The transmission through periodic arrays of subwavelength H-fractal apertures in a gold film at infrared wavelengths is investigated numerically. H-fractal apertures support subwavelength cut-off resonances that are hybridized with surface plasmons along the sidewalls of the aperture. Enhanced transmission occurs at wavelengths that are about ten times the aperture side length. The highly subwavelength size scale of the H-fractal enables an effective medium parameter description for the aperture array, which reveals a lossy plasma permittivity and a diamagnetic permeability.
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Enhanced optical transmission occurs when the transmitted optical intensity through apertures in a thin, opaque film at specific frequencies is larger than the area fraction occupied by the apertures. Consequently, at these resonant transmission frequencies, the electric field is strongly enhanced inside and near the apertures, which makes the phenomenon potentially useful for sensing applications and nonlinear optics [1,2]. The enhanced transmission can be caused by different physical mechanisms. At the visible and infrared optical frequencies, which are lower than the plasma frequency, ωp, of most metals, a periodic array of subwavelength apertures in a thin metal film can excite extended surface plasmons (SPs) that mediate the optical transmission. This grating-induced transmission resonance was observed by Ebbesen et al. in a square lattice of circular holes with radii ~1/10 of the wavelength [3,4]. A second mechanism for the enhanced transmission is the direct excitation of localized subwavelength resonances, which depends solely on the shape of the single aperture [5–7]. This is particularly relevant at the radio and microwave frequency ranges, where metals are well approximated as perfect electrical conductors (PECs) and would not support SPs. These two mechanisms can be more clearly separated for aperture arrays where the relevant dimensions, such as the array periodicity and aperture size, differ vastly. However, they may mix and lead to the enhanced transmission in some cases [8,9]. Subwavelength resonances can be supported in metallic fractal structures, such as the Koch curve and fractal tree, and have been employed in the design of small antennas . Recently, enhanced transmission through subwavelength H-fractal apertures in a metallic plate has been demonstrated at microwave frequencies .
In this letter, we show the enhanced transmission at infrared wavelengths through a square lattice of subwavelength H-fractal apertures in gold films which arises from the hybridization of the localized cut-off resonance of the aperture with SPs. The resonant transmission wavelength can be about nine times the side length of an H-fractal aperture with two scaling orders. Such subwavelength resonances are difficult to achieve using other aperture shapes, such as circles, crosses, or C’s [12–14]. An important consequence of the highly subwavelength size-scale of the H-fractal apertures is that even in the absence of any filling dielectrics, they can be well described by an effective permittivity, εeff (ω), of the lossy plasma form, and an effective permeability, μeff (ω), which is diamagnetic.
The paper is organized as follows. In Section 2, we describe the geometry of the H-fractal aperture. In Section 3, we show the enhanced optical transmission and the hybridization of a localized cut-off resonance with SPs. In Section 4, we calculate the effective medium parameters of the aperture array.
2. H-Fractal aperture
A schematic of the H-fractal is shown in Fig. 1 . An H-fractal pattern is generated from a transposed “H” of equal height and width, d. The “H” is then scaled to half of the original size, i.e. each side is of length d/2. The four half-sized “H”s are referred as the “second order” of the fractal pattern, and their centers are at the end points of the previous “H.” The side length of the fractal pattern is s. Continuing this scaling and pattern generation procedure indefinitely leads to a fractal which fills a square area without any line intersection (i.e. ), if the line width, w, is neglected.
We investigate a free-standing gold film in vacuum perforated with a square lattice of H-fractal apertures with two scaling orders as in Fig. 1, where a is the lattice constant. The dimensions of the structure are noted in the figure caption. We consider up to two orders because the feature sizes of the higher orders would be difficult to fabricate. We compute the transmission at normal incidence by solving the three-dimensional vectorial electromagnetic (EM) wave propagation with finite element methods (COMSOL Multiphysics) assuming periodic boundary conditions in the x- and y-directions. The incident wave is a plane-wave propagating along the z-direction, where the incident electric field, E0, is polarized along , and the incident magnetic field, H0, is polarized along . The metal is assumed to be gold with a relative permittivity of the plasma form:15]. The relative permeability of gold is assumed to be μm = 1.
