In order to see the control role of relative phase on the EIT effect of
probe pulse we fix the parameter p = 0.7 and plot the temporal evolution of
the pulse probe p(,) at the optical depth p = 5ns-1 for different values
of relative phase , as illustrated in figure 3.3. From figure 3.3a we can see
that the probe pulse envelope depends sensitively on the relative phase and
the influence of relative phase on the temporal evolution of pulse is a
period of 2. For 0 /2: at = 0, due to SGC making the leading
edge of pulse envelope is distorted as one considered in figure 3.2(c)

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i
MINISTRY OF EDUCATION AND TRAINING
VINH UNIVERSITY
----------
HOANG MINH DONG
INVESTIGATION OF LASER PULSE PROPAGATION
IN A THREE-LEVEL ATOMIC MEDIUM
IN THE PRESENCE OF EIT EFFECT
Specialty: Optics
Code: 62.44.01.09
A SUMMARY OF PHYSICAL DOCTORAL THESIS
NGHE AN, 2017
ii
Work completed at Vinh university
Supervisors: Prof. Dr. Dinh Xuan Khoa
Reviewer 1: Prof. Dr. Nguyen Quang Bau ..............................................
........................................................................................
Reviewer 2: Assoc. Prof. Dr Chu Dinh Thuy ..........................................
........................................................................................
Reviewer 3: Assoc. Prof. Dr Tran Hong Nhung ......................................
........................................................................................
The thesis will be presented at school-level evaluating Council at:
...h, datemonth.year 2017.
The thesis can be found at: National library of Vietnam or
Nguyen Thuc Hao information and library center, Vinh university
1
INTRODUCTION
In the past decades, the topic of laser pulses propagation without
distortion (soliton) has been attracted much research attention of scientists
because of their potential applications in optical information and data
processing. In fact, when the light pulse propagates in the resonant
medium, due to the absorption and dispersion that lead to reduction and
distortion of the signal pulse. Therefore, to obtain pulse stability, we often
use ultrashort pulses with high intensity. Moreover, most of the
applications in modern photonic devices often require low-intensity light
with high sensitivity. Therefore, reducing the absorption in the resonance
domain is an excellent solution to reduce the intensity of light pulse and
increase the operating efficiency of photonic devices.
Currently, an interesting solution is used to reduce absorption is
electromagnetic induction transparency (EIT) effect. The basis of EIT is
the result of quantum interference between the probability amplitudes
within the atomic system which is induced by laser fields. Using EIT
technique, some research groups have obtained stable laser pulses (soliton)
in EIT medium. Most recently, T. Nakajima and coworkers studied the
propagation of two short laser pulse trains in a three-level lambda-type
atomic medium under EIT conditions. They obtained laser pulses
propagating EIT medium without distortion in picosecond domain. The
initial studies of pulse propagation in EIT medium often ignore the effect
of Doppler broadening. This can only be suitable for cold atomic medium
or ultra-short laser pulses. Moreover, many applications such as optical
communications that require working with long laser pulses in nano or
micro-second domain.
In addition to quantum interference effects of the shift probability
amplitude, there is a quantum interference effect occurs between
spontaneous emission channels by the non-orthogonal orientation of the
electric dipole moment is induced by two laser fields. This interference will
create a coherent of atoms is called the coherence is generated by
spontaneous emission (spontaneously Generated Coherence - SGC).
Experimentally, SGC effects were observed for the first time by Xia and et
al 1996 in molecular sodium. The presence of the SGC makes the medium
become more transparent and narrower the line width; larger and steeper
dispersion. Furthermore, the results also show that the effect of SGC makes
an asymmetric medium so that the response of the medium is very sensitive
to the phase of laser fields.
So far, the influences of SGC and relative phase on the optical
properties of the EIT medium under steady-state regime have published.
2
However, these influences on pulse propagation effect have not been
investigated.
With the urgency of the issue of research and the reasons mentioned
above, we choose research topic "investigation of laser pulse propagation in
a three-level atomic medium in the presence of EIT effect".
The aim of the thesis is to study the effects of laser coupling
parameters and Doppler broadening on the process of laser pulse
propagation in different pulse domains. Studying on influences of the non-
orthogonal orientation of electric dipole moments and the relative phase on
the laser pulse propagation in the presence of incoherent pump.
