1. Introduction

Last year marked the 100^th anniversary since Brillouin scattering was first reported in literature. Brillouin scattering refers to the inelastic scattering of light (photons) as it interacts with acoustic waves (phonons) in a medium. The theoretical prediction of Brillouin scattering was first reported independently by Léon Brillouin in 1922 [1] and by Leonid Mandelstam in 1926 [2], and the first experimental observation was reported by Eugenii Gross in 1930 [3]. Physically, these acoustic waves are propagating density fluctuations in the medium, and the periodic nature of the density fluctuations results in a periodic change in the material’s refractive index. This periodic refractive index acts as a diffraction grating for incident light to scatter from, and the light scattered from an acoustic wave experiences a frequency shift due to the Doppler effect. This frequency shift is referred to as the Brillouin frequency shift (or simply: the Brillouin shift). Brillouin scattering can be further classified into two categories based on the origin of the density fluctuations: spontaneous Brillouin scattering if the fluctuations occur from thermal (or quantum mechanical) effects and stimulated Brillouin scattering if the fluctuations are induced by an applied electric field.

In the case of spontaneous Brillouin scattering, light scatters from phonons (density fluctuations) which are the result of thermal effects or quantum mechanical zero-point energy effects originating from the material [4]. The scattered light can lose energy (Stokes) or gain energy (Anti-Stokes) following interaction with the phonon. In Brillouin spectroscopy, this results in a spectrum which contains three fundamental peaks: a central, intense Rayleigh peak (elastic scattering) as well as the Stokes and anti-Stokes Brillouin peaks. The principle of spontaneous Brillouin scattering, in the case of Stokes shifted light, is displayed in Fig. 1(a). Using the Stokes shifted case as our example, light incident upon the acoustic wave scatters at an angle $\theta $, and the resulting wave-vector and angular frequency of the scattered light are $\vec{k}^{\prime} = \vec{k} - \vec{q}$ and $\omega ^{\prime} = \omega - {\mathrm{\Omega }_B}$ based on the laws of conservation of momentum and conversation of energy, respectively. The acoustic wave’s angular frequency ${\mathrm{\Omega }_B}$ is related to the Brillouin frequency shift ${\nu _B}$ by the relation ${\mathrm{\Omega }_B} = 2\pi {\nu _B}$. The Brillouin frequency shift ${\nu _B}$ of the scattered light is given by Eq. (1).

Here, n is the refractive index of the medium, ${V_s}$ is the speed of sound of the medium, ${\lambda _0}$ is the incident light’s wavelength in a vacuum, and $\theta $ is the angle between the incident light and scattered light. Brillouin spectroscopy provides a non-contact, label-free assessment of the mechanical properties of a medium, specifically, through the complex longitudinal modulus ${M^\ast } = M^{\prime} + iM^{\prime\prime}$. Here, $M^{\prime}$ is the storage modulus which accounts for the medium’s elastic behavior, and $M^{\prime\prime}$ is the loss modulus which accounts for the medium’s acoustic attenuation and thus viscous behavior [5]. In the case of spontaneous Brillouin spectroscopy in the backscattering (epi-) detection geometry ($\theta = 180^\circ $), the complex longitudinal modulus can be written [6,7] as shown in Eq. (2):

 

Fig. 1. Conceptual principles for (a) spontaneous Brillouin scattering and (b) impulsive stimulated Brillouin scattering. (a) In spontaneous Brillouin, the incident light with wavevector $\vec{k}$ and angular frequency $\omega $ inelastically scatters from the acoustic wave produced by the thermodynamic fluctuations in the material. The scattered light has wavevector $\vec{k}^{\prime}$ and angular frequency $\omega ^{\prime}$. (b) In impulsive Brillouin, two pulses converge in the medium and produce an interference pattern (shown on the left, in green). This interference pattern induces the acoustic waves, and these waves are then probed by another beam (shown on the right, in red).

