Abstract

We investigate, both theoretically and numerically, the asymmetric light propagation in transverse separation modulated photonic lattices. The theoretical results show that the transmission contrast η of the structure is determined only by the coupling strengths between the chirped lattice and the two boundary uniform lattice portions. The numerical studies demonstrate that η is independent of the separation modulation function of the chirped lattice, which is in good agreement with the theory.

© 2013 Optical Society of America

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  1. B. Liang, B. Yuan, and J.-C. Cheng, Phys. Rev. Lett. 103, 104301 (2009).
    [CrossRef]
  2. M. Terraneo, M. Peyrard, and G. Casati, Phys. Rev. Lett. 88, 094302 (2002).
    [CrossRef]
  3. C. Wang, C.-Z. Zhou, and Z.-Y. Li, Opt. Express 19, 26948 (2011).
    [CrossRef]
  4. M. A. Schmidt, L. Wondraczek, H. W. Lee, N. Granzow, N. Da, and P. St. J. Russell, Adv. Mater. 23, 2681 (2011).
    [CrossRef]
  5. J. Zhang, M. B. Yu, G. Q. Lo, and D. L. Kwong, IEEE J. Sel. Top. Quantum Electron. 16, 53 (2010).
    [CrossRef]
  6. M. Scalora, J. P. Dowling, C. M. Bowden, and M. J. Bloemer, J. Appl. Phys. 76, 2023 (1994).
    [CrossRef]
  7. M. D. Tocci, M. J. Bloemer, M. Scalora, J. P. Dowling, and C. M. Bowden, Appl. Phys. Lett. 66, 2324 (1995).
    [CrossRef]
  8. F. Biancalana, J. Appl. Phys. 104, 093113 (2008).
    [CrossRef]
  9. S. Lepri and G. Casati, Phys. Rev. Lett. 106, 164101 (2011).
    [CrossRef]
  10. L. Fan, J. Wang, L. T. Varghese, H. Shen, B. Niu, Y. Xuan, A. M. Weiner, and M. H. Qi, Science 335, 447 (2012).
    [CrossRef]
  11. M. W. Feise, I. V. Shadrivov, and Y. S. Kivshar, Phys. Rev. E 71, 037602 (2005).
    [CrossRef]
  12. H. Kurt, D. Yilmaz, A. E. Akosman, and E. Ozbay, Opt. Express 20, 20635 (2012).
    [CrossRef]
  13. C. Wang, X.-L. Zhong, and Z.-Y. Li, Sci. Rep. 2, 1 (2012).
  14. A. A. Sukhorukov, Opt. Lett. 35, 989 (2010).
    [CrossRef]
  15. A. Szameit, F. Dreisow, M. Heinrich, S. Nolte, and A. A. Sukhorukov, Phys. Rev. Lett. 106, 193903 (2011).
    [CrossRef]
  16. Z. Cao, X. Y. Qi, X. Q. Feng, Z. Y. Ren, G. Q. Zhang, and J. T. Bai, Opt. Express 20, 19119 (2012).
    [CrossRef]
  17. Q. M. Li, C. T. Chan, K. M. Ho, and C. M. Soukoulis, Phys. Rev. B 53, 15577 (1996).
    [CrossRef]
  18. D. H. Dunlap, H.-L. Wu, and P. W. Phillips, Phys. Rev. Lett. 65, 88 (1990).
    [CrossRef]
  19. D. Hennig and G. P. Tsironis, Phys. Rep. 307, 333 (1999).
    [CrossRef]
  20. K. Gallo, G. Assanto, K. R. Parameswaran, and M. M. Fejer, Appl. Phys. Lett. 79, 314 (2001).
    [CrossRef]
  21. A. Szameit, F. Dreisow, T. Pertsch, S. Nolte, and A. Tünnermann, Opt. Express 15, 1579 (2007).
    [CrossRef]
  22. Y. Lahini, Y. Bromberg, Y. Shechtman, A. Szameit, D. N. Christodoulides, R. Morandotti, and Y. Silberberg, Phys. Rev. A 84, 041806 (2011).
    [CrossRef]

