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# Visualization and characterization of arbitrarily shaped pulsed-laser spots

Yann Le-Guen, Maxime Verges, Michel Hehn, Stephane Mangin, Julius Hohlfeld

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Yann Le-Guen, Maxime Verges, Michel Hehn, Stephane Mangin, Julius Hohlfeld. Visualization and characterization of arbitrarily shaped pulsed-laser spots. Physical Review Applied, 2025, 23, pp.044018. 10.1103/physrevapplied.23.044018. hal-05252332

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Distributed under a Creative Commons Attribution 4.0 International License
Visualization and characterization of arbitrarily shaped pulsed-laser spots

Yann Le-Guen,*, Maxime Verges,Michel Hehn,Stephane Mangin, and Julius Hohlfeld

Institut Jean Lamour, CNRS UMR 7198, Université de Lorraine, 54506 Vandoeuvre-lès-Nancy, France

<figure><img src="image_3.png" /></figure>

(Received 8 October 2024; revised 28 February 2025; accepted 7 March 2025; published 7 April 2025)

A novel way to represent and analyze the contours of laser-pulse-induced domains, which reflect the position at which the laser fluence drops below a certain threshold and are measured as a function of the laser-pulse energy, is introduced. It enables the visualization and quantitative characterization of arbitrarily shaped *in situ* laser spots with high accuracy.

DOI: 10.1103/PhysRevApplied.23.044018

# I. INTRODUCTION

Optical quantum information, communication and data storage [1–4], optical metrology and microscopy [5–7], optical traps, tweezers and spanners [8–10], optical material processing and patterning [11–13], all these fields of optical science and technology have been advanced or enabled by the application of structured laser beams. Those beams include self-healing Airy and Bessel (Bessel-Gauss) beams [14–16], annular offset Gauss and vortex Laguerre-Gauss beams [17–19], as well as beams with arbitrary intensity, phase, and polarization profiles generated by computer-controlled spatial light modulators [20,21]. Past and present research has mainly focused on the generation and application of those beams. However, the task to characterize their fluence profiles under conditions where small working distances or comparatively large pixel sizes render characterization by direct illumination of camera chips impossible [20] has attracted only sparse attention. Consequently, most attempts to characterize laser spots *in situ*, i.e., as they are on the surfaces of investigated samples, are still based on the experimental technique and data analysis scheme introduced by Liu [22] more than 40 years ago.

In his seminal work, Liu measured the contours of amorphous domains that resulted from pulsed laser excitation of single-crystalline silicon as a function of the laser-pulse energy. Recognizing that those contours reflect the position at which the laser fluence drops below the crystalline-to-amorphous phase transition threshold, he fitted the inverse of a presumed Gaussian fluence distribution to the measured relation between domain size and laser-pulse energy, and used the width of the Gaussian returned by the fit to determine the laser fluence. Since then, different laser-fluence threshold processes, such as laser ablation [11,19,23] and all-optical helicity-independent switching

of magnetization [24,25], have been added to the experimental toolbox and used to characterize a larger variety of spot profiles, including Laguerre-Gauss and Bessel beams [11,19].

In contrast, all analyses of the relation between domain size and pulse energy follow the recipe of Liu [22] unaltered, and are therefore subject to two severe shortcomings. First and foremost, they do not provide a direct image of the *in situ* laser spot. Second, they require knowledge of its inverse fluence distribution function, which might not have an analytical form as is the case for Bessel beams [11].

In this paper, we demonstrate that a simple but decisive modification of the way in which measured domain contours are represented and analyzed unlocks the full potential of the experimental technique of Liu [22]. It directly yields an image of the *in situ* fluence distribution of any laser spot, irrespective of its shape. It also allows determination of spot size and laser fluence without the need to involve inverse fluence distribution functions. Without loss of generality, we demonstrate the power of our novel approach with the examples of Gaussian, Laguerre-Gauss, and Bessel-Gauss spots. Details about these spots are given in the Appendix.

