Abstract

The importance of Lambertian transmission, rather than total transmission, is argued and expressions for both are given. Exact expressions for output radiance are given in terms of both total sphere area and port area. A formula for choosing sphere size to give the maximum Lambertian transmission is developed. Port fractions in the range of 0.1–0.2 are recommended.

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References

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  1. A Guide to Integrating Sphere Theory and Applications, available at Labsphere's web site, www.labsphere.com/tecdocs.aspx.
  2. A Guide to Integrating Sphere Radiomatry and Photometry, available at Labsphere's web site, www.labsphere.com/tecdocs.aspx.
  3. O. E. Miller and A. J. Sant, "Incomplete integrating sphere," J. Opt. Soc. Am. 48, 828-831 (1958).
    [CrossRef]
  4. D. G. Goebel, "Generalized integrating-sphere theory," Appl. Opt. 6, 125-128 (1967).
    [CrossRef] [PubMed]
  5. L. A. Whitehead and M. A. Mossman, "Jack o'lanterns and integrating spheres: Halloween physics," Am. J. Phys. 74, 537-541 (2006).
    [CrossRef]
  6. D. Hidovic-Rowe, J. E. Rowe, and M. Lualdi, "Markov models of integrating spheres for hyperspectral imaging," Appl. Opt. 45, 5248-5257 (2006).
    [CrossRef] [PubMed]
  7. M. Szylowski, M. Mossman, D. Barclay, and L. Whitehead, "Novel fiber-based integrating sphere for luminous flux measurements," Rev. Sci. Instrum. 77, 063102 (2006).
    [CrossRef]
  8. A. Carrasco-Sanz, S. Martin-Lopez, P. Corredera, M. Gonzalez-Herraez, and M. L. Hernanz, "High-power and high-accuracy integrating sphere radiometer: design, characterization, and calibration," Appl. Opt. 45, 511-518 (2006).
    [CrossRef] [PubMed]
  9. D. B. Chenault, K. A. Snail, and L. M. Hanssen, "Improved integrating-sphere throughput with a lens and nonimaging concentrator," Appl. Opt. 34, 7959-7964 (1995).
    [CrossRef] [PubMed]
  10. D. A. Schroeder, Astronomical Optics, 2nd ed. (Academic, 2000), Eq. 12.2.7 on p. 313.
  11. A Guide to Reflectance Coatings and Materials, available at Labsphere's web site, www.labsphere.com/tecdocs.aspx.

2006

L. A. Whitehead and M. A. Mossman, "Jack o'lanterns and integrating spheres: Halloween physics," Am. J. Phys. 74, 537-541 (2006).
[CrossRef]

D. Hidovic-Rowe, J. E. Rowe, and M. Lualdi, "Markov models of integrating spheres for hyperspectral imaging," Appl. Opt. 45, 5248-5257 (2006).
[CrossRef] [PubMed]

M. Szylowski, M. Mossman, D. Barclay, and L. Whitehead, "Novel fiber-based integrating sphere for luminous flux measurements," Rev. Sci. Instrum. 77, 063102 (2006).
[CrossRef]

A. Carrasco-Sanz, S. Martin-Lopez, P. Corredera, M. Gonzalez-Herraez, and M. L. Hernanz, "High-power and high-accuracy integrating sphere radiometer: design, characterization, and calibration," Appl. Opt. 45, 511-518 (2006).
[CrossRef] [PubMed]

2000

D. A. Schroeder, Astronomical Optics, 2nd ed. (Academic, 2000), Eq. 12.2.7 on p. 313.

1995

1967

1958

Barclay, D.

M. Szylowski, M. Mossman, D. Barclay, and L. Whitehead, "Novel fiber-based integrating sphere for luminous flux measurements," Rev. Sci. Instrum. 77, 063102 (2006).
[CrossRef]

Carrasco-Sanz, A.

Chenault, D. B.

Corredera, P.

Goebel, D. G.

Gonzalez-Herraez, M.

Hanssen, L. M.

Hernanz, M. L.

Hidovic-Rowe, D.

Lualdi, M.

Martin-Lopez, S.

Miller, O. E.

Mossman, M.

M. Szylowski, M. Mossman, D. Barclay, and L. Whitehead, "Novel fiber-based integrating sphere for luminous flux measurements," Rev. Sci. Instrum. 77, 063102 (2006).
[CrossRef]

Mossman, M. A.

L. A. Whitehead and M. A. Mossman, "Jack o'lanterns and integrating spheres: Halloween physics," Am. J. Phys. 74, 537-541 (2006).
[CrossRef]

Rowe, J. E.

Sant, A. J.

Schroeder, D. A.

D. A. Schroeder, Astronomical Optics, 2nd ed. (Academic, 2000), Eq. 12.2.7 on p. 313.

Snail, K. A.

Szylowski, M.

M. Szylowski, M. Mossman, D. Barclay, and L. Whitehead, "Novel fiber-based integrating sphere for luminous flux measurements," Rev. Sci. Instrum. 77, 063102 (2006).
[CrossRef]

Whitehead, L.

M. Szylowski, M. Mossman, D. Barclay, and L. Whitehead, "Novel fiber-based integrating sphere for luminous flux measurements," Rev. Sci. Instrum. 77, 063102 (2006).
[CrossRef]

Whitehead, L. A.

L. A. Whitehead and M. A. Mossman, "Jack o'lanterns and integrating spheres: Halloween physics," Am. J. Phys. 74, 537-541 (2006).
[CrossRef]

Am. J. Phys.

