1 Introduction

Photoacoustic spectroscopy (PAS) is a technique based on the conversion of light to acoustic energy by an absorbing sample. Following absorption of light, the sample emits a part of the energy non-radiatively, leading to local heating and thermal expansion. If the excitation source is pulsed or modulated, the periodic expansion of the sample manifests itself as a detectable sound wave [1]. Since PAS is sensitive only to non-radiative decay processes, the technique is exactly complementary to photoluminescence (e.g., fluorescence) spectroscopy. Especially in air and other gases, photoacoustic detection can be very sensitive; for example, explosives may be detected at very low concentrations from a considerable distance [2, 3]. PAS in air has also been used for medical diagnostics, monitoring volcanic activity, drug testing in athletes and greenhouse gas measurements [4–9]. Aside from chemical detection and quantification, PAS has found applications in biomedical imaging [10, 11]. The PAS signal is correlated to the power of the absorbed light, and not to transmitted light as in conventional absorption spectroscopy. Consequently, PAS can be used to quantify dilute, weakly absorbing analytes, even in strongly light-scattering media [12].

The sensitivity of PAS, especially gas-phase PAS, can be dramatically increased by resonance enhancement of the acoustic signal. This enhancement was historically obtained by placing the sample into a resonant cavity, such as a Helmholtz resonator [13]. “Open” acoustic resonators—and, in particular, quartz tuning forks—have also found many uses [4, 14]. Acoustic resonance enhancement is more difficult to achieve in liquids compared to gases, since sound is more readily attenuated and scattered, and the vibration of an open acoustic resonator is strongly damped in viscous media. Very few musical instruments would exhibit “sustain” when played under water.

In the present report, a liquid-filled high-quality wineglass is used as an open acoustic resonator, possessing an acoustic quality factor of about Q = 1,100. The PAS signal is recorded by optically measuring the vibration amplitude of the lowest acoustic mode at an antinode near the wineglass rim. This study is motivated by the desire to apply resonant all-optical photoacoustic techniques to liquid samples.

Orthophosphates are well recognized as important intermediary metabolites and as markers of water quality [15]. Aqueous orthophosphates are normally quantified using the colourimetric phosphomolybdenum blue assay, which forms, in the presence of orthophosphate, a complex absorbing strongly at around 820 nm. However, many optical detectors tend to perform poorly in this region, restricting the detection limit [16]. We aim to quantitatively detect orthophosphates using cavity-enhanced photoacoustic detection at 810 nm. A strongly absorbing organic indicator, methyl red, will also be analyzed using the same methods at 445 nm to validate the measurements.

Many transducers can be used for vibrational sensing of the wineglass, including microphones [17], optical deflection [18], fiber Bragg gratings [19, 20] and, presumably, piezoelectric transducers. In this report, a fiber Fabry–Pérot (FFP) cavity was used to detect the vibrations of the wineglass rim by measuring the strain on the fiber cavity. The transducer consists of an in-fiber cavity made from two identical fiber Bragg gratings (FBGs), which act as -25-dB mirrors at around 1,550 nm [21]. We lock a current-tunable diode laser to one of the FFP cavity resonances. For the small shifts measured here, this laser has approximately linear current/wavelength characteristics, known by the examination of its tuning curve. The acoustic frequency and amplitude of the vibration-induced strain are then obtained from the optical frequency shift of these cavity fringes, i.e., from the feedback current that is applied to the laser to maintain the locking condition. These vibration measurements were recently shown to be almost noise immune [22–24]. Since the applied modulation current changes both laser intensity and wavelength, the fringe shape is slightly distorted and the laser does not lock exactly to the peak of the cavity resonance. However, as only the resonance frequency shifts are relevant to the experiment, the absolute frequency is not required, and thus, this simplification of the setup had no adverse consequences. A similar setup previously permitted strain measurements with sub-picostrain resolution [25].

