# Generalized Heterodyne Interferometry in Kerr Materials

Arnaud Rogemont, Rajath Sawant, Aurélien Coillet, Benoit Cluzel

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Arnaud Rogemont, Rajath Sawant, Aurélien Coillet, Benoit Cluzel. Generalized Heterodyne Interferometry in Kerr Materials. Advanced Photonics Research, 2025, pp.2500020. 10.1002/adpr.202500020 . hal-05250623

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# Generalized Heterodyne Interferometry in Kerr Materials

Arnaud Rogemont, Rajath Sawant, Aurélien Coillet,* and Benoit Cluzel

Interferometry has long been used to measure the phase of light signals. Combined with a heterodyne detection scheme, it allows to simultaneously record amplitude and phase variations. In this work, these well-known techniques are exploited to evaluate the nonlinear phase induced by the optical Kerr effect during the propagation of a laser pulse in a nonlinear medium. It is shown that the nonlinear index can easily be retrieved when the accumulated phase remains small, but counter-intuitive results can be observed at higher powers. In particular, the combination of pulsed excitation and Kerr nonlinearity results in nonlinear variations of the amplitude of the interference pattern, whose shape depends on the pulse profile and the dispersion of the propagation medium.

## 1. Introduction

Heterodyne detection, a technology developed primarily for radio telecommunications, has been adapted for use at optical frequencies since the 1960s.[1] When mixed with a local oscillator, it provides a means to shift down the frequency of a given signal, enabling the measurement of both phase and amplitude of the signal. Optical heterodyne, also called coherent detection, has since been used in a variety of applications, most prominently for advanced optical communication modulation schemes,[2] high resolution spectroscopy,[3] light detection and ranging,[4] or frequency combs stabilization[5] to name just a few. Besides these applications, optical heterodyne has also been used to measure various aspects of light propagation in fibers optics and photonics integrated circuits: the chirp of long pulses,[6,7] the phase delay and transmission of electro-optic modulators (EOMs),[8] the gain dynamics in semiconductor optical amplifiers,[9] the frequency shift in Brillouin-based optical time domain reflectometer,[10] and recently the nonlinear phase shift of nanophotonics waveguides.[11,12]

This latter use of the heterodyne detection technique in optics for accurately evaluating the Kerr coefficient is of particular interest, as the few measurement techniques available to experimentalists often yield conflicting results, making the matter a

topical issue.[13] The improved sensitivity, lower input power requirement, and ease of implementation compared to Z-scan,[14] D-scan[15] or four-wave-mixing techniques[16] make the heterodyne detection scheme a compelling alternative to these techniques, capable of bringing new insights to the field. The first results reported in work done by Ibouussina et al.[11] were limited to small nonlinear phase shifts, typically well below $\pi$, and where the light pulse shape remains unchanged during propagation. In this work, we further the analysis of the self-heterodyne technique applied to the mea-

surement of the Kerr index to the cases where the nonlinear phase become larger than $\pi$, and the output pulse profile differs from the input profile. We numerically simulate the output of such an interferometer and find rich and counter-intuitive dynamics. In particular, the interference pattern amplitude is shown to evolve nonlinearly with the input power, although the Kerr effect only affects the phase of the propagating light. It is demonstrated that these nonlinearities exhibit a high degree of sensitivity to the incoming pulse profile. As such, this finding presents a novel avenue for utilizing the heterodyne interferometry approach as a versatile technique for the characterization of short pulses. The experimental demonstration of these results is finally provided, and a quantitative agreement is achieved.

## 2. Nonlinear Pulse Evolution

The evolution of the complex envelope $A(z, t)$ of a light pulse in an optical waveguide with pure Kerr nonlinearity is accurately modeled by the well-known nonlinear Schrödinger equation

$$ \frac{\partial A(z, t)}{\partial z} = -\frac{\alpha}{2} A(z, t) - i \frac{\beta_2}{2} \frac{\partial^2 A(z, t)}{\partial t^2} + i\gamma |A(z, t)|^2 A(z, t) \quad (1) $$

where $\alpha$ is the linear loss, $\beta_2$ the dispersion parameter, and $\gamma$ the effective nonlinearity of the waveguide. In the general case, the nonlinear nature of this equation implies that the calculation of the envelope at the output of the waveguide, $A(z=L, t)$, is not trivial and requires numerical simulations. In the case where both dispersive and nonlinear effects are small enough, the pulse profile remains unchanged during the propagation, and the effect of the waveguide can be described by a simple complex instantaneous transmission $T(t)$.

