Abstract

We present a detailed analysis of the transverse Anderson localization of light in a one-dimensional disordered optical lattice in the language of transversely localized and longitudinally propagating eigenmodes. The modal analysis allows us to explore localization behavior of a disordered lattice waveguide independent of the properties of the external excitation. Various localization-related phenomena, such as the periodic revival of a propagating Anderson-localized beam, are easily explained in modal language. We characterize the localization strength by the average width of the guided modes and carry out a detailed analysis of localization behavior as a function of the optical and geometrical parameters of the disordered lattice. We also show that in order to obtain a minimum average mode width, the average width of the individual random sites in the disordered lattice must be larger than the wavelength of the light by approximately a factor of two or more, and the optimum site width for the maximum localization depends on the design parameters of the disordered lattice.

© 2013 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  33. Z. D. Gaeta and C. R. Stroud, “Classical and quantum mechanical dynamics of quasiclassical state of a hydrogen atom,” Phys. Rev. A 42, 6308–6313 (1990).
    [CrossRef]
  34. A. A. Karatsuba and E. A. Karatsuba, “A resummation formula for collapse and revival in the Jaynes Cummings model,” J. Phys. A 42, 195304 (2009).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  37. A. Mafi, “Impact of lattice-shape moduli on band structure of photonic crystals,” Phys. Rev. B 77, 115140 (2008).
    [CrossRef]
  38. C. M. Soukoulis and E. N. Economou, “Electronic localization in disordered systems,” Waves random media 9, 255–269 (1999).
    [CrossRef]

2013

2012

R. G. S. El-Dardiry, S. Faez, and Ad. Lagendijk, “Snapshots of Anderson localization beyond the ensemble average,” Phys. Rev. B 86, 125132 (2012).
[CrossRef]

S. Ghosh, N. D. Psaila, R. R. Thomson, B. P. Pal, R. K. Varshney, and A. K. Kar, “Ultrafast laser inscribed waveguide lattice in glass for direct observation of transverse localization of light,” Appl. Phys. Lett. 100, 101102 (2012).
[CrossRef]

S. Karbasi, C. R. Mirr, P. G. Yarandi, R. J. Frazier, K. W. Koch, and A. Mafi, “Detailed investigation of the impact of the fiber design parameters on the transverse Anderson localization of light in disordered optical fibers,” Opt. Mater. Express 20, 18692–18706 (2012).
[CrossRef]

Y. V. Kartashov, V. V. Konotop, V. A. Vysloukh, and L. Torner, “Light localization in nonuniformly randomized lattices,” Opt. Lett. 37, 286–288 (2012).
[CrossRef]

R. Keil, Y. Lahini, Y. Shechtman, M. Heinrich, R. Pugatch, F. Dreisow, A. Tunnermann, S. Nolte, and A. Szameit, “Perfect imaging through a disordered waveguide lattice,” Opt. Lett. 37, 809–811 (2012).
[CrossRef]

S. Karbasi, C. Mirr, P. Yarandi, R. Frazier, K. W. Koch, and A. Mafi, “Observation of transverse Anderson localization in an optical fiber,” Opt. Lett. 37, 2304–2306 (2012).
[CrossRef]

S. Karbasi, T. Hawkins, J. Ballato, K. W. Koch, and A. Mafi, “Transverse Anderson localization in a disordered glass optical fiber,” Opt. Mater. Express 2, 1496–1503 (2012).
[CrossRef]

2011

2010

2009

A. A. Karatsuba and E. A. Karatsuba, “A resummation formula for collapse and revival in the Jaynes Cummings model,” J. Phys. A 42, 195304 (2009).
[CrossRef]

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

2008

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett. 100, 013906 (2008).
[CrossRef]

A. Mafi, “Impact of lattice-shape moduli on band structure of photonic crystals,” Phys. Rev. B 77, 115140 (2008).
[CrossRef]

2007

T. Schwartz, G. Bartal, S. Fishman, and M. Segev, “Transport and Anderson localization in disordered two-dimensional photonic lattices,” Nature 446, 52–55 (2007).
[CrossRef]

2005

2004

T. Pertsch, U. Peschel, J. Kobelke, K. Schuster, H. Bartelt, S. Nolte, A. Tunnermann, and F. Lederer, “Nonlinearity and disorder in fiber arrays,” Phys. Rev. Lett. 93, 053901 (2004).
[CrossRef]

2003

D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behavior in linear and nonlinear waveguide lattices,” Nature 424, 817–823 (2003).
[CrossRef]

