Abstract

Light absorption inside the grooves of metallic gratings filled with a semiconductor material can be improved by means of the electric field enhancement. To this end, the influence of grating dimensions in the electric field spectral behavior is theoretically investigated. Two conditions of cavity resonance have been analyzed separately: (1) for TE polarization (electric field parallel to the grooves) and (2) for TM polarization (magnetic field parallel to the grooves). When dimensions are chosen according to the first condition, the enhancement of TE fields is found to increase with the height-to-width ratio, and it is accompanied with a decrease in the bandwidth. The same enhancement levels can be achieved for TM fields if the second condition holds, provided that the period-to-width ratio is large enough. The simultaneous enhancement of TE and TM fields, based on a condition of surface resonance excitation, can also be accomplished. In this case, the TM response is very sensitive to changes in groove depth and width.

© 2010 Optical Society of America

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References

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  1. A. Luque and A. Martí, “The intermediate band solar cell: Progress toward the realization of an attractive concept,” Adv. Mater. 21, 1–15 (2010).
  2. B. Sopori, “Thin film silicon solar cells,” in Handbook of Photovoltaic Science and Technology, A.Luque and S.Hegedus, eds. (Wiley, 2004), pp. 307–358.
  3. H. Shpaisman, O. Niitsoo, I. Lubomirsky, and D. Cahen, “Can up- and down-conversion and multi-exciton generation improve photovoltaics?” Sol. Energy Mater. Sol. Cells 92, 1541–1546 (2008).
    [CrossRef]
  4. E. Yablonovitch and G. D. Cody, “Intensity enhancement in textured optical sheets for solar cells,” IEEE Trans. Electron Devices ED-29, 300–305 (1982).
    [CrossRef]
  5. J. M. Rodríguez, I. Tobías, and A. Luque, “Random pyramidal texture modelling,” Sol. Energy Mater. Sol. Cells 45, 241–253 (1997).
    [CrossRef]
  6. F. Čajko and I. Fedoseyev, “Photoabsorption and carrier transport modeling in thin multilayer photovoltaic cell,” Lect. Notes Comput. Sci. 5544, 755–764 (2009).
    [CrossRef]
  7. K. H. Brenner, “Aspects for calculating local absorption with the rigorous coupled-wave method,” Opt. Express 18, 10369–10376 (2010).
    [CrossRef] [PubMed]
  8. M. J. Mendes, A. Luque, I. Tobías, and A. Martí, “Plasmonic light enhancement in the near-field of metallic nanospheroids for application in intermediate band solar cells,” Appl. Phys. Lett. 95, 071105 (2009).
    [CrossRef]
  9. K. R. Catchpole and A. Polman, “Plasmonic solar cells,” Opt. Express 16, 21793–21800 (2008).
    [CrossRef] [PubMed]
  10. S. A. Maier, Plasmonics: Fundamentals and Applications (Springer, 2008).
  11. P. Hewageegana and V. Apalkov, “Quantum dot photodetectors with metallic diffraction grating: Surface plasmons and strong absorption enhancement,” Physica E (Amsterdam) 40, 2817–2822 (2008).
    [CrossRef]
  12. J. R. Andrewartha, J. R. Fox, and I. J. Wilson, “Resonance anomalies in the lamellar grating,” Opt. Acta 26, 69–89 (1979).
    [CrossRef]
  13. A. Wirgin and A. A. Maradudin, “Resonant enhancement of the electric field in the grooves of bare metallic gratings exposed to S-polarized light,” Phys. Rev. B 31, 5573–5576 (1985).
    [CrossRef]
  14. A. Wirgin and A. A. Maradudin, “Resonant response of a bare metallic grating to S-polarized light,” Prog. Surf. Sci. 22, 1–99 (1986).
    [CrossRef]
  15. A. Barbara, P. Quémerais, E. Bustarret, T. López-Ríos, and T. Fournier, “Electromagnetic resonances of sub-wavelength rectangular metallic gratings,” Eur. Phys. J. D 23, 143–154 (2003).
    [CrossRef]
  16. T. López-Ríos, D. Mendoza, F. J. García-Vidal, J. Sánchez-Dehesa, and B. Pannetier, “Surface shape resonances in lamellar metallic gratings,” Phys. Rev. Lett. 81, 665–668 (1998).
    [CrossRef]
  17. F. J. García-Vidal, J. Sánchez-Dehesa, A. Dechelette, E. Bustarret, T. López-Ríos, T. Fournier, and B. Pannetier, “Localized surface plasmons in lamellar metallic gratings,” J. Lightwave Technol. 17, 2191–2195 (1999).
    [CrossRef]
  18. R.Petit, ed., Electromagnetic Theory of Gratings (Springer-Verlag, 1980).
    [CrossRef]
  19. J. A. Kong, Electromagnetic Wave Theory (Wiley, 1986).

