## 250 years Lambert surface: does it really exist? |

Optics Express, Vol. 19, Issue 5, pp. 3881-3889 (2011)

http://dx.doi.org/10.1364/OE.19.003881

Acrobat PDF (926 KB)

### Abstract

The time-honored Lambert law is widely applied for describing the angle resolved reflectance from illuminated turbid media. We show that this law is only exactly fulfilled for a very special set of geometrical and optical properties. In contrast to what is believed so far, we demonstrate theoretically and experimentally that huge deviations from the Lambert law are ubiquitous. This finding is important for many applications such as those in biomedical optics.

© 2011 Optical Society of America

## 1. Introduction

4. M. S. Patterson, B. Chance, and B. C. Wilson, “Time-resolved reflectance and transmittance for the noninvasive measurement of tissue optical properties,” Appl. Opt. **28**, 2331–2336 (1989). [PubMed]

6. F. Martelli, S. Del Bianco, A. Ismaelli, and G. Zaccanti, *Light propagation through biological tissue and other diffusive media: theory, solutions, and software* (SPIE Press Book, 2009). [PubMed]

8. S. C. Gebhart, A. Mahadevan-Jansen, and W.-C. Lin, “Experimental and simulated angular profiles of fluorescence and diffuse reflectance emission from turbid media,” Appl. Opt. **44**, 4884–4901 (2005). [PubMed]

10. V. Backman, M. B. Wallace, L. T. Perelman, J. T. Arendt, R. S. Gurjar, M. G. Müller, Q. Zhang, G. Zonios, E. Kline, T. McGilian, S. Shapshay, T. Valdez, K. Badizadegan, J. M. Crawford, M. Fitzmaurice, S. Kabani, H. S. Levin, M. Seiler, R. R. Dasari, I. Itzkan, J. Van Dam, and M. S. Feld: “Detection of preinvasive cancer cells,” Nature **406**, 35–36 (2000). [PubMed]

8. S. C. Gebhart, A. Mahadevan-Jansen, and W.-C. Lin, “Experimental and simulated angular profiles of fluorescence and diffuse reflectance emission from turbid media,” Appl. Opt. **44**, 4884–4901 (2005). [PubMed]

11. J. Xia and G. Yao, “Angular distribution of diffuse reflectance in biological tissue,” Appl. Opt. **46**, 6552–6560 (2007). [PubMed]

## 2. Methods

6. F. Martelli, S. Del Bianco, A. Ismaelli, and G. Zaccanti, *Light propagation through biological tissue and other diffusive media: theory, solutions, and software* (SPIE Press Book, 2009). [PubMed]

*μ*, the scattering coefficient

_{a}*μ*, the phase function

_{s}*p*, often characterized by the anisotropy factor

*g*, which equals the average cosine of the scattering angle, and the refractive index

*n*[6

6. F. Martelli, S. Del Bianco, A. Ismaelli, and G. Zaccanti, *Light propagation through biological tissue and other diffusive media: theory, solutions, and software* (SPIE Press Book, 2009). [PubMed]

12. R. Michels, F. Foschum, and A. Kienle, “Optical properties of fat emulsions,” Opt. Express **16**, 5907–5925 (2008). [PubMed]

12. R. Michels, F. Foschum, and A. Kienle, “Optical properties of fat emulsions,” Opt. Express **16**, 5907–5925 (2008). [PubMed]

13. F. Martelli and G. Zaccanti, “Calibration of scattering and absorption properties of a liquid diffusive medium at NIR wavelengths. CW method,” Opt. Express **15**, 486–500 (2007). [PubMed]

## 3. Results

### 3.1. Reflectance versus polar angle

*R*(

*θ*) and assumed a turbid medium with randomly aligned microstructure. We systematically investigated the dependence of

*R*(

*θ*) on the optical and geometrical parameters and found that an exact Lambert surface is obtained for the following conditions: a) the albedo

*a*=

*μ*/(

_{s}*μ*+

_{s}*μ*) equals 1, i.e. the scattering coefficient is infinitely larger than the absorption coefficient, b) the refractive index is the same inside and outside the turbid medium, c) the angular dependence of the incident light power is ∝ cos(

