Localized Dissipative Unipolar Objects under the Condition of Stimulated Raman Scattering

The possibility of the formation of dissipative unipolar soliton pulses in an amplifying medium of Raman-active molecules has been analyzed. It has been shown that the formation of such pulses is possible under the mutual compensation of Raman enhancement and irreversible losses caused by fast relaxation in the system of electron optical transitions. Since Raman enhancement is nonlinear, the threshold duration and energy of a soliton-like object being formed are determined by the parameters of the medium.


INTRODUCTION
Dissipative optical solitons now attract great attention in studies of nonlinear processes [1][2][3][4][5][6][7][8][9][10]. They are of both fundamental and applied interest. The mutual compensation of the energy entering a nonlinear medium and irreversible energy losses can result in the formation of dissipative solitons under certain conditions. The energy can be fed by various methods, e.g., from a continuous external source or an external pulse source, which can excite the nonlinear medium to a nonequilibrium state. In the latter case, the stored energy can further ensure its supply for the formation of dissipative solitons. If the duration τ p of a dissipative soliton and the observation time exceed the characteristic relaxation times T 2 and T 1 of the dipole moment and populations of stationary states, the nonequilibrium medium after the passage of the soliton becomes thermodynamically equilibrium.
The condition T 2 ≪ T 1 is usually valid in solids. The ratio T 2 /T 1 can vary from 10 -2 to 10 -5 [11]. Consequently, it is possible to ensure the interesting condition (1) Under this condition, the pulse entering the medium can induce the transition of the initially nonequilibrium medium to another nonequilibrium metastable state. This process is accompanied by the formation of localized soliton-like objects [9,12,13]. These objects were called incoherent solitons in [14,15] and soliton-like structures and soliton-like objects in nonequilibrium dissipative media in [12,13]. These localized objects are short-lived: their lifetime is always shorter than the relaxation time T 1 according to condition (1). Therefore, these objects can be observed in media and at quantum transitions where the relaxation time T 1 is long. The relaxation time T 1 at low temperatures is: , where ω tr is the transition frequency. Hence, it is necessary to use quantum transitions with low frequencies ω tr , e.g., electron vibrational (Raman) transitions corresponding to normal vibrational modes of molecules. The relaxation time T 1 for these transitions is fairly long. In particular, the relaxation time of the populations of Raman sublevels in liquid nitrogen is 56 s [16]. This giant T 1 value with a great margin satisfies inequalities (1). It is also important that Raman transitions are forbidden in the electron dipole approximation and are fundamentally two-photon.
When the spectrum of an optical pulse covers the forbidden electron vibrational transition at the frequency ω v , stimulated Raman self-scattering occurs [17,18]. In this case, the carrier frequency of the pulse is continuously redshifted in proportion to the traveled distance. The authors of [19,20] showed that this mechanism at sufficiently long distances can generate a single-cycle or even unipolar signal.
Stimulated Raman self-scattering processes, which promote the formation of unipolar signals, can be accompanied by irreversible losses of the pulse energy at other, e.g., electron optical, quantum transitions. This can lead to the formation of localized unipolar soliton-like objects at the initial nonequilibrium populations of vibrational sublevels of molecules under the dominance of stimulated Raman scattering processes. This work is devoted to the study of this problem.

