Abstract

A theoretical analysis based on scalar diffraction theory about the recently reported focal-shift phenomena in planar metallic nanoslit lenses is presented. Under Fresnel approximation, an axial intensity formula is obtained, which is used to analyze the focal performance in the far field zone of lens. The relative focal shift is totally dependent on the Fresnel number only. The influences of the lens size, preset focal length and incident wavelength can be attributed to the change of Fresnel number. The total phase difference of the lens is approximately equal to the Fresnel number multiplied by π. Numerical simulations performed using finite-difference time-domain (FDTD) and near-far field transformation method are in agreement with the theoretical analysis. Using the theoretical formula assisted by simple numerical method, we provide predictions on the focal shift for the previous literatures.

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References

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  1. T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391(6668), 667–669 (1998).
    [CrossRef]
  2. W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424(6950), 824–830 (2003).
    [CrossRef] [PubMed]
  3. H. Shi, C. Wang, C. Du, X. Luo, X. Dong, and H. Gao, “Beam manipulating by metallic nano-slits with variant widths,” Opt. Express 13(18), 6815–6820 (2005).
    [CrossRef] [PubMed]
  4. T. Xu, C. Wang, C. Du, and X. Luo, “Plasmonic beam deflector,” Opt. Express 16(7), 4753–4759 (2008).
    [CrossRef] [PubMed]
  5. Q. Zhu, J. Ye, D. Wang, B. Gu, and Y. Zhang, “Optimal design of SPP-based metallic nanoaperture optical elements by using Yang-Gu algorithm,” Opt. Express 19(10), 9512–9522 (2011).
    [CrossRef] [PubMed]
  6. L. Verslegers, P. B. Catrysse, Z. Yu, and S. Fan, “Planar metallic nanoscale slit lenses for angle compensation,” Appl. Phys. Lett. 95(7), 071112 (2009).
    [CrossRef]
  7. Y. J. Jung, D. Park, S. Koo, S. Yu, and N. Park, “Metal slit array Fresnel lens for wavelength-scale optical coupling to nanophotonic waveguides,” Opt. Express 17(21), 18852–18857 (2009).
    [CrossRef] [PubMed]
  8. L. Verslegers, P. B. Catrysse, Z. Yu, J. S. White, E. S. Barnard, M. L. Brongersma, and S. Fan, “Planar lenses based on nanoscale slit arrays in a metallic film,” Nano Lett. 9(1), 235–238 (2009).
    [CrossRef] [PubMed]
  9. P. Ruffieux, T. Scharf, H. P. Herzig, R. Völkel, and K. J. Weible, “On the chromatic aberration of microlenses,” Opt. Express 14(11), 4687–4694 (2006).
    [CrossRef] [PubMed]
  10. M.-K. Chen, Y.-C. Chang, C.-E. Yang, Y. Guo, J. Mazurowski, S. Yin, P. Ruffin, C. Brantley, E. Edwards, and C. Luo, “Tunable terahertz plasmonic lenses based on semiconductor microslits,” Microw. Opt. Technol. Lett. 52(4), 979–981 (2010).
    [CrossRef]
  11. X. M. Goh, L. Lin, and A. Roberts, “Planar focusing elements using spatially varying near-resonant aperture arrays,” Opt. Express 18(11), 11683–11688 (2010).
    [CrossRef] [PubMed]
  12. Y. Li and E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 39(4), 211–215 (1981).
    [CrossRef]
  13. Y. Li and H. Platzer, “An experimental investigation of diffraction patterns in low Fresnel-number focusing systems,” Opt. Acta (Lond.) 30(11), 1621–1643 (1983).
    [CrossRef]
  14. W. Wang, A. T. Friberg, and E. Wolf, “Structure of focused fields in systems with large Fresnel numbers,” J. Opt. Soc. Am. A 12(9), 1947–1953 (1995).
    [CrossRef]
  15. Y. Yu and H. Zappe, “Effect of lens size on the focusing performance of plasmonic lenses and suggestions for the design,” Opt. Express 19(10), 9434–9444 (2011).
    [CrossRef] [PubMed]
  16. H. Kurt and D. S. Citrin, “Graded index photonic crystals,” Opt. Express 15(3), 1240–1253 (2007).
    [CrossRef] [PubMed]
  17. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge, 2002).
  18. K. D. Mielenz, “Computation of Fresnel integrals. II,” J. Res. Natl. Inst. Stand. Technol. 105, 589–590 (2000).
  19. D. W. Prather, M. S. Mirotznik, and J. N. Mait, “Boundary integral methods applied to the analysis of diffractive optical elements,” J. Opt. Soc. Am. A 14(1), 34–43 (1997).
    [CrossRef]
  20. Q. Chen and D. R. S. Cumming, “Visible light focusing demonstrated by plasmonic lenses based on nano-slits in an aluminum film,” Opt. Express 18(14), 14788–14793 (2010).
    [CrossRef] [PubMed]
  21. Q. Chen, “Effect of the number of zones in a one-dimensional plasmonic zone plate lens: simulation and experiment,” Plasmonics 6(1), 75–82 (2011).
    [CrossRef]

