Abstract

We describe transformation design of optical elements which, in addition to image transfer, perform useful operations. For one class of operations, including translation, rotation, mirroring and inversion, an image can be generated that is ideal in the sense of the perfect lens (combining both near- and far-field components in a flat, unit transfer function, up to the limits imposed by material imperfection). We also describe elements that perform magnification, free from geometric aberrations, even while providing free-space working distance on both the input and output sides. These magnifying elements also operate in the near- and far-field, allowing them to transfer near field information into the far field, as with the hyper lens and other related devices, however in contrast to those devices, insertion loss can be much lower, due to the matching properties accessible with transformation design. The devices here described inherently require dispersive materials, thus chromatic aberration will be present, and the bandwidth limited.

© 2007 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |

  1. J. Pendry, D. Schurig, and D. Smith, "Controlling electromagnetic fields," Science 312, 1780-2 (2006).
    [CrossRef] [PubMed]
  2. D. Schurig, J. Pendry, and D. Smith, "Calculation of material properties and ray tracing in transformation media," Opt. Express 14, 9794-9804 (2006).
    [CrossRef] [PubMed]
  3. G. Milton, M. Briane, and J. Willis, "On cloaking for elasticity and physical equations with a transformation invariant form," New J. Phys. 8, 248 (2006).
    [CrossRef]
  4. S. Cummer and D. Schurig, "One path to acoustic cloaking," New J. Phys. 9, 45 (2007).
    [CrossRef]
  5. B. Wood and J. Pendry, "Metamaterials at zero frequency," J. Phys., Condens. Matter. 19, 076208 (2007).
    [CrossRef]
  6. W. Cai, U. Chettiar, A. Kildishev, and V. Shalaev, "Optical cloaking with metamaterials," 1, 224-227 Nature Photonics (2007).
    [CrossRef]
  7. M. Silveirinha, A. Alu, and N. Engheta, "Parallel-plate metamaterials for cloaking structures," Phys. Rev. E, Stat. Nonlinear Soft Matter Phys. 75, 36603 (2007).
    [CrossRef]
  8. F. Teixeira, "Closed-form metamaterial blueprints for electromagnetic masking of arbitrarily shaped convex PEC objects," IEEE Antennas Wirel. Propag. Lett. 6, 163-4 (2007).
    [CrossRef]
  9. F. Zolla, S. Guenneau, A. Nicolet, and J. Pendry, "Electromagnetic analysis of cylindrical invisibility cloaks and the mirage effect," Opt. Lett. 32, 1069-71 (2007).
    [CrossRef] [PubMed]
  10. D. Schurig, J. Mock, B. Justice, S. Cummer, J. Pendry, A. Starr, and D. Smith, "Metamaterial electromagnetic cloak at microwave frequencies," Science 314, 977-80 (2006).
    [CrossRef] [PubMed]
  11. J. Pendry, "Negative refraction makes a perfect lens," Phys. Rev. Lett. 85, 3966-9 (2000).
    [CrossRef] [PubMed]
  12. J. Pendry and S. Ramakrishna, "Near-field lenses in two dimensions," J. Phys., Condens. Matter. 14, 8463-79 (2002).
    [CrossRef]
  13. J. Pendry, "Perfect cylindrical lenses," Opt. Express 11, 755-760 (2003).
    [CrossRef] [PubMed]
  14. Z. Jacob, L. Alekseyev, and E. Narimanov, "Optical hyperlens: far-field imaging beyond the diffraction limit," Opt. Express 14,8247-56 (2006).
    [CrossRef] [PubMed]
  15. A. Salandrino and N. Engheta, "Far-field subdiffraction optical microscopy using metamaterial crystals: theory and simulations," Phys. Rev., B, Condens, Matter Mater. Phys. 74, 75103 (2006).
    [CrossRef]
  16. D. Smith and D. Schurig, "Electromagnetic wave propagation in media with indefinite permittivity and permeability tensors," Phys. Rev. Lett. 90, 077405 (2003).
    [CrossRef] [PubMed]
  17. Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, "Far-field optical hyperlens magnifying sub-diffraction-limited objects," Science 315, 1686 (2007).
    [CrossRef] [PubMed]
  18. I. Smolyaninov, Y.-J. Hung, and C. Davis, "Magnifying superlens in the visible frequency range," Science 315, 1699-701 (2007).
    [CrossRef] [PubMed]
  19. G. Shvets, S. Trendafilov, J. Pendry, and A. Sarychev, "Guiding, focusing, and sensing on the subwavelength scale using metallic wire arrays," Phys. Rev. Lett. 99, 053903 (2007).
    [CrossRef] [PubMed]
  20. D. Schurig and D. Smith, "Sub-diffraction imaging with compensating bilayers," New J. Phys. 7, 162 (2005).
    [CrossRef]
  21. V. Shalaev, "Optical negative-index metamaterials," 1, 41-48 Nature Photonics (2007).
    [CrossRef]
  22. C. Soukoulis, S. Linden, and M. Wegener, "Negative Refractive Index at Optical Wavelengths," Science 315, 47-9 (2007).
    [CrossRef] [PubMed]
  23. H. Lezec, J. Dionne, and H. Atwater, "Negative refraction at visible frequencies," Science 316, 430-2 (200).
    [PubMed]
  24. U. Leonhardt and T. Philbin, "General relativity in Electrical Engineering," New J. Phys. 8, 247 (2006).
    [CrossRef]
  25. V. Mahajan, Optical imaging and aberrations (SPIE Optical Engineering Press, 1998).
    [CrossRef]
  26. D. Schurig and D. Smith, "Negative index lens aberrations," Phys. Rev. E, Stat. Nonlinear Soft Matter Phys. 70, 65601 (2004).
    [CrossRef]

