On the profile of pulses generated by
fiber lasers:the highly-chirped positive
dispersion regime (similariton)
Pierre-André Bélanger
Département de Physique, Génie Physique et Optique,
Centre d’Optique Photonique et Laser (COPL),
Université Laval, Québec (Qc), Canada G1K 7P4
pabelang@phy.ulaval.ca
Abstract: I model the nonlinear fiber laser using an expanded GinzburgLandau equation (GLE) which includes the self-steepening (SS) and
intrapulse Raman scattering (IRS) effects. I show that above a minimum
value of the Raman effect, it is possible to find two chirped solitary pulses
for the laser system. The smaller chirped solitary wave corresponds to the
dispersion-managed (DM) regime whereas the larger chirped solitary wave
corresponds to the so-called similariton regime.
© 2006 Optical Society of America
OCIS codes: (140.3510) Lasers, fiber;(140.4050) Mode-locked lasers;(060.5530) Pulse propagation and solitons.
References and links
1. H. A. Haus, E. P. Ippen, and K. Tamura, “Additive-pulse mode-locking in fiber lasers,” IEEE J. Quantum Electron. 30, 200–208 (1994).
2. P. -A. Bélanger, “On the profile of pulses generated by fiber lasers,” Opt. Express 13, 8089–8096 (2005).
3. F. Ö. Ilday, J. R. Buckley, W. G. Clark, and F. W. Wise, “Self-similar evolution of parabolic pulses in a laser,”
Phys. Rev. Lett. 92, 213902 (2004).
4. F. Ilday, F. Wise, and F. Kaertner, “Possibility of self-similar pulse evolution in a Ti:sapphire laser,” Opt. Express
12, 2731–2738 (2004).
5. B. Ortaç, A. Hideur, M. Brunel, C. Chédot, J. Limpert, A. Tünnermann, and F. Ö. Ilday, “Generation of parabolic
bound pulses from a Yb-fiber laser,” Opt. Express 14, 6075–6083 (2006).
6. N. Akhmediev, and A. Ankiewitz, Solitons, nonlinear pulses and beams (Chapman and Hall, London, 1997).
7. Z. Li, L. Li, G. Zhou, and K. H. Spatschek, “Chirped femtosecond solitonlike laser pulse form with self-frequancy
shift,” Phys. Rev. Lett. 89, 263901 (2002).
8. D. Anderson, M. Desaix, M. Karlsson, M. Lisak, and M. L. Quiroga-Teixeiro, “Wave-breaking-free pulses in
nonlinear optical fibers,” J. Opt. Soc. Am. B 10, 1185–1190 (1993).
9. A. Ruehl, O. Prochnow, D. Wandt, D, Kracht, B. Burgoyne, N. Godbout, and S. Lacroix, “Dynamics of parabolic
pulses in a ultrafast fiber laser,” Opt. Lett. 31, 2734–2736 (2006).
10. M. E. Fermann, V. I. Kruglov, B. C. Thomsen, J. M. Dudley, and J. D. Harvey, “Self-similar propagation and
amplification of parabolic pulses in optical fibers,” Phys. Rev. Lett. 84, 6010–6013 (2000).
11. C. Finot, G. Millot, C. Billet, and J. M. Dudley, “Experimental generation of parabolic pulses via Raman amplification in optical fiber,” Opt. Express 11, 1547–1552 (2003).
12. L. M. Zhao, D. Y. Tang, T. H. Cheng, and C. Lu, “Gain-guided solitons in dispersion-managed fiber lasers with
large net cavity dispersion,” Opt. Lett. 31, 2957–2959 (2006).
13. L. M. Zhao, D. Y. Tang, and C. Lu, “Gain-guided solitons in a positive group-dispersion fiber laser,” Opt. Lett.
31, 1788–1790 (2006).
14. I. S. Gradshteyn, and I. M. Ryzhik, Tables of Integrals, Series and Products. (Academic press, New York, 2000).
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1.
