Abstract

We use a nanofabricated gold grating to validate two-dimensional spatial harmonic analysis (2D SHA) method, also known as Fourier modal method or rigorous coupled-wave analysis under oblique incidence. The transmittance spectra of the metal grating for incident angles of 0°–30° are obtained in both experiments and simulations at the zero diffraction order. The simulations are performed with our custom software, a fully functional web-based 2D SHA tool, which has been staged at nanoHUB.org and is free to use. The simulation results are compared to experimentally measured values, and a good fit is achieved for all incident angles. Possible reasons for experiment-simulation mismatch in actual nano-plasmonic structures are also discussed.

© 2010 Optical Society of America

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References

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  1. C. B. Burckhardt, “Diffraction of a plane wave at a sinusoidally stratified dielectric grating,” J. Opt. Soc. Am. 56, 1502–1509 (1966).
    [CrossRef]
  2. F. G. Kaspar, “Diffraction by thick, periodically stratified gratings with complex dielectric-constant,” J. Opt. Soc. Am. 63, 37–45 (1973).
    [CrossRef]
  3. K. Knop, “Rigorous diffraction theory for transmission phase gratings with deep rectangular grooves,” J. Opt. Soc. Am. 68, 1206–1210 (1978).
    [CrossRef]
  4. M. G. Moharam and T. K. Gaylord, “Rigorous coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. 71, 811–818 (1981).
    [CrossRef]
  5. M. G. Moharam and T. K. Gaylord, “Rigorous coupled-wave analysis of grating diffraction-E-mode polarization and losses,” J. Opt. Soc. Am. 73, 451–455 (1983).
    [CrossRef]
  6. P. Lalanne and G. M. Morris, “Highly improved convergence of the coupled-wave method for TM polarization,” J. Opt. Soc. Am. A 13, 779–784 (1996).
    [CrossRef]
  7. G. Granet and B. Guizal, “Efficient implementation of the coupled-wave method for metallic lamellar gratings in TM polarization,” J. Opt. Soc. Am. A 13, 1019–1023 (1996).
    [CrossRef]
  8. L. F. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A 13, 1870–1876 (1996).
    [CrossRef]
  9. L. F. Li, “Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings,” J. Opt. Soc. Am. A 13, 1024–1035 (1996).
    [CrossRef]
  10. L. F. Li, “Note on the S-matrix propagation algorithm,” J. Opt. Soc. Am. A 20, 655–660 (2003).
    [CrossRef]
  11. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966–3969 (2000).
    [CrossRef] [PubMed]
  12. W. S. Cai, U. K. Chettiar, H. K. Yuan, V. C. de Silva, A. V. Kildishev, V. P. Drachev, and V. M. Shalaev, “Metamagnetics with rainbow colors,” Opt. Express 15, 3333–3341 (2007).
    [CrossRef] [PubMed]
  13. U. K. Chettiar, A. V. Kildishev, T. A. Klar, and V. M. Shalaev, “Negative index metamaterial combining magnetic resonators with metal films,” Opt. Express 14, 7872–7877 (2006).
    [CrossRef] [PubMed]
  14. V. P. Drachev, W. Cai, U. Chettiar, H. K. Yuan, A. K. Sarychev, A. V. Kildishev, G. Klimeck, and V. M. Shalaev, “Experimental verification of an optical negative-index material,” Laser Phys. Lett. 3, 49–55 (2006).
    [CrossRef]
  15. V. M. Shalaev, W. S. Cai, U. K. Chettiar, H. K. Yuan, A. K. Sarychev, V. P. Drachev, and A. V. Kildishev, “Negative index of refraction in optical metamaterials,” Opt. Lett. 30, 3356–3358 (2005).
    [CrossRef]
  16. H. K. Yuan, U. K. Chettiar, W. S. Cai, A. V. Kildishev, A. Boltasseva, V. P. Drachev, and V. M. Shalaev, “A negative permeability material at red light,” Opt. Express 15, 1076–1083 (2007).
    [CrossRef] [PubMed]
  17. S. Zhang, W. J. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. J. Brueck, “Experimental demonstration of near-infrared negative-index metamaterials,” Phys. Rev. Lett. 95, 137404 (2005).
    [CrossRef] [PubMed]
  18. A. V. Kildishev and U. K. Chettiar, “Cascading optical negative index metamaterials,” Appl. Comput. Electromagn. Soc. J. 22, 172–183 (2007).
  19. X. Ni, Z. Liu, F. Gu, M. G. Pacheco, A. V. Kildishev, and J. Borneman, “PhotonicsSHA-2D: Modeling of single-period multilayer optical gratings and metamaterials,” (2009), DOI 10254/nanohub-r6977.9.
  20. P. B. Johnson and R. W. Christy, “Optical constants of noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
    [CrossRef]
  21. V. P. Drachev, U. K. Chettiar, A. V. Kildishev, H. K. Yuan, W. S. Cai, and V. M. Shalaev, “The Ag dielectric function in plasmonic metamaterials,” Opt. Express 16, 1186–1195 (2008).
    [CrossRef] [PubMed]
  22. E. G. Loewen and E. Popov, Diffraction Gratings and Applications (Marcel Dekker, 1997).
  23. K.-P. Chen, V. P. Drachev, J. D. Borneman, A. V. Kildishev, and V. M. Shalaev, “Drude relaxation rate in grained gold nanoantennas,” Nano Lett. 10, 916–922 (2010), DOI 10.1021/nl9037246.
    [CrossRef] [PubMed]
  24. K. B. Alici and E. Ozbay, “Oblique response of a split-ring-resonator-based left-handed metamaterial slab,” Opt. Lett. 34, 2294–2296 (2009).
    [CrossRef] [PubMed]
  25. C. Menzel, C. Helgert, J. Upping, C. Rockstuhl, E. B. Kley, R. B. Wehrspohn, T. Pertsch, and F. Lederer, “Angular resolved effective optical properties of a Swiss cross metamaterial,” Appl. Phys. Lett. 95 (2009).
    [CrossRef]

