Abstract

We characterize spatial dispersion in longitudinally invariant drawn metamaterials with a magnetic response at terahertz frequencies, whereby a change in the angle of the incident field produces a shift in the resonant frequency. We present a simple analytical model to predict this shift. We also demonstrate that the spatial dispersion is eliminated by breaking the longitudinal invariance using laser ablation. The experimental results are in agreement with numerical simulations.

© 2012 OSA

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  8. I. V. Shadrivov, A. A. Zharov, and Y. S. Kivshar, “Second-harmonic generation in nonlinear left-handed metamaterials,” J. Opt. Soc. Am. B23, 529–534 (2006).
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    [CrossRef]
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2011

R. C. McPhedran, I. V. Shadrivov, B. T. Kuhlmey, and K. Y. S, “Metamaterials and metaoptics,” NPG Asia Mater.3, 100–108 (2011).
[CrossRef]

M. Choi, S. H. Lee, Y. Kim, S. B. Kang, J. Shin, M. H. Kwak, K. Y. Kang, Y. H. Lee, N. Park, and B. Min, “A terahertz metamaterial with unnaturally high refractive index,” Nature470, 369–373 (2011).
[CrossRef] [PubMed]

C. M. Soukoulis and M. Wegener, “Past achievements and future challenges in the development of three-dimensional photonic metamaterials,” Nat. Photonics5, 523–530 (2011).

J. Anthony, R. Leonhardt, A. Argyros, and M. C. J. Large, “Characterization of a microstructured zeonex terahertz fiber,” J. Opt. Soc. Am. B28, 1013–1018 (2011).
[CrossRef]

A. Wang, A. Tuniz, P. G. Hunt, E. M. Pogson, R. A. Lewis, A. Bendavid, S. C. Fleming, B. T. Kuhlmey, and M. C. J. Large, “Fiber metamaterials with negative magnetic permeability in the terahertz,” Opt. Mater. Express1, 115–120 (2011).
[CrossRef]

A. Tuniz, R. Lwin, A. Argyros, S. C. Fleming, E. M. Pogson, E. Constable, R. A. Lewis, and B. T. Kuhlmey, “Stacked-and-drawn metamaterials with magnetic resonances in the terahertz range,” Opt. Express19, 16480–16490 (2011).
[CrossRef] [PubMed]

2010

P. W. Kolb, T. D. Corrigan, H. D. Drew, A. B. Sushkov, R. J. Phaneuf, A. Khanikaev, S. H. Mousavi, and G. Shvets, “Bianisotropy and spatial dispersion in highly anisotropic near-infrared resonator arrays,” Opt. Express18, 24025–24036 (2010).
[CrossRef] [PubMed]

H. Kim, J. S. Melinger, A. Khachatrian, N. A. Charipar, R. C. Y. Auyeung, and A. Piqué, “Fabrication of terahertz metamaterials by laser printing,” Opt. Lett.35, 4039–4041 (2010).
[CrossRef] [PubMed]

C. Menzel, T. Paul, C. Rockstuhl, T. Pertsch, S. Tretyakov, and F. Lederer, “Validity of effective material parameters for optical fishnet metamaterials,” Phys. Rev. B81, 035320 (2010).
[CrossRef]

A. Tuniz, B. T. Kuhlmey, R. Lwin, A. Wang, J. Anthony, R. Leonhardt, and S. C. Fleming, “Drawn metamaterials with plasmonic response at terahertz frequencies,” Appl. Phys. Lett.96, 191101 (2010).
[CrossRef]

E. Badinter, A. Ioisher, E. Monaico, V. Postolache, and I. M. Tiginyanu, “Exceptional integration of metal or semimetal nanowires in human-hair-like glass fiber,” Materials Lett.64, 1902–1904 (2010).
[CrossRef]

2009

J. K. Gansel, M. Thiel, S. M. Rill, M. Decker, K. Bade, S. Volker, G. Freymann, S. Linden, and M. Wegener, “Gold helix photonic metamaterial as broadband circular polarizer,” Science325, 1513–1515 (2009).
[CrossRef] [PubMed]

