Influence of the Wavelength Dependence of Birefringence in the Generation of Supercontinuum and Dispersive Wave in Fiber Optics

In this paper, we perform numerical analysis about the influence of the wavelength dependence of birefringence (WDB) in the Supercontinuum (SC) and dispersive wave (DW) generation. We study different birefringence profiles such as constant, linear, and parabolic. We see that, for a linear and parabolic profile, the generation of SC practically does not change, while this does so when the constant value of the birefringence varies. Similar situation happens with the generation of dispersive waves. In addition, we observe that the broadband of the SC increases when the Stimulated Raman Scattering (SRS) is neglected for all WDB profiles.


Introduction
The SC generation has attracted a lot of attention since it has found numerous applications in the fields of telecommunication [1], optical metrology [2], ultrafast coherence spectroscopy, and biological processes [3].The mechanisms for SC generation in nonlinear fiber optics (FO) have been studied both numerically and experimentally [4][5][6].Subsequently, SC generation techniques have brought the design of tunable ultrafast laser sources [7].In addition, SC generation sources have been used as a simple way to generate multiwavelength optical sources [1].It is useful for dense wavelength-division-multiplexing (WDM) telecommunications.In the generation of SC, various processes are involved such as self-and cross-phase modulation, four-wave mixing, modulation instability, soliton fission, dispersive wave generation [8], and Raman scattering [9].All these effects can contribute to creating new frequencies within the pulse spectrum.Numerous SC generation methods have been studied to get a better understanding of the mechanisms for which it is efficiently possible to generate and develop the SC laser.Within these methods, the use of Photonic Crystal Fiber has brought a lot of interests due to its highly nonlinear optics characteristics [10,11].
The SC generation can be studied with the generalized nonlinear Schrodinger equation (NLSE), which models the propagation of optical pulses in nonlinear FO.NLSE has been used to analyze the influence of various parameters on the SC generation [12,13].In those studies, the WDB in the Supercontinuum generation has not been enough studied [14,15].In this work, we study numerically the influence of WDB in the SC and DW generation.We analyze different birefringence profiles such as constant, linear, and parabolic.Some of those profiles can be seen in references [16,17].
We show that, for a linear and parabolic profile, the generation of SC and DW do not change, while they do so as the constant value of the birefringence varies.We also found that the broadband of the SC is wider when the Stimulated Raman Scattering (SRS) is neglected for all birefringence profiles.
The paper is structured as follows.Section 2 presents the fundamental theory of nonlinear pulse propagation in birefringent fiber optics.Section 3 describes the numerical calculation of SC and DW generation for different birefringent FO profiles by using the split-step Fourier method [9] and the four-order Runge-Kutta algorithm [18].Finally, the conclusions are presented in Section 4.

Nonlinear Pulse Propagation in Birefringent Fibers
The nonlinear pulse propagation, in birefringent optical fiber, can be written as follows [9]: where   ( = 1,2) corresponds to the normalized field amplitude in the  ( = 1) or  ( = 2) direction,  =  −  1  is the relative time,  is the absolute time, and  is the propagation length.The constants   are the dispersion parameters. = Δ  /2 is the normalized birefringence parameter, where Δ = Δ 0 and Δ and  0 are the birefringence and propagation constant in vacuum, respectively.  =  2  /| 2 | is the dispersion length, with  2 being the second-order dispersion parameter and   being the initial pump pulse width.  = 0.245 is the Raman coefficient and  shock = 1/  is the optical shock time scale.  is the pump frequency.The remaining terms in (1) are where ⊗ is the convolution operator, defined, for instance, as The   functions for  = 1, 2, 3 in (3) are given by ( = 0.75,   = 0.21,   = 0.04, and with  1 = 12.2,  2 = 32, and   = 96.We ignore the SRS and   for  ≥ 4 in (1) in order to analyze the DW behavior.The dispersive wave generation is characterized by the parameters  3 =  3 /6| 2 | 0 and the soliton order  = √  / NL , where  3 is the third-order dispersion parameter. NL = 1/   is the nonlinear length, where   and  are the initial peak power of the pump and the nonlinear coefficient, respectively.

