## Abstract

For a long time, light focusing from microspheres has been intensively researched. The microsphere has been shown to be capable of generating a high intensity beam with sub-wavelength width, known as a photonic nanojet (PNJ). In this article, we present a detailed report on the properties of a new asymmetrical microstructure, consisting of a supporting stage and a spherical cap, and demonstrate precise engineering of the PNJ characteristics by simply selecting its geometrical dimensions. More importantly, we find that a single asymmetrical microstructure can generate an ultra-elongated PNJ on the shadow side and the cascade of two asymmetrical elements can generate a PNJ with a full width at half maximum (FWHM) waist down to 0.27λ. In addition, because of the presence of energy convergence regions within the second element, an ultra-narrow PNJ can be generated even when the length of the second element in the cascade is many orders of magnitude greater than the wavelength or deviates somewhat from the optimal dimensions. This offers design flexibility and manufacturing tolerance, which has not been demonstrated in the conventional microsphere design or its derivatives. We believe that these remarkable performance features make the asymmetrical structure and its cascade attractive in numerous applications.

© 2016 Optical Society of America

## 1. Introduction

Scattering by microspheres or microcylinders is already a perfectly solved problem, whose history can be traced as far back as 1908 when Gustav Mie published one of his seminal papers focusing on the analytic solution of scattering by spherical gold particles [1]. In 2000, this topic once again generated widespread attention when Lu *et al.* reported that an enhanced laser irradiation generated on the shadow side surface of the illuminated dielectric microsphere can write a subwavelength structure on a silicon surface [2]. Chen *et al.* then renamed the enhanced irradiation as “photonic nanojet” [3], which has now been widely accepted by the community. By using rigorous Mie theory combined with Debye series [4], researchers have demonstrated that the behavior of the microsphere or microcylinder is significantly different from that of the conventional solid immersion lens. This incredible characteristic enables potentially important applications in the areas of nanoparticle detection [5], sizing and manipulating nanoscale objects [6–8], optical data storage [9], and maskless direct-write nanopatterning and nanolithography [10, 11]. Moreover, the groups of Backman and Astratov, respectively, reported that a chain consisting of microspheres with the same dimension or with size dispersion can achieve low loss optical transport via the mechanism of whispering gallery modes [12] or nanojet-induced modes [13], which opens up the window for microsphere-chain based waveguides and resonators [14–18] as well as laser surgery [19, 20]. Another inspiring breakthrough was made by Wang *et al*. in 2011 when they discovered that microsphere assisted optical microscopy can capture images of objects in a virtual mode with the lateral resolution down to 50 nm under white-light illumination [21]. The following investigations, spanning from microsphere-assisted confocal microscopy [22] to a new endoscopy method by functionalizing graded-index lens with microspheres [23], have demonstrated that the microsphere or microcylinder paves a much more easy-to-handle and economical way to overcome the diffraction limit than several other efforts including super/hyper-lenses driven by surface plasmon excitation [24–27], fluorescence microscopy with molecular excitation [28–32], near-field scanning microscopy (NFSM) [33–35], and super-oscillatory lens based far-field optical microscopy [36–39].

Although the microsphere and microcylinder present exceptional characteristics such as PNJ formation, optical transport with low loss, and super-resolution imaging, some of their inherent properties limit their application. For example, the detection of intrinsic nanostructures and artificially introduced nanoparticles deeply embedded within biological cells requires a long PNJ length, whereas a rapidly convergent PNJ generated by the microsphere or microcylinder is followed with a fast divergence. To obtain a much longer PNJ, the graded-index multi-layer microsphere [40] or microellipsoid [41], the two-layer dielectric microsphere [42], the liquid-filled hollow microcylinder [43], the hemispheric shell [44], and the microaxicon with specific spatial orientation [45], have been proposed in recent years. However, the engineering difficulty and complexity in fabricating these PNJ generators as well as the inevitable side lobes in turn limit their application in the detection of deeply embedded nanostructures within cells. The side lobes are created from the interference of the unscattered incident field that passes around the edge of the generator, the diverging light of the central lobes of the PNJ, and the light that has scattered multiple times within the generator. Though a considerable number of publications aiming at exploring the mechanisms beneath the unusual imaging properties of microspheres appeared in recent years [46–48], there is no complete theoretical model that can fully explain the excellent super-resolution capability of microspheres observed in [49–54]. We believe this super characteristic of microspheres stems from the combined effects of: the evanescent waves coupling close to the surface of the microsphere and their conversion into propagating waves, the relatively high refractive index of microsphere that induces shrinkage of the illumination wavelength, and the properties of the PNJ. In a very recently published article [55], the authors experimentally validated that shrinking the waist of the PNJ of a dielectric microsphere results in higher lateral resolution, which paves a reasonable way to guide the design of super-resolution components. Specifically, the goal is to reduce the full width at half maximum (FWHM) waist of the PNJ. The narrow PNJ is vital to confocal microscopy because the imaging performance of a confocal microscope is closely related to how tightly the illumination light can be focused. Wu *et al*. reported a PNJ with a FWHM waist of 0.485 *λ* (*λ* = 0.4 μm) by engineering a microsphere with four uniformly distributed rings etched at a depth of 1.2 μm and width 0.25 μm [56]. Zhu *et al*. reported that by illuminating the curved side of a hemispheric shell possessing a high refractive index and properly chosen diameters and thickness, a PNJ with a small FWHM of 0.398 *λ* (*λ* = 0.4 μm) can be achieved [44]. Yue *et al*. demonstrated that the sub-diffraction foci with a FWHM 26.7% smaller than the diffraction limit can be achieved via a cylindrical metalens assembled by hexagonally arranged nanofibers [57]. Minin *et al.* reported that an axicon or a cuboid with proper dimensions is capable of generating a PNJ with a FWHM slightly smaller than 0.5 *λ* [58, 59]. Compared to the super-oscillation lens, the PNJs generated by the above designs have much more compressed side lobes in the immediately adjacent area of the main lobe [60–63]. However, the smallest FWHM PNJ generated by an isolated structure that has been reported [64–66] is larger than *λ*/3, which limits the further improvement of super-resolution capability of the microsphere as well as the above mentioned novel designs. Another noteworthy drawback of the microsphere, which is due to its fully curved surface, is the difficulty in handling, moving and assembling single microsphere or microsphere arrays. Wang *et al*. proposed a cantilever-combined microsphere to scan over the sample surface to form a full image with post-processing. The method works similar to NFSM and thus is inevitably limited by the low serial scanning speed [67]. Allen *et al*., Darafsheh *et al*., and Du *et al*. proposed to embed microspheres into movable thin-films to precisely align the limited field of view (FOV) within a desired location and to meet the demands of large-area inspection [68–70]. However, the thin-films with refractive index larger than 1 inevitably change the ambient of the microspheres. This necessitates large index microspheres in consideration of the refractive index contrast needed to form the PNJ. Most recently, Gu *et al*. proposed a self-assembled spherical cap optical nanoscopy for subsurface nano-imaging [71], which is similar to the wavelength-scale lens microscopy [72] and the shape-controllable microlens arrays [73]. This design provides a much more easy-to-handle way to adjust the effective FOV by moving the substrate and performing image stitching [67]. However, Gu’s design, which has only one curved surface, prevents the wavefronts inside the structure from being adequately focused, which may limit the resolution.

