Abstract

We present a rigorous analytical approach to diffraction problems posed by multilayered grating structures that incorporate metallic and dielectric elements having arbitrary shapes. The dielectric media can be either transparent or absorbing and may exhibit biaxial anisotropy, while the metallic materials may have finite or infinite conductivities. Our approach uses a modal formulation that describes each layer of the grating configuration in terms of an electrical transmission-line unit. The boundary conditions between adjacent layers are generally expressed by an interface transformer whose properties are dictated by the modal characteristics of the two layers. The electromagnetic behavior of a complex grating configuration can thus be represented by an equivalent network that has simple canonic constituents. Such a formulation of the wave scattering problem serves to systematically derive the diffracted fields everywhere. In addition, it provides a convenient scheme for stable numerical evaluations that have good convergence properties. We demonstrate the accuracy and effectiveness of this approach by examples that include comparisons with situations having exact solutions for the diffracted fields.

© 2001 Optical Society of America

Full Article  |  PDF Article

Errata

Mingming Jiang, Theodor Tamir, and Shuzang Zhang, "Modal theory of diffraction by multilayered gratings containing dielectric and metallic components: errata," J. Opt. Soc. Am. A 19, 1722-1722 (2002)
https://www.osapublishing.org/josaa/abstract.cfm?uri=josaa-19-8-1722

