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Polarization domains and instabilities in nonlinear optical fibers
Physics Letters A 182 ( 1993) 289-293 North-Holland
Polarization
PHYSICS
LETTERS
A
domains and instabilities in nonlinear optical fibers
S. Wabnitz and B. Daino Fondazione UgoBordoni, Via B. Castiglione59, 00142 Rome, Italy Received 13 August 1993; accepted for publication 17 September 1993 Communicated by D.D. Holm
We study the possibility of generating polarization wall solitary waves in nonlinear optical fibers. These walls represent a polarization switching between two domains along the fiber where the state of polarization of two intense counter-propagating beams is a stable arrangement of the field. We show that the presence of spatiotemporal chaotic instabilities set an upper bound to the maximum fiber length (or input power).
The interaction between two counterpropagating beams may lead to a variety of interesting effects in nonlinear optics. Among these, polarization multistability and instabilities have been discussed by several authors [ l-7 1. In 1987 Zakharov and Mikhailov predicted that, in analogy with the formation of domain walls in magnetic materials, the cross-interaction between two counterpropagating beams in a third order nonlinear optical medium may lead to polarization domains and walls [ 81. In a domain, the polarization states of the two waves minimize the optical energy of the field and thus form a stable arrangement. The polarization domains are topologitally stable entities, that is they enjoy a particular robustness to perturbations. A domain wall is a solitary wave that represents a switching of polarization between two distinct and stable domains. Depending on the intensity ratio between the two beams, a domain wall may either be stationary in the medium or slowly moving backwards or forward. In this work we first consider the possibility of generating polarization domain solitons by means of physically accessible boundary conditions in a medium of finite length. We demonstrate that temporally stable polarization domain walls may be formed in an isotropic nonlinear medium (e.g., an optical fiber) with both self and cross polarization interaction between the two counterpropagating beams. We then derive the conditions for polarization modulational instability and discuss the possibility of ob03759601/93/$
serving polarization controlled all-optical switching from an unstable to a stable polarization arrangement. The interaction between a forward wave, E+(Z,
T)=
(Ezx+E,+y)
and a backward E-(2,
T)=
exp(i/32-iQ+
T) ,
wave,
(E;x+E;y)
exp( -i&Z-i&
T) ,
in a third order nonlinear medium is conveniently described in terms of the Stokes vectors [ 31 S’=
(E,‘*E;
+c.c.,
IE: 12- IE;
+c.c.
,
12).
HereweassumethatS$ the coupled equations isotropic [ 9 ] nonlinear Vt+&)S+
iE;*E;
= IEz 12+ IET 12=1.From for the amplitudes E’ in an medium, one obtains [ 3,7,8 ]
=d,s+=S+xJS++2rS+xJS-,
=riS-xJS-+2S-xJS+. (a,--a,)s-=a,s(1)
Here we used the dimensionless coordinates &i(t+z) and q=i(t--z), where z=RPZ and t=RPCI: Moreover, V= c/n, P and rP are the forwhereas ward and backward beam powers, J=diag{J,, J2, J,>, J1=J3=-i(B+l), and J2= f (3B- 1). R and B describe the strength and the mechanism of the nonlinearity, respectively. In op-
06.00 0 1993 Elsevier Science Publishers B.V. All rights reserved.
289
tical fibers, R = 2m2/A.Aeff, where n2 = 3.2 x 10V2’ m2/W and Aeff is the effective core area. Moreover, the electronic nonlinearity mechanism yields B = $. For simplicity we restrict our attention here to this case, which does not limit the generality of the conclusions of this work. As shown in ref. [ 10 1, system ( 1) is not integrable by means of the inverse scattering transform, unless the self-induced nonlinear polarization rotation term is neglected [ 8 1. Nevertheless, exact solutions of eqs. ( 1) exist that are either uniform in z or traveling solitary waves. In fact, by looking for solutions that depend on a single variable (e.g., t= <+ q) one reduces eqs. ( 1) to the integrable Euler equations [ 7,111. From eqs. ( 1 ), the stability analysis of the mutual arrangements of polarizations (i.e., that travel unchanged through the medium) reveals that the only temporally stable eigenstates are two corotating circular polarizations. Whereas counter-rotating circular polarizations exhibit modulational instability (MI). In fact, set fl+ S-Z
(SF sin@&-Kz),
t l,s$
cos(at-Kz))
where 1St, 1<< 1. By inserting eq. (2 ) in eqs. ( 1) , one obtains a set of four linear algebraic equations. Nontrivial solutions of these equations imply the vanishing of a determinant, which yields the dispersion relationship (for r= 1) --
(K
2
(K2_;)_-_4++jK2_+$.
