Abstract

The multilayer hyperbolic metamaterials are known to be capable of imaging with sub-wavelength resolution. In this work performance of these “hyperbolic lenses” is analyzed in depth by employing commonly used transfer matrix method as well as the eigen-mode approach, the latter offering a clear physical insight into the operation of hyperbolic imagers and revealing their fundamental limitations. The resolution of multilayer structures is shown to decrease with the number of layers not only due to increased loss but also because of the severe suppression of large spatial frequencies caused by the cancellation between symmetric and antisymmetric eigen-modes. Additionally, the resolution is strongly affected by the granularity and fill ratio. In the end, hyperbolic metamaterials can create an image with subwavelength resolution only at very close distance to the object and hence limiting their utility.

© 2017 Optical Society of America

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References

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  1. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85(18), 3966 (2000).
    [Crossref] [PubMed]
  2. N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308(5721), 534–537 (2005).
    [Crossref] [PubMed]
  3. D. O. Melville and R. J. Blaikie, “Super-resolution imaging through a planar silver layer,” Opt. Express 13(6), 2127–2134 (2005).
    [Crossref] [PubMed]
  4. Z. Liu, S. Durant, H. Lee, Y. Pikus, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical superlens,” Nano.Lett. 7, 403–408 (2007).
    [Crossref]
  5. K. Webb and M. Yang, “Subwavelength imaging with a multilayer silver film structure,” Opt. Lett. 31(14), 2130–2131 (2006).
    [Crossref] [PubMed]
  6. P. Chaturvedi, W. Wu, V. Logeeswaran, Z. Yu, M. S. Islam, S. Wang, R. S. Williams, and N. X. Fang, “A smooth optical superlens,” Appl. Phys. Lett. 96(4), 043102 (2010).
    [Crossref]
  7. Z. Jacob, L. V. Alekseyev, and E. Narimanov, “Optical hyper-lens: far-field imaging beyond the diffraction limit,” Opt. Express 14(18), 8247–8256 (2006).
    [Crossref] [PubMed]
  8. 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]
  9. I. I. Smolyaninov, Y.-J. Hung, and C. C. Davis, “Magnifying superlens in the visible frequency range,” Science 315(5819), 1699–1701 (2007).
    [Crossref] [PubMed]
  10. J. Rho, Z. Ye, Y. Xiong, X. Yin, Z. Liu, H. Choi, G. Bartal, and X. Zhang, “Spherical hyperlens for two-dimensional sub-diffractional imaging at visible frequencies,” Nat. Commun. 1, 143 (2010).
    [Crossref]
  11. D. Lu and Z. Liu, “Hyperlenses and metalenses for far-field super-resolution imaging,” Nat. Commun. 3, 1205 (2012).
    [Crossref] [PubMed]
  12. C. Lv, W. Li, X. Jiang, and J. Cao, “Far-field super-resolution imaging with a planar hyperbolic metamaterial lens,” Europhys. Lett. 105(2), 28003 (2014).
    [Crossref]
  13. M. Iwanaga, “Hyperlens-array-implemented optical microscopy,” Appl. Phys. Lett. 105, 053112 (2014).
    [Crossref]
  14. M. Kim and J. Rho, “Metamaterials and imaging,” Nano Convergence 2, 22 (2015).
    [Crossref] [PubMed]
  15. M. Byun, D. Lee, M. Kim, Y. Kim, K. Kim, J. G. Ok, J. Rho, and H. Lee, “Demonstration of nanoimprinted hyperlens array for high-throughput sub-diffraction imaging,” Sci. Rep. 7, 46314 (2017).
    [Crossref] [PubMed]
  16. V. A. Podolskiy and E. E. Narimanov, “Near-sighted super-lens,” Opt. Lett. 30(1), 75–77 (2005).
    [Crossref] [PubMed]
  17. D. R. Smith, D. Schurig, M. Rosenbluth, S. Schultz, S. A. Ramakrishna, and J. B. Pendry, “Limitations on subdiffraction imaging with a negative refractive index slab,” Appl. Phys. Lett. 82(10), 1506–1508 (2003).
    [Crossref]
  18. R. Merlin, “Analytical solution of the almost-perfect-lens problem,” Appl. Phys. Lett. 84(8), 1290–1292 (2004).
    [Crossref]
  19. B. Zhang and J. B. Khurgin, “Eigen mode approach to the sub-wavelength imaging with surface plasmon polaritons,” Appl. Phys. Lett. 98(26), 263102 (2011).
    [Crossref]
  20. T. Li and J. B. Khurgin, “Hyperbolic metamaterials: beyond the effective medium theory,” Optica 3(12), 1388–1396 (2016).
    [Crossref]
  21. J. B. Khurgin and G. Sun, “Scaling of losses with size and wavelength in nanoplasmonics and metamaterials,” Appl. Phys. Lett. 99(21), 211106 (2011).
    [Crossref]
  22. R. F. Oulton, D. F. Pile, Y. Liu, and X. Zhang, “Scattering of surface plasmon polaritons at abrupt surface interfaces: implications for nanoscale cavities,” Phys. Rev. B 98(26), 035408 (2007).
    [Crossref]
  23. I. Malitson and M. Dodge, “Refractive-index and birefringence of synthetic sapphire,” J. Opt. Soc. Am. 62, 1405A (1972).
  24. P. B. Johnson and R.-W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370 (1972).
    [Crossref]
  25. I. Avrutsky, I. Salakhutdinov, J. Elser, and V. Podolskiy, “Highly confined optical modes in nanoscale metal-dielectric multilayers,” Phys. Rev. B 75(24), 241402 (2007).
    [Crossref]
  26. P. Berini, “Long-range surface plasmon polaritons,” Adv. Opt. Photon. 1(3), 484–588 (2009).
    [Crossref]
  27. D. Sarid, “Long-range surface-plasma waves on very thin metal films,” Phys. Rev. Lett. 47(26), 1927 (1981).
    [Crossref]
  28. L. Douillard, F. Charra, Z. Korczak, R. Bachelot, S. Kostcheev, G. Lerondel, P.-M. Adam, and P. Royer, “Short range plasmon resonators probed by photoemission electron microscopy,” Nat. Mater. 8(3), 935–940 (2008).
  29. A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor meta-materials,” Nat. Mater. 6(12), 946–950 (2007).
    [Crossref] [PubMed]
  30. Z. Zhang and S. Satpathy, “Electromagnetic wave propagation in periodic structures: Bloch wave solution of Maxwell’s equations,” Phys. Rev. Lett. 65(21), 2650 (1990).
    [Crossref] [PubMed]

