Introduction

In recent decades, surface plasmon polaritons (SPP) have attracted considerable researchers for the promising prospect in highly integrated devices or circuits of the near future. SPP wave is a collective oscillation of free electrons, which is excited at the interfaces between dielectrics and noble metals under the illumination of electromagnetic waves. Generally, this kind of oscillation can propagate along the dielectric/metal interface over a distance of several microns to a few hundred microns, while it penetrates into metal and dielectric in the direction perpendicular to the interface with intensity decaying exponentially (the skin depth is about 20 nm for metals, and about 100 nm for dielectrics). SPP wave shows many superiorities, such as local field enhancement, and shorter wavelength, making it widely used in practical applications such as biosensing [1], nanolithography [2, 3], super-resolution imaging [4], polarization manipulation, and so forth [5, 6].

Recently, many progresses have been made in near-field super-resolution focusing based on SPP [7–11]. The plasmonic lens with multiple concentric metallic rings proposed by Chen et al. achieved a focal spot of 184.4 nm (about 0.35λ i) under the illumination of a radially polarized beam, and the intensity of the focus can be enhanced by increasing the number of metallic rings [12]. Yang et al. proposed a spiral plasmonic lens with the full width at half maximum (FWHM) less than 0.3λ i, which can be used as a miniature circular polarization analyzer [13]. Nevertheless, the utilization of radially or circularly polarized incident faces a significant challenge in practical applications since the efficient focusing is only achieved when the center of the incident light perfectly aligns with the center of the lens. Consequently, lenses utilizing linear polarization light were proposed [14–16], and achieved super-resolution focusing with FWHM of the spots being 0.5λ i and 0.37λ i, respectively [17, 18]. However, most proposals emphasize on manipulation of SPP waves and the generation efficiency of SPP is always ignored.

Several efficient structures have been proposed and analysis on nanoslits has been done in some previous works [19–24]. In this paper, we mainly consider SPP generation efficiency under different compositions of nanoslit. Firstly, we theoretically analyze SPP generated at metallic nanoslit and use the estimated model to find the corresponding materials, slit width, and incident wavelength of the nanoslit with optimal SPP generation efficiency. Then, a plasmonic lens with two concentric semiannular nanoslits is proposed, and the focusing performance is greatly enhanced compared with the proposal of Fang et al. Finally, optimizations on some classical focusing structures are fulfilled according to the analysis results. Numerical simulation results convey that the focusing performances of the lenses are significantly enhanced.

Analysis on Generation Efficiency of SPP at Metallic Nanoslit

Considering a metallic nanoslit as shown in Fig. 1, the slit width is w, the refraction indexes of the materials in the slit and out the slit are n 1 and n 2, respectively. SPP can be excited at both sides of the slit under the illumination of TM (transverse magnetic) light, while there is no SPP resonance under normal incidence of TE (transverse electric) light [25]. Assuming that the intensity of electromagnetic in the slit is unitary, α +(x) and α −(x) are coefficients related to intensity of the generated SPP waves along x direction. Based on the completeness theorem for the normal modes of waveguides [26], the transverse electromagnetic fields at the metal-dielectric interface can be decomposed as a linear combination of bounded and radiative normal modes:

$$ {H}_{\mathrm{y}}=\left[{\alpha}^{+}(x)+{\alpha}^{-}(x)\right]{H}_{\mathrm{spp}}(z)+{\displaystyle \sum_{\sigma }{a}_{\sigma }(x){H_{\sigma}}^{rad}(z)} $$
(1)
$$ {E}_{\mathrm{z}}=\left[{\alpha}^{+}(x)-{\alpha}^{-}(x)\right]{E}_{\mathrm{spp}}(z)+{\displaystyle \sum_{\sigma }{a}_{\sigma }(x){E_{\sigma}}^{rad}(z)} $$
(2)
Fig. 1
figure 1

Sketch map of the SPP generation at metallic nanoslit

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where E spp and H spp are transverse electric and magnetic fields of bounded SPP mode; H σ rad and E σ rad are radiative modes [27]. Because of the modes orthogonality, α +(x) and α −(x) can be obtained through numerical integration represented as following equations:

$$ {\displaystyle {\int}_{-\infty}^{\infty }dz{H}_y\left(x,z\right)}{E}_{\mathrm{spp}}(z)=2\left[{\alpha}^{+}(x)+{\alpha}^{-}(x)\right] $$
(3)
$$ {\displaystyle {\int}_{-\infty}^{\infty }dz{E}_{\mathrm{z}}\left(x,z\right)}{H}_{\mathrm{spp}}(z)=2\left[{\alpha}^{+}(x)-{\alpha}^{-}(x)\right] $$
(4)

