## Hyperbolic metamaterial lens with hydrodynamic nonlocal response |

Optics Express, Vol. 21, Issue 12, pp. 15026-15036 (2013)

http://dx.doi.org/10.1364/OE.21.015026

Acrobat PDF (1369 KB)

### Abstract

We investigate the effects of hydrodynamic nonlocal response in hyperbolic metamaterials (HMMs), focusing on the experimentally realizable parameter regime where unit cells are much smaller than an optical wavelength but much larger than the wavelengths of the longitudinal pressure waves of the free-electron plasma in the metal constituents. We derive the nonlocal corrections to the effective material parameters analytically, and illustrate the noticeable nonlocal effects on the dispersion curves numerically. As an application, we find that the focusing characteristics of a HMM lens in the local-response approximation and in the hydrodynamic Drude model can differ considerably. In particular, the optimal frequency for imaging in the nonlocal theory is blueshifted with respect to that in the local theory. Thus, to detect whether nonlocal response is at work in a hyperbolic metamaterial, we propose to measure the near-field distribution of a hyperbolic metamaterial lens.

© 2013 osa

## 1. Introduction

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

14. M. Yan and N. A. Mortensen, “Hollow-core infrared fiber incorporating metal-wire metamaterial,” Opt. Express **17**, 14851–14864 (2009) [CrossRef] [PubMed] .

7. M. A. Noginov, Y. A. Barnakov, G. Zhu, T. Tumkur, H. Li, and E. E. Narimanov, “Bulk photonic metamaterial with hyperbolic dispersion,” Appl. Phys. Lett. **94**, 151105 (2009) [CrossRef] .

8. P. Belov and Y. Hao, “Subwavelength imaging at optical frequencies using a transmission device formed by a periodic layered metal-dielectric structure operating in the canalization regime,” Phys. Rev. B **73**, 113110 (2006) [CrossRef] .

13. 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] .

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

14. M. Yan and N. A. Mortensen, “Hollow-core infrared fiber incorporating metal-wire metamaterial,” Opt. Express **17**, 14851–14864 (2009) [CrossRef] [PubMed] .

15. C. L. Cortes, W. Newman, S. Molesky, and Z. Jacob, “Quantum nanophotonics using hyperbolic metamaterials,” J. Opt. **14**, 063001 (2012) [CrossRef] .

*i.e.*with every location

**r**in the structure a certain value for the permittivity

*ε*(

**r**) is associated. In the LRA, the effective material parameters of HMMs have been thoroughly studied [16–20

20. L. Shen, T. Yang, and Y. Chau, “50/50 beam splitter using a one-dimensional metal photonic crystal with parabo-lalike dispersion,” Appl. Phys. Lett. **90**, 251909 (2007) [CrossRef] .

33. W. Yan, M. Wubs, and N. A. Mortensen, “Hyperbolic metamaterials: nonlocal response regularizes broadband supersingularity,” Phys. Rev. B **86**, 205429 (2012) [CrossRef] .

26. S. Raza, G. Toscano, A.-P. Jauho, M. Wubs, and N. A. Mortensen, “Unusual resonances in nanoplasmonic structures due to nonlocal response,” Phys. Rev. B **84**, 121412(R) (2011) [CrossRef] .

29. S. Raza, N. Stenger, S. Kadkhodazadeh, S. V. Fischer, N. Kostesha, A.-P. Jauho, A. Burrows, M. Wubs, and N. A. Mortensen, “Blueshift of the surface plasmon resonance in silver nanoparticles studied with EELS,” Nanophotonics **2**, 131 (2013) [CrossRef] .

30. G. Toscano, S. Raza, A.-P. Jauho, M. Wubs, and N. A. Mortensen, “Modified field enhancement and extinction in plasmonic nanowire dimers due to nonlocal response,” Opt. Express **13**, 4176–4188 (2012) [CrossRef] .

32. A. I. Fernández-Domínguez, A. Wiener, F. J. García-Vidal, S. A. Maier, and J. B. Pendry, “Transformation-optics description of nonlocal effects in plasmonic nanostructures,” Phys. Rev. Lett. **108**, 106802 (2012) [CrossRef] [PubMed] .

20. L. Shen, T. Yang, and Y. Chau, “50/50 beam splitter using a one-dimensional metal photonic crystal with parabo-lalike dispersion,” Appl. Phys. Lett. **90**, 251909 (2007) [CrossRef] .

