- Email: [email protected]
Numerical analysis of all-optical switching of a fiber Bragg grating induced by a short-detuned pump pulse
Optics Communications 92 (1992) 233-239 North-Holland
OPT ICS COMMUNICATIONS
Numerical analysis of all-optical switching of a fiber Bragg grating induced by a short-detuned pump pulse Jocelyn Lauzon, Sophie LaRochelle a n d Francois Ouellette Centre d'Optique, Photonique et Lasers, D~partement de G~nie Electrique, Universit~ Laval, Quebec, G1K 7P4, Canada
Received 22 January 1992; revised manuscript received 15 April 1992
We present the numerical solutions of the coupled-modeequations describing all-opticalswitchingof a fiber grating by a pump pulse detuned from the Braggcondition. The width of the pulse is shorter than the length of the grating. The results reveal that copropagating and counter-propagatingpump-probeconfigurationslead to different switchingcharacteristics. Furthermore, an initial detuning of the probe beam is found to improve the switching. These calculations are compared to experimental results previouslypublished. 1. Introduction In recent years, fast all-optical switching in optical fibers has been a field of active research [ 1-9 ]. This interest is explained by the high nonlinear-optical material figure of merit exhibited by silica glass [ 10 ]. Indeed, because of its low absorption coefficient, the weak nonlinearity of optical glass can be compensated by the long interaction length provided by optical fibers. All-optical switching has already been achieved with several configurations of optical fibers including nonlinear Mach-Zehnder interferometers [ 1 ], nonlinear directional couplers [ 2 ] and nonlinear loop mirrors [3,4]. Other devices have also been demonstrated using soliton switching [3-5 ], polarization rotation [ 6 ], and two-mode interferometry [ 7 ]. In 1988, it was shown that coupling two guided modes by a grating could reduce the switching power required to achieve complete mode conversion [ 11 ]. Park et al. subsequently proposed an intermodal switch that used a pump beam to excite two modes of an optical fiber [8]. Through cross-phase modulation, the beating of these modes produced periodic coupling between the two similar propagating modes of a probe beam. In 1990, LaRochelle et al. showed that cross-phase modulation could be used to switch the transmission of a permanent Bragg grating written in a photosen-
sitive optical fiber [ 9 ]. The switching powers were however found to be five times higher than their predicted value. It then became apparent that the proposed theoretical analysis was incomplete, especially for a pump-pulse width of the order of the grating length. In this paper, we investigate the switching behavior o f a Bragg grating induced by a short pump pulse. The coupled-mode equations describing the evolution of the probe fields are introduced in sec. 2 for co-propagating and counter-propagating pump-probe geometries. The analytical solutions of these equations in the case of a cw pump beam are also presented. The coupled-mode equations were numerically solved to describe the switching resulting from a pump pulse and the results are shown in sec. 3. The dependence of the switching characteristics on the pump-pulse width and peak power is examined. These numerical results predict that an initially detuned probe beam would improve the switching. In sec. 4, the numerical solutions are compared to experimental observations reported earlier [ 9 ].
2. Description of the model This section introduces the set of coupled-mode equations describing the evolution of the probe beam fields in a Bragg grating perturbed by a strong pump
0030-4018/92/$05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.
233
Volume 92, number 4,5,6
OPTICS COMMUNICATIONS
pulse. The well-known steady-state solutions of these equations are also presented. The pump power required to switch the reflection of the grating is then calculated for the case of a cw pump beam. Finally, a normalized pump-pulse peak power is introduced. A Bragg grating is assumed to be written from z = 0 to z=L in an infinite single-mode optical fiber (fig. 1 ). A weak cw probe beam is incident on the grating at z = 0. The electric field of the probe can be described in terms of a forward and a backward travelling wave, Eprob e : 8f
exp[i(flz-mt) ]
+ ~ exp [ i ( - f l z - m t ) ] + c . c . ,
( 1)
where fl=2nno/2o, no is the effective index, and 4o is the probe wavelength in vacuum. In the presence of a strong pump pulse, the coupled-mode equations that describe the slowly varying components of the probe fields are [ 12,13 ], O~f
no O~f - ---- ixgb exp ( -- i2Aflz) + 2iTPp gf c 0t
O--z- d -
--
0~b
no 0~b
Oz
c Ot----ix8fexp(i2Aflz)--2iyPpgb,
(2)