ref. [ 71. In the present case, keeping the same notation of ref. [ 71, one obtains S’(7)=
(sech(7)
sin[@+(7)],
+tanh(7)
?
cos[@+(z) I) 7
S-(z)=(sech(7) sech(7)
sin[@-(z)],
cos[$-(7)])
ttanh(z),
.
(4)
where 7=
2( I +r)J1
-r2 ’
3[z-(r-l)t/(r+l)+z,]
(3)
Figure 1 shows the total temporal growth rate of the perturbation in eq. (2) versus the perturbation wavevector K and the intensity ratio r of the two waves. For equally intense beams, instability occurs for KC 2. This instability is predicted for a medium of infinite length. In a nonlinear medium of finite length L, one may expect the development of MI from two intense counter-rotating and counterpropagating beams whenever t= 2n/K>77. As we shall see in the following, this simple condition is fairly well confirmed by the numerical simulations of eqs. ( 1). In optical fibers, two different stable domains of corotating circular polarizations may be connected through a domain wall. A complete derivation of the analytical expressions of domain wall solutions of eqs. ( 1) in isotropic nonlinear media may be found in 290
Fig. 1. Plot of temporal growth rate Im{ m} versus spatial perturbation wavevector and intensity ratio r of the counterpropagating beams.
sewe
, (2)
a2
15 November 1993
PHYSICS LETTERS A
Volume 182, number 2,3
Here r=---&-’
r2+1
a= &-arcos(r)
,
where POis an arbitrary constant. Equation (4) shows that the domain wall slowly moves in the direction of the stronger beam. Note that the domain wall only exists for 2 - fi < r< 2 + ,/% Correspondingly, the maximum absolute value of the soliton velocity is 1vi I= (1 +JS)/(3+$)Vr0.366v.
The wall in eq. (4) represents a polarization switching between two corotating circular polarizations (at
Volume
z—+
—
1 82, number 2,3
oo
PHYSICS
LETTERS
A
1
5
November
1993
) and the two oppositely handed polariza-
tions (4) (at z—~oo). be (4) written into the medium byofeqs. means of in Equation a may medium ofdefines infinite a length. soliton solution Therefore it is im(1) portant to find out how the polarization domain walls physically accessible boundary conditions. Figure 2 shows the result of the numerical integration of eqs. ( 1 ) for a medium oflength z=L = 4. The initial polarizations of the two equally intense counterpropagating and corotating waves are, say, left and right circular (S~( t= 0, z) = 1 ) This forms a single uniform stable domain in the medium. At some point in time the polarization ofthe forward wave is slowly rotated at the input end z= 0 (here the rise time of the polarization modulation is equal to the transit time through the medium) from left circular to elliptical, linear (i.e., St = 1 ), elliptical and finally right circular. In the meanwhile, the polarization of the backward beam at z=L = 4 is kept right circular. —
~[ S1+
~ist
.
