Modified propagation equations for soliton transmission in a polarization-division multiplexing system

Modi®ed propagation equations for soliton

transmission in a polarization-division

multiplexing system

C . - F . C H E N , S . C H I , J . - C . D U N G

Institute of Electro-Optical Engineering, National Chiao Tung University,

Hsinchu, Taiwan 30050, China

Received 9 May; accepted 5 June 1997

The modi®ed coupled averaged propagation equations describing the orthogonally polarized soliton propagation in a random birefringent ®bre are derived. These include the third-order dispersion and Raman shift terms. Using these equations, the effects of the third-order dispersion and Raman shift terms are found to reduce the soliton in-teractions in a polarization-division multiplexing transmission system.

1. Introduction

Recently polarization-division multiplexing (PDM) has been used to increase the capacity of the soliton-based transmission system [1±6]. It has been demonstrated that the inter-action of orthogonally polarized solitons is weaker than that of parallely polarized solitons [1]. A good analytic description of PDM soliton interaction was made by the perturbation method [2±4]. The PDM system was considered by assuming the correlation length to be much shorter than the soliton period, so that the ¯uctuating local birefringence vector may be averaged over all polarization states. The soliton propagation of the PDM system can be described by the coupled averaged propagation equations (CAPE) [1, 7, 8], where the third-order dispersion and Raman shift terms are neglected. Using CAPE, De Angelis and Wabnitz [3] have numerically shown the interaction of the two orthogonal pulses. Initially, the two pulses attract each other. As the collision distance is approached, a complete polarization rotation by 90 _{for each pulse occurs in its own time slot. It has been shown}

that the collision distance of the PDM system is much larger than that of the parallely polarized system. However, the e€ects of third-order dispersion and Raman shift are not reported in a random birefringent PDM transmission system.

In this paper, we will derive the modi®ed coupled averaged propagation equations (MCAPE) which describe the soliton propagation in a PDM transmission system in-cluding the e€ects of the third-order dispersion and Raman shift. Using these equations, we will show that the interactions of solitons are reduced by the third-order dispersion and Raman shift terms.

2. The modi®ed propagation equations

The soliton propagation in a linearly birefringent ®bre can be described by the modi®ed coupled non-linear SchroÈdinger equations (MCNSE) [9]:

i@U_@Z ‡ id@U_@T ‡1₂@_@T2U₂ ÿ iC₆3@_@T3U₃‡ jUj2‡2₃jV j2   U ‡1₃V2_U_{exp…ÿiRdZ† ÿ s} R U @jUj 2 @T ‡ 1 3 @jV j2 @T ! ‡1₃V @V_@TU " # ˆ icU …1a† i@V_@Zÿ id@V_@T ‡1₂@_@T2V₂ÿ iC₆3@_@T3V₃‡ jV j2‡2₃jUj2   V ‡1₃U2_V_{exp…iRdZ† ÿ s} R V @jV j 2 @T ‡ 1 3 @jUj2 @T ! ‡1₃U@U_@TV " # ˆ icV …1b† where U and V are two polarization components of the electric ®eld envelope normalized by the electric ®eld scale Q. Z and T are normalized by dispersion length LD, and time scale

T0, respectively. Q, LDand T0 are related by

Q ˆ kjb2jAeff 2pn2T02  ₁₌₂ ; LDˆ T 2 0 jb₂j

where k is the wavelength, b₂is the second-order dispersion, Aeffis the e€ective ®bre

cross-section area, and n2 is the Kerr coecient. T0ˆ TW=1:763 and TW is the initial full

pulsewidth at the half magnitude. The coecients in Equations 1 are d ˆDbLD 2T0 ; C3ˆ b₃LD T3 0 ; sRˆT_TR 0; c ˆ aLD

where Db represents the inverse group velocity di€erence of two polarization components, b3is the third-order dispersion, a is the ®bre loss, TRis the slope of the Raman gain pro®le

at the carrier frequency. R ˆ 8pcT0=k is the normalized wave number and c is the velocity

