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Analysis of soliton switching in dispersion-decreasing fiber couplers
1 December 1999
Optics Communications 171 Ž1999. 351–364 www.elsevier.comrlocateroptcom
Full length article
Analysis of soliton switching in dispersion-decreasing fiber couplers b M.G. da Silva a , K.Z. Nobrega , A.S.B. Sombra ´
a,)
a
b
´ Laboratorio nao dos Materiais (LONLCM), Departamento de Fısica, UniÕersidade Federal do Ceara, ´ de Optica ˜ Linear e Ciencia ˆ ´ ´ Caixa Postal 6030, 60455-760 Fortaleza, Ceara, ´ Brazil Departamento de Engenharia Eletrica, Centro de Tecnologia, UniÕersidade Federal do Ceara, ´ ´ 60455-760, Fortaleza, Ceara, ´ Brazil Received 28 June 1999; received in revised form 8 September 1999; accepted 14 September 1999
Abstract We present an analytical and numerical investigation of the propagation and the switching of fundamental solitons in a two-core nonlinear fiber coupler constructed with dispersion decreasing fiber ŽDDF. with a variational method using the Lagrangian density formulation. The analytical solutions were directly obtained from the coupled nonlinear Schrodinger ¨ equations. Our simulations considering six different profiles, linear, hyperbolic, exponential, logarithm, constant and Gaussian. The transmission characteristics, the critical energy and the compression factor of first order solitons obtained by the analytical procedure agree well with the results from numerical analysis. Twin core nonlinear directional couplers that include loss are also examined. The inclusion of loss results in the increase of the critical power of the coupler and broadening of the switched pulses, while the switching characteristics deteriorate. To compensate for this worse behavior, associated to the loss, we have investigated the effect of the DDF profile on the performance of the coupler. We show that appropriate shaping of the DDF profile is quite effective to recover, almost completely, the original switching behavior associated to the lossless situation. It is concluded that the hyperbolic dispersion profile is nearly optimum to the recovery of the lossless switching behavior. q 1999 Elsevier Science B.V. All rights reserved. PACS: 42.65.Pc; 42.50.Rh; 42.79.Gn
1. Introduction All optical switching devices are attracting considerable interest as fast switching components for future high-bit-rate systems. Optical fiber couplers have been studied for their potential applications to ultrafast all optical switching processing, like optical switch w1–3x. )
Corresponding author. Fax: q55-85-287-4138; e-mail: [email protected]
In a nonlinear coupler constructed from a Kerr type medium the dependence of the nonlinear refractive index n on the laser intensity is given by the expression n s n 0 q n 2 I, where n 0 is the refractive index at low intensity and n 2 is the Kerr nonlinear coefficient w4,5x. Jensen w1x showed that varying the input light in the nonlinear coupler could lead to pulse switching between the two cores. He therefore foresaw the possible use of a nonlinear directional coupler as an optical switch. A nonlinear coupler consists of two closely spaced, parallel, single mode
0030-4018r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 Ž 9 9 . 0 0 5 5 2 - 0
M.G. da SilÕa et al.r Optics Communications 171 (1999) 351–364
352
waveguides in a material with an intensity dependent index of refraction. At low light levels, the device behaves as a linear directional coupler. Because of evanescent coupling, signals introduced into guide w1–3x transfer completely to guide w4,5x in one coupler length Lc Žsee Fig. 1.. Higher intensities induce changes in the refractive index and detune the coupler. Coupler is inhibited for input intensities above the critical intensity:
l IC s
n 2 LC
Ž 1.
where IC s PrA, P is the critical power, A is the effective mode area, l is the pump wavelength, n 2 is the nonlinear refractive index w1–7x, LC is the half-beat-length coupler of length LC s pr2 K and K is the linear coupling coefficient between adjacent guides. At the critical intensity IC , 50% of the light emerges from each waveguide. Above IC most of the light emerges from channel 1 Žsee Fig. 1.. Previous studies of soliton switching in dual core optical fibers, without loss, have shown excellent switching characteristics, with efficiencies around 96% for a wide range of input energies w4,5x. By comparing the switching behavior of fundamental, second order, and quasi solitons it was observed that the fundamental soliton has the most suitable features for optical switching w4,5x. Indeed, it has been shown that pulse breakup may be avoided when the input signal is a soliton w6–8x. Since then, soliton switching in nonlinear fiber couplers have been receiving considerable attention.
However lossless device is an idealized situation. In practice, large or small, material loss is unavoidable, especially when nonlinearities are based on absorptive process in semiconductors. The absorption, uniformly distributed over the device, will set a limit to the operation of the device w9,10x. In the following, we will examine nonlinear directional double core coupler with loss. The presence of loss is responsible for the increase of the critical power of the coupler and a strong deformation of the nonlinear transmission of the device. In this paper, we have investigated the effect of using dispersion decreasing fibers on the performance of NLDC with and without loss. Six closely shaped profiles named linear, Gaussian, exponential, hyperbolic, constant and logarithm have been considered. From our study of the dispersion profiles, we suggest the optimum profile, to recover the original performance of the nonlinear transmission, to overcome the effect of the intrinsic loss in the device. The paper is divided into several sections. In Section 2 the nonlinear directional coupler constructed from dispersion decreasing fibers ŽDDF., operating in the soliton regime is presented. Using the technique developed by Anderson and co-workers w11–14x, we convert the coupled nonlinear Schrodin¨ ger equations ŽNLSEs. into a variational problem with corresponding Lagrangian equations of motion for a finite number of degrees of freedom and obtain the switching characteristics of the device in Section 4. For both Sessions we are considering the lossless coupler. The analytical and numerical procedure used is compared is Section 5.1. In Section 5.2 the effect of loss in the coupler is studied. We discuss the effect of each profile in the presence of loss and conclude that there is an optimum profile to recover the lossless switching behavior.
2. Theoretical model
Fig. 1. Schematic of nonlinear directional coupler of length LC .
We will consider picosecond pulses propagating in the anomalous dispersion regime in a nonlinear directional coupler with twin fibers with variable normalized second-order dispersion coefficient b 2 Ž j .. The propagating of ultrashort solitons through a DDF coupler is described by the nonlinear
M.G. da SilÕa et al.r Optics Communications 171 (1999) 351–364
Schrodinger equation with a axially varying disper¨ sion profile pŽ j . w15x: i
E u1 Ej
1 q 2
pŽ j .
E 2 u1 Et 2
q < u1 < 2 u1 q Ku 2 q i a u1 s 0
Ž 2. i
E u2 Ej
1 q 2
pŽ j .
E 2 u2 Et 2
q < u 2 < 2 u 2 q Ku1 q i a u 2 s 0
Ž 3. where u1 and u 2 are the modal field amplitudes in soliton units, pŽ j . s < b 2 Ž j .rb 2 Ž0.< is the normalized GVD, a is the fiber loss. j and t are the normalized length and time in soliton units with j s zrL D and t s trT0 . Here L D s T02r< b 2 <, with pulse width T0 ŽT0 s 5.685 ps, < b 2 < s 20 ps 2rkm.. The varying GVD parameter pŽ j . is normalized by the dispersion in the input of the fiber, b 2 Ž0. s < b 2 < s 20 ps 2rkm. In a DDF of certain length, the dispersion is monotonically and smoothly decreased from an initial value to a smaller value at the end of the length according to some specified value. Provided the dispersion variation in the DDF is sufficiently gradual, soliton compression can be an adiabatic process where an input fundamental soliton pulse can be ideally compressed as it propagates, while retaining its soliton character and conserving the energy w16– 19x. Interest in DDF in nonlinear fiber optics has focused on its use to compensate for the detrimental effect of signal attenuation in long distance soliton propagation. In this paper, we compare six simple coefficient of dispersion decreasing profiles, namely linear, Gaussian, exponential, logarithm, hyperbolic and constant. These profiles expressed in terms of the parameters 1rb Žminimum value of p . and L Žlength of the coupler. are: pŽ j . s
Ž1yb . bL
p Ž j . s exp p Ž j . s exp
ž
ž
j q 1 Linear
yj L
yj 2 L2
/
ln b Exponential
/
ln b Gaussian
ž
j
p Ž j . s ln e q pŽ j . s pŽ j . s
L
Ž e1r b y e . Logarithm
L
Ž b y 1. j q L 1
b
353
/
Ž 7.
Hyperbolic
Ž 8.
Constant.
Ž 9.
Note that in these normalized profiles the dispersion coefficient p monotonically decreases from 1 to a final value of 1rb , after a length L of the coupler. The dispersion profiles are illustrated in Fig. 2. For the first order soliton Ž N s 1. the solution of Eqs. Ž2. and Ž3. Ž p s 1, K s 0, a s 0. lossless case is: u i Ž j ,t . s A i sec h Ž A it . exp Ž i A2i jr2 q i w i .
Ž 10 .
where A i and w i are arbitrary constants. A fundamental soliton pulse has been already considered as an effective input pulse type for improving switching efficiency by reducing pulse break-up in a nonlinear directional coupler. In this case the laser’s output width and power should have to be adjusted to maintain the fundamental soliton profile as the input energy changes. From the experimental point of view it is easy to adjust the laser output power, but very difficult to change the time duration. However for a certain time duration, the laser output power could be adjusted to produce an exact first order soliton pulse. These pulses could be derived from different sources, where each source could be adjusted to
Ž 4. Ž 5. Ž 6. Fig. 2. Schematics of the dispersion profiles considered.
