Is two-photon absorption a limitation - OSA Publishing
344
OPTICS LETTERS / Vol. 19, No. 5 I March 1, 1994
Is two-photon absorption a limitation to dark soliton switching? Xiaoping Yang, Yuri S. Kivshar, and Barry Luther-Davies Optical Sciences Centre, Australian National University, Canberra, ACT 0200, Australia
David R. Andersen Department of Electrical and Computer Engineering, The University of Iowa, Iowa City, Iowa 52242 Received October 25, 1993 It is shown that two-photon absorption does not always place a fundamental
limitation on soliton devices:
The
steering angle, the key characteristic of switching devices based on the propagation of dark spatial solitons, is almost preserved as the intensity of the background wave decays in the presence of weak two-photon absorption.
Various types of optical switching devices have recently been proposed based on the propagation and interaction of bright' as well as dark2 spatial solitons. The advantage of switching devices that use soliton propagation is that solitons are self-guided waves; they create perfect waveguide structures that can also guide other (weak probe) beams propagating as linear modes of the soliton-induced waveguides. This kind of the switching device provides a flexible way of connecting input and output ports. To reduce the power for the soliton formation and the switching threshold, one can use materials with higher nonlinearities than that of silica. Another way to improve nonresonant nonlinearities is to use the enhancement that occurs near two-photon resonances.3 However, in many cases an enhancement of the nonlinear Kerr coefficient is accompanied by an enhancement of the two-photon absorption (TPA) coefficient. This is the case, for example, for dark spatial solitons observed in the semiconductor ZnSe, which is known to be an instantaneous, defocusing nonlinear medium with a rather strong TPA.4' 5 Therefore, to use spatial solitons for all-optical switching, one should analyze the effect of the TPA on the propagation of those solitons. Such an analysis was presented by Silberberg6 for the case of bright solitons. It follows from his analysis, as well as from numerical studies of the effect of TPA on nonlinear directional couplers and nonlinear distributed-feedback grating devices,7 that TPA seems to be a fundamental
limitation of optical
switching devices based on an intensity-dependent refractive index. The purpose of this Letter is to analyze the effect of TPA on dark solitons.
One of
the aims of our analysis is to demonstrate that TPA does not produce a critical limitation for dark-soliton switching because the steering angle, the key characteristic of any switching device based on dark soliton propagation, is almost preserved as the intensity of the background wave decays in the presence of weak TPA. To describe the effect of TPA on dark solitons, we consider the nonlinear Schr6dinger equation modified 0146-9592/94/050344-03$6.00/0
as follows (see, e.g., Ref. 6):
.au +1 az
a 2U _U1 2 U 2 aX2 l
-iKu
2u
(1)
Here u is the normalized field amplitude, z and x are the normalized propagation and transverse coordinates, and K is the normalized TPA coefficient, K = /3/2kn2 , where k is the free-space wave vector and ,8 and n 2 are the intensity-dependent absorption and refractive-index coefficients, respectively. As is well known, in the absence of the TPA contribution (i.e., at K = 0) Eq. (1) describes the case of a defocusing Kerr nonlinearity in which dark spatial solitons may propagate on a modulationally stable background wave u = uoexp(-iuo 2z), u0 being the background intensity. The nonlinear absorption, even when small, leads to attenuation of the cw background wave, and the wave's amplitude and phase become slowly dependent on Kuo 2 (0)z according to
u0 (z) 0(z)
-
=
[1 + 2Kuo2(0)z]"12
/Z
uo2 (z') dz, =
1 ln[1 + 2KUo2 (0)z]. (2)
This slow attenuation of the background wave makes it rather difficult to apply a perturbation method immediately to analyze the dissipation-induced evolution of dark solitons, although the case of a linear absorption"lo
is an exception:
As will follow from
the subsequent analysis, the TPA-induced dynamics of a dark soliton is more complicated than that due to linear absorption. To take into account explicitly the TPA-induced evolution of the background wave, we apply the following transformation:
u(z,x) = uo(z)exp[iO(z)]v(z,x),
(3)
where u0(z) and 0(z) change according to Eqs. (2), and obtain the following equation for v:
u i1 a2V _ (I|2-V12-iK(v1 (11 - 1)v = ( 1994 Optical Society of America
2
-l)V, -
1)v, (4)
March 1, 1994 / Vol. 19, No. 5 / OPTICS LETTERS
where ; and 6 are new coordinates that are connected to z and x by the differential relations d; = 2(Z)dz and de = uo(z)dx. After such a transformation, the resulting Eq. (4) has a vanishing perturbation of - K, and it may now be treated by any variant t of perturbation theory for solitons. We note that the transformation
345
One of the main characteristics of the dark-soliton switching devices is the so-called steering angle.2 It is easy to see that the total shift of the dark soliton along the x axis is given by the relation JZ dz'uo(z') sin q(z'), so that the steering angle X may be defined through the local transverse velocity:
[Eq. (3)] is the simplest way to solyve
W(z) = tan X = uo(z)sin 0 (z).