where x is the coupling coefficient, Afl=fl-K/2 is the probe detuning from the Bragg condition, K is the grating wavevector. The last terms on the righthand side of these equations represent the cross-phase modulation induced by the pump pulse on the probe fields. The coefficient 7 is equal to 2~n2/(Aeff2o), where n2 is the nonlinear Kerr index coefficient, and Aeff is the effective area obtained from the overlap
gaussian p u m p pulse
CW p r o b e b e a m
Z=O
'
j~
'
'
)~/2n0 Z=l
periodic modulation of the refractive index Fig. t. Brag grating of length L written in an optical fiber. 234
integral between the probe and the pump modes [ 13 ]. Finally, Pp is the pump power written as ep =Po exp - 2
(Zo_ct/no)_ 1_ , Wo /A
(3)
where Zo is the initial position of the pump pulse far from the grating, Po its peak power, and Woits width. The plus or minus signs refer respectively to a copropagating or a counter-propagating configuration. Since efficient Bragg gratings can be written in short lengths of optical fibers, distortion of the pump pulse due to self-phase modulation and dispersion is neglected. For example, 4.4 m m long Bragg gratings achieving 55o/o reflection have been written in photosensitive optical fibers using an external holographic technique [14]. In their experiment, LaRochelle et al. used a 3.5 cm long grating providing 50% reflection [9 ]. Furthermore, these gratings have typically very narrow bandwidth, from 200 MHz to 42 GHz [ 14,15 ]. It is thus assumed that the pump pulse is sufficiently detuned from the Bragg condition not to be affected by the presence of the grating. Also, the probe and the pump wavelengths can be closely spaced and the dispersion of the effective index of the fiber is safely neglected. Equations (2) and (3) can easily be rewritten in terms of normalized variables, O~vf
O~f
Ogb
O~
O--Z+ ~
=i
xL
eb exp[ --i2
AflL Z] + 2 i 7L Ppgf,
0Z
- - - - i xL gf exp [i2 Aft/. Z] - 2i 7L Pp & , OT
Pp=Po
exp[- 2(Z- ( Z° +-T) )
,
(4)
where Z=z/L, Wo=wo/L, T=t/z, and z=noL/c is the transit time of the pump pulse in the grating. The normalized full-width at the 1/e points of the pump pulse is Wfw=N//-2Wo. With the boundary conditions e l ( Z = 0 ) = d'm and 8b(Z= 1 ) = 0 , the steady state solution of eq. (4) when P p = 0 is [16]
optical fiber
',
1 September 1992
St(Z) = X
8fo e x p ( - i A f l L Z )
i~2L cos [I2L ( 1 - Z ) ] + Aft/, sin [£2L( 1 - Z ) ] iI2L cos(I2L) + AflL sin (s'2L)
Volume 92, number 4,5,6
°vb(Z) =
OPTICS COMMUNICATIONS
- 8to exp(iAflLZ) xL sin [~2L( 1 - Z ) ] i£-2Lcos(QL)+AflLsin(12L) '
(5) where 122=Af12-x 2. F r o m eq. (5), the intensity reflection coefficient o f an unperturbed Bragg grating is given by R = I # b ( Z = 0 ) 12 I~fol 2
( xL ) 2 sin2 (t2L) = (t2L)2 cos2(12L ) + (AflL)2 sin2(12L ) .
(6)
In steady state, the reflection o f a probe beam will thus switch from the m a x i m u m to first zero minim u m when t2L = n. Furthermore, it can be seen from eqs. (4) that the presence o f a cw p u m p beam, Wfw>> 1, introduces an additional detuning (AflL)'=27LPp. It is thus possible to define a normalized p u m p power P,=Pp/Ps, where
Ps
=
1 September 1992
sion o f gratings switched by short pulses are presented in fig. 2. The reflection is calculated at Z = 0 and the transmission at Z = 1. The peak o f the p u m p pulse enters the grating at T = 0 and exits at T = 1. The results are shown in fig. 2a for a co-propagating configuration ( P , = 2 . 5 , Wfw=1.13, KL=2.5 and AflL= 0.0), and in fig. 2b for a counter-propagating one ( P , = 1.5, Wfw=0.42, KL= 1 and A f l L = 0 . 0 ) . In both cases, the switching is sharper for the fields that propagate in the same direction as the p u m p pulse (transmission for the co-propagating geometry and reflection for the counter-propagating one). For the 150 125 100
///ELECT'ON'
75
( x L / 2 7 L )x/ (x/IcL ) 2 + 1
is the p u m p power that will produce complete switching of an initially tuned probe when Wrw >> 1 [ 9 ]. In the case o f a short p u m p pulse, the coupledmode equations (4) do not have an analytical solutions
50 ~/TSMISSION(9~,,,,,
~
j
25 0 0
1
2
3
time T
3. Numerical results In order to describe all-optical switching o f a Bragg grating by a short p u m p pulse ( Wrw < 1 ), the set o f coupled-partial-differential equations (4) were numerically solved. These calculations were done using the method o f lines with Hermit polynomials ~. The fields distributions o f an unperturbed Bragg grating, eq. (5), were used as initial conditions. In this section, the numerical results are presented for co-propagating and counter-propagating pump-pulse geometries. The switching characteristics are also examined as a function of the initial probe detuning, and as a function o f the pump-pulse width and peak power. The evolution o f the reflection and the transmis#1 We used a fortran subroutine from the IMSL Math/Library TM, explanation on the numerical method can be found in the user's manual, April 1987, pp. 680-687.
1O0
(b) 80
60
~EFLECTION (~)
TRANSMISSION 40
20
0
I
I
I
I
0
1
2
3
time T
Fig. 2. Evolution of the reflection and the transmission of a cw probe beam incident on a Bragg grating switched by: (a) a copropagating pump pulse (/',=2.5, Wr~=l.13, xL=2.5 and ApL=0.0), or (b) a counter-propagating pump pulse (P.= 1.5, Wfw=0.42, xL= 1 and AflL=0.0). 235
Volume 92, number 4,5,6
OPTICSCOMMUNICATIONS
fields propagating in the opposite direction, the wider characteristics can be partly understood by considering that the switching results from the convolution of the pump pulse with the probe electric fields inside the grating. In all numerical calculations, it was verified that the energy was conserved and that the grating was recovering its original state after the transient oscillations. Switching of the grating transmission was studied in more details for the co-propagating geometry. Fig. 3 shows the influence of the pump-pulse peak power on the calculated switching characteristics of a grating that initially transmits 2.7% of the incident probe beam (Wfw=0.71, x L = 2 . 5 and AflL=0.0). As the pump-pulse power is increased, different parts of the pulse switch the grating transmission and reflection through successive maxima and minima. These complex interference effects result in a multiple peak structure with a maximum transmission that can exceed 100% of the injected power. The results of the calculations for different pump-pulse widths, Wfw= 0.1, 0.3, 0.5, 0.8, and 1.2, are displayed in fig. 4 for a peak power P, = 2.5 and a tuned grating of x L = 2 . 5 . For a pump-pulse width Wfw~<0.05 the switching resuited in a single peak centered at T = 1. As the pump pulse becomes wider a second peak in the transmission grows and moves to later times. The position of the peak is initially linearly proportional to the width
1 September 1992
Fig. 4. Switching of the grating transmission produced by copropagating pump pulses of peak power P.=2.5 and of widths Wfw=0.1,0.3, 0.5, 0.8 and 1.2 (xL=2.5 and ApL=0.0).
1 O0
80
SSi0N (%)
6O
4O
2O
.~ECTION 0
i J
I
0
2
,
(%) i 4
, 6
time T
Fig. 5. Switchinginduced by a counter-propagatingpump pulse (P,= 1.5, Wf~=0.42) on the reflection and transmission of an initially detuned probe beam (~p/.= -3.0) incident on a Bragg filter (xL = 1 ).
Fig. 3. Switchingof the grating transmission producedby different pump-pulsepeak powersP,=0.50, 1.49, 2.49, 3.99 and 7.47, in a co-propagating geometry (Wfw=0.71, self2.5 and a#L=0.0). 236
of the pump pulse but eventually stabilizes at T = 1.7 for pump-pulse width Wfw~>1.2. The counter-propagating geometry was used to investigate the switching behavior of the grating reflection. As was previously observed for other devices [ 11 ], an initial detuning of the probe beam was found to improve the switching. In fig. 5, it can be seen that with a detuning of A p L = - 3 . 0 the reflectivity of the grating is now switched from 1.2% to 93.9% with a full-width at half-maximum (fwhm) of 0.32 for the principal peak, comparatively to the 55% modulation with a fwhm of 0.35 observed without
Volume 92, number 4,5,6
OPTICS COMMUNICATIONS
an initial detuning (fig. 2b). The reflected probepulse results from the interaction of the pump pulse with the probe fields all along the length of the fiber grating. Therefore, the switched reflectivity can be greater than the maximum steady state value of 58% obtained for a tuned grating o f x L = 1. In fig. 6, it can be seen that increasing the pump power results once again in a multiple peak structure in the switching characteristic due to transient oscillations. The secondary peak that appears first is centered at T = 2.0. A further increase in the pump-pulse peak power causes the peak located at T = 1.0 to split in two as the peak of the pump pulse over switch the grating reflection to the next minimum. The influence of the pump-pulse width on the calculated reflection switching is shown in fig. 7. The peak centered at O
1 September 1992
T = 1.0 gradually widens and eventually merges with the growing secondary peak at T=2.0. All-optical switching of a cw probe beam by a gaussian pump pulse should ideally produce a single undistorted pulse. The numerical simulations presented in this section indicate that the switching of a Bragg grating induced by a short detuned pump pulse sometimes results in a probe beam transmission, or reflection, with a multiple peak structure or with a highly distorted pulse shape. However, if the pump-pulse peak power and width are adequately chosen these undesirable effects can be avoided. Using a co-propagating geometry, more than 50% switching in the transmission of a x L = 2 . 5 grating was obtained without multiple peaks for pump-pulse peak powers, Pn, between 2.5 and 4.0, and widths, Wfw, greater than 0.5 (figs. 3 and 4). For a detuned x L = 1 grating and a counter-propagating configuration, the reflection could be switched by more than 75% with one important peak if the pump-pulse peak power, Pn, is around 3.0 and the pulse width, Wfw, is smaller than 0.4 (figs. 6 and 7).
4. Comparison with experimental results ~3
"~
'2.
"~
Fig. 6. Switchingof the grating reflectionproduced by a counterpropagatingpump pulse for various peak powersPn=0.61, 1.82, 3.03, 4.85 and 9.10, for an initially detuned probe ( Wfw=0.42, A~L= -3.0 and xL= 1). O
~-~!
/
/
I
I ~.
Fig. 7. Switching of the grating reflection produced by counterpropagating pump pulses of peak power Pn= 3.03 and widths W~=O.14, 0.28, 0.42, 0.71 and 1.13, for an initially detuned probe (ApL= -3.0 and x.L= 1).
A numerical simulation of the experimental conditions used by LaRochelle et al. [ 9 ] was performed using the model and the numerical method presented in secs. 2 and 3. Switching of the grating transmission was calculated using a counter-propagating geometry for a grating of strength x L = 0 . 8 8 and pump-pulse width of Wfw= 0.57. The slow response time of the detector, a few nanoseconds, was also taken into account. The results of the calculations are compared with the experimental data in fig. 8. The previously proposed model is also shown in the same figure. This model, based on a cw approximation, considered the pulse to be much longer than the length of the grating. Numerical simulations that take into account the short width of the pulse thus explain the higher switching powers experimentally observed. However, there is still a disagreement at low power between the numerical results and the experimental data. It should be noted that the experimental data were not corrected to include the effect of the overlap integral between the probe and the pump mode fields. This correction factor would dis237
Volume 92, number 4,5,6
OPTICS COMMUNICATIONS
58
v
56
O 54 t"
t~
o""
/
(~"
52
50( r
v 1
2
3
4
pump pulse peak power Po (kW) Fig. 8. Comparison of numerical calculations (solid line) and experimental data (dots). The dashed curve is the previously proposed model. place the experimental points towards lower power bringing the calculated fit within the experimental error bars of 1.3%. The shape of the fit however still differs from the experimental curve. It is possible that, in the experiment, the probe beam was slightly detuned from the Bragg condition as could be induced, for example, by temperature variations. The results o f numerical calculations indicate that an initial negative detuning modifies the shape o f the curve at low power, even resulting in an initial diminution of the transmission. It is not clear that an initial decrease in the transmission could have been resolved in the experimental data presented in ref. [ 9 ]. Also, the influence of the non-uniformity of the grating [ 17 ] on the switching characteristic is not known. Despite the low power discrepancy, calculations that take into account transient effects due to a pulse width of the order o f the grating length provide a better fit to the data than the cw model.
5. Conclusion Coupled-mode equations were used to describe alloptical switching o f a Bragg grating induced by crossphase modulation. These equations were numerically solved in the case where the p u m p pulse had a width o f the order of the grating length. The results show that a co-propagating (counter-propagating) geometry produces better switching of the grating 238
1 September 1992
transmission (reflection). An initial detuning o f the probe was also found to improve the switching o f the grating reflection. High pump-pulse powers were shown to induce multiple peaks in the switching characteristics. In some cases, the switching o f the probe beam transmission could exceed the level o f the injected signal. O p t i m u m switching of the transmission was obtained in a grating of x L = 2 . 5 with p u m p pulses of peak powers P , ~ 3 . 2 5 and pumppulse widths Wfw~ 0.8. A detuned grating could be switched to a reflectivity greater than the m a x i m u m steady value. P u m p pulses o f peak powers P , ~ 3.0 and widths Wfw~<0.4 were found to produce the best switching of the grating reflection when xL = 1. This numerical analysis also provided an improved fit to experimental data previously reported.
Acknowledgements This work was supported by an industrial chair grant from the Natural Sciences and Engineering Research Council o f Canada and Qudbec Tdldphone. S. LaRochelle is also grateful to the Fonds pour la Formation des Chercheurs et l'Aide ~ la Recherche du Qu6bec for its financial support.
References [ 1] B.K. Nayar, N. Finlayson, N.J. Doran, S.T. Davey, D.L. Williams and J.W. Arkwright, Optics Lett. 16 ( 1991 ) 408. [2] S.R. Friberg, A.M. Weiner, Y. Silberberg, B.G. Sfez and P.S. Smith, Optics Lett. 13 (1988) 904. [3] M.N. Islam, E.R. Sunerman, R.H. Stolen, W. Pieibel and J.R. Simpson, Optics Lett. 15 (1989) 811. [4] K.J. Blow, N.J. Doran and B.K. Nayar, Optics Lett. 14 (1989) 754. [5] M.N. Islam, Optics Lett. 14 (1989) 1257. [6] S. Trillo, S. Wabnitz, W.C. Banyai, N. Finlayson, C.T. Seaton, G.I. Stegeman and R.H. Stolen, IEEE J. Quantum Electron. 25 (1989) 104. [7] H.G. Parks, C.C. Pohalski and B.Y. Kim, Optics Lett. 13 (1988) 776. [8]H.G. Park, S.Y. Huang and B.Y. Kim, Optics Lett. 14 (1989) 877. [9] S. LaRochelle, Y. Hibino, V. Mizrahi and G.I. Stegeman, Electron. Lett. 26 (1990) 1459. [10] G.I. Stegeman, E.M. Wright, N. Finlayson, R. Zanoni and C.T. Seaton, J. LightwaveTechnol. 6 (1988) 953.
Volume 92, number 4,5,6
OPTICS COMMUNICATIONS
[ 11 ] S. Trillo, S. Wabnitz and G.I. Stegeman, J. Lightwave Technol. 6 (1988) 971. [ 12] H.G. Winful, Appl. Phys. Lett. 46 (1985) 527. [ 13 ] G.P. Agrawal, Nonlinear fiber optics (Academic, New York, 1989). [14] G. Meltz, W.W. Morey and W.H. Glenn, Optics Lett. 15 (1989) 823.
1 September 1992
[ 15] B.S. Kawasaki, K.O. Hill, D.C. Johnson and Y. Fujii, Optics Lett. 3 (1978) 66. [ 16 ] A. Yariv, Quantum Electronics, 3rd Ed. (Wiley, New York, 1989). [ 17] V. Mizrahi, S. LaRochelle, G. I. Stegeman and J. E. Sipe, Phys. Rev. A 43 ( 1991 ) 433.
