# Simplified Kerr refractive index measurement in the visible range using phase object contrast imaging

Georges Boudebs, Julien-Bilal Zinoune, Christophe Cassagne, Mihaela Chis

► To cite this version:

Georges Boudebs, Julien-Bilal Zinoune, Christophe Cassagne, Mihaela Chis. Simplified Kerr refractive index measurement in the visible range using phase object contrast imaging. Optics and Laser Technology, 2026, 198, pp.114888. 10.1016/j.optlastec.2026.114888. hal-05504621

HAL Id: hal-05504621

https://univ-angers.hal.science/hal-05504621v1

Submitted on 11 Feb 2026

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons CC BY 4.0 - Attribution - International License
<figure><img src="image_2.png" /></figure>

Full length article

# Simplified Kerr refractive index measurement in the visible range using phase object contrast imaging

Georges Boudebs<sup>a, *, †</sup>, Julien-Bilal Zinoue<sup>a, b</sup>, Christophe Cassagne<sup>a</sup>, Mihaela Chis<sup>a, b ‡</sup>

## ARTICLE INFO

**Keywords:**

Nonlinear Refractive Index ($n_2$)

Phase object

Femtosecond Regime

Fused Silica

OPA Lasers

Z-scan Technique

## ABSTRACT

A streamlined method is presented for determining the nonlinear refractive index $n_2$ using phase objects embedded in an optical imaging configuration. The approach is particularly well suited to broadband sources such as tunable optical parametric amplifiers (OPAs), which typically exhibit non-Gaussian spatial profiles and femtosecond pulses with wavelength-dependent duration. Conventional techniques such as Z-scan require precise knowledge of intensity, pulse duration, beam waist and sample position, parameters that are difficult to control in OPA-based systems. The proposed method overcomes these constraints through a two-shot comparative measurement between a reference material (e.g., fused silica) and the sample, eliminating the need for translation stages, autocorrelation diagnostics and detailed beam characterization. By referencing the nonlinear phase shift to a well-known material, the technique removes scanning and intensity-calibration requirements, ensuring broad applicability across wavelengths and sample types. Theoretical modeling and experimental validation demonstrate high sensitivity together with a substantial reduction in experimental complexity. Results show excellent agreement with established methods, confirming the reliability of the approach. This work provides a rapid and efficient route for characterizing nonlinear optical properties in situations where conventional techniques are impractical. Supplementary Materials include data-processing scripts and numerical resources to facilitate implementation and reproducibility.

## 1. Introduction

Measurement of third-order nonlinear (NL) optical properties, particularly the NL refractive index $n_2$, is essential for the development of advanced photonic materials and devices. These properties govern key phenomena such as self-focusing, self-phase modulation, and four-wave mixing, which underpin a wide range of applications in ultrafast optics, optical switching, and signal processing.

Among established techniques, the Z-scan method [1] has been widely adopted for characterizing $n_2$ due to its good sensitivity and relative simplicity. However, its reliance on mechanical translation and on averaging multiple laser shots can be limiting, especially with unstable beam sources or when single-shot capability is desired. Furthermore, quantitative Z-scan analysis requires precise knowledge of the peak irradiance at the focus, which depends critically on three parameters: (i) pulse energy, (ii) effective pulse duration, and (iii) spatial beam profile, each introducing potential measurement uncertainty.

To address these and other constraints, numerous improved or

simplified variants of Z-scan have been developed. For instance, early work explored spatial beam modifications for surface-deformation and Kerr-like effect measurements [2], while comprehensive reviews have compared the accuracy and applicability of various nonlinear characterization methods [3]. More recent advances include time-resolved eclipsed Z-scan to separate fast electronic from slow thermal responses in liquids [4], and spectral-domain Z-scan to enable measurements on highly scattering media [5]. Although these variants overcome specific challenges, such as temporal resolution, thermal artifacts, or sample scattering, they typically retain the need for scanning or detailed beam characterization.

Building on this landscape, we introduce a simplified phase-object imaging technique that eliminates both mechanical translation and the requirement for precise beam profiling. By converting the measurement into a single-point contrast evaluation at a pre-calibrated position, the method offers a robust, rapid alternative for determining the nonlinear refractive index, particularly suited to broadband or spatially irregular beams from sources such as OPAs. Originally proposed in 2004, phase

<footnote><sup>a</sup> Univ Angers, LPHIA, SFR MATRIX, F-49000 Angers, France</footnote>

<footnote><sup>b</sup> ESAIP, CERADE, 18, rue du 8 mai 1945, 49180 St-Barthélemy d'Anjou Cedex, France</footnote>

<footnote>* Corresponding author.</footnote>

<footnote>E-mail address: georges.boudebs@univ-angers.fr (G. Boudebs).</footnote>
object-based NL imaging techniques (NIT-PO) represented a promising alternative to conventional Z-scan based methods [6,7]. Nevertheless, their adoption has remained modest, and many of the method's intrinsic advantages have yet to be fully leveraged. The present work revisits this approach and extends it into a practical, OPA-compatible method, addressing the limitations that previously hindered its broader adoption. These methods employ a phase object (PO), typically a structured optical element inserted into a 4f coherent imaging system, to modulate the wavefront of the incident beam. When a NL sample is placed at the Fourier plane, the Kerr-induced phase shift alters the intensity distribution captured by a CCD camera showing negative or positive phase contrast depending on the sign of $n_2$. This distortion can be analyzed to extract the value of $n_2$ with high sensitivity and without the need for sample translation. Moreover, the use of positive-negative phase objects has been shown to significantly enhance measurement sensitivity, particularly using negative phase plate [8]. Also, the phase-object Z-scan technique is reported to distinguish the transient thermal-lensing effect from the Kerr third-order NL refraction [9]. For dynamic studies, a time-resolved pump-probe system incorporating a phase object has been employed to investigate the temporal evolution of NL absorption and refraction [10]. While in [11] the study focused on simulation of image formation in a microscope to optimize NL medium parameters for enhancing the visibility of rectangular objects.

In this work, the use of phase objects in NL imaging configurations is further examined and validated for determining $n_2$. The method provides a rapid and straightforward means of estimating $n_2$ at a given wavelength when using a tunable OPA, even in the presence of strongly non-Gaussian spatial beam profiles. The procedure relies on only two laser shots, one through a reference material and one through the sample, by comparing the resulting image contrast in the camera plane.

The protocol offers several practical advantages. It enables measurements across a broad spectral range without the use of a translation stage and, critically, without requiring knowledge of the incident intensity at the focal plane. This is particularly beneficial for characterizing materials with OPA-generated pulses, whose wavelength-dependent durations fluctuate over several tens of femtoseconds, making accurate intensity determination challenging. Although a simplified Z-scan method using a reference medium was previously proposed [12], the present approach reduces experimental complexity even further by eliminating the need to translate the NL medium, determine the pulse duration, locate the exact sample position within the focal region, or measure the wavelength-dependent beam waist.

The technique is supported by theoretical modeling and experimental validation, emphasizing its simplicity and broad applicability. As a key benchmark, measurements performed on standard nonlinear liquids yield values in excellent agreement with literature data, as well as with conventional Z-scan characterizations experimentally carried out here for comparison. Supplementary Materials provide the MATLAB codes used at each processing step, ensuring reproducibility and facilitating implementation.

