# Light-matter interaction at rough surfaces: A morphological perspective on laser-induced periodic surface structures

Vladimir Yu Fedorov, Jean-Philippe Colombier

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Vladimir Yu Fedorov, Jean-Philippe Colombier. Light-matter interaction at rough surfaces: A morphological perspective on laser-induced periodic surface structures. Physical Review B, 2024, 110, 10.1103/physrevb.110.115438. ujm-04839268

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# Light-matter interaction at rough surfaces: A morphological perspective on laser-induced periodic surface structures

Vladimir Yu. Fedorov<sup>*</sup> and Jean-Philippe Colombier<sup>†</sup>

Université Jean Monnet Saint-Etienne, CNRS, IOGS, Laboratoire Hubert Curien UMR 5516, F-42023, Saint-Etienne, France

<figure><img src="image_2.png" /></figure>

(Received 4 June 2024; revised 17 September 2024; accepted 20 September 2024; published 30 September 2024)

We use *ab initio* electromagnetic simulations to investigate light absorption by rough surfaces in the context of the formation of laser-induced periodic surface structures. Our approach involves modeling a rough surface using a statistical description of its continuous height distribution via a corresponding correlation function. We study the influence of incident light polarization and various statistical properties of surface roughness, such as root-mean-square height and correlation length, on the distribution of absorbed laser energy. By analyzing light absorption in different layers of the surface selvedge, we elucidate how different features of surface morphology influence the shape of the resulting periodic surface structures. We show that circularly polarized laser pulses are highly sensitive to initial or progressively developing asymmetries in surface roughness.

DOI: 10.1103/PhysRevB.110.115438

## I. INTRODUCTION

When an intense ultrashort laser pulse hits the surface of a solid, it triggers a chain of events, each unfolding on a specific timescale. In the initial stage occurring on a femtosecond timescale, within the pulse duration, a strong laser field excites electrons in the material; light scatters by surface roughness and, in the case of metallic media, generates surface plasmons. Then, on a picosecond timescale, after the passage of the laser pulse, the electron-phonon thermalization stage begins. In this stage, hot excited electrons, colliding with atoms of the material, transfer their energy to the lattice, thereby heating the material. Next, on a timescale from a few picoseconds to nanoseconds, hydrodynamic processes come into play. During this period, material melting, cavitation, and explosive boiling occur, accompanied by convection and hydrodynamic instabilities that ultimately shape the surface as the material solidifies. Finally, approximately a millisecond later (corresponding to a typical kilohertz laser repetition rate) the succeeding pulse within the pulse train arrives and triggers the same chain of events again. During this iterative process, each subsequent laser pulse interacts with the surface that has been modified by the previous pulses. This interaction forms a feedback loop, which, after multiple cycles, leads to the emergence of periodic ripples known as laser-induced periodic surface structures (LIPSSs) [1–4].

The formation of LIPSSs is a universal phenomenon: They can be produced on almost all types of materials (dielectrics, semiconductors, metals). By fine-tuning parameters of LIPSSs, we can change surface properties and thereby functionalize the irradiated material. For example, since LIPSSs act as a diffraction grating, by adjusting their period we can select which wavelengths of light will be reflected and

thus change the color of materials [5–7]. In addition, with LIPSSs we can alter the wettability of the surface and create hydrophobic or hydrophilic coatings [8–10]. Also, using LIPSSs, we can control the friction coefficient of materials [11–13] or impart antibacterial properties to their surfaces [14–16].

Depending on the period, LIPSSs can be divided in two main classes: LIPSSs whose period is close to the laser wavelength are referred to as low spatial frequency LIPSSs (LSFLs), and those whose period is significantly less than the laser wavelength are referred to as high spatial frequency LIPSSs (HSFLs) [1,2,4]. Among the two, HSFLs are of particular interest because they enable surface nanostructuring with a resolution well below the optical diffraction limit [17]. Interestingly, to date HSFLs have only been achieved using ultrashort femtosecond or picosecond laser pulses.

The orientation of LIPSSs depends on material properties and can be either parallel or perpendicular to the laser polarization. The determination of which orientation prevails occurs at the initial stage of laser-matter interaction in which the anisotropy of light scattering by inevitable surface defects leads to symmetry breaking. The distribution of scattered light intensity and, consequently, the spatial modulation of local energy (fluence) imprinted on the material surface through absorption are dictated by the surface roughness. The inhomogeneous distribution of absorbed light energy acts as a seed, setting on the amplification process through the interpulse feedback loop, thereby shaping the final morphology of LIPSSs. Thus, an accurate description of surface morphology and its influence on scattering and absorption of incoming light is of paramount importance for understanding the physics underlying LIPSS generation.

