OSA's Digital Library

Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 2 — Jan. 28, 2013
  • pp: 2434–2443
« Show journal navigation

Near-field focusing of the dielectric microsphere with wavelength scale radius

Hanming Guo, Yunxuan Han, Xiaoyu Weng, Yanhui Zhao, Guorong Sui, Yang Wang, and Songlin Zhuang  »View Author Affiliations


Optics Express, Vol. 21, Issue 2, pp. 2434-2443 (2013)
http://dx.doi.org/10.1364/OE.21.002434


View Full Text Article

Acrobat PDF (2186 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

Abstract

We focus on physically analyzing the origins of the numerical aperture ( 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 when the refractive index contrast ( RIC ) between the microsphere and its surrounding medium is about from 1.5 to 1.75. The 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 1.5RIC1.75 with the proper radius r of microsphere, the microsphere can generate a near-field focal spot with lateral resolution slightly beyond λ/ 2 n s , which is also the lateral resolution limit of the dielectric microsphere. The r for each RIC can be obtained by optimizing r from 1.125λ / n o to 1.275λ / n o . Here λ , n s , and n o are the wavelength in vacuum and the refractive indices of microsphere and its surrounding medium, respectively. For the case of the near-field focusing, we also develop a simple transform formula used to calculate the new radius from the known radius of microsphere corresponding to the original illumination wavelength when the illumination wavelength is changed.

© 2013 OSA

1. Introduction

Breaking through the Abbe diffraction limit become a persistently hot topic in modern optics [1

1. J. Y. Lee, B. H. Hong, W. Y. Kim, S. K. Min, Y. Kim, M. V. Jouravlev, R. Bose, K. S. Kim, I. C. Hwang, L. J. Kaufman, C. W. Wong, P. Kim, and K. S. Kim, “Near-field focusing and magnification through self-ssembled nanoscale spherical lenses,” Nature 460(7254), 498–501 (2009). [CrossRef]

3

3. Z. Wang, W. Guo, L. Li, B. Luk'yanchuk, A. Khan, Z. Liu, Z. Chen, and M. Hong, “Optical virtual imaging at 50nm lateral resolution with a white-light nanoscope,” Nat. Commun. 2, 218 (2011). [CrossRef]

]. In 2004, Chen et al. find that the transparent dielectric microsphere with wavelength scale radius can generate a photonic nanojet (PNJ) [4

4. Z. Chen, A. Taflove, and V. Backman, “Photonic nanojet enhancement of backscattering of light by nanoparticles: a potential novel visible-light ultramicroscopy technique,” Opt. Express 12(7), 1214–1220 (2004). [CrossRef] [PubMed]

]. 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 λ/2 and can propagate over 2λ with low divergence [5

5. A. Heifetz, S. C. Kong, A. V. Sahakian, A. Taflove, and V. Backman, “Photonic Nanojets,” J Comput Theor Nanosci 6(9), 1979–1992 (2009). [CrossRef] [PubMed]

], where λ 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

6. Y. E. Geints, A. A. Zemlyanov, and E. K. Panina, “Photonic jets from resonantly excited transparent dielectric microspheres,” J. Opt. Soc. Am. B 29(4), 758–762 (2012). [CrossRef]

]. Since 2004, many authors [5

5. A. Heifetz, S. C. Kong, A. V. Sahakian, A. Taflove, and V. Backman, “Photonic Nanojets,” J Comput Theor Nanosci 6(9), 1979–1992 (2009). [CrossRef] [PubMed]

13

13. Y. Ku, C. Kuang, X. Hao, Y. Xue, H. Li, and X. Liu, “Superenhanced three-dimensional confinement of light by compound metal-dielectric microspheres,” Opt. Express 20(15), 16981–16991 (2012). [CrossRef]

] 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

4. Z. Chen, A. Taflove, and V. Backman, “Photonic nanojet enhancement of backscattering of light by nanoparticles: a potential novel visible-light ultramicroscopy technique,” Opt. Express 12(7), 1214–1220 (2004). [CrossRef] [PubMed]

16

16. D. A. Fletcher, K. E. Goodson, and G. S. Kino, “Focusing in microlenses close to a wavelength in diameter,” Opt. Lett. 26(7), 399–401 (2001). [CrossRef] [PubMed]

] 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

17. T. J. Gould, S. T. Hess, and J. Bewersdorf, “Optical nanoscopy: from acquisition to analysis,” Annu. Rev. Biomed. Eng. 14(1), 231–254 (2012). [CrossRef] [PubMed]

