1MwTAantospwtttteTsppicnf

ocamss[os

Sebbah et al. Vol. 24, No. 10 /October 2007 /J. Opt. Soc. Am. B A77

Effect of absorption on quasimodesof a random waveguide

P. Sebbah,1,2,* B. Hu,1 V. I. Kopp,3 and A. Z. Genack1

1Department of Physics, Queens College of City University of New York, Flushing, New York 11367, USA2Laboratoire de Physique de la Matière Condensée/Ccntre National de la Recherche Scientifique,

Université de Nice-Sophia Antipolis, Parc Valrose 06108 Nice Cedex 02, France3Chiral Photonics, Inc., Clifton, New Jersey 07012, USA

*Corresponding author: sebbah@unice.fr

Received April 17, 2007; accepted May 11, 2007;posted June 1, 2007 (Doc. ID 82141); published August 13, 2007

The field distribution inside a random waveguide has been measured over a spectral range in which the waveis nominally localized. In addition to spatially and spectrally isolated modes, extended field distributionsknown as necklace states are found inside random samples due to the spectral overlap. Necklace states areshown to be a superposition of multipeaked quasimodes. We performed numerical simulations that demon-strate the robustness of these spectra to variations in the random structure. We also used simulations to in-vestigate the impact of dissipation on transmission and reflection spectra and upon the field distribution ofquasimodes. © 2007 Optical Society of America

OCIS codes: 290.4210, 350.4010, 030.6600.

wdttiwac

g[asttttap

otrspottitdtwSt

. INTRODUCTIONultiple scattering and interference of partial wavesithin a scattering medium leads to wave localization.he localization of particles was originally proposed bynderson to explain the metal-insulator transition [1]nd was only later appreciated to be a general wave phe-omenon [2,3]. Initially, studies of localization focused onhe size dependence of transport properties. The averagef transport over an ensemble of conductors for weaklycattered waves closely follows the behavior of diffusingarticles and falls inversely with sample thickness, 1/L,hen inelastic processes are negligible. As the role of in-

erference increases by virtue of an increase in local scat-ering strength, or an increase in wave confinement,ransmission falls more rapidly. At the localizationhreshold, transport scales as 1/L2, whereas transport isxponentially suppressed when waves are localized [4–9].he renormalization of electron transport with increasingample size is arrested, however, by inelastic dephasingrocesses due to phonon scattering. For classical wavesropagating in static samples, the scaling of transmissions changed by absorption. This leads to an exponential de-ay of transmission, even for diffusing samples. The expo-ential falloff of transmission of classical waves is there-ore not definitive in the search for localization.

An alternative test of localization arises from the studyf fluctuations and correlation. The extent of localizationan be established, even in the presence of absorption, bymeasurement of the variance of total transmission nor-alized by its average value in an ensemble of random

amples, var�sa�=Ta / �Ta�, where Ta is the total transmis-ion coefficient for a wave incident in a transverse mode a10,11]. This may also be directly related to the variancef normalized intensity. In quasi-one-dimensionalamples, which are locally three-dimensional samples

0740-3224/07/100A77-7/$15.00 © 2

ith reflecting sides and length exceeding the transverseimensions, var�sa� is precisely equal [12] to the frac-ional correlation of intensity at the sample output be-ween points at which the field correlation function van-shes [13–15]. These parameters of propagation are onlyeakly affected by absorption, and their reduction whenbsorption is present may reflect a reduction of spatial lo-alization.

Transient measurements offer another way of distin-uishing between absorption and localization11,14,16–21]. For a given time delay in a homogeneouslybsorbing sample, average transmission is reduced by aimple exponential factor, whereas the suppression ofransport due to renormalization is unaffected by absorp-ion. Absorption has no impact on fractional renormaliza-ion or mesoscopic fluctuations in the time domain, sincehe relative weight of wave trajectories is not changed bybsorption and the amplitude of all trajectories is sup-ressed equally at a given delay from an exciting pulse.Though steady-state and time-domain measurements

f fluctuations and average transport can be used to de-ermine the degree of localization, the possibility of di-ectly observing the spatial extent of the wave is of con-iderable interest, since all statistical transportroperties are directly related to the spatial distributionf waves within a medium. Indeed a full description ofransport dynamics can be given only when both the spec-ral and spatial variation of the field are known. Once thiss known, it is possible to describe wave evolution inerms of the quasimodes of the medium, whose spatialistribution does not change in time. From this perspec-ive, transport slows down as time progresses, since theeight of modes with longer lifetimes increases [18,22].imilarly, giant fluctuations observed in steady-stateransmission in samples in which the average value is ex-