Chapter 1
PULSE PROPAGATION IN RESONANT MEDIA
1.1. Interaction between two-level atoms and light
We assume a single optical field propagating in the z direction have
form:
, , .i kz tE z t z t e c c , (1.1)
where ,z t is the envelope function, ω is the frequency of light, and k =
ω/c is the wavenumber.
1.1.1 Density Matrix Formalism
Real media often cannot be sufficiently described by a single wave
function and, in that case, the density matrix formulas is necessary,
2 *
1 1 2 11 12
2* 21 222 1 2
c c c
c c c
, (1.2)
For both mixed and pure states, the density matrix must have unit
trace ( 11 22 1 ) so that probability is conserved in our closed system.
The density matrix operator’s evolution according to the Liouville equation:
,i H
t
. (1.3)
In matrix form, the atom-field interaction Hamiltonian is:
1*
2
.d E
H
d E
. (1.4)
3
1.1.2 Atomic evolution in rotating wave approximation
In the rotating wave approximation (RWA), to simplify the
Hamiltonian, we introduce the unitary transform matrix:
1
1 0
0
i t
i kz tU e e
. (1.5)
Hamiltonian in the rotating wave basis as:
W † †RH UHU i U U
t
. (1.6)
In terms of the detuning, 2 1 , the Hamiltonian in this basis is:
12W
21
0 .
.
i kz t
R
i kz t
d E e
H
d E e
. (1.7)
In making the rotating wave approximation, we neglect the fast-
oscillating 2i t terms, assuming that the envelope function , tz varies
slowly compared to the carrier wave. Using this approximation, we find
total Hamiltonian in the RWA:
12WA
*21
*
00 . 2
.
2
R dH
d
. (1.8)
In the above, we have identified the Rabi frequency,
212 . ,, d z tz t
. (1.9)
1.1.3 Rabi flopping and pulse area
The atomic population is periodically transferred between the ground
and excited states,
222 0sin / 2t t , (1.10)
at exact resonance ( = 0). This process, known as Rabi flopping, is
illustrated in Figure 1.1. The 0 0t t is so-called pulse area, allow
determines the excited state population, 22 , at any time t.
4
Fig 1.1. Rabi flopping of the population in the excited state.
Using a pulse area is convenient for describing the pulse time
dependent envelope function, defined:
,z z d , (1.11)
Fig 1.2. The Gaussian with pulse width 0 = 1 and pulse area 0t = 2.
1.1.4. The Maxwell’s and wave equations
From Maxwell’s equations, we lead to the wave equation,
2 2
2
2 2 2 2
0
1 1E E P
c t c t
. (1.12)
1.1.5. The slowly-varying envelope approximation wave equation
Wave equation in the one-dimensional form:
2 2 2
02 2 2 2
1E E P
z c t t
, (1.13)
where P is the polarization. For a dielectric medium of two-level atoms
with N is particle density, the polarization is expressed: ˆTr ,P N d (1.14a)
12 21 21 12 ,N d d (1.14b)
5
RW RW12 21 21 12 .i kz t i kz tN d e d e (1.14c)
Taking the derivatives (1.14c) and collecting same-frequency terms in
Maxwell's wave equation (1.13), we obtained:
2 2
2 2
2 2 2
2
2 RW RW RW
0 12 12 12 122
12 2
2 .
k ik i
z z c z z
Nd i
t t
(1.15)
In the slowly-varying envelope approximation (SVEA):
2
2; ;k kz z z
(1.16a)
2
2;t t t
. (1.16b)
Similarly, the rotating-wave variables are slowly-varying by definition:
RW
RW12
12t
. (1.17)
Using the SVEA and RWA and noting that k = ω/c, 0 01 /c , we find
slowly-varying wave equation:
W12 12
0
1
2
Ri Nd
z c t c
. (1.18)
or, in terms of the Rabi frequency,
W121 2 Riz c t
, (1.19)
Here
2
12
02
N d
c
, (1.20)
is the atom-field coupling parameter.
