## Analytical theory of wave propagation through stacked fishnet metamaterials

Optics Express, Vol. 17, Issue 14, pp. 11582-11593 (2009)

http://dx.doi.org/10.1364/OE.17.011582

Acrobat PDF (327 KB)

### 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

## 1. Introduction

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]

5. M. Navarro-Cía, 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-Avila1, D. A. Genov1, 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]

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]

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]

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]

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]

5. M. Navarro-Cía, 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]

7. J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila1, D. A. Genov1, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature (London) **455**, 299–300 (2008).
[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]

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]

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

*ε*, whereas the coupled metallic strips along the orthogonal direction were supposed to form an LC resonator providing a negative

*µ*[1

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]

**95**, 137404 (2005).
[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]

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]

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]

5. M. Navarro-Cía, 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]

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. Marqués, “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. Marqués, 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]

## 2. Analysis

*iωt*). The unit cell of this structure is also shown in Fig.1(c) and (d). For propagation along the

*z*axis the structure and the field are both invariant by reflections at the

*x*=±

*a*/2 and

*y*=±

*a*/2 planes. Owing to the specific properties of electric and magnetic fields under mirror symmetry [14, pp. 267–273], these planes must be virtual perfect electric or perfect magnetic conducting (PEC or PMC) walls. Choosing x-planes as PMCs and y-planes as PECs we obtain the unit cell depicted in Fig.1(c) and (d), which implies an y-polarized mode. It is also possible to make the opposite choice for the

*x*- and

*y*- planes, which leads to the orthogonal propagating mode (which, for the considered square periodicity, has the same dispersion equation). Choosing

*x*- and

*y*-planes both as PECs (or PMCs) leads - below Wood’s anomaly - to non propagative modes, which will not be considered here. Thus, for the considered propagation, the analyzed structure is equivalent to the TEM square waveguide periodically loaded by square irises shown in Fig.1(c) and (d).

13. R. Marqués, 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]

*E*, produced by the currents flowing on the metallic screen at

^{0}_{y}*z*=0 (see Fig.1) is expanded as the following series of the TEM, TE

_{2n,2m}, and TM

*modes of the infinite TEM waveguide with appropriate coefficients:*

_{2n,2m}*x,y*of the different TE

*and TM*

_{2n,2m}*modes of the waveguide. In (1) only even modes have been considered due to the symmetry of the structure, which includes two additional symmetry planes at*

_{2n,2m}*x*=0 and at

*y*=0, respectively. From (1), using standard waveguide theory, the electric field component

*E*produced by these currents is written as

^{0}_{x}*g*(

_{nm}*x,y*)=sin(2

*nπx/a*) sin(2

*mπy/a*).

^{iβz}is propagating along the structure. The total field at

*z*=0 is given by the addition of (1) and the

*incident*field,

*E*

^{inc}

*, created by the currents on all the remaining screens at*

_{y}*z=qp*, where

*p*is the periodicity along the

*z*-axis and

*q*=±1, ±2, …. Near Wood’s anomaly, where the first higher order modes TE

_{20}and TM

_{02}in the expansion (1) are at cutoff, the attenuation constant of such modes

*z*=0 regardless of the periodicity of the structure. Thus, for not very small periodicities, where additional higher order modes may be present, the incident field at

*z*=0 can be expressed as

*E*

^{TE}

*and*

_{y}*E*

^{TM}

*in the incident field at*

_{y}*z*=0 makes our analysis different from the standard analysis of periodic structures [16, Sec. 9

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]

_{02}mode what creates a transmission band starting at Wood’s anomaly which, depending on the structural parameters, can provide a backward wave propagating along the stacked fishnet.

13. R. Marqués, 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]

*x*component of the electric field (3) is

*E*(

_{x}*x,y*)

^{0}≈0, which follows from the boundary conditions at the metallic screen, and from the specific behavior of

*g*(

_{nm}*x,y*) for

*x, y*≈0. This approximation gives the relation

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

*E*=

_{y}*E*

^{0}

*+*

_{y}*E*

^{TEM}

*+*

_{y}*E*

^{TE}

*+*

_{y}*E*

^{TM}

*is the total field at*

_{y}*z*=0, and subindex

*w*and

*h*stand for the waveguide and hole sections respectively. This condition provides the equation

*E*in a series similar to (1) and using the orthogonality properties of

_{y}*f*, suffices for the determination of all the coefficients as a function of the first coefficient

_{nm}*A*

_{0}. Once all coeffcients in (1) have been determined, the tangential magnetic field created by the currents at

*z*=0,

*H*

^{0}

*, is obtained from (1) using waveguide theory [16, Sec. 5]. Finally, the phase constant*

