| In this thesis we discuss the reflectivity and transmissivity of the probe beam controlled by the standing wave and compare them with experiments before. This theory is based on atomic coherence and interference, and consists of Electromagnetically Induced Transparency (EIT), standing wave and Photonic crystals .We use three level Lambda-type and V-type atomic system with Doppler Effect.In the first part, we achieve tunable stop-bands of optical lattice in three level Lambda-type and V-type atomic system using theories of EIT and standing wave.Taking Lambda-type atomic system for example, in Fig II-1, a weak probe beam with frequencyωpand Rabi frequencyΩppropagates in the z-direction in the presence of a strong standing wave (S.W.) pump or coupling beam of frequencyωcand Rabi frequencyΩc. Hamiltonian in atomic system isBecause the coupling beam is standing wave beam, the Rabi frequency can be written ikzikzc ccΩ=Ω1 e +Ω2e?.Ω1 is forward coupling Rabi frequency andΩ2 is backward coupling Rabi frequency.With inserting Hamiltonian into density operator motion equationsWe can get two density matrix equations Here,γij (i ,j= 1,2,3;i≠j)are the coherence decay rates between level i and level j and we defineω31 =ωp ?Δpandω32 =ωc ?Δc. In order to solve density matrix equations, we have to transfer density matrix element In (II.5),σij(i,j=1,2,3)has nothing to do with time, and in the presence of a strong standing wave, we suppose that all the population is on level 1 , namely,σ11 =1,σ22 =0,σ33 =0. With inserting (II.5) into (II.3), (II.4) pieik zeikziHereΓ3 1 =γ31?i (Δp ?kpv),Γ2 1 =γ21?i (Δp ?Δc?kpv) The strong pump intensity pattern modifies in a periodic fashion, so we have to express and as a Fourier series as a function of coordinate, i.e. spatial harmonics∑( )With inserting the Fourier series into (II.6), (II.7), and the elimination of D ( n), the equation reduces to difference equations that relate amplitude of spatial harmonics with neighbor numbers Here the coefficients are And the complex coefficients in equation (II.10) are defined byWith the equations (II.8)-(II.11) and the method of continued fractions, we can get the numerical value ofΛ( n) .As a result, we can get the values ofΛ( n)which belong to an group of atoms with a certain velocity. Considering the Doppler broadened atomic system, we have to integrate the value of to all responses of atoms with different velocities by Maxwellian distribution. Here f(v ) vep vπv= is the Maxwellian distribution and vp = 2 kT/m=2RT/M represents the most probable atomic velocity. So the susceptibility and dielectric function can be written asThe resulting pump Rabi frequency varies periodically along x, with a spatial periodicity a =λc2.According to Bloch's theorem The corresponding determinant equation requires that ka =±cosK is the one-dimensional Bloch wave vector, It is then necessary to consider a finite sample of thickness L=Na where N is the number of primitive cells or periods the stack is made of. The transfer matrix approach is ideally suited for this situation as one may introduce the entire stack transfer matrix in terms of the primitive cell matrix M simply as M ( N ) = MN. Because detM=1, it can be shownI is the unity matrix. Such a compact expression enables one to write the reflection R N and transmission TN amplitudes for an N periods stack in terms of the complex Bloch wave vector k and the elements mi j of the matrix M, namely,From which, in turn, the reflectivity, transmissivity and absorption can be readily found by calculating, respectively : R N2, In the second part, we simulate the reflectivity and transmissivity of the probe beam numerically in Lambda-type and V-type atomic system, and we compare Lambda-type with V-type. Then in Lambda-type atomic system, we change atomic density, coupling beam Rabi frequency, backward coupling beam Rabi frequency, cell length, probe beam frequency and coupling beam detuning, so that we can find some variety of reflectivity and transmissivity of the probe beam.In Fig.II-2, we find that there is completely transparency