1. Introduction

A fundamental subject in near-field optics or nano-optics has been how to create a strongly localized and strongly enhanced optical field on a probe-tip [1–3]. Strongly localized and enhanced optical fields induce many nonlinear optical phenomena and have various prospective applications such as in apertureless scanning near-field optical microscopes (al-SNOMs) [4–7]. Many ideas have been proposed to date [8–14], and the use of surface plasmon polaritons (SPPs) on a metallic surface is a promising technique for solving this problem. Recently, Stockman [15] has reported that it is possible to create a localized and enhanced optical field at a probe-tip by focusing SPPs on a tapered plasmonic waveguide. However, how to practically excite the specific mode of SPPs that propagate along the tapered cylindrical plasmonic waveguide has not yet been shown.

Present authors have proposed use of an I-shaped (dumbbell-shaped) aperture in a metallic screen that simultaneously provides both high emission intensity and small spot size and showed that this effect is due to surface SPPs excited inside the aperture [16–19]. The localization and enhancement of optical fields in this aperture is based on the control the phase velocity of SPP in the nanometric metallic gap by the gap width [20]. Recently, we have proposed a practical technique by which a formation of a localized and enhanced optical near-field on a tapered metallic probe-tip can be performed [21]. The focusing of SPPs on the tapered probe-tip is performed on the basis of the optical intensity distribution of the pyramidal or conical structure of the previously reported I-shaped apertures and long narrow apertures [17, 18]. In this probe, the probe-tip is illuminated by the SPPs that are confined and enhanced by the I-shaped aperture on the screen, creating an optical near-field of high intensity and small spot size due to the small tip-radius. In this paper, we detail computer simulations of this technique. The integration problem is solved using a volume integral equation (VIE) by a generalized minimum residual method (GMRES) iteration and fast Fourier transformation (FFT).

2. Plasmonic gap probe consisting of I-shaped aperture and pyramidal structure

The schematic structure of the proposed probe-tip is shown in Fig. 1. A uniform metallic screen (slab) with thickness w and relative complex-valued permittivity ε₁ is placed on the x-y plane. A pyramidal structure of the same metal is fabricated on this screen, and an I-shaped aperture consisting of a rectangular narrow gap-region of a_x × a_y sandwiched by two rectangular wide gap-regions of b_x × b_y is formed in the screen through the pyramidal structure. The pyramidal structure is divided into two sections by the long I-shaped aperture. The height from the screen to the tip of the left section is assumed to be larger than the height from the screen to the tip of the right section, and the left section is used as a tapered metallic probe. The base of the pyramidal structure has dimensions of B_x×B_y. In this paper, for simplicity, B_y =(2b_y + a_y)=B_x is assumed. A Gaussian beam is assumed to be normally incident to the screen from the negative z-direction below the metallic screen in region (I), and the electric field polarization at z=0 is assumed to be parallel to the x-axis. By assuming the incident beam to be a Gaussian beam, we can avoid the rather troublesome treatment of the infinite-sized problem of the metallic screen in numerical simulations [16–19], [21]. The SSPs exiting the I-shaped aperture from the incident beam are confined and enhanced in the narrow-gap region in the aperture [16, 19]. The SPPs propagate along the narrow-gap region through the pyramidal structure and along the side boundary of the left section. Finally, the SPPs reach the tip of the left section shown in Fig. 1 and are focused at the sharp tip with a small-radius. Therefore, we can expect that a near-field providing high intensity and small spot size will be created near the tip of the probe.

 

Fig. 1. Geometry of the problem.

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The scattering problem for the metallic structure shown in Fig. 1 is solved using a VIE, under the assumption of exp(jωt) time dependence [22].

Here, k ₀ = ω/c (c is velocity of light in free space, ω is angular frequency), D(x) is the total electric flux, E ⁱ(x) is the incident electric field, and Ạx̤ is the vector potential, which is expressed by the following volume integral.

