Abstract

We study experimentally and theoretically the electromagnetic field in amplitude and phase behind ball-lenses across a wide range of diameters, ranging from a millimeter scale down to a micrometer. Based on the observation, we study the transition between the refraction and diffraction regime. The former regime is dominated by observables for which it is sufficient to use a ray-optical picture for an explanation, e.g., a cusp catastrophe and caustics. A wave-optical picture, i.e. Mie theory, is required to explain the features, e.g., photonic nanojets, in the latter regime. The vanishing of the cusp catastrophe and the emergence of the photonic nanojet is here understood as the refraction limit. Three different criteria are used to identify the limit: focal length, spot size, and amount of cross-polarization generated in the scattering process. We identify at a wavelength of 642 nm and while considering ordinary glass as the ball-lens material, a diameter of approximately 10 µm as the refraction limit. With our study, we shed new light on the means necessary to describe micro-optical system. This is useful when designing optical devices for imaging or illumination.

© 2016 Optical Society of America

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References

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  1. M. V. Berry, “Rays, wavefronts and phase: a picture book of cusps,” Proc. Huygens Symp., ed. H. Blok, H.A. Ferwerda, and H.K. Kuiken, Huygens' Principle 1690–1990: Theory and Applications (Elsevier Science Publishers B.V., 1992).
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    [Crossref]
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  7. R. P. Ratowsky, L. Yang, R. J. Deri, J. S. Kallman, and G. Trott, “Ball lens reflections by direct solution of Maxwell’s equations,” Opt. Lett. 20(20), 2048–2050 (1995).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  11. 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]
  12. Z. Wang, W. Guo, L. Li, B. Luk’yanchuk, A. Khan, Z. Liu, Z. Chen, and M. Hong, “Optical virtual imaging at 50 nm lateral resolution with a white-light nanoscope,” Nat. Commun. 2, 218 (2011).
    [Crossref] [PubMed]
  13. M. X. Wu, B. J. Huang, R. Chen, Y. Yang, J. F. Wu, R. Ji, X. D. Chen, and M. H. Hong, “Modulation of photonic nanojets generated by microspheres decorated with concentric rings,” Opt. Express 23(15), 20096–20103 (2015).
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    [Crossref]
  16. B. Sick, B. Hecht, U. P. Wild, and L. Novotny, “Probing confined fields with single molecules and vice versa,” J. Microsc. 202(2), 365–373 (2001).
    [Crossref] [PubMed]
  17. M.-S. Kim, T. Scharf, and H. P. Herzig, “Small-size microlens characterization by multiwavelength high-resolution interference microscopy,” Opt. Express 18(14), 14319–14329 (2010).
    [Crossref] [PubMed]
  18. 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]
  19. M.-S. Kim, T. Scharf, S. Mühlig, C. Rockstuhl, and H. P. Herzig, “Gouy phase anomaly in photonic nanojets,” Appl. Phys. Lett. 98(19), 191114 (2011).
    [Crossref]
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    [Crossref]
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    [Crossref]

2015 (1)

2013 (1)

2011 (3)

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]

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

M.-S. Kim, T. Scharf, S. Mühlig, C. Rockstuhl, and H. P. Herzig, “Gouy phase anomaly in photonic nanojets,” Appl. Phys. Lett. 98(19), 191114 (2011).
[Crossref]

2010 (1)

2009 (1)

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]

2004 (1)

2001 (1)

B. Sick, B. Hecht, U. P. Wild, and L. Novotny, “Probing confined fields with single molecules and vice versa,” J. Microsc. 202(2), 365–373 (2001).
[Crossref] [PubMed]

2000 (2)

K. Bahlmann and S. W. Hell, “Depolarization by high aperture focusing,” Appl. Phys. Lett. 77(5), 612–614 (2000).
[Crossref]

K. Youngworth and T. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express 7(2), 77–87 (2000).
[Crossref] [PubMed]

1998 (2)

R. G. Wilson, “Ball-lens coupling efficiency for laser-diode to single-mode fiber: comparison of independent studies by distinct methods,” Appl. Opt. 37(15), 3201–3205 (1998).
[Crossref] [PubMed]

U. Vokinger, R. Dändliker, P. Blattner, and H. P. Herzig, “Unconventional treatment of focal shift,” Opt. Commun. 157(1-6), 218–224 (1998).
[Crossref]

1995 (1)

1984 (1)

1946 (1)

T. Pearcey, “The structure of an electromagnetic field in the neighbourhood of a cusp of a caustic,” Philos. Mag. 37(268), 311–317 (1946).
[Crossref]

Backman, V.

