**3. Polarization phenomena in optical fiber links**

#### **3.1. Polarization mode dispersion**

The optical fiber transmission systems are exposed to some polarization effects. Changing of tranmsission quality (e.g. transmission capacity) of an optical fiber links during high bit rate transmission is caused by Polarization Mode Dispersion (PMD), Polarization Dependent Loss (PDL), Polarization Dependent Gain (PDG).

Polarization Mode Dispersion is impairment phenomenon that limits the transmission speed and distance in high bit rate optical fiber communication systems. The impairment results from PMD is similar to chromatic dispersion impairment.

According to [5]: There always exists an orthogonal pair of polarization states output a birefringent concatenation which are stationary to first order in frequency. These two states are called Principle States of Polarization (PSP).

A diffrential delay exists between signals launched along one PSP and its orthogonal comple‐ ment. This effect is quantified by Differential Group Delay (DGD). There are many ways in which an optical fiber can become birefringent. Birefringence can arise due to an asymmetric fiber core or asymmetric fiber refractive index or can be introduced through internal stresses during fiber manufacture or through external stresses during cabling and installation. Polarization Mode Dispersion of an optical fiber link is proportional to the square root of the fiber link length (strong coupling between the orthogonally polarized signal components) or to the fiber link length (weak coupling between ones). The frequency dependent evolution of SOPs in an optical fiber link (Figure 6) is described by the following equation [6]:

$$
\frac{d\vec{\mathbf{S}}}{d\alpha} = \vec{\Omega} \times \vec{\mathbf{S}},\tag{22}
$$

where S <sup>→</sup> is the Stokesa vector, ω is angular frequency and Ω <sup>→</sup> is the PMD vector. S d Sd , (22)

where S is the Stokesa vector, is angular frequency and is the PMD vector.

**Figure 6.** State of polarization transformation through the PMD vector

6) is described by the following equation [6]:

the DGD value between the slow and fast PSP. The Probability Density Function for DGD ( r,g ) is given by [7]: 2 The pointing direction of the PMD vector is aligned to the slow PSP. The length of the PMD vector is the DGD value between the slow and fast PSP.

 

The pointing direction of the PMD vector is aligned to the slow PSP. The length of the PMD vector is

 r,g 2 <sup>2</sup> r,g <sup>e</sup> <sup>8</sup> <sup>2</sup> The Probability Density Function for DGD (τg,r) is given by [7]:

Figure 6. State of polarization transformation through the PMD vector

$$\text{PDF}\{\tau\_{g,r}\} = \frac{8}{\pi^2 \left\langle \tau\_{g,r} \right\rangle} \left( \frac{2\tau\_{g,r}}{\left\langle \tau\_{g,r} \right\rangle} \right)^2 \text{ e } \frac{\left(\frac{2\tau\_{g,r}}{\left\langle \tau\_{g,r} \right\rangle}\right)^2}{\pi} \text{ } \tag{23}$$

PDF , (23)

 

r,g

r,g 2

where τg,ris average value of DGD.

2222 xyxy 2222 xyxy


pppp 0 0

é ù + - ê ú

<sup>1</sup> pppp 0 0 , <sup>2</sup> 0 0 2p p 0

0 00 p

ë û

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

The optical fiber transmission systems are exposed to some polarization effects. Changing of tranmsission quality (e.g. transmission capacity) of an optical fiber links during high bit rate transmission is caused by Polarization Mode Dispersion (PMD), Polarization Dependent Loss

Polarization Mode Dispersion is impairment phenomenon that limits the transmission speed and distance in high bit rate optical fiber communication systems. The impairment results from

According to [5]: There always exists an orthogonal pair of polarization states output a birefringent concatenation which are stationary to first order in frequency. These two states

A diffrential delay exists between signals launched along one PSP and its orthogonal comple‐ ment. This effect is quantified by Differential Group Delay (DGD). There are many ways in which an optical fiber can become birefringent. Birefringence can arise due to an asymmetric fiber core or asymmetric fiber refractive index or can be introduced through internal stresses during fiber manufacture or through external stresses during cabling and installation. Polarization Mode Dispersion of an optical fiber link is proportional to the square root of the fiber link length (strong coupling between the orthogonally polarized signal components) or to the fiber link length (weak coupling between ones). The frequency dependent evolution of

SOPs in an optical fiber link (Figure 6) is described by the following equation [6]:

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

l

where px and py are so called transmission factors.

transmission factors px=1 and py=0 for the linear vertical polarizer.

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

**3. Polarization phenomena in optical fiber links**

po ar

M

where MR(Θ) is the rotation matrix.

**3.1. Polarization mode dispersion**

(PDL), Polarization Dependent Gain (PDG).

PMD is similar to chromatic dispersion impairment.

are called Principle States of Polarization (PSP).

x

y

x y

ar R <sup>R</sup> = M (-2Q)M M × 2Q× )M ,( (21)

2p

= (20)

Figure 7 shows Probability Density Function for DGD.

