Abstract

Angle-resolved light scattering has been used for decades to quantify the surface roughness of optical components. However, because this technique is affected by the contribution of both interfaces of the sample, it cannot be applied to transparent substrates. We show how to overcome this issue and apply these principles to the characterization of superpolished samples.

© 2006 Optical Society of America

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References

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  1. C. Amra, 'Light scattering from multilayer optics. Part A: Investigation tools,' J. Opt. Soc. Am. A 11, 197-210 (1994).
  2. C. Amra, 'Light scattering from multilayer optics. Part B: Application to experiment,' J. Opt. Soc. Am. A 11, 211-226 (1994).
  3. J. M. Bennet and L. Mattsson, Introduction to Surface Roughness and Scattering (Optical Society of America, 1989).
  4. L. Bruel, 'Etude comparative de méthodes optiques et mécanique pour caractériser les états de surface. Diffusion de la lumière par les rugosités de surface et inhomogénéités de volume dans les empilements de couches minces,' Ph.D. dissertation(Univ Aix-Marseille III, Marseille, France, 1992).
  5. P. Croce and L. Prod'homme, 'Ecarts observés dans l'interprétation des indicatrices de diffusion optique par des théories vectorielles simples,' J. Opt., (Paris) 16, 143-1511985.
  6. J. M. Elson, J. P. Rhan, and J. M. Bennet, 'Relationship of the total integrated scattering from multilayer-coated optics to angle of incidence, polarization, correlation-length, and roughness cross-correlation properties,' Appl. Opt. 22, 3207-3219 (1983).
    [PubMed]
  7. A. Sentenac and J. J. Greffet, 'Mean-field theory of light scattering by one-dimensional rough surfaces,' J. Opt. Soc. Am. A 15, 528-532 (1998).
  8. A. Sentenac, J. J. Greffet, H. Giovannini, and M. Saillard, 'Scattering from rough inhomogeneous media: splitting of surface and volume scattering,' J. Opt. Soc. Am. A 19, 727-736 (2002).
  9. P. Vincent, 'Differential method,' in Electromagnetic Theory of Gratings, Vol. 22 of Topics in Current Physics (Springer-Verlag, 1980), pp. 101-121.
    [CrossRef]
  10. A. G. Voronovitch, 'Small slope approximation in wave scattering from rough surfaces,' Sov. Phys. JETP 1, 65-70 (1985).
  11. D. Torricini and C. Amra, 'Light scattering to characterize both faces of transparent substrates: embedded and radiative transfer,' in Optical Interference CoatingsProc. SPIE 2253, 1117-1130 (1994).
  12. M. Zerrad, C. Deumié, and M. Lequime, 'Diffusion de la lumière par des substrats transparents: caractérisation d'états de surfaces,' poster presented at the Horizons de l'Optique, Toulouse, France, 3-5September 2003.
  13. C. Amra, 'First order vector theory of bulk scattering in optical multilayers,' J. Opt. Soc. Am. A 10, 365-374 (1993).
  14. C. Amra, D. Torricini, and P. Roche, 'Multiwavelength (0.45-10.6 µm) angle-resolved scatterometer or how to extend the optical window,' Appl. Opt. 32, 5462-5474 (1993).
    [PubMed]
  15. C. Deumié, 'Ellipsométrie sur champ diffus et analyse multi-échelle de la microstructure des multicouches optiques: diffusion lumineuse, microscopie à force atomique, microscopie à effet tunnel optique,' Ph.D. dissertation(Université Aix-Marseille III, Marseille, France, 1997).
  16. C. Deumié, R. Richier, P. Dumas, and C. Amra, 'Multiscale roughness in optical multilayers: atomic force microscopy and light scattering,' Appl. Opt. 35, 5583-5594 (1996).
    [PubMed]
  17. D. Torricini and C. Amra, 'Light scattering to characterize both faces of transparent substrates: radiative and embedded light,' in Optical Interference CoatingsProc. SPIE 2253, 1117-1130 (1994).
  18. P. Dumas, B. Bouffakhredine, C. Amra, O. Vatel, E. André, R. Galindo, and F. Salvan, 'Quantitative microroughness using near field microscopies and optical,' Europhys. Lett. 22, 717-722 (1993).
  19. C. Amra, C. Deumié, D. Torricini, P. Roche, and R. Galindo, 'Overlapping of roughness spectra measured in macroscopic (optical) and microscopic (AFM) bandwidths,' in Optical Interference Coatings,Proc. SPIE 2253, 614-630 (1994).

Amra, C.

D. Torricini and C. Amra, 'Light scattering to characterize both faces of transparent substrates: embedded and radiative transfer,' in Optical Interference CoatingsProc. SPIE 2253, 1117-1130 (1994).

