1290 J. Opt. Soc. Am. B/Vol. 8, No. 6/June 1991

Spatial optical solitons in planar glass waveguides

J. S. Aitchison,* Y. Silberberg, A. M. Weiner, D. E. Leaird, M. K. Oliver, J. L. Jackel, E. M. Vogel, and P. W E. Smith

Bellcore, Red Bank, New Jersey, 07701-7040

Received August 6, 1990; accepted December 19, 1990

We report experimental studies of spatial optical solitons in a nonlinear glass waveguide. Spatial solitons areself-trapped beams that propagate without diffraction. We demonstrate the formation of spatial solitons, andwe investigate interactions between them. At high intensities two-photon absorption is observed to lead to thebreakup of solitons. A beam propagation method is used to simulate the experimental conditions numerically,and the results agree well with experiment.

1. INTRODUCTION

In the early days of nonlinear optics it was proposed thatan intensity-dependent refractive index could lead to self-trapping of optical beams.' It was soon realized that sta-ble self-trapping could occur only in a two-dimensionalmedium, in which the optical beam can diffract only in onetransverse direction. In a three-dimensional medium,in which light diffracts in two transverse directions, self-trapping is not stable and leads to catastrophic self-focusing.2 Zakharov and Shabat showed the connectionbetween self-trapping and soliton theory.3 They alsoshowed that there is a complete analogy between spatialsolitons, in which the nonlinear index profile balances dif-fraction, and temporal solitons, in which nonlinear phasemodulation balances dispersion. Temporal solitons havereceived much attention since Hasegawa and Tappert pro-posed that they could be obtained in the anomalous dis-persion regime in optical fibers and applied to high-speedcommunications. 4 Temporal solitons in optical fiberswere first observed by Mollenauer et al.,5 and dispersion-less propagation of solitons over thousands of kilometershas recently been demonstrated. 6 Spatial optical soli-tons, on the other hand, have been little studied. Todemonstrate a spatial soliton, one must limit diffractionto one transverse direction. This can be achieved in aplanar optical waveguide. A series of experiments withspatial optical solitons in a multimode liquid waveguidemade from CS2 was reported recently.7 In this paper wereport in detail our experiments with spatial solitons in asingle-mode planar glass waveguide.'

There are two families of optical solitons. The mostcommonly discussed soliton, sometimes termed a "brightsoliton," takes the form of a spatial or temporal localizedwave packet. Bright temporal solitons are obtained whenthe material dispersion is anomalous and when the solitonhas a positive nonlinear index (or alternatively, with nor-mal dispersion and a negative nonlinear index). Brightspatial solitons are obtained when the nonlinear indexis positive (note that, unlike for dispersion, the sign ofdiffraction cannot be controlled). The second family ofsolitons-dark optical solitons -is obtained when the rela-tive sign of dispersion (or diffraction) and the nonlinearindex are reversed. A dark soliton is a localized depres-sion (a dark pulse) in a cw background. Both temporal9

and spatial 0 dark solitons have been demonstrated re-cently. In this paper we will limit our discussion tobright spatial solitons.

Spatial solitons are special solutions of the nonlinearSchrodinger equation that describes the paraxial propaga-tion of an optical field in a Kerr medium:

aE a2E 2 22ik - -2k 2 JE2E,az aX2 no (1)

where E is the amplitude of the electric field, n is thelinear refractive index, n2 is the nonlinear index coeffi-cient defined by n = n + n2IE 2, and k is the wave vectorin the medium. The fundamental bright soliton propa-gating along the z axis is given by

1 nO) 1/2z= -kao n2

exp( 2 z)sech (X - xo), (2)

where a is a measure of the beamwidth. The integratedintensity over the beam is

Us = fIEI2 dx = no 2 (3)

As is mentioned above, effectively two-dimensionalpropagation can be realized in a planar waveguide. Thetransverse mode size w will not be affected by the nonlin-earity if w << a. The total power then required tolaunch a fundamental soliton is

Pt= 2nown2aok

In this paper we report our studies of spatial opticalsolitons in planar glass waveguides. In Section 2 we dis-cuss the experimental setup and our main results. InSection 3 we deal with the effect of two-photon absorptionin the waveguide and how it relates to some of our observa-tions. In Section 4 we describe experiments that demon-strate the interaction between two spatial solitons and thephase dependence of their interaction. In Section 5 wereport results in an experimental highly nonlinear glasswaveguide. We discuss the implications of our results andconclude in Section 6.

