Here the behavior of periodic annular aperture arrays in a perfectly conducting film is considered as the geometry of the apertures is varied. Using a previously developed rigorous electromagnetic modal method it is shown that the far-field transmission spectrum approaches that for an array of circular apertures as the stop size approaches zero. In the case where the diameter of the apertures is significantly less than that of the period of the array, the behavior of the array changes gradually from one where the predominant features in the spectra are due to the excitation of a waveguide resonance to one exhibiting ‘extraordinary transmission’.
© 2006 Optical Society of America
The performance of metallic films containing arrays of apertures has attracted considerable recent attention, primarily as a consequence of the fact that transmission significantly higher than would be anticipated on the basis of single-aperture theory has been observed with visible light. [1, 2] The peak transmission occurs close to a grating anomaly and is facilitated by strong resonant coupling from an evanescent field inside the apertures to the surface wave excited as a diffracted order undergoes the transition from a propagating to an evanescent wave. A number of analogies from the excitation of a surface-plasmon-like wave to quantum-mechanical tunneling have been created to assist in the explanation of this phenomenon. In parallel with these developments there is renewed interest in a structure consisting of a metallic film containing an array of annular apertures originally proposed as a bandpass filter . The performance of this structure was first demonstrated experimentally with millimeter waves  and its applicability as a frequency-selective surface into the far-  and midinfrared investigated. Recent research has also demonstrated that the salient features of its performance extend to the near-infrared [6, 7] and visible  regions of the electromagnetic spectrum.
The bandpass properties of this structure hinge on the excitation of a waveguide resonance [3, 9] and can be largely independent of the excitation of surface waves. Although the presence of the array permits the efficient coupling of the fields from each aperture to plane waves, the fundamental physics underpinning its high transmission is not substantially different from that of a single annular aperture.  The position and width of the passband are largely determined by the geometry of the apertures rather than the periodic properties of the array. One of the key advantages of this structure as a frequency-selective surface is the fact that it can be designed so that the passband is well-separated from the region of the spectrum where more than one diffracted order propagates. The performance of many other frequency-selective surfaces is similarly based on the excitation of individual elements within an array.
Here a rigorous electromagnetic modal method that assumes the metal is perfectly conducting and that the array is infinite,  is used to computationally investigate the behavior of resonant arrays as the geometry of the structures within the unit element changes. It is shown that as the resonant wavelength decreases from a value significantly above that of the array period to a value less than the period, the structure changes from exhibiting a response arising from resonances of individual annular apertures, to a ‘diffractive response’ where light tunnels through the aperture at wavelengths close to diffraction anomalies. At these wavelengths the evanescent fields close to the surface of the structure facilitate coupling to the zeroth transmitted order and produce the associated ‘enhanced’ transmission.
2. Theoretical formalism
A schematic showing the structure under consideration is shown in Fig. 1. A square array of annular apertures with outer radius a and inner radius b perforate a perfectly conducting screen of thickness h sitting on a semi-infinite substrate. Since the array is assumed infinite, the electric and magnetic fields above the grid and below it in the substrate can be expressed as a discrete superposition of plane waves, while the fields within the aperture are given in terms of coaxial waveguide modes [3, 11, 12]. Rigorous electromagnetic boundary conditions for the electric and magnetic fields at the upper and lower interfaces of the screen, assuming perfect conductivity, lead to a series of linear equations which can be truncated and solved numerically for the amplitudes of the coaxial waveguide modes. From these, the amplitudes of the transmitted and reflected propagating and evanescent plane waves can be calculated. Although material losses are not considered, this is not a serious issue when considering millimeter and far-infrared radiation [13, 14], since experimental results obtained for millimeter waves and in the far-infrared have been shown to be in excellent agreement with theories assuming perfect conductivity [5, 12, 15]. Also, at much shorter wavelengths, modeling of arrays of circular apertures has shown that models assuming perfect conductivity capture the salient features and essential physics involved in their electromagnetic response [13, 14]. Furthermore, surface impedance boundary conditions can be incorporated into such models to approximate the influence of material losses. It should be emphasized that it is not the intention of this paper to simulate any specific experiment, and parameters have been chosen to highlight features of interest. It would be anticipated that the influence of finite conductivity in such highly resonant structures, particularly in the visible region of the spectrum, would be significant. Investigating this is the subject of ongoing research.
