Abstract

A mathematical method of analysis of gratings having circularly concentric grooves is presented. The main emphasis is not on the ultimate calculation of the diffracted or scattered fields but rather on reducing the problem to a tractable form with clear physical interpretation, to which the point-matching method can be applied. This is done by starting with the Rayleigh–Fano equation for scattering from rough surfaces. For circular geometry, the problem initially appears intractable. However, by a series of Laplace and inverse Laplace transforms, the problem is reduced to a tractable form.

© 1983 Optical Society of America

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References

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  1. J. Dyson, “Circular and spiral diffraction gratings,” Proc. R. Soc. London Ser. A 248, 93–106 (1958).
    [Crossref]
  2. 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); R. Petit, “Electromagnetisme-diffraction d’une onde plane monochromatique par un réseau périodique infiniment conducteur,” C. R. Acad. Sci. 257, 2018–2021 (1963); F. Toigo, A. Marvin, V. Celli, and N. R. Hill, “Optical properties of rough surfaces: general theory and the small roughness limit,” Phys. Rev. B 15, 5618–5626 (1977).
    [Crossref]
  3. J. M. Elson, “High efficiency diffraction grating theory,” Proc. Soc. Photo-Opt. Instrum. Eng. 240, 128–131 (1981).
  4. J. Vlieger and D. Bedeaux, “A phenomenological theory of light scattering by surfaces,” Physica 85A, 389–398 (1976).
  5. F. Toigo, A. Marvin, V. Celli, and N. R. Hill, “Optical properties of rough surfaces: general theory and the small roughness limit,” Phys. Rev. B 15, 5618–5626 (1977).
    [Crossref]

1981 (1)

J. M. Elson, “High efficiency diffraction grating theory,” Proc. Soc. Photo-Opt. Instrum. Eng. 240, 128–131 (1981).

1977 (1)

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

1976 (1)

J. Vlieger and D. Bedeaux, “A phenomenological theory of light scattering by surfaces,” Physica 85A, 389–398 (1976).

1958 (1)

J. Dyson, “Circular and spiral diffraction gratings,” Proc. R. Soc. London Ser. A 248, 93–106 (1958).
[Crossref]

1941 (1)

Bedeaux, D.

J. Vlieger and D. Bedeaux, “A phenomenological theory of light scattering by surfaces,” Physica 85A, 389–398 (1976).

Celli, V.

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

Dyson, J.

J. Dyson, “Circular and spiral diffraction gratings,” Proc. R. Soc. London Ser. A 248, 93–106 (1958).
[Crossref]

Elson, J. M.

J. M. Elson, “High efficiency diffraction grating theory,” Proc. Soc. Photo-Opt. Instrum. Eng. 240, 128–131 (1981).

Fano, U.

Hill, N. R.

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

Marvin, A.

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

Toigo, F.

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

Vlieger, J.

J. Vlieger and D. Bedeaux, “A phenomenological theory of light scattering by surfaces,” Physica 85A, 389–398 (1976).

J. Opt. Soc. Am. (1)

Phys. Rev. B (1)

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

Physica (1)

J. Vlieger and D. Bedeaux, “A phenomenological theory of light scattering by surfaces,” Physica 85A, 389–398 (1976).

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

J. Dyson, “Circular and spiral diffraction gratings,” Proc. R. Soc. London Ser. A 248, 93–106 (1958).
[Crossref]

Proc. Soc. Photo-Opt. Instrum. Eng. (1)

J. M. Elson, “High efficiency diffraction grating theory,” Proc. Soc. Photo-Opt. Instrum. Eng. 240, 128–131 (1981).

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Figures (2)

Fig. 1
Fig. 1

Schematic showing nomenclature for the circular grating. The angles θ and ϕ are the polar and azimuthal angles associated with the diffracted beams (there is a continuum of energy diffracted in the azimuthal direction). The angle ϕ is measured from the x ˆ direction. The dielectric constants above and below the grating are + and , respectively.

Fig. 2
Fig. 2

Schematic showing the constraint |kk′| = nd imposed by the circular grating on the diffracted field. The spatial wave number of the grating profile is d, and n is a positive integer or zero. The vectors k and k′ are wave vectors parallel to the mean grating plane that are associated with the diffracted fields. Thus the above constraint plays a role in describing the coupling between different diffracted orders.

Equations (35)

