A similar concept of spun LB fiber represents highly birefringent spun fibers (Spun HB). Spun HB fibers are manufactured by rotating of melted preform also. However, the preform is prepared as for classical PM HB fiber, e.g. Bow-Tie (Laming & Payne, 1989). Spun HB fiber transforms the input linear polarization on to the elliptical. By carefully chosen rotation rate of the preform in relation to the fiber linear birefringence rate, it is possible to attain quasi-circular birefringence with negligible residual linear birefringence (Payne et al., 1982). Generally, we may consider spun HB fibers as a type of PM HB fiber group. The advantage of spun HB fibers is considerable immunity to rising of linear birefringence by fiber bending

limitation for small radius fiber coil fabrication remains (Payne et al., 1982).

*l* . (42)

c and *l*f is fiber length

rotation of the fiber core and

the fiber.

σ

the fiber. When exceeds, the fiber may be broken.

**6.2.2 Spun low- and high-birefringent fibers**

circular fiber core. Stress components are implemented by doping of designated cladding areas with atoms of certain elements, typically boron atoms. In this way, areas with different thermal expansion coefficient are formed. After the drawn fiber cools down, the doped areas cause inner strain, which acts on the fiber core. Doped areas may have variety of shapes as shown in Fig. 13. The influence of fiber latent linear birefringence is strongly exceeded by the imposed birefringence, which temperature and bending dependence is very weak.

In order to characterize the properties of PM HB fibers the polarization crosstalk *CT* is defined (Noda et al., 1986). Polarization crosstalk, defined by relation (40), is given by logarithmic ratio of optical power of excited mode *Px* and optical power of coupled mode *Py*. The polarization crosstalk is typically lower than -40 dB for fiber length of 100 meters (Senior, 2009).

Fig. 13. Various profiles of PM HB fibers: a) elliptical, b) PANDA type, c) Bow-Tie type.

$$CT = 10\log\frac{P\_y}{P\_x}.\tag{40}$$

In the following chapters, fibers with intended polarization preserving properties are discussed. Although they exploit different principles, they may be considered as a special type of PM LB or PM HB fibers, as will be mentioned.

#### **6.2 Fibers with high circular birefringence**

The demands of telecommunication applications for polarization state preserving in fiber may be satisfactory covered by PM fibers. Since, PM fiber allows preservation of state polarization only for one certain polarization plane orientation, they are not well suited for applications in polarimetric fiber sensors. The polarization plane rotates due to the sensing quantity magnitude in case of these sensors. Therefore, the different fiber modifications were studied.

#### **6.2.1 Twisted fibers**

One of the approaches is fiber twisting, which may impose a strong circular birefringence in the core. If the rate of induced circular birefringence will be much greater than the rate of linear birefringence, relation (38) may be modified

$$
\Delta \phi = \sqrt{\phi\_{\mathbf{c}}^2 + \left(\frac{\phi\_{\mathbf{l}}}{2}\right)^2} \stackrel{\phi\_{\mathbf{c}} \approx \phi\_{\mathbf{l}}}{\equiv} \sqrt{\phi\_{\mathbf{c}}^2} = \phi\_{\mathbf{c}}.\tag{41}
$$

The influence of circular birefringence will dominate and the linear birefringence may be neglected. Since the effect of linear birefringence is canceled, twisted fibers belong to group

circular fiber core. Stress components are implemented by doping of designated cladding areas with atoms of certain elements, typically boron atoms. In this way, areas with different thermal expansion coefficient are formed. After the drawn fiber cools down, the doped areas cause inner strain, which acts on the fiber core. Doped areas may have variety of shapes as shown in Fig. 13. The influence of fiber latent linear birefringence is strongly exceeded by the imposed birefringence, which temperature and bending dependence is very weak. In order to characterize the properties of PM HB fibers the polarization crosstalk *CT* is defined (Noda et al., 1986). Polarization crosstalk, defined by relation (40), is given by logarithmic ratio of optical power of excited mode *Px* and optical power of coupled mode *Py*. The polarization crosstalk is typically lower than -40 dB for fiber length of 100 meters

Fig. 13. Various profiles of PM HB fibers: a) elliptical, b) PANDA type, c) Bow-Tie type.

*CT*

type of PM LB or PM HB fibers, as will be mentioned.

**6.2 Fibers with high circular birefringence**

linear birefringence, relation (38) may be modified

φ φ 10log . *y x P*

In the following chapters, fibers with intended polarization preserving properties are discussed. Although they exploit different principles, they may be considered as a special

The demands of telecommunication applications for polarization state preserving in fiber may be satisfactory covered by PM fibers. Since, PM fiber allows preservation of state polarization only for one certain polarization plane orientation, they are not well suited for applications in polarimetric fiber sensors. The polarization plane rotates due to the sensing quantity magnitude in case of these sensors. Therefore, the different fiber modifications

One of the approaches is fiber twisting, which may impose a strong circular birefringence in the core. If the rate of induced circular birefringence will be much greater than the rate of

> φ

 ⎛ ⎞ Δ= + ≅ = ⎜ ⎟ ⎝ ⎠

The influence of circular birefringence will dominate and the linear birefringence may be neglected. Since the effect of linear birefringence is canceled, twisted fibers belong to group

**c l l c c c** 2 2 2 . 2 φ φ

φ φ

*P* = (40)

(41)

(Senior, 2009).

were studied.

**6.2.1 Twisted fibers**

of PM LB fibers. Twisted fiber will behave as polarization plane rotator, which preserves the polarization state of the wave. The polarization plane rotation is proportional to specific rotation of the fiber core and σc and *l*f is fiber length

$$
\Delta \mathcal{a} = \sigma\_{\mathfrak{e}} l\_{\mathfrak{f}}.\tag{42}
$$

If this principle would be utilized in polarimetric fiber sensor application, the polarization rotation due to sensed physical quantity will be additive to inherent polarization rotation of the fiber.

The circular birefringence in fiber core is possible to impose by fiber twisting in a plane, which is perpendicular to fiber longitudinal axis. The rate of circular birefringence corresponds to photo-elastic properties of the core, core refractive index, fiber length and specific torsion rate (relation (23)). In order to minimize the influence of linear birefringence, the specific torsion rate should be maximized. However, this is limited by torsion limit of the fiber. When exceeds, the fiber may be broken.

The disadvantage of the twisted fiber utilization in fiber sensing applications is the temperature dependence of twisting imposed circular birefringence, due to temperature dependence of core anisotropy. The next issue is the fabrication difficulty of small fiber coils for magnetic field sensing around conductors. The bending induced linear birefringence achieves a higher magnitude for small fiber coils. Therefore, the large torsion rate of the fiber has to be used and it may exceed the torsion limit. The approximate torsion limit of common single mode fiber is 100 turns per 1 meter (Payne et al., 1982). Taking into account this limit, it is possible to fabricate fiber coils with minimal diameter of 15 cm (Laming & Payne, 1989).

### **6.2.2 Spun low- and high-birefringent fibers**

The more sophisticated approach to linear birefringence suppression was development of spun fibers. Spun fiber fabrication consists in twisting of melted preform during fiber drawing. During the fiber drawing, all fiber imperfections, as deviation from circularity and other non-uniformities, are spread out in all directions. Therefore, phase retardations, which experience propagating modes, cancel each other out. Since the twisted preform is melted, no stress induced anisotropy is present in the core after the fiber cools down. Simultaneously, the fiber is free from temperature dependency effects. Hence, the spun fiber behaves as an ideal fiber with circular cross-section core, which retain any polarization state of coupled wave, from the input to the output. In principle, spun fibers belong to group of PM LB fibers. We designate them as low-birefringent spun fibers (spun LB). Spun LB fiber exhibits only a negligible latent circular birefringence, due to limited viscosity of the preform during the drawing. The principle of spun LB fiber implies their main disadvantage, which is the sensitivity on fiber bending. This results in stress induced linear birefringence and the limitation for small radius fiber coil fabrication remains (Payne et al., 1982).

A similar concept of spun LB fiber represents highly birefringent spun fibers (Spun HB). Spun HB fibers are manufactured by rotating of melted preform also. However, the preform is prepared as for classical PM HB fiber, e.g. Bow-Tie (Laming & Payne, 1989). Spun HB fiber transforms the input linear polarization on to the elliptical. By carefully chosen rotation rate of the preform in relation to the fiber linear birefringence rate, it is possible to attain quasi-circular birefringence with negligible residual linear birefringence (Payne et al., 1982). Generally, we may consider spun HB fibers as a type of PM HB fiber group. The advantage of spun HB fibers is considerable immunity to rising of linear birefringence by fiber bending

Optical Fiber Birefringence Effects – Sources, Utilization and Methods of Suppression 145

exploited. The approach is based on reciprocity of linear birefringence. Beyond this, the circular birefringence is of non-reciprocal character, which gives usability especially in polarimetric current fiber sensors. Utilizing this fact, a compensation of linear birefringence may be performed on sensor output signal level in case of counter-propagating of two light waves in fiber. Second possibility consists in compensation of modes phase shift on fiber

Since the linear birefringence is of reciprocal character, its influence on polarization state of the wave in fiber is not dependent on the propagation direction. The wave will experience the same polarization state transformation with the same orientation, no matter the propagation direction. Conversely, the magnetic field induced circular birefringence is non-reciprocal. When the wave will propagate in one direction, it will gain a polarization rotation. When the wave will propagate in opposite direction, it will gain rotation in opposite direction. The total rotation will be the double of the rotation in one direction. Setups, which exploit the reciprocity of linear birefringence, have been demonstrated as polarimetric current sensors. An example of the sensor setup is shown in Fig. 14 (Claus &

Fig. 14. Sensor setup with compensation on output signal level a), signal processing b).

photodetectors PD1, PD2. For output voltage signals of the detectors we can deduce

1 ,1 1 2 2 ,2 2 1

*FC* = *FC*1 = *FC*2 = 0,5 and *RU*,1 = *RU*,2 = *RU*. Relation (43) transforms into

In Fig. 14a), the signal from laser source L is divided into two channels by means of fiber coupler FC. After passing FC1 and FC2, the two optical signals propagate in opposite direction through the polarizing parts and the sensing part of the fiber. The unused coupler outputs are led into immersion gel to avoid reflections.  *FC*, *FC*1, *FC*2 are split ratio of coupler FC, FC1 FC2, *T*ov+ and *T*ov- are polarizing transfer functions of the fiber path in one and in second direction, *RU*,1, *RU*,2 are responsivities of photodetectors. For correct operation, it must hold

, ,

= ⋅ ⋅⋅ ⋅⋅ (43)

level in case of back-propagating of light wave with ortho-conjugated polarization.

**6.4.1 Compensation on sensor output signal level**

Fang, 1996).

or compressing. Since the quasi-circular birefringence in the fiber originates from twisting of stress components, the temperature dependence of anisotropy is indispensable. Therefore temperature compensation has to be used in applications utilizing spun HB fibers. On the present, spun HB fibers for telecommunication and sensing application are available for wavelengths from 600 nm to 1600 nm, with attenuation in order of ones of dB⋅km-1. They may be wind on fiber coils with radius above 20 mm.

In connection with recent advances in microstructured fibers research, new possibilities of spun fiber fabrication emerge. The concept of microstructured fiber allows designing and producing of fibers with specific parameters on selected wavelength, single mode or multi mode character, polarization transformation properties and others. A development of microstructured spun fiber with six air chambers around the core and attenuation below 5 dB⋅km-1 is reported in (Nikitov et al., 2009). The possibility of circular polarization transmission has been achieved by rotating of the microstructured preform, together with magneto-optic properties preservation. The fiber coils with diameter above 2.5 mm can be fabricated for current sensing application.

## **6.3 Annealed fibers**

Drawback of twisted fibers and spun HB fibers, which is anisotropy temperature dependence, limits their applicability mainly in polarimetric current sensor applications. A method for suppression of temperature dependence of anisotropy together with the suppression of bending induced linear birefringence has been proposed and experimentally studied (Stone, 1988; Rose et al., 1996). The method utilizes annealing of fabricated fiber coil. The procedure consists in temperature treatment of the fiber coil, which is installed in ceramic labyrinth. The coil is then heated up with approximate temperature-time gradient Δ*T*/Δ*t* = 8⋅10-2 °C⋅s-1. When 850 °C is reached, the temperature is maintained for roughly 24 hours. Then, the slow cooling follows with approximate gradient Δ*T*/Δ*t* = -3⋅10-3 °C⋅s-1. Annealed fiber coil is then transferred into protective case, which is filled with low-viscosity gel in order to damp the vibrations. The annealing procedure leads to removing of bending induced stress on the fiber and the linear birefringence is greatly suppressed. Prior to the annealing procedure, the fiber jacket and buffer has to be removed, because its oxidation at the temperatures 500 - 600 °C would damage the fiber. Since the fiber jacket and buffer act as fiber strength element, their removal is difficult and fiber rupture impends. The outer layers removal is facilitated by etching in organic solvent. The oxidation proceeds then without the negative influence on the fiber cladding (Rose et al., 1996).

Due to considerable temperature stability, the annealing method is used for fabrication of fiber current sensors, which are installed in outdoor environment on high voltage systems. The need for reliable galvanic isolation and accuracy predominates the technological difficulties in fiber coil fabrication. The annealing procedure has to be carefully performed and it has to be handled a technology of fiber coil isolation from vibrations. Since the fiber strength outer layers have been removed, the fiber coils have increased sensitivity to vibrations. For outdoor installations on high voltage system an annealed fiber sensors with sensitivity variation smaller than 0.2 %, dynamic range of 80 dB and temperature range 20-80 °C were developed (Higuera-Lopez, 2002).

#### **6.4 Reciprocal compensation of linear birefringence**

In combination with fibers, which were described above and with common single mode fibers also, another perspective approach for linear birefringence suppression may be exploited. The approach is based on reciprocity of linear birefringence. Beyond this, the circular birefringence is of non-reciprocal character, which gives usability especially in polarimetric current fiber sensors. Utilizing this fact, a compensation of linear birefringence may be performed on sensor output signal level in case of counter-propagating of two light waves in fiber. Second possibility consists in compensation of modes phase shift on fiber level in case of back-propagating of light wave with ortho-conjugated polarization.