33. W. Yan, M. Wubs, and N. A. Mortensen, “Hyperbolic metamaterials: nonlocal response regularizes broadband supersingularity,” Phys. Rev. B **86**, 205429 (2012) [CrossRef] .

*potentials*associated with the metal constituents are nonlocal.

33. W. Yan, M. Wubs, and N. A. Mortensen, “Hyperbolic metamaterials: nonlocal response regularizes broadband supersingularity,” Phys. Rev. B **86**, 205429 (2012) [CrossRef] .

## 2. Dispersion relations of hyperbolic metamaterials

*d*, and thicknesses

*a*and

*b*of the dielectric and metal layers, respectively. The permittivity of the dielectric layer is

*ε*

_{d}. The metal is described in the HDM as a free-electron plasma with [21, 22, 26

26. S. Raza, G. Toscano, A.-P. Jauho, M. Wubs, and N. A. Mortensen, “Unusual resonances in nanoplasmonic structures due to nonlocal response,” Phys. Rev. B **84**, 121412(R) (2011) [CrossRef] .

23. W. L. Mochán, M. Castillo-Mussot, and R. G. Barrera, “Effect of plasma waves on the optical properties of metal-insulator superlattices,” Phys. Rev. B **15**, 1088–1098 (1987) [CrossRef] .

**86**, 205429 (2012) [CrossRef] .

*k*⊥ represents the Bloch wavevector in the direction of the periodicity,

*k||*the wavevector along the layers, and

*k*

_{0}=

*ω*/

*c*the free-space wavevector. We also introduced the derived wavevectors

26. S. Raza, G. Toscano, A.-P. Jauho, M. Wubs, and N. A. Mortensen, “Unusual resonances in nanoplasmonic structures due to nonlocal response,” Phys. Rev. B **84**, 121412(R) (2011) [CrossRef] .

**86**, 205429 (2012) [CrossRef] .

34. P. Jewsbury, “Electrodynamic boundary conditions at metal interfaces,” J. Phys. F: Met. Phys. **11**, 195–206 (1981) [CrossRef] .

*β*to zero.

## 3. Effective nonlocal material parameters

*ω*<

*ω*

_{p}, where

*i.e.*when neglecting both the finiteness of the unit cells and nonlocal response, we have the well-known dispersion relation in the LRA where

9. B. Wood, J. Pendry, and D. Tsai, “Directed subwavelength imaging using a layered metal-dielectric system,” Phys. Rev. B **74**, 115116 (2006) [CrossRef] .

*f*=

_{d}*a/d*and

*f*=

_{m}*b/d*. We note that Eq. (7) is satisfied when the unit-cell thickness is much smaller than the free-space wavelength indicated by the first inequality in Eq. (4). Obviously, when

*d*of the unit cell to zero, see also Ref. [33

**86**, 205429 (2012) [CrossRef] .

*i.e.*the nonlocal) correction term, while neglecting the other correction term Δ

_{la}. The sole justification for doing so is that the second term turns out to be negligible in comparison to the first one for the HMMs that we consider, as we illustrate numerically below. So, as an improved approximation to Eq. (6), we keep the correction term concerning

*f*

_{d}

*ε*

^{T}+

*f*

_{m}

*ε*

_{d}vanishes. In particular, neglecting the Drude damping

*γ*of the metal, the frequency

*ε*

_{d}, and inversely proportional to the dimensionless parameter

*a*is the thickness of the dielectric layer! We checked, also numerically, that our first-order Taylor expansion of the dispersion relation in the small parameter

*b*of the metal layer.

*ε*

_{d}= 1. From Eq. (11), we have the local resonance frequency

*a*

_{eff}into Eq. (11), we have

*ε*

_{d}= 1, and independent of the metal layer thickness.

## 4. Effects of nonlocal response on the dispersion curve: numerical analysis

*a*= 6nm,

*b*= 3nm, and

*ε*

_{d}= 10. We choose the metal to be Au, and describe it by only its free-electron response, with parameters

*h̄ω*

_{p}= 8.812eV,

*h̄γ*= 0.0752eV, and

*v*

_{F}= 1.39 × 10

^{6}m/s. Here, we note that it is a tough task to fabricate the HMM with a unit cell of a 3nm-thick metal layer and a dielectric layer of 6nm. However, with further progress in nano-fabrication techniques, we believe that such a HMM structure will be practically feasible in the near future. For example, very recently experimentalists have succeeded in fabricating a 3nm-thick Ag film by depositing the film on a 1nm copper seed layer [39

39. N. Formica, D. S. Ghosh, A. Carrilero, T. L. Chen, R. E. Simpson, and V. Pruneri, “Ultrastable and Atomically Smooth Ultrathin Silver Films Grown on a Copper Seed Layer,” ACS Appl. Mater. Interfaces **5**, 3048–3053 (2013) [CrossRef] [PubMed] .

