Abstract

The phenomenon of superoscillation produces oscillations that are faster than the fastest Fourier component of a system, potentially forming a local “hot spot” with a size below the diffraction limit. We show that a radially polarized Laguerre–Gaussian mode has the inherent ability to form superoscillation spots simply by controlling the incident beam size. We investigate this in detail, both numerically and experimentally. Our numerical simulations predict that lateral resolutions close to 100 nm are possible for practical confocal laser scanning microscopy with visible light. We demonstrate experimentally that superoscillation focusing can offer significant spatial resolution improvements for fluorescence imaging.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2016 (1)

2015 (2)

Y. Kozawa and S. Sato, “Numerical analysis of resolution enhancement in laser scanning microscopy using a radially polarized beam,” Opt. Express 23, 2076–2084 (2015).
[Crossref]

F. Qin, K. Huang, J. Wu, J. Jiao, X. Luo, C. Qiu, and M. Hong, “Shaping a subwavelength needle with ultra-long focal length by focusing azimuthally polarized light,” Sci. Rep. 5, 9977 (2015).
[Crossref]

2014 (3)

G. Yuan, E. T. F. Rogers, T. Roy, G. Adamo, Z. Shen, and N. I. Zheludev, “Planar super-oscillatory lens for sub-diffraction optical needles at violet wavelengths,” Sci. Rep. 4, 6333 (2014).
[Crossref]

S. Ipponjima, T. Hibi, Y. Kozawa, H. Horanai, H. Yokoyama, S. Sato, and T. Nemoto, “Improvement of lateral resolution and extension of depth of field in two-photon microscopy by a higher-order radially polarized beam,” Microscopy 63, 23–32 (2014).
[Crossref]

S. Segawa, Y. Kozawa, and S. Sato, “Demonstration of subtraction imaging in confocal microscopy with vector beams,” Opt. Lett. 39, 4529–4532 (2014).
[Crossref]

2013 (4)

E. Rogers and N. I. Zheludev, “Optical super-oscillations: sub-wavelength light focusing and super-resolution imaging,” J. Opt. 15, 094008 (2013).
[Crossref]

E. T. F. Rogers, S. Savo, J. Lindberg, T. Roy, M. Dennis, and N. I. Zheludev, “Super-oscillatory optical needle,” Appl. Phys. Lett. 102, 031108 (2013).
[Crossref]

E. Greenfield, R. Schley, I. Hurwitz, J. Nemirovsky, K. G. Makris, and M. Segev, “Experimental generation of arbitrarily shaped diffractionless superoscillatory optical beams,” Opt. Express 21, 13425–13435 (2013).
[Crossref]

A. M. H. Wong and G. V. Eleftheriades, “An optical super-microscope for far-field, real-time imaging beyond the diffraction limit,” Sci. Rep. 3, 1715 (2013).
[Crossref]

2012 (4)

E. T. F. Rogers, J. Lindberg, T. Roy, S. Savo, J. E. Chad, M. R. Dennis, and N. I. Zheludev, “A super-oscillatory lens optical microscope for subwavelength imaging,” Nat. Mater. 11, 432–435 (2012).
[Crossref]

J. Lindberg, “Mathematical concepts of optical superresolution,” J. Opt. 14, 083001 (2012).
[Crossref]

H. J. Hyvärinen, S. Rehman, J. Tervo, J. Turunen, and C. J. R. Sheppard, “Limitations of superoscillation filters in microscopy applications,” Opt. Lett. 37, 903–905 (2012).
[Crossref]

Y. Kozawa and S. Sato, “Focusing of higher-order radially polarized Laguerre–Gaussian beam,” J. Opt. Soc. Am. A 29, 2439–2443 (2012).
[Crossref]

2011 (4)

2009 (1)

2008 (1)

N. I. Zheludev, “What diffraction limit?” Nat. Mater. 7, 420–422 (2008).
[Crossref]

2007 (2)

2006 (4)

Y. Kozawa and S. Sato, “Focusing property of a double-ring-shaped radially polarized beam,” Opt. Lett. 31, 820–822 (2006).
[Crossref]

