Abstract

A theoretical method is described for calculating the angular scattering curve for any given thin-film multilayer in which the various surfaces and interfaces are assumed to be rough. The energy that is scattered in any particular direction depends on two factors. The first is solely a function of the characteristics of the ideal multilayer and the observation conditions. The second depends on the roughnesses of the various surfaces, that is, on their autocorrelation and cross-correlation functions. The expressions that are obtained are completely general because no restrictive hypothesis has been developed on the relationship that may exist between the roughnesses of the various interfaces. The principal features of the apparatus used for the scattering measurements are briefly described. Next, several results of calculation and measurement are given as illustrations. The measurements show that polished surfaces of high optical quality often show marked anisotropy.

© 1981 Optical Society of America

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References

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  1. J. M. Elson and J. M. Bennett, “Relation between the angular dependence of scattering and the statistical properties of optical surfaces,” J. Opt. Soc. Am. 69, 31–47 (1979).
    [Crossref]
  2. E. Kröger and E. Kretschmann, “Scattering of light by slightly rough surfaces on thin films including plasma resonance emission,” Z. Phys. 237, 1–15 (1970).
    [Crossref]
  3. J. M. Elson, “Infrared light scattering from surfaces covered with multiple dielectric overlayers,” Appl. Opt. 16, 2872–2882 (1977).
    [Crossref] [PubMed]
  4. S. J. Gourley and P. H. Lissberger, “Optical scattering in multilayer thin films,” Opt. Acta 26, 117–143 (1979).
    [Crossref]
  5. J. M. Eastman, “Surface scattering in optical interference coatings,” Ph.D. Thesis (University of Rochester, Rochester, New York, 1974) (unpublished).
  6. C. K. Carniglia, “Scalar scattering theory for multilayer optical coatings,” Opt. Eng. 18, 104–115 (1979).
    [Crossref]
  7. J. M. Elson, J. P. Rahn, and J. M. Bennett, “Light scattering from multilayer optics: comparison of theory and experiment,” Appl. Opt. 19, 669–679 (1980); J. M. Elson, “Angle-resolved light scattering from composite optical surfaces,” Proc. Soc. Photo-Opt. Instrum. Eng. 240, 296–305 (1980). M. L. Scott and J. M. Elson, “Multilayer light scattering and the laser gyro,” Appl. Phys. Lett. 32, 158–161 (1978); J. M. Elson, “Diffraction and diffuse scattering from dielectric multilayers,” J. Opt. Soc. Am. 69, 48–54 (1979).
    [Crossref] [PubMed]
  8. P. H. Berning, “Theory and calculations of optical thin films,” in Physics of Thin Films, G. Hass, ed. (Academic, New York, 1963), Vol. 1, pp. 69–121.
  9. F. E. Nicodemus, “Directional reflectance and emissivity of an opaque surface,” Appl. Opt. 4, 767–773 (1965); P. Baumeister, “Reflectance and transmission measurements,” Proc. Soc. Photo-Opt. Instrum. Eng. 50, 37–67 (1974).
    [Crossref]
  10. E. Kretschmann and E. Kröger, “Reflection and transmission of light by a rough surface, including results for surface-plasmon effects,” J. Opt. Soc. Am. 65, 150–154 (1975); P. Croce, “Sur l’effet des couches très minces et des rugosités sur la reflexion, la transmission et la diffusion de la lumière par un dioptre,” J. Opt. (Paris) 8, 127–139 (1977).
    [Crossref]
  11. J. M. Bennett, “Measurement of the rms roughness, autocovariance function and other statistical properties of optical surfaces using a FECO scanning interferometer,” Appl. Opt. 11, 2705–2721 (1976); J. Eastman and P. Baumeister, “The microstructure of polished optical surfaces,” Opt. Commun. 12, 418–420 (1974).
    [Crossref]
  12. P. Giacomo, “Les couches réflechissantes multidiélectriques appliquées á l’interférométrie de Fabry–Perot. Etude théorique et expérimentale des couches réelles,” Rev. Opt. Theor. Instrum. 35, 317–354, 442–467 (1956).

