March 1, 1993 / Vol. 18, No. 5 / OPTICS LETTERS 337

Vector dark solitons

Yuri S. Kivshar and Sergei K. Turitsyn

Institutfir Theoretische Physik I, Heinrich-Heine-Universitat Dusseldorf, D-4000 Diusseldorf 1, Germany

Received October 16, 1992

It is shown that a novel class of optical soliton is possible in the form of vector dark solitons. These nonlinear

waves describe bound states of two gray solitons with different background intensities in each mode that are

strongly coupled through cross-phase modulation.

As is well known, optical pulses may propagate innonlinear media, e.g., in nonlinear optical fibers,1'2

without dispersive broadening in the form of bright ordark solitons when nonlinearity, which results fromthe nonlinear refractive index, exactly compensatesfor the group-velocity dispersion (GVD). For variousoptical applications, such as ultralong-distance soli-ton transmission and switching devices, it is impor-tant to understand the nature and general features ofthe interaction between soliton pulses. The interac-tion between closely spaced solitons belonging to oneoptical mode may be understood by analyzing inter-action forces between solitons (see, e.g., Refs. 3-6).However, soliton interactions in the case of a fewcoupled optical modes seem more complicated, andthey are not well understood yet. The problem ofthe intermode interaction of solitons naturally ap-pears, e.g., in highly birefringent optical fibers, asthat of nonlinear attraction between pulses of twopolarization modes. This effect has been recentlystudied experimentally and theoretically for possibleapplications to soliton switching. 7 -9

There exist four principally different cases of the in-termode interaction between solitons: (i) two brightsolitons propagate in the anomalous dispersion re-gion (see, e.g., Ref. 7), (ii) one bright soliton in theanomalous GVD region interacts with a dark solitonin the normal GVD region (the normal case; see, e.g.,Ref. 10), (iii) one bright soliton in the normal GYDregion interacts with a dark soliton in the anomalousGVD region (the so-called inverted case; see, e.g.,Ref. 11), and (iv) two dark solitons propagate in thenormal dispersion region. All these cases may bedescribed by a system of two nonlinear Schr6dinger(NLS) equations, coupled as a result of cross-phasemodulation. In the case of attraction, partial solitonpulses may form a two-component (vector) soliton asa bound state of the pulses belonging to differentpolarization modes.1 0' 12

Most theoretical and experimental studies haveconsidered the interaction of two bright solitons7 '10or bright and dark solitons,1 0' 13' 14 but interaction oftwo dark solitons belonging to different optical modeshas not been analyzed yet. However, recent experi-mental results in which stable propagation of darksolitons has been already observed"5- 7 make thisproblem real.

The purpose of this-Letter is-to analyze the inter-action of two dark solitons coupled through cross-

phase modulation and to present a novel class ofsoliton solutions in the form of vector dark solitons.We also show that the partial dark solitons, whichare composed to form the vector dark soliton, arestrongly coupled, and in the small-amplitude limitthey are described by a single Korteweg-de Vries(KdV) equation.

The interaction of two optical modes, t, andT2 , through cross-phase modulation is governedby the system of the incoherently coupled NLSequations 18 ' 19 :

i as + i 1aP 1 - a a l

az V1 at 2 at 2

+ R(I'l 12 + OP2j2)py = 0, (1)

iaP 2 . 1 aP2 a a2__2I- + £- __az V2 at 2 at2

+ R('P2 12 + I'TIj12 )p2 = 0, (2)

where v1 and v2 are the group velocities of thetwo optical polarization modes and a (a > 0) is thecoefficient of the normal GVD, which for simplicitywe assume equal for both modes. The cross-phasemodulation coefficient o- depends on the ellipticity ofthe fiber eigenmode (see, e.g., Ref. 19) and, in partic-ular, o- = 2/3 for linearly polarized modes, . = 2 forcircular polarized modes, and, in the general case,2/3 • a- 2 for elliptical eigenmodes. With thechange of variables, (t - zlv)lto - a7, z(a/2to2) -i

