. INTRODUCTIONave propagation in random media has been an impor-
ant research topic throughout the years, gaining muchttention when the concept of localization appeared andtimulated a large amount of work. The idea of localiza-ion was first raised by Anderson1 for electrons in disor-ered solids that were drastically affected by quantumechanical wave interference.2,3 The quest to study and
ealize such effects in optics was natural, and indeed aonsiderable amount of research can be found on lightropagation in random media that include aspects of lo-alization. Using light, in lieu of electrons, for the study ofocalization adds new possibilities, mainly in the experi-
ental aspects, owing to the relatively easy measuringechniques and the direct access to the optical ”wave func-ion” via light-intensity measurement.
Properties of wave propagation in random media, in-luding localization, generally depend on the system di-ensionality. The theoretical analysis of a one-
imensional (1D) system is obviously easier than it is forigher dimensions, but is not at all trivial for experimen-al realizations in solid-state physics. In optics, however,he experimental situation is very different, and 1D waveropagation is simple. There have been many papers onocalization aspects with electromagnetic waves in theptical4–9 and the microwave10,11 regimes. We point outhe work by Berry and Klein4, which is very relevant tour present study. These authors showed in a simple butemarkable experiment that for a stack of N transparentlates with randomly varying thicknesses the transmit-ed intensity decays exponentially with N. The strikingeature of the random optical elements is that their over-ll transmissivity �N is given by a simple multiplication ofhe single-element transmissivity �, i.e., � =�N. This
eans that only the direct transmission counts, whereasny multiple reflections added in the direction of theransmitted light interfere destructively. We also mentionur work on two experimental realizations for localizationf light in optical kicked rotors, which resemble the quan-um kicked rotor that relates to Anderson localization. Inhe first case12,13 we demonstrated localization in the spa-ial frequency domain of free-space light beam propaga-ion through an array of thin sinusoidal phase gratings.n the second case14–16 we studied the spectrum (or side-ands) localization of light pulses that are repeatedlykicked” by a sinusoidal rf modulation along a fiber.
In this paper we present an experimental study of 1Docalization by means of light propagation in a randomragg-grating array fabricated into a single-mode fiber.e demonstrate the localization behavior, manifested in
he strong exponential decay of the light transmissionlong the fiber, that was measured directly after the fab-ication of each additional grating. This decay, which re-ults in high reflection, should not be confused with theuch-smaller fiber loss. A report on this finding was given
arlier.17 The theoretical analysis of such system is basedn the transfer-matrix formalism in which the system isepresented by a product of random matrices. Thesymptotic behavior of such a product results from a theo-em on products of random matrices by Furstenberg.18
his theorem ensures that under very general conditions,he elements of the matrix product and any norm of theatrix product grow exponentially with the same expo-ent, giving rise to the localization behavior. We refer theeader to a comprehensive analysis for a 1D disorderedystem given by Pendry.3 Besides the basic propagationffects in the random array, the study can have importantamifications on fiber-optic communication and gratings
005 Optical Society of America
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O. Shapira and B. Fischer Vol. 22, No. 12 /December 2005 /J. Opt. Soc. Am. B 2543
echnology. Examples are strong and broadband reflectorsnd fiber and random lasers.
The scattering elements in our system are the randomratings in single-mode fibers. Fibers are an ideal experi-ental medium for 1D light propagation with a matured
echnology of in-fiber grating fabrication. The gratingsere made to be almost identical, with the same and rela-
ively large reflection-wavelength bandwidth of a few na-ometers, obtained by fabricating short gratings. There-ore the interesting effects concerning the localizationccur within that bandwidth. Gratings are very effectivecattering elements such that we could observe the local-zation effect with a relatively small number of them,bout 50 gratings. The randomness of the scattering ele-ents enters by the random spacing between the gratings
see Fig. 1).The outline of the paper chapters is as follows: We first
ive in Section 2 a theoretical treatment of the waveropagation in randomly spaced gratings. We then com-are the wave-theory result for the light transmissivity tohe calculation obtained from the ray theory and also tohe wave propagation in an ordered fiber grating. Section
describes the experiment, starting with the setup andhe grating-fabrication system and then presents the ex-erimental results. We show transmission measure-ents, the spectra, and the transmissivity as a function of
he gratings number for the random fiber system. Theseurves are the central results of the paper, showing theocalization effect in the random-grating array. We thenompare the measured results with the theory and find aery good agreement. We end the paper with conclusionsnd remarks on the application sides.
