Quantitative phase imaging of weakly scattering objects using partially coherent illumination

Transcription

1 Quanave phase magng of weakly scaeng objecs usng paally coheen llumnaon Tan H. Nguyen 1 Chs Edwads Lynfod L. Goddad and Gabel Popescu 1* 1 Quanave Lgh Imagng Laboaoy Depamen of Eleccal and Compue Engneeng Beckman Insue of Advanced Scence and Technology Unvesy of Illnos a Ubana-Champagn Ubana Illnos USA Mco and Nanoechnology Laboaoy Depamen of Eleccal and Compue Engneeng Unvesy of Illnos a Ubana-Champagn Ubana Illnos USA * gpopescu@llnos.edu Absac: In hs pape we exend ou ecen wok on paally coheen quanave phase magng (pcqpi) fom wo-dmensonal (D) o heedmensonal (3D) magng of weakly scaeng samples. Due o he mahemacal complexy mos heoecal modelng of quanave phase mage fomaon unde paal coheence has focused on hn well-focused samples. I s unclea how hese abbeaons ae affeced by defocusng. Also as 3D QPI echnques nue o develop a bee model needs o be developed o undesand and quanfy hese abeaons when magng hcke samples. Hee usng he fs ode Bon s appoxmaon we deved a mahemacal famewok ha povdes an nuve model of mage fomaon unde vayng degees of coheence. Ou descpon povdes a clea ne beween he halo effec and phase undeesmaon defocusng and he 3D sucue of he sample unde nvesgaon. Ou esuls agee vey well wh he expemens and show ha he mcoscope objecve defnes he seng ably of he magng sysem whle he dense lens s esponsble fo he halo effec. 016 Opcal Socey of Ameca OCIS codes: ( ) Coheence; ( ) Mcoscopy; ( ) Coheence and sascal opcs; ( ) Phase eeval. Refeences and lnks 1. G. Popescu Quanave Phase Imagng of Cells and Tssues (Mcgaw-Hll 011).. T. Ikeda G. Popescu R. R. Dasa and M. S. Feld Hlbe phase mcoscopy fo nvesgang fas dynamcs n anspaen sysems Op. Le. 30(10) (005). 3. D. O. Hogenboom C. A. Dmazo T. J. Gaudee A. J. Devaney and S. C. Lndbeg Thee-dmensonal mages geneaed by quadaue nefeomey Op. Le. 3(10) (1998). 4. D. Zcha and G. Dunn An mage pocessng sysem fo cell behavou sudes n subfluen culues J. Mcosc. 179(1) 11 1 (1995). 5. M. Somekh C. See and J. Goh Wde feld amplude and phase focal mcoscope wh speckle llumnaon Op. Commun. 174(1-4) (000). 6. Y. Pak C. A. Bes K. Badzadegan R. R. Dasa M. S. Feld T. Kuabova M. L. Henle A. J. Levne and G. Popescu Measuemen of ed blood cell mechancs dung mophologcal changes Poc. Nal. Acad. Sc. U.S.A. 107(15) (010). 7. Z. Monemhaghdous P. De Gol F. Monfo Y. Emey C. Depeusnge and C. Mose Towads an ncoheen off-axs dgal hologaphc mcoscope n SPIE BOS (Inenaonal Socey fo Opcs and Phooncs015) pp E 93360E Z. Wang L. Mlle M. M H. Dng S. Unaunoa J. Roges M. U. Gllee and G. Popescu Spaal lgh nefeence mcoscopy (SLIM) Op. Expess 19() (011). 9. B. Bhadu C. Edwads H. Pham R. J. Zhou T. H. Nguyen L. L. Goddad and G. Popescu Dffa phase mcoscopy: pncples and applcaons n maeals and lfe scences Adv. Op. Phooncs 6(1) (014). 10. G. Popescu L. P. Defloes J. C. Vaughan K. Badzadegan H. Iwa R. R. Dasa and M. S. Feld Foue phase mcoscopy fo nvesgaon of bologcal sucues and dynamcs Op. Le. 9(1) (004). 11. C. J. Sheppad and S. B. Meha Phase mcoscope magng n phase space Poc. SPIE 9178A (016). 1. S. Aknoun P. Bon J. Savae B. Waelle and S. Monnee Quanave eadance magng of bologcal samples usng quadwave laeal sheang nefeomey Op. Expess 3(1) (015). 13. C. Zuo Q. Chen L. Tan L. Walle and A. Asund Tanspo of nensy phase eeval and compuaonal magng fo paally coheen felds: The phase space pespecve Op. Lases Eng (015). (C) 016 OSA 30 May 016 Vol. 4 No. 11 DOI: /OE OPTICS EXPRESS 11683

