Abstract

Four industries prepared optical coatings with a common design that permits an easy determination of cross-correlation laws between the rough interfaces in the stack. Different pairs of materials and deposition processes were used. After clarifying the differences between scalar and vector theories of light scattering caused by rough interfaces in optical multilayers, we compare the experimental values with both theories. Factors such as variations of correlation with spatial frequency, residual roughness, and slight errors in the design are taken into account for comparison with the vector theory of angular scattering. Correlation of the interface roughnesses is found to be high for practically all coatings. However, at low scatter-loss levels, scattering by localized defects in the coatings appears to dominate over the scattering caused by rough interfaces.

© 1992 Optical Society of America

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  1. J. M. Bennett, E. Pelletier, G. Albrand, J. P. Borgogno, B. Lazaridès, C. K. Carniglia, R. A. Schmell, T. H. Allen, T. Tuttle-Hart, K. H. Guenther, A. Saxer, “Comparison of the properties of titanium dioxide films prepared using various techniques,” Appl. Opt. 28, 3303–3317 (1989).
    [CrossRef] [PubMed]
  2. C. Amra, “Minimizing scattering in multilayers: technique for searching optimal realization conditions,” Laser Dam. Opt. Mater. 756, 265–271 (1987).
  3. J. M. Bennett, L. Mattsson, Introduction to Surface Roughness and Scattering (Optical Society of America, Washington D.C., 1989).
  4. R. A. Craig, G. J. Exarhos, W. T. Pawlewicz, R. E. Williford, “Interference enhanced Raman scattering from TiO2/SiO2 multilayers: measurement and theory,” Appl. Opt. 26, 4193–4197 (1987).
    [CrossRef] [PubMed]
  5. J. M. Eastman, “Surface scattering in optical interference coatings,” Ph.D. dissertation (University of Rochester, Rochester, N.Y., 1974).
  6. C. K. Carniglia, “Scalar scattering theory for multilayer optical coatings,” Opt. Eng. 18, 104–115 (1979).
  7. 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 (1989).
  8. J. M. Elson, “Angle resolved light scattering from composite optical surfaces,” in Periodic Structures, Gratings, Moiré Patterns, and Diffraction Phenomena I, C. H. Chi, ed., Proc. Soc. Photo-Opt. Instrum. Eng.240, 296–306 (1980); J. M. Elson, J. P. Rahn, J. Bennett, “Light scattering from multilayer optics: comparison of theory and experiment,” Appl. Opt. 19, 669–679 (1980); J. M. Elson, “Diffraction and diffuse scattering from dielectric multilayers,” J. Opt. Soc. Am. 69, 48–54 (1979).
    [CrossRef] [PubMed]
  9. P. Bousquet, F. Flory, P. Roche, “Scattering from multilayer thin films: theory and experiment,” J. Opt. Soc. Am. 71, 1115–1123 (1981).
    [CrossRef]
  10. F. E. Nicodemus, “Directional reflectance and emissivity of an opaque surface,” Appl. Opt. 4, 767–773 (1965).
    [CrossRef]
  11. 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]
  12. 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]
  13. E. L. Church, H. A. Jenkinson, J. M. Zavada, “Relationship between surface scattering and microtopographic features,” Opt. Eng. 18, 125–136 (1979).
  14. C. Amra, G. Albrand, P. Roche, “Theory and application of antiscattering single layers: antiscattering antireflection coatings,” Appl. Opt. 25, 2695–2702 (1986).
    [CrossRef] [PubMed]
  15. C. Amra, “Scattering distribution from multilayer mirrors: theoretical research of a design for minimum losses,” Laser Dam. Opt. Mater. 752, 594–602 (1986).
  16. C. Amra, “Scattering characterization of materials in thin film form,” Laser Dam. Opt. Mater. (to be published); 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, 12–81 (1990).
  17. 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]
  18. E. Pelletier, M. Klapisch, P. Giacomo, “Synthèse d’empilements de couches minces,” Nouv. Rev. Opt. Appl. 2, 247–54 (1971).
    [CrossRef]
  19. J. P. Borgogno, B. Lazaridès, E. Pelletier, “Automatic determination of the optical constants of inhomogeneous thin films,” Appl. Opt. 21, 4020–4029 (1982).
    [CrossRef] [PubMed]
  20. J. A. Nelder, R. Mead, “A simplex method for function minimization,” Comput. J. 7, 308–313 (1965).

