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Abstract
In this work, we introduce the concept and delineate the fundamental principles of photoacoustic interferometry (PAInt), aiming at the development of a novel methodology for the precise assessment of the speed of sound in liquid media. The PAInt apparatus integrates an intensity-modulated continuous wave laser beam at 20 MHz for the efficient generation of monochromatic photoacoustic wavefronts which interfere across the surface of a vertically displaced spherically focused piezoelectric element. In this context, the resulting interference pattern can reveal the acoustic wavelength in the liquid medium with remarkable accuracy, providing thus reliable estimations of the speed of sound in reference liquids (error ∼0.1%) such as distilled and sea water, acetonitrile, and ethanol.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
The precise assessment of the speed of sound along with the determination of the respective frequency-dependent attenuation coefficient, constitute the most important measurements for the complete acoustic characterization of materials. Focusing on the speed of sound parameter, pulse-echo based methods have been broadly applied for relevant assessments in various media including liquids [1] and tissues [2,3]. These commonly used pure acoustic approaches rely on the precise time-of-flight measurement of a short ultrasonic pulse following its reflection by a surface located at a defined distance from the source. In a different approach, quantitative imaging of compositional and structural properties of soft tissue is performed by speed of sound reconstructions [4–7]. The method has been proposed as a new diagnostic biomarker for diseases manifested by phenotypic alterations in the stiffness of tissue, especially when targeting cases where contrast to normal tissue is very low, such as in breast cancer [8]. Furthermore, ultrasound or photoacoustic (PA) based imaging reconstruction methods can be significantly improved both in terms of diagnostic accuracy and signal quantification by measuring a local speed of sound distribution in tissue [9–12]. Apart from biomedical applications, the precise determination of the speed of sound can be crucial for the assessment of quality and verification of adulteration in food and fuel, as well as, the measurement of minerals’ concentrations [13]. Specifically, the accurate acoustic characterization of edible oils or milk has been demonstrated to provide information on fraudulent compositional alterations or the presence of possibly hazardous contaminants [14–16]. Similarly, acoustic methods have been employed for the identification of induced ingredient concentration changes in biofuels with increased detection accuracy compared to standard methods [17].
In this work, we present a novel, low-cost method for the determination of the speed of sound in liquids named Photoacoustic Interferometry (PAInt), which exploits the PA waves generated during the strong absorption of intensity-modulated optical radiation at a single frequency. The emitted monochromatic spherical PA wavefronts interfere across the surface of the detector to produce an oscillating signal, revealing the acoustic wavelength (λac) in the medium for the selected modulation frequency and the speed of sound with high accuracy. Despite the fact that PA measurements have been utilized in the past for the determination of the speed of sound in liquids and layered media through the cross-correlation of the laser-induced ultrasound waves [18], the employed experimental setup required the integration of a Q-switched nanosecond laser, significantly increasing the cost of the method. On the contrary, the presented approach is inspired by the recently demonstrated frequency-domain PA microscopy systems [19,20] which use inexpensive continuous wave (CW) laser sources for the excitation of the signals, as well as, cheap electronics and data acquisition equipment for the recording of the measurements. This important advantage, in combination with the experimentally demonstrated precision of the speed of sound measurements, make PAInt a powerful alternative technique for the investigation of acoustic properties in liquid media.
