Skip to main content
SpringerLink
Log in

The extended Kubelka–Munk theory and its application to spectroscopy

  • Lecture Text
  • Published:
ChemTexts Aims and scope Submit manuscript

Abstract

The Kubelka–Munk theory is one of the main theories of light flux through homogeneous isotropic media. In this work, we used the extended solution of this theory, applied to a specimen on top of an arbitrary substrate, to obtain the overall spectral reflectance and transmittance. A complete colorimetric study can be derived from these calculations and this is shown by analyzing the effect of the different properties of the system (scattering and absorption coefficients, thickness, particle radius, surrounding medium) on its coordinates on the color space. Along with the analytical solutions to the original two-flux and the more modern four-flux models, we present a computing tool based on a Monte Carlo algorithm, which is very adequate in this context. In it, both the energy and the media are discretized, and the interaction is converted into probability of scattering and absorption. This numerical procedure also introduces new capabilities in the model, since it admits properties such as inhomogeneity in the layers, or more complex light–matter interactions, and offers solutions with temporal resolution, something applicable, for example, to pulses or transient states.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13

Similar content being viewed by others

Notes

  1. Assuming z to be the distance from the illuminated surface (positive downwards).

  2. It is worth noting that the expressions from [23] are recovered if we illuminate the system with totally collimated light (\(f=1\)).

  3. Calculated in terms of the Mie coefficients as well, see, e.g., [27, p. 120].

  4. In this case we are comparing orange and pink, which are two perfectly distinguishable colors.

References

  1. Schuster A (1905) Radiation through a foggy atmosphere. Astrophys J 21:1

    Article  Google Scholar 

  2. Kubelka P, Munk F (1931) An article on optics of paint layers. Z Tech Phys 12:593–609

    Google Scholar 

  3. Dzimbeg-Malcic V, Barbaric-Mikocevic Z, Itric K (2011) Kubelka-Munk theory in describing optical properties of paper (I). Tech Gaz 18(1):117–124

    Google Scholar 

  4. Dzimbeg-Malcic V, Barbaric-Mikocevic Z, Itric K (2012) Kubelka-Munk theory in describing optical properties of paper (II). Tech Gaz 19(1):191–196

    Google Scholar 

  5. Saunderson JL (1942) Calculation of the color of pigmented plastics. J Opt Soc Am 32(12):727–736

    Article  Google Scholar 

  6. Atherton E (1955) The relation of the reflectance of dyed fabrics to dye concentration and the instrumental approach to colour matching. J Soc Dyers Colour 71(7):389–398

    Article  CAS  Google Scholar 

  7. Hui Y (1992) Encyclopedia of food science and technology, encyclopedia of food science and technology, vol 1. Wiley, New York

    Google Scholar 

  8. Chandrasekhar S (2013) Radiative transfer. Courier corporation. Dover Publications, New York

    Google Scholar 

  9. Philips-Invernizzi B, Dupont D, Cazé C (2001) Bibliographical review for reflectance of diffusing media. Opt Eng 40(6):1082–1092

    Article  Google Scholar 

  10. Pierce PE, Marcus RT (1997) Radiative transfer theory solid color-matching calculations. Color Res Appl 22:72–87. https://doi.org/10.1002/(SICI)1520-6378(199704)22:2<72::AID-COL3>3.0.CO;2-0

    Article  Google Scholar 

  11. Kumar A, Choudhury A (2015) Principles of colour and appearance measurement, vol 2. Visual measurement of colour, colour comparison and management. Woodhead Publishing, Elsevier Science. eBook ISBN: 9781782423881

  12. Vargas WE, Niklasson GA (1997) Applicability conditions of the Kubelka-Munk theory. Appl Opt 36(22):5580–5586

    Article  CAS  Google Scholar 

  13. Fukshansky L, Kazarinova N (1980) Extension of the Kubelka-Munk theory of light propagation in intensely scattering materials to fluorescent media. J Opt Soc Am 70(9):1101–1111

    Article  CAS  Google Scholar 

  14. Mills D, Subbaswamy K (1981) II Surface and size effects on the light scattering spectra of solids, vol 19. Elsevier, Amsterdam, pp 45–137. https://doi.org/10.1016/S0079-6638(08)70201-4