X. G. Peralta, M. C. Wanke, C. L. Arrington, J. D. Williams, I. Brener, A. Strikwerda, R. D. Averitt, W. J. Padilla, E. Smirnova, and A. J. Taylor, “Large-area metamaterials on thin membranes for multilayer and curved applications at terahertz and higher frequencies,” Appl. Phys. Lett.94, 161113 (2009).
[CrossRef]

M. Walther, A. Ortner, H. Meier, U. Loffelmann, P. J. Smith, and J. G. Korvink, “Terahertz metamaterials fabricated by inkjet printing,” Appl. Phys. Lett.95, 251107–251107 (2009).
[CrossRef]

S. Zhang, Y.-S. Park, J. Li, X. Lu, W. Zhang, and X. Zhang, “Negative refractive index in chiral metamaterials,” Phys. Rev. Lett.102, 023901 (2009).
[CrossRef] [PubMed]

C. E. Kriegler, M. S. Rill, M. Thiel, E. Müller, S. Essig, A. Frölich, G. von Freymann, S. Linden, D. Gerthsen, H. Hahn, K. Busch, and M. Wegener, “Transition between corrugated metal films and split-ring-resonator arrays,” Appl. Phys. B96, 749–755 (2009).
[CrossRef]

A. Argyros, “Microstructured polymer optical fibers,” J. Lightwave Technol.27, 1571–1579 (2009).
[CrossRef]

K. B. Alici and E. Ozbay, “Oblique response of a split-ring-resonator-based left-handed metamaterial slab,” Opt. Lett.34, 2294–2296 (2009).
[CrossRef] [PubMed]

C. Menzel, R. Singh, C. Rockstuhl, W. Zhang, and F. Lederer, “Effective properties of terahertz double split-ring resonators at oblique incidence,” J. Opt. Soc. Am. B26, B143–B147 (2009).
[CrossRef]

2008

R. Singh, E. Smirnova, A. J. Taylor, J. F. O’Hara, and W. Zhang, “Optically thin terahertz metamaterials,” Opt. Express16, 6537–6543 (2008).
[CrossRef] [PubMed]

I. V. Shadrivov, A. B. Kozyrev, D. W. van der Weide, and Y. S. Kivshar, “Nonlinear magnetic metamaterials,” Opt. Express16, 20266–20271 (2008).
[CrossRef] [PubMed]

M. Silveirinha and P. Belov, “Spatial dispersion in lattices of split ring resonators with permeability near zero,” Phys. Rev. B77, 233104 (2008).
[CrossRef]

A. Boltasseva and V. M. Shalaev, “Fabrication of optical negative-index metamaterials: Recent advances and outlook,” Metamaterials2, 1–17 (2008).
[CrossRef]

N. I. Landy, S. Sajuyigbe, J. J. Mock, D. R. Smith, and W. J. Padilla, “Perfect metamaterial absorber,” Phys. Rev. Lett.100, 207402 (2008).
[CrossRef] [PubMed]

2007

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]

V. M. Shalaev, “Optical negative-index metamaterials,” J. Opt. Soc. Am.1, 41–48 (2007).

M. C. K. Wiltshire, J. B. Pendry, W. Williams, and J. V. Hajnal, “An effective medium description of ’swiss rolls’, a magnetic metamaterial,” J. Phys-Condens. Mat.19, 456216 (2007).
[CrossRef]

2006

I. V. Shadrivov, A. A. Zharov, and Y. S. Kivshar, “Second-harmonic generation in nonlinear left-handed metamaterials,” J. Opt. Soc. Am. B23, 529–534 (2006).
[CrossRef]

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science314, 977–980 (2006).
[CrossRef] [PubMed]

P. Belov, Y. Hao, and S. Sudhakaran, “Subwavelength microwave imaging using an array of parallel conducting wires as a lens,” Phys. Rev. B73, 033108 (2006).
[CrossRef]

2003

P. A. Belov, R. Marqués, S. I. Maslovski, I. S. Nefedov, M. Silveirinha, C. R. Simovski, and S. A. Tretyakov, “Strong spatial dispersion in wire media in the very large wavelength limit,” Phys. Rev. B67, 113103 (2003).
[CrossRef]

J. C. Knight, “Photonic crystal fibres,” Nature424, 847–851 (2003).
[CrossRef] [PubMed]

2000

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

1999

J. Pendry, A. Holden, D. Robbins, and W. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microw. Theory Tech.47, 2075–2084 (1999).
[CrossRef]

1990

Alici, K. B.

