Abstract

Microsphere-based microscopy systems have garnered lots of recent interest, mainly due to their capacity in focusing light and imaging beyond the diffraction limit. In this paper, we present theoretical foundations for studying the optical performance of such systems by developing a complete theoretical model encompassing the aspects of illumination, sample interaction and imaging/collection. Using this model, we show that surface waves play a significant role in focusing and imaging with the microsphere. We also show that by designing a radially polarized convergent beam, we can focus to a spot smaller than the diffraction limit. By exploiting surface waves, we are able to resolve two dipoles spaced 98 nm apart in simulation using light at a wavelength of 402.292 nm. Using our model, we also explore the effect of beam geometry and polarization on optical resolution and focal spot size, showing that both geometry and polarization greatly affect the shape of the spot.

© 2015 Optical Society of America

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2015 (1)

2014 (6)

S. Lee, L. Li, and Z. Wang, “Optical resonances in microsphere photonic nanojets,” J. Opt. 16, 015704 (2014).
[Crossref]

Y. Yan, L. Li, C. Feng, W. Guo, S. Lee, and M. Hong, “Microsphere-Coupled Scanning Laser Confocal Nanoscope for Sub-Diffraction-Limited Imaging at 25 nm Lateral Resolution in the Visible Spectrum,” ACS Nano,  8, 1809–1816 (2014).
[Crossref] [PubMed]

V. M. Sundaram and S. B. Wen, “Analysis of deep sub-micron resolution in microsphere based imaging,” Appl. Phys. Lett. 105, 204102 (2014).
[Crossref]

R. Ye, Y.-H. Ye, H. F. Ma, L. Cao, J. Ma, F. Wyrowski, R. Shi, and J.-Y. Zhang, “Experimental imaging properties of immersion microscale spherical lenses,” Sci. Rep. 4, 1–5 (2014).
[Crossref]

T. X. Hoang, X. Chen, and C. J. R. Sheppard, “Multipole and plane wave expansions of diverging and converging fields,” Opt. Express 22, 8949–8961 (2014).
[Crossref] [PubMed]

M. D. Baaske, M. R. Foreman, and F. Vollmer, “Single-molecule nucleic acid interactions monitored on a label-free microcavity biosensor platform,” Nature Nanotech. 9, 933–939 (2014).
[Crossref]

2013 (2)

2012 (5)

T. X. Hoang, X. Chen, and C. J. R. Sheppard, “Multipole theory for tight focusing of polarized light, including radially polarized and other special cases,” J. Opt. Soc. Am. A 29, 32–43 (2012).
[Crossref]

T. X. Hoang, X. Chen, and C. J. R. Sheppard, “Interpretation of the scattering mechanism for particles in a focused beam,” Phys. Rev. A 86, 033817 (2012).
[Crossref]

R. Chen, K. Agarwal, Y. Zhong, C. J.R. Sheppard, J. C.H. Phang, and X. Chen, “Complete modeling of subsurface microscopy system based on aplanatic solid immersion lens,” J. Opt. Soc. Am. A 29, 2350–2359 (2012).
[Crossref]

S. Orlov, U. Peschel, T. Bauer, and P. Banzer, “Analytical expansion of highly focused vector beams into vector spherical harmonics and its application to Mie scattering,” Phys. Rev. A 85, 063825 (2012).
[Crossref]

D. Wildanger, B. R. Patton, H. Schill, L. Marseglia, J. P. Hadden, S. Knauer, A. Schnle, J. G. Rarity, J. L. O’Brien, S. W. Hell, and J. M. Smith, “Solid Immersion Facilitates Fluorescence Microscopy with Nanometer Resolution and Sub-Ångström emitter localization,” Adv. Mater 24, OP309–OP313 (2012).
[Crossref] [PubMed]

2011 (1)

Z. Wang, W. Guo, L. Li, B. Lukyanchuk, 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]

2010 (2)

J. P. Hadden, J. P. Harrison, A. C. Stanley-Clarke, L. Marseglia, Y-L. D. Ho, B. R. Patton, J. L. O’Brien, and J. G. Rarity, “Strongly enhanced photon collection from diamond defect centres under micro-fabricated integrated solid immersion lenses,” Appl. Phys. Lett.,  97, 241901 (2010).
[Crossref]

