J. Eur. Opt. Society-Rapid Publ. 21, 9( 2025) 85
derived in this paper are dependent on the sample and on the sensitivity of the experimental setup. Most importantly the behavior will be different for different pump wavelengths.
In the subsequent analysis, we employ pulse duration’ s of t p = 10 ps, while changing both the repetition rate and the pulse energy to alter the average power P ¼ E p k. Both the pulse energy and the repetition rate increase the signal strength, illustrated in the bottom left in Figure 2, which suggests picking the highest value for each possible. But also for the average power a threshold is observed, above which the peak frequency started to fluctuate with an added drift of the peak frequency, which can be attributed to induction of heat. The repetition rate and pulse energy would be selected in such a way that the average power remains below the specified threshold, thus the determination of this threshold for a given sample is important. The trade-offs between choosing a high pulse energy or high repetition rate will be discussed in the next section.
3.2 Exponential window
So far we have looked at how the SNR can be improved by increasing the signal strength with the optimal excitation parameter choice. Another valid option to improve the SNR is to lower the noise of the signal. In the frequency analysis of signals with the fast Fourier transform( FFT), the signal is commonly windowed [ 39 ]( each time point is weighted with a weighting function, the window) to reduce the spectral leakage, but can also help to reduce the noise. For decaying signals the exponential window is a widely utilized option, because it minimizes the attenuation at its strongest point( the beginning), while ensuring a gradual decline to zero at the end. These characteristics can help improve the SNR [ 34 ] and make the adoption of this window for the exponentially damped ISBS signal an ideal choice.
Although the signal is altered through the multiplication with the exponential window, these changes can be calibrated for. Assuming the following intensity signal
I ðtÞ ¼sinð2pftÞe �a 1t ð2Þ
with I( t), f, anda 1 being the light intensity at the detector, frequency, and the damping coefficient of the signal, respectively. When this signal is then multiplied with the exponential window the resulting signal will be
I w ðtÞ ¼sinð2pftÞe �a 1t e �a 2t
¼ sinð2pftÞe �ða 1þa 2 Þt
with I w( t), a 2 being the windowed intensity signal and decay coefficient of the exponential window respectively. The linewidth of the peak in the frequency spectrum will now correspond p to the sum of both decay coefficients f ¼ ffiffiffi
3 ða1 þ a 2 Þ = p [ 40 ]. But as a 2 is known a priori the calculationp of the damping coefficient simply becomes a 1 ¼ p f = ffiffi
3 � a2.
To show the advantages of the exponential window in low SNR situations, simulations were carried out with different SNRs. For each SNR 100 signals are evaluated,
ð3Þ
ð4Þ allowing to estimate the variance of each method. The exponential window has the lowest variance both for the damping coefficient and the Brillouin frequency for low SNRs( Fig. 3, stays stable even for SNRs of �12 dB). In this low SNR situations it is challenging to determine the end of the signal, potentially leading to the inclusion of noise at the end of the time-domain signal. This makes the exponential window especially effective in situations with a low SNR or when the signal decays quickly because of large damping coefficients.
To experimentally verify that the exponential window can increase the temporal resolution while accurately measuring the viscoelastic properties at low SNRs, measurements on ethanol were performed. A reference for the Brillouin frequency and linewidth of ethanol was created by averaging 10,000 measurements( corresponds to 1000 ms and SNR of 6 dB), which where taken with a pulse energy of 41 lJ. The averaged signal was evaluated with the FFT to obtain both the Brillouin frequency of 443.75 MHz and the linewidth 17.80 MHz. Afterwards measurements are taken at a reduced pulse energy with 20 averages, corresponding to only 0.4 ms / measurement. The resulting Brillouin frequency of 440.25 MHz deviates by 0.79 % from the reference value even though the SNR is only �12.1 dB. The linewidth can also be determined accurately using the exponential window 17.22 MHz deviates only by 3.26 % compared to the reference. The exponential window thus demonstrates the ability to measure both Brillouin frequency and linewidth, indicating that using the exponential window requires a SNR as low as �12.1 dB, thereby substantially reducing the required number of averages and optical power.
4 Spatial and temporal resolution trade-off
Thus far we investigated the optimal excitation parameters for a fixed spatial resolution. Now we want to expand on the considerations when changing the spatial resolution and how this effects the temporal resolution. The achievable temporal resolution is limited by the usable repetition rate and the required number of averages to obtain pan ffiffiffiffi adequate SNR. Although the SNR increases with N through averaging, the measurement time also increases proportional to N. Therefore, it is crucial to optimize the SNR of a single measurement to minimize the required averages. Therefore, it is optimal to select the highest possible pulse energy and pick a repetition rate accordingly, which together stay below the average power threshold. The relationship between spatial and temporal resolution is constrained by the maximum allowable average power density of the sample. Once this limit is reached for a given spatial resolution, further improvement in spatial resolution necessitates a reduction in pulse energy to maintain the sample’ s integrity. This reduction in pulse energy results in a lower SNR, which must be compensated for by increasing the number of averages, thereby decreasing the temporal resolution. Consequently, enhancing spatial resolution beyond this point involves a trade-off with temporal resolution. The optimal balance between these parameters