# Flexible determination of the nonlinear refractive index or pulse duration in the visible range using Z-scan-based technique: application to fused silica

Georges Boudebs, Julien-Bilal Zinoune, Christophe Cassagne, Cyril Mauclair

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Georges Boudebs, Julien-Bilal Zinoune, Christophe Cassagne, Cyril Mauclair. Flexible determination of the nonlinear refractive index or pulse duration in the visible range using Z-scan-based technique: application to fused silica. Applied Physics B - Laser and Optics, 2025, 131 (7), pp.139. 10.1007/s00340-025-08502-2. hal-05116972

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https://univ-angers.hal.science/hal-05116972v1

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# Flexible determination of the nonlinear refractive index or pulse duration in the visible range using Z-scan-based technique: application to fused silica

Georges Boudebs<fnref n="1" /> · Julien-Bilal Zinoune<fnref n="1" /> · Christophe Cassagne<fnref n="1" /> · Cyril Mauclair<fnref n="1" />

Received: 21 April 2025 / Accepted: 26 May 2025

© The Author(s) 2025

## Abstract

We present a flexible and cost-effective Z-scan-based method, referred to as Beam Size Relative Variation (BSRV), for the determination of either the nonlinear refractive index $n_2$ or the femtosecond pulse duration $\tau$ in the visible spectral range (650–900 nm). The technique, implemented using a tunable optical parametric amplifier (OPA), accommodates arbitrary beam profiles, thus removing the constraints imposed by traditional Gaussian or circular symmetry assumptions. Spatial characterization is performed via CCD imaging, while temporal information can be inferred directly from Z-scan measurements when a reference material is used. As a proof of concept, the method is applied to fused silica, a widely used reference material in nonlinear optics. A total of 108 Z-scan measurements were conducted across the spectral range, allowing for a robust statistical analysis of the extracted parameters. A new reference value is established for fused silica: $n_2 = (1.8 \pm 0.4) \times 10^{-20} m^2/W$. Notably, the pulse durations extracted from BSRV Z-scan data are statistically validated through independent measurements based on two-photon absorption (2PA) autocorrelation, with both methods revealing consistent spectral trends. All simulation tools and analysis codes are provided in the Supplementary Information to ensure reproducibility. This approach offers a reliable and accessible framework for nonlinear optical characterization, particularly in experimental contexts where conventional autocorrelation systems may be unavailable or cost prohibitive.

## 1 Introduction

Accurately measuring laser intensity in the focal plane of a lens remains a significant challenge in optical physics and photonics, despite the critical importance of such measurements for both fundamental research and practical applications. The focal region of a laser beam is characterized by extremely high intensities and complex spatial distributions, which are influenced by diffraction effects, lens aberrations, and wavefront distortions [1, 2]. These factors complicate the precise quantification of intensity, as even minor deviations in measurement can lead to significant errors in interpreting experimental results. For instance, the spatial and temporal profiles of the laser beam must be carefully characterized to avoid underestimating or overestimating peak intensities, which are crucial for understanding nonlinear (NL) optical phenomena [3].

The need for high-precision intensity measurements is particularly pronounced in the field of NL optics, where the intensity of light directly governs material responses such as NL refractive indices ($n_2$) and NL absorption coefficients ($\beta$). These parameters are essential for characterizing the behavior of materials under intense laser irradiation and are widely used in the design and optimization of photonic devices, including optical limiters, switches, and frequency converters [4]. For example, the Z-scan technique [3], a widely used method for measuring NL optical properties, relies heavily on accurate intensity measurements to determine $n_2$ and $\beta$ with high fidelity [5]. Any inaccuracies in intensity measurement can lead to erroneous conclusions about material properties, potentially hindering the development of advanced optical technologies.

Beyond NL optics, precise laser intensity measurements are necessary in a variety of applications. In laser micromachining, for instance, the ability to control and measure intensity distributions in the focal plane is critical for achieving high precision and minimizing thermal damage to surrounding materials [6]. Similarly, in biomedical imaging techniques such as multiphoton microscopy, accurate

<footnote>Georges Boudebs
georges.boudebs@univ-angers.fr</footnote>

<footnote n="1">¹ Univ Angers, LPHIA, SFR MATRIX, 49000 Angers, France</footnote>
intensity measurements are necessary to optimize imaging resolution and reduce photodamage to biological samples [7]. Furthermore, in the field of ultrafast laser spectroscopy, precise intensity measurements enable researchers to study dynamic processes in materials with unprecedented temporal and spatial resolution [8].

Fused silica stands out as one of the most extensively studied due to its widespread use as a reference standard for characterizing NLO properties [9]. To address these challenges, a set of best practices has emerged for conducting and reporting NLO measurements [10]. Despite its apparent simplicity, accurately determining $n_2$ of fused silica remains a complex and nuanced task, as evidenced by the significant discrepancies reported in Table 1 of [10]. As discussed in [11], accounting for the nuclear (Raman) contribution to the measured NL refractive index of fused silica [12]—dominant in picosecond and nanosecond regimes—yields a bound-electronic nonlinearity of $n_2 \approx 2 \times 10^{-20} m^2/W$ in the near-infrared. These challenges not only reveal the complexities of NL optical measurements but also highlight the broader consequences of uncertain reference values for material characterization. Optical parametric amplifiers (OPAs) are widely used for wavelength-tunable ultrafast laser applications. However, their output exhibits spatiotemporal coupling effects that complicate beam characterization and focal spot optimization. Such tunable sources have therefore been primarily used in open-aperture Z-scan configurations to measure 2PA spectra at discrete wavelengths [13, 14], or alternatively by generating a white-light continuum to enable broadband excitation in a single scan [15–17]. Extending this method to materials with significantly lower nonlinearities as for silica poses a challenge, as it requires higher spectral energy densities in the WLC to achieve reliable signal levels for NL refraction characterization [18]. When tuning the wavelength ($\lambda$) at the output of an OPA, different key phenomena arise as spatial beam centroid shift where the effective beam center moves due to angular dispersion or misalignment inside the laser. Moreover, the beam may deviate from an ideal Gaussian, adopting top-hat or structured intensity distributions with a truncated by a circular aperture gaussian beams. Pulse wings or satellite peaks emerge at certain $\lambda$, altering the effective pulse duration and peak intensity. Due to chromatic dispersion, the focusing lens produces wavelength-dependent focal shifts, which reshape the intensity distribution in the focal region. Additionally, thermal effects, which can arise from high repetition rates or long pulse durations, must also be carefully managed to avoid masking the intrinsic NLO electronic response [19, 20].