2 14. L. Tan and L. Walle 3D nensy and phase magng fom lgh feld measuemens n an LED aay mcoscope Opca () (015). 15. J. C. Wyan The evoluon of nefeomey fom meology o bomedcal applcaons Poc. SPIE (016). 16. J. Manez-Caanza K. Falaggs and T. Kozack Enhanced laeal esoluon fo phase eeval based on he anspo of nensy equaon wh led llumnaon Poc. SPIE (016). 17. S. Shn Y. Km K. Lee K. Km Y.-J. Km H. Pak and Y. Pak Common-pah dffa opcal omogaphy wh a low-coheence llumnaon fo educng speckle nose Poc. SPIE (015). 18. M. H. Jenkns and T. K. Gaylod Quanave phase mcoscopy va opmzed nveson of he phase opcal ansfe fun Appl. Op. 54(8) (015). 19. K. Lee K. Km J. Jung J. Heo S. Cho S. Lee G. Chang Y. Jo H. Pak and Y. Pak Quanave phase magng echnques fo he sudy of cell pahophysology: fom pncples o applcaons Sensos (Basel) 13(4) (013). 0. X. Ou R. Hosmeye C. Yang and G. Zheng Quanave phase magng va Foue pychogaphc mcoscopy Op. Le. 38() (013). 1. C. Zuo Q. Chen W. Qu and A. Asund Nonnefeomec sngle-sho quanave phase mcoscopy Op. Le. 38(18) (013).. A. El Mallah C. Mne and F. Dubos Auomaed hee-dmensonal dee and classfcaon of lvng ogansms usng dgal hologaphc mcoscopy wh paal spaal coheen souce: applcaon o he monong of dnkng wae esouces Appl. Op. 5(1) A68 A80 (013). 3. J. N. Clak X. Huang R. Hade and I. K. Robnson Hgh-esoluon hee-dmensonal paally coheen dffa magng Na. Commun (01). 4. S. Dong R. Hosmeye R. Shadka K. Guo X. Ou Z. Ban H. Xn and G. Zheng Apeue-scannng Foue pychogaphy fo 3D efocusng and supe-esoluon macoscopc magng Op. Expess (11) (014). 5. B. Bhadu H. Pham M. M and G. Popescu Dffa phase mcoscopy wh whe lgh Op. Le. 37(6) (01). 6. C. Edwads B. Bhadu B. G. Gffn L. L. Goddad and G. Popescu Ep-llumnaon dffa phase mcoscopy wh whe lgh Op. Le. 39(1) (014). 7. L. Mandel and E. Wolf Opcal Coheence and Quanum Opcs (Cambdge Unvesy 1995). 8. T. H. Nguyen C. Edwads L. L. Goddad and G. Popescu Quanave phase magng wh paally coheen llumnaon Op. Le. 39(19) (014). 9. C. Edwads B. Bhadu T. Nguyen B. G. Gffn H. Pham T. Km G. Popescu and L. L. Goddad Effecs of spaal coheence n dffa phase mcoscopy Op. Expess (5) (014). 30. F. Zenke How I dscoveed phase as Scence 11(3141) (1955). 31. T. H. Nguyen H. Majeed C. A. Edwads M. N. Do L. L. Goddad and G. Popescu Halo-fee quanave phase magng wh paally coheen lgh Poc. SPIE (015). 3. Z. Yn T. Kanade and M. Chen Undesandng he phase as opcs o esoe afac-fee mcoscopy mages fo segmenaon Med. Image Anal. 16(5) (01). 33. T. Wlson and C. J. Sheppad The halo effec of mage pocessng by spaal fequency fleng Opk (Sug.) (1981). 34. S. B. Meha and C. J. Sheppad Usng he phase-space mage o analyze paally coheen magng sysems: bgh-feld phase as dffeenal nefeence as dffeenal phase as and spal phase as J. Mod. Op. 57(9) (010). 35. J. Dohe-Ealy C. Youassowsky A. El Mallah and F. Dubos Paal spaal coheence llumnaon n dgal hologaphc mcoscopy: quanave analyss of he esulng nose edu Poc. SPIE (016). 