The filamentation regime: underlying physics

In this section, the universal physical processes that underlie the most relevant features of the filamentation regime are discussed in detail, i.e. the self-focusing, the de-focusing by free-electrons plasma and the self-phase modulation. All these phenomena are governed by the third-order nonlinearity of the macroscopic po-larization induced by the propagating electric field.

Optical Kerr-effect

In linear optics, when light propagates through an isotropic polarizable medium, the instantaneous macroscopic polarizationPis directly proportional to the elec-tric field:

P=ε0χ⁽¹⁾E (1.1)

whereχ⁽¹⁾is the first order susceptibility. The resulting linear refractive index of the medium is:

n₀= q

1+χ⁽¹⁾ (1.2)

In presence of high optical fields, Eq.(1.1) does not hold anymore; indeed, the displacement of bound electrons in the atoms of the medium becomes so large that the resulting dipole moment, hence the macroscopic polarization, cannot be considered a linear function of the applied field. This is the optical Kerr effect, and mathematically corresponds to the series development of Eq.(1.1):

P=ε₀(χ⁽¹⁾E+χ⁽²⁾EE+χ⁽³⁾EEE+...) (1.3) All the even orders susceptibilities χ⁽²ⁿ⁾ vanish in centrosymmetric materials, such as gaseous media, thus Eq.(1.3) can be rewritten as follows:

P=ε₀(χ⁽¹⁾E+χ⁽³⁾EEE) (1.4) where the series has been truncated to the third order. The refractive index thus reads:

n= q

1+χ⁽¹⁾+χ⁽³⁾EE 'n₀+n₂I (1.5) where the productEE is identified withI=|E|². Therefore, the effect of a strong optical field is to locally modulate the refractive index by superimposing, to the constant linear termn₀, an intensity-dependent termn₂I, whose weight is deter-mined by thenonlinear refractive index:

n₂= 3χ⁽³⁾

4ε₀cn₀ (1.6)

For a 100 fs pulse centered at 800 nm, the air exhibits a nonlinear indexn₂=3.6× 10⁻¹⁹cm²/W. If one first neglects higher orders of the instantaneous polarization, the intensity-dependent refractive index change induced by the field is:

∆n'n₂I (1.7)

∆n has a positive sign, as χ⁽³⁾ is generally positive in usual media. It follows that the light experiences a reduction of its phase velocity due to the self-induced

1.1 Self-guided propagation of ultrashort pulses in gaseous media

modulation of the refractive index, increasing with the intensity.

Considering a typical transverse Gaussian intensity profile for a laser beam, ∆n will also feature a bell-shaped intensity profile, reaching its maximum at the center of the beam and rapidly decaying towards the edges. Therefore, the resulting wavefront will be curved analogously to the propagation through a focusing lens, and the beam will progressively shrink. This effect is often referred to as Kerr-lens. It is responsible for the self-focusing of intense light beams and is essential to the filamentation regime.

The third order term of the polarization is obviously always present, but, at low intensities, the linear regime the diffraction, leading to a transverse spread of the beam during its propagation, is the dominating effect. Conversely, as the intensity increases, self-focusing efficiently counteracts diffraction and, finally, overcomes it. The balance between these two opposite effects is used to define the critical powerfor self-focusing:

P_cr = 3.72λ₀²

8πn₀n₂ (1.8)

which defines the unstable equilibrium point that separate the linear diffractive regime from the catastrophic collapse of the beam. At 800 nm, a laser beam undergoes self-focusing in the air if its input power exceeds the threshold of ∼ 4 GW.

Marburger [53] proposed a semi-empirical formula for the distancez_cat which an initially collimated gaussian beam of waistw₀and wavenumberk₀=2π/λ₀will collapse if its power is larger thanP_cr:

z_c= 0.184w²₀k₀

p((P/P_cr)^1/2−0.853)²−0.0219 (1.9) This expression provides a good estimation of the onset of filamentation for a Gaussian beam in the single filament regime, although it breaks down when the input power is so higher than the critical threshold that the beam undergoes mul-tiple filamentation.

