**1. Introduction**

Holography is a method for storing and reconstructing both amplitude and phase information of a wave front. In digital holography the reconstruction process is accomplished by means of a computer (Yaroslavsky & Merzyalov, 1980) obtaining directly the phase distribution of the object wave front. Particularly with the improvement of the spatial resolution of CCD cameras and the increasing computational performance of personal computers digital holography has been widely applied in many fields such as deformation analysis (Schedin et al., 2001), object contouring (Wagner et al., 2000), microscopy (Takaki & Ohzu, 1999) and particle measurement (Murata & Yasuda, 2000). The technique of digital holography has been implemented in a configuration of an optical microscope (Schilling et al., 1997); the objective lens produces a magnified image of the object and the interference between this image and the reference beam is achieved by the integration of the microscope into one of the arms of a Mach-Zender interferometer. This configuration is called Digital Holographic Microscopy (DHM).

DHM is a powerful technique for real-time quantitative phase contrast imaging, since a single intensity image, called a hologram, allows the reconstruction of the phase shift induced by a specimen. This property of holograms offers phase-contrast techniques, which can then be used for quantitative 3D imaging (Palacios et al., 2005). Quantitative phase imaging is important because it allows the determination of the optical thickness profile of a transparent object with sub-wavelength accuracy (Yu et al., 2009). Through numerical processing of the hologram one can filter out parasitic interferences and the components of the image reconstruction: zero-order and twin image terms (Cuche et al., 2000) or to compensate for curvature introduced by the microscope objective (MO) (Pedrini et al., 2001),

*1University of Oriente, Cuba* 

<sup>\*</sup> Oneida Font1, Jorge Ricardo1, Guillermo Palacios1, Mikiya Muramatsu2, Diogo Soga2, Daniel Palacios3, José Valin4 and Freddy Monroy5

*<sup>2</sup>University of Sao Paulo, Brasil* 

*<sup>3</sup>University of Simon Bolivar, Venezuela* 

*<sup>4</sup>Polytechnic Institute "José A. Echeverría", Cuba 5National University of Colombia, Colombia* 

Alternative Reconstruction Method and Object Analysis in Digital Holographic Microscopy 185

Figure 1 shows the experimental set-up used in this work. It is a Digital Holographic Microscope designed for transmission imaging with transparent sample. The basic architecture is that of a Mach-Zehnder interferometer. A linearly polarized He-Ne laser (15 mW) is used as light source. The expanded beam from the laser is divided by the beam splitter BS1 into reference and object beams. The microscope produces a magnified image of the object and the hologram plane is located between the microscope objective MO and the

digital holographic microscopy we can consider the object wave emerging from the

Fig. 1. Experimental set-up: BE, bean expander; BS, bean splitter (the splitting ratio of BS1 and BS2 are 90/10 and 50/50 respectively); M, mirror; MO, microscope objective; S, sample;

With the combinations of the dual polarizer *P1* and *P2* the intensities are adjusted in the reference arm and the object arm of the interferometer and the same polarization state is also guaranteed for both arms improving their interference. The specimen S is illuminated by a plane wave and a microscope objective, that produces a wave front called object wave *O*, collects the transmitted light**.** A condenser, not shown, is used to concentrate the light or focus the light in order that the entire beam passes into the MO. At the exit of the interferometer the two beams are combined by beam splitter BS2 being formed at the CCD plane the interference pattern between the object wave *O* and the reference wave *R*, which is

where *R\** and *O\** are the complex conjugates of the reference and object waves, respectively. The two first terms form the zero-order, the third and fourth terms are respectively the virtual (or conjugate image) and real image, which correspond to the interference terms. The off-axis geometry is considered; for this reason the mirror M2, which reflects the reference wave, is oriented so that the reference wave reaches the CCD camera with a small incidence angle with respect to the propagation direction of the object wave. A digital hologram is recorded by the CCD camera HDCE-10 with 1024x768 square pixels of size 4.65 µm, and

*2 2 \* \* IH <sup>ξ</sup>,<sup>η</sup> O R R O RO* (1)

*,)*,

P, dual polarizer; CCD, charge coupled device.

recorded as the hologram of intensity *IH(*

*-)*. In

image plane *(x'-y')* which is at a distance *d'* from the recording hologram plane *(*

magnified image and not from the object itself (VanLigten & Osterberg, 1966).

**2. Experimental set-up** 

spherical aberration (Stadelmaier & Massig, 2000), astigmatism (Grilli et al., 2001) and anamorphism (De Nicola et al., 2005).

Compensation of aberrations is fundamental when quantitative phase determination is used in microscopic metrological applications. Several approaches have been proposed to remove the aberrations. A method proposed by Cuche (Cuche et al., 1999) used a single hologram, which involved the computation of a digital replica of the reference wave depending on two reconstruction parameters. A double-exposure technique (Ferraro et al., 2003) can compensate completely the inherent wave front curvature in quantitative phase contrast imaging, although it needs two hologram recordings with and without the sample and a subtraction procedure between the two holograms. In the paper (Colomb et al., 2006) the authors developed a method to compensate the tilt aberration by recording a hologram corresponding to a blank image to compute the first-order parameters directly from the hologram. However, the method is limited to selecting profiles or areas known to be flat in the hologram plane. A method was proposed by Miccio (Miccio et al., 2007) who performed a two-dimensional fitting with the Zernike polynomials of the reconstructed unwrapped phase, although the application of the method is limited to be the special case of thin objects.

For image reconstruction different algorithms have been developed. Among them, the Single Fourier Transform Formulation (SFTF) (Schnars, 1994), the convolution-based algorithm (CV) (Demetrakopoulos & Mittra, 1974) and the angular spectrum-based algorithm (ASA) (Yu & Kim, 2005) are most commonly used. The SFTF algorithm is fast and can be used with objects larger than a CCD. However, variation of the size of the reconstructed image as result of a change in the reconstruction depth poses problems in applications such as reconstruction of color holograms (Yamaguchi et al., 2002) and particle sizing (Pan & Meng, 2003). In contrast, the CV algorithm keeps the size of the reconstructed image the same as that of the CCD. However, it is applicable only to objects that are smaller than the CCD. In addition, when the CV algorithm is used for objects much smaller than a CCD, it degrades the image quality, since the image is represented by only a small number of pixels. The CV algorithm was extended to large objects by zero padding the holograms before reconstruction (Kreis et al., 1997). This approach, however, led to an increase in computational load.

To avoid a number of limitations of the previous algorithms, the double Fresnel-transform algorithm (DBFT) that allows the reconstruction of digital holograms with adjustable magnification was developed (Zhang et al., 2004)**.** This algorithm involves two reconstruction steps implemented by a conventional single Fourier-transform algorithm. Through the adjustment of the distance parameter in the first stages, it is possible to control the size of the reconstructed images, independent of distance and wavelength, even for objects larger than a CCD without any computational penalty.

The method of image reconstruction proposed here is similar to the DBFT algorithm, since we also use two steps for image reconstruction, but they differ in the objectives and meaning of the two steps. Our formulation is specific for digital holographic microscopy; this way the objective of the first stage is the calculation of the objects' Fourier transform plane, where the complex wavefield contains all the information about the phase and intensity of object wavefield. Then, the second step is to reconstruct the complex amplitudes of the image wavefield starting from the objects' Fourier transform plane. We will describe and show advantages of the proposed method.
