## Super Talbot effect in indefinite metamaterial |

Optics Express, Vol. 19, Issue 16, pp. 15297-15303 (2011)

http://dx.doi.org/10.1364/OE.19.015297

Acrobat PDF (2904 KB)

### Abstract

The Talbot effect (or the self-imaging effect) can be observed for a periodic object with a pitch larger than the diffraction limit of an imaging system, where the paraxial approximation is applied. In this paper, we show that the super Talbot effect can be achieved in an indefinite metamaterial even when the period is much smaller than the diffraction limit in both two-dimensional and three-dimensional numerical simulations, where the paraxial approximation is *not* applied. This is attributed to the evanescent waves, which carry the information about subwavelength features of the object, can be converted into propagating waves and then conveyed to far field by the metamaterial, where the permittivity in the propagation direction is negative while the transverse ones are positive. The indefinite metamaterial can be approximated by a system of thin, alternating multilayer metal and insulator (MMI) stack. As long as the loss of the metamaterial is small enough, deep subwavelength image size can be obtained in the super Talbot effect.

© 2011 OSA

## 1. Introduction

1. J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, “Extremely low frequency plasmons in metallic mesostructures,” Phys. Rev. Lett. **76**(25), 4773–4776 (1996). [CrossRef] [PubMed]

4. D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science **305**(5685), 788–792 (2004). [CrossRef] [PubMed]

*k*,

_{x}*k*, and

_{z}*k*

_{0}are the wave vectors in transverse direction

*x*, propagation direction

*z*and in free space, respectively. From Eq. (1) we have

*ε*< 0 and

_{z}*ε*> 0, there is no cutoff for any frequency

_{x}*k*in the metamaterial. In other words, the indefinite metamaterial can convert the evanescent waves which would normally decay in conventional material into propagating waves. Compared with the circular dispersion relation in natural material (for example, air), the dispersion curve for an indefinite metamaterial is in hyperbolic form, where

_{x}*k*does not have an upper limit and can vary in (-∞, + ∞). In particular, when

_{x}5. Z. Lu, C. A. Schuetz, S. Shi, C. Chen, G. P. Behrmann, and D. W. Prather, “Experimental demonstration of self-collimation in low index contrast photonic crystals in the millimeter wave regime,” IEEE Trans. Microw. Theory Tech. **53**(4), 1362–1368 (2005). [CrossRef]

10. D. Mehuys, W. Streifer, R. G. Waarts, and D. F. Welch, “Modal analysis of linear Talbot-cavity semiconductor lasers,” Opt. Lett. **16**(11), 823–825 (1991). [CrossRef] [PubMed]

11. J. Azaña, “Spectral Talbot phenomena of frequency combs induced by cross-phase modulation in optical fibers,” Opt. Lett. **30**(3), 227–229 (2005). [CrossRef] [PubMed]

12. K. Patorski, “The self-imaging phenomenon and its applications,” Prog. Opt. **27**, 1–108 (1989). [CrossRef]

13. L. Liu, “Lau cavity and phase locking of laser arrays,” Opt. Lett. **14**(23), 1312–1314 (1989). [CrossRef] [PubMed]

15. M. S. Chapman, C. R. Ekstrom, T. D. Hammond, J. Schmiedmayer, B. E. Tannian, S. Wehinger, and D. E. Pritchard, “Near-field imaging of atom diffraction gratings: the atomic Talbot effect,” Phys. Rev. A **51**(1), R14–R17 (1995). [CrossRef] [PubMed]

16. R. Iwanow, D. A. May-Arrioja, D. N. Christodoulides, G. I. Stegeman, Y. Min, and W. Sohler, “Discrete Talbot effect in waveguide arrays,” Phys. Rev. Lett. **95**(5), 053902 (2005). [CrossRef] [PubMed]

17. Y. Wang, K. Zhou, X. Zhang, K. Yang, Y. Wang, Y. Song, and S. Liu, “Discrete plasmonic Talbot effect in subwavelength metal waveguide arrays,” Opt. Lett. **35**(5), 685–687 (2010). [CrossRef] [PubMed]

18. Y. Zhang, J. Wen, S. N. Zhu, and M. Xiao, “Nonlinear Talbot effect,” Phys. Rev. Lett. **104**(18), 183901 (2010). [CrossRef] [PubMed]

*without using the paraxial approximation*.