1.1.6. Inhomogeneous broadening
To calculate for this inhomogeneous broadening, we need to average
ρ12 through the polarization in Maxwell's equation:
12 12g d . (1.21)
The appropriate distribution function, g(), can be derived from the
Maxwell-Boltzmann distribution for gases. In terms of the detunings, it is:
201 exp p p
mm
g
kvkv
, (1.22)
6
where 2 /k is the wavenumber, 2 /m Bv k T m is the most probable
velocity that corresponds to a Doppler width D given
02 ln 2 /m pD v c (1.23)
1.2. Interaction between three-level atoms and light
In this section, we consider the cascade three-level atomic medium
excited by laser field and coupling laser field as in Figure 1.3. The weak
laser drive to transitions |1|2 and the strong coupling laser field excited
transition |2|3. We denote 21 and 32 are the population decay rate of
the states 2 and 3 , respectively. We write the sum of the two fields in the carrier-envelope form as:
, , , . .p p c ci k z t i k z tp cE z t z t e z t e c c (1.24)
Fig. 1.3. The cascade three-level atom scheme excited with laser and coupling laser
field.
In the framework of semiclassical theory, the evolution of density
matrix operator ρ of the system can be represented by the following
Liouville equation:
,i H
t
. (1.25)
Here, H is the total Hamiltonian:
1 12
21 2 23
32 3
0
0
d E
H d E d E
d E
. (1.26)
The identified,
,
nm nm
n m
L , (1.27)
here, nm is the population decay rate from level m to level n , and nm
is the coherence decay rate,
1 22nm nm nm mn nm nm mn nm mnL , (1.28)
7
here, nm n m is the density matrix operators if n = m and the electric
dipole operators if n m.
1.2.1. Hamiltonian interaction in the rotating wave approximation
We make the rotating wave approximation (RWA) as before. For the
three-level cascade-system, the unitary transformation that takes us to the
rotating frame is:
2
0 0
0 1 0
0 0
p p
c c
i k z t
i t
i k z t
e
U e
e
. (1.29)
The Hamiltonian transforms as
W † †R UH UHU i U
t
, (1.30)
and is, therefore:
12
W
21 23
32
. 0
. 0 .
0 .
p p
p p c c
c c
i k z t
p
i k z t i k z tR
i k z t
c
d E e
H d E e d E e
d E e
,(1.31)
we find:
*
*
12
RWA
21 23
* *32
020
0 02 20
0 2
p
p
p p
p c
p c
c c
c
c
d
H d d
d
,(1.32)
Where, 122 ,, pp d z tz t
và 2 ,, c cc d z tz t
, is the Rabi
frequencies of the probe and coupling pulses, respectively.
1.2.2. The laser pulse propagation equations in the slowly varying
envelope approximation
In the rotational wave reference system, we derive the wave equation
for each field:
RW121 2p piz c t
, (1.33a)
8
RW231 2c ciz c t
. (1.33b)
Here, the atom-field coupling constants (also referred to as propagation
constants) are
2
12
02
p
p
N d
c
and
2
23
02
c
c
N d
c
, respectively.
1.2.3. Coherent population trapping
1.2.4. Electromagnetically induced transparency
The EIT is a quantum interfering effect that leads to the propagation
of light through the medium without being absorbed in the resonant
frequency of the atom. EIT can be explained based on the quantum
interference of the probability amplitudes occurs between the different
excited pathways in an atomic system as shown in Figure 1.4. One term,
which is due to excitation by the resonant field p only, i.e., a direct path
from state 1 to state 2 ; An additional term, which is due to the presence
of the second field c , i.e., an indirect path from state 1 to state 2 to
state 3 and back to the state 2 . Hence, the total transition amplitude
vanishes that leading to the transparency for the probe laser beam at the
resonant frequency.
Fig. 1.4. Two excited pathways from the ground state 1 to the excited state 2 :
directly via the 1 2 pathway, or indirectly via 1 2 3 2 pathway.
1.3. Physical of three-level atomic system
1.3.1. The Rb atomic
Rb atom has two natural isotopes: 85Rb is stable isotope occupy 72%, while
unstable isotope 87Rb is 28%.
1.3.2. The fine structure
1.3.3. The hyperfine structure
9
Energy level diagram of 87Rb atom is shown as in Figure 1.5.
Fig. 1.5. Fine and hyperfine energy level diagram of an 87Rb atom.
