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Black and white vector solitons in weakly birefringent optical fibers
Volume 132, number 8,9
PHYSICS LETTERS A
24 October 1988
BLACK AND WHITE VECTOR SOLITONS IN WEAKLY BIREFRINGENT OPTICAL FIBERS D.N. CHRISTODOULIDES Department of Computer Science and ElectricalEngineering, Lehigh University, Bethlehem, PA 18015, USA Received 5 July 1988; accepted for publication 10 August 1988 Communicated by D.D. Holm
It is shown that bright and dark solitons can propagate simultaneously in a single-mode fiber with weak birefringence.
It is well known that nonlinear dispersive waves in polarization preserving single-mode fibers obey the so-called nonlinear Schrodinger equation [1,2]. This equation exhibits bright soliton solutions in the anomalous dispersion region of a fiber [1] while in the regime of normal dispersion it admits dark pulse soliton solutions [2]. After their first experimental observation [3], bright solitons have been the subject of numerous experimental and theoretical investigations. Very recently, dark soliton behavior has been observed for the first time experimentally [4,5] and in effect an earlier claim [6] (based on threedimensional arguments), stating that dark soliton propagation in optical fibers is impossible, has been dismissed. In a recent theoretical study [7], Agrawal found that the process of cross-phase modulation can have a profound effect on the properties of the socalledmodulationalinstability [8] andasaresulthe conjectured that bright solitons are possible even in the normal dispersion regime ofan optical fiber. Even though his conjecture contradicts common wisdom, in this Letter it will be shown that simultaneous propagation of bright and dark solitons is indeed possible (through that the process of is cross-phase modulation) provided the fiber weakly birefringent. In a previous paper [9] it was found that the tensor character of the x ~ nonlinearity in optical fibers together with the effect of birefringence can lead to a new class of optical solitons which were called “vector solitons” because they involve the two polarization components. Moreover, it was shown that these latter processes can play an important role
in reshaping these vector solitons and in some cases they can induce an odd-pulse wave (in one of the two polarizations) which is to some extent relevant to the pure antisymmetric dark soliton [2,5]. Let us consider a single-mode fiber with weak birefringence. The fiber is assumed to be straight (no twists) so as to avoid any effects arising from optical activity [10]. If u is the field envelope of the polarization along the ~ axis of the birefringent fiber and v the envelope along the f~ axis, then u and v obey the following pair ofcoupled evolution equations [9,11 131: —
2
ôz
+
—
2 &r 2c + ~~2~e” exp ( 4ipz) —
E( iUi2+ ~ 1v12)u ] = 0,
2
i~ az
—
+
~
2
ôi~
+ ~ ~ 2~ ~
~
[(IvV+~ u12)v
2c (4ipz)] = 0,
(1)
where w 0 = 27tc/)~0is the carrier angular frequency of the light field, ~ its wavelength, n2 theis nonlinear 2fl/3w2 the fiberKerr discoefficient of glass, fl~ =i3 persive coefficient evaluated at w 0 [11—13],z is the propagation distance along the fiber and r= t— Z/Vg) is a time coordinate frame moving at the group speed v8 of the wave. In deriving eq. (1) it was assumed that the two polarizations exhibit the same group yelocity [9,11,12]. This latter assumption can be justified if the fiber birefringence is weak and if the temporal duration of the wave involved is rather long
0375-960l/88/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
451
Volume 132, number 8,9
PHYSICS LETTERS A
so as one can ignore “walk-off” effects. The quantity p is related to the birefringence of the fiber, i.e., I3~—I3~w 0
2
It
(n~—n~)= (n~—n,~) (2) where /i~and f3~,are the propagation constants associated with the x—y axes of the birefringent fiber and n~and n~the respective refractive indices. The —
~
~—
,
presence of the cross-phase modulation process is evident in the nonlinear part of eq. (1). It can be readily shown that eq. (1) admits the following bright and dark vector soliton (strictly speaking it is a solitary wave):
u=U0 sech(r/r0) exp[ia—p)z] v=V0tanh(t/r0) exp[i(a+p)z]
(3)
,
,
(4)
where 2~fl~ — — =
4p
2oflg 4it (fix — ~2y)’
(5)
Uo2=?~(a3p),
(6)
V~=
(7)
itfl2
Itfl2
(a+p)
.