2 Experimental

Diagrams of the experimental setup are shown in Fig. 1. For experiments quantifying aqueous orthophosphates, an 810-nm diode laser (JDSU, SDL-2372) was chosen as a current-modulated excitation source, emitting approximately 0.8–1.3 W of linearly polarized light. When investigating methyl red, a 445-nm diode laser (Wicked Lasers, S3 Arctic Series, 700 mW) was used for excitation and was intensity-modulated using a mechanical chopper (Photon Technology International, OC-4000). In both cases, the same high-quality “crystal” wineglass serves as both a resonating acoustic cavity and sample holder. The bowl of the glass has a diameter ~9.5 cm, height of 136 mm and approximate rim thickness of 1 mm, with a volume of about 500 mL. When modulated laser light is absorbed by a non-fluorescing analyte within the glass, a pressure wave at the same modulation frequency is formed. By modulating the light intensity at the fundamental frequency of the wineglass, enhancement of the photoacoustic signal by the resonant cavity can be achieved.

Fig. 1
figure 1

a Photograph of the photonic wineglass setup, including dimensions height, H, and width, W = 2R of the wineglass bowl. During experiments, a lead foam box is placed over the apparatus to reduce ambient acoustic noise. b Schematic of the FFP transducer system locked using the Pound–Drever–Hall technique. For an unlocked laser, a = error signal, b = fringe spectrum. The 10-Hz scan ramp can be applied to the laser driver through the servo amplifier when searching for FFP cavity resonances. Abbreviations: FFP fiber Fabry–Pérot, DFB distributed feedback, PD photodiode, BT bias tee, LPF low-pass filter, LF low frequency, HF high frequency

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Several steps are taken to maximize the signal-to-noise ratio. To reduce the dissipation of acoustic energy into the surroundings, the foot of the glass is fastened to a heavy mass using two rubber-padded clamps. This mass is separated by acoustic foam from a heavy concrete slab, which also rests on acoustic foam. While running experiments, a soundproof lead foam box is placed over the setup to dampen external noise. Since it was not obvious to what extent the phosphomolybdenum blue complex dissipates absorbed energy by fluorescence, we added potassium iodide as a fluorescence quenching agent. However, we found no measurable effect on the photoacoustic signal.

Acoustic detection is accomplished using a fiber Fabry–Pérot (FFP) cavity, as was described previously [22]. The fiber transducer was attached to the wineglass rim using clear lacquer at a 90° azimuthal angle from the laser. It consists of two fiber Bragg gratings of −25 dB transmittance, creating an optical cavity 20 mm in length (center wavelength 1,549.6 nm; finesse ~1,000; free spectral range ~4.6 GHz). This detection method was previously developed by Gagliardi and co-workers [26] and was used in our group as a photonic guitar pickup [22]. As shown in Fig. 1b, a distributed feedback (DFB) diode laser (Avanex A1905 LMI) is used to interrogate cavity length changes in the vibration sensor. The 1,550-nm laser has a linewidth of 2 MHz and is frequency-modulated near 7 MHz. With the Pound–Drever–Hall (PDH) locking technique, an error function is generated from light reflected by the cavity, which is antisymmetric about the optical cavity resonance. This signal gives information on both the direction and magnitude of peak shift, and thus the cavity length change [27]. A portion of the feedback correction signal was applied to a ×20 gain acoustic amplifier (Texas Instruments, LME49710) and then sampled by Adobe Audition 3.0 software for recording and analysis [22]. Our PDH technique is appropriate for strain measurements of up to 30 kHz and is robust against noise resulting from laser intensity fluctuations [23].

Solutions of phosphomolybdenum blue and methyl red were prepared for analysis. Methyl red solutions of 1.13 mM–360 nM were made by adding methyl red powder (Sigma-Aldrich) to 95 % ethanol containing 1.0 g/L KOH, yielding a clear, red–orange solution with no precipitate. As methyl red is a colourimetric pH indicator, keeping a constant pH was important. An interrogation wavelength (445 nm) close to the isosbestic point near 450 nm was chosen to reduce pH effects.

