Y. Chen and J. Atai Vol. 14, No. 9 /September 1997 /J. Opt. Soc. Am. B 2365

Femtosecond soliton pulses in birefringentoptical fibers

Yijiang Chen

Optical Sciences Centre, Australian National University, Canberra 2601, Australia

Javid Atai*

MESA Research Institute, Department of Electrical Engineering, University of Twente,7500 AE Enschede, The Netherlands

Received November 27, 1996; revised manuscript received March 10, 1997

We consider femtosecond soliton-pulse propagation in a birefringent optical fiber where rapidly oscillatingterms, the difference in polarization dispersions, and the difference in group velocities of the two polarizationcomponents have to be taken into account. We demonstrate the existence of a novel class of linearly polarizedsoliton states (with the linear polarization ranging from 0 to 2p). We also find the elliptically polarized solitonstates, which do not appear to be acceptable to the coupled nonlinear Schrodinger equations describing thepulse evolution in the birefringent fiber when the different dispersions between the two polarizations are ig-nored and the group-velocity difference is taken into account. More importantly, the corresponding stabilityanalysis reveals that within certain operating regions the fast soliton can be stable and the slow soliton can beunstable, whereas in the others the fast soliton is unstable and the slow soliton is stable, in contrast to thosereported earlier by neglecting different polarization dispersions. On the other hand, both the linearly polar-ized soliton states and the elliptically polarized soliton states are found to be unstable. This indicates that forhigh-capacity coherent soliton communication in the femtosecond regime, the pulse must be launched alongeither the slow or the fast axis of a practical polarization-maintaining fiber. Finally, the potential applica-tions of weakly unstable linearly polarized soliton states for ultrafast soliton switching are discussed. © 1997Optical Society of America [S0740-3224(97)01809-2]

1. INTRODUCTIONAs a result of the 1970’s technological revolution in reduc-ing material loss, optical fiber has been playing a crucialrole for long-distance, high-capacity information trans-mission in the global information superhighway network.To prevent loss of transmitted information, each indi-vidual pulse must be kept unchanged over the propaga-tion distance. This can be achieved either by tuning thecarrier frequency of the pulse to operate at the zero-dispersion frequency of the monomode fiber for low-power(linear) operation1 or by offsetting the dispersion withself-phase modulation manifested at high power.2,3 Inpractice, however, it is difficult to maintain circular sym-metry or isotropy of the fiber because of various imperfec-tions such as geometry, stress, and twisting, which arerandomly distributed along the fiber. The single-mode fi-ber actually supports two modes polarized orthogonallyalong the two principal axes thanks to the birefringenceintroduced by imperfections. This z-dependent birefrin-gence, varying both in magnitude and in orientation,leads to the wandering of the polarization state and thebroadening of the pulse, and consequently deterioratesthe information transmitted in the coherent communica-tion system. To overcome these adverse effects, thepolarization-maintaining fiber was developed.4,5 In sucha fiber the randomly distributed birefringence is over-whelmed by the built-in birefringence and the fiber haswell-defined principal axes. In the limit of low-power(linear) operation the initial polarization state can be re-

0740-3224/97/0902365-08$10.00 ©

tained along the fiber when a light pulse (e.g., picosecond)is launched along one of the principal axes. For nonlin-ear operation the picosecond soliton launched along one ofthe principal axes can also propagate stably with the po-larization state maintained along the fiber. Within thispicosecond regime, the two orthogonally polarized solitonpulses can in effect be locked together to propagate in thesame group velocity along the fiber as long as the centralfrequency of the pulses shifts just enough in opposite di-rections such that through group-velocity dispersion, thesoliton along the fast axis slows and the soliton along theslow axis accelerates.6–8

To improve information-carrying capacity, the propaga-tion of shorter pulses such as femtosecond pulses istempting and desirable. In the linear birefringent fiber,however, the femtosecond pulse transmission poses greatdifficulty and seems to be impractical since the broaden-ing of a pulse by the dispersion is inversely proportionalto the pulse width, and therefore short pulses at the fem-tosecond scale will broaden quickly over a short distance.9

