Abstract

A theory concerning the relation between the root-mean-square (rms) roughness of a plane surface and its specular reflectance at normal incidence has been reported previously for the case when the roughness is small compared to the wavelength of light. In the present paper the theory is extended with certain restrictions to shorter wavelengths. The inadequacy of parameters such as the rms roughness, the rms slope, and the autocovariance length for describing the reflectance in the shorter wavelength region is discussed. Particular attention is given to the problem of determining the distribution of heights of the surface irregularities from reflectance measurements at normal incidence. It is shown that for many surfaces, designated as normal surfaces, this distribution may be readily determined. Simple models for both normal and abnormal surfaces are used to illustrate the behavior of the reflectance in both cases and the consequent precautions necessary to obtain accurate height distributions. The role of the phase change due to roughness in determining the height distribution is also discussed.

© 1963 Optical Society of America

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References

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  1. V. Twersky, J. Appl. Phys. 22, 825 (1951); J. Appl. Phys. 24, 659 (1953).
    [Crossref]
  2. M. A. Biot, J. Appl. Phys. 28, 1455 (1957); J. Appl. Phys. 29, 998 (1958).
    [Crossref]
  3. S. O. Rice, Commun. Pure Appl. Math. 4, 351 (1951).
    [Crossref]
  4. W. S. Ament, Proc. IRE 41, 142 (1953).
    [Crossref]
  5. M. Born and E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1959), Sec. 8.3.2.
  6. H. Davies, Proc. Inst. Elec. Engrs. (London) 101, 209 (1954).
  7. H. E. Bennett and J. O. Porteus, J. Opt. Soc. Am. 51, 123 (1961).
    [Crossref]
  8. H. E. Bennett, J. Opt. Soc. Am. 53, 1389 (1963).
    [Crossref]
  9. For a theory which includes multiple reflections see V. Twersky, J. Opt. Soc. Am. 52, 145 (1962).
    [Crossref] [PubMed]
  10. The vectors kd and ki are of magnitude 2π/λ, so that the magnitude of k for specularly reflected radiation at normal incidence is 4π/λ.
  11. It can be shown that at normal incidence (1/k)[k·n(r)/N·n(r)] differs from unity by a term of the order of α tanβ, where α is the semivertex angle of the cone of acceptance and tanβ is the maximum slope of the surface at r, i.e., β=cos−1N·n.
  12. Note that r still terminates in the actual surface ∑ and not in the plane of integration.
  13. In the strict sense the averages indicated here are to be taken over an ensemble of statistically identical surfaces. However, for practical applications where the scale of roughness along the surface is very small compared to the illuminated surface area, the ensemble average of a function defined on the surface will be assumed to be equal to the average of the function over the actual surface on which reflectance measurements are to be made.
  14. See, for example, A. M. Mood, Introduction to the Theory of Statistics (McGraw-Hill Book Company, Inc., New York, 1950), p. 75.
  15. With infinite limits under the assumption that correlation between z and z′ is effectively zero, i.e., Dj(s,z,z′)−D(z)D(z′)=0, except for values of s small compared to the xy dimensions of the surface. Edge effects are thus neglected.
  16. Actually, since 〈z〉=〈z′〉=0 as a result of our choice of origin in the MSL, A(s)=〈zz′〉 in the present case, so that the second term of Eq. (10) is zero. The following development proceeds more naturally if it is included, however.
  17. When F does not depend on s, its Fourier transform will be written simply Ff(t,t′).
  18. With a defined by Eq. (17) it can be seen from Eqs. (12) and (13) that the leading term in the expansion of the incoherent reflectance Ri/R0 at normal incidence in powers of 1/λ is 16π4σ2a2α2/λ4, independent of the functional form of A(s). Note, however, that specification of A(s) is necessary if the leading term is to be expressed in terms of m.
  19. As given by Eq. (15). In a rigorous mathematical sense the rms slope of this surface is not defined.
  20. Davies (Ref. 6) has demonstrated that, for some surfaces at least, the incoherent reflectance at very short wavelengths is confined to a cone whose semivertex angle is of the order of σ/a. It cannot be concluded that this is always the case since, as we have seen, these parameters do not necessarily determine the behavior of the incoherent reflectance at very short wavelengths. The present example, for instance, does not behave in accordance with Davies’ result.
  21. See, for example, H. Cramér, Mathematical Methods of Statistics (Princeton University Press, Princeton, New Jersey, 1946), Sec. 21.11.
  22. The repetition can be verified mathematically by noting that when σ1/λ is very small, Eqs. (29) and (30) reduce essentially to Eqs. (23) and (24) as applied to Elementary Surface 2 alone. However when (a2/λ)2 is sufficiently large, Eq. (29) and the first term of Eq. (30) combine to produce Eq. (23) as applied to Elementary Surface 1 alone, and the second term of Eq. (30) is just the incoherent reflectance from this surface as given by Eq. (24).
  23. See, for example, P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill Book Company, Inc., New York, 1953), p. 458.
  24. J. M. Bennett, J. Opt. Soc. Am. 52, 1314 (1962).

