Mitigating thermal and nonlinear lensing for optimized performance in nonlinear optical scattering

Nonlinear optical light scattering is used in a wide range of applications, from screening organic electro-optic materials to analyzing the structure and chemistry of buried interfaces in colloidal and biological systems. It is also applied in real-time studies, such as monitoring nucleation and crystal growth. Given that NLO effects are intrinsically weak, experiments require high laser power to achieve measurable signals. However, the onset of nonlinear effects, such as thermal and Kerr lensing, limits the maximum laser power that can be applied, which in turn limits the signal strength. In this study, we systematically investigated the impact of thermal and Kerr lensing on hyper-Rayleigh scattering experiments in pure water and a methanolic solution of para-nitro aniline, using an amplified femtosecond laser system with medium-range repetition rates (100–500 kHz). Through systematic evaluation of focusing optics, cuvette path length, and laser repetition rate and carefully managing these effects we achieved significant gains in signal strength, allowing sub-millisecond measurements for pure water and para-nitroaniline, while avoiding photodegradation of the organic compound. These findings present a practical route to harness the available power more effectively, by balancing intensity and undesired lensing effects. This optimization is crucial for advancing the capabilities of nonlinear optical scattering techniques, enabling time-resolved measurements across a wide range of chemical and biological systems, from aggregation and nucleation to the dynamic measurement of adsorption, surface potentials, hydrogen-bonding networks. Our results also extend to biomedical applications such as real-time volumetric imaging of tissue morphology, label-free monitoring of cellular metabolism and dynamic processes in living tissues and organisms.

The advent of laser sources has enabled the field of nonlinear optics (NLO) to rise to prominence (Maurya et al., 2019; Clays and Persoons, 1991). When high-intensity laser pulses interact with matter, they induce nonlinear optical responses, enabling applications in fields such as optical processing, data storage, and telecommunications. Concomitantly, these developments significantly expanded the role of NLO techniques in biological and medical sciences, where they enable non-invasive, label-free probing of complex systems such as membrane and lipid structures in live cells (Friedl et al., 2007), collagen in tissue engineering constructs (Rehberg et al., 2011), and protein structures in brain tissue (Lim, 2019). To gain insight into NLO properties of materials, Hyper-Rayleigh scattering (HRS) has been used as a key investigative tool for in-solution and gas phase characterization (Das, 2006; Vivas et al., 2020; Roke and Gonella, 2012; Enrigue and Eduardo, 2013; Van Steerteghem et al., 2017). HRS is a non-invasive scattering technique in which scattered light at double the excitation frequency is generated in response to a high intensity light source. In 1991, Clays and Persoons introduced HRS as a method for measuring the first hyperpolarizability (β) of organic chromophores in solution (Clays and Persoons, 1991). HRS allows determining β for a wide scope of species ranging from dipolar and octopolar molecules to ionic molecules such as biological chromophores (Clays and Persoons, 1991; Shelton, 2013). As a second-order NLO process, HRS is inherently sensitive to symmetry. Within the electric dipole approximation, HRS light can only be generated by non-centrosymmetric species in the absence of an electric field (Das, 2006; Roke and Gonella, 2012; Hendrickx et al., 1998; Williams et al., 2015; Kielich and Kozierowski, 1974). HRS is also regularly employed to study (supra)molecular symmetry. For non-centrosymmetric scatterers, symmetry information can be derived from their response to polarized light (Verbiest et al., 2009; Chan and Wong, 2012). This enables the study of symmetry, molecular orientation, supramolecular assemblies, micelle formation, dissociation processes, and even third-order responses (Das, 2006; Van Steerteghem et al., 2017; Ghosh et al., 2003; Dok et al., 2021; Gomopoulos et al., 2013; Macias-Romero et al., 2014). These capabilities extend to real-time, in-situ investigations of nanoparticles (Senapati et al., 2011; Sauerbeck et al., 2014; Khoury et al., 2018; De Coene et al., 2020; de Coene et al., 2019; Legat et al., 2024), metal-organic frameworks (Van Cleuvenbergen et al., 2018), polymers (Moris et al., 2020; Moris et al., 2021), pharmaceuticals (Dok et al., 2021), and biological processes (Gomopoulos et al., 2013; Macias-Romero et al., 2014). Such applications demand high temporal resolution, which is usually achieved by strong focusing of intense pulsed laser light in the near-infrared region. This introduces challenges, as above a certain threshold thermal and nonlinear effects arise, limiting the maximum usable laser power. Femtosecond laser pulses with high repetition rates, typically around 80 MHz, are a popular choice for HRS measurements. High repetition rates enable high HRS signal levels through signal accumulation over many pulses, while femtosecond pulses ensure high peak power at modest pulse energies. This allows the pulse energy to remain low enough to avoid higher-order effects such as Kerr lensing or supercontinuum generation, provided that weak focusing is maintained (Dok et al., 2021; Gomopoulos et al., 2013; Macias-Romero et al., 2014; Campo et al., 2009). However, at these high repetition rates, strong thermal effects can arise due to heat accumulation leading to thermal lensing and sample degradation (Agiotis and Meunier, 2022; Qi et al., 2019; Mian et al., 2002; Saytashev et al., 2012). Alternatively, several studies, including the seminal first report of HRS of molecules in solution by Clays and Persoons, have employed Q-switched Nd:YAG laser sources, typically producing nanosecond laser pulses with significantly higher pulse energies (>1 mJ) at low repetition rates (several Hz) (Clays and Persoons, 1991). While the high pulse energies help achieve scattering intensities with signal-to-noise (S/N) ratios similar to high repetition rate systems, they are often accompanied by undesired nonlinear effects, which limits the usable laser power and therefore the HRS signal level (Das, 2006; Dadap et al., 1995; Kaatz and Shelton, 1996). Sample damage provides an additional practical limitation in HRS experiments, particularly for organic and biological compounds. Importantly, the dominant damage mechanism depends on the laser regime. In low repetition rate laser systems, high pulse energies can result in plasma-mediated damage, whereas in high repetition rate systems damage typically arises from cumulative heating or electron-induced chemical decomposition (Saytashev et al., 2012; Zhao et al., 2019). The latter is especially critical for biological systems, where cumulative heating can lead to denaturation (Macias-Romero et al., 2014; Saytashev et al., 2012; Zhao et al., 2019; Macias-Romero et al., 2016).

