Abstract

We investigated four different approximation models for describing the polychromatic reflectance and transmittance of a slab with a randomly rough boundary while taking into account the coherent and the incoherent scattering of the rough boundary. Comparisons with experiments (an etched-silicon wafer) show that approximation models that apply a two-scale roughness to the randomly rough boundary and that take into account the coherent and the incoherent scattering yield better agreement and extend the range of validity of the approximation to shorter wavelengths.

© 1999 Optical Society of America

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References

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  1. P. Beckman, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, Oxford, 1963).
  2. F. G. Bass, I. M. Fuks, Wave Scattering from Statistically Rough Surfaces (Pergamon, Oxford, 1979).
  3. G. T. Ruck, D. E. Barrick, W. D. Stuart, C. K. Krichbaum, Radar Cross Section Handbook, Vols. 1 and 2 (Plenum, New York, 1970).
  4. L. Tsang, J. A. Kong, R. T. Shin, Theory of Microwave Remote Sensing (Wiley, New York, 1985).
  5. J. M. Eastman, “Scattering by all-dielectric multilayer bandpass filters and mirrors for lasers,” Phys. Thin Films 6, 167–226 (1978).
  6. T. P. Baryshewa, G. P. Golubev, I. Kh. Kaufmann, “Effect of surface roughness on thin-film optical properties,” Opt. Spectrosc. (USSR) 75, 634–636 (1992) [Opt. Spectrosk. 70, 1082–1085 (1991)].
  7. I. Ohlídal, K. Navrátil, F. Lukeš, “Reflection of light by a system of nonabsorbing isotropic film–nonabsorbing isotropic substrate with randomly rough boundaries,” J. Opt. Soc. Am. 61, 1630–1639 (1971).
    [CrossRef]
  8. J. M. Elson, J. P. Rahn, J. M. 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]
  9. C. Amra, “Light scattering from multilayer optics. I. Tools of investigation,” J. Opt. Soc. Am. A 11, 197–210 (1994).
    [CrossRef]
  10. M. Giovannini, C. Amra, “Scattering-reduction effect with overcoated rough surfaces: theory and experiment,” Appl. Opt. 36, 5574–5579 (1997).
    [CrossRef] [PubMed]
  11. R. Swanepoel, “Transmission and reflection of an absorbing thin film on an absorbing substrate,” S. Afr. Tydskr. Fis. 12(4) , 149–159 (1989).
  12. R. Swanepoel, “Determination of surface roughness and optical constants of inhomogeneous amorphous silicon film,” J. Phys. E 17, 896–903 (1984).
    [CrossRef]
  13. B. J. Stagg, T. T. Charalampopoulos, “Surface-roughness effects on the determination of optical properties of materials by the reflection method,” Appl. Opt. 30, 4113–4118 (1991).
    [CrossRef] [PubMed]
  14. A. Ghader-Aziz, “Zum Einfluss der Oberflächenrauhigkeit auf die elektromagnetischen Reflexions und Transmissionseigenschaften von geschichteten Festkörpern,” Ph.D. dissertation (University of Technology, Graz, Austria, 1996).
  15. J. M. Thériault, G. Boivin, “Modèle quantitatif pour l’étude des couches minces diffusantes,” Can J. Phys. 66, 390–395 (1988).
    [CrossRef]
  16. Z. Yin, M. S. Tan, F. W. Smith, “Determination of the optical constants of diamond films with a rough growth surface,” Diamond Rel. Mater. 5, 1490–1496 (1996).
    [CrossRef]

1997 (1)

1996 (1)

Z. Yin, M. S. Tan, F. W. Smith, “Determination of the optical constants of diamond films with a rough growth surface,” Diamond Rel. Mater. 5, 1490–1496 (1996).
[CrossRef]

1994 (1)

1992 (1)

T. P. Baryshewa, G. P. Golubev, I. Kh. Kaufmann, “Effect of surface roughness on thin-film optical properties,” Opt. Spectrosc. (USSR) 75, 634–636 (1992) [Opt. Spectrosk. 70, 1082–1085 (1991)].

1991 (1)

1989 (1)

R. Swanepoel, “Transmission and reflection of an absorbing thin film on an absorbing substrate,” S. Afr. Tydskr. Fis. 12(4) , 149–159 (1989).