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While Brillouin spectroscopy had been initially utilized for material characterization [8,9], Brillouin spectroscopic techniques have emerged as powerful tools for biological and biomedical applications [5,10–12]. Advances in the instrumentation of Brillouin systems were important to bring spontaneous Brillouin into the realm of biology and biomedicine. One of the most impactful advances for Brillouin systems towards biology was the introduction of a confocal Brillouin microscopy using a virtually imaged phased array (VIPA) based spectrometer [13]. By utilizing a VIPA [14] in the spectrometer design, the acquisition speed for each measurement was significantly improved over the Fabry-Perot interferometer based spectrometers [15,16] that had been regularly used at that time, and this was imperative for live biological imaging. Alongside spontaneous Brillouin spectroscopy, stimulated Brillouin scattering (SBS) techniques have also gained considerable interest for biomedical applications due to recent advancements in instrumentation which have made biological imaging feasible [17–19]. However, SBS microscopy does have a limitation of continuous scanning of one of the excitation wavelengths which requires time for each acquisition. However, this limitation is circumvented with impulsive stimulated Brillouin scattering which takes advantage of time-resolved measurements.

Impulsive stimulated Brillouin scattering (ISBS) is a type of stimulated Brillouin technique which utilizes the transient grating spectroscopy technique [20]. The principle of ISBS is displayed in Fig. 1(b). In ISBS, two pump pulses (with wavelength ${\lambda _{pump}}$) are first directed into the sample with an intersecting angle of $2{\phi _{pump}}$ between the two pulses. As the pulses converge into the medium, an interference pattern is produced which modulates the dielectric properties at the fringes of high intensity. This induces a transient density grating in the material due to electrostriction (or thermal absorption in the case of an absorbing material) which also results in a transient variation in the refractive index in the material. The fringe spacing $\mathrm{\Lambda }$ of this grating is given by Eq. (3):

Here, ${\lambda _{pump}}$ is the wavelength of the pump laser and ${\phi _{pump}}$ is half of the intersecting angle between the two pump lasers. After the transient grating has been produced, the grating is probed with a continuous wave laser (with wavelength ${\lambda _{probe}}$) where part of the probe is diffracted by the propagating acoustic wave and intensity modulated by the acoustic frequency. The expected frequency will be dependent on the type of excitation as well as the choice of ISBS detection scheme. In the homodyne detection scheme, the modulation frequency due to electrostrictive excitation is ${\nu _2} = 2{V_s}/\mathrm{\Lambda }$ and the modulation frequency due to thermal excitation is ${\nu _1} = {V_s}/\mathrm{\Lambda }$, where ${V_s}$ is the speed of sound of the medium and $\mathrm{\Lambda }$ is the fringe spacing from Eq. (3). In the heterodyne detection scheme, the modulation frequency due to electrostriction has an expected frequency of ${\nu _1}$. For either ISBS frequency ${\nu _1}$ or ${\nu _2}$, the ISBS frequency multiplied by the material’s refractive index n is directly proportional to the spontaneous Brillouin frequency for a selected choice of ${\phi _{pump}}$ and $\theta $ for the systems [21].

While spontaneous Brillouin spectroscopy has experienced many years of focused developments for biological and biomedical studies, ISBS has only recently seen instrumentation developments towards biological applications [21–26]. This began in 2015 when ISBS was first introduced to flow cytometry studies [21], and this is significant as it demonstrated the capability of ISBS for taking viscoelastic measurements in vascular-like environments. Following this, the first ISBS imaging study was reported which conceptually introduced ISBS microscopy for biological applications [22]. In this study, ISBS was demonstrated to distinguish between different liquid samples in a partitioned cell as well as image liquids in a narrow spatial geometry in a microfluidic device. Shortly thereafter, the lateral spatial resolution in ISBS microscopy was shown to be improved to a cellular scale by the selection of the 4f optics in the system [24]. In addition to this instrumentation advancement towards cellular studies, the researchers also demonstrated ISBS microscopy’s capability of distinguishing between hydrogels of differing stiffness, and this is significant as the hydrogel systems can be tuned and have a structure that can mimic biological tissues [27].

