight). As
is illustrated, the resonant hole gets rapidly broader and is hard to measure
above 30 K. For instance, at 27 K a hole width of ~ 500 MHz is observed while
at 2.5 K the observed width is dominated by the instrumental resolution given by
the frequency jitter of the free running diode laser, which is ~ 20 MHz as
measured by a 300 MHz FSR confocal Fabry-Pérot interferometer (Coherent
model 240). From a deconvolution it is found that the homogeneous linewidth at
2.5 K is 4 MHz. This linewidth is much narrower than the 30 MHz linewidth of
81
20 ppm ruby in zero field [12]. The linewidth in ruby is dominated by Cr3+
electron spin flip-flops in the environment of a Cr3+ centre [13].
The inset shows three different hole-burning experiments together with their
respective fits. At 2.5 K, the linewidth is defined by a Gaussian fit reflecting the
laser jitter limitation and at higher temperatures Voigt profiles i.e. the
convolution of the Gaussian laser line shape with a Lorentzian were used.
Figure 3.5. Transient spectral hole-burning in the R1(±3/2) line of a laboratory created emerald (0.0017% Cr(III) per weight) at 2.5, 10, 16, 20 and 27 K. The laser wavelength was at 682.4 nm. The inset shows transient spectral holes
(solid lines) with their respective Voigt profiles (dashed lines).
82
Luminescence line narrowing (FLN) experiments were measured in two ways. A
Fabry-Pérot interferometer and a monochromator were used for temperatures
up to 80 and 80 to 170 K, respectively.
High resolution FLN experiments conducted with a Fabry-Pérot interferometer
are displayed in Figure 3.6. Conducting the experiments with a Fabry-Pérot
provides us with better results in comparison with a monochromator since the
resolution of the former is of the order of tens of MHz whereas for the
monochromator it is about 20 GHz.
The laser, operating single mode, resonantly excites Cr3+ ions to the 2E(2Ā)
level in the same way as is illustrated in Figure 3.3 (see diagram to the left)
which then luminesce to the doublet 4A2 levels to give the observed spectrum.
Figure 3.6 shows a representative FLN spectrum for three different free spectral
ranges (FSRs) (20, 60 and 120 GHz). These experiments were conducted from
30 up to 80 K. For the spectra with a 20 GHz FSR, good results were obtained
by fitting the linewidths with the convolution of two Lorentzian profiles
(instrumental linewidth and 2 homogeneous linewidth). The instrumental
contribution was found to be approximately 1 GHz. For the spectra measured at
60 and 120 GHz FSR, the linewidth was expected to be determined by a triple
function of the inhomogeneous broadening, emission and absorption lineshape
since these measurements were done at higher temperatures. The
inhomogeneous width was found to be 27 GHz as measured in luminescence at
liquid helium temperature.
83
Figure 3.6. Fluorescence line narrowing in a laboratory created emerald (0.04% Cr(III) per weight) for 20, 60 and 120 GHz FSR at 30, 42.5 and 56.5 K, respectively. The dashed lines represent the Voigt fits.
Resonant FLN experiments with a monochromator were undertaken in the
temperature range of 80 to 180 K for the same emerald (0.04% Cr(III) per
weight). A representative spectrum is illustrated in Figure 3.7. Above 90 K the
homogeneous linewidth is much larger than the inhomogeneous width (27 GHz)
and consequently the linewidths can be measured by conventional
luminescence experiments. The splitting of the 4A2 ground state (1.79 cm-1) is
resolved in the R1 line of diluted emerald as is seen in Figure 3.7.
84
Figure 3.7. Fluorescence line narrowing (FLN) with a monochromator in a laboratory created emerald (Cr(III) 0.04% per weight) for 2.5, 61, 121 and 161 K. At 2.5 K (red line) the
4A2 ground state splitting can be observed and the R2
line is not populated. The inset to the left represents two spectra at 2.5 and 101 K (solid lines) with their respective Voigt fits (dashed lines). The inset to the right illustrates the spectrum of the single mode laser working at 682.46
nm.
Luminescence experiments were obtained for both a very dilute lab created
emerald (0.0017% Cr(III) per weight) and the more concentrated emerald
sample (0.04% Cr(III) per weight) from the lowest temperature up to 260 K. The
sample was excited with the 532 nm light of a Nd:YAG laser.
