Abstract

The Rosencwaig-Gersho equation for the photoacoustic signal is recast in a manner that emphasizes the crucial role thermal wave interference plays in the production of the photoacoustic signal. This formalism is then used to suggest a technique for extracting thermal information from the structure in the photoacoustic signal resulting from thermal wave interference. Experimental measurements illustrating this technique are presented.

© 1982 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. C. A. Bennett, R. R. Patty, Appl. Opt. 20, A60 (1981).
    [CrossRef] [PubMed]
  2. C. A. Bennett, R. R. Patty, (in press, J. Photoacoustics).
  3. H. S. Carslaw, J. C. Jaeger, Conduction of Heat in Solids (Oxford U.P., Oxford, 1959), p. 64.
  4. P. M. Morse, K. U. Ingard, Theoretical Acoustics (McGraw-Hill, New York, 1968), p. 479.
  5. A. Rosencwaig, A. Gersho, J. Appl. Phys. 47, 64 (1976).
    [CrossRef]
  6. F. A. McDonald, Am. J. Phys. 48, 41 (1980).
    [CrossRef]
  7. F. A. McDonald, G. C. Wetsel, J. Appl. Phys. 49, 2313 (1976).
    [CrossRef]
  8. M. Born, E. Wolf, Principles of Optics (Macmillan, New York, 1964), p. 323.

1981 (1)

1980 (1)

F. A. McDonald, Am. J. Phys. 48, 41 (1980).
[CrossRef]

1976 (2)

F. A. McDonald, G. C. Wetsel, J. Appl. Phys. 49, 2313 (1976).
[CrossRef]

A. Rosencwaig, A. Gersho, J. Appl. Phys. 47, 64 (1976).
[CrossRef]

Bennett, C. A.

C. A. Bennett, R. R. Patty, Appl. Opt. 20, A60 (1981).
[CrossRef] [PubMed]

C. A. Bennett, R. R. Patty, (in press, J. Photoacoustics).

Born, M.

M. Born, E. Wolf, Principles of Optics (Macmillan, New York, 1964), p. 323.

Carslaw, H. S.

H. S. Carslaw, J. C. Jaeger, Conduction of Heat in Solids (Oxford U.P., Oxford, 1959), p. 64.

Gersho, A.

A. Rosencwaig, A. Gersho, J. Appl. Phys. 47, 64 (1976).
[CrossRef]

Ingard, K. U.

P. M. Morse, K. U. Ingard, Theoretical Acoustics (McGraw-Hill, New York, 1968), p. 479.

Jaeger, J. C.

H. S. Carslaw, J. C. Jaeger, Conduction of Heat in Solids (Oxford U.P., Oxford, 1959), p. 64.

McDonald, F. A.

F. A. McDonald, Am. J. Phys. 48, 41 (1980).
[CrossRef]

F. A. McDonald, G. C. Wetsel, J. Appl. Phys. 49, 2313 (1976).
[CrossRef]

Morse, P. M.

P. M. Morse, K. U. Ingard, Theoretical Acoustics (McGraw-Hill, New York, 1968), p. 479.

Patty, R. R.

C. A. Bennett, R. R. Patty, Appl. Opt. 20, A60 (1981).
[CrossRef] [PubMed]

C. A. Bennett, R. R. Patty, (in press, J. Photoacoustics).

Rosencwaig, A.

A. Rosencwaig, A. Gersho, J. Appl. Phys. 47, 64 (1976).
[CrossRef]

Wetsel, G. C.

F. A. McDonald, G. C. Wetsel, J. Appl. Phys. 49, 2313 (1976).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Macmillan, New York, 1964), p. 323.

Am. J. Phys. (1)

F. A. McDonald, Am. J. Phys. 48, 41 (1980).
[CrossRef]

Appl. Opt. (1)

J. Appl. Phys. (2)

A. Rosencwaig, A. Gersho, J. Appl. Phys. 47, 64 (1976).
[CrossRef]

F. A. McDonald, G. C. Wetsel, J. Appl. Phys. 49, 2313 (1976).
[CrossRef]

Other (4)

M. Born, E. Wolf, Principles of Optics (Macmillan, New York, 1964), p. 323.

C. A. Bennett, R. R. Patty, (in press, J. Photoacoustics).

H. S. Carslaw, J. C. Jaeger, Conduction of Heat in Solids (Oxford U.P., Oxford, 1959), p. 64.

P. M. Morse, K. U. Ingard, Theoretical Acoustics (McGraw-Hill, New York, 1968), p. 479.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1
Fig. 1

Thermal waves in each of the three regions of the photoacoustic cell. Note that the direction of positive x is to the left. Light is incident from the right.

