Abstract

In this paper we report on the development and optical properties of nanostructured gradient index microlenses with good chromatic behavior. We introduce a new fabrication concept for the development of large diameter nanostructured gradient index microlenses based on quantized gradient index profiles and the use of nanostructured meta-rods. We show a dependence of the quality of performance on the number of refractive index levels and the lens diameter. Measurements carried out at 633 and 850nm show good optical properties and similar focal lengths for both wavelengths.

©2012 Optical Society of America

Full Article  |  PDF Article
OSA Recommended Articles
Large elliptical nanostructured gradient-index microlens

Ryszard Buczynski, Adam Filipkowski, Andrew J. Waddie, Bernard Piechal, Jedrzej Nowosielski, Dariusz Pysz, Ryszard Stepien, and Mohammad R. Taghizadeh
Appl. Opt. 55(1) 89-94 (2016)

Nanostructured elliptical gradient-index microlenses

F. Hudelist, J. M. Nowosielski, R. Buczynski, A. J. Waddie, and M. R. Taghizadeh
Opt. Lett. 35(2) 130-132 (2010)

Nanostructured gradient index microaxicons made by a modified stack and draw method

Adam Filipkowski, Bernard Piechal, Dariusz Pysz, Ryszard Stepien, Andrew Waddie, Mohammad R. Taghizadeh, and Ryszard Buczynski
Opt. Lett. 40(22) 5200-5203 (2015)

More Recommended Articles

References

  • View by:
  • Article Order
  • |
  • Year
  • |
  • Author
  • |
  • Publication
  1. C. Gómez-Reino, M. V. Perez, and C. Bao, Gradient-index Optics: Fundamentals and Applications, (Springer, Berlin, 2002).
  2. C. Gomez-Reino, M. V. Perez, C. Bao, and M. T. Flores-Arias, “Design of GRIN optical components for coupling and interconnects,” Laser Photon. Rev. 2(3), 203–215 (2008).
    [Crossref]
  3. L. Fu, X. Gan, and M. Gu, “Characterization of gradient-index lens-fiber spacing toward applications in two-photon fluorescence endoscopy,” Appl. Opt. 44(34), 7270–7274 (2005).
    [Crossref] [PubMed]
  4. R. Buczynski, V. Baukens, T. Szoplik, A. Goulet, N. Debaes, A. Kirk, P. Heremans, R. Vounckx, I. Veretennicoff, and H. Thienpont, “Fast optical thresholding with an array of optoelectronic transceiver elements,” IEEE Photon. Technol. Lett. 11(3), 367–369 (1999).
    [Crossref]
  5. S. Sinzinger and J. Jahns, Microoptics (Wiley-VCH, Weinheim, 2003).
  6. GRINTECH GmbH website: < www.grintech.de >.
  7. Y. Huang and S.-T. Ho, “Superhigh numerical aperture (NA > 1.5) micro gradient-index lens based on a dual-material approach,” Opt. Lett. 30(11), 1291–1293 (2005).
    [Crossref] [PubMed]
  8. F. Hudelist, R. Buczynski, A. J. Waddie, and M. R. Taghizadeh, “Design and fabrication of nano-structured gradient index microlenses,” Opt. Express 17(5), 3255–3263 (2009).
    [Crossref] [PubMed]
  9. F. Hudelist, J. M. Nowosielski, R. Buczynski, A. J. Waddie, and M. R. Taghizadeh, “Nanostructured elliptical gradient-index microlenses,” Opt. Lett. 35(2), 130–132 (2010).
    [Crossref] [PubMed]
  10. J. M. Nowosielski, R. Buczynski, F. Hudelist, A. J. Waddie, and M. R. Taghizadeh, “Nanostructured GRIN microlenses for Gaussian beam focusing,” Opt. Commun. 283(9), 1938–1944 (2010).
    [Crossref]
  11. A. Sihvola, Electromagnetic Mixing Formulas and Applications (The Institution of Electrical Engineers, London, 1999).
  12. D. Pysz, I. Kujawa, P. Szarniak, M. Franczyk, R. Stepien, and R. Buczynski, “Multicomponent glass fiber optic integrated structures,” Proc. SPIE 5951, 595102, 595102-15 (2005).
    [Crossref]
  13. A. Ortega-Monux, J. G. Wanguemert-Perez, I. Molina-Fernandez, E. Silvestre, and P. Andres, “Enhanced accuracy in fast-Fourier-based methods for full-vector modal analysis of dielectric waveguides,” IEEE Photon. Technol. Lett. 18(10), 1128–1130 (2006).
    [Crossref]
  14. A. Sagan, S. Nowicki, R. Buczynski, M. Kowalczyk, and T. Szoplik, “Imaging phase objects with square-root, Foucault, and Hoffman real filters: a comparison,” Appl. Opt. 42(29), 5816–5824 (2003).
    [Crossref] [PubMed]
  15. A. Boivin and E. Wolf, “Electromagnetic field in the neighborhood of the focus of a coherent beam,” Phys. Rev. 138(6B), B1561–B1565 (1965).
    [Crossref]

