**3. Multi-wavelength digital holography (MWDH)**

It is well known that one main application of **h**olographic **i**nterferometry (HI) is the generation of a fringe pattern corresponding to contours of constant elevation with respect to a reference plane [2]. These contour fringes can be used to determine the shape of a macroscopic or microscopic three-dimensional object.

There exist three main techniques to create holographic contour interferograms: (a) The two-illumination-point method, (b) the two-refractive-index technique, which is generally not practical because we have to change the refractive index of the medium where the object is located, and (c) Multi-wavelength method, which was adopted in this section [6, 8, 11]. For large height profiles (larger than several microns) 2D topography using single wavelength holographic approach is not appropriate since phase unwrapping has limitations especially for sharp edge variations. As shown in **Figure 7**, the axial displacement of an image recorded with wavelength *λ*<sup>1</sup> and reconstructed with another wavelength *λ*2, with respect to the image recorded and reconstructed with *λ*2, is [2, 6, 15–23].

$$
\Delta d\_x = z \frac{|\lambda\_1 - \lambda\_2|}{\lambda\_2}. \tag{19}
$$

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

**Figure 7.**

*The path difference of the light rays on their way from the source to the surface and from the surface to the hologram.*

This means that the phase shift depends on the distance z between the object and the hologram plane. The height jump between two adjacent fringes in the reconstructed image is

$$
\Delta H = \mathbf{z}(\Delta \boldsymbol{\rho} = (n+1) \times 2\pi) - \mathbf{z}(\Delta \boldsymbol{\rho} = n \times 2\pi) = \frac{\lambda\_1 \lambda\_2}{2|\lambda\_1 - \lambda\_2|} = \frac{\Lambda}{2},\tag{20}
$$

where Λ is known as the synthetic wavelength. For larger deformations along the *z* direction, the phase changes could be hundreds of multiples of 2π. Such large fringe densities may lead to difficulty in determining the object phase using the single wavelength technique. However, multi-wavelength illumination encodes the object height in terms of 2π multiples of the synthetic wavelength, which is generally much longer than either fundamental wavelength. This allows larger object deformations to be measured by multi-wavelength illumination as if illuminated by the single wavelength method, where the "single" wavelength is now given by the synthetic wavelength Λ. Typically, synthetic wavelengths can range from few microns to 10's of microns [16, 18]. The topographic resolution is typically on the order of 1/100 of Λ and the vertical measurement range can reach several Λ's by employing phase unwrapping for heights larger than Λ [24]. However, much longer or shorter synthetic wavelengths to measure millimeter-scale features can also be performed. **Figure 8** shows the advantages of using MWDH to extend the vertical measurement range without phase ambiguity [6]. MWDH may be used to quantify surface topography and displacement measurements for both fixed objects and time-varying objects [17, 18]. It is worth noting that DHM has a lot of applications in living cells [25–30], neural science [31], tissue analysis [32], particle tracking [33–36], and MEMS analysis [37–39].

In multiwavelength DH, both holograms are reconstructed separately at the correct fundamental wavelengths, *λ*<sup>1</sup> or *λ*2. From the resulting reconstructed complex amplitudes Γ*<sup>λ</sup>*<sup>1</sup> ð Þ *ξ*, *η* and Γ*<sup>λ</sup>*<sup>2</sup> ð Þ *ξ*, *η* the phases are calculated as:

$$\log\_{\lambda\_{\mathbb{L}^2}}(\xi,\eta) = \arctan\left(\mathrm{Im}\Gamma\_{\mathbb{A},2}(\xi,\eta) / \mathrm{Re}\,\Gamma\_{\mathbb{A},2}(\xi,\eta)\right). \tag{21}$$

The synthetic wavelength phase image is now calculated directly by pixel-wise subtraction of the fundamental wavelength hologram phases

$$
\Delta \boldsymbol{\varrho} = \begin{cases}
\boldsymbol{\varrho}\_{\dot{\lambda}\_1} - \boldsymbol{\varrho}\_{\dot{\lambda}\_2} & \text{if } \boldsymbol{\varrho}\_{\dot{\lambda}\_1} \ge \boldsymbol{\varrho}\_{\dot{\lambda}\_2}, \\
\boldsymbol{\varrho}\_{\dot{\lambda}\_1} - \boldsymbol{\varrho}\_{\dot{\lambda}\_2} + 2\pi & \text{if } \boldsymbol{\varrho}\_{\dot{\lambda}\_1} < \boldsymbol{\varrho}\_{\dot{\lambda}\_2}.
\end{cases} \tag{22}$$

