Abstract

If a single optical fiber is used for both delivery and collection of light, two major factors affect the measurement of collected light: (1) the light transport in the medium that describes the amount of light that returns to the fiber and (2) the light coupling to the optical fiber that depends on the angular distribution of photons entering the fiber. We focus on the importance of the latter factor and describe how the efficiency of the coupling depends on the optical properties of the medium. For highly scattering tissues, the efficiency is well predicted by the numerical aperture (NA) of the fiber. For lower scattering, such as in soft tissues, photons arrive at the fiber from deeper depths, and the coupling efficiency could increase twofold to threefold above that predicted by the NA.

© 2003 Optical Society of America

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References

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  1. T. P. Moffitt, S. A. Prahl, “Sized-fiber reflectometry for measuring local optical properties,” IEEE J. Quantum Electron. 7, 952–958 (2001).
    [CrossRef]
  2. S. L. Jacques, “Reflectance spectroscopy with optical fiber devices and transcutaneous bilirubinometers,” in Biomedical Optical Instrumentation and Laser-Assisted Biotechnology, A. M. Verga Scheggi, S. Martellucci, A. N. Chester, R. Pratesi, eds. (Kluwer Academic, Dordrecht, The Netherlands, 1996), pp. 83–94.
    [CrossRef]
  3. B. W. Pogue, G. Burke, “Fiber-optic bundle design for quantitative fluorescence measurement from tissue,” Appl. Opt. 37, 7429–7436 (1998).
    [CrossRef]
  4. D. R. Braichotte, J. F. Savary, P. Monnier, H. E. van den Bergh, “Optimizing light dosimetry in photodynamic therapy of early stage carcinomas of esophagus using fluorescence spectroscopy,” Lasers Surg. Med. 19, 340–346 (1996).
    [CrossRef]
  5. M. Canpolat, J. R. Mourant, “Particle size analysis of turbid media with a single optical fiber in contact with the medium to deliver and detect white light,” Appl. Opt. 40, 3792–3799 (2001).
    [CrossRef]
  6. D. R. Braichotte, G. A. Wagnieres, R. Bays, P. Monnier, H. E. van den Bergh, “Clinical pharmacokinetic studies of photofrin by fluorescence spectroscopy in the oral cavity, the esophagus and the bronchi,” Cancer 75, 2768–2778 (1995).
    [CrossRef] [PubMed]
  7. M. Sinaasapel, H. J. C. M. Sternborg, “Quantification of the hematoporphyrin derivative by fluorescence measurement using dual-wavelength excitation and dual-wavelength detection,” Appl. Opt. 32, 541–548 (1993).
    [CrossRef]
  8. J. Wu, M. S. Feld, R. P. Rava, “Analytical model for extracting intrinsic fluorescence in turbid media,” Appl. Opt. 32, 3585–3595 (1993).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  10. W. B. Pogue, T. Hasan, “Fluorophore quantitation in tissue-simulating media with confocal detection,” IEEE J. Quantum Electron. 2, 959–964 (1997).
  11. L. S. Saidi, “Transcutaneous optical measurement of hyperbilirubinemia in neonates,” Ph.D. dissertation (Rice University, Houston, Tex., 1992).
  12. J. W. Pickering, C. J. M. Moes, H. J. C. M. Sterenborg, S. A. Prahl, M. J. C. van Gemert, “Two integrating spheres with an intervening scattering sample,” J. Opt. Soc. Am. A 9, 621–631 (1992).
    [CrossRef]
  13. E. Hecht, Optics, 3rd ed. (Addison-Wesley, Reading, Mass., 1998), pp. 111–121.
  14. L. Wang, S. L. Jacques, L. Zheng, “MCML—Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47, 131–146 (1995).
    [CrossRef] [PubMed]
  15. S. A. Prahl, S. L. Jacques, “Monte Carlo simulations,” http://omlc.ogi.edu/software/mc/ .
  16. S. A. Prahl, “Light transport in tissue,” Ph.D. dissertation (University of Texas, Austin, Tex., 1988).
  17. M. Young, Optics and Lasers: Including Fibers and Optical Waveguides, 4th rev. ed. (Springer-Verlag, New York, 1992), p. 251.
  18. W. F. Cheong, S. A. Prahl, A. J. Welsh, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26, 2166–2185 (1990).
    [CrossRef]
  19. Optical fiber catalog, CeramOptec Industries, Inc., http://www.ceramoptec.com/ .
  20. Melles Griot product catalog, page 4.13 (1999).
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    [CrossRef] [PubMed]
  22. R. Bays, G. Wagnieres, D. Robert, D. Braichotte, J.-F. Savary, P. Monnier, H. van den Bergh, “Clinical determination of tissue optical properties by endoscopic spatially resolved reflectometry,” Appl. Opt. 35, 1756–1766 (1996).
    [CrossRef] [PubMed]
  23. M. Keijzer, S. L. Jacques, S. A. Prahl, A. J. Welch, “Light distributions in artery tissue: Monte Carlo simulations for finite-diameter laser beams,” Lasers Surg. Med. 9, 148–154 (1989).
    [CrossRef] [PubMed]
  24. B. C. Wilson, “Measurement of tissue optical properties: methods and theories,” in Optical-Thermal Response of Laser Irradiated Tissue, A. J. Welch, M. J. C. van Gemert, eds. (Plenum, New York, 1995), pp. 233–274.
    [CrossRef]
  25. S. L. Jacques, “Modeling light transport in tissue,” in Biomedical Optical Instrumentation and Laser-Assisted Biotechnology, A. M. Verga Scheggi, S. Martellucci, A. N. Chester, R. Pratesi, eds. (Kluwer Academic, Dordrecht, The Netherlands, 1996), pp. 21–32.
    [CrossRef]

