Abstract

A mathematical model for photon behavior within a spherical integrating-cavity absorption meter (ICAM) that does not depend on the assumption of a homogeneous energy density within the cavity has been developed. Explicit expressions for the proportion of emitted or reflected photons that survive a single transit across the cavity, the average number of collisions with the wall per photon, and the average path length per photon, are derived for an absorbing nonscattering medium. Monte Carlo modeling shows that operation of the ICAM is essentially unaffected by scattering, in agreement with the experimental observations of Fry et al. [Appl. Opt. 31, 2055 (1992)]. Calculations for the performance of the absorption meter as a function of the cavity diameter, the absorption coefficient of the medium, and the reflectivity of the cavity are presented.

© 1995 Optical Society of America

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References

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  1. P. Elterman, “Integrating cavity spectroscopy,” Appl. Opt. 9, 2140–2142 (1970).
    [CrossRef] [PubMed]
  2. E. S. Fry, G. W. Kattawar, R. M. Pope, “Integrating cavity absorption meter,” Appl. Opt. 31, 2055–2065 (1992).
    [CrossRef] [PubMed]
  3. E. S. Fry, G. W. Kattawar, “Measurement of the absorption coefficient of ocean water using isotropic illumination,” in Ocean Optics IX, M. A. Blizard, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 925, 142–148 (1988).
  4. J. A. Jacquez, H. F. Kuppenheim, “Theory of the integrating sphere,” J. Opt. Soc. Am. 45, 460–470 (1955).
    [CrossRef]
  5. D. G. Goebel, “Generalized integrating-sphere theory,” Appl. Opt. 6, 125–128 (1967).
    [CrossRef] [PubMed]
  6. J. T. O. Kirk, “A Monte Carlo study of the nature of the underwater light field in, and the relationships between optical properties of, turbid yellow waters,” Aust. J. Mar. Freshwater Res. 32, 517–532 (1981).
    [CrossRef]
  7. J. T. O. Kirk, “Monte Carlo modeling of the performance of a reflective tube absorption meter,” Appl. Opt. 31, 6463–6468 (1992).
    [CrossRef] [PubMed]
  8. T. J. Petzold, Volume Scattering Functions for Selected Waters, SIO Ref. 72–78 (Scripps Institution of Oceanography, San Diego, Calif., 1972).
  9. L. N. M. Duysens, “The flattening of the absorption spectrum of suspensions, as compared to that of solutions,” Biochim. Biophys. Acta 19, 1–12 (1956).
    [CrossRef] [PubMed]
  10. J. T. O. Kirk, Light and Photosynthesis in Aquatic Ecosystems, 2nd ed. (Cambridge U. Press, Cambridge, 1994), pp. 64–70.

1992

1981

J. T. O. Kirk, “A Monte Carlo study of the nature of the underwater light field in, and the relationships between optical properties of, turbid yellow waters,” Aust. J. Mar. Freshwater Res. 32, 517–532 (1981).
[CrossRef]

1970

1967

1956

L. N. M. Duysens, “The flattening of the absorption spectrum of suspensions, as compared to that of solutions,” Biochim. Biophys. Acta 19, 1–12 (1956).
[CrossRef] [PubMed]

1955

Duysens, L. N. M.

L. N. M. Duysens, “The flattening of the absorption spectrum of suspensions, as compared to that of solutions,” Biochim. Biophys. Acta 19, 1–12 (1956).
[CrossRef] [PubMed]

Elterman, P.

Fry, E. S.

E. S. Fry, G. W. Kattawar, R. M. Pope, “Integrating cavity absorption meter,” Appl. Opt. 31, 2055–2065 (1992).
[CrossRef] [PubMed]

E. S. Fry, G. W. Kattawar, “Measurement of the absorption coefficient of ocean water using isotropic illumination,” in Ocean Optics IX, M. A. Blizard, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 925, 142–148 (1988).

Goebel, D. G.

Jacquez, J. A.

Kattawar, G. W.

E. S. Fry, G. W. Kattawar, R. M. Pope, “Integrating cavity absorption meter,” Appl. Opt. 31, 2055–2065 (1992).
[CrossRef] [PubMed]

E. S. Fry, G. W. Kattawar, “Measurement of the absorption coefficient of ocean water using isotropic illumination,” in Ocean Optics IX, M. A. Blizard, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 925, 142–148 (1988).

Kirk, J. T. O.

