**2. Theoretical analysis**

#### **2.1 Biot's law**

The foundation of detection of analytes concentration by using optical polarimetry trace back to the observation of Biot's in early nineteenth century [13]. The mathematical equation for the optical interaction of linearly polarized light with optically active specimen was described as below which is also called Biot's law;

$$\left[\mathbf{a}\right]\_{\lambda}^{\mathrm{T}} = \frac{\mathbf{a}}{\mathbf{L}\mathbf{C}}\tag{1}$$

The term ½ � *<sup>α</sup> <sup>T</sup> <sup>λ</sup>* is known as specific rotation of the OAM under consideration for specific wavelength of light (λ) at temperature (T). Where, α is the detected rotation of plane of polarization of light, L is the path length of the sample under test and C is the concentration of sample.

#### **2.2 Analytical treatment of interference of linearly polarized light beams**

Suppose two monochromatic light beams with same frequency originating from single source represented by the electric field vector E! <sup>1</sup> and E! <sup>2</sup> and interfere at the point of observation P. Let the light beams are generalized by the equations below however, physical waves will be represented by their real parts only

$$
\overrightarrow{\mathbf{E}}\_{\mathbf{1}} = \overrightarrow{\mathbf{E}}\_{0\mathbf{1}} \mathbf{e}^{-\mathbf{i}(\alpha \mathbf{t} - \mathbf{k}\_{1} \mathbf{r} - \phi\_{1})} \tag{2}
$$

$$
\overrightarrow{\mathbf{E}}\_2 = \overrightarrow{\mathbf{E}}\_{02} \mathbf{e}^{-\mathbf{i}(\mathbf{a}\mathbf{t} - \mathbf{k}\_2, \mathbf{r} - \phi\_2)} \tag{3}
$$

Where ω is the angular frequency of the monochromatic light wave, r is the position vector of incident point P, k1 and k2 are the wave vectors and ϕ<sup>1</sup> and ϕ<sup>2</sup> are the phase differences of beam 1 and 2, respectively. According to the principle of superposition, the irradiance at point 'P' can be calculated as:

$$\mathbf{I}\_{\mathbf{P}} = \overrightarrow{\mathbf{E}}\_{1} + \overrightarrow{\mathbf{E}}\_{2} \tag{4}$$

The time averaged irradiance is proportional to the square of the amplitude of the electric field, i.e., for linear, homogeneous, isotropic dielectric medium;

$$\mathbf{I} = \epsilon \mathbf{v} \langle \mathbf{E}^2 \rangle\_\mathbf{T} \tag{5}$$

Where *ϵ* is the permittivity of the medium and *υ* is the velocity of light in that medium. Considering the relative irradiance within the same medium except for the constant of proportionality, Eq. (4) may be transformed into;

$$\mathbf{I}\_{\mathbf{P}} = \overrightarrow{\mathbf{E}}\_{\mathbf{P}}.\overrightarrow{\mathbf{E}}\_{\mathbf{P}}\ ^\* = \left(\overrightarrow{\mathbf{E}}\_1 + \overrightarrow{\mathbf{E}}\_2\right).\left(\overrightarrow{\mathbf{E}}\_1 + \overrightarrow{\mathbf{E}}\_2\right)\ ^\* = \mathbf{I}\_1 + \mathbf{I}\_2 + \mathbf{I}\_{12} \tag{6}$$

Where,

$$\mathbf{I}\_1 = \overrightarrow{\mathbf{E}}\_1 \overrightarrow{\mathbf{E}}\_1 \overset{\*}{=} \mathbf{E}\_{01}^2 \tag{7}$$

*Optical Interferometry - A Multidisciplinary Technique in Science and Engineering*

$$\mathbf{I}\_2 = \overrightarrow{\mathbf{E}}\_2 \overrightarrow{\mathbf{E}}\_2 \overset{\*}{=} \mathbf{E}\_{02}^2 \tag{8}$$

and

$$\mathbf{I}\_{12} = \overrightarrow{\mathbf{E}}\_{1} \overrightarrow{\mathbf{E}}\_{2} \overset{\*}{\ } + \overrightarrow{\mathbf{E}}\_{2} . \overrightarrow{\mathbf{E}}\_{1} \overset{\*}{\ } = \overrightarrow{\mathbf{E}}\_{01} . \overrightarrow{\mathbf{E}}\_{02} (\mathbf{e}^{i\mathfrak{E}} + \mathbf{e}^{-i\mathfrak{E}}) = 2\overrightarrow{\mathbf{E}}\_{01} . \overrightarrow{\mathbf{E}}\_{02} \cos \delta \tag{9}$$

