Equivalence classes of Fibonacci lattices and their similarity properties

1.3 MB·

by  N. Lo Gullo

EquivalenceclassesofFibonaccilatticesandtheirsimilarityproperties
N. Lo Gullo,1,2 L. Vittadello,1 M. Bazzan,1,2 and L. Dell’Anna1,2
1Dipartimento di Fisica e Astronomia, Universita` degli Studi di Padova, Padova, Italy
2CNISM, sezione di Padova
(Dated:July28,2016)
Weinvestigate,theoreticallyandexperimentally,thepropertiesofFibonaccilatticeswitharbitraryspacings.
Differently from periodic structures, the reciprocal lattice and the dynamical properties of Fibonacci lattices
dependstronglyonthelenghtsoftheirlatticeparameters,evenifthesequenceoflongandshortsegment,the
Fibonaccistring,isthesame. Inthisworkweshowthat,byexploitingaself-similaritypropertyofFibonacci
stringsunderasuitablecompositionrule,itispossibletodefineequivalenceclassesofFibonaccilattices. We
showthatthediffractionpatternsgeneratedbyFibonaccilatticesbelongingtothesameequivalenceclasscan
6 berescaledtoacommonpatternofstrongdiffractionpeaksthusgivingtothisclassificationaprecisemeaning.
1
Furthermoreweshowthat,throughthegaplabelingtheorem,gapsintheenergyspectraofFibonaccicrystals
0
belongingtothesameclasscanbelabeledbythesamemomenta(uptoaproperrescaling)andthatthelarger
2
gapscorrespondtothestrongpeaksofthediffractionspectra. Thisobservationmakesthedefinitionofequiv-

l alence classes meaningful also for the spectral, and therefore dynamical and thermodynamical properties of
u
quasicrystals. Ourresultsapplytothemoregeneralclassofquasiperiodiclatticesforwhichsimilarityundera
J
suitabledeflationruleisinorder.
6
2 PACSnumbers:

]
r
e I. INTRODUCTION II. GENERALIZEDFIBONACCILATTICES
h
t
o Inonedimension(1D)theparadigmofaquasicrystalisthe
t. Sincethefirstexperimentalproofoftheexistenceofsolids Fibonaccilattice(FL).TheFLisa1Dlatticewhoseadjacent
a pointshavedistancesbelongingtotheset{L,S},standingfor
lacking of translational invariance, but exhibiting a discrete
m LongandShortrespectively,whicharearrangedaccordingto
Braggdiffractionspectrum[1],thestudyofquasicrystalsat-
- tractedquitealotofattention.Theimpactofthisdiscoveryon agivensequence. Suchalatticecanbeconstructedbymeans
d
of the cut and project technique [3, 4, 21] thus obtaining for
n thescientificcommunitywassuchthatin1992theformerdef-
thecoordinatesofpointsontherealline[2](inunitsofS):
o initionofcrystalhadtobemodifiedinordertoincludethose
c structures whose diffraction pattern witnesses long range or-
1 (cid:106)n(cid:107)
[ der yet lacking translational invariance [2–4]. More gener- xη =n−1+ (1)
n η τ
2 ally, the study of quasiperiodic geometries has been recently
v thesubjectofdifferentfieldsalldevotedtothepropagationof where n is a natural positive number,(cid:98)x(cid:99) is the integer part
9 wavesthroughquasiperiodicpotentials. Thespectralproper- of x and η = S/(L − S). The most common instance
9 tiesofquasicrystalshavebeenrecentlyusedtoengineertopo- (cid:0)√ (cid:1)
found in literature is obtained for η = τ = 5+1 /2,
4
logicalpumpinginopticalwaveguides[5–7]andinultracold
7 the golden ratio. In this case the canonical FL (CFL) is ob-
gases[8,9].Engeneeringofquasiperiodicstructureshavealso
0 tained, such that the lengths are (up to a simple rescaling):
been employed in optical dielectric multilayers for resonant
. L = 1 + 1/τ = τ, and S = 1. Nevertheless it is possi-
1 transmission [10], solar energy harvesting [11], plasmonics
ble to construct Fibonacci lattices with η (cid:54)= τ (see App. A).
0
6 [12,13]andnonlinearoptics[14,15]. The distances ∆n = xηn+1 −xηn are either L = 1+1/η or
1 S =1andtheyarearrangedaccordingtotheFibonaccistring
: Dynamical and transport phenomena in this kind of struc- (FS)LSLLSLSLLSLLSLSLLSLSL···.Thelatterisany
v
tures are also radically different compared to periodic me- wordmadeoftwoletters,LandS,obtainedbymeansofthe
i
X dia[16–20]. Forusualperiodicarrangements,dynamicaland substitutionruleS → LandL → LS startingfromtheletter
r thermodynamicalpropertiesaredirectlyrelated,viatheBloch L. WenoticethataFSitselfisindependentontheparameter
a
theorem, to the geometry of the system. Quasiperiodic ge- ηanditonlydependsonthefactor1/τ.
ometries,instead,lacksoftranslationalinvariancesothatadi- Conversely,givenaninfiniteFS,acompositionrule(LS →
rectrelationbetweentheirstructureandtheirdynamicalprop- L(cid:48) and L → S(cid:48)) can be defined such that the old and the
erties is not generally known. It would be therefore very in- newstringsarethesameduetothepeculiarpropertiesofthe
terestingtofindasortofclassificationenablingonetogroup Fibonacci strings, as shown in Fig. 1. For the special case
togetherdifferentaperiodicsystemsonthebasisofsomesim- η = τk, with k a non-vanishing integer i.e. for the canoni-
ilaritybetweentheirgeometricarrangements.Inthispaperwe cal FL, this leads to a peculiar property: the new FL can be
attempttodefinesuchaclassification,showingthatquasiperi- rescaledtotheoriginalone. Thiscaseisthemostcommonly
odicstructureswhosegeometryisrelatedbyasuitablemath- encountered in literature, accompanied by the statement that
ematicaltransformationsharethemaincharacteristicsoftheir the CFL is self-similar. It should be stressed however that
reciprocallatticeandoftheirpseudo-bandstructure. this is not true for the general case η (cid:54)= τk. In this case, a
2
for irrational η we can resort to its continued fraction repre-
L S L L S L S L L S L sentationinordertosetthewantedprecisiontotheabovecon-
dition.
