## Three-dimensional optical imaging in layered media

Optics Express, Vol. 14, Issue 8, pp. 3415-3426 (2006)

http://dx.doi.org/10.1364/OE.14.003415

Acrobat PDF (1158 KB)

### 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

## 1. Introduction

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]

*e*.

*g*., objects of characteristic dimension of

*λ*/4 present in a homogeneous background medium, and inversion performed with an investigating domain of volume size 8

*λ*

^{3}(

*λ*being the wavelength of the incident field) [8

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]

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]

## 2. Theory

### 2.1. Forward scattering problem

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

*N*dipolar subunits, and the field at each subunit satisfies the following self consistent equation:

**E**

^{inc}(

**r**

_{i}) is the incident field, and

*α*(

**r**

_{j}) the polarizability of the jth subunit which meet the Claussius-Mossotti relation:

*ε*(

**r**

_{j}) is the relative permittivity of the subunit

*j*, and

*d*the size of the subunit.

**S**is a tensor which correspond to the linear response of a dipole in the system of reference,

*i*.

*e*., homogeneous space [10

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]

*i*=

*j*in Eq. (1) the contact term is take into account through the Clausius-Mossotti relation [10

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

*i*is written as

**p**(

**r**

_{i}) =

*α*(

**r**

_{i})

**E**(

**r**

_{i}), hence Eq. (1) can be written under this symbolic form:

**A̿**is a matrix (3

*N*× 3

*N*) which contains all the tensors

**S**.

**E**¯,

**E**¯

^{inc}and

**p**¯ are vector (3

*N*) which contain all the local field, the incident field, and the dipole moment, respectively. The field scattered by the objects at an arbitrary position

**r**reads as

*M*points of observation, one can use this symbolic form:

**B̿**is a matrix (3

*M*×3

*N*) and

**f**¯ a vector (3

*M*) which contains all the diffracted field. Notice that the matrices

**A̿**and

**B̿**do not depend of the incident field and of the nature of the object.

### 2.2. Inversion algorithm

**d**¯

_{l,n}is an updating direction,

*β*

_{l,n}a scalar number determined at each iteration step by minimizing the cost functional

*ℱ*

_{n}that represents the discrepancy between the data (measurements) and the scattered field corresponding to the best available estimate of the object

**p**¯

_{l,n}.

*ℱ*

_{n}is defined thus:

_{Ω}is the same inner product as defined previously but acting on vectors defined on Ω. The vector function

**g**¯

_{l;p¯}is the gradient of the cost functional ℱ with respect to

**p**¯

_{l}evaluated for the (

*n*- 1)

^{th}quantities. This gradient reads as:

**B̿**

^{†}is the transpose complex conjugate matrix of the matrix

**B̿**. Once the sources

**p**¯

_{l}are reconstructed, one can determine the fields

**E**¯

_{l}inside Ω using the Eq. (3). The polarizability a at the position

**r**

_{j}is then given by

*ε*distribution is determined easily using Eq. (2).

## 3. Numerical results

*λ*/20 while inversions were performed with a different mesh of size

*λ*/10. All presented results were obtained without any post-treatment.

### 3.1. Simple configuration: case of homogeneous background medium

*a*=

*λ*/4 and separated by a distance of

*c*=

*λ*/3 (

*λ*being the wavelength of vacuum). The relative permittivity of the cube located at

*x*≈ -0.3

*λ*is

*ε*= 2.25 while the relative permittivity of the other one located at

*x*≈ 0.3

*λ*is

*ε*= 2.25 +

*i*0.5. The illumination of the samples is as described in Fig. 1,

*i*.

*e*., 16 plane waves in the two perpendicular planes (

*x*,

*z*) and (

*y*,

*z*). The electrical field remains in the incidence plane (Fig. 1). Let us denote by

**k**

^{inc}the wavevector of the incident field and

**k**

^{d}the wavevector of the diffracted field. The investigated domain Ω is a large cube of volume 2

*λ*× 2

*λ*× 2

*λ*.

*x*,

*z*) for the left image and in the plane (

*x*,

*y*) for the right image. The second row is as for the first row but for imaginary part of the relative permittivity instead of the real part. The full line curves represent the boundaries of the actual objects.

**p**¯ is obtained only in 80 seconds by minimizing the cost functional

*ℱ*defined in Eq. (7). Therefore, the main computation time is spent to obtained the internal field

*via*Eq. (3). Hence the main advantage of this method is that Eq. (3) is used only once. In the method presented in Refs. [8

**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]

*x*,

*y*, or

*z*,

*A*= max(

**f**¯

_{l=1, ⋯, L}(

**r**

_{k})|) and

*k*= 1, … ,

*M*.

