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It is interesting to observe that Eq. (1) illustrates the inadequacy of the Rayleigh quarter-wave criterion (see Refs. 17 and 18) as applied to the reflection of light from a rough surface. This criterion may be stated thus: The image formed by an optical element does not fall seriously short of that from a perfect optical element if the difference between the longest and shortest optical paths leading to the focus does not exceed λ/4. Applied to light reflected at normal incidence from a surface having irregularities of peak-to-valley height h, the criterion becomes h≤λ/8 (see Ref. 9). It should be pointed out, however, that h is not equal to the rms roughness σ, so that the Rayleigh quarter-wave criterion does not imply that σ≤λ/8 at normal incidence. In the case of a surface whose irregularities have a random height distribution, h must be taken to be a statistical measure of the peak-to-valley height. The relationship between h and σ then depends on the distribution and shape of the surface irregularities. If a value h=2√2σ is assumed, it is seen from Eq. (1) that the relative specular reflective secular reflectance when the quarter-wave criterion is met is 0.73, which is considerably less than unity. For reflectance measurements of the highest accuracy, a roughness correction Will be necessary unless R_{s}/R_{0}≥0.999. At normal incidence then σ must be of the order of λ/400.

It is interesting to observe that Eq. (1) illustrates the inadequacy of the Rayleigh quarter-wave criterion (see Refs. 17 and 18) as applied to the reflection of light from a rough surface. This criterion may be stated thus: The image formed by an optical element does not fall seriously short of that from a perfect optical element if the difference between the longest and shortest optical paths leading to the focus does not exceed λ/4. Applied to light reflected at normal incidence from a surface having irregularities of peak-to-valley height h, the criterion becomes h≤λ/8 (see Ref. 9). It should be pointed out, however, that h is not equal to the rms roughness σ, so that the Rayleigh quarter-wave criterion does not imply that σ≤λ8 at normal incidence. In the case of a surface whose irregularities have a random height distribution, h must be taken to be a statistical measure of the peak-to-valley height. The relationship between h and σ then depends on the distribution and shape of the surface irregularities. If a value h=2√2σ is assumed, it is seen from Eq. (1) that the relative specular reflective secular reflectance when the quarter-wave criterion is met is 0.73, which is considerably less than unity. For reflectance measurements of the highest accuracy, a roughness correction Will be necessary unless R_{s}/R_{0}≥0.999. At normal incidence then σ must be of the order of λ/400.

A. E. Conrady, Applied Optics and Optical Design (Dover Publications, Inc., New York, 1957), p. 136.

See Ref. 8, pp. 576–587.

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W. F. Koehler, J. Opt. Soc. Am. 43, 743 (1953).

The coherent reflectance function at nonnormal incidence of a surface having a Gaussian height distribution can be shown to be strictly Gaussian (see Refs. 11 and 14) if it is assumed that the surface is perfectly conducting so that the intrinsic reflectance of the material of which the surface is composed is unity. For real materials the intrinsic reflectance is less than unity and polarization effects become important at nonnormal incidence. However, if the rough surface is coated with a material such as aluminum, which is highly reflecting, the coherent reflectance function at nonnormal incidence will be Gaussian to a good approximation.

R. W. Wood, Physical Optics (The Macmillan Company, New York, 1934), 3rd ed., pp. 40–41.

J. Strong, Concepts of Classical Optics (W. H. Freeman and Company, San Francisco, 1958), p. 279.