Abstract

The analytical solutions of the electromagnetic waves in the inhomogeneous cylindrical hyperlens (CH) comprising concentric cylindrical layers (CCLs) with multiple point sources located either outside the structure in the focusing process or inside the core in the magnifying process are obtained by means of Green’s function analysis. The solutions are consistent with FDTD simulation in both processes. The sub-wavelength focal spot λ/16.26 from two point sources with wavelength 465 nm is demonstrated in the CH made by alternating silver and silica CCLs. Our solutions are expected to be the efficient tools for designing the sub-wavelength focusing and imaging cylindrical hyperlens.

© 2014 Optical Society of America

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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  3. Z. Liu, H. Lee, Y. Xiong, C. Sun, X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315(5819), 1686 (2007).
    [CrossRef] [PubMed]
  4. S. A. Ramakrishna, J. B. Pendry, M. C. K. Wiltshrine, W. J. Stewart, “Imaging the near field,” J. Mod. Opt. 50, 1419–1430 (2003).
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  6. C. Ma, Z. Liu, “A super resolution metalens with phase compensation mechanism,” Appl. Phys. Lett. 96(18), 183103 (2010).
    [CrossRef]
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    [CrossRef]
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  13. E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. D. Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen, LAPACK Usersguide, 3rd ed. Philadelphia: Society for Industrial and Applied Mathematics (1999)
  14. J. P. Moreau, (2009, April 23). Bessel programs in fortran 90 (Online). Available: http://jean-pierre.moreau.pagesperso-orange.fr/ .
  15. A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, S. G. Johnson, “MEEP: A flexible free-software package for electromagnetic simulations by the FDTD methods,” Comput. Phys. Commun. 181(3), 687–702 (2010).
    [CrossRef]
  16. S. Foteinopoulou, J. P. Vigneron, C. Vandenbem, “Optical near-field excitations on plasmonic nanoparticle-based structures,” Opt. Express 15(7), 4253–4267 (2007).
    [CrossRef] [PubMed]

2013

S. Ishii, A. V. Kildishev, E. Narimanov, V. M. Shalev, V. P. Drachev, “Sub-wavelength interference pattern from volume plasmon polaritons in a hyperbolic medium,” Laser Photon. Rev. 7(2), 265–271 (2013).
[CrossRef]

2012

D. Lu, Z. Liu, “Hyperlenes and metalenses for far-field super-resolution imaging,” Nat. Commun. 3, 1–9 (2012).

2011

2010

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, S. G. Johnson, “MEEP: A flexible free-software package for electromagnetic simulations by the FDTD methods,” Comput. Phys. Commun. 181(3), 687–702 (2010).
[CrossRef]

C. Ma, Z. Liu, “A super resolution metalens with phase compensation mechanism,” Appl. Phys. Lett. 96(18), 183103 (2010).
[CrossRef]

2009

2007

S. Foteinopoulou, J. P. Vigneron, C. Vandenbem, “Optical near-field excitations on plasmonic nanoparticle-based structures,” Opt. Express 15(7), 4253–4267 (2007).
[CrossRef] [PubMed]

Z. Liu, H. Lee, Y. Xiong, C. Sun, X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315(5819), 1686 (2007).
[CrossRef] [PubMed]

2006

2003

S. A. Ramakrishna, J. B. Pendry, M. C. K. Wiltshrine, W. J. Stewart, “Imaging the near field,” J. Mod. Opt. 50, 1419–1430 (2003).

2000

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85(18), 3966–3969 (2000).
[CrossRef] [PubMed]

Alekseyev, L. V.

Bermel, P.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, S. G. Johnson, “MEEP: A flexible free-software package for electromagnetic simulations by the FDTD methods,” Comput. Phys. Commun. 181(3), 687–702 (2010).
[CrossRef]

Cheah, K. W.

Drachev, V. P.

S. Ishii, A. V. Kildishev, E. Narimanov, V. M. Shalev, V. P. Drachev, “Sub-wavelength interference pattern from volume plasmon polaritons in a hyperbolic medium,” Laser Photon. Rev. 7(2), 265–271 (2013).
[CrossRef]

Foteinopoulou, S.