Since the SPP waves propagate along the metal-dielectric interface of the slit, α +(x) and α −(x) can also been derived from α +(w/2) and α −(w/2), respectively, as shown in Eqs. (5) and (6).

$$ {\alpha}^{+}(x)={\alpha}^{+}\left(w/2\right) \exp \left[j{k}_{\mathrm{spp}}\left(x-w/2\right)\right] $$
(5)
$$ {\alpha}^{-}(x)={\alpha}^{-}\left(-w/2\right) \exp \left[-j{k}_{\mathrm{spp}}\left(x+w/2\right)\right] $$
(6)

where, k spp is propagation constant of SPP wave, which can be calculated by Eq. (7).

$$ {k}_{\mathrm{spp}}={k}_0\sqrt{\frac{\varepsilon {n}_2^2}{\varepsilon +{n}_2^2}} $$
(7)

Here, k 0 is the propagation constant of incident light in free space; ε is real part of the metal permittivity. Due to the symmetry of the metallic nanoslit, α +(x) equals α −(x) and they are close to zero at |x| < w/2 since there is no metal-dielectric interface. Thus, the SPP generation efficiency can be defined as:

$$ {e}_{\mathrm{spp}}=|{\alpha}^{+}\left(w/2\right)|{}^2+|{\alpha}^{-}\left(-w/2\right)|{}^2 $$
(8)

From the near-field distribution obtained for a perfect metallic nanoslit, the overlap integrals of Eqs. (3) and (4) are subsequently calculated with x = ±w/2, the results are given as follows:

$$ {\alpha}^{+}\left(w/2\right)={\alpha}^{-}\left(-w/2\right)=\frac{1}{j}{\left[\frac{4w\prime {n}_2^2\sqrt{\left|\varepsilon \right|}}{-\pi {n}_1\left(\varepsilon +{n}_2^2\right)}\right]}^{\frac{1}{2}}\frac{I_1}{1+\left({n}_2/{n}_1\right)w^{\prime }{I}_0} $$
(9)
$$ {I}_0={\displaystyle {\int}_{-\infty}^{\infty }du \sin {c}^2\left(\pi w^{\prime }u\right)/{\gamma}_{\mathrm{u}}} $$
(10)
$$ {I}_1={\displaystyle {\int}_{-\infty}^{\infty }du\frac{ \sin c\left(\pi w^{\prime }u\right) \cos \left(\pi w^{\prime }u\right)}{{\gamma_{\mathrm{u}}}^2+{\gamma}_{\mathrm{u}}\sqrt{n_2^2/\left(\varepsilon +{n}_2^2\right)}}} $$
(11)

where, w′ = n 2 w/λ represents the normalized slit width, both u and γ are positive satisfying u 2 + γ 2 = 1 [28]. We can simplify the integrals by substituting u and γ for cosθ and sinθ, where 0 < θ < π/2, thus I 0 and I 1 can be easily calculated through numerical integration, and the generation efficiency e spp is subsequently obtained.

Firstly, we analyze the generation efficiency of SPP under different materials. Four kinds of constructions are considered with the metals of gold and silver, respectively, as illustrated in Fig. 2. Here n = 1 represents that the dielectric is air, while n = 1.45 represents glass. The wavelengths of incident lights are 800 nm for golden nanoslit, 600 nm for silver nanoslit, and the corresponding permittivities are ε Au = −24.2 + 1.51j and ε Ag = −16.1 + 0.45j (Johnson and Christy’s experimental data). The calculation results are shown in Fig. 2.

Fig. 2
figure 2

SPP generation efficiency under different constructions, the metal is gold for a and silver for b

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Comparing Fig. 2 a with b, we can conclude that both golden nanoslit and silver nanoslit have maximum e spp with the slit widths are about 0.1λ ~ 0.15λ and the nanoslit always has the highest generation efficiency when the materials in the slit and out the slit are air and glass, respectively. The corresponding maximum e spp is about 0.9 for gold, and 1 for silver. Meanwhile, the minimum efficiency will be obtained with glass in the slit and air out the slit; it demonstrates a minimum value of ~0.25 for gold and 0.3 for silver.

