**4. Theoretical background of telecentric systems**

In a conventional lens systems the magnification changes with object position change, the image has distortion, perspective errors, image resolution loss along the field depth, and edge position uncertainty due to object border lighting geometry. However, a telecentric system, such as the one shown in **Figure 19**, provides nearly constant magnification, virtually eliminates perspective angle error (Object with large depth will not appear tilted), and eliminates radial and tangential distortion. In a bitelecentric system, both the entrance pupil (EP) and exit pupil (XP) are located at infinity. Given that double telecentric systems are afocal, shifting either the image or object does not affect magnification.

As shown in Sections 2 and 3, traditional DHM systems record a digital hologram using a MO. The object phase recovered from digital reconstruction using the Fresnel transform suffers from a parabolic phase factor introduced by the MO. The phase of the MO is superposed over the object phase, often obscuring it. Also, the phase tilt introduced by the reference beam results in linear fringes with high frequency that also obscure the real phase of the object. Numerical techniques as well as optical configurations are usually employed to compensate for both the parabolic phase curvature and the phase tilt. One well-known technique discussed in Section 2 is based on phase mask during reconstruction, which requires knowledge of the setup parameters [13, 14, 40–42]. If the object parameters are unknown a two-step method is used, in which the hologram of a flat reference surface is initially recorded, and upon reconstruction it is subtracted from that of the hologram of the real object [43]. In this section, we adopt two telecentric configurations in reflection and transmission modes to remove optically, instead of numerically, the phase curvature due to MO [44–46]. This telecentric setup can be used in a single wavelength or multiwavelength DHM configurations. It is worth noting that while operating in the nontelecentric mode, *a posteriori* numerical methods will not eliminate the phase aberration completely, as it depends on sample location in the field of view (FOV) [45].

In traditional DHM, the recorded wavefront on the CCD includes the interference of the reference wavefront and the total object wavefront. The total object phase consists of the defocused object phase on the image plane as well as the

**Figure 19.** *A double telecentric system.*

spherical (quadratic in paraxial approximation) phase due to propagation of the object wave from the image plane to the CCD. The object phase is expressed as [11, 46]:

$$
\rho(\mathbf{x}, \mathbf{y}) = \frac{jk}{2R} \left( \mathbf{x}^2 + \mathbf{y}^2 \right) + \rho\_{ab}(\mathbf{x}, \mathbf{y}), \tag{28}
$$

where *R* is the radius of curvature of the spherical curvature.

Typical multiwavelength DHM setups using telecentric configurations in reflection and transmission modes are shown in **Figure 20(a)** and **(b)**, respectively. In each setup, the telecentric system is formed by employing two achromatic lenses and an aperture stop similar to **Figure 19**. The achromatic lenses are crucial to eliminate achromatic aberration due to the use of multi-wavelength illumination. The telecentric system is set in an afocal configuration, where the back focal plane of L1 coincides with the front focal plane of L2 *f* <sup>1</sup> � *f* <sup>2</sup> , with the object placed at the front focal plane of L1, resulting in the cancelation of the spherical phase curvature normally present in traditional DHM systems.

Hence, the 3D amplitude distribution in the image space will be a scaled defocused replica of the 3D amplitude distribution of the object space due to the

**Figure 20.** *Schematics of the TMWDHM setups: (a) reflection and (b) transmission configurations.*

*Latest Advances in Single and Multiwavelength Digital Holography and Holographic Microscopy DOI: http://dx.doi.org/10.5772/intechopen.94382*

convolution with the PSF of the lens system. For each wavelength ð Þ *λ*1, *λ*<sup>2</sup> , the object wave recorded by the CCD can be expressed as [45]:

$$O(\mathbf{x}, \boldsymbol{y}) = -\frac{1}{M} \exp\left[jk\_{1,2}(2f\_2 + f\_1)\right] \times \left[O'\left(\frac{\mathbf{x}}{M}, \frac{\mathbf{y}}{M}\right) \* \tilde{P}\left(\frac{\mathbf{x}}{\lambda\_{1,2}f\_2}, \frac{\mathbf{y}}{\lambda\_{1,2}f\_2}\right)\right],\tag{29}$$

where, *O*<sup>0</sup> ∗ *P*~ is the convolution of the complex amplitude scattered by the object and the PSF, (\*) is the convolution operator, and the magnification is *M* ¼ �*f* <sup>2</sup>*= f* <sup>1</sup> [47].

### **4.1 Experimental results for single wave telecentric DHM (TDHM)**

**Figure 21** shows a custom developed user-friendly graphical user interface (GUI) for the single wave reflection Telecentric DHM (TDHM) setup similar to that shown in **Figure 20(a)**. The target object is shown in **Figure 16(b)**.

The MATLAB GUI is connected to a Lumenra LU120M CCD camera using a USB cable. The GUI is equipped with all the parameters needed to adapt to different CCD camera pixel size, laser wavelength, reconstruction distance, reflection vs. transmission mode. In this example, the laser wavelengths used is *λ* ¼ 488*nm* the CCD pixel size is 5.2 μm, the reconstruction distance is *d* = 20.2 cm. The reconstructed height is around 120 nm. It's worth noting that slight aberrations due to the optical components exist in the final computed phase. This can be automatically corrected by subtracting the reconstructed phase shape from the background phase using Zernike polynomial approximation of the residual phase as shown in the GUI.

The telecentric technique has a lot of advantages compared to a standard DHM system since the reconstruction parameters in a standard DHM are hard to obtain and need to be measured precisely to obtain the 3D phase information.

### **4.2 Experimental results for telecentric multi-wavelength DHM (TMWDHM)**

**Figure 22** shows the GUI for the reflection configuration shown in **Figure 20(a)**. The target in this experiment is a transmissive object (PMMA) on a reflective Si background (See element in blue circle in **Figure 16(a)**). The laser wavelengths used

#### **Figure 21.**

*A custom-designed GUI showing the TDHM in reflection configuration. The object is a reflective object on a reflective substrate (see Figure 16(b)).*

#### **Figure 22.**

*A custom-designed GUI showing the TMWDHM in reflective configuration. The object is a transmissive object on a reflective substrate (see Figure 16(a)).*

are *λ*<sup>1</sup> ¼ 514*:*5*nm*, *λ*<sup>1</sup> ¼ 488*nm:* The synthetic wavelength is: Λ ¼ 9*:*6*μm* and the CCD pixel size is: 5.2 μm. The reconstruction distance is: *d* = 20.2 cm. It's worth noting that a slight misalignment and/or achromatic aberration may result in one residual fringe to remain in the final computed phase. Although achromatic lenses were used, there still might be some remaining chromatic aberration, since the achromats are not perfect. That might be enough to cause the one remaining fringe of phase curvature. This can be automatically corrected by subtracting a the reonstructed phase shape from the background phase using Zernike polynomial approximation of the residual phase as shown in the GUI.
