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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 4 — Feb. 13, 2012
  • pp: 3693–3702
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Giant resonance absorption in ultra-thin metamaterial periodic structures

Avner Yanai, Meir Orenstein, and Uriel Levy  »View Author Affiliations


Optics Express, Vol. 20, Issue 4, pp. 3693-3702 (2012)
http://dx.doi.org/10.1364/OE.20.003693


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Abstract

We study the interaction of an incident plane wave with a metamaterial periodic structure consisting of alternating layers of positive and negative refractive index with average zero refractive index. We show that the existence of very narrow resonance peaks for which giant absorption - 50% at layer thickness of 1% of the incident wavelength - is exhibited. Maximum absorption is obtained at a specific layer thickness satisfying the critical coupling condition. This phenomenon is explained by the Rayleigh anomaly and by the excitation of Fabry Perot modes in the periodic layer. In addition, we investigate the modes supported by the structures for several limiting cases, and show that zero phase accumulation in the periodic metamaterial is obtained at resonance.

© 2012 OSA

1. Introduction

2. Modes of a DPS-DNG structure with infinite thickness

Figure 1 Schematic drawing showing a unit cell of the DPS-DNG periodic structure

Figure 2 The log(Re(KB)) as function of the real and imaginary parts of kZ /k0, calculated using TMM. (b) Magnetic power |Hy|2 in a unit cell (media boundary at x/λ0 = 0), for the modes with kz/k0 ≊ ±0.83 − 0.46i (c) Phase of Hy in a unit cell for the same mode as (b). (d) Eigenmodes of the same structure as (a), calculated using RCWA.

3. Resonant modes of a DPS-DNG slab with finite thickness

Figure 3 RCWA calculation of: (a) Transmission (b) Reflection and (c) Absorption curves as function of the incident wavelength and slab thickness.

We attribute this surprising result to two mechanisms, the Rayleigh anomaly and the guided mode resonance [20

20. A. Hessel and A. A. Oliner, “A new theory of Wood anomalies on optical gratings,” Appl. Opt. 4, 1275–1297 (1965). [CrossRef]

23

23. C. Genet, M. P. van Exter, and J. P. Woerdman, “Fano-type interpretation of red-shifts and red tails in hole array transmission spectra,” Opt. Commun. 225, 331–336 (2003). [CrossRef]

]. Rayleigh anomaly occurs when a propagating diffraction order in the semi-infinite incident and outgoing regions I and III is becoming an evanescent order, resulting in the redistribution of the energy within the other orders. In our case where ε = 1 and μ = 1 in both the incident and outgoing regions, the Rayleigh wavelength is equal to the period, i.e. λR = L. A guided mode resonance is the consequence of the matching of the diffracted orders to a leaky mode which is propagating along the x axis in region II. To gain further understanding of the physics behind the obtained transmission/absorption/reflection maps we plot (Fig. 4) the field distribution within the metamaterial structure and its vicinity. We consider a thin periodic metamaterial with thickness of h=0.01L. First, it can be seen from Figs. 4(a) and 4(b) that in regions I and III the diffracted wave occupies precisely a single unit cell and therefore can be identified as the Fabry Perot mode, discussed in Section 2, which is consistent with the choice of Lλ. The resonance in these regions is therefore in agreement with the Rayleigh anomaly. In region II, the excited mode is resonantly guided in the periodic layer as indicated by the fact that the mode periodicity is identical to the gratings period. This is somewhat similar to the very well known case of guided mode resonance [20

20. A. Hessel and A. A. Oliner, “A new theory of Wood anomalies on optical gratings,” Appl. Opt. 4, 1275–1297 (1965). [CrossRef]

,24

24. A. Sharon, D. Rosenblatt, and A. A. Friesem, “Resonant grating-waveguide structures for visible and near-infrared radiation,” J. Opt. Soc. Am. A 14, 2985–2993 (1997). [CrossRef]

,25

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

] where typically a waveguide mode is excited. However, here the modes satisfy the Fabry-Perot condition along the X direction rather than being conventional waveguide modes. The very narrow lineshape (of the order of ∼ 0.5ω0 × 10−6 for h/λ0 = 0.01) is explained by the sensitivity of the Fabry Perot resonance, where small change in wavelength shifts the domination to the Bragg regime over the Fabry Perot regime, as discussed in Section 2, following references [7

