Abstract

Motivated by the realization of the Dirac point (DP) with a double-cone structure for optical field in the negative-zero-positive index metamaterial (NZPIM), the lateral shift and tunneling time of photon tunneling through a frustrated total internal reflection structure containing a NZPIM barrier are investigated by employing Artman’s stationary phase method. Near the DP, the lateral shift can vary from positive to negative and the photon tunneling displays a superluminal dynamic. Because of the Hartman effect, both lateral shift and tunneling time tend to a saturation value when the barrier thickness increases. These results may lead to potential applications in integral optics and optical-based devices and also suggest analogous phenomena of valance electron in single-layered carbon graphene.

© 2013 Optical Society of America

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2013 (1)

Z. Li, H. Zhao, Y. Nie, and J. Liang, J. Appl. Phys. 113, 043714 (2013).
[CrossRef]

2012 (1)

2011 (2)

X. Chen, J. Tao, and Y. Ban, Eur. Phys. J. B 79, 203 (2011).
[CrossRef]

M. Shen, L. Ruan, X.-L. Wang, J.-L. Shi, and Q. Wang, Phys. Rev. A 83, 045804 (2011).
[CrossRef]

2010 (2)

O. Bahat-Treidel, O. Peleg, M. Grobman, N. Shapira, M. Segev, and T. Pereg-Barnea, Phys. Rev. Lett. 104, 063901 (2010).
[CrossRef]

M. Shen, L. Ruan, and X. Chen, Opt. Express 18, 12779 (2010).
[CrossRef]

2009 (8)

L.-G. Wang, Z.-G. Wang, J.-X. Zhang, and S.-Y. Zhu, Opt. Lett. 34, 1510 (2009).
[CrossRef]

N. Stander, B. Huard, and D. Goldhaber-Gordon, Phys. Rev. Lett. 102, 026807 (2009).
[CrossRef]

A. F. Young and P. Kim, Nat. Phys. 5, 222 (2009).
[CrossRef]

C. W. J. Beenakker, R. A. Sepkhanov, A. R. Akhmerov, and J. Tworzydlo, Phys. Rev. Lett. 102, 146804 (2009).
[CrossRef]

L.-G. Wang, Z.-G. Wang, and S.-Y. Zhu, Europhys. Lett. 86, 47008 (2009).
[CrossRef]

X. Chen, L.-G. Wang, and C.-F. Li, Phys. Rev. A 80, 043839 (2009).
[CrossRef]

A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, Rev. Mod. Phys. 81, 109 (2009).
[CrossRef]

H. G. Winful and C. Zhang, Phys. Rev. A 79, 023826 (2009).
[CrossRef]

2008 (4)

C. W. Beenakker, Rev. Mod. Phys. 80, 1337 (2008).
[CrossRef]

F. D. M. Haldane and S. Raghu, Phys. Rev. Lett. 100, 013904 (2008).
[CrossRef]

X. Zhang, Phys. Rev. Lett. 100, 113903 (2008).
[CrossRef]

F. Zhang, G. Houzet, E. Lheurette, D. Lippens, M. Chaubet, and X. Zhao, J. Appl. Phys. 103, 084312 (2008).
[CrossRef]

2007 (2)

O. Peleg, G. Bartal, B. Freedman, O. Manela, M. Segev, and D. N. Christodoulides, Phys. Rev. Lett. 98, 103901 (2007).
[CrossRef]

Y.-J. Xiang, X.-Y. Dai, and S.-C. Wen, J. Opt. A 9, 253 (2007).
[CrossRef]

2006 (2)

C.-F. Li, Q.-B. Zhu, G. Nimtz, X. Chen, and Y. Zhang, Opt. Commun. 259, 470 (2006).
[CrossRef]

M. I. Katsnelson, K. S. Novoselov, and A. K. Geim, Nat. Phys. 2, 620 (2006).
[CrossRef]

2004 (2)

L.-G. Wang, J.-P. Xu, and S.-Y. Zhu, Phys. Rev. E 70, 066624 (2004).
[CrossRef]

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, Science 306, 666 (2004).
[CrossRef]

2003 (2)

M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, Science 301, 200 (2003).
[CrossRef]

H. Winful, Phys. Rev. Lett. 90, 023901 (2003).
[CrossRef]

2002 (2)

H. G. Winful, Opt. Express 10, 1491 (2002).
[CrossRef]

N.-H. Liu, S.-Y. Zhu, H. Chen, and X. Wu, Phys. Rev. E 65, 046607 (2002).
[CrossRef]