In summary, it is vital to have a design that can be easily fabricated and at the same time generates a long PNJ with small FWHM waist and wide FOV. Unfortunately, similar to the classical imaging theory, a tradeoff exists among these primary targets. In this paper, we systematically explore the properties of an asymmetrical silica microstructure and show that it can achieve an ultra-elongated PNJ that is comparable to those from previously reported more complex designs [40–44]. Further, we show that by properly tuning the geometrical parameters, the cascaded asymmetrical silica microstructure will produce a stable optical transfer and a FWHM waist that approaches *λ*/4. We believe these asymmetric structures can not only replicate the functionality of microsphere based structures, but can offer new capabilities such as the detection of deeply embedded nanostructures in cells [74], the ultra-high density optical storage [75], and the realization of a label-free, ultra-high resolution integrable nanoscopy system.

## 2. Ultra-elongated PNJ

#### 2.1 Systematic simulation model

We start by defining some important parameters used to characterize the PNJ performance. As presented in Fig. 1(a), the working distance is defined as the distance between the intensity hot spot and the closest point on the surface of the PNJ generator; the length of PNJ is defined to be the same as the “diffraction length”, through which the light intensity decays to 1/e of the peak value, i.e., the hot spot; the FWHM waist is calculated for the plane that contains the hot spot if the hot spot is outside of the PNJ generator, or else it is calculated for the plane that is tangential to the surface on the shadow side of the PNJ generator [40]. In this article, all of the defined parameters refer to the central beam because typically, the intensity of the central beam in a PNJ is significantly larger than that of the side lobes [3, 4, 42–45, 60, 61]. It has been shown that as the refractive index of a microsphere decreases, the PNJ reaches further into the background [42]. Hereafter, we assume the background is air.

The lowest available index for conventional optical materials in the visible light frequency range is around 1.37 [76], which indicates the PNJ generated on the shadow side of a microsphere is at most only several wavelengths long followed by a fast divergence. To validate this, we first simulate the spatial intensity distribution of a silica microsphere with diameter of 4*λ* = 1.62 μm by the illumination of a TE plane wave with *λ* = 0.405 μm wavelength in air, as shown in Fig. 1(b). Silica is chosen here because of its long-term stability and generality in the semiconductor industry as well as its small refractive index at *λ* = 0.405 μm wavelength, namely, *n* = 1.4696. Note that the scattering property of the microsphere cannot be correctly inferred from the trajectory of the ray optics because of the small critical dimension; thus, full-wave treatment is necessary to observe numerically correct scattering behavior. Throughout this article, the wave optics module of COMSOL Multiphysics ® Modeling Software (version 5.1, COMSOL, Inc., Stockholm, Sweden) is used to model the scattering property of various PNJ generators in 2D. The simulation domain is surrounded by a perfectly matched layer (PML) with the triangular mesh size set as *λ*/25 to approximate an open domain. Triangular mesh elements with sizes equal to *λ*/30 and *λ*/28 are applied to the silica and air domains, respectively, for ensuring the accuracy of the finite element method. As can be seen from Fig. 1(b), the wavefronts passing through the surface on the bright side cannot be focused to a single point, but instead lead to a caustic curve whose cusp is outside of the surface. According to previous definition, we find that the working distance and length of the PNJ with respect to the microsphere of diameter 4*λ* are 0.27*λ* and 1.36*λ,* respectively. The scattering problem in terms of reflection and transmission of cylindrical waves at the surface of the microsphere can be analyzed by resolving the Mie solution into a Debye series, by which we can represent the amplitude of the field *R _{t}* outside of the microsphere as the superposition of infinite number of modes, namely,

*R*and

_{oo}*T*are the reflection and transmission coefficients for an incoming cylindrical wave from outside of the microsphere, respectively, and

_{oi}*R*and

_{ii}*T*are the reflection and transmission coefficients for an outgoing cylindrical wave from inside of the microsphere, respectively. Here, we designate

