Dariusz Burak Vol. 13, No. 3 /March 1996/J. Opt. Soc. Am. B 613

Steerable bright-soliton Y-junctions fromsecond-order Hermite–Gaussians

Dariusz Burak*

Institute of Fundamental Technological Research, Polish Academy of Science,Swietokrzyska 21, 00-049 Warsaw, Poland

Received March 30, 1995; revised manuscript received October 10, 1995

Steering properties of bright-soliton Y-junctions excited from second-order Hermite–Gaussian beams in anonlinear Kerr medium are investigated numerically. The numerical study is supplemented by a variationalapproach of two different types. The first one solves the Zakharov–Shabat scattering problem, and themoment method is applied to determine the trial eigensolution. The second approach utilizes the trialfunction ansatz of the second-order Hermite–Gaussian form and solves the nonlinear Schrodinger equation.Good agreement is found between numerical calculations and variational approximations.

Key words: Spatial soliton, Hermite–Gaussian beam, steering effect. 1996 Optical Society of America

1. INTRODUCTION

Self-trapped light beams (spatial solitons) have been theobject of intensive theoretical and experimental researchduring the past three decades, mainly because of thepossibility of guiding and manipulating light with lightitself.1 – 5 The propagation of an electromagnetic beamin a homogeneous Kerr-type medium is described by aSchrodinger equation with a cubic nonlinear term. Thespatial solutions evolve from nonlinear changes in the re-fractive index of the material that the light beam induces.When these refractive-index changes compensate for theeffect of diffraction, the beam becomes self-trapped andis called a spatial soliton. For a nonlinearity of the fo-cusing or defocusing type the transverse profile of a beamfollows a hyperbolic secant (bright spatial soliton) or hy-perbolic tangent (dark spatial soliton) law, respectively.The bright2 and the dark5,6 spatial solitons are usuallyobserved experimentally in the Kerr-like media, but dif-ferent types of such solutions have recently been foundin other materials.7 – 10 The breadth of area of using soli-tons for all-optical switching applications lies in the vari-ety of optical solitons and geometries that can be used.11

In particular, a new class of spatial optical switcheshas been proposed recently, whose principle of action isbased on the interaction between soliton beams (brightor dark) and the waveguide structures induced by theseinteractions.5,12 – 16 It has been predicted theoretically17

and experimentally demonstrated18 that, depending ontheir relative phase, two close spatial solitons with paral-lel or tilted initial directions can either repel, or attract, orbound each other. However, theoretical investigations ofsuch couplers were done with the assumption that beamintensities and profiles are adjusted to satisfy the puresoliton propagation condition, which is generally not metfor laser beam propagation in a nonlinear medium.

In defocusing Kerr-like materials the properties ofsteerable dark-soliton Y-junctions have recently beenstudied in the context of photonic switching devices.5,15

Also in focusing Kerr media, curved trajectories of singlelight beams were observed, mainly caused by either asym-

0740-3224/96/030613-08$06.00

metry in the initial intensity distribution19,20 or specific,nonplanar phase-front profiles.21 In the former case anasymmetric spatial modulation of an initial beam pro-file induced a nonlinear prism in a Kerr-like material,which in turn caused self-deflection of the beam in thefar-field region. However, the nonlinear material wasassumed to be thin enough to avoid self-focusing of thelight beam, and thus no solitons were formed. Steer-ing of soliton beams by the later technique employs thefact that an input beam with a spatially modulated phasesplits in a nonlinear material into many subbeams so thatmost of the input power occurs in a single spatial soliton.The orientation of this soliton beam can be controlled bythe initial beam’s amplitude- and/or phase-modulationparameter.