2017 (1)

M. Byun, D. Lee, M. Kim, Y. Kim, K. Kim, J. G. Ok, J. Rho, and H. Lee, “Demonstration of nanoimprinted hyperlens array for high-throughput sub-diffraction imaging,” Sci. Rep. 7, 46314 (2017).
[Crossref] [PubMed]

2016 (1)

2015 (1)

M. Kim and J. Rho, “Metamaterials and imaging,” Nano Convergence 2, 22 (2015).
[Crossref] [PubMed]

2014 (2)

C. Lv, W. Li, X. Jiang, and J. Cao, “Far-field super-resolution imaging with a planar hyperbolic metamaterial lens,” Europhys. Lett. 105(2), 28003 (2014).
[Crossref]

M. Iwanaga, “Hyperlens-array-implemented optical microscopy,” Appl. Phys. Lett. 105, 053112 (2014).
[Crossref]

2012 (1)

D. Lu and Z. Liu, “Hyperlenses and metalenses for far-field super-resolution imaging,” Nat. Commun. 3, 1205 (2012).
[Crossref] [PubMed]

2011 (2)

B. Zhang and J. B. Khurgin, “Eigen mode approach to the sub-wavelength imaging with surface plasmon polaritons,” Appl. Phys. Lett. 98(26), 263102 (2011).
[Crossref]

J. B. Khurgin and G. Sun, “Scaling of losses with size and wavelength in nanoplasmonics and metamaterials,” Appl. Phys. Lett. 99(21), 211106 (2011).
[Crossref]

2010 (2)

J. Rho, Z. Ye, Y. Xiong, X. Yin, Z. Liu, H. Choi, G. Bartal, and X. Zhang, “Spherical hyperlens for two-dimensional sub-diffractional imaging at visible frequencies,” Nat. Commun. 1, 143 (2010).
[Crossref]

P. Chaturvedi, W. Wu, V. Logeeswaran, Z. Yu, M. S. Islam, S. Wang, R. S. Williams, and N. X. Fang, “A smooth optical superlens,” Appl. Phys. Lett. 96(4), 043102 (2010).
[Crossref]

2009 (1)

2008 (1)

L. Douillard, F. Charra, Z. Korczak, R. Bachelot, S. Kostcheev, G. Lerondel, P.-M. Adam, and P. Royer, “Short range plasmon resonators probed by photoemission electron microscopy,” Nat. Mater. 8(3), 935–940 (2008).

2007 (6)

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor meta-materials,” Nat. Mater. 6(12), 946–950 (2007).
[Crossref] [PubMed]

I. Avrutsky, I. Salakhutdinov, J. Elser, and V. Podolskiy, “Highly confined optical modes in nanoscale metal-dielectric multilayers,” Phys. Rev. B 75(24), 241402 (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. I. Smolyaninov, Y.-J. Hung, and C. C. Davis, “Magnifying superlens in the visible frequency range,” Science 315(5819), 1699–1701 (2007).
[Crossref] [PubMed]

R. F. Oulton, D. F. Pile, Y. Liu, and X. Zhang, “Scattering of surface plasmon polaritons at abrupt surface interfaces: implications for nanoscale cavities,” Phys. Rev. B 98(26), 035408 (2007).
[Crossref]

Z. Liu, S. Durant, H. Lee, Y. Pikus, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical superlens,” Nano.Lett. 7, 403–408 (2007).
[Crossref]

2006 (2)

2005 (3)

2004 (1)

R. Merlin, “Analytical solution of the almost-perfect-lens problem,” Appl. Phys. Lett. 84(8), 1290–1292 (2004).
[Crossref]

2003 (1)

D. R. Smith, D. Schurig, M. Rosenbluth, S. Schultz, S. A. Ramakrishna, and J. B. Pendry, “Limitations on subdiffraction imaging with a negative refractive index slab,” Appl. Phys. Lett. 82(10), 1506–1508 (2003).
[Crossref]

2000 (1)

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

1990 (1)

Z. Zhang and S. Satpathy, “Electromagnetic wave propagation in periodic structures: Bloch wave solution of Maxwell’s equations,” Phys. Rev. Lett. 65(21), 2650 (1990).
[Crossref] [PubMed]

1981 (1)

D. Sarid, “Long-range surface-plasma waves on very thin metal films,” Phys. Rev. Lett. 47(26), 1927 (1981).
[Crossref]

1972 (2)

I. Malitson and M. Dodge, “Refractive-index and birefringence of synthetic sapphire,” J. Opt. Soc. Am. 62, 1405A (1972).

P. B. Johnson and R.-W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370 (1972).
[Crossref]

Adam, P.-M.

