OSA's Digital Library

Applied Optics

Applied Optics


  • Vol. 44, Iss. 11 — Apr. 10, 2005
  • pp: 2058–2071

PN approximation for frequency-domain measurements in scattering media

Gregory W. Faris  »View Author Affiliations

Applied Optics, Vol. 44, Issue 11, pp. 2058-2071 (2005)

View Full Text Article

Enhanced HTML    Acrobat PDF (3701 KB)

Browse Journals / Lookup Meetings

Browse by Journal and Year


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools



Presented here are expressions for the PN approximation for light propagation in scattering media in the frequency domain. To elucidate parametric dependencies, the derivation uses normalization of the resulting expressions to either the total interaction coefficient or the reduced total interaction coefficient. For the latter case, a set of reduced phase function coefficients are introduced. Expression of the PN approximation as a conventional eigenvalue problem facilitates computation of the eigenvalues or attenuation coefficients. This approach is used to determine the attenuation coefficients in the asymptotic regime over the full values of the scattering albedo and reduced scattering albedo (0 to 1) and all positive values of the asymmetry factor (0 to 1). Frequency-domain measurements yield a sensitivity to turbid media optical properties for reduced scattering albedos as small as 0.2. PN calculations are used to assess the magnitude of errors associated with the P1 and P3 approximations over a range of scattering albedo, phase function, and modulation frequency.

© 2005 Optical Society of America

OCIS Codes
(170.5270) Medical optics and biotechnology : Photon density waves
(170.5280) Medical optics and biotechnology : Photon migration
(290.1990) Scattering : Diffusion
(290.4210) Scattering : Multiple scattering
(290.7050) Scattering : Turbid media

Original Manuscript: August 10, 2004
Revised Manuscript: January 7, 2005
Manuscript Accepted: January 9, 2005
Published: April 10, 2005

Gregory W. Faris, "PN approximation for frequency-domain measurements in scattering media," Appl. Opt. 44, 2058-2071 (2005)