1963 (1)

1962 (2)

1961 (1)

1957 (1)

M. A. Biot, J. Appl. Phys. 28, 1455 (1957); J. Appl. Phys. 29, 998 (1958).
[Crossref]

1954 (1)

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

1953 (1)

W. S. Ament, Proc. IRE 41, 142 (1953).
[Crossref]

1951 (2)

V. Twersky, J. Appl. Phys. 22, 825 (1951); J. Appl. Phys. 24, 659 (1953).
[Crossref]

S. O. Rice, Commun. Pure Appl. Math. 4, 351 (1951).
[Crossref]

Ament, W. S.

W. S. Ament, Proc. IRE 41, 142 (1953).
[Crossref]

Bennett, H. E.

Bennett, J. M.

J. M. Bennett, J. Opt. Soc. Am. 52, 1314 (1962).

Biot, M. A.

M. A. Biot, J. Appl. Phys. 28, 1455 (1957); J. Appl. Phys. 29, 998 (1958).
[Crossref]

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1959), Sec. 8.3.2.

Cramér, H.

See, for example, H. Cramér, Mathematical Methods of Statistics (Princeton University Press, Princeton, New Jersey, 1946), Sec. 21.11.

Davies, H.

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

Feshbach, H.

See, for example, P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill Book Company, Inc., New York, 1953), p. 458.

Mood, A. M.

See, for example, A. M. Mood, Introduction to the Theory of Statistics (McGraw-Hill Book Company, Inc., New York, 1950), p. 75.

Morse, P. M.

See, for example, P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill Book Company, Inc., New York, 1953), p. 458.

Porteus, J. O.

Rice, S. O.

S. O. Rice, Commun. Pure Appl. Math. 4, 351 (1951).
[Crossref]

Twersky, V.

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1959), Sec. 8.3.2.

Commun. Pure Appl. Math. (1)

S. O. Rice, Commun. Pure Appl. Math. 4, 351 (1951).
[Crossref]

J. Appl. Phys. (2)

V. Twersky, J. Appl. Phys. 22, 825 (1951); J. Appl. Phys. 24, 659 (1953).
[Crossref]

M. A. Biot, J. Appl. Phys. 28, 1455 (1957); J. Appl. Phys. 29, 998 (1958).
[Crossref]

J. Opt. Soc. Am. (4)

Proc. Inst. Elec. Engrs. (London) (1)

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

Proc. IRE (1)

W. S. Ament, Proc. IRE 41, 142 (1953).
[Crossref]

Other (15)

M. Born and E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1959), Sec. 8.3.2.

The vectors kd and ki are of magnitude 2π/λ, so that the magnitude of k for specularly reflected radiation at normal incidence is 4π/λ.

It can be shown that at normal incidence (1/k)[k·n(r)/N·n(r)] differs from unity by a term of the order of α tanβ, where α is the semivertex angle of the cone of acceptance and tanβ is the maximum slope of the surface at r, i.e., β=cos−1N·n.

Note that r still terminates in the actual surface ∑ and not in the plane of integration.

In the strict sense the averages indicated here are to be taken over an ensemble of statistically identical surfaces. However, for practical applications where the scale of roughness along the surface is very small compared to the illuminated surface area, the ensemble average of a function defined on the surface will be assumed to be equal to the average of the function over the actual surface on which reflectance measurements are to be made.

See, for example, A. M. Mood, Introduction to the Theory of Statistics (McGraw-Hill Book Company, Inc., New York, 1950), p. 75.

With infinite limits under the assumption that correlation between z and z′ is effectively zero, i.e., Dj(s,z,z′)−D(z)D(z′)=0, except for values of s small compared to the xy dimensions of the surface. Edge effects are thus neglected.

Actually, since 〈z〉=〈z′〉=0 as a result of our choice of origin in the MSL, A(s)=〈zz′〉 in the present case, so that the second term of Eq. (10) is zero. The following development proceeds more naturally if it is included, however.

