. INTRODUCTIONulse and soliton propagations in uniform and nonuni-
orm nonlinear fiber Bragg gratings (FBGs) have beentudied theoretically [1–14] and experimentally [15,16].uch pulse propagation is modeled by the nonlinearoupled mode equations (NLCMEs). If the self-phaseodulation term is set to zero, the NLCMEs degenerate
o equations of the massive Thirring model (MTM) [17].he MTM is an integrable system and it can be solved byhe inverse scattering transform (IST) [18,19]. TheLCMEs, on the contrary are non-integrable. Neverthe-
ess, a solitary wave solution of the NLCMEs, known ashe Bragg soliton, can be obtained [1–4]. A solution for aingle Bragg soliton can be obtained by transforming aoliton solution of the MTM [4]. However, such a transfor-ation can be used only for a single soliton solution. An-
ther property of the NLCMEs is that, in the limit of low-ntensity pulses, they degenerate to the nonlinearchrödinger equation (NLSE) [6,12], which is an inte-rable equation.
Bragg solitons are solitary waves, not solitons. Whenwo Bragg solitons collide, the pulses after the interactionay change their total energy, their momentum, or theyay even disappear. The interaction between two Bragg
olitons was numerically studied [4,9–11]. In [4,10,11] itas shown that when two counter-propagating Bragg
olitons interact there is a regime of amplitude and veloc-ty in which an elastic collision occurs. In the elastic re-ime the amplitude and velocity of the solitons are almostnaffected after the collision. In [9] it was shown that, inhe low-intensity limit, the interaction between two iden-
ical Bragg solitons is similar to the interaction betweenwo NLSE solitons.
In this paper we show that, for a wide range of Braggoliton parameters, the solution of the MTM can be trans-ormed into a trial function that approximately describeshe interaction between two co-propagating Bragg soli-ons in a uniform FBG. In these cases Bragg solitons col-ide similar to true solitons that are solutions of the MTM,nd hence the two Bragg solitons emerge from a collisionith approximately the same momentum and energy as
he input solitons. The trial connection between the MTMnd the NLCME solitons is based on calculating the two-oliton solution of the MTM by using the IST. The kernelunction of the solution can be separated into two func-ions. Each function asymptotically equals to a kernelunction of a single MTM soliton. To obtain the interac-ion between two Bragg solitons, the two MTM functionsre then transformed separately by using the transforma-ion for a single Bragg soliton solution given in [4]. Weave validated that our trial function for studying the in-eraction between two Bragg solitons is in a good quanti-ative agreement with a numerical solution of NLCMEsor a broad range of parameters of the interacting solitonshat can be also realized experimentally.
The similarity between the interaction of the MTM andragg solitons that is obtained in a broad parameter re-ion is not straightforward since the self-phase modula-ion term in the NLCMEs is not negligible; i.e., the self-hase modulation term cannot be considered as merely amall perturbation of the MTM. The explicit trial functionhat is used to describe the interaction between Bragg
010 Optical Society of America
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1196 J. Opt. Soc. Am. B/Vol. 27, No. 6 /June 2010 Z. Toroker and M. Horowitz
olitons enables one to calculate approximately the shiftsn locations and phases of the pulses that occur as a resultf the collision. The connection between the MTM and theLCMEs also facilitates the understanding of physical ef-
ects such as the exchange in the soliton parameters dueo the interaction.
Since the NLCMEs are not an integrable system, in-lastic collisions between Bragg solitons can be obtained4,11]. Such interactions can change the velocity of theolitons and may add an intensity oscillation to the out-oing pulses [4]. Various outcomes of inelastic collisionsetween counter-propagating solitons were studied in11]. One of the most interesting interactions is theerger of two counter-propagating solitons into a single
ero-velocity pulse [11]. In our simulations we have alsoound a parameter regime in which the interaction be-ween Bragg solitons does not resemble the elastic inter-ction between MTM solitons. In particular, we describen interaction that causes the inversion of the propaga-ion direction of one of the interacting solitons. Such aonlinear interaction can be used to control the propaga-ion direction of a solitary wave. Unlike [11], we havetudied the interaction between co-propagating waves.he interaction between co-propagating waves ratherhan between counter-propagating waves enables one toontrol the propagation direction of one of the pulses andot only the magnitude of the propagation velocity.In a previous work it was shown numerically that in
he low-intensity limit, the interaction between two Braggolitons resembles the interaction between two solitons ofhe NLSE [9]. The Bragg solitons attract or repel eachther depending on their relative phases. However, theelative phase between two Bragg solitons is found to de-end on their initial separation. In [6,12] it was shownhat the NLSE can be used to approximate the NLCMEso analyze the propagation of low-intensity pulses. In par-icular, it was shown that a low-intensity single Braggoliton solution can be obtained by approximating NLC-Es to the NLSE. Such an approach can be also used to
tudy the interaction between two Bragg solitons. How-ver, since the parameters of the obtained NLSE dependn the soliton velocity, the approach is limited for study-ng the interaction between solitons that have the sameelocity. The approach described in this paper is used totudy the interaction of solitons with different velocities.he approach also enables us to accurately take into ac-ount the complex frequency dependence of the dispersionaused by the grating. Moreover, the one-soliton solutionbtained by using the MTM solution is not limited fornalyzing low-intensity solitons. The comparison to nu-erical solution indicates that our trial function can be
sed for some soliton parameters to study quasi-elasticnteractions between Bragg solitons with significant in-ensities.
