Abstract

We analyze the Bloch oscillations of electromagnetic waves in chirped layered structures with alternating layers of negative-index metamaterial and conventional dielectric under the condition of the zero average refractive index. We consider the case when the chirp is introduced by varying the thickness of the layers linearly across the structure. We demonstrate that such structures can support three different types of the Bloch oscillations for electromagnetic waves associated with either propagating or evanescent guided modes. In particular, we predict a novel type of the Bloch oscillations associated with coupling between surface waves excited at the interfaces separating the layers of negative-index metamaterial and the layers of the conventional dielectric.

© 2008 Optical Society of America

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  1. F. Bloch, "Uber die Quantenmechanik der Elektronen in Kristallgittern," Z. Phys. A: Hadrons Nucl. 52, 555-600 (1928).
  2. G. H. Wannier, "Wave functions and effective Hamiltonian for Bloch electrons in an electric field," Phys. Rev. 117, 432-439 (1960).
    [CrossRef]
  3. C. Waschke, H. G. Roskos, R. Schwedler, K. Leo, H. Kurz, and K. Kohler, "Coherent submillimeter-wave emission from Bloch oscillations in a semiconductor superlattice," Phys. Rev. Lett. 70, 3319-3322 (1993).
    [CrossRef] [PubMed]
  4. C. M. de Sterke, J. N. Bright, P. A. Krug, and T. E. Hammon, "Observation of an optical Wannier-Stark ladder," Phys. Rev. E 57, 2365-2370 (1998).
    [CrossRef]
  5. R. Morandotti, U. Peschel, J. S. Aitchinson, H. S. Eisenberg, Y. Silberberg, "Experimental observation of linear and nonlinear optical Bloch oscillations," Phys. Rev. Lett. 83, 4756-4759 (1999)
    [CrossRef]
  6. T. Pertsch, P. Dannberg, W. Elflein, A. Brauer, and F. Lederer, "Optical Bloch Oscillations in Temperature Tuned Waveguide Arrays," Phys. Rev. Lett. 83, 4752-4755 (1999).
    [CrossRef]
  7. V. Agarwal, J. A. del Rio, G. Malpuech, M. Zamfirescu, A. Kavokin, D. Coquillat, D. Scalbert, M. Vladimirova, and B. Gil, "Photon Bloch Oscillations in Porous Silicon Optical Superlattices," Phys. Rev. Lett. 92, 097401 (2004).
    [CrossRef] [PubMed]
  8. P. B. Wilkinson, "Photonic Bloch oscillations and Wannier-Stark ladders in exponentially chirped Bragg gratings," Phys. Rev. E 65, 056616 (2002).
    [CrossRef]
  9. R. Sapienza, P. Costantino, D. Wiersma, M. Ghulinyan, C. J. Oton, and L. Pavesi, "Optical Analogue of Electronic Bloch Oscillations," Phys. Rev. Lett. 91, 263902 (2003).
    [CrossRef]
  10. G. Lenz, I. Talanina, and C. M. de Sterke, "Bloch Oscillations in an Array of Curved Optical Waveguides," Phys. Rev. Lett. 83, 963-966 (1999).
    [CrossRef]
  11. D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, "Metamaterials and Negative Refractive Index," Science 305, 788-792 (2004).
    [CrossRef] [PubMed]
  12. J. Li, L. Zhou, C. T. Chan, and P. Sheng, "Photonic Band Gap from a Stack of Positive and Negative Index Materials," Phys. Rev. Lett. 90, 083901 (2003).
    [CrossRef] [PubMed]
  13. I. V. Shadrivov, A. A. Sukhorukov, and Yu. S. Kivshar, "Beam shaping by a periodic structure with negative refraction," Appl. Phys. Lett. 82, 3820-3822 (2003).
    [CrossRef]
  14. P. Yeh, Optical Waves in Layered Media (John Wiley & Sons, New York, 1988).
  15. I. V. Shadrivov, A. A. Sukhorukov, Yu. S. Kivshar, A. A. Zharov, A. D. Boardman, and P. Egan, "Nonlinear surface waves in left-handed materials," Phys. Rev. E 69, 016617 (2004).
    [CrossRef]
  16. A. A. Sukhorukov, I. V. Shadrivov, and Yu. S. Kivshar, "Wave scattering by metamaterial wedges and interfaces," Int. J. Num. Model. 19, 105-117 (2006).
    [CrossRef]
  17. V. G. Veselago, "The electrodynamics of substances with simultaneously negative values of epsilon and mu," Usp. Fiz. Nauk 92, 517-526 (1967) (in Russian) [English translation: Sov. Phys. Usp. 10, 509 (1968)].

