Extreme optical nonlinearities unveiled by ultrafast laser filamentation in semiconductors

Domain: neuroscience
Published: 2026-02-14
Editorial outcome: major_revision

Abstract
Sky-high optical nonlinearities make semiconductors ideal platforms for multifunctional photonic devices. The fabrication of such complex devices could greatly benefit from in-volume ultrafast laser writing for monolithic and contactless integration. Ironically, as exemplified for Si, nonlinearities act as an efficient immune system that self-protects the material from internal permanent modifications. Predicting high-intensity ultrashort-pulse propagation beyond Si is further limited by incomplete descriptions of carrier dynamics in narrow-gap materials. Here, we demonstrate that filamentation universally dictates ultrashort laser pulse propagation in various semiconductors. The effective key nonlinear parameters extracted differ markedly from past measurements with low-intensity pulses, while temporal scaling laws for these parameters are also derived. Based on these findings, appropriate temporal-spectral shaping is proposed for tailored energy deposition inside semiconductors. The effective parameters also provide predictive inputs for semiconductor backside processing, microelectronics security, and high-harmonic, supercontinuum and terahertz wave generation.

Introduction
Ultrafast laser filamentation is an extremely nonlinear propagation regime characterized by a dynamic balance between Kerr-induced self-focusing and plasma-induced defocusing1,2. In gases, the remarkable properties of filaments have led to a plethora of applications3–9. In wide-gap solids, the much higher nonlinear refractive index can be advantageously exploited for supercontinuum generation10,11, laser direct writing of elongated structures12 and fiber Bragg gratings13. However, in narrow-gap materials such as semiconductors, the understanding of ultrafast filamentation is, to date, limited to Si. In contrast with other media, nonlinear propagation effects in Si are disastrous when aiming for internal structuring. Strong low-order multi-photon absorption before the geometrical focus (i.e., prefocal absorption) coexists with filamentation and leads to delocalized energy deposition, which saturates below the modification threshold due to intensity clamping14–20. From previous studies of nonlinear propagation of femtosecond laser pulses in Si, various circumvention techniques have been devised for modifying the bulk of Si, by exploiting surface seeds21,22, hyper numerical aperture23,24, longer pulses in the picosecond18,25–27 and nanosecond28,29 regime, pulse trains30, and mid-infrared pulses31. Nevertheless, the strong nonlinear absorption and the complex temporal electron dynamics observed during filamentation in Si, together with the limitations of nonlinear propagation models in accounting for these dynamics, prohibit generalization to other semiconductors. From the literature (Fig. 1), extreme nonlinear refraction is expected in these narrow-gap media. This suggests that filaments would form at modest laser pulse energies. Moreover, nonlinear absorption is exalted for narrow band gaps32,33, which could indicate that, analogously to Si, prefocal absorption hinders localized energy deposition in the focal region.

In this article, we demonstrate that ultrafast laser filamentation dictates energy deposition in narrow-gap semiconductors. Two indirect (Si and Ge) and two direct (InP and GaAs) band-gap semiconductors have been selected (see the inset in Fig. 1). Beside technological importance due to their widespread use in microelectronics, photovoltaics, sensing, and quantum engineering, these materials exhibit cubic crystal structures, a property that minimizes anisotropy effects during nonlinear propagation. Because of their narrow band gaps (Eg < 1.5 eV), these media exhibit high linear and nonlinear refraction (n0 > 3 and n2 > 10−18 m2/W, respectively). To work in their transparency spectral range, we employ ultrashort laser pulses at a wavelength of λ = 1960 nm (photon energy of 0.63 eV)34. This corresponds to the 2-photon absorption (2PA) regime for Si and Ge, and the 3-photon absorption (3PA) regime for InP and GaAs. We observe that filamentation governs the interaction in all tested semiconductors. We exploit the reduced energy deposition with ultrashort pulses below the modification threshold to determine the three-dimensional (3D) fluence distribution in these materials, which eventually leads us to define key nonlinear interaction parameters, including the peak fluence Fp, the effective critical power for nonlinearities Peffcr, and the effective 2- and 3-photon absorption coefficients βeff2 and βeff3. Repeating the measurements for pulse durations τ = 275 fs – 25 ps, the temporal scaling laws for these parameters are determined. Ultimately, we propose temporal-spectral shaping approaches to increase energy deposition inside semiconductors.

Results
A shared filamentation behavior
Our approach to characterize filamentation in semiconductors relies on nonlinear propagation imaging. This technique, analogous to tomography as illustrated in Fig. 2a, was initially developed for characterizing light propagation in water35, and later applied to dielectrics36,37. It provides direct access to the laser energy per unit area [i.e., the fluence F(x, y, z) in three spatial dimensions]. Concerning semiconductors, this technique was successfully employed to examine nonlinear propagation of light in Si17,18,22,24,30,38, but it has been so far limited to this material. As shown by the fluence distributions in Fig. 2b, nonlinear propagation imaging is also applicable to other semiconductors. It is all the more favored by the fact that the modification threshold is not crossed (see Supplementary Note 2). For each material, increasing the input pulse energy Ein qualitatively results in a spatially extended focal zone with respect to the linear regime. This is ascribable to the competition between Kerr and plasma effects, i.e., the formation of a filament. Quantitatively, the fluence distribution under identical laser conditions depends strongly on the medium's intrinsic nonlinear refraction and absorption properties. Nevertheless, a common feature between all materials is the Ein-dependent evolutive morphology of the fluence distribution, as exemplified in Fig. 2c. For low Ein values, the propagation is linear, and the resulting fluence distribution takes the form of a grain of rice. The symmetry of this shape is broken for increased Ein, and distortions toward the prefocal region are observed. This egg morphology originates from the Kerr effect, which redistributes the fluence. When Ein is further increased, prefocal absorption gives rise to the formation of wings, and an angel morphology appears. The angel wings form an angle which follows the half-angle of the cone of light θ = arcsin(NA/n0). This morphology eventually breaks up into multiple foci for the highest Ein. This pearl necklace morphology highlights the complex focusing and defocusing dynamics of filamentation. In addition, morphologies for high Ein could be influenced by Fraunhofer diffraction patterns caused by the overfilling of the focusing optics. Notably, the position of the secondary on-axis lobes shifts nonlinearly with the input intensity (see Supplementary Note 4). The evolutive morphology in Fig. 2c has been observed for all tested semiconductors and pulse durations (see Supplementary Note 4).

Among all interaction parameters that can be extracted from the 3D fluence distributions (see hereafter), a low-hanging fruit is the maximum fluence Fmax defined as the highest value of the local fluence within the 3D spatial distribution [i.e., Fmax = max(F(x, y, z))]. As shown in Fig. 2d where τ = 900 fs, in all semiconductors, Fmax scales linearly with Ein for sub-nJ values. This is in excellent agreement with theoretical predictions in the linear regime (see Supplementary Note 3). In contrast, when Ein exceeds a medium-dependent threshold, the experimental data deviate from this regime, and Fmax saturates to a peak value Fp = max(Fmax), which also depends on the material. This behavior is a direct consequence of intensity clamping, which is a typical feature of filamentation1,2. This implies that increasing the deposited energy in semiconductors by simply increasing Ein is a strategy doomed to failure.

Temporal scaling laws
Generally speaking, internal energy deposition strongly depends on the pulse duration, as the laser intensity scales inversely with this parameter. In contrast with wide-gap solids, it was demonstrated that longer pulses lead to higher intensities inside Si18,25,26. This counter-intuitive result originates from decreased propagation nonlinearities when employing longer pulses, resulting in higher peak fluences. As shown in Fig. 3a, this trend is common to all examined semiconductors. Both the fluence saturation and the temporal dependence of Fp support the conclusion that filamentation still dominates the interaction even for the longest pulses employed (τ = 25 ps). Interestingly, in the tested pulse duration range (τ = 275 fs – 25 ps), the peak fluence Fp scales as √τ.

Besides the peak fluence Fp, we also determine key parameters related to nonlinear refraction and absorption. First, the experimental on-axis fluence profiles are compared to linear propagation calculations39,40 to extract the effective critical power Peffcr above which nonlinearities start to alter the propagation (see Supplementary Note 4 for more details on the method). While one could expect Peffcr to be a constant material property, Fig. 3b shows that this parameter decreases with the pulse duration for all tested semiconductors. Such a temporal dependence of the effective critical power was observed for ultrashort pulse propagation in air41,42 as well as fused silica43, and explained by the fact that Kerr nonlinearity includes an instantaneous and a delayed medium response—the latter becoming non-negligible for longer pulse durations. Following an analogous approach for semiconductors, we demonstrate that the experimental trends in Fig. 3b may also originate from delayed medium response (Supplementary Note 4 for more details).

Nonlinear propagation imaging has also been exploited to extract the effective multi-photon absorption coefficient. To do so, the experimentally determined nonlinear focal shift is compared with our recent theoretical approach38, which is based on a modified Marburger formula44 that accounts for power losses. The effective critical power values Peffcr in Fig. 3b are used as an input parameter for this model. While this approach was initially validated in the case of 2PA in Si, it is also applicable for all semiconductors in different multi-photon absorption regimes (Supplementary Note 4 for more details). The corresponding effective 2PA and 3PA coefficients βeff2 (for Si and Ge) and βeff3 (for InP and GaAs) both scale linearly with τ as shown in Fig. 3c.

An important result lies in the comparison between Peffcr, βeff2, βeff3, and the corresponding literature values for Pcr, β2, and β3 (see dashed lines in Fig. 3b, c). For both nonlinear refraction and absorption, the parameters deduced from nonlinear propagation imaging are orders of magnitude higher than the literature values. This can be ascribed to the methods traditionally employed for determining these nonlinear coefficients. For instance, in the most standard nonlinear optics technique (z-scan45), pulses with an energy right above the detection threshold for nonlinearities are loosely focused, which results in a change in transmission of a few percent. Therefore, the measured critical power and multi-photon absorption coefficient are valid for weakly excited media. In contrast, in our nonlinear propagation experiments with strongly ionized materials, the laser-produced plasma plays a critical role in nonlinear refraction (e.g., scattering, plasma defocusing) and absorption (e.g., free-carrier absorption, potentially leading to avalanche ionization).

Transverse integration of the fluence allows us to determine how the energy E is absorbed during propagation along the optical axis z. Examples of normalized E(z) profiles are shown in Fig. 4a, b for different Ein and τ in GaAs. For intermediate Ein values where the filament does not exhibit an angel or pearl necklace morphology, the experimental data are well-described by a sigmoid function (see Supplementary Note 4 for mathematical details). The sigmoid fits contain two key parameters. The first one is the fraction of absorbed energy fE, which is determined by the ratio between the energies before and after the interaction. The second parameter is the characteristic absorption length Labs, which is inversely proportional to the steepness of the sigmoid. Applying this fitting procedure for all conditions and materials, the dependence of fE and Labs on Ein and τ (Fig. 4c, d, respectively) is obtained. Interestingly, fE and Labs both scale logarithmically with Ein and also with 1/τ. These 3D plots show that high input intensity (i.e., high Ein and short τ) leads to a larger fraction of absorbed energy and an extended absorption zone along z. Combined with the saturation of the maximum fluence Fmax shown in Fig. 2d, this result shows that prefocal absorption governs the interaction in all tested semiconductors. Conversely, energy deposition is more localized for increased pulse durations.

Temporal-spectral shaping
Besides this first solution, consisting of increasing the pulse duration τ to improve energy deposition in semiconductors, we have explored two additional methods, both relying on temporal-spectral shaping. The first method examined involves changing the temporal sequence of spectral components, i.e., the chirp. For the same duration, pulses that are not bandwidth-limited can be up- or down-chirped, i.e., the long wavelengths arrive first or last, respectively. While this parameter plays a major role in high-field physics46, its effect is usually marginal at the moderate intensities typically involved in laser processing. Experimental studies on the surface of semiconductors showed that, in the linear absorption regime, the chirp has a moderate effect47 or no effect at all48 on the laser-produced carrier density. Analogous observations were made in the 2- and 5-photon absorption case in Si, where the plasma formation threshold is not significantly affected by the chirp49.