    Book  Google Scholar 

  15. Kubelka P (1954) New contributions to the optics of intensely light-scattering materials. Part II: nonhomogeneous layers. J Opt Soc Am 44(4):330–335

    Article  Google Scholar 

  16. Stokes GG (1860) On the intensity of the light reflected from or transmitted through a pile of plates. R Soc Lond 11:545–557

    Google Scholar 

  17. Alcaraz de la Osa R, García Alonso A, Ortiz D, González F, Moreno F, Saiz JM (2016) Extension of the Kubelka-Munk theory to an arbitrary substrate: a Monte Carlo approach. J Opt Soc Am A 33(10):2053. https://doi.org/10.1364/josaa.33.002053

    Article  CAS  Google Scholar 

  18. Pérez M, González I, Saiz J, Moreno F, González F (2009) Saturation processes of photoluminescent pigments embedded in sinterized glass. J Modern Opt 56(13):1466–1474. https://doi.org/10.1080/09500340902990015

    Article  CAS  Google Scholar 

  19. Kubelka P (1948) New contributions to the optics of intensely light-scattering materials. Part I. J Opt Soc Am 38(5):448–457

    Article  CAS  Google Scholar 

  20. Murphy AB (2006) Modified Kubelka-Munk model for calculation of the reflectance of coatings with optically-rough surfaces. J Phys D Appl Phys 39(16):3571–3581

    Article  CAS  Google Scholar 

  21. Mishchenko MI (2002) Vector radiative transfer equation for arbitrarily shaped and arbitrarily oriented particles: a microphysical derivation from statistical electromagnetics. Appl Opt 41(33):7114–7134

    Article  Google Scholar 

  22. de la Hoz E, Alcaraz de la Osa R, Ortiz D, Saiz JM, Moreno F, González F (2019) Physically meaningful Monte Carlo approach to the four-flux solution of a dense multilayered system. J Opt Soc Am A 36(2):292. https://doi.org/10.1364/josaa.36.000292

    Article  Google Scholar 

  23. Vargas WE, Niklasson GA (1997) Pigment mass density and refractive index determination from optical measurements. J Phys Condens Matter 9(7):1661–1670

    Article  CAS  Google Scholar 

  24. Wyszecki G, Stiles WS (1982) Color Science: concepts and methods, quantitative data and formulae, 2nd edn. Wiley, New York

    Google Scholar 

  25. Coppel LG, Edström P, Lindquister M (2009) Open source Monte Carlo simulation platform for particle level simulation of light scattering from generated paper structures. PRS 2009. Kuopio, Finland, pp 1–10

  26. Babar S, Weaver JH (2015) Optical constants of Cu, Ag, and Au revisited. Appl Opt 54(3):477–481

    Article  CAS  Google Scholar 

  27. Bohren CF, Huffman DR (1983) Absorption and scattering of light by small particles. Wiley, New York

    Google Scholar 

  28. Kerker M, Scheiner P, Cooke DD, Kratohvil JP (1979) Absorption index and color of colloidal hematite. J Colloid Interface Sci 71(1):176–187

    Article  CAS  Google Scholar 

  29. Mayerhöfer TG, Höfer S, Popp J (2019) Deviations from Beer’s law on the microscale—nonadditivity of absorption cross sections. Phys Chem Chem Phys 21(19):9793–9801. https://doi.org/10.1039/c9cp01987a

    Article  PubMed  CAS  Google Scholar 

Download references

Acknowledgements

This research has been supported by the Spanish Ministry of Science and Innovation (MICINN) under project PGC2018-096649-B-I00.

Author information

Authors and Affiliations

  1. Group of Optics, Department of Applied Physics, Faculty of Science, University of Cantabria, Avda. de los Castros s/n, 39005, Santander, Cantabria, Spain

    R. Alcaraz de la Osa, I. Iparragirre, D. Ortiz & J. M. Saiz

Authors
  1. R. Alcaraz de la Osa
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. I. Iparragirre
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. D. Ortiz
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. J. M. Saiz
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to J. M. Saiz.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Alcaraz de la Osa, R., Iparragirre, I., Ortiz, D. et al. The extended Kubelka–Munk theory and its application to spectroscopy. ChemTexts 6, 2 (2020). https://doi.org/10.1007/s40828-019-0097-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s40828-019-0097-0

Keywords

Search

Navigation