007 Optical Society of America

pmw[ttt[ms

tdntfTcPdivflqtidtectta

2Wgpr38heacbgrotT2g1tPm

dil

bsssam

sglb=montfia−

3Sa1tdwi�

Fr(w

A78 J. Opt. Soc. Am. B/Vol. 24, No. 10 /October 2007 Sebbah et al.

onentially small may be understood in terms of theodes of the medium, since transmission may be highhen the incident wave is resonant with a localized mode

23–25]. At the same time, in spectral regions in whichhe wave is localized on average, modes may overlap spec-rally, leading to a series of local intensity maxima withinhe sample, providing a path through which energy flows26,27]. Pendry [28,29] predicted that these necklaceodes, which have been recently observed [30–33],

hould play a significant role in electronic transport.In this paper, we report measurements of the field dis-

ribution inside an open one-dimensional microwave ran-om waveguide over a spectral range in which the wave isominally localized. In addition to spatially and spec-rally isolated modes, extended field distributions areound inside random samples due to the spectral overlap.hese field distributions arise from the hybridization ofoupled states and have been called necklace states byendry [28]. We performed numerical simulations thatemonstrate the robustness of these spectra to variationsn the random structure. We used these simulations to in-estigate the role of dissipation on transmission and re-ection spectra and upon the field distribution of theuasimodes, whether they are spectrally isolated and spa-ially localized or spectrally overlapping and multipeakedn space. In the absence of absorption, transmission me-iated by spectrally isolated modes localized near the cen-er of the waveguide would be close to unity. In the pres-nce of absorption however, localized modes near theenter of the system are strongly damped and their con-ribution to transmission is greatly reduced relative tohat of quasi-extended modes which are affected less bybsorption since they are short-lived.

. EXPERIMENTAL SETUPe consider a WR-42 �1.066 cm�0.413 cm� open rectan-

ular microwave waveguide. The empty waveguide sup-orts only a single transverse mode in the frequencyange from 13.5 to 19.5 GHz. The waveguide is filled with1 randomly oriented binary blocks [33,34]. Each block ismm thick, half of which is made of high dielectric andalf of two thin side walls used as spacers so that its di-lectric constant is close to unity. Five half-blocks of highnd low dielectric are introduced randomly in order toreate modes within the gap of the corresponding periodicinary system. A 60-cm-long empty section of the wave-uide separates the input antenna from the 28.8-cm-longandom sample. A 2-mm-wide slot is cut along the lengthf the waveguide. A sliding copper plate is pressed overhe slot to largely eliminate leakage from the waveguide.he output antenna is contained in a small box with a-mm-diameter hole pressed against the slot in the wave-uide. The box containing the antenna is translated inmm steps. At each position, field spectra are taken be-

ween 14 and 19.9 GHz in 3.5 MHz steps using a Hewlett-ackard 8720C vector network analyzer. These measure-ents have been conducted in 100 sample configurations.Losses due to dissipation in the waveguide and in the

ielectric material give rise to an overall attenuation thats essentially the same for all the modes. In contrast,eakage through the open ends of the random waveguide

roadens modes located close to the boundaries of theample more than modes deeper in the interior of theample, as seen in Fig. 1(a). Leakage through the coveredlit is negligible, whereas leakage through the receivingntenna, which is undercoupled to the waveguide, isinimized.The field inside the empty section of the waveguide re-

ults from the interference of the forward- and backward-oing waves, Ef and Eb respectively. The resulting modu-ation of the wave amplitude in the empty section variesetween a trough of Amin= �Ef�− �Eb� and a peak of Amax�Ef�+ �Eb�. Amin and Amax are obtained by fitting thisodulation with a sine function and oversampling it to

vercome the finite spatial resolution. The spectra areormalized to the amplitude of the wave directed towardhe sample, �Einc�= �Amax+Amin� /2 and the reflection coef-cient R, which is defined as the ratio of the backward-nd forward-going intensities, is given by R= �AmaxAmin�2 / �Amax+Amin�2.

