Single-shot self-interference incoherent digitalholography using off-axis configurationJisoo Hong and Myung K. Kim*Department of Physics, University of South Florida, Tampa, Florida 33620, USA*Corresponding author: mkkim@usf.eduReceived October 2, 2013; revised October 27, 2013; accepted November 5, 2013;posted November 6, 2013 (Doc. ID 198828); published November 27, 2013We propose a single-shot incoherent holographic imaging technique that adopts self-interference incoherent digitalholography (SIDH) with slight tilt of the plane mirror in the optical configuration. The limited temporal coherencelength of the illumination leads the guide-star hologram of the proposed system to have a Gaussian envelope ofelliptical ring shape. The observation shows that the reconstruction by cross correlation with the guide-star holo-gram achieves better quality than the usual propagation methods. Experimentally, we verify that the hologram and3D reconstruction can be implemented incoherently with the proposed single-shot off-axis SIDH. 2013 OpticalSociety of AmericaOCIS codes: (090.1995) Digital holography; (030.0030) Coherence and statistical optics; (100.3010) Imagereconstruction techniques.http://dx.doi.org/10.1364/OL.38.005196Holographicimagingiswell knownforretrievingboththeamplitudeandphaseinformationof anobject [1].Ironically, thoughtheuseof coherent illuminationledtotherealizationofviableholographicimaging, italsorestrictedthewideusageof holographicimaging. Thepossibilityofincoherentholographywasstudiedintheliterature [24], but it was not able to achieve an accept-able quality of hologram until recently. With the develop-ment of digital electronic devices and computer scienceover the past decade, many interesting techniques havebeen proposed to acquire holographic information underincoherent illumination [59]. Among them, the approachthat usesself-interferenceshowsfeasibleperformancefor practical applications [79]. Basically, it separatesthelightfromtheobjectintotwopathsandletsthosebeamsfromthesameobject point interferewitheachother. However, the spatial incoherence of the light fromthe object washes out the fringe of the recorded intensityimage. Instead, the complex hologram can be computa-tionallyretrievedfromthreeor four phase-shiftedim-ages. The completely incoherent imaging process ofthosetechniqueshaswidenedtheapplicationof holo-graphicimaging. RosenandBrookerhaveshownthata 3D profile of the fluorescence object can be obtainedby using this approach [7]. Recently, our group reportedthe successful achievement of holographic recordingandreconstructionofthenaturaloutdoorscenewithaholographiccamerabasedonself-interferenceincoher-ent digital holography(SIDH) [10]. However, theuseof phase shifting remains anissue that still restrictstheapplication. Thephaseshiftingrequirestheobjecttobenearlystationaryformultipleexposures. Hence,the temporal resolution is sacrificed and it is not appro-priatefor high-speedimaging. Moreover, becausetheamountofphaseshiftingvariesaccordingtothewave-length of the illumination source, a large number of ex-posuresisrequiredfor thefull-color imaging. For thecaseof ourholographiccamera, weusedeight phase-shiftedimages tocreateonefull-color hologram[10].In the conventional holography, the phase shifting couldberemovedbymanytechniquessuchasparallelphaseshifting[11], Fizeauinterferometry[12], thefractionalTalbot effect [13], and the random-phase reference wave[14]. However, those techniques require the illuminationand the reference wave to be carefully manipulated. Onthe other hand, in the holographic system based on self-interference, Kelneretal. hadproposedthemethodtorecordthe incoherent hologramby single shot usingthespatial-light modulator(SLM)[15]. Inthisscheme,oneof theseparatedbeampathsisconfiguredsimilarto the 4-f system (hence the propagation direction is in-verted), while the other path does not alter the propaga-tion. The two separated beams from a single object pointinterfere with opposite propagation directions, resultingintheoff-axisinterference.InthisLetter, wepresent asimplesingle-shot SIDHschemeusingtheconfigurationshowninFig. 