Sort:  Author  |  Year  |  Journal  |  Reset  


  1. V. V. Tuchin, ed., Handbook of Optical Biomedical Diagnostics (SPIE, Bellingham, Wash., 2002).
  2. C. Mobley, “Optical properties of water,” in Handbook of Optics, M. Bass, ed. (McGraw-Hill, New York, 1995), Vol. 1, pp. 43.43–43.56.
  3. Z. Sun, S. Torrance, F. K. McNeil-Watson, E. M. Sevick-Muraca, “Application of frequency domain photon migration to particle size analysis and monitoring of pharmaceutical powders,” Anal. Chem. 75, 1720–1725 (2003). [CrossRef] [PubMed]
  4. A. Ishimaru, “Transport equation for a partially polarized electromagnetic wave,” in Wave Propagation and Scattering in Random Media (Institute of Electrical and Electronics Engineers, New York, 1997), pp. 164–165.
  5. L. Marti-Lopez, J. Bouza-Dominguez, J. C. Hebden, S. R. Arridge, R. A. Martinez-Celorio, “Validity conditions for the radiative transfer equation,” J. Opt. Soc. Am. A 20, 2046–2056 (2003). [CrossRef]
  6. L.-H. Wang, S. L. Jacques, L.-Q. Zheng, “MCML—Monte Carlo modeling of photon transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47, 131–146 (1995). [CrossRef] [PubMed]
  7. S. R. Arridge, W. R. B. Lionheart, “Nonuniqueness in diffusion-based optical tomography,” Opt. Lett. 23, 882–884 (1998). [CrossRef]
  8. B. Chance, M. Cope, E. Gratton, N. Ramanujam, B. Tromberg, “Phase measurement of light absorption and scatter in human tissue,” Rev. Sci. Instrum. 69, 3457–3481 (1998). [CrossRef]
  9. A. E. Cerussi, B. J. Tromberg, “Photon migration spectroscopy frequency-domain techniques,” in Biomedical Photonics Handbook, T. Vo-Dinh, ed. (CRC Press, Boca Raton, Fla., 2003), pp. 22-21–22-17.
  10. M. Gerken, G. W. Faris, “High-precision frequency-domain measurements of the optical properties of turbid media,” Opt. Lett. 24, 930–932 (1999). [CrossRef]
  11. M. Gerken, G. W. Faris, “Frequency-domain immersion technique for accurate optical property measurements of turbid media,” Opt. Lett. 24, 1726–1728 (1999). [CrossRef]
  12. R. Aronson, N. Corngold, “Photon diffusion coefficient in an absorbing medium,” J. Opt. Soc. Am. A 16, 1066–1071 (1999). [CrossRef]
  13. M. Gerken, D. Godfrey, G. W. Faris, “Frequency-domain technique for optical property measurements in moderately scattering media,” Opt. Lett. 25, 7–9 (2000). [CrossRef]
  14. B. Davison, J. B. Sykes, Neutron Transport Theory (Oxford U. Press, London, 1957), pp. 116–173.
  15. A. M. Weinberg, E. P. Wigner, “Transport theory and the diffusion of monoenergetic neutrons,” in The Physical Theory of Neutron Chain Reactors (The University of Chicago, Chicago, 1958), Chap. IX, pp. 219–278.
  16. K. M. Case, P. F. Zweifel, “Numerical methods,” in Linear Transport Theory (Addison-Wesley, Reading, Mass., 1967), Chap. 8, pp. 194–229.
  17. E. Amaldi, “The production and slowing down of neutrons,” in Encyclopedia of Physics, S. Flugge, ed. (Springer-Verlag, Berlin, 1959), Vol. 38/2, pp. 504–580.
  18. W. M. Star, “Comparing the P3-approximation with diffusion theory and with Monte Carlo calculations of light propagation in a slab geometry,” SPIE Institute Series IS 5, 146–154 (1989).
  19. E. L. Hull, T. H. Foster, “Steady-state reflectance spectroscopy in the P3 approximation,” J. Opt. Soc. Am. A 18, 584–599 (2001). [CrossRef]
  20. D. A. Boas, H. Liu, M. A. O’Leary, B. Chance, A. G. Yodh, “Photon migration within the P3 approximation,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, B. Chance, ed., Proc. SPIE2389, 240–247 (1995). [CrossRef]
  21. D. A. Boas, “Diffuse photon probes of structural and dynamical properties of turbid media: theory and biomedical applications,” Ph.D. dissertation (University of Pennsylvania, Philadelphia, Pa., 1996).
  22. J.-M. Kaltenbach, M. Kaschke, “Frequency- and time-domain modelling of light transport in random media,” in Medical Optical Tomography: Functional Imaging and Monitoring, G. Müller, B. Chance, R. Alfano, S. Arridge, J. Beuthan, E. Gratton, M. Kaschke, B. Masters, S. Svanberg, P. van der Zee, eds. (SPIE, Bellingham, Wash., 1993), Vol. IS11, pp. 65–86.
  23. K. Rinzema, L. H. P. Murrer, W. M. Star, “Direct experimental verification of light transport theory in an optical phantom,” J. Opt. Soc. Am. A 15, 2078–2088 (1998). [CrossRef]
  24. A. D. Kim, “Transport theory for light propagation in biological tissue,” J. Opt. Soc. Am. A 21, 820–827 (2004). [CrossRef]
  25. J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975), pp. 88, 90, 99, 101.
  26. F. Bevilacqua, C. Depeursinge, “Monte Carlo study of diffuse reflectance at source–detector separations close to one transport mean free path,” J. Opt. Soc. Am. A 16, 2935–2945 (1999). [CrossRef]
  27. F. Bevilacqua, D. Piguet, P. Marquet, J. D. Gross, B. J. Tromberg, C. Depeursinge, “In vivo local determination of tissue optical properties: applications to human brain,” Appl. Opt. 38, 4939–4950 (1999). [CrossRef]
  28. M. Abramowitz, I. A. Stegun, “Equations 10.2.15, 10.2.17, 10.2.20, and 10.2.21,” in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (National Bureau of Standards, Washington, D.C., 1972), p. 444.
  29. J. B. Fishkin, S. Fantini, M. J. van de Ven, E. Gratton, “Gigahertz photon density waves in a turbid medium: theory and experiments,” Phys. Rev. E 53, 2307–2319 (1996). [CrossRef]
  30. B. J. Tromberg, L. O. Svaasand, T.-T. Tsay, R. C. Haskell, “Properties of photon density waves in multiple-scattering media,” Appl. Opt. 32, 607–616 (1993). [CrossRef] [PubMed]
  31. S. Fantini, M. A. Franceschini, J. B. Fishkin, B. Barbieri, E. Gratton, “Quantitative determination of the absorption spectra of chromophores in strongly scattering media: a light-emitting-diode based technique,” Appl. Opt. 33, 5204–5213 (1994). [CrossRef] [PubMed]
  32. H. C. van de Hulst, “Phase Functions,” in Multiple Light Scattering: Tables, Formulas, and Applications, Volume 2 (Academic, New York, 1980), Chap. 10, pp. 303–330.
  33. W. M. Irvine, “Multiple scattering by large particles,” Astrophys. J. 142, 1563–1575 (1965). [CrossRef]
  34. This axis transformation allows both negative and positive values to be plotted on a single logarithmic plot. The dual-signed logarithmic axis is created by use of the transformation y= sinh−1(x/2)/ln(10), where the value x= 1 gives the first decade value above or below zero. This mapping corresponds to the inverse of x= 10y− 10−y, while the mapping of a conventional logarithmic axis is the inverse of x=10y. The dual-signed logarithmic axis may be considered a generalization of the conventional logarithmic axis, the latter being the limiting case for large y. Note that this axis transformation has 10 tick marks in the first decade on either side of 0, whereas the remaining (higher) decades have 9 ticks per decade.
  35. M. Abramowitz, I. A. Stegun, “Equation 8.1.2,” in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (National Bureau of Standards, Washington, D.C., 1972), p. 332.
  36. B. Davison, J. B. Sykes, Neutron Transport Theory (Oxford U. Press, London, 1957), pp. 122–126.
  37. M. Abramowitz, I. A. Stegun, “Equation 4.6.35,” in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (National Bureau of Standards, Washington, D.C., 1972), p. 88.
  38. K. M. Case, P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass., 1967), pp. 174–180.

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited