Scientists discover how a hidden crystal pattern in advanced materials can control how they emit light, offering new ways to guide ultrafast signals using lasers.
(Nanowerk Spotlight) Modern technologies increasingly rely on materials that can process signals at ever faster speeds. But to push beyond the limits of traditional electronics, scientists are turning to light itself—not just as a carrier of information, but as a means to manipulate the quantum states of matter on extremely short timescales.
Achieving this requires more than just shining a laser on a surface. It demands materials that respond to light in ways that are nonlinear, meaning the output is not a simple reflection or absorption but a more complex transformation. In such materials, the interaction between electrons and strong, rapidly varying light fields can produce radiation at frequencies that are integer multiples of the original. This process, known as high-harmonic generation, reveals how electrons move, accelerate, and interact when pushed far from equilibrium.
To understand what happens during high-harmonic generation, imagine an electron in a solid exposed to an intense laser pulse. The electric field of the light distorts the potential landscape inside the material, pulling electrons away from their equilibrium positions and then driving them back, causing them to emit bursts of light at high frequencies.
In simple materials, this behavior can often be predicted from basic symmetry principles. But in quantum materials with complex electronic structures, the details of how electrons move—and what kinds of harmonics they produce—depend sensitively on the material’s geometry and internal symmetries.
One particularly interesting group of such materials is known as nodal line semimetals. In these systems, the energy bands that define the possible electron states do not merely touch at isolated points, as they do in graphene or Weyl semimetals. Instead, they remain in contact along extended lines in momentum space. These continuous intersections, or nodal lines, arise from specific symmetries in the crystal structure.
In some cases, these symmetries are not simple reflections or inversions but more elaborate operations that combine spatial transformations with fractional shifts in atomic position. These are known as nonsymmorphic symmetries. Their presence leads to unusual constraints on how electrons can transition between states, and by extension, on how they can emit light when driven by a laser.
The result is a material that does not behave like a conventional conductor or semiconductor under optical excitation. Instead, its response is shaped by hidden structural rules that determine which electronic transitions are allowed and which are forbidden. These rules influence not only the strength of the emitted harmonics, but also their direction, polarization, and frequency content.
Yet despite the theoretical richness of this scenario, the effects of nonsymmorphic symmetry on nonlinear optical phenomena remain largely untested, especially in two-dimensional systems where symmetry constraints can be particularly strong.
A new study, published in Physical Review Applied (“Nonlinear optical spectroscopy of nodal line semimetals”), addresses this gap. Focusing on two-dimensional nodal line semimetals from the NbSixTe2 family, the authors analyze how glide-mirror symmetry—an operation that reflects and shifts atomic positions within the unit cell—governs high-harmonic generation. They show that this symmetry strictly forbids even-order harmonics, a result that holds even though the material lacks conventional inversion symmetry.
Light interacting with a two-dimensional nodal line semimetal produces high-order harmonics. An incoming laser pulse (red, ω) excites the crystal lattice, generating odd-order harmonics such as the third (yellow, 3ω) and fifth (purple, 5ω). The glide-mirror symmetry of the material enforces strict selection rules, eliminating even harmonics and shaping the polarization and direction of the emitted light. (Image: Navdeep Rana)
In typical noncentrosymmetric materials, the absence of inversion symmetry allows both odd and even harmonics to appear. What makes the result here unusual is that even-order harmonics are entirely absent. This counterintuitive effect arises from the material’s glide-mirror symmetry, which acts like a hidden constraint that forces even harmonics to cancel out across the Brillouin zone.
Dr. Gopal Dixit of Max-Born-Institute for Nonlinear Optics and Short Pulse Spectroscopy in Berlin, a senior author on the study, explained the significance: “These results show how hidden crystal symmetries can control light–matter interactions in unexpected ways. By tuning light polarization, we can selectively enhance or suppress optical signals, opening up powerful new possibilities for ultrafast technologies.”
The researchers developed a tight-binding model of the two-dimensional material, capturing both intrachain and interchain electron motion. Electrons can hop along atomic zigzag chains or between them. Depending on the polarization of the laser, one mechanism dominates the harmonic response. When the laser is polarized along the chains, intrachain processes are the main contributors. When it is polarized across them, interchain effects dominate.
The emitted harmonics include components both parallel and perpendicular to the direction of the incoming laser field. In some cases, the harmonics emerge at right angles to the input light, a behavior reminiscent of nonlinear Hall effects. The authors attribute this to the interplay between symmetry constraints and anisotropic charge motion in the lattice.
One of the most distinctive features of the harmonic response is its angular dependence. As the laser polarization is rotated, the harmonic yield follows a twofold symmetric pattern. Each harmonic order peaks at a different angle, with the third harmonic strongest at around 135 degrees, the fifth at 90 degrees, and others at intermediate angles. The full pattern resembles a butterfly when plotted on a polar graph—a direct visual signature of the lattice anisotropy and nodal line orientation.
The study also investigates how the ellipticity of the laser, or the shape traced by its electric field vector, affects harmonic emission. With the major axis of the laser aligned along the chains, some lower-order harmonics such as the third and fifth are strongest under circular polarization, while higher-order harmonics show more complex dependencies.
The seventh harmonic, for instance, displays sharp minima and maxima at specific ellipticity values. When the laser is instead polarized across the chains, all harmonics decrease as the ellipticity increases, disappearing entirely under circular polarization.
This behavior reflects the balance between two underlying mechanisms: interband transitions, where electrons move between the conduction and valence bands, and intraband motion, where electrons accelerate within a single band. Interband processes are less sensitive to the laser’s shape, while intraband processes depend directly on how the electric field moves electrons through the material’s momentum space. The result is a spectrum where different harmonics respond differently to changes in polarization and ellipticity.
Navdeep Rana, lead author of the paper, emphasized the broader implications: “Lightwave-driven devices are the frontier of ultrafast science. Our work shows that the key to unlocking their full potential lies in the hidden symmetries of quantum materials.”
The authors’ model is based on a Dirac-Su-Schrieffer-Heeger Hamiltonian tailored to the NbSixTe2 lattice and reproduces key features seen in ab initio calculations. The framework is designed to be generalizable, providing a platform for investigating nonlinear optical responses in other low-dimensional or topologically structured materials.
By showing that symmetry—particularly nonsymmorphic glide-mirror symmetry—can impose strict selection rules even in inversion-broken systems, the study reveals a new way to think about light–matter interaction. The ability to control harmonic generation by adjusting laser polarization and ellipticity, combined with the material’s symmetry-governed constraints, suggests a clear path toward engineered nonlinear optical responses.
These findings have potential implications for ultrafast optical switches, directional light sources, and compact platforms for photonic and quantum information processing.
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