L' S' L' L' S' L' S' LetusnowconsidertwoFLsbelongingtothesameequiv-
alenceclassxηn0 andxηn1,withη1 =1+1/η0. Bydefininghn
Figure1: Thecompositionrules(LS → L(cid:48), L → S(cid:48))onasemi- (kn)andh(cid:48)n(kn(cid:48))asthenumeratoranddenominatorofthen-
infinite Fibonacci lattice reproduces another Fibonacci lattice with thrationalapproximantsof1+1/η (1+1/η ),thefollowing
0 1
differentlatticeparameters. relationshold: k = h +h(cid:48) andk(cid:48) = h . Thepositionof
n n n n n
thebrightestpeaksoftheFLxη1 arethenrelatedtothoseof
n
theFLxη0 by:
n
non-canonical Fibonacci lattice and the one obtained by ap-
plyingthecompositionrulearecharacterizedbytwodifferent Q1(kn,kn(cid:48))=η1Q0(hn,h(cid:48)n) (4)
lenghtratiosη andη becauseL(cid:48)/L (cid:54)= S(cid:48)/S. Thereforethe
1 2 In other word, althought the two Fibonacci lattices xη0 and
newlatticecannotbetransformedintotheoldonebyasimple n
rescaling. C(xηn0) = η1xηn1 cannot be rescaled one over the other (for
the general case η (cid:54)= τ), their brightest peak pattern can, as
We call C the operator corresponding to the effect of the
aconsequenceofthefactthattheyarerelatedbythecompo-
compositionruleontheFLxη. Itisnotdifficulttoshowthat
n sition rule. Also the intensities of the brightest peaks can be
C(xηn) = η1xηn1, whereη1 = 1+1/η: thecompositionrule relatedasI(η q,η )≈I(q,η )+ 1(1−I(q,η ))(forqsuch
maps a FL xη into another FL, characterized by a new ratio 1 1 0 τ 0
n thatI(q,η ) > 0.5)showingthatthepeaksofthescaledlat-
η and rescaled by η (see Fig. 1). If η = τ then η = τ 0
1 1 1 tice are even brighter than those of the original lattice. This
(recall that τ2 −τ −1 = 0) and therefore the CFL is self-
drives to the important conclusion that FL belonging to the
similar. Ifthecompositionruleisappliedk times, theinitial
sameequivalenceclasshavediffractionspectracharacterized
FLismappedinto
bythesamepatternofbrightestpeaks,andare,inthissense,
(cid:32) k (cid:33) (cid:18) (cid:19) similar.
C(k)(xηn)= (cid:89)ηi xηnk = Fk+1+ Fηk xηnk (2)
i=1
III. SIMILARITYOFDIFFRACTIONPATTERNS
where η = 1+1/η (η = η) and F are the Fibonacci
i i−1 0 k
numbers. Withthehelpofthisconceptwedefineequivalence Toquantifythedegreeofsimilaritybetweenthetwospec-
classes for FLs by means of the following equivalence rela- tra, we use the Kullback-Leibler divergence (KLD), a quan-
tion: tity useful to compare two distributions (normalized to unity
over a common support). Let us consider the diffraction
Definition - Two Fibonacci lattices xηna and xηnb are equiva- spectra I(q,ηα) and I(q,ηβ) of two arbitrary FL’s charac-
lent(xηna ∼ xηnb)iftheyarelinked,uptoaproperrescaling, terized by ηα (cid:54)= ηβ. We define the normalized spectrum:
bymeansofthecompositionruleC. Thelatticewiththemini- P(νq,η) = I(νq,η)/(cid:82)∞dkI(νk,η) where we introduced
0
mumη0suchthat1/(η0−1)isfiniteandpositiveiscalledthe ascalingparameterν.
generatoroftheequivalenceclasswhichisdenotedby[η0]. TheKLDisdefinedas:
Asimplewayoflabelingtheelementsofagivenequivalence D(η ,η ,ν)=(cid:90) ∞dkP(k,η )log(cid:18) P(k,ηα) (cid:19). (5)
classisbymeansofthecontinuedfractionrepresentationfor α β α P (νk,η )
0 β
the{η }.
i
Bydefinitiononehasthatthemoresimilarthetwodiffraction
Because of this, for quasiperiodic structures it is of lim-
spectra, the smaller the value of the KLD. We will use it to
ited practical utility to talk about the support of the diffrac-
measureif,forgivenη andη ,thereexistascalingparameter
tion pattern. It is more meaningful to describe the diffrac- α β
ν forwhichthetwospectralooksimilar.
tionspectrum(andthereciprocallattice)intermsofthepeaks
In Fig. 2 a) and b) we plot 1/D(ηα,η,ν) comparing two
whicharesignificantlyclosetoone,whichwillbereferredto 0
generators corresponding to ηa = 6/11 and ηb = 1/6 with
as brightest peaks. By means of the cut and project method 0 0
FLs obtained from them by applying the composition rule
outlined in App. A it is possible to show (see App. B)
that the intensity I(q,η) at points q = Q(h,h(cid:48)) is given C(nα) respectively na = 1 and nb = 3 times. The inten-
by sinc2(Q (h,h(cid:48))∆), where Q (h,h(cid:48)) = 2πd−1(h(1 + sities I(q,η) are evaluated by means of eq.B15 for lattices
⊥ ⊥
with N = 300 points. The maxima (minima of D) in the
1/η)−1−h(cid:48))and∆=τ(η/(η+1))/2. Thereforethebright-
two figures correspond to (η,ν) = (ηa,ηa) and (η,ν) =
estpeaksarefoundforpairs(h,h(cid:48))suchthatQ (h,h(cid:48)) ≈ 0 1 1
⊥ (ηb,ηbηbηb) respectively, in agreement with eq. (4). This
andthusfor 1 1 2 3
shows that two FL’s produce a similar diffraction pattern if
h 1 andonlyiftheycanberelatedviaEq. (2)andthereforeonly
=1+ . (3)
h(cid:48) η iftheybelongtothesameequivalenceclass.