_{io}*i*and

*o*to indicate if a mode is inside or outside the microsphere, respectively. It has been demonstrated that for the case of a microsphere with a relatively large radius, the predominant contribution to the PNJ comes from the term

*T*corresponding to the two refractions happening on the two hemispherical surfaces during the propagation of the wavefront [3]. This insight connects the physical and the geometrical optics pictures. We should keep in mind that the importance of the remaining terms in Eq. (1) increases with the decrease of radius

_{io}T_{oi}*r*; thus, for the analysis of a small microsphere, the consideration of more terms in the calculation of

*R*is necessary. Here, instead of accurately calculating the various terms in Eq. (1), we prefer to firstly assume that the term

_{t}*T*still contributes the most to the PNJ. Based on the above assumption, we can intuitively imagine that if an asymmetric structure consisting of a flat support slab and a solid hemisphere of the same radius as the microsphere is illuminated from the flat side by the same plane wave, the hot spot in the PNJ will be pushed further away from the hemisphere surface because refraction only happens on the curved surface. This leads to a longer working distance than that of the microsphere. Again, by assuming that

_{io}T_{oi}*T*dominates the PNJ, we can deduce the aberration along the propagation direction caused by the hemisphere [77], namely,

_{io}T_{oi}*η*denotes the distance between the point of incidence on the curved surface and the intersection point of the curved surface and the propagation axis.$J$and$J\text{'}$represent the incident and refractive angles of the far-off axis waves, respectively.

*u*and

*u’*denote the inclined angles of the incident and refracted waves with respect to the propagation axis, respectively.

*θ*denotes the incident angle of the paraxial wave. For the microsphere, the wavefronts propagating through the hemispherical surface on the shadow side have already undergone refraction by the hemispherical surface on the bright side, while those for the asymmetric structure only undergo one wavefront refraction, which means$\xi $for the microsphere should be smaller than that for the asymmetric structure. Hence, it is expected that the asymmetric structure generates a PNJ with a longer extension than the microsphere. Moreover, we note that from Fig. 1(b), the PNJ of the microsphere is strongly affected by the surrounding background signal because of the interference between the illumination wave and the waves leaking from the non-central area of the hemispherical surface on the shadow side. This effect would also exist in the asymmetric structure case. To reduce the effect of the background signal on the PNJ, we coat the non-working area, i.e. the flat portion of the shadow side, with a gold thin film.

To validate the above assumptions, we modeled the spatial field distribution of the asymmetrical structure that has a hemispherical surface with the same diameter as the microsphere. As can be seen in Fig. 1(c), the new design generates a longer working distance (1.53*λ*) and length of PNJ (3.50*λ*) when compared with that of the silica microsphere (0.27*λ* and 1.36*λ*, respectively). We could expect that by immersing the asymmetrical structure inside a liquid, a significantly longer PNJ can also be generated on the basis of ray optics, similar to what was done in [43]. Moreover, we can also observe that most of the background signal around the PNJ has been cleaned up because of the coated gold film and because of the supporting stage, which may result in a longer effective length of the PNJ in practice. From Eq. (2), we can also deduce that a refraction surface with a larger radius of curvature (ROC) can generate a larger aberration, which will further extend the length of the PNJ in air. Hence, we did another simulation by fixing the size of the opening in the gold film to be the same as that of Fig. 1(c), but replacing the hemispherical surface with a spherical cap that has a larger ROC, i.e., 4*λ*. Apparently, as can be viewed from Fig. 1(d), the larger ROC results in a far larger length of the PNJ (9.28*λ*), which is accompanied with an increase of FWHM waist (0.87*λ*). We continued varying the ROC of the curved surface with the clear aperture fixed. We present the resulting working distance, length of PNJ, and FWHM waist in Table 1, in which we directly observe an extension of the PNJ with the increase of ROC. Because the limiting case has an infinite ROC, we can extend the PNJ to any arbitrary length in theory. However, extending the PNJ length is inevitably accompanied by an increase of the FWHM waist and a decrease of the PNJ intensity, which should be carefully dealt with to make a desired compromise in practical use. Here, we should comment that the assumption of predominant *T*_{io}*T*_{oi} can only be applied to qualitatively guide the design of an effective PNJ generator and more terms in Eq. (1) should be calculated in order to quantitatively analyze the PNJ characteristics.

The above simulation indicates that we can obtain an arbitrarily elongated PNJ by varying the ROC of the curved surface. Here, we investigate the trends in PNJ working distance, length, and FWHM waist with respect to varying the clear aperture of the new design and the conventional microsphere. A hemispherical focusing surface is assumed in the new design. The clear aperture variation range is prescribed as 2*λ* - 6*λ*. As can be seen from Fig. 2(a), the working distances of both the conventional microsphere and the new design exhibit a strong linear relationship with the clear aperture, and the linearity for the new design in the full range is even better than that of the microsphere. The fitted lines for the two cases are *y* = 0.1209*x* – 0.1233 and *y* = 0.7022*x* – 0.5038, for the microsphere and the new design, respectively. This implies that the PNJ of the new design is pushed away at a far larger ratethan that of the microsphere with the increase of the critical design dimension, i.e., the clear aperture. By further exploring the relationship between the length of the PNJ and the clear aperture, we also observed a quasi-linearity for both the microsphere and the new design. From Fig. 2(b), we find that the relation is not as linear for the new design as it is for the microsphere, but it still can be approximated by the fitted line *y* = 0.6958*x* + 0.3375, whose slope is over 2 times larger than the fitted line *y* = 0.3093*x* + 0.0114 of the microsphere.

