Generation of OrT pulses from a zero-area pulse in coherent pulsepropagation

D. J. Kaup and L. R. Scacca*Department of Physics, Clarkson College of Technology, Potsdam, New York 13676

(Received 8 March 1979)

The generation of Or pulses in coherent pulse propagation is studied for a simple initial pulseprofile consisting of two separated square pulses with one inverted to give a total pulse area of zero.When the two pulses are close together, it is found that the threshold for lossless propagation occursat an absolute area of 21'2 ir. But as the pulses become farther separated, one finds two thresholds.The first one occurs for an absolute area of about wr and the second one at about 2r. Although one canhave lossless propagation occurring for the absolute area between r and 2r, it is accomplished onlyby generating very low amplitude Or pulses. It is only after the second threshold, 2wr, is achieved thatone can expect to find moderate to large amplitude Or pulses, and these may be accompanied byseveral low amplitude Ow pulses.

The phenomenon known as self-induced transparency(SIT) was first described by McCall and Hahn, 1' 2 and has beentreated theoretically by various authors.3 - 10 In 1974, basedon the previous theoretical work on SIT, it was shown that SITwas a completely integrable system"l" 2 and that the Max-well-Bloch equations could be solved by an "inverse scatteringtransform" (1ST).13 The application of this IST to SIT waspresented in Ref. 12, along with various examples, and for amore complete description and discussion of this IST, thereader is referred to that reference.' 2 What we shall presenthere is an extension of the initial value problem consideredin Ref. 12.

When the initial profile is monotonic, except for a singleextremum, then from the McCall-Hahn area theorem,2 weknow that the number of 27r pulses which will finally propa-gate through the medium is determined only by the area 0 ofthe initial profile. Furthermore, from the IST theory, one cancalculate, from only the initial profile, what the final ampli-tude of each of these 2r pulses would be."1",2 This is done bydetermining the bound-state eigenvalues of the Zakharov-Shabat (ZS) eigenvalue equation, which is

8rvi + itvi = qv2, (la)

a97V2 - i¢V2 = -q*vl, (lb)

where v 1 and v2 are the eigenfunctions, is the eigenvalue, andq is the potential, where

q = (1/2)Ke (2)

In Eq. (2), K iS the McCall-Hahn K, and e is the electric fieldenvelope. Also, we are using the coordinates 'r t - z/c andX = z, where z and t are the usual space-time coordinates. InRef. 12, the eigenvalues of Eq. (1) were found for three dif-ferent simple initial profiles of E, one of which was the simplesquare profile shown in Fig. 1(a), and their relation to theMcCall-Hahn area theorem was discussed. And, based onthese results, we now have a good understanding of what thefinal 27r-pulse spectrum would be, which would arise fromsuch a simple initial profile.

However, there is another wide class of initial profiles aboutwhich little is known concerning their final spectrum, andthese are the "zero-area" pulses, a simple example of whichis shown in Fig. 1(b). It is this pulse profile that we shall studyhere. In doing so, we shall often compare and contrast the

224 J. Opt. Soc. Am., Vol. 70, No. 2, February 1980

results for the profile in Fig. 1(b) (the "split pulse") with thosefor the profile in Fig. 1(a) (the "square pulse"), which wasdiscussed in Ref. 12. This comparison will be physically rel-evant, because one can consider the split pulse to have beengenerated from the square pulse by simply inverting and timedelaying the last half. So, if one would have such an experi-mental option, is there any advantage or disadvantage in doingthis, and what would be the difference in the spectrum? Inorder to answer these questions, we must turn to the ISTmethod to determine the final pulse spectrum, since theMcCall-Hahn area theorem becomes trivial for zero-areapulses.

