Abstract

We present results of experimental and theoretical studies of polarization-resolved light transmission through optical fiber with disorder generated in its germanium-doped core via UV radiation transmitted through a diffuser. In samples longer than a certain characteristic length, the power transmitted with preserved polarization is observed to be distributed over all forward-propagating modes, as evidenced by the Rayleigh negative exponential distribution of the near-field intensity at the output surface of the fiber. Furthermore, the transmitted power becomes also equally distributed over both polarizations. To describe the optical properties of the fibers with the experimentally induced disorder, a theoretical model based on coupled-mode theory is developed. The obtained analytical expression for the correlation function describing spatial properties of the disorder shows that it is highly anisotropic. Our calculations demonstrate that this experimentally controllable anisotropy can lead to suppression of the radiative leakage of the propagating modes, so that intermode coupling becomes the dominant scattering process. The obtained theoretical expressions for the polarization-resolved transmission fit very well with the experimental data, and the information extracted from the fit shows that radiative leakage is indeed small. The reported technique provides an easy way to fabricate different configurations of controlled disorder in optical fibers suitable for such applications as random fiber lasers.

© 2011 Optical Society of America

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  1. H. Cao, “Review on latest developments in random lasers with coherent feedback,” J. Phys. A 38, 10497–10535 (2005).
    [CrossRef]
  2. S. H. Simon, A. L. Moustakas, M. Stoytchev, and H. Safar, “Communication in a disordered world,” Phys. Today 54(9), 38–43 (2001).
    [CrossRef]
  3. S. E. Skipetrov, “Disorder is the new order,” Nature 432, 285–286 (2004).
    [CrossRef] [PubMed]
  4. A. Lagendijk, B. van Tiggelen, and D. Wiersma, “Fifty years of Anderson localization,” Phys. Today 62(8), 24–29(2009).
    [CrossRef]
  5. J. A. Sánchez-Gil, V. D. Freilikher, A. A. Maradudin, and I. Yurkevich, “Reflection and transmission of waves in surface disordered waveguides,” Phys. Rev. B 59, 5915–5925(1999).
    [CrossRef]
  6. E. I. Chaikina, S. Stepanov, A. G. Navarrete, E. R. Méndez, and T. A. Leskova, “Formation of angular power profile via ballistic light transport in multi-mode optical fiber with corrugated surface,” Phys. Rev. B 71, 085419 (2005).
    [CrossRef]
  7. F. Bass, V. Freilikher, and I. Fuks, “Propagation in statistically irregular waveguides—part I: average field,” IEEE Trans. Antennas Propagat. 22, 278–288 (1974).
    [CrossRef]
  8. A. A. Chabanov, M. Stoytchev, and A. Z. Genack, “Statistical signatures of photon localization,” Nature 404, 850–853(2000).
    [CrossRef] [PubMed]
  9. J. Topolancik, F. Vollmer, and B. Ilic, “Random high-Q cavities in disordered photonic crystal waveguides,” Appl. Phys. Lett. 91, 201102 (2007).
    [CrossRef]
  10. O. Shapira and B. Fischer, “Localization of light in a random-grating array in a single-mode fiber,” J. Opt. Soc. Am. B 22, 2542–2552 (2005).
    [CrossRef]
  11. C. Lu, J. Cui, and Y. Cui, “Reflection spectra of fiber Bragg gratings with random fluctuations,” Opt. Fiber Technol. 14, 97–101 (2008).
    [CrossRef]
  12. C. J. S. Matos, L. de S. Menezes, A. M. Brito-Silva, M. A. Martinez-Gámez, A. S. L. Gomes, and C. B. de Araújo, “Random laser action in the core of a photonic crystal fiber,” Opt. Photon. News 19(12), 27–27 (2008).
    [CrossRef]
  13. M. Gagné and R. Kashyap, “Demonstration of a 3 mW threshold Er-doped random fiber laser based on a unique fiber Bragg grating,” Opt. Express 17, 19067–19074 (2009).
    [CrossRef]
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    [CrossRef]
  15. N. Lizárraga, N. P. Puente, E. I. Chaikina, T. A. Leskova, and E. R. Méndez, “Single-mode Er-doped fiber random laser with distributed Bragg grating feedback,” Opt. Express 17, 395–404 (2009).
    [CrossRef] [PubMed]
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  17. J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications (Roberts, 2007).
  18. P. W. Anderson, “Absence of diffusion in certain random lattices,” Phys. Rev. 109, 1492–1505 (1958).
    [CrossRef]
  19. D. Marcuse, “Rayleigh scattering and the impulse response of optical fiber,” Bell Syst. Tech. J. 53, 705–715 (1974).
  20. B. Crosignani, A. Saar, and A. Yariv, “Coherent backscattering and localization in a single-mode fiber with random imperfections,” Phys. Rev. A 43, 3168–3171 (1991).
    [CrossRef] [PubMed]
  21. H. C. van de Hulst, Light Scattering by Small Particles(Dover, 1981).
  22. D. Marcuse, “Coupled power equations for lossy fibers,” Appl. Opt. 17, 3232–3237 (1978).
    [CrossRef] [PubMed]
  23. J. C. Dainty, “Recent developments,” in Laser Speckle and Related Phenomena, J.C.Dainty, ed. (Springer, 1984).
  24. Luc B. Jeunhomme, Single-Mode Fiber Optics (Marcel Dekker, 1990).