This paper focuses on the investigation of steerablebright-soliton Y-junctions formed from the second-orderHermite–Gaussian (HG2) initial profiles. Although theprevious numerical studies on this topic22 showed that aninitial (first- or second-order) Hermite–Gaussian profilebreaks up in the nonlinear Kerr medium into an evennumber of bright solitons and can form a Y-junction, theproperties of such structures have not been analyzed indetail. The main objective of this contribution is to studyhow the orientations (i.e., angles of propagation in a non-linear medium, or deflection angles) and the amplitudesof excited solitons are determined by the amplitude of theinitial beam profile. This is achieved both by numericaland variational solution of the Zakharov–Shabat scatter-ing problem.23 The following soliton-generation possibil-ity is demonstrated: For low incident-beam amplitudes,no solitons are found. With increasing incident ampli-tude two solitons arise, which propagate apart and forma Y-junction. A further increase of the incident-beamamplitude decreases the deflection angles, and above acertain amplitude level, solitons form a breather (boundsolution), i.e., a spatially localized (in the transversedirection) solution with internal oscillations (along thelongitudinal coordinate). The spatial period of a solitonbreather decreases with a further increase of the incident-beam amplitude. A complementary analysis of beam

1996 Optical Society of America

614 J. Opt. Soc. Am. B/Vol. 13, No. 3 /March 1996 Dariusz Burak

steering is presented involving variational approach witha second-order Hermite-Gaussian trial-function ansatz.22

One identifies the soliton Y-junction and breather, re-spectively, with two different critical points in the phasespace related to variational propagation equations. Bothvariational methods employed in this work give a verygood estimation for deflection of unbounded solitonbeams. Also the threshold incident amplitudes for a soli-ton Y-junction and breather formation can be predictedwith good accuracy.

The paper is organized as follows. Section 2 addressesthe nonlinear propagation equation and basic proper-ties of its solutions. In Section 3, properties of solitonY-junctions and breathers are investigated. In Section 4,variational analysis of HG2 beams’ propagation in a Kerrmedium is performed, and comparison with a numeri-cal solution is given. Finally, Section 5 summarizes theresults.

2. NONLINEAR PROPAGATION EQUATIONThe propagation of a beam of light in a dielectric mediumis described by Maxwell’s equations, which for the TEwave in the x–z plane reduce to the scalar wave equation

≠2E≠z02

1≠2E≠x02

1 k20n2E ­ 0 , (1)

governing the electric field E, where k0 is the free-spacewave number and n is the intensity-dependent refractiveindex, n ­ n0 1 n2jEj2, with the Kerr coefficient n2 . 0.Assuming that the solution is of the form Esx0, z0d ­V sx0, z0dexpsibz0d, where b ­ n0k0 is the linear propaga-tion wave number, one obtains the nonlinear Schrodingerequation (NSE)

2ib≠V≠z0

1≠2V≠x02 1 2k0n0n2jV j2V ­ 0 (2)

upon neglecting the term involving the second-orderderivative with respect to z0. This equation can berewritten in its famous form as

i≠U≠z

112

≠2U≠x2 1 jU j2U ­ 0 , (3)

where the transverse x and the longitudinal z coordinatesare rescaled with respect to the beam radius x ­ x0yw0 andto the Rayleigh length z ­ z0ybw2

0 , respectively,24 wherew0 is the initial beam size. The rescaled field envelopeis given by U sx, z d ­ k0w0

pn0n2 V sx, z d.

The Cauchy initial-value problem for NSE (3) wassolved first by Zakharov and Shabat, who applied theinverse scattering transform (IST).23 The N-soliton so-lution to this equation (when these solitons are wellseparated),

U sx, z d ­NP

i­12hi expf2isn2

i 2 j2i dz 2 2ijixg

3 sechs2hix 1 4hijiz d , (4)

corresponds to the number of beams N with ampli-tudes and widths 2hi propagating with angles qi [orso-called transverse velocities 2ji (Ref. 14)], where

qi ­ 2arctans2jid with respect to the z direction. IfN solitons (with amplitudes 2hi, i ­ 1, 2, . . . , N ) propa-gate along the z direction (i.e., ji ­ 0, i ­ 1, 2, . . . , N ),they form a bound solution (an example of a symmetric,centered at x ­ 0, two-soliton breather is given in Ref. 25).

The effect of an initial beam profile U0sxd ­ U sx, 0d onthe excited-soliton parameters ji, hi (i ­ 1, 2, . . . , N ) isgiven by the solution of the Zakharov–Shabat eigenvalueproblem26

≠w1

≠x­ 2iliw1 1 iU0sxdw2 , (5a)

≠w2

≠x­ iliw2 1 iUp

0 sxdw1 , (5b)

where li ­ ji 1 ihi. The initial conditions w1s2X0, lid ­expsiliX0d and w2s2X0, lid ­ 0 lead, after integrationfrom 2X0 to X0 [where X0 is large enough to cover allnonnegligible values of U0sxd], to the scattering coefficientaslid ­ w1sX0, lidexpsiliX0d. The zero location of aslid,aslid ­ 0, determines soliton parameters ji, hi,23 and canbe found iteratively by the numerical solution of Eqs. (5).

