**1. Introduction**

Digital holographic microscopy (DHM) is a very popular noninvasive testing tool due to its optical nature [1–3]. Many applications use DHM and they have been demonstrated, showing its unique focusing capability; among those applications, micro‐electro‐mechanical systems (MEMS) and micro‐opto‐electromechanical systems (MOEMS) analysis demand

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2017 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

higher topographical resolution accuracy [4, 5]. In the inspections of test objects with micro‐ scopes presenting some nondesirable phenomena, such as a limited depth of field, aberra‐ tions due to defects of optical elements inside the arrangement, optical noise, and parasitic interferences by multiple internal reflections, among others. DHM is not an exception. The microscope has a depth of focus (DOF) limited and as in bright‐field microscopy, the areas outside the DOF give out‐of‐focus and blurred amplitude. Ferraro et al. demonstrated that by DH it is possible to obtain an extended focused image (EFI) of a 3D object without any mechanical scanning, as occurs in conventional optical microscopy [6]. As the unique DHM feature of refocusing works normally, which Ferraro et al. and Colomb et al. demonstrated in their results [6, 7], we can put under inspection MEMS and MOEMS with thicknesses larger than microscope objective (MO) depth of focus. One of the most important chal‐ lenges in DHM is the reduction of noise. This is because, the lower noise, the smaller will be the measurement error. Different methods have been applied to reduce noise in digital holography. Kang obtained multiple holograms from different angles of illumination by rotating the object or the illumination source. He obtained an improved image through an averaging process [8], similar to the process applied by Baumbach et al. [2]. On the other hand, Rong et al. [9] varied the polarization angle to get an improved image. Also, Charrière et al. [10] applied a method to reduce the shot noise that consisted of averaging multiple holograms in order to get an improved phase image. The noise is reduced by low partial coherence sources. These are usually used, such as laser diodes or light emitting diodes (LEDs) [11, 12]. A disadvantage is their inability to reconstruct the wavefront object for larger reconstruction distances [13]. Here, we present two methods to reduce measure‐ ment error in DHM and DH. These methods perform an averaging procedure of phase maps reconstructed at different distances. In addition, we show compensation in the topo‐ graphical measurement of a 4.2 μm high microlens attained with classical interferometry. This error is due to the limited depth of focus of a Mirau interferometric objective (MIO). In this case, we extend the DOF of the MIO by using the numerical focusing capability of DHM. The last method is based on this tomographic capability of the DHM to reduce a ringing effect by using an ideal filter in off‐axis digital holography. We use this capability to get an enhanced image, which is obtained from the spatial averaging method between the focused image at plane (*z =* zhd0) and the first Talbot distance plane (*z =* zhd1). This dis‐ tance is determined by the period of the ringing phenomenon. Reductions of 50% of these anomalies are computed in simulation and 30% is obtained experimentally (nearly 2 nm). In addition, the simulation results show that the focusing resolution is related with the filter size.

#### **2. Extending DOF of a Mirau interferometric objective by DHM**

In this section, we present some experimental results to show how the DOF of a MIO is increased. We show a topographical measurement of a 4.2 μm high microlens attained with classical interferometry. A comparison with the proposal shows the existence of an error. This error is due to the limited depth of focus of the MO. We extend the DOF of the MO by using the tomographic capability of DHM.