Chapter 2
PROPAGATION OF PULSE IN THE INHOMOGENEOUSLY
BROADENED EIT MEDIUM
2.1. The Maxwell-Bloch equations for the pulse propagation
In this section, we consider three-level atom model as presented in
Section 1.2. Using the electric dipole- and rotating- wave approximations,
the density matrix equation (1.25) is transformed as,
*11 21 22 21 122 2p p
i i (2.1a)
* *22 21 22 32 33 12 21 32 232 2 2 2p p c c
i i i i , (2.1b)
*33 32 33 32 232 2c c
i i , (2.1c)
*12 12 12 22 11 13( ) ( )2 2p p c
i ii , (2.1d)
*23 23 23 33 22 13( )2 2c c p
i ii , (2.1e)
13 13 13 23 122 2p c
i ii , (2.1f)
where, 21p p and 32c c are the frequency detuning of the
laser field and coupling laser field. The density matrix elements ρik in Eqs.
(2.1) are restricted by 11 22 33 1 and ki ik .
10
It is convenient to transform Eqs. (2.1) and Eqs. (1.33) into the local
frame with new variables z and 0/t z c . In this frame Eqs. (2.1)
will be the same with the substitution t and z , whereas Eqs.
(1.33) are rewritten as
12( , ) 2 ( , )p pi
, (2.2a)
23( , ) 2 ( , )c ci
. (2.2b)
For the case of Doppler broadening, the frequency detuning of the laser
fields, respectively, are shifted according to as the following relations,
0 0/p p pv v c , (2.3a)
0 0/c c cv v c , (2.3b)
where, v is the velocity of the moving atom along the z axis,
0 210p p pv and 0 320c c cv are the frequency
detuning of the laser fields for an atom at rest (v = 0), respectively. Under
thermal equilibrium, velocity distribution of atoms is represented by
Maxwell-Boltzmann distribution g(∆) and Doppler width D identified as
the expression (1.22) and (1.23), respectively.
In the presence of Doppler broadening the evolution of the laser field
and coupling laser field in Eqs. (2.2) have form,
12( , ) 2 ( , , )p pi g d
, (2.4a)
23( , ) 2 ( , , )c ci g d
. (2.4b)
2.2. Numerical simulations
2.2.1. Runge-Kutta algorithm
2.2.2. Finite difference method
2.3. Pulse propagation in the inhomogeneous broadening medium
In order to study the propagation dynamics of the laser pulse
( , )p , we solve numerically Maxwell-Bloch equations (2.1), (2.2) and
(2.4) by the four-order Runge-Kutta and finite difference methods. The
atomic and laser parameters are chosen as 21 = 2π6 MHz, 32 = 2π MHz,
15 310 mN , 2921 2.53.10 .d C m ; 795p nm, 762c nm, 0p = 0.02
GHz and 0p c , as the all for a figure in this Chapter. We assume
both laser pulse and coupling laser pulse have the same temporal width 0
with a Gaussian type at the entrance of the medium,
11
2
00, t
t
e
. (2.5)
2.3.1. Pulse propagation in picosecond domain
This part, we fixed the pulse duration of τ0 = 25ps and considered the
influence of the peak intensity Ωc0 and the pulse area (Ωc0τ0) of the control
laser pulse on the propagation dynamics of the laser pulse when ignore and
taking into account the Doppler broadening.
In Fig. 2.1, in the left column we represent cases ignore the Doppler
effect (D = 0). By increasing the pulse area of the coupling laser pulse we
obtain a stable propagation pulse. When the area of the laser pulse is small
Ωc0τ0 ≤ 2,5, as shown in Figures 2.2a and 2.2b, the laser pulse is
significantly absorbed and each laser pulse is broken down into several
sub-pulses with a positive-negative amplitude. The number of modulations
at the trailing edge of the pulse increases as the propagation distance
increases. However, when the peak intensity of the coupling laser pulse
becomes larger Ωc0τ0 = 1THz (Fig. 2.1d) and accordingly the pulse areas
become larger Ωc0τ0 = 25, Then propagate of the laser pulse is almost not
distorted, EIT is established. The physical reason for this case is due to the
depth and width of the EIT window is increased when the coupling laser
intensity increase, so the influence of the medium on the pulse form in this
case, is negligible.
Similar consideration for the variation of the laser pulse envelope in
the right column in the presence of Doppler effect, with D = 3.15 GHz
which corresponds to room temperature. Compare the two figures in the
left and right columns we see that the dynamic of pulse envelope shape is
almost the same. Thus, the influence of Doppler effect in this pulse domain
is negligible and can be ignored. We can understand this because in the
domain of picoseconds, the time that the atoms exposed to laser pulses are
small, so the change velocity of atoms in each cycle of the laser pulse is
significant.
