Abstract

We emphasize the role of correlated isotropy in the study of microroughness in high-quality optical coatings. First, cross correlation between surfaces and cross coherence between scattering sources are discussed and compared. An isotropy degree of roughness is then introduced as a quantitative value to describe the angular disorder of a surface connected with the polar dependence of scattering. We show how the frequency variations of this isotropy degree allow one to solve the inverse problem and obtain a unique solution for the scattering parameters that describe structural irregularities of the stacks. Light scattering can also be used to detect an oblique growth of the materials in thin-film form. Finally, we study the sensitivity of the investigation method to the stack parameters.

© 1994 Optical Society of America

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References

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  1. J. M. Bennett, L. Mattsson, Introduction to Surface Roughness and Scattering (Optical Society of America, Washington, D.C., 1989).
  2. J. M. Eastman, “Surface scattering in optical interference coatings,” Ph.D. dissertation (University of Rochester, Rochester, N.Y., 1974).
  3. C. K. Carniglia, “Scalar scattering theory for multilayer optical coatings,” Opt. Eng. 18, 104–115 (1979).
    [CrossRef]
  4. S. J. Gourley, P. H. Lissberger, “Optical scattering in multilayer thin films,” Opt. Acta 26, 117–143 (1979).
    [CrossRef]
  5. C. Grèzes-Besset, C. Amra, B. Cousin, G. Otrio, E. Pelletier, R. Richier, “Etude de la diaphonie d’un système de démultiplexage par filtres interférentiels. Conséquences de la diffusion de la lumière par les irrégularités des surfaces optiques,” Ann. Telecommun. 43, 135–141 (1988).
  6. J. M. Elson, J. P. Rahn, J. M. Bennett, “Light scattering from multilayer optics: comparison of theory and experiment,” Appl. Opt. 19, 669–679 (1980).
    [CrossRef]
  7. C. Amra, J. H. Apfel, E. Pelletier, “Role of interface correlation in light scattering by a multilayer,” Appl. Opt. 16, 3134–3151 (1992).
    [CrossRef]
  8. J. M. Elson, J. P. Rahn, J. Bennett, “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).
    [CrossRef] [PubMed]
  9. C. Amra, “Scattering characterization of materials in thin film form,” in Laser-Induced Damage in Optical Materials 1989, H. E. Bennett, L. L. Chase, A. H. Guenther, B. E. Newnam, H. J. Soileau, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1438, 309–323 (1990).
    [CrossRef]
  10. S. Kassam, A. Duparré, K. Hehl, P. Bussemer, J. Neubert, “Light scattering from the volume of optical thin films: theory and experiment,” Appl. Opt. 31, 1304–1313 (1992).
    [CrossRef] [PubMed]
  11. C. Amra, “First-order vector theory of bulk scattering in optical multilayers,” J. Opt. Soc. Am. A 10, 365–374 (1993).
    [CrossRef]
  12. C. Amra, L. Bruel, C. Grèzes-Besset, “Comparison of surface and bulk scattering in optical multilayers,” Appl. Opt. 32, 5492–5503 (1993).
    [CrossRef] [PubMed]
  13. C. Amra, “Light scattering from multilayer optics. II. Application to experiment,” J. Opt. Soc. Am. A 11, 211–226 (1994).
    [CrossRef]
  14. C. Amra, D. Torricini, Y. Boucher, E. Pelletier, “Scattering from optical surfaces and coatings: an easy investigation of microroughness,” in Optical Thin Films and Applications, R. Herrmann, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1270, 72–81 (1990).
    [CrossRef]
  15. C. Amra, “Calculs et mesures de diffusion appliqués à l’étude de la rugosité dans les traitements optiques multicouches,”J. Opt. (Paris) 21, 83–98 (1990).
    [CrossRef]
  16. J. M. Elson, “Angle resolved light scattering from composite optical surfaces,” in Periodic Structures, Gratings, Moire Patterns, and Diffraction Phenomena I, C. H. Chi, ed., Proc. Soc. Photo-Opt. Instrum. Eng.240, 296–306 (1980).
    [CrossRef]
  17. P. Bousquet, F. Flory, P. Roche, “Scattering from multilayer thin films: theory and experiment,”J. Opt. Soc. Am. 71, 1115–1123 (1981).
    [CrossRef]
  18. C. Amra, P. Bousquet, “Scattering from surfaces and multilayer coatings: recent advances for a better investigation of experiment,” in Surface Measurement and Characterization, J. M. Bennett, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1009, 82–97 (1988).
    [CrossRef]
  19. C. Amra, C. Grèzes-Besset, P. Roche, E. Pelletier, “Description of a scattering apparatus: application to the problems of characterization of opaque surfaces,” Appl. Opt. 28, 2723–2730 (1989).
    [CrossRef] [PubMed]
  20. F. E. Nicodemus, “Directional reflectance and emissivity of an opaque surface,” Appl. Opt. 4, 767–773 (1965).
    [CrossRef]
  21. C. Amra, P. Roche, E. Pelletier, “Interface roughness cross-correlation laws deduced from scattering diagram measurements on optical multilayers: effect of the material grain size,” J. Opt. Soc. Am. B 4, 1087–1093 (1987).
    [CrossRef]
  22. P. Roche, E. Pelletier, G. Albrand, “Antiscattering transparent monolayers: theory and experiment,” J. Opt. Soc. Am. A 1, 1032–1033 (1984).
    [CrossRef]
  23. P. Roche, P. Bousquet, F. Flory, J. Garcin, E. Pelletier, G. Albrand, “Determination of interface roughness cross-correlation properties of an optical coating from measurements of the angular scattering,” J. Opt. Soc. Am. A 1, 1028–1031 (1984).
    [CrossRef]
  24. C. Amra, G. Albrand, P. Roche, “Theory and application of antiscattering single layers: antiscattering antireflection coatings,” Appl. Opt. 25, 2695–2702 (1986).
    [CrossRef] [PubMed]
  25. A. G. Dirks, H. J. Leamy, “Columnar microstructure in vapor-deposited thin films,” Thin Solid Films 47, 219–223 (1977).
    [CrossRef]
  26. C. Amra, “Scattering distribution from multilayer mirrors: theoretical research of a design for minimum losses,” in Laser-Induced Damage in Optical Materials, H. E. Bennett, A. H. Guenther, D. Milam, B. E. Newnam, eds., Natl. Inst. Stand. Technol. Spec. Publ.752, 594–602 (1986).

