## An enhanced contrast to detect bulk objects under arbitrary rough surfaces

Optics Express, Vol. 17, Issue 7, pp. 5758-5773 (2009)

http://dx.doi.org/10.1364/OE.17.005758

Acrobat PDF (391 KB)

### Abstract

We study a selective light scattering elimination procedure in the case of highly scattering rough surfaces. Contrary to the case of low scattering levels, the elimination parameters are shown to depend on the sample microstructure and to present rapid variations with the scattering angle. On the other hand, when the slope of the surface is moderated, we show that this parameters present smoother variations and little dependence to the microstructure, even when the roughness is high. These results allow an important selective reduction of the scattered light, with a basic experimental mounting and an analytical determination of the elimination parameters. Such selective scattering reduction is demonstrated by simulations and experiments and applied to the imaging of an object situated under a highly rough surface.

© 2009 Optical Society of America

## 1. Introduction

1. M. Saillard, P. Vincent, and G. Micolau, “Reconstruction of buried objects surrounded by small inhomogeneities,” Inv. Problems **16**, 1195–1208 (2000). [CrossRef]

2. W. Chew and Y. Wang, “Reconstruction of two-dimensional permittivity distribution usingthe distorted Born iterative method,” Med. Imaging, IEEE Transactions on **9**, 218–225 (1990). [CrossRef]

3. D. Ausserré and M. Valignat, “Wide-field optical imaging of surface nanostructures,” Nano Lett. **6**, 1384–1388 (2006). [CrossRef] [PubMed]

4. C. Dunsby and P. French, “Techniques for depth-resolved imaging through turbid media including coherence-gated imaging,” J. Phys. D: Appl. Phys. **36**, R207–R227 (2003). [CrossRef]

5. G. Lerosey, J. de Rosny, A. Tourin, A. Derode, and M. Fink, “Time reversal of wideband microwaves,” Appl. Phys. Lett. **88**, 101 (2006). [CrossRef]

6. C. Amra, C. Grèzes-Besset, and L. Bruel, “Comparison of surface and bulk scattering in optical multilayers,” Appl. Opt. **32**, 5492–5503 (1993). [CrossRef] [PubMed]

7. O. Gilbert, C. Deumié, and C. Amra, “Angle-resolved ellipsometry of scattering patterns from arbitrary surfaces and bulks,” Opt. Express **13**, 2403–2418 (2005). [CrossRef] [PubMed]

8. C. Amra, C. Deumie, and O. Gilbert, “Elimination of polarized light scattered by surface roughness or bulk heterogeneity,” Opt. Express **13**, 854–864 (2005). [CrossRef]

9. G. Georges, C. Deumié, and C. Amra, “Selective probing and imaging in random media based on the elimination of polarized scattering,” Opt. Express **15**, 9804–9816 (2007). [CrossRef] [PubMed]

10. G. Georges, L. Arnaud, L. Siozade, N. Le Neindre, F. Chazallet, M. Zerrad, C. Deumié, and C. Amra, “From angle-resolved ellipsometry of light scattering to imaging in random media,” Appl. Opt. **47**, 257–265 (2008). [CrossRef]

- Rigorous electromagnetic theories must be used and they predict a strong dependence of the ellipsometric parameters versus the sample microstructure. Therefore the microstructure must be known to match the optical plates for scattering reduction, which a priori shows less interest. Though experiment allows to scan the scattering reduction parameters, the 2 dimensional scan (retardation phase and analyzer angle) will only be useful for total reduction (or cancellation), not for selective reduction (except if all objects under probing are known).
- High scattering samples exhibit high variations of ellipsometric parameters versus scattering direction, which makes the experimental procedure much more complex. Indeed in some situations the parameters should be matched every 10
^{-2}degree, depending on the sample microstructure (roughness, inhomogeneity and associated correlation lengths or slopes). One solution may be found in using pixelized liquid crystals, but the resulting procedure becomes time-consuming and looses its simplicity. - Even if the incident source is perfectly monochromatic, partial polarization may occur due to the high derivative of phase terms versus scattering angle inside the receiver solid angle. This third point was previously addressed in ref. [11] and was shown to be solved when receiver solid angles are decreased.
11. C. Amra, M. Zerrad, L. Siozade, G. Georges, and C. Deumié, “Partial polarization of light induced by random defects at surfaces or bulks,” Opt. Express