As caninside zation be seen, the the medium two waves in order rotate to minimize their polarithe the two domains at the opposite ends ofthe medium. optical energy and a steady wall is formed between In fact, the plot of S~shows that the initially right circular polarization of the forward beam rotates tends to be corotating with the backward beam at the back, as z becomes larger, to left circular because it output end z=L = 4. At the same time, the plot of Sj- shows that the backward beam rotates from right
~ ~
4~
(b)
1I:~ 0 i
0
D1Ste~ce ~
locity to The left plot V~ handed and Note itinupon shows may bethat, propagation read in out agreement from medium. z=L with to eq. z=0. The tensity in tion of correspondence the appears field. r ofSt between the the that with linear beams, by the polarization changing wall theof wall a the localized the acquires components relative excitaa(4), yein-
polarization weak damped oscillations is finite imposed at the are ob served in fig. switching 2 polarization are due that to the risethat time offiber the input end z= 0. These oscillations tend to disappear whenever the rise time of the polarization switching at the input becomes larger than the propagation delay through the fiber. The MI of two counter-rotating circular polarizations may also be exploited for the generation of polarization solitons and all-optical switching. In fig. 3, at t= 0 both forward and backward beams are right
~Ce
4~
~
~
________
______________________________ -_____
~
Ce
‘~
Fig. 2. Polarization domain generation by rotating at z= 0 the forward circular polarization from left to right handed. The backward beam is left handed at z=L=4.
291
Volume 1 82, number 2,3
PHYSICS LETTERS A
0~5/
-1
~
ste~c~~
2
1 5 November 1993
~
I
Dj2
~
~
~/‘
teTlCe Fig. 3. Domain wall generation from the weak modulation at the input ofthe forward left handed circular polarization. The backward beam is right handed at z=L=2.5.
circular. As can be seen, a wall (or complete polarization switching) is written inside the medium by weakly modulating the polarization of the forward beam at z= 0, ~
-
s±(z=0, t) = (S~sin(wt), Lc~:
cos(wt) ) ,
_
-
—~/l—S~2—S~3 ,
(5 )
5
Fig. 4. Chaotic oscillations of the polarization state of the forward beam with the same input conditions as in fig. 3 and L=6.
tegrability of these equations. A rigorous characterization ofthe dynamical properties ofthe chaotic regime is beyond the scope of the present work. We are interested here in briefly discussing the implications of the above results on the conditions for the experimental observability of polarization domain walls in fibers. The analysis of the applicative potential of polarization domains and walls in op-
where 5~ ~ =
5cg~ o.oi and w = ~g it. Correspondingly, the polarization ofthe backward beam switches from left (at z=L) to right circular (at z = 0 ). This switching is rather insensitive to the precise values ofthe small seeding modulation amplitudes S13 and frequency The numerical simulations a completew.polarization switching occurs show for L that > 2. Note that this switching results from a steady state (as opposed to modulational) instability of the mitial arrangement of polarizations in the medium of finite length. On the other hand, eq. ( 3 ) (that predicts modulational instabilities for L ~ it) is a simple estimate for a medium ofindefinite length and is not expected to be very accurate near the threshold. Nevertheless, as can be seen in fig. 3, the polarization of the forward beam at L = 2. 5 exhibits small damped oscillations. Whenever L > 3, these ternporal modulations grow much larger. Finally, for L > 5, irregular oscillations occur in the output polarization components of the beams (see fig. 4, where L = 6 ). The spatiotemporal chaotic behavior that is observed in the numerical solutions of eqs. ( 1 ) for relatively large values of L is related to the nonin-
292
tical fibers for implementing polarization switches or memories (where ones and zeros are represented by domains of left or right handed circular polarizations ) will be the subject of a more extended report [ 12 ] . With a 250 m fiber at ~= 1.06 2, long L = 2spun entails that a polartim, with Aeffmay 1 0be written m ization wall into the fiber with two beams of power P~250 mW. In spite of their weak nonlinearity, fibers are well suited for observing polarization domain switching phenomena, since absorption, nonlinearity relaxation and transverse instabilities [ 1 3 ] may be neglected. On the other hand, competing nonlinear effects, such as Brillouin scattering, may be reduced by using long pulses from a multimode ( Q-switched) source. Note that the formation of polarization domains does not depend on the mutual phase relation or coherence between the two beams. Therefore, in order to increase the Brilbum threshold above the critical power for the formation of the domain walls, one may also use phase modulated cw beams at the two fiber input ends. This work was carried out in the framework of a
Volume 182, number 2,3
PHYSICS LETTERSA
telecommunication project ofthe National Research Council of Italy, and under the agreement between the Fondazione Ugo Bordoni and the Italian Post and . . . . . Telecommunications Administration. References [1]A.E. Kaplan and C.T. Law, IEEE J. Quantum Electron. QE21(1985)1529. [2]J. Yumoto and K. Otsuka, Phys. Rev. Lett. 54 (1985) 1806. [3]G. Gregori and S. Wabnitz, Phys. Rev. Lett. 56 (1986) 600; S. Wabnitz and G. Gregori, Opt. Commun. 59 (1986) 72. [4]M.V. Tratnik and J.E. Sipe, Phys. Rev. Lett. 58 (1987) 1104; Phys.Rev.A35 (1987) 2976.
