Abstract

We investigate light propagation in randomly spaced fiber gratings in a single-mode fiber and demonstrate the localization effect. Localization of light in random media resembles that of electrons in disordered solids, resulting from a subtle wave-interference formation. We measured the light transmission after each additional grating fabrication and found an exponential decay that follows the localization theory. Important features of the random array are its similarity to ordered gratings in the transmission and its reflection behavior at the long-array regime. Besides the basic interest in localization in one-dimensional systems, random-grating arrays have potential applications, utilizing the possibility of the fabrication of long structures with strong and broadband reflections.

© 2005 Optical Society of America

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  1. P. W. Anderson, "Absence of diffusion in certain random lattices," Phys. Rev. 109, 1492-1505 (1958).
    [CrossRef]
  2. A. P. A. Lee and T. V. Ramakrishnan, "Disordered electronic systems," Rev. Mod. Phys. 57, 287-337 (1985).
    [CrossRef]
  3. J. B. Pendry, "Symmetry and transport of waves in one-dimensional disordered systems," Adv. Phys. 43, 461-542 (1994).
    [CrossRef]
  4. M. V. Berry and S. Klein, "Transparent mirrors: rays, waves, and localization," Eur. J. Phys. 18, 222-228 (1997).
    [CrossRef]
  5. N. Garcia and A. Z. Genack, "Anomalous photon diffusion at the threshold of the Anderson localization transition," Phys. Rev. Lett. 66, 1850-1853 (1991).
    [CrossRef] [PubMed]
  6. D. Z. Zhang, Z. L. Li, W. Hu, and B. Y. Cheng, "Broad-band optical reflector--an application of light localization in one-dimension," Appl. Phys. Lett. 67, 2431-2432 (1995).
    [CrossRef]
  7. P. Han and H. Z. Wang, "Extension of omni directional reflection range in one-dimensional photonic crystals with a staggered structure," J. Opt. Soc. Am. B 20, 1996-2001 (2003).
    [CrossRef]
  8. J. J. Xu, H. P. Fang, and Z. F. Lin, "Expanding high reflection range in a dielectric multilayer reflector by disorder and inhomogeneity," J. Phys. D 34, 445-449 (2001).
    [CrossRef]
  9. W. Hu, Z. L. Li, B. Y. Cheng, and D. Z. Zhang, "Localization of light for dissipative and disordered one-dimensional systems," Phys. Rev. B 54, 11873-11875 (1996).
    [CrossRef]
  10. U. Kuhl and H. J. Stockmann, "Microwave transmission spectra in regular and irregular one-dimensional scattering arrangements," Physica E 9, 384-388 (2001).
    [CrossRef]
  11. V. Baluni and J. Willemsen, "Transmission of acoustic waves in a random layered medium," Phys. Rev. A 31, 3358-3363 (1985).
    [CrossRef] [PubMed]
  12. A. Rosen, B. Fischer, A. Bekker, and S. Fishman, "Optical kicked system exhibiting localization in the spatial frequency domain," J. Opt. Soc. Am. B B17, 1579-1588 (2000).
    [CrossRef]
  13. B. Fischer, A. Rosen, A. Bekker, and S. Fishman, "Experimental observation of localization in the spatial frequency domain of an optical kicked system," Phys. Rev. E 61, R4694-R4697 (2000).
    [CrossRef]
  14. B. Fischer, A. Rosen, and S. Fishman, "Localization in frequency for periodically kicked light propagation in a dispersive single mode fibers," Opt. Lett. 24, 1463-1465 (1999).
    [CrossRef]
  15. B. Fischer, B. Vodonos, S. Atkins, and A. Bekker, "Demonstration of localization in the frequency domain of mode-locked lasers with dispersion," Opt. Lett. 27, 1061-1063 (2002).
    [CrossRef]
  16. S. Atkins, A. Rosen, A. Bekker, and B. Fischer, "Evolution of localization in frequency for modulated light pulses in a recirculating fiber loop," Opt. Lett. 28, 2228-2230 (2003).
    [CrossRef] [PubMed]
  17. B. Fischer and O. Shapira, "Light propagation and localization in a randomly spaced grating array in a single-mode fiber," in Conference on Lasers and Electro-Optics (Optical Society of America, 2001).
  18. H. Furstenberg, "Noncommuting random products," Trans. Am. Math. Soc. 108, 377-428 (1962).
    [CrossRef]
  19. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1997).
  20. A. Yariv, Optical Electronics in Modern Communications, 5th ed. (Oxford U. Press, 1995).
  21. H. Matsuda and K. Ishii, "Localization of normal modes and energy transport in the disordered harmonic chain," Suppl. Prog. Theor. Phys. 45, 87 (1970).
    [CrossRef]
  22. I. Bennion, J. A. R. Williams, L. Zhang, K. Sugden, and N. J. Doran, "UV written in fiber Bragg gratings," Opt. Quantum Electron. 28, 93-135 (1996).
    [CrossRef]

