Integrating Sphere for Imperfectly Diffuse Samples*

D. K. Edwards; J. T. Gier; K. E. Nelson; R. D. Roddick

doi:10.1364/JOSA.51.001279

Journal of the Optical Society of America
Vol. 51,
Issue 11,
pp. 1279-1288
(1961)
•https://doi.org/10.1364/JOSA.51.001279

Integrating Sphere for Imperfectly Diffuse Samples

D. K. Edwards, J. T. Gier, K. E. Nelson, and R. D. Roddick

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D. K. Edwards, J. T. Gier, K. E. Nelson, and R. D. Roddick, "Integrating Sphere for Imperfectly Diffuse Samples*," J. Opt. Soc. Am. 51, 1279-1288 (1961)

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Abstract

An integrating sphere for determining spectral reflectance and transmittance as a function of angle of incidence and wavelength in the 0.33- to 2.5-μ region is described. Geometrical arrangement of sample, entrance port, and detector as well as directional characteristics of detector and sphere wall coating permit absolute or relative measurements to be made for a sample with an arbitrary reflection-distribution function.

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Equations (28)

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Symbol	Description	Typical units
A₁	Area of radiation source in sphere	cm²
A₂	Area viewed by detector	cm²
A_D	Area of detector	cm²
A_E	Area of exit slit	cm²
A_S	Irradiated area of sample	cm²
A_W	Area of sphere wall	cm²
f	Reflection-distribution function of sample	steradians⁻¹
K	Detector proportionality constant	volts/watt
$\bar{K}$	Average detector proportionality constant, Eq. (19)	volts/watt
N₁	Radiance of A₁	watts/cm² steradian
N₂	Radiance of area viewed by detector	watts/cm² steradian
N_E	Radiance of exit slit (after reflections by M9 and M10)	watts/cm² steradian
N_W	Radiance of sphere wall element	watts/cm² steradian
p	Direct irradiation factor, Eq. (13)
P	Radiant flux of source in sphere	watts
P_R	Radiant flux of source in sphere when reference sample is irradiated	watts
P_S	Radiant flux of source in sphere when sample is irradiated	watts
P_W	Radiant flux of source in sphere when wall reference area is irradiated	watts
q	Directional sensitivity factor, Eq. (20)
r_DS	Distance from detector element to sample element	cm
r_DW	Distance from detector element to sphere element	cm
V_R	Detector output when reference sample irradiated	volts
V_S	Detector output when sample irradiated	volts
V_W	Detector output when sphere wall irradiated	volts
θ_D	Polar angle of incidence of detector	radians
θ₀	Polar angle of incidence of sample	radians
θ_W	Polar angle for sphere wall element	radians
θ_S	Polar angle of reflection from sample	radians
λ	Wavelength	microns
ρ_D	Reflectance of detector
ρ_W	Reflectance for sphere wall for first reflection
${\bar{ρ}}_{W}$	Average reflectance of sphere for inter-reflections
ρ_R	Reflectance of reference
ρ_S	Reflectance of sample
ρ_WS	Average reflectance of sphere wall for first reflection when sample irradiated
ρ_WW	Average reflectance of sphere wall for reflection when wall reference irradiated
ϕ_D	Azimuthal angle of incidence of detector	radians
ϕ₀	Azimuthal angle of incidence of sample	radians
ϕ_W	Azimuthal angle for sphere wall element	radians
ϕ_S	Azimuthal angle of reflection from sample	radians
Ω_ES	Solid angle of the irradiated portion of the sample as viewed from the exit slit through the optics	steradians

		Magnitude of error
Source of error	Violation of assumption	Arbitrary f(θ,ϕ;θ′,ϕ′)	Specular sample	Perfectly diffuse sample	Comment
1. Direct irradiation of detector by sample	That source area A₂ is only on sphere wall in Eq. (3)	Eq. (15)	0	+0.01	Normally negligible
2. Directional sensitivity of detectors	That K≠K(θ,ϕ) in Eq. (2)	Eq. (21)	±0.001	−0.002	At 2 μ
3. Nonuniformity of sphere wall coating	That ρ_W is the same in Eqs. (6)–(8)	$\frac{ρ_{W S} - ρ_{W W}}{ρ_{W W}}$	See Comment	−0.004	Must avoid cracks, chips, and entrance port on sphere wall for specular sample.
4. Nondiffuseness of sphere wall coating	That N_W≠N_W(θ_W,ϕ_W) in Eq. (2)		See Comment	0	Masked by error 2 above
5. Interreflections with sample	That ρ_W is the same in Eqs. (6)–(8)		0	0	Error avoided by proper procedure
6. Shadowing by sample holder	That source area A₂ is only on sphere wall and entire sphere wall is visible to detector in Eq. (3)		0	0
7. Nonlinearity of detection system	That K is independent of N_W in Eq. (2)		±0.005	±0.005	Error more random than systematic
8. Polarization by monochromator optics	That N_E is unpolarized in Eqs. (6)–(8)				Error avoided by proper procedure
9. Incompletely unidirectional irradiation	That Ω_ES is infinitesimal			0	Small where d²ρ/dθ² is small.