2010 (1)

K.-P. Chen, V. P. Drachev, J. D. Borneman, A. V. Kildishev, and V. M. Shalaev, “Drude relaxation rate in grained gold nanoantennas,” Nano Lett. 10, 916–922 (2010), DOI 10.1021/nl9037246.
[CrossRef] [PubMed]

2009 (2)

K. B. Alici and E. Ozbay, “Oblique response of a split-ring-resonator-based left-handed metamaterial slab,” Opt. Lett. 34, 2294–2296 (2009).
[CrossRef] [PubMed]

C. Menzel, C. Helgert, J. Upping, C. Rockstuhl, E. B. Kley, R. B. Wehrspohn, T. Pertsch, and F. Lederer, “Angular resolved effective optical properties of a Swiss cross metamaterial,” Appl. Phys. Lett. 95 (2009).
[CrossRef]

2008 (1)

2007 (3)

2006 (2)

U. K. Chettiar, A. V. Kildishev, T. A. Klar, and V. M. Shalaev, “Negative index metamaterial combining magnetic resonators with metal films,” Opt. Express 14, 7872–7877 (2006).
[CrossRef] [PubMed]

V. P. Drachev, W. Cai, U. Chettiar, H. K. Yuan, A. K. Sarychev, A. V. Kildishev, G. Klimeck, and V. M. Shalaev, “Experimental verification of an optical negative-index material,” Laser Phys. Lett. 3, 49–55 (2006).
[CrossRef]

2005 (2)

V. M. Shalaev, W. S. Cai, U. K. Chettiar, H. K. Yuan, A. K. Sarychev, V. P. Drachev, and A. V. Kildishev, “Negative index of refraction in optical metamaterials,” Opt. Lett. 30, 3356–3358 (2005).
[CrossRef]

S. Zhang, W. J. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. J. Brueck, “Experimental demonstration of near-infrared negative-index metamaterials,” Phys. Rev. Lett. 95, 137404 (2005).
[CrossRef] [PubMed]

2003 (1)

2000 (1)

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966–3969 (2000).
[CrossRef] [PubMed]

1997 (1)

E. G. Loewen and E. Popov, Diffraction Gratings and Applications (Marcel Dekker, 1997).

1996 (4)

1983 (1)

1981 (1)

1978 (1)

1973 (1)

1972 (1)

P. B. Johnson and R. W. Christy, “Optical constants of noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[CrossRef]

1966 (1)

Alici, K. B.

Boltasseva, A.

Borneman, J.

X. Ni, Z. Liu, F. Gu, M. G. Pacheco, A. V. Kildishev, and J. Borneman, “PhotonicsSHA-2D: Modeling of single-period multilayer optical gratings and metamaterials,” (2009), DOI 10254/nanohub-r6977.9.

Borneman, J. D.