M.G. da SilÕa et al.r Optics Communications 171 (1999) 351–364
354
generate a specific time duration and power, necessary for a fundamental soliton. Therefore, solitons with different time duration and pump powers, from different sources, could be efficiently switched through the optical device w4,5x. We also define the final compression factor C, achieved after propagation of the optical input pulse in the DDF coupler. It is defined as the ratio of the optical pulse FWHM at the input of the coupler t 0 ŽFWHM of the < uŽ1.Ž0,t .< 2 , input pulse., to that at the output Žtransmitted. of the coupler, t i ŽFWHM of the < u i Ž LC ,t .< 2 , transmitted pulse.. t0 Ci s Ž 11 . ti where i s 1,2 for switching of pulses in channels 1 and 2.
co-workers w11–14x, which is based on the Lagrangian equations of motion for a finite number of degrees of freedom. This technique was used in several problems of optical solitons w11–14,20,21x. We introduce the Lagrangian density formulation following Refs. w20,21x. The Lagrangian of the DDF lossless coupler ŽEqs. Ž2. and Ž3. with a s 0. is: q`
Ls
LX s L1 q L 2 q K Ž u 2) u1 q u 2 u1) .
q` i
Ti s
q`
Hy`
L ,t
Li s
i
ž
2
Ž 14 .
< u1 Ž 0,t . < 2 dt
with i s 1, 2 and a fiber coupler with a length of j L . 4. Analytical analysis To study the soliton switching in the coupler, we will use the approach developed by Anderson and
E ui
u )i
1
. < 2 dt
Ž 16 .
where L i denotes:
y
Taking Eqs. Ž2. and Ž3. with no coupling Ž K s 0. or profile Ž p s 1. and no loss Ž a s 0. one has the well-known soliton solutions ŽEq. Ž10... We have analyzed numerically the soliton transmission of first order solitons through the two core DDF nonlinear directional fiber coupler ŽEqs. Ž2. and Ž3.. with dispersion profiles given by Eqs. Ž4. – Ž9.. The initial pulse at the input core is given by: u1 Ž 0,t . s A sec h Ž At . Ž 12 . u 2 Ž 0,t . s 0. Ž 13 . This system of linearly coupled NLSEs ŽEqs. Ž2. and Ž3. and Eqs. Ž4. – Ž9.. was solved numerically using the split-step method with 1024 temporal grid points taking in account the initial conditions given by Eqs. Ž12. and Ž13.. We can define the transmission Ti as a function of the pulse energies:
Ž 15 .
where the Lagrangian density is:
3. Numerical procedure
Hy` < u Ž j
X
Hy` L dt
2
y ui
Ej
E u i) 2
E ui
pŽ j .
/
Ej
1 q
Et
2
< ui < 4
Ž 17 .
where the asterisk denotes the complex conjugate and i s 1, 2. Eqs. Ž2. and Ž3. Žwith a s 0. result from the variational equations corresponding to the variational principle:
d
HHL
ž
E u i E u i) E u i E u i) , , , EjEt s 0 Ž 18 . Ej Ej Et Et
/
u i ,u )i ,
i.e., the equations:
E
dL s
d u )i
EL
Ej E
E u )i
E Ej
q
EL
Et E E u )i Et
EL y
E u )i
Ž 19 .
s0
for i s 1, 2 we obtain Eqs. Ž2. and Ž3.. For the DDF coupler we assume the pulse profiles u1Ž j ,t . and u 2 Ž j ,t . take the forms A
u1 Ž j ,t . s
(p Ž j .
sec h
ž
At pŽ j .
/
=cos u exp Ž i w q i c . u 2 Ž j ,t . s
A
(p Ž j .
sec h
ž
At pŽ j .
=sin u exp Ž i w y i c .
Ž 20 .
/ Ž 21 .
M.G. da SilÕa et al.r Optics Communications 171 (1999) 351–364
u Ž j . is the coupling angle that gives the power coupling between the two cores, c Ž j . is the relative phase. For these trial functions A is a constant of motion. However, the parameter w has no influence on the other parameters. We expect to have a nontrivial dynamics for u Ž j . and c Ž j .. However, our main choice, for the initial solutions given by Eqs. Ž20. and Ž21., is associated to the fact that the value of the total intensity of the solitons Ž< u1 < 2 q < u 2 < 2 . has no dependence on u Ž j .. It is a constant of motion during the coupler propagation. The obtained Lagrangian for the DDF coupler is given by: A3
Ec L s y2 A cos 2 u
y
3p Ej q 2 AK sin 2 u cos 2 c . Ž 22 . If we take the variation of the Lagrangian Eq. Ž22. with respect to c Ž j . according to: EL E EL y s0 Ž 23 . Ec Ej E Ž Ec Ej .
ž
)
Eu
s yK 1 y
Ej
ž
2
A2
/
6 Kp Ž j .
after a little manipulation: du
u
H0
(1 y Ž A H0L
C
2
6 Kp Ž j .
LD
.
2
sin2 w 2 u x
Kdj .
Ž 29 .
Ž 24 . Ž 25 .
P DD F s A2
H0L
C
LD
cos 2 Ž a 1 j . pŽ j .
We can obtain the Hamiltonian H for the system according to the equation: Ec EL Eu EL Hs q yL Ž 26 . Ej E Ž Ec Ej . Ej E Ž Eu Ej .
where for the critical energy:
with the help of Eq. Ž22. the obtained Hamiltonian for the coupler is:
the obtained value for a 1 is:
Hsy
A3 3p
2
sin 2 u q 2 AK sin 2 u cos 2 c .
Ž 27 .
In our case we consider that the soliton is launched in channel 1. The initial condition is then u s 0 in Eqs. Ž20. and Ž21.. The Hamiltonian will stay in the entire trajectory with this value, H Ž j . s H Ž0.. According to Eq. Ž27. we have two solutions with H s 0. The first one has sin 2 u s 0 all over the trajectory. And the second one is: cos 2 c s
A2 6 Kp
sin 2 u .
sin2 Ž 2 u .
For a coupler without dispersion profile Ž p s 1. one obtains the critical power of the reference coupler Ž PC s A2rK s 6. w20,21x. However Eq. Ž29. is not readily solved because the profile pŽ j . is present in the integral. In this situation we will calculate the energy in the DDF coupler when compared with the reference coupler Žno dispersion profile.. The peak power in the fiber 1, in the DDF coupler could be calculated using Eq. Ž20. and is given by:
/
which implies that either: sin 2 u s 0 or Eu s yK sin 2 c . Ej
The second solution is more interesting to us because the power transfer between cores 1 and 2 is described by the trajectory associated to evolution of u . Taking in account Eqs. Ž28. and Ž25. we obtain:
sy
sin2 2 u
355
Ž 28 .
cos 2 Ž a 1 LC L D .
LD LC
Ž 30 .
1 s 2
p Ž LC L D .
a1 s
dj
cosy1
ž(
p Ž LC L D . 2
/
Ž 31 .
for the reference coupler the transmission of channel 1 given by: P s A20
H0L
C
LD
cos 2 Ž a 0 j . d j
Ž 32 .
where A 0 is the amplitude of the light in the reference coupler. In this case a 0 is given by:
a0 s
LD LC
cosy1
1
ž( / 2
Ž 33 .
M.G. da SilÕa et al.r Optics Communications 171 (1999) 351–364
356
considering that at the critical energy, without loss, both couplers present the same power PDD F s P, using Eqs. Ž30. and Ž32. one has: cos 2 Ž a 1 j .
LC L D
A20 2
s
H0
A
pŽ j .
dj sF
LC L D
Ž 34 .
cos 2 Ž a 0 j . d j
H0
A
1
s E0
s A0
'F
.
Ž 35 .
Solving Eq. Ž35. with Eq. Ž34. for all the profiles one has the critical energy of the DDF couplers compared with the normal coupler. To obtain the transmission characteristic for each coupler one can compare Eq. Ž29. for the DDF coupler with the reference coupler:
T2 s sin2 u .
Ž 39 .
Ci s
2
2
(1 y Ž A rŽ 6 K . . sin 2u 2
2 0
LCrL D
H0
sy
2
sec h2
du
H0L
sy
(1 y Ž A F Ž 6 K . . sin 2u 2
At 0
ž /
1 s
p Ž 0.
2
then one has:
t0s
p Ž 0 . sec hy1 Ž 1r'2 . A
at the output of channel 1:
t1s
p Ž LC L D . sec hy1 Ž 1r'2 . A
the compression factor is given by: p Ž 0. p Ž LC L D .
1 s
1rb
sb
Ž 41 .
Kdj
using Eq. Ž35., one concludes that the first term in the left could be given by: u
Ž 40 .
t i Ž LC L D .
where we assume that all the profiles present a decreasing dispersion, where the final value of the dispersion in the coupler is 1rb ŽEqs. Ž4. – Ž9. and Fig. 2..
du
u
H0
2
t 0 Ž 0.
taking Eq. Ž20. for the input pulse at half maximum intensity:
Ci s
(1 y Ž A rŽ 6 Kp Ž j . . . sin 2u s
H0
Ž 38 .
du
u
H0
T1 s cos 2 u
For the analytical procedure we could also calculate the compression factor Ci defined in Eq. Ž11.:
for each profile we have different values for pŽ j . ŽEqs. Ž4. – Ž9.., a 1 ŽEq. Ž31.. and consequently for the F function. We can calculate now the critical energy of the DDF coupler and the reference coupler. Taking Eq. Ž20. for the DDF coupler, one has for the critical energy: E s 2 A and E0 s 2 A 0 Ž p s 1. for the reference coupler. Taking into account Eq. Ž34. one has: E
we can calculate the transmission characteristics of the coupler using Eqs. Ž37. and Ž34. and taking in account that for the coupler transmission is:
2
2
C
LD
Kdj .
Ž 37 . Eq. Ž37. is readily solved Žassuming a fixed length for the nonlinear coupler K j s constant. using a fourth order Runge–Kutta method. In this situation
5. Results and discussion 5.1. DDF coupler: numerical and analytical Fig. 3 shows the soliton transmission characteristics of channel 1 for six different profiles without loss, obtained numerically. The NLSE equations ŽEqs. Ž2. and Ž3.. were solved numerically with the input soliton given by Eqs. Ž12. and Ž13.. The soliton transmission was obtained ŽEq. Ž14.. for six different couplers constructed with different profiles ŽEqs.
