Abstract

The steps necessary to produce the Rayleigh equation that is based on the Rayleigh hypothesis from the equation that is based on the Green’s formula are shown. First a definition is given for the scattering amplitude that is true not only in the far zone of diffraction but also near the scattering surface. With this definition the Rayleigh equation coincides with the rigorous equation for the surface secondary sources that is based on Green’s formula. The Rayleigh hypothesis is equivalent to substituting the far-zone expression of the scattering amplitude into this rigorous equation. In this case it turns out to be the equation not for the sources but directly for the scattering amplitude, which is the main advantage of this method. For comparing the Rayleigh equation with the initial rigorous equation, the Rayleigh equation is represented in terms of secondary sources. The kernel of this equation contains an integral that converges for positive and diverges for negative values of some parameter. It is shown that if we regularize this integral, defining it for the negative values of this parameter as an analytical continuation from the domain of positive values, this kernel becomes equal to the kernel of the initial rigorous equation. It follows that the formal perturbation series for the scattering amplitude obtained from the Rayleigh equation and from Green’s equation always coincide. This means that convergence of the perturbation series is a sufficient condition for the scattering amplitude obtained from the Rayleigh hypothesis to be true.

© 1995 Optical Society of America

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References

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  1. Rayleigh (J. W. Strutt), The Theory of Sound, 2nd ed. (Macmillan, New York, 1945), Vol. 2, pp. 89–96.
  2. Rayleigh (J. W. Strutt), “On the dynamical theory of gratings,” Proc. R. Soc. London Ser. A 79, 399–416 (1907).
    [CrossRef]
  3. L. N. Deryugin, “Equations for coefficients of wave reflections from a periodically uneven surface,” Dokl. Akad. Nauk SSSR 87, 913–916 (1952).
  4. B. A. Lippmann, “Note on the theory of gratings,” J. Opt. Soc. Am. 43, 408 (1953).
    [CrossRef]
  5. R. Petit, J. Cadilhac, “Sur la diffraction d’une onde plane par un réseau infiniment conducteur,” C. R. Acad. Sci. Paris Ser. B 262, 468–471 (1966).
  6. R. F. Millar, “On the Rayleigh assumption in scattering by a periodic surface, II,” Proc. Cambridge Philos. Soc. 69, 217–225 (1971).
    [CrossRef]
  7. R. F. Millar, “The Rayleigh hypothesis and a related least-squares solution to scattering problems for periodic surfaces and other scatterers,” Radio Sci. 8, 785–796 (1973).
    [CrossRef]
  8. P. M. Van den Berg, J. T. Fokkema, “The Rayleigh hypothesis in the theory of reflection by a grating,” J. Opt. Soc. Am. 69, 27–31 (1979).
    [CrossRef]
  9. J. A. DeSanto, “Scattering from a perfectly reflecting arbitrary periodic surface: an exact theory,” Radio Sci. 16, 1315–1326 (1981).
    [CrossRef]
  10. W. A. Schlup, “On the convergence of the Rayleigh ansats for hard-wall scattering on arbitrary periodic surface profiles,” J. Phys. A 17, 2607–2619 (1984).
    [CrossRef]
  11. D. Maystre, J. Cadilhac, “Singularities of the continuation of fields and validity of Rayleigh’s hypothesis,” J. Math. Phys. 26, 2201–2204 (1985).
    [CrossRef]
  12. F. Toigo, A. Marvin, V. Celli, N. R. Hill, “Optical properties of rough surfaces: general theory and the small roughness limit,” Phys. Rev. B 15, 5618–5626 (1977).
    [CrossRef]
  13. A. G. Voronovich, “Small-slope approximation in wave scattering by rough surfaces,” Sov. Phys. JETP 62, 65–70 (1985).
  14. D. R. Jackson, D. P. Winebrenner, A. Ishimaru, “Comparison of perturbation theories for rough-surface scattering,” J. Acoust. Soc. Am. 83, 961–969 (1988).
    [CrossRef]
  15. A. G. Voronovich, “About the Rayleigh hypothesis for the problem of sound scattering on rough free surfaces,” Dokl. Akad. Nauk SSSR Ser. Geofiz. 273, 85–89 (1983) (in Russian).
  16. V. I. Tatarskii, “The expansion of the solution of the rough-surface scattering problem in powers of quasi-slopes,” Waves Random Media 3, 127–146 (1993).
    [CrossRef]