2010 (1)

S. K. Turitsyn, S. A. Babin, A. E. El-Taher, P. Harper, D. V. Churkin, S. I. Kablukov, J. D. Ania-Castñón, V. Karalekas, and E. V. Podivilov, “Random distributed feedback fibre laser,” Nature Photon. 4, 231–235 (2010).
[CrossRef]

2009 (3)

2008 (2)

C. Lu, J. Cui, and Y. Cui, “Reflection spectra of fiber Bragg gratings with random fluctuations,” Opt. Fiber Technol. 14, 97–101 (2008).
[CrossRef]

C. J. S. Matos, L. de S. Menezes, A. M. Brito-Silva, M. A. Martinez-Gámez, A. S. L. Gomes, and C. B. de Araújo, “Random laser action in the core of a photonic crystal fiber,” Opt. Photon. News 19(12), 27–27 (2008).
[CrossRef]

2007 (1)

J. Topolancik, F. Vollmer, and B. Ilic, “Random high-Q cavities in disordered photonic crystal waveguides,” Appl. Phys. Lett. 91, 201102 (2007).
[CrossRef]

2005 (3)

H. Cao, “Review on latest developments in random lasers with coherent feedback,” J. Phys. A 38, 10497–10535 (2005).
[CrossRef]

E. I. Chaikina, S. Stepanov, A. G. Navarrete, E. R. Méndez, and T. A. Leskova, “Formation of angular power profile via ballistic light transport in multi-mode optical fiber with corrugated surface,” Phys. Rev. B 71, 085419 (2005).
[CrossRef]

O. Shapira and B. Fischer, “Localization of light in a random-grating array in a single-mode fiber,” J. Opt. Soc. Am. B 22, 2542–2552 (2005).
[CrossRef]

2004 (1)

S. E. Skipetrov, “Disorder is the new order,” Nature 432, 285–286 (2004).
[CrossRef] [PubMed]

2001 (1)

S. H. Simon, A. L. Moustakas, M. Stoytchev, and H. Safar, “Communication in a disordered world,” Phys. Today 54(9), 38–43 (2001).
[CrossRef]

2000 (1)

A. A. Chabanov, M. Stoytchev, and A. Z. Genack, “Statistical signatures of photon localization,” Nature 404, 850–853(2000).
[CrossRef] [PubMed]

1999 (1)

J. A. Sánchez-Gil, V. D. Freilikher, A. A. Maradudin, and I. Yurkevich, “Reflection and transmission of waves in surface disordered waveguides,” Phys. Rev. B 59, 5915–5925(1999).
[CrossRef]

1991 (1)

B. Crosignani, A. Saar, and A. Yariv, “Coherent backscattering and localization in a single-mode fiber with random imperfections,” Phys. Rev. A 43, 3168–3171 (1991).
[CrossRef] [PubMed]

1978 (1)

1974 (2)

D. Marcuse, “Rayleigh scattering and the impulse response of optical fiber,” Bell Syst. Tech. J. 53, 705–715 (1974).

F. Bass, V. Freilikher, and I. Fuks, “Propagation in statistically irregular waveguides—part I: average field,” IEEE Trans. Antennas Propagat. 22, 278–288 (1974).
[CrossRef]

1958 (1)

P. W. Anderson, “Absence of diffusion in certain random lattices,” Phys. Rev. 109, 1492–1505 (1958).
[CrossRef]

Anderson, P. W.