Differential Group Delay distribution is Maxwellian distribution. Differential Group Delay can be also expressed as [8]:

$$\left\langle \mathbf{r}\_{\mathbf{g},r} \right\rangle^2 = \frac{1}{3} \left( \frac{\lambda \mathbf{L}\_{\mathbf{c}}}{\mathbf{c} \mathbf{L}\_{\mathbf{B}}} \right)^2 \left( \frac{\mathbf{L}}{\mathbf{L}\_{\mathbf{c}}} - \mathbf{1} + \mathbf{e}^{-\frac{\mathbf{L}}{\mathbf{L}\_{\mathbf{c}}}} \right) \tag{24}$$

where: λ is wavelength, c is light wave velocity in vacuum, LB is beat length and Lc is correlation length, L is optical fiber length.

**Figure 7.** Probability Density Function (PDF) for DGD; average value of DGD is 40 ps

The beat length describes the length required for SOP to rotate 2π (360 degrees). In turn, the correlation length is defined to be length at which the difference between average power of orthogonally polarized signal components is within 1/e2 .

Second order PMD is generated by a change of the PMD with frequency (wavelength) [9]:

$$\frac{\mathbf{d}\overline{\mathbf{d}}}{\mathbf{d}\mathbf{o}} = \frac{\mathbf{d}\tau\_{\mathbf{g},\mathbf{r}}}{\mathbf{d}\mathbf{o}} \mathbf{\overline{p}} + \tau\_{\mathbf{g},\mathbf{r}} \frac{\mathbf{d}\overline{\mathbf{p}}}{\mathbf{d}\mathbf{o}}\tag{25}$$

where p <sup>→</sup> is the Stokes vector pointing in the direction of the fast PSP.

Differentiating the PMD vector with respect to frequency gives two components of second order PMD. The first term on the right side of equation (25) is so called polarization dependent chromatic dispersion, it is known to cause polarization-dependent pulse compression and broadening, while the second term causes depolarization. Figure 8 illustrates changing the SOP with frequency – second order PMD.

The Probability Density Function of second order PMD is given by [10]:

$$\text{PDF}\left(\left|\overrightarrow{\Omega}\_{\text{o}}\right|\right) = \frac{32\left|\overrightarrow{\Omega}\_{\text{o}}\right|}{\pi\left\langle \left|\overrightarrow{\Omega}\_{\text{o}}\right|\right\rangle^{4}} \tanh\left(\frac{4\left|\overrightarrow{\Omega}\_{\text{o}}\right|}{\left\langle \left|\overrightarrow{\Omega}\_{\text{o}}\right|\right\rangle^{2}}\right) \text{sech}\left(\frac{4\left|\overrightarrow{\Omega}\_{\text{o}}\right|}{\left\langle \left|\overrightarrow{\Omega}\_{\text{o}}\right|\right\rangle^{2}}\right) \tag{26}$$

**Figure 8.** The effect of changing the SOP with frequency

where: λ is wavelength, c is light wave velocity in vacuum, LB is beat length and Lc is correlation

The beat length describes the length required for SOP to rotate 2π (360 degrees). In turn, the correlation length is defined to be length at which the difference between average power of

Second order PMD is generated by a change of the PMD with frequency (wavelength) [9]:

g,r

Differentiating the PMD vector with respect to frequency gives two components of second order PMD. The first term on the right side of equation (25) is so called polarization dependent chromatic dispersion, it is known to cause polarization-dependent pulse compression and broadening, while the second term causes depolarization. Figure 8 illustrates changing the

> ( ) 42 2 32 4 4 PDF tanh sec h , ww w

> > ww w

pW W W è øè ø

ur ur ur

WW W ç ÷ç ÷

æ öæ ö

ur ur ur (26)

g,r

= +t ww w

W t

<sup>→</sup> is the Stokes vector pointing in the direction of the fast PSP.

The Probability Density Function of second order PMD is given by [10]:

<sup>d</sup> <sup>d</sup> dp p , dd d

.

ur ur ur (25)

**Figure 7.** Probability Density Function (PDF) for DGD; average value of DGD is 40 ps

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

orthogonally polarized signal components is within 1/e2

SOP with frequency – second order PMD.

w

ur

W =

where p

length, L is optical fiber length.

Figure 9 shows Probability Density Function of second order PMD.

**Figure 9.** Probability Density Function (PDF) of second order PMD; average value of second order PMD is 10,0 ps2

It should be note that concatenation of two birefringent optical components (e.g. two sections of polarization-maintaining optical fiber) generates only first order PMD (Figure 10a). These birefringent sections are orientated randomly relative to each other. In turn, concatenation of three (and more) birefringent optical components generates high order PMD (Figure 10b).

 42 2 32 4 4 PDF tanh sech ,

Figure 9 shows Probability Density Function of second order PMD.

(26)

Figure 9. Probability Density Function (PDF) of second order PMD; average value of second order PMD

It should be note that concatenation of two birefringent optical components (e.g. two sections of polarization-maintaining optical fiber) generates only first order PMD (Figure

Figure 10. State of polarization evolution through rotating two birefringent optical components (a) and three birefringent optical components (b) which are orientated randomly relative to each other **Figure 10.** State of polarization evolution through rotating two birefringent optical components (a) and three birefrin‐ gent optical components (b) which are orientated randomly relative to each other

#### **3.2. Polarization Dependent Loss**

is 10,0 ps2

PMD (Figure 10b).

