January 15, 1995 / Vol. 20, No. 2 / OPTICS LETTERS 133

Dark optical bullets in light self-trapping

Yijiang Chen and Javid Atai

Optical Sciences Centre, Australian National University, Canberra, ACT 0200, Australia

Received September 15, 1994

Dark optical bullets—the stationary states of pulsed (3 1 1)-dimensional dark self-trapped beams propagating inuniform self-defocusing Kerr-law media without change in space and time—are found possible in the anomalousdispersion region, and the states are shown to be stable. This contrasts with the corresponding light bulletsin self-focusing media that collapse in Kerr-law nonlinearity. On the other hand, in the normal dispersionregion the pulsed dark self-trapped beams break as a result of the combined effects of diffraction, dispersion,and nonlinearity.

Propagation of a light beam in a nonlinear mediumhas been a subject of great interest over the pastthree decades.1 – 17 In a uniform medium the non-linearity was shown to have such an effect tooffset the diffraction of a beam, leading to thestationary propagation of a cw wave.2 – 9 In anoptical fiber in which a light beam is bounded bya linear guiding structure, nonlinearity can pro-duce a balancing effect on the dispersion of apulsed beam to yield stationary propagation of thepulse.10 – 14 In most experimental situations of lightself-trapping, pulsed lasers are used as sourcesfor excitation, especially when high-power opera-tion is involved.15,16 In these circumstances a beamof light will experience both diffraction and dis-persion, in addition to the self-focusing (defocus-ing) and self-phase modulation that result fromnonlinearity. In a self-focusing uniform mediumit was shown that nonlinearity can compensatefor both diffraction and anomalous dispersion si-multaneously, resulting in stationary propagationof bright pulsed self-trapped beams (light bulletsthat evolve without change in space and time).18

In the Kerr-law medium this stationary state oflight bullets is unstable; a perturbation will leadto collapse of the light bullets.18 Nevertheless, ina saturable nonlinear medium the collapse can bebounded, and the bounding may lead to the for-mation of stable, nondiffracting, and nondispersingbright pulses.18,19

In contrast to the self-focusing nonlinearity inwhich the bright beam can be trapped, in a self-defocusing medium a dark self-trapped beam or pulsecan result when diffraction or dispersion acts sep-arately and is balanced by the nonlinearity. Thequestion that immediately follows is whether self-defocusing nonlinearity can counterbalance diffrac-tion and dispersion simultaneously to give rise tostationary propagation of pulsed self-trapped darkbeams (that evolve without change in space andtime) when a pulsed laser is involved in the ex-periment. This is the question that we addressin this Letter.

The propagation of a pulsed light beam in a uniformself-defocusing nonlinear medium is governed by thetime-dependent wave equation

0146-9592/95/020133-03$6.00/0

2ik

í

≠z1

1vg

≠

≠t

!E 1 kD

≠2E≠t2

1 =t2E

2 2k2 n2

n0jEj2E ­ 0 , (1)

where E is the slowly varying vector envelope func-tion of the transverse electric field, k is the wavenumber, vg is the group velocity, D ­ 2≠2ky≠v2

is the dispersion coefficient, n2 is the nonlinearcoefficient, n0 is the linear refractive index, and=t

2 ­ ≠2y≠x2 describes the evolution in planar ge-ometry, while =t

2 ­ ≠2y≠x2 1 ≠2y≠y2 ­ ≠2y≠r2 1

s1yrd≠y≠r 1 s1yr2d≠2y≠f2 is associated with beampropagation in cylindrical geometry. Substitut-ing T ­ st 2 zyvgd

pkyjDj A, X ­ kAx, Y ­ kAy,

Z ­ kA2z, and e ­ sp

n2yn0 yAdexpsikA2zdE intoEq. (1), we then have the normalized nonlinearSchrodinger equation

i≠e≠Z

112

√sign

≠2e≠T 2

1≠2e≠X2

1≠2e≠Y 2

!1 e 2 jej2e ­ 0

(2)

governing the beam evolution, where sign ­ 1 or21 indicates operation in the anomalous or nor-mal dispersion region. The width of the beam isgiven by w , 1yskAd in terms of the amplitudeA. Keeping only the term ≠2y≠T 2 or ≠2y≠X2 inthe parentheses reduces Eq. (2) to the one that ad-mits of a (1 1 1)-dimensional dark temporal or spa-tial soliton solution,4,5,10,12 – 14 whereas dropping theterm ≠2y≠T 2 simplifies the equation to that accept-ing the (cw) vortex soliton solution with linear orcircular polarization.6 – 8

