15 May 1998

Ž .Optics Communications 151 1998 25–30

Localization of mode wavesin a disordered multi-waveguide system

Akira KomiyamaOsaka Electro-Communication UniÕersity, 18-8 Hatsu-cho, Neyagawa-shi, Osaka 572, Japan

Received 19 November 1997; revised 2 February 1998; accepted 4 February 1998

Abstract

Localization properties of mode waves in a disordered multi-waveguide system composed of randomly different cores arepresented. Coupled mode equations are transformed into a matrix eigenvalue problem and eigenvalues and eigenvectors arenumerically obtained. The propagation constant of the modes propagating along isolated cores is a random variable.Amplitudes of localized modes decrease exponentially away from the center of localization. The average extent of thelocalized mode depends only on the single parameter dbrk where db is the fluctuation of the propagation constants ofmodes and k is the mode coupling coefficient between neighboring cores. For a sufficiently large value of dbrk modes arestrongly localized. The mode is localized at one core. The localized position depends on the propagation constant of themode. q 1998 Elsevier Science B.V. All rights reserved.

Keywords: Localization of light; Waveguide array; Image fiber; Crosstalk

1. Introduction

Anderson has pointed out that wavefunctions for elec-trons are localized in a disordered system and conse-

w xquently no diffusion at all takes place 1 . In one-dimen-sional and two-dimensional disordered systems all statesare localized, whereas in a three-dimensional disorderedsystem both localized states and extended states exist and

w xthe mobility edge distinguishes both states 2 . Amplitudesof localized wavefunctions decrease exponentially awayfrom the center of localization. The strength of localizationis indicated by the ratio of the energy band width in adisordered system to the strength of the coupling betweenelectrons.

Since the essence of localization is interference effects,electromagnetic waves can be also localized in disordered

w x w xsystems 3,4 . One example is enhanced backscattering 5 .The diffusion constant for light does not vanish and thephenomenon is called weak localization. Another exampleis the strong localization of light in an optical waveguide

w xwith an aperiodic grating 6,7 . Resonant modes are local-

ized within a region of several wavelengths in the wave-guide.

The equation used by Anderson to describe the time-de-pendence of the probability amplitude is equal in form tothe coupled mode equation for a multi-waveguide system.It is expected that mode waves in a disordered waveguidesystem are also localized.

An image fiber composed of a large number of coresand a single cladding is a typical example of multi-wave-guide system. The crosstalk observed in the image fibercan be interpreted in terms of two power coupling coeffi-

w xcients 8,9 . One of the coupling coefficients, which isintroduced to describe a transfer process of power betweenneighboring cores, can be expressed in terms of statisticalproperties of the propagation constants of the modesw x10,11 . When the correlation length with respect to thepropagation constants is sufficiently long, the couplingcoefficient estimated from numerical solutions of coupledmode equations decreases with increasing correlationlength. It seems that the decrease is caused by the localiza-tion of mode waves within a region of several cores.

0030-4018r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved.Ž .PII S0030-4018 98 00073-X

( )A. KomiyamarOptics Communications 151 1998 25–3026

The purpose of this paper is to present localizationproperties of mode waves in disordered multi-waveguidesystems. A starting point of this discussion is the coupledmode equations. A system of differential equations istransformed into a matrix eigenvalue problem and eigen-values and eigenvectors are numerically obtained. Propaga-tion constants of the modes are random variables. It isshown that the average extent of the localized mode isindicated by the ratio of the fluctuation of the propagationconstants to the mode coupling coefficient between neigh-boring cores.

2. Coupled mode equations

We assume a multi-waveguide system composed of Ncircular cores of equal spacing and a single cladding as atwo-dimensional model of an image fiber. Cores differslightly in size. Each core supports only one mode and thecoupling between modes in the nearest neighboring coresis taken into account. We apply the boundary conditionthat dummy cores are introduced as shown in Fig. 1 andthe amplitude of the wave at both dummy cores is con-strained to zero. We have the following coupled modeequations,

dc1 eŽ b yb . z1 2syek e c , 1Ž .2d z

dcn eŽ b yb . z eŽ b yb . zn ny1 n nq1syek e c , yek e c ,ny1 nq1d z

ns2, . . . , Ny1, 2Ž .dcN eŽ b yb . zN Ny1syek e c , 3Ž .Ny1d z

where z is the distance along the fiber axis and c and bn n

are the amplitude and the propagation constant of themode in the nth core, respectively. b is a random vari-n

able with average b and variance db 2. Propagation con-stants of modes in neighboring cores are assumed to bestatistically independent of each other. k is the modecoupling coefficient between identical cores with the aver-

Ž . w xage size the mean coupling coefficient 10 .Ž .The coupled mode equations 1 are transformed into a

matrix eigenvalue problem by substituting the followingrelation,

c z sÕ eyeŽgqb1yb n. z 4Ž . Ž .n n

The eigenvalue g is a real number for a lossless system.

Fig. 1. Multi-waveguide system and dummy cores.

