Abstract

The relative merits of four methods—energy balance, summation of reflections, inversion of the irradiance-transfer matrix, and solution of the integral equation—are compared by using each to determine irradiance in a multizone true sphere and in a sphere with a flat port; in the process several new solutions are presented. Although limited in applicability, the energy-balance method is by far the most direct. For the flat-port configuration the relationships among various published expressions are established; furthermore, the curved-surface interreflection irradiance is shown to be nonuniform when the initial irradiance is restricted to a part of the curved surface.

© 1998 Optical Society of America

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References

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  1. E. Karrer, “Use of the Ulbricht sphere in measuring reflection and transmission factors,” Sci. Pap. Bur. Stand. No. 415 (1921). Karrer derives the (1-ρ)-1 form by energy balance.
    [CrossRef]
  2. J. W. T. Walsh, Photometry (Constable, London, 1958). Walsh gives a simple energy-balance derivation due to Ulbricht in 1920.
  3. E. E. N. Mascart, “Sur la mesure de l’éclairement,” Lum. Elect.28, 180–187 (1888). Mascart uses summation of reflections to obtain the (1-ρ)-1 expression for the total flux in a room with diffusely reflecting surfaces.Karrer1 cites A. Palaz in Photométrie Industrielle (Carré, Paris, 1892) to the effect that Mascart was the first to derive this expression.
  4. A. H. Taylor, “The measurement of diffuse reflection factors and a new absolute reflectometer,” J. Opt. Soc. Am. 4, 9–23 (1920).
    [CrossRef]
  5. A. H. Taylor, “Errors in reflectometry,” J. Opt. Soc. Am. 25, 51–56 (1935).
    [CrossRef]
  6. P. Moon, The Scientific Basis of Illuminating Engineering (McGraw-Hill, New York, 1936), p. 331.
  7. J. A. Jacquez, H. F. Kuppenheim, “Theory of the integrating sphere,” J. Opt. Soc. Am. 45, 460–470 (1955).
    [CrossRef]
  8. B. J. Hisdal, “Reflectance of perfect diffuse and specular samples in the integrating sphere,” J. Opt. Soc. Am. 55, 1122–1128 (1965).
    [CrossRef]
  9. D. G. Goebel, “Generalized integrating-sphere theory,” Appl. Opt. 6, 125–128 (1967).
    [CrossRef] [PubMed]
  10. H. L. Tardy, “Matrix method for integrating sphere calculations,” J. Opt. Soc. Am. A 8, 1411–1418 (1991).
    [CrossRef]
  11. H. L. Tardy, “Flat-sample and limited-field effects in integrating sphere measurements,” J. Opt. Soc. Am. A 5, 241–245 (1988). Note the omission of a prime in his Eq. (2.7) [cf. Eq. (22) of the present paper].
    [CrossRef]
  12. R. Siegel, J. R. Howell, Thermal Radiation Heat Transfer, 3rd ed. (Taylor & Francis, Washington, D.C., 1992).
  13. J. G. Symons, E. A. Christie, M. K. Peck, “Integrating sphere for solar transmittance measurement of planar and nonplanar samples,” Appl. Opt. 21, 2827–2832 (1982).
    [CrossRef] [PubMed]
  14. L. M. Hanssen, “Effects of restricting the detector field of view when using integrating spheres,” Appl. Opt. 28, 2097–2103 (1989).
    [CrossRef] [PubMed]
  15. This is known in illumination engineering as McAllister’s equilux theorem; see Ref. 6, p. 301. It can also be derived by using the contour integration theorem to replace the flat F with the cap K. The latter theorem states that the flux from a surface of uniform radiance L bounded by a contour C′ is the same as that from any other surface of uniform radiance L that is also bounded by C′ (Ref. 6, p. 312).
  16. M. W. Finkel, “Integrating sphere theory,” Opt. Commun. 2, 25–28 (1970). Finkel omits all but the first two groups of terms in Eq. (55) of the present paper.
    [CrossRef]
  17. W. Budde, C. X. Dodd, “Absolute reflectance measurements in the D/0° geometry,” Farbe 19, 94–102 (1970).

