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Expressions are obtained for the shortened Maxwell’s equations simulating the evolution of the ultrashort pulses propagating in anisotropic dipole-active crystals in stimulated Raman scattering (SRS) by polaritons. The developed theory considers the case of cubic crystals which become anisotropic due to the deformation of the dielectric constant by the linearly polarized pump wave. The pump field is approximated by a linearly polarized plane electromagnetic wave. The possibility of simultaneous propagation of pulses on both different frequencies (pump and Stokes) and different polarization (simultons) is theoretically shown. It is also shown that the expression for the gain factor g in SRS is consistent with the experimental results for the spectra of ZnS.

In general, the pulses (including solitons) are randomly polarized. However, it is desirable to have pulses with a certain (better yet, predominantly defined polarization). It stands to a reason that the polarization could also be a carrier of the information about processes that took place during the interaction. Especially it is important in the case of optical fibers [_{4} and SrWO_{4} crystals are the most perspectives for SRS generation on both stretching and bending modes of internal anionic group vibrations with the strongest SRS pulse shortening under synchronous laser pumping. The significant progress in applying the methods of SRS and CARS (Coherent Anti-Stokes Raman Spectroscopy) in recent years was achieved in biology and medicine [

However, in our opinion, there is a need for studying the processes of the generation of solitons in nonlinear processes with predetermined polarization that would allow generating the stable ultrashort pulses traveling through the medium not only at different frequencies but different polarization (for example, the soliton at one frequency but with two perpendicular polarizations). Such creation of ultrashort stable pulses could significantly increase the resolution of SRS microscopy of imaging of the molecular vibrations. In this paper, we theoretically considered the formation of solitons of such type in transient SRS by polaritons [

In this paper, we carry out our analysis assuming that the pump field is a linearly polarized plane electromagnetic wave. It is also assumed that the nonlinear medium takes the form of a layer bounded by the planes z = 0 and z = L. The pump wave

E l ( r , t ) = e l A l exp [ i ( k l z z − ω l t ) ] + c . c . (1)

propagates along the z-axis. The subscripts l, s, and p henceforth denote the pump (laser), Stokes, and polariton wave fields; ω are the frequencies, n and k are the refractive indices and the wave vectors in the unpumped medium, and e are the real unit vectors of electromagnetic fields. The medium is nonmagnetic and transparent at frequencies ω l , s . We use the Stokes and polariton fields in the form

E s ( r , t ) = ∑ μ = 1 , 2 e s ( μ ) A s ( μ ) exp [ i ( k s ⋅ r − ω s t ) ] + c . c . , E p ( r , t ) = ∑ σ = 1 , 2 , 3 e p ( σ ) A p ( σ ) exp [ i ( W ⋅ r − ω p t ) ] + c . c . , (2)

where: e s ( μ ) ⊥ k s , e s ( 1 ) ⊥ e s ( 2 ) , k s = q s n s , q s = ω s / c , W = k l − k s , e p ( 1 , 2 ) ⊥ W , e p ( 1 ) ⊥ e p ( 2 ) , e p ( 3 ) = W W , ω p = ω l − ω s .

The longitudinal component of the Stokes wave can be neglected, but this cannot be done for the polariton wave in the phonon region. It has been shown in [

The fields E s , p are interrelated via the nonlinear part of the polarization P ( r , t ) . The latter quantity has at the frequencies ω l , s , p the following forms

P l = χ l μ σ A s ( μ ) A p ( σ ) exp [ i ( ( k s z + k p ( σ ) z ) z − ω l t ) ] + γ l μ μ ′ A s ( μ ) A s ( μ ′ ) * A l exp [ i ( k l z z − ω l t ) ] + c . c . P s = χ s μ σ A l A p ( σ ) ∗ exp [ i ( k l z − k p ( σ ) z ) z − ω s t ] + γ s μ μ ′ | A l | 2 A s ( μ ′ ) exp [ i ( k s z z − ω s t ) ] + c . c . P p = χ p μ σ A l A s ( μ ) ∗ exp [ i ( W z z − ω p t ) ] + c . c . (3)

where χ l μ σ = e l i e s ( μ ) j e p ( σ ) k χ l , i j k ( ω s , ω p ) , γ l μ μ ′ = e l i e s ( μ ) j e s ( μ ′ ) j e l m γ l , i j k m ( ω l , ω s , − ω s ) , χ s μ σ = e s ( μ ) i e l j e p ( σ ) k χ s , i j k ( ω l , − ω p ) , γ s μ μ ′ = e s ( μ ) i e s ( μ ) j e l k e l m γ s , i j k m ( ω p , ω l , − ω l ) .