85
Figure 3.8. Temperature dependence of the non-selectively excited luminescence spectrum of a laboratory created emerald (0.0017% Cr(III) per weight)in the R-lines at 2.5, 120, 180, 220 and 260 K. The inset displays the spectrum at
2.5 and 220 K (solid lines) together with their Gaussian and Voigt fit (dashed lines), respectively.
Figure 3.8 illustrates the temperature dependence of the non-selectively excited
luminescence spectrum of very dilute emerald in the region of the R lines (2E →
4A2 transitions). At low temperature the R lines linewidths are obscured by
inhomogeneous broadening whereas at higher temperatures the homogeneous
linewidth dominates. In particular, the R1 line broadens from an inhomogeneous
linewidth of 27 GHz (2.66 cm-1) to a homogeneous width of 526 GHz (17.5 cm-1)
in the range of 2.5 to 260 K. The shift to the red is ~ 600 GHz (20 cm-1) for the
same temperatures range. In comparison, the ruby R1 linewidth is 360 GHz (12
cm-1) at room temperature and the total red shift between helium and room
temperature is ~90 GHz (3 cm-1). Gaussian profiles are observed at 2.5 K while
Voigt profiles are observed for higher temperatures.
86
The linewidth data of the four experiments are summarized in Figure 3.9 where
the temperature dependent contribution to the R1(±3/2) linewidth between 5.6 K
and 260 K is illustrated. The different data sets are represented with different
colours in the graph depending on the experiment conducted. The data are well
described by the non-perturbative expression for developed by Hsu
and Skinner [14] in Eq. (3.1)
Eq. (3.1)
where and are the direct and Raman processes respectively
expressed by Eqs. (1.8.b) and (1.9).
Figure 3.9. Temperature dependent contribution ΔГhom to the homogeneous linewidth Гhom of the R1(±3/2) line (circles) for emerald. The solid circles show the data for a very diluted lab created emerald (0.0017% Cr(III) per weight) and the open circles represent the data for a more concentrated emerald. Trace 1: non-perturbative
approach using Eq. (3.1). Trace 2: non-perturbative approach using Eq. (3.1) plus a phonon sideband at the lower temperature range. Trace 3: two-phonon Raman process using Eq. (1.8.b). Trace 4: direct process using Eq. (1.9). No
side phonon is considered in this fit. Trace 5: two-phonon Raman process but with 5 modes only using Eq. (3.7).
ΔГ h
om
/ M
Hz
87
A good fit at high temperature is obtained by the parameters W=-0.43, Г0(R2)=
12500 MHz and TD= 845 K. The Debye temperature (TD) is the temperature of a
crystal's highest mode of vibration. The Debye temperature is given by
Eq. (3.2)
where is Planck's constant, is Boltzmann's constant and is the Debye
frequency. According to Figure 3.10 our highest mode of vibration is at 711 nm
(589 cm-1 from the R1). This results in a Debye temperature of 845 K. We note
here that our Debye temperature is at variance from the 580 K reported in [15].
The value for Г0(R2) is calculated by assuming that the two-phonon Raman
process is the same for R1 and R2 at 80 K and also the inhomogeneous
broadening. The luminescence linewidth observed for R1 is
Eq. (3.3)
and the linewidth observed for R2 is
Eq. (3.4)
The difference between R1 and R2 linewidths yields
Eq. (3.5)
From values of and in luminescence at 80 K we calculate that is
15500 MHz, which differs from 12500 MHz used in the fit illustrated in Figure
3.9. We believe that this difference might be due to the Voigt fit at this particular
temperature, which is not very good since our R1 line in luminescence starts to
be asymmetrical at high temperatures (see Figure 3.8).
However, in contrast to ruby, the data was not well described for the lowest
temperature range. This mismatch comes possibly from the simplicity of the
Debye approximation. Spin-lattice relaxation cannot be taken into account for
the variation of measured and calculated linewidths below 8 K because it is very
88
slow [16] and according to Eq. (3.6) has little influence in the contribution to the
homogeneous broadening.
Eq. (3.6)
It is possible that there is a phonon sideband at low energy. After observing the
asymmetrical lineshape of the spectra in luminescence in Figure 3.8 and the
spectra of Figure 3.10 it appears that a low energy phonon is located at 683.05
nm (15 cm-1 separated from the peak of the R1 line). A good fit is obtained when
this is taken into consideration (see Trace 2 in Figure 3.9). It is also possible
that spectral diffusion causes extra broadening at this temperature.
