## Estimation of light penetration depth in turbid media using laser speckles |

Optics Express, Vol. 20, Issue 13, pp. 13692-13701 (2012)

http://dx.doi.org/10.1364/OE.20.013692

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### Abstract

We present novel experimental method for estimation of the light penetration depth (LPD) in turbid media based on the analysis of the speckle pattern structure. Under the certain illumination conditions this structure is strongly dependent on the penetration depth. Presented theoretical model based on the Bragg diffraction from the thick holograms allows LPD estimation if only one parameter of the material, namely refractive index, of the material is known. Feasibility of the method was checked experimentally. Experimental results obtained for variety of the materials are in good agreement with the theoretical assumptions. It was shown that qualitative LPD comparison does not require knowledge of the material properties.

© 2012 OSA

## 1. Introduction

2. L. Wang, S. L. Jacques, and L. Zheng, “MCML--Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Methods Programs Biomed. **47**(2), 131–146 (1995). [CrossRef] [PubMed]

4. A. H. Hielscher, R. E. Alcouffe, and R. L. Barbour, “Comparison of finite-difference transport and diffusion calculations for photon migration in homogeneous and heterogeneous tissues,” Phys. Med. Biol. **43**(5), 1285–1302 (1998). [CrossRef] [PubMed]

6. S. A. Carp, S. A. Prahl, and V. Venugopalan, “Radiative transport in the delta-P1 approximation: accuracy of fluence rate and optical penetration depth predictions in turbid semi-infinite media,” J. Biomed. Opt. **9**(3), 632–647 (2004). [CrossRef] [PubMed]

*e*times smaller than that at the source. However, the formula is valid only at the assumption of homogenous medium with low absorption, which greatly limits applicability area of the method.

## 2. Theory and methods

### 2.1 Method description

### 2.2 Theoretical model

*z*it will be equal to:where

*I*

_{0}is the intensity of the incident light at the surface, and

*Z*is the depth at which the light intensity is

*e*times smaller than at the surface. In other words, the parameter

*Z*is an effective LPD.

*z*can be estimated as

*z*-direction and modulated by the factor

*e*, then the increment of the amplitude of the diffracted light wave from the depth

^{-z/Z}*z*is also defined by the Eq. (3). The angular selectivity of such a hologram can be illustrated by the well-known method of Ewald spheres [9].

*n*/λ are representing Ewald spheres for all possible light waves in the air and medium, respectively. Two vectors pointing up-left are representing incident and refracted waves prior to the object rotation. Angles

*θ*and

*θ*are incidence angles in the air and in the medium, respectively. From the diagram of Fig. 1 one can see that

_{n}*δθ*and

*δθ*show the change of the incidence angles due to the object rotation. The horizontal vector pointing right is a wave diffracted from the hologram whose vector is denoted as

_{n}*K*.

*K*. If the hologram vector length was set strictly, then diffraction would have been possible only at certain angle of incidence. However, there is small uncertainty in the vector length caused by the finite thickness of the hologram. Consequently, there is some set of angles at which incident light can be diffracted. Efficiency of this diffraction depends on proximity of the incident angle to the Bragg angle. As a result of the finite hologram size in

*z*coordinate, the length of the vector

*K*has uncertainty, so that the amplitude of the reconstructed hologram will not be zero even when the end-point of the grating vector

*K*does not touch the corresponding sphere but it is located at a distance

*ζ*. Since the space-frequency domain is nothing else but the Fourier domain, the relative amplitude of the reconstructed hologram can be evaluated as:Solution of this integral isAnd the intensity of the reconstructed wave is proportional to the square modulus of this equation:Exactly the same dependence should be for the maximum of the correlation function calculated for two speckle pattern snapshots made before and after the object rotation leading to the mismatch from the Bragg condition by the parameter

*ζ*in the spatial frequency domain because these speckle patterns can be considered as an initial hologram and its weakened version reconstructed after the rotation with the lower diffraction efficiency.