_{x}*β*of the Bloch wave is obtained by imposing the condition

*H*

^{0}

*=0 at the aperture [14, p. 486]. After some cumbersome but straightforward calculations, the dispersion equation for the Bloch wave can be written in the following way:*

_{x}*x*)=sin(

*x*)/

*x*, and

*Y*

^{TE}

*and*

_{2n,2m}*Y*

^{TM}

*are the admittances of the TE and TM waveguide modes given by*

_{2n,2m}*N,M*=round(

*a/b*) [13

**17**, 5571–5579 (2009)
[CrossRef] [PubMed]

*a*and

*b*, the value of

*N,M*is usually very low (no more than five or six), which means that very few terms should be summed up in the series in (15).

## 3. Discussion and numerical results

*βp*≪1). This last equation corresponds to an infinite transmission line of periodicity

*p*, parallel admittance

*Y*, and series admittance 2

_{p}*Y*.

_{s}*Y*is the contribution of all the waveguide modes localized at the discontinuities,

*i.e*., of all the TE

*, and TM*

_{2n,2m}_{2n,2m}modes with

*n*+

*m*>1. The first and the second terms in (13) correspond to the contributions of the non-localized modes,

*i.e*., the TEM, TE

_{20}, and TM

_{02}modes. As expected, only these non localized modes contribute to the series admittance (14). The main difference between these equations and the standard equations for periodically loaded transmission lines and waveguides [16, Sec. 9] is the presence of the non-localized TE

_{20}and TM

_{02}modes in the series admittance

*Y*. As it was already mentioned, this presence is imposed by the small values of the attenuation constant (4) near Wood’s anomaly, and it provides the most relevant physical effects in stacked-fishnet structures. In particular, just at Wood’s anomaly, the attenuation constant (4) vanishes and the corresponding TM

_{s}_{02}mode admittance (17) becomes infinite. Also the series admittance

*Y*(14) becomes infinite and the phase constant

_{s}*β*vanishes. Hence, there is always a transmission band starting at Wood’s anomaly with zero phase and group velocity. Just below Wood’s anomaly, the series admittance is dominated by the TM

_{02}admittance, which is capacitive (ℑ(

*Y*

^{TM}

_{02}<0). Thus, if the parallel admittance is inductive (ℑ(

*Y*)>0), according to (18) there must be a backward-wave transmission band [17, pp. 263–264] just below Wood’s anomaly. Now we will examine under which conditions the parallel admittance is inductive. For small holes, the contribution of the localized modes (15) is always inductive [12

_{p}12. F. Medina, F. Mesa, and R. Marqués, “Extraordinary transmission through arrays of electrically small holes from a circuit theory perspective,” IEEE Trans. Microwave Theory Tech. **56**, 3108–3120 (2008).
[CrossRef]

*p*, the contribution of the two first terms in (13) is negligibly small (of the order of

*Y*

_{0}

*k*

_{0}

*p*). Therefore, for small holes and periodicities, a backward-wave transmission band is always expected just below Wood’s anomaly.

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

*C*and

_{s}*L*come from

_{s}*Y*=-iω

_{s}*C*and 1/

_{s}*Y*=-i

_{p}*ωL*.

12. F. Medina, F. Mesa, and R. Marqués, “Extraordinary transmission through arrays of electrically small holes from a circuit theory perspective,” IEEE Trans. Microwave Theory Tech. **56**, 3108–3120 (2008).
[CrossRef]

**17**, 5571–5579 (2009)
[CrossRef] [PubMed]

*LC*resonant circuits, as it was proposed in many previous works [1

**13**, 4922–4930 (2005).
[CrossRef] [PubMed]

**95**, 137404 (2005).
[CrossRef] [PubMed]

**31**1800–1802 (2005).
[CrossRef]

**32**53–55 (2007).
[CrossRef]

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 and Wirel. Compon. Lett. **13**, 51–53(2003).
[CrossRef]

**14**, 5445–5455 (2006).
[CrossRef] [PubMed]

## 4. Generalization of the analysis

*x*and

*y*axes,

*a*and

_{x}*a*, and for rectangular holes of dimensions

_{y}*b*and

_{x}*b*along these axes, the generalization of (12)–(15) is straightforward. Moreover, if the periodicity along the longitudinal direction

_{y}*p*is small, coupling between the metallic screens by higher order evanescent modes may appear. Specifically, this coupling will appear when the product

*α*is small, where

_{nm}p_{2n,2m}or TM

_{2n,2m}modes. For the above cases, the analysis leads to the following general expressions for

*Y*and

_{p}*Y*in (12):

_{s}*a*/

_{x}*a*in (20) and (21) does not play any role in the dispersion equation. However, the introduction of this factor is convenient in order to make the admittances consistent with its circuit definition, namely, the ratio between the average current and the average voltage through the waveguide. Note that (20) and (21) reduces to (13) and (14) as

_{y}*α*increases up to a value that there is no coupling through the corresponding mode.

