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Influence of pulse shape and frequency chirp on stability of optical solitons
1 October 2001
Optics Communications 197 (2001) 491±500
www.elsevier.com/locate/optcom
In¯uence of pulse shape and frequency chirp on stability of optical solitons M. Klaus a, J.K. Shaw a,b,* a
b
Department of Mathematics, Virginia Tech, Blacksburg VA 24061-0123, USA Department of Electrical and Computer Engineering, Virginia Tech, Blacksburg VA 24061-0111, USA Received 9 April 2001; received in revised form 13 July 2001; accepted 19 July 2001
Abstract We examine conditions under which certain combinations of initial pulse shape and chirp, or phase modulation, destabilize solitons in optical ®bers. Destabilization occurs when eigenvalues (EVs) of an associated Zakharov±Shabat system, which move along the positive imaginary axis with increasing chirp parameter C, either are absorbed into the lower half plane or collide with another EV. In either the absorption or collision case the corresponding soliton, which is a solution of the nonlinear Schr odinger equation with constant or periodic amplitude as a function of propagation distance, becomes unstable. We have observed for the ®rst time the emergence of an EV from the lower half plane that pursues, and collides with, an existing EV. We identify several properties of general EV evolution, as C varies, and give a heuristic criterion under which initial pulses of a certain shape experience EV absorptions only, with no collisions. Ó 2001 Elsevier Science B.V. All rights reserved. Keywords: Optical ®ber solitons; Nonlinear Schr odinger equation; Zakharov±Shabat systems; Eigenvalue collisions
1. Introduction In simplest form optical solitons are governed by the nonlinear Schr odinger equation [1], jwz b2 =2wtt
cP0 jwj2 w;
w 0; 0 1;
1
where w z; t is the slowly varying ®eld envelope of the pulse, t is local time, z is propagation length, b2 is the ®ber dispersion constant, c is the nonlinearity constant and P0 is peak power. Subscripts z and t in Eq. (1) denote partial dierentiation. It is * Corresponding author. Address: Department of Mathematics, Virginia Tech, Blacksburg VA 24061-0123, USA. Tel.: +1-540-231-5345; fax: +1-540-231-3362. E-mail address: [email protected] (J.K. Shaw).
customary to write Eq. (1) in units of dimensionless length f [1], jUf sign b2 =2Utt
2
N 2 jU j U ;
U 0; 0 1; 2
N being a normalized nonlinearity constant. When b2 < 0 (``anomalous'' dispersion) and N 1, then U f; t sech t exp jf=2 is the fundamental soliton solution of Eq. (2) whose amplitude is independent of normalized distance f. For integers N > 1, Eq. (2) supports higher order solitons which are periodic in f. Hereafter we assume b2 < 0 in Eq. (2). Of course, Eqs. (1) and (2) do not take into account ®ber attenuation, noise, higher order dispersion or higher order nonlinearities. However,
0030-4018/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 ( 0 1 ) 0 1 4 6 1 - 4
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M. Klaus, J.K. Shaw / Optics Communications 197 (2001) 491±500
broadband ®ber ampli®ers and various methods of dispersion compensation can partially overcome these eects, at least to the extent that Eq. (2) remains a useful design tool [1] for realistic ®ber communication systems, in which ``dispersionmanaged'' solitons constitute a principal transmission format. Solitons are quite stable relative to initial pulse shape in the sense that the envelope U 0; t does not have to be exactly sech t in order for a soliton to form. For example, if N is not an integer then the solution of Eq. (2) with initial pulse shape U 0; t N0 sech t, N0 > 1=2, evolves with increasing f into a ``pure'' soliton corresponding to the integer N nearest to N0 [2]. In fact, the initial shape does not have to be a hyperbolic secant (or a higher order analog) in order for U f; t to approach a soliton in the limit of large f [1]. Stability with respect to initial phase has also been considered. At issue here is the extent to which an initial ``chirp'', or phase modulation, present in the form of a real valued function p t in the initial condition U 0; t q t exp jp t;
3
can eect soliton evolution in Eq. (2). In Eq. (3) q t is a real valued pulse envelope which we will assume to be evenly symmetric, q t q t: As to the form of p t, the most important case is that of ``linear chirp'', where p t Ct2 in Eq. (3) and C is a controllable constant. The reason is that unchirped (C 0) Gaussian pulses launched under the theoretically linear (hence the term ``linear'' chirp) condition cP0 0 in Eq. (1) develop a phase modulation that is quadratic in time [1]. Thus a pulse can be ``prechirped'' by passing it through a length of dispersive ®ber so as to endow it with an approximately quadratic phase. It is known that overchirping can completely destroy the soliton character of a pulse that has perfect initial conditions except for its chirp [3±5]. That is, the solution to Eq. (2) with initial condition U 0; t N sech t exp jCt2 will not evolve with increasing f into a soliton at all, for any N, if jCj is suciently large.
But in this same connection optical prechirping has recently emerged as a critical design tool in ®ber optic systems [6,7], especially with the increased signi®cance of the chirped return to zero transmission format [8]. For some situations the chirp can be severe [9]. However, the stability of solitons relative to initial chirp has been considered for only a few dierent envelopes q t and phase functions p t. For this reason it is appropriate to examine the issue in a more comprehensive way than previously. We will show that chirp can destroy the soliton character of Eq. (2) in two distinct ways which are determined by the behavior of eigenvalues (EVs) of the Zakharov±Shabat (Z±S) system associated with Eq. (2) in inverse scattering theory (Eq. (5) below). This classi®cation was also found in Ref. [10] for the special cases where q t could be either Gaussian or hyperbolic secant and p t could be either quadratic Ct2 or one of the forms p t C sech t, p t C ln cosh bt, where b is a constant. In agreement with Ref. [10] we ®nd that EVs n n C of Z±S system move with increasing C along the positive imaginary axis and for some threshold chirp C Cth are either absorbed into the lower half plane, in one scenario, or collide with another EV in the other. In either case the soliton characteristics of Eq. (2) are altered, possibly destroyed, for C > Cth . In this paper we extend the classi®cation to cover pulse envelopes q t which can be rectangular, super-Gaussian (see Eq. (11) below) or piecewise linear shapes. Our main goals have been to establish, at least heuristically, qualitative features of pulses that do or do not induce collisions and to shed light on the overall EV evolution. In fact, by considering a much wider class of pulse shapes than earlier accounts we have discovered an interesting phenomenon in which collisions occur for centrally ¯at, steep-edged envelopes q t but are absorbed otherwise. Moreover, not only can existing EVs move with increasing C toward each other on the imaginary axis and collide; EVs can appear from the lower half plane, chase down and then collide with an existing EV. This curious behavior actually seems to be the rule, rather than exception, but no report of the phenomenon has been given in prior writ-
M. Klaus, J.K. Shaw / Optics Communications 197 (2001) 491±500
ing. In further regard to the literature, there are a few ambiguous points in some of the published papers on this subject, which we try to clarify. In Section 2 we brie¯y cover background material on Z±S systems, summarize the relevant behavior of Z±S EVs and present our ®ndings. Section 3 discusses some further details of EV evolution and other pertinent literature. The paper ends with some conclusions in Section 4. 2. Zakharov±Shabat systems The parameter N can be removed from Eq. (2) by the substitution u f; t N U f; t [1], which leads to the form (recalling b2 < 0) 2
juf 1=2utt juj u 0;
4
this is the version of Eq. (2) typically encountered in soliton theory [11,12]. In the inverse scattering method [11] the Z±S EV problem 0 m1 t jnm1 t u f; tm2 t; 5 m02 t jnm2 t u f; t m1 t; is associated with Eq. (4), where t is the independent variable, f is held ®xed but arbitrary, n is the EV parameter and the asterisk denotes the complex conjugate. To brie¯y recall the main points of inverse scattering [11] relative to Eq. (4), discrete EVs nk of Eq. (5) correspond to solitons of Eq. (4) and are independent of f; i.e., solitons are determined solely by u 0; t. The approach in this paper therefore is to set u 0; t q t exp jp t
6
in Eq. (4) and to investigate the in¯uence of the real functions q t and p t on the EVs of Eq. (5) corresponding to u 0; t. For reasons mentioned above we are mainly interested in the quadratic chirp case p t Ct2 . The real and imaginary parts of the nk correspond to relative soliton velocities and amplitudes, respectively [12]. Generically (but not always) the EVs lie on the imaginary axis [1,11], in which case eigenfunctions associated with two distinct EVs have equal velocities and can combine and form a higher order soliton. However, a pair of EVs with unequal real parts gives rise to solitons which
493
travel with dierent velocities and separate as f increases. In the collision events mentioned in Section 1, we will show that EVs nk C move towards each other on the imaginary axis with increasing C, collide and split at C Cth , with one EV going into each of the left and right half planes. Thus a collision represents the break-up or destabilization of a soliton. In the alternate scenario, EVs nk C for increasing C are sequentially absorbed at distinct threshold chirp levels. After an absorption Eq. (4) can support a soliton whose order has been decreased by 1 [10]; a fundamental (order 1) soliton thus is destroyed by an absorption. We will show that it is possible for both scenarios to occur for the same potential. Following Ref. [11] we will work with upper half plane EVs, Im n > 0, of Eq. (5) and assume that u 0; t ! 0 at least exponentially fast as jtj ! 1. In the scattering theory of Eq. (5) it is shown that vector Jost solutions w n; t and u n; t exist and satisfy w n; t u n; t
0 1 1 0
exp jnt;
t ! 1;
exp jnt;
t!