7. L. Wu, S. He, and L. F. Shen, “Band structure for a one-dimensional photonic crystal containing left-handed materials,” Phys. Rev. B 67, 235103 (2003). [CrossRef]

,13

13. L. Wu, S. He, and L. Chen, “On unusual narrow transmission bands for a multi-layered periodic structure containing left-handed materials,” Opt. Express 11, 1283–1290 (2003). [CrossRef] [PubMed]

]. The high Q-factor can be also understood by observing the magnetic field distribution of the Fabry Perot mode (Fig. 4(a)) having an odd symmetry with respect to the x-axis. In contrast, the reflected and transmitted plane waves have even magnetic field distribution, resulting in a vanishing overlap integral between them and the Fabry-Perot mode, and thus giving rise to “dark modes” which typically have very large Q-factors for structures with small inherent losses (see e.g. [26

26. V. Liu, M. Povinelli, and S. Fan, “Resonance-enhanced optical forces between coupled photonic crystal slabs,” Opt. Express 17(24), 21897–21909 (2009). [CrossRef] [PubMed]

]).To compare between the resonance and off resonance cases we calculated Hy for a slightly off resonance case (λ = 1.001L). From the results (See Fig. 4(c)) it can be seen that the magnetic field no longer builds up coherently in the unit cell.

Figure 4 Real part of the field profiles in the NIM grating, calculated using the RCWA method (a) Hy field component at resonance (λ = L), (b) Ez field component at resonance (λ = L) (saturated color scale) (c) Hy field for the off-resonance case (λ = 1.001L). The black rectangles denote the region of the DNG layer.

From Fig. 4 it is evident also that ripples with large spatial frequencies are formed at the boundaries of the DNG and DPS media. These ripples are typical for surface modes bound between DPS and DNG media [27

27. R. Ruppin, “Surface polaritons of a left-handed material slab,” J. Phys. Condens. Matter 13(9), 1811–1818 (2001). [CrossRef]

28

28. S. A. Darmanyan, M. Neviere, and A. A. Zakhidov, “Surface modes at the interface of conventional and left-handed media,” Opt. Commun. 225, 233–240 (2003). [CrossRef]

] when ε and μ simultaneously change their sign over a very small region [29

29. N. M. Litchinitser, A. I. Maimistov, I. R. Gabitov, R. Z. Sagdeev, and V. M. Shalaev, “Metamaterials: electromagnetic enhancement at zero-index transition,” Opt. Lett. 33, 2350–2352 (2008). [CrossRef] [PubMed]

], as is the case in the considered structure. To gain further insight, it is also instructive to evaluate the phase index (nph) of light propagation in the z-direction, in the resonant frequencies. The phase index is calculated according to: nph=chΔϕω, where Δϕ is the phase difference between the transmitted and incident waves, and ω = 2πc/λ. In Fig. 5, we have plotted both the absorption and the phase index as function of frequency near resonance for h=0.01L. It can be seen that at the peak of absorption, nph crosses zero (marked in Fig. 5 by the crossing of the dotted lines), indicating an asymptotically infinite phase velocity at this wavelength. This indicates again that at the absorption peak, the fields are evenly distributed between the DPS and DNG media. Since the phase velocity in both media is opposite, inside the grating there is full cancellation of the phase accumulation in the z direction.

Figure 5 Phase index (blue line, left y-axis) and absorption (green line, right y-axis) as function of ωω0 normalized by 2πc, calculated for h = 0.01L. The correspondence between the absorption peak and the zero phase index is marked by the crossing of the dashed lines.

It is well known from the theory of resonators [30

30. H. A. Haus, Waves and fields in Optoelectronics, (New Jersey, Prentice-Hall Inc., 1984).

], e.g. guided mode resonance filters [24

24. A. Sharon, D. Rosenblatt, and A. A. Friesem, “Resonant grating-waveguide structures for visible and near-infrared radiation,” J. Opt. Soc. Am. A 14, 2985–2993 (1997). [CrossRef]

25

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

], Ring Resonators [31

31. A. Yariv, “Universal relations for coupling of optical power between microresonators and dielectric waveguides,” Electron. Lett. 4, 321–322 (2000). [CrossRef]

