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OPTICS LETTERS / Vol. 36, No. 7 / April 1, 2011
Dissipative soliton resonance in a passively mode-locked fiber laser Edwin Ding,1,* Philippe Grelu,2 and J. Nathan Kutz1 1
2
Department of Applied Mathematics, University of Washington, Seattle, Washington 98195-2420, USA Laboratoire Interdisciplinaire Carnot de Bourgogne, UMR 5209 CNRS, Université de Bourgogne, 21000 Dijon, France *Corresponding author:
[email protected] Received January 21, 2011; accepted February 17, 2011; posted March 1, 2011 (Doc. ID 141492); published March 25, 2011 The phenomenon of dissipative soliton resonance (DSR) predicts that an increase of pulse energy by orders of magnitude can be obtained in laser oscillators. Here, we prove that DSR is achievable in a realistic ring laser cavity using nonlinear polarization evolution as the mode-locking mechanism, whose nonlinear transmission function is adjusted through a set of waveplates and a passive polarizer. The governing model accounts explicitly for the arbitrary orientations of the waveplates and the polarizer, as well as the gain saturation in the amplifying medium. It is shown that DSR is achievable with realistic laser settings. Our findings provide an excellent design tool for optimizing the mode-locking performance and the enhancement of energy delivered per pulse by orders of magnitude. © 2011 Optical Society of America OCIS codes: 140.4050, 060.5530.
Ultrashort lasers have a wide range of practical applications in such diverse fields as material processing, biology, medicine, telecommunications, and nuclear physics. The generation of ultrashort pulses inside a laser cavity relies on a mode-locking mechanism [1,2]. Among the various possible mechanisms, the use of nonlinear polarization evolution in fiber lasers has become one of the standards in commercially developed ultrafast devices [3]. Recent advances have shown that by operating the fiber laser entirely in the normal dispersion regime, the output pulse energy could be increased from about 0:1 nJ in the anomalous dispersion regime to over 20 nJ [4]. Further enhancement of the pulse energy is, however, severely restricted by the multipulsing instability (MPI) [5,6]. The mode-locking dynamics in such laser cavities can be theoretically modeled by the cubic-quintic Ginzburg-Landau equation (CQGLE). Originally proposed as a qualitative model [1,2,7], the CQGLE model can now be directly connected to the parameters of the experimental setup [8–10]. Using this theoretical platform while taking into account gain saturation, in this Letter we study quantitatively the mode-locking dynamics in the laser cavity shown in the top panel of Fig. 1, and demonstrate that the energy limiting MPI can be circumvented by means of the so-called dissipative soliton resonance (DSR) [11]. This allows for a significant (orders of magnitude) increase in the mode-locked pulse energy along with engineering guidelines for achieving such performance. The onset of MPI as a function of increasing intracavity energy is a well-known physical phenomenon that has been observed in a myriad of laser cavities [1]. By carefully engineering the system parameters, it is possible to circumvent MPI in favor of bifurcating to high-energy solutions of the governing system [5,12]. The development of this theoretical idea, in order to provide experimental guidelines, is the primary focus of this manuscript. For the laser cavity shown in Fig. 1, an increase in pulse energy up to a factor of 2 can be achieved by simply orienting the waveplates and polarizer appropriately [10]. However, when the full set of cavity parameters, which 0146-9592/11/071146-03$15.00/0
includes chromatic dispersion, is taken into account, it is possible to find a certain region in the laser cavity’s vast parameter space for which the mode-locked pulse becomes wider instead of splitting into multiple pulses as the cavity energy is increased. This specific method of circumventing MPI is referred to as DSR [11]. Compared to the previous works on DSR, this study presents two
Fig. 1. (Color online) Top: schematic representation of a ring cavity laser that includes quarter-waveplates (QWP), passive polarizer, half-waveplate (HWP), ytterbium-doped amplification, and output coupler. The Yb-doped section of the cavity is fused with standard single-mode fiber (SMF) and treated in a distributed fashion. The angles α1 , α2 , α3 , and αp can all be measured with reasonable accuracy. Bottom: normalized coefficients of the CQGLE as a function of α1 at α2 ¼ 0:16π, α3 ¼ 0:63π, αp ¼ 0, and K ¼ 0:1. © 2011 Optical Society of America