*k*

_{0}

*d*and

*ω*

_{p}<

*ω*< 0.6

*ω*

_{p}, both

*k*

_{0}

*d*and

*ω*= 0.1

*ω*

_{p}and

*ω*= 0.6

*ω*

_{p}, respectively. These frequencies are far away from the resonances

*ω*= 0.1

*ω*

_{p}and 0.6

*ω*

_{p}.

*ω*= 0.41

*ω*

_{p}, the local dispersion curve consists of two nearly flat lines of

*ω*= 0.465

*ω*

_{p}. Here the local dispersion curve is a hyperbola. With the nonlocal response, however, the dispersion curve becomes nearly flat lines, which can be attributed to the extremely large value of

## 5. Effects of nonlocal response on a hyperbolic metamaterial lens

8. P. Belov and Y. Hao, “Subwavelength imaging at optical frequencies using a transmission device formed by a periodic layered metal-dielectric structure operating in the canalization regime,” Phys. Rev. B **73**, 113110 (2006) [CrossRef] .

12. Z. Jacob, L. Alekseyev, and E. Narimanov, “Optical hyperlens: Far-field imaging beyond the diffraction limit,” Opt. Express **14**, 8427–8256 (2006) [CrossRef] .

*k*

_{⊥}ensures that all plane-wave components experience the same phase changes after transmission through the HMM slab, at least if reflections can be neglected. Actually, the reflections can be suppressed by appropriate choice of the thickness

*l*of the HMM lens, and even be made to vanish by choosing

*k*

_{⊥}

*l*=

*nπ*, where

*n*is a positive integer. In theory this could lead to a perfect image at

*x*= 0.

*ω*around

*l*= 36

*d*,

*i.e.*composed of 36 unit cells, in a free-space background with the two boundaries at

*x*= −

*l*and

*x*= 0. The unit cell is as in Fig. 1, and is arranged in a symmetric sandwich structure with the metal layer at the center. A line dipole source is positioned to the left of the HMM slab, with

*x*-coordinate −

*l*−

*x*

_{s}and

*y*-coordinate 0, and is represented by

**J**=

*δ*(

*x*+

*l*+

*x*

_{s})

*δ*(

*y*)

*ŷ*. We choose the distance to the HMM slab to be

*x*

_{s}= 10nm. The interaction between the current source and the HMM slab is solved by using the transfer-matrix method of Refs. [23

23. W. L. Mochán, M. Castillo-Mussot, and R. G. Barrera, “Effect of plasma waves on the optical properties of metal-insulator superlattices,” Phys. Rev. B **15**, 1088–1098 (1987) [CrossRef] .

**86**, 205429 (2012) [CrossRef] .

*λ*

_{p}that corresponds to the plasma frequency. In the local case, we have the known the dispersion of Fig. 3(a) featuring flat lines. There

*k*

_{⊥}

*l*approximately equals ≈ 4.25

*π*, indicating that reflections do exist. However, the transmission coefficients for the different wave components vary more smoothly as a function of

**k**

_{||}, thanks to the flat dispersion curve with the constant

*k*

_{⊥}. Accordingly, after propagation through the HMM from

*x*= −

*l*to

*x*= 0, an image is formed near

*x*= 0, as seen in Fig. 4(a1). The size of the image is approximately

*λ*

_{p}/2, i.e., 0.2

*λ*

_{0}with

*λ*

_{0}being the operating wavelength corresponding to

*ω*

_{0}= 0.41

*ω*

_{p}, so that the size of the image is in the subwavelength scale. By contrast, in the nonlocal case the dispersion curve turns into an ellipse. This closed dispersion curve sets an upper wavevector cutoff for the evanescent waves, and the wave components below the cutoff experience different phase changes. Accordingly, the quality of the image becomes worse, see Fig. 4(a2).

