Limitations to the determination of a Laguerre-Gauss spectrum via projective, phase-flattening measurement

0.78 MB·

by  Hammam Qassim

Limitations to the determination of a Laguerre-Gauss spectrum via projective,
phase-flattening measurement
HammamQassim,1FilippoM.Miatto,1JuanP.Torres,2MilesJ.Padgett,3EbrahimKarimi,1,∗andRobertW.Boyd1,3,4
1DepartmentofPhysics,UniversityofOttawa,150LouisPasteur,Ottawa,Ontario,K1N6N5Canada
2ICFO-InstitutdeCienciesFotoniques,08860Castelldefels(Barcelona),Spain
3SchoolofPhysicsandAstronomy,SUPA,UniversityofGlasgow,GlasgowG128QQ,UnitedKingdom
4InstituteofOptics,UniversityofRochester,Rochester,NewYork,14627,USA
∗Correspondingauthor:[email protected]
CompiledApril23,2014
4
1 Oneofthemostwidelyusedtechniquesformeasuringtheorbitalangularmomentumcomponentsofalightbeamistoflatten
0 thespiralphasefrontofamode,inordertocoupleittoasingle-modeopticalfiber.Thismethod,however,suffersfroman
2 efficiencythatdependsontheorbitalangularmomentumoftheinitialmodeandonthepresenceofhigherorderradialmodes.
Thereasonisthatoncethephasehasbeenflattened,thefieldretainsitsringedintensitypatternandisthereforeanontrivial

r superpositionofpurelyradialmodes,ofwhichonlythefundamentalonecouplestoasinglemodeopticalfiber.Inthispaper,
p westudytheefficiencyofthistechniqueboththeoreticallyandexperimentally.WefindthatevenforlowvaluesoftheOAM,a
A
largeamountoflightcanfalloutsidethefundamentalmodeofthefiber,andwequantifythelossesasfunctionsofthewaistof
thecouplingbeamoftheorbitalangularmomentumandradialindices.Ourresultscanbeusedasatooltoremovetheefficiency
2
biaswherefair-samplingloopholesarenotaconcern.However,wehopethatourstudywillencouragethedevelopmentofbetter
2 detectionmethodsoftheorbitalangularmomentumcontentofabeamoflight.
(cid:13)c 2014 OpticalSocietyofAmerica
] OCIScodes: 070.2580,230.6120,070.6042.
s
c
i
t
p 1. Introduction beamcannotberemoved.Moreover,thedetectionefficiency
o
for high OAM modes can be extremely low, making it
.
s seem like those components are very weak. This issue is
Structured light beams have wide applications in tech-
c
particularly important for those experiments that rely on
i nologies such as lithography, nanoscopy, spectroscopy,
s a high detection efficiency, for example, experiments that
y optical tweezers and quantum cryptography [1–5]. Among
aim at maximizing the heralding efficiency, or at closing a
h these, beams with helical phase fronts exp(i(cid:96)φ), where (cid:96)
p is an integer number and φ is the azimuthal angle in polar detection loophole [22,23], or at characterizing a state by
[ measuringeachofitsOAMcomponentsseparately[24,25].
coordinates, are of particular interest since they can be used
In this letter, we study projective measurements based on
2 for classical [6–8] and quantum communications [5]. These
v beams carry a well-defined value of optical orbital angular phase-flattening followed by coupling into a SMOF. We
2 momentum (OAM) (cid:96)(cid:126) per photon along the propagation di- examine our theoretical model experimentally for various
1 mode projections, and we verify the trends in coupling
rection.Duetotheseproposedapplications,therearefervent
5 efficiencies.
3 attemptstodesigninnovativedevicestogeneratesuchbeams.