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Spatial Filter

Det 2

Fig. 1. Schematic view of the apparatus used in these experi-ments: BS's, beam splitters; M's, mirrors; Det's, detectors; BS's,beam splitters.

2. EXPERIMENTAL OBSERVATIONS OFSPATIAL SOLITONSIn these experiments we use a 5-mm-long glass planarwaveguide formed in Schott B270 glass. Although thismaterial has a relatively low value of nonlinear coeffi-cient,"1 n2 = 3.4 x 10-16 cm2 W-1, it is potentially a goodchoice because of its low loss, positive n2, and compatibil-ity with a waveguide fabrication process.'2 A potassium-sodium ion exchange was used to define a surface layerwith an increase in refractive index of 0.007 over the bulkvalue of no = 1.53. The increase in refractive index arisesbecause the K+ ion has a larger electronic polarizabilitythan the Na' ion that it replaces.'3 A second ion exchangewas then used to bury the planar waveguide: this helpedto reduce scattering resulting from surface imperfections.The resulting waveguide layer was -3-4 ,um thick.

A schematic of the experimental arrangement used toobserve spatial soliton propagation is shown in Fig. 1. Acolliding-pulse mode-locked dye laser and copper-vapor-laser pumped dye amplifier arrangement' 4 was used to ex-cite the sample. This system produced 75-fs pulses withenergies of several microjoules at a repetition rate of8.6 kHz and a wavelength of 620 nm. A prism pair (notshown) was used to compensate for pulse broadening15 re-sulting from group-velocity dispersion in the amplifiersystem and the optics before the sample. The output wasspatially filtered and then shaped by using a cylindricaltelescope to produce an elliptical beam profile with a 10:1aspect ratio. This allowed efficient coupling of a wide,-15-ttm FWHM beam into the waveguide. From Eq. (4)we note that the power required for soliton formationvaries as ao0'. Thus a wide input beam, which diffractsless than a narrow beam, will require less power to form aself-trapped channel.

The TEO mode of the waveguide was excited by end-firecoupling the shaped beam, using a 5x microscope objec-tive. The output facet was imaged onto an intermediateimage plane where an aperture could be placed to trans-mit either the center or one of the wings of the outputmode profile. This plane was in turn imaged onto a TVcamera and detector. The input intensity could be variedby a stepper-motor-driven attenuator wheel under com-puter control. The transmission of the aperture was

recorded with a chopper, a power meter, and a lock-in am-plifier; the computer causes the attenuator wheel to rotateand acquires data by digitizing the lock-in output. Detec-tor 1 monitored the total input power, detector 2 the totaloutput, and detector 3 the apertured output power.

Figure 2 shows the experimentally measured time-aver-aged spatial-mode profiles at the output of the waveguideas a function of increasing input power. At low power,when self-focusing can be neglected, the output beam hasdiffracted to -4.6 times the input width, as expected fromthe -15-pum FWHM input spot, the 5-mm sample length,and the index of 1.53. As the input power is increased,self-focusing begins to become important, and the spatialwidth of the output beam decreases until, at an inputpower level of 400 kW, the output mode has collapsed todimensions similar to those of the input spot. Thus, de-spite the fact that the data are time averaged over the bell-shaped temporal profile of the pulse, we are able toobserve propagation of a nondiffracting self-trapped beam,i.e., a spatial soliton. Using Eq. (4) with a 0 = 8.5 m(=FWHM/1.76), w = 5 rm, and A = 0.62 ,um, we ascer-tained that the power required to form a soliton with thesame width as the input beam is Pt = 230 kW As we dis-cuss below, the effect of temporal averaging accounts forthe observation that a peak power of 400 kW was observedexperimentally to give the narrowest output profile. Asthe input power was further increased we observed abroadening of the mode profile, until, at power levels of

-50 0

Position (m)

50

Fig. 2. Experimentally measured time-averaged output-modeprofiles as a function of increasing input power: a, 15-,um inputspot; b, output beam profile, P = 24 kW; c, output beam profile,P = 400 kW; d, output beam profile P = 1.25 MW

