Abstract

The present paper deals with the reconstruction of three-dimensional objects from the scattered far-field. The configuration under study is typically the one used in the Optical Diffraction Tomography (ODT), in which the sample is illuminated with various angles of incidence and the scattered field is measured for each illumination. The retrieval of the sample from the scattered field is accomplished numerically by solving the inverse scattering problem. We present herein a fast method for solving the inverse scattering problem based on the Coupled Dipole Method (CDM) and applied it for complex background configuration such as buried objects in a layered medium. Numerical experiments are reported and robustness against the presence of noise in the data is analyzed.

© 2006 Optical Society of America

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  1. A. Chomik, A. Dieterlen, C. Xu, O. Haeberlé, J. J. Meyer and S. Jacquey, "Quantification in optical sectioning microscopy: a comparison of some deconvolution algorithms in view of 3D image segmentation," J. Opt. 28, 225 (1997).
    [CrossRef]
  2. J. O. Tegenfeldt, O. Bakajin, C.-F Chou, S. S. Chan, R. Austin, W. Fann, L. Liou, E. Chan, T. Duke, E. C. Cox, "Near-field Scanner for Moving Molecules," Phys. Rev. Lett. 86, 1378 (2001).
    [CrossRef] [PubMed]
  3. L. A. Ghebern, J. Hwang, and M. Edidin, "Design and optimization of a near field scanning optical microscope for imaging biological samples in liquid," Appl. Opt. 37, 3574 (1998).
    [CrossRef]
  4. A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging, Society of Industrial and Applied Mathematics, (2001).
    [CrossRef]
  5. V. Lauer, "New approach to optical diffraction tomography yielding a vector equation of diffraction tomography and a novel tomographic microscope," J. Microsc. 205, 165 (2002).
    [CrossRef] [PubMed]
  6. P. S. Carney and J. C. Schotland, "Three-dimensional total internal reflection microscopy," Opt. Lett. 26, 1072 (2001).
    [CrossRef]
  7. P. S. Carney and J. C. Schotland, "Theory of total-internal-reflection tomography," J. Opt. Soc. Am. A 20, 542 (2003).
    [CrossRef]
  8. K. Belkebir, P. C. Chaumet, and A. Sentenac, "Influence of multiple scattering on three-dimensional imaging with optical diffraction tomography," J. Opt. Soc. Am. A. 23, 586 (2006).
    [CrossRef]
  9. K. Belkebir, P. C. Chaumet, and A. Sentenac, "Superresolution in total-internal reflection tomography," J. Opt. Soc. Am. A. 22, 1889 (2005).
    [CrossRef]
  10. P. C. Chaumet, A. Sentenac, and A. Rahmani, "Coupled dipole method for scatterers with large permittivity," Phys. Rev. E 70, 036606 (2004).
    [CrossRef]
  11. A. Rahmani, P. C. Chaumet, and F. de Fornel, "Environment-induced modification of spontaneous emission: Single-molecule near-field probe," Phys. Rev A 63, 023819 (2001).
    [CrossRef]
  12. P. C. Chaumet, K. Belkebir, and A. Sentenac, "Three-dimensional sub-wavelength optical imaging using the coupled dipole Method," Phys. Rev. B,  69, 245405 (2004).
    [CrossRef]

2006 (1)

K. Belkebir, P. C. Chaumet, and A. Sentenac, "Influence of multiple scattering on three-dimensional imaging with optical diffraction tomography," J. Opt. Soc. Am. A. 23, 586 (2006).
[CrossRef]

2005 (1)

K. Belkebir, P. C. Chaumet, and A. Sentenac, "Superresolution in total-internal reflection tomography," J. Opt. Soc. Am. A. 22, 1889 (2005).
[CrossRef]

2004 (2)

P. C. Chaumet, A. Sentenac, and A. Rahmani, "Coupled dipole method for scatterers with large permittivity," Phys. Rev. E 70, 036606 (2004).
[CrossRef]