In most of our experiments, the pulses are up-chirped. In Fig. 5a–c, the propagation of 3-ps pulses in Si is displayed for up- and down-chirped pulses. For the same Ein value, the fluence distributions in Fig. 5a strongly differ between the two chirps. For down-chirped pulses, energy deposition is much more confined, as confirmed by the Fmax values in Fig. 5b for Ein ≥ 60 nJ. For both chirps, a saturation plateau is reached for Ein = 100 nJ. However, for down-chirped pulses, Fp is 2.4× higher than for up-chirped pulses—thus demonstrating that down-chirped pulses are beneficial for improving internal energy deposition. This is all the more confirmed by the fraction of absorbed energy fE in Fig. 5c, which grows ≈20% faster with Ein for down-chirped pulses in comparison with up-chirped pulses, while the on-axis dimension of the interaction is similar for both chirp configurations (Fig. 5a). Employing the same analysis to determine the effective nonlinear coefficients as in Fig. 3, we find the same effective critical power Peffcr = 1.6 ± 0.4 kW. However, the effective 2PA coefficient differs, with βeff2 = 1.8 ± 0.92 × 10−9 m/W and 3.0 ± 1.5 × 10−9 m/W for up- and down-chirped pulses, respectively, in agreement with the different nonlinear propagation behaviors in Fig. 5a.

Given that, for the measured smooth input spectrum, changing the chirp from up to down changes insignificantly the temporal intensity profile (see Supplementary Note 3), the higher peak fluence observed for down-chirped pulses can be explained by differences in ionization dynamics. Numerical studies for materials such as fused silica and MgF2 47,48,50 showed a similar asymmetry, as a consequence of multi-photon absorption being more efficient at shorter wavelengths—even for the same multi-photon order, as shown in Si51,52—whereas avalanche ionization becomes more efficient at longer wavelengths. The latter trend is directly described by the Drude model, which predicts that the inverse Bremsstrahlung heating rate scales as λ2, so free carriers gain energy more efficiently from longer wavelengths. In the down-chirped configuration, the blue spectral components (shorter wavelengths) arrive earlier in the pulse, producing a higher initial free-carrier density that enhances avalanche ionization by the subsequent red components (longer wavelengths). As a secondary mechanism, plasma-induced anomalous dispersion may further strengthen this effect by causing the blue components to catch up with the red ones, compressing the pulse and increasing its peak intensity.

The second method, which has been explored to improve energy deposition in semiconductors, involves changing the multi-photon absorption order. This can be achieved by selecting different wavelengths corresponding to distinct multi-photon absorption regimes for the same medium. Establishing the optimal wavelength for which in-volume laser–semiconductor interaction is exalted is a problem with no intuitive solutions. Increasing the multi-photon absorption order would theoretically decrease energy deposition at the focus, but also limit prefocal absorption. Conversely, reducing the multi-photon absorption order would have the opposite effect. This problem was studied theoretically in Si, where it was predicted that the increased free-carrier absorption efficiency lead to higher energy deposition at longer wavelengths16.

To experimentally evaluate the impact of multi-photon absorption order on filamentation, nonlinear propagation imaging has been performed in InP and GaAs for τ = 900 fs at a wavelength of λ = 1555 nm (2PA), and then compared to the ones at λ = 1960 nm (3PA) in Fig. 5d–f. As exemplified for GaAs in Fig. 5d, the wavelength impacts the fluence distribution, both in the linear and the nonlinear regime. Prefocal absorption is much more pronounced for 2PA than for 3PA. Due to their direct band gap, 2PA is even more efficient in InP and GaAs at λ = 1555 nm compared to Ge at λ = 1960 nm (Fig. 2). For a given wavelength, an analogous evolution of Fmax for GaAs and InP is shown in Fig. 5e as a function of Ein. This suggests that filamentation governs the interaction both in the 2PA and the 3PA regime. However, an important result is that the peak fluence Fp is more than one order of magnitude higher in the 3PA regime than in the 2PA regime, suggesting that higher multi-photon absorption orders are beneficial for tailoring energy deposition. This is consistent with results obtained in Si with 160-fs pulses in the 5-photon absorption regime31, where permanent bulk modifications were achieved due to reduced prefocal absorption and the ability to reach the modification threshold. Although mid-infrared pulses generate lower electron densities, the deposited energy is increased through more efficient inverse Bremsstrahlung absorption16. These findings indicate that high-order multi-photon absorption can, under specific conditions, relax filamentation-induced constraints, whereas at shorter wavelengths filamentation still limits energy deposition in semiconductors.

Applying the same approach as for Fig. 3b, c, the nonlinear refraction and absorption parameters Peffcr and βeff2 for λ = 1555 nm are determined. As shown in Fig. 5f, our theoretical approach successfully reproduces the experimental evolution of the nonlinear focal shift Δz as a function of Ein for InP and GaAs at both wavelengths. For both materials, Peffcr is one order of magnitude lower in the 2PA regime (57.1 ± 14.2 W for InP, 138.1 ± 69.0 W for GaAs) compared to the 3PA regime (573.7 ± 143.4 W for InP, and 1387 ± 693 W for GaAs). This is ascribable to the normal dispersion of the nonlinear index as well as the λ2 dependence of the critical power. The data points where P > Peffcr in Fig. 5f give access to βeff2 = 7.8 × 10−8 m/W for InP, and βeff2 = 7.5 × 10−8 m/W for GaAs. Similar to the measurement of βeff3 at λ = 1960 nm in Fig. 3c, these βeff2 values are orders of magnitude higher than the β2 literature values53–57, again attributable to differences in experimental determination method.

As a final remark, the three strategies to enhance energy deposition in semiconductors which consist of increasing τ (Fig. 3a), employing down-chirped pulses (Fig. 5b), and selecting λ in higher multi-photon absorption order regimes (Fig. 5e) are not mutually exclusive. To illustrate this aspect, besides the narrow-gap semiconductors investigated in this study, it is interesting to note that the filamentation-caused limitations for energy deposition also persist in wider band-gap semiconductors such as polycrystalline ZnSe (Eg ≈ 2.7 eV). Indeed, the threshold for internal modifications in this material could not be crossed with bandwidth-limited pulses with τ = 115 fs at λ = 800 nm (2PA regime)58. In contrast, by simultaneously using longer pulse durations (τ = 0.5–2 ps), down-chirped pulses, and λ = 1047 nm (3PA regime), in-volume laser writing was possible59. Apart from semiconductors, insulators exhibit even wider band gaps and lower nonlinear refractive index n2 (Fig. 1). Consequently, the filamentation-caused limitations are drastically reduced in these materials, and internal modifications can be produced in the 2PA regime as exemplified for sapphire60 and fused silica61.

To summarize, nonlinear propagation imaging has demonstrated that filamentation governs ultrafast laser–matter interactions in semiconductors. The 3D fluence distributions obtained exhibit an evolutive morphology, as well as a saturation of the maximum fluence for increased input pulse energy. While, in principle, the impact of propagation nonlinearities would decrease with increasing pulse duration, the established temporal scaling laws for effective key nonlinear refraction and absorption parameters indicate that this strategy is less beneficial than expected. Nevertheless, for all tested semiconductors, the peak fluence increases with the pulse duration. Other identified strategies for increasing the peak fluence in semiconductors include temporal-spectral shaping, which can take the form of down-chirped or long-wavelength pulses. Among the tested optimization strategies, increasing the multi-photon absorption order by using longer wavelengths proved most effective for enhancing the peak fluence, followed by increasing the pulse duration, and finally by using down-chirped pulses. This hierarchy, quantified in the present work, provides practical guidance for tailoring ultrafast laser parameters in semiconductors. The determination of effective key nonlinear coefficients summarized in the Supplementary Note 4 is essential for predicting optimal conditions for ultrafast laser writing in semiconductors. Beyond this, these coefficients also provide critical insight into other fields, enabling advances for instance in backside processing, microelectronics security, THz wave generation, high-harmonic generation, and supercontinuum generation (see Supplementary Note 4).

Methods
Samples
The Si, Ge, InP, and GaAs samples used in this study are 500 ± 25 μm thick, <100>-oriented, undoped, and two-sided polished.

Ultrafast laser irradiation
A Tm-doped fiber laser source emitting at a center wavelength of 1960 nm and a repetition rate of 50 kHz has been used. The spectral bandwidth is 29 nm [full width at half maximum (FWHM)]. The pulse duration is measured with autocorrelation and is adjustable from 275 fs to 25 ps (FWHM assuming a sech2 shape) by detuning the position of gratings in the optical compressor. In all experiments except the ones in Fig. 5a–c, the pulses are up-chirped, i.e., long wavelengths arrive before short wavelengths. The input pulse energy Ein is measured at the sample position and controlled optically by means of a half-wave plate and a linear polarizer. The beam diameter before focusing is adjusted with a Galilean telescope to ≈ 12.7 mm. The beam is focused by means of an objective lens of numerical aperture NA = 0.40 (Mitutoyo, M Plan Apo NIR 20×) mounted on a precision linear translation stage (Physik Instrumente, M-404.1DG) allowing its displacement along the optical axis with minimum incremental motion of 100 nm, and a resolution < 12 nm. For experiments at a wavelength of λ = 1555 nm, an Er-doped fiber laser source (Raydiance Inc., SmartLight 50) delivering non-chirped 900-fs pulses at a repetition rate of 1 kHz was employed. In all experiments the polarization is linear.

Nonlinear propagation imaging
To record 3D fluence distributions inside semiconductors, nonlinear propagation imaging has been used. This technique detailed in Supplementary Note 2 is based on an infrared microscope directed opposite to the incoming laser. This microscope is composed of an objective lens of numerical aperture NA = 0.85 (Olympus, LCPLN100XIR), a tube lens (Thorlabs, TTL200-S8), and an extended InGaAs camera (First Light Imaging, C-RED 2 ER 2.2 μm). This infrared camera is composed of 640 × 512 pixels with 15 μm pixel pitch, and responds linearly in the 1380–2150 nm spectral range. For improved imaging performance, this camera has been operated with a frame rate of 600 Hz, an exposure time of 1.66 ms, and a sensor temperature of −55 ∘C. For experiments at a wavelength of λ = 1555 nm, a standard InGaAs camera (Xenics, Bobcat 320) composed of 320 × 256 pixels, with 20 μm pixel pitch, is used. To optimize the dynamic range of the cameras for all input pulse energies Ein ranging from 1 pJ to 1 μJ, calibrated neutral density filters have been inserted between the objective lens and the tube lens. The procedure can be divided into three steps. First, the focal plane of the imaging objective is positioned at the exit surface of the sample under white light illumination. Second, at low input pulse energy (typically, a few pJ), the geometrical focus is positioned at the exit surface of the sample, so that it is imaged on the camera. Third, the focusing objective is moved along the optical axis in 100 nm steps around the geometrical focus position for different input pulse energies. A typical recording consists of 2000 images, with a total stage movement of 200 μm around the geometric focus position. The recorded z positions are then multiplied by the linear refractive index of the tested medium to get the actual positions in the material.

Determination of the key nonlinear optical coefficients
To ensure that the 3D fluence distributions obtained for the lowest input pulse energies Ein correspond to the linear propagation regime, propagation calculations detailed in Supplementary Note 3 have been performed with our vectorial model InFocus. In this regime, the maximum fluence reads Fmax = 2T Ein / (πw02), where T is the medium-dependent Fresnel transmission coefficient at the air–medium interface, and w0 is the beam radius at 1/e2. The corresponding calculations serve as a benchmark for the evaluation of key interaction parameters such as the effective critical power for nonlinearities Peffcr (see Supplementary Note 4), which is in turn implemented in our model based on a modified Marburger formula to determine the effective 2- and 3-photon absorption coefficients βeff2 and βeff3 (see Supplementary Note 4).

