Abstract

This paper deals with limitations and often overlooked sources of error introduced in compact double-beam goniophotometers. It is shown that relative errors in measured radiance factor, comparable to the total measurement uncertainty, can be introduced if recommended corrections are not carried out. Two different error sources are investigated, both related to the size of the detection solid angle. The first is a geometrical error that occurs when the size of the illuminated area and detector aperture are comparable to the distance between them. The second is a convolution error due to variations in radiant flux over the detector aperture, which is quantified by simulating the full 3D bidirectional reflectance distribution function (BRDF) of a set of samples with different degrees of anisotropic reflectance. The evaluation is performed for a compact double-beam goniophotometer using different detection solid angles, and it is shown that both error sources introduce relative errors of 1%–3%, depending on viewing angle and optical properties of the sample. Commercially available compact goniophotometers, capable of absolute measurements, are becoming more and more common, and the findings in this paper are therefore important for anyone using or planning to use this type of instrument.

© 2014 Optical Society of America

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References

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2013 (2)

2011 (2)

2010 (3)

2009 (1)

R. Baribeau, W. S. Neil, and E. Côté, “Development of a robot-based gonioreflectometer for spectral BRDF measurements,” J. Mod. Opt. 56, 1497–1503 (2009).
[CrossRef]

2008 (1)

P. Edström, “A two-phase parameter estimation method for radiative transfer problems in paper industry applications,” Inverse Probl. Sci. Eng. 16, 927–951 (2008).
[CrossRef]

2007 (1)

2006 (1)

D. Hünerhoff, U. Grusemann, and A. Höpe, “New robot-based gonioreflectometer for measuring spectral diffuse reflection,” Metrologia 43, S11–S16 (2006).
[CrossRef]

2005 (3)

M. Pointer, N. J. Barnes, P. J. Clarke, and M. J. Shaw, “A new goniospectrophotometer for measuring gonio-apparent materials,” Color. Technol. 121, 96–103 (2005).
[CrossRef]

G. Obein, R. Bousquet, and M. E. Nadal, “New NIST reference goniospectrometer,” Proc. SPIE 5880, 241–250 (2005).
[CrossRef]

P. Edström, “A fast and stable solution method for the radiative transfer problem,” SIAM Rev. 47, 447–468 (2005).
[CrossRef]

2004 (1)

2003 (1)

P. van Nijnatten, “An automated directional reflectance/transmittance analyser for coating analysis,” Thin Solid Films 442, 74–79 (2003).
[CrossRef]

1999 (2)

R. Drezek, A. Dunn, and R. Richards-Kortum, “Light scattering from cells: finite-difference time-domain simulations and goniometric measurements,” Appl. Opt. 38, 3651–3661 (1999).
[CrossRef]

D. Williams, “Establishment of absolute diffuse reflectance scales using the NPL reference reflectometer,” Anal. Chim. Acta 380, 165–172 (1999).
[CrossRef]

Baribeau, R.

R. Baribeau, W. S. Neil, and E. Côté, “Development of a robot-based gonioreflectometer for spectral BRDF measurements,” J. Mod. Opt. 56, 1497–1503 (2009).
[CrossRef]

Barnes, N. J.

M. Pointer, N. J. Barnes, P. J. Clarke, and M. J. Shaw, “A new goniospectrophotometer for measuring gonio-apparent materials,” Color. Technol. 121, 96–103 (2005).
[CrossRef]

Bousquet, R.

G. Obein, R. Bousquet, and M. E. Nadal, “New NIST reference goniospectrometer,” Proc. SPIE 5880, 241–250 (2005).
[CrossRef]

Campos, J.

Chorro, E.

Clarke, P. J.

M. Pointer, N. J. Barnes, P. J. Clarke, and M. J. Shaw, “A new goniospectrophotometer for measuring gonio-apparent materials,” Color. Technol. 121, 96–103 (2005).
[CrossRef]

Coppel, L. G.

Côté, E.

R. Baribeau, W. S. Neil, and E. Côté, “Development of a robot-based gonioreflectometer for spectral BRDF measurements,” J. Mod. Opt. 56, 1497–1503 (2009).
[CrossRef]

Delafosse, D.

