Abstract

We present a method based on incremental holographic multiplexing to create a refractive index ratchet distribution into a photorefractive crystal as an example for the generation principle of such complex multiperiodic lattices. The implemented technique follows a finite optical series expansion of the desired index modulation. To analyze the induced lattice, we determine the phase retardation of a probe beam at the back face of the crystal by digital holography analysis. Our result depicts a first example to optically explore the fascinating phenomena of ratchet resembling systems.

© 2012 Optical Society of America

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  1. N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, Phys. Rev. E 66, 046602 (2002).
    [CrossRef]
  2. J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, Nature 422, 147 (2003).
    [CrossRef]
  3. A. S. Desyatnikov, D. N. Neshev, Y. S. Kivshar, N. Sagemerten, D. Träger, J. Jägers, C. Denz, and Y. V. Kartashov, Opt. Lett. 30, 869 (2005).
    [CrossRef]
  4. F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, Phys. Rep. 463, 1 (2008).
    [CrossRef]
  5. M. Petrovic, D. Träger, A. Strinic, M. Belic, J. Schröder, and C. Denz, Phys. Rev. E 68, 055601(R) (2003).
    [CrossRef]
  6. T. Schwartz, G. Bartal, S. Fishman, and M. Segev, Nature 446, 52 (2007).
    [CrossRef]
  7. Z. Bouchal, Czech. J. Phys. 53, 537 (2003).
    [CrossRef]
  8. J. C. Gutiérrez-Vega and M. A. Bandres, J. Opt. Soc. Am. A 22, 289 (2005).
    [CrossRef]
  9. Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Eur. Phys. J. Special Topics 173, 87 (2009).
    [CrossRef]
  10. J. Becker, P. Rose, M. Boguslawski, and C. Denz, Opt. Express 19, 9848 (2011).
    [CrossRef]
  11. M. Boguslawski, P. Rose, and C. Denz, Phys. Rev. A 84, 013832 (2011).
    [CrossRef]
  12. S. López-Aguayo, Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Phys. Rev. Lett. 105, 013902 (2010).
    [CrossRef]
  13. C. López-Mariscal and K. Helmerson, Opt. Lett. 35, 1215 (2010).
    [CrossRef]
  14. S. Longhi, Phys. Rev. B 81, 075102 (2010).
    [CrossRef]
  15. F. Dreisow, M. Heinrich, R. Keil, A. Tünnermann, S. Nolte, S. Longhi, and A. Szameit, Phys. Rev. Lett. 105, 143902 (2010).
    [CrossRef]
  16. Q. Liang, Y. Yan, and J. Dong, Opt. Lett. 36, 2513 (2011).
    [CrossRef]
  17. C. Denz, G. Pauliat, G. Roosen, and T. Tschudi, Opt. Commun. 85, 171 (1991).
    [CrossRef]
  18. T. Salger, S. Kling, T. Hecking, C. Geckeler, L. Morales-Molina, and M. Weitz, Science 326, 1241 (2009).
    [CrossRef]
  19. P. Hänggi and F. Marchesoni, Rev. Mod. Phys. 81, 387 (2009).
    [CrossRef]
  20. Y. Taketomi, J. E. Ford, H. Sasaki, J. Ma, Y. Fainman, and S. H. Lee, Opt. Lett. 16, 1774 (1991).
    [CrossRef]
  21. C. Denz, K.-O. Müller, T. Heimann, and T. Tschudi, IEEE J. Sel. Top. Quantum Electron. 4, 832 (1998).
    [CrossRef]
  22. P. Rose, B. Terhalle, J. Imbrock, and C. Denz, J. Phys. D 41, 224004 (2008).
    [CrossRef]
  23. J.-L. Zhao, P. Zhang, J.-B. Zhou, D.-X. Yang, D.-S. Yang, and E.-P. Li, Chin. Phys. Lett. 20, 1748 (2003).
    [CrossRef]

2011

2010

C. López-Mariscal and K. Helmerson, Opt. Lett. 35, 1215 (2010).
[CrossRef]

S. López-Aguayo, Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Phys. Rev. Lett. 105, 013902 (2010).
[CrossRef]

S. Longhi, Phys. Rev. B 81, 075102 (2010).
[CrossRef]

F. Dreisow, M. Heinrich, R. Keil, A. Tünnermann, S. Nolte, S. Longhi, and A. Szameit, Phys. Rev. Lett. 105, 143902 (2010).
[CrossRef]

