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

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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 wavelength of the Stokes field. m ωL ωS 3 1 Fig.1. An atom initially in its ground state 1 is driven by a laser field with frequency ωL, which is not necessarily in near resonance any intermediate state m . Raman scattering at frequency ωS =ωL- ω31 may accur, living the atom in the final state 3 38
tr−êng §¹i häc Vinh T¹p chÝ khoa häc, tËp XXXVII, sè 2A-2008 II. PARAXIAL SOLUTION OF MAXWELL-BLOCK EQUATIONS In this approximation, we consider the laser field inside pensil-shape dispersionless medium depends only on the local time variable, i.e. τ = t – z/c, and ˆρ thus a laser pulse whose leading edge is at t = z/c leaves the atomic operator Q ( r , t ) unperturbed for τ < 0. Thus equations (3) should be extended to read ˆ ˆ < Q + (r ,τ = 0) Q (r ′,τ = 0) > = N −1δ 3 (r − r ′) (6) ˆ ˆ < F + (r ,τ ) F (r ′,τ ) > = 2Γ N −1 δ (τ − τ ′). ˆ+ The initial value for the Stokes field E S (0,τ ) is given at the input face of the medium, z=0, for all time t. This means that backward Stokes emission is explicitly ingnored. We will consider only the case that not Stokes wave is externally incident on the medium, and so we have for the initial field ˆ− ˆ+ (7) < E S (0,τ ' ) E S (0,τ " ) >= 0 , i.e., the vacuum fluctuations are not detected with a photodetecter. To resolve the set of set of Maxwell-Block equations, we consider the reflection will be from outside face of medium cylinder, because of that the dispersion is ignored inside medium cylinder. It means that the limit condition in face of medium cylinder is ignored and shows that the Fresnel number Φ =A/ λL is of the order of unity. With above consideration, the set of equations (1) will be resolved by Laplace transform, and the operation of Stokes field in three-dimension is found out: ρτ ˆ+ ρ ρ ρρ ˆρ ρρ ˆρ E S (r , τ ) = ∫ d 3 r ′K (r , r ′, τ ) Q(r ′,0 ) + ∫ d 3 r ′∫ dτ ′H (r , r ′, τ , τ ′)F (r ′, τ ′) , (8) 0 where K ( r , r ′,τ ) is the Kernels integration, given by:  kS ρ − ρ ′ 2  k S k 2 EL (τ ) e −Γτ ρρ *∗  I 0  4k 1k 2P(τ ) ( z − z ′)  [ ] 1 exp− i   K (r , r ' ,τ ) = 2 2(z − z ′)    z − z′ 2π     k ρ − ρ′ 2    k S k 2 * e −Γ(τ −τ ′)  * ρρ 1  I 0  (4k1k 2 [P(τ ) − P(τ ′)]( z − z ′)) 2  ,(9) H (r , r ′,τ ,τ ′) = EL (τ ) exp− i S  2 z − z′     z − z′ 2π   τ P(τ ) = ∫ EL (τ ′) dτ ′ 2 0 and ρ is the radial vector (x,y); I0(x) is the zero-order Bessel function, r’ in integration (9) changes in the excited by laser beam cylinder. The solution is obtained with an approximation that the laser beam propagates along z- axis under a litle angle and that the length of medium is shorter then its diameter. This 39
RAMAN STIMULATED ..., Tr. 37-42 C. V. Lanh , D. X. Khoa, H. Q. Quy, P. T. T. Van −1 approximation leads that the factor (z-z )-1 is replaced by the factor r − r ′ and then the divergence around of point z’ = z. This divergence is limited when pay attention that excited scattering around of point z’ = 0 is better than one around of point z’ = z. III. NON-PARAXIAL SOLUTION OF MAXWELL-BLOCK EQUATIONS In this case, we consider the pump laser is constant in certain interval of time, i.e. EL(r, τ ) = AL (0 ≤ τ ≤ τ L ). So equations (1) are rewritten: ∂ q(r , t ) = −(iω + Γ )q(r , t ) − ik1* AL AS (r , t ) + f (r , t ) (10.1) ∂t  2 1 ∂2  2k * * ∂ 2 ∇ − 2 2  AS (r , t ) = 2 AL 2 q (r , t ) (10.2)  cω c ∂t  ∂t  where AS (r , t ) = E S (r , t )e − i (ω S t − K S z ) + − i(ω S t− K S z ) (11) q (r , t ) = Q (r , t )e e f (r , t ) = F (r , t )e − i (ωS t − K S z ) . By the Laplace transform, from equation (10) we have:  k1 k 2 AL   r − r ′ z − z ′  2  k k*A d 3r′ r − r ′  exp− (iω S + Γ )τ −  × ∫ AS (r , t ) = S 2 L expi +  c  r − r′ 2π   c c          1  ′ z − z ′  2    r−r (12) 2 q(r ′,0)I 0  4k1 k 2 AL r − r ′ τ −   +  c c            1  r − r ′ z − z ′  2   τ + ∫ dτ ′ f (r ′,τ ′)I 0  4k1 k 2 AL r − r ′ τ − τ ′ −   . 2 +  c    c    0   r − r′ z − z′ In (12) there is a delay time τ − , describing the nature of laser − c c propagating through medium. Besides, in this solution there is not the divergence if 1 z’ → z, like in solution for the paraxial case. That because of the factor is z − z' 40