12
Fig. 2.1. Temporal evolution of the probe pulse p(,) at p = 0 (solid), 5 ns-1 (dashed), and 10 ns-1 (dotted) when τ0 = 25 ps. The peak intensity and pulse
areas are given as in figures.
2.3.2. Pulse propagation in nanosecond domain
The next, we consider the propagation of the laser pulse at the pulse
duration of τ0 = 25 ns which are approximate to the life time of the excited
state |2. The results show the time variation of the laser pulse envelope at
different optical depths, 0p , 5 and 10 ns-1 as shown in Fig. 2.2. The
pulse area and peak intensity of the control laser pulse are given in the
figure, the left column corresponds to (D = 0) and the right column
corresponds (D = 3.15 GHz).
From Fig. 2.2, show that for a small value of the pulse area (Ωc0τ0 ≤
25) the laser pulse is almost collapsed due to resonant absorption in an
atomic medium, there is no EIT effect (Figs. 2.2a and Fig. 2.2b). When the
pulse area (thus the pulse peak) of the coupling beam increases, although
13
the leading edge of the probe pulse is still distorted but the ending edge
approaches earlier transparency, as shown in Figs. 2.2c and 2.2d. This
phenomenon is due to the energy loss to prepare for the EIT formation of
the probe pulse. In particular, when increased peak intensity to Ωc0 = 200
GHz, respectively the pulse area reaches to the value of Ωc0τ0 = 5×103 (Fig.
2f), the probe pulse is almost unchanged, namely, an ideal EIT or soliton is
established.
Similar consideration for the case of the Doppler broadening as
presented in the right column of Fig. 2.1. From this figure we observe that
similar dynamics occur but in order to reach the EIT form of probe pulse,
larger intensity of coupling pulse is required, that is at the same intensity of
coupling pulse (for example, Figs. 2f and 2f1), the probe pulse approaches
to later EIT form in presence of the Doppler broadening. This behavior is
due to the Doppler broadening reduces EIT efficiency, hence the probe
laser pulse is decayed faster than the case of Doppler-free.
14
Fig. 2.2. Temporal evolution of the probe pulse p(,) at p = 0 (solid), 5 ns-1 (dashed), and 10 ns-1 (dotted) when τ0 = 25 ns. The peak intensity and pulse
areas are given as in figures.
2.3.3. Pulse propagation in microsecond domain
To see more clearly, the influence of the coupling pulse duration τ0
on the EIT formation of the probe pulse, we consider the temporal
evolution of the laser pulse p(,) when fixed pulse width τ0 = 0.25 μs as
shown in Fig. 2.3. By comparing Fig. 2.3 with Fig. 2.2, we show that the
ideal EIT effect can achieve by the increase coupling laser pulse. However,
in the long pulse region micro second is the EIT effect obtained at pulse
area larger ten times (Ωc0τ0 = 5×104) as shown in Figs. 2.3f and 2.2f.
Similar, when we compare the Figs in the left and right columns for
the case ignore and present Doppler broadening. We show that when
present Doppler broaden is laser pulse distorted and strong absorbed.
Influence the Doppler broadening is significant and can’t ignore even when
EIT pulse form obtains an idea near.
15
16
Fig. 2.3. Temporal evolution of the probe pulse p(,) at p = 0 (solid), 5 ns-1
(dashed), and 10 ns-1 (dotted) when τ0 = 0.25 µs. The peak intensity and pulse
areas are given as in figures.
2.3.3. Influence of Doppler broadening
In order to see the further influence of Doppler broadening we plot
the temporal profile of the probe laser pulse (Fig. 2.4a) and its peak
amplitude (Fig. 2.4b) for different Doppler widths. It is shown that when
the Doppler width D increases (with a moderate value of pulse area) which
lead to reducing an atomic coherence, the peak amplitude of probe pulse
decreases (Fig. 3b), whereas the trailing edge of pulse separates into
several sub-pulses with amplitudes oscillate stronger (Fig. 2.4a).
