An iterative wave-front sensing algorithm for high-contrast imaging systems *

Transcription

1 An ieaive wave-fon sensing algoihm fo high-conas imaging sysems * Jiangpei Dou,, Deqing Ren,,,3 and Yongian Zhu, aional Asonomical Obsevaoies / anjing Insiue of Asonomical Opics & Technology, Chinese Academy of Sciences, anjing 004, China Key Laboaoy of Asonomical Opics & Technology, anjing Insiue of Asonomical Opics & Technology, Chinese Academy of Sciences, anjing 004, China 3 Physics & Asonomy Depamen, Califonia Sae Univesiy ohidge, 8 odhoff See, ohidge, Califonia , USA Absac Wave-fon sensing fom focal plane muliple images is a pomising echnique fo high-conas imaging sysems. Howeve, he wave-fon eo of an opics sysem can be popely econsuced only when i is vey small. This pape pesens an ieaive opimizaion algoihm fo he measuemen of lage saic wave-fon eos diecly fom only one focal plane image. We fisly measue he inensiy of he pupil image o ge he pupil funcion of he sysem and acquie he abeaed image on he focal plane wih a phase eo ha is o be measued. Then we induce a dynamic phase o he esed pupil funcion and calculae he associaed inensiy of he econsuced image on he focal plane. The algoihm is o minimize he inensiy diffeence beween he econsuced image and he esed abeaed image on he focal plane, whee he induced phase is as he vaiable of he opimizaion algoihm. The simulaion shows ha he wave-fon of an opics sysem can be heoeically econsuced wih a high pecision, which indicaes ha such an ieaive algoihm may be an effecive way fo he wave-fon sensing fo high-conas imaging sysems. Key wods: echniques: wave-fon sensing, imaging pocessing mehods: numeical planeay sysems. Inoducion Discoveing life on anohe plane will poenially be one of he mos impoan scienific advances of his cenuy. The seach fo life equies he abiliy o deec phoons diecly fom one plane and he use of specoscopy o analyze is physical and amospheic condiions. The diec imaging of an Eah-like low-mass plane obiing is bigh pimay sa is, howeve, exemely challenging. Fo ASA s Teesial Plane Finde Coonagaph, a conas of 0-0 a an inne woking angula (IWA) disance bee han 4λ/D is equied in he visible wavelengh (Bown & Buows 990). Recenly many high-conas coonagaphs have been poposed fo he diec imaging of an Eah-like exoplane which can heoeically each 0-0 conas a a few λ/d fom a bigh sa (Ren &Seabyn 005; Guyon e al. 006; Ren &Zhu 007). Howeve, mos of exising coonagaphs can only each a conas in he ode of 0-5 ~0-7, even in he laboaoy (Kasdin e al. 004; Dou e al. 00; Ren e al. 00). One of he main conas limiaion facos comes fom he wave-fon eo induced by impefecions in boh he elescope opics and he coonagaph, * Suppoed by he aional aual Science Foundaion of China

