**Abstract**

In this Chapter, we discuss the latest advances in **d**igital **h**olography (DH) and **d**igital **h**olographic **m**icroscopy (DHM). Specifically, we study the different setup configurations such as single and multiwavelength approaches in reflection and transmission modes and the reconstruction algorithms used. We also propose two novel **t**elecentric recording configurations for single and **m**ulti-**w**avelength **d**igital **h**olographic **m**icroscopy (TMW-DHM) systems. Brief theory and results are shown for each of the experimental setups discussed. The advantages and disadvantages of the different configurations will be studied in details. Typical configuration features are, ease of phase reconstruction, speed, vertical measurement range without phase ambiguity, difficulty in applying optical and numerical post-processing aberration compensation methods. Aberrations can be due to: (a) misalignment, (b) multiwavelength method resulting in Chromatic aberrations, (c) the MO resulting in parabolic phase curvature, (d) the angle of the reference beam resulting in linear phase distortions, and (e) different optical components used in the setup, such as spherical aberration, astigmatism, coma, and distortion. We conclude that telecentric configuration eliminates the need of extensive digital automatic aberration compensation or the need for a second hologram's phase to be used to obtain the object phase map through subtraction. We also conclude that without a telecentric setup and even with post-processing a residual phase remains to perturb the measurement. Finally, a custom developed user-friendly graphical user interface (GUI) software is employed to automate the reconstruction processes for all configurations.

**Keywords:** digital holography, multi-wavelength digital holography

### **1. Introduction to digital holography (DH)**

Digital holograms are generated by recording the interference pattern of two mutually coherent beams. These two beams are the object beam and the reference beam and the recording medium is usually a CCD [1]. The digital hologram recorded on the CCD due to the interference of the object beam *EO* and the reference beam *ER* is given by

$$h(\mathbf{x}, \mathbf{y}) \propto I\_H = \left| E\_R \right|^2 + \left| E\_O \right|^2 + E\_R^\* E\_O + E\_O^\* E\_R,\tag{1}$$

where the \* notation denotes the complex conjugate. Traditionally, in analog holography the reconstruction is performed by illuminating a holographic film by the conjugate of the reference beam *E*<sup>∗</sup> *<sup>R</sup>* , and the real image is obtained from the last term of Eq. (1): j j *ER* 2 *E*∗ *<sup>O</sup> :* The first two terms on the right hand side and the third term contribute to the zero order and the virtual image, respectively. The digital reconstruction is generally performed by numerically propagating the field *E*∗ *<sup>R</sup> h x*ð Þ , *y* by the recording distance, *d* or –*d*, to reconstruct either the real or virtual images. A typical schematic of the recording and reconstruction of DHs is shown in **Figure 1**. Several numerical reconstruction algorithms have been developed for DH, although the most common are the discrete Fresnel transform, the convolution approach, and reconstruction by angular spectrum. Each of these reconstruction algorithms will be subsequently briefly described.

#### **1.1 Numerical reconstruction by discrete Fresnel transformation**

The Fresnel Transform is based on the Fresnel approximation to the Huygens-Fresnel diffraction integral, and under the paraxial approximation, i.e., *<sup>d</sup>*<sup>3</sup> > >ð Þ <sup>2</sup>*π=<sup>λ</sup>* ð Þ *<sup>ξ</sup>* � *<sup>x</sup>* <sup>2</sup> <sup>þ</sup> ð Þ *<sup>η</sup>* � *<sup>y</sup>* <sup>2</sup> h i, the reconstruction of the hologram can be approximated by the Fresnel transformation [1–6]:

$$\Gamma(\xi,\eta) = z(\xi,\eta)\mathfrak{X}\_{\mathbf{x},\eta} \left[ h(\mathbf{x},\mathbf{y}) E\_{\mathbb{R}}^\*(\mathbf{x},\mathbf{y}) w(\mathbf{x},\mathbf{y}) \right] \big|\_{k\_{\mathbf{x}} = 2\pi\xi/\operatorname{id}, k\_{\mathbf{y}} = 2\pi\eta/\operatorname{id}} \tag{2}$$