2. Materials and methods
The PAInt apparatus (Fig. 1(a)) employs a CW diode laser emitting visible radiation at wavelength $\lambda = 488\; nm$ (MDL-III-488, CNI, Changchun, China; maximum power output: 100 mW). A positive lens focuses the beam on the active aperture of a free-space acousto-optic modulator (TEM-200-50, Brimrose, MD, USA; modulation bandwidth: 50 MHz), which modulates temporally the instantaneous intensity of radiation with high efficiency (∼70%). For analog optical modulation, the acousto-optic driver requires the input of a sinusoidal voltage (selected frequency: $F = 20\; MHz$; range: 0-1 V), which is provided by an arbitrary function generator (DG5252, Rigol, Oregon, USA; max frequency: 250 MHz). The laser light is then collimated by a second positive lens and subsequently filtered through a small aperture of 5 mm in diameter to isolate exclusively the first order of the resulting diffraction pattern, which is modulated at 20 MHz (Fig. 1(b)). The radiation is subsequently guided through an optical telescope consisted of two positive lenses for beam expansion by 4 times, prior its reflection into a modified inverted optical microscope (Diaphot, Nikon, Tokyo, Japan). An objective lens (custom-made; magnification: 10X; numerical aperture: 0.25) focuses the modulated light on a square piece of black duct tape (1 by 1 cm2) placed at the bottom of an optically transparent tank which is filled with the liquid medium to be measured. The PA waves generated following the strong absorption of the focused optical radiation by the tape sample, propagate through the medium in order to be detected by a spherically focused immersion ultrasonic transducer (V373-SU, Olympus, Tokyo, Japan; central frequency: 20 MHz; -6 dB bandwidth: 13.3–32.9 MHz, focal distance: 31.3 mm). The transducer is fixed on a 3D translation system consisted of two manual XY stages (PT1/M, Thorlabs, NJ, USA) and a motorized stage for displacement in the Z axis with sub-µm precision (8M167-100, Standa, Vilinius, Lithuania). The XY manual stages are required for the confocal alignment between the acoustic and the optical focus, maximizing the sensitivity of the measurements. On the other hand, the motorized Z stage is used to change stepwise the relative vertical distance between the PA source (black tape) and the detector (transducer). The generated signal is enhanced by two low-noise RF amplifiers (TB-414-8A+, Mini-Circuits, Camberley, UK; gain: 31 dB) connected in series to provide a total gain of 62 dB, prior its transmission in an I/Q demodulator (AD8333, Analog Devices, MS, USA; bandwidth: DC to 50 MHz). The demodulator additionally receives a 4F local oscillator signal at 80 MHz from a second channel of the function generator to provide I and Q values which are recorded and stored by a data acquisition card (DAQ; PCIe-6363, National Instruments, TX, USA; maximum sampling rate: 2 MS/s) and a computer in synchronization with the motorized stage. The I and Q values are directly used to estimate the amplitude (Amp) of the PA signal at each step of the Z stage through the equation $Amp = \sqrt {{I^2} + {Q^2}} $. To enhance the signal to noise ratio level, a total number of 4 × 105 values is averaged for the determination of the final PA amplitude value at each transducer’s position in the Z axis. The typical average power on the tape sample’s plane has been measured to be less than 2 mW, ensuring that no apparent photodamage effects are observed during prolonged time intervals exceeding 10 minutes. The total time required for a speed of sound measurement using typical parameters for the Z stage displacement (500 steps of 5 µm) has been less than 8 minutes. All experimental procedures have been realized at a constant room temperature of 25 °C, while all the liquid media were in thermal equilibrium with the environment during the measurements. Control and synchronization of the PAInt apparatus has been performed using custom-developed programs. Finally, all recorded and simulation data have been analyzed and processed by means of MATLAB programming environment and Origin graphing software.
Fig. 1. Experimental parameters of the PA interferometer. a) Schematic representation of the developed PA apparatus for the measurement of the speed of sound in liquid media. b) Temporal modulation of the excitation laser’s intensity at 20 MHz as measured by a fast photodiode.
3. Results
3.1 Simulation of the PA interference effect
The working principle of PAInt for the measurement of the speed of sound in a liquid medium is based on the interference variation of the generated monochromatic spherical PA waves over the finite surface of a focused transducer, according to its relative distance from the source. Within this context, the transducer’s surface can be considered as a part of a sphere with a radius R, corresponding also to the focal distance. Let us now consider a spherical ring with infinitesimal width dz located at a distance z as measured from the top of the spherical surface (red shaded region in Fig. 2(a)). In a cross-sectional view of the transducer (Fig. 2(b)), the focal point is found at C, whereas the PA signal point source is placed in an arbitrary out of focus position D. Furthermore, we define the distance from the signal source to each point of the ring as d, the defocus distance from the focal point as α, and the radius of the ring equal to x. By applying the Pythagorean theorem on the right triangles ABC and ABD of Fig. 2(b), one can derive the following equations
which can be further combined to solve for d and eliminate x parameter as followsFig. 2. Measurement principle and simulation of PA signal interference. a) Simple 3D illustration of a monochromatic PA wave detection by a spherically focused transducer. The red-shaded ring of infinitesimal width dz corresponds to a locus of constant PA phase. b) Cross-sectional view of the spherically focused transducer presented in a). Red spot corresponds to the focal point, whereas black spot stands for the signal source, which is found in an out of focus position. c) Simulated PA amplitude modulation as a function of the relative vertical distance between the point absorber and the transducer’s surface for distilled water (initial transducer’s position: 31.3 mm; final transducer’s position: 33.8 mm; step: 1 µm). d) Normalized amplitude spectrum of the simulated PA interferogram shown in c). The characteristic peak indicates the predominant spatial frequency of the PA interferogram at ∼1.336 × 10−2 µm-1.