1; Im n > 0: 7
Since EVs are determined by u 0; t, we will henceforth set f 0. When the dependence on the EV parameter in Eq. (5) is important we will write m n; t for the vector m t. An EV n of Eq. (5) is then a complex number, Im n > 0, such that m n; t of Eq. (5) is a multiple of w n; t, as well as multiple of u n; t, and continuous for all t. In the case where u 0; t has compact support, say u 0; t 0 for jtj > d > 0, Eq. (7) yields boundary conditions for m n; t in the form m1 d m2 d 0; m1 d exp jnd;
m
m1 : m2
8
To remove the exponential in Eq. (8) it is convenient to introduce exp jnt 0 m n; t w n; t; 0 exp jnt
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so that Eq. (8) becomes w1 d w2 d 0;
w1 d 1;
w
w1 : w2 9
In view of Eq. (6) the w system satis®es w01 q t exp jp t exp 2jntw2 ;
w02
q t exp jp t exp 2jntw1 :
10
In our numerical EV calculations we solve Eq. (10) subject to initial conditions w1 d 1, w2 d 0 and ®nd EVs by applying the shooting method with respect to w1 d 0 at t d. In fact, even for the cases where q t does not have compact support, but decays at least exponentially as jtj ! 1, we use the same technique by choosing d large enough that q t is negligibly small for jtj > d. We tested our numerical code on the chirped rectangular pulse of constant height H for d < t < d (q t 0 otherwise) and p t Ct2 , since the characteristic equation is available in analytical form in this case. For C 0 the characteristic p p equation is tan 2d n2 H 2 jn 1 n2 H 2 [13] and for general C it can be expressed in terms of con¯uent hypergeometric functions; see Section 3, remark (vii). For the rectangular pulse we found essentially no numerical dierence between the EVs as calculated by the analytical and shooting methods. We now summarize our ®ndings for various pulse shapes. 2.1. Rectangular pulse To illustrate this case we calculated the EVs of Eq. (5) for initially rectangular pulses of total width 2 (d 1 in Eq. (8)), various heights and quadratic chirp p t Ct2 . All EVs are purely imaginary for C 0. The results are illustrated in Fig. 1 where the heights are H 5 in (a), H 4 in (b), H 3 in (c) and H 2 in (d). Plotted are the imaginary parts Im nk C of EVs as functions of chirp C P 0, and in the successive cases the number of initial EVs is 3, 3, 2 and 1. Whenever there are 2 or more EVs, the top two collide. In (a) the top two collide at about C 0:99 (all C values given are approximate to one decimal place) and the third EV is attacked and destroyed at C 5:4
Fig. 1. The imaginary EVs of Eq. (5) for rectangular initial pulses of half width d 1 and heights: (a) H 5, (b) H 4, (c) H 3 and (d) H 2.
by a fourth which comes out of the lower half plane at C 4. We see a ®fth EV emerge from the lower half plane at C 9:1 and go back in at C 13. In (b) the top two collide at C 1:2, while the third is rapidly caught and destroyed at C 6:8 by a fourth which comes out of the lower half plane at C 6:75. In (c) the top two collide at C 1:5 and a third is born at C 4:6, only to be reabsorbed at C 8. Finally, in (d) there is only one EV initially, but an EV comes out of the lower half plane at C 2:13 and quickly collides with the ®rst EV at C 2:17. Following a collision, members of a pair of colliding EVs develop nonzero real parts of opposite sign; in all the ®gures we plot only the imaginary parts of EVs, and only up to the point of collision or absorption. If the height H is small enough there are no collisions for the rectangle. Indeed, H p=4 is the threshold at which EVs appear for C 0 [13]. Numerically we ®nd that if p=4 < H < 1:93 (d 1) then the single EV is absorbed. For H > 1:93 all EVs are destroyed by collisions; we have numerically con®rmed this behavior for heights running up to H 10. In this connection, the ®rst C
M. Klaus, J.K. Shaw / Optics Communications 197 (2001) 491±500
495
where a collision occurs, call it C H , decreases with H. We are able to track the movement of EVs into the lower half complex plane; however, their evolution is highly nonlinear and very complicated. The physics of collisions and absorptions has been discussed in Ref. [10]. 2.2. Super-Gaussian pulse The super-Gaussian shape [1] is de®ned by q t H exp t=t0 2m ;
11
where H is the center height, t0 is a half-width parameter and m is an index that controls the degree to which the pulse has a square shape. For m 1 Eq. (11) reduces to the ordinary Gaussian and for large m the graph of Eq. (11) approaches a rectangle of height H and total width 2t0 . In addition to the advantage of being able to tailor pulse shape by the parameters in Eq. (11), superGaussian pulses have been shown to be superior to Gaussians in some respects as information carriers [14]. For m 30 we found dierences of at most 0.01 between the EVs of Eq. (11) and those of the corresponding rectangular pulses in Fig. 1. The graph of Eq. (11) for m 30 is almost indistinguishable from a rectangle. We calculated the EVs of Eq. (5) for superGaussian pulses of heights H 5, 4, 3 and 2 for indices m 1 and m 2, and half width t0 1 in all cases. Fig. 2 is the case m 1, the ordinary Gaussian. Here we ®nd no collisions, regardless of height, and all EVs are absorbed. When m 2, however, the top two EVs always collide, as illustrated in Fig. 3. The only absorption takes place when H 2, for which there is only one EV at C 0. The steeper edges in rectangular or superGaussian pulses for m > 1 generate a relatively broader Fourier spectrum. Heuristically speaking, since chirp also widens the spectrum, the combination of steep edges and chirp appears to enhance the eects of the chirp. Since the collision/absorption scenarios are opposite for super-Gaussians with indices m 1 and m 2, the question arises as to whether an m value separates the two. That is, for a given height
Fig. 2. The imaginary EVs of Eq. (5) for Gaussian initial pulses (super-Gaussian of index m 1), half width t0 1 and heights: (a) H 5, (b) H 4, (c) H 3 and (d) H 2.
Fig. 3. The imaginary EVs of Eq. (5) for super-Gaussian initial pulses of index m 2, half width t0 1 and heights: (a) H 5, (b) H 4, (c) H 3 and (d) H 2.
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M. Klaus, J.K. Shaw / Optics Communications 197 (2001) 491±500
H, one may speculate that there is a cuto value mc H such that collisions occur for m > mc and no collisions occur for m < mc . We have found such cutos mc H for a sample of heights and record them here as mc 2 4:5, mc 2:5 1:51, mc 3 1:12, mc 4 1:06, mc 5 1:05 and mc 10 1:03. A larger height H, for the same half width t0 , causes a steeper pulse edge and thus partially compensates for a smaller m. On the other hand, for H suciently small there are no collisions even for the rectangular pulse and it is therefore consistent that mc H is forced to be large for small H. Numerically we ®nd that if collisions occur for a rectangular case then they will occur for the superGaussian of suciently large m.