2009

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Eur. Phys. J. Special Topics 173, 87 (2009).
[CrossRef]

T. Salger, S. Kling, T. Hecking, C. Geckeler, L. Morales-Molina, and M. Weitz, Science 326, 1241 (2009).
[CrossRef]

P. Hänggi and F. Marchesoni, Rev. Mod. Phys. 81, 387 (2009).
[CrossRef]

2008

P. Rose, B. Terhalle, J. Imbrock, and C. Denz, J. Phys. D 41, 224004 (2008).
[CrossRef]

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, Phys. Rep. 463, 1 (2008).
[CrossRef]

2007

T. Schwartz, G. Bartal, S. Fishman, and M. Segev, Nature 446, 52 (2007).
[CrossRef]

2005

2003

J.-L. Zhao, P. Zhang, J.-B. Zhou, D.-X. Yang, D.-S. Yang, and E.-P. Li, Chin. Phys. Lett. 20, 1748 (2003).
[CrossRef]

J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, Nature 422, 147 (2003).
[CrossRef]

Z. Bouchal, Czech. J. Phys. 53, 537 (2003).
[CrossRef]

M. Petrovic, D. Träger, A. Strinic, M. Belic, J. Schröder, and C. Denz, Phys. Rev. E 68, 055601(R) (2003).
[CrossRef]

2002

N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, Phys. Rev. E 66, 046602 (2002).
[CrossRef]

1998

C. Denz, K.-O. Müller, T. Heimann, and T. Tschudi, IEEE J. Sel. Top. Quantum Electron. 4, 832 (1998).
[CrossRef]

1991

Assanto, G.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, Phys. Rep. 463, 1 (2008).
[CrossRef]

Bandres, M. A.

Bartal, G.

T. Schwartz, G. Bartal, S. Fishman, and M. Segev, Nature 446, 52 (2007).
[CrossRef]

Becker, J.

Belic, M.

M. Petrovic, D. Träger, A. Strinic, M. Belic, J. Schröder, and C. Denz, Phys. Rev. E 68, 055601(R) (2003).
[CrossRef]

Boguslawski, M.

J. Becker, P. Rose, M. Boguslawski, and C. Denz, Opt. Express 19, 9848 (2011).
[CrossRef]

M. Boguslawski, P. Rose, and C. Denz, Phys. Rev. A 84, 013832 (2011).
[CrossRef]

Bouchal, Z.

Z. Bouchal, Czech. J. Phys. 53, 537 (2003).
[CrossRef]

Christodoulides, D. N.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, Phys. Rep. 463, 1 (2008).
[CrossRef]

J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, Nature 422, 147 (2003).
[CrossRef]

N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, Phys. Rev. E 66, 046602 (2002).
[CrossRef]

Denz, C.

M. Boguslawski, P. Rose, and C. Denz, Phys. Rev. A 84, 013832 (2011).
[CrossRef]

J. Becker, P. Rose, M. Boguslawski, and C. Denz, Opt. Express 19, 9848 (2011).
[CrossRef]

P. Rose, B. Terhalle, J. Imbrock, and C. Denz, J. Phys. D 41, 224004 (2008).
[CrossRef]

A. S. Desyatnikov, D. N. Neshev, Y. S. Kivshar, N. Sagemerten, D. Träger, J. Jägers, C. Denz, and Y. V. Kartashov, Opt. Lett. 30, 869 (2005).
[CrossRef]

M. Petrovic, D. Träger, A. Strinic, M. Belic, J. Schröder, and C. Denz, Phys. Rev. E 68, 055601(R) (2003).
[CrossRef]

C. Denz, K.-O. Müller, T. Heimann, and T. Tschudi, IEEE J. Sel. Top. Quantum Electron. 4, 832 (1998).
[CrossRef]

C. Denz, G. Pauliat, G. Roosen, and T. Tschudi, Opt. Commun. 85, 171 (1991).
[CrossRef]

Desyatnikov, A. S.

Dong, J.

Dreisow, F.

F. Dreisow, M. Heinrich, R. Keil, A. Tünnermann, S. Nolte, S. Longhi, and A. Szameit, Phys. Rev. Lett. 105, 143902 (2010).
[CrossRef]

Efremidis, N. K.

J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, Nature 422, 147 (2003).
[CrossRef]

N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, Phys. Rev. E 66, 046602 (2002).
[CrossRef]

Fainman, Y.

Fishman, S.

T. Schwartz, G. Bartal, S. Fishman, and M. Segev, Nature 446, 52 (2007).
[CrossRef]

Fleischer, J. W.