Recently, amplified pico- and femtosecond laser sources with medium repetition rates in the kHz regime have emerged as a middle way (Gomopoulos et al., 2013; Macias-Romero et al., 2014; Campo et al., 2009). In this regime, the pulse energy is in the µJ range, falling between the nJ used in high-frequency systems and the mJ used in low-frequency counterparts. If employed judiciously, this results in an efficient balance between the immediate (per pulse) and total (per second) second harmonic response. By allowing enough time (µsec) between pulses for energy dissipation, the accumulation of heat and total thermal load can moreover be managed (Macias-Romero et al., 2016) (Gordon et al., 1965). Moreover, since gated detection is available in the kHz regime, multi-photon fluorescence (MPF), operating at nanosecond time scales, can be removed from the near-instantaneous HRS response in the time domain (Campo et al., 2009). Similar adaptations using medium repetition rates can be employed in widefield imaging schemes, enabling high throughput while minimizing damage to biological samples (Macias-Romero et al., 2014; Macias-Romero et al., 2016). However, these intermediate conditions do not automatically eliminate power-limiting artifacts. Depending on the experimental design, both heat accumulation and higher-order nonlinear effects can still occur and constrain the usable laser power. This makes experimental design and optimization central to fully exploiting kHz NLO measurements. While several studies have investigated laser-induced higher-order and thermal effects, often focusing on single-pulse behavior or isolated pulse parameters, these effects are seldom examined systematically in the context of practical NLO measurements, particularly when employing amplified kHz laser systems (Van Steerteghem et al., 2017; Agiotis and Meunier, 2022; Qi et al., 2019; Mian et al., 2002; Saytashev et al., 2012; Zhao et al., 2019). In this study, we address this gap by systematically investigating and mitigating the influence of thermal and nonlinear effects under medium repetition rate conditions. We focus on thermal and Kerr lensing, two phenomena that significantly impact nonlinear optical experiments, as they are prevalent under the moderate fluence and medium repetition rate conditions typically used in such setups. Thermal lensing arises from localized heating of the sample due to laser absorption, leading to a temperature-dependent change in refractive index that distorts the beam focus. Kerr lensing, on the other hand, results from the intensity-dependent refractive index changes caused by high peak powers, which can induce self-focusing of the laser beam, like in z-scan experiments. Both effects alter the effective focus, reducing the local intensity at the desired focal point and potentially shifting the beam out of alignment with the detection system. Together, these processes limit the maximum usable laser power and, consequently, the achievable signal strength and measurement fidelity. While we focus on HRS in this work, other related NLO scattering techniques, such as vibrational sum frequency scattering (SFS) and hyper-Raman scattering (HRaS), face similarly weak signals and benefit from improvements that enhance detection. These techniques have become popular for their ability to probe interfacial properties and vibrational modes inaccessible to traditional Raman and IR methods (Roke and Gonella, 2012; Johansson et al., 2018; Liu and Okuno, 2023; Milojevich et al., 2011; Denisov et al., 1987; Puente and Dissecting, 2024; Jones et al., 2024). Furthermore, optimization strategies presented here, are directly applicable to emerging widefield nonlinear imaging schemes for biological systems, where high acquisition speeds across large imaging volumes are necessary but are often limited by cumulative heating effects (Macias-Romero et al., 2014; Macias-Romero et al., 2016; Dementjev et al., 2020).

