Phase aberration correction by correlationin digital holographic adaptive opticsChanggeng Liu, Xiao Yu, and Myung K. Kim*Digital Holography and Microscopy Laboratory, Department of Physics University of South Florida, Tampa, Florida 33620, USA*Corresponding author: mkkim@usf.eduReceived 25 January 2013; revised 27 March 2013; accepted 27 March 2013;posted 27 March 2013 (Doc. ID 184105); published 19 April 2013We present a phase aberration correction method based on the correlation between the complex full-fieldand guide-starholograms in the context of digital holographic adaptive optics (DHAO). Removal of aglobal quadratic phase term before the correlation operation plays an important role in the correction.Correlation operation can remove the phase aberration at the entrance pupil plane and automaticallyrefocus the corrected optical field. Except for the assumption that most aberrations lie at or close to theentrance pupil, the presented method does not impose any other constraints on the optical systems. Thus,it greatly enhances the flexibility of the optical design for DHAO systems in vision science and micros-copy. Theoretical studies showthat the previously proposed Fourier transformDHAO(FTDHAO) is just aspecial case of this general correction method, where the global quadratic phase term and a defocus termdisappear. Hence, this correction method realizes the generalization of FTDHAO into arbitrary DHAOsystems. The effectiveness and robustness of this method are demonstrated by simulations and experi-ments. 2013 Optical Society of AmericaOCIS codes: (090.1995) Digital holography; (010.1080) Active or adaptive optics; (170.4460)Ophthalmic optics and devices; (170.0180) Microscopy.http://dx.doi.org/10.1364/AO.52.0029401. IntroductionAdaptive optics (AO) was initially presented to elimi-nateor alleviatetheimage distortionduetotheatmospheric turbulence in astronomy [1]. Nowadays,AO has become necessary for most of major ground-basedtelescopes[2,3]. Similartotheground-basedtelescopes, thehumaneyealsosuffersfrommanymonochromatic aberrations. In 1994, the ShackHartmann wavefront sensor used in astronomywas first adopted to measure the ocular aberrationsof the human eyes by Billes group [4]. The first AOsystemforvisionsciencewasassembledbyLianget al. in 1997 [5]. Using this system, the retinal im-ages at the cellular scale were obtained. Since thattime, AOin vision science has seen rapid growth withmore and more systems being developed [611]. AOhasalsodemonstratedsuccessinmicroscopy[12].TheaberrationsinducedbyvariationsofrefractiveindexthroughthesamplecanbereducedthroughtheAOsystem[13]. Atypical AOsystemincludesseveral critical hardware pieces: deformable mirror,lenslet array, and a second CCD camera in additionto the camerafor imaging. A novel AO systemwasrecently proposed to replace these hardware compo-nents with numerical processing for wavefront meas-urement and compensation of aberration through theprinciples of digital holography [1419].In the original digital holographic adaptive optics(DHAO) system, the CCD was put in the image planeof thepupil [14]. Althoughwecanobtainadirectmeasurement of the wavefront at the pupil, the imag-ing lens other than the eye lens will introduce spheri-calcurvaturethathas toberemovedbyadditionalmatching lens in the reference beam. Also, the correctguide-starhologramisdifficulttoobtain. Togetafocusedimage,numericalpropagationisnecessary.1559-128X/13/122940-10$15.00/0 2013 Optical Society of America2940 APPLIED OPTICS / Vol. 52, No. 12 / 20 April 2013Toaddresstheseissues, FouriertransformDHAO(FTDHAO) system was proposed [20]. The CCD wasput at the Fourier transform (FT) plane of the pupil,insteadoftheimageplane.Nosphericalcurvaturewas induced by the imaging lens. The CCD could di-rectly record the point