J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, Nature 422, 147 (2003).
[CrossRef]

N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, Phys. Rev. E 66, 046602 (2002).
[CrossRef]

Ford, J. E.

Geckeler, C.

T. Salger, S. Kling, T. Hecking, C. Geckeler, L. Morales-Molina, and M. Weitz, Science 326, 1241 (2009).
[CrossRef]

Gutiérrez-Vega, J. C.

Hänggi, P.

P. Hänggi and F. Marchesoni, Rev. Mod. Phys. 81, 387 (2009).
[CrossRef]

Hecking, T.

T. Salger, S. Kling, T. Hecking, C. Geckeler, L. Morales-Molina, and M. Weitz, Science 326, 1241 (2009).
[CrossRef]

Heimann, T.

C. Denz, K.-O. Müller, T. Heimann, and T. Tschudi, IEEE J. Sel. Top. Quantum Electron. 4, 832 (1998).
[CrossRef]

Heinrich, M.

F. Dreisow, M. Heinrich, R. Keil, A. Tünnermann, S. Nolte, S. Longhi, and A. Szameit, Phys. Rev. Lett. 105, 143902 (2010).
[CrossRef]

Helmerson, K.

Imbrock, J.

P. Rose, B. Terhalle, J. Imbrock, and C. Denz, J. Phys. D 41, 224004 (2008).
[CrossRef]

Jägers, J.

Kartashov, Y. V.

S. López-Aguayo, Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Phys. Rev. Lett. 105, 013902 (2010).
[CrossRef]

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Eur. Phys. J. Special Topics 173, 87 (2009).
[CrossRef]

A. S. Desyatnikov, D. N. Neshev, Y. S. Kivshar, N. Sagemerten, D. Träger, J. Jägers, C. Denz, and Y. V. Kartashov, Opt. Lett. 30, 869 (2005).
[CrossRef]

Keil, R.

F. Dreisow, M. Heinrich, R. Keil, A. Tünnermann, S. Nolte, S. Longhi, and A. Szameit, Phys. Rev. Lett. 105, 143902 (2010).
[CrossRef]

Kivshar, Y. S.

Kling, S.

T. Salger, S. Kling, T. Hecking, C. Geckeler, L. Morales-Molina, and M. Weitz, Science 326, 1241 (2009).
[CrossRef]

Lederer, F.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, Phys. Rep. 463, 1 (2008).
[CrossRef]

Lee, S. H.

Li, E.-P.

J.-L. Zhao, P. Zhang, J.-B. Zhou, D.-X. Yang, D.-S. Yang, and E.-P. Li, Chin. Phys. Lett. 20, 1748 (2003).
[CrossRef]

Liang, Q.

Longhi, S.

S. Longhi, Phys. Rev. B 81, 075102 (2010).
[CrossRef]

F. Dreisow, M. Heinrich, R. Keil, A. Tünnermann, S. Nolte, S. Longhi, and A. Szameit, Phys. Rev. Lett. 105, 143902 (2010).
[CrossRef]

López-Aguayo, S.

S. López-Aguayo, Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Phys. Rev. Lett. 105, 013902 (2010).
[CrossRef]

López-Mariscal, C.

Ma, J.

Marchesoni, F.

P. Hänggi and F. Marchesoni, Rev. Mod. Phys. 81, 387 (2009).
[CrossRef]

Morales-Molina, L.

T. Salger, S. Kling, T. Hecking, C. Geckeler, L. Morales-Molina, and M. Weitz, Science 326, 1241 (2009).
[CrossRef]

Müller, K.-O.

C. Denz, K.-O. Müller, T. Heimann, and T. Tschudi, IEEE J. Sel. Top. Quantum Electron. 4, 832 (1998).
[CrossRef]

Neshev, D. N.

Nolte, S.

F. Dreisow, M. Heinrich, R. Keil, A. Tünnermann, S. Nolte, S. Longhi, and A. Szameit, Phys. Rev. Lett. 105, 143902 (2010).
[CrossRef]

Pauliat, G.

C. Denz, G. Pauliat, G. Roosen, and T. Tschudi, Opt. Commun. 85, 171 (1991).
[CrossRef]

Petrovic, M.

M. Petrovic, D. Träger, A. Strinic, M. Belic, J. Schröder, and C. Denz, Phys. Rev. E 68, 055601(R) (2003).
[CrossRef]

Roosen, G.