We investigate discrepancies in fused silica's $n_2$ measurements using a CCD camera, custom autocorrelator, and wavelength-resolved statistical analysis. The developed non-Gaussian beam method directly measures focal peak

intensity, eliminating reference-sample calibrations. The experiments were conducted using an adapted D4$\sigma$ method [21] to simplify the simulation required for fitting the measurements and to mitigate chromatic aberration effects. This technique enables absolute measurements of the third-order refractive index [22] with high sensitivity, while optimizing the signal-to-noise ratio [23], as it leverages the continuous spatial information contained in the entire diffracted beam recorded by the CCD not just the portion transmitted through the closed aperture, as in conventional Z-scan methods. By establishing fused silica as a robust standard and providing open-access implementation codes, this work advances the precision of NL optical characterizations and offers a practical framework for ultrafast photonics, and NL metrology. Furthermore, the study highlights the potential of the Z-scan based technique as a practical tool for estimating the effective pulse duration of laser pulses emitted from an OPA.

## 2 Experimental methods

The experimental setup employed a femtosecond laser system (Pharos) generating 1030nm fundamental-wavelength pulses with adjustable durations (< 300fs) at variable repetition rates up to 25kHz, including single-shot capability. This source was coupled to a broadband hybrid optical parametric amplifier (ORPHEUS) that produced wavelength-tunable ultrashort pulses (650 – 900nm) while maintaining pulse durations below 290fs across the entire tuning range at repetition rates from single-shot to 25kHz. To ensure clean linear polarization and enable precise intensity control, a Glan-Taylor calcite polarizer (GT10-Thorlabs, 10 mm clear aperture, uncoated) was placed immediately after the OPA.

To determine the peak laser irradiance (intensity) at a lens's focal point, three essential parameters must be evaluated: (i) the average power (to calculate energy per pulse), (ii) the effective pulse duration, and (iii) the input spatial beam profile. This section presents these measurements in two parts, providing complete temporal and spatial characterization of the incident beam. As illustrated in Fig. 1a, the experimental setup is divided into two distinct sections, visually distinguished by colored lines: red for temporal characterization and blue for spatial characterization, which will be further detailed in Sect. 2.1.

### 2.1 Temporal characterization

The average power $P$ was measured using an OPHIR Thermal Sensor (model 3A-PF-12) with spectral response from 0.15 to 20µm, and ±3% power accuracy with ±2% calibration uncertainty. The average energy per pulse was then calculated as $E = P/F$, where $F$ is the laser repetition rate set to 25kHz specifically for this measurement. All subsequent
<figure><img src="image_2.png" /><figcaption>**Fig. 1** a Schematic of the experimental setup: the red colored part represents the Michelson interferometer with its scanning mirror for autocorrelation measurements, while the blue part depicts the imaging Z-scan setup. Key components include a two-photon absorption (2PA) photodiode, a digital storage oscilloscope (DSO), and the NL material (NLM). The red dashed-line mirror is removable and can optionally be replaced with a beam splitter. **b & c** 2PA detector autocorrelation traces recorded by the DSO for $\lambda$=660 nm (**b**) and $\lambda$=880 nm (**c**)</figcaption></figure>

Z-scan based experiments were conducted at sub-Hz repetition rates using an external laser trigger to eliminate thermal effects in the test sample during Z-scan measurements.

A home-made autocorrelator based on a Michelson interferometer combined with a two-photon diode was used for measuring the ultrashort laser pulses by analyzing their intensity autocorrelation (see Fig. 1a). The input pulse was split into two replicas by a beam splitter, each traveling along separate arms with a variable time delay ($\tau$) introduced by a movable mirror with adjustable scanning speed. The two pulses were recombined and focused on a gallium phosphide (GaP), 2PA detector (model EPD-440-0/2.5), wide bandwidth response (DC to 2.5GHz) and spectral sensitivity from 190nm to 570nm. The GaP semiconductor's wide bandgap (2.26 eV) enables efficient two-photon detection while minimizing linear absorption artifacts that could distort the autocorrelation trace.

The detector generates a signal proportional to the intensity squared ($I^2$) of the overlapping pulses, as 2PA depends nonlinearly on intensity. The measured signal is: $S(\tau) \propto \int_{-\infty}^{\infty} I(t)I(t-\tau)dt$ which provides the intensity autocorrelation. The signal was acquired using a digital storage oscilloscope (DSO) (70MHz bandwidth), which allows precise recording of the autocorrelation trace. The pulse duration can

be estimated if the pulse shape is known (typically a sech²). The advantages of our technical features are the dispersion-free measurement, since the two-photon detection avoids NL crystal artifacts by second harmonic generation, the fast pulse width estimation, although without phase information, and the adjustable scanning speed, which allows optimization for different pulse duration ranges.