In the above equations r0 is the temporal pulsewidth of the waves and a is an arbitrary constant of the problem. Eq. (5) shows that the pulsewidth of the vector wave is determined only by the fiber characteristics and that it does not depend on the arbitrary constant aas in ref. [9]. This is due to the rather special functional form (sech—tanh) describing this wave. In the region of anomalous dispersion (fl~ <0), where normally the nonlinear Schrodinger equation admits bright solitons, eqs. (5)—(7) show that the above vector soliton is possible ~fflx<~y (p —p. It can be shown that in this case U~>V~,i.e. the bright soliton dominates. In the regime of normal dispersion (fl~>0), eqs. (5)— (7) require that n~>fl~(p> 0) and that a> 3p. Furthermore in this latter case J’~U~ and therefore the dark component dominates the vector wave. These results may be illustrated by the following example. Consider a single-mode fiber with very low birefringence [14], flx~ ~ 5 X 1 Ø_9~This fiber is to
452
24 October 1988
be used in the visible spectrum, say at ~ = 0.62 ~.tm where its normal dispersion is approximately 5.4x 1026 m~s2. The Kerr coefficient ofglass 22 (rn/V)2 and the effective crossis n2=l.2x10 sectional area of the fiber is taken to be S-.~20 j.tm2. Using the previous data, p is equal to 2.53 X 10_2. From the above discussion let us choose a to be a~8 x 10~2so as a> 3p. Using eq. (5), 2t 0~ 1.45 Ps. Moreover it can be shown from Poynting’s theorem (~~max(th/2)(co/IL0) 1/2 [U~or V~ ]S where for glass n 1.45) that for these parameters the peak power required to support the dark soliton part of this vector wave is approximately 6.75 W and for that of the bright is —.0.26 W. Thus the required characteristics for such a wave (power and temporal pulsewidth) fall well within experimental limits. In conclusion, it has been shown that bright and dark vector solitons are possible in optical fibers. This can be achieved through the process of birefringence and that of cross-phase modulation which results from the tensor character of the x ~ nonlinearity. References [1] A. Hasegawa and F. Tappert, AppI. Phys. Lett. 23 (1973) 142. [2]A. Hasegawa and F. Tappert, AppI. Phys. Lett. 23 (1973) 171. [3] L.F. Mollenauer, R.H. Stolen and J.P. Gordon, Phys. Rev. Lett.45 (1980) 1095. [4] D. Krokel, N.J. Halas, G. Giuliani and D. Grischkowski, Phys. Rev. Lett. 60(1988)29. [5] A.M. Weiner, J.P. Heritage, R.J. Hawkins, R.N. Thurston, E.M. Kirschner, D.E. Leaird and W.J. Tomlinson, Conf. on Lasers and electro-optics (CLEO), paper PD-24, Anaheim, California, April 1988. [6] D.N. Christodoulides and R.I. Joseph, Opt. Lett. 9 (1984) 408. [7] G.P. Agrawal, Phys. Rev. Lett. 59 (1987) 880. [8] K. Tai, A. Hasegawa and A. Tomita, Phys. Rev. Lett. 56 (1986) 135. [9] D.N. Christodoulides and R.I. Joseph, Opt. Lett. 13 (1988) 53. [10] R. Ulrich and A. Simon, AppI. Opt. 18 (1979) 2241. [11] K.J. Blow, N.J. Doran and D. Wood, Opt. Lett. 12 (1987) 202. [12] A.D. Boardman and G.S. Cooper, J. Opt. Soc. Am. B 5 (1988) 403. [13] C.R. Menyuk, IEEE J. Quantum Electron. QE-23 (1987) 1974. [14] S.R. Norman, D.N. Payne, M.J. Adams and A.M. Smith, Electron. Lett. 15 (1979) 309.