2003 (2)

2002 (1)

2001 (2)

J. J. Xu, H. P. Fang, and Z. F. Lin, "Expanding high reflection range in a dielectric multilayer reflector by disorder and inhomogeneity," J. Phys. D 34, 445-449 (2001).
[CrossRef]

U. Kuhl and H. J. Stockmann, "Microwave transmission spectra in regular and irregular one-dimensional scattering arrangements," Physica E 9, 384-388 (2001).
[CrossRef]

2000 (2)

A. Rosen, B. Fischer, A. Bekker, and S. Fishman, "Optical kicked system exhibiting localization in the spatial frequency domain," J. Opt. Soc. Am. B B17, 1579-1588 (2000).
[CrossRef]

B. Fischer, A. Rosen, A. Bekker, and S. Fishman, "Experimental observation of localization in the spatial frequency domain of an optical kicked system," Phys. Rev. E 61, R4694-R4697 (2000).
[CrossRef]

1999 (1)

1997 (1)

M. V. Berry and S. Klein, "Transparent mirrors: rays, waves, and localization," Eur. J. Phys. 18, 222-228 (1997).
[CrossRef]

1996 (2)

I. Bennion, J. A. R. Williams, L. Zhang, K. Sugden, and N. J. Doran, "UV written in fiber Bragg gratings," Opt. Quantum Electron. 28, 93-135 (1996).
[CrossRef]

W. Hu, Z. L. Li, B. Y. Cheng, and D. Z. Zhang, "Localization of light for dissipative and disordered one-dimensional systems," Phys. Rev. B 54, 11873-11875 (1996).
[CrossRef]

1995 (1)

D. Z. Zhang, Z. L. Li, W. Hu, and B. Y. Cheng, "Broad-band optical reflector--an application of light localization in one-dimension," Appl. Phys. Lett. 67, 2431-2432 (1995).
[CrossRef]

1994 (1)

J. B. Pendry, "Symmetry and transport of waves in one-dimensional disordered systems," Adv. Phys. 43, 461-542 (1994).
[CrossRef]

1991 (1)

N. Garcia and A. Z. Genack, "Anomalous photon diffusion at the threshold of the Anderson localization transition," Phys. Rev. Lett. 66, 1850-1853 (1991).
[CrossRef] [PubMed]

1985 (2)

A. P. A. Lee and T. V. Ramakrishnan, "Disordered electronic systems," Rev. Mod. Phys. 57, 287-337 (1985).
[CrossRef]

V. Baluni and J. Willemsen, "Transmission of acoustic waves in a random layered medium," Phys. Rev. A 31, 3358-3363 (1985).
[CrossRef] [PubMed]

1970 (1)

H. Matsuda and K. Ishii, "Localization of normal modes and energy transport in the disordered harmonic chain," Suppl. Prog. Theor. Phys. 45, 87 (1970).
[CrossRef]

1962 (1)

H. Furstenberg, "Noncommuting random products," Trans. Am. Math. Soc. 108, 377-428 (1962).
[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]

Atkins, S.

Baluni, V.

V. Baluni and J. Willemsen, "Transmission of acoustic waves in a random layered medium," Phys. Rev. A 31, 3358-3363 (1985).
[CrossRef] [PubMed]

Bekker, A.