Polarization Dependent Loss is defined as absolute value or the relative difference between an optical component maximium and minimum transmission loss given all possible input SOPs [3]. Dichroism phenomenon is responsible for the PDL effect. Dichroism can be achived by optical fiber bending or interaction between optical beam and tilted glass plate. The PDL value can be given by the following relationship [11]:

$$\text{PDL[dB]} = 10 \log\_{10} \left( \frac{\mathbf{T}\_{\text{r,max}}}{\mathbf{T}\_{\text{r,min}}} \right) \tag{27}$$

where Tr,max and Tr,min are the maximum and minimum transmission intensities through an optical component.

The PDL can be also written as [11]:

$$\text{PDL[dB]} = 10 \log\_{10} \left( \frac{1 + \left| \overline{\Gamma} \right|}{1 - \left| \overline{\Gamma} \right|} \right) \tag{28}$$

where Γ <sup>→</sup> is the PDL vector.

This vector is equal to: T r ,max−T r ,min T r ,max + T r ,min . The pointing direction of the PDL vector is aligned to maximum transmission direction. In other words, this vector is aligned to a polarization vector that imparts the least PDL value. The cumulative PDL vector over concatenate two optical components with the PDL vectors (Γ1 →, Γ2 <sup>→</sup>) is [12]:

#### Polarization Effects in Optical Fiber Links http://dx.doi.org/10.5772/59000 139

$$
\overrightarrow{\Gamma\_{12}} = \frac{\sqrt{1-\Gamma\_{2}^{2}}}{1+\overrightarrow{\Gamma\_{1}}\overrightarrow{\Gamma\_{2}}}\overrightarrow{\Gamma\_{1}} + \frac{1+\overrightarrow{\Gamma\_{1}}\overrightarrow{\Gamma\_{2}}\left(\frac{1-\sqrt{1-\Gamma\_{2}^{2}}}{\Gamma\_{2}^{2}}\right)}{1+\overrightarrow{\Gamma\_{1}}\overrightarrow{\Gamma\_{2}}}\overrightarrow{\Gamma\_{2}}.\tag{29}
$$

If we want to calculate the resulting PDL value from PDL of each optical component we need to average over all possible orientation between Γ1 <sup>→</sup> and Γ2 <sup>→</sup> vectors [12]:

$$\left\langle \Gamma\_{12} \right\rangle = \frac{1}{2} \int\_{-1}^{1} \frac{\left\{ \Gamma\_1^2 + \Gamma\_2^2 - \Gamma\_1^2 \Gamma\_2^2 + 2\Gamma\_1 \Gamma\_2 \eta\_a + \Gamma\_1^2 \Gamma\_2^2 \eta\_a^2 \right.}{\left(1 + \Gamma\_1 \Gamma\_2 \eta\_a\right)^2} d\eta\_{a'} \tag{30}$$

where ηa is angle between Γ1 <sup>→</sup> and Γ2 <sup>→</sup> vectors.

 42 2 32 4 4 PDF tanh sech ,

Figure 9 shows Probability Density Function of second order PMD.

(26)

Figure 9. Probability Density Function (PDF) of second order PMD; average value of second order PMD

It should be note that concatenation of two birefringent optical components (e.g. two sections of polarization-maintaining optical fiber) generates only first order PMD (Figure 10a). These birefringent sections are orientated randomly relative to each other. In turn, concatenation of three (and more) birefringent optical components generates high order

Figure 10. State of polarization evolution through rotating two birefringent optical components (a) and

three birefringent optical components (b) which are orientated randomly relative to each other **Figure 10.** State of polarization evolution through rotating two birefringent optical components (a) and three birefrin‐

Polarization Dependent Loss is defined as absolute value or the relative difference between an optical component maximium and minimum transmission loss given all possible input SOPs [3]. Dichroism phenomenon is responsible for the PDL effect. Dichroism can be achived by optical fiber bending or interaction between optical beam and tilted glass plate. The PDL

10

where Tr,max and Tr,min are the maximum and minimum transmission intensities through an

10 1

maximum transmission direction. In other words, this vector is aligned to a polarization vector that imparts the least PDL value. The cumulative PDL vector over concatenate two optical

> Γ1 →, Γ2 <sup>→</sup>) is [12]:

1 æ ö + G ç ÷ <sup>=</sup> ç ÷

ç ÷ - G è ø

ur

ur (28)

. The pointing direction of the PDL vector is aligned to

PDL[dB] 10log ,

PDL[dB] 10log , <sup>T</sup> æ ö <sup>=</sup> ç ÷

r,max

(27)

T

r,min

ç ÷ è ø

is 10,0 ps2

**3.2. Polarization Dependent Loss**

optical component.

where Γ

The PDL can be also written as [11]:

<sup>→</sup> is the PDL vector.

components with the PDL vectors (

T r ,max−T r,min

T r ,max + T r,min

This vector is equal to:

PMD (Figure 10b).

a b

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

gent optical components (b) which are orientated randomly relative to each other

value can be given by the following relationship [11]:

Concatenation of N optical components with PDL gives the following result:

$$\overline{\Gamma} = \sum\_{j=1}^{N} \overline{\Gamma\_j} + \mathbf{0} \left( \Gamma^2 \right) . \tag{31}$$

Polarization Dependent Loss distribution is Rayleigh distribution (Figure 11).

