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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 15 — Jul. 18, 2011
  • pp: 13793–13805
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Multi-modal transmission of microwaves through hole arrays

James D. Edmunds, Euan Hendry, Alastair P. Hibbins, J. Roy Sambles, and Ian J. Youngs  »View Author Affiliations


Optics Express, Vol. 19, Issue 15, pp. 13793-13805 (2011)
http://dx.doi.org/10.1364/OE.19.013793


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Abstract

The microwave transmission through hole arrays in thick metal plates for both large holes (cut-off below onset of diffraction) and small holes (cut-off above onset of diffraction) have been compared through both experiment and modelling. Enhanced transmission is in part mediated by the excitation of diffractively coupled surface waves. Large holes, with cut-off below the onset of diffraction (due to the hole periodicity), are able to support multiple modes in transmission when the depth of the holes is sufficient to support quantisation in the propagation direction. Small holes, with cut-off above the onset of diffraction however only support two coupled surface modes (symmetric and anti-symmetric) below diffraction.

© 2011 OSA

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

1. J. Zenneck, “Über die Fortpflanzung ebener elektromagnetischer Wellen längs einer ebenen Leiterfläche und ihre Beziehung zur drahtlosen Telegraphie,” Ann. Phys. 328(10), 846–866 (1907). [CrossRef]

,2

2. A. Sommerfeld, “Ueber die Fortpflanzung elektrodynamischer Wellen längs eines Drahtes,” Ann. Phys. 67, 233–290 (1899).

] and Sommerfeld [3

3. A. Sommerfeld, “Über die Ausbreitlung der Wellen in der drahtlosen Telegraphie,” Ann. Phys. 28(4), 665–736 (1909). [CrossRef]

]. 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

4. H. M. Barlow and A. L. Cullen, “Surface waves,”Proc. IEE 100, 329–347 (1953).

,5

5. C. C. Cutler, “Genesis of the corrugated electromagnetic surface (corrugated wave-guide),”IEEE Antennas and Propagation Society International Symposium 3, 1456–1459 (1994).

]. 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

6. J. B. Pendry, L. Martín-Moreno, and F. J. García-Vidal, “Mimicking surface plasmons with structured surfaces,” Science 305(5685), 847–848 (2004). [CrossRef] [PubMed]

]. 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

7. A. P. Hibbins, B. R. Evans, and J. R. Sambles, “Experimental verification of designer surface plasmons,” Science 308(5722), 670–672 (2005). [CrossRef] [PubMed]

]. The current authors and others have recently explored further resonant sub-wavelength structured surfaces that support bound surface waves [8

8. S. A. Maier, S. R. Andrews, L. Martín-Moreno, and F. J. García-Vidal, “Terahertz surface plasmon-polariton propagation and focusing on periodically corrugated metal wires,” Phys. Rev. Lett. 97(17), 176805 (2006). [CrossRef] [PubMed]

11

11. M. J. Lockyear, A. P. Hibbins, and J. R. Sambles, “Microwave surface-plasmon-like modes on thin metamaterials,” Phys. Rev. Lett. 102(7), 073901 (2009). [CrossRef] [PubMed]

].

In waveguide theory [12

12. D. M. Pozar, Microwave Engineering, (John Wiley and Sons Inc., 2005)

] 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, fc for the lowest order waveguide mode, the TE1,1 mode is given by Eq. (1).

fc(1,1)=1.841c2πa.
(1)

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 TE1,1 waveguide mode, fc(1,1,N)is given approximately by Eq. (2).

fc(1,1,N)c2πε(1.841a)2+(NπL)2.
(2)

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, fcabove the onset of diffraction,fdiff, 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
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.
illustrates the unit cell of the array and the coordinate system used in all of the modelling and experiments discussed in this paper.

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 (fc<fdiff)

For an array of large holes the cut-off frequency,fc, may be set below the onset of diffraction,fdiff, i.e. (fc<fdiff). 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
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.
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.

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 kx the modes are asymptotic to the light line and at large values of kxtend towards a limit frequency. At large values of kxthe 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 kx 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 (kg/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
Fig. 3 Schematic representation of dispersion of surface modes when fc<fdiff. 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.
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/λg = 54.55 GHz and are band folded back into the radiative light cone with possibility of coupling with an incident photon.

