Balanced Heterodyne Brillouin Spectroscopy Towards Tissue Characterization

We present the first application of balanced heterodyne detection, established in quantum optics, to Brillouin spectroscopy in biological tissues. The balanced detection cancels out a large part of the influence of the laser noise. The center wavelength is 1064nm and the frequency resolution is 30MHz. The frequency resolution and the accessible frequency range are only limited by the measurement time and the laser tuning range (30GHz), respectively. In this way, we were able to measure the Brillouin shift in acetone, water, gummy bears, glass, and even porcine eye lenses.


I. INTRODUCTION
The analysis of scattered light provides a significant amount of information about the scattering medium. The scattered light contains different frequency components, depending on the underlying scattering process, e.g. Rayleigh scattering, i.e. elastic scattering on small particles, and Brillouin scattering, i.e. inelastic scattering on propagating sound waves. Rayleigh scattering results in a line around the excitation frequency, whereas Brillouin scattering comes with a frequency shift, depending on the speed of sound and refractive index of the investigated media, the scattering angle, and the frequency of the excitation [1].
Here, we are especially interested in the Brillouin scattering, as the frequency shift as well as the linewidth of the Brillouin scattered light relate to material properties as, e.g. bulk modulus and bulk viscosity and hence rheological properties of biological tissue as e.g. the eye, lens, cornea, or even cells [2]. The possibilities to investigate the Brillouin scattered signal vary from Brillouin spectroscopy, cf. e.g. [3], [4], to Brillouin microscopy, cf. e.g. [2], [5]- [7], and can be done for spontaneous, cf. e.g. [3], [4], and stimulated, cf. e.g. [7], [8], Brillouin scattering. Here, we are concerning The associate editor coordinating the review of this manuscript and approving it for publication was Jingang Jiang . the Brillouin spectroscopy for spontaneous Brillouin scattering, but the presented method can be extended to stimulated Brillouin scattering and also Brillouin microscopy.
There are two possible experimental approaches to determine the Brillouin scattered signal, optical filter methods [3], [4], [9]- [14] and optical mixing or beating methods [15]- [18]. The first kind spectrally decomposes the scattered light by using some elaborate and complex filter [4], [12]- [14], [19]- [21]. These methods suffer from a spectral ambiguity and limited resolution due to the limited Q value and difficulties in determining the absolute frequency of the Brillouin shift [15], [22]- [24]. The optical mixing or beating methods are technologies that are transferred from radio technology. These techniques come in handy when the signal frequency of interest is unknown or, as it is the case in Brillouin scattering, in a frequency regime (e.g. optical frequencies), i.e. THz ± GHz (optical carrier ± Brillouin shift), which is difficult to directly detect with, e.g. photodiodes. Here, the signal is mixed with a local oscillator (LO) creating a beat signal, which contains the signal of interest, in this case, the Brillouin shifted signal. This transfers the signal from THz ± GHz to the more easily accessible MHz regime, depending on the frequency difference of signal and LO. In contrast to the so far performed beating methods [15]- [18], [24]- [31], in the here presented method  The pump mode entering the cuvette is denoted byâ p . It is scattered on sound waves inside the sample resulting in the Brillouin scattered signal. Additionally, we have a vacuum mode entering the cuvette which is combined with the reflected pump and the Brillouin mode, resulting in the signal modeâ S . This combination of both fields is equivalent to the combination of two modes on a beam splitter. The origin of the beam splitting reflectively, which is in general close to 0, is linked to the Brillouin scattering process and the signal mode is almost completely composed of the vacuum mode.
of balanced heterodyne Brillouin spectroscopy the signal and the LO interfere on a 50:50 beam splitter (BS), then both outputs of the BS are detected with photodiodes and the difference signal of both diodes is measured. This is a very common technology in radio frequency detection and quantum optics [32]- [41] and we apply it to Brillouin spectroscopy. By scanning the frequency of the LO, we are able to investigate a large frequency range and study the Brillouin shift of very different materials varying from acetone over water to porcine lenses, gummy bear (Haribo), and glass (SF10 glass, Schott AG, Germany), which are presented in the following.

