Abstract

We show that nonreciprocal optical Bloch-like oscillations can emerge in transversely magnetized waveguide arrays in the presence of an effective index step between the waveguides. Normal modes of the system are shown to acquire different wavenumbers in opposite propagation directions. Significant differences in phase coherence and decoherence between these normal modes are presented and discussed. Nonreciprocity is established by imposing unequal vertical refractive index gradients at the substrate/core and core/cover interfaces in the presence of transverse magnetization.

© 2010 Optical Society of America

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    [CrossRef] [PubMed]
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2009

S. Longhi, Phys. Rev. Lett. 103, 123601 (2009).
[CrossRef] [PubMed]

2008

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimany, Phys. Rev. Lett. 100, 103904 (2008).
[CrossRef] [PubMed]

1999

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, Phys. Rev. Lett. 83, 4756 (1999).
[CrossRef]

T. Pertsch, P. Dannberg, W. Elflein, A. Bräuer, and F. Lederer, Phys. Rev. Lett. 83, 4752 (1999).
[CrossRef]

N. Bahlmann, M. Lohmeyer, O. Zhuromskyy, H. Dötsch, and P. Hertel, Opt. Commun. 161, 330 (1999) and references therein.
[CrossRef]

1998

1994

T. Dekorsy, P. Leisching, K. Köhler, and H. Kurz, Phys. Rev. B 50, 8106 (1994).
[CrossRef]

1980

1934

C. Zener, Proc. R. Soc. London A 145, 523 (1934).
[CrossRef]

Aitchison, J. S.

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, Phys. Rev. Lett. 83, 4756 (1999).
[CrossRef]

Bahlmann, N.

N. Bahlmann, M. Lohmeyer, O. Zhuromskyy, H. Dötsch, and P. Hertel, Opt. Commun. 161, 330 (1999) and references therein.
[CrossRef]

Bräuer, A.

T. Pertsch, P. Dannberg, W. Elflein, A. Bräuer, and F. Lederer, Phys. Rev. Lett. 83, 4752 (1999).
[CrossRef]

Christodoulides, D. N.

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimany, Phys. Rev. Lett. 100, 103904 (2008).
[CrossRef] [PubMed]

Dannberg, P.

T. Pertsch, P. Dannberg, W. Elflein, A. Bräuer, and F. Lederer, Phys. Rev. Lett. 83, 4752 (1999).
[CrossRef]

Dekorsy, T.

T. Dekorsy, P. Leisching, K. Köhler, and H. Kurz, Phys. Rev. B 50, 8106 (1994).
[CrossRef]

Dötsch, H.

N. Bahlmann, M. Lohmeyer, O. Zhuromskyy, H. Dötsch, and P. Hertel, Opt. Commun. 161, 330 (1999) and references therein.
[CrossRef]

Eisenberg, H. S.

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, Phys. Rev. Lett. 83, 4756 (1999).
[CrossRef]

Elflein, W.

T. Pertsch, P. Dannberg, W. Elflein, A. Bräuer, and F. Lederer, Phys. Rev. Lett. 83, 4752 (1999).
[CrossRef]

El-Ganainy, R.

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimany, Phys. Rev. Lett. 100, 103904 (2008).
[CrossRef] [PubMed]

Feit, M. D.

Fleck, J. A.

Hertel, P.

N. Bahlmann, M. Lohmeyer, O. Zhuromskyy, H. Dötsch, and P. Hertel, Opt. Commun. 161, 330 (1999) and references therein.
[CrossRef]

Köhler, K.

T. Dekorsy, P. Leisching, K. Köhler, and H. Kurz, Phys. Rev. B 50, 8106 (1994).
[CrossRef]

Kurz, H.

T. Dekorsy, P. Leisching, K. Köhler, and H. Kurz, Phys. Rev. B 50, 8106 (1994).
[CrossRef]

Lederer, F.

T. Pertsch, P. Dannberg, W. Elflein, A. Bräuer, and F. Lederer, Phys. Rev. Lett. 83, 4752 (1999).
[CrossRef]

U. Peschel, T. Pertsch, and F. Lederer, Opt. Lett. 23, 1701(1998).
[CrossRef]

Leisching, P.

T. Dekorsy, P. Leisching, K. Köhler, and H. Kurz, Phys. Rev. B 50, 8106 (1994).
[CrossRef]

Lohmeyer, M.

