Abstract

A novel solution is presented for the problem of three-dimensional electromagnetic scattering of a plane wave from a doubly periodic infinite array of perfectly conducting bodies. A set of fictitious spatially periodic and properly modulated patches of magnetic current is used to simulate the scattered field. These patch currents are of dual polarization and have complex amplitudes. The electromagnetic field radiated by each of the periodic patch currents is expressed as a double series of Floquet modes. The complex amplitudes of the fictitious patch currents are adjusted to render the tangential electric field zero at a selected set of points on the surface of any of the scatterers. The procedure is simple to implement and is applicable to arrays composed of smooth but otherwise arbitrary perfectly conducting scatterers. Results are given and compared with an analytic approximation.

© 1990 Optical Society of America

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References

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  1. J. P. Montgomery, “Scattering by an infinite periodic array of thin conductors on a dielectric sheet,” IEEE Trans. Antennas Propag. AP-23, 70–75 (1975).
    [CrossRef]
  2. C. C. Chen, “Transmission through a conducting screen perforated periodically with apertures,” IEEE Trans. Microwave Theory Tech. MTT-18, 627–632 (1970).
    [CrossRef]
  3. T. Cwik, R. Mittra, “The cascade connection of planar periodic surfaces and lossy dielectric layers to form an arbitrary periodic screen,” IEEE Trans. Antennas Propag. AP-35, 1397–1405 (1987).
    [CrossRef]
  4. C. C. Chen, “Transmission of microwave through perforated flat plates of finite thickness,” IEEE Trans. Microwave Theory Tech. MTT-21, 1–6 (1973).
    [CrossRef]
  5. G. H. Derrick, R. C. McPhedan, L. C. Botten, “Theory of crossed gratings,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, New York, 1980), Chap. 7.
  6. G. H. Derrick, R. C. McPhedan, D. Maystre, M. Nevière, “Crossed gratings: a theory and its applications,” Appl. Phys. 18, 39–52 (1979).
    [CrossRef]
  7. Y. Leviatan, A. Boag, A. Boag, “Generalized formulations for electromagnetic scattering from perfectly conducting and homogeneous material bodies—theory and numerical solution,” IEEE Trans. Antennas Propag. 36, 1722–1734 (1988).
    [CrossRef]
  8. A. Boag, Y. Leviatan, A. Boag, “Analysis of two-dimensional electromagnetic scattering from periodic grating of cylinders using a hybrid current model,” Radio Sci. 23, 612–624 (1988).
    [CrossRef]
  9. A. Boag, Y. Leviatan, A. Boag, “Analysis of two-dimensional electromagnetic scattering from non-planar periodic surfaces using a strip current model,” IEEE Trans. Antennas Propag. 37, 1437–1446 (1989).
    [CrossRef]
  10. A. Boag, Y. Leviatan, A. Boag, “Analysis of diffraction from echelette gratings, using a strip-current model,” J. Opt. Soc. Am. A 6, 543–549 (1989).
    [CrossRef]
  11. A. Hessel, “General characteristics of traveling-wave antennas,” in Antenna Theory, R. E. Collin, F. J. Zucker, eds. (McGraw-Hill, New York, 1969), Chap. 19.
  12. F. J. Harris, “On the use of windows for harmonic analysis with the discrete Fourier transform,” Proc. IEEE 66, 51–83 (1978).
    [CrossRef]
  13. V. Twersky, “Multiple scattering by double-periodic planar array of obstacles,” J. Math. Phys. 16, 633–643 (1975).
    [CrossRef]

1989 (2)

A. Boag, Y. Leviatan, A. Boag, “Analysis of two-dimensional electromagnetic scattering from non-planar periodic surfaces using a strip current model,” IEEE Trans. Antennas Propag. 37, 1437–1446 (1989).
[CrossRef]

A. Boag, Y. Leviatan, A. Boag, “Analysis of diffraction from echelette gratings, using a strip-current model,” J. Opt. Soc. Am. A 6, 543–549 (1989).
[CrossRef]

1988 (2)

Y. Leviatan, A. Boag, A. Boag, “Generalized formulations for electromagnetic scattering from perfectly conducting and homogeneous material bodies—theory and numerical solution,” IEEE Trans. Antennas Propag. 36, 1722–1734 (1988).
[CrossRef]

A. Boag, Y. Leviatan, A. Boag, “Analysis of two-dimensional electromagnetic scattering from periodic grating of cylinders using a hybrid current model,” Radio Sci. 23, 612–624 (1988).
[CrossRef]

1987 (1)

T. Cwik, R. Mittra, “The cascade connection of planar periodic surfaces and lossy dielectric layers to form an arbitrary periodic screen,” IEEE Trans. Antennas Propag. AP-35, 1397–1405 (1987).
[CrossRef]