*y*at the HMM boundary

*x*= 0, and also for the case without the HMM slab. It is seen that the electric-field intensity is nearly vanishing in the absence of the HMM slab, since the evanescent components vanish after traveling a distance of

*l*, which is of the order of one free-space wavelength. With the HMM slab in place, the electric-field intensity in the local case shows a subwavelength image that peaks at

*y*= 0, with a full width at half maximum (FWHM) of only 0.42

*λ*

_{p}. With nonlocal response, however, the electric-field intensity distribution for

*y*= ±0.55

*λ*

_{p}.

*k*

_{⊥}

*l*≈ 5

*π*, and this value of nearly an integer times

*π*indicates that reflections are nearly zero. As a result, the hydrodynamic Drude model predicts a better focusing performance of the HMM slab at

*y*at

*x*= 0 is shown. In the local case, the electric-field intensity shows several peaks with the strongest two at

*y*= ±0.33

*λ*. In the nonlocal case, the electric-field intensity is peaked at

_{p}*y*= 0 with a subwavelength FWHM of only 0.4

*λ*

_{p}.

## 6. Detecting nonlocal response by near-field measurement

*x*= 0. The images in Fig. 4(c1,c2) correspond to hypothetical measurements with infinitely small detectors. In experiments, the measured near-field signal will rather be an area-averaged electric-field intensity where

*D*is the finite detection area of the detector. Let us now assume that we have a near-field detector with detection area in the

*yz*-plane, with a square shape of size 40nm × 40nm, touching the HMM interface at

*x*= 0. For the same light source interacting with the HMM slab as in Fig. 4, we depict in Fig. 5 the calculated

*I*

_{av}as a function of the

*y*-coordinate of the center of the detector. It is seen that local and nonlocal response models give significantly different predictions for the measured signal, and thus the differences between the two models survive detection area averaging. The single peaks at 0.41

*ω*

_{p}for local response and at 0.465

*ω*

_{p}for nonlocal response are naturally broader than in Fig. 4(c1,c2), also due to the area averaging. Interestingly, as the frequency increases from 0.41

*ω*

_{p}to 0.465

*ω*

_{p}, the distribution of

*I*

_{av}becomes broader in the LRA, but narrower in the hydrodynamic Drude model.

## 7. Conclusions

27. J. A. Scholl, A. L. Koh, and J. A. Dionne, “Quantum plasmon resonances of individual metallic nanoparticles,” Nature **483**, 421–427 (2012) [CrossRef] [PubMed] .

29. S. Raza, N. Stenger, S. Kadkhodazadeh, S. V. Fischer, N. Kostesha, A.-P. Jauho, A. Burrows, M. Wubs, and N. A. Mortensen, “Blueshift of the surface plasmon resonance in silver nanoparticles studied with EELS,” Nanophotonics **2**, 131 (2013) [CrossRef] .

27. J. A. Scholl, A. L. Koh, and J. A. Dionne, “Quantum plasmon resonances of individual metallic nanoparticles,” Nature **483**, 421–427 (2012) [CrossRef] [PubMed] .

29. S. Raza, N. Stenger, S. Kadkhodazadeh, S. V. Fischer, N. Kostesha, A.-P. Jauho, A. Burrows, M. Wubs, and N. A. Mortensen, “Blueshift of the surface plasmon resonance in silver nanoparticles studied with EELS,” Nanophotonics **2**, 131 (2013) [CrossRef] .

36. L. Stella, P. Zhang, F. J. García-Vidal, A. Rubio, and P. García-González, “Performance of nonlocal optics when applied to plasmonic nanostructures,” J. Phys. Chem. C **117**, 8941–8949 (2013) [CrossRef] .

## Acknowledgments

## References and links

1. | D. R. Smith and D. Schurig, “Electromagnetic wave propagation in media with indefinite permittivity and permeability tensors,” Phys. Rev. Lett. |

2. | I. I. Smolyaninov, “Vacuum in a Strong Magnetic Field as a Hyperbolic Metamaterial,” Phys. Rev. Lett. |

3. | H. N. S. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, and V. M. Menon, “Topological transitions in metamaterials,” Science |

4. | Z. Jacob, I. Smolyaninov, and E. Narimanov, “Broadband Purcell effect: Radiative decay engineering with metamaterials,” Appl. Phys. Lett. |

5. | A. N. Poddubny, P. A. Belov, G. V. Naik, and Y. S. Kivshar, “Spontaneous radiation of a finite-size dipole emitter in hyperbolic media,” Phys. Rev. A |

