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Temperature dependent measurements with a compact fiber coupled sensor for trace gas detection in the near-infrared based on tuning fork enhanced interferometric photoacoustic spectroscopy are presented. The temperature effects on the sensor have been investigated in a range from T = −41°C to T = 107°C, in particular the influence on the resonance frequency and the Q-factor of the micro tuning fork. The refined sensor head contains a combination of a silicon tuning fork and an acoustic off-beam resonator and permits methane detection with a detection limit of S = (3.85 ± 0.01) ppm. The functional capability of a numerical model for the optimization of acoustic off-beam resonators in COMSOL Multiphysics® is presented.
© 2013 Optical Society of America
1. Introduction
Since several years, optical technologies gain influence in research as well as in industrial applications. Besides the essential field of integrated optics, optical sensing gets more important and makes its way from the research laboratories into application. Standardized technologies like absorption spectroscopy and photoacoustic spectroscopy (PAS) play the mayor role on the optical sensor market at the moment. Due to their high cost, laser based detection systems are used in a few custom applications only where high accuracy, low detection limits, and highly selective measurements are needed. Current research aims for solutions to reduce the sensor costs or enable new applications, for example by reducing the sensor size or power consumption.
For PAS, a miniaturized sensor solution was presented in 2005 by A. Kosterev et al. [1]. The use of a quartz tuning fork (TF) with its high quality factor as sound sensitive element made measurement volumes of a few cubic centimeters possible and offered a rugged and sensitive sensor. The QEPAS (quartz-enhanced photoacoustic spectroscopy) technique is the focus of several investigations in former and recent studies [2,3].
In this work, the basic QEPAS idea of using a highly resonant sound detector for PAS is combined with two further developments. To enhance the photoacoustically generated sound wave, an off-beam acoustic resonator (AR) is used [4]. This special resonator design offers more flexibility in the mechanical sensor layout and ensures a small sensor head design. Instead of using the piezoelectric properties of quartz for the readout of the TF’s deflection, an optical readout is realized with a compact interferometer. The combination of the mentioned schemes lead to a small fiber coupled sensor head without electrical connections. All needed excitation energy and information is sent and received by standard telecom single-mode fibers. A detailed description of the sensor setup can be found in section 2.
Several environmental parameters can affect the signal of a gas sensor; such as pressure, mechanical vibrations, acoustic background noise, strong electromagnetic fields or temperature. The pressure dependence of TF-based sensors has already been investigated by Dong et al. [5]. Influences of mechanical vibrations or acoustic background noise can be neglected due to the high resonance frequency of the used TFs as mechanical vibrations are typically located at considerably lower frequencies and the acoustical background noise follows an 1/f-dependence, leading to very low values above 10 kHz [1]. The influence of electromagnetic fields on the sensor signal can also be neglected, as, due to the optical readout, the sensor head contains no electrical components. The remaining environmental parameter is temperature. Fundamental investigations on the influence of temperature changes on the sensor signal will be will be the topic of section 3.
Methane was chosen as target gas, as it is important for several applications like biogas production, explosion prevention, or general work safety. Calibration measurements of the presented sensor along with a brief comparison to other optical sensing techniques are shown in section 4.
2. Experimental
Optical readout enables fully fiber coupled sensor heads without electrical connections. Therefore, the described sensor is divided in two parts, connected by three SMF 28 single-mode telecom fibers. Figure 1 shows a schematic drawing of the experimental setup. The left part of Fig. 1 contains all electronic devices and lasers as well as the data acquisition. For optical excitation a fiber coupled single-mode DFB-diode laser around λ = 1650 nm with an output power of P = 9.95 mW is used (NEL Laser Diodes NLK1U5EAAA). It is driven by a combined temperature and current controller (Thorlabs ITC110). A frequency generator (Agilent U2761A) is used for sinusoidal modulation of the laser current at half the resonance frequency of the TF for 2-f-modulation spectroscopic measurements. Methane excitation is done at λ = 1650.959 nm where the absorption cross section is S = 1.278 ∙ 10−21 cm−1/(molecule cm−2) [6].
