Multi-transmitter aperture synthesis with Zernike based aberration correction

Transcription

1 Mult-transmtter aperture synthess wth Zerne based aberraton correcton Bahadr Guntur,, Davd J Rabb, 2 and Douglas F Jameson 2 Lousana State Unversty, Dept of Electrcal & Computer Engneerng, Baton Rouge, Lousana 783, USA 2 Ar Force Research Lab, Sensors Drectorate, Wrght-Patterson AFB, Oho 45433, USA bahadr@ecelsuedu Abstract: Mult-transmtter aperture synthess ncreases the effectve aperture n coherent magng by shftng the bacscattered specle feld across a physcal aperture or set of apertures Through proper arrangement of the transmtter locatons, t s possble to obtan specle felds wth overlappng regons, whch allows fast computaton of optcal aberratons from wavefront dfferences In ths paper, we present a method where Zerne polynomals are used to model the aberratons and hgh-order aberratons are estmated wthout the need to do phase unwrappng of the dfference fronts 22 Optcal Socety of Amerca OCIS codes: (9995 Dgtal holography; (3 Image reconstructon technques References and lns N J Mller, M P Derng, and B D Duncan, Optcal sparse aperture magng, Appl Opt 46(23, (27 2 D Rabb, D Jameson, A Stoes, and J Stafford, Dstrbuted aperture synthess, Opt Express 8(, (2 3 B Guntur, N J Mller, and E A Watson, Camera phasng n mult-aperture coherng magng, Opt Express 2(, (22 4 D J Rabb, D F Jameson, J W Stafford, and A J Stoes, Mult-transmtter aperture synthess, Opt Express 8(24, (2 5 R A Muller and A Buffngton, Real-tme correcton of atmosphercally degraded telescope mages through mage sharpenng, J Opt Soc Am 64(9, 2 2 (974 6 R G Paxman and J C Marron, Aberraton correcton of specled magery wth an mage sharpness crteron, n Statstcal Optcs, Proc SPIE 976, (988 7 J R Fenup and J J Mller, Aberraton correcton by maxmzng generalzed sharpness metrcs, J Opt Soc Am A 2(4, (23 8 S T Thurman and J R Fenup, Phase-error correcton n dgtal holography, J Opt Soc Am A 25(4, (28 9 R J Noll, Zerne polynomals and atmospherc turbulence, J Opt Soc Am 66(3, 27 2 (976 D Rabb, J W Stafford, and D F Jameson, Non-teratve aberraton correcton of a multple transmtter system, Opt Express 9(25, (2 M P Rmmer and J C Wyant, Evaluaton of large aberratons usng a lateral-shear nterferometer havng varable shear, Appl Opt 4(, 42 5 (975 2 G Harbers, P J unst, and G W R Lebbrandt, Analyss of lateral shearng nterferograms by use of Zerne polynomals, Appl Opt 35(3, (996 3 S Ouda, T Nomura, amya, and H Myashro, Hgh-precson analyss of a lateral shearng nterferogram by use of the ntegraton method and polynomals, Appl Opt 39(28, (2 4 F Da, F Tang, X Wang, P Feng, and O Sasa, Use of numercal orthogonal transformaton for the Zerne analyss of lateral shearng nterferograms, Opt Express 2(2, (22 # $5 USD Receved 6 Aug 22; revsed 2 Oct 22; accepted 2 Nov 22; publshed 8 Nov 22 (C 22 OSA 9 November 22 / Vol 2, No 24 / OPTICS EXPRESS 26448

