## Circuit modeling of the transmissivity of stacked two-dimensional metallic meshes

Optics Express, Vol. 18, Issue 13, pp. 13309-13320 (2010)

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### Abstract

This paper presents a simple analytical circuit-like model to study the transmission of electromagnetic waves through stacked two-dimensional (2-D) conducting meshes. When possible the application of this methodology is very convenient since it provides a straightforward rationale to understand the physical mechanisms behind measured and computed transmission spectra of complex geometries. Also, the disposal of closed-form expressions for the circuit parameters makes the computation effort required by this approach almost negligible. The model is tested by proper comparison with previously obtained numerical and experimental results. The experimental results are explained in terms of the behavior of a finite number of strongly coupled Fabry-Pérot resonators. The number of transmission peaks within a transmission band is equal to the number of resonators. The approximate resonance frequencies of the first and last transmission peaks are obtained from the analysis of an infinite structure of periodically stacked resonators, along with the analytical expressions for the lower and upper limits of the pass-band based on the circuit model.

© 2010 Optical Society of America

## 1. Introduction

1. E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. **58**, 2059–2062 (1987). [CrossRef] [PubMed]

2. S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. **58**, 2486–2489 (1987). [CrossRef] [PubMed]

*per se*due to their practical applications in layered optical systems. For instance, although extremely thin metal layers are highly reflective at optical frequencies, the superposition of a number of these layers separated by optically thick transparent dielectric slabs has been shown to generate high transmissivity bands [3

3. M. Scalora, M. J. Bloemer, A. S. Pethel, J. P. Dowling, C. M. Bowden, and A. S. Manka, “Transparent, metallodielectric, one-dimensional, photonic band-gap structures,” J. Appl. Phys. **83**, 2377–2383 (1998). [CrossRef]

4. M. R. Gadsdon, J. Parsons, and J. R. Sambles, “Electromagnetic resonances of a multilayer metal-dielectric stack,” J. Opt. Soc. Am. B **26**, 734–742 (2009). [CrossRef]

5. S. Feng, J. M. Elson, and P. L. Overfelt, “Transparent photonic band in metallodielectric nanostructures,” Phys. Rev. B **72**, 085117 (2005). [CrossRef]

6. M. C. Larciprete, C. Sibilia, S. Paoloni, and M. Bertolotti, “Accessing the optical limiting properties of metallodielectric photonic band gap structures,” J. Appl. Phys. **93**, 5113–5017 (2003). [CrossRef]

7. I. R. Hooper and J. R. Sambles, “Some considerations on the transmissivity of thin metal films,” Opt. Express **16**, 17249–17256 (2008). [CrossRef] [PubMed]

3. M. Scalora, M. J. Bloemer, A. S. Pethel, J. P. Dowling, C. M. Bowden, and A. S. Manka, “Transparent, metallodielectric, one-dimensional, photonic band-gap structures,” J. Appl. Phys. **83**, 2377–2383 (1998). [CrossRef]

4. M. R. Gadsdon, J. Parsons, and J. R. Sambles, “Electromagnetic resonances of a multilayer metal-dielectric stack,” J. Opt. Soc. Am. B **26**, 734–742 (2009). [CrossRef]

6. M. C. Larciprete, C. Sibilia, S. Paoloni, and M. Bertolotti, “Accessing the optical limiting properties of metallodielectric photonic band gap structures,” J. Appl. Phys. **93**, 5113–5017 (2003). [CrossRef]

8. C. A. M. Butler, J. Parsons, J. R. Sambles, A. P. Hibbins, and P. A. Hobson, “Microwave transmissivity of a metamaterial-dielectric stack,” Appl. Phys. Lett. **95**, 174101 (2009). [CrossRef]

## 2. Stacked grids and unit cell model

*β*

_{0}for air-filled sections and

*β*for dielectric-filled sections) and characteristic impedances (

_{d}*Z*

_{0}and

*Z*), are known in closed form. The expressions for those parameters are

_{d}*ω*is the angular frequency and

*c*the speed of light in vacuum. Note that, due to losses,

*Z*and

_{d}*β*are complex quantities with small (low-loss regime) but non-vanishing imaginary parts.

_{d}*Z*, in Fig. 2(b). As mentioned before, these lumped elements account for the effect of below-cutoff higher-order modes excited at the mesh plane. A relatively sophisticated numerical code could be used to determine these parameters. In such case, however, no special advantage would be obtained from our circuit analog, apart from a different point of view and some additional physical insight. However, for those frequencies making the size of the unit cell,

_{g}*λ*, electrically small, accurate estimations for

_{g}*Z*are available in the literature. For

_{g}*w*≪

_{m}*λ*the grid mainly behaves as an inductive load with the following impedance for normal incidence [15

_{g}15. O. Luukkonen, C. Simovski, G. Granet, G. Goussetis, D. Lioubtchenko, A. V. Raisanen, and S. A. Tretyakov, “Simple and analytical model of planar grids and high-impedance surfaces comprising metal strips or patches,” IEEE Trans. Antennas Propag. **56**, 1624–1632 (2008). [CrossRef]

*R*=

_{g}*λ*/(

_{g}*σw*).

