1. Introduction

The observation of electromagnetic scattering including Rayleigh, Brillouin, Raman, and Mie, has proven to be useful in many diagnostic applications ranging from combustion and aerodynamics to atmospheric and environmental sensing [1]. The spectral broadening, frequency shift, and intensity of these scattering types in a medium are dependent on many fundamental thermodynamic properties of the medium including temperature, density, species composition, and velocity. Therefore, retrieval of these properties is possible by fitting the scattering spectra to theoretical models. Moreover, these aforementioned scattering types are non-resonant, meaning the scattering occurs regardless of the wavelength of the applied electromagnetic field, reducing constraints when selecting a laser source for diagnostics. Advances in modeling specifically in the Rayleigh-Brillouin scattering (RBS) regime have provided methods to retrieve the temperature and density of the medium through thermal broadening of the spectra and the Doppler shift of the Brillouin peaks (which is related to the speed of sound in the medium). For instance, the Tenti S6 model [2] made significant contributions in modeling spontaneous RBS while the Pan S7 [3] model extended the previous work to coherent RBS in gases.

Often, the desired signals overlap spatially and are too close spectrally to be separated from parasitic scatterings and reflections by conventional bandpass filters or spectrometers. For instance, Brillouin scattering is separated from Rayleigh and Mie scattering by only a few GHz in water and is even closer in air, making the experimental interrogation challenging. In addition, Mie scattering, resulting from surfaces or larger particles, often overwhelms the desired signal such as Rayleigh, Brillouin, or Raman scattering. In general, there are three major types of filtering: spectral, temporal, and spatial. Among spectral filtering techniques, several have been developed and utilize absorption features of atomic or molecular vapors to remove the unwanted spectral features. Notably, Filtered Rayleigh Scattering (FRS) demonstrated using the absorption peaks of iodine as a spectral filter to suppress Mie and other background scattering, enabling effective analysis of Rayleigh [4]. Iodine absorption features conveniently overlap the second harmonic output ($532$ nm) of Nd:YAG lasers, making FRS readily available to a broad audience of researchers. Another method, that uses atomic vapor resonances to select spectral features via temporal filtering, is Slow Light Imaging Spectroscopy (SLIS) [5]. SLIS takes advantage of the reduced group velocities due to the steep refractive index gradients near resonance absorption features to temporally separate light of different frequencies. This technique was recently demonstrated to measure the shift in Brillouin peaks in water for spatially resolved 2D thermometry applications [6].

The steep refractive index gradients may also be used to spatially, rather than temporally, separate a desired spectral feature through dispersion. In its simplest sense, dispersion occurs when light experiences frequency-dependent spatial separation. This is often accomplished through optical gratings or prisms, however, it can also be done using the changes in refractive index near an atomic resonance. For instance, a prism-shaped mercury-vapor-based dispersion filter was used to resolve rotational Raman lines [7]. Dispersion was also used in an iodine cell to detect the Doppler shift [8]. Rubidium vapor cells were previously used in a study by Starling, et.al. where several beams as close as 50 MHz were spatially separated and resolved by a camera [9]. Recently, Rekhy, et.al. demonstrated dispersion of laser scattering from air using an atomic cesium vapor cell [10]. This article extends the previous work to the realm of scattering from liquids, shows good separation of both closely spaced and far-off-resonance features, and anticipates using the technique in water LIDAR applications, specifically its usability for temperature and salinity measurements. In general, this work presents a unique diagnostic capability for a simultaneous course and fine spectroscopy of scattered spectra. Among other applications, this method can be used to separate the spectrally-shifted light scattered from laser-induced gratings, augmenting the already existing laser-induced gratings spectroscopy (LIGS) or laser-induced thermal acoustics (LITA) techniques [11,12].

