Amplification and ellipticity enhancement of high-order harmonics in a neonlike x-ray laser dressed by an IR field

In Khairulin et al. [Sci. Rep. 12, 6204 (2022)] we proposed a method for amplifying a train of sub-femtosecond pulses of circularly or elliptically polarized extreme ultraviolet radiation, composed by high-order harmonics of an infrared (IR) laser field, in a neonlike active medium of a plasma-based x-ray laser, additionally irradiated with a replica of a fundamental frequency laser field used to generate harmonics. Here we present the analytical theory and numerical study of an amplification process of an individual harmonic with either preservation of its polarization state or ellipticity enhancement. We also analyze in detail the spectral-temporal and polarization properties of a sub-femtosecond pulse train formed by a set of elliptically or circularly polarized harmonics during the propagation through the medium. We discuss also the possibility of an experimental implementation of the suggested technique in an active medium of an x-ray laser based on neonlike ${\mathrm{Ti}}^{12+}$ ions irradiated by an IR laser field with a wavelength of 3.2 µm.

Figure 1

(a) Relevant energy levels of ${\mathrm{Ti}}^{12+}$ ions corresponding to the states $|1\rangle =|3p^1{S}_0,J=0,M=0\rangle , |2\rangle =|3s^1{P}_1,J=1,M=0\rangle , |3\rangle =|3s^1{P}_1,J=1,M=1\rangle$, and $|4\rangle =|3s^1{P}_1,J=1,M=-1\rangle$ (the states $|2\rangle , |3\rangle , |4\rangle$ are energy degenerate in the absence of the IR field). (b) Local-time dependence of the frequencies of inverted transitions $|1\rangle \leftrightarrow |2\rangle$ (blue solid curve) and $|1\rangle \leftrightarrow |3\rangle , |4\rangle$ (red dashed curve) calculated according to Eq. (4) under the action of the IR field with intensity $1\times {10}^{17}\mathrm{W}/\mathrm{c}{\mathrm{m}}^2$, which is assumed in Figs. 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17. The transition frequencies are normalized to the lasing frequency in the absence of the IR field, ${\omega }_{12}^{\left(0\right)}=\left({\mathrm{E}}_1^{\left(0\right)}-{\mathrm{E}}_{2,3,4}^{\left(0\right)}\right)/\hslash$, shown by black dotted curve. (c) Normalized IR field strength corresponding to panel (b).

Figure 2

(a) Dependences of (i) the wavelength of the modulating field for which the condition (8) is satisfied (blue solid curve, left vertical axis) and (ii) ionization rates from the resonant states of ${\mathrm{Ti}}^{12+}$ ions normalized to the half-width of the transition line in the absence of a field (other curves, right vertical axis) on the intensity of the modulating field. Red dash-dotted curve corresponds to the ionization rate from the $|1\rangle$ state, green dashed curve—from the $|2\rangle$ state, and the identical (overlapping) black and turquoise dashed curves—to the ionization rates from the $|3\rangle$ and $|4\rangle$ states. (b) The frequencies of the induced gain lines of the active medium, ${\omega }_k={\overline{\omega }}_{tr}^{\left(z\right)}+2k\mathrm{\Omega }$, for different $-14\le k\le 14$ under the condition (8) as a function of the intensity of the modulating field [as the intensity changes, the wavelength of the modulating field varies along the blue curve in Fig. 2]. The frequencies of the induced gain lines are normalized to the inverted transition frequency in the absence of the modulation, ${\omega }_{21}^{\left(0\right)}=\left({\mathrm{E}}_1^{\left(0\right)}-{\mathrm{E}}_{2,3,4}^{\left(0\right)}\right)/\hslash$. Bold dash-dotted black curve corresponds to the central frequency of the inverted transition for $z$-polarization component of the XUV field, ${\overline{\omega }}_{tr}^{\left(z\right)}$. Colored thin curves correspond to the sidebands in the gain spectrum. The colors are indicated in the figure legend.

Figure 3

Dependences of the effective gain coefficients (6b) and (7b) for the XUV radiation of $z$ polarization (a) and $y$ polarization (b), on the $k$ number, which corresponds to the detuning of the radiation frequency from resonance with a time-averaged $|1\rangle \leftrightarrow |2\rangle$ transition frequency. Effective gain factors are normalized to the gain in the absence of modulation, $g_{\mathrm{total}}$. Here $P_{\mathrm{\Omega }}^{\left(z\right)}\approx 12.57$ and $P_{\mathrm{\Omega }}^{\left(y\right)}\approx 13.57$.

Figure 4

Top panel: Space-time dependence of ellipticity, σ, of 153rd harmonic of the modulating field ($k=10$). At the entrance to the medium, $\sigma \left(x=0\right)=0.3$ ($y$-polarization dominates). Green solid and black dash-dotted curves show the positions of the envelope maxima of the $z$- and $y$-polarization components of the amplified radiation, respectively. Blue dashed curve shows the maximum of the total intensity of the polarization components. For visibility, we don't show the interval $0\le \tau \le 50$ fs, where the ASE dominates over the amplified signal. Bottom panel: Ellipticity of the XUV radiation at the maximum of the total intensity of its polarization components as a function of the length of the medium, $x$.