Although there have been a number of experiments on the influence of surface quality on LIPSS formation, a thorough and comprehensive study in this area is clearly lacking. It was shown that the surface roughness directly affects the period of LIPSSs and thus can alter certain surface

<footnote>*Contact author: v.y.fedorov@gmail.com</footnote>

<footnote>†Contact author: jean.philippe.colombier@univ-st-etienne.fr</footnote>
functionality [15,18]. The formation of consistent LIPSSs can be significantly enhanced by polishing the surface in the direction perpendicular to the expected orientation of LIPSSs [19]. In contrast, the deposition of thin films through pulse vapor deposition provides a flexible method for preventing an aligned scratch and adjusting the initial nanostructures of the topography, which in turn influences the feedback process significantly [20].

In theoretical studies, the importance of surface roughness in explaining the origin of LIPSSs has become clear, starting with the pioneering works of Sipe et al. [21], who linked the appearance of LIPSSs with the interference of the incident laser radiation with surface electromagnetic waves generated by scattering at a rough surface. Later, the crucial role of light interaction with surface inhomogeneities in the formation of LIPSSs was confirmed using ab initio electromagnetic simulations, free from the limitations of Sipe et al.'s theory [22–25]. However, in all of these simulations the surface roughness was considered simply to be a discrete binary mask—a collection of identical randomly distributed subwavelength scatterers on top of a flat surface. Meanwhile, any rough surface possesses a continuous distribution of heights, which requires description in terms of statistical properties like the height probability distribution and the height correlation function. In this paper we apply such a statistical description of surface roughness in order to simulate the scattering and subsequent absorption of ultrashort laser pulses by rough surfaces. We study how various statistical properties of the surface roughness affect the distribution of the absorbed light energy. In particular, we demonstrate that different layers of surfaces with a continuous height distribution scatter light in different directions. This observation allows us to trace which surface layers are responsible for the formation of certain features of the LIPSS shape. Finally, we explore how light polarization affects the distribution of absorbed energy on rough surfaces. We show that circularly polarized laser pulses allow us to detect initial or gradually developing asymmetries in surface roughness.

## II. MODEL OF THE SURFACE ROUGHNESS

A rough surface can be described by the roughness function $R(x, y)$, which defines the random deviations of the surface height relative to some reference plane [26]. With its help, the statistical properties of the surface roughness can be expressed using a correlation function $C(X, Y)$ of the form

$$C(X, Y) = \frac{\langle R(x, y)R(x + X, y + Y) \rangle}{\sigma^2}, \quad (1)$$

where $\sigma = \sqrt{\langle R^2 \rangle}$ is the root-mean-square (rms) height and $\langle \cdots \rangle$ denotes the spatial averaging. The correlation function $C(X, Y)$ describes the spatial coherence between different surface points separated by a distance $d = \sqrt{X^2 + Y^2}$.

Since the variety of existing surface types is enormous, the choice of a proper correlation function is not an easy task. Even when considering the same material, its surface roughness can vary greatly depending on whether it is natural or artificially processed. The surface relief depends on residual stresses, the presence of fatigue cracks, and thermal expansion. Thus, the assumption that a rough surface has a certain height distribution depends largely on the origin of

the surface being considered. Moreover, sometimes, the very description by the correlation function may not be applicable, for example, for surfaces with a fractal structure.

For simplicity, in our simulations we assume that the surface roughness can be described by the Gaussian correlation function

$$C(X, Y) = \sigma^2 \exp \left( - \frac{X^2 + Y^2}{\xi^2} \right), \quad (2)$$

where $\xi$ is the correlation length. Clearly, some surfaces cannot possess a Gaussian height distribution because of their inherent asymmetry. For example, surfaces which are gradually worn down by some abrasive mechanical process may start life with a Gaussian distribution but end up with the distribution skewed toward negative heights. This is because the wearing process will preferentially remove peaks of large positive height while leaving large negative heights relatively unaffected. In turn, a process of surface formation consisting of a large number of random independent events, such as shot blasting and electrochemical polishing, will produce a Gaussian-like height distribution by the action of the central limit theorem. For a more in-depth discussion of the applicability of the Gaussian correlation function to surface roughness, we recommend Ref. [26] and the references therein.

To demonstrate the impact of the choice of correlation function on our results, we also consider the exponential correlation function

$$C(X, Y) = \sigma^2 \exp \left( - \frac{\sqrt{X^2 + Y^2}}{\xi} \right). \quad (3)$$

As we will show below, our results do not change qualitatively when we replace one correlation function with another.

