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

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 CO₂ 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-$Q$, piezoelectric crystal with a resonance on the order of $443$ kHz. The photoacoustic cell has a design analogous to a hemispherical laser resonator and can be adjusted to have a longitudinal resonance to match that of the detector crystal. The grating frequency, the length of the resonator, and the crystal must all have matched frequencies; thus, three resonances are used to advantage to produce sensitivity that extends to the parts-per-quadrillion level.

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 $\mathrm{\Lambda }$ along with the frequency difference in the two beams $\mathrm{\Omega }$ obeys the rule $\mathrm{\Omega }\mathrm{\Lambda }=\mathrm{\hspace{0.17em}2}\pi c;$ that is, the grating is tuned so that it moves at the sound speed with each antinode in the optical beam pattern moving synchronously with its photoacoustically generated compressive wave. It is shown here that, when a moving grating is produced in a cavity with two parallel reflecting surfaces resembling a laser resonator, two resonance conditions exist for generation of the photoacoustic effect—the first one when the grating motional speed matches the sound speed and the second one when twice the length of the cavity is an integral number of wavelengths of the sound wave. In addition, the method makes use of a resonant piezoelectric crystal as one of the reflecting surfaces of the detection cell so that the effects of three matched resonances are used to advantage to produce high sensitivity.

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

$$\left({\nabla }^2-\frac{1}{c^2}\frac{{\partial }^2}{\partial t^2}\right)p(𝐱,t)=-\frac{\beta }{C_P}\frac{\partial H(𝐱,t)}{\partial t},$$ [1]

where $c$ is the sound speed, $t$ is the time, $\beta$ is the coefficient of thermal expansion, $C_P$ is the specific heat capacity, and $H$ is the optical energy deposited per unit volume and time. Consider a pair of phase-coherent, frequency-shifted optical beams directed at an angle to each other. The intensity of the combined beams at their point of intersection (6) is given by

$$I_0(𝐱,t)={\epsilon }_0{c}_{L}n{|E|}^2[1+\cos (𝐪\cdot 𝐱-\mathrm{\Omega }t)],$$ [2]

where ${\epsilon }_0$ is the vacuum permittivity; $c_L$ is the speed of light; $n$ is the index of refraction; $E$ is the electric field amplitude; $𝐪$ is given by $𝐪={𝐤}_1-{𝐤}_2\text{,}$ where ${𝐤}_1$ and ${𝐤}_2$ are the propagation vectors for the two beams; and the frequency $\mathrm{\Omega }$ is given by $\mathrm{\Omega }={\omega }_1-{\omega }_2$, where ${\omega }_1$ and ${\omega }_2$ are the frequencies (or frequency shifts) of the beams. Eq. 2 describes a grating oscillating at a frequency $\mathrm{\Omega }\text{,}$ moving in space with a fringe spacing $\mathrm{\Lambda }$ given by

$$\mathrm{\Lambda }=\frac{\lambda }{2\sin (\theta /2)}\text{,}$$ [3]

where $\lambda$ is wavelength of the laser beam and $\theta$ is the intersection angle between the two beams. The speed of the moving grating $c_g$ can thus be found from

$$c_g=\frac{\lambda \mathrm{\Omega }}{\theta },$$ [4]

for small $\theta$. In experiments the speed of the grating can be matched to the sound speed by carefully tuning the crossing angle $\theta$ between two incident beams.

Transformation of Eq. 1 into the frequency domain with $H(𝐱,t)=\overline{\alpha }{I}_0(𝐱,t)$, where $\overline{\alpha }$ is the optical absorption coefficient, gives a Helmholtz equation for the frequency domain pressure $\stackrel{\sim }{p}$ for a grating moving in the $z$ direction as

$$\left(\frac{{\partial }^2}{\partial z^2}+\frac{{\omega }^2}{c^2}\right)\stackrel{\sim }{p}=-\frac{i\overline{\alpha }\beta \overline{I}\omega }{2C_P}\times {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{e}^{-i\omega t}\left[e^{i(qz-\mathrm{\Omega }t)}+e^{-i(qz-\mathrm{\Omega }t)}\right]dt,$$ [5]

where $\overline{I}={\epsilon }_0{c}_{L}n{|E|}^2\text{.}$ Evaluation of the time integral in Eq. 5 gives

