Abstract

A method is presented for analyzing electromagnetic scattering from an echelette grating separating two contrasting homogeneous media and illuminated by a plane wave. The reduction of the general problem to a consideration of the fields over a suitably selected period, referred to as the unit cell, is facilitated by the Floquet theorem. The solution, in the p-polarization case (electric field parallel to the grooves), uses sets of spatially periodic and properly modulated fictitious electric-current strips to simulate the field scattered by the grating boundary surface and the field penetrated through the surface. In the s-polarization case (magnetic field parallel to the grooves), which is not examined in this paper, sets of fictitious magnetic-current strips, instead of electric ones, should be used. The fields radiated by the current strips are expressed in terms of Floquet modes and are adjusted to fit the continuity conditions for the tangential components of the electric and magnetic fields at a finite number of points on the grating surface within the unit cell. Special attention is given to the behavior of the fields at the corners. The procedure is simple to perform and is applicable to gratings of arbitrary cross section. Perfectly conducting gratings are treated as reduced cases of the general procedure. Results are given and compared with existing data. The efficiency of the suggested method is demonstrated.

© 1989 Optical Society of America

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References

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  1. R. Petit, “Diffraction of a plane wave by metal gratings,” Rev. Opt. Theor. Instrum. 45, 249–276 (1966) (in French).
  2. H. A. Kalhor, A. R. Neureuther, “Numerical method for analysis if diffraction gratings,”J. Opt. Soc. Am. 61, 43–48 (1971).
    [Crossref]
  3. S. Jovicevic, S. Sesnic, “Diffraction of parallel- and perpendicular-polarized wave from an echelette grating,” Appl. Opt. 1, 421–429 (1962).
  4. D. Marcuse, “Exact theory of TE-wave scattering from blazed dielectric gratings,” Bell Syst. Tech. J. 55, 1295–1317 (1976).
  5. M. G. Moharam, T. K. Gaylord, “Diffraction analysis of dielectric surface-relief gratings,”J. Opt. Soc. Am. 72, 1385–1392 (1982).
    [Crossref]
  6. J. Strong, Concepts of Classical Optics (Freeman, San Francisco, Calif., 1958).
  7. A. Boag, Y. Leviatan, A. Boag, “Analysis of two-dimensional electromagnetic scattering from non-planar periodic surfaces using a strip current mode,” IEEE Trans. Antennas Propag. (to be published).
  8. Y. Leviatan, A. Boag, A. Boag, “Analysis of TE scattering from dielectric cylinders using a multifilament magnetic current mode,”IEEE Trans. Antennas Propag. AP-36, 1026–1031 (1988).
    [Crossref]
  9. Y. Leviatan, A. Boag, “Analysis of electromagnetic scattering from dielectric cylinders using a multifilament current mode,”IEEE Trans. Antennas Propag. AP-35, 1119–1127 (1987).
    [Crossref]

1988 (1)

Y. Leviatan, A. Boag, A. Boag, “Analysis of TE scattering from dielectric cylinders using a multifilament magnetic current mode,”IEEE Trans. Antennas Propag. AP-36, 1026–1031 (1988).
[Crossref]

1987 (1)

Y. Leviatan, A. Boag, “Analysis of electromagnetic scattering from dielectric cylinders using a multifilament current mode,”IEEE Trans. Antennas Propag. AP-35, 1119–1127 (1987).
[Crossref]

1982 (1)

1976 (1)

D. Marcuse, “Exact theory of TE-wave scattering from blazed dielectric gratings,” Bell Syst. Tech. J. 55, 1295–1317 (1976).

1971 (1)

1966 (1)

R. Petit, “Diffraction of a plane wave by metal gratings,” Rev. Opt. Theor. Instrum. 45, 249–276 (1966) (in French).

1962 (1)

Boag, A.

Y. Leviatan, A. Boag, A. Boag, “Analysis of TE scattering from dielectric cylinders using a multifilament magnetic current mode,”IEEE Trans. Antennas Propag. AP-36, 1026–1031 (1988).
[Crossref]

Y. Leviatan, A. Boag, A. Boag, “Analysis of TE scattering from dielectric cylinders using a multifilament magnetic current mode,”IEEE Trans. Antennas Propag. AP-36, 1026–1031 (1988).
[Crossref]

Y. Leviatan, A. Boag, “Analysis of electromagnetic scattering from dielectric cylinders using a multifilament current mode,”IEEE Trans. Antennas Propag. AP-35, 1119–1127 (1987).
[Crossref]

A. Boag, Y. Leviatan, A. Boag, “Analysis of two-dimensional electromagnetic scattering from non-planar periodic surfaces using a strip current mode,” IEEE Trans. Antennas Propag. (to be published).

A. Boag, Y. Leviatan, A. Boag, “Analysis of two-dimensional electromagnetic scattering from non-planar periodic surfaces using a strip current mode,” IEEE Trans. Antennas Propag. (to be published).

Gaylord, T. K.

Jovicevic, S.

Kalhor, H. A.

Leviatan, Y.

Y. Leviatan, A. Boag, A. Boag, “Analysis of TE scattering from dielectric cylinders using a multifilament magnetic current mode,”IEEE Trans. Antennas Propag. AP-36, 1026–1031 (1988).
[Crossref]

Y. Leviatan, A. Boag, “Analysis of electromagnetic scattering from dielectric cylinders using a multifilament current mode,”IEEE Trans. Antennas Propag. AP-35, 1119–1127 (1987).
[Crossref]

A. Boag, Y. Leviatan, A. Boag, “Analysis of two-dimensional electromagnetic scattering from non-planar periodic surfaces using a strip current mode,” IEEE Trans. Antennas Propag. (to be published).