## 2. Theoretical considerations

The general scheme of beam propagation inside an imaging system is described in detail in [12,13]. The method presented here can be applied to arbitrary beam profiles, we assume that the electric field at the object plane is, $O(x,y)$, where $x, y$ are the spatial coordinates. Let $S(u, v)$ be the spatial spectrum of $O$: $S(u, v) = \mathcal{F}[O(x, y)] = \int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} O(x, y) \exp[-j2\pi(ux + vy)]dxdy$, 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 $H(u, v) = \exp(j2\pi z' \sqrt{1 - (\lambda u)^2 - (\lambda v)^2} / \lambda)$ [14]. Then the field amplitude at $z'$ is obtained by computing the inverse Fourier transform: $E(x, y, z') = F^{-1}[S(u, v)H(u, v)]$. 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: $t_l(x, y) = \exp[-j\pi(x^2 + y^2)/(\lambda f)]$ [14]. 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 $z$ in the focal plane 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 [1]:

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

where $q(u, v, z) = \beta L_{\text{eff}}I(u, v, z)$ with $L_{\text{eff}} = [1 - \exp(-\alpha L)]/\alpha$ 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 $I(u, v, z) \propto |S(u, v, z)|^2$. $\Delta\phi_{\text{NL}}^{\text{eff}}(u, v, z) = 2\pi n_2 L_{\text{eff}} I_{\text{eff}}(u, v, z)/\lambda$ 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 $\phi_{\text{NL0}}^{\text{eff}} = \Delta\phi_{\text{NL}}^{\text{eff}}(0, 0, 0)$, 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 pure NL refractive materials, we denote the peak NL phase shift induced by the beam center ($at z = 0$) $\phi_{\text{NL0}}^{\text{eff}}$ as $\phi_0$. We consider hereafter the low NL excitation regime ($q_0 \approx 0$; $\phi_0 < 1$).

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:

$$\omega_x = 2\sqrt{\frac{\int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} I(x, y)(x - \bar{x})^2 dxdy}{\int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} I(x, y)dxdy}} \quad (2)$$

where

$$\bar{x} = \frac{\int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} I(x, y) dxdy}{\int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} I(x, y) dxdy} \quad (3)$$

is the centroid of the beam distribution. The size of the beam is defined as the mean value of $\omega_x$ and $\omega_y$. The beam size relative variation (BSRV) of the output beam is defined as $BSRV = (\omega_{NL} - \omega_L)/\omega_L$, where $\omega_{NL}$ and $\omega_L$ denote the sizes associated with the NL and linear profiles, respectively, as measured on the CCD. Details of the analysis procedure, including MATLAB code and the second-moment method, are provided in the Supplementary Information (Section E, Beam Size Analysis in Ref. [12]).

## 3. Preliminary experimental results

Image acquisition is performed using a Thorlabs CC126MU monochrome cooled CMOS camera. An example of the input beam profiles along with the recorded BSRV Z-scan pattern is shown in Fig. 2 where the NL optical refraction was investigated in the fused silica using 650nm and 900nm pulsed beam with a 3.73mm sample thickness ($L$). The OPA delivered pulses with a duration of $\tau_{FWHM} = 161$ fs at 650nm and a repetition rate below 0.5Hz, which rules out any thermal or electrostrictive contributions. The laser system (Pharos, Light Conversion) provided a pulse-to-pulse energy stability better than 1% RMS, as specified in its datasheet and verified by our preliminary measurements. Linear polarization was maintained using a Thorlabs polarizer (LPNIR050-MP2, 650–2000nm). Use of a Glan polarizer, as in [12], was avoided to prevent temporal pulse splitting. Autocorrelation measurements with the home-built setup described in [12] confirmed that the alternative polarizer produces significantly shorter pulse durations than previously reported, by suppressing the double-pulse formation associated with calcite Glan polarizers. In the presence of mixed $p$ and $s$
polarization components, the ordinary and extraordinary waves exhibit substantial group-velocity mismatch, resulting in temporal walk-off and separation into two sub-pulses. This effect, illustrated in Fig. 1(c) of [12] in the infrared range, is absent with the polarizer employed in the current configuration. The average power was measured at 25kHz to be 10mW for both acquisitions, corresponding to a pulse energy of about E = 0.4µJ (one pulse per acquisition at one z position). The beam was truncated by a nearly circular aperture with radius $R_a = 1.76mm$. It is noteworthy that for the same average power, the 650nm beam yields a nearly twofold higher focal intensity than the 900nm beam, all other parameters being equal. This intensity ratio is logically corroborated by the BSRV signal response, which is directly proportional to $I_0$. The incident intensity cannot be precisely determined using the method detailed in Ref. [12] (Eq. (C-1) in the Supplementary Information, Section C: Calculation of the Central Intensity in the focus). This is due to the presence of the phase object at the center of the beam, which diffracts light in the lens's focal plane. The degree of diffraction depends on the object's dimensions and its inherent linear phase shift, $\phi_L$. The phase object, shown in Fig. 1(a), is a circular plate comprising a glass substrate with a transparent dielectric disc of radius $L_p = 0.5mm$. The disc has a specific thickness and refractive index so that it retards the phase of the incident light by $\phi_L(\lambda)$ radians relative to the surrounding area. The value of this linear phase shift will be estimated later in this study.

The first objective is not to remeasure $n_2$ of the fused silica, which has been precisely detailed in a previous study [12], but rather to investigate the wavelength-dependent influence on the induced image contrast of the phase object as a function of the sample's longitudinal position z. The Z-scan profiles in Fig. 2(c) (data as circles) and the corresponding fits (solid lines) establish the positional reference ($z = 0$) for the two extreme wavelengths used in this study (red at 650nm and black at 900nm). This calibration accounts for the wavelength-dependent shift in the focal position, an effect caused primarily by chromatic aberration and variations in the almost Gaussian beam diameter at the input circular aperture. To facilitate a direct comparison, the profiles have been shifted to be centered around the same point, which defines $z = 0$ for each respective wavelength.

Using the same image acquisitions, the induced contrast in the phase object image can be plotted as a function of the sample's longitudinal position, z (see Fig. 3). This reveals how the contrast evolves as the NL medium experiences increasing light intensity around the focal plane region. The contrast, C, is quantified using the Michelson definition:

$$C = \frac{I_c - I_{out}}{I_c + I_{out}} \qquad (4)$$

where $I_c$ is the average intensity within the central disk of the phase object, and $I_{out}$ is the average intensity in a surrounding annular region. In practice, this calculation was performed using a custom routine that, for a given image, defines the central phase disk using its known co-ordinates and radius, and an outer annulus extending to $\sqrt{2} \times L_p$ (Fig. 3 (c)). The routine then computes the average intensities in both regions and applies Eq. (4) to yield the contrast value. The circle's center co-ordinates are determined using another custom MATLAB function (provided in the Supplementary Material-2), which calculates the geometric center and radius from four points manually selected along the phase object's visible borders in the first acquired image. These co-ordinates are saved and subsequently loaded for each image in the z-stack images to ensure consistent calculation of the Michelson contrast throughout the scan. The resulting contrast scan is presented in Fig. 3(d). Representative images are shown in Fig. 3(a) and 3(b) for the two extreme wavelengths generated by the OPA. These images were captured around $z = 0$, where the contrast is sufficiently high to ensure visibility. The C = f(z) curve exhibits a bell-like shape, characterized by a central maximum around $z = 0$ and wings that asymptotically approach zero without necessarily reaching it because the beam illuminating the circular aperture is Gaussian, with the phase object positioned at its peak. Consequently, the background intensity $I_{out}$ remains slightly lower than the central intensity $I_c$ even in the linear regime (far from the focal plane). Furthermore, the contrast in the wings of the curve ($|z| \gg 0$) is lower at $\lambda = 900nm$ than that at $\lambda = 650nm$. This difference arises because the Gaussian beam profile is wider at 900nm, as is visually apparent from the higher spatial extent of the beam in the images shown in Fig. 3(a) and (b). A larger beam waist reduces the intensity gradient across the phase object's edges, thereby lowering the measured contrast in the linear images.