36. J. Dohe-Ealy C. Youassowsky A. E. Mallah and F. Dubos Quanave assessmen of nose edu wh paal spaal coheence llumnaon n dgal hologaphc mcoscopy Op. Le. 41(1) (016). 37. A. Kalyanov N. Talaykova and V. Ryabukho Fomal heoy of dffa phase mcoscopy Poc. SPIE (014). 38. J. C. Peuccell L. Tan and G. Babasahs The anspo of nensy equaon fo opcal pah lengh ecovey usng paally coheen llumnaon Op. Expess 1(1) (013). 39. A. Ahmad V. Dubey G. Sngh V. Sngh and D. S. Meha Quanave phase magng of bologcal cells usng spaally low and empoally hgh coheen lgh souce Op. Le. 41(7) (016). 40. Y. I. Nesees and T. E. Gueyev Paally coheen as-ansfe-fun appoxmaon J. Op. Soc. Am. A 33(4) (016). 41. J. W. Goodman Sascal Opcs (John Wley & Sons 015). 4. J. W. Goodman Sascal Opcs (Wley-Inescence 1985). 43. D. Gabo Mcoscopy by esuced wave-fons n Poceedngs of he Royal Socey of London A: Mahemacal Physcal and Engneeng Scences (The Royal Socey 1949) pp N. Sebl Thee-dmensonal magng by a mcoscope J. Op. Soc. Am. A () (1985). 1. Inodu Quanave phase magng (QPI) has developed no a key opc n he feld of bomedcal magng as offes nnsc nfomaon on efacve ndex and sample opogaphy (see [1] and he efeences heen). The quany of nees n QPI s he opcal pah lengh vaaon (C) 016 OSA 30 May 016 Vol. 4 No. 11 DOI: /OE OPTICS EXPRESS 11684

3 acoss he mage feld. Ths paamee s defned as he agumen of a coss-coelaon and hus can be exaced fom nefeomec expemens. The nefeomec modales can be dvded no he adonal (e.g [ 7].) and common-pah (e.g [8 4].) subgoups. The lae subgoup geneaes he efeence feld by spaally fleng he oal feld emegng fom he sample. Hence possesses vey hgh sably o amben nose. A ecen maage beween common pah and whe-lgh mehods has fuhe suppessed he speckle phenomenon allowng hghly sensve measuemens [8] [5] [6].An mpoan benef of hese common pah mehods s he ably o be decly deployed fom commecally avalable mcoscopes snce only he oal feld elayed o he oupu po of he mcoscope s equed. In QPI he phase φ ( ) s defned as (see Eq. (4).3-34 n [7]) he agumen of he empoal coss-coelaon fun beween he oal feld U = U + U s and a efeence feld U evaluaed a zeo delay τ = 0 [8].e. φ ( ) = ag J ( ) whee * J ( ) =Γ ( 0) = U( ) U( ). Ths quany can be eeved usng any of afoemenoned nefeomec magng modales. Fgue 1 shows he dffa phase mcoscopy (DPM) seup [9] used n hs pape. Moe nfomaon on how o oban J ( ) fom nensy measuemens n DPM can be found n he Appendx A. In ecen publcaons [8 9] we showed ha n common-pah QPI modales he mpefe of he pnhole ogehe wh he paally coheen popey of he llumnaon feld may geneae naccuacy n he phase measuemens. The ne effec s an undeesmaon of he phase n egons of lage laeal exen han he coheence aea and eoneous negave values a he edges. In phase as mcoscopy hs phenomenon s usually efeed o as he halo effec [30]. A necessay don fo ecodng a halo fee mage whou pos-pocessng s he use of an llumnaon souce wh a coheence aea ha s lage han he feld of vew [9]. Ohewse he daa wll be affeced by he halo effec and phase undeesmaons. Howeve o ou knowledge mos sudes on hese afacs ypcally assumed hn well-focused samples [31 40] whee he sample s