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Chirp in forward stimulated Brillouin quasi-solitons
1 February 1995
OPTICS COMMUNICATIONS Optics Communications 114( 1995) 309-3 14
Chirp in forward stimulated Brillouin quasi-solitons Jiri Teichmann, Daniel Brabant D&artement de Physique, UniversitP de Montreal, P.O. Box 6128, Station A, Montrkai, Canada
Received 8 February 1994; revised version received 3 1August 1994
Abstract The copropagation of pump, Stokes, anti-Stokes and acoustical quasi-solitons in a uniform nonlinear medium is studied, taking into account transversal diffraction and related self- and cross-modulation in the case of small reciprocal Fresnel numbers. The corresponding spatial distribution of frequency chirp is calculated.
1. Introduction It has been recently recognised, both theoretically [ 1,2] and experimentally [ 3 1, that the regime of forward stimulated Brillouin scattering (FSBS) can be realised under full pump depletion provided the scattered waves can interact for a sufficient period of time. Such a situation takes place when the initial optical pump pulse as well as the scattered Stokes and antiStokes pulses have a soliton-like energy distribution in time and copropagate in the same or nearly the same direction. Similarly to the case of transient solitons, generated during stimulated Raman scattering, the transverse effects due to diffraction and mutual nonlinear interaction between copropagating pulses play an important role in the deformation of pulse phases and in reshaping of their energy distribution in all three dimensions. The mechanism of pulse reshaping has been previously described in Refs. [ 4-8 1. It has been found that the steepness of the initial radial distribution of energy of the pump as well as the magnitude of the Fresnel number are the most important parameters in the reshaping process. The most critical regions are the cylindrical shells having radii close to the waist thickness of the Gaussian profile (of any
order) of the pump. In the vicinity of these shells, important phase variations take place. It has already been shown in the case of SRS, both analytically [ 4,7] and using numerical simulations [ 5,6], and in the case of SBS [ 81, that radial energy currents (outwards via diffraction effects, inwards via nonlinearity) develop, leading to radial redistribution of energy inside each pulse and to the formation of Fresnel rings. The phase variations result in radially and axially dependent chirp in optical pulses.
2. The analytical formulation Following standard procedure [ 9, lo] in developing the SBS equations, we assume parallel propagation of pulses and linear polarisation of optical waves. We neglect thermal effects and we restrict ourselves to the case of phonon excitation by electrostriction only. The resulting coupled envelope equations (SWEA) read:
0030-4018/95/$09.50 0 1995 Elsevier Science B.V. All rights reserved SSDIOO30-4018(94)00597-4
310
J. Teichmann. D. Brabant /Optics Communications 114 (1995) 309-314
and the acoustic wave amplitude U,(9,p)= = 9
[EsU,exp(
-iAkz)
L
-E,Usexp(iAkz)]
,
(Ia)
$.
(if)“Ur)(q,p).
(3)
We assume that the conditions of energy and momentum conservation are satisfied, ws=wL-wu, wA=tiL+au, ks=kL-ku, kA=kL+kU, &=o. The zeroth-order system (Ak= 0, oj= 0) aEt"(ap)/all=~~[ES"(~,~)
=
[ELU;exp(
-'Em'
-iAkz)]
,
(lb)
s
A
=
EA A
A
,
p)lh=
au;‘%,
admits (ELEtSEAEt)
exp( -iAkz)
,
(IdI
where Ej( r, t) are the amplitudes of the electric fields of the pump (L), Stokes wave (S) and the anti-Stokes pulse (A); CJp(r, t) is the amplitude of the acoustic wave; c is the electrostriction coefficient; VT is the transversal Laplacian, Ol, and j=L, S, A, U are the dissipation coefficients. Assuming that the phase velocities of the pump, Stokes and the anti-Stokes waves are nearly identical, we transform the system ( 1) into the retarded frame of reference with variables q=z, r = t-z/v,_. We introduce dimension-less quantities, normalising all speeds to the pump group velocity vi_, acoustical amplitude U, to the length of propagation L, field amplitudes to the initial amplitude of the pump, K0 and the coupling coefficients to L2. We introduce the normalized radius p=r/r,,, r. being the waist thickness of the pump pulse and the normalised Fresnel number Fj = 1/J; = 2kjri/L. If the reciprocal Fresnel number, 4, is sufficently small, J CK 1, the system ( 1) can be solved using the perturbation method proposed in Ref. [4]. We develop the field amplitudes into power series in f; cf=fL=.&=fA): Ej(q>P)=
f ?I=0
(da)
(if)“E,‘“)(o,p),
(4b)
j=L,
P)
&A,
(2)
> UkO)(rt,p)
aEaP)(rl,p)larl=KAE!O’(rl,P)
+Ei”(rl,
- 2
~~“)*(wH >
= --~sE~~~(rl, P) U;“‘*(%
)
[ELUpexp(iAkz)]
9
-Ei"'(rl,~)
U$"'h~)
WO’(rl, P)larl
$+‘$-+‘:+cx~ (
into
--G[E!~)(v>
,
(4c)
P) E&“*(rl, P)
P) E!“*(rl, P) I >
several particular
solutions.