239
OPT ICS COMMUNICATIONS
Numerical analysis of all-optical switching of a fiber Bragg grating induced by a short-detuned pump pulse Jocelyn Lauzon, Sophie LaRochelle a n d Francois Ouellette Centre d'Optique, Photonique et Lasers, D~partement de G~nie Electrique, Universit~ Laval, Quebec, G1K 7P4, Canada
Received 22 January 1992; revised manuscript received 15 April 1992
We present the numerical solutions of the coupled-modeequations describing all-opticalswitchingof a fiber grating by a pump pulse detuned from the Braggcondition. The width of the pulse is shorter than the length of the grating. The results reveal that copropagating and counter-propagatingpump-probeconfigurationslead to different switchingcharacteristics. Furthermore, an initial detuning of the probe beam is found to improve the switching. These calculations are compared to experimental results previouslypublished. 1. Introduction In recent years, fast all-optical switching in optical fibers has been a field of active research [ 1-9 ]. This interest is explained by the high nonlinear-optical material figure of merit exhibited by silica glass [ 10 ]. Indeed, because of its low absorption coefficient, the weak nonlinearity of optical glass can be compensated by the long interaction length provided by optical fibers. All-optical switching has already been achieved with several configurations of optical fibers including nonlinear Mach-Zehnder interferometers [ 1 ], nonlinear directional couplers [ 2 ] and nonlinear loop mirrors [3,4]. Other devices have also been demonstrated using soliton switching [3-5 ], polarization rotation [ 6 ], and two-mode interferometry [ 7 ]. In 1988, it was shown that coupling two guided modes by a grating could reduce the switching power required to achieve complete mode conversion [ 11 ]. Park et al. subsequently proposed an intermodal switch that used a pump beam to excite two modes of an optical fiber [8]. Through cross-phase modulation, the beating of these modes produced periodic coupling between the two similar propagating modes of a probe beam. In 1990, LaRochelle et al. showed that cross-phase modulation could be used to switch the transmission of a permanent Bragg grating written in a photosen-
sitive optical fiber [ 9 ]. The switching powers were however found to be five times higher than their predicted value. It then became apparent that the proposed theoretical analysis was incomplete, especially for a pump-pulse width of the order of the grating length. In this paper, we investigate the switching behavior o f a Bragg grating induced by a short pump pulse. The coupled-mode equations describing the evolution of the probe fields are introduced in sec. 2 for co-propagating and counter-propagating pump-probe geometries. The analytical solutions of these equations in the case of a cw pump beam are also presented. The coupled-mode equations were numerically solved to describe the switching resulting from a pump pulse and the results are shown in sec. 3. The dependence of the switching characteristics on the pump-pulse width and peak power is examined. These numerical results predict that an initially detuned probe beam would improve the switching. In sec. 4, the numerical solutions are compared to experimental observations reported earlier [ 9 ].
2. Description of the model This section introduces the set of coupled-mode equations describing the evolution of the probe beam fields in a Bragg grating perturbed by a strong pump
0030-4018/92/$05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.
233
Volume 92, number 4,5,6
OPTICS COMMUNICATIONS
pulse. The well-known steady-state solutions of these equations are also presented. The pump power required to switch the reflection of the grating is then calculated for the case of a cw pump beam. Finally, a normalized pump-pulse peak power is introduced. A Bragg grating is assumed to be written from z = 0 to z=L in an infinite single-mode optical fiber (fig. 1 ). A weak cw probe beam is incident on the grating at z = 0. The electric field of the probe can be described in terms of a forward and a backward travelling wave, Eprob e : 8f
exp[i(flz-mt) ]
+ ~ exp [ i ( - f l z - m t ) ] + c . c . ,
( 1)
where fl=2nno/2o, no is the effective index, and 4o is the probe wavelength in vacuum. In the presence of a strong pump pulse, the coupled-mode equations that describe the slowly varying components of the probe fields are [ 12,13 ], O~f
no O~f - ---- ixgb exp ( -- i2Aflz) + 2iTPp gf c 0t
O--z- d -
--
0~b
no 0~b
Oz
c Ot----ix8fexp(i2Aflz)--2iyPpgb,
(2)