2 5 J W Goodman, Introducton to Fourer Optcs (Roberts and Company, 24 6 J D Schmdt, Numercal Smulaton of Optcal Wave Propagaton (SPIE, 2 7 R C Gonzalez and R E Woods, Dgtal Image Processng (Prentce Hall, 27 8 J W Goodman, Specle Phenomena n Optcs: Theory and Applcatons (Roberts and Company, 2 Introducton Coherent magng s a growng area of research where objects are maged usng laser lght The reflected wavefront s captured by an magng system some dstance away Dgtal holography s used to capture the optcal phase of the lght ncdent on the recever For typcal dffuse objects the reflected lght creates fully developed specle at the recever, whch lmts magng system performance and complcates many approaches currently used to determne atmospherc aberratons Coherent aperture synthess s a technque whch enables hgh-resoluton magery by syntheszng a large aperture through the combnaton of mage data from multple smaller subapertures [, 2] Each sub-aperture captures the pupl feld usng some holographc magng technque, then the measured sub-aperture felds are placed n a common pupl plane correspondng to the physcal locatons of the sub-apertures, and the composte pupl plane feld s Fourer transformed, thus formng a dgtal mage Usng a sharpness measure (appled on the mage formed, the nter-aperture aberratons (ncludng pston, tp, tlt, rotaton, and shft and ntra-aperture aberratons (such as defocus, astgmatsm, coma can be corrected [3] The mult-transmtter aperture synthess dea has been recently proposed n [4] By usng multple transmtters at dfferent locatons, and turnng on one transmtter at a tme, an aperture captures shfted pupl felds Ths effectvely turns a sngle, physcal aperture nto a multaperture magng system, where coherent aperture synthess technque could be used to obtan hgh-resoluton magery The technque requres that target moton orthogonal to the lne of sght over the course of the measurements s ether nown or s small wth respect to the aperture s resoluton; whle the system s also senstve to pston moton on the order of the wavelength the phase error can readly be found and corrected due to the resultng overlappng pupl data When the apertures are sparsely dstrbuted, the aberratons are estmated by defnng and optmzng a sharpness measure on the object mage [5 8] The aberratons are typcally modeled wth Zerne polynomals [9], and the problem s defned as calculatng the optmal weghts of Zerne polynomals to maxmze the sharpness measure Whle ths approach has been demonstrated to be effectve n mult-transmtter aperture synthess [4], the downsde s the computatonal complexty of the requred optmzaton process Instead of formng a set of sparsely dstrbuted apertures, the transmtter locatons can be placed close enough so that the pupl feld captured by the shfted apertures overlap [] Overlapped aperture data s used to estmate the aberratons common to all of the aperture realzatons based on technques smlar to lateral shearng nterferometry Essentally, the wavefront dfference n the overlap regon s derved n terms of aberraton parameters, whch are then estmated from the measured data The technque s based on the assumpton that the statc and atmospherc aberratons present n the aperture asde from pston are constant across the measurements; ths requres that all measurements are recorded wthn the atmospherc coherence tme n order to reconstruct the dynamc atmospherc aberratons In [], the dea s demonstrated wth low-order aberratons by modelng the aberratons wth bvarate polynomals Instead of bvarate polynomals, Zerne polynomals could be used to model optcal aberratons Whle Zerne polynomals form a complete orthogonal bass on a crcular pupl, ther shfted dfferences are not necessarly lnearly ndependent In the lateral shearng nterferometry lterature, the dfference front has been modeled n varous ways, ncludng Zerne polynomals [], Zerne polynomals wth largest possble ellptcal doman [2] or largest # $5 USD Receved 6 Aug 22; revsed 2 Oct 22; accepted 2 Nov 22; publshed 8 Nov 22 (C 22 OSA 9 November 22 / Vol 2, No 24 / OPTICS EXPRESS 26449