_{nm}p*p*, the introduction of this additional coupling substantially improves the quantitative agreement with numerical simulations. New dispersion diagrams for a stacked fishnet structure with rectangular holes have also been computed. The results are shown in Fig.6. This Figure shows that there is again a satisfactory qualitative and quantitative agreement with numerical simulations.

**56**, 3108–3120 (2008).
[CrossRef]

**17**, 5571–5579 (2009)
[CrossRef] [PubMed]

*n*=

*a*and the transmission line admittance, defined as the ratio between the average current and the average voltage through the rectangular waveguide of length

_{x}/b_{x}*t*, is

*Y*=

_{h}*Y*TE

_{10}(

*b*), where

*y*/b_{x}*Y*

_{TE10}is the wave admittance of the fundamental TE

_{10}mode excited in the waveguide formed by the hole in the screen. After solving the circuit of Fig.2(c), it turns out that this circuit can still be represented by the elemental circuit of Fig.2(a), provided the original series and parallel admittances

*Y*and

_{s}*Y*are substituted by the following new series and parallel admittances:

_{p}**17**, 5571–5579 (2009)
[CrossRef] [PubMed]

## 5. Conclusions

## Acknowledgments

## References and links

1. | S. Zhang, W. Fan, K. J. Malloy, and S. R. J. Brueck, “Near-infrared double negative metamaterials,” Opt. Express |

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. |

3. | M. Beruete, M. Sorolla, and I. Campillo “Left-handed extraordinary optical transmission through a photonic crystal of subwavelength hole arrays,” Opt. Express |

4. | M. Beruete, I. Campillo, M. Navarro-Cía, F. Falcone, and M. Sorolla “Molding left- or right-handed metamaterials by stacked cutoff metallic hole arrays,” IEEE Trans. Microwave Theory Tech. |

5. | M. Navarro-Cía, M. Beruete, M. Sorolla, and I. Campillo, “Negative refraction in a prism made of stacked subwavelength hole arrays,” Opt. Express |

6. | G. Dolling, Ch. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden “Low-loss negative-index metamaterial at telecommunication wavelengths,” Opt. Lett. |

7. | J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila1, D. A. Genov1, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature (London) |

8. | G. Dolling, M. Wegener, C. M. Soukoulis, and S. Linden, “Negative-index metamaterial at 780 nm wavelength,” Opt. Lett. |

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) |

10. | D. R. Smith, W. J. Vier, D. C. Padilla,, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. |

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. |

12. | F. Medina, F. Mesa, and R. Marqués, “Extraordinary transmission through arrays of electrically small holes from a circuit theory perspective,” IEEE Trans. Microwave Theory Tech. |

13. | R. Marqués, F. Mesa, L. Jelinek, and F. Medina “Analytical theory of extraordinary transmission through metallic diffraction screens perforated by small holes,” Opt. Express |

14. | J. D. Jackson, |

15. | R. Gordon, “Bethe’s aperture theory for arrays,” Phys. Rev. A |

16. | R. F. Collin |

17. | S. Ramo, J. R. Whinnery, and T. Van Duzer, |

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 and Wirel. Compon. Lett. |

**OCIS Codes**

(160.1245) Materials : Artificially engineered materials

(050.2065) Diffraction and gratings : Effective medium theory

(160.3918) Materials : Metamaterials

(160.5298) Materials : Photonic crystals

**ToC Category:**

Metamaterials

**History**

Original Manuscript: May 5, 2009

Revised Manuscript: June 17, 2009

Manuscript Accepted: June 17, 2009

Published: June 25, 2009

**Citation**

R. Marqués, L. Jelinek, F. Mesa, and F. Medina, "Analytical theory of wave propagation through stacked fishnet metamaterials," Opt. Express **17**, 11582-11593 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-14-11582

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### References

- 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]
- 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]
- 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).
- 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]
- 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]
- 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]
- G. Dolling, M. Wegener, C. M. Soukoulis, and S. Linden, "Negative-index metamaterial at 780 nm wavelength," Opt. Lett. 32, 53-55 (2007). [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]
- 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]
- 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]
- 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]
- J. D. Jackson, Classical Electrodynamics, (Wiley, New York 1999), 3rd Ed.
- R. Gordon, "Bethe’s aperture theory for arrays," Phys. Rev. A 76, 053806 (2007). [CrossRef]
- 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.
- 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]

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