Abstract

Expressions relating the roughness of a plane surface to its specular reflectance at normal incidence are presented and are verified experimentally. The expressions are valid for the case when the root mean square surface roughness is small compared to the wavelength of light. If light of a sufficiently long wavelength is used, the decrease in measured specular reflectance due to surface roughness is a function only of the root mean square height of the surface irregularities. Long-wavelength specular reflectance measurements thus provide a simple and sensitive method for accurate measurement of surface finish. This method is particularly useful for surface finishes too fine to be measured accurately by conventional tracing instruments. Surface roughness must also be considered in precise optical measurements. For example, a non-negligible systematic error in specular reflectance measurements will be made even if the root mean square surface roughness is less than 0.01 wavelength. The roughness of even optically polished surfaces may thus be important for measurements in the visible and ultraviolet regions of the spectrum.

© 1961 Optical Society of America

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References

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  1. Rayleigh, Nature 64, 385 (1901).
    [Crossref]
  2. F. Jentzsch, Z. tech. Physik 7, 310 (1926).
  3. H. Hasunuma and J. Nara, J. Phys. Soc. Japan 11, 69 (1956).
    [Crossref]
  4. W. E. K. Middleton and G. Wyszecki, J. Opt. Soc. Am. 47, 1020 (1957).
    [Crossref]
  5. R. S. Hunter, J. Opt. Soc. Am. 36, 178 (1946).
    [Crossref]
  6. J. Guild, J. Sci. Instr. 17, 178 (1940).
    [Crossref]
  7. E. A. Ollard, J. Electrodepositor’s Tech. Soc. 24, 1 (1949).
  8. J. Halling, J. Sci. Instr. 31, 318 (1954).
    [Crossref]
  9. R. W. Wood, Physical Optics (The Macmillan Company, New York, 1934), 3rd ed., p. 41.
  10. H. Davies, Proc. Inst. Elec. Engrs. 101, 209 (1954).
  11. Davies refers to this function as the autocorrelation function. However, we shall use the term autocovariance function as a more appropriate name when the function in question is not normalized. For a discussion of the properties of such functions see J. H. Laning and R. H. Battin, Random Processes in Automatic Control (McGraw-Hill Book Company, Inc., New York, 1956); and R. B. Blackman and J. W. Tukey, The Measurement of Power Spectra (Dover Publications, New York, 1958).
  12. E. Engelhard, Natl. Bur. Standards Circ. No. 581,  1 (1957).
  13. H. E. Bennett and W. F. Koehler, J. Opt. Soc. Am. 50, 1 (1960).
    [Crossref]
  14. F. W. Preston, Trans. Opt. Soc. (London) 23, 141 (1922).
    [Crossref]
  15. W. F. Koehler and W. C. White, J. Opt. Soc. Am. 45, 1011 (1955).
    [Crossref]
  16. W. F. Koehler, J. Opt. Soc. Am. 43, 743 (1953).
    [Crossref]
  17. S. Bochner and K. Chandrasekharan, Fourier Transforms (Princeton University Press, Princeton, New Jersey, 1949), p. 67.

1960 (1)

1957 (2)

E. Engelhard, Natl. Bur. Standards Circ. No. 581,  1 (1957).

W. E. K. Middleton and G. Wyszecki, J. Opt. Soc. Am. 47, 1020 (1957).
[Crossref]

1956 (1)

H. Hasunuma and J. Nara, J. Phys. Soc. Japan 11, 69 (1956).
[Crossref]

1955 (1)

1954 (2)

J. Halling, J. Sci. Instr. 31, 318 (1954).
[Crossref]

H. Davies, Proc. Inst. Elec. Engrs. 101, 209 (1954).

1953 (1)

1949 (1)

E. A. Ollard, J. Electrodepositor’s Tech. Soc. 24, 1 (1949).

1946 (1)

1940 (1)

J. Guild, J. Sci. Instr. 17, 178 (1940).
[Crossref]

1926 (1)

F. Jentzsch, Z. tech. Physik 7, 310 (1926).

1922 (1)

F. W. Preston, Trans. Opt. Soc. (London) 23, 141 (1922).
[Crossref]

1901 (1)

Rayleigh, Nature 64, 385 (1901).
[Crossref]

Battin, R. H.