A question that immediately follows is what will happenfor femtosecond pulses that propagate in the nonlinear bi-refringent fiber, i.e., whether the stationary propagationof the femtosecond soliton pulses is possible with two po-larization components locked together to move in a uni-form group velocity along the fiber. Clearly, this is apractical, important issue to be addressed because thepossibility of stationary femtosecond soliton-pulse propa-gation in the nonlinear birefringent fiber suggests an ap-

1997 Optical Society of America

2366 J. Opt. Soc. Am. B/Vol. 14, No. 9 /September 1997 Y. Chen and J. Atai

parent advantage over the linear pulse and is accompa-nied by a destructive broadening effect even withoperation within the zero-dispersion window because ofresidual dispersion. Such an advantage of nonlinearpulse propagation over the linear pulse propagation inthe femtosecond regime is absent for the fiber operationin the picosecond regime.

2. GOVERNING EQUATIONSThe propagation of a light wave in the optical fiberfollows the wave equation. For the electric field of theform E(x, y, z, t) 5 c(x, y)exp(iv t)@Ex(t, z)exp(ibxz )x1 Ey(t, z)exp(ibyz)y#, the evolution of the two orthogo-nally polarized envelope pulses Ex and Ey , in terms of thenormalized quantities u 5 (kn2 /ubx9uAeff)

1/2t0Ex andv 5 (kn2 /ubx9uAeff)

1/2t0Ey , is described by the couplednonlinear Schrodinger equations:

iS ]u

]Z1 d

]u

]T D 11

2

]2u

]T2 1 ~ uuu2 1 cpuvu2!u

1 ~1 2 cp!v2u* exp~2iDZ ! 5 0, (1a)

iS ]v

]Z2 d

]v

]T D 1rd

2

]2v

]T2 1 ~cpuuu2 1 uvu2!v

1 ~1 2 cp!u2v* exp~iDZ ! 5 0, (1b)

where the nonlinear coefficient n2 5 3 3 10216 cm2/Wand cp 5 2/3 for silica fiber, the effective core area Aeff' pr0

2 with r0 as the fiber radius, k 5 v/c 5 2p/l is thewave number in free space, t0 is a measurement of thepulse width, Z 5 zubx9u/t0

2 5 zp/2z0 with z0 5 pt02/2ubx9u

as the soliton period, T 5 @t 2 0.5(bx8 1 by8)z#/t0 , D5 4(bx 2 by)z0 /p 5 8z0 /Lb ' dR (R 5 8pct0 /l), Lb5 2p/(bx 2 by) is the birefringent beat length, rd5 by9/bx9 , d 5 (bx8 2 by8)t0/2ubx9u ' pt0B/Dxl2 with thebirefringence B 5 (bx 2 by)/k and the dispersion co-efficient Dx 5 2pcubx9u/l

2 measured in picosecond/nanometer/kilometer units,3 and bx,y8 5 dbx,y /dv, bx,y95 d2bx,y /dv2.

For a high-birefringent fiber with large B and/or longpulse operation (large t0), the final terms of Eq. (1) on theleft side are negligible as D(;Bt0

2) 5 8z0 /Lb @ 1.10

This corresponds to the operation of pulses in picosecondor larger for typical values7 of B 5 1026 to 1025. Whenthese rapidly oscillating terms were neglected and thedispersion coefficients of the two polarizations were as-sumed to be identical (rd 5 by9 /bx9 5 1), Eq. (1) wasshown6 to admit the soliton solution of the form

u 5 ~1 1 cp!21/2 exp@i0.5~1 1 d2!Z 2 idT#sech T,(2a)

v 5 u exp~i2dT !, (2b)

in addition to the fast soliton u 5 0, v 5 exp@i0.5(11 d2)Z 1 idT#sech T and the slow soliton u5 exp@i0.5(1 1 d2)Z 2 idT#sech T, v 5 0 solutions.Within this (picosecond for B 5 1026 to 1025) operationregime and rd 5 1, Eq. (1) also supports a family of therotating vector solitary waves with different peak inten-sities in each polarization7,11–13 that has Eq. (2) as its spe-cial case. In fact, by including the rapidly oscillating

terms but retaining rd 5 1, Eq. (1) can have the counter-part solutions for this family of rotating vector solitarywaves, which are referred to as coupled-soliton states.14