**Figure 11.** Probability density function (PDF) of PDL; average value of PDL is 0.5 dB

In the presence of PMD the PDL distribution is closed to Maxwellian distribution. It is important to note that in the case of single mode fiber the orthogonal SOPs pairs at the input lead to orthogonal output SOPs pairs, although the input SOP is not maintained in general. But, when the optical fiber link includes PDL the SOPs are no longer orthogonal. Moreover, polarization effects due to interaction between PMD and PDL can significantly impair optical fiber transmission systems. The accumulative PMD and PDL impairment is more dangerous for lightwave communication systems than a pure PMD or PDL impairment.

#### **3.3. Polarization Dependent Gain**

Another polarization effect which is closely related to PDL is PDG.This phenomenon is present in optical amplifiers (first of all in Semiconductor Optical Amplifier). Polarization dependent gain can be defined as absolute value or the relative difference between an optical amplifier maximium gain (Gmax) and minimum one (Gmin):

$$\text{PDG[dB]} = 10 \log\_{10} \left( \frac{\text{G}\_{\text{max}}}{\text{G}\_{\text{min}}} \right) \tag{32}$$

Polarization Hole Burning phenomenon is responsible for the PDG effect. It is important to know that PDG effect is observed for linear polarized optical signals which are amplified. Polarization Dependent Gain for circular polarization can be neglected [13].

### **4. Modeling of polarization phenomena**

The analysis of impact of optical fiber polarization properties on optical signal transmissions requires a detailed description of polarization effects. The most popular approaches are using an optical fiber links modeling based on homogeneous polarization segments and electro‐ magnetic wave propagation equations.

In general, an optical fiber exhibits axially-varing birefringence and can be represented by a series of short and homogeneous polarization segments. Each polarization is described as the randomly rotated polarization elements. These polarization elements are characterised by birefringence (i.e. PMD) or dichroism (i.e. PDL).

The birefringent element can be represented by retarder (phase shifter). In terms of the Jones matrix this element is described by:

$$\mathbf{J}\_{\rm DGD} = \begin{bmatrix} \mathbf{e}^{\frac{\mathbf{j}\cdot\Phi}{2}} & \mathbf{0} \\\\ \mathbf{0} & \mathbf{e}^{-\mathbf{j}\frac{\Phi}{2}} \end{bmatrix} \prime \tag{33}$$

where ϕ is is the total phase shift between the polarization signal components (two polarization modes).

In terms of the Mueller matrix the birefringent element is given by:

#### Polarization Effects in Optical Fiber Links http://dx.doi.org/10.5772/59000 141

$$\mathbf{M}\_{\rm DGD} = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & \cos(\phi) & -\sin(\phi) \\ 0 & 0 & \sin(\phi) & \cos(\phi) \end{bmatrix} \tag{34}$$

The value of phase shift can be given by:

polarization effects due to interaction between PMD and PDL can significantly impair optical fiber transmission systems. The accumulative PMD and PDL impairment is more dangerous

Another polarization effect which is closely related to PDL is PDG.This phenomenon is present in optical amplifiers (first of all in Semiconductor Optical Amplifier). Polarization dependent gain can be defined as absolute value or the relative difference between an optical amplifier

10

Polarization Hole Burning phenomenon is responsible for the PDG effect. It is important to know that PDG effect is observed for linear polarized optical signals which are amplified.

The analysis of impact of optical fiber polarization properties on optical signal transmissions requires a detailed description of polarization effects. The most popular approaches are using an optical fiber links modeling based on homogeneous polarization segments and electro‐

In general, an optical fiber exhibits axially-varing birefringence and can be represented by a series of short and homogeneous polarization segments. Each polarization is described as the randomly rotated polarization elements. These polarization elements are characterised by

The birefringent element can be represented by retarder (phase shifter). In terms of the Jones

j 2

f -

j 2

f

e 0 J , 0 e

é ù ê ú <sup>=</sup> ê ú ê ú ê ú ë û

where ϕ is is the total phase shift between the polarization signal components (two polarization

DGD

In terms of the Mueller matrix the birefringent element is given by:

<sup>G</sup> PDG[dB] 10log , <sup>G</sup> æ ö <sup>=</sup> ç ÷

Polarization Dependent Gain for circular polarization can be neglected [13].

max

(32)

(33)

min

è ø

for lightwave communication systems than a pure PMD or PDL impairment.

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

**3.3. Polarization Dependent Gain**

maximium gain (Gmax) and minimum one (Gmin):

**4. Modeling of polarization phenomena**

birefringence (i.e. PMD) or dichroism (i.e. PDL).

magnetic wave propagation equations.

matrix this element is described by:

modes).

$$
\Phi = \texttt{\texttt{\texttt{\\_}}\texttt{\\_}} \cdot \texttt{\texttt{\\_}}\texttt{\\_}\tag{35}
$$

where τg,r is DGD and ω is angular frequency.

