**4.3.1 Similar objects in random arrangement**

We consider a sample of mouse blood cell as an example of random distribution of similar objects and use the proposed methodology to determine the diameter of cells. Figure 15a shows the hologram recorded with the experimental set-up.

Applying Eq. (14) the Fraunhofer pattern is obtained, figure 15b. As predicted by theory, the Fraunhofer diffraction pattern has a 'spotty' interference pattern, with a central peak and intensity in the diffraction plane that shows random fluctuations on a general background.

The radial intensity distribution *IR*(*r*), figure 16-upper, is measured by scanning the Fraunhofer diffraction pattern along radial lines (Palacios et al., 2001).

This method for determining *di* is more accurate that their direct measure over the image plane. With the knowledge that the distance between the two bar in the object plane is *do =* 

(a) (b)

We consider objects with regular forms and two different forms of spatial distribution: randomly and periodically distributed. The objects' parameters can be determined by diffraction pattern manipulation in a simple and accurate way. This is an example of objects

We consider a sample of mouse blood cell as an example of random distribution of similar objects and use the proposed methodology to determine the diameter of cells. Figure 15a

Applying Eq. (14) the Fraunhofer pattern is obtained, figure 15b. As predicted by theory, the Fraunhofer diffraction pattern has a 'spotty' interference pattern, with a central peak and intensity in the diffraction plane that shows random fluctuations on a general

The radial intensity distribution *IR*(*r*), figure 16-upper, is measured by scanning the

Fig. 14. Digital hologram, scale bar 445 m (a), Fraunhofer pattern, scale bar 2 mm-1 (b)

**4.3 Analysis of objects according to their distribution** 

analysis that is important to biological and materials sciences.

shows the hologram recorded with the experimental set-up.

 (a) (b) Fig. 15. Digital hologram (a), Fraunhofer pattern, scale bar *2 mm-1* (b).

Fraunhofer diffraction pattern along radial lines (Palacios et al., 2001).

**4.3.1 Similar objects in random arrangement** 

background.

*10* 

*m*, *MT = 0.89/0.010 = 89*.

Fig. 16. Upper- the radial intensity curve *IR*(*r*), lower- spectrum of *IR*(*r*).

For each *r* value the intensity *IR*(*r*) is the result of averaging the intensity values *IFDP(u,v)*  along the circumference from *0º* to *360º*, mathematically this operation can be represented by the expression,

$$I\_R\left(r\right) = r^3 \xrightarrow{\sum\_{\theta=1}^{360} I\left(\mathbf{x}' = \mathbf{C}\_x\left(r, \theta\right), \mathbf{y}' = \mathbf{C}\_y\left(r, \theta\right), d' = D\right)} \tag{15}$$

where, *Cr N r x* , / 2 cos , *Cr N r y* , / 2 sin , with *0º < 360º* and *0<r N/2*. The spatial coordinates in the Fraunhofer diffraction pattern are defined on basis of the Fraunhofer diffraction pattern pixel resolution *Δu*, which is determined directly from the Fresnel diffraction formula at the reconstruction distance *z = D*. In this way, the radial distance *u=ju, j=0,1,…,Np,* where *Np* is the points number of the radial intensity curve and *u= D/M.* In the frequency spectrum, the spatial frequency is *fu=j/Npu, j=0,1,..Np.*

The spectral analysis of the radial intensity curve *IR*(*r*) is carried out by the calculation of the square of the modulus of its 1D Fourier transform. In the resulting spectrum, figure 16 lower, the harmonic components are seen. As seen, only one fundamental harmonic that characterizes the diameter (*ro = 6.3 m*) of the mouse blood cell appears.

### **4.3.2 Similar objects in periodical arrangement**

In this section we consider a sample of periodically hexagonal structures inscribed on a plastic material with an ion beam as an example of regularly repeated identical objects*.* 

In figure 17 is shown the hologram and intensity image reconstruction with the parameters for a hexagonal real space lattice.

