# Optical limiting and nonlinear optical properties of Cr2O3 and WO3 based polymer nanocomposites

O. Muller, M. Guerchoux, Pierre Gibot, L. Merlat, D. Spitzer

► To cite this version:

O. Muller, M. Guerchoux, Pierre Gibot, L. Merlat, D. Spitzer. Optical limiting and nonlinear optical properties of Cr2O3 and WO3 based polymer nanocomposites. Optics Continuum, 2022, 1 (11), pp.2389. 10.1364/OPTCON.474445. hal-04245171

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# Optical limiting and nonlinear optical properties of Cr₂O₃ and WO₃ based polymer nanocomposites

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O. MULLER*<fnref n="1" />, M. GUERCHOUX<fnref n="1" />, P. GIBOT<fnref n="2" />, L. MERLAT<fnref n="1" /> AND D. SPITZER<fnref n="2" />

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**Abstract:** The coming decades will be the scene of a massive deployment of laser systems both in civil society and for military purposes. Emerging new technologies have to face strong challenges on their ability to become more efficient and faster in response time. Whilst nothing can travel faster than the speed of light, photons technologies like lasers might become ubiquitous and swarms of lasers emitting harmful radiations will become the center of a growing interest. At the other end of the segment, the complementary activity is to enable for efficient, tough versatile protection methods toward the laser weaponry. Nonlinear optical nanomaterials can serve this purpose when properly embedded in a solid medium, resulting in nanocomposite passive optical limiting filters. The present study relies on the passive laser protection concept where materials with nonlinear optical properties are used, that are self-activated when the incident laser threat is above a certain level of intensity. Nanomaterials are tailored for optical limitation due to their versatility, their strong nonlinear absorption properties and their uniform transmittance characteristics over a broad spectral range, the latter property avoiding color distortions to the observation device. The optical limiting behavior of templated transition metals nanoparticles, Cr₂O₃ and WO₃ in a PMMA host at the wavelength of 1064 nm in the nanosecond regime is discussed. The optical filter were produced by chemical synthesis from the bulk. The optical limiting properties were characterized using an adequate custom-made optical setup and the third order nonlinear parameters, namely the nonlinear absorption coefficients and refractive indices were measured by the Z-scan method. The optical limiting performance improvement is clearly demonstrated for the PMMA/Cr₂O₃ filter bearing out a laser protection level of OD=2.0, a factor of 2 larger than the pure PMMA filter. A significant blue shift in the nonlinear activation threshold energy occurs when WO₃ or Cr₂O₃ are embedded in a PMMA host as the values have been subsequently pulled down from 500 µJ (pure PMMA) to 65 µJ and 23µJ, respectively. Z-scan measurements highlighted a self-defocusing effect as a result of a negative nonlinearity. Nonlinear refractive indexes in the order of n₂ = -1.0 X 10⁻¹⁵ cm<fnref n="2" />/W were calculated for the PMMA/Cr₂O₃ and PMMA/WO₃ systems and n₂ = -2.2 X 10⁻¹⁷ cm<fnref n="2" />/W for the pure PMMA. A nonlinear absorption coefficient as high as β = 163 cm/GW was measured for the PMMA/Cr₂O₃ optical limiting filter while the one for the pure PMMA lies 2 orders of magnitude behind. It is suggested that the PMMA/Cr₂O₃ optical system undergoes reverse saturable absorption enhanced by excited state absorption (ESA/RSA). Besides, it is believed that multi-photons absorption (MPA) occurs in PMMA/WO₃ or pure PMMA.

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<footnote n="1">¹Laboratory for Radiation Interaction with Matter, French-German Research Institute of Saint-Louis (ISL), 5 rue du Général Cassagnou, 68301 Saint-Louis, France</footnote>

<footnote n="2">²Laboratoire des Nanomatériaux pour Systèmes Sous Sollicitations Extrêmes (NS3E), CNRS/ISL/UNISTRA UMR 3208, French-German Research Institute of Saint-Louis (ISL), 5 rue du Général Cassagnou, 68301 Saint-Louis, France</footnote>

<footnote>* Corresponding author: olivier.muller@isl.eu; phone +33 3 89 69 58 59; www.isl.eu</footnote>
## 1. Introduction

Recent advances in laser technology and the miniaturization of optical devices have ushered in a new era of efficiency for the military as well as for the civilian laser systems. Visible and IR light receptors are capable of detecting electromagnetic radiation at various intensity levels and at various wavelengths in the spectral region from approximately 400 nm to 5000 nm. Examples of such light receptors include but are not limited to the human eye and optical detectors/sensors which produce a response whenever illuminated. The human eye, optical detectors/sensors and photo-receptors can be damaged by exposure to high intensity lasers. For example, optical detectors can be exposed beyond their capabilities and destroyed by either continuous or short duration exposure to a laser beam. Similarly, the retina of the eye can be damaged by being exposed to a laser beam for only a brief period of time.