Introduction
A nonlinear laser system is a complex device that requires extensive and careful numerical simulations to optimize its operation. Distributing the linear and nonlinear elements along the axis
of the laser results in a nonlinear differential equation [1]. The solitary pulse solution of this
approximate distributed model gives an average behaviour of the pulse profile of the system
which can be helpful when carrying out numerical simulations. In a previous paper [2], I have
shown that the spectrum of the chirped solitary pulse of the distributed GL model yields a realistic temporal profile of the unchirped pulse generated by the laser. As a matter of fact, in a DM
fiber laser operating in the negative average dispersion regime, the predicted pulse width (see
Eq. (6) in Ref. [2]) was confirmed by several experimental results. This particular operating
point (ΓL = 2π , β = 1) also implies that all these systems will generate pulses with the same
peak power for any average negative dispersion. For the positive average dispersion regime, no
specific operating point emerges from this model. Hence, in this paper, after extending the GL
distributed model by including higher-order nonlinear terms such as the SS and IRS effects, I
shall show that the resulting solution of this equation leads to two specific values for the chirp
parameter β . The solitary pulse having the largest chirp parameter β has a spectral profile typical of the profile observed experimentally and numerically in the so-called similariton regime
[3-5].
2.
The distributed laser model
The typical fiber laser cavity consists of two concatenated fiber segments with normal dispersion, gain and anomalous dispersion followed by a mode-locking mechanism. In order to study
numerically this complex nonlinear system, a master differential equation [6] has been derived
and is commonly known as the extended nonlinear Schroedinger equation (ENLS) where the
different physical effects are uniformly distributed along the propagation axis x. Here, in order
to study the propagation
of very short pulses, I have included the third-order dispersion (TOD)
term (β3 ), SS ωγ0 and the IRS (TR ) effects. The second order disperion (SOD) term β2 is
complex β2 = β̄ − 2igT02 , where β̄ is the group velocity dispersion (GVD), T0 is the inverse
gain bandwidth, g stands for the gain and l for the loss. The nonlinear parameter γ is complex (γ = γ0 (1 + iε0 )) where γ0 is the Kerr nonlinearity and where the small saturable absorber
parameter ε0 stands for an approximation of the mode-locking mechanism. Following the presence of gain in this ENLS, this master equation can be called an extended Ginzburg-Landau
equation (EGLE) and is given by:
iVx +
β2
i γ0
iβ3
Vττ − i(g − l)V − γ |V |2 V +
Vτττ − (|V |2 V )τ + γ0 TR (|V |2 )τ V = 0
2
6
ω0
(1)
As it is the case for the GL differential equation, the EGLE supports a chirped solitary wave
solution [7] and is given by:
V (τ , x) = V0 {sech [α (τ + bx)]}1−iβ exp [i (aτ − Γx)]
(2)
The main difference between the solitary wave given by Eq. (2) and the usual solitary wave of
the GLE comes from the phase shift term (a) relative to the central frequency ω0 . In appendix
A, it is shown how the six charateristic parameters (V0 , β , α , b, Γ, a) of the solitary pulse are
related to the several internal parameters of the laser system. I also show, in appendix A, that
eight real equations can be formally derived for the six parameters. However two complex
compatibility relations impose certain restrictions on the internal parameters. I suppose, at this
stage, that those restrictions can be supported by the dynamic operation of the laser system.
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In a previous analysis [2] of such a system where the SS and the IRS effects were not included
in the model, I was able to find out that the solitary pulse had a chirp parameter β ∼
= 1 to explain
most of the experimental observations when the average dispersion is negative β̄ L < 0 . For
positive average dispersion β̄ L > 0 , no fixed operation point for the chirp parameter β could
be found neither from the model nor from the experiments. Here this extended model yields a
direct access to a unique chirp parameter for the different operating regimes. This result comes
from the analysis of Eq. (A3) after seperating the real and imaginary parts and assuming that
β3 is real. When this is carried out, the following characteristic equation can be deduced for the
chirp parameter β :
β β2 +9
TR ω0 =
(3)
4 (β 2 − 1)
According to Eq. (3), three chirp parameters β can be calculated from this third-order algebric
relation. Here, in the framework of the laser system under study, the IRS term TR ω0 must be
positive and the chirp parameter must also be positive [see Eq. (A8)]. Hence, as shown in Fig.
