## Diffuse imaging and radius dependent frequency correlations in strongly scattering media |

Optics Express, Vol. 22, Issue 11, pp. 13330-13342 (2014)

http://dx.doi.org/10.1364/OE.22.013330

Acrobat PDF (2884 KB)

### Abstract

A new probe of multiple scattering material is demonstrated experimentally. Light from a tunable wavelength source is focused to a point on the surface of an opaque slab. A fraction of this light penetrates into the slab, is multiply scattered, and reemerges at the surface creating a surface speckle pattern. The full spatial and frequency speckle can be easily and quickly recorded using a CCD and an acoustooptical tunable filter. Both the average intensity and frequency correlations of intensity are analyzed as a function of the distance to the source. This method is demonstrated experimentally for white paint. The resulting model yields information about both the static and dynamic transport properties of the sample. The technique has prospects for both static and time resolved diffuse imaging in strongly scattering materials. The setup can be easily used as an add-on to a standard bright field microscope.

© 2014 Optical Society of America

## 1. Introduction

1. Merriam Webster online, http://www.merriam-webster.com/.

2. T. Durduran, R. Choe, W. B Baker, and A. G. Yodh, “Diffuse optics for tissue monitoring and tomography,” Reports on Progress in Physics **73**(7), 076701 (2010). [CrossRef]

3. A. P. Gibson, J. C. Hebden, and S. R. Arridge, “Recent advances in diffuse optical imaging,” Phys. Med. Biol. **50**, R1–R43 (2005). [CrossRef] [PubMed]

4. F. F. Jobsis, ”Noninvasive, infrared monitoring of cerebral and myocardial oxygen, sufficiency and circulatory parameters,” Science **198**(4323), 1264–1267 (1977). [CrossRef] [PubMed]

5. M. S. Patterson, B. Chance, and B. C. Wilson, “Time resolved reflectance and transmittance for the non-invasive measurement of tissue optical properties,” Appl. Opt **28**(12), 2331–2336 (1989). [CrossRef] [PubMed]

6. M. S. Patterson, J. D. Moulton, Br. C. Wilson, K. W. Berndt, and J. R. Lakowicz, “Frequency-domain reflectance for the determination of the scattering and absorption properties of tissue,” Appl. Opt. **30**(31), 4474–4476 (1991). [CrossRef] [PubMed]

7. P. M. Johnson, A. Imhof, B. P. J. Bret, J. Gómez Rivas, and A. Lagendijk, “Time-resolved pulse propagation in a strongly scattering material,” Phys. Rev. E **68**(1), 016604 (2003). [CrossRef]

8. A. Z. Genack and J. M. Drake, “Relationship between optical intensity, fluctuations and pulse propagation in random media,” EPL (Europhysics Letters) **11**(4), 331–336 (1990). [CrossRef]

9. O. L. Muskens and A. Lagendijk, “Method for broadband spectroscopy of light transport through opaque scattering media,” Opt. Lett. **34**, 395–397 (2009). [CrossRef] [PubMed]

## 2. Experiment

### 2.1. Measurement

*ρ*< 10 microns) and a high intensity level used for large radii (

*ρ*> 10 microns). The high intensity level was roughly ten times the low level and saturated the camera for small radii. For each condition, a measurement was taken at each of three different locations of the sample to test for consistency and to estimate the uncertainty. At the end of the paint measurements a second reference measurement was made. This gave a check to ensure that no systematic drift of the laser occurred during the measurement.

### 2.2. Data analysis

*I*(

*,*

**ρ**_{i}*ω*) where

_{j}*= (*

**ρ**_{i}*x*,

*y*)

*is the vector position of the*

_{i}*i*pixel measured from the center of the laser focus and

^{th}*ω*is the

_{j}*j*discrete optical frequency where

^{th}*ω*= 2

_{j}*πc/λ*. In our notation we will use bold font for vectors and normal font for the scalar magnitude.

_{j}*I*(

*,*

**ρ**_{i}*ω*) can be obtained for both analyzer polarizations, parallel ‖ and perpendicular ⊥ to the polarization of the incoming light.

