Abstract

This tutorial gives an overview of the transverse Anderson localization of light in one and two transverse dimensions. A pedagogical approach is followed throughout the presentation, where many aspects of localization are illustrated by means of a few simple models. The tutorial starts with some basic aspects of random matrix theory and light propagation through and reflection from a random stack of dielectric slabs. Transverse Anderson localization of light in one- and two-dimensional coupled waveguide arrays is subsequently established and discussed. Recent experimental observations of localization and image transport in disordered optical fibers are discussed. More advanced topics, such as hyper-transport in longitudinally varying disordered waveguides, the impact of nonlinearity, and propagation of partially coherent and quantum light, are also examined.

© 2015 Optical Society of America

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2014 (6)

A. Mafi, S. Karbasi, K. W. Koch, T. Hawkins, and J. Ballato, “Transverse Anderson localization in disordered glass optical fibers: a review,” Materials 7, 5520–5527 (2014).

M. Leonetti, S. Karbasi, A. Mafi, and C. Conti, “Light focusing in the Anderson regime,” Nat. Commun. 5, 4534 (2014).
[Crossref]

M. Chen and M.-J. Li, “Observing transverse Anderson localization in random air line based fiber,” Proc. SPIE 8994, 89941S (2014).

M. Leonetti, S. Karbasi, A. Mafi, and C. Conti, “Observation of migrating transverse-Anderson localizations of light in nonlocal media,” Phys. Rev. Lett. 112, 193902 (2014).
[Crossref]

M. Leonetti, S. Karbasi, A. Mafi, and C. Conti, “Experimental observation of disorder induced self-focusing in optical fibers,” Appl. Phys. Lett. 105, 171102 (2014).
[Crossref]

S. Karbasi, R. J. Frazier, K. W. Koch, T. Hawkins, J. Ballato, and A. Mafi, “Image transport through a disordered optical fibre mediated by transverse Anderson localization,” Nat. Commun. 5, 3362 (2014).
[Crossref]

2013 (5)

D. J. Richardson, J. M. Fini, and L. E. Nelson, “Space-division multiplexing in optical fibres,” Nat. Photonics 7, 354–362 (2013).
[Crossref]

M. Segev, Y. Silberberg, and D. N. Christodoulides, “Anderson localization of light,” Nat. Photonics 7, 197–204 (2013).
[Crossref]

S. Karbasi, K. W. Koch, and A. Mafi, “Image transport quality can be improved in disordered waveguides,” Opt. Commun. 311, 72–76 (2013).
[Crossref]

S. Karbasi, K. W. Koch, and A. Mafi, “Multiple-beam propagation in an Anderson localized optical fiber,” Opt. Express 21, 305–313 (2013).
[Crossref]

S. Karbasi, K. W. Koch, and A. Mafi, “Modal perspective on the transverse Anderson localization of light in disordered optical lattices,” J. Opt. Soc. Am. B 30, 1452–1461 (2013).
[Crossref]

2012 (12)

S. Karbasi, C. R. Mirr, P. G. Yarandi, R. J. 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, C. R. Mirr, R. J. Frazier, P. G. Yarandi, 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. Express 20, 18692–18706 (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]

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]

T. van der Beek, P. Barthelemy, P. M. Johnson, D. S. Wiersma, and A. Lagendijk, “Light transport through disordered layers of dense gallium arsenide submicron particles,” Phys. Rev. B 85, 115401 (2012).

A. F. Abouraddy, G. Di Giuseppe, D. N. Christodoulides, and B. E. A. Saleh, “Anderson localization and colocalization of spatially entangled photons,” Phys. Rev. A 86, 040302(R) (2012).
[Crossref]

T. Sperling, W. Bührer, C. M. Aegerter, and G. Maret, “Direct determination of the transition to localization of light in three dimensions,” Nat. Photonics 7, 48–52 (2012).
[Crossref]

S. Ghosh, B. P. Pal, R. K. Varshney, and G. P. Agrawal, “Transverse localization of light and its dependence on the phase-front curvature of the input beam in a disordered optical waveguide lattice,” J. Opt. 14, 075701 (2012).
[Crossref]

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

L. Levi, Y. Krivolapov, S. Fishman, and M. Segev, “Hyper-transport of light and stochastic acceleration by evolving disorder,” Nat. Phys. 8, 912–917 (2012).
[Crossref]

