## Profile estimation for Pt submicron wire on rough Si substrate from experimental data |

Optics Express, Vol. 20, Issue 19, pp. 21678-21686 (2012)

http://dx.doi.org/10.1364/OE.20.021678

Acrobat PDF (1624 KB)

### Abstract

An efficient forward scattering model is constructed for penetrable 2D submicron particles on rough substrates. The scattering and the particle-surface interaction are modeled using discrete sources with complex images. The substrate micro-roughness is described by a heuristic surface transfer function. The forward model is applied in the numerical estimation of the profile of a platinum (Pt) submicron wire on rough silicon (Si) substrate, based on experimental Bidirectional Reflectance Distribution Function (BRDF) data.

© 2012 OSA

## 1. Introduction

1. F. González, G. Videen, P. J. Valle, J. M. Saiz, J. L. de la Peña, and F. Moreno, “Light scattering computational methods for particles on substrates,” J. Quant. Spectrosc. Radiat. Transf. **70**(4-6), 383–393 (2001). [CrossRef]

5. M. Karamehmedović, P.-E. Hansen, and T. Wriedt, “An efficient scattering model for PEC and penetrable nanowires on a dielectric substrate,” J. Eur. Opt. Soc. Rapid Publ. **6**, 11021 (2011). [CrossRef]

6. T. A. Germer, G. W. Mulholland, J. H. Kim, and S. H. Ehrman, “Measurement of the 100 nm NIST SRM® 1963 by laser surface light scattering,” *Advanced Characterization Techniques for Optical, Semiconductor, and Data Storage Components*, Angela Duparré and Bhanwar Singh, Eds., Proc. SPIE **4779**, 60–71 (2002).

10. B. W. Bell and W. S. Bickel, “Single fiber light scattering matrix: an experimental determination,” Appl. Opt. **20**(22), 3874–3879 (1981). [CrossRef] [PubMed]

*et al.*[5

5. M. Karamehmedović, P.-E. Hansen, and T. Wriedt, “An efficient scattering model for PEC and penetrable nanowires on a dielectric substrate,” J. Eur. Opt. Soc. Rapid Publ. **6**, 11021 (2011). [CrossRef]

11. M. Karamehmedović, P.-E. Hansen, and T. Wriedt, “A fast inversion method for highly conductive submicron wires on a substrate,” J. Eur. Opt. Soc. Rapid Publ. **6**, 11039 (2011). [CrossRef]

11. M. Karamehmedović, P.-E. Hansen, and T. Wriedt, “A fast inversion method for highly conductive submicron wires on a substrate,” J. Eur. Opt. Soc. Rapid Publ. **6**, 11039 (2011). [CrossRef]

5. M. Karamehmedović, P.-E. Hansen, and T. Wriedt, “An efficient scattering model for PEC and penetrable nanowires on a dielectric substrate,” J. Eur. Opt. Soc. Rapid Publ. **6**, 11021 (2011). [CrossRef]

11. M. Karamehmedović, P.-E. Hansen, and T. Wriedt, “A fast inversion method for highly conductive submicron wires on a substrate,” J. Eur. Opt. Soc. Rapid Publ. **6**, 11039 (2011). [CrossRef]

**6**, 11039 (2011). [CrossRef]

## 2. Experimental setup

_{0}= 325 nm. Using reference Scanning Electron Microscopy (SEM) and Atomic Force Microscopy (AFM) measurements, shown in Figs. 1a )–1c), the length, width and height of the SW are estimated to λ = 98.86 μm ≈304λ

_{0}, 𝓌: = 2

*a*

_{SEM}= 537.5 nm ≈1.65λ

_{0}and 𝒽: = 2

*b*

_{AFM}= 562.5 nm ≈1.73λ

_{0}, respectively.

