## Design of high-transmission metallic meander stacks with different grating periodicities for subwavelength-imaging applications |

Optics Express, Vol. 19, Issue 4, pp. 3627-3636 (2011)

http://dx.doi.org/10.1364/OE.19.003627

Acrobat PDF (1530 KB)

### Abstract

When replacing a bulk negative index material (NIM) with two resonant surfaces that allow for surface plasmon polariton (SPP) propagation it is possible to recreate the same near-field imaging effects as with Pendry’s perfect lens. We show that a metallic meander structure is perfectly suited as such a resonant surface due to the tunability of the short (SRSPP) and long range surface plasmon (LRSPP) frequencies by means of geometrical variation. Furthermore, the Fano-type pass band between the SRSPP and LRSPP frequencies of a single meander sheet retains its dominant role when being stacked. Hence, the pass band frequency position, which is determined by the meander geometry, controls also the pass band of a meander stack. When building up stacks with different periodicities the pass band shifts in frequency for each sheet in a different way. We rigorously calculate the spectra of various meander designs and show that this shift can be compensated by changing the remaining geometrical parameters of each single sheet. We also present a basic idea how high- transmission stacks with different periodicities can be created to enable energy transfer at low loss over practically arbitrary distances inside such a stack. The possibility to stack meander sheets of varying periodicity might be the key to far field superlenses since a controlled transformation of evanescent modes to traveling wave modes of higher diffraction order could be enabled.

© 2011 OSA

## 1. Introduction

1. V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ε and μ,” Sov. Phys. Usp. **10**(4), 509–514 (1968). [CrossRef]

*ε*and a negative permeability

*μ*are able to transmit evanescent electromagnetic waves without exponential damping. Furthermore, light would bend in the “wrong direction” and hence exhibit a negative index of refraction. In 2000 Pendry [2

2. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. **85**(18), 3966–3969 (2000). [CrossRef] [PubMed]

*n*= −1, which he called superlens, would be able to restore the evanescent source fields perfectly in the image plane. However, this image is still non-magnified and cannot be used for conventional non-scanning microscopy. Among others, one approach to realize a magnifying superlens incorporates an anisotropic metamaterial crystal with hyperbolic dispersion [3

3. A. Salandrino and N. Engheta, “Far-field subdiffraction optical microscopy using metamaterial crystals: Theory and simulations,” Phys. Rev. B **74**(7), 075103 (2006). [CrossRef]

5. L. V. Alekseyev and E. Narimanov, “Slow light and 3D imaging with non-magnetic negative index systems,” Opt. Express **14**(23), 11184–11193 (2006). [CrossRef] [PubMed]

6. S. Durant, Z. Liu, J. M. Steele, and X. Zhang, “Theory of the transmission properties of an optical far-field superlens for imaging beyond the diffraction limit,” J. Opt. Soc. Am. B **23**(11), 2383 (2006). [CrossRef]

7. Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science **315**(5819), 1686 (2007). [CrossRef] [PubMed]

8. S. Maslovski and S. Tretyakov, “Phase conjugation and perfect lensing,” J. Appl. Phys. **94**(7), 4241 (2003). [CrossRef]

11. C. R. Simovski, A. J. Viitanen, and S. A. Tretyakov, “Resonator mode in chains of silver spheres and its possible application,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. **72**(6), 066606 (2005). [CrossRef]

12. H. Schweizer, L. Fu, H. Gräbeldinger, H. Guo, N. Liu, S. Kaiser, and H. Giessen, “Negative permeability around 630 nm in nanofabricated vertical meander metamaterials,” Phys. Status Solidi., A Appl. Mater. Sci. **204**(11), 3886–3900 (2007). [CrossRef]

12. H. Schweizer, L. Fu, H. Gräbeldinger, H. Guo, N. Liu, S. Kaiser, and H. Giessen, “Negative permeability around 630 nm in nanofabricated vertical meander metamaterials,” Phys. Status Solidi., A Appl. Mater. Sci. **204**(11), 3886–3900 (2007). [CrossRef]

13. L. Fu, H. Schweizer, T. Weiss, and H. Giessen, “Optical properties of metallic meanders,” J. Opt. Soc. Am. B **26**(12), B111 (2009). [CrossRef]

13. L. Fu, H. Schweizer, T. Weiss, and H. Giessen, “Optical properties of metallic meanders,” J. Opt. Soc. Am. B **26**(12), B111 (2009). [CrossRef]

## 2. Numerical simulation models and analysis methods

*t*, a meander depth

*D*and the periodicity

*P*

_{x}. To achieve inversion symmetry along the propagation direction of the incident light the condition

*W*

_{r}=

*P*

_{x}/2 -

*t*has to be fulfilled.

