# Deep Subwavelength-Scale Light Focusing and Confinement in Nanohole-Structured Mesoscale Dielectric Spheres

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## Abstract

**:**

## 1. Introduction

## 2. Dielectric Microspheres without Hole

_{s}= 1.5λ, 2.5λ, 3.5λ and 4.5λ, which were numerically modeled and simulated. For λ = 600 nm, the simulated dielectric spheres had a diameter of D

_{s}= 0.9, 1.5, 2.1, and 2.7 µm, respectively. We selected a refractive index of n =1.5, because in an optical band, many commonly used dielectric materials have a refractive index nearly equal to 1.5, such as glass, PMMA, fused silica, etc. [17]. We chose the popular spherical shape of a microparticle for the photonic nanojet formation [4]. The dielectric microspheres were modeled and simulated by using the commercial software COMSOL Multiphysics, which is based on the finite elements method (FEM). In this simulation, a non-uniform mesh was employed to reduce the computational cost and the Perfect Matched Layer (PML) was applied as the boundary condition. The incident light was assumed to be a plane wave that propagates along the z-axis, with a linear polarization along the y-axis. A schematic diagram for the simulated dielectric microsphere is given in Figure 1a.

_{x}, S

_{y}, S

_{z}), the focal volumes V (which were obtained by volume integration inside the FWHM focal spot) and the maximum light intensity I

_{max}, in the focal spot. After combining Figure 2 and Table 1, we found that, with an increase in the sphere diameter from D

_{s}= 1.5λ to 3.5λ, the focal volume V decreased from 0.057λ

^{3}to 0.051λ

^{3}, while the maximum light intensity I

_{max}increased from 16.1I

_{0}to 61.1I

_{0}(I

_{0}is the light intensity of the incident light). Furthermore, when the sphere diameter was increased to D

_{s}= 4.5λ, the focal spot split into two parts, with the major part of the focal spot expanding into the surrounding medium, and a smaller part staying inside the microsphere [2,4]. Thus, for the microspheres with a refractive index of n = 1.5, when the sphere had a smaller diameter (D

_{s}< 4.5λ), the light incident on the sphere was focused on the shadow surface (the surface of the particle that is opposite to the irradiated side [4]) of the sphere. When the sphere was large enough (D

_{s}≥ 4.5λ), PNJ formed beyond the shadow surface of the sphere [2,4], and the second focal spot formed inside the sphere and close to its shadow surface, due to light reflection at the inner side of the shadow surface [4]. It could be noted that, in this case, the maximal field intensity was slightly reduced (Table 1).

## 3. Nanohole-Structured Dielectric Microspheres

_{s}= 3.5λ and a refractive index of n = 1.5, which were simulated and analyzed. To make the structure clear, schematic diagrams for the simulated microspheres with a through hole and a blind hole are depicted in Figure 1b,c, respectively.

#### 3.1. Microspheres with a through Hole

_{h}= λ/5, λ/10, and λ/15 were simulated; light intensity around the simulated microspheres are shown in Figure 3. The corresponding focal spot properties, including the FWHM focal spot sizes (S

_{x}, S

_{y}, S

_{z}), the focal volumes V, and the maximum light intensity I

_{max}, in the focal spot, are shown in Table 2. After comparing Figure 3a,b with Figure 2e,f, we found that the focal spot sizes (S

_{x,y,z}) and focal volume V of the λ/5-nanohole-structured microsphere were even larger than that of the microspheres without a nanohole, which could be explained as the weakening of the light focusing capability of the dielectric microsphere, due to the comparatively large λ/5-sized hole. When the hole size was reduced to be smaller than λ/10, the focal spot sizes and focal volume were reduced considerably, as shown in Figure 3c–h and Table 2. Finally, the features of the FWHM focal spot of the microspheres with λ/15-sized hole are plotted in Figure 3g, where the green solid lines, the green dashed lines, and the gray solid lines represent the contour lines with a value of 0.5I

_{max}, 0.8I

_{max}, and 0.9I

_{max}, respectively. A comparison of the data in Table 1 and Table 2 shows that, with a hole of λ/15 diameter, the maximum field intensity near the shadow surface of the particle is increased by nearly two times, with a significant decrease in the field localization volume, compared to the unstructured particle.

