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----- FREQUENTLY ASKED QUESTIONS -----
What is the simpolator?
It is a pedagogic tool to help you understand tunnel ionization of atoms by strong laser fields. Classical Newtonian trajectories are launched at selected times and move subsequently in the laser field only. Phases are calculated as action integrals and the trajectories are added coherently (unless the calculation mode is set to classical) to calculate the photoelectron momentum distribution. The plotted image can be viewed as a two-dimensional slice through the cylindrically symmetric three-dimensional momentum distribution from an atom. The symmetry axis is the laser field direction (polarization axis), chosen horizontal in the plot. The distribution shows various interference patterns that are due to interference of different types of trajectories.
How do I use it?
The panel at the bottom shows a sinusoidal time-dependent electric field, which can be understood as a section of a strong laser pulse. Click to select those quarter cycles of the field where you want to allow ionization. By choosing "direct", or "long"/"short" rescattering trajectories, you can include electrons that scatter once from the parent ion (only scattering at the first return is currently supported). Click the run button to evaluate the result. Illumination of the simpolator logo in the bottom right corner indicates that the simulation is running. The rest should be self-explanatory.
What is the underlying theory?
At each moment within the selected time intervals, ionization is implemented such that many electron trajectories are launched, all with zero initial position and zero velocity component along the field. The initial transverse momenta v are distributed according to the adiabatic tunneling formula exp(-2*(2Ip+v2)1.5/(3|E(t)|)), where E(t) is the electric field strength at ionization time t. The expression takes automatically care of different weights for different ionization times. The ionization potential Ip = 0.5 au = 13.6 eV of the hydrogen atom is used always. For the coherent summation of trajectories it is required that every trajectory carries a complex amplitude. The initial amplitude is taken as the square root of the tunneling weight (see above) times exp(iIpt) according to the time-evolution of the bound state pior to ionization (assuming field-free evolution of the bound state before ionization). The trajectories then evolve according to Newton's law for classical particles including only the force due to the laser field while neglecting the force of the parent ion on the outgoing electron [Corkum, Burnett, Brunel, PRL 62, 1259 (1989)]. Electrons with zero initial transverse momentum move along a straight line, which implies that they can return to the initial position, depending on the phase of the laser field at the time of ionization. In physical terms, returning electrons can scatter elastically from the parent ion ("rescattering") [Paulus et al., J. Phys. B 27, L703 (1994)]. To incorporare this phenomenon, an additional set of trajectories is launched at the time of return, using a uniform distribution of scattering angles. The overall probability of rescattering is adjustable using the slider in the settings panel. The scattered electrons follow again Newtonian trajectories in the laser field only. Along every trajectory, the accumulated phase is calculated according to the time integral of the kinetic energy ("action integral") [Bian et al., PRA 84, 043420 (2011)]. Finally, the complex amplitudes of all trajectories ending up in the same momentum bin are added coherently and the modulus square gives the signal at each momentum bin. These bins are small squares in the momentum plane. Their size is set by the simpolator according to the laser parameters. (Depending on the settings, however, interferences in the high-energy region are not always well resolved, causing Moiré artefacts, see below.) Finally, the distribution is normalized by multiplication with a global factor such that the second highest value is set to unity.
What can I learn from it?
Depending on which trajectories you select, various types of interference phenomena show up in the calculated momentum distributions:
• Choosing direct electrons from two adjacent quarter cycles with opposite signs produces "intracycle" interference.
► Choose settings for intracycle interference
• Choosing direct electrons from two quarter cycles with a relative delay of one laser period produces rings of constant energy ("ATI rings"). Each ATI ring corresponds to a specific number of absorbed photons.
► Choose settings for ATI rings
• Including long and short rescattering trajectories from one quarter cycle leads to rings of interference between these two types of trajectories.
► Choose settings for short-long interference
• Choosing all trajectories from one half cycle gives rise to interference between direct and rescattered electrons ("holography").
► Choose settings for holography
You forgot to put units on the momentum axes.
Yes, agreed. The momenta are in atomic units. Therefore, a momentum of p = 1 corresponds to an electron with energy p2/2 = 0.5 = 13.6eV.
Does the code calculate results in real time or does it just show pre-calculated data?
Calculations are done on the fly. Note that the Newtonian trajectories are not numerical but follow analytical expressions because only the force by the laser field is accounted for, while the binding force is neglected ("simple man's model"). Thus the numerical work lies only in the sampling of the trajectories and in the coherent summation. Unless you use an old browser, the simpolator makes use of parallel computing by launching several "webworkers" that run in parallel threads. By clicking the settings button, you will see a display showing the number of threads. (This is browser and machine dependent.)
Sometimes I see small circular patterns at high energies. They look like raindrops that sit on my graph. What are they?
These are visual artefacts known as Moiré patterns. They appear when the resolution of the bitmap is not enough to resolve fine interference structures. The Moiré patterns are almost unavoidable when the picture exhibits many ATI rings because the ring spacing in momentum space becomes smaller and smaller with increasing momentum. The simpolator lets you check this easily: if you hold down the mouse button on the green scale slider and move it slightly, the upper half of the plot is replotted with lower resolution. This resolution stays until you release the mouse button. If patterns change upon change of resolution, it is an indication that you are seeing Moiré patterns. Note however, that this does not work well on mobile devices (or small window size) because - for speed reasons - the simpolator uses very low resolution when moving the scale slider in such a situation.
For some parts of the field I cannot choose rescattering. Why not?
Electrons launched in a quarter cycle with ascending field strength do not return to the parent ion. Therefore, only direct trajectories are selectable in these quarter cycles.
What is the programming language? Can I see the source code?
The simulation is written in javascript and embedded in an html webpage. Therefore it runs in every browser no matter whether it is opened on a PC or mobile device. Desktop browsers give you the possibility to view the source code of this webpage. However, you might see not the actual code but only the wrapper page, which contains an "iframe" link to the actual simpolator page named "simpolatorbackbone.html". Please follow that link and view the source code there. You can easily download the simpolator files to your computer and run the simpolator offline. (You need "simpolator.html", "simpolatorbackbone.html", "simpolator.css" and the folder "images" with the files "blowuparrow.png", "gear2.png", "simpolator.png", "simpolator2.png", "sizedown.png".)
----- RARELY ASKED QUESTIONS -----
What is the reason for the peculiar design of the simpolator?
The retro look reminds us that the underlying theory is really simple and robust. Despite its simplicity, it can support the interpretation of experiments even in present-day research.
Does the simpolator offer bachelor projects?
Yes. Possibly it makes sense to develop an SFA (strong-field approximation) mode of the simulation to improve the description of quantum mechanical aspects of strong-field ionization.
Can I change and/or redistribute the code?
You can do with it whatever you want, but you are asked to acknowledge the author (Manfred Lein, University of Hannover) in an appropriate manner if you distribute the code or if you use it for teaching purposes.
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