References

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  1. E. N. Glytsis, T. K. Gaylord, “Three-dimensional (vector) rigorous coupled-wave analysis of anisotropic grating diffraction,” J. Opt. Soc. Am. A 7, 1394–1415 (1990).
    [CrossRef]
  2. W.-P. Huang, “Coupled-mode theory for optical waveguides: an overview,” J. Opt. Soc. Am. A 11, 963–983 (1994).
    [CrossRef]
  3. M. G. Moharam, E. B. Grann, D. Q. Pommet, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A 12, 1068–1076 (1995).
    [CrossRef]
  4. M. G. Moharam, D. A. Pommet, E. B. Grann, T. K. Gaylord, “Stable implementation of the rigorous coupled-wave analysis for surface-relief gratings: enhanced transmittance matrix approach,” J. Opt. Soc. Am. A 12, 1077–1086 (1995).
    [CrossRef]
  5. S. Peng, G. M. Morris, “Efficient implementation of rigorous coupled-wave analysis for surface-relief gratings,” J. Opt. Soc. Am. A 12, 1087–1096 (1995).
    [CrossRef]
  6. S. M. Norton, T. Erdogan, G. M. Morris, “Coupled-mode theory of resonant-grating filters,” J. Opt. Soc. Am. A 14, 629–639 (1997).
    [CrossRef]
  7. D. M. Pai, K. A. Awada, “Analysis of dielectric gratings of arbitrary profiles and thicknesses,” J. Opt. Soc. Am. A 8, 755–762 (1991).
    [CrossRef]
  8. K. Matsumoto, K. Rokushima, “Three-dimensional rigorous analysis of dielectric grating waveguides for general cases of oblique propagation,” J. Opt. Soc. Am. A 10, 269–276 (1993).
    [CrossRef]
  9. L. Li, “Multilayer modal method for diffraction gratings of arbitrary profile, depth, and permittivity,” J. Opt. Soc. Am. A 10, 2581–2591 (1993).
    [CrossRef]
  10. L. Li, “Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings,” J. Opt. Soc. Am. A 13, 1024–1035 (1996).
    [CrossRef]
  11. T. Tamir, S. Zhang, “Modal transmission-line theory of multilayered grating structures,” J. Lightwave Technol. 14, 914–927 (1996).
    [CrossRef]
  12. S. Kaushik, “Vector Fresnel equations and Airy formula for one-dimensional multilayer and surface-relief gratings,” J. Opt. Soc. Am. A 14, 596–609 (1997).
    [CrossRef]
  13. L. Li, “Reformulation of the Fourier modal method for surface-relief gratings made with anisotropic materials,” J. Mod. Opt. 45, 1313–1334 (1998).
    [CrossRef]
  14. M. Guglielmi, A. A. Oliner, “Multimode network description of a planar periodic metal-strip grating at a dielectric interface,” IEEE Trans. Microwave Theory Tech. 37, 534–552 (1989).
    [CrossRef]
  15. J. Y. Andersson, L. Lundqvist, “Grating-coupled quantum-well infrared detectors: theory and performance,” J. Appl. Phys. 71, 3600–3610 (1992).
    [CrossRef]
  16. C. Wan, J. A. Encinar, “Efficient computation of generalized scattering matrix for analyzing multilayered periodic structures,” IEEE Trans. Antennas Propag. 43, 1233–1242 (1995).
  17. C. M. Shiao, S. T. Peng, “Distribution of current in-duced on metal-strip gratings by plane wave,” IEEE Trans. Microwave Theory Tech. 46, 883–885 (1998).
    [CrossRef]
  18. M. Nevière, D. Maystre, J. P. Laude, “Perfect blazing for transmission gratings,” J. Opt. Soc. Am. A 7, 1736–1739 (1990).
    [CrossRef]
  19. 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]
  20. K. K. Choi, The Physics of Quantum Well Infrared Photodetectors (World Scientific, Singapore, 1997).
  21. L. P. Rokhinson, C. J. Chen, K.-K. Choi, D. C. Tsiu, G. A. Vawter, L. Yan, M. Jiang, T. Tamir, “Optimization of blazed quantum grid infrared photodetectors,” Appl. Phys. Lett. 75, 3701–3703 (1999).
    [CrossRef]
  22. S. T. Peng, T. Tamir, H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).
    [CrossRef]
  23. S. T. Peng, “Rigorous formulation of scattering and guidance by dielectric grating waveguides: general case of oblique incidence,” J. Opt. Soc. Am. A 6, 1869–1883 (1989).
    [CrossRef]
  24. S. Zhang, T. Tamir, “Rigorous theory of grating-assisted couplers,” J. Opt. Soc. Am. A 13, 2403–2413 (1996).
    [CrossRef]
  25. T. Tamir, S. Zhang, “Resonant scattering by multilayered gratings,” J. Opt. Soc. Am. A 14, 1607–1616 (1997).
    [CrossRef]
  26. R. Magnusson, T. K. Gaylord, “Equivalence of multiwave coupled-wave theory and modal theory of periodic-media diffraction,” J. Opt. Soc. Am. 68, 1777–1779 (1978).
    [CrossRef]
  27. R. E. Collin, Field Theory of Guided Waves (IEEE Press, Piscataway, N.J., 1990), Sec. 6.1, p. 411.
  28. R. Petit, Electromagnetic Theory of Gratings (Springer-Verlag, Berlin, 1980), Sec. 6.2.1, p. 174.
  29. L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A 13, 1870–1876 (1996).
    [CrossRef]

1999

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]

L. P. Rokhinson, C. J. Chen, K.-K. Choi, D. C. Tsiu, G. A. Vawter, L. Yan, M. Jiang, T. Tamir, “Optimization of blazed quantum grid infrared photodetectors,” Appl. Phys. Lett. 75, 3701–3703 (1999).
[CrossRef]

1998

C. M. Shiao, S. T. Peng, “Distribution of current in-duced on metal-strip gratings by plane wave,” IEEE Trans. Microwave Theory Tech. 46, 883–885 (1998).
[CrossRef]

L. Li, “Reformulation of the Fourier modal method for surface-relief gratings made with anisotropic materials,” J. Mod. Opt. 45, 1313–1334 (1998).
[CrossRef]

1997

1996

1995

1994

1993

1992

J. Y. Andersson, L. Lundqvist, “Grating-coupled quantum-well infrared detectors: theory and performance,” J. Appl. Phys. 71, 3600–3610 (1992).
[CrossRef]

1991

1990

M. Nevière, D. Maystre, J. P. Laude, “Perfect blazing for transmission gratings,” J. Opt. Soc. Am. A 7, 1736–1739 (1990).
[CrossRef]