1991 (1)

1989 (1)

1988 (1)

1982 (1)

1970 (2)

M. W. Finkel, “Integrating sphere theory,” Opt. Commun. 2, 25–28 (1970). Finkel omits all but the first two groups of terms in Eq. (55) of the present paper.
[CrossRef]

W. Budde, C. X. Dodd, “Absolute reflectance measurements in the D/0° geometry,” Farbe 19, 94–102 (1970).

1967 (1)

1965 (1)

1955 (1)

1935 (1)

1921 (1)

E. Karrer, “Use of the Ulbricht sphere in measuring reflection and transmission factors,” Sci. Pap. Bur. Stand. No. 415 (1921). Karrer derives the (1-ρ)-1 form by energy balance.
[CrossRef]

1920 (1)

Budde, W.

W. Budde, C. X. Dodd, “Absolute reflectance measurements in the D/0° geometry,” Farbe 19, 94–102 (1970).

Christie, E. A.

Dodd, C. X.

W. Budde, C. X. Dodd, “Absolute reflectance measurements in the D/0° geometry,” Farbe 19, 94–102 (1970).

Finkel, M. W.

M. W. Finkel, “Integrating sphere theory,” Opt. Commun. 2, 25–28 (1970). Finkel omits all but the first two groups of terms in Eq. (55) of the present paper.
[CrossRef]

Goebel, D. G.

Hanssen, L. M.

Hisdal, B. J.

Howell, J. R.

R. Siegel, J. R. Howell, Thermal Radiation Heat Transfer, 3rd ed. (Taylor & Francis, Washington, D.C., 1992).

Jacquez, J. A.

Karrer, E.

E. Karrer, “Use of the Ulbricht sphere in measuring reflection and transmission factors,” Sci. Pap. Bur. Stand. No. 415 (1921). Karrer derives the (1-ρ)-1 form by energy balance.
[CrossRef]

Kuppenheim, H. F.

Mascart, E. E. N.

E. E. N. Mascart, “Sur la mesure de l’éclairement,” Lum. Elect.28, 180–187 (1888). Mascart uses summation of reflections to obtain the (1-ρ)-1 expression for the total flux in a room with diffusely reflecting surfaces.Karrer1 cites A. Palaz in Photométrie Industrielle (Carré, Paris, 1892) to the effect that Mascart was the first to derive this expression.

Moon, P.

P. Moon, The Scientific Basis of Illuminating Engineering (McGraw-Hill, New York, 1936), p. 331.

Peck, M. K.

Siegel, R.

R. Siegel, J. R. Howell, Thermal Radiation Heat Transfer, 3rd ed. (Taylor & Francis, Washington, D.C., 1992).

Symons, J. G.

Tardy, H. L.

Taylor, A. H.

Walsh, J. W. T.

J. W. T. Walsh, Photometry (Constable, London, 1958). Walsh gives a simple energy-balance derivation due to Ulbricht in 1920.

Appl. Opt. (3)

Farbe (1)

W. Budde, C. X. Dodd, “Absolute reflectance measurements in the D/0° geometry,” Farbe 19, 94–102 (1970).

J. Opt. Soc. Am. (4)

J. Opt. Soc. Am. A (2)

Opt. Commun. (1)

M. W. Finkel, “Integrating sphere theory,” Opt. Commun. 2, 25–28 (1970). Finkel omits all but the first two groups of terms in Eq. (55) of the present paper.
[CrossRef]

Sci. Pap. Bur. Stand. (1)

E. Karrer, “Use of the Ulbricht sphere in measuring reflection and transmission factors,” Sci. Pap. Bur. Stand. No. 415 (1921). Karrer derives the (1-ρ)-1 form by energy balance.
[CrossRef]

Other (5)

J. W. T. Walsh, Photometry (Constable, London, 1958). Walsh gives a simple energy-balance derivation due to Ulbricht in 1920.

E. E. N. Mascart, “Sur la mesure de l’éclairement,” Lum. Elect.28, 180–187 (1888). Mascart uses summation of reflections to obtain the (1-ρ)-1 expression for the total flux in a room with diffusely reflecting surfaces.Karrer1 cites A. Palaz in Photométrie Industrielle (Carré, Paris, 1892) to the effect that Mascart was the first to derive this expression.

P. Moon, The Scientific Basis of Illuminating Engineering (McGraw-Hill, New York, 1936), p. 331.

R. Siegel, J. R. Howell, Thermal Radiation Heat Transfer, 3rd ed. (Taylor & Francis, Washington, D.C., 1992).

This is known in illumination engineering as McAllister’s equilux theorem; see Ref. 6, p. 301. It can also be derived by using the contour integration theorem to replace the flat F with the cap K. The latter theorem states that the flux from a surface of uniform radiance L bounded by a contour C′ is the same as that from any other surface of uniform radiance L that is also bounded by C′ (Ref. 6, p. 312).