In turn, the power spectral density $W(K_x, K_y)$ characterizes the average distribution of the spatial frequency components of a randomly fluctuating surface profile. It is related to the correlation function $C(X, Y)$ by the Fourier transform as follows:

$$W(K_x, K_y) = \frac{1}{(2\pi)^2} \iint_{-\infty}^{\infty} C(X, Y) e^{-i(K_x X + K_y Y)} dX dY, \quad (4)$$

where $K_x$ and $K_y$ are the spatial frequencies.

To generate a rough surface numerically, we introduce a spectral function $F(K_x, K_y)$ which we obtain from the power spectral density $W(K_x, K_y)$ by modifying its amplitude and phase by a sequence of normally distributed random numbers [27,28]. This new function allows us to calculate the roughness $R(x, y)$ using the following inverse Fourier transform:

$$R(x, y) = \iint_{-\infty}^{\infty} F(K_x, K_y) e^{i(K_x x + K_y y)} dK_x dK_y, \quad (5)$$

The function $F(K_x, K_y)$ is constructed as follows [27,28]. For all positive frequencies $K_x > 0$ and $K_y > 0$ we define $F(K_x, K_y)$ as

$$F(K_x, K_y) = \sqrt{L_x L_y W(K_x, K_y)} \frac{\eta_1 + i\eta_2}{\sqrt{2}}, \quad (6)$$
<figure><img src="image_3.png" /><figcaption>FIG. 1. Surface roughness $R(x, y)$ for the (a)–(c) Gaussian and (d)–(f) exponential correlation functions with rms height $\sigma = 50$ nm and two different correlation lengths, $\xi = 100$ in (b) and (e) and 200 nm in (c) and (f). The line plots in (a) and (d) show the corresponding cross sections at $y = 0$.</figcaption></figure>

and for zero and Nyquist frequencies $[K_x = 0, K_y = 0$ or $K_x = \max(K_x), K_y = \max(K_y)]$ we define it as

$$F(K_x, K_y) = \sqrt{L_x L_y W(K_x, K_y)} \eta_3. \qquad (7)$$

Here $L_x$ and $L_y$ are the lengths of the x and y numerical domains, while each of $\eta_1$, $\eta_2$, and $\eta_3$ denotes a sequence of normally distributed random numbers in the range [0,1] with zero mean and unity standard deviation. Since the surface roughness $R(x, y)$ must be a real-valued function, $F(K_x, K_y)$ have to satisfy the following additional conditions:

$$F(-K_x, -K_y) = F^*(K_x, K_y), \qquad (8)$$

$$F(-K_x, K_y) = F^*(K_x, -K_y). \qquad (9)$$

In other words, $F(K_x, K_y)$ must be “conjugate symmetric” about the origin; i.e., the reflection of $F(K_x, K_y)$ at any point relative the origin must be its complex conjugate.

Thus, in order to obtain the surface roughness function $R(x, y)$ with the desired statistical properties we first select a correlation function $C(X, Y)$. Then, using the Fourier transform, we calculate the power spectral density $W(K_x, K_y)$. Afterward, we construct the function $F(K_x, K_y)$, and finally, applying the inverse Fourier transform to $F(K_x, K_y)$, we calculate $R(x, y)$.

Figure 1 shows the typical surface roughness $R(x, y)$ generated by the above approach for the Gaussian [Eq. (2)] and exponential [Eq. (3)] correlation functions. For both cases we use the same seed for the random numbers, the rms height

$\sigma = 50$ nm, and we consider two different correlation lengths, $\xi = 100$ and 200 nm. In Figs. 1(a)–1(c) we see that for the Gaussian correlation function the correlation length $\xi$ determines the average scale of surface inhomogeneities. In turn, Figs. 1(d)–1(f) show that for the exponential correlation function faster decay of the spatial coherence results in a surface with very rapidly varying small-scale perturbations superimposed on the Gaussian roughness. In this case the correlation length $\xi$ determines the number of these perturbations and the sharpness of their peaks.

### III. NUMERICAL MODEL

To study the interaction of ultrashort laser pulses with rough surfaces we numerically solve the following system of Maxwell equations using the finite-difference time-domain (FDTD) method [29]:

$$\vec{\nabla} \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}, \qquad \vec{\nabla} \times \vec{H} = \frac{\partial \vec{D}}{\partial t}, \qquad (10)$$

where $\vec{E}(x, y, z, t)$ and $\vec{H}(x, y, z, t)$ are the electric and magnetic field vectors; $\vec{B} = \mu_0 \vec{H}$, with $\mu_0$ being the vacuum permeability; and $\vec{D} = \varepsilon_0 \varepsilon(\omega) \vec{E}$, where the tilde means the temporal spectrum, $\varepsilon_0$ is the vacuum permittivity, and $\varepsilon(\omega)$ is the frequency-dependent permittivity of the medium.