E. N. Glytsis, T. K. Gaylord, “Three-dimensional (vector) rigorous coupled-wave analysis of anisotropic grating diffraction,” J. Opt. Soc. Am. A 7, 1394–1415 (1990).
[CrossRef]

1989

M. Guglielmi, A. A. Oliner, “Multimode network description of a planar periodic metal-strip grating at a dielectric interface,” IEEE Trans. Microwave Theory Tech. 37, 534–552 (1989).
[CrossRef]

S. T. Peng, “Rigorous formulation of scattering and guidance by dielectric grating waveguides: general case of oblique incidence,” J. Opt. Soc. Am. A 6, 1869–1883 (1989).
[CrossRef]

1978

1975

S. T. Peng, T. Tamir, H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).
[CrossRef]

Andersson, J. Y.

J. Y. Andersson, L. Lundqvist, “Grating-coupled quantum-well infrared detectors: theory and performance,” J. Appl. Phys. 71, 3600–3610 (1992).
[CrossRef]

Awada, K. A.

Bertoni, H. L.

S. T. Peng, T. Tamir, H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).
[CrossRef]

Chen, C. J.

L. P. Rokhinson, C. J. Chen, K.-K. Choi, D. C. Tsiu, G. A. Vawter, L. Yan, M. Jiang, T. Tamir, “Optimization of blazed quantum grid infrared photodetectors,” Appl. Phys. Lett. 75, 3701–3703 (1999).
[CrossRef]

Choi, K. K.

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

Choi, K.-K.

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]

L. P. Rokhinson, C. J. Chen, K.-K. Choi, D. C. Tsiu, G. A. Vawter, L. Yan, M. Jiang, T. Tamir, “Optimization of blazed quantum grid infrared photodetectors,” Appl. Phys. Lett. 75, 3701–3703 (1999).
[CrossRef]

Collin, R. E.

R. E. Collin, Field Theory of Guided Waves (IEEE Press, Piscataway, N.J., 1990), Sec. 6.1, p. 411.

Encinar, J. A.

C. Wan, J. A. Encinar, “Efficient computation of generalized scattering matrix for analyzing multilayered periodic structures,” IEEE Trans. Antennas Propag. 43, 1233–1242 (1995).

Erdogan, T.

Gaylord, T. K.

Glytsis, E. N.

E. N. Glytsis, T. K. Gaylord, “Three-dimensional (vector) rigorous coupled-wave analysis of anisotropic grating diffraction,” J. Opt. Soc. Am. A 7, 1394–1415 (1990).
[CrossRef]

Grann, E. B.

Guglielmi, M.

M. Guglielmi, A. A. Oliner, “Multimode network description of a planar periodic metal-strip grating at a dielectric interface,” IEEE Trans. Microwave Theory Tech. 37, 534–552 (1989).
[CrossRef]

Huang, W.-P.

Jiang, M.

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]

L. P. Rokhinson, C. J. Chen, K.-K. Choi, D. C. Tsiu, G. A. Vawter, L. Yan, M. Jiang, T. Tamir, “Optimization of blazed quantum grid infrared photodetectors,” Appl. Phys. Lett. 75, 3701–3703 (1999).
[CrossRef]

Kaushik, S.

Laude, J. P.

Li, L.

Lundqvist, L.

J. Y. Andersson, L. Lundqvist, “Grating-coupled quantum-well infrared detectors: theory and performance,” J. Appl. Phys. 71, 3600–3610 (1992).
[CrossRef]

Magnusson, R.

Matsumoto, K.

Maystre, D.

Moharam, M. G.

Morris, G. M.

Nevière, M.

Norton, S. M.

Oliner, A. A.

M. Guglielmi, A. A. Oliner, “Multimode network description of a planar periodic metal-strip grating at a dielectric interface,” IEEE Trans. Microwave Theory Tech. 37, 534–552 (1989).
[CrossRef]

Pai, D. M.

Peng, S.

Peng, S. T.