where x is the coupling coefficient, Afl=fl-K/2 is the probe detuning from the Bragg condition, K is the grating wavevector. The last terms on the righthand side of these equations represent the cross-phase modulation induced by the pump pulse on the probe fields. The coefficient 7 is equal to 2~n2/(Aeff2o), where n2 is the nonlinear Kerr index coefficient, and Aeff is the effective area obtained from the overlap
gaussian p u m p pulse
CW p r o b e b e a m
Z=O
'
j~
'
'
)~/2n0 Z=l
periodic modulation of the refractive index Fig. t. Brag grating of length L written in an optical fiber. 234
integral between the probe and the pump modes [ 13 ]. Finally, Pp is the pump power written as ep =Po exp - 2
(Zo_ct/no)_ 1_ , Wo /A
(3)
where Zo is the initial position of the pump pulse far from the grating, Po its peak power, and Woits width. The plus or minus signs refer respectively to a copropagating or a counter-propagating configuration. Since efficient Bragg gratings can be written in short lengths of optical fibers, distortion of the pump pulse due to self-phase modulation and dispersion is neglected. For example, 4.4 m m long Bragg gratings achieving 55o/o reflection have been written in photosensitive optical fibers using an external holographic technique [14]. In their experiment, LaRochelle et al. used a 3.5 cm long grating providing 50% reflection [9 ]. Furthermore, these gratings have typically very narrow bandwidth, from 200 MHz to 42 GHz [ 14,15 ]. It is thus assumed that the pump pulse is sufficiently detuned from the Bragg condition not to be affected by the presence of the grating. Also, the probe and the pump wavelengths can be closely spaced and the dispersion of the effective index of the fiber is safely neglected. Equations (2) and (3) can easily be rewritten in terms of normalized variables, O~vf
O~f
Ogb
O~
O--Z+ ~
=i
xL
eb exp[ --i2
AflL Z] + 2 i 7L Ppgf,
0Z
- - - - i xL gf exp [i2 Aft/. Z] - 2i 7L Pp & , OT
Pp=Po
exp[- 2(Z- ( Z° +-T) )
,
(4)
where Z=z/L, Wo=wo/L, T=t/z, and z=noL/c is the transit time of the pump pulse in the grating. The normalized full-width at the 1/e points of the pump pulse is Wfw=N//-2Wo. With the boundary conditions e l ( Z = 0 ) = d'm and 8b(Z= 1 ) = 0 , the steady state solution of eq. (4) when P p = 0 is [16]
optical fiber
',
1 September 1992
St(Z) = X
8fo e x p ( - i A f l L Z )
i~2L cos [I2L ( 1 - Z ) ] + Aft/, sin [£2L( 1 - Z ) ] iI2L cos(I2L) + AflL sin (s'2L)
Volume 92, number 4,5,6
°vb(Z) =
OPTICS COMMUNICATIONS
- 8to exp(iAflLZ) xL sin [~2L( 1 - Z ) ] i£-2Lcos(QL)+AflLsin(12L) '
(5) where 122=Af12-x 2. F r o m eq. (5), the intensity reflection coefficient o f an unperturbed Bragg grating is given by R = I # b ( Z = 0 ) 12 I~fol 2
( xL ) 2 sin2 (t2L) = (t2L)2 cos2(12L ) + (AflL)2 sin2(12L ) .
(6)
In steady state, the reflection o f a probe beam will thus switch from the m a x i m u m to first zero minim u m when t2L = n. Furthermore, it can be seen from eqs. (4) that the presence o f a cw p u m p beam, Wfw>> 1, introduces an additional detuning (AflL)'=27LPp. It is thus possible to define a normalized p u m p power P,=Pp/Ps, where
Ps
=
1 September 1992
sion o f gratings switched by short pulses are presented in fig. 2. The reflection is calculated at Z = 0 and the transmission at Z = 1. The peak o f the p u m p pulse enters the grating at T = 0 and exits at T = 1. The results are shown in fig. 2a for a co-propagating configuration ( P , = 2 . 5 , Wfw=1.13, KL=2.5 and AflL= 0.0), and in fig. 2b for a counter-propagating one ( P , = 1.5, Wfw=0.42, KL= 1 and A f l L = 0 . 0 ) . In both cases, the switching is sharper for the fields that propagate in the same direction as the p u m p pulse (transmission for the co-propagating geometry and reflection for the counter-propagating one). For the 150 125 100
///ELECT'ON'
75
( x L / 2 7 L )x/ (x/IcL ) 2 + 1
is the p u m p power that will produce complete switching of an initially tuned probe when Wrw >> 1 [ 9 ]. In the case o f a short p u m p pulse, the coupledmode equations (4) do not have an analytical solutions
50 ~/TSMISSION(9~,,,,,
~
j
25 0 0
1
2
3
time T
3. Numerical results In order to describe all-optical switching o f a Bragg grating by a short p u m p pulse ( Wrw < 1 ), the set o f coupled-partial-differential equations (4) were numerically solved. These calculations were done using the method o f lines with Hermit polynomials ~. The fields distributions o f an unperturbed Bragg grating, eq. (5), were used as initial conditions. In this section, the numerical results are presented for co-propagating and counter-propagating pump-pulse geometries. The switching characteristics are also examined as a function of the initial probe detuning, and as a function o f the pump-pulse width and peak power. The evolution o f the reflection and the transmis#1 We used a fortran subroutine from the IMSL Math/Library TM, explanation on the numerical method can be found in the user's manual, April 1987, pp. 680-687.
