Wavelength-scaled laser filamentation in solids and plasma-assisted subcycle light-bullet generation in the long-wavelength infrared

We theoretically investigate mid- and long-wavelength infrared laser filamentation in solids, revealing extreme self-steepening and optical shock, multi-octave spanning supercontinuum generation, and, most importantly, subcycle light-bullet generation. Light-bullet generation in solids at long wavelengths depends critically on plasma, which is in sharp contrast to that in air where the arrest of collapse for light-bullet generation is dominated by radiation loss. Our method for subcycle light-bullet generation in solids at long wavelengths may open new possibilities for extreme ultrafast nonlinear optics including high-order-harmonic and attosecond-pulse generation.

Figure 1

Calculated optical-field ionization rates are shown in each graph as a function of intensity for the full Keldysh optical-field ionization rate (red dotted lines), the multiphoton ionization rate (black solid lines), and the tunneling ionization rate (blue dashed lines). Rates are shown for ZnSe at central wavelengths of (a) $\lambda =4$, (b) $\lambda =6$, (c) $\lambda =8$, and (d) $\lambda =10\mu \mathrm{m}$.

Figure 2

Calculated plasma densities resulting from a laser pulse with a Gaussian temporal profile and a FWHM pulse duration of 60 fs in ZnSe. Plasma is generated due to both optical-field ionization and collisional ionization. Plasma densities are calculated with the complete Keldysh ionization rate (labeled as black solid lines, full model) and with the multiphoton and tunneling ionization rates (labeled as red dashed lines, MPI-TI model). Input wavelengths $\lambda$ and input peak intensities $I_0$ are (a) $\lambda =4$, $I_0=5.7\times {10}^{11}$, (b) $\lambda =6$, $I_0=3\times {10}^{11}$, (c) $\lambda =8$, $I_0=1.8\times {10}^{11}$, and (d) $\lambda =10\mu \mathrm{m}$, $I_0=1\times {10}^{11}\mathrm{W}/{\mathrm{cm}}^2$. Laser-intensity profiles are shown in magenta and labeled on the right vertical axis.

Figure 3

(a) Peak intensities and (b) peak plasma densities vs propagation distance, for central wavelengths of $\lambda =4$ (black dashed lines), $\lambda =6$ (red solid lines), $\lambda =8$ (blue dotted lines), and $\lambda =10\mu \mathrm{m}$ (green solid lines with circles).

Figure 4

(a) Calculated energies in pulses at each wavelength during propagation and (b) the inverse bremsstrahlung cross section in ZnSe, ${\sigma }_B$.

Figure 5

Spatially averaged spectral profiles vs propagation distance are shown for (a) $\lambda =4$, (b) $\lambda =6$, (c) $\lambda =8$, and (d) $\lambda =10\mu \mathrm{m}$. The color bars show spectral intensity in dB.

Figure 6

Normalized on-axis spectral profiles at the input (red dashed lines) and at labeled positions of the shortest on-axis pulse duration (black solid lines) are shown with (a) $\lambda =4$, (b) $\lambda =6$, (c) $\lambda =8$, and (d) $\lambda =10\mu \mathrm{m}$.

Figure 7

Spatiotemporal intensity profiles at labeled propagation distances of shortest pulse duration are shown for (a) $\lambda =4$, (b) $\lambda =6$, (c) $\lambda =8$, and (d) $\lambda =10\mu \mathrm{m}$. The corresponding on-axis lineouts of the intensity profiles are shown for (e) $\lambda =4$, (f) $\lambda =6$, (g) $\lambda =8$, and (h) $\lambda =10\mu \mathrm{m}$, at the input (red dotted lines) and the position of shortest pulse duration (black solid lines). The color bars show intensity in $\mathrm{W}/{\mathrm{cm}}^2$, and on-axis pulse durations are labeled with arrows.

Figure 8

On-axis spectral profiles vs propagation distance are shown for (a) $\lambda =6$, (b) $\lambda =8$, and (c) $\lambda =10\mu \mathrm{m}$. The color bars show spectral intensity in dB, and arrows point to high-frequency components that may be due to resonant radiation. (d) Phase-matching calculations for resonant radiation for $\lambda =6\mu \mathrm{m}$ (red solid lines), $\lambda =8\mu \mathrm{m}$ (blue solid lines with circles) and $\lambda =10\mu \mathrm{m}$ (green dashed lines). Dashed vertical lines with colors matching fundamental wavelengths are shown for wavelengths that are phase matched for resonant radiation ($\mathrm{\Delta }{k}_{RR}=0$ rad/m).

