1. Introduction

There have been many studies lately in exploiting possibilities of concentrating and channeling light wave through a metal film perforated with periodic subwavelength-sized holes or slits [1,2], owing to its many potential applications. In addition, the anomalous transmission behavior was found to be associated not only with apertures but also with structures on continuous metal films [3–7]. Surface plasmon polariton (SPP) was promoted to elucidate these anomalous far-field optical properties [8]. The generation of SPPs at single nanoholes and nanohole arrays have been studied with scanning near-field optical microscopy (SNOM) [9–12]. Using multi-wavelength scattering-type scanning near-field optical microscopy (s-SNOM) [13], we have directly observed near-field amplitude and phase of SPPs launched by an ordered Ag elliptical nanohole array and decoupling of the SPPs through another similar nanohole array [14]. This observation directly proves polarization and wavelength excitation constraint of anisotropic-shaped nanoholes and their directional propagation properties. This study extends the previous work in ordered circular nanohole arrays reported by Devaux et al. [15]. Recently, Lalanne and coworkers have proposed that in addition to SPPs near-field radiative and evanescent field components also play a key role in the near-field analysis of surface waves launched at nanoslit apertures, although only limited phase information was revealed [16,17]. Their study has further rendered the necessity of revealing the detailed behaviors of SPPs in near field.

Within two-dimensional metallic perforated structures, near-field images of the surface electromagnetic waves recorded by SNOM are very complicated [8,10,14,15]. The characteristics of shape resonance of individual nanostructures and periodicity-induced dispersion are usually mingled together in real space. Furthermore, the tip-induced effects of SNOM are expected to play an important role in the recorded images. These factors therefore make the interpretation of these characteristics rather difficult. Although previous works have suggested that Fourier analysis may help to unravel surface waves in near-field images [18,19], no systematic work has been reported so far. In this study, we utilized s-SNOM to record near-field images of surface electromagnetic waves launched on silver films with single nanhole and nanohole arrays. Systematic Fourier analysis was performed to examine the origins of different surface plasmon waves and to identify the contribution of the tip of s-SNOM.

2. Experimental method

The detailed description of s-SNOM has been described in detail previously [13]. Briefly, a single-mode diode-pump solid state laser emitting at 532 nm serves as the light source. The setup is a heterodyne interferometer which contains one frequency-shifted reference beam (Δ=80 MHz) by an acoustic-optic modulator and one excitation beam focused to the tip apex of an atomic force microscope (AFM). The p-polarized beam is sent to the tip at θ=60° with respect to the normal to the sample surface. The AFM is operated in non-contact mode with a dithering amplitude of ~40 nm and a dithering frequency of ~300 kHz (Ω). A PtIr coated silicon tip with a nominal radius less than 30 nm was used for topographic and near-field imaging wherein the sample was scanned by a nano-positioning stage. The collected backscattering radiation from the tip apex is recombined with the reference beam for detection by a fast photodiode. A wide-band lock-in amplifier locks the detector signal at Δ±nΩ frequency as the sample is scanned, yielding both optical amplitude and phase images of the n-th harmonic signal. The samples used in this study are Ag films with a thickness of 200 nm on glass substrates with single nanoholes or nanohole arrays produced by focused-ion beam. The ion dosage was carefully adjusted to produce high steepness of these nanoholes. The created structures were simultaneously examined by a scanning electron microscope to determine their dimensions.

 

Fig. 1. Two scattering field contributions of surface plasmon waves: (a) the one generated by the nanohole and (b) the one generated by the tip. k ₀, the wave vector of the incident light wave; k_SC, the wave vector of the scattering light wave; k_SPW, the wave vector of the surface plasmon wave.

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3. Results and discussion

In the consideration of the geometry of the s-SNOM measurements on the single-nanohole sample, the scattered field is expected to contain three components [20,21], depicted in Fig. 1. The first one is the scattering field from the induced polarization of the complex of the tip apex and the sample surface, E ₁=A ₁, and therefore is invariable during the sample scan over the flat surface of the Ag film. The incident light wave, on the other hand, can create a propagating surface plasmon wave (SPW) on the Ag film from a distant nanohole. The second field component, E ₂, is the result of scattering of this SPW from the tip apex. It is therefore sensitive to the relative position from the nanohole and the tip apex, r, and can be expressed as