1. Basi-cally, the optical setupof the proposedtechnique isalmost thesameas the conventional SIDHsetups of[9,10]exceptthattheplanemirrorM1isslightlytiltedfromtheon-axisposition. Different fromtheresult of[15], whichrecords a Fourier hologram, our methodFig. 1. Optical setup of the proposed off-axis SIDH system. BS,beam splitter;M1, planemirror with tilt;M2, curved mirror.5196 OPTICSLETTERS/ Vol.38, No. 23 /December 1, 20130146-9592/13/235196-04$15.00/0 2013 OpticalSociety of Americarecords the hologram in the form of a Fresnel hologram,meaning an off-axis hologram with finite object distance.Moreover, compared to the setup in [15], our setuprequires the combination of different mirrorsonly onceaftertheseparationof beams, hencebeingeasierandmore flexibleinimplementing thesystem.Similar toSIDH, the proposedscheme creates theinterference only by two copies of beams emanated fromthe same point of the object. If the illumination is coher-ent, theintensityof interferencecorrespondingtoonepointof the object willbeur ArojQr; ro Lr; nj2; (1)where Q is a quadratic phase functionand L is a linearphase function that is induced by the tilt of M1 (n is thenormal vector to the surface of M1). Here, ro and r denotethe points on the object and CCD sensor, respectively. Ais a complex amplitude that is related to the intensity oftheobjectpoint. However, theinterferenceoftwolowcoherencefieldshas aGaussianenvelopethatdependson the amount of phase difference between two interfer-ing beams [16]. Hence, the entire interferogram exhibit-ing responsesfromallthe objectpointswillbeUr ZZroAro2 PrQ L

Q

Ldro; (2)where Pr represents the envelope function incorporat-ing the short temporal coherence length of the illumina-tion source (or the object point itself for the self-luminouscase)andtheoptical pathdifferencecausedby thetiltof mirrorM1. Pr isexpressedbyPr exp r2: (3)Here, r is the phase difference between two copiesof beams at r, and is a constant related to the temporalcoherencelengthoftheillumination. Thein-phaseliner 0 comes fromthe intersectionof the inclinedplanewavefront reflectedfromM1andthesphericalwavefront from M2. Hence the shape of Pr will look likea part of theellipticalring.An example of the interferogram obtained from a pointsource object is presented in Fig. 2(a). The optical setupwas configured following the setup shown in Fig. 1 withd0 80 mm, d1 200 mm, and d2 330 mm. The focallengthsoftheconvexlensandcurvedmirrorM2were100and600mm, respectively. Thecenter wavelengthofLEDusedforilluminationwas625nm. Weinsertedanadditional lens (focal length: 100mm) infront oftheCCDtogiveflexibilitytothesetup, andtheCCD(Thorlabs DCU223M; resolution, 1024 768; pixel pitch,4.65 m 4.65 m) was 130 mm distant from the lens. Weused the same configuration for all the experiments con-ducted throughout this Letter except for d0. As in usualoff-axisholographic imaging, a 1 or 1 order spectralcomponent canbeextractedintheangular spectrumdomainwiththeappropriatefilter. TheinverseFouriertransformof the extracted spectral component givesthe complex hologram of the object as shown inFigs. 2(c)and2(d).We can apply the usual propagation methods such asangular spectrum or Fresnel propagation to this retrievedcomplex hologram to obtain the reconstructed image ofthe object. However, the short temporal coherencelength restricts the valid phase information of thiscomplexholograminsideanareawheretheenvelopefunctionislargerthannoise. Asaresult, thecomplexhologram of the point source object is an imperfect quad-ratic phase function as shown in Fig. 2(d). If we considerthiscomplexhologramasaguide-starhologramofthegiven system, a cross correlation with it will be anotheroptionfor the reconstructionmethod:Ir hrh