**16**, 372–383 (2008). [CrossRef]

12. E. Popov and M. Neviere, “Grating theory: new equations in Fourier space leading to fast converging results for TM polarization,” J. Opt. Soc. Am. A **17**, 1773–1784 (2000). [CrossRef]

13. L. Li, “Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings,” J. Opt. Soc. Am. A **13**, 1024–1037 (1996). [CrossRef]

14. L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A 1**3**, 1870–1876 (1996). [CrossRef]

16. C. Amra, “Light scattering from multilayer optics. I. Tools of investigation,” J. Opt. Soc. Am. A **11**197–210 (1994). [CrossRef]

## 2. Principles of the technique

*with wave vector*

**E***can be written*

**k***= (*

**ρ***x*,

*y*,

*z*) and

*A*and

_{s}*A*the components along the

_{p}*s*and

*p*polarization directions (Fig. 1), which can be written

**and**

*e*_{s}**the unit vectors along the directions**

*e*_{p}*s*and

*p*. We note

*δ*the phase difference between

*p*and

*s*,

**and**

*e*_{s}**, inducing a phase delay**

*e*_{p}*δ*

^{*}=

*δ*

^{*}

_{p}-

*δ*

^{*}

_{s}, and an analyser making the angle

*ψ*

^{*}with the

*s*direction, are introduced in the mounting (Fig. 2), the projection of the field on the analyzer direction is

**and**

*e*_{s}**, but Eq. (5) is slightly more complicated in these cases. We note**

*e*_{p}*f*the transformation that connects

**and**

*A**A*′:

*ζ*, the complex number defined as

*f*(

**) =0,that is to say for**

*A**ζ*= -

*A*/

_{s}*A*, which corresponds to an analyzer angle such as

_{p}

**A**_{1}and

**A**_{2}corresponds to sources 1 and 2 alone and

**A**_{12}to their interaction. Notice that quantities present in Eq. (10) are vectorial and complex fields that comprise the phase in their complex argument (cf Eq. (2) and Eq. (3)). Notice also that

**A**_{12}is defined as the difference between the total scattered field

*and the sum (*

**A***1+*

**A**

**A**_{2}) of the fields of the scattering sources 1 and 2 taken alone; all field being solutions of Maxwell equations in the presence of the 2 sources (case of field

*) or in the presence of one or the other source (case of fields*

**A**

*A*_{1}or

**A**_{2}).

*f*(

**A**_{1}) = 0, the remaining signal will then be

*f*(

**A**_{2}+

**A**_{12}) (

*f*being a linear transformation). Such procedure allows for instance, the reduction or the elimination of the light scattered by an interface in order to enhance the contrast of an object situated underneath. Since the ellipsometric properties of the field to eliminate must be known, the procedure generally requires the use of a scattering modeling.

## 3. Description of the rough surface

*z*=

*h*(

*x*,

*y*) (cf Fig. 1 for the axes). The

*roughness*of the surface is defined by

*S*is the area of the considered part of the surface,

*h*

_{0}the mean value of

*h*, and

*= (*

**r***x*,

*y*).

*correlation length*of the rough surface is defined as the width at half height of the autocorrelation function Γ(

*) of*

**r***h*(

*),*

**r***slope w*of the rough surface as the root mean square of the derivative of the surface height:

*R*

_{q}/

*w*[18

18. I. Ohlidal and K. Navratil, “Analysis of the basic statistical properties of randomly rough curved surfaces by shearing interferometry,” Appl. Opt. **24**, 2690–2695 (1985). [CrossRef] [PubMed]

*h*(

*x*,

*y*)).

*μ*=

*μ*

_{0}) and isotropic medium of refractive index

*n*. The superstate refractive index is considered equal to 1. All exemples presented in this paper consider non absorbing media since the main interest is imaging under the rough interface, however the procedure and the theoretical expression of the parameters are also valid for absorbing media.