. ISOLATED QUASIMODESpectra between 14 and 18.5 GHz of the normalized fieldmplitude along a random realization are shown in Fig.(a). Four spatially localized and spectrally isolated fea-ures corresponding to exponentially localized quasimo-es are seen in Fig. 1(a). Their localization length �,hich is the characteristic length of the exponential decay

n space of the wavefunctions seen in Fig. 1(b), �E�x��exp�−�x−x�� /��, is 19±1 mm. This value of the localiza-

ig. 1. (a) Spectra of the field amplitude at each point along aandom sample normalized to the amplitude of the incident field.b) Log of the field magnitude along the sample for an evanescentave (dotted curve) and two localized modes.

tdatrtchpstemmtfimnrp�otttstpptda

4IlsbaTplea�tS11pvseb

Bd

w�LFtsMabpip

5Wbm

Frtamp

Sebbah et al. Vol. 24, No. 10 /October 2007 /J. Opt. Soc. Am. B A79

ion length is typical and comparable to the penetrationepth of the evanescent wave off-resonance between 15nd 17 GHz, as shown in Fig. 1(b), but it is larger thanhe 12 mm evanescent penetration depth found in the pe-iodic system for the same frequency range. The localiza-ion length � increases near the edges of the gap of theorresponding periodic binary system. These quasimodesave a Lorentzian spectral shape with a width that de-ends on their proximity to the nearest boundary of theample. Indeed, the contribution to the linewidth due tohe leakage through the open ends of the system varies asxp�−2x /��, where x is the distance of the peak of theode to the nearest boundary. One mode at 18 GHz isuch broader than the others as a result of its closeness

o the input of the sample. When absorption is absent, theeld amplitude rises exponentially to the peak value forodes in the first half of the sample but first falls expo-

entially for modes in the second half [24]. The amplitudeeaches a peak value of �exp�L /2−�x� /� for a modeeaked at a distance of L /2−�x, but for modes peaked atL /2+�x�, the amplitude falls exponentially up to a depthf 2�x and then rises exponentially to give a peak ampli-ude �exp�L /2−3�x� /� [24]. Localized modes observed inhe first half of the sample are therefore generally largerhan those that peaked in the second half. In our mea-urements, the amplitude is further reduced by absorp-ion [25,35]. This is illustrated in Fig. 1(b), where an ex-onential fall of the amplitude is seen even for modeseaked near the input of the sample. As a result, modes inhe second part of the sample are usually stronglyamped, and modes peaked near the output of the samplere often not observed.

. MULTIPEAKED QUASIMODESn other sample configurations, spectral lines may over-ap, resulting in a multipeaked field distribution such ashown in Figs. 2(a) and 2(b). In such cases, the field maye appreciable far into the second half of the sample. Wenalyzed the modal structure of this field distribution.he top view representation of the logarithm of the am-litude, shown in Fig. 2(c) accentuates the damped oscil-ations of the evanescent wave deep into the system. Anxtra ridge associated with the introduction of anotherntinode along the length of the sample and an additionalphase shift is added each time the frequency is tuned

hrough a Lorentzian line associated with a quasimode.ingle additional ridges are added near 15.3 and6.1 GHz, while three ridges are added between 15.6 and5.8 GHz. To find the quasimodes associated with thehase jumps, a sum of five Lorentzian lines plus a slowlyarying polynomial of the second order representing theum of the evanescent wave and the tail of distant lines atach position inside the system was fit to the field spectraetween 15 and 16.2 GHz:

E��,x� = �n=1

5 An�x� + iBn�x�

�n�x� + i�� − �n�x��+ �

m=0

2

Cm�x��� − �0�m.

�1�

y applying an iterative double least-squares fit proce-ure independently at each position x inside the sample,

e find that the central frequencies �n�x� and linewidthn�x� are independent of x. This confirms that eachorentzian term of the sum corresponds to a quasimode.igure 3 shows the amplitude spectrum at each point in

he sample for these quasimodes. Mode 1 is broad as a re-ult of its closeness to the input of the sample. Mode 2 andode 3 are spatially extended and multipeaked. These

re the quasimodes from which the extended field distri-ution or necklace state predicted by Pendry are com-osed [28,29] and may be envisioned as a hybridization ofsolated modes [26]. Extended states with up to foureaks have been observed in our systems.