The objective of this work is to synthesize nonlinear optical polymer nanocomposite filters based on $Cr_2O_3$ and $WO_3$ bearing out optical limiting properties. The use of nonlinear optical materials offers the possibility to design passive protection filters, i.e., those activated by the incoming radiation itself, in contrast to active systems, trickier in their elaboration, where the input signal has to be controlled by an additional element [1]. Obviously speaking, an optical limiting filter must fulfill the following conditions: possess a fast response time, exhibit a low nonlinear threshold, be neutral in color, present a low insertion loss (i.e., a high linear transmittance) and cover a large spectral range. Poly(methyl methacrylate) (PMMA) is considered as the host material in this study.

Solid-state optical limiting nanocomposite filters consist frequently of PMMA as the host material, which is a widely used polymeric material in the fields of defense and aerospace as well as in the field of laser protection, see e.g. [2-5]. PMMA is the most common organic glass, it exhibits excellent properties such as small chromatic dispersion, high linear transmittance in the VIS, NIR and the SWIR part of the spectrum. Metal oxide nanomaterials formed from transition metals like $Cr_2O_3$ and $WO_3$ are known to be p-type and n-type semiconductors, respectively. Chromium and tungsten, referred in the group 6B elements have interesting electronic band structures ruling out their chemical and physical properties. Actually, they form partially filled d-orbitals which will extend up to the conduction band and even exceed the Fermi level, so they present a complete overlap of the conduction band by the valence band. Accordingly, interband electronic transitions can be readily assumed in the VIS up to the NIR part of the electromagnetic spectrum in the case of $Cr_2O_3$ and $WO_3$. Furthermore, these appropriate interband transitions will promote (nonlinear) absorption effects and contribute to the dielectric function of the materials which outlines the manner the electromagnetic radiations act on the electrons involved. Regarding their promising electronic properties, $Cr_2O_3$ and $WO_3$ should be reasonably considered as good candidates for optical limiting applications where nonlinear absorption effects are the basic prerequisites. Although the research studies relating on the electronic and optical properties of $Cr_2O_3$ and $WO_3$ [6-9] (among others), and those reporting on their optical limiting behavior [10-20] are numerous, to the best of the authors knowledge there is no work describing the nonlinear optical properties as well as the optical limiting behavior of nanocomposites made of $Cr_2O_3$ and $WO_3$ nanofillers embedded in a PMMA matrix.

This work focuses on the nonlinear optical effects resulting from pulsed laser radiations in the near infrared at the wavelength of 1064 nm at the nanosecond timescale. Actually, harmful radiations are often due to laser target designators emitting invisible beams (to the eyes) at the wavelength of 1064 nm [21,22]. In section 2 we expose the experimental principles of this work. In section 3, resulting from nonlinear transmittance measurements, the filters performance are given in terms of their global nonlinear attenuation. In a further step, by means of Z-scan assessments we study the nonlinear absorption and the nonlinear refraction of the optical limiters. Section 4 gives the conclusion of this work.
## 2. Experimental section

### 2.1 Chemical synthesis

Templated $Cr_2O_3$ and $WO_3$ nanoparticles were synthesized following the procedure described in [20, 23, 24]. Succinctly, the synthesis of chromium (III) oxide and tungsten(VI) oxide nanoparticles were carried out as follow: metal salts (Cr, W) were dissolved in an aqueous dispersion of nanosized $SiO_2$ particles (Ludox®) according to a silica/metal precursor weight ratio of 2/3. The resulting mixtures were stirred for 1 h to achieve homogeneous suspensions, placed in an oven at 80 °C to obtain total evaporation of the aqueous phase and then grounded in a mortar. The powdered samples were calcinated 2 h at a heating rate of 2 °C/min at 550 °C and 600 °C for the Cr-based and W-based composites, respectively. The pristine Cr- and W-based composites were washed twice with hydrofluoric acid (10 wt%) to remove the silica nanoparticles. The ceramic materials were separated to the acid phase via centrifugation and washed with distilled water then once with acetone. Finally, the green (Cr) and yellow (W) oxide materials were dried in an oven at 100 °C overnight. Methyl methacrylate (MMA) monomer (5 mL, 4.7 g, purchased from Carl Roth) and the composites nanoparticles were mixed in 20 mL melted glass. The same mass fraction was used for $Cr_2O_3$ and $WO_3$: mother solutions with a nanoparticle load of 0.025 wt% were prepared and subsequently diluted to obtain the final mass fraction of 0.0025 wt%. Both resulting solutions were mixed in an ultrasonic bath during 10 min. A micro-magnetic bar was added to homogenize the solutions during the reaction. Then, the AIBN initiator was added at 0.4wt%. To efficiently disperse the AIBN, a Vortexer was used during 1 min. In a further step, septum was used to keep closed mixtures during the reaction. These latter were put in an ice bath to avoid the MMA monomer from evaporating before it was degased with nitrogen for one minute. During the process the solutions were surmounted with a nitrogen bag. Mixtures were then heated up, maintained at 50 °C in an oil bath and simultaneously stirred. When solutions appeared to be too viscous, the micro-magnetic bar was removed and the melted glass was put in an oven at 50°C until a solid bulk was obtained. The glass was opened to allow the unreacted MMA monomer (mainly located in the top layer above the polymer) to escape. Finally, the melted glass was broken and separated from the solid polymer before it was polished to obtain optical-grade quality filters. The 2.5 mm thick resulting laser protection filters are shown on figure 1.