The shortened equations for the amplitudes A l , s , p ( μ , σ ) are obtained from Maxwell’s equations by the standard procedure [

∂ A l ∂ z + 1 v l z ∂ A s ∂ t = i 2 π ω l c n l cos θ l z ( χ l μ σ A s ( μ ) A p ( σ ) + γ l μ μ ′ A s ( μ ) A s ( μ ′ ) * A l ) , (4)

∂ A s ( μ ) ∂ z + 1 v s z ( μ ) ∂ A s ( μ ) ∂ t = i 2 π ω s c n s ( μ ) cos θ s z ( μ ) ( χ s μ σ A l A p ( σ ) ∗ + γ s μ μ ′ | A l | 2 A s ( μ ′ ) ) , (5)

σ = 1 , 2 , 3

2 i W z ∂ A p ( σ ) ∗ ∂ z − i W e p ( σ ) z ∂ A p ( 3 ) ∗ ∂ z + i 2 ω p ε p ( σ ) ∗ c 2 ∂ A p ( σ ) ∗ ∂ t + ( W 2 − k p 2 ∗ ) A p ( σ ) ∗ = 4 π q p 2 χ p μ σ A s ( μ ) A l ∗ (6)

− i W ( e p ( 1 ) z ∂ A p ( 1 ) ∗ ∂ z + e p ( 2 ) z ∂ A p ( 2 ) ∗ ∂ z ) + i d A p ( 3 ) * d z ( W z − W e p ( 3 ) z ) + i 2 ω p ε p ( 3 ) ∗ c 2 ∂ A p ( 3 ) ∗ ∂ t − k p 2 * A p ( 3 ) * = 4 π q p 2 χ p μ 3 A s ( μ ) A l * , (7)

Note, that in (4) σ = 1 , 2 .

Given the strong absorption we have [

| W ( A p ( σ ) ) − 1 ∂ A p ( σ ) ∂ z | ≈ | ω p c 2 ( A p ( σ ) ) − 1 ∂ A p ( σ ) ∂ t | ≪ | W 2 − k p 2 * | ,

and we can, therefore, neglect in (6) and (7) the terms with the derivatives after which these equations yield

A p ( σ ) * = 4 π χ p μ σ A s ( μ ) A l * / ( μ 2 − ε p * ) , A p ( 3 ) * = − 4 π χ p μ 3 A s ( μ ) A l * / ε p * , μ = W / q p , σ = 1 , 2. (8)

Substituting the obtained expressions in (6) and (7), we arrive at a system of two differential equations for A l , s ( μ )

∂ A l ∂ z + 1 v l z ∂ A l ∂ t = i 2 π ω l c n l cos θ l z { γ ¯ l μ σ | A s ( μ ) | 2 A l + γ l μ μ ′ A s ( μ ) A s ( μ ′ ) * A l } , (9)

∂ A s ( μ ) ∂ z + 1 v s z ( μ ) ∂ A s ( μ ) ∂ t = i 2 π ω s c n s ( μ ) cos θ s z ( μ ) { γ ¯ s μ σ | A l | 2 A s ( μ ) + γ s μ μ ′ | A l | 2 A s ( μ ′ ) } , (10)

where

γ ¯ s μ σ = 4 π χ s μ σ ( χ p μ σ μ 2 − ε p * − χ p μ 3 ε p * ) , γ ¯ l μ σ = 4 π χ l μ σ ( χ p μ σ μ 2 − ε p − χ p μ 3 ε p ) , ε p = ε ′ p + i ε ″ p ,

ε ′ p = ε p 0 + ∑ f s f v f 2 ( v f 2 − v p 2 ) / [ ( v f 2 − v p 2 ) 2 + γ f 2 v p 2 ] ,

ε ″ p = − ∑ f s f v f 2 γ f v p / [ ( v f 2 − v p 2 ) 2 + γ f 2 v p 2 ] ,

s f is the oscillator strength of the o-f transition.

To do that we bring the system (9, 10) to unitless form.

