Journal of Applied Mathematics and Physics, 2018, 6, 405-417 http://www.scirp.org/journal/jamp ISSN Online: 2327-4379 ISSN Print: 2327-4352
Nonstationary Stimulated Raman Scattering by Polaritons in Continuum of Dipole-Active Phonons
Vladimir Feshchenko1, Galina Feshchenko2
1Dawson College, Montreal, Canada 2Vanier College, Montreal, Canada
How to cite this paper: Feshchenko, V. Abstract and Feshchenko, G. (2018) Nonstationary Stimulated Raman Scattering by Polaritons The system of equations simulating the processes of nonstationary stimulated in Continuum of Dipole-Active Phonons. Raman scattering (SRS) with the excitation of polar optical phonons is ob- Journal of Applied Mathematics and Phys- tained. This system is found by applying such standard methods as the non- ics, 6, 405-417. stationary theory of perturbations, which resulted in the equations for the https://doi.org/10.4236/jamp.2018.62038 amplitudes of probabilities to find the discrete system in certain state, and Received: August 12, 2017 slowly-varying amplitudes for the electromagnetic waves. It has been shown Accepted: February 24, 2018 that the obtained system includes, as extremes, the case of classical interaction Published: February 27, 2018 between electromagnetic field and resonant medium (including the “area
Copyright © 2018 by authors and theorem”), and the one related with SRS on optical phonons. We have con- Scientific Research Publishing Inc. ducted both theoretical and numerical investigation of simplified system as- This work is licensed under the Creative suming that the amplitudes of all electromagnetic waves (laser, Stokes, and Commons Attribution International polariton) were real (there was no destructive spatial-temporal phase modula- License (CC BY 4.0). tion). Only low-order nonlinear processes are considered. It is shown that this http://creativecommons.org/licenses/by/4.0/ system can be reduced to Sine-Gordon equation. This system can also be sim- Open Access plified to the equation that simulates the motion of physical pendulum from upper equilibrium position. The numerical study of nonstationary SRS when the electromagnetic field of laser radiation and Stokes excite both polariton emission and the continuum of dipole-active phonons has been carried out. The evolution of the intensity of the polariton wave as function of the length of nonlinear medium has been numerically analyzed.
Keywords Nonstationary Stimulated Raman Scattering, Polaritons
1. Introduction
Over the past two decades, the field of polaritons (exciton-like, plasmon-like,
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V. Feshchenko, G. Feshchenko
and phonon-like) has been substantially developed [1] [2]. For instance, the theory of exciton-polaritons provides the successful explanation of physical phenomena in microcavities, which is essential for the design of polariton lasers [3] [4]. The spectroscopy of complex semiconductor layer structures is related with the theory of plasmon-polaritons [5] [6]. The progress in infrared spec- troscopy of ionic crystals is based on theory of phonon-polaritons [7] [8] [9]. Surface polaritons are considered in details in [10]. However, given the com- plexity of the equations simulating multi-photon processes in nonlinear media and accepted simplifications (the method of slowly-varying amplitudes, the giv- en field approximation, phase capture, etc.) some results still remain unforeseen [11] [12] [13] [14]. In this paper, we made an attempt to obtain and solve the system of equations that would properly describe the processes of nonstationary SRS in ionic crystals by phonon-polaritons. The Section 2 of our paper is based on technique developed in [15] where the equations for the “nonresonant” am- plitude of probabilities to find the particle at certain level were approximate
whereas the equations for the “resonant” amplitudes of probabilities a1 and a2 were exact. Despite the fact that the obtained system was cumbersome, we ma- naged to bring it to the compact form. In Section 3, we have compared the theo- retical and experimental gain factors [16]-[25]. In Section 4, we found the sim- plified system of equations, showed that this system corresponds to the extreme cases studied previously (see [26] [27]), and reduced the system to Sine-Gordon equation [28]. The numerical analysis of the system has been carried out as well.
2. Basic Principles and Equations
In this paper we assume that two optical pulses with frequencies ω1 (the fre-
quency of laser pump) and ω2 (that of Stokes) propagate in crystal at the an- gles θ1,2 with respect to z-axis, which is perpendicular to the input surface of
the medium. The laser wave ω1 and Stokes ω2 during SRS generate
ω312= ωω − as well as the polar optical phonon ωωf ≡ 21 in the vicinity of
which falls ω3 (ωω3= 21 +∆ ω) .
