Speed of sound measurement and mapping in transparent materials by impulsive stimulated Brillouin microscopy

Impulsive stimulated Brillouin scattering (ISBS) is a variant of stimulated Brillouin scattering, which can overcome the shortcomings of the long acquisition time of traditional Brillouin microscopy. We introduce the difference between ISBS and other Brillouin microscopies in calculating longitudinal modulus. The Brillouin frequency shift obtained by ISBS is only related to the system parameters and the speed of sound (SOS) in the sample, not to the refractive index. Non-contact SOS measurement of homogeneous samples is an important application of Brillouin scattering, used in the early study of Brillouin spectroscopy and the mechanical properties of liquids. However, the measurement requires prior knowledge of the sample refractive index, which limits the measurement of the unknown refractive index sample. Here, we propose a method to measure the SOS based on ISBS, which in principle avoids the need for refractive index parameters. The SOS of several liquids are measured and compared with the standard values. The mean relative standard deviation is 1.13%. Moreover, we measure the SOS of a mixture of ethanol and water to demonstrate an application of measuring SOS without refractive index information. We also demonstrate the high spatial resolution of ISBS with a methanol-filled PDMS sample.


Introduction
Brillouin microscopy (BM) is an emerging elastic microscopic imaging technique that enables label-free, noncontact three-dimensional mapping of intracellular and extracellular biomechanical information with spatial resolution up to the optical diffraction limit [1,2].Unlike mechanical techniques such as atomic force microscopy and magnetic bead twisting, conventional BM is based on the interaction of light with spontaneous phonons, which is spontaneous Brillouin scattering (SpBS) [3].The longitudinal modulus mapping of the sample is determined by measuring the optical frequency shift of the scattered light.Considering the large scattering angle (90 • or 180 • ) of the light received by the conventional Brillouin microscope [3,4], the Brillouin frequency shift can reach GHz.So far, BM has been widely used in biological and biomedical research.
However, conventional BM has shortcomings such as low signal to noise, and slow acquisition speed.In addition, due to the unrestricted propagation direction of spontaneous phonons, there is an additional broadening of the Brillouin spectrum [5].Recently, line-scanning BM has been developed to improve acquisition speed greatly [6,7].To overcome the limit, stimulated Brillouin scattering (SBS) has been used in microscopy.In SBS, coherent mechanical waves are amplified by the process of pump and probe beam interaction.Recently, pulsed SBS can achieve a higher signal to noise than spontaneous Brillouin scattering [8].Whether based on SpBS or SBS, the relationship between Brillouin frequency shift and longitudinal modulus is determined [3].The linear relationship between the longitudinal modulus obtained by BM and the Young modulus determined by AFM can be verified.Therefore, the Brillouin microscope can quantify the mechanical properties of the sample with the knowledge of refractive index and density.
Laser-induced dynamic grating [9] was initially developed for studying photon-phonon interaction around the 1980s.Ultrashort pump pulses excite coherent phonons, and the elastic acoustic wave is detected by the probe beam.This kind of Brillouin scattering is called impulsive stimulated Brillouin scattering (ISBS).In 2017, ISBS was used in microscopy [10].Recently, ISBS has enabled rapid viscoelastic measurements of PDMS samples with an acquisition time of sub-ms [11,12].The relationship between the Brillouin frequency domain and longitudinal modulus in ISBS is different from SpBS and SBS.In this paper, we prove that in ISBS, the SOS of the medium in probe volume can be calculated directly from the Brillouin frequency shift and the system parameters.While in SpBS and SBS, the refractive index information of the sample is required to determine the SOS from the Brillouin frequency shift.
Because the SOS can be derived directly from the Brillouin frequency shift, ISBS provides a noncontact method for measuring SOS for transparent samples with high spatial resolution.Usually, the SOS of liquid can be measured by a sound velocity meter or resonance interferometry, neither of which is suitable for biological samples.Using photoacoustic microscopy, the SOS can be obtained by calculating the time of flight of the photoacoustic signal.However, it needs prior knowledge of the physical distance between the detector and the location of the excitation beam [4].ISBS does not introduce the time-of-flight method, so there is no prior requirement for sample size.
This manuscript introduces the principles of ISBS and other Brillouin scattering methods.The relationship between the frequency shift of ISBS and the longitudinal modulus is given.The SOS of several liquids are measured by ISBS and compared with the standard values.To illustrate that the SOS can be measured without refractive index information, we measure the SOS of ethanol and water mixtures at different molar fractions.Finally, the SOS mapping between PDMS and ethanol is given.