$$\left(\frac{{\partial }^2}{\partial z^2}+k^2\right)\stackrel{\sim }{p}=-\frac{i\pi \overline{\alpha }\beta \overline{I}\omega }{C_P}\left[e^{iqz}\delta (\omega +\mathrm{\Omega })+e^{-iqz}\delta (\omega -\mathrm{\Omega })\right],$$ [6]

where $k=\omega /c\text{.}$ A Green’s function for a longitudinal resonator of length $L$ can be found (7) as

$$G(z,z\prime ,\omega )=\frac{2}{L}\sum _{n=1}^{\mathrm{\infty }}\frac{\cos k_{n}z\cos k_{n}z\prime }{k^2-k_n^2},$$ [7]

where $k_n=n\pi /L\text{.}$ Integration of the source term in Eq. 6 over the Green’s function gives an expression for $\stackrel{\sim }{p}\text{,}$ which, when Fourier transformed back into the time domain, gives the photoacoustic pressure as

$$p(z,t)=\frac{\overline{\alpha }\beta \overline{I}\mathrm{\Omega }}{C_p}\sum _{n=1}^{\mathrm{\infty }}\frac{\cos k_{n}z}{{\left(\frac{\mathrm{\Omega }}{c}\right)}^2-{\left(\frac{n\pi }{L}\right)}^2}\{\cos \left(\mathrm{\Omega }t\right)\times [\frac{\cos \left[(k_n+q)L\right]-1}{\left(k_n+q\right)L}-\frac{\cos \left[(k_n-q)L\right]-1}{\left(k_n-q\right)L}]+\sin \left(\mathrm{\Omega }t\right)[\frac{\sin \left[(k_n-q)L\right]}{\left(k_n-q\right)L}+\frac{\sin \left[(k_n+q)L\right]}{\left(k_n+q\right)L}]\}.$$ [8]

It can be seen from Eq. 8 that there are resonances when $\mathrm{\Omega }/c=n\pi /L$ and $k_{n}L=qL$, the first condition coming from the factor immediately following the summation sign in the first line of Eq. 8 and the second one from the sinc$[(k_n-q)L/2]$ function that multiplies the $\sin (\mathrm{\Omega }t)$ term. The combination of these equalities corresponds to matching the frequency and wavelength of the grating so that the grating moves at the sound speed and the resonator is an integral number of half wavelengths of the grating (derivation in Theory). On resonance the other terms in the expression for $p$ are either small or zero. Note that the prefactor in Eq. 8 contains a factor of $\mathrm{\Omega }\text{,}$ which on first inspection would indicate a signal increase with $\mathrm{\Omega }$; however, on resonance, when damping is included in the wave equation, this factor cancels with a factor of $\mathrm{\Omega }$ in the damping term in the denominator, as is well known from the response of harmonic systems.

The experimental apparatus used to generate the grating is shown in Fig. 1. A $10.6$-$\mu$m beam from a continuous CO₂ laser (Parallax Tech, Inc.; model $3800$) is directed into two acousto-optic light modulators (IntraAction, Inc.; model AGM$406$FB$21$) designed to operate at ∼$40\text{MHz.}$ The deflected beams from the light modulators are sent into a photoacoustic cell with the crossing angle adjusted to a value estimated from Eq. 4 by setting $\mathrm{\Omega }$ to the crystal resonance frequency and $c$ as the sound speed. Interference of the two frequency-shifted beams gives a grating with a power of approximately $1$ W. The angular position of Mirror $1$ mounted on a motorized rotation stage (Newport, Inc.; model $495$CC) with an angular precision of $0.001$° is finely controlled to tune the intersection angle $\theta$ to optimize the signal output. The power of the laser after exiting the acoustic cell is monitored by a thermal detector (Thorlabs, Inc.; model S$314$C). The reference for the radio frequency lock-in amplifier (Stanford Research, Inc.; model $844$) is generated by sending signals from the two function generators (Agilent, Inc.; model $33258$) into a mixer (Analog Devices; model $8342$) whose output is fed to an amplifier–low-pass filter (Stanford Research, Inc.; model SR $650$) set to pass the difference frequency in the range of $400$ kHz and to reject the $40$-MHz signals from the function generators and the $80$-MHz sum frequency. The signal output from the lock-in amplifier is recorded on an oscilloscope (Tektronix, Inc.; model DPO$4014$B) and stored in a computer.

Fig. 1.

