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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 13 — Jun. 21, 2010
  • pp: 13309–13320
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Circuit modeling of the transmissivity of stacked two-dimensional metallic meshes

Chandra S. R. Kaipa, Alexander B. Yakovlev, Francisco Medina, Francisco Mesa, Celia A. M. Butler, and Alastair P. Hibbins  »View Author Affiliations


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

The use of periodic structures to control electromagnetic wave propagation and energy distribution is nowadays a common practice in optics and microwaves research. Since the introduction of photonic band-gap structures (PBG’s) by the end of 1980’s [1

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

, 2

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

], hundreds of papers have been published exploring the theoretical challenges and practical realizations of such kind of structures. Although most of the published papers dealt with 3-D periodic distributions of refraction index, 1-D periodic structures have also attracted a lot of interest in the optics community. The analysis of 1-D structures requires much less computational resources, while such structures still exhibit many of the salient features observed in 3-D photonic crystals. Moreover, 1-D periodic structures are interesting 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

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]

]. Although Fabry-Pérot (FP) resonances can be invoked as the underlying mechanism behind this enhanced transmissivitty, it will be explained in this paper that PBG theory can also be used if the number of unit cells is large (each unit cell involves a thin metal film together with a thick dielectric slab). When the number of unit cells is finite, the transmission spectrum for each transmission band presents a number of peaks equal to the number of FP resonators that can be identified in the system. (Totally transparent bands without peaks have also been reported [5

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

], although that interesting case will not be considered in this paper). The highest frequency peak is associated with a low field density inside the metal films, while the lowest frequency peak corresponds to a situation where field inside the metal layers is relatively strong (the possibility of achieving field enhancement inside a nonlinear region using those stacked structures has been explored in [6

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]

]). However, all these interesting properties are lost at lower frequencies, below a few dizaines of THz. This is because electromagnetic waves inside metals at optical frequencies exist in the form of evanescent waves (the real part of the permittivity of a metal at optical frequencies is relatively large and negative, the imaginary part being smaller or of the same order of magnitude). These evanescent waves provide the necessary coupling mechanism between successive dielectric layers (Fabry-Pérot resonators) separated by metal films. At lower frequencies, metals are characterized by their high conductivities (or equivalently, large imaginary dielectric constants), in such a way that almost perfect shielding is expected even for extremely thin films a few nanometers thick [7

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]

]. Therefore, the method reported in [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

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

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]

] cannot be used in practice to enhance transmission at microwave or millimeter-wave frequencies.

In this paper, our first goal here is to show how a circuit model, whose parameters are analytically known, reasonably accounts for the experimental and numerical results reported in [8

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]

]. As stated above, this methodology is a common practice in microwave engineering and the reader can find a systematic and elegant description in a relatively recent book by S. Tretyakov [20

20. S. Tretyakov, Analytical modeling in applied electromagnetics, (Artech House, 2003).

]. Apart from avoiding lengthy and cumbersome computations, the circuit modeling provides additional physical insight and, most importantly, a methodology to design devices based on the physical phenomena described by the model. The circuit approach is also used to extract some general features of the transmission frequency bands through the analysis of an infinite structure with periodically stacked unit cells along the direction of propagation. The relation between the finite and the infinite structures is studied in the light of the equivalent circuit modeling technique.

Fig. 1. (a) Exploded schematic (the air gaps between layers are not real) of the five stacked copper grids separated by dielectric slabs used in the experiments reported in [8]. This is an example of the type of structure for which the model in this paper is suitable. (b) Top view of each metal mesh.