In order to gain more detailed information about thenonlinear beam propagation, I integrated the NSE ina finite transverse domain f2X0, X0g using the split-step fast-Fourier-transform scheme27 with a mesh dx ø0.06 and dz ø 0.01. Numerical accuracy was checked byrepetition of the simulations for different grid sizes.

3. STEERING PROPERTIES OFA SOLITON Y-JUNCTIONIn the origin of the presented work lies the fact thatsuch structures as bright-soliton Y-junctions can be ex-cited from symmetric and real incident profiles, whoseenvelopes satisfy the zero-area condition28

Z 1`

2`

U0sxddx ­ 0 . (6)

Three examples of initial profiles satisfying the above con-straint were discussed: Gaussian and hyperbolic secantenvelopes with spatially modulated amplitudes and anexact two-soliton solution of the NSE.13 Moreover, someinteresting analogies between propagation of higher-orderHermite–Gaussians and higher-order optical solitonshave recently been demonstrated numerically.22 In thispaper I perform detailed analytical and numerical studiesof properties of spatial solitons excited by the most inter-esting, from the point of view of practical applications,initial HG2 profile

U0sxd ­ q0s21 1 x2dexp

√2

x2

2

!, (7)

where q0 stands for an initial amplitude. In Appendix AI followed Siegman29 and showed that function (7) aroseas the second-order Hermite-Gaussian solution to thelinear Schrodinger equation [i.e., Eq. (3) without the lastterm describing Kerr nonlinearity] evaluated in its waist[i.e., at z ­ 0; see Eq. (A6)]. Condition (6) is obeyed as aconsequence of orthogonality of Hermite polynomials [seeEq. (A7)].

Dariusz Burak Vol. 13, No. 3 /March 1996/J. Opt. Soc. Am. B 615

Fig. 1. Soliton parameters excited by the second-order Her-mite–Gaussian beam [Eq. (7)]. The imaginary parts (solidcurves, the left-hand-side axis) and the real parts (dottedcurves, the right-hand-side axis) of eigenvalues li, i ­ 1 . . . 4(corresponding to the amplitudes and the deflection angles,li ­ 2arctan 2ji, of solitons, respectively), are plotted as afunction of the incident-beam amplitude q0.

A. Numerical ResultsResults of my calculations are presented in Fig. 1, inwhich the imaginary and the real parts of li are shown,respectively, as a function of the incident-beam amplitudeq0. For small incident-beam amplitudes q0, no solitonsare present in the spectrum of the IST. For amplitudeshigher than the threshold (numerical) value

qth0 ø 0.86 (8)

(where the superscript th indicates threshold), two soli-tons are excited by the incident beam [Eq. (7)] and form astructure that is analogous to the linear Y-junction split-ter. These zero-amplitude solitons (h1 ­ h2 ­ 0) aredeflected on the angles q1,2 ­ 653.7± (j1,2 ø 60.68) withrespect to the z direction. With an increasing incident-beam amplitude q0 the excited-soliton amplitudes in-crease (solid curves in Fig. 1), and the total deflectionbetween solitons Dq ­ arctans2j1d 2 arctans2j2d de-creases (dotted curves in Fig. 1); thus a Y-junction be-comes narrower. This is the bright solitons’ steeringeffect. In the far field the solution of the NSE is in thiscase given by the sum of two solitons, as described byEq. (4). An example of such a Y-junction is shown inFig. 2(a), in which the incident HG2 beam with ampli-tude q0 ø 1.67 initially undergoes self-focusing, forminga vertex, and then breaks up into two bright solitons.These solitons propagate apart with the same amplitudesand are deflected on the same angles but with oppositesign. This symmetry must be the case, since the symme-try of the solution is the conserved quantity. In this caseshown in Fig. 2(a) the soliton beams excited in a nonlinearmedium carry the maximal amount of total energy of theincident beam (nearly 99.1%, as can be calculated fromdata shown in Fig. 1; see also Ref. 30). Since the influ-ence of radiation modes in this case is almost negligible,Eq. (4) (with N ­ 2) provides an approximate analyti-cal expression for a soliton Y-junction valid everywhereexcept the vertex area. For the numerical value