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, 2007–2009 (2010).
[Crossref] [PubMed]

2009 (4)

S. H. Goh, C. J. R Sheppard, A. C. T. Quah, C. M. Chua, L. S. Koh, and J. C. H. Phang, “Design considerations for refractive solid immersion lens: application to subsurface integrated circuit fault localization using laser induced techniques,” Rev. Sci. Instrum. 80, 013703 (2009).
[Crossref] [PubMed]

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- Assembled Nanoscale Spherical Lenses,” Nature (London) 460, 498–501 (2009).
[Crossref]

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

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

2008 (1)

2003 (1)

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901 (2003).
[Crossref] [PubMed]

2001 (1)

2000 (1)

S. Quabis, R. Dorn, M. Eberler, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7 (2000).
[Crossref]

1999 (1)

1993 (2)

B. R. Johnson, “Theory of morphology-dependent resonances: shape resonances and width formulas,” J. Opt. Soc. Am. A 10, 343–352 (1993).
[Crossref]

W. C. Chew and Y. M. Wang, “Efficient Ways to Compute the Vector Addition Theorem,” J. Elect. Wave. Appl. 7, 651–665 (1993).
[Crossref]

1981 (1)

1974 (1)

A. J. Devaney and E. Wolf, “Multipole expansions and plane wave representations of the electromagnetic field,” J. Math. Phys. 15, 234–244 (1974).
[Crossref]

1969 (1)

H. M. Nussenzveig, “High-frequency scattering by a transparent sphere. I. Direct reflection and transmission,” J. Math. Phys. 10, 82–124 (1969).
[Crossref]

1959 (1)

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. Roy. Soc. London Ser. A 253, 358–379 (1959).
[Crossref]

Agarwal, K.

R. Chen, K. Agarwal, Y. Zhong, C. J.R. Sheppard, J. C.H. Phang, and X. Chen, “Complete modeling of subsurface microscopy system based on aplanatic solid immersion lens,” J. Opt. Soc. Am. A 29, 2350–2359 (2012).
[Crossref]

T. X. Hoang, R. Chen, K. Agarwal, C. J. R. Sheppard, and X. Chen, “Imaging with annular focusing through a dielectric interface,” in Proceedings of Focus On Microscopy 2014, Sydney, Australia2014 April 13–16.

Arfken, G. B.

G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists, 5 ed. (Academic Press, 2000).

Baaske, M. D.

M. D. Baaske, M. R. Foreman, and F. Vollmer, “Single-molecule nucleic acid interactions monitored on a label-free microcavity biosensor platform,” Nature Nanotech. 9, 933–939 (2014).
[Crossref]

Backman, V.

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

Banzer, P.

S. Orlov, U. Peschel, T. Bauer, and P. Banzer, “Analytical expansion of highly focused vector beams into vector spherical harmonics and its application to Mie scattering,” Phys. Rev. A 85, 063825 (2012).
[Crossref]

Barbastathis, G.

Bauer, T.

S. Orlov, U. Peschel, T. Bauer, and P. Banzer, “Analytical expansion of highly focused vector beams into vector spherical harmonics and its application to Mie scattering,” Phys. Rev. A 85, 063825 (2012).
[Crossref]

Bonod, N.

Born, M.

M. Born and W. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 2005).

Bose, R.

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- Assembled Nanoscale Spherical Lenses,” Nature (London) 460, 498–501 (2009).
[Crossref]

Cao, L.

R. Ye, Y.-H. Ye, H. F. Ma, L. Cao, J. Ma, F. Wyrowski, R. Shi, and J.-Y. Zhang, “Experimental imaging properties of immersion microscale spherical lenses,” Sci. Rep. 4, 1–5 (2014).
[Crossref]

Cárabe, A.

Chen, L.

Chen, R.

