University of California, Santa Cruz, CA 95064, USA; b The Institute of Optics, University of Rochester, Rochester, New York 14627, USA ABSTRACT

Transcription

1 High-speed chael demixig by scaig iterferometric focusig with biary trasmissio matrix Xiaodog Tao* a, Dare Bodigto b, Marc Reiig a ad Joel Kubby a a W.M. Keck Ceter for Adaptive Optical Microscopy, Jack Baski School of Egieerig, Uiversity of Califoria, Sata Cruz, CA 95064, USA; b The Istitute of Optics, Uiversity of Rochester, Rochester, New York 467, USA ABSTRACT I this paper, we demostrate a fast biary itesity modulatio based o the measuremet of the biary TM. For each correctio, the biary TM was calculated based o measuremets of the itesity chage at the target with a series of iput masks. After preloadig the measuremet masks, the DMD ca ru at full speed durig measuremet. The system allows dyamic focusig at.5 Hz with 04 iput modes, ad more tha 60 times itesity ehacemet. We demostrate focusig light through a highly dyamic scatterig sample, a live drosophila embryo. Keywords: Times Roma, image area, acroyms, refereces. INTRODUCTION As light propagates through biological tissues, it ca be refracted, scattered ad absorbed, limitig the imagig resolutio ad depth. To correct the refractive aberratio, adaptive optics (AO) has bee extesively ivestigated for applicatios i optical microscopy []. As the imagig depth icreases, AO becomes less effective for focusig light i the sample. Multiple scatterig becomes a domiat factor limitig the image depth. However the amplitude of elastic scatterig loss is oted to be a order of magitude or more tha that of absorptio. Overcomig the elastic scatterig ca dramatically exted the light peetratio depth. Thaks to the determiistic process of scatterig, there exists a liear relatioship betwee the iput modes ad the output modes of the wavefrot, which ca be described by a trasmissio matrix (TM) [, 3]. To geerate the desired optical field through scatterig media, TM ca be measured by usig a LC-SLM ad a full-field iterferometric measuremet. The the light trasmissio ca be completely cotrolled ad the scatterig media ca act as a les to trasfer the image [4, 5]. To focus the beam through scatterig tissue at a sigle or multiple output chaels, iterative optimizatio methods have bee demostrated [6-8]. To obtai optimized focusig, thousads of degrees of freedom of the icidet wavefrot eed to be modulated ad measured. Recetly aother high speed light modulator, the digital micromirror device (DMD), has bee used to compesate scatterig i biological tissue [9-3]. It cotais millios of fast switchable micromirrors to modulate the itesity of the light based o its two states. A commercially available DMD has frame rates up to khz ad could have more tha oe millio of pixels. The first demostratio of biary itesity modulatio usig a DMD was made by Mosk s group, where a sequetial iterative algorithm was used to focus light through turbid media [9]. The lower efficiecy of itesity modulatio compared with phase modulatio ca be compesated by the large umber of available chaels o the DMD. Phase modulatio o a DMD usig a hologram was also proposed at the expese of phase resolutio loss [0-]. The resolutio ca be further improved by a superpixel-based spatial amplitude ad phase modulatio []. A geetic algorithm has also bee used i biary itesity modulatio to improve the performace of the optimizatio process [3]. I this paper, we demostrate a fast biary itesity modulatio based o the measuremet of the biary TM. For each correctio, the biary TM was calculated based o measuremets of the itesity chage at the target with a series of iput masks. After preloadig the measuremet masks, the DMD ca ru at full speed durig measuremet. Compared with the optimizatio method, o feedback iformatio is eeded durig the measuremet. The proposed method oly requires oe measuremet for each iput mode. The total time for a sigle correctio is oly 75ms for 04 iput modes. The direct itesity modulatio used i this paper has much higher light efficiecy (60%) [0,]. This is more suitable for applicatios that are sesitive to optical power loss. We demostrate focusig light through a highly dyamic scatterig sample, a live drosophila embryo.