$$ A(L, t) = \bar{T}(t)A(0, t) \qquad \bar{T}(t) = T(t)e^{i\varphi_{\text{NL}}(t)} \quad (2) $$

$$ T(t) = \exp(-\alpha L) \qquad \varphi_{\text{NL}}(t) = \gamma |A(0, t)|^2 L_{\text{eff}} \quad (3) $$

A. Rogemont, R. Sawant, A. Coillet, B. Cluzel
Laboratoire Interdisciplinaire Carnot de Bourgogne
UMR 6303 CNRS-Université Bourgogne Europe
9 Avenue Alain Savary, 21000 Dijon, France
E-mail: aurelien.coillet@u-bourgogne.fr

The ORCID identification number(s) for the author(s) of this article can be found under https://doi.org/10.1002/adpr.202500020.

© 2025 The Author(s). Advanced Photonics Research published by Wiley-VCH GmbH. This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

DOI: 10.1002/adpr.202500020
A visual illustration of the effect of the nonlinear phase shift $\varphi_{NL}$ at the optical oscillation scale is provided in Figure 2a. The condition for this approximation to hold true is that the effective length $L_{\text{eff}} = [1 - \exp(-\alpha L)]/\alpha$ remains much shorter than both the nonlinear length $L_{\text{NL}} = 1/\gamma|A|^2$ and the dispersion length $L_D = \tau^2/\beta_2$, with $\tau$ being the duration of the pulse.

**3. Self-heterodyne and Kerr Nonlinearity**

Since self-heterodyne detection is sensitive to both the phase and amplitude of light, it is perfectly suited to measure the nonlinear phase shift $\varphi_{NL}$ when the previously discussed approximations and Equation (3) are valid. The demodulation process performed in a lock-in amplifier expectedly outputs the nonlinear phase averaged over the pulse profile, showcasing a linear variation with the input power, and allowing for the direct measurement of the effective nonlinearity.[11] Outside of this approximation though, the deformation of the pulse under the combined effects of dispersion and nonlinearity make the analysis of the heterodyne interference pattern much more complex, and this is precisely what we address in this article.

The self-heterodyne setup for the measurement of Kerr-induced phase shifts is depicted in **Figure 1**. It is based on an all-fiber Mach–Zehnder interferometer in which the signal arm goes through the sample waveguide where it undergoes the nonlinear transformation described by Equation 1, which condenses into the complex transmission of Equation (3) when the low power and low dispersion approximation holds. The reference arm of the Mach–Zehnder interferometer has its frequency shifted by $\varepsilon = 2\pi \times 110$ MHz thanks to an acousto-optic modulator (AOM), as illustrated in **Figure 2b**. A delay-line is also included in the reference arm to synchronize both pulses at the output coupler. A balanced photodiode (BPD) detects the interference beatnote and removes the continuous-wave (CW) component, so that only the oscillations of interest $x(t)$ are sent to the lock-in amplifier (see Figure 2d for a conceptual description of such a signal). The lock-in amplifier both generates the frequency shift $\varepsilon$ signal driving the AOM and uses it to demodulate the heterodyne beatnote signal coming from the BPD,

<figure><img src="image_2.png" /><figcaption>**Figure 2.** Conceptual analysis of the relevant signals: a) Amplitude of the optical field at the femtosecond scale, showing the Kerr-induced nonlinear phase shift $\varphi_{NL}$. b) Optical spectra of the pulse trains, through the sample in red and through the AOM in green. c) Power ramp at the input of the setup, and d) interference signal $x(t)$ (red) and its enveloppe (pink) compared to the local oscillator (green). The phase difference $\theta$ between these latter signals is due to $\varphi_{NL}$. For readability purposes, the various time scales have not been respected.</figcaption></figure>

producing either in-phase (I) and quadrature (Q) signals, or amplitude (R) and phase (θ) signals.

This detection system is excited with a mode-locked pulsed laser delivering 3 ps duration pulses at a repetition rate of $f_{\text{rep}} = 80$ MHz. The output power is a moderate 20 mW, corresponding to a peak power of roughly 40 W, sufficient to generate measurable nonlinear phase shifts in highly nonlinear waveguides. To evaluate such phases shifts as a function of the input power, an intensity EOM driven by a slow (1 kHz) voltage ramp is placed after the laser, effectively varying the input power periodically. The input power is measured simultaneously with a tap and a photodiode ($D_0$).

