1. Introduction

Achieving super resolution beyond the Abbe diffraction limit (~λ/2) is a hot topic in modern optics and nanophotonics [1]. In the past decades, a couple of near-field and far-field techniques were proposed to surpass the Abbe diffraction limit, such as the solid immersion lens (SIL) [2,3], the scanning near-field optical microscope [4], the superlens/hyperlens [5–7], and the optical super oscillations [8]. Among these techniques, the SIL is arguably a simple approach to achieve sub-diffraction-limited resolution for attractive applications in data storage and subsurface microscopy [3,9].

With the rapid development of SIL technique, micro/nanoscale solid immersion lenses (μ/n-SILs) and optical microspheres have recently drawn extensive attention due to their outstanding performance in projecting wavelength-scale imaging beyond the diffraction limit [10–18]. For instance, dielectric microspheres under uniform plane wave (PW) illumination have the capability to generate near-field focal spots or far-field photonic nanojets with lateral resolution beyond λ/2n_s (n_s is the refractive index of microspheres) [16,18]. Alternatively, hemispherical n-SILs under lensed plane wave (LPW) illumination (i.e., the PW is focused by a lens) was reported to generate a near-field focal spot with a ~18.6% reduction in the full-width-at-half-maximum (FWHM) compared to optical microspheres [13].

In this work, we would like to report a super-resolution strategy based on pure/hybrid n-SILs under dark-field (DF) illumination. DF illumination is a well-known high-resolution technique in optical imaging and laser measurement [19,20]. The mechanism of enhanced resolution under DF illumination is similar to the angular-spectrum technique employed in bright-field illumination [21], i.e., an annular-aperture is used to tailor incident angles for wave-vectors reconstruction. Combining DF microscope with conventional microscale SILs has been first proposed in 2013 to measure the resonance fluorescence from a single quantum dot [22]. In the current work, we will theoretically discuss the enhanced resolution by combing the DF illumination with pure/hybrid n-SILs. There is no optical imaging or laser measurement involved.

2. Methods

We employed a commercial three-dimensional finite-difference time-domain (FDTD) software package, “Lumerical FDTD Solutions”, to simulate the optical field distributions of pure/hybrid n-SILs under DF illumination. The DF source was constructed by two Gaussian beams with a slight difference of size in the x-y plane and a phase difference of π [23,24]. The electric field distributions of the outer and inner Gaussian beams at source plane (z = 0) can be written as $E_{\text{outer}}={\text{A}}_{\text{0}}{e}^{-(x^2+y^2)/w_0{}^2}{e}^{-i\omega t}$ and $E_{\text{inner}}={\text{A}}_{\text{0}}{e}^{-(x^2+y^2)/w_0{}^2}{e}^{-i(\omega t+\pi )}$. Here, A₀ is the magnitude and w₀ is the beam waist. The π-shift phase will give rise to a destructive interference in the overlapping region (i.e., dark area) of two Gaussian beams. By setting different virtual numerical apertures (NAs) and an identical focusing position in the property of two Gaussian beams, a DF light source can be generated. In our calculations on pure/hybrid n-SILs, the transversal size of outer and inner Gaussian beams is set as 6000 nm and 5830 nm respectively. The NA of outer and inner Gaussian beams is set as NA_outer = 0.9 and NA_inner = 0.895, respectively. The ideal focusing position for both Gaussian beams is z = −1453 nm. In FDTD calculations, perfectly matched layer (PML) boundary condition was used along all three dimensions. A 10 nm grid size (Δx, Δy, Δz) was set in the whole calculation space and a 5 nm fine mesh around the n-SILs was used. The time step Δt satisfies the Courant stability limit $\Delta t\le 1/v\sqrt{{(\Delta x)}^{-2}+{(\Delta y)}^{-2}+{(\Delta z)}^{-2}}$. The FDTD calculation is converged when the auto shutoff value on the PML boundaries reaches 10⁻⁵.

3. Results and discussion

3.1 Pure n-SILs under DF illumination

Figure 1(a) shows the schematic of the focusing model of n-SILs under DF illumination. The outer and inner transversal sizes of the ring aperture in the DF condenser lens are 2.5 cm and 2.43 cm, corresponding to NA_outer = 0.9 and NA_inner = 0.895, respectively. n-SIL-І and n-SIL-ІІ represent the typical face-down and face-up n-SILs which are made up of standard hemispheres with a diameter of D. f₀ is the ideal focusing distance of the DF condenser lens. Δz is the distance from the plane of interest to the ideal focal plane of the DF condenser lens. The free-space wavelength of x-polarized incident beam is 532 nm. Figures 1(b) and 1(c) compare the spatial electric-field intensity distributions in the y-z plane for DF illumination (NA_outer = 0.9, NA_inner = 0.895) and LPW illumination (NA = 0.9) without using n-SILs. DF illumination can bring a 16.3% improvement in the resolution than LPW illumination.

