1. Introduction

Microsphere-assisted super-resolution imaging shows a huge potential application in nanostructure imaging [1–5]. The technique can break Abbe’s diffraction limit and has the advantages of real-time dynamic imaging under white light illumination [6,7]. The super-resolution imaging properties of dielectric microspheres lenses are widely explained by the super-focusing effect of photonic nanojets [8–14]. A photonic nanojet is generated at the shadow-side surface of a dielectric microsphere upon illumination. It has a waist smaller than the traditional diffraction limit and can extend over several optical wavelength ranges. In microsphere imaging experiments, some observed phenomena were related to the photonic nanojet effects. Studies showed that the distance between the microsphere and the position of maximum electric intensity determined the super-resolution imaging ability of microspheres [15]. Yang reported that the resolution of a microsphere depended on the waist of its photonic nanojet, and the best resolution could be obtained when its photonic nanojet had a minimum waist [16]. Previous studies demonstrated that magnified images could be observed over a range of focal image positions (RFIP) [17, 18]. Lai et al. reported that there would be a RFIP for microsphere because more than one focus position could get clear images [19]. The RFIP is a distance that the objective lens can capture clear images in that range. A long RFIP means a longer working distance, which will make it easy to focus the images. Yang et al. pointed out first that the virtual image could be observed over a focal depth of several microns. As the waist of a photonic nanojet could also maintain beyond the diffraction limit for several optical wavelengths, and such behavior was in line with the property of a “photonic nanojet” [20]. However, there still lacks a systematic study on how the length of the photonic nanojet and the RFIP are related. In this paper, we use BaTiO₃ glass (BTG) microspheres of different sizes to image Blu-ray disc (BD) samples and hexagonally close-packed (hcp) 960-nm-diameter polystyrene (PS) microsphere arrays. Our experimental results show that the focal image position of BDs ranges from 4 to 25 μm when the diameter of microspheres increases from 5 to 50 μm. The RFIP of hcp PS microsphere arrays also increases as the BTG microsphere size increases. Moreover, the Talbot effect [21, 22] is also observed for the hcp PS microsphere array samples, and the Talbot distance relates to the size of microspheres. The Talbot distance is about 3.5 and 6 μm for BTG microspheres with a diameter of 10 and 20 μm, respectively. Numerical simulations demonstrate that the length of photonic nanojets of BTG microspheres depends on the diameter of BTG microspheres. The theoretical RFIP calculated by using the position of a photonic nanojet as a series of focal lengths match well the experimental ones. We propose that the RFIP is mainly controlled by the length of photonic nanojets. As the image magnification of BTG microspheres reaches the maximal values in the diameter range of 6-10 µm [23], the research results will be useful in the design of a high-resolution imaging system with a longer working distance. Moreover, the Talbot effect observed in microsphere imaging will have potential applications in three-dimensional nanofabrication.

2. Experimental

Figure 1(a) illustrates the schematic of the experimental setup. In our experiments, a Leica microscope (DM 2500 M) was used in reflected illumination (centered at 540 nm)mode and equipped with a CCD camera to record results. Samples were observed by a microscope objective (50 × NA = 0.75). We used the same equipment and parameters when observing the RFIP. In this experiment, two types of samples were used. The first type was a commercial BD with its protection layer peeled off before use. The second one was a single-layered hcp PS microsphere array with a diameter of 960 nm, and a 20-nm-thick gold film was deposited onto the surface of the PS microsphere array by evaporation. The single-layered hcp PS microsphere array was deposited onto a clean glass substrate by a gravity-assisted convective self-assembly method [24]. BTG microspheres in the diameter range of 5–50 µm were then spread on the top of the samples. Then, a layer of polydimethylsiloxane (PDMS) was spin coated on the BTG microspheres and let the microspheres fully immersed in the PDMS layer [23, 25–27]. Finally, we used BTG microspheres of different sizes to image the two types of samples. The scanning electron microscope (SEM) images of the samples were taken by TEOL YSM-5610LV. Figure 1(b) is the SEM image of a BD. Figure 1(b) shows that the BD sample consists of 200 nm stripes separated by 100 nm grooves. Figure 1(c) shows the SEM image of a 960-nm-diameter PS microsphere array. The SEM image reveals that the PS microsphere array is hexagonally close-packed.