2 which foms saic speckle noises suounding he sa image (Ren & Wang 006). The local speckles ae much bighe han he plane image because of he lage bigh diffeence beween he plane and is paen sa, hus making he diec imaging of an Eah-like exoplane impossible. Fo he space-based obsevaion, he wave-fon eo changes vey slowly and high S/ can be achieved by inceasing he exposue ime. Fo gound-based obsevaion, he saic wave-fon eo is one of he majo eo souces, alhough dynamic wave-fon eo ha is induced by he amospheic ubulence and is changing fom ime o ime dominaes he imaging pefomance. Fo an exeme adapive opics sysem dedicaed o he diec imaging of an exoplane, boh eos need o be coeced efficienly. In Seabyn s ecen pape, a phase eieval wave-fon sensing algoihm has been poposed o coec he saic speckle and hee planes aound he sa of HR 8799 have been imaged (Seabyn e al., 00). To emove he wave-fon eo induced speckles, one pomising appoach is called he speckle nulling echnique, which can be ealized by using a defomable mio (DM) o induce phase on he pupil plane of he coonagaphic sysem. Wih a specific phase povided by he DM, i can heoeically ceae a local high-conas zone on he focal plane ha can be seved as a discovey aea fo he exoplane diec imaging (Malbe 995). Based on he speckle emoving echnique, an exa-conas impovemen of 0 - ~0-3 is expeced o gain afe eliminaing he speckle noise suounding he sa image. The mos impoan pocedue fo he high-conas zone coecion is o pecisely measue he wave-fon eo of he sysem. In ecen papes, diffeen algoihms wee pesened o econsuc he wave-fon diecly fom muli-images aken on he focal plane of an opics sysem. Fisly, hese algoihms can wok only when he oiginal wave-fon eo of an opics sysem is no oo lage, due o he appoximaion ha was inoduced on eihe he phase eo o be measued o he phase povided by he DM (Bode & Taub 006; Give on e al. 007; Dou e al. 009). Secondly, he DM defomaion mus be chosen caefully o ceae muli-images uncoelaed o each ohe, so ha he denominao D in above algoihms will no be zeo. Ohewise, hee will be no soluion fo he above algoihm, in which case he wave-fon can neve be popely econsuced. Thidly, in he above algoihm he phase povided by he DM is calculaed hough he so called influenial funcion model, which may fuhe induce unnecessay eos in he acual pocess of he wave-fon sensing. To ovecome hese poblems, his pape pesens an ieaive opimizaion algoihm fo wave-fon measuemen diecly fom one single focal plane image. Fisly, he inensiy of he image aken on he pupil plane of he opics sysem is measued and is elecic field magniude can diecly be calculaed. Then we inoduce a dynamic phase o he pupil and pefom he Fouie ansfomaion o calculae he associaed inensiy of he econsuced focal plane image. Such a econsuced image will be compaed wih he acual focal plane image ha includes he acual wave-fon eo of he opics sysem. The algoihm is o minimize he inensiy diffeence beween he econsuced image and he acual focal plane image of he opics sysem, whee he induced wave-fon eo is as he vaiable. In each sep of he ieaive opimizaion pocedue, he induced phase, as he only vaiable of he opimizaion algoihm, will change o make he inensiy of he wo images appoximaing o each ohe. In he whole pocedue, hee is no appoximaion on he phase, which may guaanee a high pecision fo he measuemen of elaively lage wave-fon eos. Fo demonsaion pupose, a numeical simulaion is pefomed based on he gaphical use ineface (GUI) of he Opimizaion Tool in Malab.

3 Simulaion esuls show ha he wave-fon of an opics sysem can be heoeically econsuced wih a high pecision, which shows ha such an ieaive algoihm may be an effecive way fo he wave-fon sensing fo a high-conas imaging sysem. The ouline of he pape is as following. In Secion, he pinciple of he wave-fon sensing algoihm is poposed. In Secion 3, we pesen he numeical simulaion of he ieaive opimizaion algoihm. The summay and conclusions ae given in Secions 4.. PRICIPLE OF THE WAVE-FROT SESIG ALGORITHM Recen laboaoy ess have demonsaed ha he acual coonagaph can povide a conas in he ode of 0-5 ~0-7. Fuhe impovemen is limied by he speckle noise ha is induced fom he wave-fon eo of he coonagaphic sysem. In his secion, we conside a geneal opics sysem wih a cicula enance pupil. Fo simpliciy, he sysem will be opeaed a a monochomaic wavelengh. Fo an opics sysem wih wave-fon eos, he elecic field of he elecomagneic wave a he pupil plane of he sysem can be expessed: i (u, E pupil(u, =A(u,e φ, () whee A(u, epesens he pupil funcion of he opics sysem; Φ(u, epesens he oiginal wave-fon o phase eo of he opics sysem ha will induce spo-like speckles suounding he bigh sa image on he focal plane. We fisly pu a CCD camea on he pupil plane of he opics sysem and measue he inensiy of he pupil image. Then he magniude of he elecic field on he pupil o he so called pupil funcion can be achieved diecly fom such inensiy and is given as: A ( u, = I pupil, () whee I pupil epesens he esed inensiy of he pupil image. Then we induce a dynamic phase o he esed pupil funcion and he econsuced elecic field on he pupil can be expessed: i (u, E (u, =A (u,e Ψ, (3) whee ψ is he induced dynamic wave fon. Since he saligh is much bighe han ha of he plane, he plane image ha is much less in inensiy han he sa image is negligible duing he wave-fon sensing pocess wihou causing any significan eo. The elecic field of he saligh on he focal plane is he Fouie ansfom of he abeaed elecic field on he pupil plane of he sysem. Then he econsuced elecic field on he focal plane can be expessed as: E focal_ (x, =F[E (u,], (4) whee F epesens he Fouie ansfom of he associaed funcion. The poin spead funcion (PSF) of he saligh on he focal plane is squae of he complex modulus of he abeaed elecic field. The inensiy of he econsuced PSF image is given as: focal _ ( x, ) I ( x, = E y. (5)