Bahlmann, K.

K. Bahlmann and S. W. Hell, “Depolarization by high aperture focusing,” Appl. Phys. Lett. 77(5), 612–614 (2000).
[Crossref]

Blattner, P.

U. Vokinger, R. Dändliker, P. Blattner, and H. P. Herzig, “Unconventional treatment of focal shift,” Opt. Commun. 157(1-6), 218–224 (1998).
[Crossref]

Brown, T.

Chen, R.

Chen, X. D.

Chen, Z.

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

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]

Dändliker, R.

U. Vokinger, R. Dändliker, P. Blattner, and H. P. Herzig, “Unconventional treatment of focal shift,” Opt. Commun. 157(1-6), 218–224 (1998).
[Crossref]

Deri, R. J.

Guo, H.

Guo, W.

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

Han, Y.

Hecht, B.

B. Sick, B. Hecht, U. P. Wild, and L. Novotny, “Probing confined fields with single molecules and vice versa,” J. Microsc. 202(2), 365–373 (2001).
[Crossref] [PubMed]

Heifetz, A.

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]

Hell, S. W.

K. Bahlmann and S. W. Hell, “Depolarization by high aperture focusing,” Appl. Phys. Lett. 77(5), 612–614 (2000).
[Crossref]

Herzig, H. P.

M.-S. Kim, T. Scharf, S. Mühlig, C. Rockstuhl, and H. P. Herzig, “Gouy phase anomaly in photonic nanojets,” Appl. Phys. Lett. 98(19), 191114 (2011).
[Crossref]

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]

M.-S. Kim, T. Scharf, and H. P. Herzig, “Small-size microlens characterization by multiwavelength high-resolution interference microscopy,” Opt. Express 18(14), 14319–14329 (2010).
[Crossref] [PubMed]

U. Vokinger, R. Dändliker, P. Blattner, and H. P. Herzig, “Unconventional treatment of focal shift,” Opt. Commun. 157(1-6), 218–224 (1998).
[Crossref]

Hong, M.

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

Hong, M. H.

Huang, B. J.

Ji, R.

Kallman, J. S.

Khan, A.

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

Kim, M.-S.

Kong, S.-C.

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]

Li, L.

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

Li, Y.

Liu, Z.

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

Luk’yanchuk, B.

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

Mühlig, S.

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]

M.-S. Kim, T. Scharf, S. Mühlig, C. Rockstuhl, and H. P. Herzig, “Gouy phase anomaly in photonic nanojets,” Appl. Phys. Lett. 98(19), 191114 (2011).
[Crossref]

Novotny, L.

B. Sick, B. Hecht, U. P. Wild, and L. Novotny, “Probing confined fields with single molecules and vice versa,” J. Microsc. 202(2), 365–373 (2001).
[Crossref] [PubMed]

Pearcey, T.

T. Pearcey, “The structure of an electromagnetic field in the neighbourhood of a cusp of a caustic,” Philos. Mag. 37(268), 311–317 (1946).
[Crossref]

Ratowsky, R. P.

Rockstuhl, C.

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]

M.-S. Kim, T. Scharf, S. Mühlig, C. Rockstuhl, and H. P. Herzig, “Gouy phase anomaly in photonic nanojets,” Appl. Phys. Lett. 98(19), 191114 (2011).
[Crossref]

Sahakian, A. V.

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]

Scharf, T.

Sick, B.

B. Sick, B. Hecht, U. P. Wild, and L. Novotny, “Probing confined fields with single molecules and vice versa,” J. Microsc. 202(2), 365–373 (2001).
[Crossref] [PubMed]

Sui, G.

Taflove, A.

Trott, G.

Vokinger, U.

U. Vokinger, R. Dändliker, P. Blattner, and H. P. Herzig, “Unconventional treatment of focal shift,” Opt. Commun. 157(1-6), 218–224 (1998).
[Crossref]

Wang, Y.

Wang, Z.

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

Weng, X.

Wild, U. P.

B. Sick, B. Hecht, U. P. Wild, and L. Novotny, “Probing confined fields with single molecules and vice versa,” J. Microsc. 202(2), 365–373 (2001).
[Crossref] [PubMed]

Wilson, R. G.