2001 (2)

T. P. Moffitt, S. A. Prahl, “Sized-fiber reflectometry for measuring local optical properties,” IEEE J. Quantum Electron. 7, 952–958 (2001).
[CrossRef]

M. Canpolat, J. R. Mourant, “Particle size analysis of turbid media with a single optical fiber in contact with the medium to deliver and detect white light,” Appl. Opt. 40, 3792–3799 (2001).
[CrossRef]

1998 (1)

1997 (1)

W. B. Pogue, T. Hasan, “Fluorophore quantitation in tissue-simulating media with confocal detection,” IEEE J. Quantum Electron. 2, 959–964 (1997).

1996 (3)

1995 (2)

D. R. Braichotte, G. A. Wagnieres, R. Bays, P. Monnier, H. E. van den Bergh, “Clinical pharmacokinetic studies of photofrin by fluorescence spectroscopy in the oral cavity, the esophagus and the bronchi,” Cancer 75, 2768–2778 (1995).
[CrossRef] [PubMed]

L. Wang, S. L. Jacques, L. Zheng, “MCML—Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47, 131–146 (1995).
[CrossRef] [PubMed]

1993 (2)

1992 (2)

S. T. Flock, S. L. Jacques, B. C. Wilson, W. M. Star, M. J. C. van Germet, “Optical properties of Intralipid: a phantom medium for light propagation studies,” Lasers Surg. Med. 12, 510–519 (1992).
[CrossRef] [PubMed]

J. W. Pickering, C. J. M. Moes, H. J. C. M. Sterenborg, S. A. Prahl, M. J. C. van Gemert, “Two integrating spheres with an intervening scattering sample,” J. Opt. Soc. Am. A 9, 621–631 (1992).
[CrossRef]

1990 (1)

W. F. Cheong, S. A. Prahl, A. J. Welsh, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26, 2166–2185 (1990).
[CrossRef]

1989 (1)

M. Keijzer, S. L. Jacques, S. A. Prahl, A. J. Welch, “Light distributions in artery tissue: Monte Carlo simulations for finite-diameter laser beams,” Lasers Surg. Med. 9, 148–154 (1989).
[CrossRef] [PubMed]

Bays, R.

R. Bays, G. Wagnieres, D. Robert, D. Braichotte, J.-F. Savary, P. Monnier, H. van den Bergh, “Clinical determination of tissue optical properties by endoscopic spatially resolved reflectometry,” Appl. Opt. 35, 1756–1766 (1996).
[CrossRef] [PubMed]

D. R. Braichotte, G. A. Wagnieres, R. Bays, P. Monnier, H. E. van den Bergh, “Clinical pharmacokinetic studies of photofrin by fluorescence spectroscopy in the oral cavity, the esophagus and the bronchi,” Cancer 75, 2768–2778 (1995).
[CrossRef] [PubMed]

Braichotte, D.

