Abstract

This work analyzes the electromagnetic wave propagation through periodically stacked fishnets from zero frequency to the first Wood’s anomaly. It is shown that, apart from Fabry-Perot resonances, these structures support two transmission bands that can be backward under the appropriate conditions. The first band starts at Wood’s anomaly and is closely related to the well-known phenomena of extraordinary transmission through a single fishnet. The second band is related to the resonances of the fishnet holes. In both cases, the in-plane periodicity of the fishnet cannot be made electrically small, which prevents any attempt of homogenization of the structure along the fishnet planes. However, along the normal direction, even with very small periodicity transmission is still possible. An homogenization procedure can then be applied along this direction, thus making that the structure can behave as a backward-wave transmission line for such transmission bands. Closed-form design formulas will be provided by the analytical formulation here presented. These formulas have been carefully validated by intensive numerical computations.

© 2009 Optical Society of America

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  1. S. Zhang,W. Fan, K. J. Malloy, and S. R. J. Brueck, "Near-infrared double negative metamaterials," Opt. Express 13, 4922-4930 (2005).
    [CrossRef] [PubMed]
  2. S. Zhang, W. 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]
  3. M. Beruete, M. Sorolla, and I. Campillo "Left-handed extraordinary optical transmission through a photonic crystal of subwavelength hole arrays," Opt. Express 14, 5445-5455 (2006).
    [CrossRef] [PubMed]
  4. M. Beruete, I. Campillo, M. Navarro-Cıa, F. Falcone, and M. Sorolla "Molding left- or right-handed metamaterials by stacked cutoff metallic hole arrays," IEEE Trans. Microwave Theory Tech. 55, 1514-1521 (2007).
  5. M. Navarro-Cıa, M. Beruete, M. Sorolla, and I. Campillo, "Negative refraction in a prism made of stacked subwavelength hole arrays," Opt. Express 16, 560-566 (2008).
    [CrossRef] [PubMed]
  6. G. Dolling, Ch. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden "Low-loss negative-index metamaterial at telecommunication wavelengths," Opt. Lett. 31, 1800-1802 (2005).
    [CrossRef]
  7. J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, "Three-dimensional optical metamaterial with a negative refractive index," Nature (London) 455, 299-300 (2008).
    [CrossRef]
  8. G. Dolling, M. Wegener, C. M. Soukoulis, and S. Linden, "Negative-index metamaterial at 780 nm wavelength," Opt. Lett. 32, 53-55 (2007).
    [CrossRef]
  9. T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, "Extraordinary optical transmission through sub-wavelength hole arrays," Nature (London) 391, 667-669 (1998).
    [CrossRef]
  10. D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, "Composite medium with simultaneously negative permeability and permittivity," Phys. Rev. Lett. 84, 4184-4187 (2000).
    [CrossRef] [PubMed]
  11. A. Mary, S. G. Rodrigo, F. J. Garcia-Vidal, and L. Martin-Moreno, "Theory of negative-refractive-index response of double-fishnet structures," Phys. Rev. Lett. 101, 103902 (2008).
    [CrossRef] [PubMed]
  12. F. Medina, F. Mesa, and R. Marques, "Extraordinary transmission through arrays of electrically small holes from a circuit theory perspective," IEEE Trans. Microwave Theory Tech. 56, 3108-3120 (2008).
    [CrossRef]
  13. R. Marques, F. Mesa, L. Jelinek, and F. Medina "Analytical theory of extraordinary transmission through metallic diffraction screens perforated by small holes," Opt. Express 17, 5571-5579 (2009)
    [CrossRef] [PubMed]
  14. J. D. Jackson, Classical Electrodynamics, (Wiley, New York 1999), 3rd Ed.
  15. R. Gordon, "Bethe’s aperture theory for arrays," Phys. Rev. A 76, 053806 (2007).
    [CrossRef]
  16. R. F. CollinField Theory of Guided Waves, (IEEE Press, New York 1991), 2nd Ed.,
  17. S. Ramo, J. R. Whinnery, and T. Van Duzer, Fields and Waves in Communication Electronics, (Wiley, New York 1994), 3rd Ed., pp. 263-264.
  18. G. V. Eleftheriades, O. Siddiqui, and A. K. Iyer, "Transmission line models for negative refractive index media and associated implementations without excess resonators," IEEE Microwave Wirel. Compon. Lett. 13, 51-53 (2003).
    [CrossRef]

2009 (1)

2008 (4)