spread function (PSF) of thesystem, facilitating the determination of the correctguide-star hologram. Inaddition, withsome modifica-tions, low coherence or even incoherent light sourcemay be incorporated [2126]. Notwithstanding theseadvantagesovertheoriginalDHAO,thecorrectionmethodinFTDHAOhas significant constraint intheopticalconfiguration. Inthispaper,wepresenta more general and flexible correction method.FTDHAO becomes a special case of this generalizedmethod. It is realizedthroughthe correlationbetweenthecomplexfull-fieldhologramandtheguide-starhologram after removal of a global quadratic phaseterm. This correlation operation can eliminate boththe aberration at the entrance pupil and the defocusterm, obtainingacorrectedandfocusedimage, nomatter where the CCDis placed. Except for theassumptionthattheoptical aberrationsmainlylieat or close to the pupil plane, the correlation methoddoes not set any other requirement on the optical sys-tem. Therefore, it will greatly improve the flexibilityof theoptical designforAOinvisionscienceandmicroscopy. The correlation method cannot onlymaintainthe merits possessed by FTDHAO, butalsocanbeappliedfor anyDHAOsystems. It isworthnotingthat asimilar methodwas usedinincoherent DHAO[23,24]. However, in principle,it is different fromthemethodpresentedinthispaper. Correlation operation used in incoherentDHAO results incorrectedintensityinsteadofcor-rectedcomplexamplitude. Theobservationsontheglobal phasetermandthedefocustermpresentedinthis paper were not showninthe methodforincoherent DHAO [23,24].Section2presents adetailedmathematical de-scriptionofthiscorrectionmethod. Inthissection,thesamplingrequirements arealsodiscussed. InSection 3, three simulation examples are given.Correspondingtothesimulations, theexperimentsare described and discussed in Section 4. The majorconclusions are summarized in Section 5.2. PrinciplesAtypicalDHAOprocessincludesphaseaberrationmeasurement,full-fieldimaging,andimagecorrec-tion. The phase aberration is retrieved from a guide-star hologram while the full-field image is obtainedfromafull-fieldhologramthatisdistortedbytheaberration. The image is recovered by removingthemeasuredphaseaberrationfromthedistortedfull-fieldimage[14,20]. Inthispaper, wetreatthecorrectionfromadifferentpointofviewbytakingcorrelationof thecomplexfull-fieldhologramwiththe complex guide-star hologram. Although thederivation is based on a two-lens system, the gener-alizationoftheconclusiontoarbitraryoptical sys-tems is straightforward. The coordinates adoptedfor this two-lens systemareillustratedinFig. 1.For the purpose of brevity, one dimension is adoptedin the derivation. Assume the pupil of the lens L1 istheentrancepupil of thesystem. Theaberration-freepupilfunctionisrepresentedbyP(x1),andthephaseaberrationatthepupil isdenotedby (x1).ThefocallengthsofthelensL1andL2aref1andf2, respectively.Distancesd1, d2,andd3areasde-finedinFig. 1. TheamplitudePSFof thissystemis given byG(x3; x0) =_ dx2exp_ jd3(x3 x2)2_exp_jf2x22__exp_ jd2x22__ dx1A(x0) exp_ jd1(x1 x0)2_P(x1)(x1) exp_jf1x21_exp_ jd2x21_exp_j2d2x1x2__= exp_ jd3x23_exp_ jd1x20_A(x0)_ dx1P(x1)(x1) exp_j_ 1d11d21f1_x21_exp_j2d1x0x1___ dx2exp_ jd2x22_exp_ jd3x22_exp_jf2x22_exp_j21d2x2x1_exp_j2d3x2x3__= exp_ jd3x23_exp_ jd1x20_A(x0)_ dx1P(x1)(x1) exp_j_ 1d11d21f1_x21_exp_j2d1x0x1___ dx2exp_j_ 1d21d31f2_x22_exp_j2_x1d2 x3d3_x2__; (1)Fig. 1. Coordinates for a two-lens optical system.20 April 2013 / Vol. 52, No. 12 / APPLIED OPTICS 2941where a prefactor is dropped. A(x0) is the strength ofthe point source at x0 of the sample plane, and is thewavelength of the illumination. To simplifyEq. (1),we define andas =11d11d21f1and =11d21d31f2: (2)Then, Eq. (1) can be rewritten asG(x3; x0) = exp_ jd3x23_exp_ jd1x20_A(x0)_ dx1P(x1)(x1) exp_jx21_ exp_j2d1x0x1___ dx2exp_jx22_exp_j2_x1d2 