II. GENERAL IDEA
This paper aims to transfer the well-established technology of balanced heterodyne detection to Brillouin spectroscopy. Balanced heterodyne detection is a mixing technique, where two laser beams (signal and LO), are overlapped on a 50:50 beam splitter and detected with two photodiodes, cf. Fig. 1. The difference of the currents measured with the two photodiodes is the measurand, which is further processed. In contrast to this, the so far published mixing or beating methods used in Brillouin spectroscopy, utilize only the signal of one photodiode. Here, the signal and LO are either overlapped in the probe, see e.g. [28]- [30] or a strong pump is sent into the probe creating the Brillouin scattered signal, which is then overlapped with the LO on a beam splitter afterward, see e.g. [24]- [26]. In the latter case, while working with only one photodiode it is possible to use a highly unbalanced beam splitter. This ensures that a large part of the signal is sent into the detection step and, as the LO power is usually not limited technically, it can be increased, ensuring that the LO is far stronger than the signal in the detection step. However, this implies that the noise of the laser influences the measurement directly.
The benefit of our presented method is, that we overlap the Brillouin scattered light with the LO on a 50:50 beam splitter and detect the outputs of both ports with photodetectors, cf. Fig. 1 and [42]. In this way, we are able to detect all of the Brillouin scattered signal, which is incident on the heterodyne beam splitter. Additionally, measuring the difference of both photodiodes in this balanced setup has the benefit that the LO noise from our measurement within the detection bandwidth is removed. The two modes incident on the 50:50 beam splitter are the LOâ LO and the signal field where ρ B is the reflectance resulting from the scattering cross-section of the Brillouin scattering process andâ B and a vac are the annihilation operators of the Brillouin scattered mode and vacuum mode, respectively. The signal modeâ S is created inside the sample or cuvette, cf. Fig. 2, via the Brillouin scattering inside the sample and an overlap of the scattered mode with a vacuum mode. This overlap is similar to the overlap of two modes on a beam splitter. Note, that a priori knowledge of ρ B , and hence of the scattering crosssection σ B , is not necessary in the detection process. Moreover, it is in principle possible to extract the absolute value of σ B from the measured Brillouin signal after the measurement, through a direct reference of the signal to the vacuum noise. The output fields of the beam splitter, which are incident on the two photodetectors, read aŝ respectively, cf. [42]. The sign difference between the output fields results from the energy conservation which reflects one possible phase relation between the two input fields. The resulting difference current is Usingâ n = α n + δâ n exp(iω n t) where n = LO, B, vac (LO mode, Brillouin mode, and vacuum mode, respectively). α n are the coherent amplitudes, ω n are the angular frequencies, and δâ n and δâ † n are the field fluctuations. Additionally, we only consider terms linear in the coherent amplitude of the LO α LO and linear in the fluctuations of the Brillouin scattered (δâ B and δâ † B ) and vacuum mode (δâ vac and δâ † vac ). This is reasonable as the coherent amplitudes of the Brillouin scattered and the vacuum mode are much smaller than the amplitude of the LO, and hence, they do not contribute to the signal. Therefore, we obtain the difference current Hence, the detected difference current scales with the coherent amplitude of the LO and contains the fluctuations of the Brillouin scattered and vacuum mode. The frequency of the Brillouin scattered signal is given by the Brillouin shift specific to the sample and the incident pump frequency, and hence, is fixed. The beating frequency of interest is ω = ω LO − ω B . The detectors used measure only the difference current with a specific bandwidth BW around ω. As BW is approximately 30 MHz, only a part of the spectrum is detected by looking at one ω. Therefore, the frequency of the LO is varied in order to scan a larger frequency range, and with that, the full Brillouin peak. The frequency of the vacuum mode actually uniformly stretches over the whole spectrum, hence, the contribution of the vacuum mode to the final signal is the same for all difference frequencies ω LO − ω vac . Due to the limited BW , only frequency components of the vacuum similar to ω B are detected.