(1), when TR ω0 > 1.65, two positive chirp parameters β are found for a given value of the IRS
effect. Assuming that the
√saturable absorber parameter ε0 is very small, we can deduce from
Eq. (A11) that for β < 2, the average dispersion must be negative. Then, according to Eq.
(3), DM operation in the negative average dispersion regime is always possible if th IRS term
is larger than 4 (TR ω0 ≥ 4). Assuming that the Raman term TR = 3 f s and 5 f s respectively,
the Raman parameter TR ω0 are epproximately equal to 3.64 and 6 for a fiber laser operating at
1550nm whereas for a laser operating at a wavelength of 1030nm, the Raman parameter is 5
and 9 respectively. When TR ω0 < 4, two operating
modes of the laser can be achieved in the
same positive average dispersion regine β̄ L > 0 :one with a low chirp parameter β and the
other corresponding to a large value of the chirp parameter β . For the case corresponding to
TR ω0 > 4, the operation of the laser is in the DM negative average dispersion regime or in a
highly-chirped positive dispersion regime. The case corresponding to the situation where β = 0
is simply the pure solitonic regime, which occurs in the anolamous dispersion region. Figure
(1) summarises well what can be deduced from Eq. (3) and more specifically, we show the
operating regimes corresponding to the case where TR ω0 = 5.
10
9
8
7
T ω
R 0
6
5
DM−REGIME
SIMILARITON
4
3
2
1
SOLITON
0
0
√
2
β̄ < 0 ←− −→ β̄ > 0
5
10
15
20
25
β
Fig. 1. Chirp parameter β as a function of the IRS effect TR ω0 . The different operating
regimes can be seen for TR ω0 = 5. The solitonic regime corresponding to the case where
β = 0 is also included in the figure.
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3.
The highly-chirped (similariton) regime
For a realistic IRS parameter (TR ω0 > 5), Eq. (3) always yields one large possible value of
the chirp parameter β . The chirped solitary pulse will be obviously highly-chirped for large
values of β . As from now, the discussion will be restricted to this situation and I will show
that this solitary wave solution has most of the features of the so-called similariton operating
regime which occurs for large positive average dispersion in a fiber laser. Figure (2) shows the
amplitude of the spectrum [(a) and (c)] of the solitary pulse (Eq. (2) of Ref. [2]) for different
values of the chirp parameter β . It is to be noted that the amplitude of the various depicted
spectra, while being arbitrary, ensures that all the shown spectra have the same energy. The
spectrum tends to a nearly rectangular shape as the value of chirp parameter β increases. This
profile appears to be close to the shape of the similariton previously measured and calculated
ν
[9] in a fiber laser where a clear and steepest decay is observed near ν = 2f . We also depict in
Fig. (2) the phase profile [(b) and (d)] for each value of β .
(b)
(a)
1
10
β =0
8
¯
¯
¯
¯
¯V̂ (ν)¯ 6
β =1
0
−1
β =2
φ(τ )
β =4
−2
4
−3
2
−4
0
−1.5
−1
−0.5
0
ν
α
0.5
1
1.5
(c)
3
−2
0
2
ατ
4
(d)
1
β =10
0
β =20
¯
¯
¯
¯
¯V̂ (ν)¯ 2
−5
−4
−1
φ(τ )
−2
β =40
β =60
1
−3
−4
0
−15
−10
−5
0
ν
α
5
10
15
−5
−1.5
−1
−0.5
0
ατ
0.5
1
1.5
Fig. 2. Spectral [(a) and (c)] (given by Eq. (2) of Ref. [2]) and phase distribution ((b) and
(d)) for different values of β ranging from 0 − 60. The amplitude of the various depicted
spectra, while being arbitrary, ensures that all the shown spectra have the same energy.