Before entering the detailed discussion of our method and results, it is worth mentioning that the thresholds for laser-pulse-induced amorphous-to-crystalline phase transitions [26], for laser ablation [26,27], and for all-optical helicity-independent switching of magnetization [24,25] all depend on the temporal laser-pulse profile in nontrivial ways. Thus, they are ill suited for simultaneous characterization of spatial and temporal laser-pulse profiles, and their applicability for proper characterization of spatial fluence profiles is restricted to laser pulses with spatially invariant temporal profiles.

# II. THEORETICAL BACKGROUND

In general, the contour of laser-pulse-induced domains that cover the area where the laser fluence exceeds a certain

<footnote>*Contact author: yann.le-guen@univ-lorraine.fr</footnote>

<footnote>†Contact author: jkh16030@gmail.com, julius.hohlfeld@univ-lorraine.fr</footnote>
threshold, $F_{th}$, is governed by the maximum laser-pulse fluence, $F_0$, and the fluence distribution function of the laser spot, $f(\mathbf{r})$, through

$$F_{th} = F_0 f(\mathbf{r}_{th}), \qquad (1)$$

where $\mathbf{r}_{th}$ represents all the vectors $\mathbf{r}$ for which Eq. (1) is fulfilled. By definition, $f(\mathbf{r})$ is normalized to unity and determines the area of the laser spot:

$$A_s = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} f(\mathbf{r}) \, dx \, dy = 2\pi \int_{0}^{\infty} r f_r(r) \, dr. \quad (2)$$

This, in turn, relates the maximum and threshold fluences to the corresponding pulse and threshold energies, denoted as $E_p$ and $E_{th}$, respectively:

$$F_0 = \frac{E_p}{A_s} \quad \text{and} \quad F_{th} = \frac{E_{th}}{A_s}. \qquad (3)$$

In Eq. (2), the radial distribution

$$f_r(r) = \int_{0}^{2\pi} f(r, \phi) \, d\phi / (2\pi), \qquad (4)$$

with $r = |\mathbf{r}|$, embodies the azimuthal average of $f(\mathbf{r})$ and fully characterizes circularly symmetric laser spots.

Conventional analyses of the measured relations between domain size and laser-pulse energy solve Eq. (1) for $|\mathbf{r}_{th}(E_p)|$ [11,19,22–25]. Here, we follow a different approach. We use Eqs. (3) to rewrite Eq. (1) as

$$\frac{1}{E_p} = \frac{1}{E_{th}} f(\mathbf{r}_{th}), \qquad (5)$$

which makes it self-evident that a plot of $1/E_p$ vs $\mathbf{r}_{th}$ directly yields an image of the in situ fluence distribution function of the laser spot scaled by $1/E_{th}$. The spatial extent of this image is restricted to regions where $f(\mathbf{r}) \ge F_{th}/F_0$. Since experimentally accessible maximum fluences are finite, the extent of those regions and the minimum values of $F_{th}/F_0$ are limited to finite values as well. Thus, the requirement that $A_s$ has to be determined by integrating the fluence distribution function $f(\mathbf{r})$ over an infinite plane, or at least the whole area where $f(\mathbf{r})$ is nonzero [cf. Eq. (2)], can never be met for in situ fluence distributions derived from measured domain contours.

To resolve this conundrum, a theoretical distribution function, which presumably best describes the actual fluence distribution, is usually fitted to the measured data and the limitless integral of the fitted function used as a measure of $A_s$. The validity of the spot size thus obtained depends on the correct choice of the assumed function, which involves educated guesses about the behavior of this function outside the imaged region. Nevertheless, the

task to identify the optimal theoretical distribution function can be significantly simplified by a priori knowledge of the spot symmetry. In particular, it is much easier to identify the best-suited one-dimensional radial distribution $f_r(r)$ that fully characterizes a circularly symmetric spot than the optimum two-dimensional distributions $f(r, \phi)$ needed to describe noncircularly symmetric spots.

