# Accurate pulse-to-pulse stability measurement for femtosecond lasers

Michele Natile, Gautier Parize, Florent Guichard, Marc Hanna, Patrick Georges

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Michele Natile, Gautier Parize, Florent Guichard, Marc Hanna, Patrick Georges. Accurate pulse-to-pulse stability measurement for femtosecond lasers. Optics Express, 2025, 33 (10), pp.20410. 10.1364/oe.557987. hal-05336222

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# Accurate pulse-to-pulse stability measurement for femtosecond lasers

MICHELE NATILE,<fnref n="1" />,*, ID GAUTIER PARIZE,<fnref n="1" />,<fnref n="2" /> FLORENT GUICHARD,<fnref n="1" /> MARC HANNA,<fnref n="2" /> ID AND PATRICK GEORGES<fnref n="2" /> ID

**Abstract:** Short-term energy stability metrology of pulsed lasers is the source of widespread misleading comparisons. The most used value for this characterization in commercial products and laboratory applications is the rms pulse-to-pulse stability (P-to-P). For frequency-specific analysis, the relative intensity noise (RIN) power spectral density is measured and the integrated RIN over a certain bandwidth is given. In principle, integrated RIN and P-to-P correspond to the same value. In practice, the different experimental setups often cause results to differ significantly. In this paper, we detail the specifics of measurement setups for short-term stability characterization and focus on measurements with stability better than 1% rms. The sensitivity of the P-to-P approach with respect to the oscilloscope specifications is shown by measuring a commercial femtosecond amplifier using three different oscilloscopes. A simple modified setup, including a low-pass filter at the laser repetition rate to turn the photodetected pulse train into a quasi-sinusoidal response, is proposed. This approach allows to overcome the oscilloscope limitation in standard P-to-P setups, allowing to obtain reliable measurements lower than 1% rms with a resolution of the order of 0.1%.

Published by Optica Publishing Group under the terms of the Creative Commons Attribution 4.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

## 1. Introduction

Laser short-term energy stability has been recognized as a key parameter since its very first observation in laser physics through the study of the intensity noise [1].

Relative intensity noise (RIN) is a widespread concept used to quantify this noise. This function and its related measurement techniques have been developed originally for CW lasers and deeply explored in the context of laser diode technology [2]. For ultrafast lasers, specific theoretical models, experimental methods and techniques were studied 20 years ago, coupled to the phase noise [3,4]. These models are considered today as a standard basis for the intensity noise characterization, appearing in ISO procedures [5] and ultrafast laser community reference books [6]. Nevertheless, the most used parameter to characterize the short-term stability of a pulsed laser is the root-mean-square (rms) pulse-to-pulse stability (P-to-P). This is calculated as the standard deviation divided by the mean of an often unspecified number of pulse amplitudes, measured using a photodiode, a digital oscilloscope, and its maximum function. The simplicity of the approach and the easy access to the instrument required for the setup explains its widespread use.

Theoretically, the RIN integrated over a minimum frequency related to the measurement time and the Nyquist frequency of the laser is equal to the P-to-P in the same measurement time. In practice, the different measurement setups and data processing can result in substantial differences. Recently, Smith et al. [7] reported on RIN measurement techniques in the frequency and time domain corresponding to the RIN and P-to-P measurements. The different experimental setup

<footnote n="1">¹Amplitude, 11 Avenue de Canteranne, Cité de la Photonique, 33600 Pessac, France</footnote>

<footnote n="2">²Université Paris-Saclay, Institut d'Optique Graduate School, CNRS, Laboratoire Charles Fabry, 91127 Palaiseau, France</footnote>

<footnote>*michele.natile@amplitude-laser.com</footnote>
elements are discussed in this work and experimental results are reported for super-continuum sources exhibiting a stability of several % rms.