tr−êng §¹i häc Vinh T¹p chÝ khoa häc, tËp XXXVII, sè 2A-2008 1 replaced by the Green function of argument. This solution is genaralized for r − r′ space-time description of the arbitrary laser field. IV. STOKES INTENSITY IN THREE-DIMENSION The average intensity of Stokes fied at output face of medium z = L is given [2,3,4]: cA I S (ρ ,τ ) = < E s− (ρ , L,τ )E s+ (ρ , L,τ ) > , ˆ ˆ (13) 2πηω s with considering only the case that no Stokes wave is external incident on the medium, and so all atoms are in initial state. From (6), (7), (8), (9) and (13) we have:  −1 3 2 τ c N ∫ d r' K (r, r ′,τ ) + 2ΓN −1 ∫ d 3 r ′∫ dτ ′ H (r, r ′,τ ,τ ′) . 2 I S (ρ , τ ) = (14)  2πηω ρ   0 Now we discuss in the transient limit and the steady-state limit. 4.1. Stokes intensity in transient limit When the scattering time τ L is much less than the collisional dephasing time i.e. [3], (15) Γτ L < gL from (8), (9) and (14) the intensity Stokes wave at output face z = L is Φ 2 E L (τ ) 2 { } 1 exp [16k1 k 2 LP(τ )] 2 . I TR (ρ ,τ ) = (16) 8πA P(τ ) This result is similar to one of the one-dimension, byond the factor Φ2. 4.2. Stokes intensity in steady-state limit When the time τ L is much larger than the collisional dephasing time [3], i.e., (17) Γτ L >> gL from (8), (9) and (17) the intensity Stokes wave at output face z = L is Φ 2 e gL (18) I SS = . 1 A ( 4π gL ) 2 This result is similar to one of the one-dimension, byond the factor Φ2. V. DISCUSSION In three-dimensional approach, the Stokes intensities in two cases of limit are found. It is interesting that their expressions are different to one of the one- 41
RAMAN STIMULATED ..., Tr. 37-42 C. V. Lanh , D. X. Khoa, H. Q. Quy, P. T. T. Van dimension by factor Φ2 (Fresnel number), which relates to structure of medium. It is true for the general case too, when the value of Γτ L is arbitrary, for that the Stokes intensity is given by Φ 2 1D I S (ρ ,τ ) = I S ( L, τ ) (19) A where I S D (L,τ ) is the Stokes intensity when Φ = 1 (in one-dimentional approach). 1 Have in mind that all results are found out for the initiation Raman scattering at high-gains in the absence of an input Stokes field. REFERENCES [1] M. Trippenbach and Rzazewski, Stimulated Raman Scattering of Colored Chaotic Light, Optic. Society of America, Vol. 1, 671, 1984. [2] M. G. Raymer and L. A. Westling, Quantum theory of Stokes generation with a multimode laser, J. Opt. Soc. Am.B, Vol.2, No.9, 1417, 1985. [3] Dinh Xuan Khoa, Chu Van Lanh and Tran Manh Hung, Intensity of stimulated Raman scattering under quantum theory view, Proc. XXVIIth NSTP, Cualo, August 2-6, 2002. [4] D. Homoelle et al, Conical three-photon-excited stimulated hyper-Raman scattering, Phys. awRev. A, 72, 011802-2, 2005. TãM T¾T T¸N X¹ RAMAN C¦ìNG BøC TRONG GÇN §óNG BA CHIÒU Trong bµi nµy chóng t«i giíi thiÖu lý thuyÕt t¸n x¹ Raman c−ìng bøc trong gÇn ®óng ba chiÒu. C−êng ®é sãng Stokes ®· ®−îc tÝnh to¸n vµ th¶o luËn trong hai tr−êng hîp giíi h¹n lµ giíi h¹n thêi gian ng¾n vµ giíi h¹n thêi gian dµi. (a) Physical Department of Vinh University (b) Institute for Applied Physics, MISTT (c) Master student of Optical course 14th. 42

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