Figure 9

On-axis temporal profiles vs propagation distance are shown for (a) $\lambda =4$, (b) $\lambda =6$, (c) $\lambda =8$, and (d) $\lambda =10\mu \mathrm{m}$. The color bars show intensity in $\mathrm{W}/{\mathrm{cm}}^2$.

Figure 10

(a) Spatially averaged FWHM pulse durations and (b) temporally averaged FWHM beam diameters vs propagation distance for $\lambda =4$ (black dashed lines), $\lambda =6$ (red circles), $\lambda =8$ (blue dot-dash lines), and $\lambda =10\mu \mathrm{m}$ (green solid lines).

Figure 11

Spatiotemporal intensity profiles are shown for $\lambda =8\mu \mathrm{m}$ at (a) $z=0$, (b) $z=0.2$, (c) $z=0.4$, (d) $z=0.5$, (e) $z=0.6$, and (f) $z=0.8$ cm. A light bullet persists from $z=0.4$ to $z=0.6$ cm. The color bars show intensity in $\mathrm{W}/{\mathrm{cm}}^2$, and spatially averaged pulse durations are labeled with arrows where light bullets propagate.

Figure 12

(a) Peak intensity vs propagation distance with $\lambda =8\mu \mathrm{m}$ without plasma (red dashed lines) and with plasma (black solid lines). Spatiotemporal intensity profiles without plasma are shown at (b) $z=0$, (c) $z=0.2$, (d) $z=0.4$, (e) $z=0.6$, and (f) $z=0.8$ cm. The color bars show intensity in $\mathrm{W}/{\mathrm{cm}}^2$.

Figure 13

(a) Spatially averaged FWHM pulse durations and (b) temporally averaged FWHM beam diameters, with plasma (black dashed lines) and without plasma (red dot-dashed lines). Calculations are performed with $\lambda =8\mu \mathrm{m}$ and an input power of $4P_{cr}$.

Figure 14

Spatially averaged spectral profiles vs propagation distance are shown with $\lambda =8\mu \mathrm{m}$ and an input power of $4P_{cr}$, (a) with plasma and (b) without plasma. The color bars show spectral intensity in dB.

Figure 15

Calculations of (a) refractive index, (b) GVD parameter, and (c) GVD parameter zoomed-in near zero GVD are shown with a plasma density of $\rho \sim 6\times {10}^{18}{\mathrm{cm}}^{-3}$ (black solid lines) and without plasma (red dashed lines). In (b) and (c), the normal-GVD regime appears above the green horizontal line (GVD parameter $>$ 0) and the anomalous-GVD regime appears below (GVD parameter $<$ 0).

Figure 16

(a) Peak intensities and (b) peak plasma densities vs propagation distance, with $\lambda =8\mu \mathrm{m}$ and input powers and input pulse durations of $4P_{cr}$, 60 fs (black dashed lines); $6P_{cr}$, 60 fs (red solid lines); $8P_{cr}$, 60 fs (blue dotted lines); and $4P_{cr}$, 120 fs (green solid lines with circles).

Figure 17

On-axis temporal profiles (left column) and spectral profiles (right column) are shown at the input and at propagation distances where the pulse duration is minimized, as labeled. For each calculation, $\lambda =8\mu \mathrm{m}$ and input powers and input pulse durations are (a),(e) $4P_{cr}$, 60 fs; (b), (f) $6P_{cr}$, 60 fs; (c),(g) $8P_{cr}$, 60 fs; and (d),(h) $4P_{cr}$, 120 fs. On-axis pulse durations are labeled with arrows.

Figure 18

(a) Spatially averaged FWHM pulse durations and (b) temporally averaged FWHM beam diameters vs propagation distance are shown at $\lambda =8\mu \mathrm{m}$ and input powers and input FWHM pulse durations of $4P_{cr}$, 120 fs (green solid lines); $4P_{cr}$, 60 fs (black dashed lines); $6P_{cr}$, 60 fs (red closed circles); and $8P_{cr}$, 60 fs (blue dot-dashed lines).