A. Mary, S. G. Rodrigo, F. J. Garcia-Vidal, and L. Martin-Moreno, "Theory of negative-refractive-index response of double-fishnet structures," Phys. Rev. Lett. 101, 103902 (2008).
[CrossRef] [PubMed]

F. Medina, F. Mesa, and R. Marques, "Extraordinary transmission through arrays of electrically small holes from a circuit theory perspective," IEEE Trans. Microwave Theory Tech. 56, 3108-3120 (2008).
[CrossRef]

M. Navarro-Cıa, M. Beruete, M. Sorolla, and I. Campillo, "Negative refraction in a prism made of stacked subwavelength hole arrays," Opt. Express 16, 560-566 (2008).
[CrossRef] [PubMed]

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, "Three-dimensional optical metamaterial with a negative refractive index," Nature (London) 455, 299-300 (2008).
[CrossRef]

2007 (3)

G. Dolling, M. Wegener, C. M. Soukoulis, and S. Linden, "Negative-index metamaterial at 780 nm wavelength," Opt. Lett. 32, 53-55 (2007).
[CrossRef]

M. Beruete, I. Campillo, M. Navarro-Cıa, F. Falcone, and M. Sorolla "Molding left- or right-handed metamaterials by stacked cutoff metallic hole arrays," IEEE Trans. Microwave Theory Tech. 55, 1514-1521 (2007).

R. Gordon, "Bethe’s aperture theory for arrays," Phys. Rev. A 76, 053806 (2007).
[CrossRef]

2006 (1)

2005 (3)

2003 (1)

G. V. Eleftheriades, O. Siddiqui, and A. K. Iyer, "Transmission line models for negative refractive index media and associated implementations without excess resonators," IEEE Microwave Wirel. Compon. Lett. 13, 51-53 (2003).
[CrossRef]

2000 (1)

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, "Composite medium with simultaneously negative permeability and permittivity," Phys. Rev. Lett. 84, 4184-4187 (2000).
[CrossRef] [PubMed]

1998 (1)

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, "Extraordinary optical transmission through sub-wavelength hole arrays," Nature (London) 391, 667-669 (1998).
[CrossRef]

Beruete, M.

Brueck, S. R. J.

S. Zhang, W. 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]

S. Zhang,W. Fan, K. J. Malloy, and S. R. J. Brueck, "Near-infrared double negative metamaterials," Opt. Express 13, 4922-4930 (2005).
[CrossRef] [PubMed]

Campillo, I.

Dolling, G.

Ebbesen, T. W.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, "Extraordinary optical transmission through sub-wavelength hole arrays," Nature (London) 391, 667-669 (1998).
[CrossRef]

Eleftheriades, G. V.

G. V. Eleftheriades, O. Siddiqui, and A. K. Iyer, "Transmission line models for negative refractive index media and associated implementations without excess resonators," IEEE Microwave Wirel. Compon. Lett. 13, 51-53 (2003).
[CrossRef]

Enkrich, Ch.

Falcone, F.

M. Beruete, I. Campillo, M. Navarro-Cıa, F. Falcone, and M. Sorolla "Molding left- or right-handed metamaterials by stacked cutoff metallic hole arrays," IEEE Trans. Microwave Theory Tech. 55, 1514-1521 (2007).

Fan, W.

S. Zhang,W. Fan, K. J. Malloy, and S. R. J. Brueck, "Near-infrared double negative metamaterials," Opt. Express 13, 4922-4930 (2005).
[CrossRef] [PubMed]

S. Zhang, W. 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]

Garcia-Vidal, F. J.

A. Mary, S. G. Rodrigo, F. J. Garcia-Vidal, and L. Martin-Moreno, "Theory of negative-refractive-index response of double-fishnet structures," Phys. Rev. Lett. 101, 103902 (2008).
[CrossRef] [PubMed]

Ghaemi, H. F.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, "Extraordinary optical transmission through sub-wavelength hole arrays," Nature (London) 391, 667-669 (1998).
[CrossRef]

Gordon, R.

R. Gordon, "Bethe’s aperture theory for arrays," Phys. Rev. A 76, 053806 (2007).
[CrossRef]

Iyer, A. K.

G. V. Eleftheriades, O. Siddiqui, and A. K. Iyer, "Transmission line models for negative refractive index media and associated implementations without excess resonators," IEEE Microwave Wirel. Compon. Lett. 13, 51-53 (2003).
[CrossRef]

Jelinek, L.

Lezec, H. J.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, "Extraordinary optical transmission through sub-wavelength hole arrays," Nature (London) 391, 667-669 (1998).
[CrossRef]

Linden, S.