Figure 2(c) graphs the calculated FWHM waists of the PNJs in the plane of maximal field amplitude for the conventional microsphere and the new design. Immediately, we find that the conventional microsphere yields narrower FWHM waists than the new design for all the clear apertures, which indicates there is a compromise between the FWHM and the sum of working distance and length of PNJ [42]. Moreover, we find that the third, the seventh, and the tenth points in Fig. 2(c) for the microsphere are anomalies because their FWHM waists are obviously smaller than those of their neighbors. By checking the intensity distribution pattern with respect to the seventh point, we found that the hot spot is located on the surface of the microsphere. This indicates that the evanescent waves around the shadow side surface of the microsphere significantly contribute to the PNJ, resulting in a narrower FWHM waist [3]. For the third and tenth points, the corresponding working distances are also smaller than those of their surrounding neighbors. In contrast, for the new design, the PNJ has been pushed away from the hemispherical surface; thus, the effect of the evanescent waves on the PNJ is negligible. This may explain the larger FWHM waist in addition to the propagation wave interference after the surface refraction.

## 3. Optical transport

Recently, low loss and stable optical transport through a chain of microspheres with the same or varied size has been validated and understood as the effect of whispering-gallery modes (WGMs) [12] or periodically focused modes (PFMs) [14], which paves an efficient way to build coupled resonator optical waveguides and other photon-manipulation devices for effective guiding or filtering light. Generally, in a microsphere-chain, the radius is the only degree of freedom (DOF) that can be adjusted to control the photons for a given material and excitation. Here, in this section, we will show that the cascade of our asymmetrical designs can also guide light. Moreover, our design has three independent DOFs; the length of the supporting stage serves as another important DOF in addition to the clear aperture and the radius of curvature, whose effects were explored in the previous section. For simplicity, we will limit our study to 2 DOFs and only consider the case of a hemispherical focusing surface. To gain insight into how light behaves in a chain of the asymmetrical structures, we start by applying a geometrical optics approximation to each interface and inside of each structure. Although the geometrical optics approximation cannot fully characterize the photon behavior at the microscale, it provides a qualitative physical explanation of the observations [78].

#### 3.1 Optical waves in the cascaded structure

Consider a cascaded structure with two asymmetrical elements, which is illuminated by a monochromatic TE plane wave with the wavelength *λ* = 0.405 μm and with the unit intensity from the left boundary of the computational domain towards the right along the principal axis. Rays that can escape from the curved surface of the first asymmetrical structure should fulfill the relation *θ* ≥ arccos(1/*n _{s}*) to avoid total internal reflection, where

*θ*and

*n*

_{s}are the central angle and the refractive index of the structure, respectively, as shown in Fig. 3(a). Under this restriction, it is not difficult to deduce that

*α*is defined as

*α*=

*β*– (90° –

*θ*). Here,

*β*is the refractive angle with respect to the first hemispherical surface. By plotting the curve for

*α*given by Eq. (3) in Fig. 3(b), we can find that

*α*is a monotonically decreasing function of the central angle

*θ*. Further, a quick calculation reveals that the maximal incident angle

*α*on the flat surface of the second asymmetrical element exactly equals the minimum central angle

_{max}*θ*. For that case, the maximum of the refraction angle

_{min}*γ*is 29.90°, which is far smaller than the critical angle for total internal reflection inside the supporting stage,

_{max}*γ*

_{t}= 42.88°. Hence, the optical waves output from the first asymmetrical structure can be firmly confined in the supporting stage of the second element.

#### 3.2 Focus at the end of the cascaded structure

Although the output rays from the first asymmetrical element can be confined into the second element, the propagation angles of the optical rays inside the second element differ from each other. They are given by

where α is defined in Eq. (3). The various propagation angles of the rays inside the second element complicate the focusing property of the output hemispherical surface. To analyze the characteristics of the focusing, we follow the path of a specific ray during its propagation inside the cascaded structure, as shown in Fig. 4(a).Here, the analysis can be simplified by considering the axial symmetry of the cascaded structure. An optical ray corresponding to a central angle *θ* is expected to reach a point on the flat surface of the second element with a radial offset Δ that is given by

*r*is the radius of the hemispherical surface of the first element. The optical wave inside the second element undergoes up and down total internal reflection periodically. Assuming the width of the supporting stage is 2

*r*, the pitch

*ζ*along the principal axis is given bywhere

*γ*is defined in Eq. (4). Whether an optical ray can be focused at the output of the second element depends on the relative position of the circular surface of the second element with respect to the path of optical ray. Specifically, in the case as shown in Fig. 4(a), focus happens when the ray is bent by the final curved surface towards the principal axis. Actually, no analytic method currently exists that can be used to predict the total focusing property of the cascaded structure. Instead, we provide an approximate approach (see Appendices A-G) to analyze the total focusing property by discretizing the space of the central angle

*θ*and formulating three separate inequality systems that account for the focusing property of each ray with respect to a specific

*θ*. Note that in the practical case of plane wave illumination, the intensity distribution is not equally distributed in

*θ*, but rather in the cos(

*θ*), i.e., the axial position. Thus, we discretize the range of cos(

*θ*) into 10

^{3}sampling points, and each point corresponds to a specific optical ray. We define a parameter called the convergence ratio, which is the ratio of the number of convergent rays escaping from the shadow side of the cascaded structure to the total number of rays under consideration (see Appendices B-F for the detailed mathematical derivation about ray focusing property). The convergence ratio depends on angles and thus on the aspect ratio

*L/r*of the structure rather than on

*L*or on

*r*individually. As can be viewed from Fig. 4(b), the convergence ratio starts at its maximum value of 67.1% when the length of supporting stage equals zero because most of the rays fulfill the first inequality system in Eq. (7). The convergence ratio continually decreases and reaches its minimal value of 1.4% at the point 0.332. As discussed in the Appendix G, at this value, no rays satisfy the first or third inequality systems and very few rays satisfy the second inequality system. The three inequality systems have their own dominant regions, as marked in Fig. 4(b). For larger values of

