iEmission Properties of Radiative Chiral
Nematic Liquid Crystals
This dissertation is submitted for the degree of
Doctor of Philosophy
by
Themistoklis K. Mavrogordatos
Centre of Molecular Materials for Photonics and Electronics
Photonics Doctoral Training Centre, Division B, Department of Engineering
Jesus College, University of Cambridge
Supervisor: Prof. Timothy D. Wilkinson
Cambridge
February 2014 (Rev. August 2014)
Contents
Declaration v
Acknowledgments vi
Abstract vii
Introduction x
1 The mechanics of liquid crystals 1
1.1 Nematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Chiral nematics . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2 Optical properties of LC cells 22
2.1 The Belyakov formulation . . . . . . . . . . . . . . . . . . . . 22
2.1.1 Kinematical approximation and exact solution . . . . . 22
2.1.2 Formation of the Boundary Value Problem . . . . . . . 28
2.2 The de Vries formulation . . . . . . . . . . . . . . . . . . . . . 35
2.3 Oblique incidence . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.4 The Berreman method . . . . . . . . . . . . . . . . . . . . . . 52
2.5 Optical feedback in nematic LC slabs . . . . . . . . . . . . . . 56
ii
CONTENTS iii
3 The density of photon states 67
3.1 The rôle and the calculation of the DOS . . . . . . . . . . . . 67
3.1.1 The quarter-wave stack . . . . . . . . . . . . . . . . . . 84
3.2 The DOS in radiative chiral nematics . . . . . . . . . . . . . . 89
3.2.1 Inclusion of absorption . . . . . . . . . . . . . . . . . . 90
3.2.2 Inclusion of amplification . . . . . . . . . . . . . . . . . 97
4 The Jaynes-Cummings model 111
4.1 Hamiltonian, states and equations of motion . . . . . . . . . . 112
4.2 Photonic Density of States revisited . . . . . . . . . . . . . . . 118
4.3 Main results for radiative chiral nematic LCs . . . . . . . . . . 121
4.4 A 2D photonic crystal and the DOS . . . . . . . . . . . . . . . 124
4.5 Discussion of results . . . . . . . . . . . . . . . . . . . . . . . 130
4.6 More on the Fano-Anderson model . . . . . . . . . . . . . . . 136
5 Adaptive pumping of radiative LC cells 145
5.1 The method of adaptive pumping . . . . . . . . . . . . . . . . 148
5.1.1 Sample preparation . . . . . . . . . . . . . . . . . . . . 150
5.1.2 Pumping process . . . . . . . . . . . . . . . . . . . . . 151
5.2 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
6 Suggestions for further research 166
6.1 Decay dynamics in the Jaynes-Cummings model . . . . . . . . 166
6.2 The basic principles of the Resolvent Method . . . . . . . . . 169
6.3 Atomic radiation in free space . . . . . . . . . . . . . . . . . . 173
6.4 Atomic radiation in cavities and waveguides . . . . . . . . . . 177
6.5 Atomic radiation in a 1D periodic medium . . . . . . . . . . . 188
iv CONTENTS
Synoptic conclusions 192
Related publications 195
Declaration
This dissertation is the result of my own work and includes nothing which is
the outcome of work done in collaboration except where specifically indicated
in the text. No part of this thesis has been submitted for any other qual-
ification. This dissertation contains 21 figures and less than 47.300 words,
therefore it does not exceed the word limit for the respective Degree Com-
mittee.
Themistoklis K. Mavrogordatos
February 2014 (Revised August 2014)
v
Acknowledgments
I would like to thank my supervisor, Prof. Timothy D. Wilkinson for his
guidance, honesty and respect to freedom of opinion and expression. Very
valuable help I also received from Dr. Stephen M. Morris throughout my PhD
project, in both the theoretical and experimental parts of this work. In the
latter, my colleague Simon Wood contributed significantly with an unusual
perseverance and ardour. During my PhD studies, I had many stimulating
and challenging discussions with most of the members of the CMMPE group,
particularly with Dr. Flynn Castles. Some of the conclusive arguments con-
stitute part of this thesis. In that regard, the academic collaboration with
Prof. Vladimir A. Belyakov consisted a source of inspiration and rigour.
His book on optics of periodic media, alongside relevant papers are refer-
enced extensively throughout the thesis. As fas as more practical matters
are concerned, I would like to acknowledge financial support from the Onassis
Foundation as well as from the EPSRC, towards my maintenance costs and
tuition fees, respectively. Finally, on a more personal note, I am grateful to
my friend Marcin Malinowski, who was the first physics student I supervised
in Cambridge. He taught me to pay more attention to detail, and helped me
remaining focussed and determined.
vi
Abstract
In this work, we calculate the density of photon states (DOS) of the normal
modes in dye-doped chiral nematic liquid crystal (LC) cells in the presence
of various loss mechanisms. Losses and gain are incorporated into the trans-
mission characteristics through the introduction of a small imaginary part
in the dielectric constant perpendicular and along the director, for which we
assume no frequency dispersion. Theoretical results are presented on the
DOS in the region of the photonic band gap for a range of values of the loss
coefficient and different values of the optical anisotropy. The obtained values
of the DOS at the photonic band gap edges predict a reversal of the domi-
nant modes in the structure. Our results are found to be in good agreement
with the experimentally obtained excitation thresholds in chiral nematic LC
lasers. The behaviour of the DOS is also discussed for amplifying LC cells
providing an additional insight to the lasing mechanism of these structures.
We subsequently investigate the spontaneous emission properties, under the
assumption that the electronic transition frequency is close to the photonic
edge mode of the structure (resonance). We take into account the transition
broadening and the decay of electromagnetic field modes supported by the
so-called `mirror-less' cavity. We employ the Jaynes-Cummings Hamiltonian
vii
viii ABSTRACT
to describe the electron interaction with the electromagnetic field, focusing
on the mode with the diffracting polarization in the chiral nematic layer. As
known in these structures, the density of photon states, calculated via the
Wigner method, has distinct peaks on either side of the photonic band gap,
which manifests itself as a considerable modification of the emission spec-
trum. We demonstrate that, near resonance, there are notable differences
between the behaviour of the density of states and the spontaneous emission
profile of these structures. In addition, we examine in some detail the case of
the logarithmic peak exhibited in the density of states in 2D photonic struc-
tures and obtain analytic relations for the Lamb shift and the broadening of
the atomic transition in the emission spectrum. The dynamical behaviour
of the atom-field system is described by a system of two first order differ-
ential equations, solved using the Green's function method and the Fourier
transform. The emission spectra are then calculated and compared with
experimental data.
Finally, we detail a new technique for the pumping of dye lasers which re-
quires no moving parts or flushing mechanisms and is applicable to both solid
state and liquid based devices. A reconfigurable hologram is used to control
the position of incidence of a pump beam onto a dye laser and significant
increases in device lifetimes are achieved. The technique is also applied to
wavelength tune a dye laser. This offers access to higher repetition rates and
larger average output powers. With higher repetition rate pump lasers it is
feasible that the approach could allow such organic lasers to reach operating
frequencies on the order of MHz. The unique nature of the adaptive pumping
method also allows precise control of the spatial wave-front and configura-
ix
tion of the pumping wave which allows greater versatility and functionality
to be realised. It is possible to envisage that novel pump beam profiles that
optimise propagation through the medium could also be demonstrated.
Introduction
Photonic band edge lasing has been extensively demonstrated in dye-doped
chiral nematic liquid crystal (LC) cells following the first pioneering exper-
imental realisation in 1980
†
. Feedback is provided by the modulation of
the refractive index as the director precesses continually and orthogonally
along the optical axis forming a helical structure with pitch. Optical gain,
on the other hand, is provided through the addition of a gain medium such
as a laser dye. In recent years, these lasers have attracted interest because
of their remarkable emission characteristics in the form of broadband wave-
length tuneability, narrow linewidth emission, and high slope efficiencies.
In the first two chapters we review the basic mechanical and optical proper-
ties of liquid crystals, aiming to build the required background to understand
the enhancement of radiation from these molecular structures. The observed
lasing behaviour near the edges of the photonic band gap can be explained
by the drastic changes in the density of photon states (DOS). Divergence in
the DOS at the band-edges in chiral nematic LCs was shown theoretically
and experimentally who considered the fluorescence characteristics from a
†
I.P. Il'chishin, E.A. Tikhonov, V.G. Tishchenko and M.T. Shpak, Generation of a
tunable radiation by impurity cholesteric liquid crystals, JETP Lett., Vol. 32, 24-27 (1980).
x
xi
dye-doped chiral nematic LC in the region of the photonic band gap. Subse-
quently, for laser emission it has been shown that, in analyzing the case of a
Fabry-Pérot (FP) resonator, the threshold gain can be related directly to the
DOS. Furthermore, according to the space-independent rate equations the
slope efficiency can be shown to be inversely proportional to the threshold
energy and therefore directly proportional to the DOS.
Losses will be important in the design of optimised practical systems: they
will affect the DOS and hence the threshold gain and the slope efficiency.
In this work we calculate the effect of losses and gain directly on the DOS.
We modify the analysis of previous works to include imaginary parts in the
parallel and perpendicular dielectric coefficients. We will show that small
losses have appreciable and unexpected effects on the DOS, such as a re-
versal of the dominant edge mode, and hence also on the threshold gain.
The modified DOS is investigated for various cell thicknesses and optical
anisotropies. We then correlate our theoretical results with experimentally
determined excitation thresholds, finding good agreement. The behaviour of
the DOS corresponding to stimulated emission close to the lasing threshold
condition is discussed.
In order to introduce losses in to the transmission coefficient it is first nec-
essary to construct a formulation for the DOS in terms of the transmission
coefficient. For a light wave propagating along the helix of a chiral nematic
LC film there are two eigenmodes corresponding to elliptically polarized plane
waves with an opposite sense of rotation. Their polarization is wavelength
dependent. The probability of photon emission by an excited fluorescent
molecule, as obtained by Dirac's rule is the product of the DOS and the
xii INTRODUCTION
square of the matrix elements corresponding to the coupling of the electric
field with the electric dipole moment of the gain medium. Furthermore,
we consider a a two-level system coupled to a quantum harmonic oscillator,
frequently described with the Jaynes-Cummings (JC) Hamiltonian in which
only `resonant' terms feature. Such a consideration is permissible in the case
of near resonance and weak coupling. Both conditions are satisfied for spon-
taneous emission in these periodic structures for small detuning. We put the
analysis of spontaneous emission from 2D photonic crystals on a firmer basis
providing analytical results, and explore in more detail the fluorescence prop-
erties in chiral nematic LCs, outlining common features that are attributed to
resonance. Moreover, the discrepancy between the experimentally obtained
emission spectra and the theoretically calculated DOS is addressed. Such a
consideration aims to further the understanding of spontaneous and induced
emission from these distributed feedback resonators.
Despite the advantages of organic solid-state dye lasers, liquid-dye lasers
considerably outperform the solid-state organics when it comes to pulse du-
ration. They can operate with pulses as short as 10 fs, and the pulse length
can be made so long as to allow continuous-wave (CW) outputs. Solid-state
dye lasers, which consist of an organic dye that is dispersed into a solid ma-
trix, are not so versatile and are restricted to pulse durations are typically no
longer than 10 ns. CW operation is generally prohibited because of bleach-
ing issues caused by thermal degradation effects and also the accumulation of
triplet excitons. In the case of triplet excitons, which possess an absorption
band that overlaps the stimulated emission spectrum, suppression of the gain
in the active region can occur, resulting in laser action being switched off.
xiii
In order to ensure that the population of triplet states is dissipated between
the pulses, repetition rates of the pump laser are usually restricted to less
than 10 kHz. However, even if the repetition rate is low enough that triplet
excitons do not build-up it is still not necessarily the case that thermal effects
are absent.
In order to combat these thermal and excited-state absorption effects, a so-
lution is to remove the excited dye molecules out of the pump volume in
sufficient timescales so as to maintain laser action. For liquid dye lasers,
this involves a constant convective circulation of the organic species using a
jet-stream with velocities on the order of 10s of ms
−1
to replenish the dye
molecules and thus avoid the bleaching effects that ultimately shut down
laser action. Alternatively, for solid-state organic dye lasers, methodologies
that mimic the approach of `flushing out' the excited dye molecules have
been demonstrated such as mechanically rotating a disc-shaped active gain
medium at high speeds to constantly refresh the dye molecules. A common
feature of the techniques that have been demonstrated to date require phys-
ically translating the active region in some way relative to the pump. This,
therefore, requires the use of moving parts or fluid flow. In our work, we
demonstrate that a potentially elegant solution in which the pump beam
is moved relative to the solid-state organic dye laser using adaptive optics,
which bypasses the need for any moving parts as the position of the pump
beam is adjusted through the diffraction of light.
Finally, a chapter on further investigation is included, with emphasis on the
Resolvent Method that can be employed to analyze spontaneous emission
from radiative chiral nematic liquid crystals (where parallels are drawn with
xiv INTRODUCTION
atomic radiation in cavities and waveguides, with reference to the DOS). In
particular, the knowledge of the DOS for periodic structures of finite length
can help the understanding of spontaneous emission dynamics, where the
departure from an exponential-decay behaviour is prominent.
Chapter 1
The mechanics of liquid crystals
In this introductory chapter we will focus more on the mechanics of deforma-
tions of the liquid crystals of interest (nematics and chiral nematics), rather
than their on optical properties. Such a discussion is deemed necessary, in
order to understand the properties of the molecular structures that will form
the resonating cavities in both the theoretical and experimental aspects of
this work.
1.1 Nematics
Nematic liquid crystals are substances which are both microscopically and
macroscopically homogeneous in the undeformed state. The anisotropy of the
medium is attributed to the anisotropic spatial orientation of the molecules.
This anisotropy is fully defined-in the great majority of known nematics-by
specifying a unit vector n along one particular direction for each point in the
medium. This vector is called the (molecular) director. The properties of
1
2 CHAPTER 1. THE MECHANICS OF LIQUID CRYSTALS
ordinary nematics, for every volume element, are invariant under a change
in sign of all spatial co-ordinates (inversion). Hence, the state of a nematic
crystal is determined by the specification of the director n alongside the usual
quantities (mass density ρ, pressure p and velocity v) for a liquid. In equi-
librium, a nematic liquid crystal at rest under no external forces (including
the forces exerted by the walls) is uniform and n is constant throughout its
volume. In a deformed nematic, the direction of n varies slowly in space,
such that the characteristic dimensions of the deformation are much greater
than the molecular dimensions (as a result, the derivatives ∂ni/∂xk can be
regarded as small quantities).
The total free energy density of a deformed liquid crystal, being a scalar
quantity itself, can contain only scalar combinations of the components of
n and its derivatives. The scalar combinations linear in the first derivatives
are the true scalar divn and the pseudoscalar n · curln. Considering the
bulk properties of the material, only the latter should be retained, as the
former is transformed into a surface integral by virtue of Gauss's theorem.
The true scalars that are quadratic to the first derivatives of the director
can be found by considering the four-rank tensor
∂nk
∂xi
∂nl
∂xm
and forming the
invariant quantities resulting from contraction of some of its indices as well
as from multiplication with n. Bearing in mind that n2 ≡ 1 we can form
the invariant quantities (n × curln)2, ∂nk
∂xi
∂nk
∂xi
,
∂nk
∂xk
∂ni
∂xi
and
∂nk
∂xi
∂ni
∂xk
. The
difference of the last two quantities can be written in the form [1]
∂nk
∂xk
∂ni
∂xi
− ∂nk
∂xi
∂ni
∂xk
=
∂
∂xi
(
ni
∂nk
∂xk
− nk ∂ni
∂xk
)
, (1.1)
which is a divergence that again contributes to the total free energy by an
1.1. NEMATICS 3
integral over the surface of the body, and therefore will not be considered.
The invariant
∂nk
∂xi
∂nk
∂xi
= (n · curln)2 + (divn)2 (1.2)
and (n · curln)2 will be considered as the independent ones. Last, there is
the quadratic in the first derivatives pseudoscalar (n · curln) · divn. Since
∇n2 = 0 ⇒ (n · ∇)n = −n × curln, the vector (n · ∇)n is perpendicular to
curln, hence [(n · ∇)n] · curln = 0.
Taking into account the above, the deformation energy of a nematic substance
is given by an expression quadratic in the derivatives of n, its general form
being
Fd = F − F0 = c(n · curln) · divn + d(n · curln)
+
1
2
K1(divn)
2 +
1
2
K2(n · curln)2 + 1
2
K3(n× curln)2.
(1.3)
In order to meet the condition of equivalence between the directions n and
−n we must set d = 0 in the expression for the free energy. Moreover, the
presence of a pseudoscalar for a medium containing planes as a symmetry
element implies that the coefficients of these terms must be pseudo-scalars
since the free energy is a true scalar. Therefore, on reflection from such a
plane we will have c = −c. Hence, we are left with the relation
Fd = F − F0 = 1
2
K1(divn)
2 +
1
2
K2(n · curln)2 + 1
2
K3(n× curln)2, (1.4)
where the last term can be also written as
1
2
K3[(n · ∇)n]2 by virtue of the
relation n× curln = 4n2 − (n · ∇)n, since n2 ≡ 1.
4 CHAPTER 1. THE MECHANICS OF LIQUID CRYSTALS
In chiral nematics there is no center of inversion among their symmetry
elements, unlike nematics. The free energy density for a liquid crystal with
an absence of a center of symmetry, where the directions n and −n are
equivalent, is [2]
F = F0 +
1
2
K1(divn)
2 +
1
2
K2(n · curln + q0)2
+
1
2
K3(n× curln)2.
(1.5)
The inclusion of the pseudo-scalar n · curln causes a fundamental change
in the equilibrium state of the medium, such that it is no longer uniform
in space, as in nematics. The equilibrium state for a chiral nematic liquid
crystal corresponds to a distribution of directions for n such that the free
energy (1.5) attains its minimum value of zero. Hence we obtain the equations
divn = 0, n · curln = −q0 and n× curln = 0, with solution nx = cos(q0z),
ny = sin(q0z) and nz = 0.
The orientational symmetry of a chiral nematic liquid crystal is periodic in
one direction in space (z-axis) so that the correlation function can be written
as ρ1,2 = ρ1,2(z, r12). The vector n acquires its original value after a step
of length 2pi/q0 along the z-axis. However, since the direction n and −n
are equivalent the actual spatial period of the crystal is pi/q0. The above
considerations are valid only if the period of the helicoidal structure is large
in comparison with molecular dimensions. The condition is in fact satisfied
since in most chiral nematic liquid crystals pi/q0 ∼ 10−7 m.
In order to determine the static deformations for nematics we start from the
general thermodynamic condition of equilibrium: minimization of the func-
tional
∫
FdV subject to the auxiliary condition n2 = 1. Using the method
1.1. NEMATICS 5
of undetermined Lagrange coefficients, we equate to zero the variation
δ
∫ {
F − 1
2
λ(r)n2
}
dV =
∫ {
∂F
∂ni
δni +
∂F
∂(∂kni)
∂kδni − λ(r)niδni
}
dV =∫ {
∂F
∂ni
− ∂k ∂F
∂(∂kni)
− λ(r)ni
}
δnidV +
∮
∂F
∂(∂kni)
dni dfk,
(1.6)
where in the last step we have applied integration by parts resulting in a
surface integral (boundary term). Setting δn = 0 at the boundaries, we have
∫
(H + λn) · δn dV = 0, (1.7)
with Hi = ∂kΠik−(∂F )/(∂ni) and Πik = ∂F/∂(∂kni). The vector H is called
the molecular field, and tends to straighten out the direction of n [1]. Since
the variation δn is arbitrary the equilibrium equation is H = −λn and by
virtue of n2 = 1 we obtain λ = −H ·n. Therefore, the equilibrium condition
is written as h ≡ H − (n ·H)n = 0. Once more, by virtue of the unit
modulus of the director, the vector h is perpendicular to n. If a nematic is in
equilibrium such that h = 0, and moves as a whole with a constant spatial
velocity, then the equation for the director expresses the fact that each liquid
crystal particle moves in space with its own fixed n. This is expressed by the
equation
dn
dt
=
∂n
∂t
+ (v · ∇)n = 0. (1.8)
Outside of equilibrium, the right handside of the above equation will contain
terms than depend on the transverse part of the molecular field, h and the
spatial derivatives of the body, forming a tensor-in the first non-vanishing
6 CHAPTER 1. THE MECHANICS OF LIQUID CRYSTALS
hydrodynamic approximation-that can be divided into symmetric and anti-
symmetric parts, vik and Ωik respectively. The general form for the equation
of motion of the director is [3]
dni
dt
= Ωiknk + λ(δil − ninl)nkvkl + hi/γ (1.9)
Contraction of the right handside of the above with ni yields Ωkinkni +
λ(nlnkvkl − nininlnkvkl) + nihi/γ = 0, since nihi = 0, nini = 1, and a
contraction of a symmetric and an antisymmetric tensor yields zero. Hence,
the above form is justified, since n2 = 1⇒ n · (∂n/∂t) = 0. In the above, we
define vik =
1
2
(∂ivk + ∂kvi) and Ωik =
1
2
(∂ivk − ∂kvi). In (1.9) the last term
represents the relaxation of the director towards equilibrium under the action
of the molecular field, the middle term the orienting effect of the velocity
gradient of the director, and the first term the rotation of the director under
uniform rotation of the nematic as a whole. All coefficients featuring are
kinetic (the coefficient γ has dimensions of viscosity).
The equation for the time derivative of the velocity reads
ρ
[
∂vi
∂t
+ (v · ∇)vi
]
= ∂kσik. (1.10)
In order to establish the form of the stress tensor, we invoke the energy
conservation law in hydrodynamics, written as
∂
∂t
(
1
2
ρv2 + E
)
+ divQ = 0, (1.11)
where E is the internal energy density and Q is the energy flux density. The
former can be written as E = E0(ρ, S) + Ed, where E0(ρ, S) pertains to the
1.1. NEMATICS 7
undeformed homogeneous medium and Ed is attributed to the distortion of
the field n(r). By using the thermodynamic relations(
∂E
∂S
)
ρ,n
= T,
(
∂E
∂ρ
)
S,n
= µ
with µ being the chemical potential, we can expand the time derivative using
the chain rule, as follows
∂
∂t
(
1
2
ρv2 + E
)
=
1
2
v2
∂ρ
∂t
+ ρv
∂v
∂t
+
(
∂E
∂ρ
)
S,n
∂ρ
∂t
+
(
∂E
∂S
)
ρ,n
∂S
∂t
+
(
∂Ed
∂t
)
ρ,S
=
1
2
v2
∂ρ
∂t
+ ρv
∂v
∂t
+ µ
∂ρ
∂t
+ T
∂S
∂t
+
(
∂Ed
∂t
)
ρ,S
.
(1.12)
The last term can be recast in the following form
(
∂Ed
∂t
)
ρ,S
=
(
∂Ed
∂ni
)
ρ,S
∂ni
∂t
+
(
∂Ed
∂(∂kni)
)
ρ,S
∂k
∂ni
∂t
=
(
∂Ed
∂ni
)
ρ,S
∂ni
∂t
+ Πki∂k
∂ni
∂t
=
(
∂Ed
∂ni
− ∂kΠki
)
∂ni
∂t
+ ∂k
(
Πki
∂ni
∂t
)
= −h · ∂n
∂t
+ ∂k
(
Πki
∂ni
∂t
)
,
(1.13)
where we have written h instead of H since n · (∂n/∂t) = 0. Since
∂n
∂t
=
dn
dt
− (v · div)n
we write
(
∂Ed
∂t
)
ρ,S
= hi(vk∂kni + Ωik − λviknk)−Nkhk + ∂k
(
Πki
∂ni
∂t
)
, (1.14)
8 CHAPTER 1. THE MECHANICS OF LIQUID CRYSTALS
where the last term is a total divergence. We now note that
viknkhi =
1
2
(∂ivk + ∂kvi)nkhi =
1
2
(∂kvi)(nkhi + nihk)
=
1
2
{∂k[vi(nkhi + nihk)]} − 1
2
vi∂k(nkhi + nihk)
(1.15)
and that
Ωiknkhi =
1
2
(∂ivk − ∂kvi)nkhi = −1
2
(∂kvi)(nkhi − nihk)
= −1
2
{∂k[vi(nkhi − nihk)]}+ 1
2
vi∂k(nkhi − nihk),
(1.16)
so that we can write
(
∂Ed
∂t
)
ρ,S
= −Givi − h
2
γ
+ ∂k
(
Πki
∂ni
∂t
− 1
2
vi(nkhi − nihk) + λ
2
vi(nkhi + nihk)
)
,
(1.17)
with
Gi = −hk∂ink + 1
2
∂k(nihk − nkhi)− λ
2
∂k(nihk + nkhi). (1.18)
In a similar fashion to the partial time derivative considered above we have
(∂iEd)ρ,S =
(
∂Ed
∂nk
)
ρ,S
∂ink +
(
∂Ed
∂(∂lnk)
)
ρ,S
∂i(∂lnk)
=
(
∂Ed
∂nk
)
ρ,S
∂ink +
(
∂Ed
∂(∂lnk)
)
ρ,S
∂l(∂ink)
=
(
∂Ed
∂nk
− ∂lΠlk
)
∂ink + ∂l (Πlk∂ink) = −h · ∂in + ∂l (Πlk∂ink)
= −h · ∂in + ∂k (Πkl∂inl) ,
(1.19)
1.1. NEMATICS 9
where we have interchanged the order of differentiation and we have replaced
once more H by h since n∂in = 0. The first term of Gi, then, can be recast
in the form
− hk∂ink = (∂iEd)ρ,S − ∂k (Πkl∂inl) , (1.20)
so that we can write
Gi = ∂kσ
(r)
ik + (∂iEd)ρ,S, (1.21)
where
σ
(r)
ik = −Πkl∂inl +
1
2
(nihk − nkhi)− λ
2
(nihk + nkhi). (1.22)
As noted in [1], the definition of the tensor σ
(r)
ik is not unique, since the
tensor σ
′(r)
ik = σ
(r)
ik + ∂lχilk with χilk = −χikl would produce the same vector
Gi, as the order of differentiation ∂k∂l with respect to the indices k, l can be
interchanged and the tensor χˆ is antisymmetric in k and l (the contraction of
a symmetric and an antisymmetric tensor over a pair of indices is zero). The
choice of the tensor χikl is arbitrary and will be of use when symmetrizing
σ
(r)
ik . Assuming that the latter has been symmetrized we can write
(∂kσ
(r)
(ik))vi = ∂k(σ
(r)
(ik)vi)− σ(r)(ik)(1/2)(∂ivk + ∂kvi) = ∂k(σ(r)(ik)vi)− σ(r)(ik)vik
so that
10 CHAPTER 1. THE MECHANICS OF LIQUID CRYSTALS
(
∂Ed
∂t
)
ρ,S
= −Nivi + σ(r)(ik)vik − (∂iE)ρ,Svi
+ ∂k
(
Πki
∂ni
∂t
− 1
2
vi(nkhi − nihk) + 1
2
vi(nkhi − nihk) + σ(r)(ik)vi
)
.
(1.23)
We write the third term as
−(∂iE)ρ,Svi = −vi∂iE + µvi∂iρ+ viT∂iS = (−vi∂kE + µvi∂kρ+ viT∂kS)δik
= −∂k(Eviδik) + Evikδik + µ∂i(ρvi)− µρvikδik + T∂i(Svi)− TSvikδik.
(1.24)
We now return to (1.12) and substitute expressions (1.10), (1.23) as well as
the mass and entropy continuity equations ∂ρ/∂t = −div(ρv) and ∂S/∂t =
−div(Sv), respectively. In the presence of dissipative processes, the latter is
written
∂S/∂t = −div(Sv + q/T )− 2R/T, (1.25)
where q is the heat flux density and R is the mechanical energy dissipation
rate due to frictional forces. We then notice that
T∂i(Svi) = 2R− T ∂S
∂t
− T∂i
(qi
T
)
= 2R− T ∂S
∂t
+
qi
T
∂iT + ∂iqi
µ
∂ρ
∂t
= −µ∂i(ρvi)
ρvi
∂vi
∂t
= vi∂kσik − ρvivk∂kvi
1
2
v2
∂ρ
∂t
= −1
2
v2∂i(ρvi) = −1
2
∂i(v
2ρvi) + ρvivk∂ivk
= −1
2
∂i(v
2ρvi) + ρvivk∂kvi.
(1.26)
1.1. NEMATICS 11
Collecting terms, we obtain
∂
∂t
(
1
2
ρv2 + E
)
= (σ
(r)
(ik) − pδik − σik)vik −Nihi +
qi
T
∂iT + ∂i(−Qi), (1.27)
where p = ρµ + TS − E = Φ + TS − E is the pressure (Φ is the Gibbs free
energy) and Q is the energy flux density. The components of this vector will
be fully determined after having symmetrized σ
(r)
ik , as assumed throughout
in our derivations. In order to calculate explicitly the antisymmetrical part
of this tensor we use the fact that
Bik =
∂Ed
∂ni
+ Πli∂lnk − Πkl∂inl (1.28)
is symmetric in the suffixes i and k. We will prove this indirectly as follows.
Let us consider an infinitesimal rotation of the co-ordinates through an angle
δφ. The change in the director then is δni = ijkδφjnk = aiknk, where ijk
is the completely antisymmetric tensor, and aik is an antisymmetric tensor
(in the suffixes i and k), as a result of contraction of ijk. The change in the
tensor ∂kni is δ(∂kni) = δ(∂k)ni + ∂k(δni) = akl∂lni + ail∂knl. The quantity
Ed is a scalar, hence invariant under the infinitesimal arbitrary rotation.
δEd =
∂Ed
∂ni
δni + Πkiδ(∂kni) =
∂Ed
∂ni
aiknk + Πki(akl∂lni + ail∂knl)
=
∂Ed
∂ni
aiknk + Πliaik∂lnk + Πklaki∂inl = aikBik = 0.
(1.29)
In view of the arbitrariness of aik we infer that the non-zero tensor Bik must
indeed be symmetric. The tensor σ
(rn)
ik = σ
(r)
ik − (λ/2)(nihk + nkhi) can be
written as
12 CHAPTER 1. THE MECHANICS OF LIQUID CRYSTALS
σ
(rn)
ik = −Πkl∂inl +
1
2
(nihk − nkhi)
= −Πkl∂inl + ∂F
∂ni
nk + ni∂l(Πlk)− nk∂l(Πli)
=
∂Ed
∂ni
nk + ∂l(niΠlk − nkΠli) + Πli∂lnk − Πkl∂inl = Bik − Πkl∂inl
(1.30)
In the above we have used the theorem of small increments. As known
from thermodynamics, knowledge of one of the thermodynamic potentials:
internal energy E, Helmholtz free energy F , Gibbs free energy Φ and the
heat function W , is sufficient to determine the other variables, provided that
the above are expressed as functions of S and V , V and T , P and T , S and
P respectively. If there are small parameters ζi that determine a state of
the system apart from its volume, the expression for the differential of the
internal energy must involve terms proportional to the differentials dζi as
below [4]
dE = TdS − PdV +
∑
i
Ziζi,
where Mi are the conjugate variables, functions of the state of the body.
Since the transformation of the other thermodynamic potentials does not
affect the parameters ζi, similar terms will be added to the other potentials,
therefore
dF = −SdT − PdV +
∑
i
Ziζi.
We conclude that Ed is the same as Fd, except that the elastic moduli K1−3
are expressed in terms of the density end entropy, instead of the temperature.
1.1. NEMATICS 13
Based on (1.30), the antisymmetric part of σrik can be written as a divergence
σ
(r)
ik − σ(r)ki = 2∂lφikl,
where φikl = (1/2)(niΠlk − nkΠli) is a tensor antisymmetric under exchange
of the first pair of suffixes. From this result follows that the moment of forces
over the whole volume of the body
Mik =
∮
(σ
(r)
il xk − σ(r)kl xi)dfl +
∫
(σ
(r)
ki − σ(r)ik )dV
is represented as an integral over the surface alone, since by virtue of Gauss's
theorem the last term in the above can be written as
∫
(σ
(r)
ki − σ(r)ik )dV = 2
∫
∂lφikldV = 2
∮
φikldfl
The definition relation of the force components Fi = ∂kσik implies that the
definition of σik is not unique, since any other tensor of the form σ¯ik =
σik+∂lχikl with χikl = −χilk would produce the same component Fi (since we
can interchange the order of differentiation). Selecting χikl = φkli+φilk−φikl
we can produce the symmetric tensor χsikl = (1/2)(φkli + φilk − φikl + φilk +
φkli − φkil) = φkli + φilk. Hence, the required symmetrical tensor σ(r)(ik) is
produced by the unsymmetrical σ
(r)
ik as below
σ
(r)
(ik) =
1
2
(σ
(r)
ik + σ
(r)
ki ) + ∂l(φkli + φilk) = −
λ
2
(nihk + nkhi)
− 1
2
(Πkl∂inl − Πil∂knl)− 1
2
∂l[(Πik + Πki)nl − Πklni − Πilnk].
(1.31)
Comparing (1.27) with the energy conservation law
14 CHAPTER 1. THE MECHANICS OF LIQUID CRYSTALS
∂
∂t
(
1
2
ρv2 + E
)
+ divQ = 0
we identify the following terms [1]
2R = σ′ikvik +Nihi −
1
T
qi∂iT
Qi =
(
W +
1
2
v2
)
vi − Πik[−vl∂lnk + Ωlinl + λnl(vkl − nknmvlm)]
+
1
2
(nihk − nkhi) + λ
2
(nihk + nkhi)− σ′ikvk − κik∂kT,
(1.32)
where σ′ik = σ
(r)
(ik) − pδik − σik is the viscous part of the stress tensor, qi =
−κik∂kT with κik being the thermal conductivity tensor andW = p+E is the
heat function. The first equation in (1.32) determines the entropy increase
due to dissipative processes (equal to the rate of dissipation of mechanical
energy). The last term relates to the increase of total entropy of the nematic
by irreversible processes of heat conduction. Also, the coefficient λ does not
feature in this relation since the corresponding term quantifies a transport
phenomenon not of a dissipative nature. The force density for a nematic
medium in motion is
Fi = −∂ip+ ∂kσ(r)(ik) + ∂kσ′ik ≡ ∂ip+ F (r)i + F ′i . (1.33)
If a medium is at rest in equilibrium, even if deformed, F = F′ = 0 and h = 0.
In that case Gi = 0 therefore F
(r)
i = −(∂iEd)ρ,S and Fi = ∂ip − (∂iEd)ρ,S =
−∂i(p + Ed) = 0, on the assumption that the elastic moduli are constant,
independent of ρ and S. From the thermodynamical definition of pressure
we have ∂ip = µ∂iρ + ρ∂iµ − ∂iE + S∂iT + T∂iS and from the familiar
1.1. NEMATICS 15
relation dE = TdS+µdρ+ (dEd)ρ,S, ∂iE = T∂iS+µ∂iρ+ (∂iEd)ρ,S we find
Fi = −∂ip− (∂iEd)ρ,S = −ρ∂iµ− S∂iT . Hence, if T =constant, the presence
of equilibrium requires µ = constant.
If one considers a closed system whose state is described by the variables
xa, a = 1, 2...N . Their equilibrium values are determined by the condition
that the entropy of the system is a maximum in statistical equilibrium. The
relevant conditions are Xa = −∂S/∂xa = 0. If the system is in a state near
to equilibrium, xa differ only slightly from their equilibrium values and Xa
are small. Processes occur that tend to restore equilibrium, hence xa will
be functions of time, with their rate of change being x˙a = −
∑
b γabXb. Os-
anger's principle states that the coefficients γab (called kinetic coefficients)
are symmetric under the exchange of the suffixes a and b. The rate of change
of entropy is S˙ =
∑
a(∂S/∂xa)x˙a = −
∑
aXax˙a =
∑
a,b γabXaXb. Consider-
ing now deformations in nematic liquid crystals, in a weak departure from
equilibrium, under the usual hydrodynamic approximation, σ′ik is a linear
function of vik ([5], [1]). The general form of such a linear dependence is
σ′ik = ηiklmvlm.
From the symmetry properties of the tensors σ′ik and vik, as well as from
the symmetry of the kinetic coefficients (here 2R/T is the rate of change of
entropy, x˙a are taken to be the components of σ
′
ik, hence the correspond-
ing conjugate variables Xa are identified with the components of the tensor
−vlm/T ) we obtain
ηiklm = ηkilm = ηikml = ηlmik.
16 CHAPTER 1. THE MECHANICS OF LIQUID CRYSTALS
The rank-four tensor ηiklm (called the viscosity tensor) is constructed only
from the unit tensor δik and the molecular director. The five independent
linear combinations of that kind are
ninknlnm, ninkδlm + nlnmδik, ninlδkm + nknlδim + ninmδkl + nknmδil,
δikδlm, δilδkm + δklδim.
(1.34)
Hence, the viscosity tensor will comprise of five independent terms. In the
following expression the above terms feature in reverse order so that the
stress tensor can be written as
σ′ik = η12vik + (η2 − η1)δikvll + (η4 + η1 − η2)(δiknlnmvlm + ninkvll)
+ (η3 − 2η1)(ninlvkl + nknlvil) + (η5 + η1 + η2 − 2η3 − 2η4)ninknlnmvlm.
(1.35)
The dissipative coefficients are presented here in a different form to the orig-
inal given by Leslie (1966) and Parodi (1970). Selecting the z− axis to be
parallel to the molecular director, we can write the dissipative function as
2R = 2η1
(
vαβ − 1
2
δαβvγγ
)2
+ η2v
2
αα
+ 2η3v
2
az + 2η4vzzvαα + η5v
2
zz +
1
T
[
κ‖(∂zT )2 + κ⊥(∂αT )2
]
+
h2
γ
,
(1.36)
where κik = diag(κ⊥, κ⊥, κ‖), α, β, γ = 1, 2 for the x and y components,
respectively. Since the rate of change of entropy must be always positive
η(1,2,3,5) > 0, κ(⊥,‖) > 0, γ > 0 and the discriminant in the trinomial η2v2αα +
2η4vzzvαα+η5v
2
zz, where vαα is considered as a variable and vzz as a parameter,
1.1. NEMATICS 17
or the reverse. Hence the condition η2η5 > η
2
4 must also be upheld. Similar
conditions are derived in [6] The number of viscosity coefficients may be
reduced if we regard the flow as incompressible, in which case div(v) =
vll = 0. Then the second term in the expression (1.35) for the stress tensor
vanishes. In the third term of the same expression, the first summand does
not contribute to dissipation since δiknlnmvlmvik = nlnmvlmvii = 0. The same
applies for the term −pδik featuring in the complete stress tensor σik and
resulting in the redefinition of the pressure regarded as an unknown function
of co-ordinates and time and not as a thermodynamic variable linked to others
through a state equation [1]. The viscous stress tensor for an incompressible
nematic fluid is the simplified as below
σ′ik = 2η1vik + (η3− 2η1)(ninlvkl +nknlvil) + (η¯3 + η1 + η2− 2η3)ninknlnmvlm,
(1.37)
containing three independent viscosity coefficients (we have defined η¯3 =
η2 + η5 − 2η4). The dissipative function is now written as
2R = 2η1
(
vαβ − 1
2
δαβvγγ
)2
+ (η2 + η5 − 2η4)v2αα
+ 2η3v
2
az +
1
T
[
κ‖(∂zT )2 + κ⊥(∂αT )2
]
+
h2
γ
= 2η1
(
vαβ − 1
2
δαβvγγ
)2
+ η¯3v
2
αα
+ 2η3v
2
az +
1
T
[
κ‖(∂zT )2 + κ⊥(∂αT )2
]
+
h2
γ
,
(1.38)
since vαα + vzz = 0.
18 CHAPTER 1. THE MECHANICS OF LIQUID CRYSTALS
1.2 Chiral nematics
As we mentioned earlier, chiral nematic liquid crystals do not have a centre
of inversion among their symmetry elements. The fact that no use of the
presence of a centre of inversion was made in deriving the equations of motion
and equilibrium for ordinary nematics, means that the general equations are
also valid for chiral nematics. However, there are a number of differences
since the free energy Fd contains a term linear in the derivatives (the pseudo-
scalar ncurln). The molecular field h must be redefined and there is now a
difference between the isothermal and adiabatic values of the modulus K2. In
what follows we consider the adiabatic elastic moduli as definite functions of ρ
and S. Finally, there is a substantial change in the hydrodynamic equations
for chiral nematics, when compared to those of nematics, in that further
terms feature in the dissipative part of the equations as follows
σ′ik = (σ
′
ik)nematic + µ1(niklm + nkilm)nm∂lT,
Ni = (Ni)nematic + ν1iklnk∂lT,
ql = (ql)nematic + ν2lkinkhi + µ2(lmink + lmkni)nmvik.
(1.39)
These terms were first introduced by Leslie [7] in an attempt to explain the
Lehmann effect (rotation of the chiral nematic structure as a consequence
of heat flow). The presence of the completely antisymmetric tensor creates
a pseudo-tensor and pseudo-vectors, removing the symmetry under spatial
inversion. Terms that are true tensors or vectors (like ni∂kT + nk∂iT ) are
precluded by virtue of the invariance under a change in the sign of n. Sim-
ilarly, a term proportional to ∂iT in Ni is impossible because it is invariant
1.2. CHIRAL NEMATICS 19
under the change in the sign of n whereas dni/dt and hence Ni would have
to change sign. The coefficients in (1.39) are connected by relations derived
by Osanger's principle. As before, S˙ = 2R/T and we identify x˙a with the
quantities σ′ik, ql and Ni. Then, the conjugate variables are the quantities
−vik/T , ∂lT/T 2 and −hi/T respectively. As seen from their positions in
(1.9), (1.10) and (1.25), the quantities σ′ik are even and the quantities qi and
Ni are odd under time reversal. If two thermodynamic variables xa and xb
have the same (opposite) parity under time reversal then the corresponding
kinetic coefficients are symmetric (antisymmetric) under exchange of the in-
dices a and b. Odd parity under time reversal signifies an odd correlation
function. Expanding the expression x˙a = −
∑
b γabXb we have

