Abstract

In this paper, the Rayleigh hypothesis in the theory of reflection by a grating is investigated analytically. Conditions are derived under which the Rayleigh hypothesis is rigorously valid. A procedure is presented that enables the validity of the Rayleigh hypothesis to be checked for a grating whose profile can be described by an analytic function. As examples, we consider some grating profiles described by a finite Fourier series. Numerical results are then presented.

© 1979 Optical Society of America

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References

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  1. Lord Rayleigh and J. W. Strutt, Proc. Roy. Soc. Lond. A,  79, 399–416 (1907).
  2. B. A. Lippmann, J. Opt. Soc. Am.,  43, 408 (1953).
    [Crossref]
  3. R. Petit and M. Cadilhac, C. R. Acad. Sci. B,  262, 468–471 (1966).
  4. R. F. Millar, Proc. Camb. Philos. Soc,  65, 773–791 (1969).
    [Crossref]
  5. M. Nevière and M. Cadilhac, Opt. Commun.,  2, 235–238 (1970).
    [Crossref]
  6. R. F. Millar, Proc. Camb. Philos. Soc,  69, 175–188 (1971).
    [Crossref]
  7. It has been pointed out that the results of the invalidity of the Rayleigh hypothesis for these types of profile can be done by an elementary extension of the work of Petit and Cadilhac.3 We remark that the analyticity of the Rayleigh solution (and hence the validity) for some type of grating profile cannot be derived from their work.
  8. R. Petit, Revue d’Optique,  6, 249–276 (1966).
  9. The conception of an analytic function is related to complex variables. We define that a function of a real variable is analytic when it can be expanded in a convergent power series (Ref. 12, p. 83).
  10. We remark that only uniqueness is investigated; no attempt is made to prove an existence theorem.
  11. The subdomain B may not exist. In that case the Rayleigh hypothesis never holds.
  12. E. C. Titchmarsh, The theory of functions, 2nd ed., (Oxford University, New York, 1939).
  13. One exception has to be made. It is possible that the series (1) is a finite one. Then, the Rayleigh hypothesis is true. This can occur in the case of dealing with a Neumann boundary condition and a certain angle of incidence.For instance, we refer to the Maréchal and Stroke position of the triangular grating [A. Maréchal and G. W. Stroke, C. R. Acad Sci. (Paris),  249, 2042 (1959)].The same kind of exception also apply to gratings of rectangular profile.

1971 (1)

R. F. Millar, Proc. Camb. Philos. Soc,  69, 175–188 (1971).
[Crossref]

1970 (1)

M. Nevière and M. Cadilhac, Opt. Commun.,  2, 235–238 (1970).
[Crossref]

1969 (1)

R. F. Millar, Proc. Camb. Philos. Soc,  65, 773–791 (1969).
[Crossref]

1966 (2)

R. Petit and M. Cadilhac, C. R. Acad. Sci. B,  262, 468–471 (1966).

R. Petit, Revue d’Optique,  6, 249–276 (1966).

1959 (1)

One exception has to be made. It is possible that the series (1) is a finite one. Then, the Rayleigh hypothesis is true. This can occur in the case of dealing with a Neumann boundary condition and a certain angle of incidence.For instance, we refer to the Maréchal and Stroke position of the triangular grating [A. Maréchal and G. W. Stroke, C. R. Acad Sci. (Paris),  249, 2042 (1959)].The same kind of exception also apply to gratings of rectangular profile.

1953 (1)

1907 (1)

Lord Rayleigh and J. W. Strutt, Proc. Roy. Soc. Lond. A,  79, 399–416 (1907).

Cadilhac, M.

M. Nevière and M. Cadilhac, Opt. Commun.,  2, 235–238 (1970).
[Crossref]

R. Petit and M. Cadilhac, C. R. Acad. Sci. B,  262, 468–471 (1966).

Lippmann, B. A.

Maréchal, A.