In this work, we study two representative systems: pure water, a critical solvent in biological and chemical contexts, and a methanolic solution of para-nitroaniline (pNA), a prototypical chromophore with a strong second-order NLO response. In the water sample, undesired effects were due to a combination of nonlinear lensing and to a lesser extent to thermal lensing, while thermal effects were the dominant issue in the pNA solution. Starting from the equations describing both effects, we explored the influence of fluence and scattering volume, by adjusting the focusing optics, repetition rate and path length, and by clearly defining a “quadratic range” (QR), i.e., the range of input power over which the HRS signal follows the expected quadratic dependence. This allowed us to minimize undesired lensing effects, while fully exploiting the available laser power to maximize the nonlinear optical scattering signal. Finally, a comparison in terms of the onset of lensing effects and scattering efficiency is presented for the different experimental parameters.

HRS measurements were performed using an amplified Pharos PH1 femtosecond laser from LightConversion (Vilnius, Lithuania). The optical setup was configured to measure at a single detection angle of 90° to the incoming beam as shown in Figure 1.

The samples were excited using femtosecond laser pulses with a pulse width of 220 fs at a fundamental wavelength of 1030 ± 10 nm. The repetition rate delivered through the main output of the laser system can be varied from single shot to 600 kHz, while an emission at 76 MHz can be achieved through the secondary output. The latter beam was coupled into the main beam line through a series of mirrors. The repetition rate and the intensity of the incident laser light were controlled with the laser controller software. Placing two attenuator systems in series, consisting of a half-wave plate and polarizer combination (in P-direction), on the main line immediately after the laser system allowed for further power reduction while also allowing finer control over the incoming beam intensity. The control over the beam polarization was performed with a final half-wave plate and was set to S- polarization throughout these measurements. To remove SHG components from the incoming beam, generated by the optics in the beam path, a long-pass filter was placed before the beam was focused on the center of a silica quartz cuvette (10 mm or 1 mm optical path length) containing the sample. The laser beam with an average diameter of 3.6 mm was focused with a variety of lenses in order to optimize the optical setup. The resulting spot size and Rayleigh length for every lens are summarized in Table 1. The plano-convex cylindrical lenses used were positioned horizontally, so the incoming beam was focused only on the vertical axis, while the horizontal axis retained the input beam diameter (see inset Figure 1). The resulting scattered light was collected using an achromatic collection lens (f = 50.00 mm and 12.5 mm diameter, PAC13AR.15, Newport) and further imaged onto an intensified charge-coupled device (iCCD) camera (iStar DH334-18U, Andor Oxford Instruments) through a spectrophotometer (Kymera 328i, Andor Oxford Instruments). A low pass filter placed after the collection lens blocked any scattered light at the fundamental wavelength. The collimated light was then passed through a mounted dove prism rotated by 45° to rotate the image of the focus by 90°. Finally, a plano-convex focusing lens with a focal length of 100 mm was used to focus the collected light onto the slit of the spectrometer.

The collection system resulted in a total magnification of the image on the iCCD by 2.2 times. Since the collection system remained fixed, swapping out focusing optics in different optimization experiments can result in a less optimal overlap between the excitation volume and the collection lens leading to lowered collection efficiency. However, as the quadratic range (our primary parameter for detecting the onset of lensing effects) is independent of the collection optics, we opted to keep the collection system unchanged throughout our experiments.

Within the spectrometer, the collected light is further split into its different wavelength components by the grating (600 L/mm; Blaze 500 nm) before being directed to the iCCD. The use of an iCCD enabled operation in a gated detection mode to greatly increase the signal-to-noise (S/N) ratio for low intensity HRS response. This was achieved by synchronizing the detector trigger with the incoming laser pulse through a photodiode located at the laser exit. Once triggered, the intensifier, acting as an electronic gate, allowed amplification of the photoelectrons produced by the photocathode upon contact with photons entering the camera, thus preventing measurement of dark noise between pulses. The optical gate width used during the measurements was 15 ns with an average irising time of 0.14 ns (We refer to an irising time as the time delay between the center and the edge of the intensifier to achieve 63% of its final value).

Data acquisition and analysis

The obtained spectra are fitted to a linear combination of a gaussian and a linear fit to subtract the MPF contribution and background from the HRS signal as shown in Equation 1 below.