σ′ik
q1
Ni
 = −

γ11 γ12 γ13
γ21 γ22 γ23
γ31 γ32 γ33


−vik
T
∂lT
T 2
−hi
T

, (1.40)
where according to the symmetry under time reversal γ12 = −γ21 and γ23 =
γ32. Comparing with (1.39) we have [µ2(lmink + lmkni)nm]T = [µ1(niklm +
nkilm)nm]T
2 ⇒ µ2 = µ1T , since lmk = klm and −ν2lkinkT = ν1iklnkT 2 ⇒
ν2 = ν1T since lki = −ikl. The final form of (1.39) consequently reads
σ′ik = (σ
′
ik)nematic − µ1[ni(n×∇T )k + nk(n×∇T )i],
N = N
nematic
+ ν1n×∇T,
q = q
nematic
+ ν1Tn× h + 2µ1Tn× (vn),
(1.41)
20 CHAPTER 1. THE MECHANICS OF LIQUID CRYSTALS
where (vn)i = viknk.
From the equations above we note the dependence of the stress tensor and
the molecular field on the temperature gradient. This gradient (contained in
the term of the form n ×∇T ) gives rise to twisting moments acting on the
director and on the mass of the liquid crystal (responsible for the Lehmann
rotation). Conversely, the molecular field which accompanies a rotation of the
director relative to the liquid alongside the liquid velocity gradients generate
a heat flux.
Having determined the mechanical properties of nematic and chiral nematic
liquid crystals under deformations, we will now proceed to study the optical
properties of the resonators comprised of these structures in their equilibrium
configurations.
Bibliography
[1] L. D. Landau and E. M. Lifshitz, Theory of Elasticity, Vol. 7, ch. 6,
Butterworth-Heinemann, Oxford, 1999.
[2] F. C. Frank, I. Liquid crystals. On the theory of liquid crystals, Discuss.
Faraday Soc., 25, pp. 19-28 (1958).
[3] F. M. Leslie, Some constitutive equations for liquid crystals, Arch. Ro-
tation. Mech. An., 28, pp.265-283 (1968).
[4] L. D. Landau and E. M. Lifshitz, Statistical Physics, Vol. 5, ch. 2,
Butterworth-Heinemann, Oxford, 1999.
[5] L. D. Landau and E. M. Lifshitz, Fluid Mechanics, Vol. 6, ch. 6,
Pergamon Press, Oxford, 1966.
[6] D. Forster, T. C. Lubensky, P. C. Martin, J. Swift, and P. S.
Pershan, Hydrodynamics of Liquid Crystals, Phys. Rev. Lett.26, pp.
1016-1019 (1971).
[7] F. M. Leslie, Some Thermal Effects in Cholesteric Liquid Crystals,
Proc. R. Soc. Lond. A, 307, pp. 359-372 (1968).
21
Chapter 2
Optical properties of LC cells
In this chapter we will review the basic optical properties of nematics and
chiral nematics with a particular emphasis on formulating boundary value
problems. We will consider the viewpoints of various workers analysing op-
tical problems in anisotropic media, and we will aim to demonstrate a con-
vergence of their methods regarding the nature of optical feedback in these
periodic media.
2.1 The Belyakov formulation
2.1.1 Kinematical approximation and exact solution
The eigenwaves corresponding to light propagation along the helix axis are
solutions of the wave equation
∂2E
∂z2
=
ε(z)
c2
∂2E
∂t2
. (2.1)
The solutions of the above are sought not in the form of a plane wave, as in
22
2.1. THE BELYAKOV FORMULATION 23
a homogeneous medium, but in the form of a Bloch wave
†
E(z, t) =
∑
s
Es exp[i(K + sτ)z − iωt]
In the general case of a medium with one dimensional periodicity, an infinite
number of amplitudes Es are non zero. In a chiral nematic medium the
dielectric tensor assumes the form
ε(z) =

ε¯[1 + δ cos(τz)] ±ε¯δ sin(τz) 0
±ε¯δ sin(τz) ε¯[1 + δ cos(τz)] 0
0 0 ε⊥
 . (2.2)
(the two signs in the components of the above tensor correspond to left and
right handed chirality of the helical structure), which can be also expressed
in terms of Fourier series as
ε(r) =
∑
s=0,±1
εs exp(isτ · r).
The Fourier coefficients are written as
ε0 =

ε¯ 0 0
0 ε¯ 0
0 0 ε⊥
 , ε1 = ε∗−1 = ε¯δ2

1 ∓i 0
∓i −1 0
0 0 0
 ,
where ε¯ =
ε‖ + ε⊥
2
is the mean dielectric constant, δ =
ε‖ − ε⊥
ε‖ + ε⊥
is the dielec-
tric anisotropy, and k0, e0, k1, e1 are the wave and polarization vectors for
the incident and scattered waves respectively.
†
In relation with the previous chapter, we point out that τ = 2q0.
24 CHAPTER 2. OPTICAL PROPERTIES OF LC CELLS
It is instructive to invoke the kinematical approximation for scattering of
light in chiral nematic media, according to which the scattering cross-section
for a sample is [1]
dσ(k0, e0; k1, e1)
dΩk1
=
(
ω2
4pic2
)2 ∣∣∣∣∫ [e†1 (ε− ε¯I) e0] exp[i(k0 − k1) · r]dr∣∣∣∣2 ,
(2.3)
where
τ =
2pi
(p/2)
=
4pi
p
is the reciprocal lattice vector of the chiral nematic. By virtue of the specific
form of the dielectric tensor given above, rearranging (2.3) yields
dσ(k0, e0; k1, e1)
dΩk1
=
(
ω2
4pic2
)2 ∣∣∣∣∣∑
s
e†1εse0
∫
exp[i(k0 − k1) · r + sτ · r]dr
∣∣∣∣∣
2
.
(2.4)
In the limit of an infinite sample, the integral in the above relation is pro-
portional to a delta function δ(k0 − k1 + sτ ) and the scattering directions
are given by the kinematical Bragg condition
sin θ =
sλ
p
It is evident from (2.4) that only first order Bragg-reflection occurs. In an
analogous fashion to X-ray diffraction, the structure amplitude role is played
by the quantity
F (k0, e0; k1, e1) = e
†
1εse0
which is polarisation dependent and describes the scattering amplitude from
a chiral nematic layer with thickness equal to a half helical pitch [1].
2.1. THE BELYAKOV FORMULATION 25
The two waves are written as a superposition of two circularly polarized plane
waves of the form
E(z, t) = exp(−iωt)[E+n+ exp(iK+z) + E−n− exp(iK−z)], (2.5)
with n± = 1√2(x + iy) being the two unit circular polarization vectors. The
wave-vectors satisfy the Bragg condition K+−K− = τ alongside the secular
equation produced by substituting (2.5) into (2.1). The substitution yields
the system of equations
[1− (η
+)2]E+ + δE− = 0,
[1− (η−)2]E− + δE+ = 0,
(2.6)
where
η± =
K±
κ
, κ = (ω/c)ε¯.
The compatibility condition generates the secular equation
[
1− (η+)2] [1− (η−)2]− δ2 = 0.
The eigenvalues and ratios of values of eigenvectors, respectively, read
K+j =
τ
2
± k±, (2.7)
with
k± = κ
√
1 +
( τ
2κ
)2
±
√(τ
κ
)2
+ δ2
and
26 CHAPTER 2. OPTICAL PROPERTIES OF LC CELLS
ξj =
(
E−
E+
)
j
= δ[(η−j )
2]−1. (2.8)
The eigensolutions are numbered in the following way: j = 1, 4 for signs +
and - before k+ in (2.7) and j = 2, 3 for signs + and - before k− in (2.7).
Depending on the sign of K+ and K− in equation (2.5), there are either two
waves with circular polarizations of opposite handedness propagating in the
same direction (corresponding to same signs of K+ and K−) or two waves
with circular polarization of the same handedness propagating in opposite
directions (corresponding to opposite signs of K+ and K−). Changes in the
sign of the wave-vector in the phase vector of a circular wave is equivalent
to the simultaneous change of the propagation direction and handedness
of the circular polarization of the wave. The existence of eigensolutions
where the constituent plane wave solutions propagate in opposite directions
is indicative of reflection from the spatial structure of the chiral nematic.
From the defining equations for the wave-vectors we deduce that the pair
of solutions (1,4) containing k+ corresponds to two non-diffracting modes
propagating in opposite directions, while the pair (2,3) containing k− relates
to modes experiencing diffraction in the chiral nematic.
For small values of the dielectric anisotropy δ (assume δ  1) we will now
focus on the frequency dependence of the eigenvalues. For the solution 1,
K+ and K− have the same signs for all frequencies whereas for the ratio
of amplitudes we have ξ1 ∼ δ except for the high frequency region ω/c 
τ/δ where ξ1 → 1. Similar conclusions hold for solution 4 describing a
wave propagating in the opposite direction. The wave-vector in this limit
is κ
√
1 + δ ' κ(1 + δ/2) The dispersion relation for solutions 1 and 4 is
2.1. THE BELYAKOV FORMULATION 27
linear, resembling that of a wave propagating in a homogeneous medium.
Solution 1, therefore, corresponds to a circularly polarized wave rotating
with the opposite handedness from that of the chiral nematic helix, within
the accuracy of the small dielectric anisotropy δ, for all frequencies outside
the region ω/c > τ/δ. For ω/c τ/δ the solution corresponds to a linearly
polarized standing wave with rotating plane of polarization matching the
handedness of the chiral nematic helix. Depending on the frequency, the wave
vectors K+ and K− maybe either real with the same signs, real with opposite
signs or complex quantities in the region ωB/
√
1 + δ < ω < ωB/
√
1− δ,
where ωB = τc/(2
√
ε¯) denotes the center of the Bragg band (called also
hereinafter forbidden band, stop band, photonic band-gap, or simply band-
gap). Hence, no wave can propagate in this frequency range if their circular
polarization matches the handedness of the chiral nematic helix. The above
property is defined as selective reflection In the forbidden band, |ξ2,3| = 1
independently of δ. Outside the band-gap, |ξ2,3| ∼ δ except in the high
frequency limit. Consequently, solution 2 corresponds to a wave circularly
polarised in the sense of the helix. In the limit of very short wavelengths that
we have considered above, K− ' κ(1 − δ/2). Solution 3 exhibits the same
behaviour, with the difference that the corresponding wave propagate in the
opposite direction outside the forbidden region. We should remark here that
only first order reflection is possible, since |s| = 1 in the Fourier components
of the dielectric tensor.
28 CHAPTER 2. OPTICAL PROPERTIES OF LC CELLS
2.1.2 Formation of the Boundary Value Problem
Having obtained the eigensolutions, we now proceed to address the problem
of reflection and transmission for a chiral nematic sample of finite thickness.
We will consider initially the case of normal incidence onto a plane parallel
sample of thickness L with the helical axis (called also optical axis) normal to
the plate surface. The boundary conditions requiring continuous tangential
electric and magnetic field components at the boundary reduce to continu-
ous electric and magnetic fields on the sample surfaces, since the waves are
transverse. Hence, the boundary conditions attain the form

Ei(0) + Er(0) = E(0)
Et(L) = E(L)
{curl[Ei(z) + Er(z)]}z=0 = [curlE(z)]z=0
[curlEt(z)]z=L = [curlE(z)]z=L.
(2.9)
The electric fields of the incident (Ei), reflected (Er),transmitted (Et) waves
as well as the wave propagating inside the sample are given by the following
relations.
Ei(z, t) = exp[i(qiz − ωt)](E+i n+ + E−i n−),
Er(z, t) = exp[−i(qiz + ωt)](E+r n+ + E−r n−),
Et(z, t) = exp[i(qiz − ωt)](E+t n+ + E−t n−),
E = exp(−iωt)
4∑
j=1
E+j [exp(iK
+
j z)n+ + ξj exp(iK
−
j z)n−],
(2.10)
where qi = ω
√
ε/c is the wave-vector outside the chiral nematic. When waves
2.1. THE BELYAKOV FORMULATION 29
propagate in the positive (negative) direction of the z-axis, the unit vectors
n+ and n− describe left (right) and right (left)-handed circular polarizations,
respectively. Substituting (2.10) into the boundary conditions we obtain the
following system for the amplitudes of the eigenwaves
∑
j
(1 + η+j )E
+
j = 2E
+
i ,∑
j
exp(iK+j L)(1− η+j )E+j = 0,∑
j
ξj(1 + η
−
j )E
+
j = 2E
−
i ,∑
j
ξj exp(iK
−
j L)(1− η−j )E+j = 0.
(2.11)
On determining these we can then find the reflected and transmitted wave
amplitudes as follows
E+r =
1
2
∑
j
ξj(1− η−j )E+j ,
E+t =
1
2
∑
j
exp[i(K+j − qi)L](1 + η+j )E+j ,
E−r =
1
2
∑
j
(1− η+j )E+j ,
E−t =
1
2
∑
j
ξj exp[i(K
−
j − qi)L](1 + η−j )E+j .
(2.12)
If light is incident from the outside half-space onto the half-space filled with
a chiral nematic medium, then only two eigensolutions can be excited in the
sample corresponding to propagation into the depth of the chiral nematic
layer, i.e. solutions 1 and 2 (assuming a small imaginary part in ε¯ the fields of
solutions 3 and 4 would increase infinitely towards the direction of increasing
30 CHAPTER 2. OPTICAL PROPERTIES OF LC CELLS
depth).
Extending our treatment to samples that are not thick (δL/p ∼ 1) we note
that if we neglect light reflection at the CLC boundaries assuming that the
dielectric constant of the ambient space is equal to ε¯ up to the small dielec-
tric anisotropy, then three eigensolutions are excited: two correspond to the
diffracting wave and one one to the non-diffracting wave propagating along
the direction of incidence. Hence the problem reduces into solving a sys-
tem of three linear equations. Furthermore, the assumption of no boundary
reflection allows the separation of the eigensolutions such that the incident
wave with diffracting circular polarization excites only solutions 2 and 3,
while the wave with the non-diffracting polarization excites solution 1 only.
Neglecting boundary reflection introduces an error of the order of δ, consis-
tent with retaining only terms of the lowest order in δ in the equations for
the amplitudes of the diffracting eigensolutions 2 and 3.
Hereinafter, we will consider only the diffracting eigenwaves and replace the
subscript index 2 (3) by +(-). Then, after assuming that η++ = η
+
− = 1 and
η−+ = η
+
+ = −1, the last two equations of the system (2.11) take the form
E++ + E
+
− = E
+
i ,
ξ+ exp(iK
+
+L)E
+
+ + ξ− exp(iK
+
−L)E
−
+ = 0.
(2.13)
From the first two equations of (2.12) we obtain the expressions for the
reflected and the transmitted wave of the diffracting polarization
E+r = ξ+E
+
+ + ξ−E
+
− ,
E+t = exp[i(K
+
+ − qi)L]E++ + exp[i(K+− − qi)L]E−+ .
(2.14)
2.1. THE BELYAKOV FORMULATION 31
Solving system (2.13) for the amplitudes of the diffracting eigensolutions, and
replacing in the expression for E+r and E
+
t above, we obtain the following
expressions for the complex amplitude reflection and transmission coefficient
r =
E+r
Ee
=
iδ sin(k−L)(
k−τ
q2
)
+ i
[(
τ
2q
)2
+
(
k−
q
)2
− 1
] ,
t =
E+t
Ee
=
exp(iqL)
(
k−τ
q2
)
(
k−τ
q2
)
+ i
[(
τ
2q
)2
+
(
k−
q
)2
− 1
] .
(2.15)
The squared modulus of the above quantities yields what is usually called the
(intensity) reflection and transmission coefficients for the diffracting eigen-
wave, denoted by R+ and T+ respectively.
R+ =
δ2 sin(k−L)2(
k−τ
q2
)2
+
[(
τ
2q
)2
+
(
k−
q
)2
− 1
]2 ,
T+ =
∣∣∣∣exp(iqL)(k−τq2
)∣∣∣∣2(
k−τ
q2
)2
+
[(
τ
2q
)2
+
(
k−
q
)2
− 1
]2 .
(2.16)
The secular equation reads
tan(k−L) =
ik−
τ
q2(
τ
2q
)2
+
(
k−
q
)2
− 1
(2.17)
The solutions of the above transcendental equation can be written in the
form ωEM = ω
0
EM(1 + i∆). For a sufficiently small ∆ and under the condi-
32 CHAPTER 2. OPTICAL PROPERTIES OF LC CELLS
tion =(Lk−)  1 the values ω0EM coincide with the zeros of the reflection
coefficient R of a non-absorbing chiral nematic sample. The real and imagi-
nary part of the eigenfrequencies are determined by the conditions
k−L = npi, ∆ = −2
5δ(npi)2
(Lδτ)3
. (2.18)
Also
E(ωn, z, t) = i exp(−iωnt)
{
n+ exp
(
iτz
2
)
sin
(npiz
L
)
+
1
δ
n− exp
(−iτz
2
)[((
τ
2q
)2
+
(
npi
Lq
)2
− 1
)
sin
(npiz
L
)
− iτnpi
Lq2
cos
(npiz
L
)]}
.
(2.19)
For a non-absorbing CLC, the only source of the edge-mode amplitude decay
is energy leakage through the sample surfaces. Applying relation (2.19) for
z = 0, L we obtain
Eout =
τnpi
q2Lδ
for the leaking wave amplitude at the sample surfaces. The decrease in the
edge-mode electromagnetic energy per unit time is equal to the energy flow
of the outgoing waves given by the Poynting vector and is proportional to
E · E∗zˆ. The edge-mode lifetime can therefore be obtained as
τm =
∫ L
0
|E(ωEM , z, t)|2 dz
d
dt
∫ L
0
|E(ωEM , z, t)|2dz
∝ L
(
Lδ
pn
)2
The amplitudes of the eigenwaves excited in the structure read [2]
2.1. THE BELYAKOV FORMULATION 33
E++ = −Ei exp(−ik−L)
(
τ
2q
)2
+
(
k−
q
)2
− k
−τ
q2
− 1
2
{
k−τ
q2
cos(k−L) + i
[(
τ
2q
)2
+
(
k−
q
)2
− 1
]
sin(k−L)
} ,
(2.20)
E++ = Ei exp(ik
−L)
(
τ
2q
)2
+
(
k−
q
)2
+
k−τ
q2
− 1
2
{
k−τ
q2
cos(k−L) + i
[(
τ
2q
)2
+
(
k−
q
)2
− 1
]
sin(k−L)
} .
(2.21)
Solving the inhomogeneous system for an excitation with a frequency close
to that of an edge-mode, we obtain
E
EM(+)
± = ±E++ [1 + tan(k−L)],
where EEM± satisfy the homogeneous system. We can immediately see that
E
EM(+)
+
E
EM(+)
−
= −1.
For a thick chiral nematic layer, it can be shown that the amplitudes E+± are
very good approximations to EEM± .
In the presence of absorption the mean dielectric constant is written as  =
0(1+ iγ), where γ  1 in most situations. Under the condition =(Lk−) 1
mentioned above, the reflection and transmission coefficients at the reflection
minima read [2]
34 CHAPTER 2. OPTICAL PROPERTIES OF LC CELLS
R =
(a3γ)2
[(npi)2 + (a3γ)2]2
,
T =
(npi)4
[(npi)2 + (a3γ)2]2
.
(2.22)
Then, the absorption of light from the chiral nematic structure can be written
as
Γ = 1−R− T = 2(npi)
2a3γ
[(npi)2 + a3γ]2
,
with a =
τLδ
4
. The absorption attains the maximum value of Γ = 1/2 for
(npi)2 = a3γ. Under the presence of amplification, with γ < 0, |γ|  1, rela-
tions (2.22) hold, but now the coefficients become divergent for γ = −(npi)
2
a3
.
These values correspond to the lasing threshold amplification values for the
edge-modes numbered by n. As it can be deduced from the above relation,
the threshold values of |γ| are inversely proportional to the third power of the
chiral nematic layer thickness. The minimum threshold value corresponds to
the dominant edge-mode with n = 1. Thus, as a zero approximation to
the numerical solution of the solvability condition, the frequency of the first
maximum for enhanced absorption Γ as well as for the first point where Γ
diverges, coincide with that of the first zero of the reflection coefficient. This
is impractical for a collinear pumping configuration, unless the pumping fre-
quency coincides with the high-frequency edge of the reflection band and the
lasing frequency with the low frequency edge.
2.2. THE DE VRIES FORMULATION 35
2.2 The de Vries formulation
We will now summarize the main results of the de Vries theory for the optical
properties of chiral nematic layers, as presented in his seminal monograph
published in 1951. De Vries[3] transformed Maxwell's equations from the
Cartesian into a rotating co-ordinate system, whose unit vectors are defined
by
ξˆ = cos
(τz
2
)
xˆ + sin
(τz
2
)
yˆ
ηˆ = − sin
(τz
2
)
xˆ + cos
(τz
2
)
yˆ
ζˆ = zˆ,
(2.23)
where the vector ηˆ is identical to the molecular director. Applying Faraday-
Maxwell and Ampère-Maxwell law for an non-magnetic anisotropic medium
(in our case uniaxial) we obtain
curl(curlE) = −µ0 ∂
∂t
(curlH) = −µ0ε0 ∂
2
∂t2
D
where it is understood that the dielectric tensor contains only relative (di-
mensionless)dielectric constants. If we assume a transverse field E = E(z)
then for the dielectric tensor (2.2) we have div(E) = 0 and we can thus write
in Cartesian co-ordinates
1
c2
∂2Dx
∂t2
=
∂2Ex
∂z2
,
1
c2
∂2Dy
∂t2
=
∂2Ey
∂z2
,
Ez = 0, Dz = 0.
(2.24)
36 CHAPTER 2. OPTICAL PROPERTIES OF LC CELLS
In the rotating co-ordinate system we can write Dξ = ε⊥Eξ and Dη = ε‖Eη.
The fields are transformed between the two frames according to the relation
Ex
Ey
 = R−1 (τz
2
)Eξ
Eη

where R
(τz
2
)
is the rotation matrix corresponding to a rotation of the axes
by τz/2. Equations (2.24), then, acquire the form
ε⊥
c2
∂2Eξ
∂t2
=
∂2Eξ
∂z2
− τ ∂Eη
∂z
− τ
2
4
Eξ
ε‖
c2
∂2Eη
∂t2
=
∂2Eη
∂z2
− τ ∂Eξ
∂z
− τ
2
4
Eη
(2.25)
A trial solution representing an elliptically polarized wave of the form
Eξ
Eη
 = exp [2pii( t
T
− mz
λ
)]A
iB

produces the system of equations

ε⊥
c2
A = m
2
λ2
A+ 2Bm
λp
+
A
p2
ε‖
c2
B = m
2
λ2
B + 2Am
λp
+
B
p2
(2.26)
The compatibility condition of the above system of equations yields an alge-
braic equation for the `effective refractive index', m, which encompasses the
characteristics of propagation in the chiral nematic medium. The equation
reads
m4 −m2
(
ε⊥ + ε‖ + 2
λ2
p2
)
+
(
ε⊥ − λ
2
p2
)(
ε‖ − λ
2
p2
)
= 0 (2.27)
2.2. THE DE VRIES FORMULATION 37
As expected, this equation is symmetric in the interchange of ε⊥ and ε‖.
De Vries introduced the reduced quantities λ′ = λ/(p
√
ε¯) and m′ = m/
√
ε¯.
Then, the eigenvalue equation (2.27) can be written as
m′4 −m′2(1 + λ′2) + (1− δ − λ′2)(1 + δ − λ′2) = 0 (2.28)
From the eigenvectors, one obtains the degree of ellipticity", f , as below
f =
B
A =
1− δ −m′2 − λ′2
2m′λ′
(2.29)
As the quartic eigenvalue equation has two roots, the ellipticity will also be
given by two expressions with very different behaviour, corresponding to the
diffracting and the non-diffracting eigenwave. The roots of the characteristic
algebraic equation can be readily expressed as
m2± = 1 + λ
′2 ±
√
4λ′2 + δ2 (2.30)
As we can immediately deduce, the solution m− becomes imaginary for |λ′2−
1| < δ. This inequality defines the Bragg-band, the same range for which
k− is imaginary, when the eigenvalue equation is produced in the Cartesian
frame of co-ordinates. In that range of wavelengths, having m′− = −iµ, the
eigenwave can be written as
Eξ
Eη
 = exp(−2piµz
λ
)
cos
(
2pit
T
)A−
iB−

The above expression corresponds to a linearly polarized wave in the ξ − η
frame. At the edges of the Bragg-band, when λ′ → √1± δ, the ellipticity
38 CHAPTER 2. OPTICAL PROPERTIES OF LC CELLS
diverges to 0 (+∞), corresponding to a linearly polarized wave being perpen-
dicular to (or, respectively, following) the molecular director. The magnetic
field will be found with the help of Ampère-Maxwell law, according to which,
in the Cartesian frame, we have
∂Bx
∂t
=
∂Ey
∂z
,
∂By
∂t
= −∂Ex
∂z
(2.31)
The above can be written as
∂
∂t
R(−τz
2
)Bξ
Bη
 = zˆ× ∂
∂z
R(−τz
2
)Eξ
Eη

