## Abstract

In order to provide a guide for the design and optimization of bowtie slot antennas in the visible and near infrared spectral regime, their optical properties have been investigated with emphasis on geometry and materials. Although primarily theoretical, experimental investigations for reduced thickness cases are also included. As characterized by their field patterns, two types of resonances are discussed: *plasmonic* and *Fabry-Pérot-like* resonances. These resonance types show a linear dependence on aperture perimeter and film thickness, respectively, while showing a complementary behavior with near independence of the other respective parameter. Metal properties, as in the Drude model, are also considered. Various metals with respectively different skin depths are studied, showing a nearly linear dependence of the resonance wavelength on skin depth.

©2008 Optical Society of America

## 1. Introduction

Nanostructures composed of noble metals with typical dimensions around the wavelength of light can support optical resonances. On resonance, such structures are able to concentrate an incident light field into a small volume with orders of magnitude of intensity enhancement. They are, therefore, referred to as optical antennas [1-8]. In contrast to lenses and mirrors, optical resonant antennas can provide near-field light spots with dimensions determined mainly by the structure feature size rather than by the diffraction limit. Thus, by properly designing optical antennas, a sub-diffraction range light source can be created [6, 9-11]. In order to gain the strong field confinement and enhancement, one can introduce sharp features to resonant metallic nanostructures. Following this principle, optical bowtie antennas and dipole antennas with tiny gaps have been designed and utilized in near-field scanning optical microscopy and white light generation [5, 6] and thus are useful for numerous applications.

Compared to wire antennas, slot antennas have the advantage of robustness as well as greater control of the radiation pattern. Bowtie slot antennas, which are composed of bowtie shaped apertures in a metal film, are shown schematically in Fig. 1. They provide particularly strong field confinement due to the sharp edge design. For normal incident light with polarization perpendicular to the gap (along *x* direction), the electric field can be strongly concentrated in the small gap region. These antenna designs have attracted considerable attention for achieving sub-diffraction resolution for light sources and near-field optical imaging [1, 2, 10-15]. However, in addition to the influence of the sharp edge design and the small feature size, the field enhancement of optical antennas shows a strong dependence on resonance properties[3, 5, 6, 8, 15]. In this paper, we analyze the optical resonances of bowtie slot antennas with dependence on geometry and metal properties. This analysis should provide a guide for the design and optimization of slot antennas for a variety of applications. Two types of resonances are identified based on their respective field patterns. Here, they are called *plasmonic* and *Fabry-Pérot-like* resonances. These resonances have a complementary dependence on geometry, with the plasmonic resonance showing a linear dependence on aperture perimeter, and the Fabry- Pérot-like resonance on film thickness. The resonance types are also nearly independent of the other respective parameter. Beyond the study of geometry, the resonance dependence on metal properties is considered, as from the Drude model. The various metals considered each have a different respective skin depth, and this leads to an observed near linear dependence of resonance wavelength on skin depth.

## 2. Two types of resonances

As shown in Fig. 1, the geometry of a bowtie slot antenna is defined by the thickness of the metal *t*, the outline lengths of the bowtie slot antenna *a* and *b*, the bowtie flare angle *θ* and the gap size *d*. The light is normally incident with linear polarization perpendicular to the gap, as shown. In order to characterize the resonance properties of bowtie slot antennas, a finite integration time domain algorithm (CST Microwave Studio) is used to simulate transmission spectra and field distributions of antennas. The frequency dependent dielectric functions of metals are described by the Drude model, which offers a good correspondence to the measured data over a broad spectral range and is given by

where *ω _{p}* is the plasma frequency and

*γ*is the damping frequency. Table 1 shows the

*ω*and

_{p}*γ*by fitting the Drude model to the experimentally obtained data of Ref. [16].

Figure 2 shows a typical transmission spectrum of a bowtie slot antenna in an optically opaque free standing gold film. The substrate effect in a real case can be taken into account with an effective refractive index of the surrounding medium [17]. Two transmission resonances are observed in the visible and near-infrared spectral regime. In order to gain physical insight into these resonances, the electric field and current distributions at resonant wavelengths are calculated and shown in Fig. 3 (a)-(d). For the resonance in the longer wavelength range at 1.4 *µ*m, the electric field distribution inside the gap indicates that charges accumulate at the sharp edge of the bowtie shaped aperture. This dipole-like charge distribution determines the strong field enhancement in the gap area. The current density distribution shows that the induced current flows around the aperture edge [see Fig. 3 (a) and (b)]. Since this dipole-charge oscillation is similar to that of nanoparticles at fundamental plasmonic resonances [18], we refer to this resonance as a *plasmonic* resonance (labelled ’P’). In contrast to the plasmonic resonance, for the resonance in the short wavelength range at 700 nm, the field distribution shows that charges are accumulated at the upper and lower interface of the aperture gap area. The electric field in the gap along the *z* axis has the profile of a standing wave with antinodes at each interface [see Fig. 3 (c) and (d)]. This is analogous to the behaviour of a Fabry-Pérot resonator; we, therefore, refer to it as a *Fabry-Pérot-like* resonance (labelled ’FP’).