N. Matsapey, J. Faucheu, M. Flury, and D. Delafosse, “Design of a gonio-spectro-photometer for optical characterization of gonio-apparent materials,” Meas. Sci. Technol. 24, 065901 (2013).
[CrossRef]

Ding, Z.

Drezek, R.

Dunn, A.

Edstrom, P.

Edström, P.

Faucheu, J.

N. Matsapey, J. Faucheu, M. Flury, and D. Delafosse, “Design of a gonio-spectro-photometer for optical characterization of gonio-apparent materials,” Meas. Sci. Technol. 24, 065901 (2013).
[CrossRef]

Ferrero, A.

Flury, M.

N. Matsapey, J. Faucheu, M. Flury, and D. Delafosse, “Design of a gonio-spectro-photometer for optical characterization of gonio-apparent materials,” Meas. Sci. Technol. 24, 065901 (2013).
[CrossRef]

Geiser, M.

Ginsberg, I. W.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, and T. Limperis, Geometrical considerations and nomenclature for reflectance, (National Bureau of Standards, 1977).

Grusemann, U.

D. Hünerhoff, U. Grusemann, and A. Höpe, “New robot-based gonioreflectometer for measuring spectral diffuse reflection,” Metrologia 43, S11–S16 (2006).
[CrossRef]

Hauer, K.-O.

A. Höpe and K.-O. Hauer, “Three-dimensional appearance characterization of diffuse standard reflection materials,” Metrologia 47, 295–304 (2010).
[CrossRef]

Höpe, A.

A. Höpe and K.-O. Hauer, “Three-dimensional appearance characterization of diffuse standard reflection materials,” Metrologia 47, 295–304 (2010).
[CrossRef]

D. Hünerhoff, U. Grusemann, and A. Höpe, “New robot-based gonioreflectometer for measuring spectral diffuse reflection,” Metrologia 43, S11–S16 (2006).
[CrossRef]

Hsia, J. J.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, and T. Limperis, Geometrical considerations and nomenclature for reflectance, (National Bureau of Standards, 1977).

Hünerhoff, D.

D. Hünerhoff, U. Grusemann, and A. Höpe, “New robot-based gonioreflectometer for measuring spectral diffuse reflection,” Metrologia 43, S11–S16 (2006).
[CrossRef]

Ikonen, E.

Limperis, T.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, and T. Limperis, Geometrical considerations and nomenclature for reflectance, (National Bureau of Standards, 1977).

Luisa Hernanz, M.

Manoocheri, F.

Martinez-Verdu, F.

Matsapey, N.

N. Matsapey, J. Faucheu, M. Flury, and D. Delafosse, “Design of a gonio-spectro-photometer for optical characterization of gonio-apparent materials,” Meas. Sci. Technol. 24, 065901 (2013).
[CrossRef]

Modest, M. F.

M. F. Modest, Radiative Heat Transfer, 2nd ed. (Academic, 2003).

Nadal, M. E.

G. Obein, R. Bousquet, and M. E. Nadal, “New NIST reference goniospectrometer,” Proc. SPIE 5880, 241–250 (2005).
[CrossRef]

Neil, W. S.

R. Baribeau, W. S. Neil, and E. Côté, “Development of a robot-based gonioreflectometer for spectral BRDF measurements,” J. Mod. Opt. 56, 1497–1503 (2009).
[CrossRef]

Neuman, M.

Nevas, S.

Nicodemus, F. E.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, and T. Limperis, Geometrical considerations and nomenclature for reflectance, (National Bureau of Standards, 1977).

Obein, G.

G. Obein, R. Bousquet, and M. E. Nadal, “New NIST reference goniospectrometer,” Proc. SPIE 5880, 241–250 (2005).
[CrossRef]

Perales, E.

Pointer, M.

M. Pointer, N. J. Barnes, P. J. Clarke, and M. J. Shaw, “A new goniospectrophotometer for measuring gonio-apparent materials,” Color. Technol. 121, 96–103 (2005).
[CrossRef]

Pons, A.

Rabal, A.

Richards-Kortum, R.

Richmond, J. C.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, and T. Limperis, Geometrical considerations and nomenclature for reflectance, (National Bureau of Standards, 1977).