Ibanescu, M.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, S. G. Johnson, “MEEP: A flexible free-software package for electromagnetic simulations by the FDTD methods,” Comput. Phys. Commun. 181(3), 687–702 (2010).
[CrossRef]

Ishii, S.

S. Ishii, A. V. Kildishev, E. Narimanov, V. M. Shalev, V. P. Drachev, “Sub-wavelength interference pattern from volume plasmon polaritons in a hyperbolic medium,” Laser Photon. Rev. 7(2), 265–271 (2013).
[CrossRef]

Jacob, Z.

Joannopoulos, J. D.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, S. G. Johnson, “MEEP: A flexible free-software package for electromagnetic simulations by the FDTD methods,” Comput. Phys. Commun. 181(3), 687–702 (2010).
[CrossRef]

Johnson, S. G.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, S. G. Johnson, “MEEP: A flexible free-software package for electromagnetic simulations by the FDTD methods,” Comput. Phys. Commun. 181(3), 687–702 (2010).
[CrossRef]

Kildishev, A. V.

S. Ishii, A. V. Kildishev, E. Narimanov, V. M. Shalev, V. P. Drachev, “Sub-wavelength interference pattern from volume plasmon polaritons in a hyperbolic medium,” Laser Photon. Rev. 7(2), 265–271 (2013).
[CrossRef]

Lee, H.

Z. Liu, H. Lee, Y. Xiong, C. Sun, X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315(5819), 1686 (2007).
[CrossRef] [PubMed]

Li, G.

Li, J.

Liu, H.

Liu, Z.

D. Lu, Z. Liu, “Hyperlenes and metalenses for far-field super-resolution imaging,” Nat. Commun. 3, 1–9 (2012).

C. Ma, Z. Liu, “A super resolution metalens with phase compensation mechanism,” Appl. Phys. Lett. 96(18), 183103 (2010).
[CrossRef]

Z. Liu, H. Lee, Y. Xiong, C. Sun, X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315(5819), 1686 (2007).
[CrossRef] [PubMed]

Lu, D.

D. Lu, Z. Liu, “Hyperlenes and metalenses for far-field super-resolution imaging,” Nat. Commun. 3, 1–9 (2012).

Ma, C.

C. Ma, Z. Liu, “A super resolution metalens with phase compensation mechanism,” Appl. Phys. Lett. 96(18), 183103 (2010).
[CrossRef]

Narimanov, E.

S. Ishii, A. V. Kildishev, E. Narimanov, V. M. Shalev, V. P. Drachev, “Sub-wavelength interference pattern from volume plasmon polaritons in a hyperbolic medium,” Laser Photon. Rev. 7(2), 265–271 (2013).
[CrossRef]

Z. Jacob, L. V. Alekseyev, E. Narimanov, “Optical hyperlens: far-field imaging beyond the diffraction limit,” Opt. Express 14(18), 8247–8256 (2006).
[CrossRef] [PubMed]

Oskooi, A. F.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, S. G. Johnson, “MEEP: A flexible free-software package for electromagnetic simulations by the FDTD methods,” Comput. Phys. Commun. 181(3), 687–702 (2010).
[CrossRef]

Pendry, J. B.

S. A. Ramakrishna, J. B. Pendry, M. C. K. Wiltshrine, W. J. Stewart, “Imaging the near field,” J. Mod. Opt. 50, 1419–1430 (2003).

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85(18), 3966–3969 (2000).
[CrossRef] [PubMed]

Podolskiy, V. A.

Ramakrishna, S. A.

S. A. Ramakrishna, J. B. Pendry, M. C. K. Wiltshrine, W. J. Stewart, “Imaging the near field,” J. Mod. Opt. 50, 1419–1430 (2003).

Roundy, D.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, S. G. Johnson, “MEEP: A flexible free-software package for electromagnetic simulations by the FDTD methods,” Comput. Phys. Commun. 181(3), 687–702 (2010).
[CrossRef]

Shalev, V. M.