*L/r*, the convergence ratio curve presents a sharp increase until it reaches a local maximal value of 54.6% at the point log

_{10}(

*L*/

*r*) = 0.724, after which it declines quickly and finally stabilizes around 33%. Note that in reality, the convergence ratio cannot predict the intensity of the foci because the intensity is affected not only by the focusing of the rays but also by interference effects, i.e., the phases of the various rays. Thus, in order to obtain the pattern and intensity distribution of the foci, a full-wave treatment is still required. Nevertheless, the convergence ratio can guide us to design the length of the cascaded structure to some extent. For example, because the convergence ratio near the point log

_{10}(

*L*/

*r*) = 0.332 is pretty small, we should carefully choose

*L*/

*r*to ensure that we avoid this region in order for us to achieve efficient energy focusing. For smaller values of

*L*/

*r*, we can generate a bright spot with short PNJ length. As validated in Appendix G, the range 0.332 to 1.500 leads to elongated PNJs that have weak intensity because this region is dominated by the rays fulfilling the second inequality system and thus have very small inclined angles with respect to the principal axis. To validate this insight, we did full-wave simulations corresponding to the points log

_{10}(

*L*/

*r*) = 0, 0.332 and 0.724 with the radius

*r*fixed at 1 μm and at 3 μm. The

*r*= 1 μm cases are shown in Figs. 4(c)–4(e) and the corresponding

*r*= 3 μm cases are shown in Figs. 14(a)–14(c) of Appendix G. As shown in Fig. 4(c), the

*L*= 0 case indeed results in a bright foci outside the cascaded structure, but at the same time there is energy leaking from the non-central area of the curved surface. The difference in the intensity patterns between Fig. 4(c) and Fig. 14(a) illustrates how geometrical optics breaks down for small critical dimensions. Optical diffraction becomes important. Diffraction also affects the shape of the foci. In particular, a foci still exists for the design corresponding to the minimal convergence ratio point, as shown in Fig. 4(d). However, compared with the foci in Fig. 4(c), the intensity of the foci in Fig. 4(d) is much weaker, which, to some extent, still validates the use of the convergence ratio to guide the design. For the case

*L*= 10

^{0.724}

*r*in Fig. 4(e), there exists intensity maxima inside the hemisphere’s circumference, a very long tail extended along the positive direction of the principal axis because of the second governing inequality system, and a small PNJ generated near the hemisphere, which is a little bit different from the r = 3 μm case shown in Fig. 14(c). The effect of optical diffraction may also result in noticeable PNJs for other geometries with low convergence ratio values. The above analysis shows that unlike the microsphere, the length of the supporting stage of the asymmetrical structure also plays a vital role on the focusing property. Thus, it is possible to fix the radius and simply adjust the length of the supporting stage to get the desired foci. This flexibility makes realizing the structure much easier compared to the microsphere.

Because the second element is capable of confining the optical waves, it is natural to think that for a short chain consisting of 2 or 4 elements, the asymmetrical structure may result in lower loss than the microsphere. However, the loss may increase when the number of asymmetrical elements in a chain is large if total internal reflection fails in the following contact areas due to the asymmetry. Fortunately, in Fig. 4, we have demonstrated that the length *L* of the supporting stage of each element is vital to the focusing property; hence, we can adjust both *L* and radius of each element to reduce the overall scattering loss. Figure 5(a) presents the intensity distribution for a chain consisting of four elements where the first two elements have shorter supporting stages than the last two elements. The radius of the hemisphere is set as 4*λ*. Obviously, by directly observing the image contrast, we can find that the optical transport process is accompanied by a strong loss. The loss level can be characterized by comparing the highest intensity of the 1st PNJ with that of the 2nd PNJ, as shown in Fig. 5(d). The stronger loss will result in a smaller peak value of the 2nd PNJ.

Now let us consider replacing the two elements at the end of the chain in Fig. 5(a) by three identical elements and redo the simulation under the same illumination condition. We then clearly observe that the scattering loss is remarkably reduced in Fig. 5(b), as is also demonstrated in Fig. 5(e). Moreover, the chain consisting of 5 identical elements also results in an output PNJ with a sub-wavelength FWHM waist and an ultra-long extension up to 7.41λ, which is far better than the performance in Fig. 5(a). By adding three more elements at the end of the chain presented in Fig. 5(b), we find that the intensity of the output PNJ is also reduced but with nearly the same loss level as that of the 5-element chain, as presented in Fig. 5(f). The length of the 3rd PNJ reaches 9.14λ, but the FWHM waist increases to 0.967λ, which may due to the fact that the PNJ is pushed away from the surface of the cascaded structure so that the evanescent waves cannot contribute significantly to the PNJ. We believe by tailoring the supporting stage length of each element in a chain, it is possible to realize ultra-low loss optical transport in a similar manner as a disordered microsphere-chain [16]. In fact, in the next section, we show that we can achieve very long optical transport with just a two element cascade by using the supporting stage of the second element as a multimode waveguide.

## 4. Realizing ultra-narrow PNJ

Recently, Yang *et al*. experimentally observed that the super-resolution imaging property of a dielectric microsphere is governed by the waist of its PNJ [55]. Although such a claim appears to be slightly controversial in the scope of dielectric microsphere based super-resolution imaging [79], reducing the FWHM waist of the illumination source is still vital to some optical imaging setups, such as confocal microscopy. The smallest reported FWHM in air for a PNJ of any dielectric microsphere or more complex isolated structure is larger than *λ*/3 [64–66]. In this section, we report a cascade of two asymmetrical elements with different supporting stage lengths that can generate a PNJ with a FWHM waist consistently smaller than *λ*/3 in air, and moreover, the smallest waist that we have observed approaches *λ*/4.

