**5. Polarizing components**

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

The Split-Step Fourier Method obtains an aproximate solution by assuming that in propagating

( ) ( ) () z h

Propagation from z to z+h is carried out in two steps. In the first step linear effects only (L1≠0,

Figure 13 shows schematic illustration of the Split-Step Fourier Method. Fiber length is split

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

N exp z' dz' exp h . <sup>2</sup> <sup>+</sup> æ ö æ ö À + +À =À» ç ÷ ç ÷

L2≠0, N=0) are taken into account. In the first step vice versa (L1=0, L2=0, N≠0).

z

into a large number of small sygments of width h.

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

nonlinear operator is evaluated on the time domain.

( ) () 1 2 E T,z h NL L E T,z , + » (59)

( ) 1 1 L exp h , = l (60)

( ) 2 2 L exp h , = l (61)

zh z

è ø è ø <sup>ò</sup> (62)

an optical pulse over a small distance h optical effects are independent [19].

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

effects. It is a function of z alone.

where:

Optical polarizing components belong to a class of optical components characterized by the modyfication of some polarization properties of light wave. Optical polarizing components are very useful for optical fiber communication technologies. Some of them are used for PMD and PDL compensating, Polarization Division Multiplexing transmission technique and measurement procedures. Ones of the most important optical polarizing components for modern, high capacity optical communication solutions are: polarization controller, polariza‐ tion attractor, polarization scrambler and polarization effects emulator.

#### **5.1. Polarization controller**

The polarization controller is an optical component which allows one to modify the polariza‐ tion state of light. The polarization controller is used to change polarized (or unpolarized) light into any well-defined SOP. Typically, the polarization controller consists of rotated retarders (wave plates). We can distingush the polarization controllers which are based on: two rotated quarter-wave plates, two rotated quarter-wave plates and one rotated half-wave plate or one rotated quarter-wave plate and one rotated half-wave plate. Figure 14 presents structure of polarization controller which is based on two rotated quarter-wave plates and distribution of SOPs at the polarization controller output port.

**Figure 14.** Polarization controller based on two rotated quarter-wave plates; structure of polarization controller (a), output SOPs distribution on the Poincaré sphere (b)

This polarization controller changes an arbitrary SOP into the other arbitrary SOP.

Figure 15 shows structure of polarization controller which is based on two rotated quarterwave plates, one rotated half-wave plate and distribution of SOPs at the polarization controller output port.

This polarization controller is similar to above one. It transforms an arbitrary SOP into the other arbitrary SOP. Finally, Figure 16 presents structure of polarization controller which is based on one rotated quarter-wave plate, one rotated half-wave plate and distribution of SOPs at the polarization controller output port.

**Figure 15.** Polarization controller based on two rotated quarter-wave plates and one rotated half-wave plate; structure of polarization controller (a), output SOPs distribution on the Poincaré sphere (b)

**Figure 16.** Polarization controller based on one rotated quarter-wave plate and one rotated half-wave plate; structure of polarization controller (a), output SOPs distribution on the Poincaré sphere (b)

This type of polarization controller only transforms linear polarization into an arbitrary SOP. We would expect flowed operation of this polarization controller with the other input SOP. This case is shown in Figure 17.

**Figure 17.** Distribution of SOPs at the polarization controller output port for circular input SOP

#### **5.2. Polarization attractor**

**Figure 16.** Polarization controller based on one rotated quarter-wave plate and one rotated half-wave plate; structure

**Figure 15.** Polarization controller based on two rotated quarter-wave plates and one rotated half-wave plate; structure

This type of polarization controller only transforms linear polarization into an arbitrary SOP. We would expect flowed operation of this polarization controller with the other input SOP.

of polarization controller (a), output SOPs distribution on the Poincaré sphere (b)

of polarization controller (a), output SOPs distribution on the Poincaré sphere (b)

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

**Figure 17.** Distribution of SOPs at the polarization controller output port for circular input SOP

This case is shown in Figure 17.

