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.