Long-range propagation of ultrafast ionizing laser pulses in a resonant nonlinear medium

We study the propagation of 0.05–1 TW power, ultrafast laser pulses in a 10-m-long rubidium vapor cell. The central wavelength of the laser is resonant with the $D_2$ line of rubidium, and the peak intensity is in the ${10}^{12}$–${10}^{14}\mathrm{W}/{\mathrm{cm}}^2$ range, enough to create a plasma channel with single-electron ionization. We observe the absorption of the laser pulse for low energy, a regime of transverse confinement of the laser beam by the strong resonant nonlinearity for higher energies and the transverse broadening of the output beam when the resonant nonlinearity ceases due to the valence electrons all being removed during ionization. We compare experimental observations of the transmitted pulse energy and the transverse fluence profile with the results of computer simulations modeling pulse propagation. We find a qualitative agreement between theory and experiment that corroborates the validity of our propagation model. While the quantitative differences are substantial, the results show that the model can be used to interpret the observed phenomena in terms of self-focusing and channeling of the laser pulses by the saturable, resonant nonlinearity.

Figure 1

Sketch of the experimental setup. $E_{\text{in}}$ and $E_{\text{out}}$ are the input and output energy meters, respectively. Cameras in the virtual laser line correspond to the vapor chamber entrance (C1), center (C2), and exit (C3). Transmitted light distribution is detected by the pickoff camera after the imaging telescope (T). Images below the cameras are illustrative images from a single pulse measurement. Optical paths are not drawn to scale; the optical path from the final focusing telescope of the Ti:Sa laser to the vapor source entrance (and the C1 camera) is $\approx 40$ m.

Figure 2

Width parameter $\sigma$ from a Gaussian fit for transmitted laser pulses, $\mathcal{N}=6.6\times {10}^{14}{\mathrm{cm}}^{-3}$. Mean values of binned data shown with asymmetric standard deviation indicated. Insets depict pickoff camera images for selected shots with (a) $E_{\text{in}}=15.6\mathrm{mJ}$, (b)–(d) $E_{\text{in}}=26\mathrm{mJ}$, (e) $E_{\text{in}}=38\mathrm{mJ}$, and (f) $E_{\text{in}}=100\mathrm{mJ}$, each also marked by the arrows. The aspect ratio of the insets corresponds to the detector physical aspect ratio; color coding of individual images is unique, scaled to individual image maxima. Inset (g) depicts the virtual exit camera image (C3) for $E_{\text{in}}=38\mathrm{mJ}$, drawn to the same spatial scale.

Figure 3

(a) Width parameter $\sigma$ for transmitted pulse and virtual exit camera image as a function of input pulse energy for $\mathcal{N}=6.6\times {10}^{14}{\mathrm{cm}}^{-3}$ vapor density shots. (b) Transmitted laser pulse energy (left axis) and peak fluence (right axis). The points depict averages of binned data, while error bars mark the error of the mean. Vertical dotted lines mark the approximate domain boundaries, which are labeled as ST for subthreshold domain, B for breakthrough domain, CB for confined beam domain, and AT for asymptotic transparency domain.

Figure 4

Plot identical to Fig. 3 for $\mathcal{N}=4.895\times {10}^{14}{\mathrm{cm}}^{-3}$ vapor density shots. The subthreshold domain is not captured by the dataset.

Figure 5

(a) Electronic levels of the rubidium atom that are included in the model and their numbering. Three excited states are resonantly accessible from the ground state; ionization leads to level loss from each of the levels. (b) Measured spectrum of the ionizing laser oscillator before amplification and interaction with the vapor. The resonance wavelengths included in the model are marked; lines $5P_{3/2}\iff 5D_{3/2}$ and $5P_{3/2}\iff 5D_{5/2}$ are a closely spaced doublet, difficult to resolve on this scale.

Figure 6

Transmitted pulse energy vs input pulse energy for the measurements and the corresponding simulations. Labels (A)–(C) correspond to the parameter set labels of Table 1. Solid blue line: simulation with atomic resonances; dashed black line: reduced theory simulation (no resonances); red symbols: binned experimental data averages with error bars showing the error of the mean (mostly smaller in size than the symbol marking the points).

Figure 7

Transmitted beam width vs input pulse energy for the measurements and corresponding simulations. Labels (A)–(C) correspond to the parameter set labels of Table 1. Solid blue line: simulation with atomic resonances; dashed black line: reduced theory simulation (no resonances); red symbols: binned experimental data averages with error bars showing the error of the mean (mostly smaller in size than the symbol marking the points). Dashed horizontal line marks the beam width measured on the virtual exit camera C3.

Figure 8

Pulse fluence (in $\mathrm{J}/{\mathrm{cm}}^2$) and atomic ionization probability profiles as a function of propagation distance $z$ and transverse radius $r$. (a) $\mathcal{F}(r,z)$ and (b) $P_{\text{ion}}$ from full theory with resonance $E_{\text{in}}=20\mathrm{mJ}$ pulse. (c) $\mathcal{F}(r,z)$ and (d) $P_{\text{ion}}$ reduced theory calculation, $E_{\text{in}}=20\mathrm{mJ}$ pulse. (e) $\mathcal{F}(r,z)$ and (f) $P_{\text{ion}}$ reduced theory calculation, $E_{\text{in}}=80\mathrm{mJ}$ pulse. Data were taken from simulation set (C). “Plasma channel” marks the area with complete conversion to ${\mathrm{Rb}}^{1+}$ ions near axis.

Figure 9

Transmitted pulse on-axis fluence in $\mathrm{J}/{\mathrm{cm}}^2$ as a function of input energy for the three simulation series. Labels (A)–(C) correspond to the parameter set labels of Table 1. Solid blue line: simulation; red symbols: binned experimental data averages with error bars showing the error of the mean (mostly smaller in size than the symbol marking the points).

Figure 10

(a) Transmitted beam width vs input pulse energy for the original simulation [(C)] and the two simulations with modified Gaussian beam parameters [(D) and (E)]. Binned experimental data with the error of the mean are also plotted. (b) The same quantities plotted with respect to the transmitted pulse energy. (c) 2D histogram of the input beam parameters obtained for $\mathcal{N}=6.6\times {10}^{14}{\mathrm{cm}}^{-3}$ density shots with arrows pointing to the mean values (C) and perturbed parameter simulation values (D) and (E).

Figure 11

The propagation and self-focusing of a low-energy, $E_{\text{in}}=0.04\mathrm{mJ}$ pulse of parameter set (C). (a) Fluence $\mathcal{F}(r,z)$, (b) on-axis fluence $\mathcal{F}(r=0,z)$ and ionization probability $P_{\text{ion}}(r=0,z)$, and (c) ionization probability $P_{\text{ion}}(r,z)$.

Figure 12

The propagation of an $E_{\text{in}}=16\mathrm{mJ}$ pulse for simulation parameter set (C). (a) Beam width $\sigma$ as a function of propagation distance $z$. Vertical dotted lines delimit the regions that can be associated with the confined-beam (CB) and subthreshold (ST) domains. Dashed line marks the width parameter $w\left(z\right)$ for the unperturbed Gaussian beam. (b) Pulse energy spectrum at the vapor entrance $z=0$ m and at $z=1$ m inside the vapor, drawn to scale. (c) Normalized pulse energy spectrum at the two ends of the vapor source, $z=0$ and 10 m.