Abstract

We show that transformation optics can be applied to extend the functionalities of conventional optical devices. In particular, geometrically compressing the input facet of any conventional optical elements can extend the input spatial frequency bandwidth. As an example, we design a Fourier lens that can transform the image to its reciprocal space and operate for incident light of subwavelength profile. An explicit design employing metal–dielectric layers is given for realization.

© 2009 Optical Society of America

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  1. J. B. Pendry, D. Schurig, and D. R. Smith, Science 312, 1780 (2006).
    [CrossRef] [PubMed]
  2. U. Leonhardt, Science 312, 1777 (2006).
    [CrossRef] [PubMed]
  3. D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. PendryA. F. Starr, and D. R. Smith, Science 314, 977 (2006).
    [CrossRef] [PubMed]
  4. M. Rahm, S. A. Cummer, D. Schurig, J. B. Pendry, and D. R. Smith, Phys. Rev. Lett. 100, 063903 (2008).
    [CrossRef] [PubMed]
  5. H. Chen, B. Hou, S. Chen, X. Ao, W. Wen, and C. T. Chan, Phys. Rev. Lett. 102, 183903 (2009).
    [CrossRef] [PubMed]
  6. A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, Phys. Rev. Lett. 99, 183901 (2007).
    [CrossRef] [PubMed]
  7. A. V. Kildishev and E. E. Narimanov, Opt. Lett. 32, 3432 (2007).
    [CrossRef] [PubMed]
  8. Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, Science 315, 1686 (2007).
    [CrossRef] [PubMed]
  9. Y. Xiong, Z. Liu, and X. Zhang, Appl. Phys. Lett. 94, 203108 (2009).
    [CrossRef]
  10. A. V. Kildishev and V. M. Shalaev, Opt. Lett. 33, 43 (2008).
    [CrossRef]
  11. S. Han, Y. Xiong, D. Genov, Z. Liu, G. Bartal, and X. Zhang, Nano Lett. 8, 4243 (2008).
    [CrossRef]
  12. X. Zhang and Z. Liu, Nat. Mater. 7, 435 (2008).
    [CrossRef] [PubMed]
  13. S. Anantha Ramakrishna, J. B. Pendry, M. C. K. Wiltshire, and W. J. Stewart, J. Mod. Opt. 50, 1419 (2003).
  14. J. B. Pendry and S. Anantha Ramakrishna, J. Phys. Condens. Matter 15, 6345 (2003).
    [CrossRef]
  15. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).
  16. D. Mendlovic and H. M. Ozaktas, J. Opt. Soc. Am. A 10, 1875 (1993).
    [CrossRef]
  17. U. Levy, M. Abashin, K. Ikeda, A. Krishnamoorthy, J. Cunningham, and Y. Fainman, Phys. Rev. Lett. 98, 243901 (2007).
    [CrossRef] [PubMed]
  18. D. Bergmann, Phys. Rep. 43, 377 (1978).
    [CrossRef]
  19. J. B. Johnson and R. W. Christy, Phys. Rev. B 6, 4370 (1972).
    [CrossRef]

2009 (2)

H. Chen, B. Hou, S. Chen, X. Ao, W. Wen, and C. T. Chan, Phys. Rev. Lett. 102, 183903 (2009).
[CrossRef] [PubMed]

Y. Xiong, Z. Liu, and X. Zhang, Appl. Phys. Lett. 94, 203108 (2009).
[CrossRef]

2008 (4)

M. Rahm, S. A. Cummer, D. Schurig, J. B. Pendry, and D. R. Smith, Phys. Rev. Lett. 100, 063903 (2008).
[CrossRef] [PubMed]

S. Han, Y. Xiong, D. Genov, Z. Liu, G. Bartal, and X. Zhang, Nano Lett. 8, 4243 (2008).
[CrossRef]

X. Zhang and Z. Liu, Nat. Mater. 7, 435 (2008).
[CrossRef] [PubMed]

A. V. Kildishev and V. M. Shalaev, Opt. Lett. 33, 43 (2008).
[CrossRef]

2007 (4)

