Focal plane invariant algorithm for digital reconstruction of holograms recorded in the near diffraction zone

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1 Focl plne invint lgoithm fo digitl econstuction of hologms ecoded in the ne diffction zone L. Yoslvs,. Ben-Dvid, Dept. of Intedisciplin Studies, Fcult of Engineeing, Tel Aviv Univesit, Tel Aviv, Isel ABSTRACT Fo digitl econstuction of hologms ecoded in ne diffction zone, Discete Fesnel is used s discete model of wve popgtion. In pplictions whee econstuction of hologms in mn obsevtion plnes is equied, conventionl implementtion of Discete Fesnel Tnsfom econstucts imges of the object plne sections tht e shifted with espect to ech othe depending on plne distnce fom the hologm plne. In the ppe, novel epesenttion of Discete Fesnel Tnsfom, Focl Plne Invint Discete Fesnel Tnsfom is suggested nd illustted which secues econstuction of imges utomticll ligned with espect to the sme opticl is. The tnsfom invince llows to substntill educe computtion buden wheneve onl eltivel smll potion of the econstucted wve font hs to be nlzed nd to ese nlsis of the econstucted wve font s function of distnce to the object. Kewods: Digitl hologph, Discete Fesnel Tnsfom. ITRODUCTIO. ITEGRAL FRESEL TRASFORM It is well nown tht wve popgtion on distnce D between object plne (, nd obsevtion plne ( f f, is descibed b pil Fesnel ppoimtion α ( f f (, ( f + ( f, ep iπ dd (, λd of the Kichhoff integl, whee (, is comple mplitude of wve font in the object plne, ( f, f α is comple mplitude of the wve font in the obsevtion plne, λ is wve length nd D is distnce between the object nd obsevtion plnes. This integl tnsfom pls fundmentl ole in mthemticl fomultion of ecoding nd econstuction of hologms nd modeling coheent imging sstems. The pesent ppe ddesses the poblem of dequte discete epesenttion of this integl tnsfom tht cn be used fo compute snthesis nd econstuction of hologms. To this gol, it is convenient to wite the tnsfom in λ D, / λd f / λ D, f / λd : educed dimensionless coodintes ( / nd ( α [ ] ( f f (, ep iπ( f + ( f, dd (, b This integl tnsfom is efeed to s two dimensionl integl Fesnel tnsfom. Accoding to this definition, -D integl Fesnel tnsfom is sepble to two one - dimensionl tnsfoms. This llows us to confine ouselves in wht follows to consideing -D integl Fesnel tnsfom:

2 [ ] ( f ( ep iπ ( f α d. ( F Integl Fesnel tnsfom is ve closel connected with Fouie tnsfom. The m be educed one to nothe b ep iπ nd men of phse modultion of object wve font nd its Fesnel tnsfom spectum with functions ( ep( iπf clled chip-functions: ( f ep( iπf ( ep( iπ ep( i πf α d F (3 Fesnel tnsfom is convolution tnsfom. Theefoe it cn lso be lined with Fouie tnsfom though the f cn be found s invese Fouie tnsfom convolution theoem. Fesnel tnsfom α ( of wve font ( ( f α ( p ( p ep( i πfp F α CHIRP dp (4 F F of poduct of wve font Fouie spectum α ( p nd Fouie tnsfom ( p CHIRP F ( p ep( i ep( iπp d ep( iπp ep iπ ( p CHIRP of the chip-function : [ ] π d. (5 The esult of Fesnel tnsfom of ect-function, <, ect( (6, othewise. is function f F F [ ] finc ( ; F ect ep iπ ( f df ep( iπ ep( iπf ep( iπf F F df (7 We will efe to it s finc-function. It is n nlog of sinc-function of Fouie tnsfom: ( πf f + F sin sinc ( ; F ect ep( iπf df F F πf. (8 Two dimensionl nlogs of Eqs. -8 follow stightfowdl. In digitl snthesis nd econstuction of hologms, integl Fesnel tnsfom hs to be epesented in discete fom. This epesenttion should pllel tht of wve fonts used fo snthesis nd nlsis of hologms. In wt follows, we povide deivtion of Discete Fouie Tnsfom nd suggest its modifiction tht secues focl plne invint econstuction of hologms ecoded in ne diffction zone. This fetue llows to use efficient puned FFT