(2.32)
so that
Bξ
Bη
 = R(τz
2
)∫ ∂
∂z
R(−τz
2
) Eη
−Eξ
 dt = −i T
2pi

Eξ
τ
2
+
∂Eη
∂z
Eη
τ
2
− ∂Eξ
∂z

(2.33)
given that ξˆ × ηˆ = zˆ. Finally
Bξ
Bη
 = A
c
(−i)
(
λ
p
+mf
)
m+
λ
p
f
 exp [2pii( tT − mzλ
)]
, (2.34)
where c is the speed of light in the vacuum. As explained in [3] the quantity
r = m′+λ′f plays the role of the normal refractive index, as it is nearly unity
apart from the region where λ′ ' 1. Furthermore, from the system (2.26)
defining the eigenvectors, we also obtain
2.2. THE DE VRIES FORMULATION 39
1
f
=
1 + δ −m′2 − λ′2
2m′λ′
=
2m′λ′
1− δ −m′2 − λ′2
and hence we can write
(λ′ +m′f)(λ′f +m′) = f
(
λ′2 +
m′λ′
f
+m′λ′f +m′2
)
= f, (2.35)
after substituting the two equivalent expressions for the ellipticity. After the
above considerations, the magnetic field component can be written as
Bξ = −
√
ε¯
rc
Eη, Bη =
r
c
√
ε¯Eξ
We can see once more that the parameter r (through m and f) encompasses
the optical transmission properties of the chiral nematic in our consideration.
In an isotropic medium r = 1; this value is approached by the non-diffracting
eigenwave, and by the diffracting one in the wavelength region considerably
far from the stop-band. De Vries argues that the negative sign should be
chosen for m− in the region λ′ >
√
1 + δ , as follows. The time averaged
Poynting vector in the rotating co-ordinate system is given by the relation
〈S〉 = µ0
2c
<{E×B∗} = µ0
2c
<{EξB∗η − EηB∗ξ}zˆ
=
µ0
2c
<
{√
ε¯
(
r∗A2 + 1
r∗
B2
)}
zˆ.
(2.36)
Outside the Bragg band r ∈ R, hence the sign of the Poynting vector (and
hence the direction of power flow) will be determined by r. In the region
λ′ >
√
1 + δ when m < 0 then f > 0, since the numerator is always negative.
In addition, |λ′f | > |m| since λ
′2 +m′2 + δ − 1
2|m′| > |m
′| ⇒ m′2 < λ′2 + δ − 1.
40 CHAPTER 2. OPTICAL PROPERTIES OF LC CELLS
On the other hand, by the definition of m− we have m2 < 1 + λ′2 − 2λ′ + δ,
since for a, b ∈ R+, a > b, √a2 + b2 > √(a− b)2. It follows then that
m2 < (λ′−1)2+δ < (λ′−1)(λ′+1)+δ = λ′2−1+δ, as required. From the above
argumentation, we infer that m must be negative in that region, otherwise r
would be negative, resulting in a power flow in the opposite (−z) direction.
We should also notice that since m = m− is imaginary in the stop-band, f
and r are also imaginary and as a consequence
(
r∗A2 + 1
r∗
B2
)
∈ I. This
result comes as well to verify that there is no propagation of electromagnetic
power inside the band-gap, as anticipated.
We will now formulate a boundary value problem in exactly the same fash-
ion as in [1],[2], but for the solution of Maxwell's equations in a rotating
co-ordinate system inside the chiral nematic, in order to obtain an expres-
sion for the reflection and the transmission coefficient of the structure. The
components of the incident and reflected waves propagating in the medium
outside the liquid crystal cell (assumed to have a refractive index of n0 =
√
ε¯)
read
Ei(r)x
E
i(r)
y
 =
 ei(r)x
ie
i(r)
y
 exp [2pii( t
T
− mz
λ
)]
, (2.37)
B
i(r)
x[y] = ∓(±)
√
ε¯
c
E
i(r)
y[x],
where it is understood that the sign choice follows the field component and
the parenthesis follow the type of wave (incident or reflected). In the above
relations we have assumed that the refractive index of the medium surround-
ing the crystal is the same as the the root of its mean dielectric constant,
in order to avoid boundary reflection and hence conversion of the eigenwave
2.2. THE DE VRIES FORMULATION 41
from one type to another (the reader should have been already familiar with
that assumption). Imposing the boundary conditions for continuity of the
tangential components of the electric and the magnetic fields at the interface
z = 0, where ξˆ = xˆ and ζˆ = yˆ we have
eix + e
r
x = A, eiy + ery = B = fA,
eix − erx = rA, eiy − ery =
f
r
A.
(2.38)
The first two equations derive from equating the x and y components of
the electric fields, respectively, whereas the relations in the last line derive
from equating the y and x components of the magnetic field. From the four
relations above, the following amplitudes and ratios can be deduced.
2eix = A(1 + r), 2eiy =
Af(1 + r)
r
,
2erx = A(1− r), 2eiy = −
Af(1− r)
r
,
(2.39)
and
eix
eiy
= −e
r
x
ery
= λ′ +
m′
f
,
r =
erx
eix
= −e
r
y
eiy
=
1 + r
1− r .
(2.40)
In our analysis it is assumed that only the eigenwave with the diffracting
polarization results as a result of the incidence of the elliptically polarized
wave Ei. It is exactly the same assumption that we met in (2.13) and (2.14)
when the summation index assumed values j = 2, 3, again corresponding to
the diffracting polarization. The difference lies on the fact that the ratio of
the component amplitudes of the incident wave, generating the diffracting
eigenwave, is fixed in Belyakov's approach (and equal to 1 in modulus), in de
42 CHAPTER 2. OPTICAL PROPERTIES OF LC CELLS
Vries it is a function of the wavelength and equal to f/r. This ratio, however
is close to unity for the wavelength region of interest (approximately equal to
1 + δ/(2λ)). On reflection from the boundary crystal-surrounding medium,
the sign of m, r, and consequently of f changes (consistent with having the
power flux in the −z direction). At z = L = Np we have also ξˆ = xˆ and
ζˆ = yˆ. Applying the continuity boundary conditions we obtain
A exp
(
−2piimL
λ
)
+A′ exp
(
+
2piimL
λ
)
= etx exp
(
−2pii
√
ε¯L
λ
)
,
f
[
A exp
(
−2piimL
λ
)
+A′ exp
(
+
2piimL
λ
)]
= ety exp
(
−2pii
√
ε¯L
λ
)
,
r
[
A exp
(
−2piimL
λ
)
−A′ exp
(
+
2piimL
λ
)]
= etx exp
(
−2pii
√
ε¯L
λ
)
,
f
r
[
A exp
(
−2piimL
λ
)
−A′ exp
(
+
2piimL
λ
)]
= ety exp
(
−2pii
√
ε¯L
λ
)
.
(2.41)
Incorporating the phase factors in redefined complex amplitudes, as
C = A exp
(
−2piimL
λ
)
, C ′ = A′ exp
(
+
2piimL
λ
)
,
Tx(y) = etx(y) exp
(
−2pii
√
ε¯L
λ
) (2.42)
we obtain
C ′
C =
r− 1
r + 1
,
Tx
C =
2r
r + 1
,
Ty
C =
2f
r + 1
. (2.43)
Let us assume a wave of the diffracting circular polarization with complex
amplitude of a given modulus. We will first consider the x component. The
amplitude of the first reflected wave from the interface reads eix(1−r)/(1+r).
2.2. THE DE VRIES FORMULATION 43
The amplitude of the eigenwave then will be A = 2/(r+1). At the boundary
z = L, the complex amplitude will have accumulated a phase factor s =
exp (2piim′
√
ε¯L/λ). This phase factor is a measure of the absorption when
m− is imaginary. On reflection, the amplitude will now be sA(r− 1)/(r+ 1)
and at the first interface, s2A(r− 1)/(r + 1). After N boundary reflections,
the amplitude reflection coefficient reads
R = e
r
total
eix
=
1− r
(1 + r)
− 1− r
(1 + r)
2r
(1 + r)
2
(1 + r)
s2
− 1− r
(1 + r)
2r
(1 + r)
2
(1 + r)
(
1− r
1 + r
)2
s4 − ...
=
1− r
1 + r
(
1− 4r
(1 + r)2
s2
N∑
k=0
vk
)
,
(2.44)
where v =
(
s2(1− r)
1 + r
)2
.
In the limit N →∞, the above sum converges since |v| < 1. The final result
is
R = 1− r
1 + r
(
1− 4r
(1 + r)2
s2
1
1− v
)
=
1− r
1 + r
(
1− 4rs
2
(1 + r)2 − s2(1− r)2
)
=
1− r
(1 + r)
(1 + r)2(1− s2)
(1 + r)2 − s2(1− r)2 =
(1− r2)(1− s2)
(1 + r)2 − s2(1− r)2
(2.45)
We note that this formula is immediately comparable with the standard result
derived for the amplitude reflection coefficient of a Fabry-Pérot resonator.
RFP = r + tt′r′ exp (2iδ)
1− r′2 exp (2iδ) =
r[1− exp(2iδ)]
1− r2 exp (2iδ) , (2.46)
44 CHAPTER 2. OPTICAL PROPERTIES OF LC CELLS
where we identify the parameters s = exp(iδ), r = (1 − r)/(1 + r), r′ = −r,
t = 2/(1 + r) and t′ = 2r/(1 + r). We also note that the familiar condition
r2 + tt′ = 1 is satisfied. Had we considered the component ey, then the
expressions for t and t′ would have been interchanged, as well as the ones for
r and r′ (since f changes sign for the reflected wave), leaving the expression
for R unaltered apart from a change of sign. This is to be expected, as upon
consideration of a circularly polarized wave with the diffracting polarization,
the reflected wave is also of the same polarization and propagates in the
opposite direction, hence the components of the Stokes vector acquire a phase
difference of pi. Using the same correspondence, the amplitude transmission
coefficient, then, is readily available as
T = tt
′ exp (iδ)
1− r2 exp (2iδ) =
4rs
(1 + r)2 − s2(1− r)2 . (2.47)
Outside the region of total reflection s2 = cos(2δ)+i sin(2δ) and the intensity
reflection coefficient reads
R = |R|2 = (1− r
2)2 sin2 δ
(1− r2)2 sin2 δ + 4r2 . (2.48)
Inside the stop-band s is real and r2 becomes imaginary. The reflection
coefficient, then, reads
Rb = |Rb|2 = (1− r
2)2(1− s2)2
(1− r2)2(1− s2)2 − 16s2r2 . (2.49)
2.3. OBLIQUE INCIDENCE 45
2.3 Oblique incidence
We will now outline the main results of dynamic theory of light scattering
in chiral nematic liquid crystal samples pertaining to oblique propagation
of the eigenwaves, as detailed in [1]. Oblique incidence and propagation
presents a number of differences from the case of normal incidence: there
are higher order reflections with corresponding frequency bands centered at
multiples of the Bragg frequency, there exist frequency regions in which every
polarization is reflected and the polarization properties are in general more
complicated, affected both by diffractive scattering and birefringence. In the
two-wave approximation of the dynamic diffraction theory, one assumes two
eigenwaves with wave-vectors k0 and k1 = k0 + τ . The total field inside the
chiral nematic reads E(r, t) = [E0 exp (ik0 · r) + E1 exp (ik1 · r)] exp (−iωt).
Then, Maxwell's equations c2curl(curlE) = −εˆ(r)∂2E/∂t2 reduce in the
system

(
1− k
2
0
κ2
)
E0 +
εˆτ√
ε¯
E1 = 0
εˆ−τ√
ε¯
E0 +
(
1− k
2
1
κ2
)
E1 = 0,
(2.50)
where εˆτ,−τ are the Fourier components of the dielectric tensor as defined
previously and
κ =
ω
c
√
ε¯
(
1− δ
2
cos2 θ
)
(2.51)
is the mean value of the wave-vector propagating at an angle pi/2 − θ to
the optical axis (coinciding here with the z-axis). Analyzing the fields in
components along the σ axis (perpendicular to the plane of incidence) and
46 CHAPTER 2. OPTICAL PROPERTIES OF LC CELLS
two axes pi, pi1 perpendicular to the direction of propagation of the two waves
in the plane of incidence results in the following four equations
(
1− k
2
0
κ2
+
δ
2
cos2 θ
)
Eσ0 −
δ
2
Eσ1 +
iδ
2
sin θEpi1 = 0,(
1− k
2
0
κ2
− δ
2
cos2 θ
)
Eσ0 −
δ
2
sin2 θEpi1 −
iδ
2
sinθEσ1 = 0,
−δ
2
Eσ0 +
iδ
2
sin θ +
(
1− k
2
0
κ2
+
δ
2
cos2 θ
)
Eσ1 = 0,
iδ
2
sin θEσ0 −
δ
2
sin2 θEpi0 +
(
1− k
2
0
κ2
− δ
2
cos2 θ
)
Epi1 = 0.
(2.52)
Equating the determinant of the system matrix to zero yields the following
equation
u4 − (2∆2 + 2n2 − 1)u2 − (2n2 + 1)∆2 + 2n2∆ = 0, (2.53)
in which
u =
k21 − k20
κ2δ(1 + sin2 θ)
, n =
cos2 θ
1 + sin2 θ
, ∆ =
2q2 − k20 − k21
κ2δ(1 + sin2 θ)
. (2.54)
As shown in [1], the parameter ∆ measures the deviation of the angle of
incidence (and propagation) from the angle dictated by the Bragg condition.
This constant can be recast in the more instructive form
∆ =
2 sin θ(2κ sin θ − τ)
κδ(1 + sin2 θ)
. (2.55)
The non-trivial case where ∆ = 0 coincides within an accuracy of the order
of δ to the Bragg condition sin θ = λ/p. Equation (2.53) has four solutions
2.3. OBLIQUE INCIDENCE 47
uj = ±
√√√√∆2 + n2 − 1
2
±
√(
∆2 + n2 − 1
2
)
+ (2n2 + 1)∆2 − 2n2∆−∆4,
(2.56)
which are enumerated as in the case of normal propagation: j = 1, 4 for the
sign “ + ” in front of the square brackets, and signs “+,−” in front of the
braces, respectively; j = 2, 3 for the sign “−” in front of the square brackets,
and signs “+,−" in front of the braces, respectively. The eigensolutions of
(2.52) are then written in the form
E(r, t) = [E0j exp (ik0j · r) + E1j exp (ik1j · r)] exp (−iωt), (2.57)
with
E0j = E
σ
0jσˆ + E
pi
0jpˆi0,
E1j = E
σ
1jσˆ + E
pi1
1j pˆi1
(2.58)
and
Eσ0j = a1j = (∆− n+ uj)[(∆− uj)2 − n2],
Epi0j = a2j = −i sin θ(∆ + n+ uj)[(∆− uj)2 − n2],
Eσ1j = a3j = (∆− n− uj)(∆ + uj − n2),
Eσ1j = a4j = −i sin θ(∆ + n− uj)(∆ + uj − n2),
k0j = q + qδ
∆ + n+ uj
2(1 +m) sin θ
zˆ, k1j = k0j + τ,
(2.59)
where q makes an angle pi/2−θ with the z-axis. As we can see, the eigensolu-
tions are superpositions of two waves with elliptical polarization in contrast
48 CHAPTER 2. OPTICAL PROPERTIES OF LC CELLS
to normal incidence, where the eigenpolarizations are circular.
We will now formulate once more a boundary value problem whereby an
elliptically polarized wave
Ei(r, t) = Ee exp [i(q · r− ωt)]eˆα,β, (2.60)
with
eˆα,β = cosασˆ + sinα exp (iβ)pˆi0, (2.61)
is incident obliquely upon the sample surface at a given z = z1. The upper
surface of the cell is located at z = z1 + L. The amplitudes of the reflected
and transmitted waves are sought in the forms Er = Erσσˆ + E
r
pipˆi1 and E
t =
Etσσˆ+E
t
pi 0ˆ, respectively. It is assumed that the reflected (transmitted) wave
has the polarization of the eigenwave with wave-vector k1(0). Imposing the
familiar boundary conditions of continuity for the tangential components of E
and B yields the system of equations for the determination of the eigenwave
amplitudes
∑
j
a1jC
′
j = E
i
σ,
∑
j
a2jC
′
j = E
i
pi∑
j
a3jC
′
j exp (i
kjL) = 0,
∑
j
a4jC
′
j exp (i
kjL) = 0,
(2.62)
while for the reflected/transmitted waves we have
2.3. OBLIQUE INCIDENCE 49
∑
j
a3jC
′
j exp (iτz1) = E
r
σ,
∑
j
a4jC
′
j exp (iτz1) = E
r
pi,
exp [i(q · zˆ)L]
∑
j
a1jC
′
j exp (i
kjL) = E
t
σ,
exp [i(q · zˆ)L]
∑
j
a2jC
′
j exp (i
kjL) = E
t
pi,
(2.63)
where C ′j = Cj exp [i(k0j·zˆ)z1] and 
kj = qδ(∆+n+uj)/[2(1+m) sin θ] is the
aforementioned diffractive correction of the order of δ. The system matrix
determinant of (2.62) is
D =
∣∣∣∣∣∣∣∣∣∣∣∣
a11 a12 a13 a14
a21 a22 a23 a24
γ1a31 γ2a32 γ3a33 γ4a34
γ1a41 γ2a42 γ3a43 γ4a44
∣∣∣∣∣∣∣∣∣∣∣∣
, (2.64)
with γj = exp [i(∆ + uj + n)l] and l = (δqL)/[2(1+n) sin θ]. The σ polarized
component of the reflected wave is obtained by the familiar relation
Erσ =
Ei
D
∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣
a11 a12 a13 a14 cosα
a21 a22 a23 a24 sinα exp (iβ)
γ1a31 γ2a32 γ3a33 γ4a34 0
γ1a41 γ2a42 γ3a43 γ4a44 0
a31 a32 a33 a34 0
∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣
. (2.65)
Applying the numerical Newton-Raphson method to determine the solutions
of the secular equation |D| = 0 we have found that the imaginary parts of
ωEM are identical to the expressions in (2.18) for propagation angles lying
50 CHAPTER 2. OPTICAL PROPERTIES OF LC CELLS
from 0o − 30o to normal. This result is consistent with the constant lasing
threshold curve obtained also numerically in [4].
(a) (b)
(d)
(e) (f)
(c)
Figure 2.1: Transmission (T) and Reflection (R) coefficients for σ and pi polarizations in oblique incidence.
In (a, b) θ → pi/2, in (c, d, e) θ = pi/3 and in (f) θ = pi/4. In all cases δ = 0.1 and L = 30p.
2.3. OBLIQUE INCIDENCE 51
The other components of the reflected and transmitted waves are obtained
by similar relations, by modifying the elements of the last row in the previ-
ous matrix, as follows: for Erpi by a4j, for E
t
σ by γja1j and for E
t
pi by γja2j.
For crystals in which l  1 the kinematic approximation presented in the
beginning of the section cannot account for the formation of reflection bands.
In our consideration, the only eigensolutions with non-zero coefficients are
those corresponding to wave attenuation inside the crystal length [1]. Ev-
idently, the damping of eigenwaves is due to diffraction scattering (for real
εˆ(r)) and is due to the fact that the wave-numbers k0 and k1 are complex in a
certain range of ∆, where propagation is forbidden. From the solution of the
eigenvalue equation (2.53) we can distinguish three different cases regarding
light reflection based on the values of ∆. If ∆ is such that all the solutions
uj are real and the waves pass un-attenuated from the crystal which presents
a small reflectivity. The other case is such that the two solutions u2,3 are
imaginary (and conjugates) while the others (u1,4) are real. Then, one of
the waves (2,3) is attenuated and the other is amplified exponentially within
the crystal. The incident wave with elliptical polarization corresponding to
(2,3) experiences total reflection in the Bragg band and the orthogonally
polarized (incident) wave excites the eigenwaves (1,4) that contribute only
slightly to reflection. As unpolarized light can always be analyzed into two
given orthogonal axes with zero correlation, the reflectivity of the sample for
unpolarized light in that frequency (or angle) region will be approximately
1/2. Finally, there exists a region of ∆ where all solutions are complex in
conjugate pairs. In that range, incident light of any polarization is almost
completely reflected and the reflectivity of the sample approaches unity. For
52 CHAPTER 2. OPTICAL PROPERTIES OF LC CELLS
a thick crystal, two parameters γj are exponentially large and the other two
exponentially small (corresponding to negative and positive imaginary parts,
respectively, from the complex conjugate pairs). As we can deduce from
(2.65), then, Etσ = E
t
pi = 0, because the rows in the corresponding determi-
nants are linearly dependent, and D 6= 0.
Plots of the transmission and reflection coefficients as a function of frequency
(for oblique incidence) are given in Figure 2.1 for the σ and pi eigenpolariza-
tions. We can deduce that the stop-band broadens with increasing deviation
from normal incidence, and is also centered at higher frequencies. This result
holds also for other distributed feedback structures, as we will see in the next
chapter.
2.4 The Berreman method
The problem of oblique propagation in chiral nematics can be also addressed
by the matrix method proposed by Berreman[5] in 1972 that we will briefly
outline. After writing Faraday-Maxwell and Ampère-Maxwell law in the
absence of sources in a 6x6 matrix form, and linking the electric field with
the electric displacement and the magnetic field with the magnetic induction
through a matrix relation as well (of the form (D,B) = M(ε0E, µ0H)) we
end up in the following equation for transverse waves with a time dependence
exp(−iωt) with ky = 0
∂zψi =
iω
c
∆ijψj, (2.66)
where ψ = (Ex, µ0cHy, Ey,−µ0cHz) and ∆ij = f(kxc/ω, εij, µij, ρij). The
2.4. THE BERREMAN METHOD 53
penultimate arguments are components of the magnetic permeability tensor
and the last ones components of the optical-rotation tensors. If the matrix
elements ∆ij are independent of (or vary very slowly with) z over some
short interval δz equation (2.66) will have four periodic solutions of the form
ψj(δz) = exp(iqjδz)ψj(0). Substituting this form in (2.66) we obtain the
eigenvalue equation
∣∣∣∆− qc
ω
I
∣∣∣ = 0 (2.67)
and subsequently determine the eigenvectors ψj. If the matrix ∆ is indepen-
dent of z over some finite range h, as it happens with dielectric stacks then
(2.66) can be integrated to give ψi(z + h) = Pij(h)ψj(z). This is familiar
expression from one of the postulations of the Bloch-Floquet theorem for
waves in periodic media [6] with h being the spatial period of the structure.
The matrix in the above relation can be written as
P(h) = exp
(
iωh∆
c
)
=
∞∑
n=0
(
iωh∆
c
)n
. (2.68)
In many numerical problems, when (ωh/c) 1 only the first few terms from
the above expansion are retained. A secular equation for the eigenvalues q
similar to (2.66) can be formed as
|P (h)− exp(iqh)I| = 0. (2.69)
The matrix P has the obvious symmetry property P(kh) = [P(h)]k.
Focussing on chiral nematic liquid crystals, the dielectric tensor has the form
(2.2) and the corresponding differential propagation matrix is
54 CHAPTER 2. OPTICAL PROPERTIES OF LC CELLS
∆(z) =

0 1− Ξ
2
ε‖
0 0
ε¯[1 + δ cos(τz)] 0 ε¯δ sin(τz) 0
0 0 0 1
ε¯δ sin(τz) 0 ε¯[1− δ cos(τz)]− Ξ2 0

, (2.70)
where Ξ = (c/ω)kx = (c/ω)k sin θ and θ is the angle between the z axis
and the direction of propagation. Having constructed the matrix ∆ we can
obtain numerical solutions for the field amplitudes in oblique propagation
(when Ξ 6= 0). For Ξ = 0, as noted in [5] the eigenvectors ψ(z) should have
a form such that Ex and cµ0Hy at z = 0 are the same as Ey and −cµ0Hx at
z = p/4 apart from a phase factor (p is the helix pitch). This spiral symmetry
is reflected in the eigenvectors
ψ = exp(iqz)

A exp
(
i τ
2
z
)
+B exp
(−i τ
2
z
)
cµ0[A
′ exp
(
i τ
2
z
)
+B′ exp
(−i τ
2
z
)
]
A exp
[
i τ
2
(z − pi/τ)]+B exp [−i τ
2
(z − pi/τ)]
cµ0{A exp
[
i τ
2
(z − pi/τ)]+B exp [−i τ
2
(z − pi/τ)]}

= exp
[
i
(
q +
τ
2
)
z
]

A
cµ0A
′
−iA
−icµ0A′
+ exp
[
i
(
q − τ
2
)
z
]

B
cµ0B
′
−iB
−icµ0B′
 . (2.71)
It is evident from the above that the eigenvalue q is the Bloch wave-number
which is to be determined. Substituting in (2.66) we obtain
2.4. THE BERREMAN METHOD 55
exp
[
i
(
q +
τ
2
)
z
]

A′
µ0c(ε¯A+ ε¯δB)
−iA′
−iµ0c(ε¯A+ ε¯δB)

+ exp
[
i
(
q − τ
2
)
z
]

B′
µ0c(ε¯B + ε¯δA)
iB′
iµ0c(ε¯B + ε¯δA)

=
c
ω
(
q +
τ
2
)
exp
[
i
(
q +
τ
2
)
z
]

A
cµ0A
′
−iA
−icµ0A′

+
c
ω
(
q − τ
2
)
exp
[
i
(
q − τ
2
)
z
]

B
cµ0B
′
−iB
−icµ0B′
 .
(2.72)
Equating the corresponding vector components from both handsides we ob-
tain the following system of equations

A′ =
(
q + τ
2
)
c
ω
A
B′ =
(
q − τ
2
)
c
ω
B
ε¯A+ ε¯δB =
(
q + τ
2
)
c
ω
A′ =
[(
q + τ
2
)
c
ω
]2
A
ε¯B + ε¯δA =
(
q − τ
2
)
c
ω
B′ =
[(
q − τ
2
)
c
ω
]2
B.
(2.73)
56 CHAPTER 2. OPTICAL PROPERTIES OF LC CELLS
The compatibility condition of the above system yields the eigenvalue quartic
equation with solutions
q = ±
√
τ 2
4
+ k2ε¯± kτ ε¯
(
1 +
δ2k2
τ 2
)
= km±, (2.74)
with k = ω/c and m being the `effective refractive index' in the de Vries
formulation (see (2.27)).
2.5 Optical feedback in nematic LC slabs
We will conclude this chapter by treating laser radiation produced by liquid
crystal samples as a feedback mechanism, based on the treatment in [4].
We consider a homogeneous liquid crystal cell B of thickness d which is
unbounded in the x and y directions, as shown in Figure 2.2.
The slab B is surrounded by the regions A and C; we assume that the elec-
tromagnetic fields in A and B are superpositions of forward and backward-
travelling waves, whereas in region C only forward-travelling waves can exist.
The monochromatic wave incident upon the slab from the medium A is rep-
resented by a column vector
E =
 Es0 exp(−ik0 · r)
Ep0 exp[−i(k0 · r + δ0)]
 , (2.75)
where k0 is the wavevector in medium A and the indices s(p) refer to waves
polarized in (perpendicular to) the plane of incidence. The phase shift δ0
denotes the arbitrary ellipticity of the incident wave. The incident wave is
partially transmitted in medium B with a complex amplitude following the
2.5. OPTICAL FEEDBACK IN NEMATIC LC SLABS 57
n0
n‖, n⊥
n0 β0
1
A
B
C
25
34 6 3∗
4∗
βs
βp
k0
d
x
z
Figure 2.2: Schematic diagram of ray propagation in a nematic liquid crystal slab.
vector relation
E21 = T21E, (2.76)
where T21 is the transmission matrix of the interface 1− 2. Hereinafter the
order of suffixes in our expressions is indicative of the propagation direction
of the wave that is travelling from the point denoted by the outer to the
point denoted by the inner suffix. The resultant forward-propagating wave
field at point 2 is written as
E2 = E21 + R21P23R34E3, (2.77)
whereR34 is the reflection matrix of the interface 3−4, P23 is the propagation
matrix from point 3 to point 2, R21 is the reflection matrix of the interface
58 CHAPTER 2. OPTICAL PROPERTIES OF LC CELLS
2 − 1 and E3 is the forward propagating wave field at point 3. By virtue
of the homogeneity of the sample in the x and y directions the difference
between the waves at points 3 and 3∗ can only amount to a phase shift
δ33∗ = k0x(x3 − x3∗), therefore
E3 = E3∗ exp(−iδ33∗) = P3∗2E2 exp(−iδ33∗). (2.78)
Substituting (2.76) and (2.77) into (2.78) we obtain
E3 = P3∗2E2 exp(−iδ33∗)(T21E + R21P23R34E3)⇒
E3 = [I−P3∗2R21P23R34 exp(−iδ33∗)]−1P3∗2T21 exp(−iδ33∗)E.
(2.79)
On the other hand, combining (2.79) and (2.80) the outgoing wave at point
4 can be written as
E4 = T43E3 = T43[I−P3∗2R21P23R34 exp(−iδ33∗)]−1P3∗2T21 exp(−iδ33∗)E.
(2.80)
T21 Σ P3∗2 exp(−iδ33∗) T43
R34P23R21
E E21 E2 E3 E4
+
Figure 2.3: Feedback loop illustrating propagation of light in a liquid crystal cell.
The above relation can be construed in terms of the output function of the
feedback loop outlined in Figure 2.3. Feedback is determined by the matrix
2.5. OPTICAL FEEDBACK IN NEMATIC LC SLABS 59
product P3∗2R21P23R34 while the presence of an amplifying (or absorbing)
medium in the liquid crystal is quantified through the propagation parame-
ter P3∗2 (and P23). We draw attention on the fact that feedback is positive
(denoted by the sign + on summation), hence the system may become un-
stable and the output unbounded with an arbitrarily small applied input
(probe field). The singularity condition can be written in terms of the ma-
trix F = I − P3∗2R21P23R34 exp(−iδ33∗) as |F| = 0. The role of the probe
field can be played by any fluctuation. As noted in [4], the condition |F| = 0
must hold exactly. This means that when the gain exceeds its threshold
value, lasing is quenched by saturation of the output wave. The inequality of
gain exceeding losses is only meaningful under the condition that the density
of photonic states in the resonating cavity must be sufficiently high for the
quenched mode to dominate over adjacent modes reaching threshold. Al-
though no assumption has been made for the properties of the slab B so far,
we will now focus on the case of a homeotropically oriented nematic liquid
crystal cell, where an analytical expression for the transfer matrices can be
easily derived. We also assume that the liquid crystal cell is doped with a
small amount of gain medium such that the values of the principal refractive
indices are not altered when the medium is excited. If we suppose that the
molecular director coincides with the unit direction along the z- axis (n ≡ zˆ),
then the refractive indices for the s and p polarized waves are given by the
expressions
ns = n⊥,
n2p(βp) =
ε‖ε⊥
ε2‖ cos
2(βp) + ε2⊥ sin
2(βp)
,
(2.81)
60 CHAPTER 2. OPTICAL PROPERTIES OF LC CELLS
where βp is the angle between the z-axis and the wavevector of the p polarized
wave. The reflection matrices are obtained by virtue of the continuity of the
tangential components of the electromagnetic field and are given by the well-
known Fresnel formuli
R34 = R21 =
−
sin(β0 − βs)
sin(β0 + βs)
0
0
tan(β0 − βp)
tan(β0 + βp)
 , (2.82)
where β0 is the angle between the z-axis and the direction of propagation
of the wave in the homogeneous surrounding media A and C with refractive
index n0. The angles β0,p,s are related by Snell's law as follows
n0 sin β0 = ns sin βs = np(βp) sin βp.
By virtue of the symmetry of the problem and the uniformity of cell in the
xy plane, the wave propagation matrices are diagonal. The matrix elements
of P23 will have the form of exp(−iks,pr23) for the s, p-polarized waves. If we
select the points 2 and 3 to be such that r2,3 ‖ ks, then the matrix element
for the p-wave will contain the phase shift δ25 = k0(x2 − x5) sin β0 (as we
have written r23 = r25 + r53 = r53 + (x2 − x5)xˆ) since the x-component
of the wavevector is continuous across all regions (by virtue of the con-
servation of momentum in the x-direction). Also, |r23| = (d/ cos βs) and
|r53| = (d/ cos βp) so that the propagation matrices P3∗2 = P23 have the
form
2.5. OPTICAL FEEDBACK IN NEMATIC LC SLABS 61
exp
[
− d
cos βs
(
χs + i
ns
n0
k0
)]
0
0 exp
[
− d
cos βp
(
χp + i
np(βp)
n0
k0
)
− iδ25
]
 .
(2.83)
The above expression takes into account the the wavenumber has an imagi-
nary part χs,p, where χs,p > 0(< 0) corresponds to absorption (amplification).
The experimentally measured coefficient is the one pertaining to the intensity
of the wave, αs,p = 2χs,p. If the gain medium is anisotropic, then αp depends
on the angle βp. In the case of weak absorption (or gain), where the condition
|n0αs,p/(2k0)|  1 holds, the coefficients αp,s can be determined by consid-
ering the magnitude of the Poynting vector for each of the eigenwaves of the
structure, which is proportional to ep,s(Xep,s), where X = diag(χ⊥, χ⊥, χ‖).
The above inequality condition is necessary in order to write the components
of the complex dielectric permittivity tensor as a sum where χs,p appear only
in the imaginary part, up to the first order. As derived in [6], the p- eigen-
wave belongs to the xz plane and the s-eigenwave is along the y-axis. More
specifically, ep = (n
2
‖ cos βp, 0,−n2⊥ sin βp), es = yˆ, therefore
αs = α⊥
αp(βp) =
α‖ sin2 βp
sin2 βp +
(
n‖
n⊥
)4
cos2 βp
+
α‖ cos2 βp
sin2 βp +
(
n‖
n⊥
)4
sin2 βp
, (2.84)
where α⊥,‖ = 2χ⊥,‖. Also we have
62 CHAPTER 2. OPTICAL PROPERTIES OF LC CELLS
δ33∗ = k0x(x3 − x3∗) = −(2d tan βs)k0 sin β0 = −(2d tan βs)k0ns sin βs
n0
= −[2d(tan βs + tan βp − tan βp)]k0np sin βp
n0
= −(2d tan βp)k0np sin βp
n0
− 2k0(x∗3 − x6) sin β0
= −(2d tan βp)k0np sin βp
n0
− 2k0(x2 − x5) sin β0
= −2
(
k0d tan βp
np sin βp
n0
+ δ25
)
(2.85)
Hence, the lasing condition |F | = 0 is equivalent to two scalar equations
obtained by equating each diagonal element of the resulting 2x2 matrix to
zero. The conditions for the two waves s, p can be recast as follows
P3∗2R21P23R34 exp(−iδ33∗) = I
⇒

sin2(β0 − βs)
sin2(β0 + βs)
exp
(
− 2χsd
cos βs
)
exp
[
−2ik0d
(
ns
n0
1
cos βs
− sin
2 βs
cos βs
)]
= 1
tan2(β0 − βs)
tan2(β0 + βs)
exp
(
− 2χpd
cos βp
)
exp
(
−2ik0dnp(βp)
n0
)
= 1
⇒