## 3. Resonance geometry dependence

The gap size plays an important role in the field confinement and enhancement of bowtie slot antennas and determines the spatial size of the concentrated light spot. The gap size dependence of the optical properties has been studied [2, 15]. In this section, we numerically study the resonance properties of antennas in free standing gold films. The gap size *d* and flare angle *θ* of the antennas are kept constant as 40 nm and 90°, respectively. The aperture perimeter and film thickness are used as tuning parameters. Figure 4(a) shows the calculated transmission spectra of bowtie slot antenna arrays with various aperture perimeters. The film thickness is identical for each case. The aperture perimeter *L* is varied from 0.93 *µ*m to 2.38 µm in steps of about 480 nm by adjusting the outline dimensions length of *a* and *b*. The periods of the antenna arrays *p _{x}* and

*p*are increased from 350 nm to 650 nm in steps of 100 nm, corresponding to the increased aperture perimeter. The plot in Fig. 4(b) shows the resonance wavelengths of P and FP resonances as functions of aperture perimeter. The family of curves in Fig. 5(a) shows the calculated transmission spectra of bowtie slot antenna arrays for various film thicknesses. The aperture perimeter and the periods of the arrays are identical in all cases with

_{y}*L*=1.41

*µ*m and

*p*=

_{x}*p*=450 nm. The film thickness is varied from 150 nm to 300 nm in steps of 50 nm. Additionally, Fig. 5(b) shows the resonance wavelengths of P and FP resonances as functions of film thickness.

_{y}## 4. Plasmonic resonance

The P resonance appears in a longer wavelength range in the transmission spectra compared to the FP resonance. It is referred as a plasmonic resonance because it has a representative field distribution of the fundamental plasmonic resonance of nanostructures [18-20]. As shown in Fig. 4(a), when the aperture perimeter increases from about 0.9 *µ*m to 2.4 *µ*m, the resonance wavelength shifts nearly linearly from 1.0 *µ*m to 2.3 *µ*m. This dependence can be interpreted from the effective wavelength concept introduced in optical dipole antennas. In this picture, the resonance wavelength of nanorods has been shown to have a linear dependence on their geometric length [8]. In the language of plasmonics, these resonances correspond to the dipolar surface plasmon polariton excitation by the electric field of the incident light. The resonance wavelength is determined by a linear restoring force due to the generated surface polarization. Thus, the resonance wavelength is proportional to the effective oscillation path length of the conduction electrons, which is, in the nanorod case, the geometric length of the rod. In analogy, the surface polarization excitation in bowtie slot antennas results in an oscillation of the conduction electrons around the aperture edge. Therefore, the resonance wavelength shows a linear dependence on the aperture perimeter, which corresponds to an effective oscillation path length of the conduction electrons. As in the case of nanorods [8], the wavelength of P resonance of bowtie slot antennas can be expressed as *λ _{res}*=

*a*

_{1}+

*a*

_{2}

*L*, with

*L*being the aperture perimeter,

*a*

_{1}and

*a*

_{2}being the coefficients which are determined by geometry, and the properties of the metal and surrounding medium.

As shown in Fig. 5, the wavelength of P resonance is nearly independent of the film thickness. This is consistent with the explanation by the dipole-like charge oscillation excited around the aperture edge. From the electric field distribution shown in Fig. 3(b), we conclude that the accumulated charges are distributed homogenously along the *z* direction. The increasing thickness does not change the oscillation path of the conduction electrons around the aperture edge and the coefficients *a*
_{1} and *a*
_{2}, which are the crucial parameters to determine the wavelength of P resonance. Only in the case of thin films of several tens of nanometers (spectra not shown), the coefficients *a*
_{1} and *a*
_{2} are influenced strongly by the film thickness, and the P resonance shifts to a shorter wavelength range as the film thickness increases.

This linear dependence of resonance wavelength on the aperture perimeter is valid for the plasmonic resonances regardless of the type of nanoaperture. In rectangular apertures, for example, the resonance dependence on the so-called ’aspect ratio’[21, 22] is, in fact, a dependence on the aperture perimeter variation at the fixed length of the short axis.