(13)
the problem of nonvanishing boundary conditions for a dark soliton; however, the perturbation-indu cedi evolution of dark solitons may also be analyzed Ieither in the small-amplitude limit' or with the h .elp
The important conclusion based on Eq. (13) is the following: When the dark soliton propagates in the
of the so-called regularized
the function sin 0 (z) grows slowly, keeping, at least
integrals of motion."
The dark soliton of Eq. (4) in the absence of T'PA has the form
v(;,)= cos tanhZ + isinX, with
-q= cos X,
Z = -q(e - f,
fl = sin k,
(6) and such a solution is characterized by the inter rnal ess phase angle '0(Is I ' wr/2)that describes the darknzess of the soliton through the relation lvi2
IV12 cos2 (k 1 -cosh 2 Z
(7)
=
_12
If the TPA coefficient K is small, we expect the s ton features of the solution [Eqs. (5) and (6)] to domi-
nate, so that the structure's evolution may be trea ted adiabatically as in Eq. (5) with the angle (k slo,wly dependent on so that
presence for small
of TPA on a decaying background
uo(z),
/ (0), the product [Eq. (13)] almost con-
stant. This simply means that the steering angles for switching devices based on dark-soliton propagation are almost preserved in a Kerr nonlinear medium in the presence of TPA. From the physical point of view, this important property simply follows from the nature of nonlinear absorption: the background intensity decays faster than the central minimum in the soliton, forcing the soliton contrast to change. To confirm our analysis, we have checked the results [Eqs. (2) and (12)] of the adiabatic approximation by comparing them with a numerical solution of Eq. (1). Figure 1 shows the evolution of the background uo(z),the soliton parameter sin 0(z), and the 1.0 0.8
',
Z = cos k(4{
-
f
0.6
d;' sinck(i2)
(8)
1v V1 +-) 0.4
To find the internal dynamics of the dark soliton, we use a Hamiltonian approach for the perturbation theory for solitons.' 2
the Hamiltonian of the system [Eq. (4)] is a conserved quantity
given by
H=f dd
0.2
Without the TPA, i.e., at K = 0,
2
av-+ -22
-
.
0.0 ' 0.0
(9)
1.0 2.0 Propagation Distance (Kz)
3.0
(a)
1.0
It follows from Eq. (4) at K # 0 that TPA leads to slow
attentuation, and the Hamiltonian [Eq. (9)] changes
0.8
according to
dH
=iKf f de(IV12 -1)
(Vaa
-v*
0.6
(10) 0.4
Substituting the solution [Eqs. (5) and (6)], with H calculated for the soliton in Eq. (5) and H8 = (4/3) cos3 q$,into Eq. (10) and returning to the primary variables z and x, we find the equation for the soliton angle:
dz = - Kuo 2(z)sin(2k),
(11)
dz 3
which may be easily integrated to give k (z) = tan-'{tan 0 (0)[1 + 2Kuo2 (0)z]h3 } .