Here, g(x ∣ x’) is a three-dimensional free-space Green’s function given by

The volume integral region V in Eq. (2) represents the entire space, and ⎕_r(x) represents the distribution of relative permittivity, where ⎕_r(x)/⎕₀ = ⎕₁ of metal of the screen and ⎕_r(x)/⎕₀=1 in the aperture and free space. Since the region in which [⎕_r(x’) - ⎕₀⎕ is nonzero is finite and the integral region V has a finite volume, it is possible to solve Eq. (1) numerically by the well-established method of moments. To obtain the solution, the entire region of the problem is divided into small discretized cubes of size δ_x × δ_y × δ_z, and Eq. (1) is discretized by the method of moments using roof-top functions as basis and testing functions. The resultant system of linear equations is then solved by iteration using a GMRES iteration with an FFT [22, 23]. As the numerical evaluation is long, extraneous, and can be found in the literature [22–24], the details are omitted in this paper.

The incident Gaussian beam, which propagates in the positive z-direction, can be expressed as follows [25]:

where

and

and E ₀ and W ₀ are a constant amplitude and spot size at z=0, respectively. The validity of the code was checked by confirming that the code gives a reasonably accurate solution compared with the rigorous solution for a dielectric sphere and by confirming the energy conservation for a lossless problem under a paraxial approximation.

In the simulation, the wavelength λ=573 nm, and the complex permittivities of the metallic screen and the surrounding free space are fixed and given as ⎕_r(x)/⎕₀=ε₁= -12.4 - j0.85 (Silver) and ⎕_r(x)/⎕₀=1.0, respectively. The parameters of the incident wave are given by E ₀=1 and W ₀=λ. The electric incident vector Eⁱ(x) at z=0 is fixed parallel to the x-axis in Fig. 1. Throughout this paper, all lengths are normalized by the wavenumber k ₀=2π/λ. The screen thickness is given by k ₀ w = 0.7 (approximately 0.13λ) with a tip-probe height of k ₀ h = 1.2 (approximately 0.19λ). The size of the screen used in the numerical evaluation is given by k ₀ C_x= k ₀ C_y=21.2 (3.4λ) and the size of discretized cubes is given by k0δ_x= k ₀δ_y= k ₀δz = 0.1 (0.016λ).

3. General characteristics of optical intensity distribution

We first show the typical numerical simulation results of the proposed probe. The parameters of the I-shaped aperture are given by k ₀a_x × k ₀a_y = 0.4 × 0.4 (0.064λ × 0.064λ) and k ₀b_x × k ₀b_y = 1.0 × 1.8 (0.16λ × 0.29λ), and the height difference between the two pyramidal sections is given by k ₀ g=0.4 (0.064λ). The two-dimensional distributions of the optical intensities ∣E(0, k ₀ y, k ₀ z)∣² and ∣E(k ₀ x, 0, k ₀ z)∣² inside the aperture and the pyramidal structure,i.e., the intensity distributions on the y-z and x-z planes in Fig. 1, are shown in Figs. 2(a) and 2(b), respectively. Notice that the intensity scale-range normalized by the incident intensity, i.e., ∣E ₀∣²=1, is 0 - 80 in Figs. 2(a) and 2(b). In these figures, the standing wave (interference between incident and reflected guided waves) and the enhancement of guided optical waves can be observed along the narrow gap-region inside the aperture and pyramidal structure. These results show that the SPPs are excited and enhanced in the narrow gap-region of the I-shaped aperture and propagate along the narrow gap-region in the pyramidal structure. We find that the optical intensity is apparently localized and enhanced at the tip of the left pyramidal section, i.e., the probe-tip, as shown in Fig. 2(b).

 

Fig. 2. Two-dimensional optical intensity distributions on the (a) y-z and (b) x-z planes. The positions of the planes are shown by the broken lines in the upper insets. The intensity scale-range normalized by the incident intensity is 0 - 80. (λ=573 nm, ε₁= -12.4 - j0.85).

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Fig. 3. Two-dimensional optical intensity distributions just above the probe-tip on planes parallel to the x-y plane and placed at positions (a) k ₀ z=k ₀ w + k ₀ h + 0.05 and (b) k ₀ z=k ₀ w + k ₀ h + 0.15. The intensity scale-range normalized by the incident intensity is 0-1000. (λ=573 nm, ε₁= -12.4 - j0.85).