These coupled-soliton states are, however, unstable in apractical fiber for soliton transmission with the dispersionD . 0.2 ps/nm/km (Ref. 15) although they may becomestable in extreme situations when D , 0.1 ps/nm/kmfor the birefringence B 5 1024 or D , 0.01 ps/nm/km forB 5 1025 at the wavelength l 5 1.55 mm, i.e., when d5 d/AD/4 5 2Ad/R 5 a@B/(2Dlc)#1/2 . 1.14

For a small birefringence and/or a short pulse, espe-cially in the femtosecond regime, the rapidly oscillatingterms of Eq. (1) have to be included in the analysis. Moreimportantly, we must be aware that the two polarizationshave different dispersions bx9 Þ by9 or rd Þ 1 in general,and this difference between bx9 and by9 can be large espe-cially when the carrier frequency v approaches the zero-dispersion frequency point or dispersion D is small.5,16

By including the exponentially varying terms and consid-ering the case in which rd Þ 1, we will show that thetrapped-soliton solutions of Eq. (1) can differ significantlyfrom those with the rapidly oscillating terms neglectedbecause of the emergence of novel classes of femtosecondsoliton states that have no counterpart in the picosecondoperation regime.

3. TRAPPED STATIONARY FEMTOSECONDSOLITON STATESTo derive the soliton states of Eq. (1), we seek thetraveling-wave solutions of the form

u~T, Z ! 5 Nf @~T 2 Z/vp!N#exp@i~g1Z 1 h1T !#, (3a)

v~T, Z ! 5 mNg@~T 2 Z/vp!N#exp@i~g2Z 1 h2T !#, (3b)

which substituted into Eq. (1) yields a set of equationsdetermining vp , g j , h j , f(s), and g(s) @s 5 (T2 Z/vp)N]. In Eqs. (3a) and (3b), m 5 1 for linear po-larization, m 5 i for elliptical polarization, and N is aconstant introduced for normalization. Solving the re-sulting equations governing vp , g j , h j , we arrive at

h1 5 h2 5 h 5 22d/~1 2 rd!, (3c)

g1 5 g 2 D/4, g2 5 g 1 D/4, (3d)

vp 5 2~1 2 rd!/@d~1 1 rd!#, (3e)

with g an arbitrarily chosen propagation factor. Theequations governing f(s) and g(s) are

1

2

d2f

ds2 2 f 1 ~f 2 1 ag2!f 5 0, (4a)

rd

2

d2g

ds2 2 kg 1 ~af 2 1 g2!g 5 0, (4b)

where a 5 1 for m 5 1, a 5 2cp 2 1 for m 5 i,

k 5 @~rd/2!h2 2 dh 1 g 1 D/4#/~1/2h2 1 dh 1 g 2 D/4!

5 @2d2/~1 2 rd!2 1 g 1 D/4#/

@2d2rd /~1 2 rd!2 1 g 2 D/4#,

Y. Chen and J. Atai Vol. 14, No. 9 /September 1997 /J. Opt. Soc. Am. B 2367

and the constant N has been taken to be

N2 5 1/2h2 1 dh 1 g 2 D/4

5 2d2rd /~1 2 rd!2 1 g 2 D/4

so that the coefficient in front of the second term of Eq.(4a) is normalized to unity. For rd 5 1, Eq. (3) producesvp 5 0. This means the femtosecond solitons demon-strated below cannot exist when the difference in the po-larization dispersions is ignored. In other words, it iscrucial to recognize that bx9 Þ by9 for obtaining the follow-ing femtosecond soliton solutions in a birefringent fiber byincluding the rapidly oscillating terms. In effect, rdÞ 1 is the necessary condition for Eq. (4) to admit solitonsolution with linear polarization (a 5 1) and nonzero bi-refringence (B Þ 0 or k Þ 1). Also note from Eqs. (3a)–(3c) that locking of the two orthogonally polarized compo-nents for femtosecond solitons requires a frequency shiftof the two components by exactly the same amount in thesame direction, whereas locking of the two components ofthe picosecond solitons needs the frequency shift of thetwo components in opposite directions [see Eq. (2)].6 Thereason behind this locking of the two components of thefemtosecond solitons with the same central frequencyshift h1 5 h2 5 h lies in the different group-velocityvariations introduced to the two orthogonally polarizedpulses by the same amount of central frequency shift be-cause of different polarization dispersions (rd Þ 1). Theeffective group-velocity change resulting from the fre-quency shift h introduced to the pulse evolving on theslow axis is proportional to h, whereas that resulting fromh introduced to the pulse evolution on the fast axis is pro-portional to rdh. Capitalizing on this difference, the cen-tral frequency shift h can thus be adjusted such that thetwo pulses are locked together to propagate with the samegroup velocity when the group-velocity difference 2d be-tween the two orthogonally polarized pulses is balancedby a central frequency shift. This locking mechanism ofthe femtosecond soliton pulses contrasts strikingly withthat in picosecond operation.6,7

