Abstract

After reviewing the requirements, which have to be satisfied by a metamaterial-based subwavelength imaging system, a thin film lens is reported herein. The material of the lens is a composite of spherical Ag nanoparticles embedded in SiO2 host material. The image of the lens is calculated by solving Maxwell equations with the transfer matrix method. The procedure applies the Maxwell–Garnett mixing rule and high-frequency effective medium theory to calculate the electromagnetic parameters of the composite material. The formula of the composite material, the optimum working frequency, and the thicknesses of the layers are determined by minimizing the absolute difference of the field distribution in the source and image planes. The details of the design procedure are presented, and optimized configurations obtained under different constraints are discussed. The main advantage of the composite lens is that it can eliminate the hot spots present in the images of metallic superlens.

© 2014 Optical Society of America

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2014

A. Darafsheh, N. I. Limberopoulos, J. S. Derov, D. E. Walker, and V. N. Astratov, “Advantages of microsphere-assisted super-resolution imaging technique over solid immersion lens and confocal microscopies,” Appl. Phys. Lett. 104, 061117 (2014).
[CrossRef]

2013

Y. Cotte, F. Toy, P. Jourdain, N. Pavillon, D. Boss, P. Magistretti, P. Marquet, and C. Depeursinge, “Marker-free phase nanoscopy,” Nat. Photonics 7, 113–117 (2013).
[CrossRef]

T. Xu, A. Agrawal, M. Abashin, K. J. Chau, and H. J. Lezec, “All-angle negative refraction and active flat lensing of ultraviolet light,” Nature 497, 470–474 (2013).
[CrossRef]

2012

D. Lu and Z. Liu, “Hyperlenses and metalenses for far-field super-resolution imaging,” Nat. Commun. 2, 1205 (2012).
[CrossRef]

2011

Z. Wang, W. Guo, L. Li, B. Luk’yanchuk, A. Khan, Z. Liu, Z. Chen, and M. Hong, “Optical virtual imaging at 50 nm lateral resolution with a white-light nanoscope,” Nat. Commun. 2, 219 (2011).
[CrossRef]

R. Hegde, Z. Szabó, H. Yew-Li, E. P. L. Y. Kiasat, and W. J. R. Hoefer, “The dynamics of nanoscale superresolution imaging with the superlens,” IEEE Trans. Microwave Theor. Tech. 59, 2612–2623 (2011).
[CrossRef]

2010

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, 1148 (2010).
[CrossRef]

H. Liu, B. Wang, E. S. Leong, P. Yang, Y. Zong, G. Si, J. Teng, and S. A. Maier, “Enhanced surface plasmon resonance on a smooth silver film with a seed growth layer,” ACS Nano 4, 3139–3146 (2010).
[CrossRef]

2009

W. E. Moerner, “Eyes on super-resolution,” Nat. Photonics 3, 368–369 (2009).
[CrossRef]

S. W. Hell, R. Schmidt, and A. Egner, “Diffraction-unlimited three-dimensional optical nanoscopy with opposing lenses,” Nat. Photonics 3, 381–387 (2009).
[CrossRef]

S. Kawata, Y. Inouye, and P. Verma, “Plasmonics for near-field nano-imaging and superlensing,” Nat. Photonics 3, 388–394 (2009).
[CrossRef]

2008

A. Moroz, “Electron mean free path in a spherical shell geometry,” J. Phys. Chem. C 112, 10641–10652 (2008).
[CrossRef]

2007

V. M. Shalaev, “Optical negative-index metamaterials,” Nat. Photonics 1, 41–48 (2007).
[CrossRef]

G. Dolling, M. Wegener, C. M. Soukoulis, and S. Linden, “Negative-index metamaterial at 780  nm wavelength,” Opt. Lett. 32, 53–55 (2007).
[CrossRef]

H. J. Lezec, J. A. Dionne, and H. A. Atwater, “Negative refraction at visible frequencies,” Science 316, 430–432 (2007).
[CrossRef]

S. W. Hell, “Far-field optical nanoscopy,” Science 316, 1153–1158 (2007).
[CrossRef]

2006

2005

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

P. Mallet, C. A. Guérin, and A. Sentenac, “Maxwell–Garnett mixing rule in the presence of multiple scattering: derivation and accuracy,” Phys. Rev. B 72, 014205 (2005).
[CrossRef]

