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Coupled solitons in waveguides with cascaded second order nonlinearity
1 August 1994
OPTICS COMMUNICATIONS Optics Communications
ELSEVIER
110 (1994) 75-79
Coupled solitons in waveguides with cascaded second order nonlinearity Mirodaw
A. Karpierz, Maciej Sypek
Institute of Physics, Warsaw Universityof Technology, Koszykowa 75, 00-662 Warszawa, Poland Received 3 1 March 1994
Abstract
The propagation of two pulses in optical waveguides with second order nonlinearity is. analyzed. Analytical solutions in the form of coupled solitons are presented and it is shown by utilizing the Beam Propagation Method that they are stable at long distances.
I. Introduction Nonlinear guided-wave phenomena has been considered as an attractive method of realizing high speed all-optical switching devices and transmission systems [ 11. One of the major problem, which should be overcome is to find suitable materials with high third-order susceptibility. Recently, a cascaded second-order nonlinear process has been applied to create intensity-dependent phase changes of the wave propagating in optical waveguides. This kind of nonlinear effect was proposed in implementation of all-optical devices (among others nonlinear MachZehnder interferometers and nonlinear directional couplers) for continuous waves as well as for solitons [ 2-51. The cascaded second order nonlinearity can be effectively utilized in integrated optical structures based on noncentrosymmetrical crystals. On the other hand, esterberg and Margulis [6] have discovered the possibility of frequency doubling in optical fibers. This phenomena of symmetry forbidden conversion is currently investigated experimentally and analyzed theoretically [ 7- 12 1. It seems to be very promising to apply the cascaded nonlinear effect in optical fibers
prepared for efficient second harmonic generation. In previously analyzed devices with cascaded second order nonlinearity the investigation was concentrated on the behavior of the fundamental wave, while the second harmonic field was necessary only to obtain effective phase changes of the fundamental wave. However, the cross-phase modulation of two waves with different frequencies leads to soliton pairing phenomena [ 13-191. In this paper, second order nonlinearity is a source of a novel class of coupled pulses with fundamental and second harmonic frequencies propagating without changing of theirs shapes. Two pulses mutually exchange their energy due to the second harmonic generation (w + o) and difference frequency generation (20 - w). This two processes compensate themselves mutually and the energy carried by each pulse is constant. On the other hand, the cascaded secondorder nonlinear process is a source of the intensitydependent phase change which leads to the dispersion compensation of both pulses in the optical waveguide. In this paper three types of such coupled solitons are presented and their propagation is found to be stable at long distances.
0030-4018/94/$07.00 @ 1994 Elsevier Science B.V. All rights reserved SSDI0030-4018(94)00301A
16
M.A. Karpierz, M. Sypek /Optics Communications 110 (1994) 75-79
2. Theory
U2 = Azexp[-id(l
It is assumed that two electromagnetic waves propagate in lossless optical waveguide: fundamental at frequency o and second harmonic 20. Then the electric transversal field of the light wave can be expressed in the form:
where K is a constant, into the form: i&A,
+ q
+ez%(x,v)
U,(z,t)exp[i(ot
- i(2N2-
Nl)kz]
(2)
where x (2) is a nonlinear susceptibility. Hence, the wave equation for two considered frequencies is taken in the form: + J-&)U,
+ tDI$U,
i(&
+ &&)Uz
+ kD2$U2
+ q&A2
= yU;&exp(-iaz),
Al = A2
0.