A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, Phys. Rev. Lett. 99, 183901 (2007).
[CrossRef] [PubMed]

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, Science 315, 1686 (2007).
[CrossRef] [PubMed]

A. V. Kildishev and E. E. Narimanov, Opt. Lett. 32, 3432 (2007).
[CrossRef] [PubMed]

U. Levy, M. Abashin, K. Ikeda, A. Krishnamoorthy, J. Cunningham, and Y. Fainman, Phys. Rev. Lett. 98, 243901 (2007).
[CrossRef] [PubMed]

2006 (3)

J. B. Pendry, D. Schurig, and D. R. Smith, Science 312, 1780 (2006).
[CrossRef] [PubMed]

U. Leonhardt, Science 312, 1777 (2006).
[CrossRef] [PubMed]

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. PendryA. F. Starr, and D. R. Smith, Science 314, 977 (2006).
[CrossRef] [PubMed]

2003 (2)

S. Anantha Ramakrishna, J. B. Pendry, M. C. K. Wiltshire, and W. J. Stewart, J. Mod. Opt. 50, 1419 (2003).

J. B. Pendry and S. Anantha Ramakrishna, J. Phys. Condens. Matter 15, 6345 (2003).
[CrossRef]

1993 (1)

1978 (1)

D. Bergmann, Phys. Rep. 43, 377 (1978).
[CrossRef]

1972 (1)

J. B. Johnson and R. W. Christy, Phys. Rev. B 6, 4370 (1972).
[CrossRef]

Abashin, M.

U. Levy, M. Abashin, K. Ikeda, A. Krishnamoorthy, J. Cunningham, and Y. Fainman, Phys. Rev. Lett. 98, 243901 (2007).
[CrossRef] [PubMed]

Anantha Ramakrishna, S.

S. Anantha Ramakrishna, J. B. Pendry, M. C. K. Wiltshire, and W. J. Stewart, J. Mod. Opt. 50, 1419 (2003).

J. B. Pendry and S. Anantha Ramakrishna, J. Phys. Condens. Matter 15, 6345 (2003).
[CrossRef]

Ao, X.

H. Chen, B. Hou, S. Chen, X. Ao, W. Wen, and C. T. Chan, Phys. Rev. Lett. 102, 183903 (2009).
[CrossRef] [PubMed]

Bartal, G.

S. Han, Y. Xiong, D. Genov, Z. Liu, G. Bartal, and X. Zhang, Nano Lett. 8, 4243 (2008).
[CrossRef]

Bergmann, D.

D. Bergmann, Phys. Rep. 43, 377 (1978).
[CrossRef]

Chan, C. T.

H. Chen, B. Hou, S. Chen, X. Ao, W. Wen, and C. T. Chan, Phys. Rev. Lett. 102, 183903 (2009).
[CrossRef] [PubMed]

Chen, H.

H. Chen, B. Hou, S. Chen, X. Ao, W. Wen, and C. T. Chan, Phys. Rev. Lett. 102, 183903 (2009).
[CrossRef] [PubMed]

Chen, S.

H. Chen, B. Hou, S. Chen, X. Ao, W. Wen, and C. T. Chan, Phys. Rev. Lett. 102, 183903 (2009).
[CrossRef] [PubMed]

Christy, R. W.

J. B. Johnson and R. W. Christy, Phys. Rev. B 6, 4370 (1972).
[CrossRef]

Cummer, S. A.

M. Rahm, S. A. Cummer, D. Schurig, J. B. Pendry, and D. R. Smith, Phys. Rev. Lett. 100, 063903 (2008).
[CrossRef] [PubMed]

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. PendryA. F. Starr, and D. R. Smith, Science 314, 977 (2006).
[CrossRef] [PubMed]

Cunningham, J.

U. Levy, M. Abashin, K. Ikeda, A. Krishnamoorthy, J. Cunningham, and Y. Fainman, Phys. Rev. Lett. 98, 243901 (2007).
[CrossRef] [PubMed]

Fainman, Y.

