Fig. 1. Schematic scheme of linear and nonlinear diffraction in normal and conical incident configurations. (a) Pump beam shining on a linear dielectric grating in the in-plane normal incidence configuration. (b) Pump beam shining on a linear dielectric grating in the off-plane conical incidence configuration. (c) Pump beam shining on a PPLN nonlinear grating in the in-plane normal incidence configuration. The poling period of the PPLN sample is 6.97 µm. (d) Pump beam shining on a PPLN nonlinear grating in the off-plane conical incidence configuration. (e) Schematic experimental setup to study conical NRND.

Fig. 2. (a) Three-dimensional view of the polar angles and azimuthal angles for the transmitted FW (θ, φ₁) and the generated SHW (β, φ₂) laser beams when the pump laser beam is obliquely incident from air at an angle (α, φ₁) upon the PPLN nonlinear grating. k₁ and k₂ are the wave numbers of the transmitted FW laser and the generated SHW laser within the PPLN nonlinear grating, respectively. (b) Schematic illustration of the phase-matching conditions of the NRND SHG process in the PPLN crystal when the pump laser beam is obliquely incident upon the PPLN nonlinear grating.

Fig. 3. (a)–(d) Schematic illustration showing NRND phenomena at various cone-angle incidence by rotating the PPLN thin plate but keeping the pump laser beam incident along the Z^′ axis of laboratory coordinate system. (e)–(i) Experimental NRND patterns at different conical incident angles. Z^′_C-S is the distance between the crystal and the detection screen.

Fig. 4. Enlarged images of the experimental NRND patterns at different cone angles of incidence. For the distribution lines of diffraction points on the left and right sides, the green dashed line is the line connecting each order, and the white dashed line is the horizontal line relative to the laboratory coordinate system where the m = 0 order is located. (a₁)–(a₄) NRND spots located on the left-hand sides of the central spot under different incident angles (defined as negative diffraction orders). (b₁)–(b₄) NRND spots located on the right-hand sides of the central spot under different incident angles (defined as positive diffraction orders). Inset, expanded view of (b₄). ΔY is the deviation of each order of diffracted spot from the horizontal line of the m = 0 order in the Y^′ axis of the laboratory coordinate system, while d is the distance between each order of diffracted spot and the m = 0 order.

Fig. 5. Theoretical and experimental comparison of the diffraction angular distribution σ_m between multiple-order NRND diffraction spots and m = 0 order diffraction spots under various cone angles of incidence. A series of ordered NRND spots that correspond to the five nonlinear diffraction orders, including |m| = 1 (red), |m| = 2 (orange), |m| = 3 (yellow), |m| = 4 (light blue), and |m| = 5 (dark blue) in the neighborhood of the transmitted fundamental laser spot. (a) Negative NRND diffraction spots located on the left-hand sides of the central spot. (b) Positive NRND diffraction spots located on the right-hand sides of the central spot.

Fig. 6. Analysis of the deviation angle ψ_m of NRND diffracted spots at different diffraction orders from the horizontal line of the m = 0 order along the Y^′ axis of the laboratory coordinate system with different incident angles. (a) α = −10°, φ₁ = 0°; (b) α = −11.169°, φ₁ =−26.302°; (c) α = −14.106°, φ₁ = −44.561°; (d) α = −17.964°, φ₁ = −55.734°.

Fig. 7. Analysis of the diffraction spot powers of different NRND diffraction orders generated by the pump light passing through PPLN at different incident angles. (a) α = −10°, φ₁ =0°; (b) α = −11.169°, φ₁ = −26.302°; (c) α = −14.106°, φ₁ = −44.561°; (d) α = −17.964°, φ₁ = −55.734°.