] and prism coupling of surface plasmon polaritons [32

32. H. Raether, Surface plasmons on smooth and rough surfaces and on gratings, (Springer-Verlag, New York, 1988).

], that an optimal coupling condition known as the “critical coupling” exists. Under the critical coupling condition, coupling losses and internal losses of the resonator (in our case it is the periodic layer) are equal, resulting in an absorption peak and zero transmission. In Fig. 6(a), the absorption as a function of the layer thickness is plotted, calculated using the same parameters as above for the wavelength of, λ = (1 + 6 × 10−7)L. It can be seen that for this wavelength, the optimal coupling into the periodic layer is obtained around h/L ≈ 0.01. For smaller or larger values of h/L the internal losses in the resonator are detuned from their optimal value due to variations in mode confinement and the absorption in the periodic layers is reduced. Furthermore, it can be seen from Fig. 6(b) that the resonant absorption peak near h/L = 0.01, is a sharp peak in a slowly varying envelope of the absorption. This slowly varying envelope is the consequence of interference of waves inside Region II, similar to interference effects in a conventional lossy homogeneous dielectric slab, and is not the consequence of a transverse resonant mode (e.g. the transverse Fabry – Perot mode discussed above).

Figure 6 Absorption as function of the periodic layer thickness, calculated for λ = (1 + 6 × 10−7)L. (a) is an inset of (b), showing a magnified view of the absorption around the absorption peak (h ≈ 0.01L).

In order to verify if critical coupling is indeed obtained, we fitted the transmitted and reflected intensities to a Fano profile, using the temporal coupled mode theory (TCMT) model described in [30

30. H. A. Haus, Waves and fields in Optoelectronics, (New Jersey, Prentice-Hall Inc., 1984).

,33

33. S. Fan, W. Suh, and J. D. Joannopoulos, “Temporal coupled-mode theory for the Fano resonance in optical resonators,” J. Opt. Soc. Am. A 20, 569–572 (2003). [CrossRef]

]. In order to extract the coupling and internal loss coefficients, we fitted the reflected field amplitude r obtained from the RCWA simulation to the function r=rD+1/τj(ωωres)+(1/τ+1/γ)(rD+jtD) and the transmitted field amplitude t to the function t=jtD+1/τj(ωωres)+(1/τ+1/γ)(rDjtD). In this model, rD and tD are the direct (off resonance) transmission and reflection coefficients respectively, ωres is the resonant frequency, τ is the decay time in the resonator due to coupling and γ is the decay time due to internal losses in the resonator. In Fig. 7, fitting results for h=0.01L are shown, with γ = 1.49 × 106L/(2πc) and τ = 1.22 × 106L/(2πc). This indeed indicates that the resonator is near critical coupling (γ is of the order of τ) [30

30. H. A. Haus, Waves and fields in Optoelectronics, (New Jersey, Prentice-Hall Inc., 1984).

,33

33. S. Fan, W. Suh, and J. D. Joannopoulos, “Temporal coupled-mode theory for the Fano resonance in optical resonators,” J. Opt. Soc. Am. A 20, 569–572 (2003). [CrossRef]

].

Figure 7 (a) Reflection and (b) Transmission power intensities as function of the incident frequency normalized by 2πc. The red curve is the RCWA simulation result, while the blue curve is the Fano fit based on TCMT model.

Until now, we considered non-dispersive and lossy DNG and DPS media. While making the analysis of the structure easier, the above assumed media violate the causality restrictions given by the Kramers-Kroning relations. To remove this deficiency, we now repeat the RCWA calculation, but assume that both the permittivity and permeability functions of the DNG medium are given by Drude dispersion models, and that the DPS media in Region II is vacuum, namely: ε1 = 1, μ1 = 1, ε2=1ωp2/(ω2iωγε) and μ2=1ωp2/(ω2iωγμ). The material parameters are: ωp=22πc/L, γε = 1 × 10−3 ωp and γμ = 1 × 10−2 ωp. As can be observed from Fig. 8 the resonance near λ0L is very similar to the resonance in the dispersionless case discussed above.

Figure 8 RCWA calculation of: (a) Transmission (b) Reflection and (c) Absorption as a function of λλ0 and the normalized slab thickness. Calculation is made assuming a Drude dispersive model of the DNG medium.

4. Conclusions

We study the interaction of light with an infinite metamaterial periodic structure consisting of alternating layers of positive and negative refractive index. The modes supported by the structure are found using a full-vectorial electromagnetic calculation. We show that very high absorption of about 50% and sharp resonance are obtained for ultrathin periodic structures, down to ∼two orders of magnitude thinner than the vacuum wavelength. This finding is attributed to the excitation of a Fabry Perot mode in the structure as well as for the existence of Rayleigh anomaly.