2010 (2)

A. Luque and A. Martí, “The intermediate band solar cell: Progress toward the realization of an attractive concept,” Adv. Mater. 21, 1–15 (2010).

K. H. Brenner, “Aspects for calculating local absorption with the rigorous coupled-wave method,” Opt. Express 18, 10369–10376 (2010).
[CrossRef] [PubMed]

2009 (2)

M. J. Mendes, A. Luque, I. Tobías, and A. Martí, “Plasmonic light enhancement in the near-field of metallic nanospheroids for application in intermediate band solar cells,” Appl. Phys. Lett. 95, 071105 (2009).
[CrossRef]

F. Čajko and I. Fedoseyev, “Photoabsorption and carrier transport modeling in thin multilayer photovoltaic cell,” Lect. Notes Comput. Sci. 5544, 755–764 (2009).
[CrossRef]

2008 (3)

P. Hewageegana and V. Apalkov, “Quantum dot photodetectors with metallic diffraction grating: Surface plasmons and strong absorption enhancement,” Physica E (Amsterdam) 40, 2817–2822 (2008).
[CrossRef]

H. Shpaisman, O. Niitsoo, I. Lubomirsky, and D. Cahen, “Can up- and down-conversion and multi-exciton generation improve photovoltaics?” Sol. Energy Mater. Sol. Cells 92, 1541–1546 (2008).
[CrossRef]

K. R. Catchpole and A. Polman, “Plasmonic solar cells,” Opt. Express 16, 21793–21800 (2008).
[CrossRef] [PubMed]

2003 (1)

A. Barbara, P. Quémerais, E. Bustarret, T. López-Ríos, and T. Fournier, “Electromagnetic resonances of sub-wavelength rectangular metallic gratings,” Eur. Phys. J. D 23, 143–154 (2003).
[CrossRef]

1999 (1)

1998 (1)

T. López-Ríos, D. Mendoza, F. J. García-Vidal, J. Sánchez-Dehesa, and B. Pannetier, “Surface shape resonances in lamellar metallic gratings,” Phys. Rev. Lett. 81, 665–668 (1998).
[CrossRef]

1997 (1)

J. M. Rodríguez, I. Tobías, and A. Luque, “Random pyramidal texture modelling,” Sol. Energy Mater. Sol. Cells 45, 241–253 (1997).
[CrossRef]

1986 (1)

A. Wirgin and A. A. Maradudin, “Resonant response of a bare metallic grating to S-polarized light,” Prog. Surf. Sci. 22, 1–99 (1986).
[CrossRef]

1985 (1)

A. Wirgin and A. A. Maradudin, “Resonant enhancement of the electric field in the grooves of bare metallic gratings exposed to S-polarized light,” Phys. Rev. B 31, 5573–5576 (1985).
[CrossRef]

1982 (1)

E. Yablonovitch and G. D. Cody, “Intensity enhancement in textured optical sheets for solar cells,” IEEE Trans. Electron Devices ED-29, 300–305 (1982).
[CrossRef]

1979 (1)

J. R. Andrewartha, J. R. Fox, and I. J. Wilson, “Resonance anomalies in the lamellar grating,” Opt. Acta 26, 69–89 (1979).
[CrossRef]

Andrewartha, J. R.

J. R. Andrewartha, J. R. Fox, and I. J. Wilson, “Resonance anomalies in the lamellar grating,” Opt. Acta 26, 69–89 (1979).
[CrossRef]

Apalkov, V.

P. Hewageegana and V. Apalkov, “Quantum dot photodetectors with metallic diffraction grating: Surface plasmons and strong absorption enhancement,” Physica E (Amsterdam) 40, 2817–2822 (2008).
[CrossRef]

Barbara, A.