(12)
Equations (2), (3), (5), and (12) totally describe dark soliton evolution in the presence of TPA.
0.2 0.0 ' 0.0
1.0 2.0 Propagation Distance (Kz )
3.0
(b)
Fig. 1. Evolution of the background amplitude, uo(z), the normalized dark soliton eigenvalue, sin qS(z), and the transverse velocity of the dark soliton, W(z), along the Kerr medium with TPA at K = 0.05. The solid curves are from Eqs. (2) and (12), and the symbols are from a numerical simulation of Eq. (1) for (a) 0(0) = 0.2nr and (b) q5(0)= 0.17r.
OPTICS LETTERS / Vol. 19, No. 5 / March 1, 1994
346
(a)
10
-40
-20
0 X
20
40
~32^t i
-40
-20
0 X
20
40
switching device based on dark spatial solitons is the only one yet found that is mostly insensitive to TPA. In fact, the approach that we have presented here is rather general and can be applied to different problems involving dark-soliton propagation on varying backgrounds. In particular, it permits the analysis of the instability of a fundamental dark soliton in the presence of gain used to compensate the two-photon absorption.
(b)
10
o
°
k1 5
5 -40
-20
0 X
20
40
-40
-20
0 X
20
40
Fig. 2. Contour plots demonstrating the effect of TPA on a dark soliton for K = 0.05 and O0() = 0.2vn (left-hand column) and O0() = 0.1v- (right-hand column) based on (a) an adiabatic approximation given by Eqs. (2) and (12) and (b) a numerical simulation of Eq. (1).
transverse
velocity W(z) = uo(z)sin 0 (z) for two dif-
ferent initial values of the soliton angle sb(0). The analytical results (solid curves) based on Eqs. (2) and (12) are in perfect agreement with the results of the numerical simulations (symbols), and, as may be seen, the steering angle is almost preserved, provided that 4.(0) is small. Small deviations of the numerical data from the adiabatic relationship are caused by transition radiation that slightly changes the intensity of the background (see Fig. 2). Finally, it is important to compare the results [Eq. (12)] for TPA with the corresponding result for linear absorption described by the contribution -iyu on the right-hand side of Eq. (1) instead of -iKjul 2 u. Applying the same approach as above, we find that this time the background decays exponentially, u0 (z) = uo(0)exp(-yz), and the soliton phase 0 does not change, k(z) = O(0), so that the contrast of the dark soliton (- cos2 0) is always preserved and, therefore, the transverse velocity W(z) decays exponentially in direct proportion to uo(z) (see Refs. 8-10).
In conclusion, we have analyzed the effect of TPA on the propagation of dark optical solitons. It has been pointed out that TPA does not seem to produce a drastic limitation to optical switching based on dark spatial solitons, and the transverse velocity of a dark soliton is almost preserved as the intensity of the background wave decays in the presence of weak TPA. Such a property makes the effects of TPA and linear absorption quite different for darksoliton switching devices compared with other nonlinear optical switching devices, and thus it seems that a
13
The research of Yuri S. Kivshar was supported by the Australian Photonics Cooperative Research Centre, and the research of David R. Andersen was supported in part by National Science Foundation grant ECS-9105660. In addition, David R. Andersen acknowledges stimulating discussions and support while he was a visiting fellow at the Laser Physics Centre, Australian National University. Xiaoping Yang and Barry Luther-Davies are also with the Laser Physics Centre, Australian National University.