Schematic of the experimental apparatus used for high-frequency photoacoustic trace detection. The two light paths from the acousto-optic light modulators (AOM) are matched to ensure that the two $10.6$-$\mu$m laser beams are phase coherent. The beams intersect in the acoustic cell (AC) with the power monitored by a thermal detector (TD). The intersection angle can be tuned by Mirror $1$ (M$1$), which is mounted on a motorized precision rotation stage. The reference to the lock-in amplifier is generated by feeding the outputs from the two function generators (FG) to a mixer and using a low-pass filter (LPF) so that only the difference frequency from the mixer is sent to the lock-in amplifier.

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 SF₆ is less than $1$ $\mu$s (8). Acoustic scanning was conducted to find the response spectrum of the transducer. Specifically, two acousto-optic modulators were driven by frequency-scanning voltages with one slightly below $40$ MHz and the other slightly above $40$ MHz. The response of the transducer to acoustic waves of varying frequencies was recorded on a lock-in amplifier. In the first experiments the transducer was a LiNbO₃ crystal (Valpe Fisher; $11.7$ mm diameter$\times 4.6$ mm thick) mounted in a housing with three set screws ground to sharp points in an arrangement where the transducer is supported halfway between the two metallized surfaces, at a velocity node for a longitudinal resonance mode. Copper wires with diameters of $350$ $\mu$m whose ends were reduced in diameter by etching with dilute nitric acid were mounted in the transducer housing, one connected to the housing ground touching the front surface of the crystal and the other one touching the back surface of the crystal connected to the center pin of a Bayonet Neill–Concelman connector at the end of the transducer housing. The $Q$ of the LiNbO₃ transducer, determined by scanning the frequency of the grating, was measured to be $\mathrm{5,800.}$ The signal from the transducer is fed to a high-input impedance, low-noise amplifier (Stahl Electronics, Inc.; model PRE$3$) followed by a second amplifier (Femto, Inc.; model HVA-$200$M-$40$-F) whose output was sent to the lock-in amplifier with a time constant of $1$ s and a filter slope of 6 dB per octave.

The interior of the photoacoustic cell is a cylindrical cavity $25.4$ mm in diameter by $19$ mm long designed to hold the transducer at one end and a passive reflector (either a flat aluminum disk or a concave glass surface) at its other end. A micrometer is attached to the passive reflector so the length of the cell can be varied. By varying the cell length, a series of resonance modes was observed (Fig. S1). The cell is equipped with a pressure gauge (MKS, Inc.; model $10$XX$08$) and two ZnSe windows with apertures of $17$ mm for entrance and exit of the laser beams. The cell has a sealed injection port at the top for addition of gases. Gas mixtures were made up barometrically, using high-purity SF₆ (Advanced Fluorinated Products; 99.8%) and N₂ (PurityPlus, Inc.; HP$4.0$, $99.9\%$) or Ar (PurityPlus, Inc.; UHP$5.0$, $99.999\%$), and stored in stainless steel tanks that had been heated under high vacuum. Gas samples were injected into the evacuated photoacoustic cell with a hypodermic syringe to a pressure of $1$ atm.

Fig. S1.

Transducer voltage output vs. incremental cell length. The cell length was continuously increased using a micrometer attached to a motorized rotation stage.

The exact resonance frequency is determined with a dilute mixture of SF₆ in N₂ in the cell by scanning the two function generators over the course of ∼$5$ min and noting the peak output of the lock-in amplifier. The intersection angle of the two laser beams is varied in a series of experiments around the calculated value to optimize the signal magnitude, following which, gas mixtures with varying mole fractions are injected into the photoacoustic cell and the lock-in amplifier output is recorded as the frequencies are scanned. A plot of output signal vs. mole fraction for three gases that absorb at $10.6$ $\mu$m is shown in Fig. 2. The detection limit of SF₆ in N₂ using the LiNbO₃ crystal is $10$ parts per trillion. Fig. 2, together with the molar absorption coefficient of SF₆ measured with an infrared spectrophotometer (Jasco, Inc.; model FT/IR-$4100$) indicates an absorption coefficient of $1.2\times {\mathrm{\hspace{0.17em}10}}^{-9}$ cm⁻¹ at the detection limit.

Fig. 2.

