Enhanced magnification factors in super-resolution imaging using stacked dual microspheres

The enhanced super-resolution imaging by using SDMs is first experimentally demonstrated. Figure 2(a) shows an image taken with the inverted optical microscopy without any microsphere. No features can be observed in the image. That is because the stripes and grooves in the sample surface is smaller than the imaging resolution of 0.61λ/NA ≈ 523 nm [40] with the applied 60× objective. After a primary microsphere with a diameter of 20.0 μm was used, the DVD tracks can be clearly observed (figure 2(b)). Furthermore, the secondary microsphere with a diameter of 72.0 μm was placed under the primary one. A further magnified image with well-recognized DVD tracks was obtained (figure 2(c)). This indicates that the SDMs can enhance super-resolution imaging.

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What is the mechanism of the enhanced magnification factor in SDM imaging? A ray tracing diagram of the SDM imaging is illustrated in figure 2(d). From the diagram, one can see that the primary microsphere first forms a virtual image, which corresponds to the image obtained in figure 2(b). This is consistent with previous single sphere super-resolution imaging [11, 15, 16]. Once the secondary sphere is applied, a real image can be obtained.

According to the Guassian lens equation [41], we have the following two equations valid:

$$\begin{equation}\frac{1}{{{l_1}}} = \frac{1}{{{f_1}}} - \frac{1}{{0.5{D_1}}}\end{equation} \tag{ 1 }$$

$$\begin{equation}\frac{1}{{{l_2}}} = \frac{1}{{{f_2}}} - \frac{1}{{0.5({D_1} + {D_2}) + d - {l_1}}}\end{equation} \tag{ 2 }$$

where l₁ and l₂ are the image distances, f₁ and f₂ the focal lengths, D₁ and D₂ the diameters of the primary and secondary microspheres, respectively, and d is the separation distance between two microspheres. Moreover, the magnification factor M for the primary and secondary microspheres can be given as:

$$\begin{equation}{M_1} = - \frac{{{l_1}}}{{0.5{D_1}}},\end{equation} \tag{ 3 }$$

and

$$\begin{equation}{M_2} = \frac{{{l_2}}}{{0.5({D_1} + {D_2}) + d - {l_1}}}.\end{equation} \tag{ 4 }$$

The overall magnification factor of the SDM imaging can then be given as

$$\begin{equation}M = {M_1} \times {M_2}.\end{equation} \tag{ 5 }$$

In equations (3)–(5), the focal lengths f₁ and f₂ are needed. When the size of an object is much smaller than that of microsphere radius, paraxial approximation [42] can be applied and the focal length f can be estimated as f ≈ nD/4(n−1), where n is the relative refractive index of the microsphere. However, in microsphere-assisted super-resolution imaging, the focal length of microspheres is supposed to be shorter because of the coupling of evanescent waves [43]. Figure 2(e) shows the light intensity distribution of a photonic nanojet. The position at which the light intensity reaches its maximum value is taken as the focal point. The distance between the center of the microsphere and the focal point can then be taken as the focal distance f in the FDTD simulation. A series of simulation was conducted for silica microshpheres with different diameters. The obtained focal length is shown in figure 2(f). One can see that f obtained in the FDTD simulation is smaller than that obtained through paraxial approximation. By linearly fitting the obtained simulation results with respect to D, f as a function of D of silica microspheres can be estimated as:

$$\begin{equation}f = 0.75D - 1.63.\end{equation} \tag{ 6 }$$

The above obtained relationship is applied to determine the focal lengths for the silica microspheres.

According to equations (1)–(6), one can see that M in SDM imaging is directly related to D₁, D₂, and d. In this section, the influence of the three impact factors on M in SDM imaging is investigated. First, primary microspheres with three different diameters of 28 μm, 49 μm, and 71 μm were used for the same secondary microsphere of D₂= 112 μm. Three sample images are shown in figures 3(a)–(c). It clearly shows that the magnification factor decreases with increasing D₁ (figure 3(d)). Similarly, experiments were conducted with a fixed primary microsphere of D₁= 39 μm and changing D₂. The results are shown in figures 3(e)–(g) for D₂= 51 μm, 71 μm, and 121 μm, respectively. The magnification factor M increases with increasing D₂. Moreover, the obtained values of M in these two series of experiments are compared with the theoretically calculated values, as shown in figures 3(d) and (h). In the figures, the experimentally measured values agree well with the theoretically predicted ones, as shown in the red solid curves.