In our simulations we use a calculation grid with sizes $L_x = L_y = 12.2$ µm and $L_z = 0.9$ µm in the x, y, and z directions, respectively, with the corresponding step sizes $\Delta x = \Delta y = 10$ nm and $\Delta z = 5$ nm. To avoid nonphysical reflections, at each end of the grid we place convolutional perfectly matched layers 0.1 µm thick.

For the incoming laser radiation we use a plane wave propagating along the z direction, whose temporal envelope $A(t)$ is given by one period of a $\sin^2$ function: $A(t) = \sin^2(\pi t/4\tau_0)$, where $\tau_0$ is the laser pulse duration. In the case of a laser pulse polarized in the x direction, we excite the x component of the electric field: $E_x(t) = A(t) \cos(\omega_0 t)$, where $\omega_0 = 2\pi c_0/\lambda_0$ is the central frequency with $\lambda_0$ and $c_0$ are the central wavelength and the speed of light in vacuum. To model a circularly polarized laser pulse we also add the y component of the field with a relative phase shift of $\pi/2$: $E_y(t) = A(t) \sin(\omega_0 t)$. To preserve the pulse energy, for circularly polarized pulses we divide $E_x$ and $E_y$ by $\sqrt{2}$. In our simulations we use ultrashort laser pulses with duration $\tau_0 = 50$ fs and central wavelength $\lambda_0 = 1.03$ µm.

At a distance of 0.6 µm below the radiation plane [as measured relative to the reference plane of the roughness function $R(x, y)$], we place a semi-infinite stainless-steel medium with a rough surface. Stainless steel is particularly interesting in the context of LIPSSs for a number of reasons: It is a cheap material with high relevance for industrial applications; LIPSSs on the surface of stainless steel are well contrasted and appear in a large window of laser parameters; the incoming laser energy is strongly confined in the region of energy deposition, resulting in LIPSS patterns that reflect the interference and local field enhancement effects very well; and the optical properties of stainless steel are well known over a wide spectral range. To model the dispersive response of stainless steel we apply the auxiliary differential equation method [29] and assume the Drude permittivity $\varepsilon(\omega) = 1 - \omega_p^2/(\omega^2 + i\omega\gamma)$,
with plasma frequency $\omega_p = 19.2 \times 10^{15}$ 1/s and damping rate $\gamma = 9.15 \times 10^{15}$ 1/s [30]. Under these parameters the complex refractive index $n = n' + in''$ of stainless steel at the central frequency $\omega_0$ has real and imaginary parts equal to $n' = 3.02$ and $n'' = 3.51$. The corresponding skin depth is $l = 1/(2n''\omega_0/c_0) = 23.34$ nm.

In our studies we are mainly interested in the distribution of the laser pulse energy absorbed on the surface: The surface areas that have absorbed a sufficiently large amount of energy will later be extruded after the thermalization and hydrodynamic stages of the laser-matter interaction. Therefore, any asymmetry in the distribution of the absorbed energy serves as a seed for LIPSS growth. To estimate the amount of energy absorbed on the surface let us first consider the power $P$ delivered by the laser pulse per unit time per unit volume of the material: $P = \bar{E} \cdot \vec{J}$, where $\bar{E}$ is the electric field of the laser pulse and $\vec{J}$ is the current of the excited electrons [31]. For the current $\vec{J}$ we can write Ohm's law in the form $\vec{J} = \sigma \bar{E}$, where the conductivity $\sigma$, taken at the central frequency $\omega_0$, can be expressed through the imaginary part of the permittivity $\epsilon = \epsilon' + i\epsilon''$ as $\sigma = \epsilon_0\omega_0\epsilon''$. Finally, we can write $\epsilon'' = 2n'n''$, where $n'$ and $n''$ are the real and imaginary parts of the refractive index $n = n' + in'' = \sqrt{\epsilon}$. Combining all these expressions together, we obtain $P = 2\epsilon_0n'n''\omega_0E^2$. Then, to obtain the spatial distribution of the energy $Q$ delivered by the laser pulse per unit volume we have to integrate $P$ over time: $Q(x, y, z) = 2\epsilon_0n'n''\omega_0 \int_{-\infty}^{\infty} E(x, y, z)^2 dt$. Thus, the electric field $\bar{E}$, obtained during the FDTD simulations, allows us to calculate the desired distribution of the absorbed energy $Q$ in the entire material below its surface.