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Two-dimensional distributions of the localized and enhanced optical intensities on planes parallel to the x-y plane just above the probe-tip, i.e., ∣E(k ₀ x, k ₀ y, k ₀ w + k ₀ h + 0.05)∣² and ∣E(k ₀ x, k ₀ y, k ₀ w + k ₀ h + 0.15)∣², are shown in Figs. 3(a) and 3(b) respectively. It is found that the enhanced intensity very close to the probe-tip shown in Fig. 3(a) exceeds 103 times the incident intensity. The approximate full-width at half-maximum (FWHM) value normalized by the wavenumber k ₀ of the optical intensity shown in Fig. 3(a) is given by k ₀×FWHM ≈ 0.15 (0.024λ) in the both the x- and y-directions. Since the maximum value in Fig. 3(b) is about only 40 times the incident intensity, the decay of the localized and enhanced intensity in the z-direction is faster than those in the x- and y-directions from Figs. 3(a) and 3(b). In order to investigate the structure of the localized and enhanced near-field just above the probe-tip in detail, the intensities of the x-, y-, and z-components of the electric field shown in Fig. 3(a), i.e., ∣E_x(k ₀ x, k ₀ y, k ₀ w + k ₀ h + 0.05)∣², ∣E_y(k ₀ x, k ₀ y, k ₀ w + k ₀ h + 0.05)∣², and ∣E_z(k ₀ x, k ₀ y, k ₀ w + k ₀ h + 0.05)∣², are shown in Figs. 4(a), 4(b), and 4(c), respectively. From the results shown in Fig. 4, we find that the localized and enhanced near-field just above the probe-tip mainly consists of the z-component. The polarization due to the incident wave, i.e., x-direction at z=0, vanishes in the localized and enhanced near-field just above the probe-tip. The fundamental characteristics of the intensity distributions shown in Figs. 2–4 do not change for the examples shown hereafter. Next, we investigate the dependence of the localized and enhanced near-field just above the probe-tip on some fundamental parameters of the probe.

 

Fig. 4. Two-dimensional intensity distributions of the (a) x-component, (b) y-component, and (c) z-component of the total electric field just above the probe-tip on plane parallel to the x-y plane placed at k ₀ z = k ₀ w + k ₀ h + 0.05. The normalized intensity scale-range is 0-1000. (λ=573 nm, ε₁= -12.4 - j0.85).

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Notice that the metallic probe proposed in this paper does not require an external light that directly illuminate the probe-tip from outside of the system to create the localized and enhanced near-field on the probe-tip. Necessary source_is the incident light into the aperture. This characteristic will result in low background noise when this probe is applied to al-SNOMs [21].

4. Probe formed of only left section of pyramidal structure

Next, we consider a case in which the height difference between the divided pyramidal sections is given by k ₀ g = 1.2 (0.19λ). Other parameters are the same as those in Figs. 2–4. In this case, there is no right section of the pyramidal structure shown in Fig. 1 on the screen. This simple structure may have advantages from a manufacturing point of view.

Two-dimensional optical intensity distributions on the y-z and x-z planes inside the aperture and pyramidal structure are shown in Figs. 5(a) and 5(b), respectively. The normalized intensity scale-range is 0 - 80 in Figs. 4(a) and 4(b). We can see that the focused intensity near the probe-tip is smaller than that for the case of k ₀ g=0.4 shown in Fig. 4(a). The two dimensional distribution of optical intensity just above the probe-tip ∣E(k ₀ x, k ₀ y, k ₀ w + k ₀ h + 0.05)∣² is shown in Fig. 5(c). The intensity of the localized and enhanced near-field close to the probe-tip shown in Fig. 5(c) is about 500 times larger than the incident intensity. However, this value is smaller than that for the case of k ₀ g=0.4. Therefore, the guidance of SPPs along the gap inside the pyramidal structure is a factor in making a strongly enhanced near-field just above the probe-tip.

 

Fig. 5. Two-dimensional optical intensity distributions on the (a) y-z and (b) x-z planes. The positions of the planes are shown by the broken lines in the upper insets. The normalized intensity scale-range is 0 - 80 in (a) and (b). (c) Two-dimensional optical intensity distribution just above probe-tip on plane parallel to the x-y plane and placed at a position k ₀ z = k ₀ w + k ₀ h + 0.05. The normalized intensity scale-range is 0-1000 in (c). (λ=573 nm, ε₁= -12.4 - j0.85).

