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.

© 2007 Optical Society of America

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References

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  1. W. Demtröder, Laser Spectroscopy (Springer-Verlag, 2002).
  2. K. H. Michaelian, Photoacoustic Infrared Spectroscopy (Wiley-Interscience, 2003).
    [CrossRef]
  3. B. Baumann, B. Kost, H. G. Groninga, and M. Wolff, "Eigenmode analysis of photoacoustic sensors via finite element method," Rev. Sci. Instrum. 77, 044901 (2006).
    [CrossRef]
  4. M. Wolff, H. G. Groninga, B. Baumann, B. Kost, and H. Harde, "Resonance investigations using PAS and FEM," Acta Acust. 91, Suppl. 1, 99, 1477-1481 (2005).
  5. K. J. Bathe, Finite Element Procedures (Prentice Hall, 1995).
  6. COMSOL homepage, http://www.comsol.com.
  7. L. B. Kreuzer, "The physics of signal generation and detection," in Optoacoustic Spectroscopy and Detection, Y.-H.Pao, ed. (Academic, 1977), pp. 1-25.
  8. P. M. Morse and K. U. Ingard, Theoretical Acoustics (McGraw-Hill, 1968).
  9. S. Temkin, Elements of Acoustics (Wiley, 1981).
  10. VDI-Wärmeatlas, 9th ed. (Springer-Verlag, 2002).

2006 (1)

B. Baumann, B. Kost, H. G. Groninga, and M. Wolff, "Eigenmode analysis of photoacoustic sensors via finite element method," Rev. Sci. Instrum. 77, 044901 (2006).
[CrossRef]

2005 (1)

M. Wolff, H. G. Groninga, B. Baumann, B. Kost, and H. Harde, "Resonance investigations using PAS and FEM," Acta Acust. 91, Suppl. 1, 99, 1477-1481 (2005).

Bathe, K. J.

K. J. Bathe, Finite Element Procedures (Prentice Hall, 1995).

Baumann, B.

B. Baumann, B. Kost, H. G. Groninga, and M. Wolff, "Eigenmode analysis of photoacoustic sensors via finite element method," Rev. Sci. Instrum. 77, 044901 (2006).
[CrossRef]

M. Wolff, H. G. Groninga, B. Baumann, B. Kost, and H. Harde, "Resonance investigations using PAS and FEM," Acta Acust. 91, Suppl. 1, 99, 1477-1481 (2005).

Demtröder, W.

W. Demtröder, Laser Spectroscopy (Springer-Verlag, 2002).

Groninga, H. G.

B. Baumann, B. Kost, H. G. Groninga, and M. Wolff, "Eigenmode analysis of photoacoustic sensors via finite element method," Rev. Sci. Instrum. 77, 044901 (2006).
[CrossRef]

M. Wolff, H. G. Groninga, B. Baumann, B. Kost, and H. Harde, "Resonance investigations using PAS and FEM," Acta Acust. 91, Suppl. 1, 99, 1477-1481 (2005).

Harde, H.

M. Wolff, H. G. Groninga, B. Baumann, B. Kost, and H. Harde, "Resonance investigations using PAS and FEM," Acta Acust. 91, Suppl. 1, 99, 1477-1481 (2005).

Ingard, K. U.

P. M. Morse and K. U. Ingard, Theoretical Acoustics (McGraw-Hill, 1968).

Kost, B.

B. Baumann, B. Kost, H. G. Groninga, and M. Wolff, "Eigenmode analysis of photoacoustic sensors via finite element method," Rev. Sci. Instrum. 77, 044901 (2006).
[CrossRef]

M. Wolff, H. G. Groninga, B. Baumann, B. Kost, and H. Harde, "Resonance investigations using PAS and FEM," Acta Acust. 91, Suppl. 1, 99, 1477-1481 (2005).

Kreuzer, L. B.

L. B. Kreuzer, "The physics of signal generation and detection," in Optoacoustic Spectroscopy and Detection, Y.-H.Pao, ed. (Academic, 1977), pp. 1-25.

Michaelian, K. H.

K. H. Michaelian, Photoacoustic Infrared Spectroscopy (Wiley-Interscience, 2003).
[CrossRef]

Morse, P. M.

P. M. Morse and K. U. Ingard, Theoretical Acoustics (McGraw-Hill, 1968).

Temkin, S.

S. Temkin, Elements of Acoustics (Wiley, 1981).

Wolff, M.

B. Baumann, B. Kost, H. G. Groninga, and M. Wolff, "Eigenmode analysis of photoacoustic sensors via finite element method," Rev. Sci. Instrum. 77, 044901 (2006).
[CrossRef]

M. Wolff, H. G. Groninga, B. Baumann, B. Kost, and H. Harde, "Resonance investigations using PAS and FEM," Acta Acust. 91, Suppl. 1, 99, 1477-1481 (2005).