_{0}, and the minimal distance of the SW to the substrate edge is greater than 1 cm ≈30769λ

_{0}. The SW can therefore be modeled accurately as a 2D cylindrical scatterer placed on an infinite half-plane substrate, see Fig. 1d). In the following, as indicated in Fig. 1d), the BRDF setup has the incidence and the measurement plane coincide with the plane of the SW cross-section σ. The angle of incidence θ

_{0}is fixed at 60° from normal, and the measured BRDF is hence a function of just one angle, namely the angle of observation in the SW cross-section plane. Such in-plane BRDF is defined, e.g., in Stover [13, Section 1.5].

*s*-polarized, and it is chopped to reduce both electrical and optical noise. The output power is regulated by a set of neutral density filters placed in front of a polarizer. On the detector side the beam passes through an analyzer before it is detected by a linear silicon detector connected to a lock-in amplifier. The distance, width and height of the detector slit are 30 cm, 0.2 mm and 3 mm, respectively. The Si substrate with the SW is mounted on a sample holder system with 6 degrees of freedom. We first measure the incident power

*P*

_{i}by moving the sample holder system out of the light path. After moving the sample holder back, the laser light is placed in the focus of a microscope equipped with a 50 × magnifying long working distance objective (Mitutoyo). Then, we scan the Si sample until we see a microscope image of the SW. The alignment of the laser beam on SW is checked before the microscope is removed and the BRDF scan is started. During the BRDF scan we measure the scattered power per unit solid angle,

*dP*

_{s}/

*d*Ω

_{s}, for a given scattering angle θ

_{s}, and we thereby obtain the BRDF signal (see Eq. (1).9) in [13], Section 1.5),

_{s}∈ [–20°,20°] from specular reflection, and varying the incident power using the filters while always maintaining the same setting on the lock-in amplifier. Since only a discrete set of filters are available, there are gaps in the resulting scattering spectra.

## 3. Forward scattering and roughness model

*j*ω

*t*). The angle of incidence and of specular reflection is θ

_{0}= π/3 (: 60°) from normal. As shown in Fig. 2b),

**r**= (

*x*,

*y*) is the vector specifying the observation point, with length |

**r**| and angle θ from the direction of specular reflection. The laser illumination is approximated by an incident TE-polarized plane wave with Gaussian amplitude modulation,where the variance σ

_{inc}

^{2}= 3.091⋅10

^{−6}m

^{2}is found by matching the bare-substrate experimental BRDF data (no SW present) close to specular reflection, and neglecting the effect of convolution with the finite-width detector slit.

**E**

^{ref},

**H**

^{ref}); here,

**E**

^{ref}=

**z**̂

*E*

^{ref}=

**z**̂Γ

_{ref}

*E*

^{inc}, where Γ

_{ref}= (

*n*

_{0}–

*n*

_{Si}) / (

*n*

_{0}+

*n*

_{Si}) ≈–0.742 +

*j*0.137 is the Fresnel reflection coefficient for TE waves incident upon the air-Si interface at λ

_{0}= 325 nm (the refractive index

*n*

_{Si}is here taken from Palik [14]). The total field in the exterior of the SW cross-section σ in the upper half-plane, that is, in ϒ

^{2}

_{+}\ σ, is the sum of the incident wave, the reflected wave, and the field (

**E**

^{sca},

**H**

^{sca}) scattered by the SW,

**E**

^{tot}=

**E**

^{inc}+

**E**

^{ref}+

**E**

^{sca}. The field scattered by the SW is approximated in ϒ

^{2}

_{+}\ σ by a linear combination of fields emitted by discrete

*z*-directed electric line currents located within the cross-section σ and radiating in the presence of the substrate,

_{1/2,Si}(⋅,

**r**’) is the half-plane Green’s function for the Helmholtz operator in the air-substrate medium, with singularity at

**r**’. Since |

*n*

_{Si}

^{2}/

*n*

_{0}

^{2}| ≈35.8 is large, we use the ‘image at a complex depth’ approximation derived, e.g., in Lindell and Alanen [15

15. I. V. Lindell and E. Alanen, “Exact Image Theory for the Sommerfeld Half-Space Problem, Part I: Vertical Magnetic Dipole,” IEEE Trans. Antenn. Propag. **32**(2), 126–133 (1984). [CrossRef]