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Figures (3)

Fig. 1
Fig. 1

Great circle between two points on a sphere illustrating the elements in Eq. (3) for the radiation exchange factor G(Ω, Ω) in a true sphere.

Fig. 2
Fig. 2

Sphere with a flat surface showing zones of area C and F and the cap of area K=A-C.

Fig. 3
Fig. 3

Branching diagram for reflections in the sphere with a flat port showing the fluxes incident on the curved and the flat surfaces (denoted by ellipses and rectangles, respectively) at each successive reflection. These fluxes are shown as fractions of the flux at first incidence on the curved surface: For radiation initially incident on the flat surface, as discussed in Subsection 4.C.1, they are fractions of the flux zϕ0 after the first reflection [Eqs. (52)]; for radiation initially incident on the curved surface, as discussed in Subsection 4.C.2, they are fractions of the initial flux ϕ0 [Eqs. (57)].

Tables (1)

Tables Icon

Table 1 Solutions for the Wall Irradiance in Three Integrating-Sphere Configurations

Equations (94)

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πL(Ω)=ρ(Ω)E(Ω).
E(Ω)=AπL(Ω)G(Ω, Ω)dA=Aρ(Ω)E(Ω)G(Ω, Ω)dA,
G(Ω, Ω)=cos ξ cos ξ/(πs2),
FdA-A=AG(Ω, Ω)dA.
Fi-j=1Ai AiAjG(Ω, Ω)dAjdAi
AiFi-j=AjFj-i.
Φd=AdEd(Ω)dAAdEd(Ω)
Ed(Ω)=E(Ω)+E0(Ω).
Ed(Ω)=ηE0(Ω)+Vρ(Ω)[E(Ω)+E0(Ω)]×G(Ω, Ω)dA.
(1-ρ0)ϕ0+i(1-ρi)AαiEi=ϕ0.
Φi,k=jρjΦj,k-1Fj-i,
E(Ω)=πL(Ω)FdA-A=ρ(Ω)E(Ω)FdA-A.
(Ei+E0i)=jMij(Ej+E0j)+E0i,
Mij=ρjFi-j.
Ei+E0i=j[(1-M)-1]ijE0j.
E(Ω)=A[E0(Ω)+E(Ω)]ρ(Ω)G(Ω, Ω)dA.
E(Ω)=f(Ω)+AR(Ω, Ω)f(Ω)dA,
f(Ω)=AE0(Ω)ρ(Ω)G(Ω, Ω)dA,
R(Ω, Ω)=m=0(-1)mDm(Ω, Ω)/m!m=0(-1)mBm/m!.
B0=1,
D0(Ω, Ω)=ρ(Ω)G(Ω, Ω),
Bm=ADm-1(Ω, Ω)dA,
Dm(Ω, Ω)=BmD0(Ω, Ω)-mADm-1(Ω, Ω)×D0(Ω, Ω)dA.
ρ¯=iαiρi.
G(Ω, Ω)=cos ξ cos ξ/[π(2R0 cos ξ)2]=1/A,
(1-ρ0)ϕ0+i(1-ρi)AαiE=ϕ0.