. THEORETICAL ANALYSIS OF LIGHTRANSMISSIVITY IN A 1DANDOM-GRATING ARRAYe present a theoretical treatment for wave propagation
n a single-mode fiber with N randomly-spaced Braggratings and calculate the transmission in the limit of N1 to obtain the localization length. This result readily
eveals that the interference among all reflected waves isestructive for the transmission, and an intuitive expla-ation is presented. We then compare this analysis withay theory, in which light is treated as lacking phaseroperty, to show that contrary to the exact wave calcula-ion, this theory results in transmission that decays as/N. Finally, we calculate the transmission of an orderedystem and compare the decay rates in both cases.
. Disordered Grating Arraytransmission calculation through a disordered chain of
ratings can be carried out by transfer matrices
Fig. 1. Random fiber-grating array.
ethods.11 The basic idea underlying such a calculationssumes that the system can be cut into slices, where-pon each can be easily evaluated. Then, by writing theransfer matrix of the complete system as a product ofhose matrices, we can apply the Furstenberg theorem18
o obtain the asymptotic behavior of the product.The grating system is described in Fig. 1. Light with
ave number k propagates along a single-mode fiber hav-ng an array of successive randomly spaced gratings. Itan be assumed that the space widths di are drawn inde-endently from the density distribution function d��di�,nd that the gratings are identical, i.e., that they have theame lengths and refractive indices.
We describe the propagation of the light in the 1D me-ium by the transfer matrix Mi that relates the ampli-udes of the forward- and backward-propagating wavesn the right side of each optical element to those on theeft side (see Fig. 2):
�an+
an−� = Mn�an−1
+
an−1−�. �1�
or optical lossless elements that are invariant underime reversal, the scattering matrix (which relates themplitudes of the ingoing waves to those of the outgoingaves via reflections and transmissions) is unitary. Then,y denoting the amplitude reflection and transmission co-fficients from both sides, for instance from the left rn , tnnd from the right r�n and t�n, we have the relations�ntn
* +rn*t�n
* =0 and �rn�2+ �tn�2= �r�n�2+ �t�n�2=1, (the aster-sk stands for complex conjugate), and the transfer matrixs given by4,19
mn =�1
tn* −
rn*
tn*
−ri
tn
1
tn
� . �2�
he transfer matrices are unimodular �det mn=1�. In ourystem we define each element as being comprised of onerating and its successive space. The gratings are takeno be identical, and the spacing between them is respon-ible for the random part. The transfer matrix of such anlement is the product of the grating transfer matrix andhe space transfer matrix. From a coupled-wave equationf the counterpropagating waves, the transfer matrix forsingle grating is given by20
ig. 2. Incident and reflected field amplitudes that define theransfer matrix of a single grating.
wegt=d
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wsdg
TteoTura�
mtt
2544 J. Opt. Soc. Am. B/Vol. 22, No. 12 /December 2005 O. Shapira and B. Fischer
mg =�cosh�SL0� − i��
Ssinh�SL0� − i
�
Ssinh�SL0�
i�
Ssinh�SL0� cosh�SL� + i
��
Ssinh�SL0�� , �3�
Tu
here L0 is the single-grating length, � is the coupling co-fficient between the counterpropagating beams in theratings, ��=�−� /� is the wave-number deviation fromhe Bragg wavelength, � is the grating period, and S����2−��2�1/2. The transfer matrix for a space of lengthis given by
i
rtoe
w
dFmfia
ws
Ttimaita
mdi= �exp�ikdi� 0
0 exp�− ikdi�� . �4�
hen the transfer matrix for a single element is the prod-ct of the two above matrices,
mi = mgmdi= ��cosh�SL0� − i
��
Ssinh�SL0��exp�ikdi� − i
�
Ssinh�SL0�exp�− ikdi�
i�
Ssinh�SL0�exp�ikdi� �cosh�SŁ0� + i
��
Ssinh�SL0��exp�− ikdi� , �5�
nd the single-element transmission and reflection coeffi-ients are given by
ti = mi�22 = �cosh�SL0� − i��
Ssinh�SL0��−1
exp�ikdi�,
�6�
ri = − mi�11/mi�22 = � exp�ikdi�sinh�SL0�/iS cosh�SL0�
− �� sinh�SL0��. �7�
i=kdi provides the random nature of the system whene have a set of such elements. It is assumed that the
pace widths di, are drawn independently from a densityistribution function d��i�. The transfer matrix for Nratings and N spaces is:
MN = m1m2 . . . mN =�1
TN* −
RN*
TN*
−RN
TN
1
TN
� . �8�
N and RN are the amplitude transmission and the reflec-ion coefficients, respectively, for the entire system. All co-fficients as well as the transfer matrix of the completeptical array are denoted in this paper by capital letters:,R ,M, compared with t ,r ,m, for one element. For these below we also denote the intensity transmissivity andeflectivity for a single grating by �= �t�2, t,,,,=� exp��,nd �= �r�2; and for the array of N gratings: �N= �TN�2 andN= �RN�2.