L. Douillard, F. Charra, Z. Korczak, R. Bachelot, S. Kostcheev, G. Lerondel, P.-M. Adam, and P. Royer, “Short range plasmon resonators probed by photoemission electron microscopy,” Nat. Mater. 8(3), 935–940 (2008).

Alekseyev, L.

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor meta-materials,” Nat. Mater. 6(12), 946–950 (2007).
[Crossref] [PubMed]

Alekseyev, L. V.

Avrutsky, I.

I. Avrutsky, I. Salakhutdinov, J. Elser, and V. Podolskiy, “Highly confined optical modes in nanoscale metal-dielectric multilayers,” Phys. Rev. B 75(24), 241402 (2007).
[Crossref]

Bachelot, R.

L. Douillard, F. Charra, Z. Korczak, R. Bachelot, S. Kostcheev, G. Lerondel, P.-M. Adam, and P. Royer, “Short range plasmon resonators probed by photoemission electron microscopy,” Nat. Mater. 8(3), 935–940 (2008).

Bartal, G.

J. Rho, Z. Ye, Y. Xiong, X. Yin, Z. Liu, H. Choi, G. Bartal, and X. Zhang, “Spherical hyperlens for two-dimensional sub-diffractional imaging at visible frequencies,” Nat. Commun. 1, 143 (2010).
[Crossref]

Berini, P.

Blaikie, R. J.

Byun, M.

M. Byun, D. Lee, M. Kim, Y. Kim, K. Kim, J. G. Ok, J. Rho, and H. Lee, “Demonstration of nanoimprinted hyperlens array for high-throughput sub-diffraction imaging,” Sci. Rep. 7, 46314 (2017).
[Crossref] [PubMed]

Cao, J.

C. Lv, W. Li, X. Jiang, and J. Cao, “Far-field super-resolution imaging with a planar hyperbolic metamaterial lens,” Europhys. Lett. 105(2), 28003 (2014).
[Crossref]

Charra, F.

L. Douillard, F. Charra, Z. Korczak, R. Bachelot, S. Kostcheev, G. Lerondel, P.-M. Adam, and P. Royer, “Short range plasmon resonators probed by photoemission electron microscopy,” Nat. Mater. 8(3), 935–940 (2008).

Chaturvedi, P.

P. Chaturvedi, W. Wu, V. Logeeswaran, Z. Yu, M. S. Islam, S. Wang, R. S. Williams, and N. X. Fang, “A smooth optical superlens,” Appl. Phys. Lett. 96(4), 043102 (2010).
[Crossref]

Choi, H.

J. Rho, Z. Ye, Y. Xiong, X. Yin, Z. Liu, H. Choi, G. Bartal, and X. Zhang, “Spherical hyperlens for two-dimensional sub-diffractional imaging at visible frequencies,” Nat. Commun. 1, 143 (2010).
[Crossref]

Christy, R.-W.

P. B. Johnson and R.-W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370 (1972).
[Crossref]

Davis, C. C.

I. I. Smolyaninov, Y.-J. Hung, and C. C. Davis, “Magnifying superlens in the visible frequency range,” Science 315(5819), 1699–1701 (2007).
[Crossref] [PubMed]

Dodge, M.

I. Malitson and M. Dodge, “Refractive-index and birefringence of synthetic sapphire,” J. Opt. Soc. Am. 62, 1405A (1972).

Douillard, L.

L. Douillard, F. Charra, Z. Korczak, R. Bachelot, S. Kostcheev, G. Lerondel, P.-M. Adam, and P. Royer, “Short range plasmon resonators probed by photoemission electron microscopy,” Nat. Mater. 8(3), 935–940 (2008).

Durant, S.

Z. Liu, S. Durant, H. Lee, Y. Pikus, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical superlens,” Nano.Lett. 7, 403–408 (2007).
[Crossref]

Elser, J.

I. Avrutsky, I. Salakhutdinov, J. Elser, and V. Podolskiy, “Highly confined optical modes in nanoscale metal-dielectric multilayers,” Phys. Rev. B 75(24), 241402 (2007).
[Crossref]

Fang, N.

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308(5721), 534–537 (2005).
[Crossref] [PubMed]

Fang, N. X.

P. Chaturvedi, W. Wu, V. Logeeswaran, Z. Yu, M. S. Islam, S. Wang, R. S. Williams, and N. X. Fang, “A smooth optical superlens,” Appl. Phys. Lett. 96(4), 043102 (2010).
[Crossref]

Franz, K. J.

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor meta-materials,” Nat. Mater. 6(12), 946–950 (2007).
[Crossref] [PubMed]

Gmachl, C.

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor meta-materials,” Nat. Mater. 6(12), 946–950 (2007).
[Crossref] [PubMed]

Hoffman, A. J.

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor meta-materials,” Nat. Mater. 6(12), 946–950 (2007).
[Crossref] [PubMed]

Howard, S. S.

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor meta-materials,” Nat. Mater. 6(12), 946–950 (2007).
[Crossref] [PubMed]

Hung, Y.-J.

I. I. Smolyaninov, Y.-J. Hung, and C. C. Davis, “Magnifying superlens in the visible frequency range,” Science 315(5819), 1699–1701 (2007).
[Crossref] [PubMed]

Islam, M. S.