] or the PNJ. Heifetz et. al [5

5. A. Heifetz, S. C. Kong, A. V. Sahakian, A. Taflove, and V. Backman, “Photonic Nanojets,” J Comput Theor Nanosci 6(9), 1979–1992 (2009). [CrossRef] [PubMed]

] claim that the smallest FWHM of the PNJ is as small as λ/3, but the basis seem to be Fig. 1(a)
Fig. 1 Schematic of a microsphere illuminated by a incident plane-wave.
of Ref. 7

7. X. Li, Z. Chen, A. Taflove, and V. Backman, “Optical analysis of nanoparticles via enhanced backscattering facilitated by 3-D photonic nanojets,” Opt. Express 13(2), 526–533 (2005). [CrossRef] [PubMed]

, where the smallest FWHM of the PNJ is 130nm for the microsphere with λ=400nm, refractive index ns=1.59, and radius r=500nm. Actually, the spot shown in the Fig. 1(a) of Ref. 7

7. X. Li, Z. Chen, A. Taflove, and V. Backman, “Optical analysis of nanoparticles via enhanced backscattering facilitated by 3-D photonic nanojets,” Opt. Express 13(2), 526–533 (2005). [CrossRef] [PubMed]

, corresponding to the λ/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 λ/2. Fletcher et. al [16

16. D. A. Fletcher, K. E. Goodson, and G. S. Kino, “Focusing in microlenses close to a wavelength in diameter,” Opt. Lett. 26(7), 399–401 (2001). [CrossRef] [PubMed]

] study the focusing of the micrsolens with wavelength scale size, but they only demonstrate that the vector diffraction theory of Richards and Wolf [18

18. B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 358–379 (1959). [CrossRef]

] overpredicts the focal spot size in the microlens with wavelength scale size. Recently, the microsphere is used in nanolithography [14

14. E. McLeod and C. B. Arnold, “Subwavelength direct-write nanopatterning using optically trapped microspheres,” Nat. Nanotechnol. 3(7), 413–417 (2008). [CrossRef] [PubMed]

, 15

15. J. Kim, K. Cho, I. Kim, W. M. Kim, T. S. Lee, and K. S. Lee, “Fabrication of plasmonic nanodiscs by photonic nanojet lighography,” Appl. Phys. Express 5(2), 025201 (2012). [CrossRef]

] and nanoscale imaging [19

19. J. J. Schwartz, S. Stavrakis, and S. R. Quake, “Colloidal lenses allow high-temperature single-molecule imaging and improve fluorophore photostability,” Nat. Nanotechnol. 5(2), 127–132 (2010). [CrossRef] [PubMed]

, 20

20. Z. Wang, W. Guo, L. Li, B. Luk'yanchuk, A. Khan, Z. Liu, Z. Chen, and M. Hong, “Optical virtual imaging at 50nm lateral resolution with a white-light nanoscope,” Nat. Commun. 2, 1–6 (2011). [CrossRef]

]. However, these authors [14

14. E. McLeod and C. B. Arnold, “Subwavelength direct-write nanopatterning using optically trapped microspheres,” Nat. Nanotechnol. 3(7), 413–417 (2008). [CrossRef] [PubMed]

, 15

15. J. Kim, K. Cho, I. Kim, W. M. Kim, T. S. Lee, and K. S. Lee, “Fabrication of plasmonic nanodiscs by photonic nanojet lighography,” Appl. Phys. Express 5(2), 025201 (2012). [CrossRef]

, 19

19. J. J. Schwartz, S. Stavrakis, and S. R. Quake, “Colloidal lenses allow high-temperature single-molecule imaging and improve fluorophore photostability,” Nat. Nanotechnol. 5(2), 127–132 (2010). [CrossRef] [PubMed]

, 20

20. Z. Wang, W. Guo, L. Li, B. Luk'yanchuk, A. Khan, Z. Liu, Z. Chen, and M. Hong, “Optical virtual imaging at 50nm lateral resolution with a white-light nanoscope,” Nat. Commun. 2, 1–6 (2011). [CrossRef]

] 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

19. J. J. Schwartz, S. Stavrakis, and S. R. Quake, “Colloidal lenses allow high-temperature single-molecule imaging and improve fluorophore photostability,” Nat. Nanotechnol. 5(2), 127–132 (2010). [CrossRef] [PubMed]