Figure 5

(a) Dependences of the intensities of the polarization components of the XUV field on the local time, τ, corresponding to Fig. 4, at different lengths of the medium: $x=0.1$ mm, 1.6 mm, 3.1 mm, 4.6 mm, and 6.1 mm. Red dash-dotted curve corresponds to the $y$-polarization component of the field, while blue solid curve shows the $z$ component. The inset shows the time dependences of ellipticity of the field at the same lengths of the medium. For a larger length, the peak of ellipticity shifts to an earlier time. The noiselike time dependence of the ellipticity at the very initial times is caused by the ASE. (b) Spectral amplitudes of the polarization components of the XUV field at the same propagation distances through the medium. Red dash-dotted curve and lower gain correspond to the $y$ component of the field; blue solid curve and higher gain correspond to the $z$ component. The curves are normalized to the peak amplitude of the Fourier transform of the $y$ component of the seeding radiation.

Figure 6

Top panel: Space-time dependence of ellipticity, σ, of 113th harmonic of the modulating field ($k=-10$). The incident radiation is circularly polarized, $\sigma \left(x=0\right)=1$. Green solid curve shows the positions of the maxima of the envelopes of the $z$- and $y$-polarized components of the field (they coincide on the scale of the figure). For visibility, we don't show the interval $0\le \tau \le 50$ fs, where the ASE dominates over the amplified signal. Bottom panel: Ellipticity of XUV radiation at the maximum of the total intensity of its polarization components as a function of the length of the medium, $x$.

Figure 7

(a) Dependences of the intensities of the polarization components of the XUV field on the local time, τ, corresponding to Fig. 6, at different lengths of the medium: $x=0$ mm (incident field), 2 mm, 4 mm, 6 mm, 8 mm, and 10 mm. Red dash-dotted curve corresponds to the $y$-polarization component of the field, while blue solid curve shows the $z$ component. The inset shows the time dependences of ellipticity of the field at the same lengths of the medium (with increasing medium length, the ellipticity decreases). The drop of ellipticity at $\tau \le 50$ fs is caused by the ASE of the medium. (b) Spectral amplitudes of the polarization components of the XUV field at the same propagation distances through the medium. Red dash-dotted curve and lower gain correspond to the $y$ component of the field; blue solid curve and higher gain correspond to the $z$ component. The curves largely overlap at all the lengths of the medium and are normalized to the peak amplitude of the Fourier transform of the $y$ component of the seeding radiation.

Figure 8

Top panel: Space-time dependence of the ellipticity, σ, of the set of the 107th, 109th, 111th, 113th, and 115th harmonics of the modulating field ($k=-13$, −12, −11, −10, and −9, respectively). The incident radiation is circularly polarized, $\sigma \left(x=0\right)=1$. Largely overlapping green and yellow curves show the positions of the maxima of the envelopes of the $z$- and $y$-polarization components of the field, respectively. Bottom panel: Ellipticity of each individual harmonic (harmonics are numbered by the index $k$) at the maximum of the total intensity of all harmonics as a function of the length of the medium, $x$.

Figure 9

Enlarged view of the fragment of the top panel of Fig. 8 corresponding to the time interval $502.7\mathrm{fs}\le \tau \le 508\mathrm{fs}$ equal in duration to the half-cycle of the modulating field. The vertical and horizontal axes are reversed. Green curve shows the shape of the pulses of the amplified radiation (time dependence of the intensity) at the entrance to the medium, $x=0$.

Figure 10

(a) Local time dependence of the total intensity of the amplified XUV radiation corresponding to Fig. 9 within a half-cycle of the IR field at different lengths of the medium: $x=0$ mm (incident field), 1.25 mm, 2.5 mm, 5 mm, 7.5 mm, and 10 mm. The time interval is chosen at the peak of the envelope of the total intensity at $x=10$ mm; see Fig. 12. With increasing length of the medium, the pulse duration grows from ${\tau }_{\mathrm{pulse}}=0.9$ fs at $x=0$ mm to ${\tau }_{\mathrm{pulse}}=1.3$ fs at $x=10$ mm. (b) Corresponding variation of ellipticity of the XUV field within an IR field half-cycle. With increasing medium length, the peak ellipticity decreases from $\sigma =1$ at $x=0$ mm to $\sigma =0.95$ at $x=10$ mm.