Wolf, E.

Wu, J. F.

Wu, M. X.

Yang, L.

Yang, Y.

Youngworth, K.

Zhao, Y.

Zhuang, S.

Appl. Opt. (1)

Appl. Phys. Lett. (2)

M.-S. Kim, T. Scharf, S. Mühlig, C. Rockstuhl, and H. P. Herzig, “Gouy phase anomaly in photonic nanojets,” Appl. Phys. Lett. 98(19), 191114 (2011).
[Crossref]

K. Bahlmann and S. W. Hell, “Depolarization by high aperture focusing,” Appl. Phys. Lett. 77(5), 612–614 (2000).
[Crossref]

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(9), 1979–1992 (2009).
[Crossref] [PubMed]

J. Microsc. (1)

B. Sick, B. Hecht, U. P. Wild, and L. Novotny, “Probing confined fields with single molecules and vice versa,” J. Microsc. 202(2), 365–373 (2001).
[Crossref] [PubMed]

J. Opt. Soc. Am. A (1)

Nat. Commun. (1)

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

Opt. Commun. (1)

U. Vokinger, R. Dändliker, P. Blattner, and H. P. Herzig, “Unconventional treatment of focal shift,” Opt. Commun. 157(1-6), 218–224 (1998).
[Crossref]

Opt. Express (6)

Opt. Lett. (1)

Philos. Mag. (1)

T. Pearcey, “The structure of an electromagnetic field in the neighbourhood of a cusp of a caustic,” Philos. Mag. 37(268), 311–317 (1946).
[Crossref]

Other (5)

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).

J. J. Stamnes, Waves in Focal Regions (Adam Hilger, 1986).

F. Träger, Handbook of Lasers and Optics (Springer, 2007).

M. Riedl, Optical Design Fundamentals for Infrared Systems (SPIE, 2001), 2nd ed.

M. V. Berry, “Rays, wavefronts and phase: a picture book of cusps,” Proc. Huygens Symp., ed. H. Blok, H.A. Ferwerda, and H.K. Kuiken, Huygens' Principle 1690–1990: Theory and Applications (Elsevier Science Publishers B.V., 1992).

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Figures (8)

Fig. 1
Fig. 1

(a) Geometrical optical parameters of a ball-lens, where EFL is effective focal length, BFL back focal length, and R the radius of the sphere. (b) FRED ray diagram emerging from the rear surface of a glass ball-lens.

Fig. 2
Fig. 2

(a) Measured x-z intensity and (b) phase distributions near the focus. The cusp catastrophe is created by a 200-µm soda lime glass ball-lens. Intensity is normalized and the surface of the ball-lens is on z = 0 plane

Fig. 3
Fig. 3

The x-z intensity distributions of the focal spots created by the polystyrene ball-lens of (a) 10-µm, (b) 15-µm, (c) 20-µm, and (d) 30-µm diameters: (upper row) linear-scale intensity and (lower row) logarithmic-scale intensity maps. For easy comparison, the spot is aligned in the middle of each image and intensities are all normalized.

Fig. 4
Fig. 4

Focal length vs sphere diameter. Lines are calculated data by Eq. (2) and markers are measured data. Open squares of black color are results from the FRED calculation and colored asterisks are from Mie simulation.

Fig. 5
Fig. 5

Measured FWHM spot size in the x-axis vs sphere diameter. The diameters are displayed in logarithmic scale in order to emphasize the photonic nanojet regime [D ≤ 10 µm].

Fig. 6
Fig. 6

The measured transverse intensity distributions of the total field components (Et = Ex + Ey) and the orthogonal field component (Ey) in the lower row: (a) D = 1 μm, (b) D = 2 μm, (3) D = 6 μm, and (d) D = 20 μm. The white scale bar indicates 1 μm.

Fig. 7
Fig. 7

Reproduced intensity distribution of Ey for the 20-μm sphere shown in Fig. 6(d) using Mie simulation, where the extinction ratio of the polarizer 1:103 is taken into account. Intensity is normalized.

Fig. 8
Fig. 8

Peak intensity ratio of Ey/Et from experiments and Mie simulations.

Equations (4)

Equations on this page are rendered with MathJax. Learn more.

EFL= nR 2(n1) ,
BFL=EFLR,
I t = | E x | 2 + | E y | 2 ,
I y = | E y | 2 .

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