Abstract

Using numerical simulations, we demonstrate the feasibility of tunable gradient refractive index optics at terahertz frequencies based on two-dimensional graded plasmonic crystals (GPCs). They consist of semiconductor rods with spatially dependent radii. In the effective medium approximation, the GPCs can be considered as effective media with a graded effective dielectric permittivity. The semiconductor rods have the Drude-type dispersion. By varying free charge carrier concentration in the rods, it is possible to tune their permittivity. In accordance to effective medium theory, the effective permittivity of the whole GPC is changed at the same time. This property is used for the demonstration of a GPC-based lens with a tunable focus, beam deflector with tunable angle of the beam deflection, and the half Maxwell-fisheye and the Luneburg lens as antennas with tunable radiation patterns. In particular, these GPCs can be made invisible to the incoming radiation by equaling the real part of the rods permittivity to the permittivity of air background.

© 2011 Optical Society of America

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2011

J. Zhu, J. Han, Z. Tian, J. Gu, Z. Chen, and W. Zhang, “Thermal broadband tunable terahertz metamaterials,” Opt. Commun. 284, 3129–3133 (2011).
[CrossRef]

N.-H. Shen, M. Massaouti, M. Gokkavas, J.-M. Manceau, E. Ozbay, M. Kafesaki, T. Koschny, S. Tzortzakis, and C. M. Soukoulis, “Optically implemented broadband blueshift switch in the terahertz regime,” Phys. Rev. Lett. 106, 037403 (2011).
[CrossRef] [PubMed]

J. Y. Ou, E. Plum, L. Jiang, and N. I. Zheludev, “Reconfigurable photonic metamaterials,” Nano Lett. 11, 2142–2144 (2011).
[CrossRef] [PubMed]

M. D. Goldflam, T. Driscoll, B. Chapler, O. Khatib, N. M. Jokerst, S. Palit, D. R. Smith, B.-J. Kim, G. Seo, H.-T. Kim, M. Di Ventra, and D. N. Basov, “Reconfigurable gradient index using VO2 memory metamaterials,” Appl. Phys. Lett. 99, 044103 (2011).
[CrossRef]

B. Vasić and R. Gajić, “Self-focusing media using graded photonic crystals: focusing, Fourier transforming and imaging, directive emission and directional cloaking,” J. Appl. Phys. 110, 053103 (2011).
[CrossRef]

T. Zentgraf, Y. Liu, M. H. Mikkelsen, J. Valentine, and X. Zhang, “Plasmonic Luneburg and Eaton lenses,” Nat. Nanotechnol. 6, 151–155 (2011).
[CrossRef] [PubMed]

A. Vakil and N. Engheta, “Transformation optics using graphene,” Science 332, 1291–1294 (2011).
[CrossRef] [PubMed]

B. Scherger, C. Jördens, and M. Koch, “Variable-focus terahertz lens,” Opt. Express 19, 4528–4535 (2011).
[CrossRef] [PubMed]

2010

V. Fedotov, A. Tsiatmas, J. H. Shi, R. Buckingham, P. de Groot, Y. Chen, S. Wang, and N. Zheludev, “Temperature control of Fano resonances and transmission in superconducting metamaterials,” Opt. Express 18, 9015–9019 (2010).
[CrossRef] [PubMed]

Y.-Y. Kao, P. C.-P. Chao, and C.-W. Hsueh, “A new low-voltage-driven GRIN liquid crystal lens with multiple ring electrodes in unequal widths,” Opt. Express 18, 18506–18518 (2010).
[CrossRef] [PubMed]

B. Vasić, G. Isić, R. Gajić, and K. Hingerl, “Controlling electromagnetic fields with graded photonic crystals in metamaterial regime,” Opt. Express 18, 20321–20333 (2010).
[CrossRef] [PubMed]