(Ad) Defining
E,( q,
P)=A,(II, P) exP]i@j(C P)l, QCS, P)=&(rl, P) exp [ iou ( II, p) 1, j= L, S, A, we can find the simplest solution by taking OS + @o - QL = 0, @A- QL+ au = 0 in the form: A,(% P) =K,(P)
tanh[Ko(p),/n
rll T (5a)
&(rl, P) =K&o(P)
sech[Ko(p),/m
rl , (5b)
AA(v,P)=-KAKo(P)
sech[&(P)~~vl, (5c)
AJ(v, P) =&K,(P)
sech[Ko(p),/~
rll , (5d)
with K=
J&,
i=S,A, %(%-KA)
Ku= J
KL(KS+KA)
'
In the zeroth order approximation, the initial phases @j are constant. As follows from Eqs. (5 ) the pump energy is represented as a dark soliton; the
J. Teichmann, D. Brabant /Optics Communications 114 (1995) 309-314
scattered waves and the acoustical wave are bright solitons with corresponding amplitudes. The first and higher order system involve the transversal Laplacian: aEt”/a?/-v:
Eg)P’= IC~(E&~‘U;‘)+E&‘)U;~) ,
+E~O’U$l’*-E$,“U$o’) aEQ”/aq-V:
(6a)
The total field amplitudes are complex, Ej(rI,P)=E,(q,P)
exp[~(rl,p)l ,
j=L,S,A
(9a)
W~P)=Q~(V~P) exp[%(u,p)l
.
(9b)
The phases Yj( II,p) are now functions of radius and propagation distance:
E&O’ ,
=K~(E~~)U~‘)*-E~‘)U~O)) aEa”/aq-v:
311
(6b) fi(‘l’P)=arctan
El”’
=K~(E~O)U~‘)+E~‘)U~O)),
-,3EJ(3) +ocf’) Ejo) _fzEj2) +O(j-4)
>
)
(loa)
(6~)
au~‘)/atl=KU(Eto)E~‘)*-E~‘)E~o) .
+E~“‘Et”*-EA”E~oP’)
(6d)
(lob)
The parameter K,(p) in the zeroth order solution (5) allows one to specify the initial radial profile of
the pump amplitude. We assume the initial radial profile to be a Gaussian of the sth order: K,(p) =K, exp( -p’“). The system (6) is solved via consecutive iterations. For the first iteration, one can take the free space Fresnel diffraction term: n
I$“(Q~)=
The spatial dependence of the phases is due to Fresnel diffraction as well as to nonlinear coupling between the pulses. The radial gradient of the phases !I$ generates a corresponding flow of energy, J,,( ?,I,p), inside the pulses, given by Jj,,(q,p)=
~d~‘V:EIO’(~,p),
j=L,S,A.
