Ptychography intensity interferometry imaging

Ptychography intensity interferometry imaging Wentao Wang, Qi Han, Hui Chen * , Yuan Yuan, and Zhuo Xu Intensity interferometry (II) e xploits the second-order correlation to acquire the spatia l frequency information of an object, which has been used to o bserv e distant stars since 19 50s. Ho weve r, due to unr elia bility of e mpl oye d imaging r econstruction algorithms, II can only imag e simple and sparse objects such as doub le stars. We here develop a method that overcomes this unreliability problem and enables imaging complex objects by combing II and a p tyc hogr aph y iterative algorithm. Different from previous p tych ogr aph y iterative-type algorithms that work only for diffractive objects using coherence light sour ces, our method obtains the objects spatial spectrum from the seco nd-order correlation of intensity fluctuation by using an incoherent source, which therefore largely simplifes the imaging pro cess. Furthermore, by intr oducing loose suppo rts in the pty cho gra phy algorithm, a high-fdelity image ca n be recovered without knowing the precise size and position of the scanning illumination, which is a strong requirement for traditional ptyc hog raph y iterative algorithm.. I. INTRODUCTION In nowad ays, there are many imaging methods whi ch assist people to give a deep vision to the surroundings. Ge nera lly, these imaging methods can be sorted i nto tw o types in ph ysics . One is the classical imaging that is based on the electromagnet ic wa ve s am plitude inte rf er - ometry (A I) , where man y wa ve s are super imposed caus- ing the phenomenon of in terf eren ce to extract informa- tion. It can b e trac ed bac k to the disco ver y of the wa ve - property of lig ht carried by Young(1800). And then, Mac h-Ze hnd er and Mich els on proposed tw o inte rfer om- eters t o observe the int erfe renc e pattern or measure t he phase shift, the first-o rder correlation of the source, by modulating the p ath difference of tw o beams splitting from a single source[1]. The o ther is int ensi ty int erfe r- ometer imagin g, whi ch ex ploits t he high -order corr ela- tion kno ws as photon bu nchi ng of the photons based on the in tens ity interferometry(II) s chem ati c[2- 4]. Sinc e the measured information is cal led correlation fu nction wh ich is the Four ier transform of the aver age dist ri- bution of the ob ject. One can r econstruct its imag e b y taking the inv ers e Fouri er transform of the correla - tion fu nction. In essence, II is the Hanbury Brow n and Twiss(HBT) effect, whi ch utilize t he seco nd-ord er cor - relation of the inte nsit y fluctuat ion of tw o detectors to induce t he angular diamete r of hot stars wit h an exce l- len t resolution(microarcsec ond)[5 -8]. And then, II was used to long bas elines imaging to reve al details acros s and outside stellar surfaces. In classical imagi ng, it is troubled by the receiving phase accuracy bro ugh t by the complex observe envi ron ment, such as turbulence. Ma ny methods are adopted to decrease the wav ef ro nt d istor- tions, su ch as adapti ve optics, whi ch correct t he defo r- mations of an incomin g wav ef ron t by deforming a m ir- ror in order to compensate for the distortion[9]. These methods bring the comp lexi ty at the sa me time and each sub-system add its ow n noise to the whole system and increase the ch ance for a fail ure to occur due to the mul- tiplicative e ffects of errors. Ho wev er t he s etup of the II is simple and less depende nt on the equ ipme nt. And the resolution of this a pproach is dep end on the a rea o f the speckles rath er t han de pending on the size o f t he imag- ing lens . The higher resolut ion requires larger and more perfect lens . This is at a high cost. Fu rth er mor e II is im mune to the eq uipm ent imperfect ions and the atmo- spheric turbulence[10-11]. When II is applied to i magin g, the main shortcom ing is the phase retri eval al gorithm. Because the al gorithm is tend t o t rap in local mi ni m um and slow to co nve rge the true im age. Phase retrieval is to reconst ruct the phase informat ion from in ten sity me