Shaw, M. J.

M. Pointer, N. J. Barnes, P. J. Clarke, and M. J. Shaw, “A new goniospectrophotometer for measuring gonio-apparent materials,” Color. Technol. 121, 96–103 (2005).
[CrossRef]

Stover, J. C.

J. C. Stover, Optical Scattering, 2nd ed. (SPIE, 1995).

van Nijnatten, P.

P. van Nijnatten, “An automated directional reflectance/transmittance analyser for coating analysis,” Thin Solid Films 442, 74–79 (2003).
[CrossRef]

Williams, D.

D. Williams, “Establishment of absolute diffuse reflectance scales using the NPL reference reflectometer,” Anal. Chim. Acta 380, 165–172 (1999).
[CrossRef]

Zhu, Y.

Anal. Chim. Acta (1)

D. Williams, “Establishment of absolute diffuse reflectance scales using the NPL reference reflectometer,” Anal. Chim. Acta 380, 165–172 (1999).
[CrossRef]

Appl. Opt. (3)

Chin. Opt. Lett. (1)

Color. Technol. (1)

M. Pointer, N. J. Barnes, P. J. Clarke, and M. J. Shaw, “A new goniospectrophotometer for measuring gonio-apparent materials,” Color. Technol. 121, 96–103 (2005).
[CrossRef]

Inverse Probl. Sci. Eng. (1)

P. Edström, “A two-phase parameter estimation method for radiative transfer problems in paper industry applications,” Inverse Probl. Sci. Eng. 16, 927–951 (2008).
[CrossRef]

J. Mod. Opt. (1)

R. Baribeau, W. S. Neil, and E. Côté, “Development of a robot-based gonioreflectometer for spectral BRDF measurements,” J. Mod. Opt. 56, 1497–1503 (2009).
[CrossRef]

J. Opt. Soc. Am. A (3)

Meas. Sci. Technol. (1)

N. Matsapey, J. Faucheu, M. Flury, and D. Delafosse, “Design of a gonio-spectro-photometer for optical characterization of gonio-apparent materials,” Meas. Sci. Technol. 24, 065901 (2013).
[CrossRef]

Metrologia (2)

A. Höpe and K.-O. Hauer, “Three-dimensional appearance characterization of diffuse standard reflection materials,” Metrologia 47, 295–304 (2010).
[CrossRef]

D. Hünerhoff, U. Grusemann, and A. Höpe, “New robot-based gonioreflectometer for measuring spectral diffuse reflection,” Metrologia 43, S11–S16 (2006).
[CrossRef]

Opt. Express (1)

Proc. SPIE (1)

G. Obein, R. Bousquet, and M. E. Nadal, “New NIST reference goniospectrometer,” Proc. SPIE 5880, 241–250 (2005).
[CrossRef]

SIAM Rev. (1)

P. Edström, “A fast and stable solution method for the radiative transfer problem,” SIAM Rev. 47, 447–468 (2005).
[CrossRef]

Thin Solid Films (1)

P. van Nijnatten, “An automated directional reflectance/transmittance analyser for coating analysis,” Thin Solid Films 442, 74–79 (2003).
[CrossRef]

Other (4)

Joint Committee for Guides in Metrology, “Evaluation of measurement data—guide to the expression of uncertainty in measurement,” (2008).

J. C. Stover, Optical Scattering, 2nd ed. (SPIE, 1995).

M. F. Modest, Radiative Heat Transfer, 2nd ed. (Academic, 2003).

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, and T. Limperis, Geometrical considerations and nomenclature for reflectance, (National Bureau of Standards, 1977).

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Figures (9)

Fig. 1.
Fig. 1.

Defining angles for bidirectional reflectance. Light incident from (θi,ϕi) and contained within the differential solid angle dωi is reflected from a surface element dAi along the direction (θr,ϕr) and into the differential solid angle dωr.

Fig. 2.
Fig. 2.

Radiative exchange between two finite surfaces. The total flux reflected from Ai and received by Ad is the sum of the exchange between each subsurface dAi and dAd.

Fig. 3.
Fig. 3.