M. V. Berry and S. Popescu, “Evolution of quantum superoscillation and optical superresolution without evanescent waves,” J. Phys. A 39, 6965–6977 (2006).
[Crossref]

P. J. S. G. Ferreira and A. Kempf, “Superoscillations: faster than the Nyquist rate,” IEEE Trans. Signal Process. 54, 3732–3740 (2006).
[Crossref]

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science 313, 1642–1645 (2006).
[Crossref]

2004 (1)

2001 (1)

2000 (4)

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

M. A. A. Neil, R. Juškaitis, T. Wilson, Z. J. Laczik, and V. Sarafis, “Optimized pupil-plane filters for confocal microscope point-spread function engineering,” Opt. Lett. 25, 245–247 (2000).
[Crossref]

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

M. G. L. Gustafsson, “Surpassing the lateral resolution limit by a factor of two using structured illumination microscopy,” J. Microsc. 198, 82–87 (2000).
[Crossref]

1999 (1)

P. D. Higdon, P. Török, and T. Wilson, “Imaging properties of high aperture multiphoton fluorescence scanning optical microscopes,” J. Microsc. 193, 127–141 (1999).
[Crossref]

1998 (1)

1994 (1)

1975 (1)

D. H. Close, “Holographic optical elements,” Opt. Eng. 14, 408–419 (1975).
[Crossref]

1959 (1)

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

1952 (1)

G. Toraldo di Francia, “Super-gain antennas and optical resolving power,” Nuovo Cimento Suppl. 9, 426–438 (1952).
[Crossref]

1872 (1)

J. W. Strutt, “On the diffraction of object-glasses,” Mon. Not. R. Astron. Soc. 33, 59–63 (1872).
[Crossref]

Adamo, G.

G. Yuan, E. T. F. Rogers, T. Roy, G. Adamo, Z. Shen, and N. I. Zheludev, “Planar super-oscillatory lens for sub-diffraction optical needles at violet wavelengths,” Sci. Rep. 4, 6333 (2014).
[Crossref]

Baumgartl, J.

M. Mazilu, J. Baumgartl, S. Kosmeier, and K. Dholakia, “Optical eigenmodes; exploiting the quadratic nature of the energy flux and of scattering interactions,” Opt. Express 19, 933–945 (2011).
[Crossref]

J. Baumgartl, S. Kosmeier, M. Mazilu, E. T. F. Rogers, N. I. Zheludev, and K. Dholakia, “Far field subwavelength focusing using optical eigenmodes,” Appl. Phys. Lett. 98, 181109 (2011).
[Crossref]

Berry, M. V.

M. V. Berry and S. Popescu, “Evolution of quantum superoscillation and optical superresolution without evanescent waves,” J. Phys. A 39, 6965–6977 (2006).
[Crossref]

M. V. Berry, “Faster than Fourier,” in Quantum Coherence and Reality; in Celebration of the 60th Birthday of Yakir Aharonov, J. S. Anandan and J. L. Safko, eds. (World Scientific, 1994), pp. 55–65.

Betzig, E.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science 313, 1642–1645 (2006).
[Crossref]

Bokor, N.

Bonifacino, J. S.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science 313, 1642–1645 (2006).
[Crossref]

Brown, T. G.

Chad, J. E.

E. T. F. Rogers, J. Lindberg, T. Roy, S. Savo, J. E. Chad, M. R. Dennis, and N. I. Zheludev, “A super-oscillatory lens optical microscope for subwavelength imaging,” Nat. Mater. 11, 432–435 (2012).
[Crossref]

Choudhury, A.

Close, D. H.

D. H. Close, “Holographic optical elements,” Opt. Eng. 14, 408–419 (1975).
[Crossref]

Courjon, D.

T. Grosjean and D. Courjon, “Smallest focal spots,” Opt. Commun. 272, 314–319 (2007).
[Crossref]

Davidson, M. W.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science 313, 1642–1645 (2006).
[Crossref]

Davidson, N.

Dennis, M.

E. T. F. Rogers, S. Savo, J. Lindberg, T. Roy, M. Dennis, and N. I. Zheludev, “Super-oscillatory optical needle,” Appl. Phys. Lett. 102, 031108 (2013).
[Crossref]

Dennis, M. R.