1994 (1)

1993 (2)

1992 (2)

S. Kassam, A. Duparré, K. Hehl, P. Bussemer, J. Neubert, “Light scattering from the volume of optical thin films: theory and experiment,” Appl. Opt. 31, 1304–1313 (1992).
[CrossRef] [PubMed]

C. Amra, J. H. Apfel, E. Pelletier, “Role of interface correlation in light scattering by a multilayer,” Appl. Opt. 16, 3134–3151 (1992).
[CrossRef]

1990 (1)

C. Amra, “Calculs et mesures de diffusion appliqués à l’étude de la rugosité dans les traitements optiques multicouches,”J. Opt. (Paris) 21, 83–98 (1990).
[CrossRef]

1989 (1)

1988 (1)

C. Grèzes-Besset, C. Amra, B. Cousin, G. Otrio, E. Pelletier, R. Richier, “Etude de la diaphonie d’un système de démultiplexage par filtres interférentiels. Conséquences de la diffusion de la lumière par les irrégularités des surfaces optiques,” Ann. Telecommun. 43, 135–141 (1988).

1987 (1)

1986 (1)

1984 (2)

1983 (1)

1981 (1)

1980 (1)

1979 (2)

C. K. Carniglia, “Scalar scattering theory for multilayer optical coatings,” Opt. Eng. 18, 104–115 (1979).
[CrossRef]

S. J. Gourley, P. H. Lissberger, “Optical scattering in multilayer thin films,” Opt. Acta 26, 117–143 (1979).
[CrossRef]

1977 (1)

A. G. Dirks, H. J. Leamy, “Columnar microstructure in vapor-deposited thin films,” Thin Solid Films 47, 219–223 (1977).
[CrossRef]

1965 (1)

Albrand, G.

Amra, C.

C. Amra, “Light scattering from multilayer optics. II. Application to experiment,” J. Opt. Soc. Am. A 11, 211–226 (1994).
[CrossRef]

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

C. Amra, “First-order vector theory of bulk scattering in optical multilayers,” J. Opt. Soc. Am. A 10, 365–374 (1993).
[CrossRef]

C. Amra, J. H. Apfel, E. Pelletier, “Role of interface correlation in light scattering by a multilayer,” Appl. Opt. 16, 3134–3151 (1992).
[CrossRef]

C. Amra, “Calculs et mesures de diffusion appliqués à l’étude de la rugosité dans les traitements optiques multicouches,”J. Opt. (Paris) 21, 83–98 (1990).
[CrossRef]

C. Amra, C. Grèzes-Besset, P. Roche, E. Pelletier, “Description of a scattering apparatus: application to the problems of characterization of opaque surfaces,” Appl. Opt. 28, 2723–2730 (1989).
[CrossRef] [PubMed]

C. Grèzes-Besset, C. Amra, B. Cousin, G. Otrio, E. Pelletier, R. Richier, “Etude de la diaphonie d’un système de démultiplexage par filtres interférentiels. Conséquences de la diffusion de la lumière par les irrégularités des surfaces optiques,” Ann. Telecommun. 43, 135–141 (1988).