One exception has to be made. It is possible that the series (1) is a finite one. Then, the Rayleigh hypothesis is true. This can occur in the case of dealing with a Neumann boundary condition and a certain angle of incidence.For instance, we refer to the Maréchal and Stroke position of the triangular grating [A. Maréchal and G. W. Stroke, C. R. Acad Sci. (Paris),  249, 2042 (1959)].The same kind of exception also apply to gratings of rectangular profile.

Millar, R. F.

R. F. Millar, Proc. Camb. Philos. Soc,  69, 175–188 (1971).
[Crossref]

R. F. Millar, Proc. Camb. Philos. Soc,  65, 773–791 (1969).
[Crossref]

Nevière, M.

M. Nevière and M. Cadilhac, Opt. Commun.,  2, 235–238 (1970).
[Crossref]

Petit, R.

R. Petit, Revue d’Optique,  6, 249–276 (1966).

R. Petit and M. Cadilhac, C. R. Acad. Sci. B,  262, 468–471 (1966).

Rayleigh, Lord

Lord Rayleigh and J. W. Strutt, Proc. Roy. Soc. Lond. A,  79, 399–416 (1907).

Stroke, G. W.

One exception has to be made. It is possible that the series (1) is a finite one. Then, the Rayleigh hypothesis is true. This can occur in the case of dealing with a Neumann boundary condition and a certain angle of incidence.For instance, we refer to the Maréchal and Stroke position of the triangular grating [A. Maréchal and G. W. Stroke, C. R. Acad Sci. (Paris),  249, 2042 (1959)].The same kind of exception also apply to gratings of rectangular profile.

Strutt, J. W.

Lord Rayleigh and J. W. Strutt, Proc. Roy. Soc. Lond. A,  79, 399–416 (1907).

Titchmarsh, E. C.

E. C. Titchmarsh, The theory of functions, 2nd ed., (Oxford University, New York, 1939).

C. R. Acad Sci. (Paris) (1)

One exception has to be made. It is possible that the series (1) is a finite one. Then, the Rayleigh hypothesis is true. This can occur in the case of dealing with a Neumann boundary condition and a certain angle of incidence.For instance, we refer to the Maréchal and Stroke position of the triangular grating [A. Maréchal and G. W. Stroke, C. R. Acad Sci. (Paris),  249, 2042 (1959)].The same kind of exception also apply to gratings of rectangular profile.

C. R. Acad. Sci. B (1)

R. Petit and M. Cadilhac, C. R. Acad. Sci. B,  262, 468–471 (1966).

J. Opt. Soc. Am. (1)

Opt. Commun. (1)

M. Nevière and M. Cadilhac, Opt. Commun.,  2, 235–238 (1970).
[Crossref]

Proc. Camb. Philos. Soc (2)

R. F. Millar, Proc. Camb. Philos. Soc,  69, 175–188 (1971).
[Crossref]

R. F. Millar, Proc. Camb. Philos. Soc,  65, 773–791 (1969).
[Crossref]

Proc. Roy. Soc. Lond. A (1)

Lord Rayleigh and J. W. Strutt, Proc. Roy. Soc. Lond. A,  79, 399–416 (1907).

Revue d’Optique (1)

R. Petit, Revue d’Optique,  6, 249–276 (1966).

Other (5)

The conception of an analytic function is related to complex variables. We define that a function of a real variable is analytic when it can be expanded in a convergent power series (Ref. 12, p. 83).

We remark that only uniqueness is investigated; no attempt is made to prove an existence theorem.

The subdomain B may not exist. In that case the Rayleigh hypothesis never holds.

E. C. Titchmarsh, The theory of functions, 2nd ed., (Oxford University, New York, 1939).

It has been pointed out that the results of the invalidity of the Rayleigh hypothesis for these types of profile can be done by an elementary extension of the work of Petit and Cadilhac.3 We remark that the analyticity of the Rayleigh solution (and hence the validity) for some type of grating profile cannot be derived from their work.

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Figures (4)

FIG. 1
FIG. 1

Grating configuration and incident wave. S1 denotes a single period of the domain zmax < z < < ∞; S2 denotes a single period of the domain zmin < z < zmax (valley of the groove).