E. T. F. Rogers, J. Lindberg, T. Roy, S. Savo, J. E. Chad, M. R. Dennis, and N. I. Zheludev, “A super-oscillatory lens optical microscope for subwavelength imaging,” Nat. Mater. 11, 432–435 (2012).
[Crossref]

Dholakia, K.

M. Mazilu, J. Baumgartl, S. Kosmeier, and K. Dholakia, “Optical eigenmodes; exploiting the quadratic nature of the energy flux and of scattering interactions,” Opt. Express 19, 933–945 (2011).
[Crossref]

J. Baumgartl, S. Kosmeier, M. Mazilu, E. T. F. Rogers, N. I. Zheludev, and K. Dholakia, “Far field subwavelength focusing using optical eigenmodes,” Appl. Phys. Lett. 98, 181109 (2011).
[Crossref]

Diao, J.

Dorn, R.

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

Eberler, M.

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

Eleftheriades, G. V.

A. M. H. Wong and G. V. Eleftheriades, “An optical super-microscope for far-field, real-time imaging beyond the diffraction limit,” Sci. Rep. 3, 1715 (2013).
[Crossref]

Ferreira, P. J. S. G.

P. J. S. G. Ferreira and A. Kempf, “Superoscillations: faster than the Nyquist rate,” IEEE Trans. Signal Process. 54, 3732–3740 (2006).
[Crossref]

Foreman, M. R.

M. R. Foreman and P. Török, “Computational methods in vectorial imaging,” J. Mod. Opt. 58, 339–364 (2011).
[Crossref]

Glöckl, O.

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

Greenfield, E.

Grosjean, T.

T. Grosjean and D. Courjon, “Smallest focal spots,” Opt. Commun. 272, 314–319 (2007).
[Crossref]

Gustafsson, M. G. L.

M. G. L. Gustafsson, “Surpassing the lateral resolution limit by a factor of two using structured illumination microscopy,” J. Microsc. 198, 82–87 (2000).
[Crossref]

Hashimoto, N.

Hell, S. W.

Hess, H. F.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science 313, 1642–1645 (2006).
[Crossref]

Hibi, T.

S. Ipponjima, T. Hibi, Y. Kozawa, H. Horanai, H. Yokoyama, S. Sato, and T. Nemoto, “Improvement of lateral resolution and extension of depth of field in two-photon microscopy by a higher-order radially polarized beam,” Microscopy 63, 23–32 (2014).
[Crossref]

Y. Kozawa, T. Hibi, A. Sato, H. Horanai, M. Kurihara, N. Hashimoto, H. Yokoyama, T. Nemoto, and S. Sato, “Lateral resolution enhancement of laser scanning microscopy by a higher-order radially polarized mode beam,” Opt. Express 19, 15947–15954 (2011).
[Crossref]

Higdon, P. D.

P. D. Higdon, P. Török, and T. Wilson, “Imaging properties of high aperture multiphoton fluorescence scanning optical microscopes,” J. Microsc. 193, 127–141 (1999).
[Crossref]

Hong, M.

F. Qin, K. Huang, J. Wu, J. Jiao, X. Luo, C. Qiu, and M. Hong, “Shaping a subwavelength needle with ultra-long focal length by focusing azimuthally polarized light,” Sci. Rep. 5, 9977 (2015).
[Crossref]

Horanai, H.

S. Ipponjima, T. Hibi, Y. Kozawa, H. Horanai, H. Yokoyama, S. Sato, and T. Nemoto, “Improvement of lateral resolution and extension of depth of field in two-photon microscopy by a higher-order radially polarized beam,” Microscopy 63, 23–32 (2014).
[Crossref]

Y. Kozawa, T. Hibi, A. Sato, H. Horanai, M. Kurihara, N. Hashimoto, H. Yokoyama, T. Nemoto, and S. Sato, “Lateral resolution enhancement of laser scanning microscopy by a higher-order radially polarized mode beam,” Opt. Express 19, 15947–15954 (2011).
[Crossref]

Huang, K.

