Abstract

The scattering of waves by multilayered periodic structures is formulated in three-dimensional space by using Fourier expansions for both the basic lattice and its associated reciprocal lattice. The fields in each layer are then expressed in terms of characteristic modes, and the complete solution is found rigorously by using a transmission-line representation to address the pertinent boundary-value problems. Such an approach can treat periodic arbitrary lattices containing arbitrarily shaped dielectric components, which may generally be absorbing and have biaxial properties along directions that are parallel or perpendicular to the layers. We illustrate the present approach by comparing our numerical results with data reported in the past for simple structures. In addition, we provide new results for more complex configurations, which include multiple periodic regions that contain absorbing uniaxial components with several possible canonic shapes and high dielectric constants.

© 2002 Optical Society of America

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References

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  1. P. Vincent, “A finite-difference method for dielectric and conducting crossed gratings,” Opt. Commun. 26, 293–296 (1978).
    [CrossRef]
  2. D. Maystre, M. Nevière, “Electromagnetic theory of crossed gratings,” J. Opt. (Paris) 9, 301–306 (1978).
    [CrossRef]
  3. G. H. Derrick, R. C. McPhedran, D. Maystre, M. Nevière, “Crossed gratings: a theory and its applications,” Appl. Phys. 18, 39–52 (1979).
    [CrossRef]
  4. R. C. McPhedran, G. H. Derrick, L. C. Botten, “Theory of crossed gratings,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, New York, 1980), pp. 227–276.
  5. M. G. Moharam, “Coupled-wave analysis of two-dimensional dielectric gratings,” in Holographic Optics: Design and Applications, I. Cindrich, ed., Proc. SPIE883, 8–11 (1988).
    [CrossRef]
  6. S. T. Han, Y.-L. Tsao, R. M. Wasler, M. F. Becker, “Electromagnetic scattering of two-dimensional surface-relief dielectric gratings,” Appl. Opt. 31, 2343–2352 (1992).
    [CrossRef] [PubMed]
  7. P. Bräuer, O. Bryngdahl, “Electromagnetic diffraction analysis of two-dimensional gratings,” Opt. Commun. 100, 1–5 (1993).
    [CrossRef]
  8. E. Noponen, J. Turunen, “Eigenmode method for electromagnetic three-dimensional profiles,” J. Opt. Soc. Am. A 11, 2494–2502 (1994).
    [CrossRef]
  9. E. B. Grann, M. G. Moharam, D. A. Pommet, “Artificial uniaxial and biaxial dielectrics with use of two-dimensional subwavelength binary gratings,” J. Opt. Soc. Am. A 11, 2695–2703 (1994).
    [CrossRef]
  10. E. B. Grann, M. G. Moharam, “Comparison between continuous and discrete subwave-length grating structures for antireflection surfaces,” J. Opt. Soc. Am. A 13, 988–992 (1996).
    [CrossRef]
  11. S. Peng, G. M. Morris, “Resonant scattering from two-dimensional gratings,” J. Opt. Soc. Am. A 13, 993–1005 (1996).
    [CrossRef]
  12. J. B. Harris, T. W. Preist, J. R. Sambles, R. N. Thorpe, R. A. Watts, “Optical response of bigratings,” J. Opt. Soc. Am. A 13, 2041–2049 (1996).
    [CrossRef]
  13. P. Lalanne, “Improved formulation of the coupled-wave method for two-dimensional gratings,” J. Opt. Soc. Am. A 14, 1592–1598 (1997).
    [CrossRef]
  14. L. Li, “New formulation of the Fourier modal method for crossed surface-relief gratings,” J. Opt. Soc. Am. A 14, 2758–2767 (1997).
    [CrossRef]
  15. M. Bagieu, D. Maystre, “Regularized Waterman and Rayleigh methods: extension to two-dimensional gratings,” J. Opt. Soc. Am. A 16, 284–292 (1999).
    [CrossRef]
  16. V. Bagnoud, S. Mainguy, “Diffraction of electromagnetic waves by dielectric cross gratings: a three-dimensional Rayleigh–Fourier solution,” J. Opt. Soc. Am. A 16, 1277–1285 (1999).
    [CrossRef]
  17. E. Popov, M. Nevière, “Maxwell equations in Fourier space: fast-converging formulation for diffraction by arbitrary shaped, periodic, anisotropic media,” J. Opt. Soc. Am. A 18, 2886–2894 (2001).
    [CrossRef]
  18. E. Popov, M. Nevière, “Arbitrary shaped periodic anisotropic media: new presentation of Maxwell’s equations in the truncated Fourier space,” in Physics, Theory, and Applications of Periodic Structures in Optics, P. Lalanne, ed., Proc. SPIE4438, 19–30 (2001).
    [CrossRef]
  19. G. Granet, J. P. Plumey, “Rigorous electromagnetic analysis of 2D resonant subwavelength metallic gratings by parametric Fourier-modal analysis,” in Physics, Theory, and Applications of Periodic Structures in Optics, P. Lalanne, ed., Proc. SPIE4438, 124–131 (2001).
    [CrossRef]
  20. L. Li, “Fourier modal method for crossed anisotropic gratings,” in Physics, Theory, and Applications of Periodic Structures in Optics, P. Lalanne, ed., Proc. SPIE4438, 132–142 (2001).
    [CrossRef]
  21. K. M. Leung, C.-H. Lin, “Modal transmission-line theory of photonic band-gap structures,” in Proceedings of the 8th Asia–Pacific Physics Conference (World Scientific, Singapore, 2001), pp. 397–402.
  22. C.-H. Lin, “Modal transmission-line theory of photonic band-gap structures,” Ph.D. dissertation (Polytechnic University, Brooklyn, N.Y., 2001).
  23. C.-H. Lin, K. M. Leung, M. Jiang, T. Tamir, “Modal transmission-line theory of composite periodic structures: II. Three-dimensional configurations,” in Proceedings of the 2001 URSI International Symposium on Electromagnetic Theory (Union Radio-Scientifique Internationale, Ghent, Belgium, 2001), pp. 335–337.
  24. L. Yan, M. Jiang, T. Tamir, K. K. Choi, “Electromagnetic modeling of quantum-well photodetectors containing diffractive elements,” IEEE J. Quantum Electron. 35, 1870–1877 (1999).
    [CrossRef]
  25. M. Jiang, T. Tamir, S. Zhang, “Modal theory of diffraction by multilayered gratings containing dielectric and metallic components,” J. Opt. Soc. Am. A 18, 807–820 (2001).
    [CrossRef]
  26. N. W. Ashcroft, N. D. Mermin, Solid State Physics (Saunders College, Philadelphia, Pa., 1976).
  27. L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A 13, 1870–1876 (1996).
    [CrossRef]
  28. T. Tamir, S. Zhang, “Modal transmission-line theory of multilayered grating structures,” J. Lightwave Technol. 14, 914–927 (1996).
    [CrossRef]
  29. K. K. Choi, The Physics of Quantum Well Infrared Photodetectors (World Scientific, Singapore, 1997).
  30. J. Mao, A. Majumdar, K. K. Choi, D. C. Tsui, K. M. Leung, C. H. Lin, T. Tamir, G. A. Vawter, “Light coupling mechanism of quantum grid infrared photodetectors,” Appl. Phys. Lett. 80, 868–870 (2002).
    [CrossRef]