3. Enhanced infrared transmission and cut-off resonance
First, we show the enhanced optical transmission and field enhancement in subwavelength H-fractal apertures. Figure 2(a) shows the (field) amplitude transmission spectrum of the apertures with a = 600 nm in a 100 nm thick gold film, which exhibits a peak of ~70% at 72.5 THz (wavelength λ = 4.13 μm), corresponding to an intensity transmission that is more than twice the aperture area fraction of 21%. The peak wavelength is ~8.6 times the side length of the aperture, s = 480 nm. The power flow is concentrated in the central slit of the H-fractal pattern [Fig. 2(b)]. Consequently, the magnitude of the electric field inside the slit is enhanced by about 25 times compared to incident E-field [Fig. 2(c)]. Also shown in Fig. 2(a) is the transmission of the simple H-shaped apertures without the set of four smaller H-shapes in the second order fractal. The transmission peak is at 132 THz and the spatial pattern of the power flow is similar to Fig. 1(b). The addition of higher orders in the fractal pattern can down-shift the fundamental resonance frequency by introducing finer spatial features to the resonant mode, which increases λ/s further. Although high order fractal patterns can support other resonances which are localized away from the central slit , in this work we focus on the lowest order, fundamental resonance.
The enhanced transmission is due to the excitation of a localized subwavelength resonance, which is largely independent of the array lattice constant [Fig. 3(a) ]. Since the lattice constants in Fig. 3(a) are significantly smaller than 4 μm, any grating-induced (i.e. extended) resonances would occur at higher frequencies. Moreover, the resonance frequency does not strongly depend on the film thickness, since the transmission peak only downshifts in frequency by < 10% for a six-fold increase in the film thickness as shown in Fig. 3(b). This implies the propagation constant inside the aperture, β, and the phase accumulation through the aperture are close to zero. The reduced transmitted amplitude for increasing film thickness is due to losses in the metal.
Since β = 0 occurs at the cut-off wavelengths for a loss-less waveguide, the resonance frequency of the H-fractal apertures can be interpreted in terms of the cut-off wavelength of an equivalent aperture waveguide. The dispersion relation for the fundamental waveguide mode of the H-fractal aperture, assuming it is infinitely long in the z-direction, is shown in Fig. 4(a) . β is normalized to k0 = ω/c, where c is the speed of light in vacuum. Periodic boundary conditions are imposed along x and y with the Bloch wavevector set to Kx = Ky = 0. At 64.5 THz (λ = 4.65 μm), Im[β] = Re[β], and the corresponding waveguide mode is essentially identical to the finite thickness resonance mode in terms of the magnitude of the Poynting vector (Fig. 4(b) and Fig. 2(b)), as well as the EM field. For a lossy waveguide, the cut-off condition is not unambiguous as in a lossless waveguide. Here, we define “cut-off” as Im[β] = Re[β], which represents a properly attenuated wave. The transmission peak frequency is higher than the cut-off frequency of 64.5 THz, due to the finite thickness of the film. As the film thickness increases, the peak frequency approaches the cut-off frequency. Thus, the resonance of the finite thickness apertures can be referred to as a “cut-off resonance.”
