**3.1.1 Determination of the distance** *D*

The distance *D* is determined only one time, and remains the same until there is some variation in the experimental set-up. Applying the (ASA) algorithm, the following steps were performed to calculate the distance *D*:


4. The reconstructed complex wave field at any plane *(x'-y')* perpendicular to the propagating *z* axis is found by,

$$\mathcal{W}\_{ASA}\left(\mathbf{x'},\mathbf{y'};\mathbf{z}\right) = \iint A\left(k\_{\xi'}k\_{\eta};\mathbf{z}\right) \exp\left[\imath\left(k\_{\xi}\mathbf{x'} + k\_{\eta}\mathbf{x'}\right)\right] dk\_{\xi} \, dk\_{\eta} \dots$$


Fig. 7. Distance D determination using *ASA*.

Larger or lower spatial frequencies of object decomposition will be represented by intensity in the focal plane that is farther or closer from optical axis or equivalently farther or closer

The distance *D* is determined only one time, and remains the same until there is some variation in the experimental set-up. Applying the (ASA) algorithm, the following steps

*;z=0)* of the hologram *IH(*

2. Filtering of the angular spectrum to suppress both the zero-order and the twin image. In this step a region of interest corresponding only to the object spectrum is selected

> 

4. The reconstructed complex wave field at any plane *(x'-y')* perpendicular to the

*,k*

varying *z* from *0* to *300 mm* with *20 mm* as incremental step. The punctual maximum

6. Between the two points around the relative maximum the incremental step is reduced to *1 mm*. The distance *D* is equal to the *z* value at which the absolute maximum of

*,*

,; 0 is obtained.

*<sup>z</sup> k kkk* .

> 

are the corresponding spatial frequencies in the

;*z*) is calculated from *A kk <sup>H</sup>*

*)* at *z = 0* is obtained by taking

 , ;0 as,

*ASA(x',y';z)|2*, is carried out

from the center of the pattern where is contained the undiffracted object wavefield.

Fig. 6. Intensity of field distribution on the focal plane.

*,k*

*,*and *k*

and the modified angular spectrum *A kkz <sup>H</sup>*

*ASA x y z A k k z i k x k x dk dk* ', ';

5. The reconstruction of the amplitude image *IASA(x',y';z)=|*

, ; , ;0 exp *<sup>z</sup>* , where <sup>222</sup>

, ; exp ' '

.

value *P(z) = [IASA(x',y';z)]max* is calculated and plotted for each *z*, figure 7.

*IASA(x',y';z)max* is reached; the value of *z = D = 173 mm* is shows in figure 7.

 

3. The new angular spectrum at plane z, *A*(*k*

**3.1.1 Determination of the distance** *D*

1. The angular spectrum *A(k*

hologram plane

its Fourier transform. *k*

were performed to calculate the distance *D*:

*-*.

*A k k z A k k ik z*

propagating *z* axis is found by,

  Because the reference wave is plane, the distance D is the physical distance from the hologram to back focal point of the objective lens.

### **3.1.2 Behavior of the wavefield near to the back focal plane**

Using the hologram of figure 5a, the behavior of the wavefield near the back focal plane is visualized. With *z* ranging from *141 mm* to *173 mm* and incremental step of *8 mm* a sequence of image reconstructions is presented in figure 8. The sequence shows the change of phase (A) and intensity (B) calculated from the expressions: arg(*FFT-1[SFTF(l,j;z)])* and *log[1+|(SFTF(l,j;z)|2]* respectively.

Fig. 8. Behavior of the wavefield in the region of the back focal plane.

From figure 8 it is corroborated that as the reconstruction plane approaches the focal plane, the phase jumps between the reference and object waves gradually disappear. These phase jumps totally disappear for z *= D = 173 mm*, where the focal plane is reconstructed. This behaviour allows us to conclude that the curvature of the wavefront has a minimum on the focal plane and it is increased as the wavefield propagates away from this plane.

Alternative Reconstruction Method and Object Analysis in Digital Holographic Microscopy 193

 

*xyd xyd xyd i* (11)

*xyd* (a). Phase image calculated with ASA, but

*<sup>B</sup>* (a) and simultaneous visualization of

*<sup>o</sup> <sup>B</sup>* (10)

 

*<sup>B</sup>* is achieved when a median filter

*DPA* can be expressed

*o)* and

(a) (b)

Taking into consideration that the reconstructed complex wavefield

*DPA*.

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

as the superposition of two contributions corresponding to the object wave front *exp(i*

*DPA xyd i i* , ; exp exp

then, the phase curvature can be compensated numerically by the calculation of the

 , ; ', '; ' arg ', '; ' exp *corr DPA o DPA B*

(a) (b)

 

Fig. 11. Pseudo 3D rendering of the phase image

As can be appreciated from figure 11b, a good fitting of

*B)*,

*xyd* ,

*DPA* 

Fig. 12. Corrected phase image , ; *corr*

using a reference hologram (b).

*DPA* 

*<sup>B</sup> (color red)* (b)

the spherical wave front *exp(i*

with a large kernel size is applied over

corrected phase image , ; *corr*

*DPA* and