Lock-in amplifier output vs. mole fraction for three gases in N₂ and for SF₆ in Ar (top curve, SF₆ I). The detection limit for SF₆ in ultrahigh-purity Ar is $750$ parts per quadrillion using the $\alpha$-BiB₃O₆ crystal and $10$ parts per trillion for SF₆ in N₂ using the LiNbO₃ transducer.

For the experiments with SF₆ in Ar, a specially fabricated $\alpha$-BiB₃O₆ crystal with a resonance frequency of $443.15$ kHz and a $Q$ of $\mathrm{10,800}$ was used. The $\alpha$-BiB₃O₆ crystal was grown by the top-seeded solution growth method with a growth period of about $3$ mo (9, 10). A plot of the output of the crystal as the grating frequency was changed is shown in Fig. 3. The resonance mode at $443.15$ kHz corresponds to motion along the longitudinal direction of the crystal that has a piezoelectric coefficient on the order of $40$ pC/N. Side peaks besides the main resonance curve appear mainly as a result of simultaneous excitation of weakened resonance modes of the crystal, which typically have severalfold lower piezoelectric constants (10). Instead of a cell with plane parallel reflectors at each end, the adjustable reflector was a concave glass reflector with a focal length equal to the cavity length. The frequencies fed to the light modulators and the laser beam crossing angle were adjusted as noted above to determine where the signal was maximal. In taking the data, the computer averaged $100$ readings at each gas concentration to give a mean and SD of the signal for each gas mixture. As shown in Fig. 2, a detection limit of $750$ parts per quadrillion of SF₆ in ultrapure Ar was found, which corresponds to an absorption coefficient of $8.1\times {\mathrm{\hspace{0.17em}10}}^{-11}$ cm⁻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 ${10}^{-12}$ m²/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.

Fig. 3.

Voltage output in arbitrary units (a.u.) vs. grating frequency for the $\alpha$-BiB₃O₆ crystal. $\alpha$-BiB₃O₆was chosen for the experiments because of its combination of a high quality factor and large value of the charge per unit force applied.

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 $14$ $\mu$V, indicating that whatever window signal is generated is not the dominant noise source. In the design of our detection system, the window signal can be totally suppressed in principle when the optical grating is created only inside the cell. For the case when part of the grating is formed within the window, the interference signal gets no amplification because the rate at which the grating moves is adjusted to match the sound speed in the gas, which is discrepant from the sound speed in the windows. In addition, the crystal is anchored in the housing by three sharp pins—acoustic transmission through the cell walls to the crystal surface is largely suppressed. Also recorded was the noise as a function of frequency near the resonance of the transducer averaged for 10 scans for a cell filled with pure argon. The result plotted in Fig. S2 exhibits a resonance behavior indicating that the thermal noise in the transducer determines the ultimate signal-to-noise ratio (SNR) in the experiment. Another point concerning the SNR in the present experiments is that calculations show that a moving optical source can achieve an optical-to-acoustic power conversion efficiency on the order of $10\%$, a relatively large figure for the photoacoustic effect (1).

Fig. S2.

Averaged noise spectrum of the detector measured with a cell filled with pure Ar and with a laser power of approximately $1$ W.

In retrospect, it can be said that one of the most successful applications of the photoacoustic effect since its discovery by Bell in $1881$ has been its use for trace gas detection. The use of modulated, continuous-wave lasers tuned to the absorptions of gases in the infrared introduced in the $1970$s convincingly showed the remarkably high detection sensitivity and respectably high selectivity of the photoacoustic effect in gases. However, in the course of investigation of the ultimate capabilities of the photoacoustic effect for trace detection, it soon became apparent that the full sensitivity of the method, which in principle is governed by the absorbed laser power and the sensitivity of the microphone, was limited by a small photoacoustic window signal, generated by an absorption of uncertain origin at the entrance and exit windows of the photoacoustic cell, and that is coherent with the modulation. Since then, the history of photoacoustic trace detection might be said to be the history of attempts to reduce the window signal through creative design of the gas cell. Various resonator designs have been explored, including the use of Helmholtz resonators, cells with entrance and exit ports at pressure nodes of the cell, baffles between the microphone and the cell windows, open longitudinal resonators, double resonators, and frequency modulation of the laser. The use of a quartz tuning fork as a detector has also been introduced, which has shown high sensitivity and immunity from the problem of the window signal (13–15). As described in numerous reviews of photoacoustic trace gas detection and resonators over the years (16–25), all of these different approaches have achieved varying degrees of success—the overall result being that the photoacoustic effect now stands as one of the premier methods of trace gas detection, possessing a simplicity of construction that makes it sufficiently robust to be used in the field, a reasonable degree of selectivity, remarkably high sensitivity, and an unprecedented linear response range.

