Efficient generation of high-order sum and difference frequencies in the xuv region by combining a weak longer-wavelength field

We demonstrate the efficient generation of high-order sum and difference frequencies in the extreme-ultraviolet (xuv) wavelength region both experimentally and theoretically. Experimentally, we combine a weak Nd:yttrium aluminum garnet (YAG) laser field $\left(1064\mathrm{nm}\right)$ with an intense Ti:sapphire laser field $(\sim 805\mathrm{nm})$. Besides the “pure” harmonics of the Ti:sapphire field, sum and difference frequencies of Ti:sapphire photons and up to two YAG photons are observed. Surprisingly, intensities of the sum and difference frequencies are comparable to or even higher than those of the pure harmonics even though the intensity of the YAG field is lower than that of the Ti:sapphire field by more than two orders of magnitude. Theoretically, we use the extended Lewenstein model to investigate high-order harmonic generation, sum-frequency generation, and difference-frequency generation using two-color laser fields. Theoretical calculations support our experimental observations and point out that choosing appropriate wavelengths for the weak combined field is crucial for the efficient generation of sum and difference frequencies. Furthermore, we discuss the effect of molecular alignment when ${\mathrm{N}}_2$ molecules are employed as a medium.

Figure 1

Harmonic spectra generated in ${\mathrm{N}}_2$ (a) without and (b, c) with YAG pulses. The polarization of the YAG pulses is (b) parallel and (c) perpendicular to that of Ti:sapphire pulses. The spectra are corrected for monochromator and EMT efficiencies. The intensity of the Ti:sapphire pulse and the effective intensity of the YAG pulse are $\sim 2\times {10}^{14}$ and $\sim 3\times {10}^{11}\mathrm{W}∕{\mathrm{cm}}^2$, respectively (see text for details).

Figure 2

The integrated intensities of pure harmonics (open squares), $+1$ peaks (open circles), and $-1$ peaks (open triangles), as a function of the central photon energy.

Figure 3

Calculated intensities of pure harmonics (solid circles), $+1$ sum frequencies (open squares), and $-1$ difference frequencies (solid triangles), as a function of the central photon energy. The calculations are performed for atomic nonlinear medium whose ionization potential corresponds to Ar. The intensities assumed for the Ti:sapphire and the YAG pulses are $1.1\times {10}^{14}$ and $1.4\times {10}^{11}\mathrm{W}∕{\mathrm{cm}}^2$, respectively (see text for details).

Figure 4

Calculated intensities of (a) 15th- and (b) 23rd-order sum (solid curves) and difference frequencies (dashed curves) as a function of the wavelengths of the weak combined field. The calculations are performed for atomic nonlinear medium whose ionization potential corresponds to Ar. The intensities assumed for the Ti:sapphire and the combined pulses are $1.1\times {10}^{14}$ and $1.4\times {10}^{11}\mathrm{W}∕{\mathrm{cm}}^2$, respectively.