Abstract

We introduce two parameters, large-scale and small-scale rms roughness, to take into account the interface properties of thin films and multilayers in the calculation of their specular reflectance and transmittance. A theoretical motivation for the introduction of these two parameters instead of a standard single rms roughness is provided. Experimental power spectral density functions of several samples are used to illustrate ways in which the parameters introduced can be evaluated.

© 2003 Optical Society of America

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References

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  1. A. V. Tikhonravov, M. K. Trubetskov, A. A. Tikhonravov, A. Duparré, “Impact of surface roughness on spectral properties of thin films and multilayers,” in Optical Interference Coatings, Vol. 63 of 2001 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 2001), pp. ThB5-1–ThB5-3.
  2. J. M. Bennett, L. Mattsson, Introduction to Surface Roughness and Scattering, 2nd ed. (Optical Society of America, Washington, D.C., 1999).
  3. H. E. Bennett, J. O. Porteus, “Relation between surface roughness and specular reflectance at normal incidence,” J. Opt. Soc. Am. 51, 123–129 (1961).
    [CrossRef]
  4. P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, London, 1963).
  5. J. M. Elson, H. E. Bennett, J. M. Bennett, “Scattering from optical surfaces,” in Applied Optics and Optical Engineering, R. R. Shannon, J. C. Wyant, eds., (Academic, New York, 1979), Vol. VII, pp. 191–244.
    [CrossRef]
  6. C. K. Carniglia, “Scalar scattering theory for multilayer optical coatings,” Opt. Eng. 18, 104–115 (1979).
    [CrossRef]
  7. A. N. Bogolubov, A. A. Tikhonravov, “Effect of varying-scale roughness on the reflectivity from a boundary between two media,” Vestnik MSU Physics and Astronomy series, 3, 27–30 (2002) (in Russian).
  8. C. K. Carniglia, D. G. Jensen, “Single-layer model for surface roughness,” Appl. Opt. 41, 3167–3171 (2002).
    [CrossRef] [PubMed]
  9. F. Abelès, “Recherches sur la propagation des ondes electromagnetique sinusoidales dans les milieux stratifiés,” Ann. Phys. 5, 596–640, 706–782 (1950).
  10. S. Furman, A. V. Tikhonravov, Basics of Optics of Multilayer Systems (Edition Frontieres, Gif-sur-Yvette, France, 1992).
  11. J. Ferre-Borrull, A. Duparré, E. Quesnel, “Procedure to characterize microroughness of optical thin films: application to ion-beam-sputtered vacuum-ultraviolet coatings,” Appl. Opt. 40, 2190–2199 (2001).
    [CrossRef]
  12. A. Duparré, J. Ferre-Borrull, S. Gliech, G. Notni, J. Steinert, J. M. Bennett, “Surface characterization techniques for determining the root-mean-square roughness and power spectral densities of optical components,” Appl. Opt. 41, 154–171 (2002).
    [CrossRef] [PubMed]
  13. D. E. Aspnes, J. B. Theeten, F. Hottier, “Investigation of effective-medium models of microscopic surface roughness by spectroscopic ellipsometry,” Phys. Rev. B 20, 3292–3302 (1979).
    [CrossRef]
  14. D. J. Bergman, “Exactly solvable microscopic geometries and rigorous bounds for the complex dielectric constant of a two-component composite material,” Phys. Rev. Lett. 44, 1285–1287 (1980).
    [CrossRef]
  15. G. W. Milton, “Bounds on the complex dielectric constant of a composite material,” Appl. Phys. Lett. 37, 300–302 (1980).
    [CrossRef]

2002 (2)

2001 (1)

1980 (2)

D. J. Bergman, “Exactly solvable microscopic geometries and rigorous bounds for the complex dielectric constant of a two-component composite material,” Phys. Rev. Lett. 44, 1285–1287 (1980).
[CrossRef]

G. W. Milton, “Bounds on the complex dielectric constant of a composite material,” Appl. Phys. Lett. 37, 300–302 (1980).
[CrossRef]

1979 (2)

D. E. Aspnes, J. B. Theeten, F. Hottier, “Investigation of effective-medium models of microscopic surface roughness by spectroscopic ellipsometry,” Phys. Rev. B 20, 3292–3302 (1979).
[CrossRef]

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

1961 (1)

1950 (1)

F. Abelès, “Recherches sur la propagation des ondes electromagnetique sinusoidales dans les milieux stratifiés,” Ann. Phys. 5, 596–640, 706–782 (1950).

