3D image reconstruction of transparent microscopic objects using digital holography

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3D image reconstruction of transparent microscopic objects using digital holography
  3D image reconstruction of transparent microscopicobjects using digital holography Francisco Palacios  a,* , Jorge Ricardo  a , Daniel Palacios  b , Edison Gonc¸alves  c ,Jose L. Valin  d , Rodrigo De Souza  c a Physics Department, Faculty of Natural Science, University of Orient, Patricio Lumumba SN, Santiago de Cuba, Cuba b Nuclear Physics Department, University Simo´ n Bolivar, Caracas, Venezuela c Engineering Mecatronic and Mechanic System Department, Polytechnical School, Sao Paulo University (USP), Brazil  d Mechanics Department, High Polytechnical Institute ‘‘Jose´  Antonio Echeverrı´ a’’, CP 1930, AP 6028, Havana, Cuba Received 20 July 2004; received in revised form 26 November 2004; accepted 26 November 2004 Abstract In this paper, we present the potentialities of the digital holography microscopy for 3D image reconstruction of transparent microscopic objects. A method for object volume reconstruction based on the capture of only one off-axishologram is discussed. We show that this technique can be efficiently used for obtaining quantitative information fromthe intensity and the phase distributions of the reconstructed field at different locations along the propagation direction.The potential of the method has been exploited by applying the 3D image reconstruction procedure to etched nucleartracks induced in the widely used polymer detector CR-39. Nowadays, confocal microscopy and atomic force micros-copy are applied to study nuclear track morphology in CR-39 detectors. The micro-holographic method developed inthis work, and applied to that particular study, constitutes a new alternative procedure that overcomes the previouslydescribed at least in technological simplicity.   2004 Elsevier B.V. All rights reserved. PACS:  07.05.Pj; 42.30.Rx Keywords:  3D phase reconstruction; Phase unwrapping; Digital holography; Holographic contouring 1. Introduction Digital holography has several features thatmake it an interesting alternative to conventionalmicroscopy. These features include an improvedfocal depth, possibility to generate 3D images,and phase contrast images [1–3]. 0030-4018/$ - see front matter    2004 Elsevier B.V. All rights reserved.doi:10.1016/j.optcom.2004.11.095 * Corresponding author. Tel.: +53 22 633011/2263; fax: 53 22632689/2042. E-mail address:  palacios@cnt.uo.edu.cu (F. Palacios).Optics Communications 248 (2005) 41–50www.elsevier.com/locate/optcom  One popular version of the experimental set-upin digital holography is the digital Fresnel hologra-phy, which uses quasi-parallel beans [4] to recordwith a CCD camera the intensity distribution of the hologram. The digital hologram is multipliedby the reference wavefield in the hologram planeand the diffracted field in the image plane is deter-mined by the usual Fresnel–Kirchoff integral [5] tocalculate the intensity and the phase distributionof the reconstructed real image. From the numer-ical standpoint, the Fresnel–Kirchoff integral canbe represented in terms of an inverse discrete Fou-rier transformation.Although Fourier transform holographic meth-od needs only one hologram for obtaining thephase value corresponding to all points of thereconstructed phase field, must contouring methoduses two holograms to obtain the phase interfero-gram and later, after unwrapping procedure, theobject shape. This is the case of the two-wave-length contouring [6], two-source contouringmethod [7], etc. In general, the determination of  phase values by Fourier method, as is necessaryto use the arctan function for extracting the phase[8], are wrapped onto the range   p  to  p  rad, andthe process of phase unwrapping, i.e., restorationof the unknown multiple of 2 p  to all pixels, mustbe carry out before the parameter can be deducedfrom the phase distribution.A wide variety of partial solution to theunwrapping procedure is discussed in the litera-ture [9,10]. However, the temporal phase unwrapping