We determine the equivalent duration of a laser pulse with a sech² envelope as a function of its full width at half maximum (FWHM) duration. The temporal envelope of the pulse is defined as:

$$P(t) = P_{\text{peak}} \text{sech}^2\left(\frac{t}{T_0}\right) \qquad (1)$$

where $P(t)$ is the laser power at time t, $P_{\text{peak}}$ is the peak power, $T_0$ is the time parameter characterizing the pulse envelope. The full temporal width of the pulse at half-maximum $\tau_{\text{FWHM}}$ is related to $T_0$ by $\tau_{\text{FWHM}} = 2T_0\cosh^{-1}(\sqrt{2})$. The equivalent time $\tau_{\text{eq}}$ is defined as the temporal width of a rectangular pulse having the same total energy $E$ as the pulse in sech². It is possible to deduce $\tau_{\text{eq}}$ as a function of $\tau_{\text{FWHM}}$, giving:
$$ \tau_{eq} = \frac{\tau_{FWHM}}{\cosh^{-1}(\sqrt{2})} = 2T_0 = \tau_{FWHM} \times 1.135 \quad (2) $$

To illustrate the traces obtained during measurement and to assess the quality of the pulses, Fig. 1b and c present two representative acquisitions at the output of the OPA. The first trace in (b) is observed in the visible spectrum at a wavelength of 660nm, while the second trace (c) is captured in the infrared spectrum at a wavelength of 880nm. These were obtained in our experimental setup, accompanied by visual representations of the raw autocorrelation data.

The experimental data (blue circular points) were fitted with a sech² model (red solid curve), yielding FWHM durations of the autocorrelation traces measured by the DSO: $\tau_{DSO} = 448$ms for trace (b) and $\tau_{DSO} = 737$ms for trace (c) (see details in Supplementary Information (SI), sections A and B). The motorized translation stage moving one of the mirrors was set to a constant speed $v = 0.1$mm/s. The experimental autocorrelation trace widths $\tau_{DSO}$ were converted to actual pulse durations using the following physical relationship: $\tau_{FWHM}(f\bar{s}) = (2 \cdot v \cdot \tau_{DSO}/c) \times 10^{15}/1.54$ using the speed of light ($c = 3 \times 10^8$ m/s). For sech²-shaped pulses, the characteristic factor due to autocorrelation of the envelope is 1.54. As shown in Fig. 1b, we measured pulse durations of $\tau_{FWHM}(660$nm) = 194f\bar{s}$ ($R^2=0.99$) and $\tau_{FWHM}(880$nm) = 319f\bar{s}$ ($R^2=0.93$). The excellent fit quality ($R^2=0.99$), see section F in the SI) for the visible wavelength confirms reliable pulse characterization, while the reduced $R^2$ value at 880nm reflects the increased difficulty in fitting the IR pulses due to their more complex temporal structure. Notably, the autocorrelation trace in the IR spectral range (see Fig. 1c), exhibits a secondary lobe in the pulse envelope, which complicates precise duration measurement. Nevertheless, our fitting procedure yields an effective pulse width that remains consistent and operationally reliable for n₂ measurements, providing superior accuracy compared to uncritical reliance on manufacturer-reported pulse durations (< 290f\bar{s}).

The dependence of pulse duration on the OPA output wavelength is shown in Fig. 2 and was systematically measured using 5nm spectral steps. The global mean pulse duration across all wavelengths was determined to be $\langle \tau_{FWHM} \rangle = 266$fs, consistent with the manufacturer's specification of $\tau_{FWHM} < 290$fs. A sixth-degree polynomial was fitted to the data (blue solid curve, Fig. 2) to 'model' the spectral dependence:

$$ \tau_{FWHM}(\lambda) = (1.4 \times 10^{29})\lambda^6 - (6.4 \times 10^{23})\lambda^5 + (1.3 \times 10^{18})\lambda^4 - (1.3 \times 10^{12})\lambda^3 + (7.6 \times 10^5)\lambda^2 - (2.4 \times 10^{-1})\lambda + (3.0 \times 10^{-8}) \quad (3) $$

The fit exhibits a mean relative error of 31% within its 95% prediction intervals (green dashed curves,

<figure><img src="image_3.png" /><figcaption>Fig. 2 Pulse duration as a function of OPA output wavelength. The blue solid curve represents a sixth-degree polynomial fit to the experimental data, while the green dashed curves indicate the 95% prediction intervals of the fit</figcaption></figure>

$\langle \tau_{FWHM} \rangle = (266 \pm 83)f\bar{s}$ [24], reflecting high wavelength-dependent variations in pulse broadening mechanisms.

## 2.2 Spatial characterization

The characterization is carried out using a Z-scan-based technique known as the D4σ method [21] when gaussian beam are used, which we shall refer to here, more appropriately for arbitrary beam profiles, as the Beam Size Relative Variation (BSRV) method. It allows accurate third-order NL measurements, as previously reported [25–28]. It is an absolute measurement technique for characterizing NL refractive index $n_2$ and 2PA coefficient $\beta$ without requiring prior calibration. It is based on analyzing beam distortion and absorption induced by NL effects during propagation through the sample. The NL sample is translated along the beam propagation axis (z-direction) through the focal region (Fig. 3). For each position, a CCD camera records the transmitted beam profile after interaction with the sample. A Thorlabs CC126MU monochrome CMOS camera was used for beam acquisition. It features a pixel pitch of 9µm (two combined 3.45µm pixels), enabling accurate spatial resolution of the beam structure. When thermally stabilized at – 15 °C, compared to ambient temperature conditions (21 °C), the dark current is significantly reduced to 0.02 electrons/pixel/second, which ensures a high signal-to-noise ratio during 100ms exposure times. Image acquisition is digitized using a 12-bit analog-to-digital converter, providing 4095 grayscale levels for precise intensity measurements across the beam profile.
<figure><img src="image_4.png" /><figcaption>Fig. 3 Chromatic dispersion: the change of the object position and its image on the CCD because of a chromatic variation in the focal length Δf. The tested material (NLM) is placed on a translation stage at z in the focal region</figcaption></figure>

As established in prior work [22], these diffracted profile variations versus the position z of the sample directly encode both the NL phase shift and absorption dynamics. To quantitatively analyze these effects, we simulate the beam propagation from object to image planes using Fourier optics principles. Our model incorporates NL medium effects via coupled evolution equations for intensity-dependent phase shift and NL absorption (employing the thin-medium approximation) [29]. The NL refractive index ($n_2$) is then determined by optimizing the agreement between measured and simulated beam size relative variation (BSRV) profiles through least-squares fitting.