Symbol	Description	Typical units
A₁	Area of radiation source in sphere	cm²
A₂	Area viewed by detector	cm²
A_D	Area of detector	cm²
A_E	Area of exit slit	cm²
A_S	Irradiated area of sample	cm²
A_W	Area of sphere wall	cm²
f	Reflection-distribution function of sample	steradians⁻¹
K	Detector proportionality constant	volts/watt
$\bar{K}$	Average detector proportionality constant, Eq. (19)	volts/watt
N₁	Radiance of A₁	watts/cm² steradian
N₂	Radiance of area viewed by detector	watts/cm² steradian
N_E	Radiance of exit slit (after reflections by M9 and M10)	watts/cm² steradian
N_W	Radiance of sphere wall element	watts/cm² steradian
p	Direct irradiation factor, Eq. (13)
P	Radiant flux of source in sphere	watts
P_R	Radiant flux of source in sphere when reference sample is irradiated	watts
P_S	Radiant flux of source in sphere when sample is irradiated	watts
P_W	Radiant flux of source in sphere when wall reference area is irradiated	watts
q	Directional sensitivity factor, Eq. (20)
r_DS	Distance from detector element to sample element	cm
r_DW	Distance from detector element to sphere element	cm
V_R	Detector output when reference sample irradiated	volts
V_S	Detector output when sample irradiated	volts
V_W	Detector output when sphere wall irradiated	volts
θ_D	Polar angle of incidence of detector	radians
θ₀	Polar angle of incidence of sample	radians
θ_W	Polar angle for sphere wall element	radians
θ_S	Polar angle of reflection from sample	radians
λ	Wavelength	microns
ρ_D	Reflectance of detector
ρ_W	Reflectance for sphere wall for first reflection
${\bar{ρ}}_{W}$	Average reflectance of sphere for inter-reflections
ρ_R	Reflectance of reference
ρ_S	Reflectance of sample
ρ_WS	Average reflectance of sphere wall for first reflection when sample irradiated
ρ_WW	Average reflectance of sphere wall for reflection when wall reference irradiated
ϕ_D	Azimuthal angle of incidence of detector	radians
ϕ₀	Azimuthal angle of incidence of sample	radians
ϕ_W	Azimuthal angle for sphere wall element	radians
ϕ_S	Azimuthal angle of reflection from sample	radians
Ω_ES	Solid angle of the irradiated portion of the sample as viewed from the exit slit through the optics	steradians

		Magnitude of error
Source of error	Violation of assumption	Arbitrary f(θ,ϕ;θ′,ϕ′)	Specular sample	Perfectly diffuse sample	Comment
1. Direct irradiation of detector by sample	That source area A₂ is only on sphere wall in Eq. (3)	Eq. (15)	0	+0.01	Normally negligible
2. Directional sensitivity of detectors	That K≠K(θ,ϕ) in Eq. (2)	Eq. (21)	±0.001	−0.002	At 2 μ
3. Nonuniformity of sphere wall coating	That ρ_W is the same in Eqs. (6)–(8)	$\frac{ρ_{W S} - ρ_{W W}}{ρ_{W W}}$	See Comment	−0.004	Must avoid cracks, chips, and entrance port on sphere wall for specular sample.
4. Nondiffuseness of sphere wall coating	That N_W≠N_W(θ_W,ϕ_W) in Eq. (2)		See Comment	0	Masked by error 2 above
5. Interreflections with sample	That ρ_W is the same in Eqs. (6)–(8)		0	0	Error avoided by proper procedure
6. Shadowing by sample holder	That source area A₂ is only on sphere wall and entire sphere wall is visible to detector in Eq. (3)		0	0
7. Nonlinearity of detection system	That K is independent of N_W in Eq. (2)		±0.005	±0.005	Error more random than systematic
8. Polarization by monochromator optics	That N_E is unpolarized in Eqs. (6)–(8)				Error avoided by proper procedure
9. Incompletely unidirectional irradiation	That Ω_ES is infinitesimal			0	Small where d²ρ/dθ² is small.

Abstract

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Figures (12)

Tables (2)

Equations (28)

Journal of the Optical Society of America