E. Devaux, J.-Y. Laluet, B. Stein, C. Genet, T. Ebbesen, J.-C. Weeber, and A. Dereux, “Refractive micro-optical elements for surface plasmons: from classical to gradient index optics,” Opt. Express 18, 20610–20619 (2010).
[CrossRef] [PubMed]

J. Neu, B. Krolla, O. Paul, B. Reinhard, R. Beigang, and M. Rahm, “Metamaterial-based gradient index lens with strong focusing in the THz frequency range,” Opt. Express 18, 27748–27757(2010).
[CrossRef]

O. Paul, B. Reinhard, B. Krolla, R. Beigang, and M. Rahm, “Gradient index metamaterial based on slot elements,” Appl. Phys. Lett. 96, 241110 (2010).
[CrossRef]

I. Smolyaninov, “Two-dimensional metamaterial optics,” Laser Phys. Lett. 7, 259–269 (2010).
[CrossRef]

P. West, S. Ishii, G. Naik, N. Emani, V. Shalaev, and A. Boltasseva, “Searching for better plasmonic materials,” Laser Photonics Rev. 4, 795–808 (2010).
[CrossRef]

Z. L. Mei, J. Bai, and T. J. Cui, “Gradient index metamaterials realized by drilling hole arrays,” J. Phys. D 43, 055404 (2010).
[CrossRef]

I. M. Pryce, K. Aydin, Y. A. Kelaita, R. M. Briggs, and H. A. Atwater, “Highly strained compliant optical metamaterials with large frequency tunability,” Nano Lett. 10, 4222–4227 (2010).
[CrossRef] [PubMed]

M.-K. Chen, Y.-C. Chang, C.-E. Yang, Y. Guo, J. Mazurowski, S. Yin, P. Ruffin, C. Brantley, E. Edwards, and C. Luo, “Tunable terahertz plasmonic lenses based on semiconductor microslits,” Microw. Opt. Technol. Lett. 52, 979–981 (2010).
[CrossRef]

J.-M. Manceau, N.-H. Shen, M. Kafesaki, C. M. Soukoulis, and S. Tzortzakis, “Dynamic response of metamaterials in the terahertz regime: blueshift tunability and broadband phase modulation,” Appl. Phys. Lett. 96, 021111 (2010).
[CrossRef]

A. Minovich, D. N. Neshev, D. A. Powell, I. V. Shadrivov, and Y. S. Kivshar, “Tunable fishnet metamaterials infiltrated by liquid crystals,” Appl. Phys. Lett. 96, 193103 (2010).
[CrossRef]

2009

T. Driscoll, H.-T. Kim, B.-G. Chae, B.-J. Kim, Y.-W. Lee, N. M. Jokerst, S. Palit, D. R. Smith, M. Di Ventra, and D. N. Basov, “Memory metamaterials,” Science 325, 1518–1521 (2009).
[CrossRef] [PubMed]

S. Xiao, U. K. Chettiar, A. V. Kildishev, V. Drachev, I. C. Khoo, and V. M. Shalaev, “Tunable magnetic response of metamaterials,” Appl. Phys. Lett. 95, 033115 (2009).
[CrossRef]

H. Tao, A. C. Strikwerda, K. Fan, W. J. Padilla, X. Zhang, and R. D. Averitt, “Reconfigurable terahertz metamaterials,” Phys. Rev. Lett. 103, 147401 (2009).
[CrossRef] [PubMed]

J. A. Dionne, K. Diest, L. A. Sweatlock, and H. A. Atwater, “PlasMOStor: a metal-oxide-Si field effect plasmonic modulator,” Nano Lett. 9, 897–902 (2009).
[CrossRef] [PubMed]

K. F. MacDonald, Z. L. Samson, M. I. Stockman, and N. I. Zheludev, “Ultrafast active plasmonics,” Nat. Photon. 3, 55–58 (2009).
[CrossRef]

X. Mao, S.-C. S. Lin, M. I. Lapsley, J. Shi, B. K. Juluri, and T. J. Huang, “Tunable liquid gradient refractive index (L-GRIN) lens with two degrees of freedom,” Lab Chip 9, 2050–2058 (2009).
[CrossRef] [PubMed]

