**2. Digital holographic microscopy (DHM)**

DHM is usually applied to determine 3D shapes of small objects, with height excursions on the order of microns (or phase excursions on the order of a few radians). Since small objects are involved, a microscope objective (MO) is often used to zoom onto a small area of the object to enhance the transverse resolution. Holograms of microscopic objects recorded with DHM setups can be numerically reconstructed in amplitude and phase using the same DH reconstruction techniques discussed in Section 1. The phase aberrations due to the MO and the tilt from the reference beam have to be corrected to obtain the topographic profile or the phase map of the object [10–12]. **Figure 4(a)** and **(b)** show a Michelson DHM in reflection and transmission configurations, respectively.

For a reflective object on a reflective surface, the height profile on the sample surface is simply proportional to the reconstructed phase distribution *φ ξ*ð Þ , *η* , through [10]:

$$h\_x(\xi, \eta) = \left(\frac{\lambda}{4\pi}\right) \rho(\xi, \eta). \tag{12}$$

For a transmissive phase object on a reflective surface, its thickness can be calculated as:

$$h\_x(\xi, \eta) = \left(\frac{\lambda}{4\pi}\right) \frac{\varrho(\xi, \eta)}{\Delta n},\tag{13}$$

where Δ*n* is the difference of the index of refraction between the transparent object material and the surrounding medium (e.g. air).

**Figure 4.** *Digital holographic microscope: (a) reflective setup, (b) transmissive setup.*

For a transmissive phase object on a transmissive surface or between transmissive surfaces, the phase change (optical thickness) can be calculated as:

$$h\_x(\xi, \eta) = \left(\frac{\lambda}{2\pi}\right) \frac{\rho(\xi, \eta)}{\Delta n}. \tag{14}$$

As stated above, in DHM we introduce a MO to increase the spatial resolution which was computed according to Eq. (3). Due to the magnification 'M' introduced by the MO the pixel size in the image plane, Δ*ξmag* scales according to:

$$
\Delta \xi\_{mag} = \frac{\Delta \xi}{M} = \frac{\lambda d}{N.\Delta \mathbf{x} \cdot M},
\tag{15}
$$

which is simply the magnification predicted by geometric imaging. This is intuitively understood by realizing that the holographic recording is now simply a recording of the geometrically magnified virtual image located at distance *d* as shown in **Figure 5**. Thus, the pixel resolution is automatically scaled accordingly. We can enhance the transverse resolution approximately to be equal to the diffraction limit 0.61*λ*/N.A of the MO, where N.A is the numerical aperture of the MO.

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

**Figure 5.**

*Generalized holographic recording geometry using a lens. The image location is governed by geometric optics and may be on either side of the lens.*

The complete reconstruction algorithm is governed by the equation [9–14]:

$$\begin{split} \Gamma(m,n) &= \underbrace{\mathbf{A}e^{-j\left[\frac{\pi}{M}\left(m^{2}\Delta\_{t}^{2} + n^{2}\Delta\_{l}^{2}\right)\right]}}\_{\text{Quadratic Phase due to MO}}\mathbf{z}(m,n) \\ &\times \mathbf{S}\_{\mathbf{x},\mathbf{y}}\left\{\underbrace{\mathbf{A}\_{R}e^{-j\frac{2\pi}{\lambda}\left(\sin\theta\_{k}k\Delta\mathbf{x} + \sin\theta\_{l}l\Delta\mathbf{y}\right)}}\_{E\_{h}^{\*}}\mathbf{h}(k,l)\boldsymbol{\omega}(k,l)\right\}\_{m,n} \end{split} \tag{16}$$