2.4. Hyperbolic secant pulse As in the ordinary Gaussian case m 1 we ®nd no collisions for initial pulses of hyperbolic secant shape and p t Ct2 , regardless of the height. The graphs of Im nk C are similar enough to the Gaussians in Fig. 2 that it is not necessary to plot them. To provide a sense of the behavior we note that the potential q t 5 sech t has ®ve EVs for C 0 which are absorbed sequentially at the threshold C values of roughly 0.5, 2, 4, 6 and 8. Note that the hyperbolic secant decays less rapidly than the Gaussian, which we know does not induce collisions. 3. Additional remarks and details
2.3. Tent shaped pulse The jump discontinuities at d in the rectangular pulse are not a factor in whether there are collisions. Indeed, the super-Gaussians are analytically continuous, although the shooting method scheme discussed below Eq. (10) places numerical discontinuities at d. However, for the super-Gaussians we ®nd that moving d further out from a suciently remote point, so that the potential is essentially 0, has no eect on the EV calculation. Collisions appear to be caused instead by a steep slope or rapid drop-o of sucient height. To test this further, we computed EVs for a continuous piecewise linear tent potential on d; d de®ned by a line segment connecting 0; H to d1 ; h, h < H and 0 < d1 < d, and then a segment connecting d1 ; h to d; 0 for t > 0 (and symmetrically for t < 0). When d1 and h are close to d and H, respectively, then the tent is approximately rectangular and we indeed ®nd small variation in the EVs between the two cases. In particular, there are collisions for the tent, which is both numerically and analytically continuous. By varying the intermediate heights h and locations d1 , we can shape the tent to some extent. Interestingly, for most cases we considered there were collisions until h dropped to the point where the tent became roughly triangular. That is, even for only moderately steep edges we still observed collisions.
In this section we provide some further details concerning the general evolutionary properties of EVs of Eq. (5). We discuss these features brie¯y here; a fuller and more rigorous account will appear elsewhere. (i) Sign of C: In the foregoing discussion we took C P 0. However, if m1 t; m2 t is an eigenvector of system (5) for C and f, then since q t is symmetric it is easy to show that m2 t; m1 t is an eigenvector for C and f. Thus the EVs corresponding to C and C are the same. The chirp sign is taken to be positive in Ref. [3] but negative in Ref. [4]. (ii) Eventual eradication of EVs: In the cases discussed in Section 2 we saw numerically that all EVs eventually collided or were absorbed, so that no EVs remained for C suciently large. In the cases considered in Refs. [3±5] it certainly appears that all EVs are eventually eradicated in this way, but to our knowledge no veri®cation has been given. A proof can be given based on the integral equations equivalent to Eq. (10) when q t vanishes outside d; d and p t Ct2 . The ®rst of Eq. (10) can be written Z t w1 t 1 exp iCs2 2insq sw2 s ds d
12 and a similar result holds for w2 . Let K0 be the least upper bound of the values of
M. Klaus, J.K. Shaw / Optics Communications 197 (2001) 491±500
Z
b a
exp
2 ix dx ;
for all real a and b, so that Z b 2 6 K0 exp ix dx a
2
and de®ne D K0 fkqk1 4dkqk1 2dkq0 k1 g, where kf k1 denotes the maximum modulus of a function f t over d; d. Using straightforward estimates involving Eq. (12), it can be shown that jw1 tj > 0 for C > C0 4D2 d 2 kqk21 expf4d 2 2 kqk1 g. Since w1 d cannot achieve the value 0 in Eq. (9), there are no EVs. We note that the C0 is too large to be of much practical use. However, this establishes for the ®rst time that suciently high chirp eventually destroys all EVs of Eq. (5), at least for potentials q t which are 0 outside an interval. Note that this argument covers eradication of EVs in the entire upper half n plane, not just on the positive imaginary axis. (iii) Ambiguities concerning imaginary EVs: If the envelope q t is rectangular and C 0, then the characteristic equation for Eq. (5), given below Eq. (1), shows that all EVs are purely imaginary; i.e., nk 0 isk , where sk is real. As seen in the ®gures the same is true for the other initial shapes q t considered in Section 2 but we have con®rmed this only numerically. Nonetheless, in the ``generic'' cases unchirped (C 0) EVs are imaginary. Indeed, this property has been claimed to hold for all symmetric [2,4], or even real [15], q t. However, the claim is not true in general. By a trial and error process we have constructed a symmetric shape q t which has a nonimaginary EV for C 0. Let q01 t be de®ned as q01 t 2, for 2 6 jtj 6 3, q01 t 5:35 for jtj 6 1, and q01 t 0 otherwise. This is not a ``one hump'' potential, but it is symmetric. By numerical computation and veri®cation by the argument principle we ®nd an EV located at approximately 4:58 0:02i for C 0. This can be con®rmed by applying the shooting method to Eq. (5). A simpler symmetric, two-hump potential with a nonimaginary EV is given by q02 t 145:85 for 1:06 < jtj < 1:07, q02 t 0 otherwise, which has an upper half plane EV located approximately at 1:4426 i. By adjusting the height and width of the narrow rect-
497
angular peaks we can vary the EV to some extent. The height and width combination given here was chosen to put the EV approximately on the line Im n 1. Some investigation reveals that there are many potentials q t in Eqs. (5) and (6) which produce nonimaginary EVs for C 0. (iv) Movement of EVs: We can say something about the way an EV n C moves about the plane as a function of C. For a ®xed largest EV n C let the power series be given by n C n0 Cn1 C 2 n2 : Assuming that q t is real and symmetric, and that n0 is an imaginary simple EV for C 0, we are able to show that n1 0 and n2 ig where g < 0; we omit the proof. If n0 ig0 , the imaginary part of n satis®es Im n C g0 C 2 g2 . Only even powers appear in the series because n C is an even function by (i). This con®rms the inverted parabolic shape of the top curves in Figs. 1±3. That is, the top EV moves down the imaginary axis initially. The property n1 0, whereby the initial slope of the imaginary part with respect to C is 0, holds for all the EVs (not just the largest). (v) Computational considerations: Let n in Eq. (5) be purely imaginary and let U t be a fundamental matrix solution of the w system (10) with U 1 I, the identity matrix. The ®rst column of U then satis®es the initial condition w1 d 1, w2 d 0 below Eq. (10) for d 1. By Eq. (5) the matrix 0 1 1 0 U t W t 1 0 1 0 is a fundamental matrix solution of Eq. (10) satisfying W 1 I. In the previous equation the asterisk denotes the complex conjugate and not the transpose. The two fundamental matrices are clearly related by U t W tU 1
for all t;
and so from t 1 we get I W 1U 1. Writing a b U 1 ; c d 1
computing W 1 and noting W 1 U 1 shows that a is real. Thus w1 1 is real valued. The
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EVs of Eq. (5) are those n for which w1 1 0. This means that, computationally, we can ®nd EVs by plotting w1 1 w1 1; ig as g sweeps through real values and simply monitoring sign changes. (vi) Events at n 0: By this we mean the process of an EV emerging from or being absorbed into the lower half plane at n 0. Fig. 1(a), where there are three EVs initially, shows three such events. That is, the ®gure shows three events where an EV is either born at n 0 or absorbed there. The EV in the ®rst event chases down and attacks an initial EV, whereas the second and third events comprise an EV birth and subsequent re-absorption. Actually, there are a few more events at n 0 for this case (rectangle, H 5) but they are not shown in the ®gure because of the large values of C involved. For rectangular heights running successively from H 1 to H 10, we record the number of initial EVs as 1, 1, 2, 3, 3, 4, 4, 5, 6, 6 and events at n 0 as 1, 1, 2, 3, 5, 8, 10, 13, 16, 20. (vii) Hypergeometric equation: For a rectangular pulse q t H , 1 6 t 6 1, and quadratically varying chirp p t Ct2 , Eq. (10) can be written as a single second order dierential equation. Dierentiating the ®rst of Eq. (10) and using the second to eliminate w2 results in w001 2iCt 2inw01 H 2 w1 0. Introducing a new function y by 2 w1 t y i Ct n =C, one can show that y satis®es zy 00 z 1=2
zy 0 z
iH 2 =4Cy z 0; 13
2
where z i Ct n =C. Eq. (13) is a con¯uent hypergeometric equation. The characteristic equation w1 1 0 (below Eq. (10)) reduces to F20 1 F1 1 F10 1F2 1 0, where ! iH 2 1 i Ct n2 ; ; F1 t F ; 4C 2 C ! 14 2 1 iH 2 3 i Ct n ; ; F2 t Ct nF 2 4C 2 C and where F ; ; is the con¯uent hypergeometric function. The relations (13) and (14) have been useful in con®rming shooting method calculations.