The cut-off resonance is a localized SP resonance that can be understood phenomenlogically as the hybridization of the intrinsic cut-off mode of the aperture with the SPs excited along the aperture walls. The resonance from the intrinsic aperture geometry is identified by modeling the metal as a PEC. As shown by the blue thick line in Fig. 4(a), the cut-off wavelength (frequency) of an H-fractal aperture waveguide in a PEC is 3 μm (100 THz), which is shorter than the case with the real metal. The excitation of SPs along the sidewalls is identified by examining the Ez component of the real metal and PEC waveguide modes, since it is parallel to the metal boundary and is a feature of SPs. Figures 4(c) and (d) show at Re(β /k0) = 0.35, Ez is non-zero and localized at the air-metal interface for the real metal waveguide mode, and is zero in the PEC case. As in other aperture waveguides [16–18], the SPs further lower the cutoff frequency of the H-fractal aperture. If the metal is replaced with a PEC, simulations show the transmission peak is indeed at 110 THz for a 100 nm thick film with the same geometric parameters in Fig. 2(a).
4. Effective medium parameters
Due to the subwavelength size scale of the H-fractal aperture, the optical transmission and reflection properties of the aperture array lend to an effective permittivity, εeff, and effective permeability, μeff, description. The dispersion relations of the H-fractal waveguide and bulk gold (inset of Fig. 4(a)) illustrate that the cut-off frequency of the H-fractal waveguide has a similar role as the plasma frequency in metals. Within the PEC approximation, Pendry et al. have shown that a periodic array of square holes in a semi-infinite metal layer has an effective ωp given by the cut-off frequency of the square hole waveguide and an effective diamagnetic response [19,20]. However, for the effective parameters to hold near the effective ωp, the hole waveguide cut-off wavelength should be much larger than the hole size and the periodicity, requiring the holes to be filled with high index dielectrics. Otherwise, the periodicity becomes comparable to the wavelength, and effective parameters are not well defined . As we have shown, the size of the H-fractal aperture is substantially subwavelength even in the absence of filling dielectrics. Thus, the fractal aperture array may constitute a new type of plasmonic metamaterial with properties controllable by geometric design.
The effective parameters of the H-fractal array are retrieved from the complex transmission and reflection coefficients at each wavelength using the transmission/reflection coefficients of a homogenous dielectric slab . Figure 5(a) shows the transmission and reflection amplitude and phase at normal incidence for an H-aperture array in 600 nm thick gold film. The transmission phase at the peak frequency is close to zero, resembling the propagation of a transverse EM wave in bulk metal at ωp (i.e., metal refractive index is ~0). The retrieved effective parameters are shown in Fig. 5(b) and are specific to the polarization due to the structural anisotropy of the fractal pattern. εxeff is of the lossy plasma form where Re(εxeff) is zero at the transmission peak frequency, and μyeff is diamagnetic with Re(μyeff) ≈0.23 at low frequencies.
Lastly, we compare the retrieved effective permittivity with the model of Pendry et al. :Fig. 5(b). The εxeff obtained from Eq. (2) agrees well with the directly retrieved values (from low frequencies to ωpeff). The complicated behavior of εxeff and Im(μyeff) < 0 at ω > ωpeff illustrates the limited validity of the effective parameters at decreasing values of λ/a .
We have shown that enhanced optical transmission in H-fractal apertures arises from a localized, cut-off resonance. The subwavelength size-scale of the H-fractals enables an effective medium parameter description of the aperture array, where εeff is of the plasma form and μeff is diamagnetic. The transmission and reflection properties of the metallic fractal apertures are controllable by the aperture geometry, such as slit length and scaling orders, and the excitation of SPs in the narrowly-spaced aperture walls. The strength of the interaction between the SPs on the sidewalls depends on the line width of the aperture. A wider line width will upshift the resonance frequency due to the diminished interaction of the SPs. Therefore, by adding higher orders to a fractal pattern and/or modifying the line width, an effective plasma permittivity can be created and tuned over a large frequency range beyond the constraints imposed by the intrinsic properties of metals or dielectrics. These effects can enable a greater flexibility in the design of SP structures.
This work was supported by the Natural Sciences and Engineering Research Council of Canada. We thank Prof. Sean Hum for his generous help in the computations and Dr. Z. H. Hang for the beneficial discussion. B. H. is grateful for the support of an Ontario Ministry of Research and Innovation Postdoctoral Fellowship.
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