**4. A method to reduce the ringing effect in phase imaging in off‐axis digital holograms**

In this section, we present a method to reduce the ringing effect of discontinuous surfaces in the reconstruction process in off‐axis digital holography. The method is based on the natural diffraction of light (Talbot effect). We previously showed that for variable grating, Talbot phe‐ nomenon is also present [22]. When you use a binary filter in order to attain the object‐wave in off‐axis digital holography, this allows an easy implementation of filtering mask in the filter‐ ing process. By using the binary filter the appearance of Gibbs phenomenon in discontinuous surfaces appear. However, such a phenomenon was possible to reduce (experimentally nearly to 2 nm) by using the unique feature that digital holography have, this is the tomographic capacity. In addition we show that the size of the binary low‐pass filter in the holographic reconstruction process is related to the focus zone. The versatility by using binary low‐pass fil‐ ter allows us to fit size according to the sample under study. It is possible with the tomographic capability chose the interest zone in axial direction to inspect the sample. This allows us while applying the low‐pass filtering process to avoid the defects that can occur on either the optical component or the sample container. The results should be of interest to readers in the areas of optical metrology, grating diffraction, digital holography, and digital holographic microscopy.

#### **4.1. Proposal of the method**

The optical setup used in the present study for recording off‐axis digital holograms is a Michelson interferometer presented in **Figure 16**. The light source is from a laser diode with a wavelength of 643 nm, which is expanded by a beam expander system (BE). This source is linearly polarized plane wavefront with short coherence (coherence length about 0.1 mm). The beam is split by a beam splitter (BS) into a reference wave *R* and an object wave *O*. The CCD camera records an off‐ axis hologram at the exit of the interferometer. This hologram (*H*) is created by the interference between the *O*(*x,y*) and *R*(*x,y*), after they reflect on the sample (S) and mirror (M), respectively. To obtain off‐axis holograms, the orientation of M is set in such a way that, the CCD is reached by *R*(*x,y*), with an incident angle θ, while *O*(*x,y*) is perpendicularly propagated with respect to the hologram plane. The distance *z* between the CCD and the object is 85 mm. The CCD is a standard black and white camera with pixel size of 4.4 μm and 8 bits of depth.

This off‐axis digital hologram *H*(*x,y*) can be expressed by the Eq. (1). A window function *W*(*f* x *,f*y ) is used to filter the term *O*(*x,y*)*R\**(*x,y*) of Eq. (1) in the frequency domain. On the other hand, the so‐called frequency spectrum filtering method [23] is applied in order to retrieve the object wavefront *O*(*f* x *,f*y ). Here,

**Figure 16.** Optical system based on a Michelson interferometer. M is a mirror, NF is a neutral filter and BE is a beam expander.

Microtopography and Thickness Measurement with Digital Holographic Microscopy Highlighting and Its... http://dx.doi.org/10.5772/66750 173

$$\mathcal{O}(f\_{\mathbf{x'}}f\_{\mathbf{y}}) = \mathcal{W}(f\_{\mathbf{x'}}f\_{\mathbf{y}}) \, \mathfrak{T}^{\pi 1} \left\{ H(\mathbf{x}, \mathbf{y}) \, R\_{\mathbf{z}}(\mathbf{x}, \mathbf{y}) \right\},\tag{12}$$

where *R*D is a digital replica of *R*, and ℑ<sup>∓</sup><sup>1</sup> stands for either the direct continuous Fourier trans‐ form or its inverse counterpart, in other contexts. Hence, the object *O*(*f* x *,f*y ) can be propagated using the approximation of the Fresnel‐Kirchhoff propagation integral as follows [16]:

$$\mathbf{x}\_{\text{test}} = \mathbf{x}\_{\text{TF}},$$

$$O(\mathbf{x}, \mathbf{y}, \mathbf{z}) = \mathfrak{R}^{\*1} \left\{ \begin{bmatrix} O(\mathbf{f}\_{\text{f}} f\_{\text{f}}) \exp(j k z\_{\text{f}}) \\ \times \exp\left[-j \pi \lambda \, z\_{\text{f}}\right] \left[ (f\_{\text{f}})^{2} + (f\_{\text{f}})^{2} \right] \end{bmatrix} \right\},\tag{13}$$