Fig. 2.4 (a) Temporal evolution of the probe pulse at optical depth p = 5 ns-1;
(b) variation of probe peak versus optical depth at different Doppler widths
when 0 = 1 ns and c0 = 10 GHz.
Chapter 3
INFLUENCES OF ELECTRIC-DIPOLE MOMENT ORIENTATION
AND RELATIVE PHASE ON PULSE PROPAGATION
3.1. Theoretical model
We consider the three-level cascade-type atomic system with nearly
equispaced levels as shown in Fig. 3.1. A weak probe field 1 with
frequency p drives the transition |1|2, while the transition |2|3 is
17
coupled by a strong field 2 with frequency c. We denote 21 and 32
being the decay rate of the states |2 and |3, respectively. An incoherent
pump with a pumping rate 2R is applied between levels |1 and |3. The
Rabi frequencies of the fields are defined as 1 122 /pd E and
2 232 /cd E , with d12 and d23 are the electric dipole matrix element,
respectively.
Fig. 3.1. Scheme of the three-level cascade-type atomic system with nearly
equispaced levels.
When the orientation between the electric dipole moments generated
by two laser fields on two translations within the multi-level atomic system
is non-orthogonal, it can produce an interference effect between the
spontaneous emission channels of those transitions. The coherence can be
created by the interference of spontaneous emission (usually called
spontaneous generated coherence: SGC), where 12 23
12 23
. os.
d dp c
d d
the
characteristic of the interference intensity, is the angle between the two
dipole moments 12d
and 23d
. Due to SGC, the optical properties of the
system depend not only on amplitudes and detuning but also the phase of
the probe and coupling fields, thus we have to treat Rabi frequencies as
complex parameters. Let p and c are the phases of the laser fields,
respectively, then we can set 1 expp pi and 2 expc ci with
p and c are real parameters, and the relative phase between the probe
and the coupling fields is p c , and then exp i . In such a
case, the density-matrix equations (1.25) have the form:
11 11 21 22 21 122 2 piR , (3.1a)
22 21 22 32 33 12 21 32 232 2p ci i , (3.1b)
33 11 32 33 32 232 2 ciR , (3.1c)
12 21 12 22 11 13 21 32 23(R ) ( ) 22 2p p c
i ii p ,(3.1d)
18
23 21 32 23 33 22 13( )2 2c c p
i ii , (3.1e)
13 32 13 23 122 2p c
i iR i . (3.1f)
The together with *ij ij (i j) and the conservation condition
11 22 33 1 . In equations (3.1), ij describe the coherence decay rates
from state |i to state |j and represented with the population decay rates ij by
12
k i l j
ij ik jl
E E E E
, (3.2)
21 32,p p c c are the detuning of the probe and coupling
fields from their relevant atomic transitions, respectively. In the case of
nearly equispaced levels, the inclusion of two coupling fields of different
frequencies would lead to the optical Bloch equation with an additional
term 21 32 232 p . If = 1, the SGC effect has to be taken into account
and the strength of SGC will vary versus ; otherwise = 0, the effect of
SGC is absent.
The same section 1.2.2, using the slowly-varying envelope and the
rotating-wave approximation, and consider in the local frame where = z
and /t z c , we also obtained the propagation equations for laser fields,
respectively:
12( , ) 2 ( , )p pi
, (3.4)
23( , ) 2 ( , )c ci
. (3.4)
We also assume that the envelope of field is the slow and at the entrance to
the medium having formed as (2.5).
3.2. Influence of SGC on the laser pulse propagation
To start with, we ignore the effect of SGC, i.e., = 0 and hence =
0, then choosing the appropriate values of coupling intensity and pulse
duration as c0 = 25 GHz and 0 = 25 ns, at which the probe pulse envelope
is undistorted during the propagation process that is the near ideal EIT
effect is established as shown in figure 3.2a.
In order to investigate the influence of SGC on the propagation effect
of laser pulse we fix = 1 and relative phase = 0, and plot
spatiotemporal evolution of the pulse probe p(,) for different values of
quantum interference parameter p. The results show that, when the SGC
presents, i.e., p 0, the probe pulse envelope is significantly distorted
19
during the propagation. The modulations of the pulse envelope increase as
the parameter p increases [see figures (3.2b)-(3.2d)]. It is also notable that
the modulations of the envelope are mainly concentrated on the leading
edge, and these modulations increase as the propagation distance increases.