. MATHEMATICAL BACKGROUNDulse propagation in FBGs is modeled by the NLCMEs
here u �z , t� is the field envelope of the forward �+� and
±
ackward (�) propagating waves, Vg is the group velocityn the absence of the grating, �s is the self-phase modula-ion coefficient, �x is the cross-phase modulation coeffi-ient, ��z� is the grating coupling strength, and ��z� is theetuning parameter [20]. For FBGs the coefficients �s andx are equal, i.e., �s=�x=�. In case that the self-odulation �s is set to zero, ��z� is set to zero, and ��z� is
et to a constant �, Eq. (1) reduces to the MTM equations,
±i�z�± + iVg−1�t�± + ��� + 2�����2�± = 0. �2�
quation (2) is solvable by the inverse scattering theory19]. The solutions are given by
�−�z,t� = −K1�z,t�e−i��z,t�
����,
�+�z,t� = −i���z − Vg
−1�t�K1�z,t��e−i��z,t�
�����, �3�
here the kernel function K1�z , t� is a solution of an inte-ral equation [19] and the function ��z , t� is defined in theppendix. The Thirring soliton solution is given by [19]
�±�z,t� = ±� �
2�� 1 ± v
1 � v�1/4
sin���sech��z,t� � i�/2�e−i�z,t�,
�4�
here
�z,t� = �� sin����z − z0 − vVgt�, �5�
�z,t� = �� cos����v�z − z0� − Vgt� + �0, �6�
nd �=1/�1−v2. The four parameters of the soliton arehe normalized velocity v �−1 v 1�, the initial position0, � �0 � ��, and the initial phase �0.
The Bragg soliton solution of the NLCMEs can be ob-ained by transforming the MTM solution as follows [4]:
u±�z,t� = ��±�z,t�ei��z,t�, �7�
here
� = �1 +1
2
1 + v2
1 − v2�−1/2
, �8�
��z,t� = 2�2�2v arctantanh��tan� �
2� . �9�
he peak power of the Bragg soliton is given by
Ip = maxz
��u+�2 + �u−�2� =�
��2�4 sin2��/2�. �10�
. TRIAL FUNCTION FOR STUDYINGNTERACTION BETWEEN TWO BRAGGOLITONS BASED ON THE TWO-OLITON SOLUTION OF THE MTM
n this work, we study quasi-elastic collisions of tworagg solitons in a uniform grating by using a trial func-
ion based on the two-soliton solution of the MTM. Al-
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Z. Toroker and M. Horowitz Vol. 27, No. 6 /June 2010/J. Opt. Soc. Am. B 1197
hough we do not give exact conditions when our trialunction is valid, we demonstrate numerically the accu-acy of the function for a broad range of soliton param-ters. The trial function enables one, for the first time tour knowledge, to study explicitly the collision of tworagg solitons in an important region of soliton param-ters that can be also realized in experiments.
We assume that the initial conditions correspond to twopatially separated solitons. We first transform the initialonditions u±�z , t=0� of the NLCMEs into initial condi-ions �±�z , t=0� of the MTM. We then calculate �±�z , t�,hich gives the interaction between the two solitons in
he MTM. Then, we transform the exact solution �±�z , t�,�0, of the MTM into a function u±�z , t� that approxi-ates the solution u±�z , t� of NLCMEs that satisfies the
nitial conditions u±�z , t=0�. The transformation from theolution of the MTM to the trial function is based on split-ing the kernel function in the MTM, K1�z , t�, into twounctions. Although each function does not correspond to
soliton, the asymptotic behavior of each function is ap-roximately equal to that of a single soliton. Therefore,e choose to transform each of the MTM kernel functions
eparately in order to describe the interaction in theLCMEs. We show that, in important cases, the error of
he trial function is small during the entire interaction.
