1. Introduction

Conventional lenses converge the incident light (such as visible light) to form focusing spots, whose full width at half maximum (FWHM) is greater than half of the illumination wavelength. To obtain smaller focusing spots and improve the lithography resolution, several methods have been developed, such as electron beam lithography and focused ion beam lithography, which are due to the significantly smaller wavelengths of the electron and ion beams [1,2]. However, these technologies show extremely low lithography speeds. Superlens and flat lens by metasurfaces also show high performance focusing and imaging, but they are disadvantages in working length (WD), manufacturing cost, or lithography speed [3,4]. An incident light passing through monolayer microspheres forms tens of thousands of sub-diffraction focusing spots at the shadow-side near-field of microspheres, which can be utilized for paralleling imaging and lithography [5,6]. The FWHM of the spots can be as small as 120 nm when the illumination wavelength is 400 nm [7]. The super performance focusing of microspheres was discovered ten years ago, and the focus spot is known as a “photonic nanojet (PNJ)” [8]. Four main structural parameters, including the light wavelength, radius and refractive index (RI) of the microsphere, and RI of the surrounding environment, are found to influence the PNJ [5–9]. Due to the super-resolution focusing properties and easy assembly of microspheres [10–12], they are widely used for super-resolution photolithography, lithography, and imaging [13–18]. In the fields of micro- and nano-fabrication and related applications, most researchers used illuminated monolayer microspheres for patterning photoresist films [13–15], while others utilized high light intensity of PNJ behind the illuminated microspheres to direct writing pattern arrays on material films, such as GeSbTe and silicon [16,17]. Various kinds of 2D/3D large-area unit patterns, from tens of nanometers to tens of micrometers, can be achieved by adjusting structural parameters of microsphere [18,19], light scattering [20–22], interaction between adjacent microspheres [23], or multiple exposures [24]. Microsphere-assisted photolithography/lithography shows great application prospects. However, all the above are contact-type/near-field microsphere-assisted photolithography/lithography, and microspheres will be discarded after laser illumination. In addition, it takes a long time to prepare monolayer microspheres. Therefore, the method for patterning is time-consuming. Furthermore, contact-type photolithography/lithography limits the pattern shapes.

To improve the condition, two suggestions have been proposed. The first is to change the microsphere lithography technology in the near field. O’ Connell et al. successfully fabricated a reused monolayer microsphere, in which monolayer microspheres were taped on transparent glue films. When used, the microsphere mask directly covered the film for photolithography or ablation [25]. Arnold et al. achieved the near-field, non-contact, direct-writing of any pattern via a single microsphere illuminated by a laser, with electrostatic repulsions between the charged surfaces of the sphere and a substrate to balance this scattering force from illumination [26]. However, these strategies are difficult to apply to pattern unity or lithography speed. Another is the far-field application of PNJs. Kong et al. first demonstrated a multilayer microsphere (up to 100 layers) with a graded RI between $\sqrt 2 $ and 1, and an elongated PNJ (from the microsphere’s shadow-surface to the point where the PNJ intensity decreases to twice the incident along the z-axis, approximately 20 λ) via a numerical study [27]. Shen et al. suggested a shell–core two-layer microsphere, which is simpler and easier to fabricate, also exhibits a PNJ with an extension of 22 λ [28]. Wu et al. increased the length of the nanojet (from the point of the peak intensity to the 1/e value of the peak intensity along the z-axis) to 57 λ by using numerical calculations [29]. Hong et al. attempted modulating PNJs generated by microspheres decorated with concentric rings, diffractive Fresnel zone plates, center-covered with a blanket-like platinum disk, or negative axicon microsphere [30–33]. Compared with PNJs generated by microspheres without these decorations or covered disk, PNJs with smaller beam waists, or tunable focal length are obtained. In addition, Chen et al. studied the focusing properties of single-layer hemisphere shells. Compared with a microsphere lens, they numerically demonstrated a longer WD, ∼3 µm, and a longer PNJ (the distance between the two points along the axis where the light intensity drops to 1/e of its peak intensity) [34]. Li et al. demonstrated that droplet micro-lenses immersed in water have an ultralong WD and beam waist of the PNJs is close to the illumination wavelength [35]. These studies are helpful for advancing the far-field lithography applications of microspheres. Further more, Hong et al. have demonstrated impressive performance in nano-imaging of microsphere lens in far-field and given insightful details to design microsphere lens meeting the corresponding requirements. [36,37] However, there is still a long road between theoretical studies and far-field lithography applications. First, to demonstrate the advantages, the focus length should be long enough to implement far-field photolithography/lithography. Second, the focusing beam should be with a narrow beam waist, close to or less than the illumination wavelength. Third, the focus spot should have a highly enhanced intensity and long focus depth, which can improve the spot size in lithography applications.