Meanwhile, the generation efficiency of SPP under different wavelength is also analyzed. The wavelength of incident beam has a range of 300 ~ 1300 nm and the calculation results of e spp at nanoslits with interfaces of Au-air, Ag-air and Au-glass, Ag-glass, respectively, are illustrated in Fig. 3. It can be seen clearly that the maximum generation efficiency will be obtained when the wavelength of illumination is about 400 nm for silver nanoslit and 600 nm for golden nanoslit; moreover, there is a relatively high generation efficiency at nanoslit with n 1 = 1 and n 2 = 1.45.

Fig. 3
figure 3

SPP generation efficiency under different incident wavelengths, a n 1 = 1, n 2 = 1 and b n 1 = 1, n 2 = 1.45

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Actually, all above generation efficiencies are obtained based on unitary electromagnetic field intensity in the nanoslit. Nevertheless, the field intensity in the slit varies with the slit width if the incident beam intensity is fixed and a wider slit usually corresponds with a stronger field intensity. Thus, there always is a tradeoff of slit width to ensure relatively high generation efficiency with the field intensity in the nanoslit is not too weak. In this work, we utilize finite-difference time-domain (FDTD) simulation tool to sweep the slit width of the plasmonic lenses based on metallic nanoslits to make the focusing performances greatly enhanced.

Plasmonic Lens with two Concentric Semiannular Nanoslits

A plasmonic lens with two concentric semiannular nanoslits is proposed, as shown in Fig. 4. The lens is composed of a golden plate with two nanoslits, substrate on a SiO2 plane. The inner radiuses of two slits are R 0 and R 1, with a difference of 0.5λ spp, and the slit width is w = 200 nm; the thickness of the golden plate is T = 150 nm. The wavelength of the linearly polarized incident light is λ i = 640 nm with an intensity of 1 a.u., and ε Au = −12.33 + 1.19j. So the corresponding wavelength of SPP wave can be calculated by Eq. (7), and the result is λ spp = 613.5 nm. Assuming that R 0 = 1200 nm, thus R 1 = 1506.8 nm. The layout of the lens is conducted in FDTD simulation tool; several perfect matched layers (PML) are placed at the boundary of the simulation region to avoid nonphysical reflections, and monitors are utilized to detect the field distribution nearby the metal-dielectric interface.

Fig. 4
figure 4

A 3D view of the proposed plasmonic lens with two concentric semiannular nanoslits

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The simulation results are illustrated in Fig. 5. We can clearly find that a tiny focal spot emerges at the center of the lens. The FWHM of the obtained spot is 198 nm (~0.3λ i), which is much less than the theoretical diffraction limit under the near-field region. Meanwhile, because of the local field enhancement, the field intensity of the spot is amplified with an enhancement factor of 5.19 with respect to the field intensity of the incident beam. The focusing performances are significantly enhanced compared to the lens with symmetry broken nanocorrals proposed by Fang et al., whose FWHM and the field enhancement factor of the focal spot are 320 nm (~0.5λ i) and 3, respectively.

Fig. 5
figure 5

Focusing performance of the proposed plasmonic lens with slit width w = 200 nm. a Electric field intensity distribution at 50 nm above the Au-air interface. b Electric field intensity distribution along y direction with x = 0

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We further simulate the lens with slit width range from 100 to 500 nm. As aforementioned, different slit widths correspond to different SPP generation efficiencies and field intensities; the sweep of the slit width is to find the tradeoff such that the intensity of the generated SPP is optimum. The sweep results are illustrated in Fig. 6; the field intensity of the focal spot shows a peak under a slit width of 325 nm, and the corresponding enhancement factor is 6.05, while the FWHM nearly remains unchanged because the wavelength of SPP wave is independent of slit width.

Fig. 6
figure 6

Focusing performances under slit width range from 100 to 500 nm. a Electric field intensity distribution along x axis with w = 325, 425, and 500 nm respectively. b Electric intensity and FWHM of the focal spot under different widths

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Optimization on Focusing Structures

As shown in Fig. 7a, the lens is proposed by Chen et al.; the up layer of the lens is a silver plate with multiple concentric metallic rings, substrate on a SiO2 plane. The radius of the inner slit is 4.6 μm, the thickness of the silver plate is 200 nm, the slit width is 135 nm, and the period of the rings is λ spp. The structure is illuminated by a radially polarized beam with a wavelength of 532 nm; the corresponding permittivity of Ag and wavelength of SPP wave are ε Ag = −11.79 + 0.37j and λ spp = 508.9 nm, respectively.