1979 (1)

G. H. Derrick, R. C. McPhedan, D. Maystre, M. Nevière, “Crossed gratings: a theory and its applications,” Appl. Phys. 18, 39–52 (1979).
[CrossRef]

1978 (1)

F. J. Harris, “On the use of windows for harmonic analysis with the discrete Fourier transform,” Proc. IEEE 66, 51–83 (1978).
[CrossRef]

1975 (2)

V. Twersky, “Multiple scattering by double-periodic planar array of obstacles,” J. Math. Phys. 16, 633–643 (1975).
[CrossRef]

J. P. Montgomery, “Scattering by an infinite periodic array of thin conductors on a dielectric sheet,” IEEE Trans. Antennas Propag. AP-23, 70–75 (1975).
[CrossRef]

1973 (1)

C. C. Chen, “Transmission of microwave through perforated flat plates of finite thickness,” IEEE Trans. Microwave Theory Tech. MTT-21, 1–6 (1973).
[CrossRef]

1970 (1)

C. C. Chen, “Transmission through a conducting screen perforated periodically with apertures,” IEEE Trans. Microwave Theory Tech. MTT-18, 627–632 (1970).
[CrossRef]

Boag, A.

A. Boag, Y. Leviatan, A. Boag, “Analysis of two-dimensional electromagnetic scattering from non-planar periodic surfaces using a strip current model,” IEEE Trans. Antennas Propag. 37, 1437–1446 (1989).
[CrossRef]

A. Boag, Y. Leviatan, A. Boag, “Analysis of two-dimensional electromagnetic scattering from non-planar periodic surfaces using a strip current model,” IEEE Trans. Antennas Propag. 37, 1437–1446 (1989).
[CrossRef]

A. Boag, Y. Leviatan, A. Boag, “Analysis of diffraction from echelette gratings, using a strip-current model,” J. Opt. Soc. Am. A 6, 543–549 (1989).
[CrossRef]

A. Boag, Y. Leviatan, A. Boag, “Analysis of diffraction from echelette gratings, using a strip-current model,” J. Opt. Soc. Am. A 6, 543–549 (1989).
[CrossRef]

A. Boag, Y. Leviatan, A. Boag, “Analysis of two-dimensional electromagnetic scattering from periodic grating of cylinders using a hybrid current model,” Radio Sci. 23, 612–624 (1988).
[CrossRef]

Y. Leviatan, A. Boag, A. Boag, “Generalized formulations for electromagnetic scattering from perfectly conducting and homogeneous material bodies—theory and numerical solution,” IEEE Trans. Antennas Propag. 36, 1722–1734 (1988).
[CrossRef]

Y. Leviatan, A. Boag, A. Boag, “Generalized formulations for electromagnetic scattering from perfectly conducting and homogeneous material bodies—theory and numerical solution,” IEEE Trans. Antennas Propag. 36, 1722–1734 (1988).
[CrossRef]

A. Boag, Y. Leviatan, A. Boag, “Analysis of two-dimensional electromagnetic scattering from periodic grating of cylinders using a hybrid current model,” Radio Sci. 23, 612–624 (1988).
[CrossRef]

Botten, L. C.

G. H. Derrick, R. C. McPhedan, L. C. Botten, “Theory of crossed gratings,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, New York, 1980), Chap. 7.

Chen, C. C.

C. C. Chen, “Transmission of microwave through perforated flat plates of finite thickness,” IEEE Trans. Microwave Theory Tech. MTT-21, 1–6 (1973).
[CrossRef]

C. C. Chen, “Transmission through a conducting screen perforated periodically with apertures,” IEEE Trans. Microwave Theory Tech. MTT-18, 627–632 (1970).
[CrossRef]

Cwik, T.

T. Cwik, R. Mittra, “The cascade connection of planar periodic surfaces and lossy dielectric layers to form an arbitrary periodic screen,” IEEE Trans. Antennas Propag. AP-35, 1397–1405 (1987).
[CrossRef]

Derrick, G. H.

G. H. Derrick, R. C. McPhedan, D. Maystre, M. Nevière, “Crossed gratings: a theory and its applications,” Appl. Phys. 18, 39–52 (1979).
[CrossRef]

G. H. Derrick, R. C. McPhedan, L. C. Botten, “Theory of crossed gratings,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, New York, 1980), Chap. 7.

Harris, F. J.

F. J. Harris, “On the use of windows for harmonic analysis with the discrete Fourier transform,” Proc. IEEE 66, 51–83 (1978).
[CrossRef]

Hessel, A.

A. Hessel, “General characteristics of traveling-wave antennas,” in Antenna Theory, R. E. Collin, F. J. Zucker, eds. (McGraw-Hill, New York, 1969), Chap. 19.

Leviatan, Y.