R. Chen, K. Agarwal, Y. Zhong, C. J.R. Sheppard, J. C.H. Phang, and X. Chen, “Complete modeling of subsurface microscopy system based on aplanatic solid immersion lens,” J. Opt. Soc. Am. A 29, 2350–2359 (2012).
[Crossref]

T. X. Hoang, R. Chen, K. Agarwal, C. J. R. Sheppard, and X. Chen, “Imaging with annular focusing through a dielectric interface,” in Proceedings of Focus On Microscopy 2014, Sydney, Australia2014 April 13–16.

Chen, X.

Chen, Z.

Z. Wang, W. Guo, L. Li, B. Lukyanchuk, 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]

Chew, W. C.

W. C. Chew and Y. M. Wang, “Efficient Ways to Compute the Vector Addition Theorem,” J. Elect. Wave. Appl. 7, 651–665 (1993).
[Crossref]

Chua, C. M.

S. H. Goh, C. J. R Sheppard, A. C. T. Quah, C. M. Chua, L. S. Koh, and J. C. H. Phang, “Design considerations for refractive solid immersion lens: application to subsurface integrated circuit fault localization using laser induced techniques,” Rev. Sci. Instrum. 80, 013703 (2009).
[Crossref] [PubMed]

Darafsheh, A.

Devaney, A. J.

A. J. Devaney and E. Wolf, “Multipole expansions and plane wave representations of the electromagnetic field,” J. Math. Phys. 15, 234–244 (1974).
[Crossref]

Devilez, A.

Dorn, R.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901 (2003).
[Crossref] [PubMed]

S. Quabis, R. Dorn, M. Eberler, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7 (2000).
[Crossref]

Duan, Y.

Duarte, F. J.

Eberler, M.

S. Quabis, R. Dorn, M. Eberler, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7 (2000).
[Crossref]

Feng, C.

Y. Yan, L. Li, C. Feng, W. Guo, S. Lee, and M. Hong, “Microsphere-Coupled Scanning Laser Confocal Nanoscope for Sub-Diffraction-Limited Imaging at 25 nm Lateral Resolution in the Visible Spectrum,” ACS Nano,  8, 1809–1816 (2014).
[Crossref] [PubMed]

Feshbach, H.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics, Pt.2 (McGraw-Hill, 1953).

Finlay, J. C.

Fletcher, D. A.

Foreman, M. R.

M. D. Baaske, M. R. Foreman, and F. Vollmer, “Single-molecule nucleic acid interactions monitored on a label-free microcavity biosensor platform,” Nature Nanotech. 9, 933–939 (2014).
[Crossref]

Gerard, D.

Goh, S. H.

S. H. Goh, C. J. R Sheppard, A. C. T. Quah, C. M. Chua, L. S. Koh, and J. C. H. Phang, “Design considerations for refractive solid immersion lens: application to subsurface integrated circuit fault localization using laser induced techniques,” Rev. Sci. Instrum. 80, 013703 (2009).
[Crossref] [PubMed]

Goodson, K. E.

Gouesbet, G.

G. Gouesbet and G. Grehan, Generalised Lorenz-Mie Theories (Springer-Verlag, 2011).
[Crossref]

Grandy, W. T.

W. T. Grandy, Scattering of Waves from Large Spheres (Cambridge University Press, 2000).
[Crossref]

Grehan, G.

G. Gouesbet and G. Grehan, Generalised Lorenz-Mie Theories (Springer-Verlag, 2011).
[Crossref]

Guardiola, C.

Guo, W.

Y. Yan, L. Li, C. Feng, W. Guo, S. Lee, and M. Hong, “Microsphere-Coupled Scanning Laser Confocal Nanoscope for Sub-Diffraction-Limited Imaging at 25 nm Lateral Resolution in the Visible Spectrum,” ACS Nano,  8, 1809–1816 (2014).
[Crossref] [PubMed]

Z. Wang, W. Guo, L. Li, B. Lukyanchuk, 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]

Hadden, J. P.