P. W. Anderson, “Absence of diffusion in certain random lattices,” Phys. Rev. 109, 1492–1505 (1958).
[CrossRef]

Ania-Castñón, J. D.

S. K. Turitsyn, S. A. Babin, A. E. El-Taher, P. Harper, D. V. Churkin, S. I. Kablukov, J. D. Ania-Castñón, V. Karalekas, and E. V. Podivilov, “Random distributed feedback fibre laser,” Nature Photon. 4, 231–235 (2010).
[CrossRef]

Babin, S. A.

S. K. Turitsyn, S. A. Babin, A. E. El-Taher, P. Harper, D. V. Churkin, S. I. Kablukov, J. D. Ania-Castñón, V. Karalekas, and E. V. Podivilov, “Random distributed feedback fibre laser,” Nature Photon. 4, 231–235 (2010).
[CrossRef]

Bass, F.

F. Bass, V. Freilikher, and I. Fuks, “Propagation in statistically irregular waveguides—part I: average field,” IEEE Trans. Antennas Propagat. 22, 278–288 (1974).
[CrossRef]

Brito-Silva, A. M.

C. J. S. Matos, L. de S. Menezes, A. M. Brito-Silva, M. A. Martinez-Gámez, A. S. L. Gomes, and C. B. de Araújo, “Random laser action in the core of a photonic crystal fiber,” Opt. Photon. News 19(12), 27–27 (2008).
[CrossRef]

Cao, H.

H. Cao, “Review on latest developments in random lasers with coherent feedback,” J. Phys. A 38, 10497–10535 (2005).
[CrossRef]

Chabanov, A. A.

A. A. Chabanov, M. Stoytchev, and A. Z. Genack, “Statistical signatures of photon localization,” Nature 404, 850–853(2000).
[CrossRef] [PubMed]

Chaikina, E. I.

N. Lizárraga, N. P. Puente, E. I. Chaikina, T. A. Leskova, and E. R. Méndez, “Single-mode Er-doped fiber random laser with distributed Bragg grating feedback,” Opt. Express 17, 395–404 (2009).
[CrossRef] [PubMed]

E. I. Chaikina, S. Stepanov, A. G. Navarrete, E. R. Méndez, and T. A. Leskova, “Formation of angular power profile via ballistic light transport in multi-mode optical fiber with corrugated surface,” Phys. Rev. B 71, 085419 (2005).
[CrossRef]

Churkin, D. V.

S. K. Turitsyn, S. A. Babin, A. E. El-Taher, P. Harper, D. V. Churkin, S. I. Kablukov, J. D. Ania-Castñón, V. Karalekas, and E. V. Podivilov, “Random distributed feedback fibre laser,” Nature Photon. 4, 231–235 (2010).
[CrossRef]

Crosignani, B.

B. Crosignani, A. Saar, and A. Yariv, “Coherent backscattering and localization in a single-mode fiber with random imperfections,” Phys. Rev. A 43, 3168–3171 (1991).
[CrossRef] [PubMed]

Cui, J.

C. Lu, J. Cui, and Y. Cui, “Reflection spectra of fiber Bragg gratings with random fluctuations,” Opt. Fiber Technol. 14, 97–101 (2008).
[CrossRef]

Cui, Y.

C. Lu, J. Cui, and Y. Cui, “Reflection spectra of fiber Bragg gratings with random fluctuations,” Opt. Fiber Technol. 14, 97–101 (2008).
[CrossRef]

Dainty, J. C.

J. C. Dainty, “Recent developments,” in Laser Speckle and Related Phenomena, J.C.Dainty, ed. (Springer, 1984).

de Araújo, C. B.

C. J. S. Matos, L. de S. Menezes, A. M. Brito-Silva, M. A. Martinez-Gámez, A. S. L. Gomes, and C. B. de Araújo, “Random laser action in the core of a photonic crystal fiber,” Opt. Photon. News 19(12), 27–27 (2008).
[CrossRef]

de S. Menezes, L.