<figure><img src="image_2.png" /><figcaption>Fig. 1: Optical limiting filters, from the left to the right respectively, pure PMMA, PMMA/$Cr_2O_3$ and PMMA/$WO_3$. Both pictures on the right highlight the Trommsdorff effect.</figcaption></figure>

Featured on both photographs on the right of figure 1, the conical texture observed on the periphery of the optical filters is due to the Trommsdorff effect [25], often occurring in free radical bulk polymerizations. In brief, the Trommsdorff effect is a side reaction leading to a dramatic reduction of the termination reaction rates due to the diffusion limitation and the local rise in viscosity of the polymerizing arrangement. As a consequence of the lack of
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reaction obstacles, the polymerization rate is significantly increased, therefore leading in uncontrollable reactions.

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## 2.2 Linear optical properties

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The linear transmittance of the optical filters as well as their linear absorbance are shown on figures 2 and 3, respectively. The measurements were carried out by using a double beam UV/VIS/SWIR spectrometer Agilent Cary 7000 in the wavelength range [250 nm-1100 nm] and [220 nm-600 nm], respectively. It is obvious to mention that the presence of Cr₂O₃ and WO₃ nanoparticles significantly affect the linear optical properties of PMMA as it can be seen on the figures 2 & 3. Nevertheless, in the wavelength range of interest, the linear transmittance of the optical filters can be considered as broadband and the value of T=90 % at 1064 nm meets the specifications for an optical limiting filter.

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

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<caption>Fig. 2: Linear transmittance spectra of PMMA/Cr₂O₃ and PMMA/WO₃ optical limiting filters. Comparison with pure PMMA.</caption>
<figure><img src="image_4.png" /><figcaption>Figure 3: Absorption spectra of PMMA/Cr₂O₃ and PMMA/WO₃ optical limiting filters. Comparison with pure PMMA.</figcaption></figure>

As it is revealed on figures 2 & 3, the PMMA/Cr₂O₃ filter shows a strong absorption in the violet around 350 nm in accordance with the greenish color of the optical filter. It is well known that the optical properties of Cr³⁺ ions in Cr₂O₃ are due to d-d electronic transitions. Actually, 2 well defined bands can be observed at 460 nm and 600 nm in agreement with the work of Liang et al. [26]. These bands may be assigned to the $^4A_{2g} \rightarrow ^4T_{1g}$ transitions of the six coordinate geometry and the $^4A_{2g} \rightarrow ^4T_{2g}$ of Cr³⁺ ions in an octahedral environment [26]. On the other hand, the PMMA/WO₃ system exhibits quasi-monotonic transmittance and absorbance behaviors which might be related to the symmetrical nature of tungsten trioxide monoclinic crystal lattice structure and the absence of specific intragap defects. The well resolved sharp edge between 300 nm and 400 nm is to be imparted to the presence of oxygen ions vacancies in the structure of WO₃ [27]. The slight bluish color of the PMMA/WO₃ filter (figure 1) is due to gap narrowing as a consequence of these O vacancies.