 

Fig. 1 (a) Modeling of n-SIL-based focusing under DF illumination. (b) and (c) Comparison of spatial electric-field intensity distributions in the y-z plane for DF illumination (NA_outer = 0.9, NA_inner = 0.895) and LPW illumination (NA = 0.9) without using n-SILs. The inset shows the cross-section profile of electric-field intensity on the ideal focal plane (z = −1453 nm).

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Figure 2 gives a comprehensive comparison of the focusing performance of the n-SIL-І and n-SIL-ІІ under DF and LPW illumination, as well as the microsphere and n-SIL-ІІ under PW illumination. For both n-SILs and microsphere, D is 1000 nm, Δz is zero, and the refractive index is 1.6 at 532 nm (e.g., polymer) [13]. Compared with previously reported two super-resolution strategies, i.e., the n-SIL-І under LPW illumination [12,13] and the microsphere under PW illumination [15], the n-SIL-І under DF illumination [Fig. 2(a)] can generate a near-field focal spot with a 10.5% [Fig. 2(c)] and 13.8% [Fig. 2(e)] improvement in the resolution, respectively. The n-SIL-ІІ under DF illumination [Fig. 2(b)] does not show any advantage than that under LPW illumination [Fig. 2(d)] in the FWHM of focal spots. This is because the focal resolution of n-SIL-ІІ is strongly dependent on the normal-incident wave vectors that are absent in the DF illumination. An extreme case is when the n-SIL-ІІ is directly under PW illumination, a much smaller near-field focal spot can be generated [Fig. 2(f)] with strong sidelobes. Overall, considering the requirements of both high resolution and low sidelobe intensity, the n-SIL-І under DF illumination is preferred for supporting a near-field focusing spot.

 

Fig. 2 (a) and (b) Spatial electric-field intensity distributions for the n-SIL-І and n-SIL-ІІ under DF illumination. (c) and (d) Spatial electric-field intensity distributions for the n-SIL-І and n-SIL-ІІ under LPW illumination. (e) and (f) Spatial electric-field intensity distributions for the microsphere and n-SIL-ІІ under PW illumination. The inset in each sub figure shows the cross-section profile of electric-field intensity along the dashed line.

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Figure 3 shows the influence of Δz on the focusing quality of n-SIL-І under DF and LPW illumination. Here, two main parameters, i.e., FWHM and contrast, are used to compare the focusing quality. The contrast of focal spots is defined as the intensity ratio between the main peak and the strongest sidelobe. From inspection of Fig. 3(a), it is clear that a positive value of Δz is beneficial to the improvement of resolution. The n-SIL-І under DF illumination always shows a higher resolution than that under LPW illumination. The best resolution is obtained when Δz is about 1.0λ. The dependence of FWHM on Δz can be briefly explained by the geometrical optics theory. When Δz is zero, the n-SIL-І is just put on the focusing position of the DF condenser lens and the near-field focal spot is generated without refraction when light is convergent inside the n-SIL. When Δz is nonzero, the NA of the n-SIL-І will be larger or smaller than that at zero due to refraction. It will result in an improved or reduced resolution. From Fig. 3(b), it is found that the highest contrast is obtained when Δz is zero. The n-SIL-І under DF illumination shows lower contrast than LPW illumination. Figure 3(c) compares the focusing performance of four selective points in Fig. 3(a). When Δz is 0.5λ, the n-SIL-І under DF illumination shows a 12.3% improvement in the resolution than that under LPW illumination. When Δz is 1.0λ, such improvement is increased up to 22.8%. The contrast on the plane of interest for DF illumination is poor due to strong sidelobes. But this is not a big problem because most energy of sidelobes is concentrated inside the n-SIL. In following calculations, a Δz value of 0.5λ was chosen for further study.

 

Fig. 3 (a) and (b) Dependences of the FWHM and contrast of focal spots on Δz for the n-SIL-І (D = 1000 nm) under DF and LPW illumination. (c) Spatial electric-field intensity distributions with cross-section profile (inset) for four selective points in (a).