1988 (1)

J. M. Thériault, G. Boivin, “Modèle quantitatif pour l’étude des couches minces diffusantes,” Can J. Phys. 66, 390–395 (1988).
[CrossRef]

1984 (1)

R. Swanepoel, “Determination of surface roughness and optical constants of inhomogeneous amorphous silicon film,” J. Phys. E 17, 896–903 (1984).
[CrossRef]

1983 (1)

1978 (1)

J. M. Eastman, “Scattering by all-dielectric multilayer bandpass filters and mirrors for lasers,” Phys. Thin Films 6, 167–226 (1978).

1971 (1)

Amra, C.

Barrick, D. E.

G. T. Ruck, D. E. Barrick, W. D. Stuart, C. K. Krichbaum, Radar Cross Section Handbook, Vols. 1 and 2 (Plenum, New York, 1970).

Baryshewa, T. P.

T. P. Baryshewa, G. P. Golubev, I. Kh. Kaufmann, “Effect of surface roughness on thin-film optical properties,” Opt. Spectrosc. (USSR) 75, 634–636 (1992) [Opt. Spectrosk. 70, 1082–1085 (1991)].

Bass, F. G.

F. G. Bass, I. M. Fuks, Wave Scattering from Statistically Rough Surfaces (Pergamon, Oxford, 1979).

Beckman, P.

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

Bennett, J. M.

Boivin, G.

J. M. Thériault, G. Boivin, “Modèle quantitatif pour l’étude des couches minces diffusantes,” Can J. Phys. 66, 390–395 (1988).
[CrossRef]

Charalampopoulos, T. T.

Eastman, J. M.

J. M. Eastman, “Scattering by all-dielectric multilayer bandpass filters and mirrors for lasers,” Phys. Thin Films 6, 167–226 (1978).

Elson, J. M.

Fuks, I. M.

F. G. Bass, I. M. Fuks, Wave Scattering from Statistically Rough Surfaces (Pergamon, Oxford, 1979).

Ghader-Aziz, A.

A. Ghader-Aziz, “Zum Einfluss der Oberflächenrauhigkeit auf die elektromagnetischen Reflexions und Transmissionseigenschaften von geschichteten Festkörpern,” Ph.D. dissertation (University of Technology, Graz, Austria, 1996).

Giovannini, M.

Golubev, G. P.

T. P. Baryshewa, G. P. Golubev, I. Kh. Kaufmann, “Effect of surface roughness on thin-film optical properties,” Opt. Spectrosc. (USSR) 75, 634–636 (1992) [Opt. Spectrosk. 70, 1082–1085 (1991)].

Kaufmann, I. Kh.

T. P. Baryshewa, G. P. Golubev, I. Kh. Kaufmann, “Effect of surface roughness on thin-film optical properties,” Opt. Spectrosc. (USSR) 75, 634–636 (1992) [Opt. Spectrosk. 70, 1082–1085 (1991)].

Kong, J. A.

L. Tsang, J. A. Kong, R. T. Shin, Theory of Microwave Remote Sensing (Wiley, New York, 1985).

Krichbaum, C. K.

G. T. Ruck, D. E. Barrick, W. D. Stuart, C. K. Krichbaum, Radar Cross Section Handbook, Vols. 1 and 2 (Plenum, New York, 1970).

Lukeš, F.

Navrátil, K.

Ohlídal, I.

Rahn, J. P.

Ruck, G. T.

G. T. Ruck, D. E. Barrick, W. D. Stuart, C. K. Krichbaum, Radar Cross Section Handbook, Vols. 1 and 2 (Plenum, New York, 1970).

Shin, R. T.

L. Tsang, J. A. Kong, R. T. Shin, Theory of Microwave Remote Sensing (Wiley, New York, 1985).

Smith, F. W.

Z. Yin, M. S. Tan, F. W. Smith, “Determination of the optical constants of diamond films with a rough growth surface,” Diamond Rel. Mater. 5, 1490–1496 (1996).
[CrossRef]

Spizzichino, A.

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

Stagg, B. J.

Stuart, W. D.

G. T. Ruck, D. E. Barrick, W. D. Stuart, C. K. Krichbaum, Radar Cross Section Handbook, Vols. 1 and 2 (Plenum, New York, 1970).

Swanepoel, R.

R. Swanepoel, “Transmission and reflection of an absorbing thin film on an absorbing substrate,” S. Afr. Tydskr. Fis. 12(4) , 149–159 (1989).

R. Swanepoel, “Determination of surface roughness and optical constants of inhomogeneous amorphous silicon film,” J. Phys. E 17, 896–903 (1984).
[CrossRef]

Tan, M. S.