E=ρ0ϕ0A(1-ρ¯).
Φ1=ρ0ϕ0
Φ2=iρiαiΦ1=ρ¯Φ1.
Φ3=iρiαiΦ2=ρ¯2Φ1.
E=ρ0ϕ0A(1+ρ¯+ρ¯2+)=ρ0ϕ0A(1-ρ¯).
Fi-j=Aj/A=αj
Mij=ρjαj.
(1-M)-1=1(1-ρ¯) ρ1α1+1-ρ¯ρ2α2ρnαnρ1α1ρ2α2+1-ρ¯ρnαnρ1α1ρ2α2ρnαnρ1α1ρ2α2ρnαn+1-ρ¯.
E0j=(ϕ0/A0)δj0,
Ei=jρjαj1-ρ¯+δij ϕ0A0δj0-ϕ0A0δi0=ρ0ϕ0A(1-ρ¯)
D0(Ω, Ω)=ρ(Ω)/A,
B1=(1/A)Aρ(Ω)dA=ρ¯,
D1(Ω, Ω)=0.
R(Ω, Ω)=ρ(Ω)/[A(1-ρ¯)].
ϕ0=AE0(Ω)dA
f(Ω)=ρ0ϕ0/A,
Φd=ρ0ρ¯VαdαVϕ0/(1-ρ¯),
F/K=C/A.
G(Ω, Ω)=G(ΩC, ΩC)G(ΩC, ΩF)G(ΩF, ΩC)G(ΩF, ΩF)=1/Ag(ΩC, ΩF)g(ΩC, ΩF)0,
FdC-F=Fg(ΩC, ΩF)dAF=αK.
E(ΩC)=F(ρFϕ0/F)g(ΩC, ΩF)dAF=ρFϕ0/C,
FdF-C=Cg(ΩC, ΩF)dAC=1.
E(ΩF)=C(ρCϕ0/C)g(ΩC, ΩF)dAC=ρCϕ0/C;
E(ΩF)=(ρCϕ0/A0)A0g(ΩC, ΩF)dAC;
E(ΩF)ρCϕ0g(Ω0, ΩF).
(1-ρF)ϕ0+(1-ρF)ρCECCαK+(1-ρC)ECC=ϕ0,
EC=ρFϕ0C(1-ρCαC-ρFρCαK).
(1-ρC)(ϕ0+CEC)+(1-ρF)ρC(ϕ0+CEC)αK=ϕ0,
EC=ρCϕ0(αC+ρFαK)C(1-ρCαC-ρFρCαK).
ΦC,1/ρFϕ0=1,
ΦC,2/ρFϕ0=x,
ΦC,3/ρFϕ0=x2+yz,
ΦC,4/ρFϕ0=x3+2xyz.
ΦC,k=(x2+yz)ΦC,k-2+xyzΦC,k-3,
ΦF,k=yΦC,k-1.
k=1ΦC,k/ρFϕ0=(1+x+x2+x3+)+(1+2x+3x2+4x3+)yz+(1+3x+6x2+10x3+)(yz)2+(1+4x+10x2+)(yz)3+(1+)(yz)4+.
k=1ΦC,k/ρFϕ0=(1-x)-1+(1-x)-2(yz)+(1-x)-3(yz)2+(1-x)-4(yz)3+(1-x)-5(yz)4+=(1-x-yz)-1.
EC=k=1ΦC,k/C=ρFϕ0C(1-ρCαC-ρFρCαK).
ΦC,0/ϕ0=1,
ΦC,1/ϕ0=x,
ΦC,2/ϕ0=x2+yz, 
EC=k=1ΦC,k/C=k=0ΦC,k-ϕ0C,
FC-C=αC
FC-F=αK,
FF-C=1,
FF-F=0.
M=MCCMCFMFCMFF=ρCαCρFαKρC0,
(1-M)-1=(1-ρCαC-ρFρCαK)-11ρFαKρC1-ρCαC.
EC+E0CEF+E0F=(1-ρCαC-ρFρCαK)-11ρFαKρC1-ρCαCE0CE0F.
EF=ρCEC.
EF=ρC(EC+E0C),
G(ΩF, ΩC)=g(ΩC, ΩF)F-1Fg(ΩC, ΩF)dAF=1/C,
D0(Ω, Ω)=ρC/AρF/CρC/C0,
B1=ρCαC,
D1(Ω, Ω)=-ρCρFC αK001,
B2=-2ρFρCαK,
D2(Ω, Ω)=0,B3=0.
f(ΩF)=(ρCϕ0/A0)A0g(ΩC, ΩF)dAC.
f(ΩF)ρCϕ0g(Ω0, ΩF).
Fg(Ω0, ΩF)Dm(ΩC, ΩF)dAF
ϕ1=ρFϕ0
ϕ1=(αCρC+αKρCρF)ϕ0
E=ϕ1A(1-αCρC-αKρF)=ϕ1A(αKδF+αCδC),
Eϕ1C(αKδF+αCδC+αKδC),
N=wρFρCαK(1-ρCαC-ρFρCαK)/(1-ρFρCαK)2.
E(Ω)=ρFρCϕ0Fg(Ω0, Ω)g(Ω, Ω)dA
E(Ω)=ρFρCϕ0αK/C.
1+w=E(Ω)E(Ω)=7+4 cos θ0+cos2 θ06(1+cos θ0)

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