In real fibers the SOPs are not preserved because of the random birefringence. The uncontrolled SOPs variable can dramatically affect the performances of telecommunication systems. This phenomenon is very important, especially for demultiplexing process for Polarization Division Multiplexing transmission system. Possibility of polarization controlling is key issue for modern optical fiber communication technologies. The optical component which can stabilize an arbitrary polarized optical signal by lossless and instantaneous interaction is polarization attractor. This type of controlling the optical signal polarization can be based on: stimulated Brillouin scattering, stimulated Raman scattering or four wave mixing phenomenon. Here we focuse on the stimulated Raman scattering for polarization attraction effect [2, 3]. An arbitrary input SOP of the optical signal is pulled (attracted) by the SOP of the propagating pump (Raman pump), so that at the fiber output the signal SOP is matched the pump SOP. The power evolution of the pump (P <sup>→</sup> ) and signal (S <sup>→</sup> ) for copumped configuration along the optical fiber link can be modeled by means of coupled equations, respectively [20]:

$$\frac{d\overline{\mathbf{P}}}{d\mathbf{z}} = -\alpha\_{\rm p}\overline{\mathbf{P}} \cdot \frac{\alpha\nu\_{\rm p}}{2\alpha\alpha\_{s}}\mathbf{g}\_{\rm R}\left(\mathbf{P}\_{0}\overline{\mathbf{S}} + \mathbf{S}\_{0}\overline{\mathbf{P}}\right) + \left(\alpha\mathbf{o}\_{\rm p}\overline{\mathbf{b}} + \overline{\mathbf{W}\_{\rm p}^{N\overline{\mathbf{L}}}}\right) \times \overline{\mathbf{P}},\tag{63}$$

$$\frac{d\vec{\mathbf{S}}}{d\mathbf{z}} = -\alpha\_s \vec{\mathbf{S}} + \frac{1}{2} \mathbf{g}\_{\text{R}} \left( \mathbf{S}\_0 \vec{\mathbf{P}} + \mathbf{P}\_0 \vec{\mathbf{S}} \right) + \left( \omega\_s \vec{\mathbf{b}} + \overline{\mathbf{W}\_s^{\text{NL}}} \right) \times \vec{\mathbf{S}},\tag{64}$$

where ωp and ω<sup>s</sup> are pump and signal carrier angular frequencies, αp and α<sup>s</sup> are optical fiber attenuation coefficients for the pump and signal wavelengths, respectively. The gR component is the Raman gain coefficient. The vector lengths P0=| P <sup>→</sup> | and S0=| S <sup>→</sup> | represent the pump and signal powers, respectively. The vector b <sup>→</sup> is the local linear birefringence vector for optical fiber. The vectors W p→ NL and W s→ NLare given by [20]:

$$\overline{\mathbf{W}\_{\text{P}}^{\text{NL}}} = \frac{2}{3} \boldsymbol{\upchi}\_{\text{P}} \left( -2 \mathbf{S}\_{\text{S},1'} - 2 \mathbf{S}\_{\text{S},2'} \mathbf{S}\_{\text{P},3} \right) \tag{65}$$

$$\overline{\mathbf{W}\_{\rm s}^{\rm NL}} = \frac{2}{3} \gamma\_{\rm s} \left( -2 \mathbf{S}\_{\rm P,1'} - 2 \mathbf{S}\_{\rm P,2'} \mathbf{S}\_{\rm S,3} \right) \tag{66}$$

where γp and γs are the nonlinear coefficients, SP,1, SP,2,SP,3, SS,1, SS,2, SS,3 are the Stokes parameters for the pump and signal, respectively.

The values of polarization attractor parameters (i.e.: pump power, pump SOP) should be accurately selected depending on expected polarization pulling.

Figure 18 shows scheme of polarization attractor based on stimulated Raman scattering.

**Figure 18.** Polarization attractor based on Raman scattering

Figures 19 and 20 demonstrate simulated examples of polarization pulling effect for pump power equals to 1 W, 2W and 5 W. The simulated polarization attractor is based on standard single mode optical fiber [21].