=
92
k,/‘ZJJx’*‘yl:Ydxdy
(4)
where ZO = m is the impedance of free space. The aim of our interest is a solution consisting of two pulses going together with the same group velocity. Therefore we put 211= 71~= v and thus we rescale the time coordinate: T = (t - z/v ). Assuming that the amplitudes U, are expressed by
ew(i(ol),
cosh2(r/ro) 1 cosh2(t/r,,)
ew(ip2),
(6)
where the width of the soliton is defined by the formula:
= yUtU,exp(i6z),
where 6 = 2 (N2 - Ni ) k is a difference of propagation constants, v, = (k d N,/dw)-’ are the group velocities, D, = k 8’N,/802 are group velocity dispersions for both frequencies, and a nonlinear coupling coefficient:
1 + K)z],
- ^JA,A, =
1 41
2(U-D2) 6
(3)
UI = Al exp[-id(
2~)A2
The above coupled equations (5 ) have analytical solutions in a form of solitary waves propagating without changes of their amplitudes shapes. There are several types of such coupled solitons and the type depends on the value and signs of the group velocity dispersions D1 and D2 and on the sign of the propagation constants difference 6. In following, three different pairs of first order coupled solitons are presented. (i)For(2Di-02)/6 0) the coupled bright solitons are obtained:
T; =
Y=
+ 6(1 +
0,
(5)
- 2iNikz]},
i(A
K)/d, - y/l;/42 =
(1)
where k = 2nlli is a wavenumber of the fundamental frequency, N, (j = 1,2) are effective refractive indices, Yj are waveguide modes envelopes, e, are unit vectors, and U, are slowly varying complex amplitudes of both waves. The nonlinear polarization describing the interaction between these waves due to the second order nonlinear process is given by:
+ Y,!Pu~,U~U, exp[2iwt
+ 6( 1 +
- N&z)]
U2(z,f)exp[i(20t-2N2kz)l,
PNL = tox’*‘{2Y~Y2U;U2exp[iot
$Ai
the set of Eq. (3) is rewritten
<
i&A2 Et = el Y,(x,y)
+ ~K)z],
'
solitons amplitudes
are given by
(8) and the constant K =
-(D,
K
was chosen in the form:
- D2)/(2D,
~ D2).
(9)
For negative group velocity dispersions (D, ,D2 < 0) the constant phases of both solitons fulfill the condition: v)2 = 2y, i (e.g. both amplitudes A, are real ), and for positive group velocity dispersions (Dl,D2 > 0) the phases fulfill the relation: ~2 = 2yl i + 7r (e.g. both amplitudes are real and the amplitude A2 is negative). (ii) For (20, - D2)/8 > 0 and for both D1 and D2 with the same sign (DIDI > 0) the coupled semi-dark solitons are obtained:
II
M.A. Karpierz, M. Sypek / Optics Communications I1 0 (1994) 75- 79
1
Al =41
exp(ipi ),
cosh2 ( 75/ro) > 1
A2 =q2
exp(b),
cosh2(r/r0)
(10)
where the soliton width TO, the amplitudes q1 and q2, and the constant K are defined by Eqs. (7), (8) and (9) respectively. For positive group velocity dispersions (Dl,Dz > 0) the phases fulfill the relation ~2 = 2y, ,, and for negative group velocity dispersions (D,, D2 < 0) the relation ~2 = 2~~ + n. (iii) For (D, + 02 ) /6 > 0 and for the group velocity dispersions DI and 02 with opposite signs ( DI 02 < 0) the coupled bright and semi-bright solitons are obtained: A,
=
sinh(rl7o)
q,
cosh2 ( r/ zo)
AZ = q2
(11)
cosh2(r/ro)
K = -(DI
K
2
4
A ,,‘.
/
2 oo.
\
/.,,,,,
J
I’
-.--
:\_ ‘,
_I
‘,
,I
‘~.__,
I -4
-2
0
2
4
TIME r/r,
Fig. 1. Normalized ampiiiudes /Aj/q,l and intensities (I = IAj/qji’ ) for analyzed coupled solitons: bright (solid line), semi-dark (dashed) and semi-bright (dotted).
where the width of the solitons is equal to
the constant
0
LOi
-05-
1
z; = Ippq,
-2
2
ew(iyll), exp(iyl2L
I -4
(12) was chosen to be
+ 202)/(2Dl
+
2021,
(13)
and the amplitudes q1 and q2 have the form defined by Eq. (8). For DI < 0 and D2 > 0 the phases fulfill the relation ~2 = FIJI,, and for DI > 0 and D2 < 0 the relation p2 = 2~1 + 7L. It should be noted that pairing of the dark and bright solitons are not allowed in Eq. (5 ) because at infinity (z + 00, where d2A/bz2 = 0) both relation A, = 0 and A/ f 0 (j tr, I) cannot be fulfilled simultaneously.