x3d3_x2__= exp_j_ 1d3d23_x23_exp_ jd1x20_A(x0)_ dx1P(x1)(x1) exp_j_1 d22_x21_ exp_j2_x3d2d3x0d1_x1_: (3)TofurthersimplifyEq. (3), wedefinethegeneralpupil function asP1(x1) = P(x1)(x1)d(x1); (4)whered(x1) = exp_j_1 d22_x21_; (5)which is the defocus term of the system. The defocusterm becomes unity if the CCD is at image plane ofthe sample. Now, Eq. (1) can be simplified asG(x3; x0) = exp_j_ 1d3d23_x23_ exp_ jd1x20_A(x0)T_ d1d2d3x3 x0_; (6)whereT_ d1d2d3x3 x0_ = FT{P1(x1)][fx=1d1_d1d2d3x3x0_; (7)whereFTdenotesFouriertransform. Thecomplexamplitudeoftheopticalfieldofanextendedobjectat theCCDplaneisobtainedbysuperpositionoftheamplitudePSFof all thesourcepoints, whichis given byO(x3) = q(x3)_ dx0 exp_ jd1(x20)_A(x0)T_ d1d2d3x3 x0_; (8)where q(x3) is given byq(x3) = exp_j_ 1d3d23_x23_: (9)This quadratic phase term appears outside the inte-grals in Eqs. (6) and (8). It plays a crucial role in theimage correction, as will be validated in the followingtwo sections. From the guide-star hologram, we canobtain the amplitude PSF given by Eq. (6). Removingq(x3) fromthe amplitude PSFand setting thesource point at origin, we can obtain a modified am-plitude PSF, as follows:G1(x3) = A0T_ d1d2d3x3_: (10)Similarly, a modified field of the extended object canbe obtained from the full-field hologram and numeri-calremovalofthequadraticphaseterm q(x3),asfollows:O1(x3) =_ dx0 exp_ jd1(x20)_A(x0)T_ d1d2d3x3 x0_:(11)Correlating this modifiedfieldwiththe modifiedamplitude PSF given by Eq. (10), we haveO1G1(x3) =A0_ _ ddx0 exp_ jd1(x20)_A(x0)T_ d1d2d3x3 x0 d1d2d3_T+_ d1d2d3_=d2d3A0d1_ _ ddx0exp_ jd1(x20)_A(x0)T_ d1d2d3x3 x0 _T+(); (12)where denotes correlation. According to the defini-tion in Eq. (7), we have2942 APPLIED OPTICS / Vol. 52, No. 12 / 20 April 2013T_x0 d1d2d3x3 _=_ dP1() exp_j2d1_x0 d1d2d3x3 __(13)andT+() =_ dxP+1(x) exp_j2d1x_: (14)Plugging Eqs. (13) and (14) into Eq. (12), the corre-lation operation results inO1 G1(x3) = d2d3A0d1_ _ ddx0 exp_ jd1(x20)_A(x0)__ dP1() exp_j2d1_x0 d1d2d3x3 ___ dxP+1(x) exp_j2d1x__= d2d3A0d1_ dx0 exp_ jd1(x20)_A(x0)_ _ dxdP+1(x)P1() exp_j2d1_x0 d1d2d3x3____ dexp_j2d1(x )__= d2d3A0_ dx0 exp_ jd1x20_A(x0)_ _ dxdP+1(x)P1() exp_j2d1_x0 d1d2d3x3__(x )= d2d3A0_ dx0 exp_ jd1x20_A(x0)_ dxP+1(x)P1(x) exp_j2d1x_x0 d1d2d3x3__= d2d3A0_ dx0 exp_ jd1x20_A(x0)_ dxP(x) exp_j2d1x_x0 d1d2d3x3__; (15)FromEq. (15), the correlationoperationremovesboth the aberration term(x1) and the defocus termd(x1), obtaining a corrected and focused image nomatter where the CCD is put. The magnification ofthis corrected image is given by (d2d3)d1.Although our derivation is based on a two-lens sys-tem, the conclusion thus rendered can be generalizedto any optical system. The difference lies in the spe-cific expressions for the defocus term d(x1) and theglobal quadratic phase termq(x3). According to theconvolution theorem, Eq. (15) can be implemented byO1 G1(x3) = IFT{FT{O1(x3)]FT+{G1(x3)]]; (16)whereIFTdenotestheinverseFouriertransform.O(x3) andG(x3) canbeobtainedthroughoff-axisholography [1419]. Eliminating the quadratic phaseterm q(x3) from O(x3) and G(x3), we can get O1(x3)andG1(x3).ToachievethefieldsO1(x3)andG1(x3)correctly, the sampling requirements have to betaken into account. Taking FTof the amplitudePSF of the source point at origin, we haveFT{G(x3;0)]=A0FT_exp_j_ 1d3d23_x23_T_ d1d2d3x3__=A0FT_exp_j_ 1d3d23__d2d3_2f2x__FT{FT{P1(x1)](fx)]=A0d2