qs bd0 ø 2.22 (9)

both eigenvalues collapse, and a further increase of q0

causes an increase of the difference 2Dh ­ 2sh1 2 h2d invalues of soliton amplitudes, whereas real parts of eigen-values l1,2 are equal to zero, j1 ­ j2 ­ 0 (see Fig. 1).This means that, instead of a Y-junction, both solitonsform a breather that propagates along the z axis (the su-perscript b denotes bound), whose period of spatial os-cillations is given by Zp ­ pysh2

1 2 h22 d, providing h1 .

h2.25 Thus for the incident-beam amplitude equal to qsbd0

(h1 ­ h2 ø 0.75) the period Zp is infinite, and the solu-tion of the NSE has the form of two solitons that propa-gate in parallel with each other, separated in space, asis shown in Fig. 2(b). The increase of q0 above the qsbd

0

level lowers the value of Zp, and the breather soliton oscil-lates, i.e., periodically changes its amplitude and width.Figure 2(c) shows the propagation of the incident HG2beam with amplitude q0 ­ 2.24. Two solitons with am-plitudes 2h1 ø 1.78 and 2h2 ø 1.29 are present in theIST spectrum, and so the approximated period of spatialoscillations is Zp ø 8.3zF .

With a further increase of the incident-beam amplitudeq0 the process of soliton excitation repeats: the thresholdvalue for the subsequent soliton Y-junction is q0 ø 2.7,and the deflection angles are l1,2 ø 660.7± (j1,2 6 0.68).The parameters of the second pair of solitons depend on

Fig. 2. Evolution of initial HG2 profiles [Eq. (7)] in a Kerrmedium for the incident-beam amplitudes (a) q0 ø 1.67, (b)q0 ø 2.22, (c) q0 ø 2.24.

616 J. Opt. Soc. Am. B/Vol. 13, No. 3 /March 1996 Dariusz Burak

q0 in a way qualitatively similar to the case for the firstone (see Fig. 1).

B. Variational Approximation to Solitons’ SteeringIn the past, analytical results concerning self-bending oflight beams with asymmetric initial amplitude profiles(e.g., right triangular) were restricted to the case whena nonlinear medium changed only the phase profile of thebeam.19,20 Recently, Cao et al. applied the conservationlaws of the NSE and obtained an analytical expression forthe angle of self-deflection of beams with spatially modu-lated phase profiles.21 However, their method cannot beused directly on the problem discussed in this paper, sincethe HG2 initial profile [Eq. (7)] possesses a plane wavefront; thus the total momentum equals zero, and the beamas a whole does not experience any deflection [i.e., bothsoliton beams, shown in Fig. 2(a), counterbalance eachother]. In the following I incorporate some results ofRef. 21 into the Desaix et al.31 variational approach toZakharov–Shabat scattering problem [Eqs. (5)].

Equations (5) can be rewritten as a variational prob-lem,

dZ 1`

2`

Ldx ­ 0 , (10)

for the Lagrangian density in the form

L ­12

√w2

dw1

dx2 w1

dw2

dx

!1 ilw1w2

1i2

fUp0 sxdw2

1 2 U0sxdw22g . (11)

The original system of equations [Eqs. (5)] is then derivedfrom Euler–Lagrange equations for each eigensolutionw1 and w2. The successful approach of the variationalmethod is based on the proper choice of trail functions,which involves parameters that may be important inthe optimization procedure. Substituting the trial func-tions into the Lagrangian density [Eq. (11)] and perform-ing integration with respect to x, one obtains a reducedLagrangian, which, after subsequent variations with re-spect to the parameters, yields an approximate solution.

