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Virtual Journal for Biomedical Optics

| EXPLORING THE INTERFACE OF LIGHT AND BIOMEDICINE

  • Editors: Andrew Dunn and Anthony Durkin
  • Vol. 8, Iss. 9 — Oct. 2, 2013
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Classical imaging theory of a microlens with super-resolution

Yubo Duan, George Barbastathis, and Baile Zhang  »View Author Affiliations


Optics Letters, Vol. 38, Issue 16, pp. 2988-2990 (2013)
http://dx.doi.org/10.1364/OL.38.002988


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Abstract

Super-resolution in imaging through a transparent spherical microlens has attracted lots of attention because of recent promising experimental results with remarkable resolution improvement. To provide physical insight for this super-resolution phenomenon, previous studies adopted a phenomenological explanation mainly based on the super-focusing effect of a photonic nanojet, while a direct imaging calculation with classical imaging theory has rarely been studied. Here we theoretically model the imaging process through a microlens with vectorial electromagnetic analysis, and then exclude the previously plausible explanation of super-resolution based on the super-focusing effect. The results showed that, in the context of classical imaging theory subject to the two-point resolution criterion, a microlens with a perfect spherical shape cannot achieve the experimentally verified sub-100 nm resolution. Therefore, there must be some other physical mechanisms that contribute to the reported ultrahigh resolution but have not been revealed in theory.

© 2013 Optical Society of America

A long-standing issue of traditional microscopy is that its resolution is limited to about half of the illumination wavelength as a result of the loss of evanescent waves during wave propagation. To break this resolution limit and achieve super-resolution, researchers have developed various approaches. One approach is to recover the evanescent waves in far-field by using negative refractive index metamaterials, which could achieve unlimited resolution in theory [1

1. J. B. Pendry, Phys. Rev. Lett. 85, 3966 (2000). [CrossRef]

]. However, because of practical difficulties such as loss, this approach has not been practically used. Another approach is to deliberately create a specific situation where only a single light emitting spot (or sparsely distributed spots) will locate in the field of view, such that the overlapping of point spread function will not occur in principle. Typical examples include stimulated emission depletion microscopy and stochastic optical reconstruction microscopy [2

2. H. Wang, C. J. R. Sheppard, K. Ravi, S. T. Ho, and G. Vienne, Laser Photon. Rev. 6, 354 (2012). [CrossRef]

]. Although being very successful in practice, these microscopy technologies share an inherent drawback: they generally require temporal and spatial scanning, which will take a long time, and thus are not very suitable for dynamic real-time imaging.

As shown in Fig. 1, two incoherent dipoles pointing in the z direction are placed on the object plane just beside the microlens (diameter D=4.74μm, refractive index n=1.46), similar to the experimental setup of the microlens imaging system [6

6. Z. Wang, W. Guo, L. Li, B. Luk’yanchuk, A. Khan, Z. Liu, Z. Chen, and M. Hong, Nat. Commun. 2, 218 (2011). [CrossRef]

]. The waves radiated from the dipoles propagate through the microlens and are collected on the collecting plane in the far-field. The interaction between the dipole radiation and the microlens is calculated by multipole expansion based on spherical harmonics and Mie scattering theory. The numerical aperture with respect to the origin of the object plane is 0.9, the same as in [6

6. Z. Wang, W. Guo, L. Li, B. Luk’yanchuk, A. Khan, Z. Liu, Z. Chen, and M. Hong, Nat. Commun. 2, 218 (2011). [CrossRef]

]. According to angular spectrum representation, the collected waves are decomposed into plane waves, which will then numerically propagate backward in the negative x direction to form a virtual image on the image plane.

Fig. 1. Configuration of image reconstruction of two incoherent dipoles. The origin of the coordinates x-y-z coincides with the center of the microlens.

Fig. 2. Snapshots of wave propagation in x-y plane for (a) λ=401.64nm and (b) λ=403.07nm, respectively. The white circle denotes the contour of the microlens. The small blue dot denotes the position of the dipole.

