Multi-modal transmission of microwaves through hole arrays

James D. Edmunds; Euan Hendry; Alastair P. Hibbins; J. Roy Sambles; Ian J. Youngs

doi:10.1364/OE.19.013793

1. Introduction

Surface waves are supported at the interface between two dissimilar materials such as a metal and a dielectric. Early theories proposing their existence were individually developed by Zenneck [1,2] and Sommerfeld [3]. Metals at microwave frequencies are near perfectly conducting and the surface waves supported are essentially only surface currents. It is known however that by structuring a metallic surface, its electromagnetic properties can be changed and increased binding of surface waves to the surface can be achieved [4,5]. The use of structured perfectly conducting surfaces in this way in order to extend plasmonic-like behaviour into the long wavelength regime has recently been proposed by Pendry et al [6]. They considered an array of deep sub-wavelength holes in a metallic substrate that provides the required boundary conditions to support a surface wave below the waveguide cut-off of the holes. The surface mode supported disperses asymptotically to the waveguide cut-off of the holes, with this limiting frequency acting as an effective surface plasma frequency analogous to that found in the ultraviolet region for non-structured metals [7]. The current authors and others have recently explored further resonant sub-wavelength structured surfaces that support bound surface waves [8–11].

In waveguide theory [12] the cut-off frequency determines whether propagating electromagnetic fields are supported. Below cut-off the electromagnetic fields are of purely evanescent character. For a cylindrical hole of infinite length the cut-off frequency, $f_{c}$ for the lowest order waveguide mode, the TE_1,1 mode is given by Eq. (1).

f_{c(1,1)} = \frac{1.841 c}{2 π a} .

Here a is the radius of the hole and c is the speed of light. The subscripts refer to the radial and circumferential quantisation and the factor 1.841 arises from the appropriate Bessel function. When a waveguide mode is supported in a hole of finite length, L the field will be quantised longitudinally. The cut-off of the Nth longitudinally quantised TE_1,1 waveguide mode, $f_{c (1,1,N)}$ is given approximately by Eq. (2).

f_{c(1,1,N)} \approx \frac{c}{2 π \sqrt{ε}} \sqrt{{(\frac{1.841}{a})}^{2} + {(\frac{N π}{L})}^{2}} .

Enhanced transmission through holes below their waveguide cut-off has been observed by Ebbesen [13] and others [14–23] if the holes are periodically arranged. They found that a regular array of holes strongly transmits close to the onset of diffraction. This transmission was found to be much greater than that predicted by Bethe [24] who states that the transmission for a single hole will scale as ${(r / λ)}^{4}$ . These Enhanced Optical Transmission (EOT) phenomena have been attributed to the excitation of diffractively coupled surface waves [25,26]. Surface waves couple together via evanescent fields in the holes leading to two possible modes of oscillation, one with symmetric and one with anti-symmetric charge distributions. This coupled oscillator system is analogous to masses on a spring or bonding/anti-bonding atomic pairs. The coupling between these surface waves, and therefore their resonant frequency is strongly dependent on the depth and size of the holes. Many previous studies of coupled surface waves supported on hole arrays have been focused on the symmetric oscillation [13–23]. However few studies have focused on the anti-symmetric resonance due to its high Q factor and proximity to the diffraction edge [27–29]. Kirilenko and Perov [27] and Lomakin and Michielsen [28] used analytical and numerical modelling techniques to study the symmetric and anti-symmetric modes supported on hole arrays observing that the anti-symmetric mode has a very high Q factor. Suckling et al. [29] experimentally studied a hexagonal array of holes in which the anti-symmetric mode was not observed due to sample inhomogeneities.

Above the waveguide cut-off, propagating fields are supported in the hole, this allows coupling together of diffractively coupled surface waves through oscillatory waveguide modes when the depth of the holes is sufficient to support discrete longitudinally quantised field solutions.

In this study two arrays of holes in the two distinct regimes have been experimentally explored and their differences highlighted and explained. One array of holes has a cut-off, $f_{c}$ above the onset of diffraction, $f_{diff}$ , the other has a cut-off below the onset of diffraction. Their transmission responses show that these two cases are very different, with the number of modes supported being dependent on both the depth of the holes and their diameter.

