Generalized Master Equation for High-Energy Passive Mode-Locking: The Sinusoidal Ginzburg–Landau Equation

IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 47, NO. 5, MAY 2011 705

Generalized Master Equation for High-EnergyPassive Mode-Locking: The Sinusoidal

Ginzburg–Landau EquationEdwin Ding, Eli Shlizerman, and J. Nathan Kutz

Abstract— A generalized master mode-locking model is pre-sented to characterize the pulse evolution in a ring cavity laserpassively mode-locked by a series of waveplates and a polarizer,and the equation is referred to as the sinusoidal Ginzburg–Landau equation (SGLE). The SGLE gives a better description ofthe cavity dynamics by accounting explicitly for the full periodictransmission generated by the waveplates and polarizer. Numer-ical comparisons with the full dynamics show that the SGLE isable to capture the essential mode-locking behaviors includingthe multi-pulsing instability observed in the laser cavity anddoes not have the drawbacks of the conventional master mode-locking theory, and the results are applicable to both anomalousand normal dispersions. The SGLE model supports high energypulses that are not predicted by the master mode-locking theory,thus providing a platform for optimizing the laser performance.

Index Terms— Ginzburg–Landau equation, master mode-locking equation, mode-locked lasers, saturable absorption, soli-tons.

I. INTRODUCTION

S INCE its first proposed use in the early 90’s, the fiberring cavity laser mode-locked by a passive polarizer has

become one of the most reliable and compact sources forrobust ultrashort optical pulses [1]–[5]. Such fiber lasers offermajor practical advantages over solid state configurations,since the light is contained in a waveguide and does not requirethe optical cavity to be carefully aligned. The possibility ofusing fibers in short-pulse laser devices has motivated researchfor nearly two decades, however, due to limitations in theenergy output made very little impact compared to their solid-state counterparts. Indeed, fiber lasers have lagged well behindthe solid-state lasers in the key performance parameters – pulseenergy and duration. Recently, new insights into pulse-propagation physics [6]–[8] have provided glimpses of order-of-magnitude increases in the pulse energy and peak power infemtosecond fiber lasers. For the first time, it is now realis-tic to design short-pulse fiber devices that compete directlywith the existing solid state lasers in energy performance

Manuscript received October 25, 2010; January 20, 2011; accepted January31, 2011. Date of current version April 1, 2011. This work was supported inpart by the National Science Foundation, under Grant DMS-1007621, and theU.S. Air Force Office of Scientific Research, under Grant FA9550-09-0174.

The authors are with the Department of Applied Mathematics, University ofWashington, Seattle, WA 98195 USA (e-mail: [email protected];[email protected]; [email protected]).

Color versions of one or more of the figures in this paper are availableonline at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/JQE.2011.2112337

while offering major practical advantages and substantiallyreduced cost. However, the design and optimization of high-performance fiber devices is impeded by the so-called multi-pulsing instability (MPI) which ultimately imposes a funda-mental limitation on a single mode-locked pulse’s energy [9].To compete directly with solid state technologies, the finalorder-of-magnitude increase in energy is still required, thusmaking it critical to understand the limiting effects of the MPI.In this manuscript, the mode-locking theory using nonlinearpolarization rotation is revisited with the aim of more accu-rately capturing the periodic transmission curve known to existfrom the nonlinear polarization dynamics. Specifically, a peri-odic transmission curve, theoretically neglected in the standardmaster mode-locking theory, can be appropriately engineeredso that the MPI can be circumvented in favor of bifurcating tohigher-energy single pulse solutions, thus achieving significantincrease in mode-locking energy performance.

The underlying mode-locking mechanism considered here isthe self-amplitude modulation created by the waveplates andpassive polarizer. When appropriately oriented, the waveplatesand polarizer act as an effective saturable absorber that createsthe intensity discrimination required for shaping the circulatingpulse in the laser cavity after each round-trip. More generally,the nonlinear polarization rotation creates oscillations in thepolarization state. The oscillation frequency is dependent uponthe intensity and its period must furthermore match the cavityperiod, otherwise the passive polarizer will attenuate it. Fromthis simple understanding of the polarization dynamics, onecan envision higher-energy pulse states that oscillate two peri-ods or more during one cavity round-trip. These mode-lockedstates pass through the passive polarizer unscathed as theirperiods are commensurate. Such mode-locked states clearlyhave higher intensities since they are required to performtwo full (intensity-dependent) oscillations in the polarizationstate per one cavity round-trip. To date, these higher-energystates have been inaccessible from a practical point of viewdue to the fact that the MPI strongly favors multi-pulsingoperation over such higher-energy states. However, recenttheoretical studies have shown that careful engineering of thenonlinear transmission curve can suppress the MPI in favor ofbifurcating to the high-energy solutions [9].

The onset of MPI as a function of increasing laser cavityenergy is a well-known physical phenomenon [4], [5] that hasbeen observed in a myriad of laser cavities [10]–[23]. One ofthe earliest theoretical descriptions of the MPI dynamics was

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by Namiki et al. [10] in which energy rate equations werederived for the averaged cavity dynamics. Recently, theoryand experiments suggested that near the MPI, both periodicand chaotic behavior could be observed as operating statesof the laser cavity for a narrow range of parameter space[11]–[13]. These studies led to a simple geometrical charac-terization of the MPI in which we generalized the energy rateequation approach [10] and developed an iterative techniquethat provides a simple geometrical description of the entiremulti-pulsing transition behavior as a function of increasingcavity energy [9]. The model captures all the key features ob-served in experiment, including the periodic and chaotic mode-locking regions [12], and it further provides valuable insightinto laser cavity engineering for maximizing performance, i.e.enhancing the mode-locked pulse energy. Specifically, it wasdemonstrated that the MPI could be circumvented in favor ofan instability that forces the mode-locking onto higher-energymode-locked states [9]. It is our aim to demonstrate how suchhigher-energy pulse solutions can be obtained in physicallyrealizable cavity designs.

The paper is outlined as follows: Sec. II discusses the theo-retical overview of the nonlinear transmission curve generatedby the nonlinear polarization rotation. Sec. III introduces thefull governing equations of the laser shown in Fig. 1. Thederivation of the SGLE which is based upon the split-stepmethod developed in previous work is also outlined in thissection. Numerical simulations of the SGLE are presented inSec. IV, showing that the model is in fact a better modelthan the conventional mode-locking models such as the cu-bic Ginzburg-Landau equation (CGLE) and the cubic-quinticGinzburg-Landau equation (CQGLE), and further demonstrat-ing the stable mode-locking of higher-energy single pulsesolutions. Sec. V concludes the paper with a highlight of thekey findings.