To determine the position of the image plane, we examine the reconstructed intensity distribution in the x-y plane, as shown in Figs. 3(a) and 3(c) for wavelengths 401.64 and 403.07 nm, respectively, with the single dipole behind the microlens. From the view of geometrical optics, the focus of a microlens is at R×n/(n2) (R is the radius of the microlens and n is its refractive index), i.e., at x=6.41μm. This estimation may be applicable to the wavelength 403.07 nm without WGM [Fig. 3(c)], but not appropriate for the wavelength 401.64 nm with WGM [Fig. 3(a)]. Alternatively, the maximum intensity position in the x axis (x=4.40μm for wavelength 401.64 nm and x=6.84μm for wavelength 403.07 nm) can be considered as the focus, since the dipole is known to be on the x axis. However, at x=4.40μm for the wavelength 401.64 nm, the maximum side-lobes are 40% of the main-lobe [Fig. 3(b)], which will cause distortion and poor contrast in wide-field imaging [8

8. C. Sheppard, Optik 48, 329 (1977).

]. To reduce side-lobes, the focus x=4.87μm adopted from [6

6. Z. Wang, W. Guo, L. Li, B. Luk’yanchuk, A. Khan, Z. Liu, Z. Chen, and M. Hong, Nat. Commun. 2, 218 (2011). [CrossRef]

] is also considered, where the maximal side-lobes decrease to about 22% of the main-lobe. Note that at the position x=3.94μm, the side-lobes are even higher than the main-lobe, which may introduce artifacts in practice.

Fig. 3. Reconstructed intensity distribution in x-y plane for (a) λ=401.64nm and (c) λ=403.07nm. Normalized intensity profile of (b) λ=401.64nm focused at x=4.87μm (blue dash–dot line), x=4.40μm (black solid line) and x=3.94μm (red dashed line), and (d) λ=403.07nm focused at x=6.84μm (black solid line), x=6.41μm (red dashed line) and x=4.87μm (blue dash–dot line).

Another notable phenomenon is that different modes of WGM have different resolution enhancement. Here we compare the first TE mode and the first TM mode appearing at the wavelengths 401.64 and 405.55 nm, respectively. The analysis in Fig. 4 demonstrates that the effective FWHM, after normalized by magnification, of wavelength 405.55 is 186 nm focused at 8.17μm (maximum intensity position), 271 nm focused at 4.25μm (the other peak intensity position), 195 nm focused at 6.41μm (geometrical focus), and 413 nm focused at 4.87μm (position adopted from [6

6. Z. Wang, W. Guo, L. Li, B. Luk’yanchuk, A. Khan, Z. Liu, Z. Chen, and M. Hong, Nat. Commun. 2, 218 (2011). [CrossRef]

]). Compared with the TE mode, the TM mode has a much larger FWHM for the given radius and refractive index in our model. Other larger wavelengths with TE and TM WGMs result in larger FWHMs than that at 401.64 nm.

Fig. 4. (a) Reconstructed intensity distribution in x-y plane for λ=405.55nm. (b) Normalized intensity profile focused at x=8.17μm (black solid line), x=6.41μm (red dashed line), x=4.87μm (blue dash–dot line), and x=4.25μm (green dashed line).

To better evaluate resolution with the golden criterion of two-point resolution, we put two emitters behind the microlens with illumination wavelength 401.64 nm. Figures 5(a) and 5(c) show the intensity distribution in x-y plane formed by two incoherent dipoles separated by distances of 150 and 100 nm, respectively. The dipoles separated by 150 nm are clearly resolved [Figs. 5(a) and 5(b)]. However, the dipoles separated by 100 nm are hardly resolved [Figs. 5(c) and 5(d)]. One may argue that two peaks can be resolved at the position x=3.94μm in the case of 100 nm separation. However, the position x=3.94μm is not the true focus (x=4.40μm), and the peaks are because of side-lobes, as illustrated in Fig. 3(a).

Fig. 5. Reconstructed intensity distribution in x-y plane for two incoherent dipoles separated by (a) 150 and (c) 100 nm. Normalized intensity profile for two incoherent dipoles separated by (b) 150 and (d) 100 nm at the focus of x=4.87μm (blue dash–dot line), x=4.40μm (black solid line), and x=3.94μm (red dashed line).

To further explore the resolution of white light illumination, we choose 110 wavelengths including all WGMs in the spectrum 400nm700nm to mimic white light. With such white light illumination, the microlens can resolve two dipoles separated 150 nm apart [Fig. 6(a)], but cannot for those separated 100 nm apart [Fig. 6(b)]. Moreover, we have further tested images formed in various focuses, and got almost the same result. Thus, the resolution of a spherical microlens with white light illumination cannot reach sub-100 nm.