*D*

_{spa}. For stacks with geometrically varying meander sheets, the geometry parameters are additionally labeled with an integer number

*i*from top to bottom according to the direction of the incident light, hence

*t*

_{i},

*D*

_{i},

*P*

_{x,i}and

*D*

_{spa,i}(Fig. 1b).

14. P. Johnson and R. Christy, “Optical Constants of the Noble Metals,” Phys. Rev. B **6**(12), 4370–4379 (1972). [CrossRef]

*ω*

_{p}= 1.37 × 10

^{16}rad/s and the scattering frequency

*ν*= 8.5 × 10

^{13}rad/s [13

13. L. Fu, H. Schweizer, T. Weiss, and H. Giessen, “Optical properties of metallic meanders,” J. Opt. Soc. Am. B **26**(12), B111 (2009). [CrossRef]

15. L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A **13**(9), 1870 (1996). [CrossRef]

16. T. Weiss, N. A. Gippius, S. G. Tikhodeev, G. Granet, and H. Giessen, “Efficient calculation of the optical properties of stacked metamaterials with a Fourier modal method,” J. Opt. A, Pure Appl. Opt. **11**(11), 114019 (2009). [CrossRef]

17. G. Granet, “Reformulation of the lamellar grating problem through the concept of adaptive spatial resolution,” J. Opt. Soc. Am. A **16**(10), 2510 (1999). [CrossRef]

## 3. Influence of the geometric parameters on the plasmonic band structure of a single meander sheet

**26**(12), B111 (2009). [CrossRef]

18. I. R. Hooper and J. R. Sambles, “Coupled surface plasmon polaritons on thin metal slabs corrugated on both surfaces,” Phys. Rev. B **70**(4), 045421 (2004). [CrossRef]

*f*-

*k*space, enabling coupling to incident radiation (diffractive coupling) [19

19. Z. Chen, I. R. Hooper, and J. R. Sambles, “Coupled surface plasmons on thin silver gratings,” J. Opt. A, Pure Appl. Opt. **10**(1), 015007 (2008). [CrossRef]

20. B. Luk’yanchuk, N. I. Zheludev, S. A. Maier, N. J. Halas, P. Nordlander, H. Giessen, and C. T. Chong, “The Fano resonance in plasmonic nanostructures and metamaterials,” Nat. Mater. **9**(9), 707–715 (2010). [CrossRef] [PubMed]

21. 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**(8), 085415 (2003). [CrossRef]

*T*) of a meander structure as a function of frequency and meander thickness

*t*for perpendicular incidence (

*k*

_{x}= 0) retrieved with RCWA. It is evident that the SRSPP mode decreases whereas the LRSPP mode remains almost constant for lower values of

*t*[18

18. I. R. Hooper and J. R. Sambles, “Coupled surface plasmon polaritons on thin metal slabs corrugated on both surfaces,” Phys. Rev. B **70**(4), 045421 (2004). [CrossRef]

*D*at a constant thickness

*t*and periodicity

*P*

_{x}as shown in Fig. 3b leads to a down-shift of the SRSPP frequency, while the LRSPP mode experiences only a slight red-shift for smaller values but then stays fairly constant. Finally, Fig. 3c shows the extinction as a function of frequency and grating periodicity

*P*

_{x}at a constant meander depth

*D*and thickness

*t*. In this case the SRSPP mode changes less with

*P*

_{x}compared to the LRSPP mode, which becomes even more emphasized for deeper corrugation

*D*.

*t*and meander depth

*D*there are two degrees of freedom available to control the pass band position, which is limited by the SRSPP/LRSPP frequencies, for a grating with fixed periodicity

*P*

_{x}. This is the key for the design of a high-transmission stack consisting of meander sheets with different periodicities anywhere in the optical domain.

