May 15, 1998 / Vol. 23, No. 10 / OPTICS LETTERS 765

Power enhancement with super-Gaussiansliding-frequency guiding filters

Benjamin P. Luce

Center for Nonlinear Studies and Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545

Received February 17, 1998

I show with numerical simulations that higher-order sliding filters, especially super-Gaussian filters, canproduce dramatic power enhancement of optical solitons, similar to that of dispersion-managed solitons. Thisapproach should allow smaller timing jitter without sacrificing the signal-to-noise ratio. 1998 OpticalSociety of America

OCIS codes: 120.2440, 260.2030.

Recently there has been great interest in dispersion-managed soliton (DMS) systems, which were intro-duced to reduce timing jitter,1 partially owing to apower enhancement (PE) effect.2 That is, the stablepulses of DMS systems have greater intensity than apure soliton of the same pulse width for the same path-average dispersion. This greater intensity allows oneto use a smaller average dispersion, which decreasestiming jitter without sacrif icing the signal-to-noiseratio. I demonstrate here that the stable pulses ofsliding-frequency f ilter (SFF) systems, also proposed toreduce timing jitter,3 can also exhibit PE at levels thatare competitive with those of DMS systems if higher-order f ilters are used. This is especially true if thefilters are super-Gaussian. This effect should allow asimilar reduction of the dispersion, and hence of the jit-ter, without sacrificing the signal-to-noise ratio. ThePE can be attributed to the need for higher intensity tobalance the strong diffusive effect of the filters.

Nonsliding Butterworth filters were proposed as analternative to SFF transmission because a smaller ex-cess gain is required, leading to a reduction in am-plified stimulated emission.4 Super-Gaussian filterswere then proposed over Butterworth filters to im-prove soliton stability further, and it was demonstratedthat these filters could be implemented with holo-graphic fiber gratings.5 Second-order Butterworthfilters were also used effectively in a long-distanceSFF transmission experiment,6 which led to anotherstudy demonstrating that sliding Butterworth filterscan lead to significant reduction of soliton interac-tions.7 Other advantages demonstrated for nonslidinghigher-order filters, such as lower minimum requiredexcess gain,4,5 appear to survive when these f ilters areslid.7 Thus SFF with higher-order f ilters can offerseveral advantages over conventional SFF transmis-sion with Fabry–Perot etalons.

To demonstrate the robustness of the PE effectreported here, I solved both the simplif ied propagationequation (SPE) for SFF transmission8 and the damped(and undamped) nonlinear Schrodinger equation withdiscrete amplification and filtering. I call these theSPE model and the discrete model, respectively.

In standard soliton units the SPE for SFF transmis-sion with Gaussian and super-Gaussian filters is8

0146-9592/98/100765-03$15.00/0

≠u≠Z

­i2

≠2u≠T2 1 ijuj2u 1

12

∑a 2 h

µi

≠

≠T2 vf

∂2n∏u .

(1)

The parameter n, a positive integer, determines theorder of the filter. For n ­ 1, the f ilter is Gaussian,and the equations relating the nondimensional f ilterstrength h, sliding rate vf

0 (where vf0 ­ dvf ydZ),

and excess gain a to the parameters of a Fabry–Perotetalon (which have a Gaussian response at the f ilterpeaks) are given elsewhere.8

The damped nonlinear Schrodinger equation is

≠u≠Z

­i2

≠2u≠T 2 1 ijuj2u 2 gu , (2)

where g determines the loss rate. At each amplif ier,to discretely amplify and filter pulses in a mannerexactly corresponding to that of the SPE, I applied thefrequency-domain transformation

usv, Z1d ­ usv, Z2dexpΩ

12

f2g 1 a

2 hsv 2 vf d2ngZa

æ, (3)

where usv, Zd is the temporal Fourier transform ofU sT , Zd at point Z, v is the nondimensional radianfrequency, Za is the nondimensional amplif ier spacing,and Z2 and Z1 are points along the fiber just beforeand after the amplifier and the f ilter, respectively.

The simulations reported here were all carried outfor a total distance of just under 10 Mm sDZ ­ 84.4d,spanned by 39-km segments of fiber sZa ­ 0.3308d fol-lowed by ideal (noiseless) amplif iers and filters. Thefiber dispersion was chosen to be D ­ 0.55 psysnm kmd,and the fiber loss rate (when it was taken into account)was 0.21 dBykm sg ­ 2.85d. All the initial pulses cor-responded to 16-ps (FWHM) solitons, which in solitonunits corresponds to the initial condition usT , Z ­ 0d ­sechsT d.

The parameters h and vf0 were f ixed at h ­ 0.8

and vf0 ­ 0.1. These values correspond to a point

almost exactly in the middle of the stable transmission

1998 Optical Society of America

766 OPTICS LETTERS / Vol. 23, No. 10 / May 15, 1998

regime for pulses with a nondimensional pulse widthclose to 1.76 [i.e., pulses close to the form sechsT d]for Gaussian sn ­ 1d SFF filter transmission.8,9 For areason discussed below, the value of the excess gain a

was fixed at hy3 1 9s2vf0d2ys16hd ­ 0.2948 for some

of the data reported here, which happens to be theoptimal value of a for Gaussian SFF transmissiondetermined analytically.3,9,10

To calculate the PE in a simple manner, I tookadvantage of the fact that the nondimensional energyis given by Z 1`

2`

juj2dT ­ 2d2yb (4)

for a pulse with amplitude juj ­ d sechsbT d. Hence,if d is found by measurement of the peak amplitudeof u, and b is calculated from the measured FWHMpulse width Ts as b ­ 1.76yTs, the PE factor for a pulsewith profile d sechsbT d over a soliton pulse with profileb sechsbT d is

PE ­Z 1`

2`

jd sechsbT dj2dT∑Z 1`

2`

jb sechsbT dj2dT∏

21

­ d2yb2. (5)

My reason for introducing this estimate is its ease ofapplication to both numerical and experimental dataand its tendency not to include dispersive energy farfrom the pulse peak.