, 20

20. Z. Wang, W. Guo, L. Li, B. Luk'yanchuk, A. Khan, Z. Liu, Z. Chen, and M. Hong, “Optical virtual imaging at 50nm lateral resolution with a white-light nanoscope,” Nat. Commun. 2, 1–6 (2011). [CrossRef]

], 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

1. J. Y. Lee, B. H. Hong, W. Y. Kim, S. K. Min, Y. Kim, M. V. Jouravlev, R. Bose, K. S. Kim, I. C. Hwang, L. J. Kaufman, C. W. Wong, P. Kim, and K. S. Kim, “Near-field focusing and magnification through self-ssembled nanoscale spherical lenses,” Nature 460(7254), 498–501 (2009). [CrossRef]

, 21

21. M. S. Kim, T. Scharf, M. T. Haq, W. Nakagawa, and H. P. Herzig, “Subwavelength-size solid immersion lens,” Opt. Lett. 36(19), 3930–3932 (2011). [CrossRef] [PubMed]

]. Further theoretical research demonstrates that the nSIL can generate a near-field focal spot with lateral resolution beyond λ/2nsand has higher performance than the macroscopic SIL [22

22. D. R. Mason, M. V. Jouravlev, and K. S. Kim, “Enhanced resolution beyond the Abbe diffraction limit with wavelength-scale solid immersion lenses,” Opt. Lett. 35(12), 2007–2009 (2010). [CrossRef] [PubMed]

]. 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.

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, NAshould 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?

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>λ have recently been proved in theory [23

23. C. J. R. Sheppard and P. Török, “Focal shift and the axial optical coordinate for high-aperture systems of finite Fresnel number,” J. Opt. Soc. Am. A 20(11), 2156–2162 (2003). [CrossRef] [PubMed]

, 24

24. Y. Li, “Focal shifts in diffracted converging electromagnetic waves. I. Kirchhoff theory,” J. Opt. Soc. Am. A 22(1), 68–76 (2005). [CrossRef] [PubMed]

] and experiment [27

27. J. M. Yi, A. Cuche, F. de León-Pérez, A. Degiron, E. Laux, E. Devaux, C. Genet, J. Alegret, L. Martín-Moreno, and T. W. Ebbesen, “Diffraction regimes of single holes,” Phys. Rev. Lett. 109(2), 023901 (2012). [CrossRef] [PubMed]

]. Shown in Fig. 1, the x polarized and monochromatic plane wave incidence on a microsphere with radius r, refractive index ns, and center at the point O, along the +z axis. no is the refractive index of the surrounding medium. Namely RIC=ns/no. Obeying the ray tracing process, a ray with incident angle θo intersects the microsphere and the z axis at the points A, B, and F, respectively. θs and θi are the corresponding the refractive angles at the points A and B, respectively. Here θi=θo. α is the angle of the emergent ray with the z axis. Cis 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
α=2θo2θs,
(1)
d=r[cos(θoα)+sin(θoα)cotα1],
(2)
where θo and θs meet the Snell law nosinθo=nssinθs. Equation (2) is only valid for the point F outside microsphere. If the point F is inside microsphere, namely the points A and Blocating at the different side of the z axis, d=r[sinθocot(θoθs)cosθo1].

When the microsphere is considered a thick lens, the point F moves toward the microsphere with the increase of θo. For all rays, θo meets 0θo<π/2. It is easy to derive that the point F is outside the microsphere for 0θo<2θs, i.e., 0θo<2acos(ns/2no) and inside the microsphere for 2acos(ns/2no)<θo<π/2. As the focal spot outside the microsphere majorly arises from the focusing contributions of the rays with the incident angle 0θo<2θs, we define the spherical aberration S=dmaxdmin, where, in terms of Eqs. (1) and (2), dmin0 by setting θo=2acos(ns/2no)δ with a very small value δ=0.00001 in radians and dmax approximately calculated by setting θo=δ. dmax and dmin are the maximum and minimum values of d, respectively. For the microsphere, we define NA=nosinαmax, where the maximum value αmax of α can be calculated from the maximum value θomax=2acos(ns/2no)δ.