In this paper, PNJs with sub-wavelength beam waists and WD about 10 µm are demonstrated in numerical study by combining the properties of linearly polarized illumination, immersed two-layer hemisphere or immersed-engineered sing-layer hemisphere (the spherical face is partly cut parallel to the hemispherical flat surface). Compared with conventional ways of adjusting the beam shape and position of the PNJ, the highlights of our work are as follows. Firstly, we achieve PNJs with sub-wavelength FWHM and ultralong WD using a linearly polarized illumination and immersed two-layer hemisphere or immersed-engineered single-layer hemisphere. Secondly, unlike conventional light manipulating through mask or decoration fixed on an objective lens, the light manipulating we utilized is designing immersed two-layer hemisphere with low-high-low RI for each layer, and immersed single-layer hemisphere with partly cut off spherical surface, which both hardly allow incident light near optical axis contribute to the PNJs. By varying radius of inner-hemisphere, RI of each layer and surrounding environment, radius of cut off spherical surface, the PNJ can be modulated. In additionally, the hemispheres are compatibility to adjacent laser wavelengths. Furthermore, the spot size can less than 0.5 λ evaluated via a lithography simulation. The suggested hemispheres will show great potential for far-field lithography and RI applications.

2. Results and discussion

In this study, we considered a dielectric two-layer hemisphere comprising a core and a concentric shell with different refractive indices embedded in a dielectric film. Figure 1 shows a schematic of the side view of the hemisphere, illuminated by an x-polarized plane wave (amplitude of 1 V/m) from the spherical side of the hemisphere and propagating along the z-direction (the optical axis of the hemispherical lens) with a wavelength of 365 nm. A focus spot formed at the side of the plane section of hemispherical lens. The Cartesian coordinate system was adopted and the spherical center of the hemisphere was located at the coordinate origin. All films/layers used in the simulation models in this study were transparent. The key parameters of the embedded two-layer hemisphere, including radius of the inner core R_c, radius of the shell R_s, and RIs of the core/ shell/ embedding film/ surrounding material n_c, n_s, n_e, and n_l, respectively, are also shown in Fig. 1. CST Microwave studio software is utilized to build the hemisphere model and calculate light field distribution.

 

Fig. 1. Schematic of side view of the two-layer dielectric hemisphere embedded in a film, illuminated by an x-polarized, z-direction-propagating plane wave. The focus spot is shown on the right of the figure.

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Four key parameters were used to characterize the PNJ quantitatively: the focal length f (i.e., the working distance; the distance between the spherical center of the hemisphere and the focal point), the full width at half maximum (FWHM) of the focal point in the y-direction, the effective length L (i.e., the distance between two points on the z-axis among the PNJ where the intensity drops to I_P/e), and the peak intensity I_P. Additionally, I₀ is the intensity of the incident light.

2.1 Focusing properties of the immersed two-layer hemispherical lens

In our previous research, we studied the focusing characteristics of a two-layer transparent hemispherical lens. We obtained a large f, enhanced the light by approximately 80 times, and the beam waist was 432 nm, focusing length was 3.4 µm, while the outer- and inner-hemispherical radii and corresponding RIs were 3 µm, 1.8 µm, 1.8, 1.5, respectively [38]. The results were good. However, the f and beam waist values are not ideal for far-field lithography applications. To increase the WD (For this kind of hemisphere, WD and f are equal.) and narrow the beam waist size (FWHM), we selected a solid-immersion hemispherical lens with a slightly larger diameter of 9 µm for the following study.

