Abstract

A variational theory is presented that describes the diffraction properties of perfectly conducting gratings in TE polarization. By using this formalism we solve the problem of the energy distribution of light diffracted by gratings with periodic errors.

© 1982 Optical Society of America

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References

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  1. R. Petit, ed., Electromagnetic Theory of Gratings (Springer-Verlag, Heidelberg, 1980).
    [Crossref]
  2. D. S. Jones, “A critique of the variational method in scattering problems,” IEEE Trans. Antennas Propag. AP-4, 297–301 (1956).
  3. J. J. Bowman, T. B. A. Senior, and P. L. E. Uslenghi, eds., Electromagnetic and Acoustic Scattering by Simple Shapes (North-Holland, Amsterdam, 1969), pp. 37–38.
  4. D. Maystre, “A new general integral theory for dielectric coated gratings,” J. Opt. Soc. Am. 68, 490–495 (1978).
    [Crossref]
  5. W. C. Meecham, “Variational method for the calculation of the distribution of energy reflected from a periodic surface,” J. Appl. Phys. 27, 361–367 (1956).
    [Crossref]
  6. A. Roger and D. Maystre, “The perfectly conducting grating from the point of view of inverse diffraction,” Opt. Acta 26, 447–460 (1979).
    [Crossref]
  7. J. Van Bladel, Electromagnetic Fields (McGraw-Hill, New York, 1964), p. 389.
  8. J. Pavageau and J. Bousquet, “Diffraction par un réseau conducteur nouvelle méthode de résolution,” Opt. Acta 17, 469–478 (1970).
    [Crossref]
  9. H. A. Rowland, “Gratings in theory and practice,” Phil. Mag. 35, 397 (1893).
  10. A. Maréchal, “La diffusion résiduelle des surfaces polies et des réseaux,” Opt. Acta 5, 70–74 (1958).
    [Crossref]
  11. G. W. Stroke, Encyclopedia of Physics XXIX (Springer-Verlag, Berlin, 1967).
  12. A. Lohman, “Contrast transfer in the grating spectrograph,” Opt. Acta 6, 175–185 (1959).
    [Crossref]
  13. N. A. Finkelstein, C. H. Brumley, and R. J. Meltzer, “The reduction of ghosts in diffraction grating spectra,” J. Opt. Soc. Am. 42, 121–126 (1952).
    [Crossref]
  14. E. Ingelstam and E. Djurle, “The study of diffraction grating characteristics by simplified phase contrast methods,” J. Opt. Soc. Am. 43, 572–580 (1953).
    [Crossref]
  15. D. Maystre, “Sur la diffraction d’une onde plane par un reseau metallique de conductivite finie,” Opt. Commun. 6, 50–54 (1972).
    [Crossref]

1979 (1)

A. Roger and D. Maystre, “The perfectly conducting grating from the point of view of inverse diffraction,” Opt. Acta 26, 447–460 (1979).
[Crossref]

1978 (1)

1972 (1)

D. Maystre, “Sur la diffraction d’une onde plane par un reseau metallique de conductivite finie,” Opt. Commun. 6, 50–54 (1972).
[Crossref]

1970 (1)

J. Pavageau and J. Bousquet, “Diffraction par un réseau conducteur nouvelle méthode de résolution,” Opt. Acta 17, 469–478 (1970).
[Crossref]

1959 (1)

A. Lohman, “Contrast transfer in the grating spectrograph,” Opt. Acta 6, 175–185 (1959).
[Crossref]

1958 (1)

A. Maréchal, “La diffusion résiduelle des surfaces polies et des réseaux,” Opt. Acta 5, 70–74 (1958).
[Crossref]

1956 (2)

W. C. Meecham, “Variational method for the calculation of the distribution of energy reflected from a periodic surface,” J. Appl. Phys. 27, 361–367 (1956).
[Crossref]

D. S. Jones, “A critique of the variational method in scattering problems,” IEEE Trans. Antennas Propag. AP-4, 297–301 (1956).

1953 (1)

1952 (1)

1893 (1)

H. A. Rowland, “Gratings in theory and practice,” Phil. Mag. 35, 397 (1893).

Bousquet, J.