2006 (1)

A. A. Sukhorukov, I. V. Shadrivov, and Yu. S. Kivshar, "Wave scattering by metamaterial wedges and interfaces," Int. J. Num. Model. 19, 105-117 (2006).
[CrossRef]

2004 (3)

I. V. Shadrivov, A. A. Sukhorukov, Yu. S. Kivshar, A. A. Zharov, A. D. Boardman, and P. Egan, "Nonlinear surface waves in left-handed materials," Phys. Rev. E 69, 016617 (2004).
[CrossRef]

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, "Metamaterials and Negative Refractive Index," Science 305, 788-792 (2004).
[CrossRef] [PubMed]

V. Agarwal, J. A. del Rio, G. Malpuech, M. Zamfirescu, A. Kavokin, D. Coquillat, D. Scalbert, M. Vladimirova, and B. Gil, "Photon Bloch Oscillations in Porous Silicon Optical Superlattices," Phys. Rev. Lett. 92, 097401 (2004).
[CrossRef] [PubMed]

2003 (3)

J. Li, L. Zhou, C. T. Chan, and P. Sheng, "Photonic Band Gap from a Stack of Positive and Negative Index Materials," Phys. Rev. Lett. 90, 083901 (2003).
[CrossRef] [PubMed]

I. V. Shadrivov, A. A. Sukhorukov, and Yu. S. Kivshar, "Beam shaping by a periodic structure with negative refraction," Appl. Phys. Lett. 82, 3820-3822 (2003).
[CrossRef]

R. Sapienza, P. Costantino, D. Wiersma, M. Ghulinyan, C. J. Oton, and L. Pavesi, "Optical Analogue of Electronic Bloch Oscillations," Phys. Rev. Lett. 91, 263902 (2003).
[CrossRef]

2002 (1)

P. B. Wilkinson, "Photonic Bloch oscillations and Wannier-Stark ladders in exponentially chirped Bragg gratings," Phys. Rev. E 65, 056616 (2002).
[CrossRef]

1999 (3)

R. Morandotti, U. Peschel, J. S. Aitchinson, H. S. Eisenberg, Y. Silberberg, "Experimental observation of linear and nonlinear optical Bloch oscillations," Phys. Rev. Lett. 83, 4756-4759 (1999)
[CrossRef]

T. Pertsch, P. Dannberg, W. Elflein, A. Brauer, and F. Lederer, "Optical Bloch Oscillations in Temperature Tuned Waveguide Arrays," Phys. Rev. Lett. 83, 4752-4755 (1999).
[CrossRef]

G. Lenz, I. Talanina, and C. M. de Sterke, "Bloch Oscillations in an Array of Curved Optical Waveguides," Phys. Rev. Lett. 83, 963-966 (1999).
[CrossRef]

1998 (1)

C. M. de Sterke, J. N. Bright, P. A. Krug, and T. E. Hammon, "Observation of an optical Wannier-Stark ladder," Phys. Rev. E 57, 2365-2370 (1998).
[CrossRef]

1993 (1)

C. Waschke, H. G. Roskos, R. Schwedler, K. Leo, H. Kurz, and K. Kohler, "Coherent submillimeter-wave emission from Bloch oscillations in a semiconductor superlattice," Phys. Rev. Lett. 70, 3319-3322 (1993).
[CrossRef] [PubMed]

1967 (1)

V. G. Veselago, "The electrodynamics of substances with simultaneously negative values of epsilon and mu," Usp. Fiz. Nauk 92, 517-526 (1967) (in Russian) [English translation: Sov. Phys. Usp. 10, 509 (1968)].

