**3. Shot noise reduction in phase imaging of digital holograms**

In this section, we show digital holograms with shot noise when they were recorded with a CCD camera. The illumination source used in the optical set up was a commercial LED. Here, we present a technique to reduce the shot noise of the phase and amplitude images com‐ ing from a single reconstructed wavefront of the object. To attain a shot noise reduction, an averaging procedure of reconstructed images at different reconstruction distances within the range determined by the focus depth is performed. With this tomographic capacity of DHM, we ensure an improved image without quality diminution, where a noise reduction of 50% is achieved. The results were compared with results from an atomic force microscope (AFM) in order to determine the system accuracy.

#### **3.1. Experimental configuration**

#### *3.1.1. LED physical properties*

the lens was of 0.4 μm (**Figure 3(b)**). Because the DOF is limited, the areas outside the DOF give out‐of‐focus and blurred amplitudes. **Figure 3(c)** shows the amplitude distribution map which was reconstructed with the different reconstruction distance method [6, 7] using the MIO and DHM. The initial DOF = 2.09 μm is increased by a factor of 2, as the height of the microlens is about 3.8 μm. **Figure 3(d)** and **(e)** shows both the unwrapped phase images of the specimen by using phase shifting interferometry and DHM methods, respec‐ tively. **Figure 3(f)** depicts profiles that compare the phase shifting interferometry result (**Figure 3(d)**) and DHM extending the DOF numerically (**Figure 3(e)**). These profiles cor‐ respond to the black‐dashed line of the corresponding unwrapped phase map. A difference of about 77 nm is obtained between both measurement methods. The result is in agreement with the results obtained in references [6, 7] for an increased factor of 2. One has to keep in mind that the higher the factor is, the larger the measurement error (difference) will be. Here, we present a measurement system that uses DHM method in order to extend the DOF of the system. Our proposal produces better measurement accuracy due to extended DOF and the numerical reconstruction techniques. With these results, we have demonstrated

We have presented DHMM as a new reliable optical tool for performing DHM in‐line reflection configuration. In the experimental results, we have principally proved the unique refocusing capability and the amplitude and phase images of DHM. The object under test sample was a microlens of 100 μm in diameter and 4.2 μm height. With these experimental results, we have also shown that it is possible to extend the DOF of the MO by using the numerical focusing capability of DHM. In addition, we have presented not only DHMM as an alternative to obtain digital holograms without spherical aberration, but also that an easier, well‐aligned, and insensitive to external vibrations setup is reached, in comparison with the typical setups. Finally, a topographic measurement error attained with interfer‐ ometry is demonstrated and compensated with DHM, which is due to the limited depth of

**3. Shot noise reduction in phase imaging of digital holograms**

In this section, we show digital holograms with shot noise when they were recorded with a CCD camera. The illumination source used in the optical set up was a commercial LED. Here, we present a technique to reduce the shot noise of the phase and amplitude images com‐ ing from a single reconstructed wavefront of the object. To attain a shot noise reduction, an averaging procedure of reconstructed images at different reconstruction distances within the range determined by the focus depth is performed. With this tomographic capacity of DHM, we ensure an improved image without quality diminution, where a noise reduction of 50% is achieved. The results were compared with results from an atomic force microscope (AFM) in

that it is feasible to perform DHM with the MIO [18].

**2.4. Conclusion**

162 Holographic Materials and Optical Systems

focus of the MO.

order to determine the system accuracy.

Parasitic interferences and multiple reflections in optical setups are typical; one uses a low coherent source to reduce them. Here, we use a commercial ultrabrilliant LED of 5 mm in diameter, with emission in the red spectrum range. We used a calibrated i1Pro eye‐one spec‐ trophotometer of spectral range from 380 to 730 nm. The peak wavelength (λ) measured was of 630 nm, and a full width at half maximum (FWHM, Δλ) of its spectrum was of 24 nm. **Figure 4** shows the typical normalized spectrum of the LED that was used. Then, with the spectral data above mentioned (*λ*<sup>2</sup> /Δ*λ*), the coherence length was of 16.5 μm.