 

Fig. 1 (a) Schematic of the experimental setup; (b) SEM image of a BD; (c) SEM image of a single-layer hcp 960-nm-diameter PS microsphere array.

Download Full Size | PDF

3. Results and discussion

First, we use BTG microspheres of different sizes to image BD samples. Optical images of a BD using the diameter of 10 and 25 μm BTG microspheres at various focal image positions are shown in Figs. 2(a) and 2(b) respectively. The optical magnifications at these focal imagepositions are also measured. When the focal plane of the microscope moves, the image is visible over a range of focal image positions. During this range, the image changes from blur to clear and then to blur again. The RFIP is defined as the distance of the focal image positions between the first blur image and the last blur image observed. For the first and last blur images, the observed object can be still resolved. In our experiments, to reduce the measurement error, each experimental RFIP is the average of 15-20 measured data from the samples of the same parameters. Figure 2(a) reveals that the RFIP observed through the 10-μm-diameter BTG microspheres is about 6 μm. The magnifications of the 10-μm-diameter microsphere are between 3.2 × and 5.0 × . The clearest image is observed at the focal image position 11 μm below the center of the BTG microsphere [Fig. 2(a-3)], with a magnification of about 3.9 × . The RFIP of the 25-μm-diameter BTG microspheres is about 17 μm, as shown in Fig. 2(b). The magnifications are between 3.3 × and 5.0 × . The clearest image is observed at the focal image position 27 μm below the center of the BTG microsphere [Fig. 2(b-3)], and the magnification is about 3.8 × . Using BD samples as objects, we also experimentally study the RFIP and clearest image position (CIP) of other size BTG microspheres. As Allen et al. have pointed out that imaging of BD through microspheres might be different from imaging of stand-alone objects [28], 300-nm-diameter microsphere arrays with lots of defects are also observed in our experiments. The RFIP observed through the 10-µm-diameter and 20-µm-diameter BTG microspheres is about 6 and 14 µm, respectively, very close to the data of BD samples. This means that the ordering of the Blu-ray disc or the microsphere array does not affect the measured RFIPs in our experiments. The measured RFIP and CIP as a function of the diameters of BTG microspheres are summarized in Figs. 3(a) and 3(b), respectively. Figure 3 reveals that both the RFIP and CIP increase linearly as the diameter of microspheres increases. When the diameter of microspheres increases from 5 to 50 μm, the RFIP increases from 4 to 25 μm, while the CIP increases from 7 to 50 μm.

 

Fig. 2 Images of the BD observed through BTG microspheres with different diameters at various optical focal image positions: (a) 10-μm-diameter BTG microspheres; (b) 25-μm-diameter BTG microspheres. The BTG microspheres are fully immersed a PDMS layer, and the z numbers on the right corner of the figures are the relative focal image positions below the center of BTG microspheres. The position of the center of BTG microspheres is set as z~0. The scale bar is 5 μm.

Download Full Size | PDF

 

Fig. 3 (a) The experimental (red curve) and calculated (blue curve) RFIP of the BD observed through different size BTG microspheres; (b) The experimental (red curve) and calculated (blue curve) CIP of the BD observed through different size BTG microspheres.