j(d3)_exp_ j2(d3)d22x21_P1(x1); (17)where denotes the convolution operation and fx isthe spatial frequency in the horizontal direction,fx =d2d3x3: (18)Expanding Eq. (17), it becomes the spectrum of a fi-nite chirp function. The width of this spectrum canestimated as that of the general pupil function[17,27,28]. Because the sampling requirement fortheone-dimensional caseisdifferentfromthatforthetwo-dimensional case, letusnowconsiderthetwo-dimensional case. If the CCD has M N squarepixels with side length x3, then the sampling spac-ings of the spatial frequency along the horizontal andvertical dimensions are given byfx = fy =d2d3x3: (19)Thenthesamplingspacingsontwodimensionsatthe pupil plane are given by [18,19]20 April 2013 / Vol. 52, No. 12 / APPLIED OPTICS 2943x1 =d2d3Nx3and y1 =d2d3Mx3: (20)AssumethediameterofaroundpupilisDthatisestimated as the width of the image order of the holo-gram, andthewidthofzero-orderofthehologramis twice that of the image order. To recover the opti-calfieldatthepupilplane, the pupilsizeDhastosatisfy [29]D

2_d2d34x3: (21)Finally, it is worth mentioning that a special case ofthe correlation method is FTDHAO, where d2 and d3areequaltof2[20].ThenEq.(21)evolvesintotheexpression for the sampling requirement inFTDHAO.3. SimulationsIn the simulations, the focal lengths, f1 and f2, of thelens L1 and L2 are set to be 25 and 200 mm, respec-tively. Wesetd1, thedistancebetweenthesampleand the lens L1, to be 25 mm. The group 4 elements25 of USAF1951 resolution target are used to sim-ulate the amplitude of the sample, as shownbyFig. 2(a). The field of view is780 m 780 m. Thepixel pitch is 3.9 m. A random phase noise rangingfrom to simulates the phase distribution of thesample, as illustratedbyFig. 2(b). All thephaseprofiles throughout this paper are displayed inblue-white-red color map that corresponds to [; |.The wavelength of the laser beamis set to be0.633 m. We present three simulation samples, cor-responding to three different combinations of d2 andd3. In the first case, d2 is set to be 200 mm and d3 tobe 150 mm. Then is calculated as 150 mm, accord-ing to Eq. (2), and q becomes unity. The CCD is putat a defocus plane of the sample. The defocus termdisgivenbyEq. (5). ThesimulationresultsarepresentedinFig. 2. Figure2(c) istheundistortedbut defocused field at the CCD plane when no aber-rationisaddedat thepupil plane. Thesamplingspacingofthespatialfrequencyineitherdirectionis 0.031 line pairs/mm. For the purpose of compari-son, we propagate it to the image plane. The undis-torted focused image is shown in Fig. 2(d). Figure 2(e)shows the simulated phase aberration addedat the pupil plane, whichis givenby two sixth-order Zernike polynomials 4(Z26 Z46) = 4(15r620r4 6r2)[cos(2) sin(2)|. Fromthe full-fieldhologram, we can retrieve the field at the CCD planethat is distorted by this added phase aberration , asshown in Fig. 2(f). Propagating this distorted field tothefocal plane, wecanobtainthefocusedbutde-gradedimage, asshownbyFig. 2(g). TakingFT ofthedistortedfieldshownbyFig.2(f)resultsintheFig. 2. Simulation example where the defocus term dexists and the global quadratic phase term qis unity. (a),(b) Simulated am-plitude and phase. The phase maps are represented by blue-white-red color map that corresponds to [; |. (c) Optical field at the CCDplane without aberrator in place. (d) Focused image of (c). (e) Simulated phase aberration . (f) Full-field aberrated hologram at the CCDplane. (g) Focused image of (f). (h) Full-field phase profile at the pupil with aberration. (i) Guide-star hologram, i.e., the amplitude PSF ofthe system. (j) General pupil function that is the FT of (i). (k) Corrected field at the pupil. (l) Corrected image from (k).2944 APPLIED OPTICS / Vol. 52, No. 12 / 20 April 2013distorted field at the pupil, which contains both theaddedaberrationandthedefocus termd, asshown in Fig. 2(h). The spatial sampling spacing ofthisdistortedfieldis21m. Fromtheguide-starhologram, theamplitudePSFof thesystemisob-tained, which is shown in Fig. 2(i). The general pupilfunction that is the FTof the amplitude PSF is showninFig.2(j).Therootmeansquare(RMS)measure-ment error of the phase of the general pupil functionis0.97radthatcorrespondingtoabout0.15wave-lengths. Subtracting Fig. 2(j) from Fig. 2(h), we cangetthecorrectedfieldatthepupil, whichisgivenby Fig. 2(k). As described by Eq. (16), the correctedimagecanbeobtainedbytakingIFTof Fig. 2(k),whichisshowninFig. 