1993 (1)

V. I. Tatarskii, “The expansion of the solution of the rough-surface scattering problem in powers of quasi-slopes,” Waves Random Media 3, 127–146 (1993).
[CrossRef]

1988 (1)

D. R. Jackson, D. P. Winebrenner, A. Ishimaru, “Comparison of perturbation theories for rough-surface scattering,” J. Acoust. Soc. Am. 83, 961–969 (1988).
[CrossRef]

1985 (2)

D. Maystre, J. Cadilhac, “Singularities of the continuation of fields and validity of Rayleigh’s hypothesis,” J. Math. Phys. 26, 2201–2204 (1985).
[CrossRef]

A. G. Voronovich, “Small-slope approximation in wave scattering by rough surfaces,” Sov. Phys. JETP 62, 65–70 (1985).

1984 (1)

W. A. Schlup, “On the convergence of the Rayleigh ansats for hard-wall scattering on arbitrary periodic surface profiles,” J. Phys. A 17, 2607–2619 (1984).
[CrossRef]

1983 (1)

A. G. Voronovich, “About the Rayleigh hypothesis for the problem of sound scattering on rough free surfaces,” Dokl. Akad. Nauk SSSR Ser. Geofiz. 273, 85–89 (1983) (in Russian).

1981 (1)

J. A. DeSanto, “Scattering from a perfectly reflecting arbitrary periodic surface: an exact theory,” Radio Sci. 16, 1315–1326 (1981).
[CrossRef]

1979 (1)

1977 (1)

F. Toigo, A. Marvin, V. Celli, N. R. Hill, “Optical properties of rough surfaces: general theory and the small roughness limit,” Phys. Rev. B 15, 5618–5626 (1977).
[CrossRef]

1973 (1)

R. F. Millar, “The Rayleigh hypothesis and a related least-squares solution to scattering problems for periodic surfaces and other scatterers,” Radio Sci. 8, 785–796 (1973).
[CrossRef]

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]

1966 (1)

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

1953 (1)

1952 (1)

L. N. Deryugin, “Equations for coefficients of wave reflections from a periodically uneven surface,” Dokl. Akad. Nauk SSSR 87, 913–916 (1952).

1907 (1)

Rayleigh (J. W. Strutt), “On the dynamical theory of gratings,” Proc. R. Soc. London Ser. A 79, 399–416 (1907).
[CrossRef]

Cadilhac, J.

D. Maystre, J. Cadilhac, “Singularities of the continuation of fields and validity of Rayleigh’s hypothesis,” J. Math. Phys. 26, 2201–2204 (1985).
[CrossRef]

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

Celli, V.

F. Toigo, A. Marvin, V. Celli, N. R. Hill, “Optical properties of rough surfaces: general theory and the small roughness limit,” Phys. Rev. B 15, 5618–5626 (1977).
[CrossRef]

Deryugin, L. N.

L. N. Deryugin, “Equations for coefficients of wave reflections from a periodically uneven surface,” Dokl. Akad. Nauk SSSR 87, 913–916 (1952).

DeSanto, J. A.

J. A. DeSanto, “Scattering from a perfectly reflecting arbitrary periodic surface: an exact theory,” Radio Sci. 16, 1315–1326 (1981).
[CrossRef]

Fokkema, J. T.

Hill, N. R.

F. Toigo, A. Marvin, V. Celli, N. R. Hill, “Optical properties of rough surfaces: general theory and the small roughness limit,” Phys. Rev. B 15, 5618–5626 (1977).
[CrossRef]

Ishimaru, A.

D. R. Jackson, D. P. Winebrenner, A. Ishimaru, “Comparison of perturbation theories for rough-surface scattering,” J. Acoust. Soc. Am. 83, 961–969 (1988).
[CrossRef]

Jackson, D. R.

D. R. Jackson, D. P. Winebrenner, A. Ishimaru, “Comparison of perturbation theories for rough-surface scattering,” J. Acoust. Soc. Am. 83, 961–969 (1988).
[CrossRef]

Lippmann, B. A.

Marvin, A.