R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband ground-plane cloak,” Science 323, 366–369 (2009).
[CrossRef] [PubMed]

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8, 568–571 (2009).
[CrossRef] [PubMed]

H. Wallén, H. Kettunen, and A. Sihvola, “Composite near-field superlens design using mixing formulas and simulations,” Metamaterials 3, 129–139 (2009).
[CrossRef]

B. K. Juluri, S. Chin, S. Lin, T. R. Walker, L. Jensen, and T. J. Huang, “Propagation of designer surface plasmons in structured conductor surfaces with parabolic gradient index,” Opt. Express 17, 2997–3006 (2009).
[CrossRef] [PubMed]

J. H. Lee, J. Blair, V. A. Tamma, Q. Wu, S. J. Rhee, C. J. Summers, and W. Park, “Direct visualization of optical frequency invisibility cloak based on silicon nanorod array,” Opt. Express 17, 12922–12928 (2009).
[CrossRef] [PubMed]

2008

J. Han, A. Lakhtakia, and C.-W. Qiu, “Terahertz metamaterials with semiconductor split-ring resonators for magnetostatic tunability,” Opt. Express 16, 14390–14396 (2008).
[CrossRef] [PubMed]

C. Gomez-Reino, M. V. Perez, C. Bao, and M. T. Flore-Arias, “Design of GRIN optical components for coupling and interconnects,” Laser Photonics Rev. 2, 203–215 (2008).
[CrossRef]

H. Kurt, E. Colak, O. Cakmak, H. Caglayan, and E. Ozbay, “The focusing effect of graded photonic crystals,” Appl. Phys. Lett. 93, 171108 (2008).
[CrossRef]

S. Liu, J. Du, Z. Lin, R. X. Wu, and S. T. Chui, “Formation of robust and completely tunable resonant photonic band gaps,” Phys. Rev. B 78, 155101 (2008).
[CrossRef]

R. A. Pala, K. T. Shimizu, N. A. Melosh, and M. L. Brongersma, “A nonvolatile plasmonic switch employing photochromic molecules,” Nano Lett. 8, 1506–1510 (2008).
[CrossRef] [PubMed]

H.-T. Chen, J. F. O’Hara, A. K. Azad, A. J. Taylor, R. D. Averitt, D. B. Shrenkenhamer, and W. J. Padilla, “Experimental demonstration of frequency-agile terahertz metamaterials,” Nat. Photon. 2, 295–298 (2008).
[CrossRef]

2007

D. Pacifici, H. J. Lezec, and H. A. Atwater, “All-optical modulation by plasmonic excitation of CdSe quantum dots,” Nat. Photon. 1, 402–406 (2007).
[CrossRef]

E. McLeod and C. B. Arnold, “Mechanics and refractive power optimization of tunable acoustic gradient lenses,” J. Appl. Phys. 102, 033104 (2007).
[CrossRef]

A. Degiron, J. J. Mock, and D. R. Smith, “Modulating and tuning the response of metamaterials at the unit cell level,” Opt. Express 15, 1115–1127 (2007).
[CrossRef] [PubMed]

C. Min, P. Wang, X. Jiao, Y. Deng, and H. Ming, “Beam manipulating by metallic nano-optic lens containing nonlinear media,” Opt. Express 15, 9541–9546 (2007).
[CrossRef] [PubMed]

2006

H.-T. Chen, W. J. Padilla, J. M. O. Zide, A. C. Gossard, A. J. Taylor, and R. D. Averitt, “Active terahertz metamaterial devices,” Nature 444, 597–600 (2006).
[CrossRef] [PubMed]

J. G. Rivas, M. Kuttge, H. Kurz, P. H. Bolivar, and J. A. Sanchez-Gil, “Low-frequency active surface plasmon optics on semiconductors,” Appl. Phys. Lett. 88, 082106 (2006).
[CrossRef]