Figure 11

(a) Dependences of the intensities of the polarization components of the XUV field on the local time, τ, corresponding to Fig. 8, at different lengths of the medium: $x=0$ mm (incident field), 2 mm, 4 mm, 6 mm, 8 mm, and 10 mm. Red (light gray) curve corresponds to the $y$-polarization component of the field, while blue (dark gray) curve shows the $z$ component. The inset shows the time dependences of the ellipticity envelope of the field (the ellipticity at the maxima of the individual pulses) at the same lengths of the medium (with increasing medium length, the ellipticity decreases). The drop of ellipticity at $\tau \le 50$ fs is caused by the ASE. (b) Spectral amplitudes of the polarization components of the XUV field at the same propagation distances in the medium. Red dash-dotted curve and lower gain correspond to the $y$ component of the field; blue thin solid curve and higher gain correspond to the $z$ component. The curves largely overlap at all the lengths of the medium and are normalized to the peak amplitude of the Fourier transform of the $y$ component of the seeding radiation.

Figure 12

(a) Local time dependences of the total intensity (solid lavender curve, left vertical axis) and the ellipticity envelope (see the text; dash-dotted blue curve, right vertical axis) of the XUV pulse train, corresponding to Figs. 8, 9, 10, 11, amplified in 10 mm thick medium. (b) Same as in (a), but within a single half-cycle of the IR field at the maximum of the intensity envelope (in this case, the dash-dotted blue curve shows the instantaneous value of the ellipticity). The XUV intensity in (b) is normalized to its peak value, $I_{\max }=8.9\times {10}^8\mathrm{W}/\mathrm{c}{\mathrm{m}}^2$.

Figure 13

Top panel: Space-time dependence of ellipticity, σ, of the set of the 127th, 131st, 135th, 139th, and 143rd harmonics of the modulating field ($k=-3$, −1, 1, 3, and 5, respectively). At the entrance to the medium, $\sigma \left(x=0\right)=0.55$ ($z$ polarization dominates). Cyan dash-dotted and green dashed curves show the positions of the maxima of the envelopes of the $z$- and $y$-polarized components of the total harmonic field. Blue solid curve shows the maximum of the total intensity of both polarization components. Bottom panel: Ellipticity of each individual harmonic (harmonics are numbered by the index $k$) at the maximum of the total intensity (summed over all harmonics and both polarization components) as a function of the medium length, $x$.

Figure 14

Enlarged view of the fragment of the top panel of Fig. 13 corresponding to the time interval $443.5\mathrm{fs}\le \tau \le 448.9\mathrm{fs}$. The vertical and horizontal axes are reversed. Black curve shows the shape of the pulses of the amplified radiation (time dependence of the intensity) at the entrance to the medium, $x=0$.

Figure 15

(a) Dependences of the intensities of the polarization components of the XUV field on the local time, τ, corresponding to Fig. 13, at different lengths of the medium: $x=1$ mm, 3 mm, 5 mm, 7 mm, 9 mm, and 11.56 mm. Red (light gray) curve corresponds to the $y$-polarization component of the field, while blue (dark gray) curve shows the $z$ component. The inset shows the time dependences of the ellipticity envelope of the field at the same lengths of the medium (for a larger length, the peak of ellipticity shifts to an earlier time). The change in the ellipticity at the very initial times is caused by the ASE. (b) Spectral amplitudes of the polarization components of the XUV field at the same propagation distances in the medium. Red dash-dotted curve and higher gain correspond to the $y$ component of the field; blue thin solid curve and lower gain correspond to the $z$ component. The curves are normalized to the peak amplitude of the Fourier transform of the $z$ component of the seeding radiation.

Figure 16

(a) The shape of individual pulses from the pulse train shown in Fig. 15 at different lengths of the medium: $x=0$ mm (incident field), 1.4 mm, 2.75 mm, 5.5 mm, 8.25 mm, and 11.56 mm. The vertical and horizontal axes show the total intensity of the polarization components of the XUV field and the local time, τ, respectively. The time interval corresponds to the peak of the envelope of the total intensity at $x=11.56$ mm. With increasing length of the medium, the pulse duration grows from ${\tau }_{\mathrm{pulse}}=450$ as at $x=0$ mm to ${\tau }_{\mathrm{pulse}}=600$ as at $x=11.56$ mm. (b) Variation of ellipticity of the XUV field, corresponding to (a), within an IR field quarter-cycle. With increasing medium length, the peak ellipticity increases from $\sigma =0.55$ at $x=0$ mm to $\sigma =0.998$ at $x=11.56$ mm.

Figure 17

(a) Local time dependences of the total intensity (solid lavender curve, left vertical axis) and the ellipticity envelope (dash-dotted blue curve, right vertical axis) of the XUV pulse train, corresponding to Figs. 13, 14, 15, 16, amplified in 11.56 mm long medium. (b) Same as in (a), but within a single quarter-cycle of the IR field at the maximum of the intensity envelope (in this case, the dash-dotted blue curve shows the instantaneous value of the ellipticity). The XUV intensity in (b) is normalized to its peak value, $I_{\max }=6.96\times {10}^8\mathrm{W}/\mathrm{c}{\mathrm{m}}^2$.