**r**’ and at

**r**

_{̃}̃

_{ν}’ = (

*x*

_{ν}’, –

*y*

_{ν}’) +

**y**̂2

*jk*

_{0}

^{−1}(

*n*

_{Si}

^{2}–1)

^{-1/2}≈(

*x*

_{ν}’, –

*y*

_{ν}’) +

**y**̂(–0.03 +

*j*0.04), respectively. (The derivation in [15

15. I. V. Lindell and E. Alanen, “Exact Image Theory for the Sommerfeld Half-Space Problem, Part I: Vertical Magnetic Dipole,” IEEE Trans. Antenn. Propag. **32**(2), 126–133 (1984). [CrossRef]

*H*

_{0}

^{(2)}is the Hankel function of zero order and second kind, and

*k*

_{0}= 2π/λ

_{0}is the free-space wave number. The total field within the SW cross-section σ is approximated by a linear combination of fields emitted by discrete

*z*-directed electric line currents located outside the cross-section σ and radiating in the Pt-filled plane,

_{Pt}(⋅,

**r”**) is the Green’s function for the Helmholtz operator in the Pt-filled plane, with singularity at

**r”**, Φ

_{Pt}(

**r**,

**r**

_{ν}”) =

*H*

_{0}

^{(2)}(

*k*

_{Pt}|

**r**–

**r”**) / 4

*j*. The wave number

*k*

_{Pt}=

*n*

_{Pt}

*k*

_{0}, with the refractive index for Pt at 325 nm taken from [14]. Approximating the fields in terms of discrete sources instead of in terms of surface current densities simplifies the numerical implementation and accelerates the scattering computation, since the numerical integration of the densities is avoided; the classical radiation integrals are replaced with readily computable finite sums in Eq. (3) and Eq. (5). The complex source images account for the particle-surface interaction in the field scattered by the SW. The discrete sources are uniformly distributed along ellipses with semi-diameters 0.86

*a*and 0.86

*b*(interior sources and their images) and

*a*/0.86,

*b*/0.86 (exterior sources). These semi-diameters were found well-suited through numerical experimentation. The complex amplitudes

*C*

_{ν},

*D*

_{ν}of the discrete sources are computed by imposing the transmission conditions (continuity of the total tangential electric and magnetic field) at a number of discrete testing points

**t**

_{μ}distributed uniformly along the circumference of σ,

*C*

_{ν},

*D*

_{ν}, the BRDF and the near field can be computed using Eqs. (3) and (5). For the numerical results of Section 4, we use 20 interior and 20 exterior discrete sources (

*M*=

*N*= 20), as well as 20 testing points.

*I*

_{num}and the corresponding measured data

*I*

_{meas}at

*n*observation points, the two will here be compared using the following root mean square error measure:

_{0}, it holds that cos θ ≈1, and it is seen from Eq. (1) that the reflected field intensity is approximately proportional to the corresponding BRDF. We therefore approximate the surface transfer function by the ratio of the Fourier transforms of the experimental bare-substrate BRDF

*I*

_{sca,0}and the incident field intensity |

*E*

^{inc}|

^{2}. Close to specular reflection, it holds that sin θ ≈θ, so by Eq. (2) we have

*|E*

^{inc}|

^{2}(θ) ≈(max |E

^{inc}|

^{2}) exp(–(|

**r|**θ)

^{2}/σ

_{inc}

^{2}) for θ ∈ [–20°,20°], and the Fourier transform of

*|E*

^{inc}|

^{2}satisfies ℱ|

*E*

^{inc}|

^{2}(ξ) ≈(max |

*E*

^{inc}|

^{2}) σ

_{inc}π

^{1/2}|

**r**|

^{−1}exp(–ξ

^{2}σ

_{inc}

^{2}/ 4|

**r**|

^{2}). The bare-substrate BRDF

*I*

_{sca,0}is fitted by a smooth function,as shown in Fig. 3a ).