∂ A ˜ l ∂ z ˜ + 1 v ˜ l z ∂ A ˜ l ∂ t ˜ = i β l { γ ˜ l μ σ | A ˜ s ( μ ) | 2 A ˜ l + γ ˜ l μ μ ′ A ˜ s ( μ ) A ˜ s ( μ ′ ) * A ˜ l } , (11)

∂ A ˜ s ( μ ) ∂ z ˜ + 1 ν ˜ s z ( μ ) ∂ A ˜ s ( μ ) ∂ t ˜ = i β s ( μ ) { γ ˜ s μ σ | A ˜ l | 2 A ˜ s ( μ ) + γ ˜ s μ μ ′ | A ˜ l | 2 A ˜ s ( μ ′ ) } . (12)

where v ˜ l z = v l z / c , A ˜ s ( μ ) = A s ( μ ) / A 0 , A ˜ l = A l / A 0 , z ˜ = z / z 0 , t ˜ = t / τ 0 , z 0 = c τ 0 , β s ( μ ) = 2 π ω s z 0 / ( c n s ( μ ) cos θ s z ( μ ) ) , γ ˜ s μ σ = γ ¯ s μ σ A 0 2 , γ ˜ s μ μ ′ = γ s μ μ ′ A 0 2 , v ˜ s z ( μ ) = v s z ( μ ) / c , β l = 2 π ω l z 0 / ( c n l cos θ l z ) , γ ˜ l μ σ = γ ¯ l μ σ A 0 2 , γ ˜ l μ μ ′ = γ l μ μ ′ A 0 2 , τ 0 is the characteristic time related to the laser field (pump).

∂ A ˜ l ∂ z ˜ + 1 v ˜ l z ∂ A ˜ l ∂ t ˜ = i β l { γ ˜ l 1 σ | A ˜ s ( 1 ) | 2 A ˜ l + γ ˜ l 2 σ | A ˜ s ( 2 ) | 2 A ˜ l + γ ˜ l 11 | A ˜ s ( 1 ) | 2 A ˜ l + γ ˜ l 12 A ˜ s ( 1 ) A ˜ s ( 2 ) * A ˜ l + γ ˜ l 21 A ˜ s ( 2 ) A ˜ s ( 1 ) * A ˜ l + γ ˜ l 22 | A ˜ s ( 2 ) | 2 A ˜ l } , (13)

∂ A ˜ s ( 1 ) ∂ z ˜ + 1 ν ˜ s z ( 1 ) ∂ A ˜ s ( 1 ) ∂ t ˜ = i β s ( 1 ) { γ ˜ s 1 σ | A ˜ l | 2 A ˜ s ( 1 ) + γ ˜ s 11 | A ˜ l | 2 A ˜ s ( 1 ) + γ ˜ s 12 | A ˜ l | 2 A ˜ s ( 2 ) } , (14)

∂ A ˜ s ( 2 ) ∂ z ˜ + 1 ν ˜ s z ( 2 ) ∂ A ˜ s ( 2 ) ∂ t ˜ = i β s ( 2 ) { γ ˜ s 2 σ | A ˜ l | 2 A ˜ s ( 2 ) + γ ˜ s 21 | A ˜ l | 2 A ˜ s ( 1 ) + γ ˜ s 22 | A ˜ l | 2 A ˜ s ( 2 ) } . (15)

We are looking for stationary solutions as

A ˜ s ( 1 , 2 ) ( z ˜ , t ˜ ) ≡ B s 1 , s 2 ( ξ ˜ ) e i Φ s 1 , s 2 ( ξ ˜ ) and A ˜ l ( z ˜ , t ˜ ) ≡ B l ( ξ ˜ ) e i Φ l ( ξ ˜ ) (16)

where ξ ˜ ≡ t ˜ − z ˜ / ν ˜ z ; ν ˜ z is the velocity of simultaneously propagating waves at the frequencies ω l , s ; B l , s 1 , s 2 and Φ l , s 1 , s 2 are the real amplitudes and phases of the waves, respectively. Such a standard procedure of presenting the complex amplitudes of waves in terms of real and imaginary parts results in duplication of the system of (13)-(15):

d B l d ξ ˜ = − β l κ ˜ l ( γ ˜ l 12 − γ ˜ l 21 ) B s 1 B s 2 B l sin Φ , (17)

d Φ l d ξ ˜ = β l κ ˜ l [ ( γ ˜ l 1 σ + γ ˜ l 11 ) B s 1 2 + ( γ ˜ l 2 σ + γ ˜ l 22 ) B s 2 2 + ( γ ˜ l 12 + γ ˜ l 21 ) B s 1 B s 2 cos Φ ] , (18)