asure men ts without the reference beam suc h as h olo gr aph y. The ea rliest phase retrieval is G-S algorithm proposed by Gerc hberg and Saxto n[12]. Then it was developed t o Error Reduction (ER) a nd Hybrid Input-Output (HIO) algorit hms by F ienup[13-1 7]. Both tw o above algorithms are widely accepted and utiliz ed. Ho wev er , above algorithms are plagued b y some fun- damental hardships. First their field of views are t oo m uch small; sec on dl y, t hey tend t o stuck in the stag- nation when fac ing the object s with complicated struc- ture. T o im prov e the con verg enc e speed and rel iab il it y, Rodenburg pro posed the ptych ogr aphy to expand field of view[18] . Thi s method increase the obtained informa- tion by ove rlap i llum inating the object. Ev en the co mpli- cated structure can be reconstructed directly[19]. This is because t he ove rlap illumination increase the kn own ob - ject’s information and the ove rlap positions conve y the in terf eren ce whi ch help to conv erg e the un ique va lue fast. Ho wev er , this method is sensi tiv e to the errors in shape or shift of the probe[20 ]. A nd curre nt res ear che s focus on t he microsco pic whi ch belongs to the clas sical imag- ing. And the acquisition equi pme nt is so high precise and complicated that t he noise is inevitable[21] . Here we demonstrat e II wit h pt ychog rap hy to explore the second order co rrelation im aging in the near field. Fir stly , the theory is briefly deri ved to demonstrate the relation- ship between the inte nsi ty distribution and the correla - tion function. Se con dly , simul atio n and the experiment is demonstrated. The objec t is illuminated by t he inco- he rent ligh t and create the s peckle pattern. T he correla- tion function is deriv ed from the s peckle pattern. Then ∗ chenhui@mail.xjtu.edu.cn Electronic Material Research Laboratory, Key Laboratory of the Ministry of Education and International Centre for Dielectric Research, Xi’an Jiaotong University, Xi’an, China, 710049 2 1 i 2 the object’s image is recons tructed by the retrieval algo - rithm. Th irdl y, the phase re trieval algorit hm is demon- strated. At las t by introducing loose sup ports in t he pty chog rap hy algorithm, a high -fid elit y image can be re - cov ere d wi thout kno win g the precise size and position of the scanning illuminatio n, whi ch is a strong req uire ment for traditional ptych ogr aph y iterative al gorithm. II. PRINCI PLE AND SIMULAT ION A typical in tens ity inte rfe rom eter sche mati c is com - posed of tw o detecto rs and one correla tor. Two detectors placed in one coh eren t area measure the light in tens itie s from the same source. The c orrelation between these t wo detections is an ense mbl e ave rage o f the product of tw o in tens itie s I 1 and I 2 , whi ch is represented as bl ow ( I 1 ( t 1 , x 1 ) I 2 ( t 2 , x 2 ) ) = ( E 1 ( t 1 , x 1 ) E ∗ ( t 1 , x 1 ) ... ∗ E ∗ ( t 2 , x 2 ) E ∗ ( t 2 , x 2 ) ) distribution of the object in Eq (2) can be reconstructed. W e firstly give th e si mula tion about the Ptych o- graphical in tens ity in ter fer ome try show n in Fig. 1. The empl oyed so urce is thermal ligh t, whi ch is classi - fied as inc ohe rent li ght whose in tens ity is Rayleighly distributed and phase is u niformly d istributed on th e interval( − π, π ) . And it obeys the circular complex Gaussian statistical distr ibution. The ligh t illuminates the o bject and produced the speckle patter n. Then th e speckle pattern is recorded to d eri ve the autocorrelation function. At last, by the ser ies au tocorrelation functions corresponding to the illuminat ion, t he objects image is reconstructed by the phase r etrieval algorithm. 