Probing the dynamics of an optically trapped particle by phase sensitive back focal plane interferometry BasudevRoy,SambitBikasPal,ArijitHaldar,RatneshKumarGupta, NirmalyaGhosh,andAyanBanerjee DepartmentofPhysicalSciences,IISER-Kolkata 2 1 1 [email protected] 0 2 n Abstract: The dynamics of an optically trapped particle are often a determined by measuring intensity shifts of the back-scattered light from J theparticleusingpositionsensitivedetectors.Wepresentatechniquewhich 1 1 measures the phase of the back-scattered light using balanced detection in anexternalMach-Zenderinterferometerschemewhereweseparateoutand ] beat the scattered light from the bead and that from the top surface of our s c trappingchamber.Thetechniquehasimprovedaxialmotionresolutionover i intensity-based detection, and can also be used to measure lateral motion t p of the trapped particle. In addition, we are able to track the Brownian o motion of trapped 1 and 3 µm diameter beads from the phase jitter and . s show that, similar to intensity-based measurements, phase measurements c canalsobeusedtosimultaneouslydeterminedisplacementsofthetrapped i s bead as well as the spring constant of the trap. For lateral displacements, y h we have matched our experimental results with a simulation of the overall p phasecontouroftheback-scatteredlightforlateraldisplacementsbyusing [ plane wave decomposition in conjunction with Mie scattering theory. The positionresolutionislimitedbypathdriftsoftheinterferometerwhichwe 1 v havepresentlyreducedtoobtainadisplacementresolutionofaround2nm 7 for 1.1 µm diameter probes by locking the interferometer to a frequency 5 stabilizeddiodelaser. 3 2 © 2012 OpticalSocietyofAmerica . 1 OCIScodes:350.4855,120.5050,290.1350 0 2 1 Referencesandlinks : v 1. 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S.B.Pal,A.Haldar,B.Roy, andA.Banerjee,“Measurementofprobedisplacementtothethermalresolution limit in photonic force microscopy using a miniature quadrant photodetector,” http://arxiv.org/abs/1107.3427 (2011). 15. G.Volpe,G.Kozyreff, andD.Petrov,“Backscatteringpositiondetectionforphotonicforcemicroscopy,”J. Appl.Phys.,102,084701(2007b). 16. M.BornandE.Wolf,PrinciplesofOptics(PergamonPress,1989). 17. A.Banerjee,D.Das,U.D.Rapol, andV.Natarajan,“Frequencylockingoftunablediodelaserstoarubidium- stabilizedring-cavityresonator,”Appl.Opt.,43,2528–2531(2004). 18. F.Czerwinski,A.C.Richardson,andL.Oddershede, “QuantifyingNoiseinOpticalTweezersbyAllanVari- ance,”Opt.Exp.,17,13255–13269(2009). 1. Introduction A single microparticle trapped by optical tweezers can be used as a micro-probe which can be used in diverse applications. In some of these applications, the object of interest (such as DNA/RNA,molecularmotorssuchasmyosinandkinesin,singlebacteria,etc.)isattachedto the micro-probe which is held controllably in a single trap (or in some cases multiple probes trappedseparately),anditsmotioncarefullymonitoredtoyieldinformationaboutthedynamics oftheobjectofinterestincludingmicro-forcesandtorques[1,2,3,4,5,6],whileinothercases, the probe itself is used to reveal interesting information about surface topographies with nm precision[7],Brownianmotion[8,9],andtostudyfundamentalstatisticalphysicsphenomena [10]. The probe motion is manifested in changes in the forward and backward scattering pat- ternsofadetectionlaserincidentonthetrappedprobe-thescatteringpatternsbeingtypically imaged on position sensitive detectors (PSD) or quadrant photodiodes (QPD). Backscattered detectionisoftenpreferredoverforwardscatteringduetoitsrelativeindependenceofthemor- phology of the trapping apparatus, and has been shown to resolve displacements even in the