Abstract

The three-dimensional multiple multipole program (MMP) code based on the generalized multipole technique is outlined for readers who are not familiar with its concepts. This code was originally designed for computational electromagnetics. Rayleigh expansions and periodic boundary conditions are two new features that make MMP computations of arbitrary periodic structures efficient and that at the same time allow us to take advantage of the benefits of other MMP features, including surface impedance boundary conditions and a variety of available basis functions for modeling the electromagnetic field. The application of three-dimensional MMP to a simple grating of highly conducting wires with rectangular cross sections illustrates the high accuracy and the fast convergence of the method as well as the use of surface impedance boundary conditions. A more complicated biperiodic array of helical antennas demonstrates the application of thin-wire expansions in conjunction with regular MMP expansions. This model can be considered a simulation of a thin, anisotropic chiral slab with interesting characteristics.

© 1995 Optical Society of America

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References

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  1. Ch. Hafner, L. Bomholt, The 3D Electrodynamic Wave Simulator (Wiley, Chichester, UK, 1993), Chaps. 2–13, pp. 9–84.
  2. Ch. Hafner, The Generalized Multipole Technique for Computational Electromagnetics (Artech, Boston, Mass., 1990), Chaps. 7–8, pp. 157–266.
  3. G. Tayeb, “The method of fictitious sources applied to diffraction gratings,” Appl. Comput. Electromagn. Soc. J. 9, (3) 90–100 (1994).
  4. 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 (1990).
    [CrossRef]
  5. J. Li, S. Kiener, “On the solution of periodical structures with GMT,” in Proceedings of the IEEE Antennas and Propagation Society International Symposium (Institute of Electrical and Electronics Engineers, New York, 1990), pp. 610–613.
  6. J. Li, “GMT and MMP applied to the computation of general periodic structures,” Ph.D. dissertation (Eidgenössische Technische Hochschule, Zurich, 1993).
  7. Ch. Hafner, “Beitraege zur Berechnung der Ausbreitung elektromagnetischer Wellen in zylindrischen Strukturen mit Hilfe des Point-Matching Verfahrens,” Ph.D. dissertation (Eidgenössische Technische Hochschule, Zurich, 1980).
  8. A. Ludwig, “A new technique for numerical electromagnetics,” IEEE Antennas and Propagation Society Newsletter 31 (Institute of Electrical and Electronics Engineers, New York, 1989).
  9. Y. A. Eremin, N. V. Orlov, A. G. Sveshnikov, “Electromagnetic scattering analysis based on the discrete sources method,” Appl. Comput. Electromagn. Soc. J. 9, (3) 46–56 (1994).
  10. A. G. Kyurkchan, A. I. Sukov, A. I. Kleev, “The methods for solving the problems of the diffraction of electromagnetic and acoustic waves using the information on analytical properties of the scattered fields,” Appl. Comput. Electromagn. Soc. J. 9, (3) 101–111 (1994).
  11. R. S. Zaridze, D. D. Karkashadze, “Calculation of regular waveguides by method of auxiliary sources” (Tbilisi State University, Tbilisi, Russia, 1985).
  12. H. Singer, H. Steinbigler, P. Weiss, “A charge simulation method for the calculation of high-voltage fields,” IEEE Trans. Power Appar. Syst. 93, 1660–1668 (1974).
    [CrossRef]
  13. Ch. Hafner, “On the design of numerical methods,” IEEE Antennas Propag. Mag. 35, (4), 13–21 (1993).
    [CrossRef]
  14. I. N. Vekua, New Methods for Solving Elliptic Equations (North-Holland, Amsterdam, 1967), Chap. 2, pp. 103–110.
  15. J. Lochbihler, “Eine theoretische und experimentelle Untersuchung von hochleitenden Drahtgittern im Resonanzbreich,” Ph.D. dissertation (Technische Universitat München, Munich, 1993).
  16. Ch. Hafner, “Efficient MMP computation of periodic structures,”in Proceedings of the Tenth Annual Review of Progress in Applied Computational Electromagnetics (Applied Computational Electromagnetics Society, Monterey, Calif., 1994), pp. 303–310.
  17. L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1106 (1981).
    [CrossRef]
  18. S. Bassiri, “Electromagnetic waves in chiral media,” in Recent Advances in Electromagnetic Theory, H. N. Kritikos, D. L. Jaggard, eds. (Springer, New York, 1990), pp. 1–30.
    [CrossRef]
  19. P. Regli, “Automatische Wahl der sphaerischen Entwicklungsfunktionen fuer die 3D-MMP Methode,” Ph.D. dissertation (Eidgenössische Technische Hochschule, Zurich, 1992).
  20. R. Y-S. Tay, N. Kuster, “Performance of the generalized multipole technique (GMT/MMP) in antenna design and optimization,” Appl. Comput. Electromagn. Soc. J. 9, (3) 79–89 (1994).
  21. P. Leuchtmann, “MMP modelling techniques with curved line multipoles,” Appl. Comput. Electromagn. Soc. J. 9, (3) 69–78 (1994).
  22. P. Leuchtmann, L. Bomholt, “Thin wire feature for the MMP-code,” in Proceedings of the Sixth Annual Review of Progress in Applied Computational Electromagnetics (Applied Computational Electromagnetics Society, Monterey, Calif., 1990), pp. 233–240.

