High-resolution synthetic-aperture digital holography with digital phase and pupil correction

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1 High-solution synthtic-aptu digital hologaphy with digital phas and pupil coction Abbi E. Tippi, Abhishk Kuma and Jams R. Finup * Institut of Optics, Univsity of Rochst, Rochst, NY USA * finup@optics.ochst.du Abstact: A 218 mga-pixl synthtic aptu was collctd by ast scanning a CCD dtcto in a digital hologaphy imaging xpimnt. Fams w mosaickd togth using a two-stp coss-colation gistation. Phas coction using shapnss mtics w utilizd to achiv diffaction-limitd solution Optical Socity of Amica OCIS cods: ( ) Digital hologaphy; ( ) Imag constuction tchniqus. Rfncs and links 1. B. Hcht, J. Colinau, and J. C. Lhuau, Shot-ang synthtic aptu imaging at 633 nm by digital hologaphy, Appl. Opt. 41(23), (2002). 2. J. H. Massig, Digital off-axis hologaphy with a synthtic aptu, Opt. Ltt. 27(24), (2002). 3. D. Claus, High solution digital hologaphic synthtic aptu applid to dfomation masumnt and xtndd dpth of fild mthod, Appl. Opt. 49(16), (2010). 4. L. D. Wav, J. S. Fnd, and C. R. Dhainaut, Dsign considations fo multipl tlscop imaging aays, Opt. Eng. 27, (1988). 5. P. Nisnson, Spckl Imaging with th PAPA Dtcto and th Knox-Thompson Algoithm, in Diffaction- Limitd Imaging, D. M. Alloin and J.-M. Maiotti, ds. (Kluw Acadmic Publishs, 1989), pp M. Guiza-Sicaios, S. T. Thuman, and J. R. Finup, Efficint subpixl imag gistation algoithms, Opt. Ltt. 33(2), (2008). 7. R. A. Mull and A. Buffington, Ral-tim coction of atmosphically dgadd tlscop imags though imag shapning, J. Opt. Soc. Am. 64(9), (1974). 8. S. T. Thuman and J. R. Finup, Coction of anisoplanatic phas os in digital hologaphy, J. Opt. Soc. Am. A 25(4), (2008). 9. S. T. Thuman and J. R. Finup, Phas-o coction in digital hologaphy, J. Opt. Soc. Am. A 25(4), (2008). 10. A. E. Tippi and J. R. Finup, Phas-o coction fo multipl plans using a shapnss mtic, Opt. Ltt. 34(5), (2009). 11. U. Gnand, Abstact Infnc (Wily, Nw Yok, 1981). 12. A. E. Tippi and J. R. Finup, Multipl-plan anisoplanatic phas coction in a laboatoy digital hologaphy xpimnt, Opt. Ltt. 35(19), (2010). 13. A. Kozma and C. R. Chistnsn, Effcts of spckl on solution, J. Opt. Soc. Am. 66(11), (1976). 14. A. J. Pag, L. Ahnbg, and T. J. Naughton, Low mmoy distibutd constuction of lag digital hologams, Opt. Expss 16(3), (2008). 1. Intoduction Fo many imaging applications, on dsis to achiv high solution whil maintaining a wid fild-of-viw. Typically, with convntional optics, solution o fild-of-viw must b sacificd to achiv th oth. This tadoff is du pimaily to th incas in abations as th lns diamt and fild-of-viw incas. Fo optical systms that hav dmanding spcifications on both th solution and fild-of-viw, such as fo lithogaphic lnss, th numb of optical lmnts in th systm incass to compnsat and coct fo ths ffcts. Futhmo, fo focal-plan imaging applications, th numb of pixls in th dtcto aay can limit th spac-bandwidth poduct. In this pap, w discuss a laboatoy xpimnt in which w w abl to achiv both high solution and a wid fild-of-viw simultanously, all without imaging lnss in th systm, with digital hologaphic aptu synthsis by using a computational imaging appoach and xtnsiv post-dtction coction algoithms. (C) 2011 OSA 20 Jun 2011 / Vol. 19, No. 13 / OPTICS EXPRESS 12027