Braichotte, D. R.

D. R. Braichotte, J. F. Savary, P. Monnier, H. E. van den Bergh, “Optimizing light dosimetry in photodynamic therapy of early stage carcinomas of esophagus using fluorescence spectroscopy,” Lasers Surg. Med. 19, 340–346 (1996).
[CrossRef]

D. R. Braichotte, G. A. Wagnieres, R. Bays, P. Monnier, H. E. van den Bergh, “Clinical pharmacokinetic studies of photofrin by fluorescence spectroscopy in the oral cavity, the esophagus and the bronchi,” Cancer 75, 2768–2778 (1995).
[CrossRef] [PubMed]

Burke, G.

Canpolat, M.

Cheong, W. F.

W. F. Cheong, S. A. Prahl, A. J. Welsh, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26, 2166–2185 (1990).
[CrossRef]

Feld, M. S.

Flock, S. T.

S. T. Flock, S. L. Jacques, B. C. Wilson, W. M. Star, M. J. C. van Germet, “Optical properties of Intralipid: a phantom medium for light propagation studies,” Lasers Surg. Med. 12, 510–519 (1992).
[CrossRef] [PubMed]

Gardner, C. M.

Hasan, T.

W. B. Pogue, T. Hasan, “Fluorophore quantitation in tissue-simulating media with confocal detection,” IEEE J. Quantum Electron. 2, 959–964 (1997).

Hecht, E.

E. Hecht, Optics, 3rd ed. (Addison-Wesley, Reading, Mass., 1998), pp. 111–121.

Jacques, S. L.

C. M. Gardner, S. L. Jacques, A. J. Welch, “Fluorescence spectroscopy of tissue: recovery of intrinsic fluorescence from measured fluorescence,” Appl. Opt. 35, 1780–1792 (1996).
[CrossRef] [PubMed]

L. Wang, S. L. Jacques, L. Zheng, “MCML—Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47, 131–146 (1995).
[CrossRef] [PubMed]

S. T. Flock, S. L. Jacques, B. C. Wilson, W. M. Star, M. J. C. van Germet, “Optical properties of Intralipid: a phantom medium for light propagation studies,” Lasers Surg. Med. 12, 510–519 (1992).
[CrossRef] [PubMed]

M. Keijzer, S. L. Jacques, S. A. Prahl, A. J. Welch, “Light distributions in artery tissue: Monte Carlo simulations for finite-diameter laser beams,” Lasers Surg. Med. 9, 148–154 (1989).
[CrossRef] [PubMed]

S. L. Jacques, “Modeling light transport in tissue,” in Biomedical Optical Instrumentation and Laser-Assisted Biotechnology, A. M. Verga Scheggi, S. Martellucci, A. N. Chester, R. Pratesi, eds. (Kluwer Academic, Dordrecht, The Netherlands, 1996), pp. 21–32.
[CrossRef]

S. L. Jacques, “Reflectance spectroscopy with optical fiber devices and transcutaneous bilirubinometers,” in Biomedical Optical Instrumentation and Laser-Assisted Biotechnology, A. M. Verga Scheggi, S. Martellucci, A. N. Chester, R. Pratesi, eds. (Kluwer Academic, Dordrecht, The Netherlands, 1996), pp. 83–94.
[CrossRef]

Keijzer, M.

M. Keijzer, S. L. Jacques, S. A. Prahl, A. J. Welch, “Light distributions in artery tissue: Monte Carlo simulations for finite-diameter laser beams,” Lasers Surg. Med. 9, 148–154 (1989).
[CrossRef] [PubMed]

Moes, C. J. M.

Moffitt, T. P.

T. P. Moffitt, S. A. Prahl, “Sized-fiber reflectometry for measuring local optical properties,” IEEE J. Quantum Electron. 7, 952–958 (2001).
[CrossRef]

Monnier, P.

