Abstract

The temporal response of an integrating cavity is examined and compared with the results of a Monte Carlo analysis. An important parameter in the temporal response is the average distance d¯ between successive reflections at the cavity wall; d¯ was calculated for several specific cavity designs—spherical shell, cube, right circular cylinder, irregular tetrahedron, and prism; however, only the calculation for the spherical shell and the right circular cylinder will be presented. A completely general formulation of d¯ for arbitrary cavity shapes is then derived, d¯=4V/S where V is the volume of the cavity, and S is the surface area of the cavity. Finally, we consider an arbitrary cavity shape for which each flat face is tangent to a single inscribed sphere of diameter D (a curved surface is considered to be an infinite number of flat surfaces). We will prove that for such a cavity d¯=2D/3, exactly the same as d¯ for the inscribed sphere.

© 2006 Optical Society of America

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References

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  1. J. Beaulieu, A Guide to Integrating Sphere Theory and Applications (Labsphere, Inc., 1999), http://www.labsphere.com/knowledgebase.aspx.
  2. S. Bogacz, Integrating Sphere Design and Applications (SphereOptics LLC, 2004), http://www.sphereoptics.com/assets/sphere-optic-pdf/sphere-technical-guide.pdf.
  3. P. Elterman, "Integrating cavity spectroscopy," Appl. Opt. 9, 2140-2142 (1970).
    [CrossRef] [PubMed]
  4. E. S. Fry, G. W. Kattawar, and R. M. Pope, "Integrating cavity absorption meter," Appl. Opt. 31, 2055-2065 (1992).
    [CrossRef] [PubMed]
  5. A. M. Emel'yanov, V. I. Kosyakov, and B. V. Makushkin, "The use of an integrating cavity for measuring small optical absorptions," Sov. J. Opt. Technol. 45, 31-33 (1978).
  6. D. M. Hobbs and N. J. McCormick, "Design of an integrating cavity absorption meter," Appl. Opt. 38, 456-461 (1999).
    [CrossRef]
  7. 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]
  8. J. T. O. Kirk, "Point-source integrating-cavity absorption meter: theoretical principles and numerical modeling," Appl. Opt. 36, 6123-6128 (1997).
    [CrossRef] [PubMed]
  9. R. A. Leathers, T. V. Downes, and C. O. Davis, "Analysis of a point-source integrating-cavity absorption meter," Appl. Opt. 39, 6118-6127 (2000).
    [CrossRef]
  10. C. J.-Y. Lerebourg, D. A. Pilgrim, G. D. Ludbrook, and R. Neal, "Development of a point source integrating cavity absorption meter," J. Opt. A 4, S56-S65 (2002).
    [CrossRef]
  11. J. W. Pickering, C. J. M. Moes, H. J. C. M. Sterenborg, S. A. Prahl, and M. J. C. van Gemert, "Two integrating spheres with an intervening scattering sample," J. Opt. Soc. Am. A 9, 621-631 (1992).
    [CrossRef]
  12. R. M. Pope and E. S. Fry, "Absorption spectrum (380-700 nm) of pure water: II. Integrating cavity measurements," Appl. Opt. 36, 8710-8723 (1997).
    [CrossRef]
  13. E. S. Fry and J. Musser, are preparing a paper to be called "A new ultrahigh diffuse reflecting material."
  14. A. Arecchi, SphereOptics LLC, Contoocook, N. H., 03229 (personal communication, slide presentation, 2005).
  15. K. M. Case and P. F. Zweifel, Linear Transport Theory (Addison-Wesley, 1967), p. 56.

2002 (1)

C. J.-Y. Lerebourg, D. A. Pilgrim, G. D. Ludbrook, and R. Neal, "Development of a point source integrating cavity absorption meter," J. Opt. A 4, S56-S65 (2002).
[CrossRef]

2000 (1)

1999 (1)

1997 (2)

1995 (1)

1992 (2)

1978 (1)

A. M. Emel'yanov, V. I. Kosyakov, and B. V. Makushkin, "The use of an integrating cavity for measuring small optical absorptions," Sov. J. Opt. Technol. 45, 31-33 (1978).

1970 (1)

Arecchi, A.

A. Arecchi, SphereOptics LLC, Contoocook, N. H., 03229 (personal communication, slide presentation, 2005).

Beaulieu, J.