2002 (1)

J. Mao, A. Majumdar, K. K. Choi, D. C. Tsui, K. M. Leung, C. H. Lin, T. Tamir, G. A. Vawter, “Light coupling mechanism of quantum grid infrared photodetectors,” Appl. Phys. Lett. 80, 868–870 (2002).
[CrossRef]

2001 (2)

1999 (3)

1997 (2)

1996 (5)

1994 (2)

1993 (1)

P. Bräuer, O. Bryngdahl, “Electromagnetic diffraction analysis of two-dimensional gratings,” Opt. Commun. 100, 1–5 (1993).
[CrossRef]

1992 (1)

1979 (1)

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

1978 (2)

P. Vincent, “A finite-difference method for dielectric and conducting crossed gratings,” Opt. Commun. 26, 293–296 (1978).
[CrossRef]

D. Maystre, M. Nevière, “Electromagnetic theory of crossed gratings,” J. Opt. (Paris) 9, 301–306 (1978).
[CrossRef]

Ashcroft, N. W.

N. W. Ashcroft, N. D. Mermin, Solid State Physics (Saunders College, Philadelphia, Pa., 1976).

Bagieu, M.

Bagnoud, V.

Becker, M. F.

Botten, L. C.

R. C. McPhedran, G. H. Derrick, L. C. Botten, “Theory of crossed gratings,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, New York, 1980), pp. 227–276.

Bräuer, P.

P. Bräuer, O. Bryngdahl, “Electromagnetic diffraction analysis of two-dimensional gratings,” Opt. Commun. 100, 1–5 (1993).
[CrossRef]

Bryngdahl, O.

P. Bräuer, O. Bryngdahl, “Electromagnetic diffraction analysis of two-dimensional gratings,” Opt. Commun. 100, 1–5 (1993).
[CrossRef]

Choi, K. K.