Femtosecond laser amplifiers at high energy (100s µJ) and high repetition rate (>10 kHz) are today widely used for industrial and scientific applications. Proper characterization of these laser source intensity fluctuations is crucial, especially when amplifiers are followed by nonlinear stages [8,9], or secondary processes where amplitude to phase noise transfer [10,11] directly impacts the final result. Depending on the technology and the approach, P-to-P and/or integrated RIN well below 1% have been reported [12,13] and appear on commercial product datasheets. For this range, accurate measurement is sometimes challenging since RIN and P-to-P measurements are sensitive to the properties of the measurement device.

In this work, we study how the oscilloscope specifications affect the P-to-P measurement when a resolution of <1% rms is required. The definition for RIN and P-to-P measurements for femtosecond sources are first recalled. We discuss equivalences and differences between RIN and P-to-P measurement with experimental evidences based on a Yb-doped femtosecond laser amplifier at 1 MHz repetition rate. We demonstrate that oscilloscope properties can lead to a large overestimation of the P-to-P parameter. A simple measurement setup and the associated procedure is proposed and demonstrated to make the measurement more robust to the use of various oscilloscopes. Using this approach, a stability of 0.2% rms is measured for a commercial Yb-doped laser amplifier, with a resolution ≥ 0.1% rms using vastly different commercial oscilloscopes.

## 2. Preliminary considerations

### 2.1. Definitions

The theoretical equivalence between the integrated RIN over a determined frequency range and P-to-P rms is discussed here. A noisy pulse train can be described in the time domain as a sum of delta functions [14]:

$$P(t) = P_0[1 + N(t)] \sum_{n=-\infty}^{+\infty} \delta(t - nT - J(t)), \quad (1)$$

where $P_0$ is the peak power of a single pulse, $T$ is the period of the pulse train, $N(t)$ is the normalized peak power fluctuation, and $J(t)$ is the timing jitter. The phase and amplitude noises can be expressed using a frequency domain analysis. The power spectral density (PSD) of the noisy pulse train is given by

$$S_p(\omega) = F\{\langle P(t)P(t + \tau) \rangle\} = P_0^2 \sum_{n=-\infty}^{+\infty} [2\pi\delta(\omega - \omega_n) + S_N(\omega - \omega_n) + n^2\omega_n^2 S_J(\omega - \omega_n)], \quad (2)$$

where $\omega_n$ is the nth harmonic of the mode-locking frequency $\omega_n = \frac{2\pi n}{T}$, $S_N$ is the amplitude noise spectral density and $S_J$ is the phase noise spectral density. We now discard the phase spectral noise contribution $S_J$ to focus only on the amplitude noise. As a result of the properties of the PSD, the amplitude noise in time $N(t)$ rms fluctuations can be related to the integrated PSD, underlining the equivalence:

$$\sigma_N = \sqrt{\langle N(t) \rangle^2} = \sqrt{\frac{1}{\pi} \int_0^{+\infty} S_N(\omega) d\omega}. \quad (3)$$

We introduce a finite bandwidth of frequencies from $\omega_{min}$ to $\omega_{max}$ to model real measurement conditions.
* $\omega_{max}$ is coming from the bandwidth of the detection system. As we will see in the next section, the detection is limited by the setup elements to a specific frequency, that will be the maximum frequency of the RIN PSD.

* $\omega_{min}$ is linked to the measurement time $\Delta T$, according to $\omega_{min} = \frac{2}{\pi\Delta T}$.

The RIN PSD is defined as the amplitude noise $S_N$. It is the above-mentioned power laser spectrum $S_p(\omega)$ normalized by the the average power. If we isolate the spectral content in the vicinity of any harmonic of $S_p(\omega)$:

$$RIN = \frac{S_p(\omega)}{p^2} = S_N. \qquad (4)$$

The integrated value of the RIN is equal to $\sigma_N^2$. This expresses the theoretical equivalence between integrated RIN and P-to-P rms measured on the same time range $\Delta T$. We now focus on experimental measurement setups and techniques, that can cause deviations between these quantities.