#### *3.1.2. Reconstruction process*

The experimental setup used was a modified Mach‐Zehnder interferometer, as shown in **Figure 5**. When the beam is incident on the diaphragm D of a diameter of 300 μm, spatial coherence is increased. The spatial filter SP with an adjustable diaphragm to limit the source

**Figure 4.** Normalized spectrum distribution of the light intensity emitted by the commercial ultrabrilliant LED.

**Figure 5.** Optical setup implemented on a Mach‐Zehnder interferometer with a low coherent source for digital hologra‐ phic microscopy.

size creates a secondary source of low coherence. The lens L images D at the plane of the sample S and on the compensating plate (CP) when the beam is divided by BS1. The transmit‐ ted light through the specimen (S) is collected by a microscope objective (MO1) of 10× with 0.25 of numerical aperture (N.A.), which forms object wave *O*. The object wave interferes with a reference wave *R* when the light is collected by the microscope objective of 10× 0.25 N.A. (MO2) to produce a hologram. The hologram intensity (IH) is recorded by a camera; a CP was inserted in the setup to achieve interference between the two beams (Section 2.3). Also, mirror M1 was mounted on a linear displacement stage with step resolution of 1.25 μm. Mirror M2 was mounted on a piezoelectric transducer (PZT) to implement the phase‐shifting technique.

The intensity *I*(*x,y*) at the CCD sensor plane is formed by the interference of the object wave *O*(*x,y*) and the reference wave *R*(*x,y*) as Eq. (1). Mirror M2 is mounted on a PZT to calculate the phase of the initial object using the phase‐shifting technique, and to eliminate the DC terms and the virtual image of Eq. (1). The 2π phase module is calculated with the four frame algorithm [15]: (*x*, *<sup>y</sup>*; *<sup>π</sup>*/2)) \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ (*<sup>I</sup>*

$$\phi\_0(\mathbf{x}, y) = \tan^{-1} \left[ \frac{\left( I\_t(\mathbf{x}, y; 3\pi/2) - I\_t(\mathbf{x}, y; \pi/2) \right)}{\left( I\_t(\mathbf{x}, y; 0) - I\_y(\mathbf{x}, y; \pi) \right)} \right]. \tag{6}$$

The amplitude of the optical field *A*<sup>0</sup> (*x,y*) can be obtained by blocking the reference beam and recording only the diffraction object intensity in the CCD plane. Then, the object complex amplitude is as follows:

$$\mathcal{U}\_{\boldsymbol{\uprho}}(\mathbf{x},\boldsymbol{y}) = A\_{\boldsymbol{\uprho}}(\mathbf{x},\boldsymbol{y}) \exp\left[\mathrm{i}(\phi\_{\boldsymbol{\uprho}}(\mathbf{x},\boldsymbol{y}) \star \phi(\mathbf{x},\boldsymbol{y}))\right] \tag{7}$$

where the object wave is determined at the recording *(x, y)* plane, and *φ* is the phase aberra‐ tion term.

To reduce phase aberrations induced by misalignment of the optical setup and MOs, we per‐ form the reference conjugated hologram (RCH) method [19]. We obtain the phase aberration term without the presence of the test object, which can be subtracted from Eq. (7) to get the object complex amplitude:

$$\mathcal{U}\_{\boldsymbol{\vartheta}}(\mathbf{x},\boldsymbol{y}) = A\_{\boldsymbol{\vartheta}}(\mathbf{x},\boldsymbol{y}) \times \exp\left[\ \boldsymbol{i}(\phi\_{\boldsymbol{\vartheta}}(\mathbf{x},\boldsymbol{y}) + \phi(\mathbf{x},\boldsymbol{y}) - \phi(\mathbf{x},\boldsymbol{y})) \right]. \tag{8}$$

The angular spectrum method (AS) is performed to calculate the object wavefront at any other plane (*ξ, η*), in order to refocus it by Eqs. (4) and (5).