To create a 5-ppm stock solution of phosphomolybdenum blue, an aqueous molybdate solution (5 mL, 0.104 M Na2MoO4 (Sigma-Aldrich), 5 M H2SO4) was added to a phosphate stock solution (25 mL, 10 ppm KH2PO4) and stirred until no longer turbid. Ascorbic acid powder (Sigma-Aldrich, 0.2 g, 1.14 mmol) was added as a reducing agent [28]. The quantities were selected such that orthophosphate was the limiting reagent. Deionized water was added to 50 mL and mixed by inversion. The solution was heated at 110 °C for about 10 min to accelerate the complex formation, yielding a clear, dark blue liquid with no precipitate. The resulting [PMo V4 Mo VI8 O40]−7 complex is formed quantitatively, containing a single orthophosphate ion. Deionized water was used for all steps of this synthesis, and for subsequent dilutions [29]. Volumetric pipettes were used to load samples into the wineglass, as the acoustic resonance frequency is strongly dependent on the liquid volume in the glass.

3 Results and discussion

3.1 Characterization of the acoustic resonator by mechanical excitation

A “sommelier-quality” wineglass was selected as our resonating acoustic cavity, mainly because such a resonator retains a high Q-factor when filled with liquid and because there exists a large body of work on wineglass acoustics [17, 18, 30–33].

When excited at its fundamental frequency ν 0, the wineglass vibration mode can be described as a lateral quadrupole. As glass is a rigid material, the circumference of the rim is not compressed or stretched during resonance; rather, it warps between two ellipses whose primary axes are perpendicular to each other [17]. This warping motion is most pronounced at the rim, but extends to all heights of the wineglass bowl. The quadrupolar mode has four nodes and four antinodes, where the site of excitation corresponds to an antinode [33]. When a wineglass rim is gently tapped, higher harmonic frequencies are also excited. For example, the rim deforms into a rounded triangular shape for the first harmonic (six nodes) and a rounded square shape for the second harmonic (eight nodes) [18, 33]. For an empty glass, all frequencies can be estimated using Young’s modulus and the density of glass, as well as the wineglass dimensions [17, 33].

By tapping the empty wineglass with a rubber mallet, the fundamental and harmonic frequencies were determined and compared with those calculated from two simple models. In 1983, French calculated the frequencies using an energy balance, approximating the wineglass as a cylinder with one rigid, but deformable rim and one fixed rim [17]:

$$v_{n - 2} = \frac{1}{12\pi }\left( {\frac{3Y}{\rho }} \right)^{1/2} \frac{a}{{R^{2} }}\left( {\frac{{(n^{2} - 1)^{2} + (R/H)^{4} }}{{1 + 1/n^{2} }}} \right)^{1/2}$$
(1)

Here, R and H represent the radius and height of the bowl, a corresponds to the wall thickness, while Y and ρ are Young’s modulus and the glass density, respectively. An even simpler model by Apfel assumes that the cylinder has two deformable rims and gives the frequencies as [33]:

$$v_{n - 2} = \left( \frac{n}{2} \right)^{2} \frac{a}{{\pi R^{2} }}\left( {\frac{Y}{3\rho }} \right)^{1/2}$$
(2)

In our case, the values for thickness and radius are a ≈ 1 mm and 2R = 85–95 mm, depending on the height. Setting the speed of a longitudinal acoustic wave in glass as (Y/ρ)1/2 = 5,200 m/s [33], and R = 47.5 mm, we fit a to obtain good agreement with four experimentally measured frequencies ν 0 (at n = 2) to ν 3. Note that the harmonics are not integer multiples of the fundamental, but depend instead on ~n 2, where n is half the number of nodes. Both theoretical models reproduce the experimental frequencies well, but require slightly different values for the glass thickness, a (Table 1).

Table 1 Fundamental and harmonic frequencies of an empty wineglass, in comparison with two simple models
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When liquid is added to the wineglass, the resonant frequencies decrease. This effect may be qualitatively understood using a simple energy balance: The vibrating mass increases as liquid is added, resulting in a rise in kinetic energy, while the potential energy of elastic deformation remains constant [17]. In a partially filled wineglass, the vibrational amplitude is not constant within the liquid; the volume of liquid nearest the glass bowl contributes more significantly to pitch lowering than the liquid near the center [31]. In the present experiments, this observation implies that the photoacoustic signal also shows a dependence on the radial distance from the wall of the glass. Sound generated near the wineglass wall couples more strongly into vibrational motion of its rim, compared to sound generated near the center of the wineglass bowl (see Sect. 3.4).