1960 (1)

G. H. Wannier, "Wave functions and effective Hamiltonian for Bloch electrons in an electric field," Phys. Rev. 117, 432-439 (1960).
[CrossRef]

1928 (1)

F. Bloch, "Uber die Quantenmechanik der Elektronen in Kristallgittern," Z. Phys. A: Hadrons Nucl. 52, 555-600 (1928).

Agarwal, V.

V. Agarwal, J. A. del Rio, G. Malpuech, M. Zamfirescu, A. Kavokin, D. Coquillat, D. Scalbert, M. Vladimirova, and B. Gil, "Photon Bloch Oscillations in Porous Silicon Optical Superlattices," Phys. Rev. Lett. 92, 097401 (2004).
[CrossRef] [PubMed]

Aitchinson, J. S.

R. Morandotti, U. Peschel, J. S. Aitchinson, H. S. Eisenberg, Y. Silberberg, "Experimental observation of linear and nonlinear optical Bloch oscillations," Phys. Rev. Lett. 83, 4756-4759 (1999)
[CrossRef]

Bloch, F.

F. Bloch, "Uber die Quantenmechanik der Elektronen in Kristallgittern," Z. Phys. A: Hadrons Nucl. 52, 555-600 (1928).

Boardman, A. D.

I. V. Shadrivov, A. A. Sukhorukov, Yu. S. Kivshar, A. A. Zharov, A. D. Boardman, and P. Egan, "Nonlinear surface waves in left-handed materials," Phys. Rev. E 69, 016617 (2004).
[CrossRef]

Brauer, A.

T. Pertsch, P. Dannberg, W. Elflein, A. Brauer, and F. Lederer, "Optical Bloch Oscillations in Temperature Tuned Waveguide Arrays," Phys. Rev. Lett. 83, 4752-4755 (1999).
[CrossRef]

Bright, J. N.

C. M. de Sterke, J. N. Bright, P. A. Krug, and T. E. Hammon, "Observation of an optical Wannier-Stark ladder," Phys. Rev. E 57, 2365-2370 (1998).
[CrossRef]

Chan, C. T.

J. Li, L. Zhou, C. T. Chan, and P. Sheng, "Photonic Band Gap from a Stack of Positive and Negative Index Materials," Phys. Rev. Lett. 90, 083901 (2003).
[CrossRef] [PubMed]

Coquillat, D.

V. Agarwal, J. A. del Rio, G. Malpuech, M. Zamfirescu, A. Kavokin, D. Coquillat, D. Scalbert, M. Vladimirova, and B. Gil, "Photon Bloch Oscillations in Porous Silicon Optical Superlattices," Phys. Rev. Lett. 92, 097401 (2004).
[CrossRef] [PubMed]

Costantino, P.

R. Sapienza, P. Costantino, D. Wiersma, M. Ghulinyan, C. J. Oton, and L. Pavesi, "Optical Analogue of Electronic Bloch Oscillations," Phys. Rev. Lett. 91, 263902 (2003).
[CrossRef]

Dannberg, P.

T. Pertsch, P. Dannberg, W. Elflein, A. Brauer, and F. Lederer, "Optical Bloch Oscillations in Temperature Tuned Waveguide Arrays," Phys. Rev. Lett. 83, 4752-4755 (1999).
[CrossRef]

de Sterke, C. M.