(viii) Other forms of chirp: It is natural to ask how replacing the linear chirp p t Ct2 in Eq. (3) with something else might change the evolution of EVs. As mentioned earlier, this may be more academic than practical since the commonly used Gaussian pulse induces a quadratically dependent phase. Nevertheless we have carried out a few computations and found rather interesting behavior. We calculated EVs corresponding to rectangular pulses of various heights H in the cases where 3 p t Cjtj and p t Ct4 . For H 3 we start of course with two EVs for C 0 as in Fig. 1(a). For both the cubic jtj3 and quartic t4 dependencies we do not observe collisions between the EVs. The lowest EV is absorbed at C 3:6 for the cubic and at C 4 for the quartic. The largest EV persists for very large C in both cases; it is not absorbed until after C 100 for the quartic. For H 4 there are three initial EVs. For both the cubic and quartic cases we ®nd that the second and third EVs collide at about C 2:4, whereas the largest EV is again absorbed only for very large C. Note that this behavior is completely dierent from the quadratic p t Ct2 case. As mentioned in Section 1, in Ref. [10] the authors consider the forms p t Ct2 , p t C sech t, p t C ln cosh bt for the phase, the latter because it corresponds to an integrable solution of Eq. (5). (In this connection see the book [16], which includes the example of Ref. [10].) We have duplicated the result in Ref. [10] for an initial pulse u 0; t 2 sech t exp iC sech t. However, Ref. [10] reports an EV collision for u 0; t 2 sech t exp iCt2 , while (as mentioned in Section 2) we detect no collisions at all in either this case or when the envelope q t is an ordinary (m 1 in Eq. (11)) Gaussian. (ix) Other literature: In the previously cited article [5] Anderson shows that the variational method provides a very useful description of the critical chirp beyond which solitons are destroyed for the Gaussian. This was independently con®rmed in a paper of Belov [17]. Desaix et al. have also used the variational procedure [18] to directly estimate EVs of Eq. (5), and independent work of Kaup and Malomed [19] addresses the same problem. Analogous to Eqs. (13) and (14) is a di-
M. Klaus, J.K. Shaw / Optics Communications 197 (2001) 491±500
rect calculation of the scattering data of Eq. (5) for an unchirped hyperbolic secant input pulse [20]. In terms of related literature, there has been a great deal of work on the Z±S system Eq. (5) corresponding to the nonlinear Schr odinger equation in the semi-classical limit, 2
ieU1 1=2e2 Utt jU j U ;
0 < e 1:
In a recent account [21], which contains numerous references, the writers note that the small parameter appears in both the governing equation and the initial phase condition. Roughly speaking, the problem they consider is an asymptotic one that involves very large chirp and very high nonlinear parameter N in Eq. (2). (x) Additional references (added during revision): We thank a referee for bringing Refs. [22,23] to our attention. In Ref. [22] the authors study the EVs of Z±S systems for rectangular, Gaussian and hyperbolic secant pulse shapes and a saturating quadratic chirp p t at2 = b ct2 . They plot the migration of EVs for various ®xed chirp levels C as functions of pulse center height H and are mainly concerned with the dependence on C of the area phenomenon, that is, the appearance of EVs at Cdependent threshold values of H H C. There are interesting nonmonotonicities including the fact that the order of appearance (as H increases) of the EVs is reversed for chirped pulses. For example, EV # 1 of the unchirped rectangular pulse appears before EV # 1 of the chirped cases, but the order of appearance of EV # 2 is reversed. For suciently high chirp the imaginary parts of the ®rst and second EVs eventually merge as H increases; this is consistent with the collision phenomenon described in this paper and in Ref. [10]. Indeed, the merging depicted in Fig. 1 (where c 0) of Ref. [22] can be inferred from our Fig. 1 if one thinks of C as ®xed and varies H. The numerical results of Ref. [22] are validated by a WKB analysis for small chirp levels. A portion of the content of our remark (v), concerning the reality of w1 1, appears in an appendix in Ref. [22]. The authors of Ref. [22] were apparently also aware of some of the hypergeometric formulas in our remark (vii). Ref. [23] considers the rectangular pro®le with phase dependence p t kjtj, k constant. The EVs
499
can be computed in closed form in this scenario. The authors plot real and imaginary parts of EVs and record EV collisions, as well as appearances and re-absorptions at n 0. Fig. 1(a) of Ref. [23] indicates a critical chirp level beyond which EV merging occurs. The pertinent k value is a little less than 2. It is very interesting that our Fig. 1(a) shows roughly the same phenomenon for collisions, namely that they all occur after a C value of something less than 2, even though the form of the phase dependence p t is dierent from Ref. [23]. Fig. 1(a) of Ref. [23] also shows the absorption of a single EV for the rectangular case followed by a sequence of appearance and re-absorption events at n 0 for a fairly low pulse height (H 1:6). Note that this is completely dierent from the quadratic chirp case illustrated in our Fig. 1 and remark (vi) above where we indicate a single event at n 0 for pulse heights in this range. 4. Conclusions We have demonstrated that EV collisions are common over a wide range of initial pulse shapes. Some shapes, such as Gaussians and hyperbolic secants, are immune to collisions. Other shapes, such as rectangles, nearly always suer collisions. As a general principle, it appears that centrally ¯at, steep-edged pulses induce collisions. For the ®rst time we have observed the phenomenon whereby an EV emerges from the lower half plane, pursues and ``kills'' an existing EV. We have established several properties of general EV evolution, including the initial behavior (near C 0) and the eventual eradication of all EVs for C suciently large. By oering a counter-example of a symmetric pulse with a nonimaginary EV, we have clari®ed an ambiguous claim that has appeared in the literature. References [1] G.P. Agrawal, Nonlinear Fiber Optics, second ed., Academic Press, New York, 1995. [2] J. Satsuma, N. Yajima, Initial value problems of onedimensional self-modulation of nonlinear waves in dispersive media, Prog. Theor. Phys., Suppl. 55 (1974) 284±306.
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[3] K.J. Blow, D. Wood, The evolution of solitons from nontransform limited pulses, Opt. Commun. 58 (1986) 349±354. [4] C. Desem, P.L. Chu, Eect of chirping on soliton propagation in single-mode optical ®bers, Opt. Lett. 11 (1986) 248±250. [5] D. Anderson, M. Lisak, T. Reichel, Asymptotic propagation properties of pulses in a soliton-based optical-®ber communication system, J. Opt. Soc. Am. B 5 (1988) 207±210. [6] A. Sahara, T. Inui, T. Komukai, H. Kubota, M. Nakazawa, 40-Gb/s transmission over a transoceanic distance in a dispersion managed ®ber using a modi®ed inline synchronous modulation method, J. Lightwave Technol. 18 (2000) 1364±1373. [7] F. Favre, D. Le Guen, T. Georges, Experimental evidence of psuedoperiodical soliton propagation in dispersionmanaged links, J. Lightwave Technol. 17 (1999) 1032± 1036. [8] N. Bergano, Undersea ®beroptic cable system, Opt. Photon. News 11 (2000) 20±25. [9] A.S. Gouveia-Neto, A.S.L. Gomes, J.R. Taylor, Experimental study of the evolution of chirped soliton pulses in single mode optical ®bers, Opt. Commun. 15 (1987) 383± 386. [10] A.I. Maimistov, Y.M. Sklyarov, In¯uence of regular phase modulation on formation of optical solitons, Sov. J. Quant. Electron. 17 (1987) 500±504. [11] M. Ablowitz, H. Segur, Solitons and the Inverse Scattering Transform, SIAM, Philadelphia, 1981. [12] S. Novikov, S.V. Manikov, L.P. Pitaevskii, V.E. Zakharov, Theory of Solitons: The Inverse Scattering Method, Consultants Bureau, New York, 1984.
