# Nonlinear flying focus pulses for ultrafast 3D nonlinear microscopy

Tigran Mansuryan, Nicholas Bagley, Remy Boulesteix, Yago Arosa, Katarzyna Krupa, Benjamin Wetzel, Bruno P Chaves, Stefan Wabnitz, Vincent Couderc, Alejandro Aceves, et al.

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Tigran Mansuryan, Nicholas Bagley, Remy Boulesteix, Yago Arosa, Katarzyna Krupa, et al. Non-linear flying focus pulses for ultrafast 3D nonlinear microscopy. Optica, 2025, 12 (8), pp.1192. 10.1364/optica.561661. hal-05224753

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# Nonlinear flying focus pulses for ultrafast 3D nonlinear microscopy

TIGRAN MANSURYAN,<fnref n="1" /> ID NICHOLAS BAGLEY,<fnref n="2" /> REMY BOULESTEIX,<fnref n="3" /> YAGO AROSA,<fnref n="4" /> ID

KATARZYNA KRUPA,<fnref n="5" /> ID BENJAMIN WETZEL,<fnref n="1" /> ID BRUNO P. CHAVES,<fnref n="1" /> STEFAN WABNITZ,<fnref n="6" /> ID

VINCENT COUDERC,<fnref n="1" />,* ID ALEJANDRO ACEVES,<fnref n="2" /> ID AND ALESSANDRO TONELLO<fnref n="1" /> ID

Received 7 March 2025; revised 18 June 2025; accepted 18 June 2025; published 28 July 2025

Ultrafast 3D nonlinear multiphoton imaging holds great promise for the visualization of complex biological structures. However, its practical implementation remains constrained, primarily due to the limited longitudinal scanning speed achievable with tightly focused laser pulses within the sample. In this work, we propose a 3D nonlinear imaging concept that harnesses a spatiotemporal Kerr self-focusing process for depth-resolved imaging without mechanical scanning. We exploit Townes solitons to generate ultrafast nonlinear flying focus pulses that are well-suited for nonlinear fluorescence imaging. A key feature of this approach is the power-dependent longitudinal displacement of the self-focusing point within the sample. This displacement, combined with the power-limiting effects from conical-wave emissions at multiple wavelengths, enables precise control and calibration of the beam's focal position in three dimensions. Consequently, by implementing a temporal pulse encoding, either by splitting and delaying an initial pump pulse or by introducing an inter-pulse amplitude modulation, we can induce multiple self-focusing events at different time bins. This allows us to probe multiple axial planes in a single laser shot. As a result, a full 3D image can be reconstructed from a single 2D transverse scan, significantly accelerating imaging speed. Additionally, the stable nonlinear propagation of these filaments leads to collinear supercontinuum generation in the form of self-guided light beams. Overall, our method presents a pathway for ultrafast, high-resolution 3D nonlinear imaging. © 2025 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

https://doi.org/10.1364/OPTICA.561661

## 1. INTRODUCTION

Nonlinear imaging is a rapidly developing and well-established technique for optical biopsy and disease diagnosis. It relies on multiphoton absorption, achieved by using high peak-power ultrashort pulse lasers [1]. For instance, its advent enabled neuroscientists to image calcium distributions and study neurodegenerative diseases and cancer [2–4]. High spatial resolution can be achieved with single-mode laser sources on a wide range of samples, producing artifact-free images [5]. The pump pulse, typically in the femtosecond regime, is directed into the microscope and subsequently focused directly onto the sample at high power density. Images are then acquired by a two-dimensional spatial transverse scan. The key advantage of this technique is its high longitudinal resolution and inherent optical sectioning, thanks to the combination of tight beam focusing and nonlinear multiphoton absorption [2]. This technique has also been used to image samples at depth, in spite of their diffusivity. In previous works, the light scattering problem

has been mitigated by using ultrashort infrared pulses for three- and four-photon absorption [6–8] and employing custom-built laser sources with emission wavelengths up to 1.7 and 1.8 µm [9]. Consequently, nonlinear fluorescence imaging with sub-µm resolution at millimeter depths became possible [10–12], generating data matrices and 3D visualization of the sample.