In order to test our results on a real case, we performed a
Sincehandh(cid:48) areintegers,theaboveconditioncanbesatis- diffraction experiment on two quasiperiodic diffraction grat-
fiedexactlyonlyifη isarationalnumber. Ontheotherhand ings prepared using a photorefractive direct laser writing
3
D
Figure 3: (Color online). a) Comparison between the theoretical
diffractionpatternfortheFLxη0a (ηa = 0.5454, solidbluecurve)
andtheexperimentaloneproducnedby0aFLC(cid:16)xη0a(cid:17)(solidredcurve,
n
N = 300 lines and L = 23 µm and S = 17 µm, ηa = 2.83).
1
ThetheoreticalspectrumhasbeenrescaledinaccordancewithEq.
(4) and corrected for the structure factor contribution. b) The in-
verse of KL divergence (D−1(ηa,η,ν)) between the (normalized
Figure 2: (Color online) Inverse of KL divergence 1/D(ηa,η,ν) 1
0 over the interval) experimental diffraction pattern and the theoreti-
comparing the diffraction spectra of the generator of a given class
caldiffractionpatternsfordifferentgeneratorsanddifferentscaling.
xηn0 with another Fibonacci lattice, for two choices of η0. a)η0a = Here∆ηa =η−ηaand∆ν =ν−1/ηa. Themaximum(minima
6/11;b)ηb =1/6.Here∆ηa =η−ηa,∆νa =ν−ηaand∆ηb = 0 0
0 1 1 ofD)isat(η,ν)=(ηa,1/ηa).
η−ηb,∆νb =ν−ηbηbηb.Themaxima(minimaofD(η ,η ,ν)) 0 0
1 1 2 3 α β
areobtainedat∆ηa,b = 0and∆νa,b = 0,indicatingthatthetwo
spectra with the higest degree of similarity corresponds to lattices
ηaxη1a =C(cid:16)xη0a(cid:17)andηbηbηbxη3b =C(3)(xη0b)respectively,i.e.the arangeofη0 andscalingfactorν. InFig. 3b)weshowitex-
1 n n 1 2 3 n n plicitlyforthegratingwithηaanditisclearthatthemaximum
1
firstandthethirdelementoftherespectiveequivalenceclasses. c) ofD−1 (minimumofD)isfoundatη = ηα andν = η−1.
DirectcomparisonofthetwodiffractionspectrafortwoFLsxηn0aand Similarresultsareobtainedforthegrat0ingηb0. 1
ηaxη1a.ThemostprominentpeaksofthediffractionpatternI(q,ηa)
1 n 0
correspondwiththoseofthe(rescaled)spectrumI(ηaq,ηa).
1 1
IV. ENERGYSPECTRACOMPARISON
We have therefore shown that all the FL belonging to the
(DLW) technique [25, 26]. We used three gratings made up
sameequivalenceclasshavediffractionspectrathat,although
ofN =300linesallwritteninthesamesubstrate: (a)aperi-
not equals, are characterized by a similar pattern of bright
odicgratingwithspacingL=23µm;twoFibonaccigratings
with (b) L = 23µm and S = 17µm (ηa = 17/6) and (c) peaks. Thisfindingisofcrucialimportancenotonlyinscat-
L = 23µmandS = 15µm(ηb = 15/8)1respectively. Sofar tering phenomena but also in transport ones. In fact in a re-
3 cent work [23] a method to unambiguously link the gaps in
weconsideredpointlattices,butrealstructuresareconstitued
the integrated density of states to the brightest peaks in the
bysomephysicalentity(basis)arrangedonthepointsofour
diffraction pattern of the underlying potential has been pro-
quasi-periodic Fibonacci lattice. (For a detailed description
posed.Thisismoregeneralhasithasbeenshowninaseminal
of the experimental set up see App. E). For these cases, the
paperbyLuck[22]. LetusconsiderforexampletheHamilto-
diffraction pattern is given by the sum in Eq. (B15) multi-
nianforaparticleina1Dlattice:
plied by the square modulus of a structure factor. The latter,
in general, does not posses any scaling property and there-
(cid:126)2 d2
foreitisnecessarytocorrectforitwhencomparingdifferent Hˆ = − +V(x) (6)
2 dx2
lattices. We did this experimentally by using the data of the
(cid:90)
periodicgratingtoextractaphenomenologicalexpressionfor V(x) = −V dyf(x−y)(cid:88)δ(y−x ) (7)
0 n
the structure factor as a function of q. In Fig. 3 a) we com-
n
pare the experimental data relative to the grating ηb with the
1
theoretical diffraction pattern obtained from the generator of where x are the local minima of the potential and f(x) is
n
the corresponding equivalence class, ηb = 1/6. We observe introduced to account for the detailed shape of the potential
0
that,oncethespectrahavebeenrescaledinqfollowingeq.(4) minima (V > 0). We will consider the case x = xη ac-
0 n n
andcorrectedinordertotakeintoaccountthestructurefactor cordingtothequasi-periodicsequenceofEq.1. InRef.[23]
contributiontotheintensityofthepeaks,themostprominent it has been shown that it is possible to label the energy gaps
diffraction features of the generator can be found in the ex- bymeansofthebrightestpeaksofthediffractionspectrum.In
perimentaldataatthecorrectq positions. Thedegreeofsim- particularonehastoconsiderthepseudo-momentaqatwhich
ilaritybetweenthespectrumofthegeneratorandtheexperi- the square of the Fourier transform of V(x) acquires values
mentaloneisconfirmedbytheKLdivergenceDbetweenthe greater than a given threshold. This effectively corresponds
experimentaldatapointsandthetheoreticaldiffractionspec- tochoosewhichfreestatesareeffectivelycoupledbythepo-
trum(withtheinclusionofthestructurefactor)calculatedfor tential and, therefore, where wider gaps open in the single
4
shownthatitispossibletogroupdifferentFibonaccilattices
intoequivalenceclasseswhoseelementssharethemainstruc-
tural and dynamical properties as witnessed by their diffrac-
tionspectraandtheenergygaps. Theseresultsshowthatthe
concept of equivalence classes for FLs has not only a geo-
metrical meaning but also an important role in the scatter-
I ing,dynamicalandthermodynamicalpropertiesofthesystem,
contained in the energy spectrum. It is worth stressing once
again that this is a consequence of the self-similarity of FSs
0.
under the composition rule and that FLs belonging to differ-
entequivalenceclassescannotberescaledoneovertheother.