2001

M. I. Stockman, S. V. Faleev, and D. J. Bergman, “Localization versus delocalization of surface plasmons in nanosystems: can one state have both characteristics?” Phys. Rev. Lett. 87, 167401 (2001).
[CrossRef]

2000

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

1999

C. M. Soukoulis and E. N. Economou, “Electronic localization in disordered systems,” Waves random media 9, 255–269 (1999).
[CrossRef]

1997

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

1991

S. John, “Localization of light,” Phys. Today 44(5), 32–40 (1991).
[CrossRef]

1990

Z. D. Gaeta and C. R. Stroud, “Classical and quantum mechanical dynamics of quasiclassical state of a hydrogen atom,” Phys. Rev. A 42, 6308–6313 (1990).
[CrossRef]

1989

H. De Raedt, Ad. Lagendijk, and P. de Vries, “Transverse localization of light,” Phys. Rev. Lett. 62, 47–50 (1989).
[CrossRef]

1985

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

P. W. Anderson, “The question of classical localization: a theory of white paint?” Philos. Mag. B 52, 505–509 (1985).
[CrossRef]

1984

S. John, “Electromagnetic absorption in a disordered medium near a photon mobility edge,” Phys. Rev. Lett. 53, 2169–2172 (1984).
[CrossRef]

1983

T. A. Lenahan, “Calculation of modes in an optical fiber using the finite element method and EISPACK,” Bell Syst. Tech. J. 62, 2663–2694 (1983).

1982

J. B. Pendry, “Off-diagonal disorder and 1D localization,” J. Phys. C 15, 5773–5778 (1982).
[CrossRef]

1981

C. M. Soukoulis and E. N. Economou, “Off-diagonal disorder in one-dimensional systems,” Phys. Rev. B 24, 5698–5702(1981).
[CrossRef]

1980

J. H. Eberly, N. B. Narozhny, and J. J. Sanchez-Mondragon, “Periodic spontaneous collapse and revival in a simple quantum model,” Phys. Rev. Lett. 44, 1323–1326 (1980).
[CrossRef]

1958

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

Abouraddy, A. F.

Anderson, P. W.

P. W. Anderson, “The question of classical localization: a theory of white paint?” Philos. Mag. B 52, 505–509 (1985).
[CrossRef]

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

Avidan, A.

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett. 100, 013906 (2008).
[CrossRef]

Ballato, J.

Bartal, G.

T. Schwartz, G. Bartal, S. Fishman, and M. Segev, “Transport and Anderson localization in disordered two-dimensional photonic lattices,” Nature 446, 52–55 (2007).
[CrossRef]

Bartelt, H.

T. Pertsch, U. Peschel, J. Kobelke, K. Schuster, H. Bartelt, S. Nolte, A. Tunnermann, and F. Lederer, “Nonlinearity and disorder in fiber arrays,” Phys. Rev. Lett. 93, 053901 (2004).
[CrossRef]

Belic, M. R.

D. M. Jovic, Y. S. Kivshar, C. Denz, and M. R. Belic, “Anderson localization of light near boundaries of disordered photonic lattices,” Phys. Rev. A 83, 033813 (2011).
[CrossRef]

Bergman, D. J.

M. I. Stockman, S. V. Faleev, and D. J. Bergman, “Localization versus delocalization of surface plasmons in nanosystems: can one state have both characteristics?” Phys. Rev. Lett. 87, 167401 (2001).
[CrossRef]

Berry, M. V.

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

Chabanov, A. A.

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

Christodoulides, D. N.

L. Martin, G. Di Giuseppe, A. Perez-Leija, R. Keil, F. Dreisow, M. Heinrich, S. Nolte, A. Szameit, A. F. Abouraddy, D. N. Christodoulides, and B. E. A. Saleh, “Anderson localization in optical waveguide arrays with off-diagonal coupling disorder,” Opt. Express 19, 13636–13646 (2011).
[CrossRef]

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett. 100, 013906 (2008).
[CrossRef]

D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behavior in linear and nonlinear waveguide lattices,” Nature 424, 817–823 (2003).
[CrossRef]

De Raedt, H.

H. De Raedt, Ad. Lagendijk, and P. de Vries, “Transverse localization of light,” Phys. Rev. Lett. 62, 47–50 (1989).
[CrossRef]

de Vries, P.

H. De Raedt, Ad. Lagendijk, and P. de Vries, “Transverse localization of light,” Phys. Rev. Lett. 62, 47–50 (1989).
[CrossRef]

Denz, C.