where *λ* is the wavelength, *k* is the wave number and *z*<sup>i</sup> is the reconstruction distance. The reconstructed object wavefront *O*(*x,y*) provides the amplitude image *A*<sup>i</sup> (*x, y*)=|*O*<sup>i</sup> (*x,y*)|2 and the phase image *φ*<sup>i</sup> ( *x,y*)*=tg*‐1 (*imag*(*O*<sup>i</sup> (*x,y*))/*real*(*O*<sup>i</sup> (*x,y*))) of the object. The topography *T*(*x,y*) of the specimen is computed from the reconstructed phase *φ*( *x,y*), by the reflection configura‐ tion *T*(*x,y*) *= φ*( *x,y*)/2*k*.

#### *4.1.1. The Fourier filtering process*

The filtering process is a well‐known technique outlined by Cuche et al. [23]. However in this proposal the *DC* term is eliminated by the subtraction of the intensity of *R* and *O*, recorded independently, in Eq. (1). Edges and other sharp transitions (such as noise) in the digital hologram contribute significantly to the high‐frequency content of its Fourier transform [16]. We considered three types of low‐pass filters: ideal (ILPF), Butterworth (BtwLPF), and Gaussian (GLPF). These three filters cover the range from very sharp (ideal) to very smooth (Gaussian) filter functions. When we use the ILPF, a ringing effect appears at the sharp transitions zone of the reconstructed image due to transfer function *W* in Eq. (12); which is a *sinc* function. In addition, the filter size (cut‐off frequency) is directly related to the period of the ringing [24, 25]. On the other hand, since the transfer function of the Gaussian filter is also a Gaussian function, it will have a blurring behavior, but no ringing effect. When you use the BtwLPF of second order, the ringing effect and the blurring is similar to that observed in the Gaussian filtering. Then the ringing increased, according to the order of the filter as we commented in [22].

#### *4.1.2. The Talbot effect*

**Figure 16.** Optical system based on a Michelson interferometer. M is a mirror, NF is a neutral filter and BE is a beam

surfaces appear. However, such a phenomenon was possible to reduce (experimentally nearly to 2 nm) by using the unique feature that digital holography have, this is the tomographic capacity. In addition we show that the size of the binary low‐pass filter in the holographic reconstruction process is related to the focus zone. The versatility by using binary low‐pass fil‐ ter allows us to fit size according to the sample under study. It is possible with the tomographic capability chose the interest zone in axial direction to inspect the sample. This allows us while applying the low‐pass filtering process to avoid the defects that can occur on either the optical component or the sample container. The results should be of interest to readers in the areas of optical metrology, grating diffraction, digital holography, and digital holographic microscopy.

The optical setup used in the present study for recording off‐axis digital holograms is a Michelson interferometer presented in **Figure 16**. The light source is from a laser diode with a wavelength of 643 nm, which is expanded by a beam expander system (BE). This source is linearly polarized plane wavefront with short coherence (coherence length about 0.1 mm). The beam is split by a beam splitter (BS) into a reference wave *R* and an object wave *O*. The CCD camera records an off‐ axis hologram at the exit of the interferometer. This hologram (*H*) is created by the interference between the *O*(*x,y*) and *R*(*x,y*), after they reflect on the sample (S) and mirror (M), respectively. To obtain off‐axis holograms, the orientation of M is set in such a way that, the CCD is reached by *R*(*x,y*), with an incident angle θ, while *O*(*x,y*) is perpendicularly propagated with respect to the hologram plane. The distance *z* between the CCD and the object is 85 mm. The CCD is a standard

This off‐axis digital hologram *H*(*x,y*) can be expressed by the Eq. (1). A window function

) is used to filter the term *O*(*x,y*)*R\**(*x,y*) of Eq. (1) in the frequency domain. On the other hand, the so‐called frequency spectrum filtering method [23] is applied in order to retrieve the

black and white camera with pixel size of 4.4 μm and 8 bits of depth.

expander.