As has been illustrated in Fig. 4, a chain consisting of two asymmetrical elements illuminated by a plane wave parallel to the principal axis can confine the light waves by the principle of total internal reflection from the geometrical optics point of view. For an optical ray corresponding to a specific central angle *θ*, it will constructively or destructively interfere with the other rays, which may lead to a series of hot spots (constructive interference) and dark specks inside the cascaded structure along the principal axis because of the geometrical symmetry. To validate our intuition, we investigated a cascaded structure consisting of a fixed geometry asymmetrical element (*r* = 1μm and *L* = 1μm) and an *L*-varied *r*-fixed (*r* = 1μm) element. Figure 6(a) illustrates the spatial intensity distribution with respect to the case of *L* = 3 μm. Here, we clearly find an output PNJ with 0.4λ FWHM waist and two chains of sequential hot spots inside the second asymmetrical element. The difference in the lengths and pattern shapes of the two chains stem from the interference of the diffracted waves with different tilt angles and amplitudes due to the refraction on the curved surface of the first asymmetrical element and the flat surface of the second element. In this example, we observed that the spatial field inside the second element is inhomogeneous and is not periodically distributed; thus, by changing the length *L* of the second element we can change the pattern shape of the output PNJ. We then set the length of the supporting stage of the second element as 5.8 μm and present the result in Fig. 6(b). Obviously, we can observe that the pattern shape of the output PNJ, which has a FWHM waist of 0.31λ, is different from that in Fig. 6(a).

The ultra-narrow FWHM waist makes it possible to achieve super-resolution that goes beyond the limits of the conventional dielectric microsphere. Interestingly, by comparing Figs. 6(a) and 6(b), we can find that there are some regions, as marked by the white two-way arrows, in which the energy tends to converge to hot spots along the extension direction of the second element. In this paper, we call these regions as energy convergence regions (ECRs). Here, we should give the definition of an ECR in detail. As shown in Fig. 6(c), there is a pattern consisting of several pairs of bright specks in between two chains of hot spots. We mark the pair of bright specks that has the largest separation in *x* by a red two-way arrow. We then mark the next pair of bright specks to the right with a yellow two-way arrow. The yellow arrow sets the left margin of the ECR. We then mark the hot spot that has the largest peak value in the right chain of hot spots by a red line, after which the location of the second largest hot spot to the left in the same chain is set as the right margin of the ECR, as marked by a yellow line. In summary, an ECR is a region in which constructive interference patterns are converging to form a focus.

We pick out the intensities along the transverse cross section of each pair of bright specks for the ECR shown in Fig. 6(c) and present the results in Fig. 6(d). We find that the central lobe of the intensity curve tends to increase while the side lobes tend to decrease along the z direction. This phenomenon, which can be understood as an increase of the total light energy stored in the central lobe area of each pair of bright specks along the z direction from mathematical point of view, distinctly makes the name ECR appropriate. By comparing Figs. 6(a) and 6(b), we can find that the lengths of the ECRs are nearly the same. The only difference happens in the field amplitude and this difference is due to the different paths of the back reflection waves from the curved surfaces of the second elements. By making a further comparison, we observed that the location where the output PNJ in Fig. 6(a) is generated happens to be inside the ECR in Fig. 6(b), which enables us to believe that there must be a relationship between the output PNJ and the ECR.

Actually, as has already been discussed in the above sections, the high optical intensity in the ECR stems from the constructive interference of some of the diffracted waves. This indicates that a noticeable portion of these waves must be reflected by the upper or lower boundaries of the second element and thereafter would be propagating towards the principal axis in the ECR. Hence, similar to the analysis for the rays working in the third inequality system presented in Appendix F, continuously moving a curved surface inside the ECR is predicted to result in PNJs with small FWHM waists. Figure 7(a) presents the spatial intensity distribution corresponding to a structure with a length *L* = 7.4 μm, in which four ECRs as marked by Δ_{1} to Δ_{4} are clearly observed. We vary the length *L* from 5.45 μm to 6.3 μm with a 0.05μm step to ensure that the curved surface of the second element is always within the fourth ECR, as shown in Fig. 7(b). For each position, we find that there is a PNJ on the shadow side of the cascaded structure. We then calculate the FWHM waists of the output PNJs and plot them in Fig. 7(c), in which we can clearly see that super-resolution always exists when the second curved surface is placed inside this ECR. Moreover, in the first half of the ECR, the FWHM waists are nearly all smaller than *λ*/3 and the smallest one, which is 0.272λ, even approaches *λ*/4. Because each scatter plot data point in Fig. 7(c) corresponds to the case where the curved output surface is inside an ECR, the intensity of the generated central beam is significantly larger than that of the side lobes.

The above results indicate that the length of the ECR determines the tolerance for the length of the supporting stage of the second element to ensure super-resolution. The existence of this tolerance is vital for actual fabrication because the variation in the fabrication process prevents the true geometrical dimensions from equaling the design values. Ideally, we would want the length of ECR to be as long as possible so that we have a large fabrication tolerance. As can be seen in Fig. 7(a), the length of the ECR increases as *z* increases. The supporting stage is a multimode waveguide. Thus, it is no surprise that the calculated intensity pattern in it is similar to the one inside a multimode interferometer (MMI) [80–83]. However, unlike the typical MMI, the chains of hot spots inside the cascaded structure are reproduced at aperiodic intervals along the propagation direction because of the non-uniform input illumination. As can be viewed in Fig. 7(a), the first asymmetric element focuses the light and thus there is a distribution of diffracted waves along the *x* direction on the flat surface of the second asymmetric element. Although the locations and sizes of the ECRs are aperiodic, the ability to guide waves in the support structure gives us hope that we can engineer a PNJ with an ultra-narrow FWHM waist using an ultra-long second asymmetrical element and thereby achieve PNJ transport across ultra-long distances.

