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Photo-vibrational spectroscopy of solid and liquid chemicals using laser Doppler vibrometer

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Abstract

Photoacoustic/photothermal spectroscopy is an established technique for trace detection of chemicals and explosives. However, prior sample preparation is required and the analysis is conducted in a sealed space with a high-sensitivity microphone or a piezo sensor coupled with a lock-in amplifier, limiting the technique to applications in a laboratory environment. Due to the aforementioned requirements, traditionally this technique may not be suitable for defense and security applications where the detection of explosives or hazardous chemicals is required in an open environment at a safe standoff distance. In this study, chemicals in various forms (membrane, powder and liquid) were excited by an intensity-modulated quantum cascade laser (QCL), while a laser Doppler vibrometer (LDV) based on the Mach-Zehnder interferometer was applied to detect the vibration signal resulting from the photocoustic/photothermal effect. The photo-vibrational spectrum obtained by scanning the QCL’s wavelength in MIR range, coincides well with the corresponding spectrum obtained using typical FTIR equipment. The experiment demonstrated that the LDV is a capable sensor for applications in photoacoustic/photothermal spectroscopy, with potential to enable the detection of chemicals in open environment at safe standoff distance.

© 2016 Optical Society of America

1. Introduction

The ability to rapidly detect and identify hazardous materials is a key enabling technology which would facilitate security screening in the global effort against terrorism. Current priority in research and development focuses on technology to perform remote sensing of hazardous materials in the open environment without any sample preparation. In order to detect and identify threats among the diverse range of chemicals, an adaptable sensor capable of distinguishing hazardous materials from other substances is required.

Photoacoustic spectroscopy (PAS) is identified as a suitable technique for this application. Since the photoacoustic effect was discovered in late nineteen century by A. G. Bell [1], it has been developed into a sensitive spectroscopic technology for evaluation of solid [2], liquid and gaseous [3] chemicals. In recent years, the use of laser as a light source to illuminate chemical substances and induce photoacoustic effect has become increasingly popular. Conventional PAS require prior preparation of analyte samples, and the measurement is conducted within a well isolated photoacoustic cell. The response signal is captured either by a highly sensitive microphone, or any other forms of sensor located within close proximity or even in direct contact with the sample. Hence, commercial PAS equipment does not fulfill the situational requirements for hazardous material screening applications in defense and security areas. In addition, it is extremely challenging to conduct PAS in the open environment at a standoff distance, due to the tremendous transmission losses of acoustic waves in air [4]. Even with a high-sensitivity microphone coupled with a large parabolic sound reflector, the maximum detection distance reported is only 12 meters in the laboratory environment [5].

Two recent developments, however hold promise to overcome the inherent limitations associated with PAS and revolutionize the research area of standoff detection of chemicals. Firstly, technological advances in quantum cascade laser (QCL) systems enabled its use as a practical excitation source due to its high power, broadly tunable wavelength in mid-IR range and compact size over other light sources such as optical parametric oscillator (OPO) [6]. Secondly, noncontact optical detection techniques, such as thermal imaging and laser interferometry became an established method for standoff detection of photoacoustic/photothermal signal [7] in open environment.

Cho et al. reported a work utilizing QCL and optical parametric oscillator (OPO) as the excitation source placed at a distance of 0.5m away from an explosive target, coupled with a coherent detection system using a CW probe laser at 1550nm wavelength, and realized a 5-meter detection range. By measuring the optical phase shifts due to photoacoustic/photothermal effects at several wavelengths, a very rough absorbance spectrum related to the explosive material was obtained [8]. The improvement in light source and detection technologies bring about flexibility to measure different type of materials without the limitations of environment and space, enabling the technique to be feasibly applied in applications for national security.

In this research, a QCL and a laser Doppler vibrometer (LDV) [9, 10] were applied as the light source and sensor respectively, for the photo-vibrational spectroscopy of solid and liquid chemicals utilizing the photoacoustic/photothermal effect. The amplitude of QCL was modulated by an optical chopper.

Three types of solids and liquid materials were evaluated:

  • 1. Polytetrafluoroethylene (PTFE) tape with 0.075mm thickness;
  • 2. Sulfamic acid powder on an aluminum plate;
  • 3. Acetone in the liquid state.

