Four-dimensional visualization of single and multiple laser filaments using in-line holographic microscopy

It is shown, both through simulations and experiments, that the in-line holographic microscopy technique can be used to retrieve very small refractive-index perturbations caused during the filamentation of ultrashort laser pulses. This technique provides the possibility of having spatially and temporally (four dimensions) resolved measurements of refractive-index changes, down to 10${}^{-4}$, from objects with diameters as small as 10 μm. Moreover, we demonstrate the power of the technique in discriminating multiple filaments in a precise quantitative way.

Figure 1

Typical holographic diffraction images for a probe perturbed by a cylindrical structure with (a) $\Delta n>0$ (Δκ = 0), (b) $\Delta n<0$ (Δκ = 0), and (c) Δκ ≠ 0 $(\Delta n=0)$.

Figure 2

Schematic representation of the wave-front retrieval algorithm (see text).

Figure 3

(a) Simulated input amplitude distributions (in-line holograms) corresponding to the phase object shown in (b) and (d); [(b)–(e)] simulated (initial) and retrieved phase and amplitude distributions by using i-HOM technique at two different propagation distances, $z$ = 0 and $z$ = 2 mm. (f) Simulated (initial) and retrieved phase change by using only one diffraction pattern at $z$ = 2 mm.

Figure 4

(a) Simulated amplitude distribution (in-line holograms) for various propagation distances with the addition of 2% random noise; [(b)–(e)] simulated and retrieved amplitude and phase distribution at two different propagation distances, $z$ = 0 and $z$ = 2 mm.

Figure 5

(a) Imaging in a typical microscope system. Plane waves are transformed to spherical ones. (b) Imaging using a bitelecentric optical system; plane waves are still plane after exiting the system. (c) Experimental setup of the pump-probe in-line holographic microscopy utilizing a cascade of bitelecentric optical systems.

Figure 6

(a) Experimental in-line holograms remotely captured along the propagation of the probe beam, (b) retrieved phase and corresponding refractive-index change (Abel transform), and (c) respective electron-density distribution in the plasma string.

Figure 7

(a) Experimental in-line holograms of a propagating filament for various pump-probe delays ($k$ is the pump wave vector). (b) Retrieved refractive-index distribution.

Figure 8

(a) Retrieved phase information as a function of the number of input in-line holograms. (b) Number of iterations to reach convergence (under the same constraint) versus the number of input in-line holograms.

Figure 9

The simulated and retrieved phase change, corresponding to two refractive structures, located on the same plane along the $z$ axis and separated by 10 (a), 15 (b), and 20 μm (c).

Figure 10

In-line holograms recorded at $z$ = 1.5 mm, corresponding to two filaments located on the same plane along the propagation direction of the probe beam, $z$, separated by ∼145 μm (a) and ∼72 μm (b), along the $y$ axis. [(c) and (d)] Retrieved phase change profiles corresponding to (a) and (b), respectively.

Figure 11

(a) Phase distribution of the numerically propagated wave front as a function of the propagation distance $z$ and (b) normalized phase change as a function of propagation distance. The peaks indicate the position of the filaments along the $z$ axis.