[13] Z.V. Lewis, Semiclassical solutions of the Zakharov± Shabat scattering problem for phase modulated potentials, Phys. Lett. A 112 (1985) 99±103. [14] D. Anderson, M. Lisak, Propagation characteristics of frequency-chirped super-Gaussian optical pulses, Opt. Lett. 11 (1986) 569±571. [15] V.E. Zakharov, A.B. Shabat, Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media, Sov. Phys. J. Exp. Theor. Phys. 34 (1972) 62±69. [16] N.N. Akhmediev, A.A. Ankiewicz, Solitons: Nonlinear Pulses and Beams, Chapman and Hall, London, 1997. [17] M.N. Belov, In¯uence of the initial phase modulation on the formation of an optical soliton, Sov. J. Quant. Electron. 17 (1987) 1033±1034. [18] M. Desaix, D. Anderson, M. Lisak, Variational approach to the Zakharov±Shabat scattering problem, Phys. Rev. E 50 (1994) 2253±2256. [19] D.J. Kaup, B.A. Malomed, Variational principle for the Zakharov±Shabat equations, Physica D 84 (1995) 319±328. [20] J.W. Miles, An envelope soliton problem, SIAM J. Appl. Math. 41 (1981) 227±230. [21] J.C. Bronski, J.N. Kutz, Numerical simulation of the semiclassical limit of the focusing nonlinear Schr odinger equation, Phys. Lett. A 254 (1999) 325±336. [22] L.V. Hmurcik, D.J. Kaup, Solitons created by chirped initial pro®les in coherent pulse propagation, J. Opt. Soc. Am. 69 (1979) 597±604. [23] D.J. Kaup, J. El-Reedy, B. Malomed, Eect of chirp on soliton production, Phys. Rev. E 50 (1994) 1635± 1637.
Optics Communications 197 (2001) 491±500
www.elsevier.com/locate/optcom
In¯uence of pulse shape and frequency chirp on stability of optical solitons M. Klaus a, J.K. Shaw a,b,* a
b
Department of Mathematics, Virginia Tech, Blacksburg VA 24061-0123, USA Department of Electrical and Computer Engineering, Virginia Tech, Blacksburg VA 24061-0111, USA Received 9 April 2001; received in revised form 13 July 2001; accepted 19 July 2001
Abstract We examine conditions under which certain combinations of initial pulse shape and chirp, or phase modulation, destabilize solitons in optical ®bers. Destabilization occurs when eigenvalues (EVs) of an associated Zakharov±Shabat system, which move along the positive imaginary axis with increasing chirp parameter C, either are absorbed into the lower half plane or collide with another EV. In either the absorption or collision case the corresponding soliton, which is a solution of the nonlinear Schr odinger equation with constant or periodic amplitude as a function of propagation distance, becomes unstable. We have observed for the ®rst time the emergence of an EV from the lower half plane that pursues, and collides with, an existing EV. We identify several properties of general EV evolution, as C varies, and give a heuristic criterion under which initial pulses of a certain shape experience EV absorptions only, with no collisions. Ó 2001 Elsevier Science B.V. All rights reserved. Keywords: Optical ®ber solitons; Nonlinear Schr odinger equation; Zakharov±Shabat systems; Eigenvalue collisions
1. Introduction In simplest form optical solitons are governed by the nonlinear Schr odinger equation [1], jwz b2 =2wtt
cP0 jwj2 w;
w 0; 0 1;
1
where w z; t is the slowly varying ®eld envelope of the pulse, t is local time, z is propagation length, b2 is the ®ber dispersion constant, c is the nonlinearity constant and P0 is peak power. Subscripts z and t in Eq. (1) denote partial dierentiation. It is * Corresponding author. Address: Department of Mathematics, Virginia Tech, Blacksburg VA 24061-0123, USA. Tel.: +1-540-231-5345; fax: +1-540-231-3362. E-mail address: [email protected] (J.K. Shaw).
customary to write Eq. (1) in units of dimensionless length f [1], jUf sign b2 =2Utt
2
N 2 jU j U ;
U 0; 0 1; 2
N being a normalized nonlinearity constant. When b2 < 0 (``anomalous'' dispersion) and N 1, then U f; t sech t exp jf=2 is the fundamental soliton solution of Eq. (2) whose amplitude is independent of normalized distance f. For integers N > 1, Eq. (2) supports higher order solitons which are periodic in f. Hereafter we assume b2 < 0 in Eq. (2). Of course, Eqs. (1) and (2) do not take into account ®ber attenuation, noise, higher order dispersion or higher order nonlinearities. However,
0030-4018/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 ( 0 1 ) 0 1 4 6 1 - 4
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M. Klaus, J.K. Shaw / Optics Communications 197 (2001) 491±500
broadband ®ber ampli®ers and various methods of dispersion compensation can partially overcome these eects, at least to the extent that Eq. (2) remains a useful design tool [1] for realistic ®ber communication systems, in which ``dispersionmanaged'' solitons constitute a principal transmission format. Solitons are quite stable relative to initial pulse shape in the sense that the envelope U 0; t does not have to be exactly sech t in order for a soliton to form. For example, if N is not an integer then the solution of Eq. (2) with initial pulse shape U 0; t N0 sech t, N0 > 1=2, evolves with increasing f into a ``pure'' soliton corresponding to the integer N nearest to N0 [2]. In fact, the initial shape does not have to be a hyperbolic secant (or a higher order analog) in order for U f; t to approach a soliton in the limit of large f [1]. Stability with respect to initial phase has also been considered. At issue here is the extent to which an initial ``chirp'', or phase modulation, present in the form of a real valued function p t in the initial condition U 0; t q t exp jp t;
3
can eect soliton evolution in Eq. (2). In Eq. (3) q t is a real valued pulse envelope which we will assume to be evenly symmetric, q t q t: As to the form of p t, the most important case is that of ``linear chirp'', where p t Ct2 in Eq. (3) and C is a controllable constant. The reason is that unchirped (C 0) Gaussian pulses launched under the theoretically linear (hence the term ``linear'' chirp) condition cP0 0 in Eq. (1) develop a phase modulation that is quadratic in time [1]. Thus a pulse can be ``prechirped'' by passing it through a length of dispersive ®ber so as to endow it with an approximately quadratic phase. It is known that overchirping can completely destroy the soliton character of a pulse that has perfect initial conditions except for its chirp [3±5]. That is, the solution to Eq. (2) with initial condition U 0; t N sech t exp jCt2 will not evolve with increasing f into a soliton at all, for any N, if jCj is suciently large.