[ (E’“))2+(fE”))2a~(q~P) I
0
This allows one to calculate all other amplitudes, including the diffracted amplitude of the acoustic pulse up to the desired order. Thus the next iteration reads:
ap
,
(8a)
n
E&')=,!?~')+Ks
d?'(E!O'U~')-~.I')U~O))
(8b)
dq’ (E~“‘U~“+~~‘W~o))
(8~)
s 0
n
E&,$')+KA 0 ”
UC” P
dq’ (E~op’I?~‘) -&“E&o’
=Ku 0
+E~“‘,??‘t” -,!?~“E~“O’) .
(8d)
.
However, the phases Yjare functions of r as well. The angular frequency variation across the pulses can be obtained from the expression:
,,$o) _,2EJ(2) +Ea”)U;‘)-@)U;o))
(11)
J
(7)
+Ocf”)
)I .
(12)
The resulting analytical expression for 60, is rather cumbersome to be presented here. The frequency variations in Eq. ( 12) are due mainly to the crossphase modulation (XPM), which is excited by the nonlinear cross product between the pump, Stokes and acoustic waves. The XPM is radially dependent and increases with the propagation distance. Fig. 1 shows the evolution of the radial current JL for the super-Gaussian (s= 2) pump pulse. The parameter p=4flKo. A change in the direction of the energy flow in the neighbourhood of p= 1 is clearly visible. Fig. 2 shows the frequency chirp for the pump wave and the Gaussian profile s= 1. The chirp of the
312
J. Teichmann, D. Brabant/Optics Communications114 (199s) 309-314
Fig. I. The radial energy flow in the super-Gaussian pump pulse (S=2),~o=~o[Kv(Ks-KA)]“*.
Fig. 3. Chirp &(q, pulse,p=O.Ol.
p) for the Stokes wave, Gaussian (s= 1)
s=2
p=O.Ol
Fig. 2. Chirp &( q,p) for the pump wave, Gaussian (s= 1) pulse, p=O.Ol, -0.7181cbJK~pSO.463.
corresponding Stokes wave, I&, is shown in Fig. 3. Fig. 4 represents the chirp variations for the superGaussian (s=2) laser pulse. The chirp of the corresponding excited Stokes quasi-soliton is shown in Fig. 5. The frequency variations are enhanced, for any value of s, in shells p N 1, where Fresnel rings are formed [4-81.
Fig. 4. Chirp &( v,p) for the super-Gaussian pump wave (s=2), p=O.Ol, -0.38~6~JK~p10.454.
3. Conclusion Using a three-dimensional model, we have studied the copropagation of short pulses generated during the stimulated Brillouin scattering and excited by a soliton-like pump field with a radical Gaussian (rth order) distribution of energy. Similarly to the case of
J. Teichmann, D. Brabant /Optics Communications 114 (199s) 309-314
h/‘Kvsp
s=2
p=OOl
0
J ‘0
Fig. 5. Chirp Gws(q, p) for the Stokes wave and the super-Gaussian pulse (s=2), p=O.Ol, O.OI&&K@I 1.589. K= (@-K~)/ (us-K,+)=lS.lS.