## 4. Properties and design of meander stacks

### 4.1 Properties of meander stacks with two sheets having the same periodicity P_{x}

*i*= 2 as a function of the distance between the meander sheets

*D*

_{spa}and the frequency

*f*for different meander depths

*D*. The FP modes that exist in the structure can be predicted with a modified cavity equation [22] (black dashed lines in Fig. 4):In this equation

*c*is the speed of light,

*m*represents the order of the FP mode,

*θ*is the angle of incidence, and

*φ(f)*describes the frequency-dependent phase shift at the silver surface.

*D*(Fig. 4a), FP modes prevail in the optical response of the stack as expected. For increasing

*D*a red shift of both SRSPP and LRSPP modes is observed with a stronger shift on the SRSPP side (Fig. 4b). With increasing depth

*D*(Fig. 4c) the coupling between FP and plasmon modes becomes stronger and exhibits highest transmittance in the frequency range where the single meander structure already shows its SRSPP/LRSPP associated pass band. With further increasing meander depth (Fig. 4d) the coupling becomes weaker again. For all meander depths

*D*an anticrossing between local SPP (white dashed lines) and FP modes can be observed at the points of their intersection, which we interpret as cavity plasmon polaritons in analogy to cavity polaritons occurring in micro resonators [23

23. C. Weisbuch, M. Nishioka, A. Ishikawa, and Y. Arakawa, “Observation of the coupled exciton-photon mode splitting in a semiconductor quantum microcavity,” Phys. Rev. Lett. **69**(23), 3314–3317 (1992). [CrossRef] [PubMed]

*D*the plasmon pass band of the single meander sheet also governs the transmission pass band in the double meander stack.

*f*= 600 THz, λ = 500 nm) is perpendicularly incident from the top onto a 100 nm thick layer of chromium with a subwavelength slit (

*w*= 100 nm). Behind the aperture, a double meander structure with

*P*

_{x}= 400 nm,

*t*= 20 nm,

*D*= 50 nm and

*D*

_{spa}= 200 nm follows. As expected, the meander sheets behave as coupled resonant surfaces and create a near-field deep subwavelength image of the slit similar to the perfect lens. This first image has a FWHM of 126 nm, which is shown in the upper left inset of Fig. 5. However, the slit can also be imaged by a second focus with a FWHM of 346 nm almost two wavelengths behind the stack (lower right inset). This can still be considered as subwavelength imaging but occurs in a more usable distance behind the lens.

### 4.2 Design and properties of meander stacks with two sheets and different periodicities P_{x}

*D*and thickness

*t*the SRSSP and LRSPP frequencies shift down in frequency for higher periodicities

*P*

_{x}. To achieve high transmission in a meander stack with sheets of different periodicities

*P*

_{x}, the remaining geometrical parameters

*t*and

*D*have to be altered in order to compensate for this change (compare Figs. 2a and 2b). For instance, in order to match the pass bands of two sheets with the periodicities

*P*

_{x,1}= 250 nm and

*P*

_{x,2}= 500 nm, the values for

*t*

_{1}/

*D*

_{1}have to be decreased / increased whereas the values for

*t*

_{2}/

*D*

_{2}have to be increased / decreased, respectively.

*P*

_{x,1}= 250 nm and

*P*

_{x,2}= 500 nm we are using the combination

*t*

_{1}= 20 nm,

*D*

_{1}= 25 nm,

*t*

_{2}= 14 nm and

*D*

_{2}= 40 nm to obtain a pass band around

*f*= 820 THz (λ = 365 nm) in the final structure. The transmittance of the overall stack as a function of frequency

*f*and

*D*

_{spa}is shown in Fig. 6a . It is evident that the overall pass band has been shifted in frequency and momentum compared to the meander stack from the previous section (Fig. 4c). The dispersion diagram of a particular meander stack with

*D*

_{spa,1}= 110 nm (Fig. 6b) shows features similar to the low-periodicity meander structure (not shown) but the influence of the FP modes is clearly visible. It is also important that for a given frequency the transfer function can be extended to higher

*k*

_{x}values to obtain resonant transmission for various directions, hence demonstrating

*k*-filtering.

