The system of equations simulating the processes of nonstationary stimulated Raman scattering (SRS) with the excitation of polar optical phonons is obtained. This system is found by applying such standard methods as the nonstationary theory of perturbations, which resulted in the equations for the amplitudes of probabilities to find the discrete system in certain state, and slowly-varying amplitudes for the electromagnetic waves. It has been shown that the obtained system includes, as extremes, the case of classical interaction between electromagnetic field and resonant medium (including the “area theorem”), and the one related with SRS on optical phonons. We have conducted both theoretical and numerical investigation of simplified system assuming that the amplitudes of all electromagnetic waves (laser, Stokes, and polariton) were real (there was no destructive spatial-temporal phase modulation). Only low-order nonlinear processes are considered. It is shown that this system can be reduced to Sine-Gordon equation. This system can also be simplified 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.
Over the past two decades, the field of polaritons (exciton-like, plasmon-like, and phonon-like) has been substantially developed [
In this paper we assume that two optical pulses with frequencies ω 1 (the frequency of laser pump) and ω 2 (that of Stokes) propagate in crystal at the angles θ ˜ 1 , 2 with respect to z-axis, which is perpendicular to the input surface of the medium. The laser wave ω 1 and Stokes ω 2 during SRS generate ω 3 = ω 1 − ω 2 as well as the polar optical phonon ω f ≡ ω 21 in the vicinity of which falls ω 3 ( ω 3 = ω 21 + Δ ω ) .
We apply the standard equations for the amplitude of probability a k of finding the system in state with energy E k [
i ℏ ∂ a k ∂ t = ∑ l e i w k l t V k l a l , (1)