D. M. Jovic, Y. S. Kivshar, C. Denz, and M. R. Belic, “Anderson localization of light near boundaries of disordered photonic lattices,” Phys. Rev. A 83, 033813 (2011).
[CrossRef]

Di Giuseppe, G.

Dreisow, F.

Eberly, J. H.

J. H. Eberly, N. B. Narozhny, and J. J. Sanchez-Mondragon, “Periodic spontaneous collapse and revival in a simple quantum model,” Phys. Rev. Lett. 44, 1323–1326 (1980).
[CrossRef]

Economou, E. N.

C. M. Soukoulis and E. N. Economou, “Electronic localization in disordered systems,” Waves random media 9, 255–269 (1999).
[CrossRef]

C. M. Soukoulis and E. N. Economou, “Off-diagonal disorder in one-dimensional systems,” Phys. Rev. B 24, 5698–5702(1981).
[CrossRef]

El-Dardiry, R. G. S.

R. G. S. El-Dardiry, S. Faez, and Ad. Lagendijk, “Snapshots of Anderson localization beyond the ensemble average,” Phys. Rev. B 86, 125132 (2012).
[CrossRef]

Faez, S.

R. G. S. El-Dardiry, S. Faez, and Ad. Lagendijk, “Snapshots of Anderson localization beyond the ensemble average,” Phys. Rev. B 86, 125132 (2012).
[CrossRef]

Faleev, S. V.

M. I. Stockman, S. V. Faleev, and D. J. Bergman, “Localization versus delocalization of surface plasmons in nanosystems: can one state have both characteristics?” Phys. Rev. Lett. 87, 167401 (2001).
[CrossRef]

Fishman, S.

T. Schwartz, G. Bartal, S. Fishman, and M. Segev, “Transport and Anderson localization in disordered two-dimensional photonic lattices,” Nature 446, 52–55 (2007).
[CrossRef]

Frazier, R.

Frazier, R. J.

S. Karbasi, C. R. Mirr, P. G. Yarandi, R. J. Frazier, K. W. Koch, and A. Mafi, “Detailed investigation of the impact of the fiber design parameters on the transverse Anderson localization of light in disordered optical fibers,” Opt. Mater. Express 20, 18692–18706 (2012).
[CrossRef]

Gaeta, Z. D.

Z. D. Gaeta and C. R. Stroud, “Classical and quantum mechanical dynamics of quasiclassical state of a hydrogen atom,” Phys. Rev. A 42, 6308–6313 (1990).
[CrossRef]

Genack, A. Z.

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

Ghosh, S.

S. Ghosh, N. D. Psaila, R. R. Thomson, B. P. Pal, R. K. Varshney, and A. K. Kar, “Ultrafast laser inscribed waveguide lattice in glass for direct observation of transverse localization of light,” Appl. Phys. Lett. 100, 101102 (2012).
[CrossRef]

Hawkins, T.

Heinrich, M.

Hofmann, P.

John, S.

S. John, “Localization of light,” Phys. Today 44(5), 32–40 (1991).
[CrossRef]

S. John, “Electromagnetic absorption in a disordered medium near a photon mobility edge,” Phys. Rev. Lett. 53, 2169–2172 (1984).
[CrossRef]

Jovic, D. M.

D. M. Jovic, Y. S. Kivshar, C. Denz, and M. R. Belic, “Anderson localization of light near boundaries of disordered photonic lattices,” Phys. Rev. A 83, 033813 (2011).
[CrossRef]

Kar, A. K.

S. Ghosh, N. D. Psaila, R. R. Thomson, B. P. Pal, R. K. Varshney, and A. K. Kar, “Ultrafast laser inscribed waveguide lattice in glass for direct observation of transverse localization of light,” Appl. Phys. Lett. 100, 101102 (2012).
[CrossRef]

Karatsuba, A. A.

A. A. Karatsuba and E. A. Karatsuba, “A resummation formula for collapse and revival in the Jaynes Cummings model,” J. Phys. A 42, 195304 (2009).
[CrossRef]

Karatsuba, E. A.

A. A. Karatsuba and E. A. Karatsuba, “A resummation formula for collapse and revival in the Jaynes Cummings model,” J. Phys. A 42, 195304 (2009).
[CrossRef]

Karbasi, S.

Kartashov, Y. V.

Keil, R.

Kivshar, Y. S.