The usual setup implementing this technique uses a 4f system (more generally an afocal setup if the magnification needs to be different from 1 as in [30]). We present here an imaging concept analogous to the classical 4f setup but utilizing only a single lens instead of two. This modification significantly simplifies both experimental alignment and numerical processing while effectively better accounting for chromatic dispersion effects. As illustrated in Fig. 3, the simplified configuration positions the object at 2f in front of the lens, with the image forming at 2f behind it, while the NL medium remains near the Fourier plane as standard. This alternative geometry was specifically designed to minimize spatial delocalization effects caused by lens chromatic aberration. Geometric ray tracing in Fig. 3 demonstrates the behavior for two representative wavelengths: blue rays for visible light and red rays for infrared. A chromatic focal plane shift of Δf at the NL medium translates to 2Δf displacement at the object plane and to corresponding 2Δf shift at the image plane (CCD position). Compared to the previously used afocal configuration [21], this approach significantly reduces alignment complexity. The original 4f system requires coordinated adjustment of four degrees of freedom (two lenses) while single-lens system needs only three adjustments for precise imaging.

### 2.2.1 Theoretical considerations

The general scheme of beam propagation inside an imaging system is described in detail in [21]. The method presented here can be applied to arbitrary beam profiles, we assume that the electric field at the object plane is, $E(x, y)$, where x, y are the spatial coordinates. Let $S(u,v)$ be the spatial spectrum of $E$: $S(u,v) = \mathcal{F}[E(x,y)] = \int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} E(x,y)\exp[-j2\pi(ux + vy)]dxdy$, where $\mathcal{F}$ denotes the Fourier transform operation, u and v are the normalized spatial frequencies. S is propagated over distance $z'$, by considering the transfer function of the wave propagation phenomenon $H(u, v) = \exp\left(j\frac{2\pi z' \sqrt{1-(\lambda u)^2-(\lambda v)^2}}{\lambda}\right)$ [31].

Then the field amplitude at $z'$ is obtained by computing the inverse Fourier transform: $E(x, y, z') = \mathcal{F}^{-1}[S(u, v)H(u, v)]$. To calculate the output beam after passing through a lens with focal length $f$, we apply the phase transformation related to the thin lens thickness variation: $t_L(x, y) = \exp[-j\pi(x^2 + y^2)/(\lambda f)]$ [31]. The first propagation is performed at a distance $z' = 2f$ from the object plane to the first lens. Then we propagate the beam up to the sample located at z using $z' = f + z$ in H. Next the NL response of the material is taken into account before continuing the propagation up to the image plane (with $z' = f - z$). We assume cubic nonlinearity and a thin NL medium of thickness L exhibiting linear absorption defined by $\alpha(m^{-1})$, 2PA defined by $\beta(m/W)$ and NL refraction defined by $n_2(m^2/W)$. In these conditions the transmittance of the sample is described as [3]:

$$T(u, v, z) = \{e^{\alpha L} [1 + q(u, v, z)]\}^{-\frac{1}{2}} \exp[j\Delta\varphi_{\text{NL}}^{\text{eff}}(u, v, z)] \quad (4)$$

where $q(u, v, z) = \beta L_{\text{eff}} I(u, v, z)$ with $L_{\text{eff}} = (1 - e^{-\alpha L})/\alpha$ and $I(u, v, z)$ denotes the intensity of the laser beam within the sample. The latter quantity is related to the input electric field by $I(u, v, z) \propto |S(u, v, z)|^2$. $\Delta\varphi_{\text{NL}}^{\text{eff}}(u, v, z) = 2\pi n_2 L_{\text{eff}} I_{\text{eff}}(u, v, z)/\lambda$ is the NL phase shift where is defined the effective intensity $I_{\text{eff}}(u, v, z) = I(u, v, z)\log[1 + q(u, v, z)]/q(u, v, z)$. The on-axis NL absorption and phase shift at the focus are $q_0 = \beta L_{\text{eff}} I_0$ and $\phi_{\text{NL0}}^{\text{eff}} = \Delta\varphi_{\text{NL}}^{\text{eff}}(0, 0, 0)$, respectively, where $I_0$ is the temporal peak focal on-axis intensity. In the specific case where both $\alpha$ and $\beta$ are negligible—as is typical for glass materials—we denote the peak NL phase shift induced by the beam center ($atz = 0$) $\phi_{\text{NL0}}^{\text{eff}}$ as $\varphi_0$. We consider hereafter the low excitation regime ($q_0 \approx 0; \varphi_0 < 1$).

The beam size measurement is performed using the second order moment of the intensity profile $I(x, y)$. For example, the beam radius in the x direction is:
$$ \omega_x = 2 \sqrt{\frac{\int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} I(x, y)(x - \bar{x})^2 dx dy}{\int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} I(x, y) dx dy}}, \quad (5) $$

where

$$ \bar{x} = \frac{\int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} I(x, y) x dx dy}{\int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} I(x, y) dx dy} \quad (6) $$

is the centroid of the beam distribution. The beam waist at the entry of the setup is defined as the mean value of $\omega_x$ and $\omega_y$. The beam size relative variation is defined as $BSRV = (\omega_{NL} - \omega_L)/\omega_L$, where $\omega_{NL}$ and $\omega_L$ denote the beam sizes associated with the NL and linear profiles, respectively, as measured on the CCD (see SI, Beam Size Analysis—2nd Moment Method in Section E).