P. Chaturvedi, W. Wu, V. Logeeswaran, Z. Yu, M. S. Islam, S. Wang, R. S. Williams, and N. X. Fang, “A smooth optical superlens,” Appl. Phys. Lett. 96(4), 043102 (2010).
[Crossref]

Iwanaga, M.

M. Iwanaga, “Hyperlens-array-implemented optical microscopy,” Appl. Phys. Lett. 105, 053112 (2014).
[Crossref]

Jacob, Z.

Jiang, X.

C. Lv, W. Li, X. Jiang, and J. Cao, “Far-field super-resolution imaging with a planar hyperbolic metamaterial lens,” Europhys. Lett. 105(2), 28003 (2014).
[Crossref]

Johnson, P. B.

P. B. Johnson and R.-W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370 (1972).
[Crossref]

Khurgin, J. B.

T. Li and J. B. Khurgin, “Hyperbolic metamaterials: beyond the effective medium theory,” Optica 3(12), 1388–1396 (2016).
[Crossref]

B. Zhang and J. B. Khurgin, “Eigen mode approach to the sub-wavelength imaging with surface plasmon polaritons,” Appl. Phys. Lett. 98(26), 263102 (2011).
[Crossref]

J. B. Khurgin and G. Sun, “Scaling of losses with size and wavelength in nanoplasmonics and metamaterials,” Appl. Phys. Lett. 99(21), 211106 (2011).
[Crossref]

Kim, K.

M. Byun, D. Lee, M. Kim, Y. Kim, K. Kim, J. G. Ok, J. Rho, and H. Lee, “Demonstration of nanoimprinted hyperlens array for high-throughput sub-diffraction imaging,” Sci. Rep. 7, 46314 (2017).
[Crossref] [PubMed]

Kim, M.

M. Byun, D. Lee, M. Kim, Y. Kim, K. Kim, J. G. Ok, J. Rho, and H. Lee, “Demonstration of nanoimprinted hyperlens array for high-throughput sub-diffraction imaging,” Sci. Rep. 7, 46314 (2017).
[Crossref] [PubMed]

M. Kim and J. Rho, “Metamaterials and imaging,” Nano Convergence 2, 22 (2015).
[Crossref] [PubMed]

Kim, Y.

M. Byun, D. Lee, M. Kim, Y. Kim, K. Kim, J. G. Ok, J. Rho, and H. Lee, “Demonstration of nanoimprinted hyperlens array for high-throughput sub-diffraction imaging,” Sci. Rep. 7, 46314 (2017).
[Crossref] [PubMed]

Korczak, Z.

L. Douillard, F. Charra, Z. Korczak, R. Bachelot, S. Kostcheev, G. Lerondel, P.-M. Adam, and P. Royer, “Short range plasmon resonators probed by photoemission electron microscopy,” Nat. Mater. 8(3), 935–940 (2008).

Kostcheev, S.

L. Douillard, F. Charra, Z. Korczak, R. Bachelot, S. Kostcheev, G. Lerondel, P.-M. Adam, and P. Royer, “Short range plasmon resonators probed by photoemission electron microscopy,” Nat. Mater. 8(3), 935–940 (2008).

Lee, D.

M. Byun, D. Lee, M. Kim, Y. Kim, K. Kim, J. G. Ok, J. Rho, and H. Lee, “Demonstration of nanoimprinted hyperlens array for high-throughput sub-diffraction imaging,” Sci. Rep. 7, 46314 (2017).
[Crossref] [PubMed]

Lee, H.

M. Byun, D. Lee, M. Kim, Y. Kim, K. Kim, J. G. Ok, J. Rho, and H. Lee, “Demonstration of nanoimprinted hyperlens array for high-throughput sub-diffraction imaging,” Sci. Rep. 7, 46314 (2017).
[Crossref] [PubMed]

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]

Z. Liu, S. Durant, H. Lee, Y. Pikus, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical superlens,” Nano.Lett. 7, 403–408 (2007).
[Crossref]

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308(5721), 534–537 (2005).
[Crossref] [PubMed]

Lerondel, G.

L. Douillard, F. Charra, Z. Korczak, R. Bachelot, S. Kostcheev, G. Lerondel, P.-M. Adam, and P. Royer, “Short range plasmon resonators probed by photoemission electron microscopy,” Nat. Mater. 8(3), 935–940 (2008).

Li, T.

Li, W.

C. Lv, W. Li, X. Jiang, and J. Cao, “Far-field super-resolution imaging with a planar hyperbolic metamaterial lens,” Europhys. Lett. 105(2), 28003 (2014).
[Crossref]

Liu, Y.

R. F. Oulton, D. F. Pile, Y. Liu, and X. Zhang, “Scattering of surface plasmon polaritons at abrupt surface interfaces: implications for nanoscale cavities,” Phys. Rev. B 98(26), 035408 (2007).
[Crossref]

Liu, Z.

D. Lu and Z. Liu, “Hyperlenses and metalenses for far-field super-resolution imaging,” Nat. Commun. 3, 1205 (2012).
[Crossref] [PubMed]

J. Rho, Z. Ye, Y. Xiong, X. Yin, Z. Liu, H. Choi, G. Bartal, and X. Zhang, “Spherical hyperlens for two-dimensional sub-diffractional imaging at visible frequencies,” Nat. Commun. 1, 143 (2010).
[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]

Z. Liu, S. Durant, H. Lee, Y. Pikus, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical superlens,” Nano.Lett. 7, 403–408 (2007).
[Crossref]

Logeeswaran, V.

