3. Numerical results

The split-step Fourier method is used for solving Eqs. (6) and (7) [14]. The values of using parameters are the following: β<sup>1</sup> = 1, β<sup>1</sup> = �1 ps/km; β<sup>1</sup> = 0; <sup>β</sup><sup>2</sup> <sup>=</sup> �25 ps<sup>2</sup> /km, α = 0.3, γ = 1.6 1/(W-km), TR = 3 fs. The value of β<sup>1</sup> = 1 corresponds to the twisted fiber with 6 turns/m. The meaning of the change of the sign of the constant β is for representing the change of the twist direction. For simulations, the 30 ps input pulse with 40 W power in the input fiber was used, and the Gaussian noise was added on the pulse. Modulation instability (MI) effect breaks up the pulse generating a set of solitons; the highest soliton were traced in this set [8].

When a linearly polarized pulse is introduced to the fiber input, we can see the influence of the walk-off effect between circularly right- and left-polarized components on ellipticity, see Figure 1. The fiber optic length used varies from 1 to 1.5 km. The results for the simulations are present in Figure 1, the result for β<sup>1</sup> = 1 is shown by open circles, for β<sup>1</sup> = �1 is shown by closed circles, and for β<sup>1</sup> = 0 is shown by squared. The important result for the dependence of polarization behavior can be observed clearly. The polarization behavior with β<sup>1</sup> = 1, but in this special case, we consider α = 1, that is, the parallel Raman gain is equal to the perpendicular, α = 1, see Figure 2. It can be seen that in this case, the effect of the ellipticity change of the soliton along the fiber is not presented. The results obtained are based on Eq. (10), and these results show that some ellipticity change must be presented on the circular birefringence and difference between Raman parallel and perpendicular gain. It can be observed that the ellipticity of the highest soliton does not coincide with the ellipticity of the input pulse (there is an exception for the case where β1 = 0).

development of the process of the soliton formation. As we can see from the previous results, the polarization of the soliton at the end of the soliton formation process does not coincide with the polarization of the input pulse. The process of generation of solitons by the effect of modulation instability is complex, and the stochastic process depends on the noise. It was calculated by the ellipticity of solitons generated in the process of the effect of modulation instability for different noise imposed on the input pulse. The ellipticity of the highest solitons was found to be randomly distributed. In Figure 3, the number of solitons that were generated with different ellipticity when linearly polarized input pulse was used is showed; the total of number of calculation was 150. As you can see, the polarizations of solitons are mostly concentrated close to the linear one; however, solitons with a wide range of the polarization ellipticity can also appear. Figure 3 shows that the

Ellipticity vs. fiber length for linear input pulse, particularly for the case when parallel Raman gain is equal to

Ellipticity vs. fiber length for a soliton with linearly polarized pulse at the entrance for β<sup>1</sup> = 1 (closed circles),

Polarization Properties of the Solitons Generated in the Process of Pulse Breakup in Twisted Fiber…

Figure 2.

77

Figure 1.

β<sup>1</sup> = 1 (open circles), and β<sup>1</sup> = 0 (squared).

DOI: http://dx.doi.org/10.5772/intechopen.81574

perpendicular Raman gain.

For the special case when using elliptical input polarization, the results show a similar behavior of the polarization ellipticity to make the energy transform from the slow to fast circularly polarized component. With these results, it can be said that the soliton with a fast circular polarization component propagates in the fiber with stable polarization. From the case of Eq. (10), we can say that this equation describes the polarization evolution of the soliton, but it is not applicable for

Polarization Properties of the Solitons Generated in the Process of Pulse Breakup in Twisted Fiber… DOI: http://dx.doi.org/10.5772/intechopen.81574

## Figure 1.

in parenthesis of right side are the effects of Self Phase Modulation (SPM) and Cross Phase Modulation (XPM), and finally the terms in key of right side are the Stimu-

The vector soliton can be approximated by the next equations (not taking into

β<sup>2</sup> j j h � <sup>p</sup> �, (8)

β<sup>2</sup> j j h � <sup>p</sup> �: (9)

j j <sup>A</sup>þð Þ<sup>z</sup> <sup>¼</sup> Acosð Þ<sup>θ</sup> sech A tð Þ � <sup>t</sup><sup>0</sup> <sup>=</sup> ffiffiffiffiffiffiffi

j j <sup>A</sup>�ð Þ<sup>z</sup> <sup>¼</sup> Acosð Þ<sup>θ</sup> sech A tð Þ � <sup>t</sup><sup>0</sup> <sup>=</sup> ffiffiffiffiffiffiffi

And finally applying the perturbation method [16] to Eqs. (6) and (7), we can define the ratio between powers of circularly left- and right-polarized components

From Eq. (10), we can see that the change of the polarization ellipticity of the vector soliton along the fiber may occur only in the presence of circular birefrin-

The split-step Fourier method is used for solving Eqs. (6) and (7) [14]. The

sponds to the twisted fiber with 6 turns/m. The meaning of the change of the sign of the constant β is for representing the change of the twist direction. For simulations,