A two-layer hemisphere with the outer and inner radii and corresponding RI values of 4.5 µm, 2.8 µm, 1.57, and 1.44, respectively, was embedded in a film with RI n_e. The RI of the surrounding environment of the embedded hemisphere is given by n_l. The illumination wavelength was 365 nm. The intensity profiles along the z-direction and yoz plane of the PNJs for different n_e and n_l values are presented in Fig. 2.

 

Fig. 2. Intensity profiles along the z-direction and yoz plane of PNJs for different n_e and n_l values: (a) n_e = 1, n_l = 1; (b) n_e = 1.38, n_l = 1; (c) n_e = 1.38, n_l = 1.33. R_c, R_s, n_c, and n_s are 2.8 µm, 4.5 µm, 1.44, and 1.57, respectively.

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When the two-layer hemisphere is in air, the f and FWHM values of the PNJ are 5.7 µm and 422 nm, respectively, as shown in Fig. 2(a). The value of f is slightly larger than the radius of the outer hemisphere, and much longer than that of the single-layer hemisphere shell reported by Chen et al. [34]. However, the FWHM of the two-layer hemisphere is larger than the illumination wavelength, while the FWHM of the single-layer hemispherical shell can realize half of the illumination wavelength. It indicates that the surrounding environment influences the focusing properties of the two-layer hemisphere, therefore the PNJs formed under several discrete n_e and n_l values were studied for better focusing results. As shown in Fig. 2(b), when n_e and n_l are 1.38 and 1, respectively, two main peaks appears. Here, the values of f and FWHM of the focus point on the left side are 6.9 µm and 294 nm, respectively. Meanwhile, the f and FWHM values of that on the right side are 9.0 µm and 352 nm, respectively. These values are much better than those obtained from the two-layer hemisphere in air; in particular, the FWHM of the left focus point decreased to 294 nm, approximately 0.80 λ. As shown in Fig. 2(c), when n_e and n_l are 1.38 and 1.33, respectively, the f and FWHM values of the left focus point are 9.6 µm and 287 nm, respectively. The f and FWHM of the right focus point are 13.2 µm and 396 nm, respectively. Compared with the hemisphere in the case of n_e = 1.38 and n_l = 1, a much larger f value is obtained, while the FWHM remains nearly constant. The focus intensity is also high (more than 70 times the incident light intensity I_p). Moreover, the effective lengths L are 3.0 µm (8.32 λ), 6.2 µm (17.10 λ), and 8.5 µm (23.37 λ) in Figs. 2(a)–(c), respectively. The length 23.37 λ of the PNJ is long compared with that of the previous reports.