C. J. S. Matos, L. de S. Menezes, A. M. Brito-Silva, M. A. Martinez-Gámez, A. S. L. Gomes, and C. B. de Araújo, “Random laser action in the core of a photonic crystal fiber,” Opt. Photon. News 19(12), 27–27 (2008).
[CrossRef]

El-Taher, A. E.

S. K. Turitsyn, S. A. Babin, A. E. El-Taher, P. Harper, D. V. Churkin, S. I. Kablukov, J. D. Ania-Castñón, V. Karalekas, and E. V. Podivilov, “Random distributed feedback fibre laser,” Nature Photon. 4, 231–235 (2010).
[CrossRef]

Fischer, B.

Freilikher, V.

F. Bass, V. Freilikher, and I. Fuks, “Propagation in statistically irregular waveguides—part I: average field,” IEEE Trans. Antennas Propagat. 22, 278–288 (1974).
[CrossRef]

Freilikher, V. D.

J. A. Sánchez-Gil, V. D. Freilikher, A. A. Maradudin, and I. Yurkevich, “Reflection and transmission of waves in surface disordered waveguides,” Phys. Rev. B 59, 5915–5925(1999).
[CrossRef]

Fuks, I.

F. Bass, V. Freilikher, and I. Fuks, “Propagation in statistically irregular waveguides—part I: average field,” IEEE Trans. Antennas Propagat. 22, 278–288 (1974).
[CrossRef]

Gagné, M.

Genack, A. Z.

A. A. Chabanov, M. Stoytchev, and A. Z. Genack, “Statistical signatures of photon localization,” Nature 404, 850–853(2000).
[CrossRef] [PubMed]

Gomes, A. S. L.

C. J. S. Matos, L. de S. Menezes, A. M. Brito-Silva, M. A. Martinez-Gámez, A. S. L. Gomes, and C. B. de Araújo, “Random laser action in the core of a photonic crystal fiber,” Opt. Photon. News 19(12), 27–27 (2008).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications (Roberts, 2007).

Harper, P.

S. K. Turitsyn, S. A. Babin, A. E. El-Taher, P. Harper, D. V. Churkin, S. I. Kablukov, J. D. Ania-Castñón, V. Karalekas, and E. V. Podivilov, “Random distributed feedback fibre laser,” Nature Photon. 4, 231–235 (2010).
[CrossRef]

Ilic, B.

J. Topolancik, F. Vollmer, and B. Ilic, “Random high-Q cavities in disordered photonic crystal waveguides,” Appl. Phys. Lett. 91, 201102 (2007).
[CrossRef]

Jeunhomme, Luc B.

Luc B. Jeunhomme, Single-Mode Fiber Optics (Marcel Dekker, 1990).

Kablukov, S. I.

S. K. Turitsyn, S. A. Babin, A. E. El-Taher, P. Harper, D. V. Churkin, S. I. Kablukov, J. D. Ania-Castñón, V. Karalekas, and E. V. Podivilov, “Random distributed feedback fibre laser,” Nature Photon. 4, 231–235 (2010).
[CrossRef]

Karalekas, V.

S. K. Turitsyn, S. A. Babin, A. E. El-Taher, P. Harper, D. V. Churkin, S. I. Kablukov, J. D. Ania-Castñón, V. Karalekas, and E. V. Podivilov, “Random distributed feedback fibre laser,” Nature Photon. 4, 231–235 (2010).
[CrossRef]

Kashyap, R.

Lagendijk, A.

A. Lagendijk, B. van Tiggelen, and D. Wiersma, “Fifty years of Anderson localization,” Phys. Today 62(8), 24–29(2009).
[CrossRef]

Leskova, T. A.

N. Lizárraga, N. P. Puente, E. I. Chaikina, T. A. Leskova, and E. R. Méndez, “Single-mode Er-doped fiber random laser with distributed Bragg grating feedback,” Opt. Express 17, 395–404 (2009).
[CrossRef] [PubMed]

E. I. Chaikina, S. Stepanov, A. G. Navarrete, E. R. Méndez, and T. A. Leskova, “Formation of angular power profile via ballistic light transport in multi-mode optical fiber with corrugated surface,” Phys. Rev. B 71, 085419 (2005).
[CrossRef]

Lizárraga, N.

Lu, C.

C. Lu, J. Cui, and Y. Cui, “Reflection spectra of fiber Bragg gratings with random fluctuations,” Opt. Fiber Technol. 14, 97–101 (2008).
[CrossRef]

Maradudin, A. A.