III. EXPERIMENTAL SETUP
The experimental setup contains two lasers as main laser and LO laser (Innolight Diabolo (linewidth ≈1 kHz over 100 ms) and Coherent Mephisto S (linewidth <1 kHz over 100 ms), respectively), both operating around 1064 nm with a frequency stability of 2 MHz/min. Moreover, the Mephisto S has a 30 GHz tuning range. This tuning range, which is achieved by applying a voltage onto the laser crystal changing the crystal temperature, is used to vary the beating frequency in the balanced heterodyne detector in order to obtain the Brillouin spectrum. Please note, that the general method is not limited to the wavelength of 1064 nm, but can be applied for any arbitrary wavelength, and hence, the optimal wavelength in regard of the Brillouin shift and Brillouin gain for a specific probe can be used. Furthermore, if the probe is sensible to phototoxicity or a specific imaging depth is necessary, this can also be taken into account for the used wavelength. It was chosen here however, as the two used laser sources and the photo diodes were available in the laboratory. Both lasers are split into two beams with a half-waveplate and a polarizing beam splitter (PBS). This results in adjustable splitting ratios, and hence, the laser power in each beam can be changed during the experiment if needed. Note that  more combinations of half-waveplate and PBS are used in the setup to optimize the laser power and polarization throughout the experiment. However, for the sake of simplicity, they are not shown in Fig. 3. One beam of each laser is used to overlap both lasers on a fast photodiode (beat note diode in Fig. 3, Thorlabs GmbH DET08C/M), specified for measuring frequencies up to 5 GHz. The determination of the beat frequency from 10 MHz to 5 GHz is done with an oscilloscope (Rohde & Schwarz RTO 1024) and sent to a computer, cf. Fig. 4. Fig. 4 shows the determined beating frequency from 2 GHz to 4 GHz detected during the measurement of glass (SF10) depending on the voltage applied to the laser crystal of the Mephisto S. This applied voltage changes the crystal temperature, and hence, the laser frequency of the Mephisto S, and therefore, the beating frequency. The resulting linear fit (f (V ) = a · V + f 0 , with a = (3.5330 ± 0.0028) GHz/V and f 0 = (−8.938 ± 0.009) GHz), with a linear correlation coefficient of R 2 = 0.9999, is used to determine the beat frequency above 5 GHz. To ensure, that this frequency estimation reflects the actual beating frequency, we start the measurement in a regime in which the fast photodiode still reliably detects the beating frequency, as the exact voltage dependence of the frequency changes due to the temperature and humidity of the laboratory. The main laser is focused into the sample acting as the probe for the Brillouin scattering. The incoming beam has a power of approximately 72 mW with a beam waist (radius) of 0.277 mm in front of the focusing lens corresponding to an average intensity of 0.299 MW/m 2 . The focusing lens is an objective lens with a focal length of 5.8 mm, a magnification of 20 and, a numerical aperture of 0.35, resulting in a focus waist (radius) of 7.1 µ m and a Rayleigh range of 148 µ m in air. For the values in the medium, these values must be multiplied by the corresponding refractive index of the medium. Neglecting any losses, the intensity in the focused spot in air is 455 MW/m 2 . Note that the intensity in the medium scales with 1/n 2 , since the focus waist depends linearly and consequently the focus area quadratically on the refractive index. The power in the signal beam (≈ 50 µ W) originates almost entirely from Fresnel reflection on optical surfaces in the beam path. Before entering the sample, the beam passes a chopper, periodically blocking and unblocking the incoming laser beam with a frequency of 20 Hz. In this way, we reference the measured variance with an open chopper to the measured variance with a closed chopper during the measurement, ensuring that slow laser power fluctuations are not affecting the result. The quarter-waveplate between PBS and sample is used to rotate the polarization of the incoming and reflected beam to deflect the backscattered light with the PBS to the heterodyne detector. Here, care was taken to avoid the direct reflex from the sample surface, e.g. a cuvette, entering the heterodyne detector, as this would mask any Brillouin scattered signal. This is done by aligning the direct reflex from the sample surface in a way, that it is not back-reflected into the PBS, and hence, not reflected into the heterodyne detector. This is reasonable, for two different reasons. First, we avoid the direct reflex from the sample surface, since it contains a much higher percentage of Rayleigh signal and lower percentage of Brillouin signal, simply due to the higher amount