_{a}*θ*). Figure 1 shows

*R*(

*θ*) divided by cos(

*θ*) using these three conditions for an isotropic phase function (black curve). We note that the beam profile including the beam diameter does not influence the results because the total reflectance is considered. Within the statistics of the Monte Carlo simulations the reflectance is 1 for all

*θ*-angles. We calculated

*R*(

*θ*)/cos(

*θ*) for different phase functions [14], e.g. for high anisotropy values (

*g*> 0.9) or obtained from Mie theory, using the three conditions and found that the Lambert law was always valid. For the special case of isotropic scattering and for specific polar angles the result can be validated with data from the literature [15]. In Fig. 1

*R*(

*θ*)/cos(

*θ*) is also shown for a Henyey-Greenstein phase function with an anisotropy factor of

*g*= 0.5 (red curve).

*R*(

*θ*)/cos(

*θ*) for

*a*= 1/1.01 (green curve; i.e. with a non-zero absorption coefficient), for perpendicular incident light (brown curve), and for a mismatch of the refractive indices (

*n*= 1.0 outside and

*n*= 1.4 in the turbid medium, blue curve). A finite absorption coefficient of the turbid medium results in a smaller reflectance at all angles. The decrease is smaller for large polar angles indicating that the photons emitted at these angles travel a shorter path through the scattering medium, which results in a lower probability for absorption. In principle, the smaller the depth of a considered photon the higher the probability for emission without further scattering interaction and, thus, the higher the relative probability for reflectance at large polar angles. This is caused by the fact that for a photon which is located deeper in the scattering medium the probability for being emitted from the medium at large polar angles (without further scattering) is decreased more than at small polar angles due to the exponential dependence on the photon path.

*R*(

*θ*)/cos(

*θ*) decreases for larger polar angles (compare Fig. 1) due to the deeper light penetration (i.e. the longer paths in the turbid medium) and the Fresnel-reflection, respectively. Thus, any alteration of the three conditions results in a non-Lambertian reflectance.

*R*(

*θ*)/cos(

*θ*) on the distance to an incident continuous wave point beam having an infinitely small diameter (

*δ*(

*r*), Fig. 2a) and on the time after an incident pulsed beam having an infinitely small pulse duration (

*δ*(

*t*), Fig. 2b) by applying the Monte Carlo method. We employed the three standard conditions for obtaining the Lambert surface using

*μ*′

*= 1 mm*

_{s}^{−1}and applied an isotropic phase function. The refractive index inside and outside the scattering medium was 1.4.

*R*(

*θ*)/cos(

*θ*)-curves at three different distance ranges to the incident point beam exhibit significant differences, see Fig. 2a. The penetration depth of photons that are emitted at short distances are smaller compared to the photons that are emitted at large distances. Thus, for the latter case more photons are emitted at smaller angles, see Fig. 1.

*R*(

*θ*)/cos(

*θ*) is increased for short times and is decreased at long times.

### 3.2. Reflectance versus polar and azimuthal angles

*ϕ*were summed up. It could be seen that, when the above mentioned three conditions are not fulfilled, the assumption of a Lambertian surface is incorrect, but the deviations might be tolerable in some situations. However, in many applications the dependence of the emitted light on the polar and on the azimuthal angles

*R*(

*θ*,

*ϕ*) is important. Figure 3 shows

*R*(

*θ*,

*ϕ*) for three positions having different distances from the incident point beam. Reflectance data for the whole reflectance half sphere (0° ≤

*θ*≤ 90°, 0° ≤

*ϕ*≤ 360°) are depicted. Note that for the polar plots the reflectance is not divided by cos(

*θ*). The optical properties of the turbid medium were

*μ*′

*= 1 mm*

_{s}^{−1},

*μ*= 0.001 mm

_{a}^{−1}using an isotropic phase function. A perpendicular incident continuous wave point beam and refractive index matching was assumed. The figures show that

*R*(

*θ*,

*ϕ*) is very different from that of a Lambertian surface. At small distances almost all light is emitted in one

*ϕ*-direction (

*ϕ*= 180°). This is caused by single scattering, which is dominant at small distances, because the single scattered light can only be emitted at

*ϕ*= 180° due to geometrical constraints, compare Fig. 3b. For larger distances

*R*(

*θ*,

*ϕ*) becomes more rotationally symmetric (Fig. 3c) which is caused by multiple scattered photons. However, even at a distance of 4.9 mm the reflectance is still anisotropic, compare Fig. 3d. We note that considering only the

*θ*-dependance the reflectance is much closer to the Lambert law for all three distances due to the summing up of reflectance values for the

*ϕ*-angles.

