Fig. 1. (a) Geometry of collective SL modes for two overlapped beams. (For generality, a nonzero wavelength difference of these two beams is assumed.) In the presented coordinate system, the xy plane is chosen to be the (k₀₁, k₀₂) plane with the x axis along the bisector of k₀₁ and k₀₂ and the z axis along k₀₁ × k₀₂. Beam I or beam II is said to be s-polarized when ±a_0α is along the s (z-axis) direction and to be p-polarized when ±a_0α∥p_α is located in the xy plane. Other linear polarization states of beam I or II are described by the polarization angle 90° ≥ β_α ≥ −90°, which is the angle from s to ±a_0α. (b) Relative orientation between the polarization directions of the two laser beams and the scattered light, where a_s is confined within the plane perpendicular to k_s (the polarization plane of the scattered light), and the angle between a_0α and this polarization plane is φ_α. On this polarization plane, e_∥ is defined as the unit vector along the projection of a₀₁, e_⊥ is a unit vector perpendicular to e_∥, and the angle between the projection of a₀₁ and a₀₂ is δ_⊥.

Fig. 2. κcU/I15 vs λ_B − λ₀ for SL modes of two overlapping beams with the same intensity (I₀₁ = I₀₂ and I₁₅ ≡ I₀₁/10¹⁵ W/cm²) and the same vacuum wavelength (λ₀ = 351 nm) at different crossing angles for (a) θ_s = 0°, φ_s = 0°, (b) θ_s = 0°, φ_s = 60°, (c) θ_s = 180°, φ_s = 60°, and (d) θ_s = 180°, φ_s = 0°. The plasma condition n_e = 0.06 n_c, T_e = 2.8 keV, T_e/T_i = 3.5, and zero flow velocity in a He plasma is taken.

Fig. 3. Upper and lower bounds of κc/κcU when β₁ and β₂ are varied at each out-of-plane angle φ_s. Two crossing laser beams with the same wavelength are assumed.

Fig. 4. κc/κcU vs φ_s for SL modes of two beams with (a) and (b) θ_h = 30° and (c) and (d) θ_h = 60°.

Fig. 5. Achievable κ_cl_amp/I₁₅w_b vs φ̃s for (a) β₁ = β₂ = 0, (b) β₁ = −β₂ = 45°, and (c) β₁ = β₂ = 90°, where φ̃s as the angle from the bisector of k₀₁ and k₀₂ (x direction) to n_s, is equal to φ_s for SL modes with θ_s = 0, and is equal to φ_s ± 180° for SL modes with θ_s = 180°. The condition λ₀ = 351 nm, n_e = 0.06 n_c, T_e = 2.5 keV, T_e/T_i = 3.5, and zero flow velocity for He plasma is taken.

Fig. 6. Possible directions of k_s for SL modes of two crossing beams with different wavelength differences for beam crossing angles (a) θ_h = 30° and (b) θ_h = 60°. The (k₀₁, k₀₂) plane corresponding to φ_s = 0 is indicated by the dashed curve, on which θ_s = 0 as marked by the black circle corresponds to the bisector direction of k₀₁ and k₀₂, and θ_s = θ_h as marked by the black diamond corresponds to the direction of k₀₂. The indicated angle α_⊥ from the (k₀₁, k₀₂) plane to the (k_s, k₀₂) plane can be used to denote different SL modes for specified Δλ₀ and θ_h. The example of a He plasma with conditions λ₀₁ = 351 nm, n_e = 0.06 n_c, T_e = 2.8 keV, T_e/T_i = 3.5, and zero flow velocity is taken.

Fig. 7. κcU vs λ_B − λ₀₁ for SL modes of two crossing beams with different wavelength differences and crossing angles. The contributions of beams I and II are shown by dashed and dotted curves, respectively, for the example of α_⊥ = 180° in (b). The example of a He plasma with conditions λ₀₁ = 351 nm, n_e = 0.06 n_c, T_e = 2.8 keV, T_e/T_i = 3.5, and zero flow velocity is taken.

Fig. 8. Maps of κ_cl_amp/w_bI₁₅ vs the direction of k_s for SL modes of two crossing beams with different combinations of β₁ and β₂ at (a)–(c) θ_h = 30° and (d)–(f) θ_h = 60°. The direction of view is taken along −ŷ, making the k_s loop for Δλ₀ = 0 appear as a unit circle on the maps. For θ_h = 30°, the k_s loops corresponding to Δλ₀ equal to 0, 0.2, 0.3, and 0.4 nm, which encircle the direction of k₀₂ (marked by the black diamonds), are shown by the cyan curves, while for θ_h = 60°, the k_s loops corresponding to Δλ₀ equal to 0, 0.2, 0.4, and 0.6 nm are displayed. The example of a He plasma with conditions λ₀₁ = 351 nm, n_e = 0.06 n_c, T_e = 2.8 keV, T_e/T_i = 3.5, and zero flow velocity is taken.