Y. Krivolapov, L. Levi, S. Fishman, M. Segev, and M. Wilkinson, “Super-diffusion in optical realizations of Anderson localization,” New J. Phys. 14, 043047 (2012).
[Crossref]

S. Fishman, Y. Krivolapov, and A. Soffer, “The nonlinear Schrödinger equation with a random potential: results and puzzles,” Nonlinearity 25, R53–R72 (2012).
[Crossref]

2011 (5)

D. Čapeta, J. Radić, A. Szameit, M. Segev, and H. Buljan, “Anderson localization of partially incoherent light,” Phys. Rev. A 84, 011801(R) (2011).
[Crossref]

S. Ghosh, G. P. Agrawal, B. P. Pal, and R. K. Varshney, “Localization of light in evanescently coupled disordered waveguide lattices: dependence on the input beam profile,” Opt. Commun. 284, 201–206 (2011).
[Crossref]

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]

Y. Lahini, Y. Bromberg, Y. Shechtman, A. Szameit, D. N. Christodoulides, R. Morandotti, and Y. Silberberg, “Hanbury Brown and Twiss correlations of Anderson localized waves,” Phys. Rev. A 84, 041806(R) (2011).
[Crossref]

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]

2010 (5)

S. Ghosh, R. K. Varshney, B. P. Pal, and G. Monnom, “A Bragg-like chirped clad all-solid microstructured optical fiber with ultra-wide bandwidth for short pulse delivery and pulse reshaping,” Opt. Quantum Electron. 42, 1–14 (2010).
[Crossref]

J.-H. Han, J. Lee, and J. U. Kang, “Pixelation effect removal from fiber bundle probe based optical coherence tomography imaging,” Opt. Express 18, 7427–7439 (2010).
[Crossref]

A. Szameit, Y. V. Kartashov, P. Zeil, F. Dreisow, M. Heinrich, R. Keil, S. Nolte, A. Tunnermann, V. A. Vysloukh, and L. Torner, “Wave localization at the boundary of disordered photonic lattices,” Opt. Lett. 35, 1172–1174 (2010).
[Crossref]

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2009 (1)

A. D. Lagendijk, B. van Tiggelen, and D. S. Wiersma, “Fifty years of Anderson localization,” Phys. Today 62(8), 24–29 (2009).
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2008 (6)

H. Hu, A. Strybulevych, J. H. Page, S. E. Skipetrov, and B. A. van Tiggelen, “Localization of ultrasound in a three-dimensional elastic network,” Nat. Phys. 4, 945–948 (2008).
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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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A. S. Pikovsky and D. L. Shepelyansky, “Destruction of Anderson localization by a weak nonlinearity,” Phys. Rev. Lett. 100, 094101 (2008).
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X. Chen, K. L. Reichenbach, and C. Xu, “Experimental and theoretical analysis of core-to-core coupling on fiber bundle imaging,” Opt. Express 16, 21598–21607 (2008).
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2007 (2)

K. L. Reichenbach and C. Xu, “Numerical analysis of light propagation in image fibers or coherent fiber bundles,” Opt. Express 15, 2151–2165 (2007).
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T. Schwartz, G. Bartal, S. Fishman, and M. Segev, “Transport and Anderson localization in disordered two-dimensional photonic lattices,” Nature 446, 52–55 (2007).
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2006 (2)

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2005 (1)

A. Mafi and J. V. Moloney, “Shaping modes in multicore photonic crystal fibers,” IEEE Photon. Technol. Lett. 17, 348–350 (2005).

2004 (2)

K. Y. Bliokh and V. D. Freilikher, “Localization of transverse waves in randomly layered media at oblique incidence,” Phys. Rev. B 70, 245121 (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).
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2002 (1)

N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, “Discrete solitons in photorefractive optically induced photonic lattices,” Phys. Rev. E 66, 046602 (2002).
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2000 (3)

G. Kopidakis and S. Aubry, “Discrete breathers and delocalization in nonlinear disordered systems,” Phys. Rev. Lett. 84, 3236 (2000).
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1999 (4)

F. Scheffold, R. Lenke, R. Tweer, and G. Maret, “Localization or classical diffusion of light?” Nature 398, 206–207 (1999).
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Y. Fink, D. J. Ripin, S. Fan, C. Chen, J. D. Joannopoulos, and E. L. Thomas, “Guiding optical light in air using an all-dielectric structure,” J. Lightwave Technol. 17, 2039–2041 (1999).
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1997 (4)