Data availability
The data that support the results of this article are available within the Primary Manuscript or Supplementary Information.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "The authors have addressed all issues that were pointed out in my review. Therefore, I recommend the paper for publication.", "reviewer_3": "As indicated in my original report, the paper by Chambonneau et al. addresses important scientific and engineering questions related to the development of new technologies based on in-volume interactions in semiconductors. In particular, the experimental work is carried out in configurations that are especially relevant to the challenge of 3D writing.\nThe revisions have produced a version that is convincing in some respects, notably through clearer definitions of the manipulated quantities (e.g., maximum fluence) and improved use of terminology. However, it remains less convincing with regard to the major conceptual concerns raised in the first round of review, as discussed below. Moreover, the additional material introduced in response to reviewers’ comments, although informative in itself, tends at times to overload an already dense manuscript without significantly enhancing the clarity or impact of the conclusions.\nOverall, I again recognize the substantial experimental effort and the merit of compiling such a large body of measurements. Nevertheless, I must reaffirm my initial impression: the work does not present a sufficiently novel conceptual contribution, nor does it offer a decisive advance in understanding or a clear technological direction. I therefore believe that the manuscript would be more appropriate for a more specialized journal.\nIn the present report, I do not revisit all points of disagreement with the authors but focus on a few key concerns that remain and that justify my overall assessment.\n----------\nFirst, although I continue to question the appropriateness of grouping all observed effects under the term “filamentation,” I note and welcome the removal of the earlier claim of “universality,” which was not well supported and appeared overstated. Unfortunately, this change has been accompanied by the new claim of “Extreme optical nonlinearities,” which I find again somewhat artificial and even confusing once one considers how these values are obtained. The authors appropriately refer to “effective” nonlinear coefficients, but the extreme values they report arise primarily from the use of simplified retrieval models. While these effective coefficients are of interest for the particular experimental conditions studied here, they do not contradict the literature values obtained by standard methods and cannot be easily extrapolated to other conditions.\nConsequently, I do not consider it reasonable to use them to draw quantitative conclusions or make claims of broad applicability to different configurations such as looser focusing or waveguide-based applications (THz generation, high harmonics, supercontinuum, etc.).\nRegarding the chirp dependence, I again acknowledge the interesting observation of a sensitivity to chirp, but I do not find in the added material more convincing evidence that chirp can be considered as a critical control parameter. In particular, the data do not demonstrate a sufficiently broad spectral coverage to fully support the discussions. Also, the observed symmetry of the temporal profiles appears to result from the reconstruction procedure itself, which retains only the quadratic phase term in Eq. S10 without measurement.\nFinally, a major concern raised by reviewers in the first round was related to the absence of a meaningful quantitative metric to evaluate progress in addressing the challenges of 3D writing in semiconductors under nonlinear propagation effects. The metric introduced in the revision, the fluence Fp , seems to me entirely inappropriate. What matters here is the absorbed energy density (or maybe ionization level), not the maximum achievable fluence inside the material. To make this error obvious, one can consider the limiting case of a continuous, low-intensity radiation in the transparency domain of a semiconductor: in such a situation F_p can become arbitrarily large while the nonlinear absorption—and thus energy deposition—tends toward zero. This conceptual inconsistency undermines the reported ratios of improvement and the hierarchy of parameters (pp. 6–7). I would in fact strongly recommend suppressing these new sections, as they are misleading and add confusion rather than clarity."}, "author_response": ""}, {"round": 2, "reviewer_comments": {"reviewer_3": "As a preliminary remark, I observe that the authors’ reply does not reassess the conceptual framing of the manuscript, nor does it provide additional experimental evidence that would reinforce the claim of wide applicability. The authors predominantly expand on their disagreements with the comments, yet do not seem to have considered a deeper or more comprehensive re-evaluation of the underlying issues.\nAlthough the response concentrates on the few main points raised by previous report, none convincingly affects my overall evaluation. Again, my position does not concern the intrinsic interest of the study or the merit of the dataset, but reflects instead the high-level standards of originality and impact expected for this journal. From this perspective, the breadth of impact and the significance of the advances presented remain, in my view, insufficiently substantiated.\nRegarding author response to R3 – Foreword\nI appreciate the authors’ efforts in providing a point-by-point reply. However, I note that the general concerns previously raised regarding conceptual novelty and/or understanding remain insufficiently addressed. The response focuses on individual remarks on the selection of deficiencies but does not directly engage with the broader issues which have been expressed.\nRegarding author response to R3 – P1\nI maintain that the Marburger formula, even when refined to account for nonlinear losses, while effective in describing the observations under the specific conditions of this study, cannot substitute to a rigorous nonlinear propagation model to support fully quantitative conclusions. Without a clear demonstration of the validity of the reported nonlinear coefficients in more general scenarios, they cannot be considered as contradicting others found in the literature.\nAlthough the authors point out that supercontinuum generation is shown in S17, these observations remain relatively qualitative and do not directly validate the measured nonlinear coefficients. To substantiate claims of broad applicability, one would reasonably expect a range of different experiments (e.g., SC, THz, harmonic generation, material modification, ….) combined/compared with modeling that consistently supports the general relevance of the reported coefficients.\nRegarding author response to R3 – P2\nThe authors’ explanation regarding spectral dependencies offers some clarity that could improve the paper. Accordingly, I agree that even a relatively narrow spectrum may potentially lead to noticeable variations. However, the question of potential pulse asymmetry remains insufficiently addressed. While the quadratic phase term is indeed an appropriate simplification to describe the effect introduced by the stretcher, my initial remark was related to the spectral phase prior to the stretcher. Assuming an initially flat spectral phase leads naturally to a symmetric amplitude profile, but this assumption is not demonstrated. Figure R2-1 does not help to provide an unambiguous confirmation of symmetry in the conditions studied, and additional characterization would be required to fully support this point.\nRegarding author response to R3 – P3\nThe response contains a certain inconsistency: the authors state that the relevance of the chosen parameters extends well beyond 3D writing, yet they justify their choice primarily by referring to studies on 3D writing or material modification. While I agree that using a directly measurable metric is advantageous, my concerns remain regarding its general relevance for a broad range of nonlinear optics experiments.\nThe issue I initially raised with a metric that can become arbitrarily large even in a regime insufficient to trigger any nonlinear response has not been convincingly addressed. Referring to works involving CW-induced modifications does not clarify how the proposed metric can be consistently used to identify an optimum in the present context. Therefore, I must respectfully but firmly disagree with the authors’ assertion that my earlier remarks were “incorrect” and I must maintain my position regarding the limited relevance of the reported ratios.", "reviewer_4": "Response to Reviewers\nContents\nResponse to Reviewer #1 2\nReviewer #1, Foreword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2\nReviewer #1, Point 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2\nReviewer #1, Point 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2\nReviewer #1, Point 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3\nReviewer #1, Point 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4\nReviewer #1, Point 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4\nReviewer #1, Point 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5\nReviewer #1, Point 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5\nReviewer #1, Point 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5\nReviewer #1, Point 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6\nReviewer #1, Point 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6\nReviewer #1, Point 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7\nReviewer #1, Point 12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7\nResponse to Reviewer #2 9\nReviewer #2, Foreword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9\nReviewer #2, Point 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9\nReviewer #2, Point 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9\nReviewer #2, Point 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10\nReviewer #2, Point 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11\nReviewer #2, Point 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11\nReviewer #2, Point 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11\nReviewer #2, Point 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12\nReviewer #2, Point 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12\nReviewer #2, Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13\nResponse to Reviewer #3 14\nReviewer #3, Foreword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14\nReviewer #3, Point 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14\nReviewer #3, Point 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14\nReviewer #3, Point 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15\nReviewer #3, Point 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16\nReviewer #3, Point 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16\nReviewer #3, Point 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17\nReviewer #3, Point 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18\nReviewer #3, Point 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19\nReviewer #3, Point 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19\nReviewer #3, Point 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20\nReviewer #3, Point 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21\nReviewer #3, Point 12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22\nReviewer #3, Point 13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23\nReviewer #3, Point 14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23\nReviewer #3, Point 15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25\nReferences 26\n1\nResponse to Reviewer #1\nReviewer #1, Foreword\nReviewer #1: This manuscript systematically investigates ultrashort laser pulse propagation in four kinds of\nnarrow-gapsemiconductors, andrevealsthetemporalscalinglawsforkeynonlinearparameters, whichstrongly\ndiffer from standard measurements with low-intensity pulses. Temporal-spectral shaping is finally proposed to\noptimize energy deposition inside semiconductors. These results lay the foundations for selectively tailoring\nsemiconductors by ultrafast laser writing. Overall, the paper is interesting but raises some questions and de-\nmands several major improvements, which are listed below.\nAuthors: We sincerely thank Reviewer #1 for the thorough and constructive review. We greatly appreciate\nthepositiveassessmentofourwork, particularlyregardingthesystematicinvestigationofultrashortlaserpulse\npropagation in four narrow-gap semiconductors and the potential of temporal–spectral shaping to control en-\nergydeposition. Theinsightfulcommentshavebeeninvaluableinimprovingtheclarityandrigoroftherevised\nmanuscript. We address each concern point by point below. For clarity, reviewer comments are shown in blue,\nour responses in green, and modifications in the revised manuscript in red.\nReviewer #1, Point 1\nReviewer #1: Two indirect (Si and Ge) and two direct (InP and GaAs) bandgap semiconductors have been\ninvestigated. It would be better to explain the reasons for choosing these specific semiconductors.\nAuthors: Beyond demonstrating that filamentation-induced limitations in Si generalize to other semiconduc-\ntors, we selected these four materials for two main reasons. First, they are all technologically important and\nwidely used in diverse applications, such as microelectronics, photovoltaics, sensors, and quantum engineering.\nSecond, all these narrow-gap materials are transparent at a wavelength of λ = 1960 nm. At this wavelength\ndifferent material-depedent multi-photon absorption orders dominate (2-photon absorption for Si and Ge, and\n3-photon absorption for InP and GaAs). Finally, all exhibit cubic crystal structures (diamond cubic for Si and\nGe; zincblende forInPandGaAs), whichis akey featurethatreducesanisotropyandpolarizationdependence\nduring nonlinear propagation.