du_1994

D. Du, X. Liu, G. Korn, J. Squier, and G. Mourou, “Laser-induced breakdown by impact ionization in SiO2 with pulse widths from 7 ns to 150 fs”, Appl. Phys. Lett. 64, 3071-3073 (1994)

Source record · Catalogue · Review register

DOI: 10.1063/1.111350

Source kind: measurement

Recorded technique: Single-shot laser-induced breakdown of fused silica (Corning 7940, 0.15 mm thick, both sides polished, scratch/dig 20-10, cleaned in methanol) with 780-nm pulses of 150 fs to 7 ns from a 10-Hz Ti:sapphire chirped-pulse-amplification system (the pulse length varied by the compressor), focused inside the sample by an f = 25 cm lens to a 26-um FWHM Gaussian spot (measured in situ; Rayleigh length ~2 mm); thin samples to avoid self-focusing; each site irradiated once (1-on-1). Threshold from the plasma emission (photomultiplier) and the drop of the transmitted energy, fitted with Eq. (1); at nanosecond pulses confirmed by visual inspection (p. 3071).

Materials in structured entries: SiO2

Dataset quantities: None recorded.

Contents: 0 datasets; 1 formula.

Curation notes

Journal pages 3071-3073 = pdf pages 1-3. Identity checked against the first page. No tables: every threshold is in Fig. 2; the text gives the model’s parameters, which are recorded as the formula thornber_alpha and in these notes.

WHAT WAS MEASURED. The breakdown threshold fluence F_th against pulse width (Fig. 2, p. 3072: ‘From 7 ns to about 10 ps, the breakdown threshold follows the scaling law of F_th proportional to sqrt(tau_p) ... when the pulse width becomes shorter than a few picoseconds, the threshold starts to increase’), bulk (focused inside a 0.15-mm plate: Du et al. note that breakdown may have occurred ‘on the surface first’, p. 3073), single shot, criterion the onset of plasma emission and of the transmission loss. The figure also shows earlier bulk (Smith, Opt. Eng. 17, 489 (1978)) and surface (Stokowski, Milam and Weber 1978) data for longer pulses. The only threshold printed in the text: ‘at 1 ps, the rms field threshold is about 5 x 10^7 V/cm’ (p. 3072; a field, for which there is no quantity). Fig. 1 (p. 3071): plasma emission and transmitted energy against incident energy at 300 fs, with Eq. (1), U_t = k U for U <= U_th and k U_th [1 + ln(U/U_th)] above (a Gaussian pulse in space and time, the avalanche taking the whole pulse). Fig. 3 (p. 3073): the impact ionization rate per unit length alpha against the rms field (experiment and Eq. 3). Figures not digitised.