F. Toigo, A. Marvin, V. Celli, N. R. Hill, “Optical properties of rough surfaces: general theory and the small roughness limit,” Phys. Rev. B 15, 5618–5626 (1977).
[CrossRef]

Maystre, D.

D. Maystre, J. Cadilhac, “Singularities of the continuation of fields and validity of Rayleigh’s hypothesis,” J. Math. Phys. 26, 2201–2204 (1985).
[CrossRef]

Millar, R. F.

R. F. Millar, “The Rayleigh hypothesis and a related least-squares solution to scattering problems for periodic surfaces and other scatterers,” Radio Sci. 8, 785–796 (1973).
[CrossRef]

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

Petit, R.

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

Rayleigh,

Rayleigh (J. W. Strutt), “On the dynamical theory of gratings,” Proc. R. Soc. London Ser. A 79, 399–416 (1907).
[CrossRef]

Rayleigh (J. W. Strutt), The Theory of Sound, 2nd ed. (Macmillan, New York, 1945), Vol. 2, pp. 89–96.

Schlup, W. A.

W. A. Schlup, “On the convergence of the Rayleigh ansats for hard-wall scattering on arbitrary periodic surface profiles,” J. Phys. A 17, 2607–2619 (1984).
[CrossRef]

Tatarskii, V. I.

V. I. Tatarskii, “The expansion of the solution of the rough-surface scattering problem in powers of quasi-slopes,” Waves Random Media 3, 127–146 (1993).
[CrossRef]

Toigo, F.

F. Toigo, A. Marvin, V. Celli, N. R. Hill, “Optical properties of rough surfaces: general theory and the small roughness limit,” Phys. Rev. B 15, 5618–5626 (1977).
[CrossRef]

Van den Berg, P. M.

Voronovich, A. G.

A. G. Voronovich, “Small-slope approximation in wave scattering by rough surfaces,” Sov. Phys. JETP 62, 65–70 (1985).

A. G. Voronovich, “About the Rayleigh hypothesis for the problem of sound scattering on rough free surfaces,” Dokl. Akad. Nauk SSSR Ser. Geofiz. 273, 85–89 (1983) (in Russian).

Winebrenner, D. P.

D. R. Jackson, D. P. Winebrenner, A. Ishimaru, “Comparison of perturbation theories for rough-surface scattering,” J. Acoust. Soc. Am. 83, 961–969 (1988).
[CrossRef]

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

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

Dokl. Akad. Nauk SSSR (1)

L. N. Deryugin, “Equations for coefficients of wave reflections from a periodically uneven surface,” Dokl. Akad. Nauk SSSR 87, 913–916 (1952).

Dokl. Akad. Nauk SSSR Ser. Geofiz. (1)

A. G. Voronovich, “About the Rayleigh hypothesis for the problem of sound scattering on rough free surfaces,” Dokl. Akad. Nauk SSSR Ser. Geofiz. 273, 85–89 (1983) (in Russian).

J. Acoust. Soc. Am. (1)

D. R. Jackson, D. P. Winebrenner, A. Ishimaru, “Comparison of perturbation theories for rough-surface scattering,” J. Acoust. Soc. Am. 83, 961–969 (1988).
[CrossRef]

J. Math. Phys. (1)

D. Maystre, J. Cadilhac, “Singularities of the continuation of fields and validity of Rayleigh’s hypothesis,” J. Math. Phys. 26, 2201–2204 (1985).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Phys. A (1)

W. A. Schlup, “On the convergence of the Rayleigh ansats for hard-wall scattering on arbitrary periodic surface profiles,” J. Phys. A 17, 2607–2619 (1984).
[CrossRef]

Phys. Rev. B (1)

F. Toigo, A. Marvin, V. Celli, N. R. Hill, “Optical properties of rough surfaces: general theory and the small roughness limit,” Phys. Rev. B 15, 5618–5626 (1977).
[CrossRef]

Proc. Cambridge Philos. Soc. (1)

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

Proc. R. Soc. London Ser. A (1)

Rayleigh (J. W. Strutt), “On the dynamical theory of gratings,” Proc. R. Soc. London Ser. A 79, 399–416 (1907).
[CrossRef]

Radio Sci. (2)

R. F. Millar, “The Rayleigh hypothesis and a related least-squares solution to scattering problems for periodic surfaces and other scatterers,” Radio Sci. 8, 785–796 (1973).
[CrossRef]