As optical readout laser, another DFB-diode laser around λ = 1570 nm (EM4 AA1401-190600-063-PM900-FCA-NA) with an optical output power of P = 63 mW is applied, driven by a second ITC110. The detection of the optical readout signal is realized by an InGaAs photodiode with adjustable preamplifier (Thorlabs PDA10CS-EC). For all measurements described below, an amplification setting of g = 10dB is used corresponding to a current to voltage gain of G = 2.38 ∙ 103 V/A. The resulting voltage is fed into a lock-in-amplifier (Stanford Research Systems SR830) detecting in 2-f-mode. Measurement control and data acquisition is done by LabView® software and a PC.
A sketch of the sensor head is shown in the right part of Fig. 1. Three single-mode fibers enter the housing at one side, which allows a compact sensor head design. This is possible due to the special interferometer design and the off-beam resonator setup. All optical elements are implemented into a brass housing with a length of only l = 45 mm, a width of w = 28 mm, and a height of h = 10 mm. Figure 2 shows a 3D-sketch of the optical elements and the acoustic off-beam resonator. The silicon TF is glued to the brass off-beam resonator. It is cut out of silicon (111) with a diamond precision saw (Bühler Isomet 4000). A typical resonance curve is shown in Fig. 3 which contains a sketch of the TF, too. This measurement was done at room temperature and ambient pressure with the setup presented in detail by Köhring et al. in [7]. The square root of a Lorentian was fitted to the data to get the TF’s resonance properties, leading to a resonance frequency of f = (18798.3 ± 0.5) Hz and a Q-factor of Q = 7063 ± 76. No acoustical resonator was attached to the TF during this measurement; therefore shifting of its resonance is negligible. The only additional damping effect results from the mounting of the TF in the interferometer. This fact will be important for the results of the temperature dependent measurements.
Fig. 2 3D-sketch of the sensing elements. A cube beam splitter with two prisms serves as interferometer for the silicon TF (grey). Three optical fibers with attached GRIN lenses enter the sensor from the lower left (light blue). For constructional reasons, the AR below the TF is rotated by 22.5° from the TF axis (ocher).
Fig. 3 Resonance curve of the free TF without AR. The inlet shows a sketch of the TF with the parameters: l = 6.8 mm, w = 710 µm, t = 400 µm, g = 380 µm.
The excitation light is sent through the off-beam resonator. A small sonic output with a diameter of dso = 300 µm is placed above of the resonator to connect the TF to the amplified acoustic wave inside the off-beam resonator. The TF is too long to be entirely mounted directly on top of the resonator. To omit undesirable effects of the overhanging part of the TF on the acoustic wave, the excitation light axis is shifted by an angle of α = 22.5° with respect to the TF. This configuration ensures optimum signal amplification. Up to now, off-beam resonators are only used for quartz TFs with their fundamental resonance at f = 32.768 kHz [8]. Therefore, the resonator parameters for the 18.8 kHz silicon TF had to be determined. This was realized with a simple 2-dimensional model in COMSOL Multiphysics® using the acoustics module and acoustics solid interaction. Due to the GRIN lens used for collimation, The inner resonator diameter was chosen to di = 1 mm as the beam waist of the collimated excitation laser beam leaving the GRIN lens can be expected to be smaller than this value. For manufacturing-orientated reasons, the resonator wall thickness in TF direction was set to tw = 200 µm and the distance between the TF and the resonator was set to d = 100 µm. The diameter of the sonic output was chosen to be dso = 300 µm according to the spacing of the TF prongs. Figure 4(a) shows the pressure field for the acoustic resonator with optimized length Lopt. At Lopt = 11.6 mm the acoustic pressure between the TFs prongs reaches its maximum. The distribution of the sound field inside the resonator resembles well the expected behavior. Firebaugh et al. derived the sound field of an on-beam AR using a COMSOL Multiphysics® model [9]. Because of the strong disturbance of the acoustic wave induced by the TF and the spacings in the middle of the resonator system, two separated maxima arise besides the center of the resonator. As the disturbance is much smaller in the case of the shown off-beam resonator, the field reaches its maximum nearly at the center of the tube. Furthermore, in Fig. 4(b) it can be seen that some part of the pressure wave leaves the resonator at the sonic output, reaching the TF’s prongs to stimulate their deflection.