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Furthermore, the impact of the separation distance d between the two microspheres on magnification factor is investigated. In the experiment, the diameters of the primary and the secondary microspheres are 16.0 μm and 51.0 μm, respectively. The magnified images are shown in figures 4(a)–(d) for increasing separation distance d of 0 μm, 5 μm, 10 μm, and 15 μm, respectively. It clearly shows that M decreases with increasing d. Once again, the experimentally obtained M agrees well with that of the theoretically predicted results (figure 4(e)).

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In SDM imaging, whether or not the secondary microsphere helps to enhance the overall magnification M compared to the single microsphere imaging is strongly related to the combination of D₁ and D₂. Here the secondary magnification factor with different combinations of D₁ and D₂ is determined. To do so, the magnification factor M₁ in single microsphere imaging with different values of D₁ was first obtained, as shown in figure 5(a). One can see that the experimentally measured value M₁ decreases with increasing D₁, as predicted by equation (3) (red solid curve in figure 5(a)).

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For each single primary microsphere, the colloidal probes with different sizes of microspheres were applied to implement SDM imaging. With the acquired images, M was obtained for each combination of D₁ and D₂. The secondary magnification factor M₂ can be achieved as M₂ = M/M₁. The obtained M₂ as a function of D₁ and D₂ is shown in figure 5(b). The transparent surface in the figure corresponds to the theoretically estimated M₂ as a function of D₁ and D₂ when d = 0μm. From the figure, one can see that M₂ increases with decreasing D₁ and increasing D₂. Therefore, a combination of a smaller primary microsphere and a larger secondary microsphere is favorable to achieve an enhanced M. Otherwise, a decreased M may be obtained. Moreover, one can see that when the secondary magnification factor M₂ is larger than 1, one can obtain the enhanced overall magnification compared to that of the single microsphere imaging, regardless of the magnification factor of the applied objective lenses in optical microscopes. If an objective lens with a higher magnification factor is applied, a further improved overall imaging magnification will be achieved.

The size of FOV is also an import factor in super-resolution imaging. The FOV for microsphere based super-resolution imaging is defined as the diameter of the circular area in which distinguishable features with no clear distortions can be obtained in the acquired images. The size of FOV in single microsphere and SDM imaging are presented in figure 6(a). In the figure, the green star points are measured FOV in single microsphere imaging. The data points of all other symbols correspond to the measured FOV in SDM imaging with different values of D₂. One can see that the FOV in single sphere imaging increases with D₁, which is consistent with that reported by others [23, 44]. Assuming super-resolution imaging can only be achieved when the distance between sample surface and the glass sphere is within one wavelength, the area in which the vertical distance in between the sample surface to the microspheres is less than one wavelength increases with microsphere diameters. As a result, FOV increases with D₁.

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From the above result, one can see that the SDM imaging enlarges FOV, especially for smaller D₁. Moreover, a larger value of D₂ exhibits an increased FOV for a given magnitude M, as shown in figure 6(b). For example, when M is equal to 2, the FOV is about 6 μm for D₂= 51 μm. However, FOV is doubled when D₂ increases to 112 μm.

Although a much enhanced M has been demonstrated in SDM imaging, readers have to note that the image quality, as well as imaging resolution for images presented in this paper is not optimized due to the limited experimental resources in our laboratory. We compared the imaging resolution of SDM with that of single microsphere imaging (data not shown) and found that there the resolution has not been significantly improved. As a result, this paper we mostly focus on the magnification factor and FOV. The image quality and imaging resolution can be optimized through the following two approaches. First, an upright optical microscope can be used for SDM imaging. Here we used an inverted optical microscope. Since the DVD sample is partially transparent, the sample cannot be well illuminated. This causes reduced image quality. Second, microspheres with higher refractive index can be used in SDM imaging, which will present improved imaging resolution, as studied somewhere else [25, 31].