physics.md

Physics coverage

Processes

The fundamental processes included in Ptarmigan are:

• photon emission (nonlinear Compton scattering)
• electron-positron pair creation (nonlinear Breit-Wheeler)

These provide the building blocks for higher order processes, like electromagnetic showers.

Process	$e^\pm$	polarization $\gamma$	laser	QED	Modes classical	mod. clas.
$e^\pm \to e^\pm \gamma$	averaged (initial), summed (final)	arbitrary	LP / CP	LMA / LCFA	LMA / LCFA	LCFA
$\gamma \to e^+ e^-$	summed	arbitrary	LP / CP	LMA / LCFA	n/a*	n/a*

LP = linear polarization, CP = circular polarization

*Pair creation has no classical equivalent.

Particle tracking

Particles (electrons, positrons and photons) are tracked as they travel on classical trajectories through the laser pulse. Tracking stops automatically when the particle is deemed to have left the strong-field region (which may occur at different times for different particles). The classical trajectory is determined either by the Lorentz force equation (in LCFA mode) or the relativistic ponderomotive force equation (in LMA mode): see here for details.

QED events occur pseudorandomly along the particle's trajectory with a frequency that is determined by the relevant probability rate. When these events occur, the original particle may be recoiled or destroyed, and new particles created. These new particles are tracked in the same way as the original ones, except that they are created inside the strong-field region at the specific location in space and time.

Ptarmigan tracks one particle at a time, queueing any newly created ones, until the entirety of the incident beam and all its daughter products have been processed. Each particle in the simulation is given a unique ID.

LMA or LCFA

Ptarmigan models particle dynamics and strong-field QED events using one of two approximations:

• the locally monochromatic approximation (LMA)
• the locally constant crossed field approximation (LCFA).

Important

The LMA is available for a₀ ≤ 20 and η = χ / a₀ ≤ 2.

Note

The LCFA is available for arbitrary values of a₀ and η.

In the LMA, the laser pulse is treated as slices of monochromatic plane wave with defined amplitude and frequency. Particle trajectories are only defined at the cycle-averaged level, using the quasimomentum $q^\mu$, and not at scales smaller than the laser wavelength. The strong-field QED rates account for interference at this scale.

In the LCFA, the laser pulse is treated as slices of constant, crossed field. Particle trajectories are defined at all timescales, using the kinetic momentum $\pi^\mu$. Strong-field QED events are assumed to occur instantaneously, which is reasonable if the formation length is much smaller than the laser wavelength.

The electron trajectory and a photon emission event, as viewed in the LMA (left) and LCFA (right). Adapted from this paper.

The picture above shows the conceptual differences between the LMA and the LCFA for a photon emission event. In the LCFA, the fast oscillation is resolved at the level of the classical trajectory. This imprints itself on the angular structure of the radiation as the emitted photon is strongly beamed in the (instantaneous) forward direction. An equivalent structure emerges in the LMA due to quasimomentum conservation, because the fast oscillation is folded into the QED rates. However, if the photon formation length is comparable in size to the laser wavelength, this angular structure is modified by interference effects in a way that the LCFA cannot capture.

The LMA is default mode in Ptarmigan. To select the LCFA instead, ensure that the control block contains:

control:
    ... # other keys
    lcfa: true

If this key is absent (or false), the LMA is selected instead.

QED or classical

In QED, photon emission (and pair creation) are stochastic events and occur randomly along the particle trajectory. As an alternative, Ptarmigan can model laser-beam collisions using classical electrodynamics, which is deterministic. The electron does not recoil at individual emission events. Instead, energy loss is accounted for by a radiation reaction force, here in the Landau-Lifshitz prescription.

The choice of QED or classical is orthogonal to the choice of LMA or LCFA, i.e. interference effects in the classical domain can be accessed using:

control:
    ... # other keys
    classical: true
    lcfa: false

If classical is absent (or set to false), Ptarmigan uses the full QED model.

A quantum-corrected classical model

Classical electrodynamics overpredicts the amount of radiated energy, because the spectrum does not contain a cutoff at the electron energy, $\omega' < E / \hbar$. Correcting the spectrum for this cutoff, and reducing the radiated power and radiation-reaction force accordingly, results in a "modified classical" model:

control:
    ... # other keys
    classical: gaunt_factor_corrected
    lcfa: true

This model is only available under the LCFA. As it uses a deterministic radiation-reaction force, it does not include stochastic effects.

Validity

Ptarmigan is a single-particle + prescribed fields code and it therefore ignores space-charge effects (negligible for relativistic bunches) and depletion of the laser pulse.

In LMA mode, expect Ptarmigan to be accurate for laser pulses with $N \gtrsim 4$ and $w_0 \gtrsim 2 \lambda$, where $N$ is the number of cycles corresponding to the pulse duration and $w_0$ is the waist (focal spot size). There is no particular restriction on $a_0$, unless modelling classical RR using the LMA. In this case, energy losses must be small per cycle and therefore it is necessary to have $a_0^2 \eta_e \lesssim 30$.

LCFA mode begins to work when $a_0 \gtrsim 5$ and is a good substitute for LMA mode when $a_0 \gtrsim 10$. Even so, radiation spectra $dN/d\omega'$ are only accurate for photon energies satisfying $\omega' / (\gamma_0 m) > 2 \eta_e / (1 + a_0^2 + 2 \eta_e) \sim 2 \chi_e / a_0^3$.