A. Soliton States with Linear PolarizationsThe simplest solutions that Eq. (4) admits are slow andfast soliton states that can be expressed analytically:

f~s ! 5 A2 sech A2s, g 5 0, (5a)

f 5 0, g~s ! 5 A2k sech A2k/rds. (5b)

The exact mixed solutions of Eq. (4) for f Þ 0 and g Þ 0are obtainable only numerically because Eq. (4) is not in-tegrable except for a 5 rd 5 1. For a 5 1 of the linearpolarization, Eq. (4) is found to support a class of solu-tions with f and g characterized by one hump in each in-tensity profile [Fig. 1(a)]. This class of soliton solutionsbifurcates from the slow soliton at a critical valuek 5 kulb , transits continuously with decreasing ampli-tude of f and increasing amplitude of g as k varies, andeventually merges with the fast soliton at another bifur-cation point k 5 kvlb [Fig. 1(b)]. These linearly polarizedsoliton states disappear when rd 5 1 is assumed for non-zero birefringence B Þ 0 (or k Þ 1). In other words, thedifference in bx9 and by9 must be taken into account to re-

veal the linearly polarized femtosecond soliton states inthe birefringent fiber of Eq. (4). As an example, Fig. 1(a)shows the field profiles of the linearly polarized solitonstate at u 5 tan21@g(0)/f (0)# 5 32° or f(0) 5 1.2 (corre-sponding to k 5 1.017158) and rd 5 0.95. The relation-ship between peak amplitudes f(0) and g(0) of the solitonstates (including those of elliptical polarization with aÞ 1 discussed later) is governed by

g2~0 ! 5 k 2 af 2~0 ! 1 $@k 2 af 2~0 !#2 2 f 4~0 !

1 2 f 2~0 !%1/2, (6)

which is derived from an invariant of Eq. (4):

12 S d f

ds D 2

1rd

2 S dgds D 2

2 f 2 2 kg2 112

f 4 112

g4

1 af 2g2 5 const. (7)

Figure 1(b) is the graphic description of the relationshipbetween f(0) and g(0) of the linearly polarized solitonstates for different values of k at the value rd 5 0.95. Ask increases from kulb to kvlb , f(0) decreases from A2 to 0and g(0) increases from 0 to A2kvlb (51.4265 for rd5 0.95).

Fig. 1. (a) Field profiles of the linearly polarized soliton state atk 5 1.017158 (corresponding to the polarization angle u 5 32°)and rd 5 0.95 with the solid curves from numerical solution andthe dotted curves from the analytical approximation. (b) Thepaired relationships between peak amplitudes f(0) and g(0), kand f(0), and k and u of the linearly polarized soliton states atrd 5 0.95.

2368 J. Opt. Soc. Am. B/Vol. 14, No. 9 /September 1997 Y. Chen and J. Atai

The critical values ku 5 kulb and kv 5 kvlb at which thesoliton states (including a Þ 1 of the elliptical polariza-tion) bifurcate from the slow and the fast solitons [e.g., forthose shown in Fig. 1(b) at rd 5 0.95 acquired numeri-cally] can actually be obtained analytically by the pertur-bation method. For ku , we substitute f 5 A2 sech A2sand g 5 gp into Eq. (4) to yield

rd

2d2gp

ds2 2 kgp 1 2a sech2~A2s !gp 5 0, (8)

f 5 A1 sech~s/s0!, (12a)

g 5 A2 sech~s/s0!, (12b)

where the relation between A1 and A2 , with the help ofEq. (6), is given by

A22 5 2A1

2 1 k 1 @k2 2 2~k 2 1 !A12#1/2. (12c)