A. Barbara, P. Quémerais, E. Bustarret, T. López-Ríos, and T. Fournier, “Electromagnetic resonances of sub-wavelength rectangular metallic gratings,” Eur. Phys. J. D 23, 143–154 (2003).
[CrossRef]

Brenner, K. H.

Bustarret, E.

A. Barbara, P. Quémerais, E. Bustarret, T. López-Ríos, and T. Fournier, “Electromagnetic resonances of sub-wavelength rectangular metallic gratings,” Eur. Phys. J. D 23, 143–154 (2003).
[CrossRef]

F. J. García-Vidal, J. Sánchez-Dehesa, A. Dechelette, E. Bustarret, T. López-Ríos, T. Fournier, and B. Pannetier, “Localized surface plasmons in lamellar metallic gratings,” J. Lightwave Technol. 17, 2191–2195 (1999).
[CrossRef]

Cahen, D.

H. Shpaisman, O. Niitsoo, I. Lubomirsky, and D. Cahen, “Can up- and down-conversion and multi-exciton generation improve photovoltaics?” Sol. Energy Mater. Sol. Cells 92, 1541–1546 (2008).
[CrossRef]

Cajko, F.

F. Čajko and I. Fedoseyev, “Photoabsorption and carrier transport modeling in thin multilayer photovoltaic cell,” Lect. Notes Comput. Sci. 5544, 755–764 (2009).
[CrossRef]

Catchpole, K. R.

Cody, G. D.

E. Yablonovitch and G. D. Cody, “Intensity enhancement in textured optical sheets for solar cells,” IEEE Trans. Electron Devices ED-29, 300–305 (1982).
[CrossRef]

Dechelette, A.

Fedoseyev, I.

F. Čajko and I. Fedoseyev, “Photoabsorption and carrier transport modeling in thin multilayer photovoltaic cell,” Lect. Notes Comput. Sci. 5544, 755–764 (2009).
[CrossRef]

Fournier, T.

A. Barbara, P. Quémerais, E. Bustarret, T. López-Ríos, and T. Fournier, “Electromagnetic resonances of sub-wavelength rectangular metallic gratings,” Eur. Phys. J. D 23, 143–154 (2003).
[CrossRef]

F. J. García-Vidal, J. Sánchez-Dehesa, A. Dechelette, E. Bustarret, T. López-Ríos, T. Fournier, and B. Pannetier, “Localized surface plasmons in lamellar metallic gratings,” J. Lightwave Technol. 17, 2191–2195 (1999).
[CrossRef]

Fox, J. R.

J. R. Andrewartha, J. R. Fox, and I. J. Wilson, “Resonance anomalies in the lamellar grating,” Opt. Acta 26, 69–89 (1979).
[CrossRef]

García-Vidal, F. J.

F. J. García-Vidal, J. Sánchez-Dehesa, A. Dechelette, E. Bustarret, T. López-Ríos, T. Fournier, and B. Pannetier, “Localized surface plasmons in lamellar metallic gratings,” J. Lightwave Technol. 17, 2191–2195 (1999).
[CrossRef]

T. López-Ríos, D. Mendoza, F. J. García-Vidal, J. Sánchez-Dehesa, and B. Pannetier, “Surface shape resonances in lamellar metallic gratings,” Phys. Rev. Lett. 81, 665–668 (1998).
[CrossRef]

Hewageegana, P.

P. Hewageegana and V. Apalkov, “Quantum dot photodetectors with metallic diffraction grating: Surface plasmons and strong absorption enhancement,” Physica E (Amsterdam) 40, 2817–2822 (2008).
[CrossRef]

Kong, J. A.

J. A. Kong, Electromagnetic Wave Theory (Wiley, 1986).

López-Ríos, T.

A. Barbara, P. Quémerais, E. Bustarret, T. López-Ríos, and T. Fournier, “Electromagnetic resonances of sub-wavelength rectangular metallic gratings,” Eur. Phys. J. D 23, 143–154 (2003).
[CrossRef]

F. J. García-Vidal, J. Sánchez-Dehesa, A. Dechelette, E. Bustarret, T. López-Ríos, T. Fournier, and B. Pannetier, “Localized surface plasmons in lamellar metallic gratings,” J. Lightwave Technol. 17, 2191–2195 (1999).
[CrossRef]

T. López-Ríos, D. Mendoza, F. J. García-Vidal, J. Sánchez-Dehesa, and B. Pannetier, “Surface shape resonances in lamellar metallic gratings,” Phys. Rev. Lett. 81, 665–668 (1998).
[CrossRef]

Lubomirsky, I.

