**2. Instrumentation and data analysis**

The block diagram of the MZI used in our experiments is illustrated in **Figure 2**.

The details are described in a previous publication [10]. Briefly, a coherent 632.8 nm light beam from a He‐Ne laser is passed through a Glan‐Thompson prism and is converted into vertically polarized light. The polarized light beam was collimated by using collimating lens before entering the basic unit of the Mach‐Zehnder interferometer. After passing the half mir‐ ror [HM1], incident light was divided into two beams, forming the reference arm and the test arm of the MZI. A sample was interposed on the test arm and was half hidden by a movable mask to produce the reference part on the sample. The deformation of the sample under cur‐ ing is obtained from the variation of the part irradiated with UV light compared to the part hidden by a mask on the same sample. In order to observe the deformation caused by the curing reaction, the sample was submitted to the sequence of *masking‐irradiating‐mask remov‐ ing‐recording*. Movement of this mask is controlled with the precision of a micrometer. Details of the operation are provided in [10]. These two laser beams encounter with each other at the half mirror [HM2] to produce interference patterns which were recorded on a charge‐coupled device (CCD) camera. The data were subsequently analyzed using a computer.

**Figure 2.** The block diagram of the Mach‐Zehnder interferometer (MZI) used in this study for in situ monitoring the deformation induced by photocuring polymers. M, reflecting mirror; HM, half‐mirror; GTP, Glan‐Thompson prism; SF, spatial filter; CL, collimating lens.

### **2.1. Basic theory of light interference phenomena**

reflecting mirrors. The coherent light beam from a laser after collimation was divided into two beams: the reference beam and the test beam on which the sample is interposed. These two beams serve as two arms, the reference and the test arms of the interferometry. The presence of an object on the test arm will result in the difference in optical path length, thereby changing the interference pattern of the laser at the half mirror [HM2]. The fringe patterns can be moni‐ tored and recorded either along the direction of the reference beam or the test beam. Compared to other interferometers like Michelson, the separation of the two arms of MZI can provide a wide application due to large and freely accessible working space though the optical alignment is relatively difficult. Taking advantage of this spacious working place, MZI has been utilized for various experiments: electron interferometer functioning in high magnetic field [3], flow visualization and flow measurements [4], for sensing applications [5]. Furthermore, optoflu‐ idic Mach‐Zehnder interferometer for sensitive, label‐free measurements of refractive index of fluids was also developed [6]. The unique structure of Mach‐Zehnder has also been utilized for optical communication as a modulator [7]. On the other hand, a lot of efforts have been made to fabricate microscale optical systems including Mach‐Zehnder interferometer modulators using polymeric materials [8, 9]. In this chapter, we focus on studies on the local deformation in poly‐ meric systems undergoing photocuring by ultraviolet (UV) light. Since the polymer mixture undergoes transition from liquid to solid by the reaction and at the same time phase separation takes place, the deformation (shrinkage and/or swelling) would affect the phase separation pro‐ cess and the resulting morphology. Mach‐Zehnder interferometry would be useful to monitor

the extent of deformation in the nanometer scales during the reaction.

**2. Instrumentation and data analysis**

**Figure 1.** Basic unit of Mach‐Zehnder interferometry (MZI).

26 Optical Interferometry

The block diagram of the MZI used in our experiments is illustrated in **Figure 2**.

The details are described in a previous publication [10]. Briefly, a coherent 632.8 nm light beam from a He‐Ne laser is passed through a Glan‐Thompson prism and is converted into Two monochromatic planar waves E1 and E2 traveling along the two arms of the Mach‐Zehnder interferometer can be expressed by the following wave equation in complex form:

$$E\_1 = E\_{v0} \exp(i\,\phi\_1) \tag{1}$$

$$E\_z = E\_{0z} \exp(i\,\phi\_z) \tag{2}$$

Here *E* and *ϕ* represent respectively the amplitudes and the phases of the two waves and *i* is the imaginary number.