pmregime[11].However,thehighsensitivityobtainedindisplacementsensingusingPSDsor QPDs is for probe motion in the radial direction, i.e. the in the direction perpendicular to the trappingbeam.Formotionintheaxialdirection,theresolutionismuchlowersinceinthiscase, theintrinsicresolutionofQPDorPSDisnolongerobtained,themotionbeingdeterminedby measuringthechangeinthetotalamountoflightincidentonallthefourquadrantsoftheQPD (orallthepixelsofaPSD),andnotthedifferencebetweenpairsofquadrants(pixelsforaPSD). In this paper, we experimentally demonstrate a technique where the phase of the back- reflectedbeamismeasuredasafunctionofprobemotioninsteadoftheintensity.Ithasbeen shownearlier[12]thatphasemeasurementoftheinterferencepatternformedbysuperposition ofthebackscatteredlightfromtheprobeandtheglasstrappingchambercanbeusedtoquantify probeaxialmotion.Weimprovethesensitivityofaxialmotiondetectionbyimprovingthesig- naltonoiseoftheinterferencesignalbyusingbalanceddetectioninanexternalMach-Zender interferometer configuration. In addition, we show that even the radial motion of the probe causes a change in the phase of the interference signal, which can thus be used to calibrate radialmotionaswell.Tocalibratethechangeofphasewithradialmotion,weperformasim- ulation using plane wave decomposition in conjunction with general Mie scattering theory to findoutthephaseshiftasafunctionofprobedisplacementintheradialdirection.Ourmethod isalsosensitivetoBrownianmotionofthetrappedprobe-whichweareabletoquantifyfor a given trap stiffness from the phase jitter of the interference pattern. Also, as is the case in intensity-baseddetection,afouriertransformofthephasejitteryieldsthecornerfrequencyof theprobe,andthusthespringconstantorstiffnessoftheopticaltrap[13]. 2. Materialsandmethods 2.1. Experimentalapparatusandmethods OuropticaltweezersapparatusisdevelopedaroundaZeissAxiovertA1invertedfluorescence microscope as shown in Fig. 1. A 100X, 1.4 N.A. oil immersion microscope objective (Zeiss plano-apochromat, infinity corrected) is used to couple the beams into the sample chamber. Polystyrene beads of size 1.1 or 3 µm (which serve as our micro-probes) immersed in wa- ter are trapped using a focused single transverse mode 1064 nm solid state laser (Lasever LSR1064ML) beam of 600 mW power and specified M2 quality factor of 1.2. An acousto- opticmodulator(AOM)isinsertedinthebackfocalplaneofthetraptomovethepositionof theproberadiallyinacontrolledmanner.Apairofconvexlensesareusedtoimagetheplane of the AOM on to the back aperture of the microscope objective to ensure that any angular deflectionattheAOMgetsdirectlymappedtoanangulardeflectionatthebackapertureofthe objectivewithoutthebeamwalkingoff.TheefficiencyoftheAOMisaround50%forthefirst order,sothatthemaximumpoweravailablefortrappingisaround180mWafterotherlossesin thecouplingoptics.Detectionisperformedusingaseparatelaserbeam(TopticaDL100,wave- length 780 nm, max power 75 mW) whose power is kept low (around 1 mW finally) enough soastonotmodifythetrappingpotential.Thetrappinganddetectionbeamsarecombinedat theinputofthemicroscopeusingapolarizingbeamsplitter.Thedetectionlaserbeamisback- reflectedfromtheprobeandcollectedatthemicroscopesideportusingadichroicmirrorwhich hasverylowtransmission(butveryhighreflection)at1064nmbutabout50%transmissionat 780nm.Typically,thisback-reflectedlightisincidentonaminiaturequadrantphotodetector [14]toquantifytheradialmotionoftheprobe,anddeterminethepowerspectrumoftheprobe motionfortrapcalibration.Thesamplechamberconsistsofaglassslidecoatedwithgold(top surface of the chamber) and a cover slip of 160 µm thickness (bottom surface facing the