Circuit modeling of the transmissivity of stacked two-dimensional metallic meshes.
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.
- SourceAvailable from: Alexander B. Yakovlev[Show abstract] [Hide abstract]
ABSTRACT: Following our recently introduced analytical tools to model and design conformal mantle cloaks based on metasurfaces [Padooru et al., J. Appl. Phys. 112, 034907 (2012)], we investigate their performance and physical properties when excited by an electric line source placed in their close proximity. We consider metasurfaces formed by 2-D arrays of slotted (meshes and Jerusalem cross slots) and printed (patches and Jerusalem crosses) sub-wavelength elements. The electromagnetic scattering analysis is carried out using a rigorous analytical model, which utilizes the two-sided impedance boundary conditions at the interface of the sub-wavelength elements. It is shown that the homogenized grid-impedance expressions, originally derived for planar arrays of subwavelength elements and plane-wave excitation, may be successfully used to model and tailor the surface reactance of cylindrical conformal mantle cloaks illuminated by near-field sources. Our closed-form analytical results are in good agreement with full-wave numerical simulations, up to sub-wavelength distances from the metasurface, confirming that mantle cloaks may be very effective to suppress the scattering of moderately sized objects, independent of the type of excitation and point of observation. We also discuss the dual functionality of these metasurfaces to boost radiation efficiency and directivity from confined near-field sources.Journal of Applied Physics 11/2012; 112:104902. · 2.21 Impact Factor
- [Show abstract] [Hide abstract]
ABSTRACT: An analytical model is presented for the analysis of multilayer wire media loaded with 2-D arrays of thin material terminations, characterized in general by a complex surface conductivity. This includes the cases of resistive, thin metal, or graphene patches and impedance ground planes. The model is based on the nonlocal homogenization of the wire media with additional boundary conditions (ABCs) at the connection of thin (resistive) material. Based on charge conservation, new ABCs are derived for the interface of two uniaxial wire mediums with thin imperfect conductors at the junction. To illustrate the application of the analytical model and to validate the new ABCs, we characterize the reflection properties of multilayer absorbing structures. It is shown that in such configurations the presence of vias results in the enhancement of the absorption bandwidth and an improvement in the absorptivity performance for increasing angles of an obliquely incident TM-polarized plane wave. The results obtained using the analytical model are validated against full-wave numerical simulations.IEEE Transactions on Antennas and Propagation 01/2012; 60(12):5727-5742. · 2.33 Impact Factor
- [Show abstract] [Hide abstract]
ABSTRACT: Frozen modes arising in stacked subwavelength hole arrays are studied in detail. Their origin is proved to be connected with the interaction between the extraordinary transmission resonance and the Fabry–Perot cavity mode. The analysis is done for various situations that differ in metal plate thicknesses and sizes and shape of the holes. Dispersion results and finite-stack transmission spectra are in good agreement, both showing the features indicating hybridization. The boundaries of the hybridization are found in terms of the geometrical parameters. The effect of the number of stacked plates on the transmission has been demonstrated. Finally, it is shown that the group index of refraction ng in the considered finite structures can be larger than 200. The obtained estimates of ng, which are based on dispersion and transmission results, well coincide with each other.Physical review. B, Condensed matter 05/2013; 87(20). · 3.66 Impact Factor
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
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
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
1. E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys.
Rev. Lett. , 58, 2059–2062 (1987).
2. S. John,“Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev.
Lett. , 58, 2486–2489 (1987).
3. M. Scalora, M. J. Bloemer, A. S. Pethel, J. P. Dowling, C. M. Bowden, and A. S. Manka,“Trans-
parent, metallo-dielectric, one-dimensional, photonic band-gap structures,” J. Appl. Phys., 83,
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).
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–
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-
20. CST Microwave Studio CST GmbH, Darmstadt, Germany, 2008, http://www.cst.com.
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 , 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 ). 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
. 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 . 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 . 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 ). Even the extraordinary transmission phenomena observed
through perforated metal films (which are associated with the resonant excitation of
bound surface waves ) 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
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
. 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 . 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 . 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
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 . 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
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
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
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
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 :
Zg= jωLg ; Lg=η0λg
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  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 ), 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
5 10 15
Fig. 3. Transmissivity (|T|2) of the stacked grids structure experimentally and
numerically studied in . 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  (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).
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 , 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  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  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
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  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  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 . 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
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
0 6.35 12.7 19.0525.4 31.7538.1
distance along Z mm
44.4550.8 57.15 57.15
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
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 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, ).
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 :
cosh(γtd) = cos(kdtd)+ jZd
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.
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 .
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 
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
0 20 40 60 80 100 120 140 160 180
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 .
be analytically and accurately estimated from Bloch analysis  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).
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
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.
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.