**Figure 19.** Simulated examples of polarization pulling; distribution of polarized signals at the attractor input port (a), distribution of output signal SOPs at the output attractor port for pump power 1 W (b)

**Figure 20.** Simulated examples of polarization pulling; distribution of output signal SOPs at the output attractor port for pump power 2 W (a), distribution of output signal SOPs at the output attractor port for pump power 5 W (b)

It should be note that for stimulated Raman scattering the proper Raman polarization pulling and amplification for optical fiber communication systems may be simultaneously achieved.

#### **5.3. Polarization scrambler**

**Figure 18.** Polarization attractor based on Raman scattering

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

single mode optical fiber [21].

Figures 19 and 20 demonstrate simulated examples of polarization pulling effect for pump power equals to 1 W, 2W and 5 W. The simulated polarization attractor is based on standard

**Figure 19.** Simulated examples of polarization pulling; distribution of polarized signals at the attractor input port (a),

**Figure 20.** Simulated examples of polarization pulling; distribution of output signal SOPs at the output attractor port for pump power 2 W (a), distribution of output signal SOPs at the output attractor port for pump power 5 W (b)

distribution of output signal SOPs at the output attractor port for pump power 1 W (b)

It is known that, d ue to the random nature of polarization mode coupling in an optical fiber several polarization effects (PMD, PDL, PDG) may occur that lead to impairments in long haul and high bit rate optical fiber transmission systems. Polarization scrambling the states of polarization has been shown to be technique that can reduction polarization impairments or the reduction of measurement uncertainly. A polarization scrambler actively changes the SOPs using polarization modulation method. In generally, the polarization scrambler configuration consists of rotating retarders (wave plates) or phase shifting elements. Furthermore, it is often necessary that the scrambler output SOPs are distributed uniformly on the entire Poincarè sphere. The spherical radial distribution function is very useful tool for the SOPs distribution analysis on the Poincarè sphere [22]. The spherical radial distribution function is the modified form of the well known plane radial distribution function. The spherical radial distribution function is defined as follows [22]:

$$\mathbf{g(d)} = \frac{\mathbf{K(d)} \mathbf{A\_T}}{\mathbf{A(d)} \mathbf{K\_T}},\tag{67}$$

where K(d) is the total number of pairs of the points separated by a given range of radial distances (d, d+Δd), A(d) is area of the sphere between two circles cd and cd+Δd, KT is the total number of pairs of the points on the sphere; KT is equal to N2 -N, where N is number of points on the sphere, AT is area of the sphere (Figure 21).

**Figure 21.** Example of spherical radial distribution function calculation for one reference point

The value of radial distance d changes from 0 to π-Δd, step is equal to Δd. The "great circle" distance between two points nn and nk (Figure 21), whose coordinates are (Θn, ϕn) and (Θk, ϕk), is given by so called Haversine formula:

$$\mathbf{d}\_{n,\mathbf{k}} = 2\mathbf{R}\arcsin\left(\sqrt{\sin^2\left(\frac{\Theta\_\mathbf{n}-\Theta\_\mathbf{k}}{2}\right) + \sin\left(\Theta\_\mathbf{n}\right)\sin\left(\Theta\_\mathbf{k}\right)\sin^2\left(\frac{\Phi\_\mathbf{k}-\Phi\_\mathbf{n}}{2}\right)}\right),\tag{68}$$

where R is the sphere radius.

We can distinguish three typical theoretical distributions: uniform (Figure 22a), random (Figure 23a) and clustered (Figure 24a). In turn, Figures 22b, 23b and 24b show the spherical radial distribution as a function of the distance d for the uniform, random and clustered distribution, respectively [22].