*W*(*f* x *,f*y

object wavefront *O*(*f*

x *,f*y ). Here,

**4.1. Proposal of the method**

172 Holographic Materials and Optical Systems

When a monochromatic wavefront is plane and illuminates a linear grating of period *p*, mul‐ tiple identical images of the original grating are observed along the propagation axis of the light. These images are formed without any lenses on multiples of the Rayleigh distance (z*tdu*). This phenomenon is known as the Talbot effect or self‐imaging and it is due to the diffraction of light when pass through the grating [26]. The Talbot distance is located at:

 *<sup>z</sup>td<sup>u</sup>* <sup>=</sup> *up*<sup>2</sup> \_\_\_ *<sup>λ</sup>* , (14)

where *u =* 1, 2, 3,… denotes the Talbot plane order (TPO) and λ is the illumination light wave‐ length. When *u* is odd the self‐imaging has 180° phase shift and contrast reversal [16]. The same result can be present in a thin phase attenuated sinusoidal grating as we presented in [22].

#### **4.2. Simulations**

A simulation was performed to get an off‐axis digital hologram. According to the optical scheme (**Figure 16**), two plane waves of equal intensities have been considered to interfere in a Michelson interferometer for attaining off‐axis digital hologram. We design a synthetic object, which consists of three horizontal bars and three vertical bars etched on a thin chromium film (100% reflective) deposited in a glass substrate. The reflectance of the film is of 6.25% and the thickness of 0.7π rad. (**Figure 17(a)** and **(b)**). In real world the reflectivity of 100% of an object is not possibly reached, but the simulation allows us to design synthetic objects with 100% reflectivity. The size of this object is of 1200 × 1200 pixels; the distance between the object and the CCD plane is of *z =* 84.4 mm. We assumed a red wavelength of 643 nm from a laser, and pixel size of 4.4 μm × 4.4 μm of the CCD according to the real parameters. If we consider a wavelength of 643 nm then the sample thickness will be of 112.5 nm by applying the topographic formula that was mentioned at the end of Section 4.1. We have considered the standard deviation (STD) as a measure of the axial resolution and amplitude improve‐ ment (arbitrary unit [a.u.]) in all sections. **Figure 17(c)** present the synthetic off‐axis digital hologram.

We start reconstructing the object wavefront by performing the Eq. (13) with a reconstruction distance (*z = z*td0) of 84.4 mm. To remove the DC term and the virtual image, we perform the Fourier filtering process mentioned in Section 4.1.1. **Figure 18(a)** and **(b)** presents the ampli‐ tude and phase images, respectively, by using an ILPF with a cut‐off frequency of radius of 100 pixels. As we have been mentioned, **Figure 18(a)** shows a profile where the ringing effect appear with a period of *u* = 53 μm*.* This profile is comparable to a variable transmittance func‐ tion (VTF) grating simulated, according to Goodman [16] and confirmed in previous simula‐ tions [22] with the 180° phase shift and contrast reversal of the field distribution at the first TPO (z*td1*) property. Next a second reconstruction was performed at *z = z* + *z*t*d*<sup>1</sup> = 88.7 mm with the same size of the filter. It should be noted that normally the reconstruction of the object wavefront is done only at one distance; this is the focusing distance *z*. However, to reduce the ringing effect we need to perform an averaging operation between phase and amplitude reconstructed images at different distances *z*ht0 and *z*ht1 that is: *<sup>A</sup>*(*x*, *<sup>y</sup>*) <sup>=</sup> [*A*0(*x*, *<sup>y</sup>*) <sup>+</sup> *<sup>A</sup>*1(*x*, *<sup>y</sup>*)] <sup>⁄</sup>2. (15)

$$\begin{aligned} \phi(\mathbf{x}, y) &= \ [\phi\_l(\mathbf{x}, y) \star \phi\_l(\mathbf{x}, y)] / \mathbf{\hat{z}}\\ A(\mathbf{x}, y) &= \ [A\_l(\mathbf{x}, y) \star A\_l(\mathbf{x}, y)] / \mathbf{\hat{z}} \end{aligned} \tag{15}$$

**Figure 17.** (a) Amplitude and (b) phase test object used to perform numerical simulation, and (c) off‐axis digital hologram recorded at *z* = 84.4 mm from the object test.

