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# Photoacoustic trace detection of gases at the parts-per-quadrillion level with a moving optical grating

Edited by David A. Weitz, Harvard University, Cambridge, MA, and approved June 1, 2017 (received for review April 11, 2017)

## Significance

The photoacoustic effect refers to the generation of sound through a process of optical heat deposition followed by thermal expansion, resulting in a local pressure increase that produces outgoing acoustic waves. In the linear acoustic regime, a unique property of the photoacoustic effect in a geometry with symmetry in one dimension is that when the optical source moves at the speed of sound, the amplitude of the acoustic wave increases linearly in time without bound. Here, the application of this effect to trace gas detection is described, using an optical grating that moves at the sound speed inside of a resonator equipped with a resonant piezoelectric crystal detector, yielding detection limits in the parts-per-quadrillion range.

## Abstract

The amplitude of the photoacoustic effect for an optical source moving at the sound speed in a one-dimensional geometry increases linearly in time without bound in the linear acoustic regime. Here, use of this principle is described for trace detection of gases, using two frequency-shifted beams from a CO_{2} laser directed at an angle to each other to give optical fringes that move at the sound speed in a cavity with a longitudinal resonance. The photoacoustic signal is detected with a high-

One of the surprising features of the photoacoustic effect (1⇓⇓–4) as pointed out by Gusev and Karabutov (1) is that in a one-dimensional geometry, an optical beam moving in an absorbing medium at the sound speed generates a traveling compressive wave whose amplitude, in the linear acoustics limit, increases in direct proportion to the irradiation time—without bound. Here, we report the application of this principle for trace detection of gases in a scheme where a pair of frequency-shifted laser beams produced by two acousto-optic modulators operating at slightly different frequencies are combined in space to produce a moving optical grating tuned to an absorption of an infrared active gas. The angle between the two beams is adjusted so that the fringe spacing of the grating

The theory of operation of the detector is based on solution of the wave equation for pressure for a moving optical grating in a resonator. The wave equation for the photoacoustic effect in an inviscid fluid when heat conduction effects can be ignored (1, 5) is given by**2** describes a grating oscillating at a frequency

Transformation of Eq. **1** into the frequency domain with **5** gives**6** over the Green’s function gives an expression for

It can be seen from Eq. **8** that there are resonances when **8** and the second one from the sinc*Theory*). On resonance the other terms in the expression for **8** contains a factor of

The experimental apparatus used to generate the grating is shown in Fig. 1. A _{2} laser (Parallax Tech, Inc.; model **4** by setting

Two different transducers were used in the experiments. It is noted here that the resonance frequency of selected crystals should be lower than the inverse of the vibrational relaxation time of target molecules, which for SF_{6} is less than _{3} crystal (Valpe Fisher; _{3} transducer, determined by scanning the frequency of the grating, was measured to be

The interior of the photoacoustic cell is a cylindrical cavity _{6} (Advanced Fluorinated Products; 99.8%) and N_{2} (PurityPlus, Inc.; HP

The exact resonance frequency is determined with a dilute mixture of SF_{6} in N_{2} in the cell by scanning the two function generators over the course of ∼_{6} in N_{2} using the LiNbO_{3} crystal is _{6} measured with an infrared spectrophotometer (Jasco, Inc.; model FT/IR-^{−1} at the detection limit.

For the experiments with SF_{6} in Ar, a specially fabricated _{3}O_{6} crystal with a resonance frequency of _{3}O_{6} crystal was grown by the top-seeded solution growth method with a growth period of about _{6} in ultrapure Ar was found, which corresponds to an absorption coefficient of ^{−1}. The inherent linearity of the method described here is governed by the linearity of the electronics as well as the small magnitude of elastic compliance of the piezoelectric crystals, which is on the order of ^{2}/N (11, 12). Fig. 2 dictates that under the compressive acoustic forces found in the photoacoustic trace detection experiments, the crystal exhibits extremely small strains so that linear response over a large range of concentrations is guaranteed.