#### **6.4.1 Compensation on sensor output signal level**

144 Recent Progress in Optical Fiber Research

or compressing. Since the quasi-circular birefringence in the fiber originates from twisting of stress components, the temperature dependence of anisotropy is indispensable. Therefore temperature compensation has to be used in applications utilizing spun HB fibers. On the present, spun HB fibers for telecommunication and sensing application are available for wavelengths from 600 nm to 1600 nm, with attenuation in order of ones of dB⋅km-1. They

In connection with recent advances in microstructured fibers research, new possibilities of spun fiber fabrication emerge. The concept of microstructured fiber allows designing and producing of fibers with specific parameters on selected wavelength, single mode or multi mode character, polarization transformation properties and others. A development of microstructured spun fiber with six air chambers around the core and attenuation below 5 dB⋅km-1 is reported in (Nikitov et al., 2009). The possibility of circular polarization transmission has been achieved by rotating of the microstructured preform, together with magneto-optic properties preservation. The fiber coils with diameter above 2.5 mm can be

Drawback of twisted fibers and spun HB fibers, which is anisotropy temperature dependence, limits their applicability mainly in polarimetric current sensor applications. A method for suppression of temperature dependence of anisotropy together with the suppression of bending induced linear birefringence has been proposed and experimentally studied (Stone, 1988; Rose et al., 1996). The method utilizes annealing of fabricated fiber coil. The procedure consists in temperature treatment of the fiber coil, which is installed in ceramic labyrinth. The coil is then heated up with approximate temperature-time gradient Δ*T*/Δ*t* = 8⋅10-2 °C⋅s-1. When 850 °C is reached, the temperature is maintained for roughly 24 hours. Then, the slow cooling follows with approximate gradient Δ*T*/Δ*t* = -3⋅10-3 °C⋅s-1. Annealed fiber coil is then transferred into protective case, which is filled with low-viscosity gel in order to damp the vibrations. The annealing procedure leads to removing of bending induced stress on the fiber and the linear birefringence is greatly suppressed. Prior to the annealing procedure, the fiber jacket and buffer has to be removed, because its oxidation at the temperatures 500 - 600 °C would damage the fiber. Since the fiber jacket and buffer act as fiber strength element, their removal is difficult and fiber rupture impends. The outer layers removal is facilitated by etching in organic solvent. The oxidation proceeds then

Due to considerable temperature stability, the annealing method is used for fabrication of fiber current sensors, which are installed in outdoor environment on high voltage systems. The need for reliable galvanic isolation and accuracy predominates the technological difficulties in fiber coil fabrication. The annealing procedure has to be carefully performed and it has to be handled a technology of fiber coil isolation from vibrations. Since the fiber strength outer layers have been removed, the fiber coils have increased sensitivity to vibrations. For outdoor installations on high voltage system an annealed fiber sensors with sensitivity variation smaller than 0.2 %, dynamic range of 80 dB and temperature range

In combination with fibers, which were described above and with common single mode fibers also, another perspective approach for linear birefringence suppression may be

without the negative influence on the fiber cladding (Rose et al., 1996).

20-80 °C were developed (Higuera-Lopez, 2002).

**6.4 Reciprocal compensation of linear birefringence**

may be wind on fiber coils with radius above 20 mm.

fabricated for current sensing application.

**6.3 Annealed fibers**

Since the linear birefringence is of reciprocal character, its influence on polarization state of the wave in fiber is not dependent on the propagation direction. The wave will experience the same polarization state transformation with the same orientation, no matter the propagation direction. Conversely, the magnetic field induced circular birefringence is non-reciprocal. When the wave will propagate in one direction, it will gain a polarization rotation. When the wave will propagate in opposite direction, it will gain rotation in opposite direction. The total rotation will be the double of the rotation in one direction.

Setups, which exploit the reciprocity of linear birefringence, have been demonstrated as polarimetric current sensors. An example of the sensor setup is shown in Fig. 14 (Claus & Fang, 1996).

Fig. 14. Sensor setup with compensation on output signal level a), signal processing b).

In Fig. 14a), the signal from laser source L is divided into two channels by means of fiber coupler FC. After passing FC1 and FC2, the two optical signals propagate in opposite direction through the polarizing parts and the sensing part of the fiber. The unused coupler outputs are led into immersion gel to avoid reflections. The optical signals are sensed by photodetectors PD1, PD2. For output voltage signals of the detectors we can deduce

$$\begin{aligned} \mathbf{U}\_1 &= R\_{\mathbf{U},1} \cdot FC\_1 \cdot T\_{\mathbf{ov}}^\* \cdot FC\_2 \cdot FC \cdot P\_{\mathbf{o}\prime} \\ \mathbf{U}\_2 &= R\_{\mathbf{U},2} \cdot FC\_2 \cdot T\_{\mathbf{ov}}^\* \cdot FC\_1 \cdot FC \cdot P\_{\mathbf{o}\prime} \end{aligned} \tag{43}$$

where *P*o is the input optical power. *FC*, *FC*1, *FC*2 are split ratio of coupler FC, FC1 FC2, *T*ov+ and *T*ov- are polarizing transfer functions of the fiber path in one and in second direction, *RU*,1, *RU*,2 are responsivities of photodetectors. For correct operation, it must hold *FC* = *FC*1 = *FC*2 = 0,5 and *RU*,1 = *RU*,2 = *RU*. Relation (43) transforms into

Optical Fiber Birefringence Effects – Sources, Utilization and Methods of Suppression 147

rotation mirror (FRM). It consists of Faraday rotator, mounted inside a permanent magnet, flat mirror and collimator (for fiber optic application). The magnetic field magnitude, rotator dimension and its assembly is properly adjusted, that a light wave polarization is rotated with angle 45° while passing the rotator. Mirror allows reflection in perpendicular direction

Consider a birefringent fiber with a fast axis and a slow axis, Fig. 15. The modes of light wave, which travels in fiber, gain a phase shift. The mode in slow axis is retarded, while the mode in fast axis travels faster. At the fiber output, the wave is collimated to rotator. It

rotation non-reciprocity. The wave is coupled back into the fiber. Now, the wave propagates in fiber in backward direction. The wave mode, which traveled previously along the fast axis, travels now along the slow axis. Conversely, the mode, which traveled previously along the slow axis, travels now along the fast axis. The total phase shift of the modes is equalized and the original polarization state is restored. It should be mentioned, although the difference of average refractive indices will no be constant in time (for example by fiber manipulation), the final phase shift will remain zero thanks to reciprocal compensation. The temperature stability of the method is considerable also. However, this is true only when the temperature of FRM is stable, since the Verdet constant of rotator in

Though the principle is not applicable for telecommunication purposes, it may be utilized in applications, where the polarization state preservation is desirable. This is often required in erbium-doped fiber amplifiers or tunable fiber lasers. Fiber optic sensors are another field of application. In case of fiber interferometers a polarization state of waves incoming from reference arm and sensing arm has to be preserved in order to interfere. Therefore, FRMs are used in both arms of interferometer. Fiber optic current sensors are another example of usage of FRM (Drexler & Fiala, 2008). Fiber current sensors exploit polarization rotation of the guided wave due to the magnet field actuation. Since the circular birefringence owing to Faraday magneto-optic effect is of non-reciprocal character, it will not be compensated during the backward propagation. Moreover, the polarization rotation will be double, which improves the sensitivity of the sensor. The example of fiber optic polarimetric current sensor utilizing FRM is shown in Fig. 16 (Drexler & Fiala, 2009). Laser beam from laser source L is collimated by means of collimator C and linear polarized by means of polarizer P. Beam passes a non-polarizing beam splitter NBS and it its collimated

Fig. 16. Fiber optic polarimetric current sensor utilizing Faraday rotation mirror.

/2 = 45° during a single pass. Then the wave is

θ

= 90° due to the

θ

reflected back. After the second rotator pass, the rotation angle is

and collimator serves for fiber coupling.

rotates the wave polarization with angle

FRM is temperature dependent.

$$\begin{aligned} \, ^\circ \boldsymbol{L} &= \boldsymbol{R}\_{\mathbf{U}} \cdot \boldsymbol{F} \mathrm{C}^3 \cdot \boldsymbol{T}\_{\mathbf{ov}}^\* \\ \, ^\circ \boldsymbol{L} &= \boldsymbol{R}\_{\mathbf{U}} \cdot \boldsymbol{F} \mathrm{C}^3 \cdot \boldsymbol{T}\_{\mathbf{ov}}^\* \end{aligned} \tag{44}$$

In signal processing block, as shown in Fig. 14b), the normalized difference is computed

$$LI = \frac{LI\_1 - LI\_2}{LI\_1 + LI\_2} = \frac{R\_\text{U} \cdot FC^3 \cdot \left(T\_{\text{ov}}^\bullet - T\_{\text{ov}}^\bullet\right)}{R\_\text{U} \cdot FC^3 \cdot \left(T\_{\text{ov}}^\bullet + T\_{\text{ov}}^\bullet\right)} = \frac{T\_{\text{ov}}^\bullet - T\_{\text{ov}}^\bullet}{T\_{\text{ov}}^\bullet + T\_{\text{ov}}^\bullet} \tag{45}$$

Considering the presence of reciprocal linear birefringence only, the equality *T*ov+ = *T*ovholds. The output signal according to (45) will be zero. Since the magnetic field induced circular birefringence is non-reciprocal, the equality of polarizing transfer functions will not hold. Hence, the system will be responsive on varying rotation due to induced circular birefringence. Sensor utilizing above described principle dispose of considerable temperature stability and vibration insensitivity. More detailed description and sensor properties may be found in (Fang et al., 1994; Wilsch et al., 1996).

#### **6.4.2 Orthogonal conjugation compensation**

The reciprocity of undesirable birefringences in optical fiber may be used for their compensation exploiting the polarization ortho-conjugation of the wave modes. The method involves the back-propagation of light wave with conjugated modes through the same section of birefringent fiber.

As it has been stated above, imposing stress on fiber (pressure, bending) leads to origin of linear birefringent fiber core with two refractive indices, one lying in *x* axis direction – *nx* and second lying in *y* axis direction – *ny*. We may designate the axis as the fast fiber axis, with lower refractive index, and the slow fiber axis, with higher refractive index. However, the orientation of the fast and slow axes system towards the geometrical coordinate system of the fiber changes along, in dependence on the bending or pressure force orientation. The modes propagating with different refractive indices gain a phase shift to each other, which results in wave polarization transformation. Since the instantaneous magnitudes of refractive indices in fast and slow axis may vary due to rate of bending or compressing, the resulting phase difference relies on average magnitudes of the refractive indices in both axes.

In order to restore the original polarization state, the wave at the output of the birefringent fiber has to be reflected and coupled back in the fiber together with modes conjugation. To accomplish this, Faraday polarization rotator and flat mirror is exploited, as shown in Fig. 15. The whole device is called ortho-conjugation reflector (OCR) or often Faraday

Fig. 15. Principle of Faraday rotation mirror.

*U R FC T U R FC T* =⋅ ⋅

In signal processing block, as shown in Fig. 14b), the normalized difference is computed

3

. *U U R FC T T T T U U U R FC T T T T* ⋅⋅− − − = = = + ⋅⋅+ +

Considering the presence of reciprocal linear birefringence only, the equality *T*ov+ = *T*ovholds. The output signal according to (45) will be zero. Since the magnetic field induced circular birefringence is non-reciprocal, the equality of polarizing transfer functions will not hold. Hence, the system will be responsive on varying rotation due to induced circular birefringence. Sensor utilizing above described principle dispose of considerable temperature stability and vibration insensitivity. More detailed description and sensor

The reciprocity of undesirable birefringences in optical fiber may be used for their compensation exploiting the polarization ortho-conjugation of the wave modes. The method involves the back-propagation of light wave with conjugated modes through the same

As it has been stated above, imposing stress on fiber (pressure, bending) leads to origin of linear birefringent fiber core with two refractive indices, one lying in *x* axis direction – *nx* and second lying in *y* axis direction – *ny*. We may designate the axis as the fast fiber axis, with lower refractive index, and the slow fiber axis, with higher refractive index. However, the orientation of the fast and slow axes system towards the geometrical coordinate system of the fiber changes along, in dependence on the bending or pressure force orientation. The modes propagating with different refractive indices gain a phase shift to each other, which results in wave polarization transformation. Since the instantaneous magnitudes of refractive indices in fast and slow axis may vary due to rate of bending or compressing, the resulting phase difference relies on average magnitudes of the refractive indices in both

In order to restore the original polarization state, the wave at the output of the birefringent fiber has to be reflected and coupled back in the fiber together with modes conjugation. To accomplish this, Faraday polarization rotator and flat mirror is exploited, as shown in Fig. 15. The whole device is called ortho-conjugation reflector (OCR) or often Faraday

3

1

2

1 2

1 2

properties may be found in (Fang et al., 1994; Wilsch et al., 1996).

**6.4.2 Orthogonal conjugation compensation**

Fig. 15. Principle of Faraday rotation mirror.

section of birefringent fiber.

axes.

**U ov**

3

3

**- U ov**

, .

=⋅ ⋅ (44)

(45)

**+**

( ) ( ) **- + - + U ov ov ov ov**

**U ov ov ov ov**

**- + - +**

rotation mirror (FRM). It consists of Faraday rotator, mounted inside a permanent magnet, flat mirror and collimator (for fiber optic application). The magnetic field magnitude, rotator dimension and its assembly is properly adjusted, that a light wave polarization is rotated with angle 45° while passing the rotator. Mirror allows reflection in perpendicular direction and collimator serves for fiber coupling.