Davies refers to this function as the autocorrelation function. However, we shall use the term autocovariance function as a more appropriate name when the function in question is not normalized. For a discussion of the properties of such functions see J. H. Laning and R. H. Battin, Random Processes in Automatic Control (McGraw-Hill Book Company, Inc., New York, 1956); and R. B. Blackman and J. W. Tukey, The Measurement of Power Spectra (Dover Publications, New York, 1958).

Bennett, H. E.

Bochner, S.

S. Bochner and K. Chandrasekharan, Fourier Transforms (Princeton University Press, Princeton, New Jersey, 1949), p. 67.

Chandrasekharan, K.

S. Bochner and K. Chandrasekharan, Fourier Transforms (Princeton University Press, Princeton, New Jersey, 1949), p. 67.

Davies, H.

H. Davies, Proc. Inst. Elec. Engrs. 101, 209 (1954).

Engelhard, E.

E. Engelhard, Natl. Bur. Standards Circ. No. 581,  1 (1957).

Guild, J.

J. Guild, J. Sci. Instr. 17, 178 (1940).
[Crossref]

Halling, J.

J. Halling, J. Sci. Instr. 31, 318 (1954).
[Crossref]

Hasunuma, H.

H. Hasunuma and J. Nara, J. Phys. Soc. Japan 11, 69 (1956).
[Crossref]

Hunter, R. S.

Jentzsch, F.

F. Jentzsch, Z. tech. Physik 7, 310 (1926).

Koehler, W. F.

Laning, J. H.

Davies refers to this function as the autocorrelation function. However, we shall use the term autocovariance function as a more appropriate name when the function in question is not normalized. For a discussion of the properties of such functions see J. H. Laning and R. H. Battin, Random Processes in Automatic Control (McGraw-Hill Book Company, Inc., New York, 1956); and R. B. Blackman and J. W. Tukey, The Measurement of Power Spectra (Dover Publications, New York, 1958).

Middleton, W. E. K.

Nara, J.

H. Hasunuma and J. Nara, J. Phys. Soc. Japan 11, 69 (1956).
[Crossref]

Ollard, E. A.

E. A. Ollard, J. Electrodepositor’s Tech. Soc. 24, 1 (1949).

Preston, F. W.

F. W. Preston, Trans. Opt. Soc. (London) 23, 141 (1922).
[Crossref]

Rayleigh,

Rayleigh, Nature 64, 385 (1901).
[Crossref]

White, W. C.

Wood, R. W.

R. W. Wood, Physical Optics (The Macmillan Company, New York, 1934), 3rd ed., p. 41.

Wyszecki, G.

J. Electrodepositor’s Tech. Soc. (1)

E. A. Ollard, J. Electrodepositor’s Tech. Soc. 24, 1 (1949).

J. Opt. Soc. Am. (5)

J. Phys. Soc. Japan (1)

H. Hasunuma and J. Nara, J. Phys. Soc. Japan 11, 69 (1956).
[Crossref]

J. Sci. Instr. (2)

J. Guild, J. Sci. Instr. 17, 178 (1940).
[Crossref]

J. Halling, J. Sci. Instr. 31, 318 (1954).
[Crossref]

Natl. Bur. Standards Circ. No. 581 (1)

E. Engelhard, Natl. Bur. Standards Circ. No. 581,  1 (1957).

Nature (1)

Rayleigh, Nature 64, 385 (1901).
[Crossref]

Proc. Inst. Elec. Engrs. (1)

H. Davies, Proc. Inst. Elec. Engrs. 101, 209 (1954).

Trans. Opt. Soc. (London) (1)

F. W. Preston, Trans. Opt. Soc. (London) 23, 141 (1922).
[Crossref]

Z. tech. Physik (1)

F. Jentzsch, Z. tech. Physik 7, 310 (1926).

Other (3)

Davies refers to this function as the autocorrelation function. However, we shall use the term autocovariance function as a more appropriate name when the function in question is not normalized. For a discussion of the properties of such functions see J. H. Laning and R. H. Battin, Random Processes in Automatic Control (McGraw-Hill Book Company, Inc., New York, 1956); and R. B. Blackman and J. W. Tukey, The Measurement of Power Spectra (Dover Publications, New York, 1958).

R. W. Wood, Physical Optics (The Macmillan Company, New York, 1934), 3rd ed., p. 41.

S. Bochner and K. Chandrasekharan, Fourier Transforms (Princeton University Press, Princeton, New Jersey, 1949), p. 67.

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Figures (3)

Fig. 1
Fig. 1

Relative reflectance of a finely ground glass surface as a function of wavelength. Circles indicate experimental points. The dotted curve was calculated from Eq. (5) and the solid curve from Eq. (1). Both curves coincide above R/R0 = 0.90.