Fig. 6. Images formed by two dipoles separated by (a) 150 and (b) 100 nm. The focus is chosen at x=4.34μm, where the intensity of the white light is maximal.

It should be emphasized that in our calculation some realistic factors that have been ignored may offer real reasons for the experimentally observed high resolution in the record. For example, our model only considers perfect spherical shape, while in reality surface roughness may play an important role in near-field imaging. Moreover, the gold-coated fishnet anodic aluminum oxide sample was used in experiment [6

6. Z. Wang, W. Guo, L. Li, B. Luk’yanchuk, A. Khan, Z. Liu, Z. Chen, and M. Hong, Nat. Commun. 2, 218 (2011). [CrossRef]

], but the possible surface plasmon resonance and quantum or nonlocal effects induced by the periodic metallic sample are completely ignored in our calculation.

This research was supported by the National Research Foundation Singapore through the Singapore MIT Alliance for Research and Technology’s BioSystems and Micromechanics Inter-Disciplinary Research program, and Nanyang Technological University (M4081153, M4080806).

References

1.

J. B. Pendry, Phys. Rev. Lett. 85, 3966 (2000). [CrossRef]

2.

H. Wang, C. J. R. Sheppard, K. Ravi, S. T. Ho, and G. Vienne, Laser Photon. Rev. 6, 354 (2012). [CrossRef]

3.

X. Li, Z. Chen, A. Taflove, and V. Backman, Opt. Express 13, 526 (2005). [CrossRef]

4.

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, and L. J. Kaufman, Nature 460, 498 (2009). [CrossRef]

5.

D. R. Mason, M. V. Jouravlev, and K. S. Kim, Opt. Lett. 35, 2007 (2010). [CrossRef]

6.

Z. Wang, W. Guo, L. Li, B. Luk’yanchuk, A. Khan, Z. Liu, Z. Chen, and M. Hong, Nat. Commun. 2, 218 (2011). [CrossRef]

7.

A. Heifetz, J. J. Simpson, S.-C. Kong, A. Taflove, and V. Backman, Opt. Express 15, 17334 (2007). [CrossRef]

8.

C. Sheppard, Optik 48, 329 (1977).

OCIS Codes
(100.6640) Image processing : Superresolution
(290.0290) Scattering : Scattering
(290.4020) Scattering : Mie theory
(350.3950) Other areas of optics : Micro-optics
(350.5730) Other areas of optics : Resolution

ToC Category:
Image Processing

History
Original Manuscript: April 25, 2013
Revised Manuscript: July 14, 2013
Manuscript Accepted: July 15, 2013
Published: August 6, 2013

Virtual Issues
Vol. 8, Iss. 9 Virtual Journal for Biomedical Optics

Citation
Yubo Duan, George Barbastathis, and Baile Zhang, "Classical imaging theory of a microlens with super-resolution," Opt. Lett. 38, 2988-2990 (2013)
http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=ol-38-16-2988


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References

  1. J. B. Pendry, Phys. Rev. Lett. 85, 3966 (2000). [CrossRef]
  2. H. Wang, C. J. R. Sheppard, K. Ravi, S. T. Ho, and G. Vienne, Laser Photon. Rev. 6, 354 (2012). [CrossRef]
  3. X. Li, Z. Chen, A. Taflove, and V. Backman, Opt. Express 13, 526 (2005). [CrossRef]
  4. 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, and L. J. Kaufman, Nature 460, 498 (2009). [CrossRef]
  5. D. R. Mason, M. V. Jouravlev, and K. S. Kim, Opt. Lett. 35, 2007 (2010). [CrossRef]
  6. Z. Wang, W. Guo, L. Li, B. Luk’yanchuk, A. Khan, Z. Liu, Z. Chen, and M. Hong, Nat. Commun. 2, 218 (2011). [CrossRef]
  7. A. Heifetz, J. J. Simpson, S.-C. Kong, A. Taflove, and V. Backman, Opt. Express 15, 17334 (2007). [CrossRef]
  8. C. Sheppard, Optik 48, 329 (1977).

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