C. Amra, P. Roche, E. Pelletier, “Interface roughness cross-correlation laws deduced from scattering diagram measurements on optical multilayers: effect of the material grain size,” J. Opt. Soc. Am. B 4, 1087–1093 (1987).
[CrossRef]

C. Amra, G. Albrand, P. Roche, “Theory and application of antiscattering single layers: antiscattering antireflection coatings,” Appl. Opt. 25, 2695–2702 (1986).
[CrossRef] [PubMed]

C. Amra, “Scattering distribution from multilayer mirrors: theoretical research of a design for minimum losses,” in Laser-Induced Damage in Optical Materials, H. E. Bennett, A. H. Guenther, D. Milam, B. E. Newnam, eds., Natl. Inst. Stand. Technol. Spec. Publ.752, 594–602 (1986).

C. Amra, D. Torricini, Y. Boucher, E. Pelletier, “Scattering from optical surfaces and coatings: an easy investigation of microroughness,” in Optical Thin Films and Applications, R. Herrmann, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1270, 72–81 (1990).
[CrossRef]

C. Amra, P. Bousquet, “Scattering from surfaces and multilayer coatings: recent advances for a better investigation of experiment,” in Surface Measurement and Characterization, J. M. Bennett, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1009, 82–97 (1988).
[CrossRef]

C. Amra, “Scattering characterization of materials in thin film form,” in Laser-Induced Damage in Optical Materials 1989, H. E. Bennett, L. L. Chase, A. H. Guenther, B. E. Newnam, H. J. Soileau, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1438, 309–323 (1990).
[CrossRef]

Apfel, J. H.

C. Amra, J. H. Apfel, E. Pelletier, “Role of interface correlation in light scattering by a multilayer,” Appl. Opt. 16, 3134–3151 (1992).
[CrossRef]

Bennett, J.

Bennett, J. M.

J. M. Elson, J. P. Rahn, J. M. Bennett, “Light scattering from multilayer optics: comparison of theory and experiment,” Appl. Opt. 19, 669–679 (1980).
[CrossRef]

J. M. Bennett, L. Mattsson, Introduction to Surface Roughness and Scattering (Optical Society of America, Washington, D.C., 1989).

Boucher, Y.

C. Amra, D. Torricini, Y. Boucher, E. Pelletier, “Scattering from optical surfaces and coatings: an easy investigation of microroughness,” in Optical Thin Films and Applications, R. Herrmann, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1270, 72–81 (1990).
[CrossRef]

Bousquet, P.

Bruel, L.

Bussemer, P.

Carniglia, C. K.

C. K. Carniglia, “Scalar scattering theory for multilayer optical coatings,” Opt. Eng. 18, 104–115 (1979).
[CrossRef]

Cousin, B.

C. Grèzes-Besset, C. Amra, B. Cousin, G. Otrio, E. Pelletier, R. Richier, “Etude de la diaphonie d’un système de démultiplexage par filtres interférentiels. Conséquences de la diffusion de la lumière par les irrégularités des surfaces optiques,” Ann. Telecommun. 43, 135–141 (1988).

Dirks, A. G.

A. G. Dirks, H. J. Leamy, “Columnar microstructure in vapor-deposited thin films,” Thin Solid Films 47, 219–223 (1977).
[CrossRef]

Duparré, A.

Eastman, J. M.

J. M. Eastman, “Surface scattering in optical interference coatings,” Ph.D. dissertation (University of Rochester, Rochester, N.Y., 1974).

Elson, J. M.

Flory, F.

Garcin, J.

Gourley, S. J.

S. J. Gourley, P. H. Lissberger, “Optical scattering in multilayer thin films,” Opt. Acta 26, 117–143 (1979).
[CrossRef]

Grèzes-Besset, C.

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

C. Amra, C. Grèzes-Besset, P. Roche, E. Pelletier, “Description of a scattering apparatus: application to the problems of characterization of opaque surfaces,” Appl. Opt. 28, 2723–2730 (1989).
[CrossRef] [PubMed]

C. Grèzes-Besset, C. Amra, B. Cousin, G. Otrio, E. Pelletier, R. Richier, “Etude de la diaphonie d’un système de démultiplexage par filtres interférentiels. Conséquences de la diffusion de la lumière par les irrégularités des surfaces optiques,” Ann. Telecommun. 43, 135–141 (1988).