6. | T. Tumkur, G. Zhu, P. Black, Y. A. Barnakov, C. E. Bonner, and M. A. Noginov, “Control of spontaneous emission in a volume of functionalized hyperbolic metamaterial,” Appl. Phys. Lett. |

7. | M. A. Noginov, Y. A. Barnakov, G. Zhu, T. Tumkur, H. Li, and E. E. Narimanov, “Bulk photonic metamaterial with hyperbolic dispersion,” Appl. Phys. Lett. |

8. | P. Belov and Y. Hao, “Subwavelength imaging at optical frequencies using a transmission device formed by a periodic layered metal-dielectric structure operating in the canalization regime,” Phys. Rev. B |

9. | B. Wood, J. Pendry, and D. Tsai, “Directed subwavelength imaging using a layered metal-dielectric system,” Phys. Rev. B |

10. | X. Li, S. He, and Y. Jin, “Subwavelength focusing with a multilayered Fabry-Perot structure at optical frequencies,” Phys. Rev. B |

11. | A. Salandrino and N. Engheta, “Far-field subdiffraction optical microscopy using metamaterial crystals: Theory and simulations,” Phys. Rev. B |

12. | Z. Jacob, L. Alekseyev, and E. Narimanov, “Optical hyperlens: Far-field imaging beyond the diffraction limit,” Opt. Express |

13. | Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science |

14. | M. Yan and N. A. Mortensen, “Hollow-core infrared fiber incorporating metal-wire metamaterial,” Opt. Express |

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16. | J. Elser, V. Podolksiy, I. Salakhutdinov, and I. Avrutsky, “Nonlocal effects in effective-medium response of nanolayered metamaterials,” Appl. Phys. Lett. |

17. | A. Chebykin, A. Orlov, A. Vozianova, S. Maslovski, Y. S. Kivshar, and P. Belov, “Nonlocal effective medium model for multilayered metal-dielectric metamaterials,” Phys. Rev. B |

18. | A. Chebykin, A. Orlov, C. Simovski, Y. S. Kivshar, and P. Belov, “Nonlocal effective parameters of multilayered metal-dielectric metamaterials,” Phys. Rev. B |

19. | L. Shen, T. Yang, and Y. Chau, “Effect of internal period on the optical dispersion of indefinite-medium materials,” Phys. Rev. B |

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21. | F. Bloch, “Bremsvermögen von Atomen mit mehreren Elektronen,” Z. Phys. A |

22. | A. D. Boardman, |

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30. | G. Toscano, S. Raza, A.-P. Jauho, M. Wubs, and N. A. Mortensen, “Modified field enhancement and extinction in plasmonic nanowire dimers due to nonlocal response,” Opt. Express |

31. | G. Toscano, S. Raza, S. Xiao, M. Wubs, A.-P. Jauho, S. I. Bozhevolnyi, and N. A. Mortensen, “Surface-enhanced Raman spectroscopy (SERS): nonlocal limitations,” Opt. Lett. |

32. | A. I. Fernández-Domínguez, A. Wiener, F. J. García-Vidal, S. A. Maier, and J. B. Pendry, “Transformation-optics description of nonlocal effects in plasmonic nanostructures,” Phys. Rev. Lett. |

33. | W. Yan, M. Wubs, and N. A. Mortensen, “Hyperbolic metamaterials: nonlocal response regularizes broadband supersingularity,” Phys. Rev. B |

34. | P. Jewsbury, “Electrodynamic boundary conditions at metal interfaces,” J. Phys. F: Met. Phys. |

35. | R. C. Monreal, T. J. Antosiewicz, and S. P. Apell, “Plasmons do not go that quantum,” arXiv:1304.3023 (2013). |

36. | L. Stella, P. Zhang, F. J. García-Vidal, A. Rubio, and P. García-González, “Performance of nonlocal optics when applied to plasmonic nanostructures,” J. Phys. Chem. C |

37. | T. Teperik, P. Nordlander, J. Aizpurua, and A. G. Borisov, “Quantum plasmonics: Nonlocal effects in coupled nanowire dimer,” arXiv:1302.3339 (2013). |

38. | K. Andersen, K. L. Jensen, and K. S. Thygesen, “Hybridization of quantum plasmon modes in coupled nanowires: From the classical to the tunneling regime,” arXiv:1304.4754 (2013). |