To understand the contribution of light to the PNJ and reason for reduced FWHM of PNJ in Fig. 2(b), we performed a geometrical optics analysis. The light trajectories of the yoz plane in the suggested lenses are shown in Fig. 3, which is accomplished with Comsol software, and in which only the propagation beams that are always along positive direction of the optical axis are considered, which can approximately represent the true focusing properties of the hemisphere. The structural parameters of the lenses in Figs. 3(a) and (b) correspond to those in Figs. 2(a) and (b). The light paths comprise two main parts, selected as path A and path B for analysis. Regarding path A, the light refracts into the outer-hemisphere and then directly exits the hemisphere from the outer-hemisphere flat surface. For path B, the light refracts into the outer-hemisphere, then directly refracts into the inner-hemisphere, and finally exits the hemisphere from the inner-hemisphere flat surface. In addition, there is a specific light path that passes points C’ and C. The critical point C is located at the interface of the outer-hemisphere, inner-hemisphere, and hemispherical flat surface. When n_e and n_l are both equal to 1, the heights (along the y-axis, coordinate value) of points C and C’ are 2800 nm and 3920 nm, respectively. The focus position (FP) of the light at different heights from path A is located between 136 and 2772 nm, not dwelling among the PNJ in Fig. 2(a). Therefore, the main contribution to the PNJ shown in Fig. 2(a) is from path B and multiple light rays are refracted/reflected in the inner- and outer-hemisphere. When n_e is increased to 1.38, the heights of points C and C’ were 2800 nm and 3165 nm. More light passes via path A and the FP of the light at different heights from path A is between 3.0 µm and 11.9 µm, while the FP of the light at different heights from path B is on the negative z-axis. Therefore, the main contribution to the PNJ in Fig. 2(b) may have been from path A. When n_e and n_l are 1.38 and 1.33, respectively, the heights of C and C’ remains as 2800 nm and 3165 nm, respectively. The FP of the light at different heights from path A lengthens to 5.6–16.0 µm, while the FP of the light at different heights from path B remains on the negative z-axis. Thus, the main contribution to the PNJ of Fig. 2(c) may have been from path A. In fact, when n_e increases to 1.21, the outgoing light beam, which passe through edge of inner-hemisphere flat surface, already starts to diverge. With further increasing of n_e, more outgoing light beam, from edge to center of inner-hemisphere flat surface, gradually gets divergent. While n_e increases to 1.36, the propagating beam near the optical axis also gets divergent. In addition, compared with the hemisphere corresponding to Fig. 2(a), the hemisphere corresponding to Fig. 2(b) is comparable to a focusing lens with high numerical aperture (NA) and embeds in higher RI film, which both could narrow the beam waist when the WD is not long. Therefore, compared with Fig. 2(a), smaller FWHM of PNJ is obtained in Fig. 2(b). As value of n_l will not fundamentally change the convergence or divergence of the outgoing light, ray tracing for n_l of 1.33 are not presented.

 

Fig. 3. Main light paths of illumination passing through the hemisphere: (a) n_e = 1 and n_l = 1; (b) n_e = 1.38 and n_l = 1. R_c, R_s, n_c, and n_s are 2.8 µm, 4.5 µm, 1.44, and 1.57, respectively.

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2.2 Focusing properties of the immersed-engineered single-layer hemispherical lens

Based on the analysis in last paragraph of Section 2.1, we believe that the working of an immersed-engineered single-layer hemisphere (an immersed hemisphere, whose spherical area is partly cut off along a flat parallel to the hemisphere flat surface), because of converged light beam only passing through the remaining spherical surface of the single-layer hemisphere, is similar to that of the embedded two-layer hemisphere. The side view of the engineered single-layer hemisphere embedded in a transparent film is shown on the left of Fig. 4. To compare the focusing properties with those of the embedded two-layer hemisphere, the radius and RI of the engineered single-layer hemisphere are 4.5 µm and 1.57, respectively, and n_e is 1.38. The cutting plane radius is given by H, and the thickness of the remaining hemisphere is H_s. The embedded film and RI of the surrounding environment are n_e and n_l, respectively. The intensity profiles along the z-direction and yoz plane of the PNJs for different n_e and n_l values are presented in Fig. 4.

 

Fig. 4. Schematic of engineered single-layer dielectric hemisphere, illuminated by a plane wave, and key parameters of the PNJ.

