Engineering & Science
Keldysh Parameter (γ_K): Multiphoton vs. Tunneling Ionization Formula
Calculate Leonid Keldysh’s strong-field adiabaticity parameter γ_K = 5.1232 * E_ph * SQRT(m_eff * I_p) / F to classify ultrafast laser excitation into multiphoton (γ_K > 1) or optical tunneling (γ_K < 1) regimes.
One dimensionless number unifies multiphoton absorption and Zener optical tunneling!
In 1965, Leonid V. Keldysh proved that multiphoton ionization and strong-field quantum tunneling are opposite limits of a single process governed by the ratio of tunneling time to optical cycle period: γ_K = ω * SQRT(2 * m* * I_p) / (e * F)! Expressed in ultrafast laser laboratory units—photon energy E_ph = ħω (eV), reduced effective mass m_eff = m*/m_0 (1.0 for free-electron atomic ionization), ionization potential or bandgap I_p (eV), and peak laser electric field F (V/nm)—the prefactor SQRT(2 * m_0 * e) / (ħ * 10^9) equals 5.1232 (eV^-3/2 · V/nm): γ_K = 5.1232 * E_ph * SQRT(m_eff * I_p) / F!
Compare an 800 nm Ti:Sapphire femtosecond pulse (E_ph = 1.55 eV) ionizing hydrogen (m_eff = 1.0, I_p = 13.6 eV): at moderate field F = 15 V/nm, γ_K = 1.952 > 1.5 (perturbative multiphoton ionization); cranking peak field to F = 45 V/nm drives γ_K down to 0.651 < 1 (attosecond optical-field tunneling regime, foundation of High-Harmonic Generation)!
| 800 nm Ti:Sapph Field F (Eph=1.55 eV, Ip=13.6 eV) | Keldysh Parameter γ_K | Strong-Field Ionization Regime | Physical Mechanism |
|---|---|---|---|
| F = 15.0 V/nm (~3 × 10^13 W/cm^2) | 1.952 | Multiphoton Ionization | Simultaneous 9-Photon Absorption |
| F = 25.0 V/nm (Crossover Intensity) | 1.171 | Non-Adiabatic Crossover | Mixed Multiphoton + Barrier Suppression |
| F = 45.0 V/nm (~2.7 × 10^14 W/cm^2) | 0.651 | Optical Tunneling | Quasi-Static Barrier Tunneling / HHG |
Keldysh Strong-Field Adiabaticity Parameter γ_K
Computes γ_K from photon energy B2 (eV), effective mass ratio B3 (m*/m_0), ionization/bandgap energy B4 (eV), and peak electric field B5 (V/nm).
=IF(OR(B2 <= 0, B3 <= 0, B4 <= 0, B5 <= 0), "Invalid Laser Parameters", ROUND(5.1232 * B2 * SQRT(B3 * B4) / B5, 3))Ionization Regime Classifier (Tunneling vs. Crossover vs. Multiphoton)
Classifies strong-field interaction using standard attosecond physics thresholds (< 0.8 optical tunneling, 0.8–1.5 crossover, > 1.5 multiphoton).
=IF(OR(B2 <= 0, B3 <= 0, B4 <= 0, B5 <= 0), "Invalid Laser Parameters", IF(5.1232 * B2 * SQRT(B3 * B4) / B5 < 0.8, "Optical Tunneling", IF(5.1232 * B2 * SQRT(B3 * B4) / B5 <= 1.5, "Non-Adiabatic Crossover", "Multiphoton Ionization")))Try the example
Use the original values or change them to test your understanding.