Thegeneratorofaclassis,inthissense,thesimpleststructure
giving a diffraction pattern which contains the main features
common to all of the other elements of the class. Although
Figure 4: (Color online). Plot of the energy level spacings of the
wefocusedontheFibonaccilattices,ourargumentsapplyto
Hamiltonian in Eq.(7) for a potential V(x) having minima at the
themoregeneralclassofquasicrystalsforwhichdeflationor
points of FLs xη with (blue dots) η = ηa, (red crosses) ηa. Be-
lowwecomparenthegapswiththebrightest0peaksofI(q/ηa,1ηa). inflationrulescanmaptheinitiallatticesintoasimilarones.
1 1
particlespectrum. OneoftheresultspresentedinRef.[23]is Acknowledgments
thatthisisequivalenttosetathresholdtotheintensityofthe
peaks in the Bragg spectrum of the lattice. This can be eas- NL and LD acknowledge financial support from MIUR,
ilyseenbyconsideringtheFouriertransformofthepotential throughFIRBProjectNo. RBFR12NLNA 002. LVandMB
V(x),namelyV(q) = (cid:82) eıxqV(x)dxwhosesquaremodulus acknowledge financial support from Universit degli studi di
isgivenby: Padova through Chip & CIOP project No. CPDA120359.
The authors are thankful to Prof. Camilla Ferrante and Dr.
|V(q)|2 =S(q)I(q,η) (8)
Nicola Rossetto, from the Dipartimento di Chimica, Univer-
whereS(q)isthesquareoftheFouriertransformoff(x)and sit di Padova, for providing access to the direct laser writing
I(q,η)isgivenbyEq.B15.Itisclearfromwhatshownabove setup. The authors thank J. Settino for providing the energy
andconfermedbytheexperimentonthediffractionpatterns, spectraofaparticleinaFibonaccilikelattice.
that,apartfromthecontributionoftheactualformofthepo-
tential(whichplaysaroleanalogoustothestructurefactorin
diffractionexperiments),theenergypseudo-bandstructurein AppendixA:GeneralizedFibonaccilatticesfromcutand
reciprocalspacehasthesameshape(uptoarescaling)forthe projectmethod
lattices beloging to a given class. As an example, we com-
puted the spectra of Eq. (7) in the case of Gaussian wells, The Fibonacci lattices we considered in the main text can
namely f(x) = e−x2/2σ2 for a system with N = 100 min- beconstructedbymeansofthecutandprojecttechnique.One
ima and x = xη with η = ηa,ηa and η = ηb,ηb. We possibleconstructionhasbeenpresentedinref.[21]Weprefer
n n 0 1 0 3
choose V = 12, σ = 0.1. In Fig. 4 we plot the energy toresorttoamorestandardoneandinwhatfollowswewill
0
level spacing for lattices characterized by ηa = 6/11 (blue generalizetheonegiveninref. [3].
dots) and ηa = 17/6 (red crosses) both b0elonging to the Letusintroduceatwo-dimensionalperiodiclatticeIp and
0 2
equivalence class [η0a]. The momenta on the x-axis serve as its lattice vectors e1 and e2 such that any point of the lat-
areferencewithrespecttothefreeparticledispersionrelation tice can be written as p = n1e1 + n2e2 with n1,n2 ∈ Z.
((cid:15)k = k2/2,V(x) = 0) to show where the potential V(x) Furthermore we introduce the line lτ whose unit vector is
opens the gaps. After rescaling the momenta for the lattices ˆlτ = (cos(θτ),sin(θτ)) and the unit vector orthogonal to it
withη = ηa byν = ηa wecanclearlyseethatthegapsap- ˆl⊥ = (sin(θ ),−cos(θ )) such that tan(θ ) = τ−1 where
1 a 1 τ √τ τ τ
pearatthesamepoints. Ontheotherhandthesepointscorre- τ = (1 + 5)/2. The canonical Fibonacci lattice is con-
spondtothebrightestpeaks,whereI(q/ηa,ηa)calculatedby structed by projecting on the line l the points of a square
1 1 τ
Eq.(B15)issizeable.Similarresultsareobtained(notshown) lattice(e ·e = 0,|e | = |e |)thepointswhoseVonro¨ıcell
1 2 1 2
for the equivalence class [ηb] with ηb = 1/6 by considering iscutbythelineitself. Letusnoticethatwiththisprocedure
0 0
thetwolatticescharacterizedbyηb andηb =15/8,underthe thedifferentpointsareunambiguouslynumberedontheline
0 3
scalingνb =η1bη2bη3b. lτ onceanoriginandadirectionhavebeenchosen.Wearego-
ingtoconstructourFibonaccilatticesusingthisdefinitionbut
allowingthetwo-dimensionallatticetobegenericasinFig.5.
V. CONCLUSIONS Nevertheless we will see that in order to obtain a Fibonacci
lattice, namely a one-dimensional set of points whose dis-
In conclusion, we investigated the diffraction spectra of tances are distributed according to the Fibonacci strings and
FL’s in the general case η = S/(L−S) (cid:54)= τ. We have with the wanted ratio between long and short segments, we
5
willhavetorestrictthesetoftheallowedtwodimensionallat-
tices. Followingthediscussioninref.[3]inorderfortheline
γˆl tocuttheVonro¨ıcellcenteredatpointpithastointersect
τ
the secondary diagonal of the cell (joining the northwest to
southeastpointofthecell). Thediagonalslieonlinesparallel
toδ(e −e )andwhosepointsaregivenbyδ(e −e )+ne
1 2 1 2 1
with m ∈ Z. Their intersection with the line γˆl occurs at
√ τ
points(cid:0)n, n(cid:1)awherea=(τ2/ 1+τ2)A /(e −e )·ˆl⊥
τ τ2 u.c. 1 2 τ
andA =(e ∧e )·zˆisthe(oriented)areaoftheunitcell
u.c. 1 2
ofthelattice.
