1. Introduction

Breaking through the Abbe diffraction limit become a persistently hot topic in modern optics [1–3]. In 2004, Chen et al. find that the transparent dielectric microsphere with wavelength scale radius can generate a photonic nanojet (PNJ) [4]. The key properties of the PNJ are that it is a non-evanescent and propagating beam with smallest lateral full-width at half-maximum (FWHM) smaller than the Abbe diffraction limit $\lambda /2$ and can propagate over $\sim 2\lambda$ with low divergence [5], where $\lambda$ is the wavelength in vacuum. Moreover, most of authors think that the PNJ is not the near-field focal spot and can extend to the outer near-field region of the microsphere [6]. Since 2004, many authors [5–13] investigate both in theoretically and experimentally the generating mechanism and the applications of the PNJ and the effects of the physical parameters of microsphere on the PNJ. For the field of the imaging, detecting, and lithography with nanoscale resolution, it is the most important to achieve the smallest focal spot size, i.e., the highest resolution. However, the PNJ can appear for a wide range of radii of the microsphere if the refractive index contrast ($RIC$) between the microsphere and its surrounding medium is less than about 2:1, and it is easy to optimize the length of the PNJ with respect to the radius of the microsphere and the $RIC$, but most of literatures [4–16] do not clearly indicate what is the highest lateral resolution with the microsphere and what is the optimal parameters of the microsphere to realize the highest lateral resolution. In this paper, the lateral resolution of the microsphere (i.e., equivalent to the optical system) is defined by the smallest FWHM of the near field focal spot [17] or the PNJ. Heifetz et. al [5] claim that the smallest FWHM of the PNJ is as small as $\sim \lambda /3$, but the basis seem to be Fig. 1(a) of Ref. 7, where the smallest FWHM of the PNJ is 130nm for the microsphere with $\lambda \text{=}400nm$, refractive index $n_s\text{=}1.59$, and radius $r\text{=}500nm$. Actually, the spot shown in the Fig. 1(a) of Ref. 7, corresponding to the $\sim \lambda /3$ FWHM, is the near-field focal spot instead of the PNJ. Most of authors hence majorly focus how to generate the PNJ with smallest FWHM beyond the Abbe diffraction limit $\lambda /2$. Fletcher et. al [16] study the focusing of the micrsolens with wavelength scale size, but they only demonstrate that the vector diffraction theory of Richards and Wolf [18] overpredicts the focal spot size in the microlens with wavelength scale size. Recently, the microsphere is used in nanolithography [14, 15] and nanoscale imaging [19, 20]. However, these authors [14, 15, 19, 20] do not tell us how to determine the actual values of the refractive index and the radius of the microsphere in the experiments. In addition, in the experiments [19, 20], all the samples are adjacent to the microsphere, so the focal spot of the microsphere should be the near-field focal spot instead of the PNJ. With the development of the micro/nano-fabrication technology, the solid immersion lens (SIL) of wavelength scale size (called the nSIL) is recently fabricated and its focusing abilities are validated in experiment [1, 21]. Further theoretical research demonstrates that the nSIL can generate a near-field focal spot with lateral resolution beyond $\lambda /2n_s$and has higher performance than the macroscopic SIL [22]. We think intuitively that the microsphere might have the similar function of the nSIL under the case of the plane wave illumination instead of the convergent wave required by the SIL.

 

Fig. 1 Schematic of a microsphere illuminated by a incident plane-wave.

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Therefore, the existing literatures on the focusing of the microsphere are quite abundant, but they do not resolve the problems what is the highest lateral resolution with the microsphere and what is the optimal parameters of the microsphere to realize the highest lateral resolution. The natural question arises whether we can achieve higher lateral resolution with the microsphere compared with the existing literatures and how to get it. These problems are very important for researchers to clearly understand the lateral resolution limit of the microsphere and to design the optimal refractive index and the radius of the microsphere under the case of various illumination wavelengths and applications. In order to address this concern, we abandon the complex Mie theory and only focus on the two basic problems, i.e., numerical aperture ($NA$) and the spherical aberration, of the microsphere. There are two reasons. The first reason is that $NA$ and the aberration are critical factors affecting the focusing properties of an optical system. To achieve maximum lateral resolution, $NA$should be as high as possible, whereas the aberration is inverse. The second reason is the well known fact that there is seriously spherical aberration for a single macroscopic sphere. For the microsphere with wavelength scale radius, whether is there spherical aberration? If it exist, how to reduce the spherical aberration? What is the conditions of the microsphere without spherical aberration?