We next evaluate the product of those N random uni-odular matrices of Eq. (8) to obtain the overall system
ransmission. The asymptotic behavior of MN can be ob-ained using the Furstenberg theorem18 on the product of
andom matrices, stating that under very general condi-ions, the elements of the matrix product and any normalf the matrix product grow exponentially with the samexponent:
1
Nlog�mN . . . m1u� → log�m��u
u�d���dv�� �, �9�
here
v� =� v���d��
dd��� �10�
efines the probability distribution of u, and =arg�u�.or the transfer matrix given in Eq. (2), and in fact forore a general case, Furstenberg’s conditions are satis-ed, as shown by Matsuda and Ishii.21 Then from Refs. (9)nd (10) the exponent is given by
limN→
1
Nln �N = − ln�1/��, �11�
here � is the transmission of a single grating and theystem overall transmission is
�N = exp− N ln�1/��� = �N. �12�
his simple result reveals an interesting property of theransmission through a set of randomly spaced scatterersn a 1D system by showing that it includes only from the
ultiplication of the single gratings transmission withoutll multiple reflections. Of course, reflections were takennto consideration in the above calculation, but asymp-otically the result teaches us that all multiple reflectionsre canceled.
BAIfdctmrhh=bcsdpgcttcallloglo
CIfitdofibttbfs
Fr0asymptotic behavior of the transmission spectrum.
Fr0asymptotic behavior of the transmission spectrum.
Fess
Fntt
O. Shapira and B. Fischer Vol. 22, No. 12 /December 2005 /J. Opt. Soc. Am. B 2545
. Numerical Simulation for a Disordered Gratingrray
n the previous subsection we obtained the transmissivityor a large number of disordered gratings and found it toecay exponentially with the number of gratings. Here weompare this analytical result with a numerical simula-ion of the transmission. Figures 3 and 4 depict the trans-itted intensity spectrum after 1000 and 5000 gratings,
espectively. The gratings were taken to be identical andave the following properties: centered at 1540 nm andave a coupling coefficient �=185 m−1, and a length L0.385 mm. The transmissivity for a single grating at theand center is 0.022 dB. The distances between two suc-essive gratings were chosen randomly from the interval0–1� mm. The figures also present the transmissionpectrum given by Eq. (12), with the wavelength depen-ent transmission � of a single grating having the samearameters as those given above. In both cases, a veryood agreement was obtained between the analytical cal-ulation and the numerical simulation. The smoother na-ure of the longer array is obvious, as the averaging ac-ion over many gratings is more uniformly spread. Onean view the array output spectrum as being composed ofll grating pairs making many random Fabry–Perot eta-ons. The output is the collective spectra, which graduallyose their individual Fabry–Perot characteristic as theight passes more gratings. Figure 5 shows the evolutionf the transmitted intensity at the band center after eachrating. Here, too, the good agreement between the ana-ytical calculation and the numerical simulation is wellbserved.
. Destructive Interference of the High-Order Reflectionst is possible to consider the total transmission as an in-nite sum of waves formed by multiple reflections andransmissions consisting of different optical paths andifferent phases. Figure 6 is an example of a system builtut of six randomly spaced gratings. The figure exempli-es three waves with exactly the same overall path lengthut with a different number of transmissions and reflec-ions. Nevertheless, owing to the phase difference be-ween the reflection coefficient of the forward- andackward-propagating waves, the two upper waves inter-ere constructively; however, the third wave interferes de-tructively with the upper two. The fascinating result of
ig. 6. Three waves with paths of the same length but differentumbers of reflections, resulting in constructive interference be-ween the two upper waves but destructive interference with thehird.
ig. 3. (Color online) Transmission-spectrum simulation of 1000andomly spaced gratings with a single-grating transmissivity of.022 dB at the band center. The continuous curve shows the
ig. 4. (Color online) Transmission-spectrum simulation of 5000andomly spaced gratings with a single-grating transmissivity of.022 dB at the band center. The continuous curve shows the
ig. 5. (Color online) Transmissivity at the band center afterach grating. Simulation and the asymptotic behavior (thetraight line) given by the theory for a single-grating transmis-ivity of 0.022 dB.