chaacezed by a ansmsson fun T ( ). To ou knowledge s no ye clea how hese effecs vay when he sample s nehe well-focused no a good focus plane can be found. The lae case usually happens when he hckness of he sample s no small enough compaed o he deph of feld. In hs pape we deve a model descbng he mage fomaon of common-pah QPI unde paally coheen llumnaon when he sample s chaacezed by s 3D suscepbly fun Χ ( ). The opcal sysem scans axally (z-de) hough he volume of he objec and phase measuemens a mulple z-seps ae ecoded. Ou deved model necs he phase measuemen o he suscepbly of he sample. Also smla o he D case explans he halo and phase-undeesmaon afacs and povdes a genealzaon o he D poblem. We show ha unde he Bon sengs he measued phase s a hgh-pass veson of an deal phase defned on he suscepbly of he sample. The hghpass fleng kenel descbes he halo and phase-undeesmaon afacs. Moe mpoanly only elaes o he dense of he magng sysem and nvaan o defocusng. Theefoe he halo and phase-undeesmaon afacs ae unchanged due o defocusng. These effecs can be seen fom he daa n Fg. when scannng he sample ove s z-sack ove a ange of [-0 0] μm. Fgues (a) (c) and (e) show 3D phase measuemens fo a anspaen quaz mcoplla 0 μm wde and 13 nm hck usng he DPM seup shown n Fg. 1. The egon unde he nfluence of he halo does no seem o boaden o blu due o defocusng. Fgues (e)-(f) capue hs obsevaon by showng he x-z coss-ses ove he ene z- ange. Agan he sample ges blued due o defocusng bu he halo emans unaffeced. (C) 016 OSA 30 May 016 Vol. 4 No. 11 DOI: /OE OPTICS EXPRESS 11685

4 Fg. 1. A dffa phase mcoscopy seup whee wo eplcas of he oal feld a he oupu po of he mcoscope ae geneaed by a dffa gang. One of hem s jugaed o he camea plane. The ohe s low-pass fleed by a physcal pnhole o ceae a efeence feld. Inefeence fnges of hese wo felds ae ecoded by he camea and he measued phase of nees φ ( ) s obaned usng he Hlbe ansfom. Fo moe deals see [9].. Theoy Hee we popose a mahemacal analyss ha explans hese phenomena fo a 3D sample chaacezed by a efacve ndex fun n( ). We assume ha he objec s weakly scaeng so ha he fs-ode Bon appoxmaon s applcable. Agan he efeence feld s geneaed by D spaally fleng he oal feld emanang fom he sample.e. U ( z) = U ( z) d Uo( z) whee z s he axal coodnae. Fom now on we wll use he o subscp o denoe he spaal fleng opeaon o geneae he efeence. The man esul of ou calculaon s ha he phase measued wh paally coheen llumnaon φ s gven by he followng equaon ( ) ( ) ( ) h ( ) ( ) ( 3 ) ( ) h ( ) φ ϕ ϕ v = ϕ δ v. (1) whee ϕ ( ) s he deal phase expeced n dons of pefec coheence ( ) h s he spaal coelaon fun assocaed wh he llumnang feld δ ( 3) epesens he 3D Dac dela-fun and v denoes he 3D voluon opeaon ove he 3D coodnae. I can be seen fom Eq. (1) ha ou measuemen φ ( ) s a hgh-pass fleed veson of ( 3 ϕ ( ) whee he hgh-pass kenel s gven by δ ) ( ) ( ). Moe mpoanly he kenel h n spe of beng a fun of only pefoms fleng n he ansvese coodnae.e. h ( ) S( ) S ( 0) δ ( z) (See he Appendx D fo moe deals). Hee S ( ) s he wo-dmensonal Foue ansfom of he apeue nensy also call he muual nensy fun of he llumnaon [41 4] evaluaed a he mage plane. By he cenal odnae heoem we have S ( 0) S( ) d he spaal powe specum of he llumnaon evaluaed a zeo ansvese spaal fequency k 0. Consequenly he halo effec does no = h change ove he axal dmenson. Fully coheen llumnaon coesponds o a fun h h = 1 δ z. As a esul and φ( ) ϕ( ) s. ha s unfom n x-y namely ( ) ( ) ( ) whch ndcaes ha he measuemen s deal (up o an nsgnfcan san). The halo s absen n hs case. (C) 016 OSA 30 May 016 Vol. 4 No. 11 DOI: /OE OPTICS EXPRESS 11686