. NUMERICAL SIMULATIONSe have also performed numerical simulations based on

oth a one-dimensional transfer and a scattering matrixodel that includes waveguide dispersion and absorption.

ig. 2. (a) Spectra of the field amplitude at each point along aandom sample with spectrally overlapping peaks normalized tohe amplitude of the incident field. (b) Log of the field magnitudelong the sample for an evanescent wave (dotted curve) and aultipeak quasimode. (c) Top view of (a) in logarithmic

resentation.

W=asdadploa

npsa1iwteqiWrmsltaTtamtttTfifii

Fs

Fb

A80 J. Opt. Soc. Am. B/Vol. 24, No. 10 /October 2007 Sebbah et al.

aveguide dispersion gives n���=�n2− �� /2a�2, where a1.066 cm for the frequency range covered, in which onlysingle mode propagates in the waveguide. The expres-

ion for n��� may be applied to either the high or low in-ices of the materials making up the random sample, n1nd n2, respectively. Finite absorption coefficient is intro-uced in order to broaden mode linewidths so that it isractical to study the modes with moderate spectral reso-ution. Comparison with measurements of the frequenciesf the first mode at the two band edges of the bandgap inperiodic structure of binary elements gives n =1.67 and

ig. 3. Quasimodes obtained after decomposition of field distri-ution of Fig. 2.

1

2=1.08 for the refractive indices of the first and secondortions of these elements, respectively. We point out thatcattering is higher in the waveguide than it would be inir, because of waveguide dispersion. For instance, at7 GHz the effective index contrast inside the waveguides n1� /n2�=2.10/0.48=4.38, instead of n1 /n2=1.55, which itould be in empty space. An index of refraction of less

han unity indicates that the wavelength is longer than inmpty space at frequencies not much above the cutoff fre-uency, which is 14.07 GHz in air, 13.03 GHz in the low-ndex material, and 8.44 GHz in the high-index material.

e further compare simulation with measurements inandom structures. The ceramic binary blocks have beenodeled by equivalent low- and high-index 4-mm-thick

labs. Although the numerical sample is constructed fol-owing an actual sample configuration, variations in thehickness of the structural elements of the sample ofbout 3% are not taken into account in the simulation.he absorption rate within the sample has been adjusted

o fit the experimental data. Despite the lack of precisionnd the limited sensitivity of the detection in the experi-ent, the congruence between simulation and experimen-

al spectra is striking when the absorption rate used inhe simulation is chosen to be 8.0�107 s−1, which is closeo the experimental value of 6.7�107 s−1 (see Section 6).his attests to the low sensitivity of the electromagneticeld distribution to slight variations in the sample con-guration. When the experiment is reproduced by produc-

ng another sample with the same ordering of elements

ig. 4. Spectra of (a) experiment and (b) simulation for theame sample configuration as in Fig. 2 (larger frequency range).

bttrssb[

6Fstmlmctdw1bdretwt

tc

fsbilhmddslspcs

wtfltt−eamasitctr

Fc

Sebbah et al. Vol. 24, No. 10 /October 2007 /J. Opt. Soc. Am. B A81

ut with the blocks reshuffled, we find that the field struc-ure is not changed substantially. The similarity betweenhe experiment and the simulation has also been reportedecently in 2D microwave experiments [36]. This stabilityeems to be characteristic of random systems and demon-trates the fidelity of the wave in these systems, as haseen found in the perturbation of dynamical systems37,38].