Let us now consider that the ring is consisted of N detection elements distributed with a linear density ρ, which is given by the following equation
Solving Eq. (4) for N and eliminating x parameter through Eq. (1), we derive an analytical expression for the number of detection elements as a function of z according to the equation below
Furthermore, the phase φ of the PA wave when reaching the ring’s elements will be equal to
where λac corresponds to the acoustic wavelength in the propagation medium.In this manner, the total PA pressure detected by the arbitrary ring i (pi) can be directly written as
where Ni and φi correspond to the number of elements and the detected phase for the ring i respectively, as given by the Eqs. (5) and (6).By considering a linear response of the transducer, the detected PA amplitude will be calculated through the summation of the individual contributions from m concentric rings, covering the whole spherical surface with maximum thickness k at the center of the detector. In this case, m is given by the ratio of k to a finite value of ring’s thickness Δz, determining the accuracy of the simulation. Therefore, the final expression for the detected PA amplitude P will be
and can be directly used to investigate the PA interference effect as a function of the defocus distance α. The numerical values of the parameters involved in the simulation were set as follows: $R\; = \; 31.3\; mm$ (transducer’s focal distance); $k\; = \; 0.1614\; mm$ (maximum thickness at transducer’s center for an active area of 0.25 inches in diameter); ${\lambda _{ac}}\; = \; 0.7485\; mm$ (acoustic wavelength for a 20 MHz PA wave propagating in distilled water at 25 °C); $\Delta z\; = \; 1\; \mu m$ (ring’s width); $\rho \; = \; 100\; elements/mm$ (linear density of elements). Additionally, a total defocus distance α of 2500 µm was covered using steps of 1 µm. The predicted PA amplitudes were also weighted according to their attenuation due to geometrical spreading, in order to provide a more accurate representation of the interference effect. By plotting the normalized detected PA amplitude versus the vertical transducer’s displacement, we can generate the graph presented in Fig. 2(c), indicating an oscillatory behavior of the signal with a gradually decreasing peak to peak value, as expected, due to defocusing and geometric attenuation of the PA waves. Finally, to verify that the spatial period of the predicted oscillations in Fig. 2(c) coincides with the acoustic wavelength λac in the propagation medium, we have additionally generated a normalized amplitude spectrum through the Fast Fourier Transform (FFT) of the signals (Fig. 2(d)). The graph demonstrates clearly a sharp peak centered at ∼1.336 × 10−2 µm-1, which corresponds with high accuracy to the expected acoustic wavelength in the medium (74.85 µm). Despite its simplicity, this simulation has demonstrated the possibility of determining the speed of sound in a liquid medium by measuring the respective acoustic wavelength λac through a PAInt apparatus operating at a selected well-defined excitation frequency.3.2 PAInt measurements in various liquid media
Having delineated the basic principles of PAInt for the measurement of the speed of sound by simulating the behavior of the detected PA amplitude as a function of the source-detector relative distance, we proceeded to the application of the proposed method in four liquid media, namely sea and distilled water, acetonitrile and ethanol. Figure 3(a) shows a typical PA interferogram for the case of distilled water, which has been recorded using a total vertical displacement of the transducer equal to 2500 µm (500 steps of 5 µm). The plot is found in very good agreement with the respective simulation shown in Fig. 2(c), as the PA amplitude demonstrates a clear sinusoidal modulation while the transducer is gradually moved away from the signal source. Furthermore, it is observed that the PA amplitude oscillates with a progressively lower peak-to-peak value around a DC voltage of approximately 12 mV, corresponding to the offset of the demodulation device when no PA signal is received. Aiming to provide a direct comparison among the PA interferograms for the four investigated liquid media, we have generated a plot (Fig. 3(b)) presenting only their initial part (first 200 µm of the transducer’s vertical displacement), normalizing also the values in the datasets to be between 0 and 1.