2 Lag synthtic aptus can b fomd by combining togth multipl fams of digital hologaphy data, thby incasing th solution and th spac-bandwidth poduct of th systm [1 3]. As th synthtic aptu bcoms lag, th systm tolanc quimnts bcom mo dmanding. Without pop knowldg and alignmnt of fams within th synthtic aptu, sidual wavfont os will occu du to latal and longitudinal pupil gomty mismatchs [4]. Claus [3] usd an itativ appoach to stimat fam-to-fam phas offsts. To avoid abations du to a dift in th fnc bam (local oscillato) phas lativ to th objct bam, phas-o coction algoithms a ndd. To coct fo th axial position of th plan of th objct, which may vay in dpth acoss its sufac, dynamic focusing must b pfomd whn th objct dpth xcds th dpth of fild. Massig coctd fo dfocus [2]. Bint t al. [1] showd that optical abations such as dfocus and astigmatism w psnt aft foming an imag fom a synthtic aptu. Compnsation fo abations in [1] w found manually with us input. W dscib a laboatoy xpimnt in which w assmbld a lag, 218 mga-pixl synthtic-aptu digital hologam. Ou synthtic aptu constuction psnts significant incas in pixl count compad to cnt synthtic aptus such as Rf [3]. This dmonstats a non-tival advancmnt in synthtic aptu digital hologaphy, givn th dmands on th systm tolanc and stability of th xpimnt, as wll as th gat ffcts of wd-angl anisoplanatic phas os. W dvlopd phas-o coction algoithms (which qui no us intvntion) to coct fam-by-fam piston os du to fnc bam dift, fnc bam location os, local focusing os, and oth high-od os. W show quantitativ sults, dmonstating th diffaction-limitd imags capabilitis upon implmnting ths tchniqus. Ths algoithms coct fo pupil mapping os which would othwis caus spac-vaiant abations ov th aptu fo a wid fild-of-viw and allviat th tight data collction quimnts, making th synthsis of lag aptus mo pactical and affodabl. 2. Expimntal stup and data acquisition Th xpimntal stup was configud in an off-axis, lnslss Foui tansfom gomty, as illustatd in Fig. 1. Light fom an agon-ion las with an output pow of 450 mw, opating at a wavlngth of 514 nm was spilt into two ams, a fnc bam and an objct illumination bam. In th fnc am, nutal dnsity filts ducd th intnsity of th bam and a 5X micoscop objctiv focusd th light into th nd of a 3.5 μm singl-mod fib. Th output of th fib was placd in th plan of th objct and svd as a diffaction- Fig. 1. Expimntal stup fo off-axis, lnslss Foui tansfom digital hologaphy synthtic aptu imaging. B/S, bamsplitt; M, mio; SF, micoscop objctiv plus pinshol spatial filt; Obj, micoscop objctiv and Syn. Ap, width of synthtic aptu aft ast scanning CCD. limitd fnc point souc. Though a collction of mios, a scond bam was positiond to flood illuminat th objct at an off-axis angl of 18 dgs. Th objct usd in this (C) 2011 OSA 20 Jun 2011 / Vol. 19, No. 13 / OPTICS EXPRESS 12028