*ζ*does depend on the incident angle. It is easy to see that when the object is rotated by the angle of

*δθ*, the refracted beam is also changed by the angle of

*δθ*. In this case the end-point of the corresponding vector follows the arc shown by a thick line in Fig. 1. Unlike wave vectors corresponding to the light waves, vector of the hologram cannot be rotated since hologram is fixed in respect to the object. If the initial point of the hologram vector

_{n}*K*is fixed at the end-point of the wave vector corresponding to the refracted incident beam, then the end-point of the hologram vector also follows the arc when the object is rotated. In this case it goes out of the sphere of possible positions for the reconstructed wave, which leads to the Bragg condition mismatch.

*K*is congruent to the arc which shows the track of the left end-point of this vector. The mismatch parameter can be found as the change of the horizontal coordinate of the either right or left end-point since they are the same. It is easier to calculate the coordinate of the left vector’s end. As one can see, the horizontal coordinate of this end-point is defined as:Shift of the end-point caused by the change of the illumination angle on

*δθ*can be defined through differentiation of the Eq. (7) with respect to the angle:Substitution of Eq. (8) to Eq. (6) provides equation which describes the dependence of the amplitude of the correlation function on the illumination angle.

## 3. Experiment

### 3.1 Experimental setup

^{−5}radians. The setup was assembled so that the backlash was minimal, and the vector of the translation stage movement was perpendicular to the lever at the initial moment of time.

^{−5}radians, 39 - with the illumination angle changed by 17.4 ∙10

^{−5}radians and so on. In other words, dependence of the correlation function maximum from the illumination angle is calculated with better statistics in the region of small angle changes, where dependence is the strongest.

### 3.2 Results and discussion

*C*

_{0}is the autocorrelation function amplitude,

*Z*is the effective LPD,

*ζ*is a term accounting the alteration of inner speckles caused by the illumination angle change,

*δθ*is the angular step of the measurements,

*A*is the slope of the linear decline corresponding to the surface speckles, and

*B*is the constant. Fitting of this model to the experimental data becomes much easier task if following variable transformations are done:Substitution of Eq. (10) into Eq. (9) gives us following model:where

*α*,

*β*,

*A*and

*B*are the model parameters, and

*δθ*is the independent variable.

*A*,

*B*, and

*α*. First two parameters are representing influence of the surface speckles. Using these parameters one can exclude their influence on the experimental data and analyze separately the behavior of inner speckles as a function of the illumination angle. In its turn, the parameter

*α*is needed for estimation of the LPD. With known refractive index of the studied material one can calculate penetration depth using the following equation:But even without knowledge of the refractive index the reciprocal value of the dimensionless parameter

*α*allows for qualitative comparison of the LPD in semitransparent materials. Note that, according to Eq. (12), coefficient 1/

*α*have the same order of magnitude as the ratio of effective penetration depth to the wavelength of the illuminating beam. Dependence of the correlation function amplitude from the illumination angle for speckle pattern formed by the light scattered inside the material and the reciprocal of

*α*parameter for semitransparent materials are presented in Fig. 4 and Table 1 , respectively.

*α*for the metal, one can see that the graph representing this material is a straight line. Thus, experimental data confirms that there is no LPD in metal. As for semitransparent materials, it is clear from both Fig. 4 and Table 1 that the largest LPD is for the Teflon and the smallest one is for the cardboard. It is interesting to note that at constant angular resolution increasing of the LPD diminishes the peak amplitude of the graphs in Fig. 4. This fact is easy to explain, since the magnitude of the speckle pattern variation is directly proportional to the penetration depth, and less angular steps are required for complete decorrelation. This means that chosen angular resolution can be insufficient for some LPD values, at which decorrelation occurs even after one rotational step. Thus, the system resolution should be optimized with respect to the optical properties of the material. There is possibility that at the very large LPD values the speckle pattern formed on the CMOS matrix surface by the light experienced multiple scatterings inside the material will be no longer dependent on the illumination angle. In other words, it is possible that there is maximal LPD value evaluable by our method. However, this speculation should be checked experimentally.