is recognized as the best by its sim-plicity and the fact that it is guarantee to pro-duce a correct unwrapping of phase mapscontaining physical boundaries of the object. Intemporal phase unwrapping [11] the 3D phasefield is generated making holograms capture dur-ing the time and the phase unwrapping considerat  t  direction for phase accumulation. In thiscase the 2D unwrapping is not performed, onlyalong  t -axis the unwrapping works, but manyholograms must be captured. This method whenapplied for shape contouring or dynamic defor-mation is fast compared with any other methodthat uses 2D unwrapping and the results are notexposed to corrupting unwrapping mistake. Inthis case the unwrapping procedure takes intoconsideration the phase difference for obtainingthe phase image.Interesting applications of digital holographyrely on the possibility of carrying out wholereconstruction of the recorded wave front andconsequently, the determination of the phasedistribution at any arbitrary plane located be-tween the object and the recording plane, andalong the object reconstructed image. Grilliet al. [12] use the digital holography possibilityof reconstructing the image field at differentplanes from the object to find the object beamphase form and to show how the experimentalset-up affects this, but they do not found the3D object shape.In this work, we employ a variant of unwrap-ping algorithm that shares the same basic princi-ple with the temporal phase unwrappingtechnique but uses as  t -axis the reconstructeddepth in the image field. This is the principalidea applied in this work for to obtain the 3Dphase image corresponding to a nuclear track in-duced in the CR-39 detector as microscopicobject.The organization of the paper is as follows. InSection 2, the general principle of digital hologra-phy is presented. The experimental set-up isdescribed in Section 3. The 3D image reconstruc-tion algorithm is discussed in Section 4. The resultsand discussion are discussed in Section 5. 2. General principles of digital holography Holography is a method that allows reconstruc-tion of whole optical wave fields. The principle of recording and reconstruction of holograms isshown in Fig. 1. The light coming from the object under test and a reference wave interfere in thehologram plane.In classical holography the reconstruction iscarry out by illumination of the hologram intensitywith the reference wave. A virtual and a real imageof the object are reconstructed.In digital holography this reconstruction pro-cess is performed by multiplication of the storeddigital hologram with the numerical descriptionof the reference wave and by the convolution of  42  F. Palacios et al. / Optics Communications 248 (2005) 41–50  the result with the impulse response function. Thediffracted field in the image plane is given by theRayleigh–Sommerfeld diffraction formula w ð  x 0 ,  y  0 Þ ¼  1i k Z Z   h ð n , g Þ r  ð n , g Þ  f  ð  x 0 ,  y  0 , n , g Þ  cos H  d  x  d  y   ð 1 Þ with  f  ð  x 0 ,  y  0 , n , g Þ ¼  exp ð i k  q Þ q  ð 2 Þ and q  ¼  ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d  0 2 þ ð n    x 0 Þ 2 þ ð g    y  0 Þ 2 q   ,  ð 3 Þ where  d  0 is the reconstruction distance, i.e., the dis-tance backward measured from the hologramplane  n  –  g  to the image plane,  h ( n , g ) is the recordedhologram,  r ( n , g ) represents the reference wavefield,  k   denotes the wave number and  k  is the wave-length of the laser source. Due to the small anglesbetween the hologram normal and the rays fromthe hologram to the image points, the obliquityfactor cos H  can normally be set cos H  = 1.Different approaches for solution of Eq. (1)have been proposed [13]. The Fresnel approxima-tion replaces the factor  q  by the distance  d  0 inthe denominator of Eq. (2) and the square rootin the argument of the exponential function is re-placed by the first terms of a binomial expansion.This approximation is valid since the dimensionof the CCD chip is small in comparison to the dis-tance  d  0 [13]. With these approximations Eq. (1)takes the form: w ð  x 0 ,  y  0 ; d  0 Þ ¼  exp ½ i p d  0 k ð m 2 þ  