2003

K. L. Kelly, E. Coronado, L. L. Zhao, and G. C. Schatz, “The optical properties of metal nanoparticles: the influence of size, shape, and dielectric environment,” J. Phys. Chem. B 107, 668–677 (2003).
[CrossRef]

2000

R. Ruppin, “Evaluation of extended Maxwell–Garnett theories,” Opt. Commun. 182, 273–279 (2000).
[CrossRef]

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

D. R. Smith, W. Padilla, D. C. Vier, S. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184 (2000).
[CrossRef]

1991

V. Ossenkopf, “Effective-medium theories for cosmic dust grains,” Astron. Astrophys. 251, 210–219 (1991).

1982

D. E. Aspnese, “Local-field effects and effective-medium theory: a microscopic perspective,” Am. J. Phys. 50, 704–709 (1982).
[CrossRef]

1974

U. Kreibig, “Electronic properties of small silver particles: the optical constant and their temperature dependence,” J. Phys. F 4, 999–1014 (1974).
[CrossRef]

1962

H. Ehrenreich and H. R. Philipp, “Optical properties of ag and cu,” Phys. Rev. 128, 1622–1629 (1962).
[CrossRef]

1947

L. Lewin, “The electrical constants of a material loaded with spherical particles,” J. Inst. Electr. Eng. 94, 65–68 (1947).
[CrossRef]

Abashin, M.

T. Xu, A. Agrawal, M. Abashin, K. J. Chau, and H. J. Lezec, “All-angle negative refraction and active flat lensing of ultraviolet light,” Nature 497, 470–474 (2013).
[CrossRef]

Agrawal, A.

T. Xu, A. Agrawal, M. Abashin, K. J. Chau, and H. J. Lezec, “All-angle negative refraction and active flat lensing of ultraviolet light,” Nature 497, 470–474 (2013).
[CrossRef]

Aspnese, D. E.

D. E. Aspnese, “Local-field effects and effective-medium theory: a microscopic perspective,” Am. J. Phys. 50, 704–709 (1982).
[CrossRef]

Astratov, V. N.

A. Darafsheh, N. I. Limberopoulos, J. S. Derov, D. E. Walker, and V. N. Astratov, “Advantages of microsphere-assisted super-resolution imaging technique over solid immersion lens and confocal microscopies,” Appl. Phys. Lett. 104, 061117 (2014).
[CrossRef]

Atwater, H. A.

H. J. Lezec, J. A. Dionne, and H. A. Atwater, “Negative refraction at visible frequencies,” Science 316, 430–432 (2007).
[CrossRef]

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, 1148 (2010).
[CrossRef]

Blaikie, R. J.

Bohren, C. F.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-VCH Verlag GmbH & Co. KGaA, 2004).

Born, M.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University, 1999).

Boss, D.

Y. Cotte, F. Toy, P. Jourdain, N. Pavillon, D. Boss, P. Magistretti, P. Marquet, and C. Depeursinge, “Marker-free phase nanoscopy,” Nat. Photonics 7, 113–117 (2013).
[CrossRef]

Chau, K. J.

T. Xu, A. Agrawal, M. Abashin, K. J. Chau, and H. J. Lezec, “All-angle negative refraction and active flat lensing of ultraviolet light,” Nature 497, 470–474 (2013).
[CrossRef]

Chen, Z.

Z. Wang, W. Guo, L. Li, B. Luk’yanchuk, A. Khan, Z. Liu, Z. Chen, and M. Hong, “Optical virtual imaging at 50 nm lateral resolution with a white-light nanoscope,” Nat. Commun. 2, 219 (2011).
[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, 1148 (2010).
[CrossRef]

Coronado, E.

K. L. Kelly, E. Coronado, L. L. Zhao, and G. C. Schatz, “The optical properties of metal nanoparticles: the influence of size, shape, and dielectric environment,” J. Phys. Chem. B 107, 668–677 (2003).
[CrossRef]

Cotte, Y.

Y. Cotte, F. Toy, P. Jourdain, N. Pavillon, D. Boss, P. Magistretti, P. Marquet, and C. Depeursinge, “Marker-free phase nanoscopy,” Nat. Photonics 7, 113–117 (2013).
[CrossRef]

Darafsheh, A.