The solid lines in Fig. 3(d) represent the results of our numerical simulation (see the program 'sim_Zscan_f phi_L.m' in the Supplementary Materials SM-1). This code models the imaging process to elucidate how NL optical effects and wavefront shaping influence the measurable contrast. It is important to note that the spatial profiles in Fig. 3(a) and (b) deviate from an ideal distribution due to the truncation of the quasi-Gaussian beam by the circular aperture. The beam waist illuminating the aperture of radius $R_a$ was determined to be $\omega_0 = 3$ mm at 650nm and

<figure><img src="image_3.png" /><figcaption>Fig. 1. Characterization of the nonlinear imaging system. (a) Cross-sectional geometry of the phase object. (b) Schematic of the experimental imaging setup, identifying the nonlinear material (NLM), the focusing lens (L), the translation stage (TS), and CCD camera. (c) Simulated nonlinear image, revealing the phase contrast of the object.</figcaption></figure>
<figure><img src="image_4.png" /><figcaption>Fig. 2. (a), (b) Linear regime images of the phase objects at 650 nm and 900 nm, respectively. The phase object, positioned within the circular aperture, is visible due to diffraction at its edges. (c) Experimental Z-scan profiles (circles) and fits (solid lines), centered at z = 0, showing the BSRV for a fused silica reference sample.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>Fig. 3. (a), (b) Nonlinear regime images of the phase object at 650 nm and 900 nm, respectively. (c) Central intensity profile of the image in (a), with double arrows indicating the regions used for calculating $I_c$ (central disk) and $I_{out}$ (surrounding annulus). (d) Measured Z-scan contrast profiles for a fused silica sample (squares: 650 nm, circles: 900 nm) compared with theoretical simulations (solid lines).</figcaption></figure>

$\omega_0 = 4.2$ mm at 900nm. The corresponding peak intensities at z = 0 were measured as $I_0 = 65$ GW/cm² at 650nm and $I_0 = 33$ GW/cm² at 900nm. The values of the linear phase shift, $\phi_L$, that provide the best fit to the experimental data at each wavelength are indicated in the inset of Fig. 3 (d). These were calculated from a reference phase of $\phi_{ref} = 60^\circ$ measured at $\lambda_{ref} = 1064$ nm, using the relationship $\phi_L(\lambda) = \lambda_{ref} \phi_{ref}/\lambda$ to scale the phase for other wavelengths. As previously done for the BSRV profiles, the contrast curves have been shifted to center their maximum value. Furthermore, our numerical simulation (see next section) reveals that the maximum contrast does not always coincide with the z = 0 position (corresponding to the absence of the phase object) but is variable and depends strongly on the value of $\phi_L$.

## 4. Numerical simulation

To account for the circular phase object, the circular aperture, and the approximately Gaussian shape of the incident laser beam, the complex optical field at the output of the phase object is expressed as:

$$O(x, y) = \text{circ}(\frac{r}{R_a}) \exp[i\phi_L(\lambda)\text{circ}(\frac{r}{L_p})] \exp(-\frac{r^2}{\omega_0(\lambda)^2}) \quad (5)$$

where $r = \sqrt{x^2 + y^2}$ denotes the radial coordinate. A key characteristic of the measurement system is its input-output linearity, which describes the extent to which the output signal is a direct, proportional representation of the input quantity. Maintaining this linearity across the intended operating range is crucial for ensuring measurement accuracy
and simplifying the calibration process. A linear system is easy to calibrate (a two-point calibration often suffices) and its behavior is easy to model. To streamline the measurement of the NL refractive index $n_2$, a systematic investigation of the system's linearity is required. This involves characterizing the response across different parameters, including the sample position $z$, the wavelength $\lambda$, the linear phase shift $\phi_L$ of the object (Eq. (5)), and the induced NL phase shift $\phi_0$. Additional factors, such as optical magnification, numerical aperture, and sensor characteristics, are equally critical for a comprehensive sensitivity analysis. So, given the extensive and complex nature of this multi-parameter study, a full presentation is not feasible here. In the following sections, we will focus on illustrating the most significant results, specifically those obtained with parameter values that closely match our experimental imaging system as previously described.

The results in Fig. 4 were generated using a numerical model that simulates NL beam propagation through a phase-object-based imaging system. The model can be easily adapted from the program 'sim.Zscan.f.phi.L.m' provided in the Supplementary Materials SM-1. For Fig. 4 (a), the wavelength ($\lambda$) was varied at a fixed $I_0$, while for Fig. 4(b), the intensity ($I_0$) was varied at a fixed $z$. The key fixed parameters for the optical setup are as follows: an imaging configuration with a lens of focal length $f = 25$cm and a magnification of $G = 1$; a central dielectric disc of radius $L_p = 0.5$ mm, imparting a linear phase shift $\phi_L(\lambda)$ scaled from a reference value of $\pi/3$ radians (60°) at $\lambda_{ref} = 1064$ nm; the object is placed within a circular aperture of radius $R_a = 1.76$ mm, illuminated by a Gaussian beam with a waist of $\omega_0 = 3$ mm at the input plane; the NL sample, 3.73 mm thick with linear index $n = 1.5$, is considered as “thin” having a NL refractive index of $n_2 = 1.9 \times 10^{-20} \text{ m}^2/\text{W}$; the NL absorption coefficients were set to zero ($\alpha = 0, \beta = 0$) to isolate the purely refractive Kerr effect. The contrast in the image plane is calculated using Eq. (4) (via the 'compute_circularcontrast.m' function provided in the Supplementary Materials SM-4). The simulated contrast profiles in Fig. 4 (a), calculated for a fixed peak intensity of $I_0 = 50 \text{ GW/cm}^2$ across three wavelengths, reveal that the maximum contrast does not occur at the focal plane ($z = 0$) as might be assumed. Furthermore, as shown in Fig. 4 (b), obtained by varying $I_0$ from 0 to 100 GW/cm$^2$, the contrast $C$ depends linearly on the induced NL phase shift $\phi_0$ for all wavelengths and at the specific $z$ positions detailed in the inset. Finally, the sensitivity of the technique, defined by the slope of $C = f(\phi_0)$, is not constant but varies with both the wavelength and the sample's longitudinal position. The most significant practical implication of these findings is that they enable a highly simplified efficient method for determining $n_2$. For a fixed sample position $z$, wavelength, pulse duration, and focal beam waist, the $n_2$ of an unknown material can be deduced by comparing its induced phase object contrast to that of a reference material with a

known $n_2$. Crucially, this requires only two laser shots, one for the sample and one for the reference. Care must be taken, however, to account for the focus shift caused by differences in sample thickness, which can introduce non-negligible measurement error (see Fig. A1 in the Appendix A, which shows the simulation of an extreme case considered in section 6 of the experimental results).