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:

$$\left({\nabla }^2-\frac{1}{c^2}\frac{{\partial }^2}{\partial t^2}\right)p(𝐱,t)=-\frac{\beta }{C_p}\frac{\partial H(𝐱,t)}{\partial t}.$$ [S1]

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

$$\left({\nabla }^2+k^2\right)\stackrel{\sim }{p}(𝐱,\omega )=-\frac{i\beta \omega }{C_p}\stackrel{\sim }{H}(𝐱,t).$$ [S2]

For the moving optical grating investigated here, the heating function can be written as

$$H(𝐳,t)=\overline{\alpha }\overline{I}\left(1+\cos (\mathrm{𝐪𝐳}-\mathrm{\Omega }t)\right).$$ [S3]

Thus the source term in Eq. S1, expressed in the frequency domain, is

$$s(z,\omega )=-\frac{i\overline{\alpha }\omega \beta \overline{I}}{C_p}\int e^{-i\omega t}\cos (qz-\mathrm{\Omega }t)dt=-\frac{i\overline{\alpha }\omega \beta \overline{I}}{2C_p}\int \left[e^{i(qz-\mathrm{\Omega }t-\omega t)}+e^{-i(qz-\mathrm{\Omega }t+\omega t)}\right]dt=-\frac{i\overline{\alpha }\omega \beta \overline{I}\pi }{C_p}\left[e^{iqz}\delta (\omega +\mathrm{\Omega })+e^{-iqz}\delta (\omega -\mathrm{\Omega })\right].$$ [S4]

The Green’s function for a one-dimensional longitudinal resonator of length $L$ is given by

$$G(z,z\prime ,\omega )=\frac{2}{L}\sum _{n=1}^{\mathrm{\infty }}\frac{\cos (k_{n}z)\cos (k_{n}z\prime )}{k^2-k_n^2}.$$ [S5]

The integration of the Green’s function $G$ with the source term $s$ gives the pressure in the frequency domain as

$$\stackrel{\sim }{p}(z,\omega )={\int }_0^{L}G(z,z\prime ,\omega )s(z\prime ,\omega )dz\prime =-\frac{i2\overline{\alpha }\omega \beta \overline{I}\pi }{C_{p}L}\sum _{n=1}^{\mathrm{\infty }}\frac{\cos (k_{n}z)}{k^2-k_n^2}[\delta (\omega +\mathrm{\Omega }){\int }_0^L\cos (k_{n}z\prime )e^{iqz\prime }dz\prime +\delta (\omega -\mathrm{\Omega }){\int }_0^L\cos (k_{n}z\prime )e^{-iqz\prime }dz\prime ].$$ [S6]

Here, we denote ${\stackrel{\sim }{p}}_1(z,\omega )=\cos (k_{n}z)/(k^2-k_n^2){\int }_0^L\cos (k_{n}z\prime )e^{iqz\prime }dz\prime$ and ${\stackrel{\sim }{p}}_2(z,\omega )=\cos (k_{n}z)/(k^2-k_n^2){\int }_0^L\cos (k_{n}z\prime )e^{-iqz\prime }dz\prime$. It is easy to show that ${\stackrel{\sim }{p}}_1$ and ${\stackrel{\sim }{p}}_2$ are complex conjugates of each other. The first of these, ${\stackrel{\sim }{p}}_1$, can be evaluated as

$${\stackrel{\sim }{p}}_1(z,\omega )=\frac{1}{2i}\frac{\cos (k_{n}z)}{k^2-k_n^2}\left[\frac{e^{iL(q+k_n)}-1}{q+k_n}+\frac{e^{iL(q-k_n)}-1}{q-k_n}\right].$$ [S7]