Hehl, K.

Kassam, S.

Leamy, H. J.

A. G. Dirks, H. J. Leamy, “Columnar microstructure in vapor-deposited thin films,” Thin Solid Films 47, 219–223 (1977).
[CrossRef]

Lissberger, P. H.

S. J. Gourley, P. H. Lissberger, “Optical scattering in multilayer thin films,” Opt. Acta 26, 117–143 (1979).
[CrossRef]

Mattsson, L.

J. M. Bennett, L. Mattsson, Introduction to Surface Roughness and Scattering (Optical Society of America, Washington, D.C., 1989).

Neubert, J.

Nicodemus, F. E.

Otrio, G.

C. Grèzes-Besset, C. Amra, B. Cousin, G. Otrio, E. Pelletier, R. Richier, “Etude de la diaphonie d’un système de démultiplexage par filtres interférentiels. Conséquences de la diffusion de la lumière par les irrégularités des surfaces optiques,” Ann. Telecommun. 43, 135–141 (1988).

Pelletier, E.

C. Amra, J. H. Apfel, E. Pelletier, “Role of interface correlation in light scattering by a multilayer,” Appl. Opt. 16, 3134–3151 (1992).
[CrossRef]

C. Amra, C. Grèzes-Besset, P. Roche, E. Pelletier, “Description of a scattering apparatus: application to the problems of characterization of opaque surfaces,” Appl. Opt. 28, 2723–2730 (1989).
[CrossRef] [PubMed]

C. Grèzes-Besset, C. Amra, B. Cousin, G. Otrio, E. Pelletier, R. Richier, “Etude de la diaphonie d’un système de démultiplexage par filtres interférentiels. Conséquences de la diffusion de la lumière par les irrégularités des surfaces optiques,” Ann. Telecommun. 43, 135–141 (1988).

C. Amra, P. Roche, E. Pelletier, “Interface roughness cross-correlation laws deduced from scattering diagram measurements on optical multilayers: effect of the material grain size,” J. Opt. Soc. Am. B 4, 1087–1093 (1987).
[CrossRef]

P. Roche, P. Bousquet, F. Flory, J. Garcin, E. Pelletier, G. Albrand, “Determination of interface roughness cross-correlation properties of an optical coating from measurements of the angular scattering,” J. Opt. Soc. Am. A 1, 1028–1031 (1984).
[CrossRef]

P. Roche, E. Pelletier, G. Albrand, “Antiscattering transparent monolayers: theory and experiment,” J. Opt. Soc. Am. A 1, 1032–1033 (1984).
[CrossRef]

C. Amra, D. Torricini, Y. Boucher, E. Pelletier, “Scattering from optical surfaces and coatings: an easy investigation of microroughness,” in Optical Thin Films and Applications, R. Herrmann, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1270, 72–81 (1990).
[CrossRef]

Rahn, J. P.

Richier, R.

C. Grèzes-Besset, C. Amra, B. Cousin, G. Otrio, E. Pelletier, R. Richier, “Etude de la diaphonie d’un système de démultiplexage par filtres interférentiels. Conséquences de la diffusion de la lumière par les irrégularités des surfaces optiques,” Ann. Telecommun. 43, 135–141 (1988).

Roche, P.

Torricini, D.

C. Amra, D. Torricini, Y. Boucher, E. Pelletier, “Scattering from optical surfaces and coatings: an easy investigation of microroughness,” in Optical Thin Films and Applications, R. Herrmann, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1270, 72–81 (1990).
[CrossRef]

Ann. Telecommun. (1)

C. Grèzes-Besset, C. Amra, B. Cousin, G. Otrio, E. Pelletier, R. Richier, “Etude de la diaphonie d’un système de démultiplexage par filtres interférentiels. Conséquences de la diffusion de la lumière par les irrégularités des surfaces optiques,” Ann. Telecommun. 43, 135–141 (1988).

Appl. Opt. (8)

J. Opt. (Paris) (1)

C. Amra, “Calculs et mesures de diffusion appliqués à l’étude de la rugosité dans les traitements optiques multicouches,”J. Opt. (Paris) 21, 83–98 (1990).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (4)

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

Opt. Acta (1)

S. J. Gourley, P. H. Lissberger, “Optical scattering in multilayer thin films,” Opt. Acta 26, 117–143 (1979).
[CrossRef]

Opt. Eng. (1)

C. K. Carniglia, “Scalar scattering theory for multilayer optical coatings,” Opt. Eng. 18, 104–115 (1979).
[CrossRef]

Thin Solid Films (1)

A. G. Dirks, H. J. Leamy, “Columnar microstructure in vapor-deposited thin films,” Thin Solid Films 47, 219–223 (1977).
[CrossRef]

Other (7)

C. Amra, “Scattering distribution from multilayer mirrors: theoretical research of a design for minimum losses,” in Laser-Induced Damage in Optical Materials, H. E. Bennett, A. H. Guenther, D. Milam, B. E. Newnam, eds., Natl. Inst. Stand. Technol. Spec. Publ.752, 594–602 (1986).