## IV. RESULTS AND DISCUSSION

First, let us study the distribution of the absorbed energy $Q(x, y, z)$ in different layers of the rough surface exposed to an x-polarized laser pulse. To model the rough surface we take the Gaussian correlation function given in Eq. (2). This function depends on two parameters: the rms height $\sigma$ and the correlation length $\xi$. Before choosing some specific values of $\sigma$ and $\xi$, we should note that the roughness is not an intrinsic property of the surface and depends on the wavelength of the incident light or, more precisely, on the scale of the surface irregularities relative to the light wavelength. A surface that appears fairly flat to far-infrared light may appear extremely rough to ultraviolet light. In principle, the size of the surface irregularities depends on the preliminary surface treatment and can be controlled over a wide range. However, to observe HSFLs, the surface must be well polished so that the surface roughness is on a subwavelength scale. Therefore, as a reference point, we choose a roughness with rms height $\sigma = 50$ nm and correlation length $\xi = 100$ nm, which are approximately 10 times smaller than the laser wavelength.

Figure 2(a) shows the $y = 0$ cross section of the corresponding roughness function $R(x, y)$. In turn, Figs. 2(b)–2(d) show the $xy$ slices of $Q$ in three different surface layers, $z = +70, 0, -70$ nm [see the dashed lines in Fig. 2(a)]. Note that in all absorbed energy maps presented in this paper, in order to better examine small details, we plot only the central part of the calculation grid with a size of $4 \times 4$ µm$^2$,

<figure><img src="image_4.png" /><figcaption>FIG. 2. (a) The $y = 0$ cross section of the rough surface. The dashed lines mark the $z = \pm 70$ nm levels. The surface has the Gaussian correlation function with rms height $\sigma = 50$ nm and correlation length $\xi = 100$ nm. (b)–(d) The distribution of absorbed energy $Q$ in three surface layers, $z = +70, 0, -70$ nm, respectively. (e) The distribution of absorbed energy $Q_i$ integrated over all surface layers along $z$; the double arrow shows the direction of laser polarization. (f)–(i) The spatial power spectra of the corresponding $Q$ distributions. The arrows in (i) mark the characteristic spectral patterns known as type-r and type-s features.</figcaption></figure>

while the full grid size is $12 \times 12$ µm$^2$. The white areas in Figs. 2(b)–2(d) correspond to the regions outside of the
material, where $Q = 0$. We observe that, as a result of near-field enhancement, the maxima of the absorbed energy $Q$ are concentrated in the vicinity of the sharp edges of the surface.

Figures 2(f)–2(h) show the spatial spectra of the $Q$ slices in the spatial-frequency coordinates $k_x$ and $k_y$ normalized by the wave number $k_0 = \omega_0/c_0$. These spatial spectra give us an idea of typical sizes and orientations of the absorbed energy spots. The spectral components of $Q$ at spatial frequencies close to or less than $k_0$ correspond to the large spots whose characteristic sizes are comparable to or exceed the laser wavelength $\lambda_0$. They are responsible for the formation of LSFLs. In contrast, the spectral components of $Q$ at spatial frequencies significantly greater than $k_0$ correspond to the small-scale subwavelength spots responsible for generation of HSFLs. For example, in Figs. 2(f)–2(h) we see that the highest-frequency components of the spectra lie in the region of $5k_0$. Therefore, we can conclude that in all three considered layers the distribution of the absorbed energy $Q$ has morphological features with a minimum size approximately 5 times smaller than the laser wavelength $\lambda_0$.

Figures 2(f)–2(h) also show that the spectral distributions of the absorbed energy $Q$ in different layers of the surface are not isotropic. The spectrum corresponding to the $z = +70$ nm layer is elongated in the $k_y$ direction [see Fig. 2(f)], while the spectrum corresponding to the $z = -70$ nm layer is elongated in the perpendicular $k_x$ direction [see Fig. 2(h)]; the spectrum for the $z = 0$ layer represents the transition between the two previous ones. Therefore, taking into account that smaller features in the spatial domain correspond to higher spectral frequencies, we can conclude that the spots of absorbed energy $Q$ in different surface layers have, on average, different orientations. At the top edge of the surface the spots of the absorbed energy are mainly stretched in the $x$ direction, parallel to the laser polarization, while at the bottom edge they line up in the perpendicular $y$ direction. In order to explain this result we refer to our recent study in which we showed that a single nanobump or nanocavity, irradiated by linearly polarized laser light, has an asymmetric scattering pattern similar to that of a dipole [30]. In particular, we showed that nanobumps and nanocavities scatter light predominantly in mutually perpendicular directions: nanobumps along the laser polarization and nanocavities in the perpendicular one (see Fig. 3 in [30]). In turn, as shown in Fig. 1(a), any rough surface can be considered a collection of peaks (bumps) and valleys (cavities), located at the upper and lower edges of the surface, respectively. Here the upper (lower) edge corresponds to the surface heights in the $+z$ ($-z$) half-space, looking toward the incoming laser radiation (toward the material bulk). Therefore, at the upper edge of the surface, consisting of peaks, light is mainly scattered parallel to the direction of polarization, while at the lower edge of the surface, populated by valleys, the main direction of light scattering is perpendicular to the polarization direction. Thus, the upper and lower edges of the rough surface tend to form LIPSSs with mutually perpendicular orientations.