PHYSICS LETTERS A
24 October 1988
BLACK AND WHITE VECTOR SOLITONS IN WEAKLY BIREFRINGENT OPTICAL FIBERS D.N. CHRISTODOULIDES Department of Computer Science and ElectricalEngineering, Lehigh University, Bethlehem, PA 18015, USA Received 5 July 1988; accepted for publication 10 August 1988 Communicated by D.D. Holm
It is shown that bright and dark solitons can propagate simultaneously in a single-mode fiber with weak birefringence.
It is well known that nonlinear dispersive waves in polarization preserving single-mode fibers obey the so-called nonlinear Schrodinger equation [1,2]. This equation exhibits bright soliton solutions in the anomalous dispersion region of a fiber [1] while in the regime of normal dispersion it admits dark pulse soliton solutions [2]. After their first experimental observation [3], bright solitons have been the subject of numerous experimental and theoretical investigations. Very recently, dark soliton behavior has been observed for the first time experimentally [4,5] and in effect an earlier claim [6] (based on threedimensional arguments), stating that dark soliton propagation in optical fibers is impossible, has been dismissed. In a recent theoretical study [7], Agrawal found that the process of cross-phase modulation can have a profound effect on the properties of the socalledmodulationalinstability [8] andasaresulthe conjectured that bright solitons are possible even in the normal dispersion regime ofan optical fiber. Even though his conjecture contradicts common wisdom, in this Letter it will be shown that simultaneous propagation of bright and dark solitons is indeed possible (through that the process of is cross-phase modulation) provided the fiber weakly birefringent. In a previous paper [9] it was found that the tensor character of the x ~ nonlinearity in optical fibers together with the effect of birefringence can lead to a new class of optical solitons which were called “vector solitons” because they involve the two polarization components. Moreover, it was shown that these latter processes can play an important role
in reshaping these vector solitons and in some cases they can induce an odd-pulse wave (in one of the two polarizations) which is to some extent relevant to the pure antisymmetric dark soliton [2,5]. Let us consider a single-mode fiber with weak birefringence. The fiber is assumed to be straight (no twists) so as to avoid any effects arising from optical activity [10]. If u is the field envelope of the polarization along the ~ axis of the birefringent fiber and v the envelope along the f~ axis, then u and v obey the following pair ofcoupled evolution equations [9,11 131: —
2
ôz
+
—
2 &r 2c + ~~2~e” exp ( 4ipz) —
E( iUi2+ ~ 1v12)u ] = 0,
2
i~ az
—
+
~
2
ôi~
+ ~ ~ 2~ ~
~
[(IvV+~ u12)v
2c (4ipz)] = 0,
(1)
where w 0 = 27tc/)~0is the carrier angular frequency of the light field, ~ its wavelength, n2 theis nonlinear 2fl/3w2 the fiberKerr discoefficient of glass, fl~ =i3 persive coefficient evaluated at w 0 [11—13],z is the propagation distance along the fiber and r= t— Z/Vg) is a time coordinate frame moving at the group speed v8 of the wave. In deriving eq. (1) it was assumed that the two polarizations exhibit the same group yelocity [9,11,12]. This latter assumption can be justified if the fiber birefringence is weak and if the temporal duration of the wave involved is rather long
0375-960l/88/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
451
Volume 132, number 8,9
PHYSICS LETTERS A
so as one can ignore “walk-off” effects. The quantity p is related to the birefringence of the fiber, i.e., I3~—I3~w 0
2
It
(n~—n~)= (n~—n,~) (2) where /i~and f3~,are the propagation constants associated with the x—y axes of the birefringent fiber and n~and n~the respective refractive indices. The —
~
~—
,
presence of the cross-phase modulation process is evident in the nonlinear part of eq. (1). It can be readily shown that eq. (1) admits the following bright and dark vector soliton (strictly speaking it is a solitary wave):
u=U0 sech(r/r0) exp[ia—p)z] v=V0tanh(t/r0) exp[i(a+p)z]
(3)
,
,
(4)
where 2~fl~ — — =
4p
2oflg 4it (fix — ~2y)’
(5)
Uo2=?~(a3p),
(6)
V~=
(7)
itfl2
Itfl2
(a+p)
.