Inspecting the summary of linewidth data provided in Figure 3.9 another
discrepancy was found at higher temperatures. It can be observed that the
empty circles (FLN with monochromator and luminescence in more
concentrated emerald) are not fitted by the non-perturbation function. Attempts
were made to fit these points by using pseudo-local phonons. The spectra in
Figure 3.10 exhibits peaks at 691, 697, 702, 707 and 711 nm (184, 324, 408,
509 and 589 cm-1) in emerald. The two-Raman process with 5 modes described
the data as represented by the trace 5 in Figure 3.9. The formulae used in this
case was
Eq. (3.7)
where are the coupling constants, are the centre frequencies of each
phonon and i are all the local modes. We believe that these points do not match
with the rest of the data because these experiments were conducted in a more
concentrated sample where energy transfer occurs. The solid circles in Figure
3.9 represent the same experiments in a more dilute emerald (0.0017% Cr(III)
per weight). Good agreement with the data was obtained in this case. In the
diluted emerald no energy transfer happens and as a consequence narrower
linewidths were obtained.
The 2E → 4T2 energy gap in emerald is about 400 cm-1 [16] compared to 2300
cm-1 in ruby. Excitation into any higher state relaxes quickly to 2E (4T1 and 4T2
89
states relax by non-radiative transitions to the 2E excited state) and 2E → 4A2
line emission is observed. As 2E → 4A2 transition is spin-forbidden, the 2E level
has a long lifetime (~1.7 ms) [15, 17, 18]. When increasing the temperature, the
4T2 population grows from 2E and the 4T2 → 4A2 broad band emission line is
observed. The separation between 4T2 and 2E states in these two crystals
results in a large difference between the lifetimes of the metastable levels at
room temperature (~6 10-5 s for emerald and ~3 10-3 s for ruby) [17]. The
lifetime of 4T2 is found to be 10-12 s. We assume that the 2T1 level between 2E
and 4T2 has little contribution to the homogeneous broadening [17, 18].
Figure 3.10. Vibrational side band of the R1 line in diluted emerald. A side phonon is located at 683.05 nm. Other 5 phonons can be observed next to the R1 line.
In conclusion, the low temperature range up to 40 K seems to be dominated by
the direct one-phonon process between Ē and 2Ā of the excited state 2E plus a
two-phonon Raman process by a low energy phonon whereas at higher
temperatures the two-phonon Raman process plays the most important role.
683.05 nm
90
3. 2. Slow light in ruby by transient spectral hole-burning
3.2.1. Crystal structure and orientation
Ruby is a very important material in the development of the optical
spectroscopy of impurity ions in insulators and wide bandgap semiconductors.
Ruby was also the first material to display laser action in the visible spectrum as
demonstrated in 1960 by Maiman [19]. The laser action within a ruby rod was
induced by a flash tube. Coherent laser emission took place after the utilisation
of a short pulse of light to pump the ruby rod. The ruby laser is of significant
importance since is the basis of current solid state technology, such as the
Nd:YAG laser.
Ruby is the pink-to-red variety of the mineral corundum and its chemical formula
is Al2O3:Cr(III). It has a trigonal crystalline structure crystallising in the space
group D63d - R3c. In ruby Cr3+ replaces Al3+, which are octahedrally coordinated
by oxygen atoms [20]. Ruby is the second hardest material that occurs
naturally.
There are several crystal growing techniques to grow ruby in the laboratory,
such as the Czochralski [21], floating zone [22], flame fusion [23] and
Bagdasarov methods [24]. The crystal we used was grown by Bagdasarov
method. In the Bagdasarov method the material is placed in a crucible and is
melted by moving the crucible through the heating zone. This method allows
repeated crystallization when other chemical impurities of the raw material are
required.
91
Figure 3.11. Crystal structure of corundum.
In order to cut the crystals perpendicular to the c-axis the orientation of the
crystal was determined by the Laue method. Figure 3.12 shows Laue
photographs in 180° backscattering geometry of the crystallographic plane
perpendicular to the c axis for 20 ppm Verneuil ruby and 130 ppm Bagdasarov
ruby.