1 5 November 1993
[5] A.L. Gaeta, R.W. Boyd, JR. Ackerhalt and P.W. Milonni, Phys. Rev. Lett. 58 (1987) 2432. [6] S. Trillo and S. Wabnitz, Phys. Rev. A 36 ( 1987) 3881. [7] D. David, D.D. Holm and MV. Tratnik, Phys. Rep. 187 ( 1990) 281. [8] V.E. Zakharov and A.V. Mikhailov, Pis’ma Zh. Eksp. Teor. Fiz.45 (1987) 279 [JETPLett.45 (1987) 349]. [9] A.V. Mikhailov and S. Wabnitz, Opt. Lett. 1 5 (1990) 1055. [10] D.D. Tskhakaya, Teor. Mat. Phys. 81(1989) 1119 [Teor. Mat. Fiz. 81(1989) 154]. [ll]A.P. Veselov, Dokl. Akad. Nauk SSSR 270 (1983) 1298 [Soy. Math. Dokl. 27 (1983) 740]. [12] S. Wabnitz and B. Daino, Generation of polarization switching domain walls in a nonlinear optical medium, submitted for publication to J. Opt. Soc. Am. B (1993). [13]W.J.FirthandC.Parè,Opt.Lett. 13(1988)1096.
293
Polarization
PHYSICS
LETTERS
A
domains and instabilities in nonlinear optical fibers
S. Wabnitz and B. Daino Fondazione UgoBordoni, Via B. Castiglione59, 00142 Rome, Italy Received 13 August 1993; accepted for publication 17 September 1993 Communicated by D.D. Holm
We study the possibility of generating polarization wall solitary waves in nonlinear optical fibers. These walls represent a polarization switching between two domains along the fiber where the state of polarization of two intense counter-propagating beams is a stable arrangement of the field. We show that the presence of spatiotemporal chaotic instabilities set an upper bound to the maximum fiber length (or input power).
The interaction between two counterpropagating beams may lead to a variety of interesting effects in nonlinear optics. Among these, polarization multistability and instabilities have been discussed by several authors [ l-7 1. In 1987 Zakharov and Mikhailov predicted that, in analogy with the formation of domain walls in magnetic materials, the cross-interaction between two counterpropagating beams in a third order nonlinear optical medium may lead to polarization domains and walls [ 81. In a domain, the polarization states of the two waves minimize the optical energy of the field and thus form a stable arrangement. The polarization domains are topologitally stable entities, that is they enjoy a particular robustness to perturbations. A domain wall is a solitary wave that represents a switching of polarization between two distinct and stable domains. Depending on the intensity ratio between the two beams, a domain wall may either be stationary in the medium or slowly moving backwards or forward. In this work we first consider the possibility of generating polarization domain solitons by means of physically accessible boundary conditions in a medium of finite length. We demonstrate that temporally stable polarization domain walls may be formed in an isotropic nonlinear medium (e.g., an optical fiber) with both self and cross polarization interaction between the two counterpropagating beams. We then derive the conditions for polarization modulational instability and discuss the possibility of ob03759601/93/$
serving polarization controlled all-optical switching from an unstable to a stable polarization arrangement. The interaction between a forward wave, E+(Z,
T)=
(Ezx+E,+y)
and a backward E-(2,
T)=
exp(i/32-iQ+
T) ,
wave,
(E;x+E;y)
exp( -i&Z-i&
T) ,
in a third order nonlinear medium is conveniently described in terms of the Stokes vectors [ 31 S’=
(E,‘*E;
+c.c.,
IE: 12- IE;
+c.c.