#### **2.1. Experimental configuration**

higher topographical resolution accuracy [4, 5]. In the inspections of test objects with micro‐ scopes presenting some nondesirable phenomena, such as a limited depth of field, aberra‐ tions due to defects of optical elements inside the arrangement, optical noise, and parasitic interferences by multiple internal reflections, among others. DHM is not an exception. The microscope has a depth of focus (DOF) limited and as in bright‐field microscopy, the areas outside the DOF give out‐of‐focus and blurred amplitude. Ferraro et al. demonstrated that by DH it is possible to obtain an extended focused image (EFI) of a 3D object without any mechanical scanning, as occurs in conventional optical microscopy [6]. As the unique DHM feature of refocusing works normally, which Ferraro et al. and Colomb et al. demonstrated in their results [6, 7], we can put under inspection MEMS and MOEMS with thicknesses larger than microscope objective (MO) depth of focus. One of the most important chal‐ lenges in DHM is the reduction of noise. This is because, the lower noise, the smaller will be the measurement error. Different methods have been applied to reduce noise in digital holography. Kang obtained multiple holograms from different angles of illumination by rotating the object or the illumination source. He obtained an improved image through an averaging process [8], similar to the process applied by Baumbach et al. [2]. On the other hand, Rong et al. [9] varied the polarization angle to get an improved image. Also, Charrière et al. [10] applied a method to reduce the shot noise that consisted of averaging multiple holograms in order to get an improved phase image. The noise is reduced by low partial coherence sources. These are usually used, such as laser diodes or light emitting diodes (LEDs) [11, 12]. A disadvantage is their inability to reconstruct the wavefront object for larger reconstruction distances [13]. Here, we present two methods to reduce measure‐ ment error in DHM and DH. These methods perform an averaging procedure of phase maps reconstructed at different distances. In addition, we show compensation in the topo‐ graphical measurement of a 4.2 μm high microlens attained with classical interferometry. This error is due to the limited depth of focus of a Mirau interferometric objective (MIO). In this case, we extend the DOF of the MIO by using the numerical focusing capability of DHM. The last method is based on this tomographic capability of the DHM to reduce a ringing effect by using an ideal filter in off‐axis digital holography. We use this capability to get an enhanced image, which is obtained from the spatial averaging method between the focused image at plane (*z =* zhd0) and the first Talbot distance plane (*z =* zhd1). This dis‐ tance is determined by the period of the ringing phenomenon. Reductions of 50% of these anomalies are computed in simulation and 30% is obtained experimentally (nearly 2 nm). In addition, the simulation results show that the focusing resolution is related with the

**2. Extending DOF of a Mirau interferometric objective by DHM**

In this section, we present some experimental results to show how the DOF of a MIO is increased. We show a topographical measurement of a 4.2 μm high microlens attained with classical interferometry. A comparison with the proposal shows the existence of an error. This error is due to the limited depth of focus of the MO. We extend the DOF of the MO by using

filter size.

158 Holographic Materials and Optical Systems

the tomographic capability of DHM.

The CCD camera records a hologram *I*(*x, y*) on the recording plane (*x, y*). This hologram is the interference between the reference (*R*) and the object (*O*) waves. The hologram is magnified by the lens of the microscope (L) (**Figure 1**) and it is given by

$$I(\mathbf{x'}, \mathbf{y'}) \ = \left| O(\mathbf{x'}, \mathbf{y'}) \right|^2 + \left| R(\mathbf{x'}, \mathbf{y'}) \right|^2 + O(\mathbf{x'}, \mathbf{y'}) \mathbf{R}^\prime(\mathbf{x'}, \mathbf{y'}) + O^\prime(\mathbf{x'}, \mathbf{y'}) \mathbf{R}(\mathbf{x'}, \mathbf{y'}), \tag{1}$$

where the first two terms are the DC term, and the last ones represent the real and the virtual images respectively, while *\** denotes the complex conjugated.

#### **2.2. Reconstruction of the hologram**

Some considerations we need to clarify for the proposal. We must avoid the use of object specimen of low reflectivity because the reference wave cannot be attenuated. The last con‐ sideration we have to care about is the dark zone due to obstruction of the reference mirror in the MIO [14].