3 lgoithms when onl smll egion of inteest hs to be nlzed in the econstucted wve font nd simplifies utomtic locting objects in depth.. DISCRETE FRESEL TRASFORM AS A DISCRETE REPRESETATIO OF THE FRESEL ITEGRAL Signl discetiztion is signl epnsion ove set of bsis functions. It is commonl ccepted in digitl hologph tht wve fonts e epesented in discete fom s _D s of thei smples using shift bsis functions tht e obtined fom mothe function b its shifts b multiple of n intevl clled discetiztion intevl. Tnsfomtion enel of integl Fesnel tnsfom ep[ iπ ( f ] is shift invint. evetheless, in the deivtion of its discete epesenttion, it is useful to ccount fo possible bit shifts of signl nd its Fesnel spectum smples positions with espect to the signl nd spectum coodinte sstems. Let discetiztion nd econstucted bsis functions used fo discete epesenttion of wve fonts be ( ( ϕ ( ϕ [ ( + u Δ ], nd ( d ( d ϕ ( f ϕ [ f ( + v Δf ], espectivel, whee Δ nd Δ f e discetiztion intevls in coodintes nd f, espectivel, nd e coesponding intege indices, nd u nd v e bit shifts of object wve font senso of tht in the obsevtion plne, espectivel. Then discete epesenttion of the integl Fesnel tnsfom (Eq. is obtined s ( d { [( ( ] } ~ iπ + u Δ + v Δf ϕ ( f ep( iπf Φ( f s ep[ i πfs ] α ep df, (9 whee { } e smples tht econstuct the input wve font mplitude ( ( ( [ ( + u Δ] ϕ, (9b s ( + u Δ ( + v Δf, { α } e smples of Fesnel tnsfom spectum ( f α : ( ( d f ϕ [ f ( + v Δf ] α α df (9c nd ~ ( ( f ϕ ( ep( iπ ep( i πf d Φ ( ( ϕ ( modulted b chip-function ( is Fouie Tnsfom of signl econstuction bsis function ep iπ. ( Fo ϕ ( is function tht is usull compctl concentted ound point within n intevl of bout Δ, ep( iπ within this intevl nd one cn egd Φ ~ ( f s n ppoimtion to fequenc esponse of hpotheticl signl econstuction device ssumed in the signl discete epesenttion. With this esevtion, fo the discete epesenttion of Fesnel integl onl the fist multiplie in Eq. 9 is used. To complete the deivtion, intoduce educed vibles: / ( Δ / Δf μ (

4 nd w uμ v μ. ( Then, using the eltionship ΔΔf between the numbe of signl smples nd discetiztion intevls Δ nd Δ f, obtin fo smples of the integl Fesnel tnsfom spectum finll: (, w ( μ / μ + w α μ ep iπ, (3 whee multiplie / is intoduced fo nomliztion puposes. Eq. 3 cn be egded s discete epesenttion of the Fesnel integl tnsfom. We will efe to it s to Shifted Discete Fesnel Tnsfom (SDFT. Its pmete μ pls ole of the distnce (focl pmete in the Fesnel ppoimtion of Kichhof s integl. Pmete w is combined shift pmete of discetiztion ste shifts in signl nd Fesnel Tnsfom domins. It fequentl is consideed equl to zeo, nd Discete Fesnel Tnsfom (DFT is commonl defined s (, w ( μ / μ α μ ep iπ (4 Shifted Discete Fesnel Tnsfom SDFT ( μ,w is quite obviousl connected with discete Fouie Tnsfom: ( μ, w μ( μ + w α ep iπ ep iπ ( wμ ep i π (5 μ Fom this eltionship one cn conclude tht SDFT ( μ,w is invetible nd tht the invese SDFT ( μ,w is defined s α μ (, w (, w ( μ / μ + w μ ep iπ. (6 In mn cses of digitl econstuction of Fesnel hologm, onl mgnitude of the econstucted wve field (imge is needed. If we emove fom Eqs. 3 eponentil phse tems tht do not depend on, we ive t Ptil Discete Shifted Fesnel Tnsfom defined s (, w μ ( wμ α μ with its invese tnsfom s: ep iπ epiπ (, w (, w ( wμ μ. (7 α μ ep iπ epiπ μ. (8