2 2 2 = ( I 1 ( t 1 , x 1 ) )( I 2 ( t 2 , x 2 ) ) (1 + | γ 12 | ) (1) whe re E i ( t i , x i ) and E ∗ ( t i , x i ) ( i = 1 , 2 ) a re a pa ir of conj uga te ele c tron ic field va ria ble s, I i ( t i , x i ) is the instant aneous intensit y a t ti me t i at d et ecto r d i , γ 12 is th e m utua l co rr ela tion fun cti on of light inte n si ty bet we en poi nt s x 1 and x 2 , and () den ot es tim e ensemb le average. A cc ord in g to t he van Cit te rt -Z er ni ke t he o re m, corre la tio n f unc tio n γ 12 is t he Fourier transf orm of in tens ity distribution functi on O ( α, β ) o f the object, FIG. 1. The arrangement of the simulation. The source is random and obeys the circular complex Gaussian statistical distribution. The probe mov es along the X direction to scan the object. The CCD captures the spec kle pattern after ever y illumination. jφ rr O ( α,β ) exp [ j 2 π (Δ xα +Δ yβ )] dαdβ γ (Δ x, Δ y ) = e − λz tt I ( α, β ) dα dβ . (2) By Eqs (1) and (2), t he correlation function of tw o de - tectors can b e written as: ( I 1 ( t 1 , x 1 ) I 2 ( t 2 , x 2 ) ) ∝ |F { O ( α, β ) }| . (3) where O ( α, β ) is the inte nsity distribution funct ion of the object , (Δ x, Δ y ) is the relative position o f th e tw o detectors, φ is the phase dif ference induced by dif fer ent optical lengt h and z is the distance bet ween object and observer surfac e. Fro m Eq (3), it is equi val ent to perfect plane wa ve illuminating objects in first-order inte rfer ence configuration. T herefore, in II, s imple man ufact ures are feasible to obtain the diffraction pattern. Ho wev er , what measured is t he s quare of t he auto correlation function, losing the phase informati on, whic h could not re co ver the o bject b y mea ns of dire ct inve rse- Four ier transfor m of Eq (2). H ence, phase r etrieval algorithm is n ecessary to obtain t he image of the o bject. The main appr oaches that generally applied in phase -re trie val algorithm ar e Cau chy- Rie man n [25] and it erative Fouri er ph ase rec ov- ery [26]. In this pa per, ptyc hog raph ica l reconstruc tion algorithm is adopted as one of the iterative Four ier phase reco ver y [2 6]. By p hase retrieval, the inte nsit y FIG. 2. (a) is the reconstructed image with 1000 ER itera- tions ; (b) is the reconstructed image with 1000 ER iterations with beta=0.7 ; (c) is the reconstructed ima ge after 10 PII iterations . (d), (e) and (f) ar e the relative residual at each iteration cor responding to the above retrieval algor it hm in reciprocal space, r esp ect ivel y. In the s imua ltio n, a comparison about three phas e retrieval al gorithms is show n in Fi g. 2. Here speckle pattern are record ed to deriv e the au tocorrelation func- tion. Then the au tocorrelation function is se nt into ER, HIO and Pty to reco nstruct the image, resp ecti vely . The 3 result is sh own in Fig . 2. (a) is the reconstruc ted image after 1000 ER iterations . Though the rough sk etch is clearly sho wn, one cannot d istinguish the details; (d) is the corres ponding relative residual curve at eac h iteration in reciprocal space. It can be seen that it drops qu ickl y within 100 iteratio ns and then tu rns out to be steady aro und 0.03 within foll owin g iterations. T hat means ev en increasi ng the iteration num ber the quali ty of th e reconstruction can not b e imp rov ed. The HIO is called to reconstruct the object, the reconstructed image is show n in Fi g. 2(b) and the rela tive residu al cur ve is sho wn in Fig. 2(e). The reconstructed image is so blurred that nothing can be obtained. The relative residual drops quic kly to 0.1 around 50 iterations. At last, the p tychg rap hy in tens ity correlation imaging is called, the reconst ructed results are sho wn in Fig. 2(c) and Fig. 2( f). The i mage is rec over ed after 10 iterat ions and can be clearly distin guished. And the corres ponding relative residual cur ve drops and stays 0.03 a fter 6 iterations. III. EXPERIMENT The sch emat ic of the experiment is sho wn in F ig. 5. The l aser emp loyed is solid -state and wa ve le ngt h is 45 7nm(MG: 85-BLS-601 ). The laser in cide nt on the reverse-telescop e group(Lens1 and Lens 2) to be 10 times amplified, then th e ampl ified beam whose diameter is about 20mm in cide nts on the rot ating ground glass (G.G) to produce the pseudothermal li ght. The volt age on the ground glass is 1 2v. The spe ed of the grou nd glass is 0.6 rpm(round per miniute ). Then the pseu dothermal li ght propagates to the pro be to illuminating the object. The distance between the G.G and the probe is 300mm. So the coherence length on the probe plane is about 6 . 