C. Denz, G. Pauliat, G. Roosen, and T. Tschudi, Opt. Commun. 85, 171 (1991).
[CrossRef]

Rose, P.

M. Boguslawski, P. Rose, and C. Denz, Phys. Rev. A 84, 013832 (2011).
[CrossRef]

J. Becker, P. Rose, M. Boguslawski, and C. Denz, Opt. Express 19, 9848 (2011).
[CrossRef]

P. Rose, B. Terhalle, J. Imbrock, and C. Denz, J. Phys. D 41, 224004 (2008).
[CrossRef]

Sagemerten, N.

Salger, T.

T. Salger, S. Kling, T. Hecking, C. Geckeler, L. Morales-Molina, and M. Weitz, Science 326, 1241 (2009).
[CrossRef]

Sasaki, H.

Schröder, J.

M. Petrovic, D. Träger, A. Strinic, M. Belic, J. Schröder, and C. Denz, Phys. Rev. E 68, 055601(R) (2003).
[CrossRef]

Schwartz, T.

T. Schwartz, G. Bartal, S. Fishman, and M. Segev, Nature 446, 52 (2007).
[CrossRef]

Sears, S.

N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, Phys. Rev. E 66, 046602 (2002).
[CrossRef]

Segev, M.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, Phys. Rep. 463, 1 (2008).
[CrossRef]

T. Schwartz, G. Bartal, S. Fishman, and M. Segev, Nature 446, 52 (2007).
[CrossRef]

J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, Nature 422, 147 (2003).
[CrossRef]

N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, Phys. Rev. E 66, 046602 (2002).
[CrossRef]

Silberberg, Y.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, Phys. Rep. 463, 1 (2008).
[CrossRef]

Stegeman, G. I.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, Phys. Rep. 463, 1 (2008).
[CrossRef]

Strinic, A.

M. Petrovic, D. Träger, A. Strinic, M. Belic, J. Schröder, and C. Denz, Phys. Rev. E 68, 055601(R) (2003).
[CrossRef]

Szameit, A.

F. Dreisow, M. Heinrich, R. Keil, A. Tünnermann, S. Nolte, S. Longhi, and A. Szameit, Phys. Rev. Lett. 105, 143902 (2010).
[CrossRef]

Taketomi, Y.

Terhalle, B.

P. Rose, B. Terhalle, J. Imbrock, and C. Denz, J. Phys. D 41, 224004 (2008).
[CrossRef]

Torner, L.

S. López-Aguayo, Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Phys. Rev. Lett. 105, 013902 (2010).
[CrossRef]

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Eur. Phys. J. Special Topics 173, 87 (2009).
[CrossRef]

Träger, D.

A. S. Desyatnikov, D. N. Neshev, Y. S. Kivshar, N. Sagemerten, D. Träger, J. Jägers, C. Denz, and Y. V. Kartashov, Opt. Lett. 30, 869 (2005).
[CrossRef]

M. Petrovic, D. Träger, A. Strinic, M. Belic, J. Schröder, and C. Denz, Phys. Rev. E 68, 055601(R) (2003).
[CrossRef]

Tschudi, T.

C. Denz, K.-O. Müller, T. Heimann, and T. Tschudi, IEEE J. Sel. Top. Quantum Electron. 4, 832 (1998).
[CrossRef]

C. Denz, G. Pauliat, G. Roosen, and T. Tschudi, Opt. Commun. 85, 171 (1991).
[CrossRef]

Tünnermann, A.

F. Dreisow, M. Heinrich, R. Keil, A. Tünnermann, S. Nolte, S. Longhi, and A. Szameit, Phys. Rev. Lett. 105, 143902 (2010).
[CrossRef]

Vysloukh, V. A.

S. López-Aguayo, Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Phys. Rev. Lett. 105, 013902 (2010).
[CrossRef]

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Eur. Phys. J. Special Topics 173, 87 (2009).
[CrossRef]

Weitz, M.

T. Salger, S. Kling, T. Hecking, C. Geckeler, L. Morales-Molina, and M. Weitz, Science 326, 1241 (2009).
[CrossRef]

Yan, Y.

Yang, D.-S.

J.-L. Zhao, P. Zhang, J.-B. Zhou, D.-X. Yang, D.-S. Yang, and E.-P. Li, Chin. Phys. Lett. 20, 1748 (2003).
[CrossRef]

Yang, D.-X.

J.-L. Zhao, P. Zhang, J.-B. Zhou, D.-X. Yang, D.-S. Yang, and E.-P. Li, Chin. Phys. Lett. 20, 1748 (2003).
[CrossRef]

Zhang, P.