K.-P. Chen, V. P. Drachev, J. D. Borneman, A. V. Kildishev, and V. M. Shalaev, “Drude relaxation rate in grained gold nanoantennas,” Nano Lett. 10, 916–922 (2010), DOI 10.1021/nl9037246.
[CrossRef] [PubMed]

Brueck, S. R. J.

S. Zhang, W. J. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. J. Brueck, “Experimental demonstration of near-infrared negative-index metamaterials,” Phys. Rev. Lett. 95, 137404 (2005).
[CrossRef] [PubMed]

Burckhardt, C. B.

Cai, W.

V. P. Drachev, W. Cai, U. Chettiar, H. K. Yuan, A. K. Sarychev, A. V. Kildishev, G. Klimeck, and V. M. Shalaev, “Experimental verification of an optical negative-index material,” Laser Phys. Lett. 3, 49–55 (2006).
[CrossRef]

Cai, W. S.

Chen, K. -P.

K.-P. Chen, V. P. Drachev, J. D. Borneman, A. V. Kildishev, and V. M. Shalaev, “Drude relaxation rate in grained gold nanoantennas,” Nano Lett. 10, 916–922 (2010), DOI 10.1021/nl9037246.
[CrossRef] [PubMed]

Chettiar, U.

V. P. Drachev, W. Cai, U. Chettiar, H. K. Yuan, A. K. Sarychev, A. V. Kildishev, G. Klimeck, and V. M. Shalaev, “Experimental verification of an optical negative-index material,” Laser Phys. Lett. 3, 49–55 (2006).
[CrossRef]

Chettiar, U. K.

Christy, R. W.

P. B. Johnson and R. W. Christy, “Optical constants of noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[CrossRef]

de Silva, V. C.

Drachev, V. P.

Fan, W. J.

S. Zhang, W. J. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. J. Brueck, “Experimental demonstration of near-infrared negative-index metamaterials,” Phys. Rev. Lett. 95, 137404 (2005).
[CrossRef] [PubMed]

Gaylord, T. K.

Granet, G.

Gu, F.

X. Ni, Z. Liu, F. Gu, M. G. Pacheco, A. V. Kildishev, and J. Borneman, “PhotonicsSHA-2D: Modeling of single-period multilayer optical gratings and metamaterials,” (2009), DOI 10254/nanohub-r6977.9.

Guizal, B.

Helgert, C.

C. Menzel, C. Helgert, J. Upping, C. Rockstuhl, E. B. Kley, R. B. Wehrspohn, T. Pertsch, and F. Lederer, “Angular resolved effective optical properties of a Swiss cross metamaterial,” Appl. Phys. Lett. 95 (2009).
[CrossRef]

Johnson, P. B.

P. B. Johnson and R. W. Christy, “Optical constants of noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[CrossRef]

Kaspar, F. G.

Kildishev, A. V.

K.-P. Chen, V. P. Drachev, J. D. Borneman, A. V. Kildishev, and V. M. Shalaev, “Drude relaxation rate in grained gold nanoantennas,” Nano Lett. 10, 916–922 (2010), DOI 10.1021/nl9037246.
[CrossRef] [PubMed]

V. P. Drachev, U. K. Chettiar, A. V. Kildishev, H. K. Yuan, W. S. Cai, and V. M. Shalaev, “The Ag dielectric function in plasmonic metamaterials,” Opt. Express 16, 1186–1195 (2008).
[CrossRef] [PubMed]

A. V. Kildishev and U. K. Chettiar, “Cascading optical negative index metamaterials,” Appl. Comput. Electromagn. Soc. J. 22, 172–183 (2007).

H. K. Yuan, U. K. Chettiar, W. S. Cai, A. V. Kildishev, A. Boltasseva, V. P. Drachev, and V. M. Shalaev, “A negative permeability material at red light,” Opt. Express 15, 1076–1083 (2007).
[CrossRef] [PubMed]

W. S. Cai, U. K. Chettiar, H. K. Yuan, V. C. de Silva, A. V. Kildishev, V. P. Drachev, and V. M. Shalaev, “Metamagnetics with rainbow colors,” Opt. Express 15, 3333–3341 (2007).
[CrossRef] [PubMed]

U. K. Chettiar, A. V. Kildishev, T. A. Klar, and V. M. Shalaev, “Negative index metamaterial combining magnetic resonators with metal films,” Opt. Express 14, 7872–7877 (2006).
[CrossRef] [PubMed]