S. Atkins, A. Rosen, A. Bekker, and B. Fischer, "Evolution of localization in frequency for modulated light pulses in a recirculating fiber loop," Opt. Lett. 28, 2228-2230 (2003).
[CrossRef] [PubMed]

B. Fischer, B. Vodonos, S. Atkins, and A. Bekker, "Demonstration of localization in the frequency domain of mode-locked lasers with dispersion," Opt. Lett. 27, 1061-1063 (2002).
[CrossRef]

B. Fischer, A. Rosen, A. Bekker, and S. Fishman, "Experimental observation of localization in the spatial frequency domain of an optical kicked system," Phys. Rev. E 61, R4694-R4697 (2000).
[CrossRef]

A. Rosen, B. Fischer, A. Bekker, and S. Fishman, "Optical kicked system exhibiting localization in the spatial frequency domain," J. Opt. Soc. Am. B B17, 1579-1588 (2000).
[CrossRef]

Bennion, I.

I. Bennion, J. A. R. Williams, L. Zhang, K. Sugden, and N. J. Doran, "UV written in fiber Bragg gratings," Opt. Quantum Electron. 28, 93-135 (1996).
[CrossRef]

Berry, M. V.

M. V. Berry and S. Klein, "Transparent mirrors: rays, waves, and localization," Eur. J. Phys. 18, 222-228 (1997).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1997).

Cheng, B. Y.

W. Hu, Z. L. Li, B. Y. Cheng, and D. Z. Zhang, "Localization of light for dissipative and disordered one-dimensional systems," Phys. Rev. B 54, 11873-11875 (1996).
[CrossRef]

D. Z. Zhang, Z. L. Li, W. Hu, and B. Y. Cheng, "Broad-band optical reflector--an application of light localization in one-dimension," Appl. Phys. Lett. 67, 2431-2432 (1995).
[CrossRef]

Doran, N. J.

I. Bennion, J. A. R. Williams, L. Zhang, K. Sugden, and N. J. Doran, "UV written in fiber Bragg gratings," Opt. Quantum Electron. 28, 93-135 (1996).
[CrossRef]

Fang, H. P.

J. J. Xu, H. P. Fang, and Z. F. Lin, "Expanding high reflection range in a dielectric multilayer reflector by disorder and inhomogeneity," J. Phys. D 34, 445-449 (2001).
[CrossRef]

Fischer, B.

S. Atkins, A. Rosen, A. Bekker, and B. Fischer, "Evolution of localization in frequency for modulated light pulses in a recirculating fiber loop," Opt. Lett. 28, 2228-2230 (2003).
[CrossRef] [PubMed]

B. Fischer, B. Vodonos, S. Atkins, and A. Bekker, "Demonstration of localization in the frequency domain of mode-locked lasers with dispersion," Opt. Lett. 27, 1061-1063 (2002).
[CrossRef]

B. Fischer, A. Rosen, A. Bekker, and S. Fishman, "Experimental observation of localization in the spatial frequency domain of an optical kicked system," Phys. Rev. E 61, R4694-R4697 (2000).
[CrossRef]

A. Rosen, B. Fischer, A. Bekker, and S. Fishman, "Optical kicked system exhibiting localization in the spatial frequency domain," J. Opt. Soc. Am. B B17, 1579-1588 (2000).
[CrossRef]

B. Fischer, A. Rosen, and S. Fishman, "Localization in frequency for periodically kicked light propagation in a dispersive single mode fibers," Opt. Lett. 24, 1463-1465 (1999).
[CrossRef]

B. Fischer and O. Shapira, "Light propagation and localization in a randomly spaced grating array in a single-mode fiber," in Conference on Lasers and Electro-Optics (Optical Society of America, 2001).

Fishman, S.

A. Rosen, B. Fischer, A. Bekker, and S. Fishman, "Optical kicked system exhibiting localization in the spatial frequency domain," J. Opt. Soc. Am. B B17, 1579-1588 (2000).
[CrossRef]

B. Fischer, A. Rosen, A. Bekker, and S. Fishman, "Experimental observation of localization in the spatial frequency domain of an optical kicked system," Phys. Rev. E 61, R4694-R4697 (2000).
[CrossRef]

B. Fischer, A. Rosen, and S. Fishman, "Localization in frequency for periodically kicked light propagation in a dispersive single mode fibers," Opt. Lett. 24, 1463-1465 (1999).
[CrossRef]

Furstenberg, H.

H. Furstenberg, "Noncommuting random products," Trans. Am. Math. Soc. 108, 377-428 (1962).
[CrossRef]

Garcia, N.