2. Stacked grids and unit cell model

An example of the kind of structures analyzed in this paper is given in Fig. 1. The system is composed by a set of stacked metallic grids printed on dielectric slabs. This is the multilayered structure fabricated and measured by some of the authors of this paper in [8

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]

]. Five copper grids, printed on a low-loss dielectric substrate using a conventional photo-etching process, are stacked to produce an electrically thick block, whose transmission characteristics at microwave frequencies are the subject of this study. The copper cladding thickness is tm = 18µm, and the thickness of each of the low-loss dielectric slabs (Nelco NX9255) separating copper meshes is td = 6.35mm. The relative permittivity of the dielectric material is εr ≈ 3. The loss tangent used in our simulations is tan δ = 0.0018. The lattice constant of the grid is λg = 5.0mm, and the side length of square holes is wh = 4.85mm (thus the metallic strips conforming the mesh are wm = 0.15mm wide). When a y-polarized (or x-polarized wave) uniform transverse electromagnetic plane wave normally impinges on the structure, the fields are identical for each of the unit cells of the 2-D periodic system. Taking into consideration the symmetry of the unit cell and the polarization of the impinging electric field, a single unit cell such as that shown in Fig. 2 can be used in the analysis. Thus, we have a number of uniform sections equivalent to parallel-plate waveguides, filled with air or with the above mentioned dielectric material, separated by diaphragm discontinuities. This is a typical waveguide problem with discontinuities, as those commonly considered in microwave engineering practice [11

11. R. E. Collin, Field Theory of Guided Waves (IEEE Press, 1991).

]. Since a single transverse electromagnetic (TEM) mode is assumed to propagate along the uniform waveguide sections (higher-order modes operate below their cutoff frequencies, or equivalently, it is assumed a non-diffacting regime), the circuit model shown in Fig. 2(b) gives an appropriate description of the physical system in Fig. 2(a). The shunt reactances in this circuit account for the effect of the below-cutoff higher-order modes scattered by each of the discontinuities. This model is valid provided the attenuation factor of the first higher-order mode generated at the discontinuities is large enough to ensure the interaction between successive discontinuities through higher-order modes can be neglected. The first higher-order modes that can be excited by the highly symmetrical holes under study are the TM02 and TE20 parallel-plate waveguide modes (TM/TE stands for transverse magnetic/electric to the propagation direction). The cutoff wavelength for these modes is λc = λg. The attenuation factor for frequencies not too close to cutoff (fc ≈ 60GHz for the air-filled waveguides and 34.7GHz for the dielectric-filled sections) is αTM02=αTE202π(εrλg) . Since λg = 5.0mm and the separation between the perforated screens is 6.35mm, the amplitude of the higher-order modes excited by each discontinuity at the plane of adjacent discontinuities is clearly negligible. Thus, the simple circuit in Fig. 2(b) should be physically suitable for our purposes as long as the interaction between adjacent diaphragms takes place, exclusively, through the transverse electromagnetic waves represented by the transmission line sections.

Fig. 2. (a) Transverse unit cell of the 2-D periodic structure corresponding to the analysis of the normal incidence of a y-polarized uniform plane wave on the structure shown in Fig. 1 (pec stands for perfect electric conductor, and pmc stands for perfect magnetic conductor). (b) Equivalent circuit for the electrically small unit cell (λg meaningfully smaller than the wavelength in the dielectric medium surrounding the grids); Z 0 and β 0 are the characteristic impedance and propagation constant of the air-filled region (input and output waveguides); Zd and βd are the same parameters for the dielectric-filled region (real for lossless dielectric and complex for lossy material). (c) Unit cell for the circuit based analysis of an infinite periodic structure.

The parameters of the transmission lines in Fig. 2(b), propagation constants (β 0 for air-filled sections and βd for dielectric-filled sections) and characteristic impedances (Z 0 and Zd), are known in closed form. The expressions for those parameters are

β0=ωc;βd=εr(1jtanδ)β0
(1)
Z0=μ0ε0;Zd=μ0ε01εr(1jtanδ)
(2)

where ω is the angular frequency and c the speed of light in vacuum. Note that, due to losses, Zd and βd are complex quantities with small (low-loss regime) but non-vanishing imaginary parts.

Unfortunately, no exact closed-form expressions are available for the reactive loads, Zg, 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, λg, electrically small, accurate estimations for Zg are available in the literature. For wmλg the grid mainly behaves as an inductive load with the following impedance for normal incidence [15

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]

]:

Zg=jωLg;Lg=η0λg2πcln[csc(πwm2λg)]
(3)

where η0=μ0ε0377Ω is the free-space impedance. Ohmic losses can also be taken into account using the surface resistance of the metal (copper), since the skin effect penetration depth, δs=2(ωμ0σ) , is much smaller than the thickness of the metal strips in our case. This resistance, series connected with the inductance in (3), is given by Rg = λg/(σwmδs).