To validate the above hypothesis, we firstly simulate the intensity pattern for a cascaded structure with *L* = 30 μm, as presented in Fig. 8(a) and Table 2. The lengths of the ECRs initially undergo a rapid increase for the first five ECRs marked by Δ_{1} to Δ_{5}, but then start to decrease beginning with the sixth ECR. This suggests that the ECR characteristics repeat along the propagation direction to some extent. We also found that the maximal intensities in the fifth and sixth ECRs are smaller than those of the first four ones (as shown in Table 2), which indicates that destructive interference is prominent. Under this circumstance, putting the curved surface of the second element in the corresponding ECR may either lead to a dark PNJ or result in a wide-FWHM PNJ. This should be carefully dealt with by adjusting the hemispherical radius and the length of the asymmetrical elements when being applied in practice. Moreover, we should also notice that for *L* = 30 μm, the location of the curved surface of the second element happens to be outside of an ECR, which leads to a weak PNJ, as can be viewed from the right of Fig. 8(a). We then simulate another case for *L* = 49.3 μm to ensure the location of the curved surface of the second asymmetric element is inside an ECR, as shown in Fig. 8(b). A PNJ with a FWHM waist 0.286λ, is generated on the shadow side of the cascaded structure again. Thus, we have generated an ultra-narrow PNJ at a distance of hundreds of wavelengths from the first element. By comparing the first seven ECRs for the *L* = 30 μm and *L* = 49.3 μm cases, we found that they almost have the same lengths and only differ in their intensities, which again validates our statement that we can engineer the properties of the PNJ by controlling the device dimensions.

The goal of our device designs was to improve the focusing properties of light. The obtained ultra-narrow PNJ will be directly applicable to confocal microscopy. There are connections between the light focusing and the light collection properties of a microstructure and this is a subject for future work.

## 5. Conclusion

An asymmetrical microstructure consisting of a supporting stage and a spherical cap has been investigated and design guidelines for precisely engineering the PNJ properties by choosing the microstructure’s clear aperture and radius of curvature have been presented. We firstly validated that a single asymmetrical microstructure can generate an ultra-elongated PNJ on the shadow side, which cannot be achieved by the conventional dielectric microsphere with a refractive index smaller than two. By using a chain of tight-binding asymmetric microstructures, stable optical transport similar to a microsphere-chain is observed. We believe that by optimizing the geometrical dimensions of each element, the chain is also capable of achieving ultra-low loss optical transport similar to the disordered microsphere-chain. Moreover, we found that the cascade of two asymmetrical elements enables the generation of a PNJ with a FWHM waist down to 0.27 *λ*. Further, the length of the second element in the chain can be very long, which enables a simple design for low loss PNJ transport, and can have significant tolerance, which can improve manufacturability. All of these features are difficult or impossible to achieve using the conventional microsphere or its derivatives. We believe that these remarkable performance characteristics will enable the designed asymmetrical structure and its cascade to be applied in various emerging applications including inspecting nanoparticles located deep inside a cell, realizing low loss fiber and waveguide mode converters, trapping and precisely manipulating nanoscale objects, and super-resolution imaging.

## Appendix A Focus at the end of the cascaded structure

The focusing properties of the cascaded structure consisting of two elements depend heavily on the relative position and the radius of curvature of the curved surface of the second element with respect to the waves that are propagating inside. For simplicity, we will consider only the case of the hemispherical surface here. Further, we will use a geometrical optics approximation to gain some initial insights, but as discussed in the main text, a full-wave treatment is necessary to obtain the correct focusing behavior. As can be viewed from Fig. 4(a) of the main text, a ray escaping from the first element hits the flat surface of the second element with an offset Δ, after which it undergoes periodically up and down reflection inside the second element with a pitch *ζ* governed by Eq. (6) of the main text. Hence, analyzing the focusing properties of the cascaded structure is equivalent to analyzing the refraction of light waves at different points inside a single period and inside the initial fraction of a period region by continuously varying the aspect ratio of the geometrical dimensions, i.e., *L*/*r*. Here, the convergent rays can be categorized into three distinct groups, which are listed below. Without loss of generality, we will consider a ray initially propagating in the region above the principal axis because the same results hold by symmetry for a ray that is initially in the region below the principal axis. Our goal is to determine the convergence ratio, i.e. the fraction of rays that is bent towards the principal axis after the final hemispherical surface.

## Appendix B The first governing inequality system

As illustrated in Fig. 9, if *L* + *r* is smaller than Δcot(*γ*), the ray inside the second element will be refracted by the hemispherical surface towards the principal axis provided that the incident angle *φ _{i}* with respect to the hemispherical surface does not exceed the total internal reflection angle

*φ*. Hence, the governing inequalities are:

_{t}*φ*depends on the intersection point,

_{t}*I*, on the second hemispherical surface.

## Appendix C Rays crossing the principal axis inside the hemisphere

Consider a ray for which $L\le \Delta \mathrm{cot}\gamma <L+r$. This ray has crossed the principal axis at a point to the right of the center of the hemisphere and therefore the ray will refract away from the principal axis (see Fig. 10). We can now focus on the cases for which $L>\Delta \mathrm{cot}\gamma $.