The final shape of LIPSSs results from the accumulation of absorbed energy in all surface layers. Therefore, the ultimate quantity of relevance for predicting light-surface coupling is the absorbed energy $Q_i$ integrated over all $z$ layers: $Q_i(x, y) = \int_{-\infty}^{\infty} Q(x, y, z) dz$. Figure 1(e) shows the distribution of this

<figure><img src="image_5.png" /><figcaption>FIG. 3. (a) and (b) The distributions of integrated absorbed energy $Q_i$ and (c) and (d) their spectra for rough surfaces described by the Gaussian (top) and exponential (bottom) correlation functions with rms height $\sigma = 50$ nm and correlation length $\xi = 100$ nm. The double arrow shows the direction of laser polarization.</figcaption></figure>

integrated absorbed energy $Q_i$, where we see that the regions of high losses are mainly elongated along the $x$ direction. Therefore, we can expect that in this particular case the resulting LIPSS will be oriented parallel to the laser polarization. In turn, in Fig. 1(i) we see that the spatial spectrum of $Q_i$ has a very recognizable shape and contains spectral features known as “type r” and “type s” [3]. The above analysis of absorbed energy spectra across various surface layers leads to the conclusion that type-r features primarily originate from the upper edge of the surface, where the scattering of light is governed by surface peaks. Conversely, type-s features predominantly arise from the lower layers of the surface, where light is scattered by the surface valleys.

To understand how different types of correlation functions affect light absorption, let us consider two rough surfaces with the Gaussian [Eq. (2)] and exponential [Eq. (3)] correlation functions. Figure 3 shows the difference between the corresponding distributions of the integrated absorbed energy $Q_i$ and their spatial spectra. We can see that compared to the case of the Gaussian correlation function, the distribution of $Q_i$ at the surface with the exponential correlation function has many more fine details. This difference in the absorbed energy distributions is reflected in their spatial spectra, where we see that although both spectra are qualitatively very similar, the one corresponding to the exponential correlation function has much higher spectral components in the $k_y$ direction. This observation is expected since, as demonstrated earlier, the higher frequencies along $k_y$ arise from light scattered at the peaks in the top edge of the rough surface. As depicted in Fig. 1, for the surface with the exponential correlation function, these peaks are more concentrated and smaller in scale.
<figure><img src="image_6.png" /></figure>

<figure><img src="image_7.png" /><figcaption>FIG. 4. (a) The distributions of integrated absorbed energy $Q_i$ and (b) their spectra for rough surfaces described by the Gaussian correlation function with different values of the rms height $\sigma$ and the correlation length $\xi$. The double arrow shows the direction of laser polarization.</figcaption></figure>

Next, we consider a surface with the Gaussian correlation function and examine how changes in its rms height $\sigma$ and correlation length $\xi$ affect the absorbed energy $Q_i$. Figure 4 shows the distributions of $Q_i$ and their spectra for different

values of $\sigma$ and $\xi$. In Fig. 4(b) we can see that an increase in the correlation length $\xi$ leads to the compression of the spectra in the $k_y$ direction. This compression corresponds to an increase in the scale of the absorbed energy spots in the $y$ direction [see Fig. 4(a)]. Interestingly, at $\xi = 200$ nm, the spectra become equally elongated in the $k_x$ and $k_y$ directions, which means that the average sizes of the $Q_i$ spots in the $x$ and $y$ directions become the same [see the right columns in Figs. 4(a) and 4(b)]. Thus, with the increase in the correlation length, we observe a transition from the distribution of absorbed energy oriented along the laser polarization to a specklelike pattern.

Meanwhile, Fig. 4 shows that the increase of the rms height $\sigma$ leads to an increase in the total absorption: The spatial distributions of $Q_i$ and their spectra become more intense at higher $\sigma$ values. Additionally, in Fig. 4(b) we see that with an increase of $\sigma$ the type-s spectral features [see Fig. 2(i)] gradually spread out in the $k_y$ direction. In the case of the smallest $\xi$ ($\xi = 50$ nm) and the largest $\sigma$ ($\sigma = 100$ nm) the type-s features completely disappear. With such $\sigma$ and $\xi$, the surface consists of very narrow (small correlation length) and very deep (large rms height) inhomogeneities. As a result, little light penetrates to the bottom of such elongated pits, where it can be scattered by the surface valleys in the right direction.