F. M. Izrailev, T. Kottos, A. Politi, and G. P. Tsironis, “Evolution of wave packets in quasi-one-dimensional and one-dimensional random media: diffusion versus localization,” Phys. Rev. E 55, 4951–4963 (1997).
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1993 (3)

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1992 (2)

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1991 (3)

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1990 (1)

I. S. Graham, L. Piche, and M. Grant, “Experimental evidence for localization of acoustic waves in three dimensions,” Phys. Rev. Lett. 64, 3135–3138 (1990).
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1989 (1)

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1987 (2)

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1985 (1)

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1984 (1)

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1982 (2)

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1981 (2)

A. Douglas Stone and J. D. Joannopoulos, “Probability distribution and new scaling law for the resistance of a one-dimensional Anderson model,” Phys. Rev. B 24, 3592–3595 (1981).

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1980 (2)

S. S. Abdullaev and F. K. Abdullaev, “On propagation of light in fiber bundles with random parameters,” Radiofizika 23, 766–767 (1980).

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1979 (1)

E. Abrahams, P. W. Anderson, D. C. Licciardello, and T. V. Ramakrishnan, “Scaling theory of localization: absence of quantum diffusion in two dimensions,” Phys. Rev. Lett. 42, 673–675 (1979).
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1978 (1)

1976 (1)

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1972 (1)

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1960 (1)

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1958 (1)

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1956 (1)

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Supplementary Material (1)

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Figures (45)

Figure 1
Figure 1

This figure shows that the eigenvectors of the ordered matrix M defined in Eq. (1) are extended over the entire element-position domain. Eigenvectors Vj(1),Vj(2),Vj(3), and Vj(4) are plotted in (a), and Vj(200) is plotted in (b) as a function of the element position j.

Figure 2
Figure 2

Similar to Fig. 1, except the matrix M is slightly randomized according to Eq. (2) and the eigenvectors are localized.

Figure 3
Figure 3

Similar to Figs. 1 and 2, except the matrix M is strongly randomized according to Eq. (3) and the eigenvectors are strongly localized.

Figure 4
Figure 4

Probability distribution of the eigenvector widths for two separate cases of the weak disorder related to Eq. (2), and the strong disorder related to Eq. (3).

Figure 5
Figure 5

(a) Periodic array of two different dielectric materials identified with refractive indices n1 and n2. (b) is similar, except the thickness of the layers is chosen randomly.

Figure 6
Figure 6

The optical transmission through a stack of 200 glass–air layers is plotted versus k0Λ for (a) periodic layer thickness and (b) random layer thickness. Λ is the periodicity in case (a), while the thickness of each layer in case (b) is chosen from a uniform random distribution in the range [0,2Λ].

Figure 7
Figure 7

Sketch of the field amplitudes, refractive indices, and geometry related to Eqs. (5) and (6).

Figure 8
Figure 8

Same as Fig. 6, except with 10,000 layers.

Figure 9
Figure 9

(a) Transmission and reflection of light incident on a periodic stack of dielectrics. (b) Same, except the thickness of each dielectric layer is randomly selected.

Figure 10
Figure 10

Relative optical power transmission is plotted as a function of the normalized frequency for oblique incidence at a 45° angle for the TE polarization for (a) periodic dielectric stack and (b) random dielectric stack.

Figure 11
Figure 11

Relative optical power transmission is plotted as a function of the normalized frequency for oblique incidence at a 45° angle for the TM polarization for (a) periodic dielectric stack and (b) random dielectric stack.

Figure 12
Figure 12

Relative optical power transmission is plotted as a function of the normalized frequency for oblique incidence at an 85° angle for the TE polarization for (a) periodic dielectric stack and (b) random dielectric stack.

Figure 13
Figure 13

Relative optical power transmission is plotted as a function of the normalized frequency for oblique incidence at an 85° angle for the TM polarization for (a) periodic dielectric stack and (b) random dielectric stack.

Figure 14
Figure 14

Relative optical power transmission is plotted as a function of the incidence angle of TM polarized light on a random dielectric stack for (a) k0Λ=1 and (b) k0Λ=5. Perfect transmission is observed at Brewster’s angle θB=56.3°.

Figure 15
Figure 15

Relative optical power transmission is plotted as a function of the incidence angle of TE polarized light on a random dielectric stack for (a) k0Λ=1 and (b) k0Λ=5.