sin2(β0 − βs)
sin2(β0 + βs)
exp
(
− 2χsd
cos βs
)
exp
(
−2ik0dns
n0
cos βs
)
= 1
tan2(β0 − βs)
tan2(β0 + βs)
exp
(
− 2χpd
cos βp
)
exp
(
−2ik0dnp(βp)
n0
)
= 1.
(2.86)
Exactly as in the case of a Fabry-Pérot etalon, since the left handside of the
above equations is unity, we must impose the (phase) conditions
k0d
ns
n0
cos βs = mpi,
k0d
np(βp)
n0
cos βp = npi,
(2.87)
2.5. OPTICAL FEEDBACK IN NEMATIC LC SLABS 63
where n,m ∈ N.
For these real frequencies (eigenfrequencies) there exist negative values of
χs,p which correspond to lasing. In the case of normal incidence, βs,p = 0
and αs = αp = α⊥. Since β0 → 0 and βs → 0 simultaneously, from the
continuity of kx we have
dβs
dβ0
=
n0
ns
cos β0
cos βs
→ n0
n⊥
. We also apply l'Hôpital's
rule for the matrix element R2111 of the diagonal, to obtain
lim
β0→0
sin(β0 − βs)
sin(β0 + βs)
= lim
β0→0
1− dβs
dβ0
1 +
dβs
dβ0
cos(β0 − βs)
cos(β0 + βs)
=
n⊥ − n0
n⊥ + n0
, (2.88)
so that we have the expression for the threshold gain
αs,p = α⊥ =
1
d
ln
[(
n⊥ − n0
n⊥ + n0
)2]
< 0. (2.89)
In the general case of oblique incidence where β0 ∈
(
0, pi
2
)
, the threshold gain
is obtained by the following relations for the two polarizations
αs(β0) = α⊥(β0) = −
cos
[
arcsin
(
n0 sin β0
n⊥
)]
d
×
× ln
sin
2
(
arcsin
(
n0 sin β0
n⊥
)
+ β0
)
sin2
(
arcsin
(
n0 sin β0
n⊥
)
− β0
)
 ,
αp(β0) = −cos βp
d
ln
[
tan2(β0 + βp)
tan2(β0 + βp)
](
KD
A
sin2 βp +
1
B
cos2 βp
)
,
(2.90)
where A = sin2 βp + (n‖/n⊥)4 cos2 βp, B = cos2 βp + (n⊥/n‖)4 sin2 βp, KD =
(α‖/α⊥) and βp = βp(β0) = arcsin
[
n0n‖ sin β0
√
n20n
2
‖ + n
2
0(n
2
‖ − n2⊥) sin2 β0
]
.
64 CHAPTER 2. OPTICAL PROPERTIES OF LC CELLS
Consequently, lasing is in principle possible in any direction β0 associated
with a unique discrete set of discrete wavelengths and threshold gain coeffi-
cients. When the thickness of the LC slab is small compared to the pumped
surface, laser radiation can be produced within a broad spectral regime pro-
vided that the gain coefficients can vary within a sufficiently wide range (in
practice, gain is provided only in a limited range of wavelengths, considered
to be the FWHM of the luminescence spectrum of the gain medium). If the
gain is very low, the lasing condition can still be satisfied for the case of
the so-called sliding (or `leaky') modes (where β0 → pi/2). Then, the phase
condition (2.87) yields a number of (discrete) wavelengths for propagation in
the free space (taking into account that ns = n⊥ and np → n‖ ), according
to the relations
2pi
λs
dn⊥
√
1−
(
n0
n⊥
)2
= mpi ⇒ λs(m) = 2d
m
n⊥
√
1−
(
n0
n⊥
)2
,
2pi
λp
dn‖
√
1−
(
n0
n‖
)2
= npi ⇒ λp(n) = 2d
n
n‖
√
1−
(
n0
n‖
)2
.
(2.91)
It is obvious that the leaky modes can escape only through the edges of the
substrate (in the form evanescent waves) if the condition for total internal
reflection is satisfied on the substrate-air interface. Finally, as noted in [4],
the presence of sliding modes is a significant channel of energy loss responsible
for quenching band-edge lasing in chiral nematic LCs.
In this chapter we have detailed the principal analytical treatments employed
in bibliography to analyse light propagation in nematic and chiral nematic
2.5. OPTICAL FEEDBACK IN NEMATIC LC SLABS 65
liquid crystal cells. Conclusions and formulae derived herein will serve as a
basic means to develop our quantitative methods in the subsequent sections.
Bibliography
[1] V. A. Belyakov, Diffraction Optics of Complex-Structured Periodic Me-
dia, ch. 2, Springer-Verlag, New York, 1992.
[2] V. A. Belyakov and S. V. Semenov, Optical Edge Modes in Photonic
Liquid Crystals, JETP, 109, pp. 687-699 (2009).
[3] Hl. de Vries, Rotatory Power and Other Optical Properties of Certain
Liquid Crystals, Acta Cryst., 4, pp. 219-226 (1951).
[4] S. P. Palto, Lasing in Liquid Crystal Thin Films, JETP, 103, pp.472-
479 (2006).
[5] Dwight. W. Berreman, Optics in Stratified and Anisotropic Media:4x4
Matrix Formulation, J. Opt. Soc. Am. , 62, pp. 502-510 (1972).
[6] A. Yariv and P. Yeh, Photonics: Optical Electronics in Modern Com-
munications, ch. 12, 6th ed., Oxford University Press, US, 2006.
66
Chapter 3
The density of photon states
3.1 The rôle and the calculation of the DOS
In this chapter we will outline the main results of the analytic calculation of
the DOS from the general treatment of 1D periodic media presented in [1]-[2],
and we will subsequently focus on chiral nematics, where the aforementioned
method has been applied to interpret experimental results [3].
Combining Maxwell's equations in the absence of charge density ρ (which
implies also that the electrostatic potential φ is zero) and current density J,
the vector potential A is the solution of the equation
curl(curlA) +
ε(r)
c2
∂2
∂t2
A = 0. (3.1)
The solution is sought in the form Ak(r, t) = ak(r, t) exp(−iωkt). Substitu-
tion in the above yields the Helmholtz eigenvalue equation
curl[curlak(r)]− ω
2
kε(r)
c2
ak(r) = 0. (3.2)
67
68 CHAPTER 3. THE DENSITY OF PHOTON STATES
The amplitudes ak(r) obey the transverse gauge condition div[ε(r)ak(r)] = 0
(called also a ε⊥ vector) and satisfy the orthogonality relation
∫
ak
∗(r′)ak(r)d3k =
↔
δ ε⊥ (r
′ − r). (3.3)
The inhomogeneous Helmholtz equation for the vector potential in the pres-
ence of J reads
curl[curlA(r, t)] +
(r)
c2
∂
∂t2
A(r, t) = µ0J, (3.4)
assuming µ = 1. The problem is solved with the help of the so-called dyadic
Green's function
↔
G which satisfies the inhomogeneous wave equation
curl[curl
↔
G (r, t; r
′, t′)]+
ε(r)
c2
∂2
∂t2
↔
G (r, t; r
′, t′) = δ(t−t′)↔δ ε⊥ (r− r′), (3.5)
with the time conditions
↔
G (r, t; r
′, t′)
∣∣
t=t′ = 0,
∂
∂t
↔
G (r, t; r
′, t′)
∣∣
t=t′ = c
2
↔
δ ε⊥ (r− r′)
(3.6)
Green's function is often written in terms of the propagator
↔
D (r, t; r′, t′)
as
↔
G (r, t; r′, t′) = u(t − t′)
↔
D (r, t; r′, t′), where u(t − t′) is the Heaviside
function. The function
↔
D (r, t; r′, t′) satisfies the same time conditions as
↔
G (r, t; r′, t′).
The propagator can be written as an expansion in terms of the normal mode
amplitudes. Considering the time translation invariance and the parity of
the propagator, we can write
3.1. THE RÔLE AND THE CALCULATION OF THE DOS 69
↔
D (r, t; r
′, t′) =
∫
bk(r
′)ak(r) sin[ωk(t− t′)]d3k =
∫
ak
∗(r′)ak(r)d3k. (3.7)
The coefficients bk(r
′) can be determined by the time derivative and the
orthogonality conditions as follows
∂
∂t
↔
D (r, t; r
′, t′)
∣∣
t=t′ = c
2
↔
δ ε⊥ (r− r′)
=
∫
bk(r
′)ak(r)ωkd3k = c2
∫
ak
∗(r′)ak(r)d3k.
(3.8)
By virtue of the orthogonality relation
∫
ak′
∗(r)ak(r)dV = δ(k− k′), (3.9)
we obtain the relation bk(r
′) =
c2
ωk
ak
∗(r′).
Then, the dyadic Green's function becomes
↔
G (r, t; r
′, t′) = c2u(t− t′)
∫
ak
∗(r)ak(r)
sin[ωk(t− t′)]
ωk
d3k. (3.10)
A current density (hence a radiating time varying electric dipole moment)
will be immersed in its own electric field emitted at earlier times which has
been Bragg-reflected in a periodic medium. The dyadic Green's function
will be used to address the problem, since it contains all information about
the boundary conditions and the dielectric permittivity profile. The rate
of change of kinetic energy of a system of charged particles is equal (and
opposite) to the rate of change of work done by the moving charges against
a surrounding electric field. Denoting the latter by P (t) we have [4]
70 CHAPTER 3. THE DENSITY OF PHOTON STATES
P (t) = −
∑
a
qaEva = −
∫
J(r, t) · E(r, t)dV. (3.11)
Since E = −∂A
∂t
we have
P (t) =
∫
J(r, t) · E(r, t)dV =
∫
J(r, t) · ∂A
∂t
(r, t)dV, (3.12)
where
A(r, t) = µ0
∫ +∞
−∞
∫
V
↔
G (r, t; r
′, t′)J(r′, t)dt′dV ′
= µ0
∫ t
−∞
∫
V
↔
D (r, t; r
′, t′)J(r′, t)dt′dV ′.
(3.13)
By virtue of Leibniz's rule we have
∂
∂t
∫ t
−∞
∫
V
↔
D (r, t; r
′, t′)J(r′, t)dV ′dt′
=
∫ t
−∞
∫
V
∂
↔
D
dt
(r, t; r′, t′)J(r′, t)dV ′dt′ +
∫
V
↔
D (r, t; r
′, t)J(r′, t)dV ′
=
∫ t
−∞
∫
V
∂
↔
D
dt
(r, t; r′, t′)J(r′, t)dV ′dt′,
(3.14)
since the propagator has a time dependence of the form sin[ωk(t−t′)]. Hence,
the power can be written as
P (t) = µ0c
2
∫
k
∫ t
−∞
∫
V
∫
V
[J(r, t) · ak(r)][J(r′, t′) · a∗k(r′)]×
× cos[ωk(t− t′)]dt′ dV dV ′ d3k.
(3.15)
The total energy emitted by the localized source up to time T is
3.1. THE RÔLE AND THE CALCULATION OF THE DOS 71
U(T ) = µ0c
2
∫ T
−∞
P (t)dt
= µ0
c2
2
∫
k
∣∣∣∣∫ T−∞
∫
V
J(r, t) · ak(r) exp (−iωkt)dV dt
∣∣∣∣2 d3k,
(3.16)
where we have written cos[ωk(t− t′)] = R(exp[ωk(t− t′)]), taken the R sign
out of the integral since J is real, and changed the limits
∫ T
−∞ dt
∫ t
−∞ dt
′ =
1
2
∫ T
−∞
∫ T
−∞ dtdt
′
, since the area of the triangle is half that of the parallelogram
with edges
(−∞,−∞), (−∞, T ), (T, T ), (T,−∞)
We will now draw parallels between the above procedure and the total exci-
tation probability in the context of time-dependent perturbation theory [5]
in the so-called `first quantization' (where the fields are non-quantized). Ac-
cording to the standard first-order result, the transition probability (which is
the modulus squared of the first order coefficient cfi(t)) between the bound
atom states |i〉 (initial) and |f〉 (final) reads
|cfi(t)|2 = 1~2
∣∣∣∣∫ t−∞ Vfi(t′) exp(iωfit′)dt′
∣∣∣∣2 , (3.17)
where Vfi(t) = 〈f |Vˆ (t)|i〉 are the matrix elements of the time-dependent
perturbation (of the first order of smallness), and the term exp(iωfit
′) is due
to the inherent time dependence of the stationary states. If we assume a
time-dependent perturbation operator of the form
Vˆ (t) = [Fˆ exp(−iω0t) + Gˆ exp(iω0t)]u(t), (3.18)
72 CHAPTER 3. THE DENSITY OF PHOTON STATES
with ∂Fˆ /∂t = ∂Gˆ/∂t = 0. The Hermicity condition of the operator Vˆ (t)
imposes that Gˆ = Fˆ † whence Gfi = F ∗if . Inserting the expression Vˆfi(t) =
{Ffi exp[i(ωfi − ω0)t] + F ∗if exp[i(ωfi + ω0)t]}u(t) in (3.17) we obtain
|cfi(t)|2 ∼= 1~2 |Ffi|
2
sin2
(
ωfi − ω0
2
t
)
(
ωfi − ω0
2
)2 , (3.19)
where we have assumed ωfi → ω0 (resonance) so that the term containing
the factor sin[(ωif +ω0)t/(ωif +ω0)] is negligible in comparison to sin[(ωif −
ω0)t/(ωif − ω0)].
We note that
lim
t→∞
sin2 at
pita2
= δ(a), (3.20)
so that for t→∞ (in the steady state) the transition probability reads
P(t) = |cfi(t)|2 = 2~2 |Ffi|
2 pitδ(ωfi − ω0). (3.21)
The transition rate then is
Γ =
dP
dt
=
2
~2
|Ffi|2 piδ(ωfi − ω0). (3.22)
In our case, the radiating dipole located at the position r = r0 corresponds
to a current density
J(r, t) = ω0d cos(ω0t)δ(r− r0)u(t). (3.23)
Taking the dot product with the normalized eigenwave amplitudes ak(r) we
produce the scalar quantity
3.1. THE RÔLE AND THE CALCULATION OF THE DOS 73
V˜ = F˜ [exp(−iω0t) + exp(iω0t)]u(t), (3.24)
where F˜ ∝ d ·ak(r) plays the rôle of the operator Fˆ , and in our case F˜ = G˜.
We also write ωk instead of ωfi, as we are dealing with unbound photon
states with momentum ~k. Neglecting the term containing exp[i(ωk + ω0)t],
the emission energy, according to (3.16) reads
U(t) =
∫
k
F˜ 2
sin2
(
ωfi − ω0
2
t
)
(
ωfi − ω0
2
)2 d3k . (3.25)
Similarly, according to (3.16), the emitted energy for the localized dipole
reads
U(t) = µ0
c2ω20d
2
2
∫
k
∣∣∣dˆ · ak(r)∣∣∣2 δ(r− r0)sin
2
[(
ωk − ω0
2
)
t
]
(
ωk − ω0
2
)2 d3k. (3.26)
Hence, for t→∞, the emission rate can be written according to (3.20) as
P (t→∞) = µ0pic
2
4
ω20d
2
∫
k
∫
V
∣∣∣dˆ · ak(r)∣∣∣2 δ(r− r0)δ(ωk − ω0)dV d3k
= µ0
pic2
4
ω20d
2
∫
k
∣∣∣dˆ · ak(r0)∣∣∣2 δ(ωk − ω0)d3k.
(3.27)
In the free space (where ε(r) = 1), the modes are plane waves
ak(r) = (2pi)
−3/2 exp(−ik · r)eˆk, (3.28)
74 CHAPTER 3. THE DENSITY OF PHOTON STATES
where eˆk is the unit polarization vector. The waves, being transverse, sat-
isfy k · ek = 0. These solutions immediately also satisfy the orthogonal-
ity/normalization conditions (3.3), (3.9). The dispersion relation is k = ω/c,
so that we can write the `volume element' in the k-space as d3k = dΩkk
2dk.
The power-emitted by a dipole in free space is
P 3D
free
= µ0
c2pi
4 · 8pi3cω
2
0d
2
∫
k
k2δ(k − k0)dk
∫ ∣∣∣dˆ · eˆk∣∣∣2 dΩk
= µ0
k20ω
2
0d
2
32pi2c
∫
sin2(θk)dΩk,
(3.29)
where we have assumed that the electric dipole is oriented along the z− axis
(in the k-space), so that ek ≡ θˆk (and zˆ · θˆk = sin θk). The final result can
be recast in the familiar form
P 3D
free
=
1
12
µ0ω
4
0d
2
pic
, (3.30)
in which the multiplicative factor of 2 accounts for the two independent
polarizations (degrees of freedom) of the plane wave.
Let us now return to the expression (3.22), and consider explicitly sponta-
neous emission of an atom in a medium with a non-uniform dielectric profile.
We will distinguish two states: the state
∣∣1, {0}〉, corresponding to the ex-
cited state of the atom with the photon occupation number equal to 0, and
the state
∣∣0, {1}k〉, corresponding to the ground state with one photon in the
transverse mode with wave-vector k. Assuming a continuous distribution of
modes in the k− space, the spontaneous emission probability in the steady
state can be written as
†
†
Hereinafter in this chapter, the vectors will denote (formally) vector operators.
3.1. THE RÔLE AND THE CALCULATION OF THE DOS 75
P
total
=
1
~2
∫
k
∣∣〈0, {1}k∣∣d · Ek(r0)∣∣1, {0}〉∣∣2 sin
2
(
ωk − ω0
2
t
)
(
ωk − ω0
2
)2 dVk
=
pi
~2
∫ ∣∣〈0, {1}k∣∣d · Ek(r0)∣∣1, {0}〉∣∣2 sin
2
(
ωk − ω0
2
t
)
(
ωk − ω0
2
)2 dVkdωkdωk.
(3.31)
We define ρ(ωk) =
dVk
dωk
as the Density of Photon States (DOS), and also
change variables in the above integral to α = (ωk − ω0)/2, so that:
P
total
=
2pi
~2
∫ ∣∣〈0, {1}k∣∣d · Ek(r0)∣∣1, {0}〉∣∣2 tsin2(αt)
pitα2
ρ(2α + ω0)dα
t→∞
=
2pi
~2
∫ ∣∣〈0, {1}k∣∣d · Ek(r0)∣∣1, {0}〉∣∣2 tδ(α)ρ(2α + ω0)dα
=
2pi
~2
∣∣〈0, {1}k∣∣d · Ek(r0)∣∣1, {0}〉∣∣2 tρ(ω0),
(3.32)
yielding the spontaneous transition rate
Γ =
2pi
~2
∣∣〈0, {1}k∣∣d · Ek(r0)∣∣1, {0}〉∣∣2 ρ(ω0). (3.33)
The above expression is known as Fermi's Golden Rule [1]. As we have
mentioned, this is a result derived by applying first order perturbation theory
for the small perturbation d · Ek(r0) in the Hamiltonian. Hence, the result
is said to be valid for weak coupling between the atom and the field. An
analogous result can be derived classically from (3.16), according to which
the emission rate normalized to the free space rate is
76 CHAPTER 3. THE DENSITY OF PHOTON STATES
γ
classical
= ω0 |d · ak(r0)|2 dk
dω
∣∣∣∣
ω0
(3.34)
and the quantity
dk
dω
∣∣∣∣
ω0
is the DOS, defined as the number of wave-numbers k
per unit frequency ω, and consequently is the reciprocal of the group velocity,
vg(ω) =
1
ρ(ω)
=
dω
dk
. This is the velocity of the pulse envelope moving
through the one-dimensional (1D) `potential' n(x).
Let us assume a lossless and dispersion-less refractive index profile n(x),
non-zero over an interval [0, L]. The Helmholtz equation (3.2) reduces to
d2a
dx2
+
ω2k
c2
ε(x)a(x) = 0. (3.35)
The DOS can be obtained by extracting the dispersion relation k = k(ω)
from the above equation, solved by numerical methods. It can be also ob-
tained by a more elegant technique (known as the Wigner method) based
on the complex transmission coefficient of a structure, as detailed below.
The complex transmission coefficient of any structure can be written as
t(ω) = x(ω) + iy(ω) =
√
T exp(iφ), where φ = arctan[y(ω)/x(ω)]. Here,
φ is the total phase accumulated as the light propagates through the po-
tential. It can be also written as φ = kL, where L is the physical path
traversed by the light waves (for normal propagation it coincides with the
thickness of the structure). Therefore, the dispersion relation can be written
as tan(kL) = y(ω)/x(ω). Differentiating with respect to ω we obtain
d
dω
[tan(kL)] =
L
dω
(
x(ω)
y(ω)
)
⇒ 1
cos2(kL)
dk
dω
=
1
L
y′x− x′y
x2
, (3.36)
3.1. THE RÔLE AND THE CALCULATION OF THE DOS 77
where the prime denotes differentiation with respect to ω. Since 1/ cos2 θ =
1 + tan2 θ, solving the above for dk/dω we obtain
ρ(ω) =
dk
dω
=
1
L
y′x− x′y
x2 + y2
. (3.37)
We note that the above derivation is independent of the specific form of
n(x) and that the transmission coefficient can be easily obtained in periodic
media with the help of transfer matrix methods. More precisely, if we write
the general solution of (3.35) as a superposition of right and left-propagating
waves, written as u±(x) = f±(x) exp(±ikx), where f±(x) are real envelope
functions, then the column vectors u =
u+
u−

at x = 0, L are related by
u(0) =
1
r
 = Mu(L) =
A B
C D
u(L) =
A B
C D
t
0
 , (3.38)
where M is the transfer matrix. Under time reversal, due to the symmetry
of the scattering system we have
u(0)+∗
u(0)−∗
 = M
u(L)+∗
u(L)−∗
 . (3.39)
Also, we have
u(0)−∗
u(0)+∗
 = M∗
u(L)−∗
u(L)+∗
 , (3.40)
and combining the two we obtain the condition
M∗ =
0 1
1 0
M
0 1
1 0
 . (3.41)
78 CHAPTER 3. THE DENSITY OF PHOTON STATES
Hence, the final form of the transfer matrix reads
M =
1t r
∗
t∗
r
t
1
t∗
 . (3.42)
By virtue of the energy conservation, we also obtain the condition |r|2+|t|2 =
R + T = 1⇒ |M| = 1 (i.e. the transfer matrix is unimodular).
From (3.42), we deduce that the eigenvalues of M satisfy the equation
µ2 +
(
1
t
+
1
t∗
)
µ+
1− |r|2
|t|2 = 0⇒ µ
2 + 2<
(
1
t
)
+ 1 = 0. (3.43)
Since the matrix is unimodular, the two eigenvalues µ± satisfy the relation
µ+µ− = 1. Furthermore, the refractive index is periodic n(x) = n(x+ d), so
that Bloch's theorem is applicable, according to which only the phase of the
wavefunction (u± in our case) changes on moving from one cell to the other.
Denoting by β the phase (called also the Bloch's phase), we can then write
MuB = exp(±iβ)uB, (3.44)
where uB are called the Bloch eigenfunctions. Evidently,
µ+Bµ
−
B = exp(iβ) exp(−iβ) = 1. (3.45)
Substituting the eigenvalues µ±B into (3.43) we find that cos β = <(1/t). We
know from the Cayley-Hamilton theorem that a square matrix is a root of its
characteristic polynomial. Hence, the transfer matrix satisfies the equation
M2 − 2M cos β + I = 0. (3.46)
3.1. THE RÔLE AND THE CALCULATION OF THE DOS 79
Will now prove by induction that
MN =
1
sin β
{sin(Nβ)M− sin[(N − 1)β]I} . (3.47)
For N = 1 we have the trivial relationM = M. For N = 2 the above relation
yields
M2 =
1
sin β
[sin(2β)M− sin βI] = 2 cos βM− I, (3.48)
which is the eigenvalue equation. Let us assume that the equality holds for
N . We will prove it is true for N + 1. Multiplying the equation for N byM
from the right, we have
M(N+1) =
1
sin β
{
sin(Nβ)M2 − sin[(N − 1)β]M}
=
1
sin β
{sin(Nβ)[2 cos βM− I]− sin[(N − 1)β]M}
=
1
sin β
{
(2 sin[(N − 1)β] cos2 β + 2 cos[(N − 1)β] cos β sin β − sin[(N − 1)β])M− sin(Nβ)I}
=
1
sin β
{
(sin[(N − 1)β](2 cos2 β − 1)) + sin(2β) cos[(N − 1)β]M− sin(Nβ)I}
=
1
sin β
{(sin[(N − 1)β] cos(2β) + sin(2β) cos[(N − 1)β]M− sin(Nβ)I}
=
1
sin β
{sin[(N + 1)]βM− sin(Nβ)I} ,
(3.49)
which we need to show. We note that the same result can be obtained via
Chebyshev's identity applied to calculate MN . Identifying the components
of MN with those of the matrix
80 CHAPTER 3. THE DENSITY OF PHOTON STATES