U. Levy, M. Abashin, K. Ikeda, A. Krishnamoorthy, J. Cunningham, and Y. Fainman, Phys. Rev. Lett. 98, 243901 (2007).
[CrossRef] [PubMed]

Genov, D.

S. Han, Y. Xiong, D. Genov, Z. Liu, G. Bartal, and X. Zhang, Nano Lett. 8, 4243 (2008).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

Greenleaf, A.

A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, Phys. Rev. Lett. 99, 183901 (2007).
[CrossRef] [PubMed]

Han, S.

S. Han, Y. Xiong, D. Genov, Z. Liu, G. Bartal, and X. Zhang, Nano Lett. 8, 4243 (2008).
[CrossRef]

Hou, B.

H. Chen, B. Hou, S. Chen, X. Ao, W. Wen, and C. T. Chan, Phys. Rev. Lett. 102, 183903 (2009).
[CrossRef] [PubMed]

Ikeda, K.

U. Levy, M. Abashin, K. Ikeda, A. Krishnamoorthy, J. Cunningham, and Y. Fainman, Phys. Rev. Lett. 98, 243901 (2007).
[CrossRef] [PubMed]

Johnson, J. B.

J. B. Johnson and R. W. Christy, Phys. Rev. B 6, 4370 (1972).
[CrossRef]

Justice, B. J.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. PendryA. F. Starr, and D. R. Smith, Science 314, 977 (2006).
[CrossRef] [PubMed]

Kildishev, A. V.

Krishnamoorthy, A.

U. Levy, M. Abashin, K. Ikeda, A. Krishnamoorthy, J. Cunningham, and Y. Fainman, Phys. Rev. Lett. 98, 243901 (2007).
[CrossRef] [PubMed]

Kurylev, Y.

A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, Phys. Rev. Lett. 99, 183901 (2007).
[CrossRef] [PubMed]

Lassas, M.

A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, Phys. Rev. Lett. 99, 183901 (2007).
[CrossRef] [PubMed]

Lee, H.

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, Science 315, 1686 (2007).
[CrossRef] [PubMed]

Leonhardt, U.

U. Leonhardt, Science 312, 1777 (2006).
[CrossRef] [PubMed]

Levy, U.

U. Levy, M. Abashin, K. Ikeda, A. Krishnamoorthy, J. Cunningham, and Y. Fainman, Phys. Rev. Lett. 98, 243901 (2007).
[CrossRef] [PubMed]

Liu, Z.

Y. Xiong, Z. Liu, and X. Zhang, Appl. Phys. Lett. 94, 203108 (2009).
[CrossRef]

S. Han, Y. Xiong, D. Genov, Z. Liu, G. Bartal, and X. Zhang, Nano Lett. 8, 4243 (2008).
[CrossRef]

X. Zhang and Z. Liu, Nat. Mater. 7, 435 (2008).
[CrossRef] [PubMed]

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, Science 315, 1686 (2007).
[CrossRef] [PubMed]

Mendlovic, D.

Mock, J. J.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. PendryA. F. Starr, and D. R. Smith, Science 314, 977 (2006).
[CrossRef] [PubMed]

Narimanov, E. E.

Ozaktas, H. M.

Pendry, J. B.

M. Rahm, S. A. Cummer, D. Schurig, J. B. Pendry, and D. R. Smith, Phys. Rev. Lett. 100, 063903 (2008).
[CrossRef] [PubMed]

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. PendryA. F. Starr, and D. R. Smith, Science 314, 977 (2006).
[CrossRef] [PubMed]

J. B. Pendry, D. Schurig, and D. R. Smith, Science 312, 1780 (2006).
[CrossRef] [PubMed]

S. Anantha Ramakrishna, J. B. Pendry, M. C. K. Wiltshire, and W. J. Stewart, J. Mod. Opt. 50, 1419 (2003).

J. B. Pendry and S. Anantha Ramakrishna, J. Phys. Condens. Matter 15, 6345 (2003).
[CrossRef]

Rahm, M.