Figure 16
Figure 16

1D dielectric waveguide using (a) periodic dielectric stack, which can have very low leakage over a narrow range of wavelengths in the bandgaps corresponding to the specific incident wavelength and incident angle, and (b) random dielectric stack, which can operate over a broad range of wavelengths and angles, but with slightly more leakage compared with a periodic waveguide of the same number of layers optimized to operate in the center of the bandgap.

Figure 17
Figure 17

One-dimensional array of N identical single-mode optical fibers, referred to as a coupled waveguide array.

Figure 18
Figure 18

Propagation through a waveguide coupled array for the case of (a) periodic array, (b) disordered array, and (c) highly disordered array. A higher level of disorder results in a more localized propagation.

Figure 19
Figure 19

Propagation constants of a random coupled waveguide array, where the modes are numbered from 1 to N=201, sorted in descending order of the value of the propagation constant.

Figure 20
Figure 20

Mode profiles of the random coupled waveguide array of Fig. 17 are plotted. Subfigures (a), (b), (c), (d), (e), and (f) correspond to mode numbers 1, 25, 99, 101, 170, and 201. The modes whose propagation constants belong to the region near the top edge of the band in Fig. 19 are highly localized with no or few oscillations as in (a) and (b). The modes with propagation constants near the middle of the band spread over many waveguides and oscillate as in (c) and (d). The modes with propagation constants near the bottom of the band are highly localized and oscillate so rapidly that the sign of the mode profile flips between adjacent waveguides as in (c) and (d).

Figure 21
Figure 21

The mode width is plotted against the propagation constant of each mode. The results are averaged over 1000 simulations for (a) rjunif[0.006,0.006] and (b) rjunif[0.01,0.01].

Figure 22
Figure 22

This figure shows the expansion of the x-space beam width in (a) disorder-free lattice, (b) weakly disordered lattice, and (c) strongly disordered lattice. Three sample realizations of the random waveguide are shown for each disorder level. The disorder-free lattice shows ballistic expansion, while disorder-induced localization is apparent in disordered samples.

Figure 23
Figure 23

Same as Fig. 22, except the beam width is calculated in the k-space. The k-space beam width for diffractive propagation in the disorder-free periodic lattice of (a) remains unchanged.

Figure 24
Figure 24

Same as Fig. 22, except the x-space beam width is averaged over 100 independent statistical realizations of the disordered waveguides. The error bars signify the one standard deviation for the beam width over the 100 samples.

Figure 25
Figure 25

Same as Fig. 23, except the k-space beam width is averaged over 100 independent statistical realizations of the disordered waveguides. The error bars signify the one standard deviation for the beam width over the 100 samples.

Figure 26
Figure 26

A coupled array of optical fibers is placed on a hexagonal lattice. Actual simulations in this section are performed on a hexagonal lattice of N=817 fibers.

Figure 27
Figure 27

Each subfigure shows the density plot of the elements of an eigenvector of the effective propagation constant matrix B on the element-position domain of the hexagonal lattice. The density plot shows only the absolute value of the elements of each eigenvector. In the language of coupled mode theory, each density plot signifies the intensity distribution of a supermode of the entire coupled fiber lattice. This figure relates to the nonrandom deterministic situation, where all fibers are identical and all coupling strengths to the nearest neighbors are equal. Note that the eigenvectors spread over nearly the entire lattice.

Figure 28
Figure 28

This figure is similar to Fig. 27, except the nearest neighbor coupling strengths are randomized (off-diagonal disorder). Note that each eigenvector is localized in a certain region on the lattice, as expected from the transverse Anderson localization.

Figure 29
Figure 29

This figure is similar to Fig. 27, except the propagation constants of the individual fibers are randomized (diagonal disorder). Note that each eigenvector is localized in a certain region on the lattice similar to Fig. 28, as expected from the transverse Anderson localization.

Figure 30
Figure 30

(a) Sketch of the transversely random and longitudinally invariant dielectric medium for observation of the transverse Anderson localization. (b) Cross section of a Gaussian beam that is coupled to the disordered waveguide. The intensity distribution shows that the beam goes through an initial expansion and eventually localizes to a stable width, as expected from the transverse Anderson localization.

Figure 31
Figure 31

Random mixture of the PS and PMMA fiber strands.