*R*(

*θ*)/cos(

*θ*) for large polar angles is mainly due to the refractive index mismatch between Intralipid (

*n*≈ 1.33) and air (

*n*= 1.0), compare Fig. 1.

### 3.3. Reflectance from turbid media with anisotropic light propagation

*R*(

*θ*,

*ϕ*) for soft wood. For the simulations we used a model for describing the anisotropic light propagation in wood that we have recently developed [16

16. A. Kienle, C. D’Andrea, F. Foschum, P. Taroni, and A. Pifferi, “Light propagation in dry and wet soft wood,” Opt. Express **16**, 9895–9906 (2008). [PubMed]

17. A. Kienle and R. Hibst, “Light guiding in biological tissue due to scattering,” Phys. Rev. Lett. **97**, 018104 (2006). [PubMed]

*d*= 30

*μ*m, filled with air, see Fig. 5a). The phase functions and the scattering coefficients for the different incident angles relative to the cylinder axis were calculated with an analytical solution of the Maxwell equations [18] and stored for use in the Monte Carlo program [16

16. A. Kienle, C. D’Andrea, F. Foschum, P. Taroni, and A. Pifferi, “Light propagation in dry and wet soft wood,” Opt. Express **16**, 9895–9906 (2008). [PubMed]

*g*= 0.9). Originally, we used this model to describe the spatially resolved reflectance and compared it successfully to measurements [16

16. A. Kienle, C. D’Andrea, F. Foschum, P. Taroni, and A. Pifferi, “Light propagation in dry and wet soft wood,” Opt. Express **16**, 9895–9906 (2008). [PubMed]

*R*(

*θ*,

*ϕ*) for the reflectance, see Fig. 5b. A strong anisotropic

*R*(

*θ*,

*ϕ*) is obtained from the Monte Carlo simulations which is mainly due to the scattering characteristics of the cylinders. When an infinitely long cylinder is perpendicularly illuminated, light is scattered only in the plane which is perpendicular to the cylinder axis.

*x*-axis.

## 4. Discussion and conclusion

*a*=

*μ*/(

_{s}*μ*+

_{s}*μ*) equals 1, b) the refractive index is the same inside and outside the turbid medium, c) the angular dependence of the incident light power is ∝ cos(

_{a}*θ*).

*μ*m) with the exception of interference effects like speckles or coherent back scattering [23

23. F. Voit, J. Schäfer, and A. Kienle, “Light scattering by multiple wpheres: comparison between Maxwell theory and radiative-transfer-theory calculations,” Opt. Lett. **34**, 2593–2595 (2009). [PubMed]

*R*(

*θ*,

*ϕ*) is examined at a certain location and a certain time after the irradiation of a

*δ*(

*t*,

*r*)-source (having a certain incident direction) onto a turbid medium with aligned microstructure, delivers even more involved reflectance characteristics. Further, for the above presented results unpolarized light was assumed. Implementation of light propagation of polarized light gives additional features which can be used for investigation of the microstructure of turbid media. Therefore, 250 years after Johann Heinrich Lambert published his famous law the investigation of the angular light reflectance from turbid media is a more vivid research field than ever before.

