Abstract

A mathematical model has been developed for photon behavior within a spherical integrating-cavity absorption meter with a point light source at the center of the cavity. Explicit expressions for the average number of collisions with the wall per photon, the average path length per photon, and the transmittance of the cavity containing a water sample relative to that containing pure water are derived for an absorbing nonscattering medium. Monte Carlo modeling shows that the operation of the point-source integrating-cavity absorption meter is essentially unaffected by scattering. Calculations of the performance of the absorption meter as a function of the cavity diameter, absorption coefficient of the medium, and the reflectivity of cavity are presented.

© 1997 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. J. T. O. Kirk, “Modeling the performance of an integrating-cavity absorption meter: theory and calculations for a spherical cavity,” Appl. Opt. 34, 4397–4408 (1995).
    [CrossRef] [PubMed]
  4. J. T. O. Kirk, “The assessment and prediction of optical water quality,” in Thirteenth Federal Convention Australian Water Wastewater Association, Vol. 89 (Institution of Engineers, Canberra, Australia, 1989), 504–507.

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Figures (4)

Fig. 1
Fig. 1

Transmittance as a function of cavity diameter and reflectivity at two different values of absorption coefficient.

Fig. 2
Fig. 2

Transmittance as a function of absorption coefficient at three different values of reflectivity. Diameter, 100 mm. (a) a = 0.0–1.0 m-1, (b) a = 0.0–10.0 m-1.

Fig. 3
Fig. 3

Average photon path length as a function of absorption coefficient at three different values of reflectivity. Diameter, 100 mm.

Fig. 4
Fig. 4

Measured absorption coefficient (○) calculated from transmittance with Eq. (11) as a function of true absorption coefficient. For comparison the corresponding curve calculated for an ICAM with photons emitted from the wall3 is also shown (■). The straight line corresponds to a (true).

Equations (17)

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Ps=12a2r21-exp-2ar2ar+1,
CF=Ps1-ρPs,
lF = 1a1 - Ps1 - ρPs.
ρ exp-ar N0Ps1 - ρPs.
CF=1N0exp-arN0+ρ exp-arN0Ps1-ρPs=exp-ar1-ρPs.
1a1-exp-arar+1.
lF=1a1-exp-ar2-ρ1-Ps1-ρPs+exp-2arar+1.
T=exp-ar1-ρPsWexp-aWr1-ρPs,
P=4aVF0,
Pin=Pwater+Pwall.
Pinsample=4aVF0+F0A1-ρ,
Pinpure water=4aWVF0W+F0WA1-ρ.
T=4aWV+A1-ρ4aV+A1-ρ.
a=4/3aWr+1-ρ4/3r1T-31-ρ4r.
TAB=exp-aAr1-ρPsBexp-aBr1-ρPsA,
ρ=TAB exp-aBr-exp-aArTAB exp-aBrPsA-exp-aArPsB.
Δ1-ρ1-ρ=exp-aA+aBrPsB-PsATABTAB exp-aBrPsA-exp-aArPsBTAB exp-aBrPsA-1-exp-aArPsB-1ΔTABTAB

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