Fig. 3. PAInt measurements in various liquid media. a) Experimentally obtained PA interferogram at 20 MHz for the determination of the speed of sound in distilled water following a total transducer’s displacement equal to 2500 µm with a step of 5 µm. b) Initial parts of the normalized PA interferograms for liquid samples of sea and distilled water, acetonitrile and ethanol, indicating noteworthy differences in their detected spatial periods (acoustic wavelengths). c) Normalized amplitude spectra of the recorded PA interferograms for the liquid media mentioned in b), following a total transducer’s displacement of 2500 µm. The characteristic peaks indicate the predominant spatial frequency of the PA interferogram for each sample. d) Progress of the speed of sound estimation for the measured liquids as a function of the transducer’s displacement.
The plot demonstrates explicitly the remarkable differences in the spatial periods of oscillations among the different liquids, corresponding to the acoustic wavelengths λac for the 20 MHz excitation frequency. The higher acoustic wavelength is qualitatively observed for sea water (black line), whereas the lowest one is for the ethanol sample (green line). Subsequently, we attempted to provide a precise estimation of the acoustic wavelength for each case, by calculating the Fast Fourier Transform (FFT) of the full PA interferograms consisted of 500 data points, generating also the respective normalized amplitude spectra as shown in Fig. 3(c). The amplitude spectrum for each investigated liquid medium is characterized by a sharp peak corresponding to the predominant spatial frequency of the PA interferogram. More specifically, the peak spatial frequency was estimated at 1.298 × 10−2 µm-1 (${\lambda _{ac}} = \; 77.04\; \mu m$) for sea water, 1.336 × 10−2 µm-1 (${\lambda _{ac}} = \; 74.85\; \mu m$) for distilled water, 1.569 × 10−2 µm-1 (${\lambda _{ac}} = \; 63.75\; \mu m$) for acetonitrile and 1.711 × 10−2 µm-1 (${\lambda _{ac}} = \; 58.45\; \mu m$) for ethanol. The respective progress of the speed calculation for each liquid medium as a function of the transducer’s vertical displacement is shown explicitly in Fig. 3(d). The curves are generated in real time during the measurements by multiplying the excitation frequency of 20 MHz by the estimated λac using the data points collected up to the current step. All curves tend to demonstrate a very steep increase during the first 100-200 µm of the transducer’s displacement, which is typically followed by a small drop and a subsequent stabilization at a well-defined speed value, especially after 1500 µm. By using the universal wave equation $v\; = \; {\lambda _{ac}}f$ ($v$: sound speed in the liquid; $\lambda$ac: acoustic wavelength; f: temporal modulation frequency of the laser beam), the peak spatial frequency measurements presented in Fig. 3(c) resulted in a final speed of sound estimation at 1541 m/s for sea water, 1497 m/s for distilled water, 1275 m/s for acetonitrile and 1169 m/s for ethanol.
In addition, we have provided an estimation of the measurements’ accuracy by repeating the experimental procedures 5 times for each liquid sample under identical conditions. The results for the average speed of sound are graphically shown in the bar plot of Fig. 4, with the error bars representing ± one standard deviation. The calculated values have been 1540.8 ± 2.5 m/s for sea water, 1496.7 ± 1.7 m/s for distilled water, 1279 ± 6.2 m/s for acetonitrile and 1171.4 ± 3.8 m/s for ethanol, which lead to a coefficient of variation (CV) range between 0.11 (distilled water) and 0.48% (acetonitrile), confirming thus the good precision of the proposed technique summarizes the extracted speed of sound results through PAint and provides a direct comparison of the measurements with reference values from the current literature, validating the followed experimental procedures (Table 1).
Fig. 4. Average speed of sound values for the investigated liquid media, as estimated from five sequential measurements. Error bars correspond to ± one standard deviation.
Table 1. Speed of sound estimations for the investigated liquid media and comparison with reference values. The speed range for sea water and ethanol refers to temperatures between 20 and 30 °C. Reference speeds for distilled water and acetonitrile have been recorded at 25 °C.