The question to be addressed here is whether thesystem is still amenable to the soliton solution whenall three terms involving second-order differentialsaccounting for the simultaneous occurrence of the dis-persion and diffraction are present, in addition to theself-defocusing nonlinear effect. We find that, in thecase of anomalous dispersion (sign ­ 11), Eq. (1) or(2) can admit of the soliton solution as a consequenceof the balance among dispersion, diffraction, and self-defocusing nonlinearity. The polarization and theamplitude of the stationary states of the pulsed darkbeams are found to follow the form

1995 Optical Society of America

134 OPTICS LETTERS / Vol. 20, No. 2 / January 15, 1995

Fig. 1. Radial field profiles of the dark solitons in 1,2, and 3 transverse dimensions (solid curves). The an-alytical approximations for the two-dimensional (2d) andthree-dimensional (3d) cases are identified by the dashedcurves.

e ­ fscos u 1 i sin u cos fdx 6 sin u sin fy gcsrd ,(3)

where r ­ sX2 1 Y 2 1 T 2d1/2, u ­ cos21sTyrd andf ­ tan21sYyXd are the variables in spherical coordi-nates, and csrd is subject to the solution of the scalarequation

12

"d2c

dr21

sd 2 1dr

dc

dr2

sd 2 1dr2

c

#1 c 2 c3 ­ 0 ,

(4)

where d represents the transverse dimensions 1,2, and 3 that correspond to black solitons in (1 11) dimensions,4,5 to cw vortex solitons in (2 1 1)dimensions6 – 8 or (2 1 1)-dimensional spatiotemporalsolitons (planar black soliton in pulsed form), andto (3 1 1)-dimensional dark optical bullets (vortexsolitons in pulsed form), respectively. The field pro-files of the solutions csrd for the cases of d ­ 1,2, 3 are illustrated graphically in Fig. 1. With in-creasing dimensionality d, the width of the beamtends to expand. In the case of d ­ 1 the solu-tion to Eq. (4) has the exact analytical expressionc ­ tanhsrd (with r ­ Y ),4 whereas the cases of d .1 admit of only approximate analytical expressionsthat are c ­ tanhsry1.301d (with r ­

pX2 1 Y 2 ) for

d ­ 2 and tanh(ry1.664) for d ­ 3. We derive theseapproximate expressions by exploiting the variationprinciple or the invariant character of the system:

H ­Z `

0

"Édc

dr

É 2

1d 2 1

r2sc2 2 1d

1 0.5sc2 2 1d2

#rsd21ddr . (5)

The approximations for the cases of d ­ 2 and d ­ 3are identified by the dashed curves in Fig. 1, which

apparently agree well with those from numerical so-lutions.

In addition to the variation of the radial field pro-files of dark self-trapped beams with transversedimensionality, their polarization character alsovaries with dimension. The general polarizationstates of Eq. (3) for the dark optical bullets—thestationary states of the pulsed vortex self-trappedbeams—reduce to the special cases of (1 1 1)-dimensional planar black soliton when u ­ 0,4 thecw vortex soliton when u ­ py2,6 and the spatiotem-poral (or pulsed planar spatial) black soliton whenf ­ 0.

The existence of the stationary states of pulseddark self-trapped beams discussed thus far ensures anecessary condition for the observation of the steady-state evolution. The distant propagation of dark op-tical bullets in an experiment requires stability. Toreveal the stability of the dark-optical-bullet state,we conduct a linear stability analysis by substitutinge ­ fscos u 1 i sin u cos fdx 6 sin u sin fy gfcsrd 1dcsr, u, f, Zdg into Eq. (2). Linearizing, we obtaina pair of linearized equations in terms of the real andimaginary parts of the perturbations dcsr, u, f, Zd ­fusr, u, fd 1 ivsr, u, fdgexpsVZd :

Vuf ­ L0vf , Vvf ­ 2L1uf , (6a)

where

L0 ­ 212

D 2 1 1 c2 , L1 ­ L0 1 2c2 ,

D ­1r2

≠

≠r

√r2 ≠

≠r

!1

1r2 sin u

≠

≠u

√sin u

≠

≠u

!