H. Shpaisman, O. Niitsoo, I. Lubomirsky, and D. Cahen, “Can up- and down-conversion and multi-exciton generation improve photovoltaics?” Sol. Energy Mater. Sol. Cells 92, 1541–1546 (2008).
[CrossRef]

Luque, A.

A. Luque and A. Martí, “The intermediate band solar cell: Progress toward the realization of an attractive concept,” Adv. Mater. 21, 1–15 (2010).

M. J. Mendes, A. Luque, I. Tobías, and A. Martí, “Plasmonic light enhancement in the near-field of metallic nanospheroids for application in intermediate band solar cells,” Appl. Phys. Lett. 95, 071105 (2009).
[CrossRef]

J. M. Rodríguez, I. Tobías, and A. Luque, “Random pyramidal texture modelling,” Sol. Energy Mater. Sol. Cells 45, 241–253 (1997).
[CrossRef]

Maier, S. A.

S. A. Maier, Plasmonics: Fundamentals and Applications (Springer, 2008).

Maradudin, A. A.

A. Wirgin and A. A. Maradudin, “Resonant response of a bare metallic grating to S-polarized light,” Prog. Surf. Sci. 22, 1–99 (1986).
[CrossRef]

A. Wirgin and A. A. Maradudin, “Resonant enhancement of the electric field in the grooves of bare metallic gratings exposed to S-polarized light,” Phys. Rev. B 31, 5573–5576 (1985).
[CrossRef]

Martí, A.

A. Luque and A. Martí, “The intermediate band solar cell: Progress toward the realization of an attractive concept,” Adv. Mater. 21, 1–15 (2010).

M. J. Mendes, A. Luque, I. Tobías, and A. Martí, “Plasmonic light enhancement in the near-field of metallic nanospheroids for application in intermediate band solar cells,” Appl. Phys. Lett. 95, 071105 (2009).
[CrossRef]

Mendes, M. J.

M. J. Mendes, A. Luque, I. Tobías, and A. Martí, “Plasmonic light enhancement in the near-field of metallic nanospheroids for application in intermediate band solar cells,” Appl. Phys. Lett. 95, 071105 (2009).
[CrossRef]

Mendoza, D.

T. López-Ríos, D. Mendoza, F. J. García-Vidal, J. Sánchez-Dehesa, and B. Pannetier, “Surface shape resonances in lamellar metallic gratings,” Phys. Rev. Lett. 81, 665–668 (1998).
[CrossRef]

Niitsoo, O.

H. Shpaisman, O. Niitsoo, I. Lubomirsky, and D. Cahen, “Can up- and down-conversion and multi-exciton generation improve photovoltaics?” Sol. Energy Mater. Sol. Cells 92, 1541–1546 (2008).
[CrossRef]

Pannetier, B.

F. J. García-Vidal, J. Sánchez-Dehesa, A. Dechelette, E. Bustarret, T. López-Ríos, T. Fournier, and B. Pannetier, “Localized surface plasmons in lamellar metallic gratings,” J. Lightwave Technol. 17, 2191–2195 (1999).
[CrossRef]

T. López-Ríos, D. Mendoza, F. J. García-Vidal, J. Sánchez-Dehesa, and B. Pannetier, “Surface shape resonances in lamellar metallic gratings,” Phys. Rev. Lett. 81, 665–668 (1998).
[CrossRef]

Polman, A.

Quémerais, P.

A. Barbara, P. Quémerais, E. Bustarret, T. López-Ríos, and T. Fournier, “Electromagnetic resonances of sub-wavelength rectangular metallic gratings,” Eur. Phys. J. D 23, 143–154 (2003).
[CrossRef]

Rodríguez, J. M.

J. M. Rodríguez, I. Tobías, and A. Luque, “Random pyramidal texture modelling,” Sol. Energy Mater. Sol. Cells 45, 241–253 (1997).
[CrossRef]

Sánchez-Dehesa, J.

