Q-plate enabled spectrally diverse orbital-angular-momentum conversion for STED microscopy: supplementary material 

L. YAN1, P. GREGG1, E. KARIMI2, A. RUBANO3, L. MARRUCCI3, R. BOYD2,4 ANDS. RAMACHANDRAN1,*

1 Department of Electrical and Computer Engineering, Boston University, Boston, MA 02215 USA 2 Department of Physics, University of Ottawa, 25 Templeton, Ottawa, Ontario, K1N 6N5 Canada 3Dipartimento di Fisica, Università di Napoli Federico II and CNR‐SPIN via Cintia, 80126 Napoli, Italy 4 Institute of Optics, University of Rochester, Rochester, NY 14627, USA *Corresponding author: sidr@bu.edu

Published       

This document provides supplementary information to “Q‐plate enabled spectrally diverse orbital‐angular‐momentumconversionforSTEDmicroscopy,”http://dx.doi.org/10.1364/optica.2.000900.© 2015 Optical Society of America

http://dx.doi.org/10.1364/optica.2.000900.s001 

1. Mode conversion of q=1/2 plateEquation (2) in the main text shows the output of a left/rightcircularly polarized Gaussian beam passing through a q‐plate,allowingustousethecompletebasisofcircularpolarizationsσ andσ to linearlydecomposeanystateofpolarization(SOP) tocalculatethecorrespondingoutput.Particularly,foraq=1/2‐plateinwhichthebirefringenceismaximalatacertainwavelength,i.e.

∆δ 2m 1 π, form ∈ integer(and ignoring the constantphase offsetD0), we obtain for different input polarizations, allpossiblevortexmodes,whoseanalyticalformsarewellknown[1]:

∝ ∙ √ (S1)

∝ ∙ √ (S2) ∗ ∝ ∙ (S3) ∗ ∝ ∙ (S4)

whereTM01 andTE01 are the radially andazimuthallypolarizedcylindricalvectormodes,andOAM*(r)aretheso‐calledspin‐orbitanti‐aligned orbital angularmomentum (OAM)modes since thehelical phase front ( ) and the polarization ( ) have theoppositehandedness.Theseare,however,notthedesiredbeamsforSTEDmicroscopy,becausetheyresultinnon‐zerointensityatthecenterwhenfocusedwithahighnumericalaperture(NA)lens[2].Usingahalf‐waveplate,OAM*(r)canbereadilyconvertedintoOAM(r)thatarespin‐orbitaligned,andwhichpreservethecentralnullasrequiredforSTED–weperformthisoperation inall themicroscopy experiments described in this report. Fig. S1 showscameraimagesofthepolarizationprojectionsofeachofthemodesweexcited,confirmingthatallpolarizationpatternspredictedbyEqs. (2, S1‐S4)may be experimentally realizedwith ourq‐plate

FigureS1CameraimagesofconvertedTE01,TM01,OAM+andOAM-modesthroughaq=1/2plate(andahalf‐waveplateforOAM(r)),withandwithoutpassingthroughalinearpolarizerplacedinfrontof the camera. The circles with arrows indicate the differenttransmissionaxisorientationofthepolarizer.

setup.Notethat,becauseOAMmodesarecircularlypolarized,theprojected intensity patterns do not change as the polarizer isrotated. An interesting feature of the q‐plate is that switchingbetweentheradially,azimuthally,andcircularlypolarizedvortexmodesisachievedonlybyalteringtheinputSOP[3].

2. Phase retardation and group index difference Lightpropagatinginamediumaccumulatesphasethatisgivenby

∙ ,wherek=n2S/O is thewavenumber in thatmedium,and n is the refractive index of the medium and usually adispersive quantity, and d is the thickness of the q‐plate. Forbirefrigentanddispersivemediumlikeq‐plates,wemodelthetwopolarizations as twomodes in analogy to optical fibers [4]. ThephaseretardationΔ inEq.(1‐4)actingasadetuningparameter,afterTaylorexpansionatsomecentral frequencyZ0 (thecentralfrequency is chosen such that ∆δ 2m 1 π, form ∈integer,isgivenby

∙ ! ∙⋯ (S5)

wheretheprimesrepresentderivativeswithrespecttoZ.Thesederivatives are related tophysical quantities such asdifferentialgroupindex ,differentialgroupvelocity(vg)dispersion(GVD),etc.asthefollowing:

∆ ′ ′′ (S6)