An approximate solution to the Zakharov–Shabat scat-tering problem [Eqs. (5)] for positive-defined initial pro-files, U0sxd . 0, 2` , x , `, was found in Ref. 30, and ananalytical expression for the excited-soliton amplitude 2h

was obtained in the limit h ,, 1. The eigensolutions w1

and w2 were found in the form

w1sx, ld ­ Bexps2ilxdhcos bsxd 2 hfBsxdsin bsxd2 Asxdcos bsxdgj , (12a)

w2sx, ld ­ 2iw1s2x, ld , (12b)

where l ­ j 1 ih, B is the amplitude (B ­ 1 in Ref. 30),and the functions bsxd, Asxd, and Bsxd are defined as

bsxd ­Z x

2`

U0syddy , (13a)

Asxd ­Z x

2`

2ybsydsinf2bsydgdy , (13b)

Bsxd ­Z x

2`

2ybsydcosf2bsydgdy . (13c)

in Eq. (13a), U0 stands for an initial beam profile. Sincethe function U0sxd given by Eq. (7) changes its sign, wecannot use directly Eqs. (12) and (13) in our approach.However, eigensolutions (12) can be considered as anatural choice of the trial-function ansatz but with ap-propriately redefined initial profile U0sxd. According tothe moment method,21 the input beam profile

U0sxd ­ exps2ijxdU0sxd , (14)

with U0sxd given by Eq. (7), would excite [under the propercondition; see Eqs. (8) and (14) in Ref. 21] the spatialsoliton beam, whose deflection angle can be found as

tan qdef ­PM

­ 2j , (15)

where the mass M and the transverse momentum P aregiven by

M ­Z 1`

2`

jU0sxdj2dx , (16a)

P ­12i

Z 1`

2`

0@Up0

≠U0

≠x2 U0

≠Up0

≠x

1Adx . (16b)

Thus it is natural to identify the deflection angle qdef

with propagation angle q and the quantities 2j and 2h

with, respectively, transverse velocity and amplitude ofeach soliton lobe [shown, e.g., in Fig. 2(a)]. Within theIST approach, soliton parameters are determined as thezero location of the scattering coefficient, asld ­ 0 ­w1sx, ldexpsilxd, evaluated at infinity, x ­ ` [compareEqs. (5)]. Therefore the excited-soliton amplitude can beexpressed as [see Eq. (12)]

h ­cos b

B sin b 2 A cos b, (17)

where the numbers b, A, and B are the integrals inEqs. (13) evaluated at x ø `. In this way one has todetermine by a variational method only the transversevelocity 2j of an excited soliton. This is achieved by therequirement of stationarity of the reduced Lagrangian,R1`

2` Ldx, with respect to variations in the amplitude B .31

Unfortunately, since the calculation seems to be too cum-bersome to be done analytically, one needs to perform itnumerically. Trial eigensolution (12) does not take intoaccount the 6j symmetry as is the case in Fig. 1 (theright-hand-side axis). Nevertheless, numerical calcula-tions confirmed that if l1 ­ j 1 ih solves the Zakharov–Shabat scattering problem, the eigenvalue l2 ­ 2j 1 ih

does, too.The threshold for a Y-junction formation can be de-

termined from the condition h ­ 0.28 This leads to thenumerical values of the threshold amplitude, qsvd

0 ø 0.853[see Eq. (8)], and the threshold transverse velocities,jsvd ø 60.684, which approximate very well exact nu-merical values found in Subsection 3.A. In Fig. 3(a) acomparison between numerical and variational solutionsof Eqs. (5) is shown for incident amplitudes in the rangebetween the thresholds for the soliton Y-junction [Eq. (8)]and the breather [Eq. (9)] formation. The result coincidefor small amplitudes of the excited solitons, 2h ,, 1, and

Dariusz Burak Vol. 13, No. 3 /March 1996/J. Opt. Soc. Am. B 617

Fig. 3. Amplitudes (the left-hand-side axis) and transverse ve-locities (the right-hand-side axis) of excited solitons as obtainedfrom numerical and variational solutions of Eqs. (5). Solid anddashed lines: numerically obtained values of 2h and 2j, respec-tively. Solid diamonds and circles: variational results for 2hand 2j, respectively. The subscripts IST and v in h and j referto IST and variational approaches, respectively.

start to differ for higher values of h. This must be thecase, since expression (13) was obtained as a power-seriesexpansion of eigensolutions w1 and w2 with respect to l

and is valid only in its linear limit (see Ref. 30 for details).Nevertheless, the presented variational approach predictsconsiderably well the dependence of transverse velocitiesand amplitudes of excited solitons on the incident HG2beam’s amplitude.