The primary method for microscope imaging of a sample at different depths involves the longitudinal displacement of the sample's mechanical support, which can be a time-consuming task for high-resolution 3D imaging. However, there are also alternative techniques of spatiotemporal pulse-shaping for creating laser pulses with controlled structures and thus circumventing slow mechanical displacements. The flying focus technique is one such methods, which has been extensively developed over the years [13–16]. It is based on the chirp-based control of femtosecond pulses: by using highly dispersive focusing elements, one may create multiple foci at different longitudinal positions along the sample; each focus being associated with a different carrier wavelength. As

<footnote n="1">¹Université de Limoges, XLIM UMR CNRS 7252, 123 Avenue A. Thomas, 87060 Limoges, France</footnote>

<footnote n="2">²Southern Methodist University, 6425 Boaz Lane, Dallas, Texas 75205, USA</footnote>

<footnote n="3">³Université de Limoges, IRCER, UMR CNRS 7315, 12 Rue Atlantis, 87068 Limoges, France</footnote>

<footnote n="4">⁴NaFoMat, Departamento de Física Aplicada, Universidade de Santiago de Compostela, E-15782, Santiago de Compostela, Spain</footnote>

<footnote n="5">⁵Institute of Physical Chemistry, Polish Academy of Sciences, ul. Kasprzaka 44/52, 01-224 Warsaw, Poland</footnote>

<footnote n="6">⁶DIET, Sapienza University of Rome, Via Eudossiana 18, 00184 Rome, Italy</footnote>

<footnote>*vincent.couderc@xlim.fr</footnote>
a consequence, the pulse's chirp controls the arrival time of each wave packet. The use of highly chromatic diffractive elements, combined with group-velocity dispersion, is essential for this technique. This principle was initially demonstrated with static diffractive elements; the initial pump spectrum was divided into distinct parts, thus preventing the use of ultrashort pulses for imaging the sample [13,17,18]. Shortly thereafter, such a drawback was bypassed, and the chromatic flying focus was transformed into an ultrafast pulse-based flying focus at a constant velocity by exploiting geometric aberrations. An example is the use of an axiparabola [19–21], which extends the generation of “diffraction-free” beams to broadband laser pulses. However, such a practical solution is static, and it lacks the capability for ultrafast reconfiguration, thus limiting the scope of the system. In 2015, Sun et al. developed an adaptive system enabling the programing of the group velocity of different wave packets with an adaptable diffractive liquid crystal spatial light modulator and a deformable mirror [22,23]. In 2023, Ambat et al. demonstrated ultrashort flying focus pulses with a programable trajectory focal point by combining an axiparabola with a deformable mirror [16]. All these techniques use linear wave-packet deformations, such as diffraction and dispersion, to achieve spatiotemporal control over an optical pulse, ultimately creating multiple foci in both space and time.

However, nonlinear optics can offer new approaches for spatiotemporal beam reshaping and wave-packet management. When the intensity of a pulsed laser beam exceeds the self-focusing threshold, the refractive index of matter can be significantly altered. This effect may completely compensate for, or even overtake, diffraction over several diffraction lengths. Highly localized power densities may, in turn, induce a variety of nonlinear optical processes, with the consequent modification of the entire spatiotemporal shape of a laser beam [24–30]. In nonlinear optical materials with positive cubic nonlinear susceptibility, the Kerr effect leads to an intensity-dependent increase in the refractive index. In this case, self-focusing occurs in the absence of saturation processes, typically leading to a beam collapse accompanied by a catastrophic material breakdown and beam filamentation [29–32]. However, the presence of significant spectral broadening (i.e., supercontinuum generation) and the additional contribution of free charges and plasma may in turn produce a defocusing effect, thus counteracting wave collapse. This sequence of events may occur multiple times along the propagation path, leading to a spatiotemporal longitudinal modulation of the pump beam [26]. It is well-known that the propagation length that is required for beam self-focusing shrinks as the peak power grows larger. The position evolution of the self-focusing point versus power and beam waist can be well described analytically by the Marburger formula [33]. It is also interesting to note that the final transverse dimension of a self-trapped beam may be as small as a few micrometers. Spatial beam quality is also essential: high-quality input pump beams may lead to good spatial beam quality of the generated polychromatic filament, which is highly suitable for applications; whereas low-quality beams generally lead to more erratic multi-filamentation, thus limiting practical implementations.

