Effects of higher-order nonlinear processes on harmonic-generation phase matching

We experimentally and theoretically reinvestigate the significant third-harmonic enhancement via plasma generation using time-resolved methods with two-color, copropagating femtosecond laser pulses. By measuring plasma densities with digital in-line holography and using carrier-resolved unidirectional pulse propagation equations to model the experiment, we show that a fifth-order nonlinearity contributes to third-harmonic generation mainly via phase matching. In particular, our study confirms that self- and cross-phase modulation including higher-order nonlinear processes can play a crucial role in harmonic-generation phase matching, which has recently been proposed by Weerawarne $etal.$ [Phys. Rev. Lett. 114, 093901 (2015)] and should be potentially applicable to high-order harmonic generation.

Figure 1

(a) Pump-probe experimental setup used for harmonic energy measurements. BS: beam splitter; M1, M2, M3, M4: mirrors; TS: translational stage; $\lambda /2$: half-wave plate; L1, L2: $f=20$-cm lenses; DM: dichroic mirror; CM: curved mirror (for collimation); GR: grating; FLT: filters; PMT/SPEC: photomultiplier tube or fiber spectrometer. (b) Digital in-line holography setup used for plasma density measurements. M5: mirror; L3, L4: $f=10$-cm and $f=30$-cm lenses ($4f$ imaging); CCD: charge-coupled device camera communicating with computer.

Figure 2

(a), (b) Measured TH energy vs delay between harmonic driver and control beams. The harmonic driver energy is fixed at $70\mu \mathrm{J}$ and the control beam energy is set to 70, 150, and $200\mu \mathrm{J}$. ($\parallel$) - parallel polarization, ($\perp$) - perpendicular polarization. (c) Measured spectra of TH with 70 and 200-$\mu \mathrm{J}$ control beam energies [($\parallel$) configuration], at $+500$-fs delay. Spectra are normalized to the peak value at a $200\text{−}\mu \mathrm{J}$ control beam energy. In (c), the amplitude with the harmonic driver only is increased by $\times 10$ for visual clarity.

Figure 3

(a)–(d) Third-harmonic mode profiles with a $150\text{−}\mu \mathrm{J}$ control beam energy and different delays, as labeled. The CCD gain for image (a) has been increased by $\times 112$ for visual clarity. The harmonic driver mode is shown at 5 mm after the focus with (e) the harmonic driver only, (f) at zero delay, and (g) at +5-ps delay.

Figure 4

Calculations of TH energy (at 5 cm after the geometrical focus) at various delays with an input energy of $70\mu \mathrm{J}$ for the 1180-nm harmonic driver, and 70, 150, or $200\mu \mathrm{J}$ for the 800-nm control beam. For each calculation $n_2=1.2\times {10}^{-19}{\mathrm{cm}}^2/\mathrm{W}$, and (a) $n_4=0.5\times {10}^{-33}$, (b) $n_4=0.5\times {10}^{-33}$ (with delayed Raman response), (c) $n_4=0$, and (d) $n_4=-1.0\times {10}^{-33}{\mathrm{cm}}^4/{\mathrm{W}}^2$.

Figure 5

Calculated spatiospectral profiles at 5 cm after the geometrical focus, with a control beam energy of $150\mu \mathrm{J}$ and a harmonic driver energy of $70\mu \mathrm{J}$. Here $n_2=1.2\times {10}^{-19}{\mathrm{cm}}^2/\mathrm{W}$ and $n_4=0.5\times {10}^{-33}{\mathrm{cm}}^4/{\mathrm{W}}^2$. Profiles are shown for delays of (a) $-200$ and (b) 200 fs. The harmonic driver, TH, and control are labeled at 1180, 393, and 800 nm, respectively. In positive delay, the harmonic driver signal spreads radially due to plasma and the TH signal is enhanced.

Figure 6

(a)–(c) show calculated peak plasma densities vs propagation distance with $n_2=1.2\times {10}^{-19}{\mathrm{cm}}^2/\mathrm{W}$ and $n_4=0.5\times {10}^{-33}{\mathrm{cm}}^4/{\mathrm{W}}^2$ for 70-, 150-, and $200\text{−}\mu \mathrm{J}$ control beam energies, respectively, at zero delay between the control beam and the harmonic driver beam. Experimentally measured plasma densities using ILH and Abel inversion are shown at zero delay between the control and harmonic driver beams with control beam energies of (d) 200, (e) 150, and (f) 70 $\mu \mathrm{J}$. Plasma distributions are shown for zero delay and a delay of 1584 fs between the ILH probe and the harmonic driver or control beam. The color bar indicates plasma density in ${\mathrm{cm}}^{-3}$.

Figure 7

Two intensity autocorrelation trace lineouts are shown, which are used to measure the FWHM pulse durations of the control beam and the harmonic driver. Deconvolution yields a FWHM pulse duration of 65 fs for the control beam and 125 fs for the harmonic driver. The inset displays the traces as recorded on the CCD camera, where (a) and (b) correspond to the control beam and harmonic driver, respectively.

Figure 8

Calculations of TH energy at various delays with an input energy of $70\mu \mathrm{J}$ for the 1180-nm harmonic driver, and 70, 150, or $200\mu \mathrm{J}$ for the 800-nm control beam. For each calculation, $n_2=0.8\times {10}^{-19}{\mathrm{cm}}^2/\mathrm{W}$ and (a) $n_4=0.3\times {10}^{-33}$, (b) $n_4=0.1\times {10}^{-33}$, (c) $n_4=0$, and (d) $n_4=-0.5\times {10}^{-33}{\mathrm{cm}}^4/{\mathrm{W}}^2$. The TH energy is calculated at 5 cm after the geometrical focus.

Figure 9

Calculations of TH energy at various delays with an input energy of $70\mu \mathrm{J}$ for the 1180-nm harmonic driver, and 70, 150, or $200\mu \mathrm{J}$ for the 800-nm control beam. For each calculation, $n_2=2.5\times {10}^{-19}{\mathrm{cm}}^2/\mathrm{W}$ and (a) $n_4=0.5\times {10}^{-33}$, (b) $n_4=1.0\times {10}^{-33}$, (c) $n_4=0$, and (d) $n_4=-1.5\times {10}^{-33}{\mathrm{cm}}^4/{\mathrm{W}}^2$. The TH energy is calculated at 5 cm after the geometrical focus.

Figure 10

Calculated third-harmonic energy vs propagation distance for $n_2=1.2\times {10}^{-19}{\mathrm{cm}}^2/\mathrm{W}$ and (a) $n_4=0.5\times {10}^{-33}{\mathrm{cm}}^4/{\mathrm{W}}^2$ at $-200$ fs delay, (b) $n_4=0.5\times {10}^{-33}{\mathrm{cm}}^4/{\mathrm{W}}^2$ at 200 fs delay, (c) $n_4=0{\mathrm{cm}}^4/{\mathrm{W}}^2$ at $-200$ fs delay, and (d) $n_4=0{\mathrm{cm}}^4/{\mathrm{W}}^2$ at 200 fs delay. The input energy of the 1180-nm harmonic driver was fixed at $70\mu \mathrm{J}$, and the energy of the 800-nm control beam was set to 70, 150, or $200\mu \mathrm{J}$. The geometrical focus of the harmonic driver is at 6 cm.