BASIC EQUATIONS
The electric field E of an unipolar pulse with the duration τ p propagating in the z direction in an isotropic insulator containing Raman-active molecules satisfies the wave equation (2) Here, c is the speed of light in vacuum and P e and P R are the polarization electron optical and Raman responses, respectively.
The duration τ p ~ 10 -13 s certainly satisfies the inequality where s -1 is the characteristic frequency of electron optical transitions.
If the left inequality in Eq. (1) is satisfied, the polarization response of electron optical transition can be treated as linear in the electric field strength of the pulse and [13] (4) Here, χ and η are the noninertial and inertial components of the electric susceptibility of the medium, the latter being due to the phase relaxation of electron optical transitions. They can be estimated as [12,13] and , where d is the characteristic dipole moment of allowed transitions, n is the concentration of molecules responsible for the linear susceptibility χ, and ħ is the reduced Planck constant.
Under the described conditions, the populations of quantum levels involved in electron optical transitions hardly change and correspond to thermodynamic equilibrium of the electron optical subsystem.
The Raman polarization response is given by the standard expression [46] (5) where , α is the electric polarizability of a Raman-active molecule, q is the displacement of atoms in the molecule from the equilibrium position, and n R is the concentration of Raman-active molecules.
The dynamic parameters of a Raman transition at the frequency ω v satisfy the well-known equations [46] ( 6) where w is the population difference between the excited and ground Raman sublevels (w = 1 and -1 if the excited and ground sublevels are populated, respectively), M is the reduced mass of the molecule, and the relaxation terms are neglected because the duration of the pulse is much shorter than the relaxation times T 1R and T 2R for the Raman transition. It is noteworthy that the ground Raman sublevel coincides with the ground electron level in the molecule.
The frequency ω v ~ 10 12 s -1 and the characteristic durations τ p taken above satisfy the inequality (7) The spectral width of the unipolar pulse is δω 1 /τ p . Consequently, the inequality (7) should be considered as the condition mentioned above that the spectrum of the signal includes the Raman frequency. As a result, this inequality ensures the most favorable conditions for stimulated Raman self-scattering.
The second term on the left-hand side of the first equation in Eqs. (6) can be neglected under the condition (7). In this case, the solution of system (5) has the form [19,20] where (9) is the initial (at ) population difference between Raman sublevels, and .
Under the condition (7), the displacement q is negligibly small; i.e., q = 0 can be set with a high accuracy [19]. Then, according to Eqs. (5) and (8), The substitution of Eqs. (4) and (10) into Eq. (2) yields the equation (11) where is the noninertial part of the refractive index of the medium.
The two terms on the right-hand side of Eq. (11) are proportional to small parameters and . Under these conditions, the unidirectional propagation approximation for the pulse along the z axis at a velocity close to is applicable [22,24,25,27,30]. In this approximation, Eq. (11) takes the form (12) where (13) The last, diffusion, term on the right-hand side of Eq. (12) describes irreversible losses caused by phase relaxation on electron optical transitions. The first term describes a nonlocal nonlinear source caused by stimulated Raman self-scattering at the inverted initial populations of Raman quantum sublevels. Because of this term, Eq. (12) does not meet the general electric area conservation rule: = const [47]. However, this nonlinear diffusion term is relatively small in the approximation (7). Thus, the violation of the rule is a consequence of the taken approximation and does not mean a physical contradiction.
Multiplying Eq. (12) by E, using Eq. (9), and integrating with respect to t, we obtain (14) Nonlinear integrodifferential equations (12) and (14) describe the propagation of the pulse in the medium of Raman-active molecules in the presence of irreversible losses to electron optical quantum transitions.