References 1. F. Reynaud and A. Barthelemy, Europhys. Lett. 12, 401 (1990); Barthelemy, Opt. 2. B. Luther-Davies (1992); X. Yang,
M. Shalaby, F. Reynaud, and A. Lett. 17, 778 (1992). and X. Yang, Opt. Lett. 17, 1755 B. Luther-Davies, and W. Kroli-
kowski, Int. J. Nonlinear Opt. Phys. 2, 1 (1992). 3. G. I. Stegeman, C. T. Seaton, and R. J. Zanoni, Thin Solid Films 152, 231 (1987); P. N. Townsend, G. L.
Baker, N. E. Schlotter, C. F. Klausner, and S. Etemad, Appl. Phys. Lett. 53, 1782 (1988). 4. G. R. Allan, S. R. Skinner, D. R. Andersen, Smirl, Opt. Lett. 16, 156 (1991).
and A. L.
5. M. Sheik-Bahae, D. J. Hagan, and E. W. Van Stryland,
Digest of International Quantum ElectronicsConference (Optical Society of America, Washington, D.C., 1990), paper QTUM3. 6. Y. Silberberg, Opt. Lett. 15, 1005 (1990). 7. V. Mizahi, K. W. DeLong, G. I. Stegeman, M. A. Saifi,
and M. J. Andrejco, Opt. Lett. 14, 1140 (1989); K W. DeLong, K. B. Rochford, and G. I. Stegeman, Phys. Lett. 55, 1823 (1989).
Appl.
8. J. A. Giannini and R. I. Joseph, IEEE J. Quantum Electron. 26, 2109 (1990). 9. M. Lisak, D. Andersen, and B. A. Malomed, Opt. Lett. 16, 1936 (1991).
10. Yu. S. Kivshar, Phys. Rev. A 42, 1757 (1990); see also the review paper, IEEE J. Quantum Electron. 28, 250 (1993). 11. I. M. Uzunov and V. S. Gerdjikov, Phys. Rev. A 47, 1582 (1993). 12. Yu. S. Kivshar, and B. A. Malomed, Rev. Mod. Phys. 61, 763 (1989).
13. Y. Chen and J. Atai, Opt. Lett. 16, 1933 (1991); Y. Chen, Phys. Rev. A 45, 6922 (1992).
OPTICS LETTERS / Vol. 19, No. 5 I March 1, 1994
Is two-photon absorption a limitation to dark soliton switching? Xiaoping Yang, Yuri S. Kivshar, and Barry Luther-Davies Optical Sciences Centre, Australian National University, Canberra, ACT 0200, Australia
David R. Andersen Department of Electrical and Computer Engineering, The University of Iowa, Iowa City, Iowa 52242 Received October 25, 1993 It is shown that two-photon absorption does not always place a fundamental
limitation on soliton devices:
The
steering angle, the key characteristic of switching devices based on the propagation of dark spatial solitons, is almost preserved as the intensity of the background wave decays in the presence of weak two-photon absorption.
Various types of optical switching devices have recently been proposed based on the propagation and interaction of bright' as well as dark2 spatial solitons. The advantage of switching devices that use soliton propagation is that solitons are self-guided waves; they create perfect waveguide structures that can also guide other (weak probe) beams propagating as linear modes of the soliton-induced waveguides. This kind of the switching device provides a flexible way of connecting input and output ports. To reduce the power for the soliton formation and the switching threshold, one can use materials with higher nonlinearities than that of silica. Another way to improve nonresonant nonlinearities is to use the enhancement that occurs near two-photon resonances.3 However, in many cases an enhancement of the nonlinear Kerr coefficient is accompanied by an enhancement of the two-photon absorption (TPA) coefficient. This is the case, for example, for dark spatial solitons observed in the semiconductor ZnSe, which is known to be an instantaneous, defocusing nonlinear medium with a rather strong TPA.4' 5 Therefore, to use spatial solitons for all-optical switching, one should analyze the effect of the TPA on the propagation of those solitons. Such an analysis was presented by Silberberg6 for the case of bright solitons. It follows from his analysis, as well as from numerical studies of the effect of TPA on nonlinear directional couplers and nonlinear distributed-feedback grating devices,7 that TPA seems to be a fundamental
limitation of optical
switching devices based on an intensity-dependent refractive index. The purpose of this Letter is to analyze the effect of TPA on dark solitons.