2011

2010

2009

Y. J. Jung, D. Park, S. Koo, S. Yu, and N. Park, “Metal slit array Fresnel lens for wavelength-scale optical coupling to nanophotonic waveguides,” Opt. Express 17(21), 18852–18857 (2009).
[CrossRef] [PubMed]

L. Verslegers, P. B. Catrysse, Z. Yu, and S. Fan, “Planar metallic nanoscale slit lenses for angle compensation,” Appl. Phys. Lett. 95(7), 071112 (2009).
[CrossRef]

L. Verslegers, P. B. Catrysse, Z. Yu, J. S. White, E. S. Barnard, M. L. Brongersma, and S. Fan, “Planar lenses based on nanoscale slit arrays in a metallic film,” Nano Lett. 9(1), 235–238 (2009).
[CrossRef] [PubMed]

2008

2007

2006

2005

2003

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424(6950), 824–830 (2003).
[CrossRef] [PubMed]

2000

K. D. Mielenz, “Computation of Fresnel integrals. II,” J. Res. Natl. Inst. Stand. Technol. 105, 589–590 (2000).

1998

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391(6668), 667–669 (1998).
[CrossRef]

1997

1995

1983

Y. Li and H. Platzer, “An experimental investigation of diffraction patterns in low Fresnel-number focusing systems,” Opt. Acta (Lond.) 30(11), 1621–1643 (1983).
[CrossRef]

1981

Y. Li and E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 39(4), 211–215 (1981).
[CrossRef]

Barnard, E. S.

L. Verslegers, P. B. Catrysse, Z. Yu, J. S. White, E. S. Barnard, M. L. Brongersma, and S. Fan, “Planar lenses based on nanoscale slit arrays in a metallic film,” Nano Lett. 9(1), 235–238 (2009).
[CrossRef] [PubMed]

Barnes, W. L.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424(6950), 824–830 (2003).
[CrossRef] [PubMed]

Brantley, C.

M.-K. Chen, Y.-C. Chang, C.-E. Yang, Y. Guo, J. Mazurowski, S. Yin, P. Ruffin, C. Brantley, E. Edwards, and C. Luo, “Tunable terahertz plasmonic lenses based on semiconductor microslits,” Microw. Opt. Technol. Lett. 52(4), 979–981 (2010).
[CrossRef]

Brongersma, M. L.

L. Verslegers, P. B. Catrysse, Z. Yu, J. S. White, E. S. Barnard, M. L. Brongersma, and S. Fan, “Planar lenses based on nanoscale slit arrays in a metallic film,” Nano Lett. 9(1), 235–238 (2009).
[CrossRef] [PubMed]

Catrysse, P. B.

L. Verslegers, P. B. Catrysse, Z. Yu, J. S. White, E. S. Barnard, M. L. Brongersma, and S. Fan, “Planar lenses based on nanoscale slit arrays in a metallic film,” Nano Lett. 9(1), 235–238 (2009).
[CrossRef] [PubMed]

L. Verslegers, P. B. Catrysse, Z. Yu, and S. Fan, “Planar metallic nanoscale slit lenses for angle compensation,” Appl. Phys. Lett. 95(7), 071112 (2009).
[CrossRef]

Chang, Y.-C.