. ROLE OF ABSORPTIONor spectrally isolated localized modes in a finite openystem, the decay rate of the wave energy is the sum ofhe leakage and absorption rates, �=�l+�a. When theode is peaked within the interior of the sample, the

eakage rate is exponentially small. The decay rate of theode is then dominated by absorption. To check this, we

ompare the total decay rate and the absorption rate ofhe localized mode centered at 17.54 GHz in Fig. 1(a). Theecay rate obtained by measuring the time evolutionithin the sample of a narrowband pulse centered at7.54 GHz is �=5.6�107 s−1. The absorption rate is giveny the ratio of the net flux into the sample, which is theifference between the incident flux and the sum of theeflected and transmitted fluxes, and the steady-statelectromagnetic energy in the sample, which is propor-ional to 0

L��x�E2dx [33]. This gives �a=6.7�107 s−1,hich equals the total decay rate � within the experimen-

al uncertainty.We find both in experiment and in numerical simula-

ions that transmission drops much faster with an in-reasing absorption for isolated narrow resonances than

ig. 5. Transmission spectra for the same configuration as in Foefficient R and (d) log�1−R� for increasing absorption.

or broad overlapping modes. Transmission and reflectionpectra for the sample realization shown in Fig. 4 haveeen computed using numerical simulations for increas-ng values of the absorption and are shown in Fig. 5. Bothinear and logarithmic plots of transmission are shown toighlight the rapid disappearance of the narrowbandodes in contrast to the relative robustness of the wide

ouble-peaked mode. Figure 6 shows the computed fieldistribution inside the system for the three values of ab-orption of Fig. 5. As absorption increases, the longer-ived modes, which are remote from the boundaries, aretrongly damped, whereas the spectrally wide double-eak mode is only weakly affected. As absorption is in-reased, the energy stored in the long-lived modes is dis-ipated inside the sample.

While most of the energy is reflected from the samplehen the incident wave is off-resonance, energy dissipa-

ion when wave is on-resonance results in dips in the re-ection spectrum, as seen in Figs. 5(c) and 5(d). Indeed,he reflection coefficient, which may be written in terms ofhe transmission T and dissipated energy A �R=1−TA�, is essentially equal to R�1−A, since T1 for mod-rate absorption and sufficiently long sample [Figs. 5(a)nd 5(b)]. Modes peaked within the sample are thereforeore readily detected in reflection than in transmission,

s long as absorption is small. For stronger absorption [aseen in Figs. 5(c) and 5(d) for �=200�106 s−1], sensitivityn reflection is lost, because the spatially integrated in-ensity of localized waves falls below that for the evanes-ent wave. The impact of resonant absorption on reflec-ion is then small compared to the broad reduction in theeflection background due to the evanescent wave [see

(a) linear and (b) semi-logarithmic representation; (c) reflection

ig. 2 in

Ficeii

wadlpi

tstcsn

cscl

7Iqlcssttowittepsitscepltbgftimda

AWaTdsYec

R

Fi

A82 J. Opt. Soc. Am. B/Vol. 24, No. 10 /October 2007 Sebbah et al.

ig. 5(d)]. In contrast to dips in reflection associated withsolated quasimodes, which eventually decrease with in-reasing absorption, dips in wide multipeaked lines arenhanced [Fig. 5(c)]. Their detectability is enhanced withncreasing absorption, because T is maintained while A isncreased.

It has been shown recently [34] that the amplitude andidth of a dip in the reflection spectrum associated withn isolated mode depends on its position inside the ran-om system, its spatial extent, i.e., the localizationength, and on the degree of dissipation. It is thereforeossible to retrieve the position of a resonance localizednside the sample from reflection measurements [34].

This experimental system we have investigated offershe possibility of varying the density of states within thepectral range of the photonic bandgap of the periodic sys-em. Only when long-range order is disrupted are defectsreated deep within the pseudogap. Thus the number oftates within the pseudogap is directly related to theumber and type of defects introduced. Surprisingly, in-

ig. 6. Magnitude of field spectra for the same configuration asn Fig. 2 for increasing absorption.

reasing the disorder can lead to an increase in the den-ity of states and hence to increased spectral overlap withoncomitant broadening of the quasimodes and reducedocalization.