II. PRINCIPAL OF OPERATION: PERIODIC TRANSMISSION

A large number of theoretical models have been proposedto quantify the mode-locking process and stability observed inring cavity lasers. The master mode-locking equation proposedby Haus, which is also known as the cubic Ginzburg-Landauequation (CGLE), is the most well-known and recognizedmodel to date [1], [4], [5], [24]. This equation treats all lumpedcomponents of the cavity in a distributed fashion and hencedescribes the averaged pulse propagation in the laser. Theadvantages of such an averaged model is clear: it allows one tomake analytic progress in understanding the stability, trends,and operating regimes of a laser cavity without having toperform extensive numerical simulations on the discrete com-ponents [1], [4], [5], [24]. Ultimately, the averaging methodproposed by Haus allows for fundamental understanding ofthe interplay and nonlinear dynamics of the various physicaleffects in the laser cavity. As such, it merits serious consider-ation in developing a theoretical basis for mode-locked laserdynamics. The CGLE captures two key features of the mode-locking process: (i) the energy budget of the pump due to thesaturated gain, and (ii) the self-amplitude modulation (modeledby a cubic nonlinear gain) that results from the effects of

SpectralFilter

QWP QWP HWPPolarizingIsolator

OutputCoupler

Yb-doped SMFPump

980 nm WDMCoupler

α1

α2

α3

αp

Fig. 1. Experimental configuration of a generic ring cavity laser that includesquarter-waveplates (QWP), passive polarizer, half-waveplate (HWP), spectralfilter, ytterbium-doped amplification and output coupler. The Yb-doped sectionof the cavity is fused with standard single-mode fiber and treated in adistributed fashion. The birefringence K of the system can be adjusted bya pair of polarization controllers (not shown in the figure). The angles α1, α2,α3, and αp can all be measured with reasonable accuracy.

the waveplates and polarizer. The solutions of the CGLE arestable only in a small range of cubic gain values where thenonlinear growth can be balanced by other cavity effects [5],[25], [26]. As a result, the equation is often augmented witha quintic term to provide a saturation to the cubic growthwhich is expected in any physically realizable system. Theresulting equation is called the cubic-quintic Ginzburg-Landauequation (CQGLE) [5], [27], and has been studied extensivelyby Akhmediev, Soto-Crespo and co-workers [28]–[33].

Both the CGLE and CQGLE are phenomenological models,and a direct quantitative comparison with experiment is notpossible. Leblond, Sanchez and co-workers were the pioneersin establishing a quantitative connection between theory andexperiment. Their study on a ytterbium-doped double-cladfiber laser was the first work that justified the use of theCGLE in ring cavity lasers by relating the laser settingsto the master mode-locking theory [34], [35]. Specifically,the CGLE accounted explicitly for the polarization dynamicsas well as the orientations of the waveplates and polarizer(the mode-locking elements) through the coefficients of theequation. Recently, they developed a split-step type methodto model the pulse dynamics in the ANDi laser [6]–[8], [36],and the explicit dependence of the CQGLE coefficients onthe waveplate/polarizer settings was derived [37]. The set ofCQGLE coefficients was later generalized by Ding and Kutzto include the fiber birefringence and an arbitrary alignment ofthe polarizer with the eigenaxis of the fiber [38], thus giving abetter description of the underlying mode-locking dynamics.However, in all of these models, a Taylor series expansionis enacted and the first few terms retained in deriving eitherthe CGLE or CQGLE. Such a truncation destroys the periodictransmission curve associated with such laser cavities.

We extend the CQGLE to capture the sinusoidal transmis-sion observed in a generic ring cavity laser (See Fig. 1) whichwas previously derived [36], [38]. Specifically, the Taylor se-ries truncation leading to the CGLE and CQGLE is not appliedand a more general, and quantitative, mode-locking model is

DING et al.: SINUSOIDAL GINZBURG–LANDAU EQUATION 707

Power

Tra

nsm

issi

on

A

B

C1

2

Fig. 2. (Color online.) Red dash-dot line and the blue dashed line representthe transmission of the SGLE and the CQGLE, respectively. The black solidlines denote the saturating gain curves. The intersection of the gain andtransmission curve represent a stable mode-locked pulse. The green circles arethe maximum power allowed in the laser cavity. When energy is increased,gain curve 1 shifts towards gain curve 2, causing the mode-locked pulses Aand B to exceed the maximum power allowed. In the CQGLE, the solutionjumps to the next most energetically favorable configuration of a double-pulse solution (not shown). A high-energy single-pulse solution (point C)can exist in the SGLE model.

derived. We refer to this equation as the sinusoidal Ginzburg-Landau equation (SGLE) since it accounts for the full non-linear and periodic transmission function exhibited in realsystems. Indeed, the main contribution of this work is to showthat the SGLE model is able to support high-energy pulseswhich are not captured by the master mode-locking theory.

To illustrate the key idea of the SGLE model, considerthe generic transmission curves shown in Fig. 2 whichsummarizes the main findings in the recent work of Liet al. [9]. The red dash-dot line and the blue dashed linerepresent the typical transmission function of the SGLE andthe CQGLE, respectively. The black solid lines, on the otherhand, denote the saturating gain curves in the cavity [4], [9],[10]. The intersection of the gain curve and the transmissioncurve describes a mode-locked solution where the cavityenergy is in equilibrium. The points A and B along thegain curves represent the single mode-locked solution of theCQGLE and SGLE respectively. When the saturation energyis increased, the power of the mode-locked pulse acquires alarger value which in turn shifts gain curve 1 towards gaincurve 2. Eventually the power of the mode-locked solutionexceeds the the maximum value allowed (the green circles)in the laser system as computed from numerical simulations.In the case of CQGLE, the solution jumps to the nextmost energetically favorable configuration of a multi-pulsesolution (not shown). For the SGLE, however, the solution isexpressed as a stable single high-energy pulse at the point C.This example demonstrates that the CQGLE fails to capturethis solution.

The advantages of incorporating of the full sinusoidal trans-mission in the mode-locking model is two-fold. First, sincethe transmission is not approximated by series expressions,the SGLE is able to give a more accurate description of theunderlying mode-locking dynamics. Second, as shown above,high-energy pulses are possible only when a full analysisof the sinusoidal transmission curve is used [9]. Thus the

SGLE model can serve as a design tool to maximize theenergy output by adjusting the orientations of the waveplatesand polarizer. The SGLE is, to our best knowledge, the firstaveraged mode-locking equation that explicitly accounts forthe sinusoidal transmission curve. Although analytical resultsare non-trivial due to the complexity of the equation, the SGLEis relatively easy to analyze with efficient numerical algorithmsand reduction techniques [39]–[41].

