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Journal of the Optical Society of America

Journal of the Optical Society of America

  • Vol. 31, Iss. 3 — Mar. 1, 1941
  • pp: 213–222

The Theory of Anomalous Diffraction Gratings and of Quasi-Stationary Waves on Metallic Surfaces (Sommerfeld’s Waves)

U. FANO  »View Author Affiliations

JOSA, Vol. 31, Issue 3, pp. 213-222 (1941)

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The theory of Wood’s anomalous diffraction gratings, which was developed some years ago, has been reexamined in order to visualize its physical meaning. Each wave diffracted by a grating is identified through the component of its “wave vector” tangential to the grating. Surface waves similar to those found in total internal reflection are included (§2). The amplitudes of these waves can be calculated by successive approximations (§3). One feature of the anomalies is connected with the infinite dispersion of spectra at grazing emergence (§4). Emphasis is put on the existence of polarized quasi-stationary waves which represent an energy current rolling along the surface of a metal (§5). These waves can be strongly excited on the surface of metallic gratings under critical conditions depending also on the profile of the grooves; secondary interference phenomena arise then in the observed spectra (§6). The connection of the quasi-stationary surface waves with the wireless ground waves is discussed (§7). A general formulation is introduced to discuss the significance of the approximation used (Appendix).

U. FANO, "The Theory of Anomalous Diffraction Gratings and of Quasi-Stationary Waves on Metallic Surfaces (Sommerfeld’s Waves)," J. Opt. Soc. Am. 31, 213-222 (1941)

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  1. R. W. Wood, Phil. Mag. 4, 393 (1902); 23, 310 (1912); Phys. Rev. 48, 928 (1935); J. Strong, Phys. Rev. 49, 291 (1936).
  2. Lord Rayleigh, Phil. Mag. 14, 60 (1907).
  3. U. Fano, Ann. d. Physik 32, 393 (1938).
  4. This phenomenon is closely analogous to the so-called "Dellen" in the diffraction of molecular rays (discovered by Stern and Frisch), whose explanation in connection with superficial adsorption phenomena was first given by Lennard-Jones and Devonshire (Nature 137, 969 (1936), Proc. Roy. Soc. A158, 253 (1937)) and then further developed (reference 3).
  5. Lord Rayleigh, Proc. Roy. Soc. A79, 399 (1907).
  6. Reference 2, Appendix 1. See also a paper by T. Westerdijk, Ann. d. Physik 36, 696 (1939).
  7. The zero-order approximation (ζ(x) = 0) determines Ap8+(0). and A0-(0). according to Fresnel's formulae.
  8. The balance of intensity is determined by interference affecting only the central image which, within this approximation, acts as an infinitely powerful bank on which the various diffracted waves draw independently of one another. Here, as in the case of the "Dellen" (reference 4), to consider that anomalous dark bands are due to a "lack" of radiation arising from strong absorption by other processes is equivalent to assuming that the approximation method diverges.
  9. The equation determining the proper value of the momentum is: [Equation], if the magnetic vector is parallel to the surface, and: [Equation], if the electric vector is parallel to the surface.
  10. The apparent spectral width of the bands might not be due entirely to absorption of superficial waves within the metal. Second-order diffraction, i.e., the interaction with other diffracted waves, is also effective as an absorption (see Appendix).
  11. Lord Rayleigh, Phil. Mag. 14, 60 (1907).
  12. See discussion byR. W. Wood, Phys. Rev. 48, 928 (1935).
  13. U. Fano, Phys. Rev. 50, 573 (1936).
  14. H. Weyl, Ann. d. Physik 60, 481 (1919).
  15. See reference 14, p. 404. When the magnetic vector is parallel to the grating, it is: [Equation].
  16. [Equation].
  17. Cn′ n, is determined by an algebraic formula in the case of a finite system of linear equations.
  18. Rayleigh's method involves expansion of exponentials in powers of the depth of the grooves ζ(x), or better, of its ratio to the transversal wavelength.
  19. See, e.g., the article by A. Sommerfeld in Ph. Frank-R. von Mises, Die Differentialgleichungen der Mathematischen Physik (Vieweg, Braunschweig, 1935), second edition, Vol. II, p. 876.
  20. One can see that the additional damping to be added to γ is approximately proportional to the square of the depth of the grooves.
  21. The influence of the grating on the energy traveling within a superficial wave is not too close, however, because the wave stretches up to an appreciable distance from the grating into vacuum.
  22. When it is possible to bring the matrix Cn′ n into a diagonal form, represented by a set of values γm, that is, to solve the homogeneous system: [Equation], the matrix Cn′ n, is given by the formula: [Equation]. Wood's anomalies are represented by anomalously small values of one of the quantities γm. Within Rayleigh's first approximation the γm, are factors of the 2 X 2 determinants of diagonal squares of Cn′ n. It is easy to discuss the order of magnitude of the quantities xn(m). at different stages of Rayleigh's approximation.

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