The phase of these two waves can be written as

$$
\phi\_1 = \, k \, n\_0 \, \mathbf{L}\_1 \tag{3}
$$

$$
\phi\_2 = k \,\text{\textquotedblleft}\_0 \text{L}\_2 \tag{4}
$$

where *k* and *n*<sup>0</sup> are respectively the wavenumber of the incident light and the refractive index of the sample before curing. *L*<sup>1</sup> and *L*<sup>2</sup> are respectively the path length of light traveling along the reference and the test arms. The phase difference between the two beams can be expressed by

$$
\Delta \phi = \begin{pmatrix} \phi\_2 - \phi\_1 \end{pmatrix} = k \begin{pmatrix} L\_2 - L\_1 \end{pmatrix} \tag{5}
$$

where *n*<sup>0</sup> , the refractive index of air was set equal unity. If the length of two arms is set equal *L*<sup>1</sup> = *L*<sup>2</sup> , *Δϕ* = 0. There is no interference for this particular case. In the presence of a sample with refractive index *n* and the thickness *d* interposed on the test arm, the phase of the wave traveling along this direction becomes

$$
\phi\_2 = k \dots n\_0 \dots \left( L\_2 - d \right) + k \dots n \dots d \tag{6}
$$

The phase difference in the presence of the sample becomes:

$$
\Delta \phi = \left(\phi\_2 - \phi\_1\right) \\
= k(n-1)d + k\left(L\_2 - L\_1\right) \tag{7}
$$

For the case *L*<sup>1</sup> = *L*<sup>2</sup> , the phase difference becomes

$$
\Delta \phi = \begin{array}{c}
\mathbf{k}(\mathbf{r} - \mathbf{1}) \; d \; \tag{8}
$$

Therefore, the optical path length (OPL) of the sample *before* irradiation is

$$\text{OPL}\_{\text{bohou}} = \frac{\Delta \phi\_{\text{bohou}}}{k} = \text{ ( $n-1$ )} \, d \tag{9}$$

In general, both the refractive index and the thickness of the sample are varied by the reaction:

$$\text{In general, both the refractive index and the thickness of the sample are varied by the reaction:}$$

$$\text{OPL}\_{\text{air}} = \left(\frac{\Delta \phi\_{\text{air}}}{k}\right) = (\text{u} + \Delta n)(\text{d} + \Delta d) - 1. (\text{d} + \Delta d) \tag{10}$$

Since both *Δn* and *Δd* are small, (*Δn* . *Δd* ) can be neglected, leading to the final result

Applications of Mach-Zehnder Interferometry to Studies on Local Deformation of Polymers Under Photocuring http://dx.doi.org/10.5772/64611 29

$$\text{OPLD} = \left( \text{OPL}\_{\text{after}} - \text{OPL}\_{\text{before}} \right) = \Delta d \dots (n-1) + \Delta n \dots d \tag{11}$$

For the case, the change in refractive index is negligible, *Δn* ≅ 0, the optical path length can be approximately expressed as

$$\text{OPLD} \quad \text{\#} \quad \Delta d \text{ .(u-1)}\tag{12}$$

If the initial thickness (before curing) of the sample is *d*<sup>0</sup> , from definition, the deformation *ε* (either shrinkage or swelling) is given by the below equation:

$$
\varepsilon = \frac{Ad}{d\_0} = \frac{\text{OPLD}}{(n-1)\ d\_0} = \frac{\text{OPLD}}{(n-1)}\frac{d}{d\_0} \tag{13}
$$

The OPLD on the left‐hand side can be obtained from MZI experiments. Therefore, if the change in refractive index *Δn* before and after the curing reaction can be directly measured using some instrument like prism coupler [11], the change in the sample thickness *Δd* can be obtained.

#### **2.2. Data analysis**

Here *E* and *ϕ* represent respectively the amplitudes and the phases of the two waves and *i* is

*ϕ*<sup>1</sup> = *k n*<sup>0</sup> *L*<sup>1</sup> (3)

*ϕ*<sup>2</sup> = *k n*<sup>0</sup> *L*<sup>2</sup> (4)

along the reference and the test arms. The phase difference between the two beams can be

*Δϕ* = (*ϕ*<sup>2</sup> − *ϕ*1) = *k*(*L*<sup>2</sup> − *L*1) (5)

*ϕ*<sup>2</sup> = *k* . *n*<sup>0</sup> . (*L*<sup>2</sup> − *d*) + *k* . *n* . *d* (6)

*Δϕ* = (*ϕ*<sup>2</sup> − *ϕ*1) = *k*(*n* − 1)*d* + *k*(*L*<sup>2</sup> − *L*1) (7)