## References and links

1. | J. H. Lambert, |

2. | H. Gross, |

3. | S. Georghiades, P. N. Belhumeur, and D. J. Kriegman, “From few to many: illumination cone models for face recognition under variable lighting and pose,” IEEE Trans. Pattern Anal. Mach. Intell. |

4. | M. S. Patterson, B. Chance, and B. C. Wilson, “Time-resolved reflectance and transmittance for the noninvasive measurement of tissue optical properties,” Appl. Opt. |

5. | A. Kienle, L. Lilge, M. S. Patterson, R. Hibst, R. Steiner, and B. C. Wilson, “Spatially-resolved absolute diffuse reflectance measurements for non-invasive determination of the optical scattering and absorption coefficients of biological tissue,” Appl. Opt. |

6. | F. Martelli, S. Del Bianco, A. Ismaelli, and G. Zaccanti, |

7. | D. J. Durian, “Influence of boundary reflection and refraction on diffusive photon transport,” Phys. Rev. E |

8. | S. C. Gebhart, A. Mahadevan-Jansen, and W.-C. Lin, “Experimental and simulated angular profiles of fluorescence and diffuse reflectance emission from turbid media,” Appl. Opt. |

9. | D. C. Adler, Y. Chen, R. Huber, J. Schmitt, J. Connolly, and J. G. Fujimoto, “Three-dimensional endomicroscopy using optical coherence tomography,” Nature Photon. |

10. | V. Backman, M. B. Wallace, L. T. Perelman, J. T. Arendt, R. S. Gurjar, M. G. Müller, Q. Zhang, G. Zonios, E. Kline, T. McGilian, S. Shapshay, T. Valdez, K. Badizadegan, J. M. Crawford, M. Fitzmaurice, S. Kabani, H. S. Levin, M. Seiler, R. R. Dasari, I. Itzkan, J. Van Dam, and M. S. Feld: “Detection of preinvasive cancer cells,” Nature |

11. | J. Xia and G. Yao, “Angular distribution of diffuse reflectance in biological tissue,” Appl. Opt. |

12. | R. Michels, F. Foschum, and A. Kienle, “Optical properties of fat emulsions,” Opt. Express |

13. | F. Martelli and G. Zaccanti, “Calibration of scattering and absorption properties of a liquid diffusive medium at NIR wavelengths. CW method,” Opt. Express |

14. | R. Graaff, J. G. Arnoudse, F. F. M. de Mul, and H. W. Jentink, “Similarity relations for anisotropic scattering in absorbing media,” Opt. Eng. |

15. | D. C. Van der Hulst, |

16. | A. Kienle, C. D’Andrea, F. Foschum, P. Taroni, and A. Pifferi, “Light propagation in dry and wet soft wood,” Opt. Express |

17. | A. Kienle and R. Hibst, “Light guiding in biological tissue due to scattering,” Phys. Rev. Lett. |

18. | H. A. Yousif and E. Boutros, “A FORTRAN code for the scattering of EM plane waves by an infinitely long cylinder at oblique incidence,” Comput. Phys. Commun. |

19. | Y. Sun, “Statistical ray method for deriving reflection models of rough surfaces,” J. Opt. Soc. Am. A |

20. | P. Beckmann, A. Spizzichino, and A. Norwood, |

21. | A. Ishimaru, |

22. | M. I Mishchenko, L. D. Travis, and A. A. Lacis, |

23. | F. Voit, J. Schäfer, and A. Kienle, “Light scattering by multiple wpheres: comparison between Maxwell theory and radiative-transfer-theory calculations,” Opt. Lett. |

**OCIS Codes**

(170.3660) Medical optics and biotechnology : Light propagation in tissues

(290.0290) Scattering : Scattering

(290.1990) Scattering : Diffusion

(290.7050) Scattering : Turbid media

**ToC Category:**

Scattering

**History**

Original Manuscript: November 30, 2010

Revised Manuscript: January 27, 2011

Manuscript Accepted: February 2, 2011

Published: February 14, 2011

**Virtual Issues**

Vol. 6, Iss. 3 *Virtual Journal for Biomedical Optics*

**Citation**

Alwin Kienle and Florian Foschum, "250 years Lambert surface: does it really exist?," Opt. Express **19**, 3881-3889 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-5-3881