3. Results and conclusions Normalized amplitudes lAj/q, 1and intensities Zj = IAj/qjJ’ of coupled solitons introduced in Eqs. (6), ( 10) and ( 11) are presented in Fig. 1. The well-known solitons obtained in the third-order nonlinear media have sech or tanh type shapes whereas the coupled solitons presented in this paper are based on the sech square function. The important property of the coupled solitons is that their width is determined by the
waveguide parameters: group velocity dispersions Dj and propagation constants difference 6. For typical silica waveguides the group velocity dispersion is of the order 10 ps’/km for the wavelengths in the range of the lower losses. Thus to obtain the soliton width ro N 10 ps it is required (N2 - N,) N lo-” which means that a very precious quasi phase matching between both frequencies is necessary. Another difference between analyzed coupled solitons and the solitons in the third-order nonlinear media is dependence of their peak intensity and total energy on the pulse width. The maximal peak intensity of coupled bright and semi-bright solitons (proportional to q,’ defined by Eq. (8) ) depends on the pulse width as ~~~ and the energy of both bright and semibright solitons is proportional to 70~ while in classical solitons is proportional respectively to to2 (peak intensity) and to SO’ (energy). Therefore for short pulses the third order nonlinear effect will dominate while for long pulses the solitons are created due to the second-order nonlinearity and the third order nonlinear effect can be neglected. The stability of introduced coupled solitons were investigated by standard spline-step Beam Propagation Method and it was observed that the coupled soli-
78
M.A. Karpierz, M. Sypek / Optics Communications 110 (I 994) 75 79
Fig. 3. The same as Fig. 2 at longer distances. Fig. 2. Propagation of coupled pulses investigated by BPM. The normalized distance Dz/si , time 7/70 and intensities I = IAj/qjl’ are used. The initially launched pulse is times wider than the proper bright soliton solution.
v’?ro
tons do not change their shapes at long distances. Fig. 2 presents the envelope amplitude evolution of the pulses initially launched different from the coupled solitons solution. The presented result were obtained for the waveguide with the group velocity dispersions DI = Dz = D and positive propagation constants difference 6 > 0 , which causes that the constant K = 0 and condition to obtain the bright coupled solitons (Eq. (6) ) are fulfilled. The initially launched pulses have the shapes defined by Eq. (6) but for the amplitude qj/2 and the width v’??ra (i.e. with the same energies but wider then bright coupled solitons). The amplitude of observed pulse shape oscillation decays at large distances as it is showed in Fig. 3. Fig. 4 presents the comparison of the pulse shapes at the beginning distances of the waveguide (with maximal and minimal amplitudes from Fig. 2) with the pulse shapes observed at larger distances (from Fig. 3). This comparison shows that pulses tend to the coupled soliton solution. Coupled solitons solution requires identical group velocities for both frequencies (VI = 212 ) and insignificantly small phase mismatch condition (6 < k and 6 # 0). The equality of group velocities can be obtained in several ways, among others by using different waveguide modes with the same or different polarization (i.e. e i = e 2 or e i i e 2) and/or for the wavelengths chosen at both sides of the zero-dispersion of group velocity. The required waveguide modes prop-
/, -4
-2
0
I
I
2
4
TIME r/r0 Fig. 4. Comparison of pulses from Fig. 2 (dashed lines) and Fig. 3 (solid lines) of maximal and minimal amplitudes with the bright soliton (dotted line).
erties are obtained by adjustation the refractive index distribution in optical waveguide for chosen wavelengths and required relation between signs of the group velocity dispersions. The quasi phase matching conditions (6 < k ) generally is more difficult to fulfill. It could be for example obtained by application of periodic modulation of the x”’ tensor, and then 6 = 2 ( N2 - N, ) * G, where G is a grating constant. The second harmonic generation in silica fibers is described by this kind of phase matching. Assuming the group velocity dispersion D = 20 ps2/km and soliton width ro = 10 ps the unit of normalized distance < = Dz/& used in Figs. 2, 3, corresponds to 200 m. For the nonlinear susceptibility observed in silica fibers prepared for second harmonic generation (x (2) N 2.2 x lo-” m/V mea-
MA. Karpierz, M. Sypek / Optics Communications 110 (1994) 75- 79
sured in Ref. [ 93 ) the calculated pulse power q* = 6 mW and for nonlinear susceptibility of KTP this pulse power is equal to q* = 5 x 1O-‘o W.