2 . MATERIALS AND METHODS. Measuremet of the biary TM Light trasport through scatterig medium ca be characterized by a TM, which coects electric fields from the icomig ad outgoig chaels []. To focus the light through scatterig medium at oe output chael, the electric field at the output chael, E out, ca be cosidered as the sum of the cotributios from all the iput chaels, which is give by, = = E out KE kei () E is the iput electric field vector, where e i is the electric field at the th iput chael. K is the TM for a sigle output chael, where k coects the th iput chael ad the output chael. Whe usig a biary amplitude modulator, the goal is to ope the iput chaels which ca geerate costructive iterferece at the target ad block the other chaels. The oly iformatio required is the biary state of each elemet i the trasmissio matrix, i.e. whether their relative phases are i the rage of π. Figure. Priciple of biary itesity modulatio. The combiatio of the referece ad Hadamard basis is displayed o the DMD. The amplitude at the output,, ca be used to calculate the biary TM. Higher referece itesity gives a more accurate E RH measuremet. By scaig the speckles aroud the target, the maximum referece itesity is achieved at the target before measuremet of the biary TM. Whe a plae wave e 0 reflects from a biary amplitude modulator, such as the DMD, the electric field after the modulator E B, ca be writte as where A [ a K ] T a N E B = Ae 0 () = is the biary modulatio vector. The electric field at the output ca be writte as Eout = KEB = KAe 0 (3) We first defie a referece electric field at the output chael, whe all chaels are tured o, E Re f = KARef e 0, where A [ ] T Ref K = (4) The to achieve costructive iterferece at the output, we eed to block the chaels which have destructive iterferece with E Ref. That is to ope the chaels with output phase i the rage of (φ Ref π/, φ Ref +π/) ad block the

3 other chaels as show i Fig. (b), where φ Ref is the phase of the referece. The resultig electric field at the output the becomes E out = KAk e 0 (5) Sice the iput electric field e 0 is a plae wave, the biary modulatio vector A K is also a biary TM which cotais biary iformatio of the TM based o its relative phase to the referece electric field. For simplicity, we alig the referece phasor E Ref alog the real axis. Therefore the elemets of A K, ca be calculated as Re( ke0) 0 a = (6) 0 Re( ke0) < 0 where Re() is the real part of a complex vector. To obtai Re(k e 0 ), we chose the Hadamard basis as the iput basis because of its orthogoal property. The output electric field for differet Hadamard modes is give by [ EH EHN ] K[ H L H N ] e 0 L = (7) where E H is the output electric field for the th Hadamard mode. H is the th Hadamard mode defied as a Nx vector, where N is the umber of chaels. Therefore Re(Ke 0 ) ca be calculated by T Re( Ke0) = Re( EH) Re( EHN ) H H N N L L (8) where [] T is the traspose of the matrix. Although it is hard to directly measure Re(E H ), it ca be estimated by the itesity of the sum of E H ad E Ref at the output chael. Because values of the elemets i the Hadamard matrix H are either - or, the additio of A ref ad a Hadamard mode ca be obtaied after the DMD, where the elemets of ( ARef H ) ( ) 0 EBI = ARef + H e (9) + are either 0 or, which ca be perfectly modulated o the DMD. The output electric field is the summatio of the referece electric field E Ref ad the electric field for a Hadamard mode E H at the output chael as show i Fig.. ERH = EH + ERef (0) The the relatioship betwee Re(E H ) ad ERH ca be calculated accordigly as Re( EH ) β ERH ERef () If ERef is sufficietly larger tha E H. From Eq. (8) ad Eq. (), a vector J based o the itesity measuremet is obtaied, which ca be used to determie the fial biary TM, A k. NE Ref T J = ERH E L RHN H L HN γ Re( Ke0) () where γ = is a costat ad N is the umber of the chaels. Sice half of the chaels are tured o, the β threshold T ca be selected so that the sum of matrix elemets A k is equal to N/. The fial result ca be calculated as a j T = 0 j < T. (3)