The most common configurations to obtain *I(x, y)* in DHM are from either on‐line or off‐axis. In both cases, we need to tilt the sample (OS) modifying the interference angle between (*R*) and (*O*)*.* In the case when this angle is ~0°, the configuration is in‐line. This alignment is more accu‐ rate than its off‐axis counterpart but in the classical way, one needs more images recorded by the CCD camera. We eliminate DC term and the virtual image of Eq. (1), by applying the phase‐ shifting technique. The filtered hologram *I* F (*x, y*) and the wavefront of the object *U*(*x, y*), respec‐ tively, calculated from four *π/2* phase‐shifted images *I* 1 –*I* 4 with the four frame algorithm [15]:

$$I\_p(\mathbf{x}, y) \quad \text{= OR } \exp\left[i(\phi\_A + \phi\_o)(\mathbf{x}, y)\right],\tag{2}$$

$$\mathcal{L}(\mathbf{x}, y) = I\_r(\mathbf{x}, y) \left( \mathbb{R}^\* \, \exp \left[ -i \, \phi\_A(\mathbf{x}, y) \right] \right), \tag{3}$$

where *φ*A is a distribution of system aberrations. Once that remaining aberration of *U*(*x, y*) is compensated for, the numerical object wavefront is reconstructed.

**Figure 1.** Scheme of the MIO which shows the optical path. L, lens of the microscope; STM, beam splitter mirror; OS, object; CG, compensating glass; RM, reference mirror.

#### *2.2.1. Numerical reconstruction of the wavefront*

The wavefront of object *U*(*x, y*) is propagated numerically along the optical axis to the image plane (*ξ, η*), defined by the numerical distance *d*, according to the angular spectrum method [16].

$$\mathcal{U}(\xi,\eta) = \mathfrak{F}^{-1}\left\{ \exp\left[ikd\left(1-\alpha\lambda-\beta\lambda\right)^{1/2}\right] \left[\mathfrak{F}\cdot\mathcal{U}\_{\mathfrak{d}}(\mathbf{x},y)\right]\_{\left(\mathbf{a},\emptyset\right)}\right\}\_{\left(\xi,\eta\right)}\tag{4}$$

where (*ξ, η*) are the spatial variables, (α,β) are the spatial frequencies, and ℑ denotes a two‐ dimensional continuous Fourier transformation.

The discrete form of Eq. (4) is written as

$$\mathcal{U}(m\triangle\xi, n\Delta\eta) = FF \, T^{-1}\left\{ \exp\left[ikd(1-\lambda\,r^2-\lambda\,s^2)\right]FFT\left[\mathcal{U}\_0(k,l)\right]\_{\left(\nu,s\right)}\right\}\_{\left(m,s\right)}\tag{5}$$

where FFT is the fast Fourier transform operator, *Δξ* and *Δη* are the sampling intervals at the observation plane (pixel size), and *r*, *s*, *m*, and *n* are integers (−*N*/2 ≤ *m*, *n* ≤ *N*/2).

The reconstructed wavefront *U* (*ξ, η*) = *O* (*ξ, η*) exp[*iφ*<sup>o</sup> (*ξ, η*)] provides the amplitude *O*(*ξ, η*) and phase images *φ*<sup>o</sup> (*ξ, η*) of the object. From the reconstructed phase distribution *φ*<sup>o</sup> (*ξ, η*), the specimen topography is determined for the reflection configuration *T*(*ξ, η*)*= φ*<sup>o</sup> (*ξ, η*)/2*k*.

#### **2.3. Experimental results**

In **Figure 2** is presented the schematic of the digital holographic Mirau microscope (DHMM). The proposed method was carried out using a He‐Ne laser of λ = 633 nm in wavelength. The beam is filtered spatially with a spatial filter (SP). The beam goes through a Nikon 50X MIO with a numeric aperture, NA = 0.55. The hologram of sample (S) is imaged on the CCD camera plane by the tube lens (TL). The intensity hologram is recorded by a Pixelink ™ digital camera of CMOS 1280 × 1024 pixels, 8 bits, with a pixel size of 6.7 μm × 6.7 μm. The sample holder is attached with a piezoelectric transducer (PZT) to perform the phase‐shifting technique. In addition, this sample holder is attached on an *x, y, z* displacement and θ rotation stage to per‐ form the sample tilt, which is necessary for the off‐axis configuration recording.

**Figure 2.** Schematic diagram of the digital holographic Mirau microscope (DHMM).

Now we present one real application of the usefulness of the tomographic capability of DHM. Typically, the MIO is used as a white light scanning and surface profiler in interfer‐ ometry [14, 17]. In this section, we use this MIO with the DHM and compare the results with interferometry results.