SOLITON SOLUTION AND ITS ANALYSIS
An exact nontrivial analytical solution of Eq. (14), as well as Eq. (12), can hardly be found. For this reason, an approximate soliton-like solution is sought. To this end, it is desirable to approximate sin 2 (θ/2) in Eq. (14) by an appropriate polynomial. A Taylor series near θ = 0 is inappropriate for this purpose because the angular range 0 ≤ θ ≤ 4π/3 is of interest, as will be shown below. In this range, the approximation sin 2 (θ/2) ≈ F(θ) = pθ 2 -qθ 3 can be used with the constants p and g to be determined from the two conditions: (i) the polynomial reaches a maximum at the point θ m = π and (ii) this maximum is equal to unity. As a result, p = 3/π 2 and q = 2/π 3 . The resulting approximation in the angular range 0 ≤ θ ≤ 4π/3 has the form (15) It is seen in Fig. 1 that the approximation (15) in the angular range 0 ≤ θ ≤ 4π/3 is quite satisfactory.
In the absence of the last, integral, term, Eq. (16) is a reaction-diffusion equation. However, this term is important and cannot be omitted.
With the method considered in [9], the stationary solution of Eq. (16) can be found in the form of a propagating pulse (17) where , In Eqs. (17)- (20), the above expressions for a and b are used.
The electric field of the unipolar soliton-like pulse is expressed from Eqs. (17), (18), and (9) in the form (21) Here, the constants τ p and v are the duration and velocity of the pulse, respectively.
The substitution of Eq. (21) into the expression I = cE 2 /(4πn 0 ) for the signal intensity gives where . Figure 2 shows (a) the profile of the intensity (22) of the field of the unipolar pulse, as well as (b) the profile of the translational velocity of atoms (24) and (c) the profile of the population difference between the Raman sublevels (25), which are induced by this pulse.
A horizontal plateau appears in the velocity profile immediately after the passage of the pulse; it is easily explained in terms of the first equation of the system (6). Indeed, the term (restoring force) is neglected in this equation under the condition (7). Then, this equation has the form , which gives the mentioned plateau . The lifetime of the plateau can be estimated as . The term in the first equation of the system (6) becomes significant beginning with times after the passage of the pulse; this term is responsible for free oscillations corresponding to optical molecular modes.
In the absence of irreversible losses (for conservative solitons), the population difference w after the passage of the pulse returns to its initial value [25,30]. As seen in Fig. 2, under the effect of irreversible phase relaxation in the system of electron optical transitions, the population difference no longer returns to the initial value. In the central part of the pulse, Raman-excited molecules transit to the ground state. Immediately after the passage of the pulse in the medium, only about one-fourth of molecules on No. 1 2022 SAZONOV average return to the excited vibrational state. As a result, the average population difference becomes , which corresponds to another, metastable, state of molecules with the lifetime . Thus, the pulse, transferring the most part of nonequilibrium Raman-active molecules to the ground Raman sublevel takes the energy from the unit volume of the medium. This energy income is compensated by losses caused by irreversible phase relaxation in the equilibrium system of electron optical transitions.
The energy of the pulse proportional to (26) satisfies the equation (see Eq. (14) at ) Here, the first/second term corresponds to the energy supply to/loss from the pulse. The soliton-like object described above is due to the balance of these two processes. The further analysis is performed with the approximation given by Eq. (15) and under the assumption of approximate self-similarity, i.e., under the assumption that θ is given by Eq. (17) with the substitution   , , where is a certain function and τ 0 is the duration of the pulse generally different from τ p (see Eq. (19)). Summarizing, Eq. (27) is reduced to (28) where (29) (30) Figure 3 shows the plots of at τ 0 > τ c and τ 0 < τ c . In the former case, the point A = A 2 is an attractor if A > A 1 at the entrance to the medium. Here, At A < A 1 , the point A = 0 is an attractor.
In the latter case (τ 0 < τ c ), the point A = 0 is the only attractor. Thus, the found dissipative unipolar soliton-like object can be formed under two threshold conditions (32) These conditions are consistent with Eqs. (19) and (18), respectively, because τ p > τ c and . The substitution of τ 0 = τ p and Eq. (19) into Eq. (31) gives and . Consequently, the point given by Eq. (18) is an attractor in this case at A > A 1 . This property is important evidence of the stability of the dissipative unipolar object under consideration.
Thus, the unipolar pulse satisfying the conditions (32) should be formed at the entrance to the Raman-active medium. Then, it can be transformed in the medium to the pulse with the duration, electric field strength, and intensities given by Eqs. (19)-(21), respectively. Numerous methods to generate unipolar electromagnetic signals are known (see, e.g., reviews [45,48]). In our case, a significant part of the spectrum of the pulse with the duration τ p ~ 10 -13 s lies in the terahertz range. Consequently, this pulse can be referred to as a terahertz unipolar pulse. Such a pulse can be generated, e.g., by splitting of a bipolar terahertz signal into two unipolar pulses with opposite polarities [49]. One of these pulses under the conditions (32) can be used as an initial pulse at the entrance to the Raman-active medium.
Some numerical estimates are given below. For media with fast phase relaxation with the typical parameters [50] T 2 ~ 10 -13 s, ~ 10 15 s -1 , and χ ~ 0.1, we obtain ~ 10 -18 s. Then, 1 0 -27 s 2 /cm. Taking in addition cm 2 , s -1 , cm -3 , and g [20], we find cm -1 . As a result, the duration of the dissipative soliton-like object is s, as expected above. The ( ) Q A duration τ p should be several times longer than the relaxation time T 2 (see Eq. (1)). Note also that this duration of the pulse is much shorter than the phase relaxation time s [46] on the Raman transition in agreement with expectation.
The characteristic length of the pulse with the found duration τ p in the propagation direction is l || c τ p ~ 10 -3 cm. The relative difference of the velocity of the pulse from can be estimated as (see Eq. (20)). Thus, the velocity of the soliton-like pulse only slightly differs from the linear velocity . With the above parameters, we have cm 3 /(erg s). Then, Eq. (23) gives the maximum intensity I m ~ 10 12 W/cm 2 . Such high intensities are quite achievable in real experiments. The power of the pulse with the aperture d p ~ 1 mm is W, and its energy is W Ñ τ p~ 1 mJ.
The above parameters satisfy the inequality l || ≫ d p . The diffraction length under the considered conditions is estimated as cm. The onedimensional approximation under consideration can be used at these distances. Then, the observation time of the process under study is estimated as s. This estimate, as well as the estimates presented above for τ p and T 2 , certainly satisfies the condition (1).

CONCLUSIONS
To summarize, it has been shown that dissipative unipolar soliton pulses can be formed at the inverted initial populations of Raman sublevels. It is important that irreversible losses in the system of Raman sublevels are negligibly low at the taken observation time. These losses caused by phase relaxation are significant on other quantum transitions, e.g., electron optical transitions with the equilibrium populations of quantum states. The mutual compensation of these losses and the energy transferred from the nonequilibrium Raman subsystem makes it possible to form solitonlike pulses.
Since Raman enhancement is nonlinear, the dissipative soliton-like object is formed under the threshold conditions (32). At the same time, the initial state of the medium and its final metastable state to which it transits after the passage of the pulse are relatively stable because the expected observation time of the process of propagation of pulses is much shorter than the irreversible relaxation times in the system of Raman sublevels.
In agreement with expectation, such parameters of the considered dissipative object such as the amplitude, duration, and velocity are unambiguously deter-θ π 0 < < 4 /3