## 5. F-P like resonance

The FP resonance is the Fabry-Pérot-like resonance since the electric field distribution at resonance shows a standing wave characteristic similar to that of a Fabry-Pérot cavity. Figure 5 shows that the wavelength of this resonance shifts from 0.6*µ*m to 0.8 *µ*m with increasing film thickness from 0.15 *µ*m to 0.3 *µ*m. The same type of resonance has been observed in metallic gratings with sub-wavelength slits as well as in other types of apertures [23-27]. Often, the wavelength of the resonance is thought to be determined by the Fabry-Pérot resonance condition as in *λ*=2*n _{eff}t*/

*m*, where

*n*is the effective refractive index and

_{eff}*m*an integer. However, our studies indicate that

*n*is a complicated parameter which depends on structure geometry and material properties.

_{eff}In contrast to the film thickness, the aperture perimeter has a comparatively smaller influence on the spectral position of FP resonance. Figure 4 shows that the wavelength of the FP resonance shifts from 0.6 *µ*m to 0.8 *µ*m when the aperture perimeter is varied from 0.9 *µ*m to 2.4 *µ*m. The field distribution shows that the FP resonance is mainly film thickness dependent. Therefore, the aperture perimeter has less influence on the field distribution.

For slot antennas, the FP resonance appears at a shorter wavelength than the plasmonic resonance; therefore, slot antennas resonant at the FP resonance in thicker metal films have been applied in the visible and ultraviolet regime [13, 14]. Generally, both the P and the FP resonances coexist for the slot antennas in an optically thick metal film, and higher order FP resonance modes can be excited at an even shorter wavelength range. However, in addition to the film thickness dependence, the wavelength of the FP resonance is influenced by the nanoaperture cut-off wavelength [28], which is determined by the P resonance in slot antenna cases. This helps to explain the transmission variation of apertures of different shapes and with the same open area [9, 11], because the different shapes support different resonant properties.

## 6. Experiment

The resonance properties of bowtie slot antennas with different aperture perimeters are investigated experimentally. Considering the realization practicality by the current nanofabrication technology, antennas in a thin metal film are chosen as the object of our investigation. The periodic bowtie slot antenna arrays with various aperture perimeters are fabricated by electron beam lithography and dry etching in a 30-nm-thick gold film on a quartz substrate. The flare angle and gap size of the bowtie slot antennas are kept constant as *θ*=90° and *d*=80 nm, respectively. The aperture perimeter of the antennas are tuned from 1.46 *µ*m to 3.6 *µ*m in steps of about 500 nm by varying the outline dimensions of *a* and *b*. The periods of the arrays for the antennas with different outline dimensions are set as *p _{x}*=

*p*, which are increased from 600 nm to 1000 nm in steps of 100 nm, corresponding to the increased aperture perimeter. Figure 6 shows a typical electron scanning microscope image of a bowtie slot antenna array, where the outline dimension are

_{y}*a*=500 nm,

*b*=580 nm, and

*p*=

_{x}*p*=800 nm. The measured transmission spectra of the various antenna arrays obtained with a Fourier transform infrared spectrometer are shown in Fig. 7(a). The individual spectra are up-shifted for clarity. The results show that in a thin gold film, only the plasmonic resonances can be excited in the visible and near infrared regime (FP resonances are excited at shorter wavelength range). With an increased aperture perimeter, the resonance wavelength is red shifted. Figure 7(b) shows the resonance wavelength of antennas as a function of aperture perimeter. A linear fit can be obtained between the resonance wavelength and the aperture perimeter and supports on effective length explanation.

_{y}## 7. Metal dependence and RLC model

The dependence of the plasmonic resonance on the utilized metal is also numerically studied for a fixed structure geometry. The Drude model is employed to describe the optical properties of metals in the visible and near infrared regime. Figure 8(a) shows the transmission spectra of bowtie slot antennas in various metals. The geometry of the antennas are identical for each metal as *d*=40 nm, *θ*=90°, and *t*=30 nm, *a*=*b*=300 nm, and *p _{x}*=

*p*=450 nm. The resonance positions and amplitudes vary for the different metals. In order to study the metal influence on the antenna properties, these dependencies are discussed in terms of conductivity and skin depth, which are derived from the Drude model.