3 xpimnt was a USAF chom-on-glass ba tagt. Flat whit paint was applid on th chom sid of th tagt as a mans of poviding diffus flction, and that sid of th tagt facd away fom th illumination bam and th cama, so th objct was sn though th back of th glass plat. A potion of th USAF tagt was maskd, such that th ffctiv objct aa was 4.7 cm by 6.3 cm. Th USAF tagt was positiond nomal to th optical axis of th systm, making it paalll to th dtcto; mio flctions fom th glass substat and th mio-lik chom sufac flctd away fom th dtcto du to th off-axis illumination angl. Th distanc fom th objct to th dtcto was z = 3.32 m. Th dtcto was a Rtiga 2000R Q-Imaging CCD cama with a pixl pitch of 7.2 μm in both dimnsions and an aa of 1200 x 1600 pixls (8.64 mm x mm). Th CCD was mountd on two tanslation stags. Th hoizontal tanslation stag was mountd to th tabl, whil th vtical stag hld th cama. In od to acqui all th fams ncssay fo th full-aptu imag, th cama was ast scannd, lft to ight, to gath a ow of fams stating at th bottom lft-hand con of th aptu aay, and thn subsqunt ows of fams w gathd bottom to top. W always acquid fams in th sam diction to minimiz hystsis and mchanical backlash of th tanslation stags. Th tanslation btwn fams was about 833 pixls hoizontally and about 555 pixls vtically giving an appoximatly 50% ovlap btwn fams; th dundant data fom th ovlapping aas svd to hlp align and gist fams sinc th accuacy and pcision of th tanslation stags usd in this xpimnt povd to unaccptabl fo alignmnt quimnts. Th xposu tim fo ach fam was 80 ms. Th total acquisition tim fo 21 x 21 fams (ov 800 Mpixls, of which ¾ w dundant), includd a two scond wait and sttl tim fo th tanslation stags aft moving to ach location, a dak fam xposu (fo calibation), an objct-only fam and a fnc-only fam (fo diagnostics), and a hologaphy fam fo ach position, was 5 hous and 58 minuts und automatic comput contol. 3. Rgistation and mosaicking of a lag synthtic aptu Th hologam intnsity in th dtcto plan is dt * * I R H R H RH HR (1) wh R is th fnc fild and H is th objct fild. A hologaphic imag can b constuctd by coving th fild in th dtcto plan by standad tchniqus (Foui tansfom, zo out all but th dsid tm and invs Foui tansfom) and thn digitally popagating th sulting fild to th imag plan. Figu 2(a) (d) shows th imag fom a singl CCD fam; Fig. 2(a) shows th nti fild of viw of th imag, whil Fig. 2(b) (d) show zoomd-in gions of th imag to mphasiz th fin solution lmnts of th USAF tagt. Fo a singl fam, th finst solvd bas w (goup, lmnt) (1,1) in th hoizontal diction and (1,3) in th vtical diction, cosponding to 2.00 lin-pais (lp) p mm and 2.52 lp/mm, spctivly. Aptu synthsis was pfomd by fist xtacting th dtcto-plan complx fild of ach individual fam and thn mosaicking togth thos filds. W found it impotant to gist and mosaic th xtactd filds ath than th hologam intnsitis bcaus th latt s quasi-piodic fing pattn, along with piston phas changs btwn fams, mad fam-to-fam gistation of hologams vy challnging. Fam-to-fam gistation must b don with gat accuacy, and Appndix A shows that to kp spac-vaiant phas os du to fam tanslation to 1/10 wav, on must gist th fams accuatly to within 1/7 pixl (about on micon fo ou CCD). (C) 2011 OSA 20 Jun 2011 / Vol. 19, No. 13 / OPTICS EXPRESS 12029

4 Fig. 2. Imag constuction of (a) (d) Singl fam; () (h) Initial mosaic; (i) (l) Optimizd mosaic; (m) (p) Rfnc phas coction; (q) (t) High od phas coction ov nti FOV; (u) (x) High od phas coction ov ROI using M 2. Column x 6500 pixl aa of nti imag; Column x 2500 pixl subst of Col. 1; Column x 650 pix subst of Col 2; Column x 160 pixl subst of Col 3. (C) 2011 OSA 20 Jun 2011 / Vol. 19, No. 13 / OPTICS EXPRESS 12030

