Abstract

Measurements of angular scattering due to surface roughness were taken from a 24-layer dielectric mirror and compared to theory. In addition, the top surface roughness of the multilayer stack is analyzed from Talystep profilometer measurements. These roughness data are used to obtain a roughness spectral density function to be used in a vector multilayer scattering theory. The theory uses three multilayer stack models to incorporate possible effects of different degrees of correlation between interfaces of the stack. It was found that the angular scattering calculated using the experimentally obtained roughness spectral density function agreed remarkably well with the measured angular scattering data. This is especially true if care is taken to differentiate between particulate and roughness scattering. For the sake of comparison, the angular scattering from an aluminum film is also given, and differences from scattering from the multilayer mirror are noted.

© 1980 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. M. Elson, Low Efficiency Diffraction Grating Theory, Technical Report AFWL-TR-210 (Air Force Weapons Laboratory, Kirtland Air Force Base, N.M., 1976).
  2. J. M. Elson, Appl. Opt. 16, 2882 (1977).
    [CrossRef]
  3. For a discussion of Eqs. (2), (13), and related areas, see J. M. Elson, J. M. Bennett, J. Opt. Soc. Am. 69, 31 (1979).
    [CrossRef]
  4. J. S. Bendat, A. G. Piersol, Random Data: Analysis and Measurement Procedures (Wiley, New York, 1971), pp. 69–90.
  5. C. K. Carniglia, Opt. Eng. 18, 104 (1979) (scalar theory); D. L. Mills, A. A. Maradudin, Phys. Rev. B 12, 2943 (1975) (vector theory).
    [CrossRef]
  6. J. M. Bennett, Michelson Laboratory; unpublished data.
  7. J. Ebert, H. Pannhorst, H. Küster, H. Welling, Appl. Opt. 18, 818 (1979).
    [CrossRef] [PubMed]
  8. Our standard Lambertian surface was prepared from White Reflectance Coating 6080. This product is manufactured by Eastman Organic Chemicals, Eastman Kodak Co., Rochester, N.Y. 14650. Detailed information is available in Kodak publication JJ-32.
  9. J. M. Elson, J. Opt. Soc. Am. 69, 48 (1979).
    [CrossRef]

1979 (4)

1977 (1)

Bendat, J. S.

J. S. Bendat, A. G. Piersol, Random Data: Analysis and Measurement Procedures (Wiley, New York, 1971), pp. 69–90.

Bennett, J. M.

Carniglia, C. K.

C. K. Carniglia, Opt. Eng. 18, 104 (1979) (scalar theory); D. L. Mills, A. A. Maradudin, Phys. Rev. B 12, 2943 (1975) (vector theory).
[CrossRef]

Ebert, J.

Elson, J. M.

J. M. Elson, J. Opt. Soc. Am. 69, 48 (1979).
[CrossRef]

For a discussion of Eqs. (2), (13), and related areas, see J. M. Elson, J. M. Bennett, J. Opt. Soc. Am. 69, 31 (1979).
[CrossRef]

J. M. Elson, Appl. Opt. 16, 2882 (1977).
[CrossRef]

J. M. Elson, Low Efficiency Diffraction Grating Theory, Technical Report AFWL-TR-210 (Air Force Weapons Laboratory, Kirtland Air Force Base, N.M., 1976).

Küster, H.

Pannhorst, H.

Piersol, A. G.

J. S. Bendat, A. G. Piersol, Random Data: Analysis and Measurement Procedures (Wiley, New York, 1971), pp. 69–90.

Welling, H.

Appl. Opt. (2)

J. Opt. Soc. Am. (2)

Opt. Eng. (1)

C. K. Carniglia, Opt. Eng. 18, 104 (1979) (scalar theory); D. L. Mills, A. A. Maradudin, Phys. Rev. B 12, 2943 (1975) (vector theory).
[CrossRef]

Other (4)

J. M. Bennett, Michelson Laboratory; unpublished data.

J. M. Elson, Low Efficiency Diffraction Grating Theory, Technical Report AFWL-TR-210 (Air Force Weapons Laboratory, Kirtland Air Force Base, N.M., 1976).

Our standard Lambertian surface was prepared from White Reflectance Coating 6080. This product is manufactured by Eastman Organic Chemicals, Eastman Kodak Co., Rochester, N.Y. 14650. Detailed information is available in Kodak publication JJ-32.

J. S. Bendat, A. G. Piersol, Random Data: Analysis and Measurement Procedures (Wiley, New York, 1971), pp. 69–90.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (15)

Fig. 1
Fig. 1

Surface roughness autocorrelation function vs lag length ρ. To calculate this plot, actual Talystep profilometer roughness height data were taken from the top surface of the multilayer stack used in this work. Talystep stylus had a 1-μm radius and 2-mg loading.

Fig. 2
Fig. 2

Theoretical specular reflectance vs angle of incidence for s-polarized (squares) and p-polarized (circles) light incident on a 24-layer stack. Details of the stack are given in the text.