In this paper, we restrict ourselves to investigation on the focusing properties of the microsphere in the near-field region. We focus on physically analyzing the origins of the $NA$ and the spherical aberration of the microsphere with wavelength scale radius. We demonstrate that the microsphere naturally has negligible spherical aberration and high $NA$ within a small range of $RIC$, which is helpful to determine the related parameters of the microsphere with the near-field focal spot of high lateral resolution. The reason of the negligible spherical aberration of the microsphere is due to the spherical aberration compensation arising from the positive spherical aberration caused by the surface shape of the microsphere and the $RIC$ and the negative spherical aberration caused by the focal shifts [23–25] due to the wavelength scale dimension of the microsphere. We show that, only within the approximate region of $1.5\le RIC\le 1.75$ with the proper radius of the microsphere, the microsphere can generate the near-field focal spot with lateral resolution slightly beyond $\lambda /2n_s$, which is also the lateral resolution limit of the dielectric microsphere. For simplicity, hereafter we call the microsphere with the near-field focal spot of lateral resolution beyond $\lambda /2n_s$ as the optimal microsphere (OMS). Compared with the nSIL, a prominent characteristic of the OMS is that the illuminating light is a plane wave instead of the convergent wave required by the SIL, which means that one can use the arrays of OMSs to image or detect sample in wide range. Moreover, the microspheres not only have comparable resolution with the nSIL, but also can avoid the fabrication difficulty faced by the nSIL because the microspheres with various radii and materials can be obtained commercially [26]. In addition, as the focusing of the microsphere of wavelength scale are strongly affected by the diffraction effects accompanying the scattering and by the interference of the waves transmitted through and refracted by a microsphere [6], it is needed to make a very delicate balance of the interferences by precisely designing the radius of the microsphere and the $RIC$. We expect that, different for the macroscopic SIL, although the microsphere is assumed to have same $n_s$ for different wavelengths of the illuminating lights, the radius of the OMS is wavelength dependent. In order to deal with this problem, we develop the transform formula used to calculate the new radius from the known radius of microsphere corresponding to the original illumination wavelength when the wavelength of the illuminating light is changed. The work in this paper is important for the high resolution imaging and nanolithography based on the microsphere.

2. Analyses on the spherical aberration and $NA$ of the microsphere

In this paper, the principle of the ray tracing and the definitions of the $NA$ and the spherical aberration are used. It is firstly needed to indicate that the positive values of ray tracing in the diffraction of the small circular aperture with radius $r>\lambda$ have recently been proved in theory [23, 24] and experiment [27]. Shown in Fig. 1, the $x$ polarized and monochromatic plane wave incidence on a microsphere with radius $r$, refractive index $n_s$, and center at the point $O$, along the $+z$ axis. $n_o$ is the refractive index of the surrounding medium. Namely $RIC=n_s/n_o$. Obeying the ray tracing process, a ray with incident angle ${\theta }_o$ intersects the microsphere and the $z$ axis at the points $A$, $B$, and $F$, respectively. ${\theta }_s$ and ${\theta }_i$ are the corresponding the refractive angles at the points $A$ and $B$, respectively. Here ${\theta }_i={\theta }_o$. $\alpha$ is the angle of the emergent ray with the $-z$ axis. $C$is the intersection point of the $z$ axis makes with the microsphere and $d$ is the distance from the point $C$ to the point $F$. It is noted that the polarization of the incident plane wave is not considered in the ray-optical analysis and is only considered in the later calculations basing on the vector Kirchhoff theory and the finite-difference time-domain (FDTD) method.