F. J. García-Vidal, J. Sánchez-Dehesa, A. Dechelette, E. Bustarret, T. López-Ríos, T. Fournier, and B. Pannetier, “Localized surface plasmons in lamellar metallic gratings,” J. Lightwave Technol. 17, 2191–2195 (1999).
[CrossRef]

T. López-Ríos, D. Mendoza, F. J. García-Vidal, J. Sánchez-Dehesa, and B. Pannetier, “Surface shape resonances in lamellar metallic gratings,” Phys. Rev. Lett. 81, 665–668 (1998).
[CrossRef]

Shpaisman, H.

H. Shpaisman, O. Niitsoo, I. Lubomirsky, and D. Cahen, “Can up- and down-conversion and multi-exciton generation improve photovoltaics?” Sol. Energy Mater. Sol. Cells 92, 1541–1546 (2008).
[CrossRef]

Sopori, B.

B. Sopori, “Thin film silicon solar cells,” in Handbook of Photovoltaic Science and Technology, A.Luque and S.Hegedus, eds. (Wiley, 2004), pp. 307–358.

Tobías, I.

M. J. Mendes, A. Luque, I. Tobías, and A. Martí, “Plasmonic light enhancement in the near-field of metallic nanospheroids for application in intermediate band solar cells,” Appl. Phys. Lett. 95, 071105 (2009).
[CrossRef]

J. M. Rodríguez, I. Tobías, and A. Luque, “Random pyramidal texture modelling,” Sol. Energy Mater. Sol. Cells 45, 241–253 (1997).
[CrossRef]

Wilson, I. J.

J. R. Andrewartha, J. R. Fox, and I. J. Wilson, “Resonance anomalies in the lamellar grating,” Opt. Acta 26, 69–89 (1979).
[CrossRef]

Wirgin, A.

A. Wirgin and A. A. Maradudin, “Resonant response of a bare metallic grating to S-polarized light,” Prog. Surf. Sci. 22, 1–99 (1986).
[CrossRef]

A. Wirgin and A. A. Maradudin, “Resonant enhancement of the electric field in the grooves of bare metallic gratings exposed to S-polarized light,” Phys. Rev. B 31, 5573–5576 (1985).
[CrossRef]

Yablonovitch, E.

E. Yablonovitch and G. D. Cody, “Intensity enhancement in textured optical sheets for solar cells,” IEEE Trans. Electron Devices ED-29, 300–305 (1982).
[CrossRef]

Adv. Mater. (1)

A. Luque and A. Martí, “The intermediate band solar cell: Progress toward the realization of an attractive concept,” Adv. Mater. 21, 1–15 (2010).

Appl. Phys. Lett. (1)

M. J. Mendes, A. Luque, I. Tobías, and A. Martí, “Plasmonic light enhancement in the near-field of metallic nanospheroids for application in intermediate band solar cells,” Appl. Phys. Lett. 95, 071105 (2009).
[CrossRef]

Eur. Phys. J. D (1)

A. Barbara, P. Quémerais, E. Bustarret, T. López-Ríos, and T. Fournier, “Electromagnetic resonances of sub-wavelength rectangular metallic gratings,” Eur. Phys. J. D 23, 143–154 (2003).
[CrossRef]

IEEE Trans. Electron Devices (1)

E. Yablonovitch and G. D. Cody, “Intensity enhancement in textured optical sheets for solar cells,” IEEE Trans. Electron Devices ED-29, 300–305 (1982).
[CrossRef]

J. Lightwave Technol. (1)

Lect. Notes Comput. Sci. (1)

F. Čajko and I. Fedoseyev, “Photoabsorption and carrier transport modeling in thin multilayer photovoltaic cell,” Lect. Notes Comput. Sci. 5544, 755–764 (2009).
[CrossRef]

Opt. Acta (1)

J. R. Andrewartha, J. R. Fox, and I. J. Wilson, “Resonance anomalies in the lamellar grating,” Opt. Acta 26, 69–89 (1979).
[CrossRef]

Opt. Express (2)

Phys. Rev. B (1)

A. Wirgin and A. A. Maradudin, “Resonant enhancement of the electric field in the grooves of bare metallic gratings exposed to S-polarized light,” Phys. Rev. B 31, 5573–5576 (1985).
[CrossRef]