Here, A is the integrated area of the Gaussian peak, w is the Gaussian width parameter, x c is the peak center and mx + b describes the linear background. The HRS intensity, corresponding to area A of the gaussian fit, was obtained and plotted against the average input power of the fundamental beam. For each incident power, three replicate spectra were recorded on the same sample. Each spectrum was fitted using Equation 1, with the HRS intensity corresponding to the mean value of A. The error bars shown in the HRS intensity versus the power plots represent the combined uncertainty from the spectral fit and the replicate variation, propagated according to standard error-propagation rules. Since a quadratic relationship is expected between the scattering intensity and incoming beam power for a second-order process, the HRS intensity vs. laser power data were fitted to a quadratic function. We define the “Quadratic Range” (QR) as the set of input powers over which the data closely follows this quadratic trend. After some threshold, deviations from quadraticity appear due to lensing effects. To capture this threshold, a sequential fitting procedure was performed by an algorithm in Mathematica software starting from data points at low laser power. The fitting procedure continued until a deviation between the fitted curve and the observed data exceeded 7.5%. This deviation threshold of 7.5% was selected and validated as an intermediate empirical criterion through a sensitivity analysis comparing 5%, 7.5%, and 10% thresholds available in Supplementary Material. This fit was used to determine the QR. To account for outliers and errors due to low S/N, we defined the end of the QR at the third data point where the deviation exceeded the 7.5% threshold. Additionally, since we measured with very small power increments, some initial few data points at low incoming powers (below 3 mW for PCXL and 20 for CyL) were excluded from the fit to ensure correct algorithmic operation.

Materials preparation

Para-nitroaniline (≥ 99%, extra pure) and Methanol (ACS spectrophotometric grade, >99.9%) were purchased from Sigma-Aldrich. Methanol was further purified through disposable syringe filters (Chromafil®, Glass fiber + Polyester filter, 0.20 µm pore size) purchased through Macherey-Nagel. Using the filtered methanol, a 10 mM pNA solution was prepared and used throughout the measurement, unless mentioned otherwise. The measurement cuvettes (High Precision Cell (QS), Light path 10 × 10 mm and 1 × 10 mm) used for the measurements were purchased through HellmaAnalytics and Starna GmbH. Ultrapure water used throughout the measurements was obtained from a MILLI-Q Reference A + System.

We used a systematic approach to investigate the onset of thermal and nonlinear lensing effects in order to optimize HRS signal strength. Two different systems were selected for measurements. As described above, we used pure water and a methanolic pNA solution. Consistent with approaches adopted in previous studies, scattering experiments were performed at medium-range frequencies (100–500 kHz), delivering femtosecond pulses with µJ pulse energies at a wavelength of 1030 nm (Gomopoulos et al., 2013; Macias-Romero et al., 2014; Campo et al., 2009). In the following sections, we first establish baseline spectra and define the quadratic range (QR) as the operating window free of lensing distortions. We then test how focusing optics and collection overlap affect both signal and QR. Next we vary repetition rate to disentangle peak-power-driven Kerr lensing from average-power-driven thermal lensing. Finally we mitigate both effects using cylindrical focusing and reduced path length, and summarize the resulting performance gains.

During the initial experiments, the fundamental laser pulses were focused on the sample to a spot diameter of ∼ 27 µm using a focusing lens with a 75 mm focal length. To avoid thermal and nonlinear effects, the initial average power used was kept low at 22 mW (0.044 µJ per pulse). During the measurements, the scattered light was collected and spectrally resolved, leading to typical spectra illustrated in Figure 2. The obtained spectrum of 10 mM pNA solution in methanol shows two distinguishable spectral features. The first sharp peak at 515 nm corresponds to the HRS response, while the second is a broader, partially overlapping emission band attributed to MPF, which can complicate the analysis. To derive the corrected HRS peak without the interference of MPF, a gaussian fit combined with a linear fit is used to subtract the fluorescence background. The pNA spectrum shown results from a gated measurement, allowing the separation of the MPF peak from the HRS peak by shifting the gate opening for a fixed gate width of 15 ns. The suppression of the fluorescence signal by means of gate shifting and changing the gate width of the iCCD camera is shown in Supplementary Material. As a comparison, measurements of pNA were conducted at a repetition rate of 76 MHz without gated detection with an exposure time of 10 s, while using the same optical setup. The resulting spectrum of pNA is shown in Supplementary Material. Although synchronizing the gate opening with the pulse reduced noise and significantly diminished the fluorescence signal, it does not fully suppress the fluorescence due to the short fluorescence lifetime of pNA. Gated measurements can exclude the slow components of MPF. However, the fast components cannot be fully suppressed in the time-domain, hence spectral deconvolution is still required. HRS is also observed for pure water, albeit with much lower efficiency compared to pNA at the same average incoming power of 22 mW. To measure the HRS of water with sufficient S/N ratio, exposure times up to 30 s were required. To achieve shorter exposure times (e.g., for in-situ studies), higher laser powers must be used, but these often trigger undesired lensing effects above a certain threshold (Campo et al., 2009; Kaatz and Shelton, 1996).