2. Methods

The structure under study is that of an array of unfilled cylindrical holes, arranged in a square lattice of 5.5 mm pitch in a metallic plate. Figure 1 illustrates the unit cell of the array and the coordinate system used in all of the modelling and experiments discussed in this paper.

Fig. 1 Unit cell of the sample and coordinate system illustrating the plane of incidence, angle of incidence, θ, hole diameter, a, and hole depth, h.

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To understand the role of surface eigenmodes in the transmission of hole arrays, a modal matching technique has been employed to calculate the dispersion of eigenmodes on perfect electrically conducting (PEC) cylindrical hole arrays. This technique is based on earlier work [30,31]. Briefly, the time independent complex electric fields in the vacuum regions either side of the holes are represented as a two-dimensional Fourier-Floquet expansion of diffracted orders with x and y components of the form (Eqs. (3)a and 3b)),

\sum_{m_{1}, m_{2}} A^{m_{1}, m_{2}} \exp (i (k_{x} + \frac{2 m_{1} π}{d}) x) \exp (i (\frac{2 m_{2} π}{d}) y) \exp (- i k_{z}^{m_{1}, m_{2}} z),

\sum_{m_{1}, m_{2}} B^{m_{1}, m_{2}} \exp (i (k_{x} + \frac{2 m_{1} π}{d}) x) \exp (i (\frac{2 m_{2} π}{d}) y) \exp (i k_{z}^{m_{1}, m_{2}} z),

described by wavevectors (

k_{x}^{}, 0, k_{z}^{m_{1}, m_{2}}

), where the integer pairs (m₁, m₂) denote the diffracted orders from the grating of pitch d, and

A^{m_{1}, m_{2}}

and

B^{m_{1}, m_{2}}

are the complex amplitudes in the regions above and below the array, respectively. Inside the holes, the electric fields are represented by cylindrical waveguide modes. The electric field of first order (

{TE}_{1, 1}

) mode is therefore represented by Bessel functions of the form (Eq. (4)),

(C^{1, 1} \exp (i q_{z}^{1, 1} z) + D^{1, 1} \exp (- i q_{z}^{1, 1} z)) J_{1} (\frac{1.841}{a} r) \frac{\cos (φ)}{r},

where the factors C ^1,1 and D ^1,1 describe the complex electric field amplitudes of the decaying and reflected waves within the cavity, respectively, and ϕ is the azimuthal angle in the x-y plane. The z components of the wavevectors outside and inside the cavities are (Eqs. (5)a and 5b)),

k_{z}^{m_{1}, m_{2}} = \sqrt{k_{0}^{2} + {(k_{x} + \frac{2 m_{1} π}{d})}^{2} + {(\frac{2 m_{2} π}{d})}^{2}}

and

q_{z} = \sqrt{k_{0}^{2} + {(\frac{1.841}{a})}^{2}},

where k ₀ is the vacuum wavenumber. We can obtain the z components of the electric field in the three regions of space, and subsequently expressions for the magnetic field H, through the free space Maxwell’s relations

\nabla \cdot E = 0

and

\nabla \times E = - μ_{0} \partial H / \partial t

. This gives the x and y components of the electric and magnetic fields in all regions in terms of the set of unknowns A, B, C and D. To solve for these unknowns, we can apply the appropriate boundary conditions for the tangential components of the fields at the vacuum-sample interfaces (i.e.

z = 0

and

z = h

): the electric field must be continuous over the entire unit cell, while the magnetic field components are continuous only at the hole aperture. This yields sets of continuity equations relating the complex field amplitudes. It is then straight forward to solve the continuity equations for the coefficients, thereby obtaining the eigenmodes of the structure, including their dispersion as a function of k_x.

One of the most useful aspects of this modal approach is that one can include or exclude different diffracted orders and waveguide modes from the calculation. As the simplest solution, we can limit ourselves to considering only the 1st order waveguide mode in the cavity and zeroth order diffraction (i.e. m₁ = m₂ = 0 only) as in refs [6,32]. This simple model allows the general behaviour of surface modes, including their asymptotic frequencies, to be qualitatively understood. In order to get a more accurate description of these modes, including perturbations introduced due to the presence of Brillouin zone boundaries, we can incorporate higher orders in the Floquet expansion in Eqs. (3)a and 3b.