M.G. da SilÕa et al.r Optics Communications 171 (1999) 351–364
Fig. 3. Switching characteristics of channel 1 obtained from the numerical solution of Eqs. Ž2. – Ž9. with K s1, LC s p r2, b s 2, a s 0. The case b s1 identifies the reference nonlinear coupler.
Ž4. – Ž9... In this figure the reference coupler Žreference, a s 0, b s 1, p s 1. was also included. In this simulation one has b s 2 for all the six profiles. One concludes that for all the profiles there is a decrease of the critical energy. In Fig. 4 one has the critical energy of all couplers as a function of the b parameter obtained from the numerical calculation. The energy is compared with the critical energy of the reference coupler. From the figure one can conclude that the increase in b Ždecrease in the profile pŽ j .. lead to the decrease of the critical energy. The decrease is stronger for the constant profile and is
Fig. 4. Critical energy obtained numerically ŽEqs. Ž2. – Ž9. and Ž14.. for the DDF coupler.
357
Fig. 5. Switching characteristics of channel 1 obtained analytically from Eqs. Ž34., Ž35., Ž37. and Ž38. with K s1, LC s p r2, b s 2, a s 0. The case b s1 identifies the reference nonlinear coupler.
less pronounced for the Gaussian profile. The sequence of decreasing critical energy is given by Gaussian, logarithm, linear, exponential, hyperbolic and constant. In Fig. 5 one has the transmission of channel 1 for the same couplers, studied in Fig. 3, obtained from the analytical procedure. The transmission was obtained from Eq. Ž38., using Eq. Ž37. together with Eq. Ž34. for the F function. For all the profiles the value of b s 2 was used. The analytical result ŽFig. 5. for transmission is in good agreement with the numerical results ŽFig. 3.. In Fig. 6 one has the critical energy of all couplers as a function of the
Fig. 6. Critical energy obtained analytically ŽEqs. Ž34., Ž35., Ž4. – Ž9.. for the DDF coupler.
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M.G. da SilÕa et al.r Optics Communications 171 (1999) 351–364
b parameter obtained from the analytical calculation, using Eqs. Ž34. and Ž35.. The decrease of the critical energy is in good agreement with the numerical results ŽFig. 4., where the constant profile present the higher reduction of the critical energy and the Gaussian profile present the lower reduction of this energy. For b f 4 there is an inversion of the behavior of the Gaussian and Logarithm profile Žsee Fig. 6.. For the numerical calculation Žsee Fig. 4. this tendency of inversion is observed but not detected up to values of b f 5. In Fig. 7 and Fig. 7Ža. one has
the compression factor for channels 1 and 2 for the DDF coupler with b s 2 obtained numerically and analytically. The numerical values obtained from Eqs. Ž2. and Ž3., Eqs. Ž4. – Ž9. and 11 are represented by the curves. In the analytical procedure the result obtained in Eq. Ž41. shows that the compression factor Ci s b is independent of the profile, and is function only on the b factor. In Fig. 7 this result is represented by two lines at C s 1 and C s 2. Results in Fig. 7 present a good agreement between the analytical and numerical calculations. For the refer-
Fig. 7. Compression factor C1 ŽEq. Ž11.. for the DDF coupler Ž b s 2, a s 0. and reference coupler Ž b s 1, a s 0. obtained numerically and analytically Žsee text.. Ž1. Incident Žchannel 1. and transmitted pulses in channel 1 for: the DDF coupler with constant and hyperbolic profile Ž b s 2 and ErEC s 2. and the reference coupler Ž b s 1 and ErEC s 2. Žassociated to Fig. 7.. Ža. Compression factor C2 ŽEq. Ž11.. for the DDF coupler Ž b s 2, a s 0. and reference coupler Ž b s 1, a s 0. obtained numerically and analytically Žsee text.. Ža.1. Incident Žchannel 1. and switched pulses in channel 2 for the DDF coupler with constant and hyperbolic profile Ž b s 2 and ErEC s 0.65. and the reference coupler Ž b s 1 and ErEC s 0.65. Žassociated to Fig. 7Ža...
M.G. da SilÕa et al.r Optics Communications 171 (1999) 351–364
ence coupler one can see that C s 1 for energy higher that the critical one, however for the DDF couplers the compression goes to a value around C f 2 for high pump power showing a strong shaping effect which result in compression of the pump pulse. This effect is less pronounced in the constant profile where the compression is around 1.8. In Fig. 7.1 one has the transmitted pulse in channel 1 for the input energys 2 EC , associated to Fig. 7, for the reference coupler and the couplers with constant and hyperbolic profile. From this figure one can conclude that the switched pulse on channel 1 present a strong compression effect Ž C f 2. comparing to the input
359
pulse for the hyperbolic profile without any time distortion of the pulse, the integrity of the soliton was preserved. Looking for channel 2 ŽFig. 7Ža.., at low pump power, the switched pulse show an increase of time duration Ž C - 1.. The broadening is stronger for the reference coupler. The hyperbolic profile present C f 1 for energy below the critical one. In Fig. 7Ža.1. one has the switched pulses in channel 2 for the input energy s 0.65EC , associated to Fig. 7Ža., for the reference coupler and the couplers with constant and hyperbolic profile. The hyperbolic profile is given a switched pulse that is very close to the input pulse and the constant profile is
Fig. 8. Compression factor C1 ŽEq. Ž11.. for the DDF coupler Ž b s 3.5, a s 0. and reference coupler Ž b s 1, a s 0. obtained numerically and analytically Žsee text.. Ž1. Incident Žchannel 1. and transmitted pulses in channel 1 for the DDF coupler with constant and hyperbolic profile Ž b s 3.5 and ErEC s 2. and the reference coupler Ž b s 1 and ErEC s 2. Žassociated to Fig. 8.. Ža. Compression factor C2 ŽEq. Ž11.. for the DDF coupler Ž b s 3.5, a s 0. and reference coupler Ž b s 1, a s 0. obtained numerically and analytically Žsee text.. Ža.1. Incident Žchannel 1. and switched pulses in channel 2 for the DDF coupler with constant and hyperbolic profile Ž b s 3.5 and ErEC s 0.65. and the reference coupler Ž b s 1 and ErEC s 0.65. Žassociated to Fig. 8Ža...
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M.G. da SilÕa et al.r Optics Communications 171 (1999) 351–364
still compressing the pulse. In Fig. 8 one has the same calculation but with b s 3.5 Žthe decrease of the dispersion is to a lower value.. The numerical calculation for the compression factor for all the DDF couplers is going to a limit value of C f 3.5 which is good agreement with the analytical value. The exception is for the constant profile which present a lower compression Ž C f 2.7. factor. In Fig. 8.1 one has the transmitted pulse in channel 1 for the input energys 2 EC , associated to Fig. 8, for the reference coupler and the couplers with constant and hyperbolic profile. Looking for the channel 2, at low pump power, the switched pulse present strong com-
pression. The constant profile present compression factor around 3. In Fig. 8Ža.1. one has the switched pulses for E s 0.65EC . The switched pulse for the constant profile present start presenting deformations, with the presence of a pedestal around the compressed soliton. 5.2. DDF coupler with loss Fig. 9 shows the energy transmission characteristics of channel 1 as a function of the optical loss a . In this situation we are solving numerically Eqs. Ž2. and Ž3. for p s 1, b s 1 Žreference coupler. and loss
Fig. 9. Switching characteristics of channel 1 obtained from the numerical solution of Eqs. Ž2. – Ž9. with K s 1, LC s pr2, b s 1, a s 0, 0.2, 1, 5, 10, 15 dBrKm. Ža. Compression factor C1 , associated to Fig. 9 ŽEq. Ž11.. for the coupler with K s 1, LC s pr2, b s 1, a s 0, 0.2, 1, 5, 10, 15 dBrKm obtained numerically Žsee text.. Žb. Compression factor C2 , associated to Fig. 9 ŽEq. Ž11.. for the coupler with K s 1, LC s pr2, b s 1, a s 0, 0.2, 1, 5, 10, 15 dBrKm obtained numerically Žsee text..
M.G. da SilÕa et al.r Optics Communications 171 (1999) 351–364
Ž1rKm. given by a s 0, 0.046, 0.23, 1.151, 2.302, 3.453 kmy1 Žin a ŽdBrKm. one has 0, 0.2, 1, 5, 10, 15, respectively. with input pulses given by 12 and 13 and transmission calculated with Eq. Ž14.. From this figure is clear that the critical energy EC is observed to increase with increasing a and the switching characteristics deteriorate w10x. Our conclusion is that the loss uniformly distributed over the device deteriorate its operational characteristics. These results show excellent agreement with those obtained by Chen et al. w9x. Our previous numerical and analytical results ŽFigs. 3–6. show that the DDF profile lead to the decrease of the critical energy of the coupler. This is a strong suggestion to recover the normal switching of the coupler that was degenerated by the loss ŽFig. 9.. In this situation we expect that decreasing profile for dispersion will recover the normal behavior in transmission of the device. We will compare the DDF coupler with profiles given by Eqs. Ž4. – Ž9. in the presence of loss. The effect, that the decrease of the dispersion always decrease the critical energy EC was shown in Figs. 3 and 4 where we did a numerical calculation of Eqs. Ž2. and Ž3. with no loss Ž a s 0. considering the six profiles ŽEqs. Ž4. – Ž9... The main question is: In the presence of loss, all of these profiles could reduce EC and at the same time could recover the switching
361
Fig. 11. Percentual difference between the nonlinear transmission of each profile and the transmission of the lossless case Ž DT s ŽTreference yTprofile .=100%. for channel 1 as a function of pump energy. ŽRelative to Fig. 10..
characteristic of the coupler that deteriorates because of the loss? To answer this question we choose the loss factor, a ŽdBrKm. s 15, which is a strong loss compared with the loss one has in a basic telecommunication fiber Ž0.2 dBrKm.. From Fig. 9 one can conclude that this loss in the guide lead to an increase of the critical energy from EC to f 1.075EC and the switching characteristics are quite degenerated. In Fig. 9Ža. and Fig. 9Žb. one has the compression factor for the switched pulses associated to Fig. 9. From Fig. 9Ža. one can conclude that the presence of loss is leading to strong broadening of the switched pulses. For 15 dBrKm loss, a broadening factor of 30% is observed for the switched pulses in channel 1. From Fig. 9Žb. the switched pulse in channel 2 already present some broadening even in the lossless Table 1 Dispersion profiles associated with Fig. 10 Ž a ŽdBrKm. s15. and maximum difference obtained with the lossless case Ž ps1, a ŽdBrKm. s 0.