Upon tapping the wineglass containing 300 mL of solvent (deionized water or 95 % ethanol) with a rubber mallet, several resonant modes and their decays were observed (Fig. 2a, b). The figures show acoustic beats at all frequencies, but with different periods. In our case, beating occurs because the transducer breaks the cylindrical symmetry of the wineglass and, consequently, the degeneracy of the rim vibrations is lifted. This gives rise to coupled vibrational modes with frequencies that differ by about 0.25 Hz at 385 Hz (1.5 Hz at 922 Hz; 5 Hz at 1,785 Hz). Similar effects have been observed before and were then attributed to imperfections in the glass wall [30]. The decay of the fundamental mode at ν 0 = 385 Hz was recorded and fit to an exponential envelope function, giving a ring-down time of τ = 0.49(1) s (Fig. 2b). With this ring-down time, we determine Q = 2πν 0 τ = 1.15(2) × 103 [34].

Fig. 2
figure 2

Time evolution of a the frequency spectrum and b the logarithm of the acoustic power obtained after tapping a wineglass containing 300 mL water. The beating pattern is a result of mechanical coupling between two near-degenerate vibrational modes of the wineglass

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3.2 Characterization of the acoustic resonator by optical excitation

The fundamental frequency and Q-factor of the filled wineglass were validated by irradiating either an absorbing liquid or a graphite spot painted on the outside of the wineglass. The laser height was kept constant when exciting either the graphite spot or absorbing liquid, to best duplicate experimental conditions. The cavity resonance was then characterized by tuning the laser modulation frequency (Fig. 3) near ν 0. The vibrational amplitude of the photoexcited wineglass can be described as a damped driven oscillator [35], subjected to a periodically modulated force F 0 and characterized by a mass, m and a damping rate γ. Letting z equal the displacement from equilibrium,

Fig. 3
figure 3

Acoustic frequency sweeps of the photoexcited wineglass containing a 300 mL of deionized water, excited at black graphite spot, b 300 mL of 5 ppm phosphomolybdenum blue with 100 mM KI. The data are fit to Eq. (3), giving ν 0 = 385.20(4) Hz and γ = 1.04(3) in the former case and ν 0 = 375.40(1) Hz and γ = 0.841(5) in the latter case

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$$\frac{{F_{0} }}{m}\cos \omega_{f} t = \omega_{0}^{2} z + 2\gamma \frac{{{\text{d}}z}}{{{\text{d}}t}} + \frac{{{\text{d}}^{2} z}}{{{\text{d}}t^{2} }}$$
(3)

where ω 0 is the angular frequency of the undamped oscillator. The solution to this differential equation is z(t) = A(ν) sin (ω f t − α), where

$$A(\nu ) = \frac{{A_{0} }}{{\sqrt {\pi^{2} \left( {\nu^{2} - \nu_{0}^{2} } \right)^{2} + \gamma^{2} \nu^{2} } }}$$
(4)

with A 0  = F 0/4πm. The frequency at which A(ν) is maximal, \(\nu_{\hbox{max} } = \left[ {\nu_{0}^{2} - \gamma^{2} /2\pi^{2} } \right]^{1/2}\), can be approximated as ν max ≈ ν 0 since \(\gamma < < \nu_{0}\). Equation (4) is used to fit the data in Fig. 3 to obtain the resonance frequency, ν 0 and the amplitude decay rate, γ = (2τ)−1. The Q-factor is calculated as:

$$Q = 2\pi \nu_{0} \tau = \frac{{\pi \nu_{0} }}{\gamma }$$
(5)

The analysis for a wineglass containing 300 mL of water and excited at the graphite spot gives Q = 1.16(3) × 103, in excellent agreement with the value given in Sect. 3.1. Upon repeating the measurement using 300 mL of 5 ppm phosphomolybdenum blue solution, Q = 1.40(1) × 103 is obtained. The higher value is partly due to the slightly lower resonance frequency and partly due to the longer decay time.

Further optimizations included adjusting the incident beam height to maximize the path length, improving the detector location (2 mm below the rim, circumferential attachment) and optimizing the azimuthal angle between beam and detector. For excitation near the fundamental frequency, a 90° angle was selected, placing the detector near an antinode.

