How do you study a laser firing for one-quadrillionth of a second?

I’m grateful to Mukund Thattai, at the National Centre for Biological Sciences, Bengaluru, for explaining many of the basic concepts at work in the following article.

An important application of lasers today is in the form of extremely short-lived laser pulses used to illuminate extremely short-lived events that often play out across extremely short distances. The liberal use of ‘extreme’ here is justified: these pulses last for no more than one-quadrillionth of a second each. By the time you blink your eye once, 100 trillion of these pulses could have been fired. Some of the more advanced applications even require pulses that last 1,000-times shorter.

In fact, thanks to advances in laser physics, there are branches of study today called attophysics and femtochemistry that employ such fleeting pulses to reveal hidden phenomena that many of the most powerful detectors may be too slow to catch. The atto- prefix denotes an order of magnitude of -18. That is, one attosecond is 1 x 10-18 seconds and one attometer is 1 x 10-18 metres. To quote from this technical article, “One attosecond compares to one second in the way one second compares to the age of the universe. The timescale is so short that light in vacuum … travels only about 3 nanometers during 1 attosecond.”

One of the more common applications is in the form of the pump-probe technique. An ultra-fast laser pulse is first fired at, say, a group of atoms, which causes the atoms to move in an interesting way. This is the pump. Within fractions of a second, a similarly short ‘probe’ laser is fired at the atoms to discern their positions. By repeating this process many times over, and fine-tuning the delay between the pump and probe shots, researchers can figure out exactly how the atoms responded across very short timescales.

In this application and others like it, the pulses have to be fired at controllable intervals and to deliver very predictable amounts of energy. The devices that generate these pulses often provide these features, but it is often necessary to independently study the pulses and fine-tune them according to different applications’ needs. This post discusses one such way and how physicists improved on it.

As electromagnetic radiation, every laser pulse is composed of an electric field and a magnetic field oscillating perpendicular to each other. Of these, consider the electric field (only because it’s easier to study; thanks to Maxwell’s equations, what we learn about the electric field can be inferred accordingly for the magnetic field as well):

Credit: Peter Baum & Stefan Lochbrunner, LMU München Fakultät für Physik, 2002

The blue line depicts the oscillating electric wave, also called the carrier wave (because it carries the energy). The dotted line around it depicts the wave’s envelope. It’s desirable to have the carrier’s crest and the envelope’s crest coincide – i.e. for the carrier wave to peak at the same point the envelope as a whole peaks. However, trains of laser pulses, generated for various applications, typically drift: the crest of every subsequent carrier wave is slightly more out of step with the envelope’s crest. According to one paper, it arises “due to fluctuations of dispersion, caused by changes in path length, and pump energy experienced by consecutive pulses in a pulse train.” In effect, the researcher can’t know the exact amount of energy contained in each pulse, and how that may affect the target.

The extent to which the carrier wave and the envelope are out of step is expressed in terms of the carrier-envelope offset (CEO) phase, measured in degrees (or radians). Knowing the CEO phase is crucial for experiments that involve ultra-precise measurements because the phase is likely to affect the measurements in question, and needs to be adjusted for. According to the same paper, “Fluctuations in the [CEO phase] translate into variations in the electric field that hamper shot-to-shot reproducibility of the experimental conditions and deteriorate the temporal resolution.”

Ignore all the symbols and notice the carrier wave – especially how its peak within the envelope shifts with every next pulse. The offset between the two peaks is called the carrier-envelope offset phase. Credit: HartmutG/Wikimedia Commons, CC BY-SA 3.0

This is why, in turn, physicists have developed techniques to measure the CEO phase and other properties of propagating waves. One of them is called attosecond streaking. Physicists stick a gas of atoms in a container, fire a laser at it to ionise them and release electrons. The field to be studied is then fired into this gas, so its electric-wave component pushes on these electrons. Specifically, as the electric field’s waves rise and fall, they accelerate the electrons to different extents over time, giving rise to streaks of motion – and the technique’s name. A time-of-flight spectrometer measures this streaking to determine the field’s properties. (The magnetic field also affects the electrons, but it suffices to focus on the electric field for this post.)

This sounds straightforward but the setup is cumbersome: the study needs to be conducted in a vacuum and electron time-of-flight spectrometers are expensive. But while there are other ways to measure the wave properties of extreme fields, attosecond streaking has been one of the most successful (in one instance, it was used to measure the CEO phase at a shot frequency of 400,000 times per second).