11

r2 sin u

≠2

≠f2

for the general case of d ­ 3 in spherical coordinates,and f ­ cos u, sin u cos f, or sin u sin f. In termsof vf , the linearized equation reads

L1L0vf ­ 2V2vf . (6b)

It is straightforward to show that operator L1 hasthe lowest eigenvalue of 0.9419 (corresponding tothe symmetric eigenfunction independent of u andf). L1 is therefore positive-definite, and the inverseoperator L1

21 exists and is positive-definite. Theeigenvalue V is then either real or imaginary, andthe variational principle can be applied to yield thesmallest value V2:

V2 ­ 2minkvf j L0vf l

kvf j L121vf l

, (7)

where k f1j f2l ­R

f1f2pdV and the denominator is a

positive quantity because of the positive nature ofL1

21. The value of V2 is now decided by the valueof the numerator G ­ minkvf jL0vf l of Eq. (7). IfG , 0, real growth rate V exists, implying that thedark-optical-bullet state is unstable, whereas G .

0 indicates that all V are imaginary and that thestationary state is stable.

The minimum value of G can be shown to be equiv-alent to solving the equation20

January 15, 1995 / Vol. 20, No. 2 / OPTICS LETTERS 135

(a) (b) (c)Fig. 2. Demonstration of the breakdown of an initially excited (d ­ 2) spatiotemporal dark soliton during the evolutionin the uniform self-defocusing medium operating in the normal dispersion region (sign ­ 21) for (a) Z ­ 0, (b) Z ­ 2,and (c) Z ­ 4. In the anomalous dispersion region (sign ­ 1), the initial excitation demonstrated in (a) remainsunchanged with the propagation distance.

L0vf ­ lvf (8)

for minimum eigenvalue l determined from the con-dition of normalization kvf j vfl ­ 1 with f ­ cos u,sin u cos f, or sin u sin f. The analysis of the op-erator L0 for the specified f reveals that the minimumeigenvalue for L0 is l ­ 0. This leads to G $ 0 orV2 # 0, which indicates that the dark optical bulletsare stable and should be observed in an experiment.

Following a similar analysis, we can show that thespatiotemporal dark soliton [a planar spatial blacksoliton in pulsed form with d ­ 2 and f ­ 0 inEqs. (3) and (4)] is also stable since the operator L1

for the case of d ­ 2 is positive-definite, with thesmallest eigenvalue of 0.5655, and the correspondingleast eigenvalue for L0 that is subject to f ­ cos u

or sin u is l ­ 0. These results of a linear stabilityanalysis are consistent with those from direct numer-ical simulation of Eq. (2).

In a self-defocusing medium, pulsed (3 1 1)-dimensional or (2 1 1)-dimensional self-trappeddark soliton beams can be observed when the pulseexperiences anomalous dispersion (sign ­ 1), asdiscussed above. One might then wonder whathappens when the pulsed beam operates in the nor-mal dispersion region. In the case of normal dis-persion (sign ­ 21), there is no stationary state toEq. (1) or (2) because the pulsed beam experiencesthe self-defocusing effect spatially, but it “feels” theself-focusing or self-phase-modulation effect tempo-rally. Any well-shaped beam initially launched intothe medium (sign ­ 21) will break, such as thatillustrated in Fig. 2 for the (d ­ 2) spatiotemporalcase. Initially, a dark pulsed beam is launched intothe medium, as demonstrated in Fig. 2(a) for Z ­ 0.After the beam propagates to Z ­ 2, two humps ofquite substantial magnitude develop [Fig. 2(b)]. AtZ ­ 4 the initially well-shaped beam is completelydisintegrated [Fig. 2(c)].

In conclusion, the existence of dark opticalbullets—the stationary states of pulsed (3 1 1)-dimensional dark self-trapped beams propagatingin uniform self-defocusing Kerr-law media withoutchange in space and time—in anomalous dispersionregion has been predicted. The states are found to

be stable. This situation contrasts with that of thecorresponding light bullets in self-focusing mediathat collapse in Kerr nonlinearity.18 In the normaldispersion region, however, no stationary states arepossible, and a well-shaped input beam breaks downas a result of the combined effects of diffraction,dispersion, and nonlinearity.

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