We apply the standard equations for the amplitude of probability ak of
finding the system in state with energy Ek [29] [30]: ∂ ak iwkl t i = ∑e,Vakl l (1) ∂t l
11Φ −Φ i(Φ +Φ ) =−µ ε iimm +−α′ εε mn Vkl kl∑∑ m (ee) kl m n (e where 22m mn, , +ei(−Φ+Φmn) ++ ee ii( Φ−Φ mn) ( −Φ−Φ mn) )
Φ=mmmωϕt − kz + , Φmm = −Φ− , εεmm= − ,
µkl is the dipole moment of the transition kl→ ; ωεmmm,,k and ϕm are the frequencies, z-components of wave vectors, real “slowly-varying amplitudes”,
and phases of the interacting waves, accordingly; αkl′ is the tensor of the com- binational scattering.
Then we use (1) to find al (l ≠ 1, 2 ) .
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iiΦmm−Φ iΦm 1itωωll12 ee 1i t e aal=∑∑µεlm1ee++12µεl m 22 mmωωl11+− ml ωω m ωωl2+ m −Φ ii(Φ +Φ ) (−Φ +Φ ) e1i m emn emn ++αεε′ itωl1 + a21∑ l mne ωωl2− m 2 mn, ωωωωωωl11++ mn l −+ mn (2) eeii(Φmn −Φ ) (−Φmn −Φ ) 1 ei(Φmn +Φ ) ++ +αεε′ itωl 2 a1 ∑ l2 mne ωωωωωωl11+− mn l −− mn 2 mn, ωωωl2 ++ mn ei(−Φ+Φmn) ee ii(Φ−Φmn) (−Φ−Φmn) +++ ≠ al2 ,( 1, 2 ) ωωωωωωωωωl222−+ mn l +− mn l −− mn
To obtain the exact equations for a1,2 we substitute (2) into (1):
∂a1 222 −iθ i =−⋅+⋅+⋅+{ r1ε 1 r 2 ε 2 r 3 ε 3[ r1 '+− + r 2] ⋅εεε123 e +[rr 1−+ + 2] ∂t iθ 4 4 4 22 22 ⋅εεε1 2 3e3 +rrrr 1 ⋅+ ε 1 3 2 ⋅+ ε 2 3 3 ⋅+ ε 3 4 12 ⋅ εε 1 2 + r 4 13 ⋅ εε 1 3
1 −+ϕ − 22 it∆ω i( kz33) ′ ik( 21 k) z +⋅ra423ε 2 ε 3} 1 − e µ12 ε 3 e++[ α12r 5e] εε 1 2 2 ⋅i(ϕϕ12− ) + ⋅+⋅+⋅εε2 εε 23 ε i(−+ kz33ϕ ) errr 631 1 3 6 32 2 3 7e 3
33ik( 21−− k) z i(ϕϕ 1 2) ik( 21−− k) z i(ϕϕ 1 2) +rr812,+−ε 1 ε 2 ee+⋅ 821,εε 1 2 ee (3)
ik( −− k) z i(ϕϕ) ik( −− k22 k) z i(ϕ + ϕϕ −) +⋅rr9eeεεε2221 1 2+ 10ee ⋅εεε 12 3 2 31 a 123 123 2
∂a −∆ω 1 ikz( −φ ) ik( −− k) z i(ϕϕ) ip 2 =−eit µ ⋅ ε e33 +[ α′ +⋅ 5ee] εε 12 21 ∂t 2 21 3 21 1 2 + ⋅+εε2 ⋅+⋅⋅ εε 23 ε ikz( 33−ϕ ) + ⋅εε3ik( 12− k) z ppp631 1 3 6 32 2 3 7e3 p 812,+ 1 2 e
i(ϕϕ21−) 32ik( 12−− k) z i(ϕϕ 21) ik( 12−− k) z i(ϕϕ 21) ⋅e +pp 812,− ⋅εε 12 ee+⋅ 9εεε123 ee
2i(22 k321+− k kzi) (ϕϕ 1 −− 2 ϕ 3) 222 +p10 ⋅εεε123 e e ap1−{ 1 ⋅+⋅+⋅ε 1 p 2 ε 2 p 3 ε 3 −iiθθ44 +[ pp12+− + ]εεε123 e+[ pp 12−+ +]εεε123 e + p 3 1 ⋅+ ε 1 p 3 2 ⋅ ε 2 4 22 22 22 +⋅+⋅+⋅+⋅pp343ε 3 12 εε 1 2 p 4 13 εε 1 3 p 4 23 εε 2 3} a 2
1 ω ≡ µµ l1 where ri ∑ 11ll 22; 2 l≠1,2 ωωli1 − 1′ 111 r1;± ≡∑ µα11ll++ 2 l≠1,2 ωωωωωωωωωlll131± ∓∓ 121 ± 132 ±± 1′ 111 r2;± ≡∑ αµ11ll++ 2 l≠1,2 ωωωωωωlll13 ∓∓ 1 2 11± 1′′ 141 r3;i≡∑ αα11ll++ 4 l≠1,2 ωl1+− 22 ωω il 11 ω l ω i 1′′ 11 r4ij ≡+∑ αα11l l l≠1,2 ω++ ωω ω −+ ωω l11 ijl ij 1 12 +++; ω+− ωω ω −− ωω ω l111 ijl ijl