( t eS2z 4Il1,2 2Rtl/a exp ± -2-) U. V

where

2v 1 V2

V1 + V2

V2 - V1

2vlV2(3)

one obtains the system of two dimensionlessequations,

. aU a2U + + 2)U= O-a aT2 ± (1U2 ± o-1V12)U= 0

.aV -2 .JY+(If rU12)V = 0,ae aT2 12+0

(4)

(5)

which describe coupled optical polarization modes-operating in the normal GVD region. For o- # 1,

0146-9592/93/050337-03$5.00/0 © 1993 Optical Society of America

338 OPTICS LETTERS / Vol. 18, No. 5 / March 1, 1993

0 ZFig. 1. Intensities of the two components, IU12 and 1112,of the vector dark soliton, with U0 and V0 being theasymptotic values of the intensities of the orthogonalpolarization modes.

these equations are not integrable,20 but for o- = 1they have a set of integrals of motions so that one maynaturally expect to find the integrability property andexact soliton solutions as in the case of anomalousGVD.2 1

For the case of anomalous GVD, exact soliton solu-tions to the system [Eqs. (4) and (5)] may be obtainedby a simple substitution, and they are either of equalamplitudes, U = V, or of partial orthogonal polariza-tions, U = 0, V = O and U 0 O, V= O. However, inthe case of the normal GVD, a general dark soli-ton solution may have different intensities for twopolarization modes. It means that such a solutioncannot be obtained by simple substitutions like thosementioned above. However, we have found that thevector dark soliton does exist in the form of two par-tial dark solitons excited on different cw backgrounds.This solution may be written in the form

U = Uo(cos 01 tanh Z + i sin 01)exp(i01), (6)

V = Vo(COS 0(2 tanh Z + i sin 0 2)exp(iE 2 ), (7)

where Z = Y({r + 1/W - TO) and

01 = klr + (Uo2 + a-V02 + k12)f,

02 = k2r + (VO2 + a-U02 + k22)6,

and the parameters Uf, Vlw, n, W, k, and k2 arecoupled by the following relations:

UO COS 01 = Vo COS 0 2 , v2 == 2 U0 cos 2 A1 ,

(8)

W-il = U o - + a- sinCO1 02) + (k1 +k 2),2 COS '12

k2 - ki = U0 + -2sin(qS1 - 02)COS 102

(9)

The pulse intensities in each mode may be calculatedto be

I2(1 -cosh ' )

IV12 = U02 (1 - cosh2 Z) (10)

Solutions (6) and (7) describe two coupled dark (grayor black; see the terminology in Ref. 22) solitons inthe case for which the asymptotic values of the cwbackgrounds are different, i.e., IUI U- Uo and 111 - VOprovided that Irl - - (see Fig. 1).

The partial dark solitons described by the solu-tions (6) and (7) are indeed strongly coupled owingto mutual trapping. To demonstrate such a trap-ping analytically, we use a variational (Lagrangian)approach14 to introduce a time delay into one of thepulse components, e.g., by putting

V(Z)I 2 I 0 I[ cosh 2(Z (11)

Then the pulse interaction results in the systemLagrangian from the integral,

f dr(1 U12 - V02)(1U12 - U0

2),

which yields the effective interaction energy betweenthe soliton pulses,

Ueff(A) = -4V2U0 3 a- Cos3 1 cosh A(A - tanh A).v 1 + - sinh3AAah)

The effective interaction energy [Eq. (12)] has a senseeither for small o- (small intermode coupling) whenA -~ 1 or for small A (small time delay) when a- - 1,and it corresponds to an effective attraction of partialdark solitons.