1
tN
r∗N
t∗N
rN
tN
1
t∗N
 , (3.50)
we obtain
1
tN
=
1
t
sin(Nβ)
Nβ
− sin[(N − 1)β]
Nβ
rN
tN
=
r
t
sin(Nβ)
Nβ
.
(3.51)
For a N− period structure we can express the real and imaginary parts of
tN = xN + iyN =
√
TN exp(iφN) in terms of the constituents of t = x+ iy as
follows
xN =
x sin(Nβ) sin β − (x2 + y2) sin β sin[(N − 1)β]
sin2(Nβ)− 2x sin(Nβ) sin[(N − 1)β] + (x2 + y2) sin2[(N − 1)β]
yN =
y sin(Nβ) sin β
sin2(Nβ)− 2x sin(Nβ) sin[(N − 1)β] + (x2 + y2) sin2[(N − 1)β]
(3.52)
The Bloch-phase determines the pass-bands and stop-bands of the infinite-
length structure, which are very close to the bands of the N− period po-
tential mirror, exactly as it happens with chiral nematics. The Bloch phase
is real when |<{1/t}| ≤ 1 and complex when |<{1/t}| > 1 with β = iθ for
<{1/t} > 1 and β = pi+iθ for <{1/t} < −1. In the pass-bands, the intensity
transmission coefficient TN varies sinusoidally with the Bloch phase. Taking
the squared modulus of the second equation from (3.51) we obtain
1− |tN |2
|tN |2 =
1− |t|2
|t|2
sin(Nβ)
Nβ
⇒ TN = |tN |2 = 1 + sin
2(Nβ)
sin2 β
(
1
|t|2 − 1
)
,
(3.53)
3.1. THE RÔLE AND THE CALCULATION OF THE DOS 81
which shoes that TN is periodic in β with period pi/N . In the pass-band,
β traverses a distance of pi. In the first pass-band β ∈ [0, pi] in the second
β ∈ [pi, 3pi] and so forth. If we make the substitution zN = yN/xN , then the
DOS for the N− period structure can be calculated as follows
ρN =
dkN
dω
=
1
L
d
dω
arctan(zN) =
1
L
z′N
1 + z2N
, (3.54)
with L = Nd being the total length of the structure. Since the expressions
for xN and yN have the same denominator, we can write
zN =
y sin(Nβ)
x sin(Nβ)− (x2 + y2) sin[(N − 1)β] . (3.55)
Since we have defined cos β = <(1/t) we can use the scalar quantities η1 ≡
cos β = x/|t|2 and η2 = y/|t|2 to recast zN into the simpler form
zN =
z cos β sin(Nβ)
cosβ sin(Nβ)− sin[(N − 1)β] =
z cos β sin(Nβ)
cos(Nβ) sin(β)
= z cot β tan(Nβ),
(3.56)
where z = y/x. Substitution into the expression (3.54) yields
ρN =
1
L
1
2
[
sin(2Nβ)
sin β
] [
η′2 +
η1η2η
′
1
1− η21
]
−N η2η
′
1
1− η21
cos2(Nβ) + η22
[
sin(Nβ)
sin β
]2 . (3.57)
In many cases, the unit cell of the periodic structure has the form of a series
of steps, namely n(x) = ni, where x ∈ [xi−1, xi), with i = 1, 2...m and x0 = 0,
xm = d. We assume that ni do not depend on the frequency. The transfer
matrix for such a potential will be a product of matrices of two types, exactly
as in the case of light propagation in a LC slab we examined in the previous
82 CHAPTER 3. THE DENSITY OF PHOTON STATES
chapter. The first is the so-called `discontinuity' matrix ∆ij providing infor-
mation regarding reflection and transmission between the interfaces i and j.
The second type is a propagation matrix P(pi), where pi = nibi(ω/c) is the
phase accumulated as the light wave propagates from left to right traversing
a distance bi = |xi − xi−1| where the refractive index ni is constant. The
aforementioned matrices have the form
∆ij =
δ+ij δ−ij
δ−ij δ
+
ij
 , (3.58)
where δ±ij =
1
2
(1± ni/nj) and
P(pi) =
exp(ipi) 0
0 exp(−ipi)
 . (3.59)
Comparison between the form of (3.42) and ∆ij allows the identification
δ+ij = 1/tij and δ
−
ij = rij/tij. For normal incidence the elements are given by
the familiar expressions
tij =
2nj
ni + nj
,
rij = −rji = ni − nj
ni + nj
(3.60)
We define the double-transmission and double-reflection coefficients on going
across interfaces 1 and 2 as follows
R12 = −r12r21 =
(
n1 − n2
n1 + n2
)2
,
T21 = t21t12 =
4n1n2
(n1 + n2)2
.
(3.61)
3.1. THE RÔLE AND THE CALCULATION OF THE DOS 83
In order to construct the transfer matrix M for a two layer unit-cell with a
step-like refractive index profile
n(x) =
n1, x ∈ [0, a)n2, x ∈ [a, a+ b) . (3.62)
The matrix propagating from left to right is
M−1 = P(p)∆12P(q)∆21 ⇒
M = ∆12P(−q)∆21P(−p),
(3.63)
since P−1(p) = P(−p) and ∆ij = ∆−1ji . Here we define p = n1aω/c and
q = n2bω/c, with d = a + b being the period of the stack. Performing the
calculations we obtain
1
t
= M11 = δ
+
12δ
+
21 exp[−i(p + q)] + δ−12δ−21 exp[−i(p− q)]
r
t
= M21 = δ
+
12δ
−
21 exp[−i(p + q)] + δ−12δ+21 exp[−i(p− q)],
(3.64)
with M22 = M
∗
11 and M12 = M
∗
21.
For a two-layer unit cell, we thus have
1
t
=
1
4n1n2
[
(n1 + n2)
2 exp[−i(p + q)]− (n1 − n2)2 exp[−i(p− q)]
]
=
(n1 + n2)
2
4n1n2
exp[−i(p + q)]
[
1−
(
n1 − n2
n1 + n2
)2
exp(2iq)
]
⇒ t = T21 exp[−i(p + q)]
1−R12 exp(2iq) .
(3.65)
The real and imaginary parts of t for an arbitrary two-layer unit cell read
84 CHAPTER 3. THE DENSITY OF PHOTON STATES
x =
1
2
(t+ t∗)
=
1
2
T12
exp[−i(p + q)][1−R12 exp(2iq)] + exp[i(p + q)][1−R12 exp(−2iq)]
[1−R12 exp(−2iq)][1−R12 exp(2iq)]
= T12
cos(p + q)−R12 cos(p− q)
1− 2R12 cos(2q) +R212
.
(3.66)
Likewise,
y =
1
2i
(t− t∗)
=
1
2i
T12
exp[i(p + q)][1−R12 exp(−2iq)]− exp[−i(p + q)][1−R12 exp(2iq)]
[1−R12 exp(−2iq)][1−R12 exp(2iq)]
= T12
sin(p + q)−R12 sin(p + q)
1− 2R12 cos(2q) +R212
.
(3.67)
3.1.1 The quarter-wave stack
We will now focus on the special case of a quarter-wave stack (QWS), for
which n1a = n2b = λ0/4 = pic/(2ω0). This case results in a significant
simplification, as we can deduce from the expressions for x and y, since then
p = q = piω/(2ω0). Therefore, we can write
η1 =
cos
(
pi
ω
ω0
)
−R12
T12
, η2 =
sin
(
pi
ω
ω0
)
T12
,
η′1 = −
pi
ω0
sin
(
pi
ω
ω0
)
T12
, η′2 =
pi
ω0
cos
(
pi
ω
ω0
)
T12
,
(3.68)
required for determining ρN . The latter is normalized to the bulk group
3.1. THE RÔLE AND THE CALCULATION OF THE DOS 85
velocity, defined as the period divided by the time required by the light to
traverse it neglecting reflections. For a QWS this quantity is
v
bulk
=
1
ρ
bulk
=
a+ b
an1
c
+
bn2
c
= c
(
1
n1
+
b
a
1
n1
)
= c
(
1
n1
+
1
n2
)
. (3.69)
The peaks of ρN and TN nearly coincide, with the approximation improving
with increasing N . In the left pass-band, the latter occur at βN = mpi, with
m ∈ 0, 1, ...N − 1 and at these values of the Bloch phase the denominator of
ρN attains nearly its smallest value. The approximation improves for large N
since then η2 → 0. Since at these values of β, sin(2Nβ) = 0, the numerator
is proportional to N . At these approximate maxima, the expression for the
DOS is
ρmaxN
∼= − 1
Nd
N
η2η
′
1
1− η21
∣∣∣∣
β=mpi
N
. (3.70)
For a QWS, the period of the structure can be written as
d = a+ b =
pic
2ω0
(
1
n1
+
1
n2
)
=
pi
ρ
bulk
ω0
. (3.71)
We can also write
sin2
(
pi
ω
ω0
)
= 1− (T12 cos β + 1− T12)2
= T 212(1− cos β)2 − 2T12(1− cos β)
= 4T 212 sin
4
(
β
2
)
− 4T12 sin2
(
β
2
) (3.72)
and
86 CHAPTER 3. THE DENSITY OF PHOTON STATES
1− η21 = sin2 β = 4 sin2
(
β
2
)
cos2
(
β
2
)
. (3.73)
The DOS maxima for a QWS are then
ρλ/4
max
∣∣
β=mpi
N
= ρ
bulk
1− T12 sin2
(mpi
2N
)
T12 cos2
(mpi
2N
) . (3.74)
For the closest edge to the band-gap (called the `long-wavelength edge' below)
m = N−1 (corresponding to the last maximum in the pass-band), and when
N →∞ we can keep in our expression terms up to the second order as below
ρλ/4
max
∣∣
β=
(N−1)pi
N
= ρ
bulk
1− T12 cos2
( pi
2N
)
T12 sin
2
( pi
2N
) ' ρ
bulk
1− T12
(
1− 1
2
( pi
2N
)2)2
T12
( pi
2N
)2
' ρ
bulk
1− T12 + T12
( pi
2N
)2
T12
( pi
2N
)2 = ρbulkR12 + T12
( pi
2N
)2
T12
( pi
2N
)2
∝ R12
T12
N2 ∝ (n1 − n2)
2
n1n2
N2.
(3.75)
Regarding the latter symmetric expression with respect to n1, n2 as a func-
tion of n2 (without loss of generality) with fixed n1 < n2 then the above
function increases with increasing difference n2−n1. Keeping fixed the prod-
uct n1n2 we find that for N sufficiently large, the maximum value of the
DOS at the LWE is determined by the product (n1 − n2)N = (∆n)N . In-
creasing any of these terms of the product leads to enhanced feedback in the
distributed Bragg-structure. The above conclusions are verified by Figure
3.1 that follows.
3.1. THE RÔLE AND THE CALCULATION OF THE DOS 87
(a) (b)
(c) (d)
(e) (f)
(g) (h)
Figure 3.1: Quarter Wave Stack. In (a), (b) Intensity Reflection Coefficient and DOS respectively for
n1 = 1, n2 = 2 and N = 10. In (c), (d) Intensity Reflection Coefficient and DOS respectively for n1 = 1,
n2 = 2 and N = 15. In (e), (f) Intensity Reflection Coefficient and DOS respectively for n1 = 1, n2 = 2
and N = 10 for oblique propagation in the structure such that k⊥ = (
√
2/2)k0. In (g) Intensity Reflection
for n1 = 1 − 0.05i, n2 = 2 − 0.05i and N = 10 and in (h) Maximum value of DOS as a function of the
number of layers N for a stack with n1 = 1, n2 = 2 (black curve) and for a stack with n1 = 1, n2 = 2.5
(red curve).
88 CHAPTER 3. THE DENSITY OF PHOTON STATES
At the mid-gap (mg) frequency, the Bragg-phase is βmg = pi+ iθmg, such that
cos β has there an extremum. This implies that at ω = ω0, η
′
1 = 0. For the
Bragg-phase we write
cos βmg = cos(pi + iθmg) = − cosh θmg = cospi −R12
T12
⇒
cosh θmg =
1 +R12
T12
=
n21 + n
2
2
2n1n2
=
1
2
(
n2
n1
+
n1
n2
)
.
(3.76)
Without loss of generality we consider again n2 > n1 and set r = exp(a) =
n2/n1 > 1, so that
cosh θmg =
exp(a) + exp(−a)
2
= cosh(a)⇒ θmg = a = ln
(
n2
n1
)
= ln r.
(3.77)
The DOS at the midgap frequency (where η′1 = 0) is
ρN
∣∣
η′1=0
= − 1
L
η′2
2
sinh(2Nθmg)
sinh θmg
cosh2Nθmg + η22
(
sinhNθmg
sinh θmg
)2 . (3.78)
For N →∞ we can approximate sinh(Nθmg) = 12 exp(Nθmg) and then write
the DOS as
ρN = − 1
Nd
1
4
exp(2Nθmg)
sinh θmg
η′2
1
4
(
exp(2Nθmg) + η
2
2
exp(2Nθmg)
sinh2(Nθmg)
) = − 1
Nd
η′2 sinh θmg
η22 + sinh
2 θmg
.
(3.79)
For a QWS, in the mid-gap where η2 = 0, expression (3.78) can be recast in
the simpler form
3.2. THE DOS IN RADIATIVE CHIRAL NEMATICS 89
ρ
λ/4
N = −
1
2Nd
pi
ω0T12
2 sinh(Nθmg) cosh(Nθmg))
cosh2(Nθmg)
=
ρ
bulk
NT12
sinh(Nθmg)
cosh(Nθmg) sinh θmg
∝ 1
NT12
rN − r−N
rN + r−N
.
(3.80)
As N →∞ then (rN − r−N)/(rN + r−N)→ 1 and we obtain the asymptotic
behaviour 1/N .
3.2 The DOS in radiative chiral nematics
For chiral nematic liquid crystals Dirac's rule can be stated to account for
the modification of the fluorescence of dye molecules embedded in the so-
called `mirror-less' cavity with a given photonic density of states (DOS). The
relative fluorescence intensity, then reads [3]
Ij =
ρj
ρ
iso
∫ L
0
〈|e∗j dˆ|2〉cndz∫ L
0
〈|e∗i dˆ|2〉isodz
, (3.81)
where j = 1(2) denotes the diffracting (non-diffracting) eigenwave, the brack-
ets denote the spatial average over the orientational distribution of the transi-
tion dipole moment. If we assume that the fluorescent molecules are homoge-
nously distributed in the LC slab, the spatial dependence of the eigenwaves
is not important and the integrals can be omitted. After having calculated
the DOS for each of the eigenwaves, we need to determine the orientational
average of the squared projections of the transition dipole moment d on the
unit vectors eˆi. If θ is the angle between the aforementioned vectors, we
90 CHAPTER 3. THE DENSITY OF PHOTON STATES
characterize the degree of order of the transition dipole moment by the order
parameter Sd defined as follows
Sd =
3
2
〈cos2 θ〉 − 1
2
, (3.82)
in a similar fashion to the orientational order parameter in uniaxial nematic
liquid crystals (average of the second Legendre polynomial). It is found in
[3] that
〈|ei · d|2〉 = 2
3
f 2i −
1
2
f 2i + 1
Sd +
1
3
, (3.83)
where fi is the degree of ellipticity for each eigenwave, as defined in the pre-
vious chapter. In order to calculate the transmission properties of the helical
structure we use the relative dielectric anisotropy δ at optical frequencies
and the reduced wavelength λ′ that we have defined in the previous chapter.
Without loss of generality we may assume that p > 0 which corresponds
to a right-handed helical structure [3]. The dielectric anisotropy at optical
frequencies is an alternative measure of the birefringence of the LC host and
an implicit indication of feedback strength in an analogous way to the mirror
reflectivities in a FP resonator.
3.2.1 Inclusion of absorption
When considering absorption in a dye-doped LC laser configuration we must
keep in mind that the same host is an absorbing medium for the pump beam
and simultaneously for the spontaneous emission of the fluorescent dyes. For
the dye emission (as well as for the pumping beam propagation inside the
3.2. THE DOS IN RADIATIVE CHIRAL NEMATICS 91
active medium), one has to take into account various loss mechanisms in-
cluding linear absorption, light scattering from imperfections, cavity losses
due to light escaping from the cavity and Förster resonance energy trans-
fer when two chromophores are involved [6]. While spontaneous emission in
free space is considered as randomly polarized, we assume that fluorescence
is coupled to a particular normal mode with specific polarization proper-
ties in the presence of a chiral nematic LC medium. Both losses and gain
can be incorporated into our analysis through introducing a small imaginary
part to the dielectric constants parallel and perpendicular to the director, as
γ‖ and γ⊥ , respectively, altogether amassed to the dimensionless constant
γ = γ‖ + γ⊥ for which we assume no frequency dispersion and |γ|  1,
following a similar treatment to [7]. For a complex dielectric constant, the
diffraction coefficient for the diffracting eigenwave (for convenience we drop
the suffix j = 1) is modified to
T ∝ A(m,λ) exp(ikNp)
1− r2 exp(2ikNp) = A(m,λ)T
′(λ,m,N), (3.84)
where k = ka(λ) + ikb(λ) =
2pim
λ
and r = ra(λ) + irb(λ) = −
n− λ
p
f −m
n+
λ
p
f +m
,
with
f =
λp
[
ε⊥ + iγ⊥
λ2
− 1
p2
−
(m
λ2
)]
2m
=
2m
λp
[
ε‖ + iγ‖
λ2
− 1
p2
−
(m
λ2
)] . (3.85)
92 CHAPTER 3. THE DENSITY OF PHOTON STATES
Based on the de Vries formulation the modified `effective' refractive index is
the solution of the algebraic equation
m4 −m2
[
ε‖ + ε⊥ + iγ + 2
(
λ
p
)2]
+
[
ε⊥ + iγ⊥ −
(
λ
p
)2][
ε‖ + iγ‖ −
(
λ
p
)2]
= 0.
(3.86)
The real and imaginary parts of T ′(λ,m,N) can be written as
X ′ = − cos(kaNp)r2a + 2rarb sin(kaNp) + cos(kaNp)r2b + exp(2kbNp) cos(kaNp),
Y ′ = sin(kaNp)r2a + 2rarb cos(kaNp)− sin(kaNp)r2b + exp(2kbNp) sin(kaNp),
(3.87)
respectively, where we have omitted common real prefactors and wavelength
independent terms. The expression for A(m,λ) can be obtained from the
boundary conditions in the glass-chiral nematic and chiral nematic-glass in-
terface [8] and in our case reads
A(m,λ) =
λ
p
f +m
λ
p
f +m+ n
, (3.88)
where n =
√
ε¯. It is then possible to calculate the DOS of the chiral ne-
matic structure using (4.19). If we consider only the diffracting wave, the
eigenfield to which emission is coupled is E1 = exp(ikz)eˆ1 (where eˆ is the
unit polarization vector, which is wavelength dependent as we have shown).
Regarding the sign of the small additive imaginary term to the dielectric
constant, γ > 0 and γ < 0 correspond to losses and gain, respectively. As
we have no prior knowledge to assume otherwise, we assume that the losses,
3.2. THE DOS IN RADIATIVE CHIRAL NEMATICS 93
etc., are the same for both the dielectric constants parallel and perpendicular
to the director, i.e., γ⊥ = γ‖. This is the case when spontaneous emission
is isotropic in terms of polarization and it is perhaps reasonable to assume
that the stimulated absorption and emission are also isotropic. A similar
approach is also followed in [9], where equal small imaginary parts are added
to the ordinary and extraordinary refractive indices of the chiral nematic LC
slab in order to account for gain in an active cell and therefore calculate the
corresponding transmission coefficient of a defect mode structure.
If we assume an isotropic absorption of the dye we can see directly from
equation (3.81) that the relative fluorescent intensities are then only directly
proportional to the DOS when the spatial dependence of the normal modes
is deemed unimportant in the calculation of the orientational average of the
electric dipole moment of the gain medium [3]. On the other hand, one can
consider that absorption is characterized by dichroism as in [10] because the
dye molecules (usually rod-like with a length exceeding that of the liquid
crystal molecules) tend to adopt to some extent the local nematic order.
However, in our analysis we will not take into account dichroism in the ab-
sorption/fluorescence of the gain medium as we will assume the dye order
parameter Sd → 0, pertaining to the case of an isotropic dye. In that case, the
relative intensity contributions for each eigenmode are directly proportional
to the DOS. When taking into account losses and gain we employ relation
(3.84) to calculate the transmission coefficient and then equation (4.19) for
the corresponding DOS of the diffracting eigenwave.
Examples of the normalized DOS for the diffracting eigenwave as a function
of the reduced wavelength are depicted in Figure 3.2 for different values of δ
94 CHAPTER 3. THE DENSITY OF PHOTON STATES
0.85 0.9 0.95 1 1.05 1.1 1.15
0
5
10
15
|λ′|=|λ/(np)|
ρ
1
/ρ
is
0.85 0.9 0.95 1 1.05 1.1 1.15
0
2
4
6
8
|λ′|=|λ/(np)|
ρ
1
/ρ
is
0.85 0.9 0.95 1 1.05 1.1 1.15
0
1
2
3
4
|λ′|=|λ/(np)|
ρ
1
/ρ
is
0.85 0.9 0.95 1 1.05 1.1 1.15
0
1
2
3
4
|λ′|=|λ/(np)|
ρ
1
/ρ
is
(a) (b)
(c) (d)
SWE LWE
Figure 3.2: Theoretical results of the normalized DOS of the eigenwave E1(ρ1/ρ
iso
) for a chiral nematic
cell as a function of reduced wavelength for different values of the loss coefficient and the optical anisotropy.
For all plots, the cell thickness was fixed at L = 30p. (a) n = 1.581, relative dielectric anisotropy at optical
frequencies δ = 0.1, loss coefficient γ = 2γ‖ = 0.0002, (b)n = 1.581, relative dielectric anisotropy at optical
frequencies δ = 0.1, loss coefficient γ = 2γ‖ = 0.002 (c)n = 1.541, relative dielectric anisotropy at optical
frequencies δ = 0.0526, loss coefficient γ = 2γ‖ = 0.0002(d)n = 1.541, relative dielectric anisotropy at
optical frequencies δ = 0.0526, loss coefficient γ = 2γ‖ = 0.002.
3.2. THE DOS IN RADIATIVE CHIRAL NEMATICS 95
and the loss coefficient γ. Figures 3.2(a) and 3.2(b) are for the same optical
anisotropy but with different magnitudes of the losses. On the other hand,
Figures 3.2(c) and 3.2(d) show the profile of the DOS for the same values of
the loss coefficient but a smaller optical anisotropy. As it can be seen from
the plots, the DOS decreases dramatically with increasing losses, regardless
of the magnitude of the optical anisotropy, which is a result that can be
shown for a FP resonator in the presence of losses [6]. Additionally, the
DOS is larger at the band-edges for a greater optical anisotropy with the
same number of pitches and loss coefficient. In this example, as an order of
magnitude increase in the loss coefficient (from γ = 0.0002 to γ = 0.002)
results in a reduction of the DOS by a factor of 2. The decrease in the DOS
has also been noted when an imaginary term is included only in the dielectric
constant parallel to the director [6]. In such a case, the short wavelength edge
(SWE) is unaffected by the loss factor as opposed to the long wavelength edge
(LWE) whose value is diminished.
What is interesting to note is the reversal of the maximum value of the
normalized DOS between the SWE and LWE. Unlike the DOS in a quarter-
wave stack [2], the DOS profile of a non-absorbing chiral nematic LC is
non-symmetric with respect to the centre of the stop-band.
Figure 3.3 shows the ratio of the difference of DOS at the SWE and the LWE
to the isotropic DOS as a function of γ. For small values of γ, the maximum
value of ρ occurs at the SWE. As γ increases, the difference in ρ at the two
edge modes (EM) approaches zero before the LWE dominates for large losses.
Therefore, when choosing the optimum EM for laser emission it is important
to consider the losses in addition to the projection of the optical field onto the
96 CHAPTER 3. THE DENSITY OF PHOTON STATES
0.00 0.01 0.02
-0.5
0.0
0.5
1.0
 
 
(
S
W
E
 
-
 
L
W
E
)
/
i
s
o
loss coefficient, 
Figure 3.3: The asymmetry in the DOS at the first edge modes either side of the photonic band gap of a
chiral nematic LC. A plot of the difference of the maximum DOS between the short and long wavelength
edges (ρ
SWE
− ρ
LWE
) of the eigenwave E1 that is normalized to isotropic DOS as a function of the loss
coefficient. In this case n = 1.581, the optical anisotropy is δ = 0.1 and the cell thickness is L = 30p.
3.2. THE DOS IN RADIATIVE CHIRAL NEMATICS 97
transition dipole moment of the laser dye. It should be noted that the profile
is symmetric with respect to the centre of the stop band occurring at λ′ = 1
for an infinitely thick non-absorbing cell and that the SWE and LWE are
located at λ′ =
√
1− δ and λ′ = √1 + δ , respectively [3]. For a finite cell,
however, the position of the SWE is displaced towards shorter wavelengths
(and the position of the LWE is displaced towards longer wavelengths) [3].
As losses become more dominant, we find that the DOS peaks broaden and
are further shifted, in direct analogy with the behaviour of the resonance
peak in damped steady state oscillating systems.
The variation of the value of the maximum DOS at either wavelength edge for
different loss coefficients in cells with varying thickness is considered in Figure
3.4. The position of that maximum is found to shift to smaller values of the
pitch with greater optical anisotropy, which is an indication of enhanced
feedback. In particular, in this case we can observe that, when losses are
present, the position of the maximum value DOS of the LWE is L = 40p for
a cell with optical anisotropy δ = 0.1 [Figure 3.4(b)], which is reduced to the
value of L = 35p for a 25% increase in δ [Figure 3.4(d)]. These results also
highlight the interchange between the SWE and LWE as the value of γ is
increased. For example, in Figure 3.4(a) the SWE exhibits the largest value
of ρ for all values shown of the pitch. However, when sufficiently large losses
are incorporated, the LWE exhibits the largest DOS for sufficiently large N .
The same occurs for a larger optical anisotropy (birefringence) although the
maximum DOS is found to be slightly higher. Regarding the variation of the
maximum value of the DOS with cell thickness the relation
98 CHAPTER 3. THE DENSITY OF PHOTON STATES
Figure 3.4: Theoretical plots of the maximum normalized DOS (ρ
Max
/ρ
iso
) of the eigenwave E1 at the two
wavelength edges either side of the stop-band as a function of the chiral nematic LC. The cell thickness is
L = Np. (a) for n = 1.581 δ = 0.1, and γ = 2γ‖ = 0.0002, (b) for n = 1.581 δ = 0.1, and γ = 2γ‖ = 0.006
(c) for n = 1.603 δ = 0.125, and γ = 2γ‖ = 0.0002, (d) for n = 1.603 δ = 0.125, and γ = 2γ‖ = 0.006.
The arrows mark the number of pitches where the peak DOS is attained for each wavelength edge. The
short-wavelength and long-wavelength edges are shown as solid and dashed lines, respectively.
3.2. THE DOS IN RADIATIVE CHIRAL NEMATICS 99
ρ
max
∝ L2 exp(−βL) (3.89)
has previously been proposed to pertain to the experimental data of the
energy-excitation threshold of cells as a function of L = Np with N be-
ing the length of the cell and beta the collective absorption coefficient [6].
The general trend suggested by equation (3.89) is indeed vindicated by our
theoretical findings: the deviation from the parabolic profile is more osten-
sible with increasing losses and a maximum is observed whose position and
value depend on both the loss coefficient and the optical anisotropy. The
parabolic dependence of the DOS at the wavelength of the photonic band
edges is a known result for large N in quarter-wave stacks [2] as well as for
non-absorbing chiral nematic LCs [6].
Experimentally, the dependence of the DOS upon the number of pitches for
different optical anisotropies is verified, albeit indirectly, by determining the
excitation threshold as a function of cell thickness, as depicted in Figure
3.5(a). As shown in [6] the excitation threshold is inversely proportional to
ρ for a Fabry-Pérot resonator. The figure shows the excitation threshold as
a function of the cell thickness for two different LC laser samples. These
measurements were obtained using two different laser samples: one consist-
ing of the nematic LC E7 (δ = 0.13 and ∆n = 0.2 at 25oC) and the other
with the nematic LC mixture E49 (δ = 0.15 and ∆n = 0.25 at 25oC). The
optical anisotropies were determined from the refractive indices parallel and
perpendicular to the director of the nematic sample at a fixed wavelength of
λ0 = 589.6 nm using an Abbe refractometer. Both mixtures were doped with
a high twisting power chiral dopant (BDH1281, Merck KGaA) and the laser
100 CHAPTER 3. THE DENSITY OF PHOTON STATES
Figure 3.5: (a) Experimental results of the dependence of the laser excitation threshold on the cell thickness
for a low optical anisotropy (δ = 0.13) (closed squares) and a high optical anisotropy (δ = 0.15) (open
circles) LC host. (see A. D. Ford, PhD thesis, 2006) (b) Theoretical plots of the DOS for the long-
wavelength band-edge for two different chiral nematic LCs for the same optical anisotropies as those used
in the experiment. The loss coefficient in this case is γ = 0.0084 for n = 1.608 and n = 1.627 respectively.
3.2. THE DOS IN RADIATIVE CHIRAL NEMATICS 101
dye DCM (Exciton). The samples were capillary filled into wedge cells that
were fabricated in-house and were coated with a rubbed polyimide alignment
layer to ensure that a Grandjean texture was obtained after filling. Each cell
was optically excited by the second harmonic of an Nd:YAG laser (Polaris II,
New Wave Research) at positions within the wedge cell for which the dimen-
sions of the cell gap were known. In both cases the emission wavelength and
the pitch (p ' 350 nm) were the same but the main difference was the optical
anisotropy of the nematic host. The profile of the plot resembles the inverse
of the dependence of the DOS on the number of pitches confined within the
device (c.f., Figure 3.4) as suggested in [6]. The minimum in the excita-
tion threshold is found to occur at smaller cell thicknesses (corresponding
to a smaller N required for the maximum value of the DOS) for the higher
optical anisotropy compound (E49) than that obtained for the laser sample
with a lower optical anisotropy (E7). In this case, the minimum excitation
threshold is found to occur at L = 12.5 µm for the sample consisting of E7
(δ = 0.13) and is shifted to L = 10 µm when the LC host is replaced with E49
(δ = 0.15). For the purposes of comparison, theoretical curves for the DOS
for two different samples with the same optical anisotropy as E7 and E49 are
shown in Figure 3.5(b) for a suggested value of γ = 0.0084. Theoretically, we
find that the number of pitches required to obtain the maximum DOS for the
higher optical anisotropy sample is reduced from N = 32 to N = 28, which
is in good agreement with the experimental results whereby the difference in
the number of pitches for the two lasers is found to be ∆N ' 5.
102 CHAPTER 3. THE DENSITY OF PHOTON STATES
3.2.2 Inclusion of amplification
We will now take into consideration the gain of the active material in the
process of laser emission. For an amplifying medium with γ < 0 we find
that the DOS of various EMs diverges at particular values of γ, which are
the lasing excitation gain coefficients for these modes. The behaviour is
consistent with the case of an active medium filling a FP resonator, where
the DOS diverges at threshold [11].
In Figure 3.6 we depict the DOS of an amplifying chiral nematic LC for
different values of γ. For |γ|  1 and for sufficiently thick cells, the threshold
gain coefficient γth/n
2
is found to be inversely proportional to δ2 as well as
to L3, as we have shown in the previous chapter. As the threshold value for
a chiral nematic LC with a given thickness is approached, the DOS rapidly
diverges suggesting a theoretically infinite group time of photons residing in
the resonating cavity. In actuality, the DOS does not reach infinity due to
the finite lifetime of the excited state or due to the collision with a phonon
[10]-[12]. Its value depends on the microscopic properties of the sample and
the duration of the pumping. If the gain factor exceeds slightly the threshold
value the DOS is still very high and the phase derivative changes sign [11].
Such values of γ belong to the regime of superamplification. As the gain
coefficient is further increased the EM that first attained threshold (in one
of the wavelength edges) is now quenched and laser action is triggered for
successive EMs [Figure 3.6(c)]. Hence, as the author notes in [10] and we
have mentioned in the previous chapter, the region of superamplification
does not only correspond to the case where losses are slightly higher than
the active medium gain but also to the case in which they are marginally
3.2. THE DOS IN RADIATIVE CHIRAL NEMATICS 103
overcompensated. That behaviour can be traced also in the transmission and
reflection coefficients in the region of anomalous absorption for amplifying
chiral nematic LCs and other DFB structures [7]-[12].
Our results are in good agreement with the behaviour of the threshold val-
ues obtained from the maximisation of transmittance and the solvability
condition of the system produced by satisfying the continuity conditions of
tangential electric field components in a boundary value problem formula-
tion for light propagation in chiral nematic LCs [7]. As we also observe in
Figure 3.6(d), a smaller gain constant is required for the DOS of the lasing
mode to diverge in a chiral nematic LC with greater birefringence, which is
an indication of enhanced feedback.
Using the semi-classical laser theory, once the threshold gain has been
identified, it can be directly linked to the critical population inversion ∆N e
th
via [12]
γ
th
∝ g
th
= ∆eN
th
λ2
8pin20τsp
g(ω), (3.90)
where g(ω) is the linewidth function, n0 is the refractive index of the medium
away from resonance, and τ
sp
is the spontaneous lifetime of the radiative
transition. For dye-doped cells, the population inversion is equal to threshold
gain divided by the molecular cross section for spontaneous emission, which
is experimentally determined from the luminescence spectrum (under low
pumping) of the fluorescent dyes normalized to the ratio Φ/(8τ
sp
), where Φ
is the fluorescence quantum yield [13].
The vector potential operator in a cavity sustaining transverse modes is ex-
pressed as [15]
104 CHAPTER 3. THE DENSITY OF PHOTON STATES
Figure 3.6: The behaviour of the DOS in the presence of gain. Normalized DOS of the eigenwave E1 for
a chiral nematic LC cell with thickness L = 30p, n = 1.523, δ = 0.03 and (a) γ = 2γ‖ = −0.0304, (b)
γ = 2γ‖ = −0.0320 , (c) γ = 2γ‖ = −0.0540 (d) n = 1.541, δ = 0.05206 and γ = 2γ‖ = −0.0134.
Aˆ(x) =
∑
k,λ=1,2
√
~
20ωkV
[
ak,λ exp(ik · x)eˆkλ + a†k,λ exp(ik · x)eˆ∗kλ
]
, (3.91)
where a time dependence exp(−iωkt) is assumed and λ = 1, 2 denotes the
polarization of the plane waves in the cavity(two degrees of freedom). This
operator has non-zero matrix elements only for each mode where the photon
(occupancy) number differs by one, consistent with first-order perturbation
theory. In the dipole approximation, if Nk,λ photons are present in a cavity
then the emission rate is
3.2. THE DOS IN RADIATIVE CHIRAL NEMATICS 105
Γi→f =
piωk
0V
〈∣∣eˆ · d∣∣2〉(Nk,λ + 1)δ(Ei − Ef − ~ω), (3.92)
in agreement with (3.34). The above is still a first-order perturbative result
in the context of the second quantization, whereby the number of photons
has been extracted by the ladder operator ak,λ acting on the initial state
with Nk,λ photons. In the framework of the first-quantization, the role of the
photon number in stimulated emission is played by the DOS (as a distribution
function) multiplied by the total number of photons in the cavity (of all
frequencies), thus representing the selective amplification of the field in a
resonator with Bragg-reflection (which is the case in chiral nematics). Hence,
although the DOS diverges for particular values of amplification (threshold
values) in the presence of a gain medium, formula (3.81) holds within first-
order perturbation theory.
We consider a system with two weakly interacting parts. The parts have
definite energies at a particular instant, denoted by E and  respectively. If
the energy is measured again after some time interval τ = ∆t, the values
E ′, ′ are obtained and are different in general from E, . We will use the
results of first-order perturbation theory to determine the order of magnitude
of the most probable value of the difference E ′ + ′ − E − . According to
relation (3.19) the transition probability from a state with energy E to a state
with energy E ′ under the action of a weak time-independent perturbation
(ω = 0)is proportional to [5]
sin2
[
(E − E ′)t
2~
]
(E ′ − E)2 , (3.93)
106 CHAPTER 3. THE DENSITY OF PHOTON STATES
from which we infer that the most probable value of the energy difference
E − E ′ is of the order of ~/t. Applying this result to our system with the
two weakly interacting parts, we obtain the relation
|E + − E ′ − ′|∆t ∼ ~. (3.94)
The quantities , ′ are related to the measuring particle/apparatus and are
supposed to be known exactly, so that |∆E−∆E ′| ∼ ~/τ . The same conclu-
sion can be reached from another standpoint, by considering the transition
of one system from one state to another, under the action of a perturbation.
If E0 + 0 is some energy level of the system (`atom' and field photon of the
initial frequency) and E,  are the energies of the two parts into which the
system decays (`atom' and photon of the final frequency), then we find again
|E0 − E ′ − ′|τ ∼ ~, (3.95)
where τ is the reciprocal of the decay rate (called also the lifetime of the
state). Since the sum E +  is the estimate of the energy of the system
before it decays, the above result shows that the energy of a system is some
quasi-stationary state can be determined only up to ~/τ . This quantity is
inversely proportional to the spectral width of emission. The experimentally
determined emission spectral width (full width at half-maximum, FWHM)
of both fluorescence and lasing can be directly related to the DOS via the
uncertainty principle ∆τ∆ω ∝ pi, bearing in mind that the DOS is linked
to the characteristic time ∆τ as [11]-[14] ρ = ∆τ/L ∝ pi/(L∆ω). As the
behaviour of the DOS can also reflect mode quenching in lasing structures
in the regime beyond the threshold gain, we can immediately infer that the
3.2. THE DOS IN RADIATIVE CHIRAL NEMATICS 107
FWHM of the lasing radiation profile, corresponding to the dominant mode,
will decrease in the super-amplification regime.
In conclusion, in this chapter we have used the concept of the density of pho-
ton states to study amplified emission in chiral nematic LC samples doped
with a gain medium, within the semi-classical first order perturbation treat-
ment encompassed in Dirac's rule (most commonly referred to as Fermi's
Golden Rule). Absorption and gain were introduced phenomenologically and
were linked to the inhibition and enhancement of the optical feedback in these
structures, respectively, through the density of states. In the next chapter we
will examine the enhancement of spontaneous emission in these resonators,
using one of the most common models employed in quantum optics.
Bibliography
[1] J. P. Dowling and C. M. Bowden, Atomic transition rates in in-
homogeneous media with applications to photonic band structures, Phys.
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[2] J. M. Bendickson, J. P. Dowling, and M. Scalora, Ana-
lytic expressions for the electromagnetic mode density in finite, one-
dimensional, photonic band-gap structures Phys. Rev. E 53, pp. 4107-
4121 (1996).
[3] J. Schmidtke and W. Stille, Fluorescence of a dye-doped cholesteric
liquid crystal film in the region of the stop-band: theory and experiment,
Eur. Phys. J. B 31, pp. 179-194 (2003).
[4] L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields,
Vol. 2, ch. 4, Pergamon Press, 1971.
[5] L. D. Landau and E. M. Lifshitz, Quantum Mechanics (Non-
relativistic theory), Vol. 3, ch. 6, Elsevier, 2007.
108
BIBLIOGRAPHY 109
[6] M. F. Moreira, S. Relaix, W. Cao, B. Taheri, and P. Palffy-
Muhoray, Liquid Crystal Microlasers, Chap. 12, Transworld Research
Network, Kerala, India, 2010.
[7] V. A. Belyakov and S. V. Semenov, Optical Edge Modes in Pho-
tonic Liquid Crystals, JETP, 109, pp. 687-699 (2009).
[8] Hl. de Vries, Rotatory Power and Other Optical Properties of Certain
Liquid Crystals, Acta Cryst., 4, pp. 219-226 (1951).
[9] Y. Zhou, Y. Huang, Z. Ge, L.-P. Chen, Q. Hong, T. X. Wu, and
S.-T. Wu, Enhanced photonic band edge laser emission in a cholesteric
liquid crystal resonator Phys. Rev. E 74, 061705 (2006).
[10] S. P. Palto, Lasing in Liquid Crystal Thin Films, JETP, 103, pp.472-
479 (2006).
[11] S. P. Palto, Liquid Crystal Microlasers, Chap. 8, Transworld Research
Network, Kerala, India, 2010.
[12] A. Yariv and P. Yeh, Photonics: Optical Electronics in Modern Com-
munications, ch. 12, 6th ed., Oxford University Press, US, 2006.
[13] L. M. Blinov, Lasers on cholesteric liquid crystals: Mode density and
lasing threshold JETP Lett. 90, 166-171 (2009).
[14] S. P. Palto, L. M. Blinov, M. I. Barnik, V. V. Lazarev, B.
A. Umanskii, and N. M. Shtykov, Photonics of liquid-crystal struc-
tures: A review Crystallogr. Rep. 56, pp. 667-697 (2011).
110 BIBLIOGRAPHY
[15] V. B. Berestetskii, E. M. Lifshitz, and L. P. Pitaevskii, Quan-
tum Electrodynamics, Vol. 4, ch. 5, Elsevier (Singapore), 2008.
Chapter 4
The Jaynes-Cummings model
As we have seen in the previous chapter, the concept of the density of photon
states (DOS) is regularly employed to study the photonic crystal properties
that determine emission and absorption of electromagnetic radiation of a
given frequency from guest atoms [1]. For the case of (dye-doped) chiral ne-
matic liquid crystal (LC) films, which constitute a representative example of
partial 1D photonic crystals, the link between the behaviour of the DOS and
fluorescence under weak-coupling conditions and in the context of first-order
perturbation theory is attempted in [2], where Dirac's rule (often referred to
as `Fermi's golden rule') is employed to calculate the photon emission rate.
With regard to the interaction of a gain medium and the electromagnetic field
in the resonant cavity, a two-level system coupled to a quantum harmonic
oscillator is frequently described with the Jaynes-Cummings (JC) Hamilto-
nian in which only `resonant' terms feature [3], [4], [5]. Such a consideration
is permissible in the case of near resonance and weak coupling [3]. Both
conditions are satisfied for spontaneous emission in these periodic structures
111
112 CHAPTER 4. THE JAYNES-CUMMINGS MODEL
for small detuning [4]. It should be mentioned here, that through a change
of basis, a much greater range of coupling strength and detuning values can
be allowed [4]. In [5] the JC model is employed to describe emission in the
vicinity of the saddle point of 2D photonic crystals, where the DOS exhibits
a logarithmic peak. In order to analyze the behavior of the DOS near its
logarithmic peak and at the edge of the band gap, one can resort to a critical
point analysis through an expansion in the region of the saddle point [1].
In this chapter, we put the analysis of spontaneous emission from 2D pho-
tonic crystals on a firmer basis providing analytical results, and explore in
more detail the fluorescence properties in chiral nematic LCs, outlining com-
mon features that are attributed to resonance. Moreover, the discrepancy
between the experimentally obtained emission spectra and the theoretically
calculated DOS is addressed. Such a consideration aims to further the under-
standing of spontaneous and induced emission from these distributed feed-
back resonators. The case of chiral nematic LCs is selected because these
structures have the additional advantage of allowing an exact analytic solu-
tion of Maxwell's equations [6].
4.1 Hamiltonian, states and equations of mo-
tion
We will now outline the basic formulation of an atom-field interaction in
a resonant environment, within the framework of the JC model. In many
cases, the frequency of the electric field in a resonant structure is near the
transition frequency of a two-level system. Such a two-level system is called
4.1. HAMILTONIAN, STATES AND EQUATIONS OF MOTION 113
an `atom' for convenience [7]. The JC Hamiltonian for the atom-field system
is written in the form [3], [5]
Hˆ =
1
2
~ω10σˆz + ~
∑
κ
ωκaˆ
†
κaˆκ + i
∑
κ
(µκaˆ
†
κσˆ− − µ∗κaˆκσˆ+), (4.1)
where
µκ = (d · ekl)ω10
√
2pi~
V εωκ
, (4.2)
is the atom-field coupling constant, ω10 is the atomic transition frequency,
ωκ is the electromagnetic mode frequency, σz is the inversion operator, σˆ± =
(σˆx + σˆy)/2 are the raising and lowering operators for Pauli matrices acting
on qubit states, aˆ†κ & aˆκ are the bosonic creation and annihilation operators
for the κth mode, d = er10 is the transition dipole moment, ekl is the unit field
polarization vector and ε is the frequency independent dielectric constant
of the medium with volume V . In writing the Hamiltonian in the form
(4.1), we assume that the energies of the upper and lower states of the atom
are equal and opposite, i.e., E1 = −E0 = (1/2)~ω10. The first term in
the interaction part of the JC Hamiltonian in the so-called `rotating field'
approximation corresponds to the electronic transition from the lower to
the upper level with the absorption of a photon from the field mode, while
the second term represents the reverse process, i.e., the electronic transition
from the upper to the lower atomic level and the emission of a photon that
contributes into the mode field. The remaining terms in the interaction
Hamiltonian (of the form Hˆint = G(σˆ+ + σˆ−)(E∗aˆ†+Eaˆ)) that do not feature
in equation (4.1) correspond to non-resonant, virtual processes [7], namely to
the electronic transition upwards accompanied by the emission of one photon
114 CHAPTER 4. THE JAYNES-CUMMINGS MODEL
and the transition downward accompanied by the absorption of one photon.
Due to their smallness, these terms are usually omitted [7]. We consider the
atom-field system to be described by the wavefunction
∣∣ψ〉 = exp(−iω10t/2)[c1(R, t)∣∣1, {0}〉+∑
κ
exp(iδκt)c
κ
0(R, t)
∣∣0, {1}κ〉] ,
(4.3)
where δκ = ω10−ωκ. This expression is a superposition of the state
∣∣1, {0}〉,
corresponding to the excited state of the atom with the photon occupation
number equal to 0, and the state
∣∣0, {1}κ〉, corresponding to the ground state
with one photon in the κth mode. Hence, the mechanism of spontaneous
emission is quantified. In order to account for cavity losses and linewidth
broadening, we introduce phenomenologically a small imaginary part to the
field and transition frequencies, respectively, as in [5], such that
ωκ = ω´κ − iγ and ω10 = ω´10 − iγ10. (4.4)
Expressions for the imaginary part of the edge mode frequencies (eigenfre-
quencies) in the case of a chiral nematic LC layer can be found in [6]. The
appearance of such an imaginary contribution is related to the energy leak-
age through the surfaces of the LC film. Substituting the wavefunction of
relation (4.3) into the time-dependent Schrödinger equation yields a system
of equations of motion for the time varying amplitudes c1(t) and c
κ
0(t). For
the left-hand side of Schrödinger's equation applied for the state
∣∣ψ〉 in (4.3),
we have
4.1. HAMILTONIAN, STATES AND EQUATIONS OF MOTION 115
Hˆ
∣∣ψ〉 = exp(−iω10t/2)[1
2
~ω10c1(R, t)
∣∣1, {0}〉
− 1
2
~ω10
∑
κ
exp(iδκt)c
κ
0(R, t)
∣∣0, {1}κ〉+ ~∑
κ
ωκ exp(iδκt)c
κ
0(R, t)
∣∣0, {1}κ〉
+ i
∑
κ
µκc1(R, t)
∣∣0, {1}κ〉− i∑
κ
µ∗κ exp(iδκt)c
κ
0(R, t)
∣∣1, {0}〉].
(4.5)
The right-hand side of Schrödinger's equation reads
i~
∂ψ
∂t
= i~ · exp(−iω10t/2)dc1
dt
(R, t)
∣∣1, {0}〉
+
1
2
~ω10 · exp(−iω10t/2)c1(R, t)
∣∣1, {0}〉
+ exp(−iω10t/2)
[∑
κ
i~ · exp(iδκt)dc
κ
0
dt
(R, t)
∣∣0, {1}κ〉
− ~(ω10 − ωκ) exp(iδκt)cκ0(R, t)
∣∣0, {1}κ〉]
+
1
2
~ω10 exp(−iω10t/2)
∑
κ
exp(iδκt)c
κ
0(R, t)
∣∣0, {1}κ〉
= i~ · exp(−iω10t/2)dc1
dt
(R, t)
∣∣1, {0}〉+ 1
2
~ω10 exp(−iω10t/2)c1(R, t)
∣∣1, {0}〉
+ exp(−iω10t/2)
[∑
κ
i~ · exp(iδκt)dc
κ
0
dt
(R, t)
∣∣0, {1}κ〉
+ ~ωκ exp(iδκt)cκ0(R, t)
∣∣0, {1}κ〉]
− 1
2
~ω10 · exp(−iω10t/2)
∑
κ
exp(iδκt)c
κ
0(R, t)
∣∣0, {1}κ〉.
(4.6)
Upon cancelling the common terms, one obtains
116 CHAPTER 4. THE JAYNES-CUMMINGS MODEL
i
∑
κ
µκc1(R, t)
∣∣0, {1}κ〉− i∑
κ
µ∗κ exp(iδκt)c
κ
0(R, t)
∣∣1, {0}〉
= i~
dc1
dt
(R, t)
∣∣1, {0}〉+∑
κ
i~ · exp(iδκt)dc
κ
0
dt
(R, t)
∣∣0, {1}κ〉.
After taking the inner product with the states
(∣∣0, {1}κ〉)† and (∣∣1, {0}〉)†
we arrive at the equations of motion presented in [5]