Spontaneous Brillouin scattering
Spontaneous Brillouin scattering (SpBS) is caused by pressure waves originating from thermally driven density fluctuations in a sample.The Brillouin peak has a frequency shift proportional to the SOS of the sample.The frequency shift Ω B of the scattered light at the scattering angle θ is n is the refractive index at the focal point.V is the SOS at the focal point, and λ is the wavelength of the laser source.The Brillouin frequency shift can be converted to elastic longitudinal modulus by: where ρ is the density.Usually, Brillouin scattering collects backscattered light (θ = 180 • ).The laser wavelength is known, and the elastic longitudinal modulus can be quantified if the density and refractive index are known.Currently, most BM assumes that the ratio of refractive index to the square root of density n/ √ ρ is approximately constant in biological tissues and cells [3,13].Both theoretical and experimental results show that the changes in refractive index and density do not substantially affect the modulus estimation [3].The refractive index and density of biological samples can be determined by estimated constant values or the biphase ideal mixing model [14][15][16].Recently, new techniques have been developed to measure the Brillouin frequency shift and the refractive index accurately, such as dual photon-phonon scattering [17] and BM combined with optical diffraction tomography [18].The expression of frequency shift and longitudinal modulus of SBS is the same as that of spontaneous Brillouin scattering.

ISBS
The optical path geometry of ISBS is shown in figure 1(a).Two pulses of pump laser with a certain angle in the propagation direction excite coherent sound wave.The probe laser interacts with the excited sound wave.Specifically, the pump and probe laser are focused on the sample through the same grating and 4f system, respectively.In this manuscript, heterodyne ISBS is considered, in which the scattered light of one probe beam interacting with phonons coincides with another probe beam in space.The time-domain variation of the scattered light is measured, and its amplitude is modulated.The Brillouin frequency shift and linewidth are related to the elasticity and viscosity of the material, respectively.Because of the small Brillouin frequency shift of ISBS, the beat signal of inelastic scattering light and elastic scattering light can be directly detected by the oscilloscope.The method that combines a pulsed pump and continuous probe not only retains the advantage of strong scattered signals stimulated by pulse laser but also avoids the defect of slow measurement speed of delay time scanning.
The reference light interferes with the scattered light, so the measurement signal on the detector can be expressed as [19] where Ω B , α and V are the frequency, attenuation coefficient, and the SOS respectively.Since the heterodyne signal is linearly related to the diffraction efficiency, the detector measurement signal can directly reflect the acoustic oscillation information.The heterodyne interference signal is measured by the photodetector, and then the spectral signal of the heterodyne ISBS system is obtained by Fourier transform: It can be seen that the spectral characteristics of Brillouin scattered light can reflect the propagation characteristics of sound waves in the sample.
We give the calculation of the SOS.The experiments are performed in a cuvette.The pump beam is divided into two beams by the grating and imaged on the sample by a 4f system.As shown in figure 1(b), when the pump beam (Green arrows) incident into a medium, the wave vector difference k x between the two beams in the X direction at the focal point can be calculated where ω pump is the frequency of the pump laser, c is the speed of the laser, The window refractive index of the cuvette is n 1 and the refractive index of the liquid is n 2 .θ is the incidence angle, θ ′ is the refraction angle at the air-window interface and θ ′ ′ is the refraction angle at the window-liquid interface.k x does not contain the refractive index of the medium.The pump beam passes through the gating, so ω pump sin θ is a constant [20], which is Λ 0 is the grating pitch.f 1 and f 2 are the focal lengths of the front and rear lenses of the 4f system, respectively.Therefore, the wave vector difference in the X direction is a constant According to the ISBS principle, the wave vector of the phonon is the wave vector difference of the pump laser in the X direction [9], and the wavelength of the phonon in the sample is T Le et al The wavelength of phonons excited by ISBS is constant, independent of the refractive index of the sample, so the SOS is only related to the system parameters.The modulation of refractive index by sound wave causes the probe light (Red arrows) to undergo Bragg diffraction and get a Brillouin frequency shift Ω B .The SOS expression is Pump and probe light interact and pass through the liquid-window-air interface.This also causes the optical path to deflect (not shown in figure 1(b)) but does not affect the experimental results.
The real part of the complex longitudinal mode is related to the speed of sound (SOS) and the density of the material.Therefore, according to the spectral information measured by heterodyne ISBS and the density of the material, the memory modulus M ′ can be obtained to characterize the elastic characteristics of the material.
The expression of the acoustic attenuation coefficient is where ∆ B is the linewidth of the Brillouin spectrum.Through heterodyne ISBS spectral information, the loss modulus can be calculated to represent the viscous characteristics of the material M ′ ′ is related to the longitudinal viscosity η of the material, and the relationship between the attenuation coefficient α and the longitudinal viscosity η is The SOS and the attenuation coefficient can be calculated with the setup parameters and spectral information.With the material density knowledge, the storage modulus and loss modulus of the elastic longitudinal mode can be further deduced, which is different from other Brillouin techniques.As described in section 2.1, SpBS, for example, often assumes that the ratio of refractive index to the square root of density n/ √ ρ is approximately constant.This has been supported by theoretical and experimental data.For ISBS, we consider it acceptable to use the estimated density values to calculate the longitudinal modulus, and in the case of cornea, the range of density variation is about 1.6% [13].However, more data and experiments are needed to confirm this.