V. P. Drachev, W. Cai, U. Chettiar, H. K. Yuan, A. K. Sarychev, A. V. Kildishev, G. Klimeck, and V. M. Shalaev, “Experimental verification of an optical negative-index material,” Laser Phys. Lett. 3, 49–55 (2006).
[CrossRef]

V. M. Shalaev, W. S. Cai, U. K. Chettiar, H. K. Yuan, A. K. Sarychev, V. P. Drachev, and A. V. Kildishev, “Negative index of refraction in optical metamaterials,” Opt. Lett. 30, 3356–3358 (2005).
[CrossRef]

X. Ni, Z. Liu, F. Gu, M. G. Pacheco, A. V. Kildishev, and J. Borneman, “PhotonicsSHA-2D: Modeling of single-period multilayer optical gratings and metamaterials,” (2009), DOI 10254/nanohub-r6977.9.

Klar, T. A.

Kley, E. B.

C. Menzel, C. Helgert, J. Upping, C. Rockstuhl, E. B. Kley, R. B. Wehrspohn, T. Pertsch, and F. Lederer, “Angular resolved effective optical properties of a Swiss cross metamaterial,” Appl. Phys. Lett. 95 (2009).
[CrossRef]

Klimeck, G.

V. P. Drachev, W. Cai, U. Chettiar, H. K. Yuan, A. K. Sarychev, A. V. Kildishev, G. Klimeck, and V. M. Shalaev, “Experimental verification of an optical negative-index material,” Laser Phys. Lett. 3, 49–55 (2006).
[CrossRef]

Knop, K.

Lalanne, P.

Lederer, F.

C. Menzel, C. Helgert, J. Upping, C. Rockstuhl, E. B. Kley, R. B. Wehrspohn, T. Pertsch, and F. Lederer, “Angular resolved effective optical properties of a Swiss cross metamaterial,” Appl. Phys. Lett. 95 (2009).
[CrossRef]

Li, L. F.

Liu, Z.

X. Ni, Z. Liu, F. Gu, M. G. Pacheco, A. V. Kildishev, and J. Borneman, “PhotonicsSHA-2D: Modeling of single-period multilayer optical gratings and metamaterials,” (2009), DOI 10254/nanohub-r6977.9.

Loewen, E. G.

E. G. Loewen and E. Popov, Diffraction Gratings and Applications (Marcel Dekker, 1997).

Malloy, K. J.

S. Zhang, W. J. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. J. Brueck, “Experimental demonstration of near-infrared negative-index metamaterials,” Phys. Rev. Lett. 95, 137404 (2005).
[CrossRef] [PubMed]

Menzel, C.

C. Menzel, C. Helgert, J. Upping, C. Rockstuhl, E. B. Kley, R. B. Wehrspohn, T. Pertsch, and F. Lederer, “Angular resolved effective optical properties of a Swiss cross metamaterial,” Appl. Phys. Lett. 95 (2009).
[CrossRef]

Moharam, M. G.

Morris, G. M.

Ni, X.

X. Ni, Z. Liu, F. Gu, M. G. Pacheco, A. V. Kildishev, and J. Borneman, “PhotonicsSHA-2D: Modeling of single-period multilayer optical gratings and metamaterials,” (2009), DOI 10254/nanohub-r6977.9.

Osgood, R. M.

S. Zhang, W. J. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. J. Brueck, “Experimental demonstration of near-infrared negative-index metamaterials,” Phys. Rev. Lett. 95, 137404 (2005).
[CrossRef] [PubMed]

Ozbay, E.

Pacheco, M. G.

X. Ni, Z. Liu, F. Gu, M. G. Pacheco, A. V. Kildishev, and J. Borneman, “PhotonicsSHA-2D: Modeling of single-period multilayer optical gratings and metamaterials,” (2009), DOI 10254/nanohub-r6977.9.

Panoiu, N. C.

S. Zhang, W. J. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. J. Brueck, “Experimental demonstration of near-infrared negative-index metamaterials,” Phys. Rev. Lett. 95, 137404 (2005).
[CrossRef] [PubMed]

Pendry, J. B.

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966–3969 (2000).
[CrossRef] [PubMed]

Pertsch, T.

C. Menzel, C. Helgert, J. Upping, C. Rockstuhl, E. B. Kley, R. B. Wehrspohn, T. Pertsch, and F. Lederer, “Angular resolved effective optical properties of a Swiss cross metamaterial,” Appl. Phys. Lett. 95 (2009).
[CrossRef]

Popov, E.