5 Fg.. Expemenal (lef) and smulaed (gh) hckness measuemens n nanomees fo 3D PC-QPI magng of 0 μm wdh 13 nm hck mcopllas a NAc = (a) and (b) show he hckness ecoveed fom φ ( ) a he sample plane. (c) and (d) ae hckness measuemens a +10 μ m fom he sample plane (fowad scaeng). (e) (f) show he xz coss-se fo he hckness measuemens a y = 0.0 μm. The halo and phase edu can be seen fo all hese z-seps. On he ohe hand fo ncoheen llumnaon.e. h ( ) = δ (3) ( ) we have φ ( ) 0 whch esablshes he mpossbly of phase measuemen unde fully ncoheen llumnaon. All nemedae cases beween hese wo lms esul n a phase mage wh low fequency aenuaon n shaply ansen aeas.e. edges lnes ec whle pesevng hgh fequency nfomaon. Ths s he souce of he halo effec commonly known n phase as mcoscopy [30]. Moe mpoanly can be seen ha he kenel does no vay along he zdmenson and s no affeced by defocusng. Equaon (1) looks que smla o s counepa Eq. (7) n [31]. Howeve hee ae fundamenal dffeences beween hem. Fs Eq. (1) ells ha he measued phase φ s lnealy elaed o he deal phase ϕ a each 3D coodnae. Meanwhle Eq. (7) n [31] φ ( ) ϕ ( ) eϕ h ( ) povdes a non- lnea elaon beween he measued phase φ and he deal phase ϕ a each ansvese coodnae. Sed he deal phase n hs pape s a fun of he suscepbly whch s a genealzaon of he deal phase n [31] whch s he agumen of he sample ansmsson. We fuhe show he vegence of hs deal phase o he agumen of he sample ansmsson fo hn sample n Appendx B. #6749 (C) 016 OSA Receved 7 Ap 016; evsed 10 May 016; acceped 11 May 016; publshed 19 May May 016 Vol. 4 No. 11 DOI: /OE OPTICS EXPRESS 11687

6 Now le us skech ou a poof fo Eq. (1). Fo moe deals please efe o he Appendx. We sa by expessng he oal and efeence feld as U = U + Us and Uo = Uo + U so. Then he coss-coelaon fun becomes * * ( ) = U + U U + U ( ) = ( ) + ( ) + ( ) + ( ) J s o so J o Js o J so Js so () whee J o Js o J so J s so ae coss-coelaon funs evaluaed a zeo-delay τ = 0 beween he ncden feld & s low-pass veson he scae feld & he ncden low-pass veson he ncden feld & he scaeng low-pass veson he scae feld & s low-pass veson especvely. Ignong he J sso ( ) em due o s much smalle amplude he followng esuls hold (see he Appendx C D E fo sho poofs). ( ) = S( 0) Jo (3) ( ) = ( 0) ( ) Jso S ϕ (4) ( ) ( 0) ϕ ( ) ( ) Jso ;0 = S v h. (5) Noe ha among hese ems he fs one J o only elaes o he llumnaon. The sed and hd ems ae magnay; hey cay he sample nfomaon hough he phase quany ϕ ( ) whch fuhe elaes o he suscepbly of he sample X ( ). The fac ha hese wo ems have oppose sgns ells ha hey cancel ou each ohe causng he phaseundeesmaon afac. Usng hese ems Eq. (1) can be poven easly usng ( ) ( ) ( ) = ac an { Jso ( ) + Jso ( ;0) Jo ( )} ( 3 = ϕ( ) v δ ) ( ) h ( ). ag J 0 = ac an Im J Re J 3. Expemens To valdae ou model we pefomed QPI measuemens and