John W. Pickering, Christian J. M. Moes, H. J. C. M. Sterenborg, Scott A. Prahl, and Martin J. C. van Gemert, "Two integrating spheres with an intervening scattering sample," J. Opt. Soc. Am. A 9, 621-631 (1992)
Two integrating spheres placed so that the exit port of one and the entry port of the other are adjacent, with only a sample intervening, will permit the simultaneous determination of the reflectance and the transmittance of the sample. Such a geometry permits measurements to be made as the sample undergoes some external stimulation, such as heat, pressure, or a chemical change. To determine the sample reflectance and the transmittance from the measured values of irradiance within each sphere requires the calculation of the exchange of light through the sample between the spheres. First the power collected by a detector situated in the wall of an integrating sphere is calculated as a function of the area and the reflectance of the wall, the holes, the sample, and the detector for both diffuse and collimated light incident upon the sample and for a sample located at either the exit port (reflectance) or the entry port (transmittance) of the sphere. Next, by using the single-sphere equations, we calculate the effect of the multiple exchange of light between two integrating spheres arranged so that the sample is placed between them. In all the cases of two integrating spheres the power detected is greater than or equal to that for the single sphere and depends on both the reflection and the transmission properties of the sample. Additionally, the effect of a baffle placed between the sample and the detector or of a nonisotropic detector is to reduce the power detected.
John W. Pickering, Scott A. Prahl, Niek van Wieringen, Johan F. Beek, Henricus J. C. M. Sterenborg, and Martin J. C. van Gemert Appl. Opt. 32(4) 399-410 (1993)
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Diffuse reflection factor of the sample with diffuse incident light
Rc
Collimated (specular) reflection factor of the sample with collimated incident light
Rcd
Diffuse reflection factor of the sample with collimated incident light
Td
Diffuse transmission factor of the sample with diffuse incident light
Tc
Collimated transmission factor of the sample with collimated incident light
Tcd
Diffuse transmission factor of the sample with collimated incident light
ℜ
Radius of the sphere
A = 4πℜ2
Total sphere area
αA
Area of the sphere wall
δ
Area of the detector
s
Area of the sample
α = 1 − (δ/A + s/A + h/A)
Area of the sphere wall relative to the total sphere area
h
Area of the holes
When a second sphere is introduced the properties of this sphere are labeled with a prime (e.g., m′, A′).
Table 2
Calculation of Detector Power for a Single Sphere with Light Initially Incident upon the Sphere Wall
Reflection
Wall Reflection
Sample Reflection
Wall Collection
Sample Collection
Lost through Holes
Detector Collection
1
mP
0
αmp
2
mαmp
3
⋮
⋮
⋮
⋮
⋮
⋮
⋮
n
⋮
⋮
⋮
⋮
⋮
⋮
⋮
For the sake of simplifying the equations and the table, we neglect the contribution of the light reflected from the detector [r(δ/A)] since it is negligible compared with the contributions of the light reflected from the walls (mα) and that from the sample [Rd(s/A)]. For any row the sum of the reflected light (columns 2 and 3) will equal the sum of the collected and the lost light (columns 4–7). The total power detected is the sum of the terms in column 7 (for n = 1 to ∞).
Table 3
Calculation of Detector Power for a Double Spheres with Light Initially Incident upon the Sample Walla
Sphere 1
Sphere 2
Exchange
Sample Collection
Detector Collection
Sample Collection
Detector Collection
Comments
0
↘
Let
0′
↙
Let
1
↘
Let
and
1′
↙
⋮
⋮
⋮
⋮
⋮
⋮
n
↘
n′
↙
⋮
⋮
⋮
⋮
⋮
⋮
A portion (Td) of the light collected by the sample in each sphere is transmitted (in the direction of the arrows) to the other sphere, where some of it is incident upon the detectors (columns 3 and 5). The total power detected in the reflectance sphere is the sum of the terms in column 3, while the total power detected in the transmittance sphere is the sum of the terms in column 5.
For the sake of simplicity the equations for the baffle neglect the small contribution to the power of the sphere that is due to the reflected light from the detector since this light [r(δ/A)] is negligible relative to the light reflected by the wall or sample. Also note that T = (s/A) {Td/1 − [mα + Rd(s/A) + r(δ/A)]} and T′ = (s/A′) {Td/1 − [m′α′ + Rd(s/A′) + r′(δ′/A′)]}.
Diffuse reflection factor of the sample with diffuse incident light
Rc
Collimated (specular) reflection factor of the sample with collimated incident light
Rcd
Diffuse reflection factor of the sample with collimated incident light
Td
Diffuse transmission factor of the sample with diffuse incident light
Tc
Collimated transmission factor of the sample with collimated incident light
Tcd
Diffuse transmission factor of the sample with collimated incident light
ℜ
Radius of the sphere
A = 4πℜ2
Total sphere area
αA
Area of the sphere wall
δ
Area of the detector
s
Area of the sample
α = 1 − (δ/A + s/A + h/A)
Area of the sphere wall relative to the total sphere area
h
Area of the holes
When a second sphere is introduced the properties of this sphere are labeled with a prime (e.g., m′, A′).
Table 2
Calculation of Detector Power for a Single Sphere with Light Initially Incident upon the Sphere Wall
Reflection
Wall Reflection
Sample Reflection
Wall Collection
Sample Collection
Lost through Holes
Detector Collection
1
mP
0
αmp
2
mαmp
3
⋮
⋮
⋮
⋮
⋮
⋮
⋮
n
⋮
⋮
⋮
⋮
⋮
⋮
⋮
For the sake of simplifying the equations and the table, we neglect the contribution of the light reflected from the detector [r(δ/A)] since it is negligible compared with the contributions of the light reflected from the walls (mα) and that from the sample [Rd(s/A)]. For any row the sum of the reflected light (columns 2 and 3) will equal the sum of the collected and the lost light (columns 4–7). The total power detected is the sum of the terms in column 7 (for n = 1 to ∞).
Table 3
Calculation of Detector Power for a Double Spheres with Light Initially Incident upon the Sample Walla
Sphere 1
Sphere 2
Exchange
Sample Collection
Detector Collection
Sample Collection
Detector Collection
Comments
0
↘
Let
0′
↙
Let
1
↘
Let
and
1′
↙
⋮
⋮
⋮
⋮
⋮
⋮
n
↘
n′
↙
⋮
⋮
⋮
⋮
⋮
⋮
A portion (Td) of the light collected by the sample in each sphere is transmitted (in the direction of the arrows) to the other sphere, where some of it is incident upon the detectors (columns 3 and 5). The total power detected in the reflectance sphere is the sum of the terms in column 3, while the total power detected in the transmittance sphere is the sum of the terms in column 5.
For the sake of simplicity the equations for the baffle neglect the small contribution to the power of the sphere that is due to the reflected light from the detector since this light [r(δ/A)] is negligible relative to the light reflected by the wall or sample. Also note that T = (s/A) {Td/1 − [mα + Rd(s/A) + r(δ/A)]} and T′ = (s/A′) {Td/1 − [m′α′ + Rd(s/A′) + r′(δ′/A′)]}.