Abelès, F.

F. Abelès, “Recherches sur la propagation des ondes electromagnetique sinusoidales dans les milieux stratifiés,” Ann. Phys. 5, 596–640, 706–782 (1950).

Aspnes, D. E.

D. E. Aspnes, J. B. Theeten, F. Hottier, “Investigation of effective-medium models of microscopic surface roughness by spectroscopic ellipsometry,” Phys. Rev. B 20, 3292–3302 (1979).
[CrossRef]

Beckmann, P.

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, London, 1963).

Bennett, H. E.

H. E. Bennett, J. O. Porteus, “Relation between surface roughness and specular reflectance at normal incidence,” J. Opt. Soc. Am. 51, 123–129 (1961).
[CrossRef]

J. M. Elson, H. E. Bennett, J. M. Bennett, “Scattering from optical surfaces,” in Applied Optics and Optical Engineering, R. R. Shannon, J. C. Wyant, eds., (Academic, New York, 1979), Vol. VII, pp. 191–244.
[CrossRef]

Bennett, J. M.

A. Duparré, J. Ferre-Borrull, S. Gliech, G. Notni, J. Steinert, J. M. Bennett, “Surface characterization techniques for determining the root-mean-square roughness and power spectral densities of optical components,” Appl. Opt. 41, 154–171 (2002).
[CrossRef] [PubMed]

J. M. Elson, H. E. Bennett, J. M. Bennett, “Scattering from optical surfaces,” in Applied Optics and Optical Engineering, R. R. Shannon, J. C. Wyant, eds., (Academic, New York, 1979), Vol. VII, pp. 191–244.
[CrossRef]

J. M. Bennett, L. Mattsson, Introduction to Surface Roughness and Scattering, 2nd ed. (Optical Society of America, Washington, D.C., 1999).

Bergman, D. J.

D. J. Bergman, “Exactly solvable microscopic geometries and rigorous bounds for the complex dielectric constant of a two-component composite material,” Phys. Rev. Lett. 44, 1285–1287 (1980).
[CrossRef]

Bogolubov, A. N.

A. N. Bogolubov, A. A. Tikhonravov, “Effect of varying-scale roughness on the reflectivity from a boundary between two media,” Vestnik MSU Physics and Astronomy series, 3, 27–30 (2002) (in Russian).

Carniglia, C. K.

C. K. Carniglia, D. G. Jensen, “Single-layer model for surface roughness,” Appl. Opt. 41, 3167–3171 (2002).
[CrossRef] [PubMed]

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

Duparré, A.

Elson, J. M.

J. M. Elson, H. E. Bennett, J. M. Bennett, “Scattering from optical surfaces,” in Applied Optics and Optical Engineering, R. R. Shannon, J. C. Wyant, eds., (Academic, New York, 1979), Vol. VII, pp. 191–244.
[CrossRef]

Ferre-Borrull, J.

Furman, S.

S. Furman, A. V. Tikhonravov, Basics of Optics of Multilayer Systems (Edition Frontieres, Gif-sur-Yvette, France, 1992).

Gliech, S.

Hottier, F.

D. E. Aspnes, J. B. Theeten, F. Hottier, “Investigation of effective-medium models of microscopic surface roughness by spectroscopic ellipsometry,” Phys. Rev. B 20, 3292–3302 (1979).
[CrossRef]

Jensen, D. G.

Mattsson, L.

J. M. Bennett, L. Mattsson, Introduction to Surface Roughness and Scattering, 2nd ed. (Optical Society of America, Washington, D.C., 1999).

Milton, G. W.

G. W. Milton, “Bounds on the complex dielectric constant of a composite material,” Appl. Phys. Lett. 37, 300–302 (1980).
[CrossRef]

Notni, G.

Porteus, J. O.

Quesnel, E.

Spizzichino, A.

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, London, 1963).

Steinert, J.

Theeten, J. B.