## 4. Theoretical determination of parameters ∣*A*_{p}∣/∣*A*_{s}∣ and *δ*

_{p}

_{s}

### 4.1. General case: differential method

22. H. Giovannini and C. Amra, “Scattering-reduction effect with overcoated rough surfaces: theory and experiment,” Appl. Opt. **36**, 5574–5579 (1997). [CrossRef] [PubMed]

23. H. Giovannini, M. Saillard, and A. Sentenac, “Numerical study of scattering from rough inhomogeneous films,” J. Opt. Soc. Am. A **15**, 1182–1191 (1998). [CrossRef]

13. L. Li, “Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings,” J. Opt. Soc. Am. A **13**, 1024–1037 (1996). [CrossRef]

12. E. Popov and M. Neviere, “Grating theory: new equations in Fourier space leading to fast converging results for TM polarization,” J. Opt. Soc. Am. A **17**, 1773–1784 (2000). [CrossRef]

14. L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A 1**3**, 1870–1876 (1996). [CrossRef]

23. H. Giovannini, M. Saillard, and A. Sentenac, “Numerical study of scattering from rough inhomogeneous films,” J. Opt. Soc. Am. A **15**, 1182–1191 (1998). [CrossRef]

### 4.2. Case of rough surfaces with large correlation lengths: local reflections model

*θ*is linked to the incident angle

*θ*

_{i}and to the local angle

*θ*

_{local}(Fig. 3) by

*A*and

_{p}*A*, of the field after reflection can be written as a function of the incident field amplitudes,

_{s}*A*

_{0p}and

*A*

_{0s}, and the complex reflection coefficients,

*r*and

_{p}*r*,

_{s}*r*and

_{p}*r*are the Fresnel coefficients,

_{s}*β*

_{1}and

*β*

_{2}are defined by

*A*∣/∣

_{p}*A*∣ = ∣

_{s}*r*

_{p}*A*

_{0p}/∣

*r*

_{s}*A*

_{0s}∣ ratio is then

*δ*is given by

*r*coefficient is real and positive, whatever the local reflection angle, and the

_{s}*r*coefficient is real but vanishes and changes of sign when the local incidence angle,

_{p}*θ*

_{local}, goes beyond the Brewster angle, that is to say when

*r*/

_{p}*r*ratio is then real and positive when the local incidence angle is less than the Brewster angle, which implies

_{s}*δ*= arg(

*A*

_{0p}/

*A*

_{0s}), and real and negative beyond, which implies

*δ*= arg(

*A*

_{0p}/

*A*

_{0s}) + 180°. If we assume an incident field linearly polarized at 45° of the

*s*direction, the scattering pattern for

*δ*is then a step from 0° to 180°, with a scattering angle corresponding to the step

*n*= 1.5) and an incidence of 50°, the step occurs at

*θ*= 62.62°.

*δ*(

*θ*) follows a similar pattern but the step is replaced by a smoother transition from 0° to 180°.

### 4.3. Case of rough surfaces with low roughness: first order electromagnetic theory

16. C. Amra, “Light scattering from multilayer optics. I. Tools of investigation,” J. Opt. Soc. Am. A **11**197–210 (1994). [CrossRef]

*γ*(

**-**

*σ*

*σ*_{0}),

*γ*(

**) defined as**

*σ**h̃*(

**) is the Fourier transform of the surface profile**

*σ**h*(

*x*,

*y*) and

*S*is the area of the light spot.

*σ*_{0}is the projection on the (

*Oxy*) plane of the

**vector of the incident wave and**

*k**σ*the projection on (

*Oxy*) of the

**vector of the scattered wave in the direction**

*k**θ*. The coefficient

*C*(

**,**

*σ*

*σ*_{0}) only depends on the values of the refractive index, the polarization (

*s*or

*p*), the wavelength

*λ*, and

**and**

*σ*

*σ*_{0}. This

*C*(

**,**

*σ*

*σ*_{0}) coefficient does not depend on the particular microstructure of the sample. All the information concerning the microstructure is contained in the term

*γ*(

**-**

*σ*

*σ*_{0}). Let’s note that in the frame of the first order theory, the crossed polarization (

*s*→

*p*or

*p*→

*s*) is zero in the incidence plane.