_{m}δ_{s}*Z*are not exact and the model has some limitations (for instance, the unit cell has to be electrically small enough), the predictions of our model must be checked against experimental and/or numerical results. This will be done in the forthcoming section.

_{g}## 3. Comparison with numerical and experimental data

8. C. A. M. Butler, J. Parsons, J. R. Sambles, A. P. Hibbins, and P. A. Hobson, “Microwave transmissivity of a metamaterial-dielectric stack,” Appl. Phys. Lett. **95**, 174101 (2009). [CrossRef]

21. HFSS: High Frequency Structure Simulator based on the Finite Element Method, Ansoft Corporation, http://www.ansoft.com

8. C. A. M. Butler, J. Parsons, J. R. Sambles, A. P. Hibbins, and P. A. Hobson, “Microwave transmissivity of a metamaterial-dielectric stack,” Appl. Phys. Lett. **95**, 174101 (2009). [CrossRef]

*Z*should be different and losses higher). Nevertheless, the essential fact is not modified: we have four FP cavities strongly coupled through the square holes of each grid;

_{g}*i.e.*, four transmission line sections separated by predominantly reactive impedances. Note that this point of view is somewhat different and alternative to that sustained in [8

**95**, 174101 (2009). [CrossRef]

**95**, 174101 (2009). [CrossRef]

*Z*, are now considered to extend over a certain distance, from the position of each grid, inside the dielectrics. Under the present point of view, the thicknesses of the grids are not relevant if they are sufficiently small, and they can be considered zero for practical purposes. It is worth mentioning that the circuit model developed for the present microwave structure could also be applied to study the stacked slabs reported in [4

_{g}4. M. R. Gadsdon, J. Parsons, and J. R. Sambles, “Electromagnetic resonances of a multilayer metal-dielectric stack,” J. Opt. Soc. Am. B **26**, 734–742 (2009). [CrossRef]

18. N. Engheta, A. Salandrino, and A. Alu, “Circuit elements at optical frequencies: nanoinductors, nanocapacitors, and nanoresistorsExtraordinary transmission through arrays of,” Phys. Rev. Lett. **95**, 095504 (2005). [CrossRef] [PubMed]

19. A. Alu, M. E. Young, and N. Engheta, “Design of nanofilters for optical nanocircuits,” Phys. Rev. B **77**, 144107 (2008). [CrossRef]

**26**, 734–742 (2009). [CrossRef]

## 4. Field distributions for the resonance frequencies

*z*(

*i.e.*, voltage or, conversely, average value of the electric field). Thus, the comparison between circuit model and HFSS results can easily be carried out because our average values of electric field can be compared, after proper normalization, with the values reported in [8

**95**, 174101 (2009). [CrossRef]

*z*-direction through the center of a hole. It is worthwhile to consider how each of the four resonance modes in the first high transmissivity frequency band (labeled as A, B, C, and D in Fig. 3) is associated with a specific field pattern along the propagation direction (

*z*). The results for these field distributions are plotted in Fig. 4. The first obvious conclusion is that the circuit model, once again, captures the most salient details of the physics of the problem, with the advantage of requiring negligible computational resources. Slight differences can be appreciated around the grid positions because, in a close proximity to the grids, HFSS provides results for the near field (which plays the role of the

*microscopic*field in the continuous medium approach) while the analytical model gives a

*macroscopic*field described by the transverse electromagnetic waves. Microscopic and macroscopic fields averaged over the lattice period are comparable for sub-wavelength grids considered in this paper. Nevertheless, with independence of the model (numerical or analytical), we can see how the field values near and over each of the three internal grids are meaningfully different for each of the considered resonance (high transmission) frequencies. The field values are relatively small over each of those internal grids for mode D. For mode C we have two grids with low field levels, and for mode B only the central grid has low values of electric field. Finally, none of the internal grids have low electric field values for mode A. The effect of an imaginary impedance at the end of a transmission line section with a significant voltage excitation is to increase the apparent (or equivalent) length of that section, as it has been explained in detail in [17

17. F. Medina, F. Mesa, and D. C. Skigin, “Extraordinary transmission through arrays of slits: a circuit theory model,” IEEE Trans. Microwave Theory Tech. **58**, 105–115 (2010). [CrossRef]

**95**, 174101 (2009). [CrossRef]

## 5. Stacked grids with a large number of layers

*f*

_{LB}and

*f*

_{UB}tend to some limit values when the number of stacked layers increases. Moreover, the resonance frequency of a single slab without considering any grid load is 13.62GHz for the materials and thicknesses used to compute the values in Table 1. It suggests that the upper limit could be given by that frequency. However, the meaning of the limit value of

*f*

_{LB}(6.380GHz) is not clear. In the following we propose an easy explanation for both the lower and upper limits.