A. Boag, Y. Leviatan, A. Boag, “Analysis of diffraction from echelette gratings, using a strip-current model,” J. Opt. Soc. Am. A 6, 543–549 (1989).
[CrossRef]

A. Boag, Y. Leviatan, A. Boag, “Analysis of two-dimensional electromagnetic scattering from non-planar periodic surfaces using a strip current model,” IEEE Trans. Antennas Propag. 37, 1437–1446 (1989).
[CrossRef]

Y. Leviatan, A. Boag, A. Boag, “Generalized formulations for electromagnetic scattering from perfectly conducting and homogeneous material bodies—theory and numerical solution,” IEEE Trans. Antennas Propag. 36, 1722–1734 (1988).
[CrossRef]

A. Boag, Y. Leviatan, A. Boag, “Analysis of two-dimensional electromagnetic scattering from periodic grating of cylinders using a hybrid current model,” Radio Sci. 23, 612–624 (1988).
[CrossRef]

Maystre, D.

G. H. Derrick, R. C. McPhedan, D. Maystre, M. Nevière, “Crossed gratings: a theory and its applications,” Appl. Phys. 18, 39–52 (1979).
[CrossRef]

McPhedan, R. C.

G. H. Derrick, R. C. McPhedan, D. Maystre, M. Nevière, “Crossed gratings: a theory and its applications,” Appl. Phys. 18, 39–52 (1979).
[CrossRef]

G. H. Derrick, R. C. McPhedan, L. C. Botten, “Theory of crossed gratings,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, New York, 1980), Chap. 7.

Mittra, R.

T. Cwik, R. Mittra, “The cascade connection of planar periodic surfaces and lossy dielectric layers to form an arbitrary periodic screen,” IEEE Trans. Antennas Propag. AP-35, 1397–1405 (1987).
[CrossRef]

Montgomery, J. P.

J. P. Montgomery, “Scattering by an infinite periodic array of thin conductors on a dielectric sheet,” IEEE Trans. Antennas Propag. AP-23, 70–75 (1975).
[CrossRef]

Nevière, M.

G. H. Derrick, R. C. McPhedan, D. Maystre, M. Nevière, “Crossed gratings: a theory and its applications,” Appl. Phys. 18, 39–52 (1979).
[CrossRef]

Twersky, V.

V. Twersky, “Multiple scattering by double-periodic planar array of obstacles,” J. Math. Phys. 16, 633–643 (1975).
[CrossRef]

Appl. Phys. (1)

G. H. Derrick, R. C. McPhedan, D. Maystre, M. Nevière, “Crossed gratings: a theory and its applications,” Appl. Phys. 18, 39–52 (1979).
[CrossRef]

IEEE Trans. Antennas Propag. (4)

Y. Leviatan, A. Boag, A. Boag, “Generalized formulations for electromagnetic scattering from perfectly conducting and homogeneous material bodies—theory and numerical solution,” IEEE Trans. Antennas Propag. 36, 1722–1734 (1988).
[CrossRef]

A. Boag, Y. Leviatan, A. Boag, “Analysis of two-dimensional electromagnetic scattering from non-planar periodic surfaces using a strip current model,” IEEE Trans. Antennas Propag. 37, 1437–1446 (1989).
[CrossRef]

J. P. Montgomery, “Scattering by an infinite periodic array of thin conductors on a dielectric sheet,” IEEE Trans. Antennas Propag. AP-23, 70–75 (1975).
[CrossRef]

T. Cwik, R. Mittra, “The cascade connection of planar periodic surfaces and lossy dielectric layers to form an arbitrary periodic screen,” IEEE Trans. Antennas Propag. AP-35, 1397–1405 (1987).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (2)

C. C. Chen, “Transmission of microwave through perforated flat plates of finite thickness,” IEEE Trans. Microwave Theory Tech. MTT-21, 1–6 (1973).
[CrossRef]

C. C. Chen, “Transmission through a conducting screen perforated periodically with apertures,” IEEE Trans. Microwave Theory Tech. MTT-18, 627–632 (1970).
[CrossRef]

J. Math. Phys. (1)

V. Twersky, “Multiple scattering by double-periodic planar array of obstacles,” J. Math. Phys. 16, 633–643 (1975).
[CrossRef]

J. Opt. Soc. Am. A (1)

Proc. IEEE (1)

F. J. Harris, “On the use of windows for harmonic analysis with the discrete Fourier transform,” Proc. IEEE 66, 51–83 (1978).
[CrossRef]

Radio Sci. (1)

A. Boag, Y. Leviatan, A. Boag, “Analysis of two-dimensional electromagnetic scattering from periodic grating of cylinders using a hybrid current model,” Radio Sci. 23, 612–624 (1988).
[CrossRef]

Other (2)

A. Hessel, “General characteristics of traveling-wave antennas,” in Antenna Theory, R. E. Collin, F. J. Zucker, eds. (McGraw-Hill, New York, 1969), Chap. 19.