4. VARIATIONAL APPROACH TONONLINEAR PROPAGATIONIn this section one considers an analytical approximationto nonlinear propagation of the HG2 beam as a whole (incontrast to Section 3.B) using the variational approachdeveloped by Anderson.32 This section develops furtherthe previous research22 on variational approach to higher-order Hermite–Gaussian propagation in order to give abetter understanding of properties of soliton Y-junctions.The presented analysis is based on critical points of thephase space of variational equations.

Following the Anderson approach, the NSE can be re-stated as a variational problem with a Lagrangian in theform

L ­i2

√U p ≠U

≠z2 U

≠Up

≠z

!2

12

É≠U≠x

É21

12

jU j4 . (18)

Since the nonlinear Schrodinger Eq. (3) is valid only forsmall nonlinearities n2jEj2 ,, n0, it is not unreasonableto expect that in some cases the exact solution of the NSEcan be approximated by the solution of the propagationequation in the linear medium. Therefore one uses thesecond-order Hermite–Gaussian trial function

U sx, z d ­Asz dv3sz d

8<:21 1

"x

vsz d

#29=;exp

"2

x2

2v2szd

#, (19)

with As0d ­ q0 and vs0d ­ 1 [compare Eq. (7)]. Thischoice of the trial function follows more general treatment

of linear propagation of Hermite–Gaussians and a nonlin-ear propagation of optical solitons.33 Using trial-functionansatz (19), one can integrate Lagrangian density (18)over the transverse variable x. From this averagedLagrangian one derives the ordinary differential equa-tions32 (ODE) model for the complex beam parametersAsz d and vsz d (for a detailed description of the method,see Ref. 32):

dvdz

­i

2v2

i8

q20

√1 1

v2

vp2

!3/2√43

3201

780

v2

vp21

580

v4

vp4

21

80vp2

v22

580

vp4

v42

480

vp6

v6

!, (20)

dAdz

­i2

q20

s1v2

11

vp2

√9

162

27256

v2

vp21

964

vp2

v22

364

v4

vp4

118

vp4

v42

564

v6

vp6

!A . (21)

The constant of integration arises out of the analysis

Z `

2`

jU sx, z dj2dx ­3p

2p jAj2

5/2p

v2 1 vp2­ const. ­

3p

p q02

4,

(22)

which is related to the lowest-order conserved quantitythat the NSE obeys.23 Numerical solutions of the ODEmodel [Eqs. (20) and (21)] show that the Hamiltonian ofthe system governed by the NSE, H ­ s1y2d

R1`

2`sjUxj2 2

jU j4ddx, which for trial function (19) takes the form

H ­3pjAj2

svp2 1 v2d9/2

(5p

2 Refv2g 2jAj2

4jvj10

"Refv8g

1 jvj4

√Refv4g 1

358

jvj4

!#), (23)

where Re[v] denotes the real part of v, is conservedin variational approximation to the nonlinear beampropagation.

The dynamic system described by Eqs. (20) and (21)is determined by Eq. (20), since this equation does notdepend on the beam amplitude A. The first term onthe right-hand side of Eq. (20) corresponds to the lin-ear diffraction of the beam,24 whereas the second one de-scribes nonlinear beam refraction.

To analyze dynamic properties of Eq. (20), one uses apolar system of coordinates, vsz d ­ rsz dexpfiwsz dg, wherer and w are real functions of z . This substitution trans-forms Eq. (20) to the set of coupled ODE’s

Ùr ­12

sins2wd

24 1r

1q2

0p2

cos3/2s2wd

353

"47320

11280

sins6wdsins2wd

19

80sins10wdsins2wd

#, (24a)

Ùw ­12r

coss2wd

24 1r

2q2

0p2

cos3/2s2wd

353

"39320

1140

coss6wdcoss2wd

1180

coss10wdcoss2wd

#. (24b)

618 J. Opt. Soc. Am. B/Vol. 13, No. 3 /March 1996 Dariusz Burak

Fig. 4. Left-hand-side axis: peak amplitude of the beamsjU sxmax, z0dj in numerical (solid curve) and variational (soliddiamonds) approaches evaluated at z0 ­ 10zF versus the ampli-tude of the incident beam q0. Right-hand-side axis: the de-flection angles tan qsn,vd ­ xsn,vd

maxyz0 versus q0 in numerical(dashed curve) and variational (solid circles) approaches.