In the above equations r0 is the temporal pulsewidth of the waves and a is an arbitrary constant of the problem. Eq. (5) shows that the pulsewidth of the vector wave is determined only by the fiber characteristics and that it does not depend on the arbitrary constant aas in ref. [9]. This is due to the rather special functional form (sech—tanh) describing this wave. In the region of anomalous dispersion (fl~ <0), where normally the nonlinear Schrodinger equation admits bright solitons, eqs. (5)—(7) show that the above vector soliton is possible ~fflx<~y (p
452
24 October 1988
be used in the visible spectrum, say at ~ = 0.62 ~.tm where its normal dispersion is approximately 5.4x 1026 m~s2. The Kerr coefficient ofglass 22 (rn/V)2 and the effective crossis n2=l.2x10 sectional area of the fiber is taken to be S-.~20 j.tm2. Using the previous data, p is equal to 2.53 X 10_2. From the above discussion let us choose a to be a~8 x 10~2so as a> 3p. Using eq. (5), 2t 0~ 1.45 Ps. Moreover it can be shown from Poynting’s theorem (~~max(th/2)(co/IL0) 1/2 [U~or V~ ]S where for glass n 1.45) that for these parameters the peak power required to support the dark soliton part of this vector wave is approximately 6.75 W and for that of the bright is —.0.26 W. Thus the required characteristics for such a wave (power and temporal pulsewidth) fall well within experimental limits. In conclusion, it has been shown that bright and dark vector solitons are possible in optical fibers. This can be achieved through the process of birefringence and that of cross-phase modulation which results from the tensor character of the x ~ nonlinearity. References [1] A. Hasegawa and F. Tappert, AppI. Phys. Lett. 23 (1973) 142. [2]A. Hasegawa and F. Tappert, AppI. Phys. Lett. 23 (1973) 171. [3] L.F. Mollenauer, R.H. Stolen and J.P. Gordon, Phys. Rev. Lett.45 (1980) 1095. [4] D. Krokel, N.J. Halas, G. Giuliani and D. Grischkowski, Phys. Rev. Lett. 60(1988)29. [5] A.M. Weiner, J.P. Heritage, R.J. Hawkins, R.N. Thurston, E.M. Kirschner, D.E. Leaird and W.J. Tomlinson, Conf. on Lasers and electro-optics (CLEO), paper PD-24, Anaheim, California, April 1988. [6] D.N. Christodoulides and R.I. Joseph, Opt. Lett. 9 (1984) 408. [7] G.P. Agrawal, Phys. Rev. Lett. 59 (1987) 880. [8] K. Tai, A. Hasegawa and A. Tomita, Phys. Rev. Lett. 56 (1986) 135. [9] D.N. Christodoulides and R.I. Joseph, Opt. Lett. 13 (1988) 53. [10] R. Ulrich and A. Simon, AppI. Opt. 18 (1979) 2241. [11] K.J. Blow, N.J. Doran and D. Wood, Opt. Lett. 12 (1987) 202. [12] A.D. Boardman and G.S. Cooper, J. Opt. Soc. Am. B 5 (1988) 403. [13] C.R. Menyuk, IEEE J. Quantum Electron. QE-23 (1987) 1974. [14] S.R. Norman, D.N. Payne, M.J. Adams and A.M. Smith, Electron. Lett. 15 (1979) 309.