G. Lenz, I. Talanina, and C. M. de Sterke, "Bloch Oscillations in an Array of Curved Optical Waveguides," Phys. Rev. Lett. 83, 963-966 (1999).
[CrossRef]

C. M. de Sterke, J. N. Bright, P. A. Krug, and T. E. Hammon, "Observation of an optical Wannier-Stark ladder," Phys. Rev. E 57, 2365-2370 (1998).
[CrossRef]

del Rio, J. A.

V. Agarwal, J. A. del Rio, G. Malpuech, M. Zamfirescu, A. Kavokin, D. Coquillat, D. Scalbert, M. Vladimirova, and B. Gil, "Photon Bloch Oscillations in Porous Silicon Optical Superlattices," Phys. Rev. Lett. 92, 097401 (2004).
[CrossRef] [PubMed]

Egan, P.

I. V. Shadrivov, A. A. Sukhorukov, Yu. S. Kivshar, A. A. Zharov, A. D. Boardman, and P. Egan, "Nonlinear surface waves in left-handed materials," Phys. Rev. E 69, 016617 (2004).
[CrossRef]

Eisenberg, H. S.

R. Morandotti, U. Peschel, J. S. Aitchinson, H. S. Eisenberg, Y. Silberberg, "Experimental observation of linear and nonlinear optical Bloch oscillations," Phys. Rev. Lett. 83, 4756-4759 (1999)
[CrossRef]

Elflein, W.

T. Pertsch, P. Dannberg, W. Elflein, A. Brauer, and F. Lederer, "Optical Bloch Oscillations in Temperature Tuned Waveguide Arrays," Phys. Rev. Lett. 83, 4752-4755 (1999).
[CrossRef]

Ghulinyan, M.

R. Sapienza, P. Costantino, D. Wiersma, M. Ghulinyan, C. J. Oton, and L. Pavesi, "Optical Analogue of Electronic Bloch Oscillations," Phys. Rev. Lett. 91, 263902 (2003).
[CrossRef]

Gil, B.

V. Agarwal, J. A. del Rio, G. Malpuech, M. Zamfirescu, A. Kavokin, D. Coquillat, D. Scalbert, M. Vladimirova, and B. Gil, "Photon Bloch Oscillations in Porous Silicon Optical Superlattices," Phys. Rev. Lett. 92, 097401 (2004).
[CrossRef] [PubMed]

Hammon, T. E.

C. M. de Sterke, J. N. Bright, P. A. Krug, and T. E. Hammon, "Observation of an optical Wannier-Stark ladder," Phys. Rev. E 57, 2365-2370 (1998).
[CrossRef]

Kavokin, A.

V. Agarwal, J. A. del Rio, G. Malpuech, M. Zamfirescu, A. Kavokin, D. Coquillat, D. Scalbert, M. Vladimirova, and B. Gil, "Photon Bloch Oscillations in Porous Silicon Optical Superlattices," Phys. Rev. Lett. 92, 097401 (2004).
[CrossRef] [PubMed]

Kivshar, Yu. S.

A. A. Sukhorukov, I. V. Shadrivov, and Yu. S. Kivshar, "Wave scattering by metamaterial wedges and interfaces," Int. J. Num. Model. 19, 105-117 (2006).
[CrossRef]

I. V. Shadrivov, A. A. Sukhorukov, Yu. S. Kivshar, A. A. Zharov, A. D. Boardman, and P. Egan, "Nonlinear surface waves in left-handed materials," Phys. Rev. E 69, 016617 (2004).
[CrossRef]

I. V. Shadrivov, A. A. Sukhorukov, and Yu. S. Kivshar, "Beam shaping by a periodic structure with negative refraction," Appl. Phys. Lett. 82, 3820-3822 (2003).
[CrossRef]

Kohler, K.