The phase shift between the two polarization signal components (two polarization modes) can be also expressed as:

$$
\Phi = \mathbf{b} \cdot \mathbf{L}\_{\text{el}} \,\prime \tag{36}
$$

where b is birefringence of birefringent element and Lel is birefringent element length.

Then PDL element is described by the following Jones matrix [14]:

$$\mathbf{J}\_{\rm PDL} = \begin{bmatrix} \mathbf{e}^{-\frac{\alpha\_1}{2}} & \mathbf{0} \\ \mathbf{e}^{-\frac{\alpha\_1}{2}} & \mathbf{0} \\ \mathbf{0} & \mathbf{e}^{\frac{\alpha\_1}{2}} \end{bmatrix}. \tag{37}$$

where α<sup>l</sup> is defined as: PDL [dB]=10log10(exp(2α<sup>l</sup> )).

The Mueller matrix corresponding to equation (37) is equal to:

$$\mathbf{M}\_{\rm PDL} = \begin{bmatrix} \frac{1+\alpha^2}{2} & \frac{1-\alpha^2}{2} & 0 & 0\\ \frac{1-\alpha^2}{2} & \frac{1+\alpha^2}{2} & 0 & 0\\ 0 & 0 & \alpha & 0\\ 0 & 0 & 0 & \alpha \end{bmatrix} . \tag{38}$$

Here, value of α equals to: α = 10<sup>−</sup> PDL dB <sup>10</sup> .

Figure 12 illustrates an optical fiber link which is split into some polarization segments (rotated polarization elements).

**Figure 12.** Optical fiber link model consists of N polarization segments; Ms,1, Ms,2, Ms,n – matrix of polarization ele‐ ments, Θ1, Θ2, Θn – angle of rotation

If we take into account only the PMD effect then the matrix of a single polarization segment Ms,n is given by:

$$\mathbf{M}\_{\rm s,n} = \mathbf{M}\_{\rm R,n}(-2\Theta\_{\rm n}) \cdot \mathbf{M}\_{\rm DGD,n} \cdot \mathbf{M}\_{\rm R,n}(2\Theta\_{\rm n}).\tag{39}$$

When considering the PMD and PDL effect, matrix of a single polarization segment is can be writing as:

$$\mathbf{M}\_{\mathbf{s},\mathbf{n}} = \mathbf{M}\_{\mathbf{R},\mathbf{n}} (-\mathcal{D}\Theta\_{\mathbf{n}}) \cdot \mathbf{M}\_{\mathbf{PDL},\mathbf{n}} \cdot \mathbf{M}\_{\mathbf{DGD},\mathbf{n}} \cdot \mathbf{M}\_{\mathbf{R},\mathbf{n}} (\mathcal{D}\Theta\_{\mathbf{n}}).\tag{40}$$

The matrix MT of the whole optical fiber link which consists of N polarization segments is equal to:

$$\mathbf{M}\_{\rm T} = \mathbf{M}\_{\rm s,N} \cdot \dots \cdot \mathbf{M}\_{\rm s,3} \cdot \mathbf{M}\_{\rm s,2} \cdot \mathbf{M}\_{\rm s,1'} \tag{41}$$

Furthermore, the PDG element matrix (MPDG) can be described by the following equation:

$$
\begin{bmatrix}
\frac{\mathbf{g}^2 + 1}{2} & \frac{\mathbf{g}^2 - 1}{2} & 0 & 0 \\
\frac{\mathbf{g}^2 - 1}{2} & \frac{\mathbf{g}^2 + 1}{2} & 0 & 0 \\
0 & 0 & \mathbf{g} & 0 \\
0 & 0 & 0 & \mathbf{g}
\end{bmatrix}
\tag{42}
$$

where g is the PDG coefficient equals to g = 10 PDG dB <sup>10</sup> .

Moreover, to describe the backscattering process we treat an optical fiber link as a cascade of backscattering elements. We treat Rayleigh backscattering as many small reflections distributed over the optical fiber link. For a single element the matrix representing the round-trip propagation (fiber in forward direction, reflector, fiber in backward direction) is computed by:

$$\mathbf{M}\_{\rm Rs,1} = \mathbf{M}\_{\rm s,1}^{\rm T} \cdot \mathbf{M}\_{\rm R} \cdot \mathbf{M}\_{\rm s,1'} \tag{43}$$

where M s,1 Tis the transpose of Ms,1, and MR is the Mueller matrix of a reflection:

$$\mathbf{M}\_{\mathbb{R}} = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & -1 \end{bmatrix}. \tag{44}$$

For light propagating to the end of the N-th element the round-trip Mueller matrix has the following form [4]:

$$\mathbf{M}\_{\mathrm{Rs},\mathcal{N}} = \mathbf{M}\_{\mathrm{s},1}^{\mathrm{T}} \cdot \mathbf{M}\_{\mathrm{s},2}^{\mathrm{T}} \cdot \dots \cdot \mathbf{M}\_{\mathrm{s},\mathcal{N}}^{\mathrm{T}} \cdot \mathbf{M}\_{\mathrm{R}} \cdot \mathbf{M}\_{\mathrm{s},\mathcal{N}} \cdot \dots \cdot \mathbf{M}\_{\mathrm{s},2} \cdot \mathbf{M}\_{\mathrm{s},1}.\tag{45}$$

We can use polarization segments model for the DGD and PDL values calculation. We should take into account Jones and Muller notation.