III. GOVERNING EQUATIONS

Fig. 1 shows a schematic representation of the nonlinearpolarization rotation fiber laser for achieving stable and robustmode-locking. Such a ring cavity laser has been the subjectof experimental and theoretical investigations for almost twodecades. To describe the propagation of electric field envelopein such a laser cavity, the dominant physical effects must be ac-counted for. These include chromatic dispersion, fiber birefrin-gence, Kerr nonlinearity, cavity attenuation, bandwidth-limitedsaturable gain, and the discrete effects of the waveplates andpolarizer. Under certain choice of the angles α1, α2, α3 andαp , the waveplates and the polarizer provides an effectiveintensity discrimination (saturable absorption) to shape thepulse circulating in the cavity [37], [38]. Since the primaryfocus of this work is to show how the full description of thesaturable absorber can lead to the formation of high-energypulses, the effect of the spectral filter will not be included inthe governing equations. Readers can refer to Ref. [42] fora discussion of how spectral filtering impacts on the mode-locking dynamics.

A. Full Governing Equations

The full governing equations for modeling the pulse evo-lution in the laser shown in Fig. 1 can be divided into twoparts: (i) the intra-cavity dynamics induced by the interactionsof chromatic dispersion, Kerr nonlinearity and gain saturationetc., and (ii) the discrete application of the waveplates andpolarizer after each cavity round-trip. The intra-cavity evolu-tion is generally described by a pair of dimensionless couplednonlinear Schrödinger equations (CNLS) [43], [44]

i∂u

∂z+ D

2

∂2u

∂ t2 −K u+(|u|2+ A|v|2)u+Bv2u∗ = i Ru, (1a)

i∂v

∂z+ D

2

∂2v

∂ t2 +K v+(A|u|2+|v|2)v+Bu2v∗ = i Rv . (1b)

In the above system, u and v represent the two orthogonallypolarized electric field envelopes in an optical fiber withbirefringence K . The z coordinate denotes the propagating dis-tance normalized by the length of the cavity, and t denotes theretarded time normalized by the full-width at half-maximumof the pulse. D is the averaged group velocity dispersionof the cavity, and is positive for anomalous dispersion andnegative for normal dispersion. In the ANDi laser D is alwaysnegative [6]–[8], but we will consider both signs of D in thispaper. The material properties of the optical fiber determinethe values of nonlinear coupling parameters A (cross-phasemodulations) and B (four-wave mixing). These parameterssatisfy A + B = 1 by axisymmetry and, for the physical


system considered take on the specific values A = 2/3 andB = 1/3 [43], [44]. The dissipative terms Ru and Rv , account-ing for the saturable, bandwidth limited gain and attenuation,take the form

Rσ = G(z)(

1 + τ∂2t

)σ − �σ, (2)

with

G(z) = 2g0

1 + 1e0

∫ ∞−∞(|u|2 + |v|2)dt

. (3)

Here g0 and e0 are the nondimensional pumping strength andthe saturating energy of the gain respectively. The parameterτ characterizes the bandwidth of the pump, and � measuresthe distributed losses caused by the output coupling and thefiber attenuation.

The effect of the waveplates and passive polarizer canbe modeled by the corresponding Jones matrices [36], [38].The standard Jones matrices of the quarter-waveplate, half-waveplate and polarizer are given, respectively, by

W λ4=

(e−iπ/4 0

0 eiπ/4

), (4a)

W λ2=

( −i 00 i

), (4b)

Wp =(

1 00 0

). (4c)

These matrices are valid only when the principle axes of thedevices are aligned with the fast axis of the fiber. For arbitraryorientation α j ( j = 1, 2, 3, p) shown in Fig. 1, the matricesare modified according to

Jj = R(α j )W R(−α j ), (5)

where W is the Jones matrix of the device given in Eq. (4)and R is the rotation matrix

R(α j ) =(

cosα j − sin α j

sin α j cosα j

). (6)

The CNLS (1) together with Jones matrices (5) gives afull description of pulse propagation in the laser system.The principle of operation involves iterations of solving theCNLS over one round-trip and applying the Jones matricesof the waveplates and polarizer sequentially. As mentionedbefore, the discrete application of Jones matrices after eachcavity round-trip acts like a filter that can be tuned to controlthe mode-locking behavior. Depending on their orientations,the waveplates and the polarizer can either destabilize thefield propagating in the cavity or lock it into a particularpolarization component so that robust mode-locking can beachieved. Essentially, the waveplates and polarizer produce aperiodic transmission curve that can be engineered to poten-tially achieve higher-energy pulses [9].

B. Master Mode-Locking Equation and the SinusoidalGinzburg–Landau Model

The master mode-locking equation proposed by Haus(CGLE) was the first theoretical model used to describe themode-locking dynamics in the laser cavity shown in Fig. 1.

A quintic dissipation is usually added to the master mode-locking equation (CQGLE) to account for the robustness ofthe pulses observed in experiments, and the resulting equationtakes the form

iψz + D

2ψt t + γ |ψ|2ψ + ν|ψ|4ψ

= ig(z)(

1 + τ∂2t

)ψ − iδψ + iβ|ψ|2ψ + iμ|ψ|4ψ. (7)

Here the saturable gain is

g(z) = 2g0

1 + 1e0

∫ ∞−∞ |ψ|2dt

. (8)

The CGLE is a special case of the CQGLE with ν = μ = 0.The derivation of the CQGLE that relates the system parame-ters to the experimental settings was first given by Refs. [36],[37]. A more general set of coefficients which explicitlyaccounts for fiber birefringence and arbitrary waveplate ori-entations was recently derived using a similar approach [38].We will highlight the key results in the derivation and showhow a new master mode-locking model can be obtained fromthe previous work.

The key idea in deriving the CQGLE is to separate thelinear and nonlinear effect in the CNLS (1), assuming thatthese effects occur on a length scale much shorter than thefiber length. Ignoring the birefringence K (it will be treatedseparately), the linear terms alone yield the evolution equation

iψz + D

2ψt t = ig(z)

(1 + τ∂2

t

)ψ − i�ψ. (9)

The field envelope ψ is related to the two orthogonallypolarized components in the CNLS through the transformationu = ψ cosαp and v = ψ sin αp . Additionally, the nonlinearpart of the CNLS can be integrated analytically over one cavityround-trip. Approximating the birefringence with a matrixoperation, one can obtain a map that relates the field at thebeginning of the fiber (denoted by the subscript n) to the endof the fiber (denoted by the superscript “−”)(

u−v−

)= ei In JK JN L

(un

vn

)

= ei In

(e−i K 0

0 ei K

)(cosw sinw

− sinw cosw

)(un

vn

). (10)

Here In = |un|2 +|vn|2 = |ψn |2 is the total power of the fieldat the beginning of the fiber and w = B In sin 2(α1 − αp).The above map shows that the field undergoes an intensity-dependent polarization rotation (governed by JN L ) as it prop-agates along the fiber and receives a 2K -phase slip causedby the fiber birefringence (characterized by JK ). Applying theJones matrices (5) of the waveplates and polarizer to Eq. (10)results in the scalar map

ψn+1 = ei|ψn |2 Q (In) ψn, (11)

where the complex function Q is given by

Q = 12

{e−i K

[cos(2α2 − 2α3 − αp)+ i cos(2α3 − αp)

]

× [i cos(2α1 − αp −w)− cos(αp −w)

]

+ei K[sin(2α2 − 2α3 − αp)− i sin(2α3 − αp)

]

× [sin(αp −w)− i sin(2α1 − αp −w)

]}. (12)


Eqs. (9) and (11) are the leading order approximation tothe pulse propagation in the laser cavity shown in Fig. 1.The initial data is first evolved forward over one round-tripaccording to the linear equation (9). The nonlinear map (11)is then applied to the resulting field, and the process isrepeated. With an appropriate choice of parameters, the realpart of the transfer function Q provides an effective intensitydiscriminating mechanism to shape the circulating field. In thisprocess the intensity discrimination sifts out those intensitieswhose polarization rotation oscillation is commensurate withthe cavity length, thus forming a mode-locked pulse.