J. A. Sánchez-Gil, V. D. Freilikher, A. A. Maradudin, and I. Yurkevich, “Reflection and transmission of waves in surface disordered waveguides,” Phys. Rev. B 59, 5915–5925(1999).
[CrossRef]

Marcuse, D.

D. Marcuse, “Coupled power equations for lossy fibers,” Appl. Opt. 17, 3232–3237 (1978).
[CrossRef] [PubMed]

D. Marcuse, “Rayleigh scattering and the impulse response of optical fiber,” Bell Syst. Tech. J. 53, 705–715 (1974).

D. Marcuse, Theory of Dielectric Optical Waveguides(Academic, 1974).

Martinez-Gámez, M. A.

C. J. S. Matos, L. de S. Menezes, A. M. Brito-Silva, M. A. Martinez-Gámez, A. S. L. Gomes, and C. B. de Araújo, “Random laser action in the core of a photonic crystal fiber,” Opt. Photon. News 19(12), 27–27 (2008).
[CrossRef]

Matos, C. J. S.

C. J. S. Matos, L. de S. Menezes, A. M. Brito-Silva, M. A. Martinez-Gámez, A. S. L. Gomes, and C. B. de Araújo, “Random laser action in the core of a photonic crystal fiber,” Opt. Photon. News 19(12), 27–27 (2008).
[CrossRef]

Méndez, E. R.

N. Lizárraga, N. P. Puente, E. I. Chaikina, T. A. Leskova, and E. R. Méndez, “Single-mode Er-doped fiber random laser with distributed Bragg grating feedback,” Opt. Express 17, 395–404 (2009).
[CrossRef] [PubMed]

E. I. Chaikina, S. Stepanov, A. G. Navarrete, E. R. Méndez, and T. A. Leskova, “Formation of angular power profile via ballistic light transport in multi-mode optical fiber with corrugated surface,” Phys. Rev. B 71, 085419 (2005).
[CrossRef]

Moustakas, A. L.

S. H. Simon, A. L. Moustakas, M. Stoytchev, and H. Safar, “Communication in a disordered world,” Phys. Today 54(9), 38–43 (2001).
[CrossRef]

Navarrete, A. G.

E. I. Chaikina, S. Stepanov, A. G. Navarrete, E. R. Méndez, and T. A. Leskova, “Formation of angular power profile via ballistic light transport in multi-mode optical fiber with corrugated surface,” Phys. Rev. B 71, 085419 (2005).
[CrossRef]

Podivilov, E. V.

S. K. Turitsyn, S. A. Babin, A. E. El-Taher, P. Harper, D. V. Churkin, S. I. Kablukov, J. D. Ania-Castñón, V. Karalekas, and E. V. Podivilov, “Random distributed feedback fibre laser,” Nature Photon. 4, 231–235 (2010).
[CrossRef]

Puente, N. P.

Saar, A.

B. Crosignani, A. Saar, and A. Yariv, “Coherent backscattering and localization in a single-mode fiber with random imperfections,” Phys. Rev. A 43, 3168–3171 (1991).
[CrossRef] [PubMed]

Safar, H.

S. H. Simon, A. L. Moustakas, M. Stoytchev, and H. Safar, “Communication in a disordered world,” Phys. Today 54(9), 38–43 (2001).
[CrossRef]

Sánchez-Gil, J. A.

J. A. Sánchez-Gil, V. D. Freilikher, A. A. Maradudin, and I. Yurkevich, “Reflection and transmission of waves in surface disordered waveguides,” Phys. Rev. B 59, 5915–5925(1999).
[CrossRef]

Shapira, O.

Simon, S. H.

S. H. Simon, A. L. Moustakas, M. Stoytchev, and H. Safar, “Communication in a disordered world,” Phys. Today 54(9), 38–43 (2001).
[CrossRef]

Skipetrov, S. E.

S. E. Skipetrov, “Disorder is the new order,” Nature 432, 285–286 (2004).
[CrossRef] [PubMed]

Stepanov, S.

E. I. Chaikina, S. Stepanov, A. G. Navarrete, E. R. Méndez, and T. A. Leskova, “Formation of angular power profile via ballistic light transport in multi-mode optical fiber with corrugated surface,” Phys. Rev. B 71, 085419 (2005).
[CrossRef]

Stoytchev, M.