April 1, 2011 / Vol. 36, No. 7 / OPTICS LETTERS
important additional features that pave the way for experimental investigations. First, the parameters in the governing model are explicitly related to the waveplate/polarizer angles, allowing for a direct comparison between theory and experiment. Second, a saturable gain instead of a constant gain is used. This provides a more physically realistic picture that takes into account a finite pumping power budget [1,2]. The CQGLE is the standard theoretical model describing the averaged pulse dynamics in a ring cavity laser. The relationship between the CQGLE parameters and the cavity parameter was first derived by Komarov et al. [6,8] and later generalized by Ding and Kutz [9]. The CQGLE model in dimensionless form is given by [10] D ψ þ jψj2 ψ þ νjψj4 ψ ¼ igð1 þ τ∂2t Þψ − iδψ 2 tt þ iβjψj2 ψ þ iμjψj4 ψ: ð1Þ
The coefficient of the cubic Kerr nonlinearity has been normalized to unity without loss of generality. In the above formulation, ψ represents the complex electric field envelope circulating in the laser cavity, while z and t denote the propagating distance and the retarded time, respectively. D is the averaged cavity dispersion, which is positive for anomalous dispersion and negative for normal dispersion. Here, we only consider the latter case as normal dispersion leads to pulses with higher energies [3,4]. The parameter ν is the quintic modification to the nonlinear refractive index arising from the waveplate/ polarizer interaction with the electric field. The righthand side of the equation captures all the dissipative effects. Here, g is the gain of the laser cavity, with its bandwidth corresponding to τ−1 . The parameter δ is the total linear loss of the cavity, while β (positive) and μ (negative) provide the cubic-quintic saturable absorption required for intensity discrimination and stable pulse formation. The parameters in the CQGLE are related explicitly to the waveplate/polarizer angles (α1 , α2 , α3 , and αp , see Fig. 1) through [6,8–10] δ ¼ Γ − log jQð0Þj; γ ¼ 1 þ ImðQ0 ð0Þ=Qð0ÞÞ;
1 Q ¼ fe−iK ½cosð2α2 − 2α3 − αp Þ þ i cosð2α3 − αp Þ 2 × ½i cosð2α1 − αp − wÞ − cosðαp − wÞ þ eiK ½sinð2α2 − 2α3 − αp Þ − i sinð2α3 − αp Þ × ½sinðαp − wÞ − i sinð2α1 − αp − wÞg;
g ¼ g0 ; g¼
2g0 : 1 þ ∥ψ∥2 =e0
ν ¼ Im½ðQð0ÞQ00 ð0Þ − Q02 ð0ÞÞ=Q2 ð0Þ=2γ 2 ;
300
|ψ|
250 0 −45
200
μ ¼ Re½ðQð0ÞQ00 ð0Þ − Q02 ð0ÞÞ=Q2 ð0Þ=2γ 2 ;
150
where Γ represents the loss of the single-mode fiber (SMF) portion alone, and
45
2 1
Chirp
50
ð2eÞ
0
t
ð2cÞ
ð2dÞ
ð4bÞ
2.5
350
100
β ¼ ReðQ0 ð0Þ=Qð0ÞÞ=γ;
ð4aÞ
The first model represents a constant gain strength g0 , while the second model describes a saturable gain with e0 being the saturation energy. Figure 2 shows the stable mode-locked solution of the CQGLE as a function of cavity dispersion D, in the case of constant gain. As the dispersion approaches the critical value D ¼ −1:4, the temporal width of the mode-locked pulse tends to broaden indefinitely while the pulse amplitude approaches an upper limit, which can be calculated from the system parameters [11], thus forming a flat-top structure. As a consequence of this behavior, the pulse which R ∞ energy, jψj2 dt, tends to is measured by the L2 -norm ∥ψ∥2 ¼ −∞ increase infinitely. The pulse dynamics shown in Fig. 2
ð2aÞ
ð2bÞ
ð3Þ
with w ¼ ðjψj2 sin 2ðα1 − αp ÞÞ=3, and K being the fiber birefringence coefficient. We assume Γ ¼ 0:1 in what follows. In Eqs. (2), the derivatives are taken with respect to the field power jψj2 . The bottom panel of Fig. 1 shows the cubic-quintic nonlinearities as functions of the quarter-waveplate angle α1 , with the other parameters being fixed. Mode-locking dynamics and in particular DSR, will be explored in regions where β > 0 > μ. Other regions are considered as physically irrelevant because either the pulse will experience a blow-up in amplitude, or there is a lack of intensity discrimination [9]. Two gain models are considered in this work:
Energy
iψ z þ
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0 −1.4
0 −1
−1.35
D
−1.3
−2 −45
0
45
t
Fig. 2. (Color online) DSR in the case of constant gain. Left: pulse energy ∥ψ∥2 as a function of D, with α1 ¼ 0:7863π, α2 ¼ 0:3π, α3 ¼ αp ¼ 0, and K ¼ 0:1. The rest of the parameters are picked such that g0 − δ ¼ −0:05 and g0 τ ¼ 0:4. Right: the corresponding pulse shape (top) and frequency chirp profile (bottom) at D ¼ −1:31 (blue solid curves), D ¼ −1:38 (red dashed curves), and D ¼ −1:392 (green dash-dot curves).