1990

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

1987

R. A. Craig, G. J. Exarhos, W. T. Pawlewicz, R. E. Williford, “Interference enhanced Raman scattering from TiO2/SiO2 multilayers: measurement and theory,” Appl. Opt. 26, 4193–4197 (1987).
[CrossRef] [PubMed]

C. Amra, “Minimizing scattering in multilayers: technique for searching optimal realization conditions,” Laser Dam. Opt. Mater. 756, 265–271 (1987).

1986

C. Amra, “Scattering distribution from multilayer mirrors: theoretical research of a design for minimum losses,” Laser Dam. Opt. Mater. 752, 594–602 (1986).

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

1983

1982

1981

1979

E. L. Church, H. A. Jenkinson, J. M. Zavada, “Relationship between surface scattering and microtopographic features,” Opt. Eng. 18, 125–136 (1979).

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

1971

E. Pelletier, M. Klapisch, P. Giacomo, “Synthèse d’empilements de couches minces,” Nouv. Rev. Opt. Appl. 2, 247–54 (1971).
[CrossRef]

1965

J. A. Nelder, R. Mead, “A simplex method for function minimization,” Comput. J. 7, 308–313 (1965).

F. E. Nicodemus, “Directional reflectance and emissivity of an opaque surface,” Appl. Opt. 4, 767–773 (1965).
[CrossRef]

Albrand, G.

Allen, T. H.

Amra, C.

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. Amra, “Minimizing scattering in multilayers: technique for searching optimal realization conditions,” Laser Dam. Opt. Mater. 756, 265–271 (1987).

C. Amra, “Scattering distribution from multilayer mirrors: theoretical research of a design for minimum losses,” Laser Dam. Opt. Mater. 752, 594–602 (1986).

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, 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 (1989).

C. Amra, “Scattering characterization of materials in thin film form,” Laser Dam. Opt. Mater. (to be published); 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, 12–81 (1990).

Bennett, J.

Bennett, J. M.

Borgogno, J. P.

Bousquet, P.

P. Bousquet, F. Flory, P. Roche, “Scattering from multilayer thin films: theory and experiment,” J. Opt. Soc. Am. 71, 1115–1123 (1981).
[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 (1989).

Carniglia, C. K.

Church, E. L.

E. L. Church, H. A. Jenkinson, J. M. Zavada, “Relationship between surface scattering and microtopographic features,” Opt. Eng. 18, 125–136 (1979).

Craig, R. 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.

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]

J. M. Elson, “Angle resolved light scattering from composite optical surfaces,” in Periodic Structures, Gratings, Moiré Patterns, and Diffraction Phenomena I, C. H. Chi, ed., Proc. Soc. Photo-Opt. Instrum. Eng.240, 296–306 (1980); J. M. Elson, J. P. Rahn, J. Bennett, “Light scattering from multilayer optics: comparison of theory and experiment,” Appl. Opt. 19, 669–679 (1980); J. M. Elson, “Diffraction and diffuse scattering from dielectric multilayers,” J. Opt. Soc. Am. 69, 48–54 (1979).
[CrossRef] [PubMed]

Exarhos, G. J.

Flory, F.

Giacomo, P.

E. Pelletier, M. Klapisch, P. Giacomo, “Synthèse d’empilements de couches minces,” Nouv. Rev. Opt. Appl. 2, 247–54 (1971).
[CrossRef]

Grèzes-Besset, C.