855 μm according to Δ x = λ ∗ z/D , where λ is t he ce nter wa ve - length of inc iden t l igh t, z is the propagation distance and D is t he typical size. Finally the speckle pattern is recorded b y t he CCD whi ch locat es at 250mm after the ob ject. T he d iameter of the pro be a pplied in the ex - periment is 5mm and mo ve alon g the X direction with eac h step 0.5mm. The numbe r of the steps is 16. A nd CCD captures 1000 di ffer ent s peckle patterns at each step and the exposur e time is 20 ms. The cohe rence lengt h of the spec kle is about 23 μm ac cording t o t he above equa- tion.The co mputer di spla ys one captured s peckle pattern at step 1. The num ber of speckle pattern h ere used to calculate the autoc orrelation function is 1 000. Then the probe is set be fron t of the CCD to explore the long-distance o bservatio n just sh own in Fig. 5(b). The resu lts is sh own in Fig. 8. T he reconstructe d object is some inclined, this is because of the ob j ect set tle ment in the experim ent. Here, we are successf ul to demonstrat e ptych ogr aph ical in ten sit y int erfe rom etr y where the probe is lo cated in the fr ont of the object and FIG. 3. The schematic of the e xperiment setup. (a) the probe is in the front of the object , this setup is used for short distance, the w avel eng th of the laser is 457nm; the f ocal l ength of lens1 is 25.4mm; the focal length of lens2 is 250mm ; G.G is the rotating ground gla ss; probe is the circular hole wit h diameter 5mm; the computer displays the ca ptured speckle pattern by CCD. (b) the probe is behind of the object, this setup is used for long distance. detector respectively by employing the pseudotherm al lig ht. H owe ve r, the precise size of the probe mo vem ent is obtained from t he trans lation stage in the la b and there is so me bias about the probe mov em ent in real application. Hence we give some discussion a bout the fault tol erance o f the m ove me nt in our retrieval algorithm. IV. RECO NSTRU CTI ON M ETHOD FIG. 4. Flo wch art of the reconstr uction process with the method in the paper. The d ata flow ch ar t of the reconstruc tion is sch ema ti- cally show n in Fig. 1, w here the iterative reconst ruction is carried out after the initial guess is given to the object function: (1) The ob ject O ( r ) is illum inated b y the ligh t souce P ( r − R ) whi ch is modulat ed by the p robe P ( r ) to pro- duce t he lig ht field X ( r, R ) = O ( r ) P ( r − R ). The ob ject is assumed to b e very thin. (2) The li ght field F ( k ) on t he CCD plan e is num eri - 4 cally propagating the X ( r, R ) to the recording plane and replacing its modulus with t he squar e root of the corr e- sponding calculatin g autoco rrelation funct ion amp ( r − R ) and the phase information ke ep unc hang ed. (3) The mod ified F ( k ) is t hen propagated to the object plane to obtain t he updat ed t he object plane O n ( r ) with the fol lowi ng equation: O n ( r ) = O n − 1 ( r ) + X n ( r, R ) . (4) The above calculation is sent to the next illum inat ion probe unti l the abo ve calcu lation has made at all the il - lum ination position s. (5) The iteration w ill stop after achi eve the ma xim um iteration num ber. V. DISSCU SSIO N Here we analyse the effect of shift error for the p tych o- graphical int ensi ty i nt erf er om etr y. The ex periment setup is the same as Fig. 6(a ). The reconst ructed images are sh own in Fig. 8. When there is no shift error, we can successfully obtain the objec t image. T he reco nstructed Then increasing the shift to 20% , the reconstructed im - age is becoming l ittle blurred. Under 25% sh ift error, it is hard to distingu ish the object i nformation. At l ast it is becom ing so mess that no informatio n is distingu ished from the reconstruc ted image u nder 50% sh ift error. FIG. 5. The r econstructed image with 20 iterations under different shift e rrors: (a1) reco nstruction under no shift error; (a2) the relative residual curve under no shift error; (b1) reconstruction under 10% shi ft error; (b2) relative r esidual curve corresponding to 10% shift error; (c1) reconstr uction under 25% shift