Photoacoustic Spectroscopy







   
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Keywords:

Photoacoustic spectroscopy, Laser spectroscopy, Optoacoustics, Photoacoustics
 




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Photoacoustic Spectroscopy
Seminar by Deepak Rajput
PHYS 605 Advanced Topics: Laser Spectroscopy
July 10, 2007
Center for Laser Applications
University of Tennessee Space Institute
Tullahoma, TN 37388
Email: drajput@utsi.edu
Web: http://drajput.com


Introduction
> The photoacoustic (PA) or optoacoustic (OA) effect, i.e. the generation of acoustic waves due to the absorption of modulated electromagnetic waves, is an old effect, discovered by Bell in 1880.
> This effect is weak; only a very small fraction (<1ppm) of the absorbed optical energy is converted into acoustic energy.
PA spectroscopy in gases
> Kreuzer (1971) reported that an ultra low gas concentration can be detected by OA using an infrared laser beam as a light source.

> A sensitivity limit of a concentration of 10-8 of methane in nitrogen was demonstrated, and a limit as low as 10-13 could be expected with an improved light source.

Schematics
Theory
> First step: Optical absorption, which results in the production of excited states.
> Let?s take a two-level system, which involves the ground state and the excited state (N and N`)
> N` can be calculated using the rate equation as:

Theory cont..
> In many cases, the modulation frequency of the light is slow (~kHz or less) compared to the excited-state decay rate.
> Furthermore, the light intensity is usually weak enough so that (N>>N`) and the stimulated emission from the excited state can be neglected. (slow modulation and weak light)

Theory cont..
> Heat production rate (H) due to the excited-state density N` (which depends on position r and time t, because ? is a function of r and t) is given by:

Theory cont..
> If the deexcitation collision results in converting the excited state to the ground state, then the deexcitation energy is simply the energy of the excited state with respect to the ground state.
> Equation 3 states that heat source term for the OA signal is proportional to the product of molecular density (N), photon absorption rate ??, probability for nonradiative relaxation of the optically excited state ?An, and the heat energy released per deexcitation E`.
Theory cont..
* Equation 3 is applicable only when the modulation frequency of the light is slow compared to the excited-state decay rate.
* If this condition is not met, we cannot put
Theory cont..
> Where only the real part has physical meaning
> We may drop the constant in equation 5 since we are interested only in the modulated heat source which generates a corresponding OA signal.
> The solution of equations 4 and 5 is:


where

Theory cont..
* ? is the phase lag of the modulation of the excited-state density compared to the optical excitation, and is large when the excited state decays more slowly than the modulated rate of the light intensity.

* Note that equation 6 reduces to equation 2 in the limit when

* The heat generation term H corresponding to equation 6 is again given by equation 3.
Theory cont..
* As seen in the schematics, the next step in the theory is the generation of acoustic waves by the heat source H(r,t) of equation 3.
* Inhomogeneous wave equation relating the acoustic pressure p and the heat source H:


Theory cont..
* Equation 7 is usually solved for the sinusoidal modulation case by expressing the Fourier transform of p in terms of ?normal acoustic modes? pj which satisfy the appropriate boundary conditions. Thus



with the normal mode pj being solutions of the homogeneous wave equation, i.e.,

Theory cont..
* pj must be chosen to satisfy the boundary condition that the gradient of p normal to the cell wall vanish at the wall, since acoustic velocity is proportional to the gradient of p and must vanish at the wall.
* The resultant orthonormal modes in the cylindrical geometry are given by:


after Morse and Ingard (1968)

with a corresponding angular frequency ?j given by


Theory cont..
* Here gj is a normalization constant; L is the length and R0 the radius of the gas cell; (r,?,z) are the cylindrical coordinates of a spatial point; k, m, and n are the longitudinal, azimuthal, and radial mode numbers; Jm is a Bessel function; and ?mn is the nth solution of the equation dJm/dr = 0 at r = R0.
* The condition of vanishing pressure gradient at the cell wall requires that the acoustic pressure p(r,?) be expressed as linear combinations of eigenmodes pj of the form of equation 9 for a cylindrical geometry.
Theory cont..
* Solving the expansion coefficients Aj(?)
* Fourier transform of equation 7 is:


* Substituting equation 8 in the above equation and using the orthonormal conditions for the eigenfunctions pj, we may solve for Aj as:


Theory cont..
* Here V0 is the cell volume, Qj is the quality factor for the acoustic mode Pj ( is the complex conjugate of pj), and the integral is over the volume of the cell.
* Qj accounts for the mode damping and avoid the physically unreasonable situation of as
* Equation 12 may be further simplified for the case H being given by equations 2 and 3. In this case

Theory cont..
* We also assumed that the light beam is Gaussian, i.e.,


where a is the beam radius; beam propagates along the axis of cell so that only eigenmodes are of the form of equation

Theory cont..
* The amplitude of the lowest-order radial pressure mode (j=1) is then given by equation 12 as:

Theory cont..
* Close to resonance (?=?1+?; ? being small ), this equation reduces to:

Theory cont..
* This equation is valid for near resonance to the lowest radial mode. For the opposite case of far off-resonance (i.e. non resonant OA cell), then:

Theory cont..
Theory cont..
* Final step of the theory of OA is the detection, which is frequently done with a microphone.