1980 (1)

1979 (3)

J. M. Elson and J. M. Bennett, “Relation between the angular dependence of scattering and the statistical properties of optical surfaces,” J. Opt. Soc. Am. 69, 31–47 (1979).
[Crossref]

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

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

1977 (1)

1976 (1)

J. M. Bennett, “Measurement of the rms roughness, autocovariance function and other statistical properties of optical surfaces using a FECO scanning interferometer,” Appl. Opt. 11, 2705–2721 (1976); J. Eastman and P. Baumeister, “The microstructure of polished optical surfaces,” Opt. Commun. 12, 418–420 (1974).
[Crossref]

1975 (1)

1970 (1)

E. Kröger and E. Kretschmann, “Scattering of light by slightly rough surfaces on thin films including plasma resonance emission,” Z. Phys. 237, 1–15 (1970).
[Crossref]

1965 (1)

1956 (1)

P. Giacomo, “Les couches réflechissantes multidiélectriques appliquées á l’interférométrie de Fabry–Perot. Etude théorique et expérimentale des couches réelles,” Rev. Opt. Theor. Instrum. 35, 317–354, 442–467 (1956).

Bennett, J. M.

Berning, P. H.

P. H. Berning, “Theory and calculations of optical thin films,” in Physics of Thin Films, G. Hass, ed. (Academic, New York, 1963), Vol. 1, pp. 69–121.

Carniglia, C. K.

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

Eastman, J. M.

J. M. Eastman, “Surface scattering in optical interference coatings,” Ph.D. Thesis (University of Rochester, Rochester, New York, 1974) (unpublished).

Elson, J. M.

Giacomo, P.

P. Giacomo, “Les couches réflechissantes multidiélectriques appliquées á l’interférométrie de Fabry–Perot. Etude théorique et expérimentale des couches réelles,” Rev. Opt. Theor. Instrum. 35, 317–354, 442–467 (1956).

Gourley, S. J.

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

Kretschmann, E.

Kröger, E.

Lissberger, P. H.

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

Nicodemus, F. E.

Rahn, J. P.

Appl. Opt. (4)

J. Opt. Soc. Am. (2)

Opt. Acta (1)

S. J. Gourley and 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]

Rev. Opt. Theor. Instrum. (1)

P. Giacomo, “Les couches réflechissantes multidiélectriques appliquées á l’interférométrie de Fabry–Perot. Etude théorique et expérimentale des couches réelles,” Rev. Opt. Theor. Instrum. 35, 317–354, 442–467 (1956).

Z. Phys. (1)

E. Kröger and E. Kretschmann, “Scattering of light by slightly rough surfaces on thin films including plasma resonance emission,” Z. Phys. 237, 1–15 (1970).
[Crossref]

Other (2)

J. M. Eastman, “Surface scattering in optical interference coatings,” Ph.D. Thesis (University of Rochester, Rochester, New York, 1974) (unpublished).

P. H. Berning, “Theory and calculations of optical thin films,” in Physics of Thin Films, G. Hass, ed. (Academic, New York, 1963), Vol. 1, pp. 69–121.

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Figures (9)

Fig. 1
Fig. 1

Representation of a rough surface separating two media of permittivities 1 and 2.

Fig. 2
Fig. 2

Distribution of current in the neighborhood of the plane surface given by Eq. (3).

Fig. 3
Fig. 3

Reference coordinates for the incident and scattered wave vectors.

Fig. 4
Fig. 4

Notations used for the calculation of the field scattered by the ith interface. In this model the source of the field is a surface distribution of dipoles that creates in its surround the field (Ei, Hi) in the layer of refractive index ni and the field (Ei, Hi) in the layer of refractive index ni+1.

Fig. 5
Fig. 5

The optical arrangement of the apparatus used for the measurement of the angular scattering.

Fig. 6
Fig. 6

Plane sections of two angular scattering curves measured for two laser mirrors of similar performance but of different materials. The angular field 0 ≤ θ ≤ 90° corresponds to the half-space containing the specular reflection direction. Conversely, 90° < θ ≤ 180° is the half-space of the transmitted light.

Fig. 7
Fig. 7

Two different sections of the angular scattering curve of a single mirror corresponding to two different values of the angle α at the same point of the mirror surface. The angular field 0 ≤ θ ≤ 90° corresponds to the half-space containing the specular reflection direction. Conversely, 90° < θ ≤ 180° is the half-space of the transmitted light.