\nRevisions brought to the manuscript: To reflect both points, the original statement:\n“Two indirect (Si and Ge) and two direct (InP and GaAs) band-gap semiconductors have been selected (see the\ninset in Fig. 1). Because of their narrow band gaps (E <1.5 eV), these media exhibit high linear and nonlinear\ng\nrefraction (n >3, and n >10−18 m2/W, respectively).”\n0 2\nhas been revised to:\n“Two indirect (Si and Ge) and two direct (InP and GaAs) band-gap semiconductors have been selected (see\nthe inset in Fig. 1). Besides technological importance due to their widespread use in microelectronics, pho-\ntovoltaics, sensing, and quantum engineering, these materials exhibit cubic crystal structures, a property that\nminimizes anisotropy effects during nonlinear propagation. Because of their narrow band gaps (E < 1.5 eV),\ng\nthese media exhibit high linear and nonlinear refraction (n > 3 and n > 10−18 m2/W, respectively).” (Pri-\n0 2\nmary Manuscript, page 2)\nReviewer #1, Point 2\nReviewer #1: What is the specific definition of fluence? What the difference between the maximum fluence\nF and the peak fluence F ? Additionally, how to measure the F experimentally in this work?\nmax p max\nAuthors: The reviewer is right, and we agree that defining terms is essential for the clarity of our manuscript.\nThe fluence F(x,y,z) is defined as the laser energy per unit area. The maximum fluence F is the highest\nmax\nfluence over all three spatial dimensions, and reads\nF =max(F(x,y,z)). (R1)\nmax\nx,y,z\n2\nFor linear propagation, F is proportional to the input pulse energy E . However, in the nonlinear propaga-\nin\ntion regime, as F saturates for increased pulse energy E [see the plateaus in Fig. 2(d)], the peak fluence\nmax in\nF is introduced as the highest F value among all input pulse energies E , and reads\np max in\nF =max(F )= max (F(x,y,z)). (R2)\np max\nEin x,y,z,Ein\nRevisions brought to the manuscript: To clarify these definitions, the original statement:\n“This technique, analogous to tomography as illustrated in Fig. 2(a), was initially developed for characterizing\nlight propagation in water [34], and later applied to dielectrics [35, 36]. Concerning semiconductors, it was\nsuccessfully employed to examine nonlinear propagation of light in Si [17, 18, 22, 24, 30, 37], but it has been so\nfar limited to this material.”\nhas been revised to:\n“This technique, analogous to tomography as illustrated in Fig. 2(a), was initially developed for characterizing\nlight propagation in water [34], and later applied to dielectrics [35, 36]. It provides direct access to the laser\nenergy per unit area [i.e., the fluence F(x,y,z)] in three spatial dimensions. Concerning semiconductors, this\ntechnique was successfully employed to examine nonlinear propagation of light in Si [17, 18, 22, 24, 30, 37], but\nit has been so far limited to this material.” (Primary Manuscript, page 2)\nThe original statement:\n“Among all interaction parameters that can be extracted from the 3D fluence distributions (see hereafter), a\nlow-hanging fruit is the maximum fluence F .”\nmax\nhas been revised to:\n“Among all interaction parameters that can be extracted from the 3D fluence distributions (see hereafter), a\nlow-hanging fruit is the maximum fluence F defined as the highest value of the local fluence within the 3D\nmax\nspatial distribution [i.e., F =max (F(x,y,z))].” (Primary Manuscript, pages 2 and 3)\nmax x,y,z\nThe original statement:\n“In contrast, when E exceeds a medium-dependent threshold, the experimental data deviate from this regime,\nin\nand F saturates to a peak value F , which also depends on the material.”\nmax p\nhas been revised to:\n“In contrast, when E exceeds a medium-dependent threshold, the experimental data deviate from this regime,\nin\nand F saturates to a peak value F = max (F ), which also depends on the material.” (Primary\nmax p Ein max\nManuscript, page 3)\nReviewer #1, Point 3\nReviewer #1: How to obtain the normalized fluence distributions shown in Figure 2b based on the nonlinear\npropagation imaging in Figure 2a?\nAuthors: ThefluencedistributionsinFig.2(b)arereconstructedfromastackofxy imagesrecordedatsucces-\nsive on-axis positions z using the nonlinear propagation imaging setup shown in Fig. 2(a). For each z-position,\nthe transmitted light intensity map is recorded, background-subtracted, and converted into absolute fluence\nusing calibration with a reference measurement of the pulse energy. The resulting 3D fluence distribution\nF(x,y,z) is then normalized at each z-plane to its maximum value F (z) to emphasize the relative spatial\nmax\ndistribution, independent of energy fluctuations. These normalized slices are assembled to produce the plots in\nFig. 2(b). Detailed image processing steps and the normalization procedure are provided in the Supplementary\nInformation (Section S2.1), along with a schematic workflow in new Fig. S1.\nRevisions brought to the manuscript: To clarify how to obtain the fluence distributions from nonlinear\npropagation imaging, and to provide more technical details concerning the image processing, we have added\na dedicated section in the Supplementary Information (S2.1 Nonlinear propagation imaging technique), which\nincludes a new figure (Fig. S1).\nThe original statement:\n“This technique is based on an infrared microscope directed opposite to the incoming laser.”\nhas been revised to:\n“This technique detailed in Supplementary Information, Section S2.1 is based on an infrared microscope directed\n3\nopposite to the incoming laser.” (Primary Manuscript, Materials & methods)\nReviewer #1, Point 4\nReviewer #1: Figures2d, 4band4edemonstratethattheF vs. E satisfypower-lawrelationship. What\nmax in\nis the fitted power exponent? Why?\nAuthors: Actually, in these figures [now, Figs. 2(d), 5(b) and 5(e) in the revised manuscript], the saturation\nfor high input pulse energy values implies that a power-law function—or, more generally, any mathematical\nfunction which tends to infinity when E → +∞—will fail to fit the whole set of F data. Nevertheless, in\nin max\nthe linear propagation regime (i.e., for low E values), F scales linearly with E , which can be viewed as a\nin max in\npower-law relationship with an exponent of 1. This is highlighted by the dashed lines in these figures, which do\nnot result from a fitting procedure, but instead come from calculations detailed in Supplementary Information,\nSection S3.2, where we show for a Gaussian fluence distribution that\n2T E\nF = F in, (R3)\nmax πw2\n0\nwhereT =1−(1−n )2/(1+n )2 istheFresneltransmissioncoefficientattheair–mediuminterfaceatnormal\nF 0 0\nincidence, n is the linear refractive index of the material, and w is the beam radius at 1/e2.\n0 0\nRevisions brought to the manuscript: To clarify this aspect, the original statement:\n“The dashed lines correspond to the linear propagation regime.”\nhas been revised to:\n“The dashed lines correspond to calculations in the linear propagation regime (see mathematical details in Sup-\nplementary Information, Section S3).” (Primary Manuscript, Fig. 2 caption)\nThe following statement has been added to the caption of the new Fig. 5:\n“The dashed lines in (b) and (e) correspond to calculations in the linear propagation regime (see mathematical\ndetails in Supplementary Information, Section S3).” (Primary Manuscript, Fig. 5 caption)\nReviewer #1, Point 5\nReviewer #1: Figure 4a demonstrates a significant difference in energy deposition between up- and down-\nchirped pulses at the same E at line 254. However, the subsequent analysis lacks a direct explanation for this\nin\nphenomenon.\nAuthors: We agree with Reviewer #1, and we have added a clearer and more explicit explanation for the\nchirp dependence. This explanation, which also addresses Reviewer #3, Point 14, relies on the fact that the\nmeasured temporal profiles of up- and down-chirped pulses are almost identical, highlighting that the observed\ndifferences in detected fluence are due to the different arrival sequence of the spectral components rather than\npulse-shape distortions.\nRevisions brought to the manuscript: To clarify this aspect, the original statement:\n“The chirp-dependence of filamentation may originate mainly from two physical phenomena. First, the ioniza-\ntion dynamics could differ between up- and down-chirped pulse. Calculations showed that a similar asymmetry\nexists for fused silica and MgF [44, 45, 47]. This is ascribable to the increased efficiency of multi-photon\n2\nabsorption for shorter wavelengths (even for the same multi-photon absorption order [48, 49]), while avalanche\nionization is more efficient for longer wavelengths. Therefore, in the down-chirped configuration where the\n“blue” spectral components arrive before the “red” ones, the produced free-carrier density is higher. The second\nphenomenon which could play a role is nonlinear dispersion induced by the plasma. Indeed, the interaction\nresults in extended plasma channels along the optical axis, which may show anomalous dispersion, meaning that\nthe “blue” can catch up with the “red” part of the pulse, in turn resulting in shorter duration—and thus, higher\nintensity.”\nhas been revised to:\n“Given that, for the measured smooth input spectrum, changing the chirp from up to down changes insignif-\nicantly the temporal intensity profile (see Supplementary Information, Section S3.4), the higher peak fluence\nobserved for down-chirped pulses can be explained by differences in ionization dynamics. Numerical studies for\n4\nmaterials such as fused silica and MgF [44, 45, 47] showed a similar asymmetry, as a consequence of multi-\n2\nphoton absorption being more efficient at shorter wavelengths—even for the same multi-photon order, as shown\nin Si [48, 49]—whereas avalanche ionization becomes more efficient at longer wavelengths. The latter trend\nis directly described by the Drude model, which predicts that the inverse Bremsstrahlung heating rate scales as\nλ2, so free carriers gain energy more efficiently from longer-wavelength fields. In the down-chirped configura-\ntion, the blue spectral components (shorter wavelengths) arrive earlier in the pulse, producing a higher initial\nfree-carrier density that enhances avalanche ionization by the subsequent red components (longer wavelengths).\nAs a secondary mechanism, plasma-induced anomalous dispersion may further strengthen this effect by causing\nthe blue components to catch up with the red ones, compressing the pulse and increasing its peak intensity.”\n(Primary Manuscript, page 5)\nThe new Section titled “S3.4 Chirped temporal profiles” has been added to the Supplementary Information.\nThis Section includes the new Fig. S8 showing fine characterizations of the input laser spectrum as well as\ncalculationsofthetemporalintensityprofileintheup-anddown-chirpconfigurations,comparedtobandwidth-\nlimited pulses.\nReviewer #1, Point 6\nReviewer #1: Regarding the explanation of Figure 4b, how to determine the factor of 2.4 in F between up-\np\nand down-chirped pulses at line 259?\nAuthors: The factor of 2.4 is obtained by taking the ratio between the peak fluence values determined for\ndown-chirped pulses (Fdown =1.17 J/cm2) and up-chirped pulses (Fup =0.49 J/cm2).\np p\nRevisions brought to the manuscript: The peak fluence values now appear in the corresponding figure\n[Fig. 5(b) in the revised manuscript]. The factor of 2.4 is highlighted in Fig. 5(b).\nReviewer #1, Point 7\nReviewer #1: What is the theoretical basis for the wavelength selection during the switching of multiphoton\nabsorption orders at line 318?\nAuthors: The theoretical basis for the wavelength selection first relies on the band gap E of the considered\ng\nmaterials (InP: E = 1.34 eV ; GaAs: E = 1.43 eV). We acknowledge and correct an error in the originally\ng g\nsubmittedmanuscript,wherethebandgapofInPwasincorrectlygivenasE =1.24eVinsteadofE =1.34eV.\ng g\nSecond, the photon energy is ℏω =0.63 eV at a wavelength of λ=1960 nm, and ℏω =0.80 eV at a wavelength\nof λ=1555 nm. The multi-photon absorption order N is given by\nN =1+⌊E /ℏω⌋, (R4)\ng\nwhere⌊⌋denotesthefloorfunction,ℏisthereducedPlanckconstant,ω =2πc/λisthelaserangularfrequency,\nand c is the speed of light in vacuum. As a result, for λ = 1960 nm, N = 2 for Si and Ge, and N = 3 for InP\nand GaAs. In Fig. 5(d)–(f) of the revised Primary Manuscript, λ = 1555 nm which results in N = 2 for InP\nand GaAs.\nRevisions brought to the manuscript: The band gap value of InP has been changed to E = 1.34 eV in\ng\nFig. 1, as well as in Table S1 (Supplementary Information), where Ref. [S12] has been added.\nReviewer #1, Point 8\nReviewer #1: The paper proposes three methods of optimizing energy deposition including increasing pulse\nduration,usingadown-chirpedpulseandachievingahighermultiphotonabsorptionorder. Isthereahierarchy\nor relative priority among these methods? Furthermore, are there any quantitative metrics that can be used to\nevaluate the degree of energy deposition optimization?\n5\nAuthors: This is a very interesting point from Reviewer #1, and we are grateful for this comment. Using the\npeak fluence F as the optimization metric, the three methods can be ranked as follows:\np\n1. Increasing the multi-photon absorption order. The most impactful strategy is to shift to longer\nexcitation wavelengths, thereby increasing the MPA order. From the data in the revised manuscript,\nFig.5(e),using3-photonabsorptioninsteadof2-photonabsorptionincreasesF byafactorof15.0inInP\np\nand 17.6 in GaAs.\n2. Increasing the pulse duration. Optimizing the pulse duration ranks second. From Fig. 3(a), using\nτ =25 ps instead of τ =275 fs increases F by factors of 8.7 (Si), 4.6 (Ge), 5.9 (InP), and 4.0 (GaAs).\np\n3. Using down-chirped pulses. Chirp control ranks third. From Fig. 3(b), using down-chirped instead of\nup-chirped pulses increases F by a factor of 2.4 in Si for τ =3 ps.\np\nThis hierarchy is now explicitly highlighted in the conclusion. As a side note, if the final application concerns\nthe formation of permanent modifications inside the material (e.g., ultrafast laser writing), the modification\nthreshold or modification volume would be a more relevant optimization metric than F .