N. Bahlmann, M. Lohmeyer, O. Zhuromskyy, H. Dötsch, and P. Hertel, Opt. Commun. 161, 330 (1999) and references therein.
[CrossRef]

Longhi, S.

S. Longhi, Phys. Rev. Lett. 103, 123601 (2009).
[CrossRef] [PubMed]

Makris, K. G.

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimany, Phys. Rev. Lett. 100, 103904 (2008).
[CrossRef] [PubMed]

Morandotti, R.

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, Phys. Rev. Lett. 83, 4756 (1999).
[CrossRef]

Musslimany, Z. H.

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimany, Phys. Rev. Lett. 100, 103904 (2008).
[CrossRef] [PubMed]

Pertsch, T.

T. Pertsch, P. Dannberg, W. Elflein, A. Bräuer, and F. Lederer, Phys. Rev. Lett. 83, 4752 (1999).
[CrossRef]

U. Peschel, T. Pertsch, and F. Lederer, Opt. Lett. 23, 1701(1998).
[CrossRef]

Peschel, U.

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, Phys. Rev. Lett. 83, 4756 (1999).
[CrossRef]

U. Peschel, T. Pertsch, and F. Lederer, Opt. Lett. 23, 1701(1998).
[CrossRef]

Silberberg, Y.

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, Phys. Rev. Lett. 83, 4756 (1999).
[CrossRef]

Zener, C.

C. Zener, Proc. R. Soc. London A 145, 523 (1934).
[CrossRef]

Zhuromskyy, O.

N. Bahlmann, M. Lohmeyer, O. Zhuromskyy, H. Dötsch, and P. Hertel, Opt. Commun. 161, 330 (1999) and references therein.
[CrossRef]

Appl. Opt.

Opt. Commun.

N. Bahlmann, M. Lohmeyer, O. Zhuromskyy, H. Dötsch, and P. Hertel, Opt. Commun. 161, 330 (1999) and references therein.
[CrossRef]

Opt. Lett.

Phys. Rev. B

T. Dekorsy, P. Leisching, K. Köhler, and H. Kurz, Phys. Rev. B 50, 8106 (1994).
[CrossRef]

Phys. Rev. Lett.

S. Longhi, Phys. Rev. Lett. 103, 123601 (2009).
[CrossRef] [PubMed]

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimany, Phys. Rev. Lett. 100, 103904 (2008).
[CrossRef] [PubMed]

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, Phys. Rev. Lett. 83, 4756 (1999).
[CrossRef]

T. Pertsch, P. Dannberg, W. Elflein, A. Bräuer, and F. Lederer, Phys. Rev. Lett. 83, 4752 (1999).
[CrossRef]

Proc. R. Soc. London A

C. Zener, Proc. R. Soc. London A 145, 523 (1934).
[CrossRef]

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Figures (2)

Fig. 1
Fig. 1

Plot of NR Δ β ( nr ) versus normal mode β showing a nearly uniform Δ ( Δ β ( nr ) ) between consecutive normal modes. The insets show (a) a schematic depiction of the waveguide array highlighting the effective index progression between adjacent waveguides and (b) the power distribution of the first normal mode of the array.

Fig. 2
Fig. 2

Beam-propagation simulations of (a) Bloch mode oscillations for the waveguide array with film index 2.35 and substrate index 2.25, showing a BO period of 945 μm for a 1-μm-wide input beam. Simulated BOs in (b) forward and (c) backward directions. The strength of Δ ( Δ β ( nr ) ) has been artificially enhanced by a factor of 20 to highlight the difference in the BO period, shown with dashed lines.

Equations (6)

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

i d a n f , b d z + δ β f , b n a n f , b + κ f , b ( a n 1 f , b + a n + 1 f , b ) = 0.
ε ^ = ( ε 0 i g 0 ε 0 i g 0 ε ) .
( ε x 1 ε x + y 2 β 2 + ω 2 μ 0 ε 0 ε β g ε x + β ε · x g ε 2 ) H y = 0.
Δ β ( nr ) = 2 Re d x d y ( x H y ) H y * ( i g / ε 2 ) d x d y | H y | 2 ε 1 ,
δ β m + 1 , m f δ β m + 1 , m b = β m + 1 f β m f ( β m + 1 b β m b ) = Δ β m + 1 ( nr ) Δ β m ( nr ) = Δ ( Δ β ( nr ) ) m + 1 , m ,
i d a n b d z + ( δ β f Δ ( Δ β n ( nr ) ) ) n a n b + κ b ( a n 1 b + a n + 1 b ) = 0.

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