Setup
The heterodyne ISBS spectral measurement system is shown in figure 2. The pump laser source is a Nd:YVO4 solid pulsed laser with 532 nm wavelength and less than 10 ps pulse width (Huaray Laser; PINE-532-15), the pulse repetition rate is 10 kHz, and the maximum single pulse energy is 100 µJ.The pump laser is focused on the TG through the cylindrical mirror CL with a focal length of 150 mm.The cylindrical mirror compresses the pump light size in the longitudinal direction to improve the pulse energy density, while ensuring that enough interference fringes are generated in the transverse direction to enhance the excitation efficiency of the sound wave.Probe laser output 780 nm continuous wave, the seed laser is distributed Bragg reflection laser (Thorlabs; DBR780PN), transmitted through a polarization-maintaining fiber to a conical amplifier (New Focus; TA-7613) for power amplification, with a maximum output of about 350 mW.The probe laser is focused on the TG through a 125 mm focal length spherical lens L1, and the pump laser is incident on the TG through the short-pass dichroic lens DM (Thorlabs, DMSP650), and then focused on the sample measurement plane through achromatic lenses L2 and L3.Because the transmittance of the DM at 532 nm is not 100%, a portion of the pump laser is reflected by the DM and received by the photodetector PD1 (Thorlabs, PDA10A-EC, 150 MHz) as a trigger source for the oscilloscope.Through the 4f system, the pump laser excites sound waves with a fixed wavelength on the sample and one of the probe lasers is attenuated by ND2.
After filtering the pump light and stray light by bandpass filters, heterodyne signal light carrying acoustic information is detected by photodetector PD2 (New Focus, 1601FS-AC, 30 kHz-1 GHz).The signal is detected by a high-pass filter (Mini-Circuits, BHP-50+, 41-800 MHz) and two cascade amplifiers (Mini-Circuits, ZFL-1000+, 1 GHz), after about 250 times amplification, through a high sampling rate oscilloscope (Agilent Technologies, DSO9254A, 2.5 GHz) to collect and store.The average calculation process of the heterodyne signal is also completed on the oscilloscope.The ISBS spectrum is obtained by Fourier transform after zero processing of the acquired time domain signal to reduce the interval of spectral sampling points.The Brillouin frequency shift is located according to the maximum peak value, and the full width of the half-height is calculated as the Brillouin line width by interpolation from the sampling point corresponding to the half-height.The transverse (x direction) diameter of the pump focus is 472 µm, and the longitudinal (y direction) diameter is 16 µm.The transverse and longitudinal diameters of the probe focus are 13 µm and 14 µm, respectively.More details can be found in our previous work [12].