dc1
dt
(R, t) = −∑κ µ∗κ exp(iδκt)cκ0(R, t)
dcκ0
dt
(R, t) = exp(−iδκt)µκc1(R, t)
. (4.7)
The system is reduced to
dc1
dt
(R, t) = −
∫ t
0
g(R, t− τ)c1(R, τ)dτ , (4.8)
with the Green's function taking the form of
g(R, t) = u(t)
∑
κ
|µκ|2 exp(iδκt) ≡ βu(t)
∫ ω2
ω1
exp [i(ω10 − ω)t]
ω − iγ ρl(R, ω)dω ,
in which ρl(R, ω) is the local density of photon states, u(t) is the unit step
function and, assuming periodic boundary conditions,
β ∝ ω
2
10|r10|2
εLS
L
2pi
∝ ω
2
10|r10|2
εS
.
Here, we will derive an expression for the Fourier transform of the Green's
function for the determination of the emission spectrum. Applying the
4.1. HAMILTONIAN, STATES AND EQUATIONS OF MOTION 117
Fourier transform to both sides of (4.8) and invoking the convolution theo-
rem, as well as the conjugate symmetry of the coefficients under time reversal,
we obtain
F
{
dc1
dt
}
(R, t) = −F{g(R, t)}F{c1(R, t)}
⇒ −i(Ω− ω10)c˜1(Ω− ω10) = −g˜(R,Ω− ω10)c˜1(Ω− ω10)
⇒ −i(Ω− ω´10 + iγ10)c˜1(Ω− ω10) = −g˜(R,Ω− ω10)c˜1(Ω− ω10)
Hence
c˜1(Ω− ω10) = 1
γ10 − i(Ω− ω´10) + g˜(R,Ω− ω10) , (4.9)
where
f˜(ω) =
∫ ∞
0
f(t)eiωtdt. (4.10)
Also, we have
F{g(R, t)} = β
∫ ωmax
ωmin
F{u(t) exp (At)}
ω − iγ ρl(R, ω)dω (4.11)
hence
g˜(R,Ω−ω10) = −β
∫ ωmax
ωmin
ρl(R, ω)
(ω − iγ)[(<{A} − ={Ω− ω10}) + i(={A}+ <{Ω− ω10})]dω,
(4.12)
where A = γ10 − γ + i(ω´10 − ω).
The integral above can be recast in the form
118 CHAPTER 4. THE JAYNES-CUMMINGS MODEL
g˜(R,Ω− ω10) = −iβ
∫ ωmax
ωmin
ρl(R, ω)
(ω − iγ)(ω − Ω− iγ)dω. (4.13)
The emission spectrum is then determined by the relation
S(Ω) = 2<{c˜1(Ω− ω10)} = 2[γ10 + Σ2(Ω)]
[Ω− ω´10 + Σ1(Ω)]2 + [γ10 + Σ2(Ω)]2 , (4.14)
with
c˜1(Ω− ω10) =
∫ ∞
0
c1(t)e
i(Ω−ω10)tdt
and
Σ1 = β|eˆkl(R)rˆ10|2<
{∫ ωmax
ωmin
ρ(ω)
(ω − iγ)(ω − Ω− iγ)dω
}
, (4.15)
Σ2 = β|eˆkl(R)rˆ10|2=
{∫ ωmax
ωmin
ρ(ω)
(ω − iγ)(ω − Ω− iγ)dω
}
, (4.16)
where ρ(ω) is the density of photon states (DOS). The term Σ1 is linked
to the Lamb shift, while the term Σ2 is related to the broadening of the
transition between the two states of the system.
4.2 Photonic Density of States revisited
We will next consider the characteristic case of chiral nematic LCs, which are
partial 1D photonic crystals, where the DOS can be derived from the trans-
mission properties of a layer with finite thickness [2], enabling the emission
spectrum to be predicted using equations (4.15).
4.2. PHOTONIC DENSITY OF STATES REVISITED 119
For the extraction of the transmission coefficient and subsequently the den-
sity of photon states in chiral nematic LCs, we proceed as follows. We con-
sider a boundary value problem formulated such that two plane waves of
the diffracting polarization are incident on a chiral nematic LC layer. The
assumption of no boundary reflection allows the separation of eigenpolariza-
tions, introducing an error of the order of the relative dielectric anisotropy
[8]. By demanding a continuous tangential component of the electric and
magnetic field at the layer-glass interface, we formulate a system, the solu-
tion of which yields the transmission coefficient for light of diffractive circular
polarization. The transmission coefficient for a layer with N full precessions
of the molecular director, and hence thickness L = Np, reads [6]
T =
exp
(
iτL
2
) (qτ
k2
)
qτ
k2
cos(qL) + i
[( τ
2k
)2
+
( q
k
)2
− 1
]
sin(qL)
, (4.17)
where
q = k
√
1 +
( τ
2k
)2
−
√(τ
k
)2
+ δ2.
In these expressions, τ = (4pi)/p, k = (ω/c)0, with 0 = (‖ + ⊥)/2 being
the average dielectric constant where ‖ and ⊥ are the relative dielectric
constants parallel and perpendicular to the director, respectively, and δ =
(‖ − ⊥)/(‖ + ⊥) the relative dielectric anisotropy, p the helical pitch and
c the speed of light in the vacuum. This relation is a different expression
to the one given in [2], where Maxwell's equations are solved in a frame
rotating with the molecular director. Omitting common real prefactors and
frequency independent terms, the real and imaginary parts, respectively, of
120 CHAPTER 4. THE JAYNES-CUMMINGS MODEL
the transmission coefficient (equation (4.17)) read
X =
(qτ
k2
)
cos(qL),
Y = −
[( τ
2k
)2
+
( q
k
)2
− 1
]
sin(qL).
(4.18)
The normalized DOS can be written as we have shown in the previous
chapter
ρ =
c
Np
√
0
X
dY
dω
− Y dX
dω
X2 + Y 2
. (4.19)
Focusing in the region of the band gap, we can write: q = iq˜. The real and
imaginary parts become, respectively,
X˜ =
[( τ
2k
)2
−
(
q˜
k
)2
− 1
]
sinh(q˜L),
Y˜ =
(
q˜τ
k2
)
cosh(q˜L).
(4.20)
One can easily verify that in this region
ρ˜ =
c
Np
√
0
X˜
dY˜
dω
− Y˜ dX˜
dω
X˜2 + Y˜ 2
=
c
Np
√
0
X
dY
dω
− Y dX
dω
X2 + Y 2
= ρ.
Hence, we find that the expressions in relation (4.18) can also be used in-
side the band gap. Letting now N → ∞ ⇒ L → ∞ and approximating
sinh(q˜L) = cosh(q˜L) ' (1/2) exp(q˜L), we reproduce the asymptotic be-
haviour ρ ∼ (1/N) in the middle of the band-gap, in a similar fashion to
a QWS, as we have shown in the previous chapter. The low frequency band-
edge for a chiral nematic LC is given by the formula ω0 = ωc/
√
1 + δ where
ωc = (2pic)/p is the center of the reflection band [6]. Similarly to a QWS,
4.3. MAIN RESULTS FOR RADIATIVE CHIRAL NEMATIC LCS 121
also, we find that for a chiral nematic the value of the DOS at a wavelength
edge is a quantity increasing with the product Nδ, which is an indication
of feedback-strength in the cavity. In our treatment, we approximate the
edge mode frequency in the long wavelength edge with ω0. For a layer with
finite thickness, this approximation increases in validity as Nδ is apprecia-
bly higher than unity [6]. The expressions in (4.15) should be averaged over
all possible orientations of the dipole moment and from this procedure we
obtain the pertinent transition dipole order parameter, as shown in [2]. In
what follows, we will assume that the dipole order parameter is zero; this
corresponds to an isotropic distribution of the dyes i.e., absence of prefer-
ential alignment. We also consider a uniform distribution of the fluorescent
molecules in the chiral nematic host, so that our results are not affected by
the spatial distribution of the eigenmodes. In [2] it is shown that using these
assumptions
〈|eˆkl(R)rˆ10|2〉 = 1/3.
4.3 Main results for radiative chiral nematic
LCs
Equations (4.15) allow the calculation of the emission spectrum, subject to
the the determination of the DOS in the resonating structure.
First of all, Figure 4.1 depicts the DOS profile for a chiral nematic LC
layer with different values of thickness and relative dielectric anisotropy, for
the low frequency edge. It is shown that the value of the DOS increases with
increasing product of thickness and relative dielectric anisotropy. Both these
quantities determine distributed feedback within the structure. The DOS
122 CHAPTER 4. THE JAYNES-CUMMINGS MODEL
0.96 0.97 0.98 0.99 1.00 1.01
0
10
20
30
n
o
r
m
a
l
i
z
e
d
 
D
O
S
 
(a)
0.96 0.97 0.98 0.99 1.00 1.01
0
5
10
15
20
25
n
o
r
m
a
l
i
z
e
d
 
D
O
S
 
(b)
Figure 4.1: (a) Theoretically obtained normalized DOS using (4.19) for N = 65 precessions of the director
and δ = 0.091. (b) Theoretically obtained normalized DOS using equation (4.19) for N = 40 precessions
and δ = 0.13.
4.3. MAIN RESULTS FOR RADIATIVE CHIRAL NEMATIC LCS 123
exhibits a distinct peak at the short frequency edge alongside some minor
resonance peaks decreasing in magnitude with decreasing frequency. Similar
results have been also reported elsewhere [2] and appear in the graphs of the
previous chapter. The DOS value for the dominant edge mode diverges to
infinity for a given relative dielectric anisotropy and N →∞.
The DOS calculated for a sample with N = 40 precessions of the molecu-
lar director and relative dielectric anisotropy δ = 0.13 is now employed to
calculate the emission profile from equation (4.14). Figure 4.2 shows a com-
parison between the theoretically obtained fluorescence spectrum and that
measured experimentally for a cell with thickness L ' 12 µm (N ' 40) con-
sisting of the chiral nematic LC mixture E49 doped with the fluorescent dye
DCM. The LC sample was illuminated by a frequency-doubled continuous-
wave Nd:YAG laser producing an almost-monochromatic beam of λ = 532nm
which was strongly attenuated by two cross-polarizers. The beam was subse-
quently converted to the circular polarization of the opposite handedness to
that of the sample, in order to excite the fluorescent dye uniformly (see [2]),
and then collimated and focussed onto the LC sample. Radiation collected
from the sample was collimated by an achromatic lens and then coupled to
an optical fibre. The fluorescence spectrum was obtained using a universal
serial bus spectrometer with a resolution of 0.3 nm. There is a good agree-
ment between theoretical and experimental results, from which we deduce
that the spontaneous emission spectrum exhibits significant differences from
the pattern dictated by the DOS presented in Figure 4.1.
However, by accounting for the emission spectrum using the approach dis-
cussed herein appears to provide a better match with experimental observa-
124 CHAPTER 4. THE JAYNES-CUMMINGS MODEL
0.96 0.97 0.98 0.99 1.00 1.01
1
2
3
f
l
u
o
r
e
s
c
e
n
c
e
 
i
n
t
e
n
s
i
t
y
 
(
a
r
b
.
 
u
n
i
t
s
)
(a)
2.95 3.00 3.05 3.10 3.15
0.4
0.6
0.8
1.0
 
n
o
r
m
a
l
i
z
e
d
 
e
m
i
s
s
i
o
n
 
i
n
t
e
n
s
i
t
y
(10
15
Hz)
(b)
Figure 4.2: In all the figures that follow, we use the symbol ω instead of Ω that appears in the relations for
the radiation terms, for convenience. (a) Theoretically obtained emission spectrum for a chiral nematic
LC with a gain medium for N = 40 precessions of the director, δ = 0.13, γ10 = 1.25 · 10−4 · ω0 and
γ = 9.36 · 10−4 · ω0. In this case ω′10 = 1.06 · ω0. (b) Experimentally determined fluorescence spectrum
obtained from a chiral nematic LC sample doped with DCM (with a fluorescence peak in the LC host at
λmax ' 580nm) and N ≈ 40 full precessions of the molecular director. For (a) the integration limits in
Equations (4.14),(4.15) are ωmin = 0.96 · ω0 and ωmax = 1.01 · ω0. Here, β = 1.05 · 1029[SI]
4.3. MAIN RESULTS FOR RADIATIVE CHIRAL NEMATIC LCS 125
tions. Our results demonstrate that we ought to account additionally for the
relative position of the transition frequency with respect to the edge mode
location in order to describe more accurately the emission spectrum from
these periodic structures. At large oscillator strengths of the atomic tran-
sition, there is a Fano-resonant mechanism between the discrete spectrum
of the atomic transition and the continuum of photon states in the chiral
nematic feedback structure, occurring when the atomic transition frequency
lies in the region of the continuum. The same mechanism is associated with
the splitting of the fluorescence into two components in the region of the
logarithmic singularity due to the saddle point in the dispersion curve of a
2D photonic crystal [5].
Figure 4.3 depicts the transition broadening term Σ2 (equation (4.16)) nor-
malized by the same arbitrary constant, for two different values of the cavity
losses. We find that apart from the change in magnitude of the term with
decreasing losses, there is also a change in the relative height of the first two
edge-mode peaks. Moreover, we quantify the effect of resonance for a chiral
nematic LC resonator in which the feedback properties are enhanced.
Figure 4.4 depicts the emission profile calculated from equation (4.14) for
small and large detuning, i.e.,varying the frequency offset between the elec-
tronic transition and the dominant edge mode. Our theoretical results show
that the first two peaks which correspond to the two edge modes closest to
the band gap are less pronounced than those in the DOS profile, when under
the condition of exact resonance. This can be inferred from Figures 4.1(a),
(b) and Figure 4.4(a). Their magnitude also decreases with increasing detun-
ing (Figure 4(b)). The presence of residual attenuation due to a variety of
126 CHAPTER 4. THE JAYNES-CUMMINGS MODEL
0.96 0.97 0.98 0.99 1.00 1.01
0
5
10
15
20
n
o
r
m
a
l
i
z
e
d
 
 
(
a
r
b
.
 
u
n
i
t
s
)
(a)
0.96 0.97 0.98 0.99 1.00 1.01
0
5
10
15
20
25
n
o
r
m
a
l
i
z
e
d
 
 
(
a
r
b
.
 
u
n
i
t
s
)
(b)
Figure 4.3: Transition broadening Σ2 term as a function of frequency for a chiral LC with a gain medium,
for N = 60 precessions of the director, δ = 0.13 and two different values of cavity losses. In (a) γ =
5.55 · 10−4 · ω0 and in (b) γ = 3.70 · 10−4 · ω0.
4.3. MAIN RESULTS FOR RADIATIVE CHIRAL NEMATIC LCS 127
0.97 0.98 0.99 1.00
5
10
15
20
25
30
35
e
m
i
s
s
i
o
n
 
i
n
t
e
n
s
i
t
y
 
(
a
r
b
.
 
u
n
i
t
s
)
(a)
0.97 0.98 0.99 1.00
0.5
1.0
1.5
2.0
e
m
i
s
s
i
o
n
 
i
n
t
e
n
s
i
t
y
 
(
a
r
b
.
 
u
n
i
t
s
)
(b)
Figure 4.4: Theoretically obtained emission spectrum for a chiral nematic LC with a gain medium for
N = 65 precessions of the director, δ = 0.091, γ10 = 1.25 · 10−4 · ω0, γ = 7.02 · 10−4 · ω0 for two different
detuning values. In (a) ω′10 = 1.005 · ω0 and in (b) ω′10 = 1.05 · ω0. In (a), ωmin = 0.95 · ω0 and
ωmax = 1.001 · ω0 whereas in (b), ωmin = 0.97 · ω0 and ωmax = 1.001 · ω0. Here, β = 3 · 1028[SI].
128 CHAPTER 4. THE JAYNES-CUMMINGS MODEL
mechanisms, such as scattering from long range thermal fluctuations of the
molecular director and absorption from excited atomic levels, inhibits the
feedback mechanism inside the resonator, which is manifested as a decrease
in the DOS [10]. This result can also be demonstrated for a Fabry-Pérot
resonator [10].
4.4 A 2D photonic crystal and the DOS
We will now address a particular case in which one can derive analytic expres-
sions for the Lamb shift and the transition broadening featuring in equation
(4.14). Unlike 1D photonic crystals, where the DOS displays Van Hove sin-
gularities at the band extrema, spontaneous emission in 2D crystals is not
enhanced at these points despite the fact that the group velocity assumes
zero values there. The exact dispersion relationship depends on the lattice of
the periodic structure and the polarization of the modes considered. Here it
is assumed that the atomic transition frequency is close to a saddle point (P1
type) in one of the branches of the photonic band spectrum, irrespective of
the lattice and the emission direction from the periodic structure. Therefore,
our findings will pertain to the general case. It is known that near the sad-
dle point in the dispersion curve of a 2D photonic crystal, the DOS exhibits
a logarithmic divergence [1], [5]. We start by calculating the integral (the
expression for the DOS has a minus sign in front of the logarithm that we
omit here and restore in the final expressions)
I =
∫ +∞
−∞
f(ω)dω, with f(ω) =
log
(|a(ω0)(ω − ω0)|)
(ω − ω1)(ω − ω2) , (4.21)
4.4. A 2D PHOTONIC CRYSTAL AND THE DOS 129
R

ω0
ω1 = iγ
ω2 = Ω + iγ
C
CR
Re(ω)
Im(ω)
Figure 4.5: The integration contour for the application of the residue theorem in (4.22).
where a(ω0) has dimensions of Hz
−1
and is related to the specific photonic
crystal properties and the expansion near the saddle point [5]. We will find,
however, that our final results are independent of this factor. Since the
residue theorem is applied for single-valued functions, we consider the branch
of the logarithm defined by log(z) = log |z|+ iθ,−pi
2
≤ θ < 3pi
2
.
The integration contour is shown in Figure 4.5.
According to the residue theorem, we have
∫ R
ω0+
f(ω)dω +
∫
CR
f(ω)dω +
∫ ω0−
−R
f(ω)dω +
∫
C
f(ω)dω
= 2pii
{
log[a(ω2 − ω0)]
ω2 − ω1 +
log[a(ω1 − ω0)]
ω1 − ω2
}
, (4.22)
where ω1 = iγ and ω2 = Ω + iγ. We also have
130 CHAPTER 4. THE JAYNES-CUMMINGS MODEL
∣∣∣∣∫
C
log[a(ω − ω0)]
(ω − ω1)(ω − ω2)dω
∣∣∣∣ = ∣∣∣∣∫ pi
0
log(aeiθ)
(ω0 + eiθ − ω1)(ω0 + eiθ − ω2)ie
iθdθ
∣∣∣∣
≤ | log(a) | +pi
(| ω0 − ω1 | −)(| ω0 − ω2 | −)pi→ 0 ,
as → 0, since  log → 0 when → 0. For R sufficiently large, we also have
∣∣∣∣∫
CR
log[a(ω − ω0)]
(ω − ω1)(ω − ω2)dω
∣∣∣∣ = ∣∣∣∣∫ pi
0
log[a(Reiθ − ω0)]
(Reiθ − ω1)(Reiθ − ω2)iRe
iθdθ
∣∣∣∣
≤ | log[a(R + ω0)] | +pi
(R− | ω1 |)(R− | ω2 |)piR→ 0 ,
since
logR
R
→ 0 when R→∞. Hence, we deduce that
∫ +∞
−∞
f(ω)dω =
∫ +∞
−∞
log |a(ω − ω0)|
(ω − ω1)(ω − ω2)dω + ipi
∫ ω0
−∞
dω
(ω − ω1)(ω − ω2)
=
2pii
ω2 − ω1 log
(
ω2 − ω0
ω1 − ω0
)
, (4.23)
since
∫ ω0
−∞
dω
(ω − ω1)(ω − ω2) =
1
ω2 − ω1 log
(
ω2 − ω0
ω1 − ω0
)
. (4.24)
We must note here that the behaviour of the DOS far from the saddle point
may be different (usually we assume ρ ∝ ω far from the critical point). In that
case, the upper integration limit is replaced by the Compton frequency [5],
[7], ωc =
mc2
~
∼= 1021 Hz, which is many orders of magnitude higher than the
frequencies in the visible part of the electromagnetic spectrum. Therefore,
the integration to infinity can be justified. We conclude that
4.4. A 2D PHOTONIC CRYSTAL AND THE DOS 131
∫ +∞
−∞
log |a(ω − ω0)|
(ω − ω1)(ω − ω2)dω =
pii
ω2 − ω1 log
(
ω2 − ω0
ω1 − ω0
)
=
pii
Ω
log
(
Ω + iγ − ω0
iγ − ω0
)
.
(4.25)
As we have selected the particular branch of the logarithmic function with
−pi
2
≤ θ < 3pi
2
, the real part of the resulting integral will have a discontinuity
since the phase of the logarithm varies between −pi and 0 for an argument
selection −pi ≤ θ < pi. This is also understood from the fact that in order for
the identity log(z1z2) = log(z1)+log(z2) to be applied, then arg(z1)+arg(z2)
must lie within the phase range of the chosen branch-otherwise there is an
offset by 2pi. The same discontinuity, linked to the Lamb shift would have
been exhibited if we had chosen any other branch cut outside our integration
contour. The Lamb shift and the transition broadening term, in this case,
read
Σ1 = −β′|eˆklrˆ10|2<
{∫ +∞
−∞
log |a(ω − ω0)|
(ω − iγ)(ω − Ω− iγ)dω
}
, (4.26)
Σ2 = −β′|eˆklrˆ10|2=
{∫ +∞
−∞
log |a(ω − ω0)|
(ω − iγ)(ω − Ω− iγ)dω
}
, (4.27)
where we have assumed that the normalization term β in equations (4.15) is
modified by some parameters particular to the expression of the DOS for a
photonic crystal [5], to yield β′. We can deduce that for 2D photonic crystals
in the saddle point of the dispersion function, resonance is associated with
a Lamb dip in the fluorescence spectrum and a split in the real part of the
Fourier transform of the Green's function.
The split (term Σ1) here is due to the behaviour of the complex logarithmic
function; however, as we can observe in Figures 4.6(a) ,(b), the magnitude of
132 CHAPTER 4. THE JAYNES-CUMMINGS MODEL
0.98 0.99 1.00 1.01 1.02
-5
0
5
10
15
(
H
z
)
(a)
0.98 0.99 1.00 1.01 1.02
1.00
1.25
1.50
1.75
2.00
2.25
2.50
(
H
z
)
(b)
Figure 4.6: (a) The Lamb shift (Σ1) as a function of frequency for a 2D photonic crystal with a logarithmic
singularity in the DOS at ω0. (b) Transition broadening (Σ2) as a function of frequency for a photonic
crystal with a logarithmic singularity in the DOS at ω0. In all cases above, ω′10 = 1.001 · ω0 and γ =
1.25 · 10−4 · ω0
4.5. DISCUSSION OF RESULTS 133
that term is significantly lower than the broadening term Σ2 of the transition∣∣1, {0}〉→ ∣∣0, {1}κ〉.
The Lamb dip essentially vanishes for larger detunings, as we can see in the
emission spectra of Figures 4.7(a), (b). A Lamb shift has also been reported
inside the (complete) photonic band gap for hydrogenic atoms embedded in
1D periodic structures [11].
4.5 Discussion of results
Concerning the validity of our results, we note that the rotating wave approx-
imation we relied upon, requires a small detuning from the atomic transition
frequency, in order for the `non-resonant' terms to be much smaller than the
`resonant' ones when averaging over a time period of the order of 1/ω in the
interaction picture [3],[7]. This constraint is met by selecting an appropriate
upper and lower integration limit in equations (4.15) and by the presence
of the function log |a(ω − ω0)| in equations (4.26) ensuring that the major
contribution to the integral originates from the region |ω−ω10|  (ω+ω10).
Likewise, the most significant contribution in the emission from chiral nemat-
ics will be from the first two edge mode peaks on the same side of the band
gap, for Nδ  1. Regarding the the Generalized Rotating Wave Approxi-
mation (GRWA) in [3], one can effect the change of basis from | ∓ x,N〉 to
the adiabatic eigenstates |Ψ±, N〉 through a unitary transformation using the
operator (for simplicity we denote by ω0 the electromagnetic-field frequency
and by Ω the atomic transition frequency)
134 CHAPTER 4. THE JAYNES-CUMMINGS MODEL
0.950 0.975 1.000 1.025 1.050
0
2
4
6
8
10
 
f
l
u
o
r
e
s
c
e
n
c
e
 
i
n
t
e
n
s
i
t
y
 
(
a
.
u
.
)
(a)
0.90 0.95 1.00 1.05 1.10
0
2
4
6
8
10
12
14
 
f
l
u
o
r
e
s
c
e
n
c
e
 
i
n
t
e
n
s
i
t
y
 
(
a
.
u
.
)
(b)
Figure 4.7: (a) Emission spectrum for a 2D photonic crystal with a logarithmic singularity in the DOS at
ω0, as a function of frequency for ω′10 = 1.002 · ω0. (b) Emission spectrum for the photonic crystal as a
function of frequency for ω′10 = 1.005 · ω0. In all cases above, β′ = 3 · 1027[SI], γ = 1.25 · 10−4 · ω0 and
γ10 = 1.25 · 10−5 · ω0.
4.5. DISCUSSION OF RESULTS 135
Dˆ
(
µ
ω0
σˆz
)
= exp
[
− µ
ω0
σˆz(aˆ
† − aˆ)
]
. (4.28)
As we can readily verify, Dˆ†Dˆ = I. We apply this transformation to the
general Hamiltonian
Hˆ = ω0aˆ
†aˆ+
1
2
Ωσˆx+µσˆx(aˆ
†+aˆ) = ω0aˆ†aˆ+
1
2
Ωσˆx+µ(σˆ−aˆ†+σˆ+aˆ+σˆ+aˆ†+σˆ−aˆ),
(4.29)
where σˆ± = (1/2)(σˆz ∓ iσˆy) are the ladder operators in the basis of σˆx, and
µ is the interaction constant. This Hamiltonian, then, is transformed to
Hˆ ′ = ω0aˆ†aˆ+
1
2
Ωσˆx exp
[
−2µ
ω0
σˆz(aˆ
† − aˆ)
]
= ω0aˆ
†aˆ+
1
2
Ωσˆx + Hˆ1,x + Hˆ1,y,
(4.30)
Using the properties of the Pauli matrices: σˆ2x = I and σˆxσˆz = −iσˆy, and
setting uˆ = − µ
ω0
(aˆ†− aˆ) we expand the second term in (4.29) and we obtain
σˆx exp(−σˆzuˆ) = σˆx
(
I+
+∞∑
κ=1
uˆ(2κ)
(2κ)!
)
− σˆxσˆz
+∞∑
κ=1
uˆ(2κ−1)
(2κ− 1)! , (4.31)
so that the last two terms in the transformed Hamiltonian have the form
Hˆ1,x =
Ωσˆx
2
+∞∑
κ=1
(
2µ
ω0
(aˆ† − aˆ)
)2κ
(2κ)!
,
Hˆ1,y =
iΩσˆy
2
+∞∑
κ=1
(
2µ
ω0
(aˆ† − aˆ)
)2κ−1
(2κ− 1)! .
(4.32)
136 CHAPTER 4. THE JAYNES-CUMMINGS MODEL
Carrying out the GRWA by keeping the energy-conserving terms, the con-
tribution of Hˆ1,x is neglected since the only terms that do not have a rapid
time-dependence contain only powers of the number operator resulting in
zero net-excitation of the oscillator. Also, Hˆ1,y reduces to the coupling term
Ω
(
µ
ω0
)
[σˆ−aˆ†f(aˆ†aˆ) + σˆ+f ∗(aˆ†aˆ)aˆ], (4.33)
which is a generalization of the term µ(σˆ−aˆ† + σˆ+aˆ) in the standard RWA.
At this point, we ought to mention that the two-level system approach is
certainly a considerable simplification for fluorescent dyes. These complex
molecules have states with many vibrational and rotational levels determining
their spectra. In this analysis, mechanisms such as triple state generation and
resonant energy transfer have also been ignored. Under the assumption of
very small detuning, the effect of these phenomena can be quantified through
the introduction of an imaginary part in the mode frequency. A model of
coherent control of fluorescence would be that of a three-level system where
the transition |2〉 → |1〉 between the two upper levels is driven by a laser field
and that the spontaneous emission transitions |2〉 → |1〉 and |1〉 → |0〉 are
precluded by either symmetry of the presence of the band-gap [12]. In one
of these configurations the levels |0〉 and |1〉 have the same symmetry (hence
there is no allowed dipole-transition between them). We also assume that the
dipole allowed transition occurs between the levels |2〉 and |1〉 and that the
transition frequency ω21 is far inside the band-gap, such that the transition
|2〉 → |1〉 will create a photon-atom bound state whose radiative lifetime
is the same as the two photon |1〉 → |0〉 transition. In this configuration
|1〉 → |0〉 is a dipole allowed transition and it is assumed that ω21 is near
the band edge. The Hamiltonian of the system is Hˆ = HˆA + HˆF + HˆI + HˆL,
4.5. DISCUSSION OF RESULTS 137
where
HˆA = ~
2∑
j=0
ωjσˆjj,
HˆR = ~
2∑
λ=1
∑
k
ωkaˆ
†
kλaˆkλ,
HˆI = i~
2∑
λ=1
∑
k
gkλ(aˆ
†
kλσˆ02 − σˆ20aˆkλ),
HˆL = id21E0{σˆ21 exp [i(ω21t+ φ)]− σˆ21 exp [−i(ω21t+ φ)]},
(4.34)
where σˆij = |i〉〈j| are fermionic operators (the energy of the jth level is ~ωj)
and
gkλ =
ω20d20
~
(
~
2ε0ωkV
)1/2
eˆk,λ · dˆ20, (4.35)
is the atom-field coupling constant for the |2〉 → |0〉 transition (in this ex-
pression, the hat denotes a unit vector). The last term in the Hamiltonian
represents the interaction of the atom with the laser field. We assume that
the latter is sufficiently strong such that it is treated classically in the context
of the first quantization, with the pre-factor in front of the square-brackets
being ~ΩR, where ΩR is the Rabi frequency, much larger than the detuning.
Recently, rare-earth-doped nanocrystals have been used as the gain medium
hosted in chiral nematic LCs [13]. In this case, the assumption of a two-level
atom interacting with the electromagnetic field maybe much more appro-
priate to describe spontaneous emission from the resonator. Employing the
DOS in the JC model is then expected to lead to a better match between the-
ory and experiment. Moreover, for a further insight to spontaneous emission
138 CHAPTER 4. THE JAYNES-CUMMINGS MODEL
from such periodic structures, we could resort to the resolvent method [7], in
which the matrix elements of the time evolution operator can be calculated.
We will detail the fundamental concepts and techniques of this method in
the last chapter.
4.6 More on the Fano-Anderson model
We relate now our analysis to the Fano-Anderson model, pertaining to the
interaction of a discrete state
∣∣d〉 with a continuum described by a set of
states
∣∣k〉. As before, the `atom'-field Hamiltonian in a frame rotating with
frequency ωd assumes the form
H =
∫
~δω(k)
∣∣k〉〈k∣∣dk + ~∫ [v(k)∣∣d〉〈k∣∣+ v∗(k)∣∣k〉〈d∣∣] dk, (4.36)
where the second (Hermitian) interaction term accounts for a `coloured-
coupling' (dependent on the wavenumber). Evidently, this Hamiltonian is
Hermitian. Likewise, the wavefunction of the system is written as
∣∣ψ〉 =
cd(t)
∣∣d〉 + ∫ c(k, t)v(k)dk and the expansion coefficients satisfy the coupled
equations of motion (obtained by Schrödinger's equation)