An example of the input beam profile along with the recorded Z-scan pattern is shown in Fig. 4 where the NL optical refraction was investigated in the fused silica using an 800nm pulsed beam with a 3.73mm sample thickness (L). The OPA delivered pulses of $\tau_{FWHM} = 266fs$ at less than 0.5Hz repetition rate ensuring the absence of thermal or electrostriction effects. The average power was measured at 25kHz to be 13840µW, corresponding to a pulse energy of $E = 0.55\mu J$ (one pulse per acquisition at one z position). The input beam radius ($\omega_e$) was 2.13mm, measured following Eqs. 5 and 6 (see Matlab code in BEAM SIZE ANALYSIS—2nd MOMENT METHOD in SI, section E). The beam was truncated by an iris diaphragm with an effective radius of 2.43mm, producing a nearly circular aperture. Using $f = 25cm$, the peak intensity in the material reached $I_0 = 75GW/cm^2$ calculated from images of the linear-regime measurements (red squares in Fig. 4b) according to Eq.C-1 in SI, section C. The NL phase shift $\varphi_0 = 0.405 \pm 0.012rad$ was obtained from the fitting (blue solid curve in Fig. 4b), giving a NL refractive index of $n_2 = (1.8 \pm 0.7) \times 10^{-20}m^2/W$. NL absorption was negligible, falling below the instrument's sensitivity threshold. The overall uncertainty was estimated at 36%, accounting

for contributions from both pulse duration and laser power uncertainties.

It is important to note that the spatial profile in Fig. 4a is of suboptimal quality, deviating from an ideal Gaussian distribution due to non-circularity and asymmetric features. Furthermore, the wavelength tuning of the OPA induces a shift of the beam centroid inside the circular aperture that truncates the beam, which complicates the intensity calculation. While spatial filtering could theoretically enhance beam quality, its implementation remains impractical for wavelength-tunable systems due to several critical limitations. Beam-pointing variations during spectral scanning would necessitate not only constant realignment but also frequent replacement of the spatial filter pinhole to accommodate different wavelengths. Each wavelength adjustment would require both selecting an appropriately sized pinhole and precisely realigning the spatial filter assembly. This dual requirement of hardware modification and optical realignment makes the process prohibitively time-consuming while significantly increasing the risk of the pinhole optical damage. To address these issues, we developed a robust intensity determination method (see SI, section C) for calculating the central focal intensity ($I_0$) that remains valid for arbitrary beam profiles. Section D of the SI describes the noise removal thresholding protocol, which should be applied to raw beam images to ensure accurate peak intensity determination. If the background is not properly removed using a mask adapted to the object at the input of our setup, the resulting FFT (see SI, Section C) would exhibit a central peak that overestimates the true intensity in the lens' focal plane. This arises physically because the Fourier transform of a constant (background) yields a delta function and Eq. C-1 (SI, section C) clearly demonstrates that $I_0(W/m^2)$ increases with the spectral intensity maximum of $I_s = |\mathcal{F}\{O_{xy}\}|^2$. Conversely, if the intensity corresponding to the actual beam profile shown in Fig. 4a ($I_0=76GW/cm^2$), is not properly measured, the resulting values would be inaccurate. For instance, assuming a pure Gaussian beam with

<figure><img src="image_5.png" /><figcaption>**Fig. 4** a The object defined from the linear regime image (the star is the centroid of the profile). Relative variation of the beam size of the image as a function of the position of the sample z in the focal plane: the black dots are the data, and the blue line is the fit. The data in the red squares determine the linear regime. See other Z-scan parameters in the text</figcaption></figure>
the same beam waist ($\omega_e = 2.13\,\text{mm}$), the measured intensity would be overestimated ($I_0 = 2\pi P_{\text{tot}}\omega_e^2/(\lambda f)^2 \approx 130\,\text{GW/cm}^2$). In contrast, assuming a uniform circular (top-hat) profile with the same radius would lead to an underestimation of the intensity: $I_0 = \pi P_{\text{tot}}\omega_e^2/(\lambda f)^2 \approx 65\,\text{GW/cm}^2$. The total power is obtained using $P_{\text{tot}} = E/\tau_{\text{eq}}$ where the equivalent pulse duration $\tau_{\text{eq}} = \tau_{\text{FWHM}} \times 1.135$ assuming a $sech^2$ pulse.

## 2.2.2 Simulation and fitting techniques: why imaging matters

A simulation program, along with its associated functions, is provided in SI, Section E. This Matlab program accounts for various key components, including spatial beam parameters, NL material properties, and lens characteristics. It enables the generation of customizable beam profiles—such as Gaussian, circular, or Gaussian truncated by a circular aperture—and simulates their propagation through the optical system (Fig. 3). The Z-scan parameters are fully configurable, and the simulation performs an iterative loop over multiple z positions, computing the imaging patterns and corresponding beam dimensions at each step. Figure 5 presents numerical simulations of the BSRV measurements performed under identical experimental conditions relative to Fig. 4 ($I_0 = 76\,\text{GW/cm}^2$, $\lambda = 800\,\text{nm}$, $n_2 = 1.8 \times 10^{-20}\,\text{m}^2/\text{W}$, $f = 25\,\text{cm}$, $L = 3.73\,\text{mm}$) while varying only the incident beam profile shape, revealing significant differences in the NL response. The blue curve shows the BSRV for a pure Gaussian beam having a beam waist $\omega_G = 3.3\,\text{mm}$. The red curve corresponds to a purely circular aperture (top-hat beam with radius $R_c = 2.42\,\text{mm}$). The black curve represents the truncated Gaussian beam with

$\omega_e \approx 2.2\,\text{mm}$ (using the beam size second moment analysis Eqs. 5 and 6), demonstrating that the system's sensitivity strongly depends on the input beam profile despite identical NL phase shifts, thus highlighting the critical importance of precise beam characterization for accurate Z-scan measurements in NL optical materials. By defining the sensitivity as the ratio between the induced phase shift and the peak-to-valley signal difference $\Delta\omega_{pv}$: $S = \varphi_0/\Delta\omega_{pv}$, we obtain $S = 0.33$ for the Gaussian beam (blue curve), $S = 0.15$ for the top-hat profile (red curve), and an intermediate value $S = 0.19$ for the mixed one. The simulation code is made openly available, enabling full reproducibility of our results and allowing researchers to investigate how different beam profiles affect Z-scan measurements under various experimental conditions. In real experiments where beam profiles inevitably deviate from ideal Gaussian or circular distributions, we accurately define the input object by imaging the circular aperture under collimated illumination, taking the square root of the acquisition, and applying the preprocessing background removal and noise reduction (SI, section D) before calculating the focused peak intensity $I_0(W/m^2)$, (SI, Section C). This procedure, used for Fig. 4's input definition prior to experimental data fitting, enables proper field propagation through our section E-based algorithm in SI, while accounting for both the significant spatial pointing deviations occurring across the OPA's 650–900nm output range and the lens chromatic aberrations as it will be discussed in the next paragraph.