Figure 19

Spatiotemporal intensity profiles are shown for $\lambda =8\mu \mathrm{m}$ at labeled propagation distances with input powers and input FWHM pulse durations of (a)–(d) $6P_{cr}$, 60 fs; (e)–(h) $8P_{cr}$, 60 fs; and (i)–(l) $4P_{cr}$, 120 fs. The color bars show intensity in $\mathrm{W}/{\mathrm{cm}}^2$, and selected spatially averaged pulse durations are labeled with arrows.

Figure 20

Normalized on-axis intensity profiles are shown with $\lambda =8\mu \mathrm{m}$ and an input power of $4P_{cr}$ for various propagation distances as labeled: (a) $z=0$, (b) $z=0.1$, (c) $z=0.2$, (d) $z=0.3$, (e) $z=0.4$, (f) $z=0.5$, (g) $z=0.6$, (h) $z=0.7$, and (i) $z=0.8$ cm.

Figure 21

Normalized on-axis electric fields are shown for (a) the XFROG reference (Ref.) pulse, (b) $\lambda =8\mu \mathrm{m}$ pulse, $z=0.4$ cm, plasma on, (c) $\lambda =8\mu \mathrm{m}$ pulse, $z=0.6$ cm, plasma on, (d) $\lambda =8\mu \mathrm{m}$ pulse, $z=0.4$ cm, plasma off, and (e) $\lambda =8\mu \mathrm{m}$ pulse, $z=0.6$ cm, plasma off.

Figure 22

XFROG spectrograms (normalized) are shown for the 8 $\mu \mathrm{m}$ beam with (a) plasma on, $z=0.4$ cm, (b) plasma off, $z=0.4$ cm, (c) plasma on, $z=0.6$ cm, and (d) plasma off, $z=0.6$ cm. Here the delay axis corresponds to the offset between the reference pulse and the signal pulse (to be analyzed), where a delay of 0 fs means the reference field at 0 fs is overlapped with the signal field at 0 fs (see examples of signal fields in Fig. 21). More positive delays show wavelength content at the trailing edge of the pulse.

Figure 23

On-axis intensity profiles (carrier in magenta [light-gray] solid lines and envelope in black dotted lines) and on-axis plasma density (blue [dark-gray] solid lines) are shown at labeled propagation distances of maximum plasma density with (a),(e) $\lambda =4\mu \mathrm{m}$, $z=0.45$ cm, (b),(f) $\lambda =6\mu \mathrm{m}$, $z=0.35$ cm, (c),(g) $\lambda =8\mu \mathrm{m}$, $z=0.4$ cm, and (d),(h) $\lambda =10\mu \mathrm{m}$, $z=0.45$ cm.

Figure 24

On-axis normalized electric-field temporal profiles are shown at labeled propagation distances of maximum plasma density with central wavelengths of (a) $\lambda =4$, (d) $\lambda =6$, (g) $\lambda =8$, and (j) $\lambda =10\mu \mathrm{m}$. The temporal phase of each field is shown for each central wavelength in (b) $\lambda =4$, (e) $\lambda =6$, (h) $\lambda =8$, and (k) $\lambda =10\mu \mathrm{m}$. Corresponding XFROG spectrograms (normalized) are shown for central wavelengths of (c) $\lambda =4$, (f) $\lambda =6$, (i) $\lambda =8$, and (l) $\lambda =10\mu \mathrm{m}$. Here the delay axis corresponds to the offset between the reference pulse and the signal pulse (to be analyzed), where a delay of 0 fs means the reference field at 0 fs is overlapped with the signal field at 0 fs. More positive delays show wavelength content at the trailing edge of the pulse.

Figure 25

Peak intensities vs propagation distance are shown with collisional ionization and absorption (a)–(d) turned on and (e)–(h) turned off, and fractional density coefficient $\alpha$ set to (a),(e) $\alpha =1$, (b),(f) $\alpha =0.5$, (c),(g) $\alpha =0.1$, and (d),(h) $\alpha =0.05$. Results are shown for central wavelengths of $\lambda =4$ (black dashed lines), $\lambda =6$ (red solid lines), $\lambda =8$ (blue dotted lines), and $\lambda =10\mu \mathrm{m}$ (green solid lines with circles).