## 2. Super Talbot effect demonstrated in 2D simulations

### 2.1 Super Talbot effect in an indefinite metamaterial

*λ*

_{0}is much larger than the period

*D*, the paraxial approximation cannot be applied any more. In our case, light propagates in the indefinite metamaterial behind the input grating structure can be expressed in Fourier series form aswhere

*k*is expressed by Eq. (2). Replacing

_{z}*k*by

_{x}*mq*in Eq. (2) and then plugging Eq. (2) into Eq. (3), we get

_{x}*ε*= −1 and

_{z}*ε*= 1) can be calculated approximately as 100nm, while the Talbot distance measured in the simulation is around 117 nm. The difference may come from that the ratio of the wavelength

_{x}*λ*

_{0}and the period

*D*is only around 6.3 while Eq. (6) is based on the condition that

*D*= 2

*µ*m and

*d*= 100nm) in a conventional medium and in an indefinite metamaterial, respectively. As shown in Fig. 2(a) , the Talbot effect can be observed in a conventional material (for example in the air) under paraxial approximation, where the incident wavelength is still 630nm. If the medium behind the grating is replaced by an indefinite metamaterial with

*ε*= −1 and

_{z}*ε*= 1, the Talbot carpet pattern still can be clearly observed (shown in Fig. 2(b)). The feature size of the squared hot spot in Fig. 2(b) along the transverse direction measured is approximately as 160nm, which is much smaller than that in the air, 240nm. The result is within our expectation since the high spatial-frequency waves can be conveyed to far field in the indefinite metamaterial, as mentioned previously.

_{x}### 2.2 Super Talbot effect in a multilayer metal-insulator (MMI) stack

21. D. R. Smith and D. Schurig, “Electromagnetic wave propagation in media with indefinite permittivity and permeability tensors,” Phys. Rev. Lett. **90**(7), 077405 (2003). [CrossRef] [PubMed]

23. R. Nielsen, M. Thoreson, W. Chen, A. Kristensen, J. Hvam, V. Shalaev, and A. Boltasseva, “Toward superlensing with metal-dielectric composites and multilayers,” Appl. Phys. B **100**(1), 93–100 (2010). [CrossRef]

24. Y. Xiong, Z. Liu, and X. Zhang, “Projecting deep-subwavelength patterns from diffraction-limited masks using metal-dielectric multilayers,” Appl. Phys. Lett. **93**(11), 111116 (2008). [CrossRef]

25. C. H. Gan and P. Lalanne, “Well-confined surface plasmon polaritons for sensing applications in the near-infrared,” Opt. Lett. **35**(4), 610–612 (2010). [CrossRef] [PubMed]

26. D. Bergman, “The dielectric constant of a composite material—a problem in classical physics,” Phys. Rep. **43**(9), 377–407 (1978). [CrossRef]

26. D. Bergman, “The dielectric constant of a composite material—a problem in classical physics,” Phys. Rep. **43**(9), 377–407 (1978). [CrossRef]

27. B. Wood, J. B. Pendry, and D. P. Tsai, “Directed subwavelength imaging using a layered metal-dielectric system,” Phys. Rev. B **74**(11), 115116 (2006). [CrossRef]

*η*is the thickness ratio of the two layers and defined by

*ε*and

_{m}*ε*are the permittivity of the metal and the insulator, respectively. By carefully choosing two suitable materials and a thickness ratio

_{i}*η*,

*ε*

_{Π}and

*ε*

_{⊥}could have opposite signs.

_{2}were selected as the metal and insulator, respectively. The structure we investigated is similar as shown in Fig. 1(a), where the material behind the grating was replaced by the MMI stack composed of Ag and SiO

_{2}thin layers and they were assumed to be infinite in

*y*-axis. The alternating layers were stacked in the

*x*direction and each layer has a thickness of 5nm. In the simulations, periodic boundary conditions were implemented for the boundaries parallel to the

*z*-axis. The wavelength of the incident TM-polarized plane wave

*λ*

_{0}= 630nm, the period of the grating

*D*= 94nm and the duty cycle = 50%. At the wavelength of interest, the permittivity of Ag is

*ε*

_{Ag}= −15.69 +

*i*1.06 and that of SiO

_{2}is

*ε*

_{SiO2}= 2.12. As

*η*= 1, from Eq. (7) the effective permittivity of the MMI stack is calculated to be