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Figure 4 investigates the influences of D and substrate material on the resolution of our proposed strategy. On one hand, Fig. 4(a) shows that the FWHM is sensitive to D and nearly follows a periodical fluctuation relationship with a period of about 340 nm. Considering the effective wavelength inside the n-SIL is 332 nm, such fluctuation is due to the periodical change of electric-field amplitude at the air/n-SIL interface, i.e., the n-SIL interface will be periodically located at the position where the incident plane wave shows maximum/minimum electric-field amplitude. On the other hand, we demonstrated again that the n-SIL-І under DF illumination always shows a higher resolution than that under LPW illumination. When D is 1200 nm, the n-SIL-І under DF illumination shows the smallest FWHM of 134 nm with a resolution improvement of 11.8% than LPW illumination [Fig. 4(b)].

 

Fig. 4 (a) Dependence of the FWHM of focal spots on D for the n-SIL-І (Δz = 0.5λ) under DF and LPW illumination. (b) Spatial electric-field intensity distributions with cross-section profile (inset) for selected points (D = 1200 nm) in (a). (c) Dependence of the FWHM of focal spots on the material of substrate. (d) Spatial electric-field intensity distributions for selected point in (c). (e) Evolution of the FWHM and the electric-field intensity of the focal spot during a propagation distance of Z_prop away from the output surface (dashed line) in (d).

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Figure 4(c) indicates that the substrate can also sensitively influence the FWHM of focal spots. For either n-SIL-І-DF or n-SIL-І-LPW strategy, SiO₂ shows a higher resolution than the Si and Au substrates due to its higher transmission and better impedance match (less refractive index difference) at the n-SIL/substrate interface. The higher reflection at the n-SIL/Si or n-SIL/Au interface causes the focal spot located inside the n-SIL, and thus a lower resolution on the plane of interest is obtained. As shown in Fig. 4(d), a clear near-field deep focal spot can be observed around the n-SIL-I/SiO₂ interface. Figure 4(e) depicts the evolution of the FWHM and the electric-field intensity of the focal spot during a propagation distance of Z_prop inside the SiO₂ substrate. Here we define Z_eff as the “depth of focus”, where the electric-field intensity of the focal spot is decreased to one-half of that at the output surface of n-SIL-І. The n-SIL-І-DF strategy with the SiO₂ substrate shows a large Z_eff of about 280 nm, accompanying with a resolution reduction by only 24%.

3.2 Hybrid n-SILs under DF illumination

We propose a hybrid n-SIL which is constructed by embedding a cylindrical nanorod with a diameter of 2r into the n-SIL-I. The nanorod is assumed to be no absorption with a higher refractive index. Since the effective wavelength inside the material is inversely proportional to the refractive index of material, it is expected that a higher resolution can be obtained when a nanorod with a higher refractive index is embedded into the n-SIL. For instance, as shown in Fig. 5(a), when the nanorod is made up of GaP with a refractive index of 3.49 at 532 nm, there is an optimal range of r (45-55 nm) for the hybrid n-SIL on the SiO₂ substrate to generate a high-resolution near-field focal spot under DF illumination. The best resolution was obtained when r = 47 nm, where the FWHM is only 66 nm (~λ/8). Such resolution has surpassed the Abbe diffraction limit in GaP nanorod (~λ/7), where the effective wavelength in GaP nanorod is shorten to λ/n_GaP and the Abbe diffraction limit in the GaP nanorod reaches ~λ/2n_GaP. As a comparison, the dependence of the FWHM on r for a pure GaP nanorod under PW illumination is presented in Fig. 5(b). When r = 53 nm, the pure nanorod shows the highest intensity and a resolution of 76 nm that is very close to the Abbe diffraction limit in the GaP nanorod.

 

Fig. 5 (a) Dependences of the FWHM and intensity of focal spots on r for a hybrid n-SIL (Δz = 0.5λ, D = 1200 nm) on the SiO₂ substrate under DF illumination. The hybrid n-SIL (see inset) is constructed by embedding a cylindrical GaP nanorod into the n-SIL-І. (b) Dependence of the FWHM and intensity of focal spots on r for a pure GaP nanorod on the SiO₂ substrate under PW illumination. (c) and (d) Spatial electric-field intensity distributions for selected points in (a) and (b). (e) and (f) Evolution of the FWHM and the electric-field intensity of the focal spot during a propagation distance Z_prop away from the output surface (dashed line) in (c) and (d).