D. R. Braichotte, J. F. Savary, P. Monnier, H. E. van den Bergh, “Optimizing light dosimetry in photodynamic therapy of early stage carcinomas of esophagus using fluorescence spectroscopy,” Lasers Surg. Med. 19, 340–346 (1996).
[CrossRef]

R. Bays, G. Wagnieres, D. Robert, D. Braichotte, J.-F. Savary, P. Monnier, H. van den Bergh, “Clinical determination of tissue optical properties by endoscopic spatially resolved reflectometry,” Appl. Opt. 35, 1756–1766 (1996).
[CrossRef] [PubMed]

D. R. Braichotte, G. A. Wagnieres, R. Bays, P. Monnier, H. E. van den Bergh, “Clinical pharmacokinetic studies of photofrin by fluorescence spectroscopy in the oral cavity, the esophagus and the bronchi,” Cancer 75, 2768–2778 (1995).
[CrossRef] [PubMed]

Mourant, J. R.

Pickering, J. W.

Pogue, B. W.

Pogue, W. B.

W. B. Pogue, T. Hasan, “Fluorophore quantitation in tissue-simulating media with confocal detection,” IEEE J. Quantum Electron. 2, 959–964 (1997).

Prahl, S. A.

T. P. Moffitt, S. A. Prahl, “Sized-fiber reflectometry for measuring local optical properties,” IEEE J. Quantum Electron. 7, 952–958 (2001).
[CrossRef]

J. W. Pickering, C. J. M. Moes, H. J. C. M. Sterenborg, S. A. Prahl, M. J. C. van Gemert, “Two integrating spheres with an intervening scattering sample,” J. Opt. Soc. Am. A 9, 621–631 (1992).
[CrossRef]

W. F. Cheong, S. A. Prahl, A. J. Welsh, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26, 2166–2185 (1990).
[CrossRef]

M. Keijzer, S. L. Jacques, S. A. Prahl, A. J. Welch, “Light distributions in artery tissue: Monte Carlo simulations for finite-diameter laser beams,” Lasers Surg. Med. 9, 148–154 (1989).
[CrossRef] [PubMed]

S. A. Prahl, “Light transport in tissue,” Ph.D. dissertation (University of Texas, Austin, Tex., 1988).

Rava, R. P.

Robert, D.

Saidi, L. S.

L. S. Saidi, “Transcutaneous optical measurement of hyperbilirubinemia in neonates,” Ph.D. dissertation (Rice University, Houston, Tex., 1992).

Savary, J. F.

D. R. Braichotte, J. F. Savary, P. Monnier, H. E. van den Bergh, “Optimizing light dosimetry in photodynamic therapy of early stage carcinomas of esophagus using fluorescence spectroscopy,” Lasers Surg. Med. 19, 340–346 (1996).
[CrossRef]

Savary, J.-F.

Sinaasapel, M.

Star, W. M.

S. T. Flock, S. L. Jacques, B. C. Wilson, W. M. Star, M. J. C. van Germet, “Optical properties of Intralipid: a phantom medium for light propagation studies,” Lasers Surg. Med. 12, 510–519 (1992).
[CrossRef] [PubMed]

Sterenborg, H. J. C. M.

Sternborg, H. J. C. M.

van den Bergh, H.

van den Bergh, H. E.

D. R. Braichotte, J. F. Savary, P. Monnier, H. E. van den Bergh, “Optimizing light dosimetry in photodynamic therapy of early stage carcinomas of esophagus using fluorescence spectroscopy,” Lasers Surg. Med. 19, 340–346 (1996).
[CrossRef]

D. R. Braichotte, G. A. Wagnieres, R. Bays, P. Monnier, H. E. van den Bergh, “Clinical pharmacokinetic studies of photofrin by fluorescence spectroscopy in the oral cavity, the esophagus and the bronchi,” Cancer 75, 2768–2778 (1995).
[CrossRef] [PubMed]

van Gemert, M. J. C.

van Germet, M. J. C.

S. T. Flock, S. L. Jacques, B. C. Wilson, W. M. Star, M. J. C. van Germet, “Optical properties of Intralipid: a phantom medium for light propagation studies,” Lasers Surg. Med. 12, 510–519 (1992).
[CrossRef] [PubMed]

Wagnieres, G.