_{j}*ρ̄*and width Δ

*ρ*such that

*ρ*= |

_{i}*| is radius of the*

**ρ**_{i}*i*pixel. Each frequency range has a width Δ

^{th}*ω*about a given average frequency

*ω̄*such that |

*ω*−

_{j}*ω̄*| ≤ Δ

*ω*/2. The ranges are chosen to be large enough to provide good statistics, but narrow enough so that the optical properties of the material do not vary substantially within the range. In this way fluctuations due to speckle are assumed to arise from independent mesoscopic realizations of the same macroscopic material parameters allowing for standard statistical methods to be applied.

*averaged over a given frequency band*

**ρ**_{i}*ω̄*±Δ

*ω*/2, and the average intensity for a given radius range

*r̄*±Δ

*r*/2 and frequency band

*ω̄*±Δ

*ω*/2, The error bar on

*Ī*(

*ρ̄*,

*ω̄*) is its standard deviation divided by the square root of the number of speckle spots in the ring defined by

*ρ̄*± Δ

*ρ*/2. The area of the speckle spot in pixel units is determined from the full width half maximum of the spatial autocorrelation each image.

*is calculated by*

**ρ**_{i}*δω*is given by the discrete frequency difference

_{k}*ω*

_{j+k}−

*ω*. This function is then averaged over all pixels in the range Δ

_{j}*ρ*to give the average autocorrelation function,

*δω*is frequency independent. By definition,

_{k}*C̄*

_{ρ̄,ω̄}, (0) = 1. When presenting the experimental data, some of the above expressions will be subscripted with ‖ and ⊥ for the parallel and perpendicular analyzer orientations respectively. Nonlinear least squares fitting routines from Matlab were used to fit the processed data to theoretical predictions.

## 3. Theory

### 3.1. Analytic expressions

*I*(

_{s}**r**,

*t*,

**ŝ**), the intensity per unit solid angle for a given position

**r**at time

*t*and direction

**ŝ**[10]. The equation includes the propagation and scattering properties of electromagnetic waves, but does not include interference effects such as speckle, enhanced backscattering, or localization. It can however be used to deduce speckle statistics since it gives informaton about average light path lengths and times of flight.

*D*is the diffusion constant,

*τ*is the absorption time, and

_{a}*S*(

**r**,

*t*) is the source function [11

11. M. C. W. van Rossum and Th. M. Nieuwenhuizen, “Multiple scattering of classical waves: microscopy, mesoscopy, and diffusion,” Rev. Mod. Phys. **71**(1), 313–371 (1999). [CrossRef]

*U*(

**r**,

*t*) is much larger than the diffuse flux of energy in any direction and the absorption time is small compared to the mean free time between scattering events. The diffusion equation has been applied effectively to a wide variety of multiple scattering problems providing that the surface behavior is treated with care. For short paths near the surface and for extreme absorption this approximation breaks down, but for the low absorbing samples and long paths, as will be important for this article, the diffusion approximation is valid.

**= 0 where**

*ρ***is the 2d position vector on the surface of the sample, and a depth of**

*ρ**z*=

*z*. We therefore assume a delta function source of the form

_{p}*S*(

**r**,

*t*) =

*δ*(

**)**

*ρ**δ*(

*z*) Θ(

_{p}*t*), where

**r**= (

*ρ*,

_{x}*ρ*,

_{y}*z*) and Θ(

*t*) is the step function. We rescale all length units by the mean free path (

*ρ*⇒

*ρ*/

*l*,

*z*⇒

*z/l*) and time units by

*D*=

*l*

^{2}/ (3

*τ*) to simplify the expressions. The method of images yields an analytic solution for the average intensity propagator from the source to position (

_{mf}**,**

*ρ**z*): where

*dz*≡ (1 − 2

_{n}*n*)