As we known, NA and the aberration are critical factors affecting the focusing properties of an optical system. To achieve maximum lateral resolution, NAshould 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<λ/2no. Shown in Fig. 2
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.
, as the validity of the ray tracing is restricted in r>λ, the RIC should be bigger than 1.5 for S<λ/2no. In this case, both S and NA decrease with the increase of RIC. A tradeoff between S and NA is needed. Moreover, due to θomax2acos(ns/2no), 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/ns, 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 RIC1.75, NA0.85 (see Fig. 2). Therefore, within the approximate range of 1.5RIC1.75, the microsphere has not only small S, but also high NA by nature.

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

23. C. J. R. Sheppard and P. Török, “Focal shift and the axial optical coordinate for high-aperture systems of finite Fresnel number,” J. Opt. Soc. Am. A 20(11), 2156–2162 (2003). [CrossRef] [PubMed]

25

25. S. Guo, H. Guo, and S. Zhuang, “Analysis of imaging properties of a microlens based on the method for a dyadic Green’s function,” Appl. Opt. 48(2), 321–327 (2009). [CrossRef] [PubMed]

], 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

18. B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 358–379 (1959). [CrossRef]

] 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

28. J. J. Stamnes, Waves in Focal Regions (Taylor & Francis Group, 1986), p456.

]. Li [24

24. Y. Li, “Focal shifts in diffracted converging electromagnetic waves. I. Kirchhoff theory,” J. Opt. Soc. Am. A 22(1), 68–76 (2005). [CrossRef] [PubMed]

] 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/λaNAa and the small circular aperture with the numerical aperture NAa 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, NAa of the small circular aperture is different from the NA of the microsphere defined before. λa is the wavelength λ in vacuum for the small aperture and is majorly determined by the wavelength in the microsphere (i.e., λa=λ/ns) 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

24. Y. Li, “Focal shifts in diffracted converging electromagnetic waves. I. Kirchhoff theory,” J. Opt. Soc. Am. A 22(1), 68–76 (2005). [CrossRef] [PubMed]

]
Ex(z)=A4{(NAa)24cos12Ω[4+(1cosΩ)cosΩ][exp(jKτ)τ2(11jKτ)]θ=Ω,jK0Ωχ(θ,z)(1+cosθ)cos12θexp(jKτ)τsinθdθ}
(3a)
Ey(z)=0,
(3b)
Ez(z)=A4(NAa)4cos12Ω(zf+cosΩ)(1cosΩ)cosΩ[exp(jKτ)τ2(11jKτ)]θ=Ω,
(3c)
with τ=[(1+z/f)22(z/f)(1cosθ)]1/2, χ(θ,z)=1+τ1[1+(z/f)][1(jKτ)1], K=kf =2πN/(NAa)2, and NAa=r/f=sinΩ. A is the amplitude of the incident x polarized plane wave.

We set NAa=0.965 and utilize Eq. (3) to calculate the distribution of light intensity along the +z axis [see Fig. 3(a)
Fig. 3 Focal shifts along the z axis of the small circular aperture with radius r and NAa=0.965, where z=0 denotes the position of the geometrical focus and f is the focal length. (a) blue solid line: r=1.25λa; red dashed line: r=3λa; green dash-dotted line: r=5λa. (b) r=1.25λa 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.
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=1.25λa; red dashed line: r=3λa; green dash-dotted line: r=5λa). Moreover, the focal shifts are not obvious when r>5λa for the small circular aperture, which means that the focal shifts might be small for the microspheres with r>5λa=5λ/ns.

In order to examine the contributions of each ray on the focal shifts for a given aperture, we set NAa=0.965 and r=1.25λa and divide equally the wave front at the aperture into five zones within the maximum aperture angle Ω [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=nosinαmax 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., nosinαmax) defined before. In terms of Fig. 3(b), we can find that, the small aperture used in Ref. 24

24. Y. Li, “Focal shifts in diffracted converging electromagnetic waves. I. Kirchhoff theory,” J. Opt. Soc. Am. A 22(1), 68–76 (2005). [CrossRef] [PubMed]

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

24. Y. Li, “Focal shifts in diffracted converging electromagnetic waves. I. Kirchhoff theory,” J. Opt. Soc. Am. A 22(1), 68–76 (2005). [CrossRef] [PubMed]

when the above definition S=dmaxdmin 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<λ/2no 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 1.5RIC1.75. We expect that, only within 1.5RIC1.75, 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>λ 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 1×105 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 1.5RIC1.75. 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

18. B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 358–379 (1959). [CrossRef]

, 22

22. D. R. Mason, M. V. Jouravlev, and K. S. Kim, “Enhanced resolution beyond the Abbe diffraction limit with wavelength-scale solid immersion lenses,” Opt. Lett. 35(12), 2007–2009 (2010). [CrossRef] [PubMed]

]. 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 λ used in FDTD are shown in Fig. 4
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 no: (a)-(c) and (e) no=1 and (d) no=1.34. The parameter λ: (a)-(c) and (e)λ=400nm, and (d) λ=355nm. The parameterRIC: (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.
caption.