Investigations on the intrinsic absorption edge of the optical limiting filters in figure 3 enable to determine the optical band gap of the materials according to the absorption spectrum fitting method using the Tauc model well described in [28]. Taking into account indirect band to band transitions, the Tauc plot, i.e. the plot $(\text{Absorbance} / \lambda)^{1/m}$ versus $1/\lambda$ is expected to show a linear behavior in the high energy region which corresponds to a strong absorption near the absorption edge. By extrapolating the linear portion of this straight line to zero absorption edge provides the optical energy band gap, Eg of the polymer based optical filters. It is observed that the best fitting procedure is obtained for $m = 2$, therefore consistent with allowed indirect $\pi^* - \pi$ electronic transitions in material structures for which the top of the valence band is not at the same wave vector point as the bottom of the conduction band. The value of Eg is easily deduced by multiplying the extrapolated abscissa by 1239.83 (the combination of the Planck's constant and the velocity of light). The results can be seen on the figure 4 and the Eg values are summarized in table 1.
<figure><img src="image_5.png" /><figcaption>Figure 4: Assessment of the optical band gaps. Tauc plots for PMMA/Cr₂O₃, PMMA/WO₃ and pure PMMA optical limiting filters.</figcaption></figure>

<table><caption>Table 1: Optical band gaps for PMMA based optical limiting filters.</caption><thead><tr><th></th><th>pure PMMA</th><th>PMMA/Cr<sub>2</sub>O<sub>3</sub></th><th>PMMA/WO<sub>3</sub></th></tr></thead><tbody><tr><td>E<sub>g</sub> (eV)</td><td>4.6</td><td>3.7</td><td>3.5</td></tr></tbody></table>

It appears noteworthy that pure PMMA exhibits the highest Eg value of 4.6 eV in consistency with its insulator character. Our experimental value is in accordance with the one determined by Jasim et al., Eg=4.6 eV [29], and the one obtained in [30], Eg=4.9 eV. Our experimental findings on figure 4 and table 1 indicate that the optical band gap of pure PMMA is decreased due to the presence of semiconducting nanoparticles like Cr₂O₃ and WO₃. All synthesized nanocomposites were characterized by a lower optical energy gap Eg than the matrix polymer material as already observed by Matysiak et al. [31]. Indeed, energy band gaps of 3.7 eV and 3.5 eV stem from our experimental work in the case of PMMA/Cr₂O₃ and PMMA/WO₃, respectively. To the best appreciation of the authors, there is a strict lack of Eg data in the literature for PMMA nanocomposites made of Cr₂O₃ or WO₃ loads. Albeit our values cannot be strictly compared to pure compounds, Abdullah et al. calculated a value Eg=3.2 eV for as-grown Cr₂O₃ nanostructures [32], whereas regarding WO₃ thin films, Sivakumar et al. obtained Eg values ranging from 2.7 eV to 3.1 eV [27], both of which are in harmony with our findings.

## 2.3 Optical setup and measurement procedure

The laser system used for this study is a Q-switched Nd-YAG (Quantel) working at the wavelength of 1064 nm with a repetition rate fixed to 1 Hz and a pulse width of nearly 4 ns. The experimental setup to assess the nonlinear transmittance is sketched in figure 5 and the detailed experimental process is given in [1].
<figure><img src="image_6.png" /><figcaption>Fig. 5: Experimental setup used to study the optical limiting behavior of nonlinear filters. Details on the setup given in [1].</figcaption></figure>

One part of the incident laser beam is taken from the main setup right before the Keplerian telescope (see figure 5) and directed toward the Z-scan experimental setup shown in figure 6. Precise informations on the measurement procedure can be read in [1].

<figure><img src="image_7.png" /><figcaption>Fig. 6: From [1]. Z-scan optical setup in an open and close aperture scheme. L₄, plano-convex lens, f₄ = 200 mm; NDF, neutral density filters; aperture hole, 200 µm.</figcaption></figure>

### 3. Results and discussion

#### 3.1 Nonlinear optical transmittance

The assessment of the transmittance as a function of the laser incident energy, in off-resonant conditions (i.e., 1064 nm), reveals the co-existence of 2 regimes governed by a linear behavior and a nonlinear one, respectively. The experimental results depicted in figure 7 at the wavelength of λ=1064 nm are supported by theoretical fitting sessions displayed in solid lines. The laser input energy was varied from 1 nJ up to ca. 4 mJ. The linear regime, where the transmittance is constant, spreads over several decades of input energies is followed by the nonlinear regime characterized by a decrease of the optical filters transmittance. It is to be noticed that the height of the plateau giving the transmittance of each filter at 1064 nm is in good agreement with the spectrometer transmittances given in figure 2. The optical limiting threshold, is a crucial parameter since it describes the efficiency of each optical limiting filter and is defined as the energy where the transmittance drops to 3 dB of its initial value. This parameter is obtained by a direct lecture on the graphs of figure 7 and should be appropriately compared with the energy thresholds resulting from theoretical fitting whose depending on
225 more complex constraints. Readily, those are 23 µJ, 65 µJ and 500 µJ for the PMMA/Cr₂O₃,
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227 Obviously, PMMA/Cr₂O₃ optical filter shows the most favorable optical limiting
228 performance. Indeed, it presents the sharpest decrease of the transmittance, as high as 8 dB
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230 expected, the worse achievement concerns the pure PMMA optical filter. At the highest
231 incident energy, a direct lecture on figure 7 brings the global filters attenuation to be OD=2.0,
232 OD=1.0 and OD=0.9, for the PMMA/Cr₂O₃, PMMA/WO₃ and pure PMMA, respectively.