. Untilnow,possiblesolutionsincludespiralphaseplates[9],
1
computer-generated holograms imprinted onto spatial light
0
2. Theoreticalanalysis
4 modulators(holographicapproach)[10,11],modeconverters
1 (cylindrical lenses) [12], q-plates (nonuniform liquid crystal In our analysis we use Laguerre-Gauss (LG) modes, which
v: plates) [13,14], and some types of OAM-sorters [15,16]. arecharacterizedbytwoindices:theradialindex p(nonneg-
i These solutions are practical and widely used in various ex- ative integer) and the azimuthal number (cid:96) (integer), which
X
perimentalrealizations,andareimplementedbothinclassical areassociatedtothenumberofradialnodesandtotheOAM
r andquantumregimes.However,withtheexceptionofmode value, respectively. The LG modes are a complete and or-
a
converter and a hologram with an intensity mask [17,18], thonormalfamilyofsolutionsoftheparaxialwaveequation,
the above methods do not generate a pure Laguerre-Gauss i.e.(inDiracnotation)(cid:104)p(cid:48),(cid:96)(cid:48)|p,(cid:96)(cid:105) = δ δ ,andinthepo-
p(cid:48),p (cid:96)(cid:48),(cid:96)
mode[19,20].Insomecases,thereverseprocesscanbeused sitionrepresentationatthepupiltheyaregivenby
todetect thespectrumofOAMofanunknownbeam,where
eaaftcehrmitosdaeziismcuotuhpalledphtoasaesidnegpleenmdeondceeophtaiscablefiebnerfl(aStMtenOeFd). LGp,(cid:96)(r,φ):= (cid:115)πw22|((cid:96)|p+1+p!|(cid:96)|)! (cid:32)wr (cid:33)|(cid:96)|e−wr220L|p(cid:96)|2wr22 e−i(cid:96)φ,
0 0 0
Suchamethod,wasfirstintroducedbyMairetal.[21]inthe
(1)
quantum domain and then used commonly in the classical
regime.Thistechniquemightsoundaccurate,butaswewill wherer,φarethetransversecylindricalcoordinates,w isthe
0
show, its shortcoming is that the OAM bandwidth that can beam waist radius at the pupil and L(cid:96)(.) is the generalized
p
be measured has a bias that depends on the characteristics Laguerre polynomial. The devices listed above can gener-
of the beam. As a consequence, the bias for an unknown ateLGmodeswithlimitedfidelity.Themostconvenientand
1
1.0 1.0 1.0 0.7
0.8 |F0,`|2 ` 0.8 |F1,`|2 0.8 ⌘0` 0.6 ⌘1`
` 0.5
0.6 01 0.6 0 0.6 0.4
0.4 234 0.4 12 0.4 00..23
0.2 5 0.2 0.2 0.1
0.0 0.5 1 1.5 2 0.0 0.5 1 1.5 2 0.0 0.2 0.6 1 1.4 0.0 0.2 0.6 1 1.4
⇢ ⇢ �/a �/a
0 0
Fig.1.(Coloronline)Intensityofthe p = 0(left)and p = 1 Fig. 2. (Color online) Coupling efficiency for projective
(right)modesattheinputtothefiber.Asaneffectofdiffrac- measurementsforthemodesinFig.1.Foragivenchoiceof
tion,localmaximaattheperipheryga√inintensityas pand|(cid:96)| optics,thecouplingefficiencyshowsabiasdependentonthe
increase. Hereρ is inunits ofa0 = ( 2λf)/(πw0), whichis orderofthetransversemodes.Thehorizontalaxisisscanned
thenaturalscalingfactorinthefar-fieldofthelens. by changing w , as a is inversely proportional to w . The
0 0 0
shadedboxindicatestheregionlimitedbytheactiveareaof
theSLMs(seeexperimentalsection).
commonlyusedmethodistheholographicapproach,withan
embedded intensity masking. However, a mode-cleaning fil-
ter cavity can be used to increase fidelity of the generated
where σ is the beam waist radius of the SMOF Gaussian
mode[26,27].
mode.ItisworthmentioningthatthemodeofaSMOFcanbe
approximatedwithaGaussianbeam.Herewegivetheresults
A. ProjectingonLGmodes
for p=0and p=1:
To perform a projective measurement, the mode LG (in
p,(cid:96)
our case generated by an SLM) is imaged onto a differ- η(cid:96) = |(cid:96)|!2 A2|(cid:96)|+1B (5)
ent conjugate mode, LG∗ , and the resulting field is prop- 0 (2|(cid:96)|)!
p(cid:48),(cid:96)(cid:48)
agated and coupled into a SMOF in the far-field, which se- (|(cid:96)|+1)!|(cid:96)|!
η(cid:96) = A2|(cid:96)|+1B(A2+B2(|(cid:96)|+1))2, (6)
lects only the near Gaussian component. Imaging onto an 1 4(2+3|(cid:96)|)(2|(cid:96)|)!
