December 15, 1991 / Vol. 16, No. 24 / OPTICS LETTERS

Absorption and amplification of dark solitons

Yijiang Chen and Javid Atai

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

Received July 22, 1991

Propagation of dark solitons in nonlinear media that include gain and loss is investigated. Two-photon absorp-

tion is shown to lead to dark-soliton broadening and attenuation, whereas gain (such as that arising from

stimulated Raman scattering) is shown to lead to their narrowing and amplification. The relations describing

the adiabatic evolution of dark solitons in the weakly perturbed media are presented. In comparison with fun-

damental bright solitons, dark solitons are shown to be less sensitive to the perturbations. Stationary propaga-

tion of dark solitons is also found to be possible in the presence of both absorption and amplification, but it

differs from that in the passive lossless nonlinear media.

Optical solitons have been investigated extensivelyover the past decades, largely because of their poten-tial applications in optical processing, switching,and optical transmission.' Similar to their brightcounterpart, dark solitons have two varieties, tem-poral and spatial. Temporal dark solitons refer tostable distortionless dark pulses propagating in thepositive group-velocity dispersion (GVD) regionwith GVD balanced by nonlinear phase modula-tion, whereas spatial dark solitons denote stableself-guided dark beams traveling in uniform self-defocusing nonlinear media. Despite differentphysical origins, the mathematical identities ofthese two varieties are the same-they are station-ary solutions to the two-dimensional nonlinearSchrodinger equation with passive, lossless uniformself-defocusing nonlinearity. In practice, however,material loss may be unavoidable, especially whennonlinearity is associated with two-photon absorp-tion.2 5 For example, the first experimental obser-vation of the fundamental dark spatial soliton wasconducted in the material ZnSe, which has an in-stantaneous defocusing nonlinear component ofthe nonlinear refractive index at wavelength A =532 nm owing to the dispersive change associatedwith two-photon absorption.5'6 On the other hand,self-trapping may occur at Raman Stokes frequen-cies7'8; the self-guided dark beams (or dark pulses)will then experience a constant gain under condi-tions of strong Raman pumping with negligibledepletion. Accordingly, it appears necessary toexamine the effect of two-photon absorption andRaman amplification on propagation of dark soli-tons to know how they evolve in the presence of gainand loss as well as the effect of the perturbations incomparison with their bright counterpart.

The propagation of a light beams is governed byMaxwell's equations, which for a TE wave reduce tothe scalar wave equation. In the normalized quan-tities of the field e [related to the actual electricfield E by e = G/2E exp(-iknoz)], the transversedistance X = kV\/ox, and the longitudinal dis-tance Z = 0.5kz, the equation reads

Oe 1 a2 2e 2

L + - aX2±+(-1 +ia2)le e-iae =0 (1)

which corresponds to the refractive-index profile ofa medium

n2 = no 2 _ non2jEj + ia2non 2lEI 2_ ian o , (2)

where no is the linear refractive index of the medium,k is the wave number in free space, the second andthird terms on the right-hand side of the equationaccount for self-defocusing and two-photon absorp-tion, respectively, and the term involving a repre-sents Raman gain contribution with no depletion.In going from Maxwell's equations to Eq. (1), wehave left out the term concerning the second deriva-tive with respect to z on the basis of the slowly vary-ing approximation. Although Eq. (1) is derived forspatial dark solitons, it is also valid for temporaldark solitons with a modified interpretation of nota-tions. For temporal dark solitons, for example, thenormalized spatial coordinate X is replaced by thenormalized time coordinate T, and e is then inter-preted as the normalized pulse amplitudes.