smulaons unde vaous degees of spaal coheence and defocusng. To change he coheence of he llumnaon we vay he numecal apeue of he dense NA. Smalle values of NA gves naowe spaal powe specum of he llumnaon S ( k ) and hence laeally boade h and vce vesa. Fou dffeen values of NA wee used: and We used quaz pllas as samples he efacve ndex of whch s a he cenal wavelengh 574 nm. The suoundng medum s a wh efacve ndex of n = 1. These paamees ae used fo smulaon usng Eq. (1). We obaned DPM measuemens a seveal z-seps and wh dffeen values of he numecal apeue. The sep sze s se o 0.57 μm whch s abou 4.6 mes of he plla s hckness. Fo each value of NA z-seps n he ange of [ ] μm fom he sample plane wee acqued. Fgues (b) (d) and (f) show ou measuemens and smulaon esuls fo he phase measuemens a dffeen z-posons fo he case of NA = I can be seen ha he smulaon has an excellen ageemen wh he expemens shown n Fgs. (a) (c) and (e). Fgue 3 shows expemenal and smulaed coss-seal pofles fo he hckness measuemens a dffeen values of NA fo he sample a wo focal planes. The hegh pofles fm he po cluson ha as he sample s scanned hough focus he objec blus bu he halo emans san. In ou (6) (C) 016 OSA 30 May 016 Vol. 4 No. 11 DOI: /OE OPTICS EXPRESS 11688

7 smulaon we use a Gaussan pofle fo he spaal powe specum S ( k ) wh sandad devaon of β NA. Good ageemen beween he smulaed and measued pofles can be seen a he sample plane n Fgs. 3(a) and 3(b). Howeve a 10 μm fom he sample plane n Fgs. 3(c) and 3(d) he smulaon exhbs moe sgnfcan modulaon ( ngng ) han he expemenal esuls. Ths can be abued o vaous smoohng effecs n he opcal seup and he fac ha ou smulaon s based on monochomac lgh whle n he acual expemen he llumnaon has fne bandwdh aound hs fequency. Theefoe he dffa ngng s washed ou n expemens due o he combnaon of dffeen dffa kenels a dffeen opcal fequences Fg. 3. Compason beween he expemenal and smulaed pofles fo 13 nm quaz pllas fo dffeen values of NA a he plane of sample (a) (b) and a 10.0 μm fom he plane of sample (c) and (d). Fgue 4 llusaes he ndependence of defocusng whch s due o he low-pass fleng pefomed by he mcoscope objecve as well as he halo effec whch s he esuls of low spaal coheence. Fgue 4(a) shows hee dffeen x-z coss-ses of he phase measuemen φ ( x y = 0 z) fo hee dffeen values of NA. The phase undeesmaon and halo effecs can be seen n he sed and he hd cases. To ge an nsgh of how hese effecs vay wh espec o deph z we ake he 1D Foue ansfom φ ( kx y = 0 z) of he coss-ses a z = 0.0 μm and z = 15.0 μm. We fs fnd he egons n he spaal speca ha ae affeced by he halo only defocusng only o boh. Fom Eq. (1) can be seen ha ( 3 he band-pass kenel δ ) ( ) h ( ) acs as a hgh-pass fle suppessng low-fequency componens of he deal phase ϕ ( ). Hence he spaal fequency doman can be dvded no egons. Regon 1 s only affeced by he defocusng and he fequency specum of he objec χ. Regon s affeced by all buons ncludng defocusng low-fequency suppesson and χ. I s clea fom Fg. 4(b) ha fo each z-poson he amplude speca n (C) 016 OSA 30 May 016 Vol. 4 No. 11 DOI: /OE OPTICS EXPRESS 11689