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d 2 k N ( k ) f ( k , k ) ψ ( k - k , q + - q - ) = g ( k ) ψ ( - k , - ( ω c ) - q - ) ,
E ( ρ , z ) = ( ω / c ) - 1 d 2 k N ( k ) × exp [ i ( k · ρ + q + z ) ] ( k ˆ q + - z ˆ k ) / + .
ψ ( K , Q ) = d 2 ρ exp ( i K · ρ ) exp [ i Q ζ ( ρ ) ] ,
ψ p ( K , Q ) = 2 π δ ( K y ) - d x e i K x x exp ( i Q h cos α x ) ,
δ ( y ) = 1 2 π - d x exp ( i x y ) .
exp ( i Q h cos α x ) = n = - i n J n ( Q h ) exp ( i n α x ) .
ψ p ( K , Q ) = ( 2 π ) 2 δ ( K y ) n = - i n J n ( Q h ) δ ( K x + n α ) .
n = - i n J n [ ( q + - q ) h ] N ( k ) f ( k , k ) = m = - g ( k ) ( - i ) m J m [ ( ω c + q - ) h ] × δ ( k x - m α ) δ ( k y ) ,
ψ c ( K , Q ) = d 2 ρ exp ( i K · ρ ) exp ( i Q h cos α ρ ) ,
ψ c ( K , Q ) = 2 π 0 d ρ ρ J 0 ( K ρ ) exp ( i Q h cos α ρ ) .
exp ( i Q h cos α ρ ) = J 0 ( Q h ) + 2 n = 1 i n J n ( Q h ) cos ( n α ρ ) .
ψ c ( K , Q ) = ( 2 π ) 2 J 0 ( Q h ) δ ( K x ) δ ( K y ) + 4 π n = 1 i n J n ( Q h ) 0 d ρ ρ J 0 ( K ρ ) cos ( n α ρ ) .
I n = 0 d ρ ρ J 0 ( K ρ ) cos ( n α ρ ) .
I n = d d ( n α ) 0 d ρ J 0 ( K ρ ) sin ( n α ρ )
= d d ( n α ) H [ ( n α ) - K ] [ ( n α ) 2 - K 2 ] 1 / 2 ,
I n ( K , α ) = δ ( K - n α ) [ ( n α ) 2 - K 2 ] 1 / 2 - n α H ( n α - K ) [ ( n α ) 2 - K 2 ] 3 / 2 .
N ( k ) f ( k , k ) J 0 [ ( q + - q - ) h ] + 2 n = 1 i n d 2 k × J n [ ( q + - q - ) h ] N ( k ) f ( k , k ) I n ( k - k , α ) = g ( 0 ) J 0 [ ( ω c + q - ) h ] δ ( k x ) δ ( k y ) + 2 g ( k ) m = 1 ( - i ) m J m [ ( ω c + q - ) h ] I m ( k , α ) .
L { F ( x ) } = 0 d x e - s x F ( x ) ,
I n ( u , v ) = 1 n 2 { δ [ ( u ) 1 / 2 - ( v ) 1 / 2 ] ( v - u ) 1 / 2 - ( v ) 1 / 2 H [ ( v ) 1 / 2 - ( u ) 1 / 2 ] ( v - u ) 3 / 2 } .
L [ I n ( u , v ) ( v ) 1 / 2 ] = 1 n 2 e - v s d v { δ [ ( u ) 1 / 2 - ( v ) 1 / 2 ] ( v ) 1 / 2 ( v - u ) 1 / 2 - H [ ( v ) 1 / 2 - ( u ) 1 / 2 ] ( v - u ) 3 / 2 } .
δ [ ( u ) 1 / 2 - ( v ) 1 / 2 ] ( v ) 1 / 2 = 2 δ ( u - v )
L [ I n ( u , v ) ( v ) 1 / 2 ] = 2 ( s ) 1 / 2 n 2 ( π ) 1 / 2 exp ( - u s ) .
L - 1 { L [ I n ( u , v ) ( v ) 1 / 2 ] ( s ) 1 / 2 } = 2 ( π ) 1 / 2 n 2 L - 1 [ exp ( - u s ) ] ,
L - 1 [ exp ( - u s ) ] = δ ( u - v ) .
N ( k ) f ( k , k ) J 0 [ ( q + - q - ) h ] + 2 n = 1 d 2 k J n × [ ( q + - q - ) h ] N ( k ) f ( k , k ) δ ( k - k - n α ) n α = g ( 0 ) J 0 [ ( ω c + q - ) h ] δ ( k x ) δ ( k y ) + 2 m = 1 ( - 1 ) m J m [ ( ω c + q - ) h ] g ( k ) δ ( k - m α ) m α .
e x + ( ρ , z ) = e x 0 ( ρ , z ) d 2 k e x + ( k ) exp [ i ( k · ρ + q + z ) ]
e x - ( ρ , z ) = d 2 k e x - ( k ) exp [ i ( k · ρ - q - z ) ]
e x + ( ρ , 0 ) - e x - ( ρ , 0 ) = i ω c m y 0 ( ρ ) - x p z 0 ( ρ ) ,
p x 0 ( ρ ) = - 0 ζ ( ρ ) [ + e x + ( ρ , z ) - - e x - ( ρ , z ) ] d z
m y 0 ( ρ ) = - 0 ζ ( ρ ) [ b y + ( ρ , z ) - b y - ( ρ , z ) ] d z .
d 2 k N + ( k ) [ q + q - cos ( ϕ - ϕ ) + k k ] ( q + - q - ) × ψ ( k - k , q + - q - ) = + [ ( q 0 q - cos ϕ - k 0 k ) cos σ ( + ) 1 / 2 ( q 0 + q - ) + ( ω / c ) q - sin ϕ sin σ ( q 0 + q - ) ] ψ ( k 0 - k , - q 0 - q - ) .
d 2 k M + ( k ) cos ( ϕ - ϕ ) ψ ( k - k , q + - q - ) q + - q - = ( cos ϕ sin σ - cos θ 0 sin ϕ cos σ ) q 0 + q + × ψ ( k 0 - k , - q 0 - q - ) ,
ψ ( K , Q ) = d 2 ρ exp ( i K · ρ ) exp [ i Q ζ ( ρ ) ] .
E + ( ρ , z ) = d 2 k exp [ i ( k · ρ + q + z ) ] N + ( k ) + ( ω / c ) [ k ˆ q + - z ˆ k ]
E + ( ρ , z ) = d 2 k exp [ k ( k · ρ + q + z ) ] ( k ˆ × z ˆ ) M + ( k )