Wagnieres, G. A.

D. R. Braichotte, G. A. Wagnieres, R. Bays, P. Monnier, H. E. van den Bergh, “Clinical pharmacokinetic studies of photofrin by fluorescence spectroscopy in the oral cavity, the esophagus and the bronchi,” Cancer 75, 2768–2778 (1995).
[CrossRef] [PubMed]

Wang, L.

L. Wang, S. L. Jacques, L. Zheng, “MCML—Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47, 131–146 (1995).
[CrossRef] [PubMed]

Welch, A. J.

C. M. Gardner, S. L. Jacques, A. J. Welch, “Fluorescence spectroscopy of tissue: recovery of intrinsic fluorescence from measured fluorescence,” Appl. Opt. 35, 1780–1792 (1996).
[CrossRef] [PubMed]

M. Keijzer, S. L. Jacques, S. A. Prahl, A. J. Welch, “Light distributions in artery tissue: Monte Carlo simulations for finite-diameter laser beams,” Lasers Surg. Med. 9, 148–154 (1989).
[CrossRef] [PubMed]

Welsh, A. J.

W. F. Cheong, S. A. Prahl, A. J. Welsh, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26, 2166–2185 (1990).
[CrossRef]

Wilson, B. C.

S. T. Flock, S. L. Jacques, B. C. Wilson, W. M. Star, M. J. C. van Germet, “Optical properties of Intralipid: a phantom medium for light propagation studies,” Lasers Surg. Med. 12, 510–519 (1992).
[CrossRef] [PubMed]

B. C. Wilson, “Measurement of tissue optical properties: methods and theories,” in Optical-Thermal Response of Laser Irradiated Tissue, A. J. Welch, M. J. C. van Gemert, eds. (Plenum, New York, 1995), pp. 233–274.
[CrossRef]

Wu, J.

Young, M.

M. Young, Optics and Lasers: Including Fibers and Optical Waveguides, 4th rev. ed. (Springer-Verlag, New York, 1992), p. 251.

Zheng, L.

L. Wang, S. L. Jacques, L. Zheng, “MCML—Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47, 131–146 (1995).
[CrossRef] [PubMed]

Appl. Opt. (6)

Cancer (1)

D. R. Braichotte, G. A. Wagnieres, R. Bays, P. Monnier, H. E. van den Bergh, “Clinical pharmacokinetic studies of photofrin by fluorescence spectroscopy in the oral cavity, the esophagus and the bronchi,” Cancer 75, 2768–2778 (1995).
[CrossRef] [PubMed]

Comput. Methods Programs Biomed. (1)

L. Wang, S. L. Jacques, L. Zheng, “MCML—Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47, 131–146 (1995).
[CrossRef] [PubMed]

IEEE J. Quantum Electron. (3)

T. P. Moffitt, S. A. Prahl, “Sized-fiber reflectometry for measuring local optical properties,” IEEE J. Quantum Electron. 7, 952–958 (2001).
[CrossRef]

W. F. Cheong, S. A. Prahl, A. J. Welsh, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26, 2166–2185 (1990).
[CrossRef]

W. B. Pogue, T. Hasan, “Fluorophore quantitation in tissue-simulating media with confocal detection,” IEEE J. Quantum Electron. 2, 959–964 (1997).

J. Opt. Soc. Am. A (1)

Lasers Surg. Med. (3)

M. Keijzer, S. L. Jacques, S. A. Prahl, A. J. Welch, “Light distributions in artery tissue: Monte Carlo simulations for finite-diameter laser beams,” Lasers Surg. Med. 9, 148–154 (1989).
[CrossRef] [PubMed]

D. R. Braichotte, J. F. Savary, P. Monnier, H. E. van den Bergh, “Optimizing light dosimetry in photodynamic therapy of early stage carcinomas of esophagus using fluorescence spectroscopy,” Lasers Surg. Med. 19, 340–346 (1996).
[CrossRef]

S. T. Flock, S. L. Jacques, B. C. Wilson, W. M. Star, M. J. C. van Germet, “Optical properties of Intralipid: a phantom medium for light propagation studies,” Lasers Surg. Med. 12, 510–519 (1992).
[CrossRef] [PubMed]

Other (10)

Optical fiber catalog, CeramOptec Industries, Inc., http://www.ceramoptec.com/ .

