Abstract

The design of integrating cavity absorption meters of general geometry is analyzed for cases in which the incident illumination of the cavity is spatially uniform and isotropic, such as the meter of Fry et al. [Appl. Opt. 31, 2055 (1992)]. The analysis by Kirk [Appl. Opt. 34, 4397 (1995)] for the probability of photon survival in a spherical meter is extended to general geometries. An estimate of the effect of the shape of the cavity on the estimated absorption coefficient is given.

© 1999 Optical Society of America

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References

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  1. E. S. Fry, G. W. Kattawar, R. M. Pope, “Integrating cavity absorption meter,” Appl. Opt. 31, 2055–2065 (1992).
    [CrossRef] [PubMed]
  2. R. M. Pope, E. S. Fry, “Absorption spectrum (380–700 nm) of pure water. II. Integrating cavity measurements,” Appl. Opt. 36, 8710–8723 (1997).
    [CrossRef]
  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, “Point-source integrating-cavity absorption meter: theoretical principles and numerical modeling,” Appl. Opt. 36, 6123–6128 (1997).
    [CrossRef] [PubMed]
  5. K. M. Case, F. de Hoffmann, G. Placzek, Introduction to the Theory of Neutron Diffusion, (Los Alamos National Laboratory, Los Alamos, N.M., 1953), Vol. 1, pp. 17–42.
  6. K. M. Case, P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass., 1967), p. 27.
  7. G. K. Kristiansen, B. Tollander, L. I. Tiren, “Tables related to the mean square chord length in right parallelepipeds,” Nukleonik 10, 45–47 (1967).
  8. I. Carlvik, “Collision probabilities for finite cylinders and cuboids,” Nucl. Sci. Eng. 30, 150–151 (1967) and (Stockholm, Sweden, 1967).
  9. S. Wolfram, The Mathematica Book, 3rd ed. (Wolfram Media and Cambridge U. Press, Cambridge, UK, 1996).
  10. J. T. O. Kirk, Light & Photosynthesis in Aquatic Ecosystems, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1994), Table 3.2.
  11. J. T. O. Kirk, Marine Optics, P.O. Box 117, Murrumbateman, NSW 2582, Australia (personal communication, 1998).
  12. J. T. O. Kirk, “Monte Carlo modeling of the performance of a reflective tube absorption meter,” Appl. Opt. 31, 6463–6468 (1992).
    [CrossRef] [PubMed]

1997 (2)

1995 (1)

1992 (2)

1967 (2)

G. K. Kristiansen, B. Tollander, L. I. Tiren, “Tables related to the mean square chord length in right parallelepipeds,” Nukleonik 10, 45–47 (1967).

I. Carlvik, “Collision probabilities for finite cylinders and cuboids,” Nucl. Sci. Eng. 30, 150–151 (1967) and (Stockholm, Sweden, 1967).

Carlvik, I.

I. Carlvik, “Collision probabilities for finite cylinders and cuboids,” Nucl. Sci. Eng. 30, 150–151 (1967) and (Stockholm, Sweden, 1967).

Case, K. M.

K. M. Case, P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass., 1967), p. 27.

K. M. Case, F. de Hoffmann, G. Placzek, Introduction to the Theory of Neutron Diffusion, (Los Alamos National Laboratory, Los Alamos, N.M., 1953), Vol. 1, pp. 17–42.

de Hoffmann, F.

K. M. Case, F. de Hoffmann, G. Placzek, Introduction to the Theory of Neutron Diffusion, (Los Alamos National Laboratory, Los Alamos, N.M., 1953), Vol. 1, pp. 17–42.

Fry, E. S.

Kattawar, G. W.

Kirk, J. T. O.

Kristiansen, G. K.

G. K. Kristiansen, B. Tollander, L. I. Tiren, “Tables related to the mean square chord length in right parallelepipeds,” Nukleonik 10, 45–47 (1967).

Placzek, G.

K. M. Case, F. de Hoffmann, G. Placzek, Introduction to the Theory of Neutron Diffusion, (Los Alamos National Laboratory, Los Alamos, N.M., 1953), Vol. 1, pp. 17–42.

Pope, R. M.

Tiren, L. I.

G. K. Kristiansen, B. Tollander, L. I. Tiren, “Tables related to the mean square chord length in right parallelepipeds,” Nukleonik 10, 45–47 (1967).

Tollander, B.

G. K. Kristiansen, B. Tollander, L. I. Tiren, “Tables related to the mean square chord length in right parallelepipeds,” Nukleonik 10, 45–47 (1967).

Wolfram, S.

S. Wolfram, The Mathematica Book, 3rd ed. (Wolfram Media and Cambridge U. Press, Cambridge, UK, 1996).

Zweifel, P. F.

K. M. Case, P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass., 1967), p. 27.

Appl. Opt. (5)

Nucl. Sci. Eng. (1)

I. Carlvik, “Collision probabilities for finite cylinders and cuboids,” Nucl. Sci. Eng. 30, 150–151 (1967) and (Stockholm, Sweden, 1967).

