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1 October 1996
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OPTICS COMMUNICATIONS
ELSEVIER
Optics Communications 130 (1996) 245-248
Dual-frequency pulses in fiber lasers J.M. Soto-Crespo a~‘, V.V. Afanasjev b, N.N. Akhmediev b, G.E. Town c a Institute de dptica, C.S.I.C., Serrano 121, 28006 Madrid, Spain b Optical Sciences Centre, The Australian National University, Canberra 0200 ACT, Australia c Department of Electrical Engineering, The University of Sydney, Sidney, NSW 2006, Australia
Received 16 February 1996; accepted 28 March 1996
Abstract We analyze pulse propagation in a so&on fiber laser, modelled by the complex Schriidinger-Ginzburg-Landau equation. It is shown that this equation admits two previously unknown forms of stable stationary solutions. The first of them is a
dual-frequency pulse, which propagates as a single unit, but has two symmetric peaks in the spectrum. The second one is a moving pulse, with the asymmetric double-hump spectrum.
There is a large interest in soliton generation in actively and passively mode-locked fiber lasers [ l71. Several types of mode-locked lasers have been demonstrated, e.g., figure-of-eight laser [ 2,3], polarization mode-locked laser [4-6,8], and slidingfrequency laser [7]. Up to date, the mode-locked soliton fiber lasers can generate pulses with a wide range of durations, pulse powers, and repetition rates. However, the dynamics and stability of such lasers are yet to be fully understood. Theoretical studies of the soliton fiber lasers are based upon the nonlinear Schrodinger-GinzburgLandau equation (GLE). Different forms of this equation have been used, including the cubic GLE [ 91, cubic GLE with saturation [ 10,111, quintic GLE [ 12-161, as well as more complicated models [ 16181. All these models (except the cubic GLE without saturation) describe the stable pulse propagation, i.e. both the background and the pulse are stable. However, the exact analytical solutions are known only in
1E-mail address:
[email protected].
some particular cases [ 10,13,15], so any new forms of solutions are of practical interest. In this Letter we present the results of numerical studies which show that at least two new forms of pulse-like solution have been missed in previous analysis. In particular, we discover that besides the wellknown solution in the form of bell-shaped pulse (we call it the plain pulse), the GLE has stable stationary solutions in the form of composite pulses with a dual-frequency spectrum. Moreover, these two solutions can coexist in the space of parameters with the plain pulses. We write the quintic GLE in the form [ 14,161:
44 + @tt + w21cI + 4A4~ = iS$ + iel+12@ + i/3ht + i&b14$,
(1)
where z is the propagation distance, t is the retarded time, $ is the normalized envelope of the electric field, 6 is the linear gain coefficient, p is the spectral filtering, E is the nonlinear gain or loss (which describes, e.g., fast saturable absorption), p is the saturation of the nonlinear gain, and Y is the saturation of the nonlinearity.
0030-4018/96/$12.00 Copyright @ 1996 Elsevier Science B.V. All rights reserved. PII SOO30-4018(96)00279-9
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The stationary pulse solution (plain pulse) to Eq. ( 1) has been found numerically for arbitrary values of the parameters and analytically if some condition on the parameters is imposed (see Ref. [ 151 and references therein). Another type of solution, which may coexist in the parameter space with the plain pulse, is the front, also known as a shock wave (see, e.g., Ref. [ 191) . The front is an interface between the background and the continuous wave (CW) . Note that in the general case, the front moves (is uniformly translated), i.e. the solution can be presented in the form F( t - uz ). In other words, the CW spreads over the background (front velocity is positive) or background spreads over the CW (front velocity is negative). An important physical parameter of the front is the wave vector of the CW adjacent to it. Generally, it is nonzero, which means that two fronts with opposite orientation cannot match each other directly. As the front is not a stationary solution, it has been traditionally thought that it is not relevant to laser applications, where only stationary pulses are of interest. However, for the given set of parameters, the front velocity determines the stability of pulses, as the pulse solution can be stable only if the front velocity is negative. If it is positive, any pulse decays into two fronts which move in opposite directions. In the frame of the model ( 1) , this propagation is unlimited. On the other hand, in real systems the increase of the pulse duration is limited by the gain saturation. This mechanism may be responsible for the formation of the rectangular pulses with the noise structure in fiber lasers (see, e.g., Ref. [ 31 for details). We studied the stability of the plain pulse solution of Eq. ( 1) numerically and these results are presented elsewhere [ 201. In particular, we have found the threshold of the pulse stability. Near the threshold, we observed the formation and stable propagation of two new forms of stationary solution, which we call the composite pulse (CP) and the moving pulse (MP). The intensity profile of the CP and Fourier transform which we obtained from numerical simulations are shown in Fig. 1. These functions for the plain pulse which exists for the same values of parameters are also given for comparison. The CP consists of two fronts and a domain boundary between them. The study of the phase profile shows that there is an energy flow from the CP center to the fronts. Comparison with
Communications I30
(1996) 245-248
_,’
0
-10
-5
-4
I
\
\
\ 5
0 Time
-2
0 Frequency
2
10
4
Fig. 1. (a) The shapes and (b) the spectra of the composite pulse (solid lines), and the plain pulse (dashed lines). The values of parameters for this simulation are E = 1.75, 6 = -0.1, p = 0.5, p = -0.6, v = -0.1.