2005

2004

J. G. Rivas, P. H. Bolivar, and H. Kurz, “Thermal switching of the enhanced transmission of terahertz radiation through subwavelength apertures,” Opt. Lett. 29, 1680–1682 (2004).
[CrossRef] [PubMed]

A. V. Krasavin and N. I. Zheludev, “Active plasmonics: controlling signals in Au/Ga waveguide using nanoscale structural transformations,” Appl. Phys. Lett. 84, 1416–1418 (2004).
[CrossRef]

2003

X. Hu, Q. Zhang, Y. Liu, B. Cheng, and D. Zhang, “Ultrafast three-dimensional tunable photonic crystal,” Appl. Phys. Lett. 83, 2518–2520 (2003).
[CrossRef]

D. A. Mazurenko, R. Kerst, J. I. Dijkhuis, A. V. Akimov, V. G. Golubev, D. A. Kurdyukov, A. B. Pevtsov, and A. V. Sel’kin, “Ultrafast optical switching in three-dimensional photonic crystals,” Phys. Rev. Lett. 91, 213903 (2003).
[CrossRef] [PubMed]

2002

S. W. Leonard, H. M. van Driel, J. Schilling, and R. B. Wehrspohn, “Ultrafast band-edge tuning of a two-dimensional silicon photonic crystal via free-carrier injection,” Phys. Rev. B 66, 161102(2002).
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2001

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H. Tao, A. C. Strikwerda, K. Fan, W. J. Padilla, X. Zhang, and R. D. Averitt, “Reconfigurable terahertz metamaterials,” Phys. Rev. Lett. 103, 147401 (2009).
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Summers, C. J.

Sweatlock, L. A.

J. A. Dionne, K. Diest, L. A. Sweatlock, and H. A. Atwater, “PlasMOStor: a metal-oxide-Si field effect plasmonic modulator,” Nano Lett. 9, 897–902 (2009).
[CrossRef] [PubMed]

Tamma, V. A.

Tao, H.

H. Tao, A. C. Strikwerda, K. Fan, W. J. Padilla, X. Zhang, and R. D. Averitt, “Reconfigurable terahertz metamaterials,” Phys. Rev. Lett. 103, 147401 (2009).
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H.-T. Chen, J. F. O’Hara, A. K. Azad, A. J. Taylor, R. D. Averitt, D. B. Shrenkenhamer, and W. J. Padilla, “Experimental demonstration of frequency-agile terahertz metamaterials,” Nat. Photon. 2, 295–298 (2008).
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J. Zhu, J. Han, Z. Tian, J. Gu, Z. Chen, and W. Zhang, “Thermal broadband tunable terahertz metamaterials,” Opt. Commun. 284, 3129–3133 (2011).
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Tzortzakis, S.

N.-H. Shen, M. Massaouti, M. Gokkavas, J.-M. Manceau, E. Ozbay, M. Kafesaki, T. Koschny, S. Tzortzakis, and C. M. Soukoulis, “Optically implemented broadband blueshift switch in the terahertz regime,” Phys. Rev. Lett. 106, 037403 (2011).
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J.-M. Manceau, N.-H. Shen, M. Kafesaki, C. M. Soukoulis, and S. Tzortzakis, “Dynamic response of metamaterials in the terahertz regime: blueshift tunability and broadband phase modulation,” Appl. Phys. Lett. 96, 021111 (2010).
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J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8, 568–571 (2009).
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Nat. Photon.

H.-T. Chen, J. F. O’Hara, A. K. Azad, A. J. Taylor, R. D. Averitt, D. B. Shrenkenhamer, and W. J. Padilla, “Experimental demonstration of frequency-agile terahertz metamaterials,” Nat. Photon. 2, 295–298 (2008).
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Figures (11)

Fig. 1
Fig. 1

GRIN devices considered in the paper: (a) lens, (b) beam deflector, (c) half Maxwell-fisheye, and (d) Luneburg lens. The color maps are saturated for better contrast in (c) and (d).