M.-K. Chen, Y.-C. Chang, C.-E. Yang, Y. Guo, J. Mazurowski, S. Yin, P. Ruffin, C. Brantley, E. Edwards, and C. Luo, “Tunable terahertz plasmonic lenses based on semiconductor microslits,” Microw. Opt. Technol. Lett. 52(4), 979–981 (2010).
[CrossRef]

Chen, M.-K.

M.-K. Chen, Y.-C. Chang, C.-E. Yang, Y. Guo, J. Mazurowski, S. Yin, P. Ruffin, C. Brantley, E. Edwards, and C. Luo, “Tunable terahertz plasmonic lenses based on semiconductor microslits,” Microw. Opt. Technol. Lett. 52(4), 979–981 (2010).
[CrossRef]

Chen, Q.

Q. Chen, “Effect of the number of zones in a one-dimensional plasmonic zone plate lens: simulation and experiment,” Plasmonics 6(1), 75–82 (2011).
[CrossRef]

Q. Chen and D. R. S. Cumming, “Visible light focusing demonstrated by plasmonic lenses based on nano-slits in an aluminum film,” Opt. Express 18(14), 14788–14793 (2010).
[CrossRef] [PubMed]

Citrin, D. S.

Cumming, D. R. S.

Dereux, A.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424(6950), 824–830 (2003).
[CrossRef] [PubMed]

Dong, X.

Du, C.

Ebbesen, T. W.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424(6950), 824–830 (2003).
[CrossRef] [PubMed]

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391(6668), 667–669 (1998).
[CrossRef]

Edwards, E.

M.-K. Chen, Y.-C. Chang, C.-E. Yang, Y. Guo, J. Mazurowski, S. Yin, P. Ruffin, C. Brantley, E. Edwards, and C. Luo, “Tunable terahertz plasmonic lenses based on semiconductor microslits,” Microw. Opt. Technol. Lett. 52(4), 979–981 (2010).
[CrossRef]

Fan, S.

L. Verslegers, P. B. Catrysse, Z. Yu, and S. Fan, “Planar metallic nanoscale slit lenses for angle compensation,” Appl. Phys. Lett. 95(7), 071112 (2009).
[CrossRef]

L. Verslegers, P. B. Catrysse, Z. Yu, J. S. White, E. S. Barnard, M. L. Brongersma, and S. Fan, “Planar lenses based on nanoscale slit arrays in a metallic film,” Nano Lett. 9(1), 235–238 (2009).
[CrossRef] [PubMed]

Friberg, A. T.

Gao, H.

Ghaemi, H. F.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391(6668), 667–669 (1998).
[CrossRef]

Goh, X. M.

Gu, B.

Guo, Y.

M.-K. Chen, Y.-C. Chang, C.-E. Yang, Y. Guo, J. Mazurowski, S. Yin, P. Ruffin, C. Brantley, E. Edwards, and C. Luo, “Tunable terahertz plasmonic lenses based on semiconductor microslits,” Microw. Opt. Technol. Lett. 52(4), 979–981 (2010).
[CrossRef]

Herzig, H. P.

Jung, Y. J.

Koo, S.

Kurt, H.

Lezec, H. J.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391(6668), 667–669 (1998).
[CrossRef]

Li, Y.

Y. Li and H. Platzer, “An experimental investigation of diffraction patterns in low Fresnel-number focusing systems,” Opt. Acta (Lond.) 30(11), 1621–1643 (1983).
[CrossRef]

Y. Li and E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 39(4), 211–215 (1981).
[CrossRef]

Lin, L.

Luo, C.

M.-K. Chen, Y.-C. Chang, C.-E. Yang, Y. Guo, J. Mazurowski, S. Yin, P. Ruffin, C. Brantley, E. Edwards, and C. Luo, “Tunable terahertz plasmonic lenses based on semiconductor microslits,” Microw. Opt. Technol. Lett. 52(4), 979–981 (2010).
[CrossRef]

Luo, X.