S. H. Simon, A. L. Moustakas, M. Stoytchev, and H. Safar, “Communication in a disordered world,” Phys. Today 54(9), 38–43 (2001).
[CrossRef]

A. A. Chabanov, M. Stoytchev, and A. Z. Genack, “Statistical signatures of photon localization,” Nature 404, 850–853(2000).
[CrossRef] [PubMed]

Topolancik, J.

J. Topolancik, F. Vollmer, and B. Ilic, “Random high-Q cavities in disordered photonic crystal waveguides,” Appl. Phys. Lett. 91, 201102 (2007).
[CrossRef]

Turitsyn, S. K.

S. K. Turitsyn, S. A. Babin, A. E. El-Taher, P. Harper, D. V. Churkin, S. I. Kablukov, J. D. Ania-Castñón, V. Karalekas, and E. V. Podivilov, “Random distributed feedback fibre laser,” Nature Photon. 4, 231–235 (2010).
[CrossRef]

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles(Dover, 1981).

van Tiggelen, B.

A. Lagendijk, B. van Tiggelen, and D. Wiersma, “Fifty years of Anderson localization,” Phys. Today 62(8), 24–29(2009).
[CrossRef]

Vollmer, F.

J. Topolancik, F. Vollmer, and B. Ilic, “Random high-Q cavities in disordered photonic crystal waveguides,” Appl. Phys. Lett. 91, 201102 (2007).
[CrossRef]

Wiersma, D.

A. Lagendijk, B. van Tiggelen, and D. Wiersma, “Fifty years of Anderson localization,” Phys. Today 62(8), 24–29(2009).
[CrossRef]

Yariv, A.

B. Crosignani, A. Saar, and A. Yariv, “Coherent backscattering and localization in a single-mode fiber with random imperfections,” Phys. Rev. A 43, 3168–3171 (1991).
[CrossRef] [PubMed]

Yurkevich, I.

J. A. Sánchez-Gil, V. D. Freilikher, A. A. Maradudin, and I. Yurkevich, “Reflection and transmission of waves in surface disordered waveguides,” Phys. Rev. B 59, 5915–5925(1999).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

J. Topolancik, F. Vollmer, and B. Ilic, “Random high-Q cavities in disordered photonic crystal waveguides,” Appl. Phys. Lett. 91, 201102 (2007).
[CrossRef]

Bell Syst. Tech. J. (1)

D. Marcuse, “Rayleigh scattering and the impulse response of optical fiber,” Bell Syst. Tech. J. 53, 705–715 (1974).

IEEE Trans. Antennas Propagat. (1)

F. Bass, V. Freilikher, and I. Fuks, “Propagation in statistically irregular waveguides—part I: average field,” IEEE Trans. Antennas Propagat. 22, 278–288 (1974).
[CrossRef]

J. Opt. Soc. Am. B (1)

J. Phys. A (1)

H. Cao, “Review on latest developments in random lasers with coherent feedback,” J. Phys. A 38, 10497–10535 (2005).
[CrossRef]

Nature (2)

S. E. Skipetrov, “Disorder is the new order,” Nature 432, 285–286 (2004).
[CrossRef] [PubMed]

A. A. Chabanov, M. Stoytchev, and A. Z. Genack, “Statistical signatures of photon localization,” Nature 404, 850–853(2000).
[CrossRef] [PubMed]

Nature Photon. (1)

S. K. Turitsyn, S. A. Babin, A. E. El-Taher, P. Harper, D. V. Churkin, S. I. Kablukov, J. D. Ania-Castñón, V. Karalekas, and E. V. Podivilov, “Random distributed feedback fibre laser,” Nature Photon. 4, 231–235 (2010).
[CrossRef]

Opt. Express (2)

Opt. Fiber Technol. (1)

C. Lu, J. Cui, and Y. Cui, “Reflection spectra of fiber Bragg gratings with random fluctuations,” Opt. Fiber Technol. 14, 97–101 (2008).
[CrossRef]

Opt. Photon. News (1)

C. J. S. Matos, L. de S. Menezes, A. M. Brito-Silva, M. A. Martinez-Gámez, A. S. L. Gomes, and C. B. de Araújo, “Random laser action in the core of a photonic crystal fiber,” Opt. Photon. News 19(12), 27–27 (2008).
[CrossRef]