ob- jective),withthepolystyreneprobesuspensionsandwichedbetween.Thegoldcoatedslideis used to enhance the back-reflection from the surface which is required to compensate for the lowreflectanceofglassgiventhatthepowerofthedetectionlaserisquitelow.About25µlof thesampleisusedatadilutionof1:10000. The light at the microscope back-focal plane consists of back-scattered light (at 780 nm) from the trapped probe and unscattered light from different regions of the sample chamber. However,inourexperiments,wetrapthebeadclosetothetopslideandseparateoutthescat- teredandunscatteredcomponentsbyaconfocalarrangementconsistingofaspatialfilterused incombinationwithtwoapertures(Fig.1).Thespatialfiltercomprisesofafocusinglensand apinholeofdiameter10µm.Thefocallengthofthelensiscarefullychosensothatthewaist size at the focal spot is close to the pinhole diameter. The lens focuses the backscattered in- tensity pattern on the pinhole to produce an Airy pattern at the pinhole output. The diameter of the detection laser is also kept slightly larger than the probe so that the edges of the beam passbytheprobeandarereflectedbythetopslidedirectly.ThecentralregionoftheAirypat- ternthuscorrespondstolightbackscatteredfromthetrappedprobe(whichisclosetothefocus ofthedetectionlaser),whilethecomparativelydiffuseouterringscorrespondtothereflected light from the gold surface. Then, using two more apertures as shown in Fig. 1, a portion of SAMPLECHAMBER OUTPUTOFPINHOLE OBJECTIVE PINHOLE PZTMIRROR BACK-SCATTERED BS1 DETECTIONLIGHT SIGNAL GEN. DICHROIC MIRROR APERTURE1 DLAESTEERCTION REFERENCE BEAM SPLITTER APERTURE2 BS2 PD2 BACKFOCALPLANE OFOBJECTIVE MIRROR AOM PD1 A B LOCK-INAMPLIFIER TRAPPINGLASER Fig.1:Schematicoftheexperiment.Key:PD1:Photodiode1,PD2:Photodiode2,BS1:Beam splitter1,BS2:Beamsplitter2,SignalGen.:SignalGenerator.ThepinholegeneratesanAiry pattern - the central portion of which contains scattered light from the trapped probe, while thediffuseringcontainsunscatteredreflectionfromthetopgoldslideofthesamplechamber. Aperture 1 picks off the probe signal while Aperture 2 picks off a portion from the ring. The trappinglaserisat1064nmandthedetectionlaserisat780nm. thelightfromtheouterringscanbeseparatedfromthecentralpattern.Weproceedtobeatthe twoseparatedcomponentsinaMach-Zenderinterferometerwherethepathlengthofoneofthe arms is modulated by a piezo mirror to obtain interference fringes on both photodiodes PD1 andPD2(ThorlabsDET110).Thesignalsonthephotodiodesareoutofphaseby180o,sothe differencegivestwicetheindividualsignalamplitude.Thisiswellknownasbalanceddetection andimprovesourfringecontrast,andthusthesensitivityofthephasemeasurementbyafactor of two over standard backscattered interferometry. The difference signal is obtained using a lock-in amplifier (SR630), which also measures the phase of the output signal with reference tothephaseofthedrivingsignaltothepiezomirror.Thefinaloutputofthelock-inisshownin Fig.2(a).Now,thephaseoftheoutputsignalfromtheinterferometerchangeswhentheprobe moveswithrespecttothetopslide,witha2π phasechangeforaxialprobedisplacementofa unitwavelengthofthedetectionlaser.However,theestimationofphasechangeforaxialmo- tionofthetrappedprobewithrespecttothetopslideneededacalibrationoftheaxialdistance moved. This could not be performed in our apparatus due to the lack of a three dimensional piezoelectric motional stage for our microscope. The probe was instead moved transversely (i.e. in the radial direction) by the AOM, and we used the fact that there is a phase change eveninthiscasesincetheaxialdepthactuallyvariesduetothecurvatureofthebead.Fringes