#### *3.1.3. Focus depth and averaging method*

A limitation in DHM is a limited depth of focus. High magnifications are achievable for inves‐ tigating microobjects with this technique. However, higher is the required magnification, and lower is the focus depth system. As the geometrical DOF of an imaging system is related to the sampling rate, this DOF is defined as a function of the pixel size and the N.A. of the MO:

$$\text{DOF} = \frac{\Lambda \xi}{M^2 \text{N.A.'}} \tag{9}$$

where *M* is the magnification.

size creates a secondary source of low coherence. The lens L images D at the plane of the sample S and on the compensating plate (CP) when the beam is divided by BS1. The transmit‐ ted light through the specimen (S) is collected by a microscope objective (MO1) of 10× with 0.25 of numerical aperture (N.A.), which forms object wave *O*. The object wave interferes with a reference wave *R* when the light is collected by the microscope objective of 10× 0.25 N.A. (MO2) to produce a hologram. The hologram intensity (IH) is recorded by a camera; a CP was inserted in the setup to achieve interference between the two beams (Section 2.3). Also, mirror M1 was mounted on a linear displacement stage with step resolution of 1.25 μm. Mirror M2 was mounted on a piezoelectric transducer (PZT) to implement the phase‐shifting technique. The intensity *I*(*x,y*) at the CCD sensor plane is formed by the interference of the object wave *O*(*x,y*) and the reference wave *R*(*x,y*) as Eq. (1). Mirror M2 is mounted on a PZT to calculate the phase of the initial object using the phase‐shifting technique, and to eliminate the DC terms and the virtual image of Eq. (1). The 2π phase module is calculated with the four frame

algorithm [15]:

tion term.

*φ*<sup>0</sup>

164 Holographic Materials and Optical Systems

amplitude is as follows:

*U<sup>a</sup>*

object complex amplitude:

*U*<sup>0</sup>

The amplitude of the optical field *A*<sup>0</sup>

(*x*, *y*) = tan<sup>−</sup><sup>1</sup>

(*x*, *y*) = *A*<sup>0</sup>

(*x*, *y*) = *A*<sup>0</sup>

plane (*ξ, η*), in order to refocus it by Eqs. (4) and (5).

DOF <sup>=</sup> \_\_\_\_\_\_\_ *Δξ*

*3.1.3. Focus depth and averaging method*

[ (*I* 4

(*x*, *y*; 3*π*/2) −*I*

1 (*x*, *y*; 0) −*I* 3

recording only the diffraction object intensity in the CCD plane. Then, the object complex

(*x*, *y*) exp  [ *i*(*φ*<sup>0</sup>

where the object wave is determined at the recording *(x, y)* plane, and *φ* is the phase aberra‐

To reduce phase aberrations induced by misalignment of the optical setup and MOs, we per‐ form the reference conjugated hologram (RCH) method [19]. We obtain the phase aberration term without the presence of the test object, which can be subtracted from Eq. (7) to get the

The angular spectrum method (AS) is performed to calculate the object wavefront at any other

A limitation in DHM is a limited depth of focus. High magnifications are achievable for inves‐ tigating microobjects with this technique. However, higher is the required magnification, and lower is the focus depth system. As the geometrical DOF of an imaging system is related to the sampling rate, this DOF is defined as a function of the pixel size and the N.A. of the MO:

*M*<sup>2</sup> N.A.