As we known, the low aberration and high $NA$ is the precondition of the optical system with high lateral resolution, whereas there is seriously spherical aberration for a single macroscopic sphere. Let's first examine the spherical aberration and $NA$ of the microsphere. Basing on the geometrical relations shown in Fig. 1, we can obtain the following formulae

When the microsphere is considered a thick lens, the point $F$ moves toward the microsphere with the increase of ${\theta }_o$. For all rays, ${\theta }_o$ meets $0\le {\theta }_o<\pi /2$. It is easy to derive that the point $F$ is outside the microsphere for $0\le {\theta }_o<2{\theta }_s$, i.e., $0\le {\theta }_o<2\text{a}\cos (n_s/2n_o)$ and inside the microsphere for $2\text{a}\cos (n_s/2n_o)<{\theta }_o<\pi /2$. As the focal spot outside the microsphere majorly arises from the focusing contributions of the rays with the incident angle $0\le {\theta }_o<2{\theta }_s$, we define the spherical aberration $S=d_{\max }-d_{\min }$, where, in terms of Eqs. (1) and (2), $d_{\min }\approx 0$ by setting ${\theta }_o=2\text{a}\cos (n_s/2n_o)-\delta$ with a very small value $\delta =0.00001$ in radians and $d_{\max }$ approximately calculated by setting ${\theta }_o=\delta$. $d_{\max }$ and $d_{\min }$ are the maximum and minimum values of $d$, respectively. For the microsphere, we define $NA=n_o\sin {\alpha }_{\max }$, where the maximum value ${\alpha }_{\max }$ of $\alpha$ can be calculated from the maximum value ${\theta }_{o\max }=2\text{a}\cos (n_s/2n_o)-\delta$.

As we known, $NA$ and the aberration are critical factors affecting the focusing properties of an optical system. To achieve maximum lateral resolution, $NA$should be as high as possible, whereas the aberration is inverse. In order to assure that $S$ has no significant effects on the focusing properties, the minimum requirement for $S$ is $S<\lambda /2n_o$. Shown in Fig. 2 , as the validity of the ray tracing is restricted in $r>\lambda$, the $RIC$ should be bigger than 1.5 for $S<\lambda /2n_o$. In this case, both $S$ and $NA$ decrease with the increase of $RIC$. A tradeoff between $S$ and $NA$ is needed. Moreover, due to ${\theta }_{o\max }\approx 2\text{a}\cos (n_s/2n_o)$, more rays will directly intersect with the $z$ axis at the inside of the microsphere with the increase of $RIC$. As the effective wavelength in the microsphere is decreased a factor of $1/n_s$, the spherical aberration among the points $F$ inside microsphere is bigger for a given $r$. So it is improper to choose too high $RIC$ when one wants to achieve imaging with high lateral resolution. By FDTD method, this qualitative analysis is also confirmed by our numerical calculations performed with the various combinations of $r$ and $RIC$ with the range from 1.76 to 2. When $RIC\le 1.75$, $NA\ge 0.85$ (see Fig. 2). Therefore, within the approximate range of $1.5\le RIC\le 1.75$, the microsphere has not only small $S$, but also high $NA$ by nature.

 

Fig. 2 Relations between the spherical aberration $S$ (blue solid curve), the numerical aperture NA (green dashed curve) and the refractive index contrast $RIC$ between the microsphere and its surrounding medium.

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In vector diffraction theories, researchers note that the diffraction focus is shifted the geometrical focus and moves toward the optical system with small Fresnel number [23–25], i.e., the focal shifts. The focal shifts only happen in the case of the small Fresnel number and cannot be predicted by the classical vector diffraction theory developed by Richards and Wolf [18] because of the Debye approximation used being only valid for the large Fresnel number that require the focus be many wavelengths away from the aperture [28]. Li [24] defines the optical system with radius larger than one and smaller than ten wavelengths as a nonconventional optical system, where the definition of the Fresnel number is $N=r/{\lambda }_{a}N{A}_a$ and the small circular aperture with the numerical aperture $N{A}_a$ and radius $r$ is investigated. So, the microsphere is the nonconventional optical system with small $N$. It is noted that, because of the difference between the small aperture and the microsphere, $N{A}_a$ of the small circular aperture is different from the $NA$ of the microsphere defined before. ${\lambda }_a$ is the wavelength $\lambda$ in vacuum for the small aperture and is majorly determined by the wavelength in the microsphere (i.e., ${\lambda }_a=\lambda /n_s$) for the microsphere, respectively. Next, we first focus on the focal shifts of the small circular aperture, where vector Kirchhoff theory is used and the three components of the electric field distribution along the $z$ axis are [24]

We set $N{A}_a=0.965$ and utilize Eq. (3) to calculate the distribution of light intensity along the $+z$ axis [see Fig. 3(a) and 3(b)]. In Fig. 3(a) and 3(b), $z=0$ denotes the position of the geometrical focus and $f$ is the focal length. Figure 3(a) shows that the focal shifts decrease with the increase of $r$ (blue solid line: $r=1.25{\lambda }_a$; red dashed line: $r=3{\lambda }_a$; green dash-dotted line: $r=5{\lambda }_a$). Moreover, the focal shifts are not obvious when $r>5{\lambda }_a$ for the small circular aperture, which means that the focal shifts might be small for the microspheres with $r>5{\lambda }_a=5\lambda /n_s$.

