2000 J. Opt. Soc. Am. B/Vol. 11, No. 10/October 1994

Dynamics of optical-vortex solitonsin perturbed nonlinear media

Yijiang Chen and Javid Atai

Optical Sciences Centre, Australian National University, Canberra, Australia

Received January 10, 1994; revised manuscript received April 19, 1994

We examine the perturbation effect of loss (or gain) and two-photon absorption on the evolution of optical-vortex solitons. For a small perturbation the vortex soliton is shown to follow an adiabatic evolution. Theanalytical expression for describing the evolution is presented. We also find that the intensity-dependentloss resulting from the two-photon absorption can be counterbalanced by the constant gain (resulting fromRaman effect), to lead to stable stationary propagation of the optical-vortex solitons in the perturbed media.

INTRODUCTIONSelf-trapping of a light beam in a uniform medium has re-ceived a great deal of attention' following the pioneeringresearch of Chiao et al. in the early 1960's.' Some of theinteresting advances made on the subject recently are thetheoretical predictions4 '5 and experimental observation6

of the optical-vortex soliton in a self-defocusing Kerrmedium. Since this formation of the optical-vortex soli-ton occurs in a bulk medium, it then opens an intriguingnew research frontier and also presents exciting new op-portunities to control light with light.7

In the context of optical waveguide theory the optical-vortex soliton is a self-guided light pattern trapped in apassive, lossless, self-defocusing Kerr medium, and thisstationary propagation of the optical-vortex soliton (orthe induced graded-index optical fiber7) was shown tooccur only in the ideal nonlinear medium.5 In reality,however, the material loss is unavoidable. For example,the reported observation of the optical-vortex soliton wasconducted in absorbing liquid.6 Furthermore, to reducethe power required for soliton formation, one is temptedto use a material with higher nonlinearities that occurnear the two-photon resonance of the material.8 9 Thisenhancement of the nonlinear coefficient is usually ac-companied by two-photon absorption.'0' 2 On the otherhand, self-trapping of a vortex soliton may occur at theRaman Stokes frequency'3 ; the optical-vortex soliton willthen experience a constant gain under the condition ofstrong Raman pumping with negligible depletion.'3 Be-cause of these possible imperfections inherent in the ma-terial, it appears necessary to examine the perturbationeffects resulting from material loss, two-photon absorp-tion, and gain on the evolution of the optical-vortex soli-tons in a practical (perturbed) self-defocusing nonlinearmedium.

ANALYTICAL SOLUTION OFOPTICAL-VORTEX SOLITONS

The propagation of a light beam obeys Maxwell's equa-tions. Within this framework three trapped modalpatterns can be responsible for the formation of theoptical-vortex solitons, namely, TE, TM,3 5 and quasi-

circularly polarized HE21 or HE12 (Ref. 4) patterns. Un-der the weak guiding approximation, which is valid fora small refractive-index change induced by nonlinearity,the field of the trapped light patterns can be describedby one scalar wave equation. With the perturbationsincorporated, the scalar wave equation reads as

ae 2 e 1 e _ 2e1i + + - r _ e + y2e + (-1 + ia2)e 2e - iae

=0,. (1)

where a2 is the two-photon absorption coefficient, a > 0represents gain, and a < 0 signifies material loss; Z =z/2,8 and y 2 = k2n0

2- /82, with k being the wave number

in free space, n. the linear refractive index, and 83 thepropagation. In the absence of two-photon absorption(a 2 = 0) and gain or loss (a = 0), Eq. (1) was numericallyshown5 to admit of the optical-vortex-soliton solution, as isillustrated by the solid curve in Fig. 1. For this specificstationary solution the invariants (which are the quan-tities independent of the propagation distance z) of sys-tem (1) in the absence of the perturbations (a = a 2 = 0)are fixed constants. For example, the complementarypower

(2)Pc = f (y2- le12 )rdr

and the Hamiltonian

H = r FI de 12 le12 1 22

J o L d r r2 2 1 - y j d(3)

are two invariants. In addition to being an invariant (ora conserved quantity) of Eq. (1), Hamiltonian H is sta-tionary with respect to a small perturbation in field eabout the soliton solution. In other words, Eq. (1) witha2 = a = 0 for the soliton solution is equivalent to thevariational principle e5H = 0. This characteristic of Hprovides us a way to obtain an approximate analyti-cal expression for the optical-vortex soliton. Since theone-dimensional analog of Eq. (1) admits of a hyperbolic-tangent solution, we use