Guenther, K. H.

Jenkinson, H. A.

E. L. Church, H. A. Jenkinson, J. M. Zavada, “Relationship between surface scattering and microtopographic features,” Opt. Eng. 18, 125–136 (1979).

Klapisch, M.

E. Pelletier, M. Klapisch, P. Giacomo, “Synthèse d’empilements de couches minces,” Nouv. Rev. Opt. Appl. 2, 247–54 (1971).
[CrossRef]

Lazaridès, B.

Mattsson, L.

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

Mead, R.

J. A. Nelder, R. Mead, “A simplex method for function minimization,” Comput. J. 7, 308–313 (1965).

Nelder, J. A.

J. A. Nelder, R. Mead, “A simplex method for function minimization,” Comput. J. 7, 308–313 (1965).

Nicodemus, F. E.

Pawlewicz, W. T.

Pelletier, E.

Rahn, J. P.

Roche, P.

Saxer, A.

Schmell, R. A.

Tuttle-Hart, T.

Williford, R. E.

Zavada, J. M.

E. L. Church, H. A. Jenkinson, J. M. Zavada, “Relationship between surface scattering and microtopographic features,” Opt. Eng. 18, 125–136 (1979).

Appl. Opt.

F. E. Nicodemus, “Directional reflectance and emissivity of an opaque surface,” Appl. Opt. 4, 767–773 (1965).
[CrossRef]

J. P. Borgogno, B. Lazaridès, E. Pelletier, “Automatic determination of the optical constants of inhomogeneous thin films,” Appl. Opt. 21, 4020–4029 (1982).
[CrossRef] [PubMed]

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]

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

R. A. Craig, G. J. Exarhos, W. T. Pawlewicz, R. E. Williford, “Interference enhanced Raman scattering from TiO2/SiO2 multilayers: measurement and theory,” Appl. Opt. 26, 4193–4197 (1987).
[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]

J. M. Bennett, E. Pelletier, G. Albrand, J. P. Borgogno, B. Lazaridès, C. K. Carniglia, R. A. Schmell, T. H. Allen, T. Tuttle-Hart, K. H. Guenther, A. Saxer, “Comparison of the properties of titanium dioxide films prepared using various techniques,” Appl. Opt. 28, 3303–3317 (1989).
[CrossRef] [PubMed]

Comput. J.

J. A. Nelder, R. Mead, “A simplex method for function minimization,” Comput. J. 7, 308–313 (1965).

J. Opt. (Paris)

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.

Laser Dam. Opt. Mater.

C. Amra, “Minimizing scattering in multilayers: technique for searching optimal realization conditions,” Laser Dam. Opt. Mater. 756, 265–271 (1987).

C. Amra, “Scattering distribution from multilayer mirrors: theoretical research of a design for minimum losses,” Laser Dam. Opt. Mater. 752, 594–602 (1986).

Nouv. Rev. Opt. Appl.

E. Pelletier, M. Klapisch, P. Giacomo, “Synthèse d’empilements de couches minces,” Nouv. Rev. Opt. Appl. 2, 247–54 (1971).
[CrossRef]

Opt. Eng.

E. L. Church, H. A. Jenkinson, J. M. Zavada, “Relationship between surface scattering and microtopographic features,” Opt. Eng. 18, 125–136 (1979).

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

Other

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 (1989).

J. M. Elson, “Angle resolved light scattering from composite optical surfaces,” in Periodic Structures, Gratings, Moiré Patterns, and Diffraction Phenomena I, C. H. Chi, ed., Proc. Soc. Photo-Opt. Instrum. Eng.240, 296–306 (1980); J. M. Elson, J. P. Rahn, J. Bennett, “Light scattering from multilayer optics: comparison of theory and experiment,” Appl. Opt. 19, 669–679 (1980); J. M. Elson, “Diffraction and diffuse scattering from dielectric multilayers,” J. Opt. Soc. Am. 69, 48–54 (1979).
[CrossRef] [PubMed]

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,” Laser Dam. Opt. Mater. (to be published); 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, 12–81 (1990).