D. M. Jovic, Y. S. Kivshar, C. Denz, and M. R. Belic, “Anderson localization of light near boundaries of disordered photonic lattices,” Phys. Rev. A 83, 033813 (2011).
[CrossRef]

Klein, S.

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

Kobelke, J.

T. Pertsch, U. Peschel, J. Kobelke, K. Schuster, H. Bartelt, S. Nolte, A. Tunnermann, and F. Lederer, “Nonlinearity and disorder in fiber arrays,” Phys. Rev. Lett. 93, 053901 (2004).
[CrossRef]

Koch, K. W.

Konotop, V. V.

Lagendijk, A. D.

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

Lagendijk, Ad.

R. G. S. El-Dardiry, S. Faez, and Ad. Lagendijk, “Snapshots of Anderson localization beyond the ensemble average,” Phys. Rev. B 86, 125132 (2012).
[CrossRef]

I. M. Vellekoop, Ad. Lagendijk, and A. P. Mosk, “Exploiting disorder for perfect focusing,” Nat. Photonics 4, 320–322(2010).
[CrossRef]

H. De Raedt, Ad. Lagendijk, and P. de Vries, “Transverse localization of light,” Phys. Rev. Lett. 62, 47–50 (1989).
[CrossRef]

Lahini, Y.

R. Keil, Y. Lahini, Y. Shechtman, M. Heinrich, R. Pugatch, F. Dreisow, A. Tunnermann, S. Nolte, and A. Szameit, “Perfect imaging through a disordered waveguide lattice,” Opt. Lett. 37, 809–811 (2012).
[CrossRef]

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett. 100, 013906 (2008).
[CrossRef]

Lederer, F.

T. Pertsch, U. Peschel, J. Kobelke, K. Schuster, H. Bartelt, S. Nolte, A. Tunnermann, and F. Lederer, “Nonlinearity and disorder in fiber arrays,” Phys. Rev. Lett. 93, 053901 (2004).
[CrossRef]

D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behavior in linear and nonlinear waveguide lattices,” Nature 424, 817–823 (2003).
[CrossRef]

Lee, P. A.

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

Lenahan, T. A.

T. A. Lenahan, “Calculation of modes in an optical fiber using the finite element method and EISPACK,” Bell Syst. Tech. J. 62, 2663–2694 (1983).

Leonetti, M.

Lopez, C.

Mafi, A.

Martin, L.

Mirr, C.

Mirr, C. R.

S. Karbasi, C. R. Mirr, P. G. Yarandi, R. J. Frazier, K. W. Koch, and A. Mafi, “Detailed investigation of the impact of the fiber design parameters on the transverse Anderson localization of light in disordered optical fibers,” Opt. Mater. Express 20, 18692–18706 (2012).
[CrossRef]

Moloney, J. V.

Morandotti, R.

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett. 100, 013906 (2008).
[CrossRef]

Mosk, A. P.

I. M. Vellekoop, Ad. Lagendijk, and A. P. Mosk, “Exploiting disorder for perfect focusing,” Nat. Photonics 4, 320–322(2010).
[CrossRef]

Narozhny, N. B.

J. H. Eberly, N. B. Narozhny, and J. J. Sanchez-Mondragon, “Periodic spontaneous collapse and revival in a simple quantum model,” Phys. Rev. Lett. 44, 1323–1326 (1980).
[CrossRef]

Nolte, S.

Pal, B. P.

S. Ghosh, N. D. Psaila, R. R. Thomson, B. P. Pal, R. K. Varshney, and A. K. Kar, “Ultrafast laser inscribed waveguide lattice in glass for direct observation of transverse localization of light,” Appl. Phys. Lett. 100, 101102 (2012).
[CrossRef]

Pendry, J. B.

J. B. Pendry, “Off-diagonal disorder and 1D localization,” J. Phys. C 15, 5773–5778 (1982).
[CrossRef]

Perez-Leija, A.

Pertsch, T.

T. Pertsch, U. Peschel, J. Kobelke, K. Schuster, H. Bartelt, S. Nolte, A. Tunnermann, and F. Lederer, “Nonlinearity and disorder in fiber arrays,” Phys. Rev. Lett. 93, 053901 (2004).
[CrossRef]

Peschel, U.

T. Pertsch, U. Peschel, J. Kobelke, K. Schuster, H. Bartelt, S. Nolte, A. Tunnermann, and F. Lederer, “Nonlinearity and disorder in fiber arrays,” Phys. Rev. Lett. 93, 053901 (2004).
[CrossRef]

Pozzi, F.