Microtopography and Thickness Measurement with Digital Holographic Microscopy Highlighting and Its... http://dx.doi.org/10.5772/66750 175

**Figure 18.** Reconstructed images from the hologram of **Figure 3(c)**. (a) and **(b)** Reconstructed amplitude and phase images at *z* = *z*td0 respectively, (c) and (d) improved amplitude and phase images, respectively, after proposal is applied.

From **Figure 18(c)** and **(d)**, we can see the both improved images amplitude and phase map, obtained with this proposal. A comparison between the corresponding images obtained at a single distance of reconstruction *z* and our proposed method is done. We can note a clearly noiseless images reconstructed by our proposal. The improvement with this procedure is an average decrease of STD of 64% in the amplitude image and 47% in phase image. The percent‐ age is an average between the two zones delimited by the dashed rectangles in **Figure 18**. The STD of the region of interest is represented in each reconstructed images. As far as we know this is a new and useful method to reduce the ringing phenomenon.

In **Figure 19(a)** and **(b)**, we show line profiles taken from white‐dashed line shown in **Figures 17(a)**–**(b)** and **18(a)**–**(d)**. Also, we have included profiles from reconstructed images at the first TPO and profiles from reconstructed images when Gaussian and Butterworth filters were used in the filtering process. We can appreciate the periodic property as a result of using an ideal filter not only at *z*ht0 but also at *z*ht1. We can note this behavior in both ampli‐ tude and phase distributions. Additionally, and in agreement with simulations performed in [22], we can observe not only a phase shift of 180° and reversal amplitude attained in the reconstruction at *z*ht1. An unfavorable anomaly, but usual, phase, and amplitude variances are presented in boundaries (edges) due to single diffraction order that does not superpose with other diffraction orders [27]. Due to that, when the averaging is performed between reconstructed images at *z*ht0 and *z*ht1, we cannot compensate the ringing in this zone. On

**Figure 17.** (a) Amplitude and (b) phase test object used to perform numerical simulation, and (c) off‐axis digital hologram

A simulation was performed to get an off‐axis digital hologram. According to the optical scheme (**Figure 16**), two plane waves of equal intensities have been considered to interfere in a Michelson interferometer for attaining off‐axis digital hologram. We design a synthetic object, which consists of three horizontal bars and three vertical bars etched on a thin chromium film (100% reflective) deposited in a glass substrate. The reflectance of the film is of 6.25% and the thickness of 0.7π rad. (**Figure 17(a)** and **(b)**). In real world the reflectivity of 100% of an object is not possibly reached, but the simulation allows us to design synthetic objects with 100% reflectivity. The size of this object is of 1200 × 1200 pixels; the distance between the object and the CCD plane is of *z =* 84.4 mm. We assumed a red wavelength of 643 nm from a laser, and pixel size of 4.4 μm × 4.4 μm of the CCD according to the real parameters. If we consider a wavelength of 643 nm then the sample thickness will be of 112.5 nm by applying the topographic formula that was mentioned at the end of Section 4.1. We have considered the standard deviation (STD) as a measure of the axial resolution and amplitude improve‐ ment (arbitrary unit [a.u.]) in all sections. **Figure 17(c)** present the synthetic off‐axis digital

We start reconstructing the object wavefront by performing the Eq. (13) with a reconstruction distance (*z = z*td0) of 84.4 mm. To remove the DC term and the virtual image, we perform the Fourier filtering process mentioned in Section 4.1.1. **Figure 18(a)** and **(b)** presents the ampli‐ tude and phase images, respectively, by using an ILPF with a cut‐off frequency of radius of 100 pixels. As we have been mentioned, **Figure 18(a)** shows a profile where the ringing effect appear with a period of *u* = 53 μm*.* This profile is comparable to a variable transmittance func‐ tion (VTF) grating simulated, according to Goodman [16] and confirmed in previous simula‐ tions [22] with the 180° phase shift and contrast reversal of the field distribution at the first

the same size of the filter. It should be noted that normally the reconstruction of the object wavefront is done only at one distance; this is the focusing distance *z*. However, to reduce the ringing effect we need to perform an averaging operation between phase and amplitude