### III. METHODOLOGY

#### A. Experimental setup

A scheme of the setup used to measure the presented data is shown in Fig. 1. The laser beam comprises laser pulses of 320-fs duration, 1030-nm wavelength, and variable pulse energy that are emitted by a regeneratively amplified femtosecond ytterbium-doped potassium gadolinium tungstate (Yb:KGW) laser (Light Conversion, PHAROS). The energy of individual laser pulses is calculated by dividing the laser power, measured via a thermopile sensor and optical power meter (Newport, 919P-003-10 and 843-R), by the laser-pulse repetition rate. The laser spot size on the sample is controlled by the distance between the sample and a planoconvex focusing lens. Its focal length was 1 m when generating the Gaussian and the Bessel-Gauss spots, but 0.5 m for the large Laguerre-Gauss spot. The Laguerre-Gauss or Bessel-Gauss spot shapes were obtained by inserting a spiral vortex wave plate (Vortex Photonics, V-1030-20-1) or two identical axicons (Thorlabs, AX2520), mounted with their individual tips pointing against and along the beam direction and a distance of ≈ 5 mm between their flat bases, into the beam path before the focusing lens. The distances

<figure><img src="image_4.png" /><figcaption>FIG. 1. Scheme of the experimental setup and coordinate system. The two insets show examples of images of a typical in situ domain (green frame) and of the ex situ intensity distribution of the beam on the camera chip (red frame). Images are not to scale.</figcaption></figure>
of the vortex plate and of the nearest axicon to the focusing lens were about 55 cm and 15 cm, respectively.

The illuminated samples contain a 10-nm-thick layer of amorphous GdFeCo, a perpendicularly magnetized material that exhibits all-optical helicity-independent switching (AOS) of its magnetization, which is a threshold fluence effect [24]. Contours of laser-pulse-induced domains of reversed magnetization are derived from calibrated images obtained via linear magneto-optical Kerr-effect microscopy using a green-light-emitting diode (Thorlabs, M530L4) for continuous-wave illumination of the sample at an angle of incidence of $\alpha \approx 45^{\circ}$, a best-form lens (Thorlabs, LBF254-050) of 50-mm focal length to image the sample onto a monochrome scientific CMOS camera with a 5.04-µm pixel size (Thorlabs, CS2100M-USB), and a personal computer to store the acquired images with 16-bit resolution.

The use of a large angle of observation, here $\alpha_{obs} = \alpha$, was motivated solely by the need of other experiments to detect in-plane magnetizations via the longitudinal magneto-optical Kerr effect, and caused the sample to appear shortened by $\cos \alpha_{obs} \approx 1/\sqrt{2}$ along x in the images (see Fig. 1). Proper calibration factors, determined by measuring the number of pixels by which features in the image shifted in response to displacements of the sample along x and y, were used to correct for these linear image distortions in all figures shown herein. Using a large $\alpha_{obs}$ also forced us to sacrifice image resolution for depth of field. While this was acceptable for the investigation of the relatively large domains presented here, smaller angles of observation will have to be used for studies of smaller domains. When imaging the large domains written by the Laguerre-Gauss beam, we reduced the magnification by doubling the focal length of the imaging lens to 100 mm.

Before its excitation by any laser pulse, the sample is exposed to an external magnetic saturation field, generated by means of an electromagnet (GMW Associates, 5201) that is energized by a computer-controlled current supply (KEPCO, BOP 5-20DL), to ensure that its magnetic state is uniform and of a predefined direction.

The setup also encompasses two flipper mirrors, which can be used to steer the laser beam directly onto the camera. They are positioned such that they lead to an ex situ laser spot on the chip of the camera that is identical to the in situ laser spot on the sample. We achieved this feat in the following way. We first minimized the laser spot size on the sample by putting the focusing lens into a position where the laser-pulse energy required to damage the sample was minimum. Then we activated the flipper mirrors and minimized the size of the laser spot on the camera chip by varying the camera position. When steering the laser beam directly into the camera, several neutral density filters with a total optical density of 8 were inserted into its path to avoid overloading the camera.