Now, let us consider a surface with the Gaussian correlation function of the form

$$C(X, Y) = \sigma^2 \exp \left( -\frac{X^2}{\xi_x^2} - \frac{Y^2}{\xi_y^2} \right), \quad (11)$$

with the rms height $\sigma = 50$ nm and two different correlation lengths $\xi_x$ and $\xi_y$ along the $x$ and $y$ directions. Here we still use $x$-polarized incident laser pulses. The correlation function given by Eq. (11) describes a surface with initially broken symmetry where the scale of the inhomogeneities in one direction is larger than in another. In other words, this surface initially has a ripplelike structure in the $x$ or $y$ direction. It may correspond to a surface subjected to mechanical polishing along one specific direction, or it may represent a surface that has already been irradiated by a small number of preceding laser pulses. Figure 5 shows the distributions of integrated absorbed energy $Q_i$ together with their spectra for different values of $\xi_x$ and $\xi_y$.

Let us first take a look at the case with the largest $\xi_x$ ($\xi_x = 200$ nm) and $\xi_y$ decreasing from 200 to 50 nm (the right columns in Fig. 5). This case describes a transition from a surface with isotropic roughness to one where surface inhomogeneities, being compressed in the $y$ direction, turn out to be elongated in the $x$ direction, parallel to the polarization. The spectra in Fig. 5(b) show that in this case the decrease in $\xi_y$ leads to higher spectral components in the $k_y$ direction, which in the real domain corresponds to the compression of the absorbed energy spots along $y$.

Now, let us consider the scenario where $\xi_y = 200$ nm is the largest and $\xi_x$ decreases from 200 to 50 nm (the top rows in Fig. 5). Here we see a transition from the isotropic roughness to the one where the surface inhomogeneities, being compressed in the $x$ direction, become elongated along $y$ in the direction perpendicular to the laser polarization. As in the previous case, the spectra in Fig. 5(b) show that decreasing
<figure><img src="image_8.png" /><figcaption>FIG. 5. (a) The distributions of integrated absorbed energy $Q_i$ and (b) their spatial spectra for rough surfaces described by an asymmetric Gaussian correlation function with fixed rms height $\sigma = 50$ nm and different correlation lengths $\xi_x$ and $\xi_y$ along the x and y directions. The double arrow shows the direction of laser polarization.</figcaption></figure>

$\xi_x$ leads to the appearance of higher-frequency components, but this time in the y direction. The corresponding distribution of the absorbed energy becomes elongated along y.

<figure><img src="image_9.png" /><figcaption>FIG. 6. (a)–(c) The distributions of integrated absorbed energy $Q_i$ and (d)–(f) their spectra for x-polarized (top), y-polarized (middle), and circularly polarized (bottom) laser pulses. The arrows show the direction of each laser polarization.</figcaption></figure>

The rest of Fig. 5 confirms the previously observed trends: As surface irregularities extend further along either the x or y direction, the absorbed energy spots become increasingly elongated in the corresponding direction. Thus, any asymmetry in surface roughness, whether it originated as an inherent feature of the surface or arose after laser exposure, will be magnified by subsequent laser pulses. This mechanism constitutes the foundation of the interpulse feedback loop, which ultimately leads to the formation of LIPSSs from even the slightest initial symmetry breaking. Moreover, the random nature of the surface roughness ensures that no two LIPSS patterns are perfectly alike. This highlights the individuality of each imprint left by exposure to light, evoking the distinctiveness of a fingerprint, which offers anticounterfeiting abilities.

Finally, we study how the polarization of the incoming light affects the distribution of the integrated absorbed energy $Q_i$. Figure 6 shows the distributions of $Q_i$ and their spectra for x-polarized, y-polarized, and circularly polarized laser pulses. In Figs. 6(a), 6(b) 6(d), and 6(e) we see that in the case of x- and y-polarized laser pulses the shapes of the $Q_i$ distributions and their spectra are the same but rotated by 90° relative to each other. In other words, as expected,
the orientation of LIPSSs follows the direction of the laser polarization. However, Figs. 6(c) and 6(f) show that in the case of circular polarization the distribution of $Q_i$ does not show any preferred orientation and the corresponding spatial spectrum is symmetrical about the origin. This can be explained by the fact that for circularly polarized light, when the electric field vector rotates, it continuously passes through all possible orientations. The absence of a preferred orientation in the spectral domain means that the distribution of the absorbed energy in real space is isotropic. In Fig. 6(c) we see that the spots of the absorbed energy line up in a circlelike pattern. Therefore, we can expect that the resulting LIPSSs will exhibit the same radial symmetry.

