Finite element calculation of photoacoustic signals

Bernd Baumann; Marcus Wolff; Bernd Kost; Hinrich Groninga

doi:10.1364/AO.46.001120

Finite element calculation of photoacoustic signals

Bernd Baumann, Marcus Wolff, Bernd Kost, and Hinrich Groninga

Applied Optics
Vol. 46,
Issue 7,
pp. 1120-1125
(2007)
•https://doi.org/10.1364/AO.46.001120

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Bernd Baumann, Marcus Wolff, Bernd Kost, and Hinrich Groninga, "Finite element calculation of photoacoustic signals," Appl. Opt. 46, 1120-1125 (2007)

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Optics & Photonics Topics
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The topics in this list come from the OSA Optics and Photonics Topics applied to this article.
Previously assigned OCIS codes
- Numerical approximation and analysis (000.4430)
- Spectrometers and spectroscopic instrumentation (120.6200)

About this Article
History
- Original Manuscript: June 22, 2006
- Revised Manuscript: October 19, 2006
- Manuscript Accepted: November 1, 2006
- Published: February 12, 2007

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Abstract

Aprocedure for the numerical calculation of photoacoustic signals is introduced. It is based on the finite element method and uses an expansion of the signal into acoustic eigenmodes of the measuring cell. Loss is included by the incorporation of quality factors. Surface and volume loss effects attributable to viscosity and thermal conductivity are considered. The method is verified for cylindrical cells with excellent accordance. The application to photoacoustic cells of unconventional shape yields good agreement with experimental data.

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Absorption cylinder	Diameter	D _A = 26 mm
	Length	L _A = 82 mm
Resonance cylinder	Diameter	D _R = 11 mm
	Length	L _R = 10–320 mm (adjustable)
Diminution	Diameter	D _D = 8.9 mm
	Length	L _D = 2 mm

Description	Mode	Analytical Result
$N_{j}$	100	2
	010	$\frac{\sqrt{2}}{J_{1} (π α_{10})} {(1 - \frac{1}{π^{2} {α_{10}}^{2}})}^{- 1 / 2}$
	001	$\frac{1}{J_{0} (π α_{01})}$
$\int_{S_{C}} \| {p_{j} \|}^{2} d S$	100	$2 π R_{C} (2 R_{C} + L_{C})$
	010	$2 π R_{C} (R_{C} + L_{C} {(1 - \frac{1}{π^{2} {α_{10}}^{2}})}^{- 1})$
	001	$2 π R_{C} (R_{C} + L_{C})$
$\int_{S_{C}} {\| \nabla_{t} p_{j} \|}^{2} d S$	100	$2 π^{3} \frac{R_{C}}{L_{C}}$
	010	$2 π (π^{2} {α_{10}}^{2} + \frac{L_{C}}{R_{C}} {(1 - \frac{1}{π^{2} {α_{10}}^{2}})}^{- 1})$
	001	$2 π^{3} {α_{01}}^{2}$

Mode	Description	Analytical Result	Finite Element Result
010	f [Hz]	7.7142 × 10²	7.7137 × 10²
	Q _j ^{s _η} ^a	1.8074 × 10³	1.8072 × 10³
	Q _j ^{s _κ} ^b	1.0446 × 10⁴	1.0445 × 10⁴
	Q _j ^v ^c	1.9883 × 10⁶	1.9884 × 10⁶
100	f [Hz]	1.0530 × 10³	1.0530 × 10³
	Q _j ^{s _η}	2.5727 × 10³	2.5727 × 10³
	Q _j ^{s _κ}	1.0413 × 10⁴	1.0413 × 10⁴
	Q _j ^v	1.4567 × 10⁶	1.4567 × 10⁶
001	f [Hz]	1.6054 × 10³	1.6053 × 10³
	Q _j ^{s _η}	3.9708 × 10³	3.9706 × 10³
	Q _j ^{s _κ}	1.8572 × 10⁴	1.8571 × 10⁴
	Q _j ^v	9.5543 × 10⁵	9.5544 × 10⁵

L _R [mm]	Q _j ^exp	Q _j ^FE	Q _j ^FE∕Q _j ^exp
10	116	404	3.48
	127	281	2.22
20	96	198	2.07
40	98	153	1.55
60	86	133	1.55
80	75	120	1.59
100	60	110	1.84
120	55	103	1.88
140	52	97	1.86

Absorption cylinder	Diameter	D _A = 26 mm
	Length	L _A = 82 mm
Resonance cylinder	Diameter	D _R = 11 mm
	Length	L _R = 10–320 mm (adjustable)
Diminution	Diameter	D _D = 8.9 mm
	Length	L _D = 2 mm

Abstract

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Figures (7)

Tables (4)

Equations (19)

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