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Figures (22)

Fig. 1
Fig. 1

Spectral dependence of the specular and diffuse reflectances for the three-absentee layer design. The thin-film design is 2H 2L 2H at the wavelength of 632.8 nm, which is indicated by the short vertical lines that connect each curve with its label. The indices of refraction for the H and L layers are 2.35 and 1.46, respectively, and 1.52 for the substrate. The diffuse reflectance was calculated by using the scalar model with a roughness of 0.5 nm.

Fig. 2
Fig. 2

Upward and downward scattered waves from one interface in a three-absentee layer stack.

Fig. 3
Fig. 3

Contribution of the waves scattered from each interface; δ, the interface roughness assumed to be 0.5 nm; ppm, parts in 10−6.

Fig. 4
Fig. 4

Spectral variations of total scattering Dv by reflection in the case of a 2H 2L 2H stack. The calculation is performed with vector theory at normal illumination and assumes that all roughness spectra in the stack are Gaussian functions with δg = 10 nm. Curves A and B are calculated with Lg = 1000 nm and 10 nm, respectively. Curves A0 and B0 correspond to the case of uncorrelated surfaces (α = 0); A1 and B1 correspond to correlated surfaces (α = 1). R is the reflection coefficient of the stack. Notice that total scattering Dv is strongly dependent on the value of correlation length Lg. In each pair the α = 0 curve and the α = 1 curve are nearly in phase opposition. Moreover, as the correlation length increases (further described in the text), the α = 1 curve tends to vary exactly in phase with the reflection coefficient R.

Fig. 5
Fig. 5

Angular variations of the C′(θ) coefficient in the case of a 2H 2L 2H stack (curve A) and H LH LH 6L HL HL H stack (curve B). Correlation is assumed to be perfect. The vertical units are arbitrary. It is clear from curve (B) that the angular variations of C′(θ) cannot be neglected for a Fabry–Perot stack (due to scattering rings); for this coating, most of the scattering is at large angles. The calculation is performed at the design wavelength λ0.

Fig. 6
Fig. 6

Spectral dependence of the total scattering predicted with the vector theory and the modified scalar theory for (a) the three-absentee layer design (2H 2L 2H) and (b) and (c) the Fabry–Perot design (H LH LH 6L HL HL H). The solid curves, identified with unprimed labels, are the total scattering Dv obtained by integrating the vector model over all angles in the incident medium. The dashed curves, identified with primed labels, are from the modified scalar model Dv′ = Ds′. The subscripts 0 and 1 indicate total uncorrelation (α = 0) and perfect correlation (α = 1). R and T are the reflection and transmission coefficients of the stacks. Curves A were calculated with Lg = 1000 nm and curves B with Lg = 10 nm. The roughness δg is (a) 10 nm, (b) 3 nm, and (c) 30 nm. All roughness spectra in the stack are assumed to be Gaussian functions. The agreement between Dv and Dv′ completely fails at harmonic wavelengths for small correlation lengths, especially for complex stacks [see Fig. 6(c)].

Fig. 7
Fig. 7

Schematic view of the ratio C(θ)/e, where C(θ) is the grating period responsible for scattering at direction θ and e is the layer thickness. This ratio leads to a cutoff frequency (see Fig. 8).

Fig. 8
Fig. 8

Influence of cutoff frequency σc on the angular scattering curve of a three-absentee layer stack. Both materials are assumed to have the same parameter σc. The case σc = ∞ is that of perfect correlation [α(σ) = 1, where σ = (2π/λ) sin θ].