The eigenvector zŽm. corresponding to the mth eigenvalueg is a column vector,m

zŽm.s ÕŽm. . 5Ž .Ž .n

The normalization condition for the eigenvector is

< Žm. < 2Õ s1. 6Ž .Ý nn

The field of the mth mode of the entire waveguide systemis given by

cŽm. z szŽm. eyeŽg mqb 1. z . 7Ž . Ž .The evolution of optical fields in the multi-waveguidesystem is expressed as follows:

c z s V cŽm. z . 8Ž . Ž . Ž .Ý mm

The expansion coefficient V is determined from them

initial condition at zs0.

3. Numerical results

Cores of the image fiber used for the measurement ofw xthe crosstalk in Ref. 10 have a diameter of about 5 mm

on average, spacings between cores are about 8 mm andŽ .the numerical aperture NA of the fiber is 0.24. Cores can

support six LP modes at wavelengths in the red. The LP12

mode is the highest order mode. The coupling between theLP modes gives rise to the power transfer between12

neighboring cores. We are interested in effects of thedisorder of the system on propagation properties of theLP mode and assume in our model that a diameter of the12

core is 5 mm on average, the spacing between cores is 8mm and each core supports only one LP mode. Then, the12

mean coupling coefficient k has the value of 8.68=10y4

mmy1 at a wavelength of 0.633 mm.A view of the propagation of light in an ordered

waveguide system composed of identical cores is shown inŽ .Fig. 2 a . Light launched into the central core leaks out

into neighboring cores and spreads up to the outermostcores. Light reflected there is not perfectly focused into asingle core and is extended over the entire system. Thebehavior of light in a disordered waveguide system com-posed of randomly different cores in size differs strongly

Ž .from that for the ordered system as shown in Fig. 2 b .Light propagates along the illuminated core and does notalmost leak out into neighboring cores. In this example thefluctuation of the core diameter is assumed to be d aras2%. Then the fluctuation of the propagation constant of theLP mode is dbrks5=10y4 where k is the wavenum-12

ber.When one core of the ordered waveguide system is

illuminated, modes including lower order LP and LP01 11

modes are excited. Since these modes do not almost leakout into neighboring cores at a short distance, a view of the

( )A. KomiyamarOptics Communications 151 1998 25–30 27

Ž .Fig. 2. Views of the propagation of light in a the orderedŽ .waveguide system and b the disordered waveguide system.

Ž .propagation of light differs from Fig. 2 a . However thesemodes also leak out into neighboring cores at large dis-tance and spread up to the outermost cores. Consequentlylight launched into one core is always extended over theentire system. Since lower order LP and LP modes in a01 11

disordered waveguide system are more strongly concen-trated into one core than the highest order LP mode, light12

propagates along the illuminated core. The concentrationof light into one core is caused by localized mode waveswhich are eigenstates of a disordered waveguide system.Although in this paper we treat the coupling between theLP modes we derive the general properties of localized12

mode waves also applicable to lower order LP and LP01 11

modes.Modes of an ordered system are extended over the

entire system. The amplitude of the lowest order mode forŽ .Ns21 is shown in Fig. 3 a as an example. Although

amplitudes of high order modes oscillate with the corenumber and local peaks appear in the distributions, thepower of the mode is scattered over the entire system. In adisordered system, the power of the mode is concentratedinto a region of several cores. The amplitude of the modelocalized at the central core of a disordered waveguide

Ž .system is shown in Fig. 3 b . The amplitude decreasessteeply away from the center of localization. We can alsofind the modes localized at other cores. The localizedposition depends on the propagation constant of the mode.

The eigenvalues for the ordered system are given by

mpg s2kcos . 9Ž .m ž /Nq1

The propagation constants of the extended modes changeŽ .sinusoidally with mode number as shown in Fig. 4 a . The

width of the distribution is independent of the number ofcores. When the propagation constant b fluctuates largelyn

compared to the width, modes are strongly localized. Assuch an example, a distribution of the propagation constant

y4 Ž .b for dbrks5=10 is shown in Fig. 4 b . Then

distribution of the propagation constants of the localizedmodes in the disordered system differs from that for the

Ž .ordered system as shown in Fig. 4 c . The numbers in thefigure indicate the localized positions which are decidedfrom the mode patterns. Several examples of the localizedposition are shown. For example, the number 3 for thesixth mode shows that the mode is localized at the thirdcore. The propagation constant of the sixth localized modeis found to be almost equal to the propagation constant bn

of the third core. Thus the propagation constant of thelocalized mode is in fair agreement with the propagationconstant of the isolated mode propagating along the corepositioned at the center of localization.

Ž .Fig. 3. Amplitudes of modes in a the ordered waveguide systemŽ . Ž y4 .and b the disordered waveguide system dbrks5=10 .

( )A. KomiyamarOptics Communications 151 1998 25–3028

Ž .Fig. 4. Propagation constants of a the extended modes in theŽ .ordered waveguide system, b the modes propagating along iso-

Ž .lated cores and c the localized modes in the disordered wave-Ž y4 . Ž .guide system dbrks5=10 . In c , the numbers indicate

the localized positions of the modes.