5 Fig. 3. Amplitud of th mosaic of 21 x 21 (12100 by pixls) fams aft initial cosscolation of singl adjacnt fam and avaging ovlap gions. Fig. 4. Amplitud of th nti 21 x 21 synthtic aptu (12100 by pixls) aft cosscolation gistation of ovlapping fam aas. W found that th spckls in th objct bam at th dtcto changd somwhat ov th long data collction tim, contibuting to os whn gisting ach fam within th lag aptu. To minimiz th accumulation of gistation os acoss th synthtic aptu in ou initial mosaic, w gistd fams in a spial-lik pattn, stating in th cnt and spialing out countclockwis (vn though th fams w gathd in a ast scan od), analogous to what is don with phas constuctos [5]. Ou aly pliminay sults showd that a spial gistation was mo ffctiv than stating in on con of th aay and gisting in a ast scan od. Futh impovmnts in th gistation appoach should b possibl. Th mosaicking of th lag synthtic aptu was a two-stp pocss, consisting of an initial fam gistation basd on th coss-colation of th nw fam with a singl adjacnt fam, followd by th coss-colation of th nw fam with th ovlap gions fom sval fams. Using th a pioi knowldg that th is ~50% ovlap btwn adjacnt fams, th gistations w pfomd using an fficint subpixl coss-colation [6] stimating th x and y tanslations btwn th fams to within a tnth of a pixl and th piston phas. Th st of imags fomd using ths initial positions and phass fo th mosaic a shown in Fig. 2() (h). Th impovmnt ov a singl fam in solvd bas of th USAF tagt was fom (1,1) to (2,4) hoizontally and fom (1,3) to (2,5) vtically. Not that an impovmnt by (n bas, m lmnts) is quivalnt to a solution impovmnt by a lina n /6 facto of 2 m. (C) 2011 OSA 20 Jun 2011 / Vol. 19, No. 13 / OPTICS EXPRESS 12031

6 Th initial mosaic fomd by avaging all ovlapping fams, but only taking into account a singl adjacnt fam in th gistation pocss, is shown in Fig. 3. This full aptu taks into account th tanslation and phas infomation fom th initial cosscolations btwn adjacnt fams. Th ovall mosaic had a blocky, patchwok-lik appaanc instad of th xpctd unifom spckl pattn. Th dak gions indicat dstuctiv intfnc btwn filds fom diffnt fams and val os in th piston phas stimats and/o gistation os. To impov th unifomity and accuacy of th mosaic whn avaging aas of ovlap btwn fams, w pfomd a scond coss-colation. Now, instad of instad of cosscolating ach subsqunt individual fam with on adjacnt nighbo, w pfomd a coss-colation in th aa of ovlap btwn th singl fam and th avagd sum of pvious fams. This scond stp taks into account multipl fams of data and allows fo adjustmnts in th tanslations and piston phas of fams fo btt agmnt within th total synthtic aptu. Figu 4 shows th sulting full aptu aft both th initial mosaicking and th finmnt, claly showing impovd unifomity and dmonstating th nd fo this finmnt. Th imag aft this find mosaicking is shown in Fig. 2(i) (l). W notd a futh incas in solution aft this sconday alignmnt of fams: (3,2) is solvd fo hoizontal bas and (3,1) is solvd fo vtical bas. 4. Phas coction and imag constuction Th full-aptu complx fild fom th objct in th dtcto plan was usd to constuct th fin-solution, wid-fild-of-viw imag. W applid phas coction tchniqus to th fild to compnsat fo sidual phas os that main aft th piston phas coction pfomd duing mosaicking. Th phas coction algoithms w usd w cocting (i) th location of th fnc point souc, (ii) sidual high-od phas os, and (iii) local gion phas os du to vaiations in axial dpth. 4.1 Rfnc coction Th complx fild tivd in th dtcto plan aft windowing out oth hologam tms is H HR *. W wish to divid out th complx conjugat of th fnc fild, * R i ik xp xp (ignoing th fnc amplitud), wh,, x y k x y is th phas coction du to th fnc souc location applid at th dtcto plan, and x, y x x y y z, (2) x, y, z is th position of th fnc point souc, with th oigin,, 0 x y z at th cnt of th dtcto plan and k = 2. Bcaus a Fsnl tansfom is insufficintly accuat fo this wid-aptu, wid fild-of-viw imag, w usd angula spctum H x, y to th imag plan is givn by popagation of x y F F g, H x, y xp ik x, y xp i2 z 1 f f A z; H x, y xp ik x, y wh z is th distanc fom th dtcto plan to th imag plan and F (3) dnots a Foui tansfom. W found that ou masumnt of th fnc souc location was insufficintly accuat, lading to sidual abations. To dtmin and coct that o, w maximizd th shapnss of th imag [7, 8] (C) 2011 OSA 20 Jun 2011 / Vol. 19, No. 13 / OPTICS EXPRESS 12032