. CONCLUSIONn conclusion, we have measured the spatial profile ofuasimodes within a random waveguide and have ana-yzed the way they are affected by dissipation. Twolasses of quasimodes can be identified. The first class ispectrally isolated and exponentially peaked within theample. When the localization center is near the center ofhe sample, these modes are so long lived that their life-ime is dominated by absorption even for moderate levelsf absorption. A second class of quasimodes is associatedith spectrally overlapping modes. The field distribution

s then multipeaked, with the number of peaks equalinghe number of spectrally overlapping quasimodes. Whenhe excitation frequency falls within the linewidth of sev-ral overlapping modes, several quasimodes are excited toroduce a multipeaked field distribution or necklacetate. Each of the quasimodes in the superposition form-ng the necklace state is itself multipeaked. Because ofhe rapid flow of energy in these states, these modes arehort lived and only weakly affected by absorption. Be-ause short-lived modes are spectrally broad and spatiallyxtended, their contribution to transmission is greaterer mode than that of localized modes. Further, sinceong-lived modes are more strongly suppressed by absorp-ion than short-lived quasimodes are, the relative contri-ution of quasi-extended quasimodes in transmissionrows with increasing absorption. Thus broader spectraleatures become more prominent so that the variance ofransmitted intensity falls and the degree of spatial local-zation within the sample is reduced. The role of these

ultipeaked quasimodes is therefore essential to an un-erstanding of propagation in samples in which the aver-ge level is narrower than the spacing between levels.

CKNOWLEDGMENTSe thank Valentin Freilikher for useful discussions and

cknowledge the experimental help of Jerome Klosner.his work was supported by the National Science Foun-ation under grant number DMR-0538350, by a Profes-ional Staff Congress—College of City University of Nework Award, by the Centre National de la Recherche Sci-ntifique (PICS #2531), and the Groupement de Recher-hes IMCODE.

EFERENCES1. P. W. Anderson, “Absence of diffusion in certain random

lattices,” Phys. Rev. 109, 1492–1505 (1958).2. C. Kittel, Introduction to Solid State Physics (Wiley, 2004).3. J.-P. Fouque, J. Garnier, G. Papanicolaou, and K. Solna,

Wave Propagation and Time Reversal in Randomly LayeredMedia (Stochastic Modelling and Applied Probability)(Springer, 2007).

4. S. John and M. J. Stephen, “Wave propagation andlocalization in a long-range correlated random potential,”Phys. Rev. B 28, 6358–6368 (1983).

5. S. John, “Electromagnetic absorption in a disordered

1

1

1

1

1

1

1

1

1

1

2

2

2

2

2

2

2

2

2

2

3

3

3

3

3

3

3

3

3

Sebbah et al. Vol. 24, No. 10 /October 2007 /J. Opt. Soc. Am. B A83

medium near a photon mobility edge,” Phys. Rev. Lett. 53,2169–2172 (1984).

6. P. W. Anderson, “The question of classical localization—atheory of white paint?” Philos. Mag. B 52, 505–509 (1985).

7. A. Z. Genack, “Optical transmission in disordered media,”Phys. Rev. Lett. 58, 2043–2046 (1987).

8. A. Z. Genack, N. Garcia, and W. Polkosnik, “Long rangeintensity correlation in random media,” Phys. Rev. Lett. 65,2129–2132 (1990).

9. D. S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righini,“Localization of light in a disordered medium,” Nature 390,671–673 (1997).

0. M. Stoytchev and A. Z. Genack, “Measurement of theprobability distribution of total transmission in randomwaveguides,” Phys. Rev. Lett. 79, 309–312 (1997).

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

2. A. A. Chabanov and A. Z. Genack, “Statistics of themesoscopic field,” Phys. Rev. E 72, 055602(R) (2005).

3. P. Sebbah, B. Hu, A. Z. Genack, R. Pnini, and B. Shapiro,“Spatial field correlation: the building block of mesoscopicfluctuations,” Phys. Rev. Lett. 88, 123901 (2002).

4. A. A. Chabanov, B. Hu, and A. Z. Genack, “Dynamiccorrelation in wave propagation in random media,” Phys.Rev. Lett. 93, 123901 (2004).

5. S. H. Chang, A. Taflove, A. Yamilov, A. Burin, and H. Cao,“Numerical study of light correlations in a random mediumclose to the Anderson localization threshold,” Opt. Lett. 29,917–919 (2004).

6. R. L. Weaver, “Anderson localization in the time domain:numerical studies of waves in two-dimensional disorderedmedia,” Phys. Rev. B 49, 5881–5895 (1994).

7. O. I. Lobkis and R. L. Weaver, “Complex modal statistics ina reverberant dissipative body,” J. Acoust. Soc. Am. 108,1480–1485 (2000).