C. M. Shiao, S. T. Peng, “Distribution of current in-duced on metal-strip gratings by plane wave,” IEEE Trans. Microwave Theory Tech. 46, 883–885 (1998).
[CrossRef]

S. T. Peng, “Rigorous formulation of scattering and guidance by dielectric grating waveguides: general case of oblique incidence,” J. Opt. Soc. Am. A 6, 1869–1883 (1989).
[CrossRef]

S. T. Peng, T. Tamir, H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).
[CrossRef]

Petit, R.

R. Petit, Electromagnetic Theory of Gratings (Springer-Verlag, Berlin, 1980), Sec. 6.2.1, p. 174.

Pommet, D. A.

Pommet, D. Q.

Rokhinson, L. P.

L. P. Rokhinson, C. J. Chen, K.-K. Choi, D. C. Tsiu, G. A. Vawter, L. Yan, M. Jiang, T. Tamir, “Optimization of blazed quantum grid infrared photodetectors,” Appl. Phys. Lett. 75, 3701–3703 (1999).
[CrossRef]

Rokushima, K.

Shiao, C. M.

C. M. Shiao, S. T. Peng, “Distribution of current in-duced on metal-strip gratings by plane wave,” IEEE Trans. Microwave Theory Tech. 46, 883–885 (1998).
[CrossRef]

Tamir, T.

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]

L. P. Rokhinson, C. J. Chen, K.-K. Choi, D. C. Tsiu, G. A. Vawter, L. Yan, M. Jiang, T. Tamir, “Optimization of blazed quantum grid infrared photodetectors,” Appl. Phys. Lett. 75, 3701–3703 (1999).
[CrossRef]

T. Tamir, S. Zhang, “Resonant scattering by multilayered gratings,” J. Opt. Soc. Am. A 14, 1607–1616 (1997).
[CrossRef]

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

S. Zhang, T. Tamir, “Rigorous theory of grating-assisted couplers,” J. Opt. Soc. Am. A 13, 2403–2413 (1996).
[CrossRef]

S. T. Peng, T. Tamir, H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).
[CrossRef]

Tsiu, D. C.

L. P. Rokhinson, C. J. Chen, K.-K. Choi, D. C. Tsiu, G. A. Vawter, L. Yan, M. Jiang, T. Tamir, “Optimization of blazed quantum grid infrared photodetectors,” Appl. Phys. Lett. 75, 3701–3703 (1999).
[CrossRef]

Vawter, G. A.

L. P. Rokhinson, C. J. Chen, K.-K. Choi, D. C. Tsiu, G. A. Vawter, L. Yan, M. Jiang, T. Tamir, “Optimization of blazed quantum grid infrared photodetectors,” Appl. Phys. Lett. 75, 3701–3703 (1999).
[CrossRef]

Wan, C.

C. Wan, J. A. Encinar, “Efficient computation of generalized scattering matrix for analyzing multilayered periodic structures,” IEEE Trans. Antennas Propag. 43, 1233–1242 (1995).

Yan, L.

L. P. Rokhinson, C. J. Chen, K.-K. Choi, D. C. Tsiu, G. A. Vawter, L. Yan, M. Jiang, T. Tamir, “Optimization of blazed quantum grid infrared photodetectors,” Appl. Phys. Lett. 75, 3701–3703 (1999).
[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]

Zhang, S.

Appl. Phys. Lett.

L. P. Rokhinson, C. J. Chen, K.-K. Choi, D. C. Tsiu, G. A. Vawter, L. Yan, M. Jiang, T. Tamir, “Optimization of blazed quantum grid infrared photodetectors,” Appl. Phys. Lett. 75, 3701–3703 (1999).
[CrossRef]

IEEE J. Quantum Electron.

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]

IEEE Trans. Antennas Propag.

C. Wan, J. A. Encinar, “Efficient computation of generalized scattering matrix for analyzing multilayered periodic structures,” IEEE Trans. Antennas Propag. 43, 1233–1242 (1995).

IEEE Trans. Microwave Theory Tech.