7 2,, (4) S I g,, as a function of th fnc souc location, wh I, is th intnsity of th imag and is a scala quantity. Fo this wok, w usd a valu of = 0.6 which is favoabl fo th cas of a singl spckl alization [9]. (If > 1, thn w would maximiz S. ) W implmntd this using a conjugat gadint nonlina optimization outin using analytic gadints of S with spct to th th-dimnsional coodinat position of th fnc point souc. Divations of ths analytic gadints can b found in Appndix B. Figu 2(m) (p) shows th constuctd imag aft optimizing and cocting fo th fnc point souc location. Th solvabl bas aft th appopiat fnc wavfont w (3,2) fo hoizontal bas and (3,3) fo vtical bas. 4.2 High-od phas-o coction In addition to applying a phas coction that spcifically cocts fo wavfont os latd to th fnc souc, w found it ncssay to includ an additional stp to coct fo oth phas os in th systm. This may coct any sidual os fom th mosaicking pocss, focus coction to locat th plan of bst focus fo th imag, tc. W modl th complx fild in th dtcto plan aft fnc coction, as wh 1 sidual phas o. W found th H1 H xp i x, y (5) H is th complx fild with phas os, H is th idal fild and xy, is th that maximizs th shapnss of th imag using a using th analytic gadint of S [10]. conjugat gadint sach ov Fo this sidual phas coction, w usd th mthod of sivs [11, 12], in which w convolv th point-by-point vsion of th analytic gadint with a Gaussian knl. Impotant fo daling with vy lag aays, th mthod of sivs quis lss computational tim and mmoy stoag than optimizing ov cofficints of polynomial xpansions of th phas stimats, which nd stoag of polynomial basis functions fo fficincy. Using a bootstapping mthod, w statd with a knl having a standad dviation of 2500 pixls and ducd th siz of th Gaussian knl vy fiv itations (i , 2000, 1500, 1000, 500, 250), nding with a final knl with a standad dviation of 250 pixls. This allows th algoithm to fist optimiz ov lag, low-fquncy global phas os and thn vntually coct fo phas os that a mo local on a fam-by-fam basis, allowing fo only smooth phas maps and avoiding stagnation duing th optimization pocss. Visibl in Fig. 5, which shows th phas o stimat aft maximizing th shapnss of th imag, a low-od focus coction and individual-fam high-od coction. An impotant thing to not is that th sivs mthod of shapnss optimization had no xplicit knowldg of th acquisition o mosaicking pocdus; by optimizing th shapnss in th imag plan, th algoithm was abl to stimat a phas o that shows chaactistics that a consistnt with th fam-by-fam natu of th data in th dtcto plan. Figu 2(q) (t) shows th futh impovd imag aft this phas coction is applid; hoizontal bas (3,5) w solvd, as w vtical bas (3,5). (C) 2011 OSA 20 Jun 2011 / Vol. 19, No. 13 / OPTICS EXPRESS 12033