Fig. 10. Switching characteristics of channel 1 obtained from the numerical solution of Eqs. Ž2. – Ž9. with K s1, LC s p r2, a s15 dBrKm for the constant, exponential, Gaussian, hyperbolic, linear and logarithm profiles, respectively and b s1, a s 0 for the reference coupler.
Profile
b
Maximum difference Ž%.
Constant Exponential Gaussian Hyperbolic Linear Logarithm
1.15 1.89 7.38 1.70 2.35 3.75
5.11 2.91 5.60 2.47 3.41 3.87
362
M.G. da SilÕa et al.r Optics Communications 171 (1999) 351–364
case device. However the presence of loss lead to an increase of the broadening. Our main interest here is to find a dispersion profile that could recover the original critical energy and the characteristics of switching of the coupler. To do this we solve Eqs. Ž2. and Ž3. with loss Ž a ŽdBrKm. s 15. and considering the DDF profiles ŽEqs. Ž4. – Ž9... We did a large number of numerical simulations for each profile to find the exact value of b to decrease the critical power from f 1.075EC to EC . In Fig. 10 one has
the nonlinear transmission for each profile comparing with the original behavior of the device Ž a s 0, p s 1, b s 1 lossless case. as a reference. From Fig. 10 one can conclude that the worse behavior is associated to the constant profile. It is quite distant from the lossless behavior. The exponential profile Ž b s 1.89., hyperbolic profile Ž b s 1.70. and the linear profile Ž b s 2.35. gave reasonable behavior. However when comparing with the lossless case Ž b s 1, a s 0. one can notice that the number of
2q
Fig. 12. Compression factor C1 , associated to Fig. 10 ŽEq. Ž11.. for the DDF coupler with K s 1, LC s pr2, a s 15 dBrKm Žthe value of b associated to each profile is given in Fig. 10.. Data obtained numerically Žsee text.. Ž1. Incident Žchannel 1. and transmitted pulses in channel 1 for the DDF coupler with constant and hyperbolic profile Ž b s 1.15, b s 1.70 and ErEC s 2. and the reference coupler Ž b s 1 and ErEC s 2. Žassociated to Fig. 12.. Ža. Compression factor C 2 , associated to Fig. 10 ŽEq. Ž11.. for the DDF coupler with K s 1, LC s pr2, a s 15 dBrKm Žthe value of b associated to each profile is given in Fig. 10.. Data obtained numerically Žsee text. and reference coupler Ž b s 1, a s 0.. Ža.1. Incident Žchannel 1. and switched pulses in channel 2 for the DDF coupler with constant and hyperbolic profile Ž b s 1.15 and b s 1.70 and ErEC s 0.65. and the reference coupler Ž b s 1 and ErEC s 0.65. Žassociated to Fig. 12Ža...
M.G. da SilÕa et al.r Optics Communications 171 (1999) 351–364
peaks and the position of the peaks of transmission does not fit with the reference case. However the hyperbolic profile Ž b s 1.70. is almost coincident with the lossless device Ž p s 1, a s 0.. We can conclude that it is quite possible to project a dispersion decreasing profile to recompose the intrinsic loss effect on the switching characteristics of the NLDC. To have an exact idea of the real recomposition of the nonlinear transmission with the dispersion decreasing profile, we measure the percentual difference between the transmission of each profile and the transmission of the lossless case in Fig. 10. In Fig. 11 one has this percentual difference between the transmission of the lossless case and transmission of the each profile as a function of pump power. From Fig. 11 is quite clear that the differences in transmission for the constant profile Ž b s 1.15. is around 5% around 1.4 EC and decreases to 1% around 2 EC . The Gaussian profile Ž b s 7.38. presents differences around 6% for 1.1 EC . For this scale the maximum difference in the hyperbolic profile Ž b s 1.70. is around 2.47% in Fig. 11. In Table 1 one has the b values for each profile and the maximum error observed in Figs. 10 and 11. From this table one can conclude that to recover the same critical power of the lossless case, the Gaussian profile present the highest value of the b parameter. The lowest error, when comparing with the lossless case is presented by the hyperbolic profile where the difference in transmission is below 2.47%. In Fig. 12 and Fig. 12Ža. one has the compression factor in channels 1 and 2 associated to Fig. 10. In Fig. 12 the lossless case Ž p s 1, a s 0. present compression around 1. As discussed before, with the presence of loss Ž a ŽdBrKm. s 15. the switched pulses present broadening ŽFig. 9Ža. and Fig. 9Žb.. around 30%. Fig. 12 shows the effect of loss and dispersion profile together associated to Fig. 10 Žall the couplers with the same critical energy.. One can conclude that the constant profile was not able to recover the original temporal profile of the input pulse, it present broadening around 20%, C f 0.83. However the hyperbolic profile present a good recovery of the pulse shape. For the hyperbolic profile a slight compression around 22% is observed Ž C f 1.29.. The Gaussian profile present a strong shaping effect with compression factor going up to 5.5. In Fig. 12.1 one has the transmitted pulse in channel 1 for the input
363
energys 2 EC , associated to Fig. 12, for the reference coupler and the couplers with constant and hyperbolic profile. From this figure is clear the broadening of the transmitted pulse from constant DDF coupler. The transmitted pulse from the hyperbolic DDF coupler present a quite similar shape with the reference pulse with a little compression. The same kind of effect was observed for channel 2 ŽFig. 12Ža. and Fig. 12Ža.1... Looking for the channel 2, at low pump power, the switched pulse present temporal broadening. In Fig. 12Ža.1. one has the switched pulses for E s 0.65EC . The switched pulse from the hyperbolic DDF coupler present the closest shape compared with the reference coupler. One can conclude that with the decreasing dispersion profile one can recover almost perfectly the original switching behavior of the NLDC, like the transmission characteristics and the original temporal shape of the original coupler that was destroyed by the intrinsic optical loss of the device. 6. Conclusions We present an analytical and numerical investigation of the propagation and the switching of fundamental solitons in a two-core nonlinear fiber coupler constructed with dispersion decreasing fiber ŽDDF. with a variational method using the Lagrangian density formulation. The analytical solutions were directly obtained from the coupled nonlinear Schrodin¨ ger equations. Our simulations studies six different profiles, linear, hyperbolic, exponential, logarithm, constant and Gaussian. The transmission characteristics, the critical energy and the compression factor of first order solitons obtained by the analytical procedure agree well with the results from numerical analysis. Twin core nonlinear directional couplers that include loss is also examined. The inclusion of loss results in the increase of the critical energy of the coupler and broadening of the switched pulses, while the switching characteristics deteriorate. To compensate for this worse behavior, associated to the loss, we have investigated the effect of the DDF profile on the performance of the coupler. We have shown that appropriate shaping of the DDF profile is quite effective to recover, almost completely, the original switching behavior associated to the lossless situation. It is concluded that the hyperbolic disper-
364
M.G. da SilÕa et al.r Optics Communications 171 (1999) 351–364
sion profile is nearly optimum to the recovery of the lossless switching behavior. Comparing both techniques we can say that the Lagrangian formulation provide very good description of soliton switching in the DDF coupler. The use of this technique to the study of switching of ultrashort optical pulses in optical devices will be very useful. The study of soliton switching in DDF nonlinear fiber couplers provides possibilities for achieving, high efficiency in ultrafast all-optical signal processing, especially for optical switches and optical transistors.
Acknowledgements We thank CNPq ŽConselho Nacional de Desen., FINEP volvimento Cientıfico e Tecnologico ´ ´ ŽFinanciadora de Estudos e Projetos. and FUNCAP ŽFundac¸ao ˜ Cearense de Amparo a Pesquisa. for the financial support.