8 egon 1 ae almos dencal fo each value of NA ndcang ha he dense apeue only affecs he low-fequency ange of he measued phase. Defocusng and deph seng only elae o he numecal apeue of he objecve. Fgues 4(c)-4(e) show amplude speca fo he 3 dffeen values of z a NA = and especvely. Noe ha he speca ae almos he same fo egon fo all 3 values of z ndcang ha he low3 fequency suppesson due o he spaal coheence kenel δ ( ) ( ) h ( ) s essenally nvaan o he deph z. Fg. 4. (a) Thee dffeen x-z coss-ses of fo 3 dffeen values of he numecal apeues of he dense. (b) Amplude specum a z = 0.0 um and z = 15.0 um fo ee values of NA. (c)-(e) Amplude speca fo 3 dffeen values of z a NA = and especvely. Nex we expand ou analyss o hck weakly scaeng sample. Fgue 5(a) shows x-y and x-z coss-ses of a smulaed squaed mcoplla of dmenson μm3. Hee he hckness s 5 μm. The plla has efacve ndex of The suoundng meda has he efacve ndex of Usng he cenal wavelengh of μm he oal phase shf geneaed by hs plla s.19 ad. Hee we have nenonally chosen he hckness and he efacve ndex so ha he oal phase shf s less han π o avod any possble phase wappng. Fgue 5(b) shows one x-z and hee x-y coss-ses of he deal phase ϕ ( ) usng s fomula gven n Eq. (9). The x-z one s evaluaed hough he cene of he plla a he plane y = 0 μm. The x-y ones ae evaluaed a hee dffeen planes z = 0 z = 0 z = 0 μm denoed n he x-z coss-se. I can be seen ha hs he oal phase shf of.19 ad can be obseved n all hee x-y coss-ses wh dffeen amoun of defocusng. Fgue 5(c) shows he x-z coss-ses of he measued phase unde dffeen numecal apeues of he dense NA = and especvely. Dashed black ecangles denoe he egons coespondng o he locaon of he plla. Obvously he halo and phase-unde esmaon ge wose a lage values of he dense apeue NA. These #6749 (C) 016 OSA Receved 7 Ap 016; evsed 10 May 016; acceped 11 May 016; publshed 19 May May 016 Vol. 4 No. 11 DOI: /OE OPTICS EXPRESS 11690

9 effecs ae decoupled fom he defocusng as dscussed n he pevous se. The code fo ou smulaon can be obaned a hps://ghub.com/hnguyn/3d_halo_modelng.g. Fg. 5. (a) x-y and x-z coss-ses of a smulaed mcoplla of dmensons 30 x 30 x 5 µm3. (b) x-z and 3 x-y coss ses of he deal phase ϕ. The dashed ecangle denoes he locaons of he mcoplla. The nex hee coss-ses ae evaluaed a hee dffeen planes z = 0 µm z = 0 µm and z = 0 µm denoed by whle lnes n he fs x-z cossse especvely. (c) x-z coss-ses of he measued phase φ evaluaed a hee dffeen values of NA namely and Concluson In sum we have developed a model o quanfy he halo effec and phase edu n 3D pcqpi expemens. Ou model elaes sample hckness spaal coheence defocusng and feld popagaon o obseved effecs n he fnal measuemen. The fomalsm s geneal and applcable o hn samples as well as hck weakly scaeng objecs. Appendx A: Exacng J ( ) fom he nefeence fnges The DPM nensy mage I ( ) = U ( ) + U ( ) exp ( k xo x ) capued by he camea can be wen as = I ( ) + I ( ) + J ( ) exp ( k xo x ) + J ( ) exp ( k xo x ). Hee k xo s he spaal wave veco geneaed by he DPM gang. Noe ha he fs wo ems n he expanson of J ( ) exs a base-band whle he hd em and fouh em ae ceneed aound k = [ ± k xo 0] n he spaal fequency doman. Theefoe J ( ) can be obaned by applyng a band-pass fle on I ( ) so ha s bandwdh maches o ha of he fouh componen followed by shfng he emanng specum no baseband. Ths s he pncple of off-axs hologaphy poposed by Gabo [43]. #6749 (C) 016 OSA Receved 7 Ap 016; evsed 10 May 016; acceped 11 May 016; publshed 19 May May 016 Vol. 4 No. 11 DOI: /OE OPTICS EXPRESS 11691