1994 (5)

G. Tayeb, “The method of fictitious sources applied to diffraction gratings,” Appl. Comput. Electromagn. Soc. J. 9, (3) 90–100 (1994).

Y. A. Eremin, N. V. Orlov, A. G. Sveshnikov, “Electromagnetic scattering analysis based on the discrete sources method,” Appl. Comput. Electromagn. Soc. J. 9, (3) 46–56 (1994).

A. G. Kyurkchan, A. I. Sukov, A. I. Kleev, “The methods for solving the problems of the diffraction of electromagnetic and acoustic waves using the information on analytical properties of the scattered fields,” Appl. Comput. Electromagn. Soc. J. 9, (3) 101–111 (1994).

R. Y-S. Tay, N. Kuster, “Performance of the generalized multipole technique (GMT/MMP) in antenna design and optimization,” Appl. Comput. Electromagn. Soc. J. 9, (3) 79–89 (1994).

P. Leuchtmann, “MMP modelling techniques with curved line multipoles,” Appl. Comput. Electromagn. Soc. J. 9, (3) 69–78 (1994).

1993 (1)

Ch. Hafner, “On the design of numerical methods,” IEEE Antennas Propag. Mag. 35, (4), 13–21 (1993).
[CrossRef]

1990 (1)

1981 (1)

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1106 (1981).
[CrossRef]

1974 (1)

H. Singer, H. Steinbigler, P. Weiss, “A charge simulation method for the calculation of high-voltage fields,” IEEE Trans. Power Appar. Syst. 93, 1660–1668 (1974).
[CrossRef]

Adams, J. L.

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1106 (1981).
[CrossRef]

Andrewartha, J. R.

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1106 (1981).
[CrossRef]

Bassiri, S.

S. Bassiri, “Electromagnetic waves in chiral media,” in Recent Advances in Electromagnetic Theory, H. N. Kritikos, D. L. Jaggard, eds. (Springer, New York, 1990), pp. 1–30.
[CrossRef]

Boag, A.

Bomholt, L.

Ch. Hafner, L. Bomholt, The 3D Electrodynamic Wave Simulator (Wiley, Chichester, UK, 1993), Chaps. 2–13, pp. 9–84.

P. Leuchtmann, L. Bomholt, “Thin wire feature for the MMP-code,” in Proceedings of the Sixth Annual Review of Progress in Applied Computational Electromagnetics (Applied Computational Electromagnetics Society, Monterey, Calif., 1990), pp. 233–240.

Botten, L. C.

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1106 (1981).
[CrossRef]

Craig, M. S.

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1106 (1981).
[CrossRef]

Eremin, Y. A.

Y. A. Eremin, N. V. Orlov, A. G. Sveshnikov, “Electromagnetic scattering analysis based on the discrete sources method,” Appl. Comput. Electromagn. Soc. J. 9, (3) 46–56 (1994).

Hafner, Ch.

Ch. Hafner, “On the design of numerical methods,” IEEE Antennas Propag. Mag. 35, (4), 13–21 (1993).
[CrossRef]

Ch. Hafner, “Beitraege zur Berechnung der Ausbreitung elektromagnetischer Wellen in zylindrischen Strukturen mit Hilfe des Point-Matching Verfahrens,” Ph.D. dissertation (Eidgenössische Technische Hochschule, Zurich, 1980).

Ch. Hafner, L. Bomholt, The 3D Electrodynamic Wave Simulator (Wiley, Chichester, UK, 1993), Chaps. 2–13, pp. 9–84.