3 possble crcular doman [3] n the overlap regon, and numercal orthogonal polynomals [4] In ths paper, we present a method to estmate aberratons for mult-transmtter aperture synthess The aberratons are modeled wth Zerne polynomals and the dfference front s drectly used wthout any polynomal fttng unle the method n [] We segment the dfference front nto sub-regons; each sub-regon has an unnown phase offset, whch s added to the set of lnear equatons to be solved Zerne coeffcents that model the aberratons are estmated along wth the phase offsets wthout the need to do 2D phase unwrappng of the dfference front The estmated Zerne coeffcents are then used to compensate for the modeled aberratons 2 Proposed method As llustrated n Fg and descrbed n [4,], the modeled mult-transmtter system llumnates a scene wth one transmtter at a tme and ndependently captures the bacscattered felds n the pupl plane Because of the shft n the transmtter locatons, the bacscattered target feld U b (x,y s shfted by the same amount (n the reverse drecton On the other hand, the phase error s statc because the sensor s fxed Suppose that we measure two pupl-plane felds, U (x,y and U 2 (x,y, each wth a dfferent transmtter locatons: U (x,y=p(x,yexp( j2πw e (x,yu b (x x,y y, =,2, ( where P(x,y s the pupl functon, W e (x,y s the wavefront error, and (x,y and (x 2,y 2 are the shft amounts n the bacscattered felds due to transmtter locatons Let W b (x,y be the wavefront of U b (x,y, then the detected wavefronts are W (x,y=w e (x,y+w b (x x,y y, =,2 (2 The object wavefront s not well defned and may contan several branch ponts, so t s desrable to calculate the aberraton wavefront ndependent of the target wavefront It s possble to numercally remove the dependence on the bacscattered wavefront by frst regsterng the wavefronts W (x,y accordng to the shft amounts The regstraton can be acheved based on calbrated measurement of the shfts due to transmtter locatons, as well as by regsterng the pupl plane specle ntensty returnng from the target [] By tang the dfference between the wavefronts n the overlappng area (accomplshed by multplyng one complex feld by the complex conjugate of the other, the wavefront dfference (the dfference front becomes ΔW(x,y = W (x + x,y + y W 2 (x + x 2,y + y 2 (3 = (W e (x + x,y + y +W b (x,y (W e (x + x 2,y + y 2 +W b (x,y = W e (x + x,y + y W e (x + x 2,y + y 2 In other words, after regstraton, the dfference between the phase maps of measured felds s essentally due to the wavefront error The goal s to estmate the wavefront error and compensate for that Note that ΔW(x,y may have phase wraps although not explctly wrtten n (3, and we wll explan how to handle phase wrappng ssue shortly We model optcal aberratons wth Zerne polynomals Let W e (x,y= a Z (x,y, where Z (x,y s a Zerne polynomal and a s the unnown coeffcent correspondng to Z (x,y Then, the wavefront dfference becomes ΔW(x,y = a Z (x + x,y + y a Z (x + x 2,y + y 2 (4 = a (Z (x + x,y + y Z (x + x 2,y + y 2 = a ΔZ (x,y, # $5 USD Receved 6 Aug 22; revsed 2 Oct 22; accepted 2 Nov 22; publshed 8 Nov 22 (C 22 OSA 9 November 22 / Vol 2, No 24 / OPTICS EXPRESS 2645

4 ,, Coherent sensor Transmtter array Fg Mult-transmtter system The scene s llumnated wth one transmtter at a tme A shft of Δx n the transmtter locaton results n a shft of Δx n the bacscattered feld (a (b Fg 2 (a Wavefront dfference between two overlappng apertures (b Sub-regons wth unform phase offset where we defned ΔZ (x,y Z (x + x,y + y Z (x + x 2,y + y 2 At ths pont, one may propose to solve for a by formng a lnear set of equatons for all postons (x,y There are, however, two mportant ssues Frst, the dfference front ΔZ (x,y does not form an orthogonal bass, resultng n a set of equatons that are not lnearly ndependent The problem could be allevated by usng multple measurements correspondng to transmtter separatons along dfferent drectons gvng addtonal overlappng regons, whch could result n an over- or well-determned system necessary for the proposed approach to successfully calculate the desred aberraton coeffcents Second, the dfference front ΔW(x, y needs to be phase unwrapped Loong at Fg2(a, we see that there could be phase jumps of 2π from one regon to another That s, the equaton (4 s not vald for all pxels n the overlap regon On the other hand, the overlap regon could be dvded nto sub-regons, where there s no phase wrap as llustrated n Fg2(b; and wthn each sub-regon, we could wrte (4 wth a constant but unnown phase offset In sub-regon m, the wavefront dfference at the th poston n that sub-regon s ΔW (m (x (m,y (m =α (m + a ΔZ (m (x (m,y (m, (5 where α (m s the unnown phase offset, ΔZ (m (x (m,y (m s the dfference between Zerne polynomals, and a s the coeffcent of the th Zerne polynomal Let N m be the number of pxels n sub-regon m and M be the number of sub-regons, then we form the followng lnear set of equatons: # $5 USD Receved 6 Aug 22; revsed 2 Oct 22; accepted 2 Nov 22; publshed 8 Nov 22 (C 22 OSA 9 November 22 / Vol 2, No 24 / OPTICS EXPRESS 2645