where V k l = − 1 2 μ k l ∑ m ε m ( e i Φ m + e − i Φ m ) − 1 2 α ′ k l ∑ m , n ε m ε n ( e i ( Φ m + Φ n ) + e i ( − Φ m + Φ n ) + e i ( Φ m − Φ n ) + e i ( − Φ m − Φ n ) ) , Φ m = ω m t − k m z + φ , Φ m = − Φ − m , ε m = ε − m ,
μ k l is the dipole moment of the transition k → l ; ω m , k m , ε m and φ m are the frequencies, z-components of wave vectors, real “slowly-varying amplitudes”, and phases of the interacting waves, accordingly; α ′ k l is the tensor of the combinational scattering.
Then we use (1) to find a l ( l ≠ 1 , 2 ) .
a l = 1 2 ℏ ∑ m μ l 1 ε m e i ω l 1 t ( e i Φ m ω l 1 + ω m + e − i Φ m ω l 1 − ω m ) a 1 + 1 2 ℏ ∑ m μ l 2 ε m e i ω l 2 t ( e i Φ m ω l 2 + ω m + e − i Φ m ω l 2 − ω m ) a 2 + 1 2 ℏ ∑ m , n α ′ l 1 ε m ε n e i ω l 1 t ( e i ( Φ m + Φ n ) ω l 1 + ω m + ω n + e i ( − Φ m + Φ n ) ω l 1 − ω m + ω n + e i ( Φ m − Φ n ) ω l 1 + ω m − ω n + e i ( − Φ m − Φ n ) ω l 1 − ω m − ω n ) a 1 + 1 2 ℏ ∑ m , n α ′ l 2 ε m ε n e i ω l 2 t ( e i ( Φ m + Φ n ) ω l 2 + ω m + ω n + e i ( − Φ m + Φ n ) ω l 2 − ω m + ω n + e i ( Φ m − Φ n ) ω l 2 + ω m − ω n + e i ( − Φ m − Φ n ) ω l 2 − ω m − ω n ) a 2 , ( l ≠ 1 , 2 ) (2)
To obtain the exact equations for a 1 , 2 we substitute (2) into (1):
i ℏ ∂ a 1 ∂ t = − { r 1 ⋅ ε 1 2 + r 2 ⋅ ε 2 2 + r 3 ⋅ ε 3 2 + [ r 1 ' + + r 2 − ] ⋅ ε 1 ε 2 ε 3 e − i θ + [ r 1 − + r 2 + ] ⋅ ε 1 ε 2 ε 3 e i θ + r 3 1 ⋅ ε 1 4 + r 3 2 ⋅ ε 2 4 + r 3 3 ⋅ ε 3 4 + r 4 12 ⋅ ε 1 2 ε 2 2 + r 4 13 ⋅ ε 1 2 ε 3 2 + r 4 23 ⋅ ε 2 2 ε 3 2 } a 1 − e i Δ ω t { 1 2 μ 12 ε 3 e i ( − k 3 z + φ 3 ) + [ α ′ 12 + r 5 ] ε 1 ε 2 e i ( k 2 − k 1 ) z
⋅ e i ( φ 1 − φ 2 ) + [ r 6 31 ⋅ ε 1 2 ε 3 + r 6 32 ⋅ ε 2 2 ε 3 + r 7 ⋅ ε 3 3 ] e i ( − k 3 z + φ 3 ) + r 8 12 , + ε 1 3 ε 2 e i ( k 2 − k 1 ) z e i ( φ 1 − φ 2 ) + r 8 21 , − ⋅ ε 1 ε 2 3 e i ( k 2 − k 1 ) z e i ( φ 1 − φ 2 ) + r 9 ⋅ ε 1 ε 2 ε 3 2 e i ( k 2 − k 1 ) z e i ( φ 1 − φ 2 ) + r 10 ⋅ ε 1 ε 2 ε 3 2 e i ( k 1 − k 2 − 2 k 3 ) z e i ( φ 2 + 2 φ 3 − φ 1 ) } a 2 (3)