The system of Eqs. (24) possesses two critical points.34

The first critical point is located in the finite part of thesr, wd plane,

rc1 ­320

p2

51q20

, wc1 ­ 0 , (25a)

and the other one at infinity,

1rc2

­ 0, wc2 ­p

4. (25b)

Since eigenvalues of Eqs. (24) at point (25a) are purelyimaginary, l1,2 ø 60.02iq4

0, all trajectories initially closeto src1, wc1d remain in the small neighborhood of src1, wc1dand rotate around it infinitely many times.34 This corre-sponds to propagation of the soliton breather in the Kerrmedium, characterized by spatial oscillations of beamparameters.25 Moreover, the period of spatial oscilla-tions, Zp ­ 2pyl ø 100pyq4

0, decreases with an increaseof q0, which qualitatively resembles the result found nu-merically in Section 3. Second critical point (25b) de-scribes an infinite increase of the beam width during thepropagation. The numerical solution of the ODE model[Eqs. (20) and (21)] shows that, within the variational ap-proach with Hermite–Gaussian trial function (19), twotypes of solution belong to this category. The first oneis when the beam diffracts completely in the nonlinearmedium (there are no discrete eigenvalues in the spec-trum of the IST). The second type of solution corre-sponds to the soliton Y-junction and takes (in the far field)the form of Eq. (4) (with parameters taken from Fig. 1).The analysis of the H [see Eq. (23)] values shows that,for H , 0, the dynamics in the phase plane sr, wd of theODE model is determined by critical point (25a), i.e., bothsolitons form a breather. For H . 0, trajectories in thephase plane sr, wd tend to critical point (25b), and a solitonY-junction is formed. The threshold between these twofixed structures is given by H ­ 0 (Ref. 35) and corre-sponds, within the IST approach, to formation of the solu-tion shown in Fig. 2(b) [see Eq. (9)]. Substituting initialconditions As0d ­ q0, vs0d ­ 1 into Hamiltonian Eq. (23),

one obtains from the condition H ­ 0 the variational es-timation to the highest value of an incident amplitude q0

for a Y-junction excitation

qsvd0 ­ 8

s5

51p

2ø 2.11 , (26)

where the superscript v stands for the variational ap-proach. One sees that the variational result approxi-mates very well an exact IST prediction [see Eq. (9)].

In order to compare more systematically the results ofthe direct, numerical solution of the NSE with the numeri-cal solution of the ODE model, I traced the evolution ofthe peak amplitudes jU sxmax, z dj and the locations 6xmax

of the peak amplitudes in the plane sx, z d for different in-cident beams’ amplitudes q0 in both methods. Straight-forward calculation shows that the maxima of the sidelobes of the trial function jU sxmax, z dj, where U sx, z d isgiven by trial-function ansatz (19), are located at

xsvdmax ­

6

24jvj4 1 sRefv2gd2 1p

jvj8 2 jvj4sRefv2gd2 1 sRefv2gd4

Refv2g

351/2

,

(27)

where the superscript (v) in xsv dmax stands for the variational

approach. The results obtained at the plane z ­ 10zF ­z0 are shown in Fig. 4.

The values of the peak amplitudes jU sxmax, z dj(the left-hand-side axis) and the ratio xsn,vd

maxyz0 [trans-verse velocities tan qsn,vd] versus the incident-beamamplitudes in both approaches are shown in the rangeof q0, where the steering effect occurs [ i.e., for q0 , qs bd

0].

For incident amplitudes below the threshold for a solitonY-function, predictions of both methods coincide. Forhigher values of q0 the variational approach predicts thatthe beam undergoes nonlinear diffraction [trajectories inthe phase space sr, wd tend to the critical point src2, wc2d],and so its amplitude decreases. In this case the IST pre-dicts that two solitons propagate bisymmetrically in non-linear medium [cf. Figs. 1 and 2(a)], and this leads to thediscrepancy in values of jU sxmax, z dj, which is clearly seenin Fig. 4. However, the variational method preservesvery well an actual value of transverse velocity tan q

over the whole range of incident amplitudes q0 shownin Fig. 4.

