P. N. den Outer,1
Th. M. Nieuwenhuizen,1
and Ad Lagendijk2
1Van der Waals-Zeeman Laboratorium der Universiteit van Amsterdam, Valckenierstraat 65, 1018 XE Amsterdam, The Netherlands
2Van der Waals-Zeeman laboratorium der Universiteit van Amsterdam, Valckenierstraat 65, 1018 XE Amsterdam, The Netherlands, and Fundamenteel Onderzoek der Materie Instituut voor Atoom-en Molecuulfysica, Kruislaan 407, 1098 SJ Amsterdam, The Netherlands
When a small object is placed in a multiple-scattering medium the stationary diffusion equation can be used to derive the disturbance in the transmitted and backscattered light intensity. The diffusion equation will describe the intensity outside and inside the object. The object is characterized by a size, a diffusion constant, and an absorption length. In this way absorbing objects as well as nonabsorbing objects can be treated. The results are derived for two and three dimensions. Experiments are performed on suspended titanium dioxide particles in glycerine, wherein objects could be placed. There is good agreement between theory and experiment. This work shows that with the use of continuous light sources, it may be possible to recover the location of objects accurately inside a diffusive scattering medium.
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Fits of the Calculated Line Shape to Experimental Data Obtained from a Transmission Experiment with an Absorbing Roda
Experimental Normal Position (mm)
Fitted Normal Position (mm)
Absorption Strength D1/αa
Experimental Position Relative to First Position (mm)
Fitted Position Relative to First Position (mm)
0.75 ± 0.05
1.23 ± 0.02
0.95 ± 0.03
0
0
1.40 ± 0.05
1.91 ± 0.02
0.99 ± 0.03
0.65 ± 0.07
0.68 ± 0.03
2.30 ± 0.05
2.78 ± 0.02
1.04 ± 0.03
1.55 ± 0.07
1.55 ± 0.03
3.50 ± 0.05
3.91 ±0.02
1.10 ± 0.03
2.75 ± 0.07
2.68 ± 0.03
4.95 ± 0.05
5.28 ± 0.02
1.11 ± 0.03
4.20 ± 0.07
4.05 ± 0.03
6.30 ± 0.05
6.67 ± 0.02
0.96 ± 0.03
5.55 ± 0.07
5.44 ± 0.03
7.80 ± 0.05
8.2 ± 0.1
0.8 ± 0.1
7.05 ± 0.07
7.0 ± 0.1
9.30 ± 0.05
9.8 ± 0.1
0.6 ± 0.1
8.55 ± 0.07
8.6 ± 0.1
In all cases the scattering contribution is neglected, p = 0; the size of the rod Ø = 350 μm, and the diffusion constant inside the rod D2 ≪ D1. The absorption parameter α was used as a free-fitting parameter.
Table 2
Fits of the Calculated Line Shape to Transmission Experiment Dataa
Experimental Transversal Displacement (mm)
Fitted Transversal Displacement (mm)
2.00 ± 0.02
2.05 ± 0.03
4.00 ± 0.02
4.04 ± 0.06
5.00 ± 0.02
5.02 ± 0.07
7.00 ± 0.02
7.03 ± 0.09
9.00 ± 0.02
9.1 ± 0.1
11.00 ± 0.02
11.1 ± 0.2
Experimental displacements of an absorbing rod parallel to the slab are compared with those obtained from the corresponding fits. The distance between the rod and the boundary (x position) is 4.30 ± 0.02 mm, and the thickness of the cell is 10.0mm. The scattering contribution is neglected (p = 0), and the diffusion constant inside the rod is small (D2 ≪ D1) In all cases the relative absorption strength D1/αa = 1.4 ± 0.1.
Table 3
Fits of the Calculated Line Shape to Experimental Data Obtained from a Backscatter Experiment with an Absorbing Roda
The size of the rod Ø = 350 μm, and the thickness of the cell is 4 mm. The scattering contribution is neglected (p = 0). The diffusion constant inside the rod D2 ≪ D1.
The last line in the table illustrates the sensitivity for small displacements of the rod. Two successive scans are taken at 0.650 and 0.655 mm from the boundary. The displacement of only 50 μm is reproduced by the corresponding fits.
Tables (3)
Table 1
Fits of the Calculated Line Shape to Experimental Data Obtained from a Transmission Experiment with an Absorbing Roda
Experimental Normal Position (mm)
Fitted Normal Position (mm)
Absorption Strength D1/αa
Experimental Position Relative to First Position (mm)
Fitted Position Relative to First Position (mm)
0.75 ± 0.05
1.23 ± 0.02
0.95 ± 0.03
0
0
1.40 ± 0.05
1.91 ± 0.02
0.99 ± 0.03
0.65 ± 0.07
0.68 ± 0.03
2.30 ± 0.05
2.78 ± 0.02
1.04 ± 0.03
1.55 ± 0.07
1.55 ± 0.03
3.50 ± 0.05
3.91 ±0.02
1.10 ± 0.03
2.75 ± 0.07
2.68 ± 0.03
4.95 ± 0.05
5.28 ± 0.02
1.11 ± 0.03
4.20 ± 0.07
4.05 ± 0.03
6.30 ± 0.05
6.67 ± 0.02
0.96 ± 0.03
5.55 ± 0.07
5.44 ± 0.03
7.80 ± 0.05
8.2 ± 0.1
0.8 ± 0.1
7.05 ± 0.07
7.0 ± 0.1
9.30 ± 0.05
9.8 ± 0.1
0.6 ± 0.1
8.55 ± 0.07
8.6 ± 0.1
In all cases the scattering contribution is neglected, p = 0; the size of the rod Ø = 350 μm, and the diffusion constant inside the rod D2 ≪ D1. The absorption parameter α was used as a free-fitting parameter.
Table 2
Fits of the Calculated Line Shape to Transmission Experiment Dataa
Experimental Transversal Displacement (mm)
Fitted Transversal Displacement (mm)
2.00 ± 0.02
2.05 ± 0.03
4.00 ± 0.02
4.04 ± 0.06
5.00 ± 0.02
5.02 ± 0.07
7.00 ± 0.02
7.03 ± 0.09
9.00 ± 0.02
9.1 ± 0.1
11.00 ± 0.02
11.1 ± 0.2
Experimental displacements of an absorbing rod parallel to the slab are compared with those obtained from the corresponding fits. The distance between the rod and the boundary (x position) is 4.30 ± 0.02 mm, and the thickness of the cell is 10.0mm. The scattering contribution is neglected (p = 0), and the diffusion constant inside the rod is small (D2 ≪ D1) In all cases the relative absorption strength D1/αa = 1.4 ± 0.1.
Table 3
Fits of the Calculated Line Shape to Experimental Data Obtained from a Backscatter Experiment with an Absorbing Roda
The size of the rod Ø = 350 μm, and the thickness of the cell is 4 mm. The scattering contribution is neglected (p = 0). The diffusion constant inside the rod D2 ≪ D1.
The last line in the table illustrates the sensitivity for small displacements of the rod. Two successive scans are taken at 0.650 and 0.655 mm from the boundary. The displacement of only 50 μm is reproduced by the corresponding fits.