1. Introduction

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, ω_p, 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 [10]. Recently, enhanced transmission through subwavelength H-fractal apertures in a metallic plate has been demonstrated at microwave frequencies [11].

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, ε^eff (ω), of the lossy plasma form, and an effective permeability, μ^eff (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 $2d\times 2d$ square area without any line intersection (i.e. $s\to 2d$), if the line width, w, is neglected.

 

Fig. 1 Schematic of a unit cell in a square lattice of H-fractal apertures. For our simulations, the central line length is d = 300 nm, the line width is w = 30 nm, and the lattice constant is a. For the fractal aperture of two orders, the side length is s = 480 nm.

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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 $\widehat{x}$, and the incident magnetic field, H₀, is polarized along $\widehat{y}$. The metal is assumed to be gold with a relative permittivity of the plasma form:

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 [11], in this work we focus on the lowest order, fundamental resonance.

 

Fig. 2 (a) The transmission spectrum (solid line) of a 100 nm thick gold film with the H-fractal apertures (two scaling orders) with a = 600 nm at normal incidence and the transmission spectrum (dash line) of H-apertures without the second fractal order. (b) The magnitude of the normalized time-averaged Poynting vector at the transmission peak wavelength of 4.13 μm in the middle of the gold film. (c) The normalized magnitude of the electric field in the y-z plane through the center slit of the fractal aperture, where the normalization is taken with respect to the incident E-field.

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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.

 

Fig. 3 The transmission of the gold film with the H-fractal apertures at normal incidence for (a) varying lattice constants, a=600, 800, 1000 nm, and (b) film thicknesses, h=100, 300, 600 nm.

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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 K_x = K_y = 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.”

 

Fig. 4 (a) The ω-β relation for the aperture waveguide. The frequency is f = ω/2π, and β is the propagation constant along z and is normalized to the vacuum wavenumber k₀. The black solid and red dash lines are the real and the imaginary parts of β, respectively. The blue thick line is the real part of β for the PEC case. The inset is the ω-β relation for an EM wave in gold. (b) The magnitude of the normalized time-averaged Poynting vector of the waveguide mode at f = 64.5 THz for the real metal. (c) The normalized magnitude of E_z of the waveguide mode for the real metal. (d) The normalized magnitude of E_z of the waveguide mode for the PEC case.

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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 E_z 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, E_z 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, ε^eff, and effective permeability, μ^eff, 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 ω_p 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 ω_p, 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 [21]. 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 [22]. 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 ω_p (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. ε_x^eff is of the lossy plasma form where Re(ε_x^eff) is zero at the transmission peak frequency, and μ_y^eff is diamagnetic with Re(μ_y^eff) ≈0.23 at low frequencies.

 

Fig. 5 (a) The complex reflection and transmission coefficients of the 600 nm thick metallic film with the fractal apertures at normal incidence. (b) The effective permittivity and permeability retrieved from the reflection and transmission coefficients. The black and red open squares are, respectively, the real and imaginary parts of the effective permittivity calculated from Eq. (2). The wavelength is normalized to the lattice constant a = 600nm.

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Lastly, we compare the retrieved effective permittivity with the model of Pendry et al. [19]:

5. Conclusions

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 ε^eff is of the plasma form and μ^eff 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.