16. C. Amra, “Light scattering from multilayer optics. I. Tools of investigation,” J. Opt. Soc. Am. A **11**197–210 (1994). [CrossRef]

*θ*

_{i}in

*p*polarization, there is an angle

*θ*

_{b}for which the

*C*coefficient vanishes and changes of sign. This angle, called pseudo-Brewster angle, has the value

*n*= 1.5) and an incidence of 50°, the p polarized scattered light vanishing and the corresponding polarimetric phase shift step, occur for a scattering angle

*θ*

_{b}= 63.98°. The slight difference with the value obtained by the local reflections model can be attributed to the diffraction process and is not contradictory since both methods have different domains of validity.

## 5. Case of surfaces with high slopes

*R*

_{q}≪

*λ*), the ellipsometric parameters, ∣

*A*∣/∣

_{p}*A*∣ and

_{s}*δ*, vary slowly as a function of

*θ*and are independent of the rapid variations of the intensity which form the speckle [9

9. G. Georges, C. Deumié, and C. Amra, “Selective probing and imaging in random media based on the elimination of polarized scattering,” Opt. Express **15**, 9804–9816 (2007). [CrossRef] [PubMed]

7. O. Gilbert, C. Deumié, and C. Amra, “Angle-resolved ellipsometry of scattering patterns from arbitrary surfaces and bulks,” Opt. Express **13**, 2403–2418 (2005). [CrossRef] [PubMed]

25. L. Arnaud, G. Georges, C. Deumié, and C. Amra, “Discrimination of surface and bulk scattering of arbitrary level based on angle-resolved ellipsometry: Theoretical analysis,” Opt. Commun. **281**, 1739–1744 (2008). [CrossRef]

*A*∣/∣

_{p}*A*∣ and

_{s}*δ*, as a function of the scattering angle, in the case of a rough dielectric sample with an high slope (

*w*= 100%, i.e. 45°). Calculation were performed with the differential method and results given by the approximate first order theory are also presented Fig. 4. It can be observed that the parameters vary with a large amplitude at the speckle scale, even though the roughness is moderated (

*R*

_{q}= 100 nm). The elimination of such polarized field requires the adjustment of the analyzer angle and phase retarder for every speckle grain, which could more easily be done using specific devices, such as pixelized liquid crystal matrices for instance.

## 6. Case of surfaces with moderated slopes

### 6.1. Difference between a low slope surface and a high slope surface for a same roughness value

*δ*and the ∣

*A*∣/∣

_{p}*A*∣ ratio in the same conditions as for Fig. 4, but for a low slope:

_{s}*w*= 2% (or 1.15°). Even though the roughness is the same,

*R*

_{q}= 100 nm, the oscillations are small compared to the case of the high slope surface. The results presented here, not only present few oscillations, but are also in good accordance with those of first order theory.

### 6.2. Roughness influence in the case of a moderated slope

*A*∣/∣

_{p}*A*∣ ratio, simulated by the differential method and plotted as a function of

_{s}*θ*, for several surfaces with the same slope

*w*= 15% and for increasing roughness. The results obtained by the local reflections model and by the first order theory are also presented Fig. 8.

*R*

_{q}= 1 nm, which corresponds to a super-polished glass sample, to

*R*

_{q}= 500 nm, a roughness of the same order of the wavelength, for which all the incident light is scattered (no specular beam remaining).

*R*

_{q}= 1 nm) the results are in perfect accordance with those of first order theory. When the roughness increases (

*R*

_{q}= 20 nm), rapid oscillations appear in the ∣

*A*∣/

_{p}*A*∣ pattern, but remain of limited amplitude. When the roughness is still increased (

_{s}*R*

_{q}from 50 to 500 nm) we observe that the amplitude of the oscillations doesn’t increase significantly and that the average position of the curve moves slowly from the one corresponding to the first order theory to the one corresponding to the local reflections model.

*μ*m, which is less than usual spot light diameters. It is possible that the profile length has an influence on the oscillations amplitude, but the few calculations we made to answer this question doesn’t show a clear tendency and suggest a weak influence of the length.

