tr−êng §¹i häc Vinh T¹p chÝ khoa häc, tËp XXXVII, sè 2A-2008 37 RAMAN STIMULATED SCATTERING IN THREE-DIMENSIONAL APPROACH Chu Van Lanh (a) , Dinh Xuan Khoa (a) , Ho Quang Quy (b) , Pham Thi Thuy Van (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 )( * )( ),(),().,( zkti L zkti LL LLLL etrEetrEtr −−− == ωω ε ρ ρ ρ 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 L ω and scatters a photon at Stokes frequency 31 ωωω −= LS , 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: ),( ˆ ),( ˆ ),( ˆ ),( ˆ ),( ˆ ),( ˆ ),( ˆ .2 ).,( ˆ 1 * 1 )( * * 2 )( 2 2 2 2 trFtrEtrEiktrQtrQ t etrQtrE tc k etrE tc SL zkti L S zkti S SSSS ρρρρρ ρρρ +−Γ−= ∂ ∂ ∂ ∂ =       ∂ ∂ −∇ + −−−− + ωω ω (1) where ),( ˆ trE L ρ is the intensity operator of laser field with slowly-varying envelope approximation, ),( ˆ trE S ρ + is the intensity operator of Stokes field with slowly-varying envelope approximation, dependent on frequency S ω , ),( ˆ trQ ρ is the atomic-transition operator, which describes the relation between two states 1 and 3 (see Fig. 1), Q ˆ Γ is the term describing damping of ),( ˆ trQ ρ at a collisional dephasing rate NhËn bµi ngµy 23/4/2008. Söa ch÷a xong 12/6/2008. C. V. Lanh , D. X. Khoa, H. Q. Quy, P. T. T. Van RAMAN STIMULATED , Tr. 37-42 38 Γ , ),( ˆ trF ρ is the quatum statistical Langevin operator describing the collisional- induced fluctuations, c k L L ω = , c k S S ω = are the wave numbers of the laser field and Stokes field, respectively, and k 1 , k 2 are the coupling constants given by: c kN k ddk S SmLm m mm * 1 2 11 13 2 1 .2 ], 11 [ ωπ ωωωω η η = + + − = ∑ − (2) with c is the light velosity, and >=< jdid ij / ˆ / is the atomic dipole matrix element. The atomic and Langevin operators have property: ).()(2),( ˆ ),( ˆ )()'0,( ˆ )0,( ˆ 31 31 rrttNtrFtrF rrNrQrQ ′ − ′ −Γ=> ′′ < ′ −=> ′ < −+ −+ ρρρρ ρ ρ ρ ρ δδ δ (3) Two important quantities presenting in the resolution of this set of equations are: Raman gain coefficient 2 1 21 ),(2 trEkkg L ρ − Γ= (4) and Fresnel number Φ = L A S λ (5) with S λ is the wavelength of the Stokes field. ‌‌ 1 ‌‌ 3 ‌‌ ω L ‌‌ ω S m Fig.1. An atom initially in its ground state 1 is driven by a laser field with frequency ω L , which is not necessarily in near resonance any intermediate state m . Raman scattering at frequency ω S = ω L - ω 31 may accur, living the atom in the final state 3 tr−êng §¹i häc Vinh T¹p chÝ khoa häc, tËp XXXVII, sè 2A-2008 39 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 thus a laser pulse whose leading edge is at t = z/c leaves the atomic operator ),( ˆ trQ ρ unperturbed for τ < 0. Thus equations (3) should be extended to read ).(2),( ˆ ),( ˆ )()0,( ˆ )0,( ˆ 1 31 ττδττ δττ ′ −Γ=> ′ < ′ −=>= ′ =< −+ −+ NrFrF rrNrQrQ (6) The initial value for the Stokes field ),0( ˆ τ + S E 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 < 0)",0( ˆ )',0( ˆ >= +− ττ SS EE , (7) 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/ L λ 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: ( ) ( ) ( ) ),( ˆ ,,,0, ˆ ,,),( ˆ 0 33 ττττττ τ ′′′′′′ + ′′′ = ∫ ∫∫ + rFrrHdrdrQrrKrdrE S ρρρρρρρρρ , (8) where ),,( τ rrK ′ is the Kernels integration, given by: ( ) ( ) ( ) [ ] ( ) ( ) ( ) ( ) ( ) [ ] ( ) ( ) ( ) ∫ ′′ =             ′ − ′ −         ′ − ′ − − ′ − = ′′       ′ −         ′ − ′ − − ′ − = ′ −Γ− Γ− ∗ τ ττ τ τττ ττ ρρ τ π ττ τ ρρ π τ τ 0 2 2 1 210 2 * * 2 2 1 210 2 * 2 )(4 2 exp 2 ,,, )(4 2 exp 2 ),',( dEP zzPPkkI zz k i zz e E kk rrH zzPkkI zz k i zz e Ekk rrK L S L S S LS ρρ ρ ρ ,(9) and ρ is the radial vector (x,y); I 0 (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 C. V. Lanh , D. X. Khoa, H. Q. Quy, P. T. T. Van RAMAN STIMULATED , Tr. 37-42 40 approximation leads that the factor (z-z) -1 is replaced by the factor 1− ′ − rr 