Phys. Rev. Lett. (1)

T. López-Ríos, D. Mendoza, F. J. García-Vidal, J. Sánchez-Dehesa, and B. Pannetier, “Surface shape resonances in lamellar metallic gratings,” Phys. Rev. Lett. 81, 665–668 (1998).
[CrossRef]

Physica E (Amsterdam) (1)

P. Hewageegana and V. Apalkov, “Quantum dot photodetectors with metallic diffraction grating: Surface plasmons and strong absorption enhancement,” Physica E (Amsterdam) 40, 2817–2822 (2008).
[CrossRef]

Prog. Surf. Sci. (1)

A. Wirgin and A. A. Maradudin, “Resonant response of a bare metallic grating to S-polarized light,” Prog. Surf. Sci. 22, 1–99 (1986).
[CrossRef]

Sol. Energy Mater. Sol. Cells (2)

H. Shpaisman, O. Niitsoo, I. Lubomirsky, and D. Cahen, “Can up- and down-conversion and multi-exciton generation improve photovoltaics?” Sol. Energy Mater. Sol. Cells 92, 1541–1546 (2008).
[CrossRef]

J. M. Rodríguez, I. Tobías, and A. Luque, “Random pyramidal texture modelling,” Sol. Energy Mater. Sol. Cells 45, 241–253 (1997).
[CrossRef]

Other (4)

R.Petit, ed., Electromagnetic Theory of Gratings (Springer-Verlag, 1980).
[CrossRef]

J. A. Kong, Electromagnetic Wave Theory (Wiley, 1986).

S. A. Maier, Plasmonics: Fundamentals and Applications (Springer, 2008).

B. Sopori, “Thin film silicon solar cells,” in Handbook of Photovoltaic Science and Technology, A.Luque and S.Hegedus, eds. (Wiley, 2004), pp. 307–358.

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Figures (10)

Fig. 1
Fig. 1

Perfectly conducting grating illuminated by a TE polarized wave.

Fig. 2
Fig. 2

Plots of the squared magnitude of the normalized electric field for d = 1.1 c and different depth to width ratios. The inset shows an enlarged view of the plot for h / c = 4 near the maximum of the normalized electric field ( 0.308   eV ) .

Fig. 3
Fig. 3

FWHM as a function of h / c ratio, where h and c are the depth and width of the grooves, respectively.

Fig. 4
Fig. 4

Maximum intensity versus h / c ratio for different values of d / c ratio, where d is the period of the grating and c is the width of the grooves.

Fig. 5
Fig. 5

Resonance energies versus h / c ratio for different values of d / c ratio.

Fig. 6
Fig. 6

Field enhancement-bandwidth product, i.e., | E z p | 2 / | E z i | 2 × FWHM , for structures with different d / c ratios in terms of the h / c ratio (see text).

Fig. 7
Fig. 7

Application of the criterion for finding TE field resonant enhancement: squared magnitude of the TM normalized electric field (see Subsection 3A).

Fig. 8
Fig. 8

Shifting the resonance energy by changing the groove dimensions according to the scaling proposed in Subsection 3A for aspect ratios h / c = 4 and 5. In both cases, a ratio d / c = 1.1 is assumed.

Fig. 9
Fig. 9

Application of the criterion for finding TM field resonant enhancement: maximum intensity of the TM electric field in terms of d / c ratio for different values of h / c ratio.

Fig. 10
Fig. 10

Behavior of the (modulus squared) electric field at ( 0 , 0.5 h ) as a function of h and c and assuming d = λ r / ν 1 : the TE and TM responses have been averaged (a logarithmic scale is employed).

Equations (36)

