**Phase curvature compensation:**

It should be noted from figure 9b that mixed phase images *<sup>o</sup>* and *<sup>B</sup>* appear, which are related with the phase of the objects and the quadratic constant phase factor *S* respectively. Both phase images are shown more clearly in figure 10.

Fig. 10. Pseudo 3D rendering of the phase image *DPA* presented in figure 9b.

Due to the slow variation of *<sup>B</sup>*, it can be considered as a background contribution to the phase image of the objects *<sup>o</sup>*, thus the problem of phase curvature compensation can be treated as a problem of phase image background subtraction. This way, alternative image background subtraction methods can be used. A procedure (Ankit & Rabinkin, 2007), that consists of applying a median filter with a large kernel size to the phase image represents a quick and simple way to obtain *<sup>B</sup>*, figure 11.

Calculating the square modulus and the argument of Eq. (9) in figure 9 provide the

intensity (a) and phase (b) image reconstruction respectively at *z = D = 173 mm*.

(a) (b)

It should be noted from figure 9b that mixed phase images

Both phase images are shown more clearly in figure 10.

Fig. 10. Pseudo 3D rendering of the phase image

*<sup>B</sup>*, figure 11.

Due to the slow variation of

quick and simple way to obtain

phase image of the objects

Fig. 9. Reconstruction of the intensity image (a) and the phase image (b) at *z =D=173 mm*

treated as a problem of phase image background subtraction. This way, alternative image background subtraction methods can be used. A procedure (Ankit & Rabinkin, 2007), that consists of applying a median filter with a large kernel size to the phase image represents a

related with the phase of the objects and the quadratic constant phase factor *S*

*<sup>o</sup>* and 

*DPA* presented in figure 9b.

*<sup>B</sup>*, it can be considered as a background contribution to the

*<sup>o</sup>*, thus the problem of phase curvature compensation can be

*<sup>B</sup>* appear, which are

respectively.

**3.1.3 Intensity and phase image reconstruction** 

**Phase curvature compensation:** 

Fig. 11. Pseudo 3D rendering of the phase image *<sup>B</sup>* (a) and simultaneous visualization of *DPA* and *<sup>B</sup> (color red)* (b)

As can be appreciated from figure 11b, a good fitting of *<sup>B</sup>* is achieved when a median filter with a large kernel size is applied over *DPA*.

Taking into consideration that the reconstructed complex wavefield *DPA* can be expressed as the superposition of two contributions corresponding to the object wave front *exp(io)* and the spherical wave front *exp(iB)*,

$$\Psi\_{\rm DPA}\left(\mathbf{x}',\mathbf{y}';d'\right) \propto \exp(i\phi\_o)\exp(i\phi\_\oplus) \tag{10}$$

then, the phase curvature can be compensated numerically by the calculation of the corrected phase image , ; *corr DPA xyd* ,

$$\phi\_{\rm DPA}^{\rm corr}\left(\mathbf{x}', y'; d'\right) \equiv \phi\_o\left(\mathbf{x}', y'; d'\right) = \arg\left[\psi\_{\rm DPA}\left(\mathbf{x}', y'; d'\right) \exp\left(-i\,\phi\_{\rm B}\right)\right] \tag{11}$$

In figure (12a) is presented the corrected phase image calculated by Eq. (11).

Fig. 12. Corrected phase image , ; *corr DPA xyd* (a). Phase image calculated with ASA, but using a reference hologram (b).

Alternative Reconstruction Method and Object Analysis in Digital Holographic Microscopy 195

In this section we study microscopic objects with regular forms starting from their Fraunhofer diffraction patterns obtained with DHM. Two types of analysis are considered: (i) analysis of objects according to their spatial distribution and (ii) analysis of individual

From Eq. (5) it can be seen that the wave field at the back focal plane õ*(u,v)* is proportional to

a quadratic phase curvature factor that causes a phase error if the optical Fourier

Using DHM it is possible to find the exact Fourier Transform of objects at the back focal

the well-known parameters of the experimental design presented in figure 1, i.e. *d'*, *D* and *f*,

2 2 , exp <sup>2</sup>

Multiplying Eq. (12) by Eq. (5), the constant phase factor is eliminated and the exact Fourier

 , ,; , , , *<sup>f</sup> BFP SFTF o o uv uvz D S uv ox* 

Calculating the intensity distribution from Eq. (13), the object's Fraunhofer diffraction

Eq. (14) offers a powerful tool in microscopic analysis because the Fourier Transform plane can be manipulated and different techniques of Fourier optics can be applied digitally, such

The knowledge of system magnification is important when quantitative relations between lineal dimensions of the enlarged image and the microscopic object have to be known. In

The image and the back focal planes are related by a Fourier transformation, thus lineal distance in the image plane can be extracted by the reciprocal of the corresponding lineal distance in the focal plane. In the working conditions with the capture of one hologram, figure 14a, the Fraunhofer pattern is reconstructed, figure 14b. A micrometric scale

The magnification of the system *MT = di/do* can be determined by the relation between two distances *di* and *do* in the image and object plane respectively. We determine the distance *di = 0.89 mm* between two bar in the image plane as the reciprocal value of the measured distance *df = 1.12 mm* between two contiguous diffraction points on the Fraunhofer pattern.

DHM the total system magnification depends on where the camera CCD is placed.

 (12)

*y f* (13)

2 *<sup>i</sup> i k S uv u v ff f Dd*

*BFP* is obtained,

 

<sup>2</sup> , , *FDP BFP I uv uv* (14)

*(u,v)*, can be expressed through

, which represents

the Fourier transform of the objects except for the spherical wave front *S*

**4. Microscopic object analysis using DHM** 

**4.1 Fourier transformation at the back focal plane** 

plane. The complex conjugate of the constant phase factor *S*

transformation is computed (Poon, 2007).

as pattern recognition, image processing and others.

with 100 lines per mm was used as object.

Transform at back focal plane

pattern *IFDP(u,v)* is obtained,

**4.2 System magnification** 

*Mitutoyo*

objects.

For comparison figure 12b shows the phase image calculated from the same hologram and reconstructed with the ASA, but using a reference hologram (Colomb et al., 2006).