*ε*= −6.79 +

_{z}*i*0.53 and

*ε*= 4.9 +

_{x}*i*0.05. Even the material loss is considered, the super Talbot effect is still obvious as shown in Fig. 3(a) . Figure 3(b) plots the cross-sectional profile of the power in

*z*-axis (where

*x*= 52nm, the horizontal white solid line in the figure). Each of the peaks numbered from 1 to 7 of the curve represents the location of one Talbot imaging plane where the self-imaging phenomenon occurs. The intensity gradually decays due to the attenuation of the MMI stack. The distance between the two adjacent peaks is the Talbot distance

*Z*

_{T}, which is approximately measured as 152nm. The Talbot distance calculated by Eq. (6) is around 110nm. Both the effective medium theory and the long wavelength approximation may contribute to the difference. If the attenuation of the MMI stack is negligible, deep subwavelength image size can be achieved in the super Talbot effect. At point 1, where

*z*= 40nm (along the first dashed vertical line as shown in Fig. 3(a)), the full-width half-maximum (FWHM) of one image hot spot is measured as 35nm (or 0.056

*λ*

_{0}). At another Talbot imaging plane, where

*z*= 642nm (point 5 in Fig. 3(b), along the second dashed vertical line in Fig. 3(a)), a subwavelength image size of 55nm (or 0.087

*λ*

_{0}) still can be achieved. Comparing the cross-sectional power profiles of point 1 and point 5 (which are not shown in the paper), the contrast of the power at point 5 is not as sharp as that of point 1 and the quality of the image is worse, which are attributed to the loss of high-order diffraction light in the MMI stack.

## 3. Super Talbot effect demonstrated in 3D simulations

*y*-axis is also periodic, the super Talbot effect in a 3D configuration needs to be explored. Here we mimic a simple scenario in nanolithography, where light is incident from a chrome (Cr) mask with periodic subwavelength holes, as shown in Fig. 4(a) . The incident plane wave is polarized in

*x*-axis and has a wavelength of 630nm. The parameters of the hole-array are as below: hole diameter 2

*r*= 80nm and the periods along the

*x*and

*y*axes are identical as

*D*= 150nm. For simplicity, behind the Cr (

*ε*

_{Cr}= −6.3 +

*i*31.2) mask is an indefinite metamaterial which is assumed to be lossless and the permittivity components are

*ε*= −4, and

_{z}*ε*=

_{x}*ε*= 1. The origin of the whole system is defined at the center of the hole (marked with the red solid circle) with

_{y}*z*= 0 at the interface between the Cr mask and the indefinite metamaterial.

^{2}distribution of the vertical

*z*-

*y*plane at

*x*= 0 is depicted, and similarly, Fig. 4(c) for the horizontal

*z*-

*x*plane at

*y*= 0. Clearly, the super Talbot effect can be obtained in both the vertical and horizontal planes. The Talbot distance

*Z*

_{T}(as shown in Fig. 4(c)) measured is about (300 ± 15) nm, which agrees with the result from Eq. (6), calculated as 300nm. The asymmetric pattern distributions as shown in Fig. 4(b) and 4(c) may be attributed to the polarization dependent effect. Figure 4(d) shows the image in one of the Talbot imaging planes (

*x*-

*y*plane,

*z*= 365nm). The positions of the hot spots coincide with those of the periodic holes on the Cr mask. The size of each hot spot (FWHM) is approximately 40nm-by-40nm (or 0.0635

*λ*

_{0}-by-0.0635

*λ*

_{0}). Besides the integer self-imaging planes, we also observed some fractional Talbot imaging planes which locate between two adjacent integer Talbot planes [28

28. J. Winthrop and C. R. Worthington, “Theory of Fresnel images. I. Plane periodic objects in monochromatic light,” J. Opt. Soc. Am. **55**(4), 373 (1965). [CrossRef]

*z*=

*p*/

*qZ*

_{T}(

*p*and

*q*are prime integers). One representative fractional Talbot imaging plane with

*z*≈2/3

*Z*

_{T}was shown in Fig. 4(e). Comparing with the integer Talbot imaging plane [Fig. 4(d)], the image on the fractional Talbot imaging plane as shown in Fig. 4(e) has the same spatial frequency but is shifted about 0.5 periods (75nm) along