Let us note that when such a split pulse is incident on amedium, three things in general could happen. First, a certainamount of the total pulse's energy will be absorbed by themedium, analogous to the consequences of Beer's law in thelinear limit.12 Second, each half of the split pulse could havea part of itself reshaped into one or more 27r pulses. However,owing to symmetry, we shall see that this will never happenfor this symmetric split pulse. Third, owing to the non-linearities present in the medium, one could have a resonantinteraction between each half of the split pulse, resulting inone or more single losslessly propagating pulses called "07rpulses". 8' 14 (A zero-area pulse is just that, i.e., any pulse witha zero area. A 07r pulse is a member of a very special class ofzero-area pulses having an exact shape and oscillation thatallows it to propagate losslessly.) These 07r pulses willtherefore exist only when both halves of the split pulse arepresent. In the presence of only one-half of the split pulse,we in effect have only a square pulse and no such resonanceis possible, which allows only 27r pulses to be produced.

A general analysis of this problem has just recently beengiven by Hopf and Shakir.15 They have looked at the generalfeatures resulting from several different zero-area profileshapes, and our results tend to verify their general conclusions.However, by using analytical techniques instead of numericalones, we have been able to obtain several results beyond whatthey obtained. In particular, we note in general two thresh-olds for 07r-pulse production, and also the unsuspected pro-duction of low-amplitude 07r pulses, even for moderate sepa-rations. This latter would usually be undesirable in mostapplications.

As was done by Hopf and Shakir,1 5 we can determine theOr-pulse spectrum by finding the bound-state spectrum1 2 of

224

0

(a) kp

i-- 27p b -

(b)

- m -

FIG. 1. Square-pulse profile, Fig. 1(a), considered in Ref. 12, and thesplit-pulse profile considered in this paper, where q = (1/2)KE.

Eq. (1). Whereas Hopf and Shakir had to solve the eigenvalueproblem numerically, owing to the general form of their pulseprofiles, we are able to follow a much simpler procedure byrestricting our attention to only the simple split-pulse profileshown in Fig. 1(b). For this profile, there is a closed formsolution for the "transmission coefficient"'12 of Eq. (1), theroots of which for P in the upper-half P plane, will give thebound-state eigenvalues. For this profile, one can show thatthis transmission coefficient is

a(P) = f2() - e2irdg2(P), (3)

where

fU) =- eiTp[(l + t/X)e-iTpA + (1 - t/X)eilpA], (4a)2

FIG. 2. Positive real parts (K of the eigenvalue (tK) for Eq. (1) for thesplit-pulse profile shown in Fig. 1(b) when d = 0, versus the initial absolutearea 00. The heavy lines indicate the physically relevant values of (K (seetext).

split-pulse profile can generate only 0w pulses, and never any27r pulses. How we can obtain the above deduction will nowbe outlined. First, when q in Eq. (1) is real, then one can showthat the roots of a (t) = 0 for t in the upper half t plane areeither purely imaginary (OK = i'7K) or must occur in conjugatepairs,'131 6 where vK+1 = -k with OK = tK + INK and (Kid 0. Thus if any roots exist where (K # 0, then they mustoccur in these conjugate pairs. Now, one can readily verifyfrom Eqs. (3) and (4), that no roots of a(P) exist for purelyimaginary A, since then f becomes real and g imaginary, givinga(irj) to be positive definite. Thus no roots can exist for (K

= O. SO K # 0 for all roots, and all roots must then occur inthese conjugate pairs. This is basically the same result ob-tained by Hopf and Shakir15 in using numerical methods forsolving Eq. (1). However, some of their eigenvalues were soclose to the imaginary axis that they could not determine ifthe real part (K was actually zero or not. Here we have thatthe real part must definitely be nonzero. Also, we note thatin certain regions for some branches (see Figs. 2 and 7), we also

g(M) = ei7rK- (eiTpx -e-ipX)2X

(4b)

and

X2 = Q2+ -2, (5)

where Q is the maximum value of (1/2)KE. Note that f and gare even functions of X, and thus it matters not which branchwe choose for X in Eq. (5).

Now, one can find the bound-state spectrum for the split-pulse profile by simply finding the zeros of a (a) for P in theupper half tplane. Although we have now restricted our at-tention to only one profile shape, one should realize that abound-state spectrum is in general quite insensitive1 2 to theactual shape, depending mostly on macroscopic quantitiessuch as the area, absolute area, widths, separation distances,etc. Thus any results for this split-pulse profile will in generalalso be applicable to a profile of another shape, for corre-sponding values of these macroscopic quantities.