As in previous works [37], [38], the effect of the nonlineartransfer function Q can be averaged into the evolution bytaking a continuous limit of Eq. (11), yielding the differentialequation

ψz =(

i |ψ|2 + log Q(|ψ|2))ψ. (13)

The coefficients in the CGLE and CQGLE (7) can be relatedto the experimental settings by expanding the logarithmic termin the above equation in power series of the power |ψ|2(assuming it is small), truncating the result at a desired powerand combining the new equation with the linear evolution (9).These coefficients are explicitly calculated as

δ = � − log |Q(0)|, (14a)

γ = 1 + Im(Q′(0)/Q(0)

), (14b)

β = Re(Q′(0)/Q(0)

), (14c)

ν = Im[(

Q(0)Q′′(0)− Q′2(0))/Q2(0)

]/2, (14d)

μ = Re[(

Q(0)Q′′(0)− Q′2(0))/Q2(0)

]/2. (14e)

In the case of the master mode-locking model (CGLE) thequintic coefficients ν and μ are identically zero.

In the master mode-locking equation proposed by Haus(ν = μ = 0), the cubic dissipation β is always positiveso that self-amplitude modulation (intensity discrimination) ispossible. There is, however, only a small range of β valuesthat allows stable mode-locking to occur [25], [26]. Outsidethis range the cubic gain is either too low for pulse formationor too large so that the pulse amplitude blows up to infinity.In reality, the pulse governed by the full equations (1) and (5)can never blow up since it always experiences a net loss uponpassing through the waveplates and polarizer. The CQGLEprevents the pulse from blowing up by saturating the cubicgain with a quintic loss (μ < 0). This is a better descriptionof the saturable absorption process which makes the CQGLE amore physically relevant model than the master mode-lockingequation. In the regime where both the cubic and quinticdissipation are positive, there is no higher order saturationin the model to prevent the collapse of pulses, and thus theCQGLE becomes suspect as it admits a large number ofunphysical behaviors.

It is immediately clear then, that there is always the possi-bility of unphysical blowup no matter how many terms in theexpansion of the right-hand-side of Eq. (13) are kept. Also, asmentioned in the introduction, a series representation of thetransmission function can lead to elimination of the importanthigh-energy solution branches since the fundamental periodic

transmission function is destroyed by the Taylor series. Thisfact has motivated our current consideration of retaining thefull logarithmic term rather than using its Taylor series in theaveraged model. The combination of Eqs. (9) and (13) givesthe more appropriate evolution equation

iψz + D

2ψt t + |ψ|2ψ

= ig(z)(

1 + τ∂2t

)ψ − i�ψ + i(log Q)ψ. (15)

This equation will be referred to as the sinusoidal Ginzburg-Landau equation (SGLE). The name of the equation followsfrom the sinusoidal structure of the transfer function given inEq. (12). As will be seen in the next section, incorporation thefull transfer function leads to a more accurate mode-lockingmodel which has no unphysical blowing-up solutions. Moreimportantly, it predicts potentially high-energy solutions thatare not captured in either the cubic or quintic master mode-locking theories.

IV. COMPARISON OF THE SGLE WITH DIFFERENT

MODE-LOCKING MODELS

In this section, we will first establish that the SGLE is avalid mode-locking model that exhibits the essential dynamicsin the nonlinear polarization rotation laser. Then we willshow that the SGLE model is able to reproduce the high-energy pulses of the full governing system, solutions whichare precluded from the master mode-locking theory or CQGLEmodel.

A. Stable Mode-Locking and Multi-Pulsing

We first compare the mode-locking performance governedby the full, lumped mode-locking system (1) (CNLS inconjunction with the Jones matrices) and the SGLE (15).Fig. 3 demonstrates the self-starting behavior of the laser froma white-noise initial condition in the anomalous dispersionregime (D > 0) with a particular waveplate/polarizer setting.In the left panel of the figure, the pumping strength is set atg0 = 1, and the initial white-noise is dynamically locked intoa stable stationary pulse after several hundred cavity round-trips in both the full system (top) and the SGLE (bottom).The temporal location at which the pulse forms is completelyarbitrary since a random initial data is used. When the energyinjected into the system is increased by increasing the gaing0, the laser undergoes the commonly observed phenomenonknown as multi-pulsing instability (MPI) [9]. In this situation,the initial condition quickly evolves into two or more pulseswith identical energies, depending on the strength of thegain. It can be seen that the SGLE is capable of capturingqualitatively the MPI that occurs in the full system (rightpanel) at a high gain level (e.g. g0 = 2.7).

Qualitative matching between the SGLE model and the fullevolution is also achieved in the normal dispersion regime(D < 0), as shown in Fig. 4. For the parameters considered,a single mode-locked pulse can be formed in both the fullgoverning equations and the SGLE at g0 = 1 (left panel).When the gain is increased to g0 = 3, the laser cavity no


−100

10

0

5000

1

tz 1030

50

0

2000

tz

|�|

−100

10

0

5000

1

tz

|�|

0

2

|�|

−200

20

0

2000

tz

0

2

|�|

Fig. 3. (Color online.) Numerical simulations of the full governing system(top, recall that |ψ |2 = |u|2 + |v|2) and the SGLE (bottom) at differentgain values g0 in the anomalous dispersion regime. Left: Stable single-pulseevolution starting from white-noise at g0 = 1. Right: The initial white-noisequickly evolves into two identical pulses per cavity round-trip at g0 = 2.7. Therest of the parameters in the simulations are D = 0.4, α1 = 0, α2 = 0.82π ,α3 = 0.1π , αp = 0.45π , K = 0.1, � = 0.1, e0 = 1, and τ = 0.1.