SRS [ 71, the initial soliton-like pulses of the scattered Stokes and anti-Stokes waves, the acoustic wave and the dark soliton pump become unstable due to radial redistribution of their energies. This redistribution of energy is caused in the first place by Fresnel diffraction, which is proportional to fc [ exp( -pa) 1. The corresponding pulse deformation grows with larger values of the reciprocal Fresnel number f and with larger order (s) of the Gaussian profile. The nonlinear evolution within the pulses and the nonlinear cross-phase modulation are related to the nonlinear coupling terms in Eq. ( 1). There are several mechanisms involved in the process: The nonuniform distribution of pulse energies in the pulse cross section induces radially nonuniform phase variations. Cylindrical shells of different radii propagate with different speed. Due to the nonlinearities of the propagation equations, each shell in each of the pulses interacts nonlinearly with shells in their neighborhood. Due to strong coupling between pulses, shells from pump and scattered pulses interact mutually. As a consequence of these interactions, phase surfaces, assumed to be planes at the entrance into the nonlinear medium, became radially deformed. Radial phase gradients
313
induce radial energy flow in pulses, outward due to diffraction and inward due to nonlinear coupling [ 471. XPM induced chirp is radially dependent. Different cylindrical shells have different spectral broadening: red or blue shifted, depending on p, q and the initial shape of the pump in the radial direction. The radial steepness of the initial pump profile is the primary parameter in the process, together with the magnitude of the Fresnel number. The most rapid change of the frequency takes place in the vicinity of the shells having radius p= 1, i.e. equal to the waist thickness of the Gaussian profile of any order where c [ exp( -pa) ] goes through zero. The phase changes and the induced chirp are more pronounced in the Stokes pulses and for higher order Gaussian profiles. Although our analytical description covers only the initial phase of the pulse evolution, it can be concluded that the described processes grow with the propagation distance. We have examined other solutions to the system ( 1) with similar results [ lo]. Similar deformation of optical pulses was observed in Ref. [ 111 in the case of free propagation in resonant degenerate atomic vapor. It must be noted that the described pulse reshaping is fundamentally different from the collapse of Langmuir solitons in plasmas, e.g. Ref. [ 12 1.
Acknowledgement
This work was supported in part by a research grant from NSERC of Canada.
References [ 1] A. Corvo and A. Gavrielides, J. Appl. Phys. 63 (1988) 5220. [2] P. Russell, D. Culverhouse and F. Farahi, IEEE J. Quantum Electron. 27 (1991) 836. [ 31 P. Russell, D. Culverhouse and F. Farahi, Electron. Lett. 26 (1990) 1195. [4] J. Teichmann, Bull. Am. Phys. Sot. 33 ( 1988) 1240. [ 5 ] J. Teichmann and F.P. Mattar, AIP Conf. Proc. 19 1 ( 1991) 529. [6] F.P. Mattar, C.M. Burgess, Y. Claude and J. Teichmann, AIPConf. Proc. 191 (1989) 164,219,249. [ 71 J. Teichmann, A. Roussel, S. Ethier and M. Teichmann, Can. I. Phys. (1990) 1296. [ 8 ] D. Brabant and J. Teichmann, 60’ Con& de I’ACFAS (1992).
J. Teichmann, D. Brabant /Optics Communications 114 (1995) 309-314
314
[ 9 ] A. Yariv, Quantum Electronics (Wiley, New York, 1975 )
[ lO]D. Brabant and J. Teichmann, 62” Congrks de 1’ACFAS (1994).
[ 111 P. Kupecek, M. Comte, J.P. Marinier, J.P. Babuel-Peyrissac and C. Bardin, Optics Comm. 65 (1988) 306.
[ 121 A.I. Dyachenko, V.E. Zakharov, A.M. Rubenchik, R.I. Sagdeev and V.I. Shvets, Soviet Phys. JETP 67 ( 1988) 5 13.
OPTICS COMMUNICATIONS Optics Communications 114( 1995) 309-3 14
Chirp in forward stimulated Brillouin quasi-solitons Jiri Teichmann, Daniel Brabant D&artement de Physique, UniversitP de Montreal, P.O. Box 6128, Station A, Montrkai, Canada
Received 8 February 1994; revised version received 3 1August 1994
Abstract The copropagation of pump, Stokes, anti-Stokes and acoustical quasi-solitons in a uniform nonlinear medium is studied, taking into account transversal diffraction and related self- and cross-modulation in the case of small reciprocal Fresnel numbers. The corresponding spatial distribution of frequency chirp is calculated.
