tzortzakis_2001

S. Tzortzakis, L. Sudrie, M. Franco, B. Prade, A. Mysyrowicz, A. Couairon and L. Berge, “Self-Guided Propagation of Ultrashort IR Laser Pulses in Fused Silica”, Phys. Rev. Lett. 87, 213902 (2001)

Source record · Catalogue · Review register

DOI: 10.1103/physrevlett.87.213902

Source kind: measurement

Recorded technique: 160-fs, 800-nm pulses (200 kHz) focused by an f = 80 mm lens near the entrance face of moving fused-silica plates (2, 5, 10 mm); filament diameter imaged along z with a 20x objective and CCD, autocorrelation (BBO) and spectrum of the output. Simulated with an extended nonlinear Schrodinger equation for the envelope coupled to a multiphoton-plus-avalanche rate equation (Eqs. 1-2).

Materials in structured entries: SiO2

Dataset quantities: atom_density, avalanche_coefficient, band_gap, critical_power, electron_trapping_time, group_delay_dispersion, multiphoton_ionization_cross_section, nonlinear_index, refractive_index

Contents: 9 datasets; 1 formula.

Curation notes

Journal pages 213902-1 to -4 = PDF pages 1-4. Identity validated. The library text’s ‘mJ’ for the input energy is ‘uJ’ in the pdf (a Symbol-font mu): ‘E_in = 2 uJ’ (Fig. 1 caption, p. 213902-2); 2 uJ in 160 fs is 12.5 MW, about 5 P_cr as stated.

MEASURED (no curated quantity): filamentation requires P > ~3 P_cr; filament diameter 10-40 um over several mm (Fig. 1c); peak intensity ‘of the order of 10^13 W/cm2’ inferred from the size; the 160-fs pulse splits into two ~60-fs subpulses 120 fs apart (autocorrelation, Fig. 2a; spectral fringes 16 nm apart, Fig. 2b); less than 1 % of the energy in the visible continuum (p. 213902-2).

SIMULATION INPUTS (p. 213902-1 and -3; datasets assumed): P_cr = lambda0^2/(2 pi n0 n2) ~ 2.3 MW ‘with n0 = 1.453 being the linear refraction index and n2 = 3.2 x 10^-16 cm2/W the nonlinear refraction index for fused silica’ (no source); k’’ = 361 fs2/cm; the inverse-bremsstrahlung cross section sigma = 2.78 x 10^-18 cm2 [5] (Mlejnek, Wright and Moloney, Opt. Lett. 23, 382 (1998), which the paper’s introduction lists among the atmospheric-filamentation references [1-7]); rho_at = 2.1 x 10^22 atoms/cm3, rho_at/rho_c ~ 12; U_i = 7.6 eV [24] (Tohmon et al., Phys. Rev. B 39, 1337 (1989)), so K = 5; sigma_K = 1.3 x 10^-55 s^-1 cm^2K/W^K (no source); alpha = sigma/(n0^2 U_i); tau_r ~ 150 fs [23] (Audebert et al. 1994); input waist 30 um (measured), FWHM 160 fs. The Kerr term is instantaneous (no Raman response in Eq. 1).

CHECKS (2026-09-26): lambda0^2/(2 pi n0 n2) = 2.19 MW with the printed n0 and n2, not ‘~2.3 MW’ (5 % high; hamam_2018 gets 2.19 MW from the same n2); rho_at/rho_c = 12.06 with rho_c = omega0^2 eps0 m_e/e^2 (printed ~12); 7.6 eV/1.55 eV = 4.90, K = 5; the Drude cross section e^2 tau_c/(m_e n0 eps0 c (1 + omega0^2 tau_c^2)) equals 2.78e-18 cm2 for tau_c = 4.69 fs (or 0.038 fs) with n0 = 1.453, the paper states no tau_c.

MODEL RESULTS: I_max = 1.5e13 W/cm2 computed; filament length Delta z ~ 1 cm from Delta z ~ ln[P_in/P(z)]/(beta^(K) I_max^(K-1)) (p. 213902-3); figures only otherwise.