*Δϕ* = *k*(*n* − 1) *d* (8)

In general, both the refractive index and the thickness of the sample are varied by the reaction:

The phase difference in the presence of the sample becomes:

, the phase difference becomes

OPLbefore <sup>=</sup> *<sup>Δ</sup> <sup>ϕ</sup>* \_\_\_\_\_\_ before

Therefore, the optical path length (OPL) of the sample *before* irradiation is

*Δ ϕ* \_after

Since both *Δn* and *Δd* are small, (*Δn* . *Δd* ) can be neglected, leading to the final result

, the refractive index of air was set equal unity. If the length of two arms is set equal

, *Δϕ* = 0. There is no interference for this particular case. In the presence of a sample with refractive index *n* and the thickness *d* interposed on the test arm, the phase of the wave traveling

and *L*<sup>2</sup>

are respectively the wavenumber of the incident light and the refractive

are respectively the path length of light traveling

*<sup>k</sup>* <sup>=</sup> (*<sup>n</sup>* <sup>−</sup> <sup>1</sup> ) *<sup>d</sup>* (9)

*<sup>k</sup>* ) <sup>=</sup> (*<sup>n</sup>* <sup>+</sup> *<sup>Δ</sup>n*)(*<sup>d</sup>* <sup>+</sup> *<sup>Δ</sup>d*) <sup>−</sup> 1.(*<sup>d</sup>* <sup>+</sup> *<sup>Δ</sup><sup>d</sup>* ) (10)

the imaginary number.

28 Optical Interferometry

where *k* and *n*<sup>0</sup>

expressed by

where *n*<sup>0</sup>

For the case *L*<sup>1</sup>

*L*<sup>1</sup> = *L*<sup>2</sup>

The phase of these two waves can be written as

index of the sample before curing. *L*<sup>1</sup>

along this direction becomes

= *L*<sup>2</sup>

OPLafter = (

The interference patterns obtained for a polymer film under in situ photocuring on the test arm of the MZI are recorded by using a CCD camera. Though the laser beam was passed through the spatial filter to select the best part of the beam and was subsequently collimated before entering the MZI unit, the interference patterns are slightly affected by the spatial dis‐ tribution of the laser intensity. This effect can be removed by performing some correction assuming that the shape of the laser beam is Gaussian [10].

The interferograms obtained before and after this correction for the intensity distribution of a He–Ne laser (NEC, 1 mW) in the case a polystyrene/poly(vinyl methyl ether) PS/PVME (30/70) blend was used as sample are, respectively, illustrated in **Figures 3** and **4**. To reduce noise, the 2D data (480 pixel × 640 pixel) were divided into 48 horizontal strips with the dimension (10 pixel × 640 pixel) for each strip. Data along the *y*‐axis for each strip were then averaged to provide 1D data as shown in **Figure 3(a)**.

In general, the real part of the intensity of an interferogram is a periodic function of distance and can be expressed in 1D as follows:

$$I(\mathbf{x}) = \mathcal{a}(\mathbf{x}) \cos \left[ \phi(\mathbf{x}) \right] \tag{14}$$

where *a*(*x*) and *ϕ*(*x*) are the amplitude and phase of the signal, respectively.

The imaginary part *J*(*x*) of the intensity can be calculated by using Hilbert transform [10, 12]. From these calculations, the amplitude *a*(*x*) and the phase [*ϕ*(*x*)] are obtained:

$$a(\mathbf{x}) = \sqrt{I^2(\mathbf{x}) + I^2(\mathbf{x})} \tag{15}$$

$$\text{and } \varphi(\mathbf{x}) = \tan^{-1}\left(\frac{l(\mathbf{x})}{l(\mathbf{x})}\right) \tag{16}$$

Finally, the OPLD can be obtained for the left‐hand side of Eq. (12).

**Figure 3.** (a) Interferogram obtained at 20°C for a PSA/PVME (30/70) blend; (b) 1D data obtained by averaging along the *y* direction for each 10 pixels indicated by the two arrows in (a). The peripheral of the intensity distribution in (b) is affected by the Gaussian beam.

**Figure 4.** (a) The interferogram shown in **Figure 4** after the Hilbert transformation; (b) the 1D intensity distribution after averaging along the *y*‐axis as described for **Figure 3**.