Fig. 2
Fig. 2

Tuning of real parts of rod permittivities for two working frequencies ω 1 (solid curves) and ω 2 (dashed curves) by changing free charge carrier concentration N.

Fig. 3
Fig. 3

Simulation results for the z component of electric field at 4 THz : (a) GPC diverging lens, (b) invisible GPC lens, (c) GPC focusing lens, and (d) free-space beam.

Fig. 4
Fig. 4

Distributions of the electric field magnitudes normalized by the value of the electric field magnitude for the free-space beam along the optical axis: the GPC focusing lens, the GPC diverging lens, and the invisible GPC lens. The dashed curves denote the cases without losses while the solid curves stand for the cases when losses in semiconductor rods are included.

Fig. 5
Fig. 5

Simulation results for the z component of electric field at 4 THz : (a) GPC deflector with upward beam deflection, (b) invisible GPC deflector, (c) GPC deflector with downward beam deflection, and (d) free-space beam.

Fig. 6
Fig. 6

Angular distributions of the electric field magnitudes: the GPC deflector with upward deflection, the GPC deflector with downward deflection, and the invisible GPC deflector. Dashed curves denote the cases without losses while the solid curves stand for the cases when losses in semiconductor rods are included.

Fig. 7
Fig. 7

Simulation results for the z component of electric field at 5 THz : (a) GPC half Maxwell-fisheye, (b) GPC half Maxwell-fisheye with lower directivity, (c) invisible GPC half Maxwell-fisheye, and (d) point source alone. The color maps are saturated for better contrast.

Fig. 8
Fig. 8

Angular distributions of the electric field magnitudes normalized by the value of the electric field magnitude for the point source: the GPC half Maxwell-fisheye, the GPC fisheye with a lower directivity, and the invisible GPC half Maxwell-fisheye. The dashed curves stand for the cases without losses while the solid curves denote the cases when losses in semiconductor rods are included.

Fig. 9
Fig. 9

Simulation results for the z component of electric field at 5 THz : (a) GPC Luneburg lens, (b) GPC Luneburg lens with lower directivity, (c) invisible GPC Luneburg lens, and (d) point source alone. The color maps are saturated for better contrast.

Fig. 10
Fig. 10

Angular distributions of the electric field magnitudes normalized by the value of the electric field magnitude for the point source: the GPC Luneburg lens, the GPC Luneburg lens with lower directivity, and the invisible GPC Luneburg lens. The dashed curves stand for the cases without losses while the solid curves denote the cases when losses in semiconductor rods are included.

Fig. 11
Fig. 11

The FOM factor of semiconductor rods as a function of free charge carrier concentration in them and their permittivity.

Equations (14)

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

n ( y ) = n 0 1 ± g 2 y 2 ,
n ( y ) = n 0 + sin ( ϕ ) L y ,
n ( ρ ) = 2 1 + ( ρ R ) 2 ,
n ( ρ ) = 2 ( ρ R ) 2 ,
n eff = ε host 1 + f ( ε inc ε host ) ε host ,
ε inc ( ω ) = ε i ω p 2 ω ( ω + j γ c ) ,
g 2 = { α ( ε inc ε host ) ε host for diverging lens , α ( ε inc ε host ) ε host for focusing lens .
n eff = ε host + f ( ε inc ε host ) 2 ε host .
sin ( ϕ ) = β L ( ε inc ε host ) 2 ε host .
r ( ρ ) = a n 2 ( ρ ) ε host π ( ε inc const ε host ) ,
Re ( ε inc ( ω 2 , N ) ) = Re ( ε inc ( ω 1 , N ) ) ,
N = ω 2 2 + γ c 2 ω 1 2 + γ c 2 N .
N = 5.76 × 10 14 × T 3 / 2 × exp ( 0.13 / ( k B T ) ) ,
FOM = | Re ( ε inc ( ω , N ) ) Im ( ε inc ( ω , N ) ) | .

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