0740-3224/94/102000-04$06.00 ©1994 Optical Society of America

Y. Chen and J. Atai

Vol. 11, No. 10/October 1994/J. Opt. Soc. Am. B 2001

eY

0 2 4 6 8 10Yr

Fig. 1. Field profile of the optical-vortex soliton with the solidcurve representing the numerical solution, the dashed curvethe analytical hyperbolic-tangent approximation, and the dottedcurve the Gaussian approximation obtained from the variationalapproach.

e = A tanh(r/r 0 ) (4)

as the trial function. The boundary condition at infin-ity requires A = y (which can be varied by changing thepropagation constant /8 in y2 = k2n 2 - 82). Substitu-tion of Eq. (4) into Eq. (3) yields

H = Y2 [(1 + y2 r02 /2)(2 ln 2- 0.5)/3 - ln(yr0 /V2) + H],

(5)

where Ho is the integration constant independent ofr0 . The variation of H with respect to r gives r =1.301VF2/y. We thus have the hyperbolic-tangent ap-proximation of variation solution e = y tanh(yr/1.301V-2),which, as is shown by the dashed curve in Fig. 1, is ingood agreement with that obtained from the numericalsolution. It should be mentioned here that by numericalexperiment Swartzlander and Law5 also produced an ex-pression of a hyperbolic-tangent function to approximatethe vortex-soliton solution with width r = 1.270v'2/y,which is 2.4% smaller than that acquired by the varia-tional approach.

It is interesting to see what happens if we use theGaussian trial function e2

= y2[1 - exp(-r 2 /r02 )] to

approximate the optical-vortex-soliton solution. TheHamiltonian H for this ansatz is

H = y2(I - ln yr. + y2 r0

2/8 + H.), (6)

with I = 2 - x exp(x)/[exp(x) - 1]dx. Variation ofequation (6) now gives r = 2 /y. This approximation ofGaussian field profile is illustrated in Fig. 1 by the dot-ted curve. The agreement with the numerical solutionis acceptable, although not nearly so good as with thehyperbolic-tangent function.

The accuracy of the analytical approximations to thenumerical solution is also reflected in the invariants Pcand H. The complementary powers from the hyperbolic-tangent and the Gaussian approximations read as Pc =2.35 and Pc = 2, respectively, which are 28% and 38%smaller than the value Pc = 3.25 from the numerical solu-tion. The difference in H is virtually negligible. For ex-

ample, the H value calculated with the hyperbolic-tangent(Gaussian) approximation differs from that of the numeri-cal solution by only 0.4% (1.2%) with integration in Eq. (3)performed from r = 0 to the limit r = rmax = 11. This er-ror decreases with increasing rma.

ADIABATIC EVOLUTION OF OPTICAL-VORTEX SOLITONS IN PERTURBED MEDIA

In the unperturbed system discussed above the beamwidth r and the background amplitude A of the vor-tex soliton are constants, and so are the complemen-tary power P, and Hamiltonian H. However, in a prac-tical (perturbed) nonlinear medium all these quantities(A, r, Pc, and H) in general vary with the propagationdistance z. If we still employ the analytical expressionof hyperbolic function A(z)tanh[r/rp(z)] (as in the unper-turbed system), it can be shown from Eq. (1) that A(z) andr,(z) in the perturbed system change with the propaga-tion distance in the form of

A2(Z) A2(0)exp(2aZ) (7a)1 + A2(0)[exp(2aZ) - 1]a2 /a

rp(Z) = r,(0)exp(-aZ)x {1 + [exp(2aZ) - 1]3.385a2 /[arp 2 ()] 112 . (7b)

The corresponding H varies with Z, following H(Z) =H(0)A2 (Z)/A2 (0), whereas the complementary powerP0 (Z) = Pc(0) (which in numerical simulation may varyslightly with propagation distance owing to the devi-ation of the exact field profile from the hyperbolic-tangent function). Figure 2 compares Eq. (7) with thenumerical solution of Eq. (1) for four distinctive cases.Figure 2(a) is for the case of constant loss (2 = 0 anda = -0.01), Fig. 2(b) is for the case of two-photon ab-sorption (a 2 = 0.01 and a = 0), Fig. 2(c) is for the case ofconstant gain (a2 = 0 and a = 0.01), and Fig. 2(d) is for acombination of two-photon absorption and gain (a 2 = 0-01

and a = 0.01). In all these cases the analytical results(solid curves) are in good agreement with those from nu-merical solutions (dotted curves). Figure 3 illustratesthe field profiles of the analytical expression of Eqs. (7)(dashed curves) in comparison with those obtained by thenumerical simulation (solid curves) at distance Z = 10 fora = -0.01, a 2 = 0 in 3(a) and for a = 0, a 2 = 0.01 in 3(b).The agreement between the analytical approximation andthe numerical simulation is good. Note that in the pres-ence of loss or two-photon absorption the soliton widthincreases, whereas the soliton background decreases withthe propagation distance as shown in Figs. 2(a) and 2(b).This implies that in a practical nonlinear medium theoptical-vortex soliton induces a tapered graded-index op-tical fiber rather than a uniform fiber.7