One of
the aims of our analysis is to demonstrate that TPA does not produce a critical limitation for dark-soliton switching because the steering angle, the key characteristic of any switching device based on dark soliton propagation, is almost preserved as the intensity of the background wave decays in the presence of weak TPA. To describe the effect of TPA on dark solitons, we consider the nonlinear Schr6dinger equation modified 0146-9592/94/050344-03$6.00/0
as follows (see, e.g., Ref. 6):
.au +1 az
a 2U _U1 2 U 2 aX2 l
-iKu
2u
(1)
Here u is the normalized field amplitude, z and x are the normalized propagation and transverse coordinates, and K is the normalized TPA coefficient, K = /3/2kn2 , where k is the free-space wave vector and ,8 and n 2 are the intensity-dependent absorption and refractive-index coefficients, respectively. As is well known, in the absence of the TPA contribution (i.e., at K = 0) Eq. (1) describes the case of a defocusing Kerr nonlinearity in which dark spatial solitons may propagate on a modulationally stable background wave u = uoexp(-iuo 2z), u0 being the background intensity. The nonlinear absorption, even when small, leads to attenuation of the cw background wave, and the wave's amplitude and phase become slowly dependent on Kuo 2 (0)z according to
u0 (z) 0(z)
-
=
[1 + 2Kuo2(0)z]"12
/Z
uo2 (z') dz, =
1 ln[1 + 2KUo2 (0)z]. (2)
This slow attenuation of the background wave makes it rather difficult to apply a perturbation method immediately to analyze the dissipation-induced evolution of dark solitons, although the case of a linear absorption"lo
is an exception:
As will follow from
the subsequent analysis, the TPA-induced dynamics of a dark soliton is more complicated than that due to linear absorption. To take into account explicitly the TPA-induced evolution of the background wave, we apply the following transformation:
u(z,x) = uo(z)exp[iO(z)]v(z,x),
(3)
where u0(z) and 0(z) change according to Eqs. (2), and obtain the following equation for v:
u i1 a2V _ (I|2-V12-iK(v1 (11 - 1)v = ( 1994 Optical Society of America
2
-l)V, -
1)v, (4)
March 1, 1994 / Vol. 19, No. 5 / OPTICS LETTERS
where ; and 6 are new coordinates that are connected to z and x by the differential relations d; = 2(Z)dz and de = uo(z)dx. After such a transformation, the resulting Eq. (4) has a vanishing perturbation of - K, and it may now be treated by any variant t of perturbation theory for solitons. We note that the transformation
345
One of the main characteristics of the dark-soliton switching devices is the so-called steering angle.2 It is easy to see that the total shift of the dark soliton along the x axis is given by the relation JZ dz'uo(z') sin q(z'), so that the steering angle X may be defined through the local transverse velocity:
[Eq. (3)] is the simplest way to solyve
W(z) = tan X = uo(z)sin 0 (z).
(13)
the problem of nonvanishing boundary conditions for a dark soliton; however, the perturbation-indu cedi evolution of dark solitons may also be analyzed Ieither in the small-amplitude limit' or with the h .elp
The important conclusion based on Eq. (13) is the following: When the dark soliton propagates in the
of the so-called regularized
the function sin 0 (z) grows slowly, keeping, at least
integrals of motion."