P. C. Chaumet, K. Belkebir, and A. Sentenac, "Three-dimensional sub-wavelength optical imaging using the coupled dipole Method," Phys. Rev. B,  69, 245405 (2004).
[CrossRef]

2003 (1)

2002 (1)

V. Lauer, "New approach to optical diffraction tomography yielding a vector equation of diffraction tomography and a novel tomographic microscope," J. Microsc. 205, 165 (2002).
[CrossRef] [PubMed]

2001 (3)

P. S. Carney and J. C. Schotland, "Three-dimensional total internal reflection microscopy," Opt. Lett. 26, 1072 (2001).
[CrossRef]

J. O. Tegenfeldt, O. Bakajin, C.-F Chou, S. S. Chan, R. Austin, W. Fann, L. Liou, E. Chan, T. Duke, E. C. Cox, "Near-field Scanner for Moving Molecules," Phys. Rev. Lett. 86, 1378 (2001).
[CrossRef] [PubMed]

A. Rahmani, P. C. Chaumet, and F. de Fornel, "Environment-induced modification of spontaneous emission: Single-molecule near-field probe," Phys. Rev A 63, 023819 (2001).
[CrossRef]

1998 (1)

1997 (1)

A. Chomik, A. Dieterlen, C. Xu, O. Haeberlé, J. J. Meyer and S. Jacquey, "Quantification in optical sectioning microscopy: a comparison of some deconvolution algorithms in view of 3D image segmentation," J. Opt. 28, 225 (1997).
[CrossRef]

Austin, R.

J. O. Tegenfeldt, O. Bakajin, C.-F Chou, S. S. Chan, R. Austin, W. Fann, L. Liou, E. Chan, T. Duke, E. C. Cox, "Near-field Scanner for Moving Molecules," Phys. Rev. Lett. 86, 1378 (2001).
[CrossRef] [PubMed]

Bakajin, O.

J. O. Tegenfeldt, O. Bakajin, C.-F Chou, S. S. Chan, R. Austin, W. Fann, L. Liou, E. Chan, T. Duke, E. C. Cox, "Near-field Scanner for Moving Molecules," Phys. Rev. Lett. 86, 1378 (2001).
[CrossRef] [PubMed]

Belkebir, K.

K. Belkebir, P. C. Chaumet, and A. Sentenac, "Influence of multiple scattering on three-dimensional imaging with optical diffraction tomography," J. Opt. Soc. Am. A. 23, 586 (2006).
[CrossRef]

K. Belkebir, P. C. Chaumet, and A. Sentenac, "Superresolution in total-internal reflection tomography," J. Opt. Soc. Am. A. 22, 1889 (2005).
[CrossRef]

P. C. Chaumet, K. Belkebir, and A. Sentenac, "Three-dimensional sub-wavelength optical imaging using the coupled dipole Method," Phys. Rev. B,  69, 245405 (2004).
[CrossRef]

Carney, P. S.

Chan, E.

J. O. Tegenfeldt, O. Bakajin, C.-F Chou, S. S. Chan, R. Austin, W. Fann, L. Liou, E. Chan, T. Duke, E. C. Cox, "Near-field Scanner for Moving Molecules," Phys. Rev. Lett. 86, 1378 (2001).
[CrossRef] [PubMed]

Chan, S. S.

J. O. Tegenfeldt, O. Bakajin, C.-F Chou, S. S. Chan, R. Austin, W. Fann, L. Liou, E. Chan, T. Duke, E. C. Cox, "Near-field Scanner for Moving Molecules," Phys. Rev. Lett. 86, 1378 (2001).
[CrossRef] [PubMed]

Chaumet, P. C.