,
12).
HereweassumethatS$ the coupled equations isotropic [ 9 ] nonlinear Vt+&)S+
iE;*E;
= IEz 12+ IET 12=1.From for the amplitudes E’ in an medium, one obtains [ 3,7,8 ]
=d,s+=S+xJS++2rS+xJS-,
=riS-xJS-+2S-xJS+. (a,--a,)s-=a,s(1)
Here we used the dimensionless coordinates &i(t+z) and q=i(t--z), where z=RPZ and t=RPCI: Moreover, V= c/n, P and rP are the forwhereas ward and backward beam powers, J=diag{J,, J2, J,>, J1=J3=-i(B+l), and J2= f (3B- 1). R and B describe the strength and the mechanism of the nonlinearity, respectively. In op-
06.00 0 1993 Elsevier Science Publishers B.V. All rights reserved.
289
tical fibers, R = 2m2/A.Aeff, where n2 = 3.2 x 10V2’ m2/W and Aeff is the effective core area. Moreover, the electronic nonlinearity mechanism yields B = $. For simplicity we restrict our attention here to this case, which does not limit the generality of the conclusions of this work. As shown in ref. [ 10 1, system ( 1) is not integrable by means of the inverse scattering transform, unless the self-induced nonlinear polarization rotation term is neglected [ 8 1. Nevertheless, exact solutions of eqs. ( 1) exist that are either uniform in z or traveling solitary waves. In fact, by looking for solutions that depend on a single variable (e.g., t= <+ q) one reduces eqs. ( 1) to the integrable Euler equations [ 7,111. From eqs. ( 1 ), the stability analysis of the mutual arrangements of polarizations (i.e., that travel unchanged through the medium) reveals that the only temporally stable eigenstates are two corotating circular polarizations. Whereas counter-rotating circular polarizations exhibit modulational instability (MI). In fact, set fl+ S-Z
(SF sin@&-Kz),
t l,s$
cos(at-Kz))
where 1St, 1<< 1. By inserting eq. (2 ) in eqs. ( 1) , one obtains a set of four linear algebraic equations. Nontrivial solutions of these equations imply the vanishing of a determinant, which yields the dispersion relationship (for r= 1) --
(K
2
(K2_;)_-_4++jK2_+$.
ref. [ 71. In the present case, keeping the same notation of ref. [ 71, one obtains S’(7)=
(sech(7)
sin[@+(7)],
+tanh(7)
?
cos[@+(z) I) 7
S-(z)=(sech(7) sech(7)
sin[@-(z)],
cos[$-(7)])
ttanh(z),
.
(4)
where 7=
2( I +r)J1
-r2 ’
3[z-(r-l)t/(r+l)+z,]
(3)
Figure 1 shows the total temporal growth rate of the perturbation in eq. (2) versus the perturbation wavevector K and the intensity ratio r of the two waves. For equally intense beams, instability occurs for KC 2. This instability is predicted for a medium of infinite length. In a nonlinear medium of finite length L, one may expect the development of MI from two intense counter-rotating and counterpropagating beams whenever t= 2n/K>77. As we shall see in the following, this simple condition is fairly well confirmed by the numerical simulations of eqs. ( 1). In optical fibers, two different stable domains of corotating circular polarizations may be connected through a domain wall. A complete derivation of the analytical expressions of domain wall solutions of eqs. ( 1) in isotropic nonlinear media may be found in 290
Fig. 1. Plot of temporal growth rate Im{ m} versus spatial perturbation wavevector and intensity ratio r of the counterpropagating beams.
sewe
, (2)
a2
15 November 1993
PHYSICS LETTERS A
Volume 182, number 2,3
Here r=---&-’
r2+1
a= &-arcos(r)
,
where POis an arbitrary constant. Equation (4) shows that the domain wall slowly moves in the direction of the stronger beam. Note that the domain wall only exists for 2 - fi < r< 2 + ,/% Correspondingly, the maximum absolute value of the soliton velocity is 1vi I= (1 +JS)/(3+$)Vr0.366v.