SLM is described by taking the product of the two modes,
ci.oeo.rLdGinpa,(cid:96)te(rs⊥.T)LheGf∗pa(cid:48),r(cid:96)-(cid:48)fi(re⊥ld)dwishterribeurt⊥ionstbanecdosmfoerstahpeotlryannosmveirasle- w√here A = 2/(1 + σa202) and B = 2/(1 + σa202), and a0 =
Gaussianfunctiongivenbya2D-Fouriertransform: ( 2λf)/(πw0) is the natural scaling factor at the fiber. No-
tice that it is only the ratio σ/a that matters, as it should
(cid:104) (cid:105) 0
Fp,(cid:96)(ρ,ϕ)=FT LGp,(cid:96)(r⊥)LG∗p(cid:48),(cid:96)(cid:48)(r⊥) , (2) be.TheseresultsareshowninFig.2,whereitispossibleto
seethatthehighestcouplingefficiencyfordifferentmodesis
whereFT standsforthe2D-Fouriertransform,andρandϕ
achieved for different values of the waist at the fiber, which
arethecylindricalcoordinatesinthefarfield.Thefactthata
canbetunedbyadjustingthefocallengthoftheFourierlens.
SMOF only supports the TEM mode limits this technique
00
to the case in which (cid:96)(cid:48) = (cid:96). Moreover, as oscillating radial
phaseswouldalterthecouplingtotheSMOF,wealsochoose
B. Projectingonspiralmodes
p(cid:48) = p.Duetotheabsenceofanyangulardependenceafter
thephaseflatteningstage,the2D-FouriertransformFT can Analternateandlessdesirablesolutionthatweexploreonly
theoreticallyistoprojectontoapurelyspiralfieldei(cid:96)ϕ(which
besimplifiedintotheHankeltransformoforderzero,i.e.
canbeimplementedwithapitchforkhologramonanSLM),
F (ρ,ϕ)= 2πeλiπfρ2 (cid:90) ∞rdr|LG (r )|2J (cid:32)2πrρ(cid:33). (3) whereby the effect is to simply cancel out the spiral phase
p,(cid:96) iλf p,(cid:96) ⊥ 0 λf fromaninitialLGmode,inwhichcasethefieldatthefiberis
0 (cid:104) (cid:105)
givenbyF (ρ,ϕ)=FT LG (r )ei(cid:96)φ .Thisequationcan
p,(cid:96) p,(cid:96) ⊥
In Fig. 1 we show some examples of transverse intensity at
besolvedanalytically,recallthattheEq.(2)hasananalytical
thefiberforseveralvaluesof pand(cid:96).Noticethatthebeams
solutiononlyonceavalueof pisspecified.However,forthis
haveaGaussian-likeshapewithlocalmaximaattheperiph- specificcasethecouplingefficiencyisgivenby
ery,whichgiverisetoaringedpatterninthetransverseplane.
Amsov|(cid:96)e|satnodthpeboeuctoemr reinlgasrg.eTr,hitsheisbreealamteidntteontshiteyedffisetcrtibtuhtaitona η(cid:96) =(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)N(cid:96)(cid:88)p (−1)j(cid:32)p(cid:33)a(cid:96)fp,(cid:96)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)2 (7)
largerphase-flatteneddoughnutbeamisturnedintoasmaller p (cid:12)(cid:12) p j j j (cid:12)(cid:12)
(cid:12) j=0 (cid:12)
andweakercentralspotatthefarfield,whichhasbeenstud-
iedanddiscussedforspecialcasesin[28,29]. with
The coupling efficiency to a SMOF, then, is given by the
(cid:89)p
overlapoftheGaussianmodesupportedbythefiberandthe fp,(cid:96) = (cid:0)|(cid:96)|+k+kH(j−k)(cid:1)
far-fielddistributioncalculatedin(2): j
k=1
√
η(cid:96) = πσ22 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90)0∞ρdρ(cid:90)02πdϕ F(cid:96)(ρ,ϕ) e−σρ22(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)2, (4) a(cid:96)j =(cid:0)1+(σ2/πaσ0)/2a(cid:1)0|2(cid:96)|+j+1,
2
0.5 ⌘` 0.35 ⌘`
0 0.30 1
0.4
0.25
0.3 0.20
0.2 0.15
0.10
0.1
0.05
0.0
0.2 0.6 1 1.4 0.2 0.6 1 1.4
�/a �/a
0 0
Fig.4.(Coloronline)Experimentallymeasuredoverallcou-
pling efficiency for the modes shown in Fig. 1: (left) (cid:96) =
0...5 and (right) (cid:96) = 0...2. The shaded regions indicate
adomaininwhichtheeffectivebeamsizeexceedstheactive
areaoftheSLM,resultinginunreliabledata.