5 , Aperture,, Transmtter array (h (a (b Fg 3 (a Actual mult-transmtter system (b Equvalent mult-aperture system ΔW ( (x (,y( ΔW ( (x ( N,y ( N ΔW (2 (x (2,y(2 ΔW (2 (x (2 N 2,y (2 N 2 ΔW (M (x (M,y (M ΔW (M (x (M N M,y (M N M ΔZ ( (x(,y( ΔZ( (x(,y( ΔZ ( (x( N,y ( N ΔZ ( (x( N,y ( N ΔZ (2 (x(2,y(2 ΔZ(2 (x(2,y(2 ΔZ (2 (x(2 N 2,y (2 N 2 ΔZ (2 (x(2 N 2,y (2 N 2 = ΔZ (M (x (M,y (M ΔZ (M (x(m,y (M ΔZ (M (x (M N M,y (M N M ΔZ (M (x(m N M,y (M N M α ( α (2 α (M a a (6 The unnown parameters α (,,α (M,a,,a can be estmated n a number of ways; n ths paper they are estmated usng the QR decomposton based pseudo-nverse operaton (the bacslash operator n MATLAB The Zerne coeffcents a,,a are then used to correct for the aberratons n each aperture Note that the matrx equaton (6 s not lmted to two transmt realzaton of the aperture If there are more than two transmtters, the sub-regons from any par of apertures are ncluded n ths matrx equaton 3 Expermental results We present several experments to demonstrate the method A three-transmtter magng system shown n Fg 3(a s smulated The smulaton models an optcally dffuse object at range, L, whch s flood llumnated by a coherent laser source of wavelength λ The complex-valued feld reflected off the object s modeled wth ampltude equal to the square root of the object s ntensty reflectance and phase a unformly dstrbuted random varable over π to π The complex-valued feld n the recever plane, subject to the paraxal approxmaton, s gven by the Fresnel dffracton ntegral [5] The Fresnel dffracton ntegral s numercally evaluated usng the angular spectrum propagaton method [6] The object plane and recever pupl planes n the smulaton conssted of N = computatonal grds wth dentcal 82μm sample spacngs n both planes The optcal wavelength, λ, s 55μm, and the range, L, from # $5 USD Receved 6 Aug 22; revsed 2 Oct 22; accepted 2 Nov 22; publshed 8 Nov 22 (C 22 OSA 9 November 22 / Vol 2, No 24 / OPTICS EXPRESS 26452

