Page 1

Circuit modeling of the

transmissivity of stacked

two-dimensional metallic meshes

Chandra S. R. Kaipa, Alexander B. Yakovlev

Dept. of Electrical Engineering, University of Mississippi, University, MS

38677-1848, USA

Francisco Medina, Francisco Mesa

Microwave Group, University of Seville, Seville, 41012, Spain

Celia A. M. Butler, and Alastair P. Hibbins

Electromagnetic Materials Group, School of Physics, University of Exeter,

Exeter, Devon, EX4 4QL, UK

medina@us.es

Abstract:

stacked two-dimensional (2-D) conducting meshes is analyzed using

simple circuit-like models. The most relevant parameters of the

circuit are accurately determined in closed form, while losses are

reasonably estimated. The physical mechanisms behind measured

and computed transmission spectra are easily explained making use

of the circuit analog. 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-P´ erot 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.

The transmission of electromagnetic waves through

© 2010 Optical Society of America

OCIS codes: (050.1950) Diffraction and gratings; (050.6624) Subwavelength

structures ; (050.2230) Fabry-P´ erot.

References and links

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2377–2383 (1998).

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).

5. S. Feng, J. M. Elson, and P. L. Overfelt, “Transparent photonic band in metallodielectric nanos-

tructures,” Phys. Rev. B , 72, 085117 (2005).

Page 2

6. M. C. Larciprete, C. Sibilia, S. Paoloni, and M. Bertolotti, “Accessing the optical limiting prop-

erties of metallo-dielectric photonic band gap structures,” J. Appl. Phys., 93, 5113–5017 (2003).

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

Opt. Express, 16, 17249–17256 (2008).

8. C. A. M. Butler, J. Parsons, J. R. Sambles, A. P. Hibbins, and P. A. Hobson, “Microwave trans-

missivity of a metamaterial-dielectric stack,” Appl. Phys. Lett. , 95, 174101 (2009).

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

3rd International Congress on Advanced Electromagnetic Materials in Microwaves and Optics,

(London, UK, 2009), pp. 671–673.

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) , 391, 667–669 (1998).

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

12. B. A. Munk, Frequency Selective Surfaces: Theory and Design (Wiley, 2000).

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

Physics, 7, 37–55 (1967).

14. R. Sauleau, Ph. Coquet, J. P. Daniel, T. Matsui, and N. Hirose, “Study of Fabry-P´ erot cavities

with metal mesh mirrors using equivalent circuit models. Comparison with experimental results

in the 60 GHz band,” Int. J. of Infrared and Millimeter Waves, 19, 1693–1710 (1998)

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 com-

prising metal strips or patches,” IEEE Trans. Antennas Propagat., 56, 1624–1632 (2008).

16. F. Medina, F. Mesa, and R. Marqu´ es, “Extraordinary transmission through arrays of electrically

small holes from a circuit theory perspective,” IEEE Trans. Microwave Theory Tech. , 56, 3108–

3120 (2008).

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).

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

19. HFSS: High Frequency Structure Simulator based on the Finite Element Method, Ansoft Corpo-

ration, http://www.ansoft.com

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

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, 2],

hundreds of papers have been published exploring the theoretical challenges and prac-

tical 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 exhibits many

of the salient features observed in 3-D photonic crystals. Moreover, 1-D periodic struc-

tures 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 fre-

quencies, the superposition of a number of these layers separated by optically thick

transparent dielectric slabs has been shown to generate high transmissivity bands [3, 4].

Although Fabry-P´ erot (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], 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

Page 3

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]). 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 neces-

sary coupling mechanism between successive dielectric layers (Fabry-P´ erot 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]. Therefore, the method reported in [3, 4, 6] cannot be used in practice to enhance

transmission at microwave or millimeter-wave frequencies.

However, the authors have recently proposed physical systems that mimic the ob-

served behavior of stacked metal-dielectric layers at optical frequencies, but in the mi-

crowave region of the spectrum [8, 9]. In these systems, the metal films found in optical

experiments are substituted by perforated metal layers (2-D metallic meshes). The re-

sulting metal-dielectric stacked structure is shown in Fig. 1. In this work, the period of

the distribution of square holes and the holes themselves are small in comparison with

the free-space wavelength of the radiation used in the experiments and simulations.