## B. Method to image actual fluence distributions

Images of in situ fluence distribution functions were derived from measured domain images in the following way. First, a background image of the homogeneously magnetized state of the sample was taken before its excitation by a laser pulse and subtracted from the actual domain image. The magnitude of each resulting difference image was then normalized to unity and the contours at the midpoint of 0.5, $C_{0.5}$, were taken as a measure of $f(\mathbf{r}_{th} + \Delta\mathbf{r}_{beam})$ for the actual energy of the laser pulse, where $\Delta\mathbf{r}_{beam}$ denotes variations of the spot position due to beam-pointing instabilities of our laser. Note that $C_{0.5}$ is deliberately chosen since it is the only contour that mimics the actual domain without being affected by the finite point spread function of our imaging system. The contours were shifted by $-\Delta\mathbf{r}_{beam}$ to remove the impact of varying spot positions. Because of the lack of proper equipment to characterize $\Delta\mathbf{r}_{beam}$ independently, the values of $-\Delta\mathbf{r}_{beam}$ were selected such that the overlap between the areas enclosed by in situ and ex situ contours was maximized. Lastly, images of actual in situ fluence distribution functions were obtained by assigning the ratio of threshold and laser-pulse energy, $E_{th}/E_p$, as local amplitude to the position-corrected two-dimensional contours.

## C. Method to image radial fluence distributions

Radial fluence distribution functions $f_r(r)$, scaled by $1/E_{th}$, were derived from measured domain contours by plotting $1/E_p$ versus equivalent radii, $r_{eq,i}(E_p)$. The latter correspond to the radii of circles that enclose the same area as the ith contour of the domains observed after excitation by a laser pulse of energy $E_p$, denoted by $A_{C_{0.5,i}}(E_p)$,

$$r_{eq,i}(E_p) = \pm \sqrt{A_{C_{0.5,i}}(E_p)/\pi} \quad (6)$$

Note that this experimental approach is justified only for (nominally) circularly symmetric spots. Moreover, only those domain contours that close about the spot center should be used to determine $r_{eq,i}(E_p)$ in order to ensure that the data reflect the azimuthal average of the actual fluence distribution as postulated.

The definition of equivalent radii $r_{eq,i}(E_p)$ via Eq. (6) implies that their magnitudes are governed solely by the area of written domains, irrespective of their position on the sample. This allowed us to reduce the impact of laser noise on their values by simply averaging the individual radii derived from multiple domain images taken for each laser-pulse energy. In this study, all reported experimental radii $r_{eq,i}(E_p)$ represent average values derived from 10 domain images (five for each of the two initial magnetization orientations, up and down).
<figure><img src="image_5.png" /><figcaption>FIG. 2. (a) The *ex situ* and (b) *in situ* fluence distribution functions of our Gaussian spot, measured via direct camera illumination and reconstructed from pulse-energy-dependent domain images, respectively.</figcaption></figure>

# IV. RESULTS AND DISCUSSION

## A. Fluence distribution of our Gaussian spot

In Fig. 2(a) we show *ex situ* and in Fig. 2(b) *in situ* fluence distribution functions of our Gaussian spot, measured via direct camera illumination and reconstructed from the contours of written domains measured for various laser-pulse energies, respectively.

The excellent agreement between the *in situ* and *ex situ* spot profiles is accentuated in Fig. 3, where we compare exemplary contours of domains, measured for the Gaussian spot at distinct $E_{th}/E_p$ ratios, to the contours of

the corresponding laser intensity distribution in the *ex situ* spot images with matching ratios of intensity to maximum intensity, $I/I_{max}$. The two contour sets agree with each other to the level of the slight imperfections of our laser spot.

## B. Fluence distribution of our Laguerre-Gauss spot

Our first attempts to obtain similar agreement between *in situ* and *ex situ* spot profiles for Laguerre-Gauss beams failed, as illustrated in Fig. 4. Figure 4(a) displays the most circular symmetric fluence distribution that could be obtained on the camera by aligning the vortex wave plate to the beam path of our laser when emitting a continuous pulse train at 100 kHz repetition rate. Figure 4(b) shows the corresponding domain image obtained by simply removing the steering mirrors and illuminating the sample by a single laser pulse. It points to a noncircularly symmetric fluence distribution on the sample.