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5. Long narrow aperture

The authors have shown that a localized and enhanced near-field can be created on the tip of the pyramidal structure even for the case of a long narrow aperture, not an I-shaped aperture [17]. This shape of aperture also has advantages from a manufacturing point of view. The same idea can be applied to the structure proposed in this paper.

Results obtained using a simple long narrow aperture are shown in Figs. 6(a) - (c) for a case in which the height difference between the pyramidal sections is given by k ₀ g=0.4.

This case correspond to the case of the I-shaped aperture, where parameters are given by k ₀a_x × k ₀a_y = 0.4 × 0.4 (0.064λ × 0.064λ) and k ₀b_x × k ₀b_y = 0.4 × 1.8 (0.064λ × 0.29λ). Two-dimensional optical intensity distributions on the y-z and x-z planes inside the aperture and pyramidal structure are shown in Figs. 6(a) and 6(b), respectively. The two-dimensional optical intensity distribution just above the probe-tip ∣E(k ₀ x, k ₀ y, k ₀ w + k ₀ h + 0.05)∣² is shown in Fig. 6(c). It is found that the intensity of the enhanced near-field very close to the probe-tip shown in Fig. 6(c) is about 800times larger than the incident intensity. This value is smaller than that for the case of the I-shaped aperture. Therefore, the enhancement of the field by the I-shaped aperture is a factor in making a strongly enhanced near-field.

 

Fig. 6. Two-dimensional optical intensity distributions on the (a) y-z and (b) x-z planes. The positions of the planes are shown by the broken lines in the upper insets. The normalized intensity scale-range is 0-80 in (a) and (b). (c) Two-dimensional optical intensity distribution just above probe-tip on plane parallel to the x-y plane and placed at a position k ₀ z=k ₀ w + k ₀ h + 0.05. The normalized intensity scale-range is 0-1000 in (c). (λ=573 nm, ε₁= -12.4 - j0.85).

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The results for the case of the long narrow aperture with k ₀ g=1.2 are shown in Figs. 7(a) – (c). This structure is the simplest among the structures considered in this study from a manufacturing point of view. Notice that normalized intensity scale-range is 0 - 40 in Figs. 7(a) and (b). Unfortunately, the enhanced intensity is only about 80 times larger than the incident intensity, i.e., the enhancement is not strong.

 

Fig. 7. Two-dimensional optical intensity distributions on the (a) y-z and (b) x-z planes. The positions of the planes are shown by the broken lines in the upper insets. The normalized intensity scale-range is 0-40 in (a) and (b). (c) Two-dimensional optical intensity distribution just above probe-tip on plane parallel to the x-y plane and placed at a position k ₀ z=k ₀ w + k ₀ h + 0.05. The normalized intensity scale-range is 0-1000 in (c). (λ=573 nm, ε₁= -12.4 - j0.85).

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6. Other values of relative permittivity for probe and pyramidal structure

Thus far, we have only considered the case in which the relative permittivity is given by ε₁= -12.4-j0.85 (Silver, λ= 573 nm). Next, we consider other materials. We first consider the case of ε₁= -34.5-j8.5 (Aluminum, λ= 488 nm). The results are shown in Figs. 8(a)–(c). In this case, the enhanced intensity is about 100 times larger than the incident intensity, i.e., much smaller than that in the case of ε₁= -12.4-j0.85.

 

Fig. 8. Two-dimensional optical intensity distributions on the (a) y-z and (b) x-z planes. The positions of the planes are shown by the broken lines in the upper insets. The normalized intensity scale-range is 0-80 in (a) and (b). (c) Two-dimensional optical intensity distribution just above probe-tip on plane parallel to the x-y plane and placed at a position k ₀ z=k ₀ w + k ₀ h + 0.05. The normalized intensity scale-range is 0-1000 in (c). (λ=488 nm, ε₁= -34.5 - j8.5).

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We next consider the case of ε₁= -7.2-j20.5 (Titanium, λ=1300 nm). The results are shown in Figs. 9(a)–(c). In this case, the enhancement is weak from these results, the large value of the real part and small value of the imaginary part of the absolute values of relative permittivity are also a factor in obtaining a strong enhancement of the near-field on a probe-tip.