**Figure 22.** Theoretical uniform distribution; SOPs distribution on the Poincarè sphere (a), spherical radial distribution function versus radial distance (b)

**Figure 23.** Theoretical random distribution; SOPs distribution on the Poincarè sphere (a), spherical radial distribution function versus radial distance (b)

**Figure 24.** Theoretical clustered distribution; SOPs distribution on the Poincarè sphere (a), spherical radial distribution function versus radial distance (b)

For the uniform distribution (Figure 22b) the peaks on the g(d) curve provides information about the mean distance of the following neighbouring points (SOPs) on the Poincarè sphere. In the case of the random distribution (Figure 23b) the value of g(d) is close to 1. For the clustered distribution (Figure 24b) the localization of the first minimum on the g(d) curve provide information about the mean dimension (diameter) of the clusters. The location of the first lower peaks on the curve indicates the mean distance between clusters.

The analysis of the distribution of SOPs generated by polarization scramblers shows that SOPs distribution is clustered for three (and less) rotating retarders and for four (and less) phase shifting elements. For four and more rotating retarders and for five and more phase shifting elements random distribution is obtained [22].

## **5.4. Polarization effects emulator**

() () 2 2 n k k n

(68)

n,k n k d 2Rarcsin sin sin sin sin , 2 2 æ ö æ ö Q -Q æ ö f -f <sup>=</sup> ç ÷ ç ÷ +Q Q ç ÷ è ø è ø è ø

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

We can distinguish three typical theoretical distributions: uniform (Figure 22a), random (Figure 23a) and clustered (Figure 24a). In turn, Figures 22b, 23b and 24b show the spherical radial distribution as a function of the distance d for the uniform, random and clustered

**Figure 22.** Theoretical uniform distribution; SOPs distribution on the Poincarè sphere (a), spherical radial distribution

**Figure 23.** Theoretical random distribution; SOPs distribution on the Poincarè sphere (a), spherical radial distribution

where R is the sphere radius.

distribution, respectively [22].

function versus radial distance (b)

function versus radial distance (b)

You know well that polarization effects due to interaction between PMD and PDL can significantly impair optical fiber transmission systems. When PMD and PDL are both present they interact must be studied together. Emulating of PMD and PDL is one way to test and verify new transmission systems in the presence of PMD and PDL effects. Polarization effects emulation play a useful role, since it is possible to examine a large ensemble of system states far more rapidly than in a test bed with commercially available fiber optics. The polarization effects emulation devices can be split into two groups: statistical and deterministic emulators. Devices which are intended to mimic the random statistical behavior of a long single mode fiber links are termed as statistical emulators. Statistical polarization effect emulators should accurately reproduce the statistics of the polarization effect that a signal would see on a real link, as well as have good stability and repeatability. Devices that map the polarization effect space to the emulator settings and predictably generate the desired values are generally termed as deterministic emulators.

We typically have PMD and PDL statistical emulators and PDL deterministic emulators. Through pure statistical nature of the PMD effect, the PMD deterministic emulators should not be used for an optical communication systems testing.

Each statistical emulator that realistically simulates real optical fiber links should fulfil two criteria [6]:


In [23] is demonstrated that 15 rotated polarization elements (e.g. sections of polarization maintaining fiber) realistically simulate the DGD and PDL distribution and BAC of real optical fiber links. Below, results for statistical emulator consisting of 15 rotated polarization elements. Figure 25 shows the histograms of DGD and PDL. For the statistical PMD emulator the DGD distribution is always indistinguishable from theoretical Maxwellian distribution. In turn, the PDL distribution (in the presence of PMD) is similar to Maxwellian distribution [24].

The theoretical and normalized ACF for both PMD and PDL vectors is shown in Figure 26. **Figure 25.** Statistical distribution of DGD values (a) and PDL values (b)

Figure 25. Statistical distribution of DGD values (a) and PDL values (b)

Figure 26. Theoretical normalized frequency autocorrelation function for PMD

(Ts) and parallel coeffcient (Tp) to the plane of incidence [25]:

 s t 2 s s i

 <sup>2</sup> s t p p i

(70)

 i

(71)

is t

(69)

n cos T t , cos

n cos T t , cos

Dummy Text where:

2cos t , cos n cos

s

The theoretical and normalized ACF for both PMD and PDL vectors is shown in Figure 26.