<figure><img src="image_3.png" /><figcaption>**Figure 1.** a) Scheme of the heterodyne Mach–Zehnder interferometer. EOM–Electro-optic modulator, AOM–Acousto-optic modulator, LO–local oscillator, $D_0$, $D_1$, and $D_2$–Photodiode. b) Picture of the actual experiment.</figcaption></figure>

<figure><img src="image_4.png" /></figure>
## 4. Analytical and Numerical Analysis

In this context, and assuming the peak input pulse power $P_{in}$ is low enough, the output signal from an ideal, infinite-bandwidth BPD would be

$$x(t) = P_{in}E(t)^2T(t) \cos(\epsilon t - \varphi_{NL}(t)) \quad (4)$$

where $E(t)$ is the normalized envelope of the input pulse. Due to its bandwidth limitations, the output of the real BPD is averaged with a timescale that sits between the pulse duration $\tau \approx 1$ ps and $1/\epsilon \approx 10$ ns. The actual output from the BPD can therefore be written as

$$x(t) = P_{in} \cos(\epsilon t) \langle T(t') E(t')^2 \cos(\varphi_{NL}(t')) \rangle + P_{in} \sin(\epsilon t) \langle T(t') E(t')^2 \sin(\varphi_{NL}(t')) \rangle \quad (5)$$

where the average is performed over the fast time $t'$ whose timescale is shorter than the pulse duration. In contrast, $t$ represents the BPD bandwidth limited time.

The lock-in amplifier demodulates this signal by multiplying it with the local oscillator at frequency $\epsilon$ and averaging over a longer duration, typically 1 µs in our case. The in-phase $I$, quadrature $Q$, amplitude $R$, and angle $\theta$ output signals from the lock-in amplifier are

$$\begin{aligned} I &= \langle T(t') P_{in} E(t')^2 \cos(\varphi_{NL}(t')) \rangle \\ Q &= \langle T(t') P_{in} E(t')^2 \sin(\varphi_{NL}(t')) \rangle \\ R &= \sqrt{I^2 + Q^2} \qquad \theta = \arctan \frac{I}{Q} \end{aligned} \quad (6)$$

If the phase variations are only due to the Kerr effect, and small enough that they do not change the shape of the pulse, then the power transmission $T$ no longer depends on time $t'$, and the phase shift is proportional to the pulse profile: $\varphi_{NL}(t') = \gamma P_{in} E(t')^2 L_{eff}$. Plugging this expression within Equation (6) and using a Taylor expansion for $|\varphi_{NL}| \ll 1$ leads to the following expression for $R$ and $\theta$

$$R \approx TP_{in} \langle E^2(t') \rangle \qquad \theta \approx \gamma P_{in} L \frac{\langle E(t')^4 \rangle}{\langle E(t')^2 \rangle} \quad (8)$$

Hence, for low nonlinear phase shifts, both the amplitude $R$ and angle $\theta$ at the output of the lock-in amplifier are proportional to the input peak power $P_{in}$. The angle $\theta$ in particular differs from the CW case by a factor that depends only on the pulse profile: $\kappa = \langle E(t')^4 \rangle / \langle E(t')^2 \rangle$. This factor $\kappa$ is necessarily lower than 1, and its value for a few typical pulse shapes and for our experimental laser source are given in **Figure 3**.

Figure 2 illustrates all relevant signals conceptually, without adherence to actual scale, to preserve legibility. Figure 2a displays the field oscillations within a single pulse in the presence of a Kerr-induced phase shift $\varphi_{NL}$. Figure 2b plots the spectra of the pulse trains passing through the sample (red) and through the AOM (green). The spectral lines of the latter have been shifted by the local oscillator frequency $\epsilon$. Figure 2c shows the optical intensity of the input signal at the millisecond scale, with both the repetition rate and the power ramp visible. Finally,

<figure><img src="image_5.png" /></figure>

<table><caption>**Figure 3.** Top: Amplitude (red) and phase (purple) of the input pulse as retrieved from the FROG measurement. Bottom: table of $\kappa$ values for different pulse shapes, the experimental one being retrieved from the FROG data.</caption><thead><tr><th>Pulse</th><th>sech</th><th>Gaussian</th><th>Super-Gaussian</th><th>Exp.</th></tr></thead><tbody><tr><td>κ</td><td>2/3 ≈ 0.667</td><td>√2/2 ≈ 0.707</td><td>1/√3 ≈ 0.841</td><td>0.698</td></tr></tbody></table>

Figure 2d plots the signal detected by the BPD $x(t)$ in red, showing the pulse train, the beatnote interference at $\epsilon$ frequency, and the power ramp. The lock-in amplifier demodulates this signal at the frequency $\epsilon$, effectively retrieving the amplitude and phase with respect to the local oscillator (green) of the envelope (pink) of $x(t)$. In the presence of a Kerr-induced optical phase, the demodulated signal becomes out of phase with the local oscillator, by an angle $\theta$.