Acta Acust. (1)

M. Wolff, H. G. Groninga, B. Baumann, B. Kost, and H. Harde, "Resonance investigations using PAS and FEM," Acta Acust. 91, Suppl. 1, 99, 1477-1481 (2005).

Rev. Sci. Instrum. (1)

B. Baumann, B. Kost, H. G. Groninga, and M. Wolff, "Eigenmode analysis of photoacoustic sensors via finite element method," Rev. Sci. Instrum. 77, 044901 (2006).
[CrossRef]

Other (8)

W. Demtröder, Laser Spectroscopy (Springer-Verlag, 2002).

K. H. Michaelian, Photoacoustic Infrared Spectroscopy (Wiley-Interscience, 2003).
[CrossRef]

K. J. Bathe, Finite Element Procedures (Prentice Hall, 1995).

COMSOL homepage, http://www.comsol.com.

L. B. Kreuzer, "The physics of signal generation and detection," in Optoacoustic Spectroscopy and Detection, Y.-H.Pao, ed. (Academic, 1977), pp. 1-25.

P. M. Morse and K. U. Ingard, Theoretical Acoustics (McGraw-Hill, 1968).

S. Temkin, Elements of Acoustics (Wiley, 1981).

VDI-Wärmeatlas, 9th ed. (Springer-Verlag, 2002).

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Figures (7)

Fig. 1
Fig. 1

Experimental setup of the photoacoustic sensor.

Fig. 2
Fig. 2

Typical FE mesh of a T cell.

Fig. 3
Fig. 3

Experimental (upper) versus finite element (lower) response functions for various T cells. From right to left the peaks correspond to the L R = 10, 10, 20, 40, 60, 80, 100, 120, 140, 180, 200, 240, 320, 440 mm T cells. The vertical axis depicts |p| in arbitrary units.

Fig. 4
Fig. 4

Experimental response function (dashed curve) and corresponding fit (solid curve) for the L R = 40 mm (top) and L R = 80 mm (bottom) T cell.

Fig. 5
Fig. 5

(Color online) Intensity distribution of the laser beam.

Fig. 6
Fig. 6

Finite element response function for T cells of 120, 180, and 240 mm resonance cylinder lengths. The vertical axis depicts |p| in arbitrary units.

Fig. 7
Fig. 7

Experimental (upper) versus finite element (lower) response functions for various T cells. Microphone characteristic is considered in the numerical results. From right to left the peaks correspond to the L R = 10 , 10, 20, 40, 60, 80, 100, 120, 140, 180, 200, 240, 320, 440 mm T cells. The vertical axis depicts | p | in arbitrary units.

Tables (4)

Tables Icon

Table 1 Dimensions of the Photoacoustic T Cell

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Table 2 Normalization Constants and Surface Integrals for Cylindrical Cells a

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Table 3 Eigenfrequencies and Q -Factors of Cylinder Cell

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Table 4 Experimental versus Finite Element Q -Factors of T Cell

Equations (19)

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2 p ( r , ω ) + k 2 p ( r , ω ) = i ω γ 1 c 2 H ( r , ω ) ,
p ( r , ω ) = j A j ( ω ) p j ( r ) .
2 p ( r ) + k 2 p ( r ) = 0 .
p n = 0 ,
V C p i * p j d V = V C δ i j ,
A j ( ω ) = i   A j ω ω 2 ω j 2 + i ω ω j / Q j ,
A j = α ( γ 1 ) V C V C p j * I d V .
1 Q j v = ω j c [ l η + ( γ 1 ) l κ ] ,
l η = 4 3 η ρ c , l κ = κ ρ c p c ,
1 Q j s κ = 1 2 ( γ 1 ) d κ V C S C | p j | 2 d S
1 Q j s η = 1 2 ( c ω j ) 2 d η V C S C | t p j | 2 d S
d κ = 2 κ c p ρ ω j , d η = 2 η ρ ω j .
1 Q j = i 1 Q j i .
[ 1 r r ( r r ) + 1 r 2 2 φ 2 + 2 z 2 ] p ( r , φ , z ) + k 2 p ( r , φ , z ) = 0 .
p j ( r , φ , z ) = N j J m ( π α m n r R C ) cos ( m φ ) cos ( π l z L C ) ,
f j = ω j 2 π = c 2 ( l L C ) 2 + ( α m n R C ) 2 .
| N j | 2 = V C [ V C [ J m ( π α m n r R C ) cos ( m φ ) cos ( π l z L C ) ] 2  d V ] 1 .
A ( f ) = i   A j f f 2 f j 2 + i f f j / Q j
I ( r ) = I 0 exp [ - 2 ( r w ) 2 ] .

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