Abstract

The relative specular reflectance of various aluminized ground-glass surfaces has been measured at normal incidence. All of the samples tested had a Gaussian reflectance curve except for the most finely ground surfaces which could be prepared. Since the observed reflectance always became negligible at sufficiently short wavelengths, recent theoretical results could be used to determine the height distribution of the surface irregularities directly from the specular reflectance data. In most cases it was Gaussian. Since the specular reflectance of aluminized ground glass falls off very nearly exponentially with decreasing wavelength, the exponent being proportional to 1/λ<sup>2</sup>, scatter plates of this material make excellent reflection filters with very good rejection characteristics and little decrease in the energy passed in the desired wavelength region. The limiting wavelength for appreciable specular reflection at normal incidence is roughly ¼ of the average particle size of the grinding powder used, so that these scatter plates can be designed for use not only in the far infrared but also at shorter wavelengths extending nearly to the visible region of the spectrum.

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  1. Lord Rayleigh (J. W. Strutt), Nature 64, 385 (1901).
  2. J. W. French, Trans. Opt. Soc. (London) 18, 8 (1917).
  3. J. W. French, Trans. Faraday Soc. 18, 42 (1922).
  4. F. W. Preston, Trans. Opt. Soc. (London) 23, 141 (1922).
  5. A. J. Dalladay and F. Twyman, Trans. Opt. Soc. (London) 23, 165 (1922).
  6. A. J. Dalladay, Trans. Opt. Soc. (London) 23, 170 (1922).
  7. W. H. Willott, J. Soc. Glass Tech. 34, 77 (1950).
  8. F. Twyman, Prism and Lens Making (Hilger and Watts Ltd., London, 1952), pp. 49–65.
  9. F. Jentzsch, Z. Tech. Physik 7, 310 (1926).
  10. W. E. Knowles Middleton and G. Wyszecki, J. Opt. Soc. Am. 47, 1020 (1957).
  11. H. Hasunuma and J. Nara, J. Phys. Soc. Japan 11, 69 (1956).
  12. S. Tanaka, N. Noguchi, M. Watanabe, and T. Takahashi, J. Appl. Phys. (Japan) 31, 216 (1962).
  13. H. E. Bennett and J. O. Porteus, J. Opt. Soc. Am. 51, 123 (1961).
  14. H. Davies, Proc. Inst. Elec. Engrs. 101, 209 (1954).
  15. J. O. Porteus, J. Opt. Soc. Am. 53, 1394 (1963).
  16. 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 Rs/R0≥0.999. At normal incidence then σ must be of the order of λ/400.
  17. 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 Rs/R0≥0.999. At normal incidence then σ must be of the order of λ/400.
  18. A. E. Conrady, Applied Optics and Optical Design (Dover Publications, Inc., New York, 1957), p. 136.
  19. See Ref. 8, pp. 576–587.
  20. H. E. Bennett and W. F. Koehler, J. Opt. Soc. Am. 50, 1 (1960).
  21. W. F. Koehler, J. Opt. Soc. Am. 43, 743 (1953).
  22. 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.
  23. R. W. Wood, Physical Optics (The Macmillan Company, New York, 1934), 3rd ed., pp. 40–41.
  24. J. Strong, Concepts of Classical Optics (W. H. Freeman and Company, San Francisco, 1958), p. 279.

Bennett, H. E.

H. E. Bennett and J. O. Porteus, J. Opt. Soc. Am. 51, 123 (1961).

H. E. Bennett and W. F. Koehler, J. Opt. Soc. Am. 50, 1 (1960).

Conrady, A. E.

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 Rs/R0≥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.

Dalladay, A. J.

A. J. Dalladay and F. Twyman, Trans. Opt. Soc. (London) 23, 165 (1922).

A. J. Dalladay, Trans. Opt. Soc. (London) 23, 170 (1922).

Davies, H.

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

French, J. W.

J. W. French, Trans. Opt. Soc. (London) 18, 8 (1917).

J. W. French, Trans. Faraday Soc. 18, 42 (1922).

Hasunuma, H.

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

Jentzsch, F.

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

Koehler, W. F.

H. E. Bennett and W. F. Koehler, J. Opt. Soc. Am. 50, 1 (1960).

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

Middleton, W. E. Knowles

W. E. Knowles Middleton and G. Wyszecki, J. Opt. Soc. Am. 47, 1020 (1957).

Nara, J.

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

Noguchi, N.

S. Tanaka, N. Noguchi, M. Watanabe, and T. Takahashi, J. Appl. Phys. (Japan) 31, 216 (1962).

Porteus, J. O.

H. E. Bennett and J. O. Porteus, J. Opt. Soc. Am. 51, 123 (1961).

J. O. Porteus, J. Opt. Soc. Am. 53, 1394 (1963).

Preston, F. W.

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

Rayleigh, Lord

Lord Rayleigh (J. W. Strutt), Nature 64, 385 (1901).

Strong, J.

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

Takahashi, T.

S. Tanaka, N. Noguchi, M. Watanabe, and T. Takahashi, J. Appl. Phys. (Japan) 31, 216 (1962).

Tanaka, S.

S. Tanaka, N. Noguchi, M. Watanabe, and T. Takahashi, J. Appl. Phys. (Japan) 31, 216 (1962).

Twyman, F.

F. Twyman, Prism and Lens Making (Hilger and Watts Ltd., London, 1952), pp. 49–65.

A. J. Dalladay and F. Twyman, Trans. Opt. Soc. (London) 23, 165 (1922).

Watanabe, M.

S. Tanaka, N. Noguchi, M. Watanabe, and T. Takahashi, J. Appl. Phys. (Japan) 31, 216 (1962).

Willott, W. H.

W. H. Willott, J. Soc. Glass Tech. 34, 77 (1950).

Wood, R. W.

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

Wyszecki, G.

W. E. Knowles Middleton and G. Wyszecki, J. Opt. Soc. Am. 47, 1020 (1957).

Other (24)

Lord Rayleigh (J. W. Strutt), Nature 64, 385 (1901).

J. W. French, Trans. Opt. Soc. (London) 18, 8 (1917).

J. W. French, Trans. Faraday Soc. 18, 42 (1922).

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

A. J. Dalladay and F. Twyman, Trans. Opt. Soc. (London) 23, 165 (1922).

A. J. Dalladay, Trans. Opt. Soc. (London) 23, 170 (1922).

W. H. Willott, J. Soc. Glass Tech. 34, 77 (1950).

F. Twyman, Prism and Lens Making (Hilger and Watts Ltd., London, 1952), pp. 49–65.

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

W. E. Knowles Middleton and G. Wyszecki, J. Opt. Soc. Am. 47, 1020 (1957).

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

S. Tanaka, N. Noguchi, M. Watanabe, and T. Takahashi, J. Appl. Phys. (Japan) 31, 216 (1962).

H. E. Bennett and J. O. Porteus, J. Opt. Soc. Am. 51, 123 (1961).

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

J. O. Porteus, J. Opt. Soc. Am. 53, 1394 (1963).

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 Rs/R0≥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 Rs/R0≥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.

H. E. Bennett and W. F. Koehler, J. Opt. Soc. Am. 50, 1 (1960).

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.

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