J. Pavageau and J. Bousquet, “Diffraction par un réseau conducteur nouvelle méthode de résolution,” Opt. Acta 17, 469–478 (1970).
[Crossref]

Brumley, C. H.

Djurle, E.

Finkelstein, N. A.

Ingelstam, E.

Jones, D. S.

D. S. Jones, “A critique of the variational method in scattering problems,” IEEE Trans. Antennas Propag. AP-4, 297–301 (1956).

Lohman, A.

A. Lohman, “Contrast transfer in the grating spectrograph,” Opt. Acta 6, 175–185 (1959).
[Crossref]

Maréchal, A.

A. Maréchal, “La diffusion résiduelle des surfaces polies et des réseaux,” Opt. Acta 5, 70–74 (1958).
[Crossref]

Maystre, D.

A. Roger and D. Maystre, “The perfectly conducting grating from the point of view of inverse diffraction,” Opt. Acta 26, 447–460 (1979).
[Crossref]

D. Maystre, “A new general integral theory for dielectric coated gratings,” J. Opt. Soc. Am. 68, 490–495 (1978).
[Crossref]

D. Maystre, “Sur la diffraction d’une onde plane par un reseau metallique de conductivite finie,” Opt. Commun. 6, 50–54 (1972).
[Crossref]

Meecham, W. C.

W. C. Meecham, “Variational method for the calculation of the distribution of energy reflected from a periodic surface,” J. Appl. Phys. 27, 361–367 (1956).
[Crossref]

Meltzer, R. J.

Pavageau, J.

J. Pavageau and J. Bousquet, “Diffraction par un réseau conducteur nouvelle méthode de résolution,” Opt. Acta 17, 469–478 (1970).
[Crossref]

Roger, A.

A. Roger and D. Maystre, “The perfectly conducting grating from the point of view of inverse diffraction,” Opt. Acta 26, 447–460 (1979).
[Crossref]

Rowland, H. A.

H. A. Rowland, “Gratings in theory and practice,” Phil. Mag. 35, 397 (1893).

Stroke, G. W.

G. W. Stroke, Encyclopedia of Physics XXIX (Springer-Verlag, Berlin, 1967).

Van Bladel, J.

J. Van Bladel, Electromagnetic Fields (McGraw-Hill, New York, 1964), p. 389.

IEEE Trans. Antennas Propag. (1)

D. S. Jones, “A critique of the variational method in scattering problems,” IEEE Trans. Antennas Propag. AP-4, 297–301 (1956).

J. Appl. Phys. (1)

W. C. Meecham, “Variational method for the calculation of the distribution of energy reflected from a periodic surface,” J. Appl. Phys. 27, 361–367 (1956).
[Crossref]

J. Opt. Soc. Am. (3)

Opt. Acta (4)

A. Maréchal, “La diffusion résiduelle des surfaces polies et des réseaux,” Opt. Acta 5, 70–74 (1958).
[Crossref]

A. Lohman, “Contrast transfer in the grating spectrograph,” Opt. Acta 6, 175–185 (1959).
[Crossref]

A. Roger and D. Maystre, “The perfectly conducting grating from the point of view of inverse diffraction,” Opt. Acta 26, 447–460 (1979).
[Crossref]

J. Pavageau and J. Bousquet, “Diffraction par un réseau conducteur nouvelle méthode de résolution,” Opt. Acta 17, 469–478 (1970).
[Crossref]

Opt. Commun. (1)

D. Maystre, “Sur la diffraction d’une onde plane par un reseau metallique de conductivite finie,” Opt. Commun. 6, 50–54 (1972).
[Crossref]

Phil. Mag. (1)

H. A. Rowland, “Gratings in theory and practice,” Phil. Mag. 35, 397 (1893).

Other (4)

J. Van Bladel, Electromagnetic Fields (McGraw-Hill, New York, 1964), p. 389.

R. Petit, ed., Electromagnetic Theory of Gratings (Springer-Verlag, Heidelberg, 1980).
[Crossref]

J. J. Bowman, T. B. A. Senior, and P. L. E. Uslenghi, eds., Electromagnetic and Acoustic Scattering by Simple Shapes (North-Holland, Amsterdam, 1969), pp. 37–38.