*<sup>A</sup>*(*x*, *<sup>y</sup>*) <sup>=</sup> [*A*0(*x*, *<sup>y</sup>*) <sup>+</sup> *<sup>A</sup>*1(*x*, *<sup>y</sup>*)] <sup>⁄</sup>2. (15)

= 88.7 mm with

TPO (z*td1*) property. Next a second reconstruction was performed at *z = z* + *z*t*d*<sup>1</sup>

reconstructed images at different distances *z*ht0 and *z*ht1 that is: *<sup>φ</sup>*(*x*, *<sup>y</sup>*) <sup>=</sup> [*φ*0(*x*, *<sup>y</sup>*) <sup>+</sup> *<sup>φ</sup>*1(*x*, *<sup>y</sup>*)] <sup>⁄</sup><sup>2</sup>

recorded at *z* = 84.4 mm from the object test.

**4.2. Simulations**

174 Holographic Materials and Optical Systems

hologram.

**Figure 19.** Comparison between profiles measured a long white dashed line of **Figures 3(a)**, **(b)** and **4(a)**–**(d)**. **(a)** Profiles of amplitude distributions and (b) profiles of phase distributions.

the other hand, we observe a loss of lateral resolution in reconstructed images (principally in phase distribution) when BtwLPF and GLPF are applied in the filtering process. This is important to know as phase distribution is directly related with thickness and, in our case, sample topography.

Nevertheless we can see a slight difference between proposal and the BtwLPF of second order in the profiles comparison. However, an advantage of ideal low‐pass filter is that have the possibility to increase the tomographic resolution. This property is due to pixel size, magnification, and numerical aperture of the optical imaging system as demonstrated by Dubois et al. [13] and in our case, the filter size. To illustrate the determination of tomo‐ graphic zone (TZ), in **Figure 20(a)** we have plotted the intensity evolution with the recon‐ struction distance on a line profile from the **Figure 18(a)**, where the profile zone is marked with a white‐dashed line. The starting image is defocused by ‐10 to 10 mm. We can see that the TZ is of 3.2 mm by using the ideal filter and 16 mm by using the Butterworth one. Then we say that ideal filter is better to determine a focus zone than Butterworth. This result per‐ mit us not only to determine the best and most accuracy reconstruction distance zone to pre‐ vent measurement errors [28], but also to adjust the tomographic capability with respect to the sample thickness to reduce the defects that can occur on either the optical component or the sample container. Also this capacity helps us to control the resolution of plane scanning in a tomographic scheme [13]. **Figure 20(c)** presents the intensity evolution of the line profile in the zone marked with white‐dashed line as **Figure 20(a)** but with filter size of 200 pixels of radius. This shows a smaller focus zone than a filter size of 100 pixels of radius (**Figure 20(a)**) evidencing the above mentioned.

Microtopography and Thickness Measurement with Digital Holographic Microscopy Highlighting and Its... http://dx.doi.org/10.5772/66750 177

**Figure 20.** Evolution of intensities determined on a profile line when reconstruction distance is increased. Using (a) an ILPF of 100 pixels of radius, (b) a BtwLPF of 100 pixels of radius and (c) an ILPF of 200 pixels of radius.