2010 (2)

F. Hudelist, J. M. Nowosielski, R. Buczynski, A. J. Waddie, and M. R. Taghizadeh, “Nanostructured elliptical gradient-index microlenses,” Opt. Lett. 35(2), 130–132 (2010).
[Crossref] [PubMed]

J. M. Nowosielski, R. Buczynski, F. Hudelist, A. J. Waddie, and M. R. Taghizadeh, “Nanostructured GRIN microlenses for Gaussian beam focusing,” Opt. Commun. 283(9), 1938–1944 (2010).
[Crossref]

2009 (1)

2008 (1)

C. Gomez-Reino, M. V. Perez, C. Bao, and M. T. Flores-Arias, “Design of GRIN optical components for coupling and interconnects,” Laser Photon. Rev. 2(3), 203–215 (2008).
[Crossref]

2006 (1)

A. Ortega-Monux, J. G. Wanguemert-Perez, I. Molina-Fernandez, E. Silvestre, and P. Andres, “Enhanced accuracy in fast-Fourier-based methods for full-vector modal analysis of dielectric waveguides,” IEEE Photon. Technol. Lett. 18(10), 1128–1130 (2006).
[Crossref]

2005 (3)

2003 (1)

1999 (1)

R. Buczynski, V. Baukens, T. Szoplik, A. Goulet, N. Debaes, A. Kirk, P. Heremans, R. Vounckx, I. Veretennicoff, and H. Thienpont, “Fast optical thresholding with an array of optoelectronic transceiver elements,” IEEE Photon. Technol. Lett. 11(3), 367–369 (1999).
[Crossref]

1965 (1)

A. Boivin and E. Wolf, “Electromagnetic field in the neighborhood of the focus of a coherent beam,” Phys. Rev. 138(6B), B1561–B1565 (1965).
[Crossref]

Andres, P.

A. Ortega-Monux, J. G. Wanguemert-Perez, I. Molina-Fernandez, E. Silvestre, and P. Andres, “Enhanced accuracy in fast-Fourier-based methods for full-vector modal analysis of dielectric waveguides,” IEEE Photon. Technol. Lett. 18(10), 1128–1130 (2006).
[Crossref]

Bao, C.

C. Gomez-Reino, M. V. Perez, C. Bao, and M. T. Flores-Arias, “Design of GRIN optical components for coupling and interconnects,” Laser Photon. Rev. 2(3), 203–215 (2008).
[Crossref]

Baukens, V.

R. Buczynski, V. Baukens, T. Szoplik, A. Goulet, N. Debaes, A. Kirk, P. Heremans, R. Vounckx, I. Veretennicoff, and H. Thienpont, “Fast optical thresholding with an array of optoelectronic transceiver elements,” IEEE Photon. Technol. Lett. 11(3), 367–369 (1999).
[Crossref]

Boivin, A.

A. Boivin and E. Wolf, “Electromagnetic field in the neighborhood of the focus of a coherent beam,” Phys. Rev. 138(6B), B1561–B1565 (1965).
[Crossref]

Buczynski, R.