error; (c2) relative residual curve corre- sponding to 25% shift error; (d1) reconstruction under 50% shift error; (d 2) relative residual curve corresponding to 50% shift error. Ho wev er it is hard to control the shift er ror with long dis- tance observation in re ali ty . Here we prov ide a sol ution to sol ve this trouble. In r econstructing process, some looses are given to enlarge the support under 25% and 50% s hift error resp ecti vel y. The result is sho wn in Fig. 9. Fo r 25% shift error, when we enlarg e loose to 5, the reconstructed image is becom ing evid ent. It is clearly d istinguish under 10 and 15 enlarge res pec tive ly. T o enlarge loose to 20, the reconstruct ed image becomes so mess to distinguish the deta ils. F or 50 % shift error, when enlarge loose to 15 and 20 resp ecti vel y, the reconstructed images are c learly distinguish the d etails. Her e w e can conclude that when there is shift error, it is possible to adjust the loose of the support to reconstruct the evide nt im age. FIG. 6. The reconstructed images under different support looses with 20 iterations. The first row is corresponding to the 25% shift error and the second ro w corresponding to the 50% shift error. (a1) , ( a2), (a3) are reconstructed images corresponding to support loose = 0, 5, 15 respectively unde r 25% shift error; (a4) the relative residual curve under above three situati ons; (b1) , (b2), (b3) are reconstructed images corresponding to support loose = 0, 5, 15 respectively unde r 50% shift error; (b4) the relative residual curve under a bove three situations. FIG. 7. The reconstructed images under different support looses with 20 iterations. The first row is corresponding to the 25% shift error and the second ro w corresponding to the 50% shift error. (a1) , (a2), (a3) are reconstructed images corresponding to support loose = 0, 5, 15 respectively unde r 25% shift error; (a4) the relative residual curve under above three situati ons; (b1) , (b2), (b3) are reconstructed images corresponding to support loose = 0, 5, 15 respectively unde r 50% shift error; (b 4) the relative residual curve under above three situations. VI. CONC LUSIO N In sum mar y, w e hav e propo sed and demonstrated an imaging method, termed p tych ogra phi cal in tens ity i nte r- fe ro me try . This a pproach reconstruc t a sample i mage from ma ny speckle patterns with the inco her ent illumi - nation. It is the fi rst time that proposes and realises the ptyc hog raph ica l int ensi ty correlat ion imaging in the lab. The use of the pty chog rap hy i nte nsit y int erfe rom e- 5 try framework for reconstructing a sample image is new and may pr ovi de an al ternative sol ution fo r t he existing Ch eren kov Tele scop e Array to obser ve the long d istance stars. Sev era l adv an tage s associated with the reported method is stated as below . First ly, it increases the rel iab ili ty when reconstructing t he complicated object that the tra- ditional phase retrieval algorithm cannot coo perate. Sec- on dl y, it imp rov e the error tolerat ion of the probe shif t b y adjusting the lo ose of the sup port. At l ast the this method is very effi cie nt in terms of c ompution cost. The solution typically conv erg es with 5-20 loops. In our ex - periment, we used 11 - 21 loops to reco nstruct the images. There are seve ral limitations as sociated with the reporte d pty chog rap hy in tens ity i nt erf er ome tr y. Firstly the auto- correlation is deri ved from large numb er of speckle pat- terns. This i ncr ease t he coll ecting time and computional time. In the exp eriment, we found that aut ocorrelation der ive d from 10 speckle patterns can reconst ruct the sam - ple at least show n in Fig. 7. Second ly the pixel size of CCD e mplo yed in t he expe riment is 6 . 45 μm , this rest ricts the measured speckle size and the resolution of the sa m- ple is restricted in hence. Thirdl y the mi ni mum exposure time of the C CD is about 0 . 1 ms , whcih is mu ch longer than the coher ent t ime of thermal source( f s ). Hence it canno t c apture the in tens ity fluct uations of thermal source. 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