Figure 3.12. Laue photograph of 20 ppm ruby to the left and 130 ppm (Bagdasarov) to the right of the plane perpendicular to c.
70
06
50
60
05
50
50
04
50
40
03
50
30
02
50
20
01
50
10
05
00
110010501000950900850800750700650600550500450400350300250200150100500
70
06
50
60
05
50
50
04
50
40
03
50
30
02
50
20
01
50
10
05
00
110010501000950900850800750700650600550500450400350300250200150100500
92
3.2.2. Results and discussion
3.2.2.1. Experiment
Minimal inhomogeneous broadening is advantageous for the generation of slow
light by spectral hole-burning in ruby as it allows high optical densities with low
chromium (III) concentrations. Therefore, the inhomogeneous broadening of
ruby crystals was measured with different chromium (III) concentrations grown
by a range of methods (Verneuil, Czochralski, Bagdasarov, hydrothermal). It
appeared that gross macroscopic strain broadening affects some crystals,
especially the flame fusion (Verneuil) grown crystals. The best results were
obtained for Bagdasarov ruby. An inhomogeneous linewidth of 2 GHz (FWHM)
was obtained as is illustrated in Figure 3.13. The spectrum was measured in
fluorescence-excitation mode by observing R1 - luminescence whilst scanning
the laser over 30 GHz. The separation between the R1 splitting corresponds to
11.49 GHz.
Figure 3.13. Resonance luminescence excitation of the R1 line in Bagdasarov ruby. The various isotopic lines are also labelled.
93
Different linewidths ranging from 2 to 10 GHz were obtained depending on the
position in the crystal. For the slow light experiments, a 130 ppm Bagdasarov
ruby of 2.3 mm thickness was used. The set up for the slow light experiment is
schematically illustrated in Figure 2.10. In particular, a 750 μs burn pulse was
applied followed by a weak 50 ns probe pulse with 10 μs delay (i.e. at 760 μs).
Figure 3.14 shows transmission spectra of a 2.3 mm Bagdasarov ruby (130
ppm) along with its change in absorbance ΔA upon hole-burning. A magnetic
field of 9 mT parallel to the c axis was applied. When an external magnetic field
is applied, the ground state doublets ±1/2(Ē) and ±3/2(2Ā) and the 2E excited
state level Ē are split. Therefore, upon hole-burning in the 2Ā(2E) Ē(4A2) line
two side holes are expected, separated from the resonant hole by ±(3ggs-
gex)µBBIIcR R. Considering that ggs=1.98 and gex=2.44, we obtain side holes
separated by ±445 MHz from the resonant hole. The transitions R1(±3/2) and
R1(±1/2) are resolved for the various stable isotopes of chromium (III).
Figure 3.14. a) Transmission (red line) spectrum in α-polarisation of a 2.3 mm Bagdasarov ruby (130 ppm Cr(III)) in BIIc=9 mT and change in absorbance ΔA (blue line). The black line shows the spectrum without hole-burning. b)
Same as in a) but over an extended frequency range. Chromium (III) isotopes that are responsible for the various R1(±3/2) and R1(±1/2) lines are denoted.
a) b)
94
Naturally, there is a change in absorbance (A) after a burn pulse. It is obvious
that cross-relaxation happens between the 4A2 levels of all the isotopes i.e.
cross-relaxation between resonant and non-resonant ions [25].
The decay of the spectral hole burnt by the 750 μs burn pulse was studied by
the application of a 50 kHz burst of 1 μs probe pulses as is illustrated in Figure
3.15.
Figure 3.15. 50 kHz burst of 1 μs probe pulses after a 750 μs burn pulse.
It appears that the initial hole decay is faster in the case when an external
magnetic field BIIc is applied. This is caused by a change in spin-lattice
relaxation in the excited state levels +1/2 and -1/2.
Figure 3.16 summarizes the decay data. It follows that the hole in zero field
decays approximately with the lifetime of the excited state. The hole in BIIc=9
mT initially decays on a timescale of 100 μs due to spin-lattice relaxation
between the +1/2 and -1/2 split levels.
95
Figure 3.16. Hole depth decay when a magnetic field BIIc=9 mT is present (blue markers) and in zero field (red markers). For the hole depth decay in a magnetic field the fit parameters are 1/τ1=0.29 ms
-1 and 1/τ1=10.05 ms
-1
whilst for zero field 1/τ=0.29 ms-1
.