. Trial Function to Study Quasi-Elastic Collision ofwo Bragg Solitonshe kernel function K1�z , t� for the two-soliton interaction
n the MTM can be separated into two functions, f1�z , t�nd f2�z , t�:
K1�z,t� = f1�z,t� + f2�z,t�, �11�
here the functions f1�z , t� and f2�z , t� are given in Eqs.A5) and (A6) in the Appendix. The two functions dependn the parameters of the two Thirring solitonsvi ,zi,0 ,�i ,�i,0� �i=1,2�.
When v1�v2 it can be shown that, before the interac-ion �t→−�� when the two solitons are spatially sepa-ated,
f1�z,t → − �� ��
2�1 sin��1�e−i1�z,t�sech1�z,t� − i
�1
2 ,
�12�
f2�z,t → − �� ��
2�2 sin��2�e−i�2�1+2�z,t��sech2�z,t�
− i�2
2 . �13�
herefore, each of the functions f1�z , t� and f2�z , t� is pro-ortional to a soliton solution of the MTM at t→−�.We assume that the two solitons have different veloci-
ies and that the propagation velocity of soliton 2 is fasterhan that of soliton 1. With this choice soliton 2 is the fastoliton and is located before the interaction to the left ofhe slow soliton 1. At t→�, sufficiently large after the in-eraction so that the solitons are well spatially separated,he functions f and f can be approximated as follows:
1 2
f1�z,t → �� ��
2�1 sin��1�e−i1�z,t�
��2
��2�sech1�z,t� − i
�1
2 , �14�
f2�z,t → �� ��
2�2 sin��2�e−i�2�1+2�z,t��
���1�
�1sech2�z,t� − i
�2
2 , �15�
here
1�z,t → �� = ��1 sin��1��z − z1,0 −ln�1/�1�
��1 sin��1� − v1Vgt� ,
�16�
2�z,t → �� = ��2 sin��2��z − z2,0 +ln�1/�2�
��2 sin��2� − v2Vgt� .
�17�
quations (14) and (15) show that, at a sufficiently largeime after the interaction, where the two solitons are welleparated spatially, the solution �±�z , t� is a superpositionf the input solitons but with a shift in positions �zi �i1,2� and in phases ��i caused by the interaction and isiven by
�z1 =− ln�1/�1�
��1 sin��1�, �18�
�z2 =ln�1/�2�
��2 sin��2�, �19�
��1 = arg��2�, �20�
��2 = arg�− �1�, �21�
here �1 and �2 are defined in Eq. (A15) in the Appendix.In the region of the interaction the functions f1�z , t� and
2�z , t� are not proportional to soliton solutions of theTM. Nevertheless, Eqs. (12)–(15) and the connection be-
ween a single soliton solution in the MTM and theLCMEs given in Eqs. (8) and (9) gave us the motivation
o approximate the two-soliton solution u±�z , t� of theLCMEs as follows:
u−�z,t� = −G1�z,t�e−i��z,t�
����,
u+�z,t� = −iG2�z,t�e−i��z,t�
�����, �22�
here
G �z,t� = � f �z,t�ei�1�z,t� + � f �z,t�ei��2�z,t�+2�1�, �23�
1 1 1 2 2
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1198 J. Opt. Soc. Am. B/Vol. 27, No. 6 /June 2010 Z. Toroker and M. Horowitz
e assume that, before the interaction, the two NLCMEolitons are spatially separated and hence the coefficientsi and �i�z , t� �i=1,2� can be calculated from the param-ters of the input solitons before the interaction by usingqs. (8) and (9). We also note that the shifts in the soliton
ocations and phases as a result of the interaction thatere calculated for the MTM remain the same for theragg solitons when the trial function is accurate.We have solved numerically the NLCMEs and com-
ared the result u±�z , t� to the trial function u±�z , t�, whichas obtained by transforming the corresponding two-
oliton solution of the MTM. We shall show in Section 4hat in a wide-parameter regime, despite the difference inoth equations, the solution �±�z , t� of the MTM can besed to approximate the interaction between two Braggolitons u±�z , t�. The approximate result exhibitshirring-like soliton collision behavior such as the ex-hange in the soliton parameters due to the interaction.ince the NLCMEs are not an integrable system we couldnd parameter regimes in which the Bragg soliton inter-ction is not like a Thirring soliton interaction.