Marcuse, D.

D. Marcuse, “Exact theory of TE-wave scattering from blazed dielectric gratings,” Bell Syst. Tech. J. 55, 1295–1317 (1976).

Moharam, M. G.

Neureuther, A. R.

Petit, R.

R. Petit, “Diffraction of a plane wave by metal gratings,” Rev. Opt. Theor. Instrum. 45, 249–276 (1966) (in French).

Sesnic, S.

Strong, J.

J. Strong, Concepts of Classical Optics (Freeman, San Francisco, Calif., 1958).

Appl. Opt. (1)

Bell Syst. Tech. J. (1)

D. Marcuse, “Exact theory of TE-wave scattering from blazed dielectric gratings,” Bell Syst. Tech. J. 55, 1295–1317 (1976).

IEEE Trans. Antennas Propag. (2)

Y. Leviatan, A. Boag, A. Boag, “Analysis of TE scattering from dielectric cylinders using a multifilament magnetic current mode,”IEEE Trans. Antennas Propag. AP-36, 1026–1031 (1988).
[Crossref]

Y. Leviatan, A. Boag, “Analysis of electromagnetic scattering from dielectric cylinders using a multifilament current mode,”IEEE Trans. Antennas Propag. AP-35, 1119–1127 (1987).
[Crossref]

J. Opt. Soc. Am. (2)

Rev. Opt. Theor. Instrum. (1)

R. Petit, “Diffraction of a plane wave by metal gratings,” Rev. Opt. Theor. Instrum. 45, 249–276 (1966) (in French).

Other (2)

J. Strong, Concepts of Classical Optics (Freeman, San Francisco, Calif., 1958).

A. Boag, Y. Leviatan, A. Boag, “Analysis of two-dimensional electromagnetic scattering from non-planar periodic surfaces using a strip current mode,” IEEE Trans. Antennas Propag. (to be published).

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Figures (6)

Fig. 1
Fig. 1

General problem of scattering from an echelette grating.

Fig. 2
Fig. 2

Simulated equivalence for region I.

Fig. 3
Fig. 3

Simulated equivalence for region II.

Fig. 4
Fig. 4

Schematic location of the fictitious sources in the simulated equivalence for region I.

Fig. 5
Fig. 5

Grating efficiencies PnI/Pinc versus θinc for the case of a symmetric metallic grating of unit slope and period p = 1.25 μm at λ = 0.546 μm.

Fig. 6
Fig. 6

Plots of boundary-condition errors (a) ΔEbc and (b) ΔHbc versus x along a half-period for various numbers of current strips Na and Nb at normal incidence for the cage of a symmetric penetrable grating of unit slope and period p = 1.25 μm separating dielectric half-spaces of μII = μI = μ0 and II = 3I = 30 at λ = 0.546 μm.

Tables (2)

Tables Icon

Table 1 Comparison of Space-Harmonic Expansion Coefficients Obtained by Three Different Methodsa

Tables Icon

Table 2 Grating Efficiencies PnI/Pinc and PnII/Pinc and Errors for Various Angles of Incidencea

Equations (29)

Equations on this page are rendered with MathJax. Learn more.

E inc = u ^ y exp [ - j ( k x inc x + k z inc z ) ] ,
E r = u ^ y i = 1 N r E i y r ,
J i r = u ^ y I i r δ ( z - z i r ) exp [ - j k x inc ( x - x i r ) ] n = - f ( x - x i r - n p ) ,
E i y r = - k r η r I i r 2 n = - n = a n k z n r exp { - j [ k x n ( x - x i r ) + k z n r z - z i r ] } ,
k x n = k x inc + 2 π n p ,
k z n r = ( k r 2 - k x n 2 ) 1 / 2 ,
a n = 1 p - p / 2 p / 2 f ( x ) exp ( j 2 π n p x ) d x .
f ( x ) = 0.42 + 0.5 cos ( 2 π x s ) + 0.08 cos ( 4 π x s ) .
n ^ × ( E I - E II ) = - n ^ × E inc             on C ,
n ^ × ( H I - H II ) = - n ^ × H inc             on C ,
[ Z ] I = V ,
[ Z ] = [ [ Z e I ] [ Z e II ] [ Z h I ] [ Z h II ] ] ,
I = [ I I I II ] ,
V = [ V e V h ] .
I = [ Z ] - 1 V ,
[ Z e I ] I I = V e ,
z < z i r             for all i
z > z i r             for all i ,
E r = u ^ y n = - n = F n r exp [ - j ( k x n x k z n r z i r ) ] ,
F n r = - k r η r a n 2 k z n r i = 1 N r I i r exp [ j ( k x n x i r k z n r z i r ) ] .
Δ E b c = n ^ × ( E I + E inc - E II ) on C E inc max ,
Δ H b c = n ^ × ( H I + H inc - H II ) on C H inc max .
Δ P = | P inc - n P n I - n P n II | P inc ,
P inc = k z inc k I η I
P n r = k z n r k r η r F n r 2
k x inc = k 0 sin θ inc
k z inc = k 0 cos θ inc .
z c ( x ) = { x , 0 x p / 2 - x , - p / 2 < x < 0 .
z a r ( x ) = d a r + z c ( x ) ,

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