To quantify the slow light experiments we have also measured the hole shape
after the 750 μs burn pulse. This is shown in Figure 3.17. The observed hole
width of 50 MHz is limited by laser jitter and spectral diffusion.
Figure 3.17. Transient hole-burning in the presence of a magnetic field of BIIc=9 mT in 130 ppm Bagdasarov ruby. The blue line is a Gaussian fit. The hole width is 50 MHz. The hole is measured with a 1.5 ms delay after a 750 μs
burn pulse.
96
Figure 3.18 presents the time delay obtained for the probe pulse after hole-
burning with respect to the probe pulse without hole-burning. A delay of 10.8 ns
is observed for the probe that follows a burn pulse with minimal distortion.
Figure 3.18. Delay of the probe pulse 10 μs after a 750 μs burn pulse (red line) in comparison to the normalized (green line) and non-normalized probe pulse without hole-burning (blue line). Experimental parameters: Гinh = 1.9
GHz, FWHM (pulse duration of input pulse) = 59 ns, A (absorbance before burning) = 1.53, Spectral hole width = 44 MHz, ΔA (change in absorbance) = 1.09.
97
3.2.2.2. Simulations
The linear spectral filter theory as outlined in chapter 1 (see section 1.5.2.3)
was applied and a Mathematical code of Professor Rebane and translated by
Riesen to Matlab (see Appendix 5) was used.
As seen in Figure 3.19.a larger delays are obtained when the absorbance is
higher. Also, when the absorbance is increased the pulse width after hole
burning gets broader with respect to the pulse width without burn pulse as seen
in Figure 3.19.b. For these simulations, we used the following input values: Гinh
= 2 GHz, pulse width = 50 ns and hole width = 40 MHz.
Figure 3.19. a) Delay of a Gaussian probe pulse versus ΔA/A for different absorbances A=0.5, 1, 1.5 and 2. Hole width = 40, pulse width = 50 ns, Гinh = 2 GHz. b) Ratio of the widths of delayed and non-delayed pulse versus ΔA/A
for A=0.5 and 1. Hole width = 40, pulse width = 50 ns, Гinh = 2 GHz.
Another simulation was run to calculate the delay time of the probe pulse as a
function of the hole width as is depicted in Figure 3.20. As expected the delay of
the probe pulse decreases with increasing hole width. Calculation were
undertaken for 50 and 100 ns pulse duration of the probe pulse and different
hole depths ΔA. Larger delays are obtained for narrower hole widths and deep
a) b)
98
holes ΔA. For broader hole widths there is no difference in delay between
longer and shorter pulses.
Figure 3.20. Delay versus hole width for different changes in absorbance and pulse duration. Pulse width = 50 and 100 ns, A=1.5, Гinh = 2 GHz.
When the hole gets very narrow, probe pulses of 50 ns start to be distorted, i.e.
they lose their Gaussian lineshape as is illustrated in Figure 3.21.
99
Figure 3.21. Delay of the probe pulse (red line) after hole-burning in comparison with the normalized (blue line) and non-normalized non-delayed pulse (green line). The dashed lines are the Gaussian fits for each pulse. Гinh = 2 GHz, A
= 1.5, ΔA=1, Pulse width = 50 ns, hole width = 10 MHz.
A calculation to approximate the hole width to the delay of 10.8 ns observed in
the experiment of Figure 3.18 is presented in Figure 3.22. A value of 45 MHz
corresponds to 10.8 ns. By comparing the hole width (50 MHz) experimentally
observed with the hole width calculated theoretically, we deduce that the
broadening observed in experiments is due to laser jitter and/or spectral
diffusion.
100
Figure 3.22.Hole width versus delay of the probe pulse. A = 1.53, Гinh = 1.9 GHz, ΔA = 1.09, pulse width = 59 ns.
3.3. References
1. Antsiferov, V. V.; Free running emerald laser; in Technical Physics, 45,
1085-1087 (2000).
2. Morosin, B.; Structure and thermal expansion of Beryl; in Acta Cryst., 28,
1899-1903 (1972).
3. Deer Howie Zussman, Rock forming minerals, Volume 1 (Ortho- and ring
silicates); Longmans (1962).