D. Wildanger, B. R. Patton, H. Schill, L. Marseglia, J. P. Hadden, S. Knauer, A. Schnle, J. G. Rarity, J. L. O’Brien, S. W. Hell, and J. M. Smith, “Solid Immersion Facilitates Fluorescence Microscopy with Nanometer Resolution and Sub-Ångström emitter localization,” Adv. Mater 24, OP309–OP313 (2012).
[Crossref] [PubMed]

J. P. Hadden, J. P. Harrison, A. C. Stanley-Clarke, L. Marseglia, Y-L. D. Ho, B. R. Patton, J. L. O’Brien, and J. G. Rarity, “Strongly enhanced photon collection from diamond defect centres under micro-fabricated integrated solid immersion lenses,” Appl. Phys. Lett.,  97, 241901 (2010).
[Crossref]

Harrison, J. P.

J. P. Hadden, J. P. Harrison, A. C. Stanley-Clarke, L. Marseglia, Y-L. D. Ho, B. R. Patton, J. L. O’Brien, and J. G. Rarity, “Strongly enhanced photon collection from diamond defect centres under micro-fabricated integrated solid immersion lenses,” Appl. Phys. Lett.,  97, 241901 (2010).
[Crossref]

Heifetz, A.

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

Hell, S. W.

D. Wildanger, B. R. Patton, H. Schill, L. Marseglia, J. P. Hadden, S. Knauer, A. Schnle, J. G. Rarity, J. L. O’Brien, S. W. Hell, and J. M. Smith, “Solid Immersion Facilitates Fluorescence Microscopy with Nanometer Resolution and Sub-Ångström emitter localization,” Adv. Mater 24, OP309–OP313 (2012).
[Crossref] [PubMed]

Ho, Y-L. D.

J. P. Hadden, J. P. Harrison, A. C. Stanley-Clarke, L. Marseglia, Y-L. D. Ho, B. R. Patton, J. L. O’Brien, and J. G. Rarity, “Strongly enhanced photon collection from diamond defect centres under micro-fabricated integrated solid immersion lenses,” Appl. Phys. Lett.,  97, 241901 (2010).
[Crossref]

Hoang, T. X.

T. X. Hoang, X. Chen, and C. J. R. Sheppard, “Multipole and plane wave expansions of diverging and converging fields,” Opt. Express 22, 8949–8961 (2014).
[Crossref] [PubMed]

T. X. Hoang, X. Chen, and C. J. R. Sheppard, “Rigorous analytical modeling of high-aperture focusing through a spherical interface,” J. Opt. Soc. Am. A 30, 1426–1440 (2013).
[Crossref]

T. X. Hoang, X. Chen, and C. J. R. Sheppard, “Multipole theory for tight focusing of polarized light, including radially polarized and other special cases,” J. Opt. Soc. Am. A 29, 32–43 (2012).
[Crossref]

T. X. Hoang, X. Chen, and C. J. R. Sheppard, “Interpretation of the scattering mechanism for particles in a focused beam,” Phys. Rev. A 86, 033817 (2012).
[Crossref]

T. X. Hoang, “Focusing light through spherical interface for subsurface microscopy,” Ph.D. dissertation (2014).

T. X. Hoang, R. Chen, K. Agarwal, C. J. R. Sheppard, and X. Chen, “Imaging with annular focusing through a dielectric interface,” in Proceedings of Focus On Microscopy 2014, Sydney, Australia2014 April 13–16.

Hong, B. H.

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- Assembled Nanoscale Spherical Lenses,” Nature (London) 460, 498–501 (2009).
[Crossref]

Hong, M.