But in this same connection optical prechirping has recently emerged as a critical design tool in ®ber optic systems [6,7], especially with the increased signi®cance of the chirped return to zero transmission format [8]. For some situations the chirp can be severe [9]. However, the stability of solitons relative to initial chirp has been considered for only a few dierent envelopes q t and phase functions p t. For this reason it is appropriate to examine the issue in a more comprehensive way than previously. We will show that chirp can destroy the soliton character of Eq. (2) in two distinct ways which are determined by the behavior of eigenvalues (EVs) of the Zakharov±Shabat (Z±S) system associated with Eq. (2) in inverse scattering theory (Eq. (5) below). This classi®cation was also found in Ref. [10] for the special cases where q t could be either Gaussian or hyperbolic secant and p t could be either quadratic Ct2 or one of the forms p t C sech t, p t C ln cosh bt, where b is a constant. In agreement with Ref. [10] we ®nd that EVs n n C of Z±S system move with increasing C along the positive imaginary axis and for some threshold chirp C Cth are either absorbed into the lower half plane, in one scenario, or collide with another EV in the other. In either case the soliton characteristics of Eq. (2) are altered, possibly destroyed, for C > Cth . In this paper we extend the classi®cation to cover pulse envelopes q t which can be rectangular, super-Gaussian (see Eq. (11) below) or piecewise linear shapes. Our main goals have been to establish, at least heuristically, qualitative features of pulses that do or do not induce collisions and to shed light on the overall EV evolution. In fact, by considering a much wider class of pulse shapes than earlier accounts we have discovered an interesting phenomenon in which collisions occur for centrally ¯at, steep-edged envelopes q t but are absorbed otherwise. Moreover, not only can existing EVs move with increasing C toward each other on the imaginary axis and collide; EVs can appear from the lower half plane, chase down and then collide with an existing EV. This curious behavior actually seems to be the rule, rather than exception, but no report of the phenomenon has been given in prior writ-
M. Klaus, J.K. Shaw / Optics Communications 197 (2001) 491±500
ing. In further regard to the literature, there are a few ambiguous points in some of the published papers on this subject, which we try to clarify. In Section 2 we brie¯y cover background material on Z±S systems, summarize the relevant behavior of Z±S EVs and present our ®ndings. Section 3 discusses some further details of EV evolution and other pertinent literature. The paper ends with some conclusions in Section 4. 2. Zakharov±Shabat systems The parameter N can be removed from Eq. (2) by the substitution u f; t N U f; t [1], which leads to the form (recalling b2 < 0) 2
juf 1=2utt juj u 0;
4
this is the version of Eq. (2) typically encountered in soliton theory [11,12]. In the inverse scattering method [11] the Z±S EV problem 0 m1 t jnm1 t u f; tm2 t; 5 m02 t jnm2 t u f; t m1 t; is associated with Eq. (4), where t is the independent variable, f is held ®xed but arbitrary, n is the EV parameter and the asterisk denotes the complex conjugate. To brie¯y recall the main points of inverse scattering [11] relative to Eq. (4), discrete EVs nk of Eq. (5) correspond to solitons of Eq. (4) and are independent of f; i.e., solitons are determined solely by u 0; t. The approach in this paper therefore is to set u 0; t q t exp jp t
6
in Eq. (4) and to investigate the in¯uence of the real functions q t and p t on the EVs of Eq. (5) corresponding to u 0; t. For reasons mentioned above we are mainly interested in the quadratic chirp case p t Ct2 . The real and imaginary parts of the nk correspond to relative soliton velocities and amplitudes, respectively [12]. Generically (but not always) the EVs lie on the imaginary axis [1,11], in which case eigenfunctions associated with two distinct EVs have equal velocities and can combine and form a higher order soliton. However, a pair of EVs with unequal real parts gives rise to solitons which
493
travel with dierent velocities and separate as f increases. In the collision events mentioned in Section 1, we will show that EVs nk C move towards each other on the imaginary axis with increasing C, collide and split at C Cth , with one EV going into each of the left and right half planes. Thus a collision represents the break-up or destabilization of a soliton. In the alternate scenario, EVs nk C for increasing C are sequentially absorbed at distinct threshold chirp levels. After an absorption Eq. (4) can support a soliton whose order has been decreased by 1 [10]; a fundamental (order 1) soliton thus is destroyed by an absorption. We will show that it is possible for both scenarios to occur for the same potential. Following Ref. [11] we will work with upper half plane EVs, Im n > 0, of Eq. (5) and assume that u 0; t ! 0 at least exponentially fast as jtj ! 1. In the scattering theory of Eq. (5) it is shown that vector Jost solutions w n; t and u n; t exist and satisfy w n; t u n; t
0 1 1 0
exp jnt;
t ! 1;
exp jnt;
t!
1; Im n > 0: 7
Since EVs are determined by u 0; t, we will henceforth set f 0. When the dependence on the EV parameter in Eq. (5) is important we will write m n; t for the vector m t. An EV n of Eq. (5) is then a complex number, Im n > 0, such that m n; t of Eq. (5) is a multiple of w n; t, as well as multiple of u n; t, and continuous for all t. In the case where u 0; t has compact support, say u 0; t 0 for jtj > d > 0, Eq. (7) yields boundary conditions for m n; t in the form m1 d m2 d 0; m1 d exp jnd;
m
m1 : m2
8
To remove the exponential in Eq. (8) it is convenient to introduce exp jnt 0 m n; t w n; t; 0 exp jnt
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M. Klaus, J.K. Shaw / Optics Communications 197 (2001) 491±500
so that Eq. (8) becomes w1 d w2 d 0;
w1 d 1;
w
w1 : w2 9
In view of Eq. (6) the w system satis®es w01 q t exp jp t exp 2jntw2 ;
w02
q t exp jp t exp 2jntw1 :
10
In our numerical EV calculations we solve Eq. (10) subject to initial conditions w1 d 1, w2 d 0 and ®nd EVs by applying the shooting method with respect to w1 d 0 at t d. In fact, even for the cases where q t does not have compact support, but decays at least exponentially as jtj ! 1, we use the same technique by choosing d large enough that q t is negligibly small for jtj > d. We tested our numerical code on the chirped rectangular pulse of constant height H for d < t < d (q t 0 otherwise) and p t Ct2 , since the characteristic equation is available in analytical form in this case. For C 0 the characteristic p p equation is tan 2d n2 H 2 jn 1 n2 H 2 [13] and for general C it can be expressed in terms of con¯uent hypergeometric functions; see Section 3, remark (vii). For the rectangular pulse we found essentially no numerical dierence between the EVs as calculated by the analytical and shooting methods. We now summarize our ®ndings for various pulse shapes. 2.1. Rectangular pulse To illustrate this case we calculated the EVs of Eq. (5) for initially rectangular pulses of total width 2 (d 1 in Eq. (8)), various heights and quadratic chirp p t Ct2 . All EVs are purely imaginary for C 0. The results are illustrated in Fig. 1 where the heights are H 5 in (a), H 4 in (b), H 3 in (c) and H 2 in (d). Plotted are the imaginary parts Im nk C of EVs as functions of chirp C P 0, and in the successive cases the number of initial EVs is 3, 3, 2 and 1. Whenever there are 2 or more EVs, the top two collide. In (a) the top two collide at about C 0:99 (all C values given are approximate to one decimal place) and the third EV is attacked and destroyed at C 5:4
Fig. 1. The imaginary EVs of Eq. (5) for rectangular initial pulses of half width d 1 and heights: (a) H 5, (b) H 4, (c) H 3 and (d) H 2.
by a fourth which comes out of the lower half plane at C 4. We see a ®fth EV emerge from the lower half plane at C 9:1 and go back in at C 13. In (b) the top two collide at C 1:2, while the third is rapidly caught and destroyed at C 6:8 by a fourth which comes out of the lower half plane at C 6:75. In (c) the top two collide at C 1:5 and a third is born at C 4:6, only to be reabsorbed at C 8. Finally, in (d) there is only one EV initially, but an EV comes out of the lower half plane at C 2:13 and quickly collides with the ®rst EV at C 2:17. Following a collision, members of a pair of colliding EVs develop nonzero real parts of opposite sign; in all the ®gures we plot only the imaginary parts of EVs, and only up to the point of collision or absorption. If the height H is small enough there are no collisions for the rectangle. Indeed, H p=4 is the threshold at which EVs appear for C 0 [13]. Numerically we ®nd that if p=4 < H < 1:93 (d 1) then the single EV is absorbed. For H > 1:93 all EVs are destroyed by collisions; we have numerically con®rmed this behavior for heights running up to H 10. In this connection, the ®rst C
M. Klaus, J.K. Shaw / Optics Communications 197 (2001) 491±500
495
where a collision occurs, call it C H , decreases with H. We are able to track the movement of EVs into the lower half complex plane; however, their evolution is highly nonlinear and very complicated. The physics of collisions and absorptions has been discussed in Ref. [10]. 2.2. Super-Gaussian pulse The super-Gaussian shape [1] is de®ned by q t H exp t=t0 2m ;
11
where H is the center height, t0 is a half-width parameter and m is an index that controls the degree to which the pulse has a square shape. For m 1 Eq. (11) reduces to the ordinary Gaussian and for large m the graph of Eq. (11) approaches a rectangle of height H and total width 2t0 . In addition to the advantage of being able to tailor pulse shape by the parameters in Eq. (11), superGaussian pulses have been shown to be superior to Gaussians in some respects as information carriers [14]. For m 30 we found dierences of at most 0.01 between the EVs of Eq. (11) and those of the corresponding rectangular pulses in Fig. 1. The graph of Eq. (11) for m 30 is almost indistinguishable from a rectangle. We calculated the EVs of Eq. (5) for superGaussian pulses of heights H 5, 4, 3 and 2 for indices m 1 and m 2, and half width t0 1 in all cases. Fig. 2 is the case m 1, the ordinary Gaussian. Here we ®nd no collisions, regardless of height, and all EVs are absorbed. When m 2, however, the top two EVs always collide, as illustrated in Fig. 3. The only absorption takes place when H 2, for which there is only one EV at C 0. The steeper edges in rectangular or superGaussian pulses for m > 1 generate a relatively broader Fourier spectrum. Heuristically speaking, since chirp also widens the spectrum, the combination of steep edges and chirp appears to enhance the eects of the chirp. Since the collision/absorption scenarios are opposite for super-Gaussians with indices m 1 and m 2, the question arises as to whether an m value separates the two. That is, for a given height
Fig. 2. The imaginary EVs of Eq. (5) for Gaussian initial pulses (super-Gaussian of index m 1), half width t0 1 and heights: (a) H 5, (b) H 4, (c) H 3 and (d) H 2.