### 2.2.3 Dispersion of the glass

To determine how the focal length $f$ of a lens made of BK7 glass changes when transitioning from a wavelength of

<figure><img src="image_6.png" /><figcaption>Fig. 5 a Image of the truncated Gaussian object in the linear regime at the output of the imaging system. b- Simulated beam size -relative variation (BSRV) as a function of z. The blue solid line corresponds to a pure Gaussian beam,</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>the red line to a circular top-hat beam, and the black line to the truncated Gaussian beam. All other Z-scan parameters match the experimental conditions reported in Fig. 4</figcaption></figure>
650 nm to 900 nm, we need to consider the dispersion of the glass, which refers to the variation of the refractive index (n) with wavelength. At 650 nm, the refractive index of BK7 is approximately (n<sub>650</sub> ≈ 1.514) while at 900 nm, the refractive index of BK7 is approximately(n<sub>900</sub> ≈ 1.509). Using paraxial approximation, the focal length of a thin lens is given by:1/f = (n-1)(1/R<sub>1</sub> - 1/R<sub>2</sub>), where R<sub>1</sub> and R<sub>2</sub> are the radii of curvature of the lens surfaces. If we denote f<sub>650</sub> as the focal length at 650 nm and f<sub>900</sub> as the focal length at 900 nm, we can write: f<sub>900</sub>/f<sub>650</sub> = (n<sub>650</sub> - 1)/(n<sub>900</sub> - 1) ≈ 1.011. With f<sub>650</sub> = 25 cm the variation is therefore 2.6 mm. The question here is whether these matters using BSRV method. One possible solution is to use an achromatic lens. When this option is not available and to ensure proper alignment, fine focus adjustment can be performed at a wavelength of 775 nm, chosen at the center of the wavelength range to minimize its impact. Care must be taken to position the CCD camera at an equal distance from the imaging lens as the object plane (see Fig. 3), in order to preserve a well-defined and fixed magnification of unity.

The effect of this 775nm alignment on the BSRV was simulated, accounting for chromatic aberration at the two extreme wavelengths (650 nm and 900 nm). To model the chromatic focal shift in a thin BK7 glass lens (radius R<sub>1</sub>=250 mm) by evaluating its wavelength-dependent refractive index using the Sellmeier equation (see the refractive index of BK7 glass using the Sellmeier equation in SI section E). The reference focal length (f<sub>ref</sub> = R<sub>1</sub>/[2(n<sub>ref</sub> - 1)]) is calculated at λ = 775nm (mid-range wavelength), with n<sub>ref</sub> obtained via [32]. For any target wavelength, the program computes the actual focal length (f) and the resulting defocusing error (Δf = f<sub>ref</sub> - f), which arises from dispersion. The equivalent focal length (f<sub>eq</sub> = f<sub>ref</sub> - Δf) is then derived to quantify the chromatic aberration effect. This approach enables systematic analysis of focal plane deviations across the operational spectrum for our optical system using the

program detailed in SI section E where the propagation
distances and equivalent focal length must be adjusted. In
Fig. 6, the results are shown using the same parameters
as in Figs. 4, 5 (I₀ = 75GW/cm², n₂ = 1.9 × 10⁻²₀m²/W,
f ≈ 25cm, L = 3.73mm, ωG = 3.3mm, R_c = 2.42mm).
The red curve in Fig. 6a is the reference BSRV profile
at λ = 775nm. The blue one is the profile obtained when
switching to 650nm while keeping the object and CCD fixed
(see Fig. 3). We observe an increase in Δωpv as well as in
φ₀ = 2πn₂I₀L/λ, since this quantity is inversely proportional
to wavelength. However, the sensitivity S remains constant
at 0.19. A similar trend applies at 900 nm (black solid
line), where S remains unchanged, but both Δωpv and φ₀
decrease. In Fig. 6b, the profiles include chromatic defocus-
ing effects (dashed lines: blue for 650nm, black for 900nm)
compared to the ideal case where object-CCD alignment is
readjusted (Δf = 0, solid lines). Despite this, Δωpv remains
nearly identical, with S staying stable at 0.19. In summary,
under the experimental conditions detailed here, the results
demonstrate that chromatic aberration introduces negligible
measurement error, even with untreated standard lenses. A
slight focal shift is observed, but it does not affect the meas-
urement sensitivity, which is primarily related to the spatial
profile of the incident beam.