J. A. DeSanto, “Scattering from a perfectly reflecting arbitrary periodic surface: an exact theory,” Radio Sci. 16, 1315–1326 (1981).
[CrossRef]

Sov. Phys. JETP (1)

A. G. Voronovich, “Small-slope approximation in wave scattering by rough surfaces,” Sov. Phys. JETP 62, 65–70 (1985).

Waves Random Media (1)

V. I. Tatarskii, “The expansion of the solution of the rough-surface scattering problem in powers of quasi-slopes,” Waves Random Media 3, 127–146 (1993).
[CrossRef]

Other (1)

Rayleigh (J. W. Strutt), The Theory of Sound, 2nd ed. (Macmillan, New York, 1945), Vol. 2, pp. 89–96.

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Equations (44)

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A : Δ E + k 2 E = φ ( R ) , B : E [ R ( r ) ] = 0 , R ( r ) = r + e ζ ( r ) , C : Radiation conditions for z ζ ( r ) .
G ( r , z ; r , z ) = - exp ( i k R ) 4 π R , = exp [ i q ( r - r ) + i ν ( q ) z - z ] 8 i π 2 ν ( q ) d 2 q , R = [ ( r - r ) 2 + ( z - z ) 2 ] 1 / 2 , ν ( q ) = { k 2 - q 2 for k q i q 2 - k 2 for k q ,
A : Δ G ( R , R ) + k 2 G ( R , R ) = δ ( R - R ) , B : Radiation conditions .
E ( R ) = E 0 ( R ) + G [ R , R ( r ) ] f ( r ) d 2 r ,
E 0 ( R ) = G ( R , R ) φ ( R ) d 3 R
f ( r ) = 3 E ( R ) R = R ( r ) · n ( r ) { 1 + [ 2 ζ ( r ) ] 2 } 1 / 2
n ( r ) = e - 2 ζ ( r ) { 1 + [ 2 ζ ( r ) ] 2 } 1 / 2
E ( R ) = E 0 ( R ) + exp [ i q r + i ν ( q ) z ] 8 i π 2 ν ( q ) d 2 q d 2 r f ( r ) × exp { - i qr + i ν ( q ) [ z - ζ ( r ) - z ] } .
S ( q , z ) = 1 8 i π 2 ν ( q ) d 2 r f ( r ) × exp { - i qr + i ν ( q ) [ z - ζ ( r ) - z ] } ,
E ( R ) = E 0 ( R ) + exp [ i qr + i ν ( q ) z ] ν ( q ) S ( q , z ) d 2 q .
E 0 ( R ) = exp [ i q 0 r - i ν ( q 0 ) z ] ν ( q 0 ) = E 0 ( R , q 0 )
E 0 [ R ( r ) ] + G [ R ( r ) , R ( r ) ] f ( r ) d 2 r = 0.
E 0 [ R ( r ) ] + exp [ i qr + i ν ( q ) ζ ( r ) ] ν ( q ) S [ q , ζ ( r ) ] d 2 q = 0.
S [ q , ζ ( r ) ] = 1 8 i π 2 ν ( q ) d 2 r f ( r ) × exp { - i qr + i ν ( q ) [ ζ ( r ) - ζ ( r ) - ζ ( r ) ] } ,
S ˜ ( q , q 0 ) = 1 8 i π 2 d 2 r f ( r ) exp [ - i qr - i ν ( q ) ζ ( r ) ] ν ( q )             for z > max { ζ ( r ) } .
E ( R ) = E 0 ( R , q 0 ) + exp [ i qr + i ν ( q ) z ] ν ( q ) S ˜ ( q , q 0 ) d 2 q .
exp [ i qr + i ν ( q ) ζ ( r ) ] ν ( q ) S ˜ ( q , q 0 ) d 2 q + E 0 [ R ( r ) , q 0 ] = 0.
exp [ i ν ( q ) ζ ( r ) ] = exp [ - ν ζ ( r ) ] ,