When F does not depend on s, its Fourier transform will be written simply Ff(t,t′).

With a defined by Eq. (17) it can be seen from Eqs. (12) and (13) that the leading term in the expansion of the incoherent reflectance Ri/R0 at normal incidence in powers of 1/λ is 16π4σ2a2α2/λ4, independent of the functional form of A(s). Note, however, that specification of A(s) is necessary if the leading term is to be expressed in terms of m.

As given by Eq. (15). In a rigorous mathematical sense the rms slope of this surface is not defined.

Davies (Ref. 6) has demonstrated that, for some surfaces at least, the incoherent reflectance at very short wavelengths is confined to a cone whose semivertex angle is of the order of σ/a. It cannot be concluded that this is always the case since, as we have seen, these parameters do not necessarily determine the behavior of the incoherent reflectance at very short wavelengths. The present example, for instance, does not behave in accordance with Davies’ result.

See, for example, H. Cramér, Mathematical Methods of Statistics (Princeton University Press, Princeton, New Jersey, 1946), Sec. 21.11.

The repetition can be verified mathematically by noting that when σ1/λ is very small, Eqs. (29) and (30) reduce essentially to Eqs. (23) and (24) as applied to Elementary Surface 2 alone. However when (a2/λ)2 is sufficiently large, Eq. (29) and the first term of Eq. (30) combine to produce Eq. (23) as applied to Elementary Surface 1 alone, and the second term of Eq. (30) is just the incoherent reflectance from this surface as given by Eq. (24).

See, for example, P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill Book Company, Inc., New York, 1953), p. 458.

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Figures (9)

Fig. 1
Fig. 1

Vectors used in the theory, showing their relationship to the rough surface ∑ and the mean surface level (MSL). The separation of ∑ and the MSL is exaggerated for clarity.

Fig. 2
Fig. 2

Reflectance at normal incidence from a normal surface. The coherent and incoherent reflectances are plotted from Eqs. (23) and (24), respectively, which are derived from a model surface consisting of facets of random size, shape, and height parallel to the MSL. The approximate incoherent reflectance for this model as given by Eq. (25) is also shown. This curve clearly demonstrates the inability of the autocovariance function to represent the reflectance at short wavelengths.

Fig. 3
Fig. 3

Reflectance at normal incidence from an abnormal surface having two different scales of roughness. The coherent and incoherent reflectances are plotted from Eqs. (29) and (30), respectively. The curves were obtained by superimposing on the roughness model of Fig. 2 the same type of roughness but with different statistical parameters.

Fig. 4
Fig. 4

Reflectance for the model surface of Fig. 3 with the instrumental acceptance solid angle reduced by a factor of 103.

Fig. 5
Fig. 5

Density function of surface heights for normal and abnormal surfaces. The density function labeled ζ=0 produces the normal reflectance behavior of Fig. 2. Surfaces having density functions corresponding to the curves labeled ζ=η and ζ=2η show abnormal reflectance behavior, as indicated in Figs. 6 and 7.

Fig. 6
Fig. 6

Reflectance at normal incidence from an abnormal surface having the density function ζ=η in Fig. 5. Except for the difference in density functions, the roughness model is the same as in Fig. 2.

Fig. 7
Fig. 7

Reflectance at normal incidence from an abnormal surface having the density function ζ=2η in Fig. 5. Except for the difference in density functions, the roughness model is the same as in Fig. 2.

Fig. 8
Fig. 8

Nominal density function of surface heights Dnom as given by Eq. (35) and the actual density function D for a hypothetical surface having an asymmetric density function. The coherent reflectance from such a surface is shown in Fig. 9.

Fig. 9
Fig. 9

Coherent reflectance from a surface having the density functions in Fig. 8. Note that this reflectance is produced by either of the density functions in Fig. 8 and that there is no way to distinguish between them by reflectance measurements alone.