J. Mao, A. Majumdar, K. K. Choi, D. C. Tsui, K. M. Leung, C. H. Lin, T. Tamir, G. A. Vawter, “Light coupling mechanism of quantum grid infrared photodetectors,” Appl. Phys. Lett. 80, 868–870 (2002).
[CrossRef]

L. Yan, M. Jiang, T. Tamir, K. K. Choi, “Electromagnetic modeling of quantum-well photodetectors containing diffractive elements,” IEEE J. Quantum Electron. 35, 1870–1877 (1999).
[CrossRef]

K. K. Choi, The Physics of Quantum Well Infrared Photodetectors (World Scientific, Singapore, 1997).

Derrick, G. H.

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

R. C. McPhedran, G. H. Derrick, L. C. Botten, “Theory of crossed gratings,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, New York, 1980), pp. 227–276.

Granet, G.

G. Granet, J. P. Plumey, “Rigorous electromagnetic analysis of 2D resonant subwavelength metallic gratings by parametric Fourier-modal analysis,” in Physics, Theory, and Applications of Periodic Structures in Optics, P. Lalanne, ed., Proc. SPIE4438, 124–131 (2001).
[CrossRef]

Grann, E. B.

Han, S. T.

Harris, J. B.

Jiang, M.

M. Jiang, T. Tamir, S. Zhang, “Modal theory of diffraction by multilayered gratings containing dielectric and metallic components,” J. Opt. Soc. Am. A 18, 807–820 (2001).
[CrossRef]

L. Yan, M. Jiang, T. Tamir, K. K. Choi, “Electromagnetic modeling of quantum-well photodetectors containing diffractive elements,” IEEE J. Quantum Electron. 35, 1870–1877 (1999).
[CrossRef]

C.-H. Lin, K. M. Leung, M. Jiang, T. Tamir, “Modal transmission-line theory of composite periodic structures: II. Three-dimensional configurations,” in Proceedings of the 2001 URSI International Symposium on Electromagnetic Theory (Union Radio-Scientifique Internationale, Ghent, Belgium, 2001), pp. 335–337.

Lalanne, P.

Leung, K. M.

J. Mao, A. Majumdar, K. K. Choi, D. C. Tsui, K. M. Leung, C. H. Lin, T. Tamir, G. A. Vawter, “Light coupling mechanism of quantum grid infrared photodetectors,” Appl. Phys. Lett. 80, 868–870 (2002).
[CrossRef]

C.-H. Lin, K. M. Leung, M. Jiang, T. Tamir, “Modal transmission-line theory of composite periodic structures: II. Three-dimensional configurations,” in Proceedings of the 2001 URSI International Symposium on Electromagnetic Theory (Union Radio-Scientifique Internationale, Ghent, Belgium, 2001), pp. 335–337.

K. M. Leung, C.-H. Lin, “Modal transmission-line theory of photonic band-gap structures,” in Proceedings of the 8th Asia–Pacific Physics Conference (World Scientific, Singapore, 2001), pp. 397–402.

Li, L.

Lin, C. H.

J. Mao, A. Majumdar, K. K. Choi, D. C. Tsui, K. M. Leung, C. H. Lin, T. Tamir, G. A. Vawter, “Light coupling mechanism of quantum grid infrared photodetectors,” Appl. Phys. Lett. 80, 868–870 (2002).
[CrossRef]

Lin, C.-H.

C.-H. Lin, “Modal transmission-line theory of photonic band-gap structures,” Ph.D. dissertation (Polytechnic University, Brooklyn, N.Y., 2001).

K. M. Leung, C.-H. Lin, “Modal transmission-line theory of photonic band-gap structures,” in Proceedings of the 8th Asia–Pacific Physics Conference (World Scientific, Singapore, 2001), pp. 397–402.

C.-H. Lin, K. M. Leung, M. Jiang, T. Tamir, “Modal transmission-line theory of composite periodic structures: II. Three-dimensional configurations,” in Proceedings of the 2001 URSI International Symposium on Electromagnetic Theory (Union Radio-Scientifique Internationale, Ghent, Belgium, 2001), pp. 335–337.

Mainguy, S.

Majumdar, A.

J. Mao, A. Majumdar, K. K. Choi, D. C. Tsui, K. M. Leung, C. H. Lin, T. Tamir, G. A. Vawter, “Light coupling mechanism of quantum grid infrared photodetectors,” Appl. Phys. Lett. 80, 868–870 (2002).
[CrossRef]

Mao, J.