Transformation of the pressure back into the time domain yields

$$p(z,t)=\frac{1}{2\pi }\int \stackrel{\sim }{p}(z,\omega )e^{i\omega t}d\omega =-\frac{i\overline{\alpha }\omega \beta \overline{I}}{C_{p}L}\int [{\stackrel{\sim }{p}}_1(z,\omega )\delta (\omega +\mathrm{\Omega })e^{i\omega t}+{\stackrel{\sim }{p}}_2(z,\omega )\delta (\omega -\mathrm{\Omega })e^{i\omega t}]d\omega =\frac{i\overline{\alpha }\beta \overline{I}\mathrm{\Omega }}{C_{p}L}\left[{\stackrel{\sim }{p}}_1(z,-\mathrm{\Omega })e^{-i\mathrm{\Omega }t}-{\stackrel{\sim }{p}}_2(z,\mathrm{\Omega })e^{i\mathrm{\Omega }t}\right].$$ [S8]

We then denote $p_1=i\overline{\alpha }\beta \overline{I}\mathrm{\Omega }/(C_{p}L){\stackrel{\sim }{p}}_1(z,-\mathrm{\Omega })e^{-i\mathrm{\Omega }t}$ and $p_2=-i\overline{\alpha }\beta \overline{I}\mathrm{\Omega }/(C_{p}L){\stackrel{\sim }{p}}_2(z,\mathrm{\Omega })e^{i\mathrm{\Omega }t}$. It can be seen that $p_1$ and $p_2$ are also complex conjugates of each other so that $p=p_1+p_2=2$Re$(p_1)$.

The real part of $p_1$ can be readily calculated to give the photoacoustic pressure distribution within the resonator as

$$p(z,t)=\frac{\overline{\alpha }\beta \overline{I}\mathrm{\Omega }}{C_p}\sum _{n=1}^{\mathrm{\infty }}\frac{\cos (k_{n}z)}{{\left(\frac{\mathrm{\Omega }}{c}\right)}^2-{\left(\frac{n\pi }{L}\right)}^2}\left\{\text{sinc}\left(\frac{L(q+k_n)}{2}\right)\times \sin \left[\mathrm{\Omega }t-\frac{L(q+k_n)}{2}\right]+\text{sinc}\left(\frac{L(q-k_n)}{2}\right)\times \sin \left[\mathrm{\Omega }t-\frac{L(q-k_n)}{2}\right]\right\}.$$ [S9]

By expanding the terms $\sin [\mathrm{\Omega }t\pm \frac{L(q+k_n)}{2}]$, Eq. S9 can be rewritten as

$$p(z,t)=\frac{\overline{\alpha }\beta \overline{I}\mathrm{\Omega }}{C_p}\sum _{n=1}^{\mathrm{\infty }}\frac{\cos (k_{n}z)}{{\left(\frac{\mathrm{\Omega }}{c}\right)}^2-{\left(\frac{n\pi }{L}\right)}^2}\left\{\cos \left(\mathrm{\Omega }t\right)\left[\frac{\cos \left[(k_n+q)L\right]-1}{\left(k_n+q\right)L}-\frac{\cos \left[(k_n-q)L\right]-1}{\left(k_n-q\right)L}\right]+\sin \left(\mathrm{\Omega }t\right)\left[\frac{\sin \left[(k_n-q)L\right]}{\left(k_n-q\right)L}+\frac{\sin \left[(k_n+q)L\right]}{\left(k_n+q\right)L}\right]\right\}.$$ [S10]

Verification of the Solution.

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

$$LHS=\left(\frac{{\partial }^2}{\partial z^2}-\frac{1}{c^2}\frac{{\partial }^2}{\partial t^2}\right)p(z,t)=\frac{\overline{\alpha }\beta \overline{I}\mathrm{\Omega }}{C_p}\sum _{n=1}^{\mathrm{\infty }}\cos (k_{n}z)\left\{\text{sinc}\left(\frac{L(q+k_n)}{2}\right)\sin \left[\mathrm{\Omega }t-\frac{L(q+k_n)}{2}\right]+\text{sinc}\left(\frac{L(q-k_n)}{2}\right)\sin \left[\mathrm{\Omega }t-\frac{L(q-k_n)}{2}\right]\right\},$$ [S11]

(LHS, left-hand side), which essentially indicates a summation of an infinite series of standing waves in the resonator.

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

$$RHS=\frac{\overline{\alpha }\beta \overline{I}\mathrm{\Omega }}{C_p}\sin \left(\mathrm{\Omega }t-qz\right),$$ [S12]

which indicates a traveling wave. To find its relationship with the $LHS$, we expand the $RHS$ in the following way:

$$\sin \left(\mathrm{\Omega }t-qz\right)=\sum _{n=1}^{\mathrm{\infty }}{f}_n(t)\sqrt{\frac{2}{L}}\cos (k_{n}z).$$ [S13]

Note that the basis function $g_n(z)=\sqrt{\frac{2}{L}}\cos (k_{n}z)$ satisfies the orthonormality condition; that is,

$${\int }_0^L{g}_n(z)g_m(z)dz={\delta }_{nm},$$ [S14]

where ${\delta }_{nm}$ is the Kronecker delta function.