2. Experimental methods

Figure 1 shows the optical setup applied to this high-resolution study of laser scattering from liquid water using a dispersive atomic medium. A continuous wave Ti:Sapphire laser (M-squared SolsTiS) was used for the generation of narrow linewidth (< $5$ MHz) tunable coherent radiation in the near-infrared. Laser output power was kept constant at $1.8$ W. As the atomic vapor of choice for this study was rubidium, the central wavelength of interest for this study was set to match the $D_2$ transition between its $5^2 S_{1/2}$ ground state and $5^2 P_{3/2}$ excited state near $780$ nm. The laser wavelength was monitored at all times by routing a small portion of the output beam into a fiber-coupled wavemeter using a variable beam splitter cube (Thorlabs VA5-PBS252). The wavemeter used in this study was the HighFinesse Angstrom WS7-60 wavemeter with an absolute accuracy of $60$ MHz. The diameter of the vertically polarized laser beam after the beam splitter and beam expander (BE) was $7$ mm. The beam was focused into the water chamber with a $125$ mm focusing plano-convex lens. The water was filtered to have less than 0.2-micron particle size after the chamber was thoroughly cleaned with solvents and baked. The water temperature was monitored using a thermocouple attached to the outside wall of the chamber and remained constant at $T_{W}$ = $20$ $^{\circ }$C. The scattered light was collimated at a collection angle of $\theta _{s}$ = $90~^{\circ }$ with a $100$ mm aspheric plano-convex lens before being sent through the rubidium vapor prism cell. The prism cell had a center path length of $80$ mm, $25$ mm diameter, Brewster’s windows angled at $55~^{\circ }$32$'$, and no buffer gas. After the prism cell, the scattered light was imaged onto a camera sensor with a $500$ mm focal length lens, resulting in a 5X magnification. The camera used was a Basler acA 1920-40 um camera with a $780$ nm bandpass filter (Thorlabs FL780-10, $10$ nm FWHM) placed in front of it to mitigate background light interference. The selected camera has a Sony IMX249 CMOS sensor with a quantum efficiency at $780.2$ nm of $\sim 20 {\%}$. The sensor has a physical size of $11.3$ mm $\times$ $7.1$ mm and a pixel size of $5.86$ $\mu$m. A sample image showing a full frame collected for one of the test cases with three spectral peaks dispersed and spatially separated is presented in the inset of Fig. 1. The rubidium prism cell was heated using self-adhesive polyimide flexible heaters by Omega Engineering, Inc. The cell temperature was monitored via thermocouples. All thermocouple measurements were later corrected by a fixed positive bias of $4.5~^{\circ }$C due to ambient air cooling.

 

Fig. 1. Schematic of the experimental setup with a sample frame shown in the inset.

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The goal of this experimental investigation was to demonstrate ultra-high-resolution spectroscopy of tightly spaced features using a dispersive atomic medium, as well as to determine the capabilities and limitations of such a system. The temperature of the side arm determines the vapor pressure of the rubidium in the cell. During this campaign, a broad range of rubidium cell and side arm temperatures were tested to optimize the trade-off between dispersion and absorption, as both effects are dependent on the vapor pressure of the atomic vapor. The temperature range for the cell and side arm varied between $148.5~^{\circ }$C and $177.1~^{\circ }$C, corresponding to rubidium vapor pressures of $3.4\cdot 10^{-3}$ Torr and $1.4\cdot 10^{-2}$ Torr respectively. Rubidium number densities for these temperatures were calculated to be between $\sim 7.8\cdot 10^{19}$ m$^{-3}$ and $\sim 3.0\cdot 10^{20}$ m$^{-3}$.

3. Theory and numerical simulation

3.1 Scattering from liquid water

In this study, the medium of interest that scattered the laser light was liquid water. In general, scattering from a liquid medium has a similar structure to the one from atmospheric air, with several key differences. When the thermal motion of the scattering centers (molecules) dominates and they move independently of one another, such as in dilute gases, the spectral line shape is predominantly Gaussian, and gives information about temperature. In liquids and liquid mixtures, the scattering centers are affected by one another due to their relative motion, higher density, and spatial distribution, leading to correlated motion. Therefore, the spectral structure widths and relative intensities give information about the transport and thermodynamic properties. The total scattered spectrum considered for numerical simulations presented here was modeled as 6 distinct components: Mie scattering from particles - elastic, narrow-band scattering without any frequency shift; Rayleigh scattering - caused by isobaric entropy fluctuations - has no spectral shift due to negligibly small Doppler effect of flow motion and negligibly small broadening; Stokes and Anti-Stokes components of Brillouin scattering - caused by scattering due to the isentropic pressure (acoustic) fluctuations establishing a continuous spatial grating - has both spectral broadening and shift; Stokes and Anti-Stokes components of Raman Scattering - an inelastic process involving an exchange of energy with the medium through excitation or de-excitation of vibrational and/or rotational states - also spectrally broadened and shifted [13–16].