J. Mao, A. Majumdar, K. K. Choi, D. C. Tsui, K. M. Leung, C. H. Lin, T. Tamir, G. A. Vawter, “Light coupling mechanism of quantum grid infrared photodetectors,” Appl. Phys. Lett. 80, 868–870 (2002).
[CrossRef]

Maystre, D.

M. Bagieu, D. Maystre, “Regularized Waterman and Rayleigh methods: extension to two-dimensional gratings,” J. Opt. Soc. Am. A 16, 284–292 (1999).
[CrossRef]

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

D. Maystre, M. Nevière, “Electromagnetic theory of crossed gratings,” J. Opt. (Paris) 9, 301–306 (1978).
[CrossRef]

McPhedran, R. C.

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

R. C. McPhedran, G. H. Derrick, L. C. Botten, “Theory of crossed gratings,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, New York, 1980), pp. 227–276.

Mermin, N. D.

N. W. Ashcroft, N. D. Mermin, Solid State Physics (Saunders College, Philadelphia, Pa., 1976).

Moharam, M. G.

Morris, G. M.

Nevière, M.

E. Popov, M. Nevière, “Maxwell equations in Fourier space: fast-converging formulation for diffraction by arbitrary shaped, periodic, anisotropic media,” J. Opt. Soc. Am. A 18, 2886–2894 (2001).
[CrossRef]

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

D. Maystre, M. Nevière, “Electromagnetic theory of crossed gratings,” J. Opt. (Paris) 9, 301–306 (1978).
[CrossRef]

E. Popov, M. Nevière, “Arbitrary shaped periodic anisotropic media: new presentation of Maxwell’s equations in the truncated Fourier space,” in Physics, Theory, and Applications of Periodic Structures in Optics, P. Lalanne, ed., Proc. SPIE4438, 19–30 (2001).
[CrossRef]

Noponen, E.

Peng, S.

Plumey, J. P.

G. Granet, J. P. Plumey, “Rigorous electromagnetic analysis of 2D resonant subwavelength metallic gratings by parametric Fourier-modal analysis,” in Physics, Theory, and Applications of Periodic Structures in Optics, P. Lalanne, ed., Proc. SPIE4438, 124–131 (2001).
[CrossRef]

Pommet, D. A.

Popov, E.

E. Popov, M. Nevière, “Maxwell equations in Fourier space: fast-converging formulation for diffraction by arbitrary shaped, periodic, anisotropic media,” J. Opt. Soc. Am. A 18, 2886–2894 (2001).
[CrossRef]

E. Popov, M. Nevière, “Arbitrary shaped periodic anisotropic media: new presentation of Maxwell’s equations in the truncated Fourier space,” in Physics, Theory, and Applications of Periodic Structures in Optics, P. Lalanne, ed., Proc. SPIE4438, 19–30 (2001).
[CrossRef]

Preist, T. W.

Sambles, J. R.

Tamir, T.

J. Mao, A. Majumdar, K. K. Choi, D. C. Tsui, K. M. Leung, C. H. Lin, T. Tamir, G. A. Vawter, “Light coupling mechanism of quantum grid infrared photodetectors,” Appl. Phys. Lett. 80, 868–870 (2002).
[CrossRef]

M. Jiang, T. Tamir, S. Zhang, “Modal theory of diffraction by multilayered gratings containing dielectric and metallic components,” J. Opt. Soc. Am. A 18, 807–820 (2001).
[CrossRef]

L. Yan, M. Jiang, T. Tamir, K. K. Choi, “Electromagnetic modeling of quantum-well photodetectors containing diffractive elements,” IEEE J. Quantum Electron. 35, 1870–1877 (1999).
[CrossRef]

T. Tamir, S. Zhang, “Modal transmission-line theory of multilayered grating structures,” J. Lightwave Technol. 14, 914–927 (1996).
[CrossRef]

C.-H. Lin, K. M. Leung, M. Jiang, T. Tamir, “Modal transmission-line theory of composite periodic structures: II. Three-dimensional configurations,” in Proceedings of the 2001 URSI International Symposium on Electromagnetic Theory (Union Radio-Scientifique Internationale, Ghent, Belgium, 2001), pp. 335–337.

Thorpe, R. N.

Tsao, Y.-L.

Tsui, D. C.

J. Mao, A. Majumdar, K. K. Choi, D. C. Tsui, K. M. Leung, C. H. Lin, T. Tamir, G. A. Vawter, “Light coupling mechanism of quantum grid infrared photodetectors,” Appl. Phys. Lett. 80, 868–870 (2002).
[CrossRef]

Turunen, J.

Vawter, G. A.