Fig. 7
figure 7

a A 3D view of plasmonic lens with multiple concentric metallic rings. b Electric field intensity distribution at 50 nm above the Ag-air interface. c Electric field intensity distribution along x axis. d Electric field intensity of focal spot under different slit widths

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The focal results are shown in Fig. 7b and c, an annular focal spot is obtained at the center of the lens, the maximum field intensity of the spot is 58.6 a.u., and the FWHM is 160 nm (~0.3λ i). Figure 7d is the electric field intensity of the annular spot under different slit width, the value reaches to a maximum of 69.8 a.u. with w = 275 nm, and the intensity decreases when the slit width away from 275 nm. Thus, the intensity of the focal spot is improved with a percentage of 19.1 % under the optimization of the slit width.

Figure 8a is the plasmonic lens with Archimedes-spirally distributed of T-shape nanoslits designed by Liu et al. The structure is composed of a silver layer, substrate on a SiO2 plane, and the T-shape nanoslits Archimedes-spirally distribute on the silver layer. Every T-shape nanoslit is made of two perpendicular rectangular nanoslits with a length of 200 nm and a width of 50 nm. The radius of every T-shape slit is:

$$ r={r}_0+\frac{\varphi }{2\pi }{\lambda}_{\mathrm{spp}} $$
(12)
Fig. 8
figure 8

a A 3D view of plasmonic lens with Archimedes-spirally distributed of T-shape nanoslits. b Electric field intensity distribution at 50 nm above the Ag-air interface. c Electric field intensity distribution along x axis. d Electric field intensity of focal spot under different slit widths

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where r 0 = 1266 nm is the initial radius of Archimedes spiral; φ is the angle of the corresponding slit. The incident beam is a linearly polarized light with a wavelength of 650 nm, ε Ag = −19.49 + 0.46j, and λ spp = 633 nm.

Simulation results are shown in Fig. 8b and c; a strong focal spot emerges at the center of the Archimedes spiral. The field intensity of the obtained spot is 4.8 a.u., and the FWHM is 221 nm (~0.34λ i). Figure 8d is the sweep of the slit width range from 20 ~ 140 nm, the optimal slit width is 60 nm, and the corresponding field enhancement factor is 5; the sweep results also illustrates that the focal intensity of the spot has little change with width range from 40 ~ 70 nm, while it decreases quickly when w > 70 nm or w < 40 nm.

Focusing under the Metal-SiO2 Interface

As aforementioned, the generation efficiency of SPP is always higher in metal-SiO2 interface than that in metal-air interface. Meanwhile, according to Eq. (7), the wavelength of SPP wave at interface of metal-SiO2 is also shorter than that in metal-air because of the greater refraction index of SiO2. Here, we consider the proposal in this work and the lens with Archimedes-spirally distributed of T-shape nanoslits designed by Liu et al. An extra SiO2 plane is placed above the metal layer, and the slit widths of two considered lenses are w = 200 nm and w = 60 nm, respectively. The other simulation parameters are the same as previous work, and the results are shown in Fig. 9.

Fig. 9
figure 9

Focusing performance lenses under metal-air and metal-SiO2 interface. a Lens with two semiannular nanoslits proposed in this work. b Lens with Archimedes-spirally distributed of T-shape nanoslits proposed by Liu et al.

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As for the lens proposed in this paper, the field intensity of focal spot under Au-SiO2 interface is 7.69 a.u. and the FWHM of the obtained focal spot is 132 nm (~0.2λ i) as illustrated in Fig. 9a, while the results under Au-air interface are 5.19 a.u. for field intensity and 198 nm for FWHM, so the field enhancement factor and FWHM of the spot are both much improved with percentages of 48.2 and 33.3 %, respectively. From the Fig. 9b, we can clearly find that the field intensity and FWHM of the focal spot under Ag-SiO2 interface are 13.5 a.u. and 159 nm (~0.24λ i) and it shows an improvement with percentages of 170 % for field intensity and 28.1 % for FWHM, compared with the results obtained by Ag-air interface.

Conclusions

In conclusion, we theoretically analyze the generation efficiency of SPP at metallic nanoslit and a novel lens with two concentric semiannular nanoslits is proposed. The field enhancement factor and FWHM of the proposal are 5.19 and 198 nm (~0.3λ i), breaking the theoretical diffraction limit, and the field intensity can be improved to 6.05 with an optimal slit width of 325 nm. Meanwhile, some other classical focusing structures are also improved by optimizing the generation efficiency of SPP at nanoslit; the field intensities are consequently greatly enhanced. Finally, the metal-air interfaces of the lenses are changed into metal-SiO2, and both the field intensities and FWHM of the focal spot are significantly enhanced, which is also consistent with the theoretical analysis. The new proposal and the optimization of the lenses show much prospect in applications such as nanolithography, dense storage, optical probe, and so forth.