of light in a vacuum. However, in a real communication ®bre, the orientation of ®bre birefringence randomly varies with a correlation length which is typically of length 100 m or so [10]. When the soliton wavelength, k, is at 1:55 lm, the index of refraction, Dn, between the two polarizations varies in the range 5  10ÿ9 _{to 8  10}ÿ4_{, Db is found in the}

range 1:7  10ÿ2 _{to 2:7  10}3_{ps km}ÿ1_{. If we take T}

Wˆ 3 ps and b2ˆ ÿ0:255 ps2kmÿ1

…LD 11:4 km†, we can ®nd that Rd ˆ …4pDn=k†
LD Dn  1011 is much greater than

unity over the entire range of Dn. Hence, the terms with the factor eiRdZ _{are rapidly}

varying and can be neglected.

To derive modi®ed coupled averaged propagation equations …MCAPE† for a PDM soliton system, we rewrite Equations 1 into a single-vector equation:

i@W_@Z ‡ idr1@W_@T ‡1₂@ 2_W @T2ÿ i C3 6 @3_W @T3‡ 5 6jWj2W ‡ 1 6…W‡r1W†r1W ÿ 1 2sR @ @TjWj2   W ÿ1 6sR @ @T…W‡r1W†   r1W ÿ1₃sR _@T@ ‰W…r2W†‡Š   r2W ˆ icW …2†

where W ˆ …U; V †t is the polarization state envelope vector, W‡_{ˆ …W}_†t_{, r}

1ˆ 1_{0 ÿ1}0

 

and r2ˆ 0 1_{1 0}

 

. The last two terms on the left-hand side of Equation 2 are transformed from Raman shift terms.

We assume that the polarization axes of the ®bre periodically undergo a sudden rotation h, which can take any value from 0 to 2p, and can be represented by an arbitrary rotation on the Poincare sphere [7, 8]. When the rotation h occurs, we also assume that a random phase / is added to the phase di€erence between the polarization state envelopes U and V . The total transformation is given by

U0

V0

 

ˆ _{ÿ sin h e}cos h _ÿi/ sin h e_{cos h} i/

 

U V  

…3† where the angles h and / are assumed to be uniformly distributed random variables on the Poincare sphere. We have

i@W @Z ‡ idr @W @T ‡ 1 2 @2_W @T2ÿ i C3 6 @3_W @T3‡ 5 6jWj2W ‡ 1 6…W‡rW†rW ÿ 1 2sR @ @TjWj2   W ÿ1₆sR _@T@ …W‡rW†   rW ÿ1₃sRRÿ1 _@T@ ‰RWW‡R‡r‡2Š   r2RW ˆ icW …4† where

R ˆ _{ÿ sin h e}cos h _ÿi/ sin h e_{cos h}i/

 

and

r ˆ Rÿ1_r

1R ˆ cos 2h ÿ sin 2h e ÿi/

ÿ sin 2h eÿi/ _{ÿ cos 2h}

 

The second term on the left-hand side of Equation 4 can be ignored because d varies randomly between positive and negative. Averaging Equation 4 over h and / on the Poincare sphere, we obtain the modi®ed coupled averaged propagation equations (MCAPE) for a PDM soliton system:

i@u @Z‡ 1 2 @2_u @T2ÿ i C3 6 @3_u @T3‡ juj2‡ jvj2   u

ÿ sRu_@T@ juj2ÿ1₂sRu_@T@ jvj2ÿ1₂sRv_@T@ …vu† ˆ icu …5a†

i_@Z@v‡1_2@T@2v₂ÿ iC₆3_@T@3v₃‡ juj 2‡ jvj2v

ÿ sRv_@T@ jvj2ÿ1₂sRv_@T@ juj2ÿ1₂sRu_@T@ …uv† ˆ icv …5b†

where u ˆp9=8U, v ˆp9=8V . The last three terms on the left-hand side of Equations 5 describe the averaged Raman e€ect, in which the ®rst term is the self-frequency shift (SFS) term, and the other two are the cross-frequency shift (XFS) terms.