To determine the onset of such undesired effects, experiments were conducted for water and pNA at different incoming laser powers as illustrated in Figure 3. For these experiments, the incoming beam was focused on the sample using a focusing lens with 50 mm focal length, resulting in a beam waist of ∼ 18 µm. The frequency of the incoming laser beam was kept at 500 kHz during the measurements, resulting in a maximum attainable fluence of 0.1 J/cm 2 at the focus for an average power of 120 mW. In the absence of thermal or higher-order effects, a quadratic dependence of the HRS intensity on the incoming laser power is expected. In both water and pNA this QR is limited to relatively low powers, above which clear deviations become evident.

Throughout the manuscript, data points that fall within the QR are represented as filled circles. Points outside of this range will be displayed as crosses. The measurements for the pure water system shown in Figure 3 indicate that the QR is valid up to an average power of around 112 mW and a fluence of 0.086 J/cm 2. The pNA solution exhibited a much more limited QR, with significant deviations starting around 18 mW and showing a more pronounced departure from the expected quadratic dependence than water.

Similar measurements were performed for Rhodamine 6G dissolved in methanol, a highly fluorescent organic dye molecule showing a strong MPF response (see Supplementary Figure ESI 3.1). Although MPF is a third-order nonlinear optical process, the MPF response also depends quadratically on the incoming laser power. Deviations from the quadratic fit for Rhodamine 6G began around 16 mW, similar to the pNA solution. This implies that the QR is mainly determined by methanol, the solvent independent of the solute.

The applied conditions (500 kHz and f = 50 mm) show a very low detection efficiency compared to previous reports (Gomopoulos et al., 2013). The detection of the second harmonic response for both water and the pNA solution with sufficient S/N was only possible with a time resolution of 1 s, within the QR. In contrast, Gomopoulos et al. achieved millisecond time resolution for the HRS measurements of water at an average power of 118 mW. Using a similar system focusing the laser beam down to a spot size of 50 μm at a repetition rate of 200 kHz, they obtained a significantly higher detection efficiency (Gomopoulos et al., 2013). To achieve similar standards or even improving the scattering efficiency while minimizing the lensing effects several series of experiments were performed. To this end, we investigated the influence of focal length on the QR for plano-convex lenses (Figure 4). Lowering the degree of focusing and the associated peak power density is expected to increase the QR by minimizing undesired thermal or higher-order effects. Using a longer focal length lowers the local intensity, which tends to reduce HRS because HRS scales with intensity squared. But the interaction volume also becomes larger, meaning more sample volume contributes to the signal. If the collection optics captures the entire focal volume, these two trends cancel, so the detected HRS intensity is expected to be roughly independent of focal length at the same average power (Campo et al., 2009). The practical advantage of a longer focal length is therefore not a higher signal at fixed power, but the possibility of extending the QR to higher powers and obtaining a higher maximum signal without lensing artifacts.

The HRS measurements for water and pNA solutions were conducted under similar conditions and are normalized with respect to exposure times. The separate scattered intensity versus power plots for the various focal lengths can also be found in the Supplementary Figures ESI 4.1–4.8. A significant decrease in scattering intensity was observed when focal lengths longer than 50 mm were used. Minor differences in collected HRS intensity are apparent when comparing 75 mm–100 mm, while the lowest response is recorded for the 150 mm focal length lens.

While the collected HRS signal should theoretically remain independent of focal length at the same incident power, our system uses a 50 mm collection lens in order to reach the desired magnification at the slit of the spectrometer (Campo et al., 2009). As discussed in the methodology section, an optimal overlap is achieved between excitation and the collection lens (f = 50 mm) for shorter focal lengths (e.g., 50 mm), while longer focal lengths reduce collection efficiency. This is also reflected in the differences between the collected HRS intensity for 50 mm and longer focal lengths.

Contrary to expectation, weaker focusing led to a smaller QR for both water and pNA. This does not mean that lensing becomes intrinsically stronger at lower intensity. Rather, thermal or Kerr lensing gradually shifts the effective focus along the optical axis. Because our collection optics image a fixed region onto the iCCD, this shift reduces the overlap between the scattering volume and the imaged region. The detected HRS signal then drops and deviates from the expected quadratic scaling, so the QR appears smaller.