A. Darafsheh, N. I. Limberopoulos, J. S. Derov, D. E. Walker, and V. N. Astratov, “Advantages of microsphere-assisted super-resolution imaging technique over solid immersion lens and confocal microscopies,” Appl. Phys. Lett. 104, 061117 (2014).
[CrossRef]

Depeursinge, C.

Y. Cotte, F. Toy, P. Jourdain, N. Pavillon, D. Boss, P. Magistretti, P. Marquet, and C. Depeursinge, “Marker-free phase nanoscopy,” Nat. Photonics 7, 113–117 (2013).
[CrossRef]

Derov, J. S.

A. Darafsheh, N. I. Limberopoulos, J. S. Derov, D. E. Walker, and V. N. Astratov, “Advantages of microsphere-assisted super-resolution imaging technique over solid immersion lens and confocal microscopies,” Appl. Phys. Lett. 104, 061117 (2014).
[CrossRef]

Dionne, J. A.

H. J. Lezec, J. A. Dionne, and H. A. Atwater, “Negative refraction at visible frequencies,” Science 316, 430–432 (2007).
[CrossRef]

Dolling, G.

Egner, A.

S. W. Hell, R. Schmidt, and A. Egner, “Diffraction-unlimited three-dimensional optical nanoscopy with opposing lenses,” Nat. Photonics 3, 381–387 (2009).
[CrossRef]

Ehrenreich, H.

H. Ehrenreich and H. R. Philipp, “Optical properties of ag and cu,” Phys. Rev. 128, 1622–1629 (1962).
[CrossRef]

Fang, N.

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

Guérin, C. A.

P. Mallet, C. A. Guérin, and A. Sentenac, “Maxwell–Garnett mixing rule in the presence of multiple scattering: derivation and accuracy,” Phys. Rev. B 72, 014205 (2005).
[CrossRef]

Guo, W.

Z. Wang, W. Guo, L. Li, B. Luk’yanchuk, A. Khan, Z. Liu, Z. Chen, and M. Hong, “Optical virtual imaging at 50 nm lateral resolution with a white-light nanoscope,” Nat. Commun. 2, 219 (2011).
[CrossRef]

Hegde, R.

R. Hegde, Z. Szabó, H. Yew-Li, E. P. L. Y. Kiasat, and W. J. R. Hoefer, “The dynamics of nanoscale superresolution imaging with the superlens,” IEEE Trans. Microwave Theor. Tech. 59, 2612–2623 (2011).
[CrossRef]

Hell, S. W.

S. W. Hell, R. Schmidt, and A. Egner, “Diffraction-unlimited three-dimensional optical nanoscopy with opposing lenses,” Nat. Photonics 3, 381–387 (2009).
[CrossRef]

S. W. Hell, “Far-field optical nanoscopy,” Science 316, 1153–1158 (2007).
[CrossRef]

Hoefer, W. J. R.

R. Hegde, Z. Szabó, H. Yew-Li, E. P. L. Y. Kiasat, and W. J. R. Hoefer, “The dynamics of nanoscale superresolution imaging with the superlens,” IEEE Trans. Microwave Theor. Tech. 59, 2612–2623 (2011).
[CrossRef]

Hong, M.

Z. Wang, W. Guo, L. Li, B. Luk’yanchuk, A. Khan, Z. Liu, Z. Chen, and M. Hong, “Optical virtual imaging at 50 nm lateral resolution with a white-light nanoscope,” Nat. Commun. 2, 219 (2011).
[CrossRef]

Huffman, D. R.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-VCH Verlag GmbH & Co. KGaA, 2004).

Inouye, Y.

S. Kawata, Y. Inouye, and P. Verma, “Plasmonics for near-field nano-imaging and superlensing,” Nat. Photonics 3, 388–394 (2009).
[CrossRef]

Jourdain, P.

Y. Cotte, F. Toy, P. Jourdain, N. Pavillon, D. Boss, P. Magistretti, P. Marquet, and C. Depeursinge, “Marker-free phase nanoscopy,” Nat. Photonics 7, 113–117 (2013).
[CrossRef]

Kawata, S.