### 4.3 Design and properties of meander stacks consisting of four sheets with different periodicities P_{x}

*P*

_{x,1}= 250 nm,

*P*

_{x,2}= 300 nm,

*P*

_{x,3}= 350 nm and

*P*

_{x,4}= 400 nm. The pass bands of the single sheets are shown in Fig. 7a . The compensation by geometrical variation of

*t*and

*D*for the lowest and highest periodicity is displayed in Fig. 7b and Fig. 7c, respectively.

*f*= 650 THz (λ = 484 nm) the optimized geometry parameters can be found in Fig. 8 along with the dispersion diagrams to show the

*k*

_{x}dependence of the pass band. Please note that due to the periodicity of the meander sheets it is possible that plasmon modes from neighboring Brillouin zones interact with the Fano-type resonance and lower the transmission in the pass band (see arrows in Fig. 8).

*f*and the distance between the sheets

*D*

_{spa}=

*D*

_{spa,1}=

*D*

_{spa,2}=

*D*

_{spa,3}. Because of the enormous computational effort to calculate a 42 µm wide stack, which would be the least common multiple of the four different periodicities, we used a lower cell length and bore with the grating irregularities at the edges. Earlier calculations showed that this estimation is sufficient and hardly changes the results. For this particular dispersion diagram we used 250 modes and a cell width of 6 µm to achieve convergence. Within this width we fit 24 unit cells with a length of

*P*

_{x,1}, 18 unit cells with

*P*

_{x,2}, 16 unit cells with

*P*

_{x,3}and 14 unit cells with

*P*

_{x,4}(principle illustration in Fig. 9a).

*D*

_{spa}as in the case of a meander stack with two sheets of same geometry (Fig. 4b) albeit with a narrower spectral width. In this case a different combination of geometrical parameters might create a broader spectral width. However, this example illustrates that it is in principle possible to resonantly transfer the plasmon field from layer to layer via a meander stack with different periodicities. The transmission behind the whole structure still accounts to 0.5 and shows that within the pass band energy can be transferred over distances larger than 3 µm as for instance at

*D*

_{spa}= 1 µm.

*D*

_{spa}= 370 nm (Fig. 9b) we also find that the whole stack allows high transmission for a range of

*k*

_{x}values, which is important for potential subwavelength applications.

*k*

_{x}values.

*k*

_{x}values until a transition from evanescent to propagating modes occurs. For that, the last sheet requires a periodicity that is smaller than the wavelength in order to enable first-order diffraction and therefore coupling from plasmon modes with originally higher momenta to the vacuum field.

## 5. Conclusion

*f*(

*D*

_{spa}) one observes anticrossing of FP and local SPP modes, which we interpret as cavity plasmon polaritons. Furthermore, we showed that the single meander SRSPP/LRSPP behavior can be changed to a large degree by variation of geometry. We demonstrated that for meander stacks the interplay of SPP and FP modes can create a pass band, which is limited by the LRSPP and SRSPP frequencies. By means of geometry variation it can be placed anywhere in the optical domain. This especially holds for stacks composed of meander sheets of varying geometry, which could be suitable for realization of

*k*-filters.

24. B. Edwards, A. Alù, M. E. Young, M. Silveirinha, and N. Engheta, “Experimental verification of epsilon-near-zero metamaterial coupling and energy squeezing using a microwave waveguide,” Phys. Rev. Lett. **100**(3), 033903 (2008). [CrossRef] [PubMed]

## Acknowledgments

## References and links

1. | V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ε and μ,” Sov. Phys. Usp. |

2. | J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. |

3. | A. Salandrino and N. Engheta, “Far-field subdiffraction optical microscopy using metamaterial crystals: Theory and simulations,” Phys. Rev. B |

4. | Z. Jacob, L. V. Alekseyev, and E. Narimanov, “Optical Hyperlens: Far-field imaging beyond the diffraction limit,” Opt. Express |

5. | L. V. Alekseyev and E. Narimanov, “Slow light and 3D imaging with non-magnetic negative index systems,” Opt. Express |

6. | S. Durant, Z. Liu, J. M. Steele, and X. Zhang, “Theory of the transmission properties of an optical far-field superlens for imaging beyond the diffraction limit,” J. Opt. Soc. Am. B |

7. | Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science |

8. | S. Maslovski and S. Tretyakov, “Phase conjugation and perfect lensing,” J. Appl. Phys. |

9. | S. Maslovski, S. A. Tretyakov, and P. Alitalo, “Near-field enhancement and imaging in double planar polariton-resonant structures,” J. Appl. Phys. |