_{y}As shown in Fig. 8(b), the resonance wavelength decreases inversely with the increasing plasma frequency of the metal. The physical origin can be understood by considering the effective path length of the induced current flow around the aperture edge due the skin depth of the metal in the visible and near-infrared regime. The skin depth of the metal is defined as *δ*=*λ*/2*πk*, where *k* is absorption coefficient of the metal. Based on the Drude model as shown in Eq. (1) and
${n}_{r}+\mathit{ik}=\sqrt{{\epsilon}_{1}+i{\epsilon}_{2}}$
, *k* can be derived as [17]

In visible and near infrared regime, the approximation:

for the skin depth can be obtained using the approximation *γ*≪*ω*≪*ω _{p}*. Figure 9(a) shows the skin depth of various metals as a function of wavelength in the visible and near infrared regime. The lines are derived from calculations based on Eq. (2), and the scattered curves are obtained from the approximation by Eq. (3) for the respective metal. The approximation shows good agreement for each metal in the spectral range from 1

*µ*m to 5

*µ*m. As mentioned in the previous section, the plasmonic resonance wavelength of bowtie slot antennas shows a linear dependence on the aperture perimeter. The skin depth determines how localized the current distributions is to the inside edge of the aperture, therefore the effective oscillation path length of the conduction electrons is skin depth dependent. For a larger skin depth, for instance, the current density is less localized to the aperture edge and the conduction electrons experience a longer effective path length. Figure 9(b) shows the wavelength of the plasmonic resonance as a function of skin depth. The resonance wavelength increases nearly linearly with increasing skin depth due to this increase in the effective path length.

The metal properties influence not only the resonance wavelength, but also the transmission efficiency. The field enhancement is associated with the resonance amplitude of the antennas, therefore, it is important to investigate the physical origin of these properties. A model for these antennas is a distributed resistor-inductor-capacitor (*RLC*) circuit as shown in Fig. 10(a), in which the losses are due to both ohmic and radiative resistances. For a bowtie slot antenna, the ohmic resistance is determined by the resistivity and skin depth of the metal as well as from the geometry of the antenna. In the Drude model, electrical resistivity *ρ* is defined as the inverse of the conductivity σ as [29]

where σ_{0}=*ε*
_{0}
*ω*
^{2}
_{p}*τ* is the *DC* conductivity and *τ*=1/*γ* is the relaxation constant. The *DC* resistivity can be written as

Considering that the geometry of bowtie slot antennas is identical for each metal, we ignore the aperture perimeter and film thickness influence and define the surface resistance *R _{s}* of antennas as

With the approximation in Eq. (3), the surface resistance of bowtie slot antennas for each metal can be estimated as *R _{s}*=

*γ*/(

*cε*

_{0}

*ω*). Figure 10(b) shows the resonance amplitude [or transmission efficiency that is extracted from Fig. 8(a)] as a function of surface resistance of metals normalized to that of the perfect electric conductor (

_{p}*PEC*). In real metals, the resonance amplitude decreases with the increased surface resistance of metal as a function of

*R*

_{0}/(

*R*

_{0}+

*R*), where

_{s}*R*

_{0}is a constant which is related to radiative resistance in the circuit for the antennas in a

*PEC*film.

## 8. Conclusion

In conclusion, we have studied the resonance dependence of bowtie slot antennas on geometry and metal properties. Two types of resonances, namely plasmonic resonances and Fabry-Pérot-like resonances, can be supported by slot antennas in the near infrared and visible spectral range and they show a complementary behavior with respect to geometry dependence. Plasmonic resonances, in the near-infrared regime, show a linear dependence on the aperture perimeter and near independence of the film thickness. In contrast, Fabry-Pérot-like resonances are mainly determined by the film thickness and appear at shorter wavelengths as compared to plasmonic resonances. Depending on the wavelength range, one can effectively tune the antenna-operating wavelength by varying aperture perimeter or film thickness. The influence of skin depth and resistivity of metals on the plasmonic resonance of slot antennas has been investigated. The resonance wavelength shows a linear increase with increasing skin depth of the metal. The higher ohmic resistivity leads to a lower resonance amplitude and lower field enhancement at the resonance. This study can guide the design and optimization of slot antennas for specific resonance frequencies and characteristics. Such designs can be optimized by choice of geometries and materials and may therefore be directly used for a variety of applications where concentration of light to subwavelength spots is needed.

## Acknowledgments

This work was financially supported by the German Federal Ministry of Education and Research (Grant No. 13N9155) and the Deutsche Forschungsgemeinschaft (Grant No. FOR557 and FOR730). T. M. and T. Z. thank the Alexander von Humboldt Foundation and the Landesstiftung Baden-Württemberg, respectively. The authors thank S. Kaiser for help in measuring the optical spectra.

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