For a given λ and RIC within 1.5RIC1.75, we have FDTD simulations by increasing the radius r with step 5nm from r>λ. 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 λ/2ns in the near-field region. For example, the microsphere with r=490nm, RIC=1.59, and λ=400nm, the lateral resolution is 120nm and slightly better than 126nm calculated by λ/2ns and ns=RIC×no=1.59. Shown in Fig. 4(e), within 1.5RIC1.75, 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 1.5RIC1.75, 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 λ/2ns. The reason is that, only within the approximate region of 1.5RIC1.75, 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 ns for different λ 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 λ=400nm, r is from 450nm to 510nm, namely from 1.125λ/no to 1.275λ/no. 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 1.125λ/no to 1.275λ/no. 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 λ (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 λ 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 actualNA is slightly bigger than the NA (i.e., nosinαmax) 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 α remain unchanged regardless of λ. 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 λ/no remains unchanged, rno/λ is invariable. In addition, the focal shifts are determined by the ratio of the r to the wavelengthλ/no.Therefore, for the fixed RIC, if the ratio rno/λ remains unchanged, the OMS can still be obtained and its new radius r can be expressed as
r=rnoλ/noλ,
(4)
where the superscript prime denotes the corresponding new parameters.

As a example, for the OMS with the λ=400nm, r=490nm, RIC=1.59 and no=1 [see Fig. 4(b)], if the new wavelength λ=355nm and no=1.34, the new radius r calculated by Eq. (4) is 325nm. Shown in Fig. 4(d), under the case of λ=355nm, r=325nm, RIC=1.59, and no=1.34, the lateral resolution along the y axis is 80nm and slightly better than 83nm calculated by λ/2ns and ns=RIC×no=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 1.5RIC1.75, 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 1.5RIC1.75 with the proper r, the OMS can be realized, namely the lateral resolution of its near-field focal spot slightly beyond λ/2ns, which is also the lateral resolution limit of the dielectric microsphere. The r for each RIC can be obtained by optimizing r from 1.125λ/no to 1.275λ/no. 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 ns 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.

Acknowledgments

This work was supported by the National Basic Research Program of China (2011CB707504), the Leading Academic Discipline Project of Shanghai Municipal Government (S30502), the National Natural Science Foundation of China (61178079 and 61137002), the Fok Ying-Tong Education Foundation, China (121010), the Foundation for the Author of National Excellent Doctoral Dissertation of PR China (201033), and the Science and Technology Commission of Shanghai Municipality (STCSM) (11JC1413300).

References and links

1.

J. Y. Lee, B. H. Hong, W. Y. Kim, S. K. Min, Y. Kim, M. V. Jouravlev, R. Bose, K. S. Kim, I. C. Hwang, L. J. Kaufman, C. W. Wong, P. Kim, and K. S. Kim, “Near-field focusing and magnification through self-ssembled nanoscale spherical lenses,” Nature 460(7254), 498–501 (2009). [CrossRef]

2.

J. J. Schwartz, S. Stavrakis, and S. R. Quake, “Colloidal lenses allow high-temperature single-molecule imaging and improve fluorophore photostability,” Nat. Nanotechnol. 5(2), 127–132 (2010). [CrossRef] [PubMed]

3.

Z. Wang, W. Guo, L. Li, B. Luk'yanchuk, A. Khan, Z. Liu, Z. Chen, and M. Hong, “Optical virtual imaging at 50nm lateral resolution with a white-light nanoscope,” Nat. Commun. 2, 218 (2011). [CrossRef]

4.

Z. Chen, A. Taflove, and V. Backman, “Photonic nanojet enhancement of backscattering of light by nanoparticles: a potential novel visible-light ultramicroscopy technique,” Opt. Express 12(7), 1214–1220 (2004). [CrossRef] [PubMed]

5.