Acknowledgments

The authors are grateful to D.R. Smith and N. Engheta for fruitful discussions, and acknowledge the support of the AFOSR and the Israeli Science Foundation (ISF). A.Y. acknowledges the support of the CAMBR fellowship.

References

1.

N. Engheta and R. Ziolkowski, Metamaterials: Physics and Engineering Explorations, (Wiley, 2006).

2.

V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ε and μ,” Sov. Phys. Usp. 10, 509 (1968). [CrossRef]

3.

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

4.

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]

5.

S. Nefedov and S. A. Tretyakov, “Photonic band gap structure containing metamaterial with negative permittivity and permeability,” Phys. Rev. E 66, 036611 (2002). [CrossRef]

6.

J. Li, L. Zhou, C. T. Chan, and P. Sheng, “Photonic band gap from a stack of positive and negative index materials,” Phys. Rev. Lett. 90(8), 083901 (2003). [CrossRef] [PubMed]

7.

L. Wu, S. He, and L. F. Shen, “Band structure for a one-dimensional photonic crystal containing left-handed materials,” Phys. Rev. B 67, 235103 (2003). [CrossRef]

8.

A. Alu and N. Engheta, “Pairing an epsilon-negative slab with a mu-negative slab: resonance, tunneling and transparency,” IEEE Trans. Antennas Propag. 51, 2558–2571 (2003). [CrossRef]

9.

D. R. Fredkin and A. Ron, “Effectively left-handed (negative index) composite material,” Appl. Phys. Lett. 81, 1753–1755 (2002). [CrossRef]

10.

Y. Yuan, L. Ran, J. Huangfu, H. Chen, L. Shen, and J. A. Kong, “Experimental verification of zero order bandgap in a layered stack of left-handed and right-handed materials,” Opt. Express 14, 2220–2227 (2006). [CrossRef] [PubMed]

11.

S. Kocaman, R. Chatterjee, N. C. Panoiu, J. F. McMillan, M. B. Yu, R. M. Osgood, D. L. Kwong, and C. W. Wong, “Observation of zeroth-order band gaps in negative-refraction photonic crystal superlattices at near-infrared frequencies,” Phys. Rev. Lett. 102, 203905 (2009). [CrossRef] [PubMed]

12.

Y. Jin, S. Xiao, N. A. Mortensen, and S. He, “Arbitrarily thin metamaterial structure for perfect absorption and giant magnification,” Opt. Express 19, 11114–11119 (2011). [CrossRef] [PubMed]

13.

L. Wu, S. He, and L. Chen, “On unusual narrow transmission bands for a multi-layered periodic structure containing left-handed materials,” Opt. Express 11, 1283–1290 (2003). [CrossRef] [PubMed]

14.

M. G. Moharam, E. B. Grann, D. A. Pommet, and T. K. Gaylord, “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]

15.

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

16.

P. Lalanne and G. Morris, “Highly improved convergence of the coupled-wave method for TM polarization,” J. Opt. Soc. Am. A 13, 779–784 (1996). [CrossRef]

17.

T. Weiss, N. A. Gippius, S. G. Tikhodeev, G. Granet, and H. Giessen, “Derivation of plasmonic resonances in the Fourier modal method with adaptive spatial resolution and matched coordinates,” J. Opt. Soc. Am. A 28, 238–244 (2011). [CrossRef]

18.

P. Yeh, Optical waves in layered media, (John Wiley & Sons, New York, 1988).

19.

S. E. Kocabas, G. Veronis, D. A. B. Miller, and S. Fan, “Modal analysis and coupling in metal-insulator-metal waveguides,” Phys. Rev. B 79, 035120 (2009). [CrossRef]

20.

A. Hessel and A. A. Oliner, “A new theory of Wood anomalies on optical gratings,” Appl. Opt. 4, 1275–1297 (1965). [CrossRef]

21.

U. Fano, “Effects of configuration interaction on intensities and phase shifts,” Phys. Rev. 124, 1866–1878 (1961). [CrossRef]

22.

M. Sarrazin, J.-P. Vigneron, and J.-M. Vigoureux, “Role of Wood anomalies in optical properties of thin metallic films with a bidimensional array of subwavelength holes,” Phys. Rev. B 67, 085415 (2003). [CrossRef]

23.

C. Genet, M. P. van Exter, and J. P. Woerdman, “Fano-type interpretation of red-shifts and red tails in hole array transmission spectra,” Opt. Commun. 225, 331–336 (2003). [CrossRef]

24.