1. Introduction It has been recently recognised, both theoretically [ 1,2] and experimentally [ 3 1, that the regime of forward stimulated Brillouin scattering (FSBS) can be realised under full pump depletion provided the scattered waves can interact for a sufficient period of time. Such a situation takes place when the initial optical pump pulse as well as the scattered Stokes and antiStokes pulses have a soliton-like energy distribution in time and copropagate in the same or nearly the same direction. Similarly to the case of transient solitons, generated during stimulated Raman scattering, the transverse effects due to diffraction and mutual nonlinear interaction between copropagating pulses play an important role in the deformation of pulse phases and in reshaping of their energy distribution in all three dimensions. The mechanism of pulse reshaping has been previously described in Refs. [ 4-8 1. It has been found that the steepness of the initial radial distribution of energy of the pump as well as the magnitude of the Fresnel number are the most important parameters in the reshaping process. The most critical regions are the cylindrical shells having radii close to the waist thickness of the Gaussian profile (of any
order) of the pump. In the vicinity of these shells, important phase variations take place. It has already been shown in the case of SRS, both analytically [ 4,7] and using numerical simulations [ 5,6], and in the case of SBS [ 81, that radial energy currents (outwards via diffraction effects, inwards via nonlinearity) develop, leading to radial redistribution of energy inside each pulse and to the formation of Fresnel rings. The phase variations result in radially and axially dependent chirp in optical pulses.
2. The analytical formulation Following standard procedure [ 9, lo] in developing the SBS equations, we assume parallel propagation of pulses and linear polarisation of optical waves. We neglect thermal effects and we restrict ourselves to the case of phonon excitation by electrostriction only. The resulting coupled envelope equations (SWEA) read:
0030-4018/95/$09.50 0 1995 Elsevier Science B.V. All rights reserved SSDIOO30-4018(94)00597-4
310
J. Teichmann. D. Brabant /Optics Communications 114 (1995) 309-314
and the acoustic wave amplitude U,(9,p)= = 9
[EsU,exp(
-iAkz)
L
-E,Usexp(iAkz)]
,
(Ia)
$.
(if)“Ur)(q,p).
(3)
We assume that the conditions of energy and momentum conservation are satisfied, ws=wL-wu, wA=tiL+au, ks=kL-ku, kA=kL+kU, &=o. The zeroth-order system (Ak= 0, oj= 0) aEt"(ap)/all=~~[ES"(~,~)
=
[ELU;exp(
-'Em'
-iAkz)]
,
(lb)
s
A
=
EA A
A
,
p)lh=
au;‘%,
admits (ELEtSEAEt)
exp( -iAkz)
,
(IdI
where Ej( r, t) are the amplitudes of the electric fields of the pump (L), Stokes wave (S) and the anti-Stokes pulse (A); CJp(r, t) is the amplitude of the acoustic wave; c is the electrostriction coefficient; VT is the transversal Laplacian, Ol, and j=L, S, A, U are the dissipation coefficients. Assuming that the phase velocities of the pump, Stokes and the anti-Stokes waves are nearly identical, we transform the system ( 1) into the retarded frame of reference with variables q=z, r = t-z/v,_. We introduce dimension-less quantities, normalising all speeds to the pump group velocity vi_, acoustical amplitude U, to the length of propagation L, field amplitudes to the initial amplitude of the pump, K0 and the coupling coefficients to L2. We introduce the normalized radius p=r/r,,, r. being the waist thickness of the pump pulse and the normalised Fresnel number Fj = 1/J; = 2kjri/L. If the reciprocal Fresnel number, 4, is sufficently small, J CK 1, the system ( 1) can be solved using the perturbation method proposed in Ref. [4]. We develop the field amplitudes into power series in f; cf=fL=.&=fA): Ej(q>P)=
f ?I=0
(da)
(if)“E,‘“)(o,p),
(4b)
j=L,
P)
&A,
(2)
> UkO)(rt,p)
aEaP)(rl,p)larl=KAE!O’(rl,P)
+Ei”(rl,
- 2
~~“)*(wH >
= --~sE~~~(rl, P) U;“‘*(%
)
[ELUpexp(iAkz)]
9
-Ei"'(rl,~)
U$"'h~)
WO’(rl, P)larl
$+‘$-+‘:+cx~ (
into
--G[E!~)(v>
,
(4c)
P) E&“*(rl, P)
P) E!“*(rl, P) I >
several particular
solutions.