#### 3.2. Microspheres with a Blind Hole

_{h}= λ/5, λ/10, and λ/40, respectively, with a hole depth of 3d

_{h}. The corner radius at the opening of the hole, as well as the fillet radius at the blind end of the hole, were both set to be r = d

_{h}/2. From Figure 4a,b, we can see that for a sphere with a hole size of d

_{h}= λ/5, the focusing capability of the sphere is weakened by the hole, compared to a sphere without a hole in Figure 1e,f. When the hole sizes are decreased below λ/10, the focused light spot was mainly confined to the blind hole, as shown in Figure 4c–h. “Hot spots” can be observed in Figure 4e,h, which were located inside the hole and near the opening of the hole in the polarization plane (z–y plane), as indicated by the contour lines at the value of 0.85I

_{max}and 0.9I

_{max}.

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Heifetz, A.; Kong, S.C.; Sahakian, A.V.; Taflove, A.; Backman, V. Photonic Nanojets. J. Comput. Theor. Nanosci.
**2009**, 6, 1979–1992. [Google Scholar] [CrossRef] [PubMed] - Minin, I.V.; Minin, O.V. Diffractive Optics and Nanophotonics: Resolution below the Diffraction Limit; Springer International Publishing: New York, NY, USA, 2016. [Google Scholar]
- Ferrand, P.; Wenger, J.; Devilez, A.; Pianta, M.; Stout, B.; Bonod, N.; Popov, E.; Rigneault, H. Direct imaging of photonic nanojets. Opt. Express
**2008**, 16, 6930–6940. [Google Scholar] [CrossRef] [PubMed] - Lukyanchuk, B.S.; Dominguez, R.P.; Minin, I.V.; Minin, O.V.; Wang, Z. Refractive index less than two: Photonic nanojets yesterday, today and tomorrow. Opt. Mater. Express
**2017**, 7, 1820–1847. [Google Scholar] [CrossRef] - Li, Y.; Xin, H.; Liu, X.; Zhang, Y.; Lei, H.; Li, B. Trapping and detection of nanoparticles and cells using a parallel photonic nanojet array. ACS Nano
**2016**, 10, 5800–5808. [Google Scholar] [CrossRef] [PubMed] - Minin, I.V.; Minin, O.V.; Pacheco-Peña, V.; Beruete, M. Subwavelength, standing-wave optical trap based on photonic jets. Quantum Electron.
**2016**, 46, 555–557. [Google Scholar] [CrossRef] - Yang, H.; Trouillon, R.; Huszka, G.; Gijs, M. Super-resolution imaging of a dielectric microsphere is governed by the waist of its photonic nanojet. Nano Lett.
**2016**, 16, 4862–4870. [Google Scholar] [CrossRef] [PubMed] - Born, B.; Krupa, J.; Gagnon, S.; Holzman, J. Integration of photonic nanojets and semiconductor nanoparticles for enhanced all-optical switching. Nat. Commun.
**2015**, 6, 8097. [Google Scholar] [CrossRef] [PubMed] - Liu, C.; Yen, T.; Minin, O.V.; Minin, I.V. Engineering photonic nanojet by a graded-index micro-cuboid. Physica E
**2018**, 98, 105–110. [Google Scholar] [CrossRef] - Shen, Y.; Wang, L.; Shen, J. Ultralong photonic nanojet formed by a two-layer dielectric microsphere. Opt. Lett.
**2014**, 39, 4120–4123. [Google Scholar] [CrossRef] [PubMed] - Yue, L.; Yan, B.; Monks, J.; Dhama, R.; Wang, Z.; Minin, O.V.; Minin, I.V. Intensity-enhanced apodization effect on an axially illuminated circular-column particle-lens. Ann. Phys.
**2018**, 530, 1700384. [Google Scholar] [CrossRef] - Yue, L.; Yan, B.; Monks, J.; Wang, Z.; Tung, N.T.; Lam, V.D.; Minin, O.V.; Minin, I.V. Production of photonic nanojets by using pupil-masked 3D dielectric cuboid. J. Phys. D Appl. Phys.
**2017**, 50, 175102. [Google Scholar] [CrossRef] - Gu, G.; Song, J.; Liang, H.; Zhao, M.; Chen, Y.; Qu, J. Overstepping the upper refractive index limit to form ultra-narrow photonic nanojets. Sci. Rep.
**2017**, 7, 5635. [Google Scholar] [CrossRef] [PubMed] - Latimer, P. Light scattering by a structured particle: The homogeneous sphere with holes. Appl. Opt.
**1984**, 23, 1844. [Google Scholar] [CrossRef] [PubMed] - Wu, M.X.; Huang, B.J.; Chen, R.; Yang, Y.; Wu, J.F.; Ji, R.; Chen, X.D.; Hong, M.H. Modulation of photonic nanojets generated by microspheres decorated with concentric rings. Opt. Express
**2015**, 23, 20096–20103. [Google Scholar] [CrossRef] [PubMed] - Isro, S.D.; Iskandar, A.A.; Kivshar, Y.S.; Shadrilov, I.V. Engineering scattering patterns with asymmetric dielectric nanorods. Opt. Express
**2018**, 26, 32624. [Google Scholar] [CrossRef] [PubMed] - Refractive Index Database. Available online: https://refractiveindex.info (accessed on 21 January 2019).
- Almeida, V.R.; Xu, Q.; Barrios, C.A.; Lipson, M. Guiding and confining light in void nanostructure. Opt. Lett.
**2004**, 29, 1209–1211. [Google Scholar] [CrossRef] [PubMed] - Barrios, C.A.; Lipson, M. Electrically driven silicon resonant light emitting device based on slot-waveguide. Opt. Express
**2005**, 13, 10092. [Google Scholar] [CrossRef] [PubMed] - Mason, D.R.; Jouravlev, M.V.; Kim, K.S. Enhanced resolution beyond the Abbe diffraction limit with wavelength-scale solid immersion lenses. Opt. Lett.
**2010**, 35, 2007–2009. [Google Scholar] [CrossRef] [PubMed]

