**2.1. Polarization ellipse equation**

All the important features of light wave follow from a detailed examination of Maxwell equations. Electromagnetic waves have two polarization along the x axis and along the y axis. The general form of polarized light wave propagating in z direction can be derived from two linear polarized components in the x and y directions [1]:

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$$\mathbf{E}\_{\mathbf{x}}\left(\mathbf{z},\mathbf{t}\right) = \mathbf{E}\_{0\mathbf{x}}\cos\left(\mathbf{r}\_{\alpha} + \boldsymbol{\phi}\_{\mathbf{x}}\right),\tag{1}$$

$$\mathbf{E}\_{\mathbf{y}}(\mathbf{z}, \mathbf{t}) = \mathbf{E}\_{0\mathbf{y}} \cos \left(\mathbf{r}\_{\alpha} + \boldsymbol{\phi}\_{\mathbf{y}}\right) \tag{2}$$

where: x and y refers to the components in the x and y directions, E0x and E0y are the real maximum amplitudes of electric field, ϕx and ϕy are the phases and τω is so called propagator, which describes the propagation of the signal component in the z-direction.

Next,equations (1) and (2) can be written as: [1]:

$$\frac{\mathbf{E\_x(z,t)}}{\mathbf{E\_{0x}}} = \cos(\tau\_\alpha)\cos(\phi\_\mathbf{x}) - \sin(\tau\_\alpha)\sin(\phi\_\mathbf{x}) \tag{3}$$

$$\frac{\mathbf{E}\_{\rm y}(\mathbf{z},t)}{\mathbf{E}\_{0\mathbf{y}}} = \cos(\tau\_{\rm eo})\cos(\phi\_{\rm y}) - \sin(\tau\_{\rm eo})\sin(\phi\_{\rm y}).\tag{4}$$

Squaring and adding (3) and (4) then yields:

$$\frac{\mathbb{E}\_{\mathbf{x}}^{2}\left(\mathbf{z},\mathbf{t}\right)}{\mathbb{E}\_{\mathbf{0}\mathbf{x}}^{2}} + \frac{\mathbb{E}\_{\mathbf{y}}^{2}\left(\mathbf{z},\mathbf{t}\right)}{\mathbb{E}\_{\mathbf{0}\mathbf{y}}^{2}} - 2\frac{\mathbb{E}\_{\mathbf{x}}\left(\mathbf{z},\mathbf{t}\right)}{\mathbb{E}\_{\mathbf{0}\mathbf{x}}} \frac{\mathbb{E}\_{\mathbf{y}}\left(\mathbf{z},\mathbf{t}\right)}{\mathbb{E}\_{\mathbf{0}\mathbf{y}}} \cos\left(\boldsymbol{\phi}\right) = \sin^{2}\left(\boldsymbol{\phi}\right),\tag{5}$$

where: ϕ=ϕy-ϕx.

Equation (5) is an ellipse equation. This equation is called the polarization ellipse.

Figure 1 shows the polarization ellipse for optical field.

The polarization ellipse presents some important parameters enabling the characterization of the state of light polarization (SOP) [2]:


**Figure 1.** Polarization ellipse for optical field

() ( ) <sup>x</sup> 0x <sup>x</sup> E z,t E cos , <sup>w</sup> = t +f (1)

<sup>y</sup> 0y ( ) <sup>y</sup> E (z,t) E cos , <sup>w</sup> = t +f (2)

τω

is so called propagator,

where: x and y refers to the components in the x and y directions, E0x and E0y are the real

( ) () () () () <sup>x</sup>

( ) ( ) ( ) ( ) ( ) <sup>y</sup>

( ) ( ) ( ) ( ) () ()

x x y y 2

Equation (5) is an ellipse equation. This equation is called the polarization ellipse.