From Eqs. (3) and (4), one can readily deduce that the

225 J. Opt. Soc. Am., Vol. 70, No. 2, February 1980

6

4

2 /3

0

42

-4 | |S o.

0 4 8 12 16 2A0

FIG. 3. Imaginary parts i7K Of the eigenvalue (¢K), for Eq. (1) for thesplit-pulse profile shown in Fig. 1(b) when d = 0, versus the initial absolutearea, 00. The dashed lines show, for comparison, the similar result for thesimple square pulse shown in Fig. 1(a).

D). J. Kaup and L. R. Scacca 225

- .

- - - -

4prK ETp

4.8-

4.4-

4.0.

3.6-

3.2-

2.8

0 /p 2Tp ITp Trp

5Tp

FIG. 6. Threshold value of the initial absolute area GC vs dfor the split-pulseprofile. The value at d = 0 is /2 7r and as d - A, it approaches 7r.

Breather no. I geneous broadening. (The only effect of the inhomogeneousbroadening on 2ir or 07r pulses is to modify the propagation

I I I _- I 0 velocity.10'11"12 Thus since it has no effect on the bound-state8 12 16 20 24 28 spectrum, we shall ignore it here.) It is also known that each

FIG. 4. Incident energy (heavy parabolic curve), energy lost due to pulsereshaping (shaded region), and the losslessly propagated energy containedin each breather versus the initial absolute area 00 for the split-pulse profilewhen d = 0.

obtain (K so small that graphically, they appear to be zero.However, our numerical results, based on the above trans-mission coefficient, do verify that they are nonzero, althoughso small that one may practically consider them to be zero.

Now, it is well known9-12 that each purely imaginary rootof a (P) corresponds to a single 2wr pulse, which is

q = (1/2)KE = 2flK/cosh( 2 n1KT - UX), (6)

where u is a constantly' 2 dependent on q1 K and the inhomo-

56 4XK2E

48

40

32

24

8l1stSoliton

0 A24 8 12 16 20 24 28

FIG. 5. Figure 3 of Ref. 12 redrawn, showing the incident energy (heavyparabolic curve), energy lost due to pulse reshaping (shaded region), andthe losslessly propagation energy contained in each soliton versus the initialabsolute area 00, for the simple square profile shown in Fig. 1(a).

conjugate pair of roots corresponds to a U7r pulse, "," wflichis

1 (ilKcos( 2 ~Kr - wx)q = -Ke := otanl 1NCS2K -W7)

2 qt K cosh(277K T -U X)J (

where w and u are again constants" dependent on 71K, (K, and

the inhomogeneous broadening. Also, in the sine-Gordonlimit, this 07r pulse becomes the sine-Gordon breather.1 6

So, in contrast to the example solved in Ref. 12, consistingof the single simple square pulse in Fig. 1(a), the split-pulseprofile in Fig. 1(b) will never generate a pure 2w pulse, andinstead will generate only 07r pulses. However, as the timedelay between the individual pulses increases (d - c), thesplit pulse then corresponds to two well-separated squarepulses incident on the medium. In this limit, we would simplyexpect the energy that is losslessly propagated to be just twicethat for a single pulse of width -r and amplitude Q. However,since the first pulse does "prepare" the system, the arrival ofthe second pulse can give rise to a photon echo, and conse-quently a form of lossless propagation at lower absolute areas.This we shall point out later.

Throughout this paper, we shall ignore the effects of ho-mogeneous broadening and other relaxations. In general,these effects need to be considered only when T, or d becomeson the order of, or larger than, the characteristic times of theseeffects. Thus these results presented here are valid in generalonly for rp and d less than these values.

Our results for the split-pulse profile are as follows. In Fig.2 for d = 0 we have plotted the first five branches of the realpart of the eigenvalues (K vs the initial absolute area

00 = K f IfE(OT) dT,

which for the profile in Fig. 1(b) is

0o = 4QrP.