#### **4.1. Jones notation**

**Figure 12.** Optical fiber link model consists of N polarization segments; Ms,1, Ms,2, Ms,n – matrix of polarization ele‐

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

If we take into account only the PMD effect then the matrix of a single polarization segment

When considering the PMD and PDL effect, matrix of a single polarization segment is can be

The matrix MT of the whole optical fiber link which consists of N polarization segments is equal

Furthermore, the PDG element matrix (MPDG) can be described by the following equation:

g 1g 1 0 0 2 2

é ù + - ê ú

g 1g 1 0 0 , 2 2 0 0 g0 0 0 0g

ë û

PDG dB

Moreover, to describe the backscattering process we treat an optical fiber link as a cascade of backscattering elements. We treat Rayleigh backscattering as many small reflections distributed over the optical fiber link. For a single element the matrix representing the round-trip propagation (fiber in forward direction, reflector, fiber in backward direction) is

<sup>10</sup> .

g= 10

2 2

2 2

where g is the PDG coefficient equals to

computed by:


Ms,n R,n n DGD,n R,n n M -2Q )×M( ×= (M 2Q .) (39)

Ms,n R,n n PDL,n DGD,n R,n n M -2Q )×M ×= M ×M (2Q ).( (40)

M M ... M M M , T s,N s,3 s,2 s,1 × × × ×= (41)

(42)

ments, Θ1, Θ2, Θn – angle of rotation

Ms,n is given by:

writing as:

to:

Differential Group Delay value can be found by the following relationship [15]:

$$\pi\_{\mathbf{g},\mathbf{r}} = \frac{\left| \text{Arg}\left(\frac{\lambda\_{\mathbf{r},1}}{\lambda\_{\mathbf{r},2}}\right) \right|}{\mathbf{do}} \tag{46}$$

where: Arg denotes the argument function, λτ,1 and λτ,2 are two eigenvalues of matrix M T(ω + dω) ⋅M T −1(ω); M T −1 is inverse matrix.

Polarization Dependent Loss in the unit of dB at angular frequency ω is given by [16]:

$$\text{PDL[dB]} = 10 \log\_{10} \left( \frac{\lambda\_{a,1}}{\lambda\_{a,2}} \right) \tag{47}$$

where: λα,1 and λα,2 are two eigenvalues of matrix M T T(ω) ⋅M T(ω), where M T <sup>T</sup> is transpose matrix.

#### **4.2. Mueller notation**

Differential Group Delay value can be expressed as the length of the PMD vector Ω <sup>→</sup> [6]:

$$\left| \mathbf{r}\_{\mathbf{g},\mathbf{r}} = \left| \overrightarrow{\boldsymbol{\Omega}} \right| = \sqrt{\boldsymbol{\Omega}\_{\mathbf{x}}^{2} + \boldsymbol{\Omega}\_{\mathbf{y}}^{2} + \boldsymbol{\Omega}\_{\mathbf{z}}^{2}} \,\tag{48}$$

The PMD vector after (n+1)-th polarization segment may be written as [6]:

$$
\overrightarrow{\dot{\Omega}}\_{\text{n}+1} = \overrightarrow{\text{D}\ddot{\Omega}}\_{\text{n}+1} + \text{MB}\overrightarrow{\Omega}\_{\text{n}}\tag{49}
$$

where Ω → <sup>n</sup> is the PMD dispersion vector of the first n polarization segments, Δ Ω → n+1 is the PMD vector of the (n+1)-th polarization segments, matrix MB represents a transformation of the PMD vector caused by the propagation through the (n+1)-th polarization segment.

The recursive relation for the PMD vector is given by:

$$
\begin{bmatrix}
\boldsymbol{\Omega}\_{\mathbf{x},\mathbf{n}+1} \\
\boldsymbol{\Omega}\_{\mathbf{y},\mathbf{n}+1} \\
\boldsymbol{\Omega}\_{\mathbf{z},\mathbf{n}+1}
\end{bmatrix} = \begin{bmatrix}
\boldsymbol{\tau}\_{\mathbf{g},\mathbf{r},\mathbf{n}+1} \\
0 \\
0
\end{bmatrix} + \begin{bmatrix}
\mathbf{m}\_{11} & \mathbf{m}\_{12} & \mathbf{m}\_{13} \\
\mathbf{m}\_{21} & \mathbf{m}\_{22} & \mathbf{m}\_{23} \\
\mathbf{m}\_{31} & \mathbf{m}\_{32} & \mathbf{m}\_{33}
\end{bmatrix} \cdot \begin{bmatrix}
\boldsymbol{\Omega}\_{\mathbf{x},\mathbf{n}} \\
\boldsymbol{\Omega}\_{\mathbf{y},\mathbf{n}} \\
\boldsymbol{\Omega}\_{\mathbf{z},\mathbf{n}}
\end{bmatrix}.\tag{50}
$$

We use 3x3 matrix in equation (50). Because we assume that SOP=1.