Fig. 9
Fig. 9

Influence of the cross-correlation coefficient α′ on the angular scattering curve of a three-absentee layer stack. All roughnesses are assumed to be identical. The curve (sub.) is that of the substrate before coating and is superimposed on that of the coating (for perfect correlation) at angles less than 35 deg.

Fig. 10
Fig. 10

Mean plane sections of angular scattering curves from the 15 black glass samples. The total scattering varies from 5.5 to 7.7 parts in 10−6 for this set of glasses.

Fig. 11
Fig. 11

Level maps of (a) scattering11 and (b) isotropy degree curve16 for one sample.

Fig. 12
Fig. 12

Roughness spectra of 15 black glass samples. The roughness varies from 0.56 to 0.65 nm for this set of glasses.

Fig. 13
Fig. 13

Angular scattering curves of three samples [(a), (b), and (c), respectively] measured after first and second cleaning. The slight differences at low angles θ are caused by the size of the angular step Δθ (see text).

Fig. 14
Fig. 14

Measurement of the angular autocorrelation F′(α) (Ref. 17) before and after coating by a (6L) SiO2 layer (ion-assisted deposition). The two curves are superimposed and indicate that correlation is high and that the roughnesses are nearly identical.

Fig. 15
Fig. 15

Measured spectral dependence of the specular and diffuse reflectance of a coated three-absentee layer design. The solid curve is the specular reflectance for an incidence angle of 30 deg. TIS, total integrated scatter, determined by measuring the intensity of the light scattered by a nearly perfect diffuse surface

Fig. 16
Fig. 16

Measurement of the angular scattering curves of the samples that were sent to Marseilles by (a) Spectra Physics Optics, (b) OCLI, (c) Balzers, and (d) those from Ojai. Except for those from Ojai, all substrates were recleaned before coating.

Fig. 17
Fig. 17

Measurements and calculation of angular scattering from samples 30 (a), 17 (b), 3 (c), and 22 (d). Curves 1 and 2 are the measurements before and after coating. In the case of perfect correlation (α′ = 1) and a design that is free of errors, these two curves should have been identical for θ ≤ 35 deg. Curves 3 were calculated with the actual design (from spectrophotometric measurements, see text), and the calculation was performed with the measured roughness spectrum of the substrate before coating and with the hypothesis that correlation is perfect (α′ = 1). The agreement between curves 2 and 3 is rather high at small angles θ; the differences at large θ can be explained with a residual spectrum (see text).

Fig. 18
Fig. 18

Influence of thickness errors on the angular scattering curve of a 2H 2L 2H stack. Curve 1 is that of the ideal design (free of errors). Curves 2 and 3 are respectively relative to a +2% and a −2% error on high-index layers. Any intermediate curve can be obtained with other thickness errors.

Fig. 19
Fig. 19

Angular scattering of an uncoated substrate (solid curve) and a coated substrate (dashed curve). The abscissa is the scatter angle relative to the specular direction. These data were obtained on a commercial instrument in the laboratory of the manufacturer. This sample is similar to samples 15 and 17 from OCLI (same materials and deposition process).

Fig. 20
Fig. 20

Angular scattering curves for 2H 2L 2H stacks. Curves 1 are the measured scattering of the uncoated substrates (a) for the black glass sample (33) with ~ 0.5-nm roughness and (b) for the silicon substrate with < 0.1-nm roughness. Curves 2 are the measured scattering of the substrates coated simultaneously with TiO2 and SiO2 using an ion-assisted reactive deposition process. Curves 3 are calculated with the corrected designs and the substrate roughnesses of (a) 0.5 nm and (b) 0.1 nm. We assumed perfect correlation for this calculation. The remaining differences between curves 2 and 3 can be explained with a residual spectrum (see text).

Fig. 21
Fig. 21

Calculated reflectance of the 2H 2L 2H design on a glass substrate for four different values of the H-layer refractive index. The H-layer indices, for the upper to the lower curves, are 2.4, 2.2, 2.0, and 1.8, respectively. The L-layer index is 1.45, and the substrate index is 1.523.