Fig. 4 Pressure field distribution for the 2D off-beam AR from the COMSOL Multiphysics® simulation. The pressure is depicted in false colors, reaching from high pressures (red) to low pressures (blue). The left part a) shows the overview, b) shows an enlarged image of the region around the sonic output.
The optical readout is realized with a miniaturized Michelson interferometer [10]. The readout laser beam (fiber b in Fig. 1 and 2) enters a beam displacer and gets divided in two parallel beams. Both beams are aimed on the TF’s polished and silver coated side. They get reflected and interfere with each other after the recombination in the beam splitter cube. In addition to the setup shown in [10], a second prism is used to guide the light beam to the second readout fiber, which connects the sensor to the photodiode. This small modification in combination with the off-beam resonator ensures that all three fibers enter the sensor head at one side. The collimation of each of the three beams is done with a GRIN lens fiber collimator (Grintech GT-SMFP-050-025-50-NC-FC/PC).
3. Thermal influences
Generally, there are several thermal effects that have an impact on sensor signals. For the presented resonant PA sensor, the temperature stabilization of the laser sources ensures that their wavelength and optical power stays constant. Since the sensor head is connected to the lasers and electronics unit via optical fibers only and no additional electrical connection for temperature monitoring is wanted, information about the temperature dependence of the sensor head is of high interest. From a previous publication it is known, that a temperature change in the surrounding of the sensing head will change the temperature of the optical elements, which modifies the dimensions due to thermal expansion as well as the refractive index of the materials. Both effects can be compensated by the process described in [10]. In addition there is a significant temperature influence on the resonant properties of the resonator system consisting of TF and AR, which will be the topic of the following section. Other thermal influences are rather small and will be neglected in the following discussion.
In order to get an overview of the processes in the resonator system during a temperature change, a heuristic model is presented which shows good agreement with the temperature dependent measurements shown later. Even if it does not form a complete theoretical understanding of each phenomenon that leads to the measured data, it is a promising approach, to be confirmed in part by the presented data and forthcoming work.
3.1 Temperature dependence of the acoustic resonator
The change of the speed of sound causes strong effects on the resonant properties of an AR [11,12]. The temperature dependent speed of sound cs in an ideal gas can be described by:
where κ is the adiabatic exponent of the gas under test, kB the Boltzmann constant, and m the molecular mass of the gas inside the resonator. For the most common case of a gas mixture filling the resonator the parameters κi and mi can be summed up under consideration of the concentration Ci of each gas. It can be seen from Eq. (1) that a temperature change, a change of molecular mass, or a change of κ have influence on the speed of sound. The relation between the resonator properties and the speed of sound is given by the optimal length Lopt of the AR which is proportional to the half of the sound wavelength λs and not as theoretically expected an integer of λs/2 for an isolated acoustic resonator:Under consideration that the length of the acoustic resonator is fixed to the previously determined value Lopt during the measurements, Eqs. (1) and (2) illustrate that lower temperatures are connected to lower resonance frequencies and higher temperatures are connected to higher resonance frequencies as well. The optimization process for the off-beam resonator, described in section 2, was done for a temperature of T = 25°C. Therefore the resonance frequency at room temperature should be near fr = 18.8 kHz.3.2 Temperature dependence of the tuning fork resonance
The temperature dependent behavior of resonant silicon structures is known from former publications. The main temperature effect on the resonance frequency for flexural type motions of crystalline resonators is the temperature change of the Young’s modulus [13]. The temperature dependence of this material parameter can be described by the model of Wachtman et al. [14,15]. This theoretical equation was confirmed by several experimental results for different materials [16,17]. According to this model, the Young’s modulus E follows the equation:
where E0 represents the Young’s modulus at a temperature of T = 0 K. From the literature a value of E0 = 169 GPa can be found for silicon (111) [18–22]. B and T0 are empirical parameters, whereat T0 should be in the range of half the Debye temperature, which has a value of Td = 645 K for silicon [23].The resonance frequency of a TF can be approximately described by that of a beam with rectangular shape following the equation [24,25]:
here k is the spring constant, meff the effective mass of the beam, ρ the density of the material, w the beam width, and l the length of the beam. The inequality of the length of the beam and the effective length is considered by the value 1.015 [25]. The insertion of Eq. (3) in Eq. (4) results in an equation for the resonance frequency of a silicon TF in dependence of the temperature:with w = 710 µm the width of the TF`s prong, l = 6.8 mm the length of the TF`s prong and a density for silicon of ρ = 2.336 g/cm3, the parameter A can be calculated to be A = 5.13 ∙ 10−2 m1/2kg-1/2.3.3 Temperature dependence of the resonator system
For an analysis of the data shown later, it is crucial to treat the sensor as a combined resonator system consisting of the silicon TF and the off-beam AR. Both are coupled to each other, what gives them the ability to share energy. It is a result of this coupling between both resonators, induced by the temperature dependent properties described above, that the performance of the combined resonator system responds significantly to a change in the sensor temperature.