i ℏ ∂ a 2 ∂ t = − e − i Δ ω t { 1 2 μ 21 ⋅ ε 3 e i ( k 3 z − ϕ 3 ) + [ α ′ 21 + p 5 ] ⋅ ε 1 ε 2 e i ( k 1 − k 2 ) z e i ( φ 2 − φ 1 ) + [ p 6 31 ⋅ ε 1 2 ε 3 + p 6 32 ⋅ ε 2 2 ε 3 + p 7 ⋅ ε 3 3 ] ⋅ e i ( k 3 z − φ 3 ) + p 8 12 , + ⋅ ε 1 3 ε 2 e i ( k 1 − k 2 ) z ⋅ e i ( φ 2 − φ 1 ) + p 8 12 , − ⋅ ε 1 ε 2 3 e i ( k 1 − k 2 ) z e i ( φ 2 − φ 1 ) + p 9 ⋅ ε 1 ε 2 ε 3 2 e i ( k 1 − k 2 ) z e i ( φ 2 − φ 1 )
+ p 10 ⋅ ε 1 ε 2 ε 3 2 e i ( 2 k 3 + k 2 − k 1 ) z e i ( φ 1 − φ 2 − 2 φ 3 ) } a 1 − { p 1 ⋅ ε 1 2 + p 2 ⋅ ε 2 2 + p 3 ⋅ ε 3 2 + [ p 1 + + p 2 − ] ε 1 ε 2 ε 3 e − i θ + [ p 1 − + p 2 + ] ε 1 ε 2 ε 3 e i θ + p 3 1 ⋅ ε 1 4 + p 3 2 ⋅ ε 2 4 + p 3 3 ⋅ ε 3 4 + p 4 12 ⋅ ε 1 2 ε 2 2 + p 4 13 ⋅ ε 1 2 ε 3 2 + p 4 23 ⋅ ε 2 2 ε 3 2 } a 2
where r i ≡ 1 2 ℏ ∑ l ≠ 1 , 2 μ 1 l μ l 1 ω l 1 ω l 1 2 − ω i 2 ;
r 1 ± ≡ 1 2 ℏ ∑ l ≠ 1 , 2 μ 1 l α ′ l 1 [ 1 ω l 1 ± ω 3 ∓ ω 1 + 1 ω l 1 ± ω 2 ∓ ω 1 + 1 ω l 1 ± ω 3 ± ω 2 ] ;
r 2 ± ≡ 1 2 ℏ ∑ l ≠ 1 , 2 α ′ 1 l μ l 1 [ 1 ω l 1 ∓ ω 3 + 1 ω l 1 ∓ ω 2 + 1 ω l 1 ± ω 1 ] ;
r 3 i ≡ 1 4 ℏ ∑ l ≠ 1 , 2 α ′ 1 l α ′ l 1 [ 1 ω l 1 + 2 ω i + 4 ω l 1 + 1 ω l 1 − 2 ω i ] ;
r 4 i j ≡ 1 ℏ ∑ l ≠ 1 , 2 α ′ 1 l α ′ l 1 [ 1 ω l 1 + ω i + ω j + 1 ω l 1 − ω i + ω j + 1 ω l 1 + ω i − ω j + 1 ω l 1 − ω i − ω j + 2 ω l 1 ] ;
r 5 ≡ 1 4 ℏ ∑ l ≠ 1 , 2 μ 1 l μ l 2 [ 1 ω l 2 + ω 1 + 1 ω l 2 − ω 2 ] ;
r 6 i j ≡ 1 2 ℏ ∑ l ≠ 1 , 2 μ 1 l α ′ l 2 [ 1 ω l 2 + ω i − ω j + 1 ω l 2 + ω i + ω j + 1 ω l 2 ] + 1 2 ℏ ∑ l ≠ 1 , 2 α ′ 1 l μ l 2 [ 1 ω l 2 + ω i + 1 ω l 2 + ω j + 1 ω l 2 − ω j ] ;
r 7 ≡ 1 4 ℏ ∑ l ≠ 1 , 2 μ 1 l α ′ l 2 [ 1 ω l 2 + 2 ω 3 + 2 ω l 2 ] + 1 4 ℏ ∑ l ≠ 1 , 2 α ′ 1 l μ l 2 ⋅ [ 2 ω l 2 + ω 3 + 1 ω l 2 − ω 3 ] ;
r 8 i j , ± ≡ 1 2 ℏ ∑ l ≠ 1 , 2 α ′ 1 l α ′ l 2 [ 1 ω l 2 ± 2 ω i + 2 ω l 2 + 2 ω l 2 + ω 1 − ω 2 + 1 ω l 2 ∓ ω i ∓ ω j ] ;