Malloy, K. J.

S. Zhang, W. 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]

S. Zhang,W. Fan, K. J. Malloy, and S. R. J. Brueck, "Near-infrared double negative metamaterials," Opt. Express 13, 4922-4930 (2005).
[CrossRef] [PubMed]

Marques, R.

R. Marques, F. Mesa, L. Jelinek, and F. Medina "Analytical theory of extraordinary transmission through metallic diffraction screens perforated by small holes," Opt. Express 17, 5571-5579 (2009)
[CrossRef] [PubMed]

F. Medina, F. Mesa, and R. Marques, "Extraordinary transmission through arrays of electrically small holes from a circuit theory perspective," IEEE Trans. Microwave Theory Tech. 56, 3108-3120 (2008).
[CrossRef]

Martin-Moreno, L.

A. Mary, S. G. Rodrigo, F. J. Garcia-Vidal, and L. Martin-Moreno, "Theory of negative-refractive-index response of double-fishnet structures," Phys. Rev. Lett. 101, 103902 (2008).
[CrossRef] [PubMed]

Mary, A.

A. Mary, S. G. Rodrigo, F. J. Garcia-Vidal, and L. Martin-Moreno, "Theory of negative-refractive-index response of double-fishnet structures," Phys. Rev. Lett. 101, 103902 (2008).
[CrossRef] [PubMed]

Medina, F.

R. Marques, F. Mesa, L. Jelinek, and F. Medina "Analytical theory of extraordinary transmission through metallic diffraction screens perforated by small holes," Opt. Express 17, 5571-5579 (2009)
[CrossRef] [PubMed]

F. Medina, F. Mesa, and R. Marques, "Extraordinary transmission through arrays of electrically small holes from a circuit theory perspective," IEEE Trans. Microwave Theory Tech. 56, 3108-3120 (2008).
[CrossRef]

Mesa, F.

R. Marques, F. Mesa, L. Jelinek, and F. Medina "Analytical theory of extraordinary transmission through metallic diffraction screens perforated by small holes," Opt. Express 17, 5571-5579 (2009)
[CrossRef] [PubMed]

F. Medina, F. Mesa, and R. Marques, "Extraordinary transmission through arrays of electrically small holes from a circuit theory perspective," IEEE Trans. Microwave Theory Tech. 56, 3108-3120 (2008).
[CrossRef]

Navarro-Cia, M.

M. Navarro-Cıa, M. Beruete, M. Sorolla, and I. Campillo, "Negative refraction in a prism made of stacked subwavelength hole arrays," Opt. Express 16, 560-566 (2008).
[CrossRef] [PubMed]

M. Beruete, I. Campillo, M. Navarro-Cıa, F. Falcone, and M. Sorolla "Molding left- or right-handed metamaterials by stacked cutoff metallic hole arrays," IEEE Trans. Microwave Theory Tech. 55, 1514-1521 (2007).

Nemat-Nasser, S. C.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, "Composite medium with simultaneously negative permeability and permittivity," Phys. Rev. Lett. 84, 4184-4187 (2000).
[CrossRef] [PubMed]

Osgood, R. M.

S. Zhang, W. 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]

Padilla, W. J.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, "Composite medium with simultaneously negative permeability and permittivity," Phys. Rev. Lett. 84, 4184-4187 (2000).
[CrossRef] [PubMed]

Panoiu, N. C.

S. Zhang, W. 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]

Rodrigo, S. G.

A. Mary, S. G. Rodrigo, F. J. Garcia-Vidal, and L. Martin-Moreno, "Theory of negative-refractive-index response of double-fishnet structures," Phys. Rev. Lett. 101, 103902 (2008).
[CrossRef] [PubMed]

Schultz, S.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, "Composite medium with simultaneously negative permeability and permittivity," Phys. Rev. Lett. 84, 4184-4187 (2000).
[CrossRef] [PubMed]

Siddiqui, O.

G. V. Eleftheriades, O. Siddiqui, and A. K. Iyer, "Transmission line models for negative refractive index media and associated implementations without excess resonators," IEEE Microwave Wirel. Compon. Lett. 13, 51-53 (2003).
[CrossRef]

Smith, D. R.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, "Composite medium with simultaneously negative permeability and permittivity," Phys. Rev. Lett. 84, 4184-4187 (2000).
[CrossRef] [PubMed]

Sorolla, M.

Soukoulis, C. M.

Thio, T.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, "Extraordinary optical transmission through sub-wavelength hole arrays," Nature (London) 391, 667-669 (1998).
[CrossRef]

Ulin-Avila, E.