Fig. 2
Fig. 2

Theoretical plot of the ratio |R(ω)| = S/Sr vs thermal thickness asl for different values of the thermal wave reflection coefficient Rb at the sample–backing boundary, where S is the photoacoustic signal at angular modulation frequency ω due to a nonthermally thick sample, and Sr is the photoacoustic signal at ω due to a thermally thick sample of the same material. Both samples are opaque, and Rg has been taken to be 0.99. Note that a s l ω.

Fig. 3
Fig. 3

Theoretical plot of the phase difference Δϕ between the sample signal S and the reference signal Sr vs thermal thickness asl for the same values of Rb and Rg for Fig. 2.

Fig. 4
Fig. 4

Experimental plots of |R(ω)| (dots) and Δϕ (triangles) vs the square root of the modulation frequency for samples of particulate carbon deposited on Millipore substrates.

Fig. 5
Fig. 5

Photoacoustic signal vs optical thickness βl (proportional to sample thickness) for modulation frequencies of 100 and 1600 Hz due to samples of particulate carbon deposited on three different substrates: Teflon, Nuclepore, and Millipore.

Fig. 6
Fig. 6

Photoacoustic signal vs optical thickness βl due to carbon particles deposited on Teflon substrates analyzed at a modulation frequency of 3000 Hz. In Fig. 4 the structure in the photoacoustic signal due to thermal wave interference is illustrated by varying the wavelength of the thermal waves generated within the sample; in Fig. 6 this structure is illustrated by varying the physical thickness (and hence the thermal thickness) of different carbon deposits analyzed at a single modulation frequency.

Equations (13)

Equations on this page are rendered with MathJax. Learn more.

R b = 1 - b 1 + b , T b = 2 1 + b , R g = 1 - g 1 + g , T g = 2 1 + g ,
b = ρ b C b k b ρ s C s k s , g = ρ g C g k g ρ s C s k s ,
β I 0 exp ( - β x ) 4 k s σ s d x .
β I 0 exp ( - β x ) 4 k s σ s T g { exp ( - σ s x ) + R b R g exp [ - σ s ( 2 l + x ) ] + + ( R b R g ) n exp [ - σ x ( 2 n l + x ) ] + } d x ,
β I 0 exp ( - β x ) 4 k s σ s T g ( R b exp [ - σ s ( 2 l + x ) ] + ( R b R g ) R b × exp [ - σ s ( 4 l - x ) ] + + ( R b R g ) n R b × exp { - σ s [ 2 ( n + 1 ) l - x ] } + ) d x .
θ = I 0 β T g 4 k s σ s ( 1 β + σ s { 1 - exp [ - ( β + σ s ) l ] } + R b exp ( - 2 σ s l ) 1 β - σ s { 1 - exp [ - ( β - σ s ) l ] } 1 - R b R g exp ( - 2 σ s l ) ) .
R ( ω ) = S S r ,
R ( ω ) = 1 - exp [ - ( β - σ s ) l ] + R b ( β + σ s β - σ s ) exp ( - 2 σ s l ) { 1 - exp [ - ( β - σ s ) l ] } 1 - R b R g exp ( - 2 σ s l ) .
R ( ω ) 1 + R b exp ( - 2 σ s l ) 1 - R b R g exp ( - 2 σ s l ) .
R ( ω ) = [ 1 + R b exp ( - 2 a s l ) 1 - R b R g exp ( - 2 a s l ) ] 2 - F R g sin 2 a s l 1 + F sin 2 a s l ,
F = 4 R b R g exp ( - 2 a s l ) [ 1 - R b R g exp ( - 2 a s l ) ] 2 ,
Δ ϕ = tan - 1 { - R b ( 1 + R g ) exp ( - 2 a s l ) sin 2 a s l 1 - R g [ R b exp ( - 2 a s l ) ] 2 + R b ( 1 - R g ) cos 2 a s l ] .
( 1 + R b 1 - R b R g )

Metrics