Quasi-phase-matched high-harmonic generation in composites of metal nanoparticles and a noble gas

We theoretically study high-harmonic generation (HHG) in a composite which consists of ellipsoidal silver nanoparticles in argon. The significant field enhancement in argon in the vicinity of metal nanoparticles allows us to use much lower incident intensities than in typical HHG experiments. A periodic modulation of the nanoparticle concentration provides quasi–phase matching, which mitigates the negative effect of the significant phase mismatch. First, we study the linear optical properties of such a composite and the field enhancement and consider the technological possibilities of creating such a composite. Then the generation of high harmonics is simulated using a propagation equation which includes field enhancement, phase mismatch, absorption of the pump beam and harmonics, and other relevant effects. Generation of harmonics with an efficiency above ${10}^{-7}$ is predicted.

Figure 1

Scheme of the setup. The near-IR pump is indicated by the arrow, and the periodic distribution of NPs with period $\Lambda$ is shown inside the container. The periodic modulation of the NP concentration with the period corresponding to the phase mismatch is schematically shown in the bottom panel.

Figure 2

Dependence of the real part of the $n_{\mathrm{eff}}\left(\lambda \right)-1$ on wavelength for the ellipticities of 0, 1.37, 3.56, and 5.8 [solid (red), dashed (green), short-dashed (blue), and dotted (magenta) curves, respectively]. The argon pressure is 20 atm and the volume filling fraction of the NPs is 0.01.

Figure 3

Dependence of the loss in wavelength for ellipticities of 0, 1.37, 3.56, and 5.8 [solid (red), long-dashed (green), short-dashed (blue), and dotted (magenta) curves, respectively]. The argon pressure is 20 atm and the volume filling fraction of the NPs is 0.01.

Figure 4

Dependence of harmonic reabsorption on argon pressure for an NP volume fraction of 0, 0.01, and 0.1 (from bottom to top, red, green, and blue curves, respectively).

Figure 5

Dependence of the phase mismatch period on the harmonic number for silver NP with an ellipticity of 1.37 [top (red) curve] and 3.56 [bottom (green) curve] and a volume filling fraction of 0.01. The argon pressure is 20 atm.

Figure 6

Localization of NPs along the horizontal axis. The minimum diameter of NPs which can be localized by a standing wave with a given intensity is shown.

Figure 7

Field enhancement as a function of NP orientation with an ellipticity equal to 3.56.

Figure 8

Renormalized differential volume as a function of enhancement for an ellipticity of 1.38 [short-dashed (blue) and dotted (magenta) curves] and of 3.56 [solid (red) and long-dashed (green) curves]. The NP orientation is random (dotted and long-dashed curves) or oriented along the electric field (short-dashed and solid curves).

Figure 9

Dependence of the HHG efficiency for the 13th harmonic on the propagation length for the 10 $\mathrm{TW}/{\mathrm{cm}}^2$, 800-nm quasi-cw pump and a composite of 20-atm argon and silver NPs with an ellipticity of 1.37 and a volume filling fraction of 0.01. Curves describe non-phase-matched [bottom (blue) curve], quasi-phase-matched [second from the bottom (green) curve], and ideally matched [top (red) curve] propagation. In addition, the thin solid black curve is the quasi-phase-matched HHG efficiency with field inhomogeneity in the hot spot taken into account.

Figure 10

Dependence of the HHG efficiency on the harmonic number. The propagation length is fixed at 0.021 mm; other parameters are the same as in Fig. 8.

Figure 11

Dependence of the HHG efficiency for the 13th harmonic on the propagation length for the 10 $\mathrm{TW}/{\mathrm{cm}}^2$, 800-nm, 7-fs pump and a composite of 20-atm argon and silver NPs with an ellipticity of 1.37 and a volume filling fraction of 0.01. Inset: Temporal profile of the 13th-harmonic electric field relative to the peak pump field.

Figure 12

Dependence of the HHG efficiency for the 41st harmonic on the propagation length for the 10 $\mathrm{TW}/{\mathrm{cm}}^2$, 800-nm quasi-cw pump and a composite of 20-atm argon and silver NPs with an ellipticity of 3.56 and a volume filling fraction of 0.01. Curves describe non-phase-matched [bottom (green) curve] and quasi-phase-matched [top (red) curve] propagation.

Figure 13

Dependence of the HHG efficiency on the harmonic number. The propagation length is fixed at 0.01 mm; other parameters are the same as in Fig. 10.