Wepointoutthatthebirefringenceinducedphaseretardation,andthereforethemode(periodic)conversionpropertyof theq‐platedescribedbyEq.(2),isdominantlycontrolledbythegroupindexdifference'ngratherthantheindexdifference'nofthetwopolarizations.,muchlikeinopticalfibers.Higherorderterms,likeGVDmaymattereitheroververy largewavelengthrangesor inwaveguides(wherethesequantitiescanbeengineered[4]).Ignoring GVD effects, and utilizing Eq. (3) from the main

manuscriptsalongwithEq.S5,S6,itbecomesclearthatthephaseretardation is oscillatory with frequency, with the oscillationperiod∆ beingdirectlyrelatedtodispersivecharacteristicsoftheliquidcrystalby[5]:

∆ ∝ ∙∆ (S7)

Thisprovidesconsiderableflexibilityindesigningtheq‐plate’smaterialaswellasgeometricparameters tocontrol∆ ,whichthencontrolsthecolorcombinationspossiblewithSTED.

3. Methods  

Definition of Extinction Ratio (ER) Equation(2)inthemaintextindicatesthatthelocalintensityatthecenterof theoutputbeamcomesonly from theGaussianmode,thusallowingustoquantifythemodalcontentoftheoutputbeam.We first set the q‐plate bias to generate a Gaussian output

(equivalent towhennoq‐plate isplaced in thebeampath),andnotethepeakintensityofsuchGaussianbeamas0dB.Thenline‐cutprofilesthroughthebeam‐centeraremeasuredatotherbiasvalues, and the ratio of the output beam center intensity to theaforementionedGaussian peak intensity is defined as extinctionratio(ER)asshowninFig.S2.Clearly,theERapproachestounity

(0dB)foraGaussianbeam,andzero(‐fdB;butpracticallylimited

bythedynamicrangeoftheimagingcamera)foranOAMbeam.

Confocal microscopy setup with a q‐plate Ahome‐builtconfocal laserscanningmicrocopysetupisusedtocharacterize the performance of the q‐plate. Laser beams ofmultiplecolors(indicatedbyO1andO2inFig.4)arecombinedintoa singlemode fiber (SMF: RGB 400, Corning) with polarizationcontrollers (F‐POL‐IL, Newport) to obtain the desired vortexmodes(which,asmentionedinS1,maybeachievedbycontrollinginput polarization). The combined beam is collimated by areflective collimator (RC08FC‐P01, Thorlabs) to minimizechromaticaberrations,andthenpassedthroughaq‐plate(q=1/2)andahalf‐waveplate(toformaspin‐orbitalignedOAMbeamthatis normally used in STED nanoscopy), and then focused by anobjectivelens(UPlanSApo60X/1.35oil,Olympus)ontothesampleplane.WemeasurethePSFoftheexcitationanddepletionbeamsbyscanning(NanoMax‐TS,Thorlabs)asampleofgoldbeads(150nmAunanoparticle,Cytodiagnostics)mountedusingGlycerolonastandard microscope cover slip. The scattered/reflected light iscollectedby amultimode fiber (M50L02S‐A,Thorlabs) that alsoacts as a confocal pinhole (although confocal is in principle notrequired in STED), and is detected by a single photon detector(SPCM‐AQR‐15,PerkinElmer).Acustom‐madeLabviewprogramtogetherwithadataacquisition(DAQ)card(PCIe6353,NationalInstruments) areused to control the system.A supercontinuumlaserwithbuilt‐inbandpassfilters(Solea,PicoQuant)andaCWTi:sapphire laser (3900S, Spectral Physics) pumped by a coherentVerdi10DlaserareusedtocharacterizethemulticolorPSFs.The q‐plate is biased by a function generator (model 20,

Wavetek)thatoutputs2kHzsinusoidalACvoltagewithzeroDCoffset.Theq‐platecanalsobebiasedthroughtheDAQtoprovidefurthersystemintegrationandmoreaccurateanddynamictuning.Alltheexperimentsdescribedinthisletterareperformedunder2kHzACbias.

Determination  of  the  bandwidth  of  the  (Gaussian)  excitation beam Wetakefree‐spacemodeimagesandmeasurethecorrespondingPSFsatdifferentbiassettings.Fig.S3showstheline‐cutprofiles,modeimagesandPSFsareobtained,andFig.S2discussedhowtoobtain the ER of a beam. The Gaussian‐ness of our beams isquantifiedbynotingthataperfectGaussianbeam,underhigh‐NAfocus, will result in a PSF that is a Jinc function. Thus, wemaydefine an overlap integral (OL) between line‐cuts of theexperimentally measured PSFs, with appropriately resized Jincfunctions(Fig.S3).