−500

50

0

10000

0.4

tz −1000

100

0

20000

1

tz

|�| |�|

−500

50

0

10000

0.4

tz

|�|

−1000

100

0

20000

1

tz

|�|

Fig. 4. (Color online.) Numerical simulations of the full governing system(top) and the SGLE (bottom) at different gain values g0 in the normaldispersion regime. Left: Stable single-pulse evolution starting from white-noise at g0 = 1. Right: The initial white-noise evolves into two identicalpulses per cavity round-trip at g0 = 3. The rest of the parameters in thesimulations are D = −0.4, α1 = 0.1π , α2 = 0.554π , α3 = 0.23π ,αp = 0.43π , K = 0.1, � = 0.1, e0 = 1, and τ = 0.2.

longer supports a stable single-pulse solution and a double-pulse configuration is observed (right panel). From Figs. 3and 4, one can see that the two main differences between thefull dynamics and the SGLE dynamics are (i) the duration ofthe transient evolution, and (ii) the mode-locked amplitude.In the case of anomalous dispersion, for instance, it takesapproximately 300 and 350 cavity round-trips for the fullsystem and the SGLE to mode-lock into a stable pulse,respectively. The mode-locked peak amplitude is 0.52 for thefull model and 0.81 for the SGLE. The discrepancy is anintrinsic nature of the averaged evolution equations (including

−400

40

0

100000

2.5

tz −40 0 400

2.5

t

|�| |�|

−40 0 400

2.5

t

|�|

−400

40

0

100000

2.5

tz

|�|

Fig. 5. (Color online.) Transition dynamics observed in the anomalousdispersion regime. Top: Simulations of the full governing equations at g0 =1.9 (left) and the corresponding pulse profile (right). Bottom: Simulations ofthe SGLE at g0 = 2.25 (left) and the corresponding pulse profile (right). Therest of the parameters in the simulations are D = 0.4, α1 = 0, α2 = 0.82π ,α3 = 0.1π , αp = 0.45π , K = 0.1, � = 0.1, e0 = 1, and τ = 0.1. The initialconditions in all the simulations are ψ(0, t) = sech 0.5t .

the CGLE and CQGLE). Nevertheless, this does not impacton the usefulness of the SGLE model as it captures the self-starting nature of the laser and is easier to analyze than thefull governing equations, i.e. it is one scalar equation versustwo coupled equations with discrete application of four Jonesmatrices.

B. Transition Dynamics

When the waveplate/polarizer angles α1, α2, α3, αp arechosen appropriately such that mode-locking is achievable,the initial data can evolve into an arbitrary number of stablepulses depending on the pumping strength g0. In general thetransition from an n-pulse solution to an (n+1)-pulse solutionis not a discrete process. Various types of transition dynamicscan be obtained by modifying the waveplate/polarizer settingsas well as other system parameters, as confirmed by theoryand experiments [9], [12], [13]. Usually these transitionalstates are difficult to capture as they often happen in a smallparameter regime, and fine tuning of the parameters is requiredto visualize them. Fig. 5 shows a typical transition state of thelaser cavity in the anomalous dispersion regime (D > 0). Thepumping strength g0 is chosen to be between the stable single-and double-pulse operation shown in Fig. 3. The transitionalstate contains a tall pulse at the origin and one small pulseon each side of it. These side pulses are developed from thebackground. The entire structure undergoes small amplitudeoscillations (at the order of 10−4). A remarkable agreementbetween the full governing equations (top) and the SGLEmode (bottom) is observed. This periodic state is stable andpersists over long propagating distance. When g0 exceedsa critical value, the tall central pulse experiences a slightdecrease in amplitude. At the same time, one of the side pulsesis attenuated while the other one is amplified to the height ofthe central pulse, thus forming a double-pulse solution. The


−1000 100

012000

0

1

tt

z

z

t

z

t

z

|�|

−1000 100

012000

0

1

tz

|�|

−1000 100

012000

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tz

|�|

Fig. 6. (Color online.) Periodic evolution observed in the full governingequations (top, g0 = 2.27), CQGLE (middle, g0 = 2.2) and SGLE(bottom, g0 = 2.2). The simulated evolutions are shown on the left andthe corresponding pseudo-color plots are shown on the right. The rest of theparameters in the simulations are D = −0.4, α1 = 0.1π , α2 = 0.554π ,α3 = 0.23π , αp = 0.43π , K = 0.1, � = 0.1, e0 = 1, and τ = 0.2. Theinitial conditions in all the simulations are ψ(0, t) = sech 0.5t .

results found here match with those in a previous study on theCQGLE [40].

The transition dynamics are more subtle in the normaldispersion regime as depicted in Fig. 6. Similar to the caseof anomalous dispersion, a periodic structure with a tallcentral pulse and two flat side pulses is quickly developed(see top panel). However, this periodic structure is unstableand eventually loses its stability after several thousand cavityround-trips. A transient chaotic evolution is then observedfollowed by a stable co-propagation of two well-separatedpulses with different amplitudes. While the CQGLE (middle)is able to give a general qualitative approximation to thetransitional state observed in the full simulation (top), theSGLE model (bottom) additionally gives a precise descriptionof the transient chaotic behavior. This feature of the SGLEdynamics clearly demonstrates the improved description of thetrue discrete cavity dynamics.

C. Unphysical Blow-up of Pulses

As mentioned in Sec. III-B, the field ψ governed by theCGLE and CQGLE may experience a blow-up (amplitudegoing to infinity) with certain waveplate/polarizer settings.This blow-up is unphysical since energy is always lost whenthe field passes through the mode-locking elements, and is aresult of the finite truncation of the right-hand-side of Eq. (13).Therefore, in order to avoid any unphysical blow-up, theparameter space should be restricted so that the coefficientof the cubic dissipation in the CGLE should be below somethreshold β+, i.e. 0 < β ≤ β+, or the cubic-quintic terms inthe CQGLE should satisfy the relationship μ < 0 < β. Again,these artificial restrictions are simply a result of the finiteTaylor series truncation of the right-hand-side of Eq. (13). It

0 10 20 30−10

−5

0

5

10

CGLE

SGLE

CQGLE

(β, μ) = (0.28, −0.062) (β, μ) = (0.049, 0.061)

T(|�|2)

−2

0

10

T(|�|2)

|�|20 10 20 30

|�|2

Fig. 7. (Color online.) Transmission curve for the CGLE (blue dash-dot line),CQGLE (red dashed line) and SGLE (green solid line) for α2 = 0.554π (left)and α2 = 0.4π (right). The rest of the parameters are α1 = 0.1π , α3 = 0.23π ,αp = 0.43π , � = 0.1, e0 = 1, and K = 0.1.

is our intent to show that such a truncation is unnecessary formodeling the laser cavity.