i
dcd(t)
dt
= ωdcd(t) +
∫
c(k, t)v(k)dk (4.37a)
i
dc(k, t)
dt
= ω(k)c(k, t) + v∗(k)cd(t) (4.37b)
Denoting the Laplace Transform of cd(t) by
c˜d(s) =
∫ ∞
0
cd(t) exp(−st)dt, (4.38)
4.6. MORE ON THE FANO-ANDERSON MODEL 139
and then applying the Laplace Transform in both handsides of relations of the
system (4.37a) we obtain (assuming that cd(t = 0) = 1 and c(k, t = 0) = 0)
c˜d(s) =
i
is− ωd − Σ(s) , (4.39)
where Σ(s) is called the `self-energy' term reading
Σ(s) =
∫ |v(k)|2
is− ω(k)dk =
∫ ω2
ω1
ρ(ω′)|v(ω′)|2
is− ω′ dω
′, (4.40)
where ρ(ω) is the density of photon states. Applying the inverse Laplace
Transform, we finally obtain
cd(t) =
1
2pi
∫ σ+i∞
σ−i∞
est
is− ωd − Σ(s)ds, (4.41)
where σ = 0+, since the poles of c˜d(s) if they exist, belong to the imaginary
axis. Using the Sokhotski-Plemelj theorem, for ω ∈ (ω1, ω2) we have
Σ(s = −iω ± 0+) = ∆(ω)∓ ipiρ(ω)|v(ω)|2, (4.42)
where
∆(ω) = P
∫ ω2
ω1
ρ(ω′)|v(ω′)|2
ω − ω′ dω
′
(4.43)
The poles sp = −iΩ of c˜d(s) correspond to bound states of the Hamiltonian,
since they lead to the exponential term exp(−iΩt) with constant magnitude,
after applying the residue theorem. These poles satisfy the conditions
Ω− ωd = ∆(Ω), ρ(Ω)|v(Ω)|2 = 0 (4.44)
140 CHAPTER 4. THE JAYNES-CUMMINGS MODEL
For an infinitely-thick chiral nematic LC sample, the density of photon states
reads
ρj =
dkj
dω
=
dkj
dλ′
dλ′
dω
= −p
√
ε¯
c
λ′2
(
1
λ′
dm′i
dλ′
− m
′
i
λ′2
)
, (4.45)
where
dmj
dλ′
=
(
1∓ 2√
4λ′2 + δ2
)
λ′
m′j
. (4.46)
The index j assumes values 1, 2 for the non-diffracting and diffracting eigen-
wave respectively. As we have seen in the different chapters, m1 takes the
value of zero, for λ′ =
√
1± δ, hence ρ1 is divergent at these wavelengths.
For a broad density of states, as the one corresponding to the non-diffracting
eigenwave, one can use the Wigner-Weisskopf approximation [15] according
to which only the pole contribution s→ 0+ is retained and we can apply the
Sokhotski-Plemelj theorem for the calculation of Σ(s), as
lim
s→0+
1
s+ i(ωk − ωd) = −iP
1
ωk − ωd + piδ(ωk − ωd). (4.47)
In that case, in the absence of poles of c˜d(s) we find that cd(t) decays expo-
nentially to zero.
In this chapter we have outlined the application of the Jaynes-Cummings
model in quantifying spontaneous emission from chiral nematic LCs, as an
alternative to the semi-classical Dirac's rule discussed previously. The linking
element between the current and the previous chapter is the photonic density
of states, showing a selective emission enhancement characteristic of the opti-
4.6. MORE ON THE FANO-ANDERSON MODEL 141
cal feedback in the liquid crystal resonator. In light of this approach, we have
tried to interpret differences between the profile of the density of states and
the experimentally measured fluorescence from these LC samples, refining
our model phenomenologically in the context of the second quantisation.
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Network, Kerala, India, 2010.
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144 BIBLIOGRAPHY
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Chapter 5
Adaptive pumping of radiative
LC cells
In the previous chapters we have included a discussion on lowering the laser
excitation threshold of dye-doped chiral nematic cells. Here, we will focus
on a pumping application where the input power is distributed to a group of
spots, therefore a consideration of the excitation threshold of liquid crystal
lasers is essential, as such a configuration is feasible only under the low-
threshold lasing properties exhibited by these molecular structures. Organic
solid-state dye lasers offer a number of advantages over their liquid-host coun-
terparts, such as compactness and their easy-to-handle set-ups. However, the
interaction time between the dye molecules and the pump beam must be min-
imized in order to circumvent bleaching issues, requiring a pulsed operation.
To attain high repetition rates or continuous wave operation - which is nec-
essary for many spectroscopic applications - excited dye molecules need to
be removed from the pump volume [1],[2]. Hitherto, this has necessitated
145
146 CHAPTER 5. ADAPTIVE PUMPING OF RADIATIVE LC CELLS
the use of mechanical parts and/or fluid flow [3]. In this chapter, we demon-
strate a new technique that involves dynamically controlling the position of
incidence of a pump beam shone onto a thin film organic dye laser using
holography, thus negating the need for moving parts and circulation of the
dye. By moving the pump beam, the interaction time remains short enough
that the triplet population is not increased and provides a means of accessing
higher repetition rates. The method also provides additional functionality
such as wavelength tuning and spatial shaping of the pump beam. Despite
the advantages of organic solid-state dye lasers, liquid-dye lasers considerably
outperform the solid-state organics when it comes to pulse duration. They
can operate with pulses as short as 10 fs, and the pulse length can be made
so long as to allow continuous-wave (CW) outputs. Solid-state dye lasers,
which consist of an organic dye that is dispersed into a solid matrix, are
not so versatile and are restricted to pulse durations are typically no longer
than 10 ns. CW operation is generally prohibited because of bleaching issues
caused by thermal degradation effects and also the accumulation of triplet
excitons. In the case of triplet excitons, which possess an absorption band
that overlaps the stimulated emission spectrum, suppression of the gain in
the active region can occur, resulting in laser action being switched off. In
order to ensure that the population of triplet states is dissipated between
the pulses, repetition rates of the pump laser are usually restricted to less
than 10 kHz. However, even if the repetition rate is low enough that triplet
excitons do not build-up it is still not necessarily the case that thermal effects
are absent.
In order to combat these thermal and excited-state absorption effects, a so-
147
lution is to remove the excited dye molecules out of the pump volume in
sufficient timescales so as to maintain laser action. For liquid dye lasers,
this involves a constant convective circulation of the organic species using a
jet-stream with velocities on the order of 10s of ms
−1
to replenish the dye
molecules and thus avoid the bleaching effects that ultimately shut down
laser action [1]. Alternatively, for solid-state organic dye lasers, methodolo-
gies that mimic the approach of `flushing out' the excited dye molecules have
been demonstrated such as mechanically rotating a disc-shaped active gain
medium at high speeds to constantly refresh the dye molecules [4]. A com-
mon feature of the techniques that have been demonstrated to date require
physically translating the active region in some way relative to the pump.
This, therefore, requires the use of moving parts or fluid flow. In this chap-
ter, we demonstrate a solution in which the pump beam is moved relative
to the solid-state organic dye laser using adaptive optics, which bypasses the
need for any moving parts as the position of the pump beam is adjusted
through the diffraction of light. In our experimental work, a chiral nematic
liquid crystal (LC) laser [5]-[8] is used as a representative thin-film organic
dye laser. As we have shown, the chiral nematic LC provides the feedback
mechanism as it possesses a 1-d photonic band gap for visible light, which
suppresses fluorescence within the band gap [9] but enhances it at the band
edges [10]-[13]. The gain medium consists of a laser dye that is based upon
the pyrromethene class, which are known to exhibit reduced triplet-triplet
absorption across the emission spectral region and high quantum efficiencies
[14]-[20] as well as LC media [21]. The LC laser was chosen for this study be-
cause of the ease with which the laser wavelength can be selected across the
148 CHAPTER 5. ADAPTIVE PUMPING OF RADIATIVE LC CELLS
visible spectrum, which is particularly important for demonstrating the addi-
tional functionality available using the adaptive pumping technique. Studies
have shown that illumination of a chiral nematic LC with high excitation
energies, and even relatively low repetition rates (e.g. 20 Hz), can lead to a
dramatic reduction in the output energy of the LC laser within a short time
[22]. The motivation for this study is to demonstrate that the repetition
rate of the LC laser can be significantly improved using an adaptive dynamic
pumping approach. Although the results demonstrated in this chapter are for
an LC laser, in principle, the technique is applicable to all organic solid-state
and even liquid dye laser systems.
5.1 The method of adaptive pumping
Below we will detail the method of holographic adaptive pumping. An inci-
dent pump beam from the solid-state laser-the excitation source-is reflected
from a spatial light modulator (SLM) displaying a computer generated holo-
gram (CGH). The reflected light from the SLM is then focussed to form the
replay field on a glass cell containing a dye-doped LC. This optically excites
the dye and triggers lasing. By changing the hologram displayed on the
SLM, it is possible to form different spot positions or different spot patterns
incident on the sample in real-time. This enables both static pumping - in
which the position of the pump beam remains fixed, and dynamic pumping-
in which the pump beam can be translated across the cell in 2D, enabling
the active gain region to be refreshed. For static pumping, a single hologram
is continuously displayed; for dynamic pumping, a sequence of holograms is
5.1. THE METHOD OF ADAPTIVE PUMPING 149
displayed on the SLM. In this study, the CGHs were generated using the
Gerchberg-Saxton (GS) phase retrieval algorithm [23]-[24]. The most com-
mon steps followed to calculate a CGH are given below [23]
(1) Sampling of the continuous intensity distribution I(x, y) in the image
plane.
(2) Square-root of the intensity samples to obtain the (sampled) amplitude
distribution |u(x, y)|.
(3) Initial superposition of a random phase to create |u(x, y)| exp[iφ(x, y)] .
(4) Calculation of the Fast Fourier Transform (FFT) of the samples of the
complex amplitude u(x, y) and extraction of the corresponding phase distri-
bution.
(5) Sampling of the intensity distribution I(x, y) in the diffraction plane.
(6) Square-root of the intensity samples to obtain the (sampled) amplitude
distribution |u′(x, y)|.
(7) Initial superposition of a random phase to create |u′(x, y)| exp[iφ′(x, y)] .
(8) Iterations until constraints are satisfied.
The superposition of a random phase distribution (as a multiplicative ex-
ponential factor) is analogous to the introduction of a diffuser between the
source and the object. A global rearrangement of the phase samples is then
necessary to eliminate first-order zeros in the amplitude distribution [25].
This algorithm is frequently used in holographic applications and has re-
cently been employed to realize optical tweezing [26].
An illustration of the operating principle of the technique used to optically
150 CHAPTER 5. ADAPTIVE PUMPING OF RADIATIVE LC CELLS
Figure 5.1: Illustration of the principle of dynamic optical pumping using computer generated holograms.
excite thin-film organic laser samples is shown in Figure 5.1.
5.1.1 Sample preparation
In our experiment, the single wavelength laser cell consisted of the nematic
mixture E49 (Merck KGaA) doped with the chiral dopant BDH1281 (Merck
KGaA) and the laser dye Pyrromethene 580 (PM-580; Exciton) so as to form
a chiral nematic liquid crystal with a short-wavelength band-edge at 574 nm.
The cell had a thickness of 13 µm which has been shown to be suitable for
low-threshold lasing. The pitch-gradient cell, of 10 µm thickness, consisted
of two different dye-doped liquid crystal mixtures which are drawn into the
cell by capillary action to form two regions gradually diffusing together and
creating a continuous pitch-gradient. Both of these mixtures comprised of
the liquid crystal E49 (Merck) and the chiral dopant BDH1281 (Merck; 3.9
wt% for red and 4.3 wt% for green). DCM (Exciton; 1.5 wt%) was added
to make the red mixture and PM580 (Exciton; 0.5 wt%) was added to make
the green mixture.
5.1. THE METHOD OF ADAPTIVE PUMPING 151
5.1.2 Pumping process
The pump source is a frequency-doubled Nd:YAG laser (CryLaS, FDSS 532-
Q) which produces 1 ns pulses at 532 nm and has a variable repetition rate (up
to 3000 Hz, although in this work it was varied from 100 - 600 Hz). For direct
control of the pump beam energy, a half-waveplate between crossed polarisers
was implemented into the set-up before the spatial light modulation. Two
lenses act as collimating optics to expand the beam so that the entire area
of the SLM was illuminated. The beam is reflected from a nematic, multi-
level phase spatial light modulator (Cambridge Correlators, SDE1280) with
a maximum input frame addressing rate of 60 Hz displaying a CGH. A beam
splitter redirects it through a 532 nm half waveplate followed by a 532 nm
quarter waveplate in order to convert the light into the appropriate circular
polarization so as to improve the penetration depth through the sample and
avoid further losses due to the reflection band of the chiral nematic LC. A
focussing lens (f = 12.5 cm) forms the replay field on the dye-doped LC
cell. Two high-pass filters (cut-off 530 nm) remove the pump beam and a
4x microscope objective with numerical aperture 0.1 is used to collect the
light over a narrow forward angle. Emission energies of the pulses emitted
from the LC laser were recorded using a calibrated pyroelectric head (PD10-
SH-V2) connected to an energy meter (USB-2, Ophir) whereas spectra were
obtained using a universal serial bus spectrometer with a resolution of 0.3
nm (HR2000, Ocean Optics). 2D and 3D beam profiles were captured using
a Spiricon beam profiler, and photographic images of the polychromatic LC
laser were taken using a high resolution digital camera (Canon, EOS 550D).
The experimental set-up detailed above is illustrated in Figure 5.2.
152 CHAPTER 5. ADAPTIVE PUMPING OF RADIATIVE LC CELLS
Figure 5.2: Experimental setup. P: polariser; C: collimating optics; BS: beam-splitter; SLM: spatial
light modulator; WPs: wave-plates-λ/2 wave-plate followed by a λ/4 waveplate; FL: focussing lens (f =
125mm); LC: dye doped chiral nematic liquid crystal sample; HPF: high-pass filter; O: objective lens (4x);
X: camera, spectrometer or beam-profiler depending on the purpose of the experiment.
5.2. MAIN RESULTS 153
5.2 Main results
In order to prove the principle of hologram-steered optical pumping, we cre-
ated a hologram that would generate a single spot in the replay field. Ex-
amples of a CGH for a 2D array of 6 pump spots at the LC cell is shown in
Figure 5.3(a). The corresponding replay field for a single and multiple spot
array is presented in Figure 5.3(b). However, our SLM was not capable of
true 2pi modulation meaning that a fraction of a conjugate image is formed
in the replay field. This is because the Fourier transform of a real function is
always 180o-symmetric about the origin, as |ψ(u, v)|2 = |ψ(−u,−v)|2, where
ψ(u, v) is the complex amplitude of light in the diffraction plane. This re-
duces the efficiency of the hologram by 50%. Hence, if a hologram displayed
on the SLM is used to generate a single first order spot, there will actually
be two first order spots in the replay field. Thus, due to conjugate-image
formation, two spatially separate pump beams were incident on the LC and
therefore two LC laser beams-corresponding to two spatially separated active
regions-were generated. The first order pump spot in the replay field at the
sample position was found to be approximately circular with an area of 0.003
mm
2
, an excitation energy threshold of 130 nJ/pulse and a corresponding ex-
citation fluence of 4.3 mJ/cm
2
.
A beam profiler (Spiricon, LW 230) was used to record the spatial profile of
the intensity emitted by the LC laser beam at a distance of 7 cm from the
cell and it was found to be near-Gaussian as shown by the two- and three-
dimensional plots in Figures 5.4(a) and 5.4(b), respectively. In this case, two
separate beams are emitted from the LC laser device corresponding to two
spatially separated active regions. From these measurements, the waist of
154 CHAPTER 5. ADAPTIVE PUMPING OF RADIATIVE LC CELLS
Figure 5.3: Hologram and replay field used for and multi-spot beam. In (a) Phase CGH (grey-scale) for
the generation of six first order spots in the replay field (768 x 768 pixels), (b) Replay field of the ideal
CGH.
the two quasi-Gaussian beams were calculated to be approximately 70 µm.
When the CGH was removed from the SLM the spots vanished, confirming
that the output from the LC cell was caused by the hologram. The emission
spectrum was recorded on a spectrometer, which was shown to occur at a
wavelength of λ = 560.2 nm for both beams, Figure 5.4(c). Additionally,
CGHs used to generate single spots in different positions and also patterns
of spots were created: Figure 5.4(d) shows a pattern of six spots (which
becomes twelve due to the presence of the conjugate image). In this case,
the CGH divides the pump beam into twelve spots of equal intensity. This is
in contrast to a previous approach using a micro lens array [27] which results
in a non-uniform distribution of energy across the array of spots due to the
spatial variation in the pump-beam energy.
Subsequently, the benefits of pumping the organic laser dynamically were
shown. A sequence of holograms displaying spots at different positions was
5.2. MAIN RESULTS 155
Figure 5.4: Optically pumping an organic laser using the replay field of a multi-level phase CGH.(a) 2D
profile for one spot, (b) 3D profile for one spot, (c) emission spectrum from a cell pumped with a one
replay field spot in four different locations, (d) 2D profile for six spots (inset: ideal replay-field pattern).
In (a,b,d) the symmetric order is displayed and in (a,b) the zero order is present alongside a second order
spot.
156 CHAPTER 5. ADAPTIVE PUMPING OF RADIATIVE LC CELLS
created with the holograms displayed at 0.5 s intervals. This dynamic pump-
ing process was then exploited to increase the repetition rate accessible to
the LC laser. To this end, the pump beam was scanned around the lasing
medium so that it was only incident on a given active region for a short pe-
riod of time, thus minimizing the interaction time between the dye and the
pump beam to avoid triplet state generation and other degradation effects.
Results obtained for pumping the LC laser in the static case with a single spot
at a fixed location over a total of 12,000 pulses is presented in Figure 5.5(a),
which highlights the degradation in performance for high repetition rates.
The data shows how the normalised pump energy decreases with the number
of pulses and is found to be more extreme for the higher repetition rates.
In particular, at a repetition rate of 600 Hz the output drops exponentially
to 40% of the initial value in approximately 2000 pulses (around 4 seconds).
Before comparing this behavior with that observed for dynamic pumping it
was necessary to determine the best frame rate for the holograms displayed on
the SLM; that is, how often should a new hologram be displayed to change the
position of the pump spots and refresh the dye. Therefore, videos displaying
different CGHs at various frame-rates were made. The LC laser outputs for
these different replay fields at a pump repetition rate of 300 Hz were then
recorded, as shown in Figure 5.5(b). It can be seen that a frame rate of
1 frame per second (fps) yields laser pulses with a significantly larger rate
of degradation of LC laser output. Subsequent experiments were performed
using a frame rate of 2 fps as this resulted in a relatively stable output on
the order of some minutes. Increasing the rate to 10 fps did not lead to any
significant improvement in the stability although it did reduce the output
5.2. MAIN RESULTS 157
a b
c d
Figure 5.5: Accessing higher repetition rates using dynamic pumping. The energy was recorded for each
pulse, but in (a-c) results are averaged over 2 s. The energies are normalized to the first pulse. (a)
Dynamic pumping at 600 Hz with different frame rates - 1 fps (), 2 fps (•) and 10 fps (blue N) (b) Static
pumping of a LC laser. (c) Dynamic pumping with 2 fps at different repetition rates. In (b) and (c): 200
Hz (), 300 Hz (•), 400 Hz (blue N), 600 Hz (pink H) (d) Total power output over 1 minute for different
repetition rates for both the static () and dynamic (•) case.
158 CHAPTER 5. ADAPTIVE PUMPING OF RADIATIVE LC CELLS
power due to the rise time of the LC laser which became more pronounced
at greater frame rates. Figure 5.5(c) demonstrates the improvement in the
performance of the LC laser for dynamic illumination. The results show that
output is almost independent of the number of pulses even for the highest
repetition rate used in this study and the output is thus significantly more
stable. When the LC laser was statically pumped with a single spot in
a fixed location at different repetition rates (Figure 5.5b), the normalized
output energy decreased with the number of pulses and the effect was more
pronounced at higher repetition rates: at 600 Hz the output dropped to
60% of its value in roughly 2000 pulses (4 s). Increasing the repetition rate
did not significantly alter the pump pulse energy. In contrast, when dynamic
pumping is employed, the LC laser output is almost independent of the pulse
number, even at the highest repetition rate that we used; the output is
significantly more stable [Figure 5.5(c)]. For example, in the case of a 400
Hz repetition rate, it took 29 minutes to decay by 10%, compared with
5s using static pumping. The measured normalized photostabilities for the
different repetition rates, corresponding to a 10% decrease in the output
pulse energy and for the pumping volume in the LC cell based upon the cell
thickness and pump spot area, are calculated to be 0.028GJ/mol (200 Hz),
0.182 GJ/mol (300 Hz), 1.19 GJ/mol(400 Hz) and 1.41 GJ/mol (600 Hz).
We also compared mean output powers over a given length of time [Figure
5.5(d)] and this was greater at all repetition rates when dynamic pumping
was employed. The mean output power for the first minute of emission
increased with repetition rate when dynamically pumped, but decreased (due
to bleaching) when statically pumped.
5.2. MAIN RESULTS 159
Figure 5.6: Multicolor laser arrays and dynamic wavelength tuning. (a) 4 spots (one zero order, two first
order and one second order) corresponding to different colours from red to green. (b) Emission spectra
from different regions of the cell, obtained by static addressing of different regions.
CGHs can also be designed such that they increase the functionality of the LC
laser by configuring the pump beam to steer the beam onto different regions
of the cell. By combining this function with a pitch-gradient LC cell - a
cell consisting of two LCs that emit at different ends of the visible spectrum
which have diffused together to form a continuum of wavelength regions -
it is possible to `tune' the wavelength of the laser or create a multi-colored
array. Changing the position of the pump beam on the cell enables a different
output wavelength to be obtained. LC lasers have previously been tuned by
deforming the molecular structure using temperature variation, electric fields,
magnetic fields, mechanical stresses [8] or UV radiation [28]-[29]. Figure
5.6(a) depicts a pattern of spots, each of which is a different colour due to the
pitch variation across the cell. Figure 5.6(b) shows the laser emission spectra
from five different regions of the cell. Each region was isolated by changing
the hologram so that the pump beam was incident on a different region. The
160 CHAPTER 5. ADAPTIVE PUMPING OF RADIATIVE LC CELLS
method presented here can be directly applied to configurations relying on
the energy transfer between two dyes [30], allowing tuneable output across
the entire visible range. In this study, we demonstrate a simple approach
to wavelength tuning and multi-chromatic outputs simultaneously, but it is
equally possible to envisage that novel pump beam profiles that optimise
propagation through the medium could also be demonstrated. This work
has focused on the behavior of an organic solid-state dye laser, however, the
same approach could be used in the field of organic semiconductor lasers [31].
In conclusion, in this chapter we describe a system for pumping an organic
solid-state dye laser using computer generated holography. This technique
provides a means with which to vary the location of the pump beam on the
active medium in real-time enabling fresh active gain regions to be addressed
in rapid succession, thus preventing build-up of triplet states and ensuring
optimum thermal management. Both static and dynamic hologram-defined
pumping have been shown to be possible, with dynamic pumping leading
to improved stability, larger average output powers and access to higher
repetition rates. With higher repetition rate pump lasers, it is feasible that
this approach could enable these organic lasers to reach operating frequencies
on the order of a MHz, without the need of mechanically moving parts.
Additionally, due to the unique nature of the adaptive pumping method, it
is possible to precisely control the spatial wavefront and configuration of the
pumping wave allowing greater versatility and functionality to be realised.
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Chapter 6
Suggestions for further research
6.1 Decay dynamics in the Jaynes-Cummings
model
This chapter we will outline the salient parts of the resolvent method and
provide some insight leading to the full treatment of atomic radiation in the
resonant environment of a photonic crystal. We will commence our discus-
sion in the framework of the JC model, where the existence of the so-called
`quasi dressed-states' (the ones leading to the oscillatory behaviour of the
spontaneous emission decay) was firstly demonstrated in [1], and we will see
how similar conclusions can be reached with the resolvent method, which is
consistent with perturbation theory. In connection to the 4th chapter, for
a(n isotropic) 3D photonic crystal, the Laplace Transform of the bound state
amplitude is
†
[1]
†
To avoid misunderstanding, in this chapter we no longer designate unit vectors by a
hat on top, but we reserve this symbol to designate operators only.
166
6.1. DECAY DYNAMICS IN THE JAYNES-CUMMINGS MODEL 167
c˜d(s) =
[
s+
∑
κ
|µκ|2 1
s+ i(ωκ − ωd)
]−1
=
[
s+
(ω10d10)
2
2~ε0V
∫ ∫
(ekλ · rˆ)2
s+ i(ωk − ωd)
dΩkdk(
2pi
L
)3
]−1
=
[
s+
(ω10d10)
2
16~ε0pi3
∫ Λ
0
∫ pi
0
2pi
sin3 θ
s+ i(ωk − ωd)k
2dθdk
]−1
=
[
s+
(ω10d10)
2
6~ε0pi2
∫ Λ
0
k2
s+ i(ωk − ωd)dk
]−1
,
(6.1)
where Λ is the cutoff in the photon wave-vector corresponding to the Comp-
ton wavelength. Near the band-edge ωc of the crystal, the dispersion relation
is written as
ωk ' ωc + ωc
k21
(k − k1)2, (6.2)
where k1 = pi/L is the first spatial frequency in the Fourier series expansion
of the dielectric profile function (L is the lattice constant). Substituting this
expression into (6.1) the following analytic expression is obtained
c˜d(s) =
(s− iδω) 12
s(s− iδω) 12 − (iβ) 32 , (6.3)
where β =
ω
7
2
10d
2
10
6pi0~c3
is related to the coupling strength and δω = ω10 − ωc
measures the detuning. Furthermore, based on the residue theorem, the
inverse Laplace transform can be also calculated in a closed form
cd(t) =
1
2pii
∫ σ+i∞
σ−i∞
c˜d(s)e
stds = f1(δ, β) exp(βx
2
1t+ iδωt)
+ f2(δ, β) exp(βx
2
2t+ iδωt) + f3(δ, β)
exp(iδt)
(βt)
3
2
,
(6.4)
168 CHAPTER 6. SUGGESTIONS FOR FURTHER RESEARCH
where we have used the asymptotic expansion of the complementary error
function and approximated for large βt as follows
erfc(
√
βx2i t) =
exp(−βx2i t2)√
piβx2i t
∞∑
n=1
(−1)n (2n− 1)!!
(2βx2i t
2)n
' exp(−βx
2
i t
2)√
piβx2i t
(−1)
2βx2i t
2
=
exp(−βx2i t2)
2
√
pi(βt)3
,
(6.5)
where i = 1, 2, 3. As can be shown in [1] x21 = i|x1|2, so that the first term
in (6.4) represent a bound atom-photon state with no exponential decay and
frequency ωc − β|x1|2. The second term (representing also a bound state) in
(6.4) depends on the relative position between the band-gap and the atomic
radiation frequency. If the latter is far inside the band-gap, then this term
vanishes (for the pertinent condition see [1]). The third term in (6.4) arises
from the branching point of the square root and yields a state responsible for
a non-exponential decay with an amplitude oscillating with the band-edge
frequency ωc.
For a 1-D (partial) photonic crystal, as chiral nematics are, and for a given
orientational distribution of the gain medium the expression (6.1) needs to
be modified as follows
c˜d(s) =
[
s+
∑
κ
|µκ|2 1
s+ i(ωκ − ωd)
]−1
=
[
s+
(ω10d10)
2
2~ε0V
∫ 〈(ekλ · rˆ)2〉
s+ i(ωk − ωd)
dk(
2pi
L
)]−1
=
[
s+
(ω10d10)
2
2~ε0S
∫ ω
max
ω
min
f(Sg)
s+ i(ω − ωd)ρ(ω)dω
]−1
,
(6.6)
where ρ(ω) is the photonic DOS, f(Sg) is a function of the order parameter
6.2. THE BASIC PRINCIPLES OF THE RESOLVENT METHOD 169
pertaining to the orientational distribution of the gain molecule (see [2]). A
numerical study of this integral can reveal the time-evolution characteristics
of the atom-photon states. For N → ∞ (where N is the number of full he-
licoidal director precessions in the LC sample) and for oblique propagation
in the structure, the spontaneous emission intensity is calculated in [4] for
t → ∞ within the JC formulation for Floquet's theorem applied for Math-
ieu's equation (In the system of equations obtained, the contribution of field
amplitudes with subscript n ± 2 is neglected in comparison to those having
suffixes n, n − 1. This is consistent with deriving the first-order Brillouin
dispersion relation).
6.2 The basic principles of the Resolvent Method
We will now approach the same problem within the context of the resolvent
method, which is consistent with perturbation theory. The method involves
the calculation of the matrix elements of the operator R(z) = 1/(zI − H)
(called the `resolvent' operator) which features in the expression for the time-
evolution operator (in this section we drop the hats denoting the operators,
for convenience)
U(t) =
1
2pii
∮
C
e−izt/~
zI −Hdz, (6.7)
where C is a closed contour around the axis of <(z) where are located all
the eigenvalues of the unperturbed Hamiltonian (which are real, since the
operator is Hermitian). For a Hamiltonian of the form H = H0 + V , the
matrix elements of the resolvent operator are written as [3]
170 CHAPTER 6. SUGGESTIONS FOR FURTHER RESEARCH
Rnm = − 1
∆(z)
∂∆(z)
∂Vmn
= −∂ ln ∆(z)
∂Vmn
, (6.8)
and the determinant ∆(z) of the matrix (zI −H) is written in the form
∆(z) = det(zI −H) = det(zI −H0)det[I − (zI −H0)−1V ]. (6.9)
Setting A = I − (zI −H0)−1V we write
detA = exp[Tr(lnA)] = exp
(
Tr
∞∑
n=1
1
n
Bn
)
, (6.10)
where B = (zI −H0)−1V . The above can be written as
detA = exp[Tr(lnA)] = exp
[ ∞∑
n=1
1
n
Sn
]
, (6.11)
where Sn = Tr(B
n). Expanding (6.9) in terms up to the second order of V ,
as it is usually done in practice, we have
∆(z) = det(zI −H) = det(zI −H0)
[
1− S1 + 1
2!
(S21 − S2)
]
, (6.12)
where
S1 = Tr(B) =
∞∑
n=0
Vnn
z − E0n
(6.13)
and
S2 = Tr(B
2) =
∑
n
∑
m
VmnVnm
(z − E0m)(z − E0n)
, (6.14)
6.2. THE BASIC PRINCIPLES OF THE RESOLVENT METHOD 171
where E0l are the eigenvalues of the unperturbed Hamiltonian. Using these
expressions, the denominator in the expression for the matrix elements of the
resolvent operator reads
∆(z) =
∏
n
(z − E0n)−
∑
m
∆m
∏
n6=m
(z − E0n)
(z − E0m)
+
1
2!
∑
m
∑
p
∆mp
∏
n 6=m,p
(z − E0n)
(z − E0m)(z − E0p)
− ...,
(6.15)
in which ∆m = Vmm, ∆mp = VmmVpp − VmpVpm,... are the principal minors
of the perturbation matrix V . This determinant can be also written as
∆(z) =
∏
n
(z − E0n)[1− F (z)], (6.16)
with
F (z) =
∑
m
Cm(z − Em)−1, (6.17)
and
Cm = ∆m − 1
2!
∑
p 6=m
∆pm
z − E0p
+ ... (6.18)
Based on the definitions (6.16) and (6.8) we write
Rnm =
∂F (z)
∂Vmn
1− F (z) (6.19)
The decay law for the excited state |b, 0〉 (where photons are absent in all
oscillators) is described by the matrix element Ub0,b0(t) = 〈b, 0|U(t)|b, 0〉, and
the corresponding matrix elements of the resolvent operator are
172 CHAPTER 6. SUGGESTIONS FOR FURTHER RESEARCH
Rb0,b0 = − 1
∆(z)
∂∆(z)
∂Vb0,b0
= −∂ ln ∆(z)
∂Vb0,b0
. (6.20)
The denominator ∆(z) is written as
∆(z) =
∏
n
∆n =
∏
n6=m
[
z − E0n −
(
Vnn +
1
2!
∑
p
VnpVpn
z − E0m
)]
,
(6.21)
since the zeros of the denominator, zi closest to E
0
n correspond to leading
terms multiplied by the large quantity ∂F (z)/∂Vnn
∣∣(z = zi). In (6.21) we
have retained terms proportional to the first order of the interaction constant
1/µ ∼= e2/~c. The numerator of Rb0,b0 is
Nb0,b0 = −∂∆(z)
∂Vb0,b0
=
(
1 +
|Vb0,b0|
z − E0b0
) ∏
n6=b0
[
z − E0n −
(
Vnn +
1
2!
∑
p
VnpVpn
z − E0m
)]
'
∏
n6=b0
[
z − E0n −
(
Vnn +
1
2!
∑
p
VnpVpn
z − E0m
)]
,
(6.22)
where in the approximation effected we have omitted terms of the second
order in the interaction constant. Hence, under this approximation only one
term remains in the denominator and therefore the matrix elements of the
time evolution operator can be written in the form
Ub0,b0 =
1
2pii
∮
C
e−izt/~Rb0,b0(z)dz
=
1
2pii
∮
C
e−izt/~
[
(z − Eb0)−
(
Vb0,b0 +
1
2!
∑
k
∣∣Vb0,gk∣∣2
z − E0gk
)]−1
.
(6.23)
6.3. ATOMIC RADIATION IN FREE SPACE 173
6.3 Atomic radiation in free space
We now consider the sum featuring in the expression above for the case of
spontaneous radiation in free space, as a limiting case of a system with a
discrete spectrum that approaches the continuum as a parameter changes.
The simplest example is that of a cubic cavity where the field is subject
to periodic boundary conditions. Letting the volume of the cavity tend to
infinity we obtain the transition from the discrete spectrum
kd =
2pi
3
√
V
(ld,md, nd), ωd =
2pic
3
√
V
√
l2d +m
2
d + n
2
d. (6.24)
In a resonant cavity (where the modes are denoted by the subscript j, and
the states of the unperturbed atomic Hamiltonian are denoted by |g〉) the
sum featuring in the expression (6.23) takes the form
S =
1
2!
∑
g,j
Vb0,gjVgj,b0
z − E0gj
=
pie2~
m2V
∑
g,j
|(pˆ · ej)bg|2
ωj(z − Eg − ~ωj) , (6.25)
since the matrix elements of the interaction Hamiltonian are
Vb0,gj = − e
m
√
2pi~
ωjV
[(pˆ · ej)eikj ·rˆ]bg (6.26)
and
∑
g
∣∣[(pˆ·ej)eikj ·rˆ]bg∣∣2 = ∑
g
〈b|(pˆ·ej)eikj ·rˆ|g〉〈g|e−ikj ·rˆ(pˆ·ej)|b〉 =
∑
g
∣∣(pˆ·ej)bg∣∣2.
(6.27)
In the expression for (6.25) the summation is over all the field oscillators
including the energy degenerate modes that differ only in the direction of
174 CHAPTER 6. SUGGESTIONS FOR FURTHER RESEARCH
propagation or polarization. Converting the sum to an integral over k with∑
k →
∫ dΩkk2dk
(2pi)3/V
and using the dispersion relation k = ω/c we can write
S =
e2~
3pic3m2
∑
g
|pbg|2
∫ ω
max
0
ωdω
z − Eg − ~ω , (6.28)
where we have integrated over all polarization directions. After having deter-
mined S, and effecting the mass renormalization ‡ by introducing an operator
Wˆ with
Wb0,b0 =
1
2µ
〈b|p2|b〉 = 1
2µ
∑
g
〈b|p|g〉〈g|p|b〉, (6.29)
we can write the denominator of the resolvent as ∆b0 = z−Eb0− (Wb0,b0 +S)
with
Wb0,b0 + S =
e2~
3pic3m2
∑
g
|p|2bg(z − Eg)
∫ ωmax
0
dω
z − Eg − ω (6.30)
In the calculation of these matrix elements appear the Cauchy integrals of
the form
I(ζ) =
∫ b
a
F (ω)
ω − ζ dω, (6.31)
defining an analytical function of ζ. This function has a cut along the segment
(a, b), is single-valued along the entire complex plane excluding this segment,
‡
We assume that the perturbation Hamiltonian is written in the form Hˆ ′ = Vˆ + Wˆ
with Wˆ = pˆ2/(2µ) being a term that accounts for the large changes occurring due to the
interaction of the atom with the field. This term has the structure of the kinetic energy
operator and the 'effective mass' µ is chosen following the form of S.
6.3. ATOMIC RADIATION IN FREE SPACE 175
and decreases monotonically with |ζ| → ∞. Using the Sokhotski-Plemelj
theorem, for ω0 ∈ (a, b) we have
lim
ζ→ω0±i0
∫ b
a
F (ω)
ω − ζ dω = ±ipiF (ω0) + P
∫ b
a
F (ω)
ω − ω0dω. (6.32)
In the interval 0 < z < ~ωm (where ~ωm is the upper limit of integration) we
obtain the following expression
∆b0 = z−Eb0− α
3pi
∑
g
|pbg|2
m2c2
(z−Eg)
[
ipiu(z−Eg)+ ln
∣∣∣∣ ~ωmz − Eg −1
∣∣∣∣], (6.33)
where α = e2/(~c) is the fine structure constant. After the analytical contin-
uation of the expression (6.33), the main contribution to the matrix element
Ub0,b0(t) will originate from the residue at the pole z = zb
U ′b0,b0(t) =
exp{−i[(Eb0 + ∆Eb)t]− Γbt/~}(
∂∆b0
∂z
)
z=zb
. (6.34)
This contribution describes the exponential decay of the excitation. The
derivative in the denominator evaluates to
(
∂∆b0
∂z
)
z=zb
= 1− α
3pi
∑
g
|pbg|2
m2c2
[
ipi+ ln
(
~ωm
Eb − Eg − 1
)
+
~ωm
~ωm + Eg − Eb
]
.
(6.35)
For the derivation of the above we have assumed that ∆b0 can be written as a
Taylor series of (z− zb) up to the first order in the region of zb. The contour
C we defined above, along which the integral (6.23) is taken is deformed
to encircle the pole zb and to circumvent the branching points 0, Eg and
176 CHAPTER 6. SUGGESTIONS FOR FURTHER RESEARCH
Em, which is the upper limit of the integral I. The dominant contribution
arises from the pole zb, which is responsible for the exponential decay of
the excitation, since the remaining terms are multiplied by the small fine
structure constant and the small parameter |pbg|2/(m2c2) ∼ (λ/a)2 ∼ 10−7),
where a is the Bohr radius and λ the radiated wavelength from the transition
b → g. We will consider now a term, which, albeit small in comparison
to U ′b0,b0(t), yields a small deviation from the exponential excitation: the
integral along a contour starting and ending at ±ρ− i∞ (respectively), and
circumventing the origin (which is a branching point)in the z-plane, can be
recast in the form
U ′′b0,b0(t) = −
~
2pi
(∫ ∞
0
e−ρt
∆−b0(ρ)
dρ−
∫ ∞
0
e−ρt
∆+b0(ρ)
dρ
)
= − ~
2pi
∫ ∞
0
∆+b0(ρ)−∆−b0(ρ)
∆+b0(ρ)∆
−
b0(ρ)
e−ρtdρ,
(6.36)
with the parametrization z = iρ along the path. For large t, the only im-
portant contribution in the integral will arise for ρ → 0. In that case, the
difference ∆+b0(ρ)−∆−b0(ρ) can be expressed, to first order in ρ, as
∆+b0(ρ)−∆−b0(ρ) =
2α
3pi
|pbg|2
m2c2
ρ, (6.37)
while the denominator of (6.36) tends to the constant
D = Eb0 +
α
3pi
∑
g
|pbg|2
m2c2
(z − Eg)
[
ipiu(Eb − Eg) + ln
∣∣∣∣ ~ωmz − Eg − 1
∣∣∣∣]. (6.38)
Since
∫∞
0
ρe−ρtdρ = 1/t2, the contribution of this term to the matrix element
Ub0,b0(t) amounts to
6.4. ATOMIC RADIATION IN CAVITIES AND WAVEGUIDES 177
U ′′b0,b0(t) =
α
3pi
|pbg|2
m2c2
~2
D2t2
, (6.39)
representing a term responsible for the departure from exponential decay.
6.4 Atomic radiation in cavities and waveguides
The interaction of an atom with the cavity modes in the first order of pertur-
bation theory shifts the energy levels of a subsystem, however the spectrum
of the system as a whole, remains discrete. For an atom in a cavity without
losses, the state
|ψ(t)〉 = exp(−iE0t/~)
[
C0|0〉+
∑
n6=0
Cn exp(iMn∆ωt)
]
, (6.40)
describes a periodical motion of the system (known as the `Poincaré cy-
cle') with period 2pi/∆ω, where |n〉 are stationary states of the unperturbed
Hamiltonian and the coefficients of the stationary states Dn = Dn(t) =
Cn exp(iMn∆ωt) are of first order of smallness. These coefficients are periodic
with period 2pi/∆ω, since their expression contains the factor exp(iMn∆Ωt)
in the integrand, for a periodic perturbation (see Chapter 3). The phase fac-
tor exp(−iE0t/~), common to all terms can be disregarded in deriving the
equations of motion for the system.As we can infer, the system has `memory'
of the initial state, which is repeated for an infinite time and determines the
motion of the system. In a cavity with a length much larger than the atomic
radiation wavelength, the value of ∆ω is very small and consequently the ini-
tial state will be reproduced after a very long time interval. Hence, radiation
178 CHAPTER 6. SUGGESTIONS FOR FURTHER RESEARCH
in a large cavity is practically aperiodic and can be compared to radiation
in free space. Another important aspect which needs to be emphasized is
that higher orders of perturbation give rise to new combination frequencies,
approaching the continuum in the infinite limit. From this discussion, we can
treat the optical cavity as a open system whose radiation spectrum is mod-
ified by the presence of Lorentz-type signatures, which are representative of
the high-quality cavity resonances. For this reason, we write the denominator
F (z) of the resolvent operator matrix element Ub0,b0(t) as
∆(z) = z − ~Ω− 1
2!
(I1 + I2), (6.41)
where
I1,2 =
∫ ρ(ω)∣∣Vb0,gk∣∣2(1,2)
z − ~ω dω =