obtainedintheinterferometerduetotransversemotionofthebeadareshowninFig.2(b).Ex- perimentswereperformedwithprobesofdiameter1.1and3µm.Forthedistancecalibration, weusedthepixel-to-distancecalibrationtoolinourmicroscopecamera(Axiovision)software and used the diameter of the probes as reference. Phase measurements as a change of probe positionareshowninFig.5.Eachdatapointwascalculatedbyaveragingover10independent measurements. (a) (b) Fig. 2: (a). Balanced detection signal obtained by subtraction of the two out-of-phase signals fromPD1andPD2,thephotodiodeskeptinthetwoarmsoftheMach-Zenderinterferometer. (b)TypicalfringesobtainedinthephasemeasurementastheAOMvoltageischangedtodis- placethetraptransverselyandcauseaphaseshiftinthebackscatteredsignalthatiscapturedin thebalanceddetectionoutput. 2.2. Theoreticalsimulation The phase change of the backscattered signal due to axial motion is quite straight-forward (unitsof2π perhalfwavelengthchangeofaxialdistance).Ontheotherhand,thesamechange whentheprobemovesradiallyisnon-trivial.Atheoreticalestimatewasthusrequiredtomatch the change in the phase of the back-scattered pattern with experimental results. While there existsliteratureontheintensityofthebackscatteredlightfieldinopticaltweezers[15],weare not aware of any study of the phase profile of the backscattered pattern. Furthermore, even Ref. [15] does not take into account the effect of the back-reflection from the top slide of a trappingchamberincalculatingtheintensitypattern.Thisisevenmorecriticalindetermination ofthephaseprofile,sincetheback-reflectedlightwouldinterferewiththelightscatteredfrom thebeaditself,thusmodifyingthephaseprofileoftheoverallbackscatteredsignal.Therefore, a model was required to enable the understanding of the alterations in phase contour as the trapped microsphere was moved transversely. To develop the model, we used a variant of the Angular Spectrum Method (also referred to as vectorial Debye diffraction theory or Debye integral)[16]tocalculatetheelectricfielddistribution. In our model, we considered a x-polarized Gaussian beam of light incident on the micro- sphere,havingtheform w (cid:18) −r2 (cid:19) (cid:18) r2 (cid:19) E(x,y,z)=E 0 exp exp −ikz−ik +iζ(z−z ) iˆ (1) 0w(z−z ) w2(z−z ) 2R(z−z ) w w w w (cid:112) wherer= x2+y2istheradialdistance,E isthepeakintensitywhichwassettounity,w is 0 0 thesizeofthewaistorthetightestspotofthebeam,z isthepositionofthewaistrelativetothe w z=0plane,w(z)isthesizeofthewaistofthebeaminthespecifiedzplane,withw(0)=w , 0 R(z)istheradiusofcurvatureofthephasefrontinthespecifiedzplane,andζ(z)istheGouy phaseshift. As is well known, a Gaussian beam can be decomposed into an infinite number of plane waveswithappropriateweightfactors.Eachoftheseplanewavesinteractwiththeparticlein accordance with the theory of Mie scattering (we consider the case when the wavelength of the light is comparable with the size of the scatterer). This decomposition was performed by applyingatwodimensionalfastfouriertransformontheincidentlightfieldatthepositionof themicrosphere. (a) (b) Fig.3:Thetheoreticalmodel (cid:20) (cid:18)n(cid:19) (cid:18)m(cid:19)(cid:21) E(k ,k )= ∑N ∑M E(x,y,z=0)e2πi kx N +ky M (2) x y n=0m=0 Here,MandNarethesizesofthe2-dimensionalarrayfortheimageofthecrosssectionof theincidentGaussianbeam.Ineachofthek-vectordirections,aplanewaveofunitmagnitude wasassumedtobegeneratedwhicheventuallyscatteredoffthemicrosphere.Theplanewave wasassumedtobexpolarizedinit’sownframeofreference(definedasx(cid:48)). E =eikze(cid:48) (3) i i Forsuchanincidentplanewave,thescatteredwaveisoftheform ∞ E = ∑E [ia N(3)−b M(3)] (4) s n n e1n n o1n n=0 where, 2n+1 E =in (5) n n(n+1) (3) (3) and a ,b are coefficients of scattering, while N and M are vector spherical harmonics n n e1n o1n withm=1[15],givenbythefollowingexpressions (1) h (ρ) (3) n N =cosφ n(n+1)sinθπ (cosθ) e e1n n ρ r d (1) [ρh (ρ)] n dρ +cosφτ (cosθ) e n θ ρ d (1) [ρh (ρ)] n (1)dρ −sinφ π (cosθ)h e (6) n n φ ρ and (3) (1) (1) M =cosφ π (cosθ)h (ρ)e − sinφτ (cosθ)h (ρ)e (7) o1n n n θ n n φ Intheseexpressions, ρ =kr, (1) withh beingthesphericalHankelfunction.Thescatteringcoefficientswerecalculatedfrom n theboundaryconditions,andaregivenbythefollowingwhenthepermeabilityofthemedium andmicrospherearethesame. mψ (mx)ψ(cid:48)(x)−ψ (x)ψ(cid:48)(mx) a = n n n n , (8) n mψ (mx)ξ(cid:48)(x)−ξ (x)ψ(cid:48)(mx) n n n n ψ (mx)ψ(cid:48)(x)−mψ (x)ψ(cid:48)(mx) b = n n n n , (9) n ψ (mx)ξ(cid:48)(x)−mξ (x)ψ(cid:48)(mx) n n n n where, x=ka= 2πnma, λ m= kp = np, km nm ψ (ρ)=ρj (ρ), n n (1) ξ (ρ)=ρh (ρ). n n Heren isrefractiveindexofthemedium,n istherefractiveindexofthemicrosphereandψ , m p n ξ aretheRicati-Besselfunctions. n Thescatterpatternscorrespondingtoeachplanewaveweretransformedfromr,θ andφ basis tothex(cid:48),y(cid:48)andz(cid:48)basisoftheplanewaveframeusingthefollowingtransformationmatrix    sinθcosφ cosθcosφ sinφ E r Es(cid:48)= sinθsinφ cosθsinφ −cosφ  Eθ  (10) cosθ −sinθ 0 E φ Thecomponentsofthescatteredfieldinthex(cid:48),y(cid:48)andz(cid:48)basiswerethentransformedintothe x,yandzbasisofthelabframeusinganappropriatecoordinatetransformation,andeventually, scatteredcontributionsfromalltheplanewaveswereaddeduptoformthefinalbackscattered patternasshowninEqn.11.  cosθ(cid:48)cosφ(cid:48) −sinφ(cid:48) sinθ(cid:48)cosφ(cid:48)  Es=∑∑E(kx(cid:48),ky(cid:48)) cosθ(cid:48)sinφ(cid:48) cosφ(cid:48) sinθ(cid:48)sinφ(cid:48) Es(cid:48) (11) kx(cid:48) ky(cid:48) −sinθ(cid:48) 0 cosθ(cid:48) Once the back-scattered field was calculated at a certain location, the field of the light re- flectedfromthetopglassslidewasalsoestimatedatthesamelocationbyusingEq.1andthe differenceinthetwofieldscalculated.Thephasecontouroftheresultantfieldwasthentaken. Inordertosimulatetheshiftinthelongitudinalpositionofthemicrospherefromthewaistof theincidentdetectionlaserbeam,weusedanaperturefunctionthatwasmovedsimilarlyacross themicrosphere.Thesizeoftheaperturefunctionwastakenasthesizeofthemicrosphereit- self.Astheaperturefunctionwasmoved,theFFTinthereciprocalplaneshiftedaswell.The greater the offset in the radial direction, the more the shift of the FFT from the center in the reciprocalplaneasdemonstratedinFig4.Therefore,thisalsoproducedascatterpatternwhich wasshiftedfromthecenterinthetransversedirection.Next,thechangeinphaseoftheresul- tant field (superposition of scattered and reflected) was determined as a function of aperture offsetbycomparingthephasesofthefringesatdifferentoffsets.Resultsofthesimulationare providedinFig5. (a) (b) (c) Fig.4:FFToftheincidentGaussianastheaperturefunctionisoffsetinthexdirectionby(a) -1µm(b)0µm(c)1µm.TheFFTplaneisassumedtobe2µmfromthefocusinzdirection. Simulation data for 1.1 mm dia probe Experimental data for 1.1 mm dia probe 200 Simulation data for 3 mm dia probe Experimental data for 3 mm dia probe mdeg) 150 Phase shift (100 50 0 0 200 400 600 800 1000 Motion of probe (nm) Fig. 5: Experimental and simulation data for phase change for light scattered off 1.1µm and 3µm microspheres (probes) for known travel in the radial direction. In the experiment, the probes are moved in the radial direction by the AOM, while in simulation, we translate the aperture across the probe. The error bars in the experimental data signify 1σ standard devi- ation.Thestandarddeviationispredominantlyduetodriftsintheinterferometerpathlength, but at low trapping powers, the Brownian motion of the trapped probe also contributes. The datapointforthehighestdisplacementofthe3µmprobehasalargeerrorbarsincetheback- scatteredsignaltonoiseitselfwaslowwiththebeadhavingbeendisplacedsignificantlyfrom thedetectionlaser. 