J. Mao, A. Majumdar, K. K. Choi, D. C. Tsui, K. M. Leung, C. H. Lin, T. Tamir, G. A. Vawter, “Light coupling mechanism of quantum grid infrared photodetectors,” Appl. Phys. Lett. 80, 868–870 (2002).
[CrossRef]

Vincent, P.

P. Vincent, “A finite-difference method for dielectric and conducting crossed gratings,” Opt. Commun. 26, 293–296 (1978).
[CrossRef]

Wasler, R. M.

Watts, R. A.

Yan, L.

L. Yan, M. Jiang, T. Tamir, K. K. Choi, “Electromagnetic modeling of quantum-well photodetectors containing diffractive elements,” IEEE J. Quantum Electron. 35, 1870–1877 (1999).
[CrossRef]

Zhang, S.

M. Jiang, T. Tamir, S. Zhang, “Modal theory of diffraction by multilayered gratings containing dielectric and metallic components,” J. Opt. Soc. Am. A 18, 807–820 (2001).
[CrossRef]

T. Tamir, S. Zhang, “Modal transmission-line theory of multilayered grating structures,” J. Lightwave Technol. 14, 914–927 (1996).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. (1)

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

Appl. Phys. Lett. (1)

J. Mao, A. Majumdar, K. K. Choi, D. C. Tsui, K. M. Leung, C. H. Lin, T. Tamir, G. A. Vawter, “Light coupling mechanism of quantum grid infrared photodetectors,” Appl. Phys. Lett. 80, 868–870 (2002).
[CrossRef]

IEEE J. Quantum Electron. (1)

L. Yan, M. Jiang, T. Tamir, K. K. Choi, “Electromagnetic modeling of quantum-well photodetectors containing diffractive elements,” IEEE J. Quantum Electron. 35, 1870–1877 (1999).
[CrossRef]

J. Lightwave Technol. (1)

T. Tamir, S. Zhang, “Modal transmission-line theory of multilayered grating structures,” J. Lightwave Technol. 14, 914–927 (1996).
[CrossRef]

J. Opt. (Paris) (1)

D. Maystre, M. Nevière, “Electromagnetic theory of crossed gratings,” J. Opt. (Paris) 9, 301–306 (1978).
[CrossRef]

J. Opt. Soc. Am. A (12)

E. Noponen, J. Turunen, “Eigenmode method for electromagnetic three-dimensional profiles,” J. Opt. Soc. Am. A 11, 2494–2502 (1994).
[CrossRef]

E. B. Grann, M. G. Moharam, D. A. Pommet, “Artificial uniaxial and biaxial dielectrics with use of two-dimensional subwavelength binary gratings,” J. Opt. Soc. Am. A 11, 2695–2703 (1994).
[CrossRef]

M. Bagieu, D. Maystre, “Regularized Waterman and Rayleigh methods: extension to two-dimensional gratings,” J. Opt. Soc. Am. A 16, 284–292 (1999).
[CrossRef]

V. Bagnoud, S. Mainguy, “Diffraction of electromagnetic waves by dielectric cross gratings: a three-dimensional Rayleigh–Fourier solution,” J. Opt. Soc. Am. A 16, 1277–1285 (1999).
[CrossRef]

P. Lalanne, “Improved formulation of the coupled-wave method for two-dimensional gratings,” J. Opt. Soc. Am. A 14, 1592–1598 (1997).
[CrossRef]

L. Li, “New formulation of the Fourier modal method for crossed surface-relief gratings,” J. Opt. Soc. Am. A 14, 2758–2767 (1997).
[CrossRef]

E. B. Grann, M. G. Moharam, “Comparison between continuous and discrete subwave-length grating structures for antireflection surfaces,” J. Opt. Soc. Am. A 13, 988–992 (1996).
[CrossRef]

S. Peng, G. M. Morris, “Resonant scattering from two-dimensional gratings,” J. Opt. Soc. Am. A 13, 993–1005 (1996).
[CrossRef]

L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A 13, 1870–1876 (1996).
[CrossRef]

M. Jiang, T. Tamir, S. Zhang, “Modal theory of diffraction by multilayered gratings containing dielectric and metallic components,” J. Opt. Soc. Am. A 18, 807–820 (2001).
[CrossRef]

J. B. Harris, T. W. Preist, J. R. Sambles, R. N. Thorpe, R. A. Watts, “Optical response of bigratings,” J. Opt. Soc. Am. A 13, 2041–2049 (1996).
[CrossRef]

E. Popov, M. Nevière, “Maxwell equations in Fourier space: fast-converging formulation for diffraction by arbitrary shaped, periodic, anisotropic media,” J. Opt. Soc. Am. A 18, 2886–2894 (2001).
[CrossRef]

