Ultrashort-pulse propagation through free-carrier plasmas

The past decade has seen frequent use of a modified nonlinear Schrödinger equation to describe ultrashort pulse propagation in materials where free-carrier plasmas are present. The optical contribution from the resulting free-current densities in this equation is often described using a classical Drude model. However, the ultimate form of this contribution in the modified nonlinear Schrödinger equation is somewhat inconsistent in the literature. We clarify this ambiguity by deriving the modified nonlinear Schrödinger equation from the classical wave equation containing a free-current density contribution. The Drude model is then used to obtain an expression for the complex free-carrier current density envelope with temporal dispersion corrections for ultrashort laser pulses. These temporal dispersion corrections to the current-density term differ, to our knowledge, from all other models in the literature in that they depend more sensitively on the value of the Drude free-carrier collision time. These corrections reduce to the current models in the literature for limiting cases. Theoretical analysis and computer simulations show that these differences can significantly affect the dynamic interactions of plasma absorption and plasma defocusing for materials with free-carrier collision times on the order of one optical cycle (or less) of the applied field.

Figure 1

The magnitude of the real (solid line) and the imaginary (dashed line) parts of the dimensionless constant $g$ as a function of the characteristic free-carrier collision time ${\tau }_c$ scaled by the optical cycle $T_{\mathit{oc}}$ of the applied field.

Figure 2

The dimensionless constant $\beta$ for model II as a function of the collision time scaled by the optical cycle of the applied field. The dotted red line represents model I, the dashed blue line represents model II, and the solid black line represents model III, shown for comparison. The grey dot indicates the collision time at which models II and III are in exact agreement for the value of $\beta =0$.

Figure 3

Photoionization rate as a function of optical intensity in fused silica according to the theory of Keldysh. The solid line shows the full Keldysh expression, while the dashed line shows the common multiphoton ionization approximation valid at low intensities. Optical and material parameters used to calculate this rate are those from Table .

Figure 4

Total transmittivity after propagating through 100 $\mu$m of fused silica shown for 10-, 20-, and 40-fs pulses. Transmittivity is shown as a function of the free-carrier collision time. The dotted red line represents model I, the dashed blue line represents model II, and the solid black line represents model III shown for comparison.

Figure 5

Selected results of a 10-fs pulse after propagating through 100 $\mu$m of fused silica with a free-carrier collision time of 0.42 fs. The dotted red line represents model I, the dashed blue line represents model II, and the solid black line represents model III. (a) and (b) show the spatial intensity and phase at the trailing edge of the pulse ($\tau =10$ fs), (c) and (d) show the temporal intensity and phase at the spatial center ($r=0$), (e) and (f) show the spectral intensity and phase at the spatial center ($r=0$), and (g) shows the peak plasma density as a function of the propagation distance.

Figure 6

Selected results of a 10-fs pulse after propagating through 100 $\mu$m of fused silica with a free-carrier collision time of 1.0 fs. The dotted red line represents model I, the dashed blue line represents model II, and the solid black line represents model III. (a) and (b) show the spatial intensity and phase at the trailing edge of the pulse ($\tau =10$ fs), (c) and (d) show the temporal intensity and phase at the spatial center ($r=0$), (e) and (f) show the spectral intensity and phase at the spatial center ($r=0$), and (g) shows the peak plasma density as a function of the propagation distance.

Figure 7

Selected results of a 10-fs pulse after propagating through 100 $\mu$m of fused silica with a free-carrier collision time of 2.6 fs. The dotted red line represents model I, the dashed blue line represents model II, and the solid black line represents model III. (a) and (b) show the spatial intensity and phase at the trailing edge of the pulse ($\tau =10$ fs), (c) and (d) show the temporal intensity and phase at the spatial center ($r=0$), (e) and (f) show the spectral intensity and phase at the spatial center ($r=0$), and (g) shows the peak plasma density as a function of the propagation distance.

Figure 8

Selected results of a 20-fs pulse after 100 $\mu$m of propagation in fused silica with a free-carrier collision time of 0.42 fs. The dotted red line represents model I, the dashed blue line represents model II, and the solid black line represents model III. (a) and (b) show the spatial intensity and phase at the trailing edge of the pulse ($\tau =20$ fs), (c) and (d) show the temporal intensity and phase at the spatial center ($r=0$), (e) and (f) show the spectral intensity and phase at the spatial center ($r=0$), and (g) shows the peak plasma density as a function of the propagation distance.