To gain a better understanding of these effects, a theoretical model for the resonant TF sensor system has to be introduced: In this model, the resonator system can be described via an exciting force, driving a damped harmonic oscillator and another damped harmonic oscillator coupled to the first one via a certain coupling coefficient. The exciting force is the photoacoustically induced sound wave, driving the AR. The micro TF is represented by the second oscillator, coupled to the acoustic one by the small sonic output. It is well known, that the amplitude A of a driven oscillation as a function of the driving frequency ω can be described by the square root of a Lorentzian function:
where Adrive is the amplitude of the driving force, ω0 is the resonance frequency and γ is the damping coefficient of the oscillator. The phase φ between the driving force and the response of the oscillator as a function of the driving frequency can be described as:As mentioned above, the resonator system can be described by a set of two driven damped harmonic oscillators. Each of them follows the Eq. (6) and (7). The main difference between them is the damping factor, which is significantly higher for the AR leading to its lower Q-factor compared to the TF. However, the theoretical treatment of this problem would be trivial if both oscillators could be investigated separately. The measurements have shown that there is an important temperature dependent coupling between both resonators that causes strong changes in the resonant properties of the whole resonator system; therefore the coupling must not be neglected.3.4 Temperature dependent measurements
The measurements with the combined resonator system were done over a broad temperature range around room temperature. To do so the sensor head was insulated and equipped with temperature stabilization. Two stages of Peltier elements and an additional water cooling allowed measurements between T = −41°C and T = 107°C. All measurements described in this section were done with a concentration of Cmethane = 5% of methane with purity of pmethane = 99.995% (Westfahlen AG) in nitrogen with purity of pnitrogen = 99,999% (Westfahlen AG) at atmospheric pressure. The gas mixture was provided by two calibrated mass flow controllers (MKS, 1179BX12CS1BV) actuated by a multi gas controller (MKS, 647C). An overall gas flow of F = 100 sccm results in a sufficiently fast gas exchange and was kept constant during the measurements.
For each temperature, a resonance curve of the sensor was recorded. The lock-in amplifier settings were a time constant of Tc = 100 ms and a slope efficiency of 24dB. The resonance frequency and Q-factor were extracted from the data by fitting the square root of a Lorentian line shape to the data of each measurement as described in section 2.
Figure 5 shows the results of the temperature dependent measurement for the quality factor of the resonator system; the results for the resonance frequency are shown in Fig. 6. The latter one contains an inlet with the same data, but a wider range on x-scale to demonstrate the behavior of the numerical fit function which will be discussed later. It is important to mention that the measured signals are proportional to the motion of the TF; therefore both diagrams show the temperature dependent resonant properties of the TF. Their temperature dependent behavior gives an idea of the way they are influenced by the coupled AR.
Fig. 5 The Q-factor of the silicon TF, measured in dependency of the sensor temperature is drawn as dots. The solid line represents the graph of a numerically adapted square root of a Lorentzian.
Fig. 6 The resonance frequency of the silicon TF, measured in dependency of the sensor temperature is drawn as dots. The solid line represents the graph of a numerically adapted function described by Eq. (8).