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1 11 r5;≡+∑ µµ12ll 4 l≠1,2 ωll21+− ωω 22 ω 1′ 1 11 r6ij ≡∑ µα12l l ++ ωωωωωωω+− ++ 2 l≠1,2 l222 ijl ijl
1′ 111 +∑ αµ12ll ++; ωωωωωω++ − 2 l≠1,2 l22 il jl 2 j 1′′ 1 21 2 1 r7;≡∑∑µα12ll++αµ12ll ⋅ + 4 ll≠≠1,2 ωl2+ 24 ωω 32 l1,2 ωωωωll2323+−
1′′ 12 2 1 r8;ij,± ≡∑ αα12l l ++ + 22 l≠1,2 ωl2± ω il ω 22122 ω l+− ωω ωl∓∓ ωω ij 1′′ 111 r9 ≡∑ αα12ll ++ l≠1,2 ωωωωωωωωωlll213++ 213 +− 212 +−
1 11 +++; ωωωωωωωlll223−+ 223 −− 2 1′′ 21 r10 ≡+∑ αα12ll 2 l≠1,2 ωll213−+ ωω ω 212 −+ ωω
21 ++; ωll2++ ωω 23 ω 213 −+ ωω 1 ω ≡ µµ l 2 θ≡(k2 +− k 31 kz) +(ϕϕϕ 1 −− 2 3); pi ∑ 22ll 22; 2 l≠1,2 ωωli2 − 1′ 111 p1;± ≡∑ µα22ll++ 2 l≠1,2 ωωωωωωωωωlll231±±± ∓∓∓ 221 232 1′ 111 p2;± ≡∑ αµll22++ 2 l≠1,2 ωωωωωωlll23 ∓∓ 2 2 21± 1′′ 141 p3;i≡∑ αα22ll++ 4 l≠1,2 ωl2+− 22 ωω il 22 ω l ω i
1 11 p5;≡+∑ µµ21ll 4 l≠1,2 ωωωωll11−+ 1 2 1′′ 11 p4ij ≡+∑ αα22l l ω++ ωω ω −+ ωω l≠1,2 l22 ijl ij
1 12 +++; ω+− ωω ω −− ωω ω l222 ijl ijl 1′ 1 11 p6ij ≡∑ µα21l l ++ ωωωωωωω−− −+ 2 l≠1,2 lijlijl111
111 ++ +; ωωωωωω−+ − lil11 jl 1 j
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1′′ 1 21 1 2 p7;≡∑∑µα2ll 1++ µαl12 l + 4 ll≠≠1,2 ωl1− 24 ωω 31 l1,2 ωωωωll1313+− 1′′ 12 2 1 p8;ij,± ≡∑ αα21l l ++ + ω ωωωωωωωω+ − ±± 22 ∓l≠1,2 l1 il 11211 l l ij 1′′ 111 p9 ≡∑ αα21ll ++ l≠1,2 ωωωωωωωωωlll113−− 113 −+ 11 −+ 2
1 11 +++; ωωωωωωωlll123++ 123 +− 1 1′′ 2 1 21 p10 ≡∑ αα21ll+++; 22 l≠1,2 ωlll113+− ωω ω 112 +− ωω ω 12 −− ωω 3 ω l 1 − ω 3 ij,= 1, 2, 3. We define the polarization induced in this process as
∗∗itωω12lli t ∗itωω12 ∗it21 P=∑ ( a11µµll ae + a22ll a e ) +++ a1122 µ ae a 2211 µ ae cc . ., (4) l≠1,2
After substituting (2) into (4) we get = ∆(m) ⋅+κε( m) Φ+(mn,,) ⋅+ (mn) P∑{( r n ) mmcos ( pccn n pcc ) m ⋅εε Φ Φ+(mn,,) ⋅ +(mn) ⋅εε Φ Φ mncos m cos n ( pssn n pss ) mnsin m sin n} (−−13) ( ) +2sP( 1 cos Φ− 2 P 2 sin Φ 21) ⋅ε + 2 sP( 1 cosθ+ P 2 sin θε) ⋅ 3 (5) (+2) +2s( P1 cos Φ+ 1 P 2 sin Φ 1) ⋅ε 2 + 4(Pg 11 − Pg 2 2) ⋅εε 12 +µ Φ θθ −Φ + Φ θθ+Φ 4{PP1 ( cos3 cos sin 3 sin) 23( sin cos cos3 sin)} ,
2 m1 mm m1 mm 2 µω ∆≡( ) ( ) −( ) κ ( ) ≡+( ) ( ) (m) ≡ il li where r( rr22 11 ), (rr11 22 ), rii ∑ 22; 2 2 l≠1,2 ωωli− m
(mn,) ( mn ,) (−− mn ,,) ( mn) ( mn ,) pccn≡− p11 − p 11 + p 22 + p 22 ,
(mn,,,,,) ( mn) (−− mn) ( mn) ( mn) pccpp≡+11 11 ++ pp 22 22 ,