Z. Yin, M. S. Tan, F. W. Smith, “Determination of the optical constants of diamond films with a rough growth surface,” Diamond Rel. Mater. 5, 1490–1496 (1996).
[CrossRef]

Thériault, J. M.

J. M. Thériault, G. Boivin, “Modèle quantitatif pour l’étude des couches minces diffusantes,” Can J. Phys. 66, 390–395 (1988).
[CrossRef]

Tsang, L.

L. Tsang, J. A. Kong, R. T. Shin, Theory of Microwave Remote Sensing (Wiley, New York, 1985).

Yin, Z.

Z. Yin, M. S. Tan, F. W. Smith, “Determination of the optical constants of diamond films with a rough growth surface,” Diamond Rel. Mater. 5, 1490–1496 (1996).
[CrossRef]

Appl. Opt. (3)

Can J. Phys. (1)

J. M. Thériault, G. Boivin, “Modèle quantitatif pour l’étude des couches minces diffusantes,” Can J. Phys. 66, 390–395 (1988).
[CrossRef]

Diamond Rel. Mater. (1)

Z. Yin, M. S. Tan, F. W. Smith, “Determination of the optical constants of diamond films with a rough growth surface,” Diamond Rel. Mater. 5, 1490–1496 (1996).
[CrossRef]

J. Opt. Soc. Am. (1)

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

J. Phys. E (1)

R. Swanepoel, “Determination of surface roughness and optical constants of inhomogeneous amorphous silicon film,” J. Phys. E 17, 896–903 (1984).
[CrossRef]

Opt. Spectrosc. (USSR) (1)

T. P. Baryshewa, G. P. Golubev, I. Kh. Kaufmann, “Effect of surface roughness on thin-film optical properties,” Opt. Spectrosc. (USSR) 75, 634–636 (1992) [Opt. Spectrosk. 70, 1082–1085 (1991)].

Phys. Thin Films (1)

J. M. Eastman, “Scattering by all-dielectric multilayer bandpass filters and mirrors for lasers,” Phys. Thin Films 6, 167–226 (1978).

S. Afr. Tydskr. Fis. (1)

R. Swanepoel, “Transmission and reflection of an absorbing thin film on an absorbing substrate,” S. Afr. Tydskr. Fis. 12(4) , 149–159 (1989).

Other (5)

A. Ghader-Aziz, “Zum Einfluss der Oberflächenrauhigkeit auf die elektromagnetischen Reflexions und Transmissionseigenschaften von geschichteten Festkörpern,” Ph.D. dissertation (University of Technology, Graz, Austria, 1996).

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

F. G. Bass, I. M. Fuks, Wave Scattering from Statistically Rough Surfaces (Pergamon, Oxford, 1979).

G. T. Ruck, D. E. Barrick, W. D. Stuart, C. K. Krichbaum, Radar Cross Section Handbook, Vols. 1 and 2 (Plenum, New York, 1970).

L. Tsang, J. A. Kong, R. T. Shin, Theory of Microwave Remote Sensing (Wiley, New York, 1985).

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Figures (6)

Fig. 1
Fig. 1

Slab with a complex refractive index N 1 that is illuminated by a polychromatic source of intensity I e . The light is incident normally either (a) upon the smooth surface or (b) upon the rough surface of the slab.

Fig. 2
Fig. 2

Measured reflectance and transmittance of an etched-silicon wafer.

Fig. 3
Fig. 3

Experimental damping factors of the rough surface of an etched-silicon wafer as calculated from Eqs. (32)–(34).

Fig. 4
Fig. 4

Reflectance of the etched-silicon wafer if the light is incident normally upon the rough boundary of the wafer: Curve (m) represents the measured reflectance, and curves (I)–(IV) represent the calculated reflectances derived from the approximation models described in the text.

Fig. 5
Fig. 5

Transmittance of the etched-silicon wafer for normally incident light. The transmittance is symmetrical, i.e., T 01 = T 12. Curve (m) represents the measured transmittance, and curves (I)–(IV) represent the calculated transmittances derived from the approximation models described in the text.

Fig. 6
Fig. 6

Reflectance of the etched-silicon wafer if the light is incident normally upon the smooth boundary of the wafer. Curve (m) represents the measured reflectance, and curves (I)–(IV) represent the calculated reflectances derived from the approximation models described in the text.