Figures 1(a) and 1(b) show the simulated evolutionof juj2 with n ­ 2 and an excess gain of a ­ 0.2948 forthe SPE and the discrete models (with zero losses inthe later case: g ­ 0), respectively. The pulses wereplotted immediately after every tenth amplifier, andthe temporal displacements of pulses that were due tosliding were removed. Growth in both the amplitudeand the pulse width of the pulses can be seen, clearlydemonstrating PE. Note that this is the opposite ofwhat would occur for a pure soliton pulse, for whichan increase in amplitude would be accompanied by adecrease in pulse width.

Figures 2(a) and 2(b) show the numerically mea-sured values of d and b and the PE given by Eq. (5)immediately after each amplif ier for the same simu-lations. At Z ­ 0 both d and b are equal to 1, cor-responding to a pure soliton with amplitude sechsT d.After the stable pulses emerge it can be seen that thepulse width (given by 1.76yb) is slightly larger than theinitial pulse width (i.e., b , 1) and that the peak am-plitude grew considerably, leading to an estimated PEof ,3, which is already comparable with that achievedby DMS systems.2 Note that, like the power-enhancedpulses in DMS systems, these pulses are quite far fromconventional solitons in their shape. In both cases,from the point of view of the spectral problem as-sociated with the unperturbed nonlinear Schrodingerequation, there is a large amount of dispersive waveenergy that is bound up into the pulse.

Figure 3 gives the main result of this Letter. Thisfigure shows the segment-averaged PE of the emer-gent stable pulses for n ­ 1 5, with the excess gainfixed at a ­ 0.2948 for both the SPE model (for which

the PE factor is constant and equal to its average) andthe discrete model (both with and without losses takeninto account). In the SPE case it can be seen that thePE grows rapidly and quite linearly with the filter-order parameter n, whereas the discrete model showssignificant but lesser PE, with the discrepancy widen-ing as n increases. This discrepancy is expected, be-cause pulses have more opportunity to relax towardpure solitons in the discrete model. It can also beseen that the inclusion of losses has virtually no ef-fect on the PE. Notice also that there is already sig-nificant PE s,1.5d for n ­ 1, which agrees well withthe value of 1.48 that is obtained for these parame-ters from an analytical prediction for nonsliding Gauss-ian filters.3 Unfortunately, this prediction is diff icultto generalize to n . 1. If a Fabry–Perot etalon filteris used, the PE for n ­ 1 drops to a few percent be-cause this f ilter has a weaker effect on the pulse spec-tral wings than Gaussian filters.

Fig. 1. Evolution of the intensity profiles juj2 for super-Gaussian filters with n ­ 2, with temporal displacementsowing to filter sliding removed, for (a) the SPE model and(b) the discrete model.

Fig. 2. Evolution of the pulse parameters d and b and thePE for super-Gaussian f ilters with n ­ 2 for (a) the SPEmodel and (b) the discrete model.

May 15, 1998 / Vol. 23, No. 10 / OPTICS LETTERS 767

Fig. 3. PE obtained with super-Gaussian filters with n ­1 5 for (circles) the SPE model, (squares) the discretemodel without losses, and (asterisks) the discrete modelwith losses.

Table 1. Minimum Values of a and PEfor the SPE Model with Initial Condition

sechsssT ddd at h5 0.8 and vf05 0.1

n amin PE

1 0.23 1.62 0.19 2.73 0.18 3.64 0.20 4.65 0.21 5.6

I also carried out simulations (not described indetail here) with higher-order f ilters modeled on theFabry–Perot transfer functions. These simulationsstill tended to exhibit substantial but smaller PE thansuper-Gaussian filters with the same curvature at thetransmission peak, presumably because they have aweaker effect on pulse spectral wings.

The excess gain value of a ­ 0.2948 used togenerate the data discussed above is probably notexactly optimal for n . 1. For these runs I held aconstant to display the apparently linear dependenceof the PE on n (in the SPE case) when all other parame-ters are f ixed and to demonstrate that super-Gaussianfilters produce much greater PE than do Gaussian

filters for the same excess gain. But I stress thatthe PE is not simply an artifact of my having chosenan excessively high excess gain. Table 1 shows theapproximate minimum value of a for n ­ 1 5, h ­ 0.8,and vf

0 ­ 0.1, required to just barely produce astable pulse from the initial condition sechsT d in theSPE model (to within ,6 0.005). These values of a

vary remarkably little with n. Table 1 also shows thecorresponding minimum values of PE, which indicatethat significant PE still occurs even at minimum gain.The linear dependence of the PE on n can also beroughly seen here. Table 1 also illustrates that theminimum excess gain for super-Gaussian filters wasslightly less than that for Gaussian filters, consistentwith previous studies.4,5,7

Recent experiments suggest11 that the best systemsmay consist of a combination of DMS and SFF tech-niques. In such hybrids the PE should probably beattributed to both the SFF and the dispersion man-agement, especially if the sliding rate and the f ilterstrengths are strong or the filters are super-Gaussian.

I thank Taras Lakoba, Sergey Burtsev, and RobertoCamassa for useful discussion. I also thank the refer-ees for some useful suggestions. This study was per-formed under the auspices of the U.S. Department ofEnergy under contract W-7405-ENG-36 and AppliedMathematical Sciences contract KC-07-01-01.

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