Influence of wave-front curvature on supercontinuum energy during filamentation of femtosecond laser pulses in water

We demonstrate that using spatially divergent incident femtosecond 1240-nm laser pulses in water leads to an efficient supercontinuum generation in filaments. Optimal conditions were found when the focal plane is placed $100–400\mu \mathrm{m}$ before the water surface. Under sufficiently weak focusing conditions [numerical aperture ($\mathrm{NA})<0.2$] and low-energy laser pulses, the supercontinuum energy generated in divergent beams is higher than the supercontinuum energy generated in convergent beams. Analysis by means of the unidirectional pulse propagation equation shows a dramatic difference between filamentation scenarios of divergent and convergent beams, that explains corresponding features of the supercontinuum generation. Under strong focusing conditions ($\mathrm{NA}⩾0.2$) and high-energy laser pulses, the supercontinuum generation is suppressed for convergent beams in contrast to divergent beams that nevertheless are shown experimentally to allow supercontinuum generation. The presented technique of the supercontinuum generation in divergent beams in water is highly demanded in a development of femtosecond optical parametric amplifiers.

Figure 1

Map of the SC energy $E_{\text{SC}}(\mathrm{\Delta }{z}_f,E)$ generated under filamentation in water (a, b) with corresponding profiles $E_{\text{SC}}\left(\mathrm{\Delta }{z}_f\right)$ at fixed laser-pulse energies $E$ (c, d): (a, c) NA = 0.05; (b, d) NA = 0.1. (c, d) Solid and dashed lines correspond to $E=20$ and $100\mu \mathrm{J}$, respectively; black and green (light gray) lines correspond to water and fused silica samples, respectively. (c) The SC energy values are presented on the left axis for water and on the right axis for fused silica. The ratio $E_{\text{SC}}^{\left(1\right)}/E_{\text{SC}}^{\left(2\right)}$ for various incoming laser energies E and focusing NA is presented in (e) and (f) for water and fused silica samples correspondingly. (e, f, h) The position of the water sample was characterized by $|\mathrm{\Delta }{z}_f|\approx 200–300\mu \mathrm{m}$, that corresponds to the most efficient SC generation in both C1 and C2.

Figure 2

(a) The experimentally detected SC spectra in logarithmic scale ${\text{log}}_{10}S\left(\lambda \right)$ under filamentation of a 100-$\mu \mathrm{J}$ laser pulse in water for various $|\mathrm{\Delta }{z}_f|$ are presented together with the water absorption (black dotted line) in the considered spectral range. C1 is presented by light green (light gray) dash-dotted and dark green (black) dashed lines; C2 is presented by blue (black) dash-dotted and rose (light gray) dotted lines. The corresponding $|\mathrm{\Delta }{z}_f|$ is indicated in the figure. The violet (light gray) area illustrates the visible spectral region for which the SC energy measurements were done. (b, c) The blue part of the SC spectra $S(\lambda$) under strong focusing conditions. The NA is indicated in the figure. The spectral peak at 413 nm in (c) is related to the third-harmonic generation. Green (light gray) and blue (black) lines correspond to C1 and C2, respectively. (d) A typical photoemission of plasma channels $I_{\text{photoemission}}$ is presented for C1 and C2, NA = 0.1 and $E\approx 70–100\mu \mathrm{J}$. $I_{\text{photoemission}}$ is normalized to its maximum, that is 0.52 and 0.11 a.u. in C1 and C2, respectively. The position of the water sample is characterized by $|\mathrm{\Delta }{z}_f|\approx 200–300\mu \mathrm{m}$, that corresponds to the most efficient SC generation in both C1 and C2. The laser-pulse energy is $E=100\mu \mathrm{J}$.

Figure 3

2D+1 modeling: Intensity evolution $I(r=0,\tau ,z)$ and plasma density $N_e(r=0,z)$ along $z$ for a 200-fs $7\text{−}\mu \text{J}$ ($5.6P_{\text{cr}}$, $a_{\text{FWHM}}=4.1\mathrm{mm}$) laser pulse (a) in C1, $\mathrm{\Delta }{z}_f=-270\mu \mathrm{m}$, and (b) in C2, $\mathrm{\Delta }{z}_f=300\mu \mathrm{m}$; spatiotemporal intensity distribution $I(\mathrm{r},\tau )$ is shown at the propagation distances (c) in C1, $1.3\mu \text{m}$, and (b) in C2, $0.3\mu \text{m}$. 3D+1 modeling: Evolution of fluence $F(x,y)$ along the filament trace in water (g, h) in C1, $\mathrm{\Delta }{z}_f=-300\mu \mathrm{m}$, and (i, j) in C2, $\mathrm{\Delta }{z}_f=300\mu \mathrm{m}$. $a_{\text{fluct}}=0.5$, ${\tau }_{\text{FWHM}}=330\text{fs}$, $E=7\mu \text{J}$, $P=5P_{\text{cr}}$, $a=3\mathrm{mm}$.

Figure 4

(a) Numerical (lines) and experimental (dots) SC spectra in logarithmic scale ${\text{log}}_{10}S\left(\lambda \right)$. The maximal amplitude of the experimentally detected blue wing of the SC is normalized to be equal to the maximal amplitude of the blue wing of the numerically calculated spectra for $\mathrm{\Delta }{z}_f=-270\mu \mathrm{m}$ [solid green (light gray) line]. Pulse parameters in the simulations for both 2D+1 and 3D+1 geometries: ${\tau }_{\text{FWHM}}=200\text{fs}$, $E=7\mu \text{J}$ ($5.6P_{\text{cr}}$), $a=4.1\mathrm{mm}$. The fluctuation amplitude in 3D+1 is $a_{\text{fluct}}=0.5$. In the experiment $E=35\mu \mathrm{J}$. Propagation distances $z$ at which ${\text{log}}_{10}S\left(\lambda \right)$ were taken are indicated in the figure. (b–e) Numerically obtained evolution of the angularly integrated SC spectrum along the laser-pulse propagation distance $z$. C1 ($\mathrm{\Delta }{z}_f=-300\mu \mathrm{m}$) and C2 ($\mathrm{\Delta }{z}_f=300\mu \mathrm{m}$) are presented in (b, c) and (d, e), respectively. (b, d) The modeling includes all the process as described in Sec. 4. (b) The impact ionization term is switched off, i.e., it is removed from the plasma density ionization rate equation in the simulation. The pulse parameters are ${\tau }_{\text{FWHM}}=330\text{fs}$, $E=7\mu \text{J}$, $P=5P_{\text{cr}}$, $a=3\mathrm{mm}$.