Substituting Eqs. (12a)–(12c) into Eq. (11) and takingvariation of H with respect to s0 and A1 or A2 result in

k 54~1 2 rd!2A1

2/3 2 rd~1 1 2rd! 2 2~1 2 rd!@rd2 2 2~2rd

2 2 3rd 1 1 !A12/3 1 4~1 2 rd!2A1

4/9#1/2

1 2 4rd

Fig. 2. Bifurcation values kulb and kvlb of the linearly polarizedsoliton states versus rd with the shaded area indicating the ex-istence region of the linearly polarized soliton states.

by keeping the terms up to gp . Equation (8) is solvable(with the solution expressible in the form of Legendrefunctions17) only when

q 2 m 5 ~k/rb!1/2 (9a)

with m an integer and

q 5 @~8a/rd 1 1 !1/2 2 1#/2. (9b)

For branching of the mixed ( f Þ 0 and g Þ 0) solitonstates with one peak in each intensity profile from theslow soliton, we have m 5 0. Substitution of Eq. (9b)into Eq. (9a) leads to the bifurcation value

ku 5rd

4 F S 8a

rd1 1 D 1/2

2 1G2

. (10a)

Similarly, substituting f 5 fp and g 5 A2k sechA2k/rdsfor kv into Eq. (4) and solving the resulting equationyields bifurcation formulas determining branching fromthe fast soliton:

q 2 m 5 ~rb /k!1/2, (9c)

q 5 @~8ard 1 1 !1/2 2 1#/2, (9d)

which gives the bifurcation value

kv 5 4rd /@~8ard 1 1 !1/2 2 1#2 (10b)

for m 5 0 of the soliton states with one hump in each in-tensity profile. For rd 5 0.95 and a 5 1 of the linearpolarization, we have ku 5 kulb 5 1.017047669 andkv 5 kvlb 5 1.017444166, identical to those obtained nu-merically from solving Eq. (4), presented in Fig. 1(b).The difference between kvlb and kulb , within which thelinearly polarized soliton states exist, increases with de-creasing rd(,1), as shown in Fig. 2.

Although Eq. (4) is not conducive to an analytical solu-tion for linearly polarized soliton states, the analytical ap-proximation to the numerical solution may be derivedfrom the variation expression

H 5 E2`

` F1

2 S d f

ds D2

1rd

2 S dg

ds D 2

1 f 2 1 kg2 21

2f 4

21

2g4 2 af 2g2Gds (11)

with ansatz

or

A12 5

3~k 2 rd!@k 2 ~4k 2 3 !rd#

8~k 2 1 !~1 2 rd!2 , (12d)

s02 5 0.5~A1

2 1 rdA22!/~A1

2 1 kA22!. (12e)

Equation (12) gives good approximation to numerical so-lutions for a fixed amplitude A1 5 f (0) or A2 5 g(0).Take A1 5 f (0) 5 1.2 or the linear polarization angleu 5 tan21 g(0)/f (0) 5 32° for example. Numerical solu-tion to Eq. (4) yields k 5 1.01715837 and A25 0.754803,whereas the analytical approximation [Eq. (12)] gives k5 1.01698768 and A2 5 0.754739, which are ,0.017%different from the exact values. The analytical approxi-mation for the field profiles is plotted in Fig. 1(a) by thedotted curves, which in comparison with numerical solu-tion (identified by the solid curves), appear to be verygood. Other values of A1 5 f(0) (varying from 0 to 6A2)or the polarization angles u (from 0 to 2p) give similargood agreement. Figure 3 illustrates the power or en-ergy P 5 *2`

` (uuu2 1 uvu2)dT 5 N*2`` ( f 2 1 g2)ds 5 NP

of the linearly polarized soliton states versus k for rd5 0.95 together with those for the fast and the slow soli-tons. The bifurcation of the linearly polarized solitonstates from slow and fast solitons is clearly demonstrated.Note that the normalized power P 5 P1 of the linearly po-larized soliton states decreases with increasing k.