L. A. Whitehead and M. A. Mossman, "Jack o'lanterns and integrating spheres: Halloween physics," Am. J. Phys. 74, 537-541 (2006).
[CrossRef]

Appl. Opt.

J. Opt. Soc. Am.

Rev. Sci. Instrum.

M. Szylowski, M. Mossman, D. Barclay, and L. Whitehead, "Novel fiber-based integrating sphere for luminous flux measurements," Rev. Sci. Instrum. 77, 063102 (2006).
[CrossRef]

Other

D. A. Schroeder, Astronomical Optics, 2nd ed. (Academic, 2000), Eq. 12.2.7 on p. 313.

A Guide to Reflectance Coatings and Materials, available at Labsphere's web site, www.labsphere.com/tecdocs.aspx.

A Guide to Integrating Sphere Theory and Applications, available at Labsphere's web site, www.labsphere.com/tecdocs.aspx.

A Guide to Integrating Sphere Radiomatry and Photometry, available at Labsphere's web site, www.labsphere.com/tecdocs.aspx.

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Figures (5)

Fig. 1
Fig. 1

Typical integrating sphere in a laboratroy application. Light from the incoming beam's footprint, i.e., from the first reflection, illuminates the exit port from only one direction and hence does not contribute to Lambertian radiance (see Section 5).

Fig. 2
Fig. 2

Integrating sphere in a remote sensing application. Light from a distant source is sent into the sphere. Emerging light passes through a transfer lens to, for example, illuminate the slit of a spectrometer.

Fig. 3
Fig. 3

Total transmission of an integrating sphere, from Eq. ((4)).

Fig. 4
Fig. 4

Lambertian component of transmission, from Eq. ((6)).

Fig. 5
Fig. 5

Port fraction that gives maximum Lambertian transmission, from Eq. ((8)).

Equations (73)

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d 1
a 1 = π d 1 2 / 4
a 1 = π D 2 2 ( 1 1 d 1 2 / D 2 ) .
f 1 a 1 / A
A = π D 2
a 1 = A f 1 ( 1 f 1 ) = a 1 ( 1 f 1 ) .
a 1 a 1
a
Φ ρ / A
W / m 2
Φ ρ 2 / ( π A )
W / m 2 sr
( 1 f )
L L = Φ ρ 2 π A [ 1 + ρ ( 1 f ) + ρ 2 ( 1 f ) 2 + ] = Φ π A ρ 2 1 ρ ( 1 f ) .
ρ 2
ρ ( 1 f )
T = ρ f + ρ ( 1 f ) ρ f + ρ 2 ( 1 f ) 2 ρ f + = ρ f 1 ρ ( 1 f ) .
L a v e = Φ T π a = Φ π a ρ f 1 ρ ( 1 f ) .
a F
Δ L F = Φ ρ / ( π a F )
L ( x , y , θ , ϕ )
( x , y )
L = 0
( θ , ϕ )
a F
L = L L + Δ L F
L = L L
Δ L F / L L = a / ( T a F )
Δ L F
L L
a F a
T L = T ρ f = ρ 2 f ( 1 f ) [ 1 + ρ ( 1 f ) + ρ 2 ( 1 f ) 2 + ] = ρ ( 1 f ) T = ρ 2 f ( 1 f ) 1 ρ ( 1 f ) ,
L L = Φ T L π a = Φ π a ρ 2 f ( 1 f ) 1 ρ ( 1 f ) .
a = a 1
f = f 1
Φ T L
Φ T L / ( π a 1 )
a 1 = A f 1 ( 1 f 1 )
T L
f max = 1 ρ ( 1 1 ρ ) ρ ,
a 1
a 2
a = a 1 + a 2 = A f 1 ( 1 f 1 ) + A f 2 ( 1 f 2 ) = A ( f f 1 2 f 2 2 )
1 A = f f 1 2 f 2 2 a .
L L = Φ π a ρ 2 1 ρ ( 1 f ) ( f f 1 2 f 2 2 ) = Φ T L π a ( 1 f 1 2 + f 2 2 f ) ( 1 f ) ,
a 2 = 0
f 2 = 0
Δ L F
f 1 = f 2 = 0.1
f 1 2 + f 2 2
f n 2
ρ 0.96
U Ω s p a s
Ω s p
a s
Ω s p a s / ( π a )
Ω s p 0.2
Ω s p / π < 10 1
mm 2
a s
0.1 mm 2
a s / a 10 3
mm 2
d 1 2 D 2 = π d 1 2 π D 2 = 4 a 1 A .
f 1 = a 1 A = 1 2 ( 1 1 d 1 2 / D 2 ) 1 d 1 2 / D 2 = ( 1 2 f 1 ) 2 = 1 4 f 1 + 4 f 1 2 d 1 2 D 2 = 4 a 1 A = 4 f 1 ( 1 f 1 ) a 1 = A f 1 ( 1 f 1 ) = a 1 ( 1 f 1 ) .
f 1 = 1 2 ( 1 1 d 1 2 / D 2 ) d 1 2 4 D 2 ( 1 + d 1 2 4 D 2 ) 1 4 ( d 1 D ) 2 ,
( 5 % , 10 % , 20 % )
d 1 / D = ( 0.44 , 0.6 , 0.8 )
f = ( 0.051 , 0.1 , 0.2 )
0 d 1 / D 1
    d T L d f = ρ 2 ( 1 2 f ) [ 1 ρ ( 1 f ) ] ρ 3 f ( 1 f ) [ 1 ρ ( 1 f ) ] 2 .
f max
ρ f max 2 + 2 ( 1 ρ ) f max ( 1 ρ ) = 0.

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