## 2.2. Measurement setups

In the literature, RIN measurements are often reported using an electrical spectrum analyzer (ESA) [15] or vector signal analyzer (VSA) [16]. These instruments allow to directly analyze amplitude and phase noises in the frequency domain. The low limit frequency is, in these cases, in the kHz range and these setups are expensive and complex tools for short-term laser energy fluctuations analysis. High resolution RIN measurements can also be done in the time domain using an oscilloscope and additional processing of the acquired data. This approach allows a more flexible choice of the measurement time and, as a consequence, RIN PSD lower frequencies below the kHz.

In Fig. 1(top), a typical RIN measurement setup using an oscilloscope is represented. The laser pulse train is measured using a photodetector, typically a photodiode. This signal is filtered via a low-pass filter cutting at Nyquist frequency, corresponding to half the laser repetition rate, and measured using the oscilloscope. The measurement setup components are widely available in optics laboratories, and the results can be easily compared with results of a the standard setup for P-to-P measurement, shown in Fig. 1(bottom). The measurement noise floor is obtained by realizing a RIN measurement with the same setup when no light is impinging on the photodetector, while normalizing using the same value for the average voltage.

<figure><img src="image_3.png" /><figcaption>**Fig. 1.** (Top) RIN oscilloscope typical measurement setup. (Bottom) P-to-P rms typical measurement setup.</figcaption></figure>

The value that we want to measure in both cases is the variation of the energy per pulse. In the P-to-P measurement, the key assumption is that the maximum voltage of the measured pulses varies proportionally with the pulse energy. This is true only if the measured pulse time response
behavior does not change from pulse to pulse. For instance, photodiode saturation for a pulsed source can affect the pulse energy linearity. Preventing this phenomenon in the case of a pulsed laser source is not necessarily easy [17]. Indeed, these dynamic properties of the response, for a given photodiode, change with multiple factors, including the irradiation wavelength, which implies that a calibration must be carried out for each specific conditions.

Characterizing low noise levels is delicate and requires measurement setups exhibiting high signal-to-noise ratio (SNR). Here, we want to compare two measurement techniques of low intensity fluctuations, and must carefully choose the components that determine the overall SNR.

The photodiode is chosen considering two parameters: its noise properties and its time response. The noise properties are usually expressed via the noise equivalent power (NEP) [18]. The time response must be fast enough to correctly resolve the pulse train. For femtosecond laser at repetition rate from the kHz to the MHz regime, photodiodes with a time response of the order of one nanosecond are usually used for this purpose. In principle, the same photodiode can be used both for the RIN and P-to-P measurement. The same impact on the measurement SNR is expected from this component in both cases. The filter is present only in the RIN measurement. It is a passive component and it can be designed to have a negligible noise with respect to the photodiode.

The oscilloscope performs analog-to-digital conversion (ADC) in both cases. The key instrument parameters are the vertical resolution, the vertical offset and the trigger jitter. The manufacturers' datasheets usually report on these parameters performances. For the two presented measurements setups, the different instrument operation mode can lead to relevant differences in measurement SNR. Requirements to obtain high performance RIN measurements are less demanding than for P-to-P measurements. In the first case, it is not required to trigger the signal to measure the data to be processed. Furthermore, the vertical resolution and offset limitations can be easily observed by a noise floor measurement. In contrast, estimation of the effective oscilloscope noise impact on the P-to-P measurement is not trivial because it is a result of a mix of the three cited parameters and the P-to-P rms is an integrated value including the measurement noise, which prevents its estimation. As a result, although it is widely used, it is difficult to assess the reliability of a P-to-P measurement, especially for noise levels below 1% rms. In addition, the exact conditions in which the measurement is made are seldom communicated.

### 3. Oscilloscope limitations

To demonstrate the limitations of the oscilloscopes for high resolution P-to-P measurements, we use the setup sketched in Fig. 2(a). Three different oscilloscopes are used for P-to-P measurements. The oscilloscopes specifications useful for our analysis are reported in Table 1.