Both Mie and Rayleigh scattering were simulated together as elastic scattering at the laser wavelength with a $5$ MHz Full Width at Half Maximum (FWHM) [6]. Considering the Rayleigh contribution being negligibly small due to a small Landau-Placzek ratio (< 0.1), the peak intensity was primarily determined by the particle loading at the time of measurement and could marginally differ from test to test [16]. The Brillouin scattering peaks are caused by the scattering from the isentropic density fluctuations that move randomly in all directions at the speed of the sound [17]. Due to the Doppler effect, there is a symmetric frequency shift $\nu _B$ with respect to the initial laser wavelength. The expression for the Brillouin frequency shift in liquid water is primarily a function of temperature ‘$T$’ and secondly a function of salinity ‘$S$’, and depends on scattering angle ‘$\theta$’, refractive index ‘$n$’, wavelength ’$\lambda$’, and speed of sound ‘$v_s$’, as shown below in Eq. (1). As a representative example, the speed of sound in $20~^{\circ }$C fresh water is 1482 m/s, and the refractive index for the same conditions is close to 1.329. In the case of 780 nm light scattering at $90^{\circ }$, the resulting Brillouin shift is expected to be close to 3.57 GHz. The spectral profile for Brillouin scattering was approximated by a Lorentzian distribution of 0.4 GHz FWHM for room temperature water [6,18,19].

Raman scattering, being the second major non-elastic scattering process considered in this work, occurs between $97$ to $131$ THz from the probe laser frequency [20]. Due to the high density of liquid water, the Raman scattering spectrum lacks a defined narrow-band structure but is rather represented by a single broadband envelope [21]. Additionally, the Raman scattering is typically non-symmetrical, meaning the Stokes component is usually stronger than the Anti-Stokes component, as thermally dictated by the population distribution [22,23]. This suggests that the Raman scattering presented in Sec. 4. is dominated by the Stokes component. Since Raman scattering is far detuned from the absorption lines of rubidium, the rubidium refractive index encountered by those spectral components was approximated as $1$ and transmittance as $100{\%}$. Figure 3(a) shows an example of a simulated scattered spectrum.

3.2 Dispersion through a prism vapor cell

The dispersion through the vapor cell can be found knowing the refractive index of the vapor medium within the cell and the Brewster angles $\theta _{b}$ of the cell windows [24]. Figure 2 defines an angle $\theta$ at each surface and refractive index in each medium the light passed through.

 

Fig. 2. Geometry of cell

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Snell’s law and geometric properties provide relations between the angles and refractive indices defined in Fig. 2, as shown in Eqs. (2)–(4).

These values can then be rearranged through the following relations to find the deflection angle $\delta$, as shown in Eqs. (5)–(7).

The deflection angle is converted into the displacement recorded by the camera by multiplication with the focal length $F$ of the lens between the cell and camera, which is a small angle approximation of a trigonometric sine function, as shown in Eq. (8).

For frequencies far detuned from resonance, the refractive index of the cell becomes that of a vacuum (n$_{\upsilon }=1$). The displacement acquired by such a far-detuned spectrum was set to be $0$ mm in all the figures below.

3.3 Modeling displacement

All the codes used for numerical simulations in this work were developed in-house and some of the core simulations, such as refractive index simulation were previously validated and published [25]. The rubidium numerical model included both the natural broadening and thermal broadening mechanisms and was partially developed based on the work by Siddons, et.al. [26]. The self-broadening (pressure broadening) mechanism was not included in the current version of the model as it was calculated to be significantly smaller than thermal broadening effects [27]. An open-access alternative developed by the groups of Hughes and Adams is available as a ready-to-use package here [28]. The frequency offsets referenced throughout this article were calculated with respect to the rubidium D2 absorption centerline located at $780.026562$ nm in air.