A. Heifetz, S. C. Kong, A. V. Sahakian, A. Taflove, and V. Backman, “Photonic Nanojets,” J Comput Theor Nanosci 6(9), 1979–1992 (2009). [CrossRef] [PubMed]

6.

Y. E. Geints, A. A. Zemlyanov, and E. K. Panina, “Photonic jets from resonantly excited transparent dielectric microspheres,” J. Opt. Soc. Am. B 29(4), 758–762 (2012). [CrossRef]

7.

X. Li, Z. Chen, A. Taflove, and V. Backman, “Optical analysis of nanoparticles via enhanced backscattering facilitated by 3-D photonic nanojets,” Opt. Express 13(2), 526–533 (2005). [CrossRef] [PubMed]

8.

A. V. Itagi and W. A. Challener, “Optics of photonic nanojets,” J. Opt. Soc. Am. A 22(12), 2847–2858 (2005). [CrossRef] [PubMed]

9.

P. Ferrand, J. Wenger, A. Devilez, M. Pianta, B. Stout, N. Bonod, E. Popov, and H. Rigneault, “Direct imaging of photonic nanojets,” Opt. Express 16(10), 6930–6940 (2008). [CrossRef] [PubMed]

10.

A. Devilez, N. Bonod, J. Wenger, D. Gérard, B. Stout, H. Rigneault, and E. Popov, “Three-dimensional subwavelength confinement of light with dielectric microspheres,” Opt. Express 17(4), 2089–2094 (2009). [CrossRef] [PubMed]

11.

M. S. Kim, T. Scharf, S. Mühlig, C. Rockstuhl, and H. P. Herzig, “Engineering photonic nanojets,” Opt. Express 19(11), 10206–10220 (2011). [CrossRef] [PubMed]

12.

D. McCloskey, J. J. Wang, and J. F. Donegan, “Low divergence photonic nanojets from Si3N4 microdisks,” Opt. Express 20(1), 128–140 (2012). [CrossRef] [PubMed]

13.

Y. Ku, C. Kuang, X. Hao, Y. Xue, H. Li, and X. Liu, “Superenhanced three-dimensional confinement of light by compound metal-dielectric microspheres,” Opt. Express 20(15), 16981–16991 (2012). [CrossRef]

14.

E. McLeod and C. B. Arnold, “Subwavelength direct-write nanopatterning using optically trapped microspheres,” Nat. Nanotechnol. 3(7), 413–417 (2008). [CrossRef] [PubMed]

15.

J. Kim, K. Cho, I. Kim, W. M. Kim, T. S. Lee, and K. S. Lee, “Fabrication of plasmonic nanodiscs by photonic nanojet lighography,” Appl. Phys. Express 5(2), 025201 (2012). [CrossRef]

16.

D. A. Fletcher, K. E. Goodson, and G. S. Kino, “Focusing in microlenses close to a wavelength in diameter,” Opt. Lett. 26(7), 399–401 (2001). [CrossRef] [PubMed]

17.

T. J. Gould, S. T. Hess, and J. Bewersdorf, “Optical nanoscopy: from acquisition to analysis,” Annu. Rev. Biomed. Eng. 14(1), 231–254 (2012). [CrossRef] [PubMed]

18.

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 358–379 (1959). [CrossRef]

19.

J. J. Schwartz, S. Stavrakis, and S. R. Quake, “Colloidal lenses allow high-temperature single-molecule imaging and improve fluorophore photostability,” Nat. Nanotechnol. 5(2), 127–132 (2010). [CrossRef] [PubMed]

20.

Z. Wang, W. Guo, L. Li, B. Luk'yanchuk, A. Khan, Z. Liu, Z. Chen, and M. Hong, “Optical virtual imaging at 50nm lateral resolution with a white-light nanoscope,” Nat. Commun. 2, 1–6 (2011). [CrossRef]

21.

M. S. Kim, T. Scharf, M. T. Haq, W. Nakagawa, and H. P. Herzig, “Subwavelength-size solid immersion lens,” Opt. Lett. 36(19), 3930–3932 (2011). [CrossRef] [PubMed]

22.

D. R. Mason, M. V. Jouravlev, and K. S. Kim, “Enhanced resolution beyond the Abbe diffraction limit with wavelength-scale solid immersion lenses,” Opt. Lett. 35(12), 2007–2009 (2010). [CrossRef] [PubMed]

23.