OPTICS LETTERS / Vol. 36, No. 7 / April 1, 2011 2.5
200
Unstable Stable
Energy
150
|ψ|
D = −1.6 0 −40
100 D = −1.2 50 D = −0.8 0
0
100
e0
200
0 t
40
0
40
2
Chirp
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0 −2 −40
t
Fig. 3. (Color online) DSR in the case of saturating gain. Left: pulse energy ∥ψ∥2 as a function of e0 at different values of D, with g0 ¼ 2:3991, τ ¼ 0:1667, and the rest of the parameters being the same as those used in Fig. 2. Right: the corresponding pulse shape (top) and frequency chirp profile (bottom) at e0 ¼ 5 (blue solid curves), e0 ¼ 58 (red dashed curves), and e0 ¼ 180 (green dash-dot curves) along the D ¼ −1:6 line, respectively.
exhibit all the essential features of DSR found in Ref. [11], including the nonzero linear frequency chirp (negative derivative of the phase) across the pulse profile. This first simulation result confirms that the DSR phenomenon is achievable with realistic waveplate/polarizer settings in the laser cavity. In order to investigate the energy limiting effects of the saturable gain dynamics Eq. (4b) on the DSR, we use both D and the saturating energy e0 as control parameters with the results summarized in Fig. 3. Usually for a fixed dispersion, the mode-locked pulse will become unstable when e0 is too large. Such an instability is usually characterized by a Hopf bifurcation and marks the onset of MPI [5]. The cavity dispersion D is the crucial factor in determining the dominant effect in the competition between DSR and MPI. Specifically, there is a critical limit D ¼ Dc , such that the system favors DSR when D < Dc and MPI when D > Dc . For the parameters considered in Fig. 3, we found (numerically) that Dc ≈ −1:2. Consider, for instance, the case where D ¼ −1:6 < Dc . The gaussian-looking pulse (blue solid curve) at low e0 values first grows in amplitude until a saturating amplitude is reached (red dashed curve), and then in width to form a high-energy, flat-top structure (green dash-dot curve) at high e0 values. The observed transformations in pulse shape and chirp profile are signatures of the DSR, although now infinite pulse energy cannot be achieved. However, the physically unrealistic infinite pulse energy solutions of the constant gain DSR were key for motivating the present work with saturating, finite energy behavior. Remarkably, the onset of MPI is not ob-
served even when the saturating energy is as large as e0 ¼ 200. On the other hand, the Gaussian-looking modelocked pulse becomes unstable long before the formation of the high-energy, flat-top pulse when D ¼ −0:8 > Dc . The simulations here show that DSR can be used as an effective mechanism to circumvent MPI, provided the dispersion is appropriately engineered. To summarize, we have studied the phenomenon of DSR in the context of the CQGLE and have extended previous findings to coefficients that can be explicitly related to the settings of the ring cavity laser depicted in Fig. 1. In addition to the constant gain model, which was studied previously [11], DSR is also achievable in the physically relevant case of saturating gain, but it is subjected to the onset of MPI if the cavity is not carefully engineered. Specifically, we found that there is a critical normal cavity dispersion above which the DSR phenomenon is favored over MPI. Our work may be used as a practical guideline for designing high-power lasers because the parameters considered relate directly to experiment and the suppression of the MPI. J. N. Kutz acknowledges support from the National Science Foundation (NSF) (grant DMS-1007621) and the U.S. Air Force Office of Scientific Research (USAFOSR) (grant FA9550-09-0174). Ph. Greulu acknowledges support from the Agence Nationale de la Recherche (grant ANR-2010-BLANC-0417-01-SOLICRISTAL). References 1. H. Haus, J. Sel. Top. Quantum Electron. 6, 1173 (2000). 2. J. N. Kutz, SIAM Rev. 48, 629 (2006). 3. A. Chong, J. Buckley, W. Renninger, and F. Wise, Opt. Express 14, 10095 (2006). 4. A. Chong, W. H. Renninger, and F. W. Wise, Opt. Lett. 32, 2408 (2007). 5. F. Li, P. K. A. Wai, and J. N. Kutz, J. Opt. Soc. Am. B 27, 2068 (2010). 6. A. Komarov, H. Leblond, and F. Sanchez, Phys. Rev. A 68, 033815 (2003). 7. J. D. Moores, Opt. Commun. 96, 65 (1993). 8. A. Komarov, H. Leblond, and F. Sanchez, Phys. Rev. E 72, 025604R (2005). 9. E. Ding and J. N. Kutz, J. Opt. Soc. Am. B 26, 2290 (2009). 10. E. Ding, E. Shlizerman, and J. N. Kutz, “A generalized master equation for high-energy passive mode-locking: the sinusoidal Ginzburg–Landau equation,” IEEE J. Quantum Electron. (to be published). 11. Ph. Grelu, W. Chang, A. Ankiewicz, J. M. Soto-Crespo, and N. Akhmediev, J. Opt. Soc. Am. B 27, 2336 (2010). 12. N. Akhmediev, J. M. Soto-Crespo, and Ph. Grelu, Phys. Lett. A 372, 3124 (2008).