Now, coming to to the PDL emulator, the simplest deterministic PDL emulator consists of tilted glass plate. The transmission coeffcients of a dielectric surface between two media were derived by Fresnel. They field the orthogonal component i.e. perpendicular coeffcient (Ts) and parallel coeffcient (Tp) to the plane of incidence [25]:

Now, coming to to the PDL emulator, the simplest deterministic PDL emulator consists of tilted glass plate. The transmission coeffcients of a dielectric surface between two media were derived by Fresnel. They field the orthogonal component i.e. perpendicular coeffcient

**Figure 26.** Theoretical normalized frequency autocorrelation function for PMD

$$\mathbf{T}\_s = \frac{\mathbf{n}\_s \cdot \cos\left(\frac{\mathbf{\tilde{x}}\_t}{\mathbf{\tilde{s}}\_i}\right) \cdot \left|\mathbf{t}\_s\right|^2}{\cos\left(\frac{\mathbf{\tilde{x}}\_i}{\mathbf{\tilde{s}}\_i}\right)} \cdot \left|\mathbf{t}\_s\right|^2 \,\mathrm{\,} \tag{69}$$

$$\mathbf{T\_p} = \frac{\mathbf{n\_s} \cdot \cos\left(\frac{\mathbf{\tilde{x}\_t}}{\mathbf{\tilde{s}\_i}}\right)}{\cos\left(\frac{\mathbf{\tilde{x}\_t}}{\mathbf{\tilde{s}\_i}}\right)} \cdot \left|\mathbf{t\_p}\right|^2,\tag{70}$$

where:

Each statistical emulator that realistically simulates real optical fiber links should fulfil two

**1.** Differential Group Delay should be Maxwellian distributed at any fixed optical frequency. This condition is also valid for the PDL effect. In the absence of the PMD the PDL distribution is Rayleigh distribution. But in the presence of PMD the PDL distribution is closed to Maxwellian distribution. Thus the PDL distribution in real optical fiber links can

**2.** Frequency AutoCorrelation Function (ACF) of PMD and PDL vectors should tend toward zero as the frequency separation increases; so called Autocorrelation Function Back‐ ground (BAC) should be lower than 10 %. Autocorrelation Function Background is defined as the mean absolute deviation of the ACF from the expected (mean) value of ACF for the frequencies larger than the autocorrelation bandwidth where the frequency autocorrelation bandwidth of the ACF is the frequency at the half of the variation of the

In [23] is demonstrated that 15 rotated polarization elements (e.g. sections of polarization maintaining fiber) realistically simulate the DGD and PDL distribution and BAC of real optical fiber links. Below, results for statistical emulator consisting of 15 rotated polarization elements. Figure 25 shows the histograms of DGD and PDL. For the statistical PMD emulator the DGD distribution is always indistinguishable from theoretical Maxwellian distribution. In turn, the

The theoretical and normalized ACF for both PMD and PDL vectors is shown in Figure 26.

The theoretical and normalized ACF for both PMD and PDL vectors is shown in Figure 26. Now, coming to to the PDL emulator, the simplest deterministic PDL emulator consists of tilted glass plate. The transmission coeffcients of a dielectric surface between two media were derived by Fresnel. They field the orthogonal component i.e. perpendicular coeffcient (Ts) and

Now, coming to to the PDL emulator, the simplest deterministic PDL emulator consists of tilted glass plate. The transmission coeffcients of a dielectric surface between two media were derived by Fresnel. They field the orthogonal component i.e. perpendicular coeffcient

PDL distribution (in the presence of PMD) is similar to Maxwellian distribution [24].

Figure 25. Statistical distribution of DGD values (a) and PDL values (b)

**Figure 25.** Statistical distribution of DGD values (a) and PDL values (b)

parallel coeffcient (Tp) to the plane of incidence [25]:

Figure 26. Theoretical normalized frequency autocorrelation function for PMD

(Ts) and parallel coeffcient (Tp) to the plane of incidence [25]:

 s t 2 s s i

 <sup>2</sup> s t p p i

(70)

 i

(71)

is t

(69)

n cos T t , cos

n cos T t , cos

Dummy Text where:

2cos t , cos n cos

s

be also approximated by a Maxwellian function.