Recently, a review of current Brillouin microscopy techniques for biological applications has been reported which provided an assessment of each technique’s performance parameters [28]. In this assessment, the spectral resolution of ISBS in the frequency domain (here, called the Fourier domain) had been characterized. However, the effects of instrumentation on tuning the linewidth (and hence the spectral resolution) of ISBS signals in the Fourier domain have not been previously reported, to the best of our knowledge. Understanding methods for tuning the spectral resolution is important, because high spectral resolution is essential in Brillouin spectroscopic techniques for resolving subtle changes in Brillouin spectra and for distinguishing between two close responses. In the case of spontaneous Brillouin spectroscopy, it has been experimentally and computationally shown that the choice of higher numerical apertures (NA) for microscope objectives leads to increased spectral broadening of the Brillouin linewidth [29]. Since higher NA objectives allow for higher spatial resolution, the choice of objective should allow for both sufficiently high spatial and spectral resolution in spontaneous Brillouin studies.

Our motivation for instrumentation methods to tune the ISBS linewidth in the Fourier domain begins with reports that studied the signal damping in the time domain [30,31]. In the theoretical framework of impulsive stimulated light scattering techniques reported by Yan and Nelson [30], computational simulations of the time domain signal of ISBS were shown for different pump and probe spot size parameters. By adjusting the pump spot size in the x-direction (along the direction of the acoustic wave propagation), the attenuation of the signal was greatly reduced. Physically, the increase in x-directional spot size allows for the generation of more interference fringes with the same fringe spacing $\mathrm{\Lambda }$ (see Eq. (3)). This allows for the generation of more acoustic wave packets at further spatial locations in the x-direction, which then propagate through the area of the probe beam at later times allowing for more of the natural acoustic signal to be probed. These computational simulations were later verified by ISBS data on methanol at two different x-directional pump spot sizes reported by Torre and coworkers [31]. However, to the best of our knowledge, the effects of changing the x-directional pump spot size for the ISBS signal in the Fourier domain have not been reported. Importantly, ISBS signal takes on a Lorentzian like profile in the Fourier domain following Fourier transformation of the damped sinusoidal signal in the time domain. Additionally, the full width at half maximum (FWHM) of a Lorentzian function in the Fourier domain relates to the damping of the damped oscillations in the time domain. By this reasoning, changes to the signal attenuation of time domain ISBS data should affect its linewidth in the Fourier domain. Therefore, we hypothesize that shaping the initial pump beam spatial profile, such that the x-directional pump spot size is tuned at the sample, will result in a tuning of the ISBS linewidth in the Fourier domain.

In this study, we report the effects of shaping the initial pump beam profile on the measured frequency and linewidth of the ISBS signal. For shaping the pump beam geometry, our setup expands the pump pulses with a two-lens configuration and focuses it into the central 4f system with a cylindrical lens, and this allows for an expansion of the initial pump beam diameter to correspond to an expansion of the x-directional pump spot size in the sample. This paper first presents our ISBS experimental setup along with the sample preparation, data processing methodologies, and pump beam expansion procedure. In the results, we first show how signal asymmetry in the Fourier domain is addressed by using artifact removal techniques in the time domain. Then, we present and evaluate our custom curve-fitting methodology for closely fitting the ISBS signal profile for obtaining accurate spectral measurements. Finally, we present the effects of modifying the initial pump beam diameter on the ISBS signal in the time and Fourier domains, and we evaluate the effects on the ISBS frequency and linewidth. We demonstrate here that ISBS linewidth tuning (and thus, spectral resolution tuning) in the Fourier domain is possible by shaping the initial pump beam geometry while maintaining the expected peak frequency value of the ISBS signal.

2. Methods

2.1 ISBS setup

ISBS measurements were taken with our experimental setup displayed in Fig. 2, which was modified from our earlier design [32,33]. Pump pulses were provided by a pulsed fiber laser (wavelength: 532 nm, 1-ns pulse duration, GLPR-10, IPG Photonics) which was operated at a 20 kHz repetition rate. A fraction of the pump pulse power was split off using the polarizing beam splitter (PBS, Fig. 2) towards the photodiode (DET10A, Thorlabs; PD, Fig. 2) to act as an oscilloscope trigger for signal collection. The probe laser was provided by a CW diode laser (wavelength: 780 nm, TEC 520, Sacher Lasertechnik) which provided an output power of 12 mW. The probe beam was collimated and expanded using a 4f setup upon leaving the laser module (f₃/f₄, Fig. 2).