C. Waschke, H. G. Roskos, R. Schwedler, K. Leo, H. Kurz, and K. Kohler, "Coherent submillimeter-wave emission from Bloch oscillations in a semiconductor superlattice," Phys. Rev. Lett. 70, 3319-3322 (1993).
[CrossRef] [PubMed]

Krug, P. A.

C. M. de Sterke, J. N. Bright, P. A. Krug, and T. E. Hammon, "Observation of an optical Wannier-Stark ladder," Phys. Rev. E 57, 2365-2370 (1998).
[CrossRef]

Kurz, H.

C. Waschke, H. G. Roskos, R. Schwedler, K. Leo, H. Kurz, and K. Kohler, "Coherent submillimeter-wave emission from Bloch oscillations in a semiconductor superlattice," Phys. Rev. Lett. 70, 3319-3322 (1993).
[CrossRef] [PubMed]

Lederer, F.

T. Pertsch, P. Dannberg, W. Elflein, A. Brauer, and F. Lederer, "Optical Bloch Oscillations in Temperature Tuned Waveguide Arrays," Phys. Rev. Lett. 83, 4752-4755 (1999).
[CrossRef]

Lenz, G.

G. Lenz, I. Talanina, and C. M. de Sterke, "Bloch Oscillations in an Array of Curved Optical Waveguides," Phys. Rev. Lett. 83, 963-966 (1999).
[CrossRef]

Leo, K.

C. Waschke, H. G. Roskos, R. Schwedler, K. Leo, H. Kurz, and K. Kohler, "Coherent submillimeter-wave emission from Bloch oscillations in a semiconductor superlattice," Phys. Rev. Lett. 70, 3319-3322 (1993).
[CrossRef] [PubMed]

Li, J.

J. Li, L. Zhou, C. T. Chan, and P. Sheng, "Photonic Band Gap from a Stack of Positive and Negative Index Materials," Phys. Rev. Lett. 90, 083901 (2003).
[CrossRef] [PubMed]

Malpuech, G.

V. Agarwal, J. A. del Rio, G. Malpuech, M. Zamfirescu, A. Kavokin, D. Coquillat, D. Scalbert, M. Vladimirova, and B. Gil, "Photon Bloch Oscillations in Porous Silicon Optical Superlattices," Phys. Rev. Lett. 92, 097401 (2004).
[CrossRef] [PubMed]

Morandotti, R.

R. Morandotti, U. Peschel, J. S. Aitchinson, H. S. Eisenberg, Y. Silberberg, "Experimental observation of linear and nonlinear optical Bloch oscillations," Phys. Rev. Lett. 83, 4756-4759 (1999)
[CrossRef]

Oton, C. J.

R. Sapienza, P. Costantino, D. Wiersma, M. Ghulinyan, C. J. Oton, and L. Pavesi, "Optical Analogue of Electronic Bloch Oscillations," Phys. Rev. Lett. 91, 263902 (2003).
[CrossRef]

Pavesi, L.

R. Sapienza, P. Costantino, D. Wiersma, M. Ghulinyan, C. J. Oton, and L. Pavesi, "Optical Analogue of Electronic Bloch Oscillations," Phys. Rev. Lett. 91, 263902 (2003).
[CrossRef]

Pendry, J. B.

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, "Metamaterials and Negative Refractive Index," Science 305, 788-792 (2004).
[CrossRef] [PubMed]

Pertsch, T.

T. Pertsch, P. Dannberg, W. Elflein, A. Brauer, and F. Lederer, "Optical Bloch Oscillations in Temperature Tuned Waveguide Arrays," Phys. Rev. Lett. 83, 4752-4755 (1999).
[CrossRef]

Peschel, U.

R. Morandotti, U. Peschel, J. S. Aitchinson, H. S. Eisenberg, Y. Silberberg, "Experimental observation of linear and nonlinear optical Bloch oscillations," Phys. Rev. Lett. 83, 4756-4759 (1999)
[CrossRef]

Roskos, H. G.