<table><caption>Table 1. Comparison of the oscilloscopes specifications</caption><thead><tr><th>Oscilloscopes</th><th>(1) Agilent<br>DSO7054A</th><th>(2) Siglent<br>SDS2102XHD</th><th>(3) Rigol<br>DS2102A</th></tr></thead><tbody><tr><td>Bandwidth</td><td>500 MHz</td><td>100 MHz</td><td>100 MHz</td></tr><tr><td>Sample rate</td><td>4 GSa/s</td><td>2 GSa/s</td><td>2 GSa/s</td></tr><tr><td>Vertical resolution</td><td>8 bits</td><td>12 bits</td><td>8 bits</td></tr><tr><td>Vertical scale offset<br>(minimum)</td><td>+/- 5 V</td><td>+/- 1.6 V</td><td>+/- 2 V</td></tr><tr><td>Noise floor RMS (in<br>optimized conditions)</td><td>360 μV</td><td>50 μV</td><td>Not<br>communicated</td></tr><tr><td>Trigger jitter</td><td>15 ps rms</td><td>10 ps rms</td><td>Not<br>communicated</td></tr></tbody></table>
<figure><img src="image_4.png" /><figcaption>**Fig. 2.** Experimental setups used for the demonstration. (a) Standard P-to-P measurement setup using the same photodetector and three different oscilloscopes. This setup is discussed in section 3. (b) Optimized P-to-P measurement setup, described and discussed in section 4.</figcaption></figure>

From the vertical resolution, we can estimate that the oscilloscope quantization noise will be of the order of some tens of µV for the three oscilloscopes. We can also see that the minimum rms noise floor is of the order of tens or hundreds of µV. As a first test, we use a digital sinusoidal signal generated by a waveform generator (Rigol DG4162) at 1 MHz, 1 V of amplitude. The signal is triggered, and the sinusoidal peak variation is analyzed over the smallest vertical scale unit, corresponding to 2 mV/div, 500 µV/div and 500 µV/div respectively for oscilloscope 1, 2 and 3. This measurement gives a rough estimation of the oscilloscopes resolution capabilities to measure a stable sinusoidal signal. From the 14-bit waveform generator datasheet we can expect that the stability of the waveform is ≤ 0.1% rms. For the three oscilloscopes, the measured rms over 1-s long measurement is 40 µV corresponding to a P-to-P of 0.04% rms.

The pulse energy stability of a commercial femtosecond amplifier (Satsuma, Amplitude) emitting femtosecond pulses at 1 MHz, 1030 nm is measured using a fast Si-based photodiode (Thorlabs, DET10A/M) and the three oscilloscopes. The optical power is adjusted to have 1 V of maximum value and the measurements are realized on the smallest vertical scale units. The optical power corresponding to 1 V is experimentally chosen to be within the photodiode linearity range, thereby avoiding saturation. The P-to-P rms is 0.2%, 0.9% and 1.1% respectively using oscilloscope 1, 2 and 3 in the Table 1. The comparison evidently highlights the impact of the oscilloscope for the P-to-P measurement. In Fig. 3, the RIN PSD and integrated noise performed with the same laser conditions are reported using the oscilloscope 1 and 2 and a commercial low-pass filter (Thorlabs, EF506). Oscilloscope 3 is not used for this measurement because of the lack of information on input noise reported in the datasheet. The laser measured integrated noise in the range [1 Hz; 500 kHz] is 0.6% rms in the first case, shown in blue on the left, and 0.25% rms for the second one, shown on the right.

This result contradicts the P-to-P measurements reported above. Let us analyze in more depth the RIN measurement parameters to understand the origin of this contradiction. The measurement noise contributions for the two cases are different. For the RIN setup, the average voltage is $V_{mean} = 44$ mV in both cases after the filter at $f_{rep}/2$ inserted between the photodiode and the oscilloscope as explained in Fig. 1. From the data sheet values, the theoretical noise floors in this case are respectively 0.8% rms and 0.1% rms for the Agilent and Siglent oscilloscope cases. We can see that the oscilloscope noise is the limiting factor for the RIN measurements in Fig. 3, explaining the difference in the integrated RIN measured values. From the RIN measurements
<figure><img src="image_5.png" /><figcaption>**Fig. 3.** RIN measurement for laser at 1 MHz. (Left) Oscilloscope 1 RIN PSD (top) and integrated RIN (bottom). (Right) Oscilloscope 2 RIN PSD (top) and integrated RIN (bottom).</figcaption></figure>

shown in Fig. 3, we can conclude that the short-term stability of the laser under test is 0.25% rms when the measurement noise floor is significantly lower than this value.