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett. 100, 013906 (2008).
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Figures (16)

Fig. 1.
Fig. 1.

Sample refractive index profiles of (a) ordered and (b) disordered slab waveguides.

Fig. 2.
Fig. 2.

Typical mode profiles of an ordered slab waveguide; each mode extends over the entire waveguide.

Fig. 3.
Fig. 3.

Typical mode profiles of disordered slab waveguides for Δη=3 (lower disorder) and Δη=10 (higher disorder), where m¯s=20. The modes are localized and the localization effect is stronger for the higher disorder.

Fig. 4.
Fig. 4.

Gaussian beam is coupled to a disordered 1D waveguide. The intensity distribution shows that the beam goes through an initial expansion and eventually localizes to a relatively stable width.

Fig. 5.
Fig. 5.

Gaussian beam excitation of the lattice same as Fig. 4, but for a longer propagation distance to observe the revival of the in-coupling field at the propagation distance of 1000λ.

Fig. 6.
Fig. 6.

Distribution of the mode widths for all the guided modes of a disordered waveguide for two different levels of disorder are compared with an ordered periodic waveguide. Each point represents the position (x axis) and the width (y axis) of the mode defined by Eqs. 1 and 2, respectively. The circles, triangles, and squares represent Δη=20, Δη=5, and Δη=0, respectively. The modes of the waveguide with a higher level of disorder (Δη=20) are more localized.

Fig. 7.
Fig. 7.

Distribution of the mode widths for all the guided modes of a disordered waveguide for two different values of the index difference Δn and identical levels of disorder Δη are compared. Each point represents the position (x axis) and the width (y axis) of the mode, where the circles and triangles represent Δn=0.1 and Δn=0.01, respectively. The modes of the waveguide with a larger index difference (Δn=0.1) are more localized.

Fig. 8.
Fig. 8.

Probability distribution of the mode widths for two separate cases of Δn=0.1 and Δn=0.01 in a disordered lattice defined by m¯s=40 and Δη=20. Each histogram is the result of averaging over 100 independent random simulations.

Fig. 9.
Fig. 9.

Average mode widths of different disordered lattices are compared in units of λ for m¯s=50 (fixed mean slab width) as a function of Δn. The results are presented for three different values of disorder: Δη=50 (squares), Δη=25 (circles), and Δη=10 (triangles).

Fig. 10.
Fig. 10.

Average mode widths of different disordered lattices are compared in units of λ for Δη=10 (fixed level of disorder) as a function of Δn. The results are presented for three different values of the mean slab width given by m¯s=10 (squares), m¯s=30 (circles), and m¯s=50 (triangles).

Fig. 11.
Fig. 11.

Average mode widths of different disordered lattices are compared in units of λ for Δη=m¯s/2 as a function of Δn. The results are presented for three different values of mean slab width given by m¯s=10 (squares), m¯s=20 (circles), and m¯s=40 (triangles).

Fig. 12.
Fig. 12.

Average mode widths of different disordered lattices are compared in units of λ as a function of d¯s for three different cases: Δn=0.02 and Δη=m¯s/2=d¯s/2Λ (squares), Δn=0.05 and Δη=m¯s/2 (circles), and Δn=0.05 and Δη=10 (triangles).

Fig. 13.
Fig. 13.

Average mode widths of different disordered lattices are plotted in units of λ for m¯s=50 (fixed mean slab width) and Δη=25 as a function of Δn. The error-bars indicate one standard deviation around the average value.

Fig. 14.
Fig. 14.

Average mode widths of different disordered lattices are plotted in units of λ for m¯s=50 (fixed mean slab width) and Δη=25 as a function of Δn. The error-bars indicate the waveguide to waveguide variability of the average mode width metric at the one standard deviation level.

Fig. 15.
Fig. 15.

Logarithmic average of the intensity of the normalized modes of the disordered lattice defined by m¯s=50 and Δη=25. The average is also carried over 100 realizations of the lattice. The log-average intensity profiles are plotted for Δn=0.1 and Δn=0.01.

Fig. 16.
Fig. 16.

Localization length for different disordered lattices are compared in units of λ for m¯s=50 (fixed mean slab width) as a function of Δn. The results are presented for three different values of disorder: Δη=50 (squares), Δη=25 (circles), and Δη=10 (triangles). The vertical axis is presented as 2×Lc in order to facilitate direct comparison with Fig. 9.

Equations (2)

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x¯=dxxI(x)
σ2=dx(xx¯)2I(x),

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