Figure 5: (Color online) Construction of a generalized Fibonacci
lattice from a 2D periodic lattice by means of the cut and project
method.Bluedotsaretheprojectionofpointsofthe2Dlatticewhose ThisintersectionpointsareinsidetheVonro¨ıcellcentered
Vonro¨icellsarecutbythelinelτ. atpoint(u,v)ifandonlyif
|(e −e )·xˆ| |(e −e )·xˆ|
u− 1 2 < naτ−1 <u+ 1 2 (A1)
2 2
|(e −e )·yˆ| |(e −e )·yˆ|
v− 1 2 < naτ−2 <v+ 1 2 (A2)
2 2
On the other hand each point of the lattice can be written as projected.Thuswecanwriteu=n (e −e )·xˆ+ne ·xˆand
1 1 2 2
n e +n e andn +n =nbecauseitisthen-thpointtobe v =n (e −e )·yˆ+ne ·yˆandaboveinequalitiesbecome
1 1 2 2 1 2 1 1 2 2
(cid:16) s (cid:17) (cid:16) s (cid:17)
n − x (e −e )·xˆ< n(aτ−1−e ·xˆ) < n + x (e −e )·xˆ (A3)
1 2 1 2 2 1 2 1 2
(cid:16) s (cid:17) (cid:16) s (cid:17)
n − y (e −e )·yˆ< n(aτ−2−e ·yˆ) < n + y (e −e )·yˆ (A4)
1 2 1 2 2 1 2 1 2
wheres =Sign((e −e )·xˆ)andsimilarlyfors .Bymeans coordinatesonthethelinel :
x 1 2 y τ
of the expression for a it is easy to prove that (a τ−1 −e ·
2
xˆ)/(e −e )·xˆ = (a τ−2−e ·yˆ)/(e −e )·yˆandthus
1 2 2 1 2
(cid:22) (cid:23)
thetwoinequalitiesareequivalenttotheinequality: n
x(cid:48) =ne ·ˆl +(e −e )·ˆl (A7)
n 2 τ 1 2 τ β
(cid:18) (cid:19) (cid:18) (cid:19)
1 n 1
n1− 2 < β < n1+ 2 (A5) Bynormalizingwithrespecttoe2·ˆlτ weeventuallyobtainthe
one-dimensionallatticeofpoints
e ·ˆl⊥ 1
β =1− 1 τ =1+r (A6)
e ·ˆl⊥ τsin(α)−cos(α)
2 τ (cid:22) (cid:23)
1 n
x =n+ (A8)
where cos(α) = e1 ·e2/(|e1||e2|) and r = |e1|/|e2|. Be- n η β
iinngeqnua1liatniesinttoegbeersnautimsfibeedritshethoatnlny1p=oss(cid:98)inβbi(cid:99)liwtyhfeorer t(cid:98)hxe(cid:99)aibsothvee η−1 =(cid:32)e1·ˆlτ −1(cid:33). (A9)
integerpartofx. Afterprojectingontol ,then-thpointhas e ·ˆl
τ 2 τ
6
In order for the above to be a Fibonacci lattice we require |(e − e ) · ˆl⊥| around the line γˆl . It is easy to see that
1 2 τ τ
β = τ which is the case for τr = (τsin(α)−cos(α)) and A(q ) = A (q ,0). We therefore turn to the calculation of
(cid:107) X (cid:107)
thusη =(τ +tan(α))/((τ −1)tan(α)−τ2). Moreoverwe thelatter. Byintroducingthemassdensityofthetwodimen-
(cid:80)
have to require that r > 0 and η > 0 which is the case for sional lattice ρ((cid:126)r) = δ((cid:126)r −(cid:126)r ) and its Fourier
tan−1(2τ +1) < α < tan−1(−τ)+π. As it can be seen transformρ((cid:126)r)=(cid:82) dk dmk1me−2ı(cid:126)r·(cid:126)kρ˜((cid:126)k)mw1emc2anwrite:
⊥ (cid:107)
fromfigs.6foranygivenη >0therecorrespondapair(r,α):
1 1 (cid:90)
A (q ,q ) = lim dk dk (B3)
ητ +1 X (cid:107) ⊥ L→∞L2∆ ⊥ (cid:107)
tan(α) = τ2 (A10) (cid:90) ∆ (cid:90) ∞
η−τ dx dx eı(cid:126)r·(q(cid:126)−(cid:126)k) ρ˜((cid:126)k)
⊥ (cid:107)
(τtan(α)−1)
−∆ −∞
r = (A11)
(cid:112)
τ 1+tan2(α) Theaboveintegralscanbecalculated:
andthereforeatwodimensionallatticewhoseprojectionon A(q )=A (q ,0)=(cid:90) dk sin(k⊥∆)(cid:90) dk ρ˜((cid:126)k)δ(k −q )
thelinelτ returnsthewantedFL: (cid:107) X (cid:107) ⊥ k⊥∆ (cid:107) (cid:107) (cid:107)
(B4)
1 (cid:106)n(cid:107)
xη =n−1+ (A12) which expresses the fact that the diffraction pattern of an in-
n η τ
finite projected quasicrystal is the convolution of the Dirac
(cid:32) (cid:33)
e ·ˆl combformedbytheperiodichigherdimensionalperiodiclat-
η−1 = 1 τ −1 , (A13) ticewiththesincfunctionintheorthogonalspace.