Acknowledgements This work was partially supported by the Polish Committee of Scientific Research (KBN) under the contract No. P 406 0 18 04. The authors would like to acknowledge useful discussions with Prof. Bozena Jaskorzyxiska.
References 111 G.I. Stegeman and E.M. Wright, Opt. and Quantum Electron. 22 (1990) 95. 121G. Assanto, G. Stegeman, M. Sheik-Bahae and E. Van Stryland, Appl. Phys. Lett. 62 (1993) 1323. [31 R. DeSalvo, D. J. Hagan, M. Sheik-Bahae, G. Stegeman, E.W. Van Stryland and H. Vanherzeele, Optics Lett. 17 ( 1992) 28. [41 R. Schick, J. Opt. Sot. Am. B 10 (1993) 1848. [51 D.C. Hutchings, J.S. Aitchison and C.N. Ironside, Optics Lett. 18 (1993) 793.
79
[61 U. Osterberg and W. Margulis, Optics Lett. 11 (1986) 516; 12 (1987) 57. 171 P. Chmela, Modem Nonlinear Optics, Part 1, eds. M. Evans and S. Kielich (Wiley, New York, 1993) p. 249. [81 D. Z. Anderson, V. Mizrahi and J.E. Sipe, Optics Lett. 16 (1991) 796. [91 H.W.K. Tom, R. H. Stolen, G.D. Aumiller and W. Pleibel, Optics Lett. 13 (1988) 512. 1101 V. Mizrahi, Y. Hibino and G.I. Stegeman, Optics Comm. 78 (1990) 283. ill1 G. Demouchy and G.R. Boyer, Optics Comm. 101 (1993) 385. [I21 MI. Dyakonov and AS. Furman, Optics Comm. 91 (1992) 489. [I31 D. Schadt and B. Jaskorzynska, J. Opt. Sot. Am. B 5 (1988) 2374. iI41 J.E. Rothenberg, Phys. Rev. A 42 (1990) 682. [I51 G.P. Agrawal, P.L. Baldeck and R.R. Alfano, Phys. Rev. A 40 (1989) 5063. [I61 V.V. Afanasjev, E.M. Dianov and V.N. Serkin, IEEE J. Quantum Electron. 25 (1989) 2656. [I71 M. Lisak, A. Hook and D. Anderson, J. Opt. Sot. Am. B 7 (1990) 810. 1181 V.L. da Silva and C.H.B. Cruz, J. Opt. Sot. Am. B 7 ( 1990) 750. 1191 S. Trillo, S. Wabnitz, E.M. Wright and G.I. Stegeman, Optics Lett. 13 (1988) 871.
OPTICS COMMUNICATIONS Optics Communications
ELSEVIER
110 (1994) 75-79
Coupled solitons in waveguides with cascaded second order nonlinearity Mirodaw
A. Karpierz, Maciej Sypek
Institute of Physics, Warsaw Universityof Technology, Koszykowa 75, 00-662 Warszawa, Poland Received 3 1 March 1994
Abstract
The propagation of two pulses in optical waveguides with second order nonlinearity is. analyzed. Analytical solutions in the form of coupled solitons are presented and it is shown by utilizing the Beam Propagation Method that they are stable at long distances.