*ϕ*is a random number taken for each component of the positions of observation and incident angles with uniform probability density in [0,2

*π*];

*u*is a real number smaller than unity that monitors the noise level. Figures 2(e-h) show the reconstruction in the presence of noise (

*u*= 10%). It appears evident that the obtained results are similar to those shown in Figs. 2(a-d).

*λ*and of section (

*λ*/4 ×

*λ*/4). The relative permittivity of this bar is

*ε*= 2.25. The two cubes are of side

*λ*/4 and of relative permittivity

*ε*= 2.25 + 0.5

*i*. The illumination as well as the observation is unchanged. Figure 3 presents results of the inversion. It is clearly shown that the "U-shaped" object is accurately retrieved even from corrupted data with a value of

*u*as high as 30%. Thus, the reconstruction method presented here is very robust against a presence of the noise in data. In addition, this fast method can provide us with a 3-D cartography showing objects that are absorbing or not. In the following sections, we will attempt to show that this method has the major advantage of being applicable to extremely complex situations, which would be tedious to carry out with a method as described in Ref. [9

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

### 3.2. Case of two semi-infinite media

#### 3.2.1. Objects above a dielectric substrate

*a*=

*λ*/4 and separated by

*c*=

*λ*/3) of the same relative permittivity but now deposited on a flat interface separating the superstrate (air,

*ε*= 1) and the substrate (glass,

*ε*= 2.25). The total reflection angle is in this case

*θ*

^{c}= 41.8°. The investigating domain is of volume (2

*λ*× 2

*λ*×

*λ*) and located above the interface.

*i*.

*e*., |

*θ*

^{inc}| <

*θ*

^{c}. In this case, Figs. 4(a) and 4(b) are similar to Figs. 2(a) and 2(b) in terms of resolution. This is due to the fact that Ewald’s spheres are identical in both configurations. However, Figs. 4(c) and 4(d) show an accurate localization of the imaginary part of permittivity. We interpret this superior resolution as being the result of a coupling effect between objects and the substrate, with multiple scattering improving the resolution [9

**22**, 1889 (2005). [CrossRef]

*i*.

*e*., |

*≈*

^{inc}| >

*≈*

^{c}. Considering the real part of permittivity, both cubes are now perfectly resolved as shown in Figs. 4(e) and 4(f). However, the objects do not lie on the surface. The result for the imaginary part is less accurate, because the absorbing cube seems to float above the surface; and, a significant portion of the imaginary part appears at the object on the right. The use of evanescent illumination thus yields to good resolution, due to the high spatial frequencies provided by the incident wave (enlarged Ewald’s sphere), yet the object is not well located.

*i*.

*e*., -80° <

*θ*

^{inc}< 80°. Furthermore, noise has been added to the diffracted field (

*u*= 10%) in order to mimic experimental conditions. Figure 5(a-d) clearly show that adding propagative and evanescent waves improves the result: both the real and the imaginary parts show excellent adequation between the actual profile and the reconstructed one. The two cubes are well located and perfectly resolved. Furthermore, Figs. 5(c) and 5(d) show only one truly absorbing cube, which is not the case in Figs 5(g) and 5(h) where only evanescent waves are used nor in Figs 4(g) and 4(h) where noiseless data are used. The effect of noise in the reconstruction, when using evanescent waves, Figs. 5(e-h) (in particular, artifacts appear at the top of the investigating box in Fig. 5(e)), is much higher than the one observed when using both propagative and evanescent waves Fig. 5(a-d). In fact, an illumination with evanescent wave contains high spatial frequencies, which are known to be sensitive to noise. Illuminating targets with both propagative and evanescent waves leads to combine the robustness and the accuracy of the reconstruction.

#### 3.2.2. Objects buried in the dielectric substrate

*ε*= 2.25). In Figs. 6 and 7 the objects are the same as in Section 3.2.1 : cubes with

*a*=

*λ*/4 separated by

*c*=

*λ*/3 with a relative permittivity

*ε*

_{l}= 2.25 + 0.5

*i*for the cube located at (-0.3

*λ*,0,-0.65

*λ*) and

*ε*

_{r}= 2.25 for the cube located at (0.3

*λ*,0, -0.65

*λ*). Illumination remains at -80° <

*θ*

^{inc}< 80° and the dimension of the investigating domain is 2

*λ*× 2

*λ*× 1.5

*λ*and includes the air-glass interface. This configuration would be more difficult to study with an optical microscope. It is typically the one that would be used for mines detection if the scattered fields were measured closed to the interface. This is not a limitation for our method, it suffices to replace the far-field tensor into the near-field one.