F. Hudelist, J. M. Nowosielski, R. Buczynski, A. J. Waddie, and M. R. Taghizadeh, “Nanostructured elliptical gradient-index microlenses,” Opt. Lett. 35(2), 130–132 (2010).
[Crossref] [PubMed]

J. M. Nowosielski, R. Buczynski, F. Hudelist, A. J. Waddie, and M. R. Taghizadeh, “Nanostructured GRIN microlenses for Gaussian beam focusing,” Opt. Commun. 283(9), 1938–1944 (2010).
[Crossref]

F. Hudelist, R. Buczynski, A. J. Waddie, and M. R. Taghizadeh, “Design and fabrication of nano-structured gradient index microlenses,” Opt. Express 17(5), 3255–3263 (2009).
[Crossref] [PubMed]

D. Pysz, I. Kujawa, P. Szarniak, M. Franczyk, R. Stepien, and R. Buczynski, “Multicomponent glass fiber optic integrated structures,” Proc. SPIE 5951, 595102, 595102-15 (2005).
[Crossref]

A. Sagan, S. Nowicki, R. Buczynski, M. Kowalczyk, and T. Szoplik, “Imaging phase objects with square-root, Foucault, and Hoffman real filters: a comparison,” Appl. Opt. 42(29), 5816–5824 (2003).
[Crossref] [PubMed]

R. Buczynski, V. Baukens, T. Szoplik, A. Goulet, N. Debaes, A. Kirk, P. Heremans, R. Vounckx, I. Veretennicoff, and H. Thienpont, “Fast optical thresholding with an array of optoelectronic transceiver elements,” IEEE Photon. Technol. Lett. 11(3), 367–369 (1999).
[Crossref]

Debaes, N.

R. Buczynski, V. Baukens, T. Szoplik, A. Goulet, N. Debaes, A. Kirk, P. Heremans, R. Vounckx, I. Veretennicoff, and H. Thienpont, “Fast optical thresholding with an array of optoelectronic transceiver elements,” IEEE Photon. Technol. Lett. 11(3), 367–369 (1999).
[Crossref]

Flores-Arias, M. T.

C. Gomez-Reino, M. V. Perez, C. Bao, and M. T. Flores-Arias, “Design of GRIN optical components for coupling and interconnects,” Laser Photon. Rev. 2(3), 203–215 (2008).
[Crossref]

Franczyk, M.

D. Pysz, I. Kujawa, P. Szarniak, M. Franczyk, R. Stepien, and R. Buczynski, “Multicomponent glass fiber optic integrated structures,” Proc. SPIE 5951, 595102, 595102-15 (2005).
[Crossref]

Fu, L.

Gan, X.

Gomez-Reino, C.

C. Gomez-Reino, M. V. Perez, C. Bao, and M. T. Flores-Arias, “Design of GRIN optical components for coupling and interconnects,” Laser Photon. Rev. 2(3), 203–215 (2008).
[Crossref]

Goulet, A.

R. Buczynski, V. Baukens, T. Szoplik, A. Goulet, N. Debaes, A. Kirk, P. Heremans, R. Vounckx, I. Veretennicoff, and H. Thienpont, “Fast optical thresholding with an array of optoelectronic transceiver elements,” IEEE Photon. Technol. Lett. 11(3), 367–369 (1999).
[Crossref]

Gu, M.

Heremans, P.

R. Buczynski, V. Baukens, T. Szoplik, A. Goulet, N. Debaes, A. Kirk, P. Heremans, R. Vounckx, I. Veretennicoff, and H. Thienpont, “Fast optical thresholding with an array of optoelectronic transceiver elements,” IEEE Photon. Technol. Lett. 11(3), 367–369 (1999).
[Crossref]

Ho, S.-T.

Huang, Y.

Hudelist, F.

Kirk, A.

R. Buczynski, V. Baukens, T. Szoplik, A. Goulet, N. Debaes, A. Kirk, P. Heremans, R. Vounckx, I. Veretennicoff, and H. Thienpont, “Fast optical thresholding with an array of optoelectronic transceiver elements,” IEEE Photon. Technol. Lett. 11(3), 367–369 (1999).
[Crossref]

Kowalczyk, M.

Kujawa, I.

D. Pysz, I. Kujawa, P. Szarniak, M. Franczyk, R. Stepien, and R. Buczynski, “Multicomponent glass fiber optic integrated structures,” Proc. SPIE 5951, 595102, 595102-15 (2005).
[Crossref]

Molina-Fernandez, I.