Figure 3 shows the process for finding a dispersion profile of the scattering through the vapor cell with different spectral components identified as follows: MRS - Mie and Rayleigh Scattering, ASBS - anti-Stokes Brillouin Scattering, SBS - Stokes Brillouin Scattering, ASRS - anti-Stokes Raman Scattering, SRS - Stokes Raman Scattering. First, the spectrum of the scattered light was determined following the theory discussed in Sec. 3.1 above and plotted as shown in Fig. 3(a). Then the spectral profile was multiplied by the transmittance across the frequency domain, removing some of the spectral peaks, see Fig. 3(b). Next, the refractive index was determined across the frequency domain and mapped to each of the scattering peaks. Following the theory outlined in Sec. 3.2, the deflection angle and the final displacement were calculated. A binning function was used to combine contributions from scattered light at different frequencies but resulting in equal displacements. A spatial profile of the laser beam was extracted from the experimental Raman peak in the displacement domain and then convolved with the rest of the domain to simulate a realistic and finite measured beam width. The appropriate intensity ratio between the combined Raman scattering and Brillouin scattering contributions, before absorption, was determined from the experimental data once, and then kept constant throughout the simulations. Due to the overwhelming Mie scattering, whose intensity is based on stochastic particle loading, the Mie-Rayleigh scattering intensity was redetermined for each simulation with minor deviation from test to test.

 

Fig. 3. Steps required to model displacement peaks. (a) Scattering plotted in the frequency domain. (b) Scattering adjusted for absorption and (c) mapped to refractive index. (d) Convolved lineshape in the displacement domain. MRS - Mie and Rayleigh Scattering, ASBS - anti-Stokes Brillouin Scattering, SBS - Stokes Brillouin Scattering, ASRS - anti-Stokes Raman Scattering, SRS - Stokes Raman Scattering.

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4. Experimental results and discussion

During this experimental investigation of the atomic vapor cell dispersion and its utility for fine spectroscopic measurements of physical properties, a wide range of rubidium cell temperatures and laser frequency offsets were tested. Rubidium cell temperature, as was noted earlier, determines the vapor number density and consequently the complex refractive index inside the cell. While higher refractive index typically results in greater dispersion, it also leads to stronger absorption of the interrogated spectrum. The wide range of laser frequency offsets between $-15$ GHz and $+15$ GHz, in addition to aiding the numerical simulation validation, allowed the identification of the most promising locations for the laser wavelength depending on the application. If an application would require a frequency offset in the optically thick region presented here for an 80-mm long cell, a shorter prism cell design may be considered. However, excessive shortening of the cell might be hindered by the large angle between the cell windows for a given cell diameter, which is required for an effective light refraction. An example of a rubidium vapor wedge cell as thin as 30 nm was previously reported by Keaveney, et.al. [29]. These experimental results were augmented by the numerical simulations to estimate uncertainty bounds, as well as to predict the system benefits for water LiDAR and thermometry applications.

Validation of the numerical model was the first step. Figure 4 shows the normalized dispersion profiles experimentally acquired through the rubidium cell heated to $148.5~^{\circ }$C overlaid with the simulated results. The probe beam was set to several frequency offsets from the center of the rubidium D$2$ transition, including (a) $5.50$ GHz, (b) $3.23$ GHz, (c) $-2.63$ GHz and (d) $-4.50$ GHz. The selection of the $0$ mm displacement was set based on the far-detuned Raman scattering that had the complex refractive index of $1$. These Raman scattered peaks were labeled SRS$+$ASRS throughout the article and experienced no motion despite shifts in the laser frequency. The intensities and locations of the Mie, Rayleigh, and Brillouin peaks varied from frequency shift to frequency shift due to the transmittance and refractive index differences across the spectrum. In all cases, the model predicted the experimental profile well, hence the model could be used for extrapolation of the experimental results to new conditions.

 

Fig. 4. Experimental and modelled spatial profiles at selected frequencies in $148.5~^{\circ }$C cell temperature. MRS - Mie and Rayleigh Scattering, ASBS - anti-Stokes Brillouin Scattering, SBS - Stokes Brillouin Scattering, ASRS - anti-Stokes Raman Scattering, SRS - Stokes Raman Scattering.