J.-L. Zhao, P. Zhang, J.-B. Zhou, D.-X. Yang, D.-S. Yang, and E.-P. Li, Chin. Phys. Lett. 20, 1748 (2003).
[CrossRef]

Zhao, J.-L.

J.-L. Zhao, P. Zhang, J.-B. Zhou, D.-X. Yang, D.-S. Yang, and E.-P. Li, Chin. Phys. Lett. 20, 1748 (2003).
[CrossRef]

Zhou, J.-B.

J.-L. Zhao, P. Zhang, J.-B. Zhou, D.-X. Yang, D.-S. Yang, and E.-P. Li, Chin. Phys. Lett. 20, 1748 (2003).
[CrossRef]

Chin. Phys. Lett.

J.-L. Zhao, P. Zhang, J.-B. Zhou, D.-X. Yang, D.-S. Yang, and E.-P. Li, Chin. Phys. Lett. 20, 1748 (2003).
[CrossRef]

Czech. J. Phys.

Z. Bouchal, Czech. J. Phys. 53, 537 (2003).
[CrossRef]

Eur. Phys. J. Special Topics

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Eur. Phys. J. Special Topics 173, 87 (2009).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron.

C. Denz, K.-O. Müller, T. Heimann, and T. Tschudi, IEEE J. Sel. Top. Quantum Electron. 4, 832 (1998).
[CrossRef]

J. Opt. Soc. Am. A

J. Phys. D

P. Rose, B. Terhalle, J. Imbrock, and C. Denz, J. Phys. D 41, 224004 (2008).
[CrossRef]

Nature

T. Schwartz, G. Bartal, S. Fishman, and M. Segev, Nature 446, 52 (2007).
[CrossRef]

J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, Nature 422, 147 (2003).
[CrossRef]

Opt. Commun.

C. Denz, G. Pauliat, G. Roosen, and T. Tschudi, Opt. Commun. 85, 171 (1991).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rep.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, Phys. Rep. 463, 1 (2008).
[CrossRef]

Phys. Rev. A

M. Boguslawski, P. Rose, and C. Denz, Phys. Rev. A 84, 013832 (2011).
[CrossRef]

Phys. Rev. B

S. Longhi, Phys. Rev. B 81, 075102 (2010).
[CrossRef]

Phys. Rev. E

M. Petrovic, D. Träger, A. Strinic, M. Belic, J. Schröder, and C. Denz, Phys. Rev. E 68, 055601(R) (2003).
[CrossRef]

N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, Phys. Rev. E 66, 046602 (2002).
[CrossRef]

Phys. Rev. Lett.

S. López-Aguayo, Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Phys. Rev. Lett. 105, 013902 (2010).
[CrossRef]

F. Dreisow, M. Heinrich, R. Keil, A. Tünnermann, S. Nolte, S. Longhi, and A. Szameit, Phys. Rev. Lett. 105, 143902 (2010).
[CrossRef]

Rev. Mod. Phys.

P. Hänggi and F. Marchesoni, Rev. Mod. Phys. 81, 387 (2009).
[CrossRef]

Science

T. Salger, S. Kling, T. Hecking, C. Geckeler, L. Morales-Molina, and M. Weitz, Science 326, 1241 (2009).
[CrossRef]

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Figures (3)

Fig. 1.
Fig. 1.

(a) Illumination sequence consisting of 10 cosine lattice beams with harmonic intensity modulation periods. (b) Pseudo-3D surface plot of the simulated intensity summation of all involved lattice beams forming a 1D ratchet distribution.

Fig. 2.
Fig. 2.

Setup of the presented method of IMT. ASLM: amplitude spatial light modulator, BS: beam splitter, Cam: camera, L: lens, λ/2: phase retardation plate, M: mirror, MO: microscope objective, P: polarizer, PH: pinhole, PSLM: phase spatial light modulator, S: shutter, SBN: Cer-doped strontium barium niobate crystal.

Fig. 3.
Fig. 3.

Analysis of a ratchet lattice induced by IMT. (a) Intensity, (b) corrected phase, (c) refractive index change determined by the average of all lines in (b) and statistical error, (d) pseudo-3D surface plot of refractive index change.

Equations (2)

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

fr(x)=j=110(1)j1sin(2πjx/g1)j,
Δnw(t)=Δnsat[1exp(t/Tw)],Δne(t)=Δn0exp(t/Te).

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