(Ad) Defining
E,( q,
P)=A,(II, P) exP]i@j(C P)l, QCS, P)=&(rl, P) exp [ iou ( II, p) 1, j= L, S, A, we can find the simplest solution by taking OS + @o - QL = 0, @A- QL+ au = 0 in the form: A,(% P) =K,(P)
tanh[Ko(p),/n
rll T (5a)
&(rl, P) =K&o(P)
sech[Ko(p),/m
rl , (5b)
AA(v,P)=-KAKo(P)
sech[&(P)~~vl, (5c)
AJ(v, P) =&K,(P)
sech[Ko(p),/~
rll , (5d)
with K=
J&,
i=S,A, %(%-KA)
Ku= J
KL(KS+KA)
'
In the zeroth order approximation, the initial phases @j are constant. As follows from Eqs. (5 ) the pump energy is represented as a dark soliton; the
J. Teichmann, D. Brabant /Optics Communications 114 (1995) 309-314
scattered waves and the acoustical wave are bright solitons with corresponding amplitudes. The first and higher order system involve the transversal Laplacian: aEt”/a?/-v:
Eg)P’= IC~(E&~‘U;‘)+E&‘)U;~) ,
+E~O’U$l’*-E$,“U$o’) aEQ”/aq-V:
(6a)
The total field amplitudes are complex, Ej(rI,P)=E,(q,P)
exp[~(rl,p)l ,
j=L,S,A
(9a)
W~P)=Q~(V~P) exp[%(u,p)l
.
(9b)
The phases Yj( II,p) are now functions of radius and propagation distance:
E&O’ ,
=K~(E~~)U~‘)*-E~‘)U~O)) aEa”/aq-v:
311
(6b) fi(‘l’P)=arctan
El”’
=K~(E~O)U~‘)+E~‘)U~O)),
-,3EJ(3) +ocf’) Ejo) _fzEj2) +O(j-4)
>
)
(loa)
(6~)
au~‘)/atl=KU(Eto)E~‘)*-E~‘)E~o) .
+E~“‘Et”*-EA”E~oP’)
(6d)
(lob)
The parameter K,(p) in the zeroth order solution (5) allows one to specify the initial radial profile of
the pump amplitude. We assume the initial radial profile to be a Gaussian of the sth order: K,(p) =K, exp( -p’“). The system (6) is solved via consecutive iterations. For the first iteration, one can take the free space Fresnel diffraction term: n
I$“(Q~)=
The spatial dependence of the phases is due to Fresnel diffraction as well as to nonlinear coupling between the pulses. The radial gradient of the phases !I$ generates a corresponding flow of energy, J,,( ?,I,p), inside the pulses, given by Jj,,(q,p)=
~d~‘V:EIO’(~,p),
j=L,S,A.
[ (E’“))2+(fE”))2a~(q~P) I
0
This allows one to calculate all other amplitudes, including the diffracted amplitude of the acoustic pulse up to the desired order. Thus the next iteration reads:
ap
,
(8a)
n
E&')=,!?~')+Ks
d?'(E!O'U~')-~.I')U~O))
(8b)
dq’ (E~“‘U~“+~~‘W~o))
(8~)
s 0
n
E&,$')+KA 0 ”
UC” P
dq’ (E~op’I?~‘) -&“E&o’
=Ku 0
+E~“‘,??‘t” -,!?~“E~“O’) .
(8d)
.
However, the phases Yjare functions of r as well. The angular frequency variation across the pulses can be obtained from the expression:
,,$o) _,2EJ(2) +Ea”)U;‘)-@)U;o))
(11)
J
(7)
+Ocf”)
)I .
(12)
The resulting analytical expression for 60, is rather cumbersome to be presented here. The frequency variations in Eq. ( 12) are due mainly to the crossphase modulation (XPM), which is excited by the nonlinear cross product between the pump, Stokes and acoustic waves. The XPM is radially dependent and increases with the propagation distance. Fig. 1 shows the evolution of the radial current JL for the super-Gaussian (s= 2) pump pulse. The parameter p=4flKo. A change in the direction of the energy flow in the neighbourhood of p= 1 is clearly visible. Fig. 2 shows the frequency chirp for the pump wave and the Gaussian profile s= 1. The chirp of the
312
J. Teichmann, D. Brabant/Optics Communications114 (199s) 309-314
Fig. I. The radial energy flow in the super-Gaussian pump pulse (S=2),~o=~o[Kv(Ks-KA)]“*.