Y. Chen and J. Atai Vol. 14, No. 9 /September 1997 /J. Opt. Soc. Am. B 2369

B. Elliptically Polarized Soliton StatesFor a 5 2cp 2 1 of the elliptical polarization, from Eq.(4) we also find the stationary solutions [with one humpin each intensity profile of f and g as shown in Fig. 5(b) onthe right] in the range of k 5 1 to the bifurcation valuekveb . The dependence of power of the soliton states withelliptical polarization on k is shown in Fig. 4(a) fora5 1/3 (cp 5 2/3 of silica fiber) and rd 5 0.95, where kveb

Fig. 3. Power of the linearly polarized soliton states togetherwith those of the slow and the fast solitons versus k for rd5 0.95. Note that by assuming rd 5 1, power Pv of the fastsoliton is always greater than Pu of the slow soliton for k . 1 ex-cept at k 5 1 when Pu 5 Pv . The dashed lines represent un-stable branches and the solid lines represent the stable branches.

Fig. 4. (a) Power of the elliptically polarized soliton states ver-sus k for rd 5 0.95, where the dashed curves represent unstablebranches and the solid line represents the stable branches. (b)Bifurcation values kveb of the elliptically polarized soliton states(a 5 1/3 or cp 5 2/3) versus rd with the existence region of thesoliton states identified by the shaded area.

5 4.91019. The normalized power P 5 Pe here in-creases with increasing k. The bifurcation value of thisclass of elliptically polarized soliton states from the fastsoliton is determined by Eq. (10b), which for a 5 1/3 andrd 5 0.95 gives k 5 kveb 5 4.91019, identical to that fromthe numerical solution. With decreasing rd the bifurca-tion value kveb for elliptically polarized soliton states in-creases and, in the limit of rd 5 1, kveb 5 4.77921 [Fig.4(b)]. The relationship between the peak amplitudesf(0) and g(0) of the elliptically polarized soliton states isgoverned by Eq. (6), which for a 5 1/3 and rd 5 0.95 isplotted in Fig. 5(a). The ratio of R 5 f(0)/g(0) for ellip-tically polarized soliton states with k . 1 can vary fromzero to slightly above unity (which is R 5 1.0082 forrd 5 0.95). Figure 5(b) is an example of the field profileof the elliptically polarized soliton state at k 5 2 and rd5 0.95.

In addition to the soliton states with one peak in theintensity profile, Eq. (4) also admits higher order solu-tions that are characterized by more than one peak in oneand/or both components f and g.12,18–20 These higher or-der solutions bifurcate from the fast and the slow solitonsat values of k determined by Eq. (9), taking m . 0 butlimited to the integer smaller than the greatest integerin q.

It should be added here that the elliptically polarizedsoliton states presented are not acceptable solutions to

Fig. 5. (a) Paired relationships between peak amplitudes f(0),and g(0), f(0) and k, and k and R 5 f(0)/g(0) of the ellipticallypolarized soliton states at rd 5 0.95. (b) Field profiles of the el-liptically polarized soliton states at k 5 2 and rd 5 0.95.

2370 J. Opt. Soc. Am. B/Vol. 14, No. 9 /September 1997 Y. Chen and J. Atai

system of Eq. (1) when the difference in polarization dis-persions (rd Þ 1) is ignored and the difference in thegroup velocities is taken into account, i.e., for the caseconsidered in Ref. 14.