4 Combining Equaion (3), (4) and (5), he inensiy of he econsuced focal plane image becomes: I ( x, iψ(, = F [ A (, ) u u v e ]. (6) The pinciple of his algoihm is oally diffeen fom peviously poposed ones (Bode & Taub 006; Give on e al. 007; Dou e al. 009). In he ieaive algoihm, only one focal plane image ahe han 3 images is used o econsuc he oiginal wave-fon. o appoximaion is inoduced on he oiginal phase eo, making i suiable fo he measuemen of much lage wave-fon eos wih a high pecision. Hee we pu he CCD camea o measue he inensiy of he saligh image on he focal plane of he acual opics sysem and i can be expessed as: I ( x, iφ ( u, = F [ A( u, e ], (7) whee I is he esed inensiy of he saligh image on he focal plane of he acual sysem; Φ is he acual phase eo of he opics sysem ha is o be measued. To measue he oiginal wave-fon eo of he opics sysem, he opimizaion algoihm is o minimize he inensiy diffeence beween he econsuced image and he acual focal plane image aken on he CCD. Hee we subac he inensiy of each pixel on he wo images and acquie he sum of he absolue value of he esidual inensiy. Once he sum of esidual inensiy becomes a minimum, he inensiy of he wo images will be appoximaed o each ohe. As a esul, he value of he induced phase will be neaby he acual phase eo ha is o be measued. The opimizaion algoihm is o minimize he following equaion: min M x= y= I ( x, I ( x,, subjec o λ / Ψ λ /, (8) assuming he CCD wih Mx pixels. Subsiuing Equaion (6) and (7) in Equaion (8), he poblem has become o minimize: min M x= y= F[ A ( u, e iψ( u, ] F[ A( u, e iφ ( u, ]. (9) Hee in his pape, we use he Zenike polynomial o epesen boh he oiginal phase eo and he induced phase: Φ= az n n ; n= n= Ψ= az, (0) n n whee a n and a ae he Zenike coefficiens; and is he ode of he Zenike polynomial of n Z n ; because each ode of he Zenike polynomial is uncoelaed wih each ohe, i can guaanee he exisence of a soluion fo Equaion (9). In he pocedue of opimizaion, we povide an iniial phase (fo insance ψ=0), as he sa of he ieaive algoihm (he fis sep). In each ieaive sep, he induced phase ψ will change owads he diecion ha makes he esidual inensiy become smalle. Wih a easonable iniial phase, he wave-fon can be econsuced vey quickly in seveal seps. Once he wave-fon eo of he opics sysem can be pecisely measued, i may be coeced by inducing an appopiaed DM and he speckle noise suounding he bigh saligh image will be expeced o

5 be effecively eliminaed. A numeical simulaion based on he ieaive algoihm will be pesened in deail in he nex secion. 3. UMERICAL SIMULATIO The feasibiliy of he wave-fon sensing algoihm discussed above can be veified by he following numeical simulaion. Fo demonsaion pupose, we ceae a disoed opics sysem by inoducing a andom phase eo o an ideal opics sysem wih a cicula pupil. Hee he oo mean squae (RMS) of he phase eo is ~ 0.4 ad, which is o be measued. And he abeaed image on he focal plane will be as he efeence image ha is o be compaed wih he econsuced image. In his pape we use a 36-ode Zenike polynomial o epesen he phase and he 36 Zenike coefficiens ae ceaed andomly, fo a geneal pupose. Since each ode of he Zenike polynomial is uncoelaed wih each ohe, i can guaanee he exisence of a soluion fo he ieaive opimizaion algoihm. The ampliude paen of he pupil funcion (image aken on he pupil), he heoeical and abeaed PSF (he saligh image on he focal plane) of he sysem ae shown in Figue. Fig. Lef: The ampliude paen of he pupil funcion; Middle: The heoeical PSF of he saligh image; Righ: The abeaed image wih a phase eo ha is o be measued. In he pocedue of he ieaive opimizaion, a dynamic phase, as he only vaiable of he opimizaion poblem, will be induced o he heoeical pupil funcion and he inensiy of associaed PSF can be calculaed diecly by using he Fouie ansfomaion (hee in his simulaion, we use a -D fas Fouie ansfomaion). In each ieaive sep, such a calculaed inensiy ha changes wih he induced phase will be subaced fom he inensiy of he abeaed image wih oiginal phase eo (he efeenced image). In he nex sep, he phase will change owads he diecion ha makes esidual inensiy become smalle. Once he esidual inensiy becomes a minimum, he ieaive pocedue will sop and he phase in he las sep will be he opimum phase ha bes appoximaes he oiginal one. Such a poblem has become a -D consained nonlinea minimizaion poblem. Hee he algoihm is based on he GUI of he Opimizaion Tool in Malab. The objec funcion of such an opimizaion algoihm is o find a minimum of esidual inensiy beween he efeence image and he econsuced image. Supposing he wave-fon eo of he sysem is wihin a wavelengh and he consain funcion of he algoihm will be epesened as -λ/ ψ(u, λ/, fo he phase ha is induced o he pupil plane. The pocedue of he opimizaion algoihm will ake seveal seps and convege vey quickly povided wih a easonable sa poin. Hee we iniialize he sa poin