Fig. 8
Fig. 8

Anisotropy of the angular scattering curve of a multilayer dielectric mirror in the half-space containing the specular reflection direction 0 ≤ θ ≤ 90°. The different curves correspond to the following values of the product BRDF |cos θ|: a, 5 × 10−4; b, 1.2 × 10−4; c, 8 × 10−5; d, 6 × 10−5; e, 4 × 10−5; f, 1 × 10−5.

Fig. 9
Fig. 9

Scattering curve for a narrow-band interference filter. Comparison of theory and experiment. The parameters used in the calculations are given in the text. (See Refs. 1 and 11.)

Equations (42)

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curl E - i ω μ 0 H = 0 ,
curl H + i ω [ 1 + ( 2 - 1 ) ( z - h ) ] E = 0.
curl H + i ω [ 1 + ( 2 - 1 ) ( z ) ] E = J ,
J = i ω ( 2 - 1 ) rect ( z - h / 2 h ) E ,
J = i ω ( 2 - 1 ) δ z E h ( x , y ) = J ( x , y ) δ z ,
curl E - i ω μ 0 H = 0 curl H + i ω E = J = J ( x , y ) δ z ,
Δ H + k 2 H = - curl J ,
Δ E + k 2 E = 1 i ω grad ( div J ) - i ω μ 0 J .
Δ H = { Δ H } + [ ( H 2 / z ) - ( H 1 / z ) ] 0 δ z + ( H 2 - H 1 ) 0 δ z
curl J = [ curl J ( x , y ) ] δ z + [ n ˆ × J ( x , y ) ] δ z ,
δ H = ( H 2 - H 1 ) 0 = - n ˆ × J ( x , y ) .
δ E = ( E 2 - E 1 ) 0 = ( 1 / i ω ) [ n ˆ div J ( x , y ) + grad J z ] .
δ H = - n ˆ × J ( x , y ) ,
δ E T = ( 1 / i ω ) grad J z             with E T = ( E x , E y ) ,
δ D z = ( 1 / i ω ) div J ( x , y ) ,
J ( x , y ) = i ω ( 2 - 1 ) h ( x , y ) E ( x , y ) ,
J = i ω h ( x , y ) [ ( 2 - ) E 2 ( 0 ) - ( 1 - ) E 1 ( 0 ) ] .
δ E i = E i - E i ,             δ H i = H i - H i ,
E i = δ H i - Y i δ E i Y i - Y i             and             E i = δ H i - Y i δ E i Y i - Y i .
d J i ( x , y ) = d J i ( r ) = J ˜ i exp [ i K ( d ) · r ] d K ,
E i ( 0 ) exp [ i K ( i ) · r ] ,
J i ( r ) = i ω ( i + 1 - i ) E i ( 0 ) h i ( r ) exp [ i K ( i ) · r ] ,
h i ( r ) = h ˜ i ( K ) exp ( i K · r ) d K ,
J ˜ i = i ω ( i + 1 - i ) E i ( 0 ) h ˜ i [ K ( d ) - K ( i ) ] .
A i = i 2 π λ ( 0 / μ 0 ) 1 / 2 ( n i + 1 2 - n i 2 ) , B i = i 2 π λ ( μ 0 / 0 ) 1 / 2 n i + 1 2 - n i 2 n i 2 n i + 1 2 n 0 2 .
δ H ˜ i ( s , s ) = A i E i ( s ) ( 0 ) h ˜ i [ K ( d ) - K ( i ) ] cos ( ϕ - α ) , δ E ˜ i ( s , s ) = 0 , δ H ˜ i ( s , p ) = - A i E i ( s ) ( 0 ) h ˜ i [ K ( d ) - K ( i ) ] sin ( ϕ - α ) , δ E ˜ i ( s , p ) = 0 , δ H ˜ i ( p , s ) = - A i E i ( p ) ( 0 ) h ˜ i [ K ( d ) - K ( i ) ] sin ( ϕ - α ) , δ E ˜ i ( p , s ) = 0 , δ H ˜ i ( p , p ) = - A E i ( p ) ( 0 ) h ˜ i [ K ( d ) - K ( i ) ] cos ( ϕ - α ) , δ E ˜ i ( p , p ) = - B i sin i 0 sin θ 0 Y i ( p ) ( 0 ) E i ( p ) ( 0 ) h ˜ i [ K ( d ) - K ( i ) ] .