. COMPARISON TO NUMERICALIMULATIONSe compare the trial function given explicitly in Eq. (22),
ˆ ±�z , t�, to the result of a numerical simulation [21],±�z , t�, that was used to solve the NLCMEs with the ini-ial condition given in Eq. (25). The initial conditionquals
u±�z,t = 0� = u±,s1�z,t = 0� + u±,s2
�z,t = 0�, �25�
here u±,si�z , t=0� �i=1,2� is the ith Bragg soliton before
he interaction. We also assume that before the interac-ion at t=0 the distance between the centers of the soli-ons is significantly longer than their spatial widths. Thenitial positions of the two Bragg solitons were chosen as1,0=30Ws and z2,0=20Ws, where Ws is the spatial fullidth at half-maximum (FWHM) of the shorter soliton.he connections between the parameters of the two Braggolitons were set to v1=0.75v2, �1=�2, and �1,0=0, wherei and �i are the parameters of ith Bragg soliton. The cou-ling and nonlinear coefficients of the grating were cho-en as �=450 m−1 and �x=�s=6.4 km−1 W−1, respectively.hese grating parameters are similar to those used in ex-eriments [15]. The NLCMEs were solved numericallyith spatial and time step-sizes of �Z=Ws /200 and �T�Z /Vg, respectively.We describe below four different examples of Bragg
oliton collisions. Then, we show a diagram in the ��2 ,v2�lane that compares the trial function to the result of aumerical simulation after the collision.
. Example I: Overlap Between Two Bragg Solitonsuring a Collision
n our first example the Bragg soliton parameters are�1 ,v1 ,�1,0�= �0.2,0.45,0� and ��2 ,v2 ,�2,0�= �0.2,0.6,0�.he input solitons have spatial FWHMs of 1.75 and 1.57
m. Hence, Ws=1.57 cm. The peak powers of the two soli-ons and their frequency offsets, relative to the centralragg frequency, are equal to 1.8 kW, 1.7 kW, and 16.2Hz, 18.1 GHz, respectively. Figure 1 shows a comparisonf the intensity calculated by using the trial function
ˆs�z , t�= �u+�z , t��2+ �u−�z , t��2 and the intensity calculatedy using a numerical simulation Is�z , t�= �u+�z , t��2�u−�z , t��2. In this example, the trial function as well as
he numerical solution are in excellent quantitativegreement before the interaction [Fig. 1(a)], during thenteraction [Figs. 1(b) and 1(c)], and after the interactionFig. 1(d)]. Figures 1(b) and 1(c) show that the two Braggolitons spatially overlap during the interaction and even-ually the fast soliton passes through the slow soliton.igure 2 compares the amplitude and phase of the trial
unction with the results of a numerical solution for theorward and backward propagating waves that are ob-ained after the interaction. This figure indicates that af-er the interaction the numerically calculated solution u±nd the trial function u± are in quantitative agreement inoth the absolute value and phase. This means that inhis case the shifts in soliton location and the phasehanges due to the interaction are approximately theame for both the Thirring and Bragg solitons. Thehanges can be therefore calculated by using Eqs.18)–(21). The shifts in location and phase of soliton 1 dueo the interaction equal −0.97Ws and 0.57�, respectively.he shifts in location and phase of soliton 2 equal 0.87Wsnd 0.17�, respectively.
. Example II: Two Bragg Solitons Interchanging Theiroles After a Collision
n the second example, the soliton parameters are theame as in the first example but with �2,0= �10/9��. Fig-re 3 shows a comparison of the intensity of the trial func-ion Is�z , t� and the numerically calculated solution Is�z , t�.he trial function and the numerically calculated solutionre in a good quantitative agreement before the interac-ion [Fig. 3(a)], during the interaction [Figs. 3(b) and 3(c)],nd after the interaction [Fig. 3(d)]. During the interac-ion the two solitons do not pass one through the othernd they interact only due to an overlap in the solitonsails. After the interaction, the two solitons exchange
10 400
2
z/Ws
Pow
er(k
W)
(a)
40 700
2
z/Ws
Pow
er(k
W)
(b)
40 700
2
z/Ws
Pow
er(k
W)
(c)
60 900
2
z/Ws
Pow
er(k
W)
(d)
ig. 1. (Color online) Comparison between the intensities calcu-ated by the trial function Is�z , t� (dashed curve) and by using aumerical simulation Is�z , t� (solid curve) before the interactiona) at t=0, during the interaction (b) at t=50 �Ws /Vg� s and (c) at=70 �Ws /Vg�, and after the interaction (d) at t=105 �Ws /Vg�,here Ws=1.57 cm in the spatial width of the solitons ands /Vg=76 ps. The soliton parameters are �1=�2=0.2, v2=0.6, v10.45, � =� =0.