J. Beaulieu, A Guide to Integrating Sphere Theory and Applications (Labsphere, Inc., 1999), http://www.labsphere.com/knowledgebase.aspx.

Bogacz, S.

S. Bogacz, Integrating Sphere Design and Applications (SphereOptics LLC, 2004), http://www.sphereoptics.com/assets/sphere-optic-pdf/sphere-technical-guide.pdf.

Case, K. M.

K. M. Case and P. F. Zweifel, Linear Transport Theory (Addison-Wesley, 1967), p. 56.

Davis, C. O.

Downes, T. V.

Elterman, P.

Emel'yanov, A. M.

A. M. Emel'yanov, V. I. Kosyakov, and B. V. Makushkin, "The use of an integrating cavity for measuring small optical absorptions," Sov. J. Opt. Technol. 45, 31-33 (1978).

Fry, E. S.

Hobbs, D. M.

Kattawar, G. W.

Kirk, J. T. O.

Kosyakov, V. I.

A. M. Emel'yanov, V. I. Kosyakov, and B. V. Makushkin, "The use of an integrating cavity for measuring small optical absorptions," Sov. J. Opt. Technol. 45, 31-33 (1978).

Leathers, R. A.

Lerebourg, C. J.-Y.

C. J.-Y. Lerebourg, D. A. Pilgrim, G. D. Ludbrook, and R. Neal, "Development of a point source integrating cavity absorption meter," J. Opt. A 4, S56-S65 (2002).
[CrossRef]

Ludbrook, G. D.

C. J.-Y. Lerebourg, D. A. Pilgrim, G. D. Ludbrook, and R. Neal, "Development of a point source integrating cavity absorption meter," J. Opt. A 4, S56-S65 (2002).
[CrossRef]

Makushkin, B. V.

A. M. Emel'yanov, V. I. Kosyakov, and B. V. Makushkin, "The use of an integrating cavity for measuring small optical absorptions," Sov. J. Opt. Technol. 45, 31-33 (1978).

McCormick, N. J.

Moes, C. J. M.

Musser, J.

E. S. Fry and J. Musser, are preparing a paper to be called "A new ultrahigh diffuse reflecting material."

Neal, R.

C. J.-Y. Lerebourg, D. A. Pilgrim, G. D. Ludbrook, and R. Neal, "Development of a point source integrating cavity absorption meter," J. Opt. A 4, S56-S65 (2002).
[CrossRef]

Pickering, J. W.

Pilgrim, D. A.

C. J.-Y. Lerebourg, D. A. Pilgrim, G. D. Ludbrook, and R. Neal, "Development of a point source integrating cavity absorption meter," J. Opt. A 4, S56-S65 (2002).
[CrossRef]

Pope, R. M.

Prahl, S. A.

Sterenborg, H. J. C. M.

van Gemert, M. J. C.

Zweifel, P. F.

K. M. Case and P. F. Zweifel, Linear Transport Theory (Addison-Wesley, 1967), p. 56.

Appl. Opt. (7)

J. Opt. A (1)

C. J.-Y. Lerebourg, D. A. Pilgrim, G. D. Ludbrook, and R. Neal, "Development of a point source integrating cavity absorption meter," J. Opt. A 4, S56-S65 (2002).
[CrossRef]

J. Opt. Soc. Am. A (1)

Sov. J. Opt. Technol. (1)

A. M. Emel'yanov, V. I. Kosyakov, and B. V. Makushkin, "The use of an integrating cavity for measuring small optical absorptions," Sov. J. Opt. Technol. 45, 31-33 (1978).

Other (5)

J. Beaulieu, A Guide to Integrating Sphere Theory and Applications (Labsphere, Inc., 1999), http://www.labsphere.com/knowledgebase.aspx.

S. Bogacz, Integrating Sphere Design and Applications (SphereOptics LLC, 2004), http://www.sphereoptics.com/assets/sphere-optic-pdf/sphere-technical-guide.pdf.