Besides affecting the spatial domain, nonlinear propagation also significantly impacts the spectral domain, with additional nonlinear conversions driven by self-phase modulation and conical wave emission. In the latter process, the combined effects of third-harmonic generation, of spectral components generated in the infrared domain by Raman scattering, and of temporal solitons,

may in turn strongly modify the spatiotemporal beam shape. In some cases, conically emitted wave packets generated at different self-focusing points may interfere with each other, resulting in the generation of fringes of different colors [26,30,34–38]. To observe these effects, the input peak power should be large enough to induce beam collapse, leading to a broadening of both the temporal and the angular beam spectrum. This permits us to satisfy the (mostly) non-collinear phase-matching conditions associated with conical emission. Besides conical wave emission, the supercontinuum generated by controlled wave collapse has a dominant on-axis component that is spatially guided by a self-focused beam: such a component will be used in the present work. Moreover, when the nonlinear length is shorter than the dispersion length, the impact of chromatic dispersion on the temporal spreading of wave packets is mitigated. The cyclical evolution in beam focusing and defocusing that is observed in bulk crystals is directly dependent on the crystal's physical properties, such as its transparency, diffusion, dielectric conductivity, and chromatic dispersion, among others.

Here we demonstrate that nonlinear propagation in Nd:YAG crystals can be used for ultrafast longitudinal focus adjustment in nonlinear fluorescence imaging. This is obtained by properly managing the initial peak power of the pump pulse. Self-focusing, accompanied by nonlinear spatial beam reshaping and supercontinuum generation, enables the spatiotemporal emission of a perfectly stable and reproducible nearly-Gaussian beam with a wide choice of wavelengths. The tight dependence of the focal point position on the initial pump peak power can be exploited to achieve interleaved imaging at different depths in the sample by using multiple time-encoded pulses with different peak powers. In other words, unlike previous works, we implemented a fully nonlinear flying focus system without exploiting chromatic dispersion [14]. Additionally, it is important to highlight that the conical waves, which are generated along the nonlinear propagation of the beam, extract energy from the pump, acting similarly to a saturable absorber, and consequently limiting the on-axis peak power density. This effect plays a key role when exploiting exclusively the on-axis component of the supercontinuum: while the choice of the initial peak power can be used to pinpoint the filament at different positions along the crystal, saturation effects will reduce intensity fluctuations observed in the on-axis supercontinuum. Moreover, nonlinear beam trapping guarantees a nearly perfect spatial shape of the output beam. In this work, we illustrate this proof-of-principle by using a pair of pulses, but the proposed technique is fully scalable and can thus be applicable to longer sequences of pulses, thereby producing 3D images at different depths.

## 2. ON AXIS SUPERCONTINUUM: EXPERIMENTS AND NUMERICAL SIMULATIONS

The experimental setup is shown in Fig. 1. We used a 1030 nm laser source with variable pulse duration obtained by linear pulse stretching between 250 fs and 100 ps, an adjustable repetition rate from 0 to 2 MHz, and a peak power upto 100 MW. This pulsed laser beam was coupled into a Michelson interferometer mount, here used to produce two distinct pulses with adjustable power and relative temporal delay. These two pulses were focused into an Nd:YAG crystal by lens L1, leading to self-focused beams whose exact longitudinal focal point depends on their respective peak powers. The beam at the crystal output is collimated by lens L2,
<figure><img src="image_3.png" /><figcaption>**Fig. 1.** (a) Schematic of the experimental setup; λ/2: half-wave plate; λ/4: quarter-wave plate; M1, M2, and M3: Rmax mirrors at 1030 nm; converging lenses: L1 (focal length 160 mm) and L2 (focal length 25 mm); Fs: filter (λ: 708 nm and Δλ: 75 nm); Nd:YAG rod (8 cm long, 5 mm of diameter); SF: spatial filter; GS: galvanometric scanner; A: attenuator. Microscope objective: Olympus XLUMPLFLN 20×. (b) Schematic representation of how a sample can be scanned at two different depth levels using pump pulses of different peak power, respectively P1 (red) and P2 (blue). (c) Principle of focal-point shift versus pulse peak power (schematic).</figcaption></figure>