4 This calculatio is based o the assumptio that the itesity of E Ref is sufficietly large. However the relatively low itesity of the referece field i the real situatio itroduces errors to Eqs. (). To overcome this issue, a referece optimizatio process is performed by usig a scaer i the system to steer the beam alog oe axis o the sample whe all the micromirrors o the DMD are o as show i Fig.. To evaluate the effect of the referece itesity o the performace of the biary itesity modulatio, a Mote Carlo simulatio was carried out by assumig a circular complex Gaussia distributio for the output electric field of each Hadamard mode. With 0 samples o the scaig lie, the error of the fial patter would be aroud.6%. Icreasig the samples durig scaig ca further decrease the error. With 50 samples, the error ca decrease to 0%. By applyig this method, we ca dramatically improve focusig through a scatterig sample. Here the patter shift could be caused by the memory effect. If the speckle patters are totally ucorrelated after mirror scaig, the itesity of the speckles still obeys the egative expoetial statistics. The above aalysis is still valid. I applicatios for laser scaig imagig systems, the existig galvaometric mirrors i the system ca be utilized for referece optimizatio, which ca reduce the cost ad complexity of the system.. System setup The experimetal setup is show i Fig.. A DMD (DLi430, 0.7 XGA, Digital Light Iovatios) with 04x768 mirrors ad.77 khz frames per secod was used as a biary itesity modulator. A HeNe laser at 633m (5-LHP- 99, CVI Melles Griot) was employed as a light source i the experimet. I order to make the itesity more uiform across the aperture of the DMD, a telescope composed of leses L (0x/0.40, Newport) ad L (f=50mm, AC54-50-A, Thorlabs) expads the beam by 6.7 times. The output beam from the telescope with a /e diameter of 0.8 mm covers the exit pupil of the DMD. A iris diaphragm I was mouted after the telescope for avoidig artifacts from the edge of the widow aperture coatig o the DMD. The icidet agle of the beam o the DMD was adjusted carefully to achieve the blaze coditio whe the reflected beam from the micro mirror lies up with the sixth order of the diffractio gratig from the DMD. The majority of the eergy is directed ito the blazed order. The other orders were blocked by the iris diaphragm I. The efficiecy of the DMD, defied as the ratio of the reflected light to the icidet light o the DMD, is aroud 60% whe all mirrors are i the full o-state. The DMD is cojugated with the galvaometer by leses L3 ( f=50mm, AC54-50-A, Thorlabs) ad L4 ( f=50mm, AC54-50-A, Thorlabs), which have a clear aperture of 4mm. Leses L5 ( f=50mm, AC54-50-A, Thorlabs) ad L6 ( f=50mm, AC54-50-A, Thorlabs) further cojugate the aperture of the galvaometer to the 0.8mm diameter rear pupil of a 0x, NA 0.3 objective O ( PL FLUOTAR 0/0.3, Leitz). The sample, mouted o a three axis aopositioig stage (NaoMax30, MELLES GRIOT), was placed i frot of the objective. The light that is scattered whe goig through the sample was collected by aother 0x, NA 0.5 objective O (PL 0/0.5, Leitz), focused by a imagig les L7 (f=00, AC54-00-A, Thorlabs) ad the split ito two orthogoal paths with a 0/90 beam splitter SB (BNP6K0550/90, RMI). 90 percet of light was fed ito a PMT (H74-0, Hamamatsu) for fast itesity measuremet. A 0μm diameter pihole (PH) was placed i frot of the PMT to collect light from a.63 μm diameter area o the object plae. Aother 0 percet of the light was captured by a CCD camera (M400, Dalsa) for moitorig the chage of the speckle patter with a 5ms exposure time. To adjust the power of the laser, a polarizer was mouted o the output port of the laser, which is ot show i the figure. HeNe Laser PMT L DMD I L O BS PH L7 L3 I L4 L5 L6 O Sample CCD Camera Scaer