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### References

- J. H. Lambert, Photometria sive de mensura et gradibus luminus, colorum et umbrae (Eberhard Klett, 1760).
- H. Gross, Handbook of Optical Systems; Volume 1: Fundamental of Technical Optics (Wiley-VCH, 2005).
- S. Georghiades, P. N. Belhumeur, and D. J. Kriegman, “From few to many: illumination cone models for face recognition under variable lighting and pose,” IEEE Trans. Pattern Anal. Mach. Intell. 23, 243–260 (2001).
- M. S. Patterson, B. Chance, and B. C. Wilson, “Time-resolved reflectance and transmittance for the noninvasive measurement of tissue optical properties,” Appl. Opt. 28, 2331–2336 (1989). [PubMed]
- A. Kienle, L. Lilge, M. S. Patterson, R. Hibst, R. Steiner, and B. C. Wilson, “Spatially-resolved absolute diffuse reflectance measurements for non-invasive determination of the optical scattering and absorption coefficients of biological tissue,” Appl. Opt. 35, 2304–2314 (1996). [PubMed]
- F. Martelli, S. Del Bianco, A. Ismaelli, and G. Zaccanti, Light propagation through biological tissue and other diffusive media: theory, solutions, and software (SPIE Press Book, 2009). [PubMed]
- D. J. Durian, “Influence of boundary reflection and refraction on diffusive photon transport,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 2, 857–865 (1994).
- S. C. Gebhart, A. Mahadevan-Jansen, and W.-C. Lin, “Experimental and simulated angular profiles of fluorescence and diffuse reflectance emission from turbid media,” Appl. Opt. 44, 4884–4901 (2005). [PubMed]
- D. C. Adler, Y. Chen, R. Huber, J. Schmitt, J. Connolly, and J. G. Fujimoto, “Three-dimensional endomicroscopy using optical coherence tomography,” Nat. Photonics 1, 709–716 (2007).
- V. Backman, M. B. Wallace, L. T. Perelman, J. T. Arendt, R. S. Gurjar, M. G. Müller, Q. Zhang, G. Zonios, E. Kline, T. McGilian, S. Shapshay, T. Valdez, K. Badizadegan, J. M. Crawford, M. Fitzmaurice, S. Kabani, H. S. Levin, M. Seiler, R. R. Dasari, I. Itzkan, J. Van Dam, and M. S. Feld, “Detection of preinvasive cancer cells,” Nature 406, 35–36 (2000). [PubMed]
- J. Xia and G. Yao, “Angular distribution of diffuse reflectance in biological tissue,” Appl. Opt. 46, 6552–6560 (2007). [PubMed]
- R. Michels, F. Foschum, and A. Kienle, “Optical properties of fat emulsions,” Opt. Express 16, 5907–5925 (2008). [PubMed]
- F. Martelli and G. Zaccanti, “Calibration of scattering and absorption properties of a liquid diffusive medium at NIR wavelengths. CW method,” Opt. Express 15, 486–500 (2007). [PubMed]
- R. Graaff, J. G. Arnoudse, F. F. M. de Mul, and H. W. Jentink, “Similarity relations for anisotropic scattering in absorbing media,” Opt. Eng. 32, 244–252 (1993).
- D. C. Van der Hulst, Multiple Light Scattering (Academic Press, 1980).
- A. Kienle, C. D’Andrea, F. Foschum, P. Taroni, and A. Pifferi, “Light propagation in dry and wet soft wood,” Opt. Express 16, 9895–9906 (2008). [PubMed]
- A. Kienle and R. Hibst, “Light guiding in biological tissue due to scattering,” Phys. Rev. Lett. 97, 018104 (2006). [PubMed]
- H. A. Yousif and E. Boutros, “A FORTRAN code for the scattering of EM plane waves by an infinitely long cylinder at oblique incidence,” Comput. Phys. Commun. 69, 406–414 (1992).
- Y. Sun, “Statistical ray method for deriving reflection models of rough surfaces,” J. Opt. Soc. Am. A 24, 724–744 (2007).
- P. Beckmann, A. Spizzichino, and A. Norwood, The Scattering of Electromagnetic Waves from Rough Surfaces (Artech House, Inc., 1987).
- A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic Press, 1978).
- M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Multiple Scattering of Light by Particles: Radiative Transfer and Coherent Backscattering (Cambridge U. Press, 2006).
- F. Voit, J. Schäfer, and A. Kienle, “Light scattering by multiple wpheres: comparison between Maxwell theory and radiative-transfer-theory calculations,” Opt. Lett. 34, 2593–2595 (2009). [PubMed]

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