The dark soliton of Eq. (4) in the absence of T'PA has the form
v(;,)= cos tanhZ + isinX, with
-q= cos X,
Z = -q(e - f,
fl = sin k,
(6) and such a solution is characterized by the inter rnal ess phase angle '0(Is I ' wr/2)that describes the darknzess of the soliton through the relation lvi2
IV12 cos2 (k 1 -cosh 2 Z
(7)
=
_12
If the TPA coefficient K is small, we expect the s ton features of the solution [Eqs. (5) and (6)] to domi-
nate, so that the structure's evolution may be trea ted adiabatically as in Eq. (5) with the angle (k slo,wly dependent on so that
presence for small
of TPA on a decaying background
uo(z),
/ (0), the product [Eq. (13)] almost con-
stant. This simply means that the steering angles for switching devices based on dark-soliton propagation are almost preserved in a Kerr nonlinear medium in the presence of TPA. From the physical point of view, this important property simply follows from the nature of nonlinear absorption: the background intensity decays faster than the central minimum in the soliton, forcing the soliton contrast to change. To confirm our analysis, we have checked the results [Eqs. (2) and (12)] of the adiabatic approximation by comparing them with a numerical solution of Eq. (1). Figure 1 shows the evolution of the background uo(z),the soliton parameter sin 0(z), and the 1.0 0.8
',
Z = cos k(4{
-
f
0.6
d;' sinck(i2)
(8)
1v V1 +-) 0.4
To find the internal dynamics of the dark soliton, we use a Hamiltonian approach for the perturbation theory for solitons.' 2
the Hamiltonian of the system [Eq. (4)] is a conserved quantity
given by
H=f dd
0.2
Without the TPA, i.e., at K = 0,
2
av-+ -22
-
.
0.0 ' 0.0
(9)
1.0 2.0 Propagation Distance (Kz)
3.0
(a)
1.0
It follows from Eq. (4) at K # 0 that TPA leads to slow
attentuation, and the Hamiltonian [Eq. (9)] changes
0.8
according to
dH
=iKf f de(IV12 -1)
(Vaa
-v*
0.6
(10) 0.4
Substituting the solution [Eqs. (5) and (6)], with H calculated for the soliton in Eq. (5) and H8 = (4/3) cos3 q$,into Eq. (10) and returning to the primary variables z and x, we find the equation for the soliton angle:
dz = - Kuo 2(z)sin(2k),
(11)
dz 3
which may be easily integrated to give k (z) = tan-'{tan 0 (0)[1 + 2Kuo2 (0)z]h3 } .
(12)
Equations (2), (3), (5), and (12) totally describe dark soliton evolution in the presence of TPA.
0.2 0.0 ' 0.0
1.0 2.0 Propagation Distance (Kz )
3.0
(b)
Fig. 1. Evolution of the background amplitude, uo(z), the normalized dark soliton eigenvalue, sin qS(z), and the transverse velocity of the dark soliton, W(z), along the Kerr medium with TPA at K = 0.05. The solid curves are from Eqs. (2) and (12), and the symbols are from a numerical simulation of Eq. (1) for (a) 0(0) = 0.2nr and (b) q5(0)= 0.17r.
OPTICS LETTERS / Vol. 19, No. 5 / March 1, 1994
346
(a)
10
-40
-20
0 X
20
40
~32^t i
-40
-20
0 X
20
40
switching device based on dark spatial solitons is the only one yet found that is mostly insensitive to TPA. In fact, the approach that we have presented here is rather general and can be applied to different problems involving dark-soliton propagation on varying backgrounds. In particular, it permits the analysis of the instability of a fundamental dark soliton in the presence of gain used to compensate the two-photon absorption.
(b)
10
o
°
k1 5
5 -40
-20
0 X
20
40
-40
-20
0 X
20
40
Fig. 2. Contour plots demonstrating the effect of TPA on a dark soliton for K = 0.05 and O0() = 0.2vn (left-hand column) and O0() = 0.1v- (right-hand column) based on (a) an adiabatic approximation given by Eqs. (2) and (12) and (b) a numerical simulation of Eq. (1).