l 2 Þ Z Z   h ð n , g Þ r  ð n , g Þ  g  ð n , g Þ  exp ½ 2i p ð nm  þ  gl Þ  d n  d g ,  ð 4 Þ where the quadratic phase function  g  ( n , g ) is theimpulse response,  g  ð n , g Þ ¼  exp ð i2 p d  0 = k Þ i k d  0  exp i p d  0 k ð n 2 þ  g 2 Þ    ð 5 Þ and the spatial frequencies are  m  ¼  x 0 d  0 k ,  l  ¼  y  0 d  0 k .The discrete finite form of Eq. (4) is obtainedthrough the pixel size ( D n , D g ) of the CCD array[14], which is different from that ( D x 0 , D  y 0 ) in theimage plane  x 0  –   y 0 and related as follows: D  x 0 ¼  d  0 k  M  D n ,  D  y  0 ¼  d  0 k  N  D g ,where  M   and  N   are the pixels number of the CCDarray in  x 0 and  y 0 directions, respectively.According to Eq. (4), the wavefield  w ( x 0 ,  y 0 , d  0 ) isdetermined essentially by the 2D Fourier trans-form of the quantity  h ( n , g ) r ( n , g )  g  ( n , g ). For rapidnumerical calculations a discrete formulation of Eq. (4) involving a 2D fast Fourier transform algo-rithm (FFT) as showed in: Fig. 1. Recording of the hologram (a) and image reconstruction (b). F. Palacios et al. / Optics Communications 248 (2005) 41–50  43  w ð m , n , d  0 Þ¼ exp ð i2 p d  0 = k Þ i k d  0  exp i p k d  0 ð m 2 D  x 0 2 þ n 2 D  y  0 2 Þ   FFT   I  H ð  j , l Þ r  ð  j , l Þ exp i p k d  0 ð  j 2 D n 2 þ l 2 D g 2 Þ    m , n , ð 6 Þ where  j  ,  l  ,  m ,  n  are integers (  M  /2 6  j  ,  l  6 M  /2),(  N  /2 6 m ,  n 6 N/2) and  I  H  is the digitalhologram.In the formulation based on Eq. (6) the recon-structed image is enlarged or contracted accordingto the depth  d  0 . An alternative approach is usefulfor keeping the size of the reconstructed imageconstant [5]. In this formulation, the wavefield w ( x 0 ,  y 0 , d  0 ) can be calculated by, w ð  x 0 ,  y  0 , d  0 Þ ¼  I   1 f I  ½ h ð n , g Þ r  ð n , g Þ I  ½  g  ð n , g Þg ,  ð 7 Þ where  I  ½  g  ð n , g Þ  is the Fourier transform of the im-pulse response, namely I  ½  g  ð n , g Þ ¼ Z   11 Z   11  g  ð n , g Þ exp ½ i2 p ð mn þ lg Þ d n d g : ð 7a Þ Taking into account the form of the impulse re-sponse in Eq. (5) we have that its Fourier trans-form is given by, I  ½  g  ð n , g Þ ¼  exp ð i2 p d  0 = k Þ exp ½ i p k d  0 ð m 2 þ  l 2 Þ : ð 8 Þ With this method the size of the reconstructed im-age does not change, i.e.,  D x 0 =  D n ,  D  y 0 =  D g  andis necessary one Fourier transform and one inverseFourier transform each to obtain one 2D recon-structed image at a distance  d  0 . Although the com-putational procedure is heavier in this casecompared to the Fresnel approximation approachof Eq. (6), this method allows for easy comparisonof the reconstructed images at different distances  d  0 since the size does not change with modifying thereconstruction distance. Furthermore, in this casewe get an exact solution to the diffraction integralas far as the sampling Nyquist theorem is notviolated. 3. Experimental set-up Fig. 2 shows the experimental set-up used inthis work, corresponding to a digital holographicmicroscope designed for transmission imagingwith transparent sample. The basic architecture isthat of a Mach-Zender interferometer. A linearlypolarized He–Ne laser (15 mW) is used as lightsource. The expanded beam from the laser is di-vided by the beam splitter (BS1) into referenceand object arms.With the combinations of the dual polarizers  P 1 and  P 2  is adjusted the intensities in the referencearm and the object arm of the interferometer andthe same polarization state is also guaranteed forboth arms improving their interference.The specimen  M  is illuminated by a plane waveand a microscope objective ( MO ) that produces awave front called object wave  O  collects the trans-mitted light. A condenser, not shown, is used tointensify the light intensity of the objectillumination.At the exit of the interferometer, the interfer-ence between the object wave  O  and the referencewave  R  creates the hologram intensity  I  h ( n , g ). Theoff-axis geometry is considered, for this reason themirror  E2 , which reflects the reference wave is ori-ented such that the reference wave reaches theCCD camera with a small incidence angle