S. Ishii, A. V. Kildishev, E. Narimanov, V. M. Shalev, V. P. Drachev, “Sub-wavelength interference pattern from volume plasmon polaritons in a hyperbolic medium,” Laser Photon. Rev. 7(2), 265–271 (2013).
[CrossRef]

Stewart, W. J.

S. A. Ramakrishna, J. B. Pendry, M. C. K. Wiltshrine, W. J. Stewart, “Imaging the near field,” J. Mod. Opt. 50, 1419–1430 (2003).

Sun, C.

Z. Liu, H. Lee, Y. Xiong, C. Sun, X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315(5819), 1686 (2007).
[CrossRef] [PubMed]

Thongrattanasiri, S.

Vandenbem, C.

Vigneron, J. P.

Webb, K. J.

Wiltshrine, M. C. K.

S. A. Ramakrishna, J. B. Pendry, M. C. K. Wiltshrine, W. J. Stewart, “Imaging the near field,” J. Mod. Opt. 50, 1419–1430 (2003).

Xiong, Y.

Z. Liu, H. Lee, Y. Xiong, C. Sun, X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315(5819), 1686 (2007).
[CrossRef] [PubMed]

Zhang, X.

Z. Liu, H. Lee, Y. Xiong, C. Sun, X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315(5819), 1686 (2007).
[CrossRef] [PubMed]

Appl. Opt.

Appl. Phys. Lett.

C. Ma, Z. Liu, “A super resolution metalens with phase compensation mechanism,” Appl. Phys. Lett. 96(18), 183103 (2010).
[CrossRef]

Comput. Phys. Commun.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, S. G. Johnson, “MEEP: A flexible free-software package for electromagnetic simulations by the FDTD methods,” Comput. Phys. Commun. 181(3), 687–702 (2010).
[CrossRef]

J. Mod. Opt.

S. A. Ramakrishna, J. B. Pendry, M. C. K. Wiltshrine, W. J. Stewart, “Imaging the near field,” J. Mod. Opt. 50, 1419–1430 (2003).

Laser Photon. Rev.

S. Ishii, A. V. Kildishev, E. Narimanov, V. M. Shalev, V. P. Drachev, “Sub-wavelength interference pattern from volume plasmon polaritons in a hyperbolic medium,” Laser Photon. Rev. 7(2), 265–271 (2013).
[CrossRef]

Nat. Commun.

D. Lu, Z. Liu, “Hyperlenes and metalenses for far-field super-resolution imaging,” Nat. Commun. 3, 1–9 (2012).

Opt. Express

Opt. Lett.

Phys. Rev. Lett.

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85(18), 3966–3969 (2000).
[CrossRef] [PubMed]

Science

Z. Liu, H. Lee, Y. Xiong, C. Sun, X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315(5819), 1686 (2007).
[CrossRef] [PubMed]

Other

M. A. Noginov and V. A. Podolskiy, eds., “Anisotropic and Hyperbolic metamaterials,” in Tutorials in Metamaterials, Florida, Taylor & Francis Group, (Academic, 2012), pp. 172.

E. N. Economou, “Time-independent Green’s functions,” in Green’s Functions in Quantum Physics, 3rd ed. Heidelberg, Germany: Springer, (Academic, 2006), pp. 13.

E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. D. Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen, LAPACK Usersguide, 3rd ed. Philadelphia: Society for Industrial and Applied Mathematics (1999)

J. P. Moreau, (2009, April 23). Bessel programs in fortran 90 (Online). Available: http://jean-pierre.moreau.pagesperso-orange.fr/ .

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Figures (4)

Fig. 1
Fig. 1

The schematic view of the cylindrical hyperlens (CH) in 2-D for (a) focusing process and (b) magnifying process.

Fig. 2
Fig. 2

The intensity in the vacuum (a) The intensity distribution of the single point source obtained by the analytical solutions with the maximum of order 110. (b) The intensity profiles of the single point source along the y-axis obtained by the analytical solutions with the maximum of order 110 and 20 labeled by the blue and green line, respectively, and the FDTD denoted by triangular points. The red line is the analytical intensity calculated by the simplified form taking into account all orders. (c) The intensity distribution of the double point source obtained by the analytical solutions with the maximum of order 110. (d) The intensity profile of the double point source along the innermost radius (r0 = 100 nm) obtained by the analytical solutions with the maximum of order 110 compared with that obtained by FDTD.