The primary advantage of this approach is twofold: it drastically minimizes the exposure of the material to intense laser pulses (compared to the hundreds of shots required for a traditional Z-scan), and, as previously noted, it eliminates the need for independent characterization of the often wavelength-dependent pulse duration and the precise spatial profile of the incident beam [12].

## 5. Simplified measurement method principle

Assuming a monochromatic illumination and a fixed position $z$ of the sample $x$ under evaluation, corresponding to a given intensity $I(z)$ in the focal plane, and considering that image contrast is linearly proportional to the induced phase shift as shown numerically in the previous section, two image acquisitions are performed: one using a well-characterized reference sample $r$ (fused silica), yielding a contrast $C_r$; one using the test sample $x$, yielding a contrast $C_x$. Under the assumption of negligible absorption and considering that the total phase shift results from the additive contributions of each constituent element within the NL medium, the NL refractive index of the test sample can be expressed as: $n_{2x} = n_{2r}L_rC_x/(L_xC_r)$, where $L_r$ and $L_x$ denote the effective interaction lengths of the reference and test samples, respectively.

This formulation highlights that both the wavelength and the intensity distribution (containing the pulse duration and the beam waist) cancel out when comparing phase shifts under identical conditions, with only the substitution of the reference material by the test compound. As a result, the ratio of contrasts directly reflects the relative NL refractive indices, normalized by the respective interaction lengths.

Furthermore, in the presence of a liquid medium of thickness $L_x$ contained within a fused glass cuvette of total face thickness $L_c$, and yielding a combined total contrast $C_{x+r}$ in the acquired image, the same approach leads to the following expression:

$$n_{2x} = \frac{n_{2r}}{L_x} \left( L_r \frac{C_{x+r}}{C_r} - L_c \right) \quad (6)$$

This relation accounts for the contribution of the cuvette material and isolates the NL response of the liquid sample by subtracting the known phase shift induced by the glass.

In cases where achieving a measurable contrast requires a variation in intensity between the acquisition with the liquid-filled cell and the

<figure><img src="image_6.png" /><figcaption>Fig. 4. (a) Measured contrast as a function of sample position $z$ for wavelengths of 650 nm, 775 nm, and 900 nm at a fixed $I_0 = 50 \text{ GW/cm}^2$ with $L = 3.73 \text{ mm}$. (b) Contrast as a function of the induced NL phase shift, $\phi_0$, for a fixed $z$ position and wavelength, as specified in the inset.</figcaption></figure>
reference measurement, the relationship can be expressed as:

$$n_{2x} = \frac{n_{2r}}{L_x} \left( L_r \frac{C_{x+r}}{C_r} \frac{E_r}{E_{x+r}} - L_c \right) \quad (7)$$

where $E_r$ and $E_{x+r}$ denote the energies required to generate the contrasts $C_r$ and $C_{x+r}$, respectively. This formulation assumes that the intensity is controlled using a polarizer-based system, while the beam waist at position $z$ and the pulse duration are held constant. Naturally, when working at a fixed wavelength where the pulse duration is constant, the energy ratio $E_r/E_{x+r}$ can be directly replaced by the equivalent ratio of average powers measured with an optical power meter (OPM), $P_r/P_{x+r}$.

The method is explicitly designed for the low-phase-shift, perturbative regime. Its key advantage is the ability to make measurements at relatively low, non-damaging intensities, using the linear response. To measure at higher intensities or larger phase shifts, a more general model (beyond Eq. 1) would be required, which is outside the scope of this simplified, calibration-free technique.

## 6. Experimental results and measurements

The first step is to experimentally verify the linearity of the measured contrast, as defined by Eq. (4) with the induced phase shift $\phi_0$. To this end, we performed acquisitions at the two extreme wavelengths (650 nm and 900 nm), at different longitudinal positions $z$, and for various incident power levels measured with the OPM OPHIR Thermal Sensor (model 3A-PF-12) at 25 kHz repetition rate. It should be noted that while power measurements can only be performed by measuring the average at 25kHz, the image acquisitions themselves were carried out at a rate below 1 Hz, as already mentioned in Section 3. Fig. 5 shows the results obtained for a 3.73 mm thick fused silica sample at several longitudinal positions $z$: before, at, and after the focal plane. Remarkably, a linear response is observed in all cases. Furthermore, the data in this figure were acquired using non-Gaussian beam profile generated by the phase object within a circular aperture, as seen in the raw images of Fig. 3. The strong linear response observed in Fig. 5 confirms the method's robustness even with realistic, non-ideal beam profiles.

As expected from the simulation shown in Fig. 4(a), the slope of the linear fit, which determines the measurement sensitivity, is found to be enhanced for negative $z$ positions (pre-focal). This particularity can be physically explained (see Fig. 5 in Ref. [15]) by the positive value of the linear phase shift, $\phi_L$. This central phase imposed by the object adds to the phase curvature of the lens thickness, effectively increasing the beam convergence before the focal plane defined by the lens alone.

The observed linear relationship between the input mean power $P$ and the system's output contrast, as confirmed in Fig. 5, validates the operation of our measurement system within its specified dynamic range. Note that the contrast is determined from a single laser shot

<figure><img src="image_7.png" /><figcaption>Fig. 5. Measured contrast versus incident mean power at 650 nm (red) and 900 nm (black) for various sample positions $z$ of the fused silica sample. Solid lines are linear fits.</figcaption></figure>

acquisition. Furthermore, the incident power is directly proportional to the induced NL phase shift $\phi_0$, as defined previously. This linearity is essential, as it confirms that the underlying physical transduction mechanism is not saturating and that the system's response is both predictable and directly quantifiable. Consequently, the measured contrast can be interpreted as a true representation of $\phi_0$, enabling a straightforward conversion into the desired physical quantity, the NL refractive index $n_2$, through the application of Eqs. (6) and (7).