 

Fig. 9. Two-dimensional optical intensity distributions on the (a) y-z and (b) x-z planes. The positions of the planes are shown by the broken lines in the upper insets. The normalized intensity scale-range is 0-12 in (a) and (b). (c) Two-dimensional optical intensity distribution just above probe-tip on plane parallel to the x-y plane and placed at a position k ₀ z=k ₀ w + k ₀ h + 0.05. The normalized intensity scale-range is 0-30 in (c). (λ=1300 nm, ε₁= -7.2-j20.5).

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7. Dependence of enhancement on sharpness of probe-tip.

It is considered that the localization and enhancement of the electromagnetic field depends on the sharpness, i.e., average radius, of the metallic probe-tip from an electromagnetic theoretical consideration [2, 3]. The entire region of the problem is divided into small discretized cubes in the numerical simulation used in this study. Therefore, the sharpness of the probe-tip is determined by the discretized cubes that constitute the probe-tip. A part of the cross section of the screen and pyramidal structure for the case of k ₀ g=0.4 on the x-z plane used in Figs. 2–9 is shown inFig. 10(a). In this case, the top of the left pyramidal section, i.e., the probe-tip, consists of only one descretized cube whose size is given by k ₀δ_x × k ₀δ_y × k ₀δ_z = 0.1×0.1×0.1. In order to dull the sharpness of the probe-tip, we consider the probe whose tip is formed of a rectangular parallelepiped that consists of 2×2×1 discretized cubes. The height-difference between the pyramidal sections is assumed to be k ₀ g=0.4. In this case, the size of the probe-tip is given by 2k ₀δ_x × 2k ₀δ_y × k ₀δ_z = 0.2×0.2×0.1 and a part of its cross-section on the x-z plane is shown in Fig. 10(b). We can investigate the dependence of the localized and enhanced near-field on the sharpness of the probe-tip by comparing the results obtained for the probe-tips shown in Figs. 10(a) and (b). The results for the structure shown in Fig. 10(b) are shown in Figs. 11(a)–(c). The field intensity inside the aperture and pyramidal structure is similar to that shown in Fig. 2. However, the maximum value of ∣E(k ₀ x, k ₀ y, k ₀ w + k ₀ h + 0.05)∣² is about 200 times larger than the incident intensity and FWHM is given by k ₀×FWHM ≈ 0.22 (0.035λ). These results show that the sharper the probe-tip is, the stronger the enhancement is.

 

Fig. 10. Cross section of the screen and pyramidal structure on the x-z plane used (a) in Figs. 2–9, and (b) that used in Fig. 11.

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Fig. 11. Two-dimensional optical intensity distributions on the (a) y-z and (b) x-z planes. The positions of the planes are shown by the broken lines in the upper insets. The normalized intensity scale-range is 0-80 in (a) and (b). (c) Two-dimensional optical intensity distribution just above probe-tip on plane parallel to the x-y plane and placed at a position k ₀ z=k ₀ w + k ₀ h + 0.05. The normalized intensity scale-range is 0-1000 in (c). (λ=573 nm, ε₁= -12.4 - j0.85).

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8. Conclusions

A metallic probe that provides a strongly localized and strongly enhanced optical near-field at the probe-tip has been proposed and investigated. Since the probe-tip is illuminated by SPPs inside the aperture, direct illumination of the probe-tip by an external light is not necessary for this probe. Necessary light source is the incident light into the aperture. It was found that the probe can create an optical near-field intensity that is about 10³ times larger than the incident intensity on the probe-tip, when appropriate parameters of the probe are used.

We also found that the following factors increase the enhancement of the optical near-field on the probe-tip: (a) Using an I-shaped aperture, (b) guidance of SPPs along the gap in the pyramidal structure, (c) adopting a metal that has a large value for the real part and a small value for the imaginary part of the absolute value of the relative permittivity, and (d) using a sharp probe-tip. However, note that these factors can only be obtained within a limited range of probe parameters. Since the localization and enhancement of the optical near-field on the probe-tip is related to the propagation of SPPs in the nanometric gap inside the aperture and pyramidal structure, the phenomena investigated in this study are not simple. From our experience of numerical simulations, results are very sensitive to the various parameters of the aperture, demonstrating the need to develop accurate computer-aided design and simulation tools for the design of the probe. The probe investigated in this paper has a wide range of applications in near-field optics and nano-optics.