As Fig. 5 shows, the TF’s Q-Factor reaches a minimum near room temperature and seems to follow an inverse Lorentian. This impression can be strengthened by use of the theory for the AR as shown in 3.1. In a first step, the frequency of the fundamental mode of oscillation in the AR is calculated for each temperature with the aid of Eq. (1) and (2). This procedure requires at least one temperature where the frequency is known to determine the proportionality factor in Eq. (2). It was extracted from the data by searching the minimum of the curve in Fig. 5. It was found to be at T = 307.9 K corresponding to a resonance frequency of fr = 18791.6 Hz which was extracted from the data shown in Fig. 6. The upper x-scale in Fig. 5 shows the resultant resonance frequencies of the AR over the measured temperature range. Due to the change in the speed of sound, the AR resonance changes drastically over a range of about 5000 Hz. This leads to the behavior of the curve in Fig. 5 and can be illustrated with an examination of the coupling between both resonators. Two cases are of main interest: The frequency where the minimum of the curve occurs and the opposite case at much higher or lower frequency, where the Q-factor reaches its maximum. In the latter case, the resonance maximum of the AR is far away from the resonance of the TF. As the AR resonance is much wider than the TF resonance, there is still a part of the excitation energy which can be transferred from the AR to the TF due to the high Q of the TF. The reverse process, routing energy from the TF back to the AR can be neglected in this case, as the very sharp TF resonance has nearly no overlap with the AR resonance and the backward coupling coefficient is very small. In the second case where both resonances overlap, energy can be transferred in an efficient way from the AR to the TF, but also in the other direction: As the sharp TF resonance frequency coincides perfectly with the AR resonance frequency, energy can also be routed back into the AR. This is the explanation, why the Q-factor of the TF recorded in the shown measurements drops to a minimum. But this fact does not imply that energy is lost at this point, as the energy is stored in the AR. The change of energy stored in the TF`s oscillation and the change of its Q is in direct connection to an inverse change of the energy stored in the AR`s oscillation. Therefore, the overall amplification of the whole resonator system stays constant over the whole temperature range. This fact will be further confirmed at the end of section 4. Corresponding to the described assumptions, the square root of a Lorentian function was fitted to the data in Fig. 5, shown in the solid line. This curve represents the inverse resonance curve of the AR. Its Q-factor can be determined to 9.28 which is in good agreement with former publications [26]. The value of the TFs Q-factor at low and high temperatures reaches equilibrium near the value of the free oscillating TF. The slightly lower Q-factor of the measurement shown in Fig. 3 can be explained by the suboptimal mounting of the TF in the measurement apparatus. The optimized mounting used in the temperature dependent measurements induces less damping and therefore ensures higher Q.
As the TF’s Q-factor is strongly influenced by the AR temperature behavior and the Q-factor of an oscillator commonly impacts the resonance properties, the temperature dependence of the resonance frequency is investigated, too. While the curve in Fig. 5 follows the square root of a Lorentian line shape, which represents the amplitude of a driven damped harmonic oscillator, the curve in Fig. 6 seems to contain an arc tangent line shape, representing the belonging phase of a driven damped harmonic oscillator. The solid line in Fig. 6 represents the graph of a numerical fit which is a linear combination of this line shape with the equation for the TF resonance frequency temperature dependency presented in 3.2. It can be seen, that the resulting function corresponds very well with the data. In order to get some confirmation for the properness of the assumptions made before, some parameters of the fit function were fixed to values that correspond to known data from the literature or own measurements. The fit function follows the equation:
The parameter k represents the loading of both parts of the function. The parameters E0, T0 and A were fixed to the values presented in section 3.2.As the functions for the temperature dependencies of the TF`s Q-factor and resonance frequency both fit well to the data, the developed model seems to be a good estimation for the physics that dominate the temperature dependence of the resonator system. Furthermore, the results confirm the applicableness of the derived COMSOL Multiphysics® model for the optimization of the AR length to the resonance frequency of TFs. Although an optimum AR should lead to a minimum in Fig. 5 at 25°C, the measured minimum at a temperature of T = 34.8 °C is a good value for the simulation.