References w1x S.M. Jensen, IEEE J. Quantum Electron. 18 Ž1982. 1580. w2x S. Trillo, S. Wabnitz, E.M. Wright, G.I. Stegman, Opt. Lett. 13 Ž1988. 672.
w3x K. Kitayama, S. Wang, Appl. Phys. Lett. 43 Ž1983. 17. w4x G.D. Peng, A. Ankiewicz, Int. J. Non-Opt. Phys. 1 Ž1992. 135. w5x M.G. da Silva, A.F. Teles, A.S.B. Sombra, J. Appl. Phys. 84 Ž4. Ž1998. 1834. w6x A.S.B. Sombra, Braz. J. Phys. 24 Ž1994. 485. w7x A.S.B. Sombra, Opt. Commun. 94 Ž1992. 92. w8x J.M. Fang, M.J. Potasek, Phys. Rev. A 45 Ž1992. 5305. w9x Y. Chen, A.W. Snyder, D.N. Payne, IEEE J. Quantum Electron. 28 Ž1992. 239. w10x K.Z. Nobrega, A.S.B. Sombra, Opt. Commun. 151 Ž1–3. Ž1998. 31. w11x D. Anderson, Phys. Rev. A 27 Ž1983. 3135. w12x A.D. Bonderson, M. Lisak, D. Anderson, Phys. Scr. 20 Ž1979. 479. w13x D. Anderson, M. Lisak, T. Reichel, J. Opt. Soc. Am. B 5 Ž1988. 207. w14x A. Berntson, D. Anderson, M. Lisak, Phys. Scr. 52 Ž1995. 544. w15x G.P. Agrawal, Nonlinear Fiber Optics, Academic, San Diego, 1995 ŽChapter 5.. w16x M.D. Pelusi, H.F. Liu, IEEE J. Quantum Electron. V. 33 Ž8. Ž1997. 1430. w17x A. Mostofi, H.H. Hanza, P.L. Chu, IEEE J. Quantum Electron. V. 33 Ž4. Ž1997. 620. w18x P.V. Mamyshev, S.V. Chernikov, E.M. Dianov, IEEE J. Quantum Electron. 10 Ž1991. 2347. w19x H.H. Hanza, P.L. Chu, B.A. Malomed, G.D. Peng, Opt. Commun. 134 Ž1997. 59. w20x P.L. Chu, B.A. Malomed, G.D. Peng, J. Opt. Soc. Am. B 10 Ž1993. 1379. w21x P.L. Chu, G.D. Peng, B.A. Malomed, Opt. Lett. 18 Ž1993. 328.
Optics Communications 171 Ž1999. 351–364 www.elsevier.comrlocateroptcom
Full length article
Analysis of soliton switching in dispersion-decreasing fiber couplers b M.G. da Silva a , K.Z. Nobrega , A.S.B. Sombra ´
a,)
a
b
´ Laboratorio nao dos Materiais (LONLCM), Departamento de Fısica, UniÕersidade Federal do Ceara, ´ de Optica ˜ Linear e Ciencia ˆ ´ ´ Caixa Postal 6030, 60455-760 Fortaleza, Ceara, ´ Brazil Departamento de Engenharia Eletrica, Centro de Tecnologia, UniÕersidade Federal do Ceara, ´ ´ 60455-760, Fortaleza, Ceara, ´ Brazil Received 28 June 1999; received in revised form 8 September 1999; accepted 14 September 1999
Abstract We present an analytical and numerical investigation of the propagation and the switching of fundamental solitons in a two-core nonlinear fiber coupler constructed with dispersion decreasing fiber ŽDDF. with a variational method using the Lagrangian density formulation. The analytical solutions were directly obtained from the coupled nonlinear Schrodinger ¨ equations. Our simulations considering six different profiles, linear, hyperbolic, exponential, logarithm, constant and Gaussian. The transmission characteristics, the critical energy and the compression factor of first order solitons obtained by the analytical procedure agree well with the results from numerical analysis. Twin core nonlinear directional couplers that include loss are also examined. The inclusion of loss results in the increase of the critical power of the coupler and broadening of the switched pulses, while the switching characteristics deteriorate. To compensate for this worse behavior, associated to the loss, we have investigated the effect of the DDF profile on the performance of the coupler. We show that appropriate shaping of the DDF profile is quite effective to recover, almost completely, the original switching behavior associated to the lossless situation. It is concluded that the hyperbolic dispersion profile is nearly optimum to the recovery of the lossless switching behavior. q 1999 Elsevier Science B.V. All rights reserved. PACS: 42.65.Pc; 42.50.Rh; 42.79.Gn
1. Introduction All optical switching devices are attracting considerable interest as fast switching components for future high-bit-rate systems. Optical fiber couplers have been studied for their potential applications to ultrafast all optical switching processing, like optical switch w1–3x. )
Corresponding author. Fax: q55-85-287-4138; e-mail: [email protected]
In a nonlinear coupler constructed from a Kerr type medium the dependence of the nonlinear refractive index n on the laser intensity is given by the expression n s n 0 q n 2 I, where n 0 is the refractive index at low intensity and n 2 is the Kerr nonlinear coefficient w4,5x. Jensen w1x showed that varying the input light in the nonlinear coupler could lead to pulse switching between the two cores. He therefore foresaw the possible use of a nonlinear directional coupler as an optical switch. A nonlinear coupler consists of two closely spaced, parallel, single mode
0030-4018r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 Ž 9 9 . 0 0 5 5 2 - 0
M.G. da SilÕa et al.r Optics Communications 171 (1999) 351–364
352
waveguides in a material with an intensity dependent index of refraction. At low light levels, the device behaves as a linear directional coupler. Because of evanescent coupling, signals introduced into guide w1–3x transfer completely to guide w4,5x in one coupler length Lc Žsee Fig. 1.. Higher intensities induce changes in the refractive index and detune the coupler. Coupler is inhibited for input intensities above the critical intensity:
l IC s
n 2 LC
Ž 1.
where IC s PrA, P is the critical power, A is the effective mode area, l is the pump wavelength, n 2 is the nonlinear refractive index w1–7x, LC is the half-beat-length coupler of length LC s pr2 K and K is the linear coupling coefficient between adjacent guides. At the critical intensity IC , 50% of the light emerges from each waveguide. Above IC most of the light emerges from channel 1 Žsee Fig. 1.. Previous studies of soliton switching in dual core optical fibers, without loss, have shown excellent switching characteristics, with efficiencies around 96% for a wide range of input energies w4,5x. By comparing the switching behavior of fundamental, second order, and quasi solitons it was observed that the fundamental soliton has the most suitable features for optical switching w4,5x. Indeed, it has been shown that pulse breakup may be avoided when the input signal is a soliton w6–8x. Since then, soliton switching in nonlinear fiber couplers have been receiving considerable attention.
However lossless device is an idealized situation. In practice, large or small, material loss is unavoidable, especially when nonlinearities are based on absorptive process in semiconductors. The absorption, uniformly distributed over the device, will set a limit to the operation of the device w9,10x. In the following, we will examine nonlinear directional double core coupler with loss. The presence of loss is responsible for the increase of the critical power of the coupler and a strong deformation of the nonlinear transmission of the device. In this paper, we have investigated the effect of using dispersion decreasing fibers on the performance of NLDC with and without loss. Six closely shaped profiles named linear, Gaussian, exponential, hyperbolic, constant and logarithm have been considered. From our study of the dispersion profiles, we suggest the optimum profile, to recover the original performance of the nonlinear transmission, to overcome the effect of the intrinsic loss in the device. The paper is divided into several sections. In Section 2 the nonlinear directional coupler constructed from dispersion decreasing fibers ŽDDF., operating in the soliton regime is presented. Using the technique developed by Anderson and co-workers w11–14x, we convert the coupled nonlinear Schrodin¨ ger equations ŽNLSEs. into a variational problem with corresponding Lagrangian equations of motion for a finite number of degrees of freedom and obtain the switching characteristics of the device in Section 4. For both Sessions we are considering the lossless coupler. The analytical and numerical procedure used is compared is Section 5.1. In Section 5.2 the effect of loss in the coupler is studied. We discuss the effect of each profile in the presence of loss and conclude that there is an optimum profile to recover the lossless switching behavior.
2. Theoretical model
Fig. 1. Schematic of nonlinear directional coupler of length LC .
We will consider picosecond pulses propagating in the anomalous dispersion regime in a nonlinear directional coupler with twin fibers with variable normalized second-order dispersion coefficient b 2 Ž j .. The propagating of ultrashort solitons through a DDF coupler is described by the nonlinear
M.G. da SilÕa et al.r Optics Communications 171 (1999) 351–364
Schrodinger equation with a axially varying disper¨ sion profile pŽ j . w15x: i
E u1 Ej
1 q 2
pŽ j .
E 2 u1 Et 2
q < u1 < 2 u1 q Ku 2 q i a u1 s 0
Ž 2. i
E u2 Ej
1 q 2
pŽ j .
E 2 u2 Et 2
q < u 2 < 2 u 2 q Ku1 q i a u 2 s 0
Ž 3. where u1 and u 2 are the modal field amplitudes in soliton units, pŽ j . s < b 2 Ž j .rb 2 Ž0.< is the normalized GVD, a is the fiber loss. j and t are the normalized length and time in soliton units with j s zrL D and t s trT0 . Here L D s T02r< b 2 <, with pulse width T0 ŽT0 s 5.685 ps, < b 2 < s 20 ps 2rkm.. The varying GVD parameter pŽ j . is normalized by the dispersion in the input of the fiber, b 2 Ž0. s < b 2 < s 20 ps 2rkm. In a DDF of certain length, the dispersion is monotonically and smoothly decreased from an initial value to a smaller value at the end of the length according to some specified value. Provided the dispersion variation in the DDF is sufficiently gradual, soliton compression can be an adiabatic process where an input fundamental soliton pulse can be ideally compressed as it propagates, while retaining its soliton character and conserving the energy w16– 19x. Interest in DDF in nonlinear fiber optics has focused on its use to compensate for the detrimental effect of signal attenuation in long distance soliton propagation. In this paper, we compare six simple coefficient of dispersion decreasing profiles, namely linear, Gaussian, exponential, logarithm, hyperbolic and constant. These profiles expressed in terms of the parameters 1rb Žminimum value of p . and L Žlength of the coupler. are: pŽ j . s
Ž1yb . bL
p Ž j . s exp p Ž j . s exp
ž
ž
j q 1 Linear
yj L
yj 2 L2
/
ln b Exponential
/
ln b Gaussian
ž
j
p Ž j . s ln e q pŽ j . s pŽ j . s
L
Ž e1r b y e . Logarithm
L
Ž b y 1. j q L 1
b
353
/
Ž 7.
Hyperbolic
Ž 8.
Constant.
Ž 9.