A. Ortega-Monux, J. G. Wanguemert-Perez, I. Molina-Fernandez, E. Silvestre, and P. Andres, “Enhanced accuracy in fast-Fourier-based methods for full-vector modal analysis of dielectric waveguides,” IEEE Photon. Technol. Lett. 18(10), 1128–1130 (2006).
[Crossref]

Nowicki, S.

Nowosielski, J. M.

F. Hudelist, J. M. Nowosielski, R. Buczynski, A. J. Waddie, and M. R. Taghizadeh, “Nanostructured elliptical gradient-index microlenses,” Opt. Lett. 35(2), 130–132 (2010).
[Crossref] [PubMed]

J. M. Nowosielski, R. Buczynski, F. Hudelist, A. J. Waddie, and M. R. Taghizadeh, “Nanostructured GRIN microlenses for Gaussian beam focusing,” Opt. Commun. 283(9), 1938–1944 (2010).
[Crossref]

Ortega-Monux, A.

A. Ortega-Monux, J. G. Wanguemert-Perez, I. Molina-Fernandez, E. Silvestre, and P. Andres, “Enhanced accuracy in fast-Fourier-based methods for full-vector modal analysis of dielectric waveguides,” IEEE Photon. Technol. Lett. 18(10), 1128–1130 (2006).
[Crossref]

Perez, M. V.

C. Gomez-Reino, M. V. Perez, C. Bao, and M. T. Flores-Arias, “Design of GRIN optical components for coupling and interconnects,” Laser Photon. Rev. 2(3), 203–215 (2008).
[Crossref]

Pysz, D.

D. Pysz, I. Kujawa, P. Szarniak, M. Franczyk, R. Stepien, and R. Buczynski, “Multicomponent glass fiber optic integrated structures,” Proc. SPIE 5951, 595102, 595102-15 (2005).
[Crossref]

Sagan, A.

Silvestre, E.

A. Ortega-Monux, J. G. Wanguemert-Perez, I. Molina-Fernandez, E. Silvestre, and P. Andres, “Enhanced accuracy in fast-Fourier-based methods for full-vector modal analysis of dielectric waveguides,” IEEE Photon. Technol. Lett. 18(10), 1128–1130 (2006).
[Crossref]

Stepien, R.

D. Pysz, I. Kujawa, P. Szarniak, M. Franczyk, R. Stepien, and R. Buczynski, “Multicomponent glass fiber optic integrated structures,” Proc. SPIE 5951, 595102, 595102-15 (2005).
[Crossref]

Szarniak, P.

D. Pysz, I. Kujawa, P. Szarniak, M. Franczyk, R. Stepien, and R. Buczynski, “Multicomponent glass fiber optic integrated structures,” Proc. SPIE 5951, 595102, 595102-15 (2005).
[Crossref]

Szoplik, T.

A. Sagan, S. Nowicki, R. Buczynski, M. Kowalczyk, and T. Szoplik, “Imaging phase objects with square-root, Foucault, and Hoffman real filters: a comparison,” Appl. Opt. 42(29), 5816–5824 (2003).
[Crossref] [PubMed]

R. Buczynski, V. Baukens, T. Szoplik, A. Goulet, N. Debaes, A. Kirk, P. Heremans, R. Vounckx, I. Veretennicoff, and H. Thienpont, “Fast optical thresholding with an array of optoelectronic transceiver elements,” IEEE Photon. Technol. Lett. 11(3), 367–369 (1999).
[Crossref]

Taghizadeh, M. R.

Thienpont, H.

R. Buczynski, V. Baukens, T. Szoplik, A. Goulet, N. Debaes, A. Kirk, P. Heremans, R. Vounckx, I. Veretennicoff, and H. Thienpont, “Fast optical thresholding with an array of optoelectronic transceiver elements,” IEEE Photon. Technol. Lett. 11(3), 367–369 (1999).
[Crossref]

Veretennicoff, I.

R. Buczynski, V. Baukens, T. Szoplik, A. Goulet, N. Debaes, A. Kirk, P. Heremans, R. Vounckx, I. Veretennicoff, and H. Thienpont, “Fast optical thresholding with an array of optoelectronic transceiver elements,” IEEE Photon. Technol. Lett. 11(3), 367–369 (1999).
[Crossref]

Vounckx, R.