*A*

_{1}= 0.8624 sr

^{−1},

*A*

_{2}= 0.004452 sr

^{−1},

*A*

_{3}= 8.219⋅10

^{−6}sr

^{−1}, σ

_{1}= 0.001933 m, σ

_{2}= 0.009941 m and σ

_{3}= 0.08051 m. To qualify this fit, the achieved RMSE of 0.4284 is comparable to the RMSE of 0.4276 obtained for the least-squares fit of log

_{10}

*I*

_{sca,0}with a truncated Fourier series up to 11th harmonic,also shown in Fig. 3a). Here γ = 11.94, the parameters α

_{0}–α

_{11}are –4.674, 1.332, 0.7167, 0.4667, 0.4301, 0.2779, 0.2851, 0.2528, 0.2198, 0.1538, 0.08371, 0.05487, and the parameters β

_{1}–β

_{11}are –0.05542, –0.05462, –0.1115, –0.1527, –0.08773, –0.0391, –0.00576, –0.00399, 0.06058, 0.06663, 0.07861. The fit function in Eq. (7) – a sum of Gaussian terms – is chosen such that its Fourier transform is readily computable, although this is not strictly required since we can use the Discrete Fourier Transform to deal with more general cases. Also, the fit function in Eq. (7) has a form similar to what is typically found in image intensity distributions for scattering by rough substrates, see, e.g., Fig. 7c) in Harvey

*et al.*[16

16. J. E. Harvey, E. C. Moran, and W. P. Zmek, “Transfer function characterization of grazing incidence optical systems,” Appl. Opt. **27**(8), 1527–1533 (1988). [CrossRef] [PubMed]

*H*is approximated by

*I*

_{sca,1,SMOOTH}for the SW on the

*smooth*substrate. The Fourier transform of

*I*

_{sca,1,SMOOTH}is then modified using the surface transfer function

*H*, ℱ

*I*

_{sca,1,ROUGH}: =

*H*ℱ

*I*

_{sca,1,SMOOTH}, and the final BRDF

*I*

_{sca,1,ROUGH}is computed by taking the inverse Fourier transform,

*I*

_{sca,1,ROUGH}= ℱ

^{−1}

*H*ℱ

*I*

_{sca,1,SMOOTH}. Figure 3b) compares the measured BRDF with the predictions obtained using the scattering model with the measured cross-section semi-diameters

*a*

_{SEM}= 268.75 nm and

*b*

_{AFM}= 281.25 nm. It is seen that including the roughness in the model improves the correspondence between the computed and the measured BRDF; the RMSE w.r.t. the experimental data is reduced from 0.6386 to 0.3750.

_{2}and a volume fraction

*f*of air in the rough interface, the effective medium layer relative permittivity ε

_{1}is given in Eq. (5).43) on p. 179 in [17],

## 4. Numerical results and discussion

*a*∈ [200 nm, 350 nm] = [0.74

*a*

_{SEM},1.30

*a*

_{SEM}],

*b*∈ [200 nm, 350 nm] = [0.71

*b*

_{AFM},1.24

*b*

_{AFM}] of cross-section semi-diameters

*a*and

*b*. Figures 4c) and 4d) present the same RMSE in top view, to better show the global minima. These are circled in black, and located at

*a*= 290 nm,

*b*= 320 nm in the rough case and at

*a*= 350 nm,

*b*= 290 nm in the smooth case. The average computation time for a single value of the RMSE is 4.0 sec and 3.8 sec in the two cases, with a single-core MATLAB® implementation on a standard PC. It is seen that including the roughness makes the inverse problem better posed, in that the global minimum, indicating the best estimate for

*a*and

*b*, is better localized. Also, it generally improves the correspondence between the computation and the measurement results, since the RMSE takes on smaller values.