4. STABILITY OF SOLITON STATESThe soliton states examined above are the eigenstates ofthe nonlinear system Eq. (1). The distance over which aneigenstate can be maintained in propagation depends onthe stability of the stationary state. A stable state guar-antees a long-distance transmission, whereas an unstablestate may find its way in device applications such asswitching. To determine the stability of the slow soliton,the fast soliton, and the linearly and the elliptically polar-ized soliton states in the femtosecond regime, we con-ducted a linear stability analysis on the eigenstate(u0 , v0) of Eq. (1). This was done by substituting u(T,Z) 5 u0(T, Z) 1 up(T, Z) and v(T, Z) 5 v0(T, Z)1 vp(T, Z) into Eq. (1) and solving the resulting equa-tions governing the perturbation functions up and vp .From the linearized equations we found the growth ratesx [up(T, Z) 5 up(T 2 Z/vp)exp(ig1Z 1 ihT 1 xZ) andvp(T, Z) 5 vp(T 2 Z/vp)exp(ig2Z 1 ihT 1 xZ)] for thelinearly and the elliptically polarized soliton states overtheir existence regions, implying that the linearly and theelliptically polarized soliton states are unstable. The in-stability of the linearly polarized soliton states resultsfrom the presence of the real growth rates x, which areshown in Fig. 6(a) for x versus the polarization angle u5 tan21@g(0)/f(0)# and in Fig. 6(b) for x versus k. The in-stability of elliptically polarized soliton states arises fromthe presence of the complex growth rates with the corre-sponding real part Re(x) versus k presented in Fig. 7.The stability character of the fast and the slow solitons issomewhat complicated. Below k , kulb (which is the bi-furcation value of the linearly polarized soliton statesfrom the slow soliton, kulb 5 1.01704 for rd 5 0.95), wefind the growth rates for the slow soliton as demonstratedin Fig. 6(b) for x versus k, but no growth rate exists forthe other values of k. For the fast soliton we find the realgrowth rates within kvlb , k , kveb (with kvlb and kvebthe bifurcation values of the linearly and the ellipticallypolarized soliton states from the fast soliton, kvlb5 1.01744 and kveb 5 4.91019 for rd 5 0.95) and thecomplex growth rates when k . kveb . The growth ratesx of the fast soliton are given in Figs. 6(b) and 7. Thisindicates that the femtosecond slow soliton is unstablewhen k , kulb(,kvlb) and stable when k . kulb , whereasthe femtosecond fast soliton is stable when k , kvlb andunstable when k . kvlb . The stability character of thefast and the slow solitons in the femtosecond regime, re-vealed here by recognizing the difference in the polariza-tion dispersions for a practical soliton transmission fiberwith D . 0.1 ps/nm/km (Ref. 15), clearly contrasts withthat discussed in Refs. 14 and 21, where rd 5 1 is as-sumed. Note that kulb , kvlb , meaning that at any valueof k (or propagation factor g for a fixed set of fiber param-eters) there is at least one femtosecond soliton state, be-ing either the slow or the fast soliton, which can evolvestably along the principal axis of the polarization-maintaining fiber for the femtosecond pulse transmission.

The stability character of the fast soliton, the slow soliton,and the linearly and the elliptically polarized solitonstates is summarized in the power versus k diagram ofFigs. 3 and 4(a) for rd 5 0.95, where the solid curves rep-resent the stable branches and the dashed curves refer tothe unstable branches.

Finally, all the stability results by linear stabilityanalysis presented are consistent with those from directlysolving Eq. (1) by the beam propagation method. Figure8 demonstrates the stable evolution of the fast soliton atk 5 1.01 (,kvlb)

22 for rd 5 0.95, where a perturbation

Fig. 6. (a) Growth rates x of the linearly polarized soliton statesversus the polarization angle u 5 tan21@g(0)/f(0)#. (b) Growthrates x of the linearly polarized soliton states, the slow soliton,and part of the fast soliton versus k for rd 5 0.95.

Fig. 7. Growth rates xe of the elliptically polarized soliton statesand xv of the fast soliton versus k for rd 5 0.95, where Re refersto the real part of the complex value x.

Y. Chen and J. Atai Vol. 14, No. 9 /September 1997 /J. Opt. Soc. Am. B 2371

initially implanted is radiated. This stable femtosecondfast soliton at k , kvlb , as well as the stable femtosecondslow soliton at k . kulb , has potential in application forlong-distance, high-capacity information transmission.On the other hand, for the unstable soliton states (withlinear or elliptical polarization), the stationary evolutioncan be derailed even by numerical noise such as thatshown in Figs. (9a) and (9b) for the linearly polarized soli-ton state of u 5 32°, where the power of the v componentof Fig. 9(b) is swapped to the u component of Fig. 9(a) atthe distance X 5 N2Z 5 165. The distance aroundwhich the power of the v (or the u) component swapsto the u (or the v) and the direction of power exchangecan depend sensitively on the perturbation initiallyimposed. For example, by imposing a perturbation(not visible in the scale of the figures) on the linearly po-larized soliton state of Figs. (9a) and (9b), i.e., initial ex-citation u 5 u0 1 0.0055 max$uu0u%sech(4s) and v 5 v01 0.0055 max$uv0u%sech(4s) with (u0 , v0) the initial soli-ton state of Figs. (9a) and (9b) and max$ % referring to themaximum values of uu0u and uv0u, the power exchange di-rection is reversed [Figs. (9c) and (9d)]. The power nowswaps from the u component of Fig. 9(c) to the v compo-nent of Fig. 9(d) at X 5 165. This means a polarizationdetector placed at X 5 165 receives completely differentstates by imposing or not imposing the initial perturba-tion. The polarization state is switched. This unstablecharacter of the linearly polarized soliton states clearlysuggests its possible application in switching.

