1. Introduction

Focusing of light into a subwavelength region is a well-known problem in nanophotonics. One of the existing solutions to overcome this problem is by focusing light with a dielectric microsphere (microball). The study on the microsphere-aided subwavelength focusing of light was first reported by Lu et al. [1]. It has been shown that 0.5 μm quartz microspheres illuminated by an excimer KrF laser beam of wavelength 248 nm focused the beam onto a silicon substrate, resulting in 100-nm-wide melted hillocks. A similar mechanism for microrelief patterning was proposed by McLeod and Arnold [2], where a 1 μm microsphere optically trapped in a 532 nm Bessel beam was guided to a desired location and burnt a pit in the substrate by focused pulsed laser light of wavelength 355 nm before being guided to the next location. Focusing of light by means of microspheres was also studied theoretically [3]. In particular, a 1 μm microsphere (refractive index $n=1.59$) illuminated by a plane wave of wavelength $\lambda =400\mathrm{nm}$ was shown to form in a plane perpendicular to the polarization axis a focal spot of diameter full width at half-maximum $(\mathrm{FWHM})=0.325\lambda$. The focal regions generated by use of microspheres were called photonic nanojects [3]. The direct experimental observation of a photonic nanojet was reported by Ferrand et al. [4], where latex microspheres of 1, 3, and 5 μm diameter and illuminated by a plane wave of wavelength 520 nm generated focal spots of diameters $0.62\lambda$, $0.52\lambda$, and $0.58\lambda$. Though having a subwavelength diameter, the experimentally derived photonic nanojects did not demonstrate that the diffraction limit was overcome. One more characteristic of the photonic nanojet, namely, its length, or depth of focus (DOF), was studied in [5,6]. It was numerically shown that the photonic nanojet could be extended by use of a gradient-index microsphere, with the refractive index varying linearly from 1.43 to 1.59, with the DOF being equal to $11.8\lambda$ [5]. An opposite task of attaining a shorter nanoject was posed in [6], where a microsphere of radius $2.5\lambda$ was illuminated by a Gaussian beam focused with a wide-angle lens of numerical aperture $\mathrm{NA}\approx 1$. In that case, the photonic nanojet was found to have a length of $\mathrm{DOF}=0.88\lambda$. In addition to microspheres, few studies have been published on generating photonic nanojets by using other dielectric microobjects, e.g., microcylinders [7] or disks [8]. It should also be noted that sphere-aided focusing is challenging because the sphere needs to be kept in place (e.g., using a light trap as in [2]). Fabrication of the focusing element on a substrate seems to be more suitable technologically. In the presence of the antireflection coating on the rear/back side of the substrate, the focusing efficiency will be higher than with the same-diameter microsphere.

In this work, we study the focusing of a linearly polarized laser beam of wavelength $\lambda =633\mathrm{nm}$ using square-profile parallelepiped steps of height 500 nm fabricated of silica (refractive index $n=1.46$) on a substrate. Such square steps with various side dimensions of 0.4, 0.5, 0.6, and 0.8 μm generate near-surface focal regions in the form of a photonic nanojet with their intensity six times that of the incident light and focal spot diameters equal to $\mathrm{FWHM}=0.44\lambda$, $0.43\lambda$, $0.39\lambda$, and $0.47\lambda$, which is below the diffraction limit of $0.51\lambda$.

2. Simulation

The simulation was performed using the finite-difference time-domain (FDTD) method implemented in the FullWave software [9]. The size of the computational cell was 0.012 μm, i.e., about $\lambda /53$. Perfectly matched layers (PMLs) were chosen as the boundary conditions. The simulation was conducted for the side $L$ of the square-profile silica step ($n=1.46$) varying from 0.4 to 0.8 μm with a 0.02 μm step. To approximate the experimental conditions, the step was assumed to be on a substrate (Fig. 1). The rectangular columns (steps) were illuminated by a linearly polarized plane wave at the wavelength of $\lambda =633\mathrm{nm}$. In order to correlate and compare the simulation results with experimental results, the height of the steps was $H=500\mathrm{nm}$.