1O0
(b) 80
60
~EFLECTION (~)
TRANSMISSION 40
20
0
I
I
I
I
0
1
2
3
time T
Fig. 2. Evolution of the reflection and the transmission of a cw probe beam incident on a Bragg grating switched by: (a) a copropagating pump pulse (/',=2.5, Wr~=l.13, xL=2.5 and ApL=0.0), or (b) a counter-propagating pump pulse (P.= 1.5, Wfw=0.42, xL= 1 and AflL=0.0). 235
Volume 92, number 4,5,6
OPTICSCOMMUNICATIONS
fields propagating in the opposite direction, the wider characteristics can be partly understood by considering that the switching results from the convolution of the pump pulse with the probe electric fields inside the grating. In all numerical calculations, it was verified that the energy was conserved and that the grating was recovering its original state after the transient oscillations. Switching of the grating transmission was studied in more details for the co-propagating geometry. Fig. 3 shows the influence of the pump-pulse peak power on the calculated switching characteristics of a grating that initially transmits 2.7% of the incident probe beam (Wfw=0.71, x L = 2 . 5 and AflL=0.0). As the pump-pulse power is increased, different parts of the pulse switch the grating transmission and reflection through successive maxima and minima. These complex interference effects result in a multiple peak structure with a maximum transmission that can exceed 100% of the injected power. The results of the calculations for different pump-pulse widths, Wfw= 0.1, 0.3, 0.5, 0.8, and 1.2, are displayed in fig. 4 for a peak power P, = 2.5 and a tuned grating of x L = 2 . 5 . For a pump-pulse width Wfw~<0.05 the switching resuited in a single peak centered at T = 1. As the pump pulse becomes wider a second peak in the transmission grows and moves to later times. The position of the peak is initially linearly proportional to the width
1 September 1992
Fig. 4. Switching of the grating transmission produced by copropagating pump pulses of peak power P.=2.5 and of widths Wfw=0.1,0.3, 0.5, 0.8 and 1.2 (xL=2.5 and ApL=0.0).
1 O0
80
SSi0N (%)
6O
4O
2O
.~ECTION 0
i J
I
0
2
,
(%) i 4
, 6
time T
Fig. 5. Switchinginduced by a counter-propagatingpump pulse (P,= 1.5, Wf~=0.42) on the reflection and transmission of an initially detuned probe beam (~p/.= -3.0) incident on a Bragg filter (xL = 1 ).
Fig. 3. Switchingof the grating transmission producedby different pump-pulsepeak powersP,=0.50, 1.49, 2.49, 3.99 and 7.47, in a co-propagating geometry (Wfw=0.71, self2.5 and a#L=0.0). 236
of the pump pulse but eventually stabilizes at T = 1.7 for pump-pulse width Wfw~>1.2. The counter-propagating geometry was used to investigate the switching behavior of the grating reflection. As was previously observed for other devices [ 11 ], an initial detuning of the probe beam was found to improve the switching. In fig. 5, it can be seen that with a detuning of A p L = - 3 . 0 the reflectivity of the grating is now switched from 1.2% to 93.9% with a full-width at half-maximum (fwhm) of 0.32 for the principal peak, comparatively to the 55% modulation with a fwhm of 0.35 observed without
Volume 92, number 4,5,6
OPTICS COMMUNICATIONS
an initial detuning (fig. 2b). The reflected probepulse results from the interaction of the pump pulse with the probe fields all along the length of the fiber grating. Therefore, the switched reflectivity can be greater than the maximum steady state value of 58% obtained for a tuned grating o f x L = 1. In fig. 6, it can be seen that increasing the pump power results once again in a multiple peak structure in the switching characteristic due to transient oscillations. The secondary peak that appears first is centered at T = 2.0. A further increase in the pump-pulse peak power causes the peak located at T = 1.0 to split in two as the peak of the pump pulse over switch the grating reflection to the next minimum. The influence of the pump-pulse width on the calculated reflection switching is shown in fig. 7. The peak centered at O
1 September 1992
T = 1.0 gradually widens and eventually merges with the growing secondary peak at T=2.0. All-optical switching of a cw probe beam by a gaussian pump pulse should ideally produce a single undistorted pulse. The numerical simulations presented in this section indicate that the switching of a Bragg grating induced by a short detuned pump pulse sometimes results in a probe beam transmission, or reflection, with a multiple peak structure or with a highly distorted pulse shape. However, if the pump-pulse peak power and width are adequately chosen these undesirable effects can be avoided. Using a co-propagating geometry, more than 50% switching in the transmission of a x L = 2 . 5 grating was obtained without multiple peaks for pump-pulse peak powers, Pn, between 2.5 and 4.0, and widths, Wfw, greater than 0.5 (figs. 3 and 4). For a detuned x L = 1 grating and a counter-propagating configuration, the reflection could be switched by more than 75% with one important peak if the pump-pulse peak power, Pn, is around 3.0 and the pulse width, Wfw, is smaller than 0.4 (figs. 6 and 7).
4. Comparison with experimental results ~3
"~
'2.
"~
Fig. 6. Switchingof the grating reflectionproduced by a counterpropagatingpump pulse for various peak powersPn=0.61, 1.82, 3.03, 4.85 and 9.10, for an initially detuned probe ( Wfw=0.42, A~L= -3.0 and xL= 1). O
~-~!
/
/
I
I ~.
Fig. 7. Switching of the grating reflection produced by counterpropagating pump pulses of peak power Pn= 3.03 and widths W~=O.14, 0.28, 0.42, 0.71 and 1.13, for an initially detuned probe (ApL= -3.0 and x.L= 1).