The vibration amplitudes obtained by LDV at different wavelengths were normalized by QCL power. The spectra obtained were compared with the corresponding FTIR spectra of each material. The results show that LDV can be a good sensor for spectroscopy [11] and display excellent potential for applications related to standoff detection [12] of chemicals.

2. Photoacoustic effect on solids and liquid

In this study, a laser Doppler vibrometer was applied to measure the periodic change of the optical path length (OPL) due to photoacoustic/photothermal effect when different materials were excited by an intensity-modulated QCL laser source with varying wavelengths in the infrared range. The variation of OPL of the probing laser beam was then converted to vibration amplitude in displacement, or velocity whose frequency equals to the modulation frequency of excitation laser.

2.1 Photoacoustic effect and OPL variation on solids

Conventional photoacoustic spectroscopic analysis using a microphone to detect pressure variation in a photoacoustic cell is a well-established analytical technique for solid chemicals. The theory behind the interpretation of photoacoustic signals was developed by McDonald and Westsel Jr [2]. for homogeneous bulk solid or liquid materials. However, for other forms of material sample such as loose powders [13, 14], thin film or even chemical residues, the conversion efficiency of laser power to signal intensity is simply too complicated to be simulated and expressed mathematically in one equation. It is also dependent upon many other factors related to the morphological, thermo-chemical, and transport properties of the chemicals, substrate and air mass [8]. The following factors were identified to contribute concurrently towards any change in OPL:

  • • Localized surface (out-of-plane) displacement of chemicals and substrate due to thermo-elastic expansion;
  • • Heat conduction from the heated surface to the surrounding air layer producing a refractive index change in the air column.
  • • Generation of outward propagating surface acoustic Rayleigh wave.
  • • Transient surface reflectance change.
  • • Thermal expansion of air gap between particles, when the powder sample is thick.

The optical phase shift of LDV due to the first two factors can be expressed as [1, 8]

Δϕ4πλLDV(ΔnaμanaΔL)4πΔTλLDV[(dndT)aμanaμsβs],
where ΔTis the temperature rise associated with periodic heating of the illuminated region by the excitation laser, andΔLis the thermo-elastic expansion of the chemical. λLDVis the wavelength of probe laser in LDV. na is the refractive index of air; μaand μs are the air and chemical thermal diffusion lengths, respectively, and βsis the linear thermal expansion coefficient of the chemical.

It is assumed that the detected vibration amplitude by LDV is proportional to the power density of excitation laser and the absorbance efficiency of chemicals. From prior experience, the following factors were also observed to affect the detected vibration amplitude:

  • • Sample form and its boundary condition;
  • • Mechanical property and thermal conductivity of substrate (suppose no heat absorbance by substrate itself);
  • • Modulation frequency of excitation laser;
  • • Modulation shape and width of excitation laser;
  • • Size and superposition extent of excitation laser spot and probe laser spot;
  • • Noise floor of LDV and environment factors that may also change the reading of LDV at the same frequency;
  • • Others.

The above-mentioned factors will linearly affect the absolute value of vibration amplitude to be measured, but it can be assumed that these factors will not affect the qualitative value of the photoacoustic spectra of the analyte which was based on the relative signal intensity at different wavelengths.

2.2 Photoacoustic effect of liquid

There are five important interaction mechanisms which can be responsible for the generation of acoustic waves in liquid [3]: (1) dielectric breakdown; (2) vaporization; (3) thermoelastic process; (4) electrostriction; and (5) the radiation pressure. Dielectric breakdown and radiation pressure can be negligible as the power density of QCL in our experiment is much lower than 106 ~1010Wcm−2. The electrostriction effect is always present due to the electronic polarizability of molecules in the sample which causes them to move into or out of regions of higher light intensity depending on their positive or negative polarizability. Electrostriction as a sound wave generation mechanism is only significant in very weakly absorbing media as it may limit the photoacoustic detection sensitivity. Vaporization of liquids is responsible for the acoustic wave generation if the laser energy density within the absorbing volume of the sample exceeds a certain threshold as determined by the thermal properties of the medium. Thermoelastic process is the most significant mechanism for sound generation in absorbing media. This mechanism is based on the transient heating of a confined volume by the absorbed laser energy, resulting in a temperature gradient and corresponding thermal expansion that leads to a strain in the sample which generates acoustic waves. It dominates the generation of acoustic waves in absorbing media when the excitation laser energy is below the vaporization threshold.

