Quasi-phase-matching of high-order-harmonic generation using polarization beating in optical waveguides

A scheme for quasi-phase-matching high-harmonic generation is proposed in which polarization beating within a hollow core birefringent waveguide modulates the generation of harmonics. The evolution of the polarization of a laser pulse propagating in a birefringent waveguide is calculated and is shown to periodically modulate the harmonic generation process. The optimum conditions for achieving quasi-phase-matching using this scheme are explored and the growth of the harmonic intensity as a function of experimental parameters is investigated.

Figure 1

Ellipticity beating for different incidence angles $\Theta$. From top to bottom: Dotted light-gray line shows $\Theta ={45}^{\circ }$, solid gray line shows $\Theta ={30}^{\circ }$ and $\Theta ={70}^{\circ }$; and dot-dashed gray line shows $\Theta ={20}^{\circ }$ and $\Theta ={80}^{\circ }$.

Figure 2

Top graph: (a) Variation of the harmonic generation efficiency, $f\left(z\right)=f\left[\varepsilon \right(z\left)\right]$, as a function $z$ for $q=27$ and $\Theta ={45}^{\circ }$. Bottom graph: (b) Real part of the of the $\widehat{x}$ component of the integrand (solid gray) and $\widehat{y}$ component (dashed black line). Assuming $L_b=2L_{c,y}$, $\Theta ={45}^{\circ }$ and $q=27$.

Figure 3

Relative HHG intensity for different phase-matching conditions assuming that $L_b=2L_{c,y}$, $q=27$, $\Theta ={20}^{\circ }$ for PBQPM. The dashed magenta line shows perfect phase-matching, dashed black line for perfect quasi-phase-matching, dot-dashed cyan line for no phase-matching. The solid blue line shows the relative amplitude squared for the $\widehat{x}$ component while the solid red line shows the relative amplitude squared for the $\widehat{y}$ component. The thick gray line shows the total HHG intensity for PBQPM.

Figure 4

PBQPM for $\Theta ={34}^{\circ }$, $q=27$, and $L_b=n{L}_{c,y}$ with $n=1,2,3,4$. The top graph shows the relative harmonic intensity as a function of propagation distance $z$ for perfect QPM (dashed black line), $L_b=4L_{c,y}$ (dot-dashed red line), $L_b=3L_{c,y}$ (solid green line), $L_b=2L_{c,y}$ (dashed blue line), and $L_b=L_{c,y}$ (dotted magenta line). The bottom four graphs show the $p=y$ polarization source term ${\Lambda }_y$ as a function of propagation distance $z$ for (b) $L_b=4L_{c,y}$, (c) $L_b=3L_{c,y}$, (d) $L_b=2L_{c,y}$, and (e) $L_b=L_{c,y}$.

Figure 5

PBQPM for $q=27$ and $L_b=2L_{c,y}$ with $\Theta ={30}^{\circ },{80}^{\circ },{45}^{\circ },{20}^{\circ }$. The top graph shows the relative harmonic intensity as a function of $z$ for perfect QPM (dashed black line) and coupling angles of $\Theta ={30}^{\circ }$ (dot-dashed red line), $\Theta ={80}^{\circ }$ (solid green line), $\Theta ={45}^{\circ }$ (dashed blue line), $\Theta ={20}^{\circ }$ (dotted magenta line). The bottom four groups show the $p=y$ polarization source term ${\Lambda }_y$ as a function of $z$ for (b) $\Theta ={30}^{\circ }$, (c) $\Theta ={80}^{\circ }$, (d) $\Theta ={45}^{\circ }$, and (e) $\Theta ={20}^{\circ }$.

Figure 6

Top graph: (a) Relative HHG intensity as a function of coupling angle $\Theta$, normalized to perfect QPM after one beat length where $L_b=2L_{c,y}$ for $q=15$ (solid red line), $q=27$ (dashed green line), $q=39$ (dotted blue line). Bottom graph: (b) Optimal angle $\widehat{\Theta }$ as a function of harmonic order $q$ for $L_b=2L_{c,y}$.