*γ*=

*α*+

*jβ*, the following dispersion equation of the periodic structure is obtained following the method reported in [20, 22]:

*γ*is purely imaginary (

*γ*=

*jβ*) as it corresponds to propagating waves in a transmission band. For other frequency values the solution for

*γ*is real, thus giving place to evanescent waves (forbidden propagation or band gaps). For a given transmission band the upper limit is given by the condition

*βt*=

_{d}*π*(

*α*= 0), namely, a phase shift of

*π*radians in the unit cell. The frequency at which this condition appears is given by cos(

*k*) = −1, sin(

_{d}t_{d}*k*) = 0, which corresponds to the frequency of resonance of a single slab without grid,

_{d}t_{d}*k*=

_{d}t_{d}*π*. This condition is fully consistent with our previous observation in the finite structure of an upper-band limit governed mostly by the thickness of the dielectric slab with no influence of the grid and with a phase shift of the field of

*π*between adjacent layers. On the other hand, the lower limit is given by the condition

*γt*) = 1 is trivially satisfied by

_{d}*γt*= 0, (

_{d}*β*=

*α*= 0); namely, a null phase shift in the unit cell, which is in agreement with our previous observation for the field pattern of the lowest-frequency peak. The frequency where the above condition appears clearly depends on the specific value of the grid impedance,

*Z*.

_{g}23. CST Microwave Studio CST GmbH, Darmstadt, Germany, 2008, http://www.cst.com.

*Z*would not be enough to account for the complex higher-order modal interactions between adjacent grids. Fortunately, the frequency region where the model proposed in this paper works properly turns out to be the most interesting region for practical purposes, provided that non-diffracting operation is required (

_{d}*i.e.*, if higher-order grating lobes are precluded).

## 6. Conclusion

## Acknowledgments

## References and links

1. | E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. |

2. | S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. |

3. | M. Scalora, M. J. Bloemer, A. S. Pethel, J. P. Dowling, C. M. Bowden, and A. S. Manka, “Transparent, metallodielectric, one-dimensional, photonic band-gap structures,” J. Appl. Phys. |

4. | M. R. Gadsdon, J. Parsons, and J. R. Sambles, “Electromagnetic resonances of a multilayer metal-dielectric stack,” J. Opt. Soc. Am. B |

5. | S. Feng, J. M. Elson, and P. L. Overfelt, “Transparent photonic band in metallodielectric nanostructures,” Phys. Rev. B |

6. | M. C. Larciprete, C. Sibilia, S. Paoloni, and M. Bertolotti, “Accessing the optical limiting properties of metallodielectric photonic band gap structures,” J. Appl. Phys. |

7. | I. R. Hooper and J. R. Sambles, “Some considerations on the transmissivity of thin metal films,” Opt. Express |

8. | C. A. M. Butler, J. Parsons, J. R. Sambles, A. P. Hibbins, and P. A. Hobson, “Microwave transmissivity of a metamaterial-dielectric stack,” Appl. Phys. Lett. |

9. | A. B. Yakovlev, C. S. R. Kaipa, Y. R. Padooru, F. Medina, and F. Mesa, “Dynamic and circuit theory models for the analysis of sub-wavelength transmission through patterned screens,” in |

10. | 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) |

11. | R. E. Collin, |

12. | B. A. Munk, |

13. | R. Ulrich, “Far-infrared properties of metallic mesh and its complementary structure,” Infrared Phys. |

14. | R. Sauleau, Ph. Coquet, J. P. Daniel, T. Matsui, and N. Hirose, “Study of Fabry-Pérot cavities with metal mesh mirrors using equivalent circuit models. Comparison with experimental results in the 60 GHz band,” Int. J. Infrared and Millim. Waves |

15. | O. Luukkonen, C. Simovski, G. Granet, G. Goussetis, D. Lioubtchenko, A. V. Raisanen, and S. A. Tretyakov, “Simple and analytical model of planar grids and high-impedance surfaces comprising metal strips or patches,” IEEE Trans. Antennas Propag. |

16. | 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. |

17. | F. Medina, F. Mesa, and D. C. Skigin, “Extraordinary transmission through arrays of slits: a circuit theory model,” IEEE Trans. Microwave Theory Tech. |

18. | N. Engheta, A. Salandrino, and A. Alu, “Circuit elements at optical frequencies: nanoinductors, nanocapacitors, and nanoresistorsExtraordinary transmission through arrays of,” Phys. Rev. Lett. |

19. | A. Alu, M. E. Young, and N. Engheta, “Design of nanofilters for optical nanocircuits,” Phys. Rev. B |

20. | S. Tretyakov, |

21. | HFSS: High Frequency Structure Simulator based on the Finite Element Method, Ansoft Corporation, http://www.ansoft.com |

22. | D. M. Pozar, |

23. | CST Microwave Studio CST GmbH, Darmstadt, Germany, 2008, http://www.cst.com. |

**OCIS Codes**

(050.0050) Diffraction and gratings : Diffraction and gratings

(050.2230) Diffraction and gratings : Fabry-Perot

(050.6624) Diffraction and gratings : Subwavelength structures

**ToC Category:**

Diffraction and Gratings

**History**

Original Manuscript: March 4, 2010

Revised Manuscript: May 26, 2010

Manuscript Accepted: June 4, 2010

Published: June 7, 2010

**Citation**

, "Circuit modeling of the transmissivity of stacked two-dimensional metallic meshes," Opt. Express **18**, 13309-13320 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-13-13309

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