C. Waschke, H. G. Roskos, R. Schwedler, K. Leo, H. Kurz, and K. Kohler, "Coherent submillimeter-wave emission from Bloch oscillations in a semiconductor superlattice," Phys. Rev. Lett. 70, 3319-3322 (1993).
[CrossRef] [PubMed]

Sapienza, R.

R. Sapienza, P. Costantino, D. Wiersma, M. Ghulinyan, C. J. Oton, and L. Pavesi, "Optical Analogue of Electronic Bloch Oscillations," Phys. Rev. Lett. 91, 263902 (2003).
[CrossRef]

Scalbert, D.

V. Agarwal, J. A. del Rio, G. Malpuech, M. Zamfirescu, A. Kavokin, D. Coquillat, D. Scalbert, M. Vladimirova, and B. Gil, "Photon Bloch Oscillations in Porous Silicon Optical Superlattices," Phys. Rev. Lett. 92, 097401 (2004).
[CrossRef] [PubMed]

Schwedler, R.

C. Waschke, H. G. Roskos, R. Schwedler, K. Leo, H. Kurz, and K. Kohler, "Coherent submillimeter-wave emission from Bloch oscillations in a semiconductor superlattice," Phys. Rev. Lett. 70, 3319-3322 (1993).
[CrossRef] [PubMed]

Shadrivov, I. V.

A. A. Sukhorukov, I. V. Shadrivov, and Yu. S. Kivshar, "Wave scattering by metamaterial wedges and interfaces," Int. J. Num. Model. 19, 105-117 (2006).
[CrossRef]

I. V. Shadrivov, A. A. Sukhorukov, Yu. S. Kivshar, A. A. Zharov, A. D. Boardman, and P. Egan, "Nonlinear surface waves in left-handed materials," Phys. Rev. E 69, 016617 (2004).
[CrossRef]

I. V. Shadrivov, A. A. Sukhorukov, and Yu. S. Kivshar, "Beam shaping by a periodic structure with negative refraction," Appl. Phys. Lett. 82, 3820-3822 (2003).
[CrossRef]

Sheng, P.

J. Li, L. Zhou, C. T. Chan, and P. Sheng, "Photonic Band Gap from a Stack of Positive and Negative Index Materials," Phys. Rev. Lett. 90, 083901 (2003).
[CrossRef] [PubMed]

Silberberg, Y.

R. Morandotti, U. Peschel, J. S. Aitchinson, H. S. Eisenberg, Y. Silberberg, "Experimental observation of linear and nonlinear optical Bloch oscillations," Phys. Rev. Lett. 83, 4756-4759 (1999)
[CrossRef]

Smith, D. R.

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, "Metamaterials and Negative Refractive Index," Science 305, 788-792 (2004).
[CrossRef] [PubMed]

Sukhorukov, A. A.

A. A. Sukhorukov, I. V. Shadrivov, and Yu. S. Kivshar, "Wave scattering by metamaterial wedges and interfaces," Int. J. Num. Model. 19, 105-117 (2006).
[CrossRef]

I. V. Shadrivov, A. A. Sukhorukov, Yu. S. Kivshar, A. A. Zharov, A. D. Boardman, and P. Egan, "Nonlinear surface waves in left-handed materials," Phys. Rev. E 69, 016617 (2004).
[CrossRef]

I. V. Shadrivov, A. A. Sukhorukov, and Yu. S. Kivshar, "Beam shaping by a periodic structure with negative refraction," Appl. Phys. Lett. 82, 3820-3822 (2003).
[CrossRef]

Talanina, I.

G. Lenz, I. Talanina, and C. M. de Sterke, "Bloch Oscillations in an Array of Curved Optical Waveguides," Phys. Rev. Lett. 83, 963-966 (1999).
[CrossRef]

Veselago, V. G.

V. G. Veselago, "The electrodynamics of substances with simultaneously negative values of epsilon and mu," Usp. Fiz. Nauk 92, 517-526 (1967) (in Russian) [English translation: Sov. Phys. Usp. 10, 509 (1968)].