A. Sharon, D. Rosenblatt, and A. A. Friesem, “Resonant grating-waveguide structures for visible and near-infrared radiation,” J. Opt. Soc. Am. A 14, 2985–2993 (1997). [CrossRef]

25.

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

26.

V. Liu, M. Povinelli, and S. Fan, “Resonance-enhanced optical forces between coupled photonic crystal slabs,” Opt. Express 17(24), 21897–21909 (2009). [CrossRef] [PubMed]

27.

R. Ruppin, “Surface polaritons of a left-handed material slab,” J. Phys. Condens. Matter 13(9), 1811–1818 (2001). [CrossRef]

28.

S. A. Darmanyan, M. Neviere, and A. A. Zakhidov, “Surface modes at the interface of conventional and left-handed media,” Opt. Commun. 225, 233–240 (2003). [CrossRef]

29.

N. M. Litchinitser, A. I. Maimistov, I. R. Gabitov, R. Z. Sagdeev, and V. M. Shalaev, “Metamaterials: electromagnetic enhancement at zero-index transition,” Opt. Lett. 33, 2350–2352 (2008). [CrossRef] [PubMed]

30.

H. A. Haus, Waves and fields in Optoelectronics, (New Jersey, Prentice-Hall Inc., 1984).

31.

A. Yariv, “Universal relations for coupling of optical power between microresonators and dielectric waveguides,” Electron. Lett. 4, 321–322 (2000). [CrossRef]

32.

H. Raether, Surface plasmons on smooth and rough surfaces and on gratings, (Springer-Verlag, New York, 1988).

33.

S. Fan, W. Suh, and J. D. Joannopoulos, “Temporal coupled-mode theory for the Fano resonance in optical resonators,” J. Opt. Soc. Am. A 20, 569–572 (2003). [CrossRef]

OCIS Codes
(050.0050) Diffraction and gratings : Diffraction and gratings
(240.0240) Optics at surfaces : Optics at surfaces
(240.6690) Optics at surfaces : Surface waves
(310.2790) Thin films : Guided waves

ToC Category:
Metamaterials

History
Original Manuscript: November 21, 2011
Revised Manuscript: January 12, 2012
Manuscript Accepted: January 18, 2012
Published: January 31, 2012

Citation
Avner Yanai, Meir Orenstein, and Uriel Levy, "Giant resonance absorption in ultra-thin metamaterial periodic structures," Opt. Express 20, 3693-3702 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-4-3693