E. S. Fry and J. Musser, are preparing a paper to be called "A new ultrahigh diffuse reflecting material."

A. Arecchi, SphereOptics LLC, Contoocook, N. H., 03229 (personal communication, slide presentation, 2005).

K. M. Case and P. F. Zweifel, Linear Transport Theory (Addison-Wesley, 1967), p. 56.

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Figures (10)

Fig. 1
Fig. 1

(Color online) Experimental data for the temporal response at 266   nm of a 5   cm diameter integrating cavity. The input pulse is 11   ns (FWHM) and is supplied to the cavity via a 225 μ m diameter quartz optical fiber. The output pulse is observed via a second identical fiber; the output pulse peak is adjusted to make it comparable to the input pulse. The decay of the output pulse gives a wall reflectivity of 0.9964.

Fig. 2
Fig. 2

(Color online) Theoretical results for an integrating cavity with a diameter of 20 cm and a wall reflectivity of 0.999. The input pulse is assumed Gaussian with a 10 ns FWHM. The cavity is assumed to be filled with media whose absorption coefficients are 0, 5 × 10 - 6 / cm , and 10 - 5 / cm , respectively. These results indicate that a measurement of the exponential decay curves would provide a determination of the absorption coefficients to the order of a few times 10 - 6 / cm .

Fig. 3
Fig. 3

(Color online) Monte Carlo simulation of the temporal decay of the irradiance at the wall of a spherical cavity, which is filled with homogeneous isotropic radiation at time t = 0 as a function of distance s traveled by the photons.

Fig. 4
Fig. 4

(Color online) Monte Carlo simulation of the irradiance at the wall of a spherical cavity in which a pulse of radiation is injected from the cavity wall with a 2 π   sr isotropic angular distribution at time t = 0 , as a function of distance s traveled by the photons as measured by a detector on the cavity wall at 90° to the incident beam. For s < 1.445 , the results are independent of ρ, consequently the individual curves in (a) are not labeled.

Fig. 5
Fig. 5

Schematic for photon flux incident on the wall of a cavity.

Fig. 6
Fig. 6

(Color online) Cross sections of the spherical shell cavity; (a) region i is used in the computation of path lengths from outer surface to outer surface where θ c θ π / 2 , (b) region ii is used in the computation of path lengths from the outer surface to the inner surface where 0 θ θ c , (c) region iii is used in the computation of path lengths from the inner surface to the outer surface 0 θ π / 2 .

Fig. 7
Fig. 7

Schematic of a right circular cylindrical cavity.

Fig. 8
Fig. 8

Cross section of cylindrical cavity at height z 0 . The source point P is on the wall a distance z 0 above the base. The end point Q is also on the wall; it is displaced from this cross section by ( z z 0 ) .

Fig. 9
Fig. 9

Base of cylindrical cavity. The source point P is on the wall a distance z 0 above this cross section. The end point Q is on the base ( z = 0 ) a distance ρ from the cylinder axis.

Fig. 10
Fig. 10

Cross section of cylindrical cavity. The source point P is on the base ( z = 0 ) at a distance ρ 0 from the axis. The end point Q is on the top ( z = H ) at a distance ρ from the cylinder axis.

Tables (2)

Tables Icon

Table 1 Time Constant τMCU for the Decay of a Homogeneous Isotropic Radiation Field in a Spherical Cavity of Diameter 2 u ( d ¯ = 4∕3)

Tables Icon

Table 2 Time Constant τMCP for the Scenario in Which a Pulse of Radiation is Injected into the Cavity from a Point on the Cavity Wall

Equations (73)