F. Qin, K. Huang, J. Wu, J. Jiao, X. Luo, C. Qiu, and M. Hong, “Shaping a subwavelength needle with ultra-long focal length by focusing azimuthally polarized light,” Sci. Rep. 5, 9977 (2015).
[Crossref]

Hurwitz, I.

Hyvärinen, H. J.

Ipponjima, S.

S. Ipponjima, T. Hibi, Y. Kozawa, H. Horanai, H. Yokoyama, S. Sato, and T. Nemoto, “Improvement of lateral resolution and extension of depth of field in two-photon microscopy by a higher-order radially polarized beam,” Microscopy 63, 23–32 (2014).
[Crossref]

Jiao, J.

F. Qin, K. Huang, J. Wu, J. Jiao, X. Luo, C. Qiu, and M. Hong, “Shaping a subwavelength needle with ultra-long focal length by focusing azimuthally polarized light,” Sci. Rep. 5, 9977 (2015).
[Crossref]

Juškaitis, R.

Kempf, A.

P. J. S. G. Ferreira and A. Kempf, “Superoscillations: faster than the Nyquist rate,” IEEE Trans. Signal Process. 54, 3732–3740 (2006).
[Crossref]

Kosmeier, S.

M. Mazilu, J. Baumgartl, S. Kosmeier, and K. Dholakia, “Optical eigenmodes; exploiting the quadratic nature of the energy flux and of scattering interactions,” Opt. Express 19, 933–945 (2011).
[Crossref]

J. Baumgartl, S. Kosmeier, M. Mazilu, E. T. F. Rogers, N. I. Zheludev, and K. Dholakia, “Far field subwavelength focusing using optical eigenmodes,” Appl. Phys. Lett. 98, 181109 (2011).
[Crossref]

Kozawa, Y.

Kurihara, M.

Laczik, Z. J.

Leuchs, G.

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

Lindberg, J.

E. T. F. Rogers, S. Savo, J. Lindberg, T. Roy, M. Dennis, and N. I. Zheludev, “Super-oscillatory optical needle,” Appl. Phys. Lett. 102, 031108 (2013).
[Crossref]

E. T. F. Rogers, J. Lindberg, T. Roy, S. Savo, J. E. Chad, M. R. Dennis, and N. I. Zheludev, “A super-oscillatory lens optical microscope for subwavelength imaging,” Nat. Mater. 11, 432–435 (2012).
[Crossref]

J. Lindberg, “Mathematical concepts of optical superresolution,” J. Opt. 14, 083001 (2012).
[Crossref]

Lindwasser, O. W.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science 313, 1642–1645 (2006).
[Crossref]

Lippincott-Schwartz, J.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science 313, 1642–1645 (2006).
[Crossref]

Luo, X.

F. Qin, K. Huang, J. Wu, J. Jiao, X. Luo, C. Qiu, and M. Hong, “Shaping a subwavelength needle with ultra-long focal length by focusing azimuthally polarized light,” Sci. Rep. 5, 9977 (2015).
[Crossref]

Makris, K. G.

Mazilu, M.

M. Mazilu, J. Baumgartl, S. Kosmeier, and K. Dholakia, “Optical eigenmodes; exploiting the quadratic nature of the energy flux and of scattering interactions,” Opt. Express 19, 933–945 (2011).
[Crossref]

J. Baumgartl, S. Kosmeier, M. Mazilu, E. T. F. Rogers, N. I. Zheludev, and K. Dholakia, “Far field subwavelength focusing using optical eigenmodes,” Appl. Phys. Lett. 98, 181109 (2011).
[Crossref]

Neil, M. A. A.