K. Belkebir, P. C. Chaumet, and A. Sentenac, "Influence of multiple scattering on three-dimensional imaging with optical diffraction tomography," J. Opt. Soc. Am. A. 23, 586 (2006).
[CrossRef]

K. Belkebir, P. C. Chaumet, and A. Sentenac, "Superresolution in total-internal reflection tomography," J. Opt. Soc. Am. A. 22, 1889 (2005).
[CrossRef]

P. C. Chaumet, A. Sentenac, and A. Rahmani, "Coupled dipole method for scatterers with large permittivity," Phys. Rev. E 70, 036606 (2004).
[CrossRef]

P. C. Chaumet, K. Belkebir, and A. Sentenac, "Three-dimensional sub-wavelength optical imaging using the coupled dipole Method," Phys. Rev. B,  69, 245405 (2004).
[CrossRef]

A. Rahmani, P. C. Chaumet, and F. de Fornel, "Environment-induced modification of spontaneous emission: Single-molecule near-field probe," Phys. Rev A 63, 023819 (2001).
[CrossRef]

Chomik, A.

A. Chomik, A. Dieterlen, C. Xu, O. Haeberlé, J. J. Meyer and S. Jacquey, "Quantification in optical sectioning microscopy: a comparison of some deconvolution algorithms in view of 3D image segmentation," J. Opt. 28, 225 (1997).
[CrossRef]

Chou, C.-F

J. O. Tegenfeldt, O. Bakajin, C.-F Chou, S. S. Chan, R. Austin, W. Fann, L. Liou, E. Chan, T. Duke, E. C. Cox, "Near-field Scanner for Moving Molecules," Phys. Rev. Lett. 86, 1378 (2001).
[CrossRef] [PubMed]

Cox, E. C.

J. O. Tegenfeldt, O. Bakajin, C.-F Chou, S. S. Chan, R. Austin, W. Fann, L. Liou, E. Chan, T. Duke, E. C. Cox, "Near-field Scanner for Moving Molecules," Phys. Rev. Lett. 86, 1378 (2001).
[CrossRef] [PubMed]

de Fornel, F.

A. Rahmani, P. C. Chaumet, and F. de Fornel, "Environment-induced modification of spontaneous emission: Single-molecule near-field probe," Phys. Rev A 63, 023819 (2001).
[CrossRef]

Dieterlen, A.

A. Chomik, A. Dieterlen, C. Xu, O. Haeberlé, J. J. Meyer and S. Jacquey, "Quantification in optical sectioning microscopy: a comparison of some deconvolution algorithms in view of 3D image segmentation," J. Opt. 28, 225 (1997).
[CrossRef]

Duke, T.

J. O. Tegenfeldt, O. Bakajin, C.-F Chou, S. S. Chan, R. Austin, W. Fann, L. Liou, E. Chan, T. Duke, E. C. Cox, "Near-field Scanner for Moving Molecules," Phys. Rev. Lett. 86, 1378 (2001).
[CrossRef] [PubMed]

Edidin, M.

Fann, W.

J. O. Tegenfeldt, O. Bakajin, C.-F Chou, S. S. Chan, R. Austin, W. Fann, L. Liou, E. Chan, T. Duke, E. C. Cox, "Near-field Scanner for Moving Molecules," Phys. Rev. Lett. 86, 1378 (2001).
[CrossRef] [PubMed]

Ghebern, L. A.

Haeberlé, O.

A. Chomik, A. Dieterlen, C. Xu, O. Haeberlé, J. J. Meyer and S. Jacquey, "Quantification in optical sectioning microscopy: a comparison of some deconvolution algorithms in view of 3D image segmentation," J. Opt. 28, 225 (1997).
[CrossRef]

Hwang, J.

Jacquey, S.

A. Chomik, A. Dieterlen, C. Xu, O. Haeberlé, J. J. Meyer and S. Jacquey, "Quantification in optical sectioning microscopy: a comparison of some deconvolution algorithms in view of 3D image segmentation," J. Opt. 28, 225 (1997).
[CrossRef]

Lauer, V.