Ch. Hafner, The Generalized Multipole Technique for Computational Electromagnetics (Artech, Boston, Mass., 1990), Chaps. 7–8, pp. 157–266.

Ch. Hafner, “Efficient MMP computation of periodic structures,”in Proceedings of the Tenth Annual Review of Progress in Applied Computational Electromagnetics (Applied Computational Electromagnetics Society, Monterey, Calif., 1994), pp. 303–310.

Karkashadze, D. D.

R. S. Zaridze, D. D. Karkashadze, “Calculation of regular waveguides by method of auxiliary sources” (Tbilisi State University, Tbilisi, Russia, 1985).

Kiener, S.

J. Li, S. Kiener, “On the solution of periodical structures with GMT,” in Proceedings of the IEEE Antennas and Propagation Society International Symposium (Institute of Electrical and Electronics Engineers, New York, 1990), pp. 610–613.

Kleev, A. I.

A. G. Kyurkchan, A. I. Sukov, A. I. Kleev, “The methods for solving the problems of the diffraction of electromagnetic and acoustic waves using the information on analytical properties of the scattered fields,” Appl. Comput. Electromagn. Soc. J. 9, (3) 101–111 (1994).

Kuster, N.

R. Y-S. Tay, N. Kuster, “Performance of the generalized multipole technique (GMT/MMP) in antenna design and optimization,” Appl. Comput. Electromagn. Soc. J. 9, (3) 79–89 (1994).

Kyurkchan, A. G.

A. G. Kyurkchan, A. I. Sukov, A. I. Kleev, “The methods for solving the problems of the diffraction of electromagnetic and acoustic waves using the information on analytical properties of the scattered fields,” Appl. Comput. Electromagn. Soc. J. 9, (3) 101–111 (1994).

Leuchtmann, P.

P. Leuchtmann, “MMP modelling techniques with curved line multipoles,” Appl. Comput. Electromagn. Soc. J. 9, (3) 69–78 (1994).

P. Leuchtmann, L. Bomholt, “Thin wire feature for the MMP-code,” in Proceedings of the Sixth Annual Review of Progress in Applied Computational Electromagnetics (Applied Computational Electromagnetics Society, Monterey, Calif., 1990), pp. 233–240.

Leviatan, Y.

Li, J.

J. Li, S. Kiener, “On the solution of periodical structures with GMT,” in Proceedings of the IEEE Antennas and Propagation Society International Symposium (Institute of Electrical and Electronics Engineers, New York, 1990), pp. 610–613.

J. Li, “GMT and MMP applied to the computation of general periodic structures,” Ph.D. dissertation (Eidgenössische Technische Hochschule, Zurich, 1993).

Lochbihler, J.

J. Lochbihler, “Eine theoretische und experimentelle Untersuchung von hochleitenden Drahtgittern im Resonanzbreich,” Ph.D. dissertation (Technische Universitat München, Munich, 1993).

Ludwig, A.

A. Ludwig, “A new technique for numerical electromagnetics,” IEEE Antennas and Propagation Society Newsletter 31 (Institute of Electrical and Electronics Engineers, New York, 1989).

McPhedran, R. C.

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1106 (1981).
[CrossRef]

Orlov, N. V.

Y. A. Eremin, N. V. Orlov, A. G. Sveshnikov, “Electromagnetic scattering analysis based on the discrete sources method,” Appl. Comput. Electromagn. Soc. J. 9, (3) 46–56 (1994).

Regli, P.

P. Regli, “Automatische Wahl der sphaerischen Entwicklungsfunktionen fuer die 3D-MMP Methode,” Ph.D. dissertation (Eidgenössische Technische Hochschule, Zurich, 1992).

Singer, H.

H. Singer, H. Steinbigler, P. Weiss, “A charge simulation method for the calculation of high-voltage fields,” IEEE Trans. Power Appar. Syst. 93, 1660–1668 (1974).
[CrossRef]

Steinbigler, H.

H. Singer, H. Steinbigler, P. Weiss, “A charge simulation method for the calculation of high-voltage fields,” IEEE Trans. Power Appar. Syst. 93, 1660–1668 (1974).
[CrossRef]

Sukov, A. I.

A. G. Kyurkchan, A. I. Sukov, A. I. Kleev, “The methods for solving the problems of the diffraction of electromagnetic and acoustic waves using the information on analytical properties of the scattered fields,” Appl. Comput. Electromagn. Soc. J. 9, (3) 101–111 (1994).

