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A Unifying Statistical Model for Atmospheric Optical Scintillation 23
On the other hand, in reference to the M(α, β, γ, ρ, Ω
) distribution, where the β parameter is
restricted to be a natural number for this particular case, we can proceed as follows: from Eq.
(59), we can obtain the moment of the Rician-shadowed distribution, given by:
m
k
(
Y
)
=
1
γ
γβ
γβ + Ω
β
∞
0
y
k
exp
−
βy
γβ + Ω
L
β−1
−Ω
y
γ
γβ + Ω
dy
=
1
γ
γβ
γβ + Ω
β
β
−1
∑
r=0
β
−1
r
1
r!
Ω
γ
γβ + Ω
r
∞
0
y
k+r
exp
−
βy
γβ + Ω
dy.
(69)
Now, from Eq. (3.381-4) of Ref. (Gradshteyn & Ryzhik, 2000),
∞
0
x
ν−1
exp
(
−
μx
)
dx =
1
μ
ν
Γ
(
ν
)
, [Re(μ) > 0, Re(ν) > 0],
(70)
we can express Eq. (69) as:
m
k
(
Y
)
=
1
γ
γβ
γβ + Ω
β
β
−1
∑
r=0
β
−1
r
1
r!
Ω
γ
γβ + Ω
r
Γ
(
k + r + 1
)
β
γβ+Ω
k+r+1
.
(71)
Finally, when performing the product of Eq. (67) by Eq. (71), we certainly obtain Eq. (27).
9. Appendix C: proof of lemma 3
For both cases, when I follows a generalized distribution, M
(G)
,ordirectlyanM distribution
if its β parameter is a natural number, then we need to solve the same integral. Thus, from Eq.
(6.592.2) of Ref. (Gradshteyn & Ryzhik, 2000),
1
0
x
λ
(1 − x)
μ−1
K
ν
a
√
x
dx =
2
−ν−1
a
−ν
Γ(ν)Γ(μ)Γ
λ + 1 −
1
2
ν
Γ
λ + 1 + μ −
1
2
ν
1
F
2
λ
+ 1 −
1
2
ν;1
−ν, λ + 1 + μ −
1
2
ν;
a
2
4
+
+
2
−1−ν
a
ν
Γ(−ν)Γ
λ + 1 +
1
2
ν
Γ(μ)
Γ
λ + 1 + μ +
1
2
ν
1
F
2
λ
+ 1 +
1
2
ν;1
+ ν, λ + 1 + μ +
1
2
ν;
a
2
4
,
Re
(
λ
)
> −
1 +
1
2
|Re
(
ν
)
|
,Re
(
μ
)
>
0;
(72)
and by making the following change of variables: x
= I
T
·x,thendx
= I
T
dx; and identifying
μ
= 1, λ =(α + k)/2, ν = α −k,wherea = 2
α/(γI
T
) and a = 2
αβ/([γβ + Ω
]I
T
) for the
M
(G)
and the M distribution, respectively; thus, the cdf associated with the M
(G)
and the
M distribution is readily found to be the expressions indicated in Eqs. (28) and (29).
10. Acknowledgment
This work was fully supported by the Spanish Ministerio de Ciencia e Innovación, Project
TEC2008-06598.
203
A Unifying Statistical Model for Atmospheric Optical Scintillation
24 Numerical Simulations / Book 2
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26 Numerical Simulations / Book 2
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206
Numerical Simulations of Physical and Engineering Processes
9
Numerical Simulation of Lasing Dynamics
in Choresteric Liquid Crystal Based
on ADE-FDTD Method
Tatsunosuke Matsui
Mie University
Japan
1. Introduction
Liquid crystals (LCs) are categorized in one class of condensed materials which show both
character of liquids and solids (crystals). Liquid-like fluidic character of LCs allows them to
show dynamic response to external stimuli such as electrical, optical and magnetic fields.
Anisotropic characters of LCs like crystals show determines the way how they respond to
external stimuli and also how they appear (de Gennes & Prost, 1995). These characteristics
are widely utilized in LC display devices. LCs are further categorized in subgroups (phases)
in terms of their degrees of order (orientational and positional) as shown in Fig. 1.