J. M. Elson, “Angle resolved light scattering from composite optical surfaces,” in Periodic Structures, Gratings, Moire Patterns, and Diffraction Phenomena I, C. H. Chi, ed., Proc. Soc. Photo-Opt. Instrum. Eng.240, 296–306 (1980).
[CrossRef]

C. Amra, P. Bousquet, “Scattering from surfaces and multilayer coatings: recent advances for a better investigation of experiment,” in Surface Measurement and Characterization, J. M. Bennett, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1009, 82–97 (1988).
[CrossRef]

J. M. Bennett, L. Mattsson, Introduction to Surface Roughness and Scattering (Optical Society of America, Washington, D.C., 1989).

J. M. Eastman, “Surface scattering in optical interference coatings,” Ph.D. dissertation (University of Rochester, Rochester, N.Y., 1974).

C. Amra, “Scattering characterization of materials in thin film form,” in Laser-Induced Damage in Optical Materials 1989, H. E. Bennett, L. L. Chase, A. H. Guenther, B. E. Newnam, H. J. Soileau, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1438, 309–323 (1990).
[CrossRef]

C. Amra, D. Torricini, Y. Boucher, E. Pelletier, “Scattering from optical surfaces and coatings: an easy investigation of microroughness,” in Optical Thin Films and Applications, R. Herrmann, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1270, 72–81 (1990).
[CrossRef]

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Figures (9)

Fig. 1
Fig. 1

(A) Three-dimensional scattering distribution from a narrow-band filter at oblique illumination (i = 30°). The design is HL HL H 6L H LH LH. The illumination wavelength is equal to the design wavelength λ0. 1, R and T characterize the specular directions of the incident, reflected, and transmitted flux. BRDF. cos θ is the flux scattered per unit of surface and solid angle, normalized to the incident flux. θ and ϕ characterize a scattering direction (θ from the sample normal, ϕ from the polar angle). Note the presence of scattering rings characteristic of the Fabry–Perot design. (B) Scattering phenomena in an optical demultiplexer. λ0, λ1, and λ2 are the wavelengths to be separated. Fi indicate optical filters, most of which are multiple-cavity Fabry–Perot filters. For each filter the angular scattering is plotted for the illumination wavelengths λ0 (dotted curves), λ1 (dashed curves), and λ2 (solid curves). The isolation rates are altered because the scattered light does not follow the specular directions given by Snell’s laws. This effect is amplified when we bring the optical elements closer to one another to obtain a compact system, which decreases the demultiplexer performances.

Fig. 2
Fig. 2

Scattering versus wavelength λ and direction θ for a narrow-band filter. The design wavelength is λ0 = 633 nm. Calculation is performed for uncorrelated surfaces (ωij = 0 for ij). The scattering rings occur at the peak wavelengths of the filter. The spectral range is (400 → 633 nm), and the angular ranges (0 → 90°) and (90 → 180°) correspond to scattering by reflection and transmission, respectively.

Fig. 3
Fig. 3

Variations in total scattering losses versus illumination incidence (i) for four polarization states of the incident and scattered light (SS, SP, PS, and PP). The design is a 15-layer multidielectric quarter-wave mirror at λ0 = 633 nm. (A) Is calculated for perfect correlation (γij = γjj = γ), (B) for uncorrelated surfaces (γij = 0 for ij).

Fig. 4
Fig. 4

Interfaces and cross-correlation coefficients αij in a multilayer stack. ni and ei are the indices and the thicknesses of the layers, respectively.

Fig. 5
Fig. 5

(A) Level maps of scattering in polar coordinates, that is, { x = θ cos ϕ y = θ sin ϕ for a given scattering level. (B) Angular function FN(θ, α) measured on a polished glass substrate for different θ angles (25, 30, 60, and 70°). The local extrema of FN are due to the presence of several privileged directions on the surface. (C) Isotropy degree curve measured for the same substrate as in (A) and (B).

Fig. 6
Fig. 6

Influence of cutoff frequency σc on the angular causal scattering from a single (2H) layer of TiO2. The case σc = ∞ is that of perfect correlation.