As a workaround, physicists from Germany and Canada recently reported in the journal Optica a simpler way, based on one change. Instead of setting up a time-of-flight spectrometer, they propose using the pushed electrons to induce an electric current in electrodes, in such a way that the properties of the current contain information about the CEO phase. This way, researchers can drop both the spectrometer and, because the electrons aren’t being investigated directly, the vacuum chamber.

The researchers used fused silica, a material with a wide band-gap, for the electrodes. The band-gap is the amount of energy a material’s electrons need to be imparted so they can ‘jump’ from the valence band to the conduction band, turning the material into a conductor. The band-gap in metals is zero: if you placed a metallic object in an electric field, it will develop an internal current linearly proportional to the field strength. Semiconductors have a small band-gap, which means some electric fields can give rise to a current while others can’t – a feature that modern electronics exploit very well.

Dielectric materials have a (relatively) large band-gap. When it is exposed to a low electric field, a dielectric won’t conduct electricity but its internal arrangement of positive and negative charges will move slightly, creating a minor internal electric field. But when the field strength crosses a particular threshold, the material will ‘break down’ and become a conductor – like a bolt of lightning piercing the air.

Next, the team circularly polarised the laser pulse to be studied. Polarisation refers to the electric field’s orientation in space, and the effect of circular polarisation is to cause the electric field to rotate. And as the field moves forward, its path traces a spiral, like so:

A circularly polarised electric field. Credit: Dave3457/Wikimedia Commons

The reason for doing this, according to the team’s paper, is that when the circularly polarised laser pulse knocks electrons out of atoms, the electrons’ momentum is “perpendicular to the direction of the maximum electric field”. So as the CEO phase changes, the electrons’ directions of drift also change. The team used an arrangement of three electrodes, connected to each other in two circuits (see diagram below) such that the electrons flowing in different directions induce currents of proportionately different strengths in the two arms. Amplifiers attached to the electrodes then magnify these currents and open them up for further analysis. Since the envelope’s peak, or maximum, can be determined beforehand as well as doesn’t drift over time, the CEO phase can be calculated straightforwardly.

(The experimental setup, shown below, is a bit different: since the team had to check if their method works, they deliberately insert a CEO phase in the pulse and check if the setup picks up on it.)

The two tips of the triangular electrodes are located 60 µm apart, on the same plane, and the horizontal electrode is 90 µm below the plane. The beam moves from the red doodle to the mirror, and then towards the electrodes. The two wedges are used to create the ‘artificial’ CEO phase. Source:

The team writes towards the end of the paper, “The most important asset of the new technique, besides its striking simplicity, is its potential for single-shot [CEO phase] measurements at much higher repetition rates than achievable with today’s techniques.” It attributes this feat to attosecond streaking being limited by the ability of the time-of-flight spectrometer whereas its setup is limited, in the kHz range, only by the time the amplifiers need to boost the electric signals, and in the “multi-MHz” range by the ability of the volume of gas being struck to respond sufficiently rapidly to the laser pulses. The team also states that its electrode-mediated measurement method renders the setup favourable to radiation of longer wavelengths as well.

Featured image: A collection of lasers of different frequencies in the visible-light range. Credit: 彭嘉傑/Wikimedia Commons, CC BY 2.5 Generic.

Amorphous topological insulators

Physicists from the Indian Institute of Science, Bengaluru, have found that topological insulators needn’t always be crystals – as has been thought.

A topological insulator is a material that conducts electricity only on its surface. Everything below, through the bulk of the material, is an insulator. An overly simplified way to understand this is in terms of the energies and momenta of the electrons in the material.

The electrons that an atom can spare to share with other atoms – and so form chemical bonds – are called valence electrons. In a metal, these electrons can have various momenta, but unless they have a sufficient amount of energy, they’re going to stay near their host atoms – i.e. within the valence band. If they do have energies over a certain threshold, then they can graduate from the valence band to the conduction band, flowing throw the metal and conducting electricity.

In a topological insulator, the energy gap between the valence band and the conduction band is occupied by certain ‘states’ that represent the material’s surface. The electrons in these states aren’t part of the valence band but they’re not part of the conduction band either, and can’t flow throw the entire bulk.