The primary reason for the modulations at leading edge of pulse with the
SGC, arises from the influence of SGC on absorption and dispersion, the
absorption peak on both sides of zero detuning and the linewidth of
absorption line become larger and narrower than those in the case of SGC
absents, therefore the dispersion curve becomes much steeper (hence
dispersion dn/d is very high) as the parameter p increases.
Fig. 3.2. Spatiotemporal evolution of the probe pulse p(,) for different values
of p = 0 (a), p = 0.3 (b), 0.7 (c) and 1 (d). Other employed parameters are = 0,
p0 = 0.05 GHz, p = c = 0, 21 = 1.232, and R =1.232.
3.3. Influence of relative phase on the laser pulse propagation
In order to see the control role of relative phase on the EIT effect of
probe pulse we fix the parameter p = 0.7 and plot the temporal evolution of
the pulse probe p(,) at the optical depth p = 5ns-1 for different values
of relative phase , as illustrated in figure 3.3. From figure 3.3a we can see
that the probe pulse envelope depends sensitively on the relative phase and
the influence of relative phase on the temporal evolution of pulse is a
period of 2. For 0 /2: at = 0, due to SGC making the leading
edge of pulse envelope is distorted as one considered in figure 3.2(c), when
20
increases the modulations of pulse envelope are smaller. Especially,
when = /2 these modulations are disappeared that is the influence of the
SGC on the propagation effect can be canceled. For /2 < : with
increasing the modulations of pulse envelope at the front edge are
gradually stronger and at = the modulations is strongest which is
similar to that when = 0. However, the oscillations at the front edge of
pulse in regions /2 < and 0 < /2 are out of phase (see figure
3.3b), and this as a consequence of variation of the absorption and
dispersion versus the relative phase with a same rule in the steady-state.
Similarly, at = 3/2 the modulations are disappeared, while at = 2 the
modulations are strongest that coincides with case of = 0.
Fig. 3.3. (a) Temporal evolution of the probe pulse p(,) versus relative phase
and (b) Temporal evolution of the probe pulse p(,) at different relative
phases when p = 0.7 and p = 5ns-1. Other parameters are similar to those in
figure 3.2.
To find the physical reason for these phenomena, we plot Im(12)
and Re(12) corresponding to the absorption and dispersion for probe field
versus relative phase in the presence of SGC and incoherent pumping, as
21
shown in figure 3.4. It is seen that Im(12) and Re(12) varies periodically
with respect to relative phase with the period of 2. When = 0 or = 2,
absorption reaches the largest value and dispersion dn/d is highest hence
the influence of SGC on propagation effect is strongest, while when =
/2 or = 3/2, both absorption and dispersion dn/d are zero hence the
influence of SGC on the propagation effect is also disappeared. Moreover,
as seen in the Fig. 3.4, at the position = /2 or = 3/2 of medium
feature be reversed from the amplifier to the absorption and opposite. This
is also a special reason lead to a cancel of the influence of the SGC when
the relative phase equal one odd number of times /2.
Fig. 3.4. Variation of Im(12) [solid line] and Re(12) [dashed line] versus
relative phase when parameter p = 0.7 and p = 5ns-1. Other parameters are
similar to those in figure 3.2.
3.3. Role of incoherent pumping
To see the role of incoherent pumping in the control of SGC and
relative phase on the propagation effect we plot the temporal evolution of
the pulse probe p(,) versus incoherent pumping rate R at the parameter
p = 0.7 when the relative phase = 0 and = , as displayed in figure 3.5.
Where we keep the optical depth p = 5ns-1 and other parameters in figure
3.5 are similar to those in figure 3.2. From figure 3.5 we can see that when
R = 0, i.e., the incoherent pumping is absent, there only is the small
modulation at the front edge of the pulse that is the influences of SGC and
relative on the EIT effect of probe pulse are very small. However, when the
incoherent pumping presents, the influences of SGC and relative phase
become very sensitive to the incoherent pumping rate. When the speed
increases of the incoherent pumping, the amplitude of the oscillation at the
front edge of the pulse increases rapidly, but the peak amplitude of the
pulse oscillates around the initial peak amplitude value p0. In addition, by
comparing Figs. 3.5a and 3.5b, we can observe the presence of relative
phase that can control oscillations at the front edge of the pulse as we have
seen in Fig. 3.4.