D. E. Aspnes, J. B. Theeten, F. Hottier, “Investigation of effective-medium models of microscopic surface roughness by spectroscopic ellipsometry,” Phys. Rev. B 20, 3292–3302 (1979).
[CrossRef]

Tikhonravov, A. A.

A. N. Bogolubov, A. A. Tikhonravov, “Effect of varying-scale roughness on the reflectivity from a boundary between two media,” Vestnik MSU Physics and Astronomy series, 3, 27–30 (2002) (in Russian).

A. V. Tikhonravov, M. K. Trubetskov, A. A. Tikhonravov, A. Duparré, “Impact of surface roughness on spectral properties of thin films and multilayers,” in Optical Interference Coatings, Vol. 63 of 2001 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 2001), pp. ThB5-1–ThB5-3.

Tikhonravov, A. V.

A. V. Tikhonravov, M. K. Trubetskov, A. A. Tikhonravov, A. Duparré, “Impact of surface roughness on spectral properties of thin films and multilayers,” in Optical Interference Coatings, Vol. 63 of 2001 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 2001), pp. ThB5-1–ThB5-3.

S. Furman, A. V. Tikhonravov, Basics of Optics of Multilayer Systems (Edition Frontieres, Gif-sur-Yvette, France, 1992).

Trubetskov, M. K.

A. V. Tikhonravov, M. K. Trubetskov, A. A. Tikhonravov, A. Duparré, “Impact of surface roughness on spectral properties of thin films and multilayers,” in Optical Interference Coatings, Vol. 63 of 2001 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 2001), pp. ThB5-1–ThB5-3.

Ann. Phys. (1)

F. Abelès, “Recherches sur la propagation des ondes electromagnetique sinusoidales dans les milieux stratifiés,” Ann. Phys. 5, 596–640, 706–782 (1950).

Appl. Opt. (3)

Appl. Phys. Lett. (1)

G. W. Milton, “Bounds on the complex dielectric constant of a composite material,” Appl. Phys. Lett. 37, 300–302 (1980).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Eng. (1)

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

Phys. Rev. B (1)

D. E. Aspnes, J. B. Theeten, F. Hottier, “Investigation of effective-medium models of microscopic surface roughness by spectroscopic ellipsometry,” Phys. Rev. B 20, 3292–3302 (1979).
[CrossRef]

Phys. Rev. Lett. (1)

D. J. Bergman, “Exactly solvable microscopic geometries and rigorous bounds for the complex dielectric constant of a two-component composite material,” Phys. Rev. Lett. 44, 1285–1287 (1980).
[CrossRef]

Other (6)

A. N. Bogolubov, A. A. Tikhonravov, “Effect of varying-scale roughness on the reflectivity from a boundary between two media,” Vestnik MSU Physics and Astronomy series, 3, 27–30 (2002) (in Russian).

S. Furman, A. V. Tikhonravov, Basics of Optics of Multilayer Systems (Edition Frontieres, Gif-sur-Yvette, France, 1992).

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, London, 1963).

J. M. Elson, H. E. Bennett, J. M. Bennett, “Scattering from optical surfaces,” in Applied Optics and Optical Engineering, R. R. Shannon, J. C. Wyant, eds., (Academic, New York, 1979), Vol. VII, pp. 191–244.
[CrossRef]

A. V. Tikhonravov, M. K. Trubetskov, A. A. Tikhonravov, A. Duparré, “Impact of surface roughness on spectral properties of thin films and multilayers,” in Optical Interference Coatings, Vol. 63 of 2001 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 2001), pp. ThB5-1–ThB5-3.

J. M. Bennett, L. Mattsson, Introduction to Surface Roughness and Scattering, 2nd ed. (Optical Society of America, Washington, D.C., 1999).

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Figures (9)

Fig. 1
Fig. 1

Substitution of an overlayer of thickness 2σ s and refractive index n found from the equation n 2 = (n 1 2 + n 2 2)/2 for the rough boundary with a small-scale roughness characterized by σ s .

Fig. 3
Fig. 3

Real (solid curve) and imaginary (dashed curve) parts of the function g r (λ/T) that accounts for the transition from the large-scale roughness limit to the small-scale roughness limit in the case of an amplitude reflection coefficient. n 1 = 1.0, n 2 = 1.49.