Fig.3.(Coloronline)Experimentalsetupforgeneratingand
Ref.[18,31].Anautomaticprogramoptimizedthecenterof
detectingphotontransversestates.AlinearlypolarizedHeNe
thehologramsonthebothSLMsandthecouplingefficiency
laserbeamisspatiallycleanedwithtwolensesandapinhole.
with the SMOF. The pixel size and active area of the SLMs
Ahalf-waveplate(HWP)optimizesthefirstorderofdiffrac-
were8µmand15.36mm×8.64mm.Thesecharacteristicsset
tion on SLM , since SLMs are polarization dependent. The
A thelimitsoftherangeofbeamwaistsandmodenumbersthat
modeLG (r )producedbySLM isthenprojectedonthe
p,(cid:96) ⊥ A couldbeinvestigated.
modeLG (r )∗ onSLM .Theresultingfarfieldiscoupled
p,(cid:96) ⊥ B Figure 4 shows the experimental results, to be compared
intoasinglemodeopticalfiber(SMOF).Weimplementtwo
with the coupling efficiency shown in Fig. 2. Aside from an
4f-system with unit magnification and a microscope objec-
overall multiplicative efficiency of about 50% (which com-
tivetoimageSLM onSLM andSLM onthemicroscope
A B B prises reflection and scattering by microscope objective and
objective.Irisesareusedtoselectthefirstorderofdiffraction
fiber), the observed data (Fig. 4) and the theoretical model
atthefar-fieldplaneofSLMs,wherehigherorderofdiffrac-
(Fig. 2) agree, especially in those regions where the SLMs
tionarewellseparated.
resolutionandactiveareadonotaffectthequalityofthegen-
erated and projected beams. The region below 0.2 σ/a is
0
(cid:113) (cid:16) (cid:17) limitedbyresolution,astoofewpixelsareused.Ontheop-
whereN(cid:96) = 2|(cid:96)|+1 Γ |(cid:96)| +1 isthenormalizationfunc-
p π(p+|(cid:96)|)!p! 2 positeendofthehorizontalaxisthebeamseventuallyfallout
tion.H(j−k)in fp,(cid:96)istheunitstepfunction:itsvalueis0for ofthetheactivearea.Theseregionsareindicatedbyashaded
j
j < k and1for j ≥ k,andΓisthegammafunction,respec- area in the figures. Due to truncation mainly induced by the
tively. As was expected, the coupling efficiency η(cid:96) depends microscopeobjective,thereasmalldeviationforthecaseof
p
on the ratio between the beam waist radius of the SMOF σ p = 1atlargebeamwaistsizewithrespecttothetheoretical
andthesizeofthefieldatthefiberpositiona . calculation. However, the spread of these curves is an indi-
0
cation that the coupling efficiency differs for different initial
modesandthatthereforethespectrumthatisultimatelymeas-
3. Experimentalresults ured is likely not representative of the true OAM distribu-
tionofthebeam.Wecandeducethatprojectivemeasurement
In order to verify the above theory, we prepared an experi-
methodsshouldbeusedwithcare.Apossiblesolutioncould
mental setup (Fig. 3) in which we examined the projective betocalibratethecouplingefficiencyfordifferenttransverse
measurement method for different sets of transverse modes
modesandthenpost-processthemeasurementdata,buteven
withvaryingbeamsizes.Alinearlypolarizedlightbeamofa
inthiscase,iftheradialdistributionoftheinitialfieldisun-
HeNe laser is spatially cleaned, and illuminates the first of
known,thebiasmaynotberemovable,astheradialdecom-
two Pluto HOLOEYE SLMs (SLM ), to generate the ini-
A positiondependsonthewaistthatischosenforthemodes.Of
tial LG (r ) mode. This is then imaged on a second SLM
p,(cid:96) ⊥ courseitisalsotruethatalinearsuperpositionofLGbeams
(SLM ) via a 4f-system with unit magnification, where the
B leads to an inaccurate result, since the projective measure-
mode is projected onto LGp,(cid:96)(r⊥)∗. We used intensity mask- ment gives a bias among projection of different pure OAM
ing to encode transverse modes with high fidelity [18,30].
states.