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Figure 5(a) shows a single-layer hemisphere without cutting, and with f and FWHM values of 10.5 µm and 512 nm, respectively. Figure 5(b) shows the result obtained from the hemispherical lens, whose cut height H is equal to the height of C’ in Figs. 2(b) and (c). As shown in Fig. 5(b), when n_e and n_l were 1.38 and 1, respectively, the f and FWHM values of the focus point are 8.7 µm and 374 nm, respectively. Compared with the single-layer hemisphere in Fig. 5(a), the engineered single-layer hemisphere exhibits a narrowed PNJ waist, while the WD remains long. In Fig. 5(c), (H) decreases to 2500 nm. Compared with the lens corresponding to Fig. 5(b), more light is focused among the lens corresponding to Fig. 5(c), and a slightly higher peak intensity and longer L are obtained. The f and FWHM values in Fig. 5(c) are 8.3 µm and 354 nm, respectively. Additionally, the values of L are 7.0 µm (19.22 λ), 5.1 µm (13.98 λ), and 7.8 µm (21.31 λ) in Figs. 5(a)–(c), respectively. The intensity was between 79 and 110. As the engineered single-layer dielectric hemisphere exhibits a higher quality PNJ with high intensity, and the f shown in Fig. 5(b) is between the two focuses in Fig. 2(b), the intensity profiles along the z-direction in Fig. 5(b) support the above conclusion regarding the contribution of light to the PNJ.

 

Fig. 5. Intensity profiles along the z-direction and yoz plane of PNJs at different cut heights (H) of single-layer hemisphere: (a) H = 0 nm; (b) H = 3165 nm; (c) H = 2.5 µm. R_s, n_s, and n_e are 4.5 µm, 1.57, and 1.38, respectively.

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2.3 Illumination wavelength compatibility of immersed-engineered single-layer hemispherical lens

Moreover, the engineered hemisphere is expected to allow a tolerance for minor variations in the illumination wavelength. To determine the characteristic, illuminated wavelengths of 400, 365, and 325 nm were selected to continue the numerical study. The cut height H of the single-layer hemisphere was 2.5 µm, and the values of Rs, n_s, n_e, and n_l of the single-layer hemisphere were 4.5 µm, 1.57, 1.38, and 1.33, respectively. Figure 6(a) shows the intensity profiles along the z-direction and yoz plane of the PNJs obtained from the illuminated single-layer engineered hemisphere embedded in a film. The illumination wavelength was 400 nm and only one main peak appeared among the PNJ in Fig. 6(a). The values of f and FWHM are 10.9 µm and 356 nm, respectively, which are good results. Further, the value of L is 12.7 µm (31.87 λ), which is long enough to carry out some near-field parallel lithography and imaging applications with the hemispherical lens. Compared with Fig. 6(a), Fig. 6(b) shows the results under smaller λ, 365 nm, in which the f and FWHM values are 11.4 µm and 343 nm, respectively, and L is 9.6 µm (26.29 λ). Figure 6(c) shows the results under an illumination wavelength of 325 nm. The f and FWHM values are 11.5 µm and 322 nm, respectively, and L is 6.6 µm (20.19 λ). Under illumination of these three wavelengths, f varied slightly, while the FWHM remains close to the illumination wavelength. This indicates that the proposed hemisphere can work at several adjacent wavelengths, which is not possible for single-layer or multilayer micro-spherical lenses.

 

Fig. 6. Intensity profiles along the z-direction and yoz plane of PNJs at different illumination wavelengths (λ) of single-layer hemisphere: (a) λ = 400 nm; (b) λ = 365 nm; (c) λ = 325 nm. The cut height H is 2500 nm. R_s, n_s, n_e, and n_l are 4.5 µm, 1.57, 1.38, and 1.33, respectively.

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Furthermore, it is easy to find that WD of the design hemisphere is sensitive to environment RI (n_l). Therefore, when the hemisphere is put in environment to be tested, the RI of environment (n_l) can be known indirectly by measuring the WD of the hemisphere. For example, when n_l is 1, WD is about 8304 nm (Fig. 5(c)). When increases to 1.33, WD increases to 11434 nm (Fig. 6(b)). The rough RI sensitivity is 9485 nm/RIU (refractive index unit), which is a nice value and much better than most of RI sensor based on PNJ and plasmonic RI sensor [39,40].