C. J. R. Sheppard and P. Török, “Focal shift and the axial optical coordinate for high-aperture systems of finite Fresnel number,” J. Opt. Soc. Am. A 20(11), 2156–2162 (2003). [CrossRef] [PubMed]

24.

Y. Li, “Focal shifts in diffracted converging electromagnetic waves. I. Kirchhoff theory,” J. Opt. Soc. Am. A 22(1), 68–76 (2005). [CrossRef] [PubMed]

25.

S. Guo, H. Guo, and S. Zhuang, “Analysis of imaging properties of a microlens based on the method for a dyadic Green’s function,” Appl. Opt. 48(2), 321–327 (2009). [CrossRef] [PubMed]

26.

http://www.microspheres-nanospheres.com/

27.

J. M. Yi, A. Cuche, F. de León-Pérez, A. Degiron, E. Laux, E. Devaux, C. Genet, J. Alegret, L. Martín-Moreno, and T. W. Ebbesen, “Diffraction regimes of single holes,” Phys. Rev. Lett. 109(2), 023901 (2012). [CrossRef] [PubMed]

28.

J. J. Stamnes, Waves in Focal Regions (Taylor & Francis Group, 1986), p456.

OCIS Codes
(050.1960) Diffraction and gratings : Diffraction theory
(260.2110) Physical optics : Electromagnetic optics
(350.3950) Other areas of optics : Micro-optics

ToC Category:
Physical Optics

History
Original Manuscript: November 21, 2012
Revised Manuscript: December 21, 2012
Manuscript Accepted: January 15, 2013
Published: January 24, 2013

Citation
Hanming Guo, Yunxuan Han, Xiaoyu Weng, Yanhui Zhao, Guorong Sui, Yang Wang, and Songlin Zhuang, "Near-field focusing of the dielectric microsphere with wavelength scale radius," Opt. Express 21, 2434-2443 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-2-2434


