Báo cáo nghiên cứu khoa học: Tán xạ raman cưỡng bức trong gần đúng ba chiều

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Báo cáo nghiên cứu khoa học: "Tán xạ raman cưỡng bức trong gần đúng ba chiều" tr−êng §¹i häc Vinh T¹p chÝ khoa häc, tËp XXXVII, sè 2A-2008 RAMAN STIMULATED SCATTERING IN THREE-DIMENSIONAL APPROACH Chu Van Lanh , Dinh Xuan Khoa , (a) (a) Ho Quang Quy , Pham Thi Thuy Van (b) (c) Abstract. In this paper we present a theory of Raman stimulated scattering in three-dimensional approach. The intensity of Stokes waves is introduced and discussed in two limit conditions, there are transient limit and steady-state limit. I. THREE-DIMENSIONAL MAXWELL-BLOCK EQUATIONS We consider a collection of indentical atoms or molecules initially in ground states, contained in a pensil-shaped volume with length L and cross-sectional area A. The atomic positions are random, but fixed, and the average number density is N (atoms cm-3). A laser with electric field ρ ρ *ρ ε (r , t ). = E L (r , t ) e i (ω Lt − k L z ) = E L (r , t ) e − i (ω Lt − k L z ) L propagates through the volume in the z direction, which is parallel to the pencil axis. As shown in Fig.1, an atom may absorb a laser photon at frequency ω L and scatters a photon at Stokes frequency ω S = ω L − ω 31 , ending up in the final state 3 . We will treat the laser field mode as a classical electromagnetic wave and assume that it does not undergo depletion or any other back reaction from the medium. On the other hand, the remaining modes of radiation field will be treated quantum mechanically, to allow for the spontaneous initiation of Raman scattering. As well as shown in previous works [1, 2, 3], we introduce a set of Maxwell- Block equations, describing Raman stimulated scattering in three-dimensional space:  2 1 ∂2  ˆ + ρ * 2.k 2 ∂ ˆ * ρ ˆρ −i (ω S t − k S z ) E L ( r , t ) Q ( r , t )e − i (ω S t − k S z ) ∇ − 2 2  E S (r , t ).e = cω S ∂t c ∂t   (1) ∂ˆρ ˆρ ˆ ρ ˆ+ ρ ˆρ Q(r , t ) = − ΓQ(r , t ) − ik1* E L (r , t ) E S (r , t ) + F (r , t ) ∂t ρ ˆ where E L ( r , t ) is the intensity operator of laser field with slowly-varying envelope ˆ+ ρ approximation, E S (r , t ) is the intensity operator of Stokes field with slowly-varying ˆρ envelope approximation, dependent on frequency ω S , Q ( r , t ) is the atomic-transition operator, which describes the relation between two states 1 and 3 (see Fig. 1), ˆρ ˆ ΓQ is the term describing damping of Q (r , t ) at a collisional dephasing rate NhËn bµi ngµy 23/4/2008. Söa ch÷a xong 12/6/2008. 37 RAMAN STIMULATED ..., Tr. 37-42 C. V. Lanh , D. X. Khoa, H. Q. Quy, P. T. T. Van ˆρ Γ , F (r , t ) is the quatum statistical Langevin operator describing the collisional- ωS ωL induced fluctuations, k L = , kS = are the wave numbers of the laser field and c c Stokes field, respectively, and k1, k2 are the coupling constants given by: 1 1 k1 = η− 2 ∑ d 3m d m1[ + ], ω m1 − ω L ω m1 + ω S m (2) 2πNη.ω S k1* k2 = c ˆ with c is the light velosity, and d ij =< i / d / j > is the atomic dipole matrix element. The atomic and Langevin operators have property: ˆρ ˆρ ρρ < Q + (r ,0) Q (r ′,0' ) > = N −1δ 3 (r − r ′) (3) ˆ ρ ˆρ ρρ < F + (r , t ) F (r ′, t ′) > = 2ΓN −1δ (t − t ′)δ 3 (r − r ′). Two important quantities presenting in the resolution of this set of equations are: Raman gain coefficient ρ2 g = 2k1 k 2 Γ −1 E L (r , t ) (4) and Fresnel number A (5) Φ= λS L with λ S is the ...