Fig. 7
Fig. 7

Influence of σc on the angular scattering from a narrow-band filter (M5 6L M5). Curve 1 is calculated with the causal model and σH = σL = 12 μm−1. Curves 2 and 3 are calculated from the βij model for βij = 1 and βij = 0, respectively.

Fig. 8
Fig. 8

(A) Angular function and (B) isotropy degree for a 13-L layer produced at normal (β = 0, curve 1) or oblique (β = 20°, curve 2) incidence deposition. The period of FN is increased from π to 2π when β ≠ 0. FN(θ, α) is calculated at θ = 7°.

Fig. 9
Fig. 9

Influence of residual roughness on the angular scattering from a single layer. All residual spectra have autocorrelation length Lg = 100 nm. Curve 1 is calculated for δg = 0, curve 2 for δg = 0.2 nm, and curve 3 for δg = 0.5 nm. All curves are the sum of the causal and residual scattering.

Tables (1)

Tables Icon

Table 1 Values of the Sensitivity x of Angular Scattering to Residual Roughness, Calculated for Different Designsa

Equations (104)

Equations on this page are rendered with MathJax. Learn more.

I ( θ , ϕ ) = i , j P C i j ( θ , ϕ ) γ i j ( θ , ϕ ) ,
C i j NN = 1 2 ( C i j SS + C i j SP + C i j PS + C i j PP ) ,
C i j SS = cos 2 θ cos 2 ϕ C i j S ( θ ) , C i j PS C i j SS = tan 2 ϕ , C i j PP = cos 2 ϕ C i j P ( θ ) , C i j SP C i j PP = tan 2 ϕ ,
I NN = I NS + I NP = 1 2 ( I SS + I PS + I SP + I PP ) I NN ( θ ) = 1 2 [ I SS ( θ , 0 ) + I PP ( θ , 0 ) ] .
D SS = D PS ,             D PP = D SP D NS = D SS ,             D NP = D PP D NN = 1 2 ( 2 D SS + 2 D PP ) = D SS + D PP .
γ i j ( σ ) = 4 π 2 h ^ i h ^ ¯ j ,
α i j ( σ ) = γ i j / γ j j             for             γ i j ( σ ) 0.
γ i j = α i j γ j j ,
α i i = 1 , α i j ( - σ ) = α ¯ i j ( σ ) α i j ( 0 )
α i j α j i = 1 Ψ i j + Ψ j i = 0 ( 2 π ) ,
α i j 2 = γ i i / γ j j γ i j 2 = γ i i γ j j , α i j = k = 0 j - i - 1 α i + k , i + k + 1 ,             for i < j .
I = j = 0 i = 0 P γ j j f i j ,
f i j = C j j l i j + 2 α i j R e [ C i j exp ( j Ψ i j ) ] s i j ,
s i j = | 0 for i j 1 for i < j .
I ( σ ) = γ s j = 0 i = 0 p α j p 2 f i j ,
σ = k sin θ | cos ϕ sin ϕ ,
γ 12 = γ f = α 12 γ f α 12 = 1 , γ 13 = γ g = α 13 γ g α 13 = 1 , γ 23 = 0 = α 23 γ g α 23 = 0 ,
α 13 = 1 + α , α 23 = α , α 12 = 1 + 1 α α 13 = α 12 α 23 .
I 0 = j = 0 p C j j γ j j , I 1 = I 0 + 2 i < j r i j γ j j Re ( C i j ) .
I = I 0 + 2 i < j β i j r i j γ j j Re ( C i j ) ,
i < j γ j j Re ( α i j C i j ) = i < j β i j r i j γ j j Re ( C i j ) ,
β i j = Re ( α i j C i j ) r i j Re ( C i j ) = ( α i j ) / r i j Re [ C i j exp ( j Ψ i j ) ] Re ( C i j ) .