 

Fig. 3 Focal shifts along the $z$ axis of the small circular aperture with radius $r$ and $N{A}_a=0.965$, where $z=0$ denotes the position of the geometrical focus and $f$ is the focal length. (a) blue solid line: $r=1.25{\lambda }_a$; red dashed line: $r=3{\lambda }_a$; green dash-dotted line: $r=5{\lambda }_a$. (b) $r=1.25{\lambda }_a$ and the wave front at the aperture being divided equally into five zones within the maximum aperture angle [see 3(c)]. Lines 1-5 correspond the five zones.

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In order to examine the contributions of each ray on the focal shifts for a given aperture, we set $N{A}_a=0.965$ and $r=1.25{\lambda }_a$ and divide equally the wave front at the aperture into five zones within the maximum aperture angle $\Omega$ [see Fig. 3(c)]. It is obviously seen from Fig. 3(b) that the rays within the low zone will cause any bigger focal shifts. However, the position ($-0.112f$) [see the blue solid line in Fig. 3(a)] of the actual focus slightly shifts toward the small circular aperture from the focusing position ($0.096f$) of the rays within the fifth zone, which means that, for the microsphere, the actual focal spot will slightly shift the focusing position of the rays corresponding to $NA=n_o\sin {\alpha }_{\max }$ and locate at the inside proximity of the rear surface of the microsphere. Meanwhile, the actual $NA$ of the microsphere is also slightly bigger than the $NA$ (i.e., $n_o\sin {\alpha }_{\max }$) defined before. In terms of Fig. 3(b), we can find that, the small aperture used in Ref. 24 is assumed an aplanatic system, but the spherical aberration actually exists. Moreover, the spherical aberration is negative for the small aperture used in Ref. 24 when the above definition $S=d_{\max }-d_{\min }$ is used, whereas the spherical aberration predicated by the ray tracing procedure is positive for the microsphere. Therefore, for the microsphere with small $r$, the minimum requirement for $S$ is assumed as $S<\lambda /2n_o$ in the before analyses, but the actual spherical aberration $S$ will be far smaller than $S$ based on the ray tracing analysis (See Fig. 2) due to the spherical aberration compensation caused by the focal shifts. On the basis of the above discussions, we conclude that, because of the spherical aberration compensation arising from the positive spherical aberration caused by the surface shape of the microsphere and the $RIC$ and the negative spherical aberration caused by the focal shifts due to the wavelength scale dimension of the microsphere, although the illuminating light is a plane wave, the microsphere with small $r$ naturally has the characteristics of negligible $S$ and high $NA$ within the above range of $1.5\le RIC\le 1.75$. We expect that, only within $1.5\le RIC\le 1.75$, the maximum lateral resolution of the microsphere might be obtained due to the low aberration and high $NA$ being the precondition of the optical system with high lateral resolution. The principle of choosing $r$ is to make $r$ as small as possible under the case of $r>\lambda$ in order to obtain the obvious focal shifts and make the focal spot of the microsphere locate at the rear surface of the microsphere, which can be realized by the FDTD method.

3. FDTD simulations and discussions

FDTD software is afforded by Lumerical Solutions, Inc. Auto-nonuniform meshing with mesh accuracy 6 and minimum mesh step 0.25nm were used to get the most accurate results, the auto-cutoff was set as $1\times {10}^{-5}$ to ensure the convergence of the obtained results, and the boundary condition is the perfectly matched layer (PML).