**Figure 1.**Schematic diagrams for the simulated dielectric sphere (

**a**) without a nanohole, (

**b**) with a through hole, and (

**c**) with a blind hole. The incident light is a plan wave that propagates along the z-axis,

**k**is the wave vector. The incident field

**E**is polarized along the y-axis. The oval-shaped zones in dark red color indicate the focal spots, and dash-dotted lines indicate the symmetrical axes for the spheres.

**Figure 2.**Light intensity distribution of a simulated microsphere with a sphere diameter of (

**a**,

**b**) D

_{s}= 1.5λ, (

**c**,

**d**) D

_{s}= 2.5λ, (

**e**,

**f**) D

_{s}= 3.5λ, and (

**g**,

**h**) D

_{s}= 4.5λ. Subfigures (

**a**,

**c**,

**e**,

**g**) are plotted in the zx-plane, which is perpendicular to the plane of polarization; subfigures (

**b**,

**d**,

**g**), and (

**h**) are plotted in the zx-plane, which is the plane of polarization. The Full Width at Half Maximum (FWHM) focal spot of the simulated microspheres are indicated by the contour lines at the value of half maximum light intensity 0.5I

_{max}, which are plotted by the solid green lines.

**Figure 3.**Light intensity (|E|

^{2}) of the simulated dielectric microspheres with a nanohole of size (

**a**,

**b**) d

_{h}= λ/5, (

**c**,

**d**) d

_{h}= λ/10, and (

**e**–

**h**) d

_{h}= λ/15. The sphere diameter and refractive index are set as D

_{s}= 3.5λ and n = 1.5. Features of the focal spot in (

**f**) are plotted in (

**g**), where the green solid lines, the gray dashed lines, and the gray solid lines indicate the contour lines at values 0.5I

_{max}, 0.8I

_{max}, and 0.9I

_{max}, respectively. For a sphere with a hole size of d

_{h}= λ/15, light intensity along a plane that is 0.001λ from the shadow surface is plotted in (

**h**), with a circle plotted in gray dashed lines, indicating the projection of the hole interface.

**Figure 4.**Light intensity (|E|

^{2}) of the simulated dielectric microspheres with a blind nanohole of the diameter (