5 Figure. Experimetal setup for iterferometric focusig by biary measuremet of the trasmissio matrix. The laser output from a HeNe laser (wavefrot legth λ=633) is expaded by the leses L ad L ad limited by a iris (I). The beam covers the whole aperture of the DMD ad is relayed by leses L3 ad L4 to a scaer. The uwated high order beam is blocked by aother iris (I). The beam is further relayed by leses L5 ad L6 ad focused o the sample by a objective les (O). The diffuse light after the sample is collected by aother objective les (O) ad focused o the PMT ad a CCD camera by les L7. The beam is divided by a 0/90 beam splitter (SB). A pihole (PH) is istalled i frot of the PMT to collect the light oly from the target. The itesity modulatio process icludes the four followig steps. The first step is the referece itesity optimizatio. A galvaometer steers the beam with the target poit at the ceter of the travel whe all mirrors are i the full o-state. The secod step is the measuremet of the biary TM. A sequece of preloaded Hadamard basis images is displayed o the DMD at the full speed of.7 khz. I the third step, the biary TM is calculated ad the data o the fial mask is trasferred to the DMD driver board. I the fial step, the ew mask is updated o the DLP ad the exposure of the camera is triggered. To automate the above process, a customized program ruig o a persoal computer (Dell Precisio T360) was developed i C++. The OpeCV library, optimized for Itel multi-core processors, was utilized to miimize the calculatio time. Whe the 04 Hadamard modes are applied durig measuremet ad the exposure time of the camera is set as 5ms, the system ca operate at 80 ms/frame as show i Fig. 3(a). Fig. 3(b) shows the time graph for oe correctio. The first ms is spet o the referece itesity optimizatio. It is followed by a iterval of 45 ms for the measuremet of the biary TM. The the calculatio ad data trasfer takes aother 8 ms. The total time for oe correctio before the camera exposure is 75ms. I Sectio 3, 04 Hadamard modes are applied. (a) (b) Sigal(V) Sigal (V) 0.9 Referece 0.8 optimizatio (ms) Biary TM measuremet (45ms) Computatio ad data trasfer (8ms) Camera exposure Time (s) Time (ms) Figure. 3. Timig graph of the sigal from PMT durig system operatio (a) ad a elarged time graph for oe correctio (b). 3. EXPERIMENTAL RESULTS To focus light through the embryo, two experimets were performed i series. First, dyamic modulatio was applied. The mask is refreshed every 80ms. The images were recorded with a exposure time of 5ms for about 5 secods. Next, we kept the same cofiguratio but the images were captured with the same mask obtaied from the first modulatio. The ehacemet with both dyamic modulatio (blue) ad sigle correctio (red) durig the first 5 secods is show i Fig. 4(a). The elarged plot durig the first 4 secods is show i Fig. 4(b). Figures 4(c) ad 4(d) show the images at 0.8s,.8s, 3.5s ad 5.04s for dyamic modulatio ad sigle modulatio, respectively. The itesity mask is also show at the lower left corer of each image. After turig o the biary itesity modulatio, a sharp focus spot was achieved at the back side of the embryo after the first correctio, as show i the first image from the left i Fig. 4(c). With a sigle modulatio, the static mask caot compesate the dyamic chage of the ier structure of the embryo. At the decorrelatio time of 3.5s, the ehacemet decays to almost oe half as show i Figs. 4(b) ad 4(d). However with dyamic modulatio, a stable focus is achieved with the ehacemet aroud 50, as show i Fig. 4(c). Durig the operatio time of 5 secods, the system ca keep the ehacemet at the mea of 50.3 ad stadard deviatio of Although the sample has a decorrelatio of 3.5 secods, the fast modulatio ability ca still beefit the itesity ehacemet as show i the images at.8s i Figs. 4(c) ad 4(d).

6 (a) 70 (b) Ehacemet Ehacemet Dyamic focusig Sigle focusig (c) Time 0.8 s.8 s Time 3.5 s 5.04 s (d) 0.8 s.8 s 3.5 s 5.04 s Figure. 4. Focusig light though a live drosophila embryo. (a) The ehacemet for sigle ad dyamic modulatio durig the first 5 secods is show as the red ad blue curves respectively. The elarged view durig the first 4 secods is show i (b). The images from the CCD camera at 0.8,.8, 3.5 ad 5.04 secods is show i (c) ad (d) with dyamic ad sigle modulatios, respectively. The lower left corer of the images shows the correspodig mask o the DMD. Scale bars, 5µm. (Media 6) 4. Coclusio We report a high-speed iterferometric focusig method to compesate dyamic scatterig i live biological tissue based o the fast measuremet capability of the biary TM. By usig the Hadamard basis, the iterferece betwee the optical fields from the referece ad each basis elemet is achieved by displayig the summatio of a matrix with all oes ad the Hadamard basis elemets o the DMD. To overcome the estimatio error caused by the low itesity of the referece field at the target, a referece optimizatio method is demostrated which ca give much more stable focusig durig two dimesioal scaig. By usig a fast DMD as the biary itesity modulator, the proposed method ca achieve a 75ms measuremet time ad a 80ms system refresh time. Although the proposed system is slower tha the phase modulatio method usig a off-axis digital holograph made with a DMD [0, ], the higher diffractio efficiecy of the proposed method would give a higher SNR durig iterferometric measuremet for multi-photo florescece imagig. The experimets with a live drosophila embryo show its advatage for maipulatio of light i live biological tissue. The ability to use the itesity from the target for calculatio of the biary TM makes it suitable for fluorescece imagig ad targetig. As a fast, simple, power-efficiet ad low-cost solutio to deliver light through biological tissue, it has potetial for a wide rage of applicatios from basic biological research to cliical ivestigatios.

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