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EIwtrdsttbpamgmnitrw(pnwsct
Frdci
2546 J. Opt. Soc. Am. B/Vol. 22, No. 12 /December 2005 O. Shapira and B. Fischer
he localization theory is that for large arrays the overallnterference of the all multireflections in the transmittedight is fully destructive. Therefore the total transmissionomprises only the wave that passes through all elementsithout being reflected.The phase difference between the reflection coefficients
or opposite wave incidence at an optical element is gen-ral. We are familiar with the opposite sign of the reflec-ion coefficients for opposite incident waves at a boundaryf two media with different refractive indices. More gen-rally, the phase difference can be tracked in the relationentioned in Subsection 2.A that r�ntn
* +rn*t�n
* =0. Forhese optical elements �rn�= �r�n�, tn= t�n and for specifichoice of the reference planes of the waves at the twoides of the element, arg�tn�=arg�t�n�=0, and then we ob-ain rn=−r�n.
. Ray Theory for a Disordered Systeme show here the simple ray theory approach that could
ave been expected to be adequate for the disordered sys-em but in fact leads to wrong results. The developmentollows the work by Berry and Klein4 given here to clarifyhe basic wave nature responsible for the localization ef-ect. Ray-theory approach is based on the assumptionhat the waves in a disordered system are incoherent andherefore can be represented as intensities rather thanmplitudes. The appropriate matrix formalism can be ob-ained for the ray theory, referring to incident and re-ected intensities. When � and � are the one-element in-ensity reflectivity and transmissivity, where for losslesscatterers �+�=1, the one-element transfer matrix is
m =�� −�2
�
�
�
−�
�
1
�� , �13�
nd for N successive elements
mN =�� −�2
�
�
�
−�
�
1
��
N
= �I +�
��− 1 1
− 1 1��N
= I + N�
��− 1 1
− 1 1� .
�14�
is the unit matrix, and the last equality is based on−1 1−1 1
�N=0 for N�2. Therefore the ray-theory transmissiv-ty for N random gratings is
�N = m22−1 =
�
� + N�1 − ��. �15�
t is a linear decay, or Ohmic like behavior for �1/TN�N (for large N), which is fundamentally different from
he exponential dependence results from localizationheory. Figure 7 graphically shows the transmissivity inhe two approaches. The different result of the ray theoryhows the distinction in regarding waves as incoherentnd averaging over random phases. This difference is ex-mplified in Section 2 C; although propagating in a ran-om medium, different light-wave paths of the same
engths “magically” give precise destructive interferences.hus the assumption that the scattered waves have nohase correlation, and therefore that they can be re-arded as incoherent, is false. Furthermore, exact waveveraging shows that all transmitted waves (except forhe one passing without any reflection) interfere destruc-ively, leading to the exponential decay of the transmis-ivity. It is also noteworthy that the ray theory does give aorrect result when the wave interference is not domi-ant. This can occur when the reflection is very small andhe system is small enough, so even small reflectionsould not accumulate.
. Comparison with Ordered Gratingst is interesting to compare the random-grating systemith ordered gratings. We have a powerful method for ob-
aining effectively long gratings, with easy creation ofandom structures that might have been regarded as aetriment, but this turns out to be an advantage with theupport of a localization effect that provides a strongransmission decay and high reflection. Then, not only arehe random-grating arrays easy to create, but distur-ances do not have much of a deteriorating effect on theirerformance. On the other hand, long ordered gratingsre hard to create and can have a detrimental environ-ental effect. The situation is even worse for ordered-
rating arrays, which are almost impossible to imple-ent, even for a small number of gratings, since theyeed precise interferometric spacing between the grat-
ngs. An additional advantage of the random arrays ishat they can easily provide very large bandwidths for theeflection, since they depend on the single-grating band-idth. The single grating can be made very short
10–100 �m long) and still be an effective scatterer, thusroviding very large wavelength bandwidths of tens of na-ometers. One can then argue that for even larger band-idths, it is possible to form point scatterers rather then
hort gratings. Such random point-scatterer arrays are, ofourse, interesting and worthy of implementation, thoughhey are not easy to make. The array needs the fabrica-
ig. 7. (Color online) Comparison between wave and ray theo-ies for the transmissivity in random gratings; the upper curveepicts the Ohmic behavior of the ray theory, whereas the lowerurve depicts the exact wave averaging (both calculated for grat-ng transmissivity of 0.022 dB).