In addition to the modal matching approach outlined above, we have also employed finite-element method (FEM) modelling of our structures using a commercial software package [33]. These full wave solutions provide verification of our modal matching model, and allow us to interpret the behaviour of our structures by means of electromagnetic field plots.

Transmission measurements have been performed in the range 40−60 GHz, using a free-space microwave set-up using matched microwave source and detector horns. Collimating mirrors are used to form a plane wave. The sample sits on a rotating turntable which allows the polar angle of incidence, θ to be varied.

3. Results and discussion

3.1. Cut-off below onset of diffraction ( $f_{c} < f_{diff}$ )

For an array of large holes the cut-off frequency, $f_{c}$ , may be set below the onset of diffraction, $f_{diff}$ , i.e. ( $f_{c} < f_{diff}$ ). In order to investigate this regime an array of holes with 4 mm diameter in a pitch of 5.5 mm will be considered.

Figure 2 shows modelling for 4 mm diameter holes in a square array of 5.5 mm pitch and 10 mm depth without inclusion of diffracted orders in the expansion of the electric fields, i.e. the reflected and transmitted electric fields are defined as only containing a specular component and therefore only non-radiative surface modes can be supported.

Fig. 2 Modal matching eigenmode solutions without inclusion of diffracted orders (only specular components of the electric fields used) for 4 mm diameter holes in a square array of 5.5 mm pitch and 10 mm depth. N indicates the longitudinal quantisation of the electric field in the z direction. Solid lines indicate positions of modes supported. Asymptotic frequencies illustrated.

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The two lowest frequency modes supported on the hole array have a dispersion that is similar to that of a surface wave on a planar plasmonic metal/dielectric interface. At small values of $k_{x}$ the modes are asymptotic to the light line and at large values of $k_{x}$ tend towards a limit frequency. At large values of $k_{x}$ the lowest order mode is asymptotic to the infinite cut-off (43.95 GHz) frequency of the holes (N = 0) which is acting as an effective or ‘spoof’ plasma frequency. The second mode asymptotes at large values of $k_{x}$ to the cut-off frequency of the first order mode (N = 1) with quantised field in the longitudinal direction (46.44 GHz). The third mode seen in Fig. 2 is asymptotic to the cut-off frequency of the second order mode with quantised field in the longitudinal direction (53.21 GHz); this mode however starts from, but is not asymptotic to the light line. There will an infinite number of these higher order longitudinally quantised modes all starting from the light line.

The influence of diffraction can be considered by taking into account the scattering from the periodicity of the array and representing these modes in a reduced zone representation. This can be done by folding the modes back into the first Brillouin zone at the Brillouin zone boundary ( $k_{g} / 2$ ). This simple approach ignores any band gaps that may open at the Brillouin zone boundaries but gives a qualitative picture of which modes may be coupled to by an incident photon.

Figure 3 shows the modes that have been folded back into the first Brillouin zone. All four of the modes that were in the dispersion plot without diffraction occur below $c / λ_{g}$ = 54.55 GHz and are band folded back into the radiative light cone with possibility of coupling with an incident photon.

Fig. 3 Schematic representation of dispersion of surface modes when $f_{c} < f_{diff}$ . Modes folded into the first Brillouin zone to represent the effect of first order diffraction. Solid lines indicate zero order surface modes, dotted line indicate diffracted surface modes. Short dash line indicates zero order light line and long dash line indicates diffracted light line.

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In order to measure experimentally the behaviour of the modes supported when the cut-off lies below the diffraction edge, cylindrical holes of 4 mm diameter were drilled into five 400 mm × 400 mm sheets of aluminium of various thicknesses in a square array of 5.5 mm pitch (4761 holes). This allows arrays of different thicknesses to be assembled by bolting the arrays together in various combinations, using guiding pins to minimise potential misalignment of holes. The dimensions of the holes lead to them being cut-off at 43.95 GHz. The samples were mounted perpendicular to a collimated microwave beam and transmission measurements were performed from 40 – 60 GHz, normalised to transmission without the sample.

