Microscopy with self-reconstructing beams

Transcription

1 Mcroscopy wth self-reconstructng beams Floran O. Fahrbach 1,, Phlpp Smon und Alexander Rohrbach 1, 1 Centre for Bologcal Sgnallng Studes (boss), Unversty of Freburg, Germany Laboratory for Bo- and ano-photoncs, Department of Mcrosystems Engneerng-IMTEK, Unversty of Freburg, Germany Correspondence should be addressed to fahrbach@mtek.de, rohrbach@mtek.de I. Image acquston and processng For the mage acquston of the human skn n fgure 5 the energy of both beams was chosen such that the sgnal-to-nose rato (SR) n the mage was optmzed. Snce the SR s proportonal to the square-root of the number of collected photons, the 1-bt dynamc range of the CCD-camera (Zess AxoCam) was fully exploted. As t can be seen n the fluorescence ntensty lnescans F( z ) of fgure 5, the maxmum ntensty (approxmately 496 counts correspondng to 1 bts) s measured at the transton between the Stratum Corneum to the Epderms, from where t falls off nearly exponentally to the rght part of the mage. Snce the SLM les n the conjugate plane of the object plane I ll (x,y, z = ) and the cross-secton of the Bessel beam s larger than that of the Gaussan beam, the dffractng SLM-area s also larger for the Bessel beam. To fnd the optmal llumnaton power, the laser power ncdent on the SLM was ncreased for both beams untl the maxmum ntensty value on the camera had been reached. For ths reason the energy n the Gaussan beam had to be smaller than that of the Bessel beam. However, an ncrease of the beam powers would not affect the exponental decay length d of the ntensty F( z ). For mage post-processng we used standard procedures,.e. the mages do not result from an average projecton of all beams, but from a maxmum selecton of each beam at lateral x-poston to enhance mage contrast. II. Standard devaton of scattered llumnaton lght In a good mcroscopy mage wth good contrast from the objects to be dentfed, the ntensty standard devaton s ˆ should be large, whereas the ntensty varatons s ˆ of all artfacts should be small. However, usually these ntensty dstrbutons cannot be separated. The am of ths secton s to show that n our case t s possble to extract the scattered ntensty from mages wth many scatterers and to thereby analyze the strength of scatterng by ntensty ATURE PHOTOICS 1

2 standard devatons s ˆ. Ths case s smlar to the case of the large spheres n the man text, where sˆ( z ) changes wth propagaton dstance z. Theoretcally and n computer smulatons, one can decompose the ntensty of an acqured mage, whch we call the mage ntensty I ( xz,, ) nto the mage ntensty wthout scatterng artfacts and the mage ntensty, whch s a result of scattered llumnaton lght: I ( xz, ) I ( xz, ) I ( xz, ). (1) I (x,z) Lghtsheet Gaussan beam Bessel beam I (x,z) I (x,z) x (μm) 3 Lghtsheet z=55.5μm x (μm) 3 Gaussan beam z=55.5μm x (μm) 3 Bessel beam z=55.5μm normalzed ntensty normalzed ntensty normalzed ntensty Fgure S1: Comparson between mages (1 st row), mages ( nd row) and mages (3 rd row) whch could be separated due to a pror nformaton about the sphercal objects. The standard varatons sˆ sˆ are plotted as whte nsets n the mages. The lne scans n the 4th row show the ntensty profles along the vertcal dashed lnes n the mages. The contrast of the mages can be compared to that of the mages. ATURE PHOTOICS

