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The Technical Program Committee has completed the
scheduling for the POEM meeting (Nanophotonics,
Nanoelectronics and Nanosensor), which will be held 24-27
May 2013 at the Wuhan National Laboratory for
Optoelectronics in Wuhan, China.
Your schedule is as follows:
Session Title: Nanoprobe and Nanoimaging V
Session Time: May 26, 2013 from 1:30 PM to 4:05 PM
Presentation Title: Optical Resolution Limit Based on
Dielectric Microsphere
Presentation No.: NSu3C.3
Presentation Time: 2:45 PM to 3:15 PM
Location: N3 Room A202 (WNLO)
Optical Resolution Limit Based on Dielectric Microsphere
Hanming Guo,* Xiaoyu Weng
Engineering Research Center of Optical Instrument and System, Ministry of Education, Shanghai Key Lab of Modern Optical System, School of
Optical-Electrical and Computer Engineering, University of Shanghai for Science and Technology, 516 Jungong Rd, Shanghai 200093, China
[email protected]
Abstract: On the basis of the concept of spherical aberration and the phenomena of focal shift, the
lateral resolution limitation and the corresponding conditions are demonstrated for the focusing of
the dielectric microsphere.
OCIS codes: (260.2110) Electromagnetic optics; (050.1960) Diffraction theory; (350.3950) Micro-optics.
1. Introduction
Recently, several groups report that dielectric microspheres and microlenses can be used for nanopattering or
nanoimaging [1-4]. Especially in 2011, Wang et. al [4] report that a new 50-nm-resolution nanoscope that uses
optically transparent microspheres as far-field superlenses (FSL) to overcome the white-light diffraction limit, which
not only provide a new and simple method for nanoimaging, but also promote more people to think about what is
the reason of nanoimaging of the mcirospheres and whether one can achieve higher lateral resolution with the
microsphere compared with the existing literatures and how to get it. However, the existing literatures on the
focusing of the microsphere 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. 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 this paper, basing on the concept of spherical aberration and the phenomena of focal shift, we
answer the above questions.
2. Spherical aberration and focal shift of the microsphere
Shown in Fig. 1, the principle of the ray tracing is 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   have recently been proved in theory [5, 6] and experiment [7]. 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.
Fig.1. Schematic of a microsphere illuminated by a incident plane-wave.
Basing on the geometrical relations shown in Fig. 1, we can obtain the formulae   2o  2 s and
d  r[cos(o   )  sin(o   ) cot   1] for the point F outside microsphere. If the point F is inside microsphere,
namely the points A and B locating at the different side of the z axis, d  r[sin o cot(o   s )  coso  1] .
As the focal spot outside the microsphere majorly arises from the focusing contributions of the rays with the
incident angle 0  o  2 s , we define the spherical aberration S  dmax  dmin , where d min  0 by setting
o  2acos(ns 2no )   with a very small value   0.00001 in radians and d max approximately calculated by
setting  o   . For the microsphere, we define NA  no sin  max , where the maximum value  max of  can be
calculated from the maximum value o max  2acos(ns 2no )   .
As we known, to achieve maximum lateral resolution, NA should be as high as possible, whereas the aberration
is inverse. In order to assure that S has no significant effects on the focusing properties, the minimum requirement
for S is S   2no . Shown in Fig. 2, RIC  ns no should be bigger than 1.5 for S   2no (note r   ). As both
S and NA decrease with the increase of RIC , a tradeoff between S and NA is needed. So it is improper to
choose too high RIC when one wants to achieve imaging with high lateral resolution. By FDTD method, this
qualitative analysis is also confirmed by our numerical calculations performed with the various combinations of r
and RIC with the range from 1.76 to 2. When RIC  1.75 , NA  0.85 (see Fig. 2). Therefore, within the
approximate range of 1.5  RIC  1.75 , the microsphere has not only small S , but also high NA by nature.
Fig.2. Relations between the spherical aberration S (solid curve), the numerical aperture NA (dashed curve) and the refractive index contrast
RIC between the microsphere and its surrounding medium.
In addition, vector diffraction theories show that the actual focus is shifted the geometrical focus and moves
toward the optical system for small Fresnel number [5,6]. The small Fresnel number N  r a NAa is the optical
system with radius larger than one and smaller than ten wavelengths. Obviously, the microsphere is the optical
system with small N ( a   ns for the microsphere). For the small circular aperture, the focal shifts is calculated
by Eq. (25) in Ref. 6 [see Fig. 3(a) and 3(b)]. Figure 3(a) shows that the focal shifts decrease with the increase of r .