*z*−

_{p}*z*− 2

*nz*

_{0}, and

*z*

_{0}is the extrapolation length in units of

*l*[12]. Equation (7) is a scalar quantity. The frequency dependent propagator can be found by Fourier transforming Eq. (7) yielding: where

13. D. Contini, F. Martelli, and G. Zaccanti, “Photon migration through a turbid slab described by a model based on diffusion approximation. i. theory,” Appl. Opt. **36**(19), 4587–4599 (1997). [CrossRef] [PubMed]

*Ī*(

*ρ̄*,

*ω̄*), the average (static) intensity from Eq. (2) can be directly compared with

*R*(

*ρ*,

*ω*= 0). The correlation function

*C̄*(

*ρ̄*,

*ω̄*,

*δω*) from Eq. (5) can be directly compared to |

_{k}*R*(

*ρ*,

*ω*= 0)|

^{2}.

*τ*. This gives: where the plus and minus signs refer to the parallel and perpendicular orientations of the analyzer respectively. Inspection of Eq. (7) reveals that this can be rewritten as where

_{p}*C*

_{‖,⊥}(

*ρ*,

*ω*) = |

*R*

_{‖,⊥}(

*ρ*,

*ω*)

_{τa,τp}|

^{2}. This simple analytic dependence on

*τ*allows analytic calculation of any polarization dependent quantity and is therefore somewhat simpler than previous descriptions of the polarization dependence of diffuse light [14

_{ap}14. L. Fernando Rojas-Ochoa, D. Lacoste, R. Lenke, P. Schurtenberger, and F. Scheffold, “Depolarization of backscattered linearly polarized light,” J. Opt. Soc. Am. A **21**(9), 1799–1804 (2004). [CrossRef]

15. E. Akkermans, P. E. Wolf, R. Maynard, and G. Maret, “Theoretical study of the coherent backscattering of light by disordered media,” J. Phys. France **49**(1), 77–98 (1988). [CrossRef]

*L*by adding an additional infinite series of image charges. This is achieved with no further calculation via the substitution

_{slab}*dz*⇒

_{n}*dz*,

_{n}*where*

_{m}*dz*,

_{n}*≡ (1 − 2*

_{m}*n*)

*z*−

_{p}*z*−2

*nz*

_{0}+ 2

*m*(

*L*+ 2

*z*

_{0}) and adding the infinite summation over

*m*such that The rest of the results for the reflectivity and correlation function follow with the same substitution.

### 3.2. Simulated speckle

*ω*over which the correlations can be calculated for a given value of

*ω̄*. This second limit affects the data for small values of

*ρ*where the frequency correlations are very wide. Simply widening Δ

*ω*indefinitely is not an option, since the real sample scattering and absorption parameters are frequency dependent.

17. D.J. Young and N.C. Beaulieu, “The generation of correlated rayleigh random variates by inverse discrete fourier transform,” Communications, IEEE Transactions on **48**(7), 1114–1127 (2000). [CrossRef]

*G*is generated and each random number associated with a point in a time series

_{i}*t*. The simulated temporal speckle amplitude

_{i}*E*(

_{sim}*ρ*,

*t*) is then generated for each value of

*ρ*and

*t*via

_{i}*E*(

_{sim}*ρ*,

*ω*) is then computed via numerical (fast) Fourier transform of

*E*(

_{sim}*ρ*,

*t*). The intensity speckle is the absolute square of this

*I*(

_{sim}*ρ*,

*ω*) = |

*E*(

_{sim}*ρ*,

*ω*)|

^{2}. This intensity speckle may be used to simulate correlation functions following the routines described in section 3.1.

## 4. Results

*I*

_{⊥}(

*ρ*,

_{x}*ρ*,

_{y}*ω*) for

*ω*= 2.99 fs

^{−1}, (

*λ*= 630 nm). The raw data (inset) shows the spatial intensity speckle with the expected exponential intensity distribution statistics. These speckles have been digitally filtered out via a low pass digital square boxcar filter of size 2.5 × 2.5

*μ*m

^{2}to reduce the spatial speckle. The isointensity lines are logarithmically distributed, demonstrating a large dynamic range for the measurement. They form nearly perfect circles about the focus point, indicating radially symmetric propagation of light in the plane of the sample surface and good alignment of the sample surface with the image plane of the microscope. The roughness of the isointensity lines is the result of incomplete filtering of the speckle and not, for the most part, of sample roughness or dark noise of the camera.