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References

  1. N. Engheta and R. Ziolkowski, Metamaterials: Physics and Engineering Explorations, (Wiley, 2006).
  2. V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ε and μ,” Sov. Phys. Usp.10, 509 (1968). [CrossRef]
  3. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett.85, 3966 (2000). [CrossRef] [PubMed]
  4. 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]
  5. S. Nefedov and S. A. Tretyakov, “Photonic band gap structure containing metamaterial with negative permittivity and permeability,” Phys. Rev. E66, 036611 (2002). [CrossRef]
  6. J. Li, L. Zhou, C. T. Chan, and P. Sheng, “Photonic band gap from a stack of positive and negative index materials,” Phys. Rev. Lett.90(8), 083901 (2003). [CrossRef] [PubMed]
  7. L. Wu, S. He, and L. F. Shen, “Band structure for a one-dimensional photonic crystal containing left-handed materials,” Phys. Rev. B67, 235103 (2003). [CrossRef]
  8. A. Alu and N. Engheta, “Pairing an epsilon-negative slab with a mu-negative slab: resonance, tunneling and transparency,” IEEE Trans. Antennas Propag.51, 2558–2571 (2003). [CrossRef]
  9. D. R. Fredkin and A. Ron, “Effectively left-handed (negative index) composite material,” Appl. Phys. Lett.81, 1753–1755 (2002). [CrossRef]
  10. Y. Yuan, L. Ran, J. Huangfu, H. Chen, L. Shen, and J. A. Kong, “Experimental verification of zero order bandgap in a layered stack of left-handed and right-handed materials,” Opt. Express14, 2220–2227 (2006). [CrossRef] [PubMed]
  11. S. Kocaman, R. Chatterjee, N. C. Panoiu, J. F. McMillan, M. B. Yu, R. M. Osgood, D. L. Kwong, and C. W. Wong, “Observation of zeroth-order band gaps in negative-refraction photonic crystal superlattices at near-infrared frequencies,” Phys. Rev. Lett.102, 203905 (2009). [CrossRef] [PubMed]
  12. Y. Jin, S. Xiao, N. A. Mortensen, and S. He, “Arbitrarily thin metamaterial structure for perfect absorption and giant magnification,” Opt. Express19, 11114–11119 (2011). [CrossRef] [PubMed]
  13. L. Wu, S. He, and L. Chen, “On unusual narrow transmission bands for a multi-layered periodic structure containing left-handed materials,” Opt. Express11, 1283–1290 (2003). [CrossRef] [PubMed]
  14. M. G. Moharam, E. B. Grann, D. A. Pommet, and T. K. Gaylord, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A12, 1068–1076 (1995). [CrossRef]
  15. L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A13, 1870–1876 (1996). [CrossRef]
  16. P. Lalanne and G. Morris, “Highly improved convergence of the coupled-wave method for TM polarization,” J. Opt. Soc. Am. A13, 779–784 (1996). [CrossRef]
  17. T. Weiss, N. A. Gippius, S. G. Tikhodeev, G. Granet, and H. Giessen, “Derivation of plasmonic resonances in the Fourier modal method with adaptive spatial resolution and matched coordinates,” J. Opt. Soc. Am. A28, 238–244 (2011). [CrossRef]
  18. P. Yeh, Optical waves in layered media, (John Wiley & Sons, New York, 1988).
  19. S. E. Kocabas, G. Veronis, D. A. B. Miller, and S. Fan, “Modal analysis and coupling in metal-insulator-metal waveguides,” Phys. Rev. B79, 035120 (2009). [CrossRef]
  20. A. Hessel and A. A. Oliner, “A new theory of Wood anomalies on optical gratings,” Appl. Opt.4, 1275–1297 (1965). [CrossRef]
  21. U. Fano, “Effects of configuration interaction on intensities and phase shifts,” Phys. Rev.124, 1866–1878 (1961). [CrossRef]
  22. M. Sarrazin, J.-P. Vigneron, and J.-M. Vigoureux, “Role of Wood anomalies in optical properties of thin metallic films with a bidimensional array of subwavelength holes,” Phys. Rev. B67, 085415 (2003). [CrossRef]
  23. C. Genet, M. P. van Exter, and J. P. Woerdman, “Fano-type interpretation of red-shifts and red tails in hole array transmission spectra,” Opt. Commun.225, 331–336 (2003). [CrossRef]
  24. A. Sharon, D. Rosenblatt, and A. A. Friesem, “Resonant grating-waveguide structures for visible and near-infrared radiation,” J. Opt. Soc. Am. A14, 2985–2993 (1997). [CrossRef]
  25. S. M. Norton, T. Erdogan, and G. M. Morris, “Coupled-mode theory of resonant-grating filters,” J. Opt. Soc. Am. A14, 629–639 (1997). [CrossRef]
  26. V. Liu, M. Povinelli, and S. Fan, “Resonance-enhanced optical forces between coupled photonic crystal slabs,” Opt. Express17(24), 21897–21909 (2009). [CrossRef] [PubMed]
  27. R. Ruppin, “Surface polaritons of a left-handed material slab,” J. Phys. Condens. Matter13(9), 1811–1818 (2001). [CrossRef]
  28. S. A. Darmanyan, M. Neviere, and A. A. Zakhidov, “Surface modes at the interface of conventional and left-handed media,” Opt. Commun.225, 233–240 (2003). [CrossRef]
  29. N. M. Litchinitser, A. I. Maimistov, I. R. Gabitov, R. Z. Sagdeev, and V. M. Shalaev, “Metamaterials: electromagnetic enhancement at zero-index transition,” Opt. Lett.33, 2350–2352 (2008). [CrossRef] [PubMed]
  30. H. A. Haus, Waves and fields in Optoelectronics, (New Jersey, Prentice-Hall Inc., 1984).
  31. A. Yariv, “Universal relations for coupling of optical power between microresonators and dielectric waveguides,” Electron. Lett.4, 321–322 (2000). [CrossRef]
  32. H. Raether, Surface plasmons on smooth and rough surfaces and on gratings, (Springer-Verlag, New York, 1988).
  33. S. Fan, W. Suh, and J. D. Joannopoulos, “Temporal coupled-mode theory for the Fano resonance in optical resonators,” J. Opt. Soc. Am. A20, 569–572 (2003). [CrossRef]

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