Transition from backward to sideward stimulated Raman scattering with broadband lasers in plasmas

The authors have no conflicts to disclose.
Conflict of Interest
Author Contributions
X. F. Li: Data curation (lead); Formal analysis (lead); Methodology (equal); Visualization (lead); Writing – original draft (lead). S. M. Weng: Conceptualization (lead); Supervision (equal); Validation (lead); Writing – review & editing (equal). P. Gibbon: Writing – review & editing (equal). H. H. Ma: Software (equal); Writing – review & editing (equal). S. H. Yew: Writing – review & editing (equal). Z. Liu: Data curation (equal); Writing – review & editing (equal). Y. Zhao: Data curation (equal); Writing – review & editing (equal). M. Chen: Writing – review & editing (equal). Z. M. Sheng: Conceptualization (equal); Supervision (equal); Validation (equal); Writing – review & editing (equal). J. Zhang: Writing – review & editing (equal).
The data that support the findings of this study are available from the corresponding author upon reasonable request.
In the coupling case, we can assume that δn(k ± Δk, ω ± Δω) ≃ δn(k, ω). Consequently, Eq. (A9) can be greatly simplified to give the following dispersion relation:ω2−3ve2k2−ωpe2=k2c2ωpe24(Π0+Π×),whereΠ0=a121D(k+k1,ω+ω1)+1D(k−k1,ω−ω1)+a221D(k+k2,ω+ω2)+1D(k−k2,ω−ω2),Π×=a1a21D(k+k1,ω+ω1)+1D(k−k1,ω−ω1)+1D(k+k2,ω+ω2)+1D(k−k2,ω−ω2),and a_i = eA_i/m_ec² and ωpe=4πe2n0/me.
In the mutual electromagnetic field of two laser beams,AL=A1cos(k1⋅x−ω1t)+A2cos(k2⋅x−ω2t)=12A1ei(k1⋅x−ω1t)+e−i(k1⋅x−ω1t)+12A2ei(k2⋅x−ω2t)+e−i(k2⋅x−ω2t),the three-wave coupling equations can be obtained from a Fourier analysis as(ω2−c2k2−ωpe2)As(k,ω)=4πe22meA1[δn(k+k1,ω+ω1)+δn(k−k1,ω−ω1)]+4πe22meA2[δn(k+k2,ω+ω2)+δn(k−k2,ω−ω2)],(ω2−3ve2k2−ωpe2)δn(k,ω)=n0e2k22me2c2A1[As(k+k1,ω+ω2)+As(k−k1,ω−ω1)]+n0e2k22me2c2A2[As(k+k2,ω+ω2)+As(k−k2,ω−ω2)].From Eq. (A2), we obtainAs(k+k1,ω+ω1)=4πe22meA1δn(k,ω)+A2δn(k+k1−k2,ω+ω1−ω2)D(k+k1,ω+ω1),where D(k,ω)=ω2−k2c2−ωpe2, and the terms δn(k + k₁ + k₂, ω + ω₁ + ω₂) and δn(k + 2k₁, ω + 2ω₁) are ignored as nonresonant terms. On defining Δk = k₁ − k₂ and Δω = ω₁ − ω₂, we can rewrite Eq. (A4) asAs(k+k1,ω+ω1)=4πe22meA1δn(k,ω)+A2δn(k+Δk,ω+Δω)D(k+k1,ω+ω1).Using the same method, we also obtainAs(k−k1,ω−ω1)=4πe22meA1δn(k,ω)+A2δn(k−Δk,ω−Δω)D(k−k1,ω−ω1),As(k+k2,ω+ω2)=4πe22meA2δn(k,ω)+A1δn(k−Δk,ω−Δω)D(k+k2,k+ω2),As(k−k2,ω−ω2)=4πe22meA2δn(k,ω)+A1δn(k+Δk,ω+Δω)D(k−k2,ω−ω2).Substituting Eqs. (A5)–(A8) into Eq. (A3), we obtain(ω2−3ve2k2−ωpe2)δn(k,ω)=n0e2k22me2c24πe22meΞ,whereΞ=A12δn(k,ω)D(k+k1,ω+ω1)+A12δn(k,ω)D(k−k1,ω−ω1)+A22δn(k,ω)D(k+k2,ω+ω2)+A22δn(k,ω)D(k−k2,ω−ω2)+A1A2δn(k+Δk,ω+Δω)D(k+k1,ω+ω1)+A1A2δn(k−Δk,ω−Δω)D(k−k1,ω−ω1)+A1A2δn(k−Δk,ω−Δω)D(k+k2,ω+ω2)+A1A2δn(k+Δk,ω+Δω)D(k−k2,ω−ω2).