*Ī*(

*ρ̄*,

*ω*) with Δ

*ρ*= 1

*μ*m and

*ω*= 2.99 × 10

^{3}THz for each of the two polarizations. The two polarizations show different behavior near the center and then merge completely beyond

*ρ*≈ 10

*μ*m. This is expected, since the light immediately exiting the sample at

*ρ*= 0 contains many paths that have undergone just one or two scattering events and are thus not yet depolarized. Away from the focus the all light paths contain many scattering events and the light is completely depolarized. No dramatic specular reflection effects were visible by eye for these samples. By definition, such effects would occur only at the focal spot, i.e. within a radius of 0.5 microns. At the smallest averaged radius, 1 micron, the data is slightly higher than the fit for parallel polarized light and lower for perpendicular. However this slight difference falls within the error bars and thus does not affect the fitting results. Enhanced backscattering in principle also slightly biases the total refection towards the parallel polarization configuration. However the effect is so small for mean free paths on the order of microns and angular apertures as large as ours that it should not alter the fitting results.

*ω*= 0 as shown in Fig. 3. The value of

*z*

_{0}was fixed at 2.42 by assuming an effective refractive index of the paint of

*n*= 1.5, while the parameters

_{eff}*A*,

*τ*,

_{a}*τ*, and

_{p}*l*were allowed to float freely. The quality of the fits for

*ρ*≤ 30

*μm*were excellent as seen in the example in Fig. 3 with

*χ*

^{2}values on the order of 1. This demonstrates the effectiveness of the diffusion model for characterizing the data. Extending the fitting range beyond

*ρ*= 30

*μm*lead to increases in

*χ*

^{2}as the fit could visibly not account for the data for the entire radius range.

*L*. In this article we use the fit to the full theoretical expression to calculate material parameters.

_{a}*τ*(in units of mean free time), decreases systematically while the depolarization time

_{a}*τ*and the mean free path

_{p}*l*remain constant. The bottom plot shows the goodness of fit for the entire range.

*ρ*=0

*μ*m, 4

*μ*m, and 10

*μ*m for the top, middle, and bottom plots respectively). The near zero crossings indicate sufficient measurement resolution for applying single speckle statistics. It is already apparent that smaller radii show correlations over a wider range. This corresponds to our theoretical expectations. The smaller radii contain more short paths that show a high degree frequency correlation. The top plot also shows one difficulty of analyzing the data. The correlations begin to span the entire range of the scan, while we know from the fit to average intensity that the parameters (the absorption time) may change considerably over this span.

*C̄*

_{ρ̄,ω̄}, (

*δω*) for several values of

_{k}*ρ̄*and

*ω̄*are plotted in Fig. 6. In the top plot, these functions drop more rapidly to zero as the radius decreases signifying a reduction in correlation and therefore longer average time of flights. This is expected due to the greater distance from the focus point. By contrast,

*C̄*

_{ρ̄,ω̄}, (

*δω*) is fairly insensitive to variations in

_{k}*ω̄*at constant

*r̄*as seen in the lower half of Fig. 6.

*τ*= 16 fs were used with no free parameters. The mean free time was calculated from

_{mf}*τ*=

_{mf}*l*/(

*c*/

*n*) with the same value of

_{eff}*n*as used previously (

_{eff}*n*= 1.5). This calculation assumes no resonances in the material, i.e. that the energy and phase velocities are equivilant. The analytical result shows qualitatively the same trend as the data. Nevertheless, the comparison at both low and high values of the radius are off by as much as a factor of two.

_{eff}*δλ*∼ 20–50 nm).