*δ*follows the same behavior than ∣

*A*∣/

_{p}*A*∣: oscillations around the mean value remain of limited amplitude as long as the slope is moderated (≲ 20%), cf. [25

_{s}25. L. Arnaud, G. Georges, C. Deumié, and C. Amra, “Discrimination of surface and bulk scattering of arbitrary level based on angle-resolved ellipsometry: Theoretical analysis,” Opt. Commun. **281**, 1739–1744 (2008). [CrossRef]

## 7. Application to surface scattering reduction: simulations

*s*and

*p*components of the field scattered by the same rough surface is represented Fig. 6. The slope of the surface is 15% and the roughness 500 nm. The incident field is linearly polarized at 45° of the s direction and has an incidence of 50°. The scattering reduction parameters, analyzer angle

*ψ*

^{*}and phase delay

*δ*

^{*}, are those obtained by the local reflection model and are represented Fig. 9. The intensities before and after reduction of the scattered light are represented in logarithmic scale. The intensity after passing through the phase retarder and projection on the analyzer is obtained by the relation

^{-4}. This result is very satisfactory since it shows the possibility of a high attenuation factor without any knowledge of the particular topography of the rough surface, as long as its slope is moderated.

*=*

**A**

**A**_{1}+

**A**_{2}+

**A**_{12}denotes the total field with 1 and 2 corresponding respectively to the rough surface and the object situated underneath, the relation

*f*(

*ζ*

_{1},

**A**_{1}) = 0 must not imply

*f*(

*ζ*

_{1},

**A**_{2}) +

*f*(

*ζ*

_{1},

**A**_{12}) = 0.

**A**_{2}). When the surface scattering reduction parameters are used to reduce scattering from the rough surface, that is

*f*(

*ζ*

_{1},

**A**_{1}) ≃ 0, we observe a reduction of the reference object by a factor 0.12 which shows that

*f*(

*ζ*

_{1},

*A*

_{2}) is far from zero (in comparison with the 0.00029 factor obtained with the rough surface). Such successful result is connected with the different polarization behaviors for the object (

**A**_{2}field) and the rough surface (

*A*_{1}field).

*=*

**A**

**A**_{1}+

**A**_{2}+

**A**_{12}and is turned after reduction of the surface scattering (

*f*(

*ζ*

_{1},

*A*

_{1}) ≃ 0) into

*f*(

*ζ*

_{1},

*) ≃*

**A***f*(

*ζ*

_{1},

**A**_{2}) +

*f*(

*ζ*

_{1},

**A**_{12}). Notice that the angular dependence of Fig. 11(a) is not exactly recovered with Fig. 11(b). Indeed, after reduction of the light scattered by the rough interface, most of the light comes from the scattering of the cylindrical inclusion, however its angular dependence is modified by the interaction with the rough interface. Mathematically it corresponds to the term

**A**_{12}and physically to the fact that the light coming from the inclusions is scattered in transmission by the rough interface.

## 8. Application to surface scattering reduction: experimental demonstration

*μ*m and a correlation length of 32

*μ*m (measured by a white light interferometer). With this roughness, the light is completely scattered (no specular reflection). The back surface of the sample is superpolished with negligible direct contribution to the scattering process.

*λ*= 632.8 nm) linearly polarized at 45° of s direction and with an angle of incidence

*θ*

_{i}= 10°. The measurements were performed with a CCD camera in the scattering direction

*θ*= 15° in the incidence plane (i.e. 25° between the direction of incidence and the direction of measurement). The applied scattering reduction parameters, determined by the local reflection model, were

*ψ*

^{*}= 133.1° and

*δ*

^{*}= 0.0°. Results are given in Fig. 12. The left and right figures were recorded with the same exposure time, before and after applying the procedure. An attenuation filter was used for the left figure in order to avoid overexposure. Comparison of these two figures highlights a reduction factor close to 10

^{-3}, which confirm the efficiency of the procedure. Notice that the light reflected by the back surface, after scattering in transmission by the front rough surface, is also reduced, since in these conditions, its parameters are close to those of the light scattered in reflection by the front rough surface.