*x*-axis. The image size (FWHM) of those hot spots is measured as 30nm-by-40nm (or 0.0476

*λ*

_{0}-by-0.0635

*λ*

_{0}). Low loss 3D indefinite metamaterials may be constructed by an array of aligned metallic nanowires (

*ε*< 0) embedded in a dielectric host (

_{m}*ε*> 0). The fabrication of the nanowire-based metamaterial is based on a well-developed technique named “template synthetic method” [29

_{d}29. H. Masuda and K. Fukuda, “Ordered metal nanohole arrays made by a two-step replication of honeycomb structures of anodic alumina,” Science **268**(5216), 1466–1468 (1995). [CrossRef] [PubMed]

32. K. L. Hobbs, P. R. Larson, G. D. Lian, J. C. Keay, and M. B. Johnson, “Fabrication of nanoring arrays by sputter redeposition using porous alumina templates,” Nano Lett. **4**(1), 167–171 (2004). [CrossRef]

## 4. Conclusion

*λ*

_{0}) can be achieved. The super Talbot effect may find a variety of applications in the fields as nanolithography and optical storage. In particular, 3D photonic crystals may be fabricated based the super 3D self-imaging phenomenon [33

33. S. Shoji and S. Kawata, “Photofabrication of three-dimensional photonic crystals by multibeam laser interference into a photopolymerizable resin,” Appl. Phys. Lett. **76**(19), 2668 (2000). [CrossRef]

## Acknowledgments

## References and links

1. | J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, “Extremely low frequency plasmons in metallic mesostructures,” Phys. Rev. Lett. |

2. | J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microw. Theory Tech. |

3. | R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science |

4. | D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science |

5. | Z. Lu, C. A. Schuetz, S. Shi, C. Chen, G. P. Behrmann, and D. W. Prather, “Experimental demonstration of self-collimation in low index contrast photonic crystals in the millimeter wave regime,” IEEE Trans. Microw. Theory Tech. |

6. | H. F. Talbot, “Facts relating to optical science,” Philos. Mag. |

7. | A. Taflove and S. C. Hagness, |

8. | K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations,” IEEE Trans. Antenn. Propag. |

9. | L. Rayleigh, “On copying diffraction-gratings, and on some phenomena connected therewith,” Philos. Mag. |

10. | D. Mehuys, W. Streifer, R. G. Waarts, and D. F. Welch, “Modal analysis of linear Talbot-cavity semiconductor lasers,” Opt. Lett. |

11. | J. Azaña, “Spectral Talbot phenomena of frequency combs induced by cross-phase modulation in optical fibers,” Opt. Lett. |

12. | K. Patorski, “The self-imaging phenomenon and its applications,” Prog. Opt. |

13. | L. Liu, “Lau cavity and phase locking of laser arrays,” Opt. Lett. |

14. | J. M. Cowley, Diffraction Physics (North-Holland, Amsterdam, 1995). |

15. | M. S. Chapman, C. R. Ekstrom, T. D. Hammond, J. Schmiedmayer, B. E. Tannian, S. Wehinger, and D. E. Pritchard, “Near-field imaging of atom diffraction gratings: the atomic Talbot effect,” Phys. Rev. A |

16. | R. Iwanow, D. A. May-Arrioja, D. N. Christodoulides, G. I. Stegeman, Y. Min, and W. Sohler, “Discrete Talbot effect in waveguide arrays,” Phys. Rev. Lett. |

17. | Y. Wang, K. Zhou, X. Zhang, K. Yang, Y. Wang, Y. Song, and S. Liu, “Discrete plasmonic Talbot effect in subwavelength metal waveguide arrays,” Opt. Lett. |

18. | Y. Zhang, J. Wen, S. N. Zhu, and M. Xiao, “Nonlinear Talbot effect,” Phys. Rev. Lett. |

19. | M. R. Dennis, N. I. Zheludev, and F. J. García de Abajo, “The plasmon Talbot effect,” Opt. Express |

20. | W. Zhang, C. Zhao, J. Wang, and J. Zhang, “An experimental study of the plasmonic Talbot effect,” Opt. Express |

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

22. | W. Cai, D. A. Genov, and V. M. Shalaev, “Superlens based on metal-dielectric composites,” Phys. Rev. B Condens. Matter |