passes through a galvanometric scanner, and is focused on the sample by a microscope objective. The high energy required to produce the self-focusing process is limited by attenuators placed in front of the microscope entrance in order to adapt the peak power required for fluorescence emission on the biological sample and avoid any damage to it. The galvanometric scanner enables a transverse scan of the sample at the focal plane of the objective. Spectral filters were also placed in the beam path to select the wavelength used for exciting the sample and obtain multiphoton-based images. We also carried out a series of numerical simulations to understand the process of supercontinuum generation by wave collapse in Nd:YAG [39,40]. Our experimental and numerical analysis confirms that the usable wavelength range can span from 600 to 1700 nm. Some of our numerical and experimental results are shown in Figs. 2(a) and 2(b), respectively. We stress that all of these wavelengths are confined on the on-axis component of the supercontinuum.

Numerical simulations can predict the beam size at different wavelengths, which provides important information for studying the transverse and longitudinal resolution of the apparatus. Notably, although the width of the self-confined beam evolves versus wavelength, it remains nearly constant between 600 and 900 nm in the visible spectrum and between 1200 and 1700 nm in the infrared, with a diameter of a few micrometers [see Fig. 2(c)]. The largest beam diameters are expected to be near the pump wavelength (1030 nm). The spectral flux density decreases significantly as the conversion frequency shifts away from the pump [Fig. 2(d)]. This evolution is also confirmed by the experimental measurements, which are reported in Fig. 2(c). The inter-pulse time separation is controlled by the relative position of the two mirrors in the Michelson interferometer, and it can be easily modified from a few femtoseconds to up to 100 picoseconds in our experiment, and in principle, up to several nanoseconds. This delay should be

large enough to temporally separate the action of each pulse and allow for separating image acquisition. We also exploited the linear polarization of the laser beam and a polarization-based interferometer with λ/4 waveplates [see Fig. 1(a)] to independently adjust each pulse peak power. We have experimentally studied the generation of the supercontinuum as a function of the pulse duration and estimated that the best pulse duration for multiphoton imaging is 250 fs. Such a choice is associated with the specific nonlinear imaging process: in multiplex-CARS microspectroscopy, longer pulse durations offer advantages, since the pump and the wideband Stokes waves should be synchronized [39,40].

### 3. FINE TUNING OF THE BEAM FILAMENT

In order to evaluate the difference in the position of the self-focusing point in the Nd:YAG crystal and, correspondingly, at the microscope output, we first numerically calculated the self-focusing length $Z_c$ versus the input beam peak power $P_{IN}$, as predicted by Marburger's formula for a collimated Gaussian beam of Rayleigh length $Z_R$:

$$Z_c = \frac{0.367 Z_R}{\sqrt{[(P_{IN}/P_{CR})^{1/2} - 0.852]^2 - 0.0219}}, \quad (1)$$

where $P_{CR}$ is the self-focusing critical power [Fig. 3(a), blue curve]. From full numerical simulations of the envelope equations, we also calculated the position where supercontinuum generation and generation of significant visible fluence occur: these are the red and green curves in Fig. 3(a) [39,40]. Numerical simulations involve solving the forward envelope equation (FEE) coupled to a Drude model for ionization and plasma generation; details can be found in Ref. [40]. The variation of $Z_c$ can extend over 2 cm for an input peak power of up to 20 MW. Due to the use of lens L1 and to the
<figure><img src="image_4.png" /><figcaption>**Fig. 2.** Self-confined supercontinuum generation obtained in Nd:YAG crystal: (a) Numerical simulations (peak power 3.0 MW). (b) Experimental results of spectrum versus pulse duration (energy per pulse: 7 µJ). The dotted-line rectangle shows the wavelength (708 nm) used for nonlinear fluorescence imaging. (c) Simulated beam radius at filament onset location (3.2 MW, 300 fs, 50 µm input Gaussian beam diameter). (d) Simulated peak spectral flux density (log scale) at filament onset location (3.2 MW, 300 fs, 50 µm input Gaussian beam diameter).</figcaption></figure>