Several simplified measurement campaigns for determining $n_2$ were conducted, using the recently established reference value for fused silica from Ref. [12] as a benchmark. One of the key outcomes of that study was the establishment of a revised reference $n_2$ value for fused silica of $(1.8 \pm 0.4) \times 10^{-20} \text{m}^2/\text{W}$, valid across the 650–900nm range. This value was validated by 108 independent measurements, providing a reliable standard for our work. The acquisitions were performed on several liquids contained in quartz cells of varying thicknesses, ranging from 1mm to 5mm. The quartz cuvettes themselves have a NL refractive index equivalent to that of our 3.73mm thick fused silica reference sample. The wavelength from our OPA laser source was changed in 10, 20 or 50 nm steps. For each wavelength, two NL regime images were acquired: one with the fused silica reference and the other with the cuvette filled with the test liquid (e.g., distilled water in Fig. 6(a)). The contrast was measured for each image, enabling the calculation of the liquid's $n_2$ using either Eq. (6) or Eq. (7) as implemented in the MATLAB script 'NRIMPO.m' provided in Supplementary Materials SM-5. The selection between these two equations was determined by the experimental protocol: Eq. (6) was applied when the same incident power was used for both the reference and the sample, whereas Eq. (7) was employed when the incident power was adjusted individually for each. This adjustment was necessary to maintain the contrast within an optimal range of 0.1 to 0.6, ensuring reliable measurements by avoiding noisy data at low contrast (< 0.1) and NL phase shifts exceeding 1rad at high contrast (> 0.6), as can be seen in Fig. 4(b). Fig. 6 shows the results of all measurements performed, at room temperature, yielding a mean NL refractive index for water of $n_2 = (2.0 \pm 0.4) \times 10^{-20} \text{m}^2/\text{W}$. The relative uncertainty of the measurement is 21% at a 95% confidence level. Naturally, the total absolute uncertainty of the water's $n_2$ is significantly larger, as it must incorporate the absolute uncertainty of the fused silica reference measurement, which was estimated at 22% in Ref. [12]. While this approach is operationally simpler, the resulting value should be regarded as an estimate that merely captures the correct order of magnitude for the liquid. One of its practical advantages, however, is that it enables us to verify the absence of any significant wavelength dependence of $n_2$ over the investigated spectral range. As a result, the slight positive slope observed in the linear fit of Fig. 6 is unlikely to carry physical meaning, since it relies on the underlying assumption that the $n_2$ of the glass reference remains constant with wavelength. In addition, the equivalent defocus induced either by the nonlinear medium thickness (1–5 mm) or by the change in wavelength may also contribute to this small slope, whose magnitude is in any case negligible.

We extended our spectral characterization to carbon disulfide ($CS_2$), contained in a 1-mm-thick cuvette, using fused silica as the reference material. The measured $n_2$ value for $CS_2$ in the femtosecond regime was found to be significantly higher than that of water. Under linearly polarized excitation, the dominant NL mechanism for $CS_2$ on the picosecond timescale is molecular reorientation (alignment) driven by the anisotropic polarizability of the molecule [16,17]. This is a slow response (picoseconds), often called the orientational or rotational contribution to the Kerr effect. In the femtosecond regime used here, this orientational contribution cannot fully develop [18]. As a result, the measured $n_2$ in our experiment mainly reflects the instantaneous electronic response, which is why its magnitude is lower than values typically reported for picosecond excitations ($n_2 \approx 3 \times 10^{-18} \text{m}^2/\text{W}$), where the slower and stronger orientational nonlinearity fully contributes.
<figure><img src="image_8.png" /><figcaption>**Fig. 6.** (a) Measured nonlinear refractive index $n_2$ of water in the fs regime as a function of wavelength. The red solid line represents a linear fit, with the green band indicating the $2\sigma$ prediction intervals. (b) Spectral measurement of $n_2$ for various liquids. The data for $CS_2$ (red circles) are fitted with a linear function (solid red line), with $2\sigma$ prediction intervals (green lines). For comparison, data for toluene (black diamonds), acetone (green +), and water (blue circles, from panel (a)) are also shown. All measured $n_2$ values remain approximately constant across the investigated spectrum.</figcaption></figure>

Given the relatively high nonlinearity of $CS_2$, it was necessary to use Eq. (7) exclusively for these measurements to adjust the incident power and avoid excessive phase shifts. The resulting $n_2(\lambda)$ for $CS_2$ was found to be constant across the 650–900nm spectrum. The mean value was determined to be $n_2 = (42 \pm 7) \times 10^{-20} m^2/W$, which corresponds to a relative uncertainty of 16% at a 95% confidence level.

Furthermore, we performed complementary measurements on other common solvents, including ethanol, methanol, toluene, and acetone. The results, compiled in Fig. 6(b), reveal that the NL refractive indices for all these liquids also remain largely constant across the investigated spectral range. Specifically, water, ethanol, methanol, and acetone exhibit $n_2$ values that fall, within experimental uncertainties, in the narrow range of $2-4 \times 10^{-20} m^2/W$, while toluene displays a stronger nonlinearity of about $10 \times 10^{-20} m^2/W$. As already noted, $CS_2$ shows the highest nonlinearity, with a value around $42 \times 10^{-20} m^2/W$. The estimated uncertainties for these measurements are approximately 25%, primarily due to the intrinsic uncertainty in the reference $n_2$ value (see Appendix B).

The observed fluctuations in the $CS_2$ measurements appear to be of the same order of magnitude as those in water and can be attributed to several factors. A primary contributor is likely the wavelength-dependent pulse duration of the OPA laser [12]. The molecular alignment response of $CS_2$ is known to be sensitive to the pulse width, and

these temporal fluctuations introduce a variability that is challenging to deconvolve from the purely electronic nonlinearity. Furthermore, an additional source of uncertainty arises from the use of Eq. (7), which inherently carries a higher propagated error compared to Eq. (6). This is due to its dependence on the ratio of powers measured with the OPM, introducing measurement uncertainties from both the reference and sample incident power measurement relative to the acquisitions.

Within the experimental uncertainties, the values obtained in this work are in good agreement with those reported in the literature [19–24]. In particular, the early measurements reported in Ref. [19] yield $n_2$ values that are generally slightly higher than those measured here, while preserving the same order of magnitude. In Refs. [20,21], $n_2$ of water was determined using two independent techniques, Z-scan and single-shot supercontinuum spectral interferometry, leading to values on the order of $1.5 \times 10^{-20} m^2/W$ in the near-UV and near-IR spectral regions. These results are fully compatible with the present measurements within experimental error. Couris et al. [22], employing both spectral shear interferometry and Z-scan techniques in the femtosecond regime at 800nm, reported an $n_2$ value for $CS_2$ of $(31 \pm 10) \times 10^{-20} m^2/W$, in good agreement with the value obtained in this study. Finally, Ref. [24], based on a power-limiting characterization method, reports $n_2$ values of approximately $2 \times 10^{-20} m^2/W$ for water and $2.7 \times 10^{-20} m^2/W$ for ethanol, which are also consistent with our results.

<caption>**Table 1**</caption>

Experimental parameters and results from conventional BSRV-Z-scan validation measurements. The measurements were performed at 650nm with a pulse duration of $\tau = 161$ fs (FWHM), a focal beam waist of $\omega_{0f} = 63$ µm, and a Rayleigh range of $Z_0 = 12$ mm. Columns list: (2) thickness of the liquid cell ($L_c$) and quartz or fused silica substrate ($L_Q$); (3) peak on-axis intensity at $z = 0$ ($I_0$), calculated using the method in SI of [12]; (4) corresponding NL phase shift ($\phi_0$); (5) $n_2$ values obtained from fitting the data to Eq. (1); (6) mean $n_2$ values for cases with multiple Z-scan measurements; and (7) the mean $n_2$ value measured with the phase object, calculated over the 650–900 nm range as shown in Fig. 6.