where k ₀ sin θ is the surface component of the wave vector of the incident light wave, i is the unit vector of the projected wave vector of the incident light wave on the sample surface, and ϕ is the phase shift caused by the creation of the SPW from the nanohole. k_SPW is the wave vector of the SPW and follows the dispersion relation of Ag surface, k_SPW=k ₀[ε ₁ ε ₂/(ε ₁+ε ₂)]^1/2 [22], where ε ₁ and ε ₂ are the dielectric constants of vacuum and silver, respectively. The dependence of A ₂(r) on r reflects the decay of the SPW during propagation. Finally, by the incident light wave, the tip apex in proximity of the Ag film also creates another SPW that propagates to the nanohole and reflects back to the tip apex. The returning SPW then produces the third field component, E ₃, by scattering from the tip apex. It is expressed as

where ϕ′ is the total phase shift caused by the creation of the SPW from the tip apex and the reflection of the SPW from the nanohole. Because the laser beam is focused at the tip apex during the sample scan, A ₁ is normally larger than A ₂(r) and A ₃(r). The resultant amplitude signal recorded in s-SNOM is then given by [20, 23, 24]

Performing Fourier transform on the amplitude image yields

where

δ ²(k) is two-dimensional delta function, A_j(k)=FT[A_j(r)](j=2,3) g ₁(k)=|k+k ₀ sinθi|/k_SPW, and g ₂(k)=|k-k ₀ sin θi|/k_SPW. rect(k/2q) means truncation to zero beyond the circle of radius q and step(k/2q) is defined as zero for k<q and one for k>q. The symbol ⊗ denotes the convolution operator. The derivation of Eq. (5) is performed in polar coordinate under the assumption that the tip is cylindrical symmetric with respect to the surface normal. The details of this derivation involve Hankel transform [25, 26] and are not given here. According to the derived results of F ₂(k) and F ₃(k), the second components of Eq. (3) exhibits two circles centered at ±k ₀ sinθi with a radius of k_SPW, while the third component exhibits one circle centered at origin with a radius of 2k_SPW. The resultant single-nanohole near-field image in k space therefore consists of three circles. The interference between the constant scattering field, E ₁, and the scattering field from the nanohole-generated SPW, E ₂, produces two k_SPW -radius circles centered at ±k ₀ sinθi. In addition, the interference between E₁ and the scattering field from the tip-generated SPW, E₃, produces one 2k_SPW -radius circle centered at origin. For single 150-nm nanohole on the Ag film, the recorded 3rd harmonic near-field amplitude image by the s-SNOM, shown in Fig. 2(a), exhibits rather complex ring-like pattern around the nanohole. After performing fast Fourier transform (FFT), three circles emerge in the k-space image, Fig. 2(b), and match with the theoretical prediction above, supporting the interpretation of near-field images obtained by s-SNOM.

 

Fig. 2. Near-field images of single nanohole: (a) amplitude image, (b) Fourier transformed image, (c) amplitude image without the tip-induced contribution, and (d) distribution of the vertical field component calculated by FDTD method. The hole diameter is 150 nm and the excitation wavelength is 532 nm. The nanoholes are marked as open black circles. The scale bar represents 1 µm.

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To further support the interpretation of the near-field amplitude image of single nanohole discussed above, numerical calculation based on finite-difference time-domain (FDTD) method was performed. The simulation, without considering the tip, was executed over a region of 16.8 µm×12.8 µm with a mesh size of 20 nm and the dielectric function of silver was taken from Ref. 27. A Gaussian beam with a 4-µm waist was used to simulate the experimental condition. Figure 2(d) shows the calculated distribution of the field component normal to the surface, E _⊥, by finite-difference time-domain (FDTD) method. Since the scattering field from the tip is sensitive to the field component normal to the sample surface [24], the calculated E _⊥ can directly reflect the observed scattering field from the tip in s-SNOM. The calculated field distribution pattern in the r-space image is similar to the one shown in Fig. 2(a). The experimental result, however, appears to be more complicated. The k-space counterpart of the calculated pattern (not shown here) is composed of only two small k_SPW -radius circles that agree with the corresponding experimental result in Fig. 2(b), while the large 2k_SPW -radius circle centered at origin is absent. This missing circle can be explained by the fact that the tip was not considered in the simulation, thus inhibiting the generation of surface plasmon wave by the electromagnetic interaction between the tip apex and the Ag film. That is, the third component in the near-field image is missing. By removing the large circle of the recorded near-field image, Fig. 2(b), and subsequently transforming it back to r-space, the resultant r-space image, Fig. 2(c), matches almost perfectly with the calculated result shown in Fig. 2(d). This consistency thus supports the theoretical analysis of the Fourier-transformed image above. Finally, the 2k_SPW -radius circle in Fig. 2(b) is isotropic while the two k_SPW -radius circles exhibit brighter along the k_x direction. The anisotropic feature of the two k_SPW -radius circles, which is also present in the calculated result, can be attributed to the dipolar property of the SPW emanating from single nanohole reported previously [28]. In contrast, since the intense interaction between the tip apex and the Ag film provokes a strong field component normal to the surface which is much larger than the in-plane component [24], thus the excited SPW radiates isotropically from the approached site of the tip apex. The single-nanohole study above therefore demonstrates that examining near-field images in k space helps to identify the origins of different surface plasmon waves, allowing for in-depth investigation of the fundamental nature of surface electromagnetic waves that may exist around nanostructures [16].