4. Calibration measurements
The calibration of the fiber coupled sensor was done with the gas mixing system described in the previous section. To ensure faster gas exchange, a flow of F = 2000 sccm was chosen. A bypass was implemented to avoid turbulent gas flow in the sensor head. Four concentrations of methane in nitrogen were chosen: 2500 ppm, 500 ppm, 250 ppm, and 50 ppm. A measurement with pure nitrogen was performed as well. The measurements were done at atmospheric pressure and room temperature, both monitored continuously by a pressure and temperature sensor. Figure 7 shows the temporal development of the sensor signal during the calibration process. Each concentration step was held for 10 min after reaching equilibrium. The measurement was done with a lock-in-amplifier time constant of Tc = 1 s and a slope efficiency of 24dB. A linear function through the point of origin was approximated to the data; it has a slope of m = (3.91 ± 0.02) µV/ppm. The corresponding line graph can be found in the inlet in Fig. 7. Each data point of this graph was calculated as average of 30 measurement points with the associated standard deviation as error bar. The standard deviation from the measurement of pure nitrogen was used to evaluate the detection limit to S = (3.85 ± 0.01) ppm. The corresponding power and bandwidth normalized noise equivalent absorption sensitivity (NEAS) is calculated to D = (4.31 ∙ 10−9 ± 1.12 ∙ 10−11) cm−1W(Hz)-1/2.
Fig. 7 Calibration curve for methane in nitrogen. The inlet shows the related calibration graph (dots) with a linear fit as solid line.
The demonstrated detection limit of the sensor is a sufficient value for several applications, even if it is far above the values of other optical sensing techniques; methane detection limits of some ppb were achieved with PAS and direct absorption spectroscopy in combination with quantum cascade lasers before [27, 28]. However, it is important to mention, that the TF-based sensor presented above achieves its performance with a NIR-diode laser, a miniaturized and rugged optical setup and a single mode fiber coupled sensor head.
As the results of the temperature dependent measurements show a strong impact of the sensor temperature on its resonant properties, additional calibration measurements were done at different temperatures. According to the more complex routing of the gas supply due to the bypass tubing and to the higher gas flow during the calibration, stable temperatures could only be realized in a range from Tmin = −21,5 °C up to Tmax = 35.7 °C. In order to reduce the measurement time, only three calibration steps were set for each calibration. Additionally, the measurements were performed with a smaller lock-in amplifier time constant of Tc = 300 ms. All other parameters used for those measurements were kept equal to these of the calibration described earlier. Figure 8 shows the detection limits derived from the data. The appendent standard deviation (1σ) of a measurement in pure nitrogen for each temperature is shown as error bars. It can easily be seen, that there is no significant change in the detection limit of the sensor in the investigated temperature range, as the fluctuations are only within the range of the error bars. This fact strengthens the assumptions in the previous section that imply that the resonator system stores the same amount of photoacoustically induced energy at each temperature.
5. Conclusion
A further step in the development of TF enhanced interferometric photoacoustic spectroscopy is demonstrated. The improved optical setup for the interferometric readout of the micro TF in combination with an off-beam AR ensures small dimensions of the sensor head. As the fiber coupling is done by telecom single-mode fibers, standard telecom laser sources can be used for gas detection. The sensor performance was demonstrated for methane and measurements were performed with a numerically optimized off-beam AR and a self manufactured silicon micro TF.
Temperature dependent measurements show a very interesting behavior of the TF’s resonance frequency and Q-factor. Most likely, these strong dependencies arise from the coupling between the two resonators. The coupling allows an energy transfer between the AR and the TF and vice versa. According to this assumption, numerical fit functions were developed and approximated to the data, showing good agreement with the experimental results. Furthermore, these measurements showed, that the numerical optimization of the acoustic off-beam resonator for the measurement frequency of f = 18.8 kHz, which was realized in COMSOL Multiphysics®, provided an efficient way to adapt the resonator dimensions to new frequencies. Calibration measurements at different temperatures showed that the effects mentioned earlier do not influence the detection limit of the sensor; this is connected to the complementary development of the amplifying properties of both resonators.
The small sensor size in combination with the fiber coupling and the temperature independent detection limit allows flexible sensor operation over large distances without considerably decreasing the sensor performance. This paves the way for more cost efficient gas sensing, as standard fiber coupled multiplexing and splitting techniques can be applied to accomplish concentration measurements at multiple measuring points with only one instrument containing laser and data acquisition.
Acknowledgments
Financial support by the BMBF under Grant BN 10468BS123456 and by the BMWi under grant KF2110319DF2 and by the Open Access Publishing Fund of Clausthal University of Technology is gratefully acknowledged.
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