Investigation of laser pulse propagation in a three - Level atomic medium in the presence of eit effect

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 = 2π6 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 = 1.232, and R =1.232. 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.