8 Fig. 5. High od phas coction ov th nti FOV. Coloba units in adians. 4.3 Rgion-of-intst phas coction A phas-o coction applid ov th nti fild of viw shapns th imag to th plan of ovall bst focus coction. Fo th xpimntal cas psntd h, vaid dfocus ov a wid fild-of-viw objct is likly sinc th dpth of fild is DOF 2 z D 2 = 680 μm, so vn a plana objct having a small tilt can hav appciabl dpth. Th dpth of fild would not b an issu fo long-ang imaging, but it was a poblm in ou laboatoy xpimnt on account of th lativly lag angl (2.3 dgs) subtndd by th synthtic aptu. It is possibl, howv, to apply a phas coction that is optimizd fo a spcific objct dpth by slcting a gion of intst (ROI). Fo ROI phas coction, th fild in th dtcto plan was fist numically popagatd to th nominal imag plan. A mask was applid in th imag plan to window out th spcific ROI and th windowd fild was thn popagatd back to th dtcto plan. This fild, H ROI, can b wittn in tms of th oiginal fild as wh zh ; ROI A ; n, A ; (6) H z M z H A is th angula spctum popagation of th fild H by a distanc z and M n is a mask dfining th n th ROI. H ROI was thn usd in th shapnss minimization outin dscibd abov. Th ROI slctd fo this xampl was a 600 x 600 pixl gion and th mask was a squa with a wightd cosin taping function of 24 pixls applid to th dg of th gion. M 1 includd two hoizontal bas fom Goup 1, Elmnt 1, and M 2 ncompassd Goup 2 and high as shown in Fig. 6. Aft finding th phas that minimizs th shapnss of a ROI, w coctd th nti synthtic aptu fo that phas o, and th coctd fild was thn popagatd using angula spctum to th imag plan. Figu 7 shows th phas-o maps fo th two ROI shapnss optimizations.compad with Fig. 5, wh th phas coction xhibitd both global phas os as wll as famby-fam coction, th pimay phas coction sn by th ROI M 1 phas stimat a smooth high-od os. Fo th imag constuctions, this ROI phas stimat placs th phas stimat found by utilizing th nti FOV. A compaison btwn th imags sulting fom ths two diffnt phas coctions is shown in Fig. 8. Fom Fig. 8(b) and (g), w s that th ROI phas stimats do impov and shapn th imag within th maskd gion. Fom Fig. 8(d) and (f), w also s that aas fa fom th ROI mask w blud du to spac-vaiant os, psumably du to a tiltd plan o sidual os in th fnc location. (C) 2011 OSA 20 Jun 2011 / Vol. 19, No. 13 / OPTICS EXPRESS 12034

9 Th imag constuction using M 2 as a slctiv ROI is shown in Fig. 2(u) (x). Th solution incas is substantial in compaison to ou pvious, intmdiat imags. Fo th Fig. 6. Enti FOV with 600 pixl maskd gions, M 1 and M 2, shown in dashd squas. Fig. 7. High od phas coction (a) using M 1 and (b) using M 2. Coloba units in adians. (C) 2011 OSA 20 Jun 2011 / Vol. 19, No. 13 / OPTICS EXPRESS 12035

10 Fig. 8. Imag constuction with ROI high-od phas coction (a) (d) using mask M 1 and () (h) using mask M 2. Subimags (b) and (f) a 160 pixl aas of (a) and (), spctivly and (d) and (h) a 160 pixl substs of th cntal aas of (c) and (g). fnc-only phas coction, th finst solvd bas in Fig. 2(m) (p) w (Goup, Elmnt) (3, 2) fo hoizontal bas, a modst impovmnt ov th (3, 5) hoizontally solvd bas that w sn aft nti-imag shapnss coction. Howv, fo th ROI phas coction, th solution incasd an nti goup (by a facto of two), with th finst solvd hoizontal bas now (4,2) and vtical bas (4, 5). This cosponds to 17.9 lin-pais (lp) p mm and 25.4 lp/mm, spctivly, quivalnt to a solution of 56 μm and 39 μm, spctivly. Th final siz of th synthtic aptu was 12,100 by 18,000 pixls (87 mm by 129 mm), about 200 mgapixls. Th thotical imag solution, givn by zd, wh D is th synthtic aptu width, is 19.5 μm hoizontally and 13.1 μm vtically. Sinc no spckl avaging has bn pfomd, th xpctd solution fo a spckld imag of a diffus objct will b ducd [13]. Fo a USAF ba tagt, w found that th solution of a spckld imag of th-ba tagt dgads by about a facto of th. Thfo, th xpctd solution of th synthtic aptu usd in this xpimnt is 59 μm fo hoizontal bas and 39 μm fo vtical bas, consistnt with ou xpimntal. Futh xpimnts a undway to poduc a gigapixl imag using ths tchniqus. 4.4 Computational quimnts Th computational quimnts a quit dmanding fo mosaicking, phas-o coction and imag fomation fo such a lag synthtic aptu. An odinay dsktop comput having a fw GB of RAM cannot handl vything in fast mmoy ncssay fo such lag aays of data. Fo th computations pfomd in this xpimnt, an IBM x3755 sv with fou AMD Opton 8224 SE pocssos and 128 GB of mmoy was usd. Th computation tim fo calculating initial coss-colations btwn adjacnt fams was 17.9 minuts. Th mosaicking of th synthtic aptu took 9 hous and 19 minuts and th phas-o stimation took oughly 3 days of computational pocssing. Paalll implmntation using a distibutd systm fo th popagation knl [14] would b on way to duc th computation tim fo th constuction pocss. 5. Conclusion W hav dmonstatd, to ths authos knowldg, th lagst diffaction-limitd synthtic aptu of digital hologaphy data publishd to dat. Th synthtic aptu was fomd by (C) 2011 OSA 20 Jun 2011 / Vol. 19, No. 13 / OPTICS EXPRESS 12036