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Tgbsia
O. Shapira and B. Fischer Vol. 22, No. 12 /December 2005 /J. Opt. Soc. Am. B 2547
ion of many random scatterers along the fiber. To reachhe asymptotic noiseless localization regime, we need aeasonable number of scatterers. This means that thetrength of one scatterer ought to be weak to allow theight to penetrate and acquire many multireflections,hich averages to zero in the transmitted light. On thether hand, the fabrication of many scatterers is moreomplicated, and therefore we need a reasonable scatter-ng strength for each element in order for us to be able tobserve the effect. There are also some disadvantages ofhe random array. The transmission and the reflectionnd their spectra are not smooth and uniform, as the av-raging in a random structure (which is limited in length)s not optimal. We need a rather long array to reach themoother asymptotic behavior. The last point in therdered–disordered gratings comparison is that gratingsre used mostly for precise filtering purposes and not onlyor reflection, and thus the ordered element is needed, un-ess we look for special filtering uses, or with special fin-erprints.
For a comparison between the disordered array and therdered structures, like long uniform gratings, we assumehem all to be of the same overall length L=NL0. We canlso extend the comparison to ordered arrays of N grat-ngs, each of length L0, with exactly the same spacing be-ween them. We note that the latter structure is almostmpossible to implement, even for low number of gratings,ecause of the subwavelength-spacing requirement.
hd
3MWtrltt
We use Eq. (3) for the transmissivity of long uniformrating, replacing L0 with L, and then compare the out-ome to the random-grating result. For the ordered-rating array we can again start with the transfer-matrixormalism, using Eqs. (5) and (8) and setting equal spac-ng, di=d. This structure includes the ordered-gratingase for d=0. Then the transfer matrix for a set of N grat-ngs is
MN = mN. �16�
N is given for a unimodular matrix by
MN = �m11UN−1�a� − UN−2�a� m12UN−1�a�
m21UN−1�a� m22UN−1�a� − UN−2�a�� ,
�17�
here UN are the Chebyshev polynomials of the secondype,
UN�a� =sin�N + 1�cos−1 a�
�1 − a2, �18�
nd= 1
2 �m11+m22�=cosh SL cos kd+�� /S sinh SL sin kd.hen
.
N
=1
�1 − a2��cosh�SL� − i��
Ssinh SL�exp�ikd� sin�Ncos−1a� − sin�N − 1�cos−1 a�− i
or large N, in all cases (the disordered array, the orderedrray with optimal spacing, and the single grating), theong grating regime �N�1� for the transmitted intensityt the central wavelength ���=0� is given by
�N � exp�− NL0S� = �N. �21�
he exponent in the ordered case depends strongly on therating spacing selection. The strongest reflection is giveny the single long grating, which has a built-in equal-pacing zero phase; regardless, such arrays are difficult tomplement. The great surprise is the result for disorderedrrays that gives exponential dependence and accordingly
igh reflectivity, although it has its own drawbacks, as weescribe in Section 3.
. EXPERIMENT: SYSTEM,EASUREMENTS, AND RESULTSe first describe the setup for the grating fabrication and
hen the system for measuring the transmission of theandomly spaced gratings. The first setup includes a UVaser that is used for grating fabrication, a set of lenseshat are used to shape the laser beam, a mask to createhe grating pattern, a moving table and controller for fi-
bitmmtdmisl
ATniitmitnfmeabmpaccg−
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oacsFsgmtwsa
1
FsAsb
2548 J. Opt. Soc. Am. B/Vol. 22, No. 12 /December 2005 O. Shapira and B. Fischer
er placement, and a single-mode fiber. The second setups comprised of an erbium-doped fiber amplifier (EDFA)hat is used as a source for the transmission measure-ent and an optical spectrum analyzer for conducting theeasurements. We then present the measured results
hat include the transmission spectral behavior and theependence on the number of gratings. We also showeasurements for deducing the one grating transmissiv-
ty, needed for the theory verification. At the end of thisection we discuss the results with a comparison with theocalization theory.
