2. Equations to analyze

polarization are necessary for some applications like supercontinuum generation. The

propagation of a pulse in a low birefringence fiber using coupled nonlinear Schrödinger equations was considered in [1–3]. In these works, the investigation concludes that the fractional pulses in each of the two polarizations trap each other and move together as one unit which is called a vector soliton. The frequency of each pulse is shifted to compensate the difference in group velocities caused by birefringence. It has been reported the experimental observation of vector solitons [4]. The vector solitons have attracted more attention in applications with linearly birefringent fibers. For the case of fibers with circular birefringence, a special case is when circular birefringence is induced when the fibers are twisted. These twisted fibers can present special advantages for some laser applications. The especial characteristic of twisted fiber is that it induces circular birefringence and eliminates the random linear birefringence [5]. An important consequence of this result is that twisted fiber is less sensitive to environmental conditions and with this we can find new useful features for nonlinear applications [6]. This helps us make the twisted fiber less sensitive to environmental conditions and provides new useful features for nonlinear applications [6]. In [7], it has been analyzed the polarization behavior of vector solitons in a circularly birefringent fiber. In this work, for analysis, we used the two coupled propagation equations in a circularly birefringent fiber that include self-phase modulation, cross phase modulation, and the soliton self-frequency shift [8]. We consider the polarization dependence of the Raman amplification unlike the previously published works [9]. We work on the equations to make a transformation to reduce them to a form of perturbed Manakov task. For our case, the equations were considered as a perturbation unlike the Manakov integrable case. For the case of the perturbation method, we can get the equations for the analysis of the development of evolution of the polarization state of pulses. An important result when analyzing the equations shows that for circularly birefringent fiber (twisted fiber), the crosspolarization Raman term leads to unidirectional energy transfer from the slow circularly polarized component to the fast one. The product of the birefringence and the amplitudes of both polarization components determine the importance of this effect. From all of the above, we can conclude that solitons with any initial polarization state

Nonlinear Optics ‐ Novel Results in Theory and Applications

will eventually develop circularly stable polarized solitons.

74

The split-step Fourier method was used for the numerical analysis of the two coupled nonlinear Schrödinger equations. The parameters of a standard fiber (SMF-28) were used with delay between left- and right-circular polarizations of 1 ps/km that corresponds to circular birefringence in a twisted fiber by 6 turns/m. Furthermore, by the numerical analysis, it is possible to analyze the polarization of solitons generated by the modulation instability effect. An input pulse of 30 ps with 40 W of power was used with a noise imposed which was launched to the fiber input. The input pulses had different polarization ellipticity from circular to linear. From the results, it was found that polarization ellipticity of solitons does not coincide with the polarization of the input pulses. An important result that was also found is that polarization ellipticity of solitons is distributed randomly, but the average polarization ellipticity is mostly circular compared to polarization ellipticity of the input pulse. In the experimental and numerical analysis, SMF-28 standard fiber twisted with 60 y 200 m of length with a pump pulse of 1–10 ns in a wavelength of about 1550 nm was used. The output signal at the fiber end is separated in circular-right and circular-left polarization. The ellipticity of the pulses is calculated with the ratio between the output pulses. The experimental results show that circularly polarized pulses in a fiber with circular birefringence (twisted fiber) are promising for the generation of supercontinuum with stable polarization and confirm the principal conclusions of the modeling propose in [8]; the polarization properties of supercontinuum are also an important issue for application [9–13].

The equations that describe self-frequency shift of picosecond pulses with linear polarization can be written as follows [14]:

$$\partial\_{\mathbf{x}} A\_{\mathbf{x}} = i\gamma \left[ T\_R \partial\_T |A\_{\mathbf{x}}|^2 \right] A\_{\mathbf{x}} \tag{1}$$

the terms Ax, γ, and TR in Eq. (1) are the envelope of the pulse with linear polarization on the x-axis, the nonlinearity, and the Raman response time, respectively. If the pulse has elliptical polarization, two polarization components have to be included if the input pulse has elliptical polarization, for this case the nonlinear effect of self-frequency shift is considered as dependent on the sum of the powers of the orthogonal components [9]. From the above, it can be said that the value of the parallel Raman gain is equal to the orthogonal Raman gain; the parallel Raman gain is when the pump and Stokes have the same linear polarization, and the orthogonal Raman gain is when the pump and Stokes are polarized orthogonally. The experimental results show that the Raman gain caused by the perpendicular component has a value of 0.3 of the Raman gain for parallel component for a small Stokes shift [15]. For this reason, we used the following equations for the self-frequency shift effect:

$$\partial\_{\mathbf{x}} A\_{\mathbf{x}} = i\gamma \left[ T\_R \partial\_T |A\_{\mathbf{x}}|^2 \right] A\_{\mathbf{x}} + i\alpha \gamma \left[ T\_R \partial\_T |A\_{\mathbf{y}}|^2 \right] A\_{\mathbf{x}} \tag{2}$$

$$i\partial\_{\mathbf{x}}A\_{\mathbf{x}} = i\gamma \left[T\_R \partial\_T |A\_{\mathbf{x}}|^2\right] A\_{\mathbf{x}} + i\alpha\gamma \left[T\_R \partial\_T |A\_{\mathbf{y}}|^2\right] A\_{\mathbf{x}} \tag{3}$$

here α ¼ α⊥=α∥, where α<sup>⊥</sup> and α<sup>∥</sup> denote, respectively, the perpendicular and parallel Raman gains.

Using circularly polarized components, we can obtain the equation for the right- and left-circularly polarized state as follows:

$$\partial\_{\mathbf{z}} A\_{+} = \frac{i\gamma T\_R}{2} \left\{ \frac{\mathbf{1} + a}{2} \partial\_t \left( \left| A\_{+} \right|^2 + \left| A\_{-} \right|^2 \right) A\_{+} + (\mathbf{1} - a) \partial\_t \left[ \text{Re} \left( A\_{+} A\_{-}^{\*} \right) \right] A\_{-} \right\} \tag{4}$$

$$\partial\_{\pi} A\_{-} = \frac{i\gamma T\_R}{2} \left\{ \frac{1+a}{2} \partial\_t \left( \left| A\_{+} \right|^2 + \left| A\_{-} \right|^2 \right) A\_{-} + (1-a) \partial\_t \left[ \text{Re} \left( A\_{+} A\_{-}^{\*} \right) \right] A\_{+} \right\} \tag{5}$$

Eqs. (4) and (5) are the coupling equations describing the self-frequency shift. Adding group velocity dispersion (GVD) and walk-off between circularly polarized components, self-phase modulation (SPM), and cross-phase modulation (XPM) terms to these equations, we have coupling equations that we analyzed analytically and numerically:

$$\begin{aligned} \partial\_t A\_+ + \beta\_1 \partial\_t A\_+ + \frac{i\beta\_2}{2} \partial\_t A\_+ &= \frac{2i\gamma}{3} \left( |A\_+|^2 + 2|A\_-|^2 \right) A\_+ \\ &- \frac{i\gamma T\_R}{2} \left\{ \frac{1+a}{2} \partial\_t \left( |A\_+|^2 + |A\_-|^2 \right) A\_+ + (1-a) \partial\_t \left[ \text{Re}\left( A\_+ A\_-^\* \right) \right] A\_- \right\} \\ \partial\_t A\_- - \beta\_1 \partial\_t A\_- + \frac{i\beta\_2}{2} \partial\_t A\_- &= \frac{2i\gamma}{3} \left( |A\_-|^2 + 2|A\_+|^2 \right) A\_- \\ &- \frac{i\gamma T\_R}{2} \left\{ \frac{1+a}{2} \partial\_t \left( |A\_+|^2 + |A\_-|^2 \right) A\_- + (1-a) \partial\_t \left[ \text{Re}\left( A\_+ A\_-^\* \right) \right] A\_+ \right\} \end{aligned} \tag{7}$$

To describe the above equations, the last two terms on the left side are the effects of Walk-off and Group Velocity Dispersion (GVD) respectively, the terms 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 Stimulated Raman Scattering effect.

The vector soliton can be approximated by the next equations (not taking into account phases),

$$|A\_{+}(z)| = A cos(\theta) sech\left[A(t - t\_{0})/\sqrt{|\beta\_{2}|}\right]|\tag{8}$$

$$|A\_{-}(z)| = A cos(\theta) sech\left[A(t - t\_0)/\sqrt{|\beta\_2|}\right].\tag{9}$$

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 as follows [7]:

$$\frac{|A\_{-}(z)|}{|A\_{+}(z)|} = \tan\left(\theta(0)\right) \exp\left[\frac{2(1-a)}{3}\gamma A^2 \frac{T\_R \beta\_1}{|\beta\_2|} z\right].\tag{10}$$

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