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Ftbia��s
Z. Toroker and M. Horowitz Vol. 27, No. 6 /June 2010/J. Opt. Soc. Am. B 1199
heir amplitudes and velocities and hence the soliton thatad a slower speed before the interaction (soliton 1) be-omes the faster soliton (soliton 2). Figure 4 compares themplitudes and the phases of the trial function and theumerically calculated wave after the interaction. Thegure indicates that, after the interaction, the amplitudesnd the phases of the numerically calculated solutions u±nd the trial function u± are in a good quantitative agree-ent. The shifts in location and phase of solitons 1 and 2
ue to the interaction that is given in Eqs. (18)–(21) arehe same as in the first example.
. Example III: Collision Between Two Bragg Solitonsauses Soliton Oscillationsn our next example the soliton parameters are�1 ,v1 ,�1,0�= �0.8,0.6,0� and ��2 ,v2 ,�2,0�= �0.8,0.8,0�. Thenput solitons have spatial FWHMs of 4.086 and 3.065
m. Hence, Ws=3.065 mm. The peak powers of the twoolitons and their frequency offsets relative to the Braggrequency equal to 25.85 kW, 21.7 kW, and 12.9 GHz, 17.2Hz, respectively. Figure 5 shows a comparison between
he intensity calculated by the trial function Is�z , t� andhe intensity Is�z , t� calculated by a numerical solution ofLCMEs. In this example, the two functions Is�z , t� and
70 900
1.5
z/Ws
|U+|[
(kW
)1/2 ]
70 900
1.5
z/Ws
|U−|[
(kW
)1/2 ]
70 90
−1
1
z/Ws
φ+/π
(rad
)
70 90
−1
1
z/Wsφ−
/π
(rad
)
(a)
(d)
(b)
(c)
ig. 2. (Color online) Comparison between the amplitudes andhases of the forward U+ and the backward U− propagatingaves that are calculated by the trial function (dashed curve)nd by the numerical solution (solid curve) at the end of the in-eraction at t=105 �Ws /Vg�, where Ws=1.57 cm and Ws /Vg76 ps. The soliton parameters are �1=�2=0.2, v2=0.6, v1=0.45,1,0=�2,0=0.
10 400
2.5
z/Ws
Pow
er(k
W)
(a)
40 700
2.5
z/Ws
Pow
er(k
W)
(b)
40 700
2.5
z/Ws
Pow
er(k
W)
(c)
60 900
2.5
z/Ws
Pow
er(k
W)
(d)
ig. 3. (Color online) Comparison between the intensity of therial function Is�z , t� (dashed curve) and the intensity calculatedy using a numerical simulation Is�z , t� (solid curve) before thenteraction at (a) t=0, during the interaction at (b) t=50 �Ws /Vg�nd (c) t=70 �Ws /Vg�, and after the interaction at (d) t=105Ws /Vg�, where Ws=1.57 cm and Ws /Vg=76 ps. The soliton pa-ameters are �1=�2=0.2, v2=0.6, v1=0.45, �1,0=0, and �2,0�10/9��.
s�z , t� are similar before the interaction [Fig. 5(a)], duringhe interaction [Figs. 5(b) and 5(c)], and after the interac-ion [Fig. 5(d)]. During the interaction the solitons inter-ct with each other only with their tails. After the inter-ction, the initially slow soliton (soliton 1) has similaroliton parameters ��2 ,v2� as the initially fast solitonsoliton 2), and the initially fast soliton (soliton 2) hasimilar soliton parameters ��1 ,v1� as the initially slowoliton (soliton 1). However, the interaction caused an os-illation in the amplitudes of the output pulses. The am-litude of the oscillation starts to decay after the solitonsontinue to propagate after the interaction. In order touantitatively study the oscillation we calculated numeri-ally after the interaction the peak power and the velocityf each soliton. To calculate the pulse velocities we had tond the exact spatial locations of the pulses at a givenime. We first calculated the average location of the pulsest time t= t1:
zcm�t1� =�−�
� zI�z,t1�dz
�−�� I�z,t1�dz
, �26�
here I�z , t�= �u+�z , t��2+ �u−�z , t��2 is the intensity. We de-ned the spatial position of soliton 1 to be z1�t1��−�
zcm�t1�zI�z , t1�dz /�−�zcm�t1�I�z , t1�dz and the spatial position
70 900
1.5
z/Ws
|U+|[
(kW
)1/2 ]
70 900
1.5
z/Ws
|U−|[
(kW
)1/2 ]
70 90
−1
1
z/Ws
φ+
/π
(rad
)
70 90
−1
1
z/Ws
φ−
/π
(rad
)
(a) (b)
(c) (d)
ig. 4. (Color online) Comparison between the amplitudes andhases of the forward U+ and the backward U− propagatingaves that are calculated by the trial function (dashed curve)nd by the numerical solution (solid curve) at the end of the in-eraction at t=105 �Ws /Vg�, where Ws=1.57 cm and Ws /Vg76 ps. The soliton parameters are �1=�2=0.2, v2=0.6, v1=0.45,1,0=0, and �2,0= �10/9��.