In contrast, this is not valid for P-to-P measurements. As already mentioned, it is not easy to discriminate the precise origin of the P-to-P overestimation, mixing different elements in oscilloscope performances. The main effects are probably related to different noise characteristics, coupled to various trigger, sampling, and bandwidth capabilities of the oscilloscopes. This can lead to overestimate the intensity fluctuations, for instance through phase-to-amplitude coupling. Another important element to discuss here is whether or not the P-to-P measured pulses are consecutive or not. In the ISO standard [5], the indication for P-to-P is to measure, if possible, 100 consecutive pulses. This definition is rarely applicable to the case of high repetition rate lasers where the small number of measured pulses does not correspond to a meaningful characterization of the short-term energy stability. The most common lab practice is to measure a certain number of non-consecutive pulses using the oscilloscope. The number of measured pulses during a defined measured time depends on the oscilloscope trigger characteristics, which affects the measured value. For these reasons, in the next section, we propose a modified setup for P-to-P measurements to reduce the sensitivity to photodiode pulse response behavior and phase-to-amplitude couplings.

# 4. Optimized P-to-P measurement method

## 4.1. Rationale

The modified setup for P-to-P measurement is shown in Fig. 2(b). The standard setup for P-to-P, presented in Fig. 2(a), is modified to reduce the oscilloscope-related measurement error. A low-pass filter at $f_{cut} = f_{rep}$ is inserted between the fast photodiode and the oscilloscope. The aim of this filter is to turn the photo-detected pulse train into a sinusoidal trace. The rationale of our method relies on the fact that a sinusoidal photodiode response allows for a more precise measurement of the pulse maximum and its variation, thereby improving the SNR of the P-to-P measurement. The improvement is directly related to the oscilloscope voltage maximum calculation resolution. As mentioned in section 2.2, the main hypothesis of the validity of this approach is to have a proportionality between the maximum variation and the pulse energy variation. Let us consider the scheme in Fig. 4. We compare a fast photodiode response behavior
to the smooth sinusoidal response corresponding to the filtered version of the same pulse train in
our optimized method, shown respectively in blue and red in Fig. 4(Top).

<figure><img src="image_6.png" /><figcaption>**Fig. 4.** Scheme explaining the low-pass filter effect to improve the pulse-to-pulse measure-
ment SNR. (Top) Typical response function of a fast photodiode (blue) for a measured 1
MHz pulse train; sinusoidal 1 MHz wave (red) corresponding to the photodiode response
after the low-pass filter. (Bottom, left) zoom section on the maximum for the fast response.
(Bottom, right) zoom section on the maximum for the sinusoidal response.</figcaption></figure>

The response time of the photodiode, τ, is equal to 5 ns (same as the photodiode used in the experiment) and the pulse train frequency is 1 MHz. The points in the plot are separated by 1 ns, corresponding to a sampling rate of 1 GHz. In Fig. 4(Bottom, left) and Fig. 4(Bottom, right) respectively, zooms on the maximum of the responses are shown. The area selected for the zooms is 20 ns in the time domain and 2% of the peak amplitude in both cases. By comparing the two zoomed maximum areas, we observe that the number of sampling points for the pulse maximum is significantly higher in the sinusoidal case compared to the exponential decay case. Specifically, the sinusoidal response has 20 sampling points, whereas the exponential decay has only 5. The oscilloscope will add noise to this perfect response, and we can expect the maximum value to fluctuate more in the case of the fast response because the maximum will effectively be taken over much fewer points.