R. Buczynski, V. Baukens, T. Szoplik, A. Goulet, N. Debaes, A. Kirk, P. Heremans, R. Vounckx, I. Veretennicoff, and H. Thienpont, “Fast optical thresholding with an array of optoelectronic transceiver elements,” IEEE Photon. Technol. Lett. 11(3), 367–369 (1999).
[Crossref]

Waddie, A. J.

Wanguemert-Perez, J. G.

A. Ortega-Monux, J. G. Wanguemert-Perez, I. Molina-Fernandez, E. Silvestre, and P. Andres, “Enhanced accuracy in fast-Fourier-based methods for full-vector modal analysis of dielectric waveguides,” IEEE Photon. Technol. Lett. 18(10), 1128–1130 (2006).
[Crossref]

Wolf, E.

A. Boivin and E. Wolf, “Electromagnetic field in the neighborhood of the focus of a coherent beam,” Phys. Rev. 138(6B), B1561–B1565 (1965).
[Crossref]

Appl. Opt. (2)

IEEE Photon. Technol. Lett. (2)

A. Ortega-Monux, J. G. Wanguemert-Perez, I. Molina-Fernandez, E. Silvestre, and P. Andres, “Enhanced accuracy in fast-Fourier-based methods for full-vector modal analysis of dielectric waveguides,” IEEE Photon. Technol. Lett. 18(10), 1128–1130 (2006).
[Crossref]

R. Buczynski, V. Baukens, T. Szoplik, A. Goulet, N. Debaes, A. Kirk, P. Heremans, R. Vounckx, I. Veretennicoff, and H. Thienpont, “Fast optical thresholding with an array of optoelectronic transceiver elements,” IEEE Photon. Technol. Lett. 11(3), 367–369 (1999).
[Crossref]

Laser Photon. Rev. (1)

C. Gomez-Reino, M. V. Perez, C. Bao, and M. T. Flores-Arias, “Design of GRIN optical components for coupling and interconnects,” Laser Photon. Rev. 2(3), 203–215 (2008).
[Crossref]

Opt. Commun. (1)

J. M. Nowosielski, R. Buczynski, F. Hudelist, A. J. Waddie, and M. R. Taghizadeh, “Nanostructured GRIN microlenses for Gaussian beam focusing,” Opt. Commun. 283(9), 1938–1944 (2010).
[Crossref]

Opt. Express (1)

Opt. Lett. (2)

Phys. Rev. (1)

A. Boivin and E. Wolf, “Electromagnetic field in the neighborhood of the focus of a coherent beam,” Phys. Rev. 138(6B), B1561–B1565 (1965).
[Crossref]

Proc. SPIE (1)

D. Pysz, I. Kujawa, P. Szarniak, M. Franczyk, R. Stepien, and R. Buczynski, “Multicomponent glass fiber optic integrated structures,” Proc. SPIE 5951, 595102, 595102-15 (2005).
[Crossref]

Other (4)

A. Sihvola, Electromagnetic Mixing Formulas and Applications (The Institution of Electrical Engineers, London, 1999).

C. Gómez-Reino, M. V. Perez, and C. Bao, Gradient-index Optics: Fundamentals and Applications, (Springer, Berlin, 2002).

S. Sinzinger and J. Jahns, Microoptics (Wiley-VCH, Weinheim, 2003).

GRINTECH GmbH website: < www.grintech.de >.

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (19)

Fig. 1
Fig. 1

(a) Schematic of quantised nanostructured GRIN microlens consisting of 101x101 “pixels” with one of 7 discrete effective refractive index level structures formed by a nanostructured composite of two soft glasses. (b) Ideal continuous GRIN microlens.

Download Full Size | PPT Slide | PDF

Fig. 2
Fig. 2

Improvement of focusing for a 7 refractive index level nanostructured GRIN microlens with a reduced diameter of (a) 500μm, (b) 250μm and (c) 140μm. Focusing behaviour of ideal continuous GRIN lenses with diameters of (d) 500μm, (e) 250μm and (f) 140μm. The wavelength of light is 850 nm.

Download Full Size | PPT Slide | PDF

Fig. 3
Fig. 3

Structured GRIN microlens composed of 100x100 nanostructured rods of 7 types

Download Full Size | PPT Slide | PDF

Fig. 4
Fig. 4

Structures of five metarods composed of two fundamental glasses NC21 and NC25 according to the calculated pseudo-random pattern to ensure a uniform effective refractive index within each of the 5 nanostructured rods.