quasi-simultaneity of plasma effects and self-focusing, which have opposite spatial effects [26], a shift of the origin of self-focusing further into the Nd:YAG crystal is observed when compared with predictions of Marburger's formula. The evolution of the numerically obtained self-focusing position follows the same trend as that of the experimental data. However, a wider focusing range is observed in the experimental data with the focusing evolution over 6 cm. We also note that the focusing position obtained with Marburger's equation is shifted compared to the supercontinuum-based simulation data and the visible fluence-based data. This shift, however, remains limited and can be explained by the nonlinear constants used in the nonlinear simulations. Marburger's formula is also a monochromatic approach with no plasma action. Thus, supercontinuum generation and plasma in simulation and experiment serve to defocus peak energy and thereby delay filament onset in reality. For completeness, the focus position was also numerically calculated for input beam diameters varying between 10 and 100 µm [Fig. 3(d)]. Additionally, different materials may lead to different beam dynamics: e.g., when Nd:YAG crystals are used instead of Nd:YAG ceramics. For example, the generation of Townes solitons can be controlled by modifying the material conductivity, leading to soliton concatenation and increasing the length of the nonlinear interaction without causing material damage or inducing instability [39–42].

In our experiments, the focus position versus input peak power can spread over more than 3 cm in a Nd:YAG ceramic and over 6 cm in a pure Nd:YAG crystal [Fig. 3(b)]. This difference is explained by an increase in Rayleigh scattering in ceramics, as well as by the variation of dielectric conductivity as a function of the sintering process [43–45]. While many other consecutive self-focusing points can follow the first one [Fig. 3(c)], such nonlinear

dynamics are not exploitable for a flying focus application for imaging, which only involves the final position of the last focusing point. In these conditions, the longitudinal distance over which we can obtain a single focusing point is therefore shortened. This distance is schematically represented in Fig. 3(b) by the red and green shaded squares: it is 1.4 and 3.6 cm for Nd:YAG ceramics and for Nd:YAG crystals, respectively. From the near-field image in the visible spectrum of a single filament obtained at 0.8 MW [Fig. 3(c1)], we may note the strong separation among the components, which are obtained by either self-phase modulation (center of the image), or by conical wave emission (ring). In the far-field image [Fig. 3(c2)], the two contributions combine, owing to the diffracting free-space propagation in air. Interference fringes are also observed on far-field images when several focusing points are obtained in the Nd:YAG crystal [Fig. 3(c3)], demonstrating the coherent nature of the frequency conversions obtained by self-phase modulation and by conical wave emission.

## 4. IMPLEMENTATION OF THE NONLINEAR FLYING FOCUS

When passing through the microscope, the longitudinal variation of the flying focus is proportionally rescaled via the imaging system: in our case (Nd:YAG crystal), it can reach a 40 µm focusing difference on the sample when using the Olympus LMPLFLN 20× microscope objective. In the first experiment, we tested our system with an Nd:YAG crystal to focus on a biological sample in a nonlinear fluorescence configuration (two-photon absorption—TPA). The sample was a mouse kidney labeled with DAPI to visualize cell nuclei and with Alexa 488 for tubules. The excitation wavelengths were selected using a bandpass filter around 708 nm
<figure><img src="image_5.png" /><figcaption>**Fig. 3.** (a) Calculated self-focusing position in Nd:YAG crystal with Marburger's formula and by using simulations of the supercontinuum and its visible fluence in order to detect filament onset (pulse duration: 300 fs, input beam diameter: 50 µm, wavelengths: 525-750 nm). (b) Experimental data of filament onset location in Nd:YAG crystal and Nd:YAG ceramic as a function of input peak power (300 fs pulse duration). (c) Images of focusing-defocusing cycles for the three values of the peak input power marked on the (b) curve; c1-c2: near and far-field images (respectively) of self-focusing, in visible domain, for the case 1; far-field image in visible domain for case 3. (d) Simulation of the filament onset location in Nd:YAG crystal as a function of input beam diameter (3.2 MW, 300 fs pulse duration).</figcaption></figure>

with a spectral width of 75 nm (Semrock FF01-708/75-25). The fluorescence emitted by the sample appears blue and green and is selected with 447/60 and 509/22 nm bandpass filters. The images are made by a scanning process at a laser repetition rate

of 300 kHz (5 µs dwell time). For a peak power of 0.8 MW, the onset of self-focusing is localized 3 mm before the end face of the Nd:YAG. Under these conditions, the image of the sample on the microscope is obtained with a focal plane located beyond the