10 400
30
z/Ws
Pow
er(k
W)
(a)
30 600
30
z/Ws
Pow
er(k
W)
(b)
50 800
30
z/Ws
Pow
er(k
W)
(c)
80 1100
30
z/Ws
Pow
er(k
W)
(d)
ig. 5. (Color online) Comparison between the intensity of therial function Is�z , t� (dashed curve) and the intensity calculatedy using a numerical simulation Is�z , t� (solid curve) before thenteraction at (a) t=0, during the interaction at (b) t=30 �Ws /Vg�nd (c) t=55 �Ws /Vg�, and after the interaction at (d) t=95Ws /Vg�. The soliton parameters are �1=�2=0.8, v2=0.8, v1=0.6,1,0=0, and �2,0=0. Ws=3.065 mm is the spatial width of thehorter soliton and W /V =14.8 ps.
s g
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ERIisstvatdttt2s
F(ta(vt
Fdctd=
1200 J. Opt. Soc. Am. B/Vol. 27, No. 6 /June 2010 Z. Toroker and M. Horowitz
f soliton 2 to be z2�t1�=�zcm�t1�� zI�z , t1�dz /�zcm�t1�
� I�z , t1�dz.igure 6 compares the numerically calculated peak powerf the soliton, I�zi�t� , t�, with the trial function, I�zi�t� , t�,or the first soliton �i=1� [Fig. 6(a)] and for the secondoliton �i=2� [Fig. 6(b)]. The figure indicates that, afterhe interaction, the output pulses emerge with small am-litude oscillations (about 1.5% for pulse 1 and 1.4% forulse 2). Nevertheless, the shifts in pulse locations andhases are approximately the same as in the correspond-ng MTM solitons. Therefore, the shifts in location andhase of pulses can be calculated by using Eqs. (18)–(21).hese equations give shifts in location and phase of pulseby −2.1Ws and 0.37�. The shifts for pulse 2 equal
.57Ws and −0.39�, respectively.
. Example IV: Nonelastic Collisionsince the NLCMEs are a non-integrable system we couldnd a parameter regime where the Bragg solitons do notaintain their initial velocities and phases after the in-
eraction. Such collisions cannot be analyzed by using aransformation of the MTM solution. When two Braggolitons collide, the pulses after the interaction mayhange their total energy, their momentum, or they mayven disappear as was studied in previous works [4,11].uch collisions cannot be analyzed by using a transforma-ion of the MTM solution.
In [11] various types of inelastic collisions between twodentical counter-propagating Bragg solitons were studiedy using a numerical simulation. The inelastic collisionan cause the merging of two pulses into a single zero-elocity pulse and it can even cause the disappearance ofhe two incident pulses. In the case of quasi-elastic inter-ctions, two pulses emerge after the interaction with ve-ocities that can be lower or faster than their initial ve-ocities. Conspicuous spontaneous symmetry breakingfter the interaction was also observed [11]. In our simu-ations we have found that, after an inelastic collision be-ween two co-propagating Bragg solitons, either twoulses emerge or two incident pulses disappear. We haveound a parameter regime in which one of the emergingulses has a lower velocity and the other has a higher ve-ocity compared to the corresponding initial velocity be-ore the interaction. The amplitude of both pulses afterhe interaction decreases. In particular, we have foundn interaction that causes the inversion of the propaga-
50 1000
30
(Vgt)/Ws
Peak
pow
er(k
W)
(a)
50 1000
30
(Vgt)/Ws
Peak
pow
er(k
W)
(b)
ig. 6. (Color online) Comparison between the peak powers ofa) soliton 1 and (b) soliton 2 as a function of normalized time af-er the collision by solving numerically the NLCMEs (solid curve)nd by using the trial function based on MTM interactiondashed curve). The soliton parameters are �1=�2=0.8, v2=0.8,1=0.6, �1,0=0, and �2,0=0. Ws=3.065 mm is the spatial width ofhe shorter soliton and W /V =14.8 ps.
