Summary: Design of the cross-shaped metal mesh grids Silvia Seidlitz June 16, 2015 Abstract Metal-mesh grids with cross-shaped apertures are applicable as bandpass filters for frequencies ranging from hundreds of GHz up to hundreds of THz. Width ∆ν, center transmittivity T(ν0 ) and position ν0 of the pass band are determined by the filter dimensions: periodicity p, cross-member length l, cross-member width w and plate thickness t. Many theoretical and experimental studies of the filter dimensions were already realized. The aim of the following section is to provide a model allowing to understand the behaviour of the filters. Based on the experimental data from past studies, a couple of equations predicting the filter’s properties as a function of the design parameters is seeked.

1 1.1

Models A rigorous model

For a non negligible thickness, a rigorous model was developed by Compton et al. [1] which is based on a model proposed by Chen. It demands the knowledge of the complete set of waveguide modes in the cross-shaped aperture. The free space fields above and below the grid are expressed as superpositions of plane waves, the so called Rayleigh or Floquet modes. Matching both sets of modes at the boundaries using the method of the moments, one has to solve aperture integrals of inner products between Rayleigh and waveguide modes. As this demands significant computational ressources, a rigorous calculation of the filter’s parameters is abandoned. An approximation of the simulation results obtained by A. Golden et al. leads to an analytical expression for the center frequency ν0 (c is the speed of light in vacuum): ν0 =

c 1.8l − 1.35w + 0.2p

(1)

This approximation is tested on the experimental results of Porterfield et al., Page et al., Melo et al. and Moeller et al. (figure 1 experimental and calculated f over P).

1

It can be deduced, that the trend of the center frequency as a function of the periodicity P is well described whereas the deviation from the experimental results is not satisfying (mostly more than 0.1/nu0 ).

1.2

An approximation for the design of the filters

The experimental results were obtained using similar ratios p/l ≈ 1.6 and p/w ≈ 5.9. Assuming a well-known inverse dependance between center-frequency and periodicity (according to the simulation result obtained by Goldon et al.), one could design filters for a desired center-frequency by scaling these filters. In order to verify this possibility, a fit of the reciprocal function to the experimental data is realised on matlab. It yields: ν0 [THz] =

110 + of f p[]

(2)

The deviation from the experimental data is less important than by using Golden’s approximation. (I will add the exact equation and deviation as soon as I’ll reach matlab)

1.3

A simple approximation: Ulrich’s Transmission Line Model

In order to achieve a qualitative understanding of the grid’s properties, a simple model is desirable. 1.3.1

The model

The model introduced by R. Ulrich consists in translating the properties of the metal meshes into an equivalent circuit of capacitances C, inductances L and resistances R on a transmission line. There are several considerations necessary to obtain the validity of the model: The wavelength of the incident radiation has to be bigger than the grid’s periodicity p as diffraction is not included in the model. Furthermore, the metal grids are considered to be infinetly thin and perfectly conducting. A dielectric film supporting the capacitive grid is not respected in the model. Ohmic losses are neglected and the incident, plane wave is supposed to be normal on the grid. 2

Then, the cross shaped metal mesh grid is treatened as the superposition of one so-called inductive and one capacitive grid (figure 2).

The origin of these names can be found regarding the transmission spectra (figure 3): the inductive grid acts as a frequency-high-pass (as infinetly extended direct currents are possible) whereas the capacitive grid operates as a frequency-low-pass (no infinetly extended direct currents possible). If inductive and capacitive grid possess the same parameters p and w, they act like complementary grids and under the assumption of infinetly thin grids, Babinet’s principle can be applied, implying that the transmittivities of both grids are complementary. In summary, an equivalent circuit has to fulfill the following equations: τ(ω) = 1 + Γ(ω)

(3)

where τ corresponds to the complex transmission and Γ to the complex reflection coefficient and because of the neglection of the losses kτ(ω)k2 + kΓ(ω)k2 = 1

(4)

τi (ω) + τc (ω) = 1

(5)

As

referring to Babinet’s principle and using equations (3) and (4), one obtains that the power transmittivities are also complementary: kτi (ω)k2 + kτc (ω)k2 = 1

(6)

In general, these equations are fulfilled by an equivalent circuit consisting of a transmission line, respresenting the free space which is shunted by two admittances, each one representing a grid, as shown in figure 4. figure 4 here For the capacitive grid, the admittance could be replaced by a conductance and for the inductive grid by an inductance. This approximation fits well the p experimental data for λ = 0.2l was proposed by Simovsky et al. An upper limit might be obtained from the Rayleigh-Woods-Anomaly: Wood discovered in 1902 that on many diffraction gratings in the case of an incident radiation polarized in the plane parallel to the grooves, narrow spectral regions with a sharp change of the diffracted energy appear. A sharp peak corresponding to these anomalies can be observed in the transmission spectrum measured by Compton et al at a wavelength equal to the period p. An explanation of Wood’s anomalies based on the existence of surface waves was given by Fano in 1938. Palmer found in 1952 that these anomalies disappear for a plate thickness much smaller than the wavelength. By increasing the grid thickness, the peaks corresponding to the anomaly are broadend and shifted to longer wavelengths. A thick plate could even lead to multiple passbands and therefore, t