2012 (4)

L. Fan, J. Wang, L. T. Varghese, H. Shen, B. Niu, Y. Xuan, A. M. Weiner, and M. H. Qi, Science 335, 447 (2012).
[CrossRef]

H. Kurt, D. Yilmaz, A. E. Akosman, and E. Ozbay, Opt. Express 20, 20635 (2012).
[CrossRef]

C. Wang, X.-L. Zhong, and Z.-Y. Li, Sci. Rep. 2, 1 (2012).

Z. Cao, X. Y. Qi, X. Q. Feng, Z. Y. Ren, G. Q. Zhang, and J. T. Bai, Opt. Express 20, 19119 (2012).
[CrossRef]

2011 (5)

S. Lepri and G. Casati, Phys. Rev. Lett. 106, 164101 (2011).
[CrossRef]

A. Szameit, F. Dreisow, M. Heinrich, S. Nolte, and A. A. Sukhorukov, Phys. Rev. Lett. 106, 193903 (2011).
[CrossRef]

C. Wang, C.-Z. Zhou, and Z.-Y. Li, Opt. Express 19, 26948 (2011).
[CrossRef]

M. A. Schmidt, L. Wondraczek, H. W. Lee, N. Granzow, N. Da, and P. St. J. Russell, Adv. Mater. 23, 2681 (2011).
[CrossRef]

Y. Lahini, Y. Bromberg, Y. Shechtman, A. Szameit, D. N. Christodoulides, R. Morandotti, and Y. Silberberg, Phys. Rev. A 84, 041806 (2011).
[CrossRef]

2010 (2)

J. Zhang, M. B. Yu, G. Q. Lo, and D. L. Kwong, IEEE J. Sel. Top. Quantum Electron. 16, 53 (2010).
[CrossRef]

A. A. Sukhorukov, Opt. Lett. 35, 989 (2010).
[CrossRef]

2009 (1)

B. Liang, B. Yuan, and J.-C. Cheng, Phys. Rev. Lett. 103, 104301 (2009).
[CrossRef]

2008 (1)

F. Biancalana, J. Appl. Phys. 104, 093113 (2008).
[CrossRef]

2007 (1)

2005 (1)

M. W. Feise, I. V. Shadrivov, and Y. S. Kivshar, Phys. Rev. E 71, 037602 (2005).
[CrossRef]

2002 (1)

M. Terraneo, M. Peyrard, and G. Casati, Phys. Rev. Lett. 88, 094302 (2002).
[CrossRef]

2001 (1)

K. Gallo, G. Assanto, K. R. Parameswaran, and M. M. Fejer, Appl. Phys. Lett. 79, 314 (2001).
[CrossRef]

1999 (1)

D. Hennig and G. P. Tsironis, Phys. Rep. 307, 333 (1999).
[CrossRef]

1996 (1)

Q. M. Li, C. T. Chan, K. M. Ho, and C. M. Soukoulis, Phys. Rev. B 53, 15577 (1996).
[CrossRef]

1995 (1)

M. D. Tocci, M. J. Bloemer, M. Scalora, J. P. Dowling, and C. M. Bowden, Appl. Phys. Lett. 66, 2324 (1995).
[CrossRef]

1994 (1)

M. Scalora, J. P. Dowling, C. M. Bowden, and M. J. Bloemer, J. Appl. Phys. 76, 2023 (1994).
[CrossRef]

1990 (1)

D. H. Dunlap, H.-L. Wu, and P. W. Phillips, Phys. Rev. Lett. 65, 88 (1990).
[CrossRef]

Akosman, A. E.

Assanto, G.

K. Gallo, G. Assanto, K. R. Parameswaran, and M. M. Fejer, Appl. Phys. Lett. 79, 314 (2001).
[CrossRef]

Bai, J. T.