3. Experimental illustration

3.1 Excitation light sources

Two QCL were used in this study. The first QCL is a widely tunable MID-IR OEM laser module (Model No. MINI-QCL) from Block Engineering. It is an air-cooled pulsed laser with tunable range from wave number of 940cm−1 to 1375cm−1. A compact-size control electronics, mini-QCL-200 was used to communicate with a laptop through a wireless router. Figure 1(a) shows how it is arranged in experimental setup. It has a pulse repetition frequency of up to 3MHz with a maximum duty cycle of 5%. Figure 1(b) shows its output CW power at different wave numbers. The equipment was applied as a CW laser source in this study and its output CW power was modulated by an optical chopper, with adjustable frequency from 1Hz to 3000Hz. Without intensity modulation using optical chopper, the vibration amplitude was found to be too small to be detectable at 3MHz excitation frequency, but the vibration amplitude could be significantly increased by tuning the frequency range to <3kHz using the optical chopper. The output laser beam is a 2 × 4mm collimated beam. To increase the power density, a 150mm focal-length parabolic mirror was applied to focus the collimating laser beam.

 figure: Fig. 1

Fig. 1 (a) Schematic layout of experimental setup; (b) CW power distribution of QCL from Block Engineering at different wavelengths; (c) Sample 1, PTFE tape (0.075mm in thickness); (d) Sample 2, deposited sulfamic acid on an aluminum plate.

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The second QCL used in this research is MIRcat-1200-1824 from Daylight Solutions. It includes two QCL modules and a co-axial red aiming laser for easy alignment. The tunable wavelength range is narrower than the first QCL but the output CW power is much higher. The maximum output CW power being >200mW. Hence, it requires water cooling during operation and the equipment size is larger. Figure 2(a) shows its power distribution at various wave numbers. The laser is suitable for chemicals in liquid or in powder form where high excitation power is needed for detectable photo-vibrational signal.

 figure: Fig. 2

Fig. 2 (a) Output CW and pulsed power of MIRcat QCL from Daylight Solutions; (b) Schematic layout of excitation and probe laser on liquid chemicals (Acetone).

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3.2 Home-developed LDV

In order to detect the photo-vibrational signal excited by QCLs, a near-field laser Doppler vibrometer was developed. The design is based on a Mach-Zehnder interferometer with a He-Ne laser at 632.8nm as its light source. The model number of laser used is JDS Uniphase 1107P. The output power from LDV sensing head is <1mW. The coherence of He-Ne laser is periodic distance-dependent. A 40MHz acousto-optic modulator was applied in the reference beam to introduce a carrier frequency. A half inch beam expander was designed to focus the laser beam onto a distance between 20cm to 2m. The reflected object beam interferes with the frequency-shifted reference beam and the output signal from the photodetector is a frequency-modulated (FM) signal with a central carrier frequency of 40MHz. The advantage of LDV is that the amplitude and frequency of vibration signal retrieved is only relevant to the frequency shift of the received laser beam. Hence, when the interferometric signal is acceptable by the demodulation system, for example >20dB, the vibration signal obtained is standoff-distance-independent. Generally the LDV is designed for measurement on diffused surface. The laser power and optical aperture are optimized for certain distance range. In case the object surface is specular, the measurement will fail because there is no enough laser power received. In this case, the incident laser beam must be perpendicular to the surface, which is not easily satisfied in real measurement. This is one of the limitations of LDV.

A home-developed demodulation system based on FPGA was used to demodulate the FM signal in order to retrieve the OPL change. The output signal can either be displacement or velocity, up to user selection. Generally, displacement is directly proportional to the phase change and velocity is proportional to the frequency shift of the probe laser beam. When the SNR of the interferometric signal output from detector is more than 45dB, the noise floor of the displacement measurement is <5pm when integration time of one second is selected. The performance is equivalent or even better than the best commercial product in the market. The high resolution ensures the best measurement performance in the photo-vibrational signal detection of various substances.