M. Rahm, S. A. Cummer, D. Schurig, J. B. Pendry, and D. R. Smith, Phys. Rev. Lett. 100, 063903 (2008).
[CrossRef] [PubMed]

Schurig, D.

M. Rahm, S. A. Cummer, D. Schurig, J. B. Pendry, and D. R. Smith, Phys. Rev. Lett. 100, 063903 (2008).
[CrossRef] [PubMed]

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. PendryA. F. Starr, and D. R. Smith, Science 314, 977 (2006).
[CrossRef] [PubMed]

J. B. Pendry, D. Schurig, and D. R. Smith, Science 312, 1780 (2006).
[CrossRef] [PubMed]

Shalaev, V. M.

Smith, D. R.

M. Rahm, S. A. Cummer, D. Schurig, J. B. Pendry, and D. R. Smith, Phys. Rev. Lett. 100, 063903 (2008).
[CrossRef] [PubMed]

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. PendryA. F. Starr, and D. R. Smith, Science 314, 977 (2006).
[CrossRef] [PubMed]

J. B. Pendry, D. Schurig, and D. R. Smith, Science 312, 1780 (2006).
[CrossRef] [PubMed]

Starr, A. F.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. PendryA. F. Starr, and D. R. Smith, Science 314, 977 (2006).
[CrossRef] [PubMed]

Stewart, W. J.

S. Anantha Ramakrishna, J. B. Pendry, M. C. K. Wiltshire, and W. J. Stewart, J. Mod. Opt. 50, 1419 (2003).

Sun, C.

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, Science 315, 1686 (2007).
[CrossRef] [PubMed]

Uhlmann, G.

A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, Phys. Rev. Lett. 99, 183901 (2007).
[CrossRef] [PubMed]

Wen, W.

H. Chen, B. Hou, S. Chen, X. Ao, W. Wen, and C. T. Chan, Phys. Rev. Lett. 102, 183903 (2009).
[CrossRef] [PubMed]

Wiltshire, M. C. K.

S. Anantha Ramakrishna, J. B. Pendry, M. C. K. Wiltshire, and W. J. Stewart, J. Mod. Opt. 50, 1419 (2003).

Xiong, Y.

Y. Xiong, Z. Liu, and X. Zhang, Appl. Phys. Lett. 94, 203108 (2009).
[CrossRef]

S. Han, Y. Xiong, D. Genov, Z. Liu, G. Bartal, and X. Zhang, Nano Lett. 8, 4243 (2008).
[CrossRef]

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, Science 315, 1686 (2007).
[CrossRef] [PubMed]

Zhang, X.

Y. Xiong, Z. Liu, and X. Zhang, Appl. Phys. Lett. 94, 203108 (2009).
[CrossRef]

S. Han, Y. Xiong, D. Genov, Z. Liu, G. Bartal, and X. Zhang, Nano Lett. 8, 4243 (2008).
[CrossRef]

X. Zhang and Z. Liu, Nat. Mater. 7, 435 (2008).
[CrossRef] [PubMed]

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, Science 315, 1686 (2007).
[CrossRef] [PubMed]

Appl. Phys. Lett. (1)

Y. Xiong, Z. Liu, and X. Zhang, Appl. Phys. Lett. 94, 203108 (2009).
[CrossRef]

J. Mod. Opt. (1)

S. Anantha Ramakrishna, J. B. Pendry, M. C. K. Wiltshire, and W. J. Stewart, J. Mod. Opt. 50, 1419 (2003).

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

J. Phys. Condens. Matter (1)

J. B. Pendry and S. Anantha Ramakrishna, J. Phys. Condens. Matter 15, 6345 (2003).
[CrossRef]

Nano Lett. (1)

S. Han, Y. Xiong, D. Genov, Z. Liu, G. Bartal, and X. Zhang, Nano Lett. 8, 4243 (2008).
[CrossRef]

Nat. Mater. (1)

X. Zhang and Z. Liu, Nat. Mater. 7, 435 (2008).
[CrossRef] [PubMed]

Opt. Lett. (2)