C. M. Shiao, S. T. Peng, “Distribution of current in-duced on metal-strip gratings by plane wave,” IEEE Trans. Microwave Theory Tech. 46, 883–885 (1998).
[CrossRef]

M. Guglielmi, A. A. Oliner, “Multimode network description of a planar periodic metal-strip grating at a dielectric interface,” IEEE Trans. Microwave Theory Tech. 37, 534–552 (1989).
[CrossRef]

S. T. Peng, T. Tamir, H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).
[CrossRef]

J. Appl. Phys.

J. Y. Andersson, L. Lundqvist, “Grating-coupled quantum-well infrared detectors: theory and performance,” J. Appl. Phys. 71, 3600–3610 (1992).
[CrossRef]

J. Lightwave Technol.

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

J. Mod. Opt.

L. Li, “Reformulation of the Fourier modal method for surface-relief gratings made with anisotropic materials,” J. Mod. Opt. 45, 1313–1334 (1998).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

S. Zhang, T. Tamir, “Rigorous theory of grating-assisted couplers,” J. Opt. Soc. Am. A 13, 2403–2413 (1996).
[CrossRef]

M. Nevière, D. Maystre, J. P. Laude, “Perfect blazing for transmission gratings,” J. Opt. Soc. Am. A 7, 1736–1739 (1990).
[CrossRef]

E. N. Glytsis, T. K. Gaylord, “Three-dimensional (vector) rigorous coupled-wave analysis of anisotropic grating diffraction,” J. Opt. Soc. Am. A 7, 1394–1415 (1990).
[CrossRef]

W.-P. Huang, “Coupled-mode theory for optical waveguides: an overview,” J. Opt. Soc. Am. A 11, 963–983 (1994).
[CrossRef]

S. Kaushik, “Vector Fresnel equations and Airy formula for one-dimensional multilayer and surface-relief gratings,” J. Opt. Soc. Am. A 14, 596–609 (1997).
[CrossRef]

S. M. Norton, T. Erdogan, G. M. Morris, “Coupled-mode theory of resonant-grating filters,” J. Opt. Soc. Am. A 14, 629–639 (1997).
[CrossRef]

T. Tamir, S. Zhang, “Resonant scattering by multilayered gratings,” J. Opt. Soc. Am. A 14, 1607–1616 (1997).
[CrossRef]

S. T. Peng, “Rigorous formulation of scattering and guidance by dielectric grating waveguides: general case of oblique incidence,” J. Opt. Soc. Am. A 6, 1869–1883 (1989).
[CrossRef]

D. M. Pai, K. A. Awada, “Analysis of dielectric gratings of arbitrary profiles and thicknesses,” J. Opt. Soc. Am. A 8, 755–762 (1991).
[CrossRef]

K. Matsumoto, K. Rokushima, “Three-dimensional rigorous analysis of dielectric grating waveguides for general cases of oblique propagation,” J. Opt. Soc. Am. A 10, 269–276 (1993).
[CrossRef]

L. Li, “Multilayer modal method for diffraction gratings of arbitrary profile, depth, and permittivity,” J. Opt. Soc. Am. A 10, 2581–2591 (1993).
[CrossRef]

L. Li, “Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings,” J. Opt. Soc. Am. A 13, 1024–1035 (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. G. Moharam, E. B. Grann, D. Q. Pommet, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A 12, 1068–1076 (1995).
[CrossRef]

M. G. Moharam, D. A. Pommet, E. B. Grann, T. K. Gaylord, “Stable implementation of the rigorous coupled-wave analysis for surface-relief gratings: enhanced transmittance matrix approach,” J. Opt. Soc. Am. A 12, 1077–1086 (1995).
[CrossRef]

S. Peng, G. M. Morris, “Efficient implementation of rigorous coupled-wave analysis for surface-relief gratings,” J. Opt. Soc. Am. A 12, 1087–1096 (1995).
[CrossRef]

Other

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

R. E. Collin, Field Theory of Guided Waves (IEEE Press, Piscataway, N.J., 1990), Sec. 6.1, p. 411.

R. Petit, Electromagnetic Theory of Gratings (Springer-Verlag, Berlin, 1980), Sec. 6.2.1, p. 174.

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Figures (10)

Fig. 1
Fig. 1

Geometry of a general grating configuration.

Fig. 2
Fig. 2

Typical grating layer or sublayer.