3 In our expermental confguraton, where the llumnaton lght propagates n the mage plane, t s possble to dentfy the objects,.e. the unstaned black mages of the spheres. These were removed from the mages, whch are shown n the frst row of Fgure S1. The dark grey values of the spheres were replaced by the average ntensty value of the whole mage, resultng to a good approxmaton n the mages, whch are shown n the second row of Fgure S1. The dfference between the and the mages are the mages and are shown n the thrd row of Fgure S1. In accordance to eq. () of the man text, the normalzed lateral standard devaton sˆ of the mage ntensty s gven by sz ( ) 1 1 sz ˆ( ) I I ( x, ) z. () I I 1 The counter =... denotes the lateral pxel number, I ( z ) s the mean value of all ntenstes I ( x, z ). Equvalently to eq. (), one can evaluate the normalzed varance sˆ for smplcty. By nsertng eq. (1) one obtans: sz ˆ( ) ( ) (, ) ( ) (, ) I zi x z I zi x z. (3) ( ) ( ) 1 1 I z I z Sortng the product terms yelds sz ˆ( ) sˆ sˆ 4 I ( zi ) I ( x, zi ) ( x, z) (4) Where t s vald to approxmate 1 1 I( x, z) I( x, z) I and We now w ant to show that t s not only possble to separate the ntenstes, by also the varances sˆ( z ). If we demand for the followng good approxmaton: then ths requres the last terms of eq. (4) to vansh: sz ˆ( ) sˆ sˆ (5) I I I ( x, z) I ( x, z) (6) 1 1 ATURE PHOTOICS 3

4 The frst term can be approxmated as I ( zi ) I, (7) snce I 1 wthn a relatve error of 1% for the performed measurements. For the second term of eq. (6) one can assume, that all summands I ( x, z) 1 result n a I ( x, z) snce there s no mage ntensty at the locatons of the objects. The mage ntensty s wholly attrbuted to the object. We fnd I ( x, z) I ( x, z) I ( x, z) I, (8) whch confrms the requrement of eq. (6). Snce eq. (5) s vald, the standard devaton s gven by: sz ˆ( ) sˆ sˆ (9) For sˆ sˆ ths results n sˆ sˆ sˆ / sˆ sˆ, (1) whch s proven by the varances sˆ( z ) plotted n Fgure S for all three llumnaton beam types. 4 ATURE PHOTOICS

5 1x1-3 8 s (z) 6 4 Lght-sheet s (z) Term 3 s (z) Term 4 4 z (μm) Term 3, Term 4-1x1-3 3.x1-3 s (z). 1.. Gaussan beam s (z) Term 3 s (z) Term 4 4 z (μm) Term 3, Term 4-1x1-3 1.x1-3 s (z).8.4. Bessel beam s (z) Term 3 s (z) Term 4 4 z (μm) Term 3, Term 4-1x1-3 8 Fgure S: Intensty varances along propagaton dstances z for three dfferent llumnaton beams: Statc Lght-Sheet (top), scanned Gaussan beam (mddle) and scanned Bessel beam (bottom). The varance sˆ ( z ) of the mage s much smaller than the varance sˆ sˆ of the mage. The z-dependent 3rd and 4th term of eq. (4) are dentcal and therefore cancel each other out (see red sold and blue dotted lnes). ATURE PHOTOICS 5

6 III. Dstance dependent phase nose and Strehl rato The electrc feld E(x,y, z ) n a plane at a poston z n the sample can be expressed by the D- Fourer transform of ts angular spectrum E ( k,k,z x y ) 1 ( kxxkyy) ( x, yz, ) ( k,k,z x y ) e dkdk x y E E (11) 4 kx kyk E ( k,k) corresponds to an annular pupl functon for the Bessel beam and a crcular pupl x y functon wth Gaussan ampltude for the Gaussan beam. Both beams are truncated by a crcular pupl wth radus A IO k. For smplcty, we approxmate for both beams constant feld ampltudes E ( k,k,z x y ) E across ther aperture and also after propagatng over the dstance z. In other words free space propagaton nvarance s assumed over z s wthn the depth of feld due to the small focusng A. As mentoned n the man text, we further assume that absorpton s small, sca >> abs, such that all changes n the beam ntensty I(z) are determned by phase changes from scatterng. Upon ( k x,k y,z ) propagaton by z the feld spectrum reads E ( k,k,z) E e and the ntegral becomes x y 1 ( k x,k y,x,y,z) ( kxxkyy) ( x, y, z) 4 e e dkxdk y Pupl E E (1) In eq. (1) we have neglected changes of the electrc felds polarzaton. At every pont (x, y, z ) of the phase (k x, k y, r) the k-vector s orented n a dfferent drecton (k x, k y ) due to scatterng,.e. due to local phase shfts from cells or beads. We further assume, that these local phase shfts are small such that a second order Taylor expanson e ½² of the phase s vald. The beam ntensty I (x=, y=, z ) along the axs then reads [1] : 1 1 (,, ) E 4 1 ( x, y, ) ( x, y, ) x y I z k k z k k z dk dk ( ) ( ) 1 ( ) k A 4 E (13) I z z I z The term (z )² s the varance of the phase due to multple scatterng. It s a functon of propagaton dstance z and reduces the central pont beam ntensty I (,, z ) along the optcal axs. It s convenent to defne a Strehl rato SR as a measure for phase aberratons. In ths study we make the Strehl rato to be a functon of propagaton dstance z, agan assumng that natural beam spreadng s neglgble small: SR( z ) I (,, z )/ I 1 ( z ) (14) 6 ATURE PHOTOICS