Relative error in measured BRDF when using the projected solid angle instead of view factor as geometric configuration factor, for a small (5mm×5mm), medium (15mm×15mm), and large (25mm×25mm) detector aperture. The resulting error is of the same order as the overall measurement uncertainty already for medium-sized apertures.

Fig. 4.
Fig. 4.

BRDF of Spectralon (SRS) and paper samples (M1–M3). The BRDF of a perfect reflecting diffuser (shaded semicircle) has been included for reference. The paper samples exhibit aniostropic reflectance with a pronounced forward scattering.

Fig. 5.
Fig. 5.

Relative error in BRDF measurement due to anisotropic scattering over a medium-sized (detection half-angle 4.7°) detector aperture. The absolute error is largest at grazing viewing angles, where the BRDFs are over- and underestimated at backscattering and forward scattering measurements, respectively. The relative error is larger for the paper samples (M1–M3), due to the combined effect of low reflectance and high anisotropy.

Fig. 6.
Fig. 6.

Relative error in measured BRDF of (a) Spectralon (SRS) and (b) paper sample (M2), due to anisotropic scattering over detection solid angle. The error is calculated for a small (1.6°), medium (4.7°), and large (7.9°) detection half-angle. The error is virtually zero when using the small aperture, whereas a large aperture results in relative errors as large as 2%–3% at grazing viewing angles. Observe the different scales of the vertical axes.

Fig. 7.
Fig. 7.

Comparison between NRC and MSU measurements on a Spectralon diffuse reflectance standard, at λ=580nm and θi=45°. The remaining systematic source of error, after geometrical correction, is 10% and is most likely introduced by the modifications of the sample holder.

Fig. 8.
Fig. 8.

Repeatability of reflectance measurements on a Spectralon reflectance target with a nominal 8/d reflectance factor of 0.99. The relative standard deviation is calculated from N=5 measurements, for different wavelengths and viewing angles and using the instrument settings described in Section 3.D.

Fig. 9.
Fig. 9.

Errors caused by (a) rotational and (b) translational misalignment of sample holder. The primed letters and dashed lines correspond to a misaligned geometry. The rotational misalignment introduces an error α in both the angle of incidence θi and viewing angle θr. The translational misalignment δ affects the viewing angle by ϵ=θrθr, and introduces in a small shift of the illuminated area.

Tables (2)

Tables Icon

Table 1. Scattering and Absorption Coefficient σa and σa, Asymmetry Factor g, and Total Reflectance Rtot of Samples Used for Investigating the Influence of Anisotropic Scattering on Measurement Uncertainty

Tables Icon

Table 2. Uncertainty Components for Radiance Factor Measurement of a Spectralon Reflectance Standard (SRS) at λ=580nm and (θi,θr)=(45,45)

Equations (16)

Equations on this page are rendered with MathJax. Learn more.

fr(θi,ϕi,θr,ϕr)=dLr(θi,ϕi,θr,ϕr)dEi(θi,ϕi),
Lr(θr,ϕr)=ωifr(θi,ϕi,θr,ϕr)dEi(θi,ϕi)dωi.
fr(θi,ϕi,θr,ϕr)Lr(θi,ϕi,θr,ϕr)Ei(θi,ϕi),
Ei=dΦidAiΦiAi,
Lr=d2ΦrdAidωrcosθrΦrAiωdcosθr,
fr(θi,ϕi,θr,ϕr)=LrEiΦrΦir2cosθrAd,
ρ=ωdfrcosθdω=Sfrcosθr2dA,
Φr=LrAiAdcosθrcosθdr2dAddAi,
FAiAd=1AiπAiAdcosθrcosθdr2dAddAi.
fr(θi,ϕi,θr,ϕr)=LrEiΦrΦi1FAiAdπ.
β=fr·π,
[uc(β)β]2=[u(ρ)ρ]2+[2u(r)r]2+[u(Ad)Ad]2+[u(θr)tanθr]2,
u(ρ)2=u(ρ)Rep2+u(ρ)Stray2+u(ρ)NL2.
u(ρ)Rep=(N1)s+2+(N1)s22N2,
u(Ad)Ad=(u(h)h)2+(u(w)w)2,
2u(θr)2=u(θd)2+u(θs)2+u(ϵ)2,

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