2(l). Comparedtothedefo-cused and distorted field in Fig. 2(f), the correlationoperation eliminates the aberration and meanwhileautomatically focuses the corrected field.In the second case, d2 is set to be 300 mm and d3 tobe200mm. Thedefocus termdbecomes unity,whichsignifiestheCCDisat theimageplaneofthe sample. However, inthis scheme, the globalquadratic phase term q is not unity, which is givenby Eq. (9). The simulation results are shown in Fig. 3.Thebaselineimage, withoutaberration inplace, isshown in Fig. 3(a). Figure 3(b) shows the image dis-tortedbytheaberration illustratedinFig. 2(e).Figure 3(c) shows the affectedfieldat the pupil.The amplitude PSFof this systemis shown inFig. 3(d). Themeasuredaberrationatthepupil isgivenbyFig. 3(e). TheRMSmeasurementerrorofthephaseofthegeneralpupilfunctionis0.91radthat corresponding to about 0.14 wavelengths.Figure 3(f) illustrates the corrected image that showsremarkableimprovementin resolutionandquality,compared to the distorted image in Fig. 3(b). In thiscase, removal of the quadratic phase term q beforethe correlation operation is found to be of significancein the correction. The effect of this term on the cor-rected image is shown in Fig. 4. Figure 4(a) shows themeasuredaberrationat thepupil whenqisnoteliminated before the correlation operation, andFig. 4(c) illustrates the corresponding corrected im-age, which is much degraded compared to Fig. 3(f),that is obtained with qremoved. If qis partiallyremoved, the recovered image becomes better, com-pared to that with quntreated. Figure 4(b) showsthemeasuredaberrationat thepupil whenqisFig. 3. Simulation example where q exists while d is unity. (a) Undistorted optical field at CCD plane. (b) Distorted field at the CCDplane. (c) Distorted field at the pupil. (d) Amplitude PSF of the system. (e) General pupil function. (f) Corrected image.Fig. 4. Demonstration of the effect of q on the corrected image.(a) Measured aberration at the pupil when qis not eliminated.(b) Measured aberration at the pupil when qis partially elimi-nated. (c) Image corrected by (a). (d) Image corrected by (b).20 April 2013 / Vol. 52, No. 12 / APPLIED OPTICS 2945partially eliminated, and Fig. 4(d) shows thecorresponding corrected image.The third simulation sample demonstrates a gen-eral case where both q and d exist. In this case, weset d2 to be 300 mm and d3 to be 150 mm. The sim-ulation results are shown in Fig. 5. Figure 5(a) showsthe distorted full field at the CCD plane that is de-focused and distorted. Note that the quadratic phaseterm qhasbeeneliminated.Thefocusedbutdis-torted image is shown in Fig. 5(b). The distorted fieldat the pupil is given by Fig. 5(c), which includes theadded aberration and defocus termd. The ampli-tude PSF of this system is shown in Fig. 5(d). Again,thequadraticphaseterm qhasbeeneliminated.Fig. 5. Simulation example where both q and d exist. (a) Distorted optical field at the CCD plane. (b) Distorted image. (c) Distortedfield at the pupil. (d) Amplitude PSF of the system. (e) General pupil function. (f) Corrected image.Fig. 6. Schematic diagramof theexperimental apparatus. S,sample; L1L3, lens; A, aberrator; BS1-4, beam splitters.Fig. 7. Experimental example where the defocus termd exists while the global quadratic phase termq is unity. (a) Hologram withoutaberration. (b) Amplitude at the CCD plane. (c) Undistorted field at the pupil. (d) Undistorted image. (e) Distorted hologram. (f) Distortedfield at the CCD plane. (g) Distorted field at the pupil. (h) Distorted image. (i) Guide-star hologram. (j) Amplitude PSF of the system.