Generation of a coherent second-harmonic beam from incoherent conical beams

Second-harmonic generation from incoherent conical beams is investigated. A theoretical description of an incoherent conical beam is provided. It is demonstrated that in the case of noncollinear phase matching, the spectrum of the second-harmonic beam narrows with the propagation in a nonlinear crystal. A simplified experiment with two intersecting fundamental beams is described. Numerical simulations of the nonlinear coupling equations are performed and good agreement with the theory is obtained.

Figure 1

Experimental setup. PDF: phase distortion film; A: aperture, L: lens; F: neutral filter.

Figure 2

Experimental angular spectra of the first (top left) and second (top right) fundamental beams as well as generated second-harmonic beam (bottom left). White line: Eq. (46). Bottom right: spectral lines for intersection planes rotated by the angle $\alpha$ with respect to $yz$ plane, Eq. (48).

Figure 3

Theoretically calculated spectra [Eq. (37)] of the second-harmonic beam. Bottom right: theoretically calculated spectral width of second-harmonic beam at $\theta =0$ ($\Delta {\theta }_x^{\mathrm{out}}$) and $\theta =\pi /2$ ($\Delta {\theta }_y^{\mathrm{out}}$). Type-I KDP crystal, fundamental wavelength ${\lambda }_1=1.064$ $\mu$m, ${\theta }_0^{\mathrm{out}}=2.7^{\circ }$, $\Delta \beta /k_0=2$ mrad, $\gamma =-28$ mrad. $z$: 200 $\mu$m (top left), 2 mm (top right), 1 cm (bottom left).

Figure 4

Numerically calculated spectra of the first (top left) and the second (top right) fundamental beams. Bottom: normalized spectra of the first (1) and the second (2) fundamental beams at ${\theta }_y^{\mathrm{out}}=0$. Type-I KDP crystal, fundamental wavelength ${\lambda }_1=1.064$ $\mu$m, ${\theta }_0^{\mathrm{out}}=2.7^{\circ }$, $\Delta \beta /k_0=2$ mrad, $d=700$ $\mu$m, $N=32$. One simulation.

Figure 5

Numerically calculated spectra of the second-harmonic beam. Type-I KDP crystal, fundamental wavelength ${\lambda }_1=1.064$ $\mu$m, ${\theta }_0^{\mathrm{out}}=2.7^{\circ }$, $\Delta \beta /k_0=2$ mrad, $\gamma =-28$ mrad. $z$: 200 $\mu$m (top left), 2 mm (top right), 1 cm (bottom left). Bottom right: numerically (gray lines) and theoretically (black lines and squares) calculated normalized spectra at ${\theta }_y^{\mathrm{out}}=0$, $z=200$ $\mu$m (1) and $z=1$ cm (2). $d=700$ $\mu$m, $L_n=20$ cm, $N=32$, average of 50 simulations.