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

criteria [6]:

ACF.

$$\mathbf{t}\_s = \frac{2\cos\left(\xi\_{\mathbf{i}}\right)}{\cos\left(\xi\_{\mathbf{i}}\right) + \mathbf{n}\_s \cdot \cos\left(\xi\_{\mathbf{i}}\right)},\tag{71}$$

$$\mathbf{t}\_p = \frac{2\cos\left(\frac{\mathbf{r}\_\mathbf{i}}{\mathbf{r}\_\mathbf{i}}\right)}{\cos\left(\frac{\mathbf{r}\_\mathbf{i}}{\mathbf{r}\_\mathbf{i}}\right) + \mathbf{n}\_s \cdot \cos\left(\frac{\mathbf{r}\_\mathbf{i}}{\mathbf{r}\_\mathbf{i}}\right)},\tag{72}$$

$$\mathfrak{k}\_{\mathfrak{k}} = \arcsin\left(\frac{\sin\left(\mathfrak{k}\_{\mathfrak{i}}\right)}{\mathfrak{n}\_{s}}\right). \tag{73}$$

For above equations (69-73): angle ξ<sup>i</sup> is the angle of incidence, angle ξ<sup>t</sup> is the angle of refraction, ns is the index of glass refraction. We assume that the index of air refraction is equal to 1.

Next, the PDL value is given by the following relation:

$$\text{PDL[dB]} = 10 \log\_{10} \left( \frac{\text{T}\_s}{\text{T}\_p} \right) . \tag{74}$$

The PDL value is strong dependent on the angle of incidence. Figure 27 presents PDL in function of this angle. These PDL values are typically for some optical components which are used for optical fiber communication technologies.

**Figure 27.** Polarization Dependent Loss value versus the angle of incidence; ns=1.75

In conclusion, polarization issues become very important especially for long haul and high bit rate lightwave communication systems for which polarization effects, first of all, polarization mode dispersion and polarization dependent loss, become limiting factor. Optical fiber polarization phenomena must be taken into account during planning, installing and monitor‐ ing optical fiber communication systems. Additionally, the fast evolution of optical fiber transmission technologies requires powerful analysis and testing tools that must provide information about all relevant polarization phenomena in optical fiber links.

#### **Author details**

```
Krzysztof Perlicki1,2*
```
Address all correspondence to: perlicki@tele.pw.edu.pl

1 Institute of Telecommunications, Warsaw University of Technology, Warsaw, Poland

2 Orange Labs, Orange Polska, Warsaw, Poland

## **References**

(74)

s 10 p

ç ÷ è ø

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

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

used for optical fiber communication technologies.

**Figure 27.** Polarization Dependent Loss value versus the angle of incidence; ns=1.75

Address all correspondence to: perlicki@tele.pw.edu.pl

2 Orange Labs, Orange Polska, Warsaw, Poland

**Author details**

Krzysztof Perlicki1,2\*

information about all relevant polarization phenomena in optical fiber links.

1 Institute of Telecommunications, Warsaw University of Technology, Warsaw, Poland

The PDL value is strong dependent on the angle of incidence. Figure 27 presents PDL in function of this angle. These PDL values are typically for some optical components which are

In conclusion, polarization issues become very important especially for long haul and high bit rate lightwave communication systems for which polarization effects, first of all, polarization mode dispersion and polarization dependent loss, become limiting factor. Optical fiber polarization phenomena must be taken into account during planning, installing and monitor‐ ing optical fiber communication systems. Additionally, the fast evolution of optical fiber transmission technologies requires powerful analysis and testing tools that must provide


#### **Simulation of Fiber Fuse Phenomenon in Single-Mode Optical Fibers Simulation of Fiber Fuse Phenomenon in Single-Mode Optical Fibers**

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