7 where s the rms phase devaton ntegrated over all refractve ndex varatons n(r). After propagaton by z, a part of the wave front wth cross secton A s retarded and dstorted many tmes resultng n the phase on the on the optcal axs (see FgureS3) z k n( x, y, z) dadz (15) A The Strehl rato of eq. (14) s a nd order Taylor expanson of a related defnton of the Strehl rato SR, whch can be determned numercally. Whereas eq. (14) s vald for phases devatons <.5, SR s applcable over a broader range of rms phase aberratons []:.5 SR' exp 1 SR (16) The expected and expermentally observed decay of the fluorescence ntensty F(z > z E ) ~ I E exp z/ d of lght propagatng through skn epderms wth decay length d s therefore: I (,, z) I exp I exp z/ d E (17) From ths t s easy to solve for the rms devaton of the phase ln S ln F / F z/ d lni E (18) whch ncreases wth propagaton dstance z. A Strehl rato SR =.5 results from a rms phase devaton =.83 correspondng to.13. For a Bessel beam wth penetraton depth d = 77μm we fnd =.83 at z = -dln(.5) = 53μm. For Gaussan beam wth A Gauss =.9 the wavefront s nearly plane n the feld of vew of the camera, for the concal phase of the Bessel beam each plane wave component propagates under the angle sn = A Bessel /n.15. Wth ncreasng dstance L = z / cos the phase (L(z)) ncreases n t rms devaton (see FgureS3). Ballstc And Dffusve Photons The al feld propagatng through the sample can be expressed by E () r E () r E () r A () r e A () r e (19) ( r ) ( r) holo sca holo sca The al llumnaton beam ntensty s the modulus square of the al feld, ATURE PHOTOICS 7

8 h () r E () r E () r E () r h () r h () r ll holo sca holo sca ( r) ( r) holo () sca () holo() sca () Ree e E r E r E r E r () The fracton of ballstc photons h holo = E holo ² decreases exponentally wth z, whle the fracton of dffusve photons h sca ncreases wth z: holo Eholo holo E h h ( z ) exp( z/ d) h z z z z e e sca ( ) Esca ( ) Esca ( ) E holo( ) Re (1) h ( z ) 1exp( z/ d) holo E It should be remarked once agan, that both photon fractons h holo and h sca are functons of E holo. In other words, the rato between ballstc and dffusve photons can be adjusted wth the spatal lght modulator (SLM), whch generates the hologram phase of E holo! x (x,z 1 ) (x,z ) (x,z 3 ) k L(z 3 ) L(z ) k +k s z k +k s k S~ F n(x,z) = n + n(x,z) 1 ( z ) 1 1 ( z ) 1 ( z ) z 1 z z 3 3 z Fgure S3: Two plane waves as part from a Bessel beam wth focusng angle sn = A IO / n. Upon propagaton through the nhomogeneous medum, the phase (x,z) becomes ncreasngly aberrated wth propagaton dstance z or L = z/cos, respectvely. Each ncdent wave vector k spreads nto a spectrum of k-vectors k + k s due to scatterng,.e. phase dstorton. Ths results n an exponental decay of the Strehl rato and of fluorescence 8 ATURE PHOTOICS

9 REFERECES 1. Snger, W., M. Totzeck, and H. Gross, Physcal Image formaton. Handbook of optcal systems, ed. H.Gross. Vol.. 5, Wenhem: Wley-VCH.. Mahajan, V.., Strehl Rato for Prmary Aberratons n Terms of Ther Aberraton Varance. Journal of the Optcal Socety of Amerca, (6): p ATURE PHOTOICS 9