e ·ˆl
2 τ Inparticularρ˜((cid:126)k)isaDiraccombpeakedatpointsk =
hh(cid:48)
whereweshiftedthewholelatticeinorderforthefirstpoint hw1+h(cid:48)w2whereweintroducedthereciprocallatticevectors
tohavecoordinatex =0onthelinel . forthedualofthetwo-dimensionalperiodiclatticeIp:
1 τ 2
2π
w = (e −e ·eˆ eˆ ) (B5)
1 l 1 1 2 2
1
2π
w = (e −e ·eˆ eˆ ) (B6)
2 l 2 2 1 1
2
(B7)
whereeˆ =e /|e |andl =|e |2−|e ·eˆ |2andsimilarlyfor
i i i 1 1 1 2
l . Itiseasytocheckthatw ·e = 2πδ . Inwhatfollows
2 i j ij
Figure6: (Coloronline)a)Valuesofr =|e1|/|e2|asafunctionof we assume that units are scaled such that e2 · ˆlτ = 1. In
η. The red dot corresponds to the point (η,r) = (τ,1) for which ordertoevaluatetheparallelandperpendicularcomponentsof
thecanonicalFibonaccilatticeisobtained. b)Valuesoftheangleα vectorsbelongingtothereciprocalspaceweneedtoevaluate
betweene1ande2asafunctionofη.Thereddotcorrespondstothe wi·ˆlτ andwi·ˆlτ⊥. Inordertodosoitisusefultorewritethe
point(η,α) =(τ,π/2)forwhichthecanonicalFibonaccilatticeis vectorse aslinearcombinationsofˆl andˆl⊥bymeansofthe
obtained. i τ τ
expressionsforη,βandtherelationbetweentanαandη.We
thusobtain:
(cid:18) (cid:19) (cid:18) (cid:19)
AppendixB:Diffractionpattern e = 1+ 1 ˆl + 1 1+ 1 ˆl⊥ (B8)
1 η τ τ η τ
Weareinterestedinthecalculationofthequantity: (cid:18) 1(cid:19)
e = ˆl − 1+ ˆl⊥ (B9)
2 τ η τ
A(q(cid:107))=Nl→im∞N1 (cid:88)eıxηnq(cid:107) (B1) Itisnoweasytocheckthat:
n 2π 2π
w ·ˆl = w ·ˆl = (B10)
wherexη aregivenbyeq.A13. Usingtheunitvectorsˆl and 1 τ d 2 τ τd
n τ 2π 2π
ˆslτ⊥imwilaerclyanfowrrtihteeavnayriapbolient(cid:126)qin=spqa(cid:107)cˆleτa+s(cid:126)rq⊥=ˆlτ⊥x.(cid:107)Bˆlτy+inxtr⊥oˆldτ⊥ucainngd w1·ˆlτ⊥ = d(cid:16)1+ η1(cid:17) w2·ˆlτ⊥ =− d (B11)
thequantity
whered=(τ +1/η). Thereforewecandefine:
A (q ,q )= lim 1 (cid:88)eıp(cid:126)n·q(cid:126) (B2) 2π (cid:18) h(cid:48)(cid:19)
X (cid:107) ⊥ N→∞N n Q(h,h(cid:48))=khh(cid:48) ·ˆlτ = d h+ τ (B12)
(cid:18) (cid:19)
where we recall that p(cid:126) are points (cid:126)r of the two dimen- 2π η h
sional periodic latticenwhich lie in an1snt2rip of width 2∆ = Q⊥(h,h(cid:48))=khh(cid:48) ·ˆlτ⊥ = d η+1 −h(cid:48) (B13)
7
Bymeansofeq.B4wecanthuswritetheintensitiesofthe where d = (τ + 1/η). By properly choosing the integers
diffractedpointsas: h and h(cid:48), any real number can be arbitrarily well approxi-
mated,showingthatthereciprocallatticeofaFLisdensein
R,contrarilytoperiodiclatticeswhichexhibitadiscreterecip-
I(q ,η)=|A(q )|2 =(cid:88)sin2(Q⊥(h,h(cid:48))∆)δ(q −Q(h,h(cid:48))) rocallattice. Moreoveritcanbeshown[3]thatthediffraction
(cid:107) (cid:107) (Q (h,h(cid:48))∆)2 (cid:107) pattern has only pure point support, lacking of a continuous
⊥
h,h(cid:48) part (according to the classification of positive measures in
(B14)
theLebesgueclassification).
where∆=τ(1+1/η)/2andweintroducetheexplicitdepen-
denceoftheintensityontheparameterηwhichcharacterizes
theFL.Asitcanbeseen,thediffractionspectrumconsistsof
AppendixC:Brightestpeaks
asetofsharppeakscenteredonadensesetofreciprocallat-
ticepoints,asbychoosingtheappropriatevaluesofhandh’,
1. Conditionforbrightestpeaks
anyqcanbeapproximatedwitharbitraryprecision. However,
notallthesepeakshavethesameintensity.
In order to find the set of points in reciprocal space cor-
Infig.7weplotI(q )asgivenbyexpressionineq.B14and
(cid:107)
responding to a strong diffracted intensity for a FL, the fol-
itsexpressioncalculatedexplicitlybyitsdefinitioneq.B1for
lowing condition on the argument of the exponential in the
a lattice of N = 300 points and η = 17/6. We can see that
expressionforthediffractionpatternhastohold:
asexpectedthepeaks’intensitiesarewellcapturedbyeq.B14
evenforfinitesystemsespeciallyforpeakscharacterizedbya
significantintensity(¿0.2). Q (h,h(cid:48))≈0 (C1)
⊥
whichissatisfiedfor
I q
1.0 h 1
≈1+ . (C2)
h(cid:48) η
0.8
Let us now consider the equivalence class [η ] and in par-
0
0.6 ticular the sequence xηn0. We can write η0 in the continued
(cid:72) (cid:76) fractionrepresentation:
0.4 1
η =a + ≡[a ,a ,a ,a ,···] (C3)
0 0 a + 1 0 1 2 3
0.2 1 a2+a3+1...