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, "Three-dimensional optical metamaterial with a negative refractive index," Nature (London) 455, 299-300 (2008).
[CrossRef]

Valentine, J.

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, "Three-dimensional optical metamaterial with a negative refractive index," Nature (London) 455, 299-300 (2008).
[CrossRef]

Vier, D. C.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, "Composite medium with simultaneously negative permeability and permittivity," Phys. Rev. Lett. 84, 4184-4187 (2000).
[CrossRef] [PubMed]

Wegener, M.

Wolff, P. A.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, "Extraordinary optical transmission through sub-wavelength hole arrays," Nature (London) 391, 667-669 (1998).
[CrossRef]

Zentgraf, T.

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, "Three-dimensional optical metamaterial with a negative refractive index," Nature (London) 455, 299-300 (2008).
[CrossRef]

Zhang, S.

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, "Three-dimensional optical metamaterial with a negative refractive index," Nature (London) 455, 299-300 (2008).
[CrossRef]

S. Zhang,W. Fan, K. J. Malloy, and S. R. J. Brueck, "Near-infrared double negative metamaterials," Opt. Express 13, 4922-4930 (2005).
[CrossRef] [PubMed]

S. Zhang, W. 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]

IEEE Microwave Wirel. Compon. Lett. (1)

G. V. Eleftheriades, O. Siddiqui, and A. K. Iyer, "Transmission line models for negative refractive index media and associated implementations without excess resonators," IEEE Microwave Wirel. Compon. Lett. 13, 51-53 (2003).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (2)

M. Beruete, I. Campillo, M. Navarro-Cıa, F. Falcone, and M. Sorolla "Molding left- or right-handed metamaterials by stacked cutoff metallic hole arrays," IEEE Trans. Microwave Theory Tech. 55, 1514-1521 (2007).

F. Medina, F. Mesa, and R. Marques, "Extraordinary transmission through arrays of electrically small holes from a circuit theory perspective," IEEE Trans. Microwave Theory Tech. 56, 3108-3120 (2008).
[CrossRef]

Nature (London) (2)

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, "Three-dimensional optical metamaterial with a negative refractive index," Nature (London) 455, 299-300 (2008).
[CrossRef]

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, "Extraordinary optical transmission through sub-wavelength hole arrays," Nature (London) 391, 667-669 (1998).
[CrossRef]

Opt. Express (4)

Opt. Lett. (2)

Phys. Rev. A (1)

R. Gordon, "Bethe’s aperture theory for arrays," Phys. Rev. A 76, 053806 (2007).
[CrossRef]

Phys. Rev. Lett. (3)

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, "Composite medium with simultaneously negative permeability and permittivity," Phys. Rev. Lett. 84, 4184-4187 (2000).
[CrossRef] [PubMed]

A. Mary, S. G. Rodrigo, F. J. Garcia-Vidal, and L. Martin-Moreno, "Theory of negative-refractive-index response of double-fishnet structures," Phys. Rev. Lett. 101, 103902 (2008).
[CrossRef] [PubMed]

S. Zhang, W. 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]

Other (3)

R. F. CollinField Theory of Guided Waves, (IEEE Press, New York 1991), 2nd Ed.,

S. Ramo, J. R. Whinnery, and T. Van Duzer, Fields and Waves in Communication Electronics, (Wiley, New York 1994), 3rd Ed., pp. 263-264.

J. D. Jackson, Classical Electrodynamics, (Wiley, New York 1999), 3rd Ed.

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Figures (7)

Fig. 1.
Fig. 1.

Periodic array of perfect conductor screens perforated with square holes: (a) front view and (b) lateral view. (c) Front and (d) lateral views of the structure unit cell or equivalent waveguide.

Fig. 2.
Fig. 2.

Unit cells of (a) the periodic transmission line corresponding to the dispersion equation (12) for a stacked fishnet with t →0, (b) the backward-wave transmission line reported and analyzed in [17, pp. 263–264] and [18], (c) the equivalent circuit proposed in Sec. 4 for thick stacked fishnet.

Fig. 3.
Fig. 3.

Dispersion diagrams for a periodic array of stacked fishnets as that shown in Fig.1 with negligible thickness t→0, periodicity p=a/3, and different values of the hole size b. Solid lines are calculated using formulas (12)–(15). Circles correspond to numerical simulations by CST Microwave Studio.

Fig. 4.
Fig. 4.