$$w(\mathbf{x}, \boldsymbol{\uprho}) = \exp\left[-j\frac{\pi}{\lambda d}(\mathbf{x}^2 + \boldsymbol{\uprho}^2)\right],\tag{3}$$

$$z(\xi,\eta) = \frac{j}{\lambda d} \exp\left(-j\frac{2\pi d}{\lambda}\right) \exp\left[-j\pi\left(\xi^2 + \eta^2\right)/\lambda d\right],\tag{4}$$

where ℑ*<sup>x</sup>*,*<sup>y</sup>*f g• is the Fourier transform operator. The intensity is calculated by squaring the optical field, i.e., *I*ð Þ¼ *ξ*, *η* j j Γð Þ *ξ*, *η* <sup>2</sup> and the phase is calculated using *φ ξ*ð Þ¼ , *η* arctan Imð Þ ½ � Γð Þ *ξ*, *η =* Re ½ � Γð Þ *ξ*, *η :* Since *x*, *y* are discretized on a CCD rectangular raster of *Nx*, *Ny* pixels of sizes Δ*x*, Δ*y*, the reconstructed image resolution in the *ξ*, *η* coordinates are given by [5–7].

$$
\Delta \xi = \lambda d / \mathcal{N}\_{\mathbf{x}} \Delta \mathfrak{x}, \\
\Delta \eta = \lambda d / \mathcal{N}\_{\mathfrak{y}} \Delta \mathfrak{y}. \tag{5}
$$

The image resolution given by Eq. (3) is considered to be "naturally scaled," such that the value of Δ*ξ* is automatically equal to the physical resolution limit imposed by the CCD sampled signal bandwidth [2, 6].

**Figure 1.** *Coordinate system for DH recording and reconstruction.* *Latest Advances in Single and Multiwavelength Digital Holography and Holographic Microscopy DOI: http://dx.doi.org/10.5772/intechopen.94382*

**Figure 2.**

*Schematic of a DH setup (a) Mach-Zehnder setup, (b) Michelson setup. MO-SF: microscope objective-spatial filter, BS: beam splitter, NF: neutral density filter, M: mirror, CL: collimating Lens.*

#### **Figure 3.**

*The recorded hologram in (a) is reconstructed via Eq. (2) to yield (b) the reconstructed image. Note that (b) contains an in-focus (virtual) image on the right, and an out-of-focus (real) image on the left. The relevant reconstruction parameters are* d *= 39 cm,* λ *= 496.5 nm,* Δ*x = 6.7 μm,* N *= 1024, with reconstructed image resolution* Δ*ξ* ¼ *28.5 μm.*

A reflection type Fresnel DH setup based upon the Mach-Zehnder interferometer is schematically shown in **Figure 2(a)**. Light from a Laser source is divided into two parts with a beam splitter. One of the beams forms the reference, while the other is reflected off the object, then both interfere on a CCD camera to form a Fresnel hologram. **Figure 2(b)** shows a Michelson type setup [8, 9]. An example of a recorded hologram of a Newport Logo recorded using an Argon laser @ 496.5 nm and its reconstruction using Fresnel transform method are shown in **Figure 3(a)** and **(b)**, respectively.

#### **1.2 Numerical reconstruction by the convolution approach**

Since the diffracted field at a distance *z* = *d* from the hologram can be expressed as

$$\Gamma(\xi,\eta) = \int\_{-\infty}^{\infty} \int\_{-\infty}^{\infty} h(\varkappa,\jmath) E\_R^\*(\varkappa,\jmath) \mathbf{g}\_{PSF}(\varkappa-\xi,\jmath-\eta) \,d\varkappa d\jmath,\tag{6}$$

the convolution approach can be written as:

$$\Gamma(\xi,\eta) = \left[h(\xi,\eta)E\_R^\*\left(\xi,\eta\right)\right] \* \mathbf{g}\_{\rm PSF}(\xi,\eta), \; \mathbf{g}\_{\rm PSF}(\xi,\eta) = \frac{j}{\lambda} \frac{\exp\left(-jk\_0\sqrt{d^2 + \xi^2 + \eta^2}\right)}{\sqrt{d^2 + \xi^2 + \eta^2}}\tag{7}$$

where the ∗ denotes convolution. Eq. (5) can be written as

$$\Gamma(\xi,\eta) = \mathfrak{S}\_{\mathbf{x},\mathbf{y}}^{-1} \{ \mathfrak{S}\_{\mathbf{x},\mathbf{y}} \left( \hbar \cdot E\_{\mathbf{R}}^{\*} \right) \cdot \mathfrak{S}\_{\mathbf{x},\mathbf{y}} \left( \mathfrak{g}\_{\mathrm{PSF}} \right) \} \equiv \mathfrak{S}\_{\mathbf{x},\mathbf{y}}^{-1} \{ \mathfrak{S}\_{\mathbf{x},\mathbf{y}} \left( \hbar \cdot E\_{\mathbf{R}}^{\*} \right) \cdot \left( G\_{\mathrm{PSF}} \right) \},\tag{8}$$

where *GPSF* <sup>¼</sup> <sup>ℑ</sup>*x*,*<sup>y</sup> gPSF* � �. Although the pixel sizes of the images reconstructed by the convolution approach are equal to that of the hologram, namely, Δ*ξ* ¼ Δ*x*, Δ*η* ¼ Δ*y*, the physical image resolution remains according to Eq. (3) and is ultimately governed by physical diffraction [5–7].

#### **1.3 Numerical reconstruction by the angular spectrum approach**

In Fourier space and across any plane the various spatial Fourier components of the complex field distribution of a monochromatic wave can be considered as plane waves traveling in different directions away from that plane. The field amplitude at any other point can be calculated by adding the weighted contributions of these plane waves, taking into account of the phase shifts they have undergone during propagation [1]. Similar to the convolution approach above, the angular spectrum approach is based on direct application of the propagation of the angular spectrum


#### **Table 1.**

*Advantages and disadvantages of several digital holography reconstruction techniques.*

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

of the field in the hologram plane. Accordingly, we define the angular spectrum of the field *<sup>h</sup>* � *<sup>E</sup>*<sup>∗</sup> *R* � � at the hologram plane as [1]:

$$\tilde{E}\_{\hbar}(k\_{\xi}, k\_{\eta}) = \mathfrak{T}\_{\mathbf{x}, \mathbf{y}} \left( h \cdot E\_{R}^{\*} \right) = \frac{1}{4\pi^{2}} \int\_{-\infty}^{\infty} \int\_{-\infty}^{\infty} \left( h \cdot E\_{R}^{\*} \right) \exp\left[ j \left( k\_{\xi}\xi + k\_{\eta}\eta \right) \right] d\xi d\eta \tag{9}$$

where *kξ*, *k<sup>η</sup>* are spatial frequency variables corresponding to *ξ*, *η*. After propagating a distance *z*, each plane wave component of the angular spectrum acquires an additional phase factor *e*�*jkzz* where

$$k\_x = \sqrt{{k\_0^2 - k\_\xi^2}^2 - k\_\eta^2}. \tag{10}$$

Therefore, the reconstructed field at a distance *z=d* becomes:

$$\Gamma(\xi,\eta) = \frac{1}{4\pi^2} \int^\infty \int^\omega \bar{E}\_h(k\_\xi, k\_\eta) \exp\left[ -j d\sqrt{k\_0^2 - k\_\xi^2 - k\_\eta^2} \right] \\ \times \exp\left[ -j \left( k\_\xi \xi + k\_\eta \eta \right) \right] dk\_\xi dk\_\eta \\ \tag{11}$$

which is similar to Eq. (5) above.

**Table 1** shows the advantages and disadvantages of the different reconstruction techniques discussed in this section.