Vladimirova, M.

V. Agarwal, J. A. del Rio, G. Malpuech, M. Zamfirescu, A. Kavokin, D. Coquillat, D. Scalbert, M. Vladimirova, and B. Gil, "Photon Bloch Oscillations in Porous Silicon Optical Superlattices," Phys. Rev. Lett. 92, 097401 (2004).
[CrossRef] [PubMed]

Wannier, G. H.

G. H. Wannier, "Wave functions and effective Hamiltonian for Bloch electrons in an electric field," Phys. Rev. 117, 432-439 (1960).
[CrossRef]

Waschke, C.

C. Waschke, H. G. Roskos, R. Schwedler, K. Leo, H. Kurz, and K. Kohler, "Coherent submillimeter-wave emission from Bloch oscillations in a semiconductor superlattice," Phys. Rev. Lett. 70, 3319-3322 (1993).
[CrossRef] [PubMed]

Wiersma, D.

R. Sapienza, P. Costantino, D. Wiersma, M. Ghulinyan, C. J. Oton, and L. Pavesi, "Optical Analogue of Electronic Bloch Oscillations," Phys. Rev. Lett. 91, 263902 (2003).
[CrossRef]

Wilkinson, P. B.

P. B. Wilkinson, "Photonic Bloch oscillations and Wannier-Stark ladders in exponentially chirped Bragg gratings," Phys. Rev. E 65, 056616 (2002).
[CrossRef]

Wiltshire, M. C. K.

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, "Metamaterials and Negative Refractive Index," Science 305, 788-792 (2004).
[CrossRef] [PubMed]

Zamfirescu, M.

V. Agarwal, J. A. del Rio, G. Malpuech, M. Zamfirescu, A. Kavokin, D. Coquillat, D. Scalbert, M. Vladimirova, and B. Gil, "Photon Bloch Oscillations in Porous Silicon Optical Superlattices," Phys. Rev. Lett. 92, 097401 (2004).
[CrossRef] [PubMed]

Zharov, A. A.

I. V. Shadrivov, A. A. Sukhorukov, Yu. S. Kivshar, A. A. Zharov, A. D. Boardman, and P. Egan, "Nonlinear surface waves in left-handed materials," Phys. Rev. E 69, 016617 (2004).
[CrossRef]

Zhou, L.

J. Li, L. Zhou, C. T. Chan, and P. Sheng, "Photonic Band Gap from a Stack of Positive and Negative Index Materials," Phys. Rev. Lett. 90, 083901 (2003).
[CrossRef] [PubMed]

Appl. Phys. Lett. (1)

I. V. Shadrivov, A. A. Sukhorukov, and Yu. S. Kivshar, "Beam shaping by a periodic structure with negative refraction," Appl. Phys. Lett. 82, 3820-3822 (2003).
[CrossRef]

Int. J. Num. Model. (1)

A. A. Sukhorukov, I. V. Shadrivov, and Yu. S. Kivshar, "Wave scattering by metamaterial wedges and interfaces," Int. J. Num. Model. 19, 105-117 (2006).
[CrossRef]

Phys. Rev. (1)

G. H. Wannier, "Wave functions and effective Hamiltonian for Bloch electrons in an electric field," Phys. Rev. 117, 432-439 (1960).
[CrossRef]

Phys. Rev. E (3)

C. M. de Sterke, J. N. Bright, P. A. Krug, and T. E. Hammon, "Observation of an optical Wannier-Stark ladder," Phys. Rev. E 57, 2365-2370 (1998).
[CrossRef]

P. B. Wilkinson, "Photonic Bloch oscillations and Wannier-Stark ladders in exponentially chirped Bragg gratings," Phys. Rev. E 65, 056616 (2002).
[CrossRef]