V. Lauer, "New approach to optical diffraction tomography yielding a vector equation of diffraction tomography and a novel tomographic microscope," J. Microsc. 205, 165 (2002).
[CrossRef] [PubMed]

Liou, L.

J. O. Tegenfeldt, O. Bakajin, C.-F Chou, S. S. Chan, R. Austin, W. Fann, L. Liou, E. Chan, T. Duke, E. C. Cox, "Near-field Scanner for Moving Molecules," Phys. Rev. Lett. 86, 1378 (2001).
[CrossRef] [PubMed]

Meyer, J. J.

A. Chomik, A. Dieterlen, C. Xu, O. Haeberlé, J. J. Meyer and S. Jacquey, "Quantification in optical sectioning microscopy: a comparison of some deconvolution algorithms in view of 3D image segmentation," J. Opt. 28, 225 (1997).
[CrossRef]

Rahmani, A.

P. C. Chaumet, A. Sentenac, and A. Rahmani, "Coupled dipole method for scatterers with large permittivity," Phys. Rev. E 70, 036606 (2004).
[CrossRef]

A. Rahmani, P. C. Chaumet, and F. de Fornel, "Environment-induced modification of spontaneous emission: Single-molecule near-field probe," Phys. Rev A 63, 023819 (2001).
[CrossRef]

Schotland, J. C.

Sentenac, A.

K. Belkebir, P. C. Chaumet, and A. Sentenac, "Influence of multiple scattering on three-dimensional imaging with optical diffraction tomography," J. Opt. Soc. Am. A. 23, 586 (2006).
[CrossRef]

K. Belkebir, P. C. Chaumet, and A. Sentenac, "Superresolution in total-internal reflection tomography," J. Opt. Soc. Am. A. 22, 1889 (2005).
[CrossRef]

P. C. Chaumet, A. Sentenac, and A. Rahmani, "Coupled dipole method for scatterers with large permittivity," Phys. Rev. E 70, 036606 (2004).
[CrossRef]

P. C. Chaumet, K. Belkebir, and A. Sentenac, "Three-dimensional sub-wavelength optical imaging using the coupled dipole Method," Phys. Rev. B,  69, 245405 (2004).
[CrossRef]

Tegenfeldt, J. O.

J. O. Tegenfeldt, O. Bakajin, C.-F Chou, S. S. Chan, R. Austin, W. Fann, L. Liou, E. Chan, T. Duke, E. C. Cox, "Near-field Scanner for Moving Molecules," Phys. Rev. Lett. 86, 1378 (2001).
[CrossRef] [PubMed]

Xu, C.

A. Chomik, A. Dieterlen, C. Xu, O. Haeberlé, J. J. Meyer and S. Jacquey, "Quantification in optical sectioning microscopy: a comparison of some deconvolution algorithms in view of 3D image segmentation," J. Opt. 28, 225 (1997).
[CrossRef]

Appl. Opt. (1)

J. Microsc. (1)

V. Lauer, "New approach to optical diffraction tomography yielding a vector equation of diffraction tomography and a novel tomographic microscope," J. Microsc. 205, 165 (2002).
[CrossRef] [PubMed]

J. Opt. (1)

A. Chomik, A. Dieterlen, C. Xu, O. Haeberlé, J. J. Meyer and S. Jacquey, "Quantification in optical sectioning microscopy: a comparison of some deconvolution algorithms in view of 3D image segmentation," J. Opt. 28, 225 (1997).
[CrossRef]

J. Opt. Soc. Am. A (1)

J. Opt. Soc. Am. A. (2)

K. Belkebir, P. C. Chaumet, and A. Sentenac, "Influence of multiple scattering on three-dimensional imaging with optical diffraction tomography," J. Opt. Soc. Am. A. 23, 586 (2006).
[CrossRef]

K. Belkebir, P. C. Chaumet, and A. Sentenac, "Superresolution in total-internal reflection tomography," J. Opt. Soc. Am. A. 22, 1889 (2005).
[CrossRef]