P. Chaturvedi, W. Wu, V. Logeeswaran, Z. Yu, M. S. Islam, S. Wang, R. S. Williams, and N. X. Fang, “A smooth optical superlens,” Appl. Phys. Lett. 96(4), 043102 (2010).
[Crossref]

Lu, D.

D. Lu and Z. Liu, “Hyperlenses and metalenses for far-field super-resolution imaging,” Nat. Commun. 3, 1205 (2012).
[Crossref] [PubMed]

Lv, C.

C. Lv, W. Li, X. Jiang, and J. Cao, “Far-field super-resolution imaging with a planar hyperbolic metamaterial lens,” Europhys. Lett. 105(2), 28003 (2014).
[Crossref]

Malitson, I.

I. Malitson and M. Dodge, “Refractive-index and birefringence of synthetic sapphire,” J. Opt. Soc. Am. 62, 1405A (1972).

Melville, D. O.

Merlin, R.

R. Merlin, “Analytical solution of the almost-perfect-lens problem,” Appl. Phys. Lett. 84(8), 1290–1292 (2004).
[Crossref]

Narimanov, E.

Narimanov, E. E.

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor meta-materials,” Nat. Mater. 6(12), 946–950 (2007).
[Crossref] [PubMed]

V. A. Podolskiy and E. E. Narimanov, “Near-sighted super-lens,” Opt. Lett. 30(1), 75–77 (2005).
[Crossref] [PubMed]

Ok, J. G.

M. Byun, D. Lee, M. Kim, Y. Kim, K. Kim, J. G. Ok, J. Rho, and H. Lee, “Demonstration of nanoimprinted hyperlens array for high-throughput sub-diffraction imaging,” Sci. Rep. 7, 46314 (2017).
[Crossref] [PubMed]

Oulton, R. F.

R. F. Oulton, D. F. Pile, Y. Liu, and X. Zhang, “Scattering of surface plasmon polaritons at abrupt surface interfaces: implications for nanoscale cavities,” Phys. Rev. B 98(26), 035408 (2007).
[Crossref]

Pendry, J. B.

D. R. Smith, D. Schurig, M. Rosenbluth, S. Schultz, S. A. Ramakrishna, and J. B. Pendry, “Limitations on subdiffraction imaging with a negative refractive index slab,” Appl. Phys. Lett. 82(10), 1506–1508 (2003).
[Crossref]

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

Pikus, Y.

Z. Liu, S. Durant, H. Lee, Y. Pikus, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical superlens,” Nano.Lett. 7, 403–408 (2007).
[Crossref]

Pile, D. F.

R. F. Oulton, D. F. Pile, Y. Liu, and X. Zhang, “Scattering of surface plasmon polaritons at abrupt surface interfaces: implications for nanoscale cavities,” Phys. Rev. B 98(26), 035408 (2007).
[Crossref]

Podolskiy, V.

I. Avrutsky, I. Salakhutdinov, J. Elser, and V. Podolskiy, “Highly confined optical modes in nanoscale metal-dielectric multilayers,” Phys. Rev. B 75(24), 241402 (2007).
[Crossref]

Podolskiy, V. A.

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor meta-materials,” Nat. Mater. 6(12), 946–950 (2007).
[Crossref] [PubMed]

V. A. Podolskiy and E. E. Narimanov, “Near-sighted super-lens,” Opt. Lett. 30(1), 75–77 (2005).
[Crossref] [PubMed]

Ramakrishna, S. A.

D. R. Smith, D. Schurig, M. Rosenbluth, S. Schultz, S. A. Ramakrishna, and J. B. Pendry, “Limitations on subdiffraction imaging with a negative refractive index slab,” Appl. Phys. Lett. 82(10), 1506–1508 (2003).
[Crossref]

Rho, J.

M. Byun, D. Lee, M. Kim, Y. Kim, K. Kim, J. G. Ok, J. Rho, and H. Lee, “Demonstration of nanoimprinted hyperlens array for high-throughput sub-diffraction imaging,” Sci. Rep. 7, 46314 (2017).
[Crossref] [PubMed]

M. Kim and J. Rho, “Metamaterials and imaging,” Nano Convergence 2, 22 (2015).
[Crossref] [PubMed]

J. Rho, Z. Ye, Y. Xiong, X. Yin, Z. Liu, H. Choi, G. Bartal, and X. Zhang, “Spherical hyperlens for two-dimensional sub-diffractional imaging at visible frequencies,” Nat. Commun. 1, 143 (2010).
[Crossref]

Rosenbluth, M.

D. R. Smith, D. Schurig, M. Rosenbluth, S. Schultz, S. A. Ramakrishna, and J. B. Pendry, “Limitations on subdiffraction imaging with a negative refractive index slab,” Appl. Phys. Lett. 82(10), 1506–1508 (2003).
[Crossref]

Royer, P.

L. Douillard, F. Charra, Z. Korczak, R. Bachelot, S. Kostcheev, G. Lerondel, P.-M. Adam, and P. Royer, “Short range plasmon resonators probed by photoemission electron microscopy,” Nat. Mater. 8(3), 935–940 (2008).

Salakhutdinov, I.

I. Avrutsky, I. Salakhutdinov, J. Elser, and V. Podolskiy, “Highly confined optical modes in nanoscale metal-dielectric multilayers,” Phys. Rev. B 75(24), 241402 (2007).
[Crossref]

Sarid, D.