r 9 ≡ 1 ℏ ∑ l ≠ 1 , 2 α ′ 1 l α ′ l 2 [ 1 ω l 2 + ω 1 + ω 3 + 1 ω l 2 + ω 1 − ω 3 + 1 ω l 2 + ω 1 − ω 2 + 1 ω l 2 − ω 2 + ω 3 + 1 ω l 2 − ω 2 − ω 3 + 1 ω l 2 ] ;
r 10 ≡ 1 2 ℏ ∑ l ≠ 1 , 2 α ′ 1 l α ′ l 2 [ 2 ω l 2 − ω 1 + ω 3 + 1 ω l 2 − ω 1 + ω 2 + 2 ω l 2 + ω 2 + ω 3 + 1 ω l 2 − ω 1 + ω 3 ] ;
θ ≡ ( k 2 + k 3 − k 1 ) z + ( φ 1 − φ 2 − φ 3 ) ; p i ≡ 1 2 ℏ ∑ l ≠ 1 , 2 μ 2 l μ l 2 ω l 2 ω l 2 2 − ω i 2 ;
p 2 ± ≡ 1 2 ℏ ∑ l ≠ 1 , 2 α ′ l 2 μ l 2 [ 1 ω l 2 ∓ ω 3 + 1 ω l 2 ∓ ω 2 + 1 ω l 2 ± ω 1 ] ;
p 3 i ≡ 1 4 ℏ ∑ l ≠ 1 , 2 α ′ 2 l α ′ l 2 [ 1 ω l 2 + 2 ω i + 4 ω l 2 + 1 ω l 2 − 2 ω i ] ;
p 5 ≡ 1 4 ℏ ∑ l ≠ 1 , 2 μ 2 l μ l 1 [ 1 ω l 1 − ω 1 + 1 ω l 1 + ω 2 ] ;
p 4 i j ≡ 1 ℏ ∑ l ≠ 1 , 2 α ′ 2 l α ′ l 2 [ 1 ω l 2 + ω i + ω j + 1 ω l 2 − ω i + ω j + 1 ω l 2 + ω i − ω j + 1 ω l 2 − ω i − ω j + 2 ω l 2 ] ;
p 6 i j ≡ 1 2 ℏ ∑ l ≠ 1 , 2 μ 2 l α ′ l 1 [ 1 ω l 1 − ω i − ω j + 1 ω l 1 − ω i + ω j + 1 ω l 1 + 1 ω l 1 − ω i + 1 ω l 1 + ω j + 1 ω l 1 − ω j ] ;
p 7 ≡ 1 4 ℏ ∑ l ≠ 1 , 2 μ 2 l α ′ l 1 [ 1 ω l 1 − 2 ω 3 + 2 ω l 1 ] + 1 4 ℏ ∑ l ≠ 1 , 2 μ l 1 α ′ 2 l [ 1 ω l 1 + ω 3 + 2 ω l 1 − ω 3 ] ;
p 8 i j , ± ≡ 1 2 ℏ ∑ l ≠ 1 , 2 α ′ 2 l α ′ l 1 [ 1 ω l 1 ∓ 2 ω i + 2 ω l 1 + 2 ω l 1 + ω 2 − ω 1 + 1 ω l 1 ± ω i ± ω j ] ;
p 9 ≡ 1 ℏ ∑ l ≠ 1 , 2 α ′ 2 l α ′ l 1 [ 1 ω l 1 − ω 1 − ω 3 + 1 ω l 1 − ω 1 + ω 3 + 1 ω l 1 − ω 1 + ω 2 + 1 ω l 1 + ω 2 + ω 3 + 1 ω l 1 + ω 2 − ω 3 + 1 ω l 1 ] ;
p 10 ≡ 1 2 ℏ ∑ l ≠ 1 , 2 α ′ 2 l α ′ l 1 [ 2 ω l 1 + ω 1 − ω 3 + 1 ω l 1 + ω 1 − ω 2 + 2 ω l 1 − ω 2 − ω 3 + 1 ω l 1 − 2 ω 3 ] ;
i , j = 1 , 2 , 3.
We define the polarization induced in this process as
P = ∑ l ≠ 1 , 2 ( a 1 ∗ μ 1 l a l e i ω 1 l t + a 2 ∗ μ 2 l a l e i ω 2 l t ) + a 1 ∗ μ 12 a 2 e i ω 12 t + a 2 ∗ μ 21 a 1 e i ω 21 t + c . c . , (4)
After substituting (2) into (4) we get