<figure><img src="image_6.png" /><figcaption>**Fig. 4.** Left: crystal pictures with self-focusing points. Nonlinear, two-photon absorption (2PA) fluorescence images of a mouse kidney labeled with DAPI and Alexa 488. Images of nuclei and tubules illuminated at 708 nm (Δλ = 75 nm) selected from the supercontinuum obtained in the Nd:YAG with Semrock FF01-708/75-25 filter. Pump peak power of 0.8 MW allowing for Townes soliton generation accompanied by large spectral broadening. (a) Off-focusing (i.e., beyond the sample) yielded blurry image reconstruction obtained for the soliton position reached for pump input power of 0.8 MW. (b) Image is in focus for a pump peak power of 1 MW.</figcaption></figure>
<figure><img src="image_7.png" /><figcaption>**Fig. 5.** Nonlinear fluorescence images of chamomile pollen. The image is obtained by measuring the autofluorescence of the sample around 509 nm. (a) Image made with the first pulse creating a Townes soliton at a position X1 in the Nd:YAG crystal and giving an image of the sample with a first focal plane (input peak power 1 MW). (b) Image made with the second pulse, delayed by 50 ps relative to the first one with a lower input power (0.8 MW). This creates a second image at X2 (X2 > X1) at a different depth in the sample due to the creation of the Townes soliton further into the crystal. (c) Image obtained by sending the two delayed pulses into the crystal, allowing for a focus at two different levels in the sample. The maximum depth difference is 40 µm on the sample. By appropriate time delay one can separate the contributions of the two pulses.</figcaption></figure>

sample, giving a blurred appearance to the surface [see Fig. 4(a)]. By further increasing the peak power (1 MW), the self-focusing point in the Nd:YAG is shifted farther away from the end face of the crystal, thus modifying the focus and the imaging process. In this case, the image could be perfectly defined with a simple modification of the initial peak power. It is also important to note that, as the peak power grows larger, the energy released by the conical waves appears after a shorter path in the crystal and with a higher intensity, which decreases the energy propagating on the central axis as a self-trapped filament. Spatial filtering of the trapped beam allows the energy of the conical wave to be rejected, thus limiting the energy transmitted to the sample. The combination of conical wave emission and spatial filtering acts as a saturable transparency leading to power-limiting action when the power of the pump beam increases. Consequently, the image obtained with the microscope is not saturated, even in the presence of a higher initial peak power, and it can easily be managed with the gain of the photomultiplier. Under our conditions and for peak powers ranging from 1 to 10 MW, the maximum flying focus pulse separation is 40 µm on the sample. We also numerically estimated that the spatial transverse resolution is close to 1 µm, and it varies by less than 10% around the focal point for wavelengths ranging from 600 to 900 nm.

## 5. DUAL DEPTH IMAGE ACQUISITION

In a second series of experiments, we used, first alternatingly and then simultaneously, two beams of different intensities to make

images of the sample at two different depth levels and with an adjustable arrival time between the pulses. The beams at the output of the Michelson interferometer are focused in the Nd:YAG crystal, thus creating focal points at two different locations in the longitudinal dimension. We used the nonlinear autofluorescence emission of chamomile pollen to test this configuration. Again, we used a 708/75 nm bandpass filter after Michelson for excitation and a 509/22 nm bandpass filter to collect the autofluorescence emission. For a peak power of 1 MW, a sharp image of the upper part of the sample is obtained [Fig. 5(a)]. Thus, only a few pollen grains are clearly visible with a good spatial resolution of 1 µm. Other pollen grains appear blurred at the back of the microscope's focal plane. In a second step and at a slightly lower power of 0.8 MW, the focusing is obtained deeper in the crystal, shifting in turn the microscope focusing position and leading to a net image of the lower part of the sample. Conversely, the upper part of the sample appears blurred [Fig. 5(b)]. In order to combine the contributions of the two pulses, we launched both of them with a controlled delay, adjusted by moving the position of one of the mirrors in the Michelson interferometer. In the presence of a suitable temporal delay, data corresponding to the two different depths can be separated by post-processing to separately obtain the image of the upper and lower parts of the sample. In the absence of signal post-processing, we obtained a superposition of the two initial images, obtained at different levels in the sample [Fig. 5(c)].

