Tunnelling or multiphoton?
A laser pulse can pull electrons out of a metal in two different ways. If its field changes slowly, electrons tunnel through the barrier the field bends down. If it oscillates too fast for that, they escape by absorbing several photons at once. The Keldysh parameter γ tells you which picture holds. Set the laser, the metal and the field enhancement at the surface, or drag the point on the map.
Laser
Surface
Where you are
Horizontal: field at the surface, βF. Vertical: wavelength. Both axes are logarithmic. Lines mark constant γ. Drag the point, or use the arrow keys when the map has focus.
What γ measures
γ compares two times: how long an electron needs to tunnel through the barrier, and how long the laser field keeps the barrier bent down. When the field is strong and slow, tunnelling finishes well within half a cycle and the current follows the instantaneous field, much like static field emission. When the field is weak or fast, the barrier flips back before tunnelling can happen, and electrons escape by absorbing photons instead.
γ² = W / (2 Up), Up = e² F² / (4 m ω²)
Here ω is the laser angular frequency, m and e the electron mass and charge, W the work function, and F the peak field at the surface, including any enhancement β.
Reading the map
The boundaries are soft. γ ≈ 1 marks the crossover, but neither picture switches off sharply. The shading at 0.5 and 2 is a guide, not a law.
Tips move you left to right. Field enhancement at a sharp emitter raises the surface field by β without raising the pulse intensity, which is how tip experiments reach the tunnelling side at modest laser power.
Simplifications. The photon count ignores the ponderomotive shift and Schottky lowering of the barrier, and the intensity assumes a linearly polarised pulse in vacuum before enhancement.
- L. V. Keldysh, Sov. Phys. JETP 20, 1307 (1965).
- R. Bormann, M. Gulde, A. Weismann, S. V. Yalunin, C. Ropers, Phys. Rev. Lett. 105, 147601 (2010).