G. W. Stroke, Encyclopedia of Physics XXIX (Springer-Verlag, Berlin, 1967).

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Figures (4)

Fig. 1
Fig. 1

Presentation of the diffraction problem. The grating is perfectly conducting, σ = ∞. The electric vector is assumed to be parallel to the grooves (TE polarization).

Fig. 2
Fig. 2

(a) Grating with groove profile y = f(x) and period d; (b) grating with groove profile y = g(x) and period d′. g(x) differs from f(x) only in a small interval.

Fig. 3
Fig. 3

TE efficiency in the different orders as a function of the normalized amplitude difference. The unperturbed grating is sinusoidal and has a normalized amplitude h1/d = 0.1. Normal incidence. Cross, efficiency in the ±2 orders; filled circle, efficiency in the 0 order; open circle, efficiency in the ±1 order (ghosts).

Fig. 4
Fig. 4

(a) Situation described by Eq. (A4). From the solution of this equation we obtain the function ϕ(x, θ), related to the surface current; (b) situation described by Eq. (A3).

Tables (5)

Tables Icon

Table 1 TE Efficiency in the −1 Order Obtained with Different Functions φ and Different Variational Formulas for Sinusoidal Gratings with Different Normalized Depths

Tables Icon

Table 2 TE Efficiency in the −1 Order Obtained from Different Trial Functions ϕ(m) and Compared with the Values given by the Integral Formalisma

Tables Icon

Table 3 TE Efficiency in the Different Propagating Orders as Obtained from the Variational and Integral Formalisms, Respectivelya

Tables Icon

Table 4 TE Efficiency in the Different Propagating Orders as Obtained from the Variational and Integral Formalisms, Respectivelya

Tables Icon

Table 5 TE Efficiency in the Different Propagating Orders as Obtained from the Variational and Integral Formalisms, Respectivelya

Equations (62)