Fig. 2
Fig. 2

Error made in reflectance measurements when surface roughness is neglected. Circles represent experimental points. The solid line was calculated from Eq. (1).

Fig. 3
Fig. 3

Error caused by surface roughness in reflectance measurements as a function of wavelength. Dotted lines indicate the surface roughnesses of typical optically polished glass surfaces calculated from the experimental data of Koehler and White.

Tables (2)

Tables Icon

Table I Roughnesses of ground glass surfaces.

Tables Icon

Table II Roughnesses of ground steel surfaces.

Equations (23)

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R s = R 0 exp [ - ( 4 π σ ) 2 / λ 2 ] ,
r d ( θ ) d θ = R 0 2 π 4 ( a / λ ) 2 ( σ / λ ) 2 ( cos θ + 1 ) 4 sin θ × exp [ - ( π a sin θ ) 2 / λ 2 ] d θ .
a = 2 σ / m .
0 Δ θ r d ( θ ) d θ = R 0 2 5 π 4 m 2 ( σ / λ ) 4 ( Δ θ ) 2 .
R = R 0 exp [ - ( 4 π σ ) 2 / λ 2 ] + R 0 2 5 π 4 m 2 ( σ / λ ) 4 ( Δ θ ) 2 .
log 10 R 0 / R = [ ( 4 π σ ) 2 / 2.303 ] ( 1 / λ 2 ) .
σ = λ ( R 0 - R ) 1 2 / 4 π R 0 1 2 .
R d = R 0 ( 4 π σ ) 2 / λ 2 .
m 2 = lim X , Y ( 1 / X ) ( 1 / Y ) - X / 2 X / 2 - Y / 2 Y / 2 ( z / x ) 2 d x d y ,
lim X , Y ( 1 / X ) ( 1 / Y ) - - ( z X Y / x ) 2 d x d y = lim X , Y ( 1 / X ) ( 1 / Y ) - - r Y 2 ( y ) [ ( z / x ) 2 r X 2 ( x ) + 2 z ( z / x ) r X ( x ) ( d r X / d x ) + z 2 ( d r X / d x ) 2 ] d x d y .
r X ( x ) = ( X / X + ξ ) [ ( x / ξ ) + ( X / 2 ξ ) + 1 ] ;             - ( X / 2 ) - ξ < x < - ( X / 2 ) = ( X / X + ξ ) ;             x < ( X / 2 ) = ( X / X + ξ ) [ - ( x / ξ ) + ( X / 2 ξ ) + 1 ] ;             ( X / 2 ) < x < ( X / 2 ) + ξ
r Y ( y ) = 1 ; y < ( Y / 2 ) = 0 ; y > ( Y / 2 ) .
m 2 = lim X , Y ( 1 / X ) ( 1 / Y ) - - ( z X Y / x ) 2 d x d y .
m 2 = lim X , Y ( 1 / X ) ( 1 / Y ) - - | [ z X Y x ] f ( u , v ) | 2 d u d v
z X Y ( x , y ) = - - [ z X Y ] f ( u , v ) e 2 π i ( x u + y v ) d u d v .
[ z X Y x ] f ( u , v ) = 2 π i u [ z X Y ] f ( u , v )
m 2 = lim X , Y ( 1 / X ) ( 1 / Y ) - - 4 π 2 u 2 [ z X Y ] f ( u , v ) 2 d u d v = 4 π 2 μ 20 { A f ( u , v ) }
A f ( u , v ) = lim X , Y ( 1 / X ) ( 1 / Y ) [ z X Y ] f ( u , v ) 2
A ( s , t ) = lim X , Y ( 1 / X ) ( 1 / Y ) - - [ z ( x , y ) r X ( x ) r Y ( y ) ] × [ z ( s - x , t - y ) r X ( s - x ) r Y ( t - y ) ] d x d y .
A ( s , t ) = lim X , Y ( 1 / X ) ( 1 / Y ) × - X / 2 X / 2 - Y / 2 Y / 2 z ( x , y ) z ( s - x , t - y ) d x d y ,
A ( s , t ) = σ 2 exp [ - ( s 2 + t 2 ) / a 2 ]
m 2 = 4 π 2 - - u 2 [ σ 2 π a 2 exp - π 2 a 2 ( u 2 + v 2 ) ] d u d v
m 2 = 2 ( σ / a ) 2 .