3.3 Experimental setup

Figure 1(a) shows the experiment setup. The QCL beam passes through an optical chopper and is focused onto material samples by a parabolic mirror. With a help of FLIR E6 infrared camera, the excitation QCL beam is superimposed with the probe laser beam on the samples. The detection distance of LDV in these experiments is about 1m. Two solid samples were tested using the first QCL at low power. The first sample is a PTFE tape with 0.075mm thickness fixed at both ends and suspended in the middle as shown in Fig. 1(c). The second sample is sulfamic acid powder that was deposited onto an aluminum substrate with a concentration of around 20µg/mm2 as shown in Fig. 1(d). The third sample is liquid acetone in a beaker as shown in Fig. 2(b). The excitation laser source used to test the liquid sample was the high power QCL from Daylight Solutions. A phenomenon was observed during the conduct of experiment, that the gasiform acetone formed due to evaporation will also absorb the excitation laser light, causing the instability in the laser energy at the liquid sample surface. Hence, liquid acetone was cooled by surrounding the sample container with ice to <5°C during the test, in order to reduce the evaporation of acetone. After the cooling of the sample, a subsequent power measurement on a laser beam that skims over the liquid surface is almost same as the direct output power of QCL, with a difference of less than 5% and greatly reduced the error of spectrum intensity measurement to an acceptable level. As the high-power QCL was applied as the excitation laser, the laser beam was only reflected to liquid acetone by a plane mirror. The probe laser must be focused and perpendicular to the liquid surface. The two laser spots are located close spatially, but not necessary required to be superimposed on liquid surface. The output power of two QCLs was measured before the experiments. The fluctuations of output power between two measurements were less than 5%. The frequency of the optical chopper was selected to avoid other vibration frequencies detected from the environment, typically at a frequency range between 300Hz to 1kHz.

4. Results and discussion

The wavenumber of QCLs was scanned at a 5cm−1 step, and the vibration amplitude (velocity) was recorded at each step. The velocity was then normalized by QCL output CW power and plotted against wavenumber. Figure 3 shows the comparison of the normalized vibration amplitude and the corresponding IR absorbance spectrum of PTFE from NICODOM IR libraries. In Fig. 3(b), the y-axis is the value of vibration velocity detected divided by QCL output power at that wavelength. Due to the membrane structure of PTFE sample and the boundary condition in the experiment, the vibration amplitude value detected was quite large. The half maximum displacement was around 48nm when chopper frequency was set as 340Hz. The reading from LDV was observed to be very stable based on one-second integration time. Two absorbance peaks at 1165cm−1 and 1205cm−1 of the measured photo-vibrational spectra were observed to coincide very well the FTIR spectra.

 figure: Fig. 3

Fig. 3 (a) IR absorbance spectrum of PTFE from NICODOM IR libraries; (b) Normalized vibration amplitude obtained by LDV at different wave numbers.

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In the second experiment, sulfamic acid deposited on an aluminum plate was tested. In order to verify that the vibration detected was from the photoacoustic/photothermal effect of tested chemical, the aluminum plate was also measured beforehand. No obvious vibrations were detected for the aluminum substrate within the scanning frequency range. The first QCL with lower power was still applied, and the vibration amplitude value was observed to be about 20 times lower than that of the PTFE tape. Hence, the readings were not as stable due to the influence of noise floor. Two-second integration time was selected on LDV at each wave number instead to compensate for the noise. Figure 4 shows the comparison of the obtained photo-vibrational spectra of sulfamic acid against the FTIR spectra from NICODOM IR libraries. The LDV can still detect stable vibration amplitude at the absorbance peak and the result still coincides well with the standard FTIR spectra.

 figure: Fig. 4

Fig. 4 (a) IR absorbance spectrum of sulfamic acid from NICODOM IR libraries; (b) Normalized vibration amplitude on sulfamic acid with aluminum substrate obtained by LDV at different wave numbers.

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In the third test, in order to excite detectable photo-vibrational signal on liquid acetone surface, the high-power QCL from Daylight solutions was applied. The chopper frequency was set at 800Hz. Figure 5 shows the comparison of the obtained photo-vibrational spectra of liquid acetone against the FTIR absorbance spectrum measured using a PerkinElmer Frontier FT-IR/MIR spectrometer in the attenuated total reflectance (ATR) mode. Peaks at same wavenumber can be observed. The wavelength resolution difference of absorbance peaks were within 5 cm−1 between the photo-vibrational spectra and the FTIR-ATR spectra.

 figure: Fig. 5

Fig. 5 (a) IR absorbance spectrum of Acetone measured by PerkinElmer Frontier FT-IR/NIR spectrometer; (b) Normalized vibration amplitude on acetone at 5°C obtained by LDV at different wave numbers.