Fig. 3
Fig. 3

The intensity in the cylindrical hyperlens (CH) with two point sources (a) The intensity distribution of the focusing CH obtained by the analytical solutions with the maximum of order 110. (b) The intensity profile along the innermost radius (r0 = 100 nm) of the focusing CH obtained by the analytical solutions with the maximum of order 110 compared with those obtained by FDTD with resolutions 2 and 4 pixel/1 nm. (c) The intensity distribution of the magnifying CH obtained by the analytical solutions with the maximum of order 110. (d) The intensity profile along the curve ( ρ = 185 nm) of the magnifying CH obtained by the analytical solutions with the maximum of order 110 compared with those obtained by FDTD with resolutions 2 and 4 pixel/1 nm.

Fig. 4
Fig. 4

Sub-wavelength focusing of two point sources (a) The analytical intensity distribution in the focusing CH with the maximum of order 220 (b) The structure equivalent to the two point sources generated by two slits in the silver screen, the slit width w = 10 nm, the thickness of the screen L = 465 nm (c) The FDTD intensity distribution with the resolution 1.5 pixel/ 1 nm of the structure in Fig. 4(b) which is excited by the x-polarized plane wave (d) The intensity profiles along the inner radius (r0 = 500 nm): the red line is from the analytical solutions shown in Fig. 4(a), the black, green and blue lines are obtained from the FDTD simulation with the resolution 0.5, 1.0 and 1.5 pixel/1 nm, respectively

Equations (14)

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( 2 + k 2 ) H z =i ε 0 ε M 0 δ( ρ ρ )δ( φ φ )/ρ,
E= E ρ e ^ ρ + E ϕ e ^ ϕ = 1 iω ε 0 ε 1 ρ H z ϕ e ^ ρ + 1 iω ε 0 ε H z ρ e ^ ϕ .
H z u =iω ε 0 ε M 0 i 4 m= H m (1) ( ρ < k ) J m ( ρ > k ) e im( ϕ ϕ ) ,
H z b,foc =iω ε 0 ε N+1 M 0 1 2π m= A m N+1 H m (1) ( ρ k N+1 ) e imϕ ,
H z b,mag =iω ε 0 ε c M 0 1 2π m= B m 0 J m ( ρ k c ) e imϕ ,
H z j =iω ε 0 ε j M 0 1 2π m= ( A m j H m (1) ( ρ k j )+ B m j J m ( ρ k j ) ) e imϕ ,
{ x m }= ( A m 1 A m j A m N+1 B m 0 B m j B m N ) t .
A m j+1 H m (1) ( r j k j+1 )+ B m j+1 J m ( r j k j+1 ) ε j ε j +1 ( A m j H m (1) ( r j k j )+ B m j J m ( r j k j ) )=0,
A m j+1 H m (1) ( r j k j+1 )+ B m j+1 J m ( r j k j+1 ) ε j ε j +1 ( A m j H m (1) ( r j k j )+ B m j J m ( r j k j ) )=0,
b 2N+1 foc = iπ 2 J m ( r N k N+1 ) s=1 K H m (1) ( ρ s k N+1 ) e im ϕ s ,
b 2N+2 foc = iπ 2 J m ( r N k N+1 ) s=1 K H m (1) ( ρ s k N+1 ) e im ϕ s ,
b 1 mag = iπ 2 ( ε c ε 1 ) H m (1) ( r 0 k c ) s=1 K J m ( ρ s k c ) e im ϕ s ,
b 2 mag = iπ 2 ( ε c ε 1 ) H m (1) ( r 0 k c ) s=1 K J m ( ρ s k c ) e im ϕ s .
H z =( ω ε 0 M 0 /4 ) H 0 (1) ( k 0 R m ( ρ,ϕ; ρ , ϕ ) ),

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