Download Full Size | PDF

Then, we use BTG microspheres of different sizes to image a single-layered 960-nm-diameter hcp PS microsphere array. Optical images of hcp PS microsphere array observed through the 10 and 20 μm in diameter BTG microspheres at various focal image positions are shown in Figs. 4(a) and 4(b), respectively. Figures 4(a) and 4(b) show that the RFIP of hcp PS microsphere array observed through the BTG microsphere of 10-μm-diameter and 20-μm-diameter is about 20.5 and 32 μm, respectively. Figure 4 reveals that the RFIP of hcp PS microsphere array also depends on the diameter of BTG microspheres, and the range increases as the diameter of BTG microsphere increases. Moreover, the Talbot images of PS microsphere arrays are also observed in experiments. The Talbot effect is the self-imaging properties of periodic structures, such as a periodic hole array [1]. The 0, 1/2, 1, … Talbot images appear beyond the BTG microsphere. For the 1/2 and 3/2 Talbot images, the intensity of the gap between two adjacent microspheres is greater than that of microspheres. For the 0 and 1 Talbot images, the intensity of the gap between two adjacent microspheres is smaller than that of microspheres. We find the Talbot distance and the position of Talbot planes also depend on the size of BTG microspheres. For a 10-μm-diameter BTG microsphere, these Talbot planes are near the center of the microspheres [Figs. 4(a-1)-4(a-6)], and the Talbot distance is about 3.5 μm. For BTG microspheres with a larger diameter of 20 μm, these Talbot planes are located far below the center of the microspheres [Figs. 4(b-2)-4(b-7)], and the Talbot distance is about 6 μm. For BTG microspheres of other sizes, the Talbot distance also increases as the size of BTG microspheres increases. Moreover, we also use the optical microscope to observe the single-layered hcp 960-nm-diameter PS microsphere array, with no BTG microspheres deposited on top of the PS microsphere array. Because the PS microsphere array is a two-dimensional periodic hcp array, a series of Talbot planes appear above the PS microsphere array, as shown in Fig. 4(c). Figures 4(c-3), 4(c-2), and 4(c-1) are the 0, 1/2, and 1 Talbot planes, respectively. We can see that the Talbot distance is about 2.5 μm. For a hexagonal array, the theoretical Talbot distance is $z_T=3{\Delta }^2/2\lambda =2.56\text{}\text{μm}$ (Δ = 960 nm, λ = 540 nm) [29]. Moreover, Talbot images from 960-nm-diameter PS microsphere array are also modeled by using a finite difference time domain (FDTD) method. The incident light is a plane wave with a wavelength of 540 nm polarized along the x-direction. The simulated result clearly indicate that the microsphere array is self-imaged and repeats at a distance of 2.7 μm, very close to the experimental Talbot distance of 2.5 μm, and the calculated one of 2.56 μm. After a BTG microsphere is deposited on the PS microspheres array, the Talbot distance and the positions of Talbot planes change. The Talbot effect is not observed in BD samples. This is because the period of a BD is only 300 nm, and the Talbot images are difficult to be observed.

 

Fig. 4 Images of the single-layered hcp 960-nm-diameter PS microsphere arrays observed through different size BTG microspheres at various focal image positions: (a) 10-μm-diameter BTG microspheres; (b) 20-μm-diameter BTG microspheres; (c) Images of the single-layered hcp PS microsphere arrays with no BTG microspheres deposited onto the PS array. The z numbers on the right corner of the figures are the relative focal image positions below (positive) or above (negative) the center of BTG microspheres. The position of the center of BTG microspheres is set as z~0. In Figs. 4(a) and 4(b), the scale bar is 5 μm; while the scale bar is 2 μm in Fig. 4(c).