S. Kawata, Y. Inouye, and P. Verma, “Plasmonics for near-field nano-imaging and superlensing,” Nat. Photonics 3, 388–394 (2009).
[CrossRef]

Kelly, K. L.

K. L. Kelly, E. Coronado, L. L. Zhao, and G. C. Schatz, “The optical properties of metal nanoparticles: the influence of size, shape, and dielectric environment,” J. Phys. Chem. B 107, 668–677 (2003).
[CrossRef]

Khan, A.

Z. Wang, W. Guo, L. Li, B. Luk’yanchuk, A. Khan, Z. Liu, Z. Chen, and M. Hong, “Optical virtual imaging at 50 nm lateral resolution with a white-light nanoscope,” Nat. Commun. 2, 219 (2011).
[CrossRef]

Kiasat, E. P. L. Y.

R. Hegde, Z. Szabó, H. Yew-Li, E. P. L. Y. Kiasat, and W. J. R. Hoefer, “The dynamics of nanoscale superresolution imaging with the superlens,” IEEE Trans. Microwave Theor. Tech. 59, 2612–2623 (2011).
[CrossRef]

Kreibig, U.

U. Kreibig, “Electronic properties of small silver particles: the optical constant and their temperature dependence,” J. Phys. F 4, 999–1014 (1974).
[CrossRef]

Lampinen, J. A.

K. V. Price, R. M. Storn, and J. A. Lampinen, Differential Evolution, A Practical Approach to Global Optimization (Springer, 2005).

Lee, H.

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

Leong, E. S.

H. Liu, B. Wang, E. S. Leong, P. Yang, Y. Zong, G. Si, J. Teng, and S. A. Maier, “Enhanced surface plasmon resonance on a smooth silver film with a seed growth layer,” ACS Nano 4, 3139–3146 (2010).
[CrossRef]

Lewin, L.

L. Lewin, “The electrical constants of a material loaded with spherical particles,” J. Inst. Electr. Eng. 94, 65–68 (1947).
[CrossRef]

Lezec, H. J.

T. Xu, A. Agrawal, M. Abashin, K. J. Chau, and H. J. Lezec, “All-angle negative refraction and active flat lensing of ultraviolet light,” Nature 497, 470–474 (2013).
[CrossRef]

H. J. Lezec, J. A. Dionne, and H. A. Atwater, “Negative refraction at visible frequencies,” Science 316, 430–432 (2007).
[CrossRef]

Li, L.

Z. Wang, W. Guo, L. Li, B. Luk’yanchuk, A. Khan, Z. Liu, Z. Chen, and M. Hong, “Optical virtual imaging at 50 nm lateral resolution with a white-light nanoscope,” Nat. Commun. 2, 219 (2011).
[CrossRef]

Limberopoulos, N. I.

A. Darafsheh, N. I. Limberopoulos, J. S. Derov, D. E. Walker, and V. N. Astratov, “Advantages of microsphere-assisted super-resolution imaging technique over solid immersion lens and confocal microscopies,” Appl. Phys. Lett. 104, 061117 (2014).
[CrossRef]

Linden, S.

Liu, H.

H. Liu, B. Wang, E. S. Leong, P. Yang, Y. Zong, G. Si, J. Teng, and S. A. Maier, “Enhanced surface plasmon resonance on a smooth silver film with a seed growth layer,” ACS Nano 4, 3139–3146 (2010).
[CrossRef]

Liu, Z.

D. Lu and Z. Liu, “Hyperlenses and metalenses for far-field super-resolution imaging,” Nat. Commun. 2, 1205 (2012).
[CrossRef]

Z. Wang, W. Guo, L. Li, B. Luk’yanchuk, A. Khan, Z. Liu, Z. Chen, and M. Hong, “Optical virtual imaging at 50 nm lateral resolution with a white-light nanoscope,” Nat. Commun. 2, 219 (2011).
[CrossRef]

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, 1148 (2010).
[CrossRef]

Lu, D.

D. Lu and Z. Liu, “Hyperlenses and metalenses for far-field super-resolution imaging,” Nat. Commun. 2, 1205 (2012).
[CrossRef]

Luk’yanchuk, B.