Sveshnikov, A. G.

Y. A. Eremin, N. V. Orlov, A. G. Sveshnikov, “Electromagnetic scattering analysis based on the discrete sources method,” Appl. Comput. Electromagn. Soc. J. 9, (3) 46–56 (1994).

Tay, R. Y-S.

R. Y-S. Tay, N. Kuster, “Performance of the generalized multipole technique (GMT/MMP) in antenna design and optimization,” Appl. Comput. Electromagn. Soc. J. 9, (3) 79–89 (1994).

Tayeb, G.

G. Tayeb, “The method of fictitious sources applied to diffraction gratings,” Appl. Comput. Electromagn. Soc. J. 9, (3) 90–100 (1994).

Vekua, I. N.

I. N. Vekua, New Methods for Solving Elliptic Equations (North-Holland, Amsterdam, 1967), Chap. 2, pp. 103–110.

Weiss, P.

H. Singer, H. Steinbigler, P. Weiss, “A charge simulation method for the calculation of high-voltage fields,” IEEE Trans. Power Appar. Syst. 93, 1660–1668 (1974).
[CrossRef]

Zaridze, R. S.

R. S. Zaridze, D. D. Karkashadze, “Calculation of regular waveguides by method of auxiliary sources” (Tbilisi State University, Tbilisi, Russia, 1985).

Appl. Comput. Electromagn. Soc. J. (5)

G. Tayeb, “The method of fictitious sources applied to diffraction gratings,” Appl. Comput. Electromagn. Soc. J. 9, (3) 90–100 (1994).

Y. A. Eremin, N. V. Orlov, A. G. Sveshnikov, “Electromagnetic scattering analysis based on the discrete sources method,” Appl. Comput. Electromagn. Soc. J. 9, (3) 46–56 (1994).

A. G. Kyurkchan, A. I. Sukov, A. I. Kleev, “The methods for solving the problems of the diffraction of electromagnetic and acoustic waves using the information on analytical properties of the scattered fields,” Appl. Comput. Electromagn. Soc. J. 9, (3) 101–111 (1994).

R. Y-S. Tay, N. Kuster, “Performance of the generalized multipole technique (GMT/MMP) in antenna design and optimization,” Appl. Comput. Electromagn. Soc. J. 9, (3) 79–89 (1994).

P. Leuchtmann, “MMP modelling techniques with curved line multipoles,” Appl. Comput. Electromagn. Soc. J. 9, (3) 69–78 (1994).

IEEE Antennas Propag. Mag. (1)

Ch. Hafner, “On the design of numerical methods,” IEEE Antennas Propag. Mag. 35, (4), 13–21 (1993).
[CrossRef]

IEEE Trans. Power Appar. Syst. (1)

H. Singer, H. Steinbigler, P. Weiss, “A charge simulation method for the calculation of high-voltage fields,” IEEE Trans. Power Appar. Syst. 93, 1660–1668 (1974).
[CrossRef]

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

Opt. Acta (1)

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1106 (1981).
[CrossRef]

Other (13)

S. Bassiri, “Electromagnetic waves in chiral media,” in Recent Advances in Electromagnetic Theory, H. N. Kritikos, D. L. Jaggard, eds. (Springer, New York, 1990), pp. 1–30.
[CrossRef]

P. Regli, “Automatische Wahl der sphaerischen Entwicklungsfunktionen fuer die 3D-MMP Methode,” Ph.D. dissertation (Eidgenössische Technische Hochschule, Zurich, 1992).

P. Leuchtmann, L. Bomholt, “Thin wire feature for the MMP-code,” in Proceedings of the Sixth Annual Review of Progress in Applied Computational Electromagnetics (Applied Computational Electromagnetics Society, Monterey, Calif., 1990), pp. 233–240.

I. N. Vekua, New Methods for Solving Elliptic Equations (North-Holland, Amsterdam, 1967), Chap. 2, pp. 103–110.

J. Lochbihler, “Eine theoretische und experimentelle Untersuchung von hochleitenden Drahtgittern im Resonanzbreich,” Ph.D. dissertation (Technische Universitat München, Munich, 1993).

Ch. Hafner, “Efficient MMP computation of periodic structures,”in Proceedings of the Tenth Annual Review of Progress in Applied Computational Electromagnetics (Applied Computational Electromagnetics Society, Monterey, Calif., 1994), pp. 303–310.