Anthony, J.

J. Anthony, R. Leonhardt, A. Argyros, and M. C. J. Large, “Characterization of a microstructured zeonex terahertz fiber,” J. Opt. Soc. Am. B28, 1013–1018 (2011).
[CrossRef]

A. Tuniz, B. T. Kuhlmey, R. Lwin, A. Wang, J. Anthony, R. Leonhardt, and S. C. Fleming, “Drawn metamaterials with plasmonic response at terahertz frequencies,” Appl. Phys. Lett.96, 191101 (2010).
[CrossRef]

Argyros, A.

Arrington, C. L.

X. G. Peralta, M. C. Wanke, C. L. Arrington, J. D. Williams, I. Brener, A. Strikwerda, R. D. Averitt, W. J. Padilla, E. Smirnova, and A. J. Taylor, “Large-area metamaterials on thin membranes for multilayer and curved applications at terahertz and higher frequencies,” Appl. Phys. Lett.94, 161113 (2009).
[CrossRef]

Auyeung, R. C. Y.

Averitt, R. D.

X. G. Peralta, M. C. Wanke, C. L. Arrington, J. D. Williams, I. Brener, A. Strikwerda, R. D. Averitt, W. J. Padilla, E. Smirnova, and A. J. Taylor, “Large-area metamaterials on thin membranes for multilayer and curved applications at terahertz and higher frequencies,” Appl. Phys. Lett.94, 161113 (2009).
[CrossRef]

Bade, K.

J. K. Gansel, M. Thiel, S. M. Rill, M. Decker, K. Bade, S. Volker, G. Freymann, S. Linden, and M. Wegener, “Gold helix photonic metamaterial as broadband circular polarizer,” Science325, 1513–1515 (2009).
[CrossRef] [PubMed]

Badinter, E.

E. Badinter, A. Ioisher, E. Monaico, V. Postolache, and I. M. Tiginyanu, “Exceptional integration of metal or semimetal nanowires in human-hair-like glass fiber,” Materials Lett.64, 1902–1904 (2010).
[CrossRef]

Belov, P.

M. Silveirinha and P. Belov, “Spatial dispersion in lattices of split ring resonators with permeability near zero,” Phys. Rev. B77, 233104 (2008).
[CrossRef]

P. Belov, Y. Hao, and S. Sudhakaran, “Subwavelength microwave imaging using an array of parallel conducting wires as a lens,” Phys. Rev. B73, 033108 (2006).
[CrossRef]

Belov, P. A.

P. A. Belov, R. Marqués, S. I. Maslovski, I. S. Nefedov, M. Silveirinha, C. R. Simovski, and S. A. Tretyakov, “Strong spatial dispersion in wire media in the very large wavelength limit,” Phys. Rev. B67, 113103 (2003).
[CrossRef]

Bendavid, A.

Boltasseva, A.

A. Boltasseva and V. M. Shalaev, “Fabrication of optical negative-index metamaterials: Recent advances and outlook,” Metamaterials2, 1–17 (2008).
[CrossRef]

Brener, I.

X. G. Peralta, M. C. Wanke, C. L. Arrington, J. D. Williams, I. Brener, A. Strikwerda, R. D. Averitt, W. J. Padilla, E. Smirnova, and A. J. Taylor, “Large-area metamaterials on thin membranes for multilayer and curved applications at terahertz and higher frequencies,” Appl. Phys. Lett.94, 161113 (2009).
[CrossRef]

Busch, K.

C. E. Kriegler, M. S. Rill, M. Thiel, E. Müller, S. Essig, A. Frölich, G. von Freymann, S. Linden, D. Gerthsen, H. Hahn, K. Busch, and M. Wegener, “Transition between corrugated metal films and split-ring-resonator arrays,” Appl. Phys. B96, 749–755 (2009).
[CrossRef]

Cai, W.

W. Cai and V. M. Shalaev, Optical Metamaterials: Fundamentals and Applications (Springer Verlag, 2009).

Charipar, N. A.

Choi, M.