J. T. O. Kirk, “Monte Carlo modeling of the performance of a reflective tube absorption meter,” Appl. Opt. 31, 6463–6468 (1992).
[CrossRef] [PubMed]

J. T. O. Kirk, “A Monte Carlo study of the nature of the underwater light field in, and the relationships between optical properties of, turbid yellow waters,” Aust. J. Mar. Freshwater Res. 32, 517–532 (1981).
[CrossRef]

J. T. O. Kirk, Light and Photosynthesis in Aquatic Ecosystems, 2nd ed. (Cambridge U. Press, Cambridge, 1994), pp. 64–70.

Kuppenheim, H. F.

Petzold, T. J.

T. J. Petzold, Volume Scattering Functions for Selected Waters, SIO Ref. 72–78 (Scripps Institution of Oceanography, San Diego, Calif., 1972).

Pope, R. M.

Appl. Opt.

Aust. J. Mar. Freshwater Res.

J. T. O. Kirk, “A Monte Carlo study of the nature of the underwater light field in, and the relationships between optical properties of, turbid yellow waters,” Aust. J. Mar. Freshwater Res. 32, 517–532 (1981).
[CrossRef]

Biochim. Biophys. Acta

L. N. M. Duysens, “The flattening of the absorption spectrum of suspensions, as compared to that of solutions,” Biochim. Biophys. Acta 19, 1–12 (1956).
[CrossRef] [PubMed]

J. Opt. Soc. Am.

Other

J. T. O. Kirk, Light and Photosynthesis in Aquatic Ecosystems, 2nd ed. (Cambridge U. Press, Cambridge, 1994), pp. 64–70.

T. J. Petzold, Volume Scattering Functions for Selected Waters, SIO Ref. 72–78 (Scripps Institution of Oceanography, San Diego, Calif., 1972).

E. S. Fry, G. W. Kattawar, “Measurement of the absorption coefficient of ocean water using isotropic illumination,” in Ocean Optics IX, M. A. Blizard, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 925, 142–148 (1988).

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Figures (6)

Fig. 1
Fig. 1

Geometric relationships relevant to distribution of scalar irradiance within the spherical cavity.

Fig. 2
Fig. 2

Filled/empty irradiance ratio (F 0/F E , ●) and average photon path length (■) as functions of cavity diameter. a = 0.03 m−1. Reflectivity 0.992.

Fig. 3
Fig. 3

Filled/empty irradiance ratio (F 0/F E , ●) and average photon path length (○) as functions of absorption coefficient. Diameter 100 mm. Reflectivity 0.992. (a) a = 0.0 to 1.0 m−1, (b) a = 0.0 to 10.0 m−1.

Fig. 4
Fig. 4

Filled/empty irradiance ratio (F 0/F E ) as a function of cavity reflectivity for waters with three different values of absorption coefficient. Diameter 100 mm.

Fig. 5
Fig. 5

Ratio of scalar irradiance at the center of the cavity to that at the periphery as a function of absorption coefficient. Diameter 100 mm. Reflectivity 0.992.

Fig. 6
Fig. 6

Ratio of measured to true absorption coefficient (a m /a, ●), and inward/outward irradiance ratio (F 1/F 0: HED theory, □; present theory, ■) as functions of absorption coefficient. Diameter 100 mm. Reflectivity 0.992 on both sides of the wall material–cavity interface.

Equations (46)