 

Fig. 2. Schematic of the ISBS experimental setup. APD – avalanche photodiode, CL – cylindrical focusing lens, DM – dichroic mirror, f₁/f₂ – pump beam expander setup, f₃/f₄ – probe beam expander setup (50 mm/75 mm lenses), f₅/f₆ – ISBS 4f telescope (150 mm/50 mm lenses), LP – long-pass filter, ND – neutral density filter, NF – 532 nm notch filter, OSC – oscilloscope, PBS – polarizing beam splitter, PD – photodiode, S – sample, SL – spherical focusing lens, TG – transmission grating.

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The pump pulses were focused in the y-direction using a 200-mm cylindrical focusing lens (CL, Fig. 2), and the probe laser was focused with a 200-mm spherical focusing lens. The two beams were combined collinearly using a long-pass dichroic mirror (DM, Fig. 2) and propagated towards the transmission grating (48 µm grating period; TG, Fig. 2) such that both beams focused onto the grating. The ±1^st order modes of both the pump and probe light were re-imaged with the central 4f telescope setup (f₅/f₆, Fig. 2) into the sample site (S, Fig. 2), and ISBS was thus conducted in the heterodyne configuration [34]. Higher order modes (±2, ± 3, etc.) along with the 0^th order mode were blocked at the Fourier plane of the 4f setup.

For collection, the probe beam was focused with a spherical focusing lens into an avalanche photodiode (APD210, MenloSystems), and a long-pass filter as well as a 532 nm notch filter (NF, Fig. 2) were placed before the detector to remove 532 nm light. The electrical signal from the avalanche photodiode was then amplified by two low-noise amplifiers (ZFL-1000LN+, Mini-Circuits). An RF terminator was also incorporated (BTRM-50+, Mini-Circuit) to mitigate signal reflections. The electrical signal was then transferred into an oscilloscope (InfiniiVision DSOX6004A, 6 GHz bandwidth, 20 GSa/s, Keysight).

2.2 Sample preparation

Samples of acetone were prepared in cuvette by pipet from the original bottle with no further preparation thereafter. Aqueous solutions consisting of citric acid, glycine, or sucrose were prepared fresh before procedures in a stock solution of 1 M in distilled water. Diluted solutions of these solutions were performed in cuvette with new pipet tips, and 1:2 dilutions were performed with distilled water. Dilutions continued until a 1:64 dilution of the 1 M stock concentration (corresponding to approximately $1.7 \times {10^{ - 2}} $ M) was made. Replicate measurements with the aqueous solutions were done with newly made solutions before the replicate procedures.

2.3 Data collection and processing

ISBS data was collected at the maximum averaging mode of the oscilloscope which was 65,536 acquisitions. For each sample tested, 10 data collections at maximum averaging were performed. Since the repetition rate of the laser was set to 20 kHz, approximately 3.3 seconds were needed for contiguous measurements to be independent of each other. Therefore, data collections were performed with a 4 second delay between contiguous collections.

ISBS data exported from the oscilloscope was processed using a custom-built MATLAB script. Firstly, the data was processed in the time domain with a spike filtering procedure for samples with a spike artifact at $t = 0$. An average of the data before $t = 0$ was taken to find the baseline value for each set, and a neighborhood of 3 points centered at the peak of the spike artifact were taken and set to the baseline.

Data are then transformed into the frequency domain with a fast Fourier transform (FFT). For each sample, the 10 contiguous measurements were averaged together in the Fourier domain to reduce noise. The script then fits a custom fitting function $F(\nu )$, similar to a pseudo-Voigt profile, to the Fourier domain data using a sum of a Lorentzian, Gaussian, and constant offset displayed in Eq. (4).

Here, ν is frequency, a_j with 1 ≤ j ≤ 3 and 4 ≤ j ≤ 6 were the fitting parameters associated with the Lorentzian and Gaussian functions, respectively, and a₇ is the constant offset. Contrasting from the standard pseudo-Voigt profile [35], the center frequencies of the Lorentzian and Gaussian were allowed to vary slightly to account for observed peak asymmetry. For the data taken from the pump beam expansion procedure, the ISBS frequency was taken from the maximum value of the profile fit with Eq. (4). The full width at half maximum (FWHM) was taken from the whole profile and used as the measure of linewidth and spectral resolution.