N. Garcia and A. Z. Genack, "Anomalous photon diffusion at the threshold of the Anderson localization transition," Phys. Rev. Lett. 66, 1850-1853 (1991).
[CrossRef] [PubMed]

Genack, A. Z.

N. Garcia and A. Z. Genack, "Anomalous photon diffusion at the threshold of the Anderson localization transition," Phys. Rev. Lett. 66, 1850-1853 (1991).
[CrossRef] [PubMed]

Han, P.

Hu, W.

W. Hu, Z. L. Li, B. Y. Cheng, and D. Z. Zhang, "Localization of light for dissipative and disordered one-dimensional systems," Phys. Rev. B 54, 11873-11875 (1996).
[CrossRef]

D. Z. Zhang, Z. L. Li, W. Hu, and B. Y. Cheng, "Broad-band optical reflector--an application of light localization in one-dimension," Appl. Phys. Lett. 67, 2431-2432 (1995).
[CrossRef]

Ishii, K.

H. Matsuda and K. Ishii, "Localization of normal modes and energy transport in the disordered harmonic chain," Suppl. Prog. Theor. Phys. 45, 87 (1970).
[CrossRef]

Klein, S.

M. V. Berry and S. Klein, "Transparent mirrors: rays, waves, and localization," Eur. J. Phys. 18, 222-228 (1997).
[CrossRef]

Kuhl, U.

U. Kuhl and H. J. Stockmann, "Microwave transmission spectra in regular and irregular one-dimensional scattering arrangements," Physica E 9, 384-388 (2001).
[CrossRef]

Lee, A. P. A.

A. P. A. Lee and T. V. Ramakrishnan, "Disordered electronic systems," Rev. Mod. Phys. 57, 287-337 (1985).
[CrossRef]

Li, Z. L.

W. Hu, Z. L. Li, B. Y. Cheng, and D. Z. Zhang, "Localization of light for dissipative and disordered one-dimensional systems," Phys. Rev. B 54, 11873-11875 (1996).
[CrossRef]

D. Z. Zhang, Z. L. Li, W. Hu, and B. Y. Cheng, "Broad-band optical reflector--an application of light localization in one-dimension," Appl. Phys. Lett. 67, 2431-2432 (1995).
[CrossRef]

Lin, Z. F.

J. J. Xu, H. P. Fang, and Z. F. Lin, "Expanding high reflection range in a dielectric multilayer reflector by disorder and inhomogeneity," J. Phys. D 34, 445-449 (2001).
[CrossRef]

Matsuda, H.

H. Matsuda and K. Ishii, "Localization of normal modes and energy transport in the disordered harmonic chain," Suppl. Prog. Theor. Phys. 45, 87 (1970).
[CrossRef]

Pendry, J. B.

J. B. Pendry, "Symmetry and transport of waves in one-dimensional disordered systems," Adv. Phys. 43, 461-542 (1994).
[CrossRef]

Ramakrishnan, T. V.

A. P. A. Lee and T. V. Ramakrishnan, "Disordered electronic systems," Rev. Mod. Phys. 57, 287-337 (1985).
[CrossRef]

Rosen, A.

S. Atkins, A. Rosen, A. Bekker, and B. Fischer, "Evolution of localization in frequency for modulated light pulses in a recirculating fiber loop," Opt. Lett. 28, 2228-2230 (2003).
[CrossRef] [PubMed]

B. Fischer, A. Rosen, A. Bekker, and S. Fishman, "Experimental observation of localization in the spatial frequency domain of an optical kicked system," Phys. Rev. E 61, R4694-R4697 (2000).
[CrossRef]

A. Rosen, B. Fischer, A. Bekker, and S. Fishman, "Optical kicked system exhibiting localization in the spatial frequency domain," J. Opt. Soc. Am. B B17, 1579-1588 (2000).
[CrossRef]

B. Fischer, A. Rosen, and S. Fishman, "Localization in frequency for periodically kicked light propagation in a dispersive single mode fibers," Opt. Lett. 24, 1463-1465 (1999).
[CrossRef]

Shapira, O.

B. Fischer and O. Shapira, "Light propagation and localization in a randomly spaced grating array in a single-mode fiber," in Conference on Lasers and Electro-Optics (Optical Society of America, 2001).

Stockmann, H. J.

