Abstract

The foci or branch points are found for a sinusoidal boundary. It is shown how they determine the range of validity of Rayleigh’s hypothesis for diffraction gratings.

© 2000 Optical Society of America

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References

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  1. R. F. Millar, “On the Rayleigh assumption in scattering by a periodic surface,” Proc. Cambridge Philos. Soc. 65, 773–791 (1969).
    [CrossRef]
  2. R. F. Millar, “On the Rayleigh assumption in scattering by a periodic surface. II,” Proc. Cambridge Philos. Soc. 69, 217–225 (1971).
    [CrossRef]
  3. P. R. Garabedian, Partial Differential Equations (Wiley, New York, 1964), pp. 641–650.
  4. R. Petit, M. Cadilhac, “Sur la diffraction d’une onde plane par un réseau infiniment conducteur,” C. R. Acad. Sci. (Paris) Ser. A, B 262, 468–471 (1966).
  5. O. P. Bruno, F. Reitich, “Numerical solution of diffraction problems: a method of variation of boundaries,” J. Opt. Soc. Am. A 10, 1168–1175 (1993).
    [CrossRef]
  6. O. P. Bruno, F. Reitich, “Numerical solution of diffraction problems: a method of variation of boundaries. II. Finitely conducting gratings, Padé approximants, and singularities,” J. Opt. Soc. Am. A 10, 2307–2316 (1993).
    [CrossRef]

1993 (2)

1971 (1)

R. F. Millar, “On the Rayleigh assumption in scattering by a periodic surface. II,” Proc. Cambridge Philos. Soc. 69, 217–225 (1971).
[CrossRef]

1969 (1)

R. F. Millar, “On the Rayleigh assumption in scattering by a periodic surface,” Proc. Cambridge Philos. Soc. 65, 773–791 (1969).
[CrossRef]

1966 (1)

R. Petit, M. Cadilhac, “Sur la diffraction d’une onde plane par un réseau infiniment conducteur,” C. R. Acad. Sci. (Paris) Ser. A, B 262, 468–471 (1966).

Bruno, O. P.

Cadilhac, M.

R. Petit, M. Cadilhac, “Sur la diffraction d’une onde plane par un réseau infiniment conducteur,” C. R. Acad. Sci. (Paris) Ser. A, B 262, 468–471 (1966).

Garabedian, P. R.

P. R. Garabedian, Partial Differential Equations (Wiley, New York, 1964), pp. 641–650.

Millar, R. F.

R. F. Millar, “On the Rayleigh assumption in scattering by a periodic surface. II,” Proc. Cambridge Philos. Soc. 69, 217–225 (1971).
[CrossRef]

R. F. Millar, “On the Rayleigh assumption in scattering by a periodic surface,” Proc. Cambridge Philos. Soc. 65, 773–791 (1969).
[CrossRef]

Petit, R.

R. Petit, M. Cadilhac, “Sur la diffraction d’une onde plane par un réseau infiniment conducteur,” C. R. Acad. Sci. (Paris) Ser. A, B 262, 468–471 (1966).

Reitich, F.

C. R. Acad. Sci. (Paris) Ser. A, B (1)

R. Petit, M. Cadilhac, “Sur la diffraction d’une onde plane par un réseau infiniment conducteur,” C. R. Acad. Sci. (Paris) Ser. A, B 262, 468–471 (1966).

J. Opt. Soc. Am. A (2)

Proc. Cambridge Philos. Soc. (2)

R. F. Millar, “On the Rayleigh assumption in scattering by a periodic surface,” Proc. Cambridge Philos. Soc. 65, 773–791 (1969).
[CrossRef]

R. F. Millar, “On the Rayleigh assumption in scattering by a periodic surface. II,” Proc. Cambridge Philos. Soc. 69, 217–225 (1971).
[CrossRef]

Other (1)

P. R. Garabedian, Partial Differential Equations (Wiley, New York, 1964), pp. 641–650.

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Equations (14)

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z-g(z)2i=pz+g(z)2.
g(z)=1-ipz+g(z)21+ipz+g(z)2.
pz+g(z)2=i.
12[z+g(z)]=p-1(i).
12[z-g(z)]=ip[p-1(i)].
z=p-1(i)+ip[p-1(i)].
p-1(i)=sin-1(-i/a)=nπ-(-1)ni sinh-1(1/a),
p[p-1(i)]=a cos[sin-1(-i/a)]=(-1)na(1+a-2)1/2.
zn=nπ+(-1)ni[(a2+1)1/2-sinh-1(1/a)].
(a02+1)1/2-sinh-1(1/a0)=-a0.
u(x, y)=φ(x, y)onC.
u(x, y)=12[f(z)+f(z)¯]=12{f(z)+f[g(z)¯]¯}=φz+g(z)2,z-g(z)2i.
f(z)=2φz+g(z)2,z-g(z)2i-f[g(z)¯]¯.
f(z)=φx(1+g)+φy(1-g)-f[g(z)¯]¯g=φx+φy+g(z){φx-φy-f[g(z)¯]¯.

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