The wall in eq. (4) represents a polarization switching between two corotating circular polarizations (at
Volume
z—+
—
1 82, number 2,3
oo
PHYSICS
LETTERS
A
1
5
November
1993
) and the two oppositely handed polariza-
tions (4) (at z—~oo). be (4) written into the medium byofeqs. means of in Equation a may medium ofdefines infinite a length. soliton solution Therefore it is im(1) portant to find out how the polarization domain walls physically accessible boundary conditions. Figure 2 shows the result of the numerical integration of eqs. ( 1 ) for a medium oflength z=L = 4. The initial polarizations of the two equally intense counterpropagating and corotating waves are, say, left and right circular (S~( t= 0, z) = 1 ) This forms a single uniform stable domain in the medium. At some point in time the polarization ofthe forward wave is slowly rotated at the input end z= 0 (here the rise time of the polarization modulation is equal to the transit time through the medium) from left circular to elliptical, linear (i.e., St = 1 ), elliptical and finally right circular. In the meanwhile, the polarization of the backward beam at z=L = 4 is kept right circular. —
~[ S1+
~ist
.
As caninside zation be seen, the the medium two waves in order rotate to minimize their polarithe the two domains at the opposite ends ofthe medium. optical energy and a steady wall is formed between In fact, the plot of S~shows that the initially right circular polarization of the forward beam rotates tends to be corotating with the backward beam at the back, as z becomes larger, to left circular because it output end z=L = 4. At the same time, the plot of Sj- shows that the backward beam rotates from right
~ ~
4~
(b)
1I:~ 0 i
0
D1Ste~ce ~
locity to The left plot V~ handed and Note itinupon shows may bethat, propagation read in out agreement from medium. z=L with to eq. z=0. The tensity in tion of correspondence the appears field. r ofSt between the the that with linear beams, by the polarization changing wall theof wall a the localized the acquires components relative excitaa(4), yein-
polarization weak damped oscillations is finite imposed at the are ob served in fig. switching 2 polarization are due that to the risethat time offiber the input end z= 0. These oscillations tend to disappear whenever the rise time of the polarization switching at the input becomes larger than the propagation delay through the fiber. The MI of two counter-rotating circular polarizations may also be exploited for the generation of polarization solitons and all-optical switching. In fig. 3, at t= 0 both forward and backward beams are right
~Ce
4~
~
~
________
______________________________ -_____
~
Ce
‘~
Fig. 2. Polarization domain generation by rotating at z= 0 the forward circular polarization from left to right handed. The backward beam is left handed at z=L=4.
291
Volume 1 82, number 2,3
PHYSICS LETTERS A
0~5/
-1
~
ste~c~~
2
1 5 November 1993
~
I
Dj2
~
~
~/‘
teTlCe Fig. 3. Domain wall generation from the weak modulation at the input ofthe forward left handed circular polarization. The backward beam is right handed at z=L=2.5.
circular. As can be seen, a wall (or complete polarization switching) is written inside the medium by weakly modulating the polarization of the forward beam at z= 0, ~
-
s±(z=0, t) = (S~sin(wt), Lc~:
cos(wt) ) ,
_
-
—~/l—S~2—S~3 ,
(5 )
5
Fig. 4. Chaotic oscillations of the polarization state of the forward beam with the same input conditions as in fig. 3 and L=6.