*l*= 2.8

*μ*m,

*τ*= 50,

_{a}*τ*= 0.6, and,

_{p}*τ*= 20 fs) with a wavelength resolution of 0.9 nm to fit the data by eye. (Constructing a fitting routine using the simulation proved to be too time consuming). The results show that indeed the discrepancies with the analytical expression can be explained in part by resolution and processing limitations. However the data still does not overlap with the theory within the given error bars. This could point to a limitation of the theory itself. For example for short light paths, which dominate the signal at small radii, the diffusion theory itself may not be applicable and a more complete radiative transfer model may be needed. This will be an important consideration when analyzing more complex structures.

_{mf}*μ*m.

## 5. Conclusions

18. M. Alfeld and J. A.C. Broekaert, “Mobile depth profiling and sub-surface imaging techniques for historical paintings - review,” Spectrochimica Acta Part B: Atomic Spectroscopy **88**(0), 211–230 (2013). [CrossRef]

## Acknowledgments

## References

1. | Merriam Webster online, http://www.merriam-webster.com/. |

2. | T. Durduran, R. Choe, W. B Baker, and A. G. Yodh, “Diffuse optics for tissue monitoring and tomography,” Reports on Progress in Physics |

3. | A. P. Gibson, J. C. Hebden, and S. R. Arridge, “Recent advances in diffuse optical imaging,” Phys. Med. Biol. |

4. | F. F. Jobsis, ”Noninvasive, infrared monitoring of cerebral and myocardial oxygen, sufficiency and circulatory parameters,” Science |

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

6. | M. S. Patterson, J. D. Moulton, Br. C. Wilson, K. W. Berndt, and J. R. Lakowicz, “Frequency-domain reflectance for the determination of the scattering and absorption properties of tissue,” Appl. Opt. |

7. | P. M. Johnson, A. Imhof, B. P. J. Bret, J. Gómez Rivas, and A. Lagendijk, “Time-resolved pulse propagation in a strongly scattering material,” Phys. Rev. E |

8. | A. Z. Genack and J. M. Drake, “Relationship between optical intensity, fluctuations and pulse propagation in random media,” EPL (Europhysics Letters) |

9. | O. L. Muskens and A. Lagendijk, “Method for broadband spectroscopy of light transport through opaque scattering media,” Opt. Lett. |

10. | A. Ishimaru, |

11. | M. C. W. van Rossum and Th. M. Nieuwenhuizen, “Multiple scattering of classical waves: microscopy, mesoscopy, and diffusion,” Rev. Mod. Phys. |

12. | J. F. de Boer, Optical fluctuations on the tranmsission and reflection of mesoscopic systems (PhD thesis, University of Amsterdam, 1995). |

13. | D. Contini, F. Martelli, and G. Zaccanti, “Photon migration through a turbid slab described by a model based on diffusion approximation. i. theory,” Appl. Opt. |

14. | L. Fernando Rojas-Ochoa, D. Lacoste, R. Lenke, P. Schurtenberger, and F. Scheffold, “Depolarization of backscattered linearly polarized light,” J. Opt. Soc. Am. A |

15. | E. Akkermans, P. E. Wolf, R. Maynard, and G. Maret, “Theoretical study of the coherent backscattering of light by disordered media,” J. Phys. France |

16. | J. Aulbach, Spatiotemporal Control of Light in Turbid Media (PhD thesis, University of Twente, 2013). |

17. | D.J. Young and N.C. Beaulieu, “The generation of correlated rayleigh random variates by inverse discrete fourier transform,” Communications, IEEE Transactions on |

18. | M. Alfeld and J. A.C. Broekaert, “Mobile depth profiling and sub-surface imaging techniques for historical paintings - review,” Spectrochimica Acta Part B: Atomic Spectroscopy |

**OCIS Codes**

(110.7050) Imaging systems : Turbid media

(170.5280) Medical optics and biotechnology : Photon migration

(290.1990) Scattering : Diffusion

(290.7050) Scattering : Turbid media

**ToC Category:**

Scattering

**History**

Original Manuscript: March 31, 2014

Revised Manuscript: May 3, 2014

Manuscript Accepted: May 16, 2014

Published: May 27, 2014

**Citation**

Patrick M. Johnson, Timmo van der Beek, and Ad Lagendijk, "Diffuse imaging and radius dependent frequency correlations in strongly scattering media," Opt. Express **22**, 13330-13342 (2014)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-11-13330