P = ∑ m { ( Δ r ( m ) ⋅ n + κ ( m ) ) ε m cos Φ m + ( p c c n ( m , n ) ⋅ n + p c c ( m , n ) ) ⋅ ε m ε n cos Φ m cos Φ n + ( p s s n ( m , n ) ⋅ n + p s s ( m , n ) ) ⋅ ε m ε n sin Φ m sin Φ n } + 2 s ( − 1 ) ( P 1 cos Φ 2 − P 2 sin Φ 2 ) ⋅ ε 1 + 2 s ( − 3 ) ( P 1 cos θ + P 2 sin θ ) ⋅ ε 3 + 2 s ( + 2 ) ( P 1 cos Φ 1 + P 2 sin Φ 1 ) ⋅ ε 2 + 4 ( P 1 g 1 − P 2 g 2 ) ⋅ ε 1 ε 2 + 4 μ { P 1 ( cos Φ 3 cos θ − sin Φ 3 sin θ ) + P 2 ( sin Φ 3 cos θ + cos Φ 3 sin θ ) } , (5)
where Δ r ( m ) ≡ 1 2 ( r 22 ( m ) − r 11 ( m ) ) , κ ( m ) ≡ 1 2 ( r 11 ( m ) + r 22 ( m ) ) , r i i ( m ) ≡ 2 ℏ ∑ l ≠ 1 , 2 μ i l 2 ω l i ω l i 2 − ω m 2 ;
p c c n ( m , n ) ≡ − p 11 ( m , n ) − p 11 ( − m , n ) + p 22 ( m , n ) + p 22 ( − m , n ) ,
p c c ( m , n ) ≡ p 11 ( m , n ) + p 11 ( − m , n ) + p 22 ( m , n ) + p 22 ( − m , n ) ,
p s s n ( m , n ) ≡ p 11 ( m , n ) − p 11 ( − m , n ) − p 22 ( m , n ) + p 22 ( − m , n ) ,
p s s ( m , n ) ≡ − p 11 ( m , n ) + p 11 ( − m , n ) − p 22 ( m , n ) + p 22 ( − m , n ) ,
p i i ( ± m , n ) ≡ 1 ℏ ∑ l ≠ 1 , 2 μ i l α ′ l i [ 1 ω l i ± ω m + ω n + 1 ω l i ∓ ω m − ω n ] ,
s ( ∓ m ) ≡ 1 2 ℏ ∑ l ≠ 1 , 2 μ 2 l μ l 1 [ 1 ω l 1 ∓ ω m + 1 ω l 2 ± ω m ] ,
P 1 ≡ Re ( a 1 ∗ a 2 e i Δ ) , P 2 ≡ Im ( a 1 ∗ a 2 e i Δ ) , Δ ≡ Δ ω t + ( k 2 − k 1 ) z + ( φ 1 − φ 2 ) ,
g 1 ≡ Re { g } , g 2 ≡ Im { g } , g ≡ 1 2 ℏ ∑ l ≠ 1 , 2 [ μ 1 l α ′ l 2 ω l 2 − ω 2 + ω 1 + μ 2 l ∗ α l 1 1 ∗ ω l 1 + ω 2 − ω 1 ] ,
θ ≡ ( k 2 − k 1 + k 3 ) z + φ 1 − φ 2 − φ 3 , n = | a 2 | 2 − | a 1 | 2 .
The system of Equations (3) can be transformed to the equations for the polarization P 1 , 2 and n (population difference):
∂ P 1 ∂ t = 1 ℏ ( B − F ) P 1 + 1 ℏ ( A − E − Δ ω − ∂ ( φ 1 − φ 2 ) ∂ t ) P 2 + 1 2 ℏ ( D − H ) + 1 2 ℏ ( D + H ) n , (6)
∂ P 2 ∂ t = 1 ℏ ( E − A − Δ ω + ∂ ( φ 1 − φ 2 ) ∂ t ) P 1 + 1 ℏ ( B − F ) P 2 + 1 2 ℏ ( G − C ) − 1 2 ℏ ( C + G ) n , (7)
∂ n ∂ t = − 2 ℏ ( H + D ) P 1 + 2 ℏ ( C + G ) P 2 − 1 ℏ ( F + B ) + 1 ℏ ( B − F ) n , (8)
To make the system (6)-(8) complete we add the equations for “slowly-varying amplitudes” ε 1 , 2 , 3 and φ 1 , 2 , 3