A numerical simulation of the experimental conditions used by LaRochelle et al. [ 9 ] was performed using the model and the numerical method presented in secs. 2 and 3. Switching of the grating transmission was calculated using a counter-propagating geometry for a grating of strength x L = 0 . 8 8 and pump-pulse width of Wfw= 0.57. The slow response time of the detector, a few nanoseconds, was also taken into account. The results of the calculations are compared with the experimental data in fig. 8. The previously proposed model is also shown in the same figure. This model, based on a cw approximation, considered the pulse to be much longer than the length of the grating. Numerical simulations that take into account the short width of the pulse thus explain the higher switching powers experimentally observed. However, there is still a disagreement at low power between the numerical results and the experimental data. It should be noted that the experimental data were not corrected to include the effect of the overlap integral between the probe and the pump mode fields. This correction factor would dis237
Volume 92, number 4,5,6
OPTICS COMMUNICATIONS
58
v
56
O 54 t"
t~
o""
/
(~"
52
50( r
v 1
2
3
4
pump pulse peak power Po (kW) Fig. 8. Comparison of numerical calculations (solid line) and experimental data (dots). The dashed curve is the previously proposed model. place the experimental points towards lower power bringing the calculated fit within the experimental error bars of 1.3%. The shape of the fit however still differs from the experimental curve. It is possible that, in the experiment, the probe beam was slightly detuned from the Bragg condition as could be induced, for example, by temperature variations. The results o f numerical calculations indicate that an initial negative detuning modifies the shape o f the curve at low power, even resulting in an initial diminution of the transmission. It is not clear that an initial decrease in the transmission could have been resolved in the experimental data presented in ref. [ 9 ]. Also, the influence of the non-uniformity of the grating [ 17 ] on the switching characteristic is not known. Despite the low power discrepancy, calculations that take into account transient effects due to a pulse width of the order o f the grating length provide a better fit to the data than the cw model.
5. Conclusion Coupled-mode equations were used to describe alloptical switching o f a Bragg grating induced by crossphase modulation. These equations were numerically solved in the case where the p u m p pulse had a width o f the order of the grating length. The results show that a co-propagating (counter-propagating) geometry produces better switching of the grating 238
1 September 1992
transmission (reflection). An initial detuning o f the probe was also found to improve the switching o f the grating reflection. High pump-pulse powers were shown to induce multiple peaks in the switching characteristics. In some cases, the switching o f the probe beam transmission could exceed the level o f the injected signal. O p t i m u m switching of the transmission was obtained in a grating of x L = 2 . 5 with p u m p pulses of peak powers P , ~ 3 . 2 5 and pumppulse widths Wfw~ 0.8. A detuned grating could be switched to a reflectivity greater than the m a x i m u m steady value. P u m p pulses o f peak powers P , ~ 3.0 and widths Wfw~<0.4 were found to produce the best switching of the grating reflection when xL = 1. This numerical analysis also provided an improved fit to experimental data previously reported.
Acknowledgements This work was supported by an industrial chair grant from the Natural Sciences and Engineering Research Council o f Canada and Qudbec Tdldphone. S. LaRochelle is also grateful to the Fonds pour la Formation des Chercheurs et l'Aide ~ la Recherche du Qu6bec for its financial support.
References [ 1] B.K. Nayar, N. Finlayson, N.J. Doran, S.T. Davey, D.L. Williams and J.W. Arkwright, Optics Lett. 16 ( 1991 ) 408. [2] S.R. Friberg, A.M. Weiner, Y. Silberberg, B.G. Sfez and P.S. Smith, Optics Lett. 13 (1988) 904. [3] M.N. Islam, E.R. Sunerman, R.H. Stolen, W. Pieibel and J.R. Simpson, Optics Lett. 15 (1989) 811. [4] K.J. Blow, N.J. Doran and B.K. Nayar, Optics Lett. 14 (1989) 754. [5] M.N. Islam, Optics Lett. 14 (1989) 1257. [6] S. Trillo, S. Wabnitz, W.C. Banyai, N. Finlayson, C.T. Seaton, G.I. Stegeman and R.H. Stolen, IEEE J. Quantum Electron. 25 (1989) 104. [7] H.G. Parks, C.C. Pohalski and B.Y. Kim, Optics Lett. 13 (1988) 776. [8]H.G. Park, S.Y. Huang and B.Y. Kim, Optics Lett. 14 (1989) 877. [9] S. LaRochelle, Y. Hibino, V. Mizrahi and G.I. Stegeman, Electron. Lett. 26 (1990) 1459. [10] G.I. Stegeman, E.M. Wright, N. Finlayson, R. Zanoni and C.T. Seaton, J. LightwaveTechnol. 6 (1988) 953.
Volume 92, number 4,5,6
OPTICS COMMUNICATIONS
[ 11 ] S. Trillo, S. Wabnitz and G.I. Stegeman, J. Lightwave Technol. 6 (1988) 971. [ 12] H.G. Winful, Appl. Phys. Lett. 46 (1985) 527. [ 13 ] G.P. Agrawal, Nonlinear fiber optics (Academic, New York, 1989). [14] G. Meltz, W.W. Morey and W.H. Glenn, Optics Lett. 15 (1989) 823.
1 September 1992
[ 15] B.S. Kawasaki, K.O. Hill, D.C. Johnson and Y. Fujii, Optics Lett. 3 (1978) 66. [ 16 ] A. Yariv, Quantum Electronics, 3rd Ed. (Wiley, New York, 1989). [ 17] V. Mizrahi, S. LaRochelle, G. I. Stegeman and J. E. Sipe, Phys. Rev. A 43 ( 1991 ) 433.
239