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1292 J. Opt. Soc. Am. B/Vol. 8, No. 6/June 1991

a

b

C

o 600 I200 600 120

the wings is correlated with the formation of the intensityprofiles, as shown in Fig. 2d. These features are not ex-plained by simple soliton theory, which predicts formationof high-order solitons at high intensities. Although high-order solitons are expected to exhibit multipeaked pro-files, they should be comparable in width with thecorresponding fundamental soliton. The experimentalprofiles at powers well above the soliton power are obvi-ously much wider than the fundamental soliton. FromFig. 3c we see that the broadening of the intensity profilesis accompanied by a significant amount of nonlinear loss.This loss is minimal for powers smaller than 400 kW,where the fundamental soliton is observed, but it grows toas much as 25% at the highest powers of 1.2 MW Severalphysical processes could lead to nonlinear loss; however,we believe that the most likely is two-photon absorption.Indeed, the absorption edge of the B270 glass is at320 nm, which means that the laser wavelength of 620 nmis within the two-photon absorption (TPA) band. Wetherefore investigated the effect of TPA on soliton propa-gation in order to check whether the effect could explainthe observed phenomena. In Section 3 we present themain results of this study. A more detailed report of thenumerical simulations was recently published.6

0

Peak Power (kW)Fig. 3. Experimental measurements of aperture transmissionversus input power: a, aperture positioned at the center of theoutput mode; b, aperture positioned on one wing of the output; c,total transmission of the waveguide.

1.25 MW, the output developed a definite three-peakeddistribution. In a previous publication we presented datashowing that the mechanism giving rise to the self-focus-ing is nonthermal in origin.8

In order to make a more quantitative measurement ofthe process, we monitored the transmission of an aper-ture placed in the image plane after the sample, as a func-tion of the input power. We expected to observe anincrease in transmission when the aperture was placed onthe center of the output and a decrease in transmissionwhen the aperture was placed on one of the wings. InFig. 3a we show the transmission through an apertureplaced to transmit the center of the output-mode profile.The aperture was selected so that its width was the sameas the effective FWHM of the input mode at the interme-diate image plane. In Fig. 3b we show the measuredtransmission of the aperture when it is located off to oneside of the central maximum. Finally, in Fig. 3c we showa measurement of the total transmission through thewaveguide (without aperture).

From Figs. 3a and 3b we see that initially we get theexpected response (an increase of transmission on theaxis and a decrease on the wings) consistent with the self-focusing of the beam. However, as the power is increasedabove 400 kW we see a deviation from this simple trend,as power is redirected toward the wings. The growth of

3. SOLITON PROPAGATION WITHTWO-PHOTON ABSORPTIONTPA is described by an intensity-dependent absorptioncoefficient a = a + L We measured the nonlinear ab-sorption coefficient in our sample to be f = 2.3 x10-2 cm/W This number was determined by measuringthe transmission of a straight channel waveguide as afunction of the input power. The channel waveguide al-lowed us to determine the intensity in the sample withouthaving to consider self-focusing effects. The details ofthe measurement are given in Appendix A.

TPA is known to have an important effect on nonlinear-optical devices.'7 Here we are interested in investigatingthe effect of TPA on solitons. To describe propagation ina medium with TPA, the nonlinear Schrodinger Eq. (1) ismodified to

2ik E 2E _ 2kf2n2 (1 + iK) IEI 2E, (5)daz O;x_ n

where the nonlinear absorption term K is defined by

K ='2O (6)2kn 2

The parameter K relates the nonlinear absorption to thenonlinear phase shift. In our waveguide K 0.033,which means that light at the intensity that induces aphase shift of 1 rad will suffer a loss of 3.3%. Note thatthe parameter K is smaller by a factor of 8v than a similarparameter defined in Ref. 17.

We performed numerical simulations of Eq. (5), usingthe beam propagation method (BPM).'8 The input modeis assumed to have a Gaussian spatial profile. Our simu-lations show'6 that TPA effects can be divided generallyinto two regimes, depending on input power. As long asthe launched beam is within or below the fundamentalsoliton power (that is, the power required to form a funda-

cb

C

0C,).coEco,C:

coI I

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C)CCZ,

-50 0 50Position

Fig. 4. Simulation of propagation of a beam with a Gaussianinitial profile, 15 ,um FWHM, and power of 5Pt in a waveguidewith a normalized TPA coefficient of K = 0.033. The total propa-gation distance is 5 mm.

-P=6

- P=2

- P=O-50 0 50

Position (m)

Fig. 5. Calculated intensity profile at the output of the 5-mmwaveguide for a 15-Am FWHM Gaussian input beam and for vari-ous input powers.

mental soliton equal in width to the input beam), the non-linear loss will tend to attenuate adiabatically and broadenthe beam as it propagates. However, at powers wherehigh-order solitons would normally be formed, TPA willcause a breakup of that soliton into two diverging funda-mental solitons. Figure 4 shows a simulation of propaga-tion for a beam with a Gaussian initial profile with apower of five times the fundamental soliton power. Theparameters were chosen to match our experimental condi-tions, and the total propagation distance is 5 mm.