10. | P. Alitalo, C. R. Simovski, A. Viitanen, and S. A. Tretyakov, “Near-field enhancement and subwavelength imaging in the optical region using a pair of two-dimensional arrays of metal nanospheres,” Phys. Rev. B |

11. | C. R. Simovski, A. J. Viitanen, and S. A. Tretyakov, “Resonator mode in chains of silver spheres and its possible application,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

12. | H. Schweizer, L. Fu, H. Gräbeldinger, H. Guo, N. Liu, S. Kaiser, and H. Giessen, “Negative permeability around 630 nm in nanofabricated vertical meander metamaterials,” Phys. Status Solidi., A Appl. Mater. Sci. |

13. | L. Fu, H. Schweizer, T. Weiss, and H. Giessen, “Optical properties of metallic meanders,” J. Opt. Soc. Am. B |

14. | P. Johnson and R. Christy, “Optical Constants of the Noble Metals,” Phys. Rev. B |

15. | L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A |

16. | T. Weiss, N. A. Gippius, S. G. Tikhodeev, G. Granet, and H. Giessen, “Efficient calculation of the optical properties of stacked metamaterials with a Fourier modal method,” J. Opt. A, Pure Appl. Opt. |

17. | G. Granet, “Reformulation of the lamellar grating problem through the concept of adaptive spatial resolution,” J. Opt. Soc. Am. A |

18. | I. R. Hooper and J. R. Sambles, “Coupled surface plasmon polaritons on thin metal slabs corrugated on both surfaces,” Phys. Rev. B |

19. | Z. Chen, I. R. Hooper, and J. R. Sambles, “Coupled surface plasmons on thin silver gratings,” J. Opt. A, Pure Appl. Opt. |

20. | B. Luk’yanchuk, N. I. Zheludev, S. A. Maier, N. J. Halas, P. Nordlander, H. Giessen, and C. T. Chong, “The Fano resonance in plasmonic nanostructures and metamaterials,” Nat. Mater. |

21. | 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 |

22. | P. Schau, K. Frenner, L. Fu, H. Schweizer, and W. Osten, “Coupling between surface plasmons and Fabry-Pérot modes in metallic double meander structures,” in |

23. | C. Weisbuch, M. Nishioka, A. Ishikawa, and Y. Arakawa, “Observation of the coupled exciton-photon mode splitting in a semiconductor quantum microcavity,” Phys. Rev. Lett. |

24. | B. Edwards, A. Alù, M. E. Young, M. Silveirinha, and N. Engheta, “Experimental verification of epsilon-near-zero metamaterial coupling and energy squeezing using a microwave waveguide,” Phys. Rev. Lett. |

**OCIS Codes**

(050.2230) Diffraction and gratings : Fabry-Perot

(050.2770) Diffraction and gratings : Gratings

(230.4170) Optical devices : Multilayers

(240.5420) Optics at surfaces : Polaritons

(240.6680) Optics at surfaces : Surface plasmons

(160.3918) Materials : Metamaterials

**ToC Category:**

Diffraction and Gratings

**History**

Original Manuscript: October 11, 2010

Revised Manuscript: December 19, 2010

Manuscript Accepted: January 21, 2011

Published: February 10, 2011

**Citation**

Philipp Schau, Karsten Frenner, Liwei Fu, Heinz Schweizer, Harald Giessen, and Wolfgang Osten, "Design of high-transmission metallic meander stacks with different grating periodicities for subwavelength-imaging applications," Opt. Express **19**, 3627-3636 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-4-3627