4. Khaibullin, R.I.; Lopatin, O.N.; Vagizov, F. G.; Bazarov, V. V.; Bakhtin, A.I.;
Aktas, B.; Nucl. Instr. And Meth. In Phys. Res. B, 206, 207 (2003).
5. Viana, R. R.; Jordt-Evangelista, H.; Magela de Costa, G; Phys. Chem.
Minerals, 29, 668 (2002).
6. Riesen, H.; Chem. Phys. Lett., 382, 578 (2003).
7. Wood, D. L.; Nassau, K.; Am. Mineral, 53, 777 (1968).
101
8. Wood, D. L.; Ferguson, J.; Knox, K.; Dillon, J. F.; J. Chem. Phys. 39, 890
(1963).
9. Rigby, N. E.; Manson, N. B.; Dubicki, L.; Troup, G. J.; Hutton, D. R.; J. Opt.
Soc. Am. B, 9, 775 (1992).
10. Wood, D. L.; J. Chem. Phys. 42, 3404, 10 (1965).
11. Riesen, H.; Journal of Phys. Chem., 115, 5364-5370 (2011).
12. Riesen, H.; Szabo, A.; Chem. Phys. Lett. 484, 181 (2010).
13. Riesen, H.; Hayward, B. F.; Szabo, A.; J. Lumin. 127, 655 (2007).
14. Hsu, D.; Skinner, J. L.; J. Chem. Phys. 83, 2107 (1985).
15. Hasan, Z.; Keany, S. T.; Manson, N. B.; J. Phys. C: Solid State Phys. 19,
6381-6387 (1986).
16. Hayward, B.; Riesen, H.; Phys. Chem. 7, 2579 (2005).
17. Kisliuk, P.; Moore, C. A.; Phys. Rev. 160, 307 (1967).
18. Quarles, G. J.; Suchocki, A.; Powell, R. C; Phys. Rev. B 38, 14 (1988).
19. Maiman, T. H.; Nature 187, 493 (1960).
20. Van der Ziel, J. P.; Phys. Rev. 9, 2846 (1974).
21. Bobert, C.; Linares, A.; J. Phys. Chem. 26, 1817 (1965).
22. Saito, M.; J. Cryst. Growth 74, 385 (1986).
23. Duker, G.W.; Kellington, C.M.; Katzmann, M.; Atwood, J.G.; Appl. Opt. 4,
109 (1965).
24. http://www.bagdasarovcrystals.com/v1/index.php?id=4&sid=14
25. Szabo, A.; Phys. Rev. B 11, 4512 (1975).
102
Chapter 4
Conclusions
Spectral hole-burning, fluorescence line narrowing and luminescence
experiments were conducted to establish the temperature dependence between
2.5 and 260 K of the R1 linewidth in emerald (Be3Al2(SiO3)6:Cr(III)). The
contribution to the homogeneous broadening is only ~70 kHz at 6 K but a
massive ~400 GHz at 260 K. Up to 80 K the direct one-phonon process
between the split levels of the 2E excited state and the presence of a phonon
sideband at low energy seems to dominate the linewidth. Above this
temperature, the two-phonon Raman process becomes important. The non-
perturbative theory explains the two-phonon Raman process very well but at
low temperature, in addition to the direct process, a low energy phonon is
necessary to explain the data.
In this thesis, slow light has been generated by transient spectral hole-burning
for the first time. A delay of 10.8 ns was obtained for a 130 ppm Bagdasarov
ruby of 2.3 mm thickness. The experimental investigation has been extended to
simulations by a mathematical code written in Mathematica by Aleksander
Rebane and translated to Matlab by Riesen. While the delay that was observed
in ruby using transient hole-burning is impressive, the hole width cannot be too
narrow. When this happens, short probe pulses are distorted and they are not
Gaussian. Matching the probe pulse width and the hole width is necessary to
observe slow light. Due to the wide linewidths at room temperature in emerald,
it seems that it is not a suitable candidate for the creation of slow light in this
crystal at high temperatures. However, f-f transitions are usually much less
susceptible to temperature dependent homogeneous broadening and hence
these systems would result in better candidates for slow light generation by
transient spectral hole-burning at higher temperatures. The present
demonstration of slow light by transient hole-burning has opened a wide field of
interesting slow light experiments in d-d and f-f transitions in solidds.