<figure><img src="image_8.png" /><figcaption>Fig. 7: Transmittance as a function of the input energy displayed in a log-log plot for the 3 filter systems under investigation. Solid lines result from theoretical fitting. Laser wavelength is 1064 nm. Further details are mentioned in the text.</figcaption></figure>

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237 It is firmly to be expected that molecular arrangements involving transition metals, like
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239 absorption (ESA/RSA) at an appropriate laser energy level. Experimental findings in several
240 research studies tend to confirm our statement [5, 33-35]. In [5], Liao et al. demonstrated that
241 the incorporation of MoS₂ nanostructures in a PMMA host offered incentives to promote
242 RSA as the nonlinear optical limiting behavior. Varma et al. in their research article [33],
243 described a superior optical limiting material based on Ti, another transition metal and
244 ascribed the nonlinear effects to multi-photons and the induced excited state absorptions.
245 Also, Sharma et al. [34, 35] investigating hybrid nanostructures of NiCo₂O₄ observed that
246 their nonlinear behavior was mediated by ESA/RSA. It is likely that ESA/RSA in
247 PMMA/Cr₂O₃ occurs in a 2 steps process: first, interband transitions from the valence band
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249 conduction band. Basically, in the case of Cr₂O₃, the valence band is mainly composed of Cr
250 3d and O 2p states, whereas the bottom of the conduction band is filled by Cr 3d states [36].
251 Hence, the photoinduced electrons may hope from the valence into the conduction band between the O 2p orbitals and the Cr 3d orbitals.

253 To verify the aforementioned claims, the transmission was numerically solved by using a theoretical model based on ESA/RSA. Briefly speaking, this model assumes that the absorption cross section is a function of the laser energy density E travelling (along z) through a nonlinear absorptive medium which can be written as:

$$257 \qquad \frac{dE}{dz} = -\sigma E N_0 - \mu_1 E^2 N_0 \qquad (1)$$

258 Where $\sigma_0$ denotes the ground state absorption cross section, $\mu_1$ is the absorption cross section of the first excited singlet state and $N_0$ is the density of state. $\sigma_0 N_0$ and $\mu_1 N_0$ are the linear and nonlinear absorption coefficients, respectively.

261 Equation (1) can be easily integrated and one obtains,

$$262 \qquad E_{out} = T_0 E_{in} / [1 + (1 - T_0) E_{in} / E_{th}] \qquad (2)$$

263 In relation (2), $E_{in}$, $E_{out}$ and $E_{th}$ represent the input, output and threshold energy or energy density, respectively. $T_0$ is the linear transmittance.

265 The equation in relation (2) assuming ESA / RSA phenomena in the molecular medium has been used for theoretical fittings of the transmittance dependence on the laser input energy of figure 7.

268 The theory explaining the nonlinear optical limiting phenomena in PMMA/WO₃ or pure PMMA is somewhat different and we believe it originate from multi-photons absorption mechanisms (MPA). By virtue of the formalism described as given example in [37], the transmittance T as a function of the input energy or energy density is given by relation (3),

$$272 \qquad T = T_0 \left[ C \exp \left\{ - \left( \frac{Ein}{Eth} \right)^{1/2} \right\} + (1-C) \left( 1 - \exp \left\{ - \left( \frac{Ein}{Eth} \right)^{-1/2} \right\} \right) \right] \qquad (3)$$

273 In (3), C designates a constant related to the nonlinear attenuation of the materials (not necessarily pure nonlinear absorption). As shown on the line plots of figure 7, it is very interesting to notice that the MPA formalism well fits with the experimental data for PMMA/WO₃ or pure PMMA. Furthermore, equations (2) and (3) allow to estimate the threshold energy $E_{th}$, as it is considered as a fitting parameter. Results are shown below in the table 2. It is worth noting that the experimental and theoretical nonlinear thresholds for optical limiting behave in the same order of magnitude except in the case of PMMA/Cr₂O₃ optical limiting filter.