The spectral full-width-half-maximum (FWHM) of the pulse is given by [2]:
h
π i
2α
arcsinh
cosh
β
π2
2
For large values of the chirp parameter β , Eq. (4) can be approximated by:
νf =
(4)
αβ
νf ∼
(5)
=
π
In a real fiber system [9], the pulse will propagate through the mode-locking device before
being ejected from the resonator. In the present work, I have modelled the saturable absorber
with a small nonlinear parameter ε0 . In appendix B, it is shown that the passage of a rectangular
spectral profile V̂0 (ν ) through the absorber will be transformed to:
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V̂ (ν ) = V̂0 exp
V̂ (ν ) =
−1.386ν 2
ν 2f
0
νf
2
νf
|ν | >
2
|ν | ≤
f or
f or
(6a)
(6b)
The spectral amplitude is shown in Fig. 3(a) and is typical of the one calculated and measured in
the laser system of Refs. [5, 9]. Now, assuming that the pulse is chirp-free outside the resonator,
the temporal profile is depicted in Fig. 3(b).
(a)
(b)
1
1
0.9
0.8
0.8
0.7
0.6
¯
¯ 0.6
¯
¯
¯V̂ (ν)¯
V (τ )
0.5
0.4
0.4
0.2
0.3
0.2
0
0.1
−0.2
0
−2
−1
0
1
2
−10
ν
νf
−5
0
5
10
τ
τf
Fig. 3. Spectral (a) profile given by Eqs. (6a) and (6b) and corresponding temporal profile
(b) calculated from the inverse Fourier transform of Eq. (6a).
For Eqs. (6a) and (6b), the calculated time-bandwidth product (TBP) is:
τ f ν f = 0.9309
(7)
The so-called similariton regime was first introduced by Ilday et al. [3] and was inspired at first
from the idea of using the parabolic pulse asymptotic solution of the nonlinear Schroedinger
equation (NLS) which was first reported by Anderson et al. [8]. However, they were well aware
that this asymptotic solution cannot be compatible with the periodic boundary condition of a
laser resonator. According to the present model, the average pulse profile inside the laser is
given by:
V (τ ) = V0 sech (ατ ) exp {−iβ ln [sech (ατ )]}
(8)
where the small frequency shift effect a has been neglected. In order to show that this profile is
more realistic that a parabolic one in such a highly-chirped regime, I shall compare it with the
results reported recently by Ruehl et al. [9]. In this paper, the authors model the laser system
using the usual split-step Fourier algorithm and they use a two-level-model to calculate the
parameters of the gain fiber. The calculated temporal phase and amplitude profile corresponds
to Fig. 1(a) of their article. The pulse width of 6.5ps corresponds to a root-mean-square (RMS)
width of 2.76ps for a Gaussian profile and to 3.34ps for the chirped pulse distribution given by
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Eq. (8) which is to be compared to their calculated RMS width of 3.1ps. For large values of
t, their phase profile is linear with a slope close to αβ = 19 whereas for the pulse distribution
given by Eq. (8), the predicted slope will be given by αβ and when combined with the result
of Eq. (5), for large β , gives:
αβ = πν f = 22
(9)
where I have used their value of ν f = 7 for their calculated spectral width. The measured spectrum appears to be shifted as inferred by the present model through the presence of the TOD.