of reflected light based on the refractive index change from air to sample. Therefore, the detectors would be saturated without being able to detect the weak Brillouin signal. Second, without the direct reflection from the sample or cuvette surface, the ratio of Rayleigh to Brillouin signal is lower and it is in fact even possible to also measure the Rayleigh scattered peak. However, as this signal is still much stronger than the Brillouin signal, we would have to reduce the signal power before sending it into the heterodyne detector, as we would otherwise overpower the photodiodes, too. This would decrease the signal-to-noise ratio of the Brillouin peaks. The LO laser entering the heterodyne detector has a power of 1.2 mW. The LO and the Brillouin scattered light are superimposed on a 50:50 beam splitter and afterwards detected by two photodiodes. Each detecor consists of a FD300S3 photodiodes from Fermionics Opto-Technology, USA, and a self-designed and self-made amplifyer. They are matched and optimized in order to achieve almost identical detectors regarding quantum efficiency, transfer functions and spectra. At this LO power level, the photodiodes have a dark noise clearance of more than 10 dB. The difference signal of the two photodiodes is the measurand. In order to obtain one Brillouin spectrum, the frequency of the LO laser is scanned over a probe specific frequency range, by changing the laser crystal temperature (−3 GHz/K), in a set number of data points, here usually 1000. The frequency scan is done from the start frequency (f 0 ) to a maximum frequency (f m ) and back to the start frequency. This scan must be slow enough to ensure, that the frequency is stable for each measurement point. For each measurement point data is averaged over 50 ms, which improves the signal-to-noise ratio, where the noise here is the vacuum noise. Due to post measurement data selection, where 500 data points are selected, for a scan from f 0 to f m , the measurement time for a single scan is approximately 0.5 min. This is later referred to as single measurement. Figures 5 to 8 show the first successful measurements of the Brillouin shift using our presented method of balanced heterodyne Brillouin spectroscopy. The presented data are gained in either a single measurement (acetone) or averaged over a number of single measurements, in the case of water over 5, the pig lens over 10, gummy bear over 180, and glass over 180 measurements. The data shown in Fig. 5 are only the unfiltered data, whereas in Figs. 6 to 8 the unfiltered data along with data filtered by a moving mean over a window of 12 neighboring data points is shown for better visualization. The fit in each figure, which is done with the unfiltered data, is a Lorentzian fit of the form

IV. RESULTS AND DISCUSSION
where a corresponds to the height of the function, ν B is the Brillouin frequency shift, ν B is the Brillouin linewidth, and c is an offset, which results from the non-vanishing power in the signal port. Note, that for the sake of simplicity a Lorentzian fit is used here instead of a Voigt profile. A direct comparison of the absolute values of the height of the fit function is not possible, as it depends on the reflectance VOLUME 10, 2022  [43] and SF10 [45]. The given errors for the experimentally measured values result from the 95% confidence interval of the specific fit. Note, that if the beating frequency exceeds 5 GHz (here Porcine lens, Gummy bear and SF10), and hence, is gained through the linear fit, the relative error resulting from the 95% confidence interval of the linear fit, is also included in the given errors through linear error propagation. † Ranging from 4.0 GHz to 4.7 GHz for capsular and central region, respectively [19]. * Data points were taken from the respective graph in [45] and the Brillouin shift was obtained from a fit according to Eq. (8) and corrected for backscattering geometry, 1064 nm and refractive index. Due to the poor spectral resolution in [45] no reasonable linewidth could be obtained.  ρ B (in Eq. (1)), the Brillouin gain of the investigated material, and the losses in the balanced heterodyne detector, e.g. the non-unity visibility which varies from measurement to measurement. Hence, these fits give information about the Brillouin shift and linewidth of the investigated material. Moreover, we use the obtained fit parameter along with the  corresponding 95 % confidence interval values in quadratic error propagation, in order to obtain an overall error for the measurement as a measure for the quality of our measured spectrum. For acetone and water, the Brillouin shifts for the respective wavelength are published. The presented results for the Brillouin shifts and linewidths in acetone and water are in good agreement with the literature [43], cf. Table 1.