<table><caption>Table 2: Optical limiting thresholds data. Experimental assessment by a lecture on figure 7 and results from theoretical modeling.</caption><thead><tr><td>Optical Limiting thresholds (μJ)</td><td>Experimental</td><td>Theoretical Fitting</td></tr></thead><tbody><tr><td>PMMA/Cr<sub>2</sub>O<sub>3</sub></td><td>23</td><td>4</td></tr><tr><td>PMMA/WO<sub>3</sub></td><td>65</td><td>72</td></tr></tbody></table>
<table><tr><td>pure PMMA</td><td>500</td><td>490</td></tr></table>

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## 3.2 Nonlinear absorption and nonlinear refraction

Nonlinear absorption and nonlinear refraction in optical materials can be both quantified using the Z-scan technique which is a sensitive and reliable characterization method. This technique is meticulously described in the work of Sheik-Bahae et al. [37]. In order to investigate the nonlinear refraction in our samples, the Z-scan in its closed aperture scheme presented on figure 6 was used. Open Z-scan transmittance measurements hinting at the nonlinear absorption coefficients were also performed. The sensitivity of the measurement depends on the aperture factor S which has to be carefully defined. The appropriate value of S in our investigations is S = 2 %. The close Z-scan experimental traces of the PMMA/Cr₂O₃ and PMMA/WO₃ optical limiting filters were produced at an irradiance level of

$I_0 = 5.10^{10} W/cm^2$ ($E_{in} = 100 \mu J$) and $I_0 = 5.10^{11} W/cm^2$ ($E_{in} = 1 mJ$) for the pure PMMA filter. The reason for this discrepancy is blindingly obvious since a Z-scan experiment has to be related to the nonlinear behavior of the material. The nonlinear transmittance assessments of figure 7 is essential since it helps to find the dichotomy between the linear and the nonlinear regime. Moreover, we carefully controlled the absence of laser damage in the filters at those irradiances (energies). Experimental and theoretical fitting results are displayed on the figure 8. Basically, the trend observed for the close Z-scan experiments of figure 8 confirms the nonlinear behavior exposed in figure 7, thereby the most pronounced nonlinear effects occur in the PMMA/Cr₂O₃ system.

<figure><img src="image_9.png" /></figure>

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Figure 8: Close Z-scan results for the PMMA/Cr₂O₃ and PMMA/WO₃ optical limiting filters at an incident laser irradiance $I_0 = 5.10^{10} W/cm^2$ and pure PMMA at an incident laser irradiance $I_0 = 5.10^{11} W/cm^2$. The solid lines denote the theoretical fittings of equation (7). Laser wavelength is 1064 nm.

The theoretical fits on figure 8 result from the equations modeling the normalized transmittance when self-focusing or self-defocusing occur whereby the nonlinear refractive index, $n_2$, might be deducted. When a third order nonlinearity is accounted for, the refractive index $n$ can be expressed as:

$$n = n_0 + n_2 \cdot I = n_0 + \Delta n \quad (4),$$

where $I$ denotes the peak irradiance, $I = I_0 = 5.10^{10} W/cm^2$ or $5.10^{11} W/cm^2$ and $n_0$ is the linear refraction coefficient.

The expression for the normalized transmittance, $T(z)$, in our case can be written as:

$$T(z) = 1 + \frac{4a}{(a^2 + 9)(a^2 + 1)} \Delta\Phi_0 - \frac{2(a^2 + 3)}{(a^2 + 9)(a^2 + 1)} \Delta\Psi_0 \quad (5).$$

In relation (5), $a$ denotes the reduced displacement and is expressed as $a = z/z_0$ ($z$ is the linear displacement). $\Delta\Phi_0$ and $\Delta\Psi_0$ are major parameters that characterize the wave distortion and the subsequent phase shift around the focal point due to nonlinear refraction and nonlinear absorption, respectively. For both terms we can write:

$$\Delta\Phi_0 = kn_2 I_0 L_{eff} \quad (6),$$

$$\Delta\Psi_0 = \beta I_0 L_{eff} / 2 \quad (7).$$

The observed peak to valley shapes of figure 8 indicate a self-defocusing effect as a result of a negative nonlinearity. Consequently, it is reasonable to argue for a thermally-induced $n_2$ refractive index change independently of the nature of the optical limiting filter.