Mait, J. N.

Mazurowski, J.

M.-K. Chen, Y.-C. Chang, C.-E. Yang, Y. Guo, J. Mazurowski, S. Yin, P. Ruffin, C. Brantley, E. Edwards, and C. Luo, “Tunable terahertz plasmonic lenses based on semiconductor microslits,” Microw. Opt. Technol. Lett. 52(4), 979–981 (2010).
[CrossRef]

Mielenz, K. D.

K. D. Mielenz, “Computation of Fresnel integrals. II,” J. Res. Natl. Inst. Stand. Technol. 105, 589–590 (2000).

Mirotznik, M. S.

Park, D.

Park, N.

Platzer, H.

Y. Li and H. Platzer, “An experimental investigation of diffraction patterns in low Fresnel-number focusing systems,” Opt. Acta (Lond.) 30(11), 1621–1643 (1983).
[CrossRef]

Prather, D. W.

Roberts, A.

Ruffieux, P.

Ruffin, P.

M.-K. Chen, Y.-C. Chang, C.-E. Yang, Y. Guo, J. Mazurowski, S. Yin, P. Ruffin, C. Brantley, E. Edwards, and C. Luo, “Tunable terahertz plasmonic lenses based on semiconductor microslits,” Microw. Opt. Technol. Lett. 52(4), 979–981 (2010).
[CrossRef]

Scharf, T.

Shi, H.

Thio, T.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391(6668), 667–669 (1998).
[CrossRef]

Verslegers, L.

L. Verslegers, P. B. Catrysse, Z. Yu, and S. Fan, “Planar metallic nanoscale slit lenses for angle compensation,” Appl. Phys. Lett. 95(7), 071112 (2009).
[CrossRef]

L. Verslegers, P. B. Catrysse, Z. Yu, J. S. White, E. S. Barnard, M. L. Brongersma, and S. Fan, “Planar lenses based on nanoscale slit arrays in a metallic film,” Nano Lett. 9(1), 235–238 (2009).
[CrossRef] [PubMed]

Völkel, R.

Wang, C.

Wang, D.

Wang, W.

Weible, K. J.

White, J. S.

L. Verslegers, P. B. Catrysse, Z. Yu, J. S. White, E. S. Barnard, M. L. Brongersma, and S. Fan, “Planar lenses based on nanoscale slit arrays in a metallic film,” Nano Lett. 9(1), 235–238 (2009).
[CrossRef] [PubMed]

Wolf, E.

W. Wang, A. T. Friberg, and E. Wolf, “Structure of focused fields in systems with large Fresnel numbers,” J. Opt. Soc. Am. A 12(9), 1947–1953 (1995).
[CrossRef]

Y. Li and E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 39(4), 211–215 (1981).
[CrossRef]

Wolff, P. A.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391(6668), 667–669 (1998).
[CrossRef]

Xu, T.

Yang, C.-E.

M.-K. Chen, Y.-C. Chang, C.-E. Yang, Y. Guo, J. Mazurowski, S. Yin, P. Ruffin, C. Brantley, E. Edwards, and C. Luo, “Tunable terahertz plasmonic lenses based on semiconductor microslits,” Microw. Opt. Technol. Lett. 52(4), 979–981 (2010).
[CrossRef]

Ye, J.

Yin, S.

M.-K. Chen, Y.-C. Chang, C.-E. Yang, Y. Guo, J. Mazurowski, S. Yin, P. Ruffin, C. Brantley, E. Edwards, and C. Luo, “Tunable terahertz plasmonic lenses based on semiconductor microslits,” Microw. Opt. Technol. Lett. 52(4), 979–981 (2010).
[CrossRef]

Yu, S.

Yu, Y.

Yu, Z.

L. Verslegers, P. B. Catrysse, Z. Yu, J. S. White, E. S. Barnard, M. L. Brongersma, and S. Fan, “Planar lenses based on nanoscale slit arrays in a metallic film,” Nano Lett. 9(1), 235–238 (2009).
[CrossRef] [PubMed]

L. Verslegers, P. B. Catrysse, Z. Yu, and S. Fan, “Planar metallic nanoscale slit lenses for angle compensation,” Appl. Phys. Lett. 95(7), 071112 (2009).
[CrossRef]

Zappe, H.