23. | R. Nielsen, M. Thoreson, W. Chen, A. Kristensen, J. Hvam, V. Shalaev, and A. Boltasseva, “Toward superlensing with metal-dielectric composites and multilayers,” Appl. Phys. B |

24. | Y. Xiong, Z. Liu, and X. Zhang, “Projecting deep-subwavelength patterns from diffraction-limited masks using metal-dielectric multilayers,” Appl. Phys. Lett. |

25. | C. H. Gan and P. Lalanne, “Well-confined surface plasmon polaritons for sensing applications in the near-infrared,” Opt. Lett. |

26. | D. Bergman, “The dielectric constant of a composite material—a problem in classical physics,” Phys. Rep. |

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

28. | J. Winthrop and C. R. Worthington, “Theory of Fresnel images. I. Plane periodic objects in monochromatic light,” J. Opt. Soc. Am. |

29. | H. Masuda and K. Fukuda, “Ordered metal nanohole arrays made by a two-step replication of honeycomb structures of anodic alumina,” Science |

30. | H. J. Fan, P. Werner, and M. Zacharias, “Semiconductor nanowires: from self-organization to patterned growth,” Small |

31. | S. Rahman and H. Yang, “Nanopillar arrays of glassy carbon by anodic aluminum oxide nanoporous templates,” Nano Lett. |

32. | K. L. Hobbs, P. R. Larson, G. D. Lian, J. C. Keay, and M. B. Johnson, “Fabrication of nanoring arrays by sputter redeposition using porous alumina templates,” Nano Lett. |

33. | S. Shoji and S. Kawata, “Photofabrication of three-dimensional photonic crystals by multibeam laser interference into a photopolymerizable resin,” Appl. Phys. Lett. |

**OCIS Codes**

(070.6760) Fourier optics and signal processing : Talbot and self-imaging effects

(160.3918) Materials : Metamaterials

**ToC Category:**

Physical Optics

**History**

Original Manuscript: April 6, 2011

Revised Manuscript: June 2, 2011

Manuscript Accepted: June 16, 2011

Published: July 26, 2011

**Citation**

Wangshi Zhao, Xiaoyue Huang, and Zhaolin Lu, "Super Talbot effect in indefinite metamaterial," Opt. Express **19**, 15297-15303 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-16-15297