Figure 5 shows the calculated output mode profile as afunction of increasing input power. At low levels of inputpower, diffraction dominates over self-focusing, and thebeam broadens as it propagates. As the input power isincreased toward the soliton power, the output begins tonarrow in space to dimensions similar to those of the

input. In this situation diffraction and self-focusing areequal, and the beam propagates in a stable self-trappedchannel. Above the fundamental soliton power the modebegins to broaden; this occurs when the high-intensitycenter of the mode experiences more intensity-dependentloss than the lower-intensity wings. As the input power isfurther increased, the wings of the spatial profile split offand form two fundamental soliton channels.

The results of the simulation shown in Figs. 4 and 5 arefor cw input power levels. In our experiment we observeda time-averaged output. Assuming that the responsetime of the nonlinearity is much shorter than the tempo-ral duration of the pulse, we can use the calculated cwresults to predict the time-averaged output. Figure 6ashows a Gaussian temporal and spatial beam at the input.Figure 6b shows the calculated output temporal-spatialprofile when the input peak power is 4.5 times that of thefundamental soliton power. Figure 7 shows the results ofintegration over the temporal pulse profile. It is clear thatthe effect of the temporal averaging is to produce a three-peaked distribution, as observed experimentally. Thecenter peak corresponds to the formation of the funda-mental soliton at intermediate intensity values within thetemporal pulse, while the peaks on either side occur at thehighest intensity values within the temporal pulse whenthe pulse splits into two fundamental soliton channels.

The simulation can also explain the experimental re-sults of Fig. 3. Initially the beam focuses down to formthe fundamental soliton. However, as the power is in-creased we observe a transfer of power from the center tothe wings as the beam breaks into two fundamental soli-tons. Finally, we observe another decrease at the wings(Fig. 3b) as the separation of the two fundamental solitons

b

Position

Fig. 6. Spatial-temporal profile of the a, input pulse and b, output pulse for a peak power of 4 .5Pt after propagating of the beam5 mm in the waveguide.

Ca)

Position (I m)Fig. 7. Calculated time-averaged profile for the output pulse ofFig. 6.

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C0

A ')

C')

ECCO

0 2 4 6

Peak Power P/Ps

Fig. 8. Simulation of the transmission through an aperture cen-tered on the optical axis versus the input peak power. The aper-ture width is equal to the input pulse width (FWHM).

I Q l (a) I (b)Na)a.

a

6 Position X o Position X

Fig. 9. Simulations of soliton interactions. Two fundamentalsolitons are launched in parallel. The relative phase is a, 0,resulting in an attractive force, and b, 7; resulting in a repul-sive force.

increases so that they move out of the aperture's field ofview. We have used the results of our BPM simulation tomodel the transmission of an aperture placed in the cen-ter of the output mode, as shown in Fig. 8. These resultsare in good agreement with those of Fig. 3a. From thesimulation we find that the peak in transmission occursnear 1.5Pt; this gives a measured value for the fundamen-tal soliton power of 275 kW This value is in good agree-ment with the value of Pt = 230 kW, which is obtainedfrom Eq. (4).

In our discussion so far we have neglected the possibil-ity of temporal reshaping of the pulses. We note, how-ever, that at the intensities used in our experiments thecombination of dispersion and self-phase modulation maylead to temporal broadening, up to approximately 50%broadening at the highest intensities used. This effect issmall enough not to obscure the experimental results, butit could lead to an increase in the estimated experimentalpeak powers. Note also that, because of the sign of dis-persion at 620 nm, the pulses are always broadened. If

these experiments were to be repeated at a wavelengthwhere the dispersion is anomalous, one should considerthe possibility for simultaneous spatial and temporal soli-tons. This point is discussed theoretically in a separatepublication.'9

4. INTERACTIONS BETWEENSPATIAL SOLITONSOne of the intriguing properties of solitons is the way inwhich they interact with each other. Solitons interact ina way that mimics the interaction of massive particles.For example, when two fundamental solitons are launchedparallel to each other they can attract or repel, dependingon the relative phase between them.20'2 ' This effect wasobserved experimentally with temporal solitons2 2 andmore recently with spatial solitons in bulk CS2.23Figure 9 shows numerical simulations of the two limitingcases for spatial solitons with relative phases of 0 and'rr rad. The interaction between solitons can be explainedqualitatively by considering the perturbation that one soli-ton exerts on the other. Consider a soliton that withoutperturbation propagates along the z axis. A second in-phase soliton next to it perturbs the background index so

b

C

-100 0 100

Position (4,m)Fig. 10. Experimental results of soliton interactions: (a) lightdistribution at the input of the waveguide, (b) output light distri--bution when the two beams do not overlap in time, (c) interactingsolitons with a r phase shift, (d) interacting solitons with 0 phaseshift.