(mn,) ( mn ,) (−− mn ,,) ( mn) ( mn ,) pssn≡− p11 p 11 −+ p 22 p 22 ,
(mn,) ( mn ,,,,) (−− mn) ( mn) ( mn) psspp≡−11 + 11 − pp 22 + 22 , (±mn, ) 1′ 11 pii ≡+∑ µαil li , ∓l≠1,2 ωωωli±+ m n ωωω li m − n
(∓m) 1 11 s ≡+∑ µµ21ll, 2 ∓l≠1,2 ωωl12 ml ωω± m
∗∆i ∗∆i P1≡ Re( aa 12 e) , P2≡ Im( aa 12 e) , ∆≡∆ωt +( k21 − kz) +(ϕϕ 1 − 2), ∗∗1 1 µα12ll′ µα21ll gg1 ≡ Re{ } , gg2 ≡ Im{ } , g ≡+∑ , 2 l≠1,2 ωll221−+ ωωω 121 +− ωω 22 θ≡(k213 −+ k kz) +−ϕϕϕ 1 2 − 3, na=21 − a. The system of Equations (3) can be transformed to the equations for the pola-
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rization P1,2 and n (population difference): ∂P 11∂−(ϕϕ) 1 =(B − FP) + A − E −∆ω − 12P ∂∂tt 12 (6) 11 +−++(D H) ( D Hn) , 22 ∂P 11∂−(ϕϕ) 2 =EA − −∆ω + 12P+( B − FP) ∂∂tt 12 (7) 11 +(G −− C) ( C + Gn) , 22 ∂n 2 2 11 =−+(HDP) +( CGP +) −( FB ++) ( BFn −) , (8) ∂t 12 To make the system (6)-(8) complete we add the equations for “slow-
ly-varying amplitudes” ε1,2,3 and ϕ1,2,3
∂∂εε1nN 112π ω 1 (+2) +=− 2,sP22ε (9) ∂∂z c t cn1
∂∂ϕϕnN2π ω+ 1+ 11ε =− 1 ( 2) ε+∆( 11) ⋅ + κε( ) 1 (2,sP12 ( r n ) 1) (10) ∂∂z c t cn1
∂∂εεnN2π ω − 2+=−− 22 2 ( 1) ε ( 2,sP) 21 (11) ∂∂z c t cn2
∂∂ϕϕnN2π ω− 2+ 22ε =− 2 ( 1) ε+∆( 22) ⋅ + κε( ) 2 (2,sP11 ( r n ) 2) (12) ∂∂z c t cn2
∂∂εε3nN 332π ω 3 +=− 4µθ(PP21 cos− sin θ) , (13) ∂∂z c t cn3
∂∂ϕϕ3nN 33 2π ω3 +=−ε3 4µ(PP12 cos θ+ sin θ) , (14) ∂∂z c t cn3
222 Ar≡⋅+⋅+⋅+1ε 1 r 2 ε 2 r 3 ε 3 [ r1+− + r 1 + r 2 + + r 2 −] ⋅εεε123 cos θ where 4 4 4 22 22 22 +rrrr331 ⋅+ε 1 2 ⋅+ ε 2 3 3 ⋅+ ε 3 4 12 ⋅ εε 1 2 + r 4 13 ⋅ εε 1 3 + r 4; 23 ⋅ εε 2 3
Br≡[ 1+− −− r 1 r 2 + + r 2 −] ⋅εεε123 sin θ , 1 ≡µ ⋅ ε θ ++ α′ εε + ⋅+⋅+⋅ εε2 εε 23 ε θ C12 3 cos[ 12rr 5] 1 2 6 31 1 3 r 6 32 2 3 r 7 3 cos 2 3 32 2 +rrr812,1+−ε ε 2 + 8 21,12 εε +⋅ 9 εεε 123 + r 10 ⋅ εεε 123 cos 2 θ ; 1 ≡µ ⋅⋅ ε θ + ⋅ εε2 + ⋅ εε 23 + ⋅ ε ⋅ θ D12 3 sin rrr 631 1 2 6 32 2 3 7 3 sin 2 2 +⋅r10εεε123 sin 2 θ ; 222 Ep≡⋅+⋅+⋅+1ε 1 p 2 ε 2 p 3 ε 3 [ p1+− + p 1 + p 2 + + p 2 −] ⋅εεε123 cos θ
4 4 4 22 22 22 +pppp31 ⋅+ε 1 3 2 ⋅+ ε 2 3 3 ⋅+ ε 3 4 12 ⋅ εε 1 2 + p 4 13 ⋅ εε 1 3 + p 4; 23 ⋅ εε 2 3
Fp≡[ 1+− −− p 1 p 2 + + p 2 −] ⋅εεε123 sin θ ; 1 ≡µ ⋅ ε θ ++⋅+ α′ εε ⋅+ εε2 ⋅+⋅ εε 23 ε θ G21 3 cos[ 21 pppp 5] 1 2 6 32 2 3 6 32 2 3 73 cos 2 3 32 2 +p812,+− ⋅ε 1 ε 2 + p 8 21, ⋅ εε 12 +⋅ pp 9 εεε 123 + 10 ⋅ εεε123 cos 2 θ ;
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1 ≡µ ⋅ ε θ + ⋅ εε2 + ⋅ εε 23 +⋅ ε θ H21 3 sin[ ppp 631 1 2 6 32 2 3 73 sin 2 2 +⋅p10ε123 ε ε sin 2 θ.