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S. H. Goh, C. J. R Sheppard, A. C. T. Quah, C. M. Chua, L. S. Koh, and J. C. H. Phang, “Design considerations for refractive solid immersion lens: application to subsurface integrated circuit fault localization using laser induced techniques,” Rev. Sci. Instrum. 80, 013703 (2009).
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D. Wildanger, B. R. Patton, H. Schill, L. Marseglia, J. P. Hadden, S. Knauer, A. Schnle, J. G. Rarity, J. L. O’Brien, S. W. Hell, and J. M. Smith, “Solid Immersion Facilitates Fluorescence Microscopy with Nanometer Resolution and Sub-Ångström emitter localization,” Adv. Mater 24, OP309–OP313 (2012).
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S. Lee, L. Li, and Z. Wang, “Optical resonances in microsphere photonic nanojets,” J. Opt. 16, 015704 (2014).
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Y. Yan, L. Li, C. Feng, W. Guo, S. Lee, and M. Hong, “Microsphere-Coupled Scanning Laser Confocal Nanoscope for Sub-Diffraction-Limited Imaging at 25 nm Lateral Resolution in the Visible Spectrum,” ACS Nano,  8, 1809–1816 (2014).
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R. Ye, Y.-H. Ye, H. F. Ma, L. Cao, J. Ma, F. Wyrowski, R. Shi, and J.-Y. Zhang, “Experimental imaging properties of immersion microscale spherical lenses,” Sci. Rep. 4, 1–5 (2014).
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R. Ye, Y.-H. Ye, H. F. Ma, L. Cao, J. Ma, F. Wyrowski, R. Shi, and J.-Y. Zhang, “Experimental imaging properties of immersion microscale spherical lenses,” Sci. Rep. 4, 1–5 (2014).
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ACS Nano (1)

Y. Yan, L. Li, C. Feng, W. Guo, S. Lee, and M. Hong, “Microsphere-Coupled Scanning Laser Confocal Nanoscope for Sub-Diffraction-Limited Imaging at 25 nm Lateral Resolution in the Visible Spectrum,” ACS Nano,  8, 1809–1816 (2014).
[Crossref] [PubMed]

Adv. Mater (1)

D. Wildanger, B. R. Patton, H. Schill, L. Marseglia, J. P. Hadden, S. Knauer, A. Schnle, J. G. Rarity, J. L. O’Brien, S. W. Hell, and J. M. Smith, “Solid Immersion Facilitates Fluorescence Microscopy with Nanometer Resolution and Sub-Ångström emitter localization,” Adv. Mater 24, OP309–OP313 (2012).
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Appl. Opt. (2)

Appl. Phys. Lett. (2)

V. M. Sundaram and S. B. Wen, “Analysis of deep sub-micron resolution in microsphere based imaging,” Appl. Phys. Lett. 105, 204102 (2014).
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J. P. Hadden, J. P. Harrison, A. C. Stanley-Clarke, L. Marseglia, Y-L. D. Ho, B. R. Patton, J. L. O’Brien, and J. G. Rarity, “Strongly enhanced photon collection from diamond defect centres under micro-fabricated integrated solid immersion lenses,” Appl. Phys. Lett.,  97, 241901 (2010).
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J. Comput. Theor. Nanosci. (1)

A. Heifetz, S.-C. Kong, A. V. Sahakian, A. Taflove, and V. Backman, “Photonic Nanojets,” J. Comput. Theor. Nanosci. 6, 1979–1992 (2009).
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J. Elect. Wave. Appl. (1)

W. C. Chew and Y. M. Wang, “Efficient Ways to Compute the Vector Addition Theorem,” J. Elect. Wave. Appl. 7, 651–665 (1993).
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J. Math. Phys. (2)

H. M. Nussenzveig, “High-frequency scattering by a transparent sphere. I. Direct reflection and transmission,” J. Math. Phys. 10, 82–124 (1969).
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J. Opt. (1)

S. Lee, L. Li, and Z. Wang, “Optical resonances in microsphere photonic nanojets,” J. Opt. 16, 015704 (2014).
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J. Opt. Soc. Am. A (4)

Nat. Commun. (1)

Z. Wang, W. Guo, L. Li, B. Lukyanchuk, 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).
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Nature (London) (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- Assembled Nanoscale Spherical Lenses,” Nature (London) 460, 498–501 (2009).
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Nature Nanotech. (1)

M. D. Baaske, M. R. Foreman, and F. Vollmer, “Single-molecule nucleic acid interactions monitored on a label-free microcavity biosensor platform,” Nature Nanotech. 9, 933–939 (2014).
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Opt. Commun. (1)

S. Quabis, R. Dorn, M. Eberler, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7 (2000).
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Opt. Express (3)

Opt. Lett. (4)

Phys. Rev. A (2)