M. Choi, S. H. Lee, Y. Kim, S. B. Kang, J. Shin, M. H. Kwak, K. Y. Kang, Y. H. Lee, N. Park, and B. Min, “A terahertz metamaterial with unnaturally high refractive index,” Nature470, 369–373 (2011).
[CrossRef] [PubMed]

Constable, E.

Corrigan, T. D.

Cummer, S. A.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science314, 977–980 (2006).
[CrossRef] [PubMed]

Decker, M.

J. K. Gansel, M. Thiel, S. M. Rill, M. Decker, K. Bade, S. Volker, G. Freymann, S. Linden, and M. Wegener, “Gold helix photonic metamaterial as broadband circular polarizer,” Science325, 1513–1515 (2009).
[CrossRef] [PubMed]

Drew, H. D.

Essig, S.

C. E. Kriegler, M. S. Rill, M. Thiel, E. Müller, S. Essig, A. Frölich, G. von Freymann, S. Linden, D. Gerthsen, H. Hahn, K. Busch, and M. Wegener, “Transition between corrugated metal films and split-ring-resonator arrays,” Appl. Phys. B96, 749–755 (2009).
[CrossRef]

Fattinger, C.

Fleming, S. C.

Freymann, G.

J. K. Gansel, M. Thiel, S. M. Rill, M. Decker, K. Bade, S. Volker, G. Freymann, S. Linden, and M. Wegener, “Gold helix photonic metamaterial as broadband circular polarizer,” Science325, 1513–1515 (2009).
[CrossRef] [PubMed]

Frölich, A.

C. E. Kriegler, M. S. Rill, M. Thiel, E. Müller, S. Essig, A. Frölich, G. von Freymann, S. Linden, D. Gerthsen, H. Hahn, K. Busch, and M. Wegener, “Transition between corrugated metal films and split-ring-resonator arrays,” Appl. Phys. B96, 749–755 (2009).
[CrossRef]

Gansel, J. K.

J. K. Gansel, M. Thiel, S. M. Rill, M. Decker, K. Bade, S. Volker, G. Freymann, S. Linden, and M. Wegener, “Gold helix photonic metamaterial as broadband circular polarizer,” Science325, 1513–1515 (2009).
[CrossRef] [PubMed]

Gerthsen, D.

C. E. Kriegler, M. S. Rill, M. Thiel, E. Müller, S. Essig, A. Frölich, G. von Freymann, S. Linden, D. Gerthsen, H. Hahn, K. Busch, and M. Wegener, “Transition between corrugated metal films and split-ring-resonator arrays,” Appl. Phys. B96, 749–755 (2009).
[CrossRef]

Grischkowsky, D.

Hahn, H.

C. E. Kriegler, M. S. Rill, M. Thiel, E. Müller, S. Essig, A. Frölich, G. von Freymann, S. Linden, D. Gerthsen, H. Hahn, K. Busch, and M. Wegener, “Transition between corrugated metal films and split-ring-resonator arrays,” Appl. Phys. B96, 749–755 (2009).
[CrossRef]

Hajnal, J. V.

M. C. K. Wiltshire, J. B. Pendry, W. Williams, and J. V. Hajnal, “An effective medium description of ’swiss rolls’, a magnetic metamaterial,” J. Phys-Condens. Mat.19, 456216 (2007).
[CrossRef]

Halevi, P.

P. Halevi, Spatial Dispersion in Solids and Plasmas (North-Holland, 1992).

Hao, Y.

P. Belov, Y. Hao, and S. Sudhakaran, “Subwavelength microwave imaging using an array of parallel conducting wires as a lens,” Phys. Rev. B73, 033108 (2006).
[CrossRef]

Holden, A.

J. Pendry, A. Holden, D. Robbins, and W. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microw. Theory Tech.47, 2075–2084 (1999).
[CrossRef]

Hunt, P. G.

Ioisher, A.

E. Badinter, A. Ioisher, E. Monaico, V. Postolache, and I. M. Tiginyanu, “Exceptional integration of metal or semimetal nanowires in human-hair-like glass fiber,” Materials Lett.64, 1902–1904 (2010).
[CrossRef]

Justice, B. J.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science314, 977–980 (2006).
[CrossRef] [PubMed]

Kang, K. Y.

