Holography was invented over a half-century ago by D. Gabor. However, holography is widely used nowadays in different fields. Concretely, the development of digital cameras and displays allowed to go further using the principles of classic holography for recording and also displaying digital holograms. The basic concept in Digital holography is the recording of the inferential field formed by the wave-field scattered by a certain object and a reference wave. By computer processing, it is possible to recover the complete complex-wave information of the field dispersed by the object, that is, both the amplitude and the phase. The method to obtain this information includes a filtering process in the Fourier space in order to isolate object complex-amplitude information encoded within the digital hologram. Once filtered, the resulting digital hologram is multiplied by a numerical reference which matches the physical reference that was used to record the digital hologram. Then, a numerical refocus based on Fresnel-Kirchhoff integral can be done, permitting to calculate the amplitude and the phase at different planes and, in particular, at the neighboring of the object (see the animation below).

Moreover, the phase-map has depth information of the object which can be used to reconstruct the 3D surface of the original object. This is one of the most useful properties of DH inasmuch as, in the known as off-axis configuration, 3D information of the object can be extracted from a one-shot experiment. Along last decade, DH has been applied to microscopy. In Digital Holographic Microscopy (DHM) what is recorded in a digital hologram is the inference between the reference beam and the propagated wavefront of the intermediate-aerial image of the object provided by a microscope objective. DHM is particularly useful in particle tracking and trapping, micro-electromechanical systems characterization, profilometry of optical systems, etc. One interesting performance of DH in Quantitative-Phase Microscopy (QPM). In this method, the phase of the object is retrieved in the reconstruction process. By using the optical path relationship, the phase-map in the reconstruction plane is converted into a deph map. Obviously, some parameters have to be known, as the wavelength of the illumination or the refraction index for the case of transmission mode DHM. The phase-map retrieval is more complicated in the case of introducing a objective lens into the system. Basically, this lens introduces a spherical-phase factor which has to be removed. Usually the removal process is carried out by means of numerical methods compensating the factor in the reconstruction plane (that implies the knowledge of the wavefront curvature). Other forms of compensation, known as physical, are based on introducing the same curvature in the reference arm.

In our group, we look for implementation of DHM design, in terms of aberrations reduction and analyzing the resolution. In Fig.X it can be seen our implementation of a telecentric DHM in which QPM can be performed. Our design consists in an infinity-corrected microscope objective arranged in telecentric configuration with a tube lens (see Fig.2).

This implementation has some advantages over the conventional configuration. On the one hand, it supresses the spherical phase-factor associated with the introduction of an objective lens in the DH system. As a consequence, in the reconstruction algorithm we only have to compensate numerically the linear term related with the tilt of the reference beam, but not the one related with the defocus aberration. Furthermore, the field recorded by the CCD camera corresponds to the same zone of the object, independently of the CCD position. Moreover, if we analyze the spectrum of the recorded hologram by using our system, it is expected an improvement of the resolution in comparison with the conventional DHM. This is easily understandable: when recording a hologram with a spherical wavefront added to the object wavefront, it produces a spatial-frequency shift. As a consequence, the spectrum of the object is spreaded along an extended zone within the frequency domain. So that, different parts of the object are "encoded" at different spatial-frequencies. The spatial-frequency shift, besides causing some distortion in the effective point spread function of the system, complicates the filtering process and, even more, could remove some frequencies in the final reconstruction of the object amplitude.

As an example of how this distortion is produced, we can see the animation below. In this animation we show the Fourier transform of a digital hologram recorded in absence of object with our system (in transmission mode). The tube lens is displaced along different positions, including the telecentric one. As is expected, the Fourier transform of a constant signal is a point (Dirac delta function). As it can be seen, this only occurs only for the telecentric configuration. On the other cases, the spherical phase factor produces a shift in the spatial-frequencies, which results on a spreading of the object spectrum.

As we said, this configuration is particularly usable for the case of QPM inasmuch as the resulting phase-map is not affected by a spherical wavefront. This reduces the complexity of the phase-retrieval procedure. In is shown that sub-wavelenght resolution is achieved in the axial direction by using the QP method. We show two examples of depht-maps obtained with our system. The samples were a USAF-1951 test target( imaged in reflection mode) and a Fresnel lens (imaged by trasmission). Finally, the telecentric configuration can be carried out in the DHM in-line setup and represents a slight variation of the standard system.