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### References

- V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ε and μ,” Sov. Phys. Usp. 10(4), 509–514 (1968). [CrossRef]
- J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85(18), 3966–3969 (2000). [CrossRef] [PubMed]
- A. Salandrino and N. Engheta, “Far-field subdiffraction optical microscopy using metamaterial crystals: Theory and simulations,” Phys. Rev. B 74(7), 075103 (2006). [CrossRef]
- Z. Jacob, L. V. Alekseyev, and E. Narimanov, “Optical Hyperlens: Far-field imaging beyond the diffraction limit,” Opt. Express 14(18), 8247–8256 (2006). [CrossRef] [PubMed]
- L. V. Alekseyev and E. Narimanov, “Slow light and 3D imaging with non-magnetic negative index systems,” Opt. Express 14(23), 11184–11193 (2006). [CrossRef] [PubMed]
- S. Durant, Z. Liu, J. M. Steele, and X. Zhang, “Theory of the transmission properties of an optical far-field superlens for imaging beyond the diffraction limit,” J. Opt. Soc. Am. B 23(11), 2383 (2006). [CrossRef]
- Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315(5819), 1686 (2007). [CrossRef] [PubMed]
- S. Maslovski and S. Tretyakov, “Phase conjugation and perfect lensing,” J. Appl. Phys. 94(7), 4241 (2003). [CrossRef]
- S. Maslovski, S. A. Tretyakov, and P. Alitalo, “Near-field enhancement and imaging in double planar polariton-resonant structures,” J. Appl. Phys. 96(3), 1293 (2004). [CrossRef]
- P. Alitalo, C. R. Simovski, A. Viitanen, and S. A. Tretyakov, “Near-field enhancement and subwavelength imaging in the optical region using a pair of two-dimensional arrays of metal nanospheres,” Phys. Rev. B 74(23), 235425 (2006). [CrossRef]
- C. R. Simovski, A. J. Viitanen, and S. A. Tretyakov, “Resonator mode in chains of silver spheres and its possible application,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 72(6), 066606 (2005). [CrossRef]
- H. Schweizer, L. Fu, H. Gräbeldinger, H. Guo, N. Liu, S. Kaiser, and H. Giessen, “Negative permeability around 630 nm in nanofabricated vertical meander metamaterials,” Phys. Status Solidi., A Appl. Mater. Sci. 204(11), 3886–3900 (2007). [CrossRef]
- L. Fu, H. Schweizer, T. Weiss, and H. Giessen, “Optical properties of metallic meanders,” J. Opt. Soc. Am. B 26(12), B111 (2009). [CrossRef]
- P. Johnson and R. Christy, “Optical Constants of the Noble Metals,” Phys. Rev. B 6(12), 4370–4379 (1972). [CrossRef]
- L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A 13(9), 1870 (1996). [CrossRef]
- T. Weiss, N. A. Gippius, S. G. Tikhodeev, G. Granet, and H. Giessen, “Efficient calculation of the optical properties of stacked metamaterials with a Fourier modal method,” J. Opt. A, Pure Appl. Opt. 11(11), 114019 (2009). [CrossRef]
- G. Granet, “Reformulation of the lamellar grating problem through the concept of adaptive spatial resolution,” J. Opt. Soc. Am. A 16(10), 2510 (1999). [CrossRef]
- I. R. Hooper and J. R. Sambles, “Coupled surface plasmon polaritons on thin metal slabs corrugated on both surfaces,” Phys. Rev. B 70(4), 045421 (2004). [CrossRef]
- Z. Chen, I. R. Hooper, and J. R. Sambles, “Coupled surface plasmons on thin silver gratings,” J. Opt. A, Pure Appl. Opt. 10(1), 015007 (2008). [CrossRef]
- B. Luk’yanchuk, N. I. Zheludev, S. A. Maier, N. J. Halas, P. Nordlander, H. Giessen, and C. T. Chong, “The Fano resonance in plasmonic nanostructures and metamaterials,” Nat. Mater. 9(9), 707–715 (2010). [CrossRef] [PubMed]
- 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(8), 085415 (2003). [CrossRef]
- P. Schau, K. Frenner, L. Fu, H. Schweizer, and W. Osten, “Coupling between surface plasmons and Fabry-Pérot modes in metallic double meander structures,” in Proc. SPIE, Vol. 7711 2010, p. 77111F.
- C. Weisbuch, M. Nishioka, A. Ishikawa, and Y. Arakawa, “Observation of the coupled exciton-photon mode splitting in a semiconductor quantum microcavity,” Phys. Rev. Lett. 69(23), 3314–3317 (1992). [CrossRef] [PubMed]
- B. Edwards, A. Alù, M. E. Young, M. Silveirinha, and N. Engheta, “Experimental verification of epsilon-near-zero metamaterial coupling and energy squeezing using a microwave waveguide,” Phys. Rev. Lett. 100(3), 033903 (2008). [CrossRef] [PubMed]

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