REPRODUCED BY: hamam_2018 (tau_rec = 1.5e-13 s, ref. 23 there); couairon_2005 and berge_2010 cite it for the loose-focusing filament.

Recorded bibliographic identity status: VALIDATED. This concerns identification of the source; it does not establish the scientific correctness of its contents.

Datasets

tzortzakis_2001/model_n2

{
  "id": "model_n2",
  "material": "SiO2",
  "table": "text",
  "quantity": "nonlinear_index",
  "unit": "cm2/W",
  "abscissa_unit": "nm",
  "use": "check",
  "parameters": {
    "part": "unknown",
    "coefficient": "self",
    "assumed": true
  }
}

1 inline points in the source record.

p. 213902-1: ‘n2 = 3.2 x 10^-16 cm2/W the nonlinear refraction index for fused silica’; no source, no pulse duration; the model’s Kerr response is instantaneous.

tzortzakis_2001/model_refractive_index

{
  "id": "model_refractive_index",
  "material": "SiO2",
  "table": "text",
  "quantity": "refractive_index",
  "abscissa_unit": "nm",
  "use": "check",
  "parameters": {
    "assumed": true
  }
}

1 inline points in the source record.

p. 213902-1: ‘n0 = 1.453 being the linear refraction index’; no source.

tzortzakis_2001/model_critical_power

{
  "id": "model_critical_power",
  "material": "SiO2",
  "table": "text",
  "quantity": "critical_power",
  "unit": "MW",
  "abscissa_unit": "nm",
  "use": "check",
  "parameters": {
    "definition": "P_cr = lambda0^2/(2 pi n0 n2), n0 = 1.453, n2 = 3.2e-16 cm2/W"
  }
}

1 inline points in the source record.

p. 213902-1: ‘P_cr = lambda0^2/2 pi n0 n2 ~ 2.3 MW’. The printed n0 and n2 give 2.19 MW (source notes).

tzortzakis_2001/model_gvd

{
  "id": "model_gvd",
  "material": "SiO2",
  "table": "text",
  "quantity": "group_delay_dispersion",
  "unit": "fs2/m",
  "scale": 100,
  "abscissa_unit": "nm",
  "use": "check",
  "parameters": {
    "assumed": true
  }
}

1 inline points in the source record.

p. 213902-3: ‘normal GVD with coefficient k’’ = 361 fs2/cm’ (scale 100 to fs2/m); no source (couairon_2005 computes the same from Malitson’s formula).

tzortzakis_2001/model_avalanche_cross_section

{
  "id": "model_avalanche_cross_section",
  "material": "SiO2",
  "table": "text",
  "quantity": "avalanche_coefficient",
  "unit": "cm2",
  "abscissa_unit": "nm",
  "use": "check",
  "reproduced_from": [
    "M. Mlejnek, E. M. Wright and J. V. Moloney, Opt. Lett. 23, 382 (1998)"
  ],
  "parameters": {
    "definition": "sigma, the inverse-bremsstrahlung cross section of Eq. (1); Eq. (2) avalanche term alpha rho |E|^2 with alpha = sigma/(n0^2 U_i)",
    "assumed": true
  }
}

1 inline points in the source record.

p. 213902-3: ‘plasma absorption with inverse Bremsstrahlung cross section sigma = 2.78 x 10^-18 cm2 [5]’; ref. 5 is among the paper’s atmospheric-filamentation references. The avalanche coefficient of Eq. (2) is alpha = sigma/(n0^2 U_i). Equal to the Drude value with tau_c = 4.69 fs (source notes).

tzortzakis_2001/model_band_gap

{
  "id": "model_band_gap",
  "material": "SiO2",
  "table": "text",
  "quantity": "band_gap",
  "unit": "eV",
  "abscissa": "none",
  "use": "check",
  "reproduced_from": [
    "R. Tohmon, H. Mizuno, Y. Ohki, K. Sasagane, K. Nagasawa and Y. Hama, Phys. Rev. B 39, 1337 (1989)"
  ],
  "parameters": {
    "kind": "assumed",
    "assumed": true
  }
}

1 inline points in the source record.