2000 (1)

L.-J. Wang, A. Kuzmich, and A. Dogariu, Nature 406, 277 (2000).
[CrossRef]

1997 (1)

P. Balcou and L. Dutriaux, Phys. Rev. Lett. 78, 851 (1997).
[CrossRef]

1993 (1)

A. M. Steinberg, P. G. Kwiat, and R. Y. Chiao, Phys. Rev. Lett. 71, 708 (1993).
[CrossRef]

1962 (1)

T. E. Hartman, J. Appl. Phys. 33, 3427 (1962).
[CrossRef]

1948 (1)

K. V. Artmann, Ann. Phys. 437, 87 (1948).
[CrossRef]

1947 (1)

F. Goos and H. Hänchen, Ann. Phys. 436, 333 (1947).
[CrossRef]

Akhmerov, A. R.

C. W. J. Beenakker, R. A. Sepkhanov, A. R. Akhmerov, and J. Tworzydlo, Phys. Rev. Lett. 102, 146804 (2009).
[CrossRef]

Artmann, K. V.

K. V. Artmann, Ann. Phys. 437, 87 (1948).
[CrossRef]

Bahat-Treidel, O.

O. Bahat-Treidel, O. Peleg, M. Grobman, N. Shapira, M. Segev, and T. Pereg-Barnea, Phys. Rev. Lett. 104, 063901 (2010).
[CrossRef]

Balcou, P.

P. Balcou and L. Dutriaux, Phys. Rev. Lett. 78, 851 (1997).
[CrossRef]

Ban, Y.

X. Chen, J. Tao, and Y. Ban, Eur. Phys. J. B 79, 203 (2011).
[CrossRef]

Bartal, G.

O. Peleg, G. Bartal, B. Freedman, O. Manela, M. Segev, and D. N. Christodoulides, Phys. Rev. Lett. 98, 103901 (2007).
[CrossRef]

Beenakker, C. W.

C. W. Beenakker, Rev. Mod. Phys. 80, 1337 (2008).
[CrossRef]

Beenakker, C. W. J.

C. W. J. Beenakker, R. A. Sepkhanov, A. R. Akhmerov, and J. Tworzydlo, Phys. Rev. Lett. 102, 146804 (2009).
[CrossRef]

Bigelow, M. S.

M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, Science 301, 200 (2003).
[CrossRef]

Boyd, R. W.

M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, Science 301, 200 (2003).
[CrossRef]

Castro Neto, A. H.

A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, Rev. Mod. Phys. 81, 109 (2009).
[CrossRef]

Chaubet, M.

F. Zhang, G. Houzet, E. Lheurette, D. Lippens, M. Chaubet, and X. Zhao, J. Appl. Phys. 103, 084312 (2008).
[CrossRef]

Chen, H.

N.-H. Liu, S.-Y. Zhu, H. Chen, and X. Wu, Phys. Rev. E 65, 046607 (2002).
[CrossRef]

Chen, X.

X. Chen, X.-J. Lu, P.-L. Zhao, and Q.-B. Zhu, Opt. Lett. 37, 1526 (2012).
[CrossRef]

X. Chen, J. Tao, and Y. Ban, Eur. Phys. J. B 79, 203 (2011).
[CrossRef]

M. Shen, L. Ruan, and X. Chen, Opt. Express 18, 12779 (2010).
[CrossRef]

X. Chen, L.-G. Wang, and C.-F. Li, Phys. Rev. A 80, 043839 (2009).
[CrossRef]

C.-F. Li, Q.-B. Zhu, G. Nimtz, X. Chen, and Y. Zhang, Opt. Commun. 259, 470 (2006).
[CrossRef]

Chiao, R. Y.

A. M. Steinberg, P. G. Kwiat, and R. Y. Chiao, Phys. Rev. Lett. 71, 708 (1993).
[CrossRef]

Christodoulides, D. N.

O. Peleg, G. Bartal, B. Freedman, O. Manela, M. Segev, and D. N. Christodoulides, Phys. Rev. Lett. 98, 103901 (2007).
[CrossRef]

Dai, X.-Y.

Y.-J. Xiang, X.-Y. Dai, and S.-C. Wen, J. Opt. A 9, 253 (2007).
[CrossRef]

Dogariu, A.

L.-J. Wang, A. Kuzmich, and A. Dogariu, Nature 406, 277 (2000).
[CrossRef]

Dubonos, S. V.

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, Science 306, 666 (2004).
[CrossRef]

Dutriaux, L.