i
α|pbg|2
3m2c2
≡ −i~γ (1)∑
n
h2n
z − ~(ωn − iΓn) (2),
(6.42)
where the case (1) corresponds to the interaction of the atom with the free
space modes (and is due to the ipiF (ω0) term in (6.32), the other term being
approximately equal to zero), and the integral (2) represents the interaction
of the atom with the cavity modes (corresponding to Lorentzian profiles).
For the case (2) we have applied the residue theorem for the integral
I ′ =
∫ ∑
n
h2n
(z − ~ω)[(ω − ωn)2 + Γ2n]
dω
=
∫ ∑
n
h2n
(z − ~ω)[ω − (ωn + iΓn)][ω − (ωn − iΓn)]dω,
(6.43)
where we have retained only the term
6.4. ATOMIC RADIATION IN CAVITIES AND WAVEGUIDES 179
2pii
∑
n
1
z − ~(ωn − iΓn)
h2n
(−2iΓn) ≡ −
∑
n
h˜2n
z − ~(ωn − iΓn) , (6.44)
that would lead to the physically meaningful exponential decay of the ex-
citation. Summing the two contributions, the denominator can be written
as
∆(z) ∼= z − ~Ω + i~γ −
∑
n
h˜2m
z − ~(ωm − iΓm) , (6.45)
where for illustrative purposes we will limit ourselves to only one cavity
mode (n = m). Equating the denominator of the resolvent to zero in order
to find the (dominant) pole contribution, the following quadratic equation is
produced
(z − E ′)(z − E ′′) = h˜2m, (6.46)
where E ′ = ~(Ω− iγ) and E ′′ = ~(ωn − iΓn). The roots of this equation are
z1,2 =
1
2
[
(E ′ + E ′′)±
√
(E ′ − E ′′)2 − 4h˜2m
]
. (6.47)
On the assumption that |h˜m|  |E ′|, |E ′′|, which holds, then
±
√
(E ′ − E ′′)2 − 4h˜2m = ±(E ′ − E ′′)
√
1− 4h˜
2
m
(E ′ − E ′′)2 (6.48)
the roots of the equation differ from E ′, E ′′ only in terms of second order in
the small parameter h˜m (since−pi < arg [(E ′ − E ′′)2]+arg
[
1− 4h˜
2
m
(E ′ − E ′′)2
]
≤
pi). Hence, the roots of the equation, written for convenience as z1,2 =
180 CHAPTER 6. SUGGESTIONS FOR FURTHER RESEARCH
~(Ω1,2 − iκ1,2) have negative imaginary parts. Finally, the decay of the exci-
tation is determined by the matrix element [3]
Ub0,b0(t) =
[Ω1 − ωn − i(κ1 − Γn)] exp[−(κ1 + iΩ1)t]
Ω1 − Ω2 − i(κ1 − κ2)
− [Ω1 − ωn − i(κ1 − Γn)] exp[−(κ1 + iΩ1)t]
Ω1 − Ω2 − i(κ1 − κ2) .
(6.49)
This is a simple illustrative case of a not purely exponential decay process,
in which there is energy transfer between the atom and the resonator. If the
coupling between the atom and the modes of free space is weak, then γ = 0,
and only the cavity losses determine the decay law of the atomic excitation.
In the case of an atom in a waveguide, the summation term in the expression
for Ub0,b0(t) becomes
∑
κ
|Vgκ,b0|2
z − ~ωκ →
g2c
2pi
∫ ∞
0
dk′
ωk′(z − ~ωκ′) , (6.50)
where the dispersion relation reads
k′ = c−1(ω2 − ω2c )
1
2 ⇒ dk
′
dω
ω[c(ω2 − ω2c )
1
2 ]−1, (6.51)
where ωc is the cut-off frequency of the waveguide. For simplicity, hereinafter
we assume that the atom is a two level system with transition frequency Ω =
(Eb−Eg)/~ and a momentum operator that takes the form pˆ = mΩr|(b〉〈g|+
b〉〈g|). We introduce the dimension-less quantities
ν = ω/Ω, νc = ωc/Ω, dν = dω/Ω, ζ(ζ
′) = z(z′)/(~Ω) (6.52)
and the integral above acquires the form
6.4. ATOMIC RADIATION IN CAVITIES AND WAVEGUIDES 181
I(ζ ′) =
g2
2pi~Ω
∫ ∞
νc
dν
(ζ ′ − νc)(ν2 − ν2c )
1
2
. (6.53)
Applying the Sokhotski-Plemelj relation we have
lim
ζ′→ζ±i0
I(ζ ′) =
g2
2pi~Ω(ζ2 − ν2c )
1
2
{
ln
[
(ζ + νc)
1
2 + (ζ − νc) 12
(ζ + νc)
1
2 − (ζ − νc) 12
]
∓ ipi
}
.
(6.54)
Since this integral has a branch cut along the segment (νv,∞) we can separate
the integral along the upper (+) and lower (-) parts of the contour C, as below
Ub0,b0(t) =
1
2pii
∮
C
e−iζΩt
G(ζ)−1
dζ ≡ 1
2pii
∫ ∞
~ωc
e−izt/~
∆
(−)
b0
dz − 1
2pii
∫ ∞
~ωc
e−izt/~
∆
(+)
b0
dz,
(6.55)
and the denominators are
∆±b0,b0(z) = z − Eb −∆Eb(z)± iΓ(z), (6.56)
where
∆Eb(z) =
g2
2pi
ln
[
(z + ~ωc)1/2 + (z − ~ωc)1/2
(z + ~ωc)1/2 − (z − ~ωc)1/2
]
(z2 − ~2ω2c )−1/2 (6.57)
and
Γ(z) = −g
2
2
(z2 − ~2ω2c )−1/2. (6.58)
From the form of the matrix element
Ub0,b0(t) = − 1
pi
∫ ∞
~ωc
Γ(z)e−izt/~
[z − Eb −∆Eb(z)]2 + Γ2b(z)
dz, (6.59)
182 CHAPTER 6. SUGGESTIONS FOR FURTHER RESEARCH
we can deduce that the decay is not strictly exponential, as it would be in
the case that ∆Eb and Γ were independent of z. The integration contour
may be deformed as follows: since the function G(ζ) does not have any zeros
in the lower half plane (lower part of the contour) and exp(−iζΩt) → 0 as
=ζ → −∞ for t > 0, the integral along this part of the contour has a zero
contribution. The upper part of the contour must be deformed such that it
circumvents the branching point ζ = νc. One part of this contour encircles
the pole ζ = ζb, that has a negative imaginary part. The pole is found by
the analytic continuation of the function
G(ζ) = ~Ω(ζ − 1)− g
2
2pi~Ω(ζ2 − ν2c )
1
2
{
ln
[
(ζ + νc)
1
2 + (ζ − νc) 12
(ζ + νc)
1
2 − (ζ − νc) 12
]
− ipi
}
(6.60)
in the upper part of the contour onto another sheet of the Riemann surface
(being consistent with our previous assumption that the deformed lower part
of the contour does not `meet' the pole). As it happens also with radiation in
free space, the pole ζb is responsible for the exponential decay of the excited
atomic state, whereas the part of integral the contour enclosing the branching
point ζ = νc accounts for small deviations from exponentiality. We will now
focus on the transition from the discrete to the continuous spectrum. The
denominator of the resolvent can be written (omitting prefactors) as
H(ζ) = (ζ − 1)− Rλ
L
∑
m
1
νm(ζ − νm) , (6.61)
where R = g2/[2pi(~Ω)2] and the subscript m runs over the eigenfrequencies.
As the residues of the integrand in Ub0,b0(t) are
6.4. ATOMIC RADIATION IN CAVITIES AND WAVEGUIDES 183
e−iζnΩt
(∂H/∂ζ)
∣∣∣∣
ζn
, (6.62)
where H(ζn) = 0, then as the roots ζn become far from unity, the residue
will diminish in magnitude, since the denominator tends to infinity (the root
is very close to the asymptotes νm). We conclude that the most important
contribution is made when ζ ∼= νm ∼= 1. The resonance frequencies are
equidistant, so that we can write
νm = 1 +
λ
L
(
m+
1
2
)
, m = 0,±1,±2... (6.63)
The second term of H(z) in the region of interest can be written as
{[
1 +
λ
L
(
m+
1
2
)][
ζ − 1− λ
L
(
m+
1
2
)]}−1
=
(
L
λ
)2
(ζ − 1)(
L
λ
)2
(ζ − 1)2 −
(
m+
1
2
)2
(ζ − 1) +
(
L
λ
)(
m+
1
2
)
(ζ − 1)(ζ − 2)
.
(6.64)
We have assumed however that
ζ − 1 ∼= νm − 1 = λ
L
(
m+
1
2
)
, (6.65)
so that the last two terms in the denominator can be recast in the form
(
m+
1
2
)2
[(ζ − 2)− (ζ − 1)] = −
(
m+
1
2
)2
(6.66)
Finally, we can write
184 CHAPTER 6. SUGGESTIONS FOR FURTHER RESEARCH
H(ζ) = (ζ − 1) +
(
pi
Rλ
L
)∑
m
(
L
λ
) (L
λ
)
(ζ − 1)pi[(
m+
1
2
)
pi
]2
−
[(
L
λ
)
(ζ − 1)pi
]2 .
(6.67)
By virtue of the series expansion of tanx (see [5])
tan(x) =
∞∑
m=0
2x[(
m+
1
2
)
pi
]2
− x2
, (6.68)
setting x = pi(L/λ)(ζ − 1) we obtain
H(ζ) = (ζ − 1) + piR tan(x), (6.69)
with derivative
∂H
∂ζ
= 1 + piR sec2(x)
dx
dζ
= 1 + pi2
RL
λ
[1 + tan2(x)]. (6.70)
In the limit L → ∞ we can disregard unity in the above and for the roots
ζn(xn) we have
tan2(xn) =
(ζ2n − 1)2
pi2R2
, (6.71)
so that
∂H
∂ζ
∣∣∣∣
(ζ=ζn)
∼= L
Rλ
[(ζn − 1)2 + (piR)2]. (6.72)
Integration along the contour C produces the sum
6.4. ATOMIC RADIATION IN CAVITIES AND WAVEGUIDES 185
∑
n
e−iΩζnt
(ζn − 1)2 + (piR)2 , (6.73)
which for large L is approximately equal to the integral
∫ +∞
−∞
e−iΩζt
(ζ − 1)2 + (piR)2
dn
dζ
dζ, (6.74)
which is of Lorentzian form. Far from ζ = 1, the roots ζn are very close to
the resonance frequencies νm. In this region the roots are also equidistant (as
are the νm). The distance between the roots gradually decreases and reaches
its minimum when ζ ∼= 1. The limits are defined by the condition ζn − 1 '
piR ⇒ |m| ' (piRL/λ). In the region −(piRL/λ) < m < (piRL/λ) there is
one additional root compared with the number of resonance frequencies in
that region. This additional mode is distributed over a distance of (2piRL/λ).
Hence, the average distance between roots in that region differs from the
distance between roots outside this region by
∣∣∣∣ dd(L/λ)(piRL/λ)−1
∣∣∣∣ = λ22piRL2 , (6.75)
which is of the second order of smallness in terms of the (small) parame-
ter λ/L. Hence, we can calculate Ub0,b0(t) assuming that the roots ζn are
equidistant and that small deviations from equidistance are regarded as a
perturbation. Up to first order in λ/L we have ζ
(0)
n = 1 +n(λ/L) and in that
case, using that Ω = 2pic/λ we write [3]
Ub0,b0(t) =
piRLe−iΩt
2c~
+∞∑
n=∞
exp (−2pinict/L)
(npi)2 +
(
pi2RL
λ
)2 . (6.76)
186 CHAPTER 6. SUGGESTIONS FOR FURTHER RESEARCH
Setting T = L/(2c) and Γ = piΩR, we can consider the above sum as the
Fourier series expansion of a periodic function with
Ub0,b0(t) =
e−iΩt
~Ω
e−Γt + eΓ(t−2T )
1− e−2ΓT , (6.77)
in the fundamental period t ∈ [0, 2T ]. In the limit L → ∞ (T → ∞) we
obtain the exponential decay
Ub0,b0(t) =
1
~Ω
e−Γt−iΩt. (6.78)
We will consider now the effect of the deviation from equidistance on the
matrix element Ub0,b0(t) by taking the difference
∆Ub0,b0(t) =
Rλ
L~Ω
+∞∑
n=−∞
[
e−iΩζnt
(ζn − 1)2 + (piR)2 −
e−iΩζ
(0)
n t
(ζ
(0)
n − 1)2 + (piR)2
]
. (6.79)
The terms in the sum can be added to give
e−iΩζ
0
nt
{
e−iΩ∆ζnt
[(
ζ
(0)
n − 1
)2
+ (piR)2
]
− (ζn − 1)2 − (piR)2
}
[
(ζ
(0)
n − 1)2 + (piR)2
] [
(ζn − 1)2 + (piR)2
] . (6.80)
Keeping terms up to the first order in ∆ζn, the numerator (pre-multiplied by
e−iΩζ
(0)
n t
) becomes
(
e−iΩ∆ζnt − 1) [(ζ(0)n − 1)2 + (piR)2]− 2 (ζ(0)n − 1)∆ζne−iΩ∆ζnt, (6.81)
while in the denominator ζ
(0)
n replaces ζn. After cancellation of common
factors, the final result (to first order in ∆ζn) reads
6.4. ATOMIC RADIATION IN CAVITIES AND WAVEGUIDES 187
∆Ub0,b0(t) =
Rλ
L~Ω
+∞∑
n=−∞
e−iΩζ
(0)
n t
 e−iΩ∆ζnt − 1(
ζ
(0)
n − 1
)2
+ (piR)2
−
2
(
ζ
(0)
n − 1
)
∆ζne
−iΩ∆ζnt[(
ζ
(0)
n − 1
)2
+ (piR)2
]2
 .
(6.82)
The numerator of the first fraction in the sum can be recast in the form
− 2ie−i(Ω∆ζnt)/2 sin
(
1
2
Ω∆ζnt
)
. (6.83)
For finite time t,
sin
(
1
2
Ω∆ζnt
)
' 1
2
(Ωt) max |∆ζn| ∼
(
λ
L
)2
. (6.84)
The second term in the numerator is also of the second order of smallness
in (λ/L)2. In the sum (6.79) the number of significant terms is approxi-
mately 2piRL/λ, as we have claimed above. Hence the sum is of first order in
(λ/L), consequently the corrections to the exponential decay due to the non-
equidistance of the spectrum decrease inversely proportional to L as L→∞.
Finally, we note that (6.77) represents a periodic function in which the ini-
tial value appears with an arbitrary accuracy, as there is a periodic exchange
or energy between the atom and the cavity modes. As L → ∞, however,
the period T tends also to infinity. As a result, the system becomes then
aperiodic (the radius of the Poincaré cycle increases infinitely.)
188 CHAPTER 6. SUGGESTIONS FOR FURTHER RESEARCH
6.5 Atomic radiation in a 1D periodic medium
In the dipole approximation, the Hamiltonian of the system comprised of the
atom and the electromagnetic field has the form
Hˆ =
pˆ2
2m
+ V (qˆ) +
∑
j
~ωj aˆ†j aˆj −
e
m
∑
j
(
pi~
QScωjLj
) 1
2
Λj(pˆ · ej)(aˆ†j + αˆj),
(6.85)
where the constant Λj quantifies the dependence of the Hamiltonian on the
position of the atom and Lj reflects the frequency dependence of the j
th
eigenwave. The denominator of Ub0,b0(t), as usual, reads
G(z) = z − ~Ω− 1
2
∑
m
|Vb0,gm|2
z − ~ωm ≡ z − ~Ω−X(ω). (6.86)
Replacing the summation over the field oscillators with integration over
the effective wavenumber k′, we find that
G(z) = z − ~Ω−
∫ |Vb0,gωk′ |2Q(ωk′)
pi(z − ~ω′k)
dk′, (6.87)
where Q is a function characteristic of the periodic structure. The integration
is carried over all branches of the dispersion curve, in particular over the
allowed frequency bands. Hence, G(z) has cuts along the allowed bands
on the real axis (according to the Sokhotski-Plemelj relations), while in the
forbidden bands it takes real values. The boundaries between the allowed
and forbidden bands define the branching points of this function. Similarly
to our discussion for the properties of atomic radiation in a waveguide, only
the upper part of the contour C will contribute to Ub0,b0(t), and the function
G(z) for z + i0 assumes the value
6.5. ATOMIC RADIATION IN A 1D PERIODIC MEDIUM 189
G(z) = z − ~Ω + iA
z
|pbg · e|2f(k, a, b)− I, (6.88)
where f is a function of the characteristic dimensions of the 1D structure and
the dispersion relation, and the integral I is taken in the sense of a principal
value at real z = ~ω. This integral defines the Lamb shift, and in most cases
it is neglected due to its smallness [3]. It can be seen that the function G(z)
has a root situated at zb = ~(Ω− iγ), which is at the analytical continuation
of the upper edge of the physical sheet downwards. As usual, the residue at
this pole is responsible for the exponential decay.
Deforming the integration contour accordingly, to encircle the pole in the
non-physical sheet of the Riemann surface, to enclose the branch cuts and
encircle the pole on the real axis in the forbidden band (giving rise to the so-
called `dynamic state'), The matrix element Ub0,b0(t) breaks down into three
components. The two of these components that decay with time correspond
to the residue at the pole of the nonphysical sheet of the Riemann surface
and the the integral over the cut. The amplitude of the third component,
corresponding to the residue at the pole in the real axis, does not depend
on time. As noted in [3], the decay of the initial state does not occur fully,
but only up to a stationary level. When the transition frequency Ω is in
the band-gap, the exponential part of the radiation decay is very small, and
the probability of the appearance of the dynamic state approaches unity.
For the case of chiral nematic liquid crystals, the knowledge of the photonic
DOS can lead to the numerical computation of the function G(z) and hence
to the properties of spontaneous emission decay in the molecular resonator.
An agreement with the results presented above is expected when L→∞.
190 CHAPTER 6. SUGGESTIONS FOR FURTHER RESEARCH
In this chapter we have presented a treatment whereby the eigenwaves along-
side the corresponding photonic density of states in chiral nematic liquid
crystals can be incorporated into the matrix elements of the resolvent oper-
ator. This suggestion follows the transition from a discrete to a continuous
spectrum when studying atomic radiation in resonant cavities and waveg-
uides, and aims to investigate departures from the Markov approximation
and the corresponding exponentially decaying temporal spontaneous emis-
sion profiles.
Bibliography
[1] S. John and T. Quang, Spontaneous Emission near the Edge of a
Photonic Bandgap, Phys. Rev. A, 50, pp. 1764-1769 (1994).
[2] J. Schmidtke and W. Stille, Fluorescence of a dye-doped cholesteric
liquid crystal film in the region of the stop-band: theory and experiment,
Eur. Phys. J. B 31, pp. 179-194 (2003).
[3] V. P. Bykov, Radiation of Atoms in a Resonant Environment, World
Scientific Publishing, Singapore, 1993.
[4] P. Carette, C. Li, J. Boyaval and B. Meziane, Spontaneous
Emission of Eu
3+
ions in a cholesteric liquid crystal, Optical and Quan-
tum Electronics 35, pp. 909-929 (2003).
[5] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and
Products, 7th Ed., Elsevier, 2007.
191
Synoptic conclusions
In this work, we have derived analytic expressions for the calculation of the
density of photon states (DOS) in dye-doped chiral nematic liquid crystal
(LC) cells in the presence of an active lossy medium. Results are presented
on the profile of the DOS as a function of the reduced wavelength for different
macroscopic parameters such as the optical anisotropy and the magnitude of
the losses. The results show that, for a fixed value of the pitch, the DOS
at the photonic band-edges decreases with increasing losses. The band-edge
that exhibits the largest DOS (e.g., short or long wavelength band-edge)
is found to depend on the magnitude of the loss coefficient; above a critical
value the long-wavelength edge exhibits the largest DOS. Using the analytical
expressions, results have been presented on the DOS as a function of the
number of pitches showing an optimal value of the pitch that corresponds
to the maximum in the density of states. These findings are correlated with
experimental results for the lasing threshold, which is found to be inversely
proportional to the maximum DOS when only losses are considered (apart
from spontaneous absorption). The behaviour of the DOS can also reflect
mode quenching in these lasing structures in the regime beyond the threshold
gain. It is understood that the consideration of the DOS proves to be an
192
193
invaluable means for understanding lasing and such an approach should be
implemented when determining the factors limiting the lasing threshold in
chiral nematic films (e.g., leaky modes) as well as when studying the mode
behaviour in more complicated configurations (e.g., defect modes).
We investigated resonance in a radiative chiral nematic LC (a characteris-
tic example of a partial distributed feedback structure), for the diffractive
polarization. We found that there is a disparity between the DOS and the
calculated emission spectrum, which is also verified experimentally. We out-
lined the main effects occurring for different detuning values through deriving
analytic results for the Lamb shift and the transition broadening following
the logarithmic divergence of the DOS in a 2D photonic crystal. We con-
clude that incorporating cavity losses alongside broadening mechanisms, and
including the effect of resonance leads to a more comprehensive treatment of
spontaneous emission from these structures.
Subsequently, we have detailed a system for pumping an organic solid-state
dye laser using computer generated holography. This technique provides a
means with which to vary the location of the pump beam on the active
medium in real-time enabling fresh active gain regions to be addressed in
rapid succession, thus preventing build-up of triplet states and ensuring op-
timum thermal management. Both static and dynamic hologram-defined
pumping have been shown to be possible, with dynamic pumping leading to
improved stability, larger average output powers and access to higher repe-
tition rates. With higher repetition rate pump lasers, it is feasible that this
approach could enable these organic lasers to reach operating frequencies on
the order of a MHz, without the need of mechanically moving parts. Ad-
194 SYNOPTIC CONCLUSIONS
ditionally, due to the unique nature of the adaptive pumping method, it is
possible to precisely control the spatial wavefront and configuration of the
pumping wave allowing greater versatility and functionality to be realised.
In this study, we demonstrate a simple approach to wavelength tuning and
multi-chromatic outputs simultaneously, but it is equally possible to envis-
age that novel pump beam profiles that optimise propagation through the
medium could also be demonstrated.
The final chapter focusses on the relevance of the resolvent method in quanti-
fying the decay of an excited atomic state in a chiral nematic. Following the
determination of the pertinent diagonal matrix element of the time evolution
operator in the case of a periodic waveguide, we can incorporate the density
of states to examine the behaviour of integral functions albeit for a structure
of finite length.
Related publications
Journal publications
(1) Th.K. Mavrogordatos, S.M. Morris, F. Castles, P.J.W. Hands, A.D. Ford,
H.J. Coles, and T.D. Wilkinson, Density of photon states in dye-doped chiral
nematic liquid crystal cells in the presence of losses and gain, Phys. Rev. E,
86, pp. 011705(1-7) (2012).
(2) Th.K. Mavrogordatos, S.M. Morris, S. M. Wood, H.J. Coles, and T.D.
Wilkinson, Spontaneous emission from radiative chiral nematic liquid crystals
at the photonic band-gap edge: An investigation into the role of the density
of photon states near resonance, Phys. Rev. E, 87, pp. 062504(1-8) (2013).
(3) S. M. Wood, Th. K. Mavrogordatos, S. M. Morris, P. J. W. Hands, F.
Castles, D. J. Gardiner, K. L. Atkinson, H. J. Coles, and T. D. Wilkinson,
Adaptive holographic pumping of thin-film organic lasers, Opt. Lett., 38, pp.
4483-4486 (2013).
195
196 RELATED PUBLICATIONS
Conference proceedings
(1) Th.K. Mavrogordatos, S. M. Morris, F. Castles, H. J. Coles, and T. D.
Wilkinson, The density of photon states in dye–doped chiral nematic liquid
crystal cells in the presence of absorption and gain, Laser Optics 2012, St.
Petersburg, Russia, June 2012.
(2) Th.K. Mavrogordatos, S. M. Morris, F. Castles, H. J. Coles, and T. D.
Wilkinson, Fluorescence and lasing characteristics of dye-doped chiral ne-
matic liquid crystal cells, (International Liquid Crystal Conference) ILCC
2012. Mainz, Germany, August 2012.