Since we operate in the non-diffracting regime, surface waves cannot be diffractively

excited to induce enhanced transmission phenomena such as those reported in [10]. Due

to the small electrical size of the lattice constant of the mesh, very poor transmission

is expected for every single grid, alike the metallic films of the above mentioned optical

systems. However, the grid provides a mechanism for excitation of evanescent fields.

If the operation frequency is low enough, as it is the case considered in this letter,

the evanescent fields are predominantly inductive (i.e., the magnetic energy stored in

the reactive fields around the grid is higher than the electric energy). Therefore, the

effective electromagnetic response of the mesh layer is similar to that of Drude metals

in the visible regime. If a number of periodically perforated metallic screens is stacked

as shown in Fig. 1, the situation resembles the original optical problem previously dis-

cussed. The difference is that, in the microwave range, the reactive fields spread around

the holes of the perforated screens while they are confined into the metallic films in the

optical range. However, if the separation between successive metallic meshes is large

enough (roughly speaking, larger than the periodicity of the mesh itself), evanescent

fields generated at each grid do not reach the adjacent ones. In such situation a full

analogy can be found between the stacked slabs in the optical system and the stacked

meshes in the microwave system. Such analogy was confirmed through the experiments

and finite-element simulations in [8]. However, periodic structures (with periodicity

along the propagation direction or along the direction perpendicular to propagation)

have been analyzed in the microwave and antennas literature for several decades us-

ing circuit models [11, 12]. Indeed, problems closely related to the one treated in this

work have been analyzed following the circuit approach in [13, 14], for instance. More

recently, 2-D periodic high-impedance surfaces have also been analyzed following the

circuit-theory approach [15]). Even the extraordinary transmission phenomena observed

through perforated metal films (which are associated with the resonant excitation of

bound surface waves [10]) have been explained in terms of circuit analogs with sur-

prisingly accurate results [16, 17]. Since circuit models provide a very simple picture of

the physical situation and demand negligible computational resources, it would be very

Page 4

interestin to explain the behavior of stacked grids in terms of circuit operation.

Our first goal here is to show how a circuit model, whose parameters can be obtained

analytically, reasonably accounts for the experimental and numerical results reported in

[8]. Apart from avoiding lengthy and cumbersome computations, this circuit modeling

provides additional physical insight and, most importantly, a methodology to design de-

vices 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 direc-

tion 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]. Five copper grids, printed on a low-loss dielectric substrate using a conven-

tional 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 di-

electric 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 simula-

tions is tanδ = 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 identi-

cal 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

Page 5

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 struc-

ture 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 (λgmeaningfully smaller than the wavelength in the dielectric medium

surrounding the grids); Z0and β0are the characteristic impedance and propaga-

tion constant of the air-filled region (input and output waveguides); Zdand 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.

unit cell such as that shown in Fig.2 can be used in the analysis. Thus, we have a num-

ber 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]. Since, for the frequencies of interest, a single trans-

verse electromagnetic (TEM) mode propagates along the uniform waveguide sections

(higher-order modes operate below their cutoff frequencies), 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 at-

tenuation 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 TM02and TE20parallel-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 frequen-

cies not too close to cutoff (fc≈ 60GHz for the air-filled waveguides and 34.7GHz for

Page 6

the dielectric-filled sections) is αTM02=αTE20≈2π/(√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. The reason is that interaction between adjacent diaphragms takes place, ex-

clusively, through the transverse electromagnetic waves represented by the transmission

line sections.

The parameters of the transmission lines in Fig. 2(b), propagation constants (β0for

air-filled sections and βdfor dielectric-filled sections) and characteristic impedances (Z0

and Zd), are known in closed form. The expressions for those parameters are

β0=ω

c

; βd=

?

?μ0

εr(1− jtanδ)β0

(1)

Z0=

?μ0

ε0

; Zd=

ε0

1

?εr(1− jtanδ)

(2)

where ω is the angular frequency and c the speed of light in vacuum. Note that, due to

losses, Zdand βdare 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 Zgare available in the

literature. For wm? λgthe grid mainly behaves as an inductive load with the following

impedance for normal incidence [15]:

Zg= jωLg ; Lg=η0λg

2πcln

?

csc

?πwm

2λg

??