E i ( 0 ) = 2 1 + Y 0 ( 0 ) / N 0 ( 0 ) m = 1 i [ cos ψ m ( 0 ) + i Y m - 1 ( 0 ) N m ( 0 ) sin ψ m ( 0 ) ] .
ψ m ( 0 ) = 2 π λ e m ( n m 2 - n 0 2 sin 2 i 0 ) 1 / 2 , N m ( 0 ) = { ( 0 μ 0 ) 1 / 2 ( n m 2 - n 0 2 sin 2 i 0 ) 1 / 2 for s - polarized incident light ( 0 μ 0 ) 1 / 2 n m 2 ( n m 2 - n 0 2 sin 2 i 0 ) 1 / 2 for p - polarized incident light , Y m - 1 ( 0 ) = Y m ( 0 ) - i N m ( 0 ) tan ψ m ( 0 ) 1 - i Y m ( 0 ) tan ψ m ( 0 ) / N m ( 0 )             with Y p ( 0 ) = N p + 1 ( 0 ) .
Y 0 = - N 0 in the medium of refractive index n 0 , Y p = N p + 1 in the medium of refractive index n p + 1 .
Y m = Y m - 1 + i N m tan ψ m 1 + i Y m - 1 tan ψ m / N m , Y m = Y m + 1 - i N m + 1 tan ψ m + 1 1 - i Y m + 1 tan ψ m + 1 / N m + 1 ,
ψ m = 2 π λ e m ( n m 2 - n 0 1 sin 2 θ 0 ) 1 / 2 , N m = { - ( 0 μ 0 ) 1 / 2 n m 2 - n 0 2 sin 2 θ 0 ) 1 / 2 for s - polarized scattered light ( 0 μ 0 ) 1 / 2 n m 2 ( n m 2 - n 0 2 sin 2 θ 0 ) 1 / 2 for p - polarized scattered light .
E m + 1 = ( cos ψ m + 1 + i Y m N m + 1 sin ψ m + 1 ) E m , E m - 1 = ( cos ψ m - i Y m N m sin ψ m ) E m .
E i ( d ) + = δ H ˜ i - Y i δ E ˜ i Y i - Y i m = i + 1 p ( cos ψ m + i Y m - 1 N m sin ψ m ) ,
E i ( d ) - = δ H ˜ i - Y i δ E ˜ i Y i - Y i m = 1 i ( cos ψ m - i Y m N m sin ψ m ) .
E i ( d ) + = 2 π λ C i + h ˜ i [ K ( d ) - K ( i ) ] ,
E i ( d ) - = 2 π λ C i - h ˜ i [ K ( d ) - K ( i ) ] .
E ( d ) ± = 2 π λ i = 0 p C i ± h ˜ i [ K ( d ) - K ( i ) ] .
E ( d ) ± 2 = 4 π 2 λ 2 | i = 0 p C i ± h ˜ i [ K ( d ) - K ( i ) ] | 2 = 4 π 2 λ 2 i = 0 p C i ± 2 h ˜ i [ K ( d ) - K ( i ) ] 2 + + 4 π 2 λ 2 i = 0 i j p j = 0 p C i ± C j ± * h ˜ i [ K ( d ) - K ( i ) ] h ˜ j * [ K ( d ) - K ( i ) ] .
ρ - = d Φ ( d ) - Φ ( i ) cos θ 0 d Ω = 4 π 2 λ 4 | N 0 N 0 ( 0 ) | { i = 0 p j = 0 p C i - C j - * γ i , j [ K ( d ) - K ( i ) ] }
N 0 / N 0 ( 0 ) = { cos θ 0 / cos i 0 for polarization s , s cos i 0 / cos θ 0 for polarization p , p 1 / cos i 0 cos θ 0 for polarization s , p cos i 0 cos θ 0 for polarization p , s .
ρ + = d Φ ( d ) + Φ ( i ) cos θ s d Ω = 4 π 2 λ 4 | N s N 0 ( 0 ) | { i = 0 p j = 0 p C i + C j + * γ i , j [ K ( d ) - K ( i ) ] }
N s / N 0 ( 0 ) = { n s cos θ s / n 0 cos i 0 for polarization s , s n s cos i 0 / n 0 cos θ s for polarization p , p n s / n 0 cos i 0 cos θ s for polarization s , p n s cos i 0 cos θ s / n 0 for polarization p , s .