Resonant x-ray second-harmonic generation in atomic gases

We explore the x-ray second-harmonic generation process induced by resonant two-photon absorption in systems with inversion symmetry. We show that this process becomes allowed in the x-ray region due to nondipole contributions. It is found that, although a plane-wave pump field generates only a longitudinal second-harmonic field, a Gaussian pump beam creates also a radially polarized transverse second-harmonic field which is stronger than the longitudinal one. Contrary to the longitudinal component, the transverse second-harmonic field with zero intensity on the axis of the pump beam can run in free space. Our theory is applied to Ar and Ne atomic vapors and predicts an energy conversion efficiency of x-ray second-harmonic generation of $3.2\times {10}^{-11}$ and $1.3\times {10}^{-12}$, respectively.

Figure 1

The dipole moment ${\mathbf{r}}_{10}$ of the $1s\to np$ transition in the atom is parallel to $\mathbf{k}$. The axis of quantization $z$ is along the photon momentum $\mathbf{k}$.

Figure 2

Two-dimensional (2D) map of the pump intensity at $t=z/c$ for Ne and Ar atomic vapors. The legend shows the intensity in $\mathrm{W}/{\mathrm{cm}}^2$.

Figure 3

Radial distribution of $I_{\rho }$ for Ne atomic vapor at $z=0.02\mathrm{m}$. Black arrows show schematically the radially polarized SHG field.

Figure 4

Two-dimensional map of the SHG intensity at $t=z/c$ of Ne atomic vapor. The photoabsorption is negleted. The legend shows the intensity in $\mathrm{W}/{\mathrm{cm}}^2$.

Figure 5

Two-dimensional map of the SHG intensity at $t=z/c$ of Ar atomic vapor. The photoabsorption is neglected. The legend shows the intensity in $\mathrm{W}/{\mathrm{cm}}^2$.

Figure 6

Energy conversion efficiencies of the transverse SH fields ${\beta }_{\rho }\left(z\right)$ and the longitudinal SH fields ${\beta }_z\left(z\right)$ in Ne and Ar atomic vapors. The photoabsorption is neglected. The vertical axixes show the energy conversion efficiencies multiplied by the factor ${10}^n$: For example, ${10}^{10}{\beta }_{\rho }$ and ${10}^{14}{\beta }_z$ for Ne.

Figure 7

Distribution of the transverse SH field $I_{\rho }$ for Ne and Ar by taking into account the photoabsorption. The legend shows the intensity in $\mathrm{W}/{\mathrm{cm}}^2$.

Figure 8

Energy conversion efficiency ${\beta }_{\rho }\left(z\right)$ for Ne and Ar by taking into account the photoabsorption. The dashed lines display ${\beta }_{\rho }\left(z\right)$ calculated by neglecting the photoabsorption (see Fig. 6). Ne: ${\beta }_{\rho }^{\max }=3.2\times {10}^{-11}$ at $z_{\max }=0.7\mathrm{mm}$. Ar: ${\beta }_{\rho }^{\max }=1.3\times {10}^{-12}$ at $z_{\max }=0.5\mathrm{cm}$.