*2.2.1. Numerical reconstruction of the wavefront*

dimensional continuous Fourier transformation.

The reconstructed wavefront *U* (*ξ, η*) = *O* (*ξ, η*) exp[*iφ*<sup>o</sup>

The discrete form of Eq. (4) is written as

*U*(*mΔξ*, *nΔη*) = *FF T*<sup>−</sup><sup>1</sup>

and phase images *φ*<sup>o</sup>

**2.3. Experimental results**

*U*(*ξ*, *η*) = ℑ<sup>−</sup><sup>1</sup>

160 Holographic Materials and Optical Systems

The wavefront of object *U*(*x, y*) is propagated numerically along the optical axis to the image plane (*ξ, η*), defined by the numerical distance *d*, according to the angular spectrum method [16].

where (*ξ, η*) are the spatial variables, (α,β) are the spatial frequencies, and ℑ denotes a two‐

{exp[*ikd*(1 − *λ r* <sup>2</sup> − *λ s* <sup>2</sup>

where FFT is the fast Fourier transform operator, *Δξ* and *Δη* are the sampling intervals at the

In **Figure 2** is presented the schematic of the digital holographic Mirau microscope (DHMM). The proposed method was carried out using a He‐Ne laser of λ = 633 nm in wavelength. The beam is filtered spatially with a spatial filter (SP). The beam goes through a Nikon 50X MIO with a numeric aperture, NA = 0.55. The hologram of sample (S) is imaged on the CCD camera plane by the tube lens (TL). The intensity hologram is recorded by a Pixelink ™ digital camera of CMOS 1280 × 1024 pixels, 8 bits, with a pixel size of 6.7 μm × 6.7 μm. The sample holder is attached with a piezoelectric transducer (PZT) to perform the phase‐shifting technique. In addition, this sample holder is attached on an *x, y, z* displacement and θ rotation stage to per‐

(*ξ, η*) of the object. From the reconstructed phase distribution *φ*<sup>o</sup>

observation plane (pixel size), and *r*, *s*, *m*, and *n* are integers (−*N*/2 ≤ *m*, *n* ≤ *N*/2).

the specimen topography is determined for the reflection configuration *T*(*ξ, η*)*= φ*<sup>o</sup>

form the sample tilt, which is necessary for the off‐axis configuration recording.

**Figure 2.** Schematic diagram of the digital holographic Mirau microscope (DHMM).

][ℑ *U*<sup>0</sup>

)]*FFT* [*U*<sup>0</sup>

(*x*, *<sup>y</sup>*)](*α*,*β*)}(*ξ*,*η*)

(*k*, *<sup>l</sup>*)](*r*,*s*)}(*m*,*n*)

(*ξ, η*)] provides the amplitude *O*(*ξ, η*)

, (4)

, (5)

(*ξ, η*),

(*ξ, η*)/2*k*.

{exp[*ikd* (1 − *αλ* − *βλ*)1/2

As the tomographic feature of DHM works normally, which was demonstrated in [18], we can put under inspection MEMS and MOEMS with thicknesses larger than MO depth of focus (DOF), as Ferraro et al. and Colomb et al. have demonstrated in their results [6, 7], where this DOF is defined by DOF *=* λ*n*m*/NA*<sup>2</sup> , where *n*m is the refractive index of the medium. We record four shifted holograms of a microlens of 100 μm in central diameter and 4.2 μm in height. **Figure 3(a)** shows a shifted hologram. The reconstruction distance at the top of

**Figure 3.** DHM using a MIO applied to a microlens topographic measurement. (a) One of the four‐recorded holograms, (b) the reconstructed amplitude image focusing the bottom zone of the object, (c) the reconstructed amplitude image focusing the whole the object, (d) unwrapped phase map applying the phase shifting interferometry method, (e) unwrapped phase map of (c), (f) comparison between profiles measured along the black dash line in both methods, interferometry (d) and DHM (e).

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

#### **2.4. Conclusion**

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 focus of the MO.