Consider a birefringent fiber with a fast axis and a slow axis, Fig. 15. The modes of light wave, which travels in fiber, gain a phase shift. The mode in slow axis is retarded, while the mode in fast axis travels faster. At the fiber output, the wave is collimated to rotator. It rotates the wave polarization with angle θ/2 = 45° during a single pass. Then the wave is reflected back. After the second rotator pass, the rotation angle is θ = 90° due to the rotation non-reciprocity. The wave is coupled back into the fiber. Now, the wave propagates in fiber in backward direction. The wave mode, which traveled previously along the fast axis, travels now along the slow axis. Conversely, the mode, which traveled previously along the slow axis, travels now along the fast axis. The total phase shift of the modes is equalized and the original polarization state is restored. It should be mentioned, although the difference of average refractive indices will no be constant in time (for example by fiber manipulation), the final phase shift will remain zero thanks to reciprocal compensation. The temperature stability of the method is considerable also. However, this is true only when the temperature of FRM is stable, since the Verdet constant of rotator in FRM is temperature dependent.

Though the principle is not applicable for telecommunication purposes, it may be utilized in applications, where the polarization state preservation is desirable. This is often required in erbium-doped fiber amplifiers or tunable fiber lasers. Fiber optic sensors are another field of application. In case of fiber interferometers a polarization state of waves incoming from reference arm and sensing arm has to be preserved in order to interfere. Therefore, FRMs are used in both arms of interferometer. Fiber optic current sensors are another example of usage of FRM (Drexler & Fiala, 2008). Fiber current sensors exploit polarization rotation of the guided wave due to the magnet field actuation. Since the circular birefringence owing to Faraday magneto-optic effect is of non-reciprocal character, it will not be compensated during the backward propagation. Moreover, the polarization rotation will be double, which improves the sensitivity of the sensor. The example of fiber optic polarimetric current sensor utilizing FRM is shown in Fig. 16 (Drexler & Fiala, 2009). Laser beam from laser source L is collimated by means of collimator C and linear polarized by means of polarizer P. Beam passes a non-polarizing beam splitter NBS and it its collimated

Fig. 16. Fiber optic polarimetric current sensor utilizing Faraday rotation mirror.

Optical Fiber Birefringence Effects – Sources, Utilization and Methods of Suppression 149

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Laming, R. I. & Payne, D. N. (1989). Electric current sensors employing spun highly

Namihira, Y. (1983). Opto-elastic constants in single-mode optical fibers. *Journal of Lightwave*

Nikitov, S., Chamorovskiy, Y., Starostin, N., Ryabko, M., Morshnev, S., Morshnev, V.

Noda, J., Okamoto, K. & Sasaki, J. (1986). Polarization-maintaining fibers and their

Payne, D. N., Barlow, A. J. & Hansen, J. J. R. (1982). Development of Low- and

Ripka, P. (Ed(s).) (2001). *Magnetic sensors and magnetometers*, Artech House,

Rose, A., Ren, Z. F. & Day, G. W. (1996). Twisting and annealing optical fiber for current

birefringent optical fibers. *Journal of Lightwave Technology*, Vol. 7, No. 12,

& Vorob'ev, I. (2009). Microstructured optical fibers for the fiber optics sensors, *Proceedings of International Conference on Materials for Advanced Technologies*,

applications. *Journal of Lightwave Technology*, Vol. 4, No. 8. pp. (1071-1089),

High-Birefringence Optical Fibers. *IEEE Transaction on Microwave Theory and*

sensors. *Journal of Lightwave Technology*, Vol. 14, No. 11, pp. (2492-2498),

Huard, S. (1997). *Polarization of Light*, John Wiley & Sons, ISBN 2-225-85327-X, Paris

*Applied Physics Letters*, No. 34, pp. (268-270), ISSN 0003-6951

*Technology*, Vol. 3, No. 5, pp. (1078-1083), ISSN 0733-8724

*Techniques*, Vol. 30, No. 4, pp. (323-334), ISSN 0018-9480

Current Sensor. *Journal of Lightwave Technology*, Vol. 14, No. 7, pp. (1664-1673),

and its application for pulsed current sensing, *Proceedings of 2009 Waveform*

Current Sensor. *Journal of Lightwave Technology*, Vol. 12, No. 10, pp. (1882-1890),

**9. References**

ISBN<PHONE_NUMBER>224, Cambridge

ISBN 978-0-471-82053-6, New York

pp. (2084-2094), ISSN 0733-8724

Singapore, June 2009

ISBN 1-58053-057-5, London

ISSN 0733-8724

ISSN 0733-8724

*Society of America*, Vol. 31, No. 7, ISSN 1862-6254

John Wiley & Sons, ISBN 0-471-39055-0, New Jersey

*Radioengineering*, Vol. 17, No. 4, pp. (101-107), ISSN 1210-2512.

ISSN 0733-8724

ISSN 0733-8724

Toronto

into the fiber by collimator C. The beam propagates through the magnetic field sensing fiber OF and exits the fiber. Collimator C collimates the beam into the FRM and collimates backward beam back into the fiber. The beam travels the fiber in opposite direction. After the collimation, part of the beam is deflected by beam splitter NBS. It passes the analyzer A and hits the photodetector PD. The output linear polarization state is perpendicular to the input linear polarization state. When magnetic field acts on the sensing part of the fiber, the polarization plane rotates. The polarization modulation is converted on intensity modulation by means of analyzer A.

In spite of the advantages of the FRM application, several drawbacks limits its usage. One of the drawbacks is temperature dependence of rotator Verdet constant (Santoyo-Mendoza & Barmenkov, 2003). Therefore the FRM unit has to be temperature stabilized. Simultaneously, it has to be shielded from outer magnetic field. Indispensable is the cost of this solution also owing to precise fabrication and adjusting of the FRM. Commercially available are FRMs in compact fiber pigtailed housing for longer wavelengths (1310 nm, 1550 nm). There are also available FRMs for shorter wavelengths (633 nm). However, they are bulky, because of the need of more powerful magnet.

## **7. Conclusion**

Due to their unique properties, single mode fibers have found a huge application potential in various fields of industry and science. They are massively exploited in telecommunication technology, control and sensor systems, industrial laser systems and they represent an unsubstitutable tool for advanced science. The requirements for specific fiber properties differ for various applications. In lot of them, a transmission of light wave with preserving of state of polarization is demanded, which is often a weak point of common fibers. However, this drawback is possible to overcome with a suitable approach, depending on demands of particular application.

In order to evaluate the possibility of polarization state distortion, various influences have to be conceived. It is also very advantageous to be able to quantify them. According to this demand, the intention of the first part of the contribution is to specify the fundamental effects, which lead to fiber birefringence. The basic relations, which allow estimating the birefringence rate are presented also.

Once the fiber birefringence occurs, in lot of cases arises a need to suppress its undesirable consequences. A various approaches, which may be utilized, are the point of the interest of the second part of the contribution. The principles of the most significant methods are described. Their advantages and disadvantages are presented also. The suitability of a selected method is given by the application requirements. They differ in cost, complexity, temperature and mechanical stability and others. Because of the limited extent of this contribution, all of the methods properties and details could not be presented. Nonetheless, the chapter may be a convenient starting point for orientation in this field and details may be found in cited reference sources.

## **8. Acknowledgement**

The contribution has been prepared with support of research project GA102/09/0314 of the Czech Science Foundation and project FEKT-S-10-13 of the Grant Agency of Brno University of Technology.

## **9. References**

148 Recent Progress in Optical Fiber Research

into the fiber by collimator C. The beam propagates through the magnetic field sensing fiber OF and exits the fiber. Collimator C collimates the beam into the FRM and collimates backward beam back into the fiber. The beam travels the fiber in opposite direction. After the collimation, part of the beam is deflected by beam splitter NBS. It passes the analyzer A and hits the photodetector PD. The output linear polarization state is perpendicular to the input linear polarization state. When magnetic field acts on the sensing part of the fiber, the polarization plane rotates. The polarization modulation is converted on intensity

In spite of the advantages of the FRM application, several drawbacks limits its usage. One of the drawbacks is temperature dependence of rotator Verdet constant (Santoyo-Mendoza & Barmenkov, 2003). Therefore the FRM unit has to be temperature stabilized. Simultaneously, it has to be shielded from outer magnetic field. Indispensable is the cost of this solution also owing to precise fabrication and adjusting of the FRM. Commercially available are FRMs in compact fiber pigtailed housing for longer wavelengths (1310 nm, 1550 nm). There are also available FRMs for shorter wavelengths (633 nm). However, they are bulky, because of the

Due to their unique properties, single mode fibers have found a huge application potential in various fields of industry and science. They are massively exploited in telecommunication technology, control and sensor systems, industrial laser systems and they represent an unsubstitutable tool for advanced science. The requirements for specific fiber properties differ for various applications. In lot of them, a transmission of light wave with preserving of state of polarization is demanded, which is often a weak point of common fibers. However, this drawback is possible to overcome with a suitable approach, depending on

In order to evaluate the possibility of polarization state distortion, various influences have to be conceived. It is also very advantageous to be able to quantify them. According to this demand, the intention of the first part of the contribution is to specify the fundamental effects, which lead to fiber birefringence. The basic relations, which allow estimating the

Once the fiber birefringence occurs, in lot of cases arises a need to suppress its undesirable consequences. A various approaches, which may be utilized, are the point of the interest of the second part of the contribution. The principles of the most significant methods are described. Their advantages and disadvantages are presented also. The suitability of a selected method is given by the application requirements. They differ in cost, complexity, temperature and mechanical stability and others. Because of the limited extent of this contribution, all of the methods properties and details could not be presented. Nonetheless, the chapter may be a convenient starting point for orientation in this field and details may

The contribution has been prepared with support of research project GA102/09/0314 of the Czech Science Foundation and project FEKT-S-10-13 of the Grant Agency of Brno University

modulation by means of analyzer A.

need of more powerful magnet.

demands of particular application.

birefringence rate are presented also.

be found in cited reference sources.

**8. Acknowledgement**

of Technology.

**7. Conclusion**

**8**

*Algiers, Algeria*

**Spun Fibres for Compensation of PMD:**

*University of Sciences and Technology Houary Boumedienne, USTHB,*

The polarization is a relative property to the vibratory nature of light. In an optical fibre, light is a combination of two vibrations of perpendicular directions. Each direction represents one mode of polarization. Indeed, the optical fibres and the components of the optical fibres present a small difference in the refractive index in the pair of the polarization states, a property called the birefringence. This last one induces a difference of propagation speed between the two modes. So, light at the output, cannot be restored more faithfully. The birefringence can change the state of polarization (SOP) of light when it crosses the fibre. In a single mode fibre, the birefringence is combined with a random coupling of polarization modes. The delay measured at the output of the fibre between the two polarization modes is called the difference of group delay DGD (measured in picoseconds). The polarization modes dispersion (PMD) results from the variation of the DGD according

The typical tolerance of a system to the PMD is roughly 10% of the bit period, which gives 40 Ps for a system of 2.5 Gb/s, 10 Ps for a system of 10 Gb/s and only 2.5 Ps for a system of 40 Gbs/s (Noé et al.,1999). The PMD is a random phenomenon and constitutes an enormous obstacle ahead of the increase of the debits from 10 Gbit/s for a part of the networks of most telecommunication companies. Several solutions have been proposed to compensate the PMD as: The electronic compensation after a direct photo-detection that can only eliminate a part of the PMD effects since the information about the polarization and the phase get lost at the detection; the second solution is the electronic compensation in a coherent receptor with diversity of polarizations, and the third one is the optical compensation in at least a differential delay section. Other solutions are proposed by the Corning society and which rely on the use of spun fibres allowing the control of the coupling of the modes, therefore reducing the PMD; thus, giving differential group delays of order of Femtoseconds. In the past decade, some considerable efforts have been made to understand the origins of the PMD and to attenuate its effects in the systems. The PMD can be reduced in a fibre with two different manners. The first one consists in minimizing the asymmetries in the refractive index profile and the constraints, which implies improvements of the industrial process in the manufacture of the fibre in order to assure a better geometry and to reduce the rate of constraints in the fiber. The second method allows the control of the coupling of the modes

**1. Introduction**

to the wavelength and the environment conditions.

**Theory and Characterization**

Lynda Cherbi and Abderrahmane Bellil

*Laboratory of Instrumentation LINS,*

## **Spun Fibres for Compensation of PMD: Theory and Characterization**

Lynda Cherbi and Abderrahmane Bellil

*Laboratory of Instrumentation LINS, University of Sciences and Technology Houary Boumedienne, USTHB, Algiers, Algeria*

## **1. Introduction**

150 Recent Progress in Optical Fiber Research

Saleh, B. A. & Teich, M. C. (1991). *Fundamentals of Photonics*, John Wiley & Sons,

Santoyo-Mendoza, F. & Barmenkov, Y. O. (2003). Faraday plasma current sensor with

Senior, J. M., (2009). *Optical Fiber Communication: Principles and Practice*, Pearson Education,

Stone, J. (1988). Stress-optic effect, birefringence, and reduction of birefringence by

Tabor, W. J. & Chen, F. S. (1968). Electromagnetic Propagation Through Materials Processing

Ulrich, R., Rashleigh, S. C. & Eickhoff, W. (1980). Bending-induced birefringence in singlemode fibers. *Optical Letters*, Vol. 5, No. 5, pp. (273 – 275), ISSN 1539-4794 Wagner, E., Dändliker, R. & Spenner, K. (Ed(s).). (1992). *Sensors, a comprehensive survey,*

Waynant, R. & Ediger, M. (2000). *Electro-Optics Handbook*, McGraw-Hill Professional,

Wilsch, M., Menke, P. & Bosselman, T. (1996). Magneto-optic current transformers

for applications in power industry, *Proceedings of 2nd Congress of Optical Sensor*

*Journal of Applied Physics*, Vol. 10, pp. (2760-2765), ISSN 0021-8979

*Volume 6 – Optical Sensors*, VCH, ISBN 3-527-26772-7, Weinheim

compensation for reciprocal birefringence induced mechanical perturbations. *Journal of Applied Research and Technology*, Vol. 1, No. 2, pp. (157–163),

annealing in fiber Fabry-Perot interferometers. *Journal of Lightwave Technology*,

both Faraday Rotation and Birefringence: Experiments with Ytterbium Orthoferrite.