In the ideal case of a = a 2 = 0, as is well known,Eq. (1) admits stationary solutions-dark-solitonsolutions,'

e = A tanh(AX)exp(-iA 2 Z), (3)

where A is an arbitrary constant and the power ofthe beam P = f IEl2dX is an invariant (a divergentquantity for dark solitons). However, when a per-turbation is imposed by the presence of the ab-sorption or gain, stationary solutions of Eq. (1) areno longer possible. The intensity of a beam thenvaries along the propagation distance. The powerof the beam now changes down the propagationdirection. From Eq. (1) it follows that

1 dP = Crp _ a2f JeJ dX.2 dZ n2il~X

(4)

Note here that dP/dZ - 0 when n2 aP - a2 f tel 4dX.This implies the possibility of stationary solutions of

0146-9592/91/241933-03$5.00/0 © 1991 Optical Society of America

1933

1934 OPTICS LETTERS / Vol. 16, No. 24 / December 15, 1991

0 2 4 z 6 8 10 0 2 4 6 8 10 0 2 4 6 8 10

(a) (b) (c)

Fig. 1. Evolution of the dark soliton amplitude A(Z) andwidth W(Z) along the normalized distance Z for (a)a = 0.01 and a 2 = 0 for Raman gain, (b) a = 0 anda2 = 0.01 for two-photon absorption, and (c) a = a 2 = 0.01for a combination of both. The solid curves depict numer-ical results of Eq. (1), and the dotted curves represent theanalytical evaluation of Eqs. (6). Note that in (c) theRaman gain is not completely counterbalanced by two-photon absorption although a = a 2 . This is because a isthe coefficient for constant gain, whereas a 2 is the coeffi-cient for the intensity-dependent loss.

Eq. (1) when both absorption and gain are presentsuch that the condition dP/dZ - 0 holds over thewhole propagation distance. We explore this possi-bility below.

First we examine the evolution of a dark solitonwhen it is initially launched into the medium givenby Eq. (1) to address a practical question of how adark soliton evolves when a nonlinear medium isperturbed by the absorption or/and the gain, as thisis often the case in practice.5 6 We solve Eq. (1) byusing the beam-propagation method. It is foundthat a dark soliton can indeed evolve adiabatically inthe perturbed media when the absorption or/andloss are weak, meaning that a dark soliton is stableenough to catch up with a change of the inducedguiding structure due to the presence of small gainor/and loss, whereas a deformation results whenperturbations are large. Accordingly, for a smallperturbation, we assume the field profile of anadiabatically evolving dark soliton to be

e(X,Z) = A(Z)tanh[X/W(Z)]exp -i I A2(Z)dZ].

(5)

This equation, substituted into Eq. (4), immediatelyyields the amplitude A(Z),

A(Z)- =A(O)exp(aZ){1 + A2(0) [exp(2aZ) - 1]a 2/Ta}1/2

and the width,

W(Z) = 1/A(Z),

(6a)

width narrowing at the rate of exp(±2aZ).7 -9However, here for a dark soliton, we find that theamplitude increases and the width W shrinks at therate of exp(±aZ), much slower than that for abright soliton. Similarly, in a medium with onlytwo-photon absorption present, Eq. (6c) shows thatan adiabatically evolving dark soliton expandsand its amplitude decreases at the rate of [1 +A2 (0)2a2Z] ±1/2 slower than that of its bright counter-part, the width and amplitude of which vary at therate of [1 + A2(0)2.7a2Z] ±1/2.4 This indicates that adark soliton is less sensitive to small perturbationsthan a bright soliton, i.e., the former is more stablethan the latter. The physical mechanism behindthis difference is the exaggerated self-focusing ordiffraction. Introduction of gain and/or absorptioninto the medium breaks the balance between self-focusing and diffraction for stationary propagationof a dark soliton. Gain gives preference to self-focusing, whereas absorption or loss leads to a pre-dominant effect of diffraction.