(*x*, *y*) × exp  [ *i*(*φ*<sup>0</sup>

2 (*x*, *<sup>y</sup>*; *<sup>π</sup>*/2)) \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ (*<sup>I</sup>*

(*x*, *<sup>y</sup>*; *<sup>π</sup>*)) ]

(*x,y*) can be obtained by blocking the reference beam and

(*x*, *y*) +*φ*(*x*, *y*))], (7)

(*x*, *y*) +*φ*(*x*, *y*) −*φ*(*x*, *y*))]. (8)

, (9)

. (6)

DHM has as a unique feature that is possible to refocus the object complex amplitude at any plane within the maximum refocus distance [6, 11]. We demonstrate that it is possible to refo‐ cus the complex amplitude at different distances within DOF (ΔDOF). The specimen physical thickness is given by

$$h = \{ \lambda \langle \Delta \phi / 2\pi \rangle / (\mathfrak{n} - \mathfrak{n}\_{\phi}) \}. \tag{10}$$

where λ is the wavelength, Δ*φ* is the phase step, and *(n − n*<sup>0</sup> ) is the index difference between the specimen's material and air.

Shot noise depends on optical power, and it follows a Poisson's statistics [20]. Then we say that a higher light intensity corresponds to a lower shot noise. A way to increase the amount of photons is performing an averaging of the reconstructed images in order to attain an improved image.

We performed an averaging process of the reconstructed images that are obtained from dif‐ ferent reconstruction distances within the system's DOF in order to get an improved image. These reconstructed images are uncorrelated with each other at specific reconstruction dis‐ tances, and computed from the same complex amplitude [21].

We think that if these reconstructed images are uncorrelated with similar standard deviations STDc , one can write the following for the standard deviation of an averaged image *σ<sup>x</sup>*¯ :

$$
\sigma\_{\rm x} = \frac{1}{\sqrt{\mathcal{C}}} \text{STD}\_{\iota'} \tag{11}
$$

where *C* is the number of images to average.

If four images are averaged, theoretically, noise reduction is of 50%.

#### **3.2. Numerical and experimental results**

In this part, the results of the recorded holograms are presented. The shutter camera enables us to reduce the exposure time down to 40 μs, with a variable gain from 0 to 17 dB in 14 increments. We do not reach a camera's full well capacity with a LED source in the setup presented, even with the maximum integration time and no electrical gain of camera param‐ eters [21]. The optical power of the intensity was measured with a photo detector. First, a comparison between a blank experimental hologram and the simulated hologram results is shown. The intensity of the blank holograms was of 6.7 × 10‐15 W/cm2 , and this corresponds to an average number of 5100 of photons per pixel. **Figure 6(a)** presents a blank phase image that is reconstructed without a phase aberration correction. A standard deviation (STD) = 12° is computed in the black square area. On the other hand, **Figure 6(b)** presents the same recon‐ struction after the RCH method was applied, with a STD = 0.7° in the same area.

We have noncorrelation among phase images reconstructed at different distances [21]. These images were obtained of the recorded experimental holograms from Eq. (4). **Figure 7** shows that when there is a difference of distance of 2 μm from one reconstructed phase image to

**Figure 6.** Reconstructed phase image from a recorded hologram without any sample. (a) Phase image without aberration correction and (b) phase image with aberration correction.

**Figure 7.** Correlation coefficient between reconstructed phase images at different reconstruction distances from the same wavefront of holograms recorded with low intensity (photons per pixel of 5100).

another one obtained from same complex wavefront, a noncorrelation exists between these images. In that case an averaging procedure can be performed of these noncorrelated images in order to get an improved image. These results validate the proposal previously demon‐ strated in reference [21].

With the Δ*d* = 2 μm calculated, we can average *C* images. In **Figure 8(a)**, the phase image is shown without any averaging. This image was reconstructed within DOF of the system with *d* = 30 μm. The STD in this phase image is of 0.69°. On the other hand, **Figure 8(b)** shows a phase image after applying the proposed averaging procedure with 10 phase images. The STD of this image is of 0.231°. We get an image improvement of 68.4% of noise reduction. This is in agreement with Eq. (11). A profile comparison is shown in **Figure 9(a)**. The profiles were taken for the marked profile as a white line in the phase images in **Figure 8**.