s g
ion direction of one of the interacting solitons. Such annteraction cannot be obtained by a transformation ofhe MTM solution. In this example the soliton para-eters are ��1 ,v1 ,�1,0�= �1.2,0.0075,0� and ��2 ,v2 ,�2,0��1.2,0.01,0�. The input solitons have a spatial FWHM ofs=3.6 mm. The peak powers of the two solitons and
heir frequency offsets, relative to the Bragg frequency,re equal to 59.77784 kW, 59.778 kW, and 5.3695 GHz,.3696 GHz, respectively. Figure 7 shows the collision be-ween the two Bragg solitons in a three-dimensional plotFig. 7(a)] and in a two-dimensional plot [Fig. 7(b)]. Thegure indicates that the interaction changes the propaga-ion direction of the second soliton (soliton 2). Figure 8hows the intensity Is as a function of the location beforehe interaction [Fig. 8(a)], during the interaction [Figs.(b) and 8(c)], and after the interaction [Fig. 8(d)]. Duringhe interaction the two solitons do not pass one throughhe other, and the interaction occurs only due to the over-ap between the soliton tails as can be seen in Figs. 8(b)nd 8(c). After the interaction, the faster soliton (soliton) reverses its direction, whereas the slow soliton (soliton) continues in its original direction. This is a new ob-erved interaction between two Bragg solitons which dif-ers from the transformed MTM function u±. Neverthe-ess, after the interaction, the solution u± still seems toonsist of two solitary waves.
. Comparison to Numerical Simulation in a Parameteregion „�2 ,v2… of the Interacting Solitons
n this subsection we demonstrate that our trial functions accurate in a broad parameter region of the interactingolitons. In these simulations the parameters of one of theolitons (soliton 2) were varied over a broad region andhe parameters of the second soliton (soliton 1) were set to1=0.75v2, �1=�2, and �1,0=0. Because elastic collisionsre also affected by the initial phase difference betweenhe Bragg solitons [11], we studied the collision for nineifferent relative phases between the two incident soli-ons. For each parameter set ��2 ,v2�, we have calculatedhe interaction for nine different phases of soliton 2, �2,0,hat were varied between 0 and 2� with a step-size of� /9. After the interaction, when the two solitons are wellpatially separated, we calculate, numerically, the veloc-
ig. 7. (Color online) (a) Three-dimensional and (b) two-imensional plots of a collision between two solitary waves cal-ulated by solving numerically the NLCMEs. After the interac-ion one of the solitons (soliton 2) changes its propagationirection. The soliton parameters are �1=�2=1.2, v2=0.01, v10.0075, � =� =0.
1,0 2,0
ipaz�pu
wt
w
btarysyssuTtgp
5WstttcNpsfisctatbiscplt
esTeTtpw
ASIlgg
Ftwad
Faceatwestdrwt
Z. Toroker and M. Horowitz Vol. 27, No. 6 /June 2010 /J. Opt. Soc. Am. B 1201
ty and the peak power of each soliton. We define the peakower of pulse i �i=1,2� at t= t1 to be Ii,p�t1�=I�zi�t1� , t1�nd the velocity to be vi�t1�= �zi�t1+�T�−zi�t1�� /�T, wherei�t1� are calculated as described in Subsection 4.C andT is the numerical step-size of the time. For each outputulse we have calculated the relative error between
ˆ ±�z , t� and u±�z , t� after the interaction by
�i = maxt0�t�t1
�� Ii�t� − Ii�t�
Ii�t��,� vi�t� − vi�t�
vi�t��� , �27�
here t0 and t1 are chosen such that the distance betweenhe solitons equals 5Ws and 10Ws, respectively.
Figure 9 demonstrates that the trial function is validhen the parameters of the soliton are varied over a
10 600
50
z/Ws
Pow
er(k
W) (a)
10 600
50
z/Ws
Pow
er(k
W) (b)
10 600
50
z/Ws
Pow
er(k
W) (c)
10 600
50
z/Ws
Pow
er(k
W) (d)
ig. 8. (Color online) Collision between two Bragg solitons at (a)=0, (b) t=320 �Ws /Vg�, (c) t=420 �Ws /Vg�, (d) t=1745 �Ws /Vg�,here Ws=3.6 mm and Ws /Vg=17.5 ps. The soliton parametersre �1=�2=1.2, v2=0.01, v1=0.0075, �1,0=�2,0=0. The arrows in-icate the propagation direction of the solitons.