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E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science 313, 1642–1645 (2006).
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E. Rogers and N. I. Zheludev, “Optical super-oscillations: sub-wavelength light focusing and super-resolution imaging,” J. Opt. 15, 094008 (2013).
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J. Baumgartl, S. Kosmeier, M. Mazilu, E. T. F. Rogers, N. I. Zheludev, and K. Dholakia, “Far field subwavelength focusing using optical eigenmodes,” Appl. Phys. Lett. 98, 181109 (2011).
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Adv. Opt. Photon. (1)

Appl. Opt. (2)

Appl. Phys. Lett. (2)

J. Baumgartl, S. Kosmeier, M. Mazilu, E. T. F. Rogers, N. I. Zheludev, and K. Dholakia, “Far field subwavelength focusing using optical eigenmodes,” Appl. Phys. Lett. 98, 181109 (2011).
[Crossref]

E. T. F. Rogers, S. Savo, J. Lindberg, T. Roy, M. Dennis, and N. I. Zheludev, “Super-oscillatory optical needle,” Appl. Phys. Lett. 102, 031108 (2013).
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IEEE Trans. Signal Process. (1)

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J. Microsc. (2)

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P. D. Higdon, P. Török, and T. Wilson, “Imaging properties of high aperture multiphoton fluorescence scanning optical microscopes,” J. Microsc. 193, 127–141 (1999).
[Crossref]

J. Mod. Opt. (1)

M. R. Foreman and P. Török, “Computational methods in vectorial imaging,” J. Mod. Opt. 58, 339–364 (2011).
[Crossref]

J. Opt. (2)

E. Rogers and N. I. Zheludev, “Optical super-oscillations: sub-wavelength light focusing and super-resolution imaging,” J. Opt. 15, 094008 (2013).
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J. Lindberg, “Mathematical concepts of optical superresolution,” J. Opt. 14, 083001 (2012).
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J. Opt. Soc. Am. A (3)

J. Phys. A (1)

M. V. Berry and S. Popescu, “Evolution of quantum superoscillation and optical superresolution without evanescent waves,” J. Phys. A 39, 6965–6977 (2006).
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Microscopy (1)

S. Ipponjima, T. Hibi, Y. Kozawa, H. Horanai, H. Yokoyama, S. Sato, and T. Nemoto, “Improvement of lateral resolution and extension of depth of field in two-photon microscopy by a higher-order radially polarized beam,” Microscopy 63, 23–32 (2014).
[Crossref]

Mon. Not. R. Astron. Soc. (1)

J. W. Strutt, “On the diffraction of object-glasses,” Mon. Not. R. Astron. Soc. 33, 59–63 (1872).
[Crossref]

Nat. Mater. (2)

N. I. Zheludev, “What diffraction limit?” Nat. Mater. 7, 420–422 (2008).
[Crossref]

E. T. F. Rogers, J. Lindberg, T. Roy, S. Savo, J. E. Chad, M. R. Dennis, and N. I. Zheludev, “A super-oscillatory lens optical microscope for subwavelength imaging,” Nat. Mater. 11, 432–435 (2012).
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Nuovo Cimento Suppl. (1)

G. Toraldo di Francia, “Super-gain antennas and optical resolving power,” Nuovo Cimento Suppl. 9, 426–438 (1952).
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Opt. Commun. (2)

T. Grosjean and D. Courjon, “Smallest focal spots,” Opt. Commun. 272, 314–319 (2007).
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S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7 (2000).
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Opt. Eng. (1)

D. H. Close, “Holographic optical elements,” Opt. Eng. 14, 408–419 (1975).
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Opt. Express (6)

Opt. Lett. (5)

Proc. R. Soc. London A (1)

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

Sci. Rep. (3)

G. Yuan, E. T. F. Rogers, T. Roy, G. Adamo, Z. Shen, and N. I. Zheludev, “Planar super-oscillatory lens for sub-diffraction optical needles at violet wavelengths,” Sci. Rep. 4, 6333 (2014).
[Crossref]

F. Qin, K. Huang, J. Wu, J. Jiao, X. Luo, C. Qiu, and M. Hong, “Shaping a subwavelength needle with ultra-long focal length by focusing azimuthally polarized light,” Sci. Rep. 5, 9977 (2015).
[Crossref]

A. M. H. Wong and G. V. Eleftheriades, “An optical super-microscope for far-field, real-time imaging beyond the diffraction limit,” Sci. Rep. 3, 1715 (2013).
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Science (1)

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science 313, 1642–1645 (2006).
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Other (2)

M. V. Berry, “Faster than Fourier,” in Quantum Coherence and Reality; in Celebration of the 60th Birthday of Yakir Aharonov, J. S. Anandan and J. L. Safko, eds. (World Scientific, 1994), pp. 55–65.