Figure 32
Figure 32

(a) Cross section of pALOF with a nearly square profile and an approximate side width of 250 μm; (b) zoomed-in SEM image of a 24-μm-wide region on the tip of pALOF exposed to a solvent to differentiate between PMMA and PS polymer components, where feature sizes are around 0.9 μm and darker regions are PMMA; and (c) experimental measurement of the near-field intensity profile of the localized beam after 60 cm of propagation through pALOF. The total side width of (c) is 250 μm, so it can be directly compared with (a). Adapted with permission from [17]. Copyright 2012 Optical Society of America.

Figure 33
Figure 33

The localized beam radius is smaller at 405 nm wavelength compared with 633 nm wavelength, for the pALOF design with an approximate 0.9 μm feature size. Adapted with permission from [63]. Copyright 2012 Optical Society of America.

Figure 34
Figure 34

The evolution of the effective beam radius plotted versus the propagation distance is shown for different values of the fill-fraction of p=40%, and p=50%, where the latter provides a lower effective beam radius and localization length. Adapted with permission from [63]. Copyright 2012 Optical Society of America.

Figure 35
Figure 35

Effective beam radius versus propagation distance for different values of fill-fraction, p, in glass disordered optical fibers with random airholes. The index difference of 0.5 between the random sites results in a very small localization radius. Adapted with permission from [63]. Copyright 2012 Optical Society of America.

Figure 36
Figure 36

Multiple-beam propagation in a 5-cm-long pALOF: (a) simulation for five beams, (b) experiment for two beams, and (c) experiment for two beams with different wavelengths. All beams are at 405 nm wavelength, except the bottom-middle beam in (c), which is at 633 nm wavelength. Adapted with permission from [18]. Copyright 2013 Optical Society of America.

Figure 37
Figure 37

(Left) Elements of a group on 1951 U.S. Air Force test target (1951-AFTT). Transported images of different numbers through a disordered optical fiber: (middle) subfigures (a)–(d) are related to group 3 on the test target, and (right) subfigures (a)–(d) are related to group 5 on the test target. Adapted with permission, copyright 2014, Nature Communications [19].

Figure 38
Figure 38

Transported images through the disordered fiber and the commercial image fibers. Images related to group 5 of the 1951-AFTT test chart in (a) pALOF, (b) FIGH-10-350S image fiber, and (c) FIGH-10-500N image fiber (experimental measurements). The scale bar in (b) is 30 μm long, and the same scale bar can be used for (a) and (c). Each fiber is approximately 5 cm long. Adapted with permission, copyright 2014, Nature Communications [19].

Figure 39
Figure 39

(a) SEM image of the glass optical fiber with random airholes reported in Ref. [78] and (b) zoomed-in SEM image of the same fiber.

Figure 40
Figure 40

The average beam width as a function of the propagation distance in a disordered coupled waveguide array is plotted for the case of highly coherent S0=100 in red, semi-coherent S0=5 in blue, and near-incoherent S0=2 in cyan. W0=2 and j0=101 have been used, and the beam widths are averaged over 100 independent simulations. The disordered coupled waveguide array is defined by β0=6, c0=0.01, N=201, and rjunif[0.1,0.1].

Figure 41
Figure 41

|Gjk(z)|2 for k=101 is plotted as a function of the output waveguide number for a disordered coupled waveguide array. The dashed blue line represents a single simulation, while the solid red line shows the result of averaging |GjM(z)|2dw over 1000 simulations.

Figure 42
Figure 42

Γj,j(2)(z) of Eq. (42) is plotted for M=51 (N=101 waveguides in the lattice) as a function of the output waveguide numbers j and j for a disorder-free coupled waveguide array for (a) z=2400 and (b) z=10,000.

Figure 43
Figure 43

Γj,j(2)(z=2400) of Eq. (42) is plotted for M=51 (N=101 waveguides in the lattice) as a function of the output waveguide numbers j and j for a disordered coupled waveguide array for (a) rjunif[0.002,0.002] and (b) rjunif[0.004,0.004]. Both density and 3D plots are presented in each case for easier comparison.

Figure 44
Figure 44

Γj,j(2)(z) of Eq. (43) is plotted for (a) M=50 and M=51 (N=101 waveguides in the lattice) and (b) M=50 and M=52, as a function of the output waveguide numbers j and j for a disorder-free N=101 coupled waveguide array and z=2400.

Figure 45
Figure 45

Γj,j(2)(z=2400) of Eq. (43) is plotted for M=51 and M=52 (N=101 waveguides in the lattice) as a function of the output waveguide numbers j and j for a disordered coupled waveguide array for (a) rjunif[0.002,0.002] and (b) rjunif[0.004,0.004]. Both density and 3D plots are presented in each case for easier comparison.

