The study, published in Frontiers in Photonics, focused on hyper-Rayleigh scattering, or HRS, a technique in which molecules in solution scatter light at precisely double the frequency of the incoming laser. Since Clays and Persoons introduced the method in 1991, it has become a workhorse for measuring the first hyperpolarizability of organic chromophores and for probing symmetry, molecular orientation, micelle formation and supramolecular assembly. But HRS is a second-order process, meaning the scattered intensity scales with the square of the laser intensity. Any effect that quietly reduces the intensity at the focus therefore exacts a punishing, quadratic toll on the signal.

Two culprits dominate. The first is thermal lensing: even weakly absorbing samples heat up under an intense beam, and because the refractive index of most liquids changes with temperature, the heated region behaves like an extra lens that shifts and smears the focus. The second is Kerr lensing, a purely nonlinear effect in which the refractive index itself depends on the instantaneous light intensity, causing the beam to self-focus when peak power climbs high enough. Both effects steal intensity from the intended focal point and can slide the scattering volume out of alignment with the detection optics. Above a certain threshold, the measured signal stops following the expected quadratic dependence on laser power, and the data become unreliable.

The research team, led by Ahmet Dok, Thibaut Legat, Yovan de Coene and Stijn Van Cleuvenbergen, worked in a relatively unexplored middle regime of laser parameters. Traditional HRS experiments use either Q-switched nanosecond lasers firing a few pulses per second with millijoule energies, or mode-locked oscillators running at roughly 80 megahertz with nanojoule pulses. The Leuven setup instead employed an amplified femtosecond laser emitting 220-femtosecond pulses at 1030 nanometres, tunable between 100 and 500 kilohertz, with microjoule pulse energies. This regime promises a balance between per-pulse and per-second nonlinear response while leaving microseconds between pulses for heat to dissipate, but the team found that neither thermal nor Kerr artifacts vanish automatically.

As test systems they chose pure water, the archetypal biological solvent, and a 10-millimolar solution of para-nitroaniline, a classic chromophore with a strong second-order response, in methanol. The key diagnostic was what they call the quadratic range: the window of input power over which the HRS signal still tracks the expected quadratic scaling, detected algorithmically as the point where deviations exceed 7.5 percent. Under their starting conditions, with a 50-millimetre focusing lens at 500 kilohertz, water remained quadratic only up to about 112 milliwatts, while the pNA solution broke down at a mere 18 milliwatts. Intriguingly, methanol solutions of pNA and of the fluorescent dye Rhodamine 6G showed nearly identical limits, revealing that the solvent, not the solute, sets the boundary.

By varying the repetition rate, the researchers could disentangle the two mechanisms. For water, lowering the rate from 500 to 200 and then 100 kilohertz expanded the quadratic range, and plotting the data against peak power revealed that breakdown always occurred at the same fluence of roughly 14 to 16 millijoules per square centimetre, the signature of Kerr lensing, which depends on peak power. But at 500 kilohertz the limit instead tracked average power, indicating a shift to thermal dominance. The methanolic pNA solution behaved differently: its limit stayed pinned at 10 to 15 milliwatts of average power regardless of repetition rate, the fingerprint of thermal lensing throughout. The distinction matters because it dictates the remedy: thermal problems call for less average power or better heat dissipation, while Kerr problems call for lower peak intensity.

The first obvious fix, simply defocusing with a longer-focal-length lens, produced a surprise. Theory predicts that if the collection optics capture the whole focal volume, the detected HRS signal should be roughly independent of focal length at fixed average power, because the drop in intensity is offset by the larger scattering volume. Instead, weaker focusing shrank the quadratic range. The explanation lies in the fixed collection geometry: lensing gradually shifts the effective focus along the optical axis, and with long focal lengths the enlarged scattering volume fills the camera’s field of view, so even small shifts push part of the signal out of the imaged region. Infrared imaging of the focus confirmed the drift, with the pNA solution showing a pronounced shift as power increased.

The winning strategy combined two geometrical tricks. Cylindrical lenses, which focus the beam only along one axis while leaving the other dimension at its original 3.6-millimetre width, slash the power density and dramatically enlarge the focal volume. In water at 100 kilohertz the quadratic range jumped from 38 milliwatts to 470 milliwatts, and in pNA from around 15 milliwatts to roughly 250 milliwatts. Shrinking the cuvette path length from 10 millimetres to 1 millimetre, with a Rayleigh length far exceeding the cell thickness, then attacked both lensing mechanisms at once, since each scales with path length. With this final configuration, water stayed quadratic up to 4148 milliwatts at 500 kilohertz, close to the laser’s full 4500-milliwatt output, while pNA remained quadratic to about 2600 milliwatts at all repetition rates.

The performance gains were striking. Relative to the starting configuration, water measurements improved roughly 3.5-fold in usable signal, while the thermally limited pNA system enjoyed a roughly 63-fold boost, nearly two orders of magnitude. Both systems reached sub-millisecond time resolution, and crucially, ultraviolet-visible spectra recorded before and after each experiment confirmed that the pNA had not photodegraded. For a technique whose weak signals have traditionally forced exposures of seconds or longer, that combination of speed and sample integrity is transformative for tracking fast processes such as nucleation, aggregation, interfacial adsorption and the shifting hydrogen-bonding networks of water.

The implications extend well beyond solution chemistry. The authors emphasize that the same design rules, governing spot size, repetition rate and interaction length, apply to vibrational sum-frequency scattering, hyper-Raman scattering and emerging wide-field nonlinear microscopy schemes, where cumulative heating and focus stability set hard limits on imaging speed and sample viability. Biomedical applications loom large: real-time volumetric imaging of tissue morphology, label-free monitoring of cellular metabolism, and dynamic studies in living organisms all depend on extracting maximal signal at minimal photodamage. By defining a stable power window with predictable scaling, this systematic taming of thermal and Kerr lensing offers photonics labs a practical recipe for pushing nonlinear optical techniques into faster, gentler and more sensitive territory.

Subject of Research: Mitigation of thermal and Kerr lensing effects to optimize hyper-Rayleigh scattering performance with kHz femtosecond lasers

Article Title: Mitigating thermal and nonlinear lensing for optimized performance in nonlinear optical scattering

Article References: Dok, A. R., Legat, T., de Coene, Y., & Van Cleuvenbergen, S. (2026). Mitigating thermal and nonlinear lensing for optimized performance in nonlinear optical scattering. Frontiers in Photonics, 7, Article 1832085. https://doi.org/10.3389/fphot.2026.1832085

Image Credits: AI Generated

DOI: 10.3389/fphot.2026.1832085

Keywords: hyper-Rayleigh scattering, nonlinear optics, Kerr lensing, thermal lensing, femtosecond lasers, photonics, para-nitroaniline, second harmonic generation, biophotonics, laser repetition rate, cylindrical lenses, time-resolved spectroscopy