The polarization ellipse presents some important parameters enabling the characterization of

**1.** Axis x and y are the initial, unrotated axes, ξ and η are a new set of axes along the rotated.

**2.** The area of polarization ellipse depends on the lengths of major and minor axes, ampli‐

**4.** The rotation angle ψ (-βp ≤ψ≤βp) is the angle between axis x and major axis ξ. This angle

0x 0y 0x 0y

**3.** The angle βp=arctg(E0y/E0x) is called the auxiliary angle (0≤ψ≤π/2).

**5.** The ellipticity is the major axis to minor axis ratio (b/a).

E z,t E z,t E z,t E z,t

E E E E

cos cos sin sin .

x x

y y

2 cos sin ,

E z,t cos cos sin sin , <sup>E</sup> w w = t f- t f (3)

<sup>E</sup> w w = t f- t f (4)

+ - f= f (5)

maximum amplitudes of electric field, ϕx and ϕy are the phases and

126 Advances in Optical Fiber Technology: Fundamental Optical Phenomena and Applications

Next,equations (1) and (2) can be written as: [1]:

0x

0y

E z,t

Squaring and adding (3) and (4) then yields:

2 2

where: ϕ=ϕy-ϕx.

2 2

Figure 1 shows the polarization ellipse for optical field.

the state of light polarization (SOP) [2]:

is called the azimuth angle.

tudes E0x and E0y and phase shift ϕ.

which describes the propagation of the signal component in the z-direction.

**6.** The angle equals to υ=arctg(b/a) is called ellipticity angle. For linearly polarized light υ=0º; for circularly polarized light |υ|=45º. In turn, for right polarized light: 0º<υ≤45º and for left polarized light:-45º≤υ<0º.

Figure 2 presents some polarization states; The phase shift ϕ is only changed.

**Figure 2.** Different shapes of the polarization ellipse as a function of phase shift

#### **2.2. Jones notation**

The light wave components in terms of complex quantities can be expressed by means of the Jones vector [1]:

$$
\begin{bmatrix} \mathbf{E}\_{\mathbf{x}} \\ \mathbf{E}\_{\mathbf{y}} \end{bmatrix} = \begin{bmatrix} \mathbf{E}\_{0\mathbf{x}} \mathbf{e}^{\mathbf{j}\phi\_{\mathbf{x}}} \\ \mathbf{E}\_{0\mathbf{y}} \mathbf{e}^{\mathbf{j}\phi\_{\mathbf{y}}} \end{bmatrix}. \tag{6}
$$

The Jones vector representation is suited to all problems related to the totally polarized light.

Table 1 gives the Jones vectors corresponding to the fundamental SOPs.


**Table 1.** Jones vectors of the fundamental SOPs

The Jones matrices for some polarization components are 2x2 matrices.

The relationship between the both output and input Jones vectors can be written as:

$$
\begin{bmatrix}
\mathbf{E}\_{\mathbf{x},\text{out}} \\
\mathbf{E}\_{\mathbf{y},\text{out}}
\end{bmatrix} = \begin{bmatrix}
\mathbf{j}\_{\text{xx}} & \mathbf{j}\_{\text{xy}} \\
\mathbf{j}\_{\text{yx}} & \mathbf{j}\_{\text{yy}}
\end{bmatrix} \cdot \begin{bmatrix}
\mathbf{E}\_{\mathbf{x},\text{in}} \\
\mathbf{E}\_{\mathbf{y},\text{in}}
\end{bmatrix}.\tag{7}
$$

Where j xx j xy j yx j yy is the Jones matrix of a polarization component.

We now describe the matrix forms for the retarder (wave plate), rotator and polarizer (diatte‐ nuator), respectively.