Equations (48)

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Rij=IrIe,  Tij=ItIe,
R=ρ01+τ01τ10ρ12 exp-2α1d11-ρ10ρ12 exp-2α1d1,
T=τ01τ12 exp-α1d11-ρ10ρ12 exp-2α1d1,
ρij=rijrij*,  τij=njni tijtij*,
α1=2ωc κ1.
R01=ρ01+τ01τ10ρ12 exp-2α1d11-ρ10ρ12 exp-2α1d1,
T01=τ01τ12 exp-α1d11-ρ10ρ12 exp-2α1d1,
R12=ρ01+τ01τ10ρ12 exp-2α1d11-ρ10ρ12 exp-2α1d1,
T12=T01,
ρij=ρijBij,
τij=τijAij,
Bij=BijC+BijI,
Aij=AijC+AijI,
B01C=exp-4πn0σλ2,
B01I=1-exp-4πn0σλ2×1-exp-πn0ταλ2,
A10C=A12C=exp-2πn0-n1σλ2,
A10I=A12I=1-exp-2πn0-n1σλ2×1-exp-πn0ταλ2,
B10C=B12C=exp-4πn1σλ2,
A01C=exp-2πn0-n1σλ2,
B01C=exp-4πn0λ2σ12+σ22,
B01I=exp-4πn0σ2λ21-exp-4πn0λ2σ12×1-exp-πn0τ1αλ2+1-exp-4πn0σ2λ2×1-exp-πn0τ2αλ2,
A10C=A12C=exp-2πn0-niλ2σ12+σ22,
A10I=A12I=exp-2πn0-n1σ2λ2×1-exp-2πn0-n1λ2σ12×1-exp-πn0τ1αλ2+1-exp-2πn0-n1σ2λ2×1-exp-πn0τ2αλ2,
B10C=B12C=exp-4πn1λ2σ12+σ22,
A01C=exp-2πn1-n0λ2σ12+σ22,
α1tn2n1 α,
A01I=1-exp-2πn0-n1σλ2×1-exp-πn2ταλ2,
A01I=exp-2πn0-n1σ2λ2×1-exp-2πn0-n1σ1λ2×1-exp-πn2τ1αλ2+1-exp-2πn0-n1σ2λ2×1-exp-πn2τ2αλ2,
B12Islab=n2n12B12Isingle boundary,
B12I=n0n121-exp-4πn1σλ2×1-exp-πn1ταλ2,
B12I=n0n12exp-4πn1σ2λ2×1-exp-4πn1σ1λ2×1-exp-πn1τ1αλ2+1-exp-4πn1σ2λ2×1-exp-πn1τ2αλ2,
B12exp=R12exp-ρ01exp2α1d1ρ12τ01τ10+ρ10R12exp-ρ01,
A01exp=A10exp=T01exp(1-ρ10ρ12B12exp exp-2αd1)τ01τ12 exp-αd1,
B01exp=R01expρ10-ρ12ρ01T01expA10exp exp-α1d1.
R01=B01Cρ01+A01CA10Cτ01τ10ρ12 exp-2α1d11-B10Cρ10ρ12 exp-2α1d1,
T01=A01Cτ01τ12 exp-α1d11-B10Cρ10ρ12 exp-2α1d1,
R12=ρ01+B12Cτ01τ10ρ12 exp-2α1d11-B12Cρ10ρ12 exp-2α1d1,
σ=0.3548 μm.
R01=B01C+B01Iρ01+A01CA10Cτ01τ10ρ12 exp-2α1d11-B10Cρ10ρ12 exp-2α1d1,
T01=A01C+A01Iτ01τ12 exp-α1d11-B10Cρ10ρ12 exp-2α1d1,
R12=ρ01+B12C+B12Iτ01τ10ρ12 exp-2α1d11-B12C+B12Iρ10ρ12 exp-2α1d1,
σ=0.3989 μm,  τ=15.72 μm,
σ1=0.3734 μm,  τ1=11.363 μm,
σ2=0.1086 μm,  τ2=4.68 μm,
B¯12=exp-4πn1λ2σ12+σ22+n0n12×exp-4πn1σ2λ2P1×1-exp-4πn1λ2σ12×1-exp-πniτ1αλ2+P21-exp-4πn1σ2λ2×1-exp-πn1τ2αλ2,
P1=3.105,  P2=1.39,
σ1=0.3424 μm,  τ1=10.59 μm,
σ2=0.2143 μm  τ2=3.58 μm,

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