D. Torricini and C. Amra, 'Light scattering to characterize both faces of transparent substrates: radiative and embedded light,' in Optical Interference CoatingsProc. SPIE 2253, 1117-1130 (1994).

P. Dumas, B. Bouffakhredine, C. Amra, O. Vatel, E. André, R. Galindo, and F. Salvan, 'Quantitative microroughness using near field microscopies and optical,' Europhys. Lett. 22, 717-722 (1993).

C. Amra, C. Deumié, D. Torricini, P. Roche, and R. Galindo, 'Overlapping of roughness spectra measured in macroscopic (optical) and microscopic (AFM) bandwidths,' in Optical Interference Coatings,Proc. SPIE 2253, 614-630 (1994).

C. Amra, 'Light scattering from multilayer optics. Part A: Investigation tools,' J. Opt. Soc. Am. A 11, 197-210 (1994).

C. Amra, 'Light scattering from multilayer optics. Part B: Application to experiment,' J. Opt. Soc. Am. A 11, 211-226 (1994).

C. Amra, 'First order vector theory of bulk scattering in optical multilayers,' J. Opt. Soc. Am. A 10, 365-374 (1993).

C. Amra, D. Torricini, and P. Roche, 'Multiwavelength (0.45-10.6 µm) angle-resolved scatterometer or how to extend the optical window,' Appl. Opt. 32, 5462-5474 (1993).
[PubMed]

C. Deumié, R. Richier, P. Dumas, and C. Amra, 'Multiscale roughness in optical multilayers: atomic force microscopy and light scattering,' Appl. Opt. 35, 5583-5594 (1996).
[PubMed]

André, E.

P. Dumas, B. Bouffakhredine, C. Amra, O. Vatel, E. André, R. Galindo, and F. Salvan, 'Quantitative microroughness using near field microscopies and optical,' Europhys. Lett. 22, 717-722 (1993).

Bennet, J. M.

Bouffakhredine, B.

P. Dumas, B. Bouffakhredine, C. Amra, O. Vatel, E. André, R. Galindo, and F. Salvan, 'Quantitative microroughness using near field microscopies and optical,' Europhys. Lett. 22, 717-722 (1993).

Bruel, L.

L. Bruel, 'Etude comparative de méthodes optiques et mécanique pour caractériser les états de surface. Diffusion de la lumière par les rugosités de surface et inhomogénéités de volume dans les empilements de couches minces,' Ph.D. dissertation(Univ Aix-Marseille III, Marseille, France, 1992).

Croce, P.

P. Croce and L. Prod'homme, 'Ecarts observés dans l'interprétation des indicatrices de diffusion optique par des théories vectorielles simples,' J. Opt., (Paris) 16, 143-1511985.

Deumié, C.

M. Zerrad, C. Deumié, and M. Lequime, 'Diffusion de la lumière par des substrats transparents: caractérisation d'états de surfaces,' poster presented at the Horizons de l'Optique, Toulouse, France, 3-5September 2003.

C. Deumié, 'Ellipsométrie sur champ diffus et analyse multi-échelle de la microstructure des multicouches optiques: diffusion lumineuse, microscopie à force atomique, microscopie à effet tunnel optique,' Ph.D. dissertation(Université Aix-Marseille III, Marseille, France, 1997).

C. Amra, C. Deumié, D. Torricini, P. Roche, and R. Galindo, 'Overlapping of roughness spectra measured in macroscopic (optical) and microscopic (AFM) bandwidths,' in Optical Interference Coatings,Proc. SPIE 2253, 614-630 (1994).

C. Deumié, R. Richier, P. Dumas, and C. Amra, 'Multiscale roughness in optical multilayers: atomic force microscopy and light scattering,' Appl. Opt. 35, 5583-5594 (1996).
[PubMed]

Dumas, P.

C. Deumié, R. Richier, P. Dumas, and C. Amra, 'Multiscale roughness in optical multilayers: atomic force microscopy and light scattering,' Appl. Opt. 35, 5583-5594 (1996).
[PubMed]

P. Dumas, B. Bouffakhredine, C. Amra, O. Vatel, E. André, R. Galindo, and F. Salvan, 'Quantitative microroughness using near field microscopies and optical,' Europhys. Lett. 22, 717-722 (1993).

Elson, J. M.

Galindo, R.

C. Amra, C. Deumié, D. Torricini, P. Roche, and R. Galindo, 'Overlapping of roughness spectra measured in macroscopic (optical) and microscopic (AFM) bandwidths,' in Optical Interference Coatings,Proc. SPIE 2253, 614-630 (1994).

P. Dumas, B. Bouffakhredine, C. Amra, O. Vatel, E. André, R. Galindo, and F. Salvan, 'Quantitative microroughness using near field microscopies and optical,' Europhys. Lett. 22, 717-722 (1993).

Giovannini, H.

Greffet, J. J.

Lequime, M.