with re-spect to the propagation direction of the objectwave.A digital hologram is recorded by a standardblack and white CCD camera (PULNIX modelTM-9701) and transmitted to a computer bymeans of a Frame Grabber card. The digital holo-gram  I  h (  j  , l  ) is an array of   M   ·  N   = 640  ·  480 8-bit- Fig. 2. Experimental set-up.44  F. Palacios et al. / Optics Communications 248 (2005) 41–50  encoded numbers that results from the 2D sam-pling of   I  h (  j  , l  ) by the CCD camera,  I  H ð  j , l Þ ¼  I  h ð n , g Þ rect  n  L  x ,  g  L  y     X  M  = 2 k  ¼  M  = 2 X  N  = 2 l ¼  N  = 2 d ð n    j D n , g    l D g Þ ,  ð 9 Þ where  j  , l   are integers,  L x  ·  L  y  = 8.91 mm  ·  6.58mm is the area of the sensitive chip, and D n  = 11.6  l m,  D g  = 13.6  l m defines the samplingintervals in the hologram plane.Holographic microscopy has been proposed invarious configurations [15], in this work we areworking with a geometry including a MO thatwas proposed for the first time in 1966 by VanLig-ten and Osterberg [16]. In digital holography asimilar approach was used by Cuche et al. [17].As shown if  Fig. 3, the optical arrangement inthe object arm is an ordinary single-lens imagingsystem.The MO produce a magnified image of theobject, and the hologram plane 0 n  (the CCDplane) is located between the MO and the imageplane (0 x 0 ), at a distance  d  0 from the image. Thissituation can be considered to be equivalent to aholographic configuration without magnificationoptics with an object wave emerging directlyfrom the magnified image and not from the ob- ject itself. For this reason the term  image holog-raphy  is sometime used to designate thisprocedure. Classical microscopy can be achievedwith this arrangement by translation of the ob- ject or the hologram plane such that the imageis focused on the CCD. 4. 3D image reconstruction In this work, nuclear tracks induced in CR-39polymeric detector are used as transparent micro-scopic object. CR-39 detector is one of the mosthomogeneous, amorphous polymer with whichone can achieve high resolution and sensitivity innuclear track recording. The crossroad of nuclearparticles and heavy ions through most of the solidsemiconductors create fine patterns of intensedamage at atomic scale. This is the srcin of nucle-ar tracks induced in solids. These tracks have to beetched chemically to make them visible with anoptical microscopy. Etching procedure consists inchemical dissolution of material along the traceof nuclear particle and a dissolution or general at-tack of detector surface. In general, etched trackshave conical forms whose dimensions depend onthe energy of particles and conditions of chemicaletching.The variation of refractive index between thenot revealed part of CR-39 detector ( n c  = 1.5)and the etched nuclear track ( n air  = 1) leads to achange of the optical path length and thereby achange of phase between the light waves passingthrough track contour and reference wave. Withdigital holography it is possible to reconstructthe wave field of the object for different reconstruc-tion distances, so that different optic sections orreconstructed 2D images can be established. 4.1. 2D image reconstruction The 2D image reconstruction is digitally andnumerically carried out from the observed holo-gram pattern, expressed by  I  h (  j  , l  ), using the aboveFresnel diffraction equation (7). The result of thecalculations is an array of complex numbers calledwave front  w ( x 0 ,  y 0 , d  0 ), which represent the com-plex amplitude of optical field in the observationplane 0 x 0  y 0 . The distance between the hologramplane 0 n , g  and the observation plane is definedby the reconstruction distance  d  0 . When the holo-gram is recorded without a MO,  d  0 must be equalto the distance (or more precisely, to the opticalpath length) between the object and the CCD toobtain in-focus reconstructed images. In holo-graphic microscopy, image focussing occurs when Fig. 3. Configuration for holographic microscopy. F. Palacios et al. / Optics Communications 248 (2005) 41–50  45
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