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

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    [CrossRef]
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2010 (4)

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

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]

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]

2009 (1)

2008 (1)

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]

2007 (1)

2006 (2)

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]

H. J. Fan, P. Werner, and M. Zacharias, “Semiconductor nanowires: from self-organization to patterned growth,” Small 2(6), 700–717 (2006).
[CrossRef] [PubMed]

2005 (4)

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]

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]

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]

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]

2004 (2)

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. 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]

2003 (2)

S. Rahman and H. Yang, “Nanopillar arrays of glassy carbon by anodic aluminum oxide nanoporous templates,” Nano Lett. 3(4), 439–442 (2003).
[CrossRef]

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]

2001 (1)

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]

2000 (1)

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]

1999 (1)

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]

1996 (2)

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]

K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations,” IEEE Trans. Antenn. Propag. 14, 302 (1996).

1995 (2)

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]

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]

1991 (1)

1989 (2)

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

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

1978 (1)

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

1965 (1)

1881 (1)

L. Rayleigh, “On copying diffraction-gratings, and on some phenomena connected therewith,” Philos. Mag. 11, 196 (1881).

1836 (1)

H. F. Talbot, “Facts relating to optical science,” Philos. Mag. 9, 401 (1836).

Azaña, J.

Behrmann, G. P.

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]

Bergman, D.

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

Boltasseva, A.

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]

Cai, W.

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]

Chapman, M. S.

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]

Chen, C.

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]

Chen, W.

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]

Christodoulides, D. N.

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]

Dennis, M. R.

Ekstrom, C. R.

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]

Fan, H. J.

H. J. Fan, P. Werner, and M. Zacharias, “Semiconductor nanowires: from self-organization to patterned growth,” Small 2(6), 700–717 (2006).
[CrossRef] [PubMed]

Fukuda, K.

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]

Gan, C. H.

García de Abajo, F. J.

Genov, D. A.

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]

Hammond, T. D.

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]

Hobbs, K. L.

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]

Holden, A. J.

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]

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]

Hvam, J.

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]

Iwanow, R.

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]

Johnson, M. B.

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]

Kawata, S.

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]

Keay, J. C.

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]

Kristensen, A.

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]

Lalanne, P.

Larson, P. R.

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]

Lian, G. D.

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]

Liu, L.

Liu, S.

Liu, Z.

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]

Lu, Z.

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]

Masuda, H.

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]

May-Arrioja, D. A.

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]

Mehuys, D.

Min, Y.

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]

Nielsen, R.

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]

Patorski, K.

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

Pendry, J. B.

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]

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

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]

Prather, D. W.

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]

Pritchard, D. E.

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]

Rahman, S.

S. Rahman and H. Yang, “Nanopillar arrays of glassy carbon by anodic aluminum oxide nanoporous templates,” Nano Lett. 3(4), 439–442 (2003).
[CrossRef]

Rayleigh, L.

L. Rayleigh, “On copying diffraction-gratings, and on some phenomena connected therewith,” Philos. Mag. 11, 196 (1881).

Robbins, D. J.

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]

Schmiedmayer, J.

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]

Schuetz, C. A.

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]

Schultz, S.

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]

Schurig, D.

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]

Shalaev, V.

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]

Shalaev, V. M.

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]

Shelby, R. A.

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]

Shi, S.

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]

Shoji, S.

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]

Smith, D. R.

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

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]

Sohler, W.

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]

Song, Y.

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Phys. Rev. B (1)

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).
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W. Cai, D. A. Genov, and V. M. Shalaev, “Superlens based on metal-dielectric composites,” Phys. Rev. B Condens. Matter 72(19), 193101 (2005).
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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).
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Figures (4)

Fig. 1
Fig. 1

(a) Schematic illustration of the simulated structure, the 1D grating is assumed to be infinite in y-axis. (b) No Talbot effect is seen in the air when the period of the grating is much smaller than the incident wavelength. (c) Periodic Talbot carpet pattern can be observed in an indefinite metamaterial (shown in H field distribution).

Fig. 2
Fig. 2

(a) Talbot effect in the air. (b) Talbot effect in an indefinite metamaterial. Both (a) and (b) are shown in power.

Fig. 3
Fig. 3

(a) Super Talbot effect in an Ag-SiO2 stack (shown in power). (b) Cross-sectional power profile along the white solid line shown in (a), where x = 52nm.

Fig. 4
Fig. 4

(a) Illustration of the structure. Hole diameter: 2r = 80nm, hole array periods: D = 150nm along x and y axes. Incident wavelength is λ 0 = 630nm. (b) The Talbot carpet pattern in the vertical z-y plane at x = 0. (c) Talbot carpet pattern in the horizontal z-x plane at y = 0. (d) One integer Talbot imaging plane. (e) One fractional Talbot imaging plane with z ≈2/3Z T. (b-e) are shown in |E|2.

Equations (7)

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k z 2 ε x + k x 2 ε z = k 0 2 ,
k z = ε x ( k 0 2 k x 2 ε z ) .
T ( x , z ) = m f m e j m q x x e j k z z ,
T ( x , z ) = m f m e j m q x x e j z k 0 ε x 1 + m 2 ε z ( q x k 0 ) 2 .
T ( x , z ) = m f m e j m q x x e j 2 π m z / ( D ε z / ε x ) .
Z T ε z ε x D ,
{ ε = ε i + η ε m 1 + η ε 1 = ε i 1 + η ε m 1 1 + η ,

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