To show that the SGLE does not have the above limitations,we study the transmission function T (|ψ|2) which can be de-fined as the sum of the linear loss and the nonlinear dissipation.The transmission function of the CGLE (Tc), CQGLE (Tcq )and the SGLE (Ts) are given by

Tc(|ψ|2) = −δ + β|ψ|2, (16a)

Tcq(|ψ|2) = −δ + β|ψ|2 + μ|ψ|4, (16b)

Ts(|ψ|2) = −� + Re(

log Q(|ψ|2)). (16c)

In the last equation, the imaginary part of the logarithmic termis neglected since it is not responsible for energy transfer.Interestingly enough, the imaginary terms act to shift theintensity-dependent index of refraction response due to thefield intensity. Shown in Fig. 7 are the transmission curvefor the CGLE (blue dash-dot line), CQGLE (red dotted line)and SGLE (green solid line) at two different quarter-waveplateangles (left: α2 = 0.554π , right: α2 = 0.4π). For bothwaveplate settings, Tc grows linearly with the instantaneouspower |ψ|2 while Ts varies periodically with |ψ|2. On theother hand, the transmission function Tcq of the CQGLE isquadratic in the field power. It curves downward at α2 =0.554π (left) and upward at α2 = 0.4π (right). Fig. 8illustrates the pulse evolutions for the full governing equations(top row), CGLE (second row), CQGLE (third row) andthe SGLE (bottom row) corresponding to the two α2 valuesconsidered in Fig. 7. At α2 = 0.554π (left panel), theCGLE demonstrates a blow-up shortly after the simulation hasstarted when the initial amplitude is sufficiently large, whereasthe evolutions in the CQGLE and the SGLE are always offinite amplitude. This is expected since the transmission ofthe CGLE Tc fails to self-saturate, meaning that the energysupplied grows with pulse intensity. Blowing-up of pulses isobserved in both the CGLE and the CQGLE at α2 = 0.4π ,since self-saturation is impossible when both the nonlinearcoefficients β and μ are positive (right panel). For both α2values considered, the SGLE does not have the problem ofunphysical blow-up since it incorporates the full transmis-sion log Q induced by the waveplates and polarizer. Thisallows for a drastically broader and more realistic range ofwaveplate/polarizer settings for which stable solutions can beachieved.


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Fig. 8. (Color online.) Pulse evolution predicted by the full governingequations (top row), CGLE (second row), CQGLE (third row), and SGLE(bottom row). The plots on the left (right) column correspond to the left(right) panel in Fig. 7. The initial conditions for the plots on the left areψ(0, t) = 2 sech 0.5t , while those for the right plots are ψ(0, t) = 6 sech 0.5t .The dispersion and pumping strength in all the simulations are D = −0.4,and g0 = 2.

D. High-Energy Mode-Locking

As illustrated in Fig. 2, the CQGLE which uses a quadraticpolynomial in the field power |ψ|2 (See Eq. (16)) to approx-imate the full sinusoidal transmission and this can severelyrestrict the maximum amplitude a single pulse can reach beforeit splits. Fig. 9 shows the tallest possible stable pulse generatedby the full governing equations (black dash-dot line), CQGLE(red dashed line) and SGLE (green solid line) at two differenthalf-waveplate angles α3. These pulses are recorded rightbefore the MPI [9] occurs when the pumping strength g0exceeds the multi-pulsing threshold. At α3 = 0.2π (left panel),the full and the SGLE model produce pulses with amplitudesof about 3.1. The total cavity energy, which is given by

E =∫ ∞

−∞|ψ|2dt, (17)

is calculated for each model. For the SGLE, the limitingpumping strength g0 = 3.58 gives a pulse energy of E = 4.52.For the full model, the tallest pulse has an energy of E = 3with pumping strength of g0 = 3.38. One can see that thelimiting g0 value for a stable single-pulse operation in thepresent case is higher than that for a double-pulse operation(g0 = 2.7) shown in Fig. 3. The energy confined in eachindividual pulse of the double-pulse solution shown in Fig. 3is E = 2.37 for the SGLE and E = 1.11 for the full governingequations, which is significantly less than the energy outputin the present case. The limiting pumping strength for theCQGLE model is only g0 = 2.51, and the resulting pulse doesnot match well with the full simulation and the SGLE since itis much shorter and wider. With the same set of parameters, themaximum energy in the single-pulse solution of the CQGLEis E = 2.87, which is only approximately 60% of the SGLEmodel.

At α3 = 0.25π (right panel), the maximum pulse amplitudeallowed in both the full and the SGLE model increase to about

α3 = 0.2 π α

3 = 0.25 π

−5 0 50

3.5

t−5 0 5

t

0

5

|�||�|

Fig. 9. (Color online.) Stable single-pulse solution with largest possibleamplitude of the full governing system (black dash-dot curve), CQGLE (reddashed curve) and the SGLE (green solid curve) at α3 = 0.2 π (left) andα3 = 0.25 π (right). The rest of the parameters are D = 0.4, τ = 0.1,e0 = 1, α1 = 0, α2 = 0.49 π , αp = 0.45 π , and K = 0.1. The initialconditions are ψ(0, t) = 0.5 sech t .

4. The corresponding pulse energies are E = 6.58 (g0 = 11.7)and E = 8.516 (g0 = 15.4) respectively, which are about twicethe total energy of the double-pulse solutions shown in Fig. 3.This is a remarkable achievement in terms of maximizing theenergy output of the laser without going through multi-pulsing.On the other hand, such a high-intensity pulse is not supportedby the CQGLE with the parameters considered. In particular,Eq. (14) reveals that the transmission Tcq of the CQGLE (c.f.Eq. (16)) is characterized by β = −0.63 and μ = 5.74. Athigh intensities, the quintic gain always dominates over thecubic loss and consequently leads to an unphysical blow-upof the solution, which is what happens in this case.

The above simulations show that, indeed, the SGLE can sup-port high-intensity pulses with enormous energies observed inthe full governing system, including those that are unpredictedby the CQGLE model. Although these simulations are donewith anomalous dispersion, similar trends are also observedin the normal dispersion regime. Given the vast parameterspace of the SGLE, it is possible to obtain other types ofinteresting mode-locking dynamics such as soliton shakingand period doubling bifurcation [9], [33], etc. A more detailedstudy in this direction will be presented in future studies.The ability to support, model and characterize high-intensityand high-energy mode-locked solutions is the key reason forour development of the proposed SGLE theory. Ultimately, itis critical in modern mode-locked lasers to understand howto achieve pulses with maximal energy. The SGLE theoryprovides a basis model for exploring such pusles.

V. CONCLUSION

Motivated by the recent work on the generation of high-energy solutions of mode-locked lasers due to a periodictransmission function [9], we follow the averaging techniquedeveloped in previous work [36]–[38] and extend the conven-tional master mode-locking equation to account explicitly forthe periodic transmission function of a laser mode-locked by acombination of waveplates and a passive polarizer (see Fig. 1).Incorporation of the full transmission in the averaging processgenerates a Ginzburg-Landau type equation with sinusoidalsaturable absorption as the governing equation of the laser,i.e. the sinusoidal Ginzburg-Landau equation (SGLE). Themodel depends explicitly on the waveplate and polarizer


settings of the cavity and a direct connection between theoryand experiment can be made. Numerical simulations showthat the SGLE is capable of capturing the essential mode-locking dynamics observed in the laser cavity, including themulti-pulsing instability (MPI) for which a single pulse splitsinto two or more pulses when the pumping strength g0 ofthe gain is increased [9]. By carefully adjusting the wave-plate/polarizer settings, which in turn alters the characteristicsof the sinusoidal transmission, one can significantly increasethe maximum g0 allowed before pulse-splitting occurs. Theresulting high-intensity single pulses can deliver significantlymore energy than previous theoretical predictions, and are thuspractically important to high-power applications. These pulsescan exist in the parameter regime that is not allowed by themaster mode-locking theory [38]. Thus the SGLE can be usedas an excellent design tool for enhancing the energy output ofthe laser.

