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

From Eq. (5) it can be seen that the wave field at the back focal plane õ*(u,v)* is proportional to the Fourier transform of the objects except for the spherical wave front *S*, which represents a quadratic phase curvature factor that causes a phase error if the optical Fourier transformation is computed (Poon, 2007).

Using DHM it is possible to find the exact Fourier Transform of objects at the back focal plane. The complex conjugate of the constant phase factor *S(u,v)*, can be expressed through the well-known parameters of the experimental design presented in figure 1, i.e. *d'*, *D* and *f*,

$$\tilde{S}\_{\phi}(u,v) = \frac{i}{\lambda f} \exp\left[\frac{-ik}{2\left(D+d'\right)}\left(u^2+v^2\right) - f\left(f+2\right)\right] \tag{12}$$

Multiplying Eq. (12) by Eq. (5), the constant phase factor is eliminated and the exact Fourier Transform at back focal plane *BFP* is obtained,

$$\mathfrak{T}\_{BFP}(\mathsf{u}, \mathsf{v}) = \mathsf{\upmu}\_{\mathrm{SFTF}}^{f}(\mathsf{u}, \mathsf{v}; \mathsf{z} = \mathsf{D}) \, \tilde{\mathsf{S}} \phi(\mathsf{u}, \mathsf{v}) = \mathfrak{T} \Big[ o(\mathsf{x}\_{o'} y\_{o'} \mathsf{z} f) \Big] \tag{13}$$

Calculating the intensity distribution from Eq. (13), the object's Fraunhofer diffraction pattern *IFDP(u,v)* is obtained,

$$I\_{FDP}\left(\mu,\upsilon\right) = \left|\mathfrak{T}\_{BFP}\left(\mu,\upsilon\right)\right|^2\tag{14}$$

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 as pattern recognition, image processing and others.

### **4.2 System magnification**

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

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 *Mitutoyo*with 100 lines per mm was used as object.

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.

Alternative Reconstruction Method and Object Analysis in Digital Holographic Microscopy 197

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

<sup>360</sup>

 , *Cr N r y* , / 2 sin 

*.* In the frequency spectrum, the spatial frequency is *fu=j/Np*

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

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

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

In the reconstructed intensity image the parameters for the hexagonal real space lattice on

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

*Ix C r y C r d D*

*u, j=0,1,…,Np,* where *Np* is the points number of the radial intensity curve and

*m*) of the mouse blood cell appears.

360 *x y*

,, ,,

 

(15)

, with *0º <* 

 

 *360º* and *0<r*

*u, j=0,1,..Np.*

 *= 120º* and the unit cell

 *N/2*.

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

3 1

the image plane are shown: diffraction angle for hexagonal crystal

parameters *a*, *b* which for hexagonal crystal meet the condition, *a = b*.

**4.3.2 Similar objects in periodical arrangement** 

*\** are shown.

*Ir r*

*R*

by the expression,

distance *u=j*

*u= D/M*

where, *Cr N r x* , / 2 cos 

characterizes the diameter (*ro = 6.3* 

for a hexagonal real space lattice.

parameters *a\**, *b\** and

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 = 10 m*, *MT = 0.89/0.010 = 89*.

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