It is worth noting that Π_× is the contribution caused by the coupling of these two beams. In the decoupling scenario, the density perturbation terms δn(k ± Δk, ω ± Δω) may cancel each other out. On the other hand, these terms may become ignorable as nonresonant terms in comparison with the term δn(k, ω). Therefore, Eq. (A11a) can be approximated asω2−3ve2k2−ωpe2=k2c2ωpe24Π0.Ignoring the temperature effect, we can combine Eqs. (A11a)–(A11c) and (A12) into the general form of Eqs. (5a)–(5c) in the theoretical model.
For two monochromatic laser beams at frequencies ω_1,2 = ω₀ ± 0.5δω, these two beams can be considered to be coupled when their instability regions with Γ > 0 overlap as shown in Fig. 1(b). Otherwise, the two laser beams can be considered to be decoupled.
In Fig. 6(a), the temporal evolution of the electric field is shown for a broadband laser beam with a relative bandwidth Δω/ω₀ = 4.0%. The Fourier transform of this electric field can perfectly reproduce the initially assumed flat-top frequency spectrum and random phase spectrum, as shown in Fig. 6(b). In our simulations, the laser beam is propagating along the x direction, and it has a Gaussian transverse profile in the y direction with a spot size w₀. Therefore, the spatial–temporal distribution of the electric field of a broadband laser can be finally expressed asE(t,y)=exp(−y2/w02)E(t),where E(t) is given by Eq. (B3). For a broadband laser beam with a relative bandwidth Δω/ω₀ = 4.0% and a spot size w₀ = 10λ, the spatial–temporal distribution of its electric field is shown in Fig. 6(c).
To precisely model the electric field of a broadband laser light, we have employed a novel method by taking the inverse Fourier transform of the amplitude–phase frequency spectrum.²⁵ Assuming that the broadband laser light has a continuous amplitude frequency spectrum f(ω), we can construct its amplitude–phase frequency spectrum F(ω) as the following complex function:F(ω)=f(ω)exp[iψ(ω)],where the phase–frequency spectrum ψ(ω) varying within −π < ψ < π is a random function of the frequency ω. Using this amplitude–phase frequency spectrum, we can obtain the electric field E(t) of a broadband laser in the time domain asE(t)=F−1[F(ω)],where F−1 denotes the inverse Fourier transform of a complex function. Actually, the electric field defined by Eq. (B3) is equivalent to that defined by Eq. (B1) if the number of monochromatic laser beams is chosen as N = TΔω, where T and Δω are the duration and bandwidth of the broadband laser beam, respectively.²⁵
In general, a broadband laser beam can be modeled as a summation of many monochromatic laser beams that have different carrier frequencies ω_i within a given bandwidth Δω as follows:^16,22–24E(t)=∑i=0NEicos(ωit+ψi),where E_i and ω_i are respectively the electric field amplitude and frequency of the ith monochromatic laser beam, and ψ_i is the random phase. If the number of monochromatic laser beams N is chosen arbitrarily, however, the frequency spectrum of the broadband laser electric field modeled in this way will deviate from the initially assumed distribution.²⁵