P. Balcou and L. Dutriaux, Phys. Rev. Lett. 78, 851 (1997).
[CrossRef]

Firsov, A. A.

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, Science 306, 666 (2004).
[CrossRef]

Freedman, B.

O. Peleg, G. Bartal, B. Freedman, O. Manela, M. Segev, and D. N. Christodoulides, Phys. Rev. Lett. 98, 103901 (2007).
[CrossRef]

Geim, A. K.

A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, Rev. Mod. Phys. 81, 109 (2009).
[CrossRef]

M. I. Katsnelson, K. S. Novoselov, and A. K. Geim, Nat. Phys. 2, 620 (2006).
[CrossRef]

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, Science 306, 666 (2004).
[CrossRef]

Goldhaber-Gordon, D.

N. Stander, B. Huard, and D. Goldhaber-Gordon, Phys. Rev. Lett. 102, 026807 (2009).
[CrossRef]

Goos, F.

F. Goos and H. Hänchen, Ann. Phys. 436, 333 (1947).
[CrossRef]

Grigorieva, I. V.

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, Science 306, 666 (2004).
[CrossRef]

Grobman, M.

O. Bahat-Treidel, O. Peleg, M. Grobman, N. Shapira, M. Segev, and T. Pereg-Barnea, Phys. Rev. Lett. 104, 063901 (2010).
[CrossRef]

Guinea, F.

A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, Rev. Mod. Phys. 81, 109 (2009).
[CrossRef]

Haldane, F. D. M.

F. D. M. Haldane and S. Raghu, Phys. Rev. Lett. 100, 013904 (2008).
[CrossRef]

Hänchen, H.

F. Goos and H. Hänchen, Ann. Phys. 436, 333 (1947).
[CrossRef]

Hartman, T. E.

T. E. Hartman, J. Appl. Phys. 33, 3427 (1962).
[CrossRef]

Houzet, G.

F. Zhang, G. Houzet, E. Lheurette, D. Lippens, M. Chaubet, and X. Zhao, J. Appl. Phys. 103, 084312 (2008).
[CrossRef]

Huard, B.

N. Stander, B. Huard, and D. Goldhaber-Gordon, Phys. Rev. Lett. 102, 026807 (2009).
[CrossRef]

Jiang, D.

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, Science 306, 666 (2004).
[CrossRef]

Katsnelson, M. I.

M. I. Katsnelson, K. S. Novoselov, and A. K. Geim, Nat. Phys. 2, 620 (2006).
[CrossRef]

Kim, P.

A. F. Young and P. Kim, Nat. Phys. 5, 222 (2009).
[CrossRef]

Kuzmich, A.

L.-J. Wang, A. Kuzmich, and A. Dogariu, Nature 406, 277 (2000).
[CrossRef]

Kwiat, P. G.

A. M. Steinberg, P. G. Kwiat, and R. Y. Chiao, Phys. Rev. Lett. 71, 708 (1993).
[CrossRef]

Lepeshkin, N. N.

M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, Science 301, 200 (2003).
[CrossRef]

Lheurette, E.

F. Zhang, G. Houzet, E. Lheurette, D. Lippens, M. Chaubet, and X. Zhao, J. Appl. Phys. 103, 084312 (2008).
[CrossRef]

Li, C.-F.

X. Chen, L.-G. Wang, and C.-F. Li, Phys. Rev. A 80, 043839 (2009).
[CrossRef]

C.-F. Li, Q.-B. Zhu, G. Nimtz, X. Chen, and Y. Zhang, Opt. Commun. 259, 470 (2006).
[CrossRef]

Li, Z.

Z. Li, H. Zhao, Y. Nie, and J. Liang, J. Appl. Phys. 113, 043714 (2013).
[CrossRef]

Liang, J.

Z. Li, H. Zhao, Y. Nie, and J. Liang, J. Appl. Phys. 113, 043714 (2013).
[CrossRef]

Lippens, D.

F. Zhang, G. Houzet, E. Lheurette, D. Lippens, M. Chaubet, and X. Zhao, J. Appl. Phys. 103, 084312 (2008).
[CrossRef]

Liu, N.-H.

N.-H. Liu, S.-Y. Zhu, H. Chen, and X. Wu, Phys. Rev. E 65, 046607 (2002).
[CrossRef]

Lu, X.-J.

Manela, O.

O. Peleg, G. Bartal, B. Freedman, O. Manela, M. Segev, and D. N. Christodoulides, Phys. Rev. Lett. 98, 103901 (2007).
[CrossRef]

Morozov, S. V.