11 mosaicking togth 21 x 21 fams of digital hologaphy data. Stps ndd to obtain accuat mosaicking includd coss-colation and intnsity optimization of gions of ovlap btwn adjacnt fams. Sval phas coction tchniqus using shapnss mtics w dvlopd and implmntd including th automatic coction of high-od phas os. Th sulting imag ags with th thotical diffaction-limitd solution of a spckl objct. 6. Appndix A: Rquimnts fo pixl gistation Th disct Foui tansfom of an objct fmn is N m, n xp 2. (7) Fpq f i mp np N mn, wh N is th total numb of pixls in th aay. A unit-amplitud point objct at pixl coodinats m,0 0 is dfind as th Konck dlta function f mn m mo, n with a Foui tansfom of F i m p N pq (8) xp 2, having a phas of 2 m p N. Fo andom displacmnts of th individual fams having a standad dviation of p pixls, th standad dviation of th phas is qual to 2m N ad. This is consistnt with o p p o th analogous situation of pupil mapping os in multipl-tlscop aays [4]. Basd on th Machal cition, th phas accuacy ndd fo diffaction-limitd solution can b appoximatd as 1/14 = wavs ms fo all phas os. Fo th standad dviation of th phas to b within th Machal cition, th maximum allowabl standad dviation of th displacmnt is givn by 1 14 mo pmax N, o p max N (14 m0 ). Fo th wos cas, th pixl at th dg of th imag is at m0 N 2, making max 17pixl. Thfo, on must gist th fams accuatly within a 1/7 pixl to mt th phas accuacy quid to mt this diffaction-limitd cition fo th wost cas. Ths thotical calculations mphasiz th quimnts fo vy accuat sub-pixl gistation on a fam-to-fam basis acoss th nti synthtic aptu in od to maintain diffaction-limitd solution ov th nti fild of viw. Not, howv, that fo points in th imag na th optical axis, m o is small and imag quality is lativly insnsitiv to tanslation os. 7. Appndix B: Divation of analytic gadint of S Th fild obtaind by angula spctum popagation of H, p x y to th imag plan is givn by Eq. (3). Th patial divativ of th shapnss mtic S givn in Eq. (4), with spct to th x-componnt of th fnc point souc location x, is givn by o Additionally, S I, x,, I, 1 1 g I x, g*, 1 g, I, 2 R g*,., x g x can b wittn as, x (9) (C) 2011 OSA 20 Jun 2011 / Vol. 19, No. 13 / OPTICS EXPRESS 12037

12 g, ik x x A z; H x, yxp ik x, y. x (10) x, y Using Eq. (10) w can wit Eq. (9) as S x and similaly fo S y, x x by z. 1, 2R A ;, xp, I z H x y ik x y ik x x A z; H x, yxp i x, y x, y placing * (11) x x by y y, and similaly fo z, placing (C) 2011 OSA 20 Jun 2011 / Vol. 19, No. 13 / OPTICS EXPRESS 12038

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