Abstract

Random roughness of an optical surface reduces its specular reflectance and transmittance by the scattering of light. The reduction in reflectance can be modeled by a homogeneous layer on the surface if the refractive index of the layer is intermediate to the indices of the media on either side of the surface. Such a layer predicts an increase in the transmittance of the surface and therefore does not provide a valid model for the effects of scatter on the transmittance. Adding a small amount of absorption to the layer provides a model that predicts a reduction in both reflectance and transmittance. The absorbing layer model agrees with the predictions of a scalar scattering theory for a layer with a thickness that is twice the rms roughness of the surface. The extinction coefficient k for the layer is proportional to the thickness of the layer.

© 2002 Optical Society of America

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References

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  1. P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, New York, 1963).
  2. C. K. Carniglia, “Scalar scattering theory for multilayer optical coatings,” Opt. Eng. 18, 104–115 (1979).
    [CrossRef]
  3. 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, 2001 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 2001), paper ThB5.
  4. 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]
  5. H. G. Tompkins, A User’s Guide to Ellipsometry (Academic, San Diego, Calif., 1993).
  6. H. A. Macleod, Thin-Film Optical Filters, 2nd ed. (Macmillan, New York, 1986).
  7. C. K. Carniglia, D. G. Jensen, G. A. Clarke, “The equivalence between surface roughness and a homogeneous layer on the surface,” presented at the OSA Annual Meeting (Providence, R. I., 22–26 Oct. 2000), Conference Program, paper ThG3.

1979 (2)

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

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]

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, New York, 1963).

Carniglia, C. K.

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

C. K. Carniglia, D. G. Jensen, G. A. Clarke, “The equivalence between surface roughness and a homogeneous layer on the surface,” presented at the OSA Annual Meeting (Providence, R. I., 22–26 Oct. 2000), Conference Program, paper ThG3.

Clarke, G. A.

C. K. Carniglia, D. G. Jensen, G. A. Clarke, “The equivalence between surface roughness and a homogeneous layer on the surface,” presented at the OSA Annual Meeting (Providence, R. I., 22–26 Oct. 2000), Conference Program, paper ThG3.

Duparré, A.

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, 2001 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 2001), paper ThB5.

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.

C. K. Carniglia, D. G. Jensen, G. A. Clarke, “The equivalence between surface roughness and a homogeneous layer on the surface,” presented at the OSA Annual Meeting (Providence, R. I., 22–26 Oct. 2000), Conference Program, paper ThG3.

Macleod, H. A.

H. A. Macleod, Thin-Film Optical Filters, 2nd ed. (Macmillan, New York, 1986).

Spizzichino, A.

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

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. 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, 2001 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 2001), paper ThB5.

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, 2001 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 2001), paper ThB5.

Tompkins, H. G.

H. G. Tompkins, A User’s Guide to Ellipsometry (Academic, San Diego, Calif., 1993).

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, 2001 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 2001), paper ThB5.

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]

Other (5)

H. G. Tompkins, A User’s Guide to Ellipsometry (Academic, San Diego, Calif., 1993).

H. A. Macleod, Thin-Film Optical Filters, 2nd ed. (Macmillan, New York, 1986).

C. K. Carniglia, D. G. Jensen, G. A. Clarke, “The equivalence between surface roughness and a homogeneous layer on the surface,” presented at the OSA Annual Meeting (Providence, R. I., 22–26 Oct. 2000), Conference Program, paper ThG3.

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, 2001 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 2001), paper ThB5.

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

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Equations (43)

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R0=na-nsna+ns2,
T0=4nansna+ns2.
R=R0+ΔRS,
ΔRS-R04πnaσ/λ2.
T=T0+ΔTS,
ΔTS-T02πna-nsσ/λ2.
rL=rao+rso exp-iφr1+raorso exp-iφr.
rao=na-nna+n,
rso=n-nsn+ns.
φr=4πnd/λ.
RL=|rL|2,
RL=rao2+2raorso cosφr+rso21+2raorso cosφr+rao2rso2.
cosφr1-φr2/2,
RLrao+rso2-raorsoφr21+raorso2-raorsoφr2.
R0=rao+rso21+raorso2.
n2=na2+ns2/2.
RL=R0+ΔRL,
ΔRL-R0nans2πd/λ2.
d=2σna/ns,
TL=T0+ΔTL.
ΔTLT02πna-nsd/λ2.
rA=ra+rs exp-iφ1+rars exp-iφ.
ra=na-Nna+N=na-n+ikna+n-ik,
rs=N-nsN+ns=n-ik-nsn-ik+ns.
φ=φr+iφi.
φi=-4πkd/λ.
RA=|ra|2+2 Rera * rs exp-iφrexpφi+|rs|2 exp2φi1+2 Rerars exp-iφrexpφi+|rars|2 exp2φi.
tA=tats exp-iφ/21+rars exp-iφ,
ta=2nana+N=2nana+n-ik,
ts=2NN+ns=2n-ikn-ik+ns.
TA=ns/na|tA|2
TA=ns/na|tats|2 expφi1+2 Rerars exp-iφrexpφi+|rars|2 exp2φi.
k=bd/λ,
RA|ra+rs|2+2Rera * rs+|rs|2φi+Imra * rsφr-Rera * rsφr2|1+rars|2+2Rerars+|rars|2φi+Imrarsφr-Rerarsφr2
TAna/ns|tats|21+φi|1+rars|2+2Rerars+|rars|2φi+Imrarsφr-Rerarsφr2
R0=ra+rs1+rars2,
T0=ns/na|tats|2|1+rars|2.
RAR01+nan φi-nana2+3ns2n2na2-ns2 kφr-nans4n2 φr2.
ΔRA4πnadλ4knna2-ns2+πnsdλ.
TAT0 1+na+ns2n φi-na-ns22n2na+nskφr+na-ns216n2φr2.
ΔTAπdλ8knna+ns-πna-ns2dλ.
d=2σ,
k=πna-ns2na+ns4ndλ.

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