Fig. 1. Schematics of several liquid crystal phases.
Cholesteric liquid crystal (CLC) is one of these sub-phases of LCs and the self-organized
formation of periodic helical structure is the most significant characters of this phase from
the viewpoint of device application. In case the pitch of the helix of CLCs is in the range of
the wavelength of visible light, they selectively reflect part of incident light in a certain
manner determined by their refractive index and the sense of helix. Figure 2 shows the
simulated transmission and reflection spectra of right-handed-circularly-polarized light
normally incident on CLC. Transmission/reflection band (stop band) can be recognized.
The central frequency and the bandwidth of the band are dependent on both refractive
indices and pitch of helix of CLC.
Numerical Simulations of Physical and Engineering Processes
208
Fig. 2. Transmission and reflection spectra simulated by Berreman’s 4 × 4 transfer matrix
method. Right-handed circularly polarized light is supposed to normally incident on 5 μm-
thick CLC with 400 nm-pitch right-handed helix. Extraordinary and ordinary refractive
indices are 1.70 and 1.50, respectively. (Inset) schematic representation of selective reflection
by CLC with right-handed helix. As an incident light, linearly or randomly polarized and in
the photonic band wavelength is assumed.
So many studies have been made to utilize this so-called “selective reflection” character of
CLC to make the reflection type display, or in other words, electronic paper. Recently, CLCs
are also extensively studied as a photonic band gap (PBG) material or photonic crystal (PC).
In 1987, Yablonovitch (Yablonovitch, 1987) and John (John, 1987) put forward the basic
concept of PBG and since then so many studies have been carried out. Electromagnetic (EM)
wave (photons) propagating in PCs composed of periodic stacking of dielectric materials
with different dielectric permittivity behaves just like de Broglie wave (electrons) travelling
in periodic Coulomb potential in crystals. Long dwell time of photon at PBG edge energy
allows strong light-matter interaction and low threshold lasing may be obtained in such
PBG system with optical gain introduced (Dowling et al, 1994). Introducing defect states in
PC can induce photon localization in PC (Joannopoulos et al., 1995).
Periodic helical structure of CLC can also work as PBG material. In 1998, Kopp and his
coworkers succeeded in obtaining band-edge lasing from dye-doped CLC (Kopp et al.,
1998). Since their pioneering work, numerous studies have been carried out from the
viewpoint of academic interest and technical applications (Coles & Morris, 2010). The
introduction of various types of defects has also been attempted in CLC as schematically
summarized in Fig. 3. Yang and his coworkers showed, based on numerical analysis, that
introduction of isotropic thin layer as a defect in the middle of CLC layer (Fig. 3 (a)) creates
the defect state (narrow transmission band) in the stop band, which can be tuned via
altering thickness or refractive index of the defect layer (Yang et al., 1999). Kopp and Genack
have numerically demonstrated that twist-defect, discontinuous phase shift of the helical
twist of CLC molecules (Fig. 3 (b)), could indeed function as a defect (Kopp & Genack, 2002).
This type of defect is unique in CLC with optical anisotropy. Lasing from twist-defect has
Numerical Simulation of Lasing Dynamics in
Choresteric Liquid Crystal Based on ADE-FDTD Method
209
been experimentally attained utilizing photopolymerized CLC polymer films (Ozaki et al.,
2003; Schmidtke et al., 2003). It has been also shown that introducing defect can contribute to
reduce lasing threshold. Other types of defects have also been introduced. It has been
numerically shown that local modulation of helical pitch of CLC as schematically shown in
Fig. 3 (c) can also introduce defect states in the stop band (Matsui et al., 2004). Multi-layer of
CLCs with different helical pitch as shown in Fig. 3 (d) has been experimentally realized,
which have been successful to achieve reduced lasing threshold (Ozaki et al., 2006;
Takanishi et al., 2007).
Fig. 3. Schematic representations of proposed defects in CLC PCs. (a) isotropic defect (b)
twist defect (c) locally modulated pitch (d) multi-layer system.