As indicated in Section 2, the microsphere with small $r$ naturally has the characteristics of negligible $S$ and high $NA$ within the approximate range of $1.5\le RIC\le 1.75$. In general, the electric field distribution on the focal plane of the optical system with high $NA$ illuminated by the linearly polarized plane wave is strongly asymmetric about the $z$ axis, with the highest lateral resolution (smallest FWHM) at the direction orthogonal to the polarization of the incident wave [18, 22]. Therefore, for the in determining the lateral resolution of the focal spot on the rear surface ($z=C$ shown in Fig. 1) of the microsphere, we consider only the electric field intensity $|E{|}^2$ along the $y$ axis ($x=0$) for the $x$ linearly polarized plane wave. The parameters of $r$, $RIC$, and $\lambda$ used in FDTD are shown in Fig. 4 caption.

 

Fig. 4 Electric field intensity ($|E{|}^2$) distribution in the $yz$ plane of the microsphere [(a)-(d)] and (e) the variation of the lateral resolution (o) along the $y$ axis and the axial resolution (*) along the $z$ axis with the $RIC$. The white circle denotes the contour of the microsphere. The parameter $n_o$: (a)-(c) and (e) $n_o=1$ and (d) $n_o=1.34$. The parameter $\lambda$: (a)-(c) and (e)$\lambda =400nm$, and (d) $\lambda =355nm$. The parameter$RIC$: (a) $RIC=1.5$, (b) and (d) $RIC=1.59$, and (c) $RIC=1.75$. The radius $r$ of the microsphere: (a) $r=510nm$, (b) $r=490nm$, (c) $r=450nm$, and (d) $r=325nm$. In (e), six sets of parameters are used: $RIC=1.5$, $r=510nm$; $RIC=1.59$, $r=490nm$;$RIC=1.63$, $r=480nm$;$RIC=1.67$, $r=470nm$; $RIC=1.7$, $r=460nm$;$RIC=1.75$, $r=450nm$.

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For a given $\lambda$ and $RIC$ within $1.5\le RIC\le 1.75$, we have FDTD simulations by increasing the radius $r$ with step 5nm from $r>\lambda$. Shown in Fig. 4(a)-4(d), the focal spot of the microsphere can be situated at its rear surface by tuning $r$. Meanwhile, the microsphere is the OMS, namely the lateral resolution (by FWHM) along the $y$ axis of the microsphere can be beyond $\lambda /2n_s$ in the near-field region. For example, the microsphere with $r=490nm$, $RIC=1.59$, and $\lambda =400nm$, the lateral resolution is $120nm$ and slightly better than $126nm$ calculated by $\lambda /2n_s$ and $n_s=RIC\times n_o=1.59$. Shown in Fig. 4(e), within $1.5\le RIC\le 1.75$, for the six sets of $RIC$ chosen arbitrarily by us, the OMS can be obtained for proper $r$. However, if the $RIC$ is outside the range of $1.5\le RIC\le 1.75$, regardless of how to tune $r$, the focal spot can be also situated at the rear surface of the microsphere or the PNJs can be formed, but the lateral resolution is hardly beyond $\lambda /2n_s$. The reason is that, only within the approximate region of $1.5\le RIC\le 1.75$, the microsphere has the characteristics of negligible $S$ and high $NA$ as discussed in Section 2. For the OMS, the axial resolution along the $z$ axis is usually dozens of nanometers [see Fig. 4(e)].

As indicated in Section 1, different for the macroscopic SIL, although the OMS is assumed to have same $n_s$ for different $\lambda$ of the illuminating lights, the $r$ of the OMS is wavelength dependent. In terms of the six sets of $RIC$ and $r$ given in the caption of Fig. 4, for $\lambda =400nm$, $r$ is from 450nm to 510nm, namely from $1.125\lambda /n_o$ to $1.275\lambda /n_o$. Morover, the bigger the $RIC$, the smaller the $r$, which means that, in fact, the $r$ for each $RIC$ can be obtained by optimizing $r$ from $1.125\lambda /n_o$ to $1.275\lambda /n_o$. Obviously, without too amount of computations, one may easily calculate the $r$ of the OMS corresponding to each $RIC$ with range from 1.5 to 1.75 for a fixed $\lambda$ (called the original illumination wavelength) by the FDTD method or the Mie theory beforehand. If one can calculate the new $r$ from the known $r$ of the OMS corresponding to the original illumination wavelength when the wavelength of the illuminating light is changed, it is convenient and interesting for applications of the OMS. In the following, we will deal with this problem.