**Figure 9(b)** presents the behavior of the STD as a function of the number of phase images *C* used in the averaging procedure. Also, a comparison between numerical simulation results and

Microtopography and Thickness Measurement with Digital Holographic Microscopy Highlighting and Its... http://dx.doi.org/10.5772/66750 167

**Figure 8.** Improvement of phase image by the proposed averaging method in DHM. (a) Reconstructed phase image within DOF of the system and (b) phase image improved from 10 averaged phase images reconstructed.

**Figure 9.** Behavior of noise phase reduction when, the averaging method is applied. (a) Profile of phase images in **Figure 8**, where a diminution of STD when 10 phase images are averaged and (b) profile of phase STD as a function of the number of phase images, where a behavior of C‐1/2 is shown in the STD reduction [21].

experimental results is presented. The simulation results were attained with a 5100 photons per pixel. The behavior of the shot noise reduction in experimental results follows the expected theoretical function defined in Eq. (11). Then we can say that the averaged images are noncor‐ related. The difference, presented between experimental and simulations results, is due to the presence of sources of noise other than shot noise, quantum efficiency, and small optical defects in the optical components. One can see that if the number of averaged images increases, the offset also decreases. Noise sources should be speckle and thermal noise of CCD camera. Both the integration time during hologram recording is not the highest and the camera's well capac‐ ity is not reached due to low illumination, therefore the noise related to the quantum efficiency of the CCD detector is the main factor which is the difference between the experimental and simulated results.

#### *3.2.1. Decrease of shot noise in amplitude images*

another one obtained from same complex wavefront, a noncorrelation exists between these images. In that case an averaging procedure can be performed of these noncorrelated images in order to get an improved image. These results validate the proposal previously demon‐

**Figure 7.** Correlation coefficient between reconstructed phase images at different reconstruction distances from the same

**Figure 6.** Reconstructed phase image from a recorded hologram without any sample. (a) Phase image without aberration

With the Δ*d* = 2 μm calculated, we can average *C* images. In **Figure 8(a)**, the phase image is shown without any averaging. This image was reconstructed within DOF of the system with *d* = 30 μm. The STD in this phase image is of 0.69°. On the other hand, **Figure 8(b)** shows a phase image after applying the proposed averaging procedure with 10 phase images. The STD of this image is of 0.231°. We get an image improvement of 68.4% of noise reduction. This is in agreement with Eq. (11). A profile comparison is shown in **Figure 9(a)**. The profiles were

**Figure 9(b)** presents the behavior of the STD as a function of the number of phase images *C* used in the averaging procedure. Also, a comparison between numerical simulation results and

taken for the marked profile as a white line in the phase images in **Figure 8**.

wavefront of holograms recorded with low intensity (photons per pixel of 5100).

correction and (b) phase image with aberration correction.

166 Holographic Materials and Optical Systems

strated in reference [21].

A principal limitation in the proposal is limited DOF. As we have already seen in Section 3.1.3, DOF is related with the sampling distance and numerical aperture of the optical system. In the system described, the theoretical DOF is of about 0.268 μm. But, in the experimental

**Figure 10.** Determination of the DOF experimentally. (a) Recorded hologram and the profile zone taken to measure the DOF and (b) evolution of intensity when reconstruction distance is increased.

reconstruction, DOF is higher than would have been expected from Eq. (9). This is because the fact that the spatial resolution (pixel size) introduced by the optical setup is limited [13]. To experimentally attain the DOF, in **Figure 10(b)**, we have plotted a line profile shown in **Figure 10(a)**, where the profile is marked with a black line. The evolution of this plot starts at 40 μm before the image is focused. After zooming on the focus zone, we can determine that the DOF is 9 μm. The test object we used was an Edmund NBS 1963A resolution card. The zone of interest corresponds to 18 double lines per mm (lpmm). The reconstruction distance was of 15 μm.

If the DOF is 9 μm and Δ*d* is of 2 μm, then we can average four images to carry out what we propose. These images have to be reconstructed at a distance within DOF to guarantee that image quality is not affected. **Figure 11** shows the image improved when the proposed method is applied. This evaluation is through STD in the area defined by the white square in each image. We attain a STD = 1.57 gray levels (GL) for an image focused without averaging (**Figure 11(a)**), and STD = 0.869 GL by applying the proposal method (**Figure 13(b)**). Then we say that the averaging process that we propose also improves the amplitude image.