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, Science 306, 666 (2004).
[CrossRef]

Nie, Y.

Z. Li, H. Zhao, Y. Nie, and J. Liang, J. Appl. Phys. 113, 043714 (2013).
[CrossRef]

Nimtz, G.

C.-F. Li, Q.-B. Zhu, G. Nimtz, X. Chen, and Y. Zhang, Opt. Commun. 259, 470 (2006).
[CrossRef]

Novoselov, K. S.

A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, Rev. Mod. Phys. 81, 109 (2009).
[CrossRef]

M. I. Katsnelson, K. S. Novoselov, and A. K. Geim, Nat. Phys. 2, 620 (2006).
[CrossRef]

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, Science 306, 666 (2004).
[CrossRef]

Peleg, O.

O. Bahat-Treidel, O. Peleg, M. Grobman, N. Shapira, M. Segev, and T. Pereg-Barnea, Phys. Rev. Lett. 104, 063901 (2010).
[CrossRef]

O. Peleg, G. Bartal, B. Freedman, O. Manela, M. Segev, and D. N. Christodoulides, Phys. Rev. Lett. 98, 103901 (2007).
[CrossRef]

Pereg-Barnea, T.

O. Bahat-Treidel, O. Peleg, M. Grobman, N. Shapira, M. Segev, and T. Pereg-Barnea, Phys. Rev. Lett. 104, 063901 (2010).
[CrossRef]

Peres, N. M. R.

A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, Rev. Mod. Phys. 81, 109 (2009).
[CrossRef]

Raghu, S.

F. D. M. Haldane and S. Raghu, Phys. Rev. Lett. 100, 013904 (2008).
[CrossRef]

Ruan, L.

M. Shen, L. Ruan, X.-L. Wang, J.-L. Shi, and Q. Wang, Phys. Rev. A 83, 045804 (2011).
[CrossRef]

M. Shen, L. Ruan, and X. Chen, Opt. Express 18, 12779 (2010).
[CrossRef]

Segev, M.

O. Bahat-Treidel, O. Peleg, M. Grobman, N. Shapira, M. Segev, and T. Pereg-Barnea, Phys. Rev. Lett. 104, 063901 (2010).
[CrossRef]

O. Peleg, G. Bartal, B. Freedman, O. Manela, M. Segev, and D. N. Christodoulides, Phys. Rev. Lett. 98, 103901 (2007).
[CrossRef]

Sepkhanov, R. A.

C. W. J. Beenakker, R. A. Sepkhanov, A. R. Akhmerov, and J. Tworzydlo, Phys. Rev. Lett. 102, 146804 (2009).
[CrossRef]

Shapira, N.

O. Bahat-Treidel, O. Peleg, M. Grobman, N. Shapira, M. Segev, and T. Pereg-Barnea, Phys. Rev. Lett. 104, 063901 (2010).
[CrossRef]

Shen, M.

M. Shen, L. Ruan, X.-L. Wang, J.-L. Shi, and Q. Wang, Phys. Rev. A 83, 045804 (2011).
[CrossRef]

M. Shen, L. Ruan, and X. Chen, Opt. Express 18, 12779 (2010).
[CrossRef]

Shi, J.-L.

M. Shen, L. Ruan, X.-L. Wang, J.-L. Shi, and Q. Wang, Phys. Rev. A 83, 045804 (2011).
[CrossRef]

Stander, N.

N. Stander, B. Huard, and D. Goldhaber-Gordon, Phys. Rev. Lett. 102, 026807 (2009).
[CrossRef]

Steinberg, A. M.

A. M. Steinberg, P. G. Kwiat, and R. Y. Chiao, Phys. Rev. Lett. 71, 708 (1993).
[CrossRef]

Tao, J.

X. Chen, J. Tao, and Y. Ban, Eur. Phys. J. B 79, 203 (2011).
[CrossRef]

Tworzydlo, J.

C. W. J. Beenakker, R. A. Sepkhanov, A. R. Akhmerov, and J. Tworzydlo, Phys. Rev. Lett. 102, 146804 (2009).
[CrossRef]

Wang, L.-G.

L.-G. Wang, Z.-G. Wang, and S.-Y. Zhu, Europhys. Lett. 86, 47008 (2009).
[CrossRef]

X. Chen, L.-G. Wang, and C.-F. Li, Phys. Rev. A 80, 043839 (2009).
[CrossRef]

L.-G. Wang, Z.-G. Wang, J.-X. Zhang, and S.-Y. Zhu, Opt. Lett. 34, 1510 (2009).
[CrossRef]

L.-G. Wang, J.-P. Xu, and S.-Y. Zhu, Phys. Rev. E 70, 066624 (2004).
[CrossRef]

Wang, L.-J.