U. Kuhl and H. J. Stockmann, "Microwave transmission spectra in regular and irregular one-dimensional scattering arrangements," Physica E 9, 384-388 (2001).
[CrossRef]

Sugden, K.

I. Bennion, J. A. R. Williams, L. Zhang, K. Sugden, and N. J. Doran, "UV written in fiber Bragg gratings," Opt. Quantum Electron. 28, 93-135 (1996).
[CrossRef]

Vodonos, B.

Wang, H. Z.

Willemsen, J.

V. Baluni and J. Willemsen, "Transmission of acoustic waves in a random layered medium," Phys. Rev. A 31, 3358-3363 (1985).
[CrossRef] [PubMed]

Williams, J. A. R.

I. Bennion, J. A. R. Williams, L. Zhang, K. Sugden, and N. J. Doran, "UV written in fiber Bragg gratings," Opt. Quantum Electron. 28, 93-135 (1996).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1997).

Xu, J. J.

J. J. Xu, H. P. Fang, and Z. F. Lin, "Expanding high reflection range in a dielectric multilayer reflector by disorder and inhomogeneity," J. Phys. D 34, 445-449 (2001).
[CrossRef]

Yariv, A.

A. Yariv, Optical Electronics in Modern Communications, 5th ed. (Oxford U. Press, 1995).

Zhang, D. Z.

W. Hu, Z. L. Li, B. Y. Cheng, and D. Z. Zhang, "Localization of light for dissipative and disordered one-dimensional systems," Phys. Rev. B 54, 11873-11875 (1996).
[CrossRef]

D. Z. Zhang, Z. L. Li, W. Hu, and B. Y. Cheng, "Broad-band optical reflector--an application of light localization in one-dimension," Appl. Phys. Lett. 67, 2431-2432 (1995).
[CrossRef]

Zhang, L.

I. Bennion, J. A. R. Williams, L. Zhang, K. Sugden, and N. J. Doran, "UV written in fiber Bragg gratings," Opt. Quantum Electron. 28, 93-135 (1996).
[CrossRef]

Adv. Phys. (1)

J. B. Pendry, "Symmetry and transport of waves in one-dimensional disordered systems," Adv. Phys. 43, 461-542 (1994).
[CrossRef]

Appl. Phys. Lett. (1)

D. Z. Zhang, Z. L. Li, W. Hu, and B. Y. Cheng, "Broad-band optical reflector--an application of light localization in one-dimension," Appl. Phys. Lett. 67, 2431-2432 (1995).
[CrossRef]

Eur. J. Phys. (1)

M. V. Berry and S. Klein, "Transparent mirrors: rays, waves, and localization," Eur. J. Phys. 18, 222-228 (1997).
[CrossRef]

J. Opt. Soc. Am. B (2)

A. Rosen, B. Fischer, A. Bekker, and S. Fishman, "Optical kicked system exhibiting localization in the spatial frequency domain," J. Opt. Soc. Am. B B17, 1579-1588 (2000).
[CrossRef]

P. Han and H. Z. Wang, "Extension of omni directional reflection range in one-dimensional photonic crystals with a staggered structure," J. Opt. Soc. Am. B 20, 1996-2001 (2003).
[CrossRef]

J. Phys. D (1)

J. J. Xu, H. P. Fang, and Z. F. Lin, "Expanding high reflection range in a dielectric multilayer reflector by disorder and inhomogeneity," J. Phys. D 34, 445-449 (2001).
[CrossRef]

Opt. Lett. (3)

Opt. Quantum Electron. (1)

I. Bennion, J. A. R. Williams, L. Zhang, K. Sugden, and N. J. Doran, "UV written in fiber Bragg gratings," Opt. Quantum Electron. 28, 93-135 (1996).
[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)

V. Baluni and J. Willemsen, "Transmission of acoustic waves in a random layered medium," Phys. Rev. A 31, 3358-3363 (1985).
[CrossRef] [PubMed]

Phys. Rev. B (1)

W. Hu, Z. L. Li, B. Y. Cheng, and D. Z. Zhang, "Localization of light for dissipative and disordered one-dimensional systems," Phys. Rev. B 54, 11873-11875 (1996).
[CrossRef]

Phys. Rev. E (1)

B. Fischer, A. Rosen, A. Bekker, and S. Fishman, "Experimental observation of localization in the spatial frequency domain of an optical kicked system," Phys. Rev. E 61, R4694-R4697 (2000).
[CrossRef]