 

Fig. 1. Schematic view of the step under study.

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The simulation results for the 0.4-, 0.6-, and 0.8-μm-wide steps are presented in Figs. 2–4. The pattern of intensity distribution in the plane $xz$, which is perpendicular to the incident beam polarization, is shown in Fig. 2, and Fig. 3 represents the intensity distribution in the plane $yz$, which is parallel to the same beam polarization. The intensity distribution along the optical axis $z$ is shown in Fig. 4. Intensity was calculated as a sum of squared electric field components ${|E_y|}^2+{|E_z|}^2$. From the intensity patterns presented in Figs. 2 and 3, it is clearly observed how the enhanced-intensity region has been generated directly behind the surface of the step. Moreover, this region’s shape resembles the geometry of the microsphere-aided photonic nanojets. In addition, the resulting photonic nanojet is also seen to be elliptic, with its focus elongated along the incoming light polarization direction due to the longitudinal intensity component found in the plane of interest [10]. From the intensity profile along the $z$ axis in Fig. 4, it is seen that if the step width is small (less than the wavelength) the maximum near-surface intensity is formed within the step. When the step’s width (square side) reaches 0.6 μm, the intensity maximum shifts outside, with further increase in the step’s width resulting in a larger focal length (distance along the $z$ axis from the step’s edge to the intensity maximum). The near-surface focal spots (Fig. 2) generated with the square steps of sides 0.4, 0.5, 0.6, and 0.8 μm are of size $\mathrm{FWHM}=0.44\lambda$, $0.42\lambda$, $0.40\lambda$, and $0.45\lambda$, respectively.

 

Fig. 2. Intensity pattern in the plane perpendicular to the incident wave polarization ($xz$) for a step of width (a) 0.4 μm, (b) 0.6 μm, and (c) 0.8 μm. White-dashed contour outlines the element’s boundary.

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Fig. 3. Intensity pattern in the plane ($yz$) parallel to the incident beam polarization for a step of width (a) 0.4 μm, (b) 0.6 μm, and (c) 0.8 μm. White-dashed contour outlines the element’s boundary.

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Fig. 4. Intensity profile along the $z$ axis when using a diffraction step of width (a) 0.4 μm, (b) 0.6 μm, and (c) 0.8 μm. The black vertical line denotes the step’s top surface.

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From Fig. 2 it is seen that there are several local maxima inside the step, which are similar to the principal mode of a stepped-index planar waveguide. For instance, for TE polarization (the $E_y$ vector is perpendicular to the plane $xz$) the principal mode is given by the following equation [11]:

From Fig. 2, the field is also seen to converge inside the step, forming a near-surface focal spot. The local maxima inside the step shown in Fig. 2(b) are concave. The reason for this is the light propagation inside the step with a larger phase speed near its edges than near the center. This can also occur because the wavefront of the plane incident wave is curved at the step edge. The wavefront gets curved in such a way that the radiation inside the step is directed from the edge toward the center, as shown in our previous study [12], where the peripheral ray was shifted from the step edge toward its center along a square-root parabola given by

Figure 5 depicts the maximum intensity in the near-step focus as a function of the step’s width $L$. The maximum value occurs for the step width of $L=0.6\mathrm{\mu m}$. At this step width the intensity maximum (focal spot) on the $z$ axis (Fig. 4) occurs outside the step.

 

Fig. 5. Near-step intensity maximum profile as a function of the step width (square side).

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3. Experiment

An array of square-profile, equal-height microparallelepipeds of varying size was fabricated by photolithography (using an excimer 193 nm ArF laser) and the ion-beam etching of silica substrates. Figures 6 and 7 depict images of the fabricated elements’ profile obtained on an atomic-force microscope (AFM) Solver Pro. From Fig. 6(b) the microrelief height is seen to be about 500 nm.