 

Fig. 3. Near-field images of nanohole arrays: (a) amplitude image with a period of 535 nm. (b) amplitude image with a period of 750 nm. (c) and (d) are the Fourier-transformed images of (a) and (b), respectively. (e) and (f) are the predicted k-space patterns of (c) and (d), respectively. The green and blue circles are the tip- and nanohole-generated contributions, respectively. Black crosses represent the positions defined by the structure function of the arrays, while the red crosses symbolize the intersection of the three circles and the black crosses.

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The analysis of the single-nanohole near-field image above has established a foundation for performing subsequent analysis in the near-field study of nanohole arrays. Similar to the case of single nanohole, there are three field components in the scattering radiation. In this case, since the SPW is generated by the whole nanohole array, the second field component is given by

where m and n are integers describing the finite holy array, a is the period of the array, x and y are unit vectors along the x and y direction, respectively, and r in this case is the positional vector of a nanohole in the array with respect to the site of the tip apex. The third field component is similarly given by

where only the scattering of the SPW by one nanohole is considered. The contribution from more than two successional scattering events is neglected here. The near-field amplitude signal is then given by

The image in the k-space is expressed as

where

where S(k) is the structure factor of the two-dimensional nanohole array. This result was derived with the use of Fourier transform shift theorem. For an infinity array, S(k) becomes a two-dimensional square array of delta functions with a period of 2π/a. The derivation above shows that the resultant k-space pattern of the near-field image of the nanohole array is the product of the k-space pattern of single nanohole and the structure factor S(k). Figures 3(a) and 3(b) show the near-field amplitude images of two square nanohole arrays with a hole diameter of 150 nm and two periods of 535 and 735 nm, respectively. Notice that the corresponding Fourier-transformed images in k space, Figs. 3(c) and 3(d), match very well with the predicted patterns, shown in Figs. 3(e) and 3(f). This consistency supports the theoretical derivation of the Fourier-transformed images of nanohole array above and further confirms the derived results of single nanohole given earlier in this report.

As a final note, in the derivation of Eq. (9), we have neglected the contribution of multiple successional scattering of surface plasmon waves. This contribution can be understood in two cases. In the case that the SPWs do not satisfy the Bragg condition set by S(k), the multiple scattering events, similar to the single scattering events, do not produce constructive interference inside the nanohole array, leaving only non-propagating localized waves [16] that certainly make no contribution to these multiple scattering events. In the other case that the SPWs satisfy the Bragg condition, the constructive inference inside the nanohole array caused by the multiple scattering events may brighten the S(k) positions that are different from those bright positions caused by the single scattering events. This is due to the additional phase delay introduced in the multiple scattering process. The detailed derivation of the multiple-scattering contribution is certainly worthy of future study to further unravel its contribution.

Both the near-field imaging results and their theoretical derivation have two important implications. First, in the near-field study of surface plasmon polaritons with s-SNOM, the tip apex always acts as the plasmon inducer as well, making the resultant near-field images rather complicated. Our study shows that performing Fourier analysis on the recorded near-field images facilitates the identification of the tip-induced contribution, setting a solid foundation for in-depth examination of the fundamental nature of surface plasmon polaritons. Second, Eq. (5a) shows the dependence of the Fourier transformed image on the phase shift of the SPW generation from single nanohole, while Eq. (5d) shows the corresponding dependence on the phase shift occurring at the tip apex. These two derived results provide a means to determine these phase shifts induced during the generation of SPW, making possible of investigating shape-dependent plasmon generation mechanism.

4. Conclusion

We have employed scattering-type scanning near-field microscopy to investigate surface waves generated by single nanohole and nanohole arrays. The detected scattering radiation has been derived and the corresponding Fourier-transformed pattern has been predicted theoretically. The transformed pattern of the recorded near-field image of single nanohole is in agreement with the prediction, providing a firmed evidence of the surface plasmon wave launched by the tip apex in the near-field measurements. Alternatively, for nanohole arrays, the transformed pattern is consistent with the product of the effective form factor of single nanohole and the structure factor of the array. This study thus demonstrates that the Fourier analysis identifies the tip-induced effects in the recorded images, making scattering-type scanning near-field microscopy a powerful tool in investigating surface plasmon waves with sub-10 nanometer resolution.