ISBN<PHONE_NUMBER>651, New York

Vol. 6, No. 7, pp. (1245-1248), ISSN 0733-8724

ISBN 978-0-07-068716-5, New York

*Technology OPTO 1996*, Germany: Leipzig, 1996

ISSN 1665-6423

ISBN 978-0-13-032681

The polarization is a relative property to the vibratory nature of light. In an optical fibre, light is a combination of two vibrations of perpendicular directions. Each direction represents one mode of polarization. Indeed, the optical fibres and the components of the optical fibres present a small difference in the refractive index in the pair of the polarization states, a property called the birefringence. This last one induces a difference of propagation speed between the two modes. So, light at the output, cannot be restored more faithfully. The birefringence can change the state of polarization (SOP) of light when it crosses the fibre. In a single mode fibre, the birefringence is combined with a random coupling of polarization modes. The delay measured at the output of the fibre between the two polarization modes is called the difference of group delay DGD (measured in picoseconds). The polarization modes dispersion (PMD) results from the variation of the DGD according to the wavelength and the environment conditions.

The typical tolerance of a system to the PMD is roughly 10% of the bit period, which gives 40 Ps for a system of 2.5 Gb/s, 10 Ps for a system of 10 Gb/s and only 2.5 Ps for a system of 40 Gbs/s (Noé et al.,1999). The PMD is a random phenomenon and constitutes an enormous obstacle ahead of the increase of the debits from 10 Gbit/s for a part of the networks of most telecommunication companies. Several solutions have been proposed to compensate the PMD as: The electronic compensation after a direct photo-detection that can only eliminate a part of the PMD effects since the information about the polarization and the phase get lost at the detection; the second solution is the electronic compensation in a coherent receptor with diversity of polarizations, and the third one is the optical compensation in at least a differential delay section. Other solutions are proposed by the Corning society and which rely on the use of spun fibres allowing the control of the coupling of the modes, therefore reducing the PMD; thus, giving differential group delays of order of Femtoseconds. In the past decade, some considerable efforts have been made to understand the origins of the PMD and to attenuate its effects in the systems. The PMD can be reduced in a fibre with two different manners. The first one consists in minimizing the asymmetries in the refractive index profile and the constraints, which implies improvements of the industrial process in the manufacture of the fibre in order to assure a better geometry and to reduce the rate of constraints in the fiber. The second method allows the control of the coupling of the modes

Spun Fibres for Compensation of PMD: Theory and Characterization 153

The fibre spun is achieved by the rotation of the preform during the drawing of the fibre (figure 1.a). In this approach, the system of drawing is the same as that of the conventional standard fibre systems (OVD) except that a rotating motor is placed on the top of the preform. When the motor is set in motion with a predetermined speed, the preform starts turning dragging the rotations of the axes of the birefringence. The rotation will end up with the end of the drawing operation. This approach is quite simple and appropriate for the pulling of the fibre at low speed. However, this is not convenient for the production of the fibre with a high speed pulling because the rotation of the motor must be at very high speed as well. To illustrate this, we consider a rotation rate of the fibre of 3 turns/min, for a drawing speed of 1 m/s, the rotating speed of the preform is

On the other hand, for a modern drawing device having a speed higher than 20m/s, the perform must turn at a speed greater than 3600 turns/mn, which is far from practical. For this reason the concept of the spun fibres has not been used in the production of fibres until the half of the nineties when methods of more adapted spinning have been proposed (Ming-Jun & Nolan, 1998). Moreover, the transmission systems as they appeared at low rate (<= 2.5 Gb/s), the PMD was not a major problem to seek fibres that perform this

Several convenient techniques have been suggested during the year 1990, for example, by (Hart et al., 1994] in order to make the fibre turn rather than the preform. Later on, this technique became the most adapted one for the manufacturing of the fibres performing the

Fig. 2. Examples of fibre rotation systems: (a) tilted wheel, (b) two wheels moving in

thus only 180 turns/min.

reduction of the PMD.

opposite directions.

reduction.

of the polarization in the fibre while spinning it during its manufacture. Indeed, the spinning has been used in the manufacture of fibres since the beginning of the 1990s, and it showed that it is an efficient technique to reduce the PMD in the fibre. First, we start by presenting in this work the spun fibres explaining their technology, their principle and their different types. Next, the description of the reduction of the PMD by using the spinning is developed by a mathematical formalism based on the theory of coupling and Jones's matrix. Moreover, the reduction of the PMD is verified in the spun fibres while applying the method of JME and the COTDR method (photon counting -Optical temporal Domain of Reflectometry) that allowed us to measure the DGD of the order of femtosecond (Cherbi et al., 2009). The comparison of the DGD found in this type of new generation of fibres with those of the standard ones, led us to confirm that the spun fibres offer effectively a smaller DGD than those of the standard fibres, emphasizing the importance of this type of fibres in the reduction of the PMD.

We present the different results already published (Cherbi et al., 2009) while using the reflectometers COTDR and POFDR (polarization- Optical Frequency Domain of Reflectometry) which are used to get the polarization characteristics of the spun fibres as the beat length and the PMD and to observe the spatial frequencies linked directly to the period of spinning.

## **2. Principle of spun fibres**

## **2.1 Technologies of the spun fibres**

There are more than two decades when the concept of the spun fibres has been proposed originally in an article published by (Barlow et al.,1981).

Fig. 1. Two approaches used to present the rotating fibres (a) turn preforms (b) turn the fibre.

of the polarization in the fibre while spinning it during its manufacture. Indeed, the spinning has been used in the manufacture of fibres since the beginning of the 1990s, and it showed that it is an efficient technique to reduce the PMD in the fibre. First, we start by presenting in this work the spun fibres explaining their technology, their principle and their different types. Next, the description of the reduction of the PMD by using the spinning is developed by a mathematical formalism based on the theory of coupling and Jones's matrix. Moreover, the reduction of the PMD is verified in the spun fibres while applying the method of JME and the COTDR method (photon counting -Optical temporal Domain of Reflectometry) that allowed us to measure the DGD of the order of femtosecond (Cherbi et al., 2009). The comparison of the DGD found in this type of new generation of fibres with those of the standard ones, led us to confirm that the spun fibres offer effectively a smaller DGD than those of the standard fibres, emphasizing the importance of this type of fibres in

We present the different results already published (Cherbi et al., 2009) while using the reflectometers COTDR and POFDR (polarization- Optical Frequency Domain of Reflectometry) which are used to get the polarization characteristics of the spun fibres as the beat length and the PMD and to observe the spatial frequencies linked directly to the period

There are more than two decades when the concept of the spun fibres has been proposed

Fig. 1. Two approaches used to present the rotating fibres (a) turn preforms (b) turn the fibre.

the reduction of the PMD.

**2. Principle of spun fibres**

**2.1 Technologies of the spun fibres**

originally in an article published by (Barlow et al.,1981).

of spinning.

The fibre spun is achieved by the rotation of the preform during the drawing of the fibre (figure 1.a). In this approach, the system of drawing is the same as that of the conventional standard fibre systems (OVD) except that a rotating motor is placed on the top of the preform. When the motor is set in motion with a predetermined speed, the preform starts turning dragging the rotations of the axes of the birefringence. The rotation will end up with the end of the drawing operation. This approach is quite simple and appropriate for the pulling of the fibre at low speed. However, this is not convenient for the production of the fibre with a high speed pulling because the rotation of the motor must be at very high speed as well. To illustrate this, we consider a rotation rate of the fibre of 3 turns/min, for a drawing speed of 1 m/s, the rotating speed of the preform is thus only 180 turns/min.

On the other hand, for a modern drawing device having a speed higher than 20m/s, the perform must turn at a speed greater than 3600 turns/mn, which is far from practical. For this reason the concept of the spun fibres has not been used in the production of fibres until the half of the nineties when methods of more adapted spinning have been proposed (Ming-Jun & Nolan, 1998). Moreover, the transmission systems as they appeared at low rate (<= 2.5 Gb/s), the PMD was not a major problem to seek fibres that perform this reduction.

Several convenient techniques have been suggested during the year 1990, for example, by (Hart et al., 1994] in order to make the fibre turn rather than the preform. Later on, this technique became the most adapted one for the manufacturing of the fibres performing the reduction of the PMD.

Fig. 2. Examples of fibre rotation systems: (a) tilted wheel, (b) two wheels moving in opposite directions.

Spun Fibres for Compensation of PMD: Theory and Characterization 155

where (,) *ne xy* is the distribution of the electric field. For a monomode fibre, n =1, 2

In absence of disruption, the two modes are degenerated and propagate with the same constant β0. In presence of the disruption term, it is supposed that the electric field E(x,y,z) is given by the linear superposition of the two non disrupted modes (Ming-Jun et al., 2002):

0 ( , , ) ( ) ( , )exp( ) *n n*

Where ( ) *A z n* are the complex coefficients describing the amplitudes and the phases of the two modes. Let's put equation (4) into equations (1) and (2) and use the relation of

*e x y e x y dxdy m n*

1 0 2

*coeur m mm m n N e h zds e ds*

1 *n n d A dA*

*dz dz*

We get the equations of the coupled modes that describe the evolution of the complex

. *dA ik A*

11 12

*k k*

21 22

*k k*

0 2

*N m n*

1/2

(6)

(7)

*dz* (8)

1 2 ( )*T AA A* (9)

(10)

 (5)

*E x y z A ze x y i z*

*n* 1,2 (3)

(4)

0 ( , , ) ( , )exp( ) *E xyz e xy i z n n*

*n*

( , ). ( , ) 0 *m m n*

Knowing that *Nm* is a constant of normalization which can be calculated as follows:

2 2

0

where A is the complex amplitude vector taking the following form:

*k*

The coupling coefficients are associated to the different types of disruptions:

orthogonality between the two modes (Ming-Jun et al., 2002):

2

and using the condition of the weak coupling:

and *k* is the matrix of the coupling coefficients.

amplitudes ( ) *A z n* :

represent the two modes of polarization.

In this case (fig 1.b), a rotating device of the fibre, is put along the way of fibrage to rotate the fibre directly. Two examples of this device are illustrated in figue 2. In the first example (fig 2.a), a wheel is in contact with the fibre and tilted with respect to its initial position, thus applying a moment of rotation to turn the fibre. In the second example (fig 2.b), two wheels are placed horizontally and are both in contact with the fibre (Blaszyk & Christoffand,2001). The two wheels move back and forth in opposite directions driving the rotational movement of the fibre. Imposing a direct motion to the fibre eliminates the problem of the preform when turning at high speed. Besides, this technique provides flexibility to control and implement different profiles of rotation for a better reduction of the PMD.

## **2.2 Theory of spun fibres**

Two approaches have been suggested to model the reduction of the PMD in the spun fibres, one of which is based on the evolution of the polarization state (Galtarossa et al., 2001; Ming-Jun &Nolan, 1998). The evolution of the vector representing the polarization dispersion is ruled by the dynamic equation which is linked to the vector of the local birefringence. While solving the dynamic equation, the vector representing the polarization's dispersion is gotten and its module gives the delay of the differential group (DGD). Another approach is based on the theory of the coupled modes of Jones' matrix (Ming-Jun et al., 2002) where the complex amplitudes of the two modes of polarization are described by the equations of the modes coupling. While solving these equations, the complex amplitudes are derived and Jones' matrix is determined and the DGD can be computed from this matrix. Basically, the two approaches give equivalent results. Our survey of the spun fibres is founded on Jones' matrix formalism, where we notice that the analytical solutions obtained are simple.

## **2.2.1 Equations of the mode coupling**

As the birefringence in the fibres used in telecommunications is generally small, the formalism based on the theory of disruption (Ming-Jun et al., 2002), can be used to describe the different mechanisms of birefringence in the single mode fibres, including the birefringence due to the distortion of the core, constraints, curvature, rotation of the fibre and torsion. In what follows, we will present the theory of the coupled modes and we will show how to implement it in the different problems of birefringence. Indeed, the small birefringence of telecommunication fibres can be treated as an anisotropic disruption to a material originally isotrope. In the condition of weak guidance, the electric field E is described by the following wave equation (Dandliker,1992):

$$
\Delta E - \mu\_0 \varepsilon\_0 \text{ \& \ E = \mu\_0 p \tag{1}
$$

Where 0 et 0 are the dielectric and magnetic constants of vacuum respectively, ε is the relative dielectric constant of the non disrupted fibre, and p is the disruption term given by the following relation:

$$p = \mathfrak{e}\_0 \,\Delta\mathfrak{e} \to \tag{2}$$

Where is the electric tensor describing the anisotropy of the medium. Without the term of disruption, the equation (1) has modal solutions of the following shape:

In this case (fig 1.b), a rotating device of the fibre, is put along the way of fibrage to rotate the fibre directly. Two examples of this device are illustrated in figue 2. In the first example (fig 2.a), a wheel is in contact with the fibre and tilted with respect to its initial position, thus applying a moment of rotation to turn the fibre. In the second example (fig 2.b), two wheels are placed horizontally and are both in contact with the fibre (Blaszyk & Christoffand,2001). The two wheels move back and forth in opposite directions driving the rotational movement of the fibre. Imposing a direct motion to the fibre eliminates the problem of the preform when turning at high speed. Besides, this technique provides flexibility to control and

Two approaches have been suggested to model the reduction of the PMD in the spun fibres, one of which is based on the evolution of the polarization state (Galtarossa et al., 2001; Ming-Jun &Nolan, 1998). The evolution of the vector representing the polarization dispersion is ruled by the dynamic equation which is linked to the vector of the local birefringence. While solving the dynamic equation, the vector representing the polarization's dispersion is gotten and its module gives the delay of the differential group (DGD). Another approach is based on the theory of the coupled modes of Jones' matrix (Ming-Jun et al., 2002) where the complex amplitudes of the two modes of polarization are described by the equations of the modes coupling. While solving these equations, the complex amplitudes are derived and Jones' matrix is determined and the DGD can be computed from this matrix. Basically, the two approaches give equivalent results. Our survey of the spun fibres is founded on Jones' matrix formalism, where we notice that the

As the birefringence in the fibres used in telecommunications is generally small, the formalism based on the theory of disruption (Ming-Jun et al., 2002), can be used to describe the different mechanisms of birefringence in the single mode fibres, including the birefringence due to the distortion of the core, constraints, curvature, rotation of the fibre and torsion. In what follows, we will present the theory of the coupled modes and we will show how to implement it in the different problems of birefringence. Indeed, the small birefringence of telecommunication fibres can be treated as an anisotropic disruption to a material originally isotrope. In the condition of weak guidance, the electric field E is

> *E Ep* 00 0

relative dielectric constant of the non disrupted fibre, and p is the disruption term given by

0 *p*

of disruption, the equation (1) has modal solutions of the following shape:

is the electric tensor describing the anisotropy of the medium. Without the term

0 are the dielectric and magnetic constants of vacuum respectively, ε is the

(1)

*E* (2)

implement different profiles of rotation for a better reduction of the PMD.