Zhang, Y.

Zhu, Q.

Appl. Phys. Lett.

L. Verslegers, P. B. Catrysse, Z. Yu, and S. Fan, “Planar metallic nanoscale slit lenses for angle compensation,” Appl. Phys. Lett. 95(7), 071112 (2009).
[CrossRef]

J. Opt. Soc. Am. A

J. Res. Natl. Inst. Stand. Technol.

K. D. Mielenz, “Computation of Fresnel integrals. II,” J. Res. Natl. Inst. Stand. Technol. 105, 589–590 (2000).

Microw. Opt. Technol. Lett.

M.-K. Chen, Y.-C. Chang, C.-E. Yang, Y. Guo, J. Mazurowski, S. Yin, P. Ruffin, C. Brantley, E. Edwards, and C. Luo, “Tunable terahertz plasmonic lenses based on semiconductor microslits,” Microw. Opt. Technol. Lett. 52(4), 979–981 (2010).
[CrossRef]

Nano Lett.

L. Verslegers, P. B. Catrysse, Z. Yu, J. S. White, E. S. Barnard, M. L. Brongersma, and S. Fan, “Planar lenses based on nanoscale slit arrays in a metallic film,” Nano Lett. 9(1), 235–238 (2009).
[CrossRef] [PubMed]

Nature

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391(6668), 667–669 (1998).
[CrossRef]

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424(6950), 824–830 (2003).
[CrossRef] [PubMed]

Opt. Acta (Lond.)

Y. Li and H. Platzer, “An experimental investigation of diffraction patterns in low Fresnel-number focusing systems,” Opt. Acta (Lond.) 30(11), 1621–1643 (1983).
[CrossRef]

Opt. Commun.

Y. Li and E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 39(4), 211–215 (1981).
[CrossRef]

Opt. Express

H. Shi, C. Wang, C. Du, X. Luo, X. Dong, and H. Gao, “Beam manipulating by metallic nano-slits with variant widths,” Opt. Express 13(18), 6815–6820 (2005).
[CrossRef] [PubMed]

P. Ruffieux, T. Scharf, H. P. Herzig, R. Völkel, and K. J. Weible, “On the chromatic aberration of microlenses,” Opt. Express 14(11), 4687–4694 (2006).
[CrossRef] [PubMed]

H. Kurt and D. S. Citrin, “Graded index photonic crystals,” Opt. Express 15(3), 1240–1253 (2007).
[CrossRef] [PubMed]

T. Xu, C. Wang, C. Du, and X. Luo, “Plasmonic beam deflector,” Opt. Express 16(7), 4753–4759 (2008).
[CrossRef] [PubMed]

Y. J. Jung, D. Park, S. Koo, S. Yu, and N. Park, “Metal slit array Fresnel lens for wavelength-scale optical coupling to nanophotonic waveguides,” Opt. Express 17(21), 18852–18857 (2009).
[CrossRef] [PubMed]

X. M. Goh, L. Lin, and A. Roberts, “Planar focusing elements using spatially varying near-resonant aperture arrays,” Opt. Express 18(11), 11683–11688 (2010).
[CrossRef] [PubMed]

Q. Chen and D. R. S. Cumming, “Visible light focusing demonstrated by plasmonic lenses based on nano-slits in an aluminum film,” Opt. Express 18(14), 14788–14793 (2010).
[CrossRef] [PubMed]

Y. Yu and H. Zappe, “Effect of lens size on the focusing performance of plasmonic lenses and suggestions for the design,” Opt. Express 19(10), 9434–9444 (2011).
[CrossRef] [PubMed]

Q. Zhu, J. Ye, D. Wang, B. Gu, and Y. Zhang, “Optimal design of SPP-based metallic nanoaperture optical elements by using Yang-Gu algorithm,” Opt. Express 19(10), 9512–9522 (2011).
[CrossRef] [PubMed]

Plasmonics

Q. Chen, “Effect of the number of zones in a one-dimensional plasmonic zone plate lens: simulation and experiment,” Plasmonics 6(1), 75–82 (2011).
[CrossRef]

Other

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge, 2002).