3. Results A comparison of simulation and experimental results are shown in Fig. 5. The experimental datapointshaveerrorbarsthataredueto:a)pathdriftsoftheinterferometer,andb)Brownian motionofthetrappedprobes.Thepathdriftisalimitingfactorofourexperimentaltechnique in the sense that it determines the minimum displacement resolution achievable. In the data shownhere,thepathdriftswereoftheorderof15degreesthatcorrespondedtoaround75nm intermsofdisplacement.Notethatthetransferfunctionbetweendisplacementandphaseshift was determined by a straight line fit to the theoretical data, the slope of which was 4.9±0.8 µm/deg for 1.1 µm diameter probes. However, the path drifts can be reduced by locking the interferometertoafrequencystabilizeddiodelaserbyasimplesideoffringelockingtechnique. Bythis,wemanagedtoreducethephasedriftstoaround40mdegoveraveragingtimesof100 ms,soastogiveapossibledisplacementresolutionofabout2nmatabandwidthof10Hzfor 1.1µmdiameterprobes.Thephasemeasurementtechniquecanthenbeimplementedbyeither scanningtheinterferometerbyamaster-slavetechnique[17],orbyusingasecondfrequency stabilized laser that can be scanned independently to produce fringes in the interferometer. It is interesting to note that the radial displacement resolution can be higher for smaller probes sincethecurvatureofsuchprobeswouldbehigherresultingingreaterphasechangeforsmaller displacements. However, the backscattering cross section would also be lower in these cases, resulting in reduced signal to noise that would serve as a check to the resolution achievable. Theinterferometerpathdriftscouldbereducedbyusingtemperaturestabilizedorfiber-based cavitiesthatcouldthusincreasetheresolutionofphasemeasurementandevenachievesub-nm resolutionindisplacementsensinginaxialaswellasradialprobemotion. The Brownian motion of the trapped probes was also detected in our measurements. This was evident from the fact that the standard deviation in the data for optical powers of 47 and 120mWcomesouttobe19.0(0.3)and16.1(0.2)degrespectively,theuncertaintiesbeingthe statisticaluncertaintiesafteraveraging10times.Whilethestandarddeviationduetopathdrifts is 15(0.2) deg, the enhanced standard deviation would have to be due to another independent process,whichfromtheinversedependenceonpower,couldonlybeBrownianmotion.From thestandardtheoryoferrors,wecouldthususethefactthatσ2 =σ2 +σ2 ,whereσ2 is tot drift bm tot thetotalmeasuredvariance,σ2 andσ2 beingthevariancesduetopathdriftsandBrownian drift bm motionrespectively.Sinceσ2 andσ2 areknownfromexperiment,σ2 ,andthusσ -the tot drift bm bm standard deviation due to Brownian motion can be determined. This comes out to be 55(5) nmand25(5)nmforopticalpowersof47and120mWrespectively.Theseresultscouldnow becomparedtotheoreticalestimatesofBrownianmotionatthegiventrappingpowers.From Ref. [18], the amplitude of Brownian motion of a trapped probe of radius r is given by the expression (cid:115) k T6πηr B δs= , (12) κ2t ave Here, κ is the spring constant of the optical trap, k is the Boltzmann constant, T is the tem- B perature,ηisthedynamicviscosityofwater,andt istheaveragingtime.Thespringconstant ave can be determined by a measurement of the corner frequency f of the trapped particle from c theexpression f ≡κ/(2πγ ) (13) c 0 where,γ =6πrη.Astandardmeasurementofcornerfrequencycanbeperformedbythepower 0 spectrummethod[13].ItiswellknownthatatrappedprobeexecutingBrownianmotionobeys asimplifiedLangevinequationsothatthepowerspectrumoftheprobemotionisaLorentzian. Now,apositionsensitivedetectororaquadrantphotodiodeisusedtorecordthedisplacement