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so that they do not interact. The intensity distribution isthen generated by the sum of two independent solitons.When the timing is adjusted so that the two pulses coin-cide in the waveguide, the two solitons interact. The sepa-ration between the solitons changes as a function of therelative phase between them. Figures 10c and 10d showthe two extreme cases observed. In the case of attractiveinteraction the two beams practically coalesce. Whenthey repel, the distance between them increases by 70%.Note that all the distributions shown in Fig. 10 are againtime averaged over the short laser pulse. However, theoutput profile is dominated by the high-intensity portionof the pulse.

-50 0 50

Position (G^m)

Fig. 11. Experimentally measured time-averaged output modeprofiles for the high-nonlinearity glass: a, input spot; b, low-power output mode; c, Pin = 41 kW; d, Pin = 110 kW

that effectively the first soliton rides on an index gradientincreasing toward the perturbing soliton. This gradientcauses the soliton to bend toward the perturbing neighbor.When the two solitons are out of phase, they interfere tocancel the field at the centerpoint between them. Noweach soliton rides on an index gradient that increasesaway from the other soliton, causing the solitons to moveaway from each other.

To observe interactions between spatial solitons, wemodified the experimental setup shown above. The laseroutput was split into two beams of equal intensity. Thetemporal delay of one of the beams could be varied bychanging one of the path lengths. A piezoelectric trans-ducer permitted fine adjustment of the path length andhence the relative phase between the two pulses. Thetwo beams were shaped by a cylindrical telescope into3 gm x 22 Am ellipses. The light emerging from the out-put side of the guide was imaged onto a TV camera.

In Fig. 10 we illustrate our main experimental results.A more detailed discussion of the spatial soliton interac-tion experiments will be published separately.2 4 Fig-ure 10a shows the light distribution at the input to thewaveguide. The two beams are launched parallel to eachother, and they are spaced by 45 m, -2 FWHM. Thepower is adjusted so that each beam by itself forms thenarrowest profile possible. In Fig. 10b we show the out-put distribution when the two beams are displaced in time

5. SPATIAL SOLITONS IN A HIGHNONLINEARITY GLASSIn this section we present experimental evidence of spa-tial soliton formation in a glass that was designed both tohave a large nonlinear coefficient and to be compatiblewith a waveguide fabrication technique.25 The startingmaterials for this glass were SiO2 to produce the glass ma-trix, K2 CO3 for an ion exchange to produce a waveguide,and TiO2 + Nb2O5 to increase the nonlinear refractiveindex. The transition elements Ti and Nb form highlypolarizable bonds, which lead to an increase in both thelinear and nonlinear refractive indices. The resultingglass had a value of n2 = 4 x i0' , cm2 /W which is morethan an order of magnitude larger than the value for puresilica. Waveguides were fabricated in this material byusing a potassium-thallium ion exchange and then buriedby using a second exchange.26

Experimentally measured time-averaged output modeprofiles for the high-nonlinearity glass are presented inFig. 11. The same general features are present here aswere observed in Fig. 2. The output mode narrows withincreasing input power, until its dimensions are similar tothose of the input spot. At power levels above the funda-mental soliton power the beam broadens. The measuredchanges were smaller than those observed with B270 be-cause of the shorter sample length, -4 mm, and its higherrefractive index, no = 1.88, which caused the beam to dif-fract less as it propagated across the sample. The lowersoliton power is a result of the order-of-magnitude-highervalue for the nonlinear refractive index. We have ob-served, however, that the magnitude of the TPA coeffi-cient in this sample is also higher than for the B270sample discussed in Section 4 and that nonlinear loss pro-cesses became important at much lower powers. This isconsistent with the prediction that a higher n2 is accompa-nied by a shift in the absorption edge to longer wave-lengths and an increase in the TPA coefficient. This facttogether with the higher linear loss of these waveguidescaused the less dramatic results. The problems associatedwith TPA could potentially be overcome by working at alonger wavelength, outside the TPA band. However, therehave been reports that nonlinear absorption can still occurat these longer wavelengths because the amorphous na-ture of glass results in a tail to the absorption edge. Ourdata illustrate one important problem in trying to exploithighly nonlinear materials, namely, that enhanced nonlin-earity is often accompanied by a markedly increased two-photon absorption coefficient.'7