**2.2 Theory of spun fibres**

analytical solutions obtained are simple.

**2.2.1 Equations of the mode coupling**

Where 0 et

Where

the following relation:

described by the following wave equation (Dandliker,1992):

$$E\_n(\mathbf{x}, y, z) = e\_n(\mathbf{x}, y) \exp(-i\beta\_0 z) \qquad \qquad n = 1, 2 \tag{3}$$

where (,) *ne xy* is the distribution of the electric field. For a monomode fibre, n =1, 2 represent the two modes of polarization.

In absence of disruption, the two modes are degenerated and propagate with the same constant β0. In presence of the disruption term, it is supposed that the electric field E(x,y,z) is given by the linear superposition of the two non disrupted modes (Ming-Jun et al., 2002):

$$E(\mathbf{x}, y, z) = \sum\_{n} A\_{n}(z) e\_{n}(\mathbf{x}, y) \exp(-i\beta\_{0}z) \tag{4}$$

Where ( ) *A z n* are the complex coefficients describing the amplitudes and the phases of the two modes. Let's put equation (4) into equations (1) and (2) and use the relation of orthogonality between the two modes (Ming-Jun et al., 2002):

$$\int e\_m(\mathbf{x}, \mathbf{y}) \cdot e\_n(\mathbf{x}, \mathbf{y}) d\mathbf{x} d\mathbf{y} = \begin{vmatrix} N\_m & m = n \\ 0 & m \neq n \end{vmatrix} \tag{5}$$

Knowing that *Nm* is a constant of normalization which can be calculated as follows:

$$N\_m = \frac{1}{2} \int \vec{e}\_m \times \vec{h}\_m^\* z \, ds = \frac{n\_{\rm ocean}}{2} \left(\frac{\varepsilon\_0}{\mu\_0}\right)^{1/2} \int e\_m^2 ds \tag{6}$$

and using the condition of the weak coupling:

$$\frac{1}{\beta\_0} \left| \frac{d^2 A\_n}{dz^2} \right| << \left| \frac{dA\_n}{dz} \right| \tag{7}$$

We get the equations of the coupled modes that describe the evolution of the complex amplitudes ( ) *A z n* :

$$\frac{dA}{dz} = \text{ik.A} \tag{8}$$

where A is the complex amplitude vector taking the following form:

$$A = \begin{pmatrix} A\_1 & A\_2 \end{pmatrix}^T \tag{9}$$

and *k* is the matrix of the coupling coefficients.

$$k = \begin{pmatrix} k\_{11} & & & k\_{12} \\ & & & \\ & & & \\ k\_{21} & & & k\_{22} \end{pmatrix} \tag{10}$$

The coupling coefficients are associated to the different types of disruptions:

Spun Fibres for Compensation of PMD: Theory and Characterization 157

2

where ∆β is the linear birefringence, and Φ is the orientation of the birefringence with

In a spun fiber, the orientation of the birefringence takes place depending on the x axis. The rotation angle accumulated is therefore a function of the fibre length 'z', which in turn is

> 0 ( ) *z*

0

*e*

*z*

0

2 ()

*i z dz*

Replacing the equation (16) in equation (15), we get the coupling coefficients matrix of the

There are two effects in this type of fibres: The rotation of the birefringence and the mechanical torsion. The rotation of the birefringence is similar to that of the rotating fibre. If

The rate of torsion is determined by the coefficients of photo - elasticity of the fibre. The torsion constraint induces the circular birefringence proportionally to the rate of torsion.

Where g is the coefficient determined by the coefficients of photo elasticity of the glass. The typical value of g for fibres in silica is 0.16. Combining both effects of rotation and torsion,

*iTz*

*e*

*i*

0

*e*

1 2

( ) *z* :

rotating fibres, describing the disruption of the birefringence,:

1 2

*k*

the rate of torsion is T, the angle is calculated by

the coupling matrix comes up with the following form:

1 2

*k*

*k*

respect to a given axis

determined by the rate of rotation

**2.3.2 Spun fibres**

**2.3.3 Twisted fibre**

2

0

*e*

*z*

0

2

*e*

  *i Tz*

2 ()

*i z dz*

(15)

(17)

*z dz* (16)

*T z* (18)

*g*.*T* (19)

(20)

*i*

*e*

0

$$k\_{mn} = \frac{k\_0}{2n\_0 N\_0} \int e\_n^\*(\mathbf{x}, \mathbf{y}) . \Delta \varepsilon(\mathbf{x}, \mathbf{y}, \mathbf{z}) . e\_m(\mathbf{x}, \mathbf{y}) d\mathbf{x} dy \tag{11}$$

Where *n*0 is the effective refractive index of both non disrupted modes.

#### **2.2.2 Jones matrix and the PMD of spun fibre**

The evolution of the local polarization along the birefringent fibre is described by the equations of the modes coupling. The total change of polarization of an input signal, after having traveled a given distance in the fibre is better described by Jones' matrix. Let's assume that the losses in the fibres are negligible, the already predefined Jones matrix, can be put under another form which is:

$$T = \begin{bmatrix} A\_1(z) & -A\_2^\*(z) \\ & \\ A\_2(z) & A\_1^\*(z) \end{bmatrix} \text{ with } \begin{vmatrix} A\_1 \end{vmatrix}^2 + \begin{vmatrix} A\_2 \end{vmatrix}^2 = 1 \tag{12}$$

The four complex elements of Jones' matrix can be gotten while integrating the equations of the coupled modes with suitable initial conditions. Once Jones' matrix is known, the PMD can be calculated easily from the elements of the matrix (Chen, 2002; Ming-Jun et al., 2002):

$$\tau = 2\sqrt{\left|\frac{dA\_1}{d\phi}\right|^2 + \left|\frac{dA\_2}{d\phi}\right|^2} \tag{13}$$

In order to describe the reduction of the PMD, we define a parameter, named reduction factor of PMD (PMDRF) as the ratio of the DGD of the spun fibres over the DGD of the standard fibre.

$$
\zeta = \frac{\pi}{\tau\_0} \tag{14}
$$

Where the used lengths for the spun fibres and standard fibres are the same. For example, if is equal to 1, the reduction of the PMD is not achieved and if is equal to 0.5, a factor of two is obtained in the reduction of the PMD.

#### **2.3 Different types of spun fibres**

The coupling coefficients matrix depends upon the dielectric tensor of the disruption. The values of these elements are determined by the type of disruption, which means that they depend on the configuration of the fibre. In this section, we describe some configurations of the fibres and we give their coupling coefficients matrix. It is important to note that the coupling matrixes in this work are expressed on the basis of the circular polarization because it is more appropriate to process the rotating fibres (Ming-Jun et al., 2002).

#### **2.3.1 The linearly birefringent fibre**

The linear birefringence is a consequence of disruptions as the distortions of the core, the asymmetry of the lateral constraints, the curvature. In the case of the linear birefringence, the coupling coefficients matrix is given by (Ming-Jun et al., 2002):

$$k = \frac{1}{2} \begin{pmatrix} 0 & \Delta \beta e^{i2\varphi} \\\\ \Delta \beta e^{-i2\varphi} & 0 \end{pmatrix} \tag{15}$$

where ∆β is the linear birefringence, and Φ is the orientation of the birefringence with respect to a given axis

#### **2.3.2 Spun fibres**

156 Recent Progress in Optical Fiber Research

The evolution of the local polarization along the birefringent fibre is described by the equations of the modes coupling. The total change of polarization of an input signal, after having traveled a given distance in the fibre is better described by Jones' matrix. Let's assume that the losses in the fibres are negligible, the already predefined Jones matrix, can

with

2 2

\*

\*

The four complex elements of Jones' matrix can be gotten while integrating the equations of the coupled modes with suitable initial conditions. Once Jones' matrix is known, the PMD can be calculated easily from the elements of the matrix (Chen, 2002; Ming-Jun et al., 2002):

> 1 2 2 *dA dA d d*

In order to describe the reduction of the PMD, we define a parameter, named reduction

0

Where the used lengths for the spun fibres and standard fibres are the same. For example, if

The coupling coefficients matrix depends upon the dielectric tensor of the disruption. The values of these elements are determined by the type of disruption, which means that they depend on the configuration of the fibre. In this section, we describe some configurations of the fibres and we give their coupling coefficients matrix. It is important to note that the coupling matrixes in this work are expressed on the basis of the circular polarization

The linear birefringence is a consequence of disruptions as the distortions of the core, the asymmetry of the lateral constraints, the curvature. In the case of the linear birefringence,

because it is more appropriate to process the rotating fibres (Ming-Jun et al., 2002).

(11)

2 2

(13)

(14)

is equal to 0.5, a factor of

as the ratio of the DGD of the spun fibres over the DGD of the

1 2 *A A* 1 (12)

( , ). ( , , ). ( , ) 2 *mn n m k k e x y x y z e x y dxdy n N*

0 0 0

**2.2.2 Jones matrix and the PMD of spun fibre**

*T*

two is obtained in the reduction of the PMD.

**2.3 Different types of spun fibres**

**2.3.1 The linearly birefringent fibre**

be put under another form which is:

factor of PMD (PMDRF)

standard fibre.

Where *n*0 is the effective refractive index of both non disrupted modes.

1 2

*A z Az*

() ()

() ()

2 1

is equal to 1, the reduction of the PMD is not achieved and if

the coupling coefficients matrix is given by (Ming-Jun et al., 2002):

*A z Az*  In a spun fiber, the orientation of the birefringence takes place depending on the x axis. The rotation angle accumulated is therefore a function of the fibre length 'z', which in turn is determined by the rate of rotation ( ) *z* :

$$\Phi = \int\_0^z \alpha(z) \, dz \tag{16}$$

Replacing the equation (16) in equation (15), we get the coupling coefficients matrix of the rotating fibres, describing the disruption of the birefringence,:

$$k = \frac{1}{2} \begin{bmatrix} 0 & \, ^{i2\left\{\alpha(z)dz} \\ & 0 & \, \Delta \beta e \, ^{0} \\\\ & & -i2\left\{\alpha(z)dz \\ \Delta \beta e \, ^{0} & & 0 \end{bmatrix} \end{bmatrix} \end{bmatrix} \tag{17}$$

#### **2.3.3 Twisted fibre**

There are two effects in this type of fibres: The rotation of the birefringence and the mechanical torsion. The rotation of the birefringence is similar to that of the rotating fibre. If the rate of torsion is T, the angle is calculated by

$$
\Phi = T \ z \tag{18}
$$

The rate of torsion is determined by the coefficients of photo - elasticity of the fibre. The torsion constraint induces the circular birefringence proportionally to the rate of torsion.

$$
\delta = \lg T \tag{19}
$$

Where g is the coefficient determined by the coefficients of photo elasticity of the glass. The typical value of g for fibres in silica is 0.16. Combining both effects of rotation and torsion, the coupling matrix comes up with the following form:

$$k = \frac{1}{2} \begin{bmatrix} \delta & & \Delta \beta \ e^{i2Tz} \\\\ \Delta \beta \, e^{-iTz} & & -\delta \end{bmatrix} \tag{20}$$

Spun Fibres for Compensation of PMD: Theory and Characterization 159

. . ( ) 2

Equation (27) indicates that the DGD progresses linearly with the length of the fibre, and the

2

We notice that for spun fibres of constant rate, the PMDRF depends upon the length of

For the functions of periodic spin, under some conditions, we can describe analytic solutions by using the theory of disruption (Chen et al., 2002) in which fibres are submitted to uniform disruptions only, or in the case of small lengths regime (typically smaller than 100 m) in order to fine down their analysis. Indeed, in this approach, the random characteristic of the variation of the disruption in case of important lengths regime is ignored. Using the initial conditions issued from the previous paragraph, the first order solutions of disruption

> 0 ( ) ( / 2) exp[ 2 ( ')] ' *z Az i*

> > ( ) exp[ 2 ( ')'] '

Based on the theory of disruption, the first order of the disruption's expansion is valid only

The validity of this solution has been tested by (chen et al., 2002). When the length of beating

beating, the theory of disruption of the first order can always be applied. For sinusoidal profiles of spin, the expression for the factor of the PMD reduction can be gotten from the solutions of the disruption equation. Let's notice that the profile of sinusoidal spin takes the

0

Where α0 is the spin amplitude , and η is the angular frequency of the spatial modulations,

( ) cos( ) *z z*

which is linked to the spin period Λ through the following relationship η=2π/Λ.

1 . This condition puts some limits on the application of equation (31) on fibres

*z i z dz*

0

  *z*

*z*

PMDRF takes the following form:

beating or the birefringence.

**2.4.2 The periodic spin function**

for A1 (z) and A2(z) are as follows:

0 ( ) ( ') ' *z z z dz*

Where

when

following form:

that have a low PMD.

is important (some meters), i.e.

2

It becomes easier to obtain the DGD by using equation (13):

*z*

(27)

(28)

1 *A z*() 1 (29)

*i z dz* (30)

(31)

1 , and the period of spin is smaller than the length of

(32)

## **2.4 Solutions of the coupled equations for different types of the spun fibres**

Generally, the matrix of the coupling coefficients depends upon the variable z, and the analytic solutions of equation (8) have no existence in the majority of the cases. The numerical integration is always used to get numerical solutions. Different methods, as the method of the finite differences, the Runge-Kutta method (Chen, 2002), can be applied to solve the equation of the coupled modes. However, in the two following special cases, we can derive the analytic solutions which will be discussed in this section.