3. Gain Factor G
Such significant characteristics of crystals as the dispersion relation and damp- ing of phonon-polaritons have been the subject of many both theoretical and experimental investigations in recent years. For instance, the experimental me- thods used in these studies include the impulsive SRS and transient grating ex- periments with femtosecond pulses [16] [17], and coherent anti-Stokes Raman Scattering (CARS) with picosecond pulses [18]. The information on the damp- ing of polaritons is contained in Raman linewidth, which can be obtained from the stimulated gain curve [19]. This method was successfully applied in [20], where the two-beam amplifier experiment SRS gain measurements were per- −1 formed in LiNbO3 for the polariton frequencies ranging from 30 to 230 cm . The measurements were carried out at liquid-nitrogen temperature in doped
congruent LiNbO3 and undoped nearly stoichiometric LiNbO3. It was the com- parison of doped and nearly stoichiometric crystals which provided new infor- mation on the number and type of low-frequency excitations determining the polariton damping over almost complete frequency range. Now we show that our equations are consistent with the experimental results presented in [20]. The theoretical treatment of the gain factor for SRS by polari- tons is based on the solutions of the coupled wave equations for the Stokes and polariton fields. In quasi-stationary case, in absence of phase modulation of any
kind, and given pump field, the system of equations for the ε 2 (Stokes) and ε3 (polariton) can be reduced to the following (see (11) and (13)): ∂ε 1 ωω22 2 =−−2*ssε ε χεε ikss22 2 i s13 (15) ∂zk2 s c2 kcs
22 ∂ε 11ωωpp 3 =−−ik2*ε ε i χ εε (16) ∂ pp223 p12 zk22ppcck
ωsp, where kn= , ε are the dielectric constants, χ are the nonlinear sp,,c sp sp, sp, coefficients. This system is the part of more complex system considered in [21]. When we assume that there is exponential amplification of the electromagnetic field with a gain factor G and after the insertion of the electromagnetic fields 1 Gz 2 s expressed as Asp, e into (15) and (16) we get the expression for G as
ν s 1 22 G ≈ χε′′ 1 , (17) nspε ′′
which is completely consistent with the expression of the gain given in [20] and [22]. Therefore, the gain factor is related to the dielectric function of the crystal, which contains the damping of the polariton.