. Grating Fabrication and Measurement Setuphe method used to fabricate the gratings is based on aear-contact exposure through a phase mask.22 The setup
s illustrated in Fig. 8. The UV laser source is an argon–on laser whose frequency is doubled by a nonlinear crys-al to give a wavelength of 248 nm and power of approxi-ately 200 mW. The beam is broadened by a concave lens
n order to produce a spot size large enough to illuminatehe slit on the mask as uniformly as possible without sig-ificantly reducing the beam intensity. Then the beam isocused in the fiber axis by a cylindrical lens in order toaximize the intensity, exposing the fiber. The beam
merging from the cylindrical lens is normally incident onslit attached to the mask and transfers only 1 mm of theeam (the slit is adjusted to pass the interval with theaximum intensity). The slit and the mask are placed ap-
roximately at the focal point of the cylindrical lens tochieve the maximum intensity on the fiber that is adja-ent to the mask. The exposing beam is then normally in-ident on the phase mask and diffracted entirely. Therating is formed by the interference between the +1 and1 diffracted orders of the phase mask. A single-mode fi-
ig. 8. (Color online) Experimental setup. The UV laser beam, apot size for the grating writing. The beam is then focused at the fiphase-grating diffraction gives two first-order waves, causing a
urements in the fiber were done in situ after each additional gruildup with the grating number.
er was put inside a high-pressure hydrogen tank for aew days to make it sensitive to a photoinduced refractivendex change.
. Light Transmission Measurementsor reproducibility of the grating spectrum, it was neces-ary to ensure that the exposure time would be similar forll gratings. Therefore the laser power was adjusted tochieve a relatively slow grating formation time in ordero render good accuracy.
The following procedure was repeated for each grating:The illumination was started.After 1 min, the illumination was stopped and the
pectrum was recorded.The automated stage controller was adjusted to move
he fiber holding stage a random distance that was largerhan the grating.
The above procedure was repeated 55 times.It is noteworthy to mention that the maximum number
f 55 gratings was due to limitations of the spectrum-nalyzer accuracy, as the transmitted light intensity de-reased with the grating number. The measurement re-ults of the spectrum as well as the intensity are shown inigs. 9–12. The experiment was carried out twice for alightly different exposure time and a slightly differentrating length (by modifying the distance between theask and the fiber). The spectra in Figs. 9 and 11 show
he detailed wavelength dependence in the grating band-idth and the gradual loss of the individual Fabry–Perot
pectrum along the propagation while acquiring the manynd random Fabry–Perot characteristics.The transmitted power measurements shown in Figs.
1 and 12 were done at the center of the grating spectrum
to a pinhole, is broadened by a lens in order to achieve a largeres by a cylindrical lens to obtain maximum intensity on the fiber.idal interfering pattern on the fiber. The light transmission mea-fabrication. This procedure allowed us to follow the localization
lignedber axsinusoating
baEftbthvmhfp�afis
CSt=cpfpm
pg
cmogmpgcmpsisrmmgomaigfn
trum m
O. Shapira and B. Fischer Vol. 22, No. 12 /December 2005 /J. Opt. Soc. Am. B 2549
and, where the transmission is minimal, and were aver-ged over 0.5 nm in Experiment No. 1 and over 0.3 nm inxperiment No. 2. The reason for this averaging stems
rom the need to overcome the random fluctuations andhe temperature and stress changes experienced by the fi-er during the experiment, causing the measured spec-rum to drift and vary. The averaging interval, on the oneand, was chosen to be large enough to suppress those en-ironmental changes, but on the other hand, as the trans-issivity magnitude varies with wavelength the intervalad to be limited to a length at which the maximum dif-
erence in transmission could be tolerated. In more ex-licit terms, if the grating minimum transmissivity is0.4 dB, then an accuracy of 1 magnitude less is toler-
ble. Furthermore, the noise caused by the optical ampli-er and the spectrum-analyzer accuracy results in a mea-ured accuracy no better then 0.02 dB.
. Experiment versus Theoryection 2 provided the theoretical asymptotic behavior ofhe transmission with the exponential decay given by �Nexp−N ln�1/���, where � is the intensity transmissionoefficient of a single grating. Therefore, in order to com-are the theoretical results with the findings obtainedrom the experiments, it is necessary to first find � at theoint in the spectrum where the transmission measure-ents were taken. This single grating value is to be com-
ared with the experimental decay rate of the completerating-array transmissivity.