For a rational number the sequence of numbers a is finite,
i
q namely η0 = [a0,a1,··· ,an]. On the other hand, if η0
10 20 30 40 50 60 is irrational, it is possible to find a rational approximation
within the wanted error by increasing the number of terms
in its continued fraction representation. It is easy to see that
Figure 7: (Color online) Comparison between the values of I(q )
(cid:107)
η =1+1/η =[1,a ,a ,···]andingeneral
usingthecutandprojectmethodandthedirectevaluationineq.B1 1 0 0 1
foralatticeofN =300pointsandη =17/6. Bluedotsarepoints η =[1,1,··· ,1,a ,a ,···]. (C4)
k 0 1
correspondingtothevalueineq.B14whereasredcrossaregivenby (cid:124) (cid:123)(cid:122) (cid:125)
eq.B1. k
Withthisnotationisstraightforwardtoseethatregardlessof
We now consider the (Fraunhofer) diffraction pattern of a thevalueofthegeneratorη0,thesequencesofanequivalence
FLxη: class will tend to a Fibonacci sequence since limk→∞ηk =
n
[1,1,1,1,···] = τ [24]. Using the continued fraction nota-
(cid:12) (cid:12)2 tion we can write a sequence of rational approximants to η0
I(q,η)=Nl→im∞N12 (cid:12)(cid:12)(cid:12)(cid:12)(cid:88)eıxηnq(cid:12)(cid:12)(cid:12)(cid:12) . (B15) taosbae0,inat1eaag01e+rs1,thae2(aaba1o2aav01e++1c1)o+nad0i,ti·o·n·C. 2Siinscseatbisofithedhifanwdehc(cid:48)hhoaovsee
n
h=s +t andh(cid:48) =s wheres andt arethen-thapprox-
n n n n n
Thisquantityisimportantbecauseitgivesadirectexperimen- imants of η namely η ≈ s /t and can easily be derived
0 0 n n
talaccesstothereciprocallatticeofourstructure.Weshallsee fromthecontinuedfractionrepresentationofη .
0
that this quantity is also encountered in the determination of It is worth stressing that if η is a rational number (η =
0 0
a(pseudo)energydispersionrelation[23]. InthecaseofFLs a/b,witha,b ∈ N)handh(cid:48) canbechosensuchthath/h(cid:48) =
thevaluesofqatwhichanon-vanishingintensityisexpected (a+b)/a. Thus, at points q = Q(m(a+b),mb) = 2mπ
m
aregivenby: (m ∈ Z) we have that I(q ,η ) = 1. On the other hand,
m 0
forirrationalη theconditionisneversatisfiedexactlybutwe
0
2π (cid:18) h(cid:48)(cid:19) can resort to the rational approximants of η to estimate the
Q(h,h(cid:48))= h+ , (B16) 0
d τ positionsatwhichthebrightestpeaksappear.
8
2. Relationbetweenpositionsofbrightestpeaks ∆I q
0.25
Letxη0 andxη1 betwoFibonaccilatticesbelongingtothe
n n
sameequivalenceclassandtheirassociatedreciprocallattices
0.20
Q (h,h(cid:48))=2πd−1(h+h(cid:48)/τ),Q (h,h(cid:48))=2πd−1(k+k(cid:48)/τ)
0 0 1 1
respectively,wheredi =τ +1/ηi. Bydefininghn (kn)and 0.15(cid:72) (cid:76)
h(cid:48) (k(cid:48)) as the numerator and denominator of the n − th
n n
rationalapproximantsof1+1/η0 (1+1/η1),thefollowing 0.10
relationsholdtruek = h +h(cid:48) andk(cid:48) = h . Byinserting
n n n n n
theserelationsintotheexpressionforQ1(k,k(cid:48))weget: 0.05
2π (cid:18) k(cid:48) (cid:19) 2π (cid:18) h (cid:19) q
Q (k ,k(cid:48)) = k + n = h +h(cid:48) + n 10 20 30 40 50 60 70
1 n n d n τ d n n τ
1 1
2πτ (cid:18) h(cid:48) (cid:19) d τ 2π (cid:18) h(cid:48) (cid:19)
= d1 hn+ τn = d01 d0 hn+ τn (Friegdurcero8s:s)(C(Io(lηo1rqo,nηl1in)e−)PIl(oqt,oηf0)(b)lfuoerdthootss)eτq−f1o(r1w−hicIh(qI,(ηq0,)η)0)an>d
d τ 0.5showingthattheintensitiesofthebrightestpeaksofthediffrac-
= d0 Q0(hn,h(cid:48)n)=η1Q0(hn,h(cid:48)n) tionpatternofaFLxηn1 generatedfromaFLxηn0 bymeansofthe
1 compositionrulearemoreintenseofthoseoftheoriginallatticeby
whereinthelastlineweusedthefactthatd τ/d =η . This afactorofτ.
0 1 1
means that the lattice obtained by applying the composition
ruleC(xηn0)=η1xηn1 hasbrightestpeaksatthesamepositions
oftheoriginallatticeonlyrescaledbyafactoreta .