I. Introduction Nonlinear guided-wave phenomena has been considered as an attractive method of realizing high speed all-optical switching devices and transmission systems [ 11. One of the major problem, which should be overcome is to find suitable materials with high third-order susceptibility. Recently, a cascaded second-order nonlinear process has been applied to create intensity-dependent phase changes of the wave propagating in optical waveguides. This kind of nonlinear effect was proposed in implementation of all-optical devices (among others nonlinear MachZehnder interferometers and nonlinear directional couplers) for continuous waves as well as for solitons [ 2-51. The cascaded second order nonlinearity can be effectively utilized in integrated optical structures based on noncentrosymmetrical crystals. On the other hand, esterberg and Margulis [6] have discovered the possibility of frequency doubling in optical fibers. This phenomena of symmetry forbidden conversion is currently investigated experimentally and analyzed theoretically [ 7- 12 1. It seems to be very promising to apply the cascaded nonlinear effect in optical fibers
prepared for efficient second harmonic generation. In previously analyzed devices with cascaded second order nonlinearity the investigation was concentrated on the behavior of the fundamental wave, while the second harmonic field was necessary only to obtain effective phase changes of the fundamental wave. However, the cross-phase modulation of two waves with different frequencies leads to soliton pairing phenomena [ 13-191. In this paper, second order nonlinearity is a source of a novel class of coupled pulses with fundamental and second harmonic frequencies propagating without changing of theirs shapes. Two pulses mutually exchange their energy due to the second harmonic generation (w + o) and difference frequency generation (20 - w). This two processes compensate themselves mutually and the energy carried by each pulse is constant. On the other hand, the cascaded secondorder nonlinear process is a source of the intensitydependent phase change which leads to the dispersion compensation of both pulses in the optical waveguide. In this paper three types of such coupled solitons are presented and their propagation is found to be stable at long distances.
0030-4018/94/$07.00 @ 1994 Elsevier Science B.V. All rights reserved SSDI0030-4018(94)00301A
16
M.A. Karpierz, M. Sypek /Optics Communications 110 (1994) 75-79
2. Theory
U2 = Azexp[-id(l
It is assumed that two electromagnetic waves propagate in lossless optical waveguide: fundamental at frequency o and second harmonic 20. Then the electric transversal field of the light wave can be expressed in the form:
where K is a constant, into the form: i&A,
+ q
+ez%(x,v)
U,(z,t)exp[i(ot
- i(2N2-
Nl)kz]
(2)
where x (2) is a nonlinear susceptibility. Hence, the wave equation for two considered frequencies is taken in the form: + J-&)U,
+ tDI$U,
i(&
+ &&)Uz
+ kD2$U2
+ q&A2
= yU;&exp(-iaz),
Al = A2
0.
=
92
k,/‘ZJJx’*‘yl:Ydxdy
(4)
where ZO = m is the impedance of free space. The aim of our interest is a solution consisting of two pulses going together with the same group velocity. Therefore we put 211= 71~= v and thus we rescale the time coordinate: T = (t - z/v ). Assuming that the amplitudes U, are expressed by
ew(i(ol),
cosh2(r/ro) 1 cosh2(t/r,,)
ew(ip2),
(6)
where the width of the soliton is defined by the formula:
= yUtU,exp(i6z),
where 6 = 2 (N2 - Ni ) k is a difference of propagation constants, v, = (k d N,/dw)-’ are the group velocities, D, = k 8’N,/802 are group velocity dispersions for both frequencies, and a nonlinear coupling coefficient:
1 + K)z],
- ^JA,A, =
1 41
2(U-D2) 6
(3)
UI = Al exp[-id(
2~)A2
The above coupled equations (5 ) have analytical solutions in a form of solitary waves propagating without changes of their amplitudes shapes. There are several types of such coupled solitons and the type depends on the value and signs of the group velocity dispersions D1 and D2 and on the sign of the propagation constants difference 6. In following, three different pairs of first order coupled solitons are presented. (i)For(2Di-02)/6
T; =
Y=
+ 6(1 +
0,
(5)
- 2iNikz]},
i(A
K)/d, - y/l;/42 =
(1)
where k = 2nlli is a wavenumber of the fundamental frequency, N, (j = 1,2) are effective refractive indices, Yj are waveguide modes envelopes, e, are unit vectors, and U, are slowly varying complex amplitudes of both waves. The nonlinear polarization describing the interaction between these waves due to the second order nonlinear process is given by:
+ Y,!Pu~,U~U, exp[2iwt
+ 6( 1 +
- N&z)]
U2(z,f)exp[i(20t-2N2kz)l,
PNL = tox’*‘{2Y~Y2U;U2exp[iot
$Ai
the set of Eq. (3) is rewritten
<
i&A2 Et = el Y,(x,y)
+ ~K)z],
'
solitons amplitudes
are given by
(8) and the constant K =
-(D,
K
was chosen in the form:
- D2)/(2D,
~ D2).