TheproductfieldisfinallycoupledtoaSMOFwithmodedi-
ameter of (cid:39) 4.8µm and a numerical aperture NA = 0.12 at
the far-field of a 20× microscope objective (f = 9 mm and
4. Conclusions
NA=0.40).Inordertonormalizethecouplingefficiencyfor
different modes, we used a Newport power meter with two In conclusion, we studied the efficiency of projective
readoutheadstorecordboththecouplingefficiencyandthe measurement as a method to characterize the transverse
powerofthefieldjustbeforethefibersimultaneously.Recall mode of a light beam. Our analysis can be summarized
thatduetotheintensitymaskingdifferentmodeshavediffer- in two important messages. The first is that although the
entgenerationanddetectionefficiencies,formoredetailssee couplingefficiencyismodal-andbeamwaist-dependent,the
3
bias that is induced might be removed in post-processing 10. V. Yu. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, Screw
after a careful calibration. Of course, issues may arise Dislocations in Light Wavefronts, J. Mod. Opt. 39, 985-990
in the context of an experiment aimed at violating Bell’s (1992).
inequalities: post-processed results could be regarded as 11. S.Ngcobo,I.Litvin,L.BurgerandA.Forbes, Adigitallaser
an artificial manipulation of the data, and detection-related foron-demandlasermodes,Nat.Commun.4,2289(2013).
12. L.Allen,M.W.Beijersbergen,R.Spreeuw,andJ.P.Woerd-
loopholes might be called into consideration. The second
man,Orbitalangularmomentumoflightandthetransforma-
message builds on the fact that the radial modal content of
tionofLaguerre-Gaussianlasermodes, Phys.Rev.A458185
theinitialbeamdependsonthewaistw thatischosenforthe
0 (1992).
decomposition, and the optimal choice (i.e. the one that re-
13. L.Marrucci,C.Manzo,andD.Paparo, Opticalspin-to-orbital
sultsintheleastnumberofradialmodes)forageneralbeam
angularmomentumconversionininhomogeneousanisotropic
couldbefoundonlyafterfurthermeasurements.However,as
media, Phys.Rev.Lett.96,163905(2006).
one then should have to calibrate for this optimal size, this
14. E.Karimi,B.Piccirillo,E.Nagali,L.Marrucci,andE.Santam-
is clearly not an ideal procedure, especially in the context ato,Efficientgenerationandsortingoforbitalangularmomen-
ofquantumoptics,wheretheremightbeascarcenumberof tum eigenmodes of light by thermally tuned q-plates, Appl.
photonsavailable).Itisourhopethatthisworkwillmotivate Phys.Lett.94,231124(2009).
the search of new and better measurement techniques for 15. G.Berkhout,M.Lavery,J.Courtial,M.Beijersbergen,andM.
OAM and more generally for the transverse radial modes of Padgett,EfficientSortingofOrbitalAngularMomentumStates
alightbeam. ofLight, Phys.Rev.Lett. 105,105601(2010).
16. M. Mirhosseini, M. Malik, Z. Shi, R.W. Boyd, Efficient sep-
arationoftheorbitalangularmomentumeigenstatesoflight,
Nat.Comm.4,2781(2013).
17. J.Leach,M.R.Dennis,J.Courtial,andM.J.Padgett,Vortex
5. Acknowledgments
knotsinlight,NewJ.Phys.7,55(2005).
TheauthorsthankProf.GerdLeuchsforfruitfuldiscussions. 18. E. Bolduc, N. Bent, E. Santamato, E. Karimi, R. W. Boyd,
H.Q., F.M.M., E.K. and R.W.B. acknowledge the support of Exact solution to simultaneous intensity and phase encryp-
tionwithasinglephase-onlyhologram, Opt.Lett. 38,3546
the Canada Excellence Research Chairs (CERC) Program.
(2013).
M.J.P.andR.W.B.weresupportedbytheDARPAInPhopro-
19. E. Karimi, G. Zito, B. Piccirillo, L. Marrucci, E. Santamato,
gram.M.J.P.supportedbytheroyalsociety.
Hypergeometric-Gaussianmodes, Opt.Lett.32,3053(2007).
20. M.R.Dennis,K.O’HolleranandM.JPadgett, SingularOp-
tics:OpticalVorticesandPolarizationSingularities,Prog.Opt.
References
53293(2009).
21. A.Mair,A.Vaziri,G.WeihsandA.Zeilinger,Entanglementof
1. T.Grosjean,D.Courjon,andC.Bainier,Smallestlithographic
theorbitalangularmomentumstatesofphotons,Nature412,
marks generated by optical focusing systems, Opt. Lett. 32,
313(2001).