p. 213902-3: ‘Band-to-band transitions are induced in silica through the gap potential U_i = 7.6 eV [24], yielding the number of photons K = 5 required for ionization.’

tzortzakis_2001/model_sigma5

{
  "id": "model_sigma5",
  "material": "SiO2",
  "table": "text",
  "quantity": "multiphoton_ionization_cross_section",
  "unit": "s-1 cm10/W5",
  "abscissa_unit": "nm",
  "use": "check",
  "parameters": {
    "photons": 5,
    "assumed": true
  }
}

1 inline points in the source record.

p. 213902-3: ‘The MPI coefficient then reads as sigma_K = 1.3 x 10^-55 s^-1 cm^2K/W^K’ with K = 5 (printed cm^2K/W^K); per atom, the rate being sigma_K |E|^2K (rho_at - rho) (Eq. 2). No source; the multiphoton absorption coefficient is beta^(K) = K hbar omega0 sigma_K rho_at.

tzortzakis_2001/model_atom_density

{
  "id": "model_atom_density",
  "material": "SiO2",
  "table": "text",
  "quantity": "atom_density",
  "unit": "cm-3",
  "abscissa": "none",
  "use": "check",
  "parameters": {
    "counts": "atoms",
    "assumed": true
  }
}

1 inline points in the source record.

p. 213902-3: ‘rho_at is the background atom density (rho_at/rho_c ~ 12 where rho_at = 2.1 x 10^22 atoms/cm3)’.

tzortzakis_2001/model_recombination_time

{
  "id": "model_recombination_time",
  "material": "SiO2",
  "table": "text",
  "quantity": "electron_trapping_time",
  "unit": "fs",
  "abscissa": "none",
  "use": "check",
  "reproduced_from": [
    "P. Audebert et al., Phys. Rev. Lett. 73, 1990 (1994)"
  ],
  "parameters": {
    "assumed": true
  }
}

1 inline points in the source record.

p. 213902-3: ‘electron recombination with a characteristic time tau_r ~ 150 fs in fused silica [23]’.

Formulas

tzortzakis_2001/propagation_model

{
  "id": "propagation_model",
  "material": "SiO2",
  "equation": "Eqs. (1)-(2), p. 213902-3",
  "printed": "dE/dz = (i/(2k)) T^-1 (d^2/dr^2 + (1/r) d/dr) E - (i k''/2) d^2E/dtau^2 + i k0 n2 T(|E|^2 E) - (sigma/2 + i k0/(2 rho_c)) T^-1 (rho E) - (beta^(K)/2) |E|^(2K-2) E (1); d rho/dtau = sigma_K |E|^2K (rho_at - rho) + alpha rho |E|^2 - rho/tau_r (2); beta^(K) = K hbar omega0 sigma_K rho_at; alpha = sigma/(n0^2 U_i); T = 1 + (i/omega0) d/dtau",
  "variable": "E: envelope (|E|^2 an intensity); tau = t - z/v_g; k = n0 k0",
  "coefficients": {
    "n0": "1.453",
    "n2": "3.2e-16 cm^2/W",
    "k2": "361 fs^2/cm",
    "sigma": "2.78e-18 cm^2",
    "rho_at": "2.1e22 atoms/cm^3",
    "U_i": "7.6 eV",
    "K": "5",
    "sigma_K": "1.3e-55 s^-1 cm^10/W^5",
    "tau_r": "150 fs"
  },
  "check": "rho_at/rho_c = 12.06 (printed ~12); K = 5 (7.6/1.55 = 4.90) (checked 2026-09-26). The filament-length estimate Delta z ~ ln[P_in/P(z)]/(beta^(K) I_max^(K-1)) ~ 1 cm with I_max = 1.5e13 W/cm2 is not recomputed (P(z) not given)."
}

Solved with a Fourier decomposition in time and Crank-Nicholson in space; Gaussian input, waist 30 um, 160 fs FWHM, E_in = 2 uJ (P_in ~ 5 P_cr).

Corrections

No dataset errata are recorded in this source record.

Formula corrections, where recorded, appear with their formulas above. Curation notes may also discuss corrections or unresolved interpretation. Consult the review register for the checks and decisions.