S. Orlov, U. Peschel, T. Bauer, and P. Banzer, “Analytical expansion of highly focused vector beams into vector spherical harmonics and its application to Mie scattering,” Phys. Rev. A 85, 063825 (2012).
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T. X. Hoang, X. Chen, and C. J. R. Sheppard, “Interpretation of the scattering mechanism for particles in a focused beam,” Phys. Rev. A 86, 033817 (2012).
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Phys. Rev. Lett. (1)

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901 (2003).
[Crossref] [PubMed]

Proc. Roy. Soc. London Ser. A (1)

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. Roy. Soc. London Ser. A 253, 358–379 (1959).
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Rev. Sci. Instrum. (1)

S. H. Goh, C. J. R Sheppard, A. C. T. Quah, C. M. Chua, L. S. Koh, and J. C. H. Phang, “Design considerations for refractive solid immersion lens: application to subsurface integrated circuit fault localization using laser induced techniques,” Rev. Sci. Instrum. 80, 013703 (2009).
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Sci. Rep. (1)

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Figures (11)

Fig. 1
Fig. 1

Imaging with Conventional Lenses: a) Physical Setup and b) System Modeling.

Fig. 2
Fig. 2

Microsphere based Microscopy.

Fig. 3
Fig. 3

Illumination with a) An incident converging wave and b) An incident plane wave.

Fig. 4
Fig. 4

Far field approximation for expressing the field in O coordinate system.

Fig. 5
Fig. 5

a) Scattering coefficients as a function of the wavelength λ: the peaks correspond to the resonant wavelengths and b) Scattering coefficients with a resonant wavelength of λ = 401.6345 nm and an off-resonant wavelength of λ = 403.07 nm.

Fig. 6
Fig. 6

Electric intensity in linear (left) and logarithmic (right) scales with λ = 401.6345 nm. Light propagates from the left to the right sides.

Fig. 7
Fig. 7

Electric intensity at focus along the transversal direction with a) λ = 401.6345 nm (off-resonance) and λ = 439.44 nm (resonance). b) λ = 403.07 nm (off-resonance) and λ = 445.5127 nm (resonance). CP and LP refer to circularly polarized and linearly polarized light, respectively.

Fig. 8
Fig. 8

Resonance with a converging wave illumination and d = 7R.

Fig. 9
Fig. 9

A plane wave illumination with a circular polarization.

Fig. 10
Fig. 10

Multipole coefficients with a) λ = 450 nm and b) λ = 445.5127 nm.

Fig. 11
Fig. 11

Images of two dipoles spaced 109 nm apart with a) λ = 445.5127 nm, b) λ = 450 nm, and c) λ = 445.5127 nm.

Equations (32)