Phys. Rep. (1)

D. Bergmann, Phys. Rep. 43, 377 (1978).
[CrossRef]

Phys. Rev. B (1)

J. B. Johnson and R. W. Christy, Phys. Rev. B 6, 4370 (1972).
[CrossRef]

Phys. Rev. Lett. (4)

U. Levy, M. Abashin, K. Ikeda, A. Krishnamoorthy, J. Cunningham, and Y. Fainman, Phys. Rev. Lett. 98, 243901 (2007).
[CrossRef] [PubMed]

M. Rahm, S. A. Cummer, D. Schurig, J. B. Pendry, and D. R. Smith, Phys. Rev. Lett. 100, 063903 (2008).
[CrossRef] [PubMed]

H. Chen, B. Hou, S. Chen, X. Ao, W. Wen, and C. T. Chan, Phys. Rev. Lett. 102, 183903 (2009).
[CrossRef] [PubMed]

A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, Phys. Rev. Lett. 99, 183901 (2007).
[CrossRef] [PubMed]

Science (4)

J. B. Pendry, D. Schurig, and D. R. Smith, Science 312, 1780 (2006).
[CrossRef] [PubMed]

U. Leonhardt, Science 312, 1777 (2006).
[CrossRef] [PubMed]

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. PendryA. F. Starr, and D. R. Smith, Science 314, 977 (2006).
[CrossRef] [PubMed]

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, Science 315, 1686 (2007).
[CrossRef] [PubMed]

Other (1)

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

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Figures (4)

Fig. 1
Fig. 1

(a) The rectangular GRIN medium is geometrically compressed at the input and is transformed into (b) a cylindrical shell. (c) Metamaterial building block for the system shown in (b). (d) Expansion (shown by the red arrow) of the paraxial regime shown in reciprocal space from circular (blue circle, the original dielectric) to elliptical (red ellipse, the metal–dielectric-based metamaterial).

Fig. 2
Fig. 2

Detail of implementation using metal–dielectric multilayers to construct the Fourier lens with an effective medium [Eq. (5)]. The 66 yellow (light gray) concentric layers are Ag ( ε = 2.25 + 0.26 , while the gray-scale map is the permittivity profile of the GRIN medium (maximum index of 1.5 for white) interlacing with them.

Fig. 3
Fig. 3

(a) Optical Fourier transform (plotted amplitude) of a rectangular input function from the device at a wavelength of 360 nm . (b) The amplitude of the Fourier transform at the lens output is shown in red (thicker gray), while the blue curve (thinner black) corresponds to the ideal one.

Fig. 4
Fig. 4

Magnitude of the amplitude transfer function at output interface R 2 for the transformed lens (red smooth curve) and the GRIN medium before transformation (blue wavy curve). A flat response is expected for an ideal Fourier lens. Only data points that fall within the lens diameter are shown.

Equations (12)

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

n ( x ) = n 1 ( 1 π 2 x 2 8 f 2 ) ,
H ( x , f ) H ( x , 0 ) exp ( i κ x ) d x ,
κ = n 1 k 0 π x 2 f ,
x = R 2 θ ,
z = ( r 2 R 1 2 ) 2 R 2 ,
n r ( r , θ ) = r R 2 n 1 ( 1 π 2 R 2 2 θ 2 8 f 2 ) ,
n θ ( r , θ ) = R 2 r n 1 ( 1 π 2 R 2 2 θ 2 8 f 2 ) ,
κ = R 2 R 1 n 1 k 0 π 2 f R 2 θ .
n r = n 1 ( 1 1 η 1 2 η δ ) ( 1 2 η ) 1 2 ,
n θ = n 1 ( 1 1 η 1 2 η δ ) ( 1 2 η ) 1 2 .
η ( r ) = 1 2 r 2 2 R 2 2 ,
n 2 ( r , θ ) = n 1 ( 1 2 R 2 2 R 2 2 + r 2 π 2 r 2 θ 2 8 f 2 ) .

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