No localization of mode waves takes place in an or-dered phase-mismatched waveguide system when differentkinds of cores are periodically arranged. The amplitude ofthe lowest order mode in the ordered mismatched systemcomposed of two kinds of cores with the same dbrk as

Ž . Ž .for Fig. 3 b is shown in Fig. 5 a as an example. Theamplitudes considerably differ in neighboring cores. Thisfact shows that distant matched cores strongly couple witheach other. Consequently light launched into one core isextended over the entire system through matched cores as

Ž .shown in Fig. 5 b .

Ž . Ž .Fig. 5. a Amplitude of the lowest order mode and b view of thepropagation of light in the ordered phase-mismatched waveguidesystem composed of two kinds of cores with dbrks5=10y4 .

The propagation constants of localized modes in a largesystem are insensitive to the boundary conditions, providedthat the center of localization is not close to the boundary,since the amplitude of the mode is small at the distantboundary. The difference between propagation constants ofmodes for different boundary conditions is a measure

Fig. 6. Difference between the propagation constants of the modesfor different boundary conditions.

( )A. KomiyamarOptics Communications 151 1998 25–30 29

Fig. 7. Average intensity distribution of the localized mode fordbrks2=10y4 .

indicating the extent of the localized mode. We calculatethe propagation constants using the boundary conditionthat the amplitude of the mode is zero at the boundaryŽ .dummy core , repeat the calculation using the periodicboundary condition and then calculate the difference be-tween the propagation constants of the modes localized atthe central core. Since these differences vary by manyorders of magnitude, the geometrical mean of the differ-ences Dg is calculated. The number of disordered systemsis one thousand. The results are shown in Fig. 6. Thedifferences decrease with the number of cores and this factshows that all modes used for the calculation are localized.Since the region of localization widens with decreasingfluctuation of the propagation constant db , the slope of thecurve becomes gentle.

Fig. 8. Dependence of the localization length on db and k .

The average intensity distribution of the localized modefor dbrks2=10y4 is shown in Fig. 7. The arithmeticmean of the intensity is calculated. In the disorderedsystem, several cores behave apparently as a single guidingregion and a mode concentrates in this region. Outside this

Ž < < .region, the intensity decreases as exp y x rj where j isthe localization length and x is the distance from thecenter of localization. The intensity distribution near theboundary is affected by the boundary condition and de-parts from the exponential decrease.

The localization length estimated from the averageintensity distribution for a sufficiently large system by amethod of curve fitting is shown in Fig. 8. The localizationlength is found to be inversely proportional to the fluctua-tion of the propagation constants db and proportional tothe mode coupling coefficient k . Consequently, the local-ization length is expressed as a function of dbrk asfollows:

constjs , 10Ž .

dbrk

where const is a constant depending on the spacing be-tween cores.

4. Conclusions

We have presented the localization properties of modewaves in a disordered multi-waveguide system composedof randomly different cores in size by numerically solvingthe coupled mode equations. In a disordered system, sev-eral cores behave apparently as a single guiding region anda mode concentrates in this region. A cladding region isformed outside this region and the amplitude of the modedecreases exponentially there. The mode is localized at onecore. The localized position depends on the propagationconstant of the mode. The average extent of the localizedmode depends on only the single parameter dbrk wheredb is the fluctuation of the propagation constants ofmodes propagating along isolated cores and k is the modecoupling coefficient between neighboring cores. For asufficiently large value of dbrk modes are strongly local-ized in narrow regions. Since localized modes also existfor a small value of dbrk , it is expected that modes arealways localized independent of the value of dbrk .

References

w x Ž .1 P.W. Anderson, Phys. Rev. 109 1958 1492.w x2 E. Abrahams, P.W. Anderson, D.C. Licciardello, T.V. Ra-

Ž .makrishnan, Phys. Rev. Lett. 42 1979 673.w x Ž .3 S. John, Phys. Rev. Lett. 53 1984 2169.w x Ž .4 P.W. Anderson, Phil. Mag. B 52 1985 505.

( )A. KomiyamarOptics Communications 151 1998 25–3030

w x5 Y. Okamura, G. Hirakawa, S. Yamamoto, Appl. Optics 34Ž .1995 5537.

w x6 T. Nakayama, K. Yakubo, M. Takano, Phys. Rev. B 47Ž .1993 9249.

w x7 M. Otaka, A. Hashimoto, T. Itoh, T. Santo, A. Yamamoto,Ž .Jpn. J. Appl. Phys. 34 1995 4502.

w x Ž .8 A. Komiyama, M. Hashimoto, Optics Comm. 107 1994 49.w x Ž .9 A. Komiyama, M. Hashimoto, Trans. IEICE E77-C 1994

1808.w x Ž .10 A. Komiyama, Trans. IEICE E79-C 1996 243.w x Ž .11 A. Komiyama, Trans. IEICE E80-C 1997 717.