Moreover, the focal shifts are not obvious when r  5a for the small circular aperture, which means that the focal
shifts might be small for the microspheres with r  5a  5 ns .
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) solid line: r  1.25a ; dashed line: r  3a ; dash-dotted line: r  5a . (b) r  1.25a and the wave front at the aperture being divided
equally into five zones within the maximum aperture angle [see 3(c)]. Lines 1-5 correspond the five zones.
To examine the contributions of each ray on the focal shifts, we divide equally the wave front at the aperture
into five zones within the maximum aperture angle  [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 ( 0.112 f ) [see the solid line in
Fig. 3(a)] of the actual focus slightly shifts toward the small circular aperture from the focusing position ( 0.096 f )
of the rays within the fifth zone, which means that, for the microsphere, the actual focal spot will slightly shift the
focusing position corresponding to NA  no sin  max and locate at the inside proximity of the rear surface of the
microsphere. Meanwhile, the actual NA of the microsphere is also slightly bigger than the NA (i.e., no sin  max )
defined before. In terms of Fig. 3(b), we can find that, the small aperture used in Ref. 6 is assumed an aplanatic
system, but the spherical aberration actually exists. Moreover, the spherical aberration is negative for the small
aperture when the above definition S  dmax  dmin is used, whereas the spherical aberration predicated by the ray
tracing procedure is positive for the microsphere. Therefore, for the microsphere, the actual spherical aberration S
will be far smaller than the above S based on the ray tracing analysis due to the spherical aberration compensation
caused by the focal shifts. So, 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, the microsphere
naturally has the characteristics of negligible S and high NA within the above range of 1.5  RIC  1.75 . And only
within 1.5  RIC  1.75 , the maximum lateral resolution of the microsphere might be obtained.
The above conclusion will further be proved by the FDTD method. Here, the x linearly polarized plane wave
with   400nm is used and others parameters are shown in Fig. 4 caption.
Shown in Fig. 4(a), the focal spot of the microsphere can be situated at its rear surface by tuning r . Figure 4(a)
shows the lateral resolution is 120nm and slightly better than 126nm calculated by  2ns and ns  RIC  no  1.59
( no  1 ). We find that within 1.5  RIC  1.75 , the optimal resolution (i.e., beyond  2ns ) can be obtained for
proper r [see Fig. 4(b)]. However, if the RIC is outside the range of 1.5  RIC  1.75 , regardless of how to tune r ,
the focal spot can be also situated at the rear surface of the microsphere can be formed, but the lateral resolution is
hardly beyond  2ns . The reason is that, only within the approximate region of 1.5  RIC  1.75 , the microsphere
has the characteristics of negligible S and high NA as discussed before.
Fig. 4. (a) Electric field intensity ( | E |2 ) distribution in the yz plane of the microsphere ( r  490nm , RIC  1.59 ) and (b) the variation of the
lateral resolution (o) along the y axis and the axial resolution () along the z axis with the RIC . In (b), six sets of parameters are used:
RIC  1.5 , 1.59, 1.63, 1.67, 1.7, and 1.75; r  510nm , 490nm , 480nm , 470nm , 460nm , and 450nm .
3. Conclusion
The dielectric microsphere naturally has negligible S and high NA within the approximate region of
1.5  RIC  1.75 , whose reason is due to the spherical aberration compensation arising from the positive spherical
aberration caused by the surface shape of the microsphere and the RIC and the negative spherical aberration caused
by the focal shifts due to the wavelength scale dimension of the microsphere. Only within the approximate region of
1.5  RIC  1.75 with the proper radius r of microsphere, the microsphere can generate a focal spot with lateral
resolution slightly beyond  2ns , which is also the lateral resolution limit of the dielectric microsphere.
This work was supported by the National Natural Science Foundation of China (61178079), the Fok Ying-Tong
Education Foundation, China (121010), and the Foundation for the Author of National Excellent Doctoral
Dissertation of PR China (201033).
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
E. Mcleod, and C. B. Arnold, “Subwavelength direct-write nanopattering using optically trapped microspheres,” Nat. Nanotech. 3, 413-417
(2008) .
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, 498-501
(2009).
J. J. Schwartz, S. Stavrakis, and S. R. Quake, “Colloidal lenses allow high-temperature single-molecule imaging and improve fluorophore
photostability,” Nat. Nanotech. 5, 127-132 (2010).
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).
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, 2156-2162 (2003).
Y. Li, “Focal shifts in diffracted converging electromagnetic waves. I. Kirchhoff theory,” J. Opt. Soc. Am. A 22, 68-76 (2005).
J. 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,” Phy. Rev. Lett. 109, 023901 (2012).