Figure 4 shows normal incidence transmission measurements and a FEM model for a sample of 9.94 mm deep holes. Below the cut-off of the holes (43.95 GHz) transmission is near zero as predicted by waveguide theory [12]. A series of modes are supported between the cut-off frequency and the onset of diffraction at 54.55 GHz which manifest themselves as peaks in the transmitted signal. The modes that are present at the edges of the transmission band close to the cut-off frequency or the diffraction edge have a high Q factor. These modes are sensitive to angle and as such are greatly affected by any beam spread (~1 − 2° for this experimental setup), this accounts for the reduced visibility of the mode at 54.55 GHz in the experiment. The modes close to the cut-off of the holes are also very sensitive in frequency to the hole size. Consequently, because of some non-uniformity in the diameter of the holes the modes are broadened and weakened. This effect manifests itself most strongly in the loss of intensity of the first order mode. Furthermore there is a degree of roughness in the cross-section of the holes which results in a mean diameter of 4.05 mm.

Fig. 4 Normal incidence transmission measurements for 4 mm diameter holes in a square array of 5.5 mm pitch, 9.94 mm thick aluminium. Fit achieved using FEM modelling also illustrated.

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The x-component of the electric fields (parallel to the incident electric field) is shown in Fig. 5 for the four transmission maxima in Fig. 4.

Fig. 5 FEM model predictions of the x-component of the electric field (parallel to the incident electric field). Amplitude plotted through the centre of the 4 mm diameter, 9.94 mm deep holes(θ = 0°).

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The two lowest order modes have similar field profiles to that seen for coupled surface waves on thin metal films with symmetric and anti-symmetric fields that have hyperbolic cosine (45 GHz) and hyperbolic sine (47.6 GHz) character. As a result of diffraction the frequency of the lowest order mode is higher than the cutoff frequency predicted by Eq. (2) which assumes a single isolated waveguide. Since these modes are above the cut-off frequency of the lowest order waveguide mode, the holes can support a mixture of propagating and evanescent fields. This results in the fields appearing to have some oscillatory character. The two higher frequency modes are oscillatory and have cosine (51 GHz) and sine (54.5 GHz) solutions as these modes are Fabry Perot resonances inside the holes. This explains why these modes are not asymptotic to the light line but only existing above the cutoff frequencies of the holes. Due to the proximity of the fourth mode to the diffraction edge, evanescent diffraction has resulted in the two nodes near the interface being forced outside of the holes.

The 9.94 mm depth sample was mounted on a stepper motor driven turntable to allow the incident angle, θ, to be varied while the transmission was measured. The dispersion of the modes was experimentally measured for p-polarised (transverse magnetic) radiation (Fig. 6 ).

Fig. 6 (a) Experimental zero order transmission measurements for the 9.94 mm thick square array of 4.05 mm diameter holes, 5.5 mm pitch, as a function of in-plane momentum, k_x. Diffracted light lines illustrated. (b) Detailed plot of resonant transmission maxima.

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Figure 6a shows that the modes are dispersive, predominantly influenced by the ( ± 1,0) diffracted light lines suggesting that it is the in-plane $\pm k_{g} {\hat{k}}_{x}$ diffraction that is the dominant mechanism in the coupling to these modes. This type of dispersion is characteristic of coupled surface waves on hole arrays [13,25,33–36]. Figure 6b shows the transmission in more detail so that the lowest three modes can be clearly distinguished. The modes closest to the diffraction edge are strongly incident angle dependent a common characteristic of surface waves.

Experimental and modelled transmission maxima for arrays of various thicknesses are shown in Fig. 7 . Samples with small hole depths only support two modes, the lower frequency symmetric and anti-symmetric coupled surface modes close to the diffraction edge. As the hole depth is increased higher order modes with the electric field quantised longitudinally become supported. These modes appear from the diffraction edge as they are diffractively coupled modes and lower in frequency as the hole depth is increased, asymptotically approaching the cut-off frequency for an infinite waveguide. Increasing the hole depth reduces the longitudinal momentum component, $k_{z}$ , i.e. Equation (2) will tend to that of Eq. (1) reducing the cut-off of these modes to that of an infinite waveguide. Since the arrays are bolted together to make even deeper holes there are inevitably small non-uniform gaps between the layers of the holes in some of the samples. This results in a loss of intensity in the background level of the transmission, accounting for some of the discrepancy between the model and experiment. These high frequency mode maxima are also lowered in frequency due to the 1 - 2° angle spread present in the incident beam.