\np\nRevisions brought to the manuscript: The following statement has been added to the conclusion:\n“Among the tested optimization strategies, increasing the multi-photon absorption order by using longer wave-\nlengths proved most effective for enhancing the peak fluence, followed by increasing the pulse duration, and\nfinally by using down-chirped pulses. This hierarchy, quantified in the present work, provides practical guidance\nfor tailoring ultrafast laser parameters in semiconductors.” (Primary Manuscript, page 7)\nIn agreement with Reviewer #2, Point 2, a new section titled “S4.5 Summary tables” has been added to\nthe Supplementary Information. This new section contains Tables S7–S10 where the peak fluence F , the ef-\np\nfective critical power Peff, and the multi-photon absorption coefficient βeff are given for different wavelengths\ncr N\nλ, multi-photon absorption orders N, pulse durations τ, and chirps. These tables which contain the results\nfrom the revised Primary Manuscript, Figs. 3 and 5, give readers a comprehensive comparison of the effects of\ndifferent regimes.\nReviewer #1, Point 9\nReviewer #1: The introduction inadequately covers the applications of ultrafast laser filamentation in gases.\nAuthors: We agree with Reviewer #1 that a detailed list of applications is not necessary in the introduction.\nWe have therefore shortened this part while keeping references to representative studies that demonstrate how\nthe fundamental understanding of filamentation has enabled important applications.\nRevisions brought to the manuscript: The original statement in the introduction:\n“In gases, the remarkable properties of filaments have led to a plethora of applications including light detection\nand ranging (LIDAR) [3], spectroscopy [4], terahertz wave generation [5], fog clearing [6], air waveguides [7, 8],\nand lightning guidance [9].”\nhas been revised to:\n“Ingases,theremarkablepropertiesoffilamentshaveledtoaplethoraofapplications[3–9]”(PrimaryManuscript,\npage 1)\nReviewer #1, Point 10\nReviewer #1: In the Materials & methods, the section of propagation calculations mainly describes the cal-\nculation process of F , Peff, βeff βeff, the title should be revised for clarity.\nmax cr 2 3\nAuthors: We agree with Reviewer #1 and have revised the title of the corresponding section to better reflect\nits content.\nRevisions brought to the manuscript: In the Materials & methods, the original title of the section:\n“Propagation calculations.”\n6\nhas been revised to:\n“Determination of the key nonlinear optical coefficients.” (Primary Manuscript, Materials & methods, page 8)\nReviewer #1, Point 11\nReviewer #1: Besides ultrafast laser writing, supercontinuum generation during ultrashort laser pulses prop-\nagation in transparent solid is also widely concerned. Could the authors conduct a discussion regarding the\nimpact of energy deposition on the supercontinuum generation?\nAuthors: We are grateful to Reviewer #1 for this particularly relevant and inspiring point. The work pre-\nsented in our manuscript is indeed not only limited to ultrafast laser writing, but is also valuable in various\nfields including supercontinuum generation (SCG). While this is not the primary goal of our study, we have\nperformed additional experiments to explore SCG in Si. These consist of transverse spectral measurements for\nvariousinputpulseenergiesE andpulsedurationsτ. Asexpected,higherE and/orlowerτ resultingreater\nin in\nspectral broadening. While most studies on SCG in Si were performed with sub-ps pulses, our measurements\nsuggest that spectral broadening persists even at single-digit ps pulse durations.\nBesides SCG, accurate determination of the nonlinear optical coefficients in semiconductors is also crucial\ninotherfieldslikebacksideprocessing, microelectronicssecurity, THzwavegeneration, andhigh-harmonicgen-\neration(HHG).Inallthesefields, aninaccurateestimationofPeff and/orβeff potentiallyresultsinsuboptimal\ncr 2,3\noutcomes.\nRevisions brought to the manuscript: Anewsectiontitled“S4.6 Applications beyond laser direct writing”\nhas been added in the Supplementary Information. In this new section, the impact of a misestimation of Peff\ncr\nand/or βeff in the aforementioned fields is discussed with corresponding references.\n2,3\nIn this new section, the additional experimental results we obtained on SCG are included in a new figure\n(Fig. S17).\nThe original statement in the conclusion:\n“The whole set of results allows us to envision future optimizations to improve the degree of control of energy\ndeposition in the bulk of semiconductors, with various potential applications including in-chip ultrafast laser\nfunctionalization.”\nhas been revised to:\n“The determination of effective key nonlinear coefficients summarized in the Supplementary Information, Sec-\ntion S4.5, is essential for predicting optimal conditions for ultrafast laser writing in semiconductors. Beyond\nthis, these coefficients also provide critical insight into other fields, enabling advances for instance in backside\nprocessing, microelectronicssecurity, THzwavegeneration, high-harmonicgeneration, andsupercontinuumgen-\neration (see Supplementary Information, Section S4.6).” (Primary Manuscript, page 7)\nThe original statement has been added to the abstract:\n“Beyond laser writing, the accurate determination of effective nonlinear parameters reported here is essential\nfor predicting and optimizing semiconductor backside processing, microelectronics security, as well as high-\nharmonics, supercontinuum, and terahertz wave generation.” (Primary Manuscript, abstract, page 1)\nReviewer #1, Point 12\nReviewer#1: JustlikeFigure3b,pleaseaddtheerrorbarsandcorrespondingmeaningsforFigures3aand3c.\nAuthors: We thank Reviewer #1 for spotting this oversight.\nRevisions brought to the manuscript: The error bars have been added to Fig. 3(a) and (c).\nTo clarify the meaning of the error bars in Fig. 3(b), the original statement:\n“The critical pulse energy E , which delimits the two propagation regimes is thus determined as the average\ncr\nbetween the highest input pulse energy for which the propagation is linear [E− = 3 nJ in Fig. S9(b)], and the\nin\nlowest input pulse energy for which the propagation is nonlinear [E+ =6 nJ in Fig. S9(b)].”\nin\n7\nhas been revised to:\n“The critical pulse energy E , which delimits the two propagation regimes is thus determined as the average\ncr\nbetween the highest input pulse energy for which the propagation is linear [E− =3 nJ in Fig. S13(b)], and the\nin\nlowest input pulse energy for which the propagation is nonlinear [E+ = 6 nJ in Fig. S13(b)]. The effective\nin\ncritical power shown in the Primary Manuscript, Fig. 3(b) is evaluated for different pulse durations τ as\nPeff =0.88TFEcr, (S15)\ncr τ\nwhere T is the Fresnel transmission coefficient. The uncertainty in Peff values is calculated as\nF cr\nε=0.88\nTF|E\ni\n+\nn\n−Ecr|\n=0.88\nTF|E\ni\n−\nn\n−Ecr|\n. (S16)” (Supplementary Information, page 8)\nτ τ\nTo clarify the meaning of the error bars in Fig. 3(a), the following sentence has been added:\n“Therefore, F values are reported with an uncertainty of ±15%.” (Supplementary Information, page 4)\np\nTo clarify the meaning of the error bars in Fig. 3(c), the following sentence has been added:\n“Accounting for the uncertainty in Peff, the error on βeff and βeff values is estimated at ±50%.” (Supplemen-\ncr 2 3\ntary Information, page 11)\n8\nResponse to Reviewer #2\nReviewer #2, Foreword\nReviewer #2: In the manuscript “The universality of filamentation-caused challenges of ultrafast laser en-\nergy deposition in semiconductors”, the authors analyze the influence of material properties, wavelength, pulse\nduration, and chirp of ultrashort laser pulses on the process of laser energy deposition in semiconductors. The\nmanuscripthasundeniablepracticalsignificancefromtheperspectiveofintegrated3Dsiliconphotonics,aswell\nasconsiderablefundamentalimportance,sincetheinteractionofultrashortlaserpulseswithsemiconductorsre-\nmainsstillinsufficientlystudied. Theworkcanbeconditionallydividedintoseveralparts: thefirstinvestigates\ntheinfluenceofthematerialonthefluenceprofilewithinthevolumeofthesemiconductor; theseconddiscusses\nthe impact of pulse duration and chirp on this distribution; and the last examines the influence of wavelength.\nThe quality of the experimental and numerical modeling work is at a very high level. The literature review is\nsufficiently modern, and the results are novel. The work is potentially of interest to a broad readership.\nIndividually, the sections mentioned above are well-organized, have clear conclusions, and apart from some\npresentation features, I have practically no comments about them. However, combined, they do not form a co-\nhesivesinglework,sometimespresentingtoomuchinformationandraisingquestionsthatarenotfullyanswered.\nAuthors: We sincerely thank Reviewer #2 for the thorough reading of our manuscript and the constructive\nfeedback. We are grateful for the positive assessment of our work, particularly regarding “the quality of the\nexperimental and numerical modeling work” and the novelty of the results. The reviewer’s insightful comments\nhavebeeninvaluableinimprovingnotonlytheclarityandpresentation,butalsothecohesivenessoftherevised\nmanuscript, ensuring that the individual sections now form a more unified narrative. We address each concern\nin a point-by-point manner below. For clarity, reviewer comments are shown in blue, our responses in green,\nand modifications in the revised manuscript in red.\nReviewer #2, Point 1\nReviewer #2: Firstly, at the beginning of the manuscript, in Fig. 1, the authors provide information on the\nbandgap, nonlinear and linear refractive indices, from which it follows that Ge differs most strongly from the\notherthreesamples,InPandGaAshavesimilarproperties,andSipropertiesaresomewhatintermediate(except\nfor the nonlinear refractive index). However, later in the text, the maximum fluence is achieved precisely in\nsilicon,andoverall,thedependenciesaremostpronouncedthere(propertiesareleastpronouncedingermanium).\nAuthors: Among all tested materials, Ge exhibits the smallest band gap, which is consistent with Fig. 3(b),\nwhere Ge shows the lowest effective critical power Peff. This means that a lower input pulse energy E and/or\ncr in\nlonger pulse duration τ is sufficient to trigger nonlinear effects in Ge. In terms of nonlinear absorption mech-\nanism, Ge behaves similarly to Si, since two-photon absorption (2PA) is the dominant process in both. In\nFig. 3(c), the effective two-photon absorption coefficient βeff is higher in Ge than in Si, consistent with the\n2\nsmaller band gap of Ge. This indicates that lower intensities are required to generate a given free-carrier den-\nsity in Ge compared to Si. However, the higher βeff also leads to stronger prefocal absorption, which depletes\n2\nthe beam before the nonlinear focusing can fully develop, and limits the achievable peak fluence F compared\np\nto Si.\nReviewer #2, Point 2\nReviewer #2: Secondly, the article varies several parameters simultaneously: wavelength, energy, duration,\nchirp, and material. As a result, after multiple readings, it creates an impression of some excessive clutter,\nand it is unclear which particular effect dominates in each case.\nBased on these two points, at present, the open question remains: which specific material properties influ-\nencethepropagationandabsorptionofultrashortlaserpulsesinthemid-IRrangeinsemiconductors? Itseems\nthereisnomonotonicdependenceonthebandgap,linear,ornonlinearrefractiveindices,althoughfromgeneral\nprinciples,onemightexpectthatalargerbandgapandlowerrefractiveindices(bothlinearandnonlinear)\nwould\nlead to increased fluence. However, the data presented by the authors do not demonstrate this. Therefore, it\nmight be advisable for the authors to consider a more comprehensive comparison of the effects of different\n9\nregimes, perhaps in the form of one or several tables, or alternatively, splitting the work into multiple publica-\ntions (even materials on silicon and/or germanium seem sufficient for a high-quality paper). As it stands, the\nnumber of samples rather confuses than adds value.\nAuthors: We have reorganized the manuscript to reduce the impression of excessive clutter.\nThe question raised by Reviewer #2 on which specific material properties influence ultrashort laser pulse\npropagation and absorption in the mid-IR is highly relevant. As noted, there is unfortunately no universal\nsingle-parameter rule, since nonlinear propagation results from the combined action of several physical effects.\nNonlinear refraction and absorption depend on the band structure, while the ratio between the laser photon\nenergy and the band gap determines the multi-photon absorption (MPA) order. The direct or indirect nature\nof the band gap also plays a role, as indirect transitions require phonon participation, making the MPA process\nless probable. Additional factors that were not varied in our study, such as dopant type, doping concentration,\nand crystal orientation, can also affect the effective nonlinear coefficients.\nDespite these material-specific quantitative differences, a key finding of our work is that all tested semicon-\nductors follow the same qualitative trends when E and τ are varied [see Figs. 2(d), 3, and 4 in the revised\nin\nPrimary Manuscript].\nWe are honored by the comment of Reviewer #2 that our results could be split into multiple high-quality\npapers. However,webelievethatkeepingallresultstogetherinasinglepublicationprovidesauniqueandcom-\nprehensiveoverviewofultrashortpulsefilamentationinsemiconductors,makingitmorevaluableasareference.\nFinally, we thank Reviewer #2 for suggesting a summary in tabular form. We have added a dedicated section\nwith summary tables to the Supplementary Information.\nRevisions brought to the manuscript: Toavoidtheimpressionofsomeexcessiveclutter,Fig.3intheorig-\ninally submitted manuscript has been split into two figures (Figs. 3 and 4 in the revised Primary Manuscript).\nA new section titled “S4.5 Summary tables” has been added to the Supplementary Information.