3.3 Noise characteristics of the system

One can characterize the type of noise affecting the measurement using an Allan–Werle diagram [36, 37]. Figure 4 shows the Allan deviation of the photoacoustic signal for a time series of measurements {x i } as a function of the sampling interval, δt, for intervals from 1 to 3,000 s.

Fig. 4
figure 4

Allan deviation of photoacoustic signal as a function of the sampling interval, δt. The data acquisition rate was 1 Hz, and the signal was generated by photoexcitation of 1.5 ppm aqueous phosphomolybdenum blue solution. The linear fit (red line) has a slope of −0.52, indicating that at short time intervals, the main noise contribution is white frequency noise [36, 37]

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$$\sigma_{\text{PAS}} (\delta t) = \sqrt {\frac{1}{{2\left( {\delta t} \right)^{2} }}\left\langle {\left( {x_{n + 2} - 2x_{n + 1} + x_{n} } \right)^{2} } \right\rangle }$$
(6)

It is apparent that the measurement uncertainty is minimized for sampling intervals of 2–3 min and that measurement noise is predominantly white frequency noise as indicated by the slope d(log σ PAS)/d(log δt) = − 0.52 for δt = 1–100 s [36, 37]. The main source of noise appears to be electronic noise, while thermal fluctuations dominate at long times. Variations of the acoustic amplitude sampled at 1-s intervals may also contribute to the noise spectrum, as well as power fluctuations of the 810-nm excitation laser. To obtain the most precise data, a time constant of 500 s is optimal. However, as a compromise between signal-to-noise ratio and experimental duration, we typically averaged for only 10 s.

3.4 Quantification of phosphomolybdenum blue and methyl red concentrations

The photoacoustic responses at the peak of the respective acoustic resonances of both phosphomolybdenum blue and methyl red solutions were recorded at different concentrations (Fig. 5). As the liquid level may vary slightly for each concentration, ν 0 is determined for each measurement. This determination of the acoustic resonance frequency is difficult for low analyte concentrations, ultimately restricting the limit of detection.

Fig. 5
figure 5

a, c Photoacoustic response following resonant excitation of a methyl red solution and c phosphomolybdenum blue solution. b, d After linearization of Eq. (8), the detection limits are calculated from the respective calibration curves (red lines) and the 3σ (99 %) confidence intervals (blue lines) as described in Ref. [39]

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While the transmitted intensity follows the Beer–Lambert law,

$$I_{\text{trans}} = I_{0} 10^{ - \,\varepsilon CW}$$
(7)

the photoacoustic signal, I PAS, scales with the absorbed light, i.e.,

$$I_{\text{PAS}} = I_{0} k_{0}\left( {1 - 10^{ - k(W)\,\varepsilon \,C\,W} } \right)$$
(8)

Here, k(W) is a dimensionless function that relates the absorbance to the amplitude of the photoacoustic signal and k 0 collects all instrument related scaling factors. This function is equal to unity only if the photoacoustic response is independent of the site of sound wave generation along the absorption path, W, and the quantum yield for the photoacoustic response is unity, i.e., radiative relaxation is negligible. In most photoacoustic systems, however, the acoustic signal depends strongly on the origin of the sound wave, and our setup is no exception. In an attempt to obtain k(W), we performed a fit of (8) to the data in Fig. 5a, c, obtaining k(W)εr = 0.090 ppm−1 and 6,275 M−1, respectively (Table 2). By comparing with ε values that are obtained using transmission-based UV–Vis at the same wavelength, we find that k(r = 95 mm) = 2.5–4 %. Transmission-based values of ε were obtained using the same samples and a conventional spectrometer and were consistent with literature data [29, 38].

Table 2 Molar absorptivities of two absorbing analytes found using both UV–Vis spectroscopy and photoacoustic spectroscopy (PAS). The photoacoustic data were analyzed using Eq. (8)
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Clearly, the detected acoustic response in our system depends on the origin of the sound wave along the radial coordinate. This is consistent with previous observations that liquid nearest the wineglass wall couples most strongly to the glass vibration [31]. Conversely, an isotropic sound wave generated in the center of the wineglass bowl should not contribute to the quadrupolar vibrational mode of the wineglass. This hypothesis is supported by a simple experiment: An absorbing target was submersed in a water-filled glass and excited at different distances from the glass wall by intensity-modulated light at the wineglass resonance frequency. It was apparent that the region near the wineglass wall contributed most strongly to the detected signal (Fig. 6).