Sort:  Author  |  Year  |  Journal  |  Reset  

References

  1. J. Y. Lee, B. H. Hong, W. Y. Kim, S. K. Min, Y. Kim, M. V. Jouravlev, R. Bose, K. S. Kim, I. C. Hwang, L. J. Kaufman, C. W. Wong, P. Kim, and K. S. Kim, “Near-field focusing and magnification through self-ssembled nanoscale spherical lenses,” Nature460(7254), 498–501 (2009). [CrossRef]
  2. J. J. Schwartz, S. Stavrakis, and S. R. Quake, “Colloidal lenses allow high-temperature single-molecule imaging and improve fluorophore photostability,” Nat. Nanotechnol.5(2), 127–132 (2010). [CrossRef] [PubMed]
  3. Z. Wang, W. Guo, L. Li, B. Luk'yanchuk, A. Khan, Z. Liu, Z. Chen, and M. Hong, “Optical virtual imaging at 50nm lateral resolution with a white-light nanoscope,” Nat. Commun.2, 218 (2011). [CrossRef]
  4. Z. Chen, A. Taflove, and V. Backman, “Photonic nanojet enhancement of backscattering of light by nanoparticles: a potential novel visible-light ultramicroscopy technique,” Opt. Express12(7), 1214–1220 (2004). [CrossRef] [PubMed]
  5. A. Heifetz, S. C. Kong, A. V. Sahakian, A. Taflove, and V. Backman, “Photonic Nanojets,” J Comput Theor Nanosci6(9), 1979–1992 (2009). [CrossRef] [PubMed]
  6. Y. E. Geints, A. A. Zemlyanov, and E. K. Panina, “Photonic jets from resonantly excited transparent dielectric microspheres,” J. Opt. Soc. Am. B29(4), 758–762 (2012). [CrossRef]
  7. X. Li, Z. Chen, A. Taflove, and V. Backman, “Optical analysis of nanoparticles via enhanced backscattering facilitated by 3-D photonic nanojets,” Opt. Express13(2), 526–533 (2005). [CrossRef] [PubMed]
  8. A. V. Itagi and W. A. Challener, “Optics of photonic nanojets,” J. Opt. Soc. Am. A22(12), 2847–2858 (2005). [CrossRef] [PubMed]
  9. P. Ferrand, J. Wenger, A. Devilez, M. Pianta, B. Stout, N. Bonod, E. Popov, and H. Rigneault, “Direct imaging of photonic nanojets,” Opt. Express16(10), 6930–6940 (2008). [CrossRef] [PubMed]
  10. A. Devilez, N. Bonod, J. Wenger, D. Gérard, B. Stout, H. Rigneault, and E. Popov, “Three-dimensional subwavelength confinement of light with dielectric microspheres,” Opt. Express17(4), 2089–2094 (2009). [CrossRef] [PubMed]
  11. M. S. Kim, T. Scharf, S. Mühlig, C. Rockstuhl, and H. P. Herzig, “Engineering photonic nanojets,” Opt. Express19(11), 10206–10220 (2011). [CrossRef] [PubMed]
  12. D. McCloskey, J. J. Wang, and J. F. Donegan, “Low divergence photonic nanojets from Si3N4 microdisks,” Opt. Express20(1), 128–140 (2012). [CrossRef] [PubMed]
  13. Y. Ku, C. Kuang, X. Hao, Y. Xue, H. Li, and X. Liu, “Superenhanced three-dimensional confinement of light by compound metal-dielectric microspheres,” Opt. Express20(15), 16981–16991 (2012). [CrossRef]
  14. E. McLeod and C. B. Arnold, “Subwavelength direct-write nanopatterning using optically trapped microspheres,” Nat. Nanotechnol.3(7), 413–417 (2008). [CrossRef] [PubMed]
  15. J. Kim, K. Cho, I. Kim, W. M. Kim, T. S. Lee, and K. S. Lee, “Fabrication of plasmonic nanodiscs by photonic nanojet lighography,” Appl. Phys. Express5(2), 025201 (2012). [CrossRef]
  16. D. A. Fletcher, K. E. Goodson, and G. S. Kino, “Focusing in microlenses close to a wavelength in diameter,” Opt. Lett.26(7), 399–401 (2001). [CrossRef] [PubMed]
  17. T. J. Gould, S. T. Hess, and J. Bewersdorf, “Optical nanoscopy: from acquisition to analysis,” Annu. Rev. Biomed. Eng.14(1), 231–254 (2012). [CrossRef] [PubMed]
  18. B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. A Math. Phys. Sci.253(1274), 358–379 (1959). [CrossRef]
  19. J. J. Schwartz, S. Stavrakis, and S. R. Quake, “Colloidal lenses allow high-temperature single-molecule imaging and improve fluorophore photostability,” Nat. Nanotechnol.5(2), 127–132 (2010). [CrossRef] [PubMed]
  20. Z. Wang, W. Guo, L. Li, B. Luk'yanchuk, A. Khan, Z. Liu, Z. Chen, and M. Hong, “Optical virtual imaging at 50nm lateral resolution with a white-light nanoscope,” Nat. Commun.2, 1–6 (2011). [CrossRef]
  21. M. S. Kim, T. Scharf, M. T. Haq, W. Nakagawa, and H. P. Herzig, “Subwavelength-size solid immersion lens,” Opt. Lett.36(19), 3930–3932 (2011). [CrossRef] [PubMed]
  22. D. R. Mason, M. V. Jouravlev, and K. S. Kim, “Enhanced resolution beyond the Abbe diffraction limit with wavelength-scale solid immersion lenses,” Opt. Lett.35(12), 2007–2009 (2010). [CrossRef] [PubMed]
  23. C. J. R. Sheppard and P. Török, “Focal shift and the axial optical coordinate for high-aperture systems of finite Fresnel number,” J. Opt. Soc. Am. A20(11), 2156–2162 (2003). [CrossRef] [PubMed]
  24. Y. Li, “Focal shifts in diffracted converging electromagnetic waves. I. Kirchhoff theory,” J. Opt. Soc. Am. A22(1), 68–76 (2005). [CrossRef] [PubMed]
  25. S. Guo, H. Guo, and S. Zhuang, “Analysis of imaging properties of a microlens based on the method for a dyadic Green’s function,” Appl. Opt.48(2), 321–327 (2009). [CrossRef] [PubMed]
  26. http://www.microspheres-nanospheres.com/
  27. J. M. Yi, A. Cuche, F. de León-Pérez, A. Degiron, E. Laux, E. Devaux, C. Genet, J. Alegret, L. Martín-Moreno, and T. W. Ebbesen, “Diffraction regimes of single holes,” Phys. Rev. Lett.109(2), 023901 (2012). [CrossRef] [PubMed]
  28. J. J. Stamnes, Waves in Focal Regions (Taylor & Francis Group, 1986), p456.

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

Figures

Fig. 1 Fig. 2 Fig. 3
 
Fig. 4
 

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited