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

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Figures (6)

Fig. 1.
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.

Fig. 2.
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.

Fig. 3.
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.

Fig. 4.
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.

Fig. 5.
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.

Fig. 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].

Tables (1)

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Table 1. Frequencies of lower (f LB) and upper (f UB) band edges with respect to the number of layers.

Equations (6)

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β 0 = ω c ; β d = ε r ( 1 j tan δ ) β 0
Z 0 = μ 0 ε 0 ; Z d = μ 0 ε 0 1 ε r ( 1 j tan δ )
Z g = j ω L g ; L g = η 0 λ g 2 π c ln [ csc ( π w m 2 λ g ) ]
cosh ( γ t d ) = cos ( k d t d ) + j Z d 2 Z g sin ( k d t d )
cosh ( γ t d ) = 1
cosh ( γ t d ) cos ( k d t d ) + j Z d 2 Z g sin ( k d t d ) = 1 .

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