To calculation the PDL value of an optical component or optical fiber link, one must determine the minimum and maximum transmission. Because of this we should take into account 4x4 matrix:

$$\begin{array}{ccccc} \mathbf{m}\_{00} & \mathbf{m}\_{01} & \mathbf{m}\_{02} & \mathbf{m}\_{03} \\ \mathbf{m}\_{10} & \mathbf{m}\_{11} & \mathbf{m}\_{12} & \mathbf{m}\_{13} \\ \mathbf{m}\_{20} & \mathbf{m}\_{21} & \mathbf{m}\_{22} & \mathbf{m}\_{23} \\ \mathbf{m}\_{30} & \mathbf{m}\_{31} & \mathbf{m}\_{32} & \mathbf{m}\_{33} \end{array} \tag{51}$$

Polarization Dependent Loss in the unit of dB is [17]:

$$\text{PDL[dB]} = 10\log\_{10}\left(\frac{\text{m}\_{00} + \sqrt{\text{m}\_{01}^2 + \text{m}\_{02}^2 + \text{m}\_{03}^2}}{\text{m}\_{00} - \sqrt{\text{m}\_{01}^2 + \text{m}\_{02}^2 + \text{m}\_{03}^2}}\right). \tag{52}$$

For an understanding of linear and, first of all, nonlinear optical effects in optical fiber links it is necessary to consider the electromagnetic wave propagation. The linear and nonlinear optical effects in an optical fiber are described by so called nonlinear Schroedinger propagation equation. The nonlinear coupled Schroedinger propagation equations governing evolution of an optical pulse consisiting of the two polarization components along a fiber link (z) are given by [18]:

$$\frac{\partial \mathbf{E}\_{\mathbf{x}}}{\partial \mathbf{z}} = -\frac{\alpha\_{\mathbf{x}}}{2} \mathbf{E}\_{\mathbf{x}} + \mathbf{j} \frac{\mathbf{B}\_{2}}{2} \frac{\partial^{2} \mathbf{E}\_{\mathbf{x}}}{\partial \mathbf{t}^{2}} + \mathbf{j} \gamma \left( \left| \mathbf{E}\_{\mathbf{x}} \right|^{2} + \frac{2}{3} \left| \mathbf{E}\_{\mathbf{y}} \right|^{2} \right) \mathbf{E}\_{\mathbf{x}} + \mathbf{j} \frac{\gamma}{3} \mathbf{E}\_{\mathbf{x}}^{\*} \mathbf{E}\_{\mathbf{y}}^{2} \tag{53}$$

$$\frac{\partial \mathbf{E\_y}}{\partial \mathbf{z}} = -\frac{\alpha\_\mathbf{y}}{2} \mathbf{E\_y} + \mathbf{j} \frac{\mathbf{B\_2}}{2} \frac{\partial^2 \mathbf{E\_y}}{\partial \mathbf{t}^2} + \mathbf{j} \eta \left( \left| \mathbf{E\_y} \right|^2 + \frac{2}{3} \left| \mathbf{E\_x} \right|^2 \right) \mathbf{E\_y} + \mathbf{j} \frac{\gamma}{3} \mathbf{E\_y^\*} \mathbf{E\_{x'}} \tag{54}$$

Where Ex, Ey are slowly varying amplitudes, αx and α<sup>y</sup> is attenuation coefficient for Ex and Ey, respectively. Moreover, β2 is second-order term of the expansion of the propagation constant (the group velocity dispersion parameter), γ is the nonlinear parameter,\* designates complex conjugation, j is imaginary unit.

222

The PMD vector after (n+1)-th polarization segment may be written as [6]:

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

The recursive relation for the PMD vector is given by:

+ +

Polarization Dependent Loss in the unit of dB is [17]:

+ +

We use 3x3 matrix in equation (50). Because we assume that SOP=1.

<sup>n</sup> is the PMD dispersion vector of the first n polarization segments, Δ

x,n 1 g,r,n 1 11 12 13 x,n y,n 1 21 22 23 y,n 31 32 33 z,n 1 z,n

é ù W W é ù <sup>t</sup> é ù é ù ê ú ê ú ê ú ê ú ê ú W= + ê ú × Wê ú ê ú ê ú ê ú ê ú ê ú ë û W W ê ú ë û ë û ê ú ë û

0 mmm

To calculation the PDL value of an optical component or optical fiber link, one must determine the minimum and maximum transmission. Because of this we should take into account 4x4

> 00 01 02 03 10 11 12 13 20 21 22 23 30 31 32 33

mmmm , mmmm

mmmm

é ù ê ú

mmmm

ë û

PDL[dB] 10log .

222 00 01 02 03 <sup>10</sup> <sup>222</sup> 00 01 02 03

m mmm

æ ö + ++ ç ÷ <sup>=</sup>

For an understanding of linear and, first of all, nonlinear optical effects in optical fiber links it is necessary to consider the electromagnetic wave propagation. The linear and nonlinear optical effects in an optical fiber are described by so called nonlinear Schroedinger propagation equation. The nonlinear coupled Schroedinger propagation equations governing evolution of an optical pulse consisiting of the two polarization components along a fiber link (z) are given

m mmm


PMD vector caused by the propagation through the (n+1)-th polarization segment.

vector of the (n+1)-th polarization segments, matrix MB represents a transformation of the

mmm 0 mmm .

where Ω

matrix:

by [18]:

→

g,r xyz t = W = W +W +W , ur (48)

<sup>W</sup>n1 n1 n + + <sup>W</sup> <sup>+</sup> MBW= ,D ur ur ur (49)

Ω

n+1 is the PMD

(50)

(51)

(52)

→

A numerical approach is necessary for the polarization and nonlinear propagation equations solution.