Results
According to the principle of ISBS, the phonon wavelength and Brillouin frequency shift are needed to calculate the SOS of the sample.According to the existing parameters, the phonon wavelength is 4.13 µm.Based on the above ISBS system and viscoelastic theoretical model, the Brillouin spectra of several groups of liquid samples are measured, each group averaged 10 times, and the mean SOS values and relative deviation (RD) from the standard values are calculated in table 1.The Brillouin frequency shift is obtained by Lorentz fitting the spectrum.V mea is the measured SOS.V sta is the standard SOS.
The mean standard deviation of SOS measurement for 6 groups of liquid samples is about 1.13%.It can be seen that the frequency shift measured by ISBS can be directly mapped to the SOS of the sample without refractive index information.To further illustrate this point, the relative deviation is plotted as a function of refractive indices in figure 3. The figure shows that there is no obvious correlation between the two variables.
In addition to the measurement of the pure product, the SOS of the ethanol-water mixture is further measured to apply the ISBS method to the characterization of the molecular aggregation properties.Ethanol is a typical amphoteric molecule, containing both hydrophobic and hydrophilic components.Intuitively, the mixture of ethanol and water appears as a homogeneous liquid, but due to the characteristics of amphoteric  molecules, the molecular binding of the ethanol and water mixing process is more complicated.For uniformly disordered systems, physical quantities such as the SOS, which depend on the proportion of components, should vary monotonically between the properties exhibited by the pure product.However, many studies of ethanol-water mixtures have shown that the behavior of the mixtures varies between disordered liquids and more complex amphiphilic molecules, exhibiting micellar-type local aggregation while exhibiting disordered system properties [22].In previous studies, various properties of their mixtures were studied by various methods, such as measurements of their refractive index [23] and fluorescence spectra [24].Brillouin spectroscopy, as a non-contact measurement method of mechanical properties, can calculate the propagation SOS waves by comparing the Brillouin frequency shift of ethanol-water mixture, to study the particle distribution characteristics from another side.Figure 4 shows the SOS of ethanol-water mixtures with different ethanol molar fractions at 20 • C. We compared the results obtained by ISBS with the published data and theory [25].The published SOS data measured by the standard phase comparison method are in good agreement with the experimental data by ISBS.The deviation between the two sets of data may be that the ambient temperature is not completely consistent.We use a simple two-state model, which can explain the main experimental characteristics of the data, but there are still deviations from the experimental values.The parameters also come from Ref [25].The SOS of water is greater than that of ethanol because the structure of water tends to be more ordered in a network structure, and therefore sound waves travel faster through it [22].If the ethanol-water mixture is a completely disordered liquid system, the SOS in pure water is larger than that of ethanol.With the increase of ethanol molar fraction, the SOS characteristics of water should gradually weaken until it becomes completely ethanol characteristics.However, as shown in the curve in the figure, the SOS of the mixture with a small amount of ethanol exceeds the SOS of pure water, and the SOS is maximum when the ethanol molar fraction is about 0.1.This is consistent with previous results [22,25].A small amount of ethanol strengthens the mesh structure of pure water, making it more 'stiff ' , so the SOS travels faster.When the fraction of ethanol is low, the water separates the ethanol molecules due to the hydrophobic groups of ethanol, so the water retains its hydrogen bond grid, and because the small number of ethanol clusters further restricts the geometry of the water molecules, causing the grid to harden, so the sound travels faster in the ethanol-water mixture at a small concentration of ethanol than in pure water.When the ethanol molar fraction increases to about 0.1, the grid structure is decomposed, and when the ethanol molar fraction further increases, the sound wave propagates in the broken water grid and ethanol cluster, and the SOS decreases until the propagation speed is consistent with that in pure ethanol, and the SOS gradually decreases [22].Therefore, ISBS spectroscopy directly verifies the molecular model of the ethanol-water mixture and becomes a powerful tool for verifying the aggregation state of molecules.
We applied ISBS to mixed samples of PDMS and ethanol to demonstrate spatial resolution.By measuring the signal intensity and the SOS, PDMS and ethanol can be distinguished from each other in figure 5.It is worth noting that the signal is weakened at the PDMS and ethanol boundary, and the Brillouin shift cannot be extracted.We believe that this is due to the momentum mismatch near the interface.When the two beams of the pump(probe) incident into different mediums, the x-direction wave vector difference of the two beams at the focal point is related to the refractive index of the two mediums.For PDMS and air, the wave vector  mismatch is large, and the signal cannot be observed.For PDMS and ethanol, the refractive index difference is small, but there is also a wave vector mismatch, which makes the interference signal weak.This is not apparent when sound waves travel in a direction parallel to the interface.In figure 5, the material in letter regions is in PDMS, which is surrounded by ethanol.The x direction is the direction of sound wave propagation, and the y direction is perpendicular to the direction of sound propagation.It can be seen that the signal intensity weakens near the interface between PDMS and ethanol, and this phenomenon occurs on a larger scale in the x direction and a smaller scale in the y direction.

Conclusion
In this manuscript, we show that the frequency shift expressions of ISBS are different from SpBS and SBS.This allows the SOS to be derived from the ISBS frequency shift without refractive index information.We demonstrate this by using 6 sets of liquid samples, without considering the refractive index, in which the mean relative deviation from the standard value of the SOS measured by ISBS is about 1.13%.This expands the application range of Brillouin spectroscopy to measure sound velocity, which is illustrated the ethanol-water mixture.The experiment shows that the SOS of the mixture is maximum when the molar fraction of ethanol is about 0.1, which is consistent with the previous conclusion [22,25].Combined with the high spatial resolution of ISBS, the measurement of sound velocity allows BM to obtain more mechanical information.

T Le et alFigure 1 .
Figure 1.(a) The excitation and detection of acoustic waves.Green represents the pump laser and red represents the probe laser.Interference patterns are created at the intersection of the pump laser, which excites sound waves in the sample.(b) Optical path diagram.Green and red arrows represent the pump and probe light.

Figure 3 .
Figure 3.The relative deviation of SOS as a function of refractive indices for typical liquids.

Figure 4 .
Figure 4. SOS of ethanol-water mixtures as functions of ethanol molar fractions at 20 • C. The line represents the calculated data by the two-state model, the blue points are the experiment data by ISBS method, and the green points are the published data.

Figure 5 .
Figure 5. Signal intensity and SOS mapping of PDMS-ethanol samples obtained by ISBS, PDMS at letters, ethanol environment.The signal becomes weaker near the boundary between the two mediums, and signal attenuation along the x direction (the direction of the sound wave) is more pronounced.

Table 1 .
calculated values of ISBS frequency shift, measured SOS, standard SOS, and relative deviation for typical liquids.