(9)
For negative group velocity dispersions (D, ,D2 < 0) the constant phases of both solitons fulfill the condition: v)2 = 2y, i (e.g. both amplitudes A, are real ), and for positive group velocity dispersions (Dl,D2 > 0) the phases fulfill the relation: ~2 = 2yl i + 7r (e.g. both amplitudes are real and the amplitude A2 is negative). (ii) For (20, - D2)/8 > 0 and for both D1 and D2 with the same sign (DIDI > 0) the coupled semi-dark solitons are obtained:
II
M.A. Karpierz, M. Sypek / Optics Communications I1 0 (1994) 75- 79
1
Al =41
exp(ipi ),
cosh2 ( 75/ro) > 1
A2 =q2
exp(b),
cosh2(r/r0)
(10)
where the soliton width TO, the amplitudes q1 and q2, and the constant K are defined by Eqs. (7), (8) and (9) respectively. For positive group velocity dispersions (Dl,Dz > 0) the phases fulfill the relation ~2 = 2y, ,, and for negative group velocity dispersions (D,, D2 < 0) the relation ~2 = 2~~ + n. (iii) For (D, + 02 ) /6 > 0 and for the group velocity dispersions DI and 02 with opposite signs ( DI 02 < 0) the coupled bright and semi-bright solitons are obtained: A,
=
sinh(rl7o)
q,
cosh2 ( r/ zo)
AZ = q2
(11)
cosh2(r/ro)
K = -(DI
K
2
4
A ,,‘.
/
2 oo.
\
/.,,,,,
J
I’
-.--
:\_ ‘,
_I
‘,
,I
‘~.__,
I -4
-2
0
2
4
TIME r/r,
Fig. 1. Normalized ampiiiudes /Aj/q,l and intensities (I = IAj/qji’ ) for analyzed coupled solitons: bright (solid line), semi-dark (dashed) and semi-bright (dotted).
where the width of the solitons is equal to
the constant
0
LOi
-05-
1
z; = Ippq,
-2
2
ew(iyll), exp(iyl2L
I -4
(12) was chosen to be
+ 202)/(2Dl
+
2021,
(13)
and the amplitudes q1 and q2 have the form defined by Eq. (8). For DI < 0 and D2 > 0 the phases fulfill the relation ~2 = FIJI,, and for DI > 0 and D2 < 0 the relation p2 = 2~1 + 7L. It should be noted that pairing of the dark and bright solitons are not allowed in Eq. (5 ) because at infinity (z + 00, where d2A/bz2 = 0) both relation A, = 0 and A/ f 0 (j tr, I) cannot be fulfilled simultaneously.
3. Results and conclusions Normalized amplitudes lAj/q, 1and intensities Zj = IAj/qjJ’ of coupled solitons introduced in Eqs. (6), ( 10) and ( 11) are presented in Fig. 1. The well-known solitons obtained in the third-order nonlinear media have sech or tanh type shapes whereas the coupled solitons presented in this paper are based on the sech square function. The important property of the coupled solitons is that their width is determined by the
waveguide parameters: group velocity dispersions Dj and propagation constants difference 6. For typical silica waveguides the group velocity dispersion is of the order 10 ps’/km for the wavelengths in the range of the lower losses. Thus to obtain the soliton width ro N 10 ps it is required (N2 - N,) N lo-” which means that a very precious quasi phase matching between both frequencies is necessary. Another difference between analyzed coupled solitons and the solitons in the third-order nonlinear media is dependence of their peak intensity and total energy on the pulse width. The maximal peak intensity of coupled bright and semi-bright solitons (proportional to q,’ defined by Eq. (8) ) depends on the pulse width as ~~~ and the energy of both bright and semibright solitons is proportional to 70~ while in classical solitons is proportional respectively to to2 (peak intensity) and to SO’ (energy). Therefore for short pulses the third order nonlinear effect will dominate while for long pulses the solitons are created due to the second-order nonlinearity and the third order nonlinear effect can be neglected. The stability of introduced coupled solitons were investigated by standard spline-step Beam Propagation Method and it was observed that the coupled soli-
78
M.A. Karpierz, M. Sypek / Optics Communications 110 (I 994) 75 79
Fig. 3. The same as Fig. 2 at longer distances. Fig. 2. Propagation of coupled pulses investigated by BPM. The normalized distance Dz/si , time 7/70 and intensities I = IAj/qjl’ are used. The initially launched pulse is times wider than the proper bright soliton solution.