976(2007).
22. T.Ve´rtesi,S.Pironio,andN.Brunner, Closingthedetection
2. W. S. Hell, Far-Field Optical Nanoscopy, Science 316, 1153
loophole in Bell experiments using qudits, Phys. Rev. Lett.
(2007).
104,060401(2010).
3. H. He, N. R. Heckenberg, and H. Rubinsztein-Dunlop,Opti-
23. A.C.Dada,J.Leach,G.S.Buller,M.J.PadgettandE.Anders-
calparticletrappingwithhighorderdoughnutbeamsproduced
usinghighefficiencycomputergeneratedholograms, J.Mod. son,Experimentalhigh-dimensionaltwo-photonentanglement
and violations of generalized Bell inequalities, Nat. Phys. 7,
Opt.42,217(1995).
677(2011).
4. L.Paterson,M.P.MacDonald,J.Arlt,W.Sibbett,P.E.Bryant,
24. J.P.Torres,A.Alexandrescu,andL.Torner, Quantumspiral
andK.Dholakia,ControlledRotationofOpticallyTrappedMi-
bandwidthofentangledtwo-photonstates, Phys.Rev.A68,
croscopicParticles,Science292,912(2001).
050301(R)(2003).
5. G.Molina-Terriza,J.P.Torres,andL.Torner,Twistedphotons,
25. H. Di Lorenzo Pires, H. C. B. Florijn, and M. P. van Exter,
Nat.Phys.3,305(2007).
Measurementofthespiralspectrumofentangledtwo-photon
6. G.Gibson,J.Courtial,M.J.Padgett,M.Vasnetsov,V.Pasko,
states, Phys.Rev.Lett.104,020505(2010).
S. M. Barnett, and S. Franke-Arnold,Free-space information
26. L.Carbone,C.Bogan,P.Fulda,A.Freise,andB.Willke,Gen-
transferusinglightbeamscarryingorbitalangularmomentum,
eration of high-purity higher-order Laguerre-Gauss beams at
Opt.Express12,5448(2004).
highlaserpower, Phys.Rev.Lett.110,251101(2013).
7. J.Wang,J.-Y.Yang,I.M.Fazal,N.Ahmed,Y.Yan,H.Huang,
27. M.Granata,C.Buy,R.Ward,andM.Barsuglia, Higher-order
Y.Ren,Y.Yue,S.Dolinar,M.Tur,andA.E.Willner,Terabit
Laguerre-Gaussmodegenerationandinterferometryforgrav-
free-spacedatatransmis-sionemployingorbitalangularmo-
itationalwavedetectors, Phys.Rev.Lett.105,231102(2010).
mentummultiplexing.Nat.Photonics6,488(2012).
28. S.Slussarenko,E.Karimi,B.Piccirillo,L.Marrucci,E.San-
8. N.Bozinovic,Y.Yue,Y.Ren,M.Tur,P.Kristensen,HHuang,
tamato, Universal unitary gate for single-photon spin-orbit
A.E.Willner,andS.Ramachandran,Terabit-scaleorbitalan-
four-dimensionalstates, Phys.Rev.A 80,022326(2009).
gularmomentummodedivisionmultiplexinginfibers,Science
29. G.Molina-Terriza,L.Rebane,J.P.Torres,L.Torner,S.Car-
340,1545(2013).
rasco,Probingcanonicalgeometricalobjectsbydigitalspiral
9. M.W.Beijersbergen,R.P.C.Coerwinkel,M.Kristensen,and
imaging,J.Europ.Opt.Soc.Rap.Public.2,07014(2007).
J.P.Woerdman,Helical-wavefrontlaserbeamsproducedwith
30. V.D’Ambrosio,F.Cardano,E.Karimi,E.Nagali,E.Santam-
aspiralphaseplate, Opt.Commun. 112,321(1994).
4
ato,L.MarrucciandF.Sciarrino, Testofmutuallyunbiased
bases for six-dimensional photonic quantum systems, Scien-
tificReports3,2726(2013).
31. D.Flamm,C.Schulze,D.Naidoo,S.Schro¨ter,A.Forbes,and
M.Duparre´, All-digitalholographictoolformodeexcitation
andanalysisinopticalfibers,J.LightwaveTechnol.31,1023
(2013).
5