∂ ε 1 ∂ z + n 1 c ∂ ε 1 ∂ t = − 2 π ω 1 N c n 1 2 s ( + 2 ) P 2 ε 2 , (9)
( ∂ φ 1 ∂ z + n 1 c ∂ φ 1 ∂ t ) ε 1 = − 2 π ω 1 N c n 1 ( 2 s ( + 2 ) P 1 ε 2 + ( Δ r ( 1 ) ⋅ n + κ ( 1 ) ) ε 1 ) , (10)
∂ ε 2 ∂ z + n 2 c ∂ ε 2 ∂ t = − 2 π ω 2 N c n 2 ( − 2 s ( − 1 ) ) P 2 ε 1 , (11)
( ∂ φ 2 ∂ z + n 2 c ∂ φ 2 ∂ t ) ε 2 = − 2 π ω 2 N c n 2 ( 2 s ( − 1 ) P 1 ε 1 + ( Δ r ( 2 ) ⋅ n + κ ( 2 ) ) ε 2 ) , (12)
∂ ε 3 ∂ z + n 3 c ∂ ε 3 ∂ t = − 2 π ω 3 N c n 3 4 μ ( P 2 cos θ − P 1 sin θ ) , (13)
( ∂ φ 3 ∂ z + n 3 c ∂ φ 3 ∂ t ) ε 3 = − 2 π ω 3 N c n 3 4 μ ( P 1 cos θ + P 2 sin θ ) , (14)
where A ≡ r 1 ⋅ ε 1 2 + r 2 ⋅ ε 2 2 + r 3 ⋅ ε 3 2 + [ r 1 + + r 1 − + r 2 + + r 2 − ] ⋅ ε 1 ε 2 ε 3 cos θ + r 3 1 ⋅ ε 1 4 + r 3 2 ⋅ ε 2 4 + r 3 3 ⋅ ε 3 4 + r 4 12 ⋅ ε 1 2 ε 2 2 + r 4 13 ⋅ ε 1 2 ε 3 2 + r 4 23 ⋅ ε 2 2 ε 3 2 ;
B ≡ [ r 1 + − r 1 − − r 2 + + r 2 − ] ⋅ ε 1 ε 2 ε 3 sin θ ,
C ≡ 1 2 μ 12 ⋅ ε 3 cos θ + [ α ′ 12 + r 5 ] ε 1 ε 2 + [ r 6 31 ⋅ ε 1 2 ε 3 + r 6 32 ⋅ ε 2 2 ε 3 + r 7 ⋅ ε 3 3 ] cos θ + r 8 12 , + ε 1 3 ε 2 + r 8 21 , − ε 1 ε 2 3 + r 9 ⋅ ε 1 ε 2 ε 3 2 + r 10 ⋅ ε 1 ε 2 ε 3 2 cos 2 θ ;
D ≡ 1 2 μ 12 ⋅ ε 3 ⋅ sin θ + [ r 6 31 ⋅ ε 1 2 ε 2 + r 6 32 ⋅ ε 2 2 ε 3 + r 7 ⋅ ε 3 3 ] ⋅ sin θ + r 10 ⋅ ε 1 ε 2 ε 3 2 sin 2 θ ;
E ≡ p 1 ⋅ ε 1 2 + p 2 ⋅ ε 2 2 + p 3 ⋅ ε 3 2 + [ p 1 + + p 1 − + p 2 + + p 2 − ] ⋅ ε 1 ε 2 ε 3 cos θ + p 3 1 ⋅ ε 1 4 + p 3 2 ⋅ ε 2 4 + p 3 3 ⋅ ε 3 4 + p 4 12 ⋅ ε 1 2 ε 2 2 + p 4 13 ⋅ ε 1 2 ε 3 2 + p 4 23 ⋅ ε 2 2 ε 3 2 ;
F ≡ [ p 1 + − p 1 − − p 2 + + p 2 − ] ⋅ ε 1 ε 2 ε 3 sin θ ;
G ≡ 1 2 μ 21 ⋅ ε 3 cos θ + [ α ′ 21 + p 5 ] ⋅ ε 1 ε 2 + [ p 6 32 ⋅ ε 2 2 ε 3 + p 6 32 ⋅ ε 2 2 ε 3 + p 7 ⋅ ε 3 3 ] cos θ + p 8 12 , + ⋅ ε 1 3 ε 2 + p 8 21 , − ⋅ ε 1 ε 2 3 + p 9 ⋅ ε 1 ε 2 ε 3 2 + p 10 ⋅ ε 1 ε 2 ε 3 2 cos 2 θ ;
H ≡ 1 2 μ 21 ⋅ ε 3 sin θ + [ p 6 31 ⋅ ε 1 2 ε 2 + p 6 32 ⋅ ε 2 2 ε 3 + p 7 ⋅ ε 3 3 ] sin θ + p 10 ⋅ ε 1 ε 2 ε 3 2 sin 2 θ .