(3)

where η0=?μ0/ε0≈ 377Ω is the free-space impedance. Ohmic losses can be taken

penetration depth, δs=?2/(ωμ0σ), is much smaller than the thickness of the metal

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.

into account using the surface resistance of the metal (copper), since the skin effect

strips in our case. This resistance, series connected with the inductance in (3), is given

3.Comparison with numerical and experimental data

As a first test for our model, in Fig. 3 we compare its predictions with the numeri-

cal and experimental results reported in [8] 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 HFSS [19]), 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

Page 7

5 10 15

Frequency (GHz)

20 2530

0

0.2

0.4

0.6

0.8

1 1

|T|2

|T|2FEM model

|T|2Experimental

|T|2Analytical

A

B

C

D

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 tanδ = 0.0018. The four resonant modes in the first band are labeled as A, B,

C, and D in the increasing order of frequency.

stop band, are predicted by the present analytical model, in agreement with the exper-

imental results in [8] (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 explana-

tion 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. Onthe 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 sim-

ulation. 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, ap-

proximately, the same frequency range for the analytical model and measured data).

Page 8

This is because the model in Fig.2 is still valid at those frequencies, except for the effects

above mentioned (Zgshould 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], 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] 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 wort mentioning 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. 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 out of the scope of this paper.

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. Of course, 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] 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

Page 9

Fig. 4. Field distributions for the four resonance modes of the four open and

coupled Fabry-P´ erot 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.

the apparent (or equivalent) length of that section, as it has been explained in detail

in [17] 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] 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.

5. Stacked grids with a large number of layers

In the previous section, a five-grid structure supported by four dielectric slabs has been

shown to exhibit four FP-like resonances corresponding to the four coupled FP res-

onators formed by the reactively-loaded dielectric slabs. We have demonstrated that

the circuit model gives a very good quantitative account of the first transmission band,

while results are qualitatively correct but quantitatively poor when the frequency in-

creases (second and further bands). We have also mentioned that the highest-frequency

peak should not be far from the resonance frequency corresponding to a single slab

being half-wavelength thick, in agreement with the theory reported in [8]. This is the

practical consequence of the observation of field patterns for the last resonant mode

within the first band. However, this is an a posteriori conclusion. Moreover, no clear

theory has been provided for the position of the first resonance (or, equivalently, for

Page 10

the bandwidth of the first transmission band), which seems to be closely related to the

geometry of the grids. The application of our model to structures having a large num-

ber of slabs (cells along the z-direction) can shed some light on the problem. Thus, for

instance, we have verified that the behavior of the field distributions for any number of

slabs follows patterns similar to those obtained for the four-slab structure. In particular,

the field pattern for the first and last resonance peaks has the same qualitative behavior

shown for modes A and D of the four-slab structure. We can say that the phase shift

from cell to cell along the z-direction is close to zero for the first mode and close to π

for the last mode (with intermediate values for all the other peaks). As an example,

the field patterns for the first and last resonance modes within the first transmission

band of a nine-slab structure (with 10 grids) is provided in Fig. 5. It is remarkable the

similarity of these plots with the field distributions reported in [4, Fig. 4] for a stacked

metal/dielectric system operating at optical wavelengths.

0 6.3512.719.0525.4 31.7538.1

distance along Z mm

44.45 50.857.15 57.15

−3

−2

−1

0

1

2

3

EyV/m

0 6.35 12.7 19.0525.4 31.7538.1

distance along Z mm

44.4550.8 57.15 57.15

−3

−2

−1

0

1

2

3

EyV/m

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. Di-

mensions of the grids and individual slabs are the same as in Fig. 4. Dielectrics

and metals are the same as well.

As the number of identical layers is increased, the number of transmission peaks also

increases (there are as many peaks as slabs) but all the peaks lie within a characteristic

frequency band whose limits are given by the electrical parameters and dimensions of

the unit cell. For instance, the values of the first and last resonance frequencies are

tabulated in Table 1 as a function of the number of slabs. The slabs and grids are

the same used in the previous figures. Inspection of Table 1 tells us that fLBand fUB

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

the number of layers.