#### **1.** Retarder

The retarder causes a phase shift of ϕ/2 along the fast (i.e. x) axis and a phase shift of-ϕ/2 along slow (i.e. y) axis. This behavior is described by [3]:

$$
\begin{bmatrix}
\mathbf{E}\_{\mathbf{x},\text{out}} \\
\mathbf{E}\_{\mathbf{y},\text{out}}
\end{bmatrix} = \begin{bmatrix}
\mathbf{e}^{\frac{\boldsymbol{\Phi}}{2}} & \mathbf{0} \\
& \mathbf{0} \\
\mathbf{0} & \mathbf{e}^{-\frac{\boldsymbol{\Phi}}{2}}
\end{bmatrix} \cdot \begin{bmatrix}
\mathbf{E}\_{\mathbf{x},\text{in}} \\
\mathbf{E}\_{\mathbf{y},\text{in}}
\end{bmatrix}.
\tag{8}
$$

For quarter-wave plate ϕ is π/2 and for half-wave plate ϕ is π.

#### **2.** Rotator

The Jones vector representation is suited to all problems related to the totally polarized light.

1 0

0 1

1 2 1 − *j*

(7)

(8)

Table 1 gives the Jones vectors corresponding to the fundamental SOPs.

128 Advances in Optical Fiber Technology: Fundamental Optical Phenomena and Applications

**State of polarization Jones vector**

The Jones matrices for some polarization components are 2x2 matrices.

The relationship between the both output and input Jones vectors can be written as:

x,out xx xy x,in y,out yx yy y,in E E j j . E E j j é ù éù é ù ê ú êú = × ê ú ê ú ë û ëû ë û

We now describe the matrix forms for the retarder (wave plate), rotator and polarizer (diatte‐

The retarder causes a phase shift of ϕ/2 along the fast (i.e. x) axis and a phase shift of-ϕ/2 along

f -

is the Jones matrix of a polarization component.

j <sup>2</sup> x,out x,in y,out <sup>j</sup> y,in <sup>2</sup> E E e 0 . E E 0 e

f

é ù é ù éù ê ú ê ú êú <sup>=</sup> ê ú× ë û ëû ê ú ê ú ë û

Linear horizontal

Linear vertical E0x =0; *E*0*<sup>y</sup>* <sup>2</sup> =1

<sup>2</sup> =1

<sup>2</sup> =1

<sup>2</sup> =1

<sup>2</sup> =1

**Table 1.** Jones vectors of the fundamental SOPs

E0y=0; *E*0*<sup>x</sup>* <sup>2</sup> =1

Linear 45º E0x=E0y; 2*E*0*<sup>x</sup>*

Linear -45º E0x=-E0y; 2*E*0*<sup>x</sup>*

Right circular E0x=E0y, ϕ=π/2; 2*E*0*<sup>x</sup>*

Left circular

Where

j xx j xy

j yx j yy

**1.** Retarder

nuator), respectively.

slow (i.e. y) axis. This behavior is described by [3]:

E0x=E0y, ϕ =-π/2; 2*E*0*<sup>x</sup>*

If the angle of rotation is Θ then the components of light emerging from rotation are written as [3]:

$$
\begin{bmatrix} \mathbf{E}\_{\rm x,out} \\ \mathbf{E}\_{\rm y,out} \end{bmatrix} = \begin{bmatrix} \cos(\Theta) & \sin(\Theta) \\ -\sin(\Theta) & \cos(\Theta) \end{bmatrix} \cdot \begin{bmatrix} \mathbf{E}\_{\rm x,in} \\ \mathbf{E}\_{\rm y,in} \end{bmatrix} \tag{9}
$$

#### **3.** Polarizer

The polarizer behavior is characterized by the transmission factor pxand py. Here, for complete transmission px=py=1 and for complete attenuation px=py=0.