## Appendix D Simplifying the analysis using the periodicity

After propagating a longitudinal distance Δcot(*γ*), the ray then undergoes up and down total internal reflection periodically, as shown in Fig. 11. Thus, we only need to analyze the focusing property of the microstructure corresponding to a single pitch. We can rewrite the length of supporting stage as *L* = Δcot(*γ*) + *mζ* + *L’*, where $m=floor\left(\frac{L-\Delta \mathrm{cot}\gamma}{\zeta}\right)$ is a nonnegative integer, *ζ* is the pitch of a ray along longitudinal direction, and *L’* is necessarily apositive quantity that is smaller than *ζ*. *L’* represents the remaining distance from the last complete period’s principal axis crossing (as marked by point *A* in Fig. 11) to the center of the hemisphere (as marked by point *C* in Fig. 11(a)). Thus, *L’* = *AC*.

## Appendix E The second governing inequality system

In this section, we analyze the focusing properties of the cascaded structure shown in Fig. 11 for the case that the intersection point *I* is inside the first or third quarter pitch of the ray. This happens when $0<L\text{'}<0.25\zeta $ or $0.5\zeta <L\text{'}<0.\text{7}5\zeta $, as illustrated in Figs. 11(a) and 11(b). In order to bend the ray *AI* or the ray *G _{0}I* towards the principal axis, the output refraction angle

*φ*

_{o}should be larger than

*φ*

_{i}+

*γ*. In conclusion, to focus the ray working in the first or third quarter pitch, the following inequalities should be fulfilled:

## Appendix F The third governing inequality system

In this section, we consider extending the supporting stage such that the intersection point *I* is now inside the second or fourth quarter pitch of the ray as shown in Fig. 12. The ray transitions from the first to second quarter pitch once *L’* > 0.25*ζ*, while the ray transitions from the third to fourth quarter pitch once *L’* > 0.75*ζ*. Also, we should bound the right edge point *O* of the second element to make sure *O* is located on the left of point *T*_{0}, which indicates that *L’*+ *r* < 0.5*ζ* or *L’*+ *r* < *ζ*. Hence, we can deduce the inequality system:

## Appendix G Simulation results for Appendices B-F

Figure 13 presents a numerical calculation of the convergence ratio curves corresponding to the different inequality systems using a step size in *L* of 0.05 *r*. Specifically, the convergence ratio curve with respect to the 1st inequality system dominates the overall convergence ratio of the cascaded structure for small *L*/*r* whereas the contributions from the 2nd and 3rd inequality systems dominate for large *L*/*r*. By parametrizing the input rays using the central angle *θ*, one can show after significant algebra that the contribution from the 1st inequality system is zero for $\frac{L}{r}>\frac{1}{n-1}=2.13$. The contribution from the 2nd inequality system is zero for *L*/*r* < 1.96. An analytical formula is not possible because the output refraction angle condition in Eq. (8) creates a transcendental equation. The contribution from the 3rd inequality system is zero for $\frac{L}{r}<\frac{\left(n+2\right)\sqrt{{n}^{4}-{n}^{2}+1}-n}{\sqrt{{n}^{2}-1}}-\sqrt{{n}^{4}-{n}^{2}+1}=2.79$. We observe these cutoff conditions in the Fig. 13 convergence ratio curves. We find a minimum in the total sum at the point log_{10}(*L*/*r*) = 0.332, i.e. *L* = 2.15 *r*. This value agrees with the cutoff condition of *L* = 2.13 *r* given the step size in *L* of 0.05 *r*. We also observe that the contribution from the 2nd inequality system starts to increase sharply in the range of 0.332 to 0.724, i.e. *L* = 2.15 *r* to *L* = 5.30 *r*, followed by a fast decrease and finally stabilizes at around 14.5%, as shown in Fig. 13. The curve with respect to the 3rd inequality system starts to increase after the point 0.446, i.e. *L* = 2.80 *r*, and converges to the fixed value of 18.3%. In the actual design of the PNJ generator, we should carefully choose *L*/*r* to ensure that we avoid the small convergence ratio region near the point log_{10}(*L*/*r*) = 0.332 because we want efficient energy focusing. Moreover, note that the second inequality system dominates the range 0.332 to 1.50 and that rays for the second inequality system are initially moving away from the principal axis before being bent by the output hemisphere to a very small inclined angle with respect to the principal axis. Thus, the PNJ generated by a design within this range is expected to be an elongated one with weak intensity. Hence, for the case where a strong and short PNJ is required, one should avoid the range dominated by the second inequality system.

To demonstrate the insights provided by the above analysis, we conducted additional simulations using full-wave analysis for three representative points shown in Fig. 13, i.e., log_{10}(*L*/*r*) = 0, 0.332 and 0.724 for *r* = 1 μm and 3 μm hemispheres. The *r* = 1 μm cases are presented in Figs. 4(c), 4(d), and 4(e) of the main text, respectively. The *r* = 3 μm cases are presented in Fig. 14. Apparently, for the case log_{10}(*L*/*r*) = 0, i.e. the second element only has a hemisphere, a short PNJ with large intensity is generated outside of the structure as shown in Fig. 14(a). For the case *L* = 10^{0.332}*r*, strong ray divergence is observed in the adjacent region of the right end of the cascaded structure in Fig. 14(b). In Fig. 14(c), we observe an extremely long but weak PNJ generated quite far away from the hemispherical surface of the cascaded structure. These observations are all in accordance with the analysis of the convergence ratios at these points. Thus, we can engineer the characteristics of the output PNJ of the cascaded structure simply by selecting its geometrical dimensions.

## Funding

Cisco Systems Inc. (gift award CG 624009).

## Acknowledgment

We are grateful to Cisco Systems Inc. for access to its Arcetri cluster.

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