Note that in these normalized profiles the dispersion coefficient p monotonically decreases from 1 to a final value of 1rb , after a length L of the coupler. The dispersion profiles are illustrated in Fig. 2. For the first order soliton Ž N s 1. the solution of Eqs. Ž2. and Ž3. Ž p s 1, K s 0, a s 0. lossless case is: u i Ž j ,t . s A i sec h Ž A it . exp Ž i A2i jr2 q i w i .
Ž 10 .
where A i and w i are arbitrary constants. A fundamental soliton pulse has been already considered as an effective input pulse type for improving switching efficiency by reducing pulse break-up in a nonlinear directional coupler. In this case the laser’s output width and power should have to be adjusted to maintain the fundamental soliton profile as the input energy changes. From the experimental point of view it is easy to adjust the laser output power, but very difficult to change the time duration. However for a certain time duration, the laser output power could be adjusted to produce an exact first order soliton pulse. These pulses could be derived from different sources, where each source could be adjusted to
Ž 4. Ž 5. Ž 6. Fig. 2. Schematics of the dispersion profiles considered.
M.G. da SilÕa et al.r Optics Communications 171 (1999) 351–364
354
generate a specific time duration and power, necessary for a fundamental soliton. Therefore, solitons with different time duration and pump powers, from different sources, could be efficiently switched through the optical device w4,5x. We also define the final compression factor C, achieved after propagation of the optical input pulse in the DDF coupler. It is defined as the ratio of the optical pulse FWHM at the input of the coupler t 0 ŽFWHM of the < uŽ1.Ž0,t .< 2 , input pulse., to that at the output Žtransmitted. of the coupler, t i ŽFWHM of the < u i Ž LC ,t .< 2 , transmitted pulse.. t0 Ci s Ž 11 . ti where i s 1,2 for switching of pulses in channels 1 and 2.
co-workers w11–14x, which is based on the Lagrangian equations of motion for a finite number of degrees of freedom. This technique was used in several problems of optical solitons w11–14,20,21x. We introduce the Lagrangian density formulation following Refs. w20,21x. The Lagrangian of the DDF lossless coupler ŽEqs. Ž2. and Ž3. with a s 0. is: q`
Ls
LX s L1 q L 2 q K Ž u 2) u1 q u 2 u1) .
q` i
Ti s
q`
Hy`
L ,t
Li s
i
ž
2
Ž 14 .
< u1 Ž 0,t . < 2 dt
with i s 1, 2 and a fiber coupler with a length of j L . 4. Analytical analysis To study the soliton switching in the coupler, we will use the approach developed by Anderson and
E ui
u )i
1
. < 2 dt
Ž 16 .
where L i denotes:
y
Taking Eqs. Ž2. and Ž3. with no coupling Ž K s 0. or profile Ž p s 1. and no loss Ž a s 0. one has the well-known soliton solutions ŽEq. Ž10... We have analyzed numerically the soliton transmission of first order solitons through the two core DDF nonlinear directional fiber coupler ŽEqs. Ž2. and Ž3.. with dispersion profiles given by Eqs. Ž4. – Ž9.. The initial pulse at the input core is given by: u1 Ž 0,t . s A sec h Ž At . Ž 12 . u 2 Ž 0,t . s 0. Ž 13 . This system of linearly coupled NLSEs ŽEqs. Ž2. and Ž3. and Eqs. Ž4. – Ž9.. was solved numerically using the split-step method with 1024 temporal grid points taking in account the initial conditions given by Eqs. Ž12. and Ž13.. We can define the transmission Ti as a function of the pulse energies:
Ž 15 .
where the Lagrangian density is:
3. Numerical procedure
Hy` < u Ž j
X
Hy` L dt
2
y ui
Ej
E u i) 2
E ui
pŽ j .
/
Ej
1 q
Et
2
< ui < 4
Ž 17 .
where the asterisk denotes the complex conjugate and i s 1, 2. Eqs. Ž2. and Ž3. Žwith a s 0. result from the variational equations corresponding to the variational principle:
d
HHL
ž
E u i E u i) E u i E u i) , , , EjEt s 0 Ž 18 . Ej Ej Et Et
/
u i ,u )i ,
i.e., the equations:
E
dL s
d u )i
EL
Ej E
E u )i
E Ej
q
EL
Et E E u )i Et
EL y
E u )i
Ž 19 .
s0
for i s 1, 2 we obtain Eqs. Ž2. and Ž3.. For the DDF coupler we assume the pulse profiles u1Ž j ,t . and u 2 Ž j ,t . take the forms A
u1 Ž j ,t . s
(p Ž j .
sec h
ž
At pŽ j .
/
=cos u exp Ž i w q i c . u 2 Ž j ,t . s
A
(p Ž j .
sec h
ž
At pŽ j .
=sin u exp Ž i w y i c .
Ž 20 .
/ Ž 21 .
M.G. da SilÕa et al.r Optics Communications 171 (1999) 351–364
u Ž j . is the coupling angle that gives the power coupling between the two cores, c Ž j . is the relative phase. For these trial functions A is a constant of motion. However, the parameter w has no influence on the other parameters. We expect to have a nontrivial dynamics for u Ž j . and c Ž j .. However, our main choice, for the initial solutions given by Eqs. Ž20. and Ž21., is associated to the fact that the value of the total intensity of the solitons Ž< u1 < 2 q < u 2 < 2 . has no dependence on u Ž j .. It is a constant of motion during the coupler propagation. The obtained Lagrangian for the DDF coupler is given by: A3
Ec L s y2 A cos 2 u
y
3p Ej q 2 AK sin 2 u cos 2 c . Ž 22 . If we take the variation of the Lagrangian Eq. Ž22. with respect to c Ž j . according to: EL E EL y s0 Ž 23 . Ec Ej E Ž Ec Ej .
ž
)
Eu
s yK 1 y
Ej
ž
2
A2
/
6 Kp Ž j .
after a little manipulation: du
u
H0
(1 y Ž A H0L
C
2
6 Kp Ž j .
LD
.
2
sin2 w 2 u x
Kdj .
Ž 29 .
Ž 24 . Ž 25 .
P DD F s A2
H0L
C
LD
cos 2 Ž a 1 j . pŽ j .
We can obtain the Hamiltonian H for the system according to the equation: Ec EL Eu EL Hs q yL Ž 26 . Ej E Ž Ec Ej . Ej E Ž Eu Ej .
where for the critical energy:
with the help of Eq. Ž22. the obtained Hamiltonian for the coupler is:
the obtained value for a 1 is:
Hsy
A3 3p
2
sin 2 u q 2 AK sin 2 u cos 2 c .
Ž 27 .
In our case we consider that the soliton is launched in channel 1. The initial condition is then u s 0 in Eqs. Ž20. and Ž21.. The Hamiltonian will stay in the entire trajectory with this value, H Ž j . s H Ž0.. According to Eq. Ž27. we have two solutions with H s 0. The first one has sin 2 u s 0 all over the trajectory. And the second one is: cos 2 c s
A2 6 Kp
sin 2 u .
sin2 Ž 2 u .
For a coupler without dispersion profile Ž p s 1. one obtains the critical power of the reference coupler Ž PC s A2rK s 6. w20,21x. However Eq. Ž29. is not readily solved because the profile pŽ j . is present in the integral. In this situation we will calculate the energy in the DDF coupler when compared with the reference coupler Žno dispersion profile.. The peak power in the fiber 1, in the DDF coupler could be calculated using Eq. Ž20. and is given by:
/
which implies that either: sin 2 u s 0 or Eu s yK sin 2 c . Ej
The second solution is more interesting to us because the power transfer between cores 1 and 2 is described by the trajectory associated to evolution of u . Taking in account Eqs. Ž28. and Ž25. we obtain:
sy
sin2 2 u
355
Ž 28 .
cos 2 Ž a 1 LC L D .
LD LC
Ž 30 .
1 s 2
p Ž LC L D .
a1 s
dj
cosy1
ž(
p Ž LC L D . 2
/
Ž 31 .
for the reference coupler the transmission of channel 1 given by: P s A20
H0L
C
LD
cos 2 Ž a 0 j . d j
Ž 32 .
where A 0 is the amplitude of the light in the reference coupler. In this case a 0 is given by:
a0 s
LD LC
cosy1
1
ž( / 2
Ž 33 .
M.G. da SilÕa et al.r Optics Communications 171 (1999) 351–364
356
considering that at the critical energy, without loss, both couplers present the same power PDD F s P, using Eqs. Ž30. and Ž32. one has: cos 2 Ž a 1 j .
LC L D
A20 2
s
H0
A
pŽ j .
dj sF
LC L D
Ž 34 .
cos 2 Ž a 0 j . d j
H0
A
1
s E0
s A0
'F
.
Ž 35 .
Solving Eq. Ž35. with Eq. Ž34. for all the profiles one has the critical energy of the DDF couplers compared with the normal coupler. To obtain the transmission characteristic for each coupler one can compare Eq. Ž29. for the DDF coupler with the reference coupler:
T2 s sin2 u .
Ž 39 .
Ci s
2
2
(1 y Ž A rŽ 6 K . . sin 2u 2
2 0
LCrL D
H0
sy
2
sec h2
du
H0L
sy
(1 y Ž A F Ž 6 K . . sin 2u 2
At 0
ž /
1 s
p Ž 0.
2
then one has:
t0s
p Ž 0 . sec hy1 Ž 1r'2 . A
at the output of channel 1:
t1s
p Ž LC L D . sec hy1 Ž 1r'2 . A
the compression factor is given by: p Ž 0. p Ž LC L D .
1 s
1rb
sb
Ž 41 .