the plain pulse solution shows that there are many similarities between these two solutions. In particular, the top part of the CP has exactly the shape of the top part of the plain pulse. The tails of the plain pulse and the CP are also very similar. To explain this fact, we compared the propagation constants of the two structures. It appears that the difference between them is around 10%. Then the similarity between the tails follows from the linearized version of Eq. ( 1) . The spectrum of the CP is also shown in Fig. 1. It has a dip in its centre and two well-separated peaks. To explain such a structure in the spectrum, let us suppose that the full spectral width of the CP is much larger than the typical width of both the front and the plain pulse. Such structure is shown in Fig. 1 by the dashed line. In this case, the CP consists mainly of two continuous waves and the spectrum of this structure has two ‘well separated peaks. The spectral separation between the two peaks is determined by the difference between the wave-numbers of the two CW components. The spectrum of the real CP is the intermediate case between the spectrum of the hypothetical “wide” CP, which has two well-separated peaks, and spectrum of the plain pulse, which has a single peak. Note, that the spectrum of the CP resembles the spectrum of the N = 2 soliton pulse at the point of compression, although these two spectra have completely different origin. This similarity may be misleading in
J.M. Soto-Crespo et al./Optics Communications 130 (1996) 245-248
“10
5
0
-5
I
I::
247
Composite
Pulse
Mow
Time
lo T
Plain
/j Pulse
5t.~..~~~~~~.~~~~~~‘.i 1.65
-4
-2
0
2
4
Frequency
Fig. 2. (a) The shapes and (b) the spectra of the moving pulse (solid lines), and the plain pulse (dashed lines). The values of parameters for this simulation are the same as in Fig. 1.
1.70 Parameter
1.75 E
Fig. 3. The energies of the plain pulse, the composite pulse, and the moving pulse versus E. The values of the parameters arc a=-O.l,p=O.5,/~=-O.~,V=-0.1. I”“”
”
”
”
I
2.0
experimental observations. Another new solution which we have found is an asymmetric moving pulse. Traditionally, it is supposed that if the coefficient /I in front of the second-order derivative term in the r.h.s. of Eq. (1) is nonzero, then only motionless and symmetric pulse-like solutions can exist. This conclusion follows, for example, from the adiabatic perturbation theory. However, perturbation theory cannot be applied if at least one of the coefficients in the r.h.s. of Eq. (I) is not small. The intensity profile and the spectrum of the MP are given in Fig. 2. The amplitude profile is very close to the profiles of the plain pulse and the composite pulse. Note that both the left- and right-moving pulses exist, and they are mirror images, due to the symmetry of Eq. ( 1) relative to the transformation t t-) --t. The MP always moves in the direction with the pulse ahead. The spectrum of the moving pulse is also asymmetric. There are two peaks of different height and a valley between them. To understand the appearance of the composite pulses and the moving pulses, let us use the idea df plain pulses and fronts as elementary “building blocks” for more complicated structures. From this point of view, the CP is the combination of a plain pulse at the centre and two fronts placed by its sides. The moving pulse (MP) can be considered as a particular case of this structure when one of these fronts is missing (see Fig. 2). This follows from the com-
;
1.8
% E e 2 1.6
-1.0
-0.8
-0.6 Parameter
-0.4
-0.2
p
Fig. 4. The ranges of existence of the plain pulses, pulses, and moving pulses on the (,u, e)-plane.
composite
parison of the energies of the plain pulse, MP, and CP (Fig. 3)) where the difference between the plain pulse and CP energies is approximately twice the difference between the plain pulse and MP energies. Note, all three types of pulses can exist at the same values of parameters and all three of them are stable. Boundaries of existence for the CP, MP and the plain pulses in ( ,u, E)-plane are shown in Fig. 4. To the best of our knowledge, these results are the first demonstration of the coexistence of three different stable stationary pulse-like solutions of the quintic GLE for the same set of parameters. This fact is not trivial. It does not follow, for example, from the perturbation analysis around the conservative limit. The range of parameters where these pulses exist is (comparatively) wide
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and they should be observed easily. We have also studied the interaction between the different possible combinations of the moving pulse, plain pulse, and the composite pulse. In particular, it has been found that the stable composite pulse can be formed after a collision between the stationary pulse and the moving pulse. In conclusion, we have studied numerically localized structures in a laser model described by the quintic Ginzburg-Landau equation near the threshold of zero velocity fronts. We have found that besides the plain pulse solution, known before, two other types of the stationary pulses with dual-frequency spectrum can exist. Either the composite pulse, moving pulse, or plain pulse may exist at the same values of parameters. To explain the formation of these pulses, we suppose that the plain pulse and simple fronts can be considered as elementary building blocks, which can form more complicated nonlinear superpositions. This work was supported by the Australian Photonics Cooperative Research Centre (APCRC). The work of JMS-C was supported by the CICyT under contract TIC95-0563-CO5-03.
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