However, what happens if a circularly polarized laser pulse interacts with a surface that initially has some asymmetry in its roughness? In this case, the breaking of symmetry should occur not due to a dedicated direction of laser polarization (circular polarization does not have a preferred direction), but due to the properties of the surface itself. In order to answer this question let us examine a rough surface with an asymmetric Gaussian correlation function defined by Eq. (11). We fix rms height $\sigma = 50$ nm and y correlation length $\xi_y = 100$ nm and consider the x correlation length values $\xi_x = 100, 200$, and $400$ nm. While $\xi_x = 100$ nm defines a surface with an isotropic roughness, the $\xi_x = 200$ and $400$ nm values correspond to surfaces whose inhomogeneities are stretched in the x direction by 2 and 4 times, respectively. Figure 7 shows the resulting distributions of the integrated absorbed energy $Q_i$ and their spectra. We can see that the increase in the x correlation length leads to the elongation of the absorbed energy spots in the x direction. Meanwhile, the corresponding spectra are compressed in the $k_x$ direction, remaining the same in the $k_y$ direction. This means that the increase in the correlation length in one direction can change the size of the absorbed energy spots only in that same direction. Moreover, we see that the compression of spectra in the $k_x$ direction is directly proportional to the increase in $\xi_x$: a 2 times larger $\xi_x$ corresponds to a 2 times narrower spectrum. Therefore, by using circularly polarized laser pulses, we can effectively test the asymmetry of surface roughness.

Interestingly, the compression of the absorbed energy spectra occurs only due to high-frequency components, which are responsible for HSFL formation. In turn, the low-frequency components responsible for LSFLs [the central ring structures in Figs. 7(d)–7(f) at frequencies close to $k_0$] do not change with the increasing asymmetry of the surface. Taking into account that HSFLs need ultrashort laser pulses to form, we can conclude that sensing of the surface roughness asymmetries using circular polarization would be possible only with picosecond or femtosecond laser pulses.

V. CONCLUSION

In conclusion, we used *ab initio* electromagnetic simulations to study the interaction of ultrashort laser pulses with rough surfaces. We investigated the spatial distribution of the absorbed laser energy in order to better understand the underlying physics of LIPSS formation. In contrast to existing studies, to model rough surfaces we applied a statistical

<figure><img src="image_10.png" /><figcaption>FIG. 7. (a)–(c) The distributions of integrated absorbed energy $Q_i$ and (d)–(f) their spectra for a circularly polarized laser pulse interacting with rough surfaces described by an asymmetric Gaussian correlation function with fixed rms height $\sigma = 50$ nm and y correlation length $\xi_y = 100$ nm but with different x correlation lengths: $\xi_x = 100$ nm (top), $\xi_x = 200$ nm (middle), and $\xi_x = 400$ nm (bottom). The arrow shows the direction of laser polarization.</figcaption></figure>

description of the distribution of their heights. This approach allowed us to investigate the distribution of the absorbed laser energy across the surface selvedge. As a result, we discovered that the high-frequency (type-s) and low-frequency (type-r) components of LIPSS morphology emerge in distinct surface layers: Type-s features predominantly appear in the upper edge, while type-r features are mainly present in the lower edge of the surface. We investigated how different statistical properties of surface roughness affect the distribution of absorbed laser energy. Our findings demonstrate that as the correlation length increases, the pattern of absorbed energy spots transitions from one aligned with the laser polarization to a specklelike distribution. Moreover, we demonstrated that an increase in the rms height of the surface roughness leads to the suppression of the type-r LIPSS features up to their complete disappearance. We showed that the lack of a dedicated direction of the electric field vector in circularly polarized pulses leads to the formation of a radially symmetric distribution of the absorbed energy. The resulting LIPSSs can be expected to inherit this symmetry and form a pattern without
any preferential orientation. The symmetry breaking induced by the roughness under circularly polarized laser pulses can be exploited to detect and measure asymmetries in surface roughness. In the case of circular polarization, changes in the scale of surface inhomogeneities lead to a directly proportional deformation of the absorbed energy spectrum. Our findings allowed us to deepen the understanding of the physics behind the formation of LIPSSs and offer valuable guidance for precision laser-induced surface fabrication, with potential

applications ranging from tailored photonics devices to advanced materials design.

## ACKNOWLEDGMENTS

This work was supported by the LABEX MANUTECHSISE (ANR-10-LABX-0075) of the Université de Lyon, within the Plan France 2030 operated by the French National Research Agency (ANR).

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