\nThis new\nsection contains Tables S7–S10 where the peak fluence F , the effective critical power Peff, and the multi-\np cr\nphoton absorption coefficient βeff are given for different wavelengths λ, multi-photon absorption orders N,\nN\npulse durations τ, and chirps. These tables, which present the results from the revised Primary Manuscript,\nFigs. 3 and 5, allow readers to make a comprehensive comparison of the effects of different regimes, and to use\nthese parameters for various applications, as suggested by Reviewer #2.\nThe original statement in the conclusion:\n“The whole set of results allows us to envision future optimizations to improve the degree of control of energy\ndeposition in the bulk of semiconductors, with various potential applications including in-chip ultrafast laser\nfunctionalization.”\nhas been revised to:\n“The determination of effective key nonlinear coefficients summarized in the Supplementary Information, Sec-\ntion S4.5, is essential for predicting optimal conditions for ultrafast laser writing in semiconductors. Beyond\nthis, these coefficients also provide critical insight into other fields, enabling advances for instance in backside\nprocessing, microelectronicssecurity, THzwavegeneration, high-harmonicgeneration, andsupercontinuumgen-\neration (see Supplementary Information, Section S4.6).” (Primary Manuscript, page 7)\nReviewer #2, Point 3\nReviewer #2: I believe that the information about the influence of chirp is somewhat excessive for the main\ntext of a data-heavy publication. Perhaps it could be moved to supplementary materials?\nAuthors: We appreciate this suggestion from Reviewer #2. However, we note that Reviewer #3 explicitly\nrequested both clarification and emphasis on chirp-related effects (Points 2 and 14). These results—shown in\ntherevisedPrimaryManuscript,Figs.5(a)–(c)—arecriticaltodemonstratethatenergydepositionisinfluenced\nnot only by the central wavelength [Figs. 5(d)–(f)] but also by the temporal ordering of spectral components.\nFor this reason, we believe retaining this material in the main text is essential to preserve the completeness of\nthe discussion and address simultaneously the points of both reviewers.\n10\nReviewer #2, Point 4\nReviewer #2: Fig. 3g,h are extremely poorly readable. Extracting information from them in their current\nform is very difficult.\nAuthors: Weacknowledgethattheoriginal3Dplotswerechallengingtointerpret. Toimproveclarity, thexy-\nintegrated measurements are now shown in a separate figure in the Primary Manuscript, and two-dimensional\nplots extracted from the 3D data have been added to the Supplementary Information. The revised manuscript\nalso retains Supplementary Videos 1 and 2, which provide rotating visualizations of the 3D plots.\nRevisions brought to the manuscript: Intherevisedmanuscript,Figs.3(a)–(c)and3(e)–(h)arepresented\nastwoseparatefigures(Figs.3and4,respectively). Figure3(d)fromtheoriginalsubmissionhasbeenremoved.\nNew figures which show 2D plots extracted from the 3D plots have been added to the Supplementary Informa-\ntion (Figs. S11 and S12). The following statement has been added to the Supplementary Information:\n“The evolution of f and L as a function of E and the corresponding fits according to Eq. (S13) are dis-\nE abs in\nplayed as two-dimensional graphs in Figs. S11 and S12, respectively.” (Supplementary Information, page 8)\nThe following statement has been added to the caption of the new Fig. 4:\n“Supplementary Videos 1 and 2 offer rotating visualization of the 3D plots in (c) and (d), respectively, and\ntwo-dimensional representations of these 3D plots are shown in Supplementary Information, Section S4.2.”\n(Primary Manuscript, Fig. 4 caption)\nReviewer #2, Point 5\nReviewer #2: Due to the large number of varying parameters and numerous figures, I recommend explicitly\nindicating the sample and the wavelength of the laser pulse on each figure. Working with the current presenta-\ntion is quite challenging.\nAuthors: WethankReviewer#2forthishelpfulsuggestionandfullyagreethatclearerfigurelabelingimproves\nreadability. IntherevisedPrimaryManuscriptandSupplementaryInformation,thematerial,laserwavelength,\nand pulse duration are now explicitly indicated on every figure where this information is relevant. This ensures\nthat readers can immediately identify the experimental conditions without referring back to the text.\nRevisions brought to the manuscript: When this was not explicit in the originally submitted manuscript,\nwe have added information about the tested material, wavelength and pulse duration. This information is now\nindicated consistently across all figures in the revised manuscript.\nReviewer #2, Point 6\nReviewer #2: In Fig. 4d, the change in wavelength clearly affects the divergence angle of the ”angel wings,”\nsee the figure below (in the pdf version of the review). Interestingly, in Figs. 4a, 2b, these angles are approx-\nimately the same. Is this caused by the change in the refractive index at the interfaces when changing the\nwavelength and, consequently, the effective focusing?\nAuthors: Once again, this is a very interesting point from Reviewer #2, and we are thankful for this. After\ncareful verification in all tested configurations, the angle of the “angel wings” follows the expectations from the\nlinear propagation regime, where the half-angle of the cone of light reads θ =arcsin(NA/n ), where NA=0.40\n0\nisthenumericalapertureofthefocusingobjectivelens,andn isthelinearrefractiveindexofthemedium. For\n0\nInP and GaAs, the wavelength difference between 1555 nm and 1960 nm results in a refractive index change\non the order of 3×10−2. As a consequence, the difference in θ between the two wavelengths is less than 0.1◦,\nwhich cannot explain the different angles of the angel wings.\nIn fact, the originally submitted manuscript contained a scale error in the images, and we apologize for this\noversight. This has been corrected in Fig. 5(d) of the revised manuscript, where the angle of the “angel wings”\n11\nin GaAs is now consistent between the two wavelengths. This correction does not affect any measured values\nof F , Peff, βeff.\np cr 2,3\nRevisions brought to the manuscript: To better visualize that the angel wings follow the half-angle of the\ncone of light, θ appears in Fig. 2(b) of the revised manuscript.\nThe following statement has been added to the caption of the new Fig. 2:\n“The vector ⃗k indicates the direction of propagation, and θ = arcsin(NA/n ) indicates the medium-dependent\n0\nhalf-angle of the cone of light.” (Primary Manuscript, Fig. 2 caption)\nThe following statement has been added:\n“The angel wings form an angle which follows the half-angle of the cone of light θ =arcsin(NA/n ).” (Primary\n0\nManuscript, page 2)\nThe images in Fig. 5(d) in the revised manuscript now show fluence distributions in GaAs to better visualize\nthe conservation of the angle of the angel wings.\nReviewer #2, Point 7\nReviewer #2: In Fig. 2b, there is a clear trend of increasing defocusing effect when varying the sample\nfrom Si → InP → GaAs → Ge. It might be worth changing the order to facilitate comparison of the impact\nregimes. Also, it seems that when changing the wavelength for the InP sample, the regime becomes similar\nto propagation in Ge. It would be advisable to align the energies presented in Figs. 2b, 4a, and 4c for better\ncomparison.\nAuthors: We agree with Reviewer #2 on the trend of increasing defocusing when varying the sample from\nSi → InP → GaAs → Ge. This order could thus indeed be considered in Fig. 2(b). However, Si and Ge are\nmedia where two-photon absorption (2PA) dominates at a wavelength of λ=1960 nm, and InP and GaAs are\nmedia where three-photon absorption (3PA) dominates at the same wavelength. Moreover, Si and Ge exhibit\nanindirectbandgap, whileInPandGaAsexhibitadirectbandgap. Forthesereasons, weprefertokeepthese\ntwo subgroups of materials side by side, and we hope Reviewer #2 will understand our choice.\nReviewer #2 also rightly recalls that when the multi-photon absorption order is decreased from 3 to 2, the\npropagation in InP, and also in GaAs [see Fig. 5(d) of the revised manuscript and Reviewer #2, Point 6]\nbecomes similar to Ge. This is due to the fact that prefocal absorption is much more pronounced for low\nmulti-photon absorption orders, and this is actually one of the key messages of our study. Nevertheless, while\n2PA dominates for Ge irradiated at λ=1960 nm, and InP or GaAs irradiated at λ=1555 nm, the comparison\nis not completely fair due to the indirect band gap of Ge and the direct band gaps of InP and GaAs. This\nimpliesthat2PAislessefficientinGethaninInPandGaAs,whichisconfirmedinthesummaryTablesS7–S10\nin the Supplementary Information (see Reviewer #2, Point 2). The enhanced absorption in InP and GaAs at\nλ = 1555 nm also implies that the employed input pulse energies E cannot be directly compared to Ge at\nin\nλ=1960 nm, as prefocal absorption governs the interaction at lower E values.\nin\nRevisions brought to the manuscript: The following statement has been added:\n“Prefocal absorption is much more pronounced for 2PA than for 3PA. Due to their direct band gap, 2PA is\neven more efficient in InP and GaAs at λ=1555 nm compared to Ge at λ=1960 nm (see Fig. 2).” (Primary\nManuscript, page 6)\nReviewer #2, Point 8\nReviewer #2: The authors use the term “filamentation,” but it seems that the length of the plasma chan-\nnelisapproximatelytheRayleighlength,especiallyatlowenergies. Isitcorrecttousethisterminthiscontext?\nAuthors: This comment is absolutely correct, and it is in agreement with Reviewer #3, Points 1 and 8. We\ncompletely agree that a clear definition was missing in the originally submitted manuscript. This definition is\nnow given at the beginning of the introduction.\nConcerning the length of the plasma channel, let us first recall that nonlinear propagation imaging is time-\nintegrated measurements, and we are thus unable to measure the plasma directly. However, we measure how\nlight has been absorbed and refracted by the plasma. In the revised manuscript, we show in Fig. 4 (Primary\nManuscript) and Fig. S12 (Supplementary Information) that the characteristic absorption length L scales\nabs\nlogarithmically with the intensity (through E and τ). Therefore, the length of the interaction zone increases\nin\nwith this parameter, but not spectacularly as in studies in air for example. This can be explained by our\nrelativelytightfocusingconditions,whichimplythattheKerreffectnotonlycompeteswithplasmadefocusing,\nbut also with diffraction. Nevertheless, resembling features were observed for looser focusing conditions. For\ninstance, in Ref. [24], the numerical aperture NA is varied from 0.30 to 2.97, and in Ref. [37] filamentation was\ncharacterized for NA=0.26\nRevisions brought to the manuscript: The original statement in the introduction:\n“Ultrafast laser filamentation is an extremely nonlinear propagation regime where nonlinear refraction competes\nwith plasma effects [1, 2].”\nhas been revised to:\n“Ultrafast laser filamentation is an extremely nonlinear propagation regime characterized by a dynamic balance\nbetween Kerr-induced self-focusing and plasma-induced defocusing [1, 2].” (Primary Manuscript, page 1)\nReviewer #2, Conclusion\nReviewer #2: In conclusion, I believe that this article undoubtedly deserves publication in Nature Commu-\nnications, but only after major revision.\nAuthors: We thank Reviewer #2 once again for all their comments, which enabled us to drastically improve\nour manuscript, and we hope that the revisions we have made will be deemed satisfactory.\n13\nResponse to Reviewer #3\nReviewer #3, Foreword\nReviewer #3: The paper by Chambonneau et al. is aiming to provide a broad vision about physical origins\nof limitation conditions in semiconductors when using ultrafast laser pulses, and therefore explain the inability,\nin many cases, to induce modification inside such materials. It treats an important scientific and engineering\nproblem for the development of 3D laser writing technologies applicable to semiconductors.\nThe manuscript is generally pleasant for reading, contains original data and is well-structured. The authors\npresent interesting discussions based on observations of nonlinear propagation effects across variety of semi-\nconductors. However, from both technical and fundamental viewpoints (explained below), the paper does not\nsucceedtoconvincinglydemonstratetruly“universal”filamentationprocesswhichwouldberesponsibleforthe\nobservedlimitations. Althoughstrongnonlinearitiesareindeedexpectedinthestudiedregimes,authors’choice\nto associate all observed effects under the same term of “filamentation” is, in my opinion, often excessive and\nleads to confusion.\nAuthors: WesincerelythankReviewer#3forthethoroughreadingofourmanuscriptandfortheconstructive\nfeedback. We are grateful for the compliments noting that our “manuscript is generally pleasant for reading,\ncontains original data and is well-structured”. We agree that the previous title may have been misleading, and\nwe have revised it so that it now stresses the novelty aspects on the extracted effective nonlinear coefficients\n(see Reviewer #3, Point 2). The insightful comments raised by the reviewer have significantly contributed\nto enhancing the clarity, cohesiveness, and overall quality of the revised manuscript. We address each of the\nconcerns point by point below. For clarity, reviewer comments are shown in blue, our responses in green,\nand modifications in the revised manuscript in red.\nReviewer #3, Point 1\nReviewer#3: Evenifthisterminologymightbejustifieddependingonachosendefinition(whichisnotclearly\ngiven in manuscript, see after), the discussions do not systematically analyze the problem in view of competing\neffects trapping the light propagation as traditionally expected for filamentation phenomena (e.g. Kerr-based\nand plasma nonlinearities in the simplest case). Filamentation in usual sense implies intensity clamping as\nresult. But intensity clamping itself can occur in absence of filamentation, and can originate from different\nnonlinear mechanisms (plasma mirror effect as an example in a different context).