The output Jones vector for a polarizer is given by [3]:

$$
\begin{bmatrix} \mathbf{E}\_{\mathbf{x},\text{out}} \\ \mathbf{E}\_{\mathbf{y},\text{out}} \end{bmatrix} = \begin{bmatrix} \mathbf{p}\_{\mathbf{x}} & \mathbf{0} \\ \mathbf{0} & \mathbf{p}\_{\mathbf{y}} \end{bmatrix} \cdot \begin{bmatrix} \mathbf{E}\_{\mathbf{x},\text{in}} \\ \mathbf{E}\_{\mathbf{y},\text{in}} \end{bmatrix}. \tag{10}
$$

The Jones matrix (Jar) a polarization component (J) rotated through an angle Θ is:

$$\mathbf{J}\_{\rm ar} = \mathbf{J}\_{\rm R}(-\Theta) \cdot \mathbf{J} \cdot \mathbf{J}\_{\rm R}(\Theta),\tag{11}$$

where JR(Θ) is the rotation matrix.

#### **2.3. Stokes parameters**

Let us introduce the S0, S1, S2 i S3 real quantities defined by the following relations [4]:

$$\mathbf{S}\_0 = \mathbf{E}\_{0\mathbf{x}}^2 + \mathbf{E}\_{0\mathbf{y}}^2 \tag{12}$$

$$\mathbf{S}\_1 = \mathbf{E}\_{0\mathbf{x}}^2 - \mathbf{E}\_{0\mathbf{y}}^2 \tag{13}$$

$$\mathbf{S}\_2 = 2\mathbf{E}\_{0\mathbf{x}}\mathbf{E}\_{0\mathbf{y}}\cos(\boldsymbol{\phi})\tag{14}$$

$$\mathbf{S}\_3 = 2\mathbf{E}\_{0\mathbf{x}}\mathbf{E}\_{0\mathbf{y}}\sin\left(\phi\right),\tag{15}$$

where E0x, E0y are the real maximum amplitudes and ϕ is the phase difference.

These quantities are called the Stokes parameters. The Stokes parameters have a physical meaning in terms of intensity. The parameter S0 represents the total intensity of light. The second parameter S1 describes the difference in the intensities of the linearly horizontal polarized light and the linearly vertical polarized light. The third parameter S2 represents the difference in the intensities of the linearly 45º polarized light and linearly-45º polarized light. The last parameter S3 represents the difference in the intensities of the right circularly polarized light and the left circularly polarized light. The Stokes parameters are real values. The Stokes representation is the most adequate representation in treating partially polarized and unpo‐ larized light problems. Moreover, Stokes representation is well suited to the definition of the Degree Of Polarization (DOP). This parameter is equal to [2]:

$$\text{DOP} = \frac{\sqrt{\mathbf{S}\_1^2 + \mathbf{S}\_2^2 + \mathbf{S}\_3^2}}{\mathbf{S}\_0},\tag{16}$$

with value between 0 (unpolarized light) and 1 (totally polarized light).

Often the normalized Stokes parameters are used to describe the light polarization: S 1 S 0 , S 2 S 0 , S 3 S 0 ; with value between-1 and 1.

Table 2 shows some Stokes vectors corresponding to the fundamental SOPs.



#### **Table 2.** Stokes vectors of the fundamental SOPs

where E0x, E0y are the real maximum amplitudes and ϕ is the phase difference.

130 Advances in Optical Fiber Technology: Fundamental Optical Phenomena and Applications

Degree Of Polarization (DOP). This parameter is equal to [2]:

; with value between-1 and 1.

S 1 S 0 , S 2 S 0 , S 3 S0

Linear horizontal

Linear vertical E0x=0, *E*0*<sup>y</sup>* <sup>2</sup> =1

Linear 45º

Linear 45º

E0x=E0y=E0, ϕ= 0, 2*E*<sup>0</sup>

E0x=E0y=E0, ϕ=π, 2*E*<sup>0</sup>

<sup>2</sup> =1

<sup>2</sup> =1

E0y=0, *E*0*<sup>x</sup>* <sup>2</sup> =1

with value between 0 (unpolarized light) and 1 (totally polarized light).

Table 2 shows some Stokes vectors corresponding to the fundamental SOPs.