From the result of Eq. (9) and from the knowledge of the pulse width, the value of β can
be estimated to be around 80 which implies an IRS term of TR ω0 ≈ 20. It is important here
to understand that the distributed model gives only an averaged information of the pulse that
propagates in the laser and no accurate information on the local temporal and phase profiles is
available. However, as pointed out and discussed in a previous paper [2], the spectrum is generally less affected by the nonlinear effect during the propagation and therefore its profile can
be very realistic of the physical situation. Finally, for very large values of the chirp parameter β
and assuming that the saturable absorber term ε0 and the frequency chirp to be very small, the
main characteristics of the chirped averaged similariton pulse can be obtained approximately
from the equations of appendix A:
(γ0 L)V02 = (ΓL)
(10)
where the total phase shift for a laser of length L is given by:
(ΓL) =
π 2 ν 2f
2
β̄ L
(11)
2
assuming that for a parabolic gain profile, the gain parameter 2gT0 can be approximated by:
2gLT02 ∼
=
0.05
νg2
(12)
where νg is the frequency gain bandwidth. Using Eqs. (A11) and (11), it can be shown that:
ν 2f
νg2
12 (ΓL)
∼
=
β
(13)
Refering again to the laser system of Ref. [9], with β̄ L = 0.013, Eq. (13) yields a phase shift
(ΓL = π ). With the estimated chirp parameter β = 80, we can predict via the use of Eq. (13) that
the gain frequency width should be 10T Hz which is close to the gain baindwidth of 12.7T Hz
used in Ref. [9]. In a recent paper by Zhao et al. [12] on the propagation of gain-guided solitons in a DM laser operating with a large positive dispersion, the authors approximately report
the experimental observation of the same typical spectrum. Finally, following the numerical
integration of the GL equation, they have obtained a similar type of spectrum [13] as the one
described analytically in the present paper.
4.
Conclusion
After including the SS and IRS terms into the GL differential equation, I have shown how to
derive a characteristic equation for the chirp parameter β . This main result defines clearly three
different operating regimes for a fiber laser system. The first regime is the solitonic regime
which appears as a unique point in Fig. (1). Two other values for the chirp parameter β can
also be found to satisfy the phase of the chirped pulse solution. The smaller one defines the DM
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regime laser operation either in anomalous or normal average dispersion regime. The anomalous dispersion regime is always possible for a realistic Raman parameter and the chirp parameter will be close to 1. The normal average dispersion regime seems to
√ be possible if the
Raman parameter is very small and if the chirp parameter is larger than 2. However, for a
laser operating in a positive dispersion regime, the chirp parameter is always large and the corresponding highly-chirped pulse appears to have the characteristics features of the so-called
similariton regime recently introduced as a modification of what is commonly known as the
temporal parabolic pulse regime [3, 4, 5, 9]. The parabolic pulse is an asymptotic solution of a
pulse propagating into an amplifier and is not compatible with the periodic conditions imposed
by a laser resonator. The parabolic pulse propagating into an amplifier of gain (g − l) will have
a RMS chirp parameter C given by:
C=
0.66 (g − l) L
β̄ L
(14)
according to Ref. [10]. For the chirped hyperbolic-secant distribution, the RMS chirp parameter,
for large β , is given by:
C=
1.22 (g − l) L
β̄ L
(15)
This result shows that the laser resonator doubles the optimal chirp that the amplifier could
have achieved.The present model has thus demonstrated that it is not necessary to introduce
the self-similar temporal parabolic pulse in order to explain the similariton regime. However,
the chirped solitary pulse used here is without doubt a self-similar pulse and hence the term
similariton [11] should continue to be used to define this high energy laser regime.
A.
Appendix A
It has already been observed [7] that the chirped hyperbolic secant ansatz given by Eq. (2) is a
solitary wave solution to the EGLE given by Eq. (1) under specific conditions. After direct substitution of ansatz (2) in Eq. (1), the resulting equation can be seperated into a time-symmetrical
part and a time-antisymmetrical part. Ansatz (2) is a solution of the symmetrical differential
equation if:
2V02 γ − γω00a
α 2 (1 − iβ ) (iβ − 2) =
(A1)
(β2 − β3 a)
and

−α 2 (1 − iβ )2 = 2 
2
Γ − a2

β2 − β33 a − i (g − l)

(β2 − β3 a)
(A2)
In order to force the antisymmetrical part to satisfy ansatz (2), the two following additional
complex conditions need to be satisfied:
2
6γ0V02
β + 9 − 2TR ω0 β + 6iTR ω0
2
β3 ω0 (β + 9)
6
β3 a2
+b
β2 a −
−α 2 (1 − iβ )2 =
β3
2
α 2 (1 − iβ ) (iβ − 2) =
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(A3)
(A4)
Received 27 November 2006; accepted 28 November 2006
11 December 2006 / Vol. 14, No. 25 / OPTICS EXPRESS 12180
These four complex relations result into eight real equations for the six characteristic parameters
(V0 , a, b, Γ, α , β ) of ansatz (2). However, Eqs. (A1) and (A3) as well as Eqs. (A2) and (A4)
impose certain relations among some of the internal and external parameters of the laser system.