Fig. 22
Fig. 22

Calculation (+ curve) and measurement of the spectral reflectance of the coated samples. Calculation was performed with the index and the thickness values of Table III; (a), (b), (c), and (d) are relative to samples 17, 30, 22, and 003, respectively.

Tables (3)

Tables Icon

Table I Comparison of Scalar and Vector Values of Total Scattering Losses from a 2H 2L 2H Stack at λo in the Limits of Correlation (α = 0 and α = 1)

Tables Icon

Table II Participants, Deposition Techniques, and Materials, with the Associated Values of Scattering, Correlation, and Residual Roughness

Tables Icon

Table III Values of Refractive Indices n(λ) = A + (B2) + (C4) (λ in nm) and of Layer Thicknesses (e in nm) of the Stack: Glass, nHe1, nLe2, nHe3, Air Calculated from Reflectance Measurements

Equations (29)

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

D s = C s ( 4 π δ s λ ) 2 ,
BRDF cos θ ( θ , ϕ ) = i - j 0 p C i C ¯ j α i j γ j j
γ ( σ ) = H T [ Γ e + Γ g ] = γ e + γ g             with σ = 2 π λ sin θ ,
Γ e ( τ ) = δ e 2 exp - τ / L e ,             Γ g ( τ ) = δ g 2 exp - ( τ / L g ) 2 ,
D v = 2 π θ = 0 π / 2 BRDF cos θ d Ω             with d Ω = sin θ d θ .
C i C ¯ j = C i j = ½ [ C i j ( S S ) + C i j ( S P ) + C i j ( P S ) + C i j ( P P ) ] ,
D s = D v = R ( 4 π δ λ ) 2 ,
δ s 2 = ( δ e 2 + δ g 2 ) = 2 π σ = 0 σ γ ( σ ) d σ ,
δ v 2 = 2 π σ = σ 1 σ = σ 2 σ γ ( σ ) d σ ,
D v = 2 π θ = 0 π / 2 C ( θ ) γ ( θ ) sin θ cos θ d θ ,
C ( θ ) = 1 cos θ i j C i C ¯ j ( θ ) α i j .
D v D v = C ( 0 ) 2 π θ = 0 π / 2 γ ( θ ) sin θ cos θ d θ ,
D v = C δ v 2 ,
C = R ( 4 π λ ) 2 C ( 0 ) = 64 ( π λ ) 4 R ,
C = C s ( 4 π λ ) 2 D s = D v ,
D s = D s [ 1 - exp - ( π L g / λ ) 2 ] ,
δ v 2 = δ g 2 [ 1 - exp - ( π L g / λ ) 2 ] .
D ( S S ) = D ( S P ) = D ( P S ) = D ( P P ) = ½ D v = ½ D s ,
h i = 1 4 π 2 a i , i - 1 * h i - 1 + g i ,
h ^ i = α i j h ^ j ,             with α i j = k = 0 i - j - 1 α i - k , i - k - 1 and             α i - k , i - k - 1 = FT ( α i - k , i - k - 1 ) ,
h ^ i = k = 0 i = 1 α i , i - k g ^ i - k .
γ i j = k = 0 i - 1 q = 0 j - 1 α i , i - k α j , j - q α i - k , j - q g γ j - q , j - q g ,
γ j - q , j - q g = g ^ j - q 2 ,
γ i j = k = 0 i - 1 α i , i - k α j , i - k γ i - k , i - k g ,
γ i j = γ i j 1 + γ i j 2 ,
δ i = f ( n i , n H , n L ) ,
δ r 2 = δ g 2 [ 1 - exp - ( π L / λ ) 2 ] .
f = λ j R j - R j , exp 2 ,
f min ( n H ( λ ) , n L ( λ ) , e 1 th , e 2 th , e 3 th ) .

Metrics