transverse
velocity W(z) = uo(z)sin 0 (z) for two dif-
ferent initial values of the soliton angle sb(0). The analytical results (solid curves) based on Eqs. (2) and (12) are in perfect agreement with the results of the numerical simulations (symbols), and, as may be seen, the steering angle is almost preserved, provided that 4.(0) is small. Small deviations of the numerical data from the adiabatic relationship are caused by transition radiation that slightly changes the intensity of the background (see Fig. 2). Finally, it is important to compare the results [Eq. (12)] for TPA with the corresponding result for linear absorption described by the contribution -iyu on the right-hand side of Eq. (1) instead of -iKjul 2 u. Applying the same approach as above, we find that this time the background decays exponentially, u0 (z) = uo(0)exp(-yz), and the soliton phase 0 does not change, k(z) = O(0), so that the contrast of the dark soliton (- cos2 0) is always preserved and, therefore, the transverse velocity W(z) decays exponentially in direct proportion to uo(z) (see Refs. 8-10).
In conclusion, we have analyzed the effect of TPA on the propagation of dark optical solitons. It has been pointed out that TPA does not seem to produce a drastic limitation to optical switching based on dark spatial solitons, and the transverse velocity of a dark soliton is almost preserved as the intensity of the background wave decays in the presence of weak TPA. Such a property makes the effects of TPA and linear absorption quite different for darksoliton switching devices compared with other nonlinear optical switching devices, and thus it seems that a
13
The research of Yuri S. Kivshar was supported by the Australian Photonics Cooperative Research Centre, and the research of David R. Andersen was supported in part by National Science Foundation grant ECS-9105660. In addition, David R. Andersen acknowledges stimulating discussions and support while he was a visiting fellow at the Laser Physics Centre, Australian National University. Xiaoping Yang and Barry Luther-Davies are also with the Laser Physics Centre, Australian National University.
References 1. F. Reynaud and A. Barthelemy, Europhys. Lett. 12, 401 (1990); Barthelemy, Opt. 2. B. Luther-Davies (1992); X. Yang,
M. Shalaby, F. Reynaud, and A. Lett. 17, 778 (1992). and X. Yang, Opt. Lett. 17, 1755 B. Luther-Davies, and W. Kroli-
kowski, Int. J. Nonlinear Opt. Phys. 2, 1 (1992). 3. G. I. Stegeman, C. T. Seaton, and R. J. Zanoni, Thin Solid Films 152, 231 (1987); P. N. Townsend, G. L.
Baker, N. E. Schlotter, C. F. Klausner, and S. Etemad, Appl. Phys. Lett. 53, 1782 (1988). 4. G. R. Allan, S. R. Skinner, D. R. Andersen, Smirl, Opt. Lett. 16, 156 (1991).
and A. L.
5. M. Sheik-Bahae, D. J. Hagan, and E. W. Van Stryland,
Digest of International Quantum ElectronicsConference (Optical Society of America, Washington, D.C., 1990), paper QTUM3. 6. Y. Silberberg, Opt. Lett. 15, 1005 (1990). 7. V. Mizahi, K. W. DeLong, G. I. Stegeman, M. A. Saifi,
and M. J. Andrejco, Opt. Lett. 14, 1140 (1989); K W. DeLong, K. B. Rochford, and G. I. Stegeman, Phys. Lett. 55, 1823 (1989).
Appl.
8. J. A. Giannini and R. I. Joseph, IEEE J. Quantum Electron. 26, 2109 (1990). 9. M. Lisak, D. Andersen, and B. A. Malomed, Opt. Lett. 16, 1936 (1991).
10. Yu. S. Kivshar, Phys. Rev. A 42, 1757 (1990); see also the review paper, IEEE J. Quantum Electron. 28, 250 (1993). 11. I. M. Uzunov and V. S. Gerdjikov, Phys. Rev. A 47, 1582 (1993). 12. Yu. S. Kivshar, and B. A. Malomed, Rev. Mod. Phys. 61, 763 (1989).
13. Y. Chen and J. Atai, Opt. Lett. 16, 1933 (1991); Y. Chen, Phys. Rev. A 45, 6922 (1992).