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6. CONCLUSIONSWe have reported experimental observations of spatial op-tical solitons in a single-mode glass waveguide. The glassthat we used for these waveguides has a low value of n2 butpermits the fabrication of good-optical-quality wave-guides. At low levels of input power the beam diffracts asit propagates across the sample. As the input power isincreased, the effect of self-focusing causes the beam tocollapse into a stable channel, and the beam propagates asa soliton. At substantially higher power levels we observea decrease in the total transmission, which we attribute totwo-photon absorption. This nonlinear absorption leadsto a breakup of the spatial channel into two fundamentalsolitons. Our results correspond well to a numericalmodel that includes the effects of TPA.

We have also reported similar effects in a high nonlin-earity glass, in which n 2 is an order of magnitude largerthan that of pure SiO 2 . This increase in n2 is accompa-nied by an increase in the magnitude of the TPA coeffi-cient. Although spatial solitons are observed, thedecrease in power required to generate them is not theorder-of-magnitude-lower value expected from the order-of-magnitude-larger nonlinear index.

Finally, by launching two parallel solitons along thewaveguide we have demonstrated the interactions of twosolitons. Attraction was demonstrated for two in-phasesolitons, and repulsion for out-of-phase solitons, in agree-ment with theoretical predictions.

Spatial solitons are completely analogous to the morefamiliar temporal solitons; they are described by the sameequation, and they exhibit the same phenomena. How-ever, the propagation and properties of spatial solitonscan be explained in terms of diffraction, waveguides, andmodes-all concepts that appeal directly to physical intu-ition. In this study, for example, we have shown how theinteraction of spatial solitons can have a simple intuitivemeaning. Of course, the investigation of spatial solitonscan uncover new facts that apply to temporal solitons aswell. Here we reported the discovery of soliton breakupowing to TPA, a phenomenon that could have implicationsfor temporal solitons and soliton systems.

APPENDIX A: EXPERIMENTALDETERMINATION OF THE TWO-PHOTONABSORPTION COEFFICIENTIn this appendix we present an experimental determina-tion of the magnitude of the TPA coefficient at 620 nm.A direct measurement from our spatial soliton data iscomplicated by the changing beam size, and hence thechanging intensity across the sample. To overcome thisproblem we used rib waveguides fabricated in B270. Thesurface of the glass was patterned with photoresist; then ametal film was evaporated over this to produce a maskwith a series of stripes through which the previously dis-cussed ion exchange process could take place. The result-ing high-index channels were then buried by a second ionexchange. The waveguides were single mode in bothtransverse directions and were 9.2 mm long.

The intensity, (z), a distance z along the propagationdirection is given by27

I(z) = Io + ao exp(-aoz)a0 + /3lo[l - exp(-ceoz)]

(Al)

Assuming that there is a Gaussian variation in intensityin both the spatial and temporal domains and further as-suming that the linear and the nonlinear absorptions areboth small, we can integrate Eq. (Al) to give

T = C(1 - R)2(1 - aoL) [1 - C(1 - R)2 , ]ff ' (A2)

where T is the transmission, Pinc is the incident peakpower, L is the length of the sample, C is the coupling tothe waveguide mode, R is the reflectivity at each wave-guide interface, and Aeff is the effective-mode cross-sectional area. Plotting T as a function of the incidentpower results in a straight line with a negative slope.The TPA coefficient can be calculated from the slope byusing Eq. (A2).

We measured the TPA coefficient, at 0.62 ,um, byend-fire coupling the colliding-pulse mode-locked dye-amplifier output into a single-mode waveguide and mea-suring the total transmission as a function of input power.From the straight-line fit to the experimental data we canestimate the magnitude of the TPA coefficient. WithAeff = 4.7 x 10-7 cm2, a = 0.1 cm-', R = 0.04, andC = 0.3 we obtain a value for the TPA coefficient for B270of , = 2.3 x 10-12 cm W-'. This corresponds to a nor-malized value of TPA, used in our numerical simulations,of K = f3/2kn2 = 0.033.

*Present address, Department of Electronics and Elec-trical Engineering, University of Glasgow, Glasgow G128QQ, Scotland.

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