All-Regions Tunable High Harmonic Enhancement by a Periodic Static Electric Field

Simulations show that a static electric field periodically distributed in space can be used to control the production of coherent light by high-order harmonic generation in a wide spectral range covering extreme-ultraviolet and soft x-ray radiation. The radiation yield is selectively enhanced due to symmetry breaking induced by a static electric field on the interaction between the driving laser and the medium. The spectral position of the enhancement is tuned by varying the periodicity of the static electric field which matches twice the coherence length of the harmonics in the desired region. We find that the static electric field strength inducing enhancement decreases for shorter wavelengths and predict an increase of more than two orders of magnitude for harmonics in the water window spectral range with a static electric field as weak as $1.12\mathrm{MV}/\mathrm{cm}$.

Figure 1

(a) Focused laser pulse linearly polarized in $x$ direction propagating in a medium with a periodically distributed static electric field also oriented in the $x$ direction. (b) Implementation of the periodic electric field: A pulsed ${\mathrm{CO}}_2$ laser is collimated and sent through a small wedge with a binary amplitude mask whose amplitude pattern generates the desired periodic field modulation after each multiple of the Talbot distance $z_T$. The wedge ensures equal field strengths of the periodic pattern.

Figure 2

Spectral power of the harmonics (radially integrated spectral energy—in Joules per photon energy unit—in eV) versus photon energy near the cutoff region after a propagation distance of 2 mm with $E_0\approx 726\mathrm{MV}/\mathrm{cm}$ and $\varphi =0$. The black line corresponds to $\beta =0$, and the other lines show the calculations for a static electric field strength such as $\approx 7.26\mathrm{MV}/\mathrm{cm}$ ($\beta =0.01$) spatially distributed along the $z$ direction, with periodicities ${\Delta }_s=100\mu \mathrm{m}$, $125\mu \mathrm{m}$, $150\mu \mathrm{m}$, $200\mu \mathrm{m}$, and $300\mu \mathrm{m}$ as indicated. The colored areas indicate the photon yield enhancement with respect to $\beta =0$ with a maximum efficiency tuned by ${\Delta }_s$.

Figure 3

Evolution of the spectral power for different harmonics as a function of the propagation distance $z$ with $\beta =0.01$ (a). The dynamics with no static electric field ($\beta =0$) is shown in (b), from where the coherence length $L_{c_q}$ is evaluated.

Figure 4

Spectral power of the harmonics (radially integrated spectral energy—in Joules per photon energy unit—in eV) versus photon energy near the cutoff region after a propagation distance of 0.3 mm with a static electric field periodicity such that ${\Delta }_s=32\mu \mathrm{m}$. In (a) the carrier-envelope phase of the driving field is $\varphi =0$; the dotted black line corresponds to $\beta =0$ (absence of static electric field), and the violet full line shows the calculations with $\beta =7.5\times {10}^{-4}$ ($1.12\mathrm{MV}/\mathrm{cm}$). In (b) the carrier-envelope phase of the driving field is $\varphi =\pi /2$, and the other parameters are as in (a), as indicated.

Figure 5

The spectral power obtained with $\beta =3\times {10}^{-4}$ ($450\mathrm{kV}/\mathrm{cm}$) is compared to the case with no electric filed ($\beta =0$). The initial phase is $\varphi =\pi /2$ and the other parameters are as in Fig. 4. The enhanced region results highly localized at about 580–600 eV.