Domain patterns of satisfactory circular symmetry, as the one shown in Fig. 4(d), could be obtained only by realigning the vortex wave plate. However, reinsertion of the steering mirrors and imaging the continuous pulse train on the camera led to the noncircularly symmetric fluence distribution shown in Fig. 4(c). Based on the observation

<figure><img src="image_6.png" /><figcaption>FIG. 3. Comparison of exemplary contours obtained from measured domains (lines) and from the camera images (symbols). Numbers give $F_{th}/F_0$ for the domains and $I/I_{max}$ for the camera images.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>FIG. 4. (a,c) Comparison of *ex situ* fluence distributions captured by the camera for two different alignments of the vortex wave plate in continuous-wave operation of the laser. (b,d) Corresponding images of domains written by a single laser pulse, which reflect the *in situ* fluence distribution on the sample. The *ex situ* and *in situ* images are not to scale.</figcaption></figure>
<figure><img src="image_8.png" /><figcaption>FIG. 5. Comparison of (a) *ex situ* and (b) *in situ* fluence distribution functions of the nominally Laguerre-Gauss laser spot of our laser beam at a 1-kHz pulse repetition rate, as revealed by the camera under continuous illumination for 10 ms, and reconstructed from pulse-energy-dependent images of domains written by 11 consecutive laser pulses, respectively. Lines represent contours at multiples of 0.05.</figcaption></figure>

of repetition rate and pulse-number-dependent changes of the laser beam direction, we suspect the resulting misalignment of the beam center with the phase singularity of the vortex wave plate to be the main cause of this problem [28].

By reducing the pulse repetition rate to 1 kHz, adjusting the vortex wave plate to optimize the intensity distribution on the camera (operating at 10-ms exposure time), and using 11 consecutive pulses to write the analyzed domains, we achieved good agreement between *ex situ* and *in situ* fluence distributions, as shown in Fig. 5.

### C. Fluence distribution of our Bessel-Gauss spot

Figure 6 shows the *ex situ* and *in situ* fluence distribution functions of our nominal Bessel-Gauss spot. As theoretically expected, both distributions show a central spot

<figure><img src="image_9.png" /><figcaption>FIG. 6. Comparison of (a) *ex situ* and (b) *in situ* fluence distribution functions of our nominally Bessel-Gauss laser spot, as revealed by the camera and reconstructed from pulse-energy-dependent domain images, respectively. Note that the color transition point is set to 0.05 in order to make the low-intensity side lobes visible.</figcaption></figure>

surrounded by concentric side lobes of much lower amplitude. The fluence of the central spot is so much higher than that of the side lobes that it started to permanently change the magnetic properties of the sample, and even to drill a hole before the contours of the domains written by the first and second side lobes were closed, respectively. In addition, both distributions show significant deviations from circular symmetry that point to imperfections of our beam-shaping optics. Nevertheless, since the imperfections of both spots are similar, e.g., the quasi-threefold modulation of the valley separating the central spot and the first side lobe, we saw no need to realign the axicons when changing the number and/or repetition rate of the laser pulses.

### D. Radial distribution of our Gaussian spot

Figure 7 shows experimental values of the radial fluence distribution measured for our Gaussian beam (symbols) in comparison to the best fit of a Gaussian [cf. Eq. (A1)]. The experimental values of $r_{eq,1}$ are derived from the outer contour of the written domains, which are simply connected before overheating and resulting thermal erasure leads to the appearance of a multidomain pattern within their center region at high fluences.

The drop in the density of data points with decreasing radius and pulse energy (also visible in Fig. 8) reflects the fact that we increased the laser-pulse energy between successive measurements by a fixed amount $\Delta E$ so that the differences of $\Delta(1/E_p) = -\Delta E/(E_p^2 + \Delta E \cdot E_p)$ increase with decreasing $E_p$.

<figure><img src="image_10.png" /><figcaption>FIG. 7. Inverse of $E_p$ vs $r_{eq,1}$, the outer radius of the domains written by our Gaussian spot (symbols). The line represents the best fit of Eq. (5) to the data under the assumption that $f_r(r) = f_G(r)$ [cf. Eq. (A1)], with $1/E_{th}$ and $\sigma$ as fit parameters. Fit results are given in the legend together with the value derived for $F_{th}$. Errors correspond to the full difference between the upper and lower 95% confidence levels.</figcaption></figure>
The stunning agreement between data and fit manifests itself in identical $R^2$ and adjusted $R^2$ values of 0.99998 and in very small uncertainties of the fitted parameters $1/E_{th}$ and $\sigma$. Under the reasonable assumption that the fit continues to describe the actual fluence distribution outside the experimentally accessible range, our method reveals the external AOS threshold fluence of our sample with an accuracy of about one part per thousand.