The overall error of the spectrum of acetone is 6 % and of water 10 %. The measured values for acetone and water lie within the range of accepted values given in [43]. Small differences may arise in the temperature dependence of the refractive index and the speed of sound, and hence the temperature dependence of the Brillouin shift and linewidth, in the specific material. This is particularly relevant since we do not know the sample temperature precisely, although we monitor the laboratory temperature in the vicinity of the sample (21.5±0.5) • C and the literature values for the Brillouin shift and linewidth are given for an unspecified temperature. Furthermore, the detection bandwidth influences the measured linewidth, if the bandwidth is not significantly smaller than the Brillouin linewidth, which is not the case here. Our presented results are not corrected in terms of the detection bandwidth. In the case of the literature values, it is not clear what the detection bandwidth is and if the values are corrected in terms of the bandwidth or not. Conclusively, the measurements in acetone and water validate our heterodyne method, which we further extended to other materials beyond liquids such as soft materials, e.g. porcine lenses and gummy bears, and solids, e.g. glass. Fig. 6 shows the ex vivo measured Brillouin spectrum of an enucleated porcine eye lens. The lens was placed inside a cuvette containing 0.9% sodium chloride solution. As the cuvette was not moved during the measurement, a movement of the porcine lens can be excluded. The data presented in Fig. 6 were averaged over 10 measurements, which corresponds to approximately 5 minutes within a total measurement time of approximately 10 minutes. The fit is performed with Eq. (8), too. The overall error of the spectrum of the porcine lens is 34 %. The measured Brillouin shift is ν B,Lens = (4.70 ± 0.06) GHz and the Brillouin linewidth is ν B,Lens = (726 ± 230) MHz. Note that, due to the low power of the Brillouin signal, the fluctuations of the spurious signal power due to parasitic interferences have a stronger influence here than in the measurement of acetone or water, resulting in additional peaks on both sides of the Brillouin peak. In order to achieve a comparable signal-to-noise ratio for the porcine lens as for acetone, an averaging over more measurements is necessary. Nevertheless, this result demonstrates the capability of our method to investigate biological material. However, as the laser power, which is sent into the porcine lens is approximately 72 mW, this technique is not yet suitable for in vivo measurements. Nonetheless, this first detection of a Brillouin shift obtained by balanced heterodyne Brillouin spectroscopy in an eye lens of a vertebrate is already a promising result for further research for a possible application of this method in ophthalmology, e.g. the investigation of accommodation or diseases. The investigation of accommodation will support the search for suitable lens refilling materials potentially restoring accommodation [44]. Moreover, the here obtained results are comparable to the results obtained in [19]. There, Brillouin scattering in a porcine eye lens was measured with a laser beam at 780 nm in dependence of the focus position inside the lens and the time after extraction showing the post mortem tissue change. Even as the used wavelength differs in this measurement from ours, the results are still comparable, as the frequency shift scales with the wavelength. According to [19], the Brillouin shift in the center of the lens is approximately 6.4 GHz and 5.45 GHz in the capsular region for 780 nm. Rescaling it to 1064 nm results in a shift of 4.7 GHz in the center of the lens and about 4.0 GHz in the capsular region. This indicates that we measured in or near the center of the lens, as our measured Brillouin shift is (4.70±0.06) GHz. However, the exact Brillouin shift in the porcine eye lens depends not only on the focus position of the probe laser inside the lens but also on the post mortem delay, as the tissue changes rapidly. The time period after extraction in the here presented measurements is not precisely known, approximately 1-1.5 days, and hence, an exact comparison to the results obtained in [19] is difficult. Nevertheless, the presented Brillouin shifts in the post mortem porcine lens still resemble the results in [19]. Furthermore, even though the exact time after enucleation is unknown, we were also able to observe some post mortem tissue changes with our setup. As the post mortem time remains unknown and the aim of this work is to present and verify this method, these data are not presented here. Figure 7 shows the Brillouin peak of a yellow gummy bear obtained by our method. The measured Brillouin shift is ν B,GB = (9.16 ± 0.09) GHz and the Brillouin linewidth is ν B,GB = (663 ± 200) MHz and the overall error of the spectrum of the gummy bear is 35 %. These values are calculated by the fit according to Eq. (8) using data averaged over 180 single measurements, which corresponds to approximately 90 minutes averaging time within a total measurement time of 3 hours. We are not aware of literature values for the Brillouin shift