6 (h -3-3 (a (b (c -3 (d (e (f Fg 4 (a to (c show the phase dfference n overlap regons (d to (e show the subregons wthn each overlap regon the receve pupl plane to the object s meters The numercal propagaton conssts of partal propagatons of meters each to avod the wraparound effects The optcal feld n the recever pupl plane s collected usng a 48mm dameter aperture wth three transmtters n the confguraton shown n Fg3(b The focused optcal felds are then aberrated usng randomly weghted Zerne polynomals To estmate the aberratons, we regster the pupl felds accordng to the transmtter locatons and tae the dfference between the wavefronts n the overlappng regons, as shown n Fg 4(a to 4(c There are three overlap regons Wthn each overlap regon, we determne the sub-regons through segmentaton Our segmentaton procedure s as follows We frst detect the edges usng the Canny edge detector [7] The edge lnes are morphologcally dlated (wth a3 3 ernel to have a wder coverage of dscontnutes Regon segments outsde the edges form the sub-regons The extracted sub-regons are shown n Fg 4(d to 4(f As seen n these sample results, the segmentaton procedure may result n over-segmentaton, however, ths s not an ssue The opposte (under-segmentaton, on the other hand, would be an ssue as multple regons wth dfferent phase offsets would be forced to have the same phase offset, whch would degrade the estmaton of the Zerne coeffcents Wth over-segmentaton, the only concern s the ntroducton of addtonal phase offset parameters to be estmated, and therefore an ncrease n the computatonal cost One possble approach to reduce computatonal cost s to dscard small sub-regons, whch should not affect the performance as the majorty of pxels are ncluded In our experments we dscard sub-regons that are less than 2 pxels, and place the remanng sub-regons nto the matrx equaton (6 In conventonal shearng nterferometry the dfference front s nterpreted as the slope or gradent of the aberraton at a gven locaton along the drecton of the shear, here we nstead use a matrx based approach to gve a system of equatons to relate the observed dfference fronts # $5 USD Receved 6 Aug 22; revsed 2 Oct 22; accepted 2 Nov 22; publshed 8 Nov 22 (C 22 OSA 9 November 22 / Vol 2, No 24 / OPTICS EXPRESS 26453

7 (h (a (b -35 (c (d Fg 5 (a Image formed at an aperture by averagng 3 specle realzatons (b Estmated phase error usng the proposed algorthm (Unts are n waves; up to ffth order Zerne polynomals are used (c Aberraton corrected frst aperture, correspondng to (a (d Composte formed by all three aberraton-corrected apertures # $5 USD Receved 6 Aug 22; revsed 2 Oct 22; accepted 2 Nov 22; publshed 8 Nov 22 (C 22 OSA 9 November 22 / Vol 2, No 24 / OPTICS EXPRESS 26454

8 RMS wavefront reconstructon error [waves] th order Zerne aberratons (h SNR= 2 3, a 3 2 7th order Zerne aberratons RMS wavefront aberraton [waves] SNR= 2, a 5 SNR= 3, a 5 SNR= 4, a 5 SNR= 2, a SNR= 4, a Fg 6 Root mean square error n the wavefront reconstructon as a functon of root mean square wavefront aberraton and those that would be expected from a partcular aberraton Through a pseudo-nverse operaton we obtan the least squares soluton mnmzng the dfference between the observed and calculated dfference fronts Ths lnear algebrac approach wthout the need for unwrappng allows the aberraton to be estmated effcently even wth a complex target scene present An mage formed usng one of the three transmt locatons s shown n Fg 5(a The phase aberraton obtaned through solvng (6 s shown n Fg5(b The aberraton s found for a sngle specle realzaton, then the aberraton correcton s appled to 3 dfferent specle realzatons, whch are fnally averaged wth the result shown n Fg5(c We can synthesze a larger aperture by placng all three pupl felds n a common plane; the resultng composte mage, averaged over 3 realzatons, s shown Fg5(d, whch has a hgher spatal resoluton than Fg5(c We also wanted to evaluate the performance of the algorthm wth nose present In addton to the phase aberratons, we corrupt the pupl feld data wth addtve whte Gaussan nose [8] Here the SNR s defned as the average rato of the sgnal ntensty to the nose ntensty for all the pxels n the coherent recever, ths would be equvalent to the number of target photo-electrons collected at each pxel for a shot nose lmted system We calculate the root mean square (RMS wavefront reconstructon error for dfferent nose amounts and wavefront aberratons In an experment, wavefront aberraton s smulated by randomly choosng the coeffcents of Zerne polynomals from a specfc range, and ts RMS value s recorded; n addton, random nose wth a specfc sgnal to nose rato s created for each realzaton The actual wavefront s degraded wth the smulated wavefront aberraton and nose; the proposed algorthm s then used to perform the restoraton, and the RMS wavefront reconstructon error s calculated We repeat the experment tmes for each sgnal to nose rato, Zerne aberraton order and coeffcent range The overall results are shown n Fg 6 For lower strength # $5 USD Receved 6 Aug 22; revsed 2 Oct 22; accepted 2 Nov 22; publshed 8 Nov 22 (C 22 OSA 9 November 22 / Vol 2, No 24 / OPTICS EXPRESS 26455