8. A. A. Chabanov, A. Z. Genack, and Z.-Q. Zhang,“Breakdown of diffusion in dynamics of extended waves inmesoscopic media,” Phys. Rev. Lett. 90, 203903 (2003).

9. S. K. Cheung, X. Zhang, Z. Q. Zhang, A. A. Chabanov, andA. Z. Genack, “Impact of weak localization in the timedomain,” Phys. Rev. Lett. 92, 173902 (2004).

0. S. E. Skipetrov and B. A. van Tiggelen, “Dynamics ofweakly localized waves,” Phys. Rev. Lett. 92, 113901 (2004).

1. M. Störzer, P. Gross, C. M. Aegerter, and G. Maret,“Observation of the critical regime near Andersonlocalization of light,” Phys. Rev. Lett. 96, 063904 (2006).

2. A. D. Mirlin, “Statistics of energy levels and eigenfunctionsin disordered systems,” Phys. Rep. 326, 259–382 (2000).

3. D. J. Thouless, “Metallic resistance in thin wires,” Phys.

Rev. Lett. 39, 1167–1169 (1977).

4. M. Ya. Azbel, “Resonance tunneling and localizationspectroscopy,” Solid State Commun. 45, 527–530 (1983).

5. V. D. Freilikher and S. A. Gredeskul, “Localization of wavesin media with one-dimensional disorder,” Prog. Opt. 30,137–203 (1992).

6. N. F. Mott, “Conduction in non-crystalline systems. IV.Anderson localization in a disordered lattice,” Philos. Mag.22, 7–29 (1970).

7. I. M. Lifshits and V. Ya. Kirpichenkov, “On tunneltransparency of disordered systems,” Zh. Eksp. Teor. Fiz.77, 989–1016 (1979); [JETP Lett. 50, 499–511 (1979)].

8. J. B. Pendry, “Quasi-extended electron states in stronglydisordered systems,” J. Phys. C 20, 733–742 (1987).

9. J. B. Pendry, “Symmetry and transport of waves in one-dimensional disordered systems,” Adv. Phys. 43, 461–542(1994).

0. V. Milner and A. Z. Genack, “Photon localization laser:low-threshold lasing in a random amplifying layeredmedium via wave localization,” Phys. Rev. Lett. 94, 073901(2005).

1. J. Bertolotti, S. Gottardo, D. S. Wiersma, M. Ghulinyan,and L. Pavesi, “Optical necklace states in Andersonlocalized 1D systems,” Phys. Rev. Lett. 94, 113903 (2005).

2. J. Bertolotti, M. Galli, R. Sapienza, M. Ghulinyan, S.Gottardo, L. C. Andreani, L. Pavesi, and D. S. Wiersma,“Wave transport in random systems: multiple resonancecharacter of necklace modes and their statistical behavior,”Phys. Rev. E 74, 035602(R) (2006).

3. P. Sebbah, B. Hu, J. M. Klosner, and A. Z. Genack,“Extended quasimodes within nominally localized randomwaveguides,” Phys. Rev. Lett. 96, 183902 (2006).

4. K. Yu. Bliokh, Yu. P. Bliokh, V. Freilikher, A. Z. Genack, B.Hu, and P. Sebbah, “Localized modes in open one-dimensional dissipative random systems,” Phys. Rev. Lett.97, 243904 (2006).

5. L. I. Deych, A. Yamilov, and A. A. Lisyansky, “Localpolariton modes and resonant tunneling of electromagneticwaves through periodic Bragg multiple quantum wellstructures,” Phys. Rev. B 64, 075321 (2001).

6. D. Laurent, O. Legrand, P. Sebbah, C. Vanneste, and F.Mortessagne, “Direct observation of localized modes in anopen disordered microwave cavity,” arXiv:cond-mat/0702657.

7. R. Schäfer, T. Gorin, T. H. Seligman, and H.-J. Stöckmann,“Fidelity amplitude of the scattering matrix in microwavecavities,” New J. Phys. 7, 1–14 (2005).

8. G. S. Ng, J. Bodyfelt, and T. Kottos, “Critical fidelity at themetal-insulator transition,” Phys. Rev. Lett. 97, 256404(2006).