 

Fig. 6. (a) AFM image of the microsteps under study with a 0.6 μm side and (b) their selected profiles. White-dashed line on (a) depicts the line along which the profile is taken.

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Fig. 7. (a) AFM image of 2.5 μm side steps and (b) an exemplified step’s profile.

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There are some different approaches to investigate focal spots generated by microobjects. For example, confocal microscopy was used in [4] to experimentally obtain a photonic nanojet. An imaging process based on fluorescent quantum dots was used in [13] to observe Bessel-like beams. To image focal spots smaller than the diffraction limit in our research the experimental study was conducted using scanning near-field optical microscopy (SNOM) on the Ntegra Spectra (NT-MDT) microscope. The steps under analysis were illuminated from the substrate’s side by a linearly polarized Gaussian beam of wavelength 633 nm with a waist radius of about 5 μm. The Gaussian beam was focused onto an individual step. Columns with sides of 0.4, 0.5, 0.6, and 0.8 μm were characterized. Intensity patterns of the focal spots formed near the steps’ output surface were obtained. The intensity was measured with a 0.015 μm step. Thus, the accuracy of the experimental measurement of the focal spot size (FWHM) was about $0.01\lambda$. An example of the resulting intensity distribution measured with the SNOM is given in Fig. 8.

 

Fig. 8. Intensity distribution near the exit surface of a 0.6 μm side step, which was experimentally obtained under illumination with a linearly polarized 633 nm laser light: (a) half-tone intensity pattern and (b), (c) intensity profiles for the minimal and maximal focal spot’s diameter.

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Unfortunately, the SNOM arrangement is designed so that the illumination light polarization plane makes an angle of 45° with the Cartesian coordinates $x$ and $y$ [Fig. 8(a)], with the slightly elliptic focal spot near the step’s exit surface appearing to be elongated along the polarization direction. The focal spot intensity profiles in Figs. 8(b) and 8(c) were measured along the minimal and maximal focal spots’ diameters, so that at the nanojet’s initial section they were found to be ${\mathrm{FWHM}}_{\min }=(0.39\pm 0.01)\lambda$ and ${\mathrm{FWHM}}_{\max }=(0.45\pm 0.01)\lambda$. In Fig. 9, the experimentally measured values of the (a) minimal and (b) maximal focal spots’ diameters at the nanojet’s cross section near the exit surface versus the varying-width steps are marked with error bars. Figure 9(a) shows the focal spot size for the FWHM intensity in a plane perpendicular to the incident beam polarization. Figure 9(b) shows the measured (bars) and simulated (curve) size of the focal spot’s transverse component in a plane parallel to the incident light polarization. The curves in Fig. 9 were obtained using the FDTD simulation, whereas the bars correspond to the SNOM-aided experiment, with the vertical lines showing the experimental error.

 

Fig. 9. (a) Minimal and (b) maximal FWHM size of the photonic nanojet’s cross section near the exit surface versus the side of the step’s square cross section. The curve shows the simulation results; the bars show the experimental results.

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From Fig. 9 it is seen that in the plane perpendicular to the incident light polarization direction, the minimal photonic nanojet’s (focal spot) diameter is achieved for a step width of 0.6 μm both in simulation, ${\mathrm{FWHM}}_{\min }=0.40\lambda$, and in experiment, ${\mathrm{FWHM}}_{\min }=0.39\lambda$. The discrepancy between the experimental and simulated curves in Fig. 9 is not larger than 24% and can be assigned to imprecisely positioning the microscope’s cantilever in the immediate step’s surface vicinity. Even so, the simulated curve is found within the range of values defined by the experimental bars. Figure 10 illustrates the SNOM-aided intensity profiles for the square step side of 0.5 μm. For comparison, the simulated intensity profiles are also presented in the plots. The plots in Fig. 10 suggest that the experimentally derived spot diameters agree with the transverse, rather than total, intensity distribution. The fact that a pyramid metal cantilever in the SNOM is primarily sensitive to the transverse intensity component, being insensitive to the longitudinal component, was earlier noted by the present authors in [14].