Equations (39)

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E = C Σ k · n ( r ) e - i k · r d S .
C = ( u ρ / 4 π l ) exp ( 2 π i l / λ ) ,
E = C k · n ( r ) N · n ( r ) e - i k · r d x d y ,
E 2 = C 2 k 2 exp [ - i k · ( r - r ) ] d x d y d x d y .
E 2 = C 2 k 2 D j ( s , z , z ) × exp [ - i k · ( r - r ) ] d x d y d z d x d y d z ,
D ( z ) = - + D j ( s , z , z ) d z .
E 2 = C 2 k 2 D ( z ) D ( z ) × exp [ - i k · ( r - r ) ] d x d y d z d x d y d z + C 2 k 2 [ D j ( s , z , z ) - D ( z ) D ( z ) ] × exp [ - i k · ( r - r ) ] d x d y d z d x d y d z .
R c = R 0 cos 2 ψ | - + D ( z ) exp [ - 4 π i ( z / λ ) cos ψ ] d z | 2 ,
R i = R 0 λ 2 [ D j ( s , z , z ) - D ( z ) D ( z ) ] × exp [ - 4 π i λ ( z - z ) - i k d · s ] d z d z d s d Ω ,
A ( s ) = - + - + z z D j ( s , z , z ) d z d z - [ - + z D ( z ) d z ] 2 .
F f ( s , t , t ) = - + - + F ( s , z , z ) × exp [ - 2 π i ( t z + t z ) ] d z d z .
R i = R 0 λ 2 [ D j - D D ] f ( s , 2 λ , - 2 λ ) × exp [ i k d · s ] d s d Ω ,
A ( s ) = - ( 2 π ) - 2 ( 2 / t t ) [ D j - D D ] f ( s , 0 , 0 ) .
σ 2 = A ( 0 ) .
m 2 = lim s 0 [ ( z - z ) / s ] 2 ,
m 2 = lim s 0 2 [ σ 2 - A ( s ) ] s 2 = - d 2 A ( s ) d s 2 | s = 0 .
a 2 = 2 σ 2 0 s A ( s ) d s .
D j ( s , z , z ) = D ( z ) δ ( z - z ) p ( s ) + D ( z ) D ( z ) [ 1 - p ( s ) ] ,
D j f ( s , t , t ) = [ D D ] f ( t + t , 0 ) p ( s ) + [ 1 - p ( s ) ] [ D D ] f ( t , t ) ,
A ( s ) = σ 2 p ( s ) .
D ( z ) = [ 1 / ( 2 π σ 2 ) 1 2 ] exp ( - z 2 / 2 σ 2 ) ,
p ( s ) = exp ( - s 2 / a 2 ) ,
R c = R 0 exp [ - ( 4 π σ / λ ) 2 ] ,
R i = R 0 { 1 - exp [ - ( 4 π σ / λ ) 2 ] } × { 1 - exp [ - ( π a α / λ ) 2 ] } ,
R i R 0 ( 16 π 2 σ 2 / λ 2 ) { 1 - exp [ - ( π a α / λ ) 2 ] } ,
D j f ( s , t , t ) = [ D j 1 f ( s , t , t ) ] [ D j 2 f ( s , t , t ) ] .
A ( s ) = σ 1 2 p 1 ( s ) + σ 2 2 p 2 ( s ) .
R c / R 0 = D j f ( s , 2 / λ , 0 ) 2 ,
R c / R 0 = exp [ - ( 4 π / λ ) 2 ( σ 1 2 + σ 2 2 ) ] .
R i R 0 = { exp [ - ( 4 π σ 1 λ ) 2 ] - exp [ - ( 4 π λ ) 2 ( σ 1 2 + σ 2 2 ) ] } × { 1 - exp [ - ( π a 2 α λ ) 2 ] } + { 1 - exp [ - ( 4 π σ 1 λ ) 2 ] } × { 1 - exp [ - ( π a 1 α λ ) 2 ] } .
D ( z ) = 1 2 ( 2 π η 2 ) 1 2 { exp [ - ( z - ζ ) 2 2 η 2 ] + exp [ - ( z + ζ ) 2 2 η 2 ] } .
σ 2 = η 2 + ζ 2 .
R c / R 0 = exp [ - ( 4 π η / λ ) 2 ] cos 2 ( 4 π ζ / λ )
R i R 0 = { 1 - exp [ - ( 4 π η λ ) 2 ] cos 2 ( 4 π ζ λ ) } × { 1 - exp [ - ( π a α λ ) 2 ] } .
D nom ( z ) = 2 cos ψ 0 ( R c R 0 ) 1 2 ( t ) cos ( 2 π z t ) d t ,
D e ( z ) = 1 2 [ D ( z ) + D ( - z ) ]
D 0 ( z ) = 1 2 [ D ( z ) - D ( - z ) ] .
D nom ( z ) = - + { [ - + D e ( z ) cos ( 2 π z t ) d z ] 2 + [ - + D 0 ( z ) sin ( 2 π z t ) d z ] 2 } 1 2 cos ( 2 π z t ) d t .
D 0 ( z ) = 0.