g; (4)where Ir is the reconstructed image, hr is the retrievedcomplexhologram, andhgistheguide-starhologram.Theoperator representstheconvolution. For easyvariation of the reconstruction plane, the syntheticguide-starhologramcanbecreatedusingEq. (2). TheinEq.(3)shouldbedeterminedinawayresultinginthebestreconstructionresult.Moredetailsregardingageneral discussionof thismethodcanbe foundin[9].We performed a preliminary experiment with theextended object to investigate the feasibility of theproposed off-axis SIDH scheme. The complex hologramof the object, which is a part of resolution target(elements 5 and 6 of group 1), was retrieved by the wholeprocess of off-axis SIDH as explained above. The relatedinformation(recordedCCDimage, angular spectrum,andthecomplexhologram)isprovidedinFig.3.Fromthe retrieved complex hologram, we use various recons-tructionmethods(angular spectrum, Fresnel propaga-tion, andcross correlationwithguide-star hologram)toobtaintheobject imageat thebest focusedplane.Fig. 2. (a) Interferogram(700 700) obtainedbythepointsource object and (b) its angular spectrum. The brighter circleis afilter usedfor extracting 1order. (c) Amplitudeand(d)phase ofthe retrieved complex hologram.December 1,2013 / Vol.38, No. 23 /OPTICSLETTERS 5197Figure4comparesthereconstructionresultsbythreepropagationmethods.Figures 5(a)5(c) showthe reconstructions of thepoint source object at the best focused plane by variousreconstructionmethods: (a)angularspectrummethod,(b)Fresnelpropagation, and(c)crosscorrelationwithguide-star hologram. Those reconstruction results corre-spond to the point spread function (PSF) of each recons-tructionmethod, whichissupposedtobeanimpulseresponse for the best image quality. We can observe thatthere will be little difference in the PSF if we apply thecrosscorrelationwithguide-star hologramfor there-construction. AsshowninFig. 5(d), alongaxis1, theunexpectedripples at the left-hand side of the peakaresuppressedbythecross-correlationmethod. How-ever, for the extended object, the quality of recons-truction is significantly improved by the cross correlationwith the guide-star hologram as shown in Fig. 4. For easycomparison, weprovidetheaverageof y-cut intensityprofiles of reconstruction results in Fig. 4 [see Fig. 5(g)].Thisisbecausethecomplexhologramoftheextendedobjectbecomesindispensablynoisierasthenumberofnonzero point sources at the object plane increases. Thisresult coincides with the discussion in [9]: the presenceof noise (or aberration) makes the cross correlation withthe guide-star hologram more effective. And the shape ofthe PSFs shown in Figs. 5(a)5(c) is asymmetric becausethevalidphaseinformationinthecorrespondingcom-plex hologramis restricted inside the narrowbandrelatedto theenvelopefunctionas showninFig.2(d).There are two ways to address this issue. One is to useaband-passfilterwithnarrowerbandwidth(theband-passfilterusedforexperimenthas10nmbandwidth).ConsideringFig. 2(d), fivefoldormorenarrowerband-widthwill recoverthephaseinformationintheentirefieldof view(FOV). Theother wayis toadopt SLM,whichcanimplementthetiltoftheplanemirrorinthesameplanewiththecurvedmirror. InadoptingSLM,thesystemparametersshouldbecarefullydesignedtoavoidthehigher-orderdiffractions inside the FOV.Theabilitytochangethefocusofthereconstructionresult is one of the important features of the holographytoobtain3Dinformationof theobject. For thecrosscorrelation with the guide-star hologram, computationalFig. 3. (a) Interferogram(700 700) obtainedfromtheex-tendedobject illuminatedbyLED(wavelength: 625nm)and(b) its angular spectrum. The brighter rectangle is a filter usedforextracting 1order. (c)Amplitudeand(d)phaseof theretrieved complexhologram.Fig. 4. Reconstructionresults fromthecomplexhologramshown in Fig.3 by using various reconstructionmethods: (a)angular spectrum,(b)Fresnel propagation, and(c) crosscor-relation with the guide-star hologram. The synthetic guide-starhologram was usedfor (c).Fig. 5. PSF of the reconstruction computed from the complexhologram shown in Fig. 2 by using various reconstruction meth-ods: (a) angular spectrum, (b) Fresnel propagation, and(c) crosscorrelation with the guide-star hologram. The images are 111 111 area cropped from the whole FOV. (d) and (e) show the nor-malized intensity profile along axes 1 and 2, respectively.