I. V. Shadrivov, A. A. Sukhorukov, Yu. S. Kivshar, A. A. Zharov, A. D. Boardman, and P. Egan, "Nonlinear surface waves in left-handed materials," Phys. Rev. E 69, 016617 (2004).
[CrossRef]

Phys. Rev. Lett. (7)

R. Sapienza, P. Costantino, D. Wiersma, M. Ghulinyan, C. J. Oton, and L. Pavesi, "Optical Analogue of Electronic Bloch Oscillations," Phys. Rev. Lett. 91, 263902 (2003).
[CrossRef]

G. Lenz, I. Talanina, and C. M. de Sterke, "Bloch Oscillations in an Array of Curved Optical Waveguides," Phys. Rev. Lett. 83, 963-966 (1999).
[CrossRef]

R. Morandotti, U. Peschel, J. S. Aitchinson, H. S. Eisenberg, Y. Silberberg, "Experimental observation of linear and nonlinear optical Bloch oscillations," Phys. Rev. Lett. 83, 4756-4759 (1999)
[CrossRef]

T. Pertsch, P. Dannberg, W. Elflein, A. Brauer, and F. Lederer, "Optical Bloch Oscillations in Temperature Tuned Waveguide Arrays," Phys. Rev. Lett. 83, 4752-4755 (1999).
[CrossRef]

V. Agarwal, J. A. del Rio, G. Malpuech, M. Zamfirescu, A. Kavokin, D. Coquillat, D. Scalbert, M. Vladimirova, and B. Gil, "Photon Bloch Oscillations in Porous Silicon Optical Superlattices," Phys. Rev. Lett. 92, 097401 (2004).
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Science (1)

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[CrossRef] [PubMed]

Usp. Fiz. Nauk (1)

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Z. Phys. A: Hadrons Nucl. (1)

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Other (1)

P. Yeh, Optical Waves in Layered Media (John Wiley & Sons, New York, 1988).

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Figures (5)

Fig. 1.
Fig. 1.

Schematic of linearly chirped one-dimensional structure with alternating layers of negative-index metamaterial and conventional dielectric.

Fig. 2.
Fig. 2.

Bandgap diagram for the TE-polarized waves in structure with alternating dielectric (εr =µr =1) and metamaterial (εl =-5, µl =-0.8) in case of zero average refractive index. Black and white areas correspond to gaps and bands, respectively. Two spectra of the excited Bloch oscillations are shown on the left. The inset shows a magnified part of the spectrum.

Fig. 3.
Fig. 3.

Field distribution in the case of surface-wave-assisted Bloch oscillations. The Wannier-Stark ladder appears for the propagation constants centered around k y0=2.47, normalized period is Ly =820.

Fig. 4.
Fig. 4.

Field distribution for the case of guided waves. The Wannier-Stark ladder appears for the propagation constants centered around k y0=1.34, period is Ly ≅100.

Fig. 5.
Fig. 5.

Field distribution for the case when waves propagate in both type of materials. The Wannier-Stark ladder appears with average propagation constant k y0=0.8, period of oscillations is Ly =210.

Equations (4)

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n ( z ) = { n r = ε r μ r z ( z i , z i + a i ) n l = ε l μ l z ( z i + a i , z i + Λ i )
Δ 2 E x ( y , z ) + n 2 ( z ) E x ( y , z ) 1 μ d μ ( z ) d z E x ( y , z ) z = 0 ,
2 cos ( K B Λ ) = 2 cos ( k zr a ) cos ( k zl b ) ( k zl μ r k zr μ l + k zr μ l k zl μ r ) sin ( k zr a ) sin ( k zl b ) ,
2 μ m 1 1 + μ m + 1 1 [ U m + 1 U m μ m + 1 U m U m 1 μ m 1 ] 1 h 2 + ε m 1 + ε m + 1 μ m 1 1 + μ m + 1 1 U m = k y 2 U m ,

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