Phys. Rev. (1)

P. W. Anderson, “Absence of diffusion in certain random lattices,” Phys. Rev. 109, 1492–1505 (1958).
[CrossRef]

Phys. Rev. A (1)

B. Crosignani, A. Saar, and A. Yariv, “Coherent backscattering and localization in a single-mode fiber with random imperfections,” Phys. Rev. A 43, 3168–3171 (1991).
[CrossRef] [PubMed]

Phys. Rev. B (2)

J. A. Sánchez-Gil, V. D. Freilikher, A. A. Maradudin, and I. Yurkevich, “Reflection and transmission of waves in surface disordered waveguides,” Phys. Rev. B 59, 5915–5925(1999).
[CrossRef]

E. I. Chaikina, S. Stepanov, A. G. Navarrete, E. R. Méndez, and T. A. Leskova, “Formation of angular power profile via ballistic light transport in multi-mode optical fiber with corrugated surface,” Phys. Rev. B 71, 085419 (2005).
[CrossRef]

Phys. Today (2)

A. Lagendijk, B. van Tiggelen, and D. Wiersma, “Fifty years of Anderson localization,” Phys. Today 62(8), 24–29(2009).
[CrossRef]

S. H. Simon, A. L. Moustakas, M. Stoytchev, and H. Safar, “Communication in a disordered world,” Phys. Today 54(9), 38–43 (2001).
[CrossRef]

Other (5)

H. C. van de Hulst, Light Scattering by Small Particles(Dover, 1981).

J. C. Dainty, “Recent developments,” in Laser Speckle and Related Phenomena, J.C.Dainty, ed. (Springer, 1984).

Luc B. Jeunhomme, Single-Mode Fiber Optics (Marcel Dekker, 1990).

D. Marcuse, Theory of Dielectric Optical Waveguides(Academic, 1974).

J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications (Roberts, 2007).

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Figures (5)

Fig. 1
Fig. 1

Experimental setup.

Fig. 2
Fig. 2

Examples of the output intensity distribution observed in some realizations with the disordered part of fiber (a)  1 cm and (b)  2 cm . The left column in each figure presents the pp-polarized distribution, and the right column presents the ps-polarized distribution. The angles of incidence are 0 ° , 2 ° , and 5 ° from the top to the bottom images.

Fig. 3
Fig. 3

Experimentally measured total co- (open symbols) and cross-polarized (solid symbols) transmission as a function of the length of the disordered part of the fiber for different polarizations of the transmitted beam. The circles correspond to an angle of incidence of 0 ° , the triangles to 2 ° , and the squares to 5 ° . Solid and dashed curves represent the theoretical fit with Eqs. (24, 25) with parameters α = 0.064 cm 1 , σ 2 = 0.1917 cm 1 .

Fig. 4
Fig. 4

The distributions, which correspond to an unconstrained random sum (shown as a dashed curve) and to a constrained random sum (shown with the solid curve) of all modes of the fiber, are compared to the experimentally observed distributions of the near-field intensity measured in co- (circles) and cross-polarized (squares) channels in a sample with L = 8 cm . The thin symbols correspond to an angle of incidence of 2 ° , and the bold symbols correspond to an angle of incidence of 5 ° .

Fig. 5
Fig. 5

Panel (a) plots the size of the speckle defined by Eqs. (6, 7, 8) with L x = 0.3 mm and L z = 2 mm as a function of the distance between the diffuser and the core of the photosensitive fiber. Panel (b) compares the values of the characteristic length ( x x ) 1 σ 3 after which all forward-propagating modes with one polarization become equally populated. It is found numerically from Eq. (16) without (solid curve) and with (circles) the delta function approximation to the order of magnitude estimate (squares) in Eq. (29). Panel (c) compares the amplitude of the radiative loss rate computed from Eq. (18) to the intermode coupling rate σ 3 . The plot shows that for the disorder patterns generated with D > 4 mm , the coupling becomes the dominant effect. This conclusion is borne out by the experimental results in Fig. 3.