[Yellow, intensity profile of (a); green, that of (b); red, that of(c)]. The direction and range of axes 1 and 2 are presented asyellow lines in (a), (b), and (c). (f) compares the y-cut intensityprofilesofFig.4(a)(yellow),(b)(green),and(c)(red).Eachgraphwasobtainedbyaveragingthey-cut intensitiesinsidetherangefromx 180to274. (g)illustratestheboundariesof the range as two yellow lines overlaid on Fig. 4(a).5198 OPTICSLETTERS/ Vol.38, No. 23 /December 1, 2013refocusing can be achieved by using the guide-starhologramcorrespondingtothepointsourcelocatedatthetarget focal plane. Toshowtherefocusingability,we have performed the experiment with the object com-posed of a set of LEDs located at different distances. AsshowninFig. 6, threeLEDsarelocatedat nearlythesamedistancewhilethefourthLEDislocatedaround10mmfarther fromtheinput apertureof thesystem.For the reconstruction, we used the synthetic guide-starholograms corresponding to the point sources located atdifferent target planes. Clearly, the reconstruction resultsin Figs. 6(a) and 6(b) show that the proposed scheme cansuccessfullyachieve computational refocusing. We alsoprovide a movie file that shows the continuous change offocusincomputationalreconstruction (Media1).In summary, we have shown that single-shot incoher-ent holography can be realized by the SIDH configurationwith the slight tilt of the plane mirror. To the best of ourknowledge, this is the first successful image recons-tructionfromtheincoherentdigitalhologramrecordedasaformof Fresnel hologram. Theproposedmethodhas an advantage that the setup is very simple and flex-ible. Theoff-axis interferencebetweentwocopies ofbeams works similarly to the usual off-axis holography;hence, byextractinga 1or 1orderspectralcompo-nent, acomplexhologramcanberetrieved. However,becauseoftheshorttemporal coherencelengthoftheilluminationsource,theguide-starhologramofthesys-tem has a reduced area of visible interference. We haveshown that the cross correlation with the guide-star holo-gramis the optional reconstruction method. We areexpecting that the use of a band-pass filter with narrowerbandwidth or SLM will much improve the reconstructionresult bymakingtheshapeoftheguide-starhologramcloserto theperfectquadraticphase function.Theresearchreportedinthis publicationwas sup-ported by the National Eye Institute of the National Insti-tutes of Health under Award No. R21EY021876. Thecontent issolelytheresponsibilityof theauthorsanddoesnotnecessarilyrepresenttheofficial viewsoftheNationalInstitutesofHealth.References1. D. Gabor, Nature 161,777 (1948).2. A. W. Lohmann, J.Opt. Soc. Am. 55, 1555 (1965).3. G. Cochran, J.Opt. Soc. Am. A 56, 1513 (1966).4. F. Dubois, L. Joannes, and J.-C. Legros, Appl. Opt. 38, 7085(1999).5. T.-C. Poon, Adv. Imaging ElectronPhys. 126, 329 (2003).6. Y. Li, D. Abookasis, andJ. Rosen, Appl. Opt. 40, 2864(2001).7. J. Rosenand G. Brooker, Nat.Photonics 2, 190(2008).8. G. Pedrini, H. Li, A. Faridian, and W. Osten, Opt. Lett. 37,713 (2012).9. M.K. Kim,Appl. Opt. 52, A117 (2013).10. M.K. Kim,Opt. Express 21,9636 (2013).11. T. Tahara, Y. Ito, Y. Lee, P. Xia, J. Inoue, Y. Awatsuji, K.Nishio, S. Ura, T. Kubota, andO. Matoba, Opt. Lett. 38,2789 (2013).12. D. Abdelsalam, B. Yao, P. Gao, J. Min, andR. Guo, Appl.Opt. 51, 4891 (2012).13. M. A. Araiza-Esquivel, L. Martnez-Len, B. Javidi, P.Andrs, J. Lancis, andE. Tajahuerce, Appl. Opt. 50, B96(2011).14. T. Nomura and M.Imbe, Opt. Lett. 35,2281 (2010).15. R. Kelner, J. Rosen, and G. Brooker, Opt. Express 21, 20131(2013).16. J. W.Goodman, StatisticalOptics(Wiley, 1985).Fig. 6. Experimental results for the object composed of fourLEDs located at different distances (three LEDs at d0 120 mm; oneLEDat d0 130 mm). (a)and(b)arerecons-truction results obtained from the complex hologram retrievedusing the proposed off-axis SIDH. (a) Best focus at three LEDs.(b) Best focus at the fourth LED. The continuous focus changebetween (a) and (b) is provided as a movie file (Media 1). (c)and (d) are directly recorded images of the object. For (c), thefocus is at three LEDs while (d) is at the fourth LED. In (c) and(d), thed0value(distancebetweentheobjectandtheinputaperture) for eachLEDis providedtoshowthepositionalrelation more clearly.December 1,2013 / Vol.38, No. 23 /OPTICSLETTERS 5199