Phys. Rev. Lett. (1)

N. Garcia and A. Z. Genack, "Anomalous photon diffusion at the threshold of the Anderson localization transition," Phys. Rev. Lett. 66, 1850-1853 (1991).
[CrossRef] [PubMed]

Physica E (1)

U. Kuhl and H. J. Stockmann, "Microwave transmission spectra in regular and irregular one-dimensional scattering arrangements," Physica E 9, 384-388 (2001).
[CrossRef]

Rev. Mod. Phys. (1)

A. P. A. Lee and T. V. Ramakrishnan, "Disordered electronic systems," Rev. Mod. Phys. 57, 287-337 (1985).
[CrossRef]

Suppl. Prog. Theor. Phys. (1)

H. Matsuda and K. Ishii, "Localization of normal modes and energy transport in the disordered harmonic chain," Suppl. Prog. Theor. Phys. 45, 87 (1970).
[CrossRef]

Trans. Am. Math. Soc. (1)

H. Furstenberg, "Noncommuting random products," Trans. Am. Math. Soc. 108, 377-428 (1962).
[CrossRef]

Other (3)

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1997).

A. Yariv, Optical Electronics in Modern Communications, 5th ed. (Oxford U. Press, 1995).

B. Fischer and O. Shapira, "Light propagation and localization in a randomly spaced grating array in a single-mode fiber," in Conference on Lasers and Electro-Optics (Optical Society of America, 2001).

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Figures (15)

Fig. 1
Fig. 1

Random fiber-grating array.

Fig. 2
Fig. 2

Incident and reflected field amplitudes that define the transfer matrix of a single grating.

Fig. 3
Fig. 3

(Color online) Transmission-spectrum simulation of 1000 randomly spaced gratings with a single-grating transmissivity of 0.022 dB at the band center. The continuous curve shows the asymptotic behavior of the transmission spectrum.

Fig. 4
Fig. 4

(Color online) Transmission-spectrum simulation of 5000 randomly spaced gratings with a single-grating transmissivity of 0.022 dB at the band center. The continuous curve shows the asymptotic behavior of the transmission spectrum.

Fig. 5
Fig. 5

(Color online) Transmissivity at the band center after each grating. Simulation and the asymptotic behavior (the straight line) given by the theory for a single-grating transmissivity of 0.022 dB .

Fig. 6
Fig. 6

Three waves with paths of the same length but different numbers of reflections, resulting in constructive interference between the two upper waves but destructive interference with the third.

Fig. 7
Fig. 7

(Color online) Comparison between wave and ray theories for the transmissivity in random gratings; the upper curve depicts the Ohmic behavior of the ray theory, whereas the lower curve depicts the exact wave averaging (both calculated for grating transmissivity of 0.022 dB ).

Fig. 8
Fig. 8

(Color online) Experimental setup. The UV laser beam, aligned to a pinhole, is broadened by a lens in order to achieve a larger spot size for the grating writing. The beam is then focused at the fiber axes by a cylindrical lens to obtain maximum intensity on the fiber. A phase-grating diffraction gives two first-order waves, causing a sinusoidal interfering pattern on the fiber. The light transmission measurements in the fiber were done in situ after each additional grating fabrication. This procedure allowed us to follow the localization buildup with the grating number.

Fig. 9
Fig. 9

(Color online) Experiment 1. Transmitted spectrum measured after (a) 3, (b) 10, (c) 25, and (d) 50 gratings.

Fig. 10
Fig. 10

(Color online) Experiment 2. Transmitted spectrum measured after (a) 3, (b) 10, (c) 25, and (d) 50 gratings.

Fig. 11
Fig. 11

(Color online) Experiment 1. Transmission measured at the grating center wavelength (at minimum transmission). The fitted straight-line slope is 0.405 dB /grating.

Fig. 12
Fig. 12

(Color online) Experiment 2. Transmission measured at the grating center wavelength (at minimum transmission). The fitted straight-line slope is 0.326 dB /grating.

Fig. 13
Fig. 13

(Color online) Experiment 1. The normalized power as measured after two gratings is similar to the spectrum of a Fabry–Perot resonator but with the envelope of the grating spectrum. The transmissivity of a single grating is obtained from the ratio between the maximum and the minimum transmission power, which in this experiment resulted in 0.43 dB .