Z. Wang, W. Guo, L. Li, B. Luk’yanchuk, A. Khan, Z. Liu, Z. Chen, and M. Hong, “Optical virtual imaging at 50 nm lateral resolution with a white-light nanoscope,” Nat. Commun. 2, 219 (2011).
[CrossRef]

Magistretti, P.

Y. Cotte, F. Toy, P. Jourdain, N. Pavillon, D. Boss, P. Magistretti, P. Marquet, and C. Depeursinge, “Marker-free phase nanoscopy,” Nat. Photonics 7, 113–117 (2013).
[CrossRef]

Maier, S. A.

H. Liu, B. Wang, E. S. Leong, P. Yang, Y. Zong, G. Si, J. Teng, and S. A. Maier, “Enhanced surface plasmon resonance on a smooth silver film with a seed growth layer,” ACS Nano 4, 3139–3146 (2010).
[CrossRef]

Mallet, P.

P. Mallet, C. A. Guérin, and A. Sentenac, “Maxwell–Garnett mixing rule in the presence of multiple scattering: derivation and accuracy,” Phys. Rev. B 72, 014205 (2005).
[CrossRef]

Marques, R.

R. Marques, F. Martin, and M. Sorolla, Metamaterials with Negative Parameters (Wiley, 2008).

Marquet, P.

Y. Cotte, F. Toy, P. Jourdain, N. Pavillon, D. Boss, P. Magistretti, P. Marquet, and C. Depeursinge, “Marker-free phase nanoscopy,” Nat. Photonics 7, 113–117 (2013).
[CrossRef]

Martin, F.

R. Marques, F. Martin, and M. Sorolla, Metamaterials with Negative Parameters (Wiley, 2008).

Melville, D. O. S.

Moerner, W. E.

W. E. Moerner, “Eyes on super-resolution,” Nat. Photonics 3, 368–369 (2009).
[CrossRef]

Moroz, A.

A. Moroz, “Electron mean free path in a spherical shell geometry,” J. Phys. Chem. C 112, 10641–10652 (2008).
[CrossRef]

Nemat-Nasser, S.

D. R. Smith, W. Padilla, D. C. Vier, S. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184 (2000).
[CrossRef]

Ossenkopf, V.

V. Ossenkopf, “Effective-medium theories for cosmic dust grains,” Astron. Astrophys. 251, 210–219 (1991).

Padilla, W.

D. R. Smith, W. Padilla, D. C. Vier, S. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184 (2000).
[CrossRef]

Pavillon, N.

Y. Cotte, F. Toy, P. Jourdain, N. Pavillon, D. Boss, P. Magistretti, P. Marquet, and C. Depeursinge, “Marker-free phase nanoscopy,” Nat. Photonics 7, 113–117 (2013).
[CrossRef]

Pendry, J. B.

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

Philipp, H. R.

H. Ehrenreich and H. R. Philipp, “Optical properties of ag and cu,” Phys. Rev. 128, 1622–1629 (1962).
[CrossRef]

Price, K. V.

K. V. Price, R. M. Storn, and J. A. Lampinen, Differential Evolution, A Practical Approach to Global Optimization (Springer, 2005).

Rho, J.

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H. Liu, B. Wang, E. S. Leong, P. Yang, Y. Zong, G. Si, J. Teng, and S. A. Maier, “Enhanced surface plasmon resonance on a smooth silver film with a seed growth layer,” ACS Nano 4, 3139–3146 (2010).
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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, 1148 (2010).
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[CrossRef]

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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, 1148 (2010).
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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, 1148 (2010).
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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, 1148 (2010).
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D. Lu and Z. Liu, “Hyperlenses and metalenses for far-field super-resolution imaging,” Nat. Commun. 2, 1205 (2012).
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V. M. Shalaev, “Optical negative-index metamaterials,” Nat. Photonics 1, 41–48 (2007).
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[CrossRef]

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www.sspectra.com/sopra.html .

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Figures (9)

Fig. 1.
Fig. 1.

(a) Metamaterial-based imaging systems with the double Gaussian excitation. (b) Fourier transform of the source and the transfer function. (c) Intensity distribution of the source and image; the highest spatial harmonic, which is passed, has a wavelength of 21.24 nm.

Fig. 2.
Fig. 2.

Quality of the image in function of the bandwidth of the transfer function.

Fig. 3.
Fig. 3.

Transverse wavenumber kz for different material parameters.