In the reconstructed intensity image the parameters for the hexagonal real space lattice on the image plane are shown: diffraction angle for hexagonal crystal  *= 120º* and the unit cell parameters *a*, *b* which for hexagonal crystal meet the condition, *a = b*.

The Fraunhofer diffraction pattern is calculated and shown in figure 18a. In figure 18b a section of the reciprocal lattice is specified and drawn through the diffraction points, the parameters *a\**, *b\** and *\** are shown.

Alternative Reconstruction Method and Object Analysis in Digital Holographic Microscopy 199

determined and by measuring the parameters *a\* = b\* = 1.15 mm* in the reciprocal lattice the

parameters, but in the image plane. The parameters *a*, *b* of real lattice in the object plane are obtained dividing *a'* and *b'* by the total system magnification *MT = 89*, i.e., *a = b = a'/MT =* 

We demonstrate in this section the potentialities of Digital Holographic Microscopy in the determination of morphological parameters of nucleated cells. The spectral analysis of the radial behaviour of the Fraunhofer diffraction pattern allows the correlation between the

As an example of application a sample of oral mucosa epithelial cell was selected. These

*m* are calculated. It can be noticed that *a'* and *b'* are the real lattice

*m* *\* = 60º* is

plane. Applying the relations (16) the diffraction angle in the reciprocal lattice

*m*. This value coincides with that obtained by AFM.

parameters *a' = b' = 863* 

*b'/MT = 9.69* 

**4.4 Calculation of nucleated cell dimensions** 

peaks observed in the spectra and lineal dimensions of the cell.

Fig. 19. Oral mucosa epithelial cell, optical microscopy image. Scale bar *20* 

the cell and (ii) the nucleus is outside the centre.

argument of Eq. (9) with *d' = 0.*

are observed in the spectrum.

Two cases were analyzed, (i) the nucleus of the cell is located approximately at the centre of

Figure 20 shows the hologram (left) and the phase image reconstruction (right) of cell with the nucleus at the centre (A) and outside of centre (B). Both holograms were captured with the apparatus of figure 1 and the phase image was obtained by the calculation of the

For both holograms the Fraunhofer pattern was calculated and is shown in figure 21 (left). Applying the spectral analysis of the Fraunhofer pattern radial intensity, the corresponding frequency spectra are obtained, figure 21 (right). In each spectrum a sequence of peaks are seen. In the case of cell with nucleus at the centre (A) four peaks appear in the spectrum, which is different for the case of the cell with nucleus outside the centre (B) where six peaks

cells have a nucleus inside a regular cytoplasm, figure 19.

Fig. 17. Digital hologram (a) and intensity image (b) of the sample. Scale bar *5m*

Fig. 18. Fraunhofer diffraction pattern (a), scale bar 2 mm-1. Over diffraction pattern it is drawn the reciprocal lattice (b). (*Inset: Representation of the real space*)

As shown in figure 18b, according to the theory of crystal diffraction, the reciprocal lattice edges of dimensions *a\** and *b\** are respectively perpendicular to the cell edges *b* and *a*, and quantities are related by the following expressions,

$$a^\* = \frac{K}{a \sin \gamma^\*} \text{ } b^\* = \frac{K}{b \sin \gamma^\*} \text{ and } \gamma^\* = 180^\circ - \gamma \tag{16}$$

where *K = L* is the constant of diffraction, where the wavelength of monochromatic radiation and *L* is the camera length, i.e., the distance from the specimen to the diffraction plane. Applying the relations (16) the diffraction angle in the reciprocal lattice *\* = 60º* is determined and by measuring the parameters *a\* = b\* = 1.15 mm* in the reciprocal lattice the parameters *a' = b' = 863 m* are calculated. It can be noticed that *a'* and *b'* are the real lattice parameters, but in the image plane. The parameters *a*, *b* of real lattice in the object plane are obtained dividing *a'* and *b'* by the total system magnification *MT = 89*, i.e., *a = b = a'/MT = b'/MT = 9.69 m*. This value coincides with that obtained by AFM.