In order to achieve the self-guided propagation of a collimated beam, which fines the filamentation regime, a dynamical balance between a focusing and a de-focusing effect must be established, in order to prevent the collapse of the beam at the nonlinear focus.

Multi-photon ionization and plasma defocusing

Due to the availability of high-power lasers emitting in this spectral region, the filamentation regime is generally investigated in the near-infrared, at 800 nm. At this wavelength, the ionization potential of major air constituents (oxygen and

Figure 1.3: Schematic of Kerr self-focusing (a) and Plasma defocusing (b).

Adapted from [1].

nitrogen) largely exceeds the energy borne by a single photon. Indeed, the simul-taneous absorption of 8 photons is required for the ionization of oxygen, which features the lowest ionization barrier. This process is highly improbable at low intensities, that is, before self-focusing, thus the beam propagates lossless con-veying the whole beam power towards the onset of filamentation.

However, when the beam collapse approaches, the optical intensity enormously increases and so does the probability associated to the multi-photon absorption, which scales as∼I^K withK =8 for air (oxygen) at 800 nm. At the optical in-tensities typically featured by the filament core, i.e. 50 TW/cm², Multi-photon ionization (MPI) dominates over the tunnel ionization, which becomes relevant above 10¹⁴ W/cm², according to the Keldysh theory [54].

As a consequence of the MPI, the bound electrons are extracted from their orbitals and a plasma channel is generated, with a typical density of 10¹⁵−10¹⁷cm⁻³. The contribution of the plasma electrons to the refractive index is negative, according to the law:

∆n_plasma= ρ

2ρ_c (1.10)

whereρis the generated plasma density andρ_c=ε₀m_eω₀²/e²is the critical plasma density above which the plasma becomes opaque (me andeare the electron mass and charge,ω₀the pulse frequency, andρ_c∼1.7×10²¹ cm⁻³at 800 nm).

Therefore, the negative index variation induced by the free electrons provides a mechanisms which counteracts the third-order Kerr-focusing, as it is depicted in Fig. 1.3. Indeed, the refractive index spatial profiles induces an opposite curva-ture of the wavefront, resulting in a negative lens that defocuses the beam. Clas-sically, filamentation is said to stem from the dynamical balance between these

1.1 Self-guided propagation of ultrashort pulses in gaseous media

Figure 1.4: Variation of the nonlinear refractive index of the major air constituents at room temperature and 1 atm, induced by a laser pulse, as a function of the optical intensity. Reprint from [2].

two effects: Kerr-focusing and plasma-defocusing. This interplay leads to several beam refocusing cycles that maintains the filament size approximately constant over several Rayleigh distances; the filament has a diameter of 100−200µm and keeps an almost fixed energy (few mJ) and intensity (50 TW/cm² [45]), a phe-nomenon known asintensity clamping.

The plasma channel is widely used as a diagnostic tool to measure the length and the strength of a filament, through the detection of the charge release [9], the acoustic shock-wave [48], or the backscattered nitrogen fluorescence [47].

Kerr-Driven Laser Filamentation

The classical picture of filamentation attributed to the plasma-induced negative index change the role of de-focusing mechanism, as it has been discussed in the previous paragraph. However, recent measurements of high-order nonlinear re-fractive indexes of common air constituents [2] suggested that they may them-selves yield a defocusing contribution. This topic is nowadays still deeply de-bated [55, 56].