#### **2.4.1 Constant spinning rate**

For a constant spinning rate, the function 'spin' (rotation) can be written as follows:

$$
\alpha = \alpha\_0 \tag{21}
$$

Where 0 is a constant. In this case, the birefringence of a fibre is estimated in only one direction with a rate 0 . For this reason, the constant spinning rate is often assigned to an unidirectional spinning. For a spun fibre, with a constant spinning rate, the integral of the coupling matrix can be calculated easily, and the coupled equations become (Hart, 1994):

$$\frac{dA\_1}{dz} = \frac{1}{2} \mathbf{i} \,\,\Delta\beta \, e^{i2\alpha\_0 z} A\_2 \tag{22}$$

$$\frac{dA\_2}{dz} = \frac{1}{2} \text{i } \Delta \mathcal{J} \, e^{i2\, a\_0 z} A\_1 \tag{23}$$

With initial conditions 1 2 *A A* (0) 1, (0) 0 . The solutions of equations (22) and (23) are:

$$A\_1 = -\frac{a\_0 - \nu}{2\nu} e^{i(a\_0 + \nu)z} + \frac{a\_0 + \nu}{2\nu} e^{i(a\_0 - \nu)z} \tag{24}$$

$$A\_2 = \frac{\Delta\beta}{4\upsilon}e^{i(-a\_0+\upsilon)z} - \frac{\Delta\beta}{4\upsilon}e^{-i(a\_0+\upsilon)z} \tag{25}$$

Where 2 2 0 1 4

Using equations (12) and (13), we find that the DGD can be expressed by a simple equation for the spinning constant:

$$\tau(z) = \frac{\gamma\_{\alpha}}{2\nu} \sqrt{\left(\Delta\beta\right)^{2} z^{2} + \left(\frac{4\alpha\_{0}}{\Delta\beta} \sin\left(\frac{\Delta\beta}{2} z\right)\right)^{2}} \tag{26}$$

Where 0 *d B d L* is the PMD of a uniform birefringent fibre without modes coupling at

the z position. The sinusoidal term of equation (26) doesn't play an important role when the fibre is sufficiently long. On the other hand, for long fibres, the DGD is given by:

$$
\tau(z) = \frac{\gamma\_{\alpha} \cdot \Delta \beta \cdot z}{2\nu} \tag{27}
$$

Equation (27) indicates that the DGD progresses linearly with the length of the fibre, and the PMDRF takes the following form:

$$
\zeta = \frac{\Delta \beta}{2\nu} \tag{28}
$$

We notice that for spun fibres of constant rate, the PMDRF depends upon the length of beating or the birefringence.

### **2.4.2 The periodic spin function**

158 Recent Progress in Optical Fiber Research

Generally, the matrix of the coupling coefficients depends upon the variable z, and the analytic solutions of equation (8) have no existence in the majority of the cases. The numerical integration is always used to get numerical solutions. Different methods, as the method of the finite differences, the Runge-Kutta method (Chen, 2002), can be applied to solve the equation of the coupled modes. However, in the two following special cases, we

**2.4 Solutions of the coupled equations for different types of the spun fibres**

For a constant spinning rate, the function 'spin' (rotation) can be written as follows:

unidirectional spinning. For a spun fibre, with a constant spinning rate, the integral of the coupling matrix can be calculated easily, and the coupled equations become (Hart, 1994):

0 1 2

0 2 2

1 2 *dA i z ie A*

1 2 *dA i z ie A*

1 2 2

2 4 4 *i z iz Ae e*

*dz*

*dz*

fibre is sufficiently long. On the other hand, for long fibres, the DGD is given by:

*z z*

2 2 0 4 ( ) sin

0 is a constant. In this case, the birefringence of a fibre is estimated in only one

0 0 0 0 () ()

0 0 ( ) ()

>

*iz iz Ae e*

Using equations (12) and (13), we find that the DGD can be expressed by a simple equation

2 2

the z position. The sinusoidal term of equation (26) doesn't play an important role when the

>

is the PMD of a uniform birefringent fibre without modes coupling at

0 . For this reason, the constant spinning rate is often assigned to an

2

1

(24)

(25)

2

*z*

0 (21)

(22)

(23)

(26)

can derive the analytic solutions which will be discussed in this section.

**2.4.1 Constant spinning rate**

With initial conditions 1 2 *A A* (0) 1, (0) 0 . The solutions of equations (22) and (23) are:

>

Where

direction with a rate

Where 2 2 0 1 4

for the spinning constant:

Where 0 *d B*

*d L*

For the functions of periodic spin, under some conditions, we can describe analytic solutions by using the theory of disruption (Chen et al., 2002) in which fibres are submitted to uniform disruptions only, or in the case of small lengths regime (typically smaller than 100 m) in order to fine down their analysis. Indeed, in this approach, the random characteristic of the variation of the disruption in case of important lengths regime is ignored. Using the initial conditions issued from the previous paragraph, the first order solutions of disruption for A1 (z) and A2(z) are as follows:

$$A\_1(z) = 1\tag{29}$$

$$A\_2(z) = (i \;/\; 2) \Delta \beta \int\_0^z \exp[-2i\Theta(z')]dz' \tag{30}$$

Where 0 ( ) ( ') ' *z z z dz*

It becomes easier to obtain the DGD by using equation (13):

$$\tau(z) = \gamma\_o \left| \int\_0^z \exp[-2i\,\Theta(z')']dz' \right| \tag{31}$$

Based on the theory of disruption, the first order of the disruption's expansion is valid only when 1 . This condition puts some limits on the application of equation (31) on fibres that have a low PMD.

The validity of this solution has been tested by (chen et al., 2002). When the length of beating is important (some meters), i.e. 1 , and the period of spin is smaller than the length of beating, the theory of disruption of the first order can always be applied. For sinusoidal profiles of spin, the expression for the factor of the PMD reduction can be gotten from the solutions of the disruption equation. Let's notice that the profile of sinusoidal spin takes the following form:

$$a(z) = a\_0 \cos(\eta z) \tag{32}$$

Where α0 is the spin amplitude , and η is the angular frequency of the spatial modulations, which is linked to the spin period Λ through the following relationship η=2π/Λ.

Spun Fibres for Compensation of PMD: Theory and Characterization 161

We suppose that ( ) *z* is a periodic function. When ( ) *z* is an even function,

zero. For multiple values of the period T, the DGD becomes *n DGD T* ( ) . For values in between, some oscillations are encrusted in the linear variation of the DGD. On the other hand, this survey based on [40] leads us to the conclusion that the dependence of the DGD of the standard fibres on the square root of their lengths, comes from the statistical nature of the random coupling of the two modes of polarization. The linear evolution of the DGD with respect to the length of the spun fibre is caused by the periodicity of the coupling induced by the spinning, thus we have a coupling mode better-controlled than in the case of standard fibres. However, it is possible that the DGD of the spun fibres follows a different evolution law in a region where the first order theory of disruption is not valid any more; for example, when the intrinsic birefringence of the fibre is high and / or the spin rate is

With the aforementioned results, it is rather simple to find the phase matching conditions for which the maximum reduction of the PMD can be obtained. In this case, the condition is

exp 2 ( ') ' 0

Equation (34) can be expressed in another way if we use the properties discussed previously for even and odd functions. We notice that when the phase matching conditions are satisfied, the evolution of the DGD along the spun fibre is periodic. The DGD doesn't

Equations (31) and (34) are valid for a whole category of periodic profiles of spin. To illustrate the way how to determine the phase matching conditions, we take an example of a sinusoidal spin profile. Such a profile is defined by equation (32). The integration of this

> 2 sin ' ( ) exp[ ] ' *z z DGD z i dz*

0 2 2 1 1 0 exp[ sin( )] ( ) 2 ( ).cos(2 ) 2 ( )sin[(2 1) ] *n n*

*ix J x J x n i J x n*

1/2 2 2 *DGD z R z I z* () () ()

1 (2 / ) ( ) (2 / ) sin(2 ) *n n J Rz J z n z*

2 0

*n*

(36)

0

The integral can be valued analytically by using the following identity:

*n*

0 0

 ; then we get the DGD by using equation (31):

0

fixed such that the PMDFR is equal to zero (chen et al., 2002):

increase anymore when the length of the fibre increases.

profile gives 0 ( ) sin( ) / *z z*

Then, we get

where

>

0

*T*

0

*i z dz* (34)

(35)

(37)

(38)

*T*

cos 2 ( ') '

*z dz* is equal to

*z dz* is equal to zero. When ( ) *z* is an odd function ,

sin 2 ( ') '

0

high.

*T*

With the analytical solution of equation (31), we are able to assert which spin parameters give right to the optimization of the PMD performances. With the first observations, we remark that, when the length of beating of a fibre is bigger than some meters, the PMDRF is independent of the beating length, and therefore of the intrinsic birefringence of the fibre. In equation (31), the only contribution to the birefringence of the fibre, comes from , and the DGD is proportional to this size. Let's note that is the PMD of the unspun fibers (non rotating). On the other hand, the PMDFR will be independent of .

This conclusion is also verified by the direct numerical integration of equation (8) with k given by equation (17). Some old fibres had beating lengths inferior to some meters; with the improvement of fibre manufacturing, the majority of these lengths were improved lately beyond some meters. The PMDRF independence from the intrinsic birefringence of the fibre, offers the advantage of simplicity in its conception because it is worthless to optimize the spin profiles for the different birefringences of the fibre. Moreover, we noticed that the DGD increases linearly when the length of the fibre increases (figure 3) despite the fact that we got some overlapping oscillations on the graph representing the variation of the DGD with respect to the distance.

Fig. 3. Evolution of the DGD along a spun fibre.

We can also separate the real and imaginary contributions of the integral (31) in order to better analyze the variation of the DGD along the fibre. We express the equation (31) in an alternative way of the DGD for one spatial period T:

$$DGD(T) = \mathcal{Y}\_o \left| \bigcap\_{0}^{T} \cos[2\Theta(z^\prime)]dz^\prime - i \prod\_{0}^{T} \sin[2\Theta(z^\prime)]dz^\prime \right| \tag{33}$$

$$DGD(T) = \mathcal{Y}\_o \left| \begin{matrix} \cos[2\Theta(z^\prime)]dz^\prime \end{matrix} \right. \quad \Theta(z) \text{ is an even function} \right.$$

$$DGD(T) = \mathcal{Y}\_o \left| \begin{matrix} \sin[2\Theta(z^\prime)]dz^\prime \end{matrix} \right.$$

We suppose that ( ) *z* is a periodic function. When ( ) *z* is an even function, 0 sin 2 ( ') ' *T z dz* is equal to zero. When ( ) *z* is an odd function , 0 cos 2 ( ') ' *T z dz* is equal to

zero. For multiple values of the period T, the DGD becomes *n DGD T* ( ) . For values in between, some oscillations are encrusted in the linear variation of the DGD. On the other hand, this survey based on [40] leads us to the conclusion that the dependence of the DGD of the standard fibres on the square root of their lengths, comes from the statistical nature of the random coupling of the two modes of polarization. The linear evolution of the DGD with respect to the length of the spun fibre is caused by the periodicity of the coupling induced by the spinning, thus we have a coupling mode better-controlled than in the case of standard fibres. However, it is possible that the DGD of the spun fibres follows a different evolution law in a region where the first order theory of disruption is not valid any more; for example, when the intrinsic birefringence of the fibre is high and / or the spin rate is high.

With the aforementioned results, it is rather simple to find the phase matching conditions for which the maximum reduction of the PMD can be obtained. In this case, the condition is fixed such that the PMDFR is equal to zero (chen et al., 2002):

$$\int\_0^T \exp\left[-2i\Theta(z')\right]dz'=0\tag{34}$$

Equation (34) can be expressed in another way if we use the properties discussed previously for even and odd functions. We notice that when the phase matching conditions are satisfied, the evolution of the DGD along the spun fibre is periodic. The DGD doesn't increase anymore when the length of the fibre increases.

Equations (31) and (34) are valid for a whole category of periodic profiles of spin. To illustrate the way how to determine the phase matching conditions, we take an example of a sinusoidal spin profile. Such a profile is defined by equation (32). The integration of this profile gives 0 ( ) sin( ) / *z z* ; then we get the DGD by using equation (31):

$$DGD(z) = \gamma\_o \left| \bigwedge\_{0}^{z} \exp[-i \frac{2a\_0 \sin(\eta z')}{\eta}] dz' \right| \tag{35}$$

The integral can be valued analytically by using the following identity:

$$\exp[-ix\sin(\theta)] = f\_0(\mathbf{x}) + 2\sum\_{n=1}^{\infty} f\_{2n}(\mathbf{x})\cos(2n\theta) - 2i\sum\_{0}^{\alpha} f\_{2n+1}(\mathbf{x})\sin[(2n+1)\theta] \tag{36}$$

Then, we get

$$DGD(z) = \gamma\_o \left[ R^2(z) + I^2(z) \right]^{1/2} \tag{37}$$

where

160 Recent Progress in Optical Fiber Research

With the analytical solution of equation (31), we are able to assert which spin parameters give right to the optimization of the PMD performances. With the first observations, we remark that, when the length of beating of a fibre is bigger than some meters, the PMDRF is independent of the beating length, and therefore of the intrinsic birefringence of the fibre. In

This conclusion is also verified by the direct numerical integration of equation (8) with k given by equation (17). Some old fibres had beating lengths inferior to some meters; with the improvement of fibre manufacturing, the majority of these lengths were improved lately beyond some meters. The PMDRF independence from the intrinsic birefringence of the fibre, offers the advantage of simplicity in its conception because it is worthless to optimize the spin profiles for the different birefringences of the fibre. Moreover, we noticed that the DGD increases linearly when the length of the fibre increases (figure 3) despite the fact that we got some overlapping oscillations on the graph representing the variation of the DGD with

We can also separate the real and imaginary contributions of the integral (31) in order to better analyze the variation of the DGD along the fibre. We express the equation (31) in an

> 0 0 ( ) cos 2 ( ') ' sin 2 ( ') ' *T T DGD T z dz i z dz*

is an even function

is an odd function

(33)

is the PMD of the unspun fibers (non

. , and the

equation (31), the only contribution to the birefringence of the fibre, comes from