## 9. Conclusion

## References and links

1. | M. Saillard, P. Vincent, and G. Micolau, “Reconstruction of buried objects surrounded by small inhomogeneities,” Inv. Problems |

2. | W. Chew and Y. Wang, “Reconstruction of two-dimensional permittivity distribution usingthe distorted Born iterative method,” Med. Imaging, IEEE Transactions on |

3. | D. Ausserré and M. Valignat, “Wide-field optical imaging of surface nanostructures,” Nano Lett. |

4. | C. Dunsby and P. French, “Techniques for depth-resolved imaging through turbid media including coherence-gated imaging,” J. Phys. D: Appl. Phys. |

5. | G. Lerosey, J. de Rosny, A. Tourin, A. Derode, and M. Fink, “Time reversal of wideband microwaves,” Appl. Phys. Lett. |

6. | C. Amra, C. Grèzes-Besset, and L. Bruel, “Comparison of surface and bulk scattering in optical multilayers,” Appl. Opt. |

7. | O. Gilbert, C. Deumié, and C. Amra, “Angle-resolved ellipsometry of scattering patterns from arbitrary surfaces and bulks,” Opt. Express |

8. | C. Amra, C. Deumie, and O. Gilbert, “Elimination of polarized light scattered by surface roughness or bulk heterogeneity,” Opt. Express |

9. | G. Georges, C. Deumié, and C. Amra, “Selective probing and imaging in random media based on the elimination of polarized scattering,” Opt. Express |

10. | G. Georges, L. Arnaud, L. Siozade, N. Le Neindre, F. Chazallet, M. Zerrad, C. Deumié, and C. Amra, “From angle-resolved ellipsometry of light scattering to imaging in random media,” Appl. Opt. |

11. | C. Amra, M. Zerrad, L. Siozade, G. Georges, and C. Deumié, “Partial polarization of light induced by random defects at surfaces or bulks,” Opt. Express |

12. | E. Popov and M. Neviere, “Grating theory: new equations in Fourier space leading to fast converging results for TM polarization,” J. Opt. Soc. Am. A |

13. | L. Li, “Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings,” J. Opt. Soc. Am. A |

14. | L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A 1 |

15. | M. Born and E. Wolf, |

16. | C. Amra, “Light scattering from multilayer optics. I. Tools of investigation,” J. Opt. Soc. Am. A |

17. | A. Voronovich, |

18. | I. Ohlidal and K. Navratil, “Analysis of the basic statistical properties of randomly rough curved surfaces by shearing interferometry,” Appl. Opt. |

19. | P. Beckmann and A. Spizzichino, |

20. | R. Petit, “Diffraction d’une onde plane par un reseau metallique,” Rev. Opt. |

21. | G. Cerutti-Maori, R. Petit, and M. Cadilhac, “Etude numérique du champ diffracté par un réseau,” CR Acad. Sci |

22. | H. Giovannini and C. Amra, “Scattering-reduction effect with overcoated rough surfaces: theory and experiment,” Appl. Opt. |

23. | H. Giovannini, M. Saillard, and A. Sentenac, “Numerical study of scattering from rough inhomogeneous films,” J. Opt. Soc. Am. A |

24. | M. Neviere and E. Popov, |

25. | L. Arnaud, G. Georges, C. Deumié, and C. Amra, “Discrimination of surface and bulk scattering of arbitrary level based on angle-resolved ellipsometry: Theoretical analysis,” Opt. Commun. |

26. | R. Azzam and N. Bashara, |

**OCIS Codes**

(120.6150) Instrumentation, measurement, and metrology : Speckle imaging

(240.5770) Optics at surfaces : Roughness

(260.2130) Physical optics : Ellipsometry and polarimetry

(290.5820) Scattering : Scattering measurements

(290.5825) Scattering : Scattering theory

(290.5855) Scattering : Scattering, polarization

**ToC Category:**

Scattering

**History**

Original Manuscript: November 7, 2008

Revised Manuscript: December 24, 2008

Manuscript Accepted: December 25, 2008

Published: March 26, 2009

**Citation**

L. Arnaud, G. Georges, J. Sorrentini, M. Zerrad, C. Deumié, and C. Amra, "An enhanced contrast to detect bulk objects under arbitrary rough surfaces," Opt. Express **17**, 5758-5773 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-7-5758