M. Zerrad, C. Deumié, and M. Lequime, 'Diffusion de la lumière par des substrats transparents: caractérisation d'états de surfaces,' poster presented at the Horizons de l'Optique, Toulouse, France, 3-5September 2003.

Mattsson, L.

J. M. Bennet and L. Mattsson, Introduction to Surface Roughness and Scattering (Optical Society of America, 1989).

Prod'homme, L.

P. Croce and L. Prod'homme, 'Ecarts observés dans l'interprétation des indicatrices de diffusion optique par des théories vectorielles simples,' J. Opt., (Paris) 16, 143-1511985.

Rhan, J. P.

Richier, R.

Roche, P.

C. Amra, D. Torricini, and P. Roche, 'Multiwavelength (0.45-10.6 µm) angle-resolved scatterometer or how to extend the optical window,' Appl. Opt. 32, 5462-5474 (1993).
[PubMed]

C. Amra, C. Deumié, D. Torricini, P. Roche, and R. Galindo, 'Overlapping of roughness spectra measured in macroscopic (optical) and microscopic (AFM) bandwidths,' in Optical Interference Coatings,Proc. SPIE 2253, 614-630 (1994).

Saillard, M.

Salvan, F.

P. Dumas, B. Bouffakhredine, C. Amra, O. Vatel, E. André, R. Galindo, and F. Salvan, 'Quantitative microroughness using near field microscopies and optical,' Europhys. Lett. 22, 717-722 (1993).

Sentenac, A.

Torricini, D.

C. Amra, D. Torricini, and P. Roche, 'Multiwavelength (0.45-10.6 µm) angle-resolved scatterometer or how to extend the optical window,' Appl. Opt. 32, 5462-5474 (1993).
[PubMed]

C. Amra, C. Deumié, D. Torricini, P. Roche, and R. Galindo, 'Overlapping of roughness spectra measured in macroscopic (optical) and microscopic (AFM) bandwidths,' in Optical Interference Coatings,Proc. SPIE 2253, 614-630 (1994).

D. Torricini and C. Amra, 'Light scattering to characterize both faces of transparent substrates: radiative and embedded light,' in Optical Interference CoatingsProc. SPIE 2253, 1117-1130 (1994).

D. Torricini and C. Amra, 'Light scattering to characterize both faces of transparent substrates: embedded and radiative transfer,' in Optical Interference CoatingsProc. SPIE 2253, 1117-1130 (1994).

Vatel, O.

P. Dumas, B. Bouffakhredine, C. Amra, O. Vatel, E. André, R. Galindo, and F. Salvan, 'Quantitative microroughness using near field microscopies and optical,' Europhys. Lett. 22, 717-722 (1993).

Vincent, P.

P. Vincent, 'Differential method,' in Electromagnetic Theory of Gratings, Vol. 22 of Topics in Current Physics (Springer-Verlag, 1980), pp. 101-121.
[CrossRef]

Voronovitch, A. G.

A. G. Voronovitch, 'Small slope approximation in wave scattering from rough surfaces,' Sov. Phys. JETP 1, 65-70 (1985).

Zerrad, M.

M. Zerrad, C. Deumié, and M. Lequime, 'Diffusion de la lumière par des substrats transparents: caractérisation d'états de surfaces,' poster presented at the Horizons de l'Optique, Toulouse, France, 3-5September 2003.

Appl. Opt. (3)

Europhys. Lett. (1)

P. Dumas, B. Bouffakhredine, C. Amra, O. Vatel, E. André, R. Galindo, and F. Salvan, 'Quantitative microroughness using near field microscopies and optical,' Europhys. Lett. 22, 717-722 (1993).

J. Opt. Soc. Am. A (5)

Proc. SPIE (3)

D. Torricini and C. Amra, 'Light scattering to characterize both faces of transparent substrates: embedded and radiative transfer,' in Optical Interference CoatingsProc. SPIE 2253, 1117-1130 (1994).

D. Torricini and C. Amra, 'Light scattering to characterize both faces of transparent substrates: radiative and embedded light,' in Optical Interference CoatingsProc. SPIE 2253, 1117-1130 (1994).

C. Amra, C. Deumié, D. Torricini, P. Roche, and R. Galindo, 'Overlapping of roughness spectra measured in macroscopic (optical) and microscopic (AFM) bandwidths,' in Optical Interference Coatings,Proc. SPIE 2253, 614-630 (1994).

Sov. Phys. JETP (1)

A. G. Voronovitch, 'Small slope approximation in wave scattering from rough surfaces,' Sov. Phys. JETP 1, 65-70 (1985).

Other (6)

M. Zerrad, C. Deumié, and M. Lequime, 'Diffusion de la lumière par des substrats transparents: caractérisation d'états de surfaces,' poster presented at the Horizons de l'Optique, Toulouse, France, 3-5September 2003.