For higher powers and nonlinear phase shifts, the Taylor expansion introduced previously no longer suffices. To take into account the full effects of the propagation of the input pulse in the Kerr media, we first numerically solve the nonlinear Schrödinger equation using a split-step Fourier method with an adaptive step size based on the nonlinear and dispersion lengths. We then numerically calculate the interference pattern as well as the output from the lock-in amplifier. The results from **Figure 4**, obtained for various input profiles (CW, hyperbolic-secant, Gaussian, and super-gaussian of order 2) and for a nonlinear phase shift reaching $4\pi$, show that both amplitude $R$ and angle $\theta$ no longer depend linearly on the input power $P_{in}$. Strikingly, the evolution of the amplitude $R$ with the input power (Figure 4c) displays oscillations whose depths depend on the shape of the initial pulse. Specifically, pulses with wider tails have stronger fluctuations compared to pulses with sharper edges, such as super Gaussian pulses. For small nonlinear phase shifts, typically below 1, the angle $\theta$ of Figure 4d shows a linear variation with the input power, with slopes that depend on the pulse shape but remain below the CW case, in agreement with the Taylor expansion discussed previously. For higher input powers and phase shifts though, the angle curves move away from this initial linear evolution, and become parallel to the CW curve at large input powers.

The shape and phase of a propagating pulse in a waveguide is influenced by both nonlinearity and dispersion, according to the nonlinear Schrödinger equation. We performed the same numerical analysis to evaluate the effect of these parameters and gather the results in **Figure 5**. Figure 5a,b displays the envelope and optical phase of the sech pulse at the output of a waveguide with zero dispersion and increasing effective nonlinearity. As expected, the nonlinear phase shift is maximum at the center
<figure><img src="image_6.png" /><figcaption>**Figure 4.** a) Output profile and b) phase profile of the laser pulses used in the simulation of the self-heterodyne setup. c) Amplitude *R* and angle *θ* of the interference pattern, at the output of the lock-in amplifier, as a function of the input peak power, for different pulse profiles.</figcaption></figure>

of the pulse and for higher nonlinearities, and the pulse profile is
slightly flattened. The output of the lock-in amplifier is shown in
Figure 5c,d. The amplitude *R* exhibits the same oscillations as
previously discussed, except for vanishing effective nonlinearity.
More specifically, the frequency of the oscillations increases
when $\gamma$ increases, which could allow for the evaluation of the
nonlinear parameter with the interference amplitude only.

The angle $\theta$ strongly depends on the effective nonlinearity in
a more intuitive fashion, since the overall slope increases as $\gamma$
increases, as expected from the analytical approximation.

For a fixed nonlinearity $\gamma = 9.5 \text{ W}^{-1} \text{ m}^{-1}$, a change in the dispersion of the waveguide can produce much more diverse effects, as is shown in Figure 5e,f. While normal dispersion only broadens and chirps the pulses, the anomalous dispersion case lead to pulse compression and splitting. The output from the lock-in amplifier can nevertheless be calculated, and the results are shown in Figure 5g,h. The change in pulse profile has a direct influence on the amplitude $R$, with oscillations that are barely marked in the normal dispersion case, and high-contrast patterns when the pulse is split into several. In this case, the amplitude $R$ of the interference pattern almost vanishes for periodic values of the input power. This dips in the $R$ signal corresponds to $\pi/2$ or $3\pi/2$ abrupt changes in the angle $\theta$ (Figure 5h), which means that the interference pattern change sign at such points.

**5. Experimental Section**

To experimentally verify our numerical analysis, we used the
setup of Figure 1 with two 50 m segments of highly nonlinear
fiber (HNLF) as reference samples, one with normal dispersion
and the other with anomalous dispersion. HNLF is indeed par-
ticularly well-suited for this purpose as it has a large nonlinear
coefficient of $\gamma \approx 11 \text{ W}^{-1} \text{ km}^{-1}$, low linear losses, and no other
nonlinear effects such as two-photon absorption or free carrier
effects, that would make the interpretation much more complex.
A mode-locked fiber laser generates the input pulses with a pro-
file that has been preliminarily fully characterized using a
frequency-resolved optical gating (FROG), as shown in Figure 3.
With a maximum average power of 15 mW injected inside the
HNLF, the input peak power $P_{\text{in}}$ reaches a maximum of 35 W.