Fig. 7 Experimental and modelled normal incidence transmission maxima as a function of hole depth for an array of 4.05 mm diameter holes, 5.5 mm pitch.

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3.2. Cut-off above onset of diffraction ( $f_{c} > f_{d i f f}$ )

Figure 8 shows the dispersion of the modes supported by 3 mm diameter holes in a square array of 5.5 mm pitch and 2 mm depth calculated by the modal matching method.

Fig. 8 Calculated eigenmode solutions without inclusion of diffracted orders for 3 mm diameter holes in a square array of 5.5 mm pitch and 2 mm depth. Solid lines indicate position of the modes supported. N indicates the longitudinal quantisation number for the electric field in the z direction.

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The small holes support a series of modes with surface wave character similar to that seen for the large holes. At large values of $k_{x}$ the lowest order mode is asymptotic close to the infinite cut-off (58.6 GHz) frequency of the holes which is acting as an effective plasma frequency. Again, the second mode also asymptotes at large values of $k_{x}$ to the cut-off frequency of the first order mode with quantised field in the longitudinal direction (95.18 GHz). The third mode seen in Fig. 8 is asymptotic to the cut-off frequency of the second order mode with quantised field in the longitudinal direction (161.04 GHz); but again as was the case for the large holes this mode starts from the light line rather than being asymptotic to it. Reducing the size of the holes changes the position of the high frequency asymptotic limit of the modes due to the change in the cutoff frequencies however the low frequency asymptote remains unchanged, with the two lowest frequency modes still being asymptotic to the light line and the higher order modes starting from the light line. The behavior is however very different when diffraction and band-folding is considered.

When first order diffraction is introduced by folding the dispersion of the modes back into the first Brillouin zone all of the modes at a frequency below $c / λ_{g}$ = 54.55 GHz are folded back into the light cone allowing radiative coupling to them (Fig. 9 ). Since for these small holes the cut-off frequency is above $c / λ_{g}$ only the two lowest order evanescent modes will remain in the normal incidence transmission spectra as the higher order modes are not asymptotic to the light line.

Fig. 9 Schematic representation of dispersion of surface modes when $f_{c} > f_{diff}$ . Modes folded into the first Brillouin zone to represent the effect of first order diffraction. Solid lines indicate zero order surface modes, dotted line indicate diffracted surface modes. Short dash line indicates zero order light line and long dash line indicates diffracted light line.

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A similar experiment as that used for the large holes was performed to observe the behaviour of small holes. Cylindrical holes of 3.1 mm diameter were drilled into 400 mm × 400 mm sheets of aluminium of various thicknesses in a square array of 5.5 mm pitch (4761 holes). This allows arrays of different thicknesses to be assembled by bolting the arrays together in various combinations. The holes have a cut-off at 56.71 GHz. The samples were mounted perpendicular to a collimated microwave beam, transmission measurements performed from 40 – 60 GHz and normalised to transmission without the sample.

Figure 10 shows the normal incidence transmission response for the 3.1 mm diameter holes. Only the two lowest order evanescent transmission modes are present as predicted. Also shown is a finite element method model fit to the data. As seen previously for the arrays with the larger holes, the hole diameter in the model has to be increased by 50 μm to account for a spread in hole size. The two modes are observed very close to the onset of diffraction as expected. Due to its proximity to the diffraction edge the modes are very sensitive to the incident angle. The anti-symmetric mode has a high Q factor and angle-spread in the beam as well as sample inhomogenities have resulted in its intensity being significantly reduced.

Fig. 10 Experimentally measured normal incidence transmission for 3.1 mm diameter holes in a square array of 5.5 mm pitch and 1.905 mm depth.