C. Deumié, 'Ellipsométrie sur champ diffus et analyse multi-échelle de la microstructure des multicouches optiques: diffusion lumineuse, microscopie à force atomique, microscopie à effet tunnel optique,' Ph.D. dissertation(Université Aix-Marseille III, Marseille, France, 1997).

J. M. Bennet and L. Mattsson, Introduction to Surface Roughness and Scattering (Optical Society of America, 1989).

L. Bruel, 'Etude comparative de méthodes optiques et mécanique pour caractériser les états de surface. Diffusion de la lumière par les rugosités de surface et inhomogénéités de volume dans les empilements de couches minces,' Ph.D. dissertation(Univ Aix-Marseille III, Marseille, France, 1992).

P. Croce and L. Prod'homme, 'Ecarts observés dans l'interprétation des indicatrices de diffusion optique par des théories vectorielles simples,' J. Opt., (Paris) 16, 143-1511985.

P. Vincent, 'Differential method,' in Electromagnetic Theory of Gratings, Vol. 22 of Topics in Current Physics (Springer-Verlag, 1980), pp. 101-121.
[CrossRef]

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Figures (13)

Fig. 1
Fig. 1

Scattering angles.

Fig. 2
Fig. 2

Light scattering in transparent substrates.

Fig. 3
Fig. 3

Light scattered by interface 1 in a transparent substrate.

Fig. 4
Fig. 4

Light scattered by interface 2 in a transparent substrate.

Fig. 5
Fig. 5

Difference between real and equivalent spectra.

Fig. 6
Fig. 6

Side view showing the separation of both faces using the geometrical approach. (a) Definition of the equivalent thickness of the sample as a function of the scattering angle. (b) Visualization of the areas illuminated and measured by the scatterometer on each interface of the sample.

Fig. 7
Fig. 7

Top view showing the separation of both faces using the geometrical approach.

Fig. 8
Fig. 8

Capture ratio for different sample thicknesses.

Fig. 9
Fig. 9

Intensity scattered I by a superpolished transparent substrate measured without the separation system. BRDF, bidirectional reflectance distribution function.

Fig. 10
Fig. 10

Intensity scattered I 1 and I 2 by each interface of a superpolished transparent substrate measured with the separation system.

Fig. 11
Fig. 11

Calculated and measured roughness spectra (higher spectra values for higher scattering angles are due to the parasite light coming from the sample side).

Fig. 12
Fig. 12

AFM image of the sample surface.

Fig. 13
Fig. 13

Comparison between the roughness spectra calculated from the AFM measurements of each interface of the sample and the measured spectra for each interface.

Equations (20)

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I ( θ , ϕ ) = C ± ( θ , ϕ ) γ ( θ , ϕ ) ,
C ( θ , ϕ ) = C ( θ ) = 1 2 ( 2 π n 0 λ ) 2 [ cos 2 θ 0 | q s | 2 + | q p | 2 ] ,
C + ( θ , ϕ ) = 1 2 ( n s 3 n 0 ) ( 2 π λ ) 2 [ cos 2 θ s | q s | 2 + | q p | 2 ] ,
q s ( θ ) = j ( 2 π λ ) 2 n 0 ( n 0 n s ) ( n 0 cos θ 0 + n s cos θ s ) 1 ,
q p ( θ ) = j ( 2 π λ ) 2 n 0 ( n 0 n s ) ( n 0 / cos θ 0 + n s / cos θ s ) 1 ,
n 0 sin θ 0 = n s sin θ s .
γ = 4 π 2 S | h ^ ( σ ) | 2 ,
δ 2 = 1 S s h 2 ( r ) d r = σ γ ( σ ) d σ ,
I 1 - = C 1 - γ 1 ,
I 1 + = C 1 + γ 1 .
I 1 t = C 1 + γ 1 ,
β = T ( n 0 n s ) 2 cos θ 0 cos θ s .
I i = T 1 R 2 I 0 ,
I 2 = T 1 R 2 C 2 γ 2 .
I 2 t = β I 2 .
I ( θ ) = I 1 - ( θ ) + I 1 t ( θ ) + I 2 t ( θ ) = C 1 - γ 1 + C 1 + γ 1 + T 1 R 2 β C 2 - γ 2 .
γ ( θ ) = I ( θ ) C 1 - ( θ ) ,
γ eq = I ( θ ) C 1 - ( θ ) = I 1 - ( θ ) + I 1 t ( θ ) + β I 2 ( θ ) C ( θ ) = γ 1 + R β C 1 + ( θ ) C 1 - ( θ ) γ 1 + T 1 R 2 β C 2 - ( θ ) C 1 - ( θ ) γ 2 .
C = S 2 2 ( face   2 ) S 1 1 ( face   1 ) ,
e = e cos θ ( n 2 sin 2 θ ) 1 / 2 .

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