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### References

- M. Saillard, P. Vincent, and G. Micolau, "Reconstruction of buried objects surrounded by small inhomogeneities," Inv. Problems 16, 1195-1208 (2000). [CrossRef]
- W. Chew and Y. Wang, "Reconstruction of two-dimensional permittivity distribution using the distorted Born iterative method," IEEE Trans. Med. Imaging 9, 218-225 (1990). [CrossRef]
- D. Ausserré and M. Valignat, "Wide-field optical imaging of surface nanostructures," Nano Lett. 6, 1384-1388 (2006). [CrossRef] [PubMed]
- C. Dunsby and P. French, "Techniques for depth-resolved imaging through turbid media including coherencegated imaging," J. Phys. D: Appl. Phys. 36, R207-R227 (2003). [CrossRef]
- G. Lerosey, J. de Rosny, A. Tourin, A. Derode, and M. Fink, "Time reversal of wideband microwaves," Appl. Phys. Lett. 88, 101 (2006). [CrossRef]
- C. Amra, C. Grèzes-Besset, and L. Bruel, "Comparison of surface and bulk scattering in optical multilayers," Appl. Opt. 32, 5492-5503 (1993). [CrossRef] [PubMed]
- O. Gilbert, C. Deumié, and C. Amra, "Angle-resolved ellipsometry of scattering patterns from arbitrary surfaces and bulks," Opt. Express 13, 2403-2418 (2005). [CrossRef] [PubMed]
- C. Amra, C. Deumie, and O. Gilbert, "Elimination of polarized light scattered by surface roughness or bulk heterogeneity," Opt. Express 13, 854-864 (2005). [CrossRef]
- G. Georges, C. Deumié, and C. Amra, "Selective probing and imaging in random media based on the elimination of polarized scattering," Opt. Express 15, 9804-9816 (2007). [CrossRef] [PubMed]
- G. Georges, L. Arnaud, L. Siozade, N. Le Neindre, F. Chazallet, M. Zerrad, C. Deumié, and C. Amra, "From angle-resolved ellipsometry of light scattering to imaging in random media," Appl. Opt. 47, 257-265 (2008). [CrossRef]
- C. Amra, M. Zerrad, L. Siozade, G. Georges, and C. Deumié, "Partial polarization of light induced by random defects at surfaces or bulks," Opt. Express 16, 372-383 (2008). [CrossRef]
- E. Popov and M. Neviere, "Grating theory: new equations in Fourier space leading to fast converging results for TM polarization," J. Opt. Soc. Am. A 17, 1773-1784 (2000). [CrossRef]
- L. Li, "Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings," J. Opt. Soc. Am. A 13, 1024-1037 (1996). [CrossRef]
- L. Li, "Use of Fourier series in the analysis of discontinuous periodic structures," J. Opt. Soc. Am. A 13, 1870-1876 (1996). [CrossRef]
- M. Born and E. Wolf, Principles of Optics, 7th ed. (Pergamon Press New York, 1999).
- C. Amra, "Light scattering from multilayer optics. I. Tools of investigation," J. Opt. Soc. Am. A 11197-210 (1994). [CrossRef]
- A. Voronovich, Wave Scattering from Rough Surfaces (Springer, 1994).
- I. Ohlidal and K. Navratil, "Analysis of the basic statistical properties of randomly rough curved surfaces by shearing interferometry," Appl. Opt. 24, 2690-2695 (1985). [CrossRef] [PubMed]
- P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Norwood, MA, Artech House, Inc., 1987).
- R. Petit, "Diffraction d’une onde plane par un reseau metallique," Rev. Opt. 45, 353-370 (1966).
- G. Cerutti-Maori, R. Petit, and M. Cadilhac, "Etude numérique du champ diffracté par un réseau," C. R. Acad. Sci. 268, 1060-1063 (1969).
- H. Giovannini and C. Amra, "Scattering-reduction effect with overcoated rough surfaces: theory and experiment," Appl. Opt. 36, 5574-5579 (1997). [CrossRef] [PubMed]
- H. Giovannini, M. Saillard, and A. Sentenac, "Numerical study of scattering from rough inhomogeneous films," J. Opt. Soc. Am. A 15, 1182-1191 (1998). [CrossRef]
- M. Neviere and E. Popov, Light Propagation in Periodic Media: Differential Theory and Design (Marcel Dekker, 2003).
- L. Arnaud, G. Georges, C. Deumié, and C. Amra, "Discrimination of surface and bulk scattering of arbitrary level based on angle-resolved ellipsometry: Theoretical analysis," Opt. Commun. 281,1739-1744 (2008). [CrossRef]
- R. Azzam and N. Bashara, Ellipsometry and Polarized Light (North-Holland, 1977).

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