of the probe using a detection laser so that the power spectrum can then be obtained. In our technique,thephasehasalinearrelationshipwiththedisplacement-thus,afouriertransform of the jitter in the phase yields a similar power spectrum as is shown in Fig. 6. A Lorentzian fit to the data yields a corner frequency of around 25 Hz, which leads to a trap stiffness of 1.3 pN/µm for an optical power of 47 mW. Then, using a=0.55µm, an integration time of t of8.3ms,temperatureTof300K,viscosityη of0.008Poise,oneobtainsthetheoretical ave estimate of Brownian motion as 49(2) nm, the error being due to the fit error in the corner frequency. This is within 1σ of our experimental estimate of Brownian motion at the same power. A similar measurement at an optical power of 60 mW yields a theoretical estimate of 21(2) nm, which is again within 1σ of the experimental measurement. At higher powers, the path drifts dominated the phase jitter and it was not possible to separate out the Brownian motionoftheprobe.Asacheckofthecornerfrequencymeasurementsusingthephasejitter, weperformedpowerspectrummeasurementsofthedisplacementjitteroftheprobeatthesame powerlevelsusingourquadrantphotodiodedetectionsystem[14],andfoundanagreementto within10%.Wecanthereforeconcludethatourphasemeasurementscanbeusedtodetermine probedisplacementstothelevelofafewnm,aswellastrapstiffnessforanopticallytrapped probe in photonic force microscopy. The technique is more sensitive to axial displacement, however,wecanuseittodetermineradialmotionaswellwiththehelpofatransferfunction determinedfromexperimentandverifiedbysimulation. 10-4 ELxopreenritmziaenn tfailt spectrum Intensity (arb. units) 111000---765 Coefficieyn0t value=s0 ± ± 9 05% Confidence Interval A =0.090995 ± 0.00416 10-8 x0 =0 ± 0 B =634.01 ± 45 10-9 4 567810 2 3 4 5678100 2 3 4 56781000 Frequency (Hz) Fig.6:Powerspectrumobtainedbyfouriertransformofthephasejitterforatrappedprobeof diameter1.1µmforanopticalpowerof47mW.ThedataisfittedtoaLorentzianusingIGOR (cid:112) fittingsoftware.Thefitparametersareshown,withthecornerfrequencygivenby f = (B) c comingouttobeabout25Hz. 4. Conclusion In conclusion, we have developed a phase sensitive interferometric technique for simultane- ouslymeasuringprobedisplacementsandtrapparametersinaphotonicforcemicroscopysetup. ThetechniqueisbasedonbalanceddetectionoftheoutputoftwoarmsofaMach-Zenderin- terferometer set up using the backscattered light from the trapped probe and its environment, andreducestheinherentlackofsensitivityinstandardbackfocalplaneinterferometryforaxial probe displacements. We have extended the technique to measure radial probe displacements andmatchedourexperimentalresultsforradialmotionwithatheoreticalsimulation.Thesim- ulation was performed using plane wave decomposition in conjunction with Mie scattering theorytofindoutthephasedistributionofthebackscatteredsignalduetoradialprobemotion, alsotakingintoaccounttheeffectsofasamplechamber.Inaddition,ourtechniqueissensitive toBrownianmotion,andcanbeusedassuchinanyexperimenttodetermineBrownianmotion optically.Thedisplacementsensitivityislimitedmostlybythepathdriftsoftheinterferometer, thatwehavecontrolledpresentlytoalevelwherewecanachievearesolutionofaround2nm for1.1µmdiameterprobes.Thiscouldbeimprovedfurtherbyusingtemperaturestabilizedor fiber-basedcavities,sothatthecapabilitiesofthetechniquecanbeextendedtoachievesub-nm resolutionforprobedisplacementinphotonicforcemicroscopy. 5. Acknowledgement ThisworkwassupportedbytheIndianInstituteofScienceEducationandResearch,Kolkata, an autonomous research and teaching institute funded by the Ministry of Human Resource Development,GovtofIndia.