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ω Here ν = s ; n is the index of refraction at Stokes frequency; s 2πc s νγν2 ε ′′ = ffp pf∑ s 2 ; s f is the oscillator’s strength of the dipole-ac- f 2 2 22 (νf−+ ν p) γν fp ′′ tive phonon ν f ; γ f is the damping constant; χ is the imaginary part of quadratic polarizability. The expression for χ′′ is given in [23]: νγν2 χχ′′ = ffp ∑ f 2 , (18) f 2 2 22 (νf−+ ν p) γν fp
1 2 1 11s ffσ where χ = , .σ is the effective cross-section of combi- f 32ν f 8π c ν l f national scattering.
In calculations we used for G in LiNbO3 the following (see [20]): 2 −8 −1 Il ≈10 MW cm ; χ f ≈10 esu ; ZnO-doped LiNbO3: ν f (cm ) = 112, 122, −1 124, 150, 163, 190, 199, 222, and 235; γ f (cm ) = 33, 20, 12, 12, 14, 12, 11, 14, −1 and 14; nearly stoichiometric LiNbO3: ν f (cm ) = 106, 125, 148, 153,167, 188, −1 198, 217, 233, and 239; γ f (cm ) = 33, 20, 10, 8, 20, 13, 3, 10, 10, and 4. The gain factor G as a function of the polariton frequency in nearly stoichiometric
LiNbO3 is shown in Figure 1. The gain factor G versus polariton frequency in
ZnO: LiNbO3 is shown in Figure 2. In both graphs the squares represent expe- rimental points ([20]), whereas the solid curves are the simulation based on (17).
4. Analysis of Basic System
In this section we provide the analysis of the system (6)-(14) assuming that the
all amplitudes of electromagnetic waves are real (ϕ1,2,3 = 0) and the waves are synchronous (θ ≈ 0) . Only low-order nonlinear processes are considered. The simplified system can be written as follows: ∂n 2 =(µε + 2, α′ ε ε ) P (19) ∂t 3 12 2 ∂P 1 2 =−+(µε2, α′ ε ε )n (20) ∂t 2 3 12
∂∂εε1nN 112π ω 1 (+2) +=− 2,sP22ε (21) ∂∂z c t cn1
∂∂εεnN2π ω − 2+=−− 22 2 ( 1) ε ( 2,sP) 21 (22) ∂∂z c t cn2
∂∂εε3nN 332π ω 3 +=− 4,µP2 (23) ∂∂z c t cn3
where n1,2,3 are linear indices of refraction n1,2,3≈ ε 1,2,3 .
The solutions for n and P2 can be expressed as 1 nP=−=cosψψ , sin , (24) 2 2
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8
7
6 gain factor G (1/cm) 5
4
3
2
1 polariton frequency v ( 1/cm) 0 0 50 100 150 200 250 300 Figure 1. Gain factor G versus polariton frequency in nearly stoichiometric
LiNbO3. Squares: experimental points ([20]); solid curve: simulation based on (17).
8
7
6 gain factor G 5 (1/cm)
4
3
2
1 polariton frequency v ( 1/cm) 0 0 50 100 150 200 250 300
Figure 2. Gain factor G versus polariton frequency in ZnO-doped LiNbO3. Squares: experimental points ([20]); solid curve: simulation based on (17).