103
Appendix 1
Voigt function used for hole-burning
Function voigt(w,x) : FitFunc
Wave w
Variable x
//CurveFitDialog/ w[0] = IGORwidth
//CurveFitDialog/ w[1] = gamma
//CurveFitDialog/ w[2] = scale
variable m, product, xp, integral
m=0
do
xp=-100+m*0.08
product=(exp(-xp^2/(w[0]^2))*(w[1]/((x-xp)^2+(w[1]/2)^2)))
integral=integral+product*0.08
m+=1
while(m<2500)
return integral*w[2]
End
104
Appendix 2
Function used for FLN monochromator and 60 and 120 Ghz scan lengths with
Fabry-Perot interferometer
Function FLN(w,x) : FitFunc
Wave w
Variable x
//CurveFitDialog/ w[0] = scale
//CurveFitDialog/ w[1] = INHwidth
//CurveFitDialog/ w[2] = gamma
Variable m, product,xp,integral
m=0
integral=0
do
xp=-0.5+m*0.0005
product=1/((xp-x)^2+(w[2]/2)^2)*exp(-xp^2/w[1]^2)*1/((xp)^2+(w[2]/2)^2)
integral=integral+product*0.0005
m+=1
while (m<2000)
return integral*w[0]
End
105
Appendix 3
Voigt function used for luminescence (RRR1RR and RRR2RR lines).
Function voigt(w,x) : FitFunc
Wave w
Variable x
//CurveFitDialog/ w[0] = IGORwidth
//CurveFitDialog/ w[1] = gamma
//CurveFitDialog/ w[2] = x0
//CurveFitDialog/ w[3] = gamma2
//CurveFitDialog/ w[4] = x02
//CurveFitDialog/ w[5] = scale
//CurveFitDialog/ w[6] = ratio
Variable m, product, xp, integral
m=0
integral=0
do
m+=1
xp=-10+m*0.005
product=exp(-(xp)^2/w[0]^2)*(1/(((x-w[2])-xp)^2+(w[1]/2)^2)+w[6]/(((x-w[4])-
xp)^2+(w[3]/2)^2))
integral=integral+product*0.005
while (m<4001)
106
return integral*w[5]
end
107
Appendix 4
Non-perturbation function used for the final data analysis.
Function nonperturbation(w,T) : FitFunc
Wave w
Variable T
//CurveFitDialog/ w[0] = W
//CurveFitDialog/ w[1] = beta
//CurveFitDialog/ w[2] = gamma0
//CurveFitDialog/ w[3] = beta1
//CurveFitDialog/ w[4] = DT
//CurveFitDialog/ w[5] = a1
//CurveFitDialog/ w[6] = w1
variable f1,f2,f3,f4,f5,f6,f7,x, f8, f9, f10,f11
variable i, lim
variable pi
i=1
lim=1000
pi=3.1415
f1=1.38e-23*w[4]*2*pi/(6.62e-34*(4*pi^2))
f2=w[4]/T
f7=0
do
108
x=i/lim+0
f3=ln((1-x)/(1+x))
f4=1/((1+w[0]*(1+3*x^2+3/2*x^3*f3))^2+(w[0]^2)*9*((pi^2)/4)*(x^6))
f5=1+(9*(pi^2)*(w[0]^2)*(x^6)*exp(x*f2)/(exp(x*f2)-1)^2)*f4
f6=ln(f5)
f7=f7+f1*f6/lim
i+=1
while (i<lim)
f8=(w[6])/(T*0.695)
f9=exp(f8)-1
f10=1/f9
f11=w[5]*f10*(f10+1)
return f7/1e6+w[1]*(1/(exp(63/(0.695*T))-1))+w[3]*(1/(exp(6/(0.695*T))-
1))+w[2]+f11
End
109
Appendix 5
Matlab code for slow light experiments. % pulse reshaping/slow light calculation following Aleks Rebane's code in
% Mahematica; translated to Matlab by Hans Riesen 2012
%Note: Matlab was written by an engineer and thus has reverse sign
%convention in Fourier transformation i.e. FFT in Matlab is DFFT in
%Mathematica nd vice versa; note that normalisation factors have to b
%adjusted
%define parametres
clear
va=4.32*10^5; % centre of inhomogeneous band/GHz
a1=2; %width of inhomogeneous distribution/GHz
a21=1.5; %absorbance before burning in log10 base
deltaA=0.25; %change in absorbance at spectral hole in log10 base
tsp=10^(-(a21-deltaA)); %transmission at hole