Nukleonik (1)

G. K. Kristiansen, B. Tollander, L. I. Tiren, “Tables related to the mean square chord length in right parallelepipeds,” Nukleonik 10, 45–47 (1967).

Other (5)

K. M. Case, F. de Hoffmann, G. Placzek, Introduction to the Theory of Neutron Diffusion, (Los Alamos National Laboratory, Los Alamos, N.M., 1953), Vol. 1, pp. 17–42.

K. M. Case, P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass., 1967), p. 27.

S. Wolfram, The Mathematica Book, 3rd ed. (Wolfram Media and Cambridge U. Press, Cambridge, UK, 1996).

J. T. O. Kirk, Light & Photosynthesis in Aquatic Ecosystems, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1994), Table 3.2.

J. T. O. Kirk, Marine Optics, P.O. Box 117, Murrumbateman, NSW 2582, Australia (personal communication, 1998).

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Figures (4)

Fig. 1
Fig. 1

Cross section of a generic integrating cavity design. The leads into and out of the cavities are fiber-optic cables. (From Fry et al.1.)

Fig. 2
Fig. 2

Survival probability for general geometry, given by Eq. (4), is evaluated along the direction vector Ω from 0 to the chord length ℓ, the distance to the opposite surface. n i is the interior unit vector normal to the surface.

Fig. 3
Fig. 3

For spherical geometry (Ω · n i ) = cos θ = μ, and the chord has a length ℓ = 2Rµ.

Fig. 4
Fig. 4

Ratio of HED to the true absorption coefficient a HED/a versus the absorption coefficient in inverse meters. The lower curve is for a cylinder with dimensions of the Fry et al.1 ICAM and the upper curve is for a sphere of equal volume.

Tables (3)

Tables Icon

Table 1 Q and Q′ Factors for Approximations (12) and (13), Respectively, with Qsphere′ = 0.4653a

Tables Icon

Table 2 Q and Q′/Qsphere′ for a Right Parallelepiped of Side Lengths X, Y, and Z, with Qsphere′ = 0.4653a

Tables Icon

Table 3 Ps for the Fry et al.1 Instrument for Absorption Values a of Water Measured by Pope and Fry2 at Wavelength λ

Equations (30)

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a=K1S1S0-K2,
F1F0=1Ps11-ρ1-ρ1-ρ1,
F¯=1a¯1-Ps1-ρPs,
Ps=2πinS exp-aΩ·nidSdΩ2πinSΩ·nidSdΩ,
Ps=πS-12πinS exp-aΩ·nidSdΩ.
Lr1, Ω1r2, Ω2=Lr2, -Ω2r1, -Ω1,
Pe=2πinS0 exp-arΩ·niadrdSdΩ2πinS0Ω·niadrdSdΩ,
Pe=4πVa-12πinS1-exp-aΩ·nidSdΩ.
¯=2πinSΩ·nidSdΩ2πinSΩ·nidSdΩ=4VS,
Pe=πSa¯-12πinS1-exp-aΩ·nidSdΩ=a¯-11-πS-12πinS×exp-aΩ·nidSdΩ.
Ps=1-a¯Pe.
Pe1-a22¯¯=1-Qa¯,
Pe1-QaV1/3,
Ps1-a¯+Qa¯2,
Ps1-a¯1-QaV1/3.
Ps=2 01 μ exp-2aRμdμ=12aR21-1+2aRexp-2aR,
Ps1-43 aR.
Pe=h-1E30-E3h-4πd2h0ddtE3t-E3t2+h21/2d2-t21/2+4πd2h0h×duh-u0ddt exp-t2+u21/2t2+u23/2 t2d2-t21/2.
h=a*Hd=a*Dy1=NIntegrateExpIntegralE3, t-ExpIntegralE3, t^2+h^2^1/2*d^2-t^2^1/2, t, 0, dy2=NIntegrateh-u*t^2*Exp-t^2+u^2^1/2*d^2-t^2^1/2/t^2+u^2^3/2, t, 0, d, u, 0, hPe=1/h*ExpIntegralE3, 0-ExpIntegralE3, h-4/Pi*h*d^2*y1-y2,
Enz=1 u-n exp-zudu.
Pe=2πxyz-1Ixy0-Ixyz+Iyz0-Iyzx+Izx0-Izxy,
Ixyz=2πxyE3z-2 0x2x-vE3v2+z21/2+v2-2xv+y2y2+v2 vE3v2+y2+z21/2dv-2 0y2y-uE3u2+z21/2+u2-2yu+x2x2+u2 uE3u2+x2+z21/2du.
F1F0=1+aHED¯1-ρ1-ρ1-ρ1,
aHEDa=1-Psa¯Ps.
aHEDa1+1-Qa¯, a¯ 1.
a1+1-Qa¯=K1S1S0-K2, a¯ 1,
a-aHEDa=-1-1+a¯Psa¯Ps-1-Qa¯, a¯ 1.
Ps1-0.0690a+0.0041a2,
Pe=4πd20du 0ddt exp-t2+u21/2t2+u23/2 t2d2-t21/2.
Pe=2r32rK1rI1r+rK0rI0r-1+r-1K1rI1r-K0rI1r+K1rI0r,

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