For the incident-beam amplitudes above the solitonbreather formation value qsv d

0 , no quantitative agreementhas been found between predictions of both methods.However, since the scope of this paper is steering of soli-ton Y-junction by HG2 beams, the analysis presented inthis section is restricted only to the case of propagation ofunbounded solitons. The detailed study of common as-pects of a linear propagation of HG2 beams and a non-linear propagation of optical solitons will be publishedelsewhere.22

5. CONCLUSIONIn conclusion, the steering properties of bright-solitonY-junctions excited by HG2 beams have been studied by

Dariusz Burak Vol. 13, No. 3 /March 1996/J. Opt. Soc. Am. B 619

IST, numerical, and variational (two types) approaches.Considered as the initial condition to the NSE, the second-order Hermite-Gaussian function excites pairs of soli-tons in a focusing Kerr medium. These solitons canform either a Y-junction or a breather, depending onthe value of the incident-beam amplitude. To analyzethe parameters of Y-junctions, I employed the resultsof the moment method in the variational formulation ofthe Zakharov–Shabat scattering problem This enabledme to predict the threshold amplitude for a Y-junctionformation and the deflection of each soliton beam for in-cident amplitude above the threshold. By using a varia-tional approximation to the beam propagation as a whole,I showed that the propagation of a soliton Y-junction anda breather correspond to two different types of trajec-tory in the phase space of variational equations. Thiswhole-beam variational approach leads to an accurateprediction of the threshold value of the incident-beam am-plitude for the soliton breather formation. The steerablesoliton Y-junction could be used for the design of photonicdevices that realize optical signal routing in communi-cation systems, which makes it important in applied aswell as in basic research.

APPENDIX AIn a linear medium, n2 ­ 0 and propagation Eq. (2) re-duces to the linear paraxial wave equation (with therescaled quantities x and z )

i≠V≠z

112

≠2V≠x2

­ 0 . (A1)

Equation (A1) possesses the following fundamental solu-tion:

V0sx, z d ­q0

vsz dexp

"2

x2

2v2sz d

#, (A2)

where the complex width is given by vsz d ­p

1 1 iz .This solution can be found, e.g., by the Fourier transformof Eq. (A1) with the initial condition given by V sx, 0d ­q0 exps2x2y2d. Assuming that the solution of Eq. (A1)has the form

Vnsx, z d ­q0

vn11sz dHen

"x

vsz d

#exp

"2

x2

2v2sz d

#, (A3)

where the functional form of Hensxyvd is initially unde-termined, and, putting it into Eq. (A1), one obtains

d2 Hensxyvddsxyvd2 2

xv

d Hensxyvddsxyvd

1 n Hensxyvd ­ 0 . (A4)

One sees that the function Hensxyvd is a Hermite polyno-mial of complex argument xyv, and it can therefore alsobe obtained from fundamental solution (A2) by means ofthe Rodriques formula,36

Hen

"x

vsz d

#­ s21dnexpfsxyvd2y2g

dn

dsxyvdn expf2sxyvd2y2g .

(A5)

Equation (A5) can be considered as a special case of amore general way of obtaining higher-order solutions of

the propagation equation, Eq. (A1). It was pointed outin Ref. 37 that since Eq. (A1) is the linear differentialequation with constant coefficients, the infinitesimal dis-placement operators ≠y≠x and ≠y≠z commute with theoperator i≠y≠z 1 1/2≠2y≠x2, and so the new solution ofEq. (A1), V 0sx, z d can be obtained from known solutionV0sx, z d by arbitrary differentiation.

The second-order solution to propagation Eq. (A1)therefore has the form

V2sx, z d ­q0

v3sz d

8<:21 1

"x

vsz d

#29=;exp

"2

x2

2v2szd

#. (A6)

Since Hermite polynomials Hensxd of real argument x[i.e., polynomials (A5) evaluated at the beam’s waist z ­0] are orthogonal with the weight function exps2x2y2d, thefollowing result holds for any solution to Eq. (A1) of thenonzero order [see Eq. (A3)]:

Z `

2`

Vnsx, 0ddx ­ 0 , n $ 1 . (A7)

ACKNOWLEDGMENTSThe author is grateful to W. Nasalski for introductionto the concept of nonlinear propagation of higher-orderHermite–Gaussians.

*Present address, Optical Sciences Center, Universityof Arizona, Tucson, Arizona 85721.

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