## 6. CONCLUSION

We have demonstrated that beam self-focusing and spectral broadening processes in an Nd:YAG crystal or ceramics can be advantageously used to implement flying-focus pulses controlled by the input optical pulse power, which are suitable for nonlinear fluorescence imaging. The technique based on beam collapse allows us to obtain, either sequentially or simultaneously, images at different depths in a biological sample. The nonlinear Kerr dynamics strongly preserve the spatial shape of the beam when we use, at the input, a Gaussian beam. Thus, the stability and repeatability of this process are good enough to obtain well-defined images in a thick sample. Supercontinuum generation combined with autofocusing can be used for multispectral imaging, from visible to infrared. With the latter, we could reduce scattering in biological tissues and enable deeper penetration into tissues and obtain sharper images. The process is accompanied by non-collinear conical wave emission; although conical waves are not directly used for imaging, they introduce a saturation in the overall nonlinear response, which permits regularizing the otherwise catastrophic beam collapse by releasing the excess of optical power density. In turn, this permits us to maintain reasonable illumination level for imaging via fluorescence measurements with photomultipliers. Beyond the specific technical solution implemented in this work, simple amplitude modulation of a train of pulses from a mode-locked femtosecond laser source could also be directly used for ultrafast 3D imaging. Indeed, a single laser source with a repetition rate ranging from a few MHz to 200 MHz, combined with an ultrafast amplitude modulator, can be used to control in real time the peak power of each pulse. Thus, for each pulse amplitude, an image can be obtained at a particular depth in the sample. This avoids the use of an interferometer to duplicate the pulses and to control their energies. The multiplication of the process of image acquisitions at different levels is relatively simple and instantaneous with a single transverse scanning process. Only the dwell
time for a pixel needs to contain all the pulses needed to make an image at each different depth level. The amplitude modulator can also be combined with wavelength selection and nonlinear imaging modalities such as second harmonic-generation or coherent anti-Stokes Raman scattering (CARS).

**Funding.** Labex Sigma malim (10-LABX-0074-01, EUR TACTIC 18-EURE-0017, Equipex ADD4P ANR-21-ESRE-0012, ANR 3Dmoc ANR-23-CE08-0021-02, ANR-24CE46-7295-01-ACT); European Research Council (GA 950618—STREAMLINE); HORIZON EUROPE European Innovation Council (GA 101185664—MULTISCOPE); Conseil Regional Nouvelle-Aquitaine (SPINAL); National Science Foundation Graduate Research Fellowship (DGE-2034834); Narodowa Agencja Wymiany Akademickiej (BPN/BFR/2024/00005); Narodowe Centrum Nauki (OPUS 2023/49/B/ST7/01482); Campus France PHC Polonium 2025 (52964RF); Postdoctoral fellowship from the Xunta de Galicia (ED481D-2024-001).

**Acknowledgment.** B. Wetzel acknowledges a grant from European Research Council. N. Bagley acknowledges the National Science Foundation Graduate Research Fellowship under Grant No. DGE-2034834. K. Krupa acknowledges the Narodowa Agencja Wymiany Akademickiej (BPN/BFR/2024/00005) and Narodowe Centrum Nauki (OPUS 2023/49/B/ST7/01482); A. Tonello acknowledges Campus France PHC Polonium 2025 (52964RF). Y. Arosa acknowledges a postdoctoral fellowship (ED481D-2024-001) from the Xunta de Galicia. S. Wabnitz acknowledges the Italian Ministry of Health under the Ricerca Finalizzata 2021 program, Italy (Project Code RF-2021-12373094).

**Disclosures.** The authors declare no conflicts of interest.

**Data availability.** Data underlying the results presented in this paper may be obtained from the authors upon reasonable request.

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