Fig. 6
figure 6

Photoacoustic response following resonant excitation of a black target held at a selected distance from the wall within the water-filled wineglass. The signal is largest when the sound is generated near the wall and decreases to approximately zero near the center of the bowl at 47 mm. A fit to a second-order polynomial is meant to guide the eye but can also be used to quantify the photoacoustic response function (see text)

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An approximately second-order polynomial dependence of the signal as a function of the radial coordinate was observed. Of course, the radial photoacoustic response function is likely a much more complicated function, considering the acoustic mode structure inside the non-spherical wineglass bowl, the acoustically reflective water surface and the symmetry-breaking absorbing target.

The differential photoacoustic signal at each point along the absorption path can now be obtained by multiplying the quadratic response function with the differential of the absorption function, i.e., (1-I trans) [see Eq. (7)]

$$\frac{{{\text{d}}I_{\text{PAS}} }}{{{\text{d}}r}} = I_{0} \varepsilon C10^{ - \varepsilon Cr} \left( {P_{1} r^{2} + P_{2} r + 1} \right)\ln (10)$$
(9)

where we obtained P 1 = 4.15 × 10−4 mm−2 and P 2 = −4.03 × 10−2 mm−1 from the quadratic fit (Fig. 6).

After integrating over the entire path length, W, one obtains a new function that explicitly contains the photoacoustic response (not shown), which may be approximated by Eq. (8), allowing the calculation of k(W). When evaluating this function for W = 95 mm, our wineglass diameter, we obtain a scaling factor of k = 11 %, i.e., a value that is larger than the k = 2.5–4 % obtained experimentally (Table 2). Given the crude assumptions necessary in measuring and modeling k(r), the discrepancy is not too surprising. Also, the above estimate assumes that the quantum yield for the photoacoustic response is unity. Nevertheless, the above experiments illustrate that for quantitative measurements, our setup has to be calibrated using a known absorber, as is common for all photoacoustic systems and, indeed, for almost all absorption spectrometers.

To determine the limits of detection, C LOD, we linearize the calibration curves (Fig. 5a, c) to obtain Fig. 5b, d. This is done by re-arranging Eq. (8) to \(- \log\left( {1 - {{I_{PAS} } \mathord{\left/ {\vphantom {{I_{PAS} } {I_{0} }}} \right. \kern-0pt} {I_{0} }}} \right) = k\varepsilon CW\). Under the assumption that, after this transformation, the measurement errors remain similar and concentration independent, the limits of detection within 3σ can be determined using the procedure outlined in [39]. They are calculated as C LOD = 8 µM for methyl red and 1.2 ppm for orthophosphate, respectively. In an earlier article, a lower C LOD = 100 ppb for phosphomolybdenum blue was reported following another photoacoustic method [40], but the statistical analysis was less rigorous. Applying the same criteria for the limit of detection, we estimate that the earlier limit of detection was closer to 300 ppb and possibly even higher.

Our setup could be improved in several ways. Measurements at lower concentrations could be made more reliable using a reference beam that serves to find the wineglass resonance using an absorbing spot on the glass wall. The resonance frequency, ν 0, could then be found even for weakly absorbing samples without moving the setup. Of course, a more powerful laser is also expected to drastically improve the data precision and lower the LOD. Finally, choosing a resonator with a smaller volume, higher resonance frequency and similar Q-factor will improve the signal-to-noise ratio and may lower the detection limit.

4 Summary and conclusion

We presented an all-optical photoacoustic spectroscopic system incorporating a fiber Fabry–Pérot detector and acoustic wineglass resonator. The resonator has been characterized with regard to its Q-factor and resonance frequencies, and the results were compared to established wineglass resonance models. Following optimization and calibration, the concentrations of aqueous phosphomolybdenum blue solution and methyl red in basic ethanol were quantitatively measured with the setup. Detection limits with a confidence interval of 3σ were found to be 1.2 ppm and 8 μM for phosphomolybdenum blue and methyl red, respectively. Ongoing work explores different types of driven acoustic resonators that exhibit “sustain” while submersed in solution, with the aim of a more compact setup and lower detection limit.