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t ¯ = d ¯ c .
E I ( t ) = E 0 e t / τ ,
E L ( t ) = ρ E 0 e t / τ .
E I ( τ ) = E 0 ρ n , E L ( τ ) = E 0 ρ n + 1 .
E I ( τ ) = E 0 e 1 , E L ( τ ) = ρ E 0 e 1 .
n = 1 ln   ρ , τ i = 1 ln ( ρ ) t ¯ = 1 ln ( ρ ) d ¯ c
E I ( t + t ¯ ) = E I ( t ) + k = 1 t ¯ k k ! d k E I ( t ) d t k .
E I ( t + t ¯ ) = E I ( t ) + k = 1 1 k ! d k E I ( t ) d ξ k .
k = 1 1 k ! d k E L ( t ) d ξ k + E I ( t ) ( 1 ρ ) = 0.
k = 1 1 k ! γ k + ( 1 ρ ) = 0.
e γ ρ = 0 ,
E I ( t ) = E 0 e γ t / t ¯ = E 0 e t   ln   ρ / t ¯ .
τ = t ¯ ln   ρ ,
E I ( τ ) = E 0 ρ n 1 , E L ( τ ) = E 0 ρ n .
n = 1 1 ln   ρ , τ i i = ( 1 1 ln ( ρ ) ) t ¯ = ( 1 1 ln ( ρ ) ) d ¯ c
d ¯ = 4 R 2 3 ( 1 β 3 1 + β 2 ) ,
d ¯ = 2 D H 2 H + D .
p u = 1 4 π d μ d φ .
d N u = n ( c Δ t ) ( μ d S ) ( 1 4 π d μ d φ ) = ( n c 4 π ) ( μ d μ d φ ) d S d t .
n V ( d ¯ / c ) S = n c d ¯ V S .
p r = 1 π  μ d μ d φ ,
d N r = ( n c d ¯ V S ) ( 1 π μ d μ d φ ) d S d t = ( n c π d ¯ V S ) ( μ d μ d φ ) d S d t .
d ¯ = 4 V S .
( V S ) cyl = 1 2 ( D H 2 H + D ) .
d ¯ = 4 ( V S ) cyl = 2 D H 2 H + D ,
i = 1 M V i = V , i = 1 M S i = S .
s i ( r ) = S i r 2 R 2 .
V i = 0 R s i ( r ) d r = S i D 6 .
V S = D 6 ,
d ¯ = 2 3 D .
p ( μ , ϕ ) d μ d ϕ = μ d μ d ϕ π ,
0 2 π d ϕ 0 1 p ( μ , ϕ ) d μ = 1.
d ¯ i j = 1 π 0 2 π d ϕ μ 2 μ 1 l μ d μ ,
sin   θ c = R 1 R 2 = D 1 D 2 β cos   θ c = ( 1 β 2 ) 1 / 2 = μ c .
l = D 2 μ ,
d ¯ I I I I = D 2 π 0 2 π d ϕ 0 μ c μ 2 d μ = 2 D 2 μ c 3 3 .
l = R 2 ( μ ( μ 2 1 + β 2 ) 1 / 2 ) = R 2 ( μ ( μ 2 μ c 2 ) 1 / 2 ) .
   d ¯ I I I = 1 π 0 2 π d ϕ μ c 1 l μ d μ = D 2 3 ( 1 μ c 3 β 3 ) .
l = R 2 { μ β + [ β 2 ( μ 2 1 ) + 1 ] 1 / 2 } .
d ¯ I I I = 1 π 0 2 π d ϕ 0 1 l μ d μ = D 2 3 ( β + 1 β 2 ( 1 μ c 3 ) ) .
d ¯ = 2 D 2 3 ( 1 β 3 1 + β 2 ) .
d A j r 2 | n ^ j Q r | r .
| n ^ i P r | r .
d p j = 1 π d A j r 2 | n ^ j Q r | r | n ^ i P r | r ,
d ¯ P j = j r d p j = 1 π j 1 r | n ^ j Q r | r | n ^ i P r | r d A j .
d ¯ i j = 1 A i i d ¯ P j d A i = 1 π A i i d A i × j 1 r | n ^ j Q r | r | n ^ i P r | r d A j .
A I = π D H ; A I I = A I I I = π D 2 4 .
n ^ I · r I r I = R ( 1 cos   φ ) [ 2 R 2 ( 1 cos   φ ) + ( z z 0 ) 2 ] 1 / 2 .
d ¯ P I = R 3 π H 2 0 2 π d φ 0 1 d ζ ( 1 cos   φ ) 2 [ 1 2 α 2 ( 1 cos   φ ) + ( ζ ν ) 2 ] 3 / 2 .