2007

S. Cummer and D. Schurig, "One path to acoustic cloaking," New J. Phys. 9, 45 (2007).
[CrossRef]

B. Wood and J. Pendry, "Metamaterials at zero frequency," J. Phys., Condens. Matter. 19, 076208 (2007).
[CrossRef]

M. Silveirinha, A. Alu, and N. Engheta, "Parallel-plate metamaterials for cloaking structures," Phys. Rev. E, Stat. Nonlinear Soft Matter Phys. 75, 36603 (2007).
[CrossRef]

F. Teixeira, "Closed-form metamaterial blueprints for electromagnetic masking of arbitrarily shaped convex PEC objects," IEEE Antennas Wirel. Propag. Lett. 6, 163-4 (2007).
[CrossRef]

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

I. Smolyaninov, Y.-J. Hung, and C. Davis, "Magnifying superlens in the visible frequency range," Science 315, 1699-701 (2007).
[CrossRef] [PubMed]

G. Shvets, S. Trendafilov, J. Pendry, and A. Sarychev, "Guiding, focusing, and sensing on the subwavelength scale using metallic wire arrays," Phys. Rev. Lett. 99, 053903 (2007).
[CrossRef] [PubMed]

C. Soukoulis, S. Linden, and M. Wegener, "Negative Refractive Index at Optical Wavelengths," Science 315, 47-9 (2007).
[CrossRef] [PubMed]

F. Zolla, S. Guenneau, A. Nicolet, and J. Pendry, "Electromagnetic analysis of cylindrical invisibility cloaks and the mirage effect," Opt. Lett. 32, 1069-71 (2007).
[CrossRef] [PubMed]

2006

Z. Jacob, L. Alekseyev, and E. Narimanov, "Optical hyperlens: far-field imaging beyond the diffraction limit," Opt. Express 14,8247-56 (2006).
[CrossRef] [PubMed]

D. Schurig, J. Pendry, and D. Smith, "Calculation of material properties and ray tracing in transformation media," Opt. Express 14, 9794-9804 (2006).
[CrossRef] [PubMed]

U. Leonhardt and T. Philbin, "General relativity in Electrical Engineering," New J. Phys. 8, 247 (2006).
[CrossRef]

J. Pendry, D. Schurig, and D. Smith, "Controlling electromagnetic fields," Science 312, 1780-2 (2006).
[CrossRef] [PubMed]

D. Schurig, J. Mock, B. Justice, S. Cummer, J. Pendry, A. Starr, and D. Smith, "Metamaterial electromagnetic cloak at microwave frequencies," Science 314, 977-80 (2006).
[CrossRef] [PubMed]

G. Milton, M. Briane, and J. Willis, "On cloaking for elasticity and physical equations with a transformation invariant form," New J. Phys. 8, 248 (2006).
[CrossRef]