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### References

- Merriam Webster online, http://www.merriam-webster.com/ .
- T. Durduran, R. Choe, W. B Baker, A. G. Yodh, “Diffuse optics for tissue monitoring and tomography,” Reports on Progress in Physics 73(7), 076701 (2010). [CrossRef]
- A. P. Gibson, J. C. Hebden, S. R. Arridge, “Recent advances in diffuse optical imaging,” Phys. Med. Biol. 50, R1–R43 (2005). [CrossRef] [PubMed]
- F. F. Jobsis, ”Noninvasive, infrared monitoring of cerebral and myocardial oxygen, sufficiency and circulatory parameters,” Science 198(4323), 1264–1267 (1977). [CrossRef] [PubMed]
- M. S. Patterson, B. Chance, B. C. Wilson, “Time resolved reflectance and transmittance for the non-invasive measurement of tissue optical properties,” Appl. Opt 28(12), 2331–2336 (1989). [CrossRef] [PubMed]
- M. S. Patterson, J. D. Moulton, Br. C. Wilson, K. W. Berndt, J. R. Lakowicz, “Frequency-domain reflectance for the determination of the scattering and absorption properties of tissue,” Appl. Opt. 30(31), 4474–4476 (1991). [CrossRef] [PubMed]
- P. M. Johnson, A. Imhof, B. P. J. Bret, J. Gómez Rivas, A. Lagendijk, “Time-resolved pulse propagation in a strongly scattering material,” Phys. Rev. E 68(1), 016604 (2003). [CrossRef]
- A. Z. Genack, J. M. Drake, “Relationship between optical intensity, fluctuations and pulse propagation in random media,” EPL (Europhysics Letters) 11(4), 331–336 (1990). [CrossRef]
- O. L. Muskens, A. Lagendijk, “Method for broadband spectroscopy of light transport through opaque scattering media,” Opt. Lett. 34, 395–397 (2009). [CrossRef] [PubMed]
- A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic Press, 1978).
- M. C. W. van Rossum, Th. M. Nieuwenhuizen, “Multiple scattering of classical waves: microscopy, mesoscopy, and diffusion,” Rev. Mod. Phys. 71(1), 313–371 (1999). [CrossRef]
- J. F. de Boer, Optical fluctuations on the tranmsission and reflection of mesoscopic systems (PhD thesis, University of Amsterdam, 1995).
- D. Contini, F. Martelli, G. Zaccanti, “Photon migration through a turbid slab described by a model based on diffusion approximation. i. theory,” Appl. Opt. 36(19), 4587–4599 (1997). [CrossRef] [PubMed]
- L. Fernando Rojas-Ochoa, D. Lacoste, R. Lenke, P. Schurtenberger, F. Scheffold, “Depolarization of backscattered linearly polarized light,” J. Opt. Soc. Am. A 21(9), 1799–1804 (2004). [CrossRef]
- E. Akkermans, P. E. Wolf, R. Maynard, G. Maret, “Theoretical study of the coherent backscattering of light by disordered media,” J. Phys. France 49(1), 77–98 (1988). [CrossRef]
- J. Aulbach, Spatiotemporal Control of Light in Turbid Media (PhD thesis, University of Twente, 2013).
- D.J. Young, N.C. Beaulieu, “The generation of correlated rayleigh random variates by inverse discrete fourier transform,” Communications, IEEE Transactions on 48(7), 1114–1127 (2000). [CrossRef]
- M. Alfeld, J. A.C. Broekaert, “Mobile depth profiling and sub-surface imaging techniques for historical paintings - review,” Spectrochimica Acta Part B: Atomic Spectroscopy 88(0), 211–230 (2013). [CrossRef]

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