Fig. 3
Fig. 3

Theoretical relative electric field intensity distribution in the 24-layer dielectric stack on a transparent quartz substrate for p-polarixed light incident from the air. Three angles of incidence are shown, and the polarizing angle for the stack is approximately 70°. Alternating high- and low-index layers are quarterwave optical thickness (except for the outer layer which is half-wave) at 30° incidence.

Fig. 4
Fig. 4

Beam intensity vs angle from beam. Measurement is performed with the aid of a density filter.

Fig. 5
Fig. 5

Actual light-scattering data taken from two different spots on the 24-layer stack, where the incident beam is at 30° from normal and p polarized with 6328-Å wavelength. Scattered light is also p polarized and measured in the plane of incidence.

Fig. 6
Fig. 6

Measured angular light scattering (solid line) from an opaque Al film is compared to theory (dashed line). Incident and scattered beams are p polarized and remain in the plane of incidence. The incident beam has a wavelength of 6328 Å at 30° incident angle.

Fig. 7
Fig. 7

Measured angular light scattering (solid line) from a 24-layer dielectric mirror compared to theory for three multilayer stack models: correlated, uncorrelated, and partially correlated. Incident and scattered beams are p polarized and remain in the plane of incidence. The incident beam has 6328-Å wavelength at 0° incidence angle.

Fig. 8
Fig. 8

Same as Fig. 7 except that the incidence angle is 10°

Fig. 9
Fig. 9

Same as Fig. 7 except that the incidence angle is 20°.

Fig. 10
Fig. 10

Same as Fig. 7 except that the incidence angle is 30°.

Fig. 11
Fig. 11

Measured angular light scattering (solid line) from a 24-layer dielectric mirror compared to theory for three multilayer stack models: correlated, uncorrelated, and partially correlated. Incident and scattered beams are s polarized and remain in the plane of incidence. The incident beam has a 6328-Å wavelength at 30° incidence angle.

Fig. 12
Fig. 12

Same as Fig. 7 except that the incidence angle is 40°.

Fig. 13
Fig. 13

Same as Fig. 7 except that the incidence angle is 50°.

Fig. 14
Fig. 14

Same as Fig. 7 except that the incidence angle is 60°.

Fig. 15
Fig. 15

Same as Fig. 7 except that the incidence angle is 70°.

Equations (20)

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

E ( r , t ) = l = 1 L + 1 d 2 k ζ l ( k - k 0 ) l ( k , k 0 ) exp [ i ( k · ρ + q z ) ] ,
ζ l ( k - k 0 ) = d 2 ρ ζ l ( ρ ) exp [ i ( k - k 0 ) · ρ ] .
d P / d Ω / P 0 = m = 1 L + 1 n = 1 L + 1 F m ( k , k 0 ) · F n * ( k , k 0 ) × ζ m ( k - k 0 ) ζ n * ( k - k 0 ) A ,
ζ m ( k - k 0 ) ζ n * ( k - k 0 ) = d 2 ρ d 2 τ × exp [ i ( k - k 0 ) · τ ] ζ m ( ρ ) ζ n ( ρ + τ ) .
g m n ( k - k 0 ) = d 2 τ exp [ i ( k - k 0 ) · τ ] G m n ( τ ) ,
g m n ( k - k 0 ) = 2 0 d τ τ J 0 ( k - k 0 τ ) G m n ( τ ) ,
d P / d Ω / P 0 = g ( k - k 0 ) | l = 1 L + 1 F l ( k , k 0 ) | 2 .
ζ m ( ρ ) = μ m ( ρ ) .
d P / d Ω / P 0 = g ( k - k 0 ) l = 1 L + 1 F ( k , k 0 ) 2 .
ζ m ( ρ ) = ζ 1 ( ρ ) + j = 2 m μ l ( ρ ) ,
G m n ( τ ) = = ζ m ( ρ ) ζ n ( ρ + τ ) = l = 1 p G l ( τ ) ,
G 1 ( τ ) = ζ 1 ( ρ ) ζ 1 ( ρ + τ ) = G ( τ ) ,
G l ( τ ) = μ l ( ρ ) μ l ( ρ + τ ) l 2.
δ rms ( m ) = ( l = 1 m δ l 2 ) 1 / 2 ,
g m n ( k - k 0 ) = l = 1 p g l ( k - k 0 ) ,
G l ( τ ) = δ l 1 2 exp ( - τ 2 / σ l 1 2 ) + δ l 2 2 exp ( - τ 2 / σ l 2 2 ) .
δ rms ( m ) = [ l = 1 m ( δ l 1 2 + δ l 2 2 ) ] 1 / 2 .
g l ( k - k 0 ) = π [ δ l 1 2 σ l 1 2 exp ( - k - k 0 2 σ l 1 2 / 4 ) + δ l 2 2 σ l 2 2 exp ( - k - k 0 2 σ l 3 2 / 4 ) ] ,
G ( τ ) = δ 11 2 exp ( - τ 2 / σ 11 2 ) + δ 12 2 exp ( - τ 2 / σ 12 2 ) .
1 P 0 d P d Ω = V m ( θ ) V c S π ,

Metrics