in probe beam resonance when there is only forward coupling beam, but completely absorbed when there is standing wave coupling beam. Fig.II-3 shows the reflectivity of the probe beam when there is standing wave coupling beam. Parameters areΔc =0,γ31 = 2π×6MHz,γ21 = 2π×1000Hz,ω31 = 2π×384.227691610THz,ω32 = 2π×384.220856 928THz, atomic density 5×109cm ?3, cell length 7.5cm, cell temperature 60 degree Centigrade, forward coupling beam Rabi frequencyΩ1 =5γ31, backward beam coupling beam Rabi frequencyΩ2 =0 andΩ2 =Ω1 respectively, probe beam Rabi frequencyΩp =10.Fig.II-4 shows V-type atomic system. Parameters Fig.II-2 transmissivity of the probe beamFig.II-3 reflectivity of the probe beam areΔc =0,γ21 = 2π×3MHz,γ23 = 2π×6MHz,ω21 = 2π×384.227691610THz,ω31 = 2π×384.220856 928THz, atomic density 5×109cm ?3, cell length 7.5cm, cell temperature 60 degree Centigrade, forward coupling beam Rabi frequencyΩ1 =5γ21, backward beam coupling beam Rabi frequencyΩ2 =0 andΩ2 =Ω1 respectively, probe beam Rabi frequencyΩp =10.In Fig.II-5, we find that there is not completely transparency in probe beam resonance when there is only forward coupling beam, and not completely absorbed when there is standing wave coupling beam. Fig.II-6 shows that the reflectivity of the probe beam in resonance in V-type less than one percent of that in Lambda-type when there is standing wave coupling beam. Fig.II-5 transmissivity of the probe beam Fig.II-6 reflectivity of the probe beam Then we compare all kinds of parameters in Lambda-type system. The parameters not mentioned next are as followsΔc =0,γ31 = 2π×6MHz,γ21 = 2π×1000Hz,ω31 = 2π×384.227691610THz,ω32 = 2π×384.220856 928THz, atomic density 5×109cm ?3, cell length 7.5cm, cell temperature 60 degree Centigrade, forward coupling beam Rabi frequencyΩ1 =5γ21, backward beam coupling beam Rabi frequencyΩ2 =Ω1, probe beam Rabi frequencyΩp =10. Probe beam and standing wave coupling beam are in the same line.With the increasing of standing wave coupling beam Rabi frequency, the reflectivity of the probe beam increases in resonance, in Fig.II-7 Fig.II-7 reflectivity of the probe beam (a)Ω1 =5γ31(b)Ω1 =7γ31(c)Ω1 =9γ31 With the increasing of cell length, the reflectivity of the probe beam stays the same in resonance, in Fig.II-8. Fig.II-8 reflectivity of the probe beam (a) L=2.5cm (b) L=5cm (c) L=7.5cmWith the increasing of atomic density, resonance absorption increases, and the probe beam is also completely absorbed off-resonance, in Fig.II-9. The reflectivity of the probe beam decreases to zero in resonance, in Fig.II-10. With the backward coupling beam coupling beam Rabi frequency decreasing, the transmissivity of the probe beam increases in resonance, in Fig.II-11, but reflectivity decreases in resonance, in Fig.II-12.With the approaching of the coupling beam frequency to the probe beam frequency, the reflectivity of the probe beam decreases to zero in resonance, but increases off-resonance, so we can see a dip in the middle, in Fig.II-13. Fig.II-11 transmissivity of the probe beam (a)Ω2 =Ω1(b)Ω2 =0 .8Ω1 Fig.II-12 reflectivity of the probe beam (a)Ω2 =Ω1(b)Ω2 =0 .8Ω1 When the probe beam and the standing wave coupling beam lie at an angle of aboutα=0.34,namely,λc cosα=λpsatisfying the Bragg condition The transmissivity of the probe beam stay the same, and the reflectivity of the probe beam decreases to zero in resonance, but increases off-resonance, so we can see a dip in the middle, in Fig.II-14.When coupling beam detuning considered with incident angleα=0.34, we can see the reflectivity of the probe beam in several kinds of detuning in Fig.II-15 similar as Fig.II-16 with incident angleα=0.5. Fig.II-14 reflectivity of the probe beam In summary, we get a theory for tunable photonic crystals achieved by EIT in a standing wave. Then we numerically simulate the theory and compare in different situation. We get similar regular pattern as an experiment before, so that we understand the especial photonic crystals more and it is good for further experiment and the development of optic communications.
|
PDF Full Text Request