Opt. Lett. (1)

Phys. Rev A (1)

A. Rahmani, P. C. Chaumet, and F. de Fornel, "Environment-induced modification of spontaneous emission: Single-molecule near-field probe," Phys. Rev A 63, 023819 (2001).
[CrossRef]

Phys. Rev. B (1)

P. C. Chaumet, K. Belkebir, and A. Sentenac, "Three-dimensional sub-wavelength optical imaging using the coupled dipole Method," Phys. Rev. B,  69, 245405 (2004).
[CrossRef]

Phys. Rev. E (1)

P. C. Chaumet, A. Sentenac, and A. Rahmani, "Coupled dipole method for scatterers with large permittivity," Phys. Rev. E 70, 036606 (2004).
[CrossRef]

Phys. Rev. Lett. (1)

J. O. Tegenfeldt, O. Bakajin, C.-F Chou, S. S. Chan, R. Austin, W. Fann, L. Liou, E. Chan, T. Duke, E. C. Cox, "Near-field Scanner for Moving Molecules," Phys. Rev. Lett. 86, 1378 (2001).
[CrossRef] [PubMed]

Other (1)

A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging, Society of Industrial and Applied Mathematics, (2001).
[CrossRef]

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Figures (9)

Fig. 1.
Fig. 1.

Sketch of the illumination and detection configuration. The observation points are in the far field zone regularly placed on a half sphere. The illumination corresponds to 16 plane wave propagating towards to the positive values of z in both planes (x,z) and (y,z) such that the electric field is in the incidence plane. The angle between the incident wave vector and the z axis ranges over -80° to 80°. The cube represents the objects under test and the background can be either homogeneous or layered medium.

Fig. 2.
Fig. 2.

Map of the reconstructed relative permittivity using our inversion scheme. The size of the investigating domain is 2λ × 2λ × 2λ. Objects under test are cubes (boundaries of these cubes are plotted in black) of side a = λ/4 and separated by c = λ/3. The actual relative permittivity of the cube is εl = 2.25+0.5i and εr = 2.25 for the left and right cube, respectively. The four left figures are obtained from noiseless data, while the right figures are obtained from a corrupted scattered field with u = 10%. a) and e) shows the real part of the relative permittivity in the plane y = 0. b) and f) shows the real part of the permittivity in the plane z = 0. c) and g) shows the imaginary part of the permittivity in the plane y = 0. d) and h) shows the imaginary part of the relative permittivity in the plane z = 0.

Fig. 3.
Fig. 3.

Same as Fig. 2 but with an inhomogeneous U-shaped object. This object is constituted by a dielectric bar (ε = 2.25) and two absorbing cubes located at the extremities (ε = 2.25 + 0.5i). Maps a), c), e) and g) are plotted in the plane y = 0; b) and f) in the plane z = 0; and d) and h) in the plane z = -0.4λ. The dashed lines in the figures a), c), e) and g) represent, the cross sections maps of b), f), d) and h) in the (x,y) plane. The figures on the right are organized as the figures on the left but with a scattered field corrupted with noise (u = 30%).

Fig. 4.
Fig. 4.

Map of the reconstructed relative permittivity using the inversion algorithm. The size of the investigating domain is 2λ × 2λ × λ. We have a = λ/4, c = λ/3, and εl = 2.25 + 0.5i, εr = 2.25, and εs = 2.25. The four left figures are obtained with only propagative waves, while the figures on the right are obtained with solely evanescent waves. a) and e) show the real part of the permittivity in the plane y = 0. b) and f) show the real part of the permittivity in the plane zλ/7 (dashed line). c) and g) show the imaginary part of the permittivity in the plane y = 0. d) and h) show the imaginary part of the permittivity in the plane z = λ/7 (dashed line).

Fig. 5.
Fig. 5.