\nAuthors: Weagreewith Reviewer#3 that, ingeneral, thereare alwaysphysicallimitsto energydepositionin\na material using ultrashort laser pulses. Reviewer #3 rightly recalls that for surfaces, the plasma mirror effect\nismainlyresponsiblefortheselimitations. InthebulkofSi,thiseffectcannotoccurbecausethelaser-produced\nplasma is underdense [R1]. The clamping effect is due to the evolution of the interaction in the material within\nthe prefocal region. This was observed in dielectrics under comparable conditions to those in our study [R2].\nRevisions brought to the manuscript: To clearly define filamentation, the original statement in the intro-\nduction:\n“Ultrafast laser filamentation is an extremely nonlinear propagation regime where nonlinear refraction competes\nwith plasma effects [1, 2].”\nhas been revised to:\n“Ultrafast laser filamentation is an extremely nonlinear propagation regime characterized by a dynamic balance\nbetween Kerr-induced self-focusing and plasma-induced defocusing [1, 2].” (Primary Manuscript, page 1)\nReviewer #3, Point 2\nReviewer #3: Furthermore, paper does not bring new technique — it reproduces previous measurements\ndoneinsiliconforothersemiconductors. Also, nonewphysicalconceptormechanismisdirectlydemonstrated.\nAlthough the dependence on parameters like the chirp of the pulses is shown and is indeed interesting, authors\nthemselvesadmittheydonotprovideaconcretesolutionoraclearnewpathtoovercometheissue. Theywrite\nthat “the whole data set allows [them] to envision future optimizations”, but such statement stays vague and\n14\ndoes not bring truly novel idea comparing to earlier reports.\nAuthors: We agree with Reviewer #3 that the technique we employ is not new, and we never claimed this.\nIn fact, in the originally submitted manuscript, we explained that nonlinear propagation imaging was initially\ndeveloped for water, adapted to dielectrics, and then applied to examine nonlinear propagation of light in Si.\nThe essential aspect of our study is the application of this simple and elegant technique to determine realistic\nnonlinear optical coefficients, which has never been done before. As noted by Reviewer #3, we did not restrict\nour study to Si, but performed experiments in different semiconductors.\nWe respectfully disagree with the following statement: “no new physical concept or mechanism is directly\ndemonstrated”. One of the key original findings of our study is that the nonlinear optical coefficients are orders\nofmagnitudehigherthanexpectedfromliteraturevaluesobtainedwiththez-scanmethod. Moreover,weestab-\nlish the temporal scaling laws for these nonlinear coefficients, and we demonstrate that longer pulse durations\nleadtohighernonlinearrefractionandabsorption—aphenomenonneverstudiedbeforeinsemiconductors. The\ntemporal dependence of the effective critical power is explained by delayed Kerr nonlinearity, previously un-\nchartedforsemiconductors. ThemodelwedevisewhichisbasedonamodifiedMarburgerformulaandallowsus\nto extract the multi-photon absorption coefficients of different orders, and simultaneously predict the nonlinear\nfocal shift, is novel. Finally, we propose three ways to increase the peak fluence F in semiconductors, which\np\nconsist of employing (1) longer pulses, (2) down-chirped pulses, and (3) higher multi-photon absorption orders.\nFinally, we agree with Reviewer #3 on the vague statement, and we have revised it accordingly. In the re-\nvised manuscript, applications beyond ultrafast laser writing are discussed, in line with Reviewer #1, Point 11\nRevisions brought to the manuscript: To put the emphasis on the extreme nonlinear coefficients de-\ntermined in our study, the originally submitted title: “The universality of filamentation-caused challenges of\nultrafast laser energy deposition in semiconductors”\nhas been revised to:\n“Extremeopticalnonlinearitiesunveiledbyultrafastlaserfilamentationinsemiconductors”(PrimaryManuscript,\npage 1)\nReviewer #3, Point 3\nReviewer #3: Especially, I find not well addressed the question on high-order multiphoton absorption in\ncomparison to a recent study on silicon using mid-IR femtosecond pulses (see ref.31). This previous study\nshouldbemorecriticallydiscussed,becauseitsuggestspossibilityforin-volumemodificationinsiliconandmay\ncontradicttheclaimofan“universal”limitationduetofilamentation. Thecurrentpapercanbeseenasdirectly\nsupporting this alternative route, or contrarily as not fully compatible with the ‘universal’ picture stressed in\nit, making the delivered message unclear on this aspect.\nAuthors: The study in Ref. [31] reports internal modifications in Si with 160-fs pulses at 4600 nm, corre-\nsponding to the 5-photon absorption regime. To the best of our knowledge, these are the first modifications\nproduced solely in the bulk of Si through a plane surface with such short pulses. This result does not con-\ntradict our conclusions; rather, it supports our finding in the revised manuscript [Fig. 5(d)–(f)] that higher\nmulti-photon absorption (MPA) orders can enhance energy deposition. In the mid-IR, the longer wavelength\nincreases the MPA order, which delays plasma defocusing and allows the fluence modification threshold to be\nreached even in the filamentation regime. Moreover, while mid-IR pulses lead to lower electron densities,\nthe deposited energy is higher because of the enhanced efficiency of inverse Bremsstrahlung absorption (IBA) [R3].\nWe thank Reviewer #3 for the suggestion to improve the discussion of our results compared to Ref. [31],\nand have incorporated this comparison into the revised manuscript.\nMore generally, there is no incompatibility between filamentation and bulk modifications. As exemplified in\nRef.[R2]forfusedsilica,modificationscanoccurinthefilamentationregimewheneverthelocalfluenceexceeds\nthe modification threshold. The specificity of semiconductors lies in their narrow band gap, which increases\nprefocal absorption at shorter wavelengths, making it far more difficult to reach the modification threshold\ncompared to wide-gap dielectrics.\nRevisions brought to the manuscript: The original statement:\n“However, an important result is that the peak fluence F is one order of magnitude higher in the 3PA regime\np\ncompared to the 2PA regime (maximum ratio of 13 and 21 for InP and GaAs, respectively), suggesting that\n15\nhigher multi-photon absorption orders are beneficial to exalt the interaction.”\nhas been revised to:\n“However, an important result is that the peak fluence F is more than one order of magnitude higher in the\np\n3PA regime compared to the 2PA regime, suggesting that higher multi-photon absorption orders are beneficial\nto tailor energy deposition. This is consistent with results obtained in Si with 160-fs pulses in the 5-photon ab-\nsorption regime [31], where permanent bulk modifications were achieved due to reduced prefocal absorption and\nthe ability to reach the modification threshold. Although mid-infrared pulses generate lower electron densities,\nthe deposited energy is increased through more efficient inverse Bremsstrahlung absorption [16]. These find-\nings indicate that high-order multi-photon absorption can, under specific conditions, relax filamentation-induced\nconstraints, whereasatshorterwavelengthsfilamentationstilllimitsenergydepositioninsemiconductors.” (Pri-\nmary Manuscript, page 6)\nReviewer #3, Point 4\nReviewer #3: Mycriticismsshouldnotbeunderstoodasaquestioningonthemeritandinterestofthework.\nThe presented data are valuable and of interest for a specialized community. However, in my view, the paper\ndoes not offer sufficiently significant advancement in terms of new mechanism, concept, or technique to make it\nappropriate for a broad audience journal as Nature Communications.\nAuthors: We emphasized in the originally submitted manuscript that the main application of our study was\nultrafast laser writing. Because of this focus, we understand that Reviewer #3 believes our manuscript is only\nvaluable for a “specialized community” and thus not appropriate for a broad audience journal such as Nature\nCommunications. In fact, the determination of effective nonlinear coefficients is applicable in numerous fields\nbeyond ultrafast laser writing. These fields include for instance backside processing, microelectronics security,\nTHzwavegeneration,high-harmonicgeneration,andsupercontinuumgeneration,whicharenowdetailedinthe\nrevised manuscript.\nRevisions brought to the manuscript: A new section titled “S4.6 Applications beyond laser direct writ-\ning” has been added in the Supplementary Information. In this section, the impact of a misestimation of Peff\ncr\nand/orβeff isdiscussedforbacksideprocessing, microelectronicssecurity, THzwavegeneration, high-harmonic\n2,3\ngeneration, and supercontinuum generation, with corresponding references.\nIn this new section, the additional experimental results we obtained on supercontinuum generation are in-\ncluded in a new figure (Fig. S17).\nThe original statement in the conclusion:\n“The whole set of results allows us to envision future optimizations to improve the degree of control of energy\ndeposition in the bulk of semiconductors, with various potential applications including in-chip ultrafast laser\nfunctionalization.”\nhas been revised to:\n“The determination of effective key nonlinear coefficients summarized in the Supplementary Information, Sec-\ntion S4.5, is essential for predicting optimal conditions for ultrafast laser writing in semiconductors. Beyond\nthis, these coefficients also provide critical insight into other fields, enabling advances for instance in backside\nprocessing, microelectronicssecurity, THzwavegeneration, high-harmonicgeneration, andsupercontinuumgen-\neration (see Supplementary Information, Section S4.6).” (Primary Manuscript, page 7)\nThe following statement has been added to the abstract:\n“Beyond laser writing, the accurate determination of effective nonlinear parameters reported here is essential\nfor predicting and optimizing semiconductor backside processing, microelectronics security, as well as high-\nharmonics, supercontinuum, and terahertz wave generation.” (Primary Manuscript, abstract, page 1)\nReviewer #3, Point 5\nReviewer #3: The general claim about broad potential impact of the findings is not clearly supported. The\nsuggested advances in understanding and controlling nonlinear propagation regimes in semiconductors remain\nvague. Authors refer to applications in sensing, quantum engineering, health care, etc., but without concrete\n16\nconnection or perspective to actual implementations.\nAuthors: Here, Reviewer #3 refers to the following original statement in the abstract:\n“Lightpropagationinsemiconductorsisthecornerstoneofemergingdisruptivetechnologiesholdingconsiderable\npotential to revolutionize telecommunications, sensors, quantum engineering, healthcare, and artificial intelli-\ngence. Sky-high optical nonlinearities make these materials ideal platforms for photonic integrated circuits.”\nAs a matter of fact, the applications in telecommunications, sensors, quantum engineering, healthcare, and\nartificial intelligence are all directly linked with semiconductor-based photonic integrated circuits (PICs):\n• Telecommunications: Silicon photonics and InP-based PICs are the backbone of high-speed optical\ntransceivers and data center interconnects.\n• Sensing: Silicon nitride (SiN) and InP PICs are used in LiDAR, spectroscopic sensors, and integrated\nbiosensing platforms.\n• Quantumengineering: GaAs,InP,andSi-basedPICssupportintegratedquantumlightsources,waveg-\nuides, and detectors.\n• Healthcare: Semiconductor PIC biosensors are used for point-of-care diagnostics, OCT imaging, and\nlab-on-chip devices.\n• Artificial intelligence: Silicon photonics is being developed for optical neural networks and photonic\ntensor processors.\nThe findings in our manuscript allow us to envision selective internal control of properties of semiconductors.\nRevisions brought to the manuscript: To clarify this aspect, the following statement in the abstract:\n“Lightpropagationinsemiconductorsisthecornerstoneofemergingdisruptivetechnologiesholdingconsiderable\npotential to revolutionize telecommunications, sensors, quantum engineering, healthcare, and artificial intelli-\ngence. Sky-high optical nonlinearities make these materials ideal platforms for photonic integrated circuits.”\nhas been revised to:\n“Light propagation in semiconductors is the cornerstone of emerging disruptive technologies, such as photonic\nintegrated circuits with the potential to revolutionize telecommunications, sensing, quantum engineering, health-\ncare,andartificialintelligence. Sky-highopticalnonlinearitiesmakethesematerialsidealplatformsforcompact,\nmultifunctional photonic devices.” (Primary Manuscript, abstract)\nThe original statement:\n“Two indirect (Si and Ge) and two direct (InP and GaAs) band-gap semiconductors have been selected (see the\ninset in Fig. 1). Because of their narrow band gaps (E <1.5 eV), these media exhibit high linear and nonlinear\ng\nrefraction (n >3, and n >10−18 m2/W, respectively).”\n0 2\nhas been revised to:\n“Two indirect (Si and Ge) and two direct (InP and GaAs) band-gap"}, "author_response": "Third response to Reviewers\nContents\nResponse to Reviewer #3 2\nReviewer #3, Foreword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2\nReviewer #3, Point 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2\nReviewer #3, Point 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3\nReviewer #3, Point 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4\nReviewer #3, Foreword\nReviewer #3: As a preliminary remark, I observe that the authors’ reply does not reassess the conceptual\nframing of the manuscript, nor does it provide additional experimental evidence that would reinforce the claim\nofwideapplicability. Theauthorspredominantlyexpandontheirdisagreementswiththecomments,yetdonot\nseem to have considered a deeper or more comprehensive re-evaluation of the underlying issues.\nAlthough the response concentrates on the few main points raised by previous report, none convincingly affects\nmy overall evaluation. Again, my position does not concern the intrinsic interest of the study or the merit of\nthe dataset, but reflects instead the high-level standards of originality and impact expected for this journal.