FigureS2Measuringtheextinctionratio(ER).

2

 

Figure S4 Overlapped PSFs of excitation and STED beams atmultiple wavelengths (at corresponding bias settings for the q‐plate).Blue traces are line‐cutprofiles along the centerofdonutPSFs, and inset numbersdescribe the STEDwavelength and thePSFextinctionratioindB.Scalebars:500nm.

We note that when the ER is better than ‐3 dB (i.e., morenegative value), the OL varies only <1% from that of a pureGaussianbeam.Nocentralintensitydropisobserved,andthesizeofthePSFonlychangesby<30nm.Incontrast,astheERdropsbelow‐7dB,acentraldipofabout22%intensitydropbecomesevident,andtheOLdropsbyabout5%.Thus,weset‐3dBasthe

ERatwhichtodescribeabeamasbeingGaussianenoughforthepurposesofSTEDmicroscopy,andwe findsuchaGaussian likebeammaybeobtainedoveraslargeabandwidthas60nm(Fig.2ofmaintext).

4. PSFs results at other wavelength combinations BesidestheresultsweshowinFig.5ofmaintext,PSFsatotherwavelength pairs are also measured. Specific wavelengths arechosenbasedoncommonlydeployedfluorescencedyesinordertotestthefeasibilityofoursystemtobeapplicableforarangeofSTED microscopy applications. Fig. S4 shows the measuredoverlappedPSFs(withfalsecolorscorrespondingtotherespectivewavelengthsofillumination)inthelateralandtheaxialplanes.Wenote the following: 1) for the STEDbeams, all pairshave

extinction ratios better than ‐13 dB, as required for STEDnanoscopy,indicatingthehighqualityofdonutPSFsgeneratedinour system across a wide spectral (RGB) range; 2) for eachwavelengthpair,PSFsoftheexcitationandSTEDbeamsarewellalignedbothinthelateralandtheaxialplane,whichaddressestheco‐alignmentchallengeofcurrentlydeployedSTEDsystems.Itisalsoexpectedthatsincetwocolorsoriginatefromthesamesinglemode fiber aperture, they will be much less susceptible toenvironmental changes compared to systems using twodistinctsources;3)otherthanchangingtheq‐platebiassettings,noopticalre‐alignmentisneededforswitchingbetweenwavelengthpairs.The transmission loss of the q‐plate is measured at different

wavelengths. Fig. S5 shows the results. The overall loss in thevisible wavelength range is less than 0.5 dB, corresponding to>89%transmission.Notethat,thisalreadylowlossvaluecanbefurtherreducedbyapplyingappropriateantireflectioncoatingstothesubstrateoftheq‐platedevice. 

FigureS3Dashedbluecurves:theidealJincfunction.Scatterlines:line‐cutprofilesofmeasuredPSFs(bottominsets).ER:extinctionratio obtained from free‐space mode image (top insets); OL:overlapintegralbetweenthescatterlineandtheJincfunction.

3

References  1 S. Ramachandran and P. Kristensen, “Optical vortices in fiber,” Nanophotonics 2 (5‐6), 455–474 (2013)  2 B. Richards and E. Wolf, ‘‘Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,’’ Proc. R. Soc. London, Ser. A 253, 358–379 (1959). 3 P. Gregg, M. Mirhosseini, A. Rubano, L. Marrucci, E. Karimi, R. W. Boyd, and S. Ramachandran, "Q‐plates as higher order polarization controllers for orbital angular momentum modes of fiber," Opt. Lett. 40, 1729‐1732 (2015) 4 S. Ramachandran, “Dispersion‐Tailored Few‐Mode Fibers: A Versatile Platform for In‐Fiber Photonic Devices,” J. Lightwave Technol. 23, 3426‐3443 (2005). 5 S. Ramachandran, Z. Wang, and M. Yan. "Bandwidth control of long‐period grating‐based mode converters in few‐mode fibers." Optics letters 27 (9), 698‐700 (2002). 

FigureS5Measuredtransmissionlossoftheq‐plateasafunctionof wavelength. The power meter reading has a fluctuation ofabout±0.06dB,manifestedastheerrorbarsshowninthefigure.

500 550 600 650 700 7500

0.2

0.4

0.6

0.8

1

Wavelength (nm)

Transm

ission loss (d

B)

4