d ¯ I I = 1 A I 0 H d ¯ P I 2 π R d z 0 = 1 H 0 H d ¯ P I d z 0 = 0 1 d ¯ P I d ν ,
d ¯ I I = R 3 π H 2 0 2 π d φ 0 1 d ζ 0 1 d ν × ( 1 cos   φ ) 2 [ 1 2 α 2 ( 1 cos   φ ) + ( ζ ν ) 2 ] 3 / 2 .
d ¯ I I = D π 0 2 π sin 2 φ 2 ( 1 + α 2 sin 2 φ 2 ) 1 / 2 d φ 8 D α 3 π
n ^ I · r II r II = R ρ   cos   φ ( R 2 + ρ 2 2 ρ R   cos   φ + z 0 2 ) 1 / 2 ,
n ^ II · r II r II = z 0 ( R 2 + ρ 2 2 ρ R   cos   φ + z 0 2 ) 1 / 2 .
d ¯ P II = α 3 8 π 0 2 π d φ 0 1 d η η ( 1 η   cos   φ ) z 0 [ 1 4 α 2 ( 1 + η 2 2 η cos φ ) + ν 2 ] 3 / 2 .
d ¯ I II = α 2 D 8 π 0 2 π d φ 0 1 d η 0 1 d ν × η ν ( 1 η   cos   φ ) [ 1 4 α 2 ( 1 + η 2 2 η cos φ ) + ν 2 ] 3 / 2 .
d ¯ I II = 2 α D 3 π α 2 D 4 π 0 2 π d φ 0 1 d η × η ( 1 η   cos   φ ) [ 4 + α 2 ( 1 + η 2 2 η cos φ ) ] 1 / 2 .
n ^ II · r III r III = n ^ III · r III r III = H ( ρ 2 + ρ 0 2 2 ρ ρ 0   cos   φ + H 2 ) 1 / 2 .
d ¯ P III = α 2 H 4 π 0 2 π d φ 0 1 d η × η [ 1 4 α 2 ( η 2 + μ 2 2 η μ   cos   φ ) + 1 ] 3 / 2 .
d ¯ II III = 1 π R 2 0 R d ¯ P III 2 π ρ 0 d ρ 0 = 2 0 1 d ¯ P III μ d μ .
d ¯ II III = α 2 H 2 π 0 2 π d φ 0 1 d η 0 1 d μ × η μ [ 1 4 α 2 ( η 2 + μ 2 2 η μ   cos   φ ) + 1 ] 3 / 2 .
d ¯ II III = 2 H 4 H π 0 2 π d φ 0 1 d η × η ( 4 + α 2 η 2 α 2 η   cos   φ ) [ 4 + α 2 η 2 + α 2 ( 1 2 η   cos   φ ) ] 1 / 2 ( 4 + α 2 η 2 sin 2 φ ) .
d ¯ w = d ¯ I I + d ¯ I II + d ¯ I III .
d ¯ B = d ¯ II I + d ¯ II III .
d ¯ T = d ¯ III I + d ¯ III II .
d ¯ II I = A I A II d ¯ I II .
P W = A I A I + 2 A II = 2 π R H 2 π R H + 2 π R 2 = H H + R = 2 α + 2 ,
P B = P T = A II A I + 2 A II = π R 2 2 π R H + 2 π R 2 = R 2 H + 2 R = 1 2 α α + 2 .
d ¯ = d ¯ W P W + d ¯ B P B + d ¯ T P T = d ¯ W P W + 2 d ¯ B P B = ( d ¯ I I + 2 d ¯ I II ) 2 α + 2 + ( A I A II d ¯ I II + d ¯ II III ) α α + 2 ,
d ¯ = 2 α + 2 d ¯ I I + ( 4 α + 2 + A I A II α α + 2 ) d ¯ I II + α α + 2 d ¯ II I II = 2 α + 2 d ¯ I I + 8 α + 2 d ¯ I II + α α + 2 d ¯ II III .
d ¯ = 2 D α + 2 + 1 π 2 D α + 2 { 0 2 π sin 2 φ 2 ( 1 + α 2 sin 2 φ 2 ) 1 / 2 d φ α 2 0 2 π d φ × 0 1 d η η ( 1 η   cos   φ ) ( 4 + α 2 + α 2 η 2 2 α 2 η   cos   φ ) 1 / 2 2 0 2 π d φ × 0 1 d η η ( 4 + α 2 η 2 α 2 η   cos   φ ) ( 4 + α 2 + α 2 η 2 2 α 2 η   cos   φ ) 1 / 2 ( 4 + α 2 η 2 sin 2 φ ) } .
d ¯ = 2 D α + 2 = 2 D H 2 H + D .
d ¯ D = H = 2 3 D .

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