6 o be ψ(u,=0 fo a geneal pupose. Figue shows he oiginal phase map o be measued and he econsuced wave-fon map by using he algoihm, especively. I clealy indicaes ha he econsuced wave-fon is vey consisen wih he oiginal one based on he algoihm. A ade-off is needed beween he accuacy of he wave-fon sensing and he velociy of he pocedue. Fo example, duing he ieaive pocedue he convegence will be vey fas fo he fis 50 seps wih a emaining phase eo of RMS in he ode of 0 - ad. Howeve, he convegence velociy will gealy decease when he emaining phase eo is vey small. Tha means we have o wai fo a long ime if an exemely high pecision fo he wave-fon sensing is needed. Hee we sop he opimizaion pocedue afe 86 seps wih an accepable accuacy. The econsuced wave-fon is of a RMS ~ ad, wih a emaining wave-fon eo in he ode of 0-3 ad (RMS). Compaing wih oiginal wave-fon, he elaive eo is 0.004%. Figue 3 shows he emaining wave-fon eo and he esidual inensiy beween he econsuced image and he abeaed saligh image of he acual opics sysem. Fig. The oiginal wave-fon map ha is o be measued (lef) and he econsuced wave-fon map based on he ieaive opimizaion algoihm (igh). Fig. 3 The emaining wave-fon eo (lef) and he esidual inensiy beween wo saligh images (igh) Since he oiginal wave-fon eo is ceaed andomly, he algoihm should be suiable fo a geneal siuaion. To esify ha, we also use ohe wave-fon maps o eplace he one ha has

7 been used above, and achieve he same esul, which gives us confidence ha he pefomance of he ieaive opimizaion algoihm is eliable and pomising fo he wave-fon measuemen fo high-conas imaging sysems. 4 SUMMARY AD COCLUSIOS Based on he ieaive opimizaion algoihm, he wave-fon eo of an opics sysem can be pecisely measued diecly fom one focal plane image, which has been demonsaed in ou numeical simulaion. The ieaive opimizaion algoihm we have poposed hee is oally diffeen fom he muli-image focal plane wave-fon sensing algoihms. Since no appoximaion is pefomed on he oiginal wave-fon eo o on he DM induced phase duing he whole pocedue, such an algoihm can be used fo he measuemen of lage wave-fon eos, which is impossible fo he peviously poposed 3-image focal plane wave-fon sensing algoihm. Alhough in his pape we have only consideed fo a monochomaic wavelengh, fo a geneal coonagaphic sysem, wave-fon a ohe wavelengh can be simply scaled accoding o he acual wavelengh. A pesen a laboaoy es sysem has been se up fo such a wave-fon sensing. We will discuss he lae esul in he fuue publicaion. Acknowledgemens We acknowledge he anonymous efeee fo his/he houghful commens and insighful suggesions ha impoved his pape gealy. This wok was funded by he aional aual Science Foundaion of China (SFC) (Gan o ) and was paially suppoed by he aional Asonomical Obsevaoies Special Fund fo Asonomy. Pa of he wok descibed in his pape was caied ou a Califonia Sae Univesiy ohidge, wih suppo fom he aional Science Foundaion unde gan ATM Refeence Bode, P. J., & Taub, W. A. 006, ApJ, 638,488 Bown, R. A., & Buows, C.J. 990, ICARUS, 87,484 Dou, J. P., Ren, D. Q., Zhu Y. T. e al.009, Science in China Seies G, 5, 84 Dou, J. P., Ren, D. Q., & Zhu, Y. T.00, RAA (Reseach Ason. Asophys.), 0, 89 Give on, A., Beliko, R., Shklan S., e al. 007, Op. Expess, 5, 338 Guyon, O., Pluzhnik, E. A., Kuchne, e al. 006, ApJ, 67, 8 Kasdin,. J., Vandebei, R. J., Liman, M. J.e al. 004, Poc. SPIE, 5487, 3 Malbe, F., Yu, J. W., &Shao, M. 995, PASP, 07, 386 Ren, D. Q., & Seabyn, E. 005, Appl. Op., 44, 7070 Ren, D. Q., & Wang, H. M. 006, ApJ, 640, 530. Ren, D. Q., & Zhu, Y. T. 007, PASP, 9, 063 Ren, D. Q., Dou, J. P., & Zhu, Y. T. 00, PASP,, 590 Seabyn, E., Mawe, D., & Buuss, R. 00, ATURE, 464, 08