(k) General pupil function. (l) Corrected image.2946 APPLIED OPTICS / Vol. 52, No. 12 / 20 April 2013TheFT ofthisamplitudePSFisgivenbyFig.5(e)that includesandd. TheRMSmeasurementerrorof thephasemaprepresentedbyFig. 5(e) is0.88 radthat corresponding to about 0.14 wave-lengths. Removing Fig. 5(e) from Fig. 5(c) and takingIFT, we can get the corrected image shown inFig. 5(f). The resolution is completely recoveredand the defocus is eliminated.4. Experimental Results and DiscussionsThe schematic diagram of the experimental setup isillustrated in Fig. 6. The focal length f1 of the lens L1is 25 mm. S represents the sample plane that is atthe back focal plane of eye lens E. Hence, d1equals25 mm. The phase aberrator A is close to the pupil ofthe lens L1. The focal length f2 of L2 is 200 mm. TheCCDhas 1024 768 pixels with the pixel pitch6.45m. Inourexperiments, HeNelaserisusedas light source. The sample under test is a positiveUSAF 1951 resolution target with a piece of Teflontape tightly attached behind. The specular reflectionis blocked by the pupil whose size is set to be 5 mm indiameter, and the CCD receives the diffuse scatteredlightfromtheTeflontape. Apieceofclearbrokenglassservesasthephaseaberrator. TheLensL3is inserted for full-field illumination. Correspondingto the three simulation cases, we present three exper-imental examples by choosing different values of d2and d3. In the first example, we set d2 to be 200 mmand d3 to be 150 mm, which indicates the CCD is at adefocus plane of the sample. The defocus term d iscalculated by Eq. (5). According to Eq. (9), the quad-raticphaseterm qbecomesunity. Asetofimagedata is shown in Fig. 7. The field of view on the sam-ple plane is 594 m 445 m. The full-fieldholo-gram, without theaberratorinplace, isshowninFig. 7(a). By the holographic process, the complex op-tical field at the CCD plane can be achieved, which isshown in Fig. 7(b) [1519]. The sampling spacing ofthe spatial frequency at the CCD plane is 0.037 linepairs/mmineitherdirection, accordingtoEq. (19).TakingFT ofthisfield, thefullopticalfieldatthepupil isobtained, asshowninFig. 7(c). Accordingto Eq. (20), the spatial sampling spacings alongthe horizontal andvertical directions are27and35 m, respectively. For the purpose of comparison,we propagate this defocused field at the CCD planeillustrated by Fig. 7(b) to the image plane, and obtaintheundistortedfocusedimageshowninFig. 7(d),servingasabaseline. Thedistortedfull-fieldholo-gramisshowninFig. 7(e), fromwhichwecangetthe distorted and defocused field at the CCD plane,as shown in Fig. 7(f). The distorted full field at thepupil is shown in Fig. 7(g), which contains the addedaberration and the defocus term. Figure 7(h) is thedistortedimage.Theguide-starhologramisshowninFig. 7(i), fromwhichwe obtaintheamplitudePSFof thesystemthat isillustratedbyFig. 7(j).Figure7(k)showsthegeneralpupilfunction. Sub-tracting Fig. 7(k) from Fig. 7(g), we get the correctedFig. 8. Experimental example where q exists while d takes unity. (a) Undistorted image. (b) Distorted image. (c) Distorted full field atthe pupil. (d) Amplitude PSF of the system. (e) Measured aberration. (f) Corrected image.Fig. 9. Experimental demonstration of the effect of q on the cor-rected image. (a) Measured aberration at the pupil when q is noteliminated. (b) Measured aberration at the pupil when q is par-tiallyeliminated.(c)Image corrected by(a). (d)Image correctedby (b).20 April 2013 / Vol. 52, No. 12 / APPLIED OPTICS 2947field at the pupil. As described by Eq. (16), the cor-rectedimagecanbeobtainedbytaking IFT ofthiscorrected field, which is shown in Fig. 7(l). Comparedto the defocused and distorted field in Fig. 7(f), thecorrelation operation eliminates both the aberrationand the defocus term.In the second example, we set d2 to be 150 mm andd3 to be 200 mm, which indicates the CCD is at theimage plane of the sample. The defocus term d dis-appearswhilethequadraticphaseterm qexists.Figure 8(a) shows the baseline image. The distortedimageisillustratedbyFig. 