4.2. *Experimental validation*

In this section we describe the experimental validation of the presented optimized method. First, we apply the proposed method to characterize the same laser source, in the same conditions than the measurements reported in section 3. The P-to-P rms is 0.2% using oscilloscope 1, 2 or 3 as reported in the first line in the Table 2, in line with the best results reported in the previous section. To further characterize the resolution capabilities of the P-to-P measurement, a deterministic modulation is applied to the output laser energy. The laser used for the tests is equipped with an acousto-optic modulator (AOM) that allows analog modulation of the output average power. We apply a sinusoidal intensity modulation at a frequency of 1 kHz, changing its amplitude from 1.2% to 3.6% peak-to-peak. The resulting laser intensity fluctuations are measured with oscilloscopes 1, 2 and 3 and reported in Table 2. In all cases, comparable values of P-to-P are measured by the three oscilloscopes. We also find the expected correspondence between peak-to-peak and rms amplitude variations for a sinusoidal applied waveform.

As a further confirmation of the improved accuracy allowed by this method, a RIN measurement
is realized using the Agilent oscilloscope and the setup presented in Fig. 2 for a peak-to-peak
<table><caption>Table 2. P-to-P measured using the proposed technique using the setup in 2(b) applying a 1 kHz sinusoidal modulation to the laser output average power with different amplitude and at 0.1, 1, 10 kHz with a fixed modulation amplitude.</caption><thead><tr><th>Modulation</th><th>(1) Agilent<br>DSO7054A</th><th>(2) Siglent<br>SDS2102XHD</th><th>(3) Rigol<br>DS2102A</th></tr></thead><tbody><tr><td>None</td><td>0.2% rms</td><td>0.2% rms</td><td>0.2% rms</td></tr><tr><td>f=1 kHz, 1.2%</td><td>0.4% rms</td><td>0.5% rms</td><td>0.5% rms</td></tr><tr><td>f=1 kHz, 2.4%</td><td>0.9% rms</td><td>0.9% rms</td><td>1.0% rms</td></tr><tr><td>f=1 kHz, 3.6%</td><td>1.3% rms</td><td>1.3% rms</td><td>1.5% rms</td></tr><tr><td>f=0.1 kHz, 3.6%</td><td>1.3% rms</td><td>1.4% rms</td><td>1.3% rms</td></tr><tr><td>f=10 kHz, 3.6%</td><td>1.1% rms</td><td>1.2% rms</td><td>1.5% rms</td></tr></tbody></table>

amplitude modulation of 3.6%. The result, reported in Fig. 5, shows an integrated RIN on the bandwidth [10 Hz; 500 kHz] of 1.3% rms, confirming the correspondence between integrated RIN and P-to-P rms.

<figure><img src="image_7.png" /><figcaption>Fig. 5. RIN PSD (top) and integrated (bottom) measurement for an applied sinusoidal waveform with an amplitude of 3.6% at 1 kHz.</figcaption></figure>

Finally, to further validate that the newly proposed setup is sensitive to noise content over a large bandwidth, the sinusoidal waveform modulation frequency is changed from 1 to 0.1 and 10 kHz, while keeping its amplitude at 3.6%. Higher frequencies can not be applied to the modulator included in the laser setup. The measured P-to-P stability for the different frequencies is comparable in all cases. The slight measured differences probably come from variations in the laser environment. The presented setup allows to achieve reliable and high sensitivity measurements of P-to-P and is efficient in removing limitations related to the oscilloscope performance discussed in the previous section. Considering the measurements reported in this section and the oscilloscope limitations described in section 3 using a stable sinusoidal waveform, we estimate that the method offers a resolution better than 0.1%, with a noise floor of the order of 0.05% rms.