DGD is proportional to this size. Let's note that

Fig. 3. Evolution of the DGD along a spun fibre.

alternative way of the DGD for one spatial period T:

*DGD T z dz z*

*DGD T z dz z*

( ) sin 2 ( ') ', ( )

( ) cos 2 ( ') ', ( )

0

0

*T*

*T*

respect to the distance.

rotating). On the other hand, the PMDFR will be independent of

$$R(z) = l\_0(2a\_0 \;/\; \eta)z + \sum\_{n=1}^{\infty} \frac{l\_{2n}(2a\_0 \;/\; \eta)}{\eta \, n} \sin(2n\eta \, z) \tag{38}$$

Spun Fibres for Compensation of PMD: Theory and Characterization 163

be achieved for different spin parameters enabling to get an optimal reduction of PMD; though in general, amplitudes of higher spins give a better reduction of PMD**.** Figure (3.a) shows the evolution of DGD along the fibre with the phase matching condition. Finally, the maximum of reduction can be reached at the zeroes of the Bessel function of order zero

While using equation (28), the PMDRF, as function of the spin rate, is represented for

Fig. 5. The factor of reduction of the PMD with respect to the spin amplitude in the case of a

We note that the PMD is reduced when the spin rate increases. For the same spin rate, PMDRF depends on the beating length. The higher the beating length is, the more reduced is the PMD. For a high PMD of the fibre (beating length <1m), a high spin rate is necessary

In figure 6, we use a beating length of 1m as example to illustrate the reduction of the PMD

Figure 6 shows that for sinusoidal spin types, the PMDRF oscillates with the spin amplitude, which is different from the case where the spin was constant. Furthermore, this figure shows that, for a sinusoidal spin, the phase matching condition can be gotten in order to come to a low PMD; on the other hand, in the case of constant spin, the phase matching doesn't exist. The phenomenon of phase matching can be explained by the mechanism of coupling of modes. The constant spin reduces the birefringence of the fibre, and causes no coupling of modes as well. For the sinusoidal spin, the variation in the rate of spin carries along the two modes of polarization to intercouple, reaching a compensation of the PMD. For some spin profile and birefringence of fibre, the conditions of phase matching are satisfied and the maximum of energy exchange occurs in order to provide a better reduction of PMD. The results of modeling indicate that the conditions of phase matching depend on the beating length, the period of the spin and the amplitude of the spin. We can use the same function of

**2.5 Reduction of the PMD for different profiles of spin**

different beating lengths in figure 5; for a constant spin rate.

constant spinning rate for different beat lengths.

for sinusoidal types of spin (Ming-Jun et al., 2002).

to reduce the PMD.

**2.5.2 Sinusoidal spin**

(equation 41).

**2.5.1 The constant spin rate**

Fig. 4*.* The PMD change factor as a function of the spin amplitude and the spin period.

We note that when 0 0 *J* (2 / ) 0 , the dominant contribution comes from the term of the linear increase of equation (38). Neglecting the oscillations term, the expression of the DGD becomes:

$$\text{DGD}(z) = \gamma\_{\alpha} l\_0 (2\alpha\_0 \;/\; \eta) z \tag{40}$$

As in the case of spun fibres with a constant rate, in absence of random disruptions, the DGD increases linearly with the length of the fibre; in contrast with the PMDRF which takes a simpler shape:

$$PMDRF = f\_0(2\alpha\_0 \; / \; \eta) \tag{41}$$

Equation (41) indicates that the PMDRF is independent from the beating length in the case of spun fibres with sinusoidal profile whose beating lengths are equal to some meters or more. When 0 0 *J* (2 / ) 0 , the linear increase term disappears, and the oscillation terms cannot be neglected any more. In this case, the DGD oscillates between 0 and a maximum value and is independent of the propagating distance. The condition where the minimum of the PMD is reached is called the condition of phase matching (figure 4).

Figure 4 illustrates the presentation in three dimensions for the graph of the PMD reduction as a function of the spin period and the spin amplitude. The phase matching condition can

(2 / ) ( ) cos[(2 1) ] (2 1)

, the dominant contribution comes from the term of the

(40)

(41)

(39)

*J I z n z n*

21 0

Fig. 4*.* The PMD change factor as a function of the spin amplitude and the spin period.

linear increase of equation (38). Neglecting the oscillations term, the expression of the DGD

0 0 *DGD z J z* ( ) (2 / )

As in the case of spun fibres with a constant rate, in absence of random disruptions, the DGD increases linearly with the length of the fibre; in contrast with the PMDRF which takes

> 0 0 *PMDRF J* (2 / )

Equation (41) indicates that the PMDRF is independent from the beating length in the case of spun fibres with sinusoidal profile whose beating lengths are equal to some meters or

cannot be neglected any more. In this case, the DGD oscillates between 0 and a maximum value and is independent of the propagating distance. The condition where the minimum of

Figure 4 illustrates the presentation in three dimensions for the graph of the PMD reduction as a function of the spin period and the spin amplitude. The phase matching condition can

the PMD is reached is called the condition of phase matching (figure 4).

, the linear increase term disappears, and the oscillation terms

We note that when 0 0 *J* (2 / ) 0

becomes:

a simpler shape:

more. When 0 0 *J* (2 / ) 0

  *n*

0

*n*

be achieved for different spin parameters enabling to get an optimal reduction of PMD; though in general, amplitudes of higher spins give a better reduction of PMD**.** Figure (3.a) shows the evolution of DGD along the fibre with the phase matching condition. Finally, the maximum of reduction can be reached at the zeroes of the Bessel function of order zero (equation 41).

## **2.5 Reduction of the PMD for different profiles of spin 2.5.1 The constant spin rate**

While using equation (28), the PMDRF, as function of the spin rate, is represented for different beating lengths in figure 5; for a constant spin rate.

Fig. 5. The factor of reduction of the PMD with respect to the spin amplitude in the case of a constant spinning rate for different beat lengths.

We note that the PMD is reduced when the spin rate increases. For the same spin rate, PMDRF depends on the beating length. The higher the beating length is, the more reduced is the PMD. For a high PMD of the fibre (beating length <1m), a high spin rate is necessary to reduce the PMD.

## **2.5.2 Sinusoidal spin**

In figure 6, we use a beating length of 1m as example to illustrate the reduction of the PMD for sinusoidal types of spin (Ming-Jun et al., 2002).

Figure 6 shows that for sinusoidal spin types, the PMDRF oscillates with the spin amplitude, which is different from the case where the spin was constant. Furthermore, this figure shows that, for a sinusoidal spin, the phase matching condition can be gotten in order to come to a low PMD; on the other hand, in the case of constant spin, the phase matching doesn't exist. The phenomenon of phase matching can be explained by the mechanism of coupling of modes. The constant spin reduces the birefringence of the fibre, and causes no coupling of modes as well. For the sinusoidal spin, the variation in the rate of spin carries along the two modes of polarization to intercouple, reaching a compensation of the PMD. For some spin profile and birefringence of fibre, the conditions of phase matching are satisfied and the maximum of energy exchange occurs in order to provide a better reduction of PMD. The results of modeling indicate that the conditions of phase matching depend on the beating length, the period of the spin and the amplitude of the spin. We can use the same function of

Spun Fibres for Compensation of PMD: Theory and Characterization 165

Fig. 7. The DGD of a spun fibre according to the length of the fibre. The amplitude of spin is 3.5 turn/m, the period of spin is 1m, the length of beating is 10m and the length of coupling

length of fiber (m)

Fig. 8. Probability density function versus the DGD of the fibre when the condition of phase matching is not verified. The amplitude of spin is 3.5 turn/m, the period of spin is 1m, the

The simplest law of evolution given by equation (42) has been verified by using a numerical modeling (Chen, 2003)**.** Figure 7 shows the results of the numerical simulation for a sinusoidal spun fibre under the non optimal conditions. As it is shown in this figure, the

In the case of standard fibres (unspun fibres ) with a random coupling mode length h, the distribution of the DGD is analog to the distribution of Maxwell, where the standard

used in the expression of PDF can be given by the following expression:

( ) ./ 3 *b*

We have proven that the Maxwell distribution is valid in the case of the spun fibres, except

*cL*

*h l*

(43)

should be corrected by the contribution of the spinning fibre. The

numerical modeling accomodates very well with the theoretical prediction.

is now under the following form

length of fibre is 500m and the length of coupling is 10m.

is 10m.

deviation

that, the parameter

modified parameter

Fig. 6. The factor of reduction of the PMD versus the spin amplitude in the sinusoidal spin profiles.

spin to get a small reduction of PMD for high lengths of beating. However, for small lengths of beating; the phase matching has a strong dependence with the length of beating.

The fact that the birefringence of the real fibres is not constant and changes randomly, it is impossible to have the phase matching for the whole birefringence while using only one sinusoidal spin. This problem can be solved by admitting spin profiles with many Fourier components. To get to this point, the concept of the use of the modulated spin in amplitude and frequency has been developed by the Corning society.

#### **2.6 Statistical evolution of the PMD of the spun fibres**

As it was mentioned in the previous sections, the spun fibres follow a linear evolution law without the random modes coupling or in the régime of short lengths. When the random mode coupling is present, it has been found that the spun fibres follow an evolution law, a function of square root, similar to that of the unspun fibres, but with a different rate depending on the spin parameters (Chen, 2002)**.** The random mode coupling can be characterized by a random variation of the birefringence axis and / or by the induced phase shift by the external constraints with an occurrence frequency of *1/h*, where *h* is called the coupling length of the modes. On the other side, a fibre of length «*l*» can be divided into (*l/h*) segments. Using this model, for a sinusoidal profile of spun fibre under no optimal conditions (no phase matching ), the DGD can be expressed under the following simple form:

$$
\pi = \zeta \gamma\_{\phi} \sqrt{l \ln l} \tag{42}
$$

We notice that the fact that the PMDRF « » is independent of the beating length when the length of beating is greater than some meters, the DGD in the régime of important lengths, and in presence of the random coupling mode, is corrected by a factor , which is the reduction induced by the fibre spinning during the process of drawing. In this case, the property of evolution of the PMD is similar to that of the fibre possessing the linear birefringence.

*mL*

1

*b b b*

*mmL mmL*

5.0,1 1,1

constant spin

Fig. 6. The factor of reduction of the PMD versus the spin amplitude in the sinusoidal spin

spin to get a small reduction of PMD for high lengths of beating. However, for small lengths

The fact that the birefringence of the real fibres is not constant and changes randomly, it is impossible to have the phase matching for the whole birefringence while using only one sinusoidal spin. This problem can be solved by admitting spin profiles with many Fourier components. To get to this point, the concept of the use of the modulated spin in amplitude

As it was mentioned in the previous sections, the spun fibres follow a linear evolution law without the random modes coupling or in the régime of short lengths. When the random mode coupling is present, it has been found that the spun fibres follow an evolution law, a function of square root, similar to that of the unspun fibres, but with a different rate depending on the spin parameters (Chen, 2002)**.** The random mode coupling can be characterized by a random variation of the birefringence axis and / or by the induced phase shift by the external constraints with an occurrence frequency of *1/h*, where *h* is called the coupling length of the modes. On the other side, a fibre of length «*l*» can be divided into (*l/h*) segments. Using this model, for a sinusoidal profile of spun fibre under no optimal conditions (no phase matching ), the DGD can be expressed under the following simple

> *hl*

(42)

» is independent of the beating length when the

, which is the reduction

in presence of the random coupling mode, is corrected by a factor

evolution of the PMD is similar to that of the fibre possessing the linear birefringence.

length of beating is greater than some meters, the DGD in the régime of important lengths, and

induced by the fibre spinning during the process of drawing. In this case, the property of

of beating; the phase matching has a strong dependence with the length of beating.

and frequency has been developed by the Corning society.

**2.6 Statistical evolution of the PMD of the spun fibres**

We notice that the fact that the PMDRF «

profiles.

PMDRF

0.0

0.2

0.4

0.6

0.8

form:

Fig. 7. The DGD of a spun fibre according to the length of the fibre. The amplitude of spin is 3.5 turn/m, the period of spin is 1m, the length of beating is 10m and the length of coupling is 10m.

Fig. 8. Probability density function versus the DGD of the fibre when the condition of phase matching is not verified. The amplitude of spin is 3.5 turn/m, the period of spin is 1m, the length of fibre is 500m and the length of coupling is 10m.

The simplest law of evolution given by equation (42) has been verified by using a numerical modeling (Chen, 2003)**.** Figure 7 shows the results of the numerical simulation for a sinusoidal spun fibre under the non optimal conditions. As it is shown in this figure, the numerical modeling accomodates very well with the theoretical prediction.

In the case of standard fibres (unspun fibres ) with a random coupling mode length h, the distribution of the DGD is analog to the distribution of Maxwell, where the standard deviation used in the expression of PDF can be given by the following expression:

$$
\sigma = (\frac{\lambda}{cL\_b})\sqrt{h.1} \ne \sqrt{3} \tag{43}
$$

We have proven that the Maxwell distribution is valid in the case of the spun fibres, except that, the parameter should be corrected by the contribution of the spinning fibre. The modified parameter is now under the following form

Spun Fibres for Compensation of PMD: Theory and Characterization 167

The PMD of the fibre under test is determined by the arithmetic mean of the '*n*' measured

0

We applied the above procedure to two types of the spun fibres in order to compare their performances. The first fibre is unidirectional of length 212m in which we noted that the rotation of the spins was only in one sense once removed it from the spool. On the other hand, for the second bi-directional of length 1Km, the rotation of the spins was in the two senses. The results gotten in the figure 9.a and the figure 9.b, show that this method has a good resolution because it permitted to measure DGD of the order of femtoseconds, and to show that this type of fibre presents effectively low DGD compared to those measured in standard fibres that are of order of the picoseconds*.* Besides, we noted that the bi-directional fibre possesses a lower DGD than that of the unidirectional one indicating thus the

Fig. 9. Representation of DGD measured of spun fibre according to the wavelength for the

length (a ) L = 290m, (b) L = 212m.