\nFromthisperspective,thebreadthofimpactandthesignificanceoftheadvancespresentedremain,inmyview,\ninsufficiently substantiated.\nRegarding author response to R3 – Foreword I appreciate the authors’ efforts in providing a point-by-point\nreply. However, I note that the general concerns previously raised regarding conceptual novelty and/or un-\nderstanding remain insufficiently addressed. The response focuses on individual remarks on the selection of\ndeficiencies but does not directly engage with the broader issues which have been expressed.\nAuthors: We thank Reviewer #3 for the continued engagement and for clearly restating the remaining con-\ncerns regarding the conceptual framing and the scope of applicability. Since the manuscript has already been\nrevisedsubstantiallyinthepreviousrounds(includingclearerdefinitionsandterminology),wedonotintroduce\nfurthermanuscriptchangesinthisresubmission;instead,weaddressthereviewer’soverarchingpointasdirectly\nas possible by clarifying here the intended scope of the work as it is stated in the manuscript. Our contribution\nis the experimental determination of effective parameters and scaling trends in the investigated regime (mid-IR\nexcitation, ultrashort pulses, and tight focusing in bulk semiconductors), where Kerr self-focusing and plasma-\ndriven effects jointly govern the interaction. These effective parameters are defined at the propagation scale\n(i.e., effective coefficients that reproduce the observed self-focusing/onset and nonlinear-loss behavior over the\npropagation path in our regime), and are intended for quantitative description and modeling of strongly driven\nnonlinearpropagation(ratherthanasareplacementforsmall-signalmaterialconstants);withinthisframework,\nthey enable comparison across materials and operating conditions and provide practical inputs for predicting\nenergy-deposition trends in related nonlinear propagation scenarios. For clarity, reviewer comments are shown\nin blue, and our responses in green.\nReviewer #3, Point 1\nReviewer #3: Regarding author response to R3 – P1\nI maintain that the Marburger formula, even when refined to account for nonlinear losses, while effective in\ndescribingtheobservationsunderthespecificconditionsofthisstudy, cannotsubstitutetoarigorousnonlinear\npropagation model to support fully quantitative conclusions. Without a clear demonstration of the validity of\nthe reported nonlinear coefficients in more general scenarios, they cannot be considered as contradicting others\nfound in the literature.\nAlthough the authors point out that supercontinuum generation is shown in S17, these observations remain\nrelatively qualitative and do not directly validate the measured nonlinear coefficients. To substantiate claims\nof broad applicability, one would reasonably expect a range of different experiments (e.g., SC, THz, harmonic\ngeneration, material modification, ....) combined/compared with modeling that consistently supports the gen-\neral relevance of the reported coefficients.\nAuthors:\nOn our developed theoretical approach compared to a rigorous nonlinear propagation model.\nReviewer #3 maintains that the Marburger formula, even when refined to account for nonlinear losses, cannot\nsubstituteforafullyrigorousnonlinearpropagationmodel“tosupportfullyquantitativeconclusions”. Weagree\nthat our approach is not intended to replace a full spatio-temporal propagation simulation; rather, it provides\nanexperimentallyanchoredroutetodeterminepropagation-scaleeffectivecoefficientsinaregimewhereacom-\nprehensive, predictive model is currently not available across the full range of pulse durations addressed here.\nAsindicatedinourfirstresponse(Reviewer#3,Point11),thisisespeciallytrueforpulses>1pswherecarrier\nand lattice dynamics evolve during the pulse and strongly couple to propagation. A rigorous 3D model would\nrequire self-consistent treatment of propagation, ionization and transport, carrier–phonon energy exchange,\n2\ntemperature-dependent material properties (e.g., band-gap renormalization), and potentially hydrodynamics,\nmaking it intrinsically multi-scale and computationally demanding. The goal of the present work is therefore\nto establish the experimentally relevant nonlinear coefficients and their temporal scaling laws that constitute\nnecessary inputs and benchmarks for such future comprehensive models. In this context, the substantial de-\nviations we report from small-signal literature values are not a matter of semantics: they reflect strong-field,\npropagation-scaleeffectivebehaviorthatisdirectlyrelevanttoquantitativemodelingintheinvestigatedregime.\nOn the claim of broad applicability. Reviewer #3 suggests that substantiating broad applicability would\nrequire a wide set of additional experiments (e.g., SC, THz, HHG, material modification) combined with mod-\neling that consistently supports the reported coefficients. We respectfully note that our claim of applicability\nis tied to the propagation regime and parameter space investigated in this study, and is supported by (i) cross-\nmaterial consistency of the observed propagation trends, (ii) systematic variations of pulse duration, chirp, and\nwavelength, and (iii) the additional supercontinuum measurements included in the Supplementary Informa-\ntion. Rather than presenting a claim of exhaustive validation across all nonlinear-optics platforms, we provide\na quantitative reference set of effective coefficients and scaling laws that can be directly used as inputs and\nbenchmarks for modeling and for designing experiments in closely related nonlinear-propagation scenarios. As\nthese coefficients enter the description of a broad range of strong-field phenomena, we outline representative\nconnections to supercontinuum generation, THz generation, high-harmonic generation, and laser modification\nin Section 4.6 of the Supplementary Information, while keeping the present work focused on the experimentally\nvalidated propagation-scale determination of the parameters.\nReviewer #3, Point 2\nReviewer #3: Regarding author response to R3 – P2\nTheauthors’explanationregardingspectraldependenciesofferssomeclaritythatcouldimprovethepaper. Ac-\ncordingly,Iagreethatevenarelativelynarrowspectrummaypotentiallyleadtonoticeablevariations. However,\nthe question of potential pulse asymmetry remains insufficiently addressed. While the quadratic phase term is\nindeed an appropriate simplification to describe the effect introduced by the stretcher, my initial remark was\nrelated to the spectral phase prior to the stretcher. Assuming an initially flat spectral phase leads naturally to\na symmetric amplitude profile, but this assumption is not demonstrated. Figure R2-1 does not help to provide\nan unambiguous confirmation of symmetry in the conditions studied, and additional characterization would be\nrequired to fully support this point.\nAuthors:\nOn potential pulse asymmetry. We thank Reviewer #3 for recognizing that the explanation we gave in the\nsecond response “offers some clarity”, and for agreeing that the spectrum we used potentially leads to “no-\nticeable variations”. Reviewer #3 also clarified their original comment concerning potential pulse asymmetry,\nwhich could be related to a non-flat spectral phase prior to the stretcher.\nWeemphasizethatwedonotassumeaninitiallyflatspectralphase,nordoweclaimthatthetemporalintensity\nprofile is perfectly symmetric. We agree that a full retrieval of the spectral phase (e.g., by FROG/SPIDER)\nwould be required to uniquely reconstruct the exact field; however, our conclusions regarding chirp effects do\nnot rely on perfect temporal symmetry. In particular, the physical interpretation discussed in the manuscript\nis driven by the temporal ordering of spectral components (up- vs down-chirp) combined with wavelength-\ndependent ionization/heating mechanisms, rather than by fine details of the temporal envelope. Moreover, any\nresidual higher-order spectral phase present upstream would be common to both chirp settings, and therefore\ncannot reasonably account for the systematic difference observed between up- and down-chirped pulses.\nTo directly assess the sensitivity to possible asymmetry, we performed a conservative test within the same\nrate-equation framework used in our first response to Reviewer #3, Point 14. The temporal evolution of the\nelectrondensityiscalculatedinFig.R3-1forasymmetric3-psGaussianpulse(red)andforanasymmetric3-ps\npulse (blue) obtained by applying a quadratic spectral phase (yielding 3 ps) to the experimentally measured\nspectral amplitude and Fourier transforming (thereby incorporating small spectral-amplitude modulations into\nthe temporal profile). As expected, small modulations in the power spectral density yield small modulations\nof the temporal intensity profile (dashed blue vs dashed red). Nevertheless, these modulations translate into\na negligible change of the ionization dynamics: the resulting electron-density curves (solid lines) are nearly\nidentical, and the ratio between the maximum electron densities produced by the two pulses differs by less\nthan 1%. We therefore conclude that a plausible degree of pulse-shape asymmetry cannot significantly affect\nthe carrier generation dynamics in our conditions and does not alter the interpretation of the observed chirp\n3\ndependence.\nFig. R3-1 Temporalevolutionofthelaserintensity(dashedlines,leftverticalaxis),andtheelectrondensityn (solid\ne\nlines, right vertical axis) for 3-ps symmetric (red) and asymmetric (blue) pulses. The pulse energy is E =100 nJ, and\nin\nthe beam radius at 1/e2 is w =2.2 µm.\nReviewer #3: Regarding author response to R3 – P3\nThe response contains a certain inconsistency: the authors state that the relevance of the chosen parameters\nextends well beyond 3D writing, yet they justify their choice primarily by referring to studies on 3D writing\nor material modification. While I agree that using a directly measurable metric is advantageous, my concerns\nremain regarding its general relevance for a broad range of nonlinear optics experiments.\nThe issue I initially raised with a metric that can become arbitrarily large even in a regime insufficient to\ntrigger any nonlinear response has not been convincingly addressed. Referring to works involving CW-induced\nmodifications does not clarify how the proposed metric can be consistently used to identify an optimum in the\npresent context. Therefore, I must respectfully but firmly disagree with the authors’ assertion that my earlier\nremarkswere“incorrect”andImustmaintainmypositionregardingthelimitedrelevanceofthereportedratios.\nAuthors:\nOn the primary focus on laser direct writing. In our manuscript (Primary Manuscript, abstract and\nintroduction; Supplementary Information, Section 4.6), we mention multiple potential applications where our\nwork could constitute a critical step. These applications include not only laser direct writing, but also backside\nprocessing, microelectronics security, THz wave generation, high-harmonic generation, and supercontinuum\ngeneration. Reviewer #3 rightly recalls that the primary focus is on laser direct writing. The main reason\nis that this is one of the most extreme case where light intensity is sufficiently high to permanently modify\nsemiconductors internally. However, our results are not restricted to the “writing” end-point: the reported\npropagation-scale effective coefficients and their scaling laws are extracted from nonlinear propagation itself\n(self-focusing, nonlinearlosses, andplasma-driveneffects)overawiderangeofinputintensitiesandpulsedura-\ntions, and therefore directly inform other strong-field nonlinear-propagation scenarios in bulk semiconductors.\nIn this sense, direct writing serves as a motivating and technologically relevant benchmark, but not as the sole\njustification for the extracted parameters.\nOn the metric that can become arbitrarily large. The example previously given by Reviewer #3 was:\n“a continuous, low-intensity radiation in the transparency domain of a semiconductor: in such a situation F\np\ncan become arbitrarily large while the nonlinear absorption—and thus energy deposition—tends toward zero”.\nWe believe that this counterexample does not address the way F is defined and used in our study. In our\np\nwork, F is a per-pulse quantity defined from ultrashort-pulse experiments as the saturation (plateau) value of\np\nF when increasing E , i.e., F = max (F ) in the nonlinear regime where F ceases to increase due\nmax in p Ein max max\nto nonlinear propagation and losses. In a regime insufficient to trigger nonlinear response (linear propagation /\nCW-like low intensity), such a saturation plateau does not occur, and therefore the “peak saturation fluence”\nF is not the relevant quantity.\np\n• The notion of “fluence” as used here is a per-pulse energy-per-area quantity. For a continuous radiation\n(I = I ), one would rather discuss intensity and absorbed power, or define an energy-per-area over a\n0\nchosen exposure time. In that case, F =I ×t grows with t, but this growth is unrelated to the per-pulse\n0\nsaturation concept captured by F .\np\n• Even considering an extremely long laser pulse (for instance, in the millisecond range), the energy depo-\nsition does not tend to zero. The pulse parameters would be defined the same way as in our manuscript\n(pulse energy E , pulse duration τ and beam radius at 1/e2 w ). The maximum fluence would read\nin 0\nF = 2E /(πw2), which is a finite metric. Nonlinear absorption—and thus energy deposition—would\nmax in 0\ndepend on the maximum intensity I = F /τ, which does not tend to zero as the increase in τ will\nmax max\nalso result in an increase in F [see Primary Manuscript, Fig. 3(a)].\nmax\nFor these reasons, F remains a robust bulk-accessible per-pulse metric in the ultrafast nonlinear-propagation\np\nregimeinvestigatedhere,especiallywheninterpretedtogetherwiththeabsorbed-energyfractionandabsorption-\nlengthobservablesreportedinourmanuscript. Weagreethatourwordinginthepreviousresponsewasstronger\nthan necessary; our intended point was that the CW limiting case is not a meaningful test of a per-pulse satu-\nration metric.\n4"}]

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