2.4 Evaluation of curve-fitting model

To evaluate the closeness of fit of the custom fitting function $\textrm{F}(\mathrm{\nu } )$, the custom fit was compared to a Lorentzian fit, a generally accepted standard model for fitting Brillouin spectra [22,24,36], which is displayed in Eq. (5).

Similar to Eq. (4), ν is frequency, a_j with 1 ≤ j ≤ 3 are the fitting parameters associated with the Lorentzian function, and a₄ is the constant offset. For the curve-fitting analysis, data was first fit using both models from Eqs. (4) and (5). The sum of squared residuals (SSR) was then computed for both models, and the closeness of fit for $F(\nu )$ compared to $L(\nu )$ was evaluated by taking a ratio of the two SSR values as shown in Eq. (6).

Here, ν_j are the discrete frequency data points for the Fourier data and ${y_j}$ are the amplitude values at those frequency values. When $SS{R_{ratio}} < 1$, this implies that the SSR of the custom fit function is less than the SSR of the Lorentzian fit, and this implies that the custom fit would fit the data closer than the Lorentzian fit. The reverse is true when $SS{R_{ratio}} > 1$, and $SS{R_{ratio}} = 1$ indicates that the test by this metric was inconclusive.

2.5 Pump beam expansion and procedure

Changes to the pump beam diameter were made with two lenses in the 4f telescope setup indicated by f₁ and f₂ in Fig. 2. This 4f telescope was constructed with optical cage equipment, and the lenses were placed in XY translation mounts in the cage. The initial pump beam diameter leaving the laser (${d_0}$) was magnified by this telescope to the new beam diameter (${d_{pump}}$) using the lens configurations presented in Table 1.

Table 1. Lenses used for each configuration and the corresponding pump diameter magnification. Positive and negative values correspond to converging and diverging lenses, respectively.

View Table

To ensure that the pump beam remained aligned after placing the lenses, the following procedure was performed for each configuration. Before placing the lenses, a collared temporary mirror was placed before the cylindrical lens directing the beam outside the experimental setup towards a second mirror external to the setup. This formed a new beam path which was twice the length of the ISBS setup itself. The beam was first aligned to the beam path using the temporary mirror and external mirror. The lenses were then placed in the f₁/f₂ telescope, and the translation knobs of the XY mounts were used to realign the beam to the external beam path. The temporary mirror was then removed allowing the beam to propagate through the ISBS system.

3. Results

3.1 Spike artifact and FFT profile symmetry

For most samples tested within our system, we observed an artifact in the form of a spike at $t = 0$ in the time domain before the acoustic response began. Figure 3(a) displays an example of the time domain ISBS signal with the spike artifact at $t = 0$ before and after the spike removal procedure in data processing as described in section 2.3. The effect the spike has on the peak profile in the frequency domain following Fourier transformation is displayed in Fig. 3(b) where the FFTs of the two signals are shown. Prior to spike filtering, the peak profile typically displays an asymmetry in the form of a raised tail on the left or right side, and Fig. 3(b) depicts an example of the tail artifact appearing on the left side. Following spike filtering, the symmetry of the peak profile significantly improved as shown in Fig. 3(b). This peak was found for all samples of aqueous solutions and distilled water that were tested. Upon further investigation, this spike was found to be present in our data likely due to pump light scattered from our cuvette into the APD which was not completely eliminated by our rejection filters. We note that the spike was not appreciable in the time domain signal for samples of acetone, and we attribute this to the high magnitude of its acoustic signal overcoming the magnitude of the spike artifact. The spike removal procedure was found to improve the accuracy of curve-fitting procedures in data processing and therefore improve the accuracy of assessing the ISBS signal.

 

Fig. 3. (a) ISBS signal of the 1 M sucrose solution in the time domain before spike filtering (black, offset of +35 mV added for clarity) and after (red). (b) FFTs of the same ISBS signals.

