**5. Simulation of coherent speckle on phase and intensity**

Due to the use of coherent optical sources, the recorded holograms and reconstructed fields contain coherent speckle patterns, as seen in the inset in **Figure 23(a)**. Speckle is produced by the coherent interference of a set of wavefronts. Mutual interference occurs when coherence is lost, where coherence is defined as the wavefront having constant phase at each frequency. A well-known mechanism for incoherence is optical roughness; when illuminated with monochromatic light the reflected (or scattered) wave consist of the contribution from many scattering points. Different scattering areas or small highlights on the object emit spherical wavelets which combine and interfere coherently resulting in a complex interference pattern known as speckle (Ref. [48–54]). This speckle generation mechanism also applies to transmission (scattering) through an optically rough phase object.

Due to variable phase shifts produced as the wavefront propagates through an optically rough object, the field leaving the object has a corrugated structure of interference. In addition, the presence of an optical diffuser before the object (which consists of small thickness variations) in transmissive configuration, has the same effect as a rough surface in reflective imaging. In this section, we seek to

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

**Figure 23.** *(a) Spatial frequency spectrum of a hologram recorded using an off-axis digital holographic setup. (b) A slice through constant spatial frequency.*

demonstrate an accurate representation of speckle in transmissive imaging through nearly transparent samples, valid for biological imaging applications. To simulate speckle, we consider the complex phasor amplitude *EO r* ! <sup>¼</sup> j j *EO <sup>e</sup> j k*! �*r* ! <sup>þ</sup>*<sup>ϕ</sup> <sup>r</sup>* ! ð Þ given by a plane wave propagating through an object which induces a spatially dependent phase shift given by *ϕ r* ! . A spatially dependent phase shift *<sup>ϕ</sup>rough <sup>r</sup>* ! is introduced to the field at the object plane to account for optical roughness/ diffuser. The optical roughness can then be represented by a combination of phasors at each location given by *Arough r* ! *<sup>e</sup> <sup>j</sup>ϕrough <sup>r</sup>* ! ð Þ, such that the object complex amplitude wave with the inclusion of optical roughness is given by

$$E\_O\left(\overrightarrow{r}\right) = E\_O\left(\overrightarrow{r'}\right) A\_{rough}\left(\overrightarrow{r'}\right) e^{j\phi\_{mph}\left(\overrightarrow{r'}\right)} = |E\_O| A\_{rough}\left(\overrightarrow{r'}\right) e^{j\left[\left(\overrightarrow{k}\cdot\overrightarrow{r'}\right) + \phi\left(\overrightarrow{r}\right) + \phi\_{mph}\left(\overrightarrow{r}\right)\right]},\tag{30}$$

where the total phase is the sum of the phase derived from the height profile *ϕ r* ! and the phase introduced by optical roughness. The amplitude contribution of speckle *Arough r* ! is computed by integrating the absorption coefficient of the material along the optical path length of the roughness. We assume that the phase contribution from each phasor are statistically independent as well as statistically independent from all other phasors such that the phase induced by each surface patch is uniformly distributed over the interval �*ϕmax*, *ϕmax* ð Þ (Ref. [49]). The maximum phase shift induced by optical roughness, *ϕmax*, is derived from the maximum height deviation of the sample roughness. If the surface is rough relative to the optical wavelength, such that each phasor can produce phase shifts of many 2π multiples, the phase shift induced by each surface patch is uniformly distributed over the interval ð Þ �*π*, *π* (Ref. [49]). The numerical propagation of the complex field then captures the coherent interference of the spherical wavelets emitted from the optically rough surface as the wavefront propagates in space.

As an example, we consider a USAF resolution target with a maximum thickness of 10 microns and random height deviations of 1 micron (10% of total height and 1.6λ for red light) due to roughness, imaged through a telecentric holographic configuration with 3x magnification. **Figure 24** shows probability density functions

**Figure 24.**

*Probability density functions of reconstructed speckle patterns. The real and imaginary components of the complex speckle field (a,b) are Gaussian distributed, the magnitude and intensity (c,d) are Rayleigh and χ*<sup>2</sup> 2 *distributed respectively, and the phase (e) is uniform.*

computed using the phase reconstruction of simulated speckle patterns; the real and imaginary components of the complex speckle field (A,B) are *i.i.d*. Gaussian random variables, such that magnitude and intensity (C,D) are Rayleigh and *χ*<sup>2</sup> 2 (negative exponential) distributed respectively, and the phase (E) is uniform. The validity of the probability density functions in **Figure 24** is well documented in the literature (Ref [48–49]).

In a typical experiment speckle can be reduced using diversity in polarization, space, frequency, or time (Ref. [49]). One of the time domain techniques is through rotating a diffuser or by using a liquid crystal based electronic speckle reducer (Ref. [55]). Another technique is to average multiple holograms or reconstructions recorded by varying the optical path length of the reference beam relative to the object beam (Ref. [56]). **Figure 25** show the reconstructed height profile averaged over increasing phase reconstruction frames, where the initial roughness distributions are assumed to be statistically independent from frame to frame due to the varying optical path length difference between the object and reference beam. **Figure 26** shows the standard deviation of the phase and height profile contribution of simulated speckle as a function of increasing averaging frames. As expected, the standard deviation decreases as 1*=* ffiffiffiffi *N* <sup>p</sup> where N is the number of averaged frames.

In this section, we have demonstrated that the distributions of the simulated speckle phase and intensity are consistent with theory and observations in the limit when the optical roughness is large relative to the optical wavelength. In addition, we have shown that the reduction of speckle standard deviation associated with averaging is as expected. While we have demonstrated an accurate and robust numerical representation of optical speckle patterns in holographic imaging, we do not seek to address speckle mitigation techniques in detail. Our goal is to mimic experimentally recorded and reconstructed holograms for realistic machine learning training not to mitigate speckle, as shown in Section 6 below. In future work we seek to explore the sensitivity of speckle statistics to the roughness of the object relative to the optical wavelength.

**Figure 25.**

*Reconstructed height profile for a telecentric configuration with a magnification of M* ¼ 3 *averaged over 1 (a), 3 (b), 10 (c), and 30 (d) frames.*

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

**Figure 26.** *Standard deviation of speckle phase and corresponding height profile as a function of number of frames averaged.*

## **6. Conclusion**

In this Chapter, we developed the theory, the reconstruction algorithms, and discussed the different experimental configurations for digital holography and digital holographic microscopy. We also showed typical experimental setups for single and multiwavelength configurations. We concluded that single wavelength setups are used for heights that do not exceed few microns while multiwavelength-based setups are used for heights that can reach 100's of microns depending on the synthetic wavelength used. We also discussed in details the two shot versus the one shot MWDH setup. Although hologram reconstruction using one-shot setup needs an extra digital correlation step, it is very well suited for dynamic objects which change relatively quickly. We also discussed briefly how Zernike polynomials are used to cancel the residual phase due to the different aberrations in the optical system. We also discussed the theory and experimental setups of novel reflection as well as transmission telecentric digital holographic microscopy configurations. The setup optically removes, without the need of any post-processing, the parabolic phase distortion caused by the microscope objective which is present in a traditional multi-wavelength digital holographic microscope. Without a telecentric setup and even with post-processing a residual phase remains to perturb the measurement. The telecentric technique has a major advantage since the reconstruction parameters needed and hard to obtain in a standard DHM do not need to be measured precisely to obtain the 3D phase information. Finally, a custom developed userfriendly GUI was employed to automate the recording and reconstruction process.

*Augmented Reality and Its Application*