**E**

^{inc}+

**E**

^{sca}and the reconstructed total electric near field

**E**

^{inc}+

**E**

^{ref}+

**E**

^{sca}, respectively, corresponding to the estimates

*a*= 290 nm,

*b*= 320 nm. We here do not make use of the reconstructed near field in the estimation of the SW cross-section, and Figs. 4e)-4f) are only shown to demonstrate the quality of the reconstructed near field. The contour of the SW cross section is visible as a locus of minima of the field in Fig. 4e); the discrete sources can also be seen in this figure. The SW cross-section itself is visible as the region where the total field vanishes in Fig. 4f). The plotted field amplitude in Fig. 4f) has been limited to 2V/m to enhance the contrast. This imaging of the particle is possible because the Pt SW is highly conductive, and the total field nearly vanishes within the cross-section σ. However, we stress that the forward model is directly applicable to arbitrary penetrable SW, and the imaging step is not necessary in the estimation of the cross-section semi-diameters

*a*and

*b*. The near-field plots in Figs. 4e) and 4f) are of resolution 100 × 100 points, and each took approx. 48 sec to compute. Table 1

a | b | ||
---|---|---|---|

rough model | 290 nm | 320 nm | |

smooth model | 350 nm | 290 nm | |

SEM measurement | 268.75 nm | - | |

AFM measurement | - | 281.25 nm | |

rel. error, rough model | 7.91% | 13.78% | average: 10.85% |

rel. error, smooth model | 30.23% | 3.11% | average: 16.67% |

*et al.*[6

6. T. A. Germer, G. W. Mulholland, J. H. Kim, and S. H. Ehrman, “Measurement of the 100 nm NIST SRM® 1963 by laser surface light scattering,” *Advanced Characterization Techniques for Optical, Semiconductor, and Data Storage Components*, Angela Duparré and Bhanwar Singh, Eds., Proc. SPIE **4779**, 60–71 (2002).

7. J. L. de la Peña, J. M. Saiz, P. J. Valle, F. González, and F. F. Moreno, “Tracking Scattering Minima to Size Metallic Particles on Flat Substrates,” Part. Part. Syst. Charact. **16**(3), 113–118 (1999). [CrossRef]

10. B. W. Bell and W. S. Bickel, “Single fiber light scattering matrix: an experimental determination,” Appl. Opt. **20**(22), 3874–3879 (1981). [CrossRef] [PubMed]

6. T. A. Germer, G. W. Mulholland, J. H. Kim, and S. H. Ehrman, “Measurement of the 100 nm NIST SRM® 1963 by laser surface light scattering,” *Advanced Characterization Techniques for Optical, Semiconductor, and Data Storage Components*, Angela Duparré and Bhanwar Singh, Eds., Proc. SPIE **4779**, 60–71 (2002).

7. J. L. de la Peña, J. M. Saiz, P. J. Valle, F. González, and F. F. Moreno, “Tracking Scattering Minima to Size Metallic Particles on Flat Substrates,” Part. Part. Syst. Charact. **16**(3), 113–118 (1999). [CrossRef]

10. B. W. Bell and W. S. Bickel, “Single fiber light scattering matrix: an experimental determination,” Appl. Opt. **20**(22), 3874–3879 (1981). [CrossRef] [PubMed]

*a*and

*b*are estimated. (In Karamehmedović

*et al.*[18

18. M. Karamehmedović, M.-P. Sørensen, P.-E. Hansen, and A. V. Lavrinenko, “Application of the method of auxiliary sources to a defect-detection inverse problem of optical diffraction microscopy,” J. Eur. Opt. Soc. Rapid Publ. **5**, 10021 (2010). [CrossRef]

16. J. E. Harvey, E. C. Moran, and W. P. Zmek, “Transfer function characterization of grazing incidence optical systems,” Appl. Opt. **27**(8), 1527–1533 (1988). [CrossRef] [PubMed]

19. S. Schröder, A. Duparré, L. Coriand, A. Tünnermann, D. H. Penalver, and J. E. Harvey, “Modeling of light scattering in different regimes of surface roughness,” Opt. Express **19**(10), 9820–9835 (2011). [CrossRef] [PubMed]