Since the sample used for the measurements presented above degraded during the year that passed before we started our investigations of Laguerre-Gauss and Bessel-Gauss spots, we had to use another sample for the latter studies. According to results obtained for a Gaussian spot, this new sample possesses an AOS fluence threshold of $F_{th} = 27.80(4)$ J/m$^2$. This value will serve as reference for the studies presented below.

### E. Radial distribution of our Laguerre-Gauss spot

As expected from the imperfections visible in the spot images shown in Fig. 5, the radial fluence distribution obtained for our *in situ* Laguerre-Gaussian spot, represented by the symbols in Fig. 8, exhibits systematic deviations from the best fit of a Laguerre-Gaussian distribution [blue dotted line in Fig. 8, Eq. (A2)]. These deviations lead to $R^2$ values of about 0.975, and the fitted AOS threshold $F_{th} = 26.3(2)$ J/m$^2$ is ≈ 6% smaller than the reference. The modified Laguerre-Gauss distribution given by Eq. (A5) yields much better agreement between data

<figure><img src="image_11.png" /><figcaption>FIG. 8. Inverse of $E_p$ vs $r_{eq,i}$, where open and closed symbols represent the inner ($r_{eq,2}$) and outer ($r_{eq,1}$) radius of the annular domain written by our Laguerre-Gauss spot. The dotted blue line and solid red line represent best fits of Eq. (5) to the data under the assumption that $f_r(r)$ equals $f_{LG}(r)$ or $f_{LG,mod}(r)$ as given by Eqs. (A2) and (A5), respectively. Fit results for $1/E_{th}$ and $\sigma$ are given in the legend together with the derived values of $F_{th}$. Errors represent the full difference between the upper and lower 95% confidence levels.</figcaption></figure>

and fit (red solid line, $R^2 \approx 0.997$, in Fig. 8) and a value of $F_{th} = 27.4(1)$ J/m$^2$ that is just 1.5% smaller than the reference.

### F. Radial distribution of our Bessel-Gauss spot

The large azimuthal variations of the side-lobe amplitudes, visible in the images of the *ex situ* and *in situ* Bessel-Gauss fluence distributions shown in Fig. 6, hinder the determination of equivalent side-lobe radii for so many pulse energies that a meaningful determination of $f_r(r)$ from plots of $1/E_p$ vs $r_{eq,i}$ is impossible. Thus, we directly fitted the assumed fluence distribution $f_{BG}(r)$ given by Eq. (A7) to the directly measured fluence distributions.

Recognizing that the spatial extent of the measured *in situ* distribution is too small to provide meaningful fit results for $\sigma_G$, we proceeded as follows. We first fitted the unnormalized *in situ* fluence distribution to Eq. (A7) with a multiplicative amplitude $A = 1/E_{th}$ as additional fit factor. From this fit, we obtained $1/E_{th} = 11.68(5)$ $\mu$J$^{-1}$ and $\sigma_B = 14.20(2)$ $\mu$m.

We then used the results from the fit of the normalized *ex situ* fluence distribution, $\sigma_B = 14.18(2)$ $\mu$m and $\sigma_G = 120(2)$ $\mu$m, to determine $A_{s,BG} = 3030(50)$ $\mu$m$^2$. Justified by the excellent agreement between the values of $\sigma_B$ returned by the fits of the *in situ* and *ex situ* fluence distributions, we combined the results of both fits to determine the AOS threshold fluence. The derived value, $F_{th} = 28.3(6)$ J/m$^3$, is compatible with our reference.

We want to stress that all information needed for applications of Bessel-Gauss beams in laser material processing (the size and shape of the central spot, the laser fluence) can be obtained from the measured *in situ* fluence distribution alone, provided that the threshold pulse energy $E_{th}$ is accurately measured and the threshold fluence $F_{th}$ of the sample is known.