To address the question of the accuracy of the ap-proximation, we compare the analytical relations (6)with numerical solutions of Eq. (1). Figure 1(a) isfor the case of Raman amplification, Fig. 1(b) for thecase of two-photon absorption, and Fig. 1(c) for acombination of both, a = a2. In all three cases, theamplitude of adiabatically evolving solitons obtainedfrom the analytical estimation (represented by thedotted curves in the figure) agrees well with thatobtained from numerical solutions (signified by thesolid curves in the figure). However, the approxima-tion for the width of Eq. (6b) slightly overestimatesthe expansion of dark solitons traveling in themedium associated with two-photon absorption[Fig. 1(b)] or the narrowing of the dark solitonspropagating in the active medium [Fig. 1(a)]. Theinteresting case is in Fig. 1(c), where according toEqs. (6) the beamwidth is predicted to remain un-changed [as indicated by the dotted line at W(Z) = 1]while the numerical solution shows a slight variationof the width near W(Z) = W(0) = 1 during the evolu-tion, i.e., intensity-dependent two-photon absorp-tion is almost balanced by constant Raman gain inthis case.

30

20

(6b) eof adiabatically evolving dark solitons, provided thatW(0) = 1/A(0), so that initially a dark soliton isexcited. In the absence of gain, Eq. (6a) reduces to

A(Z) = A(0)/[1 + A2 (0)2a 2Z]1/2, (6c)

but when two-photon absorption disappears, it sim-plifies to

A(Z) = A(0)exp(aZ). (6d)

Recall that a bright soliton evolves in an activemedium with its amplitude increasing and beam-

10

0

1.2

1.0

0.8

: 0.6

0.4

0.2

0.00 2 4 6 8 10 0 2 4 6

a 2 a28 10

(a) (b)

Fig. 2. Variation of (a) the propagation constant f8 and(b) the width W = 1/v of stationary solitary waves in self-defocusing media that include two-photon absorption andgain versus the absorption coefficient a

2 for the amplitudeA = \/a/a2 = 0.5, 1, and 1.5.

December 15, 1991 / Vol. 16, No. 24 / OPTICS LETTERS 1935

The presence of the absorption or Raman gainalone leads to decay or amplification of dark solitons.The simultaneous existence of both tends to counter-balance each other, which results in decreasingeffects of attenuation or amplification as shown inFig. 1(c). Naturally, a question then arises as towhether two-photon absorption can be completelycounteracted by gain to lead to stationary propaga-tion of dark solitary waves in the perturbed media.From Eq. (1), we find that this is indeed possible.The stationary propagation in this case follows

e(X, Z) = A tanh(vX)exp{i[13Z + y ln sech(vX)]},(7a)

where A = /-a-, v = \A, = 2a/3y, and

y = 1.5/a2 - 9/4a22 + 2, (7b)

which can readily be verified by direct substitutionof Eqs. (7) into Eq. (1). Also, direct insertion ofEqs. (7) into Eq. (4) yields dP/dZ - 0, which is thenecessary condition for stationary propagation.Figure 2 illustrates the width W = 1/v and thepropagation constant P3 versus the absorption coeffi-cient 2 for various amplitude values of A = =

0.5, 1, and 1.5. For a given A, the absolute value ofthe propagation constant 1,/1 increases with increas-ing absorption coefficient a 2 while the width W =1/v decreases with increasing a°2 . On the other hand,1,31 increases but W = 1/v diminishes with increas-ing amplitude A for a given absorption coefficient a 2.

In summary, it is shown that a dark soliton under-goes an adiabatic expansion and attenuation while

propagating in a self-defocusing nonlinear mediumassociated with two-photon absorption, but it experi-ences narrowing and amplification while travelingin an active self-defocusing nonlinear medium. Ex-pressions describing the adiabatic evolution underthe weak perturbations are given. Compared withbright solitons, dark solitons are shown to be lesssensitive to the perturbations, indicating that darksolitons are more stable than their bright counter-part. The exact solutions depicting the stationarypropagation of dark solitary waves in self-defocusingmedia that include two-photon absorption andRaman gain are also found and presented.

References

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