M. Choi, S. H. Lee, Y. Kim, S. B. Kang, J. Shin, M. H. Kwak, K. Y. Kang, Y. H. Lee, N. Park, and B. Min, “A terahertz metamaterial with unnaturally high refractive index,” Nature470, 369–373 (2011).
[CrossRef] [PubMed]

Kang, S. B.

M. Choi, S. H. Lee, Y. Kim, S. B. Kang, J. Shin, M. H. Kwak, K. Y. Kang, Y. H. Lee, N. Park, and B. Min, “A terahertz metamaterial with unnaturally high refractive index,” Nature470, 369–373 (2011).
[CrossRef] [PubMed]

Keiding, S.

Khachatrian, A.

Khanikaev, A.

Kim, H.

Kim, Y.

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Figures (7)

Fig. 1
Fig. 1

Schematic of the fabrication procedure. (a) A Zeonex preform is fed through a furnace and drawn, (b) sputtered with silver on three sides and (c) spooled into an array. (d) Each side of the array forms a longitudinally invariant U-shaped resonator. (e) Optical microscope image of the 100 μm square fiber array (top view).

Fig. 2
Fig. 2

(a) Schematic of the transmittance experiment under TM polarization (electric field directed perpendicular to the fibers) for different incident angles. (b) Experimental and numerical spectral transmittance for different angles of incidence.

Fig. 3
Fig. 3

Schematic of longitudinally invariant square resonators with (a) air everywhere, (b) Zeonex everywhere, (c) Zeonex inside the resonator, air outside the resonator. Note that (a) includes the relevant parameters: the wave-vector k forms an angle θ with respect to the normal of the longitudinal z axis of the resonators as shown, and phase-matches along z with the propagation constant β of the resonant mode. Only the magnetic field H is shown for clarity.

Fig. 4
Fig. 4

(a) Calculated frequency-dependent effective index for structures in a uniform dielectric and for our fabricated structures. (b) Color density plot for the fields of the resonant mode, neff = 1.001 at 0.65THz.

Fig. 5
Fig. 5

Measured resonant angle, compared to the simulation, for the longitudinally invariant samples. The change in resonant frequency with respect to incident angle is well predicted by the heuristic theory.

Fig. 6
Fig. 6

(a) Schematic of the laser-ablation procedure, resulting in 3-dimensional (patterned) resonators from longitudinally invariant (unpatterned) resonators. (b) Optical microscope image of the patterned and unpatterned fibers. (c) The transmittance was measured for both i) the unpatterned and ii) the patterned region of the sample, for TE polarization (electric field directed along the fibers). (d) Experimentally measured and simulated transmittance.

Fig. 7
Fig. 7

(a) Schematic of the transmittance experiment under TM polarization (electric field directed perpendicular to the patterned fibers) for different incident angles. (b) Experimental and numerical spectral transmittance for different angles of incidence. Note that in this case the fiber is ∼ 110 μm wide.

Tables (1)

Tables Icon

Table 1 Theoretical and numerical resonance frequencies as a function of angle for different uniform dielectric refractive index values. For the theory columns, the value of ω0 is taken from simulations at normal incidence.

Equations (11)

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β = n ω c sin ( θ ) ,
t 2 E m + γ E m = 0 .
γ = ( n 2 ω 2 c 2 β 2 ) .
ω ( θ ) = ω 0 1 sin 2 θ .
γ = ( n ˜ 2 ω 2 c 2 β 2 ) ,
ω ( θ ) = ω 0 1 sin 2 θ / n ˜ 2 .
n ˜ = 1 1 ω 0 2 / ω 2 = 1.33 .
δ n = n eff n ¯ eff = A ( n 2 n ¯ 2 ) E E ¯ * d A Z 0 A ( E × H ¯ * + E ¯ * × H ) z ^ d A ,
n ¯ eff = n eff air + A [ n ¯ ( x , y ) 2 1 ] | E | 2 d A 2 Z 0 A e ( E × H * ) z ^ d A ,
n eff = n eff air + A [ n ˜ 2 1 ] | E | 2 d A 2 Z 0 A e ( E × H * ) z ^ d A .
n ˜ = 1 + ( n znx 2 1 ) f ,

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