#### **4.3. Experimental results**

the other hand, we observe a loss of lateral resolution in reconstructed images (principally in phase distribution) when BtwLPF and GLPF are applied in the filtering process. This is important to know as phase distribution is directly related with thickness and, in our case,

**Figure 19.** Comparison between profiles measured a long white dashed line of **Figures 3(a)**, **(b)** and **4(a)**–**(d)**. **(a)** Profiles

Nevertheless we can see a slight difference between proposal and the BtwLPF of second order in the profiles comparison. However, an advantage of ideal low‐pass filter is that have the possibility to increase the tomographic resolution. This property is due to pixel size, magnification, and numerical aperture of the optical imaging system as demonstrated by Dubois et al. [13] and in our case, the filter size. To illustrate the determination of tomo‐ graphic zone (TZ), in **Figure 20(a)** we have plotted the intensity evolution with the recon‐ struction distance on a line profile from the **Figure 18(a)**, where the profile zone is marked with a white‐dashed line. The starting image is defocused by ‐10 to 10 mm. We can see that the TZ is of 3.2 mm by using the ideal filter and 16 mm by using the Butterworth one. Then we say that ideal filter is better to determine a focus zone than Butterworth. This result per‐ mit us not only to determine the best and most accuracy reconstruction distance zone to pre‐ vent measurement errors [28], but also to adjust the tomographic capability with respect to the sample thickness to reduce the defects that can occur on either the optical component or the sample container. Also this capacity helps us to control the resolution of plane scanning in a tomographic scheme [13]. **Figure 20(c)** presents the intensity evolution of the line profile in the zone marked with white‐dashed line as **Figure 20(a)** but with filter size of 200 pixels of radius. This shows a smaller focus zone than a filter size of 100 pixels of radius (**Figure 20(a)**)

sample topography.

176 Holographic Materials and Optical Systems

of amplitude distributions and (b) profiles of phase distributions.

evidencing the above mentioned.

In this section, we present experimental results of the recorded holograms of 1600 × 1200 pixels size. **Figure 15** shows the digital holographic setup that we implement. We use a laser diode of 643 nm in wavelength as light source. This source is a diode of low coherence (about 0.1 mm) linearly polarized plane wavefront to prevent parasitic interference and optical noise. The hologram is recorded by a CCD Pixelink™ digital camera of 1600 × 1200 pixels, 8 bits, with a pixel size of 4.4 μm × 4.4 μm. The sample holder is supporting on an *x*, *y*, *z*, displace‐ ment and θ rotation stage to perform the sample tilt, which is necessary for the off‐axis config‐ uration recording. We used an USAF 1951 resolution test target in the zone that corresponds to the 1‐1 test group as the object test. In **Figure 21**, we show a recorded digital hologram [22].

At the beginning, we reconstructed the object wavefront by performing the Eq. (13) with a reconstruction distance (*z = z*td0 ) of 85 mm. **Figure 22(a)** and **(b)** shows the amplitude and topography distributions, respectively, by using an ILPF with radius of 100 pixels (cut‐off frequency). As ILPF size is the same, as used in simulations results, then a ringing artifact with a period of *u* = 53 μm appear in the reconstructed images. A second reconstruction was

**Figure 21.** Digital experimental hologram of the USAF 1951 resolution test target.

**Figure 22.** Reconstructed images from the hologram of **Figure 7**. (a) and (b) Reconstructed amplitude and topography distributions at *z* = *z*td0 respectively, (c) and (d) improved amplitude and topography respectively after proposal is applied.

performed at *z = z + z*td1 = 89.4 mm. To reduce the ringing effect we perform the averaging proposal between phase and amplitude reconstructed images, as we previously mentioned. In **Figure 22(c)** and **(d)**, we present the enhanced images obtained with this proposal. The improved distributions contain nearly all the details of the original object. The reduction with this procedure is an average of 43.7% of STD in amplitude image and 23.1% in the topogra‐ phy. The percentage is an average between the two zones delimited by the dashed‐white rect‐ angles in **Figure 22**. **Figure 23(a)** and **(b)** presents line profiles where region is marked with a white‐dashed line shown in **Figure 22(a)**–**(d)**. Also, as simulation results, we have included profiles from reconstructed images at the first TPO and profile from reconstructed images when Butterworth filter is used in the filtering process. We can note the periodic component as a result of using an ideal filter at *z*ht0. This behavior is presented in both amplitude and phase distributions in agreement with simulation results. We assume that the percentage dif‐ ference between simulation results and experimental results is principally due to an addi‐ tional low frequency noises coming from optical devices defects, a nonperfect plane wave reference, aberrations noncompensated, and also a nonuniform thin film deposited on target. Nevertheless, we can see a difference between proposal and the BtwLPF of second order in the profiles plot exactly as in simulation performed in [22]. These differences are mainly at the transition edges, which are related to the lateral resolution.