Here, I will assume that β3 is real (it is to be pointed out that this is not the case in Ref. [7])
and as a consequence of this assumption, Eq. (A3) can be solved and allows one to fix the chirp
parameter β relative to the Raman term namely TR such as:
β β2 +9
= TR ω0
(A5)
4 (β 2 − 1)
Next, after imposing the compatibility relation between Eqs. (A1) and (A2), one finds that the
phase shift parameter a must satisfy the following relation:
"
#
ε0 β 2 − 2
α2 β 2 + 1 β 2 + 4
a2
a
(A6)
=
−
1−
3β
6 (β 2 − 1)
ω02 ω0
The phase shift parameter a is of course assumed to be much smaller than the central frequency
2
ω0 and from here I shall neglect all the contribution in ωa 2 . Therefore, within this approximation,
the rest of the external parameters are given by:
0
(g − l) β 2 + 2
Γ=
β
γ0V02
3 (g − l) β 2 + 4
=
[3β − ε0 (β 2 − 2)]
3 (g − l)
gT0 (β 2 + 1)
β̄ a β 2 + 1 β − ε0 β 2 + 2
b=−
2β [(β 2 − 2) + 3β ε0 ]
α2 =
Finally, the two residual compatibility relations read as:
β̄ 3β − ε0 β 2 − 2
2
gT0 =
2 [(β 2 − 2) + 3β ε0 ]
3β̄ β 2 + 1 β 2 + 4
β3 ω0 =
2 (β 2 − 1) [(β 2 − 2) + 3β ε0 ]
(A7)
(A8)
(A9)
(A10)
(A11)
(A12)
Notice that if the the self-steepening is not included into the model (ω0 → ∞), Eq. [A3] fixes the
chirp parameter to β ≡ 1, and as discussed in Ref. [2], this chirp parameter (β ≡ 1) is typical
of the negative DM regime.
B.
Appendix B
The propagation of a pulse V (τ , x) propagating through a saturable absorber is modelled by the
differential equation:
Vx = ε0 γ0 |V |2 V
(B1)
Here the propagation is assumed to be on a short virtual distance x0 and it can be assumed
that the pulse profile will not change significantly during this short distance. It can be shown
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1−iβ
2
[14] that the Fourier
is related to the Fourier
transform of |V | V where V = V0 [sech (ατ )]
transform of V V̂ through the following relation:
h
2 i
i
h
(1 − iβ )2 + 2πν
α
FT |V |2 V =
V̂
(B2)
(2 − iβ ) (1 − iβ )
Taking the Fourier transform of the differential equation (B1), the output spectrum V̂ can be
related to the incoming spectrum V̂i via the following relation:
"
2 #
ε0 γ0 x0V02 2πν
α
V̂ = V̂i exp
(B3)
(2 − iβ ) (1 − iβ )
For very large values of β this relation reads as
"
V̂ = V̂i exp
ε0 γ0 x0V02
4ν 2
ν 2f
!#
(B4)
where the spectral width for large values of β has been used accordingly with Eq. (5). When β
is large, the incoming spectrum V̂i is rectangular and has been normalised to 1. Assuming that
ν f is still the FWHM of the output spectrum, it is straightforward to show that:
V̂ (ν ) = V̂0 exp
V̂ (ν ) =
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−1.386ν 2
ν 2f
0
f or
f or
νf
2
νf
|ν | >
2
|ν | ≤
(B5a)
(B5b)
Received 27 November 2006; accepted 28 November 2006
11 December 2006 / Vol. 14, No. 25 / OPTICS EXPRESS 12182