In a photoacoustic resonator, any absorbing medium within the acoustic cell along the optical beam path, including the cell windows, can generate acoustic waves whose frequency is identical to that of the modulation. The signal that is generated by a laser beam in entering and exiting a conventional photoacoustic cell, referred to commonly as the “window signal,” can mask the photoacoustic signal in the gas, limiting detection sensitivity. In an experiment where the cell was filled with pure Ar, the noise at the resonance frequency recorded with the laser either on or off was measured to be identical, approximately

In retrospect, it can be said that one of the most successful applications of the photoacoustic effect since its discovery by Bell in

Somewhat surprisingly, in carrying out the experiments described here it was found that the use of a conventional closed resonator is not essential for operation of the instrument. The cavity with its two reflecting surfaces is a photoacoustic analog of a plane parallel or hemispherical laser resonator with the moving grating acting in a manner analogous to the gain medium in a laser. The requirement of only two reflecting surfaces points out an unexpected feature of the moving grating technique described here, namely, that trace gas detection can be carried out continuously, directly monitoring the atmosphere, without the usual sampling and injection into a closed cell as is the case with conventional detection cells. A further observation is that there is no evidence of electrical pickup or any effect of wind currents on the open cell: The high frequency of operation at hundreds of kilohertz means that acoustic noise, even if it were produced nearby, would be rapidly damped out as a result of viscous damping.

## Theory

In the first part of this section, we present details of the calculation of the photoacoustic pressure; then we show that when the expression for the pressure is substituted back into the original wave equation, a series expansion can be used to prove its validity. We show explicitly the resonance conditions implied in the solution for the photoacoustic pressure. Note that the same variables are used here as in the main text, and thus no further identification of the variables is given. The reader is referred to the theory part of the main text for the meanings of the different variables.

### Solution to the Wave Equation.

The photoacoustic pressure, when the heat conduction and viscosity are ignored, obeys the following wave equation:

The Fourier transform of the above equation into the frequency domain gives

For the moving optical grating investigated here, the heating function can be written as**S1**, expressed in the frequency domain, is

The Green’s function for a one-dimensional longitudinal resonator of length

The integration of the Green’s function

Here, we denote

Transformation of the pressure back into the time domain yields

We then denote

The real part of

By expanding the terms **S9** can be rewritten as

### Verification of the Solution.

We substitute the solution Eq. **S9** back into the original wave equation Eq. **S1** to give

The source term on the right-hand side (RHS) of Eq. **S1** gives

Note that the basis function

Therefore, the coefficient

It is finally found that

### Resonance Conditions.

As has been discussed in the main text, two resonance conditions are implied in Eq. **S9**, the first one being

Also note that at resonance the frequency of the acoustic wave **S18** is the typical resonance condition of a longitudinal resonator, which states that at resonance the cell length is an integral number of half wavelengths of the standing wave.

## Experiments

### Resonance Modes of the Acoustic Cell.

As predicted by Eqs. **S9** and **S18**, a series of longitudinal resonances exists in the acoustic cell. In the experiments, one side of the acoustic cell was a concave reflector attached to a micrometer which could be used to adjust the length of the cell. To investigate the longitudinal resonances of the acoustic cell, the micrometer was bonded to a motorized rotation stage (Newport, Inc.; model

### Determination of the Equivalent Absorption Coefficient.

The detection limit that was obtained from the experiment was expressed in units of mole fraction. However, it is common practice to express the detection limit in terms of the absorption coefficient. To find the corresponding absorption coefficient, the infrared spectrum for an SF_{6}/Ar mixture with a mole fraction of ^{−1}, which is the wavenumber of the CO_{2} laser output.

The molar absorption coefficient can be calculated based on the Beer–Lambert law, which states that_{6} the molar absorption coefficient at ^{−1} was calculated to be 2,655 L/(cm

### Detector Noise Spectrum.

To record the noise spectrum of the detector, the cell was first evacuated and filled with pure Ar to _{3}O_{6} crystal. With the laser power adjusted to

## Acknowledgments

Dr. Qingming Lu is acknowledged for the kind help during single-crystal growth. L.X., W.B., and G.J.D. thank Parallax Technology, Inc. for the donation of the CO_{2} laser. L.X., W.B., and G.J.D. are grateful to the US Department of Energy under Grant DE-SC0001082 for the support of this research. F.C., X.Z., and F.Y. acknowledge financial support from the State Key Laboratory of Crystal Materials at Shandong University.

## Footnotes

↵

^{1}L.X. and W.B. contributed equally to this work.- ↵
^{2}To whom correspondence should be addressed. Email: gerald_diebold{at}brown.edu.

Author contributions: G.J.D. designed research; L.X. and W.B. performed research; F.C., X.Z., and F.Y. fabricated the alpha-BiB3O6 crystal; L.X. and W.B. analyzed data; and W.B. and G.J.D wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1706040114/-/DCSupplemental.

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