## 5. Conclusion and outlook

**6**, 11021 (2011). [CrossRef]

**6**, 11039 (2011). [CrossRef]

## Acknowledgments

## References and links

1. | F. González, G. Videen, P. J. Valle, J. M. Saiz, J. L. de la Peña, and F. Moreno, “Light scattering computational methods for particles on substrates,” J. Quant. Spectrosc. Radiat. Transf. |

2. | E. Eremina, Yu. Eremin, and T. Wriedt, “Analysis of the light scattering properties of a gold nanorod on a plane surface via discrete sources method,” Opt. Commun. |

3. | E. Eremina, Yu. Eremin, and T. Wriedt, “Discrete sources method for simulation of resonance spectra of nonspherical nanoparticles on a plane surface,” Opt. Commun. |

4. | P. Albella, F. Moreno, J. M. Saiz, and F. González, “2D double interaction method for modeling small particles contaminating microstructures located on substrates,” J. Quant. Spectrosc. Radiat. Transf. |

5. | M. Karamehmedović, P.-E. Hansen, and T. Wriedt, “An efficient scattering model for PEC and penetrable nanowires on a dielectric substrate,” J. Eur. Opt. Soc. Rapid Publ. |

6. | T. A. Germer, G. W. Mulholland, J. H. Kim, and S. H. Ehrman, “Measurement of the 100 nm NIST SRM® 1963 by laser surface light scattering,” |

7. | J. L. de la Peña, J. M. Saiz, P. J. Valle, F. González, and F. F. Moreno, “Tracking Scattering Minima to Size Metallic Particles on Flat Substrates,” Part. Part. Syst. Charact. |

8. | P. Albella, F. Moreno, J. M. Saiz, and F. González, “Backscattering of metallic microstructures with small defects located on flat substrates,” Opt. Express |

9. | J. M. Saiz, J. L. de la Peña, F. González, and F. Moreno, “Detection and recognition of local defects in 1D structures,” Opt. Commun. |

10. | B. W. Bell and W. S. Bickel, “Single fiber light scattering matrix: an experimental determination,” Appl. Opt. |

11. | M. Karamehmedović, P.-E. Hansen, and T. Wriedt, “A fast inversion method for highly conductive submicron wires on a substrate,” J. Eur. Opt. Soc. Rapid Publ. |

12. | D. Colton and R. Kress, |

13. | J. C. Stover, |

14. | E. D. Palik, |

15. | I. V. Lindell and E. Alanen, “Exact Image Theory for the Sommerfeld Half-Space Problem, Part I: Vertical Magnetic Dipole,” IEEE Trans. Antenn. Propag. |

16. | J. E. Harvey, E. C. Moran, and W. P. Zmek, “Transfer function characterization of grazing incidence optical systems,” Appl. Opt. |

17. | H. Fujiwara, |

18. | M. Karamehmedović, M.-P. Sørensen, P.-E. Hansen, and A. V. Lavrinenko, “Application of the method of auxiliary sources to a defect-detection inverse problem of optical diffraction microscopy,” J. Eur. Opt. Soc. Rapid Publ. |

19. | S. Schröder, A. Duparré, L. Coriand, A. Tünnermann, D. H. Penalver, and J. E. Harvey, “Modeling of light scattering in different regimes of surface roughness,” Opt. Express |

**OCIS Codes**

(120.4290) Instrumentation, measurement, and metrology : Nondestructive testing

(110.3200) Imaging systems : Inverse scattering

**ToC Category:**

Instrumentation, Measurement, and Metrology

**History**

Original Manuscript: June 21, 2012

Revised Manuscript: July 23, 2012

Manuscript Accepted: August 7, 2012

Published: September 6, 2012

**Citation**

Mirza Karamehmedović, Poul-Erik Hansen, Kai Dirscherl, Emir Karamehmedović, and Thomas Wriedt, "Profile estimation for Pt submicron wire on rough Si substrate from experimental data," Opt. Express **20**, 21678-21686 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-19-21678