Opt. Commun. (2)

P. Vincent, “A finite-difference method for dielectric and conducting crossed gratings,” Opt. Commun. 26, 293–296 (1978).
[CrossRef]

P. Bräuer, O. Bryngdahl, “Electromagnetic diffraction analysis of two-dimensional gratings,” Opt. Commun. 100, 1–5 (1993).
[CrossRef]

Other (10)

E. Popov, M. Nevière, “Arbitrary shaped periodic anisotropic media: new presentation of Maxwell’s equations in the truncated Fourier space,” in Physics, Theory, and Applications of Periodic Structures in Optics, P. Lalanne, ed., Proc. SPIE4438, 19–30 (2001).
[CrossRef]

G. Granet, J. P. Plumey, “Rigorous electromagnetic analysis of 2D resonant subwavelength metallic gratings by parametric Fourier-modal analysis,” in Physics, Theory, and Applications of Periodic Structures in Optics, P. Lalanne, ed., Proc. SPIE4438, 124–131 (2001).
[CrossRef]

L. Li, “Fourier modal method for crossed anisotropic gratings,” in Physics, Theory, and Applications of Periodic Structures in Optics, P. Lalanne, ed., Proc. SPIE4438, 132–142 (2001).
[CrossRef]

K. M. Leung, C.-H. Lin, “Modal transmission-line theory of photonic band-gap structures,” in Proceedings of the 8th Asia–Pacific Physics Conference (World Scientific, Singapore, 2001), pp. 397–402.

C.-H. Lin, “Modal transmission-line theory of photonic band-gap structures,” Ph.D. dissertation (Polytechnic University, Brooklyn, N.Y., 2001).

C.-H. Lin, K. M. Leung, M. Jiang, T. Tamir, “Modal transmission-line theory of composite periodic structures: II. Three-dimensional configurations,” in Proceedings of the 2001 URSI International Symposium on Electromagnetic Theory (Union Radio-Scientifique Internationale, Ghent, Belgium, 2001), pp. 335–337.

R. C. McPhedran, G. H. Derrick, L. C. Botten, “Theory of crossed gratings,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, New York, 1980), pp. 227–276.

M. G. Moharam, “Coupled-wave analysis of two-dimensional dielectric gratings,” in Holographic Optics: Design and Applications, I. Cindrich, ed., Proc. SPIE883, 8–11 (1988).
[CrossRef]

K. K. Choi, The Physics of Quantum Well Infrared Photodetectors (World Scientific, Singapore, 1997).

N. W. Ashcroft, N. D. Mermin, Solid State Physics (Saunders College, Philadelphia, Pa., 1976).

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Figures (4)

Fig. 1
Fig. 1

Two-dimensional periodic variation of the dielectric constant within the xy plane. The lattice can be specified by arbitrary lattice vectors a1 and a2.

Fig. 2
Fig. 2

Convergence of diffraction efficiency versus truncation order N for several diffraction orders in the case of Example 2 reported by Li.14 The results shown in his Table 3 are indicated here by crosses.

Fig. 3
Fig. 3

Radiation incident on a 3D structure with a complex geometry: (a) cross section in the yz plane, (b) three different shapes in the xy plane for the posts considered here.

Fig. 4
Fig. 4

Normalized scattered field pscat versus θ for unpolarized waves at a wavelength of 8.3 µm, for the post shapes shown in Fig. 2(b). The cross-sectional area of the post is 4 µm2 for all three post shapes. The dimensions are a1=3.5, tc=0.1, tq=1.15, tb=2.0, ts=0.1, and d=2.6 μm. The upper region is assumed to be air. The quantum-well region is modeled as an absorbing biaxial material, with dielectric constants x=y=10.43 and z=10.43+i. The dielectric constant of the stop-etch layer is isotropic and has a value of 9.956. The substrate and all the remaining layers are also isotropic, and their dielectric constant is 11.156.