Fig. 3
Fig. 3

Equivalent electrical network elements: (a) transmission-line unit representing a single layer, (b) general junction of two transmission-line units.

Fig. 4
Fig. 4

Types of simple interfaces.

Fig. 5
Fig. 5

Types of more-complex metal–metal interfaces.

Fig. 6
Fig. 6

Equivalent transmission-line network for a typical grating configuration as in Fig. 1.

Fig. 7
Fig. 7

Equivalent transmission-line network for the metal–metal junction in Fig. 5(b).

Fig. 8
Fig. 8

Incidence on a blazed reflection grating: (a) grating geometry, (b) calculated intensities of diffracted orders for P=3 and θ=30° using N×N truncated matrices and S horizontal sublayers, (c) contours of |Hy| field amplitude.

Fig. 9
Fig. 9

Incidence on a blazed transmission grating: (a) grating geometry, (b) calculated intensities of diffracted orders for a specific case.

Fig. 10
Fig. 10

Diffraction in a grating21 forming part of a quantum-well infrared photodetector. (a) Grating geometry: all lengths are in micrometers, εg=11.1556, εa is a uniaxial tensor with εax=10.315+0.3i, and εay=εaz=10.315. (b) Normalized reflected prefl, transmitted ptr and absorbed pabs powers at λ=7.6 µm for variable period Λ and fixed spacing s=1.5 µm. (c) Intensity contours for |Ex| at the first absorption peak.

Tables (2)

Tables Icon

Table 1 Listing of Conventional (Scalar) and Generalized (Matrix) Transmission-Line Functionsa

Tables Icon

Table 2 Listing of Transformations at Simple Interfaces

Equations (72)