<table><thead><tr><th colspan="5">Z-scan @ 650nm, 161fs(FWHM), ω<sub>0f</sub> = 63μm</th><th>Phase object</th></tr><tr><th></th><th>L<sub>c</sub> + L<sub>Q</sub>(mm)</th><th>I<sub>0</sub>(GW/cm<sup>2</sup>)</th><th>φ<sub>0</sub>(rad)</th><th>n<sub>2</sub> × 10<sup>-20</sup> (m<sup>2</sup>/W)</th><th>⟨n<sub>2</sub>⟩<sub>650</sub> × 10<sup>-20</sup> (m<sup>2</sup>/W)</th><th>⟨n<sub>2</sub>⟩<sub>650-900</sub> × 10<sup>-20</sup> (m<sup>2</sup>/W)</th></tr></thead><tbody><tr><td>SiO<sub>2</sub></td><td>3.73</td><td>59</td><td>0.39</td><td>1.8 ± 0.5</td><td rowspan="2">1.9 ± 0.25</td><td rowspan="2">Not applicable</td></tr><tr><td>SiO<sub>2</sub></td><td>3.73</td><td>105</td><td>0.75</td><td>2.0 ± 0.6</td></tr><tr><td>SiO<sub>2</sub></td><td>3.73</td><td>102</td><td>0.75</td><td>2.0 ± 0.6</td><td></td><td></td></tr><tr><td>Cell (quartz)</td><td>0 + 2.25</td><td>91</td><td>0.41</td><td>2.0 ± 0.6</td><td></td><td></td></tr><tr><td>Cell (quartz)</td><td>0 + 2.25</td><td>97</td><td>0.37</td><td>1.8 ± 0.5</td><td></td><td></td></tr><tr><td>CS<sub>2</sub></td><td>1 + 2.5</td><td>1.9</td><td>0.082</td><td>40 ± 12</td><td>34 ± 7</td><td>42 ± 7</td></tr><tr><td>CS<sub>2</sub></td><td>1 + 2.5</td><td>7.9</td><td>0.26</td><td>30 ± 9</td><td></td><td></td></tr><tr><td>Methanol</td><td>5 + 2.3</td><td>37</td><td>0.60</td><td>2.7 ± 0.8</td><td>2.7 ± 0.8</td><td>2.2 ± 0.5</td></tr><tr><td>Acetone</td><td>5 + 2.3</td><td>25</td><td>0.63</td><td>4 ± 1.4</td><td>4.0 ± 1.4</td><td>4.1 ± 1.2</td></tr><tr><td>Water</td><td>5 + 2.3</td><td>32</td><td>0.54</td><td>2.6 ± 0.7</td><td>2.6 ± 0.7</td><td>2.1 ± 0.5</td></tr><tr><td>Ethanol</td><td>5 + 2.3</td><td>21</td><td>0.31</td><td>2.3 ± 0.7</td><td>2.3 ± 0.7</td><td>2.0 ± 0.5</td></tr><tr><td>Toluene</td><td>1 + 2.5</td><td>63</td><td>0.93</td><td>11.7 ± 3.4</td><td>11.7 ± 3.4</td><td>9 ± 2</td></tr></tbody></table>
Our implementation prioritizes a practical balance between sensi-
tivity, robustness, and ease of use. By employing area-matched detection
regions and operating in the linear, low-phase-shift regime, the tech-
nique reliably measures $n_2$ values down to ~2 × 10⁻²⁰ m²/W in com-
mon solvents using sub-TW/cm² intensities. The ultimate measurement
fidelity is further ensured by using a high dynamic-range, cooled CCD
detector, which maximizes the signal-to-noise ratio for a given optical
sensitivity.

To conclusively validate our measurement method, we performed a
series of conventional Z-scan measurements using the BSRV technique
[13]. The results are summarized in Table 1. The BSRV profiles were
acquired at a wavelength of 650nm with a pulse duration of $\tau = 161$ fs
(FWHM), as measured by a Michelson interferometer-based autocorre-
lator (for technical details, see [12]). The equivalent beam waist at the
focal plane was $\omega_{0f} = 63$ µm, resulting in a Rayleigh range of $Z_0 =
\pi\omega_{0f}^2/\lambda = 12$ mm. Table 1 is organized into two distinct sections: the left
side summarizes the Z-scan results along with the associated parameters,
while the right side (last column) presents the measurements obtained
using our simplified phase-object method. The details of this table is
structured as follows: column (2) specifies the thickness of the liquid cell
($L_c$) and the thickness of the quartz or fused silica substrate ($L_q$); column
(3) provides the peak on-axis intensity at $z = 0$ ($I_0$), calculated according
to the method detailed in the Supplementary Information of Ref. [12];
column (4) states the corresponding NL phase shift, $\phi_0$; column (5)
presents the $n_2$ value obtained by fitting the data to the model given by
Eq. (1) (see BSRV Z-scan profiles in the supplementary Materials SM-8);
column (6) reports the mean $n_2$ value for cases where multiple Z-scan
measurements were performed (at 650nm). The associated uncertainty
was calculated as the standard error of the weighted mean, ensuring that
more precise measurements (those with smaller uncertainties) contrib-
uted more significantly to the final result. The final column (7) gives the
average $n_2$ value measured over all available OPA output wavelengths
($\langle n_2 \rangle_{650-900}$), based on the contrast provided by the phase object when
using the reference glass and the tested liquid. Considering the mea-
surement uncertainties, the values obtained by comparing column (6),
corresponding to the Z-scan based technique, and column (7), corre-
sponding to the phase object NL imaging technique, are fully consistent.
This confirms the validity of this simplified two-shot laser method.

The femtosecond characterization across a broad wavelength range, which confirmed the $n_2(\lambda)$ stability for purely refractive materials, establishes a foundational methodology whose utility would be particularly pronounced for spectrally studying absorptive solutions. This approach would become essential when investigating solutes with linear absorption peaks within this spectral band, or when dealing with the complex interplay of negative solute and positive solvent nonlinearities as in [25]. In such scenarios, where linear and/or nonlinear absorption is present as in thin films [26] or bulk materials as in [27], traditional single-wavelength techniques struggle to deconvolve the cumulative thermal and electronic contributions. Our rapid, intensity-calibration-free method would allow for a direct spectral mapping of the effective nonlinearity, providing a means to visualize and disentangle the resonant dispersion of the solute's nonlinearity from the solvent's background. Beyond spectral dependence, the method is also compatible with polarization-resolved measurements. By rotating the incident linear polarization using a half-wave plate and repeating the contrast measurement, yielding an effective $n_2(\theta)$ without modification of the analysis and providing direct access to nonlinear anisotropy in crystal-line or oriented materials. This prospective extension to the general case of media exhibiting non-zero linear and/or nonlinear absorption ($\alpha, \beta$) and polarization-dependent responses constitutes a natural next step and is currently under investigation.

7. Conclusions

In this work, we have developed, validated, and applied a simplified phase-object technique for the rapid characterization of the NL refractive index $n_2$. The method leverages a single laser shot per sample to measure the induced contrast in the image of a phase object, providing a unified framework that integrates spatial beam profiling with NL temporal diagnostics.