STATIONARY SOLUTION OF OPTICAL-VORTEX SOLITON IN PERTURBED MEDIA

The presence of two-photon absorption or Raman gainalone leads to variation of the background and the widthof the optical-vortex soliton with the propagation dis-tance as shown in Figs. 2(b) and 2(c). Simultaneous

Y. Chen and J. Atai

2002 J. Opt. Soc. Am. B/Vol. 11, No. 10/October 1994

i l I I * . |.I2 4 6 8 1(

z

0 2

-V * . .

4 6 8z

balanced by the Raman effect to result in the stationarypropagation of the optical-vortex soliton in the perturbedmedium. From Eq. (1) we find that this is indeed pos-sible for any values of 2 > 0 and a > 0. Shown inFig. 4 are the intensity profiles of the optical-vortex soli-tons in the perturbed medium with a 2 = 0.02 and a/K =

0.02, 0.016, 0.012, 0.008, 0.004 associated with the valuesof y 2

= 1, 0.8, 0.6, 0.4, 0.2, where K = 0.9999 1. As isillustrated, the background intensity A 2 of the optical-vortex soliton in the perturbed medium varies with thecoefficients a 2 and a and is determined by AP,2 = a/a 2 .Here for self-trapping AP,2 # y2 , in contrast to the unper-turbed system for which the background intensity A2 =y2 . The difference between AP2 and y2 increases with in-creasing a and a 2, and AP 2 - Y

2 when a -h 0 and a 2 - -

The optical-vortex solitons exist in the perturbedmedium as demonstrated. The question left to be ad-dressed is their stability. We have performed a stan-dard linear stability analysis of the optical-vortex solitonsolution to Eq. (1) with a > 0 and a 2 > 0 and foundthat the optical-vortex solution in the perturbed media isstable to any perturbation. This stability characteristicof stationary solution in (2 + 1) dimensions of a self-defocusing medium involving two-photon absorption and

10

ITI,Y

0 2 4 z6 8 10

1.2

1.1

1.0

0.9

0.8

(d)

0 2i4 6 8 a 10z

Fig. 2. Evolution of the background intensity and width of theoptical-vortex soliton in a perturbed self-defocusing Kerr mediumfor (a) a = -0.01 and a2 = 0 for constant loss, (b) a = 0 anda2 = 0.01 for two-photon absorption, (c) a = 0.01 and a2 = 0 forconstant gain, and (d) a = 0.01 and a2 = 0.01 for a combination.The solid curves represent the analytical evaluation of Eq. (7),and the dotted curves stand for the numerical simulation ofEq. (1).

existence of both tends to cancel out and leads to a di-minished effect of attenuation or amplification as shownin Fig. 2(d). Naturally, the question arises whether thetwo-photon absorption effect can be completely counter-

(a)

It

Yr

(b)Fig. 3. Field profiles at distance Z = 10 in (a) a lossy mediuma = -0.01 and a2 = 0 and (b) a two-photon absorption mediuma = 0 and a2 = 0.01, where the'solid curves represent thenumerical simulation and the dashed curves the analytical ap-proximation.

(a)

A~ (0)

1.2

1.1

1.0

0.9

0.8

1.2

1.1 H1.0

0.9 -

0.8 -

1.2 -

1.1.-

1.0

0.9-

(b)

r (zIr (0)

- |

.. . . . . .

. , .. ,

Y. Chen and J. Atai

Vol. 11, No. 10/October 1994/J. Opt. Soc. Am. B 2003

Fig. 4. Intensity profiles of the optical-vortex soliton in theperturbed medium with a2 = 0.02 and a/K = 0.02, 0.016, 0.012,0.008, 0.004 associated with the values of 2 = 1, 0.8, 0.6,0.4, 0.2, where K = 0.9999 1.

gain contrasts with that in (1 + 1) dimensions consideredin Ref. 11, which was found to be stable to symmetric per-turbations but unstable to asymmetric perturbations.' 2

CONCLUSION

In conclusion, we have examined the dynamics of theevolution of the optical-vortex soliton in a practical self-defocusing Kerr medium. For a small perturbation theoptical-vortex soliton is shown to evolve adiabatically.

The analytical expression for describing the evolution isgiven. We also find the stationary solution of the optical-vortex soliton in the medium with two-photon absorptionand gain, and the vortex soliton in the perturbed mediumis shown to be stable.

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Y. Chen and J. Atai