86
J. Eur. Opt. Society-Rapid Publ. 21, 9( 2025)
Figure 3. The measured linewidth( left) and peak frequency( right) of 100 simulated signals and evaluated with the exponential window. The orange line, the upper and lower edge of the boxes correspond to the median value, the 25th and 75th percentiles respectively. The dashed line from each box extends to the minimum or maximum non-outlier value inside the distribution. The red x’ s indicate outliers, which are more than 1.5 the interquartile range away from the first or third quartile. The horizontal dashed line indicates the ground truth. frequency. With the measurement point rate of 50 kHz the theoretical frame rate for this 24 pixel image is 0.48 ms corresponding to a pixel measurement rate of 20 s / pixel, which is around two orders of magnitudes faster then current then SpBM techniques.
5 Spatial and spectral resolution trade-off
Figure 4. a) Time signal of a single-shot measurement from water and b) the corresponding frequency spectrum once evaluated without and with exponential window.
depends on the specific requirements of the experiment or application.
To illustrate these principles in practice, we conducted experiments at a spatial resolution of 40 lm. With pulse energies of E p = 35 lJ and a pulse repetition rate of 50 kHz single-shot measurements were possible without using the exponential window for data evaluation. Consecutive measurements varied less then 0.3 % of the peak frequency, which we define here as adequate measurement quality. However, the average power is very close to the maximum applicable dose. In order to not to impair the sample a 20 lJ pulse energy and 50 kHz repetition rate were chosen as parameters. Now only the data evaluation with the exponential windows leads to single-shot measurement( Fig. 4) and it is possible to acquire the peak frequency and linewidth. There is still a peak present at the frequency of hydrogel in the spectrum when no window is used, but the SNR is too low for a reliable result.
As a simple example we applied stage-scanning to record 2D images of a hydrogel cube, freshly located in water with a coarse spatial resolution of 100 lm( Fig. 5). Hydrogel mostly contains water and to increase the image contrast it was dried before it was placed inside of the water. The hydrogel and the water frequencies can be clearly distinguished, see Figure 5 on the right. While the exact frequency difference is hard to estimate, the frequency difference of 50 MHz seems reasonable, as the dried hydrogel cube is stiffer and one would expect a higher Brillouin
Increasing the spatial resolution effects also the spectral resolution. When the spatial resolution is increased, the angle of the two pump beams increases and thus the number of fringes inside the excitation volume increases as well( Eq.( 1) leading to a higher Brillouin frequency). More critically, this leads to a shorter signal duration in time. The excited phonons naturally propagate outside the excitation volume, a process that occurs more rapidly in smaller excitation volumes. For instance, with a 40 lm excitation diameter, the signal in water persists for approximately 40 ns. For a 40 lm excitation diameter, the signal in water lasts approximately 40 ns. According to the Cramer- Rao lower bound( CRLB), this short signal duration fundamentally limits the obtainable spectral resolution. Specifically, the CRLB indicates that the minimum variance of any unbiased estimator of the frequency is inversely proportional to the signal duration [ 41 ]. The spectral resolution should not be confused with the resolution of the Fourier sampling, which can be enhanced by zero padding and improves the peak determination accuracy. Hence, a small measurement volume is mainly usable in homogeneous media, where one frequency is present. If two materials are located in the measurement volume and the spectral resolution is not high enough to distinguish the corresponding two peaks, the measurement will be distorted by a shifted peak position, as the two peaks will merge into one peak with broader bandwidth.
Recently O’ Connoretal.[ 42 ] proposed a line excitation, which extends the time signal, because phonons are excited over a larger area and will propagate into the probe beam with a delay. The line excitation was realized by employing a cylindrical lens instead of the spherical lens L1 in Figure 1 to focus the pump beam on the grating. The probe beam keeps its symmetric profile, which in our case has a full width at half maximum( FWHM) of approximately