D. Sarid, “Long-range surface-plasma waves on very thin metal films,” Phys. Rev. Lett. 47(26), 1927 (1981).
[Crossref]

Satpathy, S.

Z. Zhang and S. Satpathy, “Electromagnetic wave propagation in periodic structures: Bloch wave solution of Maxwell’s equations,” Phys. Rev. Lett. 65(21), 2650 (1990).
[Crossref] [PubMed]

Schultz, S.

D. R. Smith, D. Schurig, M. Rosenbluth, S. Schultz, S. A. Ramakrishna, and J. B. Pendry, “Limitations on subdiffraction imaging with a negative refractive index slab,” Appl. Phys. Lett. 82(10), 1506–1508 (2003).
[Crossref]

Schurig, D.

D. R. Smith, D. Schurig, M. Rosenbluth, S. Schultz, S. A. Ramakrishna, and J. B. Pendry, “Limitations on subdiffraction imaging with a negative refractive index slab,” Appl. Phys. Lett. 82(10), 1506–1508 (2003).
[Crossref]

Sivco, D. L.

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor meta-materials,” Nat. Mater. 6(12), 946–950 (2007).
[Crossref] [PubMed]

Smith, D. R.

D. R. Smith, D. Schurig, M. Rosenbluth, S. Schultz, S. A. Ramakrishna, and J. B. Pendry, “Limitations on subdiffraction imaging with a negative refractive index slab,” Appl. Phys. Lett. 82(10), 1506–1508 (2003).
[Crossref]

Smolyaninov, I. I.

I. I. Smolyaninov, Y.-J. Hung, and C. C. Davis, “Magnifying superlens in the visible frequency range,” Science 315(5819), 1699–1701 (2007).
[Crossref] [PubMed]

Sun, C.

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]

Z. Liu, S. Durant, H. Lee, Y. Pikus, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical superlens,” Nano.Lett. 7, 403–408 (2007).
[Crossref]

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308(5721), 534–537 (2005).
[Crossref] [PubMed]

Sun, G.

J. B. Khurgin and G. Sun, “Scaling of losses with size and wavelength in nanoplasmonics and metamaterials,” Appl. Phys. Lett. 99(21), 211106 (2011).
[Crossref]

Wang, S.

P. Chaturvedi, W. Wu, V. Logeeswaran, Z. Yu, M. S. Islam, S. Wang, R. S. Williams, and N. X. Fang, “A smooth optical superlens,” Appl. Phys. Lett. 96(4), 043102 (2010).
[Crossref]

Wasserman, D.

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor meta-materials,” Nat. Mater. 6(12), 946–950 (2007).
[Crossref] [PubMed]

Webb, K.

Williams, R. S.

P. Chaturvedi, W. Wu, V. Logeeswaran, Z. Yu, M. S. Islam, S. Wang, R. S. Williams, and N. X. Fang, “A smooth optical superlens,” Appl. Phys. Lett. 96(4), 043102 (2010).
[Crossref]

Wu, W.

P. Chaturvedi, W. Wu, V. Logeeswaran, Z. Yu, M. S. Islam, S. Wang, R. S. Williams, and N. X. Fang, “A smooth optical superlens,” Appl. Phys. Lett. 96(4), 043102 (2010).
[Crossref]

Xiong, Y.

J. Rho, Z. Ye, Y. Xiong, X. Yin, Z. Liu, H. Choi, G. Bartal, and X. Zhang, “Spherical hyperlens for two-dimensional sub-diffractional imaging at visible frequencies,” Nat. Commun. 1, 143 (2010).
[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]

Z. Liu, S. Durant, H. Lee, Y. Pikus, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical superlens,” Nano.Lett. 7, 403–408 (2007).
[Crossref]

Yang, M.

Ye, Z.

J. Rho, Z. Ye, Y. Xiong, X. Yin, Z. Liu, H. Choi, G. Bartal, and X. Zhang, “Spherical hyperlens for two-dimensional sub-diffractional imaging at visible frequencies,” Nat. Commun. 1, 143 (2010).
[Crossref]

Yin, X.

J. Rho, Z. Ye, Y. Xiong, X. Yin, Z. Liu, H. Choi, G. Bartal, and X. Zhang, “Spherical hyperlens for two-dimensional sub-diffractional imaging at visible frequencies,” Nat. Commun. 1, 143 (2010).
[Crossref]

Yu, Z.

P. Chaturvedi, W. Wu, V. Logeeswaran, Z. Yu, M. S. Islam, S. Wang, R. S. Williams, and N. X. Fang, “A smooth optical superlens,” Appl. Phys. Lett. 96(4), 043102 (2010).
[Crossref]

Zhang, B.

B. Zhang and J. B. Khurgin, “Eigen mode approach to the sub-wavelength imaging with surface plasmon polaritons,” Appl. Phys. Lett. 98(26), 263102 (2011).
[Crossref]

Zhang, X.

J. Rho, Z. Ye, Y. Xiong, X. Yin, Z. Liu, H. Choi, G. Bartal, and X. Zhang, “Spherical hyperlens for two-dimensional sub-diffractional imaging at visible frequencies,” Nat. Commun. 1, 143 (2010).
[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]

R. F. Oulton, D. F. Pile, Y. Liu, and X. Zhang, “Scattering of surface plasmon polaritons at abrupt surface interfaces: implications for nanoscale cavities,” Phys. Rev. B 98(26), 035408 (2007).
[Crossref]

Z. Liu, S. Durant, H. Lee, Y. Pikus, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical superlens,” Nano.Lett. 7, 403–408 (2007).
[Crossref]

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308(5721), 534–537 (2005).
[Crossref] [PubMed]

Zhang, Z.