The key interest in the use of these structures as frequency-selective surfaces hinges on their high transmission (100% in the absence of any loss) at a wavelength close to the cut-off wavelength for the dominant TE(1,1) mode for normally incident plane waves. Note that symmetry considerations preclude the coupling of a normally incident plane wave to the TEM mode. For narrow annuli, the cutoff wavelength for the TE(1,1) mode occurs at a wavelength approximately equal to π(a+b)  which is in contrast to the cutoff wavelength for the lowest order TE mode (also denoted the TE(1,1) mode) in a circular waveguide of radius a which is equal to 3.41a. This means that a coaxial waveguide can support a propagating mode at a much longer wavelength than a circular waveguide of the same radius. In the context of a screen perforated with apertures separated by a distance d, this means that this resonance can be well-separated from the longest wavelength diffraction anomaly that occurs when λ=d in the absence of a substrate. Given that the largest possible aperture radius must be less than d/2, the separation between the resonant wavelength for an array of circular apertures and this diffraction anomaly is constrained by geometry to be relatively small (~1.4a), whereas the equivalent quantity for the array of annular apertures is ~4.3a. It has also been shown Ref.  that the quality factor (Q) of the resonance increases as the width of the annuli decreases and as the thickness of the screen increases. In addition, as the screen thickness increases further, Fabry-Pérot resonances appear  which have a similar origin to those seen from arrays of thick grids containing circular apertures,  metallic screens containing isolated circular and annular apertures [10, 18] and other thick diffractive structures .
One of the interesting features of the modal formulation is that it can be shown to analytically and computationally approach the equivalent formulation for an array of circular apertures in the limit b→0, i.e. as the central stop disappears.  As a consequence it is instructive to use this technique to explore the transition from the regime where the dominant enhanced transmission feature in the transmission spectrum is due to that of a ‘waveguide resonance’ to that where strong coupling to surface waves is required to produce high transmission.
3. Computational results and discussion
Figure 2 shows the transmission spectra for arrays of circular and annular apertures with the same outer radius (a=0.3d) in screens of identical thickness (h=0.4d) on a substrate of index 1. The array was illuminated with a normally incident plane wave. Note that at normal incidence, the transmission is independent of polarization and the incident field can couple to the coaxial waveguide TE and TM modes with only n=1. Typically 5 TE and 5 TM modes and 317 plane waves were included in calculations. In this and all figures, it can be seen that Rayleigh-Wood anomalies appear at the expected wavelengths of λ=d and λ=d/√2, where diffracted orders become evanescent. Other significant features in the spectrum are also apparent. The transmission spectrum for the array of annular apertures when the central stop size is large (i.e. the rings are very narrow) can be seen to have a distinct, high-Q, resonant maximum in transmission at a wavelength of 1.74d (c.f. the cutoff wavelength of the TE (1,1) mode is 1.82d). As the size of the central stop is decreased, the wavelength of the transmission resonance decreases and its Q decreases. It is also apparent that as the radius of the central stop becomes very small compared to the radius of the aperture, the transmission spectrum (shown as asterisks in Fig. 2) approaches that of the array of circular apertures of the same diameter.
In Fig. 3(a) the total transmission through an array of circular apertures is shown as a function of the wavelength and aperture size for a fixed screen thickness of 0.4d, while the transmission through an array of annular apertures of varying outer radii but constant ratio of outer radius to inner radius (a/b) is shown in Fig. 3(b). The cutoff wavelength for the corresponding TE (1,1) waveguide mode is also shown in each figure. It can be seen that where the waveguide cutoff is greater than the period of the array, d, then a transmission maximum appears at a wavelength dependent on the geometry of the individual apertures rather than that of the array.  As the size of the apertures decreases, however, the wavelength at which this maximum occurs decreases. In the case of the array of circular apertures [Fig. 3(a)], this maximum remains pinned to a value slightly larger than d. As the dimensions of the aperture decrease further to a point where the waveguide cutoff is less than d, the optical performance of the structure enters the realm of ‘extraordinary transmission’ where the peak at a wavelength slightly larger than d appears despite the fact that the field within the waveguide is evanescent. Note that, on the other hand, for the parameters chosen for Fig. 3(b) this ‘extraordinary transmission’ maximum is not apparent.
In Fig. 4 the change in the transmission spectrum as the structure undergoes the transition from an array of annular apertures toward an array of circular apertures is displayed. The radius of the apertures (a=0.2d) has been chosen so that the TE(1,1) waveguide mode cutoff is clearly greater than the period of the array for very narrow rings (λc=1.2d for b=0.19d), but the cutoff wavelength for the corresponding array of circular apertures (λc=0.52d) is much less than the period of the array. It is apparent that, as is the case with the array of circular apertures, the peak associated with this waveguide resonance persists even when it occurs at a wavelength less than the period of the array but that there is a stronger peak in transmission (‘extraordinary transmission’) close to λ≈d.
From the results of the previous section it can be seen that the relationship between the aperture geometry and the array period defines the nature of the key features in the transmission spectra of an array. In the case of an array of annular apertures, the fact that the waveguide cutoff for a coaxial waveguide can be much larger than the outer diameter of the apertures can increase the central wavelength of this resonance to values much larger than the period of the array, whereas the largest cutoff wavelength for circular apertures must lie closer to the period of the array. It can also be seen that the transmission properties of the array of annular apertures morph into those of an array of circular apertures. Similarly, by varying one parameter (the size of the central stop), the optical behavior of the structure changes from being dominated by a waveguide resonance to exhibiting ‘extraordinary transmission’ at a wavelength considerably longer than the cutoff wavelength for the waveguide.
SMO acknowledges the financial support of an Australian Postgraduate Research Award.
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