tegrability of these equations. A rigorous characterization ofthe dynamical properties ofthe chaotic regime is beyond the scope of the present work. We are interested here in briefly discussing the implications of the above results on the conditions for the experimental observability of polarization domain walls in fibers. The analysis of the applicative potential of polarization domains and walls in op-
where 5~ ~ =
5cg~ o.oi and w = ~g it. Correspondingly, the polarization ofthe backward beam switches from left (at z=L) to right circular (at z = 0 ). This switching is rather insensitive to the precise values ofthe small seeding modulation amplitudes S13 and frequency The numerical simulations a completew.polarization switching occurs show for L that > 2. Note that this switching results from a steady state (as opposed to modulational) instability of the mitial arrangement of polarizations in the medium of finite length. On the other hand, eq. ( 3 ) (that predicts modulational instabilities for L ~ it) is a simple estimate for a medium ofindefinite length and is not expected to be very accurate near the threshold. Nevertheless, as can be seen in fig. 3, the polarization of the forward beam at L = 2. 5 exhibits small damped oscillations. Whenever L > 3, these ternporal modulations grow much larger. Finally, for L > 5, irregular oscillations occur in the output polarization components of the beams (see fig. 4, where L = 6 ). The spatiotemporal chaotic behavior that is observed in the numerical solutions of eqs. ( 1 ) for relatively large values of L is related to the nonin-
292
tical fibers for implementing polarization switches or memories (where ones and zeros are represented by domains of left or right handed circular polarizations ) will be the subject of a more extended report [ 12 ] . With a 250 m fiber at ~= 1.06 2, long L = 2spun entails that a polartim, with Aeffmay 1 0be written m ization wall into the fiber with two beams of power P~250 mW. In spite of their weak nonlinearity, fibers are well suited for observing polarization domain switching phenomena, since absorption, nonlinearity relaxation and transverse instabilities [ 1 3 ] may be neglected. On the other hand, competing nonlinear effects, such as Brillouin scattering, may be reduced by using long pulses from a multimode ( Q-switched) source. Note that the formation of polarization domains does not depend on the mutual phase relation or coherence between the two beams. Therefore, in order to increase the Brilbum threshold above the critical power for the formation of the domain walls, one may also use phase modulated cw beams at the two fiber input ends. This work was carried out in the framework of a
Volume 182, number 2,3
PHYSICS LETTERSA
telecommunication project ofthe National Research Council of Italy, and under the agreement between the Fondazione Ugo Bordoni and the Italian Post and . . . . . Telecommunications Administration. References [1]A.E. Kaplan and C.T. Law, IEEE J. Quantum Electron. QE21(1985)1529. [2]J. Yumoto and K. Otsuka, Phys. Rev. Lett. 54 (1985) 1806. [3]G. Gregori and S. Wabnitz, Phys. Rev. Lett. 56 (1986) 600; S. Wabnitz and G. Gregori, Opt. Commun. 59 (1986) 72. [4]M.V. Tratnik and J.E. Sipe, Phys. Rev. Lett. 58 (1987) 1104; Phys.Rev.A35 (1987) 2976.
1 5 November 1993
[5] A.L. Gaeta, R.W. Boyd, JR. Ackerhalt and P.W. Milonni, Phys. Rev. Lett. 58 (1987) 2432. [6] S. Trillo and S. Wabnitz, Phys. Rev. A 36 ( 1987) 3881. [7] D. David, D.D. Holm and MV. Tratnik, Phys. Rep. 187 ( 1990) 281. [8] V.E. Zakharov and A.V. Mikhailov, Pis’ma Zh. Eksp. Teor. Fiz.45 (1987) 279 [JETPLett.45 (1987) 349]. [9] A.V. Mikhailov and S. Wabnitz, Opt. Lett. 1 5 (1990) 1055. [10] D.D. Tskhakaya, Teor. Mat. Phys. 81(1989) 1119 [Teor. Mat. Fiz. 81(1989) 154]. [ll]A.P. Veselov, Dokl. Akad. Nauk SSSR 270 (1983) 1298 [Soy. Math. Dokl. 27 (1983) 740]. [12] S. Wabnitz and B. Daino, Generation of polarization switching domain walls in a nonlinear optical medium, submitted for publication to J. Opt. Soc. Am. B (1993). [13]W.J.FirthandC.Parè,Opt.Lett. 13(1988)1096.
293