Since the formulas for Zg 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.

3. Comparison with numerical and experimental data

As a first test for our model, in Fig. 3 we compare its predictions with the numerical and experimental results reported in [8

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]

] for the transmissivity of the five stacked grids studied in that paper. Experimental, numerical (simulations based on the finite elements method implemented into the commercial code [21

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

]), and analytical (circuit-model predictions) results are included in this figure. We can clearly appreciate how two bands, consisting of two groups of four transmission peaks separated by a deep stop band, are predicted by the present analytical model, in agreement with the experimental results in [8

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]

] (no HFSS simulations were reported for the second band in that paper). In the frequency range where the metal mesh is reasonably expected to behave as a purely inductive grid (well below the onset of the first higher-order mode in the dielectric-filled sections, at approximately 34.7GHz for the dielectric material and cell dimensions involved in this example), the quantitative agreement between analytical and experimental/numerical data is very good. The quality of the analytical results, however, deteriorates when the frequency increases (second band). A possible explanation for the disagreement is that the inductive model is not expected to capture the behavior of the near field around the strip wires at the higher frequencies of the second transmission band (it can be conjectured that capacitive effects cannot be ignored at high frequencies). Indeed, the effect of adding a small shunt capacitance would be to slightly shift the peaks to lower frequencies, thus improving the qualitative matching to experimental results. Unfortunately, no closed form expression has been found for that capacitance. On the other hand, dielectric losses at that frequency region appears to be higher than expected from the loss tangent used in the circuit simulation (nominal value for the commercial substrate). Likely, loss tangent of the dielectric slab is much higher than supposed, in such a way that the height of transmission peaks could be adequately predicted wiht our model provided the true loss tangent is used in the simulation. In spite of these quantitative discrepancies affecting the high frequency portion of the transmission spectrum, reasonable qualitative agreement can still be observed even in the second transmission band (four transmission peaks distributed along, approximately, the same frequency range for the analytical model and measured data). This is because the model in Fig.2 is still valid at those frequencies, except for the effects above mentioned (Zg 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; 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

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]

], which is based on the interaction between the standing waves along the dielectric regions and the evanescent waves in the grid region, although compatible with it. The difference is that the evanescent fields are not considered to be exclusively confined to the interior of the holes (which are regarded in [8

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]

] as very short sections of square waveguides operating below cutoff or, equivalently, as imaginary-index regions). The reactive fields yielding the reactive load, Zg, 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

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]

]. The reason is that the narrow metal films having negative permittivity are expected to behave as lumped inductors following the theory in [18

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

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

]. Note that the model in this paper should be modified (and the transmission spectrum would be different too) if the distance between grids were much smaller than considered. In such case the interaction due to higher order modes should be incorporated in the model, but this is not a trivial task and it is beyond the scope of the present paper. However, this problem would not affect to the optical structure analyzed in [4

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]

] because in that structure only TEM waves are excited at the interfaces between metal films and dielectric slabs, and they can be taken into account in closed form. This is an important simplifying difference with respect to the problem treated in this paper.

Fig. 3. Transmissivity (|T|2) of the stacked grids structure experimentally and numerically studied in [8]. HFSS (FEM model, FEM standing for finite elements method) and circuit simulations (analytical data) are obtained for the following parameters [with the notation used in Fig. 1]: λg = 5.0mm, wm = 0.15mm, td = 6.35mm, tm = 18µm; metal is copper and the dielectric is characterized by εr = 3 and tan δ = 0.0018. The four resonant modes in the first band are labeled as A, B, C, and D in the increasing order of frequency.