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Figures 5, 6, and 7 compare experimentally the acquired peak displacements with the numerical simulations across the entire frequency domain for three cell temperatures: $148.5~^{\circ }$C, $160.7~^{\circ }$C, and $177.1~^{\circ }$C, respectively. Experimental data is represented by green (Stokes Brillouin peaks), red (Mie and Rayleigh peaks), and blue (anti-Stokes Brillouin peaks) dots at their corresponding frequencies and displacements. Theoretical curves of displacement and transmittance as a function of frequency are plotted along the experimental data. Subfigures (b) and (c) show enlarged sections of the frequency domain to highlight the degree of the experimental data agreement with the simulation as well as to introduce measurement uncertainties. Each experimental measurement had two main sources of error: vapor cell temperature fluctuations, and uncertainty in laser frequency. Fluctuations of the vapor cell temperature were recorded during the experimental campaign and were on the order of $2~^{\circ }$C for lower temperatures, and rose to $3~^{\circ }$C for higher temperatures over the course of several hours. These temperature fluctuations were accounted for in the displacement simulation by calculating the lower and upper bounds of the expected values and are shown with the yellow-shaded regions. Each experimental data point had a frequency uncertainty of $\pm$ $60$ MHz associated with the wavemeter measurement uncertainty added to the plots as well.

 

Fig. 5. Comparison of experimentally determined displacements of Mie/Rayleigh and Brillouin peaks with modeled displacements at $148.5~^{\circ }$C cell temperature. The $2~^{\circ }$C uncertainty in the vapor cell temperature is depicted on the enlarged portions of the spectrum for (b) Stokes and (c) anti-Stokes sides with yellow shading.

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Fig. 6. Comparison of experimentally determined displacements of Mie/Rayleigh and Brillouin peaks with modeled displacements at $160.7~^{\circ }$C cell temperature. The $2^{\circ }$C uncertainty in the vapor cell temperature is depicted on the enlarged portions of the spectrum for (b) Stokes and (c) anti-Stokes sides with yellow shading.

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Fig. 7. Comparison of experimentally determined displacements of Mie/Rayleigh and Brillouin peaks with modeled displacements at $177.1~^{\circ }$C cell temperature. The $3~^{\circ }$C uncertainty in the vapor cell temperature is depicted on the enlarged portions of the spectrum for (b) Stokes and (c) anti-Stokes sides with yellow shading.

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In general, experimental data matched the simulations well with significantly smaller deviations than were predicted based on the thermocouple reading uncertainty for both lower temperature cases. The temperature uncertainty that encapsulates all data points was estimated at $0.4~^{\circ }$C for those test cases. The $177.1~^{\circ }$C case, shown in Fig. 7, had the highest experimental data fluctuations in the −15 GHZ to −8 GHz frequency range that mostly fell within the expected uncertainty due to temperature. These fluctuations are explained by the excessive and random heat loading on the vapor cell from the circulating room air. Such fluctuations could be mitigated by providing an isolated enclosure to the vapor cell/heater assembly and incorporating a PID temperature controller. Additionally, a starved-type vapor cell may be utilized to limit the maximum atomic vapor number density. In the tested configuration, the largest spatial displacement of $2.65$ mm was achieved for the Mie/Rayleigh scattering peak with $1.03$ mm separation from the anti-Stokes component of Brillouin scattering, corresponding to only a $3.58$ GHz spectral shift. A sub-pixel precision in laser beam or filament position identification is routinely achieved in molecular tagging velocimetry applications using Gaussian or Voigt profile fitting algorithms [30]. Hence, the authors propose that the spectral resolution for the current approach is primarily determined by the vapor cell temperature and wavelength uncertainties rather than the camera resolution as long as the implied spatial displacement is on the order of a pixel size or more ($5.86$ $\mu$m in case of this work). While the best separation of the spectral features was achieved with the highest cell temperature, atomic vapor cells tend to degrade at such high temperatures over time. In case an application requires high signal transmittance or a long stable life span, lower temperatures and/or shorter cell lengths might be necessary.