Tables (1)

Tables Icon

Table 1 Diffraction Efficiencies (in %) for Diffraction Orders Reflected by the Circular Post Gratinga

Equations (106)

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

(ρ+R)=(ρ).
al·bj=2πδl, j forl, j=1or2,
exp(iK·R)=1
f(ρ)=KfK exp(iK·ρ),
fK=celldρΩf(ρ)exp(-iK·ρ),
celldρΩexp(i(K-K)·ρ)=δK,K
K exp(iK·(ρ-ρ))=ΩRδ(ρ-ρ+R),
f(ρ)=fb-fb1-f(ρ)fb.
fK=fbδK,0+(fa-fb)atomdρΩexp(-iK·ρ).
kK=kt+K.
yEz-zEy=iωμ0Hx,yHz-zHy=-iω0xEx,
zEx-xEz=iωμ0Hy,zHx-xHz=-iω0yEy,
xEy-yEx=iωμ0Hz,xHy-yHx=-iω0zEz,
Ez=iω01z(xHy-yHx),
Hz=-iωμ0(xEy-yEx).
zExEy=i-1ω0x1zyωμ0+1ω0x1zx-ωμ0-1ω0y1zy1ω0y1zx×HxHy,
zHxHy=i1ωμ0xy-ω0y-1ωμ0x2ω0x+1ωμ0y2-1ωμ0xyExEy.
z2ExEy
=-k02x-x1zxx-y2xy-x1zyyxy-y1zxx-k02y-x2-y1zyy
×ExEy.
z2HxHy
=-k02y-x2-yy1zy-xy+yy1zx-xy+xx1zy-k02x-xx1zx-y2
×HxHy.
z2ExEy=-k02-x2-y200-k02-x2-y2ExEy.
ExEy=exp(iκKz)exp(ikK·ρ)Ex0Ey0
κK2=k02-kK2,
kK=[(kK)x2+(kK)y2]1/2,
κK=k02-kK2ifk02>kK2ikK2-k02ifk02<kK2,
10,01.
Ex(r)Ey(r)=K exp(ikK·ρ)fK,xfK,y×exp(iκKz)+gK,xgK,yexp[iκK(t-z)],
Hy(r)-Hx(r)=0μ0 K exp(ikK·ρ)YKfK,xfK,y×exp(iκKz)-gK,xgK,yexp[iκK(t-z)],
YK=(kK)x2+κK2k0κK(kK)x(kK)yk0κK(kK)x(kK)yk0κK(kK)y2+κK2k0κK
YK=k0κK00κKk0.
z2ExEy=-k02-x2-y21-zxy1-zxy-k02-x2-zy2×ExEy.
κ2-k02+(kK)x2+(kK)y2-1-z(kK)x(kK)y-1-z(kK)x(kK)yκ2-k02+(kK)x2+z(kK)y2Ex,0Ey,0=0.
κK,12=k02-kK2,
Ex,0Ey,0=1kK(kK)x(kK)y.
κK,22=k02-kK2,
Ex,0Ey,0=1kK-(kK)y(kK)x.
ExEy=K exp(ikK·ρ)AK×fK,x exp(iκK,1z)+gK,x exp[iκK,1(t-z)]fK,y exp(iκK,2z)+gK,y exp[iκK,2(t-z)],
AK=1kK(kK)x-(kK)y(kK)y(kK)x.
Hy-Hx=0μ0 K exp(ikK·ρ)AKYK×fK,x exp(iκK,1z)-gK,x exp[iκK,1(t-z)]fK,y exp(iκK,2z)-gK,y exp[iκK,2(t-z)],
YK=k0κK,100κK,2k0.
Ex,0Ey,0=10,01.
ExEy=K exp(ikK·ρ)m(PK,m)x(PK,m)y×{fm exp(iκmz)+gm exp[iκm(t-z)]},
Hy-Hx=0μ0 K exp(ikK·ρ)m(QK,m)x(QK,m)y×{fm exp(iκmz)-gm exp[iκm(t-z)]}.
α(ρ)=KK,α exp(iK·ρ),
K,α=celldρΩα(ρ)exp(-iK·ρ).
γα(ρ)=1α(ρ)=KγK,α exp(iK·ρ),
γK,α=celldρΩ1α(ρ)exp(-iK·ρ).
mκm(PK,m)x{fm exp[iκmz]-gm exp[iκm(t-z)]}
=k0m(QK,m)x{fm exp(iκmz)-gm exp[iκm(t-z)]}-(kK)xk0KγK-K,zm[(kK)y(QK,m)y+(kK)x(QK,m)x]×{fm exp(iκmz)-gm exp[iκm(t-z)]}.