As a consequence of this work, a specific method forachieving high-energy pulses is established. Specifically, high-energy pulses are conjectured to be achieved by mode-lockingonto higher-energy, single mode-locked solution branches inthe periodic transmission curve. In order to achieve such high-energy pulses, the pernicious effects of the MPI must be cir-cumvented. The current theoretical treatment along with recentengineering insights of the transmission curve [9] illustratehow the final order of magnitude increase in pulse energy canbe achieved in order to make fiber lasers directly competitivewith leading solid state mode-locking configurations. Thetremendous benefits of fiber lasers in terms of cost, readilyavailable technology and ease of use lend importance to thetheoretical developments proposed in the current manuscript.

ACKNOWLEDGMENT

The authors acknowledge insightful conversations aboutthe sinusoidal Ginzburg–Landau equation with P. Grelu,W. Renninger, and F. Wise.

REFERENCES

[1] H. A. Haus, J. G. Fujimoto, and E. P. Ippen, “Structures for additivepulse mode locking,” J. Opt. Soc. Amer. B, vol. 8, no. 10, pp. 2068–2076,1991.

[2] K. Tamura, H. A. Haus, and E. P. Ippen, “Self-starting additive pulsemode-locked erbium fibre ring laser,” Electron. Lett., vol. 28, no. 24,pp. 2226–2228, Nov. 1992.

[3] H. A. Haus, E. P. Ippen, and K. Tamura, “Additive-pulse modelockingin fiber lasers,” IEEE J. Quantum Electron., vol. 30, no. 1, pp. 200–208,Jan. 1994.

[4] H. A. Haus, “Mode-locking of lasers,” IEEE J. Sel. Topics QuantumElectron., vol. 6, no. 6, pp. 1173–1185, Nov.–Dec. 2000.

[5] J. N. Kutz, “Mode-locked soliton lasers,” SIAM Rev., vol. 48, no. 4, pp.629–678, Nov. 2006.

[6] A. Chong, J. Buckley, W. Renninger, and F. Wise, “All-normal-dispersion femtosecond fiber laser,” Opt. Exp., vol. 14, no. 21, pp.10095–10100, 2006.

[7] A. Chong, W. H. Renninger, and F. W. Wise, “All-normal-dispersionfemtosecond fiber laser with pulse energy above 20 nJ,” Opt. Lett., vol.32, no. 16, pp. 2408–2410, 2007.

[8] A. Chong, W. H. Renninger, and F. W. Wise, “Properties of normal-dispersion femtosecond fiber lasers,” J. Opt. Soc. Amer. B, vol. 25, no.2, pp. 140–148, 2008.

[9] F. Li, P. K. A. Wai, and J. N. Kutz, “Geometrical description of theonset of multi-pulsing in mode-locked laser cavities,” J. Opt. Soc. Amer.B, vol. 27, no. 10, pp. 2068–2077, 2010.

[10] S. Namiki, E. P. Ippen, H. A. Haus, and C. X. Yu, “Energy rate equationsfor mode-locked lasers,” J. Opt. Soc. Amer. B, vol. 14, no. 8, pp. 2099–2111, 1997.

[11] J. N. Kutz and B. Sandstede, “Theory of passive harmonic mode-lockingusing waveguide arrays,” Opt. Exp., vol. 16, no. 2, pp. 636–650, 2008.

[12] B. G. Bale, K. Kieu, J. N. Kutz, and F. Wise, “Transition dynamicsfor multi-pulsing in mode-locked lasers,” Opt. Exp., vol. 17, no. 25, pp.23137–23146, 2009.

[13] Q. Xing, L. Chai, W. Zhang, and C.-Y. Wang, “Regular, period-doubling,quasi-periodic, and chaotic behavior in a self-mode-locked Ti:sapphirelaser,” Opt. Commun., vol. 162, nos. 1–3, pp. 71–74, Apr. 1999.

[14] B. C. Collings, K. Berman, and W. H. Knox, “Stable multigigahertzpulse-train formation in a short-cavity passively harmonic mode-lockederbium/ytterbium fiber laser,” Opt. Lett., vol. 23, no. 2, pp. 123–125,Jan. 1998.

[15] M. E. Fermann and J. D. Minelly, “Cladding-pumped passive harmoni-cally mode-locked fiber laser,” Opt. Lett., vol. 21, no. 13, pp. 970–972,1996.

[16] A. B. Grudinin, D. J. Richardson, and D. N. Payne, “Energy quantisationin figure eight fibre laser,” Electron. Lett., vol. 28, no. 1, pp. 67–68, Jan.1992.

[17] M. J. Guy, P. U. Noske, A. Boskovic, and J. R. Taylor, “Femtosecondsoliton generation in a praseodymium fluoride fiber laser,” Opt. Lett.,vol. 19, no. 11, pp. 828–830, 1994.

[18] M. Horowitz, C. R. Menyuk, T. F. Carruthers, and I. N. Duling, III,“Theoretical and experimental study of harmonically modelocked fiberlasers for optical communication systems,” J. Lightw. Technol., vol. 18,no. 11, pp. 1565–1574, Nov. 2000.

[19] R. P. Davey, N. Langford, and A. I. Ferguson, “Interacting solutions inerbium fibre laser,” Electron. Lett., vol. 27, no. 14, pp. 1257–1259, Jul.1991.

[20] M. J. Lederer, B. Luther-Davis, H. H. Tan, C. Jagadish, N. N.Akhmediev, and J. M. Soto-Crespo, “Multipulse operation of aTi:sapphire laser mode locked by an ion-implanted semiconductorsaturable-absorber mirror,” J. Opt. Soc. Amer. B, vol. 16, no. 6, pp.895–904, 1999.

[21] M. Lai, J. Nicholson, and W. Rudolph, “Multiple pulse operation of afemtosecond Ti:sapphire laser,” Opt. Commun., vol. 142, nos. 1–3, pp.45–49, Oct. 1997.

[22] C. Wang, W. Zhang, K. F. Lee, and K. M. Yoo, “Pulse splitting in aself-mode-locked Ti:sapphire laser,” Opt. Commun., vol. 137, nos. 1–3,pp. 89–92, Apr. 1997.