J. Li, S. Kiener, “On the solution of periodical structures with GMT,” in Proceedings of the IEEE Antennas and Propagation Society International Symposium (Institute of Electrical and Electronics Engineers, New York, 1990), pp. 610–613.

J. Li, “GMT and MMP applied to the computation of general periodic structures,” Ph.D. dissertation (Eidgenössische Technische Hochschule, Zurich, 1993).

Ch. Hafner, “Beitraege zur Berechnung der Ausbreitung elektromagnetischer Wellen in zylindrischen Strukturen mit Hilfe des Point-Matching Verfahrens,” Ph.D. dissertation (Eidgenössische Technische Hochschule, Zurich, 1980).

A. Ludwig, “A new technique for numerical electromagnetics,” IEEE Antennas and Propagation Society Newsletter 31 (Institute of Electrical and Electronics Engineers, New York, 1989).

R. S. Zaridze, D. D. Karkashadze, “Calculation of regular waveguides by method of auxiliary sources” (Tbilisi State University, Tbilisi, Russia, 1985).

Ch. Hafner, L. Bomholt, The 3D Electrodynamic Wave Simulator (Wiley, Chichester, UK, 1993), Chaps. 2–13, pp. 9–84.

Ch. Hafner, The Generalized Multipole Technique for Computational Electromagnetics (Artech, Boston, Mass., 1990), Chaps. 7–8, pp. 157–266.

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Figures (17)

Fig. 1
Fig. 1

Three different MMP models of the cross section of a rectangular cylinder with round edges. Model A, single multipole in the center of the rectangle; model B, three multipoles per quadrant; model C, seven multipoles per quadrant. Only the first quadrant is shown because the structure is symmetric with respect to the x and the y axes.

Fig. 2
Fig. 2

MMP error distribution (mismatching) along the boundary of the rectangular cylinder illustrated in Fig. 1 for a plane wave incident in the −y direction, with the electric field polarized in the x direction. Results were obtained by means of model A, with one multipole in the center with different orders. Top, model with 9 + 10 unknowns for the two parts of the symmetry decomposition; center, model with 17 + 18 unknowns; bottom, model with 65 + 66 unknowns. Only the left-hand half of the boundary is shown because of the symmetry of the errors with respect to the y axis. The matching-point numbers M, as well as the location of points A and D on the y and the x axes and the critical points B and C (see Fig. 1), are indicated.

Fig. 3
Fig. 3

MMP error distribution along the boundary of the rectangular cylinder illustrated in Fig. 1 for a plane wave incident in the −y direction, with the electric field polarized in the x direction. Results were obtained by means of model B, with three multipoles in the first quadrant with different orders. Top, 10 + 11 unknowns; center, 17 + 18 unknowns; bottom, 59 + 60 unknowns. Only the left-hand half of the boundary is shown because of the symmetry of the errors with respect to the y axis. The matching points are identical to the matching points shown in Fig. 2.

Fig. 4
Fig. 4

MMP error distribution along the boundary of the rectangular cylinder illustrated in Fig. 1 for a plane wave incident in the −y direction, with the electric field polarized in the x direction. Results were obtained by means of model C, with seven multipoles in the first quadrant with different orders. Top, 8 + 8 unknowns; center, 22 + 22 unknowns; bottom, 64 + 64 unknowns. Only the left-hand half of the boundary is shown because of the symmetry of the errors with respect to the y axis. The matching points are identical to the matching points shown in Fig. 2.

Fig. 5
Fig. 5

MMP error distribution along the boundary and estimation of the error distribution in the near field of the rectangular cylinder illustrated in Fig. 1 for a plane wave incident in the −y direction, with the electric field polarized in the x direction. Results were obtained with different models. (a) Model A with 65 + 66 unknowns (see also Fig. 2), (b) model B with 59 + 60 unknowns (see also Fig. 3). The most accurate MMP results (model C with 78 + 78 unknowns, maximum error on the boundary <0.1%) were used as a reference for these error computations. The dark areas indicate the highest errors.

Fig. 6
Fig. 6

MMP models of gratings (left-hand side) and biperiodic structures (right-hand side).