Microtopography and Thickness Measurement with Digital Holographic Microscopy Highlighting and Its... http://dx.doi.org/10.5772/66750 179

**Figure 23.** Comparison between profiles measured a long white dashed line of **Figure 8(a)**–**(d)**. (a) Profiles of amplitude distributions and (b) profiles of topography distributions.

#### **4.4. Conclusions**

performed at *z = z + z*td1 = 89.4 mm. To reduce the ringing effect we perform the averaging proposal between phase and amplitude reconstructed images, as we previously mentioned. In **Figure 22(c)** and **(d)**, we present the enhanced images obtained with this proposal. The improved distributions contain nearly all the details of the original object. The reduction with this procedure is an average of 43.7% of STD in amplitude image and 23.1% in the topogra‐ phy. The percentage is an average between the two zones delimited by the dashed‐white rect‐ angles in **Figure 22**. **Figure 23(a)** and **(b)** presents line profiles where region is marked with a white‐dashed line shown in **Figure 22(a)**–**(d)**. Also, as simulation results, we have included profiles from reconstructed images at the first TPO and profile from reconstructed images when Butterworth filter is used in the filtering process. We can note the periodic component as a result of using an ideal filter at *z*ht0. This behavior is presented in both amplitude and phase distributions in agreement with simulation results. We assume that the percentage dif‐ ference between simulation results and experimental results is principally due to an addi‐ tional low frequency noises coming from optical devices defects, a nonperfect plane wave reference, aberrations noncompensated, and also a nonuniform thin film deposited on target. Nevertheless, we can see a difference between proposal and the BtwLPF of second order in the profiles plot exactly as in simulation performed in [22]. These differences are mainly at the

**Figure 22.** Reconstructed images from the hologram of **Figure 7**. (a) and (b) Reconstructed amplitude and topography distributions at *z* = *z*td0 respectively, (c) and (d) improved amplitude and topography respectively after proposal is

transition edges, which are related to the lateral resolution.

applied.

178 Holographic Materials and Optical Systems

In this work, we present a new method to reduce the ringing effect of discontinuous sur‐ faces reconstruction in off‐axis digital holography. The technique is based on the diffractive nature of light (Talbot effect). We use an ideal filter in the filtering process in digital off‐axis holography, because it allows an easy implementation and versatility to choose frequencies of interest. The major disadvantage in using this filter is the appearance of Gibbs phenomenon in discontinuous surfaces. However, such a phenomenon was possible to reduce by using the unique feature that digital holography have, this is the tomographic capability. Experimental results have proved reductions of these anomalies, 30%. Also, we have demonstrated a better tomographic capacity by using an ideal filter than Butterworth. Numerical simulation evi‐ denced that the Talbot effect can also be present in VTF grating from 0 to 2 TPO. Also, we have shown that the size of the ideal low‐pass filter in the holographic reconstruction process is dependent on focus zone. The results should be of interest to readers in the areas of optical metrology, grating diffraction, digital holography, and digital holographic microscopy.

#### **5. Summary**

DHM give us the possibility to scan biological samples, semitransparent materials and tissues in axial direction with nanometric resolution. This is possible because the technique have a unique characteristic of numerical refocusing or tomographic capacity. In this chapter, we have shown three application of this capacity. First, we extend the DOF of the MIO in order to avoid topographical measurement error of a microlens of a 4.2 μm high. The compactness and easy‐use of the MIO that we have presented not only DHMM as a new alternative to obtain digital holograms without spherical aberration and easy tilt correction in the phase image, but also that an easier, well‐aligned, and insensitive to external vibrations setup is reached, in comparison with the typical setups. Second, we reduce the shot noise in phase and amplitude images coming from digital holograms. This reduction allows us to attain high topographic resolution comparable with an AFM results. Finally, we present a method to reduce noise in off‐axis holograms when a Fourier filtering method is applied.