This phase map is equivalent to the phase distribution of a hologram recorded with the *synthetic wavelength*

**Figure 8.** *The advantage of MWDH is that it extends the vertical measurement range without phase ambiguity.*

$$
\Lambda = \left[\frac{\lambda\_1 \lambda\_2}{|\lambda\_1 - \lambda\_2|}\right].\tag{23}
$$

At normal incidence, a *2π* phase jump corresponds to a height step of Λ/2, and the change in longitudinal distance or height Δ*z* is given by [2, 6, 9].

$$
\Delta z = \left(\frac{\Delta \rho}{2\pi}\right)\frac{\Lambda}{2} = \frac{\Delta \rho}{2\pi} \left[\frac{\lambda\_1 \lambda\_2}{2|\lambda\_1 - \lambda\_2|}\right] = \left(\frac{\Delta \rho}{2\pi}\right)\Delta H. \tag{24}
$$

Note that the transverse resolution is the same as in DH namely, Δ*ξ* ¼ *λd=N*Δ*x* is the reconstructed pixel size. Similar to DH, MWDH setup can be constructed using Mach-Zehnder or Michelson configuration as shown in **Figure 9(a)** and **(b)**, respectively. According to **Figure 9(c)** and **(d)**, we notice that the true height measurements in the Mach-Zehnder and Michelson configurations are:

$$
\Delta \varpi\_{True, Mach-Zender} = \left(\frac{\Delta \rho}{2\pi}\right) \frac{\Lambda}{2} \cos \theta,\tag{25}
$$

$$
\Delta x\_{True, Michelson} \approx \left(\frac{\Delta \rho}{2\pi}\right) \frac{\Lambda}{2},
\tag{26}
$$

respectively.

One important detail that must be considered when applying the two wavelengths technique is pixel matching. Recall from Eq. (3) that the pixel resolution Δξ of each hologram is dependent upon the fundamental recording wavelength (*λ*<sup>1</sup> or *λ*2). In order for the reconstruction to be successful, the subtraction described by Eq. (18) must be performed on a pixel-by-pixel basis, in which the pixel sizes match between each hologram (i.e. Δ*ξ*<sup>1</sup> = Δ*ξ*2). This can be accomplished by zeropadding the holograms to alter the numerical resolution according to the following procedure: One hologram is zero-padded prior to reconstruction such that its value of Δξ matches that of the second hologram. The second hologram is then either zero-padded after reconstruction, or the first hologram (which is now larger) is

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

#### **Figure 9.**

*(a) Mach-Zehnder configuration, (b) Michelson configuration, with illustration of the true height* Δ*zTrue relative to the path of phase accumulation for (c) Mach-Zehnder and (d) Michelson configurations.*

cropped, such that the total sizes of each image are again equal. If it is assumed that *λ<sup>1</sup>* > *λ<sup>2</sup>* then the degree of padding applied to both the *λ<sup>1</sup>* hologram prereconstruction and the *λ<sup>2</sup>* hologram post-reconstruction is

$$pad\ size = round\left[\frac{N}{2}\left(\frac{\lambda\_1}{\lambda\_2} - 1\right)\right],\tag{27}$$

where *pad size* is the number of zero elements to be added symmetrically to each edge of the hologram matrix, rounded to the nearest integer value.

## **3.1 Experimental results for MWDH with and without spatial heterodyning (MWDH-SH)**

An example of a 3D profile setup using the MWDH technique is shown in **Figure 10**. Since the two holograms are recorded sequentially for each wavelength, this technique needs two sequential CCD recordings (i.e. two "shots"). Obviously, this "two-shot" method will not work for dynamic objects. **Figure 11(a)** shows the

**Figure 10.** *Shape measurement using MWDH.*

**Figure 11.**

*(a) Newport logo, (b) reconstructed hologram at* λ*<sup>1</sup> = 496.5 nm, (c) wrapped phase, and (d) unwrapped phase or 3D surface profile. The two wavelengths used are:* λ *<sup>1</sup> = 496.5 nm and* λ*<sup>2</sup> = 488 nm and the synthetic wavelength is Λ = 28.5 μm.*

Newport logo test object while **Figure 11(b)** shows the reconstructed hologram of the Newport Logo at one of the wavelengths used (λ<sup>1</sup> = 496.5 nm). **Figure 11(c)** shows the wrapped phase and **Figure 11(d)** shows the unwrapped 3D surface profile.