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3.2 ISBS curve-fitting results and analysis

Curve-fitting on the frequency-domain ISBS data was conducted using the standard Lorentzian fit as well as the custom-fit function for each of the prepared aqueous solutions described in section 2.2 (with $n = 5$ replicate solutions). The ISBS peaks were first fit with the Lorentzian profile from Eq. (5), and Fig. 4(a) displays an example of the curve-fit with data from a replicate sucrose solution at 1 M. The Lorentzian profile was found to fit the peak value well, but the peak’s profile was not fit as well due to the pronounced shoulder at the base of the profile and the peak asymmetry. The custom-fit function from Eq. (4) was then used to fit the data from the aqueous solutions, and Fig. 4(b) displays the results of applying the custom-fit to the same sucrose data. The Gaussian contribution of Eq. (4) was found to improve the fit of the fitting profile at the base of the peak in the data and correct for the peak asymmetry. In addition, the custom-fit function was also found to fit the peak frequency better than or equally as well as the Lorentzian fit for all solutions tested.

 

Fig. 4. Curve-fitting results on a sample of 1 M sucrose solution while using the (a) Lorentzian fit and (b) the custom fit. Data is shown in black while the curve-fit is shown in red.

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Following the curve-fitting procedure with the two fitting functions, the goodness of fit was evaluated for each solution by computing the ratio of their SSR values with Eq. (6). The mean SSR_ratio values for all samples were found to be between 0.58 and 0.70 for these samples. Based on how SSR_ratio was previously defined, the custom-fit function has a smaller SSR value than the Lorentzian fit when the ratio is less than 1. Therefore, the custom-fit function was found to fit the data closer than the Lorentzian fit for each solution based on the results displayed in Fig. 4(c). This is in agreement with the qualitative assessment of the Lorentzian and custom-fit curve-fits to the data such as in Figs. 4(a) and 4(b). Prior to taking the mean values of each ratio, the highest and lowest SSR_ratio values were found to be 0.89 and 0.44, respectively. Therefore, the custom-fit function provided a closer fitting model to the aqueous solution data for all replicate measurements of each solution taken for this assessment. The Lorentzian curve-fit typically suffered from the broadening of its peak to account for the pronounced shoulder at the base of the profile of the data. The custom-fit function generally fit the peak profile better since the Gaussian contribution accounted for the shoulders and the asymmetry in the data. Therefore, we conclude that the custom-fit function allows for a more accurate linewidth measurement in addition to the more accurate peak frequency measurement.

3.3 Effects of shaping initial pump beam profile on signal damping and linewidth

As a beginning to the pump profile shaping studies, we first studied the dependence of the acoustic wave damping on the initial pump beam diameter (${d_{pump}}$) and verified our results with the previous literature. The pump beam diameter was expanded using the two lenses before the PBS cube (f₁/f₂, Fig. 2) with the lens configurations listed in Table 1. Figure 5(a) displays the normalized ISBS signals from measuring acetone for each pump diameter. We observed that the damping of the ISBS signal decreased with increasing ${d_{pump}}$, and vice-versa. Since our setup uses a cylindrical lens (CL, Fig. 2) to focus the pump in the y-direction, an increase in ${d_{pump}}$ corresponds to an increase of the x-directional spot size at the transmission grating and therefore a larger x-directional spot size at the sample. In the previously reported theoretical simulations [30] and experiments with methanol [31], the signal damping was found to decrease with increasing pump spot size in the x-direction. Since an increase in ${d_{pump}}$ resulted in less signal damping in our data and corresponds to a larger x-directional spot size at the focus, we conclude that the observed results in Fig. 5(a) are consistent with the results from the previous literature.

 

Fig. 5. (a) Normalized ISBS signals from acetone in the time domain with varying pump beam diameters. (b) Normalized FFTs of the same ISBS signals. (c) FWHM and (d) peak frequencies extracted from the curve-fits used to fit the acetone data (n = 3 replicate runs of beam expansion procedure), as a function of the relative pump diameter. The dashed line in (c) represents the lower bound for ISBS spectral resolution reported previously [28].