L.-J. Wang, A. Kuzmich, and A. Dogariu, Nature 406, 277 (2000).
[CrossRef]

Wang, Q.

M. Shen, L. Ruan, X.-L. Wang, J.-L. Shi, and Q. Wang, Phys. Rev. A 83, 045804 (2011).
[CrossRef]

Wang, X.-L.

M. Shen, L. Ruan, X.-L. Wang, J.-L. Shi, and Q. Wang, Phys. Rev. A 83, 045804 (2011).
[CrossRef]

Wang, Z.-G.

L.-G. Wang, Z.-G. Wang, and S.-Y. Zhu, Europhys. Lett. 86, 47008 (2009).
[CrossRef]

L.-G. Wang, Z.-G. Wang, J.-X. Zhang, and S.-Y. Zhu, Opt. Lett. 34, 1510 (2009).
[CrossRef]

Wen, S.-C.

Y.-J. Xiang, X.-Y. Dai, and S.-C. Wen, J. Opt. A 9, 253 (2007).
[CrossRef]

Winful, H.

H. Winful, Phys. Rev. Lett. 90, 023901 (2003).
[CrossRef]

Winful, H. G.

H. G. Winful and C. Zhang, Phys. Rev. A 79, 023826 (2009).
[CrossRef]

H. G. Winful, Opt. Express 10, 1491 (2002).
[CrossRef]

Wu, X.

N.-H. Liu, S.-Y. Zhu, H. Chen, and X. Wu, Phys. Rev. E 65, 046607 (2002).
[CrossRef]

Xiang, Y.-J.

Y.-J. Xiang, X.-Y. Dai, and S.-C. Wen, J. Opt. A 9, 253 (2007).
[CrossRef]

Xu, J.-P.

L.-G. Wang, J.-P. Xu, and S.-Y. Zhu, Phys. Rev. E 70, 066624 (2004).
[CrossRef]

Young, A. F.

A. F. Young and P. Kim, Nat. Phys. 5, 222 (2009).
[CrossRef]

Zhang, C.

H. G. Winful and C. Zhang, Phys. Rev. A 79, 023826 (2009).
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Figures (3)

Fig. 1.
Fig. 1.

Schematic diagram of wave propagating through the FTIR structure with a NZPIM barrier.

Fig. 2.
Fig. 2.

(a)–(c) Lateral shift S in the unit of wavelength λ (S/λ) versus the thickness of the barrier with different incident angles when the frequencies are ω=8GHz [(a)], and ω=12GHz [(b) and (c)], respectively. (d) S/λ as a function of the incident frequency for different incident angles with the barrier thickness d=3mm. (e) and (f) S/λ versus the incident angle for different barrier’s thickness when the frequencies are ω=8GHz (e) and ω=12GHz (f). The permittivity and permeability are ε1=4, μ1=1 for the PIMs.

Fig. 3.
Fig. 3.

Tunneling time [(a) and (c)] and tunneling velocity [(b) and (d)] versus the thickness of barrier with different angles, where frequencies are ω=8GHz for (a) and (b), and ω=12GHz for (c) and (d). The tunneling time versus the incident frequency (e) and the thickness in case of ω=10GHz (f) with different incident angles.

Equations (9)

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ε2(ω)=1ωep2/(ω2+iγeω),
μ2(ω)=1ωmp2/(ω2+iγmω),
M=(cos(k2zd)ik0μ2k2zsin(k2zd)ik2zk0μ2sin(k2zd)cos(k2zd)),
t(d,θ,ω)=[cosh(κd)iμ22k1z2κ22μ2k1zκsinh(κd)]1,
ϕ=arctan[μ22k1z2κ22μ2k1zκtanh(κd)]
S=μ2tanθ([κ4+k1z2κ2(μ22+1)+μ22k1z4]×sinh(2κd)2κk1z2d(μ22k1z2κ2))4μ22k1z2κ3cosh2(κd)+(μ22k1z2κ2)2κsinh2(κd).
τS=n1Ssinθc,
τϕ=[2μ2k1zκ(aμ22k1zbκ)(μ22k1z2κ2)(aκ+bk1z)]×sinh(2κd)+2bdk1zκ(μ22k1z2κ2)4μ22k1z2κ2cosh2(κd)+(μ22k1z2κ2)2sinh2(κd),
θc=arcsin|2(ωωD)ωε1μ1|,

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