Fig. 7
Fig. 7

Time average of the Poynting vector in the near field for a plane wave incident upon a grating consisting of the rectangular cylinders illustrated in Fig. 1. The plane wave is incident perpendicular to the z axis of the cylinders; the angle between the wave vector and the x axis is 45°; the magnetic field is in the z direction; the frequency is 2 × 1014 Hz; the material of the cylinders is gold (complex relative permittivity, −131.2 + 11.4i); the distance between two neighboring cylinders is 10−6 m; the horizontal length of the rectangle is 5 × 10−7 m; the vertical length of the rectangle is 4 × 10−7 m. (a) Radius of the edges of the cylinders, 10−7 m; (b) radius of the edges of the cylinders, 2.5 × 10−8 m.

Fig. 8
Fig. 8

Frequency dependence of the transmitted and the reflected zeroth-order efficiencies for the same grating and excitation as indicated in Fig. 7.

Fig. 9
Fig. 9

Frequency dependence of the relative error of the reflected and the transmitted zeroth-order efficiencies for a MMP–SIBC computation for the same grating and excitation as indicated in Fig. 7. The results shown in Fig. 8 were used as a reference for the computation of the relative error.

Fig. 10
Fig. 10

Frequency dependence of the real and the imaginary parts of the relative permittivity of gold, obtained from a simplified theoretical model. These data were used for the computation of Figs. 79.

Fig. 11
Fig. 11

Original cell of a biperiodic array of ideally conducting helical antennas, simulating an anisotropic chiral slab. The fictitious and the periodic boundaries of the cell form a cube with side length of 10−6 m. Left-hand side, one-turn helix; right-hand side, three-turn helix.

Fig. 12
Fig. 12

Zeroth-order efficiencies of the transmitted and the reflected TE waves versus frequency for a plane wave incident upon the structure defined in Fig. 11, with an angle of 45° with respect to the z axis and of 0° with respect to the x axis. The electric field of the incident wave is polarized in the y direction. The axes of the one-turn and the three-turn helices are in the x direction. Orientation of the helices, right handed.

Fig. 13
Fig. 13

Same as in Fig. 12, but the axes of the one-turn and the three-turn helices are in the y direction.

Fig. 14
Fig. 14

Same as in Fig. 12, but the axes of the one-turn and the three-turn helices are in the z direction.

Fig. 15
Fig. 15

Angle da of the rotation of the time average of the transmitted E field versus frequency for a plane wave incident upon the structure defined in Fig. 11, with an angle of 45° with respect to the z axis and of 0° with respect to the x axis. The electric field of the incident wave is polarized in the y direction. The axes of the one-turn and the three-turn helices are in the x, the y, and the z directions, respectively. Orientation of the helices, right handed. Computation is in the far field. Note that this computation makes no sense as soon as higher-order Rayleigh terms are important, i.e., above the first resonance of the grid. This fact explains the chaotic behavior of the functions at high frequencies.

Fig. 16
Fig. 16

Angle dφ between the phase of the transmitted E and H fields versus frequency for a plane wave incident upon the structure defined in Fig. 11, with an angle of 45° with respect to the z axis and of 0° with respect to the x axis. The electric field of the incident wave is polarized in the y direction. The axes of the one-turn and the three-turn helices are in the x, the y, and the z directions, respectively. Orientation of the helices, right handed. Computation is in the far field. The angle dφ is a good measure of the polarization of the transmitted plane wave. Linear polarization is characterized by dφ = 0°, whereas circular polarization is characterized by dφ = ±90°. Note that this computation makes no sense as soon as higher-order Rayleigh terms are important, i.e., above the first resonance of the grid. This fact explains the chaotic behavior of the functions at high frequencies. The angle is restricted to the interval −180° … 180°, which explains the jumps in some of the functions.

Fig. 17
Fig. 17

Sum of the angles dφ shown in Fig. 16 versus frequency.

Equations (11)

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Field = k = 1 K A k field k + Error .
r = r + l u + m v + n w ,
Field ( r ) = C Field ( r ) ,
C = exp [ i k ( r r ) ] .
C = C u l C υ m C w n ,
C u = exp ( i ku ) , C υ = exp ( i kv ) , C w = exp ( i kw ) .
f m ( x , y , z ) = exp ( i { [ k x inc + ( m π / u ) ] x + k y y + k z inc z } )
f n m ( x , y , z ) = exp ( i { [ k x inc + ( m π / u ) ] x + [ k y inc + ( n π / υ ) ] y + k z z } )
k y 2 = k 2 [ k x inc + ( m π / u ) ] 2 ( k z inc ) 2 ,
k z 2 = k 2 [ k x inc + ( m π / u ) ] 2 + [ k y inc + ( n π / υ ) ] 2 ,
k 2 = ω 2 μ .

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