The electrons within these states, i.e. on the surface, display a unique property. Their spins (on their own axis) are coupled strongly with their motion around their host atoms. As a result, theirs spins become aligned perpendicularly to their momentum, the direction in which they can carry electric charge. Such coupling staves off an energy-dissipation process called Umklapp scattering, allowing them to conduct electricity. Detailed observations have shown that the spin-momentum coupling necessary to achieve this is present only in a few-nanometre-thick layer on the surface.

If you’re talking about this with a physicist, she will likely tell you at this point about time-reversal symmetry. It is a symmetry of nature that is said to (usually) ‘protect’ a topological insulator’s unique surface states.

There are many fundamental symmetries in nature. In particle physics, if a force acts similarly on left- and right-handed particles, it is said to preserve parity (P) symmetry. If the dynamics of the force are similar when it is acting against positively and negatively charged particles, then charge conjugation (C) symmetry is said to be preserved. Now, if you videotaped the force acting on a particle and then played the recording backwards, the force must be seen to be acting the way it would if the video was played the other way. At least if it did it would be preserving time-reversal (T) symmetry.

Physicists have known some phenomena that break C and P symmetry simultaneously. T symmetry is broken continuously by the second law of thermodynamics: if you videographed the entropy of a universe and then played it backwards, entropy will be seen to be reducing. However, CPT symmetries – all together – cannot be broken (we think).

Anyway, the surface states of a topological insulator are protected by T symmetry. This is because the electrons’ wave-functions, the mathematical equations that describe some of the particles’ properties, do not ‘flip’ going backwards in time. As a result, a topological insulator cannot lose its surface states unless it undergoes some sort of transformation that breaks time-reversal symmetry. (One example of such a transformation is a phase transition.)

This laboured foreword is necessary – at least IMO – to understand what it is that scientists look for when they’re looking for topological insulators among all the materials that we have been, and will be able, to synthesise. It seems they’re looking for materials that have surface states, with spin-momentum coupling, that are protected by T symmetry.

Physicists from the Indian Institute of Science, Bengaluru, have found that topological insulators needn’t always be crystals – as has been thought. Instead, using a computer simulation, Adhip Agarwala and Vijay Shenoy, of the institute’s physics department, have shown that a kind of glass also behaves as a topological insulator.

The band theory described earlier is usually described with crystals in mind, wherein the material’s atoms are arranged in a well-defined pattern. This allows physicists to determine, with some amount of certainty, as to how the atoms’ electrons interact and give rise to the material’s topological states. In an amorphous material like glass, on the other hand, the constituent atoms are arranged randomly. How then can something as well-organised as a surface with spin-momentum coupling be possible on it?

As Michael Schirber wrote in Physics magazine,

In their study, [Agarwala and Shenoy] assume a box with a large number of lattice sites arranged randomly. Each site can host electrons in one of several energy levels, and electrons can hop between neighboring sites. The authors tuned parameters, such as the lattice density and the spacing of energy levels, and found that the modeled materials could exhibit symmetry-protected surface currents in certain cases. The results suggest that topological insulators could be made by creating glasses with strong spin-orbit coupling or by randomly placing atoms of other elements inside a normal insulator.

The duo’s paper was published in the journal Physical Review Letters on June 8. The arXiv preprint is available to read here. The latter concludes,

The possibility of topological phases in a completely random system opens up several avenues both from experimental and theoretical perspectives. Our results suggest some new routes to the laboratory realization of topological phases. First, two dimensional systems can be made by choosing an insulating surface on which suitable [atoms or molecules] with appropriate orbitals are deposited at random (note that this process will require far less control than conventional layered materials). The electronic states of these motifs will then [interact in a certain way] to produce the required topological phase. Second is the possibility of creating three dimensional systems starting from a suitable large band gap trivial insulator. The idea then is to place “impurity atoms”, again with suitable orbitals and “friendly” chemistry with the host… The [interaction] of the impurity orbitals would again produce a topological insulating state in the impurity bands under favourable conditions.

Agarwala/Shenoy also suggest that “In realistic systems the temperature scales over which one will see the topological physics … may be low”, although this is not unusual. However, they don’t suggest which amorphous materials could be suitable topological insulators.

Thanks to and its nonexistent autosave function, I had to write the first half of this article twice. Not the sort of thing I can forgive easily, less so since I’m loving everything else about it.