* If the microphone has a known frequency response, then all the various components Aj in equation 12 with frequencies ?j within in the microphone bandwidth will be detected, and suitable frequency analysis of the microphone signal should give the various Aj?s.
Theory cont..
* In case of pulsed OA excitation, boundary conditions are frequently unimportant when short-duration light pulses are used because the time needed for the acoustic wave to reach the OA cell well is roughly 30 microseconds, which?s much longer than the light pulse duration and much longer than decay times of excited states in most gases.

* Thus, interference of the generated acoustic wave and the reflected acoustic waves generally do not occur in contrast to the CW modulated case.

* However, Pulsed OA generation does produce a ?ringing? acoustic signal due to multiple reflections in the gas cell
Theory cont..
* The net heat released up to time t is:

Theory cont..
* The time dependence of p(t) for the pulsed OA signal is indicated in slide 22(b) for the case of short optical pulse duration and long thermal diffusion time ?D, given by

Instrumentation for OA Studies of Gases
Instrumentation for OA Studies of Gases
Instruments:
* Light source
* OA cell with transducer
* A means of modulating the light source (e.g., pulsing a laser or using a chopper), or modulating the sample absorption (e.g., using a modulated electric field for Stark modulation of the absorption)
Instrumentation: Light Source
Two general classes:

* Lamps, filament lamps, and glow bars

* Inexpensive, usually compact and reliable, and cover broad spectral ranges from the UV to the far IR.
* Low spectral brightness, incapability of fast modulation or switching , and necessity of an external spectral selection element like a monochromator.
Instrumentation: Light Source
* Lasers

* High spectral brightness and collimation, can be readily modulated by extracavity or by intracavity means, and are of narrow spectral linewidth.
* Expensive and limited tuning range.
OA cells for gases
Resonances in OA cells
Applications
* Measurement of weak Absorption lines
(~10-10cm-1/cell length ~10cm (Patel et al 1977)
* High sensitivity trace detection (SFRL)
* Absorption of excited states
* Chemically reactive gases
* Raman-Gain Spectroscopy (PARS) [non-linear]
PA Spectroscopy in Condensed Matter
* Two methods:

* The Gas-Coupling Method
* The Direct Coupling Method
Gas-Coupling Method
* Use of gas-phase microphone for detecting PA signals in condensed matter
* PA signal was generated by sinusoidally modulated CW light beam incident on the condensed sample, and the periodic heating of the gas at the irradiated surface of the sample generated the acoustic wave, which was detected by a gas-phase microphone.

Gas-Coupling Method
* The periodic heating of the sample occurs in the ?absorption length? ?? of the sample.
* But only the heat within a diffusion length ?s from the interface can communicate with the gas and heat up a layer of gas of length ?g (diffusion length in gas) which expands periodically, producing acoustic waves.

Gas-Coupling Method
Gas-Coupling Method
* The heat generated in the thin absorption layer of thickness is mainly conducted into the condensed sample (heat conduction into the gas is much smaller); the heat conduction equation is:

Gas-Coupling Method
* Using the ideal gas law, we obtain the amplitude ?V of the volume change of Vact:

Gas-Coupling Method
* Here Vres is the residual cell volume for lg=0, and can be due to the dead space in front of the microphone. Finally, we have:

Gas-Coupling Method
Direct-Coupling Method
* Problems with Gas-Coupling led to the invention of Direct-Coupling method (microphone signal due to acoustic vibration) .
* It involves the insertion or attachment of a transducer (usually piezoelectric) into or onto the sample without the intervention of a gas medium.
* Thus, the serious acoustic impedance mismatch from condensed matter to gas can be avoided.
Direct-Coupling Method
Two general types of PA excitation are:

* The use of a chopped or modulated CW excitation beam when the detected PA signal depends on the boundary conditions
* The use of a pulsed excitation beam when the boundary conditions frequently have no effect on the detected optoacoustic signal, especially if short-duration pulses (<1?s) at low repetition rate (~10 Hz) are used.
Direct-Coupling Method
Direct-Coupling Method
Substance
* PA or OA spectroscopy is based on OA effect.
* Generation of acoustic waves due to the absorption of a modulated EM wave.
* Can be done to analyze gas and condense matter.
* Very useful and can be used efficiently for trace detection, depth profile studies, etc. !!



Discussion


( Don?t Ask, Can?t Tell )
References
* Ultrasensitive Laser Spectroscopy by David S. Kilnger, Academic Press (1983)
* Laser Spectroscopy by R.K. Gupta, AAPT (1992)

Thank You