This sensitivity increases at longer focal lengths. Up to about 75 mm, the imaged scattering region does not fully fill the camera field of view (see Supplementary Material), so small focus shifts still leave the scattering volume within the detected area and the signal remains relatively robust. At longer focal lengths, the scattering volume fills the field of view completely, and even minor shifts push part of it outside the imaged region, causing an earlier apparent deviation from quadraticity. In short, with our fixed collection geometry, longer focal lengths can make the measurement more sensitive to lensing-induced focus shifts, reducing the observed QR, even though the local intensity is lower. We confirmed this by imaging the focus with an infrared camera, which clearly illustrates the gradual shift in focus as the power increases. These measurements are shown in Supplementary Figures ESI 5.1, 5.2. For both systems, the focus was measured at 3 mW, 30 mW and 112 mW respectively. The pNA solution shows a significant shift of the focus position after an increase in power, while this effect is less pronounced for water. These measurements show that while the collection system is independent of lensing effects thermal or nonlinear in origin, it still plays an important role in the determination of the QR during a measurement.

In a subsequent series of experiments, we explored the influence of the repetition rate in order to investigate the origin of lensing effects in both solvents. To increase our sensitivity to lensing-induced changes, we used a 150 mm focal length lens, which is more susceptible to these effects as discussed above. Reducing the frequency of the incoming laser pulses from 500 kHz to 200 kHz and finally to 100 kHz allows for more time to dissipate the heat between incoming pulses, potentially mitigating thermal effects. Remark that, for the same incident average power (P av), reducing the repetition rate (f) increases the pulse energy (E pp).

Figure 5 illustrates the HRS experiments for pNA and water at different frequencies. Reducing the frequency from 500 kHz to 200 kHz resulted in an HRS intensity increase by a factor of 2,5x for a given average power. When reducing the frequency to 100 kHz, the scattering intensity increased by a factor of 5. These observations are in line with the expected dependence of HRS on fluence (F) (quadratic) and the repetition rate (f) (linear), as described by Equation 2 (Macias-Romero et al., 2016). In this equation, the focal volume is described by the area (A) and the Rayleigh length (Rl), while the pulse width is denoted as pw.

The influence of the repetition rate on the QR is different for the methanolic pNA solution and water. For the HRS response of water, we observe an increase of the QR as a function of average power with increasing repetition rate. The QR nearly doubled from 38 mW at 100 kHz to 66 mW at a repetition rate of 200 kHz. Moreover, plotting the HRS intensity versus peak power (Supplementary Figures ESI 6.1, 6.2) reveals the boundary of the QR at similar values of 16 and 14 mJ/cm 2 for 100 kHz and 200 kHz respectively. This implies that deviations from quadraticity depend on the peak power rather than average power, indicating that Kerr lensing effects determine the QR in this case. When the peak power becomes sufficiently high, the intensity dependent changes in refractive index come into play, thus a Kerr lens is formed which is described by Equation 3 as derived by Hasnaoui et al. (2019) Rodrigues et al. (2014) Feng et al. (1995):

The focal length f K e r r is proportional to the spot size (w 0) and inversely proportional to the path length and the peak power P peak. Parameter n 2 describes the nonlinear refractive index of the medium in which the Kerr lens is formed. According to Equation 3, it is evident that the effect of Kerr lensing is minimized for longer focal lengths f Kerr. In other words, higher peak power, smaller spot size and longer path lengths result in stronger Kerr lensing. However, the dependency of the QR on the peak power was not observed when the laser pulse frequency was further increased to 500 kHz. The HRS response of water showed deviations from quadraticity around 65 mW (5.5 mJ/cm 2), at an average power comparable at 200 kHz. This suggests a shift in mechanism, from a dependence on peak power to one dominated by average power, implying that thermal lensing effects become more important at higher repetition rates.

In the case of the methanolic pNA solution, the QR boundary was reached at similar average powers around 10 mW to a maximum of 15 mW when lowering the frequency from 500 kHz to 100 kHz, despite an increase in peak power density with a factor of 5 (see Supplementary Material). This implies that thermal lensing effects determine the QR for the pNA solution in methanol for all repetition rates, as these effects depend on the heat deposited in the medium which scales with the average power (Gordon et al., 1965).

Increasing the average power will result in an increased heat buildup. However, lowering the repetition rate from 500 kHz to 100 kHz, hence increasing the time between pulses from 2 µs to 10 µs, allows for more heat dissipation in between the pulses. The balance obtained between the pulse frequency and the characteristic relaxation time (t c) of the medium will result in the final thermal lens build up. The calculations of t c for both water and methanol shown in Supplementary Material indicate that cumulative heating cannot be fully prevented at medium repetition rates. The resulting formation of a thermal lens in the solution can therefore be expressed by the steady-state approach used by Gordon et al. (Gordon et al., 1965) This leads to the following equation describing the focal length of the induced thermal lens (Gordon et al., 1965):

Equation 4 describes the effective focal length (f TL) of the thermal lens in the solution. The focal length of the thermal lens is scaled with the square of the beam waist w 0 2 and decreases with increasing absorbed power, which is determined by the average power P av of the laser beam. The thermo-optic coefficient (dn/dT) describes the change in the refractive index with temperature, while the absorption coefficient (α) and the path length (L) indicate the material’s contribution to the thermal lensing effect. A positive f TL leads to additional focusing of the incoming laser beam, while a negative value causes defocusing.