From evidence, strong nonlinear refractive effects occur in the PMMA/Cr₂O₃ system, whereas it can be seen that the PMMA/WO₃ optical limiting filter exhibits an asymmetric shape like a reduced peak and a pronounced valley. This very interesting observation points to the contribution of nonlinear absorption in accordance with the observations of Sheik-Bahae et al. in [37]. Actually, these authors demonstrated that nonlinear absorption like two-photon absorption (TPA) or multiphoton absorption suppresses the peak and enhances the valley in a close Z-scan scheme. The occurrence of TPA in PMMA/WO₃ is to be ruled out regarding its optical band gap, $E_g = 3.5$ eV, given in table 1. Yet, a MPA process like 3 photons absorption is more likely to occur at an incident photons energy of 1.1 eV (i.e., 1064 nm). As in the case of PMMA/Cr₂O₃, the close Z-scan findings for PMMA/WO₃ well match the estimated fittings of the nonlinear transmittance revealing MPA as the major nonlinear phenomenon. The $n_2$ values calculated owing to relations (5) and (6) are $n_2 = -1.6 \times 10^{-15} cm^2/W$ for PMMA/Cr₂O₃, $n_2 = -7.1 \times 10^{-16} cm^2/W$ for PMMA/WO₃ and $n_2 = -2.2 \times 10^{-17} cm^2/W$ for the pure PMMA (see also table 3).
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The thermal nature of the nonlinearity originates from laser-induced heating subsequent to the absorption of the tightly focused laser beam. Accordingly, a strong temperature gradient occurs in conjunction with a severe refractive index change leading to a thermal lensing effect as well as a markedly phase distortion $\Delta\Phi_0$ of the propagating laser radiation. Such an effect may consist of a fast process of acoustic wave propagation and a slow steady state variation of the suspension density due to the cumulative thermal heating of the absorbing area.

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It is likely that Cr₂O₃ and WO₃ nanofillers embedded in the PMMA host readily absorbs the incident laser radiations and reshuffle the energy in the surrounding medium in the form of heat. The refractive nonlinearity in PMMA/Cr₂O₃ might be connected with the cumulative contribution made by the processes of intraband transitions of free equilibrium electrons and nonlinear hyperpolarizability of bound 3d3 electrons. In a previous paper [20], the authors calculated nonlinear refractivity values $n_2 \approx 10^{-14} cm^2/W$ for Cr₂O₃ and WO₃ suspensions in chloroform, namely an order of magnitude larger than the solid-state filters of the present study under the same experimental conditions. The discrepancy is worthy if we consider that molecules embedded in a liquid present a strong contribution to the nonlinear refractive index due to molecular reorientation.

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The open Z-scan method serves to assess the nonlinear absorption coefficient, $\beta$, figure 9 exposes the experimental results. At first sight, as expected, it is noteworthy that the nanoparticle loads greatly enhance the nonlinear absorption behavior. Actually, PMMA/Cr₂O₃ and PMMA/WO₃ optical limiting filters present a broader valley jointly with a significantly deeper drop in transmittance.

<figure><img src="image_10.png" /></figure>
Figure 9: Open Z-scan traces for the PMMA/Cr₂O₃ and PMMA/WO₃ optical limiting filters at an incident laser irradiance $I_0 = 5.10^{10} W/cm^2$ and pure PMMA at an incident laser irradiance $I_0 = 5.10^{11} W/cm^2$. The solid lines denote the theoretical fittings of equation (12). Laser wavelength is 1064 nm.

The values of the nonlinear absorption coefficient $\beta$, can be easily deducted from the theory developed in [38] reporting the expression for the open-aperture normalized transmittance in an optical device:

$$T_{\text{norm}}(z) = \frac{\ln(1 + q_0(z))}{q_0(z)} \qquad (8).$$

In relation (8), $q_0(z) = \beta I(z)L_{eff}$, stands for the beam parameter that contains $\beta$ to be assessed and $I(z)$ is the magnitude of the intensity of the Gaussian laser beam travelling in the $+z$ direction. On the propagation axis for $r=0$, it can be written:

$$I(z) = \frac{I_0 e^{-\gamma}}{1 + a^2} \qquad (9).$$

In equation (9), we consider the extinction of the laser radiation in the propagating medium caused by scattering centers or defects by means of a further parameter $\gamma$. Therefore, the substitution in equation (8) brings:

$$T_{\text{norm}}(z) = \frac{(1 + a^2)^2}{\beta I_0 L_{\text{eff}} e^{-\gamma}} \ln(1 + \frac{\beta I_0 L_{\text{eff}} e^{-\gamma}}{(1 + a^2)^2}) \qquad (10).$$