Gaussian noise was added on the pulse. Modulation instability (MI) effect breaks up the pulse generating a set of solitons; the highest soliton were traced in this set [8]. When a linearly polarized pulse is introduced to the fiber input, we can see the influence of the walk-off effect between circularly right- and left-polarized components on ellipticity, see Figure 1. The fiber optic length used varies from 1 to 1.5 km. The results for the simulations are present in Figure 1, the result for β<sup>1</sup> = 1 is shown by open circles, for β<sup>1</sup> = �1 is shown by closed circles, and for β<sup>1</sup> = 0 is shown by squared. The important result for the dependence of polarization behavior can be observed clearly. The polarization behavior with β<sup>1</sup> = 1, but in this special case, we consider α = 1, that is, the parallel Raman gain is equal to the perpendicular, α = 1, see Figure 2. It can be seen that in this case, the effect of the ellipticity change of the soliton along the fiber is not presented. The results obtained are based on Eq. (10), and these results show that some ellipticity change must be presented on the circular birefringence and difference between Raman parallel and perpendicular gain. It can be observed that the ellipticity of the highest soliton does not coincide with the ellipticity of the input pulse (there is an exception for the case where

For the special case when using elliptical input polarization, the results show a similar behavior of the polarization ellipticity to make the energy transform from the slow to fast circularly polarized component. With these results, it can be said that the soliton with a fast circular polarization component propagates in the fiber with stable polarization. From the case of Eq. (10), we can say that this equation describes the polarization evolution of the soliton, but it is not applicable for

/km, α = 0.3, γ = 1.6 1/(W-km), TR = 3 fs. The value of β<sup>1</sup> = 1 corre-

values of using parameters are the following: β<sup>1</sup> = 1, β<sup>1</sup> = �1 ps/km; β<sup>1</sup> = 0;

the 30 ps input pulse with 40 W power in the input fiber was used, and the

2 1ð Þ � α

<sup>3</sup> <sup>γ</sup>A<sup>2</sup> TRβ<sup>1</sup>

� �

<sup>β</sup><sup>2</sup> j j <sup>z</sup>

: (10)

lated Raman Scattering effect.

j j A�ð Þz

Nonlinear Optics ‐ Novel Results in Theory and Applications

gence (twisted fiber, β1 is not equal to 0).

j j <sup>A</sup>þð Þ<sup>z</sup> <sup>¼</sup> tan ð Þ <sup>θ</sup>ð Þ <sup>0</sup> exp

account phases),

as follows [7]:

3. Numerical results

<sup>β</sup><sup>2</sup> <sup>=</sup> �25 ps<sup>2</sup>

β1 = 0).

76

Ellipticity vs. fiber length for a soliton with linearly polarized pulse at the entrance for β<sup>1</sup> = 1 (closed circles), β<sup>1</sup> = 1 (open circles), and β<sup>1</sup> = 0 (squared).

### Figure 2.

Ellipticity vs. fiber length for linear input pulse, particularly for the case when parallel Raman gain is equal to perpendicular Raman gain.

development of the process of the soliton formation. As we can see from the previous results, the polarization of the soliton at the end of the soliton formation process does not coincide with the polarization of the input pulse. The process of generation of solitons by the effect of modulation instability is complex, and the stochastic process depends on the noise. It was calculated by the ellipticity of solitons generated in the process of the effect of modulation instability for different noise imposed on the input pulse. The ellipticity of the highest solitons was found to be randomly distributed. In Figure 3, the number of solitons that were generated with different ellipticity when linearly polarized input pulse was used is showed; the total of number of calculation was 150. As you can see, the polarizations of solitons are mostly concentrated close to the linear one; however, solitons with a wide range of the polarization ellipticity can also appear. Figure 3 shows that the

The number of solitons with different ellipticity generated by the effect of modulation instability at linear polarization of the input pulse.

can be seen that most of the solitons have the ellipticity closer to the circular polarization than the input pulse. Average ellipticity of solitons in this case was

The number of solitons with different ellipticity generated by modulation instability at polarization of the input

Polarization Properties of the Solitons Generated in the Process of Pulse Breakup in Twisted Fiber…

DOI: http://dx.doi.org/10.5772/intechopen.81574

In Figure 6, the experimental setup is showed. For the source of signal, a continuous wave distributed feedback semiconductor laser with a wavelength of 1550 nm was used. The continuous wave signal was gated and amplified by the erbium doped fiber amplifier (EDFA) from which you can get pulses with 1–10 ns duration and a maximum peak power of about 150 W. To assure the stable polarization state, the pulses from the EDFA pass through a polarization controller (PC) and a polarizer. With the rotation of quarter wave retarder (QWR1), we can change the polarization ellipticity to be able to control the input polarization on the fiber. The output of the fiber under the test (twisted fiber) is connected to a quarter wave retarder (QWR2) and polarization beam splitter (PBS). The QWR2 and PSB convert the right and left circularly polarized component at the output fiber to orthogonally polarized linear components at the output PBS. The output of PSB (linearly polarized component) is separated in time by a delay line (10 m of SMF-28 fiber), and they come together using a 50/50 coupler to launch the same monochromator input.

The output pulses are detected and monitored by an oscilloscope. A typical

found to be about 0.95.

pulse of 0.906 and 0.906.

Figure 5.

Figure 6. Experimental setup.

79

4. Experimental setup

Figure 4.

The number of solitons with different ellipticity generated by modulation instability at polarization of the input pulse of 0.82 and 0.82.

average polarization ellipticity of solitons is 0.02. The maximum ellipticity found in this set of calculations was 0.3.

The distribution of polarization of solitons when the input pulse has ellipticity of 0.82 is showed in Figure 4. From Figure 4, it can be observed that the average soliton polarization moves toward circular polarization. For the case when the polarization of the input pulse is close to the circular polarization, the dispersion of the polarization ellipticity of solitons becomes much less, see Figure 5. For Figure 5, the polarization ellipticity of the input pulse used was equal to 0.906 and 0.906. It Polarization Properties of the Solitons Generated in the Process of Pulse Breakup in Twisted Fiber… DOI: http://dx.doi.org/10.5772/intechopen.81574