β i j = α i j r i j ( cos Ψ i j - tan c i j sin Ψ i j ) ,
β i j ( σ ) = α i j ( σ ) δ j / δ i = δ j / δ i γ i i / γ j j ,
α = i < j γ j j Re ( α i j C i j ) i < j γ j j Re ( C i j ) ,
h j = 1 4 π 2 a j * h j + 1 + g j ,
h ^ j = α j h ^ j + 1 + g ^ j ,
α j ( σ ) = F T [ a j ( r ) ] .
α i j = k = i j - 1 α k
h ^ j = α j p h ^ s + k = 0 p - j - 1 α j , j + k g ^ j + k .
γ i j = α i p α ¯ j p γ s + k = 0 p - j - 1 q = 0 p - i - 1 α i , i + q α ¯ j , j + k α i + q , j + k g γ j + k , j + k g ,
γ i j = α i j ( α j p 2 γ s + q = j p - 1 α j q 2 γ q q g ) = α i j γ j j ,
I = j = 0 i = 0 p γ j j f i j ,
f i j = C j j l i j + 2 Re ( α i j C i j ) s i j , γ j j = α j p 2 γ s + q = j p - 1 α j q 2 γ q q g = γ j c + γ j g .
I = i , j f i j γ j c + i , j f i j γ j g = I c + I g .
h ^ j = α j p h ^ p γ j j = γ j c = α j p 2 · γ s .
I c = γ s j = 0 i = 0 p α j p 2 · f i j ,
- f i j = C j j l i j + 2 Re ( α i j C i j ) s i j , - α i j = k = i j - 1 α k ,             α i i = 1.
h j ( r ) = 1 4 π 2 a j * h j + 1 ( r - d j ) ,
Ψ j ( σ ) = σ · d j .
γ j j = γ j g = k = j p - 1 α j k 2 γ k k g ,
I g = j = 0 i = 0 p γ j g f i j .
D ( τ ) = 1 l x [ h ( x ) - h ( x - τ ) ] 2 d x ,
Γ ( x ) = 1 l x h ( x ) h ( x - τ ) d x
D ( τ ) = 2 [ Γ ( 0 ) - Γ ( τ ) ] d D = - 2 d Γ ,
Γ a ( α ) = r h ( r ) h α ( r ) d r ,
h α ( r ) = h [ R α ( r ) ] ,
F ( α ) = τ Γ ( τ ) Γ α ( τ ) d τ ,
F ( α ) = F ( - α ) = F ( π + α ) , F ( π 2 - α ) = F ( π 2 + α ) , d F d α ( 0 ) = d F d α ( π 2 ) = 0.
F N ( α ) = F ( α ) / F ( 0 ) ;
d = min [ F N ( α ) ] 1 ,
γ s ( σ ) = I ( σ ) C 00 ( σ ) ,
F ( α ) = 4 π 2 σ γ ( σ ) γ [ R α ( σ ) ] d σ .
F ( α ) = 4 π 2 k 2 θ = 0 π / 2 sin θ cos θ ϕ = 0 2 π γ ( θ , ϕ ) γ ( θ , ϕ + α ) d ϕ d θ ,
h ( r ) = σ = 0 h σ ( r ) d σ ,
h σ ( r ) = σ ϕ = 0 2 π h ^ [ R ϕ ( σ 0 ) ] exp [ j R ϕ ( σ 0 ) · r ] d ϕ ,
σ 0 = σ x ,             R ϕ ( σ 0 ) = σ | cos ϕ sin ϕ .
Γ σ ( τ ) = 1 l 2 r h ( r ) h σ ( r - τ ) d r , Γ σ ( τ ) = 1 l 2 σ 2 ϕ , ϕ h ^ [ R ϕ ( σ 0 ) ] h ^ [ R ϕ ( σ 0 ) ] exp [ - j R ϕ ( σ 0 ) · τ ] × r exp { j [ R ϕ ( σ 0 ) + R ϕ ( σ 0 ) ] · r } d r d ϕ d ϕ , Γ σ ( τ ) = 1 l 2 σ 2 ϕ = 0 2 π γ [ R ϕ ( σ 0 ) ] exp [ - j R ϕ ( σ 0 ) · τ ] d ϕ . s s
δ 2 ( σ ) = Γ σ ( 0 ) = 2 π σ 2 γ ¯ ( σ ) / l 2 ,
γ ¯ ( σ ) = 1 2 π ϕ = 0 2 π γ ( σ , ϕ ) d ϕ .
δ 2 = 2 π l 2 σ σ γ ¯ ( σ ) d σ = σ δ 2 ( σ ) σ d σ .
Γ σ ( τ ) = 2 π l 2 σ 2 γ ( σ ) J 0 ( σ τ ) ,
F σ ( α ) = τ Γ σ ( τ ) Γ σ [ R α ( τ ) ] d τ , F σ ( α ) = σ 4 ϕ , ϕ γ ( σ , ϕ ) γ ( σ , ϕ ) τ exp [ - j R ϕ ( σ 0 ) · τ ] × exp [ - j R ϕ ( σ 0 ) · R α ( τ ) ] d τ .