## 5. Conclusions

The short-term pulse energy stability for ultrashort pulsed laser is often source of misleading comparisons and interpretations. We have presented a summary of how this laser parameter is
measured, explaining the technical difference between RIN and P-to-P measurements. We show an experimental evidence of the strong impact of the oscilloscope performance in standard P-to-P measurements, which prevents reliable and comparable results for intensity noises better than 1% rms. An easily applicable method to remove this problem is proposed, that consists in adding a high-order low-pass filter between the photodiode and the oscilloscope. This setup allows to alleviate the impact of oscilloscope performances on the measurement SNR, and allows P-to-P characterization with a resolution of the order of 0.1% with standard affordable oscilloscopes.

**Funding.** Agence Nationale de la Recherche (ANR-19-CE30-0001-MIRTHYX).

**Disclosures.** The authors declare no conflicts of interest.

**Data availability.** Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

**References**

1. D. McCumber, "Intensity fluctuations in the output of cw laser oscillators. i," Phys. Rev. **141**(1), 306-322 (1966).

2. K. Petermann, *Laser diode modulation and noise*, vol. 3 (Springer Science & Business Media, 1991).

3. R. Paschotta, "Noise of mode-locked lasers (part i): numerical model," Appl. Phys. B **79**(2), 153-162 (2004).

4. R. Paschotta, "Noise of mode-locked lasers (part ii): timing jitter and other fluctuations," Appl. Phys. B **79**(2), 163-173 (2004).

5. ISO, "Iso 11554:2017 : Optics and photonics — lasers and laser-related equipment — test methods for laser beam power, energy and temporal characteristics," Tech. rep., ISO (2017).

6. U. Keller and R. Paschotta, *Ultrafast lasers* (Springer, 2021).

7. C. R. Smith, R. D. Engelsholm, and O. Bang, "Pulse-to-pulse relative intensity noise measurements for ultrafast lasers," Opt. Express **30**(5), 8136-8150 (2022).

8. W. Chen, J. Fan, A. Ge, et al., "Intensity and temporal noise characteristics in femtosecond optical parametric amplifiers," Opt. Express **25**(25), 31263-31272 (2017).

9. Q. Bournet, M. Natile, M. Jonusas, et al., "Intensity noise in difference frequency generation-based tunable femtosecond mir sources," Opt. Express **31**(8), 12693-12702 (2023).

10. T. Balčiūnas, O. D. Mücke, P. Mišeikis, et al., "Carrier envelope phase stabilization of a yb: Kgw laser amplifier," Opt. Lett. **36**(16), 3242-3244 (2011).

11. E. Shestaev, D. Hoff, A. Sayler, et al., "High-power ytterbium-doped fiber laser delivering few-cycle, carrier-envelope phase-stable 100 µj pulses at 100 khz," Opt. Lett. **45**(1), 97-100 (2020).

12. P. Gierschke, C. Jauregui, T. Gottschall, et al., "Relative amplitude noise transfer function of an yb³⁺-doped fiber amplifier chain," Opt. Express **27**(12), 17041-17050 (2019).

13. P. K. Mukhopadhyay, K. Ozgoren, I. L. Budunoglu, et al., "All-fiber low-noise high-power femtosecond yb-fiber amplifier system seeded by an all-normal dispersion fiber oscillator," IEEE J. Sel. Top. Quantum Electron. **15**(1), 145-152 (2009).

14. A. M. Weiner, *Ultrafast optics* (John Wiley & Sons, 2011).

15. R. P. Scott, C. Langrock, and B. H. Kolner, "High-dynamic-range laser amplitude and phase noise measurement techniques," IEEE J. Sel. Top. Quantum Electron. **7**(4), 641-655 (2001).

16. J. Chen, J. W. Sickler, E. P. Ippen, et al., "High repetition rate, low jitter, low intensity noise, fundamentally mode-locked 167 fs soliton er-fiber laser," Opt. Lett. **32**(11), 1566-1568 (2007).

17. R. E. Vest and S. Grantham, "Response of a silicon photodiode to pulsed radiation," Appl. Opt. **42**(25), 5054-5063 (2003).

18. V. Mackowiak, J. Peupelmann, Y. Ma, et al., "Nep-noise equivalent power," Tech. rep., Thorlabs, Inc (2015).