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u i ( x , y , t ) = exp i ( α 0 x - β 0 y ) exp ( - i ω t ) e z ,
B n = - ω μ 0 2 d 0 d 1 β n j s ( x ) exp { - i [ α n x + β n f ( x ) ] } × [ 1 + f ( x ) 2 ] 1 / 2 d x ,
α n = α 0 + n K ,             K = 2 π / d , n Z , β n = { ( k 2 - α n 2 ) 1 / 2 k 2 - α n 2 0 + i ( α n 2 - k 2 ) 1 / 2 k 2 - α n 2 < 0 .
E n = B n B n * cos θ n / cos θ ,
- i ω μ 0 0 d G 0 ( x , ξ , θ ) j s ( ξ ) [ 1 + f ( ξ ) 2 ] 1 / 2 d ξ = - exp { i [ α 0 x - β 0 f ( x ) ] } .
G 0 ( x , ξ , θ ) = 1 2 i d n = - + 1 β n exp { i [ α n ( x - ξ ) + β n f ( x ) - f ( ξ ) ] } .
B n = 1 2 i d β n u n , ϕ ,
G ϕ = - w ,
δ B n = 1 2 i d β n ( δ u n , ϕ + u n , δ ϕ ) .
G δ ϕ + δ G ϕ = - δ w
G * ϕ n = u n .
ϕ n ( x , θ ) = - ϕ * ( x , - θ n ) exp ( i n K x ) .
u n , δ ϕ = G * ϕ n , δ ϕ = ϕ n , G δ ϕ = ϕ n , - δ w - δ G ϕ = - ϕ n , δ w - ϕ n , δ G ϕ ,
δ B n = 1 2 i d β n ( δ u n , ϕ ) - ϕ n , δ w - ϕ n , δ G ϕ ) .
B n = 1 2 i d β n ( u n , ϕ - ϕ n , w - ϕ n , G ϕ ) ,
B n = 1 2 i d β n u n , ϕ = 1 2 i d β n × u n , ϕ ϕ n , w - exp ( i n K x ) ϕ * , G ϕ = 1 2 i d β n × u n , ϕ 2 - exp ( i n K x ) ϕ * , G ϕ .
B - 1 ( 0 ) = tan θ 2 i π u - 1 , ϕ ,
B - 1 ( 1 ) = 2 B - 1 ( 0 ) - tan θ 2 i π ϕ - 1 , G ϕ ,
B - 1 ( 2 ) = tan θ 2 i π u - 1 , ϕ 2 ϕ - 1 , G ϕ ,
ϕ mirror = - i K cot θ
ϕ Pavageau = - i K [ f ( x ) + cot θ ] exp [ - i K 2 cot θ f ( x ) ] .
E ghost = J 1 2 [ 2 π d ( n ± 1 M ) ] ,
E ghost π 2 2 d 2 ( n ± 1 M ) 2 ,
B n = 1 2 i d β n [ u n + δ u n , ϕ - ϕ n , w + δ w - ϕ n , ( G + δ G ) ϕ ] .
B n = 1 2 i d β n ( u n , ϕ - ϕ n , w - ϕ n , G ϕ ) .
δ B n = 1 2 i d β n ( δ u n , ϕ - ϕ n , δ w - ϕ n , δ G ϕ ) .
δ G ( x , ξ , θ ) = G g g ( x , ξ , θ ) - G f f ( x , ξ , θ ) ,
G g g ( x , ξ , θ ) = 1 2 i d m = - + 1 β m exp { i [ m K ( x - ξ ) + β m g ( x ) - g ( ξ ) ] } G f f ( x , ξ , θ ) = 1 2 i d m = - + 1 β m exp { i [ m K ( x - ξ ) + β m f ( x ) - f ( ξ ) ] } .
G f g ( x , ξ , θ ) = 1 2 i d - + 1 β m exp { i [ m K ( x - ξ ) + β m f ( x ) - g ( ξ ) ] } ,
I = Ψ n , δ G ϕ = 0 d 0 d [ G g g ( x , ξ , θ ) - G f f ( x , ξ , θ ) ] × ϕ ( ξ ) Ψ n * ( x ) d x d ξ .
J = 0 d ' 0 d ' G g g ( x , ξ , θ ) ϕ ( ξ ) Ψ n * ( x ) d x d ξ .
J 1 = 0 d 0 d G g g ( x , ξ , θ ) ϕ ( ξ ) Ψ n * ( x ) d x d ξ , J 2 = 0 d d d ' G g g ( x , ξ , θ ) ϕ ( ξ ) Ψ n * ( x ) d x d ξ , J 3 = d d ' 0 d G g g ( x , ξ , θ ) ϕ ( ξ ) Ψ n * ( x ) d x d ξ , J 4 = d d ' d d ' G g g ( x , ξ , θ ) ϕ ( ξ ) Ψ n * ( x ) d x d ξ .
J 1 + J 2 = 0 d ( 0 d G g g ϕ d ξ + d d G g g ϕ d ξ ) Ψ n * d x = 0 d [ 0 d ( G g g + G g f - G g f ) ϕ d ξ + d d G g f ϕ d ξ ] Ψ n * d x = 0 d 0 d ( G g g - G g f ) ϕ Ψ n * d x d ξ + 0 d 0 d G g f ϕ Ψ n * d x d ξ .
0 d G g f ϕ d ξ = 0 d G g f ϕ d ξ ,
J 1 + J 2 = 0 d [ 0 d ( G g g - G g f ) ϕ d ξ + 0 d G g f ϕ d ξ ] Ψ n * d x .