 

Fig. 10. Intensity profiles directly behind the step surface with the square side of 0.5 μm in the plane (a) perpendicular and (b) parallel to the input beam polarization axis.

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4. Comparison with Microsphere-Aided Focusing

To evaluate the difference between the characteristics of the photonic nanojets generated by means of square-shaped steps and microspheres [1–5], we simulated focusing a linearly polarized plane wave using microspheres with diameters, respectively, equal to the square step sides of 0.4, 0.6, and 0.8 μm. The simulation parameters were analogous to those for step-aided focusing. The simulation results are shown in Figs. 11–13 (which are similar to the respective Figs. 2–4 for the steps). Table 1 shows the comparison results for the parameters of the step-aided and microsphere-aided photonic nanojets. The photonic nanojet’s DOF was calculated as the FWHM of intensity outside the step (i.e., if the maximum intensity was found inside the step, the half-maximum was counted from the surface). Besides, the step’s surface was assumed to be the left half-maximum’s boundary (Fig. 13). The photonic nanojet’s diameters were calculated near the element’s surface ($0.02\lambda$ above the surface, which amounts to a single step of grid in the FDTD method).

 

Fig. 11. Intensity pattern in the plane perpendicular to the input radiation polarization ($xz$) for a microsphere of diameter (a) 0.4 μm, (b) 0.6 μm, and (c) 0.8 μm. The microsphere’s boundary is shown as a dashed circle.

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Fig. 12. Intensity pattern in the plane parallel to the input radiation polarization ($yz$) for a microsphere of diameter (a) 0.4 μm, (b) 0.6 μm, and (c) 0.8 μm. The microsphere’s boundary is shown as a dashed circle.

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Fig. 13. Intensity profile along the $z$ axis for a sphere of diameter (a) 0.4 μm, (b) 0.6 μm, and (c) 0.8 μm. The sphere’s boundary is shown as a vertical line.

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Table 1. Comparison of the Optical Nanojet Parameters: Silica Microspheres versus Square Steps ($\mathsf{\lambda }=\mathsf{633}\mathsf{nm}$)

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Table 1 suggests that (1) in the plane perpendicular to the polarization plane, the step- and microsphere-aided focal spots have similar diameters; (2) the microsphere-aided focal spots show a wider range of variations (from $0.39\lambda$ to $0.53\lambda$) than the step-aided spots (from $0.40\lambda$ to $0.45\lambda$); except for one case, all the diameters were below the diffraction limit of $0.51\lambda$; (3) the microsphere-aided focal spots are more elliptical than the step-aided focal spots; at $D=L=0.4\mathrm{\mu m}$, the diffraction by the step results in a circular focal spot of diameter $\mathrm{FWHM}=0.44\lambda$, whereas the microsphere-aided focal spot is elliptical, measuring ${\mathrm{FWHM}}_{\min }=0.53\lambda$ and ${\mathrm{FWHM}}_{\max }=0.74\lambda$; and (4) with increasing microsphere diameter and square step’s side, the DOF (photonic nanojet’s length) measured as the intensity half-maximum behaves differently, increasing from $\mathrm{DOF}=0.69\lambda$ to $\mathrm{DOF}=2.08\lambda$ for the step and decreasing from $\mathrm{DOF}=1.97\lambda$ to $\mathrm{DOF}=0.59\lambda$ for the microsphere. It should be noted that the use of a focused Gaussian beam could change the DOF in a wide range [6].

5. Conclusions

Based on the FDTD simulation and SNOM-aided experiment with a pyramid metal cantilever having a nanohole, we have investigated the sharp focus of a linearly polarized laser light of wavelength 633 nm by means of dielectric fused-silica square steps of varying size and the same height, fabricated on a silicon substrate with $n=1.46$.

The following results have been obtained.