Laser Electric Field vs Phase

This plot shows the magnitude of the incident laser electric field (E) as a function of phase (ωt).

Though the field amplitude of the incident laser pulse is spatially non-homogeneous, in this model it is assumed so, relative to the size of the electron (dipole approximation).

Furthermore, it is assumed that the envelope of the driver laser pulse is much longer than the simulated electron dynamics and thus the waveform of the laser pulse looks like a simple AC sinusoidal wave.

Potential Energy vs Distance

This plot shows the net potential energy (U) of the electron as a function of distance (x).

The blue curve is the field-alternated Coulomb potential energy that the valence electron in an atom feels. The time-dependent field tilts the Coulomb potential back and forth, causing the electron to tunnel ionize.

The dotted line is the asymptote describing the potential energy of the electron due to the external laser field alone.

In this model, it is assumed that when the electron undergoes tunnel ionization, it jumps discretely from (0, -U_{I}) to (0, 0) and vice-versa during recombination. Here, U_{I} is the ionization energy for a given atom.

Kinetic Energy at Recombination vs (Left) Ionization Phase and (Right) Recombination Phase

This plot shows the effective kinetic energy (U_{K}/U_{P}) of the electron at recombination as a function of (left) ionization phase (θ_{i}) and (right) recombination phase (θ_{r}). U_{P} is the ponderomotive energy of the electron (average wiggle energy of the electron).

The curves peak at approximately 3.17, i.e., U_{K,max} ≈ 3.17 U_{P}, for θ_{i} = π/10 rad and correspondingly, θ_{r} ≈ 4.39 rad. This prediction was one of the successes of this simple model.

This plot also illustrates the "long" and "short" trajectories of the electron depending on θ_{i}. Trajectories for θ_{i} < π/10 rad are long and those for θ_{i} > π/10 rad are short.

Displacement vs Phase

This plot shows the effective displacement (x) of the electron as a function of phase (ωt).

In this model, it is assumed that the initial position of the electron immediately after the ionization event remains unchanged. Hence, the electron trajectories always start from zero displacement irrespective of the ionization phase.

It is observed that the trajectory of the electron with the maximum distance traveled is the one with θ_{i} = nπ, where, n = 0, 1, 2, ... These trajectories form closed orbits.

Phase Portrait (Velocity vs Displacement)

This plot shows a phase portrait (effective velocity (v) vs effective displacement (x)) describing the trajectory of the ionized electron.

It is observed that this system is not conservative, i.e., the trajectories only take the form of orbits for θ_{i} = nπ, where n = 0, 1, 2, ... For other values of θ_{i}, the trajectories are not closed. Thus, an electron, accelerated to high velocities by the time-dependent laser potential, releases its extra energy at recombination event emitting an HHG photon. The maximum velocity that the electron can attain at recombination is when θ_{i} = π/10 rad.

Recombination Phase as a Function of Ionization Phase

This plot shows the recombination phase (θ_{r}) as a function of ionization phase (θ_{i}).

It is observed that the curve is a decreasing function for 0 ≤ θ_{i} ≤ π/2, with a maximum at (0, 2π) and minimum at (π/2, π/2).

Velocity vs Phase

This plot shows the effective velocity (v) of the electron as a function of phase (ωt).

In this model, it is assumed that the initial velocity of the electron immediately after the ionization event is zero, meaning the electron is at rest. Hence, the electron trajectories always start with zero velocity irrespective of the ionization phase.

It is observed that the trajectory of the electron with the maximum velocity at recombination is the one with θ_{i} = (n + 0.1)π, where, n = 0, 1, 2, ...

Phase Portrait (Kinetic Energy vs Displacement)

This plot shows a phase potrait (effective kinetic energy (U_{K}/U_{P}) vs effective displacement (x)) describing the trajectory of the electron.

It is observed that in orbital trajectories obtained for θ_{i} = nπ, where n = 0, 1, 2, ..., the kinetic energy at recombination is zero. For other values of θ_{i}, the kinetic energy at recombination is non-zero.

Thus, this system is not conservative and the electron gains kinetic energy from the time-dependent laser potential. The maximum kinetic energy that the electron can attain at recombination is when θ_{i} = π/10 rad. This energy is emitted as radiation at recombination and contains high-harmonics of the incident laser radiation.