When $\lambda$ is given and the polarization of the plane wave is not considered, the factors affecting the focusing of the microsphere are its $r$ and the $RIC$. The above analyses and the FDTD simulation results show that the $S$ and the $NA$ explain well the near-field focusing of the microsphere and predicate the $RIC$ of the OMS. It is noted that, as indicated before, the actual$NA$ is slightly bigger than the $NA$ (i.e., $n_o\sin {\alpha }_{\max }$) defined before, which could not be calculated accurately for the microsphere. In terms of the Snell law and Eq. (2), the invariant $RIC$ can assure that both the tracks of rays in the microsphere and the value of $\alpha$ remain unchanged regardless of $\lambda$. Meanwhile, the $S$ is proportional to $r$ [see Eq. (2)] because it is calculated by $d$. If the ratio of the $S$ to the wavelength $\lambda /n_o$ remains unchanged, $r{n}_o/\lambda$ is invariable. In addition, the focal shifts are determined by the ratio of the $r$ to the wavelength$\lambda /n_o$.Therefore, for the fixed $RIC$, if the ratio $r{n}_o/\lambda$ remains unchanged, the OMS can still be obtained and its new radius $r^{\prime }$ can be expressed as

As a example, for the OMS with the $\lambda =400nm$, $r=490nm$, $RIC=1.59$ and $n_o=1$ [see Fig. 4(b)], if the new wavelength ${\lambda }^{\prime }=355nm$ and ${n^{\prime }}_o=1.34$, the new radius $r^{\prime }$ calculated by Eq. (4) is $325nm$. Shown in Fig. 4(d), under the case of ${\lambda }^{\prime }=355nm$, $r^{\prime }=325nm$, $RIC=1.59$, and ${n^{\prime }}_o=1.34$, the lateral resolution along the $y$ axis is $80nm$ and slightly better than $83nm$ calculated by $\lambda /2{n^{\prime }}_s$ and ${n^{\prime }}_s=RIC\times {n^{\prime }}_o=2.1306$. Figure 4(d) is almost identical with Fig. 4(b) except the spatial size (see the coordinates), which means that the transform Eq. (4) is suitable.

4. Conclusion

In conclusion, we focus on physically analyzing the origins of the $NA$ and the spherical aberration of the microsphere with wavelength scale radius. We demonstrate that the microsphere naturally has negligible $S$ and high $NA$ within the approximate region of $1.5\le RIC\le 1.75$, whose reason is due to the spherical aberration compensation arising from the positive spherical aberration caused by the surface shape of the microsphere and the $RIC$ and the negative spherical aberration caused by the focal shifts due to the wavelength scale dimension of the microsphere. We show that, only within the approximate region of $1.5\le RIC\le 1.75$ with the proper $r$, the OMS can be realized, namely the lateral resolution of its near-field focal spot slightly beyond $\lambda /2n_s$, which is also the lateral resolution limit of the dielectric microsphere. The $r$ for each $RIC$ can be obtained by optimizing $r$ from $1.125\lambda /n_o$ to $1.275\lambda /n_o$. As the focusing of the microsphere are strongly affected by the diffraction effects accompanying the scattering and by the interference of the waves transmitted through and refracted by a microsphere, different for the macroscopic SIL, although the OMS is assumed to have same $n_s$ for different wavelengths of the illuminating lights, the radius of the OMS is wavelength dependent. In order to deal with this problem, we develop a simple transform formula used to calculate the new radius from the known radius of OMS corresponding to the original illumination wavelength when the wavelength of the illuminating light is changed. Compared with the nSIL, the illuminating light incidence on the OMS is a plane wave instead of the convergent wave required by the SIL, which means that one can use the arrays of OMSs to image or detect sample in wide range. Moreover, the microspheres not only have comparable resolution with the nSIL, but also can avoid the fabrication difficulty faced by the nSIL because the microspheres with various radii and materials can be obtained commercially. In this paper, our physical analyses on the origins of the $NA$ and the spherical aberration of the microsphere clearly indicate what is the highest lateral resolution with the microsphere and what is the optimal parameters of the microsphere to realize the highest lateral resolution, which are very important for researchers to clearly understand the lateral resolution limit of the microsphere and to design the optimal refractive index and the radius of the microsphere under the case of various illumination wavelengths and applications. The work in this paper is important for the high resolution imaging and nanolithography based on the microsphere.