K ( r , r ) = 1 8 i π 2 d 2 q ν ( q ) × exp [ i q ( r - r ) + i ν ( q ) ζ ( r ) - ζ ( r ) ] .
K R ( r , r ) = 1 8 i π 2 d 2 q ν ( q ) exp { i q ( r - r ) + i ν ( q ) [ ζ ( r ) - ζ ( r ) ] } .
E 0 [ R ( r ) ] + K R ( r , r ) f ( r ) d 2 r = 0.
K ( r , r ) = lim 0 K ( r , r ; ) , K ( r , r ; ) = 1 8 i π 2 d 2 q ν ( q ) exp ( - q 2 ) × exp [ i q ( r - r ) + i ν ( q ) ξ ] .
K R ( r , r ) = lim 0 K R ( r , r ; ) ,             for ξ > 0 , K R ( r , r ; ) = 1 8 i π 2 d 2 q ν ( q ) exp ( - q 2 ) × exp [ i q ( r - r ) + i ν ( q ) ξ ] .
K ˜ R ( r , r ; ) 1 8 i π 2 d 2 q ν ( q ) exp ( - q 2 ) × exp [ i q ( r - r ) + i ν ( q ) ξ ] .
lim 0 K ˜ R ( r , r ; ) f ( r ) d 2 r = K ( r , r ) f ( r ) d 2 r
K ˜ R ( r , r ) lim 0 K ˜ R ( r , r ; ) = K ( r , r ) .
K ˜ R ( r , r ; ) = n = 0 ( i ξ ) n n ! M n ( r - r ; ) , M n ( r - r ; ) = d 2 q ν n - 1 8 i π 2 exp [ - q 2 + i q ( r - r ) ] .
M 2 m + 1 ( s , ) = d 2 q ν 2 m 8 i π 2 exp ( - q 2 + i qs ) = d 2 q 8 i π 2 ( k 2 - q 2 ) m exp ( - q 2 + i qs ) = ( k 2 + s 2 ) m d 2 q 8 i π 2 exp ( - q 2 + i qs ) = 1 8 i π ( k 2 + s 2 ) m exp ( - s 2 / 4 ) .
G 2 m + 1 = lim 0 i 2 m + 1 8 i π ( 2 m + 1 ) ! d 2 r × f ( r ) ξ 2 m + 1 ( k 2 + s 2 ) m exp ( - s 2 4 ) .
G 2 m + 1 = lim 0 i 2 m + 1 8 i π ( 2 m + 1 ) ! d 2 s exp ( - s 2 4 ) × ( k 2 + s 2 ) m f ( r - s ) ξ ( r , s ) 2 m + 1 .
1 4 π exp [ - s 2 4 ] δ ( s )
ξ = ζ ( r ) - ζ ( r - s ) = s i ζ ( r ) r i - 1 2 ! s i s k 2 ζ ( r ) r i r k + , f ( r - s ) = f ( r ) - s i f ( r ) r i + .
G 2 m + 1 = lim 0 i 2 m + 1 8 i π ( 2 m + 1 ) ! d 2 s × exp ( - s 2 4 ) ( k 2 + s 2 ) m × [ A i j k ( r ) s i s j s k ( 2 m + 1 ) + O ( s 2 m + 2 ) ] .
( k 2 + s 2 ) m s i s j s k ( 2 m + 1 ) = B i s i + B i k s i s k + ,
C a b = 1 - d s 1 s 1 a exp ( - s 1 2 4 ) - d s 2 s 2 b exp ( - s 2 2 4 ) ,
- d s s 2 k exp ( - s 2 4 ) = D k k + 1 / 2 D k = 2 2 k + 1 Γ ( k + 1 / 2 ) .
G 2 m + 1 = 0.
M 2 n + 1 ( r - r ) = 0.
K ( r , r ; ) = n = 0 ( i ξ ) n n ! M n ( r - r ; ) .
K ( r , r ; ) = n = 0 ( i ξ ) 2 n ( 2 n ) ! M 2 n ( r - r ; )
K ˜ R ( r , r ; ) = n = 0 ( i ξ ) 2 n ( 2 n ) ! M 2 n ( r - r ; ) .
K ˜ R ( r , r ; ) = K ( r , r ; ) .
K ˜ R ( r , r ; ) f ( r ) d 2 r = K ( r , r ; ) f ( r ) d 2 r .
K ( r , r ) = - exp [ i k R ( r , r ) ] 4 π R ( r , r ) , R ( r , r ) = [ ( r - r ) 2 + ξ 2 ( r , r ) ] 1 / 2 .

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