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E ( r ¯ ) = l = 1 m = l l [ p El m N lm ( 1 ) ( r ¯ ) + p Ml m M lm ( 1 ) ( r ¯ ) ] ,
N lm ( 1 ) ( r ¯ ) = × × [ r ¯ h l ( 1 ) ( k r ) Y l m ( θ , ϕ ) ] = r ^ l ( l + 1 ) r h l ( 1 ) ( k r ) Y l m ( θ , ϕ ) + θ ^ [ 1 r d d r ( r h l ( 1 ) ( k r ) ) θ Y l m ( θ , ϕ ) ] + ϕ ^ im sin θ 1 r d d r ( r h l ( 1 ) ( k r ) ) Y l m ( θ , ϕ ) , M lm ( 1 ) ( r ¯ ) = i k × [ r ¯ h l ( 1 ) ( k r ) Y l m ( θ , ϕ ) ] = k h l ( 1 ) ( k r ) [ θ ^ m sin θ Y l m ( θ , ϕ ) + i ϕ ^ θ Y l m ( θ , ϕ ) ] ,
Y l m ( θ , ϕ ) = ( 1 ) m [ 2 l + 1 4 π ( l m ) ! ( l + m ) ! ] 1 2 P l m ( cos θ ) exp ( im ϕ ) = c lm P l m ( cos θ ) exp ( im ϕ ) ,
P l m ( x ) = 1 2 l l ! ( 1 x 2 ) m / 2 d l + m d x l + m ( x 2 1 ) l .
E r = k l = 1 m = l l l ( l + 1 ) 2 l + 1 c lm ( p El m [ h l 1 ( 1 ) ( k r ) + h l + 1 ( 1 ) ( k r ) ] P l m ( cos θ ) ) exp ( im ϕ ) , E θ = k l = 1 m = l l l ( l + 1 ) 2 l + 1 c lm ( p El m [ h l 1 ( 1 ) ( k r ) l h l + 1 ( 1 ) ( k r ) l + 1 ] d P l m ( cos θ ) d θ m 2 l + 1 l ( l + 1 ) p Ml m h l ( 1 ) ( k r ) P l m ( cos θ ) sin θ ) exp ( im ϕ ) , E ϕ = i k l = 1 m = l l l ( l + 1 ) 2 l + 1 c lm ( m p El m [ h l 1 ( 1 ) ( k r ) l h l + 1 ( 1 ) ( k r ) l + 1 ] P l m ( cos θ ) sin θ 2 l + 1 l ( l + 1 ) p Ml m h l ( 1 ) ( k r ) d P l m ( cos θ ) d θ ) exp ( im ϕ ) .
k r L ( L + 1 ) / 2
h l ( 1 ) ( k r ) ( i ) l + 1 e i k r k r
h l 1 ( 1 ) ( k r ) + h l + 1 ( 1 ) ( k r ) 0 and l ( l + 1 ) 2 l + 1 [ h l 1 ( 1 ) ( k r ) l h l + 1 ( 1 ) ( k r ) l + 1 ] ( i ) l e i k r k r .
E r 0 , E θ e i k r r l = 1 L m = l l K l m ( θ ) exp ( im ϕ ) , E ϕ e i k r r l = 1 L m = l l G l m ( θ ) exp ( im ϕ ) ,
K l m ( θ ) = ( i ) l c lm [ p El m d P l m ( cos θ ) d θ + i m p Ml m P l m ( cos θ ) sin θ ] , G l m ( θ ) = ( i ) l 1 c lm [ m p El m P l m ( cos θ ) sin θ + i p Ml m d P l m ( cos θ ) d θ ] .
E α 1 = E θ e i k f 1 f 1 l = 1 L m = l l K l m ( α 1 ) exp ( im β 1 ) ,
E β 1 = E ϕ e i k f 1 f 1 l = 1 L m = l l G l m ( α 1 ) exp ( im β 1 ) ,
E ρ = E α 1 cos α 1 , E φ = E β 1 cos α 1 .
sin α = f 1 f sin α 1 , β = β 1 ,
E α = E ρ cos α = E α 1 cos α cos α 1 , E β = E φ cos α = E β 1 cos α cos α 1 .
E ( x , y , z ) = i k f e i k f 2 π 0 α m sin α d α 0 2 π d β [ E α ( cos β cos α sin β cos α sin α ) + E β ( sin β cos β 0 ) ] e i k [ ρ sin α cos ( β ϕ ) + z cos α ] .
E ( x , y , z ) = l = 1 L m = l l ( I l m 1 ( r ¯ ) + I l m + 1 ( r ¯ ) i ( I l m 1 ( r ¯ ) I l m + 1 ( r ¯ ) ) 2 I l m ( r ¯ ) ) ,