1
3. Relationbetweenintensitiesofbrightestpeaks
From eq.B14 we can also estimate the relation between
the intensities of the brightest peaks in the diffraction spec-
trum of two FL belonging to the same class. Using the ex- samplewiththeaidofacomputer-controlledXYstageatcon-
pression in eq.B14 and assuming k⊥∆ ≈ 0 we can write stantspeedof50µm/s. Thenominalprecisionofthetrans-
sin2(k⊥∆)/(k⊥∆)2 −1 ≈ (k⊥∆)2/9. Using the condition lation stage is 0.5µm for the conditions used in this experi-
fork⊥ ≈ 0andfollowingacalculationsimilartothattode- ment. A frequency doubled diode pumped Nd:YWO4 solid
terminedrelationbetweenthepositionsofthebrightestpeaks state laser (Coherent Verdi V5) emitting a CW beam at 532
wefindthat(k⊥1∆1)2 =(k⊥0∆0)2/τ2. Thereforewehave nmhasbeenusedaslightsourceforDLW.Thebeamwassuit-
ablyattenuatedbyaseriesofneutraldensityfiltersandsentto
1
I(η1q,η1)≈I(q,η0)+ τ(1−I(q,η0)), (C5) afocusingmicroscopeobjective(Olympus100X/0.80)sothat
thepoweraftertheobjectivewassetat17mW.Thesubstrate
meaning that the intensities of brightest peaks of the scaled usedtoengravetheopticalstructuresisaslabofphotorefrac-
lattice are more intense of those of the original lattice by a tive lithium niobate doped with iron at the nominal concen-
termproportionaltothedifferencebetweenthemaximumat- tration of 0.1 mol% in the melt. The sample was X-cut with
tainableintensityandtheintensityoftheoriginallatticeinten- dimensions(X×Y×Z)1mm×8mm×13mmandthelines
sities. werewrittenontheXface,byscanningalongtheYdirection
Infig.8weplotthequantities(bluedots)τ−1(1−I(q,η0)) with an ordinarily polarized beam. This process can induce
and (red cross) (I(η1q,η1) − I(q,η0)) for q such that extraordinaryrefractiveindexchangesaslargeas10−3inthe
I(q,η0) > 0.5 and for lattices of N = 300 sites and η0 = written lines and therefore can be used to produce arbitrary
6/11andη1 =1+1/η0respectively. diffraction structures. The diffraction pattern of these struc-
tures was measured with the help of a computer-controlled
optical diffractometer in which the sample and the detector
AppendixD:Experimentalsetup were mounted on two co-axial goniometers that were inde-
pendently controlled by a computer [26]. An optical beam
TotestexperimentallythediffractionfromFL’s,aseriesof producedbyaHe-Nelaserat632.8nmwithapowerof4mW
quasi-periodic diffraction gratings have been prepared using wasexpanded,polarizedalongtheextraordinarydirectionand
a photorefractive direct laser writing (DLW) technique [25]. finally transmitted through the sample surface, resulting in a
Thistechniqueconsistsinscanningwithafocusedlaserbeam clearlyvisiblediffractionpattern. Thispatternwasmeasured
aphotorefractivesample,engravingonitaseriesoflineswith by a Si photodiode and a lock-in amplifier and recorded on
amodifiedrefractiveindexwithrespecttotherestofthesam- the computer as a function of the detector and of the sample
ple. Thescanningmovementisperformedbytranslatingthe angle.
9
AppendixE:Experimentaldiffractionpatterns form for S(q) describes adequately the peak intensity in the
wholerangeofmeasuredvalues(seefig. 9):
1. Structurefactor
Inordertocomparetheexperimentaldatawiththetheoret- S(q)=S0e−λq−qq0 (E2)
icalcalculationweneedtotakeintoaccountthatourgratings
aremadeupofa(quasi)periodicrepetitionofaregionwitha where the parameters S ,λ,q are determined by a least
0 0
modifiedrefractiveindex,∆n(x). Thisleadstothefact,well square fit in the range q ∈ [0.5,3] µm−1 excluding the last
knownfromstandarddiffractiontheory,thatthediffractedin- peaks because the corresponding momenta where compara-
tensityinreciprocalspaceisproportionaltotheproductoftwo ble with a length scale of the order of the optical waveguide
terms: afirstone,S(q)=(cid:12)(cid:12)(cid:82) ∆n(x)eıxqdx(cid:12)(cid:12)2whichdependson width. We also notice that all measured diffraction patterns
thedetailedstructureoftherepeatedunitofthegrating(struc- drop almost to zero outside the interval q ∈ [−4,4] µm−1;
turefactor)andasecondtermduetolatticegeometry,which this is due to the fact that our lines have a width determined
isthetrueobjectofthisstudy: bythelaserwritingopticswhichisnotsmallerthan2µm,so
thatourdiffractionpatterncannotprobe|q|>3µm−1.
(cid:12) (cid:12)2
IR(q,η)=S(q)N12 (cid:12)(cid:12)(cid:12)(cid:12)(cid:88)eıxηnq(cid:12)(cid:12)(cid:12)(cid:12) =S(q)I(q,η) (E1) 2. Comparisonwiththeoreticalpatterns
n
The structure factor modulates the intensity of the lattice
In Fig. 10 we compare the theoretical diffraction pattern
diffractionpattern,complicatingthecomparisonbetweenex-
I(q,η) with the experimental data points for the grating ηb.
periments and theory. In principle S(q) could be calculated 3
Theintensityoftheexperimentalpointshasbeenrescaledto
byknowingthedetailsoftherefractiveindexprofilechanges
take into account the contribution of the structure factor of
produced by our technique. Here we used another approach
the grating and the q axis of the experimental plot has been
whichexploitsthefactoursamplesdifferonlyforthelinepo-
rescaledinordertocompareitwiththesimulation,whichcon-
sitionsequencex . Wecanusethereforetheperiodicgrating
n siders FL’s with S = 1. A similar figure is obtained for the
(Fig.9)tomeasurethefunctionS(q)directlyatthereciprocal
caseηa =17/6. Theagreementisverysatisfactory: notonly
latticepoints{qi }oftheperiodicgrating,whereI(qi )has 1
M M thepositionbutalsotheintensityofthediffractionpeaksare
localmaxima.
correctlyobtained,confirmingthatourapproachisreliable.
I q
10(cid:45)4 I(q,η b)
10(cid:45)6
10(cid:45)8(cid:72) (cid:76)
10(cid:45)10
q
q
(cid:45)2 0 2 4
Figure 10: (Color online). Experimental diffraction pattern (solid
Figure9: (Coloronline). Experimentaldiffractionpatternofthepe- redtopcurve)comparedwiththetheoreticaldiffractionpatternwith
riodicgratingwithspacingL=23µm(blue,solidcurve)andfitof theinclusionofthestructurefactor(solidbluebottomcurve)fora
thesatellitepeakintensitiesusingfunctionE2(red,dashedcurve). FibonaccigratingwithN = 300linesandL = 23 µmandS =
15µm,ηb =1.875.
3
We found that the following phenomenological functional
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