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Journal of the Optical Society of America

Journal of the Optical Society of America

  • Vol. 38, Iss. 11 — Nov. 1, 1948
  • pp: 921–928

On the Theory of Imperfect Diffraction Gratings

U. FANO  »View Author Affiliations

JOSA, Vol. 38, Issue 11, pp. 921-928 (1948)

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The “dynamical” theory of gratings originally developed by Rayleigh and Voigt is applied to derive the intensity of the light diffracted in various directions by an imperfect grating of finite area. The problem is reduced to the numerical solution of a system of linear equations by an approximation method in which “ghosts” and high order spectra are treated as perturbations of the main spectra. Current elementary theories are then seen to yield merely order of magnitude estimates of the intensity of the ghosts caused by various grating imperfections.

U. FANO, "On the Theory of Imperfect Diffraction Gratings," J. Opt. Soc. Am. 38, 921-928 (1948)

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  1. E.g., C. M. Sparrow, Astrophys. J. 49, 65 (1919).
  2. E.g., H. A. Rowland, Physical Papers (Johns Hopkins University Press, Baltimore, 1902), p. 525.
  3. E.g., P. Frank and R. V. Mises, Differentialgleichungen der Physik (Friedrich Vieweg and Sohn, Braunschweig, 1935), Vol. 2, p. 853 ff.
  4. Lord Rayleigh, Proc. Roy. Soc. (A) 79, 399 (1907).
  5. W. Voigt, Göttinger Nachrichten 40 (1911).
  6. U. Fano, Ann. d. Physik 32, 393 (1938).
  7. U. Fano, J. Opt. Soc. Am. 31, 213 (1941).
  8. A detailed analytical treatment by this method has also been given by C. T. Tai, for the case in which the grating spacing is much shorter than the radiation wave-length, see Tech. Rep. No. 28, Cruft Laboratory, Harvard University, Cambridge, Massachusetts, Jan. 15, 1948.
  9. This set includes waves reflected and transmitted in "complex directions," i.e., waves confined to the proximity of the gratinig's surface. See reference 7, p. 214. For a derivation of the composition of the diffracted light, see reference 6, p. 402.
  10. 1O. See reference 7, Appendix.
  11. See reference 1, p. 83. See also reference 14.
  12. An index applied to v or w means that these functions have the values corresponding to the value of u with the same index.
  13. In fact, if no assumption were made at this point, the calculation could be carried further, but more laboriously, taking into account interaction terms between the light diffracted from the grating and that coming from the rest of the metal surface. In that case rq and tq, should still be considered as slowly varying functions of u.
  14. No specific reference is given by Sparrow (see references 1 and 11) for this type of result. A paper by Lord Rayleigh (Phil. Mag. 37, 498 (1919) and Coll; Works, Vol. 6, p. 627) is pertinent to this question. A proof of (12) is outlined here. The coefficient ap can be written in the form [Equation] If the ruling is perfect except that each groove may be displaced from its theoretical position by a small fraction δ of the spacing L/N and if p is a multiple of N, bpm=(ap)perf exp(-2πipôm/N), where (ap)perf is the value of ap in the absence of groove displacements. Assuming, finally, that all displacements δm are distributed as random errors with a r.m.s. value σ, the expected, or mean, value of ap is the product of (ap)perf and of the expected value of exp(-2πipδm/N), namely, ∫-∞exp[-2πipδm/N-δm2/2σ2]dδm/(2π)½σ=exp(-2π2σ2p2/N2).

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