Fig. 4
Fig. 4

Real (solid curve) and imaginary (dashed curve) parts of the function g t (λ/T) that accounts for the transition from the large-scale roughness limit to the small-scale roughness limit in the case of an amplitude transmission coefficient. n 1 = 1.0, n 2 = 1.49.

Fig. 5
Fig. 5

Comparison of the effects on thin-film spectral reflectance that result from large-scale and small-scale roughness: solid curve, reflectance of the model film without interface roughness; dotted curve, reflectance in the case of large-scale roughness with σ l = 2.5 nm; dashed curve, reflectance in the case of small-scale roughness with σ s = 2.5 nm.

Fig. 6
Fig. 6

Comparison of the effects of large-scale and small-scale roughness on thin-film spectral transmittance: solid curve, transmittance of the model film without interface roughness; dotted curve, transmittance in the case of large-scale roughness with σ l = 2.5 nm; dashed curve, transmittance in the case of small-scale roughness with σ s = 2.5 nm.

Fig. 7
Fig. 7

Effects of large-scale roughness on the transmittance of a quarter-wave mirror: solid curve, transmittance of an ideal 31-layer quarter-wave mirror; dashed curve, transmittance of the same mirror in the case of large-scale roughness characterized by σ l = 5 nm at each layer interface.

Fig. 8
Fig. 8

Effects of small-scale roughness on the transmittance of a quarter-wave mirror: solid curve, transmittance of an ideal 31-layer quarter-wave mirror; dashed curve, transmittance of the same mirror in the case of small-scale roughness characterized by σ s = 5 nm at each layer interface.

Fig. 9
Fig. 9

PSD function of the bare fused-silica substrate in the spatial-frequency range 0.1–256 μm-1.

Tables (1)

Tables Icon

Table 1 Four rms Roughness Values Calculated from the PSD Functions of Three Samples Discussed in Section 5

Equations (37)

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hx=m=-NN hm exp2πix/Tm.
h-m=hm*,
σ2=m=-NN |hm|2.
r=r01+2k2n1m=-NN |hm|2-n2-1kγm,1-γm,2,
t=t01+k22n1-n2m=-NN |hm|2n1-n2-2kγm,1-γm,2.
γm,12+2π/Tm2=2π/λ2n12, γm,22+2π/Tm2=2π/λ2n22.
γm,1-γm,2kn1-n2
r=r01-2k2n12|hm|2+|h-m|2,
t=t01-0.5k2n1-n22|hm|2+|h-m|2.
r=r01-2k2n12σl2, t=t01-0.5k2n1-n22σl2,
σl2=large |hm|2
r=r01-2k2n1n2|hm|2+|h-m|2,
t=t01+0.5k2n1-n22|hm|2+|h-m|2.
r=r01-2k2n1n2σs2,
t=t01+0.5k2n1-n22σs2.
σs2=small |hm|2
n2=n12+n22/2.
grλ/T=-n2-1kγ1-γ2 =-n2-n12-λ/T21/2-n22-λ/T21/2,
gtλ/T=n1-n2-2kγ1-γ2 =n1-n2-2n12-λ/T21/2-n22-λ/T21/2.
lnλ/Tm<-ln n2
lnλ/Tm>ln n2
-ln n2lnλ/Tmln n2
σ2=2π fminfmaxPSDffdf,
σ2=σl2+σi2+σs2.
r=r1 expiφ+r2 exp-iφexpiφ+r1r2 exp-iφ.
r1=n1-n/n1+n, r2=n-n2/n+n2,
φ=2knσ,
r=r0 expia-bφ1+0.5a2-b2φ2,
r0=n1-n2/n1+n2,
a=r1-r2/r1+r2, b=1-r1r2/1+r1r2.
a-b2knσ=-2kn1σ.
n2=n12+n22/2,
r=r0 exp-2kn1σ1-2k2n1n2σ2.
C=1/rr*/t*r/t1/t*,
Bj=1/tjrj/tjrj/tj1/tj,
Cj=expiknjdj00exp-iknjdj,
C=BmCmBm-1Cm-1B1C1B0.

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