v’?ro
tons do not change their shapes at long distances. Fig. 2 presents the envelope amplitude evolution of the pulses initially launched different from the coupled solitons solution. The presented result were obtained for the waveguide with the group velocity dispersions DI = Dz = D and positive propagation constants difference 6 > 0 , which causes that the constant K = 0 and condition to obtain the bright coupled solitons (Eq. (6) ) are fulfilled. The initially launched pulses have the shapes defined by Eq. (6) but for the amplitude qj/2 and the width v’??ra (i.e. with the same energies but wider then bright coupled solitons). The amplitude of observed pulse shape oscillation decays at large distances as it is showed in Fig. 3. Fig. 4 presents the comparison of the pulse shapes at the beginning distances of the waveguide (with maximal and minimal amplitudes from Fig. 2) with the pulse shapes observed at larger distances (from Fig. 3). This comparison shows that pulses tend to the coupled soliton solution. Coupled solitons solution requires identical group velocities for both frequencies (VI = 212 ) and insignificantly small phase mismatch condition (6 < k and 6 # 0). The equality of group velocities can be obtained in several ways, among others by using different waveguide modes with the same or different polarization (i.e. e i = e 2 or e i i e 2) and/or for the wavelengths chosen at both sides of the zero-dispersion of group velocity. The required waveguide modes prop-
/, -4
-2
0
I
I
2
4
TIME r/r0 Fig. 4. Comparison of pulses from Fig. 2 (dashed lines) and Fig. 3 (solid lines) of maximal and minimal amplitudes with the bright soliton (dotted line).
erties are obtained by adjustation the refractive index distribution in optical waveguide for chosen wavelengths and required relation between signs of the group velocity dispersions. The quasi phase matching conditions (6 < k ) generally is more difficult to fulfill. It could be for example obtained by application of periodic modulation of the x”’ tensor, and then 6 = 2 ( N2 - N, ) * G, where G is a grating constant. The second harmonic generation in silica fibers is described by this kind of phase matching. Assuming the group velocity dispersion D = 20 ps2/km and soliton width ro = 10 ps the unit of normalized distance < = Dz/& used in Figs. 2, 3, corresponds to 200 m. For the nonlinear susceptibility observed in silica fibers prepared for second harmonic generation (x (2) N 2.2 x lo-” m/V mea-
MA. Karpierz, M. Sypek / Optics Communications 110 (1994) 75- 79
sured in Ref. [ 93 ) the calculated pulse power q* = 6 mW and for nonlinear susceptibility of KTP this pulse power is equal to q* = 5 x 1O-‘o W.
Acknowledgements This work was partially supported by the Polish Committee of Scientific Research (KBN) under the contract No. P 406 0 18 04. The authors would like to acknowledge useful discussions with Prof. Bozena Jaskorzyxiska.
References 111 G.I. Stegeman and E.M. Wright, Opt. and Quantum Electron. 22 (1990) 95. 121G. Assanto, G. Stegeman, M. Sheik-Bahae and E. Van Stryland, Appl. Phys. Lett. 62 (1993) 1323. [31 R. DeSalvo, D. J. Hagan, M. Sheik-Bahae, G. Stegeman, E.W. Van Stryland and H. Vanherzeele, Optics Lett. 17 ( 1992) 28. [41 R. Schick, J. Opt. Sot. Am. B 10 (1993) 1848. [51 D.C. Hutchings, J.S. Aitchison and C.N. Ironside, Optics Lett. 18 (1993) 793.
79
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