F σ ( α ) = 4 π 2 σ 4 ϕ = 0 2 π γ ( σ , ϕ ) γ ( σ , ϕ + α ) d ϕ ,
d ( σ ) = min a [ F N ( σ , α ) ] ,
F N ( θ , α ) = F σ ( α ) F σ ( 0 ) = ϕ = 0 2 π γ ( θ , ϕ ) γ ( θ , ϕ + α ) d ϕ ϕ = 0 2 π γ 2 ( θ , ϕ ) d ϕ , σ = 2 π λ sin θ .
F ( α ) = k 2 θ = 0 π / 2 F θ ( α ) sin θ cos θ d θ ,
k = 2 π / λ , F θ ( α ) = ϕ = 0 2 π I ( θ , ϕ ) I ( θ , ϕ + α ) d ϕ .
f j ( σ ) = [ 1 + ( σ σ j + 1 c ) 2 ] - 1 / 2 ,
α 01 2 C 00 + C 11 + 2 α 01 Re ( C 01 ) 0.
d j = 1 2 e j tan β | cos ζ sin ζ ,
Ψ j ( σ ) = σ · d j .
α H ( σ ) = f H ( σ ) exp ( - j σ · d H )
α L ( σ ) = f L ( σ ) exp ( - j σ · d L )
γ H ( σ ) = 1 4 π δ H 2 L H 2 exp [ - ( σ L H / 2 ) 2 ] .
δ r 2 = 2 π σ min σ max σ γ g ( σ ) d σ ,
δ r 2 = δ g 2 { exp [ - ( σ min L g / 2 ) 2 ] - exp [ - ( σ max L g / 2 ) 2 ] } δ r 2 δ g 2 { 1 - exp [ - ( π L g / λ ) 2 ] } .
I ¯ m ( θ ) = I c [ γ ¯ s ( θ ) , α H ( θ ) , α L ( θ ) ] + I g [ γ H g ( θ ) , γ L g ( θ ) , α H ( θ ) , α L ( θ ) ] ,
X ¯ ( θ ) = ( 1 / 2 π ) ϕ = 0 2 π X ( θ , ϕ ) d ϕ .
F θ ( α ) = i , j , k , l C i j ( θ ) C k l ( θ ) ϕ = 0 2 π γ i j ( θ , ϕ ) γ k l ( θ , ϕ + α ) d ϕ .
γ i j = γ i j c = α i j · α j , p 2 · γ s .
F θ ( α ) = F θ c ( α ) = g ( θ ) ϕ = 0 2 π γ s ( θ , ϕ ) γ s ( θ , ϕ + α ) d ϕ ,
g ( θ ) = i , j , k , l C i j · C k l · α i j · α k l · α j , p 2 · α l , p 2 ,
F θ c ( α ) F θ c ( 0 ) = F θ ( α ) F θ ( 0 ) F N c ( θ , α ) = F N ( θ , α ) .
F θ ( α ) = F θ c ( α ) + 4 π I g ( θ ) I ¯ c ( θ ) + 2 π I g 2 ( θ ) ,
F N ( θ , α ) = F N ( θ , α ) + B ( θ ) 1 + B ( θ ) = 1 - 1 - F N 1 + B < 1 ,
B = ξ x ( 2 + x ) ,             x = I g / I ¯ c ,             ξ = ( γ ¯ s ) 2 γ ¯ s 2 .
F N B = 1 - F N ( 1 + B ) 2 0 ,
F N α = 1 1 + B F N α < F N α ,
Eq . ( 19 ) d ( θ ) = d ( θ ) + B ( θ ) 1 + B ( θ ) ,
B ( θ ) = d ( θ ) - d ( θ ) 1 - d ( θ )
Eq . ( 20 ) x ( θ ) = I g / I ¯ c = - 1 + [ 1 + B ( θ ) / ξ ( θ ) ] 1 / 2 ,
[ γ H g ( θ ) , γ L g ( θ ) ] = h [ α H ( θ ) , α L ( θ ) ] .
I ¯ m ( θ ) = ( 1 + x ) I ¯ c = [ 1 + B ( θ ) / ξ ( θ ) ] 1 / 2 × I c [ γ ¯ s ( θ ) , α H ( θ ) , α L ( θ ) ] .
( γ H g , γ L g ) = h [ α H , k ( α H ) ] .
I = j = 0 i = 0 p γ j j f i j ,
f i j = C j j l i j + 2 Re ( α i j C i j ) s i j .
α H = α L = 1 ,             γ H g = γ L g = γ g ,
γ j j = γ s + ( p - j ) γ g = γ s + γ j g .
x = I g / I c = S ( θ ) γ g γ s ,
S ( θ ) = p - j f i j / f i j .
y = ( d - d ) / d = B 1 + B 1 - d d 1 - d d .
y / x = 1 - d d ξ ( 2 + x ) 1 + ξ x ( 2 + x ) .
{ x = θ cos ϕ y = θ sin ϕ

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