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In the experiment results shown above, the vibration amplitudes measured were normalized by the power of excitation laser. This was based on following two assumptions: (1) the temperature change of chemicals is linearly proportional to the modulated power density of excitation laser; (2) the OPL variation is linearly proportional to the temperature change. Unfortunately it may not be absolutely the case in actual applications. Many other factors can influence the linearity as well as the absolute values of the photo-vibrational amplitude. When comparing the photo-vibrational spectra results against FTIR spectra, the absorbance peaks can be observed at the same wavenumber positions, but deviations in relative intensity values between the peaks were observed. Further studies on the influence of various factors on peak intensity are necessary.

The experimental results show LDV can be a good sensor for photoacoustic/photothermal signal detection of chemicals in various forms, and the detection can be conducted in open environment. Well-resolved photo-vibrational spectra of various substances in the MIR range can be obtained when a suitable QCL is adopted as the excitation light source. The standoff distance of LDV will influence the received power of reflected beam, thus affecting the signal noise ratio (SNR) of interferometric signal, which may then increase the noise floor and reduce the sensitivity of the detection, but it will not affect the measurement value of photo-vibrational signal. The intensity and frequency of vibration signal retrieved depends only on the frequency shift of the reflected object beam, and not the received power. Hence, when the SNR of interferometric signal is acceptable by demodulation system, the intensity of vibration signal retrieved will be the same. This is the distinctive advantage over other sensors such as microphone, for which the intensity of acoustic signal detected is inversely proportional to detection distance [4]. The standoff distance of LDV is constrained by many factors, such as transmitted laser power, optical aperture, alignment of optical system, etc. However, the coherence length of laser is the main limitation as other factors are engineering issues that can be optimized during system design based on certain standoff distance. Multimode He-Ne lasers typically possess a coherence length of 20 cm, while the coherence length of visible DPSS laser is around ten to several tens of meters. The recent development of narrow linewidth 1550nm laser ignites the possibility of building a LDV with a standoff distance of several hundred meters. Hence, the proposed technique coupled with emerging developments in laser technologies is shown to be able to overcome the inherent limitations associated with PAS and make detection of hazardous materials at a realistic safe standoff distance more reliable and effective. Long standoff distance may be complicated by the presence of environment noise during detection, due to factors such as the turbulence of the air, environmental vibration, etc., and it will also magnify the inherent speckle noise of LDV. Fortunately, several methods may reduce these influences according to our experience. (1) Avoid the high noise floor in spectrum by proper selection of chopper frequency. The excitation frequency should be adjusted to certain range that can be easily separated from noise by a proper filter; (2) Increase the integration time to reduce the noise floor. Mathematically it is equivalent to a lock-in amplifier. (3) Apply differential configuration that can measure the vibration of two adjacent points. Two object beams will share the almost-same optical path. When the differential vibration amplitude obtained, the environment vibration and the influence of air turbulence will be reduced dramatically. However, this requires perfect synchronization [10] between these two channels. Preliminary research work has already shown that it is possible to detect photo-vibrational signal at 100m distance [12]. Further investigation is being conducted and will be reported soon.

5. Concluding remarks

In this study, a LDV was applied as a sensor to detect the photo-vibrational signal of materials in different forms, such as membrane, deposited powder or liquid. QCL was adopted as the excitation light source due to its high power, broadly tunable wavelength in mid-IR range and compact size over other light sources. The intensity of QCL beam was modulated by an optical chopper, while the LDV was used to detect the vibration signals due to photoacoustic effect in the illuminated area. The photo-vibrational spectra obtained by plotting the normalized vibration amplitude against the QCL output wavenumber range were compared with standard FTIR spectra. The results demonstrate that the LDV is a good non-contact sensor for photo-vibrational signal detection in open environment. It is a necessary first step in a series of developments to realize the proposed technology for standoff detection of hazardous materials in defense and security screening applications.