In Fig. 2, we have plotted only the positive values of the realpart of the zeros of a (t), (K, and since these roots occur inconjugate pairs, for every positive branch of (K, there is also

226 J. Opt. Soc. Am., Vol. 70, No. 2, February 1980

d-O

d

(8)

(9)

-

D. J. Kaup and L. R. Scacca 226

6

4.

2,

0

(a)

5

3 ' 5 (b)2

4 8 12 16 20 24 io 28

FIG. 7. Positive real part .K of the eigenvalue for the split-pulse profilewhen d = Tp [Fig. 7(a)] and when d = 4Tp [Fig. 7(b)], vs 00. The heavylines again indicate the physically relevant values of SK-

a correspondingly negative one. The significance of SK is thatit corresponds to a frequency shift12 (beat frequency) from thecentral frequency of the initial electromagnetic wave, and asseen in Eq. (7), it also corresponds to the natural oscillationfrequency of the resulting OT7 pulse, and also affects the am-plitude. In Fig. 3 we give the imaginary parts of these ei-genvalues, flK, corresponding to the first five branches shownin Fig. 2. The numbering of these branches are such that theKth branch in Fig. 2 gives (K and the same branch in Fig. 3gives rK, where the Kth eigenvalue is AK = (K + hqK. As seenfrom Eq. (7), -lK gives the width of the Ow7 pulse, and also af-fects the amplitude. Note in Fig. 3 that some of thesebranches are always negative and that some are extended intothe negative region. When rK is negative, the root of a(P)does not correspond to a 07r pulse. Only eigenvalues with i7K

> 0 correspond to 0Or pulses. We present these negativebranches here because as d becomes nonzero and large, theywill become important, as we shall see later. Returning to Fig.2, the heavy lines on the branches simply indicate the valuesof (K for which the corresponding values of '7K are positive,and therefore indicate the physically relevant parts of thebranches.

For comparison, we also show in Fig. 3 the eigenvalues (dashlines) for the square pulse [Fig. 1(a)] of width 2rp and am-plitude Q, which has been taken from Fig. 1 of Ref. 12. It isimmediately obvious that upon splitting and inverting the lasthalf of a simple square-pulse profile, the threshold initial areafor lossless propagation has been raised from 7r to 7r2/ , andthat the resulting OwT pulses have smaller values of '7K than didthe 2wr pulses from a simple square-pulse profile. This resultis a little more explicit than that obtained by Hopf andShakir,15 where they could only conclude that the thresholdinitial absolute area 00 had to be larger than about Tw. For d= 0, we find no lossless propagation until 00 is 41% larger thantheir value.

227 J. Opt. Soc. Am., Vol. 70, No. 2, February 1980

FIG. 8. Imaginary part 77K of the eigenvalue for the split-pulse profile whend = T-p, versus 00.

However, when we consider the energy that is losslesslypropagated, we find that, except for the increase in thethreshold, essentially the same amount is losslessly propa-gated in both cases. To evaluate the energy that will belosslessly propagated, we simply proceed as discussed in Sec.V of Ref. 12 and sum the imaginary parts of the eigenvalues.Since each value of '7K in Fig. 2 for the split pulse correspondsto a pair (+,K and -K), we have to double the sum in thiscase, and the result for the losslessly propagated energy isthen

8c NE =- E 77K.wx- K=1

(10)

The result for d = 0 is shown in Fig. 4, where the paraboliccurve gives the incident energy versus 00, the shaded regionrepresents the energy reversibly lost to the medium duringpulse reshaping, and the contribution of each of the first three

0 4 8 12 lb ZU do 1o

FIG. 9. Imaginary part '7K, of the eigenvalue for the split-pulse profile whend = 4T p, versus 00.

D. J. Kaup and L. R. Scacca 227

4MVE becomes approximately 2wr does a strong Or pulse start to56 p develop. And even then, it may be accompanied by several

low-amplitude Or pulses.