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

E z i ( x , y ) = exp [ i ( α 0 x χ 0 y ) ] ,
α 0 = κ ν 1   sin   θ ,     χ 0 = κ ν 1   cos   θ ,
E R ( x , y ) = E z i ( x , y ) + n = A n   exp [ i ( α n x + χ n y ) ] ,
α n = α 0 + n 2 π d ,     χ n = κ 2 ν 1 2 α n 2 .
n = N 1 N 2 | A n | 2 χ n χ 0 = 1 ,
2 E z + κ 2 ν 1 2 E z = 0.
E M ( x , y ) = m = 1 a m ϕ m ( x , y ) ,
ϕ m ( x , y ) = sin [ μ m ( y + h ) ] sin ( m π x c )     if   m   is   even ,
ϕ m ( x , y ) = sin [ μ m ( y + h ) ] cos ( m π x c )     if   m   is   odd ,
μ m = ( 2 π ν 1 λ ) 2 ( m π c ) 2
D R ( λ ) = μ 1 ( λ ) + 4 c d | χ 1 ( λ ) | | J 11 ( λ ) | 2 tan [ μ 1 ( λ ) h ] ,
D I ( λ ) = 2 c d χ 0 ( λ ) | J 10 ( λ ) | 2 tan [ μ 1 ( λ ) h ] ,
J m n ( λ ) = 1 c c / 2 c / 2 exp [ i α n ( λ ) x ] cos ( m π x c ) d x .
λ j I = 2 ν 1 ( j 2 / h 2 + 1 / c 2 ) 1 / 2 ,     j = 1 , 2 .
| E ( 0 , y ) | λ = λ j I = 8 χ 0 π | sin [ μ 1 ( λ j I ) ( y + h ) ] μ 1 ( λ j I ) | .
b 0 = 2   sinc ( α 0 c / 2 ) sin ( μ 0 h ) 1 cot ( μ 0 h ) i μ 0 c d n = [ sinc ( α n c / 2 ) ] 2 χ n ,
H z = b 0   cos [ μ 0 ( y + h ) ] ,     E x = i ω ν 1 2 b 0 μ 0   sin [ μ 0 ( y + h ) ] ,    
E y = 0.
( E M y ) y = 0 = ( E R y ) y = 0 + ,     c / 2 x c / 2 ,
E R ( x , 0 + ) = E M ( x , 0 ) ,     c / 2 x c / 2 ,
E R ( x , 0 + ) = 0 ,     d / 2 x c / 2 ,     c / 2 x d / 2.
1 c c / 2 c / 2 ( E M y ) y = 0 ϕ m ( x , 0 ) d x = 1 c c / 2 c / 2 ( E R y ) y = 0 + ϕ m ( x , 0 ) d x ,
1 2 a m μ m   cos ( μ m h ) = i χ 0 J m 0 + n = i χ n J m n A n ,
J m n = 1 c c / 2 c / 2 exp ( i α n x ) sin ( m π x c ) d x ,     if   m   is   even ,
J m n = 1 c c / 2 c / 2 exp ( i α n x ) cos ( m π x c ) d x ,     if   m   is   odd .
1 d d / 2 d / 2 E R ( x , 0 + ) exp ( i α n x ) d x = 1 d c / 2 c / 2 E M ( x , 0 ) exp ( i α n x ) d x ,
A n = δ n , 0 + c d m = 1 J m n   sin ( μ m h ) a m .
G a = H q = 1 G m q a q = H m ,     m = 1 , 2 , ,
G m q = 1 2 δ m , q μ q   cos ( μ q h ) c d sin ( μ q h ) n = i χ n J m n J q n ,    
H m = 2 i χ 0 J m 0 ,     m = 1 , 2 , .
a 1 = H 1 G 11 = 2 i χ 0 J 10 1 2 μ 1   cos ( μ 1 h ) c d sin ( μ 1 h ) n = i χ n | J 1 n | 2 .
a 1 = 4 i χ 0 J 10 / cos ( μ 1 h ) μ 1 2 i c d χ 0   tan ( μ 1 h ) | J 10 | 2 + 4 c d | χ 1 | tan ( μ 1 h ) | J 11 | 2 .
D R ( λ ) = μ 1 ( λ ) + 4 c d | χ 1 ( λ ) | | J 11 ( λ ) | 2 tan [ μ 1 ( λ ) h ] ,
D I ( λ ) = 2 c d χ 0 ( λ ) | J 10 ( λ ) | 2 tan [ μ 1 ( λ ) h ] .
λ j I = 2 ν 1 ( j 2 / h 2 + 1 / c 2 ) 1 / 2 ,     j = 1 , 2 , .
| E ( 0 , y ) | λ = λ j I = | a 1   sin [ μ 1 ( λ j I ) ( y + h ) ] | = 8 χ 0 π | sin [ μ 1 ( λ j I ) ( y + h ) ] μ 1 ( λ j I ) | .

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