Refining device architecture may realize further lowering of the lasing threshold, which
motivated us to develop numerical simulation technique for the development of the more
efficient laser device architectures. We have recently reported on numerical simulation of
lasing dynamics in CLCs (Matsui & Kitaguchi, 2010). We have employed an auxiliary
differential equation finite-difference time-domain (ADE-FDTD) method, which was first
applied to the analysis of random lasing in one-dimensional (1D) random system (Jiang &
Soukoulis, 2000). We have successfully reproduced circularly-polarized lasing in CLC at the
energy of the edge of the stop band. Moreover, as will be discussed later, we have also
shown that our computational scheme can also be utilized to search for more efficient device
architecture with reduced lasing threshold. Here we will summarize the computational
procedure of ADE-FDTD method for the analysis of lasing dynamics in CLC and show that
this technique is quite useful for the analysis of EM field dynamics in and out of CLC laser
cavity under lasing condition, which might contribute to the deep understanding of the
underlying physical mechanism of lasing dynamics in CLC.
2. Numerical simulations
In this section, numerical simulation techniques employed in this study (1) ADE-FDTD
method for the analysis of lasing dynamics and (2) Berreman’s 4 × 4 transfer matrix method for
the analysis of transmission and reflection spectra are summarized. As will be discussed, ADE-
FDTD approach enables us to analyze lasing dynamics in CLC from various viewpoints such
as time-dependent EM fields, Fourier-transformed emission spectra and snapshots of spatial-
distributions of EM fields. Berreman’s 4 × 4 transfer matrix method is traditionally employed
to simulate transmission and reflection spectra in rather simpler way.
Numerical Simulations of Physical and Engineering Processes
210
2.1 ADE-FDTD based numerical simulation of lasing dynamics
In this subsection, overview of ADE-FDTD approaches for the analysis of lasing dynamics in
various types of micro- and nano-laser systems made so far will be given first, and then
detailed numerical procedure will be given.
2.1.1 Overview of ADE-FDTD approach for the analysis of lasing dynamics in micro-
and nano-photonics systems
FDTD has been widely utilized to numerically simulate the propagation and/or localization
of EM waves in micro- and nano-photonic media (Taflove & Hagness, 2005). In order to
investigate lasing dynamics, ADE-FDTD approaches have also been developed, in which the
FDTD method is usually coupled with the rate equation in a four-level energy structure and
the equation of motion of polarization (Nagra & York, 1998) as schematically shown in
Fig. 4. As discussed above, Jiang and Soukoulis have employed ADE-FDTD method for the
analysis of random lasing in 1D random system (Jiang & Soukoulis, 2000). Numerous
groups have followed them to investigate lasing dynamics in various kinds of laser cavities
such as 2D random media (Vanneste & Sebbah, 2001), PCs (Bermel et al., 2006; Shi
& Prather, 2007), and distributed Bragg reflectors (Chang & Taflove, 2004; Redding et
al., 2008).
Fig. 4. Schematic representation of ADE-FDTD simulation scheme.
2.1.2 ADE-FDTD method for the analysis of lasing dynamics in CLCs
Here numerical procedures for the ADE-FDTD analysis of lasing dynamics will be given.
Here we deal with a 1D system, where the time-dependent EM field propagating along the
z-axis is simulated using Yee's FDTD algorithm (Yee, 1966) to solve the following Maxwell's
equations:
()
()
0
,
,
zt
zt
t
μ
∂
∇× =−
∂
H
E
(1)
() ()
()
()
0
,,
,
zt zt
zt z
tt
εε
∂∂
∇× = +
∂∂
EP
H
(2)
Numerical Simulation of Lasing Dynamics in
Choresteric Liquid Crystal Based on ADE-FDTD Method
211
where
ε
0
and
μ
0
are the dielectric permittivity and the magnetic permeability in vacuum,
respectively.