Melles Griot product catalog, page 4.13 (1999).

B. C. Wilson, “Measurement of tissue optical properties: methods and theories,” in Optical-Thermal Response of Laser Irradiated Tissue, A. J. Welch, M. J. C. van Gemert, eds. (Plenum, New York, 1995), pp. 233–274.
[CrossRef]

S. L. Jacques, “Modeling light transport in tissue,” in Biomedical Optical Instrumentation and Laser-Assisted Biotechnology, A. M. Verga Scheggi, S. Martellucci, A. N. Chester, R. Pratesi, eds. (Kluwer Academic, Dordrecht, The Netherlands, 1996), pp. 21–32.
[CrossRef]

L. S. Saidi, “Transcutaneous optical measurement of hyperbilirubinemia in neonates,” Ph.D. dissertation (Rice University, Houston, Tex., 1992).

S. L. Jacques, “Reflectance spectroscopy with optical fiber devices and transcutaneous bilirubinometers,” in Biomedical Optical Instrumentation and Laser-Assisted Biotechnology, A. M. Verga Scheggi, S. Martellucci, A. N. Chester, R. Pratesi, eds. (Kluwer Academic, Dordrecht, The Netherlands, 1996), pp. 83–94.
[CrossRef]

E. Hecht, Optics, 3rd ed. (Addison-Wesley, Reading, Mass., 1998), pp. 111–121.

S. A. Prahl, S. L. Jacques, “Monte Carlo simulations,” http://omlc.ogi.edu/software/mc/ .

S. A. Prahl, “Light transport in tissue,” Ph.D. dissertation (University of Texas, Austin, Tex., 1988).

M. Young, Optics and Lasers: Including Fibers and Optical Waveguides, 4th rev. ed. (Springer-Verlag, New York, 1992), p. 251.

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Figures (10)

Fig. 1
Fig. 1

Diagram of the possible return paths of light incident from a single optical fiber. Light that reaches the fiber face with an angle smaller than the half-angle of the acceptance cone will be guided through the fiber to the detector (R core). Light that reaches the fiber face with an angle greater than the half-angle of the acceptance cone will escape through the fiber cladding (R clad). R air is the light that leaves the medium outside the fiber, and r sp is the Fresnel reflection that is due to the fiber-tissue index of refraction mismatch. Light can also be absorbed by the tissue.

Fig. 2
Fig. 2

Diagram of the single optical fiber reflectance system. A single 600-μm optical fiber is connected to the distal end of a bifurcated fiber bundle composed of two 300-μm optical fibers. One fiber has the proximal end connected to a tungsten-halogen white lamp and the other is connected to a spectrophotometer. The distal end of the 600-μm optical fiber is placed in contact with the gel samples through a drop of water. Optical density filters are used to avoid detector saturation. For a description of the distal end acrylic support, refer to the text.

Fig. 3
Fig. 3

Setup of the integrating sphere experiment. White light guided through a 600-μm optical fiber positioned 5 mm away from the sample surface is used to illuminate a 3-mm-diameter spot on the sample. Diffuse reflectance from the sample is trapped in an 8-in.- (20-cm-) diameter integrating sphere. Light is collected by an optical fiber positioned at a 0.25-in.- (0.6-cm-) diameter port of the sphere and guided to a spectrophotometer. Spectralon standards are used to calibrate the diffuse reflectance from the samples.

Fig. 4
Fig. 4

Fraction of collected light (f core) determined by Monte Carlo (open symbols) and experiments (filled symbols) for three μ s ′ (◇, 7; □, 14; and ○, 28 cm-1) and six μ a (0.01, 0.1, 0.4, 0.9, 2.5, and 4.9 cm-1, greater μ a to the left). The fiber diameter was 600 μm and the NA was 0.22. f core (dimensionless) is plotted as a function of the dimensionless parameter X = δmfp′/d 2, where d is the fiber diameter, δ = [3 μ a a + μ s ′)]-1/2 and mfp′ equals 1/(μ a + μ s ′). Vertical bars are the standard deviation of the data for seven measurements.