9 (h RMS wavefront reconstructon error [waves] th order Zerne aberratons, a 5th order Zerne aberratons, a 7th order Zerne aberratons, a 5 5th order Zerne aberratons, a SNR n the pupl feld Fg 7 Root mean square error n the wavefront reconstructon as a functon of sgnal to nose rato n the pupl feld (h (a (b (c (d (e (f (g -5 Fg 8 A sample restoraton, where the RMS wavefront aberraton s 495 and the sgnal to nose rato s (a-(c The dfference fronts and the correspondng sub-regons are dsplayed (d Actual wavefront aberraton (e Estmated wavefront aberraton (f-(g The dfference between the actual and estmated wavefront aberratons The RMS wavefront reconstructon error s 43 Note that n (d-(f, the same colormap s used for comparson purposes # $5 USD Receved 6 Aug 22; revsed 2 Oct 22; accepted 2 Nov 22; publshed 8 Nov 22 (C 22 OSA 9 November 22 / Vol 2, No 24 / OPTICS EXPRESS 26456

10 aberratons the reconstructon error appears to be only a functon of the sgnal to rato For hgher strength aberratons the performance starts to deterorate due to the smaller and nosy segments n the overlap area In Fg 7, RMS wavefront reconstructon error s plotted aganst sgnal to nose rato A sample restoraton result s gven n Fg8 The segmented regons are apparently more fragmented due to nose; but a satsfactory restoraton can stll be acheved 4 Conclusons In ths paper we present a method to estmate aberratons, modeled wth Zerne polynomals, n a mult-transmtter magng system We choose the transmtter locatons to form overlap regons between effectve apertures and avod computatonally costly teratve optmzaton technques In order to avod 2D phase unwrappng, the dfference wavefront s segmented nto sub-regons (The segmentaton process dscards edge pxels to avod msplacement; t s possble to mnmze the amount of pxels dscarded by usng sub-pxel accurate edge detecton The phase offset n each sub-regon s not nown and s ncluded n the set of lnear equatons along wth the unnown Zerne polynomal coeffcents The estmaton s smply a pseudo-nverse operaton We reported modeled results wth ffth and seventh order aberratons The reconstructon method proposed here could be used wth mult-transmtter, multrecever sparse aperture magng systems By formng overlaps between dfferent apertures, t s possble to estmate pston/tp/tlt errors between apertures by ncludng the correspondng Zerne polynomals We use an aperture confguraton, whch results n the coverage of entre pupl area so that the system of equatons gves a soluton for the aberraton n all areas of the pupl that s represented by the Zerne coeffcents found If the confguraton dd not cover the entre pupl area (that s, when the dstance between the transmtter locatons were larger than the aperture radus, then the system would be under-determned due to some parts of the aberraton not beng accounted for n the overlap regons (typcally n the center of the aperture Smlar to other shearng nterferometry approaches, some hgher spatal frequency aberratons cannot be reconstructed when those spatal frequences are cyclcal over transmtter separatons For the case shown the separatons were maxmzed to acheve a larger effectve aperture, for stuatons where aberraton spatal frequences are of concern smaller transmtter separatons would be desrable # $5 USD Receved 6 Aug 22; revsed 2 Oct 22; accepted 2 Nov 22; publshed 8 Nov 22 (C 22 OSA 9 November 22 / Vol 2, No 24 / OPTICS EXPRESS 26457