The most popular numerical method is Split-Step Fourier Method. There is useful to write equations (53) and (54) formally in the following form [19]:

$$\frac{\partial \overline{\mathbf{E}}\left(\mathbf{T}, \mathbf{z}\right)}{\partial \mathbf{z}} = \left[\ell\_1(\mathbf{T}) + \ell\_2(\mathbf{T}) + \aleph(\mathbf{z})\right] \overline{\mathbf{E}}\left(\mathbf{T}, \mathbf{z}\right),\tag{55}$$

where: E → = Ex Ey and T=t-β1z; t is time, β<sup>1</sup> is first-order term of the expansion of the propagation constant (differential coefficient of the propagation constant with respect to optical frequency). The operators on the right side of equation (55) are linear ℓ1(T ), ℓ2(T ) and nonlinear ℵ(z). These operators have the following definitions [19]:

$$\ell\_1(\mathbf{T}) = \frac{1}{2} \begin{bmatrix} \mathbb{B}\_1 & 0 \\ 0 & -\mathbb{B}\_1 \end{bmatrix} \cdot \frac{\mathcal{O}}{\mathcal{O}\mathbf{T}} \,\prime \tag{56}$$

$$\boldsymbol{\ell}\_{2}(\mathbf{T}) = -\frac{1}{2}\mathbf{I} \left( \mathbf{j} \mathbb{B}\_{2} \frac{\hat{\sigma}^{2}}{\hat{\sigma}\mathbf{T}^{2}} + \frac{1}{3}\mathbb{B}\_{3} \frac{\hat{\sigma}^{3}}{\hat{\sigma}\mathbf{T}^{3}} \right) \boldsymbol{\prime} \tag{57}$$

$$\begin{aligned} \mathbf{N}(\mathbf{z}) &= -\mathbf{j}\frac{\mathbf{y}}{3} \begin{bmatrix} \mathbf{3}\left|\mathbf{E}\_{\mathbf{x}}\right|^{2} + 2\left|\mathbf{E}\_{\mathbf{y}}\right|^{2} & \mathbf{E}\_{\mathbf{x}}^{\*}\mathbf{E}\_{\mathbf{y}} \\ \mathbf{E}\_{\mathbf{y}}^{\*}\mathbf{E}\_{\mathbf{x}} & 3\left|\mathbf{E}\_{\mathbf{y}}\right|^{2} + 2\left|\mathbf{E}\_{\mathbf{x}}\right|^{2} \end{bmatrix} + \\\ \mathbf{j}\frac{1}{2}\begin{bmatrix} -\mathbf{j}\alpha\_{1} + \beta\_{0} & 0 \\ 0 & \mathbf{j}\alpha\_{1} - \beta\_{0} \end{bmatrix} \end{aligned} \tag{58}$$

Where β<sup>0</sup> is zeroth-order term of the expansion of the propagation constant, β3 is third-order term of the expansion of the propagation constant (third differential coefficient of the propa‐ gation constant with respect to optical frequency). The symbol I stands for the identity matrix.

The linear operators describe first-order and high-order PMD effect. It is a function of T alone. The nonlinear operator includes phenomena that do not depend on T i.e. PMD, nonlinear effects. It is a function of z alone.

The Split-Step Fourier Method obtains an aproximate solution by assuming that in propagating an optical pulse over a small distance h optical effects are independent [19].

$$\mathbf{E}\left(\mathbf{T},\mathbf{z}+\mathbf{h}\right)\approx\mathbf{N}\mathbf{L}\_1\mathbf{L}\_2\mathbf{E}\left(\mathbf{T},\mathbf{z}\right),\tag{59}$$

where:

$$\mathbf{L}\_1 = \exp(\ell\_1 \mathbf{h}),\tag{60}$$

$$\mathbf{L}\_2 = \exp\left(\ell\_2 \mathbf{h}\right),\tag{61}$$

$$\mathbf{N} = \exp\left(\int\_{\mathbf{z}}^{\mathbf{z}+\mathbf{h}} \aleph\left(\mathbf{z}'\right) \mathrm{d}\mathbf{z}'\right) \approx \exp\left(\mathbf{h}\frac{\aleph(\mathbf{z}+\mathbf{h}) + \aleph\left(\mathbf{z}\right)}{2}\right). \tag{62}$$

Propagation from z to z+h is carried out in two steps. In the first step linear effects only (L1≠0, L2≠0, N=0) are taken into account. In the first step vice versa (L1=0, L2=0, N≠0).

Figure 13 shows schematic illustration of the Split-Step Fourier Method. Fiber length is split into a large number of small sygments of width h.

**Figure 13.** Schematic illustration of the Split-Step Fourier Method

It is important to know, that the linear operators are evaluated on the Fourier domain. In turn, nonlinear operator is evaluated on the time domain.