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P = 4 a V F 0 ,
F 1 A ( 1 - ρ 1 ) = F 0 A ( 1 - ρ ) + F 0 A D + 4 a F 0 V ,
a = A ( 1 - ρ 1 ) 4 V F 1 F 0 - [ A ( 1 - ρ ) + A D 4 V ] .
a = K 1 S 1 S 0 - K 2 ,
P n = ρ n - 1 ( 1 - ρ ) ,
C E = P 1 1 + P 2 2 + P 3 3 + + P n n + = ( 1 - ρ ) + ρ ( 1 - ρ ) 2 + ρ 2 ( 1 - ρ ) 3 + + ρ n - 1 ( 1 - ρ ) n + = ( 1 - ρ ) ( 1 + 2 ρ + 3 ρ 2 + + n ρ n - 1 + ) .
C E = 1 ( 1 - ρ ) .
I ( θ ) = 2 r cos θ .
I ( θ ) = I 0 cos θ .
Φ ( θ ) = I ( θ ) 2 π sin θ Δ θ = I 0 2 π sin θ cos θ Δ θ .
Φ tot = 2 π I 0 0 π / 2 sin θ cos θ d θ = π I 0 .
P ( θ ) = Φ ( θ ) Φ tot .
l E ¯ = θ = 0 θ = π / 2 P ( θ ) l ( θ ) = θ = 0 θ = π / 2 I 0 2 π sin θ cos θ Δ θ π I 0 l ( θ ) = 2 0 π / 2 sin θ cos θ l ( θ ) d θ = 4 r 0 π / 2 sin θ cos 2 θ d θ = 4 r 3 .
l E = C E 4 r 3 = 4 r 3 ( 1 - ρ ) .
P a = 1 - P s .
P ( θ ) = exp [ - a l ( θ ) ] = exp ( - 2 a r cos θ )
P s = θ = 0 θ = π / 2 P ( θ ) exp ( - 2 a r cos θ ) = 2 0 π / 2 sin θ cos θ exp ( - 2 a r cos θ ) d θ .
P s = 1 2 a 2 r 2 [ 1 - exp ( - 2 a r ) ( 2 a r + 1 ) ] .
P a = 1 - 1 2 a 2 r 2 [ 1 - exp ( - 2 a r ) ( 2 a r + 1 ) ] .
P n = P s ( ρ P s ) n - 1 ( 1 - ρ P s ) = ρ n - 1 P s n ( 1 - ρ P s ) .
C F = P 0 0 + P 1 1 + P 2 2 + P 3 3 + + P n n + = 0 + P s ( 1 - ρ P s ) + 2 ρ P s 2 ( 1 - ρ P s ) + 3 ρ 2 P s 3 ( 1 - ρ P s ) + + n ρ n - 1 P s n ( 1 - ρ P s ) + = P s ( 1 - ρ P s ) ( 1 + 2 ρ P s + 3 ρ 2 P s 2 + + n ρ n - 1 P s n - 1 + ) = P s ( 1 - ρ P s ) ( 1 - ρ P s ) 2 = P s 1 - ρ P s .
N = N 0 exp ( - a l ) .
P ( l ) = N a δ l N 0 = exp ( - a l ) a δ l .
l a = 0 l exp ( - a l ) a d l = a 0 l exp ( - a l ) d l = 1 / a .
Path i c = j = 1 m l j P ( l j ) + W i exp ( - a W i ) = j = 1 m l j exp ( - a l j ) a δ l + W i exp ( - a W i ) = a 0 W i l exp ( - a l ) d l + W i exp ( - a W i ) = 1 a [ 1 - exp ( - a W i ) ] = P a i a ,
Path c = P ( W 1 ) Path 1 c + P ( W 2 ) Path 2 c + + P ( W 1 ) Path i c + = 1 a [ P ( W 1 ) P a 1 + P ( W 2 ) P a 2 + + P ( W i ) P a i + ] = P a a ,
Path tot = N 0 Path c + ρ P s N 0 Path c + ( ρ P s ) 2 N 0 Path c + + ( ρ P s ) n N 0 Path c + = N 0 Path c [ 1 + ρ P s + ( ρ P s ) 2 + + ( ρ P s ) n + ] = N 0 Path c 1 - ρ P s ,
l F = Path tot N 0 = Path c 1 - ρ P s .
l F = 1 a ( 1 - P s 1 - ρ P s ) ,
l F = 1 a ( 1 - C F C E )
F 0 = C F F 1 ( 1 - ρ 1 ) .
F 1 F 0 = 1 C F ( 1 - ρ 1 ) = ( 1 - ρ P s ) P s ( 1 - ρ 1 ) .
F 1 F 0 = 1 P s ( 1 1 - ρ 1 ) - ρ 1 - ρ 1 ,
F 1 F 0 = ( 1 + 4 a r 3 ) ( 1 1 - ρ 1 ) - ρ 1 - ρ 1 .
F 0 E F 0 = C E C F = 1 - ρ P s P s ( 1 - ρ ) .
F 0 W F 0 = C F W C F = P s W ( 1 - ρ P s ) P s ( 1 - ρ P s W ) .
E 0 ( X ) = 4 π L ( X ; γ ) d ω = 2 π 0 π L ( X ; γ ) sin γ d γ .
L ( X ; γ ) = L exp [ - a l ( γ ) ] .
l ( γ ) = r cos θ - d cos γ ,
cos θ = ( 1 - d 2 r 2 sin 2 γ ) 1 / 2 .
E 0 ( X ) = 2 π L 0 π exp [ - a ( r cos θ - d cos γ ) ] sin γ d γ ,
P s = 1 + 2 ( - 2 a r ) n [ 1 ( n + 1 ) ! - 1 ( n + 2 ) ! ] + ,
F ( α ) = sin 2 α .
F 1 = ( 1 - ρ 1 - ρ 1 ) F 0 E .
F 1 F 0 = f ( a r ) ( 1 1 - ρ 1 ) - ρ 1 - ρ 1 ,
F 1 F 0 = [ 4 r 3 ( 1 - ρ 1 ) ] a + ( 1 - ρ 1 - ρ 1 )

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