*z*axis for the real part of permittivity, and a good lateral separation. In addition, the imaginary part is now perfectly located on the actual cube. We explain this resolution by looking at the Ewald sphere, which is slightly enlarged with respect to the

*x*-axis and much more enlarged in the

*z*-axis direction. In the case of the scattered field corrupted with noise (

*u*= 10%), Figs. 6(e-h) and Figs. 7(e-h), show that the reconstruction is almost not altered by the presence of noise in the data. This is particularly true when observation points are located above and below the surface (Figs. 7(e-h)).

*x*,

*z*) plane. One may wonder what would be the reconstruction if the objects are placed somewhere else. This is investigated in Fig. 8 where the observation points are located above and below the surface, the illumination being the same as previously. Figure 8 presents the result of the reconstruction of four targets located in different (

*x*,

*z*) planes. All objects are well retrieved. In fact, objects can be distributed anywhere, we have only chosen objects in particular plane for sake of simplicity. In addition, the computational time is identical for all cases presented

### 3.3. Complex configuration: Case of layered medium

*θ*

^{inc}< 80°, and the observation points are located only above the surface.

*i*.

*e*. matrix

**A̿**. Once the matrix

**A̿**is built the needed computational time for solving the inverse scattering problem remains almost unchanged. Saving the matrix

**A̿**would be preferred for a repeated imaging objects present in an investigated domain of fixed size.

*u*= 10%) is presented in Figs. 9(d) and 9(e). The main effect of the noise is to perturb the map of the imaginary part of the relative permittivity. We noticed that when two cubes are located in different layers but one on top of the other, coupling occurs, thus hindering reconstruction. This type of coupling, between different objects present in different layers, deserves to be investigated more in depth.

## 4. Conclusion

## References and links

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. |

2. | J. O. Tegenfeldt, O. Bakajin, C.-F Chou, S. S. Chan, R. Austin, W. Fann, L. Liou, E. Chan, T. Duke, and E. C. Cox, “Near-field Scanner for Moving Molecules,” Phys. Rev. Lett. |

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. |

4. | A. C. Kak and M. Slaney, |

5. | V. Lauer, “New approach to optical diffraction tomography yielding a vector equation of diffraction tomography and a novel tomographic microscope,” J. Microsc. |

6. | P. S. Carney and J. C. Schotland, “Three-Dimensional Total Internal Reflection Microscopy,” Opt. Lett. |

7. | P. S. Carney and J. C. Schotland, “Theory of total-internal-reflection tomography,” J. Opt. Soc. Am. A |

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. |

9. | K. Belkebir, P. C. Chaumet, and A. Sentenac, “Superresolution in total-internal reflection tomography,” J. Opt. Soc. Am. A. |

10. | P. C. Chaumet, A. Sentenac, and A. Rahmani, “Coupled dipole method for scatterers with large permittivity,” Phys. Rev. E |

11. | A. Rahmani, P. C. Chaumet, and F. de Fornel, “Environment-induced modification of spontaneous emission: Single-molecule near-field probe,” Phys. Rev A |

12. | P. C. Chaumet, K. Belkebir, and A. Sentenac, “Three-dimensional sub-wavelength optical imaging using the coupled dipole Method,” Phys. Rev. B, |

**OCIS Codes**

(100.6640) Image processing : Superresolution

(100.6890) Image processing : Three-dimensional image processing

(180.6900) Microscopy : Three-dimensional microscopy

**ToC Category:**

Microscopy

**History**

Original Manuscript: January 17, 2006

Revised Manuscript: March 30, 2006

Manuscript Accepted: April 1, 2006

Published: April 17, 2006

**Virtual Issues**

Vol. 1, Iss. 5 *Virtual Journal for Biomedical Optics*

**Citation**

Patrick C. Chaumet, Kamal Belkebir, and Raphaël Lencrerot, "Three-dimensional optical imaging in layered media," Opt. Express **14**, 3415-3426 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-8-3415

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### References

- 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. 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]
- 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]
- A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging, Society of Industrial and Applied Mathematics, (2001). [CrossRef]
- 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]
- P. S. Carney and J. C. Schotland, "Three-dimensional total internal reflection microscopy," Opt. Lett. 26, 1072 (2001). [CrossRef]
- P. S. Carney and J. C. Schotland, "Theory of total-internal-reflection tomography," J. Opt. Soc. Am. A 20, 542 (2003). [CrossRef]
- 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]
- 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]
- 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]

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