**State of polarization Stokes vector**

These quantities are called the Stokes parameters. The Stokes parameters have a physical meaning in terms of intensity. The parameter S0 represents the total intensity of light. The second parameter S1 describes the difference in the intensities of the linearly horizontal polarized light and the linearly vertical polarized light. The third parameter S2 represents the difference in the intensities of the linearly 45º polarized light and linearly-45º polarized light. The last parameter S3 represents the difference in the intensities of the right circularly polarized light and the left circularly polarized light. The Stokes parameters are real values. The Stokes representation is the most adequate representation in treating partially polarized and unpo‐ larized light problems. Moreover, Stokes representation is well suited to the definition of the

> 222 123 0 SSS DOP , <sup>S</sup>

Often the normalized Stokes parameters are used to describe the light polarization:

+ + <sup>=</sup> (16)

The Poincaré sphere (Figure 3) is a very useful graphical tool representation of polarization in real three-dimensional space.

**Figure 3.** The Poincaré sphere and fundamental SOPs on this sphere

Each polarization is represented by a point on the Poincaré sphere (totally polarized light) or within the Poincaré sphere (partially polarized light) centered on rectangular coordinate system. Center of the Poincaré sphere represents unpolarized light. The coordinates of the point are normalized Stokes parameters. All linear SOPs lie on the equator. The right circular SOP and left one is located at the North and South Pole, respectively. Elliptically polarized states are represented everywhere else on the surface of the Poincaré sphere. The two orthog‐ onal polarizations are located diametrically opposite on the Poincaré sphere. A continuous evolution of SOP is represented on the Poincaré sphere as a continuous path on this sphere (Figure 4).

**Figure 4.** Example of a continuous evolution of SOP on the Poincaré sphere

this sphere (Figure 4).

Figure 5 presents changing the SOPs by means of the retarder and rotator. Dummy Text Figure 5 presents changing the SOPs by means of the retarder and rotator.

Figure 4. Example of a continuous evolution of SOP on the Poincaré sphere

On the Poincaré sphere the phase shift causes that the intial SOP moves to a new SOP along the same longitude line (Figure 5a). The linear, elliptically and circular SOPs can be achieved by means of a single retarder. In turn, on the Poincaré sphere the rotation by a rotator causes

Impact of an optical component (or optical system) properties on the polarization of light can be determined by constructing the Stokes vector for the input light and applying

that the intial SOP moves to a new SOP along the same latitude line (Figure 5b).

Mueller calculus, to obtain the Stokes vector of the light leaving the component:

Figure 5. The effect of changing the SOPs by the retarder (a) and the rotator (b) **Figure 5.** The effect of changing the SOPs by the retarder (a) and the rotator (b)

**2.4. Mueller notation** 

On the Poincaré sphere the phase shift causes that the intial SOP moves to a new SOP along the same longitude line (Figure 5a). The linear, elliptically and circular SOPs can be achieved by means of a single retarder. In turn, on the Poincaré sphere the rotation by a rotator causes that the intial SOP moves to a new SOP along the same latitude line (Figure 5b).

#### **2.4. Mueller notation**

Each polarization is represented by a point on the Poincaré sphere (totally polarized light) or within the Poincaré sphere (partially polarized light) centered on rectangular coordinate system. Center of the Poincaré sphere represents unpolarized light. The coordinates of the point are normalized Stokes parameters. All linear SOPs lie on the equator. The right circular SOP and left one is located at the North and South Pole, respectively. Elliptically polarized states are represented everywhere else on the surface of the Poincaré sphere. The two orthog‐ onal polarizations are located diametrically opposite on the Poincaré sphere. A continuous evolution of SOP is represented on the Poincaré sphere as a continuous path on this sphere

132 Advances in Optical Fiber Technology: Fundamental Optical Phenomena and Applications

two orthogonal polarizations are located diametrically opposite on the Poincaré sphere. A continuous evolution of SOP is represented on the Poincaré sphere as a continuous path on

**Figure 4.** Example of a continuous evolution of SOP on the Poincaré sphere

a b

**2.4. Mueller notation** 

Figure 5 presents changing the SOPs by means of the retarder and rotator.