References and links

1. A. C. Tam, “Applications of photoacoustic sensing techniques,” Rev. Mod. Phys. 58(2), 381–431 (1986). [CrossRef]  

2. F. A. McDonald and G. C. Westsel Jr., “Generalized theory of the photoacoustic effect,” J. Appl. Phys. 49(4), 2313–2322 (1978). [CrossRef]  

3. M. W. Sigrist, “Laser generation of acoustic waves in liquids and gases,” J. Appl. Phys. 60(7), R83 (1986). [CrossRef]  

4. X. Chen, L. Cheng, D. Guo, Y. Kostov, and F. S. Choa, “Quantum cascade laser based standoff photoacoustic chemical detection,” Opt. Express 19(21), 20251–20257 (2011). [CrossRef]   [PubMed]  

5. X. Chen, D. Guo, F. S. Choa, C. C. Wang, S. Trivedi, A. P. Snyder, G. Ru, and J. Fan, “Standoff photoacoustic detection of explosives using quantum cascade laser and an ultrasensitive microphone,” Appl. Opt. 52(12), 2626–2632 (2013). [CrossRef]   [PubMed]  

6. J. S. Li, B. Yu, H. Fischer, W. Chen, and A. P. Yalin, “Contributed Review: Quantum cascade laser based photoacoustic detection of explosives,” Rev. Sci. Instrum. 86(3), 031501 (2015). [CrossRef]   [PubMed]  

7. I. Suemune, H. Yamamoto, and M. Yamanishi, “Noncontact photoacoustic measurements of semiconductors with Michelson interferometry,” J. Appl. Phys. 58(1), 615–617 (1985). [CrossRef]  

8. P. S. Cho, R. M. Jones, T. Shuman, D. Scoglietti, and G. Harston, “Investigation of standoff explosives detection via photothermal/photoacoustic interferometry,” Proc. SPIE 8018, 80181T (2011). [CrossRef]  

9. P. Castellini, M. Martarelli, and E. P. Tomasini, “Laser Doppler Vibrometry: Development of advanced solutions answering to technology’s needs,” Mech. Syst. Signal Process. 20(6), 1265–1285 (2006). [CrossRef]  

10. Y. Fu, M. Guo, and P. B. Phua, “Multipoint laser Doppler vibrometry with single detector: principles, implementations, and signal analyses,” Appl. Opt. 50(10), 1280–1288 (2011). [CrossRef]   [PubMed]  

11. L. S. Marcus, E. L. Holthoff, J. F. Schill, and P. M. Pellegrino, “Photoacoustic chemical sensing: ultracompact sources and standoff detection,” Proc. SPIE 9073, 907307 (2014). [CrossRef]  

12. Y. Fu, Q. Hu, and H. Liu, “Standoff photoacoustic sensing of tracing chemicals by laser Doppler vibrometer,” Proc. SPIE 9824, 98240O (2016).

13. J. P. Monchalin, L. Bertrand, G. Rousset, and F. Lepoutre, “Photoacoustic spectroscopy of thick powdered or porous samples at low frequency,” J. Appl. Phys. 56(1), 190–210 (1984). [CrossRef]  

14. K. Y. Wong, “Signal enhancement in photoacoustic spectroscopy on powder samples,” J. Appl. Phys. 49(6), 3033–3035 (1978). [CrossRef]  

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Figures (5)

Fig. 1
Fig. 1 (a) Schematic layout of experimental setup; (b) CW power distribution of QCL from Block Engineering at different wavelengths; (c) Sample 1, PTFE tape (0.075mm in thickness); (d) Sample 2, deposited sulfamic acid on an aluminum plate.
Fig. 2
Fig. 2 (a) Output CW and pulsed power of MIRcat QCL from Daylight Solutions; (b) Schematic layout of excitation and probe laser on liquid chemicals (Acetone).
Fig. 3
Fig. 3 (a) IR absorbance spectrum of PTFE from NICODOM IR libraries; (b) Normalized vibration amplitude obtained by LDV at different wave numbers.
Fig. 4
Fig. 4 (a) IR absorbance spectrum of sulfamic acid from NICODOM IR libraries; (b) Normalized vibration amplitude on sulfamic acid with aluminum substrate obtained by LDV at different wave numbers.
Fig. 5
Fig. 5 (a) IR absorbance spectrum of Acetone measured by PerkinElmer Frontier FT-IR/NIR spectrometer; (b) Normalized vibration amplitude on acetone at 5°C obtained by LDV at different wave numbers.

Equations (1)

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Δϕ 4π λ LDV ( Δ n a μ a n a ΔL ) 4πΔT λ LDV [ ( dn dT ) a μ a n a μ s β s ],
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