相移同轴像面数字全息显微术中的复振幅重构和像差补偿

1076 CHINESE OPTICS LETTERS / Vol. 7, No. 12 / December 10, 2009Complex amplitude reconstruction and phase aberrationscompensation in phase-shifting in-line image digitalholographic microscopyZhijian Ma (GIE3DG), Hongyan Li (D3GXH3), and Xiaoxu Lu (F9EHCA)School of Information and Opto-Electronic Science and Engineering,South China Normal University, Guangzhou 510006, ChinaE-mail: Received June 4, 2009When an image digital holographic microscopy (DHM) layout is employed, the Fresnel integral cannot beused in separating the reconstructed image from its conjugate image and background. However, combiningimage plane DHM with the phase-shifting in-line technique, the complex amplitude of reconstructed imagecan be obtained without using Fresnel integral, moreover the approximate error of reconstruction calcu-lation is easily eliminated and the signal-to-noise ratio of reconstructed image is significantly improved.Since a normal incidence plane wave is used as the reference wave, the difficulty and complexity of phaseaberration and phase unwrapping of DHM are remarkably decreased.OCIS codes: 090.2880, 180.3170, 090.1000, 090.1995.doi: 10.3788/COL20090712.1076.Digital holographic microscopy (DHM)1 is a powerfuloptical metrologyfor simultaneously obtaining amplitudeand phase of a sample, and has been widely used in a vari-ety of technical fields such as profile of three-dimensional(3D) objects2,3, concentration change distribution4,measurement of particles flow field5, structures of bi-ological specimens6, micro-electro-mechanical system(MEMS)7, etc.In conventional DHMs, a microscope objective (MO)and off-axis geometry are usually used in recording, andan image sensor named charge-coupled device (CCD)array is placed between MO and the image plane. Ingeneral cases, the reconstruction of digital hologramis performed through Fresnel integral, followed by thephase aberrations compensation of curvature and tiltfor phase recovering. Some compensation methods ofphase aberrations of DHM have been reported6,810.The basic phase compensation method is to searchfor a digital reference wave for compensation of tiltaberration and a digital phase mask for compensa-tion of quadratic-phase curvature8. An amelioratedmethod is to make the adjustment implement as a semi-automated procedure9. For another, by performing atwo-dimensional (2D) fitting with Zernike polynomials,the reconstructed unwrapped phase curvature and tiltingcan be compensated6. Furthermore, by directly calcu-lating a polynomialphase mask from the hologram, phaseaberrations of DHM can also be compensated10. How-ever, these compensation methods are complicated andtime-consuming for their reconstruction being performedthrough Fresnel integral or complex iterative calculation.In this letter, we present a new phase aberrationcompensation method based on the combination of im-age plane DHM with phase-shifting in-line technique torecord and reconstruct digital hologram directly withoutusing Fresnel integral, and then a digital phase mask isemployed to perform phase aberration compensation.Figure 1 depicts the optical configuration of phase-shifting in-line geometry to record DHM of transparentobjects. The basic architecture is a Mach-Zehnder inter-ferometer (MZI). A magnified image of object imaged byMO is used as an object wave O(xH, yH). A plane wave,which is perpendicular to the CCD surface, is used as ref-erence wave R(xH,yH) in another branch. The phase ofR(xH, yH) can be shifted by a movable mirror driven bya piezoelectric ceramics. By shifting the reference wavephase sequentially by 0, pi/2, pi, and 3pi/2, four corre-sponding phase shifting in-line holograms In(xH, yH) (n= 1, 2, 3, 4) are recorded, and a four-step phase-shiftingdigital hologram is obtained as2,11IPS(xH,yH) = O(xH,yH)R(xH,yH)=(I1 I3) + j(I2 I4)/4. (1)From Eq. (1), we can see that the background | O |2+| R |2 and the virtual image RO* have been eliminatedby using phase-shifting technique.Figure 2 is the configuration for a DHM. In the figure,f is the focus of MO, do and di are the object and imagedistances from MO, respectively, then the image on theFig. 