FIG. 2
FIG. 2

Conformal transformations. sp is either a zero of dw1/ds or a singularity of either f(s) or g(s), or both.

FIG. 3
FIG. 3

hmax as a function of K when the grating profile is given by z = h / 2 h j = 1 K [ 4 / π 2 ( 2 j 1 ) 2 ] cos { 2 π ( 2 j 1 ) x / D } (= approximation to a triangular profile).

FIG. 4
FIG. 4

hmax as a function of K when the grating profile is given by z = h / 2 h j = 1 K [ 2 ( ) / π ( 2 j 1 ) ] cos { 2 π ( 2 j 1 ) x / D } (= approximation to a rectangular profile).

Equations (30)

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u r ( x , z ) = n = ρ n exp ( i α n x + i γ n z ) , z > z max ,
u R r ( x , z ) = n = R n exp { i α n x + i γ n z } , ( x , z ) ( S 1 + S 2 ) .
lim inf n | R n | 1 / n exp ( 2 π z min / D ) > 1 , lim inf n | R n | 1 / n exp ( 2 π z min / D ) > 1 ,
u i ( s ) = n = R n u n ( s ) , Im ( s ) = 0 ,
lim n | u n | 1 / n = | w 1 | , lim n | u n | 1 / n = | w 2 | 1 ,
lim sup n | R n | 1 / n | w 1 , max | < 1 , lim sup n | R n | 1 / n | w 2 , min | 1 < 1 ,
u i ( s ) = n = R n u n ( s ) , s B ,
υ 1 ( s ) = n = 0 R n w 1 n
υ 2 ( s ) = n = R n w 2 n
w 2 * ( s ) = [ w 1 ( s * ) ] 1 , s A ,
υ 1 ( w 1 ) = n = 0 R n w 1 n .
( d / d s ) [ i f ( s ) g ( s ) ] = 0 , Im ( s ) < 0 .
lim inf n | R n | 1 / n = | w 1 ( s p ) | = | exp { 2 π [ i f ( s p ) g ( s p ) ] / D } | .
lim inf n | R n | 1 / n = | w 1 ( s p * ) | 1 = | exp { 2 π [ i f ( s p ) g ( s p ) ] / D } | .
Re [ i f ( s p ) g ( s p ) + z min ] > 0 .
z = h ζ ( x ) ; z min = h ζ min , h > 0 ,
i h d ζ ( x ) / d x = 0 ,
Re [ i x p h ζ ( x p ) + h ζ min ] > 0 .
Re [ i x p h max ζ ( x p ) + h max ζ min ] = 0 .
d d h Re [ i x p h ζ ( x p ) + h ζ min ] = { Re [ ζ ( x p ) + ζ min ] , when x p is a singular point of ζ ( x ) , Re ( i d x p d h h d ζ ( x p ) d h d x p d h ζ ( x p ) + ζ min ) , when x p is a root of Eq . ( 16 ) .
d d h Re [ i x p h ζ ( x p ) + h ζ min ] = Re [ ζ ( x p ) + ζ min ] ,
Re [ i x p h ζ ( x p ) + h ζ min ] < 0 for h = h max + Δ ,
ζ ( x ) = ½ cos ( 2 π x / D )
ζ min = ½ .
ζ ( x ) = ½ j = 1 K 4 π 2 ( 2 j 1 ) 2 cos [ 2 π ( 2 j 1 ) x / D ]
ζ min = ½ j = 1 K 4 π 2 ( 2 j 1 ) 2 .
ζ ( x ) = ½ j = 1 K 2 ( ) j π ( 2 j 1 ) cos [ 2 π ( 2 j 1 ) x / D ]
ζ min = ½ + j = 1 K 2 ( ) j π ( 2 j 1 ) .
z = h / 2 h j = 1 K [ 4 / π 2 ( 2 j 1 ) 2 ] cos { 2 π ( 2 j 1 ) x / D }
z = h / 2 h j = 1 K [ 2 ( ) / π ( 2 j 1 ) ] cos { 2 π ( 2 j 1 ) x / D }