Figure 26

Peak plasma densities vs propagation distance are shown with collisional ionization and absorption (a)–(d) turned on and (e)–(h) turned off, and fractional density coefficient $\alpha$ set to (a),(e) $\alpha =1$, (b),(f) $\alpha =0.5$, (c),(g) $\alpha =0.1$, and (d),(h) $\alpha =0.05$. Results are shown for central wavelengths of $\lambda =4$ (black dashed lines), $\lambda =6$ (red solid lines), $\lambda =8$ (blue dotted lines), and $\lambda =10\mu \mathrm{m}$ (green solid lines with circles).

Figure 27

Maximum plasma densities vs wavelength are shown with fractional density coefficient $\alpha$ and collisional ionization and absorption turned on (solid red lines, stars) and turned off (dashed blue lines, solid circles) for (a) $\alpha =1$, (b) $\alpha =0.5$, (c) $\alpha =0.1$, and (d) $\alpha =0.05$.

Figure 28

Calculations are shown here with low molecular densities ($\alpha =0.05$). On-axis normalized electric-field temporal profiles are shown at labeled positions of maximum plasma density with central wavelengths of (a) $\lambda =4$, (d) $\lambda =6$, (g) $\lambda =8$, and (j) $\lambda =10\mu \mathrm{m}$. The temporal phase of each field is shown for each central wavelength in (b) $\lambda =4$, (e) $\lambda =6$, (h) $\lambda =8$, and (k) $\lambda =10\mu \mathrm{m}$. Corresponding XFROG spectrograms are shown for central wavelengths of (c) $\lambda =4$, (f) $\lambda =6$, (i) $\lambda =8$, and (l) $\lambda =10\mu \mathrm{m}$. Here the delay axis corresponds to the offset between the reference pulse and the signal pulse (to be analyzed), where a delay of 0 fs means the reference field at 0 fs is overlapped with the signal field at 0 fs.

Figure 29

An envelope-interpolated version of Fig. 7: spatiotemporal intensity profiles at labeled propagation distances of shortest pulse duration are shown for (a) $\lambda =4$, (b) $\lambda =6$, (c) $\lambda =8$, and (d) $\lambda =10\mu \mathrm{m}$. The color bars show intensity in $\mathrm{W}/{\mathrm{cm}}^2$.

Figure 30

An envelope-interpolated version of Fig. 9: on-axis temporal profiles vs propagation distance are shown for (a) $\lambda =4$, (b) $\lambda =6$, (c) $\lambda =8$, and (d) $\lambda =10\mu \mathrm{m}$. The color bars show intensity in $\mathrm{W}/{\mathrm{cm}}^2$.

Figure 31

An envelope-interpolated version of Fig. 11: spatiotemporal intensity profiles are shown for $\lambda =8\mu \mathrm{m}$ at (a) $z=0$, (b) $z=0.2$, (c) $z=0.4$, (d) $z=0.5$, (e) $z=0.6$, and (f) $z=0.8$ cm. A light bullet persists from $z=0.4$ to $z=0.6$ cm. The color bars show intensity in $\mathrm{W}/{\mathrm{cm}}^2$, and spatially averaged pulse durations are labeled with arrows where light bullets propagate.

Figure 32

An envelope-interpolated version of Fig. 12: spatiotemporal intensity profiles are shown with $\lambda =8\mu \mathrm{m}$ and an input power of $4P_{cr}$ at propagation distances of (a) $z=0$, (b) $z=0.2$, (c) $z=0.4$, (d) $z=0.6$, and (e) $z=0.8$ cm, as labeled. Here, plasma effects are turned off. The color bars show intensity in $\mathrm{W}/{\mathrm{cm}}^2$.

Figure 33

An envelope-interpolated version of Fig. 19: spatiotemporal intensity profiles are shown for $\lambda =8\mu \mathrm{m}$ at labeled propagation distances with input powers and input FWHM pulse durations of (a)–(d) $6P_{cr}$, 60 fs; (e)–(h) $8P_{cr}$, 60 fs; and (i)–(l) $4P_{cr}$, 120 fs. The color bars show intensity in $\mathrm{W}/{\mathrm{cm}}^2$, and selected spatially averaged pulse durations are labeled with arrows.