The strong coupling between partial dark solitonsmay be also confirmed in another way, by consider-ing the small-amplitude limit of the vector soliton[Eqs. (6) and (7)]. Following to the approach devel-oped for the one-component NLS equation,2 3 we lookfor solutions of the coupled system (4) and (5) in theform

U = (Uo + a)exp[i(U0 2 + a-V02)e + i.0], (13)

V= (VO + b)exp[i(V02 + a-U0

2)6 + id], (14)

assuming that a, b and derivatives of '1 and ifare small. Substituting Eqs. (13) and (14) into thesystem (4) and (5), one may obtain four coupledequations for the functions a, b, 1, and ai. Thissystem of equations may be investigated by theasymptotic method,23 and it shows that even in thesmall-amplitude limit the optical modes are stronglycoupled if waves propagate along the finite-amplitudebackgrounds. In the linear limit such excitations aredescribed by the dispersion relation

K 2U02f 2 - { K4)(K2

- 2Vn2 fi2 _

= 4a-2 U02V0

2fQ4 , (15)

with K and fl being the wave number and wavefrequency, respectively, of linear waves. In this ap-proximation, the modes dynamics may be simplifiedby introducing normal variables as linear combina-tions of a and b. This result simply means that fora- - 1 any dip on the cw background pulse of one

March 1, 1993 / Vol. 18, No. 5 / OPTICS LETTERS 339

mode will immediately create a similar dip in theother mode strongly coupled to the primary pulse.This situation drastically differs from the case ofintermode interaction of bright solitons.

Such a strong interaction between the polarizationmodes also remains valid in the nonlinear case, andin the limit of small amplitudes the mode dynamicsis described by a single KdV equation but not by thecoupled equations. For example, in the case U0 =

V0, the asymptotic expansions (which are similar tothose for the one-component NLS equation23) yieldthe simple relation a a b, and in the zero-order ap-proximation the evolution of the pulse amplitude a isgiven by the KdV equation

aa ~~~aa a3a2C a + 12Uo(1 + a-)a T - = °- , (16)

ay aT aT3

where slow variables y and T (see details in Ref. 23)are connected with the reference frame moving at thesound velocity C, with C2 = 2(1 + a)U 0

2.

The strong coupling between the partial dark pulsesmeans that optical switching involving dark pulses isnot possible in the way already proposed for brightsolitons.2 4 To support such a conclusion by a clearexample, let us consider the model of a nonlineardirectional coupler (see, e.g., Ref. 25 and referencestherein), which in dimensionless units is describedby the coupled equations

AU aU + (1U12 + a-1112)U = KV (17)

iaV a2V + (1V12 + a-IU12)V= KU. (18)ae ar2

In the case a = 1 the system (17) and (18) has exactsolutions (which may be found analogously to thebright soliton case26 ); the simplest one is

JUI2, 1112 U0

2[1 ± sin(2,/)cos(2KW)]tanh 2 (UoT),

(19)

where p is an arbitrary constant. The result[Eq. (19)] looks like that of the linear theory exceptfor the T-dependent factor describing the dark-profileenvelope. Thus, as follows from Eq. (19), the opticalswitching in this case is realized through backgroundpulses themselves but not through dark solitons.

In conclusion, we have found a new class of solu-tions of the two coupled NLS equations in the formof a vector dark soliton that describes mutual trap-ping of two dark-profile pulses. We have shown thatthese waves correspond to strongly coupled statesthat in the small-amplitude limit may be describedby a single KdV equation. We have also pointedout that a strong coupling of the partial solitonswith their backgrounds does not allow one to usedark solitons for optical switching in the way alreadyproposed for bright solitons.

The authors are indebted to K. H. Spatschek forwarm hospitality at the Institut fulr Theoretische

Physik I. This research has been supported by theAlexander von Humboldt-Stiftung.

Yuri S. Kivshar is also with the Institute forLow Temperature Physics and Engineering, 310164Kharkov, Ukraine.

Sergei K. Turitsyn is on leave from the Institute forAutomation and Electrometry, 630090 Novosibirsk,Russia.

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