Such significant characteristics of crystals as the dispersion relation and damping of phonon-polaritons have been the subject of many both theoretical and experimental investigations in recent years. For instance, the experimental methods used in these studies include the impulsive SRS and transient grating experiments with femtosecond pulses [
Now we show that our equations are consistent with the experimental results presented in [
∂ ε 2 ∂ z = i 1 2 k s ( k s 2 − ω s 2 c 2 ε ˜ s ) ε 2 − i ω s 2 2 k s c 2 χ s ε 1 ε 3 * (15)
∂ ε 3 ∂ z = i 1 2 k p ( k p 2 − ω p 2 c 2 ε ˜ p ) ε 3 − i 1 2 k p ω p 2 c 2 χ p ε 1 ε 2 * (16)
where k s , p = ω s , p c n s , p , ε ˜ s , p are the dielectric constants, χ s , p are the nonlinear
coefficients. This system is the part of more complex system considered in [
expressed as A s , p e 1 2 G s z into (15) and (16) we get the expression for G as
G ≈ ν s n s 1 ε ″ ˜ p χ ″ 2 ε 1 2 , (17)
which is completely consistent with the expression of the gain given in [
Here ν s = ω s 2 π c ; n s is the index of refraction at Stokes frequency;
ε ″ p = ∑ f s f ν f 2 γ f ν p ( ν f 2 − ν p 2 ) 2 + γ f 2 ν p 2 ; s f is the oscillator’s strength of the dipole-ac-
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 [
χ ″ = ∑ f χ f ν f 2 γ f ν p ( ν f 2 − ν p 2 ) 2 + γ f 2 ν p 2 , (18)
where χ f = 1 8 π 3 1 ℏ c 1 ν l 2 ( s f σ f ν f ) 1 2 , . σ f is the effective cross-section of combinational scattering.
In calculations we used for G in LiNbO3 the following (see [
In this section we provide the analysis of the system (6)-(14) assuming that the all amplitudes of electromagnetic waves are real ( φ 1 , 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 ∂ t = 2 ℏ ( μ ε 3 + 2 α ′ ε 1 ε 2 ) P 2 , (19)
∂ P 2 ∂ t = − 1 2 ℏ ( μ ε 3 + 2 α ′ ε 1 ε 2 ) n , (20)
∂ ε 1 ∂ z + n 1 c ∂ ε 1 ∂ t = − 2 π ω 1 N c n 1 2 s ( + 2 ) P 2 ε 2 , (21)
∂ ε 2 ∂ z + n 2 c ∂ ε 2 ∂ t = − 2 π ω 2 N c n 2 ( − 2 s ( − 1 ) ) P 2 ε 1 , (22)
∂ ε 3 ∂ z + n 3 c ∂ ε 3 ∂ t = − 2 π ω 3 N c n 3 4 μ P 2 , (23)
where n 1 , 2 , 3 are linear indices of refraction n 1 , 2 , 3 ≈ ε ˜ 1 , 2 , 3 .
The solutions for n and P 2 can be expressed as
n = − cos ψ , P 2 = 1 2 sin ψ , (24)
where ψ = 1 ℏ ∫ − ∞ t ( μ ε 3 + 2 α ′ ε 1 ε 2 ) d t .
1) The system of Equations (19)-(23) includes both the classical case of resonant interaction of electromagnetic wave with two-level system (See [
2) The extreme case α ′ = 0 also results in standard “area theorem” for ε 3 (See [
d W 3 d z = − 8 π ℏ ω 3 N c n 3 ( 1 − cos Φ ) , (25)
where Φ = μ ℏ ∫ − ∞ + ∞ ε 3 ( z , t ) d t , W 3 ≡ ∫ − ∞ + ∞ ε 3 2 ( z , t ) d t .