No. of layers

4

5

6

10

18

36

fLB(GHz)

7.004

6.780

6.664

6.468

6.380

6.380

fUB(GHz)

11.610

12.200

12.560

13.190

13.490

13.600

tend to some limit values when the number of stacked layers increases. Moreover, the

Page 11

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 fLB(6.380GHz) is not clear. In the following we propose an easy explanation

for both the lower and upper limits.

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 pe-

riodic system. This kind of periodic structures represented by means of circuit elements

are commonly analyzed in textbooks of microwave engineering (see, for instance, [18]).

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 + jβ, the following dispersion equation of the periodic structure is obtained

following the method reported in [18]:

cosh(γtd) = cos(kdtd)+ jZd

2Zgsin(kdtd)(4)

where kd= ω√εr/c. For those frequencies making the RHS of (4) greater than -1 and

smaller than +1, the solution for 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

cosh(γtd) = −1

(α =0), namely, a phase shift of π radians in the unit cell.

(5)

which is fulfilled by βtd=π

The frequency at which this condition appears is given by cos(kdtd) = −1, sin(kdtd) = 0,

which corresponds to the frequency of resonance of a single slab without grid, kdtd= 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

cosh(γtd) ≡ cos(kdtd)+ jZd

2Zgsin(kdtd) = 1 .

(6)

The condition cosh(γtd)=1 is trivially satisfied by γtd=0,

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, Zg.

Solving the dispersion equation (4) we can obtain the Brillouin diagram for any

desired band. This has been done in Fig.6 for the first transmission band of the structure

under study, which occurs at low frequencies within the limits of homogenization of the

proposed circuit model. Numerical results obtained via commercial software CST [20]

have been superimposed to verify the validity of the analytical data. It is clear that

the lower limit of the calculated transmission band coincides with the first resonance

frequency of the finite structures when the number of cells is large enough. Thus, the

range of frequencies where the peaks are expected for a finite stacked structure can

(β =α =0); namely, a null

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4

5

6

7

8

9

10

11

12

13

14

15

0 20 40 60 80 100 120 140 160 180

frequency (GHz)

Phase (degrees)

Analytical

CST

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 fi-

nite structure considered in Table 1. Numerical results were generated using the

commercial software CST [20].

be analytically and accurately estimated from Bloch analysis [18] using the proposed

circuit model. In particular, the influence of the grid impedance on the lower limit of the

transmission band can be obtained from this analysis. The same model explains why

the upper limit is solely controlled by the thickness of the slabs. Thus, our analysis gives

satisfactory qualitative and quantitative answers to our initial question of what controls

the limits of the transmission band. It is worth mentioning here that the second band

(or any higher-order band) is not just the second harmonic of the first one: a Bloch

wave analysis must be carried out to obtain the actual limits. However, for higher-

order transmission bands, the inductive grid could be a poor model that should be

corrected by a more accurate value of the loading grid impedance. However, this simple

analysis cannot be extended beyond the frequency range where multimode operation

arises in the parallel-plate waveguides connecting the grids. In such case the simple

transmission line with characteristic impedance Zd 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 (i.e., if higher-order grating lobes are precluded).

6.Conclusion

This work has shown that the study of the wave propagation along stacked metallic grids

separated by dielectric slabs can be carried out analytically with negligible computa-

tional effort making use of a simple circuit model. The circuit model remains valid even

at frequencies for which the closed-form expressions that account for the influence of the

grids are not valid; although in such a case better estimations of grid impedances are

required. The main characteristics of the transmission bands (frequencies of the lower

and upper resonances) are directly related to the behavior of the infinite 1-D periodic

photonic crystal resulting from the use of an infinite number of unit cells. In this case

Page 13

the transmission bands and the band-gaps are accurately determined by means of cir-

cuit concepts and textbook analysis methods. The model is valid in the non-diffracting

frequency region, far apart from the onset of the first grating lobe.

Acknowledgments

This work has been supported by the Spanish Ministerio de Ciencia e Innovaci´ on and

European Union FEDER funds (projects TEC2007-65376 and Consolider Ingenio 2010

CSD2008-00066), and by the Spanish Junta de Andaluc´ ıa (project TIC-4595). Fran-

cisco Medina would like to acknowledge the financial support from Spanish Ministerio

de Ciencia e Innovaci´ on (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.