Nonlinear Cherenkov radiation in rotatory nonlinear optics

Fig. 1. (a) The NCR process in optically rotatory and nonlinear optical crystals. The red and green cylinders with arrowheads represent the propagation of the fundamental and second harmonic beams, respectively. M_i represents dipoles in the propagation pathway of fundamental light, and the blue spheres represent the second harmonic of the dipole radiation at different locations. The second harmonic coherently superimposes along the Cherenkov angle θ_c. (b) The schematic diagram of the NCR angle with participation of the rotational angular velocity ${\vec{ρ}}_{1}$ . The gray dashed line represents the equi-phase plane during coupled wave propagation. The black solid line represents the second harmonic wavefront of linearly polarized light in the nonlinear Cherenkov radiation process. The blue and green dashed lines represent the second harmonic wavefronts in nonlinear Cherenkov radiation with the participation of the rotational angular velocity ${\vec{ρ}}_{1}$ along the optical axis. With the additional rotation phase $Δ φ_{\pm {\vec{ρ}}_{1}}$ in the coupled waves, the angle of the NCR has changed accordingly.

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Fig. 2. Different nonlinear Cherenkov phase matching processes with the rotational polarization direction. The optical rotation NCR processes are demonstrated by the wave vector superposition of the second harmonic wave ${\vec{k}}_{2} (θ_{c})$ , polarization wave ${\vec{k}}_{1}$ , and rotational angular velocity ${\vec{ρ}}_{i} (θ_{i})$ . Here, the different Cherenkov phase matching conditions can be expressed as (a) $| {\vec{k}}_{2} (θ_{c}) - {\vec{ρ}}_{2} (θ_{c}) | \cos θ_{c} = | 2 ({\vec{k}}_{1} - {\vec{ρ}}_{1}) |$ , (b) $| {\vec{k}}_{2} (θ_{c}) - {\vec{ρ}}_{2} (θ_{c}) | \cos θ_{c} = | 2 {\vec{k}}_{1} |$ , (c) $| {\vec{k}}_{2} (θ_{c}) - {\vec{ρ}}_{2} (θ_{c}) | \cos θ_{c} = | 2 ({\vec{k}}_{1} + {\vec{ρ}}_{1}) |$ , (d) $| {\vec{k}}_{2} (θ_{c}) | \cos θ_{c} = | 2 ({\vec{k}}_{1} - {\vec{ρ}}_{1}) |$ , (e) $| {\vec{k}}_{2} (θ_{c}) | \cos θ_{c} = | 2 {\vec{k}}_{1} |$ , (f) $| {\vec{k}}_{2} (θ_{c}) | \cos θ_{c} = | 2 ({\vec{k}}_{1} + {\vec{ρ}}_{1}) |$ , (g) $| {\vec{k}}_{2} (θ_{c}) + {\vec{ρ}}_{2} (θ_{c}) | \cos θ_{c} = | 2 ({\vec{k}}_{1} - {\vec{ρ}}_{1}) |$ , (h) $| {\vec{k}}_{2} (θ_{c}) + {\vec{ρ}}_{2} (θ_{c}) | \cos θ_{c} = | 2 {\vec{k}}_{1} |$ , and (i) $| {\vec{k}}_{2} (θ_{c}) + {\vec{ρ}}_{2} (θ_{c}) | \cos θ_{c} = | 2 ({\vec{k}}_{1} + {\vec{ρ}}_{1}) |$ .

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Fig. 3. Theoretically simulated Cherenkov second harmonic ring distributions under different phase matching types. The simulated optical rings are divided into two phase matching types: “oo-o” and “oo-e.” Each NCR process contains left circular polarization −ρ_i, right circular polarization ρ_i, and linearly polarization (ρ_i = 0) of the fundamental frequency wave and the second harmonic. (a) Theoretically simulated Cherenkov rings generated by the “oo-o” phase matching type. (b) Theoretically simulated Cherenkov rings generated by the “oo-e” phase matching type. (c) The final simulated SHG rings, containing the o- and e-polarized components of the second harmonic.

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Table 1. Cherenkov Radiation Angles of Polarized Waves and the Second Harmonic Waves with the Optical Rotation at the Fundamental Wavelength of 1030 nm

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Table 1. Cherenkov Radiation Angles of Polarized Waves and the Second Harmonic Waves with the Optical Rotation at the Fundamental Wavelength of 1030 nm

${\vec{v}}^{p}$
${\vec{v}}^{2 ω}$	$2 ω / 2 ({\vec{k}}_{1}^{o} - {\vec{ρ}}_{1})$	$ω / {\vec{k}}_{1}^{o}$	$2 ω / 2 ({\vec{k}}_{1}^{o} + {\vec{ρ}}_{1})$
$2 ω / [{\vec{k}}_{2}^{o} (θ_{c}) - {\vec{ρ}}_{2} (θ_{c})]$	11.648°	11.639°	11.631°
$2 ω / [{\vec{k}}_{2}^{o} (θ_{c})]$	11.665°	11.657°	11.648°
$2 ω / [{\vec{k}}_{2}^{o} (θ_{c}) + {\vec{ρ}}_{2} (θ_{c})]$	11.683°	11.674°	11.666°
$2 ω / [{\vec{k}}_{2}^{e} (θ_{c}) - {\vec{ρ}}_{2} (θ_{c})]$	11.718°	11.710°	11.701°
$2 ω / [{\vec{k}}_{2}^{e} (θ_{c})]$	11.736°	11.727°	11.719°
$2 ω / [{\vec{k}}_{2}^{e} (θ_{c}) + {\vec{ρ}}_{2} (θ_{c})]$	11.753°	11.745°	11.736°

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Zhongmian Zhang, Dazhi Lu, Haohai Yu, Huaijin Zhang, Yicheng Wu, "Nonlinear Cherenkov radiation in rotatory nonlinear optics," Chin. Opt. Lett. 23, 041901 (2025)

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