10 Appendx B: The deal phase ϕ ( ) q ( nβ ) z The deal phase s gven as ϕ( ) = β ( n) Xv I ( e ) ( ). k In ode o show s ne wh he sample ansmsson fo hn sample n he wo-dmensonal case le us sde a hn objec of hckness h placed aound he z = 0 plane chaacezed by he suscepbly fun ( ) ( ) ( ) ( ) ( ) Χ =Π zh n n nπ zhn n whee Π (). s he ecangula fun. We have also appoxmae n( ) n n. sample gnoe he defocusng dffa.e. ( ) h n( ) n + Fo well-focus q ( nβ ) z ( I ( ) ) k e δ ( ) 1( z) we have ϕ β whch s he defnon of phase fo he wo-dmenson case. Appendx C: Poof of Eq. (3) We have o( ) = ( ) ( ) = ( ) J z z d J z z d * ;0 U U ' ' ' '. Combne hs equaon wh Eq. (14) of [44] n whch J ( z ' z) = S ( k ) exp k. ( ' ) d k we have: ( J ( ) ( ) ( ) ( ) ) ( ) o ;0 = S k exp k. ' d k d ' = S 0 δ k d k = S ( 0 ) whch complees he poof fo Eq. (3). Appendx D: Poof of Eq. (4) The scaeed feld s gven usng he 1s-ode Bon appoxmaon as 3 U ; β X ' U ' g ' d ' whee g (). s he Geen s fun. Fo s( ) ( ) ( ) ( ) smplcy we gnoed dspeson of he sample.e. ( ) ( ) X = n n wavelengh. The sed em educes o * ( ) = U ( ) U ( '' ) '' = β ( ) ( ) ( ) ( ) 3 = β X ( ' ) J ( ' z' '' z) g( ' ) d ' d ''. J z z d so s s ndependen of X ' U ' z' g ' d ' U '' z d '' 3 * Agan Eq. (14) of [44] gves J ( ' z' '' z) = S ( ) exp.( ' '' ) + q( ) ( z' z) d whch gealy smplfes Eq. (7) o so 3 ( ) = β ( ' ) ( k) exp k. ( ' '' ) + ( k) ( ' ) k ( ' ) ' '' 3 S( 0 ) β X ( ' ) g( ' ) exp nβ( z z') d ' S( 0 ) β { Xv gexp ( nβz) }( ). J X S q z z d g d d = = (8) qz Unde paaxal appoxmaon ( ) I k ( ) (7) g e nβ Eq. (8) educes o: ( ) { { } 1 q nβ z β }( ) k ( ) ϕ( ) ( ) ( ) ( ) Jso = S 0 Xv n I e. = S 0 whee (C) 016 OSA 30 May 016 Vol. 4 No. 11 DOI: /OE OPTICS EXPRESS 1169

11 Appendx E: Poof of Eq. (5) q ( nβ ) z ϕ( ) = β ( n) Xv I ( e ) ( ). k (9) * ( ) = U( ) U ( ' ) ' = β ( ) ( ) ( ) ( ) = β ( ) ( ) ( ) J z z d so s X '' U z U '' z'' g ' '' d '' d ' * * * 3 X '' J z '' z'' g ' '' z z'' d '' d '. * * 3 Usng Eq. (14) of [44] and doppng he jugae noaon on X snce s a eal fun as well as changng he ode of negaon we have: so ( ) β ( ) ( ) ( ) ( ) β ( ) ( ) ( ) ( ) (10) { } ( ) { } { ( ) } J X S k k q k z z d k g d d * 3 = '' exp. '' + ( '') ' '' z z'' '' ' * 3 = X '' S k exp k. '' + q k ( z z'') d k g ' '' z z'' d ' d ''. (11) Unde he paaxal appoxmaon: ( ) (11) educes o: so * nβ ( z z '') g d e k n ' '' z z'' ' I β Eq. { β } ( ) ( ) q ( nβ ) z ( ) v ( ) δ ( ) q ( nβ ) z ( ) v { δ } ( ) ( ) 1 q nβ z 0 ϕ ( ) β ( ) ( ) ( ) ( ) ( ) q ( nβ )( z z'' ) 3 β ( ) ( '' ) v '' '' J = n X S k k + q k n z z d k d 3 '' exp. '' ( '') '' = n X I e S d = n { X I e S z } = S X n I e S S z = S k β ( ) v k ( 0) { v { β ( ) k } ( ) ( 0) ( ) }( ) ( 0 ){ Xv { β ( n) I e } v h}( ) = S( ){ v h}( ) k whee (1) S S as noduced n he man ex. h was defned as h ( ) = ( ) ( 0) δ ( z) Acknowledgmen Ths wok was suppoed by he Naonal Scence Foundaon (NSF) (CBET STC IIP ). Fo moe nfomaon vs hp://lgh.ece.llnos.edu/. (C) 016 OSA 30 May 016 Vol. 4 No. 11 DOI: /OE OPTICS EXPRESS 11693