E. G. Loewen and E. Popov, Diffraction Gratings and Applications (Marcel Dekker, 1997).

Rockstuhl, C.

C. Menzel, C. Helgert, J. Upping, C. Rockstuhl, E. B. Kley, R. B. Wehrspohn, T. Pertsch, and F. Lederer, “Angular resolved effective optical properties of a Swiss cross metamaterial,” Appl. Phys. Lett. 95 (2009).
[CrossRef]

Sarychev, A. K.

V. P. Drachev, W. Cai, U. Chettiar, H. K. Yuan, A. K. Sarychev, A. V. Kildishev, G. Klimeck, and V. M. Shalaev, “Experimental verification of an optical negative-index material,” Laser Phys. Lett. 3, 49–55 (2006).
[CrossRef]

V. M. Shalaev, W. S. Cai, U. K. Chettiar, H. K. Yuan, A. K. Sarychev, V. P. Drachev, and A. V. Kildishev, “Negative index of refraction in optical metamaterials,” Opt. Lett. 30, 3356–3358 (2005).
[CrossRef]

Shalaev, V. M.

Upping, J.

C. Menzel, C. Helgert, J. Upping, C. Rockstuhl, E. B. Kley, R. B. Wehrspohn, T. Pertsch, and F. Lederer, “Angular resolved effective optical properties of a Swiss cross metamaterial,” Appl. Phys. Lett. 95 (2009).
[CrossRef]

Wehrspohn, R. B.

C. Menzel, C. Helgert, J. Upping, C. Rockstuhl, E. B. Kley, R. B. Wehrspohn, T. Pertsch, and F. Lederer, “Angular resolved effective optical properties of a Swiss cross metamaterial,” Appl. Phys. Lett. 95 (2009).
[CrossRef]

Yuan, H. K.

Zhang, S.

S. Zhang, W. J. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. J. Brueck, “Experimental demonstration of near-infrared negative-index metamaterials,” Phys. Rev. Lett. 95, 137404 (2005).
[CrossRef] [PubMed]

Appl. Comput. Electromagn. Soc. J. (1)

A. V. Kildishev and U. K. Chettiar, “Cascading optical negative index metamaterials,” Appl. Comput. Electromagn. Soc. J. 22, 172–183 (2007).

Appl. Phys. Lett. (1)

C. Menzel, C. Helgert, J. Upping, C. Rockstuhl, E. B. Kley, R. B. Wehrspohn, T. Pertsch, and F. Lederer, “Angular resolved effective optical properties of a Swiss cross metamaterial,” Appl. Phys. Lett. 95 (2009).
[CrossRef]

J. Opt. Soc. Am. (5)

J. Opt. Soc. Am. A (5)

Laser Phys. Lett. (1)

V. P. Drachev, W. Cai, U. Chettiar, H. K. Yuan, A. K. Sarychev, A. V. Kildishev, G. Klimeck, and V. M. Shalaev, “Experimental verification of an optical negative-index material,” Laser Phys. Lett. 3, 49–55 (2006).
[CrossRef]

Nano Lett. (1)

K.-P. Chen, V. P. Drachev, J. D. Borneman, A. V. Kildishev, and V. M. Shalaev, “Drude relaxation rate in grained gold nanoantennas,” Nano Lett. 10, 916–922 (2010), DOI 10.1021/nl9037246.
[CrossRef] [PubMed]

Opt. Express (4)

Opt. Lett. (2)

Phys. Rev. B (1)

P. B. Johnson and R. W. Christy, “Optical constants of noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[CrossRef]

Phys. Rev. Lett. (2)

S. Zhang, W. J. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. J. Brueck, “Experimental demonstration of near-infrared negative-index metamaterials,” Phys. Rev. Lett. 95, 137404 (2005).
[CrossRef] [PubMed]

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966–3969 (2000).
[CrossRef] [PubMed]

Other (2)

X. Ni, Z. Liu, F. Gu, M. G. Pacheco, A. V. Kildishev, and J. Borneman, “PhotonicsSHA-2D: Modeling of single-period multilayer optical gratings and metamaterials,” (2009), DOI 10254/nanohub-r6977.9.

E. G. Loewen and E. Popov, Diffraction Gratings and Applications (Marcel Dekker, 1997).

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Figures (6)

Fig. 1
Fig. 1

Multilayer single-period grating studied in 2D SHA. The period is d. Each layer has a thickness h p .