ε
(z) is dielectric permittivity of the medium and should be tensor for LCs and
will be given later. P(z, t) is the polarization density, which provides a gain mechanism in
the laser system. On the basis of the classical electron oscillator (Lorentz) model, one can
obtain the following equation of motion of P(z, t) in the presence of an electric field
() ()
() ()()
2
2
2
2
,,
,,,
r
aa
c
dzt dzt
e
zt N zt zt
tm
dt
γ
ωω
γ
+Δ + = Δ
∂
PP
PE
(3)
where Δ
ω
a
= 1/
τ
21
+ 2/T
2
is the full width at half-maximum (FWHM) linewidth of the
atomic transition. T
2
is the mean time between dephasing events and
ω
a
(= 2πc/
λ
a
) is the
central frequency of emission. ΔN(z, t) (= N
1
(z, t) – N
2
(z, t)) is the difference between electron
numbers at levels 1 and 2 (Fig.
4),
γ
r
= 1/
τ
21
and
γ
c
= (e
2
/m)[
ω
a
2
/(6π
ε
0
c
3
)] is the classical rate,
e is the electron charge, m is the electron mass and c is the speed of light in vacuum.
The electron numbers at each energy level, N
0
(z, t), N
1
(z, t), N
2
(z, t) and N
3
(z, t) obey the
following rate equations.
()
()
()
33
0
32
,,
,
r
dN z t N z t
PN z t
dt
τ
=−
(4)
() ()
()
()
()
23 2
32 21
,, ,,
1
,
a
dN zt N zt d zt N zt
zt
dt dt
τω τ
=+ ⋅−
P
E
(5)
() ()
()
()
()
12 1
21 10
,, ,,
1
,
a
dN z t N z t d z t N z t
zt
dt dt
τω τ
=− ⋅−
P
E
(6)
() ()
()
01
0
10
,,
,
r
dN z t N z t
PN z t
dt
τ
=−
(7)
where
τ
32
,
τ
21
, and
τ
10
, are the lifetimes at each energy levels, and P
r
is the pumping rate of
electrons from ground state (level 0) to upper energy level (level 3) and is a controlled
variable that should be tuned by the pumping intensity in the real experiment. By coupling
these equations (1) – (7), numerical simulation of lasing dynamics can be made as
schematically shown in Fig. 4. Flow chart of ADE-FDTD algorithm for the analysis of lasing
dynamics is summarized in Fig. 5.
In order to deal with anisotropic medium like LC, dielectric permittivity should be
represented as tensor. Assuming that LC molecules are uniaxial with optical major axis
(director) along y-axis and that extraordinary and ordinary refractive indices of LCs are n
e
and n
o
, respectively, then the dielectric tensor of LCs should be represented as
()
() () ()
() () ()
() () ()
() ()
2
2
2
00
00
00
o
xx xy xz
yx yy yz e
zx zy zz
o
n
zzz
zzzz zn z
zzz
n
εεε
εε ε ε θ θ
εεε
==−
RR
(8)
Numerical Simulations of Physical and Engineering Processes
212
where R[
θ
(z)] is a rotation matrix about the z-axis and should be expressed as follows,
()
() ()
() ()
cos sin 0
sin cos 0
001
zz
zzz
θθ
θθθ
−
=
R
(9)
where
θ
(z) is the rotated angle from y-axis. By changing
θ
(z) gradually as a linear function of
z, modeling CLC with a helix can be made. Introduction of various types defects as
summarized in Fig. 3 can be easily made by modulating this
θ
(z) appropriately.
In order to excite the system, a short seed pulse should be launched. E
x
and E
y
fields are
monitored at a point in the glass until the system reaches a steady state. By Fourier-
transforming time domain signals, emission spectra can be analyzed in frequency domain.
Fig. 5. Flow chart of ADE-FDTD algorithm for the analysis of lasing dynamics.
2.2 Berreman’s 4 × 4 matrix for the analysis of transmission and reflection
spectra of CLCs
Berreman’s 4 × 4 transfer matrix has been widely utilized for the numerical analyses of the
optical transmission and reflection spectra in CLC with helical structure (Berreman, 1970).
EM fields propagating along the z-axis with frequency
ω
are given by