<figure><img src="image_7.png" /><figcaption>**Figure 5.** a–d) Influence of the nonlinearity and e–h) dispersion on the output of the lock-in amplifier. (a,e) are the output pulse profiles after propagation in the waveguide, (b,f) are their phase profile, (c,g) are the interference pattern amplitude, and (d,h) the angle $\theta$ after demodulation.</figcaption></figure>
<figure><img src="image_8.png" /><figcaption>**Figure 6.** Comparison between experiments and numerical analysis. Measured interference a) amplitude and b) angle as a function of the input power (black) in the normal dispersion case, compared to the simulations at the different stages of the fitting procedure. The red line indicates the best fit obtained with a Gaussian as the input pulse profile. c,d) corresponds to the interference amplitude and angle for the 50 m HNLF with anomalous dispersion.</figcaption></figure>

The experimental output from the lock-in amplifier is shown in black in **Figure 6**, and the amplitude *R* clearly exhibit oscillations when the input power is changed, as calculated previously. The angle *θ* also follows the same pattern as the numerical simulations of Figure 4, with a linear increase at low power, then bending toward a slightly stronger slope at high power. The linear section of this curve is ideal to obtain a first value of the effective nonlinearity of the waveguide, which yields *γ* = 10.6 W⁻¹ m⁻¹ in the normal dispersion case and 10.4 W⁻¹ m⁻¹ in the anomalous case, in good agreement with the manufacturer's own measurement at 10.9 W⁻¹ m⁻¹ and 11.6 W⁻¹ m⁻¹, respectively.

In the anomalous case, shown in Figure 6c,d, both the amplitude *R* and angle *θ* exhibit pronounced and seemingly irregular patterns. This behavior is consistent with the simulations presented in Figure 5g,h, indicating that the pulse undergoes significant modification during its propagation through the HNLF section.

To further the analysis of the experimental results, we performed the previously discussed numerical analysis using the envelope retrieved from the FROG measurement as input profile. The nonlinear coefficient *γ* and dispersion *D* parameters are then adjusted to match the experimental curves following a multistep procedure: the *κ* coefficient is first calculated from the retrieved FROG pulse profile, and used in the calculation of the effective nonlinearity *γ* from the initial slope of the measured angle *θ*, as shown in Figure 6b, dashed line. Starting with this initial guess, *γ* and *D* are adjusted using a least square algorithm to match the measured angle *θ*(*P*<sub>in</sub>) curve for all values of

<table><caption>**Table 1.** Summary of the manufacturer values for both HNLF and comparison with our measurement and their associated standard deviations.</caption><thead><tr><th></th><th><i>D</i> [ps nm<sup>-1</sup> km<sup>-1</sup>]</th><th><i>γ</i> [W<sup>-1</sup> km<sup>-1</sup>]</th></tr></thead><tbody><tr><td colspan="3"><strong>Normal dispersion HNLF</strong></td></tr><tr><td>Manufacturer</td><td>-1.2</td><td>10.9</td></tr><tr><td>This work</td><td>-1.5 ± 0.4</td><td>10.6 ± 1.0</td></tr><tr><td colspan="3"><strong>Anomalous dispersion HNLF</strong></td></tr><tr><td>Manufacturer</td><td>1.0</td><td>11.6</td></tr><tr><td>This work</td><td>1.2 ± 0.4</td><td>10.4 ± 1.0</td></tr></tbody></table>

*P*<sub>in</sub>. A second stage of adjustment is performed on *R*(*P*<sub>in</sub>) to yield the green curves of Figure 6a,b. The agreement between experimental and simulation traces is now adequate in the normal dispersion regime for both amplitude *R* and angle *θ*, demonstrating the soundness of the previously described analysis.

In the anomalous dispersion regime, the fitting procedure becomes significantly more delicate due to the extreme sensitivity to various parameters, in particular the input pulse power and duration. Nevertheless, the shapes of the *R*(*P*<sub>in</sub>) and *θ*(*P*<sub>in</sub>) curves exhibit qualitative agreement with the theoretical predictions, with pulse break-up occurring at ≈15 W of peak power. The extracted HNLF parameters are also in good agreement with the manufacturer's specifications.