5. RAMAN EFFECT AND THIRD-ORDERDISPERSIONTo demonstrate basic physics, we have, in the aboveanalysis, assumed the system to be passive and lossless.In a practical fiber operating in the femtosecond regime,the effects of intrapulse Raman scattering and third-orderdispersion should be taken into account. This intrapulseRaman scattering can cause soliton self-frequency shiftsof a short pulse at the femtosecond scale,23,24 which to-gether with third-order dispersion contributes to dissipa-tion of the pulse. By introduction of the amplificationwith bandwidth-limited gain into the system, the intra-pulse Raman scattering effect on the ultrashort pulsepropagation can, however, be suppressed.25 Further-more, the distributed amplification with bandwidth-limited gain introduced into the system (which can beachieved by doping the fiber with erbium) can counterbal-ance both the intrapulse Raman scattering and the third-order dispersion simultaneously to lead to the stablepropagation of the short pulses as reported in Refs. 26and 27. Thus the novel soliton pulses, supported by Eq.(1) for the passive and lossless system, are expected in apractical birefringent fiber incorporated with distributedamplification with bandwidth-limited gain when the dis-sipation owing to self-frequency shift and third-order dis-persion is balanced. A detailed analysis and presenta-tion by inclusion of the effects of intrapulse Ramanscattering and third-order dispersion are beyond thescope of this paper.

6. CONCLUSIONSThe propagation of femtosecond soliton pulses in birefrin-gent fiber is considered. By including the rapidly oscil-lating terms (necessary in the femtosecond or subpicosec-

Fig. 8. Stable evolution of the femtosecond fast soliton atk 5 1.01 and rd 5 0.95: (a) uuu/N, (b) uvu/N; s 5 (T 2 Z/vp)N, X 5 N2Z, and initial input is u 5 u0 1 0.2 max3 $uu0u%sech@4(s1 1.5)] and v 5 v0 1 0.2 max$uv0u%sech@4(s1 1.5)] with (u0 , v0) the soliton state and max$ % referring tothe maximum value.

Fig. 9. Unstable evolution of the linearly polarized soliton stateat k 5 1.017158 (u 5 32°) and rd 5 0.95: (a), (c) uuu/N; (b), (d)uvu/N. In (a) and (b) no initial perturbation is imposed, and in(c) and (d) initial excitation is u 5 u0 1 0.0055 max$uu0u%5 sech(4s) and v 5 v0 1 0.0055 max$uv0u%sech(4s).

2372 J. Opt. Soc. Am. B/Vol. 14, No. 9 /September 1997 Y. Chen and J. Atai

ond regime) and taking into account the difference inpolarization dispersions and in group velocities, we pre-dict the existence of the trapped femtosecond solitonpulses consisting of the two orthogonally polarized com-ponents that are locked together to propagate in the samegroup velocity when the two pulses in each polarizationshift their central frequency just enough in the same di-rection by the same amount. In addition to the fast andthe slow solitons (which exist over the whole parameterranges k . 1), the birefringent fiber is shown to supporttwo new families of the trapped soliton pulses: linearlypolarized ones and elliptically polarized ones. More im-portantly, the corresponding stability analysis revealsthat either the slow or the fast soliton can be stablewithin certain operating regions, whereas the ellipticallyand the linearly polarized soliton states are unstable.This indicates that the slow or the fast soliton has the po-tential for long-distance, high-capacity coherent solitoncommunication in the femtosecond regime, whereas theunstable linearly and elliptically polarized soliton statescan find their way in device applications such as ultrafastsoliton switching.

*Present address: Department of Electrical and Elec-tronic Engineering, University of Sydney, Sydney, Aus-tralia.

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