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Figures 5(c) and 5(d) compare the spatial electric-field intensity distributions for two selected points in Figs. 5(a) and 5(b). Due to the high refractive index of GaP nanorod, a highly-compressed near-field focal spot with acceptable sidelobe distributions can be observed for both cases. Besides a higher resolution, the hybrid n-SIL also owns other advantages compared with a pure GaP nanorod. One advantage is that the hybrid n-SIL can generate a relatively stable near-field focal spot. The intensity of the focal spot in Fig. 5(a) is less sensitive to the variation of r than that in Fig. 5(b). This is because the focal spot’s location and intensity of the hybrid n-SIL under DF illumination are mainly modulated by the n-SIL. Another advantage is that the hybrid n-SIL can generate a focal spot with less divergence and larger depth of focus. As shown in Figs. 5(e) and 5(f), the focal spot generated by the hybrid n-SIL is obviously more convergent with a predicted Z_eff of 214 nm, which is about 6-fold deeper than that of a pure GaP nanorod.

From the practical point of view, we have investigated several important factors for the application of the hybrid n-SIL strategy. Firstly, we have investigated the feasible refractive-index range of the nanorod in the hybrid n-SIL. To generate an obvious near-field focal spot, an embedded nanorod with a refractive index from 1.6 to 3.8 is acceptable. Besides GaP, other low-absorption materials with a relatively high refractive index in visible range, such as Si₃N₄, TiO₂, and ZnO [25], can be used in the experiment.

Secondly, imperfections of a nanorod in fabrication should be taken into account. Figures 6(a)-6(d) show the dependences of focal quality on the height variation, rotation angles, and cross-section ellipticity ratio of the GaP nanorod in the hybrid n-SIL. It is clear that the hybrid n-SIL strategy shows good tolerance on the imperfections of a nanorod. The FWHM and intensity of focal spots are not sensitively changed during a large modulation of parameters. For instance, when the rotation angle in either the y-z or x-z plane is 20 degrees, the hybrid n-SIL can still generate a high-resolution near-field focal spot with low divergence along the propagation direction.

 

Fig. 6 Dependences of the FWHM and intensity of focal spots on the (a) height variation ΔH, (b) rotation angle θ_y in the y-z plane, (c) rotation angle θ_x in the x-z plane, and (d) cross-section ellipticity ratio r_y/r_x of the GaP nanorod in the hybrid n-SIL (Δz = 0.5λ, D = 1200 nm, r = 47 nm) on the SiO₂ substrate under DF illumination. In (d), r_x = 47 nm. (e) and (f) Spatial electric-field intensity distributions for two selected points, i.e., θ_y = 20° in (b) and θ_x = 20° in (c).

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Thirdly, such hybrid n-SIL strategy can also be extended to μ-SILs that are relatively easy to be fabricated based on the self-assembled techniques [11]. For instance, Fig. 7(a) shows that a μ-SIL (D = 5000 nm) with a GaP microrod (r = 500 nm) embedded also works well under DF illumination to support a super-resolution near-field focal spot. The FWHM of focal spot on the plane of interest is only 57 nm (~λ/9) [Fig. 7(d)], which is much better than the pure μ-SIL under DF illumination [Figs. 7(b) and 7(e)] or the pure microrod under PW illumination [Figs. 7(c) and 7(f)].

 

Fig. 7 Comparison of spatial electric-field intensity distributions in the y-z plane for (a) a hybrid μ-SIL (Δz = 0.5λ, D = 5000 nm, r = 500 nm) under DF illumination, (b) a μ-SIL (Δz = 0.5λ, D = 5000 nm) under DF illumination, and (c) a microrod (r = 500 nm) under PW illumination. The height of microrod is 2500 nm. The substrate is SiO₂. (d)-(f) The cross-section profile of spatial electric-field intensity distributions on the plane of interest in (a)-(c).

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Finally, in practical application, the shape of the focal spot depends greatly on the incoming polarization. For instance, when the hybrid n-SIL is working under DF illumination with circular polarization, our simulation results show that an identical donut-shaped spot will be generated in both the x-z and y-z planes. A tight spot may be generated if radial polarization is used. This is because one important property of focusing a radially polarized beam is that the sidelobes of the electric intensity distribution are relatively low compared to those of linear and circular polarizations [30].

4. Conclusions

In conclusion, we have demonstrated that the face-down n-SIL under DF illumination can generate a near-field focal spot with an obvious improvement in the resolution (up to 22.8%) compared to that under LPW illumination. More importantly, a much higher resolution of ~λ/8 can be obtained when a GaP nanorod with a higher refractive index is embedded into the face-down n-SIL to construct a hybrid n-SIL. The proposed super-resolution strategy also shows advantages in supporting a stable near-field focal spot with small divergence and large depth of focus. Our studies have potential applications in data storage [3], imaging and microscopy [9–11,26–30], lithography [31,32], and luminescence engineering [33].