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The x-component of the electric fields (parallel to the incident electric field) for the two modes is plotted in Fig. 11 . The lower frequency mode (52.9 GHz) has an electric field distribution that is symmetric with peak electric field in the centre of the hole. The electric fields for the higher frequency mode can also be seen in Fig. 11 and show that the mode is anti-symmetric in character. These field profiles can be represented with hyperbolic cosine and hyperbolic sine functions respectively due to their evanescent nature. The electric fields for the anti-symmetric mode reverse in the centre of the hole, this rapid reversal of the electric field means that more energy is stored in the fields resulting in a higher resonant frequency.

Fig. 11 FEM model predictions of the x-component of the electric field (parallel to the incident electric field). Amplitude plotted through the centre of the 3.15 mm diameter, 2 mm deep holes. Interfaces illustrated by dotted lines.

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Figure 12 shows the depth dependence of the normal incidence transmission maxima for 3.1 mm diameter holes in a square lattice of 5.5 mm pitch for various hole depths.

Fig. 12 Experimental and modelled (FEM) normal incidence transmission maxima for 3.15 mm diameter holes in a square array of 5.5 mm pitch for various hole depths. Error bars represent the approximate error in determining resonant frequency.

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For small hole depths the surface waves excited on the front and back interface of the holes are strongly coupled resulting in the symmetric/anti-symmetric modes being well separated in frequency. As the depth of the holes is increased the coupling between the two coupled modes decreases and their resonant frequencies converge with the structure acting as a single interface in the limit of infinitely deep holes with only one mode supported. The anti-symmetric mode position appears too low in frequency relative to the model, this is due to the effect of beam spread and non-planar sample as any non-normal component of the incident wavefront will result in the mode being reduced to a lower frequency. The error bars represent the error in determining the peak positions from the transmission spectra. They are small relative to the scatter in the data and deviation from the model, as the main source of error is sample and experimental uncertainties. In the region where the two modes converge, the position of the modelled transmission maxima are not plotted as the modes cannot be clearly distinguished.

4. Conclusions

In summary, a transmission study of metallic hole arrays of large holes with the cut-off below diffraction, and small holes where the cut-off lies above the onset of diffraction has been presented. The large holes support a series of modes, if the depth of the holes is sufficient to support a longitudinally quantised field solution. The two lowest order modes have evanescent field character whereas the fields of the higher order modes are propagating much like those of a Fabry Perot cavity. The small holes however can only support two modes regardless of the depth of the holes. Only evanescent fields are supported resulting in a symmetric/anti-symmetric pair of coupled surface waves. The symmetric mode being the often observed enhanced transmission mechanism and the anti-symmetric mode, the higher Q mode often neglected in these types of study. The fundamental difference between these two cases is that only the two lowest-order coupled surface waves are asymptotic to the light line whereas the higher order modes start from the light line above the cut-off frequency as they have non evanescent fields within the holes. By tuning the hole size so that the cutoff frequency lies above or below the diffraction frequency, one can therefore easily change from bimodal to multimodal transmission regimes. This behavior illustrates why hole array structures make excellent tunable filters [37,38], as they can be specifically designed as either narrowband or broadband transmission filters with a relatively small change to structural dimensions.

Acknowledgments

The authors would like to thank Nick Cole for help with sample fabrication. JDE is grateful to Dstl and EPSRC for financial support through the ICASE scheme. This work is part funded by the Ministry of Defence and is published with the permission of the Defence Science and Technology Laboratory on behalf of the Controller of HMSO.

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Multi-modal transmission of microwaves through hole arrays

Abstract

1. Introduction

2. Methods

3. Results and discussion

3.1. Cut-off below onset of diffraction ( $f_{c} < f_{diff}$ )

3.2. Cut-off above onset of diffraction ( $f_{c} > f_{d i f f}$ )

4. Conclusions

Acknowledgments

References and links

Cited By

Figures (12)

Equations (7)

Optics Express

Abstract

1. Introduction

2. Methods

3. Results and discussion

3.1. Cut-off below onset of diffraction (fc<fdiff)

3.2. Cut-off above onset of diffraction (fc>fdiff)

4. Conclusions

Acknowledgments

References and links

Cited By

Figures (12)

Equations (7)

Optics Express

3.1. Cut-off below onset of diffraction ( $f_{c} < f_{diff}$ )

3.2. Cut-off above onset of diffraction ( $f_{c} > f_{d i f f}$ )