By measuring the transient birefringence of the main air components induced by an intense optical field, Loriot et al. [2] were able to identify the additional contri-bution to the molecular alignment yielded by the Kerr effect, and thus retrieve the nonlinear refractive index denoted by n_Kerr. Their results, displayed in Fig. 1.4,

Figure 1.5: Schematic of purely Kerr-driven filamentation. As the beam shrinks, due to Kerr focusing, higher orders nonlinear terms become also important, lead-ing to a dynamical balance between the contributions from positive (n₂, n₆, ...) and negative(n4, n₈, ...) Kerr terms. Dashed lines indicate the beam collapse in the absence of any defocusing mechanism.

evidenced that for N₂ and O₂n_Kerr first linearly increases, then saturates, and fi-nally drops dramatically with the intensity, reversing its sign. This trend cannot obviously be described by Eq.(1.5), thus revealing that higher orders of the instan-taneous polarization of Eq.(1.3), up to the ninth, must be included. A polynomial fit of the detected intensity-dependence allowed to infer the values of nonlinear refractive coefficients up to n₈, showing a sign alternation between consecutive terms. Obviously, a positive non-linear coefficient yields a focusing contribution, as already discussed forn₂-driven Kerr self-focusing, while negative coefficients yield the opposite contribution.

In a theoretical work conducted at GAP Biophotonics, Bejot et al. [55] imple-mented the High-Order Kerr terms (HOKE)n₄,n₆,n₈in the Non-Linear Schrödinger Equation (NLSE) for the propagation of ultrashort pulses. Based also on pump-probe experimental measurements, they concluded that the HOKE negative terms yield a considerable de-focusing effect that balances the tendency of the beam to collapse; therefore, for sub-ps pulses, the plasma appears to have a minor role in driving the filamentation [46]. The mechanism of purely Kerr-driven filamenta-tion [46] is schematically depicted in Fig. 1.5.

1.1 Self-guided propagation of ultrashort pulses in gaseous media

Figure 1.6: Frequency shift of the edges of an optical waveform, induced by self-phase modulation, according to Eq.(1.12). Dashed lines represent the Gaussian envelope.

Self-phase modulation

As depicted above, the Kerr nonlinear index, modulated by the spatial intensity gradient of the pulse, leads to self-focusing of the laser beam. Analogously, a temporal effect is induced by the Kerr nonlinear term, owing to the fact that the pulse intensity profile also varies in time. Like in the spatial coordinates, the typ-ical temporal profile of a laser pulse features a Gaussian-like intensity envelope.

Therefore, the central slice will ’see’ a higher refractive index and will be delayed with respect to its wings. This results in a spectral broadening of the pulse. This phenomenon is calledself-phase modulation, and was discovered already in the early days of nonlinear optics [57]. It owes this definition by the fact that the in-tensity dependent nonlinear refractive index modulates the phase of the waveform according to:

φ(z,t) =φ0+ω c

Z _z

0

∆n_nldz (1.11)

where φ₀ is the initial phase of the electric field, ω is the carrier frequency and

∆n_nl =n₂Iis the nonlinear refractive index change, which depends upon the opti-cal intensityI. Thus, the phase variation experienced by the pulse over a lengthz is∆φ =n₂ωIz/c, and the instantaneous frequency is modulated accordingly:

∆ω(t) =−∂ φ(t)

dt =−n₂ω∂I

∂t z

c (1.12)

Eq.(1.12) describes the self-phase modulation induced by the Kerr nonlinearity;

it depends upon the time derivative of the intensity profile in such a way that the rising edge of the electromagnetic wave generates redder frequencies, while the

Figure 1.7: Spectrum of the continuum generated by TW filaments in the air. Ab-sorption bands of typical atmospheric constituents are indicated. Reprint from [3].

trailing edge adds up bluer frequencies to the initial spectrum. The resulting opti-cal waveform is displayed in Fig. 1.6. SPM therefore yields a broadening of the initial spectrum of the laser pulse, a phenomenon commonly known as White-light generation (WL) or supercontinuum generation, as discussed in Sec. 1.1.1.

This emitted light preserves the coherence of the driving electromagnetic field;

therefore, different spectral components of the continuum emission maintain a constant phase relationship between each other. More strikingly, the coherence persists also between the WL resulting by two separated filaments of the same beam.

Fig. 1.7 displays the typical WL spectrum emitted by filaments generated by a TW laser beam propagating in the air. The continuum spans from 230 nm to more than 4µm, covering the absorption bands of many typical atmospheric trace gases such as VOCs (Volatile Organic Compounds), NOx, CO₂, H₂O, etc.