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Figures (8)

Fig. 1
Fig. 1

Schematic illustration of planar lens

Fig. 2
Fig. 2

Relative focal shift as a function of Fresnel number under Fresnel approximation

Fig. 3
Fig. 3

Focal shift as a function of lens size at different working wavelength

Fig. 4
Fig. 4

(a) Geometry of the lens made of metallic slits with different widths. (b) The squares present the phase delay at each slit (compared with the central slit) for lenses with different preset focal lengths; the lines stand for the theoretical design obtained from Eq. (3); the slit interspacing (center to center) is fixed at Δ = 200 nm.

Fig. 5
Fig. 5

(a) The simulation result of magnetic ðeld intensity pattern for metallic lens with half-size a = 2 μm, f = 2.8 μm and λ = 637 nm. The magnetic intensity of the incident field is 0.25. The slit widths: 16, 16, 16, 18, 20, 22, 24, 30, 38, 54 and 90 nm from middle to the edges. (b) Axial field intensity distributions extracted from simulation results and calculated using theoretical Eq. (5), respectively. For the purpose of comparison, both curves are normalized to the focus.

Fig. 6
Fig. 6

(a) The simulation result of magnetic ðeld intensity pattern for metallic lens with half-size a = 2 μm, f = 6 μm and λ = 637 nm. The magnetic intensity of the incident field is 0.25. The slit widths: 26, 26, 26, 28, 30, 32, 34, 40, 46, 56 and 72 nm from middle to the edges. (b) Axial field intensity distributions extracted from simulation results and calculated using theoretical Eq. (5), respectively.

Fig. 7
Fig. 7

(a) The simulation result of magnetic ðeld intensity pattern for metallic lens with half-size a = 2 μm, f = 10 μm and λ = 637 nm. The magnetic intensity of the incident field is 0.25. The slit widths: 36, 36, 36, 38, 40, 42, 44, 48, 54, 62 and 74 nm from middle to the edges. (b) Axial field intensity distributions extracted from simulation results and calculated using theoretical Eq. (5), respectively.

Fig. 8
Fig. 8

(a) The simulation result of magnetic ðeld intensity pattern for metallic lens with half-size a = 2 μm, f = infinity and λ = 637 nm. The magnetic intensity of the incident field is 0.25. All slit widths are the same: 50 nm. (b) Axial field intensity distributions extracted from simulation results and calculated using theoretical Eq. (10), respectively. The intensity distribution along the optical axis for the case of circular aperture is also calculated using Eq. (11) and shown for comparison.

Tables (2)

Tables Icon

Table 1 Focusing characteristics of planar plasmonic lenses with different sizes and focal lengths

Tables Icon

Table 2 Prediction of focal-shift for the lenses from previous literatures

Equations (14)

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

U 2 (0,z)= 1 jλz exp[jkz] a a U 1 (x,0)exp[ jk x 2 2z ]dx ,
U 1 (x,0)={ Aexp[jφ(x)]|x|<a 0else .
φ(x)=k( f f 2 + x 2 ).
φ(x)= k 2f x 2 .
I(0,z)=| U 2 (0,z) U 2 * (0,z)|= 2 A 2 |p| [ C 2 (ξ)+ S 2 (ξ)],
p= zf f ,
C(ξ)= 0 ξ cos( π 2 u 2 )du S(ξ)= 0 ξ sin( π 2 u 2 )du
ξ=a 2|p| λz .
N= a 2 λf ,
ξ= 2N|p| 1+p .
I(0,z)=| U 2 (0,z) U 2 * (0,z)|=2 A 2 [ C 2 (ξ)+ S 2 (ξ)].
I(0,z)=4 A 2 [ sin( k a 2 4z ) ] 2 .
( a f ) 2 =N λ f .
tanh( k 1 d 2 )= ε d k 2 ε m k 1

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