Kdj
using Eq. Ž35., one concludes that the first term in the left could be given by: u
Ž 40 .
t i Ž LC L D .
where we assume that all the profiles present a decreasing dispersion, where the final value of the dispersion in the coupler is 1rb ŽEqs. Ž4. – Ž9. and Fig. 2..
du
u
H0
2
t 0 Ž 0.
taking Eq. Ž20. for the input pulse at half maximum intensity:
Ci s
(1 y Ž A rŽ 6 Kp Ž j . . . sin 2u s
H0
Ž 38 .
du
u
H0
T1 s cos 2 u
For the analytical procedure we could also calculate the compression factor Ci defined in Eq. Ž11.:
for each profile we have different values for pŽ j . ŽEqs. Ž4. – Ž9.., a 1 ŽEq. Ž31.. and consequently for the F function. We can calculate now the critical energy of the DDF coupler and the reference coupler. Taking Eq. Ž20. for the DDF coupler, one has for the critical energy: E s 2 A and E0 s 2 A 0 Ž p s 1. for the reference coupler. Taking into account Eq. Ž34. one has: E
we can calculate the transmission characteristics of the coupler using Eqs. Ž37. and Ž34. and taking in account that for the coupler transmission is:
2
2
C
LD
Kdj .
Ž 37 . Eq. Ž37. is readily solved Žassuming a fixed length for the nonlinear coupler K j s constant. using a fourth order Runge–Kutta method. In this situation
5. Results and discussion 5.1. DDF coupler: numerical and analytical Fig. 3 shows the soliton transmission characteristics of channel 1 for six different profiles without loss, obtained numerically. The NLSE equations ŽEqs. Ž2. and Ž3.. were solved numerically with the input soliton given by Eqs. Ž12. and Ž13.. The soliton transmission was obtained ŽEq. Ž14.. for six different couplers constructed with different profiles ŽEqs.
M.G. da SilÕa et al.r Optics Communications 171 (1999) 351–364
Fig. 3. Switching characteristics of channel 1 obtained from the numerical solution of Eqs. Ž2. – Ž9. with K s1, LC s p r2, b s 2, a s 0. The case b s1 identifies the reference nonlinear coupler.
Ž4. – Ž9... In this figure the reference coupler Žreference, a s 0, b s 1, p s 1. was also included. In this simulation one has b s 2 for all the six profiles. One concludes that for all the profiles there is a decrease of the critical energy. In Fig. 4 one has the critical energy of all couplers as a function of the b parameter obtained from the numerical calculation. The energy is compared with the critical energy of the reference coupler. From the figure one can conclude that the increase in b Ždecrease in the profile pŽ j .. lead to the decrease of the critical energy. The decrease is stronger for the constant profile and is
Fig. 4. Critical energy obtained numerically ŽEqs. Ž2. – Ž9. and Ž14.. for the DDF coupler.
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Fig. 5. Switching characteristics of channel 1 obtained analytically from Eqs. Ž34., Ž35., Ž37. and Ž38. with K s1, LC s p r2, b s 2, a s 0. The case b s1 identifies the reference nonlinear coupler.
less pronounced for the Gaussian profile. The sequence of decreasing critical energy is given by Gaussian, logarithm, linear, exponential, hyperbolic and constant. In Fig. 5 one has the transmission of channel 1 for the same couplers, studied in Fig. 3, obtained from the analytical procedure. The transmission was obtained from Eq. Ž38., using Eq. Ž37. together with Eq. Ž34. for the F function. For all the profiles the value of b s 2 was used. The analytical result ŽFig. 5. for transmission is in good agreement with the numerical results ŽFig. 3.. In Fig. 6 one has the critical energy of all couplers as a function of the
Fig. 6. Critical energy obtained analytically ŽEqs. Ž34., Ž35., Ž4. – Ž9.. for the DDF coupler.
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b parameter obtained from the analytical calculation, using Eqs. Ž34. and Ž35.. The decrease of the critical energy is in good agreement with the numerical results ŽFig. 4., where the constant profile present the higher reduction of the critical energy and the Gaussian profile present the lower reduction of this energy. For b f 4 there is an inversion of the behavior of the Gaussian and Logarithm profile Žsee Fig. 6.. For the numerical calculation Žsee Fig. 4. this tendency of inversion is observed but not detected up to values of b f 5. In Fig. 7 and Fig. 7Ža. one has
the compression factor for channels 1 and 2 for the DDF coupler with b s 2 obtained numerically and analytically. The numerical values obtained from Eqs. Ž2. and Ž3., Eqs. Ž4. – Ž9. and 11 are represented by the curves. In the analytical procedure the result obtained in Eq. Ž41. shows that the compression factor Ci s b is independent of the profile, and is function only on the b factor. In Fig. 7 this result is represented by two lines at C s 1 and C s 2. Results in Fig. 7 present a good agreement between the analytical and numerical calculations. For the refer-
Fig. 7. Compression factor C1 ŽEq. Ž11.. for the DDF coupler Ž b s 2, a s 0. and reference coupler Ž b s 1, a s 0. obtained numerically and analytically Žsee text.. Ž1. Incident Žchannel 1. and transmitted pulses in channel 1 for: the DDF coupler with constant and hyperbolic profile Ž b s 2 and ErEC s 2. and the reference coupler Ž b s 1 and ErEC s 2. Žassociated to Fig. 7.. Ža. Compression factor C2 ŽEq. Ž11.. for the DDF coupler Ž b s 2, a s 0. and reference coupler Ž b s 1, a s 0. obtained numerically and analytically Žsee text.. Ža.1. Incident Žchannel 1. and switched pulses in channel 2 for the DDF coupler with constant and hyperbolic profile Ž b s 2 and ErEC s 0.65. and the reference coupler Ž b s 1 and ErEC s 0.65. Žassociated to Fig. 7Ža...
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ence coupler one can see that C s 1 for energy higher that the critical one, however for the DDF couplers the compression goes to a value around C f 2 for high pump power showing a strong shaping effect which result in compression of the pump pulse. This effect is less pronounced in the constant profile where the compression is around 1.8. In Fig. 7.1 one has the transmitted pulse in channel 1 for the input energys 2 EC , associated to Fig. 7, for the reference coupler and the couplers with constant and hyperbolic profile. From this figure one can conclude that the switched pulse on channel 1 present a strong compression effect Ž C f 2. comparing to the input
359
pulse for the hyperbolic profile without any time distortion of the pulse, the integrity of the soliton was preserved. Looking for channel 2 ŽFig. 7Ža.., at low pump power, the switched pulse show an increase of time duration Ž C - 1.. The broadening is stronger for the reference coupler. The hyperbolic profile present C f 1 for energy below the critical one. In Fig. 7Ža.1. one has the switched pulses in channel 2 for the input energy s 0.65EC , associated to Fig. 7Ža., for the reference coupler and the couplers with constant and hyperbolic profile. The hyperbolic profile is given a switched pulse that is very close to the input pulse and the constant profile is
Fig. 8. Compression factor C1 ŽEq. Ž11.. for the DDF coupler Ž b s 3.5, a s 0. and reference coupler Ž b s 1, a s 0. obtained numerically and analytically Žsee text.. Ž1. Incident Žchannel 1. and transmitted pulses in channel 1 for the DDF coupler with constant and hyperbolic profile Ž b s 3.5 and ErEC s 2. and the reference coupler Ž b s 1 and ErEC s 2. Žassociated to Fig. 8.. Ža. Compression factor C2 ŽEq. Ž11.. for the DDF coupler Ž b s 3.5, a s 0. and reference coupler Ž b s 1, a s 0. obtained numerically and analytically Žsee text.. Ža.1. Incident Žchannel 1. and switched pulses in channel 2 for the DDF coupler with constant and hyperbolic profile Ž b s 3.5 and ErEC s 0.65. and the reference coupler Ž b s 1 and ErEC s 0.65. Žassociated to Fig. 8Ža...
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M.G. da SilÕa et al.r Optics Communications 171 (1999) 351–364
still compressing the pulse. In Fig. 8 one has the same calculation but with b s 3.5 Žthe decrease of the dispersion is to a lower value.. The numerical calculation for the compression factor for all the DDF couplers is going to a limit value of C f 3.5 which is good agreement with the analytical value. The exception is for the constant profile which present a lower compression Ž C f 2.7. factor. In Fig. 8.1 one has the transmitted pulse in channel 1 for the input energys 2 EC , associated to Fig. 8, for the reference coupler and the couplers with constant and hyperbolic profile. Looking for the channel 2, at low pump power, the switched pulse present strong com-
pression. The constant profile present compression factor around 3. In Fig. 8Ža.1. one has the switched pulses for E s 0.65EC . The switched pulse for the constant profile present start presenting deformations, with the presence of a pedestal around the compressed soliton. 5.2. DDF coupler with loss Fig. 9 shows the energy transmission characteristics of channel 1 as a function of the optical loss a . In this situation we are solving numerically Eqs. Ž2. and Ž3. for p s 1, b s 1 Žreference coupler. and loss
Fig. 9. Switching characteristics of channel 1 obtained from the numerical solution of Eqs. Ž2. – Ž9. with K s 1, LC s pr2, b s 1, a s 0, 0.2, 1, 5, 10, 15 dBrKm. Ža. Compression factor C1 , associated to Fig. 9 ŽEq. Ž11.. for the coupler with K s 1, LC s pr2, b s 1, a s 0, 0.2, 1, 5, 10, 15 dBrKm obtained numerically Žsee text.. Žb. Compression factor C2 , associated to Fig. 9 ŽEq. Ž11.. for the coupler with K s 1, LC s pr2, b s 1, a s 0, 0.2, 1, 5, 10, 15 dBrKm obtained numerically Žsee text..