The data indicate that the onset of lensing in our experiments is governed primarily by the solvent, not the solute. This is most clearly supported by the observation that methanolic solutions of pNA and Rhodamine 6G show very similar quadratic ranges, despite their very different fluorescence and molecular properties. In other words, for methanol the QR boundary is largely determined by the medium’s response to the applied average power. The different repetition-rate dependence in water versus methanol then follows naturally from their thermal properties. At 1030 nm both liquids exhibit weak but non-zero absorption, so at the high average powers and 10 mm path length used here, cumulative heating can still build up. Because water has a higher thermal conductivity and heat capacity than methanol, it reaches a different steady-state temperature profile for the same deposited power, consistent with the characteristic thermal time scales calculated in Supplementary Material. This explains why water shows a transition toward thermal-load-dominated behavior at higher repetition rates, whereas methanol remains predominantly thermal-limited across the studied range. Besides the solvent dependence, lensing effects, both thermal or nonlinear in origin can be influenced by the degree of focusing as can be deduced from Equations 3, 4. According to these equations, increasing the spot size should reduce the undesired effects and increase the QR. Furthermore, since increasing the spot size results in reduction of the power density as well, Kerr lensing effects can be minimized further. Therefore, the following experiments made use of cylindrical lenses to achieve weaker focusing, while at the same time increasing the focal volume. Earlier studies have noted that the use of cylindrical lenses can be beneficial when using amplified systems in order to avoid undesired nonlinear and thermal effects, albeit without systematic investigation of the QR (Campo et al., 2009). To this end experiments were performed using a cylindrical lens with a focal length of 150 mm, as illustrated in Figure 6.

The detected HRS intensity is considerably lower for cylindrical lenses compared to plano-convex lenses of the same focal length. Significantly higher laser powers (∼ 9 x) are required to achieve the same signal. As the HRS intensity does not depend on the amount of focusing for the same input power, as long as all scattered light in the focus is collected, this points to a lower detection efficiency of the collection lens for the much larger focal volume (Campo et al., 2009). However, compared to the measurements with plano-convex lenses, we observe a notable increase in the QR for all measurements, that can be related to the much lower power density (∼ 7.5 x) (see ESI for peak power density relation). The experiments performed on water show a quadratic trend of the measured HRS response over the entire applied experimental range for a used repetition rate of 200 kHz and 500 kHz. If higher peak power is used, as is the case during a measurement at 100 kHz, deviations from the quadratic fit start to occur around 470 mW or when the peak power reaches 3 mJ/cm 2. As previously discussed, the boundary of the QR is primarily determined by the pulse power for the water sample at 100 kHz, while thermal effects start to dominate above 200 kHz (Figure 5), which accounts for this observation. Measurements on the pNA solution also revealed an increase in the QR compared to plano-convex lenses, showing a quadratic dependence until an average power of around 230–260 mW was reached for all frequencies. This confirms the dependency of the QR on average power for the methanolic pNA solution, in line with thermal lensing effects. From Equation 4 it is clear that strong thermal lensing effects are expected for small w 0. The much larger beam waist of the cylindrical lenses therefore reduces thermal lensing and increases the QR.

In a final series of experiments, we studied the effect of the dimensions of the scattering cell. In this case, the laser beam was focused on a thin cell with a pathlength of 1 mm instead of 10 mm. We used a cylindrical lens with a focal length of 150 mm and a Rayleigh length of ∼ 2.3 mm, significantly longer than the path length of the cuvette. Due to the dependence on path length, both Kerr and thermal lensing effects are expected to be further minimized according to Equations 3, 4. Figure 7 presents the measurements conducted for water and pNA solutions at three different frequencies. Compared to previous measurements with a 10 mm × 10 mm cell, these experiments showed a lower scattering efficiency for the same laser power (<500 mW), stemming from a lowered scattering volume due to the limited path length of the cell. However, using a thin cell allowed us to extend the QR by a significant margin. For measurements conducted at 500 kHz, the HRS response of water showed a quadratic dependence up to 4100 mW, close to the maximum available power of 4500 mW of the system. Lowering the repetition rate resulted in a reduced QR for water, with deviations from quadraticity around 3800 mW at 200 kHz and a QR of 2500 mW at 100 kHz. Consistent with previous measurements, these observations suggest that the end of QR for water is determined by nonlinear lensing effects, while measurements at 200 kHz and 500 kHz show an increasing influence of thermal effects.