Fig. 3. The imaginary EVs of Eq. (5) for super-Gaussian initial pulses of index m 2, half width t0 1 and heights: (a) H 5, (b) H 4, (c) H 3 and (d) H 2.
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M. Klaus, J.K. Shaw / Optics Communications 197 (2001) 491±500
H, one may speculate that there is a cuto value mc H such that collisions occur for m > mc and no collisions occur for m < mc . We have found such cutos mc H for a sample of heights and record them here as mc 2 4:5, mc 2:5 1:51, mc 3 1:12, mc 4 1:06, mc 5 1:05 and mc 10 1:03. A larger height H, for the same half width t0 , causes a steeper pulse edge and thus partially compensates for a smaller m. On the other hand, for H suciently small there are no collisions even for the rectangular pulse and it is therefore consistent that mc H is forced to be large for small H. Numerically we ®nd that if collisions occur for a rectangular case then they will occur for the superGaussian of suciently large m.
2.4. Hyperbolic secant pulse As in the ordinary Gaussian case m 1 we ®nd no collisions for initial pulses of hyperbolic secant shape and p t Ct2 , regardless of the height. The graphs of Im nk C are similar enough to the Gaussians in Fig. 2 that it is not necessary to plot them. To provide a sense of the behavior we note that the potential q t 5 sech t has ®ve EVs for C 0 which are absorbed sequentially at the threshold C values of roughly 0.5, 2, 4, 6 and 8. Note that the hyperbolic secant decays less rapidly than the Gaussian, which we know does not induce collisions. 3. Additional remarks and details
2.3. Tent shaped pulse The jump discontinuities at d in the rectangular pulse are not a factor in whether there are collisions. Indeed, the super-Gaussians are analytically continuous, although the shooting method scheme discussed below Eq. (10) places numerical discontinuities at d. However, for the super-Gaussians we ®nd that moving d further out from a suciently remote point, so that the potential is essentially 0, has no eect on the EV calculation. Collisions appear to be caused instead by a steep slope or rapid drop-o of sucient height. To test this further, we computed EVs for a continuous piecewise linear tent potential on d; d de®ned by a line segment connecting 0; H to d1 ; h, h < H and 0 < d1 < d, and then a segment connecting d1 ; h to d; 0 for t > 0 (and symmetrically for t < 0). When d1 and h are close to d and H, respectively, then the tent is approximately rectangular and we indeed ®nd small variation in the EVs between the two cases. In particular, there are collisions for the tent, which is both numerically and analytically continuous. By varying the intermediate heights h and locations d1 , we can shape the tent to some extent. Interestingly, for most cases we considered there were collisions until h dropped to the point where the tent became roughly triangular. That is, even for only moderately steep edges we still observed collisions.
In this section we provide some further details concerning the general evolutionary properties of EVs of Eq. (5). We discuss these features brie¯y here; a fuller and more rigorous account will appear elsewhere. (i) Sign of C: In the foregoing discussion we took C P 0. However, if m1 t; m2 t is an eigenvector of system (5) for C and f, then since q t is symmetric it is easy to show that m2 t; m1 t is an eigenvector for C and f. Thus the EVs corresponding to C and C are the same. The chirp sign is taken to be positive in Ref. [3] but negative in Ref. [4]. (ii) Eventual eradication of EVs: In the cases discussed in Section 2 we saw numerically that all EVs eventually collided or were absorbed, so that no EVs remained for C suciently large. In the cases considered in Refs. [3±5] it certainly appears that all EVs are eventually eradicated in this way, but to our knowledge no veri®cation has been given. A proof can be given based on the integral equations equivalent to Eq. (10) when q t vanishes outside d; d and p t Ct2 . The ®rst of Eq. (10) can be written Z t w1 t 1 exp iCs2 2insq sw2 s ds d
12 and a similar result holds for w2 . Let K0 be the least upper bound of the values of
M. Klaus, J.K. Shaw / Optics Communications 197 (2001) 491±500
Z
b a
exp
2 ix dx ;
for all real a and b, so that Z b 2 6 K0 exp ix dx a
2
and de®ne D K0 fkqk1 4dkqk1 2dkq0 k1 g, where kf k1 denotes the maximum modulus of a function f t over d; d. Using straightforward estimates involving Eq. (12), it can be shown that jw1 tj > 0 for C > C0 4D2 d 2 kqk21 expf4d 2 2 kqk1 g. Since w1 d cannot achieve the value 0 in Eq. (9), there are no EVs. We note that the C0 is too large to be of much practical use. However, this establishes for the ®rst time that suciently high chirp eventually destroys all EVs of Eq. (5), at least for potentials q t which are 0 outside an interval. Note that this argument covers eradication of EVs in the entire upper half n plane, not just on the positive imaginary axis. (iii) Ambiguities concerning imaginary EVs: If the envelope q t is rectangular and C 0, then the characteristic equation for Eq. (5), given below Eq. (1), shows that all EVs are purely imaginary; i.e., nk 0 isk , where sk is real. As seen in the ®gures the same is true for the other initial shapes q t considered in Section 2 but we have con®rmed this only numerically. Nonetheless, in the ``generic'' cases unchirped (C 0) EVs are imaginary. Indeed, this property has been claimed to hold for all symmetric [2,4], or even real [15], q t. However, the claim is not true in general. By a trial and error process we have constructed a symmetric shape q t which has a nonimaginary EV for C 0. Let q01 t be de®ned as q01 t 2, for 2 6 jtj 6 3, q01 t 5:35 for jtj 6 1, and q01 t 0 otherwise. This is not a ``one hump'' potential, but it is symmetric. By numerical computation and veri®cation by the argument principle we ®nd an EV located at approximately 4:58 0:02i for C 0. This can be con®rmed by applying the shooting method to Eq. (5). A simpler symmetric, two-hump potential with a nonimaginary EV is given by q02 t 145:85 for 1:06 < jtj < 1:07, q02 t 0 otherwise, which has an upper half plane EV located approximately at 1:4426 i. By adjusting the height and width of the narrow rect-
497
angular peaks we can vary the EV to some extent. The height and width combination given here was chosen to put the EV approximately on the line Im n 1. Some investigation reveals that there are many potentials q t in Eqs. (5) and (6) which produce nonimaginary EVs for C 0. (iv) Movement of EVs: We can say something about the way an EV n C moves about the plane as a function of C. For a ®xed largest EV n C let the power series be given by n C n0 Cn1 C 2 n2 : Assuming that q t is real and symmetric, and that n0 is an imaginary simple EV for C 0, we are able to show that n1 0 and n2 ig where g < 0; we omit the proof. If n0 ig0 , the imaginary part of n satis®es Im n C g0 C 2 g2 . Only even powers appear in the series because n C is an even function by (i). This con®rms the inverted parabolic shape of the top curves in Figs. 1±3. That is, the top EV moves down the imaginary axis initially. The property n1 0, whereby the initial slope of the imaginary part with respect to C is 0, holds for all the EVs (not just the largest). (v) Computational considerations: Let n in Eq. (5) be purely imaginary and let U t be a fundamental matrix solution of the w system (10) with U 1 I, the identity matrix. The ®rst column of U then satis®es the initial condition w1 d 1, w2 d 0 below Eq. (10) for d 1. By Eq. (5) the matrix 0 1 1 0 U t W t 1 0 1 0 is a fundamental matrix solution of Eq. (10) satisfying W 1 I. In the previous equation the asterisk denotes the complex conjugate and not the transpose. The two fundamental matrices are clearly related by U t W tU 1
for all t;
and so from t 1 we get I W 1U 1. Writing a b U 1 ; c d 1
computing W 1 and noting W 1 U 1 shows that a is real. Thus w1 1 is real valued. The
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EVs of Eq. (5) are those n for which w1 1 0. This means that, computationally, we can ®nd EVs by plotting w1 1 w1 1; ig as g sweeps through real values and simply monitoring sign changes. (vi) Events at n 0: By this we mean the process of an EV emerging from or being absorbed into the lower half plane at n 0. Fig. 1(a), where there are three EVs initially, shows three such events. That is, the ®gure shows three events where an EV is either born at n 0 or absorbed there. The EV in the ®rst event chases down and attacks an initial EV, whereas the second and third events comprise an EV birth and subsequent re-absorption. Actually, there are a few more events at n 0 for this case (rectangle, H 5) but they are not shown in the ®gure because of the large values of C involved. For rectangular heights running successively from H 1 to H 10, we record the number of initial EVs as 1, 1, 2, 3, 3, 4, 4, 5, 6, 6 and events at n 0 as 1, 1, 2, 3, 5, 8, 10, 13, 16, 20. (vii) Hypergeometric equation: For a rectangular pulse q t H , 1 6 t 6 1, and quadratically varying chirp p t Ct2 , Eq. (10) can be written as a single second order dierential equation. Dierentiating the ®rst of Eq. (10) and using the second to eliminate w2 results in w001 2iCt 2inw01 H 2 w1 0. Introducing a new function y by 2 w1 t y i Ct n =C, one can show that y satis®es zy 00 z 1=2
zy 0 z
iH 2 =4Cy z 0; 13
2
where z i Ct n =C. Eq. (13) is a con¯uent hypergeometric equation. The characteristic equation w1 1 0 (below Eq. (10)) reduces to F20 1 F1 1 F10 1F2 1 0, where ! iH 2 1 i Ct n2 ; ; F1 t F ; 4C 2 C ! 14 2 1 iH 2 3 i Ct n ; ; F2 t Ct nF 2 4C 2 C and where F ; ; is the con¯uent hypergeometric function. The relations (13) and (14) have been useful in con®rming shooting method calculations.