In order to show that the lateral resolution is not affected by the averaging method that here is proposed, we have plotted a line profile marked by the white lines in **Figure 11**. **Figure 12**

**Figure 11.** Reconstructed focused amplitude images. (a) Reconstructed focused amplitude image without averaging and (b) reconstructed amplitude image when the averaging process is performed with four focused amplitude images.

Microtopography and Thickness Measurement with Digital Holographic Microscopy Highlighting and Its... http://dx.doi.org/10.5772/66750 169

**Figure 12.** Comparison between profiles measured along the white lines defined in **Figure 11(a)** and **(b)**.

shows this plot and the comparison between the focused amplitude image without averaging and the improved image after applying the proposed method.

We can note, from this comparison in **Figure 12**, that there is no difference in the transition edges. On the other hand, we can clearly note the improvement on top and bottom areas from these profiles where the STD of the proposal clearly is the lowest.

#### *3.2.2. Decrease of shot noise in phase images*

**Figure 10.** Determination of the DOF experimentally. (a) Recorded hologram and the profile zone taken to measure the

reconstruction, DOF is higher than would have been expected from Eq. (9). This is because the fact that the spatial resolution (pixel size) introduced by the optical setup is limited [13]. To experimentally attain the DOF, in **Figure 10(b)**, we have plotted a line profile shown in **Figure 10(a)**, where the profile is marked with a black line. The evolution of this plot starts at 40 μm before the image is focused. After zooming on the focus zone, we can determine that the DOF is 9 μm. The test object we used was an Edmund NBS 1963A resolution card. The zone of interest corresponds to 18 double lines per mm (lpmm). The reconstruction distance was of

If the DOF is 9 μm and Δ*d* is of 2 μm, then we can average four images to carry out what we propose. These images have to be reconstructed at a distance within DOF to guarantee that image quality is not affected. **Figure 11** shows the image improved when the proposed method is applied. This evaluation is through STD in the area defined by the white square in each image. We attain a STD = 1.57 gray levels (GL) for an image focused without averaging (**Figure 11(a)**), and STD = 0.869 GL by applying the proposal method (**Figure 13(b)**). Then we

In order to show that the lateral resolution is not affected by the averaging method that here is proposed, we have plotted a line profile marked by the white lines in **Figure 11**. **Figure 12**

**Figure 11.** Reconstructed focused amplitude images. (a) Reconstructed focused amplitude image without averaging and (b) reconstructed amplitude image when the averaging process is performed with four focused amplitude images.

say that the averaging process that we propose also improves the amplitude image.

DOF and (b) evolution of intensity when reconstruction distance is increased.

168 Holographic Materials and Optical Systems

15 μm.

First, we use a 100 nm step‐wise specimen made at home of TiO2 thin film, with a refrac‐ tion index of 1.82 for a wavelength of 632.8 nm, as a phase‐calibrating gauge. The specimen was made using a Balzer B‐510 vapor deposition machine. To ensure a real and accurate measurement reference, the test object was measured with a Digital Instruments 3100 AFM. **Figure 13(a)** shows the reconstructed phase image of the step‐wise, where the reconstruction distance was of 10 μm. The STD measured in the zone enclosed by the black square is of 3.44 nm. After applying the averaging proposal of four reconstructed phase images at Δ*d =* 2 μm inside DOF, we compute a reduction of about 1.24 nm (corresponding to 0.57° of STD reduc‐ tion). **Figure 13(b)** shows the improved phase image in comparison with the predicted value of the experimental results, where the STD reduction was of 0.35 deg (**Figure 9(b)**). This is because the higher than expected SDT reduction is mainly due to a lower intensity recording than the blank recorded holograms. This lower intensity is due to the glass plates thickness located in the arms of the interferometer (4.7 mm glass of each). There are some other causes that generate noise in the phase image, such as quantum efficiency of CCD and small optical defects in the optical components, Thermal noise of CCD, among others. The noise generated by these causes is also reduced in a percentage. The evaluation zone is marked in **Figure 13(b)** by the white solid square. In **Figure 13(d)**, we present the topography measurement, which was done by AFM. Normally, this measurement needs to correct it by a numerical procedure. This is due to a tilt contribution in one or both lateral directions. In **Figure 13(c)**, we present the zone where the comparing information was taken of the improved phase image by zooming in the white dashed rectangle of **Figure 13(b)**. A profile comparison extracted from the white lines in **Figure 13(c)** and **(d)** is presented in **Figure 14**. These profiles show their topographic