The core strength of this approach lies in its significant practical
advantages over conventional methods. It accommodates arbitrary
spatial beam profiles, eliminates the need for mechanical sample
translation, and, most importantly, operates without requiring prior
knowledge of the precise incident intensity at the focus. This makes it
particularly valuable for characterizing materials with laser sources
exhibiting wavelength-dependent pulse durations and spatial profiles,
such as those from an OPA.

The technique was rigorously validated through numerical simula-
tions, which revealed the critical influence of the phase object's linear
phase shift $\phi_L$ and the sample's longitudinal position $z$ on the measure-
ment sensitivity. Experimental validation confirmed a linear relation-
ship between the measured contrast and the incident power, enabling
the direct quantification of the NL phase shift $\phi_0$. The method was
successfully applied to measure the $n_2$ of various liquids across a broad
spectral range (650–900nm). The results for water ($n_2 = (2.0 \pm 0.4) \times 10^{-20}$ m²/W) and carbon disulfide ($n_2 = (43.8 \pm 11.4) \times 10^{-20}$ m²/W) are consistent with established values and demonstrate the expected
spectral stability in the femtosecond regime. Finally, the method was
cross-validated against traditional BSRV-Z-scan measurements, con-
firming its accuracy and reliability.

By providing all necessary simulation and analysis code in the *Supplementary Materials*, we support the reproducibility and broader adoption of this technique. This work establishes a reliable, accessible, and experimentally efficient framework for NL optical characterization, especially in conditions where conventional methods are limited by intensity calibration challenges.

CRediT authorship contribution statement

**Georges Boudebs:** Writing – original draft, Validation, Supervision, Software, Resources, Project administration, Methodology, Investigation, Formal analysis, Data curation, Conceptualization. **Julien-Bilal Zinoune:** Investigation. **Christophe Cassagne:** Software, Resources, Investigation. **Mihaela Chis:** Project administration.

Funding

This work was supported by the “*Contrat de Plan État-Région*” (CPER) program FEMTOMAT (2021–2027). This research was funded through a doctoral contract awarded by the University of Angers and supplemented by financial support from ESAIP, covering half of the total expenses.

Declaration of competing interest

The authors declare that they have no known competing financial
interests or personal relationships that could have appeared to influence
the work reported in this paper.

Acknowledgment

The authors gratefully acknowledge the University of Angers for its
financial support. The authors would also like to thank Dr. Cyril Mau-
clair for useful discussions.
# Appendix A

## Estimation of Focusing Error Due to Sample thickness

This appendix estimates the measurement error introduced by not adjusting the CCD position when replacing the reference sample with a test sample of different thickness. The theoretical foundation of the focus shift is first established, followed by a numerical simulation of its impact on the measured contrast.

### Theoretical focus shift

When a plane-parallel plate of thickness L and refractive index n is inserted near the image focal plane of a thin lens, at a distance d = f - z with z << f, the axial image position shifts by an amount that is, to a first approximation, independent of the exact plate position. Replacing an air segment of length L with glass introduces an optical path difference (n-1)L, so that the effective image distance becomes: $p'_{i} = p_{i}^{(0)} - (n-1)L$, where $p_{i}^{(0)} = 2f$ in our 4f imaging configuration (see Fig. 1). Consequently, the image plane moves toward the lens by (n-1)L, irrespective of the small displacement z of the plate around the focal plane, provided that aberrations and higher-order terms are negligible.

### Numerical error estimation

<figure><img src="image_9.png" /><figcaption>**Fig. A1.** (a) Simulated contrast as a function of z for a defocus of -1.9 mm at 650 nm. (b) Relative error in the measured contrast with a CCD defocus of -1.9 mm, plotted as a function of sample position for wavelengths of 650 nm (red), 775 nm (green), and 900 nm (black).</figcaption></figure>

We assume perfect focus was initially achieved on the phase object image with the reference glass (thickness $L_r = 3.73$ mm, refractive index $n_r = 1.5$), corresponding to the CCD position $p'_{i}(\text{ref})$. Introducing a test sample of thickness $L \approx 7.5$ mm and $n = 1.5$ without refocusing induces a defocus error of $(n-1)(L - L_r) \approx -1.9$ mm. This value represents the most severe defocus case encountered in our measurements reported in Table 1. To quantify the resulting error in the measured contrast for different wavelengths, we performed a simulation using the program 'sim_Zscan_f phi_L.m' (see SM-1), setting the defocus parameter to $\text{defoc} = -1.9$ mm. The simulation parameters were: $I_0 = 65$ GW/cm$^2$, $z = [-20, 20]$ mm, (other parameters remain identical to those considered in the main text). The results, presented in Fig. A1(a), show the contrast at $\lambda = 650$ nm as a function of the test sample position for the CCD in the image plane ($\text{defoc} = 0$, red solid line) and with a defocus of -1.9 mm (dashed red line). A relatively large shift of the $C = f(z)$ profile is observed, although the position of the maximum signal remains unchanged. If the focus is not corrected, this shift leads to an underestimation of $n_2$. Panel (b) of the same figure shows the dependence of the relative measurement error $\Delta C/C$ on the sample position z for $\lambda = 650, 775, 900$ nm. The importance of image focusing for measurement quality is clearly demonstrated.

To minimize measurement errors, the sample should be positioned between z = 0 and -10 mm, an experimentally accessible range. Within this interval, the induced error preserves the correct order of magnitude of the measured $n_2$, ensuring reliable qualitative evaluation. However, for quantitative accuracy, the CCD should be mounted on a manual translation stage to correct for the image defocus, provided that the latter remains small.

In summary, this method simplifies the experimental workflow by replacing the continuous scanning required in a traditional Z-scan with a single-point measurement at a pre-calibrated position, determined through an initial one-time optimization. This approach accelerates measurements, enhances robustness for routine characterization, and makes the technique accessible to laboratories without automated translation stages.

# Appendix B. Uncertainty analysis of the measured nonlinear refractive index

Based on Eq. (6),

$$n_{2x} = \frac{n_{2r}}{L_x} \left( L_r \frac{C_{x+r}}{C_r} - L_c \right),$$

the absolute uncertainty $\Delta(n_{2x})$ is estimated by standard propagation of the relative uncertainties of each parameter.

The reference material (fused silica) was measured in Ref. [12] with a relative uncertainty of 22%. Sample interaction lengths are measured with a relative uncertainty of 1%. For contrast measurements $C_{x+r}$ and $C_r$, the dominant sources of uncertainty are the laser pulse energy stability (< 1%, experimentally verified) and CCD noise combined with sample optical quality. For high-optical-quality samples and optimal imaging conditions, an additional relative uncertainty of 1.5% per contrast measurement is estimated.