**3 Experimental results**

To begin, given the significant pulse duration variations
observed during measurements (see Fig. 2 in Sect. 2.1), we
used the average value of τ<sub>m</sub> = 266fs across the entire wave-
length range available from the OPA output. Using the BSRV
method, a total of 108 measurements were conducted, and
the resulting NL refractive index n<sub>2</sub> values are presented as a
function of the wavelength λ in Fig. 7a. The mean value was

<figure><img src="image_8.png" /><figcaption>**Fig. 6 a** The object-image alignment (with a magnification of 1) is set for $\lambda$ = 775 nm (red profile). Without moving either the object or the CCD, the BSRV profiles at 650 nm (blue) and 900 nm (black) are</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>shown; b Simulated profiles, comparing cases with and without defo-
cusing spatial shifts caused by chromatic aberration</figcaption></figure>
<figure><img src="image_10.png" /><figcaption>Fig. 7 a $n_2 = f(\lambda)$ with fixed pulse duration, considering $\tau_{FWHM} = 266\text{fs}$ Estimation of the pulse duration $\tau_{FWHM}$versus wavelength, considering the mean value of $n_2$: $n_{2m} = (1.8 \times 10^{-20})m^2/W$. Red circles-represent the data, and the green dashed lines indi-</figcaption></figure>

<figure><img src="image_11.png" /></figure>

cate the $2\sigma$ (95%) prediction intervals. The blue solid lines show the fitting (linear for a and a sixth degree polynomial for b). In gray, the values measured by autocorrelation using a two photon photodiode (Fig. 2) are reported

determined to be $n_{2m} = \langle n_2 \rangle = (1.8 \pm 0.4) \times 10^{-20} m^2/W$, where the uncertainty represents a $2\sigma(95\%)$ confidence interval. Multiple measurements were conducted at specific wavelengths (e.g., 800nm) to check measurement reproducibility. The linear regression of the data (blue line in Fig. 7a) indicates that $n_2$ remains nearly constant but decreases slightly over the measured wavelength range, in agreement with several reported studies (see e.g., [10, 12]). However, the measurement dispersion remains relatively large, which is expected since the pulse duration was assumed fixed. This variability cannot be attributed to the uncertainty in average power measurements (accurate to within $\pm 5\%$), nor to chromatic aberration (see Sect. 2.1.c), nor to geometric aberrations given our extremely low numerical aperture ($NA \sim 10^{-2}$). Moreover, all the fits used to estimate $n_2$ exhibit $R^2$ values close to 1 (see Fig F-1, Section F in the SI). A plausible explanation is that since the NL refractive response of our BSRV measurement system is intensity-dependent, this variation should correlate with pulse duration. We therefore estimated $\tau$ values under the assumption that $n_2$ remains constant in the glass transparency region, around its mean value $n_{2m}$:

$$ \tau_{FWHM} = \tau_m \cdot n_{2m}/n_2 \qquad (7) $$

The results of this estimation are plotted in Fig. 7b in red circles, along with a sixth-degree polynomial fit (blue solid line) and the $\pm 2\sigma(95\%)$ prediction intervals (green dashed lines).

$$ \tau_{c(FWHM)}(s) = (3.8 \times 10^{28})\lambda^6 - (1.8 \times 10^{23})\lambda^5 + (3.7 \times 10^{17})\lambda^4 \\ - (4.0 \times 10^{11})\lambda^3 + (2.4 \times 10^5)\lambda^2 \\ - (7.4 \times 10^{-2})\lambda + (9.7 \times 10^{-9}), \qquad (8) $$

with $\lambda$ expressed in m. For comparison, the $\tau$ values obtained from the 2PA through autocorrelation measurements (Fig. 2) are also shown in gray. Remarkably, a very good agreement is observed between the 2PA and the NL refraction fits. Both independent measurement methods reveal consistent spectral features: two valleys near 670 nm and 775 nm, and two peaks around 720 nm and 865 nm, occurring at almost identical wavelength ranges. This validates that the based Z-scan BSRV method can estimate the laser pulse duration when a reference $n_2$ value is assumed to be accurate. Specifically, we measured $\langle \tau_{c(FWHM)} \rangle = 273 \pm 80\text{fs}$ within the $\pm 2\sigma(95\%)$ confidence interval, demonstrating a measurement error variation comparable to that of the conventional 2PA method and reflecting approximately the same high wavelength-dependent variations in pulse broadening mechanisms.

It is therefore appropriate to correct both $n_2$ and $I_0$ values using the pulse duration $\tau_{c(FWHM)}$ derived from the polynomial in Eq. 8, thereby enabling a direct comparison with values obtained under the assumption of a constant pulse duration. These corrections can be implemented directly within the measurement program, or more simply by recognizing that the measured NL phase shift from the fit $\phi_0 = 2\pi n_2 LI_0/\lambda = 2\pi n_{2c} LI_{0c}/\lambda$ remains invariant. The corrected values were calculated according to Eqs. 9, and the corresponding results are presented in Fig. 8:
<figure><img src="image_12.png" /><figcaption>Fig. 8 Correction of the n₂ values using the sixth polynomial fit of τc(FWHM) shown in Fig. 7b according to Eqs. 7, 8, 9 & 10; a n₂c = f(λ); b n₂c = f(I₀c). Red circles represent the data of panel a,</figcaption></figure>

<figure><img src="image_13.png" /></figure>

and the green dashed lines indicate the 2σ (95%) prediction intervals. The blue solid lines show the linear fittings. In gray, data values and fittings considering a constant pulse duration (266 fs)

$$n_{2c} = n_2 \tau_{c(FWHM)} / \tau_m \qquad (9)$$

$$I_{0c} = I_0 \tau_m / \tau_{c(FWHM)} \qquad (10)$$

In Fig. 8a, the gray data points exhibit a slightly decreasing negative slope, whereas the corrected values (red circles) flatten this trend, rendering the $dn₂/dλ$ variation nearly negligible. Similar behavior is observed in Fig. 8b. The new linear fits (solid blue lines) of the corrected data n₂c provide a more physically consistent interpretation, demonstrating that: (i) the corrected fit yields a nearly constant n₂ versus λ dependence, in better agreement with the BGO model [33, 34] in the transparency region of the glass (see section E in SI); (ii) the results remain consistent across the full intensity range (4.3–197GW/cm²), showing constant n₂ values with nearly all data points lying within the ±2σ confidence bounds (Fig. 8b). This confirms the absence of higher-order NL effects over the investigated regime.