Z. Zhang and S. Satpathy, “Electromagnetic wave propagation in periodic structures: Bloch wave solution of Maxwell’s equations,” Phys. Rev. Lett. 65(21), 2650 (1990).
[Crossref] [PubMed]

Adv. Opt. Photon. (1)

Appl. Phys. Lett. (6)

P. Chaturvedi, W. Wu, V. Logeeswaran, Z. Yu, M. S. Islam, S. Wang, R. S. Williams, and N. X. Fang, “A smooth optical superlens,” Appl. Phys. Lett. 96(4), 043102 (2010).
[Crossref]

M. Iwanaga, “Hyperlens-array-implemented optical microscopy,” Appl. Phys. Lett. 105, 053112 (2014).
[Crossref]

D. R. Smith, D. Schurig, M. Rosenbluth, S. Schultz, S. A. Ramakrishna, and J. B. Pendry, “Limitations on subdiffraction imaging with a negative refractive index slab,” Appl. Phys. Lett. 82(10), 1506–1508 (2003).
[Crossref]

R. Merlin, “Analytical solution of the almost-perfect-lens problem,” Appl. Phys. Lett. 84(8), 1290–1292 (2004).
[Crossref]

B. Zhang and J. B. Khurgin, “Eigen mode approach to the sub-wavelength imaging with surface plasmon polaritons,” Appl. Phys. Lett. 98(26), 263102 (2011).
[Crossref]

J. B. Khurgin and G. Sun, “Scaling of losses with size and wavelength in nanoplasmonics and metamaterials,” Appl. Phys. Lett. 99(21), 211106 (2011).
[Crossref]

Europhys. Lett. (1)

C. Lv, W. Li, X. Jiang, and J. Cao, “Far-field super-resolution imaging with a planar hyperbolic metamaterial lens,” Europhys. Lett. 105(2), 28003 (2014).
[Crossref]

J. Opt. Soc. Am. (1)

I. Malitson and M. Dodge, “Refractive-index and birefringence of synthetic sapphire,” J. Opt. Soc. Am. 62, 1405A (1972).

Nano Convergence (1)

M. Kim and J. Rho, “Metamaterials and imaging,” Nano Convergence 2, 22 (2015).
[Crossref] [PubMed]

Nano.Lett. (1)

Z. Liu, S. Durant, H. Lee, Y. Pikus, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical superlens,” Nano.Lett. 7, 403–408 (2007).
[Crossref]

Nat. Commun. (2)

J. Rho, Z. Ye, Y. Xiong, X. Yin, Z. Liu, H. Choi, G. Bartal, and X. Zhang, “Spherical hyperlens for two-dimensional sub-diffractional imaging at visible frequencies,” Nat. Commun. 1, 143 (2010).
[Crossref]

D. Lu and Z. Liu, “Hyperlenses and metalenses for far-field super-resolution imaging,” Nat. Commun. 3, 1205 (2012).
[Crossref] [PubMed]

Nat. Mater. (2)

L. Douillard, F. Charra, Z. Korczak, R. Bachelot, S. Kostcheev, G. Lerondel, P.-M. Adam, and P. Royer, “Short range plasmon resonators probed by photoemission electron microscopy,” Nat. Mater. 8(3), 935–940 (2008).

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor meta-materials,” Nat. Mater. 6(12), 946–950 (2007).
[Crossref] [PubMed]

Opt. Express (2)

Opt. Lett. (2)

Optica (1)

Phys. Rev. B (3)

P. B. Johnson and R.-W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370 (1972).
[Crossref]

I. Avrutsky, I. Salakhutdinov, J. Elser, and V. Podolskiy, “Highly confined optical modes in nanoscale metal-dielectric multilayers,” Phys. Rev. B 75(24), 241402 (2007).
[Crossref]

R. F. Oulton, D. F. Pile, Y. Liu, and X. Zhang, “Scattering of surface plasmon polaritons at abrupt surface interfaces: implications for nanoscale cavities,” Phys. Rev. B 98(26), 035408 (2007).
[Crossref]

Phys. Rev. Lett. (3)

D. Sarid, “Long-range surface-plasma waves on very thin metal films,” Phys. Rev. Lett. 47(26), 1927 (1981).
[Crossref]

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

Z. Zhang and S. Satpathy, “Electromagnetic wave propagation in periodic structures: Bloch wave solution of Maxwell’s equations,” Phys. Rev. Lett. 65(21), 2650 (1990).
[Crossref] [PubMed]

Sci. Rep. (1)

M. Byun, D. Lee, M. Kim, Y. Kim, K. Kim, J. G. Ok, J. Rho, and H. Lee, “Demonstration of nanoimprinted hyperlens array for high-throughput sub-diffraction imaging,” Sci. Rep. 7, 46314 (2017).
[Crossref] [PubMed]

Science (3)

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308(5721), 534–537 (2005).
[Crossref] [PubMed]

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. I. Smolyaninov, Y.-J. Hung, and C. C. Davis, “Magnifying superlens in the visible frequency range,” Science 315(5819), 1699–1701 (2007).
[Crossref] [PubMed]

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Figures (10)

Fig. 1
Fig. 1

Symmetric and antisymmetric eigen modes in multilayers hyperbolic metamaterials contain N metal layers with thickness tm and separated by dielectric with distances td. do and di are the distance between object, image and hyperbolic metamaterials.