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. E L (r, τ ) = A L (0 L ττ ≤≤ ). So equations (1) are rewritten: ( ) ( ) ( ) ( ) trftrAAiktrqitrq t SL ,),(,, * 1 +−Γ+−= ∂ ∂ ω (10.1) ),( 2 ),( 1 2 2 * * 2 2 2 2 2 trq t A c k trA tc LS ∂ ∂ =       ∂ ∂ −∇ ω (10.2) where )( ),(),( zKti SS SS etrEtrA −− + = ω )( ),(),( z S Kt S i e etrQtrq −− = ω (11) ( ) ( ) .,, )( zKti SS etrFtrf −− = ω By the Laplace transform, from equation (10) we have: ( ) ( ) ( ) ( ) .4, 40, expexp 2 , 2 1 2 210 0 2 1 2 210 2 21 3 * 2                                ′ − + ′ − − ′ − ′ − ′′′ +                              ′ − + ′ − − ′ − ′ ×                 ′ − + ′ − −Γ+−         ′ − ′ − ′ = ∫ ∫ c zz c rr rrAkkIrfd c zz c rr rrAkkIrq c zz c rr irr c Akk i rr rd Akk trA L L S L LS S ττττ τ τω π τ (12) In (12) there is a delay time c zz c rr ′ − − ′ − − τ , describing the nature of laser propagating through medium. Besides, in this solution there is not the divergence if z’ → z, like in solution for the paraxial case. That because of the factor ' 1 z z − is tr−êng §¹i häc Vinh T¹p chÝ khoa häc, tËp XXXVII, sè 2A-2008 41 replaced by the Green function of r r ′ − 1 argument. This solution is genaralized for 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]: ( ) ( ) ( ) ><= +− τρτρ ωπ τρ ,, ˆ ,, ˆ 2 , LELE cA I ss s S η , (13) 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: ( ) ( ) ( ) .,,,2,,' 2 , 0 2 31 2 31       ′′′′ Γ+ ′ = ∫ ∫ ∫ −− τ ρ ττττ ωπ τρ rrHdrdNrrKrdN c I S η (14) Now we discuss in the transient limit and the steady-state limit. 4.1. Stokes intensity in transient limit When the scattering time L τ is much less than the collisional dephasing time i.e. [3], gL L <Γ τ (15) from (8), (9) and (14) the intensity Stokes wave at output face z = L is ( ) ( ) ( ) ( ) [ ] { } .16exp 8 , 2 1 21 2 2 τ τ τ π τρ LPkk P E A I L TR Φ = (16) This result is similar to one of the one-dimension, byond the factor Φ 2 . 4.2. Stokes intensity in steady-state limit When the time L τ is much larger than the collisional dephasing time [3], i.e., gL L >>Γ τ (17) from (8), (9) and (17) the intensity Stokes wave at output face z = L is . )4( 2 1 2 gLA e I gL SS π Φ = (18) This result is similar to one of the one-dimension, byond the factor Φ 2 . 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- C. V. Lanh , D. X. Khoa, H. Q. Quy, P. T. T. Van RAMAN STIMULATED , Tr. 37-42 42 dimension by factor 2 (Fresnel number), which relates to structure of medium. It is true for the general case too, when the value of L is arbitrary, for that the Stokes intensity is given by ( ) ( ) ,, 1 2 LI A I D SS = (19) where ( ) , 1 LI D S is the Stokes intensity when 1 = (in one-dimentional approach). 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. XXVII th 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. TóM TắT TáN Xạ RAMAN CƯỡNG BứC TRONG GầN ĐúNG BA CHIềU Trong bài này chúng tôi giới thiệu lý thuyết tán xạ Raman cỡng bức trong gần đúng ba chiều. Cờng độ sóng Stokes đã đợc tính toán và thảo luận trong hai trờng hợp giới hạn là giới hạn thời gian ngắn và giới hạn thời gian dài. (a) Physical Department of Vinh University (b) Institute for Applied Physics, MISTT (c) Master student of Optical course 14 th . . hyper -Raman scattering, Phys. awRev. A, 72, 011802-2, 2005. TóM TắT TáN Xạ RAMAN CƯỡNG BứC TRONG GầN ĐúNG BA CHIềU Trong bài này chúng tôi giới thiệu lý thuyết tán xạ Raman cỡng bức trong. này chúng tôi giới thiệu lý thuyết tán xạ Raman cỡng bức trong gần đúng ba chiều. Cờng độ sóng Stokes đã đợc tính toán và thảo luận trong hai trờng hợp giới hạn là giới hạn thời gian ngắn và giới. tr−êng §¹i häc Vinh T¹p chÝ khoa häc, tËp XXXVII, sè 2A-2008 37 RAMAN STIMULATED SCATTERING IN THREE-DIMENSIONAL APPROACH Chu Van Lanh (a) , Dinh Xuan Khoa (a) , Ho Quang Quy (b) ,