where *z m*ð Þ¼ , *<sup>n</sup> <sup>j</sup> <sup>λ</sup><sup>d</sup>* exp �*<sup>j</sup>* <sup>2</sup>*π<sup>d</sup> λ* � � exp �*jπλ<sup>d</sup> <sup>m</sup>*<sup>2</sup> *Nx* <sup>2</sup>Δ*x*<sup>2</sup> <sup>þ</sup> *<sup>n</sup>*<sup>2</sup> *Ny* <sup>2</sup>Δ*y*<sup>2</sup> h i � � , *w k*ð Þ¼ , *<sup>l</sup>* exp �*<sup>j</sup> <sup>π</sup> <sup>λ</sup><sup>d</sup> <sup>k</sup>*<sup>2</sup> <sup>Δ</sup>*x*<sup>2</sup> <sup>þ</sup> *<sup>l</sup>* 2 Δ*y*<sup>2</sup> � � � � <sup>1</sup> *<sup>D</sup>* <sup>¼</sup> <sup>1</sup> *di* <sup>1</sup> <sup>þ</sup> *do di* � �, and the focal length of the MO is: <sup>1</sup> *<sup>f</sup>* <sup>¼</sup> <sup>1</sup> *di* <sup>þ</sup> <sup>1</sup> *do* .

Aberration compensation can be performed manually using a phase mask Ψ to cancel the effects of the quadratic phase due to the MO and the linear phase due to the reference tilt. The phase mask can be written as [11].

$$\Psi = A \exp\left\{ \left. j \left[ \frac{2\pi}{\lambda} \left( m \Delta x \sin \theta\_x + n \Delta y \sin \theta\_\gamma \right) \right] \right\} \times \exp\left\{ j \left[ \frac{\pi}{\lambda D} \left( m^2 \Delta x^2 + n^2 \Delta y^2 \right) \right] \right\}, \tag{17}$$

where *θx*, *θ<sup>y</sup>* are the tilt angles of the reference beam and *D* is defined in Eq. (11). A more robust technique is to perform automatic aberration cancelation by approximating the residual phase front due to aberration using Zernike polynomials as explained in details in Refs. [13, 14], and shown in Section 4 below.

Consider the transmission setup shown in **Figure 4(b)**. A USAF 1951 resolution chart target is used as an object. The resolution of a USAF resolution chart is documented as:

$$R\_{\left[lp/mm\right]} = \mathbf{2}^{\left[G + \frac{(E-1)}{6}\right]},\tag{18}$$

where *R*½ � *lp=mm* is resolution in line pair per millimeter, *G* is the group number, and *E* is the element number. (See **Figure 6(a)**). As an example, Group 4, Elements 3 and 4 has a resolution of 20.16 *lp/mm*, and 22.62 *lp/mm*, respectively. The wavelength used is λ = 488 nm, the reconstruction distance *d* = 0.202 m, *D* = 0.14 m, the magnification is *M* ≈ 8.25, *kx*0*=k*<sup>0</sup> ¼ sin *θ<sup>x</sup>* ¼ 0*:*01307, *ky*0*=k*<sup>0</sup> ¼ sin *θ<sup>y</sup>* ¼ 0*:*01305*:* It should be noted that in practice, it is very difficult to obtain such precise parameter measurements in the laboratory. Typically, approximate measurements are made, then, varied slightly during numerical reconstruction to yield the "best focus" image. Such a process was followed to obtain the parameters listed above. In Section 4, we discuss the telecentric setups which mitigate these difficulties. **Figure 6(b)**

**Figure 6.**

*(a) Schematic of the USAF resolution target, (b) recorded hologram, (c) reconstructed hologram amplitude, (d) reconstructed phase using approximate phase mask parameters, (e) reconstructed phase using exact phase mask parameters, and (f) residual phase aberration approximation using Zernike polynomials to be subtracted from (e).*

shows the recorded hologram. **Figure 6(c)** shows the reconstructed hologram amplitude. **Figure 6(d)** shows the reconstructed phase using approximate phase mask parameters (see the circular fringes). **Figure 6(e)** shows the reconstructed phase using exact phase mask parameters (no circular fringes). **Figure 6(f)** shows the residual phase aberration approximation using Zernike polynomials to be subtracted from (e).