48 More interesting are the plots of energy losslessly propa-gated vs 00 for d = TP in Fig. 10 and for d = 4'rP in Fig. 11.What one observes is that these low-amplitude Or pulses tend

40- dp //to contribute enough energy so as to smooth out the "cusp"regions where the odd-numbered Or pulses start to contribute

32- significantly, thus giving a smoother curve for the energytransmitted versus 00.

A$ Breathef 3When d is larger than 4Tp, it becomes impractical to find24- the roots of a () numerically, owing to rapid oscillations as one

Enegy Lost to the Medium varies the parameters. However, based on the above results16- and studying the limit of a (v) as d - I, one can extrapolate

Breather m. 4 and determine what should happen. First, Figs. 2 and 7 haveindicated that as d - -, (K - 0. And for lK > 0 and fixed

8 Inc*ient Energy Breher no 1 as d - X, Eq. (3) shows that a / 2, where f is the transmis-sion coefficient for a simple square pulse of width rp [see Eq.

0 (3.2) of Ref. 12]. Thus for iUK finite, all roots of a will be4 8 12 16 20 24 28 double (a f2) and will be exactly the same as those of a

square pulse of width rp. These values of 71K are shown in Fig.FIG. 10 Incident energy (heavy parabolessly propagated energy contained 12. Looking back from Fig. 3 through Figs. 8 and 9, one canin each breather versus the initial absolute area 00 for the split-pulse profile clearly see that Fig. 12 is the limit for rlK > 0 and finite as dwhen d = TP. - O.

breathers to the losslessly propagated energy is indicated. Forcomparison, we also show the result for the square pulse in Fig.5, which is simply Fig. 3 of Ref. 12 redrawn. In order ofmagnitude, both profiles generate the same amount of losslesspropagation.

Now, let us consider what happens as we vary the time delayd. First, as d increases, the threshold value of 0o drops from7rw down to 7r, as shown in Fig. 6. This is the opposite di-rection from what one might at first suspect, because thethreshold area for a single square pulse is 7r, and therefore thethreshold area for two infinitely separated square pulsesshould be 2wr, not 7r. What is happening here will becomemore evident as we consider the eigenvalue spectrum for d= TP and 4 rP, which is shown in Figs. 7-9. In Figs. 7(a) and7(b) we give SK vs 00 for d = -r and 4-r. Again, the darkenedportions of the branches indicate the physically relevantportions of the branches where '7K > 0. Figure 8 gives U1K VSSo for d = 7p while Fig. 9 is the same for d = 4rP,. Note thatFigs. 2 and 7 show that as d increases, SK for any given branchdecreases toward zero, with all branches collapsing uniformly.Also for d = rp, the fourth branch now contributes a smallamount for 0o approximately between 12 and 20, and for d= 4-rp both the second and fourth branches are now contrib-uting, as well as the fifth branch. Since '1K is always small forthese even branches, the resulting Ow pulses will have smallamplitudes and wide widths. Also note in Figs. 3,8, and 9 thata "footlike" structure is developing on the odd branches, for'1K small and positive, as d increases. This is quite clear inFig. 9 and shows that although the threshold area is indeedapproaching wr, only low-amplitude, wide Or pulses are gen-erated until 00 t 27r, at which point the amplitude of thelargest Or pulse dramatically starts increasing. Thus formoderate to large d, we actually have two thresholds. When0 becomes approximately 7r, SIT does start to occur, but onlylow-amplitude, wide Or pulses are generated. Only when So

228 J. Opt. Soc. Am., Vol. 70, No. 2, February 1980

Now, what about these low-amplitude Or pulses?Eq. (3), assuming as d -- that we can expand AK as

AK = P~)l/d + )/d2 + . ,one can easily show that

Ad) = [K + (1/2)] r + i lnI tan[(1/4)6o] 1,

and

Ad) = (GT)/Q) cot[(1/4)o0].

According to Eqs. (11) and (12a),

From

4 8 12 16 20 24 28

FIG. 11. Incident energy (heavy parabolic curve), energy lost due to pulsereshaping (shaded region), and the losslessly propagated energy containedin each breather versus the initial absolute area 00 for the split-pulse profilewhen d = 4Tp.