κm(PK,m)x=k0(QK,m)x-(kK)xk0KγK-K,z×[(kK)y(QK,m)y+(kK)x(QK,m)x],
KδK,Kκmk0(PK,m)x
=KδK,K-(kK)xk0γK-K,z(kK)xk0(QK,m)x
-(kK)xk0KγK-K,z(kK)yk0(QK,m)y.
KδK,Kκmk0(PK,m)y
=KδK,K-(kK)yk0γK-K,z(kK)yk0(QK,m)y
-(kK)yk0KγK-K,z(kK)xk0(QK,m)x,
KδK,Kκmk0(QK,m)x
=KK-K,y-δK,K(kK)x2k02(PK,m)y
+K(kK)xk0(kK)yk0δK,K(PK,m)x,
KδK,Kκmk0(QK,m)y
=KK-K,x-δK,K(kK)y2k02(PK,m)x
+K(kK)xk0(kK)yk0δK,K(PK,m)y.
P=PxPy,Q=QxQy,
U=I-x¯γ¯zx¯-x¯γ¯zy¯-y¯γ¯zx¯I-y¯γ¯zy¯,
V=¯x-y¯2y¯x¯x¯y¯¯y-x¯2.
U Q=κP,
V P=κQ,
U V P=κ2P.
Q=V Pκ-1 or Q=U-1 Pκ.
V U Q=κ2Q.
P=V-1 Qκor P=U Qκ-1,
Λ=U V=¯x-x¯γ¯zx¯¯x-y¯2y¯x¯-x¯γ¯zy¯¯yx¯y¯-y¯γ¯zx¯¯x¯y-y¯γ¯zy¯¯y-x¯2.
Ex(r)Ey(r)(j)=mem(j)(ρ)vm(j)(z),
Hy(r)-Hx(r)(j)=mhm(j)(ρ)im(j)(z),
em(j)(ρ)=K exp(ikK·ρ)(PK,m)x(PK,m)y(j)
hm(j)(ρ)=0μ0 K exp(ikK·ρ)(QK,m)x(QK,m)y(j)
vm(j)(z)=fm(j) exp(iκm(j)z)+gm(j) exp[iκm(j)(t(j)-z)],
im(j)(z)=fm(j) exp(iκm(j)z)-gm(j) exp[iκm(j)(t(j)-z)]
m(PK,m)x(PK,m)y(j)vm(j)(t(j))=m(PK,m)x(PK,m)y(j+1)vm(j+1)(0),
m(QK,m)x(QK,m)y(j)im(j)(t(j))=m(QK,m)x(QK,m)y(j+1)im(j+1)(0).
Ptrans=RK[fK,x(a)fK,y(a)]*YK(a)fK,x(a)fK,y(a),
Prefl=RK[gK,x(s)gK,y(s)]*YK(s)gK,x(s)gK,y(s),
Pinc=[f0,x(s)f0,y(s)]*Y0(s)f0,x(s)f0,y(s).
ExEy=K exp(ikK·ρ)AK×fK,xfK,yexp(iκKz)+gK,xgK,yexp[iκK(t-z)].
exp(ikK)·ExEy=K AKfK,xgK,xfK,ygK,yexp(iκKz)exp[iκK(t-z)].
fK,xgK,xfK,ygK,y=AKfK,xgK,xfK,ygK,y.
fK,xgK,xfK,ygK,y=AKTfK,xgK,xfK,ygK,y.
Hy-Hx=0μ0 K exp(ikK·ρ)AKYKfK,xgK,xfK,ygK,y×exp(iκKz)-exp[iκK(t-z)],
YK=AKYKAKT.
KK,KγK,K=celldρΩcelldρΩ(ρ)(ρ)×exp[-i(K·ρ-K·ρ)]×K exp[iK·(ρ-ρ)],
KK,KγK,K=δK,K.
KγK,KK,K=δK,K,
κm1κK,1κm2κK,2,
fm1fK,x,fm2fK,y,gm1gK,x,
gm2gK,y,
(PK,m1)x(PK,m2)x(PK,m1)y(PK,m2)y(AK)=1kK(kK)x-(kK)y(kK)y(kK)x,
QAKYK,
·D=xKK,x exp(iK·ρ)K exp(ik·ρ)×m(PK,m)xvm(z)yKK,y exp(iK·ρ)×K exp(ikK·ρ)m(PK,m)yvm(z)-iK exp(ikK·ρ)m[(kK)x(QK,m)x×(kK)y(QK,m)y]κmvm(z).
·D=iK exp(ikK·ρ)Km(kK)xK-K,x(PK,m)x+(kK)yK-K,y(PK,m)y-δK,K[(kK)x(QK,m)x+(kK)y(QK,m)y]κmk0vm(z).
x¯¯xPx+y¯¯yPy-(x¯Qx+y¯Qy)κ¯
=[x¯¯x y¯¯y]P-[x¯ y¯]Q κ,
·H=i0μ0 K exp(ikK·ρ)m{-(kK)x(QK,m)y+(kK)y(QK,m)x+[(kK)x(PK,m)y-(kK)y(PK,m)x]κm}im(z).
[y¯-x¯]Q-[y¯-x¯]P κ=0,

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