22
Fig. 3.5. Temporal evolution of the pulse probe p(,) versus incoherent pumping
rate R at the parameter p = 0.7 and the relative phase = 0 (a), (b). The optical
depth p = 5ns-1 and other parameters are similar to those in figure 3.2.
CONCLUSIONS
In this work, we have investigated the propagation dynamics of the
laser pulse in a three-level cascade atomic medium when into account the
Doppler effect and non-orthogonal orientation of the electric dipole
moment, by numerically solving the Maxwell–Bloch equations for atoms
and fields. The main results are as follows:
For short pulse durations (ps), the EIT effect of laser pulse is easily
achieved at small coupling pulse areas, such as Ωc0τ0 = 25. In contrast, the
EIT effect for stronger pulse durations (ns or s) is established at large
laser pulse areas, such as Ωc0τ0 = 5×103.
23
The influence of Doppler broadening on the form of laser pulse is
negligible for short pulses, while for long pulses the influence of Doppler
broadening is significant which causes strong modulations in pulse tail.
Growth of Doppler width leads to increase in amplitude of this oscillation.
Moreover, the EIT effect of laser pulse for the case with Doppler
broadening is achieved at larger laser pulse areas compared to the case
without Doppler broadening.
The presence of SGC causes oscillations at the front edge of the
pulse. These oscillations increase when the propagation distance increases.
At the same optical depth, as the parameter p increases, the oscillations at
the front edge of the pulse increase.
The probe pulse envelope also depends sensitively on the relative
phase. The influence of relative phase on the temporal evolution of pulse is
a period of 2π. At the relative phase = 0, π and 2π the modulations at the
front edge of the pulse are strongest, while at = π/2 and 3π/2 these
modulations are disappeared, i.e., the influence of the SGC on the EIT
effect of probe pulse is neglected.
In the presence of incoherent pumping between |1 and |3 levels
leading to the influences of SGC and relative phase on pulse form become
more clear.
The results obtained are useful for the selection of experimental
configuration and the parameters of the experimental research of laser
effects spread EIT in pulse mode. It is also the foundation for research
applications in all-optical switches, quantum information processing, and
optical communications, etc.
The main results of the thesis have been published in 02 articles in
international journals in the Scopus and ISI category.
24
LIST OF POSTGRADUATE’S WORKS RELATED TO THE THESIS
1. Dinh Xuan Khoa, Hoang Minh Dong, Le Van Doai and Nguyen Huy
Bang, “Propagation of laser pulse in a three-level cascade
inhomogeneously broadened medium under electromagnetically
induced transparency conditions”, Optik 131 (2017) 497–505.
2. Dinh Xuan Khoa, Hoang Minh Dong, Le Van Doai and Nguyen Huy
Bang, “Influences of spontaneously generated coherence and relative
phase on propagation effect in a three-level cascade atomic medium with
incoherent pumping”, manuscript submission in J. Opt. Soc. Am. B.
3. H. M. Dong, L. V. Doai, V. N. Sau, D. X. Khoa and N. H. Bang,
“Propagation of laser pulse in a three-level cascade atomic medium
under conditions of electromagnetically induced transparency”,
Photonics Letter of Poland, Vol. 8, N 3 (2016) 73-75.
4. H. M. Dong, L. V. Doai, P. V. Trong, M. V. Luu, D. X. Khoa, V. N.
Sau and N.H. Bang, “Propagation dynamics of laser pulse in a three-
level V-type atomic medium under electromagnetically induced
transparency”, The 4th academic conference on natural science for
young scientists, master and Ph.D. Students from Asian countries
(2016) 337-344.
5. H. M. Dong, D. T. Thuy, V. N. Sau, T. M. Hung, M. V. Luu, B. D.
Thuan and T. T. Lam, “Effects of nonlinear absorption and third o-
rder dispersion on soliton propagation in optical fiber”, Photonics
Letter of Poland, Vol. 8 (3) (2016), 76-78.
6. Hoang Minh Dong, Dinh Xuan Khoa, Bui Dinh Thuan, “Ảnh hưởng của
nhiễu loạn điều kiện đầu lên lan truyền soliton quang học”, Tap̣ chı́
Nghiên cứu khoa hoc̣ và công nghệ quân sư,̣ số 29 (2014) 105-113.

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