Download Full Size | PDF

Finally, to investigate whether the RFIP is related to the length of photonic nanojets, the photonic nanojet of different size BTG microspheres is numerically simulated using a FDTD method. The electric field intensity |E|² of the photonic nanojet of microspheres in optical axis direction is plotted in Fig. 5(a). Figure 5(a) indicates that as the diameter of microspheres increases, the |E|² increases relatively, and the position of the maximum |E|² is gradually away from the center of microspheres. The length of the photonic nanojet as a function of the diameter is shown in Fig. 5(b). The photonic nanojet length is defined in the upper left corner (from the maximum intensity of the |E|² point along the optical axis to the |E|² ̸ e² point) [30]. Figure 5(b) reveals that as the diameter of the microspheres increases from 5 to 50 μm, the photonic nanojet length increases from 2.9 to 7.1 μm. According to the lens equation, $1/f=1/s+1/s\text{'}$, where $s$ and $s\text{'}$ is the distance between the object and the microsphere lens, and between the microsphere lens and the image, respectively, $f$ is the focal length of the microsphere lens, a $f$ will result in a $s\text{'}$. Lee et al. pointed out that the focal length of a nanolens could be obtained from electromagnetic wave simulation [31]. If we treat the positions of a photonic nanojet as a series of focal lengths ($f$), we can get a series of $s\text{'}$, and then the RFIP. Therefore, we can get image at a longer RFIP when the photonic nanojet is longer.

 

Fig. 5 (a) The electric field intensity |E|² of different size BTG microspheres along the vertical z axis at the center of the photonic nanojet; (b) The length of the photonic nanojet (LPNJ) as a function of the diameter of BTG microspheres. Schematic of the photonic nanojet length is shown in the upper left corner of Fig. 5(b).

Download Full Size | PDF

The calculated RFIPs as a function of BTG microsphere size are also shown in Fig. 3(a). Figure 3(a) shows that the experimental results fit the calculated values, which indicates that the RFIP of microspheres relate to the length of photonic nanojets. Then, if we use the position with the strongest electric field intensity in a photonic nanojet as the focal length, we can obtain an image position. The calculated image positions as a function of BTG microsphere size are also plotted in Fig. 3(b). The tendency of the two curves is similar, while the calculated image positions are larger than the observed CIPs. On the other hand, the image positions and magnifications estimated using geometrical optics approximation as a function of the BTG microsphere size are also calculated [32]. Although the calculated magnifications do not match with the experimental ones, we find that these calculated image positions fit well with the experimental CIPs. This is because the appearance of photonic nanojets of microspheres needs a wave explanation, and the position of a photonic nanojet with the strongest electric field intensity is shorter than the calculated focal length using ray description. However, our studies reveal that the length of a photonic nanojet and the RFIP are related, and a BTG microsphere with a longer length of photonic nanojet has a longer RFIP. As the focusing of a collimated beam and imaging properties of point sources by microspheres are not directly related [33], further studies are needed to explore this connection. As the scale unit of the microscopy is 1 µm, the measurement error in RFIP is at least 2 µm. In Fig. 3(a), the difference between the theoretical and experimental data is within 2 µm, so there is no physical reason for the crossing point at about 20 µm. For the periodic hcp microsphere array, Talbot images appear above the PS microsphere array upon illumination. After the BTG microspheres are deposited, the BTG microspheres will image both the periodic microsphere array and its Talbot images. So, for the same size BTG microsphere, the whole RFIP of periodic microsphere array samples is larger than that of the BD samples. If we remove the Talbot images from the whole RFIP, the RFIP of hcp PS microsphere array observed through the BTG microsphere of 10-μm-diameter and 20-μm-diameter is about 10 and 15 μm, respectively. The results are close to the experimental results of BD samples in Fig. 3. Moreover, the focal length of a 10-μm-diameter BTG microsphere is small, so the image is near the center of the microsphere. Its length of photonic nanojet is short, the range of focal Talbot image is short, so the Talbot distance is small. Because BTG microspheres of larger size have a longer photonic nanojet, the range of focal Talbot image is longer, so the Talbot distance is larger. A larger size BTG microsphere has a longer focal length, which will result in the best image plane away from the center of the microsphere, so the image observed through the 20-μm-diameter BTG microsphere is below the center of the microsphere.

4. Conclusions

We investigate the effect of the microsphere size on the RFIP. The results show that larger size microspheres have larger RFIP, and the best image plane is far from the center of microsphere. The length of photonic nanojet of microsphere also depends on the size of microspheres, and it plays a critical role in the RFIP. Our research results further reveal the super-resolution imaging mechanism of microspheres.