Fig. 4.
Fig. 4.

Maxwell–Garnett-type composite with spherical multilayer inclusions.

Fig. 5.
Fig. 5.

Recursive algorithm to calculate the electric permittivity of Maxwell–Garnett type composites with multilayer spherical inclusions.

Fig. 6.
Fig. 6.

Decomposition of the electric permittivity of Ag into free and bound electron contributions. In (a) the real part, while in (b) the imaginary part of the electric permittivity is shown.

Fig. 7.
Fig. 7.

Frequency-dependent electric permittivity of the SiO2 host material.

Fig. 8.
Fig. 8.

Design of the imaging device with the single-layer composite lens. (a), (c), (e) and (f) refer to the lens with thickness of 20 nm and surrounded with air, while for (b), (d), (f) and (h) the thickness of the lens is 30 nm, and it is surrounded with SiO2. The error function Ω, which is searched by the optimization procedure, is presented in (a) and (b). The markers indicate optimized results obtained with constraints on the filling factor. The transfer function of the composite lenses are presented in (c) and (d). The intensity distribution in the source and image planes are plotted in (e) and (f). For reference, the transfer function and the image without the lens are presented as well. The intensity distributions in the image plane corresponding to points 1–4 of (a) and (b) are presented in (g) and (h).

Fig. 9.
Fig. 9.

Imaging with the multilayer lens. In (a) the transfer functions and in (b) the images corresponding to the parameters presented in Table 2 are shown. The intensity distribution of the source and of the image for the lens, with parameters presented in the fourth row of the table, along with the reference image without the lens are shown in (c).

Tables (2)

Tables Icon

Table 1. Filling Factor, Frequency, and Effective Electric Permittivity of the Single-Layer Composite Lens

Tables Icon

Table 2. Design Parameters of the Multilayer Lens

Equations (25)

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

G(fx,fy)=g(x,y)ei2π(fxx+fyy)dxdy,
T(kx)=T0(kx)T1(kx)T2(kx),
T1=4ξ1ξ2(ξ1+1)(ξ2+1)eikz1d1+(ξ11)(ξ21)eikz1d1
ξ1=ϵr1kz0ϵr0kz1,ξ2=ϵr2kz1ϵr1kz2,
ξ1=μr1kz0μr0kz1,ξ2=μr2kz1μr1kz2.
g(x,y,z2)=T(fx,fy)G(fx,fy)ei2π(fxx+fyy)dxdy.
Ω=1Nxj=1Nx|g(xj,z2)g(xj,0)|,
Mm=[cos(kzmdm)iωϵrmkzmsin(kzmdm)ikzmωϵrmsin(kzmdm)cos(kzmdm)],
M=m=n+10Mm=[M11M12M21M22].
T(kx)=2M11+ωϵr0kz0M21+kzn+1ωϵrn+1(M12+ωϵr0kz0M22),
Q2(x)=1x1+2x,
ϵreff=ϵrhQ2(ζQ2(ϵrhϵri)),
an=mΨn(mx)Ψn(x)Ψn(x)Ψn(mx)mΨn(mx)ξn(x)ξn(x)Ψn(mx),
bn=Ψn(mx)Ψn(x)mΨn(x)Ψn(mx)Ψn(mx)ξn(x)mξn(x)Ψn(mx),
m=ninh=ϵriμriϵrhμrh,x=ϵrhμrhωr/c0,
αe=i3r32x3a1,αm=i3r32x3b1.
ϵreffϵrhϵreff+2ϵrh=ζr3αe,
ϵreff=x3+3iζa1x332iζa1ϵrh,μreff=x3+3iζb1x332iζb1μrh.
Ψ1(ρ)=sinρρcosρ,Ψ1(ρ)=sinρ(11ρ2)+cosρρ,ξ1(ρ)=Ψ1(ρ)i(cosρρ+sinρ),ξ1(ρ)=Ψ1(ρ)+i[sinρρ(11ρ2)cosρ],
ϵr=ϵrf+ϵrb,
ϵrf=1ωpe2ω2+iγeω.
ωpe=ω1ϵr+ϵr21ϵr,
γe=ω|ϵr|1ϵr,
ϵrb=ϵrϵrf,
γe=γebulk+vFLeff,

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