a2=log(10)*a21; %absorbance at max of band before burning in ln base
Dt1=100.0; %pulse duration of input pulse FWHM/ns
a22=exp(-a2);
%following line is from Aleks; I think the peak transmission needs to be
%corrected to: 1. absorbance on log10 base -log10(a22); 2 corrected for
%hole: -0.4; convert back to transmission 10^diff
a3=tsp-a22; %peak transmission of spectral hole (10^-0.4=0.38, exp(-2)=0.135;
%mine: a222=a2+log(10^-0.4)
%mine: a3=exp(-a222)
a4=0.10; %Width of spectral hole
%normalized inhomogeneous absorption line shape
v=4.31980*10^5:0.001:4.32020*10^5;
func01=exp(-log(2)*((v-va)./(a1./2)).^2);
figure (1)
plot(v,func01);
%Intensity transmission without spectral hole
110
figure(2)
func02=exp(-a2.*func01);
plot(v,func02);
%temporal pulse shape * Gaussian amplitude profile
t=-100:0.001:100; %nanoseconds
%%%t=t*10^(-9);
func03=exp(-(log(2)./2)*(t./(Dt1./2)).^2);
figure(3)
plot(t,func03.^2);
%evaluate Fourier transform spectrum of the incident pulse
%func03a=func03.*exp(-i*2*t*va*3.1415);
%F=fourier(func03a);
figure(4)
v1=-0.1:0.001:0.1;
denom=sqrt(1/Dt1.^2)*sqrt(2*log(4));
func04a=((1/denom.*exp(-(3.1415)^2*Dt1.^2*((v1).^2)/log(4))).^2);
plot(v1,func04a)
denom=sqrt(1/Dt1^2)*sqrt(2*log(4));
func04=1/denom.*exp(-(3.1415)^2.*Dt1.^2*((v-va).^2)/log(4));
func05=a3.*exp(-log(2).*((v-va)./(a4./2)).^2);
func06=(func02+func05);
figure(5)
plot(v,func06);
%2.Calculate causal complex response function
%2.1. Numerical list representation of the amplitude transmission frequency
%range 40 GHz
list02=func02;
list06=func06;
list02a=1/sqrt(40000)*fft(log(list02));
list02b=list02a;
%cut off negative time part
for k=20000:40001;
111
list02b(k)=0;
end
%transform back to frequency
list02c=1/sqrt(40000)*fft(list02b);
list02d=(sqrt(list02).*exp(j*2*imag(list02c)));
%combines ampl;itude with causal phase
for k99=1:1000
list02c(k99);
end
%2.4 with spectral hole
list06a=1/sqrt(40000)*fft(log(list06));
list06b=list06a;
%cut off negative time part
for k=20000:40001;
list06b(k)=0;
end
list06c=1/sqrt(40000)*fft(list06b);
list06d=sqrt(list06).*exp(j*2*imag(list06c));
%combine amplitude with causal phase
%calculate transmitted phase
list04=func04;
list07a=sqrt(40000)*ifft(list04.*list02d);
list07aa=list07a; %make copy
for k=1:20000;
list08a(k)=list07a(20000+k);
end
for k=20001:40001;
list08a(k)=list07aa(k-20000);
end
list07b=sqrt(40000)*ifft(list04.*list06d);
list07bb=list07b; %make copy
112
for k=1:20000;
list08b(k)=list07b(20000+k);
end
for k=20001:40001;
list08b(k)=list07bb(k-20000);
end
%
for i1=16000:24000;
list09ax(i1)=-500+i1/40;
list09ay(i1)=abs(list08a(i1).^2);
end
for i1=16000:24000;
list09bx(i1)=-500+i1/40;
list09by(i1)=abs((list08b(i1).^2));
end
figure(6)
plot(list09ax,list09ay,list09bx,list09by);
i1 = 16000:1:24000;
y = [list09ax(i1); list09ay(i1); list09bx(i1); list09by(i1)];
fid = fopen('list09.txt', 'wt');
fprintf(fid, '%12.8f %12.8f %12.8f %12.8f\n', y);
fclose(fid)
%figure(7)
%plot(list09bx,list09by);
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