8(b). Figure8(c)showsthedistortedfullfieldatthepupil. TheamplitudePSF of the systemis illustrated by Fig. 8(d).Figure 8(e) is the measured aberration at the pupil.The recovered image is shown in Fig. 8(f). The reso-lution and contrast are almost completely recovered.Note that the quadratic phase term q has to be re-moved before the correlation operation. The effect ofthis termonthe correctedimage is also demon-strated in this example, shown in Fig. 9. Figure 9(a)shows the measured aberration at the pupil when qisnot eliminatedbeforethecorrelationoperation,andFig.9(c)showscorrespondingcorrectedimage,which is rather blurred compared to Fig. 8(f). Whenqis partiallyremoved, therecoveredimagebe-comes better. Figure 9(b) shows the measured aber-ration at the pupil when qis partially eliminatedand Fig. 9(d) shows the corresponding correctedimage.For the third experimental sample, d2is 150 mmand d3 250 mm. This is a general case where both qanddexist. The results are showninFig. 10.Figure10(a) showsthedistortedfieldattheCCDplane that is defocused and distorted. Note that thequadratic phase term qhas beeneliminated. ThefocusedbutdistortedimageisshowninFig. 10(b).The distorted full field at the pupil is given byFig. 10(c). The amplitude PSF of this system is illus-tratedbyFig. 10(d). ItsFTisshowninFig. 10(e).SubtractingFig. 10(e) fromFig. 10(c) andtakingIFT, wecangetthecorrectedimagethatisshownin Fig. 10(f). The resolution is completely recoveredand the defocus is eliminated.5. ConclusionsIn summary, a novel correctionmethod is proposedfor the DHAO system. It is realized through the cor-relation between the complex full-field hologram andcomplex guide-star hologram. By this method, boththe aberration at the pupil and the defocus of the sys-tem can be removed, which means it is not necessaryto further propagate the corrected full field to the im-ageplane, wherevertheCCDis.It isworthnotingthat if the global phase term q does exist, it has tobe removed before the correlation operation. Other-wise, theaberrationscannot becorrectlycompen-satedfor. Althoughour derivationis basedonatwo-lenssystem, theconclusioncanbegeneralizedtoanyoptical system, if theoptical aberrationsofthe system mainly lie at or close to the pupil plane.It generalizestheFTDHAOintoarbitraryDHAOsystemsandprovidesusaguidancetodesignnewexperimental schemes for applications in AO in oph-thalmology and microscopy. The measurement errorof the phase aberration is due mainly to the deviationof the guide-star spot fromthe ideal point source. Thesize of the incident beam at the pupil in the first pas-sage for the guide-star hologram is usually set to beabout 2 mm in diameter to minimize the effect of theaberrationandgenerateasharpguidestar. Fromsimulationsandexperiments, thiserrordoesexistbutis notsevere. Also, coherent noise seemsinevi-tableif laserisusedasthelightsource. Thiscanbeaddressedbyuseof lowcoherent light source[21,22,25].Research reported in this paper was supported bythe National Eye Institute of the National Institutesof Health under Award No. R21EY021876. The con-tentissolelytheresponsibilityoftheauthorsanddoesnotnecessarilyrepresenttheofficial viewsofthe National Institutes of Health.Fig. 10. Experimental example where both q and d exist. (a) Distorted optical field at CCD plane. (b) Distorted image. (c) Distortedfield at the pupil. (d) Amplitude PSF of the system. (e) Phase map of the FT of (d). (f) Corrected image.2948 APPLIED OPTICS / Vol. 52, No. 12 / 20 April 2013References1. H. W. 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