We denote the relative uncertainties on interaction lengths and contrasts as
$$ \frac{\Delta(L_i)}{L_i} = \varepsilon_L, \quad \frac{\Delta(C_i)}{C_i} = \varepsilon_C. $$

Defining

$$ B = L_r \frac{C_{x+r}}{C_r}, $$

the relative uncertainty on $n_{2x}$ is given by

$$ \frac{\Delta(n_{2x})}{n_{2x}} = \sqrt{\left(\frac{\Delta(n_{2r})}{n_{2r}}\right)^2 + \varepsilon_L^2 + \frac{B^2(\varepsilon_L^2 + 2\varepsilon_C^2) + (\varepsilon_L L_c)^2}{(B - L_c)^2}}. $$

The denominator in the last term must not vanish; therefore, experimental configurations approaching $B \approx L_c$ should be avoided. In the common experimental regime $B \gg L_c$, the expression simplifies to

$$ \frac{\Delta(n_{2x})}{n_{2x}} \approx \sqrt{\left(\frac{\Delta(n_{2r})}{n_{2r}}\right)^2 + 2\varepsilon_L^2 + 2\varepsilon_C^2}. $$

This result shows that the uncertainty on the nonlinear refractive index of the investigated material remains close to that of the reference $n_{2r}$, since the contributions associated with length and contrast measurements are negligible in comparison.

## Appendix C. Supplementary data

Supplementary data to this article can be found online at https://doi.org/10.1016/j.optlastec.2026.114888.

## Data availability

Data will be made available on request.

All raw data, and MATLAB processing scripts used in this study are archived at the University of Angers. These materials are available from the corresponding author upon reasonable request.

## References

[1] M. Sheik-Bahae, A.A. Said, T.H. Wei, D.J. Hagan, E.W. Van Stryland, Sensitive measurement of optical nonlinearities using a single beam, IEEE J. Quantum Electron. 26 (4) (2002) 760-769.

[2] D.V. Petrov, A.S.L. Gomes, C.B. De Araújo, Reflection Z-scan technique for measurements of optical properties of surfaces, Appl. Phys. Lett. 65 (9) (1994) 1067-1069.

[3] C.B. De Araújo, A.S. Gomes, G. Boudebs, Techniques for nonlinear optical characterization of materials: a review, Rep. Prog. Phys. 79 (3) (2016) 036401.

[4] P. Qi, Q. Su, D. Lu, L. Lin, N. Zhang, W. Liu, Optical nonlinearities of alcoholic liquids under high-repetition-rate femtosecond lasers by single beam time-resolved eclipsed Z-scan, Opt. Laser Technol. 109 (2019) 643-647.

[5] W.W. Liu, X. Zeng, L. Lin, P.F. Qi, N. Zhang, Spectral domain Z-scan technique, Opt. Lasers Eng. 146 (2021) 106693.

[6] G. Boudebs, S. Cherukulappurath, Nonlinear refraction measurements in presence of nonlinear absorption using phase object in a 4f system, Opt. Commun. 250 (4-6) (2005) 416-420.

[7] G. Boudebs, S. Cherukulappurath, Nonlinear optical measurements using a 4f coherent imaging system with phase objects, Physical Review A—Atomic Molecular, and Optical Physics 69 (5) (2004) 053813.

[8] Y. Li, Y. Song, J. Yang, K. Yang, X. Zhang, Y. Wang, Measurement of third-order nonlinear refraction using a nonlinear-imaging technique with positive-negative phase object, Appl. Phys. B 92 (2) (2008) 253-257.

[9] J. Yang, Y. Song, Direct observation of the transient thermal-lensing effect using the phase-object Z-scan technique, Opt. Lett. 34 (2) (2009) 157-159.

[10] J. Yang, Y. Song, Y. Wang, C. Li, X. Jin, M. Shui, Time-resolved pump-probe technology with phase object for measurements of optical nonlinearities, Opt. Express 17 (9) (2009) 7110-7117.

[11] G. Boudebs, M. Chis, A. Monteil, Contrast increasing by third-order nonlinear image processing: a numerical study for microscopic rectangular objects, Opt. Commun. 150 (1-6) (1998) 287-296.

[12] G. Boudebs, J.B. Zinoune, C. Cassagne, C. Mauclair, Flexible determination of the nonlinear refractive index or pulse duration in the visible range using Z-scan-based technique: application to fused silica, Appl. Phys. B 131 (7) (2025) 139.

[13] G. Boudebs, V. Besse, C. Cassagne, H. Leblond, C.B. de Araújo, Nonlinear characterization of materials using the D4σ method inside a Z-scan 4f-system, Opt. Lett. 38 (2013) 2206-2208.

[14] J.W. Goodman, Introduction to fourier optics, Roberts and Company publishers, 2005.

[15] G. Boudebs, K. Fedus, Linear optical characterization of transparent thin films by the Z-scan technique, Appl. Opt. 48 (21) (2009) 4124-4129.

[16] W. Robert, Boyd, nonlinear optics, 4th ed., Academic Press, 2020.

[17] I.A. Heisler, R.R. Correia, T. Buckup, S.L. Cunha, N.P. da Silveira, Time-resolved optical Kerr-effect investigation on CS2/polystyrene mixtures, J. Chem. Phys. 123 (5) (2005).

[18] S. Kedenburg, A. Steinmann, R. Hegenbarth, T. Steinle, H. Giessen, Nonlinear refractive indices of nonlinear liquids: wavelength dependence and influence of retarded response, Appl. Phys. B 117 (3) (2014) 803-816.

[19] P.P. Ho, R.R. Alfano, Optical Kerr effect in liquids, Phys. Rev. A 20 (5) (1979) 2170.

[20] H. Wang, C. Ciret, J.L. Godet, C. Cassagne, G. Boudebs, Measurement of the optical nonlinearities of water, ethanol and tetrahydrofuran (THF) at 355 nm, Appl. Phys. B 124 (6) (2018) 95.

[21] Z.W. Wilkes, S. Varna, Y.-H. Chen, H.M. Milchberg, T.G. Jones, A. Ting, Direct measurements of the nonlinear index of refraction of water at 815 and 407 nm using single-shot supercontinuum spectral interferometry, Appl. Phys. Lett. 94 (2009) 211102.

[22] S. Couris, M. Renard, O. Faucher, B. Lavorel, R. Chaux, E. Koudoumas, X. Michaut, An experimental investigation of the nonlinear refractive index (n2) of carbon disulfide and toluene by spectral shearing interferometry and z-scan techniques, Chem. Phys. Lett. 369 (3-4) (2003) 318-324.

[23] I. Rau, F. Kajzar, J. Luc, B. Sahraoui, G. Boudebs, Comparison of Z-scan and THG derived nonlinear index of refraction in selected organic solvents, J. Opt. Soc. Am. B 25 (10) (2008) 1738-1747.

[24] L. Agiotis, M. Meunier, Optical power limiter in the femtosecond filamentation regime, Sci. Rep. 11 (1) (2021) 14270.

[25] B. Derkowska-Zielinska, K. Fedus, H. Wang, C. Cassagne, G. Boudebs, Nonlinear optical characterization of disperse orange 3, Opt. Mater. 72 (2017) 545-548.

[26] C.B. de Araújo, A. Humeau, G. Boudebs, V.D. Del Cacho, L.R. Kassab, Giant third-order nonlinearity of lead and germanium based films in the visible and in the infrared, J. Appl. Phys. 101 (6) (2007).

[27] C.B. de Araújo, E.L. Falcão-Filho, A. Humeau, D. Guichaoua, G. Boudebs, L. R. Kassab, Picosecond third-order nonlinearity of lead-oxide glasses in the infrared, Appl. Phys. Lett. 87 (22) (2005).