Spatial heterodyning technique has the ability to capture both wavelength measurements in a single composite holographic exposure [6, 9, 21]. This is accomplished by introducing a different angular tilt to the *λ*<sup>1</sup> and *λ*<sup>2</sup> reference beams. These angular tilts in the spatial domain introduce linear phase shifts in the frequency domain of the recorded composite hologram. When reconstructed, the different phase shifts result in spatially separated object locations in the image that each correspond to their respective *λ*<sup>1</sup> and *λ*<sup>2</sup> recordings. One of these reconstructed images is cropped and digitally overlaid upon the other to perform the required phase subtraction. A typical recording configuration using MWDH-SH method is shown in **Figure 12**. Since the two holograms are recorded for each wavelength at the same time using spatial heterodyning this technique needs only one CCD exposure (i.e. "one-shot") [6, 9, 21, 22]. This method is well suited for dynamic objects which change relatively quickly and only limited by the integration time of the CCD. **Figure 13** shows the reconstructed Newport logo test object. The reconstructed image resolution Δ*ξ* = 32 μm/pixel. Note that, two separate reconstructions are required (λ<sup>1</sup> and λ2) from the single hologram, although only one reconstruction is shown here.

In order to align the two phase images, a block matching algorithm (BMA) is used. After cropping the two reconstructed holograms it is necessary to slide the reference image over the target image looking for best correlation, this is shown in **Figure 14**. Given the typically rapid variation in object phase, BMA algorithms can *Latest Advances in Single and Multiwavelength Digital Holography and Holographic Microscopy DOI: http://dx.doi.org/10.5772/intechopen.94382*

**Figure 12.**

*Lab setup (Michelson configuration) for macroscopic, spatial heterodyne MWDH using coaxial beams and a single spatial filter and collimation lens. The collimation lens should ideally be achromatic at the λ<sup>1</sup> and λ<sup>2</sup> wavelengths. M1, M2: Mirrors, BS: Beam splitter, PBS: Polarizing beam splitter. The polarizer, P0, ensures λ<sup>1</sup> and λ<sup>2</sup> maintain orthogonal polarization [6, 23].*

**Figure 13.** *MWDH-SH hologram reconstruction.*

#### **Figure 14.**

*BMA slides (a) reference image over (b) the target image, (c) correlation. The best correlation occurs at: X*shift*: 0 and Y*shift*: 4 pixels.*

**Figure 15.**

*MWDH-SH phase reconstruction, (a) amplitude reconstruction, (b) phase difference, and (c) phase unwrapping for* Λ ¼ *150 μm.*

only match to within ½ pixel. Hence, BMA matching will generally underperform the two-shot method. After aligning the images, the phase difference is calculated by phase subtraction similar to the two shot technique. The example shown in **Figure 15** is for synthetic wavelength Λ ¼ 150 μm.

An alternative method of matching the two images is to introduce a phase "tilt" to either one, or both holograms during reconstruction which causes lateral shifts in the position of each image. This is typically referred to as introducing a phase mask, Ψð Þ *m*, *n* , during reconstruction, and in general can take any form, although the most commonly used are tilt phases and lens phases. Proper selection of the phase mask (typically found via multiple iterations) can position one hologram reconstruction directly over the other, and phase subtraction may then be performed in a matter analogous to the "two-shot" method previously described, including appropriate resolution matching via zero-padding. An example of phase due to tilt and due to the MO are given by Eq. (13). The hologram matrix is simply multiplied by the phase mask, Ψ, prior to reconstruction. Although the phase mask method is typically more difficult to implement, requiring multiple iterations to arrive at the correct phase mask, it does not suffer from the inherent mismatch error of up to ½ pixel, as the BMA process does. However, the overlap accuracy will now depend upon the accuracy of the modeled phase mask.

#### **Figure 16.**

*(a) Custom fabricated micro-scale objects and (b) 1951 USAF resolution chart with a* � *50 nm reflective molybdenum film sputtered on it.*

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