4. Discussion and conclusions
In this study, we have introduced for the first time to our knowledge, the fundamental principles of PAInt and demonstrated its promising capabilities for the assessment of the speed of sound in various liquid media by employing a novel experimental prototype. The tight focusing of the intensity-modulated optical beam has resulted in an excitation spot on the absorber’s surface having a diameter of ∼1 µm, which behaves as a point source of monochromatic spherical PA waves with good approximation, taking into consideration the geometrical parameters of the focused transducer (i.e., diameter and focal distance). The gradual change of the relative distance between the signal source and the transducer results in a periodic modulation of the PA amplitude, with a spatial period equal to the acoustic wavelength λac within the medium for a fixed excitation frequency, as demonstrated by the generated signal simulations. The direct determination of λac through the FFT of the recorded PA interferogram instead of measuring the temporal delay of an emitted acoustic pulse after its propagation for a well-defined distance, constitutes the most important technical difference when compared to the standard pulse-echo approaches for speed of sound estimations. Within this context, PAInt provides an alternative methodology for relevant measurements with high accuracy, low cost of the integrated components (in principle, the total budget for the equipment can be less than 2k euros), and technical simplicity of the experimental apparatus.
It is worth mentioning that the average speeds of sound as determined through PAInt, are found in excellent agreement with the respective values reported in literature using conventional methods. In particular, the sea water sample which has been taken from the Cretan Sea with a reported salinity of 39 ppt [21], is characterized by a speed of sound ranging approximately between 1527.1 m/s (20 °C) and 1550.8 m/s (30 °C) [22] under atmospheric pressure (measured average value: 1540.8 ± 2.5 m/s at 25 °C). On the other hand, the reported speed of sound for distilled water at 25 °C is 1496.65 m/s [23] which coincides perfectly with the respective estimated average value. Similarly, significant levels of agreement have been achieved for acetonitrile, with a speed of sound measured in the literature at 1278 m/s [24] versus 1279 m/s through PAIint. Finally, the speed of sound for pure ethanol has been shown to vary according to temperature between 1146 (30 °C) and 1181.9 m/s (20 °C) [25], which is once more in accordance with the corresponding PAInt measurement (1171.4 ± 3.8 m/s at 25 °C).
Despite the experimentally proven high accuracy of the PAInt method, several future technical upgrades could improve further the performance of the developed PA interferometer in terms of measurements’ repeatability and speed. First of all, the continuous movement of the transducer at a constant rate instead of a gradual stepwise displacement may enable the fast acquisition of additional data points (e.g., every 1 µm), increasing thus the precision of the measurement and expediting the experimental procedures at the level of one minute or less. As the acoustic focal zone extent for our transducer is in the order of 18 mm, the signal recording would still be possible even in the case of large displacements away from the source. In this direction, the precision of our measurements could be additionally enhanced by covering distances approximately 10 times higher than the current one. Furthermore, the precise adjustment and stabilization of the liquid medium’s temperature using proportional–integral–derivative (PID) controllers connected with specially designed immersed resistors and respective detectors, could contribute substantially in the improvement of the measurements. This important technical parameter may be crucial for future implementations of PAInt, as speed of sound is strongly dependent on temperature for many common liquids (e.g., for distilled water, 1 °C increase corresponds to ∼3.2 m/s change in the speed of sound) [26]. Additionally, the use of higher modulation frequencies combined with the integration of broadband detectors has the potential to improve the assessment of the speed of sound, as more amplitude oscillations can be recorded for a defined vertical displacement of the transducer, providing thus the predominant spatial frequency of PA signal’s modulation with higher precision. Finally, the incorporation of sophisticated, ultra-sensitive detection equipment such as lock-in amplifiers operating in the MHz regime, may upgrade further the validity and the repeatability of PAInt results, increasing significantly, however, the total budget required for the development of the experimental setup. As a future target, the proposed method could be utilized except of liquids, for the speed of sound assessment of compressible soft media including gels, polymers or even several types of biological tissues by integrating non-focused (flat) transducers in direct contact with the sample. Such applications are anticipated to highlight further the promising capabilities of the novel PAInt method towards the precise acoustic characterization of various materials.
Funding
“INNOVAPROTECT” (MIS 5030524); Hellaas CH (MIS 5002735); NSRF 2014-2020 “Bioimaging-GR” (MIS 5002755); H2020 FETOPEN project “DynAMic” (EC-GA-863203); Laserlab-Europe (EC-GA 871124).
Acknowledgments
G. J. Tserevelakis would like to thank Mr. Andreas Lemonis for his valuable support in the development of the PAInt apparatus.
Disclosures
The authors declare no potential conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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