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After verifying the dependence of the acoustic damping on the pump spot size in the time domain, the acetone signals were then Fourier transformed to the frequency domain. This allowed us to test our hypothesis which claimed that shaping the initial pump profile (and thus the x-direction spot size in the sample) will tune the linewidth in the Fourier domain. Figure 5(b) displays the normalized FFTs of the same acetone signals from Fig. 5(a). In Fig. 5(b), the linewidth of the signal was observed to consistently decrease with increasing pump diameter, and vice-versa, based on our shaping setup as given in Fig. 2. These results support and validate our hypothesis. To our knowledge, these results provide the first experimental demonstration of ISBS instrumentation tuning the signal linewidth, and subsequently the spectral resolution, of ISBS signals in the Fourier domain. These results further indicate that the spectral resolution of ISBS profiles can be improved by shaping the pump beam profile following emission from the laser.

Following the transformation of the signals to the Fourier domain, the acetone peaks from Fig. 5(b) were all fit using the custom-fitting function given by Eq. (4), which provided a closer fit to the profiles than the standard Lorentzian function. Following the fitting procedure, the FWHM and peak frequency values were extracted from the curve-fits. The FWHM values as a function of the relative pump diameter (${d_{pump}}/{d_0}$) are displayed in Fig. 5(c). In addition, the narrowest spectral resolution for the currently reported state-of-the-art for ISBS is displayed in Fig. 5(c) as a dashed line at 3 MHz [28]. The FWHM of the acetone signals was observed to decrease nonlinearly with increasing relative pump diameter. At the largest relative pump diameter ${d_{pump}}/{d_0} = 3/2$, the FWHM was found to be approximately 2.5 MHz, and this demonstrates an improvement on the current performance standard for spectral resolution in ISBS. When comparing the linewidth values from the different configurations, the FWHM obtained from the largest pump diameter was approximately 0.8 times the value extracted from the original diameter and approximately 0.4 times the value from the smallest diameter.

In addition to analyzing the behavior of the linewidth, the peak frequency values were also compared for each configuration. Figure 5(d) displays the ISBS peak frequency extracted from each curve-fit as a function of the relative pump diameter. The observed ISBS frequency value remained consistent for each of the tested configurations without significant changes. Since the expected ISBS frequency is given by ${\nu _1} = {V_s}/\mathrm{\Lambda }$ (where $\mathrm{\Lambda }$ is defined in Eq. (3)), we anticipated that the measured frequency should be the same for each configuration since the material speed of sound ${V_s}$, pump wavelength ${\lambda _{pump}}$, and pump intersecting angle ${\phi _{pump}}$ should remain unchanged. This indicates that the pump spot size can be tuned to change the linewidth, and thus spectral resolution, without changing the sample’s expected ISBS frequency measurement relating to the material speed of sound.

4. Discussion

In the recent assessment of Brillouin techniques and their performance parameters towards biological applications, the state-of-the-art spectral resolution for ISBS was reported to be between 3–6 MHz [28]. As we demonstrate in our results, shaping the pump beam spot size in the sample can allow for a linewidth of less than 3 MHz which demonstrates an improvement in spectral resolution compared to the currently reported state-of-the-art. We anticipate that further tuning of the linewidth is possible. Our current setup limitation for expanding the pump beam is the dimensions of our polarized beam splitter cube, which had been used to control the polarization of our pump pulses. The half inch sides of the cube limited our ability to further expand the pump beam diameter beyond 1.5 times the initial pump diameter. By either shifting the placement of the pump expanding lenses or by utilizing a PBS cube with 1-inch sides, we anticipate that the spectral resolution can be further improved while still maintaining control of the pump pulse polarization.