What should be especially emphasized here is that it is overly optimistic to claim accurate NL refractive index measurements based solely on a few Z-scan traces at the output of an OPA. Even with over one hundred carefully conducted measurements, our determination of the glass NL index n₂ still carries a statistical uncertainty of ±22%. To improve the reliability of the results, it would be necessary to perform around ten measurements at each wavelength, with 5 nm spectral steps, by implementing a parallel measurement system. This could be achieved by introducing a beam splitter at the entrance of the setup, directing the beam into two separate arms. One dedicated to autocorrelation measurements via two-photon absorption (2PA), and the other to NL refraction measurements using the BSRV method. However, this approach

would require repeating the Z-scan experiment more than 500 times, making the process cumbersome and time-consuming.

## 4 Discussion and limitations of the method

While the BSRV technique has proven effective under the current experimental conditions, namely, in the visible range (650–900 nm) with moderate pulse durations of a few hundred femtoseconds, several limitations must nonetheless be acknowledged.

First, although lens-induced dispersion was found to have a negligible impact in this study, its influence may become more significant when using shorter pulses or materials with higher group velocity dispersion. The accuracy of the Z-scan based method relies on a balance: the pulse duration must be short enough for the NL interaction to remain localized in space and time, yet long enough for the slowly varying envelope approximation (SVEA) to remain valid. This condition is generally fulfilled here. However, when femtosecond pulses propagate through dispersive optical elements such as a few-millimeter-thick fused silica sample, the temporal broadening induced by group velocity dispersion becomes increasingly significant as the pulse duration decreases. Below a certain pulse width, typically in the few tens of femtoseconds range, this broadening can reach levels that noticeably alter the temporal profile of the pulse. Therefore, care must be taken when interpreting results involving short pulses propagating through dispersive media.

Although this study does not focus on thin film systems, it is important to note that extracting the intrinsic nonlinear response of a film deposited on a substrate is particularly challenging in BSRV-Z-scan-based configurations,
especially when the film is submicron-thick and the substrate is significantly thicker. In such cases, nonlinear contributions from the substrate often dominate the response. While advanced analytical methods or dual-arm experimental configurations (as discussed in [35–37]) may help disentangle these contributions, their implementation requires meticulous control.

Furthermore, for absorbing materials, irreversible modifications—such as thermal damage or photoablation—can occur even at moderate fluences, especially at high repetition rates. These effects may lead to misinterpretation of the nonlinear signals, resulting in artificially high values of $n_2$ or $\beta$. To mitigate this, we recommend a three-step Z-scan procedure (linear–nonlinear–linear) to monitor such effects and detect any irreversible changes, as demonstrated in [38]. In this case, maintaining precise spatial alignment between the three scans is crucial, as even slight beam displacements (for example, caused by the insertion or removal of neutral density filters) may result in probing different regions of the film.

In summary, while the BSRV method is well suited for sub-picosecond OPA pulses in the visible spectrum, its accuracy ultimately depends on careful control of dispersion, the fidelity of the spatial beam profile, and the validity of the Kerr-only approximation. These factors must be critically evaluated when applying the method to other regimes, such as ultrashort pulses, high-energy beams, or materials with more complex structures, such as thin films or layered media.

## 5 Conclusions

This study suggests the robustness and versatility of the BSRV (Beam Size Relative Variation) Z-scan-based technique for the simultaneous measurement of $n_2$ and the effective pulse duration $\tau$ of femtosecond laser beams in the visible spectral range (650–900 nm), using an optical parametric amplifier source. By integrating CCD-based spatial beam profiling with temporal diagnostics, initially through autocorrelation and two-photon absorption detection, and ultimately through Z-scan fitting itself, our approach provides a unified framework for analyzing both spatial and temporal properties in NL refraction measurements.

Unlike traditional Z-scan techniques that assume circular or Gaussian beams, the BSRV method accommodates arbitrary spatial profiles and allows direct focal intensity measurements without the need for reference-sample calibration. The dispersion induced by the focusing lens was shown to have no measurable impact on the accuracy of $n_2$ determination, and no higher-order NL effects were detected under the experimental conditions. A key outcome is the establishment of a revised reference value for fused silica's NL refractive index: $n_2 = (1.8 \pm 0.4) \times 10^{-20} m^2/W$, valid across

the 650–900 nm range and potentially extendable within the infrared material’s transparency window.

The technique was validated across 108 independent measurements, with very good agreement observed between the values of $\tau$ obtained from NL refraction fitting and those from independent 2PA autocorrelation measurements. Both methods revealed consistent spectral features. However, its applicability relies on the availability of a suitable reference material for pulse duration estimation.

All necessary simulation programs and processing functions, including beam generation, NL propagation, and Z-scan analysis, are provided in the Supplementary Information, supporting reproducibility and broader adoption. Furthermore, the ability to estimate sub-picosecond pulse durations directly from Z-scan data presents a practical, low-cost alternative to conventional autocorrelation techniques provided that a suitable reference material is used. Altogether, the BSRV method provides a reliable, accessible, and experimentally efficient framework for NL optical characterization, especially in conditions where dispersion effects are well controlled, and the Kerr-only approximation holds.

**Supplementary Information** The online version contains supplementary material available at https://doi.org/10.1007/s00340-025-08502-2.

**Acknowledgements** The authors would like to thank SFR Matrix and the University of Angers for their financial support. G.B. acknowledges 6-month sabbatical leave from the University of Angers.

**Author contributions** Conceptualization, G.B.; Data curation, G.B.; Formal analysis, G.B.; Investigation, C.C., J.B.Z., C.M., G.B.; Methodology, G.B.; Project administration, G.B.; Resources, C.C., G.B.; Software, C.C., G.B.; Supervision, G.B.; Writing—original draft, G.B. All authors have read and agreed to the published version of the manuscript.

**Funding** Open access funding provided by Université d'Angers. This work was supported by the "Contrat de Plan État-Région" (CPER) program FEMTOMAT (2021–2027). No additional external funding was received for this study.

**Data availability** No datasets were generated or analysed during the current study.

## Declarations

**Conflicts of interest** The authors declare no competing interests.

**Open Access** This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
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