39. | N. Formica, D. S. Ghosh, A. Carrilero, T. L. Chen, R. E. Simpson, and V. Pruneri, “Ultrastable and Atomically Smooth Ultrathin Silver Films Grown on a Copper Seed Layer,” ACS Appl. Mater. Interfaces |

**ToC Category:**

Metamaterials

**History**

Original Manuscript: April 18, 2013

Revised Manuscript: May 14, 2013

Manuscript Accepted: May 15, 2013

Published: June 17, 2013

**Virtual Issues**

Hyperbolic Metamaterials (2013) *Optics Express*

**Citation**

Wei Yan, N. Asger Mortensen, and Martijn Wubs, "Hyperbolic metamaterial lens with hydrodynamic nonlocal response," Opt. Express **21**, 15026-15036 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-12-15026

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### References

- D. R. Smith and D. Schurig, “Electromagnetic wave propagation in media with indefinite permittivity and permeability tensors,” Phys. Rev. Lett.90, 077405 (2003). [CrossRef] [PubMed]
- I. I. Smolyaninov, “Vacuum in a Strong Magnetic Field as a Hyperbolic Metamaterial,” Phys. Rev. Lett.107, 253903 (2011). [CrossRef]
- H. N. S. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, and V. M. Menon, “Topological transitions in metamaterials,” Science336, 205–209 (2012). [CrossRef] [PubMed]
- Z. Jacob, I. Smolyaninov, and E. Narimanov, “Broadband Purcell effect: Radiative decay engineering with metamaterials,” Appl. Phys. Lett.100, 181105 (2012).
- A. N. Poddubny, P. A. Belov, G. V. Naik, and Y. S. Kivshar, “Spontaneous radiation of a finite-size dipole emitter in hyperbolic media,” Phys. Rev. A84, 023807 (2011). [CrossRef]
- T. Tumkur, G. Zhu, P. Black, Y. A. Barnakov, C. E. Bonner, and M. A. Noginov, “Control of spontaneous emission in a volume of functionalized hyperbolic metamaterial,” Appl. Phys. Lett.99, 151115 (2011). [CrossRef]
- M. A. Noginov, Y. A. Barnakov, G. Zhu, T. Tumkur, H. Li, and E. E. Narimanov, “Bulk photonic metamaterial with hyperbolic dispersion,” Appl. Phys. Lett.94, 151105 (2009). [CrossRef]
- P. Belov and Y. Hao, “Subwavelength imaging at optical frequencies using a transmission device formed by a periodic layered metal-dielectric structure operating in the canalization regime,” Phys. Rev. B73, 113110 (2006). [CrossRef]
- B. Wood, J. Pendry, and D. Tsai, “Directed subwavelength imaging using a layered metal-dielectric system,” Phys. Rev. B74, 115116 (2006). [CrossRef]
- X. Li, S. He, and Y. Jin, “Subwavelength focusing with a multilayered Fabry-Perot structure at optical frequencies,” Phys. Rev. B75, 045103 (2007). [CrossRef]
- A. Salandrino and N. Engheta, “Far-field subdiffraction optical microscopy using metamaterial crystals: Theory and simulations,” Phys. Rev. B74, 075103 (2006). [CrossRef]
- Z. Jacob, L. Alekseyev, and E. Narimanov, “Optical hyperlens: Far-field imaging beyond the diffraction limit,” Opt. Express14, 8427–8256 (2006). [CrossRef]
- Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science315, 1686 (2007). [CrossRef] [PubMed]
- M. Yan and N. A. Mortensen, “Hollow-core infrared fiber incorporating metal-wire metamaterial,” Opt. Express17, 14851–14864 (2009). [CrossRef] [PubMed]
- C. L. Cortes, W. Newman, S. Molesky, and Z. Jacob, “Quantum nanophotonics using hyperbolic metamaterials,” J. Opt.14, 063001 (2012). [CrossRef]
- J. Elser, V. Podolksiy, I. Salakhutdinov, and I. Avrutsky, “Nonlocal effects in effective-medium response of nanolayered metamaterials,” Appl. Phys. Lett.90, 191109 (2007).
- A. Chebykin, A. Orlov, A. Vozianova, S. Maslovski, Y. S. Kivshar, and P. Belov, “Nonlocal effective medium model for multilayered metal-dielectric metamaterials,” Phys. Rev. B84, 115438 (2011). [CrossRef]
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