d B s 1 d ξ ˜ = β s ( 1 ) κ ˜ s ( 1 ) γ ˜ s 12 B l 2 B s 2 sin Φ , (19)

d B s 2 d ξ ˜ = − β s ( 2 ) κ ˜ s ( 2 ) γ ˜ s 21 B l 2 B s 1 sin Φ , (20)

d Φ d ξ ˜ = [ β s ( 1 ) κ ˜ s ( 1 ) ( γ ˜ s 1 σ + γ ˜ s 11 ) − β s ( 2 ) κ ˜ s ( 2 ) ( γ ˜ s 2 σ + γ ˜ s 22 ) ] B l 2 + [ ( β s ( 1 ) κ ˜ s ( 1 ) γ ˜ s 12 B l 2 ) B s 2 B s 1 − ( β s ( 2 ) κ ˜ s ( 2 ) γ ˜ s 21 B l 2 ) B s 1 B s 2 ] cos Φ , (21)

where κ ˜ l ≡ v ˜ l v ˜ z / ( v ˜ z − v ˜ l z ) , κ ˜ s ( 1 , 2 ) ≡ v ˜ s z ( 1 , 2 ) v ˜ z / ( v ˜ z − v ˜ s z ( 1 , 2 ) ) , Φ ≡ Φ s 1 − Φ s 2 .

We are looking for the solitary (asymptotic) solution for solitons at ω l , s _{ }as following:

Q = B l 2 λ l 2 = B s 1 2 λ s 1 2 = B s 2 2 λ s 2 2 , (22)

where λ l 2 ≡ − β l κ ˜ l ( γ ˜ l 12 − γ ˜ l 21 ) , λ s 1 2 ≡ β s ( 1 ) κ ˜ s ( 1 ) γ ˜ s 12 and λ s 2 2 ≡ − β s ( 2 ) κ ˜ s ( 2 ) γ ˜ s 21 .

The introduction of Q allows to reduce the system of nonlinear equations (17)-(22) to

d Q d x = Q 2 sin Φ , (23)

d Φ d x = Q ( m ˜ + n ˜ cos Φ ) , (24)

where x = α ξ ˜ , α ≡ 2 λ l 2 λ s 1 λ s 2 , m ≡ [ β s ( 1 ) κ ˜ s ( 1 ) ( γ ˜ s 1 σ + γ ˜ s 11 ) − β s ( 2 ) κ ˜ s ( 2 ) ( γ ˜ s 2 σ + γ ˜ s 22 ) ] λ l 2

n ≡ [ ( β s ( 1 ) κ ˜ s ( 1 ) γ ˜ s l 12 ) λ s 2 λ s 1 − ( β s ( 2 ) κ ˜ s ( 2 ) γ ˜ s l 21 ) λ s 1 λ s 2 ] λ l 2 , m ˜ ≡ m / α , n ˜ ≡ n / α . The space-time evolution of the normalized intensities is shown in

Now we show that the system of Equations (14) and (15) is consistent with the experimental results presented, for example, in [

∂ A ˜ s ( μ ) ∂ z ˜ = i β s ( μ ) { γ ˜ s μ σ | A ˜ l | 2 A ˜ s ( μ ) + γ ˜ s μ μ ′ | A ˜ l | 2 A ˜ s ( μ ′ ) } , (25)

where A ˜ s ( μ ) = A s ( μ ) / A 0 , A ˜ l = A l / A 0 , z ˜ = z / z 0 , t ˜ = t / τ 0 , z 0 = c τ 0 , β s ( μ ) = 2 π ω s z 0 / ( c n s ( μ ) cos Θ s z ( μ ) ) , γ ˜ s μ σ = γ ¯ s μ σ A 0 2 , γ ˜ s μ μ ′ = γ s μ μ ′ A 0 2 , v ˜ s z ( μ ) = v s z ( μ ) / c , τ 0 is the characteristic time related to the laser field (pump).

The theoretical consideration of the gain factor for SRS by polaritons is based on the modeling of the quasi-stationary solutions of the coupled wave equations for the different polarizations of the Stokes (for the complete analysis see [

Δ μ μ ′ B μ ′ = − i κ B μ , μ = 1 , 2 (26)

where Δ μ μ ′ = β s ( μ ) | A ˜ l | 2 ( γ ˜ s μ μ ′ + γ ˜ s μ σ ) .