or linewidth of gummy bears as well as values for the refractive index or the speed of sound in a gummy bear. However, this result further confirms the power of our method, as we are not restricted to specific materials, but rather can investigate seemingly random materials. Moreover, this was a further extension to a detected beat frequency of up to 11 GHz. The long averaging time of almost three hours results from the strong absorption in the gummy bear, and as the signal is rather small, also probably from a small Brillouin gain. Figure 8 shows the Brillouin peak of a SF10 glass obtained by our method. This measurement demonstrates the large accessible frequency range of the presented method. The estimated Brillouin shift is ν B,SF10 = (12.937 ± 0.027) GHz and the Brillouin linewidth is ν B,SF10 = (45 ± 7) MHz. These values are calculated through the fit according to Eq. (8) using data averaged over 180 single measurements, which corresponds to approximately 90 minutes averaging time within 180 minutes total measurement time with an overall error of 20 %. A Brillouin spectrum for SF10 glass can be found in [45]. Note, that the authors used a different laser wavelength, λ L = 514.5 nm instead of λ L = 1064 nm, and a different scattering angle, θ = 90 • instead of θ = 180 • . However, as the Brillouin shift of a medium can be calculated with the knowledge of the velocity of sound v s , the refractive index of the medium n for the used laser wavelength λ L and the scattering angle θ it is also possible to correct a given Brillouin shift for another setup if the laser wavelength, refractive index and scattering angle of both setups are known. We extracted data points from the Brillouin spectrum in [45], applied the fit according to Eq. (8) and corrected it to our setup. With this procedure the expected Brillouin shift could be determined to ν B,Lit = 13.059±0.013 GHz. The discrepancy between the literature and our measured value is only about 0.1 GHz. Considering that no exact temperature was specified in [45] and that no systematic errors were considered, these values are in good agreement.
The results of the yellow gummy bear and the SF10 glass emphasize the potential of our method, as we were able to extend the estimated frequency range to beating frequencies of approximately 13 GHz, which can be extended to 30 GHz with the presented method and is much higher than the limit of 3.6 GHz beating frequency detected in other beating methods [24]. It is worth noticing, that long measurement times are problematic in the investigation of living tissues. With our setup, we were able to reduce the measurement time in comparison to other beating methods. In the case of toluene, the measurement took around one hour [25], which has a Brillouin gain comparable to acetone (cf. [43]), measured in half a minute with our method. For optical filter methods measurement times for the Brillouin shift in biological tissue as low as 0.3 s were reported [19]. However, a direct comparison is difficult due to missing data of the signalto-noise ratio or other comparable measures. Nevertheless, as the here presented results are the first proof of principle measurements, there is still room for improvement in terms of the measurement time, as this strongly depends on the investigated material, the optical quality of the setup, the laser stability, number of sampling points, and the bandwidth of the photodiodes, which will be further improved in future experiments.

V. SUMMARY AND CONCLUSION
We present the first application of balanced heterodyne detection, a method known from e.g. quantum optics, in Brillouin spectroscopy. Here, we overlap the weak Brillouin signal on a 50:50 beam splitter with a strong LO, enabling us to detect all of the Brillouin scattered signal, which is incident on the heterodyne beam splitter. The output of both beam splitter ports is detected with two photodiodes and the difference current of these photodiodes is the measurand. By measuring the difference of both photodiodes in this balanced setup the LO noise from our measurement within the detection bandwidth is removed. We present the quantum physical description of the balanced heterodyne Brillouin spectroscopy and its application in the experiment. The Brillouin peak of different materials including liquids (acetone and water), solids (glass), soft materials (gummy bears), and even biological tissues (porcine eye lenses) were successfully measured with our method. The results obtained for acetone and water validate our method, as we can reproduce the literature values for these samples. In comparison with other beating methods [15]- [18], we were able to extend the detectable beating frequency to 30 GHz, here shown up to 13 GHZ, and reduce the measurement time. The resolution of our measurement is limited by the detection bandwidth, and hence, the averaging time necessary to receive a Brillouin spectrum. Furthermore, the direct reference of our signal to the vacuum noise principally allows the estimation of the absolute value of the Brillouin scattering cross-section, however, in order to achieve this estimation, the overall quantum efficiency of the setup must be quantified.