The retrieved values for the dispersion parameter *D* and effective nonlinearity *γ* for both fibers are summed up and compared with the manufacturer's specification in **Table 1**. In the normal dispersion case, the uncertainty on the evaluation of the nonlinear parameter is actually largely dominated by the evaluation of the peak power at the input of the HNLF, while the error from the adjustment remains below 2%.

The red curves displayed on Figure 6 also illustrate the sensitivity to the input pulse profile showcased in Figure 4: they represent the best fit obtained with a Gaussian pulse instead of the actual profile, as retrieved from the FROG measurements. The mismatch is especially obvious on the amplitude signal, where the depth of the simulated oscillations does not correspond to the experimental ones. One could imagine that such a strong sensitivity to the exact shape of the input profile could be leveraged to perform the opposite measurement as initially intended: using a known nonlinear waveguide with a calibrated Kerr response, such as a given length of HNLF fiber, a dedicated and nontrivial adjustment procedure should be able to retrieve the input pulse profile from the self-heterodyne *R* and *θ* measurements. The development of such a retrieval algorithm promises to be convoluted, and is beyond the scope of the present article.

## 6. Conclusions

To better understand the outputs of the self-heterodyne setup introduced in work done by Ibnoussina et al.,[11] we have performed an analytical and numerical analysis of the amplitude and angle of the heterodyne interference pattern for an intense light pulse propagating in a nonlinear waveguide. The simulation
of both the nonlinear propagation and the self-heterodyne process confirms that it can allow for the accurate and simple measurement of the Kerr coefficient, provided that the nonlinear phase remains small. However, if the accumulated nonlinear phase reaches values in excess of $\pi$, the output signals of the lock-in amplifier exhibit counter-intuitive results: the angle $\theta$ is no longer proportional to the input power, and the amplitude $R$ displays oscillations that strongly depends on the pulse shape and nonlinearity of the waveguide. Since dispersion induces pulse deformations, especially in conjunction with strong nonlinearities, the output from the self-heterodyne setup is also strongly dependent on the dispersion of the sample waveguide.

These numerical results are confirmed by an experiment performed on a Mach–Zehnder interferometer with a piece of highly nonlinear fiber in the sample arm and an AOM on the other arm. The detected interference indeed displays the characteristic behaviors observed in simulations, and in particular the typical oscillations of the interference amplitude. A fitting procedure that makes use of the FROG-retrieved pulse profile is able to accurately reproduce the experimental measurements, and yield an effective nonlinearity coefficient in good agreement with the value given by the manufacturer's measurement. In addition, the dispersion parameter of the waveguide is also retrieved in the same fitting procedure. Finally, the observed oscillations depend on the input pulse profile, paving the way to a fully fibered, Kerr-based optical pulse characterization.

## Acknowledgements

This work was made possible by the facilities available on the SMARTLIGHT platform, which was funded by the Agence Nationale de la Recherche (EQUIPEX+ contract "ANR-21-ESRE-0040") and Region Bourgogne Franche-Comté. The authors acknowledge funding from the ANR (Agence Nationale de la Recherche) through the OCTANE project (grant agreement ANR-20-CE24-0019). In addition, this work has been supported in part in the frame of the NEUROPULS project, funded by the European Union's Horizon Europe research and innovation program under grant agreement no. 101070238. A.R. received funding from Région Bourgogne-Franche-Comté in the framework of the ICE program.

## Conflict of Interest

The authors declare no conflict of interest.

## Author Contributions

**Arnaud Rogemont:** conceptualization (equal); data curation (lead); formal analysis (lead); investigation (lead); methodology (equal); software (equal); validation (equal); visualization (lead); writing—original draft (equal); writing—review and editing (equal).
**Rajath Sawant:** data curation

(supporting); formal analysis (supporting); investigation (equal); validation (supporting); visualization (supporting); writing—original draft (supporting); writing—review and editing (supporting).
**Aurélien Coillet:** conceptualization (equal); data curation (equal); formal analysis (equal); investigation (supporting); methodology (equal); software (equal); supervision (equal); validation (equal); visualization (equal); writing—original draft (lead); writing—review and editing (lead).
**Benoit Cluzel:** conceptualization (equal); funding acquisition (lead); investigation (supporting); methodology (equal); project administration (lead); resources (lead); supervision (equal); validation (supporting); writing—original draft (equal); writing—review and editing (equal).

## Keywords

heterodyne detection, interferometry, nonlinear optics

Received: February 7, 2025

Revised: July 21, 2025

Published online:

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