The nonlinear absorption coefficients $\beta$ resulting from model fittings according to relation (10) are summarized in table 3. They vary from 1.8 cm / GW to 163 cm / GW for PMMA/WO₃ and PMMA/Cr₂O₃, respectively, while the one for pure PMMA lies 2 orders of magnitude behind. Obviously, the higher value obtained in the case of PMMA/Cr₂O₃ is to be ascribed to RSA enhanced by ESA, whilst the $\beta$ value calculated for PMMA/WO₃ validates its MPA character. It is to be pointed out that Liao et al. [5], calculated a nonlinear absorption coefficient $\beta = 297$ cm / GW in the case of a transition metal based MoS₂/PMMA nanocomposite submitted to nanosecond laser radiations at the wavelength of 532 nm in off-resonant conditions. This value is comparable to the one obtained for the PMMA/Cr₂O₃ nanocomposite. The authors in [5] suggested RSA as the major nonlinear mechanism occurring in their solid-state optical limiting filters which reasonably corroborate our statements on the PMMA/Cr₂O₃ system. One of the previous author's work [20] on the twin systems in the form of suspensions revealed a comparable $\beta$ value for the PMMA/WO₃ optical filter, whereas a pointedly lower $\beta$ value was observed for the PMMA/Cr₂O₃ suspension $\beta = 3$ cm / GW, i.e. factor of 50. This discrepancy might be clarified by the presence of intermolecular charges transfer between the PMMA host molecule and the Cr₂O₃ nanoparticle load which would have the effect of raising the contribution of the nonlinear absorption component.

<caption>Table 3: $n_2$ and $\beta$ nonlinear parameters of PMMA/Cr₂O₃, PMMA/WO₃ and pure PMMA optical limiting filters at the wavelength of 1064 nm in the nanosecond regime. Further details in the text.</caption>
<table><thead><tr><td></td><td>n<sub>2</sub> (cm<sup>-2</sup> / W)</td><td>β (cm / GW)</td></tr></thead><tbody><tr><td>PMMA/Cr<sub>2</sub>O<sub>3</sub></td><td>- 1.6 X 10<sup>-15</sup></td><td>163</td></tr><tr><td>PMMA/WO<sub>3</sub></td><td>- 7.1 X 10<sup>-16</sup></td><td>1.8</td></tr><tr><td>pure PMMA</td><td>- 2.2 X 10<sup>-17</sup></td><td>0.02</td></tr></tbody></table>

396

## 4. Conclusion

Cr₂O₃ and WO₃ templated nanoparticles were embedded in a PMMA host resulting in efficient optical limiting filters. Optical grade filters with transmittances as high as T=90 % at 1064 nm have been produced by chemical synthesis from the bulk. To the best of the author's knowledge, the nonlinear optical properties and the optical limiting behavior of Cr₂O₃ and WO₃ embedded in a PMMA matrix at the wavelength of 1064 nm in the nanosecond regime have never been reported. This work showcased the cause routing to optical limiting, more specifically, ESA/RSA in the case of the PMMA/Cr₂O₃ system and MPA for the PMMA/ WO₃ one. The optical limiting performance improvement has been demonstrated for the PMMA/Cr₂O₃ filter bearing out a laser protection level of OD=2.0, a factor of 2 larger than the pure PMMA filter. In that case, 99.99 % of the incoming laser radiations can be filtered out. A significant blue shift in the nonlinear activation threshold energy was observed when WO₃ or Cr₂O₃ are embedded in a PMMA host as the values have been subsequently pulled down from 500 µJ (pure PMMA) to 65 µJ and 23µJ, respectively. Z-scan measurements highlighted a self-defocusing effect as a result of a negative nonlinearity. Nonlinear refractive indexes in the order of n₂ = -1.0 X 10⁻¹⁵ cm²/W were calculated for the PMMA/Cr₂O₃ and PMMA/ WO₃ systems and n₂ = -2.2 X 10⁻¹⁷ cm²/W for the pure PMMA. A nonlinear absorption coefficient as high as β = 163 cm / GW has been assessed for the PMMA/Cr₂O₃ optical limiting filter. The ease of manufacture and the level of performance of the PMMA/Cr₂O₃ optical limiting filters make them suitable to be implemented as protection filters in electro optical systems against harmful laser radiations. This research work established consistent ways to promote new performance milestones within the field of laser protection.

## 5. Acknowledgements

The authors gratefully acknowledge the German Bundesamt für Ausrüstung, Informationstechnik und Nutzung der Bundeswehr (BAAINBw, Germany) and the French Direction Générale de l'Armement (DGA, France) for their financial support.

## 6. Disclosures

The authors do not declare any financial, personal or professional conflicts of interest.

## 7. Data availability

The data involved in the results presented in this study are not publicly available at this time but may be obtained from the authors upon request.

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