Fig. 2
Fig. 2

(a) The dispersion curves of the eigen modes when the superlens contains five metal slabs; (b) The profile of electric in-plane component of each eigen modes inside the superlens; (c) The absolute value of OTF when N=1, 5, 9, 13, 17, respectively; (d) The cut-off lateral wavevectors versus the number of metal layers for three different object and image distances.

Fig. 3
Fig. 3

(a) Comparison of PSF with different metal loss when the superlens contains four metal slabs; (b) Comparison of PSF when the superlens contain four and five metal slabs respectively; (c) Change of OTF with different metal loss, when N = 5, FR = 50%, P = 30nm, di = do = 30nm. λ0 is the wavelength of signal.

Fig. 4
Fig. 4

(a) Dispersion relations of the eigen modes when N=10; (b, c) The “OTF oscillations” as a function of the number of layers in the sum Eq. (9) for two values of special frequencies k = 5k0 and k = 10k0.

Fig. 5
Fig. 5

(a)–(c) The dispersion relations of eigen modes when the period P=10nm, 30nm, 50nm respectively; (d) Change of the OTF with increase in period when the metal fill ratio is set at 50%; (e) Change of the cut-off wavevector with different period when the fill ratio is set at 50%.

Fig. 6
Fig. 6

(a)–(c) are the dispersion relation of eigen modes with metal fill ratio 10%, 50%, 80% respectively; (d) The absolute value of OTF for different metal fill ratios; (e) The cut-off wavevector for different fill ratios.

Fig. 7
Fig. 7

(a)–(c) The OTF of hyperlens contains 100 metal/dielectric period for different metal loss when the thickness of meta and dielectric are tm = td = 5nm; (d) The averaged OTF of hyperlens contains 100 metal/dielectric period for different metal loss when the thickness of meta and dielectric are tm = td = 5nm.

Fig. 8
Fig. 8

(a) and (b) The averaged OTF of hyperlens contains 50 metal/dielectric periods for different metal loss when the thickness of meta and dielectric are tm = td = 5nm and tm = td = 15nm respectively; (c) and (d) The averaged OTF of hyperlens contains 100 metal/dielectric periods for different metal loss when the thickness of meta and dielectric are tm = td = 5nm and tm = td = 15nm respectively.

Fig. 9
Fig. 9

Configuration of multilayer metal/dielectric superlens.

Fig. 10
Fig. 10

Comparison of the calculated OTF with eigenmode model and TMM.

Equations (16)

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[ 2 z 2 k 2 + ( z , ω ) 0 ( ω ˜ n ) 2 c 2 ] f k n ( z ) = 0 ,
p ( r , z , t ) = e i ω t δ ( z 0 z ) k p k e i k r d k ,
2 E ( r , z , t ) ( z , ω ) 0 μ 0 2 t 2 E ( r , z , t ) = ω 2 μ 0 p ( r , z , t ) ,
E ( r , z , t ) = e i ω t 0 n = 1 N + 1 C k n ( ω ) f k n ( z ) e i k r d k ,
n = 1 N + 1 C k n [ 2 z 2 k 2 + ( z , ω ) 0 ω 2 c 2 ] f k n ( z ) = ω 2 μ 0 p k δ ( z z 0 ) .
n = 1 N + 1 C k n [ ( z , ω ) 0 ω 2 / c 2 ( z , ω ) 0 ( ω ˜ n ) 2 / c 2 ] f k n ( z ) = ω 2 μ 0 p k δ ( z z 0 ) ,
C k n = ω 2 ω 2 ω ˜ n 2 p k 0 f k n ( z 0 ) .
E ( z i ) = | k | > k 0 , n [ ω 2 ω 2 ω ˜ n 2 p k 0 f k n ( z 0 ) * ] f k n ( z i ) e i k r d k + | k | < k 0 e i k r E k ( z i ) d k .
OTF ( k , z 0 , z i ) = n = 1 N + 1 ω 2 ω 2 ω ˜ n 2 ( k ) f k n ( k , z 0 ) * f k n ( k , z i ) .
PSF ( x ) = k e i k x E k , imag d k ~ k e i k x OTF d k .
D d m = [ ( 1 2 + m ( f ) K ( k , f ) 2 d Q ( k , f ) ) ( 1 2 m ( f ) K ( k , f ) 2 d Q ( k , f ) ) ( 1 2 ) m ( f ) K ( k , f ) 2 d Q ( k , f ) ( 1 2 + m ( f ) K ( k , f ) 2 d Q ( k , f ) ) ]
D m d = [ ( 1 2 + d Q ( k , f ) 2 m ( f ) K ( k , f ) ) ( 1 2 d Q ( k , f ) 2 m ( f ) K ( k , f ) ) ( 1 2 d Q ( k , f ) 2 m ( f ) K ( k , f ) ) ( 1 2 + d Q ( k , f ) 2 m ( f ) K ( k , f ) ) ]
P d = [ exp ( K ( k , f ) t d ) 0 0 exp ( K ( k , f ) t d ) ]
P m = [ exp ( Q ( k , f ) t m ) 0 0 exp ( Q ( k , f ) t m ) ]
[ E x n + E x n ] = Nmetallayers D m d ( k , f ) P m ( k , f ) P m ( k , f ) D d m ( k , f ) [ E x 1 + E x 1 ] = [ M 11 ( k , f ) M 12 ( k , f ) M 21 ( k , f ) M 22 ( k , f ) ] [ E x 1 + E x 1 ]
OTF ( k , f ) = E img E obj = 1 M 11 ( k , f )

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