1 t ψ= µε + α′ ε ε where ∫ ( 32d 12) t . −∞ 1) The system of Equations (19)-(23) includes both the classical case of reso- nant interaction of electromagnetic wave with two-level system (See [26]) (α′ = 0) and the one that corresponds to the combinational scattering by non- polar optical phonons studied in [27] (µ = 0) .
2) The extreme case α′ = 0 also results in standard “area theorem” for ε3 (See [26]): d8WNπ ω 33=−(1 −Φ cos ) , (25) dz cn3
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µ +∞ +∞ Φ= ε ≡ ε 2 where ∫ 3 ( zt,d) t, W33∫ ( zt,d) t. −∞ −∞
3) In the absence of dispersion (nnnn123≈≈≈) the system (19)-(23) can be reduced to Sine-Gordon equation ∂2ψ = −sinψ (26) ∂∂τς zn 224πN ′ where τ =ca t − , ς = az , aC= µω3 + α , c cn2 (+−21) 22( ) ω1s εω 22−≡ sC ε 1 (the given field approximation for ε1,2 ). z 4) If we introduce a new variable τ =t − (ν is the speed of the pulse), we v can reduce the system (19)-(23) to the equation of the motion of physical pen- dulum from the position of upper unstable equilibrium: ∂2ψ 1 = ψ 22sin (27) ∂ττp n ωκ µ2 −1 1,2,3 1 −2 4πN 33 where κ1,2,3 = − , ταp =−+′C , cv cn3 ωκ(+−21) ωκ ( ) 11ss222 2 C ≡−εε21 + (the given field approximation for ε1,2 ). nn12 5) We also provided the numerical solution of (19)-(23). To do that, we brought that system to unitless form: ∂∂εε 11+n =−⋅ CP1,ε (28) ∂∂zt 1 22 ∂∂εε 22+=⋅n CP2,ε (29) ∂∂zt 2 21 ∂∂εε 33+n =−⋅ CP3, (30) ∂∂zt 32 ∂P 2 =−⋅+(CC45ε ⋅ εε ) n , (31) ∂t 3 12 ∂n =(CC45 ⋅+ε ⋅ εε ) P , (32) ∂t 3 12 2 ω (+2) 2π 10Nz(2 s ) where εε 1,2,3= 1,2,3A '0 , tt= τ 0 , z = zz0 , zc00= τ , C1 ≡ , cn1 (−1) ω 2 2π 20Nz(2 s ) 2πωµNz4 τµA ατ′ A C2 ≡ , C3 ≡ 30, C4 ≡ 00, C5 ≡ 00; cn2 cn30 A 2
τ 0 and A0 are characteristic time interval and amplitude of electromagnetic wave; N is the number of atoms in cm3; µ is the average dipole moment.
The space-time evolution of the polariton wave at frequency ω3 is shown in 22 ∗ Figure 3 ( I31= εε 3 ( zt , ) 1,max ; CCCC1= 2 = 4 = 51 = ; C3= 15 ; z= zz0 ; ∗ 19− 3 −10 −18 t= tt0 ; N ≈10 cm , τ 0 ≈10 s , µ ≈10 esu ).
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Figure 3. The space-time evolution of the normalized polariton intensity I31.
5. Conclusions
In this paper we have found the following: 1) The system of Equations (6)-(14) that models the processes of nonstatio- nary SRS by polaritons in nonlinear media consisting of polar optical phonons is obtained; 2) It has been shown that the frequency dependence of the gain factor of the Stokes matches with the experimental results; 3) The simplified system of Equations (19)-(23) (for real amplitudes of all electromagnetic waves and “low-order” nonlinear processes); 4) It is shown that the latter system could be reduced to either case of purely combinational interaction (the nonstationary SRS by nonpolar phonons ( µ = 0 )) or the classical case of nonlinear resonant (but not combinational ( α′ = 0 )) in- teraction (including “area theorem”) between the electromagnetic field and sys- tem; 5) It is also shown that (19)-(23) could be reduced to the standard Sine-Gordon equation; 6) The conditions at which the system (19)-(23) becomes the equation of the motion of physical pendulum; 7) The numerical analysis of (19)-(23) has indicated the possibility of effective conversion to infrared radiation, which could be useful for the design of wide- band frequency converters.
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