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
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.
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

12. D. M. Pozar, Microwave Engineering, (John Wiley and Sons Inc., 2005)

]. 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.

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

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, kx. Diffracted light lines illustrated. (b) Detailed plot of resonant transmission maxima.
).

Figure 6a shows that the modes are dispersive, predominantly influenced by the ( ± 1,0) diffracted light lines suggesting that it is the in-plane ±kgk^xdiffraction 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

13. T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391(6668), 667–669 (1998). [CrossRef]

,25

25. H. F. Ghaemi, T. Thio, D. E. Grupp, T. W. Ebbesen, and H. J. Lezec, “Surface plasmons enhance optical transmission through subwavelength holes,” Phys. Rev. B 58(11), 6779–6782 (1998). [CrossRef]

,33

33. HFSS, Ansoft Corporation, Pittsburgh, PA, USA.

36

36. S. Collin, C. Sauvan, C. Billaudeau, F. Pardo, J. Rodier, J. Pelouard, and P. Lalanne, “Surface modes on nanostructured metallic surfaces,” Phys. Rev. B 79(16), 165405 (2009). [CrossRef]

]. 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
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.
. 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,kz, 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.

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

The small holes support a series of modes with surface wave character similar to that seen for the large holes. At large values of kxthe 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 kx 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/λg = 54.55 GHz are folded back into the light cone allowing radiative coupling to them (Fig. 9
Fig. 9 Schematic representation of dispersion of surface modes when fc>fdiff. 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.
). Since for these small holes the cut-off frequency is above c/λg 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.

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
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.
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.

The x-component of the electric fields (parallel to the incident electric field) for the two modes is plotted in Fig. 11
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.
. 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.

Figure 12
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.
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.

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

37. J. Bravo-Abad, L. Martín-Moreno, F. J. García-Vidal, E. Hendry, and J. Gómez-Rivas, “Transmission of light through periodic arrays of square holes: From a metallic wire mesh to an array of tiny holes,” Phys. Rev. B 76(24), 241102 (2007). [CrossRef]

,38

38. G. Xiao and H. Yang, “The effect of array periodicity on the filtering characteristics of metal/dielectric photonic crystals,” J. Semicond. 32(4), 044004 (2011). [CrossRef]

], 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.

References and links

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2.

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A. Sommerfeld, “Über die Ausbreitlung der Wellen in der drahtlosen Telegraphie,” Ann. Phys. 28(4), 665–736 (1909). [CrossRef]

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S. Collin, C. Sauvan, C. Billaudeau, F. Pardo, J. Rodier, J. Pelouard, and P. Lalanne, “Surface modes on nanostructured metallic surfaces,” Phys. Rev. B 79(16), 165405 (2009). [CrossRef]

37.

J. Bravo-Abad, L. Martín-Moreno, F. J. García-Vidal, E. Hendry, and J. Gómez-Rivas, “Transmission of light through periodic arrays of square holes: From a metallic wire mesh to an array of tiny holes,” Phys. Rev. B 76(24), 241102 (2007). [CrossRef]

38.

G. Xiao and H. Yang, “The effect of array periodicity on the filtering characteristics of metal/dielectric photonic crystals,” J. Semicond. 32(4), 044004 (2011). [CrossRef]

OCIS Codes
(050.1220) Diffraction and gratings : Apertures
(050.1950) Diffraction and gratings : Diffraction gratings
(240.6690) Optics at surfaces : Surface waves
(050.6624) Diffraction and gratings : Subwavelength structures

ToC Category:
Diffraction and Gratings

History
Original Manuscript: May 5, 2011
Revised Manuscript: June 10, 2011
Manuscript Accepted: June 10, 2011
Published: July 5, 2011

Citation
James D. Edmunds, Euan Hendry, Alastair P. Hibbins, J. Roy Sambles, and Ian J. Youngs, "Multi-modal transmission of microwaves through hole arrays," Opt. Express 19, 13793-13805 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-15-13793


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References

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