G. H. Derrick, R. C. McPhedan, L. C. Botten, “Theory of crossed gratings,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, New York, 1980), Chap. 7.

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Figures (7)

Fig. 1
Fig. 1

General problem of plane-wave scattering from a doubly periodic grating of finite-sized scatterers.

Fig. 2
Fig. 2

Simulated equivalence for the region surrounding the scatterers.

Fig. 3
Fig. 3

(a) Plot of the window function f(ξ) given by Eq. (4). (b) Fourier transform of the window function f(ξ) depicted in (a).

Fig. 4
Fig. 4

Plots of max(ΔEbc) and ΔP versus rs/r for the case of a plane wave normally incident (θinc = ϕinc = 0°) upon an orthogonal array with periods d1 = d2 = 0.8λ composed of perfectly conducting spheres of radius r = 0.2λ obtained with N = 44 patch-current sources per sphere.

Fig. 5
Fig. 5

Plots of max(ΔEbc) and ΔP versus N for the case of Fig. 4 with the sources situated on a sphere of radius rs = 0.2r.

Fig. 6
Fig. 6

Magnitude of induced electric surface current Jθ versus θ for gratings of various periods and for the single-scatterer case.

Fig. 7
Fig. 7

Values of reflection and transmission efficiencies P m 0 ± / P inc at discrete mode angles (asterisks) and the corresponding plot of πσ(θ, ϕ)/[|cos θ|(kd1d2)2] versus θ for ϕ = 0° (solid curve) for the case of normal incidence (θinc = ϕinc = 0°) upon a grating with periods d1 = d2 = 2.5λ composed of perfectly conducting spheres of radius r = 0.2λ.

Tables (1)

Tables Icon

Table 1 Efficiencies P mn ± / P i n c Power-Conservation Error, and Boundary-Condition Error for the Case of a TM-Polarized Plane Wave of Wavelength λa

Equations (31)

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E inc ( r ) = E 0 inc exp ( j k inc · r ) ,
E s ( r + d p ) = exp ( j k inc · d p ) E s ( r ) , p = 1,2.
n ˆ × E s = n ˆ × E inc ,
E s ( r ) = q = 1 2 i = 1 N K q i E q i ( r ) ,
[ Z ] K = V ,
[ Z ] = [ Z 11 Z 12 Z 21 Z 22 ]
K = [ K 1 K 2 ]
V = [ V 1 V 2 ]
K = [ Z ] 1 V .
K = { [ Z ] [ Z ] } 1 [ Z ] V .
M q i = û q i K q i δ ( z z i ) exp [ j k T inc · ( r r i ) ] p = 1 2 n = f ( ξ i p n / s p ) ,
f ( ξ ) = 0.35875 + 0.48829 cos ( 2 π ξ ) + 0.14128 cos ( 4 π ξ ) + 0.01168 cos ( 6 π ξ ) ,
E q i ( r ) = × F q i ( r ) ,
F q i ( r ) = û q i m = n = a 1 m a 2 n 2 j k Z m n exp [ j k m n ± · ( r r i ) ] .
k m n ± = k T m n ± z ˆ k Z m n ,
k T m n = k T inc + m κ 1 + n κ 2
k Z m n = ( k 2 k T m n · k T m n ) 1 / 2 ,
a p l = κ p 2 π s p / 2 s p / 2 f ( ξ / s p ) exp ( j l κ p ξ ) d ξ , p = 1,2 , l
z < z i i ,
E = m = n = E m n exp ( j k m n · r ) ,
E m n = a 1 m a 2 n 2 k Z m n q = 1 2 i = 1 N K q i k m n × u ˆ q i exp ( j k m n · r i ) .
z > z i i .
E + = m = n = E m n + exp ( j k m n + · r ) ,
E m n + = δ m 0 δ n 0 E 0 inc + a 1 m a 2 n 2 k Z m n q = 1 2 i = 1 N K q i k m n + û q i exp ( j k m n + · r i ) .
Δ E b c = | n ˆ × ( E s + E inc ) | | E inc | max on S .
Δ P = | P inc m n P m n + m n P m n | P inc ,
P inc = k Z inc k η | E 0 inc | 2
P m n + = k Z m n k η | E m n ± | 2
k inc = k ( sin θ inc cos ϕ inc x ˆ + sin θ inc sin ϕ inc y ˆ + cos θ inc z ˆ ) .
J = n ˆ × ( H inc + H s ) on S
P m n ± P inc π σ ( θ m n ± , ϕ m n ± ) cos θ inc | cos θ m n ± | ( k | d 1 × d 2 | ) 2 .

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