A. Salandrino and N. Engheta, "Far-field subdiffraction optical microscopy using metamaterial crystals: theory and simulations," Phys. Rev., B, Condens, Matter Mater. Phys. 74, 75103 (2006).
[CrossRef]

2005

D. Schurig and D. Smith, "Sub-diffraction imaging with compensating bilayers," New J. Phys. 7, 162 (2005).
[CrossRef]

2004

D. Schurig and D. Smith, "Negative index lens aberrations," Phys. Rev. E, Stat. Nonlinear Soft Matter Phys. 70, 65601 (2004).
[CrossRef]

2003

J. Pendry, "Perfect cylindrical lenses," Opt. Express 11, 755-760 (2003).
[CrossRef] [PubMed]

D. Smith and D. Schurig, "Electromagnetic wave propagation in media with indefinite permittivity and permeability tensors," Phys. Rev. Lett. 90, 077405 (2003).
[CrossRef] [PubMed]

2002

J. Pendry and S. Ramakrishna, "Near-field lenses in two dimensions," J. Phys., Condens. Matter. 14, 8463-79 (2002).
[CrossRef]

2000

J. Pendry, "Negative refraction makes a perfect lens," Phys. Rev. Lett. 85, 3966-9 (2000).
[CrossRef] [PubMed]

IEEE Antennas Wirel. Propag. Lett.

F. Teixeira, "Closed-form metamaterial blueprints for electromagnetic masking of arbitrarily shaped convex PEC objects," IEEE Antennas Wirel. Propag. Lett. 6, 163-4 (2007).
[CrossRef]

J. Phys., Condens. Matter.

B. Wood and J. Pendry, "Metamaterials at zero frequency," J. Phys., Condens. Matter. 19, 076208 (2007).
[CrossRef]

J. Pendry and S. Ramakrishna, "Near-field lenses in two dimensions," J. Phys., Condens. Matter. 14, 8463-79 (2002).
[CrossRef]

Matter Mater. Phys.

A. Salandrino and N. Engheta, "Far-field subdiffraction optical microscopy using metamaterial crystals: theory and simulations," Phys. Rev., B, Condens, Matter Mater. Phys. 74, 75103 (2006).
[CrossRef]

New J. Phys.

D. Schurig and D. Smith, "Sub-diffraction imaging with compensating bilayers," New J. Phys. 7, 162 (2005).
[CrossRef]

G. Milton, M. Briane, and J. Willis, "On cloaking for elasticity and physical equations with a transformation invariant form," New J. Phys. 8, 248 (2006).
[CrossRef]

S. Cummer and D. Schurig, "One path to acoustic cloaking," New J. Phys. 9, 45 (2007).
[CrossRef]

U. Leonhardt and T. Philbin, "General relativity in Electrical Engineering," New J. Phys. 8, 247 (2006).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev. Lett.

G. Shvets, S. Trendafilov, J. Pendry, and A. Sarychev, "Guiding, focusing, and sensing on the subwavelength scale using metallic wire arrays," Phys. Rev. Lett. 99, 053903 (2007).
[CrossRef] [PubMed]

J. Pendry, "Negative refraction makes a perfect lens," Phys. Rev. Lett. 85, 3966-9 (2000).
[CrossRef] [PubMed]

D. Smith and D. Schurig, "Electromagnetic wave propagation in media with indefinite permittivity and permeability tensors," Phys. Rev. Lett. 90, 077405 (2003).
[CrossRef] [PubMed]

Science

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

I. Smolyaninov, Y.-J. Hung, and C. Davis, "Magnifying superlens in the visible frequency range," Science 315, 1699-701 (2007).
[CrossRef] [PubMed]

C. Soukoulis, S. Linden, and M. Wegener, "Negative Refractive Index at Optical Wavelengths," Science 315, 47-9 (2007).
[CrossRef] [PubMed]

D. Schurig, J. Mock, B. Justice, S. Cummer, J. Pendry, A. Starr, and D. Smith, "Metamaterial electromagnetic cloak at microwave frequencies," Science 314, 977-80 (2006).
[CrossRef] [PubMed]

J. Pendry, D. Schurig, and D. Smith, "Controlling electromagnetic fields," Science 312, 1780-2 (2006).
[CrossRef] [PubMed]

Stat. Nonlinear Soft Matter Phys.