**a**–

**c**) d

_{h}= λ/10, (

**d**–

**f**) d

_{h}= λ/10, and (

**g**–

**i**) d

_{h}= λ/40. The sphere diameter and refractive index are set as D

_{s}= 3.5λ and n = 1.5, with a hole depth of 3d

_{h}. Subfigures (

**a**,

**d**,

**g**) are plotted in the zx-plane, which is perpendicular to the polarization plane. Subfigures (

**b**,

**e**,

**h**) are plotted in the zy-plane, which is the polarization plane. To indicate the “hot spots” in subfigures (

**e**,

**h**,

**i**), contour lines at the value of 0.5I

_{max}, 0.85I

_{max}, and 0.9I

_{max}are plotted by the green solid lines, the green dashed lines, and the gray solid lines, respectively. Subfigures (

**c**,

**f**,

**i**) show the light intensity along a plane that is 0.001λ from the shadow surface of the sphere and the circles, which are plotted by gray dashed lines, indicating the projection of the hole interface in the shadow surface.

**Figure 5.**Light intensity along the two imaginary cutting lines L1 and L2, which lie on the shadow surface of the simulated microspheres. L1 is defined as y = 0 and z = D

_{s}/2 + 0.001λ, which is parallel to the x-axis and perpendicular to the polarization direction. L2 is defined as x = 0 and z = D

_{s}/2 + 0.001λ, which is parallel to the y-axis, as well as the polarization direction. The gray circles, the blue dashed lines, and the red solid lines indicate the light intensity of the 3.5λ-diameter microsphere without hole, the microsphere with a through hole of λ/15 (see Figure 3e–h), and the microsphere with a blind hole of λ/40 (see Figure 4g–i), respectively.

**Table 1.**Light focusing properties of the microspheres with a fixed refractive index of n = 1.5 and different sphere diameters (D

_{s}).

D_{s} | S_{x} | S_{y} | S_{z} | V | I_{max} |
---|---|---|---|---|---|

1.5λ | 0.33λ | 0.77λ | 0.50λ | 0.057λ^{3} | 16.1I_{0} |

2.5λ | 0.32λ | 0.78λ | 0.43λ | 0.055λ^{3} | 35.5I_{0} |

3.5λ | 0.35λ | 0.74λ | 0.49λ | 0.051λ^{3} | 61.1I_{0} |

4.5λ | 0.50λ | 0.81λ | 0.81λ | 0.173λ^{3} | 57.4I_{0} |

**Table 2.**Light focusing properties of the simulated nanohole-structured microspheres with a sphere diameter of D

_{s}= 3.5λ, refractive index of n = 1.5, and a through hole of d

_{h}= λ/15 in diameter.

D_{s} | S_{x} | S_{y} | S_{z} | V | I_{max} |
---|---|---|---|---|---|

λ/5 | 0.39λ | 0.81λ | 0.63λ | 0.064λ^{3} | 48.6I_{0} |

λ/10 | 0.25λ | 0.42λ | 0.27λ | 0.0033λ^{3} | 93.5I_{0} |

λ/15 | 0.20λ | 0.14λ | 0.25λ | 0.0012λ^{3} | 112I_{0} |

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**MDPI and ACS Style**

Cao, Y.; Liu, Z.; Minin, O.V.; Minin, I.V. Deep Subwavelength-Scale Light Focusing and Confinement in Nanohole-Structured Mesoscale Dielectric Spheres. *Nanomaterials* **2019**, *9*, 186.
https://doi.org/10.3390/nano9020186

**AMA Style**

Cao Y, Liu Z, Minin OV, Minin IV. Deep Subwavelength-Scale Light Focusing and Confinement in Nanohole-Structured Mesoscale Dielectric Spheres. *Nanomaterials*. 2019; 9(2):186.
https://doi.org/10.3390/nano9020186

**Chicago/Turabian Style**

Cao, Yinghui, Zhenyu Liu, Oleg V. Minin, and Igor V. Minin. 2019. "Deep Subwavelength-Scale Light Focusing and Confinement in Nanohole-Structured Mesoscale Dielectric Spheres" *Nanomaterials* 9, no. 2: 186.
https://doi.org/10.3390/nano9020186