3) In the absence of dispersion ( n 1 ≈ n 2 ≈ n 3 ≈ n ¯ ) the system (19)-(23) can be reduced to Sine-Gordon equation
∂ 2 ψ ∂ τ ∂ ς = − sin ψ (26)
where τ = c a ( t − z n ¯ c ) , ς = a z , a 2 = 4 π N ℏ c 2 n ¯ [ μ 2 ω 3 + α ′ C ] ,
ω 1 s ( + 2 ) ε 2 2 − ω 2 s ( − 1 ) ε 1 2 ≡ C (the given field approximation for ε 1 , 2 ).
4) If we introduce a new variable τ = t − z v ( ν is the speed of the pulse), we can reduce the system (19)-(23) to the equation of the motion of physical pendulum from the position of upper unstable equilibrium:
∂ 2 ψ ∂ τ 2 = 1 τ p 2 sin ψ (27)
where κ 1 , 2 , 3 − 1 = n 1 , 2 , 3 c − 1 v , τ p − 2 = 4 π N ℏ c [ − ω 3 κ 3 μ 2 n 3 + α ′ C ˜ ] ,
C ˜ ≡ − ω 1 κ 1 s ( + 2 ) n 1 ε 2 2 + ω 2 κ 2 s ( − 1 ) n 2 ε 1 2 (the given field approximation for ε 1 , 2 ).
5) We also provided the numerical solution of (19)-(23). To do that, we brought that system to unitless form:
∂ ε ˜ 1 ∂ z ˜ + n 1 ∂ ε ˜ 1 ∂ t ˜ = − C 1 ⋅ P 2 ε ˜ 2 , (28)
∂ ε ˜ 2 ∂ z ˜ + n 2 ∂ ε ˜ 2 ∂ t ˜ = C 2 ⋅ P 2 ε ˜ 1 , (29)
∂ ε ˜ 3 ∂ z ˜ + n 3 ∂ ε ˜ 3 ∂ t ˜ = − C 3 ⋅ P 2 , (30)
∂ P 2 ∂ t ˜ = − ( C 4 ⋅ ε ˜ 3 + C 5 ⋅ ε ˜ 1 ε ˜ 2 ) n , (31)
∂ n ∂ t = ( C 4 ⋅ ε ˜ 3 + C 5 ⋅ ε ˜ 1 ε ˜ 2 ) P 2 , (32)
where ε ˜ 1 , 2 , 3 = ε 1 , 2 , 3 / A ' 0 , t ˜ = t / τ 0 , z ˜ = z / z 0 , z 0 = c τ 0 , C 1 ≡ 2 π ω 1 N z 0 ( 2 s ( + 2 ) ) c n 1 , C 2 ≡ 2 π ω 2 N z 0 ( 2 s ( − 1 ) ) c n 2 , C 3 ≡ 2 π ω 3 N 4 μ z 0 c n 3 A 0 , C 4 ≡ τ 0 μ A 0 2 ℏ , C 5 ≡ α ′ τ 0 A 0 2 ℏ ;
τ 0 and A 0 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 ω 3 is shown in
t ∗ = t / t 0 ; N ≈ 10 19 cm − 3 , τ 0 ≈ 10 − 10 s , μ ≈ 10 − 18 esu ).
In this paper we have found the following:
1) The system of Equations (6)-(14) that models the processes of nonstationary 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 )) interaction (including “area theorem”) between the electromagnetic field and system;
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 wideband frequency converters.
Feshchenko, V. and Feshchenko, G. (2018) Nonstationary Stimulated Raman Scattering by Polaritons in Continuum of Dipole-Active Phonons. Journal of Applied Mathematics and Physics, 6, 405-417. https://doi.org/10.4236/jamp.2018.62038