Equating the determinant of the system (26) to zero, we obtain the solutions for κ

κ = i [ ( Δ 11 + Δ 22 ) ± ( Δ 11 − Δ 22 ) 2 + 4 Δ 12 Δ 21 ] / 2 . (27)

We will need the explicit expressions for the tensors χ and γ . They can be found within the framework of the microscopic theory in the dipole approximation based on the perturbation theory states [

χ i j k ( ω l , − ω p ) = χ i j k ∘ ( ω l , − ω p ) + N ℏ ∑ f ν α i j ( f ν ) P f ν k F f ( ω p ) (28)

γ i j k m = 1 ℏ v 0 ∑ f ν α i k ( f v ) α j m ( f v ) F f ( ω p ) + γ i j k m ∘ , (29)

where F f ( ω ) ≈ 2 ω t / ( ω f 2 − ω p 2 + i γ ˜ ω p ) .

The summation in (28) and (29) is over all dipole-active phonons, the frequencies of which are equal ω f − i γ ˜ f / 2 , where γ ˜ f are the attenuation constants. For example in a cubic crystal, the dipole-active phonons are triply degenerate [

χ i j k = χ ′ i j k + i χ ″ i j k , χ ′ i j k = χ i j k 0 + ∑ f χ f v f 2 ( v f 2 − v p 2 ) / [ ( v f 2 − v p 2 ) 2 + γ f 2 v p 2 ] , χ ″ i j k = − ∑ f χ f v f 2 γ f v p / [ ( v f 2 − v p 2 ) 2 + γ f 2 v p 2 ] , γ i j k = γ ′ i j k + i γ ″ i j k , γ ′ i j k = γ i j k 0 + ∑ f γ f v f 2 ( v f 2 − v p 2 ) / [ ( v f 2 − v p 2 ) 2 + γ f 2 v p 2 ] , γ ″ i j k m = − ∑ f γ f v f 2 γ f v p / [ ( v f 2 − v p 2 ) 2 + γ f 2 v p 2 ] , (30)

where χ f = ( h c / 2 π ) − 1 / 2 v l − 2 ( s f σ f / v f ) 1 / 2 , γ f = ( 8 π 2 σ f ) / ( h c v l 4 v f ) , σ f is the Raman differential cross-section per unit cell v 0 (cm^{−1}/sr).

We introduce the principal axes of the tensor Δ μ μ 1 as a whole. If we denote its principal values as Δ μ we obtain from (26) κ μ = i Δ μ . Finally, we introduce the gain g μ = 2 R e κ μ which can be expressed as

g μ = 8 π 2 ω l ω s z 0 I l c 2 n l n s cos θ ( 4 π [ K ′ μ ε ″ p − K ″ μ ( s 2 − ε ′ p ) ( s 2 − ε ′ p ) 2 + ε ″ p 2 + L ′ μ ε ″ p + L ″ μ ε ′ p ε ′ p 2 + ε ″ p 2 ] − M μ ) , μ = 1 , 2 (31)

where K μ = ∑ σ = 1 , 2 ( χ μ σ ) 2 = K ′ μ + i K ″ μ , L μ = ( χ μ 3 ) 2 = L ′ μ + i L ″ μ , I l = c n l | A l | 2 / 2 π is the pump intensity, M μ are the principal values of the tensor γ s ( μ μ ′ ) ′ ′ , θ is the scattering angle (the angle between k l and k s ( n s ( μ ) ≈ n s , cos θ s z ( μ ) ≃ cos θ )).

Formula (31) denotes two gain coefficients for Stokes waves polarized along e s ( μ ) .

To verify (31), we were using the parameters of crystals widely used in optical display and storage, optical communication network, optical detection, etc. such as ZnO [^{−1}. The red dots represent the experimental points [

In this paper, we theoretically showed that in the case of nonstationary SRS by polaritons, there is a possibility of occurrence of simultaneously propagating ultrafast stable pulses (simultons) not only at different frequencies but with different polarizations as well. This can be used in optoelectronics when creating polarization filters.

The authors declare no conflicts of interest regarding the publication of this paper.

Feshchenko, V. and Feshchenko, G. (2021) Polarization Simultons in Stimulated Raman Scattering by Polaritons. Journal of Applied Mathematics and Physics, 9, 2193-2204. https://doi.org/10.4236/jamp.2021.99139