Equations (29)

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δ ε ( r ) δ ε ( r ) = δ ε 2 | A ( r ) A * ( r ) | 2 | A ( r ) | 2 | A ( r ) | 2 δ ε 2 | μ ( r , r ) | 2 ,
μ ( r , r ) μ ( r r ) μ ( x x , 0 , z z ) μ ( 0 , y y , 0 ) .
I ( x ˜ , z ˜ ) exp [ x ˜ 2 / L x 2 z ˜ 2 / L z 2 ] ,
| μ ( x x , 0 , z z ) | 2 = exp [ ( x x S x ) 2 ] × exp [ ( z z S z ) 2 ] ,
| μ ( 0 , y y , 0 ) | 2 = 1 ( 1 + [ π L x 2 λ UV D 2 ( y y ) ] 2 ) 1 / 2 ( 1 + [ π L z 2 λ UV D 2 ( y y ) ] 2 ) 1 / 2 1 ( 1 + [ y y S y 2 ] 2 ) 1 / 2 .
S x = λ UV D 2 π n core L x 0.15 λ UV D L x ,
S y = 3 λ UV D 2 π n core L z 2 0.38 λ UV D 2 L z 2 ,
S z = λ UV D 2 π n core L z 0.15 λ UV D L z ,
δ ε ( r ) δ ε ( r ) δ ε 2 exp [ ( x x S x ) 2 ] 1 [ 1 + ( y y S y 2 ) 2 ] 1 / 2 exp [ ( z z S z ) 2 ] .
E ( r ) ν c ν ( z ) e i ( ω t β ν z ) ( E t , ν ( x , y ) + e ^ z E z , ν ( x , y ) ) .
β ν [ E t , ν ( x , y ) · E t , ν ( x , y ) ] d x d y = δ ν ν ,
d c ν ( z ) d z = ν K ν ν ( z ) c ν ( z ) e i ( β ν β ν ) z ,
K ν ν ( z ) = ω 2 2 c 2 δ ε ( r ) [ E t , ν ( x , y ) · E t , ν ( x , y ) + E z , ν ( x , y ) E z , ν ( x , y ) ] d x d y
d P ν d z = c ν * d c ν d z + c.c. ,
d P ν d z = ν h ν ν ( P ν P ν ) ,
h ν ν = δ ε 2 ω 4 π log ( 2 ) S x S y S z c 4 e S z 2 | β ν β ν | 2 / 4 × [ E t , ν ( x , y ) · E t , ν ( x , y ) + E z , ν ( x , y ) E z , ν ( x , y ) ] 2 d x d y .
α h ν ν 2 3 π k 0 2 n core 2 A ,
α I = d Ω Δ k ( e ^ scat · e ^ z ) 2 d u x d u y d u z δ ε ( r ) δ ε ( r + u ) exp [ i Δ k · u ] .
d P ν d z = α P ν + ν h ν ν ( P ν P ν ) .
P ν ( z ) = P ν ( lossless ) ( z ) × exp [ α z ] ,
P ν ( lossless ) ( z ) = A ν exp [ σ z ] ,
det [ h ν ν δ ν ν τ h ν τ + σ ] = 0 ,
P ν ( z ) = e α z × [ n c n A ν ( n ) e σ n z ] , with     c n = [ n A ν ( n ) P ν ( 0 ) ] .
P ( x ) ( z ) ν = 0 N / 2 1 P 2 ν + 1 e α z × 1 2 [ 1 + e σ 2 z ] ,
P ( y ) ( z ) ν = 1 N / 2 P 2 ν e α z × 1 2 [ 1 e σ 2 z ] .
P 2 ν + 1 ( z ) e α z × [ ( P 2 ν + 1 ( 0 ) 2 N ) e σ 3 z + 2 N ] , P 2 ν ( z ) 0 .
[ E t , ν ( x , y ) · E t , ν ( x , y ) + E z , ν ( x , y ) E z , ν ( x , y ) ] 2 d x d y ,
{ [ n core 2 ( ω 2 / c 2 ) π a 2 ] 1 x x , y y ( NA / 2 ) 4 [ n core 2 ( ω 2 / c 2 ) π a 2 ] 1 x y , y x .
( x x ) 1 σ 3 Δ n 2 n core π ω 2 S x S y S z c 2 a 2 = Δ n n core 4 λ UV 3 D 4 λ 2 a 2 L x L z 3 , ( x y ) 1 σ 2 mixing ( x x ) 1 ( NA 2 ) 4 ,

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