Fig. 2
Fig. 2

(a) Schematic of a unit cell of the structure under study; (b) SEM image of the fabricated device.

Fig. 3
Fig. 3

Transmittance spectra for both TE and TM polarizations at incident angles of 0°, 5°, 10°, 15°, 20°, 25°, and 30°. The experimental and SHA simulated spectra are both shown for comparison.

Fig. 4
Fig. 4

Convergence of 2D SHA for TE polarization. The difference spectra between 5, 15, 30, and 50 Fourier modes and the reference are plotted. The reference spectra are simulated using 2D SHA with 100 modes.

Fig. 5
Fig. 5

Convergence of 2D SHA for TM polarization. The difference spectra between 5, 15, 30, and 50 Fourier modes and the reference are plotted. The reference spectra are simulated using 2D SHA with 100 modes.

Fig. 6
Fig. 6

Average difference versus number of modes. The reference spectra are simulated using 2D SHA with 100 modes. For each number of modes the difference is averaged over the spectrum.

Tables (2)

Tables Icon

Table 1 Designed and Fitted Parameters of the Sample

Tables Icon

Table 2 Wavelengths of Different Diffraction Orders

Equations (28)

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

k ̃ x m 2 E m = ( ε k ̃ y 2 ) E m ,
k ̃ x m 2 H z m = ε ¯ 1 ( I k ̃ y ε 1 k ̃ y ) H z m ,
[ u ( N + 1 ) d ( 0 ) ] = [ T u u ( N ) R u d ( N ) R d u ( N ) T d d ( N ) ] [ u ( 0 ) d ( N + 1 ) ] ,
y E z = i ω μ 0 H x ,
x E z = i ω μ 0 H y ,
x H y y H x = i ω ε E z ,
2 E z x 2 + 2 E z y 2 + ε r k 2 E z = 0.
E z m n ( k ̃ x m 2 + k ̃ y n 2 ) = q ε n q E z m q ,
k ̃ x m 2 E m = ( ε k ̃ y 2 ) E m ,
ε = [ ε 0 ε 1 ε 2 ε 2 M ε 1 ε 0 ε 1 ε 2 M + 1 ε 2 M ε 2 M 1 ε 1 ε 0 ] .
y H z = i ω ε E x ,
ε 1 x H z = i ω E y ,
x E y y E x = i ω μ 0 H z .
ω ε 0 ε E x m = k k ̃ y H z m ,
ω ε 0 E y m = k k ̃ x m ε ¯ H z m ,
k k ̃ x m E y m k k ̃ y E x m = ω μ 0 H z m ,
ε ¯ = [ ( 1 / ε ) 0 ( 1 / ε ) 1 ( 1 / ε ) 2 ( 1 / ε ) 2 M ( 1 / ε ) 1 ( 1 / ε ) 0 ( 1 / ε ) 1 ( 1 / ε ) 2 M + 1 ( 1 / ε ) 2 M ( 1 / ε ) 2 M 1 ( 1 / ε ) 1 ( 1 / ε ) 0 ] .
k ̃ x m 2 ε ¯ H z m = ( I k ̃ y ε 1 k ̃ y ) H z m .
k ̃ x m 2 H z m = ε ¯ 1 ( I k ̃ y ε 1 k ̃ y ) H z m .
[ u ( p + 1 ) d ( 0 ) ] = [ T u u ( p ) R u d ( p ) R d u ( p ) T d d ( p ) ] [ u ( 0 ) d ( p + 1 ) ] ,
Q E ( p ) = ( E ( p + 1 ) ) 1 E ( p ) ,
Q H ( p ) = ( H ( p + 1 ) ) 1 H ( p ) ,
t 1 ( p ) = ( Q E ( p ) + Q H ( p ) ) / 2 ,
t 2 ( p ) = ( Q E ( p ) Q H ( p ) ) / 2     for   TE ,
or   t 2 ( p ) = ( Q H ( p ) Q E ( p ) ) / 2     for   TM ,
Ω ( p ) = Φ ( p ) R u d ( p 1 ) Φ ( p ) ,
R u d ( p ) = [ t 2 ( p ) + t 1 ( p ) Ω ( p ) ] [ t 1 ( p ) + t 2 ( p ) Ω ( p ) ] 1 ,
T d d ( p ) = T d d ( p 1 ) Φ ( p ) [ t 1 ( p ) + t 2 ( p ) Ω ( p ) ] 1 .

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