*DGD*

*n*

*i*

*PMD*

efficiency of the bi-directional spun fibres in the reduction of the PMD.

*n*

( )

*i*

DGD:

(a)

(b)

$$\sigma = \left[ f\_0(2\alpha\_0 \,/\, \eta) \right](\frac{\lambda}{cL\_b})\sqrt{h\lambda} \,/\sqrt{3} \tag{44}$$

This equation has been tested and validated in (chen, 2002). Figure 8 represents the probability density function 'PDF ', according to the DGD of the spun fibre, obtained by numerical calculations and Maxwell distribution equation where we confirm according to the figures that the two results converge perfectly.

When the conditions of phase matching are satisfied, the total DGD of the fibre is a periodic function, and it oscillates between the zero value and a maximal value max . For this reason, the DGD of only one segment of a fibre is linked to the average of DGD inside one period of spin. Therefore, in the regime of high lengths (*l >>h*), the total DGD can be written as follows (Ming-Jun et al., 2002)**:**

$$
\pi = \varepsilon^{\cdot} \varepsilon\_{q} \sqrt{\mathbf{l} \ / \ h} \tag{45}
$$

Where *q* is the square average of the DGD in one period of spin, and ' is a coefficient that depends on the average coupling coefficient between two segments. For a condition of phase matching (for example: 1 ' 0 2.76 / 2 ), *tours m et m* is found equal to 1.194. Besides, the DGD increases when the length of coupling of modes decreases **(**Ming-Jun et al., 1998). It is foreseeable, because under the conditions of phase matching, the DGD is minimum. Any disruption moves the fibre away from the optimal conditions, implying an increase of the PMD. Despite the fact that the DGD of the optimized spun fibres changes differently with the coupling length in comparison with the DGD of the non optimized spun fibres, the DGD always follows a Maxwell distribution, but with a modified parameter (Chen, 2002).

$$
\sigma = (\stackrel{\cdot}{\varepsilon} \varepsilon\_q \sqrt{1/h}) / \sqrt{3} \tag{46}
$$

## **3. Application of the JME method for the measurement of the PMD of the spun fibres**

We used the JME method (Derickson**,** 1998**)** in order to verify the reduction of the PMD in the spun fibres (Cherbi et al., 2006). This applied method, between 1510 to 1615 nanometers, consists in determining the DGD directly between the two main states of polarization by measuring the Jones matrix of the device under test to a set of wavelengths. In order to determine the PMD of the spun fibres, we take the following steps:

$$DGD(\lambda\_i) = \Delta \tau = \left| \frac{\arg\left(\frac{\rho\_1}{\rho\_2}\right)}{\Delta \alpha} \right| \tag{47}$$

*J h l cL*

This equation has been tested and validated in (chen, 2002). Figure 8 represents the probability density function 'PDF ', according to the DGD of the spun fibre, obtained by numerical calculations and Maxwell distribution equation where we confirm according to

When the conditions of phase matching are satisfied, the total DGD of the fibre is a periodic

the DGD of only one segment of a fibre is linked to the average of DGD inside one period of spin. Therefore, in the regime of high lengths (*l >>h*), the total DGD can be written as

> '

that depends on the average coupling coefficient between two segments. For a condition of

Besides, the DGD increases when the length of coupling of modes decreases **(**Ming-Jun et al., 1998). It is foreseeable, because under the conditions of phase matching, the DGD is minimum. Any disruption moves the fibre away from the optimal conditions, implying an increase of the PMD. Despite the fact that the DGD of the optimized spun fibres changes differently with the coupling length in comparison with the DGD of the non optimized spun fibres, the DGD

2.76 / 2 ), *tours m et m*

( / )/ 3 *q*

We used the JME method (Derickson**,** 1998**)** in order to verify the reduction of the PMD in the spun fibres (Cherbi et al., 2006). This applied method, between 1510 to 1615 nanometers, consists in determining the DGD directly between the two main states of polarization by measuring the Jones matrix of the device under test to a set of wavelengths. In order to

for a set of wavelengths 1 2, , ....

1 2

arg

is the square average of the DGD in one period of spin, and '

'

**3. Application of the JME method for the measurement of the PMD of the**

function, and it oscillates between the zero value and a maximal value max

phase matching (for example: 1 ' 0

always follows a Maxwell distribution, but with a modified parameter

determine the PMD of the spun fibres, we take the following steps:

1 and

*i* is gotten then by (Heffner**,** 1992):

( ) *DGD i*

the figures that the two results converge perfectly.

follows (Ming-Jun et al., 2002)**:**

Where *q*

**spun fibres**

Measure Jones' matrixes ( )*i JM*

 Do the product 1 ( ). ( ) *i i JM JM*

(1510 nm-1615nm)

The DGD ( )

Determine the eigen values

0 0 (2 / ) ( ) . / 3 *b*

(44)

(Chen, 2002).

*n* of the work range

*q l h* / (45)

 is found equal to 1.194.

*l h* (46)

(47)

2 of the calculated product of matrix

. For this reason,

is a coefficient

The PMD of the fibre under test is determined by the arithmetic mean of the '*n*' measured DGD:

$$\text{PMD} = \frac{\sum\_{i=0}^{n} DGD(\vec{\lambda}\_i)}{n}$$

We applied the above procedure to two types of the spun fibres in order to compare their performances. The first fibre is unidirectional of length 212m in which we noted that the rotation of the spins was only in one sense once removed it from the spool. On the other hand, for the second bi-directional of length 1Km, the rotation of the spins was in the two senses. The results gotten in the figure 9.a and the figure 9.b, show that this method has a good resolution because it permitted to measure DGD of the order of femtoseconds, and to show that this type of fibre presents effectively low DGD compared to those measured in standard fibres that are of order of the picoseconds*.* Besides, we noted that the bi-directional fibre possesses a lower DGD than that of the unidirectional one indicating thus the efficiency of the bi-directional spun fibres in the reduction of the PMD.

Fig. 9. Representation of DGD measured of spun fibre according to the wavelength for the length (a ) L = 290m, (b) L = 212m.

Spun Fibres for Compensation of PMD: Theory and Characterization 169

relation already demonstrated experimentally by COTDR in our anterior works (Cherbi et al., 2009) existing between the spatial period of the backscattered signal and the PMD of the spun fibre and given by (Chen, 2003). Even more, the COTDR method allowed us to compare the results found with those of the JME method. Afterward we will present the POFDR method which used for spun fibres (Cherbi et al., 2009; Wegmuller et al., 2005) to obtain the beat lengths of the two types of spun fibres and the spin period of the bi-

Chen( Chen, 2003) has demonstrated that the spatial period of the backscattered signals obtained from a POTDR (polarization-sensitive optical time-domain reflectometer) of the spun fibres varies linearly with the beat length of the fibre. This means that for a given beat length, the spatial period *Ts* can be used as calibration for the reduction of the PMD. A simple relation linking the spatial frequency F ( *F T* 1 / *s* ) to the beat length and the spin

> *FJ L* 0 0 2 / /( /2)

The PMD of the bi-directional spun fibre is linked to the spatial frequency in the form

*PMD c F*

Thus, the PMD of the bi-directional spun fibres can be determined directly through the measure of the spatial period as in the case of the standard fibres, while measuring the spatial period of the backscattered obtained from the reflectometers. The equation (50) shows that when the spin is zero, the spatial period converges to the one of the standard

The technique (COTDR) (Wegmuller et al., 2004) is appropriate to detect the defaults in a given fibre (sites of reflection, losses) with a spatial resolution of the decimetre order. The main difference of this reflectometer (Cherbi et al., 2009) compared to a classic OTDR (Ellison & Siddiqui, 1998) resides in the use of photon counting detector (InGaAs avalanche photodiode). It is used in the so-called gated Geiger mode, which means that the detector is only active during a short time slot. During this period, only a single photon falls in the detector and triggers an avalanche, which is then detected by electronics discriminator. Contrary to the operation of a classic detector APD in linear regime, this avalanche is no longer proportional to optical input signal power, but independent of it. The detection is therefore a binary one, either there is an avalanche, or not. In order to evaluate the incident optical power (or mean photon number) on the detector during its activation, the detection process (gate opening) must be repeated many times in order to determine the detection probability of photons with a good precision. This probability is proportional to the incident signal power that is smaller than about 40% (no detector saturation) and larger than the detector thermal noise (dark counts). This condition is satisfied in our set-up by using the

**4.1 Measure of the DGD in the spun fibres by the C-OTDR method**

variable attenuator before the excitation of the fibre.

*b* (50)

/2 (51)

directional fibre.

parameters, is given by:

(Chen, 2003):

fibres.

The researchers and inventors of the optical fibre of telecommunication systems predict the impact of PMD from the distribution of , because this results from the variation of as a function of wavelength and the conditions of the environment. On the other hand, due to this type of variation, the PMD of an optical path is expressed then statistically, as either the average or the root mean square (RMS) of *i* (Derickson**,** 1998 **).** It is interesting to determine the total PMD of a link made of a series of different spun fibres.

For this reason, we took three different lengths of spun fibres (fibre2, 3 and 4). We started by measuring, with the JME method, their PMD separately, for a given temperature while using a reference fibre (fibre1) used in calibration (table 1). Then, we connected the three fibres, and done the measurement of the total PMD in the same experimental conditions. The same procedure has been applied for the two fibres (fibres 3 and 4). We sought for the best relation of computation to determine the total PMD of a link of spun fibres, by testing the two following relations:

$$\text{PMD}\_{\text{bubble}} = \text{PMD}\_1 + \text{PMD}\_2 + \dots + \dots \dots \text{PMD}\_n \tag{48}$$

$$\text{Cor} \quad \text{PMD}\_{\text{total}} = \sqrt{\text{PMD}\_1^2 + \text{PMD}\_2^2 + ... ... ... \text{PMD}\_n^2} \tag{49}$$

With *n* the number of fibres used in the link

Our experimental results regrouped in table 1 are in very good agreement with the first relation [Cherbi et al., 2006].

Table 1. The PMD relation of the spun fibres link.

## **4. Determination of the polarization's properties of the spun fibres using the reflectometers**

The beat length of the fibre can be measured directly by the extraction of the spatial period of the backscattered signals (Wegmuller**,** 2002, 2004)**,** which permits to estimate the PMD in the single-mode fibres (Ellison et al., 1998; Chen, 2002)**.** The OFDR method is not exploited again especially for investigating of the spun fibres for the determination of its parameters and of their PMD according to the distance. In this section, we will present the

The researchers and inventors of the optical fibre of telecommunication systems predict the

a function of wavelength and the conditions of the environment. On the other hand, due to this type of variation, the PMD of an optical path is expressed then statistically, as either the

>

For this reason, we took three different lengths of spun fibres (fibre2, 3 and 4). We started by measuring, with the JME method, their PMD separately, for a given temperature while using a reference fibre (fibre1) used in calibration (table 1). Then, we connected the three fibres, and done the measurement of the total PMD in the same experimental conditions. The same procedure has been applied for the two fibres (fibres 3 and 4). We sought for the best relation of computation to determine the total PMD of a link of spun fibres, by testing

Or 22 2

Our experimental results regrouped in table 1 are in very good agreement with the first

, because this results from the variation of

1 2 .............. *PMD PMD PMD PMD totale n* (48)

1 2 .......... *PMD PMD PMD PMD totale n* (49)

Total DGD (fs) calculated with relation (48)

(Derickson**,** 1998 **).** It is interesting to

Total DGD (fs) calculated with relation (49)

as

determine the total PMD of a link made of a series of different spun fibres.

DGD(fs) measured with the step (10nm)

connected Fibres (2+3+4) 22,9437 21,99 13,16 connected fibres (3+4) 17,1985 17,17 12,25

**4. Determination of the polarization's properties of the spun fibres using the**

The beat length of the fibre can be measured directly by the extraction of the spatial period of the backscattered signals (Wegmuller**,** 2002, 2004)**,** which permits to estimate the PMD in the single-mode fibres (Ellison et al., 1998; Chen, 2002)**.** The OFDR method is not exploited again especially for investigating of the spun fibres for the determination of its parameters and of their PMD according to the distance. In this section, we will present the

impact of PMD from the distribution of

With *n* the number of fibres used in the link

Table 1. The PMD relation of the spun fibres link.

the two following relations:

relation [Cherbi et al., 2006].

reference fibre of (1 km) 98,721 Fibre 2 (212 m) 4,8223 Fibre 3 (290m) 7,4315 Fibre 4 (1 km) 9,7399

**reflectometers**

average or the root mean square (RMS) of *i*

relation already demonstrated experimentally by COTDR in our anterior works (Cherbi et al., 2009) existing between the spatial period of the backscattered signal and the PMD of the spun fibre and given by (Chen, 2003). Even more, the COTDR method allowed us to compare the results found with those of the JME method. Afterward we will present the POFDR method which used for spun fibres (Cherbi et al., 2009; Wegmuller et al., 2005) to obtain the beat lengths of the two types of spun fibres and the spin period of the bidirectional fibre.

Chen( Chen, 2003) has demonstrated that the spatial period of the backscattered signals obtained from a POTDR (polarization-sensitive optical time-domain reflectometer) of the spun fibres varies linearly with the beat length of the fibre. This means that for a given beat length, the spatial period *Ts* can be used as calibration for the reduction of the PMD. A simple relation linking the spatial frequency F ( *F T* 1 / *s* ) to the beat length and the spin parameters, is given by:

$$F = \left| I\_0 \left( 2a\_0 \mid \eta \right) \right| / \left( L\_b \; / \; \mathcal{D} \right) \tag{50}$$

The PMD of the bi-directional spun fibre is linked to the spatial frequency in the form (Chen, 2003):

$$\text{PMD} = \left(\mathbb{Z} / \mathfrak{L}\mathfrak{L}\right) \text{F} \tag{51}$$

Thus, the PMD of the bi-directional spun fibres can be determined directly through the measure of the spatial period as in the case of the standard fibres, while measuring the spatial period of the backscattered obtained from the reflectometers. The equation (50) shows that when the spin is zero, the spatial period converges to the one of the standard fibres.

#### **4.1 Measure of the DGD in the spun fibres by the C-OTDR method**