Therefore, the coefficient $f_n$ is given by

$$f_n=\sqrt{\frac{2}{L}}{\int }_0^L\sin \left(\mathrm{\Omega }t-qz\right)\cos (k_{n}z)dz=\sqrt{\frac{L}{2}}\left\{\text{sinc}\left(\frac{L(q+k_n)}{2}\right)\sin \left[\mathrm{\Omega }t-\frac{L(q+k_n)}{2}\right]+\text{sinc}\left(\frac{L(q-k_n)}{2}\right)\sin \left[\mathrm{\Omega }t-\frac{L(q-k_n)}{2}\right]\right\}.$$ [S15]

It is finally found that $LHS=RHS$, which validates the above solution.

Resonance Conditions.

As has been discussed in the main text, two resonance conditions are implied in Eq. S9, the first one being $\mathrm{\Omega }/c=n\pi /L$ and the second one $k_n=q$. These two conditions can be rewritten as

$$L=\frac{n}{2}c{T}_g,$$ [S16a]

$$L=\frac{n}{2}\mathrm{\Lambda },$$ [S16b]

where $T_g$ and $\mathrm{\Lambda }$ are the period and wavelength of the moving optical grating, and $c$ is the sound speed. Combining the above two equations yields

$$c=\frac{\mathrm{\Lambda }}{T_g},$$ [S17]

which implies that at the resonance condition the optical grating moves at the sound speed.

Also note that at resonance the frequency of the acoustic wave $f_a$ must be matched with the frequency of the optical grating $f_g$; that is, $f_a=f_g$. As such, we have the second resonance condition

$$L=n\frac{{\lambda }_a}{2},$$ [S18]

where ${\lambda }_a$ is the wavelength of the photoacoustic wave. Eq. 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 $495$CC) such that the controlled rotation could be transformed into the translation of the concave reflector with a resolution of $0.025$ mm, which enabled investigation of the signal dependence on cell length. The result is shown in Fig. S1. It can be seen that the pressure amplitude at the end of the cell varies sinusoidally as the cell length changes, and the distance between adjacent amplitude peaks is well matched to the half wavelength of the acoustic wave, which is around $0.36$ mm. It is also worth noting that the peak pressure amplitude decreases with increasing cell length, which can be ascribed to a decreased acoustic energy density within the cell and increased acoustic damping.

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 SF₆/Ar mixture with a mole fraction of $0.1\%$ and a total pressure of $1$ atm was measured. A transmittance of $50\%$ is observed at $943$ cm⁻¹, which is the wavenumber of the CO₂ laser output.

The molar absorption coefficient can be calculated based on the Beer–Lambert law, which states that

$$I=I_0{10}^{-\epsilon cl},$$ [S19]

where $I$ and $I_0$ are the intensities of the transmitted and incident light, $\epsilon$ is the molar absorption coefficient, $c$ is the molar concentration, and $l$ is the medium path length. For SF₆ the molar absorption coefficient at $943$ cm⁻¹ was calculated to be 2,655 L/(cm$\cdot$mol). The sensitivity expressed in terms of the absorption coefficient $\alpha$ can then be evaluated through

$$\alpha =c\epsilon ,$$ [S20]

where $c$ is related to the mole fraction by $c=Px/(RT)$ with $x$ being the mole fraction at the detection limit, $P$ the mixture pressure, $R$ the gas constant, and $T$ the temperature.

Detector Noise Spectrum.

To record the noise spectrum of the detector, the cell was first evacuated and filled with pure Ar to $1$ atm. The acousto-optic modulators were scanned near the resonance frequency of the $\alpha$-BiB₃O₆ crystal. With the laser power adjusted to $1$ W, the voltage output from the transducer was recorded and averaged for 10 scans. It can be seen from Fig. S2 that a peak appears near the resonance frequency of the crystal. Because the magnitude of the noise peak remained the same with the laser either turned on or turned off, this peak is ascribed to the thermal noise of the crystal itself.