10. F. P. Bolin, L. E. Preuss, R. C. Taylor, and R. J. Ference, “Refractive index of some mammalian tissues using a fiber optic cladding method,” Appl. Opt. **28**(12), 2297–2303 (1989). [CrossRef] [PubMed]

## 4. Conclusion

## Acknowledgments

## References and links

1. | K. Thyagarajan and A. Ghatak, |

2. | L. Wang, S. L. Jacques, and L. Zheng, “MCML--Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Methods Programs Biomed. |

3. | S. L. Jacques and L. Wang, “Monte Carlo modeling of light transport in tissues,” in |

4. | A. H. Hielscher, R. E. Alcouffe, and R. L. Barbour, “Comparison of finite-difference transport and diffusion calculations for photon migration in homogeneous and heterogeneous tissues,” Phys. Med. Biol. |

5. | R. A. J. Groenhuis, H. A. Ferwerda, and J. J. T. Bosch, “Scattering and absorption of turbid materials determined from reflection measurements. 1: theory,” Appl. Opt. |

6. | S. A. Carp, S. A. Prahl, and V. Venugopalan, “Radiative transport in the delta-P1 approximation: accuracy of fluence rate and optical penetration depth predictions in turbid semi-infinite media,” J. Biomed. Opt. |

7. | S. Xie, H. Li, and B. Li, “Measurement of optical penetration depth and refractive index of human tissue,” Chin. Opt. Lett. |

8. | V. V. Tuchin, |

9. | M. P. Petrov, S. I. Stepanov, and A. V. Khomenko, |

10. | F. P. Bolin, L. E. Preuss, R. C. Taylor, and R. J. Ference, “Refractive index of some mammalian tissues using a fiber optic cladding method,” Appl. Opt. |

**OCIS Codes**

(030.6140) Coherence and statistical optics : Speckle

(290.1350) Scattering : Backscattering

(290.7050) Scattering : Turbid media

**ToC Category:**

Scattering

**History**

Original Manuscript: April 12, 2012

Revised Manuscript: May 23, 2012

Manuscript Accepted: May 28, 2012

Published: June 4, 2012

**Virtual Issues**

Vol. 7, Iss. 8 *Virtual Journal for Biomedical Optics*

**Citation**

Igor S. Sidorov, Serguei V. Miridonov, Ervin Nippolainen, and Alexei A. Kamshilin, "Estimation of light penetration depth in turbid media using laser speckles," Opt. Express **20**, 13692-13701 (2012)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-20-13-13692

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### References

- K. Thyagarajan and A. Ghatak, Lasers: Fundamentals and Applications (Springer, 2010).
- L. Wang, S. L. Jacques, and L. Zheng, “MCML--Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Methods Programs Biomed.47(2), 131–146 (1995). [CrossRef] [PubMed]
- S. L. Jacques and L. Wang, “Monte Carlo modeling of light transport in tissues,” in Optical-Thermal Response of Laser-Irradiated Tissue, A. J. Welch and M. J. C. van Gemert, eds. (Plenum, 1995), Chap. 4.
- A. H. Hielscher, R. E. Alcouffe, and R. L. Barbour, “Comparison of finite-difference transport and diffusion calculations for photon migration in homogeneous and heterogeneous tissues,” Phys. Med. Biol.43(5), 1285–1302 (1998). [CrossRef] [PubMed]
- R. A. J. Groenhuis, H. A. Ferwerda, and J. J. T. Bosch, “Scattering and absorption of turbid materials determined from reflection measurements. 1: theory,” Appl. Opt.22(16), 2456–2462 (1983). [CrossRef] [PubMed]
- S. A. Carp, S. A. Prahl, and V. Venugopalan, “Radiative transport in the delta-P1 approximation: accuracy of fluence rate and optical penetration depth predictions in turbid semi-infinite media,” J. Biomed. Opt.9(3), 632–647 (2004). [CrossRef] [PubMed]
- S. Xie, H. Li, and B. Li, “Measurement of optical penetration depth and refractive index of human tissue,” Chin. Opt. Lett.1, 44–46 (2003).
- V. V. Tuchin, Tissue Optics: Light Scattering Methods and Instruments for Medical Diagnosis (SPIE Press, 2007), Chap. 6.
- M. P. Petrov, S. I. Stepanov, and A. V. Khomenko, Photorefractive Crystals in Coherent Optical Systems (Springer-Verlag, 1991).
- F. P. Bolin, L. E. Preuss, R. C. Taylor, and R. J. Ference, “Refractive index of some mammalian tissues using a fiber optic cladding method,” Appl. Opt.28(12), 2297–2303 (1989). [CrossRef] [PubMed]

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