J 3 + J 4 = d d [ 0 d ( G g g - G g f ) ϕ d ξ + 0 d G g f ϕ d ξ ] Ψ n * d x = d d [ 0 d ( G f g - G f f ) ϕ d ξ + 0 d G f f ϕ d ξ ] Ψ n * d x .
J = 0 d ( 0 d G g g Ψ n * d x + d d G f g Ψ n * d x - 0 d G g f Ψ n * d x - d d G f f Ψ n * d x ) ϕ d ξ + 0 d × 0 d G g f Ψ n * ϕ d x d ξ + ( M - 1 ) 0 d 0 d G f f Ψ n * ϕ d x d ξ .
d d G f g Ψ n * d x = 0 d G f g Ψ n * d x - 0 d G f g Ψ n * d x = 0 d G f g Ψ n * d x - 0 d G f g Ψ n * d x ,
d d G f f Ψ n * d x = 0 d G f f Ψ n * d x - 0 d G f f Ψ n * .
J = 0 d 0 d ( G g g + G f f - G f g - G g f ) ϕ Ψ n * d x d ξ + 0 d 0 d [ G f g + G g f + ( M - 2 ) G f f ] ϕ Ψ n * d x d ξ ,
I = 0 d 0 d ( G g g + G f f - 2 G f g ) ϕ Ψ n * d x d ξ + 0 d 0 d ( 2 G f g - 2 G f f ) ϕ Ψ n * d x d ξ .
δ u n , ϕ = 0 d exp ( - i n K x ) × { exp [ - i β n g ( x ) ] - exp [ - i β n f ( x ) ] } ϕ ( x , θ ) d x , ϕ n , δ w = 0 d - ϕ ( x , - θ n ) exp ( - i n K x ) × { exp [ - i β n g ( x ) ] - exp [ - i β n f ( x ) ] } d x .
g ( x ) = { h 1 d sin K x 0 x < d h 2 d sin K x d x < 2 d
0 d G ( x , ξ , θ ) ϕ ( ξ ) d ξ = - i 4 - H 0 ( 1 ) ( k r ) ϕ ( ξ ) d ξ .
0 d G * ( ξ , x , θ ) ϕ n ( ξ ) d ξ = exp { i [ n K x + β n f ( x ) ] } ,
G * ϕ n = u n .
G ( ξ , x , θ ) = 1 2 i d n = - + 1 β n exp { i [ - n K ( x - ξ ) + β n f ( x ) - f ( ξ ) ] } = 1 2 i d Σ 1 β - n exp { i [ n K ( x - ξ ) + β - n f ( x ) - f ( ξ ) ] } ,
β - n = [ k 2 - ( - n K + k sin θ ) 2 ] 1 / 2 = [ k 2 - ( - n K - k sin θ ) 2 ] 1 / 2
G ( ξ , x , θ ) = G ( x , ξ , - θ ) .
0 d G ( x , ξ , - θ ) ϕ n * ( ξ ) d ξ = exp { - i [ n K x + β n f ( x ) ] } .
0 d G ( x , ξ , θ ) ϕ ( ξ ) d ξ = - exp [ - i β 0 f ( x ) ] .
G ( x , ξ , - θ ) exp i n K ( x - ξ ) = 1 2 i d m = - + 1 β m exp { i [ m K ( x - ξ ) + n K ( x - ξ ) + β m f ( x ) - f ( ξ ) ] } = ( m + n = p ) = 1 2 i d p 1 β p - n exp { i [ p K ( x - ξ ) + β p - n f ( x ) - f ( ξ ) ] } .
sin θ n = - ( k sin θ + n K ) ,
0 d G ( x , ξ , - θ n ) ϕ n * ( ξ ) exp ( i n K ξ ) d ξ = exp [ - i β n f ( x ) ] .
- ϕ n * ( x , θ ) exp i n K x = ϕ ( x , - θ n )
ϕ n ( x , θ ) = - ϕ * ( x , - θ n ) exp ( i n K x ) ,
0 d G g f ( x , ξ , θ ) ϕ ( ξ ) d ξ = 0 d G g f ( x , ξ , θ ) ϕ ( ξ ) d ξ .
0 d G g f ( x , ξ , θ ) ϕ ( ξ ) d ξ = p = 0 M - 1 0 d G g f ( x , ξ + p d , θ ) ϕ ( ξ ) d ξ .
p = 0 M - 1 G g f ( x , ξ + p d , θ ) = 1 2 i d m [ p = 0 M - 1 exp ( - i p m K M d ) ] × 1 β m exp { i m [ K M ( x - ξ ) + β m g ( x ) - f ( ξ ) ] } ,
p = 0 M - 1 exp ( - i p m K M d ) = 1 - exp ( - 2 i π m ) 1 - exp ( - 2 i π m M ) = { M if m = j M j Z 0 otherwise ,
p = 0 M - 1 G g f = 1 2 i d m = j M 1 β m exp { i [ m K M ( x - ξ ) + β m g ( x ) - f ( ξ ) ] } = 1 2 i d j 1 β j exp { i [ j K ( x - ξ ) + β j g ( x ) - f ( ξ ) ] } = G g f ( x , ξ , θ ) ,
0 d G g f ( x , ξ , θ ) Ψ n * ( x ) d x = 0 d G g f ( x , ξ , θ ) Ψ n * ( x ) d x .