I). ,J. Kaup and L. R. Scacca 228

(K f (K + 1/2)7r/d,

UK (lid) lnItan[(1/4)o0]I,

(13a)

(13b)

and thus as d - -, there is a countable infinity of these roots,all with approximately the same value of ?1K. By Eq. (13b),71K is positive only if (1 + 2n)7r < So < 2(n + 1)r for n= 0,1,2.... Looking at Figs. 7(b) and 9, we see that this is thetrend, except that whenever Oo 2(n + 1)r, the (K curvestend to "break" by an amount of 7r/d. This is not inconsistentsince Eq. (12b) shows that the approximation (11) does breakdown for 00 close to these values. Presumably, a more exactanalysis about these points would show this breaking for 00

2(n + 1) r.

Nevertheless, this crude analysis does establish that a largenumber of these very low amplitude 07r pulses will occur as d

X, and since AK 1id, we can expect the total number ofthem to approximate d/-r,, and therefore the total energypropagated losslessly by them (-qd/-r) to be finite. In fact,inspection of Figs. 4, 10, and 11 for increasing d shows that theadditional energy transmitted tends to fill in and smooth outthe cusp regions in Fig. 13 where 00 o 2(n + 1)7r. If we wouldsuperimpose Fig. 11 onto Fig. 13, one finds that the additionalenergy transmitted, beyond that transmitted by the regularlarge-amplitude 07r pulses, is exactly those heavy-shaded re-gions in Fig. 13. And the reason why this additional energyis transmitted is related to the photon echo effect. The firstof the two square pulses in Fig. 1(b), when its area is greaterthan 7r/2, prepares the system by starting the atomic polar-ization vectors to rotate, each with a different frequency, ac-cording to the inhomogeneous broadening. The impingementof the second pulse, also with an area greater than 7r/2, thencauses a large share of these polarization vectors to come backinto resonance, allowing lossless propagation, although at avery reduced amplitude and increased width, due to the largespreading in the phases of the polarization vectors, whichoccurred before the impingement of the second pulse.

In conclusion, our results agree reasonably well with thoseof Hopf and Shakir. 15 However, we note a higher value forthe threshold than what they found, and also the existence oflow-amplitude 07r pulses. These latter tend to occur for initialabsolute areas near the various thresholds. Usually, owingto their broader widths, one could expect these low-amplitudepulses to decay away (via homogeneous broadening, sponta-

0

FIG. 12. Extrapolated values of 71K VS Oo for the split-pulse profile if onewould let d-o r. The heavy line indicates where we would expect to findthe very-low-amplitude eigenvalues (see text).

229 J. Opt. Soc. Am., Vol. 70, No. 2, February 1980

4 8 12 16 20 24 28

FIG. 13. Extrapolated values for the distribution of the incident energy inthe limit of d- a. The contribution for the two breathers is taken fromFig. 12. The remaining energy has been distributed between the "zeroamplitude" breathers (heavy shading) and the energy lost due to pulse re-shaping (light shading) upon considering the trend indicated by the sequenceof Figs. 4, 10, and 11.

neous emission, etc.) faster than the moderate to large-am-plitude 07r pulses would. But, their existence for a value ofd even as small as Tp indicates that they will in general bepresent, unless the two halves of the initial pulses are veryclose together. Of course, one should realize that techniquessuch as those used by Hopf and Shakir would not necessarilyshow these features, since a bound-state spectrum does be-come increasingly more difficult to determine numericallywhen the eigenvalues approach the real axis. And one shouldnote that our results differ from theirs mainly in this low-amplitude spectrum region.

ACKNOWLEDGMENT

Research supported in part by NSF Grant No. MCS78-03979 and ONR Contract No. N00014-76-C-0867.

*Current address: Lt. Louis R. Scacca, AFWL/ALE, Kirtland AFB,New Mexico 87117.

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230 J. Opt. Soc. Am., Vol. 70, No. 2, February 1980 0030-3941/80/020230-06$00.50 � 1980 Optical Society of America 230