2.4 Photolithography performance of suggested hemispherical lens

Finally, photolithography resolution of the suggested hemisphere (selected to correspond to that of Fig. 6(c) as an example) are evaluated via simulation. Illumination intensity values of the 325 nm laser is set as 3.16, 2.24, 2.0, and 1.92 V/m (Suppose the exposure time is 1 second, thus the corresponding exposure dose is 1.0, 0.5, 0.4, and 0.37 mJ/cm², respectively), respectively. If AR-P 3170 photoresist (AR-P 3170 photoresist has a photosensitive threshold of 40 mJ/cm².) film is positioned at the PNJ center, the relationship between the exposure dose and spot size (i.e., the PNJ peak, where the exposure dose is higher than the photosensitive threshold of the photoresist) can be obtained. As shown in Fig. 7, under the illumination intensities of 3.16, 2.24, 2.0, and 1.92 V/m, the doses of the PNJ center are found to be higher than 40 mJ/cm². The obtained spot sizes are 404, 263, 187, and 153 nm, respectively, and the obtained depths (D) of the exposed spots are 6.8, 3.4, 1.6, and 1.0 µm, corresponding to the illumination intensities of the 325 nm laser of 3.16, 2.24, 2.0, and 1.92 V/m, which are ideal for high-resolution photolithography. Owing to the large height-width ratio of the FWHM in the PNJ center, under the illumination intensity of 1.92 V/m (the corresponding exposure dose is 0.37 mJ/cm²), the spot size could be reduced to below 0.5 λ, while the D value of the spot is not very small, 1.0 µm. This indicates that the proposed hemisphere is promising for sub-half wavelength photolithography.

 

Fig. 7. Relationship of exposure dose and spot size: (a) Exposure dose values of 1, 0.5, 0.4, and 0.37 mJ/cm²; (b) Depth D of the spot under the exposure dose of 0.37 mJ/cm². The structural parameters and illumination are consistent with those in Fig. 6(c).

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Of course, the proposed hemispheres can be experimentally fabricated referring to some of the steps for multi-layer hemispherical shell in Ref. [41]. The two-layer hemisphere can be made as follows: After spin-coating photoresist (resistant to acid and alkali corrosion) film on a quartz substrate (1), uniform round holes is prepared on the photoresist film via microsphere photolithography (2). Isotropic wet etching in buffered oxide etch makes hemisphere grooves in quartz (3). After photoresist film is removed (4), polymer film with hemisphere lens arrays is fabricated via spin-coating (5). After quartz is removed via wet etching (6), outer-hemispheres are fabricated using deposition of outer-hemisphere material (7). Solid immersion material is deposited and flat layer part of the polymer is removed by dry etching (8). The suggested immersed-engineered single-layer hemisphere can be fabricated thorough referring to the fabrication process flow of the two-layer hemisphere, in which a polymer substrate, with silica film with thickness of H_s on it, is selected as the substrate, and step (7) is redundant.

3. Conclusion

In this work, the PNJs generated by a two-layer hemispherical lens embedded in a transparent film (with low-high-low RI for each layer) is demonstrated with an extended focus, narrowed beam waist, and extremely long length. Through geometrical optics analysis, it is concluded that the improved PNJ is attributed to the convergence of part of incident light beams, which pass through the spherical fringe area of the outer-hemisphere, then refract into and pass through the outer-hemisphere, and finally directly exit the hemisphere from the outer-hemisphere flat surface. As the inner-hemisphere contributed less, the two-layer hemisphere can be regarded as an immersed focusing lens with high NA. Thus ultralong WD and narrowed beam are possible for it. An engineered single-layer hemisphere embedded in a film is demonstrated to own the same focusing characteristic as the two-layer hemisphere embedded in a film. Moreover, the proposed hemisphere shows compatibility with the adjacent laser wavelengths, which is not possible for single-layer or multilayer micro-spherical lenses. Additionally, spot width less than half wavelength is realized via a numerical calculation method to evaluate the photolithography performance of the suggested hemisphere. Last but not least, the proposed hemisphere shows good RI sensing performance due to their WD sensitive to the environment (n_l). Therefore, it is concluded that these two kinds of hemispheres, with low cost, have great potential for application in far-field flexible parallel lithography for any surface pattern arrays with half-wavelength line width, and RI sensing.