Ultrashort Laser Pulse Phenomena, 2nd Edition

Solitons 597

difficulties in achieving breakdown reduction and discharge guiding over large

gaps. A solution to the problem of triggering and guiding discharges over long

distances is to maintain the ionized channel created by the UV or IR filament

through inverse Bremstrahlung (plasma heating) and photodetachment of oxy-

gen. Pulse intensities ranging from 1 to 10 MW/cm

2

have to be maintained in

the channel, for the time duration required to trigger the discharge [71].

The ability to trigger a discharge depends also on the initial electron density

deposited by the fs pulse in the filament path. Various evaluations of the elec-

tron density have been based on conductivity measurements. The values reported

vary between 10

12

cm

−3

[74], 10

14

cm

−3

[51], and 10

16

cm

−3

[75]. Measure-

ments performed with the same setups and 1-ps UV pulses (250 nm) and 100-fs

IR pulses (800 nm) [51] indicate a 20 × larger conductivity induced by the

UV filament than by an IR filament, the latter produced by a 10 × more ener-

getic pulse. The larger conductivity in the UV filament is important for laser

discharge applications and is attributed to the fact that the nonlinear ionization

is only a three-photon process compared to a 9- to 10-photon process for the IR

filaments. In both cases a diameter of the filaments of 100 µm was obtained,

implying an intensity of 1 TW/cm

2

in the UV filament versus 100 TW/cm

2

in

the IR filament. Using these intensities and typical cross sections for the mul-

tiphoton absorption one estimates electron densities that are consistent with the

conductivity measurements.

1

13.2.3. Spatial and Temporal Solitons

The GVD parameter k





of air is approximately 0.15 fs

2

/cm at 800 nm [78,79].

For a 50-fs pulse that is often used to produce filaments, this dispersion corre-

sponds to a characteristics distance [as defined by Eq. (1.128)] of L

D

≈ 160 m.

All dispersion effects of the atmosphere are thus negligible at that wavelength.

If we consider instead the dispersion of air at 248 nm, k





≈ 1fs

2

/cm and for

a 50-fs pulse L

D

≈ 25 m. Thus pulse broadening should occur over distances of

the order of 10 m with UV fs pulses. Therefore, the existence of filaments over

tens of meters requires that the pulse be trapped in space and in time.

A similar situation arises with filaments created at 800 nm with pulses of less

than 10 fs duration: the dispersion length in air is now only L

D

≈ 6 m. Using

gases other than air (such as Ar) at higher pressure the characteristic length

can be made of the order of tens of cm, and spatial-temporal soliton formation is

possible. A recent application of filaments involves compressing intense fs pulses

1

Couairon and Bergé [76] using the Keldysh formula [77] for the evaluation of the three-photon

ionization of oxygen, infer from their calculations a beam diameter of 40 µm for the UV filament

and 200 µm for the IR, leading to the same intensity in UV and IR filaments.