**Figure 13.** Topographic measurement of the TiO2 step‐wise specimen. (a) One reconstructed phase image, (b) improved phase image when the averaging process is performed, (c) corresponding zoomed area inside the white dashed rectangle in (b), and (d) numerical data extracted from AFM.

**Figure 14.** Comparison between profiles measured along the white lines defined in **Figure 13(c)** and **(d)**.

heights. The biggest difference is mainly located in the transition zones. This difference is due to differences in the lateral resolution between microscopes. It can also be improved in the upper and lower areas with topographic measured while the AFM (blue solid line), due to what has been discussed above. DHM must be reached in larger lateral resolution by using an MO with high NA., this makes our proposal is a comparable alternative method with AFM [21]. Furthermore, the method has some advantages DHM against AFM. One of them is the time to obtain topographic data, because while our proposal takes just seconds to perform the Microtopography and Thickness Measurement with Digital Holographic Microscopy Highlighting and Its... http://dx.doi.org/10.5772/66750 171

**Figure 15.** 3D TiO2 step images. (a) Topographic measurement done by AFM and (b) topographic measurement enhanced through the averaging process done by DHM.

topography, AFM measurement takes several minutes to complete this task. Another advan‐ tage is the flexibility to test an area larger than the AFM method. A last important advantage of our proposal is that it is cheaper in their application that the AFM.

Finally, **Figure 15(a)** and **(b)** present the 3D topographic map of the TiO2 step presented in **Figure 13(d)** and **(c)**, respectively. **Figure 15(a)** corresponds to the data provided by the AFM. The smaller sampling rate commented above has a better ability to detect defects in the sam‐ ple. **Figure 15(b)** corresponds to an improved topographic data obtained by DHM.

#### **3.3. Conclusion**

**Figure 13.** Topographic measurement of the TiO2

170 Holographic Materials and Optical Systems

in (b), and (d) numerical data extracted from AFM.

step‐wise specimen. (a) One reconstructed phase image, (b) improved

phase image when the averaging process is performed, (c) corresponding zoomed area inside the white dashed rectangle

heights. The biggest difference is mainly located in the transition zones. This difference is due to differences in the lateral resolution between microscopes. It can also be improved in the upper and lower areas with topographic measured while the AFM (blue solid line), due to what has been discussed above. DHM must be reached in larger lateral resolution by using an MO with high NA., this makes our proposal is a comparable alternative method with AFM [21]. Furthermore, the method has some advantages DHM against AFM. One of them is the time to obtain topographic data, because while our proposal takes just seconds to perform the

**Figure 14.** Comparison between profiles measured along the white lines defined in **Figure 13(c)** and **(d)**.

In DHM the phase information has great importance for the analysis and characterization of mate‐ rials, such as biological samples and microoptical systems. In this study we have shown a different way to get an improved topographic measurement. The proposal is based on the decrease of the shot noise in DHM. In this section, we show a proposal that is based on the averaging process of reconstructed images by the tomographic capacity of DHM within the range determined by the focus depth. We obtained an improved phase image without quality diminution, in which a noise reduction of 50% was achieved. In addition, we have been shown axial topographic measure‐ ments in agreement with the measurements made with a standard AFM.