Fig. 14
Fig. 14

(Color online) Experiment 2. The transmissivity of a single grating is obtained from the ratio between the maximum and the minimum transmission power, which in this experiment resulted in 0.35 dB .

Fig. 15
Fig. 15

(Color online) Experiment setup for fiber loss measurement. Light reflected from the tested fiber is rerouted back to the left coupler. Half of the reflection is then present at the left tap of the right coupler and is coupled to half of the transmission from the tested fiber. The right tap of the right coupler is connected to the spectrum analyzer and measures one-quarter of the reflection plus the transmission. The EDFA serves as the light source, and OSA is the optical spectrum analyzer.

Tables (1)

Tables Icon

Table 1 Summary of Experiment Results

Equations (22)

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

[ a n + a n ] = M n [ a n 1 + a n 1 ] .
m n = ( 1 t n * r n * t n * r i t n 1 t n ) .
m g = ( cosh ( S L 0 ) i Δ β S sinh ( S L 0 ) i κ S sinh ( S L 0 ) i κ S sinh ( S L 0 ) cosh ( S L ) + i Δ β S sinh ( S L 0 ) ) ,
m d i = [ exp ( i k d i ) 0 0 exp ( i k d i ) ] .
m i = m g m d i = { [ cosh ( S L 0 ) i Δ β S sinh ( S L 0 ) ] exp ( i k d i ) i κ S sinh ( S L 0 ) exp ( i k d i ) i κ S sinh ( S L 0 ) exp ( i k d i ) [ cosh ( S Ł 0 ) + i Δ β S sinh ( S L 0 ) ] exp ( i k d i ) } ,
t i = [ m i ] 22 = [ cosh ( S L 0 ) i Δ β S sinh ( S L 0 ) ] 1 exp ( i k d i ) ,
r i = [ m i ] 11 [ m i ] 22 = κ exp ( i k d i ) sinh ( S L 0 ) [ i S cosh ( S L 0 ) Δ β sinh ( S L 0 ) ] .
M N = m 1 m 2 m N = ( 1 T N * R N * T N * R N T N 1 T N ) .
1 N log m N m 1 u log m ( α ) u u d μ ( α ) d v ( θ ) γ ,
v [ θ ) = v [ θ ( α ) ] d θ ( α ) d θ d μ ( α )
lim N 1 N ln τ N = ln ( 1 τ ) ,
τ N = exp [ N ln ( 1 τ ) ] = τ N .
m = ( τ ρ 2 τ ρ τ ρ τ 1 τ ) ,
m N = ( τ ρ 2 τ ρ τ ρ τ 1 τ ) N = [ I + ρ τ ( 1 1 1 1 ) ] N = I + N ρ τ ( 1 1 1 1 ) .
τ N = m 22 1 = τ τ + N ( 1 τ ) .
M N = m N .
M N = [ m 11 U N 1 ( a ) U N 2 ( a ) m 12 U N 1 ( a ) m 21 U N 1 ( a ) m 22 U N 1 ( a ) U N 2 ( a ) ] ,
U N ( a ) = sin [ ( N + 1 ) cos 1 a ] 1 a 2 ,
M N = 1 1 a 2 { [ cosh ( S L ) i Δ β S sinh S L ] exp ( i k d ) sin ( N cos 1 a ) sin [ ( N 1 ) cos 1 a ] i κ S sinh ( S L ) exp ( i k d ) sin ( N cos 1 a ) i κ S sinh ( S L ) exp ( i k d ) sin ( N cos 1 a ) [ cosh ( S L ) + i Δ β S sinh S L ] exp ( i k d ) sin ( N cos 1 a ) sin [ ( N 1 ) cos 1 a ] } .
T N = [ M N ] 22 1 = ( 1 a 2 ) 1 2 ( cosh S L + i Δ β S sinh S L ) exp ( i k d ) sin [ N cos 1 ( a ) ] sin [ ( N 1 ) cos 1 ( a ) ] .
τ N exp ( N L 0 S ) = τ N .
M 2 = ( 1 t * r * t * r t 1 t ) [ exp ( i k d ) 0 0 exp ( i k d ) ] ( 1 t * r * t * r t 1 t ) ,

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