### V. CONCLUSIONS

We have generalized the “Simple technique for measurements of pulsed Gaussian-beam spot sizes” of Liu [22] such that it can be used to directly image the actual fluence distribution of arbitrarily shaped *in situ* laser spots. The unique capability to directly image actual *in situ* fluence distributions is essential to verify their agreement with theoretical expectations. Here, it was key for the unexpected discovery and subsequent removal of significant differences between *ex situ* and *in situ* Laguerre-Gauss spots that resulted from the use of distinct laser-pulse repetition rates and/or pulse numbers for their respective investigation.

Moreover, we introduced data-acquisition and data-analysis schemes that provide experimental access to quasi-noise-free radial *in situ* fluence distributions of circularly symmetric spots and enable the determination of
their in situ spot areas and laser fluences with high accuracy. We believe that our technique offers a robust and accurate method for characterizing in situ laser spots, with the potential to set a new standard in the field.

The data that support the findings of this study are available from the corresponding author upon reasonable request.

## ACKNOWLEDGMENTS

This work is supported by the ANR-23-CE30-0047 SLAM, the Institute Carnot ICEEL, and the interdisciplinary project LUE “MAT-PULSE”, part of the French PIA project “Lorraine Université d’Excellence” reference ANR-15-IDEX-04-LUE. This article is based upon work from COST Action CA23136 CHIROMAG, supported by COST (European Cooperation in Science and Technology). This work was supported by the ANR through the France 2030 government grants: PEPR Electronique-EMCOM (ANR-22-PEEL-0009), PEPR SPIN-Chirex (ANR-22-EXSP-0002), and PEPR SPIN-SPINMAT ANR-22-EXSP-0007. All funding was shared equally among all authors. In addition, Y.L.-G. was supported by the Magnus Ehrnrooth Foundation.

J.H., M.V., and Y.L.-G. conceived the data representation and analysis scheme; J.H. and Y.L.-G. carried out the experiments; Y.L.-G. developed the image acquisition and analysis software; M.H. produced the samples; and M.H. and S.M. acquired the funding for this work. All authors contributed to the interpretation of the results and to the preparation of the manuscript.

The authors declare no conflicts of interest regarding this article.

## APPENDIX: FLUENCE DISTRIBUTIONS AND ASSOCIATED SPOT AREAS

The fluence distributions and spot areas for Gaussian and Laguerre-Gauss spots, indicated by the indices G and LG, respectively, are given by

$$f_G(r) = \exp\left\{-\left(\frac{r}{\sigma}\right)^2\right\}, \qquad (A1)$$

$$f_{LG}(r) = \exp(1) \left(\frac{r}{\sigma}\right)^2 \exp\left\{-\left(\frac{r}{\sigma}\right)^2\right\}, \qquad (A2)$$

$$A_{s,G} = \pi\sigma^2, \qquad (A3)$$

$$A_{s, LG} = \exp(1) \pi \sigma^2 \approx 8.5397 \sigma^2, \qquad (A4)$$

where $r = |r|$. The modified fluence distribution that best describes our experimental fluence distribution for the

nominally Laguerre-Gauss spot is

$$f_{LG,mod}(r) \approx 2.4756 \left(\frac{r}{\sigma}\right)^{3/2} \exp\left\{-\left(\frac{r}{\sigma}\right)^{5/2}\right\}, \qquad (A5)$$

with an associated spot-size area of

$$A_{s, LG,mod} \approx 5.5205\sigma^2. \qquad (A6)$$

The fluence distribution and spot-size area of the Bessel-Gauss spot are given by

$$f_{BG}(r) = J_0\left(\frac{r}{\sigma_B}\right) \exp\left\{-\left(\frac{r}{\sigma_G}\right)^2\right\}, \qquad (A7)$$

$$A_{s,BG} = \pi \sigma_G^2 I_0 \left( \frac{\sigma_G^2}{2\sigma_B^2} \right), \qquad (A8)$$

where $J_0(x)$ and $I_0(x)$ denote the zeroth-order Bessel function and the zeroth-order modified Bessel function of the first kind, respectively.

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