THE MODEL (pp. 3072-3073). Eq. (2): dn_e/dt = eta(E) n_e + (dn_e/dt)_PI - (dn_e/dt)_loss, eta the avalanche coefficient; losses neglected for picosecond pulses. With photo- ionisation neglected, n_e = n_0 exp(eta t). Breakdown criterion: ‘breakdown occurs when the electron density exceeds n_th ~ 10^18 cm^-3 and an initial density of n_0 ~ 10^10 cm^-3, the breakdown condition is then given by eta tau_p = 18’ (ln 1e8 = 18.4); ‘For our experiment, it is more appropriate to use n_th ~ 1.6 x 10^21 cm^-3, the plasma critical density, hence the threshold is reached when eta tau_p ~ 30’. Checks (computed 2026-09-27): ln(1.6e21/1e10) = 25.8, not 30; the critical density eps0 m_e omega^2/e^2 at 780 nm is 1.83e21 cm^-3, not 1.6e21. eta = alpha(E) v_drift with the saturated drift velocity v_drift ~ 2 x 10^7 cm/s (Ref. 12, Hughes 1978, dc); E_th^rms(omega) = E_th^dc (1 + omega^2 tau^2)^(1/2), with the collision time tau ‘less than 0.4 fs for electrons with energy U >= 5-6 eV’ (Ref. 11, Fischetti and DiMaria 1988), so the field acts as a dc field. alpha(E) by Thornber (Eq. 3, formula thornber_alpha) with E_i = 30, E_p = 3.2 and E_kT = 0.01 MV/cm, ‘calculated from U = eEl, where U is the appropriate thermal, phonon, and ionization energy, and l is the corresponding energy relaxation length (l_kT = l_p ~ 5 A, the atomic spacing, and l_i ~ 30 A)’. The dashed curve of Fig. 3 is Eq. (3) ‘corrected by a factor of 1.7’, unexplained (‘a systematic correction, or ... breakdown occurred on the surface first ... The uncertainty of the saturation value of v_drift’). The experimental alpha of Fig. 3 comes from eta tau_p = 30 and eta = alpha v_drift. Du et al. conclude that avalanche ionization dominates down to 150 fs, with photoionisation (tunnelling, Ref. 9, ADK) supplying the seed electrons: ‘For short pulses, E ~ 10^8 V/cm, the tunneling rate is estimated to be w ~ 4 x 10^9 s^-1’; at 1 ps the field ‘will reach a value of 3.5 x 10^7 V/cm (rms) at 0.5 ps before the peak of the pulse, and w ~ 100 s^-1. During a Delta t ~ 100 fs period the electron density can reach n_e ~ n_t[1 - exp(-w Delta t)] ~ 10^11 cm^-3, where n_t ~ 10^22 is the total initial valence band electron density’ (checked: 1e22 x 100 s^-1 x 100 fs = 1e11). These order-of-magnitude estimates are not recorded as datasets. ‘In our experiment, the applied electric field is on the order of a few tens of MV/cm and higher. Under such a high field, the electrons have an average energy of ~5 eV’. Eq. (3) leads to F_th proportional to E^2 tau_p ~ 1/tau_p when E > sqrt(E_p E_i).

LATER VIEWS. Lenzner et al. 1998 (lenzner_1998, their note 18) find their data ‘inconsistent with a sqrt(I) scaling of the avalanche as proposed in Ref. [5]’ (this paper) and favour Stuart et al.’s linear scaling; Liu et al. 2005 (liu_2005) discuss the rising threshold below 1 ps against bulk focusing.

REPRODUCTIONS OF THIS PAPER: stuart_1996 cites Du et al. (another extraction’s record: listed for review).

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

Datasets

No datasets are recorded.

Formulas

du_1994/thornber_alpha

{
  "id": "thornber_alpha",
  "material": "SiO2",
  "equation": "Eq. (3), p. 3072, with the parameters of p. 3073",
  "printed": "\\alpha(E) = \\frac{eE}{U_i} \\exp\\left(-\\frac{E_i}{E(1 + E/E_p) + E_{kT}}\\right)",
  "variable": "E: the (rms) optical field, treated as a dc field; alpha: the impact-ionisation rate per unit length; eta = alpha v_drift with v_drift ~ 2e7 cm/s",
  "range": "all fields (Thornber's interpolation between the thermal, Shockley and Wolff limits); used here up to ~3e8 V/cm",
  "coefficients": {
    "E_i": "30 MV/cm",
    "E_p": "3.2 MV/cm",
    "E_kT": "0.01 MV/cm",
    "v_drift": "2 x 10^7 cm/s",
    "correction": "1.7"
  },
  "check": "The fields follow from U = e E l as the paper says (computed 2026-09-27): E_i l_i = 30 MV/cm x 30 A = 9.0 eV (U_i, not printed: the band gap), E_p l_p = 3.2 MV/cm x 5 A = 0.16 eV (a phonon energy); but E_kT l_kT = 0.01 MV/cm x 5 A = 0.5 meV, whereas kT = 25.9 meV at room temperature would give 0.52 MV/cm: the printed E_kT does not follow from the stated rule (it matters only at low fields). The curves of Fig. 3 are figures only: no printed alpha to reproduce."
}

U_i is not printed. The factor 1.7 multiplies Eq. (3) for the dashed curve of Fig. 3 that fits the data. Thornber, J. Appl. Phys. 52, 279 (1981) (Ref. 13).

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.