Figure 5. The effect of changing the SOPs by the retarder (a) and the rotator (b)

**Figure 5.** The effect of changing the SOPs by the retarder (a) and the rotator (b)

that the intial SOP moves to a new SOP along the same latitude line (Figure 5b).

Mueller calculus, to obtain the Stokes vector of the light leaving the component:

Dummy Text Figure 5 presents changing the SOPs by means of the retarder and rotator.

On the Poincaré sphere the phase shift causes that the intial SOP moves to a new SOP along the same longitude line (Figure 5a). The linear, elliptically and circular SOPs can be achieved by means of a single retarder. In turn, on the Poincaré sphere the rotation by a rotator causes

Impact of an optical component (or optical system) properties on the polarization of light can be determined by constructing the Stokes vector for the input light and applying

Figure 4. Example of a continuous evolution of SOP on the Poincaré sphere

(Figure 4).

this sphere (Figure 4).

Impact of an optical component (or optical system) properties on the polarization of light can be determined by constructing the Stokes vector for the input light and applying Mueller calculus, to obtain the Stokes vector of the light leaving the component:

$$
\vec{\mathbf{S}}\_{\text{out}} = \mathbf{M} \cdot \vec{\mathbf{S}}\_{\text{in}} \tag{17}
$$

where S → in and S → out is the input and output Stokes vector, respectively, M is the Mueller matrix of an optical component.

We now describe the Mueller matrix forms for the retarder, rotator and polarizer, respectively.

#### **1.** Retarder

The retarder causes a total phase shift ϕ between fast (x) and slow (y) axis. The Mueller matrix of the retarder is seen to be [3]:

$$\mathbf{M}\_{\text{Ret}} = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & \cos(\phi) & -\sin(\phi) \\ 0 & 0 & \sin(\phi) & \cos(\phi) \end{bmatrix} \tag{18}$$

#### **2.** Rotator

The Mueller matrix of the rotator is given [3]:

$$\mathbf{M}\_{\text{Rot}} = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & \cos\left(2\Theta\right) & \sin\left(2\Theta\right) & 0 \\ 0 & -\sin\left(2\Theta\right) & \cos\left(2\Theta\right) & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} . \tag{19}$$

Because polarization effects are described in the intensity domain the physical rotation through an angle Θ leeds to the appearance of 2Θ.

#### **3.** Polarizer

The Mueller matrix for polarizer is [3]:

$$\mathbf{M}\_{\text{polar}} = \frac{1}{2} \begin{bmatrix} \mathbf{p}\_{\text{x}}^2 + \mathbf{p}\_{\text{y}}^2 & \mathbf{p}\_{\text{x}}^2 - \mathbf{p}\_{\text{y}}^2 & 0 & 0\\ \mathbf{p}\_{\text{x}}^2 - \mathbf{p}\_{\text{y}}^2 & \mathbf{p}\_{\text{x}}^2 + \mathbf{p}\_{\text{y}}^2 & 0 & 0\\ 0 & 0 & 2\mathbf{p}\_{\text{x}}\mathbf{p}\_{\text{y}} & 0\\ 0 & 0 & 0 & 2\mathbf{p}\_{\text{x}}\mathbf{p}\_{\text{y}} \end{bmatrix} \tag{20}$$

where px and py are so called transmission factors.

Here, the transmission factors are px=1 and py=0 for the linear horizontal polarizer. In turn, the transmission factors px=1 and py=0 for the linear vertical polarizer.

The Mueller matrix (Mar) for a polarization component (M) rotated through an angle Θ is:

$$\mathbf{M}\_{\rm ar} = \mathbf{M}\_{\rm R}(-2\Theta) \cdot \mathbf{M} \cdot \mathbf{M}\_{\rm R}(2\Theta),\tag{21}$$

where MR(Θ) is the rotation matrix.