4. Field distributions for the resonance frequencies

It is important to verify if the field distribution predicted by the circuit model agrees with that provided by numerical simulations based on HFSS. Being a 3-D finite element method solver, HFSS gives information about the fields at any point within the unit cell of the structure. Certainly this is beyond the possibilities of a one-dimensional circuit model. However, the circuit model can give information about the line integral of the field along any line going from the top to the bottom metal plates of each of the parallel-plate waveguides for each particular value of 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

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]

] for the field along a line plotted in the 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]

] for a different system having a similar equivalent circuit (resonant slits in a metal screen). The above reason explains why the resonance frequencies of the modes with more highly excited discontinuities have smaller resonance frequency. In general, this discussion is compatible with that given in [8

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]

] about the distribution of peaks. However, some further details can be clarified using the circuit model; for instance, those concerning the positions of the first and last resonance and the parameters these two limits depend on. Quantitative details about the range of values where the transmission peaks should be expected will be given in the following section.

Fig. 4. Field distributions for the four resonance modes of the four open and coupled Fabry-Pérot cavities that can be associated to each of the dielectric slabs in the stacked structure in Fig. 1. The numerical (HFSS, red curves) and analytical (circuit model, blue curves) results show a very good agreement.

5. Stacked grids with a large number of layers

Fig. 5. Field distributions for the first and last resonance peaks (within the first transmission band, which has nine peaks) of a 9 slabs (10 grids) structure. Dimensions of the grids and individual slabs are the same as in Fig. 4. Dielectrics and metals are the same as well.

Table 1. Frequencies of lower (fLB) and upper (fUB) band edges with respect to the number of layers.

table-icon
View This Table

The structure with a large number of cells has a large number of resonances within a finite band. In the limit case of an infinite number of cells, instead of resonances we should have a continuous transmission band, out of which propagation is not possible (forbidden regions). This is expected from the solution of the wave equation in any periodic system. This kind of periodic structures represented by means of circuit elements are commonly analyzed in textbooks of microwave engineering (see, for instance, [22

22. D. M. Pozar, Microwave Engineering, third edition, (Wiley, 2004).

]). The unit cell of the infinite periodic structure resulting of making infinite the number of slabs of our problem is shown in Fig. 2(c). If, for simplicity, losses are ignored in the forthcoming discussion and the propagation factor for the Bloch wave is written as γ = α + , the following dispersion equation of the periodic structure is obtained following the method reported in [20

20. S. Tretyakov, Analytical modeling in applied electromagnetics, (Artech House, 2003).

, 22

22. D. M. Pozar, Microwave Engineering, third edition, (Wiley, 2004).

]:

cosh(γtd)=cos(kdtd)+jZd2Zgsin(kdtd)
(4)

where kd=ωεrc . For those frequencies making the RHS of (4) greater than −1 and smaller than +1, the solution for γ is purely imaginary (γ = ) 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

cosh(γtd)=1
(5)

cosh(γtd)cos(kdtd)+jZd2Zgsin(kdtd)=1.
(6)

Fig. 6. Brillouin diagram for the first transmission band of an infinite periodic structure (1-D photonic crystal) with the same unit cell as that used in the finite structure considered in Table 1. Numerical results were generated using the commercial software CST [23].

6. Conclusion

Acknowledgments

This work has been supported by the Spanish Ministerio de Ciencia e Innovación and European Union FEDER funds (projects TEC2007-65376 and Consolider Ingenio 2010 CSD2008-00066), and by the Spanish Junta de Andalucía (project TIC-4595). Francisco Medina would like to acknowledge the financial support from Spanish Ministerio de Ciencia e Innovación (mobility grant PR09-0405) during his stay at Queen Mary University of London, under supervision of Prof. Yang Hao. Alastair Hibbins and Celia Butler would like to acknowledge the financial support of the EPSRC (UK) and QinetiQ for supporting this work through APH’s Advanced Research Fellowship and CAMB’s Industrial CASE studentship.

References and links

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2.

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

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]

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]

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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]

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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.

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]

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]

20.

S. Tretyakov, Analytical modeling in applied electromagnetics, (Artech House, 2003).

21.

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

22.

D. M. Pozar, Microwave Engineering, third edition, (Wiley, 2004).

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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