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

Authors: Rodrigo Acuna Herrera
Journal: Mathematical Problems in Engineering
 (2017)

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## Abstract

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.


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## Body

## 1. 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 [4–6]. 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. Section2 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.


## 2. Nonlinear Pulse Propagation in Birefringent Fibers

The nonlinear pulse propagation, in birefringent optical fiber, can be written as follows [9]:(1)i∂Aj∂z+∑m≥2im-1βmjm∂Ajm∂Tm--1jbAj=-1+iτshock∂∂T1-fRNj1+fRNj2,where Aj (j=1,2) corresponds to the normalized field amplitude in the x (j=1) or y (j=2) direction, T=t-β1z is the relative time, t is the absolute time, and z is the propagation length. The constants βmj are the dispersion parameters. b=ΔkLD/2 is the normalized birefringence parameter, where Δk=Δnk0 and Δn and k0 are the birefringence and propagation constant in vacuum, respectively. LD=To2/β2 is the dispersion length, with β2 being the second-order dispersion parameter and To being the initial pump pulse width. fR=0.245 is the Raman coefficient and τshock=1/ωo is the optical shock time scale. ωo is the pump frequency. The remaining terms in (1) are(2)Nj1=Aj2+23A3-j2Aj+13Aj∗A3-j2,(3)Nj2=Ajf1⊗Aj2+f2⊗A3-j2+A3-jf3⊗AjA3-j∗+A3-jAj∗,where ⊗ is the convolution operator, defined, for instance, as(4)f1⊗Aj2z,T=∫-∞∞f1T′Ajz,T-T′2dT′.Thefs functions for s=1,2,3 in (3) are given by(5)f1t=fa+fchat+fbhbt,f2t=fahat,f3t=fbhbt+fchat2.fa=0.75, fb=0.21, fc=0.04, and(6)hat=τ12+τ22τ1τ22exp⁡-tτ2sin⁡tτ1,hbt=2τb-tτb2exp⁡-tτb,with τ1=12.2fs, τ2=32fs, and τb=96fs. We ignore the SRS and βmj for m≥4 in (1) in order to analyze the DW behavior. The dispersive wave generation is characterized by the parameters δ3=β3/6β2T0 and the soliton order N=LD/LNL, where β3 is the third-order dispersion parameter. LNL=1/Poγ is the nonlinear length, where Po and γ are the initial peak power of the pump and the nonlinear coefficient, respectively.


## 3. Numerical Results

For nonlinear pulse simulations, (1) is numerically solved by combining the split-step Fourier method [9] and the four-order Runge-Kutta algorithm [18]. The basic idea is to divide the equation into a dispersive and nonlinear operator; that is,(7)∂Aj∂z=Dj^+Nj^Aj,where(8)Dj^=i∑m≥2im-1βmjm∂Ajm∂Tm--1jbAj,(9)Nj^=i1+iτshock∂∂T1-fRNj1+fRNj2.The dispersive and nonlinear operators act together along the fiber. During numerical simulations, the fiber is divided into manyN-sections with size h; at each section each operator acts independently; that is, at section i, Nj^=0 and Dj^ acts; at the next section i+1, Dj^=0, and Nj^ acts and so on. The case(10)∂Aj∂z=Dj^Ajis solved in the Fourier domain according to(11)Ajz+h,T≈exp⁡hDj^Ajz,T.The exponential operatorexp⁡hDj^ is calculated by the mathematical prescription:(12)exp⁡hDj^Ajz,T=FT-1exp⁡hDj^-iωFTAjz,T,where FT represents the Fourier-transform operator, Dj^-iω is obtained from (8) by replacing ∂/∂T by -iω, and ω is the frequency in the Fourier domain. The other case(13)∂Aj∂z=Nj^Ajis solved by using the four-order Runge-Kutta Method [18], where the following algorithm is implemented:(14)∂Aj∂z=fz,Aj=Nj^Aj,Ajm+1=Ajm+h6k1+2k2+2k3+k4,zm+1=zm+h,for m=0,1,2,…,N and Aj0z=0=A0sech⁡T/T0, where A0=P0. The constants k1, k2, k3, and k4 are given by(15)k1=fzm,Ajm,k2=fzm+h2,Ajm+h2k1,k3=fzm+h2,Ajm+h2k2,k4=fzm+h,Ajm+hk3.For our simulations, we assume the dispersion terms are the same forx and y 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 x and y 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.Table 1
Simulation parameter.
Parameter
Value
γ
0.08
W
-
1
m
-
1
β2
-
1.3504
×
1
0
-
2
p
s
2
k
m
-
1
β3
8.2385
×
1
0
-
5
p
s
3
k
m
-
1
β4
-
9.1913
×
1
0
-
8
p
s
4
k
m
-
1
β5
1.7589
×
1
0
-
10
p
s
5
k
m
-
1
β6
-
3.8095
×
1
0
-
13
p
s
6
k
m
-
1
β7
9.4138
×
1
0
-
16
p
s
7
k
m
-
1
Po
10
k
W
T0
50
f
s
A(0,T)
P
o
s
e
c
h
T
/
T
o
Ax(0,T)
A
(
0
,
T
)
c
o
s
(
θ
); θ is the polarization angle
Ay(0,T)
A
(
0
,
T
)
s
i
n
(
θ
)Figure 1
Birefringence profiles. Profile 1:Δn=-1.25×10-6+2.25×10-9λ. Profile 2: Δn=2.35×10-6-2.25×10-9λ. Profile 3: Δn=10-7+2.25×10-11λ-8002. Profile 3: Δn=10-6-2.25×10-11λ-9002. λ is in nm.Figure 2
SC generation for different constant birefringence values.Δn=10-7,2×10-7,4×10-7,6×10-7,8×10-7, and 10-6.We also calculated the generation of SC for all profiles shown in Figure1. 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 fR=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.Figure 3
SC generation for profiles shown in Figure1. x and y polarization are the same. Polarization angle θ=45°.Figure 4
SC generation for profiles shown in Figure1, with fR=0. x and y polarization are the same. Polarization angle θ=45°.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 third-order dispersionβ3. 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 ΔνdT0 and DW peak power versus δ3 for constant birefringence profile and the ones shown in Figure 1. We set N=2, which corresponds to the second-order soliton. DW frequency shift is defined as Δνd=νd-νs, where νd is the DW frequency and νs 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 x- and y-DW polarizations have equal behavior for all WDB. From Figure 6, the normalized DW peak power almost scales linearly to δ3, 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.Figure 5
DW generation for constant birefringence profile and the ones shown in Figure1. Profile-ix=Profile-iy, with i=1,2,3,4. Polarization angle θ=45°.Figure 6
DW peak power for constant birefringence profile and the ones shown in Figure1. Profile-ix=Profile-iy, with i=1,2,3,4. Polarization angle θ=45°.


## 4. Conclusion

We have shown that the generation of SC and DW properties do not depend on the four WDB profiles seen in Figure1. 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.



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*Source: 3517407-2017-06-15.xml*