Fig. 1. Schematic of the chain selection rule for the proposed approach. Two laser pulses a₁(ω) and a₂(ω), irradiate a thin foil symmetrically at a large crossing angle 2θ, considering the normal direction of the target surface. High-order harmonics are emitted at different spatial locations at an angle α, which is determined by the conservation of energy and linear momentum through the chain selection rule. This chain selection rule is demonstrated by the phase-matching schemes (a) and (b).

Fig. 2. (a) Configuration of the PIC simulation box. The input laser field distribution before the lasers strike the target. (b) Electric field (E_z) distribution of the harmonics for two counter-rotation CP lasers after the lasers are reflected completely from the target, where the fundamental components are filtered out. The dashes denote the location of the target. The field is normalized to (3.2×10¹² V/m).

Fig. 3. The spectrum distribution of harmonics in k-space corresponding to that in Figure 2(b). Here 2ω₁₁ is the second harmonic in the direction normal to the target; 3ω₃₀, 3ω₂₁, 3ω₁₂, and 3ω₀₃ are the third harmonics emitted in different directions; 4ω₄₀, 4ω₃₁, 4ω₂₂, 4ω₁₃, and 4ω₀₄ are the fourth harmonics emitted in different directions; and 5ω₃₂ and 5ω₂₃ are the fifth harmonics emitted in different directions. The small white circles indicate the harmonics derived from the new phase matching selection rule Equation (4). The blue dashed lines indicate that same order harmonics in different directions have the same wavenumber.

Fig. 4. Electric field (E_z) distributions of the (a), (d) second harmonic (2ω₁₁), (b), (e) third harmonic (3ω₂₁), and (c), (f) fourth harmonic (4ω₃₁) in the (a)–(c) x–y plane at z = 0, and (d)–(f) are the section planes taken along the black dashed lines in (a)–(c).

Fig. 5. (a) Energy conversion efficiency for the harmonics in the reflected directions. (b) Energy conversion efficiency for the harmonics of the same order emitted at different angles.

Fig. 6. The spectrum distribution of harmonics in k-space after the lasers are reflected completely from the target, where the fundamental components are filtered out in the cases of (a) a₀ =10, (b) a₀=20, and (c) a₀=50.