Dispersion diagrams for a periodic array of stacked fishnets as that shown in Fig.1 with negligible thickness t→0, hole size b=a/6, and different values of the periodicity p. Blue solid lines (Analytical 1) are calculated using formulas (12)–(15). Red dashed lines (Analytical 2) are obtained using (20)–(21) for the parallel and series admittances. Circles correspond to numerical simulations by CST Microwave Studio.

Fig. 5.
Fig. 5.

Dispersion diagrams for a periodic array of stacked fishnets as that shown in Fig.1 with negligible thickness t→ 0, hole size b=a/2, and periodicity p=a/3. Solid lines were calculated using formulas (12)–(15). Circles correspond to numerical simulations by CST Microwave Studio.

Fig. 6.
Fig. 6.

Dispersion diagrams for a periodic array of stacked fishnets as that shown in Fig.1 with negligible thickness t→0, periodicity p=a/3, and three different rectangular hole sizes. Solid lines were calculated using formulas (12) and (20),(21). Circles correspond to numerical simulations obtained by CST Microwave Studio.

Fig. 7.
Fig. 7.

Dispersion diagrams for a periodic array of stacked fishnets as that shown in Fig.1 with periodicity p=a/3 and hole size b=a/6. Two different screen thicknesses have been considered. Solid lines correspond to our analytical formulas. Circles correspond to numerical simulations obtained by CST Microwave Studio.

Equations (28)

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

Ey0(x,y)=A0+n=1NAn0TEfn0(x,y)+m=1MA0mTMf0m(x,y)+n,m=1N,m(AnmTE+AnmTM)fnm(x,y)
fnm(x,y)=cos (2nπxa)cos(2mπya)
Ex0(x,y)=n,m=1(mnAnmTEnmAnmTM)gnm(x,y)
a=k0(λa)21 ; k0=ω ε0μ0
Eyinc=EyTEM+EyTE+EyTM
EyTEM=q0A0eiβqpeik0qp=A0cos(βp)eik0pcos(k0p)cos(βp)
EyTE=q0A10TEf10eiβqpeαqp=A10TEf10cos(βp)eαpcosh(αp)cos(βp)
EyTM=q0A01TMf01eiβqeαqp=A01TMf01cos(βp)eαpcosh(αp)cos(βp).
mn AnmTE n nm AnmTM 0 ; n , m0 .
wEyfnmdxdy=hEyfnmdxdyhEydxdy=wEydxdy
wEyfnmdxdy=a2A0(1+cos(βp)eik0pcos(k0p)cos(βp))
cos (βp)=1 + YpYs
Yp=2 i Y0 sin2(k0p2)sin(k0p) + 4 (Y20TE+Y02TM)sinh2(αp2)sinh(αp)sinc(πba)+Y
Y3=i Y0 csc (k0p)+2(Y20TE+Y02TM)csch(αp)sinc(πba)
Y=2 n=2NY2n,0TEsinc(nπba)+2m=2MY0,2mTMsinc(mπba)
+ 4 n,m=1N,M(Y2n,2mTEn2n2+m2+Y2n,2mTMm2n2+n2)sinc(nπba)sinc(mπba)
Y2n,2mTE=i Y0 (nλa)2+(mλa)21
Y2n,2mTM=iY0(nλa)2+(mλa)21
βp=2Y2Ys
ε=1ω2Lpp,μ= 2ω2Csp
Yp= 2 i axay Y0 sin2(k0p2)sin(k0p)
+ 4 axay n=1NY2n,0TEsinh2(αn0p2)sinh(αn0p)sinc(nπbxax)+4axaym=1mY0,2mTMsinh2(α0mp2)sinh(α0mp)sinc(mπbyay)
+ 8 axay n,m=1N,M(Y2n,2mTEn2n2+m2+Y2n,2mTMm2n2+m2)sinh2(αnmp2)sinh(αnmp)sinc(nπbxax)sinc(mπbyay)
Ys=i axay Y0 csc (k0p)
+ 2 axay n=1NY2n,0TEcsch(αn0p)sinc(nπbxax)+2axaym=1MY0,2mTMcsch(α0mp)sinc(mπbyay)
+ 4 axay n,m=1N,M(Y2n,2mTEn2n2+m2+Y2n,2mTMm2n2+m2)csch(αnmp)sinc(nπbxax)sinc(mπbyay).
Ys={1Ys+iYpsin(kht)+2Yh(cos(kht)1)(Yp2+Yh)2eikht(Yp2Yh)2eikht}1
Yp=(Yp2+Yh)2eikht(Yp2Yh)2eikht2Yh

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