Where,

$$\vec{\\$} = \left(\vec{k}\_1 - \vec{k}\_2\right) \vec{r} + \left(\phi\_1 - \phi\_2\right) \tag{10}$$

The I1 and I2 are the irradiance of the individual beams and I12 represents the interference term. The symbol, δ represents the total phase difference between the two waves at the point of interest P. From Eq. (9) it can be observed that maximum inference will occur if both the beams are parallel to each other and no interference will occur if both the vectors are orthogonal to each other. For maximum interference Eq. (9) can be written as;

$$\mathbf{I}\_{12} = 2\mathbf{E}\_{01}\mathbf{E}\_{02}\cos\delta = 2\sqrt{\mathbf{I}\_1\mathbf{I}\_2}\cos\delta\tag{11}$$

and total irradiance at point P will become

$$\mathbf{I\_P} = \mathbf{I\_1} + \mathbf{I\_2} + 2\sqrt{\mathbf{I\_1 I\_2}} \cos \delta \tag{12}$$

The interference is called fully constructive if cos *δ* ¼ 2*nπ where n* ¼ 0, 1, 2, … then Eq. (12) can be written as;

$$\mathbf{I\_P = I\_1 + I\_2 + 2\sqrt{I\_1 I\_2}}\tag{13}$$

Eq. (13) is called the maximum intensity or the maxima of fringes pattern (constructive interference) and can be interpreted as;

$$\mathbf{I}\_{\text{max}} = \left(\mathbf{E}\_{01} + \mathbf{E}\_{02}\right)^2 \tag{14}$$

and IP will be minimum if the term 2 ffiffiffiffiffiffiffi *I*1*I*<sup>2</sup> <sup>p</sup> cos *<sup>δ</sup>* is negative, i.e., cos *<sup>δ</sup>* ¼ �1 which is possible only if cos *δ* ¼ ð Þ 2*n* þ 1 *π* for n = 0,1,2, … , and IP will be called Imin;

$$\mathbf{I\_{min}} = \mathbf{I\_1} + \mathbf{I\_2} - 2\sqrt{\mathbf{I\_1 I\_2}} = \left(\mathbf{E\_{01} - E\_{02}}\right)^2 \tag{15}$$

If the two interfering light waves are mutually coherent then the time averaged value of cos *δ* overtime period T should not vanish and a stationary fringe pattern will be obtained in space i.e.,

$$
\langle \cos \delta \rangle\_{\rm T} = \frac{1}{T} \int\_0^T \cos \delta \,\mathrm{d}t \neq \mathbf{0} \tag{16}
$$

At different points of observation ( r!), different values of cos h i<sup>δ</sup> <sup>T</sup> will be obtained and resultantly different intensities will be obtained at different locations in space. Also ϕ<sup>1</sup> � ϕ<sup>2</sup> ð Þ should not vary in time otherwise cos h iδ ¼ 0 and no sustained

*A Review of Optical Interferometry Techniques for Quantitative Determination of Optically… DOI: http://dx.doi.org/10.5772/intechopen.104937*

interference fringe pattern will be obtained. The quality of interference fringes is quantitatively described by visibility (V):

$$\mathbf{V} = \frac{\mathbf{I}\_{\text{max}} - \mathbf{I}\_{\text{min}}}{\mathbf{I}\_{\text{max}} + \mathbf{I}\_{\text{min}}} \tag{17}$$

Where Imax and Imin represents the irradiances corresponding to maximum and adjacent minimum in the interference fringes. If one beam is incident with some small angle *θ* with respect to the plane of incidence then visibility could be defined as;

$$\mathbf{V} = \frac{2\sqrt{\mathbf{I}\_1 \mathbf{I}\_2} \cos \psi}{\mathbf{I}\_1 + \mathbf{I}\_2} \tag{18}$$

Where, I1 and I2 are the irradiances of reference and sample beams respectively and ψ is the polarization angle of the sample beam of the interferometer [14].