Measurement of the transmissivity of a single gratingan be performed by one of two methods. The first andost straightforward method is to take the result
btained from measuring the spectrum of the firstrating and normalizing it according to the spectrumeasured for a grating-free fiber (which is basically the
ower spectrum of the EDFA). This method has areat disadvantage in that it features a wide inaccuracyaused by a power drift that may occur between the twoeasurements as a consequence of fiber bending,
olarization dependent loss, and an instability of theource power. Whereas the drift caused by these effectss tolerable for most of the experiment when it isuppressed to values of �0.1 dB by maintaining aelatively constant temperature, a vibration-free environ-ent, and a minimized fiber movement during the experi-ent this is not the case when measuring the first
rating, as the minimum transmission is in mere tenthsf decibels. The second method is to measure the trans-ission of two gratings, which is a form of Fabry–Perot,
nd extracting from it the transmissivity of a single grat-ng. Although this method is less straightforward, it has areat advantage over the previous one in that all the ef-ects causing the inaccuracy of the previous method areegligible. This is because the measurement is performed
easured after (a) 3, (b) 10, (c) 25, and (d) 50 gratings.
amstmFttd
wttfiow+t
noipevtw
tfiTm−2sius−
DTgt
2550 J. Opt. Soc. Am. B/Vol. 22, No. 12 /December 2005 O. Shapira and B. Fischer
t a specific given time without necessitating measure-ent of the reference level. To achieve good accuracy, the
econd method was selected. It is now possible to derivehe transmissivity of a single grating from the spectrumeasurement of two successive gratings. Although theabry–Perot properties are simple and known, we derivehem here by using the transfer-matrixmethod. Theransfer matrix of two successive gratings with a spacingbetween them is
M2 =�1
t* −r*
t*
−r
t
1
t��exp�ikd� 0
0 exp�ikd���1
t* −r*
t*
−r
t
1
t� ,
�22�
here t1 and r1 are the single-grating amplitude-ransmission-reflection coefficients, respectively. The dis-ance between the two gratings is d, and the propagatedeld wave number is k. Then, the intensity transmissivityf the grating pair can be calculated from 1/T2= M2�22,hich together with �= �t�2, �= �r�2, �= �t�2, �2= �T2�2 and ��=1, =kd, gives the simple Fabry–Perot intensity
is the refractive index and � is the wavelength. Obvi-usly, it is difficult to extract � from the last expression, ast requires knowledge of the transmission coefficienthase and the exact distance between the gratings. How-ver, we can easily find it from the minimum of �, as itsalue does not depend on the phase �min=�2 / �2−��2. Thenhe transmission coefficient of a single grating can beritten as �=2�min
1/2 / �1+�min1/2 �.
The grating pair measurements of the normalizedransmission spectra are shown in Figs. 13 and 14 for therst two gratings from Experiments 1 and 2, respectively.he value �min is the square root of the minimum trans-issivity at the center of the grating spectrum; it is1.63 dB for Experiment 1 and −1.41 dB for Experiment. For Experiment 1, according to Eq. (22), the transmis-ivity of a single grating is −0.43 dB; for Experiment 2, its −0.35 dB. According to the localization theory these val-es are to be compared with the overall transmissionlopes that are a−0.405 dB/grating in Experiment 1 and a0.326 dB/grating for Experiment 2.
. Losshroughout the study, it was assumed that loss is negli-ible. This assumption should be confirmed experimen-ally, since although the loss of the fiber itself for such
measured after (a) 3, (b) 10, (c) 25, and (d) 50 gratings.
trum
scfc
csmswtEtfTsarbtl
Ftfi
Ftfi
FmFtrp
Fsar
Fmltttta
O. Shapira and B. Fischer Vol. 22, No. 12 /December 2005 /J. Opt. Soc. Am. B 2551
hort distances is negligible, the process of grating fabri-ation might introduce some additional loss in the gratingormation with the exposure to the UV radiation thathanges the fiber uniformity and the absorption coeffi-
ig. 13. (Color online) Experiment 1. The normalized power aseasured after two gratings is similar to the spectrum of aabry–Perot resonator but with the envelope of the grating spec-rum. The transmissivity of a single grating is obtained from theatio between the maximum and the minimum transmissionower, which in this experiment resulted in −0.43 dB.