A. E. Siegman, “How to (maybe) measure laser beam quality,” in DPSS (Diode Pumped Solid State) Lasers: Applications and Issues, M. Dowley, ed., OSA Trends in Optics and Photonics (Optical Society of America, 1998), Vol. 17, paper MQ1.

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Figures (6)

Fig. 1.
Fig. 1. (a) Peak intensity variation of the central focal spot for RP - LG 1,1 , RP - LG 3,1 , and RP - LG 5,1 beams (illustrated inset in the figure) as a function of incident beam size. (b) FWHM values for the central focal spots formed by the total intensity (solid lines) and the longitudinal component only (dashed lines) for each mode. The annotations A to D marked with arrows in (b) correspond to Figs. 2(a)2(d). (c) Peak intensity ratio for the first (innermost) side lobe of the central spot (illustrated inset in the figure) for the RP - LG p , 1 modes with NA = 1.4 and n = 1.52 .
Fig. 2.
Fig. 2. Intensity profiles at the focus for the focusing of an RP - LG 3,1 beam with incident beam size parameter β 4 σ values of (a) 1.056, (b) 0.870, (c) 0.850, and (d) 0.824, indicated as A, B, C, and D, respectively, in Fig. 1.
Fig. 3.
Fig. 3. Numerically simulated images for line objects with spacings of 110–160 nm [illustrated in (a)]. The second and third rows show the conventional, confocal PSFs and the resultant images for each object obtained by a circularly polarized Gaussian beam using an objective lens of NA = 1.4 ( n = 1.52 ) and confocal pinholes of (b) 1.0 AU and (c) 0.5 AU. The bottom two rows show the PSFs and corresponding images for superoscillation spots from RP - LG 3,1 beams with size parameters of (d) 0.850 and (e) 0.824, using confocal pinholes of 0.5 AU. An excitation wavelength of 488 nm is assumed. The scale bars in the PSF patterns represent 200 nm.
Fig. 4.
Fig. 4. (a) Schematic of the experimental setup. (b), (c) Calculated intensity distributions at (b) the focal plane and (c) the x z -plane of the converted RP - LG 3,1 beam that produces the superoscillation focal spot. The corresponding intensity distributions, measured without a confocal pinhole, are shown in (d) and (e), respectively. The scale bars represent 1 μm. The intensity profiles along x - and z -axes for the calculated (black solid) and measured (red dashed) focal spots are shown in (f) and (g).
Fig. 5.
Fig. 5. Measured PSFs (0.5 AU) for (a) LP and (b)  RP - LG 3,1 beams with superoscillation. The scale bars represent 500 nm. Intensity profiles for the measured PSFs along the x - and y -axes are shown in (c) and (d), respectively, while (e) and (f) show images of clusters of fluorescent beads with diameters of 170 nm acquired by (e) LP and (f)  RP - LG 3,1 beams. The scale bars in (e) and (f) represent 1 μm. Intensity profiles along the dashed lines in (e) and (f) are shown in (g). The black dashed and red solid lines in (c), (d), and (g) correspond to LP and RP - LG 3,1 beams, respectively.
Fig. 6.
Fig. 6. HeLa cell images acquired by the (a) LP beam (1 AU), (b) LP beam (0.5 AU), and (c)  RP - LG 3,1 beam (0.5 AU). The scale bars in (a)–(c) represent 2 μm. (d)–(f) Magnified images of the regions indicated by white rectangles in (a), (b). Normalized intensity profiles along the dashed lines in (d)–(f) are shown in (g). The black dashed, blue dashed-dotted, and red solid lines correspond to the profiles for (d)–(f), respectively. The scale bars in (d)–(f) represent 500 nm.

Equations (2)

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I ( r ) = NA 2 n 2 [ J 0 ( k NA r ) ] 2 + ( 1 NA 2 n 2 ) [ J 1 ( k NA r ) ] 2 ,
E ( ρ ) ρ w 0 exp ( ρ 2 w 0 2 ) L p 1 ( 2 ρ 2 w 0 2 ) ,

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