Similar results could be achieved using other combinations of atomic vapors and laser wavelengths, depending on the application requirements. For example, cesium has strong absorption features near $455.5$ nm, and $459.3$ nm, and potassium has strong absorption features near $404.4$ nm, and $404.7$ nm. Shorter wavelengths such as those enabled by potassium and cesium will be preferable for measurements in water due to better penetration [31].

5. Modeling of the system sensitivity and other future applications

As described earlier, the Brillouin peak frequency shift is dependent on the salinity and temperature of the water. Here the use of dispersion in an atomic vapor cell is proposed to precisely measure such shifts as a means to determine the temperature and/or salinity of liquids in general and water in particular. Using the simulation validated by this work, the minimum precision for temperature and salinity was determined. First, the frequency with the maximum change in refractive index was found for transmissions above $10{\% }$. Then, two frequencies corresponding to the upper and lower bounds of the temperature uncertainty at that central frequency were calculated. The difference in these frequencies is the system’s resolution/precision, which can be then converted to either a difference in temperature or salinity by mapping against their relationship with the Brillouin peak shift. In the experimental configuration used in this work ($90^{\circ }$ scattering angle, rubidium cell temperature uncertainty of $0.4~^{\circ }$C), frequency shifts as low as $15.5$ MHz could be measured. Such high resolution can be achieved by carefully selecting the probe laser central frequency and cell temperature ($0.0954$ GHz probe offset and $136~^{\circ }$C cell and side arm temperatures). This corresponds to water temperature and salinity changes of $2.3~^{\circ }$C and $4.9$ PPT, respectively, near room temperature and average ocean salinity ($21.1~^{\circ }$C and $35$ PPT). A further improvement in measurement precision is expected by stabilizing the rubidium cell temperature with a PID temperature controller or utilization of a starved vapor cell and collecting the signal from the backward direction ($180^{\circ }$ scattering angle), which is a typical arrangement in LiDAR applications. Mitigation of the laser frequency uncertainty and errors associated with experimental setup vibrations in the case of a liquid water LiDAR system can be achieved by selecting a laser wavelength at which both Brillouin peaks are displaced and transmitted, see Fig. 8. The arrangement shown in Fig. 8 allows one to simultaneously track any deviations in the position of both Brillouin peaks, thereby accounting for the instability of the image plane, and/or frequency drifts.

 

Fig. 8. Simulated solution with both Brillouin peaks displaced. Laser frequency offset - $0.7694$ GHz; cell temperature $160~^{\circ }$C.

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6. Summary and conclusion

This article demonstrates the use of an atomic vapor cell for high-resolution spectroscopy and high-efficiency filtering of laser scattering from liquid water. This new approach relies on a highly dispersive medium (rubidium vapor) to spatially separate different frequency components of the scattered signal near the rubidium resonance at $780$ nm. By tracking their position on the camera sensor, MHz-level changes in the spectral shifts were measured and consequently, physical properties of the liquid medium such as temperature or salinity were determined. In this work, a range of rubidium cell temperatures between $148~^{\circ }$C and $177~^{\circ }$C as well as multiple laser frequency offsets between $-15$ GHz and $+15$ GHz were tested and scattering displacements were mapped. A maximum displacement of the Brillouin peak of $2.65$ mm was recorded, which corresponds to a maximum separation from Mie and Rayleigh scattering of $1.03$ mm. The Raman scattering component was also spatially separated from all other types of scattering. Additionally, a numerical model was developed and compared against the experimental results. Overall, the model and data had good agreement within the uncertainty in the rubidium cell temperature. Combining the experimental results and validated model solutions, the minimum frequency shift detectable by the current experimental setup (system resolution) was estimated at $15.5$ MHz, with the limiting factor being attributed primarily to the vapor cell temperature uncertainty.

Overall, the proposed approach was successful at both spatially separating major scattering components for filtering purposes (Mie/Rayleigh, both Brillouin and Raman scattering), as well as resolving spectral shifts of a given component as small as $15$ MHz. Liquid water LiDAR was identified as one of the possible future applications and the ways to improve the experimental setup were suggested.