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### References

- F. González, G. Videen, P. J. Valle, J. M. Saiz, J. L. de la Peña, and F. Moreno, “Light scattering computational methods for particles on substrates,” J. Quant. Spectrosc. Radiat. Transf.70(4-6), 383–393 (2001). [CrossRef]
- E. Eremina, Yu. Eremin, and T. Wriedt, “Analysis of the light scattering properties of a gold nanorod on a plane surface via discrete sources method,” Opt. Commun.273(1), 278–285 (2007). [CrossRef]
- E. Eremina, Yu. Eremin, and T. Wriedt, “Discrete sources method for simulation of resonance spectra of nonspherical nanoparticles on a plane surface,” Opt. Commun.246(4-6), 405–413 (2005). [CrossRef]
- P. Albella, F. Moreno, J. M. Saiz, and F. González, “2D double interaction method for modeling small particles contaminating microstructures located on substrates,” J. Quant. Spectrosc. Radiat. Transf.106(1-3), 4–10 (2007). [CrossRef]
- M. Karamehmedović, P.-E. Hansen, and T. Wriedt, “An efficient scattering model for PEC and penetrable nanowires on a dielectric substrate,” J. Eur. Opt. Soc. Rapid Publ.6, 11021 (2011). [CrossRef]
- T. A. Germer, G. W. Mulholland, J. H. Kim, and S. H. Ehrman, “Measurement of the 100 nm NIST SRM® 1963 by laser surface light scattering,” Advanced Characterization Techniques for Optical, Semiconductor, and Data Storage Components, Angela Duparré and Bhanwar Singh, Eds., Proc. SPIE 4779, 60–71 (2002).
- J. L. de la Peña, J. M. Saiz, P. J. Valle, F. González, and F. F. Moreno, “Tracking Scattering Minima to Size Metallic Particles on Flat Substrates,” Part. Part. Syst. Charact.16(3), 113–118 (1999). [CrossRef]
- P. Albella, F. Moreno, J. M. Saiz, and F. González, “Backscattering of metallic microstructures with small defects located on flat substrates,” Opt. Express15(11), 6857–6867 (2007). [CrossRef] [PubMed]
- J. M. Saiz, J. L. de la Peña, F. González, and F. Moreno, “Detection and recognition of local defects in 1D structures,” Opt. Commun.196(1-6), 33–39 (2001). [CrossRef]
- B. W. Bell and W. S. Bickel, “Single fiber light scattering matrix: an experimental determination,” Appl. Opt.20(22), 3874–3879 (1981). [CrossRef] [PubMed]
- M. Karamehmedović, P.-E. Hansen, and T. Wriedt, “A fast inversion method for highly conductive submicron wires on a substrate,” J. Eur. Opt. Soc. Rapid Publ.6, 11039 (2011). [CrossRef]
- D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory. (Springer, 1998).
- J. C. Stover, Optical Scattering: Measurement and Analysis, 2nd ed. (SPIE, 1995).
- E. D. Palik, Handbook of Optical Constants of Solids. (Academic Press, 1985).
- I. V. Lindell and E. Alanen, “Exact Image Theory for the Sommerfeld Half-Space Problem, Part I: Vertical Magnetic Dipole,” IEEE Trans. Antenn. Propag.32(2), 126–133 (1984). [CrossRef]
- J. E. Harvey, E. C. Moran, and W. P. Zmek, “Transfer function characterization of grazing incidence optical systems,” Appl. Opt.27(8), 1527–1533 (1988). [CrossRef] [PubMed]
- H. Fujiwara, Spectroscopic Ellipsometry: Principles and Applications. (Wiley, 2007).
- M. Karamehmedović, M.-P. Sørensen, P.-E. Hansen, and A. V. Lavrinenko, “Application of the method of auxiliary sources to a defect-detection inverse problem of optical diffraction microscopy,” J. Eur. Opt. Soc. Rapid Publ.5, 10021 (2010). [CrossRef]
- S. Schröder, A. Duparré, L. Coriand, A. Tünnermann, D. H. Penalver, and J. E. Harvey, “Modeling of light scattering in different regimes of surface roughness,” Opt. Express19(10), 9820–9835 (2011). [CrossRef] [PubMed]

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