3. Numerical results

The typical ®bre parameters used to solve Equations 5 numerically are: soliton wavelength k ˆ 1:55 lm, b₂ ˆ ÿ0:255 ps2_kmÿ1 _{D ˆ 0:2 ps km}ÿ1_nmÿ1_{†, b}

3ˆ 0:14 ps3kmÿ1, a ˆ

0:22 dB kmÿ1_{, n}

2ˆ 3:2  10ÿ20m2Wÿ1, and TRˆ 5 fs. The e€ective ®bre cross-section is

35 lm2_{. The ®bre loss is periodically compensated by the lumped ampli®ers and the}

am-pli®cation period is assumed to be 0:25LD. To show the e€ects of the third-order

dis-persion and Raman shift terms, we consider the soliton pulsewidth TWˆ 3 ps. Equations 5

are solved by the split-step Fourier method with the initial condition u…Z ˆ 0; T † ˆ sech…T ‡ D0=2T0† and v…Z ˆ 0; T † ˆ sech…T ÿ D0=2T0† with D0ˆ 3:5TW. In the absence

of the third-order dispersion and Raman shift terms, Figs. 1a and 1b show the envelopes of juj and jvj in the PDM soliton transmission system, respectively. The two pulses attract each other in the beginning and then repel to their own time slot after the collision distance Zc 112LD. At the collision point the interaction leads to a complete polarization

rotation by 90 _{for each pulse. The polarization-state each of the two pulses can exactly}

recover its own orientation at a distance which is a multiple of 2Zc. In Figs. 2a and 2b, we

show the envelopes of juj and jvj, respectively, in the presence of the third-order dispersion and Raman shift terms. It is seen that the interaction leads to some polarization rotation which is much smaller than 90 _{at the collision distance. The polarization rotation at the}

second collision point is larger than that at the ®rst collision point. Moreover, the degree of polarization rotation of the u polarization component is smaller than that of the v polarization component. In Fig. 2, the ®rst collision distance Zc is found at about 93LD

and the second collision distance is found at about 325LD. Its period is no longer 2Zc. In

addition, we can see the obvious time delay of the two pulses, which is mainly due to the Raman shift terms. Figure 3 shows the variation of separation of two solitons as a function of normalized distance, curve (a) is obtained in the absence of the third-order dispersion and Raman shift terms and curve (b) is obtained in the presence of the third-order dispersion and Raman shift terms. For curve (a), the separation gradually reduces until the collision distance where the separation is minimum at about 7.2 ps. The oscil-lation period is 224LD. For curve (b), the ®rst minimum separation is at about 8.9 ps at a

Figure 1 The interaction of two solitons in a PDM transmission system without the third-order dispersion and Raman shift terms: (a) the envelope of juj, and (b) the envelope of jvj.

distance 93LD, the next minimum separation is at about 8.8 ps at a distance 325LDand the

maximum separation is at about 11.4 ps at a distance 215LD where the polarization

rotation is almost equal to zero. Comparing curves (a) and (b), it can be seen that the third-order dispersion and Raman shift terms reduce the variation of pulse separation at the collision distance. The ®rst collision distance will be shorter when the third-order dispersion and Raman shift terms are present. According to the degree of polarization rotation and the variation of pulse separation, we know that the third-order dispersion and Raman shift terms reduce soliton interaction.

4. Conclusion

In conclusion, we have derived the modi®ed coupled averaged propagation equations of solitons with third-order dispersion and Raman shift terms in a random birefringent PDM soliton transmission system. It is found that the polarization rotation and the variation of pulse separation will be reduced when the third-order dispersion and Raman shift terms are present; i.e., the soliton transmission is reduced in the PDM transmission system.

Figure 2 The interaction of two solitons in a PDM transmission system with the third-order dispersion and Raman shift terms: (a) the envelope of juj, and (b) the envelope of jvj.

Figure 3 The variation of separation of two solitons as a function of normalized distance: curve (a) without the third-order dispersion and Raman shift terms, and curve (b) with the third-order dispersion and Raman shift terms.

Acknowledgement

The work is partially supported by National Science Council, Republic of China, under Contract NSC 86-2811-E009-002R.

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