Download Full Size | PPT Slide | PDF

Fig. 5
Fig. 5

Schematic of the development of metarods: drawing of calibrated rods (a), assembly of hexagonal metarod preform (b), drawing final metarods (c).

Download Full Size | PPT Slide | PDF

Fig. 6
Fig. 6

Schematic of development of nanostructured GRIN lens: (a) assembly of structured lens preform, (b) drawing lens preform, (c) cutting and (d) surface polishing.

Download Full Size | PPT Slide | PDF

Fig. 7
Fig. 7

Development of nanostructured quantised GRIN lens: (a) assembly of prefrom with metarods, (b) final preform of lens structure with the diameter of 60 mm, (c) intermediate preform with the diameter of 30 mm and (d) final microlens with the diameter of 0.1 mm.

Download Full Size | PPT Slide | PDF

Fig. 8
Fig. 8

A phase contrast microscope photo of developed nanostructured GRIN lens

Download Full Size | PPT Slide | PDF

Fig. 9
Fig. 9

Propagation of light within infinite quantised nanostructured lens with diameter of 100 μm: (a) the cross-section along optical axis, (b) cross-section at focus perpendicular to the optical axis. Simulations are performed for 850 nm.

Download Full Size | PPT Slide | PDF

Fig. 10
Fig. 10

Propagation of light within infinite ideal continuous nanostructured lens with diameter of 100 μm: (a) the cross-section along optical axis, (b) cross-section at focus perpendicular to the optical axis. Simulations are performed for 850 nm

Download Full Size | PPT Slide | PDF

Fig. 11
Fig. 11

Propagation of light within 140μm long quantised nanostructured lens with diameter of 100 μm: (a) the cross-section along optical axis, (b) cross-section at focus perpendicular to the optical axis. Simulations are performed for 850 nm.

Download Full Size | PPT Slide | PDF

Fig. 12
Fig. 12

Propagation of light within 140μm long ideal continuous nanostructured lens with diameter of 100μm: (a) the cross-section along optical axis, (b) cross-section at focus perpendicular to the optical axis. Simulations are performed for 850 nm.

Download Full Size | PPT Slide | PDF

Fig. 13
Fig. 13

Propagation of light within 140 μm long quantised nanostructured lens with diameter of 100 μm: (a) the cross-section along optical axis, (b) cross-section at focus perpendicular to the optical axis. Simulations are performed for 633 nm.

Download Full Size | PPT Slide | PDF

Fig. 14
Fig. 14

Propagation of light within 140μm long ideal continuous nanostructured lens with diameter of 100μm: (a) the cross-section along optical axis, (b) cross-section at focus perpendicular to the optical axis. Simulations are performed for 633 nm.

Download Full Size | PPT Slide | PDF

Fig. 15
Fig. 15

Measurement set-up (a). Typical output image of the measured lens imaged at the lens focus captured by linear CCD camera (b). Scattering at lens border can be observed.

Download Full Size | PPT Slide | PDF

Fig. 16
Fig. 16

(a) The intensity distribution in the focal plane at working distance of 34μm for linearly polarized laser beam at the wavelength of 850 nm. Intensity profiles along (b) X and (c) Y axis are also shown.

Download Full Size | PPT Slide | PDF

Fig. 17
Fig. 17

Intensity distribution for wavelength of 850nm in the plane perpendicular to the optical axis at a distance of (a) 30μm, (b) 32μm, (c) 34μm, (d) 36μm and (e) 38μm from the output facet of the microlens.

Download Full Size | PPT Slide | PDF

Fig. 18
Fig. 18

(a) The intensity distribution in the focal plane at working distance of 40 μm (a) for linearly polarized laser beam at the wavelength of 633 nm. Intensity profiles along (b) X and (c) Y axis are also shown.

Download Full Size | PPT Slide | PDF

Fig. 19
Fig. 19

Intensity distribution for wavelength of 633nm in the plane perpendicular to the optical axis at a distance of (a) 36μm, (b) 38μm, (c) 40μm, (d) 42μm and (e) 44μm from the output facet of the microlens.

Download Full Size | PPT Slide | PDF

Equations (1)

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

n= n 0 ( 1− A 2 r 2 )

Metrics