Fig. 3. Chirp &(q, pulse,p=O.Ol.
p) for the Stokes wave, Gaussian (s= 1)
s=2
p=O.Ol
Fig. 2. Chirp &( q,p) for the pump wave, Gaussian (s= 1) pulse, p=O.Ol, -0.7181cbJK~pSO.463.
corresponding Stokes wave, I&, is shown in Fig. 3. Fig. 4 represents the chirp variations for the superGaussian (s=2) laser pulse. The chirp of the corresponding excited Stokes quasi-soliton is shown in Fig. 5. The frequency variations are enhanced, for any value of s, in shells p N 1, where Fresnel rings are formed [4-81.
Fig. 4. Chirp &( v,p) for the super-Gaussian pump wave (s=2), p=O.Ol, -0.38~6~JK~p10.454.
3. Conclusion Using a three-dimensional model, we have studied the copropagation of short pulses generated during the stimulated Brillouin scattering and excited by a soliton-like pump field with a radical Gaussian (rth order) distribution of energy. Similarly to the case of
J. Teichmann, D. Brabant /Optics Communications 114 (199s) 309-314
h/‘Kvsp
s=2
p=OOl
0
J ‘0
Fig. 5. Chirp Gws(q, p) for the Stokes wave and the super-Gaussian pulse (s=2), p=O.Ol, O.OI&&K@I 1.589. K= (@-K~)/ (us-K,+)=lS.lS.
SRS [ 71, the initial soliton-like pulses of the scattered Stokes and anti-Stokes waves, the acoustic wave and the dark soliton pump become unstable due to radial redistribution of their energies. This redistribution of energy is caused in the first place by Fresnel diffraction, which is proportional to fc [ exp( -pa) 1. The corresponding pulse deformation grows with larger values of the reciprocal Fresnel number f and with larger order (s) of the Gaussian profile. The nonlinear evolution within the pulses and the nonlinear cross-phase modulation are related to the nonlinear coupling terms in Eq. ( 1). There are several mechanisms involved in the process: The nonuniform distribution of pulse energies in the pulse cross section induces radially nonuniform phase variations. Cylindrical shells of different radii propagate with different speed. Due to the nonlinearities of the propagation equations, each shell in each of the pulses interacts nonlinearly with shells in their neighborhood. Due to strong coupling between pulses, shells from pump and scattered pulses interact mutually. As a consequence of these interactions, phase surfaces, assumed to be planes at the entrance into the nonlinear medium, became radially deformed. Radial phase gradients
313
induce radial energy flow in pulses, outward due to diffraction and inward due to nonlinear coupling [ 471. XPM induced chirp is radially dependent. Different cylindrical shells have different spectral broadening: red or blue shifted, depending on p, q and the initial shape of the pump in the radial direction. The radial steepness of the initial pump profile is the primary parameter in the process, together with the magnitude of the Fresnel number. The most rapid change of the frequency takes place in the vicinity of the shells having radius p= 1, i.e. equal to the waist thickness of the Gaussian profile of any order where c [ exp( -pa) ] goes through zero. The phase changes and the induced chirp are more pronounced in the Stokes pulses and for higher order Gaussian profiles. Although our analytical description covers only the initial phase of the pulse evolution, it can be concluded that the described processes grow with the propagation distance. We have examined other solutions to the system ( 1) with similar results [ lo]. Similar deformation of optical pulses was observed in Ref. [ 111 in the case of free propagation in resonant degenerate atomic vapor. It must be noted that the described pulse reshaping is fundamentally different from the collapse of Langmuir solitons in plasmas, e.g. Ref. [ 12 1.
Acknowledgement
This work was supported in part by a research grant from NSERC of Canada.
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