0.1 0.5 0.9
0.1
1
1.8
v2
ρ2
ig. 9. (Color online) Comparison between the trial functionnd the numerical simulation for two Bragg soliton interactionalculated for several parameters of soliton 2 ��2 ,v2�. The param-ters of the first soliton (soliton 1) were set to v1=0.75v2, �1=�2,nd �1,0=0. For each parameter set ��2 ,v2�, we have calculatedhe interaction for nine different phases of soliton 2, �2,0, whichere varied between 0 and 2� with a step-size of 2� /9. A param-ter set ��2 ,v2� that gives at the end of the interaction an errormaller than 10% for both solitons for all the nine phases �2,0hat were checked is marked with a circle. Squares representata points in which eight of the nine phases of �2,0 yielded aelative error less than 10%. Diamonds represent data points inhich less than eight phases of �2,0 yielded relative error less
han 10%.
road region. A circle represents a parameter set ��2 ,v2�hat gives an error smaller than 10% for both solitons forll the nine phases �2,0 that were checked. A square rep-esents data point in which eight of the nine phases of �2,0ielded a relative error less than 10%. A diamond repre-ents data point in which less than eight phases �2,0ielded with a relative error less than 10%. Figure 9hows that there is a broad region in which the two Braggolitons collide quasi-elastically and the trial function
ˆ ±�z , t� can explicitly and accurately describe the collision.his region is an important region of soliton parameters
hat can be realized experimentally. Moreover, in this re-ion one can explicitly calculate the shifts in locations andhases that occur as a result of the collision.
. CONCLUSIONe have studied the collision of two co-propagating Bragg
olitons in an infinite uniform FBG. We have presented arial function to describe in important cases the interac-ion between two Bragg solitons by using the known solu-ion to the interaction between two Thirring solitons. Weompared our trial function to a numerical solution of theLCMEs. We found a parameter regime in which the ex-licit function is in a good agreement with the numericalolution. Hence, the trial function enables one, for therst time, to describe explicitly the collision of two Braggolitons in an important region of soliton parameters thatan be realized experimentally. This indicates that al-hough the NLCMEs are a non-integrable system there is
parameter regime in which Bragg solitons collide likerue solitons, as occurs in the MTM. The trial connectionetween the MTM and the NLCMEs enables understand-ng important physical effects such as the exchange in theoliton parameters due to the interaction or a shift in lo-ation and in phase obtained after the interaction. The ex-licit function also enables one to calculate the shifts inocations and phases that occur when the collision be-ween the Bragg solitons is quasi-elastic.
Since the NLCMEs are not an integrable system ofquations the interaction between Bragg solitons can beubstantially different from the interaction betweenhirring solitons. Nevertheless, Bragg solitons canmerge from a collision as solitary waves in a way thathirring solitons cannot. We report on an interaction be-
ween Bragg solitons that causes an inversion of theropagation direction of one of the interacting solitaryaves.
PPENDIX: TWO-THIRRING SOLITONOLUTION
n this appendix we describe the two-Thirring soliton so-ution of the MTM obtained by the IST. We begin with theeneral solution of the equations of the MTM which isiven by
�−�z,t� = −K1�z,t�e−i��z,t�
� ,
���
w N
w
T
w
1202 J. Opt. Soc. Am. B/Vol. 27, No. 6 /June 2010 Z. Toroker and M. Horowitz
�+�z,t� = −i���z − Vg
−1�t�K1�z,t��e−i��z,t�
�����, �A1�
here
���−,�+� = − 2��
�
�K1��−, �+��2d�+, �A2�
+
1 − vi
w
ATt
R
�± =1
2�z ± Vgt�. �A3�
ow, the two-Thirring soliton solution is given by
K1�z,t� = f1�z,t� + f2�z,t�, �A4�
here the functions f �z , t� and f �z , t� are equal to
1 2
f1�z,t� = �1
1 +1
�2�ei�2−22
cosh�1 − i�1
2 �
��
2�1 sin��1�e−i1
+
ei�2−22
��1��1 +1
�2�ei�2−22� cosh�1 − i
�1
2− ln��1�� − D1�z,t�
��
2�1 sin��1�e−i1 �
−1
, �A5�
f2�z,t� = ��1ei�1−21
1 + �1ei�1−21cosh�2 − i
�2
2 ���
2�2 sin��2�e−i2
+
�1
��1��1 + �1ei�1−21�cosh�2 − i
�2
2− ln��1�� − D2�z,t�
��
2�2 sin��2�e−i2 �
−1
. �A6�
he functions D1�z , t� and D2�z , t� are defined as follows:
here i=1,2 is the soliton index before the interaction.
CKNOWLEDGMENThis work was supported by the Israel Science Founda-
ion (ISF) of the Israeli Academy of Sciences.
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