An important consideration when shaping the pump beam profile for linewidth tuning is the effect that it has on the spatial resolution at the sample site. Instrumentation choices can result in trade-offs where one parameter is improved at the expense of another. As previously described with spontaneous Brillouin scattering, a low NA objective can reduce the spectral broadening of the signal at the cost of spatial resolution in the sample [29]. In this ISBS study, expanding the initial pump profile and focusing it with a cylindrical lens in the system results in an improvement to the spectral resolution and y-directional spatial resolution but also results in a significant reduction in the x-directional spatial resolution. However, this can be remedied by the choice of optics utilized for the central 4f telescope (f₅/f₆, Fig. 2), as demonstrated previously [24]. For example, the f₆ lens in this setup could be exchanged for a shorter focal length optic (such as a 10x objective) to increase the spatial resolution in the axial and lateral directions. We note that this change will result in an increase in the pump intersecting angle ${\phi _{pump}}$ and thus increase the expected ISBS frequency. Therefore, this can be done so long as (1) the acquisition equipment has the necessary bandwidth to detect the ISBS frequency and (2) the pump laser source has a shorter pulse duration than one cycle of the acoustic wave to meet the impulsive limit condition [30].

As a future study, it would be interesting to assess the changes in the ISBS linewidth while adjusting the experimental setup to higher pump intersecting angles ${\phi _{pump}}$. By increasing ${\phi _{pump}}$ in the system, there will be an increase in the axial and lateral spatial resolution as well as the expected ISBS frequency, as discussed previously. However, the acoustic wave attenuation in a medium has typically displayed a dependence on the acoustic wave’s frequency [37]. Therefore, a change in ${\phi _{pump}}$ may result in a notable change in the ISBS signal damping in the time domain. Based on our results, this would subsequently affect the ISBS linewidth, and thus spectral resolution, in the Fourier domain. If the linewidth increases with increasing ${\phi _{pump}}$, this would indicate that the choice of optics for the central 4f system should be carefully selected to balance for both sufficiently high spatial and spectral resolution in ISBS, analogously to the choice of objective NA in spontaneous Brillouin microscopy [29].

Finally, as a technical note, an important consideration when expanding both the pump and probe beams is the subsequent expansion of the pump and probe profiles at the Fourier plane of the central 4f telescope, following the transmission grating. This can result in a horizontal overlap of the pump and probe profiles at the Fourier plane that occurs at both the +1 and -1 diffraction modes which would limit the system to strictly the heterodyne detection scheme. From our experimental observations, an advantage of using the cylindrical lens to focus the pump pulses before the grating is that the pump pulses have a vertical line spatial geometry at the Fourier plane which limits its horizontal spread. By applying the use of a cylindrical lens to vertically focus both the pump and the probe lasers onto the transmission grating, the pump and probe spatial profiles would then both have a vertical line spatial geometry in the Fourier plane. This can allow for the separation of the pump and probe profiles at the +1 and -1 diffraction modes, and therefore allow for a choice of either heterodyne or homodyne detection schemes.

5. Conclusion

In this report, we have demonstrated that the ISBS signal linewidth can be tuned by shaping the initial pump beam profile, and thus the pump spot size in the sample. Specifically, expanding the pump spot size in the x-direction was shown to decrease the FWHM value of the ISBS signal, and vice-versa. Pump diameter variation was shown to produce an ISBS signal with a FWHM of less than 3 MHz, and this improves upon the current performance standard for the spectral resolution of ISBS [28]. The ISBS peak frequency value, which relates to the speed of sound of the material, was shown to be unchanged by the pump diameter variation. In addition to these results on spectral resolution, we demonstrated the effect of spike artifact removal on ISBS signal symmetry, and we introduced a custom-fitting function for ISBS data and compared it to the standard Lorentzian fit. Removing spike artifacts at $t = 0$ in the time domain data was shown to improve the symmetry of the data in the Fourier domain. The custom-fitting function was found to provide a closer fit to the ISBS data than a Lorentzian profile, and this allowed for more accurate assessment of the ISBS peak and linewidth. Since ISBS frequency and linewidth can access the material speed of sound ${V_s}$ and acoustic attenuation similarly to spontaneous Brillouin scattering, ISBS microscopy provides a powerful alternative technique for acquiring accurate viscoelastic measurements. By combining these pump shaping methods and computational procedures with the recently reported methodologies for improving lateral resolution [24], we believe that ISBS microscopy with high accuracy, high spectral resolution, and high spatial resolution is possible. These universal improvements to understanding peak frequency shifts and linewidth shaping due to varying experimental geometries will lead to consistent spectral acquisitions across upcoming miniature Brillouin spectroscopy setups.