1. Experimental setup for recording phase-shifting in-lineDHM. BS1, BS2:beam splitters; PZT: piezoelectric transduceractuator.1671-7694/2009/121076-03 c 2009 Chinese Optics LettersDecember 10, 2009 / Vol. 7, No. 12 / CHINESE OPTICS LETTERS 1077CCD array plane can be expressed as8O(xH,yH) = |O(xH,yH)|exp jpiD(x2H +y2H) +O(xH,yH), (2)where o(xH, yH) denotes the object phase, and D is aparameter used to compensate the wavefront curvature,which is denoted as81D =1di(1 +dodi ). (3)The wavefront curvature in Eq. (2) can be compensatedby the following digital phase mask8:(xH,yH) = exp jpiD(x2H +y2H). (4)While a digital hologram is reconstructed, a digital ref-erence wave810 RD(xH, yH), which should be replicaof the reference wave R(xH, yH), is multiplied with thedigital hologram8,9. In our case, it can be written asRD(xH,yH) = ARexpj2pi (kxxH +kyyH). (5)In our study, the CCD is exactly located at the im-age plane, thus the image digital hologram of object isrecorded. In this situation, Fresnel integral cannot beused any more. However, the combination of image planedigital holographic microscopy with phase-shifting in-linetechnique induces not only the calculating process to besimplified, but also high quality reconstructed image tobe obtained. The complex amplitude of reconstructedimage can be directly described asUr(xH,yH) = (xH,yH)IPS(xH,yH)RD(xH,yH). (6)Therefore the complex amplitude is directly recon-structed in the image plane.The amplitude image and the phase-contrastimage canbe respectively calculated asI(xH,yH) = Ur(xH,yH)Ur (xH,yH), (7)O(xH,yH) = arctanIm(Ur(xH,yH)Re(Ur(xH,yH). (8)The experimental result is presented in Fig. 3. Thesample is the 25th element in Chinese standard No. 3resolution test pattern, which consists of four groups ofFig. 2. Coordinate system for image formation.Fig. 3. Reconstructed amplitude and phase of a Chinese stan-dard No. 3 resolution test pattern. (a) One of four digitalholograms; (b) amplitude-contrast image reconstructed form(a); (c) phase-contrast image reconstructed from the markedrectangle of (a); (d) 3D presentation of (c).gratings with grating constant d = 20 m. Firstly, thesample is imaged on the CCD array surface by a 4 MO, and interferes with the plane reference wave which isperpendicularly incident to the image plane, then shiftingreference wave phase sequentially by 0, pi/2, pi, and 3pi/2,respectively. Thus four phase shifting in-line hologramsare recorded, and one of them is shown in Fig. 3(a).Figure 3(b) is the reconstructed amplitude image whichis actually obtained by directly calculating the modulussquare of the four-step phase-shifting hologram, as shownin Eq. (1). Figure 3(c) shows the reconstructed phase ofthe marked white rectangle in Fig. 3(a), and Fig. 3(d)is the corresponding 3D presentation of Fig. 3(c), whichis calculated by Eq. (8). It can be clearly seen that thewavefront curvature has been compensated by the digi-tal phase mask in Eq. (4). Moreover, because a normalincidence plane wave is used as the reference wave, thedifficulty and complexity of phase unwrapping and recon-struction are obviously decreased.From above analysis and experimental results, we canconclude some advantages about the combination of theimage plane DHM with the phase-shifting in-line tech-nique for the reconstruction of complex amplitude andthe compensation of phase aberrations. Firstly, the re-constructed imagecan be quickly obtained without calcu-lating Fresnel integral since the sample is exactly imagedon the CCD target plane. Secondly, the separation ofreconstructed image from its conjugate and backgroundcan be performed easily, and the signal-to-noise ratio ofreconstructed image is significantly improved because thephase-shifting technique is employed. Thirdly, since thein-line geometry is used, instead of conventional off-axisgeometry, the tilt phase aberrations are easily eliminatedand the phase unwrapping is simplified. Furthermore, a1078 CHINESE OPTICS LETTERS / Vol. 7, No. 12 / December 10, 2009digital phase mask is useful to perform the compensa-tion of phase aberrations. Therefore the combination ofimage plane DHM with phase-shifting in-line techniqueshould be a good candidate to perform complex ampli-tude reconstruction and phase aberration compensationof DHM.This work was supported by the National Nature Sci-ence Foundation of China under Grant Nos. 60877070and 60747001.References1. T.-C. Poon, (ed.) Digital Holography and Three-Dimensional Display (Springer, New York, 2006).2. T. Zhang and I. Yama