D. Schurig and D. Smith, "Negative index lens aberrations," Phys. Rev. E, Stat. Nonlinear Soft Matter Phys. 70, 65601 (2004).
[CrossRef]

M. Silveirinha, A. Alu, and N. Engheta, "Parallel-plate metamaterials for cloaking structures," Phys. Rev. E, Stat. Nonlinear Soft Matter Phys. 75, 36603 (2007).
[CrossRef]

Other

W. Cai, U. Chettiar, A. Kildishev, and V. Shalaev, "Optical cloaking with metamaterials," 1, 224-227 Nature Photonics (2007).
[CrossRef]

H. Lezec, J. Dionne, and H. Atwater, "Negative refraction at visible frequencies," Science 316, 430-2 (200).
[PubMed]

V. Shalaev, "Optical negative-index metamaterials," 1, 41-48 Nature Photonics (2007).
[CrossRef]

V. Mahajan, Optical imaging and aberrations (SPIE Optical Engineering Press, 1998).
[CrossRef]

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 (5)

Fig. 1.
Fig. 1.

Coordinate transformations from the reference space (A) to a space corresponding to a perfect lens (B), an image translator (C) and a magnifier (D). The folded representation of the z'-axis in the reference space, and the color coding of the vertical coordinate lines display the multi-valued mapping. The gray shading indicates the region where the coordinate transformation differs from “flat” Cartesian space, or equivalently, where the material properties differ from free space.

Fig. 2.
Fig. 2.

z-coordinate transformation corresponding to a perfect lens of thickness d (solid blue), and modified to shorten the front focal length from d/2 to f (dashed blue). Transverse function used in coordinate transformation corresponding to an image rotating element (red) and a magnifying element (green). The gray shaded region indicates where these functions correspond to material properties that differ from free space.

Fig. 3.
Fig. 3.

Ray diagram for the magnifying element. Both the internal and external images (large red arrows) are magnified relative to the object (smaller red arrow). All the magnification occurs in the left half of the element. A non-aberrating refraction occurs at the element center where the internal image is formed.

Fig. 4.
Fig. 4.

Magnitude of the spatial transfer function for the magnifying element, with magnification, M, of: one (red), three (green) and ten (blue). The dependent variable is the transverse component of the wave vector at the input. (The transverse component of the wave vector at the output is reduced by the magnification factor.)

Fig. 5.
Fig. 5.

Material properties of the magnifying element. The magnitudes of the three principle values of the permittivity and permeability are represented by the intensities of the three color channels: red, green and blue (A) with M = 3 and F = 1. The scale is logarithmic as shown in the inset. The principle value that assumes large magnitudes, n 2, is most extreme at the front surface at the aperture boundary (orange dots). (B) shows the value of n 2 for various f-numbers, F, and magnifications, M.

Equations (25)

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

x i ' = f i ' ( x i )
ε i ' j ' = μ i ' j ' = det ( Λ i i ' ) 1 Λ i i ' Λ i j ' δ ij
Λ j i ' = x i ' x j '
x ' = x
y ' = y
z ' = { z d z ' < d 2 z d 2 < z ' < d 2 z + d d 2 < z '
( Λ j i ' ) = ( 1 0 0 0 1 0 0 0 1 )
( ε i ' j ' ) = ( μ i ' j ' ) = ( 1 0 0 0 1 0 0 0 1 )
x ' = x cos θ ( z ' ) y sin θ ( z ' )
y ' = x sin θ ( z ' ) y cos θ ( z ' )
x ' = m ( z ' ) x
y ' = m ( z ' ) y
τ = 2 M k z M k z + k z +
M k z = k 0 2 k x 2
k z + = k 0 2 k x 2 M 2
f min F M d
m ( z ' ) = M + M 1 d 2 z
z = f d 2 z '
x ' = ( M M 1 f z ) x
y ' = ( M M 1 f z ) y
z ' = d 2 f z
( Λ j i ' ) = M ( 1 + αz ' 0 βx ' 1 + αz' 0 1 + αz ' βy ' 1 + αz' 0 0 β α )
α = M 1 M 1 d 2
β = 1 M M 1 fM
n 1 = f d 2

Metrics