Map of the relative permittivity given by our inversion scheme when the scattered field is corrupted with noise (u = 10%). The size of the investigating domain is 2λ × 2λ × λ. We have a = λ/4, c = λ/3, and εr = 2.25 + 0.5i, εr = 2.25, and εs = 2.25. The four figures on the right are obtained with only evanescent waves, while the figures on the left are obtained with both evanescent and propagative waves. a) and e) show the real part of the permittivity in the plane y = 0. b) and f) show the real part of the permittivity in the plane z = λ/7 (dashed line). c) and g) show the imaginary part of the permittivity in the plane y = 0. d) and h) show the imaginary part of the permittivity in the plane z = λ/7.

Fig. 6.
Fig. 6.

Map of the relative permittivity when the objects are embedded in the substrate. The size of the investigating domain is 2λ × 2λ × 1.5λ. We have a = λ/4, c = λ/3, and εl = 1.5+0.5i, εr = 1.5, and εs = 2.25. Left figures (a-d) are obtained without noise and figures on the right (e-h) are obtained with a scattered field corrupted with noise (u = 10%). a) and e) show the real part of the permittivity in the plane y = 0. b) and f) show the real part of the permittivity in the (x, y) plane located at z = -0.65λ (dashed line). c) and g) show the imaginary part of the permittivity in the plane y = 0. d) and h) show the imaginary part of the permittivity in the plane (x ,y) located at z = -0.65λ (dashed line). in Figs. 6–8. Indeed, the computational time is completely independent of the shape and the distribution of the objects inside the investigating domain. It depends only of the size of the investigating domain.

Fig. 7.
Fig. 7.

Same as in Fig. 6 but with measurements carried out from both below and above the surface. Illumination of objects remains unchanged.

Fig. 8.
Fig. 8.

Same as in Fig. 7 but with four objects located at (-0.3λ, -0.3λ, -0.65λ), (-0.3λ, 0.3λ, -0.65λ), (0.3λ, -0.3λ, -0.65λ), and (0.3λ, 0.3λ, -0.65λ), with ε = 2.25, ε = 2.25+0.5i, ε = 2.25+0.5i, and ε = 2.25, respectively.

Fig. 9.
Fig. 9.

a) Sketch of the studied configuration. The dimension of the investigating domain is 2λ × 2λ × 2.2λ and a = λ/4. b) Real part of the relative permittivity in the (x, z) plane. Notice that each layer has its own color scale. c) Imaginary part of the relative permittivity in the (x,z) plane. d) and e) same as b) and c), respectively, but with noisy data (u = 10%).

Equations (11)

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E ( r i ) = E inc ( r i ) + j = 1 N S ( r i , r j ) α ( r j ) E ( r j )
α ( r j ) = 3 d 3 4 π ε ( r j ) 1 ε ( r j ) + 2 .
E ¯ = E ¯ inc + A ̿ p ¯
E ( r ) = j = 1 N S ( r , r j ) p ( r j ) .
f ¯ = B ̿ p ¯
p ¯ l , n = p ¯ l , n 1 + β l , n d ¯ l , n ,
n ( p ¯ l , n ) = W Γ l = 1 L f ¯ l B ̿ p ¯ l , n Γ 2 , with W Γ = 1 l = 1 L f ¯ l Γ 2 .
d ¯ l , n = g ¯ l , n ; p ¯ + γ l , n d ¯ l , n 1 , with γ l , n = g ¯ l , n ; p ¯ , g ¯ l , n ; p ¯ g ¯ l , n 1 ; p ¯ Ω g ¯ l , n 1 ; p Ω 2
g ¯ l , n ; p ¯ = W Γ B ̿ [ f ¯ l B ̿ p ¯ l , n 1 ] ,
α ( r j ) = l = 1 L E l * ( r j ) · p l ( r j ) l = 1 L E l ( r j ) 2 .
f ˜ l v ( r k ) = f l v ( r k ) + uA e .

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