5 An impotnt ole in digitl econstuction of Fesnel hologms is pled b discete Fouie Tnsfom of the ep i π q /. It is peiodic function discete chip-function ( fincd q ( ; q; ep iπ ep iπ To the ccuc of phse chip-function ep( i π / q. (9, function ( ; q; wve. It cn lso be egded s discete nlog of function ( ;F ( ; q; fincd coincides with sincd-function of Discete Fouie Tnsfom: fincd descibes DFT of plne finc defined in Eq. 6. When q, function ( ( fincd ;, sincd ; ; ep iπ, ( discete nlog of sinc-function fo integl Fouie Tnsfom. Fig., illusttes the behvio of fincd -function fo diffeent vlues of focussing pmete q..9 bs(fincd q.9 bs(fincd q q..5.4 q..3.. q.8 q.8 q.3 q q.8 q.8 q.3 q b Fig.. Absolute vlues of functions fincd ( ;q; ( nd fincd ( ;q; pmete q. (b fo 56 nd diffeent vlues of focusing 3. FOCAL PLAE IVARIAT DISCRETE FRESEL TRASFORMS Since shift pmete w in the definition of Shifted Discete Fesnel Tnsfom (Eq. 4 is combintion of shifts in object nd spectl domins, shift in coodinte cuses coesponding shift in tnsfom domin in coodinte f which, howeve depends on the focl pmete μ. This phenomenon cn be clel see in Fig., in the behvio of

6 fincd -function fo diffeent focl distnces. One cn be this dependence if, in the definition of the discete epesenttion of integl Fesnel tnsfom, impose n dditionl condition of smmet: α ( μ, w ( μ, w α ( onto the tnsfom (, w ( / μ w α μ ep iπ. ( of point souce δ (. This condition is stisfied when w. (3 μ Then SDFT ( μ, / μ fo such shift pmete tes the fom: (, w ( μ ( / / μ α μ ep iπ. (4 This vesion of the discete Fesnel tnsfom is efeed to s Focl Plne Invint Discete Fesnel Tnsfom (FPIDFT. It llows to eep position of objects econstucted fom Fesnel hologms to be invint to the focl distnce. The coesponding fincd-function fo focl plne invint DFT is defined s fincd ( ; q; ( q ep iπ ep iπ. (5 Fig., b illusttes behvio of fincd -function nd its invince to diffeent vlues of focussing pmete q. Simill, Focl Plne Invint Ptil Discete Fesnel Tnsfom is defined s (, w μ ( / α μ ep iπ epiπ μ ( ep iπ ep iπ (6 with its invese tnsfom s μ ( ep iπ μ α ep iπ. (7

7 The invince of the tnsfoms to focl plne distnce is useful fetue when one needs to econstuct nd nlze onl smll potion within limited egion of inteest of the econstucted wve font. In this cse one cn efficientl use in the econstuction lgoithms puned FFT lgoithms tht llows to substntill educe the numbe of opetions equie fo computtion of DFT. The invince is lso desible when loction of objects in depth is not nown nd hs to be detemined fom the econstuction of hologm. In this cse it is lso possible to substntill sve computtion time b selecting smll egion of inteest nd pefoming fst econstuction fo diffeent depth in the pocess of the sech though the depth co-odinte. Fig. illusttes geomet of hologm econstuction on diffeent depth with μ s pmete of depth coodinte. Figs. 3 nd 4 illustte esults of econstuction of two tpes of digitl hologms in diffeent focl plnes. Imges -c wee obtined using conventionl Discete Fesnel tnsfom defined b Eq. 4. Imges in d -f wee obtined using Focl Plne Invint Discete Fesnel tnsfom defined b Eq. 4. REFERECES. Jne, Emde, Loesh, Tfeln Hoehee Functionen, 6 Aufgbe, B. G. Teubne Velggesellschft, Stuttgt, 96 ((see lso L. Yoslvs, M. Eden, Fundmentls of Digitl Optics, Bihuse, Boston, G. Pedini, H. J. Tizini, Shot-Coheence Digitl Micoscop b Use of Lensless Hologphic Imging Sstem, Applied Optics-OT, 4, (, , (. 4. E. Ciche, P. Mquet, Ch. Depeusinge, Sptil filteing of zeo-ode nd twin-imge elimintion in digitl off-is hologph, Appl. Optics, v. 3, o. 3, Aug. f f μ b c f μ Fig.. Geomet of econstuction of hologms ( nd longitudinl coss sections fo econstuction with DFT (b nd FPIDFT (c.

8 b c d e f Fig. 3 Reconstuction of hologm ecoded in fom of mplitude nd phse. - c: econstuction on diffeent depth using Discete Fesnel Tnsfom (Eq.; d - f - econstuction using Focl Plne Invint Discete Fesnel Tnsfom. (Hologm ws indl povided b D. G. Pedini 3 b c d e f Fig. 4. Reconstuction of n off is hologm. - c: econstuction on diffeent depth using Discete Fesnel Tnsfom (Eq.; d - f - econstuction using Focl Plne Invint Discete Fesnel Tnsfom. The hologm ws copied fom PDF file of ppe 4

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