ient. To confirm that the exponential decay did indeed re-ult from localization and not from loss, the fiber loss waseasured after the grating fabrication. The experiment
etup is given in Fig. 15, where one end of the tested fiberas connected to a spectrum analyzer through a 3 dB at-
enuator and a 50% coupler, and the other end to theDFA through a 50% coupler. The intensity measured at
he coupler output tap was one-quarter of the reflectionrom the tested fiber plus one-quarter of the transmission.o evaluate the loss, a measurement was made of thepectrum at the coupler-output tap using a regular fibernd then repeated using the tested fiber. Assuming that aegular fiber can be used as a reference for a loss-free fi-er, the maximum measured difference between the spec-rum of the regular fiber and that of the tested fiber wasess then 0.5 dB for all wavelengths. As the total trans-
Table 1. Summary of Experiment Results
ExperimentNo.
Transmissivityof a Single
Grating(dB)
TotalTransmissivity
Slope(dB/grating)
1 −0.43 −0.4052 −0.35 −0.326
ig. 14. (Color online) Experiment 2. The transmissivity of aingle grating is obtained from the ratio between the maximumnd the minimum transmission power, which in this experimentesulted in −0.35 dB.
ig. 15. (Color online) Experiment setup for fiber loss measure-ent. Light reflected from the tested fiber is rerouted back to the
eft coupler. Half of the reflection is then present at the left tap ofhe right coupler and is coupled to half of the transmission fromhe tested fiber. The right tap of the right coupler is connected tohe spectrum analyzer and measures one-quarter of the reflec-ion plus the transmission. The EDFA serves as the light source,nd OSA is the optical spectrum analyzer.
mt
ETe
sctsestmatpE
4WwstltTtgpwtdhds
gttfibirHersbbpdt
ATt
rr
R
1
1
1
1
1
1
1
1
1
1
2
2
2
2552 J. Opt. Soc. Am. B/Vol. 22, No. 12 /December 2005 O. Shapira and B. Fischer
ission measured was �−20 dBm, it may be deducedhat the fabricated-fiber loss is negligible.
. Experimental Conclusionhe overall results show a good agreement between thexperimental results and the theory; see Table 1.
The slight difference between the transmission of aingle grating and the total transmission slope has fourauses. First, the gratings are not exactly equivalent, andherefore the transmission of a single grating (as mea-ured from the first two gratings) can deviate from the av-rage transmission of the gratings. Second, the transmis-ion slope is a consequence of a calculation that evaluateshe asymptotic behavior of the transmission. The trans-issivity fluctuates in a finite system, deviating from the
symptotic calculation. Third, measurement uncertain-ies exist, such as temperature, fiber stress, andolarization-dependent loss. Finally, the stability of theDFA plays a role.
. SUMMARYe have presented a realization of Anderson localizationith light propagating in one-dimensional randomly
paced gratings in a single mode fiber. We described theheoretical analysis and experimentally demonstrated theocalization effect. We measured the transmissivity withhe exponential decay along the disordered fiber gratings.he magnitude of the decay rate, i.e., the inverse localiza-ion length, is equal to the log of the inverse single-rating transmissivity. The total transmission is com-rised of only the wave that passes through all gratingsithout experiencing any reflections. All other transmit-
ed waves interfere destructively for the transmission. Weiscussed a ray approach that treats the waves as inco-erent owing to an averaging over random phases in theisordered array, but the approach fails to adequately de-cribe the special wave-interference nature.
We conclude with the application sides of the random-rating array. We refer to the reflection side, complemen-ary to the transmission, that can become very large withhe strong localization effect. Ordered gratings with theirltering and reflection capabilities are widely used in fi-er optics. However, it is very difficult to fabricate grat-ngs longer than a few centimeters. Random-grating ar-ays are by far easier to make, with much larger lengths.ere, the random nature becomes an advantage. How-
ver, even most important feature is that the random ar-ay can easily provide very large bandwidth reflection,ince it depends on the single-grating bandwidth that cane made very short, thus providing very large wavelengthandwidths of tens of nanometers. Another interestingossibility is the use of the array for fiber lasers. The ran-om grating can provide the pseudocavity for feedback,hus providing a kind of 1D random laser.
CKNOWLEDGMENTShis work was supported by the Israeli Science Founda-ion of the Israeli Academy of Sciences. O. Shapira can be
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