M.G. da SilÕa et al.r Optics Communications 171 (1999) 351–364
Ž1rKm. given by a s 0, 0.046, 0.23, 1.151, 2.302, 3.453 kmy1 Žin a ŽdBrKm. one has 0, 0.2, 1, 5, 10, 15, respectively. with input pulses given by 12 and 13 and transmission calculated with Eq. Ž14.. From this figure is clear that the critical energy EC is observed to increase with increasing a and the switching characteristics deteriorate w10x. Our conclusion is that the loss uniformly distributed over the device deteriorate its operational characteristics. These results show excellent agreement with those obtained by Chen et al. w9x. Our previous numerical and analytical results ŽFigs. 3–6. show that the DDF profile lead to the decrease of the critical energy of the coupler. This is a strong suggestion to recover the normal switching of the coupler that was degenerated by the loss ŽFig. 9.. In this situation we expect that decreasing profile for dispersion will recover the normal behavior in transmission of the device. We will compare the DDF coupler with profiles given by Eqs. Ž4. – Ž9. in the presence of loss. The effect, that the decrease of the dispersion always decrease the critical energy EC was shown in Figs. 3 and 4 where we did a numerical calculation of Eqs. Ž2. and Ž3. with no loss Ž a s 0. considering the six profiles ŽEqs. Ž4. – Ž9... The main question is: In the presence of loss, all of these profiles could reduce EC and at the same time could recover the switching
361
Fig. 11. Percentual difference between the nonlinear transmission of each profile and the transmission of the lossless case Ž DT s ŽTreference yTprofile .=100%. for channel 1 as a function of pump energy. ŽRelative to Fig. 10..
characteristic of the coupler that deteriorates because of the loss? To answer this question we choose the loss factor, a ŽdBrKm. s 15, which is a strong loss compared with the loss one has in a basic telecommunication fiber Ž0.2 dBrKm.. From Fig. 9 one can conclude that this loss in the guide lead to an increase of the critical energy from EC to f 1.075EC and the switching characteristics are quite degenerated. In Fig. 9Ža. and Fig. 9Žb. one has the compression factor for the switched pulses associated to Fig. 9. From Fig. 9Ža. one can conclude that the presence of loss is leading to strong broadening of the switched pulses. For 15 dBrKm loss, a broadening factor of 30% is observed for the switched pulses in channel 1. From Fig. 9Žb. the switched pulse in channel 2 already present some broadening even in the lossless Table 1 Dispersion profiles associated with Fig. 10 Ž a ŽdBrKm. s15. and maximum difference obtained with the lossless case Ž ps1, a ŽdBrKm. s 0.
Fig. 10. Switching characteristics of channel 1 obtained from the numerical solution of Eqs. Ž2. – Ž9. with K s1, LC s p r2, a s15 dBrKm for the constant, exponential, Gaussian, hyperbolic, linear and logarithm profiles, respectively and b s1, a s 0 for the reference coupler.
Profile
b
Maximum difference Ž%.
Constant Exponential Gaussian Hyperbolic Linear Logarithm
1.15 1.89 7.38 1.70 2.35 3.75
5.11 2.91 5.60 2.47 3.41 3.87
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case device. However the presence of loss lead to an increase of the broadening. Our main interest here is to find a dispersion profile that could recover the original critical energy and the characteristics of switching of the coupler. To do this we solve Eqs. Ž2. and Ž3. with loss Ž a ŽdBrKm. s 15. and considering the DDF profiles ŽEqs. Ž4. – Ž9... We did a large number of numerical simulations for each profile to find the exact value of b to decrease the critical power from f 1.075EC to EC . In Fig. 10 one has
the nonlinear transmission for each profile comparing with the original behavior of the device Ž a s 0, p s 1, b s 1 lossless case. as a reference. From Fig. 10 one can conclude that the worse behavior is associated to the constant profile. It is quite distant from the lossless behavior. The exponential profile Ž b s 1.89., hyperbolic profile Ž b s 1.70. and the linear profile Ž b s 2.35. gave reasonable behavior. However when comparing with the lossless case Ž b s 1, a s 0. one can notice that the number of
2q
Fig. 12. Compression factor C1 , associated to Fig. 10 ŽEq. Ž11.. for the DDF coupler with K s 1, LC s pr2, a s 15 dBrKm Žthe value of b associated to each profile is given in Fig. 10.. Data obtained numerically Žsee text.. Ž1. Incident Žchannel 1. and transmitted pulses in channel 1 for the DDF coupler with constant and hyperbolic profile Ž b s 1.15, b s 1.70 and ErEC s 2. and the reference coupler Ž b s 1 and ErEC s 2. Žassociated to Fig. 12.. Ža. Compression factor C 2 , associated to Fig. 10 ŽEq. Ž11.. for the DDF coupler with K s 1, LC s pr2, a s 15 dBrKm Žthe value of b associated to each profile is given in Fig. 10.. Data obtained numerically Žsee text. and reference coupler Ž b s 1, a s 0.. Ža.1. Incident Žchannel 1. and switched pulses in channel 2 for the DDF coupler with constant and hyperbolic profile Ž b s 1.15 and b s 1.70 and ErEC s 0.65. and the reference coupler Ž b s 1 and ErEC s 0.65. Žassociated to Fig. 12Ža...
M.G. da SilÕa et al.r Optics Communications 171 (1999) 351–364
peaks and the position of the peaks of transmission does not fit with the reference case. However the hyperbolic profile Ž b s 1.70. is almost coincident with the lossless device Ž p s 1, a s 0.. We can conclude that it is quite possible to project a dispersion decreasing profile to recompose the intrinsic loss effect on the switching characteristics of the NLDC. To have an exact idea of the real recomposition of the nonlinear transmission with the dispersion decreasing profile, we measure the percentual difference between the transmission of each profile and the transmission of the lossless case in Fig. 10. In Fig. 11 one has this percentual difference between the transmission of the lossless case and transmission of the each profile as a function of pump power. From Fig. 11 is quite clear that the differences in transmission for the constant profile Ž b s 1.15. is around 5% around 1.4 EC and decreases to 1% around 2 EC . The Gaussian profile Ž b s 7.38. presents differences around 6% for 1.1 EC . For this scale the maximum difference in the hyperbolic profile Ž b s 1.70. is around 2.47% in Fig. 11. In Table 1 one has the b values for each profile and the maximum error observed in Figs. 10 and 11. From this table one can conclude that to recover the same critical power of the lossless case, the Gaussian profile present the highest value of the b parameter. The lowest error, when comparing with the lossless case is presented by the hyperbolic profile where the difference in transmission is below 2.47%. In Fig. 12 and Fig. 12Ža. one has the compression factor in channels 1 and 2 associated to Fig. 10. In Fig. 12 the lossless case Ž p s 1, a s 0. present compression around 1. As discussed before, with the presence of loss Ž a ŽdBrKm. s 15. the switched pulses present broadening ŽFig. 9Ža. and Fig. 9Žb.. around 30%. Fig. 12 shows the effect of loss and dispersion profile together associated to Fig. 10 Žall the couplers with the same critical energy.. One can conclude that the constant profile was not able to recover the original temporal profile of the input pulse, it present broadening around 20%, C f 0.83. However the hyperbolic profile present a good recovery of the pulse shape. For the hyperbolic profile a slight compression around 22% is observed Ž C f 1.29.. The Gaussian profile present a strong shaping effect with compression factor going up to 5.5. In Fig. 12.1 one has the transmitted pulse in channel 1 for the input
363
energys 2 EC , associated to Fig. 12, for the reference coupler and the couplers with constant and hyperbolic profile. From this figure is clear the broadening of the transmitted pulse from constant DDF coupler. The transmitted pulse from the hyperbolic DDF coupler present a quite similar shape with the reference pulse with a little compression. The same kind of effect was observed for channel 2 ŽFig. 12Ža. and Fig. 12Ža.1... Looking for the channel 2, at low pump power, the switched pulse present temporal broadening. In Fig. 12Ža.1. one has the switched pulses for E s 0.65EC . The switched pulse from the hyperbolic DDF coupler present the closest shape compared with the reference coupler. One can conclude that with the decreasing dispersion profile one can recover almost perfectly the original switching behavior of the NLDC, like the transmission characteristics and the original temporal shape of the original coupler that was destroyed by the intrinsic optical loss of the device. 6. Conclusions We present an analytical and numerical investigation of the propagation and the switching of fundamental solitons in a two-core nonlinear fiber coupler constructed with dispersion decreasing fiber ŽDDF. with a variational method using the Lagrangian density formulation. The analytical solutions were directly obtained from the coupled nonlinear Schrodin¨ ger equations. Our simulations studies six different profiles, linear, hyperbolic, exponential, logarithm, constant and Gaussian. The transmission characteristics, the critical energy and the compression factor of first order solitons obtained by the analytical procedure agree well with the results from numerical analysis. Twin core nonlinear directional couplers that include loss is also examined. The inclusion of loss results in the increase of the critical energy of the coupler and broadening of the switched pulses, while the switching characteristics deteriorate. To compensate for this worse behavior, associated to the loss, we have investigated the effect of the DDF profile on the performance of the coupler. We have shown that appropriate shaping of the DDF profile is quite effective to recover, almost completely, the original switching behavior associated to the lossless situation. It is concluded that the hyperbolic disper-
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sion profile is nearly optimum to the recovery of the lossless switching behavior. Comparing both techniques we can say that the Lagrangian formulation provide very good description of soliton switching in the DDF coupler. The use of this technique to the study of switching of ultrashort optical pulses in optical devices will be very useful. The study of soliton switching in DDF nonlinear fiber couplers provides possibilities for achieving, high efficiency in ultrafast all-optical signal processing, especially for optical switches and optical transistors.
Acknowledgements We thank CNPq ŽConselho Nacional de Desen., FINEP volvimento Cientıfico e Tecnologico ´ ´ ŽFinanciadora de Estudos e Projetos. and FUNCAP ŽFundac¸ao ˜ Cearense de Amparo a Pesquisa. for the financial support.
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