Equations (45)

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Mi,i=1,Mi,i+1=Mi+1,i=0.1
Mi,i=1,Mi,i+1=Mi+1,i=0.1+ri,riunif[0.01,0.01],
Mi,i=1,Mi,i+1=Mi+1,i=0.1+ri,riunif[0.05,0.05].
σi=(j=1N(jj)2|Vj(i)|2j=1N|Vj(i)|2)1/2,ji=j=1Nj|Vj(i)|2j=1N|Vj(i)|2.
M=12n2[(n2+n1)eiφ(n2n1)eiφ(n2n1)eiφ(n2+n1)eiφ],φ=n1k0d.
[U2+U2]=M·[U1+U1].
τ2N=exp(2Nlog(1/τ)),τ=4n1n2(n1+n2)2.
0ϕ=(n1k0d+n2k0d)2(n1+n2)k0Λ.
k¯0=k0Λπ/(n1+n2),
(iz+β0)Aj(z)+cj+Aj+1(z)+cjAj1(z)=0,j=1,,N.
A=(A1,A2,A3,,AN1,AN),
izA+B·A=0,
izA¯+B¯·A¯=0,whereA¯=Q·A,B¯=Q·B·QT.
izA¯j(z)+β¯jA¯j(z)=0,j=1,,N,
A¯=(A¯1,A¯2,A¯3,,A¯N1,A¯N).
A¯j(z)=A¯j(0)exp[iβ¯jz],j=1,,N.
Aj(z)=k=1Nl=1NQk,jQk,lAl(0)exp[iβ¯kz],j=1,,N
=k=1NQk,jQk,101exp[iβ¯kz]
=k=1NVj(k)V101(k)exp[iβ¯kz].
Aj(z=0)=exp[(j101)24W02],j=1,,N,W0=2.
(iz+β0)Aj(z)+kNNcjkAk(z)=0,
iAz+12n0k0[T2A+k02(n2n02)A]=0.
ξ(z)=A(r)|(RR¯)2|A(r),
IPR=I2(x,y;z)dxdy(I(x,y;z)dxdy)2,
w(z)=w01+(zz0)2,z0=πw02λ,
iAz+12n0k0[T2A+k02(n2n02)A]+k0n2|A|2A=0,
(iz+βj)Aj(z)+c0[Aj+1(z)+Aj1(z)]+γ|Aj(z)|2Aj(z)=0,j=1,,N.
βjunif[β0B2,β0+B2].
σ(z)=j=1N(jj)2|Aj(z)|2,j=j=1Nj|Aj(z)|2.
Γj,k(1)(z=0)=exp[(jj0)2+(kj0)24W02]exp[(jk)2S02],j,k=1,,N,
Γj,k(1)(z)=k=1Nj=1Nbj,kVj(j)Vk(k)exp[i(β¯jβ¯k)z],
bj,k=k=1Nj=1NΓj,k(1)(z=0)Vj(j)Vk(k).
[a^j,a^k]=0,[a^j,a^k]=δkj,j,k=1,,N.
a^j(z)=k=1NGjk(z)a^k(0).
Gjk(z)=k=1NVk(k)Vj(k)exp[iβ¯kz].
Γj,j(1)(z)=k=1Nk=1NGjk*(z)Gjk(z)dwΓk,k(1)(0),
Γk,k(1)(0)=Tr{ρa^k(0)a^k(0)}.
Γk,k(1)(0)=αk*αk.
Γk,k(1)(0)=nkδkk.
I¯j(z)=k=1Nk=1NGjk*(z)Gjk(z)dwαk*αk.
I¯j(z)=k=1NGjk*(z)Gjk(z)dwnk.
I¯j(z)=|GjM(z)|2dw|αM|2,or=|GjM(z)|2dwnM.
Γj,j(2)(z)=a^j(z)a^j(z)a^j(z)a^j(z)=k=1Nk=1NGjk*(z)Gjk*(z)Gjl(z)Gjl(z)dwa^k(0)a^k(0)a^l(0)a^l(0)qs
Γj,j(2)(z)=|GjM(z)|2|GjM(z)|2dw.
Γj,j(2)(z)=|GjM(z)GjM(z)+GjM(z)GjM(z)|2dw.

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