I l m + 1 ( r ¯ ) = i k f e i k f 2 i m + 1 e i ( m + 1 ) ϕ 0 α m J m + 1 ( k ρ sin α ) [ U l m ( α ) cos α + i V l m ( α ) ] e i k z cos α sin α d α , I l m 1 ( r ¯ ) = i k f e i k f 2 i m 1 e i ( m 1 ) ϕ 0 α m J m 1 ( k ρ sin α ) [ U l m ( α ) cos α i V l m ( α ) ] e i k z cos α sin α d α , I l m ( r ¯ ) = i k f e i k f 2 i m e i m ϕ 0 α m J m ( k ρ sin α ) U l m ( α ) sin α e i k z cos α sin α d α , U l m ( α ) = e i k f 1 f 1 K l m ( α 1 ) cos α cos α 1 , V l m ( α ) = e i k f 1 f 1 G l m ( α 1 ) cos α cos α 1 .
p El m = i l f e i k f c lm l ( l + 1 ) 0 2 π 0 α 1 m ( d P l m ( cos α ) d α E α i m P l m ( cos α ) sin α E β ) e i m β sin α d α d β , p Ml m = i l f e i k f c lm l ( l + 1 ) 0 2 π 0 α 1 m ( d P l m ( cos α ) d α E β + i m P l m ( cos α ) sin α E α ) e i m β sin α d α d β .
p El m = l = 1 m = l l [ A l m lm p El m + i B l m lm p Ml m ] , p Ml m = l = 1 m = l l [ A l m lm p Ml m i B l m lm p El m ]
p ¯ El m = ( ε / ε s ) c l p El m p ¯ Ml m = ( k / k s ) d l p Ml m
p ˜ El m = a l p El m p ˜ Ml m = b l p Ml m
a l = ε ε s k s k J ^ l ( k R ) J ^ l ( k s R ) J ^ l ( k R ) J ^ l ( k s R ) H ^ l ( 1 ) ( k R ) J ^ l ( k s R ) ε ε s k s k H ^ l ( 1 ) ( k R ) J ^ l ( k s R ) , b l μ μ s k s k J ^ l ( k R ) J ^ l ( k s R ) J ^ l ( k R ) J ^ l ( k s R ) H ^ l ( 1 ) ( k R ) J ^ l ( k s R ) μ μ s k s k H ^ l ( 1 ) ( k R ) J ^ l ( k s R ) , c l = i k s k H ^ l ( 1 ) ( k R ) J ^ l ( k s R ) ε ε s k s k H ^ l ( 1 ) ( k R ) J ^ l ( k s R ) , d l = i k s k H ^ l ( 1 ) ( k R ) J ^ l ( k s R ) μ μ s k s k H ^ l ( 1 ) ( k R ) J ^ l ( k s R ) ,
p El m = E 0 k i l + 1 π ( 2 l + 1 ) l ( l + 1 ) ( δ m 1 δ m 1 ) , p Ml m = E 0 k i l π ( 2 l + 1 ) l ( l + 1 ) ( δ m 1 + δ m 1 ) .
p El m = 2 E 0 k i l + 1 π ( 2 l + 1 ) l ( l + 1 ) δ m 1 , p Ml m = 2 E 0 k i l π ( 2 l + 1 ) l ( l + 1 ) δ m 1 .
p El m = ω μ I l 2 6 π ( a l [ A lm 1 , 1 A lm 1 , 1 ] + [ C lm 1 , 1 C lm 1 , 1 ] ) , p Ml m = i ω μ I l 2 6 π ( b l [ B lm 1 , 1 B lm 1 , 1 ] + [ D lm 1 , 1 D lm 1 , 1 ] ) ,
p El m = i ω μ I l 2 6 π ( a l [ A lm 1 , 1 + A lm 1 , 1 ] + [ C lm 1 , 1 + C lm 1 , 1 ] ) , p Ml m = ω μ I l 2 6 π ( b l [ B lm 1 , 1 + B lm 1 , 1 ] + [ D lm 1 , 1 + D lm 1 , 1 ] ) .
p El m = ω μ I l 2 3 π ( a l A lm 1 , 0 + C lm 1 , 0 ) , p Ml m = i ω μ I l 2 3 π ( b l B lm 1 , 0 + D lm 1 , 0 ) .
p ¯ El m = e i k d cos α 1 p El m p ¯ Ml m = e i k d cos α 1 p Ml m
k R 37 , l 25 , l + 49 .
p El m = i l f e i k f [ π ( 2 l + 1 ) ] 1 2 l ( l + 1 ) δ m 0 0 α m d P l ( cos α ) d α sin 2 α d α , p Ml m = 0 .
Δ λ = 2 λ 0 ( n s 2 1 ) k 0 R ( H ^ l ( 1 ) ( k 0 R ) J ^ l ( k 0 R ) ) 2 .

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