Numerical Results
For nonlinear pulse simulations, ( 1) is numerically solved by combining the split-step Fourier method [9] and the fourorder Runge-Kutta algorithm [18].The basic idea is to divide the equation into a dispersive and nonlinear operator; that is, where The dispersive and nonlinear operators act together along the fiber.During numerical simulations, the fiber is divided into many N-sections with size ℎ; at each section each operator acts independently; that is, at section , N = 0 and D acts; at the next section  + 1, D = 0, and N acts and so on.The case    = D   (10) is solved in the Fourier domain according to The exponential operator exp(ℎ D ) is calculated by the mathematical prescription: where   represents the Fourier-transform operator, D (−) is obtained from (8) by replacing / by −, and  is the frequency in the Fourier domain.The other case is solved by using the four-order Runge-Kutta Method [18], where the following algorithm is implemented: Mathematical Problems in Engineering  for  = 0, 1, 2, . . .,  and  0  ( = 0) =  0 sech(/ 0 ), where  0 = √ 0 .The constants  1 ,  2 ,  3 , and  4 are given by For our simulations, we assume the dispersion terms are the same for  and  polarization and use the typical parameters shown in Table 1, where the length of propagation is 6 cm.We consider the birefringence profiles shown in Figure 1.They are linear increasing (profile 1), linear decreasing (profile 2), parabolic positive (profile 3), and parabolic negative (profile 4).We first compute the SC generation in both  and  polarization for six constant birefringence values between 10 −7 and 10 −6 .The results are shown in Figure 2. We see that the generation of SC varies as the birefringence does so.For each birefringence value, we see depths at different wavelengths.
We also calculated the generation of SC for all profiles shown in Figure 1.We display the results in Figure 3.We see that all SC generations are the same and have the same two big depths around 550 nm and 900 nm, which is different from constant birefringence profile (Figure 2).We repeated the above computation for   = 0 (no SRS).The results can be viewed in Figure 4.The broadband of the SC is wider compared to the one seen in Figure 3.In addition, the biggest depth is now around 700 nm, which tells us that SCG and depths on it are really influenced by WDB.
We finally investigated how the DW frequency can be affected by birefringence profiles.As is known, DW is a linear wave that propagates in any dispersive medium which can be generated from the disturbance of solitons due to the thirdorder dispersion  3 .The dispersive wave frequency can be obtained by a phase-matching argument requiring that the DW propagates with the same phase velocity as that of the soliton.Then, we computed the normalized DW frequency shift Δ]   0 and DW peak power versus  3 for constant  birefringence profile and the ones shown in Figure 1.We set  = 2, which corresponds to the second-order soliton.DW frequency shift is defined as Δ]  = (]  − ]  ), where ]  is the DW frequency and ]  is the soliton frequency.We performed simulations and had the results shown in Figures 5 and 6.The DW characteristics are the same for any WDB profile, but different from constant profile.In addition,  the and -DW polarizations have equal behavior for all WDB.From Figure 6, the normalized DW peak power almost scales linearly to  3 , which means that this property does not depend on WDB.The above effects can be used to control the generation of the dispersive wave created from solitons using the WDB.

Conclusion
We have shown that the generation of SC and DW properties do not depend on the four WDB profiles seen in Figure 1.We observed that there are a couple of big depths around 500 nm and 900 nm, while they are at different positions for constant birefringence profile depending on its value.The generation of DW has the same behavior for any WDB profile, but different from the constant ones.Finally, we have proven that the broadband of SC is bigger for WDB when SRS is neglected.

Figure 3 :
Figure 3: SC generation for profiles shown in Figure 1. and  polarization are the same.Polarization angle  = 45 ∘ .

Figure 4 :
Figure 4: SC generation for profiles shown in Figure 1, with   = 0.  and  polarization are the same.Polarization angle  = 45 ∘ .

3
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