Fig. 5
Fig. 5

Comparison between the experimental and theoretical (Monte Carlo) values for f core. The symbols ◇, □, and ○ represent reduced scattering coefficienties of 7, 14, and 28 cm-1 for six μ a (same as Fig. 4).

Fig. 6
Fig. 6

(a) Monte Carlo simulations of f core for three optical fiber diameters 200 μm (○), 600 μm (□), and 2000 μm (◇) for μ s ′ of 10 cm-1 (open symbols) and 20 cm-1 (filled symbols) and for μ a ranging from 0.01 to 50 cm-1. The solid curve is a hyberbolic tangent function that follows the form f core = C [1 - (1 +tanh{A[ln(X) + B]})/2]. For a fiber NA of 0.39 and the above range of optical properties, A = 0.278, B = 1.005, and C = 0.0835. (b) Same data as (a) for μ s ′ of 10 cm-1 (open symbols) plotted against the mfp′ for comparison.

Fig. 7
Fig. 7

(a) Plot of Monte Carlo simulations of the collected light as a function of the collection angle bin (θ) for three μ s ′ (70, 10, and 1 cm-1, top to bottom) and μ a of 0.005 cm-1. The curves are proportional to cos(θ)sin(θ) [see Eq. (10) in Section 5] and show the similarities of the data to this simple expression for higher scattering and the differences for low scattering. (b) Integral of (a) over θ, representing the fraction of the total incident light that couples to the fiber core (R core for a given angle). The dashed line is proportional to sin2(θ) (see text). The dotted line at θ = 15 deg and R core = 0.0266 for μ s ′ = 70 cm-1 corresponds to a 600-μm-diameter optical fiber with a NA of 0.22.

Fig. 8
Fig. 8

Monte Carlo simulations of the collection efficiency (η c ) for a fiber diameter of 600 μm in contact with a medium with an index of refraction of 1.35. (a) η c as a function of μ s ′ and (b) η c as a function of μ a for a NA of 0.39 (acceptance angle of 16.8° in the medium). (c) η c as a function of μ s ′ and (d) η c as a function of μ a for a NA of 0.22 (acceptance angle of 9.38° in the medium). Values of η c equaling 0.0835 (a) and (b) and 0.0266 (c) and (d) are shown for comparison with Eq. (10) (see text).

Fig. 9
Fig. 9

Collection efficiency η c determined by Monte Carlo simulations for anisotropies of 0.9 (○) and 0.95 (◇) plotted as a function of the η c for an anisotropy of 0.83. μ a ranged from 0.5 to 5 cm-1 and μ s ′ from 1 to 20 cm-1. Fiber diameter was 600 μm and the NA was 0.39.

Fig. 10
Fig. 10

Collection efficiency η c determined by Monte Carlo simulations as a function of the angular distribution of the launched photons. The NA of collection was fixed to 0.39. Data are for an absorption coefficient of 1 cm-1; reduced scattering coefficients of 5 cm-1 (open symbols) and 40 cm-1 (filled symbols); and the optical fiber diameters of 200 (○), 600 (□), and 2000 μm (◇).

Equations (11)

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

Pesc=P0rsp+P01-rsp0 Tr2πrdr=P0rsp+P01-rspRdiffuse,
Pcollected=P0rsp+P01-rsp0d20d2 Tr, rdAdA=P0rsp+P01-rspRcollected,
f=PcollectedPesc-P0rsp=RcollectedRdiffuse.
f=Pcollected-P0rspPesc-P0rsp=RcollectedRdiffuse.
fcore=RcoreRdiffuse.
ηc=RcoreRcollected=RcoreRcore+Rclad.
fcore=fηc=RcollectedRdiffuseRcoreRcollected=RcoreRdiffuse.
Rcore=Msample-MclearMwater Rwater.
X=δmfpd2=1μeffdμtd.
fcore=C1-1+tanhAlnX+B2,
ηc=RcoreRcollected=02πdϕ 0θacosθsinθdθ02πdϕ 0π2cosθsinθdθ=-π sin2θ|0θa-π sin2θ|0π2=sin2θa.

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