[23] H. Kitano and S. Kinoshita, “Stable multipulse generation from a self-mode-locked Ti:sapphire laser,” Opt. Commun., vol. 157, nos. 1–6, pp.128–134, Dec. 1998.

[24] H. A. Haus, J. G. Fujimoto, and E. P. Ippen, “Analytic theory of additivepulse and Kerr lens mode locking,” IEEE J. Quantum Electron., vol. 28,no. 10, pp. 2086–2096, Oct. 1992.

[25] T. Kapitula, J. N. Kutz, and B. Sandstede, “Stability of pulses in themaster mode-locking equation,” J. Soc. Opt. Amer. B, vol. 19, no. 4, pp.740–746, 2002.

[26] T. Kapitula, J. N. Kutz, and B. Sandstede, “The Evans function fornonlocal equations,” Ind. Univ. Math. J., vol. 53, pp. 1095–1126, Apr.2004.

[27] W. H. Renninger, A. Chong, and F. W. Wise, “Dissipative solitons innormal-dispersion fiber lasers,” Phys. Rev. A, vol. 77, no. 2, pp. 023814-1–023814-4, Feb. 2008.

[28] N. N. Akhmediev, V. V. Afanasjev, and J. M. Soto-Crespo, “Singularitiesand special soliton solutions of the cubic-quintic complex Ginzburg–Landau equation,” Phys. Rev. E, vol. 53, no. 1, pp. 1190–1201, Jan.1996.

[29] J. M. Soto-Crespo, N. N. Akhmediev, V. V. Afanasjev, and S. Wabnitz,“Pulse solutions of the cubic-quintic complex Ginzburg–Landau equa-tion in the case of normal dispersion,” Phys. Rev. E, vol. 55, no. 4, pp.4783–4796, Apr. 1997.

[30] N. N. Akhmediev, V. V. Afanasjev, and J. M. Soto-Crespo, “Multisolitonsolutions of the complex Ginzburg–Landau equation,” Phys. Rev. Lett.,vol. 79, no. 21, pp. 4047–4051, Nov. 1997.

[31] N. Akhmediev, J. M. Soto-Crespo, and G. Town, “Pulsating solitons,chaotic solitons, period doubling, and pulse coexistence in mode-lockedlasers: Complex Ginzburg–Landau equation approach,” Phys. Rev. E,vol. 63, no. 5, pp. 056602-1–056602-13, Apr. 2001.

[32] E. N. Tsoy, A. Ankiewicz, and N. Akhmediev, “Dynamical models fordissipative localized waves of the complex Ginzburg–Landau equation,”Phys. Rev. E, vol. 73, no. 3, pp. 036621-1–036621-10, Mar. 2006.


[33] J. M. Soto-Crespo, P. Grelu, N. Akhmediev, and N. Devine, “Soli-ton complexes in dissipative systems: Vibrating, shaking, and mixedsoliton pairs,” Phys. Rev. E, vol. 75, no. 1, pp. 016613-1–016613-9,Jan. 2007.

[34] H. Leblond, M. Salhi, A. Hideur, T. Chartier, M. Brunel, and F.Sanchez, “Experimental and theoretical study of the passively mode-locked ytterbium-doped double-clad fiber laser,” Phys. Rev. A, vol. 65,no. 6, pp. 063811-1–063811-9, Jun. 2002.

[35] M. Salhi, H. Leblond, and F. Sanchez, “Theoretical study of thestretched-pulse erbium-doped fiber laser,” Phys. Rev. A, vol. 68, no. 3,pp. 033815-1–033815-7, Sep. 2003.

[36] A. Komarov, H. Leblond, and F. Sanchez, “Multistability and hysteresisphenomena in passively mode-locked fiber lasers,” Phys. Rev. A, vol.71, no. 5, pp. 053809-1–053809-9, May 2005.

[37] A. Komarov, H. Leblond, and F. Sanchez, “Quintic complex Ginzburg–Landau model for ring fiber lasers,” Phys. Rev. E, vol. 72, no. 2, pp.025604R-1–025604R-4, Aug. 2005.

[38] E. Ding and J. N. Kutz, “Operating regimes, split-step modeling, andthe Haus master mode-locking model,” J. Opt. Soc. Amer. B, vol. 26,no. 12, pp. 2290–2300, 2009.

[39] C. R. Jones and J. N. Kutz, “Stability of mode-locked pulse solutionssubject to saturable gain: Computing linear stability with the Floquet–Fourier–Hill method,” J. Opt. Soc. Amer. B, vol. 27, no. 6, pp. 1184–1194, 2010.

[40] E. Ding, E. Shlizerman, and J. N. Kutz, “Modeling multipulsing transi-tion in ring cavity lasers with proper orthogonal decomposition,” Phys.Rev. A, vol. 82, no. 2, pp. 023823-1–023823-10, Aug. 2010.

[41] B. G. Bale and J. N. Kutz, “Variational method for mode-locked lasers,”J. Opt. Soc. Amer. B, vol. 25, no. 7, pp. 1193–1202, 2008.

[42] B. G. Bale, J. N. Kutz, A. Chong, W. H. Renninger, and F. W. Wise,“Spectral filtering for high-energy mode-locking in normal dispersionfiber lasers,” J. Opt. Soc. Amer. B, vol. 25, no. 10, pp. 1763–1770,2008.

[43] C. R. Menyuk, “Pulse propagation in an elliptically birefringent Kerrmedium,” IEEE J. Quantum Electron., vol. 25, no. 12, pp. 2674–2682,Dec. 1989.

[44] C. R. Menyuk, “Nonlinear pulse propagation in birefringent opticalfibers,” IEEE J. Quantum Electron., vol. 23, no. 2, pp. 174–176, Feb.1987.

Edwin Ding received the B.Eng. degree in mechanical engineering from theUniversity of Hong Kong, Pokfulam, Hong Kong, in 2006. He is currentlypursuing the Ph.D. degree in the Department of Applied Mathematics,University of Washington, Seattle.

Eli Shlizerman received the B.Sc. degree in mathematics and computerscience from Bar-Ilan University, Ramat Gan, Israel, in 2002, and the M.Sc.and Ph.D. degrees in applied mathematics from the Weizmann Institute,Rehovot, Israel, in 2005 and 2009, respectively.

He is currently working as an acting Assistant Professor in the Departmentof Applied Mathematics, University of Washington, Seattle. He is interestedin problems in nonlinear dynamics arising from physics and biology. Hiscurrent research interests include dimensionality reduction methods for suchdynamics.

J. Nathan Kutz received the B.S. degree in physics and mathematics from theUniversity of Washington, Seattle, in 1990, and the Ph.D. degree in appliedmathematics from Northwestern University, Evanston, IL, in 1994.

He is currently working as a Professor and Chair of Applied Mathematicsat the University of Washington.

Documents

Generalized Master Equation for High-Energy Passive Mode-Locking: The Sinusoidal Ginzburg–Landau Equation