Different behavior was observed for HRS measurements of the pNA solution. The boundary of the QR was reached at similar average power of around 2600 mW for all frequencies, consistent with the previous experiments showing the dependency of the QR on thermal lensing. However, compared to previous measurements, the QR could be extended significantly.

As summarized in Table 2, despite a modest QR for the initial 50 mm lens configuration, it yielded good detection efficiency due to optimal overlap of excitation and collection volumes. Longer focal lengths or cylindrical lenses improved the QR but reduced collection efficiency. By employing a 1 mm path length and cylindrical optics, we overcame these limitations, achieving both a significantly extended QR and higher HRS signals. At lower repetition rates, even with Kerr lensing constraints, we could increase HRS intensities enough to reach sub-millisecond time resolution, further improving the performance compared to earlier reports (Supplementary Figure ESI 10.1) (Gomopoulos et al., 2013). The results for the methanolic solution are summarized in Table 3. For pNA in methanol, the initially very limited QR could be extended tremendously, leading to improvements in HRS signal over almost two orders of magnitude. This allowed HRS measurements in the sub-millisecond regime for this solution. To confirm that no degradation of the pNA solute occurred under these optimized conditions, UV-vis spectra of the pNA solution were measured before and after each HRS experiment. As shown in Supplementary Figure ESI 11.1, no change in UV-vis spectra was observed for HRS experiments conducted within the QR.

In this study, we systematically investigated the onset of thermal and nonlinear lensing effects in HRS experiments using amplified femtosecond pulses with medium-range repetition rates, demonstrating how these effects limit the usable power range and HRS signal. Both for water and pNA in methanol, deviations from the expected quadratic dependence of the HRS intensity occurred at higher powers. Our results demonstrate that thermal lensing, which depends on average power, dominates in methanolic solutions, while Kerr effects, which depend on peak power, are also observed in pure water depending on the applied repetition rate. These effects resulted in a significant reduction of the QR for plano-convex lenses, limiting the maximal laser power that can be applied in HRS experiments. As the HRS intensity depends quadratically on the incoming power, this significantly impacts the attainable signal levels and temporal resolution of HRS measurements. To address these limitations, we explored approaches to mitigate thermal and nonlinear lensing effects, including the use of cylindrical lenses, reduced path lengths, and adjustments to the repetition rate. The use of cylindrical lenses, with their larger beam waist, extended the QR by reducing power density. Additionally, shortening the path length from 10 mm to 1 mm further minimized lensing effects. These interventions efficiently reduce the impact of both Kerr and thermal lensing, allowing higher laser powers to be used without compromising the measurement integrity. Other than interventions related to focusing, changes in repetition rate must be considered on a case-by-case basis, depending on the solvent and the dominant nonlinear effect. This distinction is evident in the different impact of repetition rate adjustments on Kerr and thermal lensing. For thermal lensing, which depends on average power, lower repetition rates allowed measurements at higher peak powers, significantly boosting the HRS signal, as observed in methanolic pNA. In contrast, for Kerr lensing, which depends on peak power, lower repetition rates led to reduced QR, as seen in water. Ultimately, systems dominated by thermal lensing (like pNA in methanol) benefited most from these optimizations, achieving a remarkable ∼63-fold improvement in signal, while Kerr-lensing-dominated systems (like water) still saw a notable ∼3.5-fold gain. We also emphasize that careful alignment of the excitation and collection volumes can still further boost signal levels, further maximizing the impact of these mitigation strategies. These optimizations demonstrate the potential for advancing nonlinear optical scattering techniques, spanning applications from HRS and SHS to hyper-Raman studies of solvents and solutions. The optimization framework presented here is therefore particularly valuable wherever signal-to-time-resolution trade-offs limit dynamic studies, and the same dependencies (repetition rate, focusing geometry, interaction length) translate as a conceptual design guideline to wide-field nonlinear imaging schemes operating in this medium-repetition-rate regime. Looking forward, the ability to achieve sub-millisecond time resolution for a range of solvents and chromophores opens new avenues in real-time probing of dynamic processes such as nucleation, aggregation, interfacial adsorption, and complex biological interactions. Especially in biological systems, the increased temporal resolution allows for dynamic processes in living systems to be probed with high fidelity while reducing the need for long exposures at damaging average powers. The mitigation strategies identified here can furthermore help define a stable power window with predictable scaling and reduced photodamage beyond scattering measurements. Thermal and Kerr lensing are also practical limits in biological nonlinear microscopy. The same dependencies translate to microscopy choices such as spot size, pulse repetition rate, and sample thickness. These design rules are increasingly relevant for parallelized and wide-field excitation schemes, where thermal management and focus stability across the field of view are critical.