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

kzn=ko(εs sin θ+nλ/Λ),n=0,±1,±2,,
Ej(x, z)=mvjm(x)ejm(z),
Hj(x, z)=mijm(x)hjm(z),
vjm(xj)=fjm exp[-ikjm(tj-xj)]+gjm exp[ikjm(tj-xj)],
ijm(xj)=Yjm{fjm exp[-ikjm(tj-xj)]-gjm exp[ikjm(tj-xj)]},
Yjm=1Zjm=kjm/ωμo(TE)ωεo/kjmγj0(TM),
eum(z)=hum(z)=exp(ikznz)=hun(z)=eun(z),
kum=kun=(k02εyu-kzn2)1/2(TE)[(k02-kzn2/εxu)εzu]1/2(TM),
edm(z)=nanm(d) exp(ikznz),
hdm(z)=nbnm(d) exp(ikznz),
E¯d=nv¯dn(x)exp(ikznz),
H¯d=ni¯dn(x)exp(ikznz),
v¯dn=manm(d)vjm,
i¯dn=mbnm(d)ijm,
epm(z)=exp(ikz0qΛ)νaνm(p)trig[νπ(z0-dp)/wp],
hpm(z)=exp(ikz0qΛ)νbνm(p) trig[νπ(z0-dp)/wp],
kpm=[k02εpy-(mπ/wp)2]1/2m=1, 2, 3,(TE)[k02εpz-(mπ/wp)2(εpz/εpx)]1/2m=0, 1, 2,(TM) .
vxj=exp[-iKj(tj-xj)]ftj+exp[iKj(tj-xj)]gtj,
ixj=Yj{exp[-iKj(tj-xj)]ftj-exp[iKj(tj-xj)]gtj},
Advtd=Ad+1v0,d+1,
Bditd=Bd+1i0,d+1,
mvdmnanm(d) exp(ikznz0)=Ej(hj, z0)=μvpμνaνμ(ρ) trig[νπ(z0-dp)/wp],forallz0,
manm(d)vdm=μvpμνaνμ(p)dpdp+wp exp(-ikznz0)Λ×trig[νπ(z0-dp)/wp]dz0,
foreveryn,
Advd=TνjApvp,
midmnbnm(d) exp(ikznz0)=Hj(hj, z0)=μipμνbνμ(p) trig[νπ(z0-dp)],fordpz0dp+wp.
μbνμ(p)ipμ=midmnbnm(d)×dpdp+wp cν trig[νπ(z0-dp/wp)]wp×exp(ikznz0)dz0,forevery ν,
Bpip=TijBdid,
μaνμ(g)vgμ=μvsμνaνμ(s)dsds+ws cν trig[νπ(z0-dg)/wg]wg×trig[νπ(z0-ds)/ws]dz0,forevery ν,
μbνμ(s)isμ
=μigμνbνμ(g)dsds+ws cν trig[νπ(z0-ds)/ws]ws×trig[νπ(z0-dg)/wg]dzd,forevery ν,
Agvg=TvjAsvs,
Bsis=TijBgig,
Rtj=Frj(R0,j+1),
f0,j+1=Tfjftj,
R0j=exp(iKjtj)Rtj exp(iKjtj).
gs=Rsfs,
ftj=exp(iKjtj)Tf,j-1ft,j-1.
f0,J+1=fc=TfJftJ.
Rtj=(Ztj-Zj)(Ztj+Zj)-1,
Ztj=Aj-1TvljZljTiljBj
Zlj=(I+Rlj)(I-Rlj)-1Zτ,
Ruj=(Yτ+Yuj)-1(Yτ-Yuj),
Yuj=TiujBj+1Y0,j+1Aj+1-1Tvuj,
Y0,j+1=Yj+1(I-R0,j+1)(I+R0,j+1)-1.
(I+R0,j+1)f0,j+1=v0,j+1=Aj+1-1Tvujvuj=Aj+1-1TvujZujiuj=Aj+1-1TvujZljTiljBjitj=Aj+1-1TvujZljTiljBjYj(I-Rtj)ftj,
Tfj=(I+R0,j+1)-1Aj+1-1TvujZljTiljBjYj(I-Rtj)
Em±(x, z)=exp(±ikmx)em(z)=exp(±ikmx)rarmϕr(z),
Hm±(x, z)=±Ym exp(±ikmx)hm(z)=±Ym exp(±ikmx)rbrmϕr(z),
ϕr(z)=exp(ikznz),withr=n=0,±1,±2,forregionswithoutPEC'sexp(ikz0pΛ)sin[νπ(z0-d)/w],r=ν=1, 2, 3,(TE)forregionswithPEC'sexp(ikz0pΛ)cos[νπ(z0-d)/w],r=ν=0,1,2,(TM)forregionswithPECs
εx(z)=rσrψr(z),
εy(z)=rτrψr(z),
γ(z)=1/εz(z)=rγrψr(z),
ψr(z)=exp(2inπz/Λ),withr=n=0,±1,±2,forregionswithoutPEC'scos[νπ(z0-d)/w],withr=ν=0, 1, 2, forregionswithPEC's.
iωμoHxm=-Eym/z,
iωμ0Hzm=Eym/x,
iωεoεy(z)Eym=Hzmx-Hxm/z.
ωμohxm=-Dam.
ωμoYmbm=kmam,
drm=kznδnmforregionswithoutPEC's(νπ/w)δνmforregionswithPEC's.
am=bm.
(ko2Py-D2)am=km2am,
prm=τn-m forregionswithoutPEC's(τ0δνm+τ|ν-m|-τν+m)/2forregionswithPEC's.
iωεoεx(z)Exm=Hym/z,
iωεoEzm=-γ(z)Hym/x,
-iωμoHym=Ezm/x-Exm/z,
ωεoPxexm=-Dbm,
ωε0am=kmYmQbm,
prm=pnm=σn-m forregionswithoutPEC'sσν ifm=0,(σ0δ0m+σm)/2if ν=0forregionswithPEC's(σ0δνm+σ|ν-m|-σν+m|)/2if ν, m0for regions with PECs.
qrm=qnm=γn-m forregionswithoutPEC'sγν ifm=0,(γ0δ0m+γm)/2if ν=0forregionswithPEC's(γ0δνm+γ|ν-m|+γν+m)/2if ν,m0for regionswith PECs.
γ0am=Qbm.
Q-1(ko2I-DPx-1D)bm=km2bm,

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