(viii) Other forms of chirp: It is natural to ask how replacing the linear chirp p t Ct2 in Eq. (3) with something else might change the evolution of EVs. As mentioned earlier, this may be more academic than practical since the commonly used Gaussian pulse induces a quadratically dependent phase. Nevertheless we have carried out a few computations and found rather interesting behavior. We calculated EVs corresponding to rectangular pulses of various heights H in the cases where 3 p t Cjtj and p t Ct4 . For H 3 we start of course with two EVs for C 0 as in Fig. 1(a). For both the cubic jtj3 and quartic t4 dependencies we do not observe collisions between the EVs. The lowest EV is absorbed at C 3:6 for the cubic and at C 4 for the quartic. The largest EV persists for very large C in both cases; it is not absorbed until after C 100 for the quartic. For H 4 there are three initial EVs. For both the cubic and quartic cases we ®nd that the second and third EVs collide at about C 2:4, whereas the largest EV is again absorbed only for very large C. Note that this behavior is completely dierent from the quadratic p t Ct2 case. As mentioned in Section 1, in Ref. [10] the authors consider the forms p t Ct2 , p t C sech t, p t C ln cosh bt for the phase, the latter because it corresponds to an integrable solution of Eq. (5). (In this connection see the book [16], which includes the example of Ref. [10].) We have duplicated the result in Ref. [10] for an initial pulse u 0; t 2 sech t exp iC sech t. However, Ref. [10] reports an EV collision for u 0; t 2 sech t exp iCt2 , while (as mentioned in Section 2) we detect no collisions at all in either this case or when the envelope q t is an ordinary (m 1 in Eq. (11)) Gaussian. (ix) Other literature: In the previously cited article [5] Anderson shows that the variational method provides a very useful description of the critical chirp beyond which solitons are destroyed for the Gaussian. This was independently con®rmed in a paper of Belov [17]. Desaix et al. have also used the variational procedure [18] to directly estimate EVs of Eq. (5), and independent work of Kaup and Malomed [19] addresses the same problem. Analogous to Eqs. (13) and (14) is a di-
M. Klaus, J.K. Shaw / Optics Communications 197 (2001) 491±500
rect calculation of the scattering data of Eq. (5) for an unchirped hyperbolic secant input pulse [20]. In terms of related literature, there has been a great deal of work on the Z±S system Eq. (5) corresponding to the nonlinear Schr odinger equation in the semi-classical limit, 2
ieU1 1=2e2 Utt jU j U ;
0 < e 1:
In a recent account [21], which contains numerous references, the writers note that the small parameter appears in both the governing equation and the initial phase condition. Roughly speaking, the problem they consider is an asymptotic one that involves very large chirp and very high nonlinear parameter N in Eq. (2). (x) Additional references (added during revision): We thank a referee for bringing Refs. [22,23] to our attention. In Ref. [22] the authors study the EVs of Z±S systems for rectangular, Gaussian and hyperbolic secant pulse shapes and a saturating quadratic chirp p t at2 = b ct2 . They plot the migration of EVs for various ®xed chirp levels C as functions of pulse center height H and are mainly concerned with the dependence on C of the area phenomenon, that is, the appearance of EVs at Cdependent threshold values of H H C. There are interesting nonmonotonicities including the fact that the order of appearance (as H increases) of the EVs is reversed for chirped pulses. For example, EV # 1 of the unchirped rectangular pulse appears before EV # 1 of the chirped cases, but the order of appearance of EV # 2 is reversed. For suciently high chirp the imaginary parts of the ®rst and second EVs eventually merge as H increases; this is consistent with the collision phenomenon described in this paper and in Ref. [10]. Indeed, the merging depicted in Fig. 1 (where c 0) of Ref. [22] can be inferred from our Fig. 1 if one thinks of C as ®xed and varies H. The numerical results of Ref. [22] are validated by a WKB analysis for small chirp levels. A portion of the content of our remark (v), concerning the reality of w1 1, appears in an appendix in Ref. [22]. The authors of Ref. [22] were apparently also aware of some of the hypergeometric formulas in our remark (vii). Ref. [23] considers the rectangular pro®le with phase dependence p t kjtj, k constant. The EVs
499
can be computed in closed form in this scenario. The authors plot real and imaginary parts of EVs and record EV collisions, as well as appearances and re-absorptions at n 0. Fig. 1(a) of Ref. [23] indicates a critical chirp level beyond which EV merging occurs. The pertinent k value is a little less than 2. It is very interesting that our Fig. 1(a) shows roughly the same phenomenon for collisions, namely that they all occur after a C value of something less than 2, even though the form of the phase dependence p t is dierent from Ref. [23]. Fig. 1(a) of Ref. [23] also shows the absorption of a single EV for the rectangular case followed by a sequence of appearance and re-absorption events at n 0 for a fairly low pulse height (H 1:6). Note that this is completely dierent from the quadratic chirp case illustrated in our Fig. 1 and remark (vi) above where we indicate a single event at n 0 for pulse heights in this range. 4. Conclusions We have demonstrated that EV collisions are common over a wide range of initial pulse shapes. Some shapes, such as Gaussians and hyperbolic secants, are immune to collisions. Other shapes, such as rectangles, nearly always suer collisions. As a general principle, it appears that centrally ¯at, steep-edged pulses induce collisions. For the ®rst time we have observed the phenomenon whereby an EV emerges from the lower half plane, pursues and ``kills'' an existing EV. We have established several properties of general EV evolution, including the initial behavior (near C 0) and the eventual eradication of all EVs for C suciently large. By oering a counter-example of a symmetric pulse with a nonimaginary EV, we have clari®ed an ambiguous claim that has appeared in the literature. References [1] G.P. Agrawal, Nonlinear Fiber Optics, second ed., Academic Press, New York, 1995. [2] J. Satsuma, N. Yajima, Initial value problems of onedimensional self-modulation of nonlinear waves in dispersive media, Prog. Theor. Phys., Suppl. 55 (1974) 284±306.
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