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### References

- J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, “Extremely low frequency plasmons in metallic mesostructures,” Phys. Rev. Lett. 76(25), 4773–4776 (1996). [CrossRef] [PubMed]
- J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microw. Theory Tech. 47(11), 2075–2084 (1999). [CrossRef]
- R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292(5514), 77–79 (2001). [CrossRef] [PubMed]
- D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science 305(5685), 788–792 (2004). [CrossRef] [PubMed]
- Z. Lu, C. A. Schuetz, S. Shi, C. Chen, G. P. Behrmann, and D. W. Prather, “Experimental demonstration of self-collimation in low index contrast photonic crystals in the millimeter wave regime,” IEEE Trans. Microw. Theory Tech. 53(4), 1362–1368 (2005). [CrossRef]
- H. F. Talbot, “Facts relating to optical science,” Philos. Mag. 9, 401 (1836).
- A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, 2005).
- K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations,” IEEE Trans. Antenn. Propag. 14, 302 (1996).
- L. Rayleigh, “On copying diffraction-gratings, and on some phenomena connected therewith,” Philos. Mag. 11, 196 (1881).
- D. Mehuys, W. Streifer, R. G. Waarts, and D. F. Welch, “Modal analysis of linear Talbot-cavity semiconductor lasers,” Opt. Lett. 16(11), 823–825 (1991). [CrossRef] [PubMed]
- J. Azaña, “Spectral Talbot phenomena of frequency combs induced by cross-phase modulation in optical fibers,” Opt. Lett. 30(3), 227–229 (2005). [CrossRef] [PubMed]
- K. Patorski, “The self-imaging phenomenon and its applications,” Prog. Opt. 27, 1–108 (1989). [CrossRef]
- L. Liu, “Lau cavity and phase locking of laser arrays,” Opt. Lett. 14(23), 1312–1314 (1989). [CrossRef] [PubMed]
- J. M. Cowley, Diffraction Physics (North-Holland, Amsterdam, 1995).
- M. S. Chapman, C. R. Ekstrom, T. D. Hammond, J. Schmiedmayer, B. E. Tannian, S. Wehinger, and D. E. Pritchard, “Near-field imaging of atom diffraction gratings: the atomic Talbot effect,” Phys. Rev. A 51(1), R14–R17 (1995). [CrossRef] [PubMed]
- R. Iwanow, D. A. May-Arrioja, D. N. Christodoulides, G. I. Stegeman, Y. Min, and W. Sohler, “Discrete Talbot effect in waveguide arrays,” Phys. Rev. Lett. 95(5), 053902 (2005). [CrossRef] [PubMed]
- Y. Wang, K. Zhou, X. Zhang, K. Yang, Y. Wang, Y. Song, and S. Liu, “Discrete plasmonic Talbot effect in subwavelength metal waveguide arrays,” Opt. Lett. 35(5), 685–687 (2010). [CrossRef] [PubMed]
- Y. Zhang, J. Wen, S. N. Zhu, and M. Xiao, “Nonlinear Talbot effect,” Phys. Rev. Lett. 104(18), 183901 (2010). [CrossRef] [PubMed]
- M. R. Dennis, N. I. Zheludev, and F. J. García de Abajo, “The plasmon Talbot effect,” Opt. Express 15(15), 9692–9700 (2007). [CrossRef] [PubMed]
- W. Zhang, C. Zhao, J. Wang, and J. Zhang, “An experimental study of the plasmonic Talbot effect,” Opt. Express 17(22), 19757–19762 (2009). [CrossRef] [PubMed]
- D. R. Smith and D. Schurig, “Electromagnetic wave propagation in media with indefinite permittivity and permeability tensors,” Phys. Rev. Lett. 90(7), 077405 (2003). [CrossRef] [PubMed]
- W. Cai, D. A. Genov, and V. M. Shalaev, “Superlens based on metal-dielectric composites,” Phys. Rev. B Condens. Matter 72(19), 193101 (2005). [CrossRef]
- R. Nielsen, M. Thoreson, W. Chen, A. Kristensen, J. Hvam, V. Shalaev, and A. Boltasseva, “Toward superlensing with metal-dielectric composites and multilayers,” Appl. Phys. B 100(1), 93–100 (2010). [CrossRef]
- Y. Xiong, Z. Liu, and X. Zhang, “Projecting deep-subwavelength patterns from diffraction-limited masks using metal-dielectric multilayers,” Appl. Phys. Lett. 93(11), 111116 (2008). [CrossRef]
- C. H. Gan and P. Lalanne, “Well-confined surface plasmon polaritons for sensing applications in the near-infrared,” Opt. Lett. 35(4), 610–612 (2010). [CrossRef] [PubMed]
- D. Bergman, “The dielectric constant of a composite material—a problem in classical physics,” Phys. Rep. 43(9), 377–407 (1978). [CrossRef]
- B. Wood, J. B. Pendry, and D. P. Tsai, “Directed subwavelength imaging using a layered metal-dielectric system,” Phys. Rev. B 74(11), 115116 (2006). [CrossRef]
- J. Winthrop and C. R. Worthington, “Theory of Fresnel images. I. Plane periodic objects in monochromatic light,” J. Opt. Soc. Am. 55(4), 373 (1965). [CrossRef]
- H. Masuda and K. Fukuda, “Ordered metal nanohole arrays made by a two-step replication of honeycomb structures of anodic alumina,” Science 268(5216), 1466–1468 (1995). [CrossRef] [PubMed]
- H. J. Fan, P. Werner, and M. Zacharias, “Semiconductor nanowires: from self-organization to patterned growth,” Small 2(6), 700–717 (2006). [CrossRef] [PubMed]
- S. Rahman and H. Yang, “Nanopillar arrays of glassy carbon by anodic aluminum oxide nanoporous templates,” Nano Lett. 3(4), 439–442 (2003). [CrossRef]
- K. L. Hobbs, P. R. Larson, G. D. Lian, J. C. Keay, and M. B. Johnson, “Fabrication of nanoring arrays by sputter redeposition using porous alumina templates,” Nano Lett. 4(1), 167–171 (2004). [CrossRef]
- S. Shoji and S. Kawata, “Photofabrication of three-dimensional photonic crystals by multibeam laser interference into a photopolymerizable resin,” Appl. Phys. Lett. 76(19), 2668 (2000). [CrossRef]

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