Biancalana, F.

F. Biancalana, J. Appl. Phys. 104, 093113 (2008).
[CrossRef]

Bloemer, M. J.

M. D. Tocci, M. J. Bloemer, M. Scalora, J. P. Dowling, and C. M. Bowden, Appl. Phys. Lett. 66, 2324 (1995).
[CrossRef]

M. Scalora, J. P. Dowling, C. M. Bowden, and M. J. Bloemer, J. Appl. Phys. 76, 2023 (1994).
[CrossRef]

Bowden, C. M.

M. D. Tocci, M. J. Bloemer, M. Scalora, J. P. Dowling, and C. M. Bowden, Appl. Phys. Lett. 66, 2324 (1995).
[CrossRef]

M. Scalora, J. P. Dowling, C. M. Bowden, and M. J. Bloemer, J. Appl. Phys. 76, 2023 (1994).
[CrossRef]

Bromberg, Y.

Y. Lahini, Y. Bromberg, Y. Shechtman, A. Szameit, D. N. Christodoulides, R. Morandotti, and Y. Silberberg, Phys. Rev. A 84, 041806 (2011).
[CrossRef]

Cao, Z.

Casati, G.

S. Lepri and G. Casati, Phys. Rev. Lett. 106, 164101 (2011).
[CrossRef]

M. Terraneo, M. Peyrard, and G. Casati, Phys. Rev. Lett. 88, 094302 (2002).
[CrossRef]

Chan, C. T.

Q. M. Li, C. T. Chan, K. M. Ho, and C. M. Soukoulis, Phys. Rev. B 53, 15577 (1996).
[CrossRef]

Cheng, J.-C.

B. Liang, B. Yuan, and J.-C. Cheng, Phys. Rev. Lett. 103, 104301 (2009).
[CrossRef]

Christodoulides, D. N.

Y. Lahini, Y. Bromberg, Y. Shechtman, A. Szameit, D. N. Christodoulides, R. Morandotti, and Y. Silberberg, Phys. Rev. A 84, 041806 (2011).
[CrossRef]

Da, N.

M. A. Schmidt, L. Wondraczek, H. W. Lee, N. Granzow, N. Da, and P. St. J. Russell, Adv. Mater. 23, 2681 (2011).
[CrossRef]

Dowling, J. P.

M. D. Tocci, M. J. Bloemer, M. Scalora, J. P. Dowling, and C. M. Bowden, Appl. Phys. Lett. 66, 2324 (1995).
[CrossRef]

M. Scalora, J. P. Dowling, C. M. Bowden, and M. J. Bloemer, J. Appl. Phys. 76, 2023 (1994).
[CrossRef]

Dreisow, F.

A. Szameit, F. Dreisow, M. Heinrich, S. Nolte, and A. A. Sukhorukov, Phys. Rev. Lett. 106, 193903 (2011).
[CrossRef]

A. Szameit, F. Dreisow, T. Pertsch, S. Nolte, and A. Tünnermann, Opt. Express 15, 1579 (2007).
[CrossRef]

Dunlap, D. H.

D. H. Dunlap, H.-L. Wu, and P. W. Phillips, Phys. Rev. Lett. 65, 88 (1990).
[CrossRef]

Fan, L.

L. Fan, J. Wang, L. T. Varghese, H. Shen, B. Niu, Y. Xuan, A. M. Weiner, and M. H. Qi, Science 335, 447 (2012).
[CrossRef]

Feise, M. W.

M. W. Feise, I. V. Shadrivov, and Y. S. Kivshar, Phys. Rev. E 71, 037602 (2005).
[CrossRef]

Fejer, M. M.

K. Gallo, G. Assanto, K. R. Parameswaran, and M. M. Fejer, Appl. Phys. Lett. 79, 314 (2001).
[CrossRef]

Feng, X. Q.

Gallo, K.

K. Gallo, G. Assanto, K. R. Parameswaran, and M. M. Fejer, Appl. Phys. Lett. 79, 314 (2001).
[CrossRef]

Granzow, N.

M. A. Schmidt, L. Wondraczek, H. W. Lee, N. Granzow, N. Da, and P. St. J. Russell, Adv. Mater. 23, 2681 (2011).
[CrossRef]

Heinrich, M.

A. Szameit, F. Dreisow, M. Heinrich, S. Nolte, and A. A. Sukhorukov, Phys. Rev. Lett. 106, 193903 (2011).
[CrossRef]

Hennig, D.

D. Hennig and G. P. Tsironis, Phys. Rep. 307, 333 (1999).
[CrossRef]

Ho, K. M.

Q. M. Li, C. T. Chan, K. M. Ho, and C. M. Soukoulis, Phys. Rev. B 53, 15577 (1996).
[CrossRef]

Kivshar, Y. S.

M. W. Feise, I. V. Shadrivov, and Y. S. Kivshar, Phys. Rev. E 71, 037602 (2005).
[CrossRef]

Kurt, H.

Kwong, D. L.

J. Zhang, M. B. Yu, G. Q. Lo, and D. L. Kwong, IEEE J. Sel. Top. Quantum Electron. 16, 53 (2010).
[CrossRef]

Lahini, Y.

Y. Lahini, Y. Bromberg, Y. Shechtman, A. Szameit, D. N. Christodoulides, R. Morandotti, and Y. Silberberg, Phys. Rev. A 84, 041806 (2011).
[CrossRef]

Lee, H. W.

M. A. Schmidt, L. Wondraczek, H. W. Lee, N. Granzow, N. Da, and P. St. J. Russell, Adv. Mater. 23, 2681 (2011).
[CrossRef]

Lepri, S.

S. Lepri and G. Casati, Phys. Rev. Lett. 106, 164101 (2011).
[CrossRef]

Li, Q. M.

Q. M. Li, C. T. Chan, K. M. Ho, and C. M. Soukoulis, Phys. Rev. B 53, 15577 (1996).
[CrossRef]

Li, Z.-Y.

C. Wang, X.-L. Zhong, and Z.-Y. Li, Sci. Rep. 2, 1 (2012).

C. Wang, C.-Z. Zhou, and Z.-Y. Li, Opt. Express 19, 26948 (2011).
[CrossRef]

Liang, B.

B. Liang, B. Yuan, and J.-C. Cheng, Phys. Rev. Lett. 103, 104301 (2009).
[CrossRef]

Lo, G. Q.

J. Zhang, M. B. Yu, G. Q. Lo, and D. L. Kwong, IEEE J. Sel. Top. Quantum Electron. 16, 53 (2010).
[CrossRef]

Morandotti, R.

Y. Lahini, Y. Bromberg, Y. Shechtman, A. Szameit, D. N. Christodoulides, R. Morandotti, and Y. Silberberg, Phys. Rev. A 84, 041806 (2011).
[CrossRef]

Niu, B.

L. Fan, J. Wang, L. T. Varghese, H. Shen, B. Niu, Y. Xuan, A. M. Weiner, and M. H. Qi, Science 335, 447 (2012).
[CrossRef]

Nolte, S.

A. Szameit, F. Dreisow, M. Heinrich, S. Nolte, and A. A. Sukhorukov, Phys. Rev. Lett. 106, 193903 (2011).
[CrossRef]

A. Szameit, F. Dreisow, T. Pertsch, S. Nolte, and A. Tünnermann, Opt. Express 15, 1579 (2007).
[CrossRef]

Ozbay, E.

Parameswaran, K. R.

K. Gallo, G. Assanto, K. R. Parameswaran, and M. M. Fejer, Appl. Phys. Lett. 79, 314 (2001).
[CrossRef]

Pertsch, T.

Peyrard, M.

M. Terraneo, M. Peyrard, and G. Casati, Phys. Rev. Lett. 88, 094302 (2002).
[CrossRef]

Phillips, P. W.

D. H. Dunlap, H.-L. Wu, and P. W. Phillips, Phys. Rev. Lett. 65, 88 (1990).
[CrossRef]

Qi, M. H.

L. Fan, J. Wang, L. T. Varghese, H. Shen, B. Niu, Y. Xuan, A. M. Weiner, and M. H. Qi, Science 335, 447 (2012).
[CrossRef]

Qi, X. Y.

Ren, Z. Y.

Russell, P. St. J.

M. A. Schmidt, L. Wondraczek, H. W. Lee, N. Granzow, N. Da, and P. St. J. Russell, Adv. Mater. 23, 2681 (2011).
[CrossRef]

Scalora, M.

M. D. Tocci, M. J. Bloemer, M. Scalora, J. P. Dowling, and C. M. Bowden, Appl. Phys. Lett. 66, 2324 (1995).
[CrossRef]

M. Scalora, J. P. Dowling, C. M. Bowden, and M. J. Bloemer, J. Appl. Phys. 76, 2023 (1994).
[CrossRef]

Schmidt, M. A.

M. A. Schmidt, L. Wondraczek, H. W. Lee, N. Granzow, N. Da, and P. St. J. Russell, Adv. Mater. 23, 2681 (2011).
[CrossRef]

Shadrivov, I. V.

M. W. Feise, I. V. Shadrivov, and Y. S. Kivshar, Phys. Rev. E 71, 037602 (2005).
[CrossRef]

Shechtman, Y.

Y. Lahini, Y. Bromberg, Y. Shechtman, A. Szameit, D. N. Christodoulides, R. Morandotti, and Y. Silberberg, Phys. Rev. A 84, 041806 (2011).
[CrossRef]

Shen, H.

L. Fan, J. Wang, L. T. Varghese, H. Shen, B. Niu, Y. Xuan, A. M. Weiner, and M. H. Qi, Science 335, 447 (2012).
[CrossRef]

Silberberg, Y.

Y. Lahini, Y. Bromberg, Y. Shechtman, A. Szameit, D. N. Christodoulides, R. Morandotti, and Y. Silberberg, Phys. Rev. A 84, 041806 (2011).
[CrossRef]

Soukoulis, C. M.

Q. M. Li, C. T. Chan, K. M. Ho, and C. M. Soukoulis, Phys. Rev. B 53, 15577 (1996).
[CrossRef]

Sukhorukov, A. A.

A. Szameit, F. Dreisow, M. Heinrich, S. Nolte, and A. A. Sukhorukov, Phys. Rev. Lett. 106, 193903 (2011).
[CrossRef]

A. A. Sukhorukov, Opt. Lett. 35, 989 (2010).
[CrossRef]

Szameit, A.

A. Szameit, F. Dreisow, M. Heinrich, S. Nolte, and A. A. Sukhorukov, Phys. Rev. Lett. 106, 193903 (2011).
[CrossRef]

Y. Lahini, Y. Bromberg, Y. Shechtman, A. Szameit, D. N. Christodoulides, R. Morandotti, and Y. Silberberg, Phys. Rev. A 84, 041806 (2011).
[CrossRef]

A. Szameit, F. Dreisow, T. Pertsch, S. Nolte, and A. Tünnermann, Opt. Express 15, 1579 (2007).
[CrossRef]

Terraneo, M.

M. Terraneo, M. Peyrard, and G. Casati, Phys. Rev. Lett. 88, 094302 (2002).
[CrossRef]

Tocci, M. D.

M. D. Tocci, M. J. Bloemer, M. Scalora, J. P. Dowling, and C. M. Bowden, Appl. Phys. Lett. 66, 2324 (1995).
[CrossRef]

Tsironis, G. P.

D. Hennig and G. P. Tsironis, Phys. Rep. 307, 333 (1999).
[CrossRef]

Tünnermann, A.

Varghese, L. T.

L. Fan, J. Wang, L. T. Varghese, H. Shen, B. Niu, Y. Xuan, A. M. Weiner, and M. H. Qi, Science 335, 447 (2012).
[CrossRef]

Wang, C.

C. Wang, X.-L. Zhong, and Z.-Y. Li, Sci. Rep. 2, 1 (2012).

C. Wang, C.-Z. Zhou, and Z.-Y. Li, Opt. Express 19, 26948 (2011).
[CrossRef]

Wang, J.

L. Fan, J. Wang, L. T. Varghese, H. Shen, B. Niu, Y. Xuan, A. M. Weiner, and M. H. Qi, Science 335, 447 (2012).
[CrossRef]

Weiner, A. M.

L. Fan, J. Wang, L. T. Varghese, H. Shen, B. Niu, Y. Xuan, A. M. Weiner, and M. H. Qi, Science 335, 447 (2012).
[CrossRef]

Wondraczek, L.

M. A. Schmidt, L. Wondraczek, H. W. Lee, N. Granzow, N. Da, and P. St. J. Russell, Adv. Mater. 23, 2681 (2011).
[CrossRef]

Wu, H.-L.

D. H. Dunlap, H.-L. Wu, and P. W. Phillips, Phys. Rev. Lett. 65, 88 (1990).
[CrossRef]

Xuan, Y.

L. Fan, J. Wang, L. T. Varghese, H. Shen, B. Niu, Y. Xuan, A. M. Weiner, and M. H. Qi, Science 335, 447 (2012).
[CrossRef]

Yilmaz, D.

Yu, M. B.

J. Zhang, M. B. Yu, G. Q. Lo, and D. L. Kwong, IEEE J. Sel. Top. Quantum Electron. 16, 53 (2010).
[CrossRef]

Yuan, B.

B. Liang, B. Yuan, and J.-C. Cheng, Phys. Rev. Lett. 103, 104301 (2009).
[CrossRef]

Zhang, G. Q.

Zhang, J.

J. Zhang, M. B. Yu, G. Q. Lo, and D. L. Kwong, IEEE J. Sel. Top. Quantum Electron. 16, 53 (2010).
[CrossRef]

Zhong, X.-L.

C. Wang, X.-L. Zhong, and Z.-Y. Li, Sci. Rep. 2, 1 (2012).

Zhou, C.-Z.

Adv. Mater. (1)

M. A. Schmidt, L. Wondraczek, H. W. Lee, N. Granzow, N. Da, and P. St. J. Russell, Adv. Mater. 23, 2681 (2011).
[CrossRef]

Appl. Phys. Lett. (2)

M. D. Tocci, M. J. Bloemer, M. Scalora, J. P. Dowling, and C. M. Bowden, Appl. Phys. Lett. 66, 2324 (1995).
[CrossRef]

K. Gallo, G. Assanto, K. R. Parameswaran, and M. M. Fejer, Appl. Phys. Lett. 79, 314 (2001).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

J. Zhang, M. B. Yu, G. Q. Lo, and D. L. Kwong, IEEE J. Sel. Top. Quantum Electron. 16, 53 (2010).
[CrossRef]

J. Appl. Phys. (2)

M. Scalora, J. P. Dowling, C. M. Bowden, and M. J. Bloemer, J. Appl. Phys. 76, 2023 (1994).
[CrossRef]

F. Biancalana, J. Appl. Phys. 104, 093113 (2008).
[CrossRef]

Opt. Express (4)

Opt. Lett. (1)

Phys. Rep. (1)

D. Hennig and G. P. Tsironis, Phys. Rep. 307, 333 (1999).
[CrossRef]

Phys. Rev. A (1)

Y. Lahini, Y. Bromberg, Y. Shechtman, A. Szameit, D. N. Christodoulides, R. Morandotti, and Y. Silberberg, Phys. Rev. A 84, 041806 (2011).
[CrossRef]

Phys. Rev. B (1)

Q. M. Li, C. T. Chan, K. M. Ho, and C. M. Soukoulis, Phys. Rev. B 53, 15577 (1996).
[CrossRef]

Phys. Rev. E (1)

M. W. Feise, I. V. Shadrivov, and Y. S. Kivshar, Phys. Rev. E 71, 037602 (2005).
[CrossRef]

Phys. Rev. Lett. (5)

D. H. Dunlap, H.-L. Wu, and P. W. Phillips, Phys. Rev. Lett. 65, 88 (1990).
[CrossRef]

A. Szameit, F. Dreisow, M. Heinrich, S. Nolte, and A. A. Sukhorukov, Phys. Rev. Lett. 106, 193903 (2011).
[CrossRef]

S. Lepri and G. Casati, Phys. Rev. Lett. 106, 164101 (2011).
[CrossRef]

B. Liang, B. Yuan, and J.-C. Cheng, Phys. Rev. Lett. 103, 104301 (2009).
[CrossRef]

M. Terraneo, M. Peyrard, and G. Casati, Phys. Rev. Lett. 88, 094302 (2002).
[CrossRef]

Sci. Rep. (1)

C. Wang, X.-L. Zhong, and Z.-Y. Li, Sci. Rep. 2, 1 (2012).

Science (1)

L. Fan, J. Wang, L. T. Varghese, H. Shen, B. Niu, Y. Xuan, A. M. Weiner, and M. H. Qi, Science 335, 447 (2012).
[CrossRef]

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Figures (3)

Fig. 1.
Fig. 1.

(a), (b) Schematic of the transverse separation modulated photonic lattices and (c) curves of the transmission contrast η versus the separation difference ΔΛ. (a) Refractive index profile of the waveguide arrays. (b) Sketch of the one-dimensional photonic lattices. I, II, III marked above (a) indicate the three lattice segments, respectively. Δn marked in (a) is the refractive index change of the lattice profile. θ1 and θ2 marked with red and black arrows in (b) refer to two half-Bragg incidence angles, respectively. The waveguide numbers of region II are from site 1 to N.

Fig. 2.
Fig. 2.

(a), (d) Calculated eigenmode profiles, and the asymmetric propagation of a broad Gaussian beam incident at half-Bragg angles (b) θ1=0.64°, (c), (f) θ2=0.45°, and (e) θ3=0.85°, respectively, through the linearly chirped lattice (region II). γ marked in (a) and (d) represents the eigenvalues normalized to CI=0.15mm1. The top and bottom rows correspond to ΔΛ=5 and 8 μm, respectively.

Fig. 3.
Fig. 3.

(a), (d) Calculated eigenmode profiles, and the asymmetric propagation of a broad Gaussian beam incident at half-Bragg angles (b) θ1=0.64°, (c), (f) θ2=0.45°, and (e) θ3=0.85°, respectively, through the Gaussian chirped lattice (region II). γ marked in (a) and (d) represents the eigenvalues normalized to CI=0.15mm1. The top and bottom rows correspond to ΔΛ=5 and 8 μm, respectively.

Equations (6)

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

iEnz+Cn,n1En1+Cn,n+1En+1+βnEn=0,
2·βn·φn+Cn,n1·φn1+Cn,n+1·φn+1=0.
φn={R0exp(iκn)+Rexp(iκn),n1,Texp(iκn),nN,
Cn,n1·un1+Cn,n+1·vn=0,
uNm={vN